Journal of Functional Analysis 256 (2009) 1–90 www.elsevier.com/locate/jfa
Phase space analysis on some black hole manifolds P. Blue a,∗ , A. Soffer b a School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh,
James Clerk Maxwell Building, King’s Buildings, Mayfield Road, Edinburgh EH9 3JZ, UK b Department of Mathematics, Rutgers University, Hill Center for the Mathematical Sciences, 110 Frelinghuysen Rd.,
Piscataway, NJ 08854-8019, USA Received 7 October 2005; accepted 3 October 2008
Communicated by H. Brezis
Abstract The Schwarzschild and Reissner–Nordstrøm solutions to Einstein’s equations describe space–times which contain spherically symmetric black holes. We consider solutions to the linear wave equation in the exterior of a fixed black hole space–time of this type. We show that for solutions with initial data which decay at infinity and at the bifurcation sphere, a weighted L6 norm in space decays like t −1/3 . This weight vanishes at the event horizon, but not at infinity. To obtain this control, we require only an loss of angular derivatives. © 2008 Elsevier Inc. All rights reserved. Keywords: Black holes; Phase space analysis; Schwarzschild metric; Decay estimates
Contents 1. 2.
3.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Structure of the paper . . . . . . . . . . . . . . . . . . . . . . . . The wave equation on the Reissner–Nordstrøm space . . . . . . . 2.1. The Reissner–Nordstrøm solution . . . . . . . . . . . . . . . . 2.2. The wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . Methods for pointwise decay . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Densities, energy conservation, and the conformal charge
* Corresponding author.
E-mail address:
[email protected] (P. Blue). 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.10.004
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
2 6 7 7 11 13 13
2
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
3.2. Sobolev estimates . . . . . . . . . . . . . . . . . . . . . . . 3.3. Local support of the trapping terms . . . . . . . . . . . 4. Relativistic considerations on the event horizon . . . . . . . 4.1. Decay on the bifurcation sphere . . . . . . . . . . . . . 5. The Heisenberg-type relation . . . . . . . . . . . . . . . . . . . . 6. Morawetz estimates . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. Preliminary bounds . . . . . . . . . . . . . . . . . . . . . . 6.2. Computation of Morawetz commutators . . . . . . . . 6.3. L2 local decay estimate . . . . . . . . . . . . . . . . . . . 7. Angular modulation . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1. Angular modulation and initial estimates . . . . . . . 7.2. Direct angular momentum bounds . . . . . . . . . . . . 8. Phase space analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1. Phase space variables, localization, and multipliers 8.2. Commutator expansions . . . . . . . . . . . . . . . . . . 8.3. Phase space estimates . . . . . . . . . . . . . . . . . . . . 8.4. Derivative bounds . . . . . . . . . . . . . . . . . . . . . . . 8.5. Phase space induction . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
17 22 26 27 30 32 33 34 39 43 44 45 51 52 56 62 76 83 88 89
1. Introduction Black holes are very important objects in relativity, but very little is known about their dynamics and interaction. Even the question of stability under small perturbations is a challenging, open problem. Any solution to these problems will require an understanding of the interaction between black holes and gravitational radiation. The structure of the vacuum Einstein equations, which govern these dynamics in the absence of matter, make it impossible to consider only radial perturbations, so that more general perturbations must be considered immediately. We hope that the study of linear, decoupled waves outside the black hole will help provide an understanding of these problems. Even the stability of the empty space was a very difficult problem. It was solved originally by Christodoulou and Klainerman [10]. A simpler proof has since been developed [30]. Decay estimates for solutions of nonlinear wave equations played an important role in both proofs. In relativity, the structure of space–time is determined by a Lorentzian pseudo-metric, g, which, in the absence of matter, satisfies the Einstein equations, Rμν = 0.
(1.1)
The Schwarzschild and Reissner–Nordstrøm solutions are singular solutions to the Einstein equations. They describe space–times, each containing a spherically symmetric black hole, and are characterized by the mass M and charge Q of the black hole. The Schwarzschild solution is the special case of the Reissner–Nordstrøm solution when Q = 0. We will restrict our attention to the exterior region, outside the black hole. (We will not deal with the supercritical case, |Q| > M, in which there is no black hole.) The boundary of the exterior region is called the event horizon. There are no singularities in the exterior region of the black hole. We discuss the geometry of the exterior region further in Section 2.1. Because the structure of the Einstein equations makes
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
3
it impossible for these to have spherically symmetric perturbations, it is expected that the study of black-hole stability will require analysis of the more general, non-spherically-symmetric, rotating, Kerr–Newman black holes. In 1957, Regge and Wheeler investigated the linear stability of the Schwarzschild black hole, and were able to reduce the problem to the study of a second-order, scalar equation [34]. (Although, this did not govern all components [43].) After appropriate transformations, this equation and the geometrically defined, scalar wave equation differ only by a multiple of the potential term appearing in each. Because of the intrinsic importance of the geometric wave equation and because of its possible applications to the study of black hole stability, we consider the wave equation defined by the Reissner–Nordstrøm metric in the exterior region of the corresponding black hole space–time. We only consider the non-super-critical case with |Q| M. We use RN to represent the second order d’Alembertian defined by the Reissner–Nordstrøm pseudo-metric. In a particular choice of coordinates, the exterior region can be written as (t, ρ∗ , θ, φ) ∈ R × M = R × R × S 2 . Far from the black hole, the t coordinate is similar to the time coordinate in R1+3 . We consider initial data, u˜ 0 and u˜ 1 , defined on the Cauchy surface t = 1, but this is equivalent to considering initial data on any hypersurface of constant t. Using this notation, we can write the initial value problem for the wave equation as RN u˜ = 0,
(1.2)
u(1) ˜ = u˜ 0 ,
(1.3)
˙˜ u(1) = u˜ 1 .
(1.4)
This equation is “decoupled,” in the sense that the solution to the wave equation does not influence the original manifold. For this equation, we prove the following result: Theorem 8.21. If u˜ is a solution to the wave equation (1.2)–(1.4), and ε > 0, then 1 2 F 2 u˜ 6 ˜ t − 13 C u0 2 2 + E[u0 , u1 ] + EC [u0 , u1 ] + L u0 L2 (M) L (M) L (M) 1 + E L u0 , L u1 2 , (1.5) 1 1 − 2 1 + ρ 2 2 u˜ 2 ˜ t − 2 C u0 2 2 + E[u0 , u1 ] + EC [u0 , u1 ] + L u0 L2 (M) ∗ L (M) L (M) 1 + E L u0 , L u1 2 , (1.6) where C is a positive constant depending upon ε only, r and ρ∗ are radial coordinates defined in Section 2, (u0 , u1 ) = (r u˜ 0 , r u˜ 1 ) are simplified functions defined in Section 2, F is a weight which vanishes near the event horizon and which is defined in Section 2, the measures in Lp (M) ˜ are defined in Section 2, E[v, w] and EC [v, w] are the energy and conformal charge and Lp (M) defined in Section 3, and L is an angular derivative operator defined in Section 7. We expect the loss of angular derivatives is necessary, but it is known that the rate of decay in time can be improved. By angular derivative loss, we mean that more derivatives are required on the initial data than are controlled on the solution. Our method for obtaining this result follows from our earlier integrated space–time estimates for the wave equation and Regge–Wheeler equation [4,5]. The original proof of the results in this paper and our previous work contained a
4
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
gap in the local decay argument.1 The results in [4,5] have been recovered in [6,8]. At about the same time as the announcement of the results in this paper, stronger decay rates were obtained, but with a loss of more derivatives [9,15]. The results in [9] give t −1 decay for the L∞ loc norm −1/2 of u˜ in regions bounded away from the event horizon and |t − |ρ∗ || decay globally, and are sufficiently general to cover the Regge–Wheeler equation and some small-data, nonlinear problems. The results in [15] give a similar decay rate for the linear wave equation, can be extended to the event horizon, and apply to initial data which does not vanish on the bifurcation sphere, where the surface t = 1 meets the event horizon. The results in this paper and in [9] require the initial data to vanish at the bifurcation sphere. This vanishing is a technical requirement and is not natural for this problem, since the Reissner–Nordstrøm manifold extends beyond the bifurcation sphere. Because we use rough initial data, it is impossible to obtain L∞ estimates. The decay in this paper and in [9,15] rely on controlling EC . We obtain bounds like EC ∼ Ct; the bound obtained in [9,15] is EC ∼ C, which corresponds to a decay of the L6 norm like t −2/3 . The EC ∼ C bound is analogous to that in Minkowski space, R1+3 . All these results rely on local decay estimates using a radial multiplier which is defined separately on each spherical harmonic. Recently, we have used a numerical solution to an ODE to provide a new proof of the needed local decay result, which applies to some large-data, nonlinear problems and does not require a spherical harmonic decomposition [8]. This has also been shown analytically, at the cost of an additional derivative [16]. The ε derivative loss occurs in the local-decay estimate, and such an estimate cannot hold if there is no derivative loss. In a preprint [7], we have shown it is possible to obtain the t −2/3 decay rate with only an loss of regularity by combining the methods in [9,15] with those in this paper. For decoupled, linear waves on black hole backgrounds, the first rigorous result, of which we are aware, was that solutions to the wave equation remain bounded if certain norms of the initial data are finite [41]. These norms used an L2 space which required the initial data to vanish near the bifurcation sphere. Later work removed this restriction and showed that solutions remained bounded even if the initial data did not vanish on the bifurcation sphere [29]. Many other results concern scattering theory. For the linear wave equation, there is existence of wave operators and asymptotic completeness for the linear wave equation [19] and the massive, Klein–Gordon equation [20]. Asymptotic completeness and global existence for the cubic semilinear wave equation have been proven for a wide class of black hole manifolds, including the Kerr–Newman solutions [1,2,26,33]. This gave a map from the t = C initial data surface to the (future) event horizon and null infinity for initial data which is compactly supported away from the bifurcation sphere. Asymptotic completeness has also been proven on a class of noncompact manifolds, including the Reissner–Nordstrøm manifolds [17]. Other results have been proven using an analogue of spherical harmonics for the nonspherical, Kerr black hole. These require smooth initial data with support away from the bifurcation sphere. The linearization of the Einstein equations around the Kerr solution has been shown to have no unstable modes [42]. For each component in such a decomposition, the L∞ loc norm of solutions to the wave equation have been shown to decay [24] and solutions to the Dirac equation decay at a rate of t −5/6 in L∞ loc [23]. The problem of fields coupled to the Einstein equation is very challenging. The main result for coupled fields is that spherically symmetric solutions to the wave equation coupled to the Einstein 1 We would like to thank the referee for bringing this to our attention. The corrected method which we use follows from arguments due to J. Sterbenz [9]. Other methods also exist [15].
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
5
equations decay along the event horizon according to an inverse cubic, Price law [13] (actually a −3 + decay rate). Decay estimates along the event horizon are important for understanding the structure of the interior region and the central singularity [12]. Following from this, a weaker decay rate has been proven for decoupled, radial, semilinear wave equations in the exterior of Reissner–Nordstrøm solutions [14]. Without using spherical symmetry, several results have been obtained using control on localized L2 norms of the solution or its derivative. For spherically symmetric initial data and sufficiently supercritical Reissner–Nordstrøm solutions, Strichartz estimates have been proven for solutions to the wave equation using localized L2 norms [39]. We obtained integrated, space– time, L2 estimates in our earlier results [4,5] (corrected in [6,8]). As noted earlier, recent results give t −1 decay for initial data which is sufficiently smooth [9,15]. We start to prove Theorem 8.21 by introducing an analogue of the conformal charge used in R1+3 [25]. However, we are prevented from completing the argument which holds for the wave equation in Euclidean space by the presence of a geodesic surface, which orbits the black hole. This surface behaves like a closed geodesic surface in a Riemannian manifold. The absence of such geodesics is a non-trapping condition which is commonly imposed in scattering theory. In the full 1 + 3-dimensional space–time, these geodesics orbit the black hole. The presence of this geodesic surface and the gravitational lensing these geodesics cause is already known [11,40]. This surface is called the photon sphere. To overcome the obstacle created by this surface, it is sufficient to prove estimates on the angular derivative of the solution in the region near the closed geodesics. We prove: Theorem 8.20. If u˜ is a solution to the wave equation (1.2)–(1.4), and ε > 0, then the simplified function u = r u˜ satisfies ∞ 1−ε L χα u2 2
L (M)
dt < C u0 2L2 (M) + E[u0 , u1 ] ,
(1.7)
1
where χα is a function with compact support near the orbiting geodesic surface, and the remaining quantities are the same as those in Theorem 8.21. This is sufficient to prove Theorem 8.21. The loss of L is responsible for the additional factors of L appearing in Theorem 8.21. Estimates on the angular derivative have already been used to prove Strichartz and pointwisein-time Lp estimates in Euclidean space [31,38] and on non-trapping manifolds [27]. Our method is to introduce a Heisenberg-type relation for the wave equation which is an analogue of the Heisenberg relation for the Schrödinger equation. There is a self-adjoint operator H which determines the time evolution of solutions to the wave equation (1.2)–(1.4). We refer to this operator as the Hamiltonian and other self-adjoint operators as observables. For the Schrödinger equation, the Heisenberg relation relates the time derivative of the expectation value of an operator u, Au to the expectation value of the Hamiltonian with the operator u, [H, A]u. For the wave equation, the Heisenberg-type relation is similar, but the expectation value of the operator is replaced by a more complicated inner product involving both u and u. ˙ If an observable satisfies certain positivity conditions for the commutator, it is called a propagation observable. The first propagation observable that we use is a radial derivative operator directed away from the geodesic surface. This is used to prove a smoothed Morawetz estimate, which is like
6
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
Theorem 8.20, but with zero powers of the angular derivative and a noncompactly supported, but decaying, localization function. A weight, g, of the radial variable is used to direct this radial derivative away from the geodesics. The main technical innovation for proving Theorem 8.20 is the introduction of improved propagation observables. We define the positive operator L by L2 = 1 − S 2 ,
(1.8)
where S 2 is the Laplace–Beltrami operator on the sphere, which acts as the derivative in the angular directions. By rescaling the argument of g by powers of L, we generate a new propagation observable, which we use to prove Theorem 8.20 with ε = 1/4. We call this rescaling by L angular modulation. To prove the result for all positive ε, we use phase space analysis. Typically, in phase space analysis, solutions to the wave equation are localized in space or frequency by the application of a localizing function to the solution or its Fourier transform. (The Fourier transform is taken in the spatial variables only and not the time variable.) This localization can also be described as applying a compactly supported multiplication operator or Fourier multiplication operator. To control additional angular derivatives, we introduce phase-space variables which are rescaled by L. These are x m = Lm ρ ∗ , ξ n = −iLn−1
(1.9) ∂ . ∂ρ∗
(1.10)
We localize a propagation observable in phase space by multiplying by compactly supported functions of the phase space variable, X(xm ) and Φ(ξ n ). We also localize with functions which are not compactly supported but which decay rapidly outside a region of a certain scale. By both rescaling the argument of g and localizing in ξ n , we can control powers of L in various regions of phase space. This means the space–time L2 norm of X(xm )Φ(ξ n )L1−ε u is controlled by the energy and L2 norm of the initial data. The scales of X(xm ) and Φ(ξ n ) are chosen so that there is no violation of the uncertainty principle. However, numerous lower order terms, or error terms, are generated by rearranging the phase space localizations. In a process of phase space induction, we use the estimate in one region of phase space to control the error terms in another. Thus, we are able to remove the X(xm ) and Φ(ξ n ) localizations and to prove Theorems 8.20 and 8.21. This heuristic is explained further in Section 8. 1.1. Structure of the paper Section 2 describes the wave equation on the Reissner–Nordstrøm solution, some common notation and transformations to simplify the equation, and discusses important regions of the geometry, including the photon sphere. Section 3 describes the energy, conformal charge, and a Sobolev estimate. In that section, we prove pointwise-in-time Lp estimates can be reduced to bounding weighted space–time integrals of solutions and their angular derivatives. Section 4 translates some of our conditions into terminology commonly used by physicists. In particular, we show that the finite L2 , energy, and conformal charge conditions, require that initial data, but not its derivatives, vanish at the bifurcation sphere.
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
7
The remainder of the paper deals with proving weighted space–time integral bounds. Section 5 provides the basic propagation observable framework, using the Heisenberg-type relation. In Section 6, we use a propagation observable, γ , to control the time integral of an L2 norm with a decaying weight. This is analogous to a smoothed Morawetz estimate. The section concludes with an L6 estimate for radial data. Section 7 introduces the new technique of angular modulation to control 3/4 angular derivatives in L2 in the desired region. In the final section, Section 8, we introduce a family of propagation observables to control 1 − angular derivatives in L2 . The propagation observables are like the one from the angular modulation argument, but are also localized in the phase space variables xm and ξ n . We refer to arguments involving such localization as phase space analysis. In Section 8.1, we define the rescaled phase space variables, several localization functions, and the propagation observables Γn,m . In Section 8.2, we introduce commutator results to allow us to rearrange phase space localizations at the expense of lower order terms. The lower order terms have 1 + 2n angular derivatives in expectation value. We prove positive commutator results localized in both phase space variables, xm and ξ n , in Section 8.3 and in only ξ n in Section 8.4. In Section 8.5, we use a finite induction on n to control the lower order terms, which allows us to prove Theorem 8.20. This allows us to prove the L6 estimate in Theorem 8.21. Throughout this paper, the notation C is used to denote constants which may vary from equation to equation. Indices are used to separate different constants within a line of an equation and to refer to constants later in a proof. In most cases, such constants are understood to be positive. 2. The wave equation on the Reissner–Nordstrøm space 2.1. The Reissner–Nordstrøm solution The Reissner–Nordstrøm solution to the Einstein equations represents the space–time outside a spherically symmetric, charged, massive body. It is the unique spherically symmetric, static,2 asymptotically flat solution to the vacuum Einstein–Maxwell equations, which govern the structure of space–time in the presence of gravity and electromagnetic fields. This solution can be represented in terms of the co-ordinates (t, r, θ, φ) by the Lorentzian pseudo-metric: ds 2 = F dt 2 − F −1 dr 2 − r 2 dθ 2 + sin2 θ dφ 2 , F =1−
Q2
2M + 2, r r
(2.1) (2.2)
where M 0 is the central mass and Q ∈ R is the central charge, both as measured by observers at infinity. As r → ∞, the terms in the metric approach those of the metric on R1+3 expressed in spherical coordinates, and the Reissner–Nordstrøm solution is said to be asymptotically flat. For r > r0 , the Reissner–Nordstrøm solution is a physically reasonable model for the region of space–time outside a spherically symmetric body with mass M, electric charge Q, and radius r0 . The polynomial r 2 F has roots at r± = M ±
M 2 − Q2 .
(2.3)
2 The static condition is redundant since spherically symmetric, vacuum solutions to the Einstein–Maxwell equations are necessarily static.
8
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
The Reissner–Nordstrøm solution is said to be subcritical, critical, or supercritical if there are two, one, or no real roots. To highlight the different behaviour, we will sometime refer to these as different Reissner–Nordstrøm solutions. Since we take the Reissner–Nordstrøm solutions as fixed background space–times, we will typically refer to these solutions as the Reissner– Nordstrøm spaces or manifolds and use the word solution to refer to solutions of the wave equation (1.2)–(1.4). In the non-super-critical case, |Q| M, the metric (2.1) defines a Lorentz manifold on the exterior region, (t, r, θ, φ) ∈ R × (r+ , ∞) × S 2 . The exterior can be extended to a larger Lorentz manifold. The maximal analytic extension is well known and can be found, with an explicit choice of coordinates, in many relativity textbooks [21,32]. The metric (2.1) clearly defines a Lorentz metric on the regions corresponding to r in (0, r− ), (r− , r+ ), or (r+ , ∞). (It also defines a Lorentz pseudo-metric on r ∈ (−∞, 0), but through the substitutions r → −r and M → −M, this is the same as considering a negative mass solution, which is rejected on physical grounds.) Regardless of Q, as r → 0+ , the curvature polynomial generated by contracting the Riemann curvature with itself diverges, Rabcd R abcd → ∞, and the solution is said to be singular as r → 0. In the intervals (r− , r+ ), since F is negative, the variable r in (2.1) plays the role of a time-like coordinate. The Schwarzschild solution, with Q = 0, has the simplest extension. In this case r− = 0, so there are only two regions to consider: the interior where 0 < r < r+ = 2M and the exterior r > r+ = 2M. Along null geodesics as r → 2M, the coordinate t must diverge to plus or minus infinity. The exterior region where r > 2M can be smoothly connected to two interior regions where r < 2M, one in the past and one in the future. The maximal analytic extension of the Schwarzschild solution consists of a past interior region connected to two exterior regions which are both connected to the same future interior region. The four regions meet along a single sphere, known as the bifurcation sphere. The boundary between these regions, on which r = 2M, are null hypersurfaces and are called event horizons. Null or time-like curves, which represent the trajectories of particles or light rays, can cross these null hypersurfaces in only one direction. Since anything can enter the future interior region, but nothing, not even light rays can escape, the future interior region is called a black hole. In the maximal analytic extension of the general subcritical case, 0 < |Q| < M, there are infinitely many copies of each of the three regions. Each intermediate region, with r ∈ (r− , r+ ), is connected to a pair of exterior regions, with r ∈ (r+ , ∞), in either the past or the future and to a pair of inner regions, with r ∈ (0, r− ), in the opposite direction. Each such pair of exterior regions is connected to a future and a past intermediate region. The pair of exterior regions and the two adjacent intermediate regions all meet on a single sphere, which we also refer to as the bifurcation sphere. Each pair of inner regions is also connected to a future and a past intermediate region. Thus, the maximal extension consists of a “ladder,” with each rung consisting of a pair of exteriors, followed by an intermediate region, followed by a pair of interior regions, followed by another intermediate region, which is finally followed by the next pair of exterior regions, which make up the next rung. In the critical case, |Q| = M, each exterior region, with r ∈ (r+ , ∞), has a past and a future inner region, with r ∈ (0, r+ ), and each inner region has a past and a future exterior region. In the supercritical case, |Q| > M, the roots r± are complex, and there is only one region with r in (0, ∞). For a non-super-critical solution, |Q| M, any hypersurface t = C is a Cauchy surface for the exterior region in which it lies. In the supercritical case, |Q| > M, geodesics can pass from the central singularity at r = 0 to infinity, and there are no Cauchy surfaces for the exterior region.
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
9
In the subcritical case, |Q| < M, since the coefficient of dr 2 vanishes linearly, the length of radial curves in the hypersurface t = C, from (t, r+ , θ φ) to (t, r, θ, φ), goes like 1
s ∼ (r − r+ ) 2
(2.4)
as r approaches r+ from above. The finiteness of this length is consistent with the bifurcation sphere being a set of regular points in the maximally extended space–time. On the other hand, in the critical case, |Q| = M, the length of such curves, from (t, r0 , θ, φ) to (t, r, θ, φ), goes like s ∼ ln(r − r+ ) − ln(r0 − r+ ), which is consistent with the absence of a bifurcation sphere in the critical case. In the critical case, the limit r → r+ along t = C hypersurfaces corresponds to approaching a point at infinity. In the subcritical case, |Q| < M, any t = C hypersurface can be smoothly continued through the bifurcation sphere to form a Cauchy surface for both exterior regions and the adjacent intermediate regions. In the Schwarzschild case, |Q| = 0, this produces a Cauchy surface for the entire maximally extended space–time. From the standard properties of Cauchy surfaces it is possible to perturb these Cauchy surfaces so that they intersect the event horizons at places other than the bifurcation sphere, while still remaining Cauchy surfaces. Such Cauchy surfaces are interesting, but fall outside the type of initial data surfaces the method of this paper can handle. Similarly, the natural conditions to impose would simply be a certain level of regularity everywhere and a certain degree of decay at infinity. As explained in Section 4, we also require additional decay at the bifurcation sphere. In the critical case, |Q| = M, the t = C hypersurfaces are inextendible. Furthermore, any perturbation of them which crosses the event horizon will fail to be a Cauchy surface for the entire exterior region. If it crosses the future even horizon, then it will fail to be a Cauchy surface for the past part of the exterior region, and similarly, if it crosses the past event horizon, it will fail to be a Cauchy surface for the future part of the exterior region. We will restrict our attention to the exterior region of the non-super-critical solutions, t ∈ R,
r > r+ = M +
M 2 − Q2 ,
(θ, φ) ∈ S 2 .
(2.5)
We will always pose our initial data on the hypersurface t = 1, which is a Cauchy surface for this exterior region. The exterior region of the Schwarzschild solution, ds 2 = F dt 2 − F −1 dr 2 − r 2 dθ 2 + sin2 θ dφ 2 , F =1− t ∈ R,
2M , r
r > r+ = 2M,
(2.6) (2.7)
(θ, φ) ∈ S 2 ,
(2.8)
is representative of all the subcritical solutions. The critical solution is similar; however, F vanishes quadratically instead of linearly towards the horizon, and this affects the rate of decay of other quantities. In studying the exterior region of the Reissner–Nordstrøm or Schwarzschild solutions it is common to introduce a new radial co-ordinate, the Regge–Wheeler tortoise co-ordinate, r∗ , defined by dr = F. dr∗
(2.9)
10
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
This co-ordinate extends from −∞ to ∞ and has the effect of “pushing the horizon to negative infinity.” The original radial co-ordinate, r, is now treated as a function of r∗ . The exterior region of the Reissner–Nordstrøm solution is represented by ds 2 = F dt 2 − F dr∗2 − r 2 dθ 2 + sin2 θ dφ 2 , F =1−
Q2
2M + 2, r r
r∗ ∈ R,
t ∈ R,
(2.10) (2.11)
(θ, φ) ∈ S 2 .
(2.12)
In the Schwarzschild case, r∗ can be expressed simply in terms of r and M as
r − 2M r∗ = r + 2M ln M
+ C∗ ,
(2.13)
= 1,
1 r − 2M lim r −1 ln = . r∗ →−∞ ∗ M 2M
(2.14)
and has simple asymptotic behaviour, lim r −1 r r∗ →∞ ∗
(2.15)
In the literature, C∗ is commonly taken, for simplicity, to be −2M log(2) [32] or −2M log(M/2), see [21]. In Eq. (2.33), we will use a particular choice of C∗ , coming from the geometry of the Reissner–Nordstrøm solution. In the general subcritical case, |Q| < M, the expression for r∗ is slightly more complicated, but the asymptotics are the same. These are
2 2 r− r+ r − r+ r − r− − + C∗ , ln ln r∗ = r + r+ − r− M r+ − r− M lim r −1 r r∗ →∞ ∗
= 1,
r+ − r− r − r+ lim r∗−1 ln = . 2 r∗ →−∞ M r+
(2.16) (2.17) (2.18)
In the critical case, |Q| = M, the expression for r∗ and the asymptotics are inverse linear, instead of logarithmic, towards the event horizon. The relevant expressions are r∗ = r + 2M ln
r −M M
lim r −1 r r∗ →∞ ∗
−
M2 + C∗ , r −M
= 1,
lim r∗ (r − M) = −M 2 .
r∗ →−∞
(2.19) (2.20) (2.21)
We conclude this subsection by describing some notation. We use RN to denote the d’Alembertian operator defined by the Reissner–Nordstrøm pseudo-metric. If u is a function in the
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
11
exterior region of the Reissner–Nordstrøm solution, we use u˙ to denote the t derivative, u to denote the r∗ derivative, and ∇S 2 u to denote the derivative with respect to the angular derivatives. (∇S 2 u is a vector tangent to S 2 .) We use S 2 to denote the Laplace–Beltrami operator on S 2 and S 2 u to denote the spherical Laplacian of the function defined on the sphere by taking t and r∗ fixed and allowing the angular variables to vary. For a function u defined in the exterior region of the Reissner–Nordstrøm solution, we use u(t, r∗ , θ, φ) to denote the value of the function at the point with coordinates (t, r∗ , θ, φ) and use u(t1 ) to denote the restriction of u to the space-like hypersurface on which t = t1 . The definition of r∗ contains a free parameter C∗ . An explicit choice of C∗ is made in the following subsection. With this choice, we use ρ∗ to denote r∗ . Since f is used to denote the derivative with respect to r∗ = ρ∗ , for a function of one variable, f , we use f [k] to denote the kth derivative of f with respect to its argument, even in the case k = 1. 2.2. The wave equation We wish to study the linear wave equation for u˜ : R × R × S 2 → C, RN u˜ = 0,
u(1) ˜ = u˜ 0 ,
∂ u(1) ˜ = u˜ 1 , ∂t
(2.22)
in the exterior region of the Reissner–Nordstrøm solution. In terms of the tortoise co-ordinate the d’Alembertian is 2
−1 ∂ −2 ∂ 2 ∂ RN = F − r −2 S 2 . −r r (2.23) ∂r∗ ∂r∗ ∂t 2 The substitution u = r u˜
(2.24)
simplifies the wave equation to ∂2 u + H u = 0, ∂t 2
u(1) = u0 ,
∂ u(1) = u1 , ∂t
(2.25)
where the operator H is composed of the following terms: H=
3
Hi ,
(2.26)
∂2 , ∂r∗2
(2.27)
i=1
H1 = −
1 dF H2 = V (r) = F , r dr H3 = VL (r)(− S 2 ), 1 VL (r) = 2 F. r We refer to V as the potential and VL as the angular potential.
(2.28) (2.29) (2.30)
12
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
The solution u is a function with domain and range u : R × R × S 2 → C. Typically, solutions to the wave equation are taken to be real-valued, since the real and imaginary parts of a complex solution are each real-valued solution. We consider complex solutions because of the possibility that the operators which we use in Section 8 may mix the real and imaginary components. In the exterior region, r > r+ , the angular potential has a single critical point at (in terms of the r variable)3 α=
3M +
9M 2 − 8Q2 . 2
(2.31)
The critical point, α, is extremely important. Geodesics at r = α and tangent to this surface will remain on the surface forever, and geodesics which approach this surface almost tangentially can approach the surface as t → ∞ or orbit the black hole arbitrarily many times before escaping to r∗ → ±∞ [11,40]. This geodesic surface at r = α is called the photon sphere. The region near r = α will also be the most difficult in which to prove decay of solution to the wave equation. This region only presents a problem for data composed of infinitely many spherical harmonics. The value of r∗ corresponding to r = α will be denoted α∗ , and we introduce the new radial co-ordinate ρ∗ = r∗ − α∗ .
(2.32)
This corresponds to taking r∗ = ρ∗ with the integration constant C∗ in Eqs. (2.13), (2.16), or (2.19) chosen so that α∗ = 0.
(2.33)
For this reason, ∂r∂∗ = ∂ρ∂ ∗ , etc. For simplicity, we typically use the measure d 3 μ = dρ∗ d 2 μS 2 ,
(2.34)
where d 2 μS 2 = sin θ dθ dφ. The space M refers to R × S 2 with the measure d 3 μ. Unless otherwise specified, all integrals in this paper are over M = R × S 2 . This measure defines L2 (M) and, more generally, Lp (M) for any p 1. Unless otherwise specified, we use · and ·,· to denote the L2 (M) norm and inner product, and we use · p to denote the Lp (M) norm. The norm of a (T S 2 -valued) vector is equal to the norm of the length, ∇S 2 ψX = (|∇S 2 ψ|)X , for any function space X. 1
The function space H 1 (M) is the collection of functions for which u and VL2 ∇S2 u are in L2 (M). This space is not particularly useful, and we typically consider functions with finite energy or conformal charge, as defined in Section 3. For an operator B acting on L2 , we use B to denote the operator norm. 3 For the critical Reissner–Nordstrøm solution, the angular potential has a second critical point at the event horizon r = M.
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
13
When dealing with the original function, u, ˜ we also use the measure d 3 μ˜ = r 2 dρ∗ d 2 μS 2 ,
(2.35)
˜ to refer to R×S 2 with this measure. This defines Lp (M). ˜ The two L2 spaces coincide, and use M in the sense that ˜ L2 (M) uL2 (M) = u ˜ .
(2.36)
For functions of a single variable, f : R → C, we also use Lp and f p to denote the standard spaces and norms with respect to the standard measure on R. The Schwartz space S refers to functions v : M → C which are infinitely differentiable and decay sufficiently rapidly at infinity that Lp
∂ ∀i, j, k ∈ N ∃Ci,j,k : ρ∗i ∂ρ
j ∗
∇Sk2 v < Ci,j,k .
(2.37)
The space S(R × M) refers to functions on v : R × M → C which are infinitely differentiable and for which, for all t ∈ R, the restricted functions u(t) : M → C and u(t) ˙ : M → C are in S. The Fourier transform on R × S 2 is defined by first making a spherical harmonic decomposition in the angular variable, and then applying the one-dimensional Fourier transform on each spherical harmonic. 3. Methods for pointwise decay In Minkowski space, R1+3 , there is a conformal charge which is conserved and which dominates t 2 r −1 ∇S 2 u2 . Using a Sobolev estimate, the decay of the angular component of the H 1 (R3 ) norm implies decay of the L6 (R3 ) norm [25]. We introduce an analogous conformal charge for the Reissner–Nordstrøm solution. The growth of this charge is dominated by the sum of localized space–time integrals of u and its angular derivative. Using a Sobolev estimate and the conformal charge, we reduce the proof of a pointwise-in-time Lp (M) estimate to bounding localized space–time integrals of a solution and its angular derivative. 3.1. Densities, energy conservation, and the conformal charge There are many formalisms for studying wave equations. In this paper, we use L2 -based integrals related to the energy. In this subsection, we will introduce various densities and show that they satisfy differential relations. From these relations, we verify the conservation of energy and find an estimate on the growth of a conformal charge, which is central in the proof of the main estimates of this paper. Definition 3.1. Given a pair of functions, (v, w) ∈ S × S, the energy, radial momentum, and angular momentum densities are defined respectively by 1 2 |w| + |v |2 + V |v|2 + VL ∇S 2 v¯ · ∇S 2 v , 2 pρ∗ [v, w] = (wv ¯ ), pω [v, w] = (w∇ ¯ S 2 v). e[v, w] =
(3.1) (3.2) (3.3)
14
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
The energy is defined to be E[v, w] =
e[v, w] d 3 μ.
(3.4)
These densities satisfy the following differential relations. Lemma 3.2. If u ∈ S(R × M) is a solution to the wave equation (2.25), then the following relations hold ∂ ∂ ˙ − ∇S 2 · VL pω u(t), u(t) ˙ , pρ∗ u(t), u(t) e u(t), u(t) ˙ − ∂t ∂ρ∗ 2 2 2 2 ∂ 1 ∂ u(t) ˙ + u (t) − V u(t) − VL ∇S 2 u(t) ˙ − 0 = pρ∗ u(t), u(t) ∂t ∂ρ∗ 2 − ∇S 2 · u (t)VL ∇S 2 u(t)
0=
1 1 − u(t)V u(t) − ∇S 2 u(t) · VL ∇S 2 u(t), 2 2
(3.5)
(3.6) (3.7)
and the energy is conserved d E u(t), u(t) ˙ = 0. dt
(3.8)
Proof. The relations are proven using the method of multipliers, in which both sides of the wave equation are multiplied by a quantity, typically a differential operator acting on u, and then the right-hand side is rearranged. From multiplying the wave equation by ∂t∂ u, we find the relation for the derivative of the energy density, 0=
∂ u¯ u¨ − u + V u + VL (− S 2 )u ∂t
∂ ˙ ˙¯ u + ∇S 2 u˙¯ · (VL ∇S 2 u) − ∇S 2 · (uV ˙¯ L ∇S 2 u) = u˙¯ u¨ + u˙¯ u − (uu ¯ ) + uV ∂ρ∗ ∂ 1 ∂ ˙ ˙¯ ) − ∇S 2 · (uV ˙¯ L ∇S 2 u) (u¯ u˙ + u¯ u + uV ¯ u + ∇S 2 u¯ · VL ∇S 2 u) − (uu 2 ∂t ∂ρ∗ ∂ ∂ ˙ − pρ [u, u] ˙ − ∇S 2 · VL pω [u, u] ˙ . = e[u, u] ∂t ∂ρ∗ ∗ =
(3.9) (3.10) (3.11) (3.12)
Since the integral of a pure spatial derivative is identically zero, integrating this result gives d ˙ is zero and that the energy is conserved. that dt E[u(t), (t)] From multiplying the wave equation by ∂ρ∂ ∗ u, we find the relation for the derivative of the radial momentum,
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
15
∂ u¯ u¨ − u + V u + VL (− S 2 )u (3.13) ∂ρ∗ = u¯ u¨ − u¯ u + u¯ V u + u¯ VL (− S 2 )u (3.14)
∂ (u¯ u) = ˙ − u˙¯ u˙ − u¯ u + u¯ V u + ∇S 2 u¯ · VL ∇S 2 u − ∇S 2 · (u¯ VL ∇S 2 u) (3.15) ∂t
0=
=
∂ 1 ˙ ∂ pρ [u, u] (u¯ u˙ + u¯ u − uV ˙ − ¯ u − ∇S 2 u¯ · VL ∇S 2 u) ∂t ∗ ∂ρ∗ 2
(3.16)
1 1 ¯ u − ∇S 2 u¯ · VL ∇S 2 u − uV 2 2
(3.17)
− ∇S 2 · (u¯ VL ∇S 2 u).
(3.18)
2
Since E[u(t), u(t)] ˙ is conserved, when we are working with a solution to the wave equation, we will commonly write E for E[u(t), u(t)]. ˙ This is determined by the initial conditions, E[u(t), u(t)] ˙ = E[u0 , u1 ]. We now define the conformal charge in terms of the energy and momentum densities. In Minkowski space, the conformal multiplier is found by conjugating the time derivative with a discrete inversion of Minkowski space which is a conformal transformation. The conformal multiplier is then used to define the conformal charge and its density by the same process which defines the energy and energy density from the time derivative. The Reissner–Nordstrøm solution does not have the same discrete conformal transformation, so we define our conformal multiplier by formally taking the Minkowski conformal multiplier and replacing the Minkowski radial variable by ρ∗ , the Reissner–Nordstrøm radial variable. Definition 3.3. The conformal multiplier, the conformal charge density for a pair of functions (v, w) ∈ S × S, and the conformal charge for the same pair are defined respectively by ∂ ∂ C = t 2 + ρ∗2 + 2tρ∗ , ∂t ∂ρ∗ eC [v, w] = t 2 + ρ∗2 e[v, w] + 2tρ∗ pρ∗ [v, w], EC [v, w] = eC [v, w] d 3 μ.
(3.19) (3.20) (3.21)
The conformal charge density can be rewritten as a manifestly positive quantity. This form is more useful for making estimates. Lemma 3.4. For any pair (v, w) ∈ S × S, 1 1 eC [v, w] = (t − ρ∗ )2 |w − v |2 + (t + ρ∗ )2 |w + v |2 4 4 1 1 + t 2 + ρ∗2 V |v|2 + t 2 + ρ∗2 VL (∇S 2 v¯ · ∇S 2 v). 2 2
(3.22) (3.23)
16
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
Proof. We start by regrouping terms involving v and w, ¯ + t 2 |v |2 (t − ρ∗ )2 |w − v |2 = t 2 |w|2 − 2t 2 wv
(3.24)
− 2tρ∗ |w|2 + 4tρ∗ wv ¯ − 2tρ∗ |v |2 + ρ∗2 |w|2 − 2ρ∗2 wv ¯ + ρ∗2 |v |2 .
(3.25) (3.26)
From this, (t − ρ∗ )2 |w − v |2 + (t + ρ∗ )2 |w + v |2 = 2 t 2 + ρ∗2 |w|2 + |v |2 + 8tρ∗ (wv ¯ ).
2
(3.27)
In Minkowski space, the conformal multiplier is the generator of a conformal symmetry. From Noether’s theorem, the charge associated with a symmetry is conserved, and, similarly, the conformal charge can be modified to generate a conserved quantity. Our conformal multiplier is not constructed from a conformal symmetry and does not generate a conserved quantity. However, it is not unreasonable to expect that the conformal charge is almost conserved in some limit. Heuristically, the change of the Reissner–Nordstrøm conformal charge should only involve the potentials, since, formally, the conformal multiplier is the same as in R1+3 , and the wave equation differs only by the presence of potentials. In R1+3 , the analogue of VL is r −2 , and the corresponding term appearing on the right-hand side of the analogue Eq. (3.28), 2VL + rVL , vanishes. Lemma 3.5. If u ∈ S(R × M) is a solution to the wave equation (2.25), then d EC u(t), u(t) ˙ = dt
t 2V + ρ∗ V |u|2 d 3 μ +
t 2VL + ρ∗ VL |∇S 2 u|2 d 3 μ. (3.28)
Proof. In this proof, we suppress the argument of e, pρ∗ , and pω . In all cases, the argument is (u, u). ˙ We multiply the wave equation by the conformal multiplier, Cu, and then apply the relations from Lemma 3.2. This gives us (3.29) 0 = (C u) ¯ u¨ − u + V u + VL (− S 2 )u 2 = t + ρ∗2 u˙¯ u¨ − u + V u + VL (− S 2 )u (3.30) + 2tρ∗ u¯ u¨ − u + V u + VL (− S 2 )u (3.31)
∂ ∂ e− pρ − ∇S 2 · (VL pω ) (3.32) = t 2 + ρ∗2 ∂t ∂ρ∗ ∗
∂ ∂ 1 ˙ + 2tρ∗ ¯ u − ∇S 2 u¯ · VL ∇S 2 u) − ∇S 2 · (u¯ VL ∇S 2 u) pρ∗ − (u¯ u˙ + u¯ u − uV ∂t ∂ρ∗ 2 (3.33)
1 1 ¯ u + ∇S 2 u¯ · VL ∇S 2 u . (3.34) − 2tρ∗ uV 2 2
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
17
By integrating these terms and then integrating by parts in the angular derivatives, we can eliminate the angular gradients and find
2 ∂ ∂ 2 e− 0= t + ρ∗ pρ − ∇S 2 · (VL pω ) d 3 μ ∂t ∂ρ∗ ∗ ∂ ∂ 1 ˙ + 2tρ∗ pρ − (u¯ u˙ + u¯ u − uV ¯ u − ∇S 2 u¯ · VL ∇S 2 u) ∂t ∗ ∂ρ∗ 2
− ∇S 2 · (u¯ VL ∇S 2 u) d 3 μ
1 1 uV ¯ u + ∇S 2 u¯ · VL ∇S 2 u d 3 μ 2 2
2 ∂ ∂ 2 = t + ρ∗ e− pρ d 3 μ ∂t ∂ρ∗ ∗
∂ ∂ 1 ˙ + 2tρ∗ pρ∗ − (u¯ u˙ + u¯ u − uV ¯ u − ∇S 2 u¯ · VL ∇S 2 u) d 3 μ ∂t ∂ρ∗ 2
1 1 − 2tρ∗ uV ¯ u + ∇S 2 u¯ · VL ∇S 2 u d 3 μ. 2 2
(3.36)
−
(3.35)
2tρ∗
(3.37) (3.38) (3.39) (3.40)
This is further simplified by integrating by parts in the radial variable and isolating pure time derivatives to leave ∂ 2 t + ρ∗2 e + 2tρ∗ pρ∗ d 3 μ (3.41) 0= ∂t (3.42) + −2te − 2ρ∗ pρ∗ d 3 μ 1 + 2ρ∗ pρ∗ + 2t (u˙¯ u˙ + u¯ u − uV ¯ u − ∇S 2 u¯ · VL ∇S 2 u) d 3 μ (3.43) 2
1 1 ¯ u + ∇S 2 u¯ · VL ∇S 2 u d 3 μ − 2tρ∗ uV (3.44) 2 2 2 d = (3.45) t + ρ∗2 e + 2tρ∗ pρ∗ d 3 μ dt (3.46) − 2t V |u|2 + VL |∇S 2 u|2 d 3 μ
−
2tρ∗
1 2 1 2 V |u| + VL |∇S 2 u| d 3 μ. 2 2
Integrating in time gives the desired result.
(3.47)
2
3.2. Sobolev estimates Our goal in this section is to bound the L6 norm by the energy, the conformal charge, and a negative power of t. The main step in this is to prove a Sobolev estimate, which, roughly
18
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
speaking, controls the L6 norm by the H 1 norm. The analogous result in Minkowski space, R1+3 , is that the spatial L6 norm can be controlled by a third of a factor of the radial component of the H 1 norm and two thirds of the angular component of the inhomogeneous H 1 norm. The energy controls the radial component of the H 1 norm, and the conformal charge controls the product of a positive power of t and the angular component of the H 1 norm. It is not obvious from the definition that the conformal charge controls the weighted L2 norm needed in the inhomogeneous part of the angular H 1 norm, but it does [25]. A similar result holds for the Reissner–Nordstrøm solution, but, because of the weight appearing in the L6 norm, we require two estimates on weighted L2 norms instead of one. The two estimates in the following lemma are the estimates needed for the Sobolev estimate. Note that the terms appearing on the right are independent of w, the second argument of EC . Lemma 3.6. There is a positive constant, C, such that for all (v, w) ∈ S × S t 2 + ρ∗2 EC [v, w] C v, 2 v , ρ∗ + 1
1 EC [v, w]t −2 C v, 2 v . ρ∗ + 1
(3.48) (3.49)
Proof. This is a smoothed version of the argument used in R1+n [25]. The ingoing and outgoing wave terms can be isolated and rearranged. We introduce the notation 2EC ,(t,ρ∗ ) [v, w] = ρ∗ w2 + tw2 + ρ∗ v2 + tv2 + w, 4ρ∗ tv = tw + ρ∗ v 2 + ρ∗ w + tv 2
(3.50) (3.51)
to denote these terms. We introduce the weighted term hu = h(ρ∗ )u, with h real-valued, into these terms to obtain 2EC ,(t,ρ∗ ) [v, w] = tw + ρ∗ v − ρ∗ hv2 + ρ∗ w + tv + thv2
+ 2ρ∗ v , ρ∗ hv − 2tv , thv − ρ∗ hv, ρ∗ hv − thv, thv.
(3.52) (3.53)
Dropping the first two terms, which are strictly positive, and integrating by parts in the following pair yields 2EC ,(t,ρ∗ ) [v, w] − v, 2ρ∗ h + ρ∗2 − t 2 h v − v, ρ∗2 + t 2 h2 v .
(3.54)
We now choose h in terms of parameters a > 0 and > 0 to be ρ∗ . ρ∗2 + a
(3.55)
a − ρ∗2 . (ρ∗2 + a)2
(3.56)
h(ρ∗ ) = The derivative is h (ρ∗ ) =
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
19
We now substitute this choice of h into the previous calculations. This gives us that − 2ρ∗ h + ρ∗2 − t 2 h − ρ∗2 + t 2 h2
a − ρ∗2 (ρ 2 + t 2 )ρ∗2 2ρ 2 − 2 ∗ 2 = − 2 ∗ + ρ∗2 − t 2 2 2 ρ∗ + a (ρ∗ + a) (ρ∗ + a)2 = −
ρ∗4 + 3aρ∗2 − t 2 a + t 2 ρ∗2 ρ 4 + t 2 ρ∗2 − 2 ∗ 2 2 2 (ρ∗ + a) (ρ∗ + a)2
= − + 2
(ρ∗2
t 2 ρ∗2 ρ∗4 3aρ 2 −t 2 a − 2 ∗ 2 − 2 − + 2 . 2 2 + a) (ρ∗ + a) (ρ∗ + a) (ρ∗2 + a)2
(3.57) (3.58) (3.59)
For ∈ (−1, 0) the first, second, and forth terms are positive. For ∈ (− 12 , 0) and ρ∗2 > a4 , the forth term dominates the third by a factor of 2. We choose sufficiently close to zero so that for ρ∗2 a/4 0 −
−t 2 a −t 2 a + V ρ∗2 + t 2 = − 2 + H2 ρ∗2 + t 2 . 2 2 2 (ρ∗ + a) (ρ∗ + a)
(3.60)
Thus, with a = 1,
2 t + ρ2 2EC [v, w] 2EC ,(t,ρ∗ ) [v, w] + v, t 2 + ρ∗2 H2 v − − 2 v, 2 ∗ v . ρ∗ + 1
(3.61)
This proves the first statement in this lemma. The second part follows trivially by dropping the ρ∗2 /(ρ∗2 + 1) term from the first estimate and dividing by t 2 . 2 We now turn to the Sobolev estimate. This controls a L6 norm by the energy and conformal charge. Once again, the energy and conformal charge take a second argument w which does not appear in the quantity estimated. Lemma 3.7. There is a positive constant, C, such that for all (v, w) ∈ S × S 1 −2 F 2 r 3 v
L6 (M)
1 1 2 C E[v, w] + EC [v, w]t −2 6 EC [v, w] 3 t − 3 .
(3.62)
Proof. We begin by proving a Sobolev estimate which controls the L6 norm by weighted H 1 norms. Following the standard argument [22,37], the proof starts with a W 1,1 (R) → L3/2 (R) estimate. We take ψ(ρ∗ , ω) ∈ S. To this function ψ, we associate a function on the sphere, I1 (ω) =
∂ dρ∗ . ψ(ρ , ω) ∗ ∂ρ
R
∗
(3.63)
Integration and the Sobolev estimates for R and S 2 can be applied. Using the Sobolev estimate in one dimension, which is equivalent to the Fundamental Theorem of Calculus, and integrating
20
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
in the angular variables, we make the estimates ψ(ρ∗ , ω) I1 (ω), 3 ψ(ρ∗ , ω) 2 I1 (ω) 12 ψ(ρ∗ , ω) , 3 ψ(ρ∗ , ω) 2 d 2 μ 2 I1 (ω) 12 ψ(ρ∗ , ω) d 2 μ 2 . S S S2
(3.64) (3.65) (3.66)
S2
We now apply the Cauchy–Schwarz inequality to get
3 ψ(ρ∗ , ω) 2 d 2 μ 2 S
S2
1 I1 (ω) d 2 μS 2
S2
2
ψ(ρ∗ , ω) 2 d 2 μ
1 2
S2
,
(3.67)
S2
and estimate the second term on the right by a spherical Sobolev estimate. Sobolev estimates on S 2 follows from using a partition of unity on S 2 into co-ordinate charts and then applying the Sobolev estimate on R2 [22]. The notation ψ( 1 ) (ρ∗ ) is introduced to denote the L1 (S 2 ) norm ω of ψ(ρ∗ , ω) with ρ∗ fixed. Using this notation, S2
1 3 2 ψ(ρ∗ , ω) 2 d 2 μ 2 ∂ ψ ∇ 2 ψ 1 (ρ∗ ) + ψ 1 (ρ∗ ) . S S ∂ρ ) ) ( ( ω ω ∗ 1
(3.68)
For this proof, we introduce exponents α and β, which are only used in this proof and the following remark and which have no relation to the value of r = α governing the location of the photon sphere. We introduce a positive, radial weight f α (ρ∗ ) and integrate with respect to dρ∗ to get
3 f α (ρ∗ ) ψ(ρ∗ , ω) 2 d 2 μS 2 dρ∗
R S2
1 ∂ 2 f α ∇S 2 ψ( 1 ) (ρ∗ ) + ψ( 1 ) (ρ∗ ) dρ∗ , ∂ρ ψ ω ω
(3.69)
1 2α 3 2 f 3 ψ 3 ∂ ψ f α ∇ 2 ψ + f α ψ 3 . S 1 1 2 ∂ρ∗ 1
(3.70)
∗
1
R
The substitution ψ = f β |v1 |4 transforms this to a H1 → L6 Sobolev estimate,
1 3 2α +β f 3 |v1 |4 3 4 f β |v1 |3 ∂ v1 + β f β−1 f |v1 |4 d 3 μ 2 ∂ρ∗ 4
2 3 α+β 3 3 α+β 4 3 |v1 | |∇S 2 v1 | d μ + f |v1 | d μ , × 4 f
(3.71)
(3.72)
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
α+ 32 β
f
21
2 3
|v1 | d μ 6
3
2
1
1 2 6 6 3 ∂ f 2 3 β 4 |v1 | d μ f |v1 | d μ ∂ρ v1 d μ + f ∗ 1
1 3 3 2α+2β−1 6 3 2 3 2 3 |v1 | d μ × f |∇S 2 v1 | d μ + f |v1 | d μ . f
2β
6
3
(3.73)
(3.74)
We now choose 3 α + β = 2β = 2α + 2β − 1 2
⇒
1 α= , 2
β = 1,
(3.75)
so that f 1/3 v1 6 can be cancelled. Contrary to the convention in the rest of this paper, for this argument we use · 2 to denote the L2 norm. Cancelling the f 1/3 v1 6 factors,we obtain the general estimate 1 3 1 2 1 f 2 ∇ 2 v1 + f 12 v1 3 . f 3 v 1 4 ∂ v 1 + f v 1 S 6 2 2 ∂ρ∗ f 2 2
(3.76)
We take f = r −2 to give a result analogous to the Sobolev estimate in R3 ,
1 −2 3 −1 2 r 3 v1 C ∂ v1 + −2F r −1 v1 r ∇ 2 v1 + r −1 v1 3 . S ∂ρ 6 2 2 2 ∗ 2
(3.77)
To complete the proof of the Sobolev estimate for the Reissner–Nordstrøm solution, the sub1 1 1 stitution v1 = F 2 v and the inequality F 2 r −1 C(1 + ρ∗2 )− 2 are used to obtain
1 1 −2 3 F 2 r 3 v C F 12 v + 1 F 12 2Mr −2 − 2Q2 r −3 v + 2F 32 r −1 u (3.78) 2 6 2 2 2 1 2 1 (3.79) × F 2 r −1 ∇S 2 v 2 + F 2 r −1 v 2 3 1 − 1 1 C F 2 v 2 + 1 + ρ∗2 2 v 2 3 (3.80) 2 1 1 × F 2 r −1 ∇S 2 v 2 + F 2 r −1 v 2 3 . (3.81) We introduce the dummy function w to act as the second argument of the energy and conformal charge. From the definition of the energy, the conformal charge, and Lemma 3.6, 1 2 F 2 v v 2 E[v, w], 2
2
1 −1 F 2 r ∇ 2 v 2 EC [v, w]t −2 , S 2 1 −1 2 − 1 2 2 2 v CE [v, w]t −2 . F 2 r v C 1 + ρ C ∗ 2 2 Substituting these into Eq. (3.81) proves the result.
2
(3.82) (3.83) (3.84)
22
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
We remark that if we repeat the same argument with f = F r −2 and v1 = v, the Sobolev type result, analogous to Eq. (3.81), would be
1 1 −2 3 F 3 r 3 v C ∂ v + −2r −1 + 6Mr −2 − 4Q2 r −3 v ∂ρ 6
(3.85)
1 2 1 × F 2 r −1 ∇S 2 v + F 2 r −1 v 3 .
(3.86)
∗
The new weighted L2 norm is also controlled by Lemma 3.6, for t > 1, although, there is no longer the additional factor of t −2 : 1 2 2 2 2 −2r −1 + 6Mr −2 − 4Q2 r −3 v 2 t + ρ∗ u v EC [v, w]. ρ2 + 1 ∗
(3.87)
Combining these results, we have control of a norm which decays less rapidly towards the event horizon but at the cost of less time decay on the right-hand side: (3.88) |v|6 F 2 r −4 d 3 μ C E[v, w] + EC [v, w] EC [v, w]2 t −2 . 3.3. Local support of the trapping terms We refer to terms of the form 2V + ρ∗ V , for both the potential and the angular potential, as trapping terms. Recall that these are the weights determining the growth of the conformal charge in Section 3.1. In this subsection, we show that the trapping terms are positive only in a finite interval of ρ∗ values. The compactly supported functions W and WL will refer to the positive part of the trapping terms. Through the conformal identity and the Sobolev estimates, this reduces the of finding pointwise, weighted L6 estimates to proving local decay estimates of the form problem 2 3 tWL |∇S 2 u|2 d 3 μ dt C. tW |u| d μ dt + One of the factors in the derivative of the potential will require careful attention, both here and in Section 6.2. We introduce it with the notation P (r). This has coefficients depending on the background metric through the parameters M > 0 and |Q| M. Definition 3.8. The polynomial P (r) is defined by P (r) = 3Mr 3 − 4 Q2 + 2M 2 r 2 + 15MQ2 r − 6Q4 .
(3.89)
We now show that the trapping term for the potential is positive only in a bounded set of ρ∗ values by computing the limit at ±∞. Lemma 3.9. The derivative of the potential, V , is given by V = −2F r −7 P (r).
(3.90)
For |ρ∗ | sufficiently large, the trapping term 2V + ρ∗ V is negative. There is a compactly supported, non-negative, bounded function, W , such that W 2V + ρ∗ V .
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
23
Proof. The derivative is computed from the definition of V in Eq. (2.28) to be
dr ∂ ∂ −1 ∂F r F V= ∂ρ∗ dρ∗ ∂r ∂r ∂ r −1 − 2Mr −2 + Q2 r −3 2Mr −2 − 2Q2 r −3 =F ∂r ∂ 2Mr −3 − 2 Q2 + 2M 2 r −4 + 6MQ2 r −5 − 2Q4 r −6 =F ∂r = −2F r −7 3Mr 3 − 4 Q2 + 2M 2 r 2 + 15MQ2 r − 6Q4 .
(3.91) (3.92) (3.93) (3.94)
From this, 2V + ρ∗ V
ρ∗ dF − 6 P (r) dr r
2M 2Q2 = 2r −1 F − − r2 r3
2M 2Q2 = 2r −1 F − − r2 r3 = 2r −1 F
(3.95)
ρ∗ P (r) r6
ρ∗ 3M 4(Q2 + 2M 2 ) 15MQ2 6Q4 . − + − r r2 r3 r4 r5
(3.96) (3.97)
To show that this is negative for sufficiently large values of |ρ∗ |, we will multiply by a positive factor and then show that the resulting quantity has a negative limit as ρ∗ → ±∞. As ρ∗ → −∞, the subcritical and critical cases must be dealt with separately. In both the subcritical and critical cases, for ρ∗ → ∞, ρr∗ → 1 and F → 1, so r 3 (2V + ρ∗ V ) → 4M − 6M = −2M. For ρ∗ → −∞, the original radial variable has limit r → r+ = M + limiting value of P (r+ ) is
(3.98)
M 2 − Q2 , and the
P M + M 2 − Q2 = −2 2M 2 − Q2 M 2 − Q2 − 4M M 2 − Q2 M 2 − Q2 .
(3.99)
In the subcritical case, the term with −ρ∗ dominates, so it is sufficient to look at P (r+ ). We estimate this as r7 (2V + ρ∗ V ) → P (r+ ) 2F (−ρ∗ ) = −2 2M 2 − Q2 M 2 − Q2 − 4M M 2 − Q2 M 2 − Q2 < 0.
(3.100) (3.101) (3.102)
In the critical case, since P (r+ ) = P (M) = 0, it is necessary to multiply by a different positive factor before taking the limit. For ρ∗ → −∞, from the asymptotic behavior of ρ∗ given in Eq. (2.21),
24
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
1 r4 r −M ρ∗ 1 P (r) (2V + ρ∗ V ) = 2M + 2F r − M r − M r − M r3 3M = 2M + ρ∗ (r − M) 3 (r − 2M) r → 2M − 3M = −M.
(3.103) (3.104) (3.105) (3.106)
From these limits, it follows that 2V + ρ∗ V is negative for sufficiently large values of |ρ∗ |. Since V and ρ∗ V are continuous, it follows that 2V + ρ∗ V can be bounded above by a compactly supported, non-negative, bounded function. In particular, if W is the positive part of 2V + ρ∗ V , then it is a upper bound satisfying these conditions. 2 In the non-super-critical case, a similar calculation of limits as ρ∗ → ±∞ shows that 2VL + ρ∗ VL is positive only on a bounded set of ρ∗ values. Lemma 3.10. For |ρ∗ | sufficiently large, 2VL + ρ∗ VL is negative, and there is a compactly supported, positive, bounded function, WL , such that WL 2VL + ρ∗ VL . Proof. As in the previous lemma, we will bound 2VL + ρ∗ VL by a compactly supported, nonnegative function by looking at the limiting behavior as ρ∗ → ±∞. For this lemma, it is simplest to separate the subcritical and critical cases, when evaluating the limits. From the definition of VL ,
∂ −2F 3M 2Q2 + 2 , VL = 3 1 − ∂ρ∗ r r r
(3.107)
so that 2VL + ρ∗ VL
3M 2Q2 2F . + 2 = 3 r − ρ∗ 1 − r r r
(3.108)
In the subcritical case, as ρ∗ → ∞, by the explicit expansion of ρ∗ given in Eq. (2.16), r3 2VL + ρ∗ VL 2F
2 2 r− r+ 3M 2Q2 r − r+ r − r− − +C 1− + 2 =r − r + log log r+ − r− M r+ − r− M r r (3.109)
2 2 r− r+ r − r+ r − r− + + O(1) (3.110) log log =− r+ − r− M r+ − r− M → −∞. In the critical case, as ρ∗ → ∞, by the expansion in (2.19),
(3.111)
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
M2 r3 r −M 3M 2Q2 − (2V + ρ∗ V ) = r − r + 2M log +C 1− + 2 2F M r −M r r
r −M = −2M log + O(1) M → −∞.
25
(3.112) (3.113) (3.114)
For ρ∗ → −∞, we use both the asymptotics of ρ∗ from Section 2.1 and the geometry of the angular potential from Section 2.2. The critical points of VL are the roots of r 2 − 3Mr + 2Q2 , which we denote by α± =
3M ±
9M 2 − 8Q2 . 2
(3.115)
Only one of these, α+ = α is in the exterior region r > r+ . We now multiply the angular trapping term by a positive term. This gives r5 1 r3 2VL + ρ∗ VL = + r 2 − 3Mr + 2Q2 2F −ρ∗ −ρ∗
(3.116)
r3 + (r − α)(r − α− ). −ρ∗
(3.117)
=
In the subcritical case, the relevant terms are ordered α− < r+ < α+ , so that in the limit ρ∗ → −∞, the original radial function has limit r → r+ , and r5 1 2VL + ρ∗ VL → (r+ − α− )(r+ − α+ ) < 0. 2F −ρ∗
(3.118)
In the critical case, the relevant terms are not distinct, α− = r+ = M < 2M = α+ , so the previous argument does not hold. Instead, Eq. (3.117) can be rearranged as r5 2VL + ρ∗ VL = r 3 − ρ∗ (r − α)(r − α− ) . 2F
(3.119)
From the asymptotics of ρ∗ in Eq. (2.19), r 3 − ρ∗ (r − α− )(r − α+ )
M2 r −M 3 − + C∗ (r − M)(r − 2M) = r − r + 2M ln M r −M
r −M = r 3 + M 2 (r − 2M) + ln (r − M)2M(2M − r) M − (r − M)(r + C∗ )(r − 2M).
(3.120) (3.121) (3.122)
On the right-hand side, the first term is a polynomial in r and vanishes at r = r+ = M, so it must vanish at least linearly. The third term also vanishes linearly. On the other hand, the second term
26
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
vanishes slower than linearly, because of the logarithmic factor. Since the second term is negative + for r < 2M, in the limit as r → r+ , the second term dominates, and the sum is negative. In all cases, the trapping term is negative as ρ∗ → ±∞, so it is bounded above by a compactly supported function. 2 Together, the results from this section show that a weighted L6 norm is controlled, pointwise in time, by weighted space–time integrals of a solution and its derivative. Lemma 3.11. There are compactly supported, non-negative, bounded functions W and WL and a positive constant C such that for all u ∈ S(R × M) satisfying the wave equation (2.25) there are the following estimates
|u|6 (t, ρ∗ , ω)F 3 r −4 d 3 μ
1/6
−2 1/6 1/3 −2/3 ˙ t ˙ C E[u0 , u1 ] + EC u(t), u(t) EC u(t), u(t) t ,
EC u(t), u(t) ˙ EC [u0 , u1 ] +
t
2τ W |u|2 + WL |∇S 2 u|2 d 3 μ dτ.
(3.123) (3.124)
1
4. Relativistic considerations on the event horizon Throughout this paper, we require the initial data (u0 , u1 ) to have finite L2 norm, energy, and conformal charge. The purpose of this section is to show that this forces the initial data to vanish as ρ∗ → −∞ and to provide a sufficient rate of decay. In Section 2.1, we noted that, for non-super-critical Reissner–Nordstrøm spaces, the hypersurface t = C is a Cauchy surface for the exterior region in which it lies. In the Schwarzschild solution, the surface t = C can be continued smoothly through the bifurcation sphere as a surface of constant t in the other exterior region. This extended surface is a Cauchy surface for the entire, maximally extended Schwarzschild solution. For the general subcritical solution, 0 < |Q| < M, there is no Cauchy surface for the maximally extended Reissner–Nordstrøm solution; however, a hypersurface t = C in one exterior region can be smoothly continued as a surface of constant t in the exterior that is paired with the first exterior region. This generates a Cauchy surface for the two exterior regions and the adjacent intermediate regions. On these extended Cauchy surfaces, it is natural to consider arbitrary initial data which vanishes sufficiently rapidly in the asymptotically flat regions, but not at the bifurcation sphere. For such initial data in the Schwarzschild space, solutions to the wave equation have been found to remain bounded [29] and, more recently, to decay in the exterior region [14]. This section quantifies how far we are from this class of initial data. We provide sufficient decay rates in terms of both the radial coordinate ρ∗ , which we have chosen to use, and the distance along the hypersurface t = C, which is defined geometrically. In the subcritical case, the limit ρ∗ → −∞ (along the hypersurface t = C) corresponds to approaching the bifurcation sphere. In the critical case, it is known that there is no bifurcation sphere and the limit ρ∗ → −∞ corresponds to approaching a point at infinity. In this section, we show that the finite L2 condition requires the initial data to vanish as ρ∗ → −∞. In the subcritical case, any square-integrable power with respect to the ρ∗ variable is sufficient to enforce the finite L2
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
27
norm condition. In terms of the geometrically defined distance from the bifurcation sphere, this corresponds to an inverse logarithmic rate, so that any positive power is also sufficient. Differentiability of the initial data, with respect to normalized derivatives, is sufficient to impose the finite energy and conformal charge conditions. In fact, the derivatives can diverge. This is consistent with the sufficient condition for the finite L2 norm that the initial data vanish like an arbitrarily small inverse power of the distance from the bifurcation sphere. In the critical case, the limit ρ∗ → −∞ corresponds to a point at infinity, since the distance along ρ∗ → −∞ diverges. In this case, for the finite L2 norm condition, it is sufficient that the initial data decay exponentially, and a similar decay rate for the derivatives is sufficient for the finite energy and conformal charge conditions. In this section, we will assume that the solutions are real-valued. 4.1. Decay on the bifurcation sphere We made several transformations and change of variables in Section 2. To determine the physical constraints imposed by finite L2 (M) norm, finite energy, and finite conformal charge, we must return to the original function u˜ = r −1 u, and use normalized vectors and their duals, ∂ , ∂t −1 ∂ , ∂R = F 2 ∂ρ∗ ∂T = F
−1 2
(4.1) (4.2)
1
dT = F 2 dt,
(4.3)
1 2
dR = F dρ∗ .
(4.4)
From this, the natural measure on (t, ρ∗ , ω) ∈ {t0 } × R × S 2 is 1
dμnormalized = F 2 r 2 dρ∗ d 2 μS 2 .
(4.5)
We begin by writing the L2 norm in terms of u˜ and the natural measure: u2L2 (M) = u ˜ 2L2 (M) ˜ =
|u| ˜ 2 r 2 dρ∗ d 2 μS 2
(4.6)
˜ M
=
1
|u| ˜ 2 F − 2 dμnormalized .
(4.7)
˜ M
From the first line, the function u˜ must vanish at the bifurcation sphere, but any decay rate |ρ∗ |(−1−)/2 for > 1 is sufficient. In the subcritical case, |Q| < M, from integrating the length of the radial vector ∂R , the distance from the bifurcation sphere, s, is given by 1
s ∼ (r − r+ ) 2 .
(4.8)
28
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
The |ρ∗ |(−1−)/2 decay rate is guaranteed by an inverse logarithmic decay rate in terms of |r −r+ | or by any positive power decay rate. Thus, if u˜ decays like any positive power of the distance from the bifurcation sphere, then the finite L2 condition holds. In the critical case, |Q| = M, the length along t = C from r0 to r is like s ∼ ln(r − r+ ) − ln(r0 − r+ ), which diverges as r → r+ . The inverse power decay rate in terms of ρ∗ corresponds to a positive power decay rate in r − r+ and hence an exponential decay rate in the distance along the initial hypersurface. We now write the energy in terms of u˜ as
|u| ˙ 2 + |u |2 + F r −2 |∇S 2 u|2 + V |u|2 d 3 μ
E[u, u] ˙ = = =
(4.9)
2 −1 2 ˙˜ + F r u˜ + u˜ + F r −2 |∇ 2 u| ˜ 2 r 2 d 3μ |u| S ˜ + V |u|
(4.10)
2 2 2 3 ˙˜ + |u˜ |2 + F r −2 |∇ 2 u| r d μ |u| S ˜
(4.11)
2F r u˜ u˜ + F 2 |u| ˜ 2 + V r 2 |u| ˜ 2 d 3 μ.
+
(4.12)
The co-ordinates t and ρ∗ are singular at the bifurcation sphere, so we must rewrite the derivatives in terms of u˜ T = ∂T u, ˜ u˜ R = ∂R u, ˜ and the four gradient ∇˜ 4 u. ˜ In terms of these quantities, the energy is E[u, u] ˙ =
−1 2 2 2 ˙˜ + F −1 |u˜ |2 + r −2 |∇ 2 u| F |u| F r dρ∗ d 2 μS 2 S ˜
dF dF ˜ 2 + F 2 + rF r + F 2 |u| |u| ˜ 2 d 3μ − F dr dr = |u˜ T |2 + |u˜ R |2 + r −2 |∇S 2 u| ˜ 2 F r 2 dρ∗ d 2 μS 2
(4.13)
+
=
|∇˜ 4 u| ˜ 2 F 2 dμnormalized . 1
(4.14) (4.15) (4.16)
t=C
In the subcritical case, |Q| < M, for the energy to be bounded, it is sufficient that, far from the bifurcation sphere, the four gradient of u˜ is square integrable and that near the bifurcation sphere, for some > 0, |∇˜ 4 u| ˜ 2 < C(r − r+ )−1+ ,
(4.17)
˜ < C(r − r+ )− 2 + 2 . |∇˜ 4 u|
(4.18)
1
Clearly, differentiability with respect to normalized derivatives is sufficient to guarantee the finiteness of the energy. In fact, the derivatives can diverge like the distance from the bifurcation sphere to the power (−1 + ). The bound on the derivative and the vanishing of u˜ on the bifurcation, could impose stronger rates of decay for u. ˜ Under condition (4.18),
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
|∂r u| ˜ = F −1 |∂ρ∗ u| ˜ 1
29
(4.19)
= F − 2 |∂R u| ˜
(4.20)
|∇˜ 4 u| ˜
(4.21)
F
− 12
< C(r − r+ )−1+ .
(4.22)
Integrating this, we find |u| ˜ < C(r − r+ ) .
(4.23)
This is consistent with the previous sufficient condition. ˜ < (r − r+ )−1+ and |u| ˜ < (r − r+ ) are sufIn the critical case, the same conditions, |∂r u| ˜ = |dr/ds||∂r u| ˜ ∼ (r − r+ )(r − r+ )−1+ ∼ (r − r+ ) . ficient. In terms of the arc length, |∂s u| Thus, exponential decay for u˜ and ∂s u˜ is sufficient to guarantee the finiteness of the L2 and energy norms. The calculations and conditions for the conformal charge are similar. To control the conformal charge of u˜ at time t = 1, it is sufficient to control the following
2 2 ˙ + |u |2 + F r −2 |∇S 2 u|2 + V |u|2 d 3 μ ρ∗ + 1 |u| 2 = ρ∗ + 1 |u˜ T |2 + |u˜ R |2 + r −2 |∇S 2 u| ˜ 2 F r 2 dρ∗ d 2 μS 2 2 + ρ∗ + 1 2F r u˜ u˜ + F 2 |u| ˜ 2 + V r 2 |u| ˜ 2 d 3μ 2 = ρ∗ + 1 |u˜ T |2 + |u˜ R |2 + r −2 |∇S 2 u| ˜ 2 F r 2 dρ∗ d 2 μS 2 + −2F rρ∗ |u| ˜ 2 d 3μ 2 = ρ∗ + 1 |u˜ T |2 + |u˜ R |2 + r −2 |∇S 2 u| ˜ 2 F r 2 dρ∗ d 2 μS 2 2ρ∗ 2 2 |u| ˜ F r dρ∗ d 2 μS 2 + − r
1 1 2ρ∗ |∇˜ 4 u| ˜ 2 ρ∗2 + 1 F 2 dμnormalized + |u| ˜2 − = F 2 dμnormalized . r t=C
(4.24) (4.25) (4.26) (4.27) (4.28) (4.29) (4.30) (4.31)
t=C
The weights in the conformal charge only differ by factors of ρ∗ . In the subcritical case, ρ∗ is logarithmic in r − r+ near the bifurcation sphere. Thus, the sufficient condition given for the energy in Eq. (4.18) is sufficient to guarantee that the conformal charge is also bounded. In the critical case, ρ∗ ∼ (r − r+ )−1 , so the stronger conditions on the derivative of u˜ and on its derivative are simply exponential requirements with lower bounds for the rate of exponential decay. In addition, in the critical case, the angular derivatives must also decay. To enforce the finite L2 norm, energy, and conformal charge conditions, in the subcritical case, it is sufficient that the function u˜ decay like an arbitrarily small power of the distance from the
30
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
bifurcation sphere and that the energy not diverge too rapidly. In the critical case, it is sufficient that u˜ and its radial derivatives decay exponentially in the distance along the initial hypersurface. 5. The Heisenberg-type relation In this section we derive a Heisenberg-type relation for the wave equation. It is based on the Heisenberg equation for the Schrödinger equation in quantum mechanics. Applying this relation with a differential operator is similar to applying the method of multipliers with the same operator; however, the Heisenberg-type relation allows a wider class of operators to be used. Since domain issues for commutators and self-adjoint operators are delicate, we note at the end of this section that this can be applied more directly for operators mapping S to S. In this paper, we only use multiplication operators and Fourier multiplication operators, for which the domain issues are also more easily understood. For the Schrödinger equation, −iψt + H ψ = 0,
(5.1)
there is the well-known Heisenberg relation for the time derivative of the expectation value of a self-adjoint operator, A, d ψ, Aψ = ψ, i[H, A]ψ . dt
(5.2)
This formulation is central to the standard interpretation of quantum mechanics which associates operators to physically observable quantities, and the expectation value to the mean observed value. This formulation was also used in the original proof of scattering for the quantum n-body problem [18,28,35,36]. We begin by defining the commutator in the form sense. Definition 5.1. If H and A are two self-adjoint operators, and D(A) ∩ D(H ) is dense in L2 (M), then the commutator [H, A] is defined to be the form Ω given by Ω(v, w) = H v, Aw − Av, H w.
(5.3)
The Heisenberg-type relation follows from the wave equation and this definition. Lemma 5.2 (Heisenberg-type relation). Suppose A is a time-independent, self-adjoint operator; u ∈ S(R × M) is a solution to the wave equation (2.25); and for all t in some time interval, I , the functions u and u˙ satisfy u(t) ∈ D(A) ∩ D(H ) and u(t) ˙ ∈ D(A). If these conditions hold, then, for t ∈ I , d u, Au ˙ − u, ˙ Au = u, [H, A]u , dt
(5.4)
where u, [H, A]u is understood to mean the quadratic form [H, A] evaluated on the pair (u, u).
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
31
Proof. The left-hand side is d u, Au ˙ − u, ˙ Au = u, ˙ Au ˙ + u, Au ¨ − u, ¨ Au − u, ˙ Au ˙ dt = u, −AH u + H u, Au.
(5.5) (5.6)
Since A is self-adjoint, d u, Au ˙ − u, ˙ Au = −Au, H u + H u, Au. dt The right-hand side is exactly the commutator.
(5.7)
2
We now introduce the concept of a propagation observable which majorates another operator. We first illustrate and then define these concepts. In this paper, our method is to find such propagation observables. A simple example of a propagation observable, A, which majorates an operator G, is one for which [H, A] = G∗ G
(5.8)
(where G∗ represents the adjoint of G). If A is a bounded operator on the energy space, then, from the Heisenberg-type relation, t2
t2 Gu dt 2
t1
d u, Au ˙ − u, ˙ Au dt dt
t1
˙ 2 )Au(t2 ) CE[u, u]. ˙ 1 )Au(t1 ) + u(t ˙ 2 u(t
(5.9)
The standard definitions of a propagation observable and of majoration are broader than this [36], and our definition is even broader. Definition 5.3. Given a pair of operators, A and G, with D(A) ∩ D(H ) ∩ D(G) dense in L2 (M), the operator A is a propagation observable which majorates G if there is a pair of bounded, nonzero operators, X1 and X2 , for which X1 + X2 = Id,
(5.10)
[H, A] = G∗ X1 G + lower order terms,
(5.11)
where “lower order terms” refers to the sum of operators R for which either: (1) R is a bounded operator and, for all u ∈ S(R × M) which solve the wave equation, ∞ u, Ru dt C, 1
or
(5.12)
32
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90 1
(2) R is an operator with domain D(R) ⊃ D((G∗ G) 2 ), and, for some positive constant, C, and 1 all u ∈ D((G∗ G) 2 ), u, Ru C u, (G∗ G)1− u .
(5.13)
If A maps S to S, even if it is not self-adjoint, then, since H maps S to S, for v ∈ S, H (Av) − A(H v) is well defined on S. Since H is self-adjoint, by a similar calculation, d u, Au ˙ − u, ˙ Au = u, (H A − AH )u . dt
(5.14)
Since products of smooth functions, integer powers of the radial derivative, and integer powers of angular derivatives map S to S, if A is of this form, we can write d u, Au ˙ − u, ˙ Au = u, [H, A]u , dt [H, A] = H A − AH.
(5.15) (5.16)
For anti-self-adjoint operators, since multiplication by i commutes with H , we can define
u, [H, A]u = u, [H, −iA]iu .
(5.17)
From this, for an anti-self-adjoint operator, we have the same Heisenberg-type relation. 6. Morawetz estimates The goal of this section is to prove bounds on weighted space–time norms of solutions. We begin by introducing a propagation observable γ , show it majorates decaying weights, and conclude with a Gronwall’s type argument to integrate the Heisenberg-type relation. These estimates are proven using a spherical harmonic decomposition. We show these estimates have a uniform nature in the spherical harmonic parameter, so that we can recover an estimate for general u. In later sections, the contribution from H2 is controlled by the local decay estimate, and we can use a uniform multiplier. In R1+3 , the radial derivative can be used as a propagation observable to control the time 2 . This can be thought of as a weighted space–time integral with the δ-function integral of |u(t, 0)| as a weight. Essentially, our propagation observable is a smooth version of the radial derivative, which leads to a smooth weight. This follows ideas in our earlier work but with corrections following [9]. The spherical Laplacian is well known to have discrete spectrum. On each spherical harmonic we use l to refer to the standard spherical harmonic parameter and l˜ to refer to the value of the square root of the Laplacian on that harmonic, hence l˜ 2 = l(l + 1). (In Sections 7 and 8, we use a slightly different definition, with l˜ 2 = 1 + l(l + 1).) We also introduce the effective potential on each spherical harmonic, given by Vl = H2 + H3 = V + l˜ 2 Vl .
(6.1)
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
33
If necessary, we will use Pl to denote the projection on to the spherical harmonic with parameter l, but typically, in this section, we will assume that u has support on only one spherical harmonic. In this case, u satisfies u¨ − u + Vl u = 0.
(6.2)
We will show in Lemma 6.3 that Vl has a single critical point, which is a maximum. We denote this critical point (αl )∗ . The Morawetz-type operator is defined on each spherical harmonic in terms of a weight gl,b,σ . The weight depends on the radial variable ρ∗ and parameters σ and b as well as the sphericalharmonic parameter l. We take σ > 1 so that gl,b,σ is bounded and, later in the argument, b sufficiently small to close the Morawetz estimate. Definition 6.1. Given σ > 1, b > 0, and l ∈ N, the Morawetz-type multiplier γl,b,σ is defined by b(ρ∗−(αl )∗ )
1 dτ, (1 + |τ |)σ
(6.3)
1 ∂ ∂ gl,b,σ (ρ∗ ) = + gl,b,σ (ρ∗ ) 2 ∂ρ∗ ∂ρ∗
(6.4)
gl,b,σ (ρ∗ ) = 0
γl,b,σ
= gl,b,σ (ρ∗ )
∂ 1 + g (ρ∗ ). ∂ρ∗ 2 l,b,σ
(6.5)
We also use the notation γ = γl,b,σ and g = gl,b,σ in this section. The projection operator onto the lth spherical harmonic is denoted Pl . When working on more than one spherical harmonic, we use γ to denote γ=
γl,b,σ Pl .
(6.6)
l
We note that as l varies, the observable γl,b,σ is simply translated. We will also show that (αl )∗ → 0 from which it follows that gl,b,σ has a limit in L∞ . 6.1. Preliminary bounds Lemma 6.2. If σ > 1 and b > 0, then there are positive constants C1 and C2 , independent of l, such that for all u ∈ S and v ∈ S v, γ u = −γ v, u,
(6.7)
γ uL2
(6.8)
−σ/2 C1 u + C2 1 + ρ∗2 u −σ/2 C1 E 1/2 + C2 1 + ρ∗2 u.
(6.9)
Proof. We work on a fixed spherical harmonic. Since u ∈ S and v ∈ S, derivatives can be moved about freely, and we have
34
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
∂ 1 ∂ g v, γ u = v, + g u 2 ∂ρ∗ ∂ρ∗
1 ∂ ∂ g−g = v, u − 2 ∂ρ∗ ∂ρ∗ = −γ v, u.
(6.10) (6.11) (6.12)
b(ρ −(α ) ) Since g = 0 ∗ l ∗ (1 + |τ |)−σ dτ , g is bounded by C2 (1 + ρ∗2 )−σ/2 . Since the integrand is continuous and decays as τ −σ for σ > 1, g is bounded by some constant C1 . Using this and the fact that u ∈ S, we find 1 (6.13) γ u = gu + g u 2 1 (6.14) gu + 2 g u −σ/2 C1 u + C2 1 + ρ∗2 (6.15) u. The constant C2 is given by the integral of (1 + |τ |)−σ from 0 to ∞. This is independent of l. The constant C1 is based on the equivalence of b(1 + b|ρ∗ − (αl )∗ |)−σ and (1 + ρ∗2 )−σ/2 . Since (αl )∗ converges to zero by Lemma 6.3, this equivalence is uniform in l. 2 6.2. Computation of Morawetz commutators (σ/2)−1
In this section, we show that the propagation observable, γ , majorates (1 + ρ∗2 )− 2 . We prove this on each spherical harmonic, and then show there is a uniform lower bound on these estimates, to get an estimate for uniform l. We first show that on each spherical harmonic, the effective potential has a unique maximum. Lemma 6.3. If l ∈ N, then the effective potential Vl has a unique critical point, which is a maximum. The maxima, (αl )∗ , converge to 0. Proof. The Reissner–Nordstrøm potentials are
1 2M Q2 V= 1− + 2 2Mr −2 − 2Q2 r −3 r r r 2 −3 = 2Mr − 2 Q + 2M 2 r −4 + 6MQ2 r −5 − 2Q4 r −6 ,
2M Q2 1 + 2 , VL = 2 1 − r r r Vl = V + l˜ 2 VL .
(6.16) (6.17) (6.18)
Our goal is to show that in the outer region, the effective potential Vl has a single critical point. The outer region is given by r > r+ with r± = M ±
M 2 − Q2 ,
(6.19)
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
35
being the zeroes of the factor 1 − 2M/r + Q2 /r 2 . We do this by showing that a positive factor times the derivative of each of the potentials increases from a nonpositive value to infinity from r = 2M to r → ∞ and that the derivatives are negative for r from the horizon at r = r+ to r = 2M. The derivatives are given by V = −2F r −7 P ,
P = 3Mr 3 − 4 Q2 + 2M 2 r 2 + 15MQ2 r − 6Q4 , (6.20)
VL = −2F r −7 PL ,
PL = r 4 − 3Mr 3 + 2Q2 r 2 ,
Vl
˜2
=V +l
VL
(6.21)
˜2
(6.22)
I = r −2 P = 3Mr − 4 Q2 + 2M 2 + 15MQ2 r −1 − 6Q4 r −2 ,
(6.23)
= −2F r
−7
Pl ,
Pl = P + l PL .
We will also use
IL = r −2 PL = r 2 − 3Mr + 2Q2 ,
(6.24)
Il = r −2 Pl = I + l˜ 2 IL .
(6.25)
Showing that Pl has a single root in the exterior region is equivalent to showing that Il has a single root in the exterior region. We start by showing that I and IL are increasing in the region r 2M. The derivative of I is ∂r I = 3M − 15MQ2 r −2 + 12Q4 r −3 = r −3 3Mr 3 − 15MQ2 r + 12Q4 .
(6.26) (6.27)
The derivative of the term in brackets, for r 2M and Q M, can be estimated by ∂r 3Mr 3 − 15MQ2 r + 12Q4 = 9Mr 2 − 15MQ2 > 0.
(6.28)
At r = 2M, the quantity in brackets in (6.27) is 24M 4 − 30M 2 Q2 + 12Q4 > 0, thus ∂r I is positive for r 2M. The derivative of IL , ∂r IL = 2r − 3M > 0, is positive for r 2M (and even for r > 3M/2). The value of these quantities at r = 2M is 3Q4 I = 6M 2 − 4 Q2 + 2M 2 + (15/2)Q2 − 2M 2 1 = 2 −2M 4 + (7/2)M 2 Q2 − (3/2)Q4 0, M IL = 4M 2 − 6M 2 + 2Q2 0.
(6.29) (6.30) (6.31)
Thus each of I and IL are increasing for r 2M, diverge to infinite as r → ∞, and are nonpositive for r = 2M. We now show that I and IL are negative for r between r+ and 2M. To show I is negative, it is sufficient to show P = r 2 I is negative. This is a cubic, and we observe a number of properties about it before drawing a conclusion. The derivative is a quadratic polynomial
36
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
∂r P = 9Mr 2 − 8 Q2 + 2M 2 r + 15MQ2 ,
(6.32)
with two roots at r=
4Q2 + 8M 2 ±
64M 4 − 71M 2 Q2 + 16Q4 . 9M
(6.33)
The discriminant is bounded above by 64M 2 − 64M 2 Q2 + 16Q4 = (8M 2 − 4Q2 )2 . Therefore, the upper root is less than 16M/9 < 2M. The discriminant is bounded from below by 9Q4 , so the lower root is bounded above by (8M 2 + Q2 )/9M < M r+ . The value of P = r 2 I at r = r+ was computed in Eq. (3.99) to be P (r+ ) = −2 2M 2 − Q2 + 2M M 2 − Q2 M 2 − Q2 0.
(6.34)
Considered as a function on the entire real line, P goes from negative infinite to a local maximum value at the lower root of the derivative, and then becomes nonpositive at r = r+ . Since P is cubic, it can have at most one critical point root for r > r+ . Since P is nonpositive at r+ and 2M and diverges as r → ∞, it must be negative for r ∈ (r+ , 2M). The function IL is much easier to estimate. It has two roots at r=
3M ±
9M 2 − 8Q2 . 2
(6.35)
For Q M, the lower root is less than or equal to M and the upper root is greater than or equal to 2M. Thus, IL is negative from r+ = M + Q2 − M 2 to 2M. We know that I and IL are negative for r between r+ = M + Q2 − M 2 and 2M, are increasing for r 2M, and go to infinite as r → ∞. Therefore, the weighted sum Il = I + l˜2 IL has the same properties and must have a single root for r > r+ . Therefore, Pl and Vl have unique roots for r > r+ . Since the quantities Il and Pl go from negative to positive, the derivative Vl goes from positive to negative, and the critical points are maxima. Since Vl l˜−2 → VL and the maximum of VL occurs at 0, if (αl )∗ denotes the maximum point of Vl , then (αl )∗ → 0. 2 We now show that the commutators uniformly bound a polynomially decaying term, for an appropriate choice of b. This value of b will always be used in γl,b,σ from now on. There are additional positive terms which are also dominated, but we are better able to take advantage of them in the later sections using a slightly modified multiplier. Lemma 6.4. If σ > 1, then there is a choice of b > 0 and a positive constant C such that for all u∈S − σ −1 u, [H, γ ]u C u, 1 + ρ∗2 2 u .
(6.36)
Proof. We work on a single spherical harmonic and sum at the end of the argument. The commutator is
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
[H, γ ] = −
37
∂ 1 ∂2 ∂ ∂3 ∂3 ∂ ∂2 ∂ − g g + g − g − g V − V g l l 2 ∂ρ∗2 ∂ρ∗ ∂ρ∗3 ∂ρ∗ ∂ρ∗ ∂ρ∗3 ∂ρ∗ ∂ρ∗2
= −2
∂ ∂ 1 g − g − gVl . ∂ρ∗ ∂ρ∗ 2
(6.37) (6.38)
We will use x to denote ρ∗ − (αl )∗ . The derivatives of g are g =
b , (1 + b|x|)σ
(6.39)
g =
−b2 σ sgn(x) , (1 + b|x|)σ +1
(6.40)
g = −b2 σ 2δ(x) +
b3 σ (σ + 1) . (1 + b|x|)σ +2
(6.41)
To estimate the contribution from the δ function, we introduce a smooth, compactly support function, χ which takes values in [0, 1] and is identically one in a neighborhood of x = 0 (this is unrelated to any other cut-off function used outside of this proof). Using integration by parts and temporarily treating u as a function of x, we make the estimates ∞ 0= 0
∂ xχ|u|2 dx ∂ρ∗
∞
∞ χ|u| dx +
=
∞
xχ |u| dx +
2
0
∞
(6.42)
2
0
0
∞
|xχ | + x 2 χ |u|2 dx +
χ|u| dx 2
0
0
u(0) 2 = −
0
∞ 2
0
∞
0
∞
|χ ||u| dx + 2
(6.45)
∂ 2χ u u dx ∂ρ∗
(6.46)
∞ ∂ 2 χ|u| dx + χ u dx. ∂ρ 2
0
0
R
|χ | + |xχ | + x 2 χ |u|2 dx +
(6.47)
∗
Applying a symmetric argument for (−∞, 0] and substituting the result for 2 2 u(0)
(6.43)
(6.44)
∗
∂ χ|u|2 dx ∂ρ∗ χ |u| dx −
=−
0
∂ 2 χ u dx, ∂ρ
0
∞
∞
∞
∂ 2xχ u¯ u dx, ∂ρ∗
R
2 R χ|u| dx, we have
∂ 2 2χ u dx. ∂ρ∗
(6.48)
38
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
We use a similar integration by parts argument to estimate (1 + b|x|)−σ −2 |u|2 . We first compute
∂ 2 g |u| dx ∂ρ∗ R
∂ u dx, = g |u|2 dx + 2g u¯ ∂ρ∗
0=
R
b3 R
(6.49)
(6.50)
R
σ (σ + 1) |u|2 dx (1 + b|x|)σ +2
2 2σ b u(0) +
2
(6.51)
σ (σ + 1) b |u|2 dx (1 + b|x|)σ +2
1
3
R
2
R
2| ∂ρ∂ ∗ u|2 2 2σ b dx . σ + 1 (1 + b|x|)σ 1
(6.52)
Substituting Eq. (6.48) into this, we find b
3 R
σ (σ + 1) |u|2 dx (1 + b|x|)σ +2
σb
2 R
1 + 2
∂ 2 2χ u dx ∂ρ
(6.53)
2| ∂ρ∂ ∗ u|2 2σ b dx, σ + 1 (1 + b|x|)σ
(6.54)
|χ | + |xχ | + x 2 χ |u|2 dx + σ b2
σ (σ + 1) 1 b |u|2 dx + 2 (1 + b|x|)σ +2
3
R
σ (σ + 1) b |u|2 dx (1 + b|x|)σ +2 R 2 2σ b |χ | + |xχ | + x 2 χ |u|2 dx
R
R
∗
3
R
+ 2σ b R
∂ 2 2 1 + 2bχ u ∂ρ dx. σ + 1 (1 + b|x|)σ ∗
(6.55)
The commutator term can now be estimated as R
∂ 2 1 2 2g u − g |u| − gVl |u|2 dx ∂ρ∗ 2 2 1 ∂ 2 2b b3 σ (σ + 1) 2 u(0) = u dx + σ b − |u|2 dx (1 + b|x|)σ ∂ρ∗ 2 (1 + b|x|)σ +2 R R 2 − gVl |u| dx
(6.56)
R
(6.57)
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
R
2 ∂ 2 2b 2 2 |χ | + |xχ | + x 2 χ |u|2 dx u dx + σ b u(0) − σ b (1 + b|x|)σ ∂ρ∗
39
(6.58)
R
− σb R
2b R
+
∂ 2 2 1 dx − gV |u|2 dx + 2bχ u l ∂ρ σ + 1 (1 + b|x|)σ ∗
(6.59)
R
∂ 2 1 1 − σ bχ u dx σ + 1 (1 + b|x|)σ ∂ρ∗
2 −σ b2 |χ | + |xχ | + x 2 χ − gVl |u|2 dx + σ b2 u(0) .
(6.60)
R
The term σ b|u(0)|2 is positive and will be ignored. The term (σ + 1)−1 (1 + b|x|)−σ − σ bχ , treated as a function of x, is independent of l, so we can choose b sufficiently small so that this term is bounded below by c(1 + b|x|)−σ . Since Vl = V + VL l˜ 2 , and the (αl )∗ converge, the value of Vl is uniformly bounded from above at (αl )∗ , and b can be chosen sufficiently small so that the quadratically vanishing quantity −σ b2 (|χ | + |xχ | + x 2 χ) − gVl is bounded below by c(|χ | + |xχ | + x 2 χ), uniformly in l. Thus there is a uniform constant, c1 , for which, R
∂ 2 1 2 2g u − g |u| − gVl |u|2 dx ∂ρ∗ 2
(6.61)
∂ 2 1 u + |χ | + |xχ | + x 2 χ |u|2 dx. σ (1 + b|x|) ∂ρ∗
c1 R
(6.62)
Applying (6.55), we have for some constant c2 R
∂ 2 1 2 1 2 2g u − g u − gVl |u| dx c2 |u|2 dx. ∂ρ∗ 2 (1 + b|x|)σ +2
(6.63)
R
Since the (αl )∗ converge, there is a uniform equivalence between (1 + b|x|)−σ −2 and (1 + ρ∗2 )−σ/2−1 . Integrating over the angular variables gives
− σ −1 u, [H, γ ]u C u, 1 + ρ∗2 2 u .
6.3. L2 local decay estimate We begin with a type of Gronwall’s estimate.
2
(6.64)
40
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
Lemma 6.5. If θ : [0, ∞) → [0, ∞) has a uniformly bounded derivative, ∈ (0, 1/3), and there are constants C1 , C2 , and T , such that for t > T t θ (τ )2 dτ C1 + C2 t θ (t)1− ,
(6.65)
1
then there is a sequence {ti } such that limi→∞ ti = ∞ and lim ti θ (ti )1− = 0.
i→∞
(6.66)
Proof. Successively stronger bounds on θ (t) will be proven. The bound on the derivative implies θ (t) and t θ (t)1− are linearly bounded above. Suppose there is no sequence {ti } on which θ (ti ) → 0, then θ (t) is bounded below by a constant. By the integral condition t θ (t)1− is bounded below by a linear function. From this θ (t)2 is bounded below by a quadratic, and its integral is bounded below by a cubic. This contradicts the linear upper bound of the integral. Therefore, there is a sequence {ti } such that ti → ∞ and θ (ti ) → 0. Since < 1/3, there exists r < 1 such that −
1− <− . 2 − r + r 1−
We now choose δ negative such that δ
− 1− ,
from which it follows that
rδ < 1 +
2δ . 1−
(6.67)
Suppose there is a decaying lower bound for sufficiently large time, i.e., that ∃K > 0, T1 > 0 ∀t > T1 : θ (t) > Kt δ , then by the integral condition (6.65) there is a positive C3 such that C3 t 1+2δ C1 + C2 t θ (t)1− . − Since δ 1− > C4 , C5 such that
−1 2 ,
(6.68)
1 + 2δ is strictly positive. Thus for sufficiently large t, there are constants C4 t 1+2δ t θ (t)1− , 2δ
C5 t 1+ 1− θ (t). If δ is sufficiently close to zero, then this contradicts the previous result that θ (t) goes to zero on a subsequence. If δ is larger in magnitude, then we use the fact that by Eq. (6.67), θ (t) C5 t rδ . This implies the originally assumed lower bound is replaced by the larger lower bound t rδ . Ren peated iterations of this process shows that θ (t) is bounded below by t r δ for any n. Since r < 1, this reduces the situation to the δ close to zero case, which led to a contradiction. Thus θ (t) can− . In particular, θ (t) cannot be not be bounded below by a function of the form Kt δ for δ 1− −
bounded below by a function of the form Kt 1− and so the desired subsequence must exist.
2
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
41
From Lemma 6.4, the propagation observable γ majorates the weight (1 + ρ∗2 )−β/2 for β > 3/2. As a consequence, solutions to the wave equation are space–time integrable in any bounded spatial region and must decay in that region. This is a regularized version of the Morawetz estimate. We also refer to this estimate as the local-decay estimate. Theorem 6.6 (Local decay estimate). If β > 3/2, then there is a positive constant CLD such that for all u ∈ S(R × M) satisfying the wave equation (2.25) there is the integral estimate ∞ 1 + ρ 2 −β/2 u2 dt CLD E 1/2 E 1/2 + u0 . ∗
(6.69)
1
Proof. First, we make a crude estimate on the norm growth of (1 + ρ∗2 )−β/2 u by d 1 + ρ 2 −β/2 u2 = d 1 + ρ 2 −β/2 u, 1 + ρ 2 −β/2 u , (6.70) ∗ ∗ ∗ dt dt −β/2 d −β/2 −β/2 −β/2 u 1 + ρ∗2 u = 2 1 + ρ∗2 u, 1 + ρ∗2 u˙ , (6.71) 2 1 + ρ∗2 dt d 1 + ρ 2 −β/2 u 1 + ρ 2 −β/2 u˙ u ˙ E u(1) . (6.72) ∗ ∗ dt Initially, we restrict β to (3/2, 2), and use the variable σ = 2(β − 1) ∈ (1, 2). From the commutator estimates Lemma 6.4, T
−β/2 2 C 1 + ρ2 u dt =
T
∗
1
1
− σ −1 u, 1 + ρ 2 2 u dt −
T 3 Hi , γ u dt. (6.73) u, 1
i=1
From integrating the Heisenberg-type relation, the integral on the right-hand side is bounded by T 1
T ∂ u, γ u ˙ − u, ˙ γ u dt u, γ u ˙ − u, ˙ γ u t=1 ∂t T γ u, u ˙ − 2u, ˙ γ u t=1 C uγ ˙ u t=1 + uγ ˙ u t=T .
(6.74) (6.75) (6.76)
From energy conservation and the bound on γ u in Lemma 6.2, these norms are bounded by −σ ˙ u t=T C1 E 1/2 E 1/2 + 1 + ρ∗2 2 u(1) uγ ˙ u t=1 + uγ −σ + C2 E 1/2 E 1/2 + 1 + ρ∗2 2 u(T ) .
(6.77) (6.78)
Since σ ∈ (1, 2), we can choose q ∈ ( σ1 + 12 , 32 ) ⊂ (1, 32 ). If p is the conjugate exponent to q and κ = 2/p, then
42
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
2 1 1 <1− = , p 3 3
2−κ q = 1, 2 σq >
(6.79) (6.80)
σ +2 . 2
(6.81)
We apply Hölder’s inequality with conjugate exponents p and q to the last expression in (6.78) to get, σ 1 + ρ 2 − 2 u2 =
|u|κ |u|2−κ 3 d μ (1 + ρ∗2 )σ
1 p pκ 3 |u| d μ
∗
2−κ pκ 2 |u| 2 q p 2 |u| σq (1 + ρ∗2 ) 2 The fact that σ q >
σ +2 2
(6.82) |u|(2−κ)q 3 d μ (1 + ρ∗2 )σ q 2 q .
1 q
(6.83)
(6.84)
implies that 1 σ −σ −2 1 1 + ρ 2 − 2 u u p 1 + ρ 2 4 u1− p . ∗
∗
(6.85)
1
Since u ˙ 2 E, the norm of u is controlled by u(t) u0 + tE 2 . The computations from the beginning of (6.73) to (6.85) give an integral inequality for the weighted norm of u, T
−σ −2 1 + ρ 2 4 u2 dt ∗
1
−σ −2 1− 1 1 + 1 1 C1 E 1/2 E 1/2 + u0 + C2 E + u0 2 2p T p 1 + ρ∗2 4 u p .
(6.86)
−σ −2
d (1 + ρ∗2 ) 4 u is uniformly bounded by Eq. (6.72), and since 1/p ∈ (0, 1/3), the Since dt Gronwall’s estimate, Lemma 6.5, applies, and there is a subsequence on which 1 1− 1 −σ −2 Ti p 1 + ρ∗2 4 u(Ti ) p → 0.
(6.87)
In inequality (6.86), the left-hand side is monotonically increasing and the right-hand side is sequentially decreasing. Taking the limit as T → ∞ gives the desired result for β ∈ (3/2, 2). Since (1 + ρ∗2 )−β/2 |u| is monotonically decreasing in β, the Lebesgue dominated convergence theorem extends the result to all β > 3/2. 2 To conclude this section, we apply this result to the conformal estimate. Since the weights in the local decay theorem will dominate any compactly supported function, if u is a radial function, the trapping terms can be integrated in time to control the growth of the conformal charge and the weighted L6 norm.
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
43
Corollary 6.7. There is a positive constant C such that if u0 and u1 are radial functions, then for all u ∈ S(R × M) satisfying the wave equation (2.25) and for all t 1 there is the following pointwise-in-time estimate
3 −4
|u| (t, ρ∗ , ω)F r 6
1 3
6
d μ
Ct
−1 3
1/2 E[u0 , u1 ] + EC [u0 , u1 ] + u0 2 . (6.88)
Proof. Since the initial conditions are radial, the solution u will remain radial, and WL |∇S 2 u|2 d 3 μ = 0.
(6.89)
Since the potential trapping term, W , is compactly supported, it can be dominated by the weights in the local decay result. From Lemma 3.11, the conformal charge is bounded by ˙ EC [u0 , u1 ] + EC u(t), u(t)
t 2τ W |u|2 d 3 μ dτ
(6.90)
2W |u|2 d 3 μ dτ
(6.91)
1 M
t EC [u0 , u1 ] + t 1 M
1 1 EC [u0 , u1 ] + tE 2 E 2 + u0 ,
(6.92)
and the L6 norm is bounded by
−2 2 −4 |u|6 (t, ρ∗ , ω)F 3 r −4 d 3 μ C E[u0 , u1 ] + EC u(t), u(t) ˙ t EC u(t), u(t) ˙ t 3 Ct −2 EC [u0 , u1 ] + E + u0 2 .
2
(6.93) (6.94)
7. Angular modulation The purpose of this section is to improve the weighted, space–time integral estimates from the previous section to control the integral of derivatives of u. To close the conformal estimate, it is necessary to estimate one angular derivative of u in L2 localized near the photon sphere r = α. This is the same as estimating two angular derivatives in expectation value. Using energy (and L2 based norms involving at most one derivative) alone, this is not possible, but we are able to bound fractional powers of the angular derivative. In this section, we control (1 − S 2 )3/4 in L2 . To do this, we introduce a propagation observable, which is an analogue of γ but is rescaled, on each spherical harmonic, by a fractional power of 1 − S 2 . In Section 8, we will modify this operator further to control 1 − derivatives. At the end of this section, we include two corollaries, which each control a full derivative, but with certain restrictions. Heuristically, the local decay estimate comes from combining the third derivative of g with the potential term gV . By rescaling g by K m , the contribution from the third derivative becomes K 3m times stronger on a region of scale K −m . Since the angular potential term already has a factor of (− S 2 ) and VL vanishes linearly, outside the region of scale K −m , the rescaled weight
44
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
g is of order one, and gVL (− S 2 ) dominates K −m (− S 2 ). The heuristic suggests that, if we take K ∼ l˜ on each spherical harmonic and m 1/2, then on all scales l˜3m is controlled. The decay of VL for large ρ∗ is not accounted for in this heuristic, so that the heuristic only holds on scales not larger than order one. Inspired by, but deviating from, the standard notation from quantum mechanics, L is notationally used to refer to the operator square root of 1 − S 2 . Definition 7.1. L is defined to√ be the operator square root of 1 − S 2 . On each spherical harmonic this acts as multiplication by 1 + l(l + 1). Since L is self-adjoint, by the spectral theorem, functions of L are defined. Furthermore, since L and ρ∗ are commuting operators, functions of both of these can be defined. Results will be stated in terms of the operator L. We will use l˜ for the eigenvalue of L on the lth spherical harmonic. In this section, it is frequently easier to work on a single spherical harmonic, treat L as ˜ prove uniform results independent of the spherical-harmonic parameter l, and then a constant l, sum to recover estimates which are not restricted to a single spherical harmonic. 7.1. Angular modulation and initial estimates A new operator γLm is introduced to give better estimates near ρ∗ = 0. The weight in the new operator is rescaled by Lm . Because the contribution from H2 = V involves no powers of L, the local decay result allows us to ignore it, and the new multiplier is centered at the peak of VL without any dependence on the spherical-harmonic parameter. Definition 7.2. The angularly modulated multiplier γLm is defined for 0 m 1/2, b > 0, and σ > 1, by m bL ρ∗
gLm =
−σ 1 + |τ | dτ
(7.1)
0 m
=
2 ρ∗ ∞ b(1+l(l+1))
l=0
(7.2)
0
∂ 1 ∂ gLm + gLm . 2 ∂ρ∗ ∂ρ∗
γLm =
−σ 1 + |τ | dτ Pl ,
(7.3)
We will take σ = 2, since there is no improvement in varying σ and it simplifies the estimates. Lemma 7.3. If 0 m 1/2, σ = 2, and b > 0, then for all u ∈ S −1 2 γLm u2 C E[u] + 1 + ρ∗2 u .
(7.4)
Proof. Since σ = 2, the function gLm is bounded uniformly on all spherical harmonics by a constant C. Equivalently, the operator gLm is bounded with norm C. The ρ∗ derivative of the
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
45
operator gLm is bounded on each spherical harmonic shell by −1 −2 1 C bl˜m F − 2 r −1 + 1 + ρ∗2 gL m l˜m 1 + bl˜m |ρ∗ |
(7.5)
(this constant is independent of the spherical harmonic shell). In terms of the operator gLm , since ˜ 12 r −1 u2 + (1 + ρ∗2 )−1 u2 , which is l˜ dominates l˜m , the norm gL m u2 is bounded by lF bounded by the energy and local decay norm. Therefore,
2 1 2 γLm u 2 gLm u + gLm u 2 −1 2 C E[u] + 1 + ρ∗2 u . 2
Summing over spherical harmonics gives the desired result.
(7.6) (7.7)
2
7.2. Direct angular momentum bounds We prove the angular modulation theorem, Theorem 7.7, in the same way as the local decay theorem, Theorem 6.6. By tracking the effect of the rescaling parameter, Lm , we are able to gain additional angular derivatives. We begin with a commutator estimate. Typically, whenever we encounter a commutator involving a ρ∗ localization, f (ρ∗ ), that term can be controlled by the local decay theorem, Theorem 6.6. For this reason, we start by ignoring the contribution to the commutator from H2 and estimate it later. The compactly supported function χ˜ in the following lemma restricts to a scale of order one so that the decay of VL can be ignored. The following lemma provides several results. Not only do estimates (7.9) and (7.10) provide lower bounds on the commutator, but also the right-hand side of (7.9) dominates the right-hand side of (7.10). Lemma 7.4. If 0 m 1/2, σ = 2, 0 < < 1, and χ˜ is a continuous, compactly supported function, then there is a choice of b, positive constants C1 , C2 , C3 , and C4 and a smooth, compactly supported, radial function χ such that for all u ∈ S
u, [H1 + H3 , γLm ]u
Lm u C1 u , (1 + Lm |ρ∗ |)2
2 m u, −gL VL L − 1 u − C2 b2 ρ∗2 χ|u|2 d 3 μ, + 1− 3 C3 u, L3m χ˜ u + (1 − ) u, −gLm VL L2 − 1 u − C4 b2 ρ∗2 χ|u|2 d 3 μ.
(7.8)
(7.9) (7.10)
Proof. We follow the same argument as in the proof of Lemma 6.4, but with the extra parameters from scaling by l˜m on each spherical harmonic. We start by working on a single spherical
46
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
harmonic. From the definition of gLm , we have gL m =
l˜m b , (1 + l˜m b|ρ∗ |)σ
(7.11)
gL m =
−σ l˜2m b2 sgn ρ∗ , (1 + l˜m b|ρ∗ |)σ +1
(7.12)
gL m = −2l˜2m b2 σ δ(ρ∗ ) + σ (σ + 1)l˜3m
b3 (1 + l˜m b|ρ∗ |)σ +2
.
(7.13)
Once again, we begin by controlling the δ function appearing in gL m . For this argument, we use χ0 , χ1 , and χ2 to denote compactly supported functions which take values in [0, 1] and are identically one in an interval of width l˜−m about the origin, supported on an interval of the same scale, and with nested domains so that χi is identically one on the support of χi−1 . From this, χ0 χ1 χ2 . We begin by noting ∞ 0= 0
∂ ρ∗ χ1 |u|2 dρ∗ ∂ρ∗
∞ =
∞ χ1 |u| dρ∗ + 2
0
(7.14)
ρ∗ χ1 |u|2 dρ∗
0
∞ + 0
∂ 2ρ∗ χ1 u¯ u dρ∗ . ∂ρ∗
(7.15)
We isolate the first term, multiply by l˜3m , note that the derivative of χi is bounded by C l˜2m |ρ∗ |χi+1 , and apply the Cauchy–Schwarz inequality, ∞
l˜3m χ1 |u|2 dρ∗
0
∞ C
l˜5m ρ∗2 χ2 |u|2 dρ∗ + 2
l˜5m ρ∗2 χ22 |u|2 dρ∗
1 2
2
1 2 ˜l m χ2 ∂ u dρ∗ . (7.16) ∂ρ ∗
0
Similarly, u(0) 2 = −
∞ 0
∞ =−
∂ χ0 |u|2 dρ∗ ∂ρ∗ χ0 |u|2 dρ∗
∞ −
0
0
∞ C
(7.17)
∂ 2χ0 u u dρ∗ ∂ρ∗
1 ∞ 2
χ1 |u| dρ∗ 2
0
0
l˜4m ρ∗2 χ1 |u|2 dρ∗
(7.18) ∞ + 0
1 2 ∂ 2 χ1 u dρ∗ . ∂ρ ∗
(7.19)
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
47
We now multiply by l˜2m 2 l˜2m u(0) C
∞
l˜3m χ1 |u|2 dρ∗
1 ∞ 2
0
l˜5m ρ∗2 χ1 |u|2 dρ∗ +
0
∞ 0
1 2 2 ˜l m χ1 ∂ u dρ∗ . ∂ρ ∗
(7.20)
Applying Eq. (7.16) to control the weighted L2 norms and applying symmetry, we have 2 2l˜2m u(0) C0
l˜5m ρ∗2 χ2 |u|2 dρ∗ +
R
R
2
˜l m χ2 ∂ u dρ∗ . ∂ρ
(7.21)
∗
We now turn to controlling the noncompactly supported weights coming from gL m . From a similar calculation,
∂ gLm |u|2 dρ∗ ∂ρ∗ R
∂ 2 = gLm |u| dρ∗ + 2gLm u u dρ∗ , ∂ρ∗
0=
R
R
l˜3m b3
(7.22)
(7.23)
R
σ (σ + 1) |u|2 dρ∗ (1 + bl˜m |ρ∗ |)σ +2
2 2σ l˜2m b2 u(0)
1
1 ∂ 2 2 2 σ (σ + 1) 2σ ˜m 2| ∂ρ∗ u| 3m 3 2 ˜ bl + |u| dρ∗ dρ∗ . l b m σ +2 m σ σ +1 (1 + bl˜ |ρ∗ |) (1 + bl˜ |ρ∗ |) R
(7.24) (7.25)
R
Substituting Eq. (7.21) and noting C0 χ2 C˜ 0 (1 + bl˜m |ρ∗ |)σ , we find R
l˜3m b3
σ (σ + 1) |u|2 dρ∗ (1 + bl˜m |ρ∗ |)σ +2
(7.26)
5m 2 2 ˜ σ b C0 bl ρ∗ χ2 |u| dρ∗ + C0 bl˜m R
+ R
1 2
R
R
l˜3m b3
2σ bC0 R
bρ∗2 l˜5m χ2 |u|2 dρ∗ +
(1 + bl˜m |ρ∗ |)σ
σ (σ + 1) |u|2 dρ∗ + (1 + bl˜m |ρ∗ |)σ +2
σ (σ + 1) |u|2 dρ∗ l˜3m b3 (1 + bl˜m |ρ∗ |)σ +2
2| ∂ρ∂ ∗ u|2
R
(7.27)
dρ∗
2| ∂ρ∂ ∗ u|2 2σ ˜m dρ∗ , l b σ +1 (1 + bl˜m |ρ∗ |)σ
(7.28) (7.29)
2σ + 2σ bC˜ 0 σ +1
R
bl˜m
2| ∂ρ∂ ∗ u|2
(1 + bl˜m |ρ∗ |)σ
dρ∗ .
(7.30)
48
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
The commutator term can now be estimated as
∂ 2 1 u − gLm |u|2 − gLm VL l˜ 2 − 1 |u|2 dρ∗ ∂ρ∗ 2
2gL m
R
= R
l˜m b
2| ∂ρ∂ ∗ u|2
(1 + bl˜m |ρ∗ |)σ
l˜3m b3
− R
R
2 σ +1
−
2 dρ∗ + l˜2m b2 σ u(0)
σ (σ + 1) |u|2 dρ∗ − m σ +2 ˜ 2(1 + bl |ρ∗ |)
| ∂ρ∂ ∗ u|2 m ˜ ˜ − 2σ bC0 l b (1 + bl˜m |ρ∗ |)σ
σ b C0 l˜5m ρ∗2 χ2 |u|2 dρ∗ −
2
R
(7.31)
(7.32) gLm VL l˜ 2 − 1 |u|2 dρ∗
(7.33)
R
2 dρ∗ + l˜2m b2 σ u(0)
gLm VL l˜ 2 − 1 |u|2 dρ∗ .
(7.34)
(7.35)
R
We choose b sufficiently small so that 2/(σ + 1) − 2σ bC˜ 0 > 1/2 and −(/3)ρ∗ VL χ2 > σ b2 C0 ρ∗2 χ2 . This choice of b can be computed explicitly from , C0 , C˜ 0 , and VL (0), but it will be independent of the spherical harmonic. Since gLm vanishes like l˜m ρ∗ in the support of χ2 and m 1/2, the coefficients on u2 are −(/3)gLm VL l˜ 2 > σ C0 l˜5m b2 ρ∗2 χ2 . This gives us sufficient control, except in the case l˜ 2 − 1 = 0, when we must subtract off an additional localization term. With these choices, we have
∂ 2 1 u − gLm |u|2 − gLm VL l˜ 2 − 1 |u|2 dρ∗ ∂ρ∗ 2
2gL m
R
| ∂ρ∂ ∗ u|2
2 dρ∗ + l˜2m σ b2 u(0) m σ ˜ (1 + bl |ρ∗ |) R
− 1− gLm VL l˜ 2 − 1 |u|2 dρ∗ − C2 b2 ρ∗2 χ2 |u|2 dρ∗ . 3
C1
l˜m b
(7.36)
R
(7.37)
R
Dropping the |u(0)|2 term and integrating over the angular variables, we obtain the first result of this lemma, estimate (7.9). We divide the remaining positive terms into three pieces. A large piece will be 1 − times −gVL (l˜ 2 − 1). Another piece will be the term with |u |2 plus /3 times the term with −gVL (l˜ 2 − 1). In a compact set of width l˜−m , by Eq. (7.16), this controls l˜3m (except for a term arising from the difference between l˜ 2 and − S 2 , but this difference is controlled by χ2 ). The remaining /3 factor of −gVL , we use outside |ρ∗ | l˜−m . In this outer region, but inside a compact set ˜ −gV (l˜ 2 − 1) dominates any continuous, compactly supported function, χ˜ , by independent of l, L 2−m ˜ > C l˜ 3m , again with an extra contribution required for the l˜ 2 − 1 = 0 case. Here, a factor of l we use the assumption m 1/2 again. From this and integration over the angular variables,
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
C1
| ∂ρ∂ ∗ u|2
d μ− 1− m σ 3 (1 + bl˜ |ρ∗ |) − C2 b2 ρ∗2 χ2 |u|2 d 3 μ l˜m b
C3
3
l˜3m χ˜ |u|2 d 3 μ + (1 − )
49
gLm VL l˜ 2 − 1 |u|2 d 3 μ (7.38)
gLm VL l˜ 2 − 1 |u|2 d 3 μ
− C4
b2 (χ2 + χ˜ )|u|2 d 3 μ.
(7.39)
Recall χ˜ denotes an arbitrary continuous, compactly supported function. In the statement of the lemma, we take χ = χ2 + χ˜ . Summing over spherical harmonics gives the desired result. 2 In the next lemma, we show the H2 commutator contribution is time integrable. Lemma 7.5. If σ 2, and b > 0, then there is a positive constant C such that for all 0 m 1/2 and for all u ∈ S u, [H2 , γLm ]u C 1 + ρ 2 −1 u2 . ∗
(7.40)
Proof. From Lemma 3.9, the derivative of the potential is −V = 2F r −7 P (r) where P is a cubic polynomial in r. Since, for ρ∗ → ∞, r ∼ ρ∗ , and, for ρ∗ → −∞, F decays exponentially, there is a constant C so that |V | < C(1 + ρ∗2 )−2 . Since |gLm | is uniformly bounded multiplication operator on each spherical harmonic, and [H2 , γLm ] = −gLm V , the result holds. 2 The previous two lemmas shows that γLm is a propagation observable which majorates powers of L. That is, ignoring terms which are space–time integrable, the commutator is bounded below by the product of powers of L and localization functions. These localization functions are functions of Lm ρ∗ . In the following theorem, the Heisenberg-type relation with γLm is integrated in time. The function χ˜ is included to restrict attention to the region where ρ∗ is of order one. In the theorem, taking the optimal value m = 1/2 gives an estimate for u, χ˜ L3/2 u. The estimate for other values of m is used later, in the phase space analysis. Theorem 7.6. There is a positive constant CAM such that if 0 m 1/2, then for all u ∈ S(R × M) satisfying the wave equation (2.25) there are the following integral estimates ∞ u , 1
2 Lm u + u, −gLm VL L2 − 1 u dt CAM E + u(1) , m 2 (1 + L |ρ∗ |) ∞
2 u, L3/2 χα u dt CAM E + u(1) .
1
(7.41)
(7.42)
50
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
Proof. Applying the Heisenberg-type identity to the operator γLm gives
u,
3
Hi , γLm u =
i=1
d u, γLm u ˙ − u, ˙ γLm u . dt
(7.43)
Using Lemma 7.3 and the fact that γLm is antisymmetric with respect to the L2 (M) norm when acting on C ∞ functions, this equation can be integrated to estimate the commutators by t
u,
1
3
˙ γLm u(t) ˙ γLm u(1) + u(t) Hi , γLm u dτ 2 u(1)
(7.44)
i=1
2 2 −1 −1 C E + 1 + ρ∗2 u(1) + 1 + ρ∗2 u(t) .
(7.45)
By the local decay result, Theorem 6.6, and Lemma 7.5, for any compactly supported function χ ∞ ∞ u, [H2 , γLm ]u dτ 1 + ρ 2 −1 u2 dτ C E + u(1)2 , ∗ 1
∞
(7.46)
1
∞ 2 −1 2 2 2 2 b ρ∗ χu dρ∗ dτ 1 + ρ∗2 u dτ C E + u(1) .
1 M
(7.47)
1
From the first estimate for [H1 + H3 , γLm ], Eq. (7.9), the following estimate holds on any sequence of times: t 1
Lm u, u + u, −gLm VL L2 − 1 u dτ (1 + Lm |ρ∗ |)2
2 2 −1 C E + u(1) + 1 + ρ∗2 u(t) 2 −1 + C E + 1 + ρ∗2 u(1) .
(7.48) (7.49) (7.50)
Since the left-hand side is monotonically increasing and the right-hand side is decreasing on a sequence, by the local decay estimate, Theorem 6.6, Eq. (7.41) holds for all t. We can take any 0 < < 1, such as = 1/2, to get the constant from (7.9). (For this argument, it is not important that the term involving gLm VL has a coefficient which can be taken arbitrarily close to 1.) From Lemma 7.4, we know that the right-hand side of (7.9) dominates the right-hand side of (7.10), and taking m = 1/2 (and = 1/2), we can conclude that ∞
2 u, L3/2 χα u dτ C E + u(1) .
1
This is the second result of this theorem.
2
(7.51)
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
51
We have proven an estimate for u, L3/2 χα u. In the later phase space analysis, it will be necessary to estimate mixed derivative norms involving ∂ρ∂ ∗ and functions of L and ρ∗ . The following corollary of Theorem 7.6 controls u in (weighted) L2 . This could have been proven directly from the Morawetz estimate without angular modulation. Corollary 7.7. There is a positive constant C such that for all u ∈ S(R × M) satisfying the wave equation (2.25) there is the following integral estimate ∞ u , 1
2 1 u dτ C E + u(1) . 1 + ρ∗2
(7.52)
In particular, if u is a solution of the linear wave equation, u¨ + H u = 0, and χ1 is a smooth, compactly supported function, then there is a C such that, ∞ 2 u , χ1 u dτ C E + u(1) ,
(7.53)
1
∞
2 2 ∂ ∂ρ χ1 u dτ C E + u(1) .
1
∗
(7.54)
Proof. Using the first term in the first estimate of Theorem 7.6 with m = 0 gives this result.
2
We also have control of the full two angular derivatives with a weight which vanishes quadratically at the photon sphere, ρ∗ = 0, and which is compactly supported. Corollary 7.8. For any continuous, compactly supported function χ˜ , there is a positive constant C such that for all u ∈ S(R × M) satisfying the wave equation (2.25) ∞
2 u, ρ∗2 χ˜ L2 u dτ C E + u(1) .
(7.55)
1
Proof. This follows from using the first estimate of Theorem 7.6 with m = 0, this time using the second term, and noting that any continuous, compactly supported function which vanishes quadratically at the photon sphere, in particular ρ∗2 χ˜ , is dominated by a multiple of −gVL . In the support of χ˜ , the difference between L2 − 1 and L2 is a constant, independent of l. Thus, the difference restricted to this interval is space–time integrable by Theorem 6.6. 2 8. Phase space analysis The phase space induction theorem, Theorem 8.20, states C L1−ε χα u dt < E + u0 2 (for any compactly supported χα ). This result fails by an ε power of L to control the growth of the conformal charge. This loss accounts for the ε loss of derivatives in the main result of this
52
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
paper, Theorem 8.21. Relative to the local decay theorem, Theorem 6.6, the angular modulation theorem, Theorem 7.6, gains 3/4 angular derivatives in L2 , and the phase space induction theorem, Theorem 8.20, gains an additional 1/4 − ε. The heuristic for the phase space induction theorem builds upon the heuristic given to justify the angular modulation theorem, 7. We crudely characterize the local at the start of Section 2 decay theorem as saying u , g u + u, (−gVL )L u dt < u, ˙ u . Working in a representation ∂ 2 of L which diagonalizes the operators −i ∂ρ∗ and L simultaneously, it makes sense to ask if |−i ∂ρ∂ ∗ | L. If this is the case, then u , g u already controls u, g L2 u. Thus, a full angular
derivative in L2 is controlled. If |−i ∂ρ∂ ∗ | L1−n , then we can multiply the local decay estimate by Ln , and the right-hand side, u, ˙ Ln u u, ˙ Lu, will remain bounded by the energy (ignoring weights in ρ∗ ). Applying the angular modulation argument with m = 1/2, one would expect to control 3/2 + n angular derivatives in expectation value, with the gain of Ln . Alternatively, if |−i ∂ρ∂ ∗ | ∼ L1−n , we expect to control L to the power 2(1 − n) (from (u )2 ) plus n (from multiplying by Ln ) plus m (from gL m ∼ Lm ). In this case, if we take m = n, then 2 + m − n = 2 angular derivatives are controlled in expectation value. Thus, the heuristic suggests the number of angular derivatives controlled in expectation value is 2 when |−i ∂ρ∂ ∗ | ∼ L1−n and 3/2 + n when |−i ∂ρ∂ ∗ | L1−n . To implement the heuristic, we must give meaning to statements localized in
ρ∗ , i ∂ρ∂ ∗ , and L. Phase space analysis identifies an L2 function, f , with its Fourier transform, F [f ], so that both can be thought of as a function of either the original or Fourier variable. By treating the Fourier variable as a multiplication operator which acts as differentiation on the original function, we say that the derivative operator is the Fourier variable. To make statements like |ρ∗ | < L−m (for the angular modulation heuristic) and |−i ∂ρ∂ ∗ | < L1−n (as in this section’s heuristic), we would like to apply compactly supported, localization functions in either ρ∗ or the Fourier variable −i ∂ρ∂ ∗ . However, if a compactly supported localization function in the original variable, X(x), and another in the Fourier variable Φ(−i ∂ρ∂ ∗ ), are both applied to an L2 function, the result (treated in the identified L2 spaces) will only have compact support in one of the variables (unless the resulting function is identically zero), depending on which localization was last applied. To overcome this obstacle, we apply slowly decaying localization functions. The associated scale of these is L−m in ρ∗ and L1−n in −i ∂ρ∂ ∗ . When 0 n m 1/2, the product of these scales, L1−m−n 1, is consistent with the uncertainty principle. Technical concerns restrict the rate of decay for the localization functions, and we introduce a parameter > 0 to quantify this decay rate. We also introduce a parameter δ > 0 to quantify the width L1−n−δ < |−i ∂ρ∂ ∗ | < L1−n associated to the heuristic statement |−i ∂ρ∂ ∗ | ∼ L1−n . The positivity of these leads to the ε loss in Theorem 8.20. An outline of this section appears in Section 1.1. 8.1. Phase space variables, localization, and multipliers Previously, we introduced a function, gLm , which was rescaled by the fractional angular derivative operator Lm . We can also think of this as a function of a rescaled radial variable Lm ρ∗ . Now, we also introduce a rescaled radial derivative. We refer to these as the phase space variables xm and ξ n , respectively. These will be rescaled separately with different parameters m and n.
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
53
Recall that we use f to denote the derivative of f with respect to ρ∗ and f [1] to denote the derivative of f for a function of one variable. This is particularly useful when we consider functions of xm = Lm ρ∗ , so that we can clearly write expressions like f (Lm ρ∗ ) = Lm f [1] (Lm ρ∗ ). Definition 8.1. The phase space variables are x m = Lm ρ ∗ ,
(8.1)
∂ . ξ n = −iLn−1 ∂ρ∗
(8.2)
Corollary 7.8 provides control on the angular energy away from ρ∗ = 0, so that the angular component of the energy only needs to be controlled in a compact set around ρ∗ = 0. This is the region near r = α. We introduce the weight χα to work in this region, although, the particular choice of χα will not matter. Definition 8.2. The function χα is defined to be a smooth, compactly supported, radial function, such that χα 1, and χα = 1 in a neighborhood of ρ∗ = 0. For the phase space variables, we have a notion of lower order, involving our choice of χα . Notation 8.3. The notation O(L1 (dt)) denotes functions of time which are integrable, and for which the L1 norm in time is bounded by a combination of constants, the energy of u, and its initial L2 (M) norm. By Theorem 6.6, Corollaries 7.7 and 7.8, this includes local decay norms, localized radial derivatives, and angular derivatives with vanishing weight at ρ∗ = 0: χα u2 = O L1 (dt) , χα u 2 = O L1 (dt) , χα ρ∗ ∇S 2 u2 = O L1 (dt) . By Theorem 7.6, the norm of 3/4 angular derivatives is also controlled, 3/4 L χα u2 = O L1 (dt) . The notation B denotes an arbitrary bounded operator that commutes with L, just as the notation C denotes an arbitrary constant. As with arbitrary constants, the value of B may vary from line to line in an argument. The notation Bi is used to refer to a bounded operator which is referred to later in the same argument. 1+2n For 0 n 1/2, the notation O(L 2 χα u2 , L1 (dt)) denotes inner products and norms of u which are either of the following form 1+2n L 2 χα u2 ,
or O L1 (dt) ,
(8.3)
or bounded by norms of this form. Note that for q 1 + 2n, this includes any term of the form q L 2 Bχα u2 .
(8.4)
54
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
Note that in particular if m 1/2, then m + n 1/2 + n and 2 2 m+n 1+2n L Bχα u = O L 2 χα u , L1 (dt) .
(8.5)
At this point, we introduce several phase space localization functions. These are functions of the phase space variables that have the bulk of their support in a compact interval of xm or ξ n values. The localizations occur in families, with X referring to xm localization, Φ referring to ξ n localization in a wide band around zero (heuristically |ξ n | 1), and Ψ referring to ξ n localization in a narrow band away from 0 (heuristically |ξ n | ∼ 1). To begin with, it is not possible to use sharp cut-off functions in xm and ξ n , so more regular functions with infinite support must be used. The small parameter determines the decay rate of the smoothed Φ, and the small parameter δ determines the width of the support of the Ψ . Definition 8.4. The weight g is defined by bx g(x) = 0
1 dτ, (1 + |τ |)2
(8.6)
where b is chosen as in Lemma 7.3. The near, smoothed-near, and far localization functions for xm are b , (1 + b|x|)2 1 X˜ ↓ (x) = , 1 + x2 g(x) X↑ (x) = x . x
X↓ (x) =
(8.7) (8.8) (8.9)
From this, X↓ (xm ), X˜ ↓ (xm ), and X↑ (xm ) are defined by the spectral theorem. We note that g [1] (x) = X↓ (x). Definition 8.5. The smooth, near localization for ξ n and two functions derived from it are defined by − 1− 4 , Φa, (x) = 1 + x 2
(8.10)
[1] (x), Φb, (x) = xΦa,
(8.11)
[1] (x) , Φc, (x) = Φa, (x) Φa, (x) + 2xΦa,
(8.12)
where 0 < < 1. (The definition of Φc, (x) requires the argument of the square root to be positive, which is shown in Lemma 8.7.) We use the notation χ(A, x) to denote the characteristic function of A which is 1 if x ∈ A and 0, otherwise.
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
55
The sharp, near localization and sharp, interval localization for ξ n are Φ|x|1 = χ [0, 1], |x| , Ψl −δ |x|1 = χ l −δ , 1 , |x| ,
(8.13) (8.14)
where 0 < δ < 1. Smooth, interval localization will be defined in Section 8.4. From these, Φa, (ξ n ), Φb, (ξ n ), Φc, (ξ n ), Φ|ξ n |1 and ΨL−δ |ξ n |1 are defined by the spectral theorem. To prove a phase space localized version of the Morawetz estimate, we introduce the operator Γn,m . The parameters n and m are restricted to 0 n m 1/2, because this is the range on which an estimate can be proven. Definition 8.6. For 0 n m 1/2, 0 < < 1, and 0 < δ < 1, the phase space induction multiplier is defined to be Γn,m = χα Φa, (ξ n )Ln− γLm Φa, (ξ n )χα 1 = L1− χα Φa, (ξ n )i g(xm )ξ n + ξ n g(xm ) Φa, (ξ n )χα . 2
(8.15) (8.16)
We now prove some preliminary results, starting with some estimates on the localization. Lemma 8.7. If 0 n 1/2, 0 < < 1, then for all v ∈ S Φc, (ξ n )v 12 Φa, (ξ n )v , 1− ∂ L ξ n Φa, (ξ n )2 v L 1+n−2 1− v + v ∂ρ . ∗
(8.17) (8.18)
Proof. The function Φc, (x) is computed explicitly as [1] Φc, (x)2 = Φa, (ξ n ) Φa, (ξ n ) + 2xΦa, (x)
− 1− − 5− 1− 2 4 4 1 + x2 − 2 2x = 1 + x2 1 + x2 4 6−2 − 4 = 1 + x2 1 + x 2 − 1− 2 1 + x2 Φa, (ξ n ) . 2
(8.19) (8.20) (8.21) (8.22) (8.23)
[1] (x)) is strictly positive, the square root function, Φc, (x) Since Φa, (ξ n )(Φa, (ξ n ) + 2xΦa, is well defined. Since Φc, (x)2 Φa, (ξ n )2 , the spectral theorem also determines that 1 Φc, (ξ n )v 2 Φa, (ξ n )v. In a representation where L and −i ∂ρ∂ ∗ both act as multiplication operators, Φa, (ξ n )2
|ξ n |−1+ = |−i ∂ρ∂ ∗ Ln−1 |−1+ , and
56
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
1− L ξ n Φa, (ξ n )2 v L1− ξ n |ξ n |−1+ v L1− |ξ n | v 1+n−2 −i ∂ v . L ∂ρ ∗
(8.24) (8.25) (8.26)
Continuing in a representation where both L and −i ∂ρ∂ ∗ act as multiplication operators, the iden1
1
tity, for positive f and g, f g f 1− + g can be applied with f = L1+n−2 and g = |−i ∂ρ∂ ∗ | to obtain 1− ∂ L ξ n Φa, (ξ n )2 v L 1+n−2 1− v + (8.27) ∂ρ v , ∗ which is Eq. (8.18).
2
Since Γn,m is a product of operators, there is a Leibniz (or product) rule for computing the commutator with another operator. The main application will be when the other operator is H or one of the subterms Hi . Lemma 8.8. If 0 n m 1/2, 0 < < 1, 0 < δ < 1, and G is a self-adjoint operator which commutes with L, then for all u ∈ S
u, [G, Γn,m ]u = u, χα Φa, (ξ n )[G, γLm ]Ln− Φa, (ξ n )χα u + u, χα G, Φa, (ξ n ) γLm Ln− Φa, (ξ n )χα u + u, χα Φa, (ξ n )γLm Ln− G, Φa, (ξ n ) χα u + u, χα Φa, (ξ n )γLm Ln− Φa, (ξ n )[G, χα ]u + u, [G, χα ]Φa, (ξ n )γLm Ln− Φa, (ξ n )χα u .
Proof. This is simply an application of the Leibniz rule for commutators.
(8.28) (8.29) (8.30) (8.31) (8.32)
2
8.2. Commutator expansions To apply the Heisenberg-type relation, it is necessary to expand commutators involving localization in ξ n . This is done through a version of the commutator expansion lemma previously used in scattering theory [36]. We consider the special case of this expansion for commutators involving localization in the phase space variables ξ n and ρ∗ or xm . These expansions are as finite order power series with an error term which involves the Fourier transform of a kth order derivative of one of the localizing functions. First, we recall the definition of the adjoint. These are iterated commutators and appear in the commutator expansion, both as terms in the finite power series expansion and as a term in the remainder. Definition 8.9. For two operators H and A, the kth commutator of A with respect to H , AdkA (H ) is defined recursively by
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
Ad1A (H ) = [H, A], AdkA (H ) = Adk−1 A (H ), A .
57
(8.33) (8.34)
Initially the commutator [Adk−1 A (H ), A] is defined only as a form on the domain of A intersect the domain of H by the formula
k−1 k−1 u, Adk−1 A (H ), A u = Ad A (H )u, Au − Au, Ad A (H )u .
(8.35)
k If Adk−1 A (H ) extends to a bounded operator, then Ad A (H ) is defined on the domain of A.
The following theorem expands the commutator of an operator, F1 , with a function of a selfadjoint operator, F2 (A). The expansion is as a power series in the adjoint and has remainder involving the L1 norm of the Fourier transform of F2[k] . Theorem 8.10 (Commutator expansion theorem). If k > 0 is an integer, A is a self-adjoint operator, F1 is a self-adjoint operator satisfying j
1. for 1 j k, AdA (F1 ) extends to a bounded operator, and F2 (x) is a smooth function satisfying 2. F [F2[k] ]1 < ∞, and [F1 , F2 (A)] is defined as a form on the domain of An , then there is a positive constant C and a bounded operator Rk such that k−1 1 [j ] j F (A)AdA (F1 ) + Rk F1 , F2 (A) = j! 2
(8.36)
j =1
in the form sense with the remainder Rk satisfying Rk C F F2[k] 1 AdkA (F1 ).
(8.37)
Consequently, [F1 , F2 (A)] defines an operator on the domain of Ak−1 . Recall that for an operator, B, we use B to denote the operator norm. Proof. This is proven in [36].
2
In the following lemma, the commutator expansion theorem is specialized to localizing functions in the phase space variables. In terms of the previous theorem, A = ξ n = −i ∂ρ∂ ∗ Ln−1 , and F1 is a function of ρ∗ or of xm . There are two applications of this lemma. The first application is the expansion of the commutator of the Hamiltonian, H , and the phase space induction multiplier, Γn,m , as a power series in the adjoint. The remainder is a lower order term, in the sense that the commutator minus the power series expansion can be multiplied by a positive power of L and still remains a bounded operator. The second application is using the expansion with no terms and using the remainder to directly estimate the entire commutator as a lower order term.
58
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
Lemma 8.11 (Commutator order reduction lemma). Let k be a positive integer, 0 n m 1/2, F1 (x) be a smooth function satisfying [j ]
1. for 1 j k, F1 ∞ ∞, and F2 (x) be a smooth function satisfying [j ]
2. for 1 j k − 1, F2 ∞ ∞, 3. F [F2[k] ]1 ∞, then there is a positive constant, Ck , depending only on k, and bounded operators (both denoted Rk ), such that k−1 j [j ] [j ] Lk(1−n) F1 (ρ∗ ), F2 (ξ n ) = Lk(1−n) F2 (ξ n ) iLn−1 F1 (ρ∗ ) + Rk ,
(8.38)
j =1
Rk Ck F F2[k] 1 F1[k] ∞ ,
(8.39)
and k−1 j [j ] [j ] Lk(1−m−n) F1 (xm ), F2 (ξ n ) = Lk(1−m−n) F2 (ξ n ) iLm+n−1 F1 (xm ) + Rk ,
(8.40)
j =1
Rk Ck F F2[k] 1 F1[k] ∞ .
(8.41)
In both cases, the remainders Rk commute with L. In both cases, the commutators extend as bounded operators applied to powers of L. Proof. The main idea in this proof is to apply the commutator expansion theorem, Theorem 8.10. The m = 0 case is a special case of the formula for general m 0. All quantities in Eqs. (8.38) and (8.40) are composed of powers of L and functions of operators which commute with L. Therefore, they preserve the spherical harmonic decomposition, and it is sufficient to prove the power series expansion of the commutator on each spherical harmonic. Recall that we use l˜ to denote the action of L on each spherical harmonic. On a fixed spherical harmonic, we take x = xm = l˜m ρ∗ and A = ξ n = −i l˜n−1 ∂ρ∂ ∗ and apply the commutator expansion theorem to find
k−1 1 [k] j F2 (ξ n )Adξ F1 (xm ) + Rk , F1 (xm ), F2 (ξ n ) = n j!
(8.42)
j =1
Rk Ck F F2[k] 1 Adkξ (F1 ). n
˜ ρ∗ , and The adjoint can be rewritten in terms of l,
∂ ∂ρ∗
as
(8.43)
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
j −1 j F1 (xm ) , ξ n Adξ F1 (xm ) = Adξ n n j −1 ˜m n−1 ∂ ˜ = Adξ F1 l ρ∗ , −i l n ∂ρ∗ ∂ j −1 = i l˜n−1 F1 l˜m ρ∗ . Ad ∂ρ∗ ξ n
59
(8.44) (8.45) (8.46)
Taking a derivative with respect to ρ∗ of a function of l˜m ρ∗ will introduce an additional factor of l˜m . For j = 1, we have ∂ Ad1ξ n F1 (xm ) = i l˜n−1 F1 l˜m ρ∗ ∂ρ∗ m+n−1 = i l˜ F1[1] l˜m ρ∗ .
(8.47) (8.48)
Each adjoint acts as a derivative, so, by induction, j [j ] j Adξ F1 (xm ) = i l˜m+n−1 F1 l˜m ρ∗ n j [j ] = i l˜m+n−1 F1 (xm ).
(8.49) (8.50)
Applying this, and multiplying by l˜k(1−m−n) , gives that on each spherical harmonic, k−1 j [j ] 1 l˜k(m+n−1) F1 (xm ), F2 (ξ n ) = l˜k(m+n−1) F2[k] (ξ n ) i l˜1−m−n F1 (xm ) + Rk , j!
(8.51)
j =1
[j ] Rk Ck F F2[k] 1 F1 ∞ .
(8.52)
Since the constant in the commutator expansion theorem, Theorem 8.10, is independent of the ˜ The result on each choice of x, A, F1 , and F2 , it follows that the constant Ck is independent of l. spherical harmonic can be extended across all harmonics without any change. In particular, the operator Rk is uniformly bounded across all spherical harmonics. 2 The commutator order reduction lemma, Lemma 8.11, is used to estimate commutators of localization in the phase space variables. Usually it is sufficient to know that the commutator is a bounded operator times negative powers of L, and that is what the following two lemmas assert for certain classes of localizing operators. The first of these lemmas applies in the common situation when the localization in ξ n is a Schwartz class function. Lemma 8.12. There is a positive constant C, such that if 0 n m 1/2, F1 has its first derivative in L∞ , and F2 is of Schwartz class, then there is a bounded operator that commutes with L, B1 , satisfying the following F1 (xm ), F2 (ξ n ) = Lm+n−1 B1 , B1 C F F2[1] 1 F1[1] ∞ .
(8.53) (8.54)
60
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
Proof. Since the Schwartz class is preserved by the Fourier transform and differentiation, the function F [F2[1] ] is Schwartz class and in L1 . The expansion to order k = 1 from the commutator order reduction lemma, Lemma 8.11, is L1−m−n F1 (xm ), F2 (ξ n ) = R1 , R1 C F F2[1] 1 F2[1] ∞
(8.55) (8.56)
Since L is bounded below by 1, and 0 n m 1/2, the operator Lm+n−1 is bounded. The operators Lm+n−1 and R1 are bounded, so they can be composed. Applying Lm+n−1 to (8.55) proves the desired result. 2 The following lemma provides a similar, commutator-order-reduction result for the smooth, near localizations for ξ n . These localizations are not Schwartz class functions, so the previous lemma does not apply. The following lemma is even sufficiently strong to control the unbounded operators F2 (x) = x and F2 (x) = xΦa, (x). Lemma 8.13. There is a positive constant C, such that if 0 n m 1/2, 0 < < 1, F1 has its first derivative in L∞ , and F2 (x) is one of the following functions: x, Φa, (x), Φb, (x), Φc, (x), [1] [2] (x), or xΦa, (x), then there is a bounded operator that commutes with L, BF1 ,F2 , satisfying Φa, the following F1 (xm ), F2 (ξ n ) = Lm+n−1 BF1 ,F2 , BF1 ,F2 C F1[1] ∞ .
(8.57) (8.58)
If the previous conditions apply, F1 also has its second derivative in L∞ , and F2 (x) = xΦa, (x), then there is a bounded operator that commutes with L, BF1 ,F2 , satisfying F1 (xm ), F2 (ξ n ) = Lm+n−1 BF1 ,F2 , BF1 ,F2 C F1[1] ∞ + F1[2] ∞ .
(8.59) (8.60)
Proof. The fundamental problem here is that the permitted F2 functions are not in Schwartz class. There are finitely many functions F2 considered in this theorem, so if a C can be found for each of them, then the largest one can be applied to make the result hold uniformly. For the function F2 (x) = x, the corresponding operator F2 (ξ n ) = ξ n = −i ∂ρ∂ ∗ Ln−1 . All the operators involved commute with the spherical harmonic decomposition, and on each spherical harmonic, the commutator can be explicitly computed as F1 (xm ), −i
∂ n−1 ∂ n−1 = F1 Lm ρ∗ , −i L L ∂ρ∗ ∂ρ∗ = iLm+n−1 F1[1] Lm ρ∗ = iLm+n−1 F1[1] (xm ).
(8.61) (8.62)
The commutator is a product of a power of L and a multiplication operator. The L∞ norm of the function is the operator norm of the associated multiplication operator.
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
61
To apply the commutator order reduction lemma, Lemma 8.11 to the remaining functions, we prove: if h ∈ C 2 (R) ∩ L2 (R) and h[2] ∈ L2 (R), then F [h]1 ∞. Since h ∈ C 2 , h[2] is well defined, and since h, h[2] ∈ L2 (R), both F [h] and F [h[2] ] are in L2 (R). Let f (ξ ) = 1 + ξ 2 F [h](ξ ) , fbig (ξ ) = f (ξ )χ [1, ∞), f (ξ ) , fsmall (ξ ) = f (ξ )χ [0, 1), f (ξ ) ,
(8.63) (8.64) (8.65)
and observe that f (ξ ) = fbig (ξ ) + fsmall (ξ ), F [h](ξ ) fbig (ξ ) + fsmall (ξ ) fbig (ξ ) + fsmall (ξ ) . 1 + ξ2 1 + ξ2 1 + ξ2
(8.66) (8.67)
From properties of the Fourier transform it is known that F [h](ξ ) + F h[2] (ξ ) = F [h](ξ ) + ξ 2 F [h](ξ ) = f (ξ ) ,
(8.68)
2 f 2, f 1 and hence that f is in L2 (R). Since fbig fbig big is in L (R), and since fsmall 1, fsmall (ξ ) 1+ξ 2
is also in L1 (R). The sum of these bounds |F [h]|, so that F [h] is in L1 (R). For the three functions, Φa, (x), Φb, (x), and Φc, (x), each of these is a power of a rational −1+ −3+ −7+ function and decays like x 2 . Since F2[1] and (F2[1] )[2] decay like x 2 and x 2 , respectively, they are each in L2 . Therefore, F [F2[1] ] is in L1 . By the same argument as in Lemma 8.12, F1 (xm ), F2 (ξ n ) = Lm+n−1 B, B C F1[1] ∞ .
(8.69) (8.70)
[1] [2] For the two function Φa, (x) and xΦa, (x), the same argument applies, except that the decay −3+ rate for these functions is stronger, x 2 . The last, and most difficult piece, is for F2 (x) = xΦa, (x). In the previous arguments, the commutator was directly estimated. In this case, it is expanded to order k = 2, and both the firstorder term and the remainder are estimated. The derivative of xΦa, (x) decays too slowly to be in L2 and may not have an L1 Fourier transform. As usual, all the operators involved commute with the spherical-harmonic decomposition. On each spherical harmonic, by Lemma 8.11,
F1 (xm ), F2 (ξ n ) = F2[1] (ξ n ) iLm+n−1 F1[1] (xm ) + L2n+2m−2 R2 .
(8.71)
The first term is a product of operators which commute with powers of L. In the first term, the operators other than powers of L give a bounded operator,
62
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
[1] F (ξ n ) −F [1] (xm ) = F [1] (ξ n )F [1] (xm ) 2 1 2 1 F2[1] ∞ F1[1] ∞ .
(8.72) (8.73)
Since F2 (x) = xΦa, (x), the function F2[1] is bounded. The second term is estimated by Lemma 8.11 as R2 = C F F2[2] 1 F1[2] ∞ .
(8.74)
Since F2 = xΦa, (x), the function F2[2] is a smooth function, which is in L2 and has its second derivative, (F2[2] )[2] , also in L2 , we conclude that F [F2[2] ] is in L1 . The commutator can be factored as
F1 (xm ), F2 (ξ n ) Lm+n−1 F2[1] (xm )F1[1] (ξ n ) + Lm+n−1 R2 .
(8.75)
Working on each spherical harmonic, we can move the powers of L to the left-hand side and bound the right-hand side 1−m−n L F1 (xm ), F2 (ξ n ) C F [1] 1
∞
+ F1[2] ∞ .
(8.76)
On each spherical harmonic there is a bounded operator, B, such that F1 (xm ), F2 (ξ n ) = Lm+n−1 B, B C F1[1] ∞ + F1[2] ∞ . Therefore, B can be extended as an operator on L2 satisfying the same condition.
(8.77) (8.78) 2
8.3. Phase space estimates In this subsection, we compute the commutators between Γn,m and the components of the Hamiltonian, Hi . This is analogous to Section 6.2 with phase space localization and additional factors of L. The H2 commutator is simply lower order. The H1 + H3 commutator dominates the radial derivative localized to small xm and powers of L localized to large xm . The commutator with H2 is computed first. Because H2 is a function of ρ∗ and contains no derivatives, all the rearrangements can be shown to be lower order by Lemma 8.13. Lemma 8.14. If 0 n m 1/2, and 0 < < 1, then for all u ∈ S(R × M) satisfying the wave equation (2.25) there is the H2 commutator estimate that
u, [H2 , Γn,m ]u = O L1 (dt) .
(8.79)
Proof. We use the Leibniz formula for commutators, Lemma 8.8, to calculate [H2 , Γn,m ] and then show the commutators are lower order by Lemma 8.13. Since χα and V commute,
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
63
[H2 , χα ] = 0, the three nonvanishing terms in the expansion of the [H2 , Γn,m ] are
u, [H2 , Γn,m ]u = u, χα Ln− V , Φa, (ξ n ) γLm Φa, (ξ n ) + Φa, (ξ n )[V , γLm ]Φa, (ξ n ) + Φa, (ξ n )γLm V , Φa, (ξ n ) χα u .
(8.80) (8.81) (8.82)
The commutator [V , γLm ] only involves the angularly modulated multiplier and was shown to be a bounded operator in the proof of Lemma 7.5. Applying additional bounded operators and powers of L yields a bounded operator multiplied by the same power of L, Ln− Φa, (ξ n )[V , γLm ]Φa, (ξ n ) = Ln− B.
(8.83)
Since n 1/2 < 3/2, by the angular modulation theorem, Theorem 7.6, the expectation value of this operator is
u, χα Ln− Φa, (ξ n )[V , γLm ]Φa, (ξ n )χα u = u, Ln− Bu = O L1 (dt) .
(8.84) (8.85)
The remaining terms are complex conjugates of each other. Only one is considered here, and the other can be dealt with in the same way. We expand the first as L
n−
∂ 1 n− Φa, (ξ n )γLm V , Φa, (ξ n ) = L Φa, (ξ n ) gLm − gLm Ln−1 B. ∂ρ∗ 2
(8.86)
Since gLm is bounded and gL m = Lm X↓ (xm ), we can write this estimate as Ln− Φa, (ξ n )γLm V , Φa, (ξ n ) ∂ 1 = L2n−1− Φa, (ξ n ) B1 + L2n−1− Lm Φa, (ξ n ) X↓ (xm )B2 ∂ρ∗ 2 ∂ B1 + Lm+2n−1− B3 . = L2n−1− Φa, (ξ n ) ∂ρ∗
(8.87) (8.88)
The expectation value of this can be estimated by the Cauchy–Schwarz inequality as u, χα Ln− Φa, (ξ n )γLm V , Φa, (ξ n ) χα u
∂ 2n−1− = − Φa, (ξ n )χα u, L B1 χα u + χα u, Lm+2n−1− B3 χα u ∂ρ∗
∂ 2n−1− m+2n−1− 2 2 L L + −C χ u B χ u χ u 1 α α ∂ρ α ∗ 2
∂ 2n−1− 2 m+2n−1− 2 2 . −C χα u + L χα u + L χα u ∂ρ∗
(8.89) (8.90) (8.91) (8.92)
Since 2n − 1 − < 0 < 3/4 and m + 2n − 1 − 1/2 < 3/2, by the angular modulation theorem, Theorem 7.6, the two terms involving powers of L are time integrable, i.e. O(L1 (dt)).
64
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
By Corollary 7.7, ∂ρ∂ ∗ χα u2 = O(L1 (dt)). Therefore, the expectation value is time integrable, u, χα Ln− Φa, (ξ n )γLm V , Φa, (ξ n ) χα u = O L1 (dt) .
2
(8.93)
The following lemma is the main result of this subsection. The lemma gives a lower bound for the commutator of H1 + H3 and Γn,m . This is like Lemma 7.4, but strengthened by n − additional factors of L. The lower bound consists of three terms. The first is localized to small xm and consists of two radial derivatives and m + n − angular derivatives. The second is localized to large xm and consists of 2 + n − angular derivatives. The third term is the sum of various rearrangement terms accumulated in the proof of the lemma. This third term is lower order. Were it not for the technicalities of rearranging the phase-space localization functions, we would be able to prove this lemma by saying that the phase-space induction multiplier Γn,m is the angular modulation multiplier with additional factors of Ln− and of phase-space localization functions. Estimates on the error terms generated by rearranging the phase-space localizations take up most of the proof. In this lemma, we require the strict positivity of Φc, (ξ n ). It is at this point that we require > 0, which limits the decay rate of Φa, (x). Lemma 8.15. If 0 < < 1, then there is a positive constant, C, such that for 0 n m 1/2, and for all u ∈ S(R × M) satisfying the wave equation (2.25) there is the H1 + H3 commutator estimate that
u, [H1 + H3 , Γn,m ]u
∂ m ∂ n− 2 m − L X↓ (xm ) − L gL VL Φa, (ξ n )χα u C Φa, (ξ n )χα u, L ∂ρ∗ ∂ρ∗ 2 1+2n + O L 2 χα u , L1 (dt) .
(8.94)
Proof. The proof consists of four main steps. First, we expand the commutator of H1 + H3 using the Leibniz rules. Second, we show the commutators involving χα are lower order. Third, we analyze the commutator involving the phase space localizations Φa, (ξ n ) and the angularly modulated multiplier γLm . Fourth, we combine the positive commutator estimates. In the second and third steps, we decompose the commutators into several pieces and then rearrange the phase space localizations. The rearrangements are lower order by Lemma 8.13. When rearranging terms, we typically write the rearranged term first, followed by commutators arising from the rearrangements, and finally lower order terms. This means that terms may appear in a radically different order than they did in the previous line of a calculation. In the third step, it is also necessary to eliminate some terms via exact cancellation and via improvements of Lemma 8.13. The third step is by far the most complicated. Step 1. The commutator is given by the Leibniz rule as the sum of five terms, [H1 + H3 , Γn,m ] = [H1 + H3 , χα ]Φa, (ξ n )Ln− γLm Φa, (ξ n )χα + χα Φa, (ξ n )Ln− γLm Φa, (ξ n )[H1 + H3 , χα ]
(8.95)
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
65
+ χα H1 + H3 , Φa, (ξ n ) Ln− γLm Φa, (ξ n )χα + χα Φa, (ξ n )Ln− γLm H1 + H3 , Φa, (ξ n ) χα
(8.96)
+ χα Φa, (ξ n )Ln− [H1 + H3 , γLm ]Φa, (ξ n )χα .
(8.97)
Since the first two terms are adjoints of each other, it is sufficient to estimate only one of them. The remaining three terms are more complicated and must be estimated together. Step 2. Since χα is a function, it commutes with VL , and we only need to determine the contri∂2 bution from − ∂ρ 2 , which can be explicitly calculated, and is found to give ∗
∂2 u, χα Φa, (ξ n )γLm L Φa, (ξ n ) − 2 , χα u ∂ρ∗
∂ n− m = u, χα Φa, (ξ n )γL L Φa, (ξ n ) −2 χ + χα u . ∂ρ∗ α n−
(8.98) (8.99)
To analyze this term, γLm is expanded into two terms so that the most significant term can be dealt with first. The slightly less common expansion γLm = ∂ρ∂ ∗ gLm − 12 gL m is used. There are a total of four terms to consider, two from expanding [H1 , χα ] times two from expanding γLm . The term that appears to be highest order is the one involving ∂ρ∂ ∗ gLm from γLm and −2 ∂ρ∂ ∗ χα 2
∂ from [− ∂ρ 2 , χα ]. It will be referred to as I0 . It can be simplified by moving one of the derivative ∗ operators through Φa, (ξ n ) (since they commute), and then moving this derivative and a factor of χα to the other side of the inner product. This gives
∂ ∂ I0 = u, χα Φa, (ξ n ) gLm Ln− Φa, (ξ n ) −2 χα u ∂ρ∗ ∂ρ∗
∂ ∂ =− χα u, Φa, (ξ n )gLm Ln− Φa, (ξ n ) −2 χα u . ∂ρ∗ ∂ρ∗
(8.100) (8.101)
We apply the Cauchy–Schwarz estimate to get 2 2 ∂ ∂ n− m |I0 | χα u + Φa, (ξ n )gL L Φa, (ξ n ) χ u , ∂ρ ∂ρ α ∗
∗
(8.102)
which is a sum of two terms. The first is ∂ρ∂ ∗ χα u2 which is O(L1 (dt)) by Corollary 7.7. We rewrite the other in terms of ξ n to get 2 |I0 | L1− Φa, (ξ n )gLm ξ n Φa, (ξ n )χα u + O L1 (dt) .
(8.103)
The remaining norm is going to be manipulated with the goal of applying Lemma 8.7 to control ξ n Φa, (ξ n )2 by powers of L. To avoid terms involving ξ n outside a commutator, we rearrange the phase-space localizations twice as
66
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
2 |I0 | L1− ξ n Φa, (ξ n )gLm Φa, (ξ n )χα u 2 + L1− Φa, (ξ n )[gLm , ξ n ]Φa, (ξ n )χα u + O L1 (dt) 2 2 L1− gLm ξ n Φa, (ξ n )2 χα u + L1− ξ n Φa, (ξ n ), gLm Φa, (ξ n )χα u 2 + L1− Φa, (ξ n )[gLm , ξ n ]Φa, (ξ n )χα u + O L1 (dt) .
(8.104) (8.105) (8.106) (8.107)
By Lemma 8.13, the commutators are of the form Lm+n−1 B, so that 2 2 |I0 | C1 L1− gLm ξ n Φa, (ξ n )2 χα u + C2 L1− Lm+n−1 BΦa, (ξ n )χα u + O L1 (dt) . (8.108) Since Φa, (ξ n ) is also a bounded operator, all the error terms from commuting are lower order terms. To deal with the first term, we can drop the bounded factor of gLm , apply Lemma 8.7 to control ξ n Φa, (ξ n )2 , and obtain
2 ∂ 2 2 1 1+n−2 1+2n χ u |I0 | C L 1− χα u + ∂ρ α + O L 2 χα u , L (dt) . ∗
(8.109)
We are left with the sum of three terms. The third is already lower order. The second, ∂ρ∂ ∗ χα u2 is time integrable by Corollary 7.7, and, hence, lower order. To control the first term, we observe that by interpolating the local decay Theorem 6.6 and Corollary 7.8, we know that a term is time integrable if it consists of the norm squared of u with L to a fractional power between 0 and 1 and with a smooth, compactly supported weight which vanishes at least linearly at ρ∗ = 0. The first term in the expression under consideration is exactly of this form. Therefore, all the terms in I0 are lower order and
∂ ∂ n− gLm L Φa, (ξ n ) −2 χα u |I0 | = u, χα Φa, (ξ n ) ∂ρ ∂ρ ∗
2 1+2n = O L 2 χα u , L1 (dt) .
∗
(8.110)
We now turn to estimating the remaining terms in (8.99). The analysis is sufficiently quick, that we do not introduce names for these terms. The term that appears to be the next highest order ∂2 involves gL m from γLm and −2 ∂ρ∂ ∗ χα from [− ∂ρ 2 , χα ]. We start estimating this term by moving ∗ most of the localization to the other side of the inner product, so that
1 bLm+n− ∂ u, χα Φa, (ξ n ) − Φa, (ξ n ) χ u 2 (1 + b|Lm ρ∗ |)2 ∂ρ∗ α
∂ 1 χα u . = − Φa, (ξ n )Lm+n− X↓ (xm )Φa, (ξ n )χα u, 2 ∂ρ∗
We have, from applying the Cauchy–Schwarz inequality,
(8.111) (8.112)
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
m+n− ∂ u, χα Φa, (ξ n ) − 1 bL Φ (ξ ) χ u a, n 2 (1 + b|Lm ρ∗ |)2 ∂ρ∗ α m+n− ∂ C L Φa, (ξ n )X↓ (xm )Φa, (ξ n )χα u χ u, ∂ρ∗ α
67
(8.113)
and then, from replacing bounded operators by a constant,
m+n− ∂ u, χα Φa, (ξ n ) − 1 bL Φa, (ξ n ) χ u 2 (1 + b|Lm ρ∗ |)2 ∂ρ∗ α
m+n− 2 ∂ 2 −C L χα u + χ u . ∂ρ∗ α
(8.114)
These terms are lower order, and
m+n− ∂ u, χα Φa, (ξ n ) − 1 bL Φa, (ξ n ) χ u 2 (1 + b|Lm ρ |)2 ∂ρ α 2 1+2n O L 2 χα u , L1 (dt) .
∗
∗
(8.115)
2
∂ The terms involving χα from [− ∂ρ 2 , χα ] are the lowest order terms and are quickly shown to ∗
be lower order. The same method as was used for the term involving gL m and −2 ∂ρ∂ ∗ χα can be applied. The calculations are
∂ n− L gLm Φa, (ξ n )χα u u, χα Φa, (ξ n ) ∂ρ∗
∂ = gLm Φa, (ξ n )χα u, Ln− Φa, (ξ n )χα u , ∂ρ∗
u, χα Φa, (ξ n ) ∂ Ln− gLm Φa, (ξ n )χ u C ∂ χα uLn− χ u α α ∂ρ∗ ∂ρ∗ 1 = O L (dt) 2 1+2n = O L 2 χα u , L1 (dt) ,
(8.116)
(8.117) (8.118) (8.119)
and m+n− u, χα Φa, (ξ n ) bL Φ (ξ )χ u a, n α m 2 (1 + b|L ρ∗ |) = Φa, (ξ n )χα u, Lm+n− X↓ (xm )Φa, (ξ n )χα u m+n− m+n− C L 2 X↓ (xm )Φa, (ξ n )χα uL 2 Φa, (ξ n )χα u m+n− 2 C L 2 χα u 2 1+2n O L 2 χα u , L1 (dt) .
(8.120) (8.121) (8.122) (8.123) (8.124)
68
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
Step 3. We now turn to the three remaining terms in Eq. (8.95). For the commutators with ∂2 Φa, (ξ n ), since − ∂ρ 2 commutes with ξ n , it is sufficient to compute the commutator with VL . ∗
No distinction is made between [H3 , Γn,m ] = [VL (L2 − 1), Γn,m ] and [VL L2 , Γn,m ] since the difference is lower order by an identical argument to that in Lemma 8.14. These commutators are expanded to order k = 3 as [1] [2] VL , Φa, (ξ n ) = Φa, (ξ n )[VL , ξ n ] + Φa, (ξ n ) [VL , ξ n ], ξ n + R3 .
(8.125)
In each of the third and fourth terms of (8.95), we make this substitution and expand γLm as a sum of two terms. This gives twelve terms, which are combined with two terms from the fifth term in (8.95), for a total of fourteen. The operator γLm is expanded as Ln− (gLm ∂ρ∂ ∗ + gL m /2) when it appears to the left of the commutator and as Ln− ( ∂ρ∂ ∗ gLm − gL m /2) when it appears to the right of the commutator. The fourteen terms are grouped as χα H3 , Φa, (ξ n ) Ln− γLm Φa, (ξ n )χα + χα Φa, (ξ n )Ln− γLm H3 , Φa, (ξ n ) χα + χα Φa, (ξ n )L
(8.126) (8.127)
[H1 + H3 , γLm ]Φa, (ξ n )χα (8.128) ∂2 − 2 , γLm Φa, (ξ n )χα + χα (I1 + I2 + I3 + I4 + I5 + I6 )χα , (8.129) ∂ρ∗
n−
= χα Φa, (ξ n )Ln−
where ∂ [1] Φa, I1 = L (ξ n )[VL , ξ n ] gLm Φa, (ξ n ) ∂ρ∗ ∂ Φa, (ξ n ) + Φa, (ξ n )gLm VL , ∂ρ∗
∂ [1] m Φ (ξ )[VL , ξ n ] , + Φa, (ξ n )gL ∂ρ∗ a, n 1 [1] [1] I2 = L2+n− −Φa, (ξ n )[VL , ξ n ]gL m Φa, (ξ n ) + Φa, (ξ n )gL m Φa, (ξ n )[VL , ξ n ] , 2 ∂ [2] (ξ n ) [VL , ξ n ], ξ n gLm Φa, (ξ n ) I3 = L2+n− Φa, ∂ρ∗
∂ [2] Φa, (ξ n ) [VL , ξ n ], ξ n , + Φa, (ξ n )gLm ∂ρ∗ 1 [2] I4 = L2+n− −Φa, (ξ n ) [VL , ξ n ], ξ n gL m Φa, (ξ n ) 2 [2] + Φa, (ξ n )gL m Φa, (ξ n ) [VL , ξ n ], ξ n ,
∂ ∂ I5 = L2+n− R3 gLm Φa, (ξ n ) + Φa, (ξ n )gLm R3 , ∂ρ∗ ∂ρ∗ 1 I6 = L2+n− −R3 gL m Φa, (ξ n ) + Φa, (ξ n )gL m R3 . 2 2+n−
(8.130) (8.131) (8.132) (8.133) (8.134) (8.135) (8.136) (8.137) (8.138) (8.139)
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
69
The leading order term, I1 , is composed of two types of terms. One type comes from the commutator [L2 VL , γLm ], and the other comes from combining the first order term in the expansion of [L2 VL , Φa, (ξ n )] with the derivative terms in γLm . The commutators in I1 can be expanded as ∂ 2+n− [1] Φa, (ξ n )(−VL ) −iLn−1 gLm Φa, (ξ n ) (8.140) I1 = L ∂ρ∗ ∂ Φa, (ξ n ) + Φa, (ξ n )gLm VL , (8.141) ∂ρ∗
∂ [1] n−1 m + Φa, (ξ n )gL . (8.142) Φ (ξ ) −VL −iL ∂ρ∗ a, n If the (−iLn−1 ) are grouped with the ∂ρ∂ ∗ to form ξ n , and the VL and the gLm are grouped together by commuting the necessary operators, then [1] I1 = L2+n− Φa, (ξ n )ξ n −VL gLm Φa, (ξ n ) + Φa, (ξ n ) −VL gLm Φa, (ξ n ) [1] + Φa, (ξ n ) −VL gLm ξ n Φa, (ξ n ) [1] + L2+n− Φa, (ξ n ) −VL , ξ n gLm Φa, (ξ n ) [1] + Φa, (ξ n )gLm ξ n Φa, (ξ n ), −VL .
(8.143) (8.144) (8.145) (8.146) (8.147)
Using Lemma 8.13 to control the terms introduced by rearranging the localizations, we find that I1 is the sum of terms involving VL gLm and lower order terms, [1] I1 = L2+n− ξ n Φa, (ξ n ) −VL gLm Φa, (ξ n ) + Φa, (ξ n ) −VL gLm Φa, (ξ n ) [1] + Φa, (ξ n ) −VL gLm ξ n Φa, (ξ n ) +L
1+2n−
B.
(8.148) (8.149) (8.150) (8.151)
We move the localization −VL gLm to the front, so that I1 can be expressed as [1] I1 = L2+n− −VL gLm Φa, (ξ n ) Φa, (ξ n ) + 2ξ n Φa, (ξ n ) [1] + L2+n− ξ n Φa, (ξ n ), −VL gLm Φa, (ξ n ) + Φa, (ξ n ), −VL gLm Φa, (ξ n ) [1] + Φa, (ξ n ), −VL gLm ξ n Φa, (ξ n ) + L1+2n− B.
(8.152) (8.153) (8.154) (8.155) (8.156)
2 . We move one The ξ n localization on the right of the first line of this expression is exactly Φc, factor of Φc, back through the −VL gLm localization to obtain
70
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
I1 = L2+n− Φc, (ξ n ) −VL gLm Φc, (ξ n ) + L2+n− −VL gLm , Φc, (ξ n ) Φc, (ξ n ) [1] + ξ n Φa, (ξ n ), −VL gLm Φa, (ξ n ) + Φa, (ξ n ), −VL gLm Φa, (ξ n ) [1] + Φa, (ξ n ), −VL gLm ξ n Φa, (ξ n )
(8.157) (8.158) (8.159) (8.160) (8.161)
+ L1+2n− B.
(8.162)
Of the terms in the previous expression, the first is a positive term we wish to keep, and the ex1+2n pectation value of the last is O(L 2 χα u2 , L1 (dt)). The other terms all involve a commutator of the form [F (ξ n ), (−VL gLm )] multiplied by L2+n− and bounded operators. The commutator [F (ξ n ), (−VL gLm )] must be decomposed since it involves a mixed type localization, VL gLm , which involves functions of ρ∗ and of xm = Lm ρ∗ . Naively applying the Leibniz rule for commutators with the decomposition into VL and gLm , gives a contribution from the commutator with gLm which is L1+m+2n− B, which is not O(L in expectation value. The decomposition
1+2n 2
χα u2 , L1 (dt))
−VL gLm = −VL ρ∗−1 ρ∗ Lm gLm L−m
(8.163)
1+2n
ensures that the commutators are all O(L 2 χα u2 , L1 (dt)) in expectation. This decomposition is possible since −VL vanishes linearly at ρ∗ = 0. Taylor’s theorem guarantees −VL ρ∗−1 is C ∞ . We expand the commutator as L2+n− F (ξ n ), −VL gLm = L2−m+n− F (ξ n ), −VL ρ∗−1 Lm ρ∗ gLm .
(8.164)
Using the Leibniz rule, we further expand this as L2+n− F (ξ n ), −VL gLm = L2−m+n− −VL ρ∗−1 F (ξ n ), X↑ (xm )2 + L2−m+n− F (ξ n ), −VL ρ∗−1 X↑ (xm )2 .
(8.165) (8.166)
[1] (ξ n ), and since both Since the F considered here are Φa, (ξ n ), Φb, (ξ n ), Φc, (ξ n ), and ξ n Φa, −1 m 2 VL ρ∗ and L ρ∗ gLm = X↑ (xm ) both have bounded first and second derivatives, we can apply Lemma 8.13 to control the two commutators and find that
L2+n− F (ξ n ), −VL gLm = L2−m+n− Lm+n−1 B + L2−m+n− Ln−1 BX↑ (xm )2 . (8.167) Since X↑ (xm )2 = BLm ρ∗ , the mixed type commutator becomes L2+n− F (ξ n ), −VL gLm = L1+2n− B + L1+2n− Bρ∗ .
(8.168)
The first term has an expectation value which is L1 . Since ρ∗ is bounded on the support of χα ,
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
2 χα u, L1+2n− Bχα u + χα u, L1+2n− Bρ∗ χα u C L 1+2n− 2 χα u 2 1+2n = O L 2 χα u , L1 (dt) .
71
(8.169) (8.170)
Therefore, all the mixed type commutators are lower order terms. Substituting this into the previous estimate on I1 , (8.157), we find that the expectation value of I1 is 2−m+n− 2 1 −VL ρ∗−1 2 X↑ (xm )Φc, (ξ n )χα u u, χα I1 χα u = L 2 2 1+2n + O L 2 χα u , L1 (dt) .
(8.171)
Using the estimate in Eqs. (8.164)–(8.170) again, we can rearrange the localization as 2−m+n− 2 1 u, χα I1 χα u C L 2 Φc, (ξ n ) −VL ρ∗−1 2 X↑ (xm )χα u 2 1+2n + O L 2 χα u , L1 (dt) .
(8.172)
A factor of C is lost here, since we are applying the triangle inequality to squared norms. The localization Φc, (ξ n ) can be replaced by 1/2 Φa, (ξ n ), so the estimate becomes 2−m+n− 2 1 u, χα I1 χα u C L 2 Φa, (ξ n ) −VL ρ∗−1 2 X↑ (xm )χα u 2 1+2n + O L 2 χα u , L1 (dt) .
(8.173)
To conclude the estimate on I1 , we commute the localization again: 2−m+n− 2 1 u, χα I1 χα u (1/2) L 2 −VL ρ∗−1 2 X↑ (xm )Φa, (ξ n )χα u 2 1+2n + O L 2 χα u , L1 (dt) .
(8.174)
The factor of 1/2 appears because, from Eq. (8.172) to Eq. (8.174), we have lost constant factors by applying the triangle inequality to squared norms. Because, at each step, we have the freedom to write |a + b|2 (1 − 1/N)|a|2 − (1 − 1/N)(4N 2 − 8N + 5)|b|2 , we can choose the constants on the leading-order, lower bound to be arbitrarily close to one. This forces us to accept large constants on the commutator terms, but they remain lower order. Applying this twice, we can choose the constant on the leading-order, lower bound to arbitrarily close to one. In particular, we can take it to be larger than 1/2. 1+2n It now remains to show that I2 , . . . , I6 are O(L 2 χα u2 , L1 (dt)). Discussion of these continues from highest order to lowest. The first of these is I2 which contains terms involving [1] gL m and ξ n Φa, (ξ n ) terms. Note that because two different expansions of γLm were used in Eq. (8.129), the signs on the two factors of gL m are different. To simplify I2 , the factors of gL m are expanded. We write I2 as
72
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
1 [1] −Φa, (ξ n )[VL , ξ n ]gL m Φa, (ξ n ) 2 [1] (ξ n )[VL , ξ n ] + Φa, (ξ n )gL m Φa,
I2 = L2+n−
1 [1] = L2+n− −Φa, (ξ n ) −iLn−1 −VL Lm X↓ (xm )Φa, (ξ n ) 2 [1] (ξ n ) −iLn−1 −VL + Φa, (ξ n )Lm X↓ (xm )Φa, i [1] (ξ n )VL X↓ (xm )Φa, (ξ n ) = L1+m+2n− −Φa, 2 [1] (ξ n )VL . + Φa, (ξ n )X↓ (xm )Φa,
(8.175) (8.176) (8.177) (8.178) (8.179) (8.180)
This leaves an expression for I2 as the difference between two terms. The goal now is to rearrange these terms so that the highest order parts from each cancel. As usual, the rearrangement [1] will introduce commutators. Naively attempting to commute Φa, (ξ n ) or Φa, (ξ n ) with gLm will 1+2n leave a commutator of the form L2m+3n− B, which is not O(L 2 χα u2 , L1 (dt)) in expectation value. An additional factor of ρ∗ ρ∗−1 is introduced and used to absorb a factor of Lm into a function of ρ∗ Lm , so that i [1] I2 = L1+2n− −Φa, (ξ n )ρ∗ Lm X↓ (xm ) VL ρ∗−1 Φa, (ξ n ) 2 [1] (ξ n )ρ∗ VL ρ∗−1 . + Φa, (ξ n )Lm X↓ (xm )Φa,
(8.181) (8.182)
Note here, that since X↓ (xm ) and (VL ρ∗−1 ) commute, they can be rearranged without introducing error terms. Functions of ρ∗ are now moved to obtain the same localizing functions, (VL ρ∗−1 ) and ρ∗ Lm X↓ (xm ) = xm X↓ (xm ), in both terms. We find i [1] I2 = L1+2n− −Φa, (ξ n ) xm X↓ (xm ) Φa, (ξ n ) VL ρ∗−1 2 [1] (ξ n ) VL ρ∗−1 + Φa, (ξ n ) xm X↓ (xm ) Φa, i [1] (ξ n ) xm X↓ (xm ) VL ρ∗−1 , Φa, (ξ n ) + L1+2n− −Φa, 2 + Φa, (ξ n )X↓ (xm )Lm Φa, (ξ n ), ρ∗ VL ρ∗−1 .
(8.183) (8.184) (8.185) (8.186)
Applying Lemma 8.13 to estimate the commutators, we find i [1] (ξ n ) xm X↓ (xm ) Φa, (ξ n ) VL ρ∗−1 I2 = L1+2n− −Φa, 2 [1] (ξ n ) VL ρ∗−1 + Φa, (ξ n ) xm X↓ (xm ) Φa, + L3n− B1 + Lm+3n− B2 .
(8.187) (8.188) (8.189)
We rewrite I2 by bringing the ξ n localization grouped to the left of all ρ∗ and Lm ρ∗ localiza[1] (ξ n )(xm X↓ (xm ))(VL ρ∗−1 ), but with opposite tion. The leading order terms are both Φa, (ξ n )Φa, signs, so they cancel, leaving
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
i [1] I2 = 0 + L1+2n− −Φa, (ξ n ) xm X↓ (xm ) , Φa, (ξ n ) VL ρ∗−1 2 [1] + Φa, (ξ n ) xm X↓ (xm ) , Φa, (ξ n ) VL ρ∗−1 +L
3n−
B1 + L
m+3n−
73
(8.190) (8.191) (8.192)
B2 .
The remaining terms are commutators which are Lm+3n− B by Lemma 8.13. Although X↓ (x) is not C 2 , we will only need to use xX↓ (x) which has its first (and even second) derivative in L∞ . Thus, I2 = Lm+3n− B1 + L3n− B2 .
(8.193)
Because the powers of L involved are all less than 1 + 2n, 2 1+2n χα u, I2 χα u = O L 2 χα u , L1 (dt) .
(8.194)
Next I3 is considered. This involves a second commutator [[VL , ξ n ], ξ n ] which can be explicitly computed as (VL )(−L2n−2 ). Making this substitution, we have ∂ [2] (ξ n ) [VL , ξ n ], ξ n gLm Φa, (ξ n ) I3 = L2+n− Φa, ∂ρ∗
∂ [2] + Φa, (ξ n )gLm Φ (ξ ) [VL , ξ n ], ξ n ∂ρ∗ a, n ∂ [2] = L2+n− Φa, (ξ n )(VL ) −L2n−2 gLm Φa, (ξ n ) ∂ρ∗
2n−2 ∂ [2] + Φa, (ξ n )gLm . Φ (ξ ) V −L ∂ρ∗ a, n L To further simplify this, it is rewritten in terms of ξ n instead of
∂ ∂ρ∗
(8.195) (8.196) (8.197) (8.198)
as
[2] (ξ n ) VL ξ n gLm Φa, (ξ n ) I3 = iL1+2n− Φa, [2] (ξ n ) VL . + Φa, (ξ n )gLm ξ n Φa,
(8.199) (8.200)
[2] (ξ n ) together and rewrite I3 as We group ξ n and Φa,
[2] I3 = iL1+2n− ξ n Φa, (ξ n ) VL gLm Φa, (ξ n ) [2] + Φa, (ξ n )gLm ξ n Φa, (ξ n ) VL [2] + iL1+2n− Φa, (ξ n ) VL , ξ n gLm Φa, (ξ n ) . −5+
(8.201) (8.202) (8.203)
[2] [2] Since xΦa, (x) is smooth and decays like x 4 as x → ∞, xΦa, (x) is a bounded function and [2] ξ n Φa, (ξ n ) is a bounded operator. The rearrangement in the last line introduces commutators which are estimated in the standard way by Lemma 8.13. Thus,
74
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
I3 = L1+2n− B1 + L3n− B2 .
(8.204)
Since the powers of L involved are bounded by 1 + 2n, the expectation value of I3 with respect 1+2n to χα u is O(L 2 χα u2 , L1 (dt)). To estimate I4 , the commutator [[VL , ξ n ], ξ n ] and the derivative gL m are explicitly expanded to show I4 is a product of bounded operators and powers of L. This direct expansion gives 1 [2] −Φa, (ξ n ) [VL , ξ n ], ξ n gL m Φa, (ξ n ) 2 [2] + Φa, (ξ n )gL m Φa, (ξ n ) [VL , ξ n ], ξ n 1 [2] (ξ n ) VL −L2n−2 Lm X↓ (xm )Φa, (ξ n ) = L2+n− −Φa, 2 [2] + Φa, (ξ n )Lm X↓ (xm )Φa, (ξ n ) VL −L2n−2
(8.208)
=L
(8.209)
I4 = L2+n−
m+3n−
(8.205) (8.206) (8.207)
B. 1+2n
Since m + 3n − < 1 + 2n, the expectation value of I4 with respect to χα u is O(L 2 χα u2 , L1 (dt)). Finally, the terms I5 and I6 involving R3 = L3n−3 B are considered. These are each the sum of two inner products, but by the symmetry and antisymmetry properties of the operators involved, it is sufficient to consider just one of the inner products for each. The expectation value of one term in I5 is considered, and the operators χα , Φa, (ξ n ), and ∂ρ∂ ∗ are moved to the left-hand side of the inner product. After such manipulations,
∂ ∂ u, χα Φa, (ξ n )L2+n− gLm R3 χα u = − Φa, (ξ n )χα u, L2+n− gLm R3 χα u . (8.210) ∂ρ∗ ∂ρ∗ We apply the Cauchy–Schwarz inequality and drop bounded operators to get u, χα Φa, (ξ n )L2+n− ∂ gLm R3 χα u ∂ρ ∗ 2+n− 3n−3 ∂ L Bχα u Φa, (ξ n ) ∂ρ χα u L ∗ 2 2 ∂ Φa, (ξ n ) χα u + L2+n− L3n−3 Bχα u ∂ρ∗ 2 ∂ 4n−1− 2 χα u . ∂ρ χα u + L ∗ By Corollary 7.7, the first term, ∂ρ∂ ∗ χα u2 , is time integrable and, hence, O(L 4n−1− χ u2 , is also O(L L1 (dt)). Since 4n − 1 − < 1+2n α 2 , the second term, L 1 L (dt)), and
2 1+2n ∂ gLm R3 χα u = O L 2 χα u , L1 (dt) . u, χα Φa, (ξ n )L2+n− ∂ρ∗
Thus, I5 is O(L
1+2n 2
χα u2 , L1 (dt)).
(8.211) (8.212) (8.213) 1+2n 2
χα u2 ,
1+2n 2
χα u2 ,
(8.214)
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
75
The term I6 is estimated in expectation value by expanding gL m = Lm X↓ (xm ) and finding the relevant power of L is m + 4n − 1 − . Since m + 4n − 1 − < 3/2, this expectation value is O(L1 (dt)). Direct calculation gives u, χα L2+n− gL m R3 χα u = u, χα L2+m+n− X↓ (xm )R3 χα u = u, χα L2+m+n− L3n−3 Bχα u = u, χα Lm+4n−1− Bχα u = O L1 (dt) .
(8.215) (8.216) (8.217) (8.218)
Thus, the contributions from I2 , . . . , I6 are lower order. Step 4. Combining the unestimated term in (8.129), and the lower bound on I1 in (8.174), and the lower order contributions from all other terms, we are left with
u, [H1 + H3 , Γn,m ]u
∂2 n− − 2 , γLm Φa, (ξ n )χα u u, χα Φa, (ξ n )L ∂ρ∗ 2+n− (/2) −VL gLm Φa, (ξ n )χα u + u, χα Φa, (ξ n )L 2 1+2n + O L 2 χα u , L1 (dt) Φa, (ξ n )χα u, Ln− H1 + (/2)H3 , γLm Φa, (ξ n )χα u 2 1+2n + O L 2 χα u , L1 (dt) .
From the angular modulation commutator estimate, Lemma 7.4, applied to L we conclude
(8.219)
(8.220)
(8.221) n− 2
Φa, (ξ n )χα u,
u, [H1 + H3 , Γn,m ]u
∂ ∂ m+n− C1 Φa, (ξ n )χα u, L X↓ (xm ) Φa, (ξ n )χα u ∂ρ∗ ∂ρ∗ + (/6) Φa, (ξ n )χα u, L2+n− −gLm VL Φa, (ξ n )χα u 2 1+2n + O L 2 χα u , L1 (dt) .
(8.222)
(8.223)
We take the from Lemma 7.4 to be the same as in this lemma. Lemma 7.4 does not require the full −gLm VL term, since there is a −(1 − (/3))gLm VL term appearing on the right of Eq. (7.9). Subtracting −(1 − /2)gLm VL from both of (7.9) and (7.10), we get control over −(/6)gLm VL . Since there is no gain from tracking the dependence further, we simply replace it by a positive constant to obtain the statement of the current lemma. 2
76
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
8.4. Derivative bounds In this subsection, we remove the localization in the xm phase space variable. The resulting estimates are localized in ξ n only. Our estimates will be in the regions corresponding to Φ|•|1 and ΨL−δ |•|1 . The first lemma shows that Γn, 1 majorates L 2
1−δ− 2
3/2+n− 2
, and the second, that Γn,n
majorates L . The first lemma is a quick application of Lemma 8.15, with Lemma 8.14 used to show the H2 contributions are lower order. Here we take m = 1/2. Lemma 8.16. If 0 < < 1 and 0 < δ < 1, then there is a positive constant C such that for 0 n 1/2 and for all u ∈ S(R × M) satisfying the wave equation (2.25) there is the commutator estimate that 3/2+n− 2 2 1+2n u, [H, Γn, 1 ]u C L 2 Φ|ξ n |1 χα u + O L 2 χα u , L1 (dt) . 2
(8.224)
Proof. By the results of the angular modulation argument, we have that for any smooth, compactly supported function, χ˜ , there is another compactly supported function χ such that −
∂ m ∂ L X↓ (xm ) − L2 gLm VL C1 L3m χ˜ − C2 ρ∗2 χ ∂ρ∗ ∂ρ∗
(8.225)
in expectation value. Taking m = 1/2 and χ˜ to be smooth, positive, compactly supported, and identically one on the support of χα , we have, from the result of Lemmas 8.15, 8.14, and 7.4, that
u, [H, Γn, 1 ]u 2
∂ 1/2 ∂ C Φa, (ξ n )χα u, Ln− − L X↓ (x 1 ) − L2 g 1 VL Φa, (ξ n )χα u 2 ∂ρ∗ (L 2 ) ∂ρ∗ 2 1 1+2n (8.226) + O L 2 χα u , L (dt) 2 1+2n C Φa, (ξ n )χα u, L3/2+n− χ˜ Φa, (ξ n )χα u + O L 2 χα u , L1 (dt) . (8.227)
We can commute the localization χ˜ through the Φa, (ξ n ) localization to get 2 3/2+n− u, [H, Γn, 1 ]u C1 L 2 Φa, (ξ n )χα u
2
(8.228)
2 2 3/2+n− 1+2n + C2 L 2 Ln−1 Bχα u + O L 2 χα u , L1 (dt) . (8.229)
Here C1 and C2 in (8.229) are not the same as those is (8.225). Since −1/2 + 3n − < 1 + 2n, the desired result holds. 2 Our next goal is a result with localization ΨL−δ |ξ n |1 . We begin by introducing new Ψ type localizations, which are used as smooth approximations to ΨL−δ |•|1 .
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
77
Definition 8.17. The function Ψ1 : [0, ∞) → [0, 1] is defined to be a smooth C ∞ function which has support on [1/2, ∞) and is identically one on [1, ∞). The function Ψ2 : [0, ∞) → [0, 1] is defined to be a C ∞ function which has support on [0, 2] and is identically 1 on [0, 1]. These are extended as even functions. The extensions are also Schwartz class, since the original functions are constant in a neighborhood of zero. The operator Ψ (L−δ , ξ n ) is defined by Ψ L−δ , ξ n = Ψ1 Lδ ξ n Ψ2 (ξ n ).
(8.230)
The following lemma allows us to replace Φ type localization with Ψ type localization and to move this replacement through xm localization. Lemma 8.18. If 0 < < 1 and 0 < δ < 1, then there is a positive constant C1 such that for 0 n m 1/2, for all v ∈ S, and for F2 either Φa, or Φc, , ΨL−δ |ξ n |1 v Ψ L−δ , ξ n v , C1 L−δ Ψ L−δ , • F2−1 (ξ n )v ξ n v.
(8.231) (8.232)
Furthermore, there is a constant C2 such that for 0 n m 1, for any function F1 with its first derivative in L∞ and for F2 either Φa, or Φc, , 1−m−n−δ L F1 (xm ), Ψ L−δ , • F2−1 (ξ n ) C2 F1[1] ∞ .
(8.233)
Proof. Since L, ∂ρ∂ ∗ , and ξ n are all commuting operators, any functions of these operators, defined by the spectral theorem, also commute. The spherical-harmonic decomposition is into orthogonal subspaces preserved by these operators. Therefore, it is sufficient to prove the first two results for functions with a single spherical-harmonic component and to prove the third result by considering the operator on the right as an operator on a single spherical harmonic. For a fixed value of l, Ψ l˜−δ , x = Ψ1 l˜δ |x| Ψ2 |x| χ [1, ∞), l˜δ |x| χ [0, 1], |x| χ [l˜−δ , ∞), |x| χ [0, 1], |x| χ l˜−δ , 1 , |x| Ψl˜−δ |x|1 .
(8.234) (8.235) (8.236) (8.237) (8.238)
Therefore, by the spectral theorem, for any function v ∈ S, on each spherical harmonic, ΨL−δ |ξ n |1 v Ψ L−δ , ξ n v .
(8.239)
Now we consider the case when F2 is either Φa, or Φc, . Both Φa, (x) and Φc, (x) are smooth, strictly positive functions, so in either case F2 (x) has a bounded inverse for |x| ∈ supp(Ψ2 ) ⊂ [0, 2], and (Ψ2 F2−1 )(ξ n ) is a bounded operator. Since all the operators involved commute and Ψ1 (l δ |x|) is bounded, (Ψ (L−δ , •)F2−1 )(ξ n ) is a well-defined, bounded operator. Again, for a fixed l,
78
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
1 ˜−δ l , ∞ , |x| 2
1 1 ˜−δ δ ˜ l χ , ∞ , l |x| 2 2 1 l˜−δ Ψ1 l˜δ |x| 2 C l˜−δ Ψ1 l˜δ |x| Ψ2 F2−1 (x) C l˜−δ Ψ l˜−δ , • F2−1 (x).
1 |x| l˜−δ χ 2
(8.240) (8.241) (8.242) (8.243) (8.244)
On each spherical harmonic, by the spectral theorem, ξ n v = |ξ n |v C L−δ Ψ L−δ , • F2−1 (ξ n )v .
(8.245)
Working on a single harmonic, we compute the commutator, −δ −1 Ψ L , • F2 (ξ n ) = Ψ1 Lδ ξ n Ψ2 F2−1 (ξ n ), L1−m−n−δ F1 (xm ), Ψ L−δ , • F2−1 (ξ n ) = L1−m−n−δ F1 (xm ), Ψ1 Lδ ξ n Ψ2 F2−1 (ξ n ) = L1−m−n−δ F1 (xm ), Ψ1 Lδ ξ n Ψ2 F2−1 (ξ n ) + L1−m−n−δ Ψ1 Lδ ξ n F1 (xm ), Ψ2 F2−1 (ξ n ) .
(8.246)
(8.247) (8.248) (8.249)
Both of these terms are now estimated. Since we are only considering one spherical harmonic, we can replace the operator L with the constant l˜ when we choose. This simplifies the discussion of the commutator involving Ψ1 (Lδ ξ n ). Since Ψ1 is C ∞ and constant outside a compact interval, it follows that Ψ1[1] is Schwartz class and that F [Ψ1[1] ]1 < ∞. The Ψ1 commutator satisfies 1−m−n−δ L F1 (xm ), Ψ1 Lδ ξ n C l˜−δ ˜l 1−m−n F1 (xm ), Ψ1 l˜δ ξ n [1] . C l˜−δ F1[1] ∞ F Ψ1 l˜δ | • | 1
(8.250) (8.251)
At this point, the derivative of the rescaled function, (Ψ1 (l δ •))[1] , is evaluated to be a scaled version of the derivative evaluated at the scaled variable, l δ Ψ1[1] (l δ •). To evaluate the Fourier transform of this, it is noted that F [f (λ•)]1 = F [f (•)]1 . Thus, the commutator satisfies 1−m−n−δ L F1 (xm ), Ψ1 Lδ ξ n Ψ2 F2−1 (ξ n ) C l˜−δ F [1] ˜l δ F Ψ [1] l˜δ | • | . 1
Since l˜ is bounded below,
∞
1
1
(8.252)
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
1−m−n−δ L F1 (xm ), Ψ1 Lδ ξ n Ψ2 F2−1 (ξ n ) C F1[1] ∞ F Ψ1[1] 1 C F1[1] ∞ .
79
(8.253) (8.254)
This completes the estimate on the first of the commutator terms in (8.248). Since Ψ2 is Schwartz class, and F2 is smooth and has bounded inverse on supp(Ψ2 ), (Ψ2 F2−1 ) is Schwartz class, and F [(Ψ2 F2−1 )[1] ]1 < ∞, so the commutator satisfies 1−m−n−δ δ L Ψ1 L ξ n F1 (xm ), Ψ2 F2−1 (ξ n ) C L1−m−n−δ F1 (xn ), Ψ2 F2−1 (ξ n ) C L1−m−n F1 (xm ), Ψ2 F2−1 (ξ n ) [1] C F1[1] ∞ F Ψ2 F2−1 1 [1] C F1 ∞ .
(8.255) (8.256) (8.257) (8.258)
This completes the estimate on the second of the commutator terms in (8.249). Thus both commutators are estimated, and the desired result holds. 2 We now prove a ΨL−δ |•|1 localized estimate. This is analogous to the Φ|•|1 localized estimate in Lemma 8.16, although, the proof is more complicated because the Ψ type localizations are more complicated to work with. Because ξ n is bounded below on the support of ΨL−δ |ξ n |1 , we can dominate more powers of L with this localization. Here, we take m = n. Lemma 8.19. If 0 < < 1 and 0 < δ < 1, then there is a positive constant C such that for 0 n 1/2 and for all u ∈ S(R × M) satisfying the wave equation (2.25) there is the commutator estimate that 2 2 1+2n u, [H, Γn,n ]u C L1−δ− 2 ΨL−δ |ξ n |1 χα u + O L 2 χα u , L1 (dt) . (8.259)
Proof. We take the result of Lemmas 8.15 and 8.14 with m = n,
∂ n ∂ n− 2 n u, [H, Γn,n ]u C Φa, (ξ n )χα u, L − L X↓ (xn ) + L −gL VL Φa, (ξ n )χα u ∂ρ∗ ∂ρ∗ 2 1+2n (8.260) + O L 2 χα u , L1 (dt) .
The term involving the derivatives is estimated first. The first step is to replace the nonsmooth X↓ (xm )1/2 with the smooth X˜ ↓ (xm ) so that 2n− L 2 X↓ (xn ) 12 ∂ Φa, (ξ n )χα u C Ln− 2 X˜ ↓ (xn ) ∂ Φa, (ξ n )χα u. ∂ρ∗ ∂ρ∗ Now ∂ρ∂ ∗ is treated as if it were a localization in operators to give
∂ ∂ρ∗
(8.261)
and is commuted to the left of the chain of
80
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
2n− L 2 X↓ (xn ) 12 ∂ Φa, (ξ n )χα u C Ln− 2 ∂ X˜ ↓ (xn )Φa, (ξ n )χα u ∂ρ∗ ∂ρ∗ n− ∂ ˜ 2 − C L Φa, (ξ n )χα u X↓ (xn ), . ∂ρ∗
(8.262) (8.263)
The commutator from moving ∂ρ∂ ∗ through X˜ ↓ (xn ) can be computed explicitly. It involves the function X˜ [1] (x) = −2x X˜ ↓ (x)2 and is bounded by ↓
2 −1 ∂ ∂ ˜ =− 1 + Ln ρ ∗ X↓ (xn ), ∂ρ∗ ∂ρ∗
(8.264)
= 2Ln xn X˜ ↓ (xn )2
(8.265)
= Ln B.
(8.266)
−1 )(ξ ) using Lemma 8.18. ConThe “localization” Ln ∂ρ∂ ∗ can be replaced by L1−δ (Ψ (L−δ , •)Φa, n tinuing from (8.263), we have
2n− L 2 X↓ (xn ) 12 ∂ Φa, (ξ n )χα u ∂ρ∗ 1−δ− −δ −1 2 Ψ L C L , • Φa, (ξ n )X˜ ↓ (xn )Φa, (ξ n )χα u − L2n− 2 Bχα u.
(8.267) (8.268)
−1 )(ξ ) is commuted back through the x localization to give The (Ψ (L−δ , •)Φa, n n
2n− L 2 X↓ (xn ) 12 ∂ Φa, (ξ n )χα u ∂ρ∗ C1 L1−δ− 2 X˜ ↓ (xn )Ψ L−δ , ξ n χα u −1 − C2 L1−δ− 2 Ψ L−δ , • Φa, (ξ n ), X˜ ↓ (xn ) Φa, (ξ n )χα u − L2n− 2 Bχα u.
(8.269) (8.270) (8.271)
−1 )(ξ ) is shown to be lower order by Lemma 8.18. The commutator involving (Ψ (L−δ , •)Φa, n From this estimate, we have
2n− L 2 X↓ (xn ) 12 ∂ Φa, (ξ n )χα u C L1−δ− 2 X˜ ↓ (xn )Ψ L−δ , ξ n χα u ∂ρ∗ − L1−δ− 2 L2n−1+δ Bχα u − L2n− 2 Bχα u.
(8.272) (8.273)
Since 2n − /2 is the power of L appearing in both error terms, and this exponent is less than (1 + 1+2n 2n)/2, the error terms are O(L 2 χα u2 , L1 (dt)). To simplify the summation of the current estimate with the next step in this proof, we introduce an additional localization by (−VL ρ∗−1 )1/2 ,
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
2n− L 2 X↓ (xn ) 12 ∂ Φa, (ξ n )χα u ∂ρ ∗
81
(8.274)
2 1+2n 1 C L1−δ− 2 X˜ ↓ (xn ) −VL ρ∗−1 2 Ψ L−δ , ξ n χα u + O L 2 χα u , L1 (dt) . (8.275) This completes the first step in this proof. Now the −gLn VL terms in (8.261) are estimated. The goal is again to replace the localization −1 )(ξ ) through the ρ localΦa, (ξ n ) by Ψ (L−δ , ξ n ). This requires commuting (Ψ (L−δ , •)Φa, ∗ n ization. A factor of Lδ is dropped to control commutator terms involving Ψ (L−δ , ξ n ) at a later stage. The term under consideration is 2− n 1 L 2 L 2 −gLn V 1/2 Φa, (ξ n )χα u L1−δ− 2 X↑ (xn ) −V ρ −1 2 Φa, (ξ n )χα u. (8.276) L L ∗ −1 )(ξ ) is introduced, and then commuted through the The bounded localization (Ψ (L−δ , •)Φa, n localization in xn . This gives
2− n L 2 L 2 −gLn V 1/2 Φa, (ξ n )χα u L 1 −1 (8.277) C Ψ L−δ , • Φa, (ξ n )L1−δ− 2 X↑ (xn ) −VL ρ∗−1 2 Φa, ξ n χα u 1 1−δ− −δ −1 2 X↑ (xn ) Ψ L (8.278) C L , • Φa, (ξ n ) −VL ρ∗−1 2 Φa, (ξ n )χα u 1 1−δ− −δ −1 2 − C L Ψ L , • Φa, (ξ n ), X↑ (xn ) −VL ρ∗−1 2 Φa, (ξ n )χα u. (8.279) −1 )(ξ ) is commuted through the localization (−V ρ −1 ). This Now the operator (Ψ (L−δ , •)Φa, n L ∗ eliminates the Φa, (ξ n ) localization, so that
2− n L 2 L 2 −gLn V 1/2 Φa, (ξ n )χα u L 1 C L1−δ− 2 X↑ (xn ) −VL ρ∗−1 2 Ψ L−δ , ξ n χα u 1 −1 − C L1−δ− 2 X↑ (xn ) Ψ L−δ , • Φa, (ξ n ), −VL ρ∗−1 2 Φa, (ξ n )χα u 1 −1 − C L1−δ− 2 Ψ L−δ , • Φa, (ξ n ), X↑ (xn ) −VL ρ∗−1 2 Φa, (ξ n )χα u.
(8.280) (8.281) (8.282) (8.283)
By Lemma 8.18, the commutator terms are lower order, error terms. Since X↑ (x) has a L∞ derivative, we conclude that the −gLn VL term in (8.261) satisfies 2− n L 2 L 2 −gLn V 1/2 Φa, (ξ n )χα u L 1 C L1−δ− 2 X↑ (xn ) −VL ρ∗−1 2 Ψ L−δ , ξ n χα u − L1−δ− 2 X↑ (xn )Ln+δ−1 B1 Φa, (ξ n )χα u 1 − L1−δ− 2 L2n+δ−1 B2 −VL ρ∗−1 2 Φa, (ξ n )χα u 1 C L1−δ− 2 X↑ (xn ) −VL ρ∗−1 2 Ψ L−δ , ξ n χα u 2 1+2n + O L 2 χα u , L1 (dt) .
(8.284) (8.285) (8.286) (8.287) (8.288) (8.289)
82
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
Eqs. (8.275) and (8.289) can be combined with the initial estimate on the expectation value of [H, Γn,n ] to produce the intermediate result that 2 1 u, [H, Γn,n ]u C L1−δ− 2 X˜ ↓ (xn ) −VL ρ∗−1 2 Ψ L−δ , ξ n χα u 2 1 + L1−δ− 2 X↑ (xn ) −VL ρ∗−1 2 Ψ L−δ , ξ n χα u 2 1+2n + O L 2 χα u , L1 (dt) .
(8.290) (8.291) (8.292) 1
The first two terms on the right both involve localization in xn acting on (−VL ρ∗−1 ) 2 × Ψ (L−δ , ξ n )χα u. The sum of the localizations is bounded below by a constant, X˜ ↓ (x) + X↑ (x) = 2
2
1 1 + x2
2
g(x) 2 + x C. x
(8.293)
Therefore, the two terms can be combined to provide the better estimate that 2 1 u, [H, Γn,n ]u C L1−δ− 2 −VL ρ∗−1 2 Ψ L−δ , ξ n χα u 2 1+2n + O L 2 χα u , L1 (dt) .
(8.294)
1
To eliminate the factor of (−VL ρ∗−1 ) 2 , a new function is introduced. This function, f , is a 1
smooth, compactly supported function and equal to the inverse of (−VL ρ∗−1 ) 2 on supp χα . Since it is bounded, it can be freely introduced into the norm. We can commute the ρ∗ localization to find u, [H, Γn,n ]u 2 2 1 1+2n C L1−δ− 2 f −VL ρ∗−1 2 Ψ L−δ , ξ n χα u + O L 2 χα u , L1 (dt) 2 1 C L1−δ− 2 Ψ L−δ , ξ n f −VL ρ∗−1 2 χα u 2 1 − C L1−δ− 2 f −VL ρ∗−1 2 , Ψ L−δ , ξ n χα u 2 1+2n + O L 2 χα u , L1 (dt) .
(8.295) (8.296) (8.297) (8.298)
From the definition of f , the product of all the ρ∗ localization reduces to χα . The new commutator terms can be estimated by Lemma 8.18 and are found to be lower order error terms, so that 2 2 u, [H, Γn,n ]u C L1−δ− 2 Ψ L−δ , ξ n χα u − L1−δ− 2 Ln+δ−1 Bχα u (8.299) 1+2n 2 + O L 2 χα u , L1 (dt) (8.300) 1−δ− −δ 2 2 1 1+2n 2Ψ L C L , ξ n χα u + O L 2 χα u , L (dt) . (8.301)
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
83
Lemma 8.18 can now be used to replace the localization in Ψ (L−δ , ξ n ) by localization in ΨL−δ |ξ n |1 . Thus, we conclude that
u, [H, Γn,n ]u 2 2 1+2n C L1−δ− 2 ΨL−δ |ξ n |1 χα u + O L 2 χα u , L1 (dt) .
2
(8.302)
8.5. Phase space induction 3/2+n−
and L1−δ− 2 , respectively. The previous section shows that Γn, 1 and Γn,n majorate L 2 2 We would like to integrate the Heisenberg-type identity to conclude that the time integral of the expectation value of these powers of L are bounded. However, our definition of majoration allows the domination to occur only in a region of phase space and the lower order terms to be unlocalized. The lower order terms involve L to the power (1 + 2n)/2. To control these terms, we use a finite induction on n, to eventually control L1−ε without phase space localization. Theorem 8.20 (Phase space induction). If ε > 0, then there is a positive constant, CPS , such that for all u ∈ S(R × M) satisfying the wave equation (2.25) there is the integral estimate that ∞ 2 1−ε L χα u(t)2 dt CPS E[u, u] ˙ + u(1) .
(8.303)
1
Proof. This is proven for ε = δ. Our argument requires 2 δ, and, since there is no gain from varying , we will take = δ/2. We induct on n to prove, simultaneously, the two statements 2 3/2+n−δ− L 2 χα u = O L1 (dt) ,
2 1−δ 1−n L χ L , ∞ , −i ∂ χα u = O L1 (dt) . ∂ρ ∗
(8.304) (8.305)
The induction will run from n = 0, in steps of size δ, as long as n
1 − 2δ + . 2
(8.306)
The base case of (8.304) follows from the angular modulation result, Theorem 7.6, which says, in the notation of this section, L3/4 χα u2 = O(L1 (dt)). The base case of (8.305) follows from the angular modulation theorem, Theorem 7.6. For n = 0, since ˜ ∞), x l˜ 1−δ χ l˜ 1−δ , ∞ , x x, l˜1−δ χ [l,
(8.307)
by the spectral theorem and Corollary 7.7, 1−δ L χ [L, ∞), −i ∂ ∂ρ
∗
2 2 χα u ∂ χα u = O L1 (dt) . ∂ρ ∗
(8.308)
84
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
The inductive step is now considered. The Morawetz-type estimates with Γn,n and Γn, 1 are 2
used. Before these estimates are proven, Γn,m u2 is shown to be bounded by the energy and a local decay term, under the condition 0 n m 1/2. To begin estimating the norm of Γn,m u, the operator Γn,m is expanded as a product of operators, and the factor of γLm is expanded as a sum of two terms. These terms can be further simplified by eliminating bounded functions. Thus, we write Γn,m u = χα Φa, (ξ n )γLm Ln− Φa, (ξ n )χα u ∂ n− m χ Φ (ξ ) g L Φ (ξ )χ u a, n α α a, n ∂ρ L ∗ 1 m n− + χα Φa, (ξ n ) L X↓ (xm )L Φa, (ξ n )χα u 2 m+n− ∂ n− χα u. Φa, (ξ n ) ∂ρ gLm L Φa, (ξ n )χα u + L ∗
(8.309) (8.310) (8.311) (8.312)
Interpolation can be used to simplify Lm+n− χα u, so that 1/2 −1 ∂ n− m Φ (ξ ) L g Φ (ξ )χ u + 1 + ρ∗2 u . Γn,m u L a, n α + C E a, n ∂ρ ∗
(8.313)
To control Φa, (ξ n ) ∂ρ∂ ∗ gLm Ln− Φa, (ξ n )χα u, we rewrite it in terms of ξ n and L and rearrange its subfactors as Φa, (ξ n ) ∂ Ln− gLm Φa, (ξ n )χα u = L1− Φa, (ξ n )ξ n gLm Φa, (ξ n )χα u ∂ρ∗ L1− gLm ξ n Φa, (ξ n )2 χα u + L1− ξ n Φa, (ξ n ), gLm Φa, (ξ n )χα u.
(8.314) (8.315) (8.316)
The first of these two norms can be estimated by Lemma 8.7. The second can be estimated using Lemma 8.13 which states that the commutator is Lm+n−1 B. Thus, ∂ Φa, (ξ n ) ∂ Ln− gLm Φa, (ξ n )χα u L 1+n−2 1− χ u + χ u α α ∂ρ∗ ∂ρ∗ m+n− + L BΦa, (ξ n )χα u.
(8.317) (8.318)
Since (1 + n − 2)/(1 − ) 1 and m + n − 1, each of the terms involving powers of L in this expression can be estimated by interpolation, and Φa, (ξ n ) ∂ Ln− gLm Φa, (ξ n )χα u C E 1/2 + 1 + ρ 2 −1 u . ∗ ∂ρ ∗
This completes the estimate on Γn,m u.
(8.319)
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
85
We can now integrate the Heisenberg-type identity to find T
u, [H, Γn,m ]u dt =
1
T
d u, Γn,m u ˙ − u, ˙ Γn,m u dt dt
(8.320)
1
= −2u, ˙ Γn,m u|T1 .
(8.321)
The real part appears here because Γn,m is composed of bounded, self-adjoint operators taking S to S (i.e. χα and Φa, (ξ n )) and of an antisymmetric operator (γLm ), so we may rewrite u, Γn,m u ˙ as −Γn,m u, u. ˙ From the Cauchy–Schwarz inequality, T
2 2 −1 −1 u, [H, Γn,m ]u dt C E + 1 + ρ∗2 u(T ) + 1 + ρ∗2 u(1) .
(8.322)
1
From the local decay theorem, Theorem 6.6, the second term on the right-hand side vanishes on a subsequence Ti → ∞, so Ti lim
i→∞
2 −1 u, [H, Γn,m ]u dt C E + 1 + ρ∗2 u(1) .
(8.323)
1
The derivative localization results, Lemmas 8.19 and 8.16 can now be applied. From condition (8.306), it follows that 1 + 2n 3/2 + n − 2δ + . From this, the condition that 2 δ, and the inductive hypothesis (8.304), the lower order terms are integrable: 1+2n L 2 χα u2 = O L1 (dt) ,
2 1+2n O L 2 χα u , L1 (dt) = O L1 (dt) .
(8.324) (8.325)
We consider the inductive step for (8.305). From inductive hypothesis (8.305) and the second, derivative-localizing result, Lemma 8.19, we have that
2 1−δ 1−n−δ L χ L −i ∂ χα u , ∞ , ∂ρ ∗
2 1−δ 1−n−δ 1−n −i ∂ χα u L , = χ L , L ∂ρ ∗
2 1−δ 1−n −i ∂ χα u L + χ L , ∞ , ∂ρ ∗ 2 = L1−δ ΨL−δ |ξ |1 χα u n
1−δ 1−n ∂ + L χ L , ∞ , −i ∂ρ
∗
2 χα u
(8.326) (8.327) (8.328) (8.329)
86
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
2 1+2n u, [H, Γn,n ]u + O L 2 χα u , L1 (dt) + O L1 (dt) .
(8.330) (8.331)
Integrating in time, we get ∞ 1−δ 1−n−δ L χ L −i ∂ , ∞ , ∂ρ 1
∗
2 χα u dt C E[u] + u(1)2 ,
(8.332)
and we have extended inductive hypothesis (8.305) to n + δ. We now consider the inductive step for (8.304). From condition (8.306), it follows that 3/2+n− < 1 − δ. From the first, derivative-localizing result, Lemma 8.16, 2
2 3/2+n− 3/2+n− L 2 χα u2 = L 2 χ 0, L1−n , −i ∂ χα u ∂ρ ∗ 2
3/2+n− 1−n ∂ χα u + L 2 χ L , ∞ , −i ∂ρ∗
2 3/2+n− 2 1−δ 1−n −i ∂ χα u . L L 2 Φ|ξ n |1 χα u + χ L , ∞ , ∂ρ ∗
(8.333) (8.334) (8.335)
Applying Lemma 8.16 and estimate (8.332), we have 3/2+n− L 2 χα u2 u, [H, Γ
n, 12 ]u
2 1+2n + O L 2 χα u , L1 (dt) + O L1 (dt) .
(8.336)
We can integrate this in time, applying (8.323) to control the expectation value of the commutator and the inductive hypothesis to control the lower order terms on the left-hand side. This extends inductive hypothesis (8.304) up to n + δ, ∞ 3/2+n− L 2 χα u2 dt C E[u] + u(1)2 .
(8.337)
1
Since the induction continues as long as condition (8.306) holds, after the last application of Eq. (8.337), ∞ 2−2δ L 2 χα u2 dt C E[u] + u(1)2 .
(8.338)
1
This proves the desired result with ε = δ. Since δ can be chosen to be arbitrarily small, so can ε. 2 Except for the loss of Lε and factors of t, this provides the control on the angular energy near the photon sphere required by Lemma 3.11 to control the weighted L6 norm. Since the
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
87
wave equation is linear and L commutes with H , Lε u is also a solution, and the energy of this function can be used to recover the additional factor of Lε u. Theorem 8.21 (Pointwise-in-time decay). If ε > 0, then there is a positive constant CPW such that for all u˜ satisfying the original wave equation (2.22) and for u = r u, ˜ there are the following pointwise-in-time estimates for t 1, 1 F 2 u˜
˜ L6 (M)
2 1 t − 3 CPW u0 2 + E[u0 , u1 ] + EC [u0 , u1 ] + L u0
1 + E L u0 , L u1 2 , (8.339) 2 −1 ρ + 1 2 u˜ 2 ˜ t − 12 CPW u0 2 + E[u0 , u1 ] + EC [u0 , u1 ] + L u0 2 ∗ L (M) 1 + E L u0 , L u1 2 ,
(8.340)
if the norms on the right-hand side are finite. Proof. If u˜ is a solution to Eq. (2.22), then u is a solution to Eq. (2.25). To begin, assume (u0 , u1 ) ∈ S × S, by the standard well-posedness theory for linear wave equations, if (u0 , u1 ) ∈ S × S, then there is a solution u ∈ S(R × M). The weighted L6 norms of u and u˜ are related by 1 6 F 2 u˜ 6 L
˜ = (M)
F 3 |u| ˜ 6 d 3 μ˜
(8.341)
F 3 r −4 |u| d 3 μ.
(8.342)
˜ M
=
Since the wave equation, Eq. (2.25), is linear and H commutes with L, if u is a solution, then so is L u. The phase space induction theorem, Theorem 8.20, can be applied to L u to give 1− 2 L χα L u = O L1 (dt) ,
(8.343)
where the quantification of the time integrability condition is with respect to Lε u not u. We write this explicitly as ∞
2 Lχα u2 dt C L u0 + E L u0 , L u1 .
(8.344)
1
This is sufficient to control the conformal charge. From Lemma 3.11, we know that it is sufficient to control the space–time integral of W |u|2 + WL |∇S 2 u|2 , with W and WL compactly supported. The term with no derivative is controlled by the local decay theorem in Theorem 6.6. The term with the angular derivatives is controlled by inequality (8.344), since we are free to choose χα identically one on the support of WL . Thus,
88
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
1 1 ˙ EC [u0 , u1 ] + CtE[u0 , u1 ] 2 E[u0 , u1 ] 2 + u0 EC u(t), u(t)
(8.345)
t + Ct
χα |∇S 2 u|2 d 3 μ dτ
(8.346)
1
1 1 EC [u0 , u1 ] + CtE[u0 , u1 ] 2 E[u0 , u1 ] 2 + u0 2 + Ct L u0 + E L u0 , L u1 .
(8.347) (8.348)
Since the conformal charge controls the weighted L6 norm by Lemma 3.7, we have 1 F 2 u˜
˜ L6 (M)
Ct
F 3 r −4 |u| d 3 μ
−1 3
1 6
(8.349)
2 1/2 u0 2 + E[u0 , u1 ] + EC [u0 , u1 ] + L u0 + E L u0 , L u1 . (8.350)
Similarly, since Lemma 3.6 controls the weighted L2 norm by the conformal charge, 1 u˜ ρ∗2 + 1 L2 (M) ˜
1 u = u, 2 ρ∗ + 1 Ct −2 EC u(t), u(t) ˙
u, ˜
Ct −2 EC [u0 , u1 ] + Ct
(8.351) (8.352) −1
2 E[u0 , u1 ] + u0 2 + L u0 + E L u0 , L u1 .
(8.353)
Using a limiting argument and the linearity of solutions to the wave equation, the same result holds as long as the norms on the initial data are finite, so we may drop the hypothesis that the initial data, (u0 , u1 ) and (u˜ 0 , u˜ 1 ), is Schwartz class. 2 Acknowledgments We would like to express our thanks to the anonymous referee. The original version of this paper had a gap in the first part of the paper and stylistic problems. We would like to thank the referee for patiently reading the earlier versions and for bringing these problems to our attention. We would also like to express our thanks to J. Sterbenz for valuable discussions and for providing a corrected version of our original argument. These results originally appeared as part of the PhD dissertation of P. Blue, “Decay estimates and phase space analysis for wave equations on some black hole metrics” [3], submitted in October 2004. Both authors would like to thank the NSF for partial support under grants NSF DMS 0100490 (P. Blue) and 0501043 (A. Soffer).
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
89
References [1] A. Bachelot, Asymptotic completeness for the Klein–Gordon equation on the Schwarzschild metric, Ann. Inst. H. Poincaré Phys. Théor. 61 (1994) 411–441. [2] A. Bachelot, J.-P. Nicolas, Equation non linéaire de Klein–Gordon dans des métriques de type Schwarzschild, C. R. Acad. Sci. Paris Sér. I Math. 316 (1993) 1047–1050. [3] P. Blue, Decay estimates and phase space analysis for wave equations on some black hole metrics, PhD thesis, Rutgers, The State University of New Jersey, New Brunswick, NJ, October 2004. [4] P. Blue, A. Soffer, Semilinear wave equations on the Schwarzschild manifold I: Local decay estimates, Adv. Differential Equations 8 (2003) 595–614. [5] P. Blue, A. Soffer, The wave equation on the Schwarzschild metric. II. Local decay for the spin-2 Regge–Wheeler equation, J. Math. Phys. 4 (2005) 012502, 9 pp. [6] P. Blue, A. Soffer, Errata for “Global existence and scattering for the nonlinear Schrödinger equation on Schwarzschild manifolds”, “Semilinear wave equations on the Schwarzschild manifold I: Local decay estimates”, and “The wave equation on the Schwarzschild metric. II: Local decay for the spin-2 Regge–Wheeler equation”, preprint, http://xxx.lanl.gov, math/0608073, 2006. [7] P. Blue, A. Soffer, Improved decay rates with small regularity loss for the wave equation about a Schwarzschild black hole, preprint, http://xxx.lanl.gov, math/0612168, 2006. [8] P. Blue, A. Soffer, A space–time integral estimate for a large data semi-linear wave equation on the Schwarzschild manifold, Lett. Math. Phys. 81 (3) (2007) 227–238. [9] P. Blue, J. Sterbenz, Uniform decay of local energy and the semi-linear wave equation on Schwarzschild space, Comm. Math. Phys. 268 (2006) 481–504. [10] D. Christodoulou, S. Klainerman, The Global Nonlinear Stability of the Minkowski Space, Princeton Math. Ser., vol. 41, Princeton Univ. Press, Princeton, NJ, 1993. [11] C.-M. Claudel, K.S. Virbhadra, G.F.R. Ellis, The geometry of photon surfaces, J. Math. Phys. 42 (2) (2001) 818– 838. [12] M. Dafermos, The interior of charged black holes and the problem of uniqueness in general relativity, Comm. Pure Appl. Math. 58 (2005) 445–504. [13] M. Dafermos, I. Rodnianski, A proof of Price’s law for the collapse of a self-gravitating scalar field, Invent. Math. 162 (2) (2005) 381–457. [14] M. Dafermos, I. Rodnianski, Small-amplitude nonlinear waves on a black hole background, J. Math. Pures Appl. (9) 84 (9) (2005) 1147–1172. [15] M. Dafermos, I. Rodnianski, The red-shift effect and radiation decay on black hole spacetimes, preprint, http://xxx.lanl.gov, gr-qc/0512119, 2005. [16] M. Dafermos, I. Rodnianski, A note on energy currents and decay for the wave equation on a Schwarzschild background, preprint, http://xxx.lanl.gov, 0710.0171, 2007. [17] S. DeBièvre, P.D. Hislop, I.M. Sigal, Scattering theory for the wave equation on noncompact manifolds, Rev. Math. Phys. 4 (1992) 575–608. [18] J. Derezi´nski, C. Gérard, Scattering Theory of Classical and Quantum N -Particle Systems, Springer, 1997. [19] J. Dimock, Scattering for the wave equation on the Schwarzschild metric, Gen. Relativity Gravitation 17 (353–369) (1985). [20] J. Dimock, B.S. Kay, Classical and quantum scattering theory for linear scalar fields on the Schwarzschild metric I, Ann. Phys. 175 (1987) 366–426. [21] G.F.R. Ellis, S.F. Hawking, The Large Scale Structure of Space–Time, Cambridge Monogr. Math. Phys., Cambridge Univ. Press, Cambridge, 1973. [22] L. Evans, Partial Differential Equations, Grad. Stud. Math., vol. 19, Amer. Math. Soc., Providence, RI, 1998. [23] F. Finster, N. Kamran, J. Smoller, S.-T. Yau, Decay rates and probability estimates for massive Dirac particles in the Kerr–Newman black hole geometry, Comm. Math. Phys. 230 (2002) 201–244. [24] F. Finster, N. Kamran, J. Smoller, S.-T. Yau, Decay of solutions of the wave equation in the Kerr geometry, Comm. Math. Phys. 264 (2) (2006) 465–503. [25] J. Ginibre, G. Velo, Conformal invariance and time decay for nonlinear wave equations. I, Ann. Inst. H. Poincaré 47 (1973) 221–261. [26] D. Häfner, Complétude asymptotique pour l’équation des ondes dans une classe d’espaces-temps stationnaires et asymptotiquement plats, Ann. Inst. H. Poincaré 51 (2001) 779–833. [27] A. Hassell, T. Tao, J. Wunsch, A Strichartz inequality for the Schrödinger equation on non-trapping asymptotically conic manifolds, Comm. Partial Differential Equations 30 (2005) 157–205.
90
P. Blue, A. Soffer / Journal of Functional Analysis 256 (2009) 1–90
[28] W. Hunziker, I.M. Sigal, The quantum N -body problem, J. Math. Phys. 41 (2000) 3448–3510. [29] B.S. Kay, R.M. Wald, Linear stability of Schwarzschild under perturbations which are nonvanishing on the bifurcation 2-sphere, Classical Quantum Gravity 4 (1987) 893–898. [30] H. Lindblad, I. Rodnianski, Global existence for the Einstein vacuum equations in wave coordinates, Comm. Math. Phys. 256 (1) (2005) 43–110. [31] S. Machihara, M. Nakamura, K. Nakanishi, T. Ozawa, Endpoint Strichartz estimates and global solutions for the nonlinear Dirac equation, J. Funct. Anal. 219 (2005) 1–20. [32] C.W. Misner, K.S. Thorne, J.A. Wheeler, Gravitation, Freeman, New York, 1973. [33] J.P. Nicolas, Nonlinear Klein–Gordon equation on Schwarzschild-like metrics, J. Math. Pures Appl. (9) 74 (1995) 35–58. [34] T. Regge, J.A. Wheeler, Stability of a Schwarzschild singularity, Phys. Rev. 108 (1957) 1063–1069. [35] I.M. Sigal, A. Soffer, The N -particle scattering problem: Asymptotic completeness for short-range systems, Ann. of Math. (2) 126 (1987) 35–108. [36] A. Soffer, On the many body problem in quantum mechanics, Astérisque 207 (1992) 109–152. [37] E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, NJ, 1971. [38] J. Sterbenz, Angular regularity and Strichartz estimates for the wave equation, Int. Math. Res. Not. 4 (2005) 187– 231. [39] J.G. Stalker, A.S. Tahvildar-Zadeh, Scalar waves on a naked-singularity background, Classical Quantum Gravity 21 (2004) 2831–2848. [40] K.S. Virbhadra, G.F.R. Ellis, Schwarzschild black hole lensing, Phys. Rev. D 62 (2000), 084003-1-8. [41] R.M. Wald, Note on the stability of the Schwarzschild metric, J. Math. Phys. 20 (1979) 1056–1058. [42] B.F. Whiting, Mode stability of the Kerr black hole, J. Math. Phys. 30 (1989) 1301–1305. [43] F.J. Zerilli, Gravitational field of a particle falling in a Schwarzschild geometry analyzed in tensor harmonics, Phys. Rev. D 3 (1970) 2141–2160.
Journal of Functional Analysis 256 (2009) 91–128 www.elsevier.com/locate/jfa
Reduced Weyl asymptotics for pseudodifferential operators on bounded domains II. The compact group case ✩ Roch Cassanas, Pablo Ramacher Georg-August-Universität Göttingen, Institut für Mathematik, Bunsenstr. 3-5, 37073 Göttingen, Germany Received 4 October 2007; accepted 28 January 2008 Available online 10 April 2008 Communicated by Paul Malliavin
Abstract Let G ⊂ O(n) be a compact group of isometries acting on n-dimensional Euclidean space Rn , and X a bounded domain in Rn which is transformed into itself under the action of G. Consider a symmetric, classical pseudodifferential operator A0 in L2 (Rn ) that commutes with the regular representation of G, and assume that it is elliptic on X. We show that the spectrum of the Friedrichs extension A of the operator res ◦ 2 A0 ◦ ext : C∞ c (X) → L (X) is discrete, and using the method of the stationary phase, we derive asymptotics for the number Nχ (λ) of eigenvalues of A equal or less than λ and with eigenfunctions in the χ -isotypic component of L2 (X) as λ → ∞, giving also an estimate for the remainder term for singular group actions. Since the considered critical set is a singular variety, we recur to partial desingularization in order to apply the stationary phase theorem. © 2008 Elsevier Inc. All rights reserved. Keywords: Pseudodifferential operators; Asymptotic distribution of eigenvalues; Compact group actions; Peter–Weyl decomposition; Partial desingularization
Contents 1. 2.
✩
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reduced spectral asymptotics and the approximate spectral projection operators . . . . . . .
This research was financed by the grant RA 1370/2-1 of the German Research Foundation (DFG). E-mail addresses:
[email protected] (R. Cassanas),
[email protected] (P. Ramacher).
0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.01.012
92 95
92
R. Cassanas, P. Ramacher / Journal of Functional Analysis 256 (2009) 91–128
3. Compact group actions and the principle of the stationary phase 4. Phase analysis and partial desingularization . . . . . . . . . . . . . . 5. Computation of the leading term . . . . . . . . . . . . . . . . . . . . . . 6. Proof of the main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
101 108 115 126 127
1. Introduction Let G ⊂ O(n) be a compact Lie group of isometries acting on Euclidean space Rn , and X a bounded open set of Rn which is transformed into itself under the action of G. Consider the regular representation of G T (k)ϕ(x) = ϕ k −1 x
(1)
in the Hilbert spaces L2 (Rn ) and L2 (X) of square-integrable functions by unitary operators. As a consequence of the Peter–Weyl theorem, T decomposes into isotypic components according to Hχ , L2 Rn =
L2 (X) =
ˆ χ∈G
res Hχ ,
ˆ χ∈G
ˆ denotes the set of irreducible characters of G, and res : L2 (Rn ) → L2 (X) is the natuwhere G ral restriction operator. The spaces Hχ are closed subspaces, and the corresponding orthogonal projection operators are given by Pχ = dχ
χ(k)T (k) dk,
(2)
G
where dχ = χ(1) is the dimension of any irreducible representation belonging to the character χ , and dk denotes the normalized Haar measure on G. In what follows, we do not assume that the boundary ∂X of X is smooth, but only that there exists a constant c > 0 such that for any sufficiently small > 0, vol(∂X) c, where (∂X) = {x ∈ Rn : dist(x, ∂X) < }, and that 0∈ / ∂X. Consider now a symmetric, classical pseudodifferential operator A0 in Rn of order 2m that commutes with the operators T (k) for all k ∈ G. Let a2m be its principal symbol, and assume that there exists a constant C0 > 0 such that a2m (x, ξ ) C0 |ξ |2m ,
∀x ∈ X, ∀ξ ∈ Rn .
(3)
2 n If we write ext : C∞ c (X) → L (R ) for the natural extension operator by zero, it turns out that under condition (3), the operator 2 res ◦ A0 ◦ ext : C∞ c (X) → L (X)
R. Cassanas, P. Ramacher / Journal of Functional Analysis 256 (2009) 91–128
93
is symmetric and lower semi-bounded, and we denote its Friedrichs extension by A. It can be shown that A has compact resolvent, and if the boundary of X is sufficiently smooth, and A0 satisfies the transmission property, the domain of A is given by D(A) = u ∈ H0m (X): A0 u ∈ L2 (X) , m where H0m (X) is the closure of C∞ c (X) in the Sobolev space H (X), so that we are in presence of a generalized Dirichlet problem. Since A leaves invariant each of the isotypic components res Hχ , the restriction of A to res Hχ gives rise to the so-called reduced operator Aχ . Its domain is D(Aχ ) = D(A) ∩ res Hχ , and its spectrum is discrete, the spectrum of A being equal to the union of the spectra of the operators Aχ . The purpose of this paper is to investigate the spectral counting function Nχ (λ) of Aχ , which is given by the number of eigenvalues of Aχ , counting multiplicities, that are less than λ ∈ R. It corresponds to the number of eigenvalues of A less than λ, and with eigenfunctions in the χ -isotypic component of L2 (X), so that
Nχ (λ) = dχ
μχ (t),
tλ
where μχ (t) denotes the multiplicity of any irreducible representation χ of dimension dχ corresponding to the character χ in the eigenspace of A with eigenvalue t. Nχ (λ) describes the distribution of eigenvalues of A, and we shall investigate its asymptotic behavior as λ → +∞ by means of the generalized theorem of the stationary phase. It will turn out that Nχ (λ) is intimately related to the representation theory of G, and the geometry of the Hamiltonian action of G on the symplectic manifold T ∗ (X). In fact, if (A1 , . . . , Ad ) is a basis of the Lie algebra g of G, let J : T ∗ (X) X × Rn → g Rd ,
(x, ξ ) → A1 x, ξ , . . . , Ad x, ξ ,
be the associated momentum map, where ·,· stands for the Euclidean scalar product in Rn , and denote by Ω0 /G = J−1 {0} /G the symplectic quotient of T ∗ (X) at level zero. This quotient is naturally related to the critical set of the phase function in question, and plays a crucial role in our reduction. Indeed, we shall prove that Nχ (λ) is asymptotically determined by a certain volume of the quotient Ω0 /G, which is symplectically diffeomorphic to T ∗ (X/G) on its smooth part [8]. Now, the major difficulty in applying the generalized stationary phase theorem in our setting stems from the fact that, due to the singular orbit structure of the underlying group action, the zero level Ω0 of the momentum map, and, consequently, the considered critical set, are in general singular varieties. In fact, if the G-action on T ∗ (X) is not free, the considered momentum map is no longer a submersion, so that Ω0 and Ω0 /G are not smooth anymore. Nevertheless, it can be shown that these spaces have a Whitney stratification into smooth submanifolds, see [20, Theorems 8.3.1 and 8.3.2], which corresponds to the stratification of T ∗ (X), and Rn into orbit types. To apply the principle of the stationary phase to our problem, we shall therefore proceed to partially resolve the singularities of Ω0 , and then apply the stationary phase theorem in the resolution space under the sole assumption that the set Sing Ω0 of points where Ω0 is not a manifold is contained in a strict vector
94
R. Cassanas, P. Ramacher / Journal of Functional Analysis 256 (2009) 91–128
subspace of T ∗ (X). This is always fulfilled for group actions that satisfy the following condition1 : if H0 is a closed subgroup of G, and if Rn(H0 ) denotes the union of all principal orbits in Rn of type G/H0 , which is an open and dense subset in Rn , then Rn \ Rn(H0 ) should be contained in a strict vector subspace of Rn . The main result of this paper is Theorem 8, which states that, as λ → +∞, one has the asymptotic formula Nχ (λ) =
−1 dχ [χ|H0 : 1] (−∞, 1] ∩ Ω0 /G λ(n−κ)/2m + O λ(n−κ−1/4)/2m , vol a2m (2π)n−κ
where dχ = χ(1), [χ|H0 : 1] is the multiplicity of the trivial representation in the restriction of χ to any principal isotropy group conjugated to H0 , and κ the common dimension of the orbits −1 of principal type. The volume of the quotient [a2m ((−∞, 1]) ∩ Ω0 ]/G is defined in Section 5. The asymptotic distribution of eigenvalues was first studied by Weyl [25] for certain second order differential operators in Rn using variational techniques. Hörmander [13] then extended these results to elliptic pseudodifferential operators on closed manifolds using the theory of Fourier integral operators. The first ones to study reduced Weyl asymptotics for elliptic operators on closed Riemannian manifolds in the presence of a compact group of isometries were Donnelly [6] together with Brüning and Heintze [3]. In the semi-classical context, reduced Weyl asymptotics and trace formulae were investigated in [7], and in [4] via coherent states. Our approach is based on the method of approximate spectral projections, first introduced by Tulovskii and Shubin [24]. Nevertheless, due to the presence of the boundary, the original method cannot be applied to our situation, and one has to use more elaborate techniques, which were subsequently developed by Feigin [9] and Levendorskii [18]. Compared to the method of Fourier integral operators, this approach gives weaker estimates for the remainder, but allows to consider non-smooth boundaries. Recently, Bronstein and Ivrii have obtained even sharp estimates for the remainder term in the case of differential operators on manifolds with boundaries satisfying the conditions specified above [2,16]. This paper is the second part of an investigation initiated in [21], which we shall refer to in the following as Part I. There, the foundations of the calculus of approximate spectral projection operators were provided, and the case of a finite group of isometries was settled. In this second part, the case of a compact group of isometries will be considered. Before we start, some comments on the results obtained might be in place. Asymptotics for the spectral counting function Nχ (λ) were obtained in [6] and [3] for general compact, isometric and effective Lie group actions using heat kernel methods; nevertheless, this approach does not allow to derive estimates for the remainder term. Using Fourier integral operator techniques, the same authors obtained rather optimal remainder estimates for compact G-manifolds in the cases where there is only one orbit type, or all orbits have the same dimension. For orthogonal actions in Rn , estimates for the remainder where obtained in [7,12] in case that the union Rn(H0 ) of all principal orbits is given by Rn \ {0}. In this paper, remainder estimates are obtained for the first time for a large class of singular group actions by partially resolving the singularities of the zero level of the momentum map Ω0 . We remark that our method could be easily adapted to the setting of Fourier integral operators as e.g. described in [12] or [7], extending previous results to more general situations, which would even yield optimal remainder estimates. 1 Examples for such group actions are given in Remark 1.
R. Cassanas, P. Ramacher / Journal of Functional Analysis 256 (2009) 91–128
95
2. Reduced spectral asymptotics and the approximate spectral projection operators In this section, we shall review some basic facts in the theory of pseudodifferential operators that will be needed in the sequel, and introduce the method of approximate spectral projection operators. For a more detailed exposition, the reader is referred to Part I, Sections 2 and 3. Let A0 be a classical pseudodifferential operator of order 2m in Rn , regarded as an operator in L2 (Rn ) n with domain C∞ c (R ). In other words, A0 can be represented by an oscillatory integral of the form A0 u(x) = ei(x−y)ξ a(x, ξ )u(y) dy dξ, ¯ where its symbol a(x, ξ ) has an asymptotic expansion of the form a(x, ξ ) ∼
a2m−j (x, ξ ) 1 − χ(ξ ) ,
j 0
χ being a compactly supported function equal to 1 in a neighborhood of zero, and the functions a2m−j are homogeneous of degree 2m − j in variable ξ . a2m is called the principal symbol m (Y × Rn ) the set of of A0 . If 0 , δ 1, and Y is an open set in Rn , let us denote by S,δ n smooth functions σ (x, ξ ) on Y × R such that for all compact sets K in Y, and all multi-indices α, β, there exist constants CK,α,β > 0 such that α β ∂ ∂ σ (x, ξ ) CK,α,β ξ m−|α|+δ|β| . ξ x
m n Let Lm ,δ (Y) be the class of pseudodifferential operators with symbols in S,δ (Y × R ). Then, 2m n as a local pseudodifferential operator, A0 ∈ L1,0 (R ), see [23, Section 3.7]. In what follows, we shall also need certain global spaces of symbols and pseudodifferential operators, which also take decay properties in x into account. They were introduced by Hörmander within the framework of Weyl calculus of pseudodifferential operators. Thus, consider on R2n the metric
δ − 2 g˜ x,ξ (y, η) = 1 + |x|2 + |ξ |2 |y|2 + 1 + |x|2 + |ξ |2 |η| , where 1 > δ 0, and put h(x, ξ ) = (1 + |x|2 + |ξ |2 )−1/2 . Definition 1. Let p be a g-continuous ˜ function. The class Γ,δ (p, R2n ), 0 δ < 1, consists ∞ 2n of all functions u ∈ C (R ) which for all multiindices α, β satisfy the estimates α β ∂ ∂ u(x, ξ ) Cαβ p(x, ξ ) 1 + |x|2 + |ξ |2 (−|α|+δ|β|)/2 . ξ x
l (R2n ) for Γ (h−l , R2n ), where l ∈ R. In particular, we shall write Γ,δ ,δ
The class Γ,δ (p, R2n ) is also denoted by S(g, ˜ p), see Part I, Definitions 1 and 3. Let now a ∈ Γ,δ (p, R2n ), 0 1 − δ < 1, and τ ∈ R. Then Au(x) =
ei(x−y)ξ a (1 − τ )x + τy, ξ u(y) dy dξ ¯
96
R. Cassanas, P. Ramacher / Journal of Functional Analysis 256 (2009) 91–128
defines a continuous operator in S(Rn ), respectively S (Rn ), see Part I, Corollary 1. In this case, a is called the τ -symbol of A, and the operator A is denoted by Opτ (a). If τ = 1/2, a is called they Weyl symbol of A, and one also writes Opw (a) for A. Pseudodifferential operators with real Weyl symbols give rise to self-adjoint operators. For τ = 0 and τ = 1 one simply obtains the usual left and right symbols, respectively. Our symbol classes will be mainly of the form S(h−2δ g, p) = Γ1−δ,δ (p, R2n ) with gx,ξ (y, η) = |y|2 + h(x, ξ )2 |η|2 , where p is a smooth, positive, g-continuous function, and 0 δ < 1/2. In what follows, l (Rn ) will denote the classes of pseudodifferential operators with symbols Π,δ (p, Rn ) and Π,δ l (R2n ), respectively. in Γ,δ (p, R2n ), and Γ,δ Consider now a bounded domain X in Rn with not necessarily smooth boundary ∂X, and let a be the left symbol of the classical pseudodifferential operator A0 . Clearly, a ∈ S(g, h−2m , Z × Rn ) for any compact set Z ⊂ Rn . By changing a outside X × Rn , we can therefore assume that a ∈ S(g, h−2m ), so that n 2m A0 ∈ Π1,0 R . Assume now that A0 satisfies the ellipticity condition (3). Lemma 1. The ellipticity condition (3) is equivalent to the existence of constants C, M > 0 such that (A0 + M1)u, u L2 (X) Cu2H m (X) ,
∀u ∈ C∞ c (X),
(4)
where . H m (X) is the norm in the Sobolev space H m (X). Proof. Since A0 + M1 is a classical symmetric pseudodifferential operator with principal symbol a2m , the implication (4) ⇒ (3) follows with [18, Lemma 13.1]. Now, let us assume that (1) is fulfilled. By compactness, there exists a constant ε > 0 such that, if Xε = {x ∈ Rn : dist(x, X) < ε}, one has a2m (x, ξ )
C0 2m |ξ | , 2
∀x ∈ Xε , ∀ξ ∈ Rn .
(5)
The restriction of A0 to Xε is of course in L2m 1,0 (Xε ) since X is bounded, and is elliptic in view of (5). It is not properly supported in general but, according to [23, Proposition 3.3], there exist an operator R with smooth kernel KR ∈ C∞ (Xε × Xε ), and an operator A1 in L2m 1,0 (Xε ) which is 2 properly supported in Xε such that, on L (Xε ), A0 = A1 + R.
R. Cassanas, P. Ramacher / Journal of Functional Analysis 256 (2009) 91–128
97
A1 is a classical pseudodifferential operator in Xε , with the same principal symbol as A0 , and is elliptic on Xε in view of (5). Applying now the Gårding inequality as stated in [11, p. 51], one deduces the existence of a constant C1 > 0 such that, for all u ∈ C∞ c (Xε ) with support in X, 1 Re (A1 + C1 1)u, u L2 (X ) u2H m (Xε ) . ε C1 Now, by the Schwarz inequality, Ru2L2 (X) = X
Ru(x) 2 dx
X
KR (x, y) 2 dy
X
u(z) 2 dz dx,
u ∈ C∞ c (X),
X
which implies that the restriction of R to L2 (X) is a bounded operator. Consequently, there exists a constant C2 > 0 such that for u ∈ C∞ c (X) 1 1 (A0 + C1 1)u, u L2 (X) u2H m (Xε ) + Re(Ru, u)L2 (X) u2H m (X) − C2 u2L2 (X) , C1 C1 and the assertion of the lemma follows.
2
∞ Next note that if A0 were properly supported, then A0 ◦ ext : C∞ c (X) → Cc (X1 ), where X1 n is some compact set in R , see [23, Proposition 3.4]. By continuity, this map would extend to a map from D (X) to D (X1 ), but in general it is not immediately clear if the restriction of A0 2m (Rn ), the to X extends to D (X). Nevertheless, as a pseudodifferential operator in the class Π1,0 operator A0 : S(Rn ) → S(Rn ) extends to a mapping from S (Rn ) to S (Rn ), see [14]. Therefore, if u ∈ L2 (X), then (A0 ◦ ext)(u) ∈ S (Rn ), and via the inclusion S (Rn ) → D (X), the operator res ◦ A0 ◦ ext extends naturally to an operator from L2 (X) to D (X). Let us now assume that A0 is symmetric, and that (3) is satisfied. Under these circumstances, the previous lemma implies that the operator 2 res ◦ A0 ◦ ext : C∞ c (X) → L (X)
is lower semi-bounded, and we shall denote its Friedrichs extension by A. It is a self-adjoint operator in L2 (X), and is itself lower semi-bounded. Its spectrum is real. The following proposition shows that A has compact resolvent, which implies that the spectrum of A is discrete, i.e. it consists of a sequence of isolated eigenvalues of finite multiplicity tending to infinity, while the essential spectrum of A is empty. Proposition 1. As an operator in L2 (X), A has compact resolvent. Moreover, D(A) ⊂ H0m (X) and (A + M)u, u L2 (X) Cu2H m (X)
∀u ∈ D(A).
(6)
m α 2 Here, H0m (X) denotes the closure of C∞ c (X) in H (X) = {u ∈ D (X): ∂ u ∈ L (X), |α| m} with respect to the Sobolev norm.
98
R. Cassanas, P. Ramacher / Journal of Functional Analysis 256 (2009) 91–128
2 ∞ ˜ Proof. Put A˜ = res ◦ A0 ◦ ext : C∞ c (X) → L (X). In view of (4), A is semi-bounded on Cc (X). ∞ (X) with respect to the norm p(v) = Let Q(A) be its form domain, that is, the completion of C c
((A˜ + M)v, v), see [22, p. 177]. Q(A) is endowed with the limit norm . Q(A) of p. Accord˜ one has D(A) ⊂ Q(A), ing to (4), Q(A) ⊂ H0m (X). Since A is the Friedrichs extension of A, and we obtain Eq. (6). Let now λ < −M. If u ∈ D(A), the Schwarz inequality yields uH m (X) C (A − λ)uL2 (X) for some constant C > 0. Thus, if v ∈ L2 (X), (A − λ)−1 vH m (X) CvL2 (X) . Therefore (A − λ)−1 is a continuous map from L2 (X) to H0m (X). But the injection H0m (X) → L2 (X) is compact by the Rellich theorem. Consequently, A must have compact resolvent. 2 Consider now a compact group of isometries G ⊂ O(n) acting on Euclidean space Rn , and assume that the bounded domain X in Rn is invariant under G. Then its boundary is G-invariant, too. Let T be the unitary representation of G in the Hilbert spaces L2 (Rn ) and L2 (X) defined in (1), and assume that the operator A0 commutes with the representation T . The G-action on X induces a Hamiltonian action of G in the cotangent bundle T ∗ (X) of X given by G × T ∗ (X) → T ∗ (X) : (k, x, ξ ) → σk (x, ξ ) = κk (x), t κk (x)−1 (ξ ) = κk (x), κk (ξ ) , where we wrote κk (x) = kx. Now, since T (k)Opτ (a)T k −1 = Opτ (a ◦ σk ),
a ∈ S(g, ˜ p),
the G-invariance of A0 is equivalent to the G-invariance of its symbol, by the uniqueness of the τ -symbol. In particular, the principal symbol a2m of A0 is invariant under σk for all k ∈ G. Since the operator A is also G-invariant, the eigenspaces of A are unitary G-modules that decompose into irreducible subspaces. The restriction of A to the isotypic component res Hχ in the Peter– Weyl decomposition of (T , L2 (X)) is called the reduced operator, and is denoted by Aχ . Its domain is D(Aχ ) = D(A) ∩ res Hχ . As explained in [4], Aχ inherits from A the property of ˆ of the spectra having compact resolvent, and the spectrum of A is equal to the union over χ in G of the operators Aχ . Our purpose in this paper is to investigate the spectral counting function Nχ (λ) of Aχ , which is given by the number of eigenvalues of Aχ , counting multiplicities, that are equal or less than λ ∈ R. It corresponds exactly the number of eigenvalues of A equal or less than λ, whose eigenfunctions belong to the χ -isotypic component of L2 (X), so that Nχ (λ) = dχ
μχ (t),
tλ
where μχ (t) denotes the multiplicity of any irreducible representation of dimension dχ corresponding to the character χ in the eigenspace of A with eigenvalue t. We shall study Nχ (λ) using the method of approximate spectral projection operators, which was first introduced by Shubin and Tulovskii, and adapted to the case of bounded domains by Levendorskii. It departs from the observation that N (λ) = tr(Eλ ),
R. Cassanas, P. Ramacher / Journal of Functional Analysis 256 (2009) 91–128
99
where Eλ = 1(−∞,λ] (A) is the spectral projector of A belonging to the value λ. The idea is then to approximate Eλ by a pseudodifferential operator Eλ . The trace of Eλ should then give a good approximation of N (λ). The approximate spectral projection operators Eλ will be constructed using Weyl quantization. In order to define them, we introduce now the relevant symbols. Thus, let aλ ∈ S(g, 1), and d ∈ S(g, d) be G-invariant symbols which, on X × {ξ : |ξ | > 1}, X = {x: dist(x, X) < }, are given by
λ 1 1− , aλ (x, ξ ) = a2m (x, ξ ) 1 + λ|ξ |−2m d(x, ξ ) = |ξ |−1 , where > 0 is some fixed constant, and in addition assume that d is positive and that d(x, ξ ) → 0 as |x| → ∞. We also define bλ (x, ξ ) = aλ x, λ1/2m ξ , which for |ξ | > λ−1/2m is given by bλ (x, ξ ) =
1 1 1 − , a2m (x, ξ ) 1 + |ξ |−2m
(7)
and actually independent of λ. We need to define smooth approximations to the Heaviside function, and to certain characteristic functions on X. Thus, let χ˜ be a smooth function on the real line satisfying 0 χ˜ 1, and χ(s) ˜ =
1 for s < 0, 0 for s > 1.
Let C0 > 0 and δ ∈ (1/4, 1/2) be constants, and put ω = 1/2 − δ. We then define the G-invariant functions χλ = χ˜ ◦ aλ + 4hδ−ω + 8C0 d h−δ ,
χλ+ = χ˜ ◦
aλ − 4hδ−ω − 8C0 d h−δ ,
0 where 0 < δ − ω < 1/2. One can then show that χλ , χλ+ ∈ S(h−2δ g, 1) = Γ1−δ,δ (R2n ) uniformly 2n in λ, see Part I, Lemma 10. Next, let U be a subset in R , c > 0, and put
U (c, g) = (x, ξ ) ∈ R2n : ∃(y, η) ∈ U, g(x,ξ ) (x − y, ξ − η) < c ; according to [18, Corollary 1.2], there exists a smoothened characteristic function ψc ∈ S(g, 1) belonging to the set U and the parameter c, such that supp ψc ⊂ U (2c, g), and ψc |U (c,g) = 1. Let now Mλ = (x, ξ ) ∈ R2n : aλ < 4hδ−ω + 8C0 d . Both Mλ and ∂X × Rn are invariant under σk for all k ∈ G, as well as (∂X × Rn )(c, h−2δ g), and Mλ (c, h−2δ g), due to the invariance of a2m (x, ξ ), and the considered metrics and symbols.
100
R. Cassanas, P. Ramacher / Journal of Functional Analysis 256 (2009) 91–128
Now, let η˜ c , ψλ,c ∈ S(h−2δ g, 1) be smoothened characteristic functions corresponding to the parameter c, and the sets ∂X × Rn and Mλ , respectively. According to Lemma 5 in Part I, we can assume that they are invariant under σk for all k ∈ G; otherwise consider G η˜ c ◦ σk dk, G ψλ,c ◦ σk dk, respectively. We then define the functions ηλ,−c (x, ξ ) = ηc (x, ξ ) =
0, (1 − η˜ c (x, ξ ))ψλ,1/c (x, ξ ), η˜ c (x, ξ ), 1,
x∈ / X, x ∈ X,
x∈ / X, x ∈ X.
Only the support of ψλ,c depends on λ, but not its growth properties, so that ηc , ηλ,−c ∈ S(h−2δ g, 1) uniformly in λ. Furthermore, since η˜ 2c = 1 on supp η˜ c , and ψλ,1/c = 1 on supp ψλ,1/2c , on has ηλ,−c = 1 on supp ηλ,−2c , which implies ηλ,−2c ηλ,−c = ηλ,−2c . Similarly, one verifies ηc η2c = ηc . We are now ready to define the approximate spectral projection operators. Definition 2. The approximate spectral projection operators of the first and second kind are defined by the equations E˜λ = Opw (ηλ,−2 )Opw (χλ )Opw (ηλ,−2 ),
Eλ = E˜λ2 (3 − 2E˜λ ),
while the ones of the third and fourth kind are given by F˜ λ = Opw η22 χλ+ ,
Fλ = F˜ λ2 (3 − 2F˜ λ ).
Both Eλ and Fλ are integral operators with kernels in S(R2n ). By [14, Lemma 7.2], this implies that Eλ and Fλ are of trace class and, in particular, compact operators in L2 (Rn ). In addition, by Theorem 2, and the asymptotic expansion (10) in Part I, one has σ τ (Eλ ), σ τ (Fλ ) ∈ S(h−2δ g, 1) uniformly in λ. On the other hand, all the involved symbols are real valued, which by general Weyl calculus implies that Opw (ηλ,−2 ), Opw (χλ ), Opw (η22 χλ+ ), and consequently also Eλ , and Fλ , are self-adjoint operators in L2 (Rn ). Let Pχ denote the orthogonal projector defined in (2) onto the isotypic component of the Peter–Weyl decomposition of (T , L2 (Rn )) corresponding to the character χ . By construction, both Eλ and Fλ commute with the projection Pχ , so that Pχ Eλ and Pχ Fλ are self-adjoint operator of trace class as well. Although the decay properties of σ τ (Eλ ), σ τ (Fλ ) are independent of λ, their supports do depend on λ, which will result in estimates for the trace of Pχ Eλ and Pχ Fλ in terms of λ that will be used in order to prove Theorem 8. In particular, the estimate for the remainder term in Theorem 8 is determined by the particular choice of the range (1/4, /1/2) for the parameter δ, which guarantees that 1 − δ > δ. The method of approximate spectral projection operators is based on variational arguments. Thus, if S is a symmetric, lower semi-bounded operator in a separable Hilbert space, and if V is a subspace contained in its domain D(S), the variational quantity N (S, V ) = sup dim L: (Su, u) < 0 ∀0 = u ∈ L L⊂V
R. Cassanas, P. Ramacher / Journal of Functional Analysis 256 (2009) 91–128
101
can be used to give a qualitative description of the spectrum of S. In our case one has Nχ (λ) = N A0 − λ1, Hχ ∩ C∞ c (X) . Indeed, the Friedrichs extension of res ◦ A0 ◦ ext : C∞ c (X) ∩ Hχ → res Hχ is given by Aχ , and the assertion follows with [18, Lemma A.2]. Now by the general theory of compact, self-adjoint operators, zero is the only accumulation point of the point spectra of Eλ and Fλ , as well as the only point that could possibly belong to the continuous spectrum. Therefore the number of eigenvalues of Eλ which are 1/2, and whose eigenfunctions belong to the isotypic component Hχ is clearly finite, and shall be denoted by NχEλ . Similarly, the number of eigenvalues of the operators Fλ which are 1/2, and whose eigenfunctions belong to the isotypic component Hχ , shall be denoted by NχFλ . As it was shown in Part I, Theorems 4 and 5, these quantities constitute upper and lower bounds for the spectral counting function Nχ (λ), namely Fλ NχEλ − C N A0 − λ1, Hχ ∩ C∞ c (X) Mχ + C for some constant C > 0. Furthermore, by Lemmata 11 and 12 of Part I one has 2 tr(Pχ Eλ · Pχ Eλ ) − tr Pχ Eλ − c1 NχEλ 3 tr Pχ Eλ − 2 tr(Pχ Eλ · Pχ Eλ ) + c2 , 2 tr(Pχ Fλ · Pχ Fλ ) − tr Pχ Fλ − c1 NχFλ 3 tr Pχ Fλ − 2 tr(Pχ Fλ · Pχ Fλ ) + c2 , for some constants ci > 0. The study of the asymptotic behaviour of Nχ (λ) is therefore reduced to an examination of the traces of Pχ Eλ and Pχ Fλ , together with their squares, and will occupy us for the rest of this paper. 3. Compact group actions and the principle of the stationary phase In this section, we shall begin to estimate the traces of Pχ Eλ and Pχ Fλ using the method of the stationary phase, in order to obtain a description of the spectral counting function Nχ (λ) as λ → +∞. As mentioned in the introduction, first order asymptotics for invariant elliptic operators were already obtained in [3,6] in the general case of effective group actions by using heat kernel methods; nevertheless, estimates for the remainder are not accessible via this approach. On the other hand, the derivation of remainder estimates within the framework of Fourier integral operators or, as we shall see, within the setting of approximate spectral projections, meets with serious difficulties when singular orbits are present. The reason for this is that, using these approaches, one is led to the study of the asymptotic behavior of integrals of the form ei(x−kx)ξ/μ a(x, ξ, k) dx dξ (8) ¯ dk, μ → 0+ , G Rn Rn n n via the generalized stationary phase theorem, where a(x, ξ, k) ∈ C∞ c (R × R × G) is an amplitude which might also depend on μ. While for free group actions, the critical set of the phase function (x − kx)ξ is a smooth manifold, this is no longer the case for general effective actions, so that, a priori, the principle of the stationary phase cannot be applied in this case. Nevertheless, in what follows, we shall show how to circumvent this obstacle by partially resolving the singularities of the critical set of the phase function in question, and in this way obtain remainder
102
R. Cassanas, P. Ramacher / Journal of Functional Analysis 256 (2009) 91–128
estimates for Nχ (λ) in the case of singular group actions. Let us begin by stating the generalized stationary phase theorem. Theorem 1 (Generalized stationary phase theorem for manifolds). Let M be a n-dimensional Riemannian manifold, ψ ∈ C∞ (M) be a real-valued phase function, a ∈ C∞ c (M), μ > 0, and set I (μ) =
eiψ(m)/μ a(m) dm, M
where dm denotes the volume form on M. Let C = {m ∈ M: ψ : T Mm → T Rψ(m) is zero} be the critical set of the phase function ψ, and assume that: (i) C is a smooth submanifold of M of dimension p in a neighborhood of the support of a; (ii) for all m ∈ C, the restriction ψ (m)|Nm C of the Hessian of ψ at the point m to the normal space Nm C is a non-degenerate quadratic form. Then, for all N ∈ N, there exists a constant CN,ψ > 0 such that N −1 n−p iψ0 /μ j (2πμ) 2 μ Lj (ψ; a) CN,ψ μN vol(supp a ∩ C) sup D l a ∞,M , I (μ) − e l2N j =0
where D l is a differential operator on M of order l, and ψ0 is the constant value of ψ on C. Furthermore, for each j there exists a constant C˜ j,ψ > 0 such that Lj (ψ; a) C˜ j,ψ vol(supp a ∩ C) sup D l a , ∞,C l2j
and, in particular, L0 (ψ; a) = C
a(x) dσC (m) eiπσψ , |det ψ (m)|Nm C |1/2
where σψ is the constant value of the signature of ψ (m)|Nm C for m in C. Proof. See [15, Theorem 7.7.5], and [5, Theorem 3.3].
2
From now on, we shall restrict ourselves to the study of tr Pχ Eλ , since the corresponding considerations for Fλ are completely analogous. Let therefore σ l (Eλ )(x, ξ ) denote the left symbol of Eλ . Since σ l (Eλ ) is G-invariant, we have n Pχ Eλ u(x) = dχ χ(k)ei(x−ky)ξ σ l (Eλ )(x, ξ )u(y) dy dξ ¯ dk, u ∈ C∞ c R . G
The kernel of Pχ Eλ , which is a rapidly decreasing function, is given by the absolutely convergent integral
R. Cassanas, P. Ramacher / Journal of Functional Analysis 256 (2009) 91–128
103
KPχ Eλ (x, y) = dχ
χ(k)ei(x−ky)ξ σ l (Eλ )(x, ξ ) dξ ¯ dk. G
Consequently, the trace of Pχ Eλ can be computed by tr Pχ Eλ =
KPχ Eλ (x, x) dx = dχ
χ(k)ei(x−kx)ξ σ l (Eλ )(x, ξ ) dx dξ ¯ dk. G
0 As already noticed, the decay properties of σ l (Eλ ) ∈ S(h−2δ g, 1) = Γ1−δ,δ (R2n ) are independent of λ, while its support does depend on λ. Indeed, as it was already explained in Part I, Eq. (51),
2 2 2 σ l (Eλ ) = ηλ,−2 χλ 3 − 2ηλ,−2 χλ + fλ + rλ ,
(9)
where rλ ∈ S(h−2δ g, hN (1−2δ) ) for arbitrary large N , and fλ ∈ S(h−2δ g, h1−2δ ), everything uniformly in λ. Moreover, in Lemma 9 we shall see that supp fλ ⊂ Ac,λ = (x, ξ ) ∈ X × Rn : aλ < c hδ−ω + d . Now, since |rλ (x, ξ )| C (1 + |x|2 + |ξ |2 )−N/2 for some constant C independent of λ and N arbitrarily large, we get the uniform bound
rλ (x, ξ ) dx dξ ¯ C;
note that the x-dependence of h(x, ξ ) is crucial at this point. In order to determine the asymptotic behaviour of tr Pχ Eλ with respect to λ, we can therefore neglect the contribution coming from rλ (x, ξ ), so that tr Pχ Eλ = dχ
2 2
2 χ(k)ei(x−kx)ξ ηλ,−2 χλ 3 − 2ηλ,−2 χλ + fλ (x, ξ ) dx dξ ¯ dk + O(1),
G
as λ goes to infinity. To apply the generalized stationary phase theorem, we introduce the new parameter μ = λ−1/2m ,
λ = μ−2m ,
and performing the change of variables Ψμ : (x, ξ ) → (x, μξ ) we obtain tr Pχ Eλ = dχ λn/2m I (λ−1/2m ) + O(1), where we set
104
R. Cassanas, P. Ramacher / Journal of Functional Analysis 256 (2009) 91–128
I (μ) =
i
e μ ψ(x,ξ,k) χ(k)σμ (x, ξ ) dx dξ ¯ dk,
G X Rn
ψ(x, ξ, k) = (x − kx)ξ, 2
2 2 χλ 3 − 2ηλ,−2 χλ + fλ ◦ Ψμ−1 . σμ = ηλ,−2
(10)
As we shall see later, there exists a compact subset K ⊂ R2n , such that σμ has support in K for all μ > 0, see (38). To get an asymptotic expansion of I (μ) as μ → 0+ via the generalized stationary phase theorem, we commence by examining the critical set C = (x, ξ, k) ∈ X × Rn × G: ψ (x, ξ, k) = 0
(11)
of the phase function ψ . After a straightforward computation we obtain C = (z, k) ∈ Ω0 × G: kz = z , where we put z = (x, ξ ), and Ω0 = (x, ξ ) ∈ X × Rn : xA, ξ = 0 for all A ∈ g , g being the Lie algebra of G. ·,· denotes the Euclidean product in Rn . Note that Ω0 is invariant under the Hamiltonian action of G on the cotangent space T ∗ (X) given by (x, ξ ) → (kx, kξ ), as well as homogeneous with respect to x and ξ . It has the following interpretation in terms of the Hamiltonian action of G on T ∗ (X). If (A1 , . . . , Ad ) is a basis of g, let J : T ∗ (X) X × Rn → g Rd ,
(x, ξ ) → A1 x, ξ , . . . , Ad x, ξ ,
be the associated momentum map, and denote by Ω0 /G = J−1 {0} /G the symplectic quotient of T ∗ (X) at level zero. This quotient is naturally related to the critical set of the phase function in question, and we shall prove that Nχ (λ) is asymptotically determined by a certain volume of the quotient Ω0 /G. Now, the major difficulty in applying the generalized stationary phase theorem in our setting stems from the fact that, due to the singular orbit structure of the underlying group action, the zero level Ω0 of the momentum map, and, consequently, the considered critical set C, are in general singular varieties. In fact, if the G-action on T ∗ (X) is not free, the considered momentum map is no longer a submersion, so that Ω0 and Ω0 /G are not smooth anymore. To circumvent this difficulty, we will partially resolve the singularities of C by constructing a partial resolution of Ω0 , which takes into account the singular orbit structure of the underlying G-action, and then apply the stationary phase theorem in the resolution space.2 2 As we shall see in Section 5,
σμ (x, ξ ) → 1{a2m 1} (x, ξ )
as μ → 0+ ,
R. Cassanas, P. Ramacher / Journal of Functional Analysis 256 (2009) 91–128
105
In what follows, we shall therefore briefly recall some basic notions of the theory of compact group actions. For a detailed exposition, we refer the reader to [1] or [17]. Let G be a compact Lie group acting locally smoothly on some n-dimensional C∞ -manifold M, and denote the stabilizer, or isotropy group, of x ∈ M by Gx = {k ∈ G: k · x = x}. The orbit of a point x ∈ M under the action of G will be denoted by Ox . Assume that M/G is connected. One of the main results in the theory of compact group actions is the following. Theorem 2 (Principal orbit theorem). There exists a maximum orbit type G/H for G on M. The union M(H ) of orbits of type G/H is open and dense, and its image in M/G is connected. Proof. See [1, Theorem IV.3.1].
2
Orbits of type G/H are called of principal type, and the corresponding isotropy groups are called principal. A principal isotropy group has the property that it is conjugated to a subgroup of each stabilizer of M. The following result says that there is a stratification of G-spaces into orbit types. Theorem 3. Let G and M be as above, K a subgroup of G, and denote the set of points on orbits of type G/K by M(K) . Then M(K) is a topological manifold, which is locally closed. Furthermore, M(K) consists of orbits of type less than or equal to type G/K. The orbit map M(K) → M(K) /G is a fiber bundle projection with fiber G/K and structure group N (K)/K. Proof. See [1, Theorem IV.3.3].
2
Let now Mτ denote the union of non-principal orbits of dimension at most τ . Theorem 4. If κ is the dimension of a principal orbit, then dim M/G = n − κ, and Mτ is a closed set of dimension at most n − κ + τ − 1. Proof. See [1, Theorem IV.3.8].
2
Here the dimension of Mτ is understood in the sense of general dimension theory. In what follows, we shall write Sing M = M − M(H ) = Mκ . Clearly, Sing M = M0 ∪ (M1 − M0 ) ∩ (M2 − M1 ) ∪ · · · ∪ (Mκ − Mκ−1 ), where Mi − Mi−1 is precisely the union of non-principal orbits of dimension i, and M−1 = ∅, by definition. Note that where 1A stands for the characteristic function of the set A. By homogeneity, a2m (0, 0) = 0, so that zero is contained in the support of 1{a2m 1} . In general, σμ is therefore not supported away from the set of singular points of C, since (0, 0) is always a singularity of Ω0 in case that 0 ∈ X.
106
R. Cassanas, P. Ramacher / Journal of Functional Analysis 256 (2009) 91–128
Mi − Mi−1 =
j
M(H i ) , j
Hji ⊂ G, dim G/Hji = i,
is a disjoint union of topological manifolds of possibly different dimensions. We apply this theory now to the case where M = Rn , and G is a compact subgroup of O(n). Definition 3. Let G/H0 be the principal orbit type of the action of G ⊂ O(n) on Rn , and denote by κ the dimension of G/H0 . Since X is open in Rn , it has the same principal orbit type than Rn . Now, even if Ω0 is not a smooth manifold, it can be shown that it has a Whitney stratification into smooth submanifolds, see [20, Theorem 8.3.1], which corresponds to the stratification of T ∗ (X) and Rn into orbit types. In particular, the strata of Ω0 are submanifolds of R2n , and Ω0 admits a principal orbit type, too. Proposition 2. Let Reg Ω0 = Ω0(H1 ) be the principal stratum of Ω0 . Then Reg Ω0 is an open dense subset of Ω0 , and a submanifold of X × Rn of codimension κ. Moreover, for z ∈ Reg Ω0 one has
0 1n ⊥ . (12) Tz (Reg Ω0 ) = (J gz) , where J = −1n 0 Futhermore, H1 is conjugated to H0 , and thus Reg Ω0 = {z ∈ Ω0 : Gz is conjugated to H0 }. In particular, if (x, ξ ) ∈ Ω0 , and if Ox or Oξ are of type G/H0 , then (x, ξ ) ∈ Reg Ω0 . To prove the proposition, we need the following. Lemma 2. Assume that (x, ξ ) ∈ Ω0 . If Ox is of principal orbit type in Rn , then Gx ⊂ Gξ . If Oξ is of principal orbit type in Rn , then Gξ ⊂ Gx . Proof. Let (x, ξ ) ∈ Ω0 , that is, ξ ∈ Nx Ox , where Nx Ox denotes the normal space to the G-orbit Ox at the point x, which is a vector subspace in Rn . Assume now that Ox is of principal type. Denote by Vε the open ε-ball in Nx Ox , and consider the linear tube G ×Gx Vε → G · Vε ,
[g, v] → gv,
around Ox , see [1, Corollary II.5.2]. By [1, Theorem IV.3.2], Gx acts trivially on Vε , and consequently also on Nx Ox , and the assertion follows. To see this directly, one can also argue as follows. Let (x, ξ ) ∈ Ω0 , so that ξ ∈ (gx)⊥ . If g ∈ Gx , then gξ ∈ (gx)⊥ . Thus (g − 1)ξ ∈ (gx)⊥ . We claim that if Ox is of principal orbit type in Rn , then (g − 1)ξ ∈ gx, which will yield (g − 1)ξ = 0, and prove the inclusion Gx ⊂ Gξ . Now, by [17, Theorem 4.19], the canonical projection π : Rn(H0 ) Rn(H0 ) /G is a smooth submersion. Since the preimage of the tangent space of a smooth manifold under a submersion is equal to the tangent space of the preimage of the considered manifold at the given point, ker dx π = gx. Moreover, since M(H0 ) is an open set of Rn , one can differentiate the relation π(gy) = π(y),
R. Cassanas, P. Ramacher / Journal of Functional Analysis 256 (2009) 91–128
107
with respect to variable y at x to obtain dgx π ◦ g = dx π . Since gx = x, dx π ◦ (g − 1) = 0. This proves that the image of g − 1 is contained in ker dx π = gx. 2 Proof of Proposition 2. The first part of the statement follows from the references previously given, while the characterization of the tangent space is obtained by observing that dim Reg Ω0 = 2n − κ. By the previous lemma, (Rn × Rn(H0 ) ) ∩ Ω0 is a non-empty open subset of Ω0 consisting of orbits of type G/H0 . As Reg Ω0 is open and dense in Ω0 , it must intersect (Rn × Rn(H0 ) ) ∩ Ω0 , and therefore consist of orbits of type G/H0 . 2 In what follows, we will denote by Sing Ω0 the complement of Reg Ω0 in Ω0 . The next lemma will provide us with a suitable parametrization of Reg Ω0 . Lemma 3. The sets {(x, ξ ) ∈ Reg Ω0 : x ∈ Sing Rn }, {(x, ξ ) ∈ Reg Ω0 : ξ ∈ Sing Rn } have measure zero in Reg Ω0 with respect to the induced volume form on Reg Ω0 . Proof. We shall show that N = {(x, ξ ) ∈ Ω0 : x ∈ Sing Rn } is a closed set in Ω0 of dimension at most 2n − κ − 1. Indeed, with M = Rn , and notations as above, N=
κ κ (x, ξ ) ∈ Ω0 : x ∈ Mi − Mi−1 = (x, ξ ) ∈ R2n : x ∈ Rn(H i ) , ξ ∈ Nx Ox , i=0
j
i=0 j (i)
where the union over j (i) ranges over all non-principal orbit types G/Hji with dim G/Hji = i. By the previous theorem, dim Rn i dim Mi n−κ +i −1, and in addition, dim Nx Ox = n−i for (Hj )
all x ∈ Rn
(Hji )
. Consequently, {(x, ξ ) ∈ R2n : x ∈ Rn
(Hji )
, ξ ∈ Nx Ox } is a subset of Ω0 of dimension
at most 2n − κ − 1. Since for orthogonal group actions there are only finitely many orbit types, the union over j (i) is finite, and the assertion of the lemma follows. 2 Finally, for future reference we note the following. Lemma 4. The set Reg C = (z, k) ∈ Reg Ω0 × G: kz = z is a smooth submanifold of dimension 2n + d − 2κ, and for (z, k) ∈ Reg C, T(z,k) Reg C = (α, Ak): α ∈ Tz Reg Ω0 , A ∈ G and (1 − k)α + Az = 0 . Proof. See [4, Lemma 3.2].
2
In particular note that if (z, k) belongs to Sing C, the complement of Reg C in C, then z must necessarily lie in Sing Ω0 . After these preliminary remarks, we are now ready for the analysis of I (μ).
108
R. Cassanas, P. Ramacher / Journal of Functional Analysis 256 (2009) 91–128
4. Phase analysis and partial desingularization We shall now start with the computation of an asymptotic formula for integrals of the form (8) via the generalized stationary phase theorem by partially resolving the singularities of the critical set C = (x, ξ, k) ∈ X × Rn × G: ψ (x, ξ, k) = 0 of the phase function ψ(x, ξ, k) = x − kx, ξ . Such a resolution will be given by a proper Ranalytic map ζ : M˜ → M of some smooth manifold M˜ onto M = Rn , inducing a transformation ˜ and ζ induces an isomorphism ζ : C˜ → C such that C˜ is a partially desingularized subvariety of M, −1 of real analytic manifolds ζ (Reg C) → Reg C, where Reg C denotes the set of nonsingular points of C. By performing such a resolution we will be led to a new phase function, whose critical set is no longer a singular variety. As before, denote by Rn(H0 ) the union of all orbits of principal type G/H0 in Rn . We will construct an explicit resolution of C˜ by constructing a resolution of Ω0 first, under the following assumption. Assumption 1. The set Sing Rn = Rn \ Rn(H0 ) is included in a strict vector subspace F of Rn of dimension r < n. Remark 1. Particular cases of Assumption 1 are as follows. (i) Transitive actions on the sphere. For any compact subgroup of O(n) acting transitively on the (n − 1)-dimensional sphere, Sing Rn = {0}. The list of compact, connected Lie groups acting transitively and effectively on spheres has been found by Montgomery and Samelson [19]. It includes all the holonomy groups of a simply-connected Riemannian manifold with an irreducible, nonsymmetric metric appearing in Berger’s list, and in particular, the group SO(n) acting on Rn . (ii) Cylindrical actions. For the group of rotations around an axis in Rn , Rnsing is equal to the rotation axis. More generally, any group conjugated to G × {1q } in O(n), where G is a compact subgroup of O(p) acting transitively on the (p − 1)-dimensional sphere, and p + q = n, is included. We begin by considering the blowing-up of M = R2n with center C = {ξ1 = · · · = ξn = 0} given by M˜ =
x, ξ, [μ] ∈ M × RPn−1 : ξi μj = ξj μi , i < j ,
together with the monoidal transformation ζM : M˜ → M,
x, ξ, [μ] → (R0 x, R0 ξ ),
with R0 ∈ O(n) such that R0 Rr × {0} = F.
R. Cassanas, P. Ramacher / Journal of Functional Analysis 256 (2009) 91–128
109
Covering M˜ with the charts M˜ j = M˜ ∩ (M × Uj ), where Uj = {[μ] ∈ RPn−1 : μj = 0}, one obtains in M˜ j the local coordinates xi ,
i = 1, . . . , n,
ηk =
μk , μj
ηj = ξj ,
k = 1, .∧. . , n,
and we write ϕ˜ j : R2n → M˜ j ,
(x, η) → x, ηj (η1 , . . . , 1, . . . , ηn ), [η1 : . . . : 1 : . . . : ηn ] .
−1 Now, the total transform of Ω0 is given by Ω˜ 0tot = ζM (Ω0 ), and contains the exceptional divisor −1 E = ζM (C), while the strict transform of Ω0 in the j th chart is locally given by
Ω˜ 0st = (x, η) ∈ R2n : AR0 x, R0 (η1 , . . . , 1, . . . , ηn ) = 0, A ∈ g . For j = r + 1, . . . , n, it is a non-singular variety, since in this case the condition (x, η) ∈ Ω˜ 0st implies that (R0 x, R0 (η1 , . . . , 1, . . . , ηn )) ∈ Reg Ω0 by Assumption 1, and Proposition 2. By ˜ To construct a partial resolution for C, functoriality, the G-action on M lifts to a G-action on M. we put N = M × G, N˜ = M˜ × G, and ζN : N˜ → N, (x, ξ, [μ], k) → (x, ξ, k). Using the coordinates introduced above, we see that the strict transform of C with respect to ζN is locally given by C˜st = (x, η, k) ∈ Ω˜ 0st × G: (k − 1)R0 x = 0, (k − 1)R0 (η1 , . . . , 1, . . . , ηn ) = 0 . For j = r + 1, . . . , n, G acts on Ω˜ 0st only with one orbit type, so that in this case C˜st must be non-singular. From now on, we shall restrict ourselves to the study of the integral I (μ) defined in (10). Since each chart M˜ j completely covers M˜ except for a set of measure zero, one has I (μ) = G
˜ ei ψj (x,η,k)/μ σ˜ μ,j (x, η)χ (k) ηjn−1 dx dη ¯ dk
(13)
R2n
for arbitrary j , where we put ψ˜ j (x, η, k) = ψ((ζM ◦ ϕ˜ j )(x, η), k), σ˜ μ,j (x, η) = (σμ ◦ ζM ◦ ϕ˜ j )(x, η), and took into account the fact that |det D(ζM ◦ ϕ˜j )(x, η)| = |ηjn−1 |. In what follows, we shall work in the chart j = n, and denote ψ˜ n and σ˜ μ,n simply by ψ˜ and σ˜ μ , respectively. Let us now introduce the new parameter3 ν = μ/ηn . Defining the new phase function4 3 The idea of introducing the new parameter ν was taken from [7, Section 6]. Nevertheless, Helffer and El-Houakmi work in spherical variables, which leads to secondary critical points that were not explicitly taken into account in their work. Our approach does not lead to secondary critical points. 4 The subscript “wk” stands for “weak transform.”
110
R. Cassanas, P. Ramacher / Journal of Functional Analysis 256 (2009) 91–128
ψwk (x, η , k) = (1 − k)R0 x; R0 (η , 1) Rn ,
ψwk : Rn × Rn−1 × G → R, and taking into account (38), we write 1 I (μ) = (2π)n
E0 Iηn (μ/ηn )|ηn |n−1 dηn , −E0
where E0 is some suitable positive number, and i Iηn (ν) = e ν ψwk (x,η ,k) σ˜ νηn (x, η , ηn )χ (k) dx dη dk.
(14)
G R2n−1
The significance of the new phase function ψwk stems from the following proposition. It will enable us to derive an asymptotic formula for Iηn (ν) as ν goes to zero by using the stationary phase theorem in the region where ηn is not small. Note that, in particular, Theorem 1 will allow us to handle the dependence of the amplitude σ˜ μ in variable μ = νηn . = 0} denote the critical set of ψ . Then Proposition 3. Let Cψwk = {ψwk wk
Cψwk = (x, η , k) ∈ Rn × Rn−1 × G: R0 x, R0 (η , 1), k ∈ Reg C . It is a smooth submanifold of Rn × Rn−1 × G of codimension 2κ. Moreover, at each point (x, η , k) of Cψwk , the transversal Hessian of ψwk defines a non-degenerate quadratic form on the normal space N(x,η ,k) Cψwk of Cψwk in Rn × Rn−1 × G. Remark 2. Note that if ψwk is regarded as a function on N˜ , that is, as a function of x, η, and k, the proposition implies that its critical set is given by the strict transform C˜st of C; moreover, its transversal Hessian does not degenerate along C˜st . Proof of Proposition 3. We shall denote by (e1 , . . . , en ) the canonical basis of Rn . With respect to the coordinates (x, η, k) one computes ∂x ψwk (x, η , k) = 0
⇔
∂k ψwk (x, η , k) = 0
⇔
∂η ψwk (x, η , k) = 0
⇔
1 − k −1 R0 (η , 1) = 0, AR0 x, R0 (η , 1) = 0, ∀A ∈ g, (1 − k)R0 x, R0 ei = 0, i = 1, . . . , n − 1.
The second equation is equivalent to the fact that (R0 x, R0 (η , 1)) ∈ Ω0 . By Assumption 1, / F , so that using Proposition 2, we obtain that our second equation is equivalent R0 (η , 1) ∈ to the fact that (R0 x, R0 (η , 1)) ∈ Reg Ω0 . Using Lemma 2, the two first equations imply that kR0 x = R0 x, and therefore imply the third one. Consequently, we obtain Cψwk = (x, η , k) ∈ Rn × Rn−1 × G: (k − 1)R0 x = 0, (k − 1)R0 (η , 1) = 0, R0 x, R0 (η , 1) ∈ Reg Ω0 .
R. Cassanas, P. Ramacher / Journal of Functional Analysis 256 (2009) 91–128
111
Next, we see immediately that Cψwk is diffeomorphic to the intersection of Reg C and (R0 × R0 )({ηn = 1}). Thus, in order to show that Cψwk is a smooth manifold, we have to prove that these two sets are transversal. Let (z, k) = (R0 x, R0 η, k) ∈ Reg C ∩ (R0 × R0 )({ηn = 1}). We need to prove that T(z,k) Reg C ⊂ (R0 × R0 )({ηn = 0}). For this purpose, consider α = (−R0 x, R0 η). This / {ηn = 0}. Moreover, we shall is an element of Tz Ω0 = J gz⊥ which satisfies (R0 × R0 )−1 (α) ∈ see later in Lemma 7 that kz = z implies (k − 1)α ∈ gz for all α ∈ Tz Ω0 . Consequently, there exists an A ∈ g such that (α, Ak) ∈ T(z,k) Reg C \ (R0 × R0 )({ηn = 0}). The dimension of Cψwk follows from Lemma 4, and the tangent space at (x, η , k) is therefore given by T(x,η ,k) Cψwk = (q, p , Ak) ∈ Rn × Rn−1 × gk: R0 (q), R0 (p , 0), Ak ∈ T(R0 x,R0 (η ,1),k) (Reg C) .
(15)
To compute the Hessian of ψwk at a point (x0 , η0 , k0 ) ∈ Cψwk , we fix a basis (A1 , . . . , Ad ) of g, and use the chart α : R2n−1 × Rd → R2n−1 × G defined by
α(x, η , s) = x, η , exp −
d
si Ai k0 .
i=1
With respect to these coordinates, the Hessian of ψwk is given by the matrix Hess ψwk (x0 , η0 , k0 ) =
∂ 2 (ψwk ◦ α) (x0 , η0 , 0) ∂Xi ∂Xj 1i,j 2n+d−1
which is a square matrix of size 2n + d − 1. Before entering the computations, we recall that by (3.17) of [4] we have J Az, Bz R2n = 0 ∀z ∈ Ω0 , ∀A, B ∈ g,
(16)
which is equivalent to Ax, Bξ Rn = Bx, Aξ Rn
∀(x, ξ ) ∈ Ω0 , ∀A, B ∈ g.
(17)
Using these identities, we obtain for all (x, η , k) ∈ Cψwk that Hess ψwk (x, η , k) is given by ⎛
0
⎜ R e , (k − 1)R e 0 i 0 j ⎝ R0 ej , k −1 Ai R0 (η , 1)
R0 ei , (k −1 − 1)R0 ej R0 ei , k −1 Aj R0 (η , 1) 0
− Aj R0 x, R0 ei
− Ai R0 x, R0 ej
− Ai R0 x, Aj R0 (η , 1)
⎞ ⎟ ⎠,
where the first diagonal block is of size n, the second of size n − 1 and the third of size d; each block has been characterized by specifying the entry of the ith line and the j th column. Let now (q, p , s) ∈ Rn × Rn−1 × Rd . We set A = di=1 si Ai . Then (q, p , s) ∈ ker Hess ψwk (x, η , k) if and only if ⎧ (a) ⎪ ⎨ (1 − k)R0 (p , 0) + AR0 (η , 1) = 0, (k − 1)R0 (q) − AR0 x = λ0 R0 (en ), (b) ⎪ ⎩ kR0 (q), Ai R0 (η , 1) − Ai R0 x, R0 (p , 0) − Ai R0 x, AR0 (η , 1) = 0, ∀i = 1, . . . , d, (c)
112
R. Cassanas, P. Ramacher / Journal of Functional Analysis 256 (2009) 91–128
for some λ0 ∈ R. Taking the scalar product of (b) with R0 (η , 1), we obtain λ0 = 0. Using (a), we find that (c) is equivalent to the fact that kR0 (q), BR0 (η , 1) = kR0 (p , 0), BR0 x for all B in g. Since kR0 x = R0 x and kR0 (η , 1) = R0 (η , 1), we see that for all B ∈ g,
kR0 (q), BR0 (η , 1) = kR0 (p , 0), BR0 x
⇔
⊥ R0 (q), R0 (p , 0) ∈ J g R0 x, R0 (η , 1) .
But then, according to Lemma 4, and Eq. (15), we deduce that α ker Hess ψwk (x, η , k) = T(x,η ,k) Cψwk , which concludes the proof of the proposition.
2
Using the preceding proposition, we are in position to apply Theorem 1 to the integral (14). Nevertheless, since the integrand in (14) also depends on the parameter ν, the derivatives of σ˜ νηn (x, η) with respect to x and η have to be examined carefully. Indeed, while the derivatives of χλ ◦ Ψμ−1 and ψλ,c ◦ Ψμ−1 behave nicely in terms of μ, the derivatives of η˜ c ◦ Ψμ−1 with respect to ξ turn out to be more delicate. Lemma 5. For all multiindices α, β, there exists a constant C > 0, which depends only on α and β, such that sup (x,η)∈X×Rn
β α ∂ ∂ σ˜ νη (x, η) C max 1, |ν|−δ(|β|+|α|) . n x η
2 Proof. With σ˜ νηn (x, η) = σνηn (x, ηn (η1 , . . . , 1)) = τνηn (x, (η1 , . . . , 1)/ν), τμ = [(ηλ,−2 χλ )2 × 2 (3 − 2ηλ,−2 χλ ) + fλ ] one computes
β α ∂ ∂ σ˜ νη (x, η) = |ν|−|α| ∂ β ∂ α τνη x, (η , 1)/ν n n x η x η (δ|β|−(1−δ)|α|)/2 Cα,β |ν|−|α| 1 + |x|2 + |η |2 + 1 /ν 2 (δ|β|−(1−δ)|α|)/2 Cα,β |ν|−δ|α| |ν|−δ|β| ν 2 + |νx|2 + |η |2 + 1 δ|β|/2 Cα,β |ν|−δ(|α|+|β|) ν 2 + |νx|2 + |η |2 + 1 . Since by (38) σμ has support in a compact set independent of μ, we obtain an estimate of order O(1) for large ν, and one of order O(ν −δ(|α|+|β|) ) for small ν. 2 β
It is interesting to note that similar bounds for ∂ξα ∂x σμ do not exist; indeed, the fact of considering only differential operators which are transversal to Reg C in the variable ξ turns out to be decisive. We can now give an asymptotic expansion for I (μ). Theorem 5. There exists a constant C > 0 independent of μ such that for all μ > 0, and all δ ∈ (1/4, 1/2), I (μ) − (2πμ)κ L0 (μ) Cμκ+1−2δ , where κ is given by Definition 3, and
R. Cassanas, P. Ramacher / Journal of Functional Analysis 256 (2009) 91–128
L0 (μ) =
1 (2π)n
χ(k)σ˜ μ (x, η , ηn ) |det ψwk (x, η , k)|N(x,η ,k) Cψwk |1/2
0|ηn |E0 Cψwk
113
dσCψwk (x, η , k)|ηn |n−κ−1 dηn .
Proof. In view of Proposition 3, we can apply Theorem 1 to the integral Iηn (ν) which was defined in (14). Consequently, for each N ∈ N, there exists a constant CN > 0 independent of ηn such that −1 κ N j |ν| Qj (ηn ) CN |ν|N sup ∂ηα ∂xβ σ˜ νηn ∞,X×Rn , Iηn (ν) − 2π|ν| |α|+|β|2N j =0
as well as constants C˜ j > 0 independent of ηn , such that Qj (ηn ) C˜ j
sup
|α|+|β|2j
α β ∂ ∂ σ˜ νη η x
n
∞,X×Rn
,
where, in particular, Q0 (ηn ) = Cψwk
χ(k)σ˜ νηn (x, η , ηn ) dσCψwk (x, η , k). (x, η , k) 1/2 |det ψwk |N(x,η ,k) Cψwk |
Now, by the previous lemma, for |ν| 1 one has sup
|α|+|β|2N
α β ∂ ∂ σ˜ νη η x
n
∞,X×Rn
c1 |ν|−2N δ ,
where c1 is some constant depending only on N . Thus, if |ν| 1, we obtain Iη (ν) − 2π|ν| κ Q0 (ηn ) n # " −1 N −1 κ N j j = Iηn (ν) − 2π|ν| |ν| Qj (ηn ) − |ν| Qj (ηn ) j =0
CN |ν|
N
sup
|α+β|2N
j =1
−1 κ N j + 2π|ν| |ν| Qj (ηn ) ∞,X×Rn
α β ∂ ∂ σ˜ νη η x
n
j =1
c2 |ν|N (1−2δ) + c3 |ν|κ
N −1
|ν|j (1−2δ)
j =1
with constants ci > 0. Next, let us fix ε > 0, and write I (μ) = J1 (μ) + J2 (μ), where
(18)
114
R. Cassanas, P. Ramacher / Journal of Functional Analysis 256 (2009) 91–128
J1 (μ) =
Iηn (μ/ηn )|ηn |n−1 dηn ,
ε|ηn |E0
J2 (μ) =
Iηn (μ/ηn )|ηn |n−1 dηn .
|ηn |ε
Since Iηn (μ) is uniformly bounded in ηn and μ, J2 (μ) c4 ε n ,
(19)
where c4 is independent of ηn and μ. Now, according to Eq. (18), if ε μ, then J1 (μ) − (2πμ)κ
Q0 (ηn )|ηn |n−1−κ dηn
ε|ηn |E0
C1
N −1
j =1
+ C2 μ
|ηn |n−1−κ−j (1−2δ) dηn
μκ+j (1−2δ) ε|ηn |E0
|ηn |n−1−N (1−2δ) dηn
N (1−2δ) ε|ηn |E0
for some constants Ci > 0. One easily computes that
|ηn |n−1−κ−j (1−2δ) dηn C3 max 1, ε n−κ−j (1−2δ) ,
ε|ηn |E0
|ηn |
n−1−N (1−2δ)
dηn C3 max 1, ε n−N (1−2δ) ,
ε|ηn |E0
so that if we take ε = μ, which ensures that |ν| 1 for J1 (μ), we obtain n−1−κ J1 (μ) − (2πμ)κ Q (η )|η | dη 0 n n n μ|ηn |E0
C1 max μκ+1−2δ , μn + C2 max μN (1−2δ) , μn . As the dimension of an orbit of G ⊂ O(n) in Rn is at most n − 1, one necessarily has κ n − 1, yielding μn = O(μκ+1 ) as μ goes to zero. Therefore, by choosing N large enough, and taking equation (19) together with κ
(2πμ)
0|ηn |μ
into account, one gets
Q0 (ηn )|ηn |n−1−κ dηn = O μn
R. Cassanas, P. Ramacher / Journal of Functional Analysis 256 (2009) 91–128
I (μ) − (2πμ)κ
Q0 (ηn )|ηn |
n−1−κ
115
dηn Cμκ+1−2δ .
0|ηn |E0
The proof of the theorem is now complete.
2
Remark 3. Note that the strict transform of the critical set C of ψ is locally given by C˜st = (x, η, k) ∈ R2n × G: R0 x, R0 (η , 1), k ∈ Reg C Cψwk × R. The first coefficient in the expansion of Theorem 5 can therefore also be expressed as L0 (μ) =
1 (2π)n
C˜st
χ(k)σ˜ μ (x, η)|ηn |n−κ−1 dσC˜ (x, η, k). (x, η , k) 1/2 |det ψwk |N(x,η ,k) Cwk |
(20)
5. Computation of the leading term In this section, we shall address the question of computing the leading coefficient L0 (μ) in the expansion of I (μ). The main result of this section is the following Proposition 4. One has L0 (μ) =
1 [χ |H0 : 1] (2π)n
Reg Ω0
σμ (z)
dσReg Ω0 (z) , vol Oz
(21)
where dσReg Ω0 is the Riemannian measure on Reg Ω0 , and vol Oz denotes the Riemannian volume of the G-orbit of z. [χ |H0 : 1] stands for the multiplicity of the trivial representation in the ˆ to a prinrestriction of any irreducible representation χ corresponding to the character χ ∈ G cipal isotropy group H0 . In particular, the integral on the right-hand side of (21) is convergent. Note that Reg Ω0 is not compact; nevertheless, the existence of the integral in (21) will be deduced on basis of the partial desingularization of C accomplished in the previous section. Let us start proving Proposition 4, and introduce first certain cut-off functions for Sing Ω0 . Definition 4. Let K be compact subset in R2n as in (38), ε > 0, and denote by vε the characteristic function of the set (Sing Ω0 ∩ K)2ε = z ∈ R2n : |z − z | < 2ε for some z ∈ Sing Ω0 ∩ K . 2n ∞ Consider further the unit ball B1 in R , and a function ι ∈ Cc (B1 ) with −2n ιε (z) = ε ι(z/ε). Clearly ιε dz = 1, supp ιε ⊂ Bε , and we define
uε = vε ∗ ιε .
ι dz = 1, and set
116
R. Cassanas, P. Ramacher / Journal of Functional Analysis 256 (2009) 91–128
One can then show that uε ∈ C∞ c ((Sing Ω0 ∩ K)3ε ), and uε = 1 on (Sing Ω0 ∩ K)ε , together with α ∂ uε Cα ε −|α| , z
where Cα is a constant which depends only on α and n, see Hörmander [15, Theorem 1.4.1]. Next, we shall prove 2n Lemma 6. Let α ∈ C∞ c (R ). Then the limit
lim
ε→0 Reg C
χ(k)[α(1 − uε )](z) dσReg C (z, k) |det ψ (z, k)|N(z,k) Reg C |1/2
exists and is finite. In particular, one has L0 (μ) =
1 lim (2π)n ε→0
Reg C
χ(k)[σμ (1 − uε )](z) dσReg C (z, k), |det ψ (z, k)|N(z,k) Reg C |1/2
(22)
where dσReg C is the Riemannian measure on Reg C. Proof. With uε as in the previous definition, let us define
i
e μ ψ(x,ξ,k) χ(k) α(1 − uε ) (x, ξ ) dx dξ ¯ dk.
Iε (μ) = G X Rn
Since (x, ξ, k) ∈ Sing C implies (x, ξ ) ∈ Sing Ω0 , a direct application of the generalized theorem of the stationary phase for fixed ε > 0 gives Iε (μ) − (2πμ)κ L0 (μ, ε) Cε μκ+1−2δ
(23)
for some δ ∈ [0, 1/2), where Cε > 0 is a constant depending only on ε, and 1 L0 (μ, ε) = (2π)n
Reg C
χ(k)[α(1 − uε )](z) dσReg C (z, k). |det ψ (z, k)|N(z,k) Reg C |1/2
If α is independent of μ, on has δ = 0. For α = σμ , the stationary phase theorem has to be applied on G × X × S n−1 , and δ ∈ (1/4, 1/2). On the other hand, applying Theorem 5 to Iε (μ) instead of I (μ), we obtain again an asymptotic expansion of the form (23) for Iε (μ), where now, according to (20), the first coefficient is given by 1 L0 (μ, ε) = (2π)n
C˜st
χ(k)[α(1 − uε ) ◦ ζM ◦ ϕ˜ n ](x, η)|ηn |n−κ−1 dσC˜ (x, η, k). (x, η , k) 1/2 |det ψwk |N(x,η ,k) Cwk |
R. Cassanas, P. Ramacher / Journal of Functional Analysis 256 (2009) 91–128
117
Since the first term in the asymptotic expansion (23) is uniquely determined, the two expressions for L0 (μ, ε) must be identical. The statement of the lemma now follows by the Lebesgue theorem on bounded convergence, by which, in particular, 1 ε→0 (2π)n
lim
C˜st
χ(k)[σμ (1 − uε ) ◦ ζM ◦ ϕ˜n ](x, η)|ηn |n−κ−1 dσC˜ (x, η, k) = L0 (μ). (x, η , k) 1/2 |det ψwk |N(x,η ,k) Cwk |
2
Remark 4. Note that existence of the limit in (22) has been established by partially resolving the singularities of the critical set C, the corresponding limit being given by the absolutely convergent integral (20). The proof of the next lemma is mainly algebraic. Lemma 7. Let α be a smooth, compactly supported function on Reg Ω0 . Then Reg C
χ(k)α(z) dσReg C (z, k) = [χ |H0 : 1] |det ψ (z, k)|N(z,k) Reg C |1/2
α(z)
Reg Ω0
dσReg Ω0 (z) . Vol Oz
Proof. The main difficulty consists in computing the determinant of the transversal Hessian, which will be accomplished by recuring to previous computations done in [4]. Thus, let (z, k) be a fixed point in Reg C, and choose an appropriate basis (A1 , . . . , Ad ) for g as follows. If κ denotes the dimension of Oz , let (A1 , . . . , Aκ ) be an orthonormal basis of (Te Gz )⊥ , (Aκ+1 , . . . , Ad ) be an orthonormal basis of Te Gz , where orthogonality is defined with respect to the scalar product A, B = tr( tAB) for arbitrary linear maps A and B in Rn . From [4] we recall that
ψ (z, k) |N(z,k) Reg C A|F ⊥ = det , det i i where A = Hess ψ(z, k) denotes the Hessian of ψ with respect to the coordinates (z, s) → (z, exp( di=1 si Ai )k), and $ F = (α, s) ∈ R
2n
× R : (k − 1)α + d
d
% si Ai z = 0 .
(24)
i=1
Next, let (B1 , . . . , Bκ ) be in g such that (B1 z, . . . , Bκ z) is an orthonormal basis of gz. For j = 1, . . . , κ, we define εj = k −1 − 1 Bj z, Ai z, Bj z , 0 (i = 1, . . . , κ). (25) εj = (J Bj z, 0),
118
R. Cassanas, P. Ramacher / Journal of Functional Analysis 256 (2009) 91–128
Then (ε, ε ) constitutes a basis of F ⊥ , see [4, Lemma 3.3]. In what follows, we shall compute A|F ⊥ in this basis. Writing αj = (kBj x, Bj ξ ) we find Aεj =
κ −1 k − 1 (1 − Πgz )αj , 0 + αj , Br z εj ,
(26)
r=1
where Πgz is the orthogonal projection onto the space gz in R2n . We state now certain relations that will be crucial for the rest of the computation. For all (z, k) ∈ Reg C, we have [k, Πgz ] = 0, [J, k] = 0.
rank (k − 1)(1 − Πgz ) ⊂ J gz.
(27) (28)
The first equality follows easily from the relations k −1 gk = g and kz = z, while the second simply says that k is symplectic as a Hamiltonian action in R2n . In order to establish (28), we differentiate the identity π(kz) = π(z) with respect to z ∈ Ω0 , and obtain (k − 1)α ∈ ker dz π = gz for all α in Tz Ω0 , where π denotes the canonical projection of R2n (H0 ) onto the quotient by G. The inclusion (28) now follows by using (12). Coming back to (26), we get Aεj =
κ κ − J k −1 − 1 αj , Br z εr + αj , Br z εr . r=1
r=1
Using (16), and the fact that (B1 z, . . . , Bκ z) is orthonormal, we obtain Aεj =
κ
κ
(1 − k) 1 − k −1 Bj x, Br ξ εr + (k − 1)Bj x, Br x − δj r εr ,
r=1
(29)
r=1
where δj r is the Kronecker symbol. In the same way we obtain Aεj
& ' κ κ −1 1 −1 = − J k − 1 βj + k + I Cj z, Br z εr + βj , Br z εr , 2 r=1
r=1
where Cj =
κ Ar z, Bj z Ar , r=1
1 βj = k −1 − 1 (−kBj ξ, Bj x) − (Cj ξ, Cj x). 2
Let now f : gz → gz be defined by f (˜z) =
κ Ar z, z˜ Ar z, r=1
∀˜z ∈ gz,
(30)
R. Cassanas, P. Ramacher / Journal of Functional Analysis 256 (2009) 91–128
119
and let Λ = (k − 1) k −1 − 1 + f |
gz
be the restriction of the map (k − 1)(k −1 − 1) + f to gz. Note that Λ plays a crucial part in the computations of [4]. Using again (16), one easily gets Aεj
=
κ & −1 k
0
r=1
' 1n ΛBj , Br z εr , 0
' κ & 0 0 ΛBj z, Br z εr − 0 1n r=1
where the matrices have an obvious meaning. Together with (29), the last equation implies that the matrix of A in the basis (ε, ε ) is given by
(1 − k)(1 − k −1 )Bj x, Bi ξ (k − 1)Bj x, Bi x − δij
k −1
0 ΛBj z, Bi z 0 011n − 0 0n ΛBj , Bi z
(31)
.
Let Λ0 be the matrix of Λ in the basis (B1 z, . . . , Bκ z). Then (31) is equal to
(1 − k)(1 − k −1 )Bj x, Bi ξ (k − 1)Bj x, Bi x − δij
k −1
1κ . 0
0 Bj z, Bi z 00 11n n − 0 0 Bj z, Bi z
0 Λ0
.
Multiplying by i, and shifting the two columns, we obtain
ψ (z, k)| N(z,k)C 0 = det(Λ) · D, det i where D = det
(k −1 − 1)Bj x, Bi x + δij (k − 1)(k −1 − 1)Bj ξ, Bi x − Bj ξ, Bi x (k − 1)Bj x, Bi x + δij
.
(32)
We are going to show that D = 1. For this, we introduce the notation U = (B1 x| . . . |Bκ x),
V = (B1 ξ | . . . |Bκ ξ ),
where Bj x is taken as a column vector in the canonical basis of Rn . U and V are therefore matrices of size n × κ. Lemma 8. For all k ∈ G we have (a) (b) (c) (d) (e)
+ t V V = 1κ ; = tV U; k commutes with U t U , V t V , U t V , and V t U ; (k − 1)U t V = (k − 1)V t U ; (k − 1)(U t U + V t V ) = k − 1.
tUU tUV
120
R. Cassanas, P. Ramacher / Journal of Functional Analysis 256 (2009) 91–128
Proof. Part (a) says that (B1 z, . . . , Bn z) is orthonormal. Part (b) comes from (17). Next, let us denote by X the matrix X = (B1 z| . . . |Bκ z). Then X t X is the matrix of Πgz in the canonical basis of R2n . Moreover,
t U U UtV t . X X= V tU V tV Therefore the property [Πgz , k] = 0, see (27), is equivalent to (c). The two last properties are more subtle. One has to note that (27) is equivalent to Πgz (k − 1)J (1 − Πgz ) = (k − 1)J (1 − Πgz ). By expressing this in terms of matrices, one easily obtains (d) and (e).
2
Coming back to the proof of Lemma 7, we rewrite Eq. (32) as D = det
t
U (k − 1)U + 1κ t V (k −1 − 1)(k − 1)U t U (k −1 − 1)U + 1 −t V U κ
= det
a c
b d
,
where we replaced k −1 by k. We claim that the blocks c and d commute. Indeed, cd = −t V U t U k −1 − 1 U − t V U, dc = −t U k −1 − 1 U t V U − t V U = −t U k −1 − 1 V t U U − t V U, by (d) of Lemma 8. By (c) of Lemma 8, (k −1 − 1) commutes with V t U , and since t U V = t V U , by (b), we get [c, d] = 0. Therefore, D = det(ad − bc). Using (a) and (d) of Lemma 8, it is then a straightforward computation to show that in fact, ad − bd = 1κ , yielding D = 1. We have thus shown the equality ψ (z, k)|
N(z,k)C 0 det = det (k − 1) k −1 − 1 | + f , gz i where the map f : gz → gz was defined in (30). The rest of the proof of Lemma 7 now follows by the argument given in [4, Section 3.3.2]. 2 To finish proving Proposition 4, we note that, as a consequence of Lemmata 6 and 7, the limit lim
ε→0 Reg Ω0
dσReg Ω0 (z) α(1 − uε ) (z) Vol Oz
2n exists for any α ∈ C∞ c (R ) and is finite. Assume now that α is non-negative. Since |uε | 1, the Lemma of Fatou implies
Reg Ω0
dσReg Ω0 (z) lim α(1 − uε ) (z) lim ε→0 ε→0 Vol Oz
Reg Ω0
dσReg Ω0 (z) α(1 − uε ) (z) < ∞, Vol Oz
R. Cassanas, P. Ramacher / Journal of Functional Analysis 256 (2009) 91–128
121
which means that α(z) Reg Ω0
2n dσReg Ω0 (z) < ∞ ∀α ∈ C∞ c R , R+ . Vol Oz
(33)
In particular, if α is taken to be equal 1 on the compact set K specified in (38), we obtain Reg Ω0
σμ (z) dσReg Ω0 (z) C Vol Oz
α(z)
Reg Ω0
dσReg Ω0 (z) <∞ Vol Oz
(34)
for some C > 0. Now, by Lemmata 6 and 7, L0 (μ) =
1 [χ |H0 : 1] lim ε→0 (2π)n
Reg Ω0
dσReg Ω0 (z) σμ (1 − uε ) (z) . Vol Oz
(35)
Since (34) implies that the integrand in (35) has an integrable majorant for arbitrary ε, we can apply the Lebesgue theorem of bounded convergence to obtain L0 (μ) =
1 [χ |H0 : 1] (2π)n
Reg Ω0
σμ (z)
dσReg Ω0 (z) . Vol Oz
2
This completes the proof of Proposition 4.
So far we have shown that tr Pχ Eλ = dχ λn/2m I (λ−1/2m ) + O(1), where I (μ) =
μκ [χ |H0 : 1] (2π)n−κ
Reg Ω0
σμ (z)
dσReg Ω0 (z) + O μκ+1−2δ , Vol Oz
(36)
2 2 χλ )2 (3 − 2ηλ,−2 χλ ) + fλ ] ◦ Ψμ−1 with λ = μ−2m . In particular, δ ∈ (1/4, 1/2), and σμ = [(ηλ,−2 the last integral exists, and is finite, so that in order to finish the computation of the leading term in the asymptotic expansion for tr Pχ Eλ , we are left with the task of examining the latter integral. To characterize the support of σμ , let us introduce the sets
Wλ = (x, ξ ) ∈ X × Rn : aλ < 0 , Bc,λ = X × Rn − Ac,λ , Ac,λ = (x, ξ ) ∈ X × Rn : aλ < c hδ−ω + d , Dc = ∂X × Rn c, h−2δ g , Fλ = (x, ξ ) ∈ X × Rn : χλ = 0 or ηλ,−2 = 0 or χλ = ηλ,−2 = 1 , RV c,λ = (x, ξ ) ∈ X × Rn : |aλ | < c hδ−ω + d ∪ (x, ξ ) ∈ Dc : x ∈ X, aλ < c hδ−ω + d . Note that −δ/2 √ Dc = (x, ξ ) ∈ R2n : dist(x, ∂X) < c 1 + |x|2 + |ξ |2 ,
122
R. Cassanas, P. Ramacher / Journal of Functional Analysis 256 (2009) 91–128
since for −2δ
h
δ (x, ξ )g(x,ξ ) (x − y, ξ − η) = 1 + |x|2 + |ξ |2
(
) |ξ − η|2 2 + |x − y| < c 1 + |x|2 + |ξ |2
to hold for some (y, η) ∈ ∂X × Rn , it is necessary and sufficient that |x − y|2 (1 + |x|2 + |ξ |2 )δ < c is satisfied for some y ∈ ∂X. Lemma 9. For sufficiently large c > 0 one has: (i) supp fλ ⊂ RV c,λ ⊂ Ac,λ ; 2 2 (ii) supp(ηλ,−2 χλ )2 (3 − 2ηλ,−2 χλ ) ⊂ Ac,λ ; 2 2 2 (iii) (ηλ,−2 χλ ) (3 − 2ηλ,−2 χλ ) = 1 on Wλ ∩ X×Rn RV c,λ . Proof. As already explained in Part I, Eq. (51), the support of fλ is contained in X×Rn Fλ , the complement of Fλ in X × Rn . Furthermore, for sufficiently large c > 0, the set X×Rn Fλ is contained in RV c,λ , which is a consequence of the inclusions X×Rn Fλ ⊂ Ac,λ ∩ X×Rn Eλ ⊂ RV c,λ ,
(37)
/ D4 , aλ < −4hδ−ω − 8C0 d}, see Part I, Lemma 16. Next, where Eλ = {(x, ξ ) ∈ X × Rn : (x, ξ ) ∈ 2 2 2 we note that (ηλ,−2 χλ ) (3 − 2ηλ,−2 χλ )(x, ξ ) must be equal 1 on Wλ ∩ X×Rn RV c,λ , since according to (37) we have the inclusion X×Rn RV c,λ ⊂ Bc,λ ∪ Eλ , and hence Wλ ∩ X×Rn RV c,λ ⊂ Eλ ⊂ {(x, ξ ) ∈ X × Rn : χλ = ηλ,−2 = 1}, due to the fact that Wλ ∩ Bc,λ = ∅. Furthermore, 2 2 (ηλ,−2 χλ )2 (3 − 2ηλ,−2 χλ )(x, ξ ) vanishes on Bc,λ , since for large c, (x, ξ ) ∈ Bc,λ implies −2δ (x, ξ ) ∈ / Mλ (1, h g), by the proof of the previous lemma. 2 Consequently, by introducing the sets *μ = Ψμ (Wμ−2m ) = (x, ξ ) ∈ X × Rn : bμ−2m < 0 , W *c,μ = Ψμ (Ac,μ−2m ) = (x, ξ ) ∈ X × Rn : bμ−2m < c hδ−ω + d ◦ Ψμ−1 , A *c,μ , *c,μ = X × Rn − A B + c,μ = Ψμ (RV c,μ−2m ) RV = (x, ξ ) ∈ X × Rn : |bμ−2m | < c hδ−ω + d ◦ Ψμ−1 ∪ (x, ξ ) ∈ X × Rn : (x, ξ/μ) ∈ Dc , bμ−2m < c hδ−ω + d ◦ Ψμ−1 , one sees that, for all μ ∈ R+ ∗, supp σμ ⊂ A˜ c,μ ⊂ K
(38)
for some sufficiently large c > 0, and some suitable compact subset K ⊂ R2n . We proceed now to split the integral in (36) into the three integrals
R. Cassanas, P. Ramacher / Journal of Functional Analysis 256 (2009) 91–128
*μ Reg Ω0 ∩W
dσReg Ω0 (z) − vol Oz
+ c,μ *μ ∩RV Reg Ω0 ∩W
dσReg Ω0 (z) + vol Oz
+ c,μ Reg Ω0 ∩RV
σμ (z)
123
dσReg Ω0 (z) , (39) vol Oz
where we made use of the fact that, since Wλ , RV c,λ are contained in Ac,λ , and Ac,λ RV c,λ ⊂ Wλ , one has Ac,λ − Wλ ∩ X×Rn RV c,λ = RV c,λ . The next lemma will show that the main contribution to L0 (μ) is actually given by the first integral in (39), provided that we make the following assumption. Assumption 2. There exists a constant c > 0 such that for sufficiently small > 0, vol(∂X) c. Furthermore, 0 ∈ / ∂X. Lemma 10. Put δ−ω n + (1) RV + d ◦ Ψμ−1 , c,μ = (x, ξ ) ∈ X × R : |b| < c h δ−ω n + (2) RV + d ◦ Ψμ−1 , c,μ = (x, ξ ) ∈ X × R : (x, ξ/μ) ∈ Dc , b < c h + c,μ = RV + c,μ ∪ RV + c,μ . Then, as μ → 0, so that RV (1)
(2)
+ (1) Reg Ω0 ∩RV c,μ
+ (2) Reg Ω0 ∩RV c,μ
dσReg Ω0 (z) 1 = O μ2δ− 2 , vol Oz δ dσReg Ω0 (z) = O μ 1+δ , vol Oz
for arbitrary δ ∈ (1/4, 1/2). Proof. Let 1A denote the characteristic function of the set A. As already noted, Ω0 is homogeneous in x and ξ , meaning that (x, ξ ) ∈ Ω0 implies (sx, tξ ) ∈ Ω0 for all s, t ∈ R. Furthermore, by Lemma 3, {(x, ξ ) ∈ Reg Ω0 : ξ ∈ Sing Rn } is a subset of measure zero in Reg Ω0 . Consequently, we can parametrize Reg Ω0 up to a set of measure zero as follows. Take z = (x, ξ ) ∈ Ω0 , ξ ∈ Rn(H0 ) , and let ξ = sη, x = rϑ be polar coordinates in Rn , and Nξ Oξ , respectively, where r, s > 0, and η ∈ S n−1 , ϑ ∈ S n−κ−1 . In this coordinates one computes then + c,μ Reg Ω0 ∩RV
dσReg Ω0 (z) vol Oz
=
1RV + c,μ (x, ξ ) Nξ O ξ
Rn(H ) 0
∞ ∞
= 0 S n−1 (H ) 0
1 O 0 Nsη sη
dσNξ Oξ (x) vol O(x,ξ )
dξ
dr dϑ n−1 n−κ−1 1RV r ds dη, + c,μ (rϑ, sη)s vol O(rϑ,sη)
(40)
124
R. Cassanas, P. Ramacher / Journal of Functional Analysis 256 (2009) 91–128
since
det g| Reg Ω0 (r, s, ϑ, η) = s n−1 r n−κ−1 dϑ dη,
where g| Reg Ω0 denotes the induced metric on Reg Ω0 , and dη and dϑ are the volume elements of S n−1 and Nξ1 Oξ = {v ∈ Nξ Oξ : v = 1}, respectively. Now, the condition b(x, ξ ) < c(hδ−ω + d)(x, ξ/μ) implies that |ξ | < c1 , see Part I, Eq. (60); here, and in what follows, ci > 0 will denote positive constants. Hence, n ε2 −δ δ + (2) RV c,μ ⊂ (x, ξ ) ∈ X × R : c0 μ |ξ | c1 , dist(x, ∂X) < c2 |ξ | μ ∪ (x, ξ ) ∈ X × Rn : |ξ | < c0 με2
⊂ (∂X)c3 μδ(1−ε2 ) × B n (c1 ) ∪ X × B n c0 με2 , where B n () denotes the ball of radius in n-dimensional Euclidean space, and 1 > ε2 > 0 will be chosen later. On the other hand, the proof of Lemma 18 in Part I implies that, for small μ, and some 0 < ε1 < 1 to be specified later,
n δ−ω + (1) RV ∪ X × B n με1 . c,μ ⊂ (x, ξ ) ∈ X × R : c4 |ξ | c1 , 1 − 1/a2m (x, ξ ) c5 μ Now, using the parametrization of Reg Ω0 specified above, one sees that for small > 0 Reg Ω0 ∩[X×B n ()]
dσReg Ω0 (z) = vol Oz
1X (x)
0 S n−1 (H ) 0
dσNsη Osη (x)
Nsη Osη
vol O(x,sη)
s n−1 ds dη = O(),
where we took into account took that vol O(x,sη) is at most of order s κ for small s, and κ n − 1. + (1) Therefore, the restriction of the integral (40) to Reg Ω0 ∩ RV c,μ can be estimated from above by 1{(x,ξ )∈X×Rn : c4 |ξ |c1 , 1−1/a2m (x,ξ ) c5 μδ−ω } (z) Reg Ω0
dσReg Ω0 (z) + O με1 . vol Oz
Now, by letting x ∈ Rn(H0 ) , ξ ∈ Nx Ox , and interchanging the roles of x and ξ , we obtain c1
X∩Rn(H
c4 N 1 O x x
0)
s n−κ−1 ds dη 1{(x ,ξ ): |1−1/a2m (x ,ξ )|c5 μδ−ω } (x, sη) dx vol O(x,sη)
= X∩Rn(H
0)
ς −1
{ς: |ς−1|c5 μδ−ω } Nx1 Ox
c6 {ς: |ς−1|c5 μδ−ω }
dς = O μδ−ω ,
1 ςa2m (x, η)
n−κ 2m
1 1[c1 ,c4 ] ((ςa2m (x, η))− 2m ) dς dη dx vol O(x,(ςa2m (x,η))−1/2m η)
R. Cassanas, P. Ramacher / Journal of Functional Analysis 256 (2009) 91–128
125
where we made the change of variables ς = |ξ |−2m /a2m (x, ξ/|ξ |) = s −2m /a2m (x, η), and used the fact that (1 + z)β − (1 − z)β = O(|z|) for arbitrary z ∈ C, |z| < 1, and β ∈ R. Note that due to the ellipticity condition (3), a2m (x, η) C0 > 0 for all η ∈ Nx1 Ox and x ∈ X. Putting ε1 = δ − ω = 2δ − 1/2 therefore yields dσReg Ω0 (z) 1 = O μ2δ− 2 . vol Oz + Reg Ω0 ∩RV c,μ (1)
Let us now assume that Assumption 2 is fulfilled. Then, for small μ, the restriction of the integral + (2) (40) to Reg Ω0 ∩ RV c,μ can be estimated from above by 1[(∂X) Reg Ω0
c3 μδ(1−ε2 )
×B n (c1 )] (z)
dσReg Ω0 (z) + O με2 vol Oz
= Rn(H ) ∩(∂X) 0
c3 μδ(1−ε2 )
Nx Ox ∩B n (c1 )
dσNx Ox (ξ ) dx + O με2 vol O(x,ξ )
δ c7 vol(∂X)c3 μδ(1−ε2 ) + O με2 = O μδ(1−ε2 ) + O με2 = O μ 1+δ , / ∂X, the integrand of the last where we put ε2 = δ/(1 + δ), and took into account that, since 0 ∈ integral over x is bounded on Rn(H0 ) ∩ (∂X)c3 μδ(1−ε2 ) by some constant independent of μ. The assertion of the lemma now follows. 2 Now, for x ∈ X, |ξ | > μ, the condition bμ−2m (x, ξ ) < 0 is equivalent to a2m (x, ξ ) < 1, due to the ellipticity condition (3), and Eq. (7). By using arguments similar to those given in the proof of the previous lemma one therefore computes dσReg Ω0 (z) dσReg Ω0 (z) dσReg Ω0 (z) + 1(−∞,1] a2m (z) vol Oz vol Oz vol Oz *μ Reg Ω0 ∩W
Reg Ω0 ∩[X×B n (μ)]
= O(μ) +
Reg Ω0
1(−∞,1] a2m [z] dσReg Ω0 /G [z]
Reg Ω0 /G
−1
= O(μ) + vol a2m (−∞, 1] ∩ Reg Ω0 /G ,
(41)
where we took into account [4, Eq. (3.37)]. Here the latter volume is defined in the sense of [10, Section 3.H.2]. This finishes the computation of the leading term. Collecting everything together, we obtain Proposition 5. As λ → +∞, one has
tr Pχ Eλ − dχ [χ|H0 : 1] vol a −1 (−∞, 1] ∩ Ω0 /G λ(n−κ)/2m = O λ(n−κ−1/4)/2m , 2m (2π)n−κ Furthermore, a similar result holds for the trace of (Pχ Eλ )2 , too.
126
R. Cassanas, P. Ramacher / Journal of Functional Analysis 256 (2009) 91–128
Proof. Since tr Pχ Eλ = dχ λn/2m I (λ−1/2m ) + O(1), the assertion follows with Theorem 5 and Proposition 4, together with Eqs. (39), (41), and Lemma 10, by taking into account that
1 1 δ , 1 − 2δ, 2δ − = . max min 1+δ 2 4 δ∈(1/4,1/2) Finally, if in all the previous computations Eλ is replaced by Eλ2 , we obtain a similar estimate for the trace of Pχ Eλ · Pχ Eλ = Pχ Eλ2 . 2 6. Proof of the main result As a consequence of Lemma 11 of Part I, and Proposition 5, we get the following. Theorem 6. Let NχEλ be the number of eigenvalues of Eλ which are 1/2 and whose eigenfunctions are contained in the χ -isotypic component Hχ of L2 (Rn ), and assume that Assumptions 1 and 2 are satisfied. Then E
N λ − dχ [χ|H0 : 1] vol a −1 (−∞, 1] ∩ Ω0 /G λ(n−κ)/2m = O λ(n−κ−1/4)/2m , 2m χ n−κ (2π) as λ → +∞. Similar estimates for the traces of F˜ λ and Fλ can be derived as well, and using Lemma 12 of Part I we obtain the following. Theorem 7. Let MχFλ be the number of eigenvalues of Fλ which are 1/2 and whose eigenfunctions are contained in the χ -isotypic component Hχ of L2 (Rn ). Under Assumptions 1 and 2 one has then F
M λ − dχ [χ|H0 : 1] vol a −1 (−∞, 1] ∩ Ω0 /G λ(n−κ)/2m = O λ(n−κ−1/4)/2m , 2m χ n−κ (2π) as λ → +∞. Proof. The proof is similar to the one of Theorem 6; in analogy to Eq. (9) one has 2 σ l (Fλ ) = η22 χλ+ 3 − 2η22 χλ+ + fλ + rλ , where rλ ∈ S −∞ (h−2δ g, 1), and fλ ∈ S(h−2δ g, h1−2δ ), everything uniformly in λ. Again we |rλ (x, ξ )| dx dξ have supp fλ ⊂ RV c,λ for sufficiently large c, and ¯ C for some constant C > 0 independent of λ, so that in order to study the asymptotic behavior of tr Pχ Fλ , we can restrict ourselves to the integral 2 χ(k)ei(x−kx)ξ η22 χλ+ 3 − 2η22 χλ+ + fλ (x, ξ ) dx dξ ¯ dk. G
An application of the method of the stationary phase then yields the desired result.
2
R. Cassanas, P. Ramacher / Journal of Functional Analysis 256 (2009) 91–128
127
We are now in position to prove our main result. In the case G = {1}, one has Ω0 = R2n , and we simply obtain [18, Theorem 13.1]. Theorem 8. Let G be a compact group of isometries in Euclidean space Rn , H0 a principal isotropy group, and X ⊂ Rn a bounded open set invariant under G. Assume that: (i) for sufficiently small > 0, vol(∂X) c, where c > 0 is a constant independent of , and 0∈ / ∂X; (ii) the set Sing Rn = Rn \ Rn(H0 ) is included in a strict vector subspace F of Rn of dimension r < n. Let further A0 be a symmetric, classical pseudodifferential operator in L2 (Rn ) of order 2m with principal symbol a2m that commutes with the regular representation of G in L2 (Rn ), and assume that A0 satisfies the ellipticity condition (3). Consider further the Friedrichs extension of the operator 2 res ◦ A0 ◦ ext : C∞ c (X) → L (X),
and denote it by A. Then A has discrete spectrum. Furthermore, if Nχ (λ) denotes the number of eigenvalues of A less or equal λ and with eigenfunctions in the χ -isotypic component res Hχ of L2 (X), and κ = dim H0 , then Nχ (λ) =
−1 dχ [χ|H0 : 1] (−∞, 1] ∩ Ω0 /G λ(n−κ)/2m + O λ(n−κ−1/4)/2m , vol a2m n−κ (2π)
where dχ denotes the dimension of any unitary irreducible representation χ corresponding to the character χ , and [χ|H0 : 1] is the multiplicity of the trivial representation in the restriction of χ to H0 . Proof. The discreteness of the spectrum was already shown in Proposition 1. Now, by Theorems 4 and 5 of Part I, there exist constants Ci > 0 independent of λ such that Fλ NχEλ − C1 N A0 − λ1, Hχ ∩ C∞ c (X) Mχ + C2 . Theorems 6 and 7 then yield the estimate
Nχ (λ) − dχ [χ|H0 : 1] vol a −1 (−∞, 1] ∩ Ω0 /G λ(n−κ)/2m = O λ(n−κ−1/4)/2m . 2m n−κ (2π)
2
Remark 5. Note that if G is a finite group, we recover exactly the first term in the asymptotic expansion for Nχ (λ) given in Part I, since in this case Ω0 = R2n , H0 = {1}, [χ|H0 : 1] = dχ , −1 −1 and κ = 0, while vol(a2m ((−∞, 1])/G) = vol(a2m ((−∞, 1]))/|G|. References [1] G.E. Bredon, Introduction to Compact Transformation Groups, Pure Appl. Math., vol. 46, Academic Press, New York, 1972.
128
R. Cassanas, P. Ramacher / Journal of Functional Analysis 256 (2009) 91–128
[2] M. Bronstein, V. Ivrii, Sharp spectral asymptotics for operators with irregular coefficients I. Pushing the limits, Comm. Partial Differential Equations 28 (2003) 83–102. [3] J. Brüning, E. Heintze, Representations of compact Lie groups and elliptic operators, Invent. Math. 50 (1979) 169– 203. [4] R. Cassanas, Reduced Gutzwiller formula with symmetry: Case of a Lie group, J. Math. Pures Appl. 85 (2006) 719–742. [5] M. Combescure, J. Ralston, D. Robert, A proof of the Gutzwiller semiclassical trace formula using coherent states decomposition, Comm. Math. Phys. 202 (1999) 463–480. [6] H. Donnelly, G-spaces, the asymptotic splitting of L2 (M) into irreducibles, Math. Ann. 237 (1978) 23–40. [7] Z. El Houakmi, B. Helffer, Comportement semi-classique en présence de symétries: action d’un groupe de Lie compact, Asymptotic Anal. 5 (2) (1991) 91–113. [8] C. Emmrich, H. Römer, Orbifolds as configuration spaces of systems with gauge symmetries, Comm. Math. Phys. 129 (1) (1990) 69–94. [9] V.I. Feigin, Asymptotic distribution of eigenvalues for hypoelliptic systems in Rn , Math. USSR Sb. 28 (4) (1976) 533–552. [10] S. Gallot, D. Hulin, J. Lafontaine, Riemannian Geometry, third ed., Springer-Verlag, Berlin, 2004. [11] A. Grigis, J. Sjöstrand, Microlocal Analysis for Differential Operators, London Math. Soc. Lecture Note Ser., vol. 196, Cambridge Univ. Press, Cambridge, 1994. [12] B. Helffer, D. Robert, Etude du spectre pour un opératour globalement elliptique dont le symbole de Weyl présente des symétries II, Amer. J. Math. 108 (1986) 973–1000. [13] L. Hörmander, The spectral function of an elliptic operator, Acta Math. 121 (1968) 193–218. [14] L. Hörmander, The Weyl calculus of pseudo-differential operators, Comm. Pure Appl. Math. 32 (1979) 359–443. [15] L. Hörmander, The Analysis of Linear Partial Differential Operators, vol. I, Springer-Verlag, Berlin, 1983. [16] V. Ivrii, Sharp spectral asymptotics for operators with irregular coefficients II. Domains with boundaries and degenerations, Comm. Partial Differential Equations 28 (2003) 103–128. [17] K. Kawakubo, The Theory of Transformation Groups, Clarendon/Oxford Univ. Press, New York, 1991. [18] S.Z. Levendorskii, Asymptotic Distribution of Eigenvalues, Kluwer Academic Publ., Dordrecht, 1990. [19] D. Montgomery, H. Samelson, Transformation groups of spheres, Ann. of Math. 44 (1943) 457–470. [20] J.P. Ortega, T.S. Ratiu, Momentum Maps and Hamiltonian Reduction, Progr. in Math., vol. 222, Birkhäuser Boston, Boston, MA, 2004. [21] P. Ramacher, Reduced Weyl asymptotics for pseudodifferential operators on bounded domains I. The finite group case, J. Funct. Anal. (2008), doi:10.1016/j.jfa.2008.02.012, in press. [22] M. Reed, B. Simon, Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-adjointness, Academic Press, New York, 1975. [23] M.A. Shubin, Pseudodifferential Operators and Spectral Theory, 2nd ed., Springer-Verlag, Berlin, 2001. [24] V.N. Tulovsky, M.A. Shubin, On the asymptotic distribution of eigenvalues of pseudodifferential operators in Rn , Math. Trans. 92 (4) (1973) 571–588. [25] H. Weyl, Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung), Math. Ann. 71 (1912) 441–479.
Journal of Functional Analysis 256 (2009) 129–148 www.elsevier.com/locate/jfa
The existence of tight Gabor duals for Gabor frames and subspace Gabor frames Deguang Han Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA Received 30 November 2007; accepted 17 October 2008
Communicated by N. Kalton
Abstract Let K and L be two full-rank lattices in Rd . We give a complete characterization for all the Gabor frames that admit tight dual of the same type. The characterization is given in terms of the center-valued trace of the von Neumann algebra generated by the left regular projective unitary representations associated with the time–frequency lattice K × L. Two applications of this characterization were obtained: (i) We are able to prove that every Gabor frame has a tight dual if and only if the volume of K × L is less than or equal to 1 . (ii) We are able to obtain sufficient or necessary conditions for the existence of tight Gabor pseudo-duals 2 for subspace Gabor frames in various cases. In particular, we prove that every subspace Gabor frame has a tight Gabor pseudo-dual if either the volume v(K × L) 12 or v(K × L) 2. Moreover, if K = αZd , L = βZd with αβ = 1, then a subspace Gabor frame G(g, L, K) has a tight Gabor pseudo-dual only when G(g, L, K) itself is already tight. © 2008 Elsevier Inc. All rights reserved. Keywords: Frames; Parseval duals; Frame representations; Gabor frames; Lattice tiling; Subspace Gabor frame; Pseudo-duals
1. Introduction Frames are generalizations of Riesz bases. Although the concept of frames for Hilbert spaces was formally introduced by Duffin and Schaeffer [7] to deal with some difficult problems in nonharmonic Fourier analysis, the idea to represent a function in terms of time–frequency shifts of a single function (atom) was originated in communication theory by D. Gabor [14] and in quantum E-mail address:
[email protected]. 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.10.015
130
D. Han / Journal of Functional Analysis 256 (2009) 129–148
mechanic theory by von Neumann [32]. Therefore it is not surprising that the Gabor frame theory has close connections with the theory of operator algebras, and that one of the key ingredients of this research will be the techniques involving operator algebra theory and geometric properties of time–frequency lattices. We point out that in the past decade, there have been many significant developments in frame theory mainly due to the fact that Gabor frames and wavelet expansions have emerged as an important research area in analysis and useful tools in applications such as signal and image processing, data compressions and control theory. Moreover, frame theory also has very close connections with many other areas in pure mathematics such as the connection with the famous Kadison–Singer’s conjecture, Bourgain–Tzafriri’s paving conjecture (cf. [1,2,4, 6,8,10–12,19,21–24,26]). In frame theory, tight frames are the ones that have attracted particular attention due to their simplicity (e.g. the canonical dual is a scalar multiple of the tight frame itself) and due to some other useful features in applications (e.g. tight frames are optimal for erasures, etc. [5,15]). When a frame itself is not a tight frame, the canonical dual frame cannot be tight. However, it is possible that tight (alternate) dual frames exist even when a given frame is not a tight one. The existence of tight dual frames for non-tight frames could be a useful feature to have for either theoretical or practical reasons. For instance, for frames induced by group representations, the existence problem of tight dual frames with the same structure is tightly related to some geometric properties of the group representations (cf. [20]). In encoding-decoding applications, due to the irregularity of the applied problem the favorite/suitable frame for engineers in encoding may not be necessarily a tight one, and quite often the conditional number of the frame operator for the frame (which is equal to the conditional number of the frame operator for the canonical dual frame) could be very large. This usually causes very unstable reconstructions (decoding). However, the conditional number of the frame operator of a tight canonical dual frame is always one. In this case a tight dual certainly could have some advantages over the canonical dual frame for the purpose of stable reconstruction (decoding). For general frames (frames for abstract Hilbert spaces) we proved in [17] that the existence of a tight dual is equivalent to the condition that the given frame can be dilated to (a scalar multiple of) an orthogonal basis (under an oblique projection) with uniform length. However, for frames with special structures (such as Gabor frames, wavelet frames and frames induced by group representations), we often require that the dual frames also have the same structure. In this case, the existence problem for tight dual frame is a much more delicate issue. For example, let G be a countable abelian group and let π be a group representation from G to the set of unitary operators on some Hilbert space H . If {π(g)ξ : g ∈ G} is a frame for H but not a tight one, then {π(g)ξ : g ∈ G} does NOT have a tight dual of the same type (tight pseudo-duals may exist when working on subspace frames). However, there are many so-called structured frames that admit tight duals of the same structure. It even can happen that a frame can have two different tight duals. This paper focuses on the so-called Gabor frames. Let K and L be two full-rank lattices in Rd , and let g(x) ∈ L2 (Rd ) and Λ = L × K. Then the Gabor (or Weyl–Heisenberg) family is the following family of functions in L2 (Rd ): G(g, Λ) = G(g, L, K) := e2πi,x g(x − κ) ∈ L, κ ∈ K . For convenience, we write gλ = gκ, = e2πi,x g(x − κ), where λ = (κ, ). If E and Tκ are the modulation and translation unitary operators defined by
D. Han / Journal of Functional Analysis 256 (2009) 129–148
131
E f (x) = e2πi,x f (x) and Tκ f (x) = f (x − κ) for all f ∈ L2 (Rd ). Then we have gκ, = E Tκ g. For a Gabor family G(g, L, K), if there exist two constants C, D > 0 such that Cf 2
f, gκ, 2 Df 2 ,
(1.1)
κ∈K, ∈L
holds for every f ∈ L2 (Rd ), then G(g, L, K) is called a Gabor frame for L2 (Rd ). The optimal constants (maximal for C and minimal for D) are called, respectively, the upper and lower frame bounds. A Gabor frame G(g, L, K) is called tight if C = D, and is called Parseval if C = D = 1. If we only require the upper frame bound condition in (1.1), then G(g, L, K) is called a Bessel sequence. When a Gabor family G(g, L, K) satisfies the condition (1.1) only for those functions f ∈ M = span G(g, L, K) (the closed linear span of G(g, L, K)), then we say that G(g, L, K) is a subspace Gabor frame. Note that since E Tκ = e2πiκ Tκ E , we have that M = span G(g, L, K) is both Eκ and T invariant for all (κ, ) ∈ K × L. In general, a closed subspace M of L2 (Rd ) is called a time–frequency shift invariant subspace for a time–frequency lattice K × L if M is both Eκ and T invariant for all (κ, ) ∈ K × L. Although not every time–frequency lattice K × L admits a Gabor frame for the entire space L2 (Rd ), every cyclic (K × L)-shift invariant subspace admits a subspace Gabor frame. The following is well known in Gabor theory (cf. [21]). Theorem 1.1. Let L = AZd and K = BZd , where A and B are non-singular d × d real matrices. Then the following are equivalent: (i) There exists a function g such that G(g, L, K) is a frame for L2 (Rd ). (ii) There exists a function g such that G(g, L, K) is complete in L2 (Rd ). (iii) |det(AB)| 1. Moreover, |det(AB)| = 1 if and only if there exists a function g such that G(g, L, K) is a Riesz basis for L2 (Rd ). In this case every Gabor frame for L2 (Rd ) must be a Riesz basis. One of the key features of frames is to allow us to have a stable representation for all the functions in the underlying space in terms of the functions in the frame sequence. Let Λ = K × L, and let G(g, Λ) be a Bessel sequence. Then the operator from L2 (Rd ) to 2 (Λ) defined by Θg (f ) =
f, gλ eλ
λ∈Λ
is a bounded linear operator, where {eλ } is the standard orthonormal basis for 2 (Λ). This operator is usually refereed as the analysis operator of G(g, Λ). It is easy to check that Θg∗ eλ = gλ ,
∀λ ∈ Λ,
132
D. Han / Journal of Functional Analysis 256 (2009) 129–148
and hence we have
f, gλ gλ , Θg∗ Θg f =
∀f ∈ L2 Rd .
(1.2)
λ∈Λ
Listed below are a few useful facts about analysis operators for Gabor Bessel sequences: • Θg∗ Θg commutes with E and Tκ for all ∈ L and κ ∈ K. • G(g, Λ) is a Gabor frame for L2 (Rd ) if and only if Θg is injective and has closed range. • G(g, Λ) is a subspace Gabor frame for a time–frequency shift invariant subspace M if and only if Θg∗ Θg is an invertible bounded operator when restricted to M. • G(g, Λ) is a Parseval subspace Gabor frame if and only if Θg∗ Θg (or equivalently, Θg Θg∗ ) is an orthogonal projection. In particular, G(g, Λ) is a Parseval Gabor frame for L2 (Rd ) if and only if Θg∗ Θg = I . In the case that G(g, Λ) is a subspace Gabor frame for M, the frame operator S := Θg∗ Θg is a positive operator on L2 (Rd ), and is also invertible when restricted to M. Throughout this paper we will use S −1 to denote the operator on L2 (Rd ) that is the inverse of S when restricted to M and 0 when restricted to M ⊥ . Since S commutes with E and Tκ for all ∈ L and κ ∈ K, so does S −1 . Thus (1.2) implies the reconstruction formula: f=
f, S −1 g λ gλ ,
∀f ∈ M,
(1.3)
λ∈Λ
where the convergence is in norm and unconditional. It can be verified that G(S −1 g, Λ) is also a subspace frame for M, which is called the canonical Gabor dual of G(g, Λ). Due to the redundance property of frames (a key difference between frames and Riesz bases), there might exist some other Gabor families, together with G(g, Λ), yielding similar reconstruction formulas. We call a Gabor family G(h, Λ) a Gabor pseudo-dual frame for G(g, Λ) if it satisfies the condition: f=
f, hλ gλ ,
∀f ∈ M.
(1.4)
λ∈Λ
Both canonical and Gabor pseudo-duals are called Gabor duals. We remark that a Gabor pseudodual G(h, Λ) could be an inside dual in the sense of h ∈ M, and it could also be an outside dual in the sense that h ∈ / M. The canonical dual is an inside dual since S −1 g ∈ M. We are interested in the Gabor pseudo-dual frames that are also tight (or Parseval). Definition 1. Let G(g, Λ) be a subspace Gabor frame for M. A Bessel family G(h, Λ) is called a tight (respectively Parseval) Gabor dual for G(g, Λ) if it is a Gabor pseudo-dual (i.e. f = λ∈Λ f, hλ gλ , ∀f ∈ M), and at the same time it is also a tight (respectively Parseval) frame for span G(h, Λ). For Gabor systems, our goal is to find conditions under which a tight Gabor dual frame exists for a given Gabor frame. Let us first examine the Parseval dual case. As we have pointed out in [17] for a Gabor frame G(g, Λ) to have a Parseval dual, it is necessary that the lower frame bound of G(g, Λ) be greater that or equal to one. We refer to this condition as the lower frame
D. Han / Journal of Functional Analysis 256 (2009) 129–148
133
bound (LFB, in short) condition. With the help of a result on multi-lattice tiling, we were able to obtain the following [17]. Theorem 1.2. Let L = AZd and K = BZd be two full-rank lattices in Rd . Then the following are equivalent: (i) |det(AB)| 12 . (ii) Every Gabor frame G(g, L, K) for L2 (Rd ) with the (LFB)-condition has a Parseval Gabor dual G(h, L, K). In the case when 1/2 < |det(AB)| < 1, while some of the Gabor frames admit tight Gabor duals, there exist some other Gabor frames which do not admit tight Gabor duals. Therefore it remains a question to find a necessary and sufficient condition for an individual Gabor frame to admit a tight Gabor dual. It is clear from Theorem 1.2 that if |det(AB)| 12 , then every Gabor frame for L2 (Rd ) admits a tight Gabor dual. However, it does not tell us whether the condition |det(AB)| 12 is necessary or not for such a property to hold. We remark that this is not a simple “rescaling” problem! Both questions were asked several times by researchers when the author was presenting the results of [17] at various conferences and seminars. The first part of this paper is aimed to settle these two problems. It turns out that these two questions are closely related. In Section 2 we will present a complete characterization for all the Gabor frames that admit tight Gabor duals (Theorem 2.2). The characterization is given in terms of the center-valued trace of the von Neumann algebra generated by the left regular projective unitary representations associated with the time–frequency lattice K × L. As an application we obtain the following result. Corollary 1.3. Let L = αZd , K = βZd , and let G(g, L, K) be a frame for L2 (Rd ) and S be its frame operator. If αβ is irrational, then G(g, L, K) has a tight Gabor dual if and only if ϕ2 1 − |det(AB)|, where G(ϕ, L, K) is any fixed Parseval subspace Gabor frame for ker(S −1 I − S −1 )⊥ (which is the closure of the range space of S −1 I − S −1 ). With the help of Theorem 2.2, we settle the second question with the following theorem. Theorem 1.4. Let L = AZd and K = BZd be two full-rank lattices in Rd . Then the following are equivalent: (i) |det(AB)| 12 . (ii) Every Gabor frame for L2 (Rd ) with the (LFB)-condition has a Parseval Gabor dual. (iii) Every Gabor frame for L2 (Rd ) has a tight Gabor dual. Due to the density requirement for the existence of Gabor frames (see Theorem 1.1), not every time–frequency lattice admits a Gabor frame for the entire space L2 (Rd ). However, subspace Gabor frames do exist for many time–frequency shift invariant subspaces for any time–frequency lattice. The study of subspace Gabor frames has been an active research topic in recent years (cf. [3,13,23,27–30]). Most frequently, the techniques involved in the study of subspace Gabor frames are quite different from the techniques that are used in studying Gabor frames for the entire space L2 (Rd ). The second part of this paper will be devoted to the investigation on the existence problem of tight Gabor pseudo-dual for subspace Gabor frames. We obtain several
134
D. Han / Journal of Functional Analysis 256 (2009) 129–148
necessary and (or) sufficient conditions for a subspace Gabor frame to admit a tight pseudo-dual of the Gabor structure. One of them is the following theorem. Theorem 1.5. Let L = AZd and K = BZd be two full-rank lattices in Rd . (i) If |det(AB)| 1/2, then every subspace Gabor frame G(g, L, K) has a (inside) tight Gabor dual. (ii) If |det(AB)| 2, then every subspace Gabor frame G(g, L, K) has a tight (usually, outside) Gabor pseudo-dual. (iii) If AB t = I , then a subspace Gabor frame G(g, L, K) has a tight Gabor pseudo-dual if and only if G(g, L, K) itself is already tight. (iv) In the case that L = αZd , K = βZd with αβ irrational and |αβ| 1, we have that if G(g, L, K) is a subspace Gabor frame satisfying S −1/2 g2 1 − |det(AB)|, then G(g, L, K) has a tight (inside) Gabor pseudo-dual. Remarks. (i) There are subspace Gabor frames that admit outsider tight Gabor pseudo-dual but do not admit inside ones. For example, assume that |det(AB)| is an integer that is bigger than 1, and G(g, L, K) is a subspace Gabor frame but not a tight subspace Gabor frame. Then, from Theorem 1.5(ii), we have that G(g, L, K) has a tight Gabor pseudo-dual. However, since the only Gabor dual inside the subspace is G(S −1 g, L, K) (see [12]) which is not tight, we have that G(g, L, K) does not have an inside tight Gabor pseudo-dual. (ii) Suppose that G(h, L, K) is a tight Gabor pseudo-dual for a subspace Gabor frame G(g, L, K) of M. If M ⊆ span G(h, L, K), then G(P h, L, K) is an inside tight Gabor pseudodual for G(g, L, K), where P is the orthogonal projection onto M. Finally we recall a lattice tiling result that will be used in both Section 2 and Section 3. Let Ω be a measurable set in Rd , and let L be a full rank lattice in Rd . We say Ω tiles Rd by L, or Ω is a fundamental domain of L, if (i) ∈L (Ω + ) = Rd a.e.; (ii) (Ω + ) ∩ (Ω + ) has Lebesgue measure 0 for any = in L. We say that Ω packs Rd by L if only (ii) holds. Equivalently, Ω tiles Rd by L if and only if
χΩ (x − ) = 1 for a.e. x ∈ Rd ,
∈L
and Ω packs Rd by L if and only if
χΩ (x − ) 1 for a.e. x ∈ Rd .
∈L
Let v(L) denote the volume of L, i.e. v(L) = |det(A)| if L = AZd . We also have that μ(Ω) = v(L) if Ω tiles by L, and μ(Ω) v(L) if Ω packs by L, where μ is the Lebesgue measure on Rd . Furthermore, if Ω packs Rd by L and μ(Ω) = v(L), then Ω necessarily tiles Rd by L.
D. Han / Journal of Functional Analysis 256 (2009) 129–148
135
Lemma 1.6. (See [17,21].) Let L, K be two full rank lattices in Rd such that v(L) N v(K). Then there exists N measurable subsets Ωi (i = 1, . . . , N ) of Rd such that: (i) each Ωi tiles Rd by K and packs Rd by L; (ii) (Ωi + ) ∩ (Ωj + ) has Lebesgue measure 0 for any , in L and all i = j . The rest of this paper will be organized as follows. In Section 2 we will present and prove our first main result (Theorem 2.2) and then use it to prove Theorem 1.4 and Corollary 1.3. We also point out that Theorem 1.2 can also be obtained from Theorem 2.2 (However, the lattice-tiling result, Lemma 1.6, is still needed in the proof.) Section 3 will be mainly devoted to proving Theorem 1.5. Part of the proof uses Theorems 1.2 and 2.2. Rd ) 2. Gabor frames for the entire space L2 (R To state the main result of this section, we need to recall a few more concepts and notations. Let B(H ) be the algebra of all bounded linear operators on a separable Hilbert space H . A ∗subalgebra M of B(H ) is a subalgebra with the property that the adjoint operator T ∗ of T belongs to M whenever T ∈ M. A ∗-subalgebra M is called a von Neumann algebra if the identity operator I is in M and if M is closed in the weak operator topology. By the double commutant theorem, a ∗-subalgebra M is a von Neumann algebra if and only if M = M , where M = {T ∈ B(H ): T S = ST , ∀S ∈ M} denotes the commutant of M. A von Neumann algebra M is called finite if every isometry in M is unitary. Two orthogonal projections P and Q in a von Neumann algebra M are said to be equivalent if there exists an operator T ∈ M such that T T ∗ = P and T ∗ T = Q. In this case we write P ∼ Q. A subprojection E of Q is an orthogonal projection such that EH ⊆ QH . We use the notation P Q if P is equivalent to a subprojection of Q in M. It is well known that if P Q and Q P , then P ∼ Q. A faithful normal trace on a von Neumann algebra M is a trace that is continuous in the weak operator topology and satisfies the condition that ρ(T ) > 0 whenever T ∈ M is a nonzero positive operator. For any finite von Neumann algebra M, there exists a unique mapping from M to its center M ∩ M satisfying the following conditions: (i) (ii) (iii) (iv)
τ (ST ) = τ (T S), ∀S, T ∈ M; τ (C) = C for each C ∈ M ∩ M ; τ (T ) is a nonzero positive whenever T ∈ M is a nonzero positive operator; τ (CT ) = Cτ (T ) if T ∈ M and C ∈ M ∩ M .
This mapping τ is called the center-valued trace of M. Moreover, if ρ is a faithful normal trace on M, then we also have (v) ρ(T ) = ρ(τ (T )) for all T ∈ M (see the proof of Theorem V.2.6 in [31]). For Λ = K × L, we define the Gabor representation π : Λ → B(L2 (Rd )) by π(λ) = E Tκ , Then G(g, L, K) = {π(κ, )g: ∈ L, κ ∈ K} and
λ = (κ, ) ∈ Λ.
136
D. Han / Journal of Functional Analysis 256 (2009) 129–148
π(λ1 )π(λ2 ) = μ(λ1 , λ2 )π(λ1 + λ2 ),
λ1 , λ2 ∈ Λ,
where μ(λ1 , λ2 ) ∈ T := {t ∈ C: |t| = 1} and is called the multiplier of π . There exists an associated left regular representation σ on 2 (Λ) defined by σ (λ)(eω ) = μ(λ, ω)eλ+ω ,
λ, ω ∈ Λ,
where {eω : ω ∈ Λ} is the standard orthonormal basis for 2 (Λ) and μ(·,·) is the multiplier of π . Let MΛ = {T ∈ B(2 (Λ)): T σ (λ) = σ (λ)T , λ ∈ Λ} be the commutant of σ (Λ). Then MΛ is a finite von Neumann algebra. In what follows, we will use τΛ to denote the unique center-valued trace on MΛ . A closed subspace M of L2 (Rd ) is called Λ-shift invariant if it is π -invariant, i.e., π(λ)M ⊆ M for all λ ∈ Λ. Given a Bessel sequence G(g, Λ) and let Θg be its analysis operator. It is routine to check that Θg (L2 (Rd )) is invariant under σ (λ) for every λ ∈ Λ. So if we use Pg to denote the orthogonal projection of 2 (Λ) onto Θg (L2 (Rd )), then Pg ∈ MΛ . We need the following lemma. Lemma 2.1. (i) If G(g, Λ) and G(ψ, Λ) are two subspace Gabor frames for the same Λ-shift invariant subspace M, then Pg ∼ Pψ in MΛ , and thus τΛ (Pg ) = τΛ (Pψ ). (ii) If G(g, Λ) is a Gabor frame for the entire space L2 (Rd ), then Pg e0 , e0 = τΛ (Pg )e0 , e0 = det(AB), where 0 = (0, 0) ∈ Λ. Proof. (i) Let S be the frame operator for G(g, Λ). Then we have Θg L2 Rd = ΘS −1/2 g L2 Rd . Thus we can assume that both G(g, Λ) and G(ψ, Λ) are Parseval subspace Gabor frames for M. So we have Pg = Θg Θg∗ , Pψ = Θψ Θψ∗ and Θg∗ Θg = Θψ∗ Θψ = Q, where Q is the orthogonal projection from L2 (Rd ) onto M. Let V = Θg Θψ∗ . Then it can be checked that V ∈ MΛ . Moreover, we have V V ∗ = Θg Θψ∗ Θψ Θg∗ = Θg Θψ∗ Θψ Θg∗ = Θg QΘg∗ = Θg Θg∗ = Pg and similarly, V ∗ V = Pψ . Hence Pg ∼ Pψ in MΛ . (ii) Note that the mapping tr(T ) = T e0 , e0 (T ∈ MΛ ) defines a faithful trace on MΛ such that tr(I ) = 1. So, by property (v) for the center-valued trace τΛ on MΛ , we have that Pg e0 , e0 = τΛ (Pg )e0 , e0 . On the other hand, since |det(AB)| = S −1/2 g2 (cf. [10,21]), Pg e0 = ΘS −1/2 g S −1/2 g and ΘS −1/2 g is an isometry, we obtain that 2 2 Pg e0 , e0 = Pg e0 2 = ΘS −1/2 g S −1/2 g = S −1/2 g = det(AB), as claimed.
2
D. Han / Journal of Functional Analysis 256 (2009) 129–148
137
By Lemma 2.1(i), we get that τΛ (Pg ) is a quantity that is independent of the choices of the subspace Gabor frame G(g, Λ) for the Λ-shift invariant subspace M. So in the rest of this paper, we can use δΛ (M) to denote τΛ (Pg ), where G(g, Λ) is any subspace Gabor frame for M. Now we are ready to state our first main result. Theorem 2.2. Let Λ = L × K and let G(g, Λ) be a frame for H := L2 (Rd ) and S be its frame operator. Assume that G(g, Λ) satisfies the lower frame bound condition (i.e., S −1 1). Then G(g, Λ) has a Parseval Gabor dual if and only if δΛ (ker(I − S −1 )⊥ ) I − δΛ (H ). Remark. In the case that |det(AB)| = 1 and G(g, L, K) is a frame for H := L2 (Rd ), we have that G(g, L, K) must be a Riesz basis and thus Pg = I . This implies that δΛ (H ) = τΛ (Pg ) = τΛ (I ) = I . Thus δΛ (ker(I − S −1 )⊥ ) I − δΛ (H ) holds only when S = I , which is consistence with the well-known fact that a Riesz basis has a Parseval dual only when the basis itself is already an orthonormal basis. Before we prove Theorem 2.2, we point out the following consequence which provides a simple characterization in the case that MΛ is a factor von Neumann algebra (i.e. MΛ ∩ M Λ = CI ). Theorem 2.3. Let Λ = L × K and let G(g, L, K) be a frame for L2 (Rd ) whose frame operator S satisfying the condition S −1 1. Assume that MΛ is a factor von Neumann algebra. Then G(g, L, K) has a Parseval Gabor dual if and only if ϕ2 1 − |det(AB)|, where G(ϕ, L, K) is any fixed Parseval subspace Gabor frame for ker(I − S −1 )⊥ . In particular, the above conclusion holds when L = αZd and K = βZd with αβ irrational. Proof. When MΛ is a factor von Neumann algebra we have that τΛ (T ) = tr(T )I , where tr(·) is the trace defined by tr(T ) = T e0 , e0 for all T ∈ MΛ . Therefore the condition δΛ (ker(I − S −1 )⊥ ) I − δΛ (H ) in Theorem 2.2 becomes tr(Pϕ ) 1 − tr(Pψ ), where G(ϕ, L, K) is any fixed Parseval subspace Gabor frame for ker(I − S −1 )⊥ and G(ψ, L, K) is any Parseval Gabor frame for the entire space L2 (Rd ). Since tr(Pψ ) = ψ2 = |det(AB)| and tr(Pϕ ) = ϕ2 , we complete the proof for the first part of the theorem. The second part follows from the fact that MΛ is a factor von Neumann algebra when L = αZd , K = βZd with αβ irrational. 2 We now come back to the proof of Theorem 2.2. We need the following two lemmas: Lemma 2.4. (See [25].) Let P and Q be two orthogonal projections in MΛ such that τΛ (P ) τΛ (Q). Then there exists an orthogonal projection R ∈ MΛ such that R Q and P ∼ R. Lemma 2.5. Let G(g, L, K) and G(h, L, K) be two Bessel sequences in L2 (Rd ) such that Θh∗ Θg = Θg∗ Θh = 0. Then there exists a Parseval subspace Gabor frame G(ϕ, L, K) for M := span G(h, L, K) such that Θϕ∗ Θg = Θg∗ Θϕ = 0.
138
D. Han / Journal of Functional Analysis 256 (2009) 129–148
Proof. Following from the proof of Theorem 2.1 in [9], we have that there exists a Parseval Gabor frame G(ϕ, L, K) for M := span G(h, L, K) such that Range(Θϕ ) = Range(Θh ). Thus we have Θϕ∗ Θg = Θg∗ Θϕ = 0.
2
Proof of Theorem 2.2. Let G(g, L, K) be a Gabor frame for L2 (Rd ) with the (LFB)-condition, and let M = ker(I − S −1 )⊥ . (⇒) Assume that G(g, L, K) has a Parseval Gabor dual, say G(ϕ, L, K). Let h = ϕ − S −1 g. Then G(h, L, K) is Bessel. Moreover, for any f ∈ L2 (Rd ) we have Θg∗ Θh f =
f, hκ, gκ,
κ∈K,∈L
=
κ∈K,∈L
=
f, ϕκ, − S −1 g κ, gκ,
f, ϕκ, gκ, −
κ∈K,∈L
κ∈K,∈L
f, S −1 g κ, gκ,
= f − f = 0. So we have Θg∗ Θh = Θh∗ Θg = 0, which also implies that ΘS∗−1 g Θh = Θh∗ ΘS −1 g = 0 and ΘS∗−1/2 g Θh = Θh∗ ΘS −1/2 g = 0 since ΘS −1 g = Θg S −1 and ΘS −1/2 g = Θg S −1/2 . Now we have I = Θϕ∗ Θϕ = (Θh + ΘS −1 g )∗ (Θh + ΘS −1 g ) = Θh∗ Θh + ΘS∗−1 g ΘS −1 g = Θh∗ Θh + S −1 . Therefore I − S −1 = Θh∗ Θh , which implies that M = Range(Θh∗ Θh ) = span G(h, L, K). By Lemma 2.5, there exists ψ ∈ M such that G(ψ, L, K) is a Parseval subspace Gabor frame for M and ΘS∗−1 g Θψ = Θψ∗ ΘS −1 g = 0. So we also have Θg∗ Θψ = Θψ∗ Θg = 0. Therefore we have Pg ⊥ Pψ , and so Pψ (I − Pg ) which implies that τΛ (Pψ ) τΛ (I − Pg ) = I − τΛ (Pg ). That is, δΛ (M) I − δΛ (L2 (Rd )). 2 d −1 −1 (⇐) Assume that δΛ (M) √I − δΛ (L (R )). Since S 1, we have that I − S is a −1 positive operator. Write B = I − S . Then B commutes with E Tκ for all (κ, ) ∈ K × L. Also note that since B is positive, we have ⊥ M = ker I − S −1 = (ker B)⊥ = Range(B). Let G(h, L, K) be a Parseval subspace Gabor frame for M (the existence of such a frame is guaranteed by the condition |det(AB)| 1). Then the assumption δΛ (M) I − δΛ (L2 (Rd )) is the same as the condition
D. Han / Journal of Functional Analysis 256 (2009) 129–148
139
τΛ (Ph ) τΛ (I − Pg ). Hence, by Lemma 2.4, there exists a subprojection Q I − Pg such that Ph ∼ Q. Let V ∈ MΛ be the partial isometry such that V V ∗ = Ph and V ∗ V = Q. Set ψ = Θh∗ V e0 . We claim that G(ψ, L, K) is a Parseval subspace Gabor frame for M such that Θg∗ Θψ = Θψ∗ Θg = 0. In fact, note that ψκ, = Θh∗ σ (κ, )V e0 = Θh∗ V σ (κ, )e0 = Θh∗ V eκ, . So for any f ∈ M, we have
f, ψκ, 2 =
κ∈K,∈L
∗ f, Θ V e0 h
κ∈K,∈L
=
κ,
2
∗ V Θh f, eκ, 2
κ∈K,∈L
= V ∗ Θh f 2 = f 2 . Thus G(ψ, L, K) is a Parseval subspace Gabor frame for M. Moreover, for any f ∈ L2 (Rd ), we have
Θg∗ Θψ f = Θg∗
f, ψκ, eκ,
κ∈K,∈L
= Θg∗
V ∗ Θh f, eκ, eκ,
κ∈K,∈L
= Θg∗ V ∗ Θh f
= 0,
where the last equality uses the facts that Θg∗ Pg⊥ = 0 and Range(V ∗ ) = Range(Q) ⊆ Range(Pg⊥ ). Thus Θg∗ Θψ = Θψ∗ Θg = 0, as claimed. Let R be the orthogonal projection from L2 (Rd ) onto M. Then we have Θψ∗ Θψ = R and BRB = B 2 = I − S −1 . So I − S −1 = B 2 = BΘψ∗ Θψ B. Since B commutes with all the E , Tκ , we have for any f ∈ L2 (Rd ) that BΘψ∗ Θψ Bf = B
Bf, ψκ, ψκ,
κ∈K,∈L
=
f, Bψκ, Bψκ,
κ∈K,∈L
=
f, (Bψ)κ, (Bψ)κ,
κ∈K,∈L ∗ = ΘBψ ΘBψ f.
140
D. Han / Journal of Functional Analysis 256 (2009) 129–148
Thus ∗ I − S −1 = BΘψ∗ Θψ B = ΘBψ ΘBψ .
Note that S −1 = ΘS∗−1 g ΘS −1 g . So we have ∗ ΘBψ = I. ΘS∗−1 g ΘS −1 g + ΘBψ
Let ϕ = S −1 g + Bψ . Then Θϕ = ΘS −1 g + ΘBψ . Since ΘS∗−1 g ΘBψ = S −1 Θg∗ Θψ B = 0 and ∗ Θ ∗ −1 = 0, we have that ΘBψ S −1 g = BΘψ Θg S
∗ ∗ Θϕ∗ Θϕ = ΘS∗−1 g ΘS −1 g ΘS∗−1 g ΘBψ + ΘBψ ΘS −1 g + ΘBψ ΘBψ ∗ = ΘS∗−1 g ΘS −1 g + ΘBψ ΘBψ = I,
which implies that G(ϕ, L, K) is a Parseval Gabor frame for L2 (Rd ). Moreover, ∗ Θϕ∗ Θg = ΘS∗−1 g + ΘBψ Θg ∗ = ΘS∗−1 g Θg + ΘBψ Θg
= I + BΘψ∗ Θg = I + 0 = I. Hence G(ϕ, L, K) is a Parseval Gabor dual of G(g, L, K).
2
To prove Corollary 1.3, we need the following lemma. Lemma 2.6. Let G(g, L, K) be a frame for L2 (Rd ) and S be its frame operator. If G(h, L, K) is a tight Gabor dual of G(g, L, K) with frame bound b, then b S −1 . Proof. From
f 2 =
f, hκ, gκ, , f
κ∈K,∈L
f, hκ, 2
κ∈K,∈L
=
√ bf ·
1/2 ·
f, gκ, 2
1/2
κ∈K,∈L
f, gκ, 2
1/2
,
κ∈K,∈L
we have that b
S −1 .
1 b
2
1 S −1
since
1 S −1
is the (optimal) lower frame bound of G(h, L, K). Thus
D. Han / Journal of Functional Analysis 256 (2009) 129–148
141
Proof of Corollary 1.3. Let S be the frame operator for the Gabor frame G(g, L, K). (⇒) Assume that G(g, L, K) has a tight Gabor dual G(h, L, K). Let b be the frame bound of G(h, L, K). Then, from Lemma 2.6, we have b S −1 . Note that G( √1 h, L, K) is a Parseval b √ √ Gabor dual of G( bg, L, K), and the frame operator for G( bg, L, K) is bS. Thus, by Theorem 2.3, we have that ϕ2 1 − |det(AB)|, where G(ϕ, L, K) is any fixed Parseval subspace −1 Gabor frame for [ker(I − S b )]⊥ . Note that if b > S −1 , then bI − S −1 is invertible and so
S −1 ker I − = ker bI − S −1 = {0}. b Therefore ⊥ −1 S −1 ⊥ ⊆ ker I − ker S · I − S −1 b whenever b S −1 , and so we obtain that ϕ2 1 − |det(AB)|, where G(ϕ, L, K) is any fixed Parseval subspace Gabor frame for [ker(S −1 · I − S −1 )]⊥ . (⇐) Now assume that ϕ2 1 − |det(AB)|, where G(ϕ, L, K) is any fixed Parseval subspace Gabor frame for [ker(S −1 I − S −1 )]⊥ . Note that
−1 S −1 −1 I −S = ker I − −1 ker S S and that S −1 · S is the frame operator for the Gabor frame G( S −1 g, L, K) which satisfies the (LFB)-condition. Thus, by Theorem 2.3, we have that the Gabor frame G( S −1 g, L, K) has a Parseval Gabor dual, which certainly implies that G(g, L, K) has a tight Gabor dual. 2 Using Theorem 2.2, and with the exact same argument as in the proof of Corollary 1.3, we have the following: Theorem 2.7. Let Λ = L × K. Let G(g, L, K) be a frame for H := L2 (Rd ) and S be its frame operator. Then G(g, L, K) has a tight Gabor dual if and only if δΛ ([ker(S −1 · I − S −1 )]⊥ ) I − δΛ (H ). Proof of Theorem 1.4. By Theorem 1.2, we only need to show that (iii) ⇒ (i). Assume that every Gabor frame for L2 (Rd ) has a tight Gabor dual. Without losing the generality, we can assume that A = I and |det(B)| 1. By Lemma 1.6, there exists a measurable subset Ω of Rd such that Ω tiles Rd by BZd and packs Rd by L. Let {Fn }∞ n=1 be a measurable partition of Ω such that μ(Fn ) > 0. Let Qn = Θg∗n Θgn , where gn = χFn . Then {Qn }∞ n=1 are mutually orthogonal
∞ projections such that n=1 Qn = I . Let {dn } be a strictly decreasing sequence such that limn→∞ dn = 1. Define S=
∞ n=1
dn Qn .
142
D. Han / Journal of Functional Analysis 256 (2009) 129–148
Then S commutes with E and Tκ for all κ ∈ K and ∈ L. Moreover S −1 = 1√and ker(I − S −1 ) = {0}. Let G(h, L, K) be a fixed Parseval frame for L2 (Rd ) and let g = Sh. Then G(g, L, K) is a Gabor frame for L2 (Rd ) and ∗ Θ√Sh = Θ√ Sh
√ ∗ √ SΘh Θh S = S.
Hence S is the frame operator for G(g, L, K). Since, by assumption, G(g, L, K) has a tight Gabor dual, we have from Theorem 2.7 that δΛ ([ker(S −1 · I − S −1 )]⊥ ) I − δΛ (H ). But ker(S −1 · I − S −1 ) = ker(I − S −1 ) = {0}. So we have ⊥ = δΛ (H ), δΛ ker S −1 · I − S −1 which implies that 2δΛ (H ) I , i.e., 2τΛ (Pg ) I . By Lemma 2.1(ii), we obtain 2det(AB) = 2Pg e0 , e0 = 2 τΛ (Pg )e0 , e0 I e0 , e0 = 1. Hence |det(AB)| 12 .
2
We end this section by explaining that how Theorem 1.2 can be obtained as a consequence of Theorem 2.2 and Lemma 1.6. Assume that |det(AB)| 12 . Then 1 − |det(AB)| 12 . Similar to the proof of Theorem 1.4, we can assume that A = I and |det(B)| 12 . So we have v(L) 2v(K). By Lemma 1.6, there exist two measurable sets Ωi (i = 1, 2) of Rd such that each Ωi tiles Rd by BZd (= K) and packs Rd by L (i = 1, 2), and (Ω1 + κ) ∩ (Ω2 + κ ) has Lebesgue measure 0 for any κ, κ in L. Let g = χΩ1 and h = χΩ2 . Then the both Gabor families G(g, L, K) and G(h, L, K) are Parseval frames for L2 (Rd ). Moreover, we also have Pg ⊥ Ph . Thus Pg + Ph I . Note that and condition (ii) implies that span G(ϕ1 , L, K) ⊥ span G(ϕ2 , L, K). Note that τΛ (Pg ) = τΛ (Ph ). So we have τΛ (Pg ) I − τΛ (Ph ), which implies that δΛ (H ) I − δΛ (H ), where H = L2 (Rd ). Hence for any Gabor frame G(ψ, L, K) for L2 (Rd ) with the (LFBC) and S being its frame operator, we have that ⊥ δΛ ker I − S −1 δΛ (H ) I − δΛ (H ), and so, by Theorem 2.2, G(ψ, L, K) for L2 (Rd ) has a Parseval Gabor dual. On the other side, assume that every Gabor frame with the (LFB)-condition has a Parseval Gabor dual. Pick any Gabor frame G(g, L, K) for L2 (Rd ) such that S −1 < 1, where S is its frame operator. Then G(g, L, K) has Parseval dual, and so by Theorem 2.2, we have δΛ (ker(I − S −1 )⊥ ) I − δΛ (H ). Note that ker(I − S −1 )⊥ = {0}⊥ = L2 (Rd ). Hence 2δΛ (H ) I , which implies that |det(AB)| 12 .
D. Han / Journal of Functional Analysis 256 (2009) 129–148
143
3. Pseudo-dual for subspace Gabor frames The main purpose of the section is to prove Theorem 1.5. Since each part of the theorem requires some preparatory lemmas, we divide the proof into several propositions (Propositions 3.2, 3.4, 3.9 and 3.11). Lemma 3.1. Assume that L = AZd and K = BZd , and |det(AB)| 1. Let G(g, L, K) be a subspace Gabor frame for M. Then there exists a Gabor frame G(h, L, K) for L2 (Rd ) such that g = P h, where P is the orthogonal projection from L2 (Rd ) onto M. Moreover, h can be chosen in a way such that G(h, L, K) has the same lower (respectively, upper) frame bounds as G(g, L, K). This is a high-dimension generalization of Theorem 4 in [9]. The proof is similar, and we include a sketch here for completeness. Proof. We first assume that G(g, L, K) is a Parseval frame for M. By Theorem 1.1, there exists Parseval Gabor frame G(h, L, K) for L2 (Rd ). From the “dilation” theorem for group-like unitary systems (cf. [9,18]), we can find a Hilbert space H , a vector η ∈ H and a representation Δ of Λ = K × L to the set of unitary operators on H satisfying the following conditions: (i) Δ(λ1 )Δ(λ2 ) = μ(λ1 , λ2 )Δ(λ1 + λ2 ) (λ1 , λ2 ∈ Λ), where μ(·,·) is the multiplier of the Gabor representation π of the time–frequency lattice Λ on the space L2 (Rd ); (ii) {π(λ)h ⊕ Δ(λ)η: λ ∈ Λ} is an orthonormal basis for L2 (Rd ) ⊕ H . By Lemma 5 in [9], there exists a vector ψ in L2 (Rd ) ⊕ H such that {(π(λ) ⊕ Δ(λ))ψ: λ ∈ Λ} is an orthonormal basis for L2 (Rd ) ⊕ H , and P ψ = g. Now let Q be the orthogonal projection from L2 (Rd ) ⊕ H onto L2 (Rd ) and let h = Qψ . Then G(h, L, K) = Q π(λ) ⊕ Δ(λ) ψ: λ ∈ Λ is a Parseval frame for L2 (Rd )(= Q(L2 (Rd ) ⊕ H )). Note that P Q = P . So we obtain g = P ψ = P Qψ = P h. Now let G(g, L, K) be an arbitrary subspace Gabor frame for M with frame operator S. Then G(S −1/2 g, L, K) is a Parseval subspace Gabor frame for M. So we can dilate G(S −1/2 g, L, K) to a Parseval Gabor frame, say G(ϕ, L, K), for L2 (Rd ). Let h = g + bP ⊥ ϕ for some b > 0. Then it can be checked that G(h, L, K) is a Gabor frame for L2 (Rd ) such that P h = g and it has the 1 . So if we pick b √ 1 −1 , then G(h, L, K) will satisfy all lower frame bound max{S −1 ,1/b2 } the requirements.
2
S
Proposition 3.2. Assume that L = AZd , K = BZd . If |det(AB)| 1/2, then every subspace Gabor frame has a tight (inside) Gabor dual. Proof. Let G(g, L, K) be a subspace Gabor frame for M. Then, by Lemma 3.1, there exists a Gabor from G(h, L, K) for L2 (Rd ) such that g = P h, where P is the orthogonal projection from L2 (Rd ) onto M. From Theorem 2.7, G(h, L, K) has a tight Gabor dual, say G(ψ, L, K). Let ϕ = P ψ. Then it can be easily checked that G(ϕ, L, K) is a tight (inside) Gabor dual for G(g, L, K). 2
144
D. Han / Journal of Functional Analysis 256 (2009) 129–148
Lemma 3.3. Assume that L = AZd and K = BZd . Let G(g, L, K) be a subspace Gabor frame for M. If |det(AB)| 2, then there exists ψ ∈ M ⊥ such that G(ψ, L, K) is an orthonormal sequence. Proof. Without losing the generality, we can assume that A = I and |det(B)| 2. So we have v(K) 2v(L). By Lemma 1.6, there exist two measurable sets Ωi (i = 1, 2) of Rd such that (i) each Ωi tiles Rd by Zd (= L) and packs Rd by K (i = 1, 2); (ii) (Ω1 + ) ∩ (Ω2 + ) has Lebesgue measure 0 for any , in L. Define ϕ1 = χΩ1 and ϕ2 = χΩ2 . Then condition (i) implies that each Gabor family G(ϕi , L, K) is an orthonormal sequence, and condition (ii) implies that span G(ϕ1 , L, K) ⊥ span G(ϕ2 , L, K). We can assume that G(g, L, K) is a Parseval subspace Gabor frame for M (since, otherwise, we will replace G(g, L, K) by the Parseval frame G(S −1/2 g, L, K) for the same subspace M). Define T : L2 (Rd ) → L2 (Rd ) by T (ϕ1 )λ = gλ ,
λ ∈ Λ = K × L,
and Tf = 0 when f ⊥ span G(ϕ1 , L, K). Then clearly T is a partial isometry with the initial space span G(ϕ1 , L, K) and the final space M. Moreover, it can be easily checked that T commutes with all the translation operators Tκ and all the modulation operators E for all κ ∈ K and ∈ L. So T is a partial isometry in the von Neumann algebra {E Tκ : κ ∈ K, ∈ L} . Since {E Tκ : κ ∈ K, ∈ L} is a finite von Neumann algebra (cf. [11]), we have that P ⊥ ∼ Q⊥ in {E Tκ : κ ∈ K, ∈ L} , where P and Q are the orthogonal projections onto M and span G(ϕ1 , L, K), respectively. Therefore there exists a partial isometry, say V , in {E Tκ : κ ∈ K, ∈ L} such that V V ∗ = P ⊥ and V ∗ V = Q⊥ . Let ψ = V ϕ2 . Then G(ψ, L, K) is an orthonormal sequence, and span G(ψ, L, K) ⊥ M. 2 Proposition 3.4. Assume that L = AZd and K = BZd . If |det(AB)| 2, then every subspace Gabor frame has a tight (usually, outside) Gabor pseudo-dual. Proof. Let G(g, L, K) be a subspace Gabor frame for M. From Lemma 3.3, there exists an orthonormal sequence G(ψ, L, K) such that M ⊥ span G(ψ, L, K). Let a = S −1 , where S = Θg∗ Θg (and it is invertible when restricted to M). Write D = aI − ΘS −1 g ΘS∗−1 g as an operator on 2 (L × K). Note that Θ
∗ ∗ −1 . S −1 g ΘS −1 g = ΘS −1 g ΘS −1 g = S
√ So D 0. Let h = Θψ De0 and ϕ = S −1 g + h, where e0 ∈ 2 (L × K) is the vector which takes value 1 at (0, 0) and 0 everywhere else. We claim that G(ϕ, L, K) is a Gabor pseudo-dual for G(g, L, K). √ In fact, a direct calculation shows that Θh = DΘψ . Thus we have
D. Han / Journal of Functional Analysis 256 (2009) 129–148
Θh Θh∗ =
√
145
√ √ √ DΘψ Θψ∗ D = DI D = D
and Θh ΘS −1 g =
√ DΘψ ΘS∗−1 g = 0,
where the last identity uses the fact that span G(ψ, L, K) ⊥ span G(S −1 g, L, K). Hence we have span G(h, L, K) ⊥ span G(S −1 g, L, K), which implies that G(S −1 g + h, L, K) is a pseudo-dual of G(g, L, K). Moreover, from ΘS −1 g+h ΘS∗−1 g+h = ΘS −1 g ΘS∗−1 g + Θh ΘS∗−1 g + ΘS −1 g Θh∗ + Θh Θh∗ = ΘS −1 g ΘS∗−1 g + 0 + 0 + D = aI, we have that ΘS∗−1 g+h ΘS −1 g+h = aP , where P is the orthogonal projection from L2 (Rd ) onto span G(S −1 g + h, L, K). Hence G(S −1 g + h, L, K) is a tight subspace frame, as claimed.
2
Lemma 3.5. Assume that L = AZd and K = BZd with AB ∗ = I . If G(g, L, K) is a subspace Gabor frame for M := span G(g, L, K), then there exists a unique function ϕ ∈ M such that G(ϕ, L, K) is a dual of G(g, L, K) (i.e., the inside dual is unique). Proof. Under the condition AB ∗ = I we have that {E Tκ : ∈ L, κ ∈ K} is an abelian group of unitary operators. Thus the lemma is a special case of Corollary 3.11 in [12]. 2 Lemma 3.6. Assume that L = AZd and K = BZd with AB ∗ = I . If G(ϕ, L, K) is a pseudodual for a subspace Gabor frame G(g, L, K), the span G(h, L, K) ⊥ span G(g, L, K), where h = ϕ − S −1 g and S is the frame operator for G(g, L, K). Proof. Let P be the orthogonal projection onto span G(g, L, K). Since G(S −1 g + h, L, K) is a pseudo-dual for G(g, L, K), it follows that G(S −1 g + P h, L, K) is also (inside) pseudo-dual for G(g, L, K). Thus, from Lemma 3.5, we have that the inside dual for G(g, L, K) is unique. So P h = 0, which implies that h ⊥ span G(g, L, K) and so span G(h, L, K) ⊥ span G(g, L, K). 2 Lemma 3.7. (See [16].) Let U be a countable abelian group of unitary operators on a Hilbert space H . Assume that {U η: U ∈ U} is a Parseval frame for H and {U ξ : U ∈ U} is a Bessel sequence. Then there exists a bounded operator V on H such that V η = ξ and V commutes with every operator in U . Lemma 3.8. Assume that L = AZd and K = BZd with AB ∗ = I . If G(h, L, K) is Bessel such that span G(h, L, K) ⊥ span G(g, L, K), then Range(Θh ) ⊥ Range(Θg ). Proof. From Lemma 3.1, we can dilate the Parseval subspace Gabor frame G(S −1/2 g, L, K) to an orthonormal basis, say G(ψ, L, K), for L2 (Rd ). Let P be the orthogonal projection onto M := span G(g, L, K). Then we have P ψ = S −1/2 g and G(P ⊥ ψ, L, K) is a Parseval frame for M ⊥ . Note that G(h, L, K) is a Bessel sequence contained in M ⊥ . So, by Lemma 3.7, there exists a bounded operator V such that V (P ⊥ ψ) = h, V (M) = {0} and V commutes with E
146
D. Han / Journal of Functional Analysis 256 (2009) 129–148
and Tκ for all ∈ L, κ ∈ K. Thus we have Θh = Θψ P ⊥ V ∗ . Also note that ΘS −1/2 g = Θψ P and Θψ∗ Θψ = I . Now for any f1 , f2 ∈ L2 (Rd ) we have Θh f1 , ΘS −1/2 g f2 = Θψ P ⊥ V ∗ f1 , Θψ Pf1 = Θψ∗ Θψ P ⊥ V ∗ f1 , Pf1 = P ⊥ V ∗ f1 , Pf1 = 0.
2
Proposition 3.9. Assume that L = AZd and K = BZd with AB ∗ = I . Then a subspace Gabor frame has a tight Gabor pseudo-dual if and only it itself is already tight. Proof. Suppose that a subspace Gabor frame G(g, L, K) has a tight Gabor pseudo-dual G(ψ, L, K). Let h = ψ − S −1 g, where S is the frame operator for G(g, L, K). Lemmas 3.6 and 3.8 imply that span G(h, L, K) ⊥ span G(g, L, K) and Range(Θh ) ⊥ Range(Θg ). So Θh ΘS∗−1 g = 0 and Θh∗ ΘS −1 g = 0. Since G(S −1 g + h, L, K) is a tight subspace Gabor frame, we have that (ΘS −1 g + Θh )∗ (ΘS −1 g + Θh ) = aP for some a > 0, where P is the orthogonal projection onto span G(S −1 g + h, L, K). Thus ΘS∗−1 g ΘS −1 g + Θh∗ Θh = aP . Let U = a1 ΘS∗−1 g ΘS −1 g and V = a1 Θh∗ Θh . Then U V = V U = 0, U = U ∗ , V = V ∗ and U + V =
P is an orthogonal projection. Hence U = a1 ΘS∗−1 g ΘS −1 g must be an orthogonal projection,
which implies that G(S −1 g, L, K) is tight. Therefore G(g, L, K) is a tight subspace Gabor frame. 2 Lemma 3.10. Assume that |det(AB)| 1 and let G(g, L, K) be a subspace Gabor frame for M and S be its frame operator. If δΛ ([ker(S −1 · PM − S −1 )]⊥ ∩ M) I − δΛ (L2 (Rd )), then G(g, L, K) has a tight Gabor pseudo-dual. Proof. Let P be the orthogonal projection onto M. By Lemma 3.1, there exists a Parseval subspace Gabor frame G(g, L, K) for M ⊥ such that G(g + bh, L, K) is a frame of L2 (Rd ), where b = √ 1 −1 . Let S˜ be the frame operator for G(g + bh, L, K). Then we have S˜ = SP + b2 P ⊥ S
and S˜ −1 = S −1 P + S −1 P ⊥ . So S˜ −1 = S −1 and ⊥ −1 ⊥ −1 = ker S · PM − S −1 ∩ M, ker S˜ · I − S˜ −1 which then implies that ⊥ I − δΛ (H ). δΛ ker S˜ −1 · I − S˜ −1 By Theorem 2.7, G(g + bh, L, K) has a tight Gabor dual, say G(ψ, L, K). Then we have
D. Han / Journal of Functional Analysis 256 (2009) 129–148
f=
f, ψλ (gλ + bhλ ),
147
f ∈ L2 (Rd ).
λ∈ K×L
Restrict f in M and apply P to both sides of the above identity, we get f=
f, (P ψ)λ gλ ,
f ∈ M.
λ∈K×L
Hence G(P ψ, L, K) is a dual of G(g, L, K). Moreover, since G(ψ, L, K) is a tight frame for L2 (Rd ) and M ⊆ L2 (Rd ), we have that G(P ψ, L, K) is a tight frame for M. 2 Proposition 3.11. Assume that L = αZd , K = βZd with αβ irrational and |αβ| 1. If G(g, L, K) is a subspace Gabor frame such that S −1/2 g2 1 − |det(AB)|, then G(g, L, K) has a tight (inside) Gabor pseudo-dual. Proof. Under the assumption, we have that MΛ is a factor von Neumann algebra. Let τΛ (T ) = tr(T )I , where tr(·) is a trace defined by tr(T ) = T e0 , e0 for all T ∈ MΛ . Similar to the proof of Theorem 2.3, we have δΛ (M) = S −1/2 g2 · I and δΛ (L2 (Rd )) = |det(AB)| · I. Thus ⊥ δΛ ker S −1 · PM − S −1 ∩ M δΛ (M) 2 = S −1/2 g · I I − det(AB) · I = I − δΛ L2 Rd . Therefore, by Proposition 3.10, G(g, L, K) has a tight (inside) Gabor pseudo-dual.
2
Acknowledgment The author would like to thank the anonymous reviewer for many helpful suggestions that helped improve the presentation of this paper. References [1] A. Aldroubi, D. Larson, Wai-Shing Tang, E. Weber, Geometric aspects of frame representations of abelian groups, Trans. Amer. Math. Soc. 356 (12) (2004) 4767–4786. [2] P. Casazza, Modern tools for Weyl–Heisenberg (Gabor) frame theory, Adv. Imag. Elect. Phys. 115 (2001) 1–127. [3] P. Casazza, O. Christensen, Weyl–Heisenberg frames for subspaces of L2 (R), Proc. Amer. Math. Soc. 129 (2001) 145–154. [4] P. Casazza, O. Christensen, A. Lindner, R. Vershynin, Frames and the Feichtinger conjecture, Proc. Amer. Math. Soc. 133 (2005) 1025–1033. [5] P. Casazza, J. Kovaˇcevi´c, Equal-norm tight frames with erasures, Adv. Comput. Math. 18 (2003) 387–430. [6] P. Casazza, J. Tremain, The Kadison–Singer problem in mathematics and engineering, Proc. Natl. Acad. Sci. USA 103 (7) (2006) 2032–2039. [7] R.J. Duffin, A.C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc. 72 (1952) 341–366. [8] M. Frank, D. Larson, Frames in Hilbert C ∗ -modules and C ∗ -algebras, J. Operator Theory 48 (2002) 273–314. [9] J.-P. Gabardo, D. Han, Subspace Weyl–Heisenberg frames, J. Fourier Anal. Appl. 7 (2001) 419–433. [10] J.-P. Gabardo, D. Han, Frame representations for group-like unitary operator systems, J. Operator Theory 49 (2003) 223–244. [11] J.-P. Gabardo, D. Han, Aspects of Gabor analysis and operator algebras, in: Advances in Gabor Analysis, in: Appl. Numer. Harmon. Anal., Birkhäuser Boston, Boston, MA, 2003, pp. 129–152.
148
D. Han / Journal of Functional Analysis 256 (2009) 129–148
[12] J.-P. Gabardo, D. Han, The uniqueness of the dual of Weyl–Heisenberg subspace frames, Appl. Comput. Harmon. Anal. 17 (2004) 226–240. [13] J.-P. Gabardo, D. Han, Balian–Low phenomenon for subspace Gabor frames, J. Math. Phys. 45 (2004) 3362–3378. [14] D. Gabor, Theory of communication, J. Inst. Electr. Engrg. (London) 93 (1946) 429–457. [15] V. Goyal, J. Kovaˇcevi´c, J. Kelner, Quantized frame expansions with erasures, Appl. Comput. Harmon. Anal. 10 (2001) 203–233. [16] D. Han, Approximations for Gabor and wavelet frames, Trans. Amer. Math. Soc. 355 (2003) 3329–3342. [17] D. Han, Frame representations and Parseval duals with applications to Gabor frames, Trans. Amer. Math. Soc. 360 (2008) 3307–3326. [18] D. Han, D. Larson, Frames, bases and group parametrizations, Mem. Amer. Math. Soc. 697 (2000). [19] D. Han, D. Larson, Wandering vector multipliers for unitary groups, Trans. Amer. Math. Soc. 353 (2001) 3347– 3370. [20] D. Han, D. Larson, Frame duality properties for projective unitary representations, Bull. London Math. Soc. 40 (2008) 685–695. [21] D. Han, Y. Wang, Lattice tiling and Weyl–Heisenberg frames, Geom. Funct. Anal. 11 (2001) 742–758. [22] D. Han, Y. Wang, The existence of Gabor bases, in: Contemp. Math., vol. 345, Amer. Math. Soc., Providence, RI, 2004, pp. 183–192. [23] C. Heil, History and evolution of the Density Theorem for Gabor frames, J. Fourier Anal. Appl. 13 (2) (2007) 113–166. [24] C. Heil, P. Jorgensen, D. Larson (Eds.), Wavelets, Frames and Operator Theory, Contemp. Math., vol. 345, Amer. Math. Soc., Providence, RI, 2004. [25] R. Kadison, J. Ringrose, Fundamentals of the Theory of Operator Algebras, vols. I and II, Academic Press, New York, 1983; Orlando, FL, 1985. [26] D. Larson, Frames and wavelets from an operator-theoretic point of view, in: Contemp. Math., vol. 228, Amer. Math. Soc., Providence, RI, 1998, pp. 201–218. [27] S. Li, Irregular sampling in a generic subspace, Sampl. Theory Signal Image Process. 3 (2) (2004) 101–117. [28] S. Li, H. Ogawa, Pseudo-duals of frames with applications, Appl. Comput. Harmon. Anal. 11 (2001) 289–304. [29] S. Li, H. Ogawa, Pseudoframes for subspaces with applications, J. Fourier Anal. Appl. 10 (2004) 409–431. [30] S. Li, H. Ogawa, Optimal noise suppression: A geometric nature of pseudoframes for subspaces, Adv. Comput. Math. 28 (2) (2008) 141–155. [31] M. Takesaki, Theory of Operator Algebras I, Springer, New York, 1979. [32] J. von Neumann, Mathematical Foundations of Quantum Mechanics, in: Princeton Univ. Press, Princeton, NJ, 1932/1945/1955.
Journal of Functional Analysis 256 (2009) 149–178 www.elsevier.com/locate/jfa
Isoperimetry and symmetrization for logarithmic Sobolev inequalities Joaquim Martín a,1 , Mario Milman b,∗ a Department of Mathematics, Universitat Autònoma de Barcelona, Spain b Department of Mathematics, Florida Atlantic University, FL, USA
Received 7 March 2008; accepted 2 September 2008 Available online 26 September 2008 Communicated by L. Gross
Abstract Using isoperimetry and symmetrization we provide a unified framework to study the classical and logarithmic Sobolev inequalities. In particular, we obtain new Gaussian symmetrization inequalities and connect them with logarithmic Sobolev inequalities. Our methods are very general and can be easily adapted to more general contexts. © 2008 Elsevier Inc. All rights reserved. Keywords: Logarithmic Sobolev inequalities; Symmetrization; Isoperimetric inequalities
Contents 1. 2.
3. 4.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gaussian rearrangements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Gaussian profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Rearrangements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Rearrangement-invariant spaces . . . . . . . . . . . . . . . . . . . . . . Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Pólya–Szegö principle is equivalent to the isoperimetric inequality
. . . . . . .
. . . . . . .
. . . . . . .
* Corresponding author.
E-mail addresses:
[email protected] (J. Martín),
[email protected] (M. Milman). URL: http://www.math.fau.edu/milman (M. Milman). 1 Supported in part by MTM2004-02299 and by CURE 2005SGR00556 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.09.001
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
150 156 156 158 159 159 162
150
J. Martín, M. Milman / Journal of Functional Analysis 256 (2009) 149–178
5. 6.
The Pólya–Szegö principle implies Gross’ inequality . Poincaré type inequalities . . . . . . . . . . . . . . . . . . . 6.1. Feissner type inequalities . . . . . . . . . . . . . . . 7. On limiting embeddings and concentration . . . . . . . . 8. Symmetrization by truncation of entropy inequalities . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
165 168 173 173 176 177
1. Introduction The classical L2 -Sobolev inequality states that |∇f | ∈ L2 Rn
⇒
∗ f ∈ Lpn Rn ,
where
1 1 1 = − . ∗ pn 2 n
Consequently, limn→∞ pn∗ = 2 and, therefore, the improvement on the integrability of f disappears as n → ∞. On the other hand, Gross [20] showed that, if one replaces dx by the Gaussian 2 measure dγn (x) = (2π)−n/2 e−|x| /2 dx, we have
f (x)2 lnf (x) dγn (x)
∇f (x)2 dγn (x) + f 2 lnf 2 . 2
(1.1)
This is Gross’ celebrated logarithmic Sobolev inequality (= lS inequality), the starting point of a new field, with many important applications to PDEs, functional analysis, probability, etc. (as a sample, and only a sample, we mention [2,7,13,24], and the references therein). The inequality (1.1) gives a logarithmic improvement on the integrability of f , with constants independent of n, that persists as n → ∞, and is best possible. Moreover, rescaling (1.1) leads to Lp variants of this inequality, again with constants independent of the dimension (cf. [20]),
f (x)p lnf (x) dγn (x)
p p Re Nf, fp + f p lnf p , 2(p − 1)
where f, g = f g¯ dγn , Nf, f = |∇f (x)|2 dγn (x), fp = (sgn(f ))|f |p−1 . In a somewhat different direction, Feissner’s thesis [17] under Gross, takes up the embedding implied by (1.1), namely W21 Rn , dγn ⊂ L2 (LogL) Rn , dγn , where the norm of W21 (Rn , dγn ) is given by f W 1 (Rn ,dγn ) = ∇f L2 (Rn ,dγn ) + f L2 (Rn ,dγn ) , 2
J. Martín, M. Milman / Journal of Functional Analysis 256 (2009) 149–178
and extends it to Lp , even Orlicz spaces. A typical result2 from [17] is given by Wp1 Rn , dγn ⊂ Lp (LogL) Rn , dγn , p 2.
151
(1.2)
The connection between lS inequalities and the classical Sobolev estimates has been investigated intensively. For example, it is known that (1.1) follows from the classical Sobolev estimates with sharp constants (cf. [3,4] and the references therein). In a direction more relevant for our development here, using the argument of Ehrhard [15], we will show, in Section 5 below, that (1.1) follows from the symmetrization inequality of Pólya–Szegö for Gaussian measure (cf. [16] and Section 4) ∇f ◦ L2 (R,dγ1 ) ∇f L2 (Rn ,dγn ) , where f ◦ is the Gaussian symmetric rearrangement of f with respect to Gaussian measure (cf. Section 2 below). The purpose of this paper is to give a new approach to lS inequalities through the use of symmetrization methods. While symmetrization methods are a well-established tool to study Sobolev inequalities, through the combination of symmetrization and isoperimetric inequalities we uncover new rearrangement inequalities and connections, that provide a context in which we can treat the classical and logarithmic Sobolev inequalities in a unified way. Moreover, with no extra effort we are able to extend the functional lS inequalities to the general setting of rearrangement-invariant spaces. In particular, we highlight a new extreme embedding which clarifies the connection between lS, the concentration phenomenon and the John–Nirenberg lemma. Underlying this last connection is the apparently new observation that concentration inequalities self-improve, a fact we shall treat in detail in a separate paper (cf. [30]). The key to our method are new symmetrization inequalities that involve the isoperimetric profile and, in this fashion, are strongly associated with geometric measure theory. In previous papers (cf. [31] and the references therein) we had obtained the corresponding inequalities in the classical case without making explicit reference to the Euclidean isoperimetric profile. Using isoperimetry we are able to connect each of the classical inequalities with their corresponding (new) Gaussian counterparts. We will show that the difference between the classical and the new Gaussian inequalities can be simply explained in terms of the difference of the corresponding isoperimetric profiles. In particular, in the Gaussian case, the isoperimetric profile is independent of the dimension, and this accounts for the fact that our rearrangement inequalities in this setting have this property. Another bonus is that our method is rather general, and amenable to considerable generalization: to Sobolev inequalities in general measure spaces, metric Sobolev spaces, even discrete Sobolev spaces. We hope to return to some of these developments elsewhere. To describe more precisely our results let us recall that the connection between isoperimetry and Sobolev inequalities goes back to the work of Maz’ya and Federer and can be easily explained by combining the formula connecting the gradient and the perimeter (cf. [27]): ∞ ∇f 1 =
Per |f | > t dt,
(1.3)
0
2 For the most part the classical work on functional lS inequalities has focused on L2 , or more generally, Lp and Orlicz spaces.
152
J. Martín, M. Milman / Journal of Functional Analysis 256 (2009) 149–178
with the classical Euclidean isoperimetric inequality n−1 1/n Per |f | > t nn |f | > t n ,
(1.4)
where n = volume of unit ball in Rn . Indeed, combining (1.4) and (1.3) yields the sharp form of the Gagliardo–Nirenberg inequality 1/n
(n − 1)n f
n ,1
L n−1 (Rn )
∇f L1 (Rn ) .
(1.5)
In [31], we modified Maz’ya’s truncation method,3 to develop a sharp tool to extract symmetrization inequalities from Sobolev inequalities like (1.5). In particular, we showed that, given any rearrangement-invariant norm (r.i. norm) · , the following optimal Sobolev inequality4 holds (cf. [32]): ∗∗ f (t) − f ∗ (t) t −1/n c(n, X)∇f ,
f ∈ C0∞ Rn .
(1.6)
An analysis of the role that the power t −1/n plays in this inequality led us to connect (1.6) to isoperimetric profile of (Rn , dx). In fact, observe that we can formulate (1.4) as Per(A) In voln (A) , 1/n
where In (t) = nn t (n−1)/n is the “isoperimetric profile” or the “isoperimetric function,” and equality is achieved for balls. The corresponding isoperimetric inequality for Gaussian measure (i.e. Rn equipped with 2 Gaussian measure dγn (x) = (2π)−n/2 e−|x| /2 dx), and the solution to the Gaussian isoperimetric problem, was obtained by Borell [11] and Sudakov–Tsirelson [34], who showed5 that Per(A) I γn (A) , with equality achieved for half-spaces,6 and where I = Iγ is the Gaussian profile7 (cf. (2.2) below for the precise definition of I ). To highlight a connection with the lS inequalities, we only note here that I has the following asymptotic formula near the origin (say t 1/2, see Section 2 below),
1 1/2 I (t) t log . t
(1.7)
3 We termed this method “symmetrization via truncation.” 4 This inequality is optimal and includes the problematic borderline “end points” of the Lp theory. 5 Erhard [14] provides an approach using symmetrization. Erhard also proves using this method a Gaussian version
of the Brunn–Minkowski inequality but only for convex bodies. This restriction remained an open problem until it was finally removed by Borell [12]. For a nice survey concerning these inequalities prior to 2002, see [22]. 6 In some sense one can consider half-spaces as balls centered at infinity. 7 In principle I could depend on n but by the very definition of half-spaces it follows that the Gaussian isoperimetric profile is dimension-free.
J. Martín, M. Milman / Journal of Functional Analysis 256 (2009) 149–178
153
As usual, the symbol f g will indicate the existence of a universal constant c > 0 (independent of all parameters involved) so that (1/c)f g cf , while the symbol f g means that f cg. With this background one may ask: what is the Gaussian replacement of the Gagliardo– Nirenberg inequality (1.5)? The answer was provided by Ledoux who showed (cf. [23]) ∞
I λf (s) ds
|∇f | dγn (x),
f ∈ Lip Rn .
(1.8)
Rn
0
In fact, following the steps of the proof we indicated for (1.5), but using the Gaussian profile instead, we readily arrive at Ledoux’s inequality. This given we were therefore led to apply our method of symmetrization by truncation to the inequality (1.8). We obtained the following counterpart of (1.6) ∗∗ t f (t) − f ∗ (t) |∇f |∗∗ (t), I (t) here f ∗ denotes t the non-increasing rearrangement of f with respect to Gaussian measure, and f ∗∗ (t) = 1t 0 f ∗ (s) ds. Further analysis showed that, in agreement with the Euclidean case we had worked out in [31], all these inequalities are in fact equivalent8 to the isoperimetric inequality9 (cf. Section 3 below): Theorem 1. The following statements are equivalent (all rearrangements are with respect to Gaussian measure): (i) Isoperimetric inequality: For every Borel set A ⊂ Rn , with 0 < γn (A) < 1, Per(A) I γn (A) . (ii) Ledoux’s inequality: For every Lipschitz function f on Rn , ∞
I λf (s) ds
0
∇f (x) dγn (x).
(1.9)
Rn
(iii) Talenti’s inequality10 (Gaussian version): For every Lipschitz function f on Rn , d (−f ) (s)I (s) ds ∗
∇f (x) dγn (x).
(1.10)
{|f |>f ∗ (s)}
8 It is somewhat paradoxical that (1.1), because of the presence of squares, needs a special treatment and is not, as far
as we know, equivalent to the isoperimetric inequality (for a partial converse in this direction cf. [24]). 9 The equivalence between (i) and (ii) in Theorem 1 above is due to Ledoux [24], see also [9]. 10 In connection with the Euclidean version of this inequality see also [26].
154
J. Martín, M. Milman / Journal of Functional Analysis 256 (2009) 149–178
(iv) Oscillation inequality (Gaussian version): For every Lipschitz function f on Rn , ∗∗ t |∇f |∗∗ (t). f (t) − f ∗ (t) I (t)
(1.11)
This formulation coincides with the corresponding Euclidean result we had obtained in [31], and thus, in some sense, unifies the classical and Gaussian Sobolev inequalities. More precisely, by specifying the corresponding isoperimetric profile we automatically derive the correct results in either case. Thus, for example, if in (1.9) we specify the Euclidean isoperimetric profile we get the Gagliardo–Nirenberg inequality, in (1.10) we get Talenti’s original inequality [35] and in (1.11) we get the rearrangement inequality of [1]. Underlying all these inequalities is the so called Pólya–Szegö principle. The Lp Gaussian versions of this principle had been obtained earlier by Ehrhard11 [16]. We obtain here a general version of the Pólya–Szegö principle (cf. [18] where the Euclidean case was stated without proof), what may seem surprising at first is the fact that, in our formulation, the Pólya–Szegö principle is, in fact, equivalent to the isoperimetric inequality (cf. Section 4). Theorem 2. The following statements are equivalent: (i) Isoperimetric inequality: for every Borel set A ⊂ Rn , with 0 < γn (A) < 1 Per(A) I γn (A) . (ii) Pólya–Szegö principle: for every Lipschitz function f on Rn , |∇f ◦ |∗∗ (s) |∇f |∗∗ (s). Very much like Euclidean symmetrization inequalities lead to optimal Sobolev and Poincaré inequalities and embeddings (cf. [29,31] and the references therein), the new Gaussian counterpart (1.11) we obtain here leads to corresponding optimal Gaussian Sobolev–Poincaré inequalities as well. The corresponding analog of (1.6) is: given any rearrangement-invariant space X on the interval (0, 1), we have the optimal inequality, valid for Lip functions (cf. Section 6 below), ∗∗ I (t) ∗ ∇f X . f LS(X) := f (t) − f (t) t
(1.12)
X
The spaces LS(X) defined in this fashion are not necessarily normed, although often they are equivalent to normed spaces.12 As a counterpart to this defect we remark that, since the Gaussian isoperimetric profile is independent of the dimension, the inequalities (1.12) are dimension-free. In particular, we note the following result here (cf. Sections 6 and 6.1 below for a detailed analysis). 11 For comparison we mention that Ehrhard’s results are formulated in terms of increasing rearrangements. 12 For the Euclidean case a complete study of the normability of these spaces has been recently given in [33].
J. Martín, M. Milman / Journal of Functional Analysis 256 (2009) 149–178
155
Theorem 3. Let X, Y be two r.i. spaces. Then, the following statements are equivalent: (i) For every Lipschitz function f on Rn f − f ∇f X .
(1.13)
Y
(ii) For every positive function f ∈ X with supp f ⊂ (0, 1/2), 1 ds f (s) f X . I (s) Y
t
Part II. Let α X and α X be the lower and the upper Boyd indices of X (see Section 2 below). If α X > 0, then the following statement is equivalent to (i) and (ii) above: ∗ I (t) . f (t) f Y t
(iii)
X
In particular, if Y is a r.i. space such that (1.13) holds, then ∗ I (t) . f Y f (t) t X If 0 = α X < α X < 1, then the following statement is equivalent to (i) and (ii) above: f Y f LS(X) + f L1 .
(iv)
In particular, if Y is a r.i. space such that (1.13) holds, then f Y f LS(X) + f L1 . To recognize the logarithmic Sobolev inequalities that are encoded in this fashion we use the asymptotic property (1.7) of the isoperimetric profile I (s) and suitable Hardy type inequalities. Our result improves upon (1.2). Corollary 1. (See Section 6.1 below.) Let X = Lp , 1 p < ∞. Then, 1
∗
f−
f
I (s) (s) s
p
ds
∇f (x)p dγn (x).
0
In particular, 1 0
p p 1 p/2 ∇f (x) dγn (x) + f (x) dγn (x). f (s) log ds s ∗
p
156
J. Martín, M. Milman / Journal of Functional Analysis 256 (2009) 149–178
In the final section of this paper we discuss briefly a connection with concentration inequalities. We refer to Ledoux [25] for a detailed account, and detailed references, on the well-known connection between lS inequalities and concentration. In our setting, concentration inequalities can be derived from a limiting case of the functional lS inequalities. Namely, for X = L∞ , (1.12) yields
∗∗ I (t) ∗ f LS(L∞ ) = sup f (t) − f (t) sup |∇f |∗∗ (t) = f Lip . t t t<1 We denote the new space Llog1/2 (∞, ∞) (cf. (7.2) below). Through the asymptotics of I (s) we see that Llog1/2 (∞, ∞) is a variant of the Bennett–DeVore–Sharpley [5] space13 L(∞, ∞) = rearrangement-invariant hull of BMO. As it was shown in [5], the definition of L(∞, ∞) is a reformulation of the John–Nirenberg inequality and thus yields exponential integrability. Llog1/2 (∞, ∞) allows us to be more precise about the level of exponential integrability implied by our inequalities. In this fashion, via symmetrization and isoperimetry we have connected the John–Nirenberg inequality with the lS inequalities. In a similar manner we can also treat the embedding into L∞ using the fact that the space L(∞, 1) = L∞ (cf. [1]). Finally, let us state that our main focus in this paper was to develop our methods and illustrate their reach, but without trying to state the results in their most general form. We refer the reader to [28] for a development of our results in the metric setting. The section headers are self-explanatory and provide the organization of the paper. 2. Gaussian rearrangements In this section we review well-known results and establish the basic notation concerning Gaussian rearrangements that we shall use in this paper. 2.1. Gaussian profile Recall that the n-dimensional Gaussian measure on Rn is defined by dγn (x) = (2π)−n/2 e−
|x|2 2
φn (x) = (2π)−n/2 e−
|x|2 2
dx1 . . . dxn .
It is also convenient to let ,
x ∈ Rn ,
13 L(∞, ∞)(Rn , dγ ) is defined by the condition n
1 sup f ∗∗ (t) − f ∗ (t) = sup 0
t 0
∗ f (s) − f ∗ (t) ds < ∞.
J. Martín, M. Milman / Journal of Functional Analysis 256 (2009) 149–178
157
and therefore
φn (x) dx = γn Rn = 1.
(2.1)
Rn
Let Φ : R → (0, 1) be the increasing function given by r Φ(r) =
φ1 (t) dt.
−∞
The Gaussian perimeter of a set is defined by Per(Ω) = φn (x) dHn−1 (x), ∂Ω
where dHn−1 (x) denotes the Hausdorff (n − 1)-dimensional measure. The isoperimetric inequality now reads Per(Ω) I γn (Ω) , where I is the Gaussian isoperimetric function given by (cf. [24,25]) I (t) = φ1 Φ −1 (t) , t ∈ [0, 1].
(2.2)
It was shown by Borell [11] and Sudakov–Tsirelson [34] that for the solution of the isoperimetric problem for Gaussian measures we must replace balls by half-spaces. We choose to work with half-spaces defined by Hr = x = (x1 , . . . , xn ): x1 < r , r ∈ R. Therefore by (2.1), r γn (Hr ) =
φ1 (t) dt.
−∞
Given a measurable set Ω ⊂ Rn , we let Ω ◦ be the half-space defined by Ω ◦ = Hr , where r ∈ R is selected so that Φ(r) = γn (Hr ) = γn (Ω). In other words, r is defined by r = Φ −1 γn (Ω) .
158
J. Martín, M. Milman / Journal of Functional Analysis 256 (2009) 149–178
It follows that Per(Ω) Per Ω ◦ = φ1 Φ −1 γn (Ω) . Concerning the Gaussian profile I we note here some useful properties for our development in this paper (cf. [24] and the references therein). First, we note that, by direct computation, we have that I satisfies I =
−1 , I
(2.3)
and, as a consequence of (2.1), we also have the symmetry I (t) = I (1 − t),
t ∈ [0, 1].
Moreover, from (2.3) we deduce that I (s) is concave, has a maximum at t = 1/2 with I (1/2) = (0) (2π)−1/2 , and since I (0) = 0, then I (s)−I = I (s) s s is decreasing; summarizing: I (s) is decreasing on (0, 1) s
and
s is increasing on (0, 1). I (s)
(2.4)
Logarithmic Sobolev inequalities are connected with the asymptotic behavior of I (t) at the origin (or at 1 by symmetry) (cf. [24]) lim
t→0
I (t) t (2 log 1t )1/2
= 1.
(2.5)
2.2. Rearrangements Let f : Rn → R. We define the non-increasing, right-continuous, Gaussian distribution function of f , by means of λf (t) = γn x ∈ Rn : f (x) > t ,
t > 0.
The rearrangement of f with respect to Gaussian measure, f ∗ : (0, 1] → [0, ∞), is then defined, as usual, by f ∗ (s) = inf t 0: λf (t) s ,
t ∈ (0, 1].
In the Gaussian context we replace the classical Euclidean spherical decreasing rearrangement by a suitable Gaussian substitute, f ◦ : Rn → R defined by f ◦ (x) = f ∗ Φ(x1 ) . It is useful to remark here that, as in the Euclidean case, f ◦ is equimeasurable with f :
J. Martín, M. Milman / Journal of Functional Analysis 256 (2009) 149–178
159
γn x: f ◦ (x) > t = γn x: f ∗ Φ(x1 ) > t = γn x: Φ(x1 ) λf (t) = γn x: x1 Φ −1 λf (t) = γ1 −∞, Φ −1 λf (t) = λf (t). 2.3. Rearrangement-invariant spaces Finally, let us recall briefly the basic definitions and conventions we use from the theory of rearrangement-invariant (r.i.) spaces, and refer the reader to [6] for a complete treatment. A Banach function space X = X(Rn ) is called a r.i. space if g ∈ X implies that all functions f with the same rearrangement with respect to Gaussian measure, i.e. such that f ∗ = g ∗ , also belong to X, and, moreover, f X = gX . The space X can then be “reduced” to a onedimensional space (which by abuse of notation we still denote by X), X = X(0, 1), consisting of all g : (0, 1) → R such that g ∗ (t) = f ∗ (t) for some function f ∈ X. Typical examples are the Lp -spaces and Orlicz spaces. We shall usually formulate conditions on r.i. spaces in terms of the Hardy operators defined by 1 Pf (t) = t
t
1 Qf (t) =
f (s) ds;
f (s)
ds . s
t
0
It is well known (see for example [6, Chapter 3]), that if X is a r.i. space, P (respectively Q) is bounded on X if and only if the upper Boyd index α X < 1 (respectively the lower Boyd index α X > 0). We notice for future use that if X is a r.i. space such that α X > 0, then the operator ˜ (t) = 1 + log 1/t 1/2 Qf
1 f (s) t
ds s(1 + log 1/s)1/2
is bounded on X. Indeed, pick α X > a > 0, then since t a (1 + log 1/t)1/2 is increasing near zero, we get a 1/2 ˜ (t) = t (1 + log 1/t) Qf ta
1 t
ds 1 f (s) a 1/2 t s(1 + log 1/s)
1 s a f (s)
ds = Qa f (t), s
t
and Qa is bounded on X since α X > a (see [6, Chapter 3]). 3. Proof of Theorem 1 The proof follows very closely the development in [31] with appropriate changes.
160
J. Martín, M. Milman / Journal of Functional Analysis 256 (2009) 149–178
(i) ⇒ (ii). By the co-area formula (cf. [27]) and the isoperimetric inequality
∇f (x) dγn (x) =
∞ 0
∞ =
φn (x) dHn−1 (x) ds
{|f |=s}
Per |f | > s ds
0
∞
I λf (s) ds.
0
(ii) ⇒ (iii). Let 0 < t1 < t2 < ∞. The truncations of f are defined by ftt12 (x) =
t − t 2 1 |f (x)| − t1 0
if |f (x)| > t2 , if t1 < |f (x)| t2 , if |f (x)| t1 .
Applying (1.9) to ftt12 we obtain, ∞
I λf t2 (s) ds t1
t2 ∇ft (x) dγn (x). 1
Rn
0
We obviously have t2 ∇ft = |∇f |χ{t 1
1 <|f |t2 }
,
and, moreover, ∞
I λf t2 (s) ds =
t 2 −t1
I λf t2 (s) ds.
t1
0
t1
(3.1)
0
Observe that for 0 < s < t2 − t1 γn f (x) t2 λf t2 (s) γn f (x) > t1 . t1
Consequently, we have t 2 −t1
I λf t2 (s) ds (t2 − t1 ) min I γn |f | t2 , I γn |f | > t1 . t1
0
For s > 0 and h > 0, pick t1 = f ∗ (s + h), t2 = f ∗ (s), then s γn f (x) f ∗ (s) λf t2 (s) γn f (x) > f ∗ (s + h) s + h. t1
(3.2)
J. Martín, M. Milman / Journal of Functional Analysis 256 (2009) 149–178
161
Combining (3.1) and (3.2) we have,
∗ f (s) − f ∗ (s + h) min I (s + h), I (s)
∇f (x) dγn (x)
{f ∗ (s+h)<|f |f ∗ (s)}
h
|∇f |∗ (t) dt,
(3.3)
0
whence f ∗ is locally absolutely continuous. Thus, 1 (f ∗ (s) − f ∗ (s + h)) min I (s + h), I (s) h h
∇f (x) dγn (x).
{f ∗ (s+h)<|f |f ∗ (s)}
Letting h → 0 we obtain (1.10). (iii) ⇒ (iv). We will integrate by parts. Let us note first that using (3.3) we have that, for 0 < s < t, s f ∗ (s) − f ∗ (t)
s min(I (s), I (t))
t−s |∇f |∗ (s) ds.
(3.4)
0
Now, f
∗∗
1 (t) − f (t) = t ∗
t
∗ f (s) − f ∗ (t) ds
0
t t 1 ∗ ∗ ∗ = s f (s) − f (t) 0 + s(−f ) (s) ds t 0
=
1 t
t
s(−f ∗ ) (s) ds
0
= A(t), where the integrated term [s(f ∗ (s) − f ∗ (t))]t0 vanishes on account of (3.4). By (2.4), s/I (s) is increasing on 0 < s < 1, thus 1 A(t) I (t)
t
I (s)(−f ∗ ) (s) ds
0
1 I (t)
t
0
∂ ∂s
{|f |>f ∗ (s)}
∇f (x) dγn (x) ds
(by (1.10))
162
J. Martín, M. Milman / Journal of Functional Analysis 256 (2009) 149–178
1 I (t)
∇f (x) dγn (x)
{|f |>f ∗ (s)}
t |∇f |∗∗ (t). I (t)
(iv) ⇒ (i). Let A be a Borel set with 0 < γn (A) < 1. We may assume without loss that Per(A) < ∞. By definition we can select a sequence {fn }n∈N of Lip functions such that fn −→ χA , and 1 L
Per(A) = lim sup∇fn 1 . n→∞
Therefore, lim sup I (t) fn∗∗ (t) − fn∗ (t) lim sup n→∞
t
n→∞
∇fn (s)∗ ds
0
lim sup n→∞
|∇fn | dγn
= Per(A).
(3.5)
As is well known fn −→ χA implies that (cf. [19, Lemma 2.1]): 1 L
fn∗∗ (t) → χA∗∗ (t), fn∗ (t) → χA∗ (t)
uniformly for t ∈ [0, 1],
and
at all points of continuity of χA∗ .
Therefore, if we let r = γn (A), and observe that χA∗∗ (t) = min(1, rt ), we deduce that for all t > r, fn∗∗ (t) → rt , and fn∗ (t) → χA∗ (t) = χ(0,r) (t) = 0. Inserting this information back in (3.5), we get r I (t) Per(A), t
∀t > r.
Now, since I (t) is continuous, we may let t → r and we find that I γn (A) Per(A), as we wished to show. 4. The Pólya–Szegö principle is equivalent to the isoperimetric inequality In this section we prove Theorem 2. Our starting point is inequality (1.10). We claim that if A is a positive Young’s function, then ∂ ∗ A (−f ) (s)I (s) A ∇f (x) dγn (x). (4.1) ∂s {|f |>f ∗ (s)}
J. Martín, M. Milman / Journal of Functional Analysis 256 (2009) 149–178
163
Assuming momentarily the validity of (4.1), by integration we get 1
A (−f ∗ ) (s)I (s) ds
A ∇f (x) dγn (x).
Rn
0
It is easy to see that the left-hand side is equal to s = Φ(x1 ), we find 1
(4.2)
A (−f ∗ ) (s)I (s) ds =
Rn
A(|∇f ◦ (x)|) dγn (x). Indeed, letting
A (−f ∗ ) Φ(x1 ) I Φ(x1 ) Φ (x1 ) dx
R
0
=
A (−f ∗ ) Φ(x1 ) I Φ(x1 ) dγn (x)
Rn
=
A ∇f ◦ (x) dγn (x),
Rn
where in the last step we have used the fact that (−f ∗ ) Φ(x1 ) I Φ(x1 ) = (f ∗ ) Φ(x1 ) Φ (x1 ) = ∇f ◦ (x). Consequently, (4.2) states that for all Young’s functions A, we have A ∇f ◦ (x) dγn (x) A ∇f (x) dγn (x), Rn
Rn
which, by the Hardy–Littlewood–Pólya principle, yields t
◦ ∗
t
|∇f | (s) ds 0
|∇f |∗ (s) ds,
0
as we wished to show. It remains to prove (4.1). Here we follow Talenti’s argument. Let s > 0, then we have three different alternatives: (a) s belongs to some exceptional set of measure zero, (b) (f ∗ ) (s) = 0, or (c) there is a neighborhood of s such that (f ∗ ) (u) is not zero, i.e. f ∗ is strictly decreasing. In the two first cases there is nothing to prove. In case alternative (c) holds then it follows immediately from the properties of the rearrangement that for a suitable small h0 > 0 we can write h = γn f ∗ (s + h) < |f | f ∗ (s) , 0 < h < h0 . Therefore, for sufficiently small h, we can apply Jensen’s inequality to obtain,
1 1 ∇f (x) dγn (x) . A ∇f (x) dγn (x) A h h {f ∗ (s+h)<|f |f ∗ (s)}
{f ∗ (s+h)<|f |f ∗ (s)}
164
J. Martín, M. Milman / Journal of Functional Analysis 256 (2009) 149–178
Arguing like Talenti [35] we thus get ∂ ∂s
∂ A ∇f (x) dγn (x) A ∂s
{|f |>f ∗ (s)}
∇f (x) dγn (x)
{|f |>f ∗ (s)}
A (−f ∗ ) (s)I (s) ,
as we wished to show. To prove the converse we adapt an argument in [1]. Let f be a Lipschitz function on Rn , and let 0 < t < 1. By the definition of f ◦ we can write f ∗ (t) − f ∗ 1− = f ∗ Φ Φ −1 (t) − f ∗ Φ(∞) ∞
|∇f ◦ |(s) ds.
= Φ −1 (t)
Thus,
f
∗∗
1 (t) − f 1− = t ∗
t
∞
|∇f ◦ |(s) ds dr.
0 Φ −1 (r)
Making the change of variables s = Φ −1 (z) in the inner integral and then changing the order of integration, we find
f
∗∗
t 1
1 (t) − f 1− = t ∗
|∇f ◦ | Φ −1 (z) Φ −1 (z) dz dr
0 r
1 =
1 |∇f | Φ −1 (z) Φ −1 (z) dz + t ◦
t
t
z|∇f ◦ | Φ −1 (z) Φ −1 (z) dz
0
1 = f (t) − f 1− + t ∗
∗
t
z|∇f ◦ | Φ −1 (z) Φ −1 (z) dz.
0
Since Φ (Φ −1 (z)) = φ1 (Φ −1 (z)) = I (z), we readily deduce that (Φ −1 (z)) =
f
∗∗
1 (t) − f 1− = f ∗ (t) − f ∗ 1− + t ∗
t 0
and consequently
1 I (z) .
1 dz, z|∇f ◦ | Φ −1 (z) I (z)
Thus,
J. Martín, M. Milman / Journal of Functional Analysis 256 (2009) 149–178
f
∗∗
1 (t) − f (t) = t ∗
t
165
1 z|∇f ◦ | Φ −1 (z) dz I (z)
0
t 1 I (t) t
t
z|∇f ◦ | Φ −1 (z) dz
since t/I (t) is increasing
0
1 = I (t) 1 = I (t) t
Φ−1 (t)
|∇f ◦ |(s)Φ (s) ds
−∞ Φ−1 (t)
|∇f ◦ |(s) dγ1 (s)
−∞
|∇f ◦ |∗ (s) ds
since γ1 −∞, Φ −1 (t) = t .
0
Summarizing, we have shown that ∗∗ t f (t) − f ∗ (t) |∇f ◦ |∗∗ (t), I (t) which combined with our current hypothesis yields ∗∗ t t |∇f ◦ |∗∗ (t) |∇f |∗∗ (t). f (t) − f ∗ (t) I (t) I (t) By Theorem 1 the last inequality is equivalent to the isoperimetric inequality. Remark 1. We note here, for future use, that the discussion in this section shows that the following equivalent form of the Pólya–Szegö principle holds: t 0
∗ (−f ∗ ) (·)I (·) (s) ds
t
|∇f |∗ (s) ds.
0
Therefore, by the Hardy–Littlewood principle, for every r.i. space X on (0, 1), (−f ∗ ) (s)I (s) ∇f X . X
(4.3)
5. The Pólya–Szegö principle implies Gross’ inequality We present a proof due to Ehrhard [15], showing that the Pólya–Szegö principle implies (1.1). We present full details, since Ehrhard’s method is apparently not well known and some details are missing in [15].
166
J. Martín, M. Milman / Journal of Functional Analysis 256 (2009) 149–178
We first prove a one-dimensional inequality which, by symmetrization and tensorization, will lead to the desired result. Let f : R → R be a Lip function such that f and f ∈ L1 . By Jensen’s inequality ∞
f (x) lnf (x) dx = f
∞ L1
−∞
−∞
|f (x)| dx lnf (x) f L1
∞
f L1 ln −∞
|f (x)| dx f (x) . f L1
We estimate the inner integral using the fundamental theorem of calculus: |f (x)| f L1 , to obtain ∞
f (x) lnf (x) dx f
L1
lnf L1 .
−∞
Applying the preceding to f 2 we get: ∞ −∞
f (x)2 lnf (x) dx 1 f 2 2 ln 2ff 1 . L L 2
Using Hölder’s inequality ff L1 f L2 f L2 , and elementary properties of the logarithm we find ∞ −∞
f (x)2 lnf (x) dx 1 f 2 2 ln 2f 2 f 2 L L L 2 f 2L2 1 = f 2L2 ln 4f 4L2 4 f 2 2 L
=
f 2L2 1 f 2L2 ln 4 4 f 2L2
+ f 2L2 lnf L2
f 2L2 + f 2L2 lnf L2
(in the last step we used ln t t). (5.1)
We apply (5.1) to u = (2πex )−1/4 f (x) = φ1 (x)1/2 f (x) and compute both sides of (5.1). The left-hand side becomes 2
∞ −∞
u(x)2 lnu(x) dx =
∞
−∞
f (x)2 lnf (x) + ln 2πex 2 −1/4 dγ1 (x)
J. Martín, M. Milman / Journal of Functional Analysis 256 (2009) 149–178
∞ = −∞
167
f (x)2 lnf (x) dγ1 (x) − 1 ln 2πf 2 L (dγ1 ) 4
1 − 4
∞
f (x)2 x 2 dγ1 (x),
−∞
while the right-hand side is equal to f 2L2
= f 2L2 (dγ ) 1
= f 2L2 (dγ ) 1 ∞ −
1 + 4 1 − 4
∞
∞ f (x) x dγ1 (x) − 2 2
−∞
∞
f (x)f (x)xφ1 (x) dx
−∞
1 f (x) x dγ1 (x) + 2
∞
2 2
−∞
f (x)2 x 2 dγ1 (x) −∞
f (x)f (x)xφ1 (x) dx.
(5.2)
−∞
We simplify the last expression integrating by parts the third integral to the right, 1 2
∞
1 f (x) x dγ1 (x) = − 2
∞
2 2
−∞
2 f (x)2 x d (2π)−1/2 e−x
−∞
1 2 ∞ = − f (x)2 x (2π)−1/2 e−x −∞ 2 ∞ 1 2 (2π)−1/2 e−x 2f (x)f (x)x + f 2 (x) dx + 2 −∞
∞ = −∞
1 f (x)f (x)xφ1 (x) dx + f 2L2 (dγ ) . 1 2
We insert this back in (5.2) and then comparing results and simplifying we arrive at ∞
f (x)2 lnf (x) dγ1 (x) f 2 2
L (dγ1 )
+ f 2L2 (dγ ) lnf 2L2 (dγ 1
1)
−∞
+
ln(2πe2 ) f 2L2 (dγ ) . 1 4
(5.3)
Let f be a Lipschitz function on Rn . We form the symmetric rearrangement f ◦ considered as a one-dimensional function. Then, (5.3) applied to f ◦ , combined with the fact that f ◦ is equimeasurable with f and the Pólya–Szegö principle, yields
168
J. Martín, M. Milman / Journal of Functional Analysis 256 (2009) 149–178
f (x)2 lnf (x) dγn (x) =
Rn
◦ 2 ◦ f (x) lnf (x) dγ1 (x)
R
f ◦ 2L2 (dγ ) + f 2L2 (dγ ) lnf 2L2 (dγ 1
+
ln(2πe2 ) 4
f 2L2 (dγ
1)
2 = ∇f ◦ (x) 2
L (dγn )
+
ln(2πe2 )
1)
1
+ f ◦ 2L2 (dγ ) lnf ◦ 2L2 (dγ
n)
n
f ◦ 2L2 (dγ
n)
4
∇f 2L2 (dγ ) + f 2L2 (dγ ) lnf 2L2 (dγ n
+
ln(2πe2 ) 4
n)
n
f 2L2 (dγ ) .
(5.4)
n
We now use tensorization to prove (1.1). Note that, by homogeneity, we may assume that f has been normalized so that f L2 (dγn ) = 1. Let l ∈ N, and let F be defined on (Rn )l = Rnl by F (x) = lk=1 f (xk ), where xk ∈ Rn , k = 1, . . . , l. The Rnl version of (5.4) applied to F , and translated back in terms of f , yields 2 ln(2πe2 ) . l f (x) lnf (x) dγn (x) l∇f 2L2 (dγ ) + n 4 Rn
Therefore, upon diving by l and letting l → ∞, we obtain f (x)2 lnf (x) dγn (x) ∇f 2 2 , L (dγ ) n
Rn
as we wished to show. 6. Poincaré type inequalities We consider L1 Poincaré inequalities first. Indeed, for L1 norms the Poincaré inequalities are a simple variant of Ledoux’s inequality. Let f be a Lipschitz function on Rn , and let m be a median14 of f . Set f + = max(f −m, 0) and f − = − min(f −m, 0) so that f −m = f + −f − . Then, + |f − m| dγn = f dγn + f − dγn Rn
Rn
Rn
∞ =
∞ λf + (s) ds +
0
= (A). 14 I.e. γ (f m) 1/2 and γ (f m) 1/2. n n
λf − (s) ds 0
J. Martín, M. Milman / Journal of Functional Analysis 256 (2009) 149–178
169
We estimate each of these integrals using the properties of the isoperimetric profile and Ledoux’s inequality (1.9). First we use the fact that I (s) s is decreasing on 0 < s < 1/2, combined with the definition of median, to find that 2λg (s)I
1 I λg (s) , 2
where g = f + or g = f − .
Consequently,
(A)
=
∞
1 2I ( 12 )
0
1 2I ( 12 )
I λf + (s) ds +
∞ 0
+
I λf − (s) ds
∇f (x) dγn (x)
+
∇f (x) dγn (x) + Rn
1 2I (1/2)
(by (1.9))
Rn
∇f (x) dγn (x).
Rn
Thus, |f − m| dγn Rn
1 2I (1/2)
∇f (x) dγn (x).
(6.1)
Rn
We now prove Theorem 3. Proof. (i) → (ii). Obviously condition (1.13) is equivalent to f − mY ∇f X , where m is a median of f . Let f be a positive measurable function with supp f ⊂ (0, 1/2). Define 1 u(x) =
f (s)
ds , I (s)
x ∈ Rn .
Φ(x1 )
It is plain that u is a Lipschitz function on Rn such that γn (u = 0) 1/2, and therefore it has 0 median. Moreover, ∇u(x) = ∂ u(x) = −f Φ(x1 ) Φ (x1 ) = f Φ(x1 ) . ∂x I (Φ(x1 )) 1
170
J. Martín, M. Milman / Journal of Functional Analysis 256 (2009) 149–178
It follows that 1
∗
u (t) =
f (s)
ds , I (s)
and |∇u|∗ (t) = f ∗ (t).
t
Consequently, from u − 0Y ∇uX we deduce that 1 ds f X . f (s) I (s) Y
t
(ii) → (i). Let f be a Lipschitz function f on Rn . Write 1/2 f (t) = (−f ∗ ) (s) ds + f ∗ (1/2). ∗
t
Thus, 1/2 ∗ f Y = f Y 2f χ[0,1/2] Y (−f ) (s) ds + f ∗ (1/2)1Y ∗
∗
Y
t
1/2 ds (−f ∗ ) (s)I (s) + 21Y f L1 I (s) Y
t
(−f ∗ ) (s)I (s) X + f L1 ∇f X
(by (6.1) and (4.3)).
Part II. Case 0 < α X . (ii) ⇒ (iii). Let 0 < t < 1/4, then
∗
2t
f (2t) t
therefore,
ds f (s) s ∗
1/2 I (s) ds f ∗ (s) , s I (s) t
J. Martín, M. Milman / Journal of Functional Analysis 256 (2009) 149–178
171
1/2 ∗ I (s) ds ∗ f (2t) + f ∗ (1/2) f (s) Y s I (s) Y
t
∗ I (t) + f ∗ (1/2) f (t) t X ∗ I (t) + f 1 f (t) t X ∗ I (t) . f (t) t
(by (ii))
X
(iii) → (ii). By hypothesis 1/2 1/2 ds ds I (t) f ∗ (s) f ∗ (s) . I (s) I (s) t Y
t
X
t
Using that (see 2.5), I (s)
s
1 log s
1 1 + log , s
0 < s < 1/2,
we have 1/2 t
1 ds I (t) 1 ds ˜ (t). 1 + log f (s) f (s) = Qf I (s) t t s 1 + log 1 t
s
˜ is a bounded operator on X (see Section 2.3), and thus we Now from α X > 0 it follows that Q are able to conclude. Part II. Case 0 = α X < α X < 1. (ii) → (iv). By the fundamental theorem of calculus and (ii), we have f
∗∗
1/2 ds ∗∗ ∗ f (s) − f (s) χ(0,1/2) Y + f ∗∗ (1/2)1Y s Y
t
1 I (s) ds f ∗∗ (s) − f ∗ (s) χ(0,1/2) (s) + f 1 s I (s) Y t ∗∗ I (t) ∗ f (t) − f (t) χ(0,1/2) (t) t + f 1 X ∗∗ I (t) ∗ f (t) − f (t) t + f 1 . X
172
J. Martín, M. Milman / Journal of Functional Analysis 256 (2009) 149–178
(iv) → (i). Let f be a Lipschitz function on Rn , let m be a median of f and let g = f − m. By hypothesis we have ∗∗ I (t) ∗ + g1 . gY g (t) − g (t) t X
From (see [1]) g ∗∗ (t) − g ∗ (t) P g ∗ (s/2) − g ∗ (s) (t) + g ∗ (t/2) − g ∗ (t), combined with the fact that
I (t) t
decreases, we get
∗ I (t) I (s) ∗ P g (s/2) − g (s) (t). P g (s/2) − g (s) (t) t s
∗
∗
Therefore,
∗∗ g (t) − g ∗ (t) I (t) P g ∗ (s/2) − g ∗ (s) I (s) (t) + g ∗ (t/2) − g ∗ (t) I (t) t X s t X X ∗ I (t) ∗ (since α X < 1). g (t/2) − g (t) t X We compute the right-hand side, t ∗ I (t) I (t) = (g (t/2) − g ∗ (t) (−g ∗ ) (s) ds t X t
X
t/2
t I (s) ∗ ds (−g ) (s) s t/2
X
t 2 ∗ (−g ) (s)I (s) ds t t/2
X
t 1 2 (−g ∗ ) (s)I (s) ds t 0
(−g ∗ ) (t)I (t)
X
X
∇f X
(by (4.3)).
gY ∇f X + g1 ∇f X
(by (6.1)).
Summarizing, we have obtained 2
J. Martín, M. Milman / Journal of Functional Analysis 256 (2009) 149–178
173
6.1. Feissner type inequalities Theorem 3 readily improves upon Feissner’s inequalities (1.2). Indeed, for the particular choice X = Lp (1 p < ∞), Theorem 3 yields 1
∗
f−
f
(s)
I (s) s
p
ds
∇f (x)p dγn (x).
0
In particular, using again the asymptotics of I (s), 0 < s < 1/2, we get 1
p p 1 p/2 f ∗ (s)p log ds ∇f (x) dγn (x) + f (x) dγn (x). s
0 p 1/2 Moreover, the space L (LogL) is the best possible among r.i. spaces Y for which the Poincaré inequality f − f Y ∇f Lp holds. The case X = L∞ , which is new is more interesting. Indeed, since I (t)/t decreases, and limt→0 I (t) t = ∞,
sup f ∗ (t) 0
I (t) <∞ t
⇔
f = 0.
But Theorem 3 ensures that
∗∗ ∗ I (t) f− f ∇f L∞ . (t) − f − f (t) t L∞
(6.2)
Furthermore, for every r.i. space Y such that f − f ∇f L∞ , Y
the following embedding holds: ∗∗ I (t) ∗ f f Y (t) − f (t) t
L∞
+ f 1 .
Notice that due to the cancellation afforded by f ∗∗ (t) − f ∗ (t), the corresponding space LS(L∞ ) is nontrivial. The relation between concentration and LS(L∞ ) will be studied in the next section. 7. On limiting embeddings and concentration Elsewhere15 (cf. [30]) we shall explore in detail the connection between concentration inequalities and symmetrization, including the self-improving properties of concentration. In this 15 In particular the method of symmetrization by truncation can be extended to this setting.
174
J. Martín, M. Milman / Journal of Functional Analysis 256 (2009) 149–178
section we merely wish to call attention to the connection between a limiting lS inequality that follows from (1.11) and concentration. We have argued that, in the Gaussian world, Ledoux’s embedding corresponds to the Gagliardo–Nirenberg embedding. In the classical n-dimensional Euclidean case the “other” borderline case for the Sobolev embedding theorem occurs when the index of integrability of the gradients in the Sobolev space, say p, is equal to the dimension, i.e. p = n. In this case, as is well known, from |∇f | ∈ Ln (Rn ) we can deduce the exponential integrability of |f |n (cf. [36]). A refinement of this result, which follows from the Euclidean version of (1.11), is given by the following inequality from [1]: ∞ 1/n 1/n ∞ n ∗∗ n ds ∗ ∇f (x) dx f (s) − f (s) . s 0
0
In this fashion one could consider the corresponding borderline Gaussian embedding that results from (1.11) when n = p = ∞. The result now reads
∗∗ I (t) f (t) − f ∗ (t) sup |∇f |∗∗ (t) = f Lip . t t t<1
sup
(7.1)
We now show how (7.1) is connected with the concentration phenomenon (cf. [25] and the references therein). For the corresponding analysis we start by combining (7.1) with (2.5)
1 1/2 I (t) ct log , t
1 t ∈ 0, , 2
to obtain f
∗∗
∗
(t) − f (t)
f Lip
1 t ∈ 0, . 2
,
(log 1t )1/2
Therefore, for t ∈ (0, 12 ], we have
f
∗∗
(t) − f
∗∗
1/2 ∗∗ ds f (s) − f ∗ (s) (1/2) = s t
|∇f | ∞
1/2 t
2 |∇f |
∞
Thus, if λ|∇f |2∞ ≺ 1,
1 (log
1 log t
1 1/2 s)
1/2 .
ds s
J. Martín, M. Milman / Journal of Functional Analysis 256 (2009) 149–178
175
1/2 1/2 1 (log 2 ) λ(f ∗∗ (t)−f ∗∗ (1/2))2 t λ|∇f |∞ dt e dt e 0
0
1/2 = 0
1 2 t λ|∇f |∞
dt < ∞.
Moreover, since f ∗∗ is decreasing we have 1 e
λ(f ∗∗ (t)−f ∗∗ (1/2))2
1 dt
1/2
eλ(f
∗∗ (1−t)−f ∗∗ (1/2))2
dt
1/2
1/2 ∗∗ ∗∗ 2 = eλ(f (t)−f (1/2)) dt. 0
This readily implies the exponential integrability of (f (t) − f ∗∗ (1/2)): eλ(f (x)−f
∗∗ (1/2))2
dγn (x) < ∞,
Rn
and, in fact, we can readily compute the corresponding Orlicz norm. In this fashion we are led to define a new space Llog1/2 (∞, ∞)(Rn , dγn ) by the condition16
∗∗ 1 1/2 ∗ f Llog1/2 (∞,∞)(Rn ,dγn ) = sup f (t) − f (t) log < ∞. t 0
(7.2)
Summarizing our discussion, we have f Llog1/2 (∞,∞)(Rn ,dγn ) ∇f L∞ (Rn ,dγn ) and 2 n Llog1/2 (∞, ∞) Rn , dγn ⊂ eL (R ,dγn ) . The scale of spaces {Llogα (∞, ∞)}α∈R+ is thus suitable to measure exponential integrability. When α = 0 we get the celebrated L(∞, ∞) spaces introduced in [5], which characterize the 16 More generally, the relevant spaces to measure exponential integrability to the power p are defined by
1 1/p sup f ∗∗ (t) − f ∗ (t) log < ∞. t
176
J. Martín, M. Milman / Journal of Functional Analysis 256 (2009) 149–178
rearrangement-invariant hull of BMO. The corresponding underlying rearrangement inequality in the Euclidean case is the following version of the John–Nirenberg lemma ∗ f ∗∗ (t) − f ∗ (t) f # (t) where f # is the sharp maximal operator used in the definition of BMO (cf. [5] and [21]). In fact, in our context the L(∞, ∞) space is connected to the exponential inequalities by Bobkov, Götze [8]. Proceeding as before we see that (compare with [8])
∗∗ 1 −1/2 ∗ ∗∗ , f (t) − f (t) |∇f | (t) log t
1 0
2
L from where if follows readily that |∇f |L ∈ e ⇒ f ∈ L(∞, ∞), and therefore if, moreover f = 0, we can also conclude that f ∈ e .
8. Symmetrization by truncation of entropy inequalities In this brief section we wish to indicate, somewhat informally, how our methods can be extended to far more general setting. Let (Ω, μ) be a probability measure space. As in the literature, we consider the entropy functional defined, on positive measurable functions, by Ent(g) = g log g dμ − g dμ log g dμ. Suppose for example that Ent satisfies a lS inequality of order 1 on a suitable class of functions, Ent(g) c Γ (g) dμ. (8.1) Here Γ is to be thought as an abstract gradient. We will make an assumption that is not made in the literature but is crucial for our method to work: we will assume that Γ is “truncation friendly,” in the sense that for any truncation of f (see Section 3) we have h2 Γ f = Γ (f )χ{h
1 <|f |h2 }
h1
(8.2)
.
While this is a non-standard assumption, as we know, the usual gradients are indeed “truncation friendly.” In order to continue we need the following elementary result that comes from [10] (Lemma 2.2): Ent(g) − log g0 g dμ, (8.3) here g0 = μ{g = 0}. Combining (8.1)–(8.3) it follows that − log fhh12 0
fhh12 dμ c
− log λf (h1 )μ h1 < f (x) h2 c
Γ (f )χ{h
dμ,
Γ (f )χ{h
dμ,
1 <|f |h2 }
1 <|f |h2 }
J. Martín, M. Milman / Journal of Functional Analysis 256 (2009) 149–178
− log λf (h2 ) (h2 − h1 )λf (h2 ) c
177
Γ (f ) dμ.
{h1 <|f |h2 }
Pick h1 = f ∗ (s + h), h2 = f ∗ (s). Then
1 ∗ f (s) − f ∗ (s + h) c s log s
Γ (f ) dμ.
{f ∗ (s+h)<|f |f ∗ (s)}
Thus, 1 (f ∗ (s) − f ∗ (s + h)) c s log s h h
Γ (f ) dμ.
{f ∗ (s+h)<|f |f ∗ (s)}
Therefore, following the analysis of Section 4, we find that, for any Young’s function A, we have
1 d A s log (−f ∗ ) (s) A Γ (f ) dμ . s ds {|f |>f ∗ (s)}
Integrating, and using the Hardy–Littlewood–Pólya principle exactly as in Section 4, we obtain the following abstract version of the Pólya–Szegö principle ∗ t
t ∗ 1 s log (−f ∗ ) (s) (r) dr Γ (f ) (r) dr. s 0
0
This analysis establishes a connection between entropy inequalities and logarithmic Sobolev inequalities via symmetrization. In particular, our inequalities extend the classical results to the setting of rearrangement-invariant spaces. For more details see [30]. References [1] J. Bastero, M. Milman, F. Ruiz, A note on L(∞, q) spaces and Sobolev embeddings, Indiana Univ. Math. J. 52 (2003) 1215–1230. [2] W. Beckner, Inequalities in Fourier analysis, Ann. of Math. 102 (1975) 159–182. [3] W. Beckner, Sobolev inequalities, the Poisson semigroup, and analysis on the sphere S n , Proc. Natl. Acad. Sci. USA 89 (1992) 4816–4819. [4] W. Beckner, M. Persson, On sharp Sobolev embedding and the logarithmic Sobolev inequality, Bull. London Math. Soc. 30 (1998) 8084. [5] C. Bennett, R. DeVore, R. Sharpley, Weak L∞ and BMO, Ann. of Math. 113 (1981) 601–611. [6] C. Bennett, R. Sharpley, Interpolation of Operators, Academic Press, Boston, MA, 1988. [7] S. Blanchere, D. Chafai, P. Fougeres, I. Gentil, F. Malrieu, C. Roberto, G. Scheffer, Sur les inégalités de Sobolev logarithmiques, Soc. Math. France, Paris, 2000, 213 pp. [8] S.G. Bobkov, F. Götze, Exponential integrability and transportation cost related to logarithmic Sobolev inequalities, J. Funct. Anal. 163 (1999) 1–28. [9] S.G. Bobkov, C. Houdre, Some connections between isoperimetric and Sobolev-type inequalities, Mem. Amer. Math. Soc. 616 (1997). [10] S.G. Bobkov, B. Zegarlinski, Entropy bounds and isoperimetry, Mem. Amer. Math. Soc. 829 (2005). [11] C. Borell, The Brunn–Minkowski inequality in Gauss space, Invent. Math. 30 (1975) 207–216.
178
[12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36]
J. Martín, M. Milman / Journal of Functional Analysis 256 (2009) 149–178
C. Borell, The Ehrhard inequality, C. R. Math. Acad. Sci. Paris 337 (10) (2003) 663–666. E.B. Davies, Heat Kernel and Spectral Theory, Cambridge Univ. Press, Cambridge, 1989. A. Ehrhard, Symétrisation dans l’espace de Gauss, Math. Scand. 53 (1983) 281–301. A. Ehrhard, Sur l’inégalité de Sobolev logarithmique de Gross, in: Séminaire de Probabilités XVIII, in: Lecture Notes in Math., vol. 1059, Springer-Verlag, Berlin, 1984, pp. 194–196. A. Ehrhard, Inégalités isopérimétriques et intégrales de Dirichlet gaussiennes, Ann. Sci. École Norm. Sup. 17 (1984) 317–332. G.F. Feissner, Hypercontractive semigroups and Sobolev’s inequality, Trans. Amer. Math. Soc. 210 (1975) 51–62. J.J.F. Fournier, Mixed norms and rearrangements: Sobolev’s inequality and Littlewood’s inequality, Ann. Mat. Pura Appl. 148 (1987) 51–76. A. Garsia, E. Rodemich, Monotonicity of certain functionals under rearrangement, Ann. Inst. Fourier 24 (1974) 67–116. L. Gross, Logarithmic Sobolev inequalities, Amer. J. Math. 97 (1975) 1061–1083. J. Kalis and M. Milman, Symmetrization and sharp Sobolev inequalities in metric spaces, preprint, 2006. R. Latala, On some inequalities for Gaussian measures, in: Proceedings of the International Congress of Mathematicians, vol. II, Science Press, Beijing, 2002, pp. 813–822. M. Ledoux, Isopérimétrie et inégalitées de Sobolev logarithmiques gaussiennes, C. R. Acad. Sci. Paris Ser. I Math. 306 (1988) 79–92. M. Ledoux, Isoperimetry and Gaussian analysis, in: Ecole d’Eté de Probabilités de Saint-Flour 1994, in: Lecture Notes in Math., vol. 1648, Springer-Verlag, Berlin, 1996, pp. 165–294. M. Ledoux, The Concentration of Measure Phenomenon, Math. Surveys, vol. 89, Amer. Math. Soc., 2001. V.G. Maz’ya, Weak solutions of the Dirichlet and Neumann problems, Trudy Moskov. Mat. Obsch. 20 (1969) 137– 172 (in Russian). V.G. Maz’ya, Sobolev Spaces, Springer-Verlag, New York, 1985. J. Martin, M. Milman, Isoperimetry and symmetrization for Sobolev spaces on metric spaces, preprint, 2008. J. Martin, M. Milman, Self-improving Sobolev–Poincaré inequalities, truncation and symmetrization, Potential Anal., in press. J. Martin, M. Milman, On the connection between concentration and symmetrization inequalities, in preparation. J. Martin, M. Milman, E. Pustylnik, Sobolev inequalities: Symmetrization and self-improvement via truncation, J. Funct. Anal. 252 (2007) 677–695. M. Milman, E. Pustylnik, On sharp higher order Sobolev embeddings, Comm. Cont. Math. 6 (2004) 1–17. E. Pustylnik, On a rearrangement-invariant function set that appears in optimal Sobolev embeddings, in press. V.N. Sudakov, B.S. Tsirelson, Extremal properties of half-spaces for spherically invariant measures, J. Soviet. Math. 9 (1978) 918; translated from Zap. Nauch. Sem. LOMI 41 (1974) 1424. G. Talenti, Inequalities in rearrangement-invariant function spaces, in: Nonlinear Analysis, Function Spaces and Applications, vol. 5, Prometheus, Prague, 1995, pp. 177–230. N. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech. 17 (1967) 473–483.
Journal of Functional Analysis 256 (2009) 179–214 www.elsevier.com/locate/jfa
On flows associated to Sobolev vector fields in Wiener spaces: An approach à la DiPerna–Lions Luigi Ambrosio a,∗ , Alessio Figalli b a Scuola Normale Superiore, Department of Mathematics, Piazza dei Cavalieri 7, Pisa, PI, Italy b Université de Nice Sophia-Antipolis, Laboratoire J.-A. Dieudonné, CNRS UMR 6621, Parc Valrose,
06108 Nice Cedex 02, France Received 7 March 2008; accepted 13 May 2008 Available online 10 June 2008 Communicated by Paul Malliavin
Abstract In this paper we extend the DiPerna–Lions theory of flows associated to Sobolev vector fields to the case of Cameron–Martin-valued vector fields in Wiener spaces E having a Sobolev regularity. The proof is based on the analysis of the continuity equation in E, and on uniform (Gaussian) commutator estimates in finite-dimensional spaces. © 2008 Elsevier Inc. All rights reserved. Keywords: Flows; Sobolev spaces; Wiener spaces
1. Introduction The aim of this paper is the extension to an infinite-dimensional framework of the theory of flows associated to weakly differentiable (with respect to the spatial variable x) vector fields b(t, x). Starting from the seminal paper [15], the finite-dimensional theory had in recent times many developments, with applications to fluid dynamics [14,20,21], to the theory of conservation laws [4,5], and it covers by now Sobolev and even bounded variation [1] vector fields, under suitable bounds on the distributional divergence of bt (x) := b(t, x). Furthermore, in the case 1,p of Wloc vector fields with p > 1, even quantitative error estimates have been found in [10]; we refer to the Lecture Notes [2] and [3], and to the bibliographies therein, for the most recent * Corresponding author.
E-mail addresses:
[email protected] (L. Ambrosio),
[email protected] (A. Figalli). 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.05.007
180
L. Ambrosio, A. Figalli / Journal of Functional Analysis 256 (2009) 179–214
developments on this subject. Our paper fills the gap, pointed out in [2], between this family of results and those available in infinite-dimensional spaces, where only exponential integrability assumptions on ∇bt have been considered so far. Before passing to the description of our results in Wiener spaces, we briefly illustrate the heuristic ideas underlying the above-mentioned finite-dimensional results. The first basic idea is not to look for pointwise uniqueness statements, but rather to the family of solutions to the ODE as a whole. This leads to the concept of flow map X(t, x) associated to b, i.e. a map satisfying ˙ x) = bt (X(t, x)). It is easily seen that this is not an invariant concept, under X(0, x) = x and X(t, modification of b in negligible sets. This leads to the concept of Lr -regular flow: we give here the definition adopted in this paper when (E, · ) is a separable Banach space endowed with a Gaussian measure γ ; in the finite-dimensional theory (E = RN ) other reference measures γ could be considered as well (for instance the Lebesgue measure [1,15]). Definition 1.1 (Lr -regular b-flow). Let b : (0, T ) × E → E be a Borel vector field. If X : [0, T ] × E → E is Borel and 1 r ∞, we say that X is a Lr -regular flow associated to b if the following two conditions hold: (i) for γ -a.e. x ∈ X the map t → bt (X(t, x)) belongs to L1 (0, T ) and t
bτ X(τ, x) dτ
X(t, x) = x +
∀t ∈ [0, T ];
(1)
0
(ii) for all t ∈ [0, T ] the law of X(t, ·) under γ is absolutely continuous with respect to γ , with a density ρt in Lr (γ ), and supt∈[0,T ] ρt r < ∞. In (1), the integral is understood in Bochner’s sense, namely
∗
t
e , X(t, x) − x =
e∗ , bτ X(τ, x) dτ
∀e∗ ∈ E ∗ .
0
It is not hard to show that (see Remark 4.2), because of condition (ii), this concept is indeed invariant under modifications of b, and so it is appropriate to deal with vector fields belonging to Lp spaces. On the other hand, condition (ii) involves all trajectories X(·, x) up to γ -negligible sets, so the best we can hope for, using this concept, is existence and uniqueness of X(·, x) up to γ -negligible sets. The second basic idea is the concept of flow is directly linked, via the theory of characteristics, to the transport equation d f (s, x) + bs (x), ∇x f (s, x) = 0 dt
(2)
d μt + div(bt μt ) = 0. dt
(3)
and to the continuity equation
The first link has been exploited in [15] to transfer well-posedness results from the transport equation to the ODE, getting uniqueness of L∞ -regular (with respect to Lebesgue measure) b-flows
L. Ambrosio, A. Figalli / Journal of Functional Analysis 256 (2009) 179–214
181
in RN (see [9] for the generalization of this approach to the case of a Gaussian measure). This is possible because the flow maps (s, x) → X(t, s, x) (here we made also explicit the dependence on the initial time s, that we kept equal to 0 in Definition 1.1) solve (2) for all t ∈ [0, T ]. Here, in analogy with the approach initiated in [1] (see also [16] for a stochastic counterpart of it, where (3) becomes the forward Kolmogorov equation), we prefer to deal with the continuity equation, which seems to be more natural in a probabilistic framework. The link between the ODE and (3) is based on the fact that any positive finite measure η in C([0, T ]; E) concentrated on solutions to the ODE is expected to give rise to a weak solution to (3) (if the divergence operator is properly understood), with μt given by the marginals of η at time t: indeed, (3) describes the evolution of a probability density under the action of the “velocity field” b. We shall call these measures η generalized b-flows. Our goal will be, as in [1,16], to transfer well-posedness informations from the continuity equation to the ODE, getting existence and uniqueness results of the Lr -regular b-flows, under suitable assumptions on b. We have to take into account an intrinsic limitation of the theory of Lr -regular b-flows that is typical of infinite-dimensional spaces (see for instance [25]): even if b(t, x) ≡ v were constant, the flow map X(t, x) = x + tv would not leave γ quasi-invariant, unless v belongs to a particular subspace of E, the so-called Cameron–Martin space H of (E, γ ), see (7) for its precise definition. So, from now on we shall assume that b takes its values in H. However, thanks to a suitable change of variable, we will treat also some non H-valued vector fields, in the same spirit as in [7, 22] (see also [17,25]). We recall that H can be endowed with a canonical Hilbertian structure ·,· H that makes the inclusion of H in E compact; we fix an orthonormal basis (ei ) of H and we shall denote by bi the components of b relative to this basis (however, all our results are independent of the choice of (ei )). With this choice of the range of b, whenever μt = ut γ Eq. (3) can be written in the weak sense as d ut dγ = bt , ∇φ H ut dγ ∀φ ∈ Cyl(E, γ ), (4) dt E
E
where Cyl(E, γ ) is a suitable space of cylindrical functions induced by (ei ) (see Definition 2.3). Furthermore, a Gaussian divergence operator divγ c can be defined as the adjoint in L2 (γ ) of the gradient along H:
c, ∇φ H dγ = − E
φ divγ c dγ
∀φ ∈ Cyl(E, γ ).
E
Another typical feature of our Gaussian framework is that L∞ -bounds on divγ do not seem natural, unlike those on the Euclidean divergence in RN when the reference measure is the Lebesgue measure: indeed, even if b(t, x) = c(x), with c : RN → RN smooth and with bounded derivatives, we have divγ c = div c − c, x which is unbounded, but exponentially integrable with respect to γ . We can now state the main result of this paper: Theorem 1.2 (Existence and uniqueness of Lr -regular b-flows). Let p, q > 1 and let b : (0, T ) × E → H be satisfying:
182
L. Ambrosio, A. Figalli / Journal of Functional Analysis 256 (2009) 179–214
(i) bt H ∈ L1 ((0, T ); Lp (γ )); q (ii) for a.e. t ∈ (0, T ) we have bt ∈ LDH (γ ; H) with T
1/q (∇bt )sym (x)q dγ (x) dt < ∞, HS
(5)
E
0
and divγ bt ∈ L1 ((0, T ); Lq (γ )); (iii) exp(c[divγ bt ]− ) ∈ L∞ ((0, T ); L1 (γ )) for some c > 0. If r := max{p , q } and c rT , then the Lr -regular flow exists and is unique in the following ˜ satisfy sense: any two Lr -regular flows X and X ˜ x) X(·, x) = X(·,
in [0, T ], for γ -a.e. x ∈ E.
Furthermore, X is Ls -regular for all s ∈ [1, Tc ] and the density ut of the law of X(t, ·) under γ satisfies
− (ut ) dγ exp T s[divγ bt ] dγ s
L∞ (0,T )
E
c . ∀s ∈ 1, T
In particular, if exp(c[divγ bt ]− ) ∈ L∞ ((0, T ); L1 (γ )) for all c > 0, then the Lr -regular flow exists globally in time, and is Ls -regular for all s ∈ [1, ∞). The symmetric matrix (∇bt )sym , whose Hilbert–Schmidt norm appears in (5), corresponds to the symmetric part of the derivative of bt , defined in a weak sense by (22): notice that, in analogy with the finite-dimensional result [8], no condition is imposed on the antisymmetric part of the derivative, which need not be given by a function; this leads to a particular function space LDq (γ ; H) (well studied in linear elasticity in finite dimensions, see [24]) which is for instance 1,q larger than the Sobolev space WH (γ ; H), see Definitions 2.4 and 2.6. Also, we will prove that uniqueness of X holds even within the larger class of generalized b-flows. Let us explain first the main differences between our strategy and the techniques used in [7, 11–13,22] for autonomous (i.e. time independent) vector fields in infinite-dimensional spaces. The standard approach for the existence of a flow consists in approximating the vector field b with finite-dimensional vector fields bN , constructing a finite-dimensional flow XN , and then passing to the limit as N → ∞. This part of the proof requires quite strong a priori estimates on the flows to have enough compactness to pass to the limit. To get these a priori estimates, the assumptions on the vector field, instead of the hypotheses (i)–(iii) in Theorem 1.2, are: bH ∈
Lp (γ ),
p∈[1,∞)
exp c∇bL(H,H) ∈ L1 (γ ) exp c|divγ b| ∈ L1 (γ )
for all c > 0, for some c > 0,
L. Ambrosio, A. Figalli / Journal of Functional Analysis 256 (2009) 179–214
183
where ∇bL(H,H) denotes the operator norm of ∇b from H to H. So, apart from the minor fact that we allow a measurable time dependence of b, the main difference between these results and ours is that we replace exponential integrability of the operator norm of ∇b by q-integrability of the (stronger) Hilbert–Schmidt norm of ∇bt (or, as we said, of its symmetric part). Let us remark for instance that, just for the existence part of a generalized b-flow, the hypothesis on divγ b could be relaxed to a one sided bound, as we did. Indeed, this assumption allows to prove uniform estimates on the density of the approximating flows, see for instance Theorem 6.1. On the other hand, the proof of the uniqueness of the flow strongly relies on the fact that one can use the approximating flows XN also for negative times. Our strategy is quite different from the above one: the existence and uniqueness of a regular flow will be proved at once in the following way. First of all, the existence of a generalized b-flow η, even without the regularity assumption (5), can be obtained thanks to a tightness argument for measures in C([0, T ]; E) and proving uniform estimates on the density of the finite-dimensional approximating flows. Then we prove uniqueness in the class of generalized b-flows. This implies as a byproduct that η is induced by a “deterministic” X, thus providing the desired existence and uniqueness result. Moreover the flexibility of this approach allows us to prove the stability of the Lr -regular flow under smooth approximations of the vector field, and thanks to the uniqueness we can also easily deduce the semigroup property. The main part of the paper is therefore devoted to the proof of uniqueness. As we already said, this depends on the well-posedness of the continuity equation (4). Specifically, we will show uniqueness of solutions ut in the class L∞ ((0, T ); Lr (γ )). The key point, as in the finitedimensional theory, is to pass from (4) to d dt
β(ut ) dγ = E
bt , ∇φ H β(ut ) dγ E
+
β(ut ) − ut β (ut ) divγ bt dγ
∀φ ∈ Cyl(E, γ ),
(6)
E
for all β ∈ C 1 (R) with β (z) and zβ (z) − β(z) bounded, and then to choose as function β suitable C 1 approximations of the positive or of the negative part, to show that the equation preserves the sign of the initial condition. The passage from (4) to (6) can be formally justified using the rule divγ (vc) = v divγ c + ∇v, c H and the chain rule ∇β(u) = β (u)∇u, but it is not always possible. It is precisely at this place that the regularity assumptions on bt enter. The finite-dimensional strategy involves a regularization argument (in the space variable only) and a careful analysis of the “commutators” (with v = ut , c = bt ) r ε (c, v) := eε c, ∇Tε v H − Tε divγ (vc) , where ε is the regularization parameter and Tε is the regularizing operator. Already in the finitedimensional theory (see [1,15]) a careful estimate of r ε is needed, taking into account some cancellation effects. These effects become even more important in this framework, where we
184
L. Ambrosio, A. Figalli / Journal of Functional Analysis 256 (2009) 179–214
use as a regularizing operator the Ornstein–Uhlenbeck operator (32) (in particular the semi group property and the fact that Tt is self-adjoint from Lp (γ ) to Lp (γ ) will play an important role). The core of our proof is indeed Section 6.2, where we obtain commutator estimates in RN independent of N , and therefore suitable for an extension, via the canonical cylindrical approximation, to E. The paper is structured as follows: first we recall the main notation needed in the paper. In Section 3 we prove the well-posedness of the continuity equation, while in Section 4 we prove existence, uniqueness and stability of regular flows. The results of both sections rely on some finite-dimensional a priori estimates that we postpone to Section 6. Finally, to apply our results also in more general situations, in Section 5 we see how our results can be extended to the case non-H-valued vector fields. 2. Main notation and preliminary results Measure-theoretic notation. All measures considered in this paper are positive, finite and defined on the Borel σ -algebra. Given f : E → F Borel and a measure μ in E, we denote by f# μ the push-forward measure in F , i.e. the law of f under μ. We denote by χA the characteristic function of a set A, equal to 1 on A, and equal to 0 on its complement. We consider a separable Banach space (E, · ) endowed with a Gaussian measure γ , i.e. in R for all e∗ ∈ E ∗ . We shall assume that γ is centered and (e∗ )# γ is a Gaussian measure non-degenerate, i.e. that E x dγ (x) = 0 and γ is not supported in a proper subspace of E. We recall (see [19]) that, by Fernique’s theorem, E exp(cx2 ) dγ (x) < ∞, whenever 2c < supe∗ 1 e∗ , x L2 (γ ) . Cameron–Martin space. We shall denote by H ⊂ E the Cameron–Martin space associated to (E, γ ). It can be defined [6,19] as
φ(x)x dγ (x): φ ∈ L2 (γ ) .
H :=
(7)
E
The non-degeneracy assumption on γ easily implies that H is a dense subset of E. If we denote by i : L2 (γ ) → H ⊂ E the map φ → E φ(x)x dγ (x), and by K the kernel of i, we can define the Cameron–Martin norm i(φ) := min φ − ψ 2 , L (γ ) H ψ∈K
whose induced scalar product ·,· H satisfies i(φ), i(ψ) H = φψ dγ
∀φ ∈ L2 (γ ), ∀ψ ∈ K ⊥ .
E
Notice also that i( e∗ , x ) ∈ K ⊥ for all e∗ ∈ E ∗ , because E
e∗ , x ψ(x) dγ (x) = e∗ , xψ(x) dγ (x) = 0
E
∀ψ ∈ K.
(8)
L. Ambrosio, A. Figalli / Journal of Functional Analysis 256 (2009) 179–214
185
Since i is not injective in general, it is often more convenient to work with the map j : E ∗ → H, dual of the inclusion map of H in E (i.e. j (e∗ ) is defined by j (e∗ ), h H = e∗ , h for all h ∈ H). The set j (E ∗ ) is obviously dense in H (for the norm · H ), and j is injective thanks to the density of H in E; furthermore, choosing φ(x) = e∗ , x in (8), we see that i( e∗ , x ) = j (e∗ ). ∗ ∗ ∗ ⊥ ∗ As a consequence the vector space { e , x : e ∈ E } is dense in K . Since i( e , x ) ( E x2 dγ )1/2 e∗ , x L2 (γ ) = i( e∗ , x )H , the inclusion of H in E is continuous, and it is not hard to show that it is also compact (see [6, Corollary 3.2.4]). This setup becomes much simpler when (E, · ) is an Hilbert space: Remark 2.1 (The Hilbert case). Assume that (E, · ) is an Hilbert space. Then, after choosing an orthonormal basis in which the covariance operator (x, y) → E x, z y, z dγ (z) is diagonal, we can identify E with 2 , endowed with the canonical basis i , and the coordinates xi of x ∈ 2 relative to i are independent, Gaussian and with variance λ2i (with λi > 0 by the non-degeneracy assumption). Then, the integrability of x2 implies that i λ2i is convergent, ei∗ = i (here we are using the Riesz isomorphism to identify 2 with its dual), ei = λi i and the Cameron–Martin space is H := x ∈ 2 :
∞ (x i )2 i=1
λ2i
<∞ .
The map j : 2 → H is given by (xi ) → (λi xi ). Let us remark that, although we constructed H starting from E, it is indeed H which plays a central role in our results; according to the Gross viewpoint, this space might have been taken as the starting point, see [6, §3.9] and Section 4.4 for a discussion of this fact. Finite-dimensional projections. The above-mentioned properties of j allow the choice of (en∗ ) ⊂ E ∗ such that (j (en∗ )) is a complete orthonormal system in H. Then, setting en := j (en∗ ), we can define the continuous linear projections πN : E → H by πN (x) :=
N
ek∗ , x
k=1
ek
N ek , x H ek for x ∈ H . =
(9)
k=1
The term “projection” is justified by the fact that, by the second equality in (9), πN |H is indeed the orthogonal projection on HN := span(e1 , . . . , eN ).
(10)
From now such a basis (ei ) of H will be fixed, and we shall denote by v i the components of v ∈ H relative to this basis. Also, for a given Borel function u : E → R, we shall denote by EN u ∗ , x . The the conditional expectation of u relative to the σ -algebra generated by e1∗ , x , . . . , eN following result follows by martingale convergence theorems, because the σ -algebra generated by ei∗ , x is the Borel σ -algebra (see also [6, Corollary 3.5.2]). Lemma 2.2. For all p ∈ [1, ∞) and u ∈ Lp (γ ) we have EN u → u γ -a.e. and in Lp (γ ).
186
L. Ambrosio, A. Figalli / Journal of Functional Analysis 256 (2009) 179–214
According to these projections, we can define the space Cyl(E, γ ) of smooth cylindrical functions (notice that this definition depends on the choice of the basis (en )). Definition 2.3 (Smooth cylindrical functions). Let Cb∞ (RN ) be the space of smooth functions in RN , bounded together with all their derivatives. We say that φ : E → R is cylindrical if ∗ φ(x) = ψ e1∗ , x , . . . , eN ,x
(11)
for some integer N and some ψ ∈ Cb∞ (RN ). If v ∈ E and φ : E → R we shall denote by ∂v φ the partial derivative of φ along v, wherever this exists. Obviously, cylindrical functions are differentiable infinitely many times in all directions: if φ is as in (11), the first order derivative is given by N ∗ ∗ ∂ψ ∗ e , x , . . . , eN , x ei , v . ∂v φ(x) = ∂zi 1
(12)
i=1
If v ∈ H the above formula becomes ∂v φ(x) =
N ∗ ∂ψ ∗ e1 , x , . . . , eN , x ei , v H , ∂zi i=1
and this allows to define the gradient of φ as an element of H: ∇φ(x) :=
N ∗ ∂ψ ∗ e , x , . . . , eN , x ei ∈ H. ∂zi 1 i=1
Gaussian divergence and differentiability along H . Let b : E → H be a vector field with bH ∈ L1 (γ ); we say that a function divγ b ∈ L1 (γ ) is the Gaussian divergence of b (see for instance [6, §5.8]) if
∇φ, b H dγ = − E
φ divγ b dγ
∀φ ∈ Cyl(E, γ ).
(13)
E
In the finite-dimensional space E = RN endowed with the standard Gaussian we have, by an integration by parts, divγ b = div b − b, x .
(14)
We recall the integration by parts formula
∂j (e∗ ) φ dγ =
E
E
φ e∗ , x dγ
∀φ ∈ Cyl(E, γ ), ∀e∗ ∈ E ∗ .
(15)
L. Ambrosio, A. Figalli / Journal of Functional Analysis 256 (2009) 179–214
187
This motivates the following definitions: if both u(x) and u(x) e∗ , x belong to L1 (γ ), we call weak derivative of u along j (e∗ ) the linear functional on Cyl(E, γ ) φ → − u∂j (e∗ ) φ dγ + uφ e∗ , x dγ . (16) E
E
As in the classical finite-dimensional theory, we can define Sobolev spaces by requiring that these functionals are representable by Lq (γ ) functions, see Chapter 5 of [6] for a more complete discussion of this topic. 1,q
Definition 2.4 (Sobolev space WH (γ )). If 1 q ∞, we say that u ∈ L1 (γ ) belongs to 1,q WH (E, γ ) if u(x) e∗ , x ∈ L1 (γ ) for all e∗ ∈ E ∗ and there exists g ∈ Lq (γ ; H) satisfying
u∂
j (e∗ )
φ dγ +
E
φ g, j e∗ H dγ =
E
uφ e∗ , x dγ
∀e∗ ∈ E ∗ , ∀φ ∈ Cyl(E, γ ). (17)
E
The condition u(x) e∗ , x ∈ L1 (γ ) is automatically satisfied whenever u ∈ Lp (γ ) for some p > 1, thanks to the fact that the law of e∗ , x under γ is Gaussian, so that e∗ , x ∈ Lr (γ ) for all r < ∞. We shall denote, as usual, the (unique) weak derivative g by ∇u and its components g, ei H by ∂i u, so that (17) becomes u∂i φ dγ + φ∂i u dγ = uφ ei∗ , x dγ ∀i 1, ∀φ ∈ Cyl(E, γ ). (18) E
E
E
We recall that a continuous linear operator L : H → H is said to be Hilbert–Schmidt if LHS , defined as the square root of the trace of Lt L, is finite. Accordingly, if Lij = L(ei ), ej H is the symmetric matrix representing L : H → H in the basis (ei ), we have that L is of Hilbert–Schmidt class if and only if i,j L2ij is convergent, and LHS =
L2ij .
(19)
i,j
The following proposition shows that bounded continuous operators from E to H are of Hilbert–Schmidt class, when restricted to H. In particular our results apply under p-integrability assumptions on ∇bt when the operator norm between E and H is used. Proposition 2.5. Let L : E → H be a linear continuous operator. Then the restriction of L to H is of Hilbert–Schmidt class and LHS CLL(E,H) , with C depending only on E and γ . Proof. By [6, Theorem 3.5.10] we can find a complete orthonormal system (fn ) of H such that 2 n fn =: C < +∞. Denoting by L the operator norm of L from E to H, we have then L2HS =
2 L(fi )2 L2 L(fi ), fj H = fi 2 = CL2 . H i,j
i
i
2
188
L. Ambrosio, A. Figalli / Journal of Functional Analysis 256 (2009) 179–214
From now on, we shall denote by Lp (γ ; H) the space of Borel maps c : E → H such that cH ∈ Lp (γ ). Given the basis (ei ) of H, we shall denote by ci the components of c relative to this basis. Definition 2.6 (The space LD(γ ; H)). If 1 q ∞, we say that c ∈ L1 (γ ; H) belongs to LDq (γ ; H) if: (a) for all h = j (e∗ ) ∈ H, the function c, h H has a weak derivative in Lq (γ ) along h, that we shall denote by ∂h c, h H , namely c, h H ∂h φ dγ + φ∂h c, h H dγ = c, h H φ e∗ , x dγ ∀φ ∈ Cyl(E, γ ); (20) E
E
E
(b) the symmetric matrices sym
(∇c)ij (x) :=
1 ∂(ei +ej ) ci + cj (x) − ∂(ei −ej ) ci − cj (x) 4
(21)
satisfy
(∇c)sym q dγ < ∞. HS
E sym
1,q
If all components ci of c belongs to WH (γ ) then the function (∇c)ij in (21) really corresponds to the symmetric part of (∇c)ij = ∂j ci , and this explains our choice of notation. However, according to our definition of LDq (γ ; H), the vector fields c in this space need not have compo1,q nents ci in WH (γ ). Moreover, from (21) we obtain that (∂i cj + ∂j ci )/2 are representable by sym q the L (γ ) functions (∇c)ij , namely
1 i c ∂j φ + cj ∂i φ dγ + 2
E
=
sym
φ(∇c)ij dγ E
1 i ∗ c ej , x + cj ei∗ , x φ dγ 2
∀φ ∈ Cyl(E, γ ).
(22)
E
Remark 2.7 (Density of cylindrical functions). We recall that Cyl(E, γ ) is dense in all spaces 1,p 1,p WH (γ ), 1 p < ∞. More precisely, if 1 p, q < ∞, any function u ∈ WH (γ ) ∩ Lq (γ ) can q be approximated in L (γ ) by cylindrical functions un with ∇un → ∇u strongly in Lp (γ ; H). In the case p = ∞, convergence of the gradients occurs in the weak∗ topology of L∞ (γ ; H). These density results can be proved first in the finite-dimensional case and then, thanks to Lemma 2.2, in the general case. Remark 2.8. In the sequel we shall use the simple rule divγ (bu) = u divγ b + b, ∇u H ,
L. Ambrosio, A. Figalli / Journal of Functional Analysis 256 (2009) 179–214
189
1,q
valid whenever divγ b ∈ Lp (γ ), u ∈ Lp (γ ), b ∈ Lq (γ ; H) and u ∈ WH (γ ). The proof is a direct consequence of Remark 2.7. 1,q
Remark 2.9 (Invariance of divγ , WH (γ ), LDq (γ )). The definitions of Gaussian divergence, Sobolev space and LD space, as given, involve the space Cyl(E, γ ), which depends on the choice of the complete orthonormal basis (ei ). However, an equivalent formulation could be given using the space Cb1 (E, γ ) of functions that are Fréchet differentiable along all directions in H, with a bounded continuous gradient: indeed, cylindrical functions belong to Cb1 (E, γ ), and since 1,∞ (γ ), thanks to Remark 2.7 the functions in this space can be well Cb1 (E, γ ) is contained in WH approximated (in all spaces Lp (γ ) with p < ∞, and with weak∗ convergence in L∞ (γ ) of gradients) by cylindrical functions. A similar remark applies to the continuity equation, discussed in the next section. 3. Well posedness of the continuity equation Let I ⊂ R be an open interval. In this section we shall consider the continuity equation in I × E, possibly with a source term f , i.e. d (ut γ ) + divγ (bt ut γ ) = f γ . dt This equation has to be understood in the weak sense, namely we require that t → absolutely continuous in I and d dt
ut φ dγ = E
(23)
E ut φ dγ
is
bt , ∇φ H ut dγ +
E
f φ dγ
a.e. in I, ∀φ ∈ Cyl(E, γ ).
(24)
E
The minimal requirement necessary to give a meaning to (24) is that u, f and |u|bH belong to L1 (I ; L1 (γ )), and we shall always make assumptions on u, f and b to ensure that these properties are satisfied. Sometimes, to simplify our notation, with a slight abuse we drop γ and write (23) just as d ut + divγ (bt ut ) = f. dt However, we always have in mind the weak formulation (24), and we shall always assume that f ∈ L1 (I ; L1 (γ )). Since we are, in particular, requiring all maps t → E ut φ dγ to be uniformly continuous in I , the map t → ut is weakly continuous in I , with respect to the duality of L1 (γ ) with Cyl(E, γ ). Therefore, if I = (0, T ), it makes sense to say that a solution ut of the continuity equation starts from u¯ ∈ L1 (γ ) at t = 0:
ut φ dγ =
lim t↓0
E
uφ ¯ dγ E
∀u ∈ Cyl(E, γ ).
(25)
190
L. Ambrosio, A. Figalli / Journal of Functional Analysis 256 (2009) 179–214
Theorem 3.1 (Well-posedness of the continuity equation). (Existence) Let b : (0, T ) × E → H be satisfying bt H ∈ L1 (0, T ); Lp (γ )
for some p > 1
(26)
and exp c[divγ bt ]− ∈ L∞ (0, T ); L1 (γ ) for some c > Tp .
(27)
Then, for any nonnegative u¯ ∈ L∞ (γ ), the continuity equation has a nonnegative solution ut with u0 = u¯ satisfying (as a byproduct of its construction) − dγ ¯ rL∞ (γ ) exp T r[div b ] (ut )r dγ u γ t ∞ L (0,T )
E
c ∀r ∈ 1, , t ∈ [0, T ]. T
(28)
(Uniqueness) Let b : (0, T ) × E → H be satisfying (26), bt ∈ LDq (γ ; H) for a.e. t ∈ (0, T ) with T
(∇bt )sym q dγ HS
1/q dt < ∞
(29)
E
0
for some q > 1, and divγ bt ∈ L1 (0, T ); Lq (γ ) .
(30)
Then, setting r = max{p , q }, if c T r the continuity equation (23) in (0, T ) × E has at most one solution in the function space L∞ ((0, T ); Lr (γ )). Definition 3.2 (Renormalized solutions). We say that a solution ut of (23) in I × E is renormalized if d β(ut ) + divγ bt β(ut ) = β(ut ) − ut β (ut ) divγ bt + fβ (ut ) dt
(31)
in the sense of distributions in I × E, for all β ∈ C 1 (R) with β (z) and zβ (z) − β(z) bounded. In the sequel we shall often use the Ornstein–Uhlenbeck operator Tt , defined for u ∈ L1 (γ ) by Mehler’s formula Tt u(x) := u e−t x + 1 − e−2t y dγ (y). (32) E
In the next proposition we summarize the main properties of the OU operator used in this paper, see Theorems 1.4.1, 2.9.1 and Proposition 5.4.8 of [6].
L. Ambrosio, A. Figalli / Journal of Functional Analysis 256 (2009) 179–214
191
Proposition 3.3 (Properties of the OU semigroup). Let Tt be as in (32). (i) Tt uLp (γ ) uLp (γ ) for all u ∈ Lp (γ ), p ∈ [1, ∞], t 0, and equality holds if u is nonnegative and p = 1. (ii) Tt is self-adjoint in L2 (γ ) for all t 0. More generally, if 1 p ∞, we have
vTt u dγ = uTt v dγ ∀u ∈ Lp (γ ), ∀v ∈ Lp (γ ). (33) E
E 1,p
(iii) For all p ∈ (1, ∞), t > 0 and u ∈ Lp (γ ) we have Tt u ∈ WH (γ ) and ∇Tt uLp (γ ;H) C(p, t)uLp (γ ) .
(34)
(iv) For all p ∈ [1, ∞] and u ∈ WH (γ ) we have ∇Tt u = e−t Tt ∇u. (v) Tt maps Cyl(E, γ ) in Cyl(E, γ ) and Tt u → u in Lp (γ ) as t ↓ 0 for all u ∈ Lp (γ ), 1 p < ∞. 1,p
In the same spirit of (16), we can now extend the action of the semigroup from L1 (γ ) to elements in the algebraic dual of Cyl(E, γ ) as follows: Tt , φ := , Tt φ ,
φ ∈ Cyl(E, γ ).
This is an extension, because if is induced by some function u ∈ L1 (γ ), i.e. , φ = E φu dγ for all φ ∈ Cyl(E, γ ), then because of (33) Tt is induced by Tt u, i.e. Tt , φ = E φTt u dγ for all φ ∈ Cyl(E, γ ). In general we shall say that Tt is a function whenever there exists (a unique) v ∈ L1 (γ ) such that Tt , φ = E vφ dγ for all φ ∈ Cyl(E, γ ). In the next lemma we will use this concept when is the Gaussian divergence of a vector field c: indeed, can be thought via the formula − E c, ∇φ H dγ as an element of the dual of Cyl(E, γ ). Our first proposition provides a sufficient condition ensuring that Tt (divγ c) is a function. Lemma 3.4. Assume that r ∈ (1, ∞) and c ∈ Lr (γ ; H). Then Tt (divγ c) is a function in Lr (γ ) for all t > 0. Proof. We use Proposition 3.3(iii) to obtain Tt (divγ c), φ = divγ c, Tt φ cH ∇Tt φH dγ C(q, t)cLr (γ ;H) φ r L (γ ) E
for all φ ∈ Cyl(E, γ ), and we conclude.
2
In the sequel we shall denote by (Λ(p))p the pth moment of the standard Gaussian in R, i.e. 1/p −1/2 p −|x|2 /2 |x| e dx . Λ(p) := (2π) R
(35)
192
L. Ambrosio, A. Figalli / Journal of Functional Analysis 256 (2009) 179–214
Proposition 3.5 (Commutator estimate). Let c ∈ Lp (γ ; H) ∩ LDq (γ ; H) for some p > 1, 1 q 2, with divγ c ∈ Lq (γ ). Let r = max{p , q } and set r ε = r ε (v, c) := eε c, ∇Tε (v) − Tε divγ (vc) .
(36)
Then, for ε > 0 and v ∈ Lr (γ ) we have
ε r
L1 (γ )
vLr (γ ) √
Λ(p)ε 1 − e−2ε
cLp (γ ;H) +
√ 2divγ cLq (γ )
+ 2(∇c)sym HS Lq (γ ) .
(37)
Finally, −r ε → v divγ c in L1 (γ ) as ε ↓ 0. Proof. The a priori estimate (37), which is indeed the main technical point of this paper, will be proved in the Section 6 in finite-dimensional spaces. Here we will just mention how the finitedimensional approximation can be performed. Let us first assume that v ∈ L∞ . Since vc ∈ Lp (γ ; H), the previous lemma ensures that r ε is a function. Keeping c fixed, we see that if vn → v strongly in Lr (γ ) then r ε (vn , c) → r ε (v, c) in the duality with Cyl(E, γ ), and since the L1 (γ ) norm is lower semicontinuous with respect to convergence in this duality, thanks to the density of cylindrical functions we see that it suffices to prove (37) when v is cylindrical. Keeping now v ∈ Cyl(E, γ ) fixed, we consider the vector fields cN :=
N
E N c i ei .
i=1
We observe that (13) gives divγ cN = EN (divγ c), while (22) gives (∇cN )sym = EN (∇c)sym . Thus, by Jensen’s inequality for conditional expectations we obtain cN Lp (γ ;H) cLp (γ ;H) and q q q sym q (∇cN ) |divγ cN | dγ |divγ c| dγ , dγ (∇c)sym HS dγ . HS E
E
E
E
∗ , x , if we choose a cylindrical test funcNow, assuming that v depends only on e1∗ , x , . . . , eM ∗ ∗ tion φ depending only on e1 , x , . . . , eN , x , with N M (with no loss of generality, because v is fixed), we get
r ε (v, c)φ dγ = E
r ε (v, cN )φ dγ sup|φ|
E
ε r (v, cN ) dγ
E
sup|φ|vLr (γ ) √
Λ(p)ε
cN Lp (γ ;H) +
1 − e−2ε + 2(∇cN )sym q HS L (γ )
√ 2divγ cN Lq (γ )
L. Ambrosio, A. Figalli / Journal of Functional Analysis 256 (2009) 179–214
193
√ Λ(p)ε sup|φ|vLr (γ ) √ cLp (γ ;H) + 2divγ cLq (γ ) −2ε 1−e sym . + 2 (∇c) HS Lq (γ ) This means that, once we know (37) in finite-dimensional spaces, we obtain that the same inequality holds in all Wiener spaces for all v ∈ L∞ (γ ). Finally, to remove also this restriction on v, we consider a sequence (vn ) ⊂ L∞ (γ ) converging in Lr (γ ) to v and we notice that, because of (37), r ε (vn , c) is a Cauchy sequence in L1 converging in the duality with Cyl(E, γ ) to r ε (v, c). The strong convergence of r ε can be achieved by a density argument. More precisely, if q > 1 (so that r < ∞), since r ε (v, c) = r ε (v − φ, c) + r ε (φ, c), by (37) and the density of cylindrical functions in Lr (γ ), we need only to consider the case when v = φ is cylindrical. In this case r ε = c, Tε ∇φ − Tε φ divγ c + c, ∇φ and its convergence to −φ divγ c is an obvious consequence of the continuity properties of Tε . In the case q = 1 (that is r = ∞), the approximation argument is a bit more involved. Since we will never consider L∞ -regular flows, we give here just a sketch of the proof. We argue ˜ c˜ ) + r ε (v, ˜ c˜ ), with v˜ and c˜ smooth and as in [21]: we write r ε (v, c) = r ε (v, c − c˜ ) + r ε (v − v, bounded with all their derivatives. Using (37) twice, we first choose c˜ so that r ε (v, c − c˜ ) is small uniformly in ε, and then, since now c˜ is smooth with bounded derivatives, it suffices to choose v˜ ˜ c˜ ) small. We can now conclude as above. 2 close to v in Ls for some s > 1 to make r ε (v − v, The following lemma is standard (both properties can be proved by a smoothing argument; for the second one, see [6, Corollary 5.4.3]): Lemma 3.6 (Chain rules). Let β ∈ C 1 (R) with β bounded. d u = f in the weak sense, then (i) If u, f ∈ L1 (I ; L1 (γ )) satisfy dt weak sense. 1,p 1,p (ii) If u ∈ WH (γ ) then β(u) ∈ WH (γ ) and ∇β(u) = β (u)∇u.
d
dt β(u) = β (u)f ,
still in the
Theorem 3.7 (Renormalization property). Let b : I × E → H be satisfying the assumptions of the uniqueness part of Theorem 3.1, with I in place of (0, T ). Then any solution ut of the continuity equation (23) in L∞ (I ; Lr (γ )), with r = max{p , q }, is renormalized. 1,r (γ ) for a.e. t, Proof. In the first step we prove the renormalized property assuming that ut ∈ WH ∞ r and that both ut and ∇ut H belong to L (I ; L (γ )). Under this assumption, Remark 2.8 gives that divγ (bt ut ) = ut divγ bt + bt , ∇ut H , therefore
d ut = −ut divγ bt + bt , ∇ut H ∈ L1 I ; L1 (γ ) . dt Now, using Lemma 3.6 and Remark 2.8 again, we get d β(ut ) = −β (ut )ut divγ bt − β (ut ) bt , ∇ut H dt
194
L. Ambrosio, A. Figalli / Journal of Functional Analysis 256 (2009) 179–214
= β(ut ) − β (ut )ut divγ bt − β(ut ) divγ bt − bt , ∇β(ut ) H
= β(ut ) − β (ut )ut divγ bt − divγ bt β(ut ) . Now we prove the renormalization property in the general case. Let us define uεt := e−ε Tε (ut ); since Tε is self-adjoint in the sense of Proposition 3.3(ii) and Tε maps cylindrical functions into d cylindrical functions, the continuity equation dt ut + divγ (bt ut ) = 0 gives, still in the weak sense of duality with cylindrical functions,
d ε ut + e−ε Tε divγ (bt ut ) = 0. dt Recalling the definition (36), we may write d ε u + divγ bt uεt = e−ε r ε + uεt divγ bt . dt t Denoting by f ε the right-hand side, we know from Proposition 3.5 that f ε → 0 in L1 ((0, T ); L1 (γ )). Taking into account that uεt and ∇uεt H belong to L∞ (I ; Lr γ ) (by Proposition 3.3(iii)), from the first step we obtain d ε β ut + divγ bt β uεt = β uεt − uεt β uεt divγ bt + β uεt f ε dt for all β ∈ C 1 (R) with β (z) and zβ (z) − β(z) bounded. So, passing to the limit as ε ↓ 0 we obtain that ut is a renormalized solution. 2 Proof of Theorem 3.1. (Existence) It can be obtained as a byproduct of the results in Section 4: Theorem 4.5 provides a generalized flow, i.e. a positive finite measure η in the space of paths Ω(E), whose marginals (et )# η at all times have a density uniformly bounded in Lr (γ ), and (e0 )# η = uγ ¯ . Then, denoting by ut the density of (et )# η with respect to γ , Proposition 4.8 shows that ut solve the continuity equation. (Uniqueness) By the linearity of the equation, it suffices to show that u¯ = 0 implies ut 0 for all t ∈ [0, T ] for all solutions u ∈ L∞ ((0, T ); Lr (γ )). We extend ut and bt to the interval I := (−1, T ) by setting ut = u¯ and bt = 0 for all t ∈ (−1, 0], and it is easy to check that this extension preserves the validity of the continuity equation (still in the weak form). 1 √ We choose, as a C approximation of the positive part, the functions βε (z) equal to 2 2 z + ε − ε for z 0, and null for z 0. Thanks to Theorem 3.7, we can apply (31) with β = βε , with the test function φ ≡ 1, to obtain d dt
βε (ut ) dγ = E
E
βε (ut ) − ut βε (ut ) divγ bt dγ ε
[divγ bt ]− dγ ,
E
+ d where we used the fact that −ε βε (z) − zβε (z) 0. Letting ε ↓ 0 we obtain that dt E ut dγ 0 in (−1, T ) in the sense of distributions. But since ut = 0 for all t ∈ (−1, 0), we obtain u+ t =0 for all t ∈ [0, T ). 2
L. Ambrosio, A. Figalli / Journal of Functional Analysis 256 (2009) 179–214
195
4. Existence, uniqueness and stability of the flow In this section we discuss the problems of existence and uniqueness of a flow associated to b : [0, T ] × E → H, and we discuss its main properties. 4.1. Existence of a generalized b-flow It will be useful, in order to establish our first existence result, a definition of flow more general than Definition 1.1. In the sequel we shall denote by Ω(E) the space of continuous maps from [0, T ] to E, endowed with the sup norm. Since E is separable, Ω(E) is complete and separable. We shall denote by et : Ω(E) → E,
et (ω) := ω(t)
the evaluation maps at time t ∈ [0, T ]. If 1 α ∞, we shall also denote by AC α (E) ⊂ Ω(E) the subspace of functions ω satisfying t ω(t) = ω(0) +
g(s) ds
∀t ∈ [0, T ]
(38)
0
for some g ∈ Lα ((0, T ); E). The function g, that we shall denote by ω, ˙ is uniquely determined up to negligible sets by (38): indeed, if t¯ is a Lebesgue point of g then e∗ , g(t¯ ) coincides with the derivative at t = t¯ of the real-valued absolutely continuous function t → e∗ , ω(t) , for all e∗ ∈ E ∗ . Definition 4.1 (Generalized b-flows and Lr -regularity). If b : [0, T ]×E → E, we say that a probability measure η in Ω(E) is a flow associated to b if: (i) η is concentrated on maps ω ∈ AC 1 (E) satisfying the ODE ω˙ = b(t, ω) in the integral sense, namely t ω(t) = ω(0) +
bτ ω(τ ) dτ
∀t ∈ [0, T ];
(39)
0
(ii) (e0 )# η = γ . If in addition there exists 1 r ∞ such that, for all t ∈ [0, T ], the image measures (et )# η are absolutely continuous with respect to γ with a density in Lr (γ ), then we say that the flow is Lr -regular. Remark 4.2 (Invariance of b-flows). Assume that η is a generalized L1 -regular b-flow and b˜ is a modification of b, i.e., for a.e. t ∈ (0, T ) the set Nt := {bt = b˜ t } is γ -negligible. Then, / Nt η-almost surely. By Fubini’s because of L1 -regularity we know that, for a.e. t ∈ (0, T ), ω(t) ∈ theorem, we obtain that, for η-a.e. ω, the set of times t such that ω(t) ∈ Nt is negligible in (0, T ). ˜ As a consequence η is a b-flow as well.
196
L. Ambrosio, A. Figalli / Journal of Functional Analysis 256 (2009) 179–214
Remark 4.3 (Martingale solutions of ODEs). We remark that the notion of generalized flow coincides with the Stroock–Varadhan’s notion of martingale solutions for stochastic differential equations in the particular case when there is no noise (so that the stochastic differential equation reduces to an ordinary differential equations), see for instance [23] and [16, Lemma 3.8]. From now on, we shall adopt the convention vH = +∞ for v ∈ E \ H. Proposition 4.4 (Compactness). Let K ⊂ E be a compact set, C 0, α ∈ (1, ∞) and let F ⊂ AC α (E) be the family defined by
T
F := ω ∈ AC (E): ω(0) ∈ K,
ω ˙ H dt C .
α
α
0
Then F is compact in Ω(E). Proof. Let us fix an integer h, and split [0, T ] in the h equal intervals Ii := [iT / h, (i + 1)T / h], i = 0, . . . , h − 1. We consider the family Fh obtained by replacing each curve ω(t) in F with the continuous “piecewise affine” curve ω h coinciding with ω at the endpoints of the intervals Ii and with constant derivative, equal to Th Ii ω(t) ˙ dt, in all intervals (iT / h, (i + 1)T / h). We will check that each family Fh is relatively compact, and that sup |ω − ωh | → 0 as h → ∞, uniformly with respect to ω ∈ F . These two facts obviously imply, by a diagonal argument, the relative compactness of F . compact: indeed, the initial points of the curve The family Fh is easily seen to be relatively lie in the compact set K, and since { I0 ω(t) ˙ dt}ω∈F is uniformly bounded in H, the compactness of the embedding of H in E shows that also the family of points {ωh (T / h)}ω∈F is relatively compact; continuing in this way, we prove that all families of points {ωh (iT / h)}ω∈F , i = 0, . . . , h − 1, and therefore the family Fh , are relatively compact. Fix ω ∈ F ; denoting by L the norm of the embedding of H in E, we have ω(t) − ωh (t)
t
iT / h
ω(τ ˙ ) − ω˙ h (τ ) dτ 2L
t
1−1/α T ω(τ ˙ )H dτ 2LC 1/α h
iT / h
for all t ∈ [iT / h, (i + 1)T / h]. This proves the uniform convergence of ωh to ω as h → ∞, as ω varies in F . Finally, we have to check that F is closed. The stability of the condition ω(0) ∈ K under uniform convergence is obvious. The stability of the second condition can be easily obtained thanks to the reflexivity of the space Lα ((0, T ); H). 2 Theorem 4.5 (Existence of Lr -regular generalized b-flows). Let b : [0, T ] × E → H be satisfying the assumptions of the existence part of Theorem 3.1. Then there exists a generalized b-flow η, Lr -regular for all r ∈ [1, c/T ]. In addition, the density ut of (et )# η with respect to γ satisfies
− exp T r[div dγ b ] (ut )r dγ γ t
L∞ (0,T )
∀t ∈ [0, T ].
(40)
L. Ambrosio, A. Figalli / Journal of Functional Analysis 256 (2009) 179–214
197
Proof. Step 1 (Finite-dimensional approximation). Let bN : [0, T ]×E → HN be defined by where biN (t, ·) := EN bit ,
N
i i=1 bN ei ,
1 i N, t ∈ [0, T ].
Arguing as in the proof of Proposition 3.5, we have the estimates T 0
1/p 1/p T p (bN )t p dγ (x) dt bt H dγ (x) dt, H
E
exp c divγ (bN )t − dγ (x) E
E
0
L∞ (0,T )
(41)
− exp c[divγ bt ] dγ (x)
L∞ (0,T )
E
.
(42)
By applying Theorem 6.1 to the finite-dimensional fields b˜ N given by the restriction of bN to [0, T ] × HN , we obtain a generalized flow σ N in HN (i.e. a positive finite measure in Ω(HN )) associated to b˜ N . Using the inclusion map iN of HN in H we obtain a generalized flow ηN := (iN )# σ N associated to bN . In addition, (42) and the finite-dimensional estimate (57) give N r − ut dγ u dγ ¯ rL∞ (γ ) exp T r[div b ] , (43) sup sup γ t t∈[0,T ] N 1
E
L∞ (0,T )
E
with uN t equal to the density of (et )# η N with respect to γ . Step 2 (Tightness and limit flow η). We call coercive a functional Ψ if its sublevel sets {Ψ C} ¯ ) is a tight family of measures, by Prokhorov theorem are compact. Since (EN uγ we can find (see for instance [23]) a coercive functional Φ1 : E → [0, +∞) such that supN E Φ1 EN u¯ dγ < ∞. We choose α ∈ (1, p) such that (p/α) c/T (this is possible because we are assuming that p T < c) and consider the functional T α dt if ω ∈ AC p (E), ˙ H Φ(ω) := Φ1 (ω(0)) + 0 ω(t) (44) +∞ if ω ∈ Ω(E) \ AC α (E). Thanks to Proposition 4.4 and the coercivity of Φ1 , Φ is a coercive functional in Ω(E). Since
Φ(ω) dηN (ω) =
Ω(E)
T
(bN )t ω(t) α dηN (ω) dt H
Φ1 (x)EN u(x) ¯ dγ (x) + E
0 Ω(E)
T Φ1 (x)EN u(x) ¯ dγ (x) +
= E
(bN )t (x)α uN (x) dγ (x) dt H t
0 E
we can apply Hölder inequality with the exponents p/α and (p/α) , (41), (42) and (43) to obtain that Φ dηN is uniformly bounded. So, we can apply again Prokhorov theorem to obtain that
198
L. Ambrosio, A. Figalli / Journal of Functional Analysis 256 (2009) 179–214
(ηN ) is tight in Ω(E). Therefore we can find a positive finite measure η in Ω(E) and a family of integers Ni → ∞ such that ηNi → η weakly, in the duality with Cb (Ω(E)). In the sequel, to simplify our notation, we shall assume that convergence occurs as N → ∞. Obviously, because of (43), η is Lr -regular and, more precisely, (40) holds. Step 3 (η is a b-flow). It suffices to show that
t 1 ∧ ω(t) − ω(0) − bs ω(s) ds dη = 0
(45)
0
Ω(E)
for any t ∈ [0, T ]. The technical difficulty is the integrand in (45), due to the lack of regularity of bt , is not continuous in Ω(E); the truncation with the constant 1 is used to have a bounded integrand. To this aim, we prove first that
t T bs (x) − cs (x)us (x) dγ (x) ds 1 ∧ ω(t) − ω(0) − cs ω(s) ds dη 0 E
0
Ω(E)
(46)
for any bounded continuous function c. Then, choosing a sequence (cn ) converging to b in L1 ((0, T ); Lp (γ ; E)), and noticing that
T
bs ω(s) − (cn )s ω(s) ds dη =
T
bs (x) − (cn )s (x)us (x) dγ (x) ds → 0,
0 E
Ω(E) 0
we can pass to the limit in (46) with c = cn to obtain (45). It remains to show (46). This is a limiting argument based on the fact that (45) holds for bN , ηN : t 1 ∧ ω(t) − ω(0) − cs ω(s) ds dη
0
Ω(E)
= lim
N →∞ Ω(E)
= lim
N →∞ Ω(E)
t 1 ∧ ω(t) − ω(0) − cs ω(s) ds dηN 0
t 1 ∧ (bN )s ω(s) − cs ω(s) ds dηN
T lim sup N →∞
0
(bN )s (x) − cs (x)uN (x) dγ (x) ds s
0 E
L. Ambrosio, A. Figalli / Journal of Functional Analysis 256 (2009) 179–214
T =
199
bs (x) − cs (x)us (x) dγ (x) ds.
0 E
In order to obtain the last equality we added and subtracted bs − cs uN s , and we used the strong convergence of bN to b in L1 ((0, T ); Lp (γ ; E)) and the weak∗ convergence of uN s to us in
L∞ ((0, T ); Lp (γ ; E)). 2 4.2. Uniqueness of the b-flow The following lemma provides a simple characterization of Dirac masses (i.e. measures concentrated at a single point), for measures in Ω(E) and for families of measures in E. Lemma 4.6. Let σ be a positive finite measure in Ω(E). Then σ is a Dirac mass if and only if (et )# σ is a Dirac mass for all t ∈ Q ∩ [0, T ]. A Borel family {νx }x∈E of positive finite measures in E (i.e. x → νx (A) is Borel in E for all A ⊂ E Borel) is made, for γ -a.e. x, by Dirac masses if and only if νx (A1 )νx (A2 ) = 0 γ -a.e. in E, for all disjoint Borel sets A1 , A2 ⊂ E.
(47)
Proof. The first statement is a direct consequence of the fact that all elements of Ω(E) are continuous maps, which are uniquely determined on Q ∩ [0, T ]. In order to prove the second statement, let us fix an integer k 1 and a countable partition (Ai ) of Borel sets with diam(Ai ) 1/k (its existence is ensured by the separability of E). By (47) we obtain a γ -negligible Borel set Nk satisfying νx (Ai )νx (Aj ) = 0 for all x ∈ E \ Nk . As a consequence, the support of each of the measures νx , as x varies in E \ Nk , is contained in the closure of one of the sets Ai , which has diameter less than 1/k. It follows that νx is a Dirac mass for all x ∈ E \ k Nk . 2 Theorem 4.7 (Uniqueness of Lr -regular generalized b-flows). Let b : [0, T ] × E → H be satisfying the assumptions of the uniqueness part of Theorem 3.1, let r = max{p , q } and assume that c rT . Let η be a Lr -regular generalized b-flow. Then: (i) for γ -a.e. x ∈ E, the measures E(η|ω(0) = x) are Dirac masses in Ω(E), and setting E η|ω(0) = x = δX(·,x) ,
X(·, x) ∈ Ω(E),
(48)
the map X(t, x) is a Lr -regular b-flow, according to Definition 1.1. (ii) Any other Lr -regular generalized b-flow coincides with η. In particular X is the unique Lr -regular b-flow. Proof. (i) We set ηx := E(η|ω(0) = x). Taking into account the first statement in Lemma 4.6, it suffices to show that, for t¯ ∈ Q ∩ [0, T ] fixed, the measures νx := E((et¯)# η|ω(0) = x) = (et¯)# ηx are Dirac masses for γ -a.e. x ∈ E. Still using Lemma 4.6, we will check the validity of (47). Since νx = δx when t¯ = 0, we shall assume that t¯ > 0. Let us argue by contradiction, assuming the existence of a Borel set L ⊂ E with γ (L) > 0 and disjoint Borel sets A1 , A2 ⊂ E such that both νx (A1 ) and νx (A2 ) are positive for x ∈ L.
200
L. Ambrosio, A. Figalli / Journal of Functional Analysis 256 (2009) 179–214
We will get a contradiction with Theorem 3.1, building two distinct solutions of the continuity equation with the same initial condition u¯ ∈ L∞ (γ ). With no loss of generality, possibly passing to a smaller set L still with positive γ -measure, we can assume that the quotient β(x) := νx (A1 )/νx (A2 ) is uniformly bounded in L. Let Ωi ⊂ Ω(E) be the set of trajectories ω which belong to Ai at time t¯; obviously Ω1 ∩ Ω2 = ∅ and we can define positive finite measures ηi in Ω(E) by η2 := β(x)χΩ2 ηx dγ (x). η1 := χΩ1 ηx dγ (x), L
L
By Proposition 4.8, both η1 and η2 induce, via the identity uit γ = (et )# ηi , a solution to the continuity equation which is uniformly bounded (just by comparison with the one induced by η) ¯ = νx (A1 )χL (x). in Lr (γ ). Moreover, both solutions start from the same initial condition u(x) On the other hand, by the definition of Ωi , u1t¯ γ is concentrated in A1 while u2t¯ γ is concentrated in A2 , therefore u1t¯ = u2t¯ . So, uniqueness of solutions to the continuity equation is violated. (ii) If σ is any other Lr -regular generalized b-flow, we may apply statement (i) to the flows σ , to obtain that for γ -a.e. x also the measures E(σ |ω(0) = x) are Dirac masses; but since the property of being a generalized flow is stable under convex combinations, also the measures 1 η + σ 1 E η|ω(0) = x + E σ |ω(0) = x = E ω(0) = x 2 2 2 must be Dirac masses for γ -a.e. x. This can happen only if E(η|ω(0) = x) = E(σ |ω(0) = x) for γ -a.e. x. 2 ˙ = bt (X) and the continuity equation is The connection between solutions to the ODE X classical: in the next proposition we present it under natural regularity assumptions in this setting. Proposition 4.8. Let η be a positive finite measure in Ω(E) satisfying: (a) η is concentrated on paths ω ∈ AC 1 (E) such that ω(t) = ω(0) + t ∈ [0, T ]; T ˙ (b) 0 Ω(E) ω(t) H dη(ω) dt < ∞. Then the measures μt := (et )# η satisfy
d dt μt
t 0
bs (ω(s)) ds for all
+ divγ (bt μt ) = 0 in (0, T ) × E in the weak sense.
∗ , x ) be cylindrical. By (a) and Fubini’s theorem, for a.e. t Proof. Let φ(x) = ψ( e1∗ , x , . . . , eN the following property holds: the maps ei∗ , ω(t) , 1 i N , are differentiable at t, with derivative equal to ei∗ , bt (ω(t)) , for η-a.e. ω. Taking (12) into account, for a.e. t we have
d dt
φ dμt = E
d dt
∗ ψ e1∗ , ω(t) , . . . , eN , ω(t) dη
Ω(E)
=
N i=1Ω(E)
∗ ∂ψ ∗ e1 , ω(t) , . . . , eN , ω(t) ei∗ , ω(t) ˙ dη ∂zi
L. Ambrosio, A. Figalli / Journal of Functional Analysis 256 (2009) 179–214
=
N i=1 Ω(E)
201
∗ ∂ψ ∗ e1 , ω(t) , . . . , eN , ω(t) ei , bt ω(t) H dη ∂zi
=
∇φ, bt H dμt . E
In the previous identity we used, to pass to the limit under the integral sign, the property lim
h→0
ei∗ ,
ω(t + h) − ω(t) = ei∗ , ω(t) ˙ h
in L1 (η), for 1 i N,
whose validity for a.e. t is justified by assumption (b). The same assumption also guarantees (see for instance [2, §3] for a detailed proof) that t → E φ dμt is absolutely continuous, so its pointwise a.e. derivative coincides with the distributional derivative. 2 4.3. Stability of the b-flow and semigroup property The methods we used to show existence and uniqueness of the flow also yield stability of the flow with respect to approximations (not necessarily finite-dimensional ones) of the vector field. In the proof we shall use the following simple lemma (see for instance [2, Lemma 22] for a proof), where we use the notation id × f for the map x → (x, f (x)). Lemma 4.9 (Convergence in law and in probability). Let F be a metric space and let fn , f : E → F be Borel maps. Then fn → f in γ -probability if and only if id × fn → id × f in law. Theorem 4.10 (Stability of Lr -regular b-flows). Let p, q > 1, r = max{p , q } and let bn , b : (0, T ) × E → H be satisfying: (i) bn → b in L1 ((0, T ); Lp (γ ; H)); q (ii) for a.e. t ∈ (0, T ) we have (bn )t , bt ∈ LDH (γ ; H) with T sup n∈N
1/q ∇(bn )t sym (x)q dγ (x) dt < ∞ HS
(49)
E
0
and divγ (bn )t and divγ bt belong to L1 ((0, T ); Lq (γ )); (iii) exp(c[divγ (bn )t ]− ) are uniformly bounded in L∞ ((0, T ); L1 (γ )) for some c T r. Then, denoting by Xn (respectively X) the unique Lr regular bn -flows (respectively b-flow) we have lim
n→∞ E
sup Xn (·, x) − X(·, x) dγ (x) = 0.
[0,T ]
(50)
202
L. Ambrosio, A. Figalli / Journal of Functional Analysis 256 (2009) 179–214
Proof. Let us denote the generalized bn -flows ηn induced by Xn , namely the law under γ of x → Xn (·, x). The uniform estimates (iii), together with the boundedness of bn H in L1 ((0, T ); Lp (γ )) imply, in view of (40), sup n∈N
n r n − exp T r div dγ b ut dγ sup γ t
L∞ (0,T )
n∈N
< ∞ ∀t ∈ [0, T ],
(51)
where unt is the density of (et )# ηn = X(t, ·)# γ with respect to γ . In addition, by the same argument used in Step 2 of the proof of Theorem 4.5 we have sup n∈N Ω(E)
Φ(ω) dηn (ω) < ∞,
where Φ is defined as in (44), with α ∈ (1, p) and Φ1 : E → [0, ∞) γ -integrable and coercive. This estimate implies the tightness of (ηn ). If η is a limit point, in the duality with Cb (Ω(E)), of ηn , the same argument used in Step 3 of the proof of Theorem 4.5 gives that η is a generalized b-flow. In addition, the uniform estimates (51) imply that η is Lr -regular. As a consequence we can apply Theorem 4.7 to obtain that η is the law of the Ω(E)-valued map x → X(·, x), and more precisely that E(η|ω(0) = x) = δX(·,x) for γ -a.e. x. Therefore, by the uniqueness of X, the whole sequence (ηn ) converges to η and Xn converge in law to X. In order to obtain that x → Xn (·, x) converge in γ -probability to x → X(·, x) we use Lemma 4.9 with F = Ω(E), so we have to show that id × Xn (·, x) converge in law to id × X(·, x). For all ψ ∈ Cb (E × Ω(E)) we have
ψ x, Xn (·, x) dγ (x) =
E
ψ e0 (ω), ω dηn →
Ω(E)
=
ψ e0 (ω), ω dη
Ω(E)
ψ x, X(·, x) dγ (x),
E
and this proves the convergence in law. Finally, by adding and subtracting x, we can prove (50) provided we show that sup[0,T ] |X(·, x) − x| ∈ L1 (γ ) and sup[0,T ] |Xn (·, x) − x| are equi-integrable in L1 (γ ). We prove the second property only, because the proof of the first one is analogous. Starting from the integral formulaT tion of the ODE, Jensen’s inequality gives sup[0,T ] |Xn (·, x) − x|α T α−1 0 bτ (Xn (τ, x)) dτ and by integrating both sides with respect to γ , Fubini’s theorem gives E
α sup Xn (·, x) − x dγ (x) T α−1
[0,T ]
T bτ α unτ dγ dτ. E 0 E
Choosing α > 1 such that (p/α) c/T (this is possible because we are assuming that c > p T ) and applying the Hölder inequality with the exponents p/α and (p/α) we obtain that sup[0,T ] |Xn (·, x) − x| are equibounded in Lα (γ ). 2
L. Ambrosio, A. Figalli / Journal of Functional Analysis 256 (2009) 179–214
203
Under the same assumptions of Theorem 4.7, for all s ∈ [0, T ] also a unique Lr -regular flow Xs : [s, T ] × E → E exists, characterized by the properties that τ → Xs (τ, x) is an absolutely continuous map in [s, T ] satisfying t Xs (t, x) = x +
bτ Xs (τ, x) dτ
∀t ∈ [s, T ]
(52)
s
for γ -a.e. x ∈ E, and the regularity condition Xs (τ, ·)# γ = fτ γ , with fτ ∈ Lr (γ ), uniformly for τ ∈ [s, T ]. This family of flow maps satisfies the semigroup property: Proposition 4.11 (Semigroup property). Under the same assumptions of Theorem 4.7, the unique Lr -regular flows Xs starting at time s satisfy the semigroup property Xs t, X r (s, x) = X r (t, x)
for γ -a.e. x ∈ E, ∀0 r s t T .
(53)
Proof. Let r, s, t be fixed. By combining the finite-dimensional projection argument of Step 1 of the proof of Theorem 4.5, with the smoothing argument used in Step 2 of the proof of Theorem 6.1 we can find a family of vector fields bn converging to b in L1 ((0, T ); Lp (γ ; H)) and satisfying the uniform bounds of Theorem 4.10, whose (classical) flows Xn satisfy the semigroup property (see (62)) Xsn t, X rn (s, x) = Xrn (t, x)
for γ -a.e. x ∈ E, ∀0 r s t T .
(54)
We will pass to the limit in (54), to obtain (53). To this aim, notice that (50) of Theorem 4.10 immediately provides the convergence in L1 (γ ) of the right-hand sides, so that we need just to show convergence in γ -measure of the left-hand sides. Notice first that the convergence in γ -measure of Xrn (s, ·) to Xr (s, ·) implies the convergence in γ -measure of ψ(Xrn (s, ·)) to ψ(Xr (s, ·)) for any Borel function ψ : E → R (this is a simple consequence of the fact that, by Lusin’s theorem, we can find a nondecreasing sequence of compact sets Kn ⊂ E such that ψ|Kn is uniformly continuous and γ (E \ Kn ) ↓ 0, and of the fact that the laws of Xrx (s, ·) are uniformly bounded in Lr (γ )), so that choosing ψ(z) := Xs (t, z), and adding and subtracting Xs (t, X n (s, x)), the convergence in γ -measure of the right-hand sides of (54) to Xs (t, X r (s, x)) follows by the convergence in γ -measure to 0 of Xsn t, Xrn (s, x) − Xs t, Xrn (s, x) . Denoting by ρn the density of the law of Xrn (s, ·), we have
1 ∧ Xsn t, Xrn (s, x) − X s t, Xrn (s, x) dγ (x)
E
=
1 ∧ Xsn (t, y) − Xs (t, y)ρn (y) dγ (y),
E
and the right-hand side tends to 0 thanks to (50) and to the equi-integrability of (ρn ).
2
204
L. Ambrosio, A. Figalli / Journal of Functional Analysis 256 (2009) 179–214
The semigroup property allows also to construct a unique family of flows Xs : [s, T ] × E × E even in the case when the assumption (27) is replaced by exp c[divγ bt ]− ∈ L∞ (0, T ); L1 (γ )
for some c > 0.
The idea is to compose the flows defined on sufficiently short intervals, with length T satisfying c > rT . It is easy to check that this family of flow maps is uniquely determined by the semigroup property (53) and by the local regularity property
Xs (t, ·)# γ γ with a density in Lr (γ ) for all t ∈ s, min{s + T , T } , s ∈ [0, T ]. Globally in time, the only property retained is Xs (t, ·)# γ γ for all t ∈ [s, T ]. 4.4. Convergence of finite-dimensional flows Assume that we are given vector fields bN : [0, T ] × RN → RN satisfying, for some p, q > 1 the assumptions (i)–(iii) of Theorem 1.2 (with E = H = RN ) relative to the standard Gaussian γN in RN , with norms uniformly bounded by constants independent of N . Let us assume that bN is a consistent family, namely the conditional expectation of the projection of (bN +1 )t on RN , given x 1 , . . . , x N , is (bN )t . Let XN : [0, T ] × RN → RN be the associated bN -flows. In this section we briefly illustrate how the stability results of this paper can be used to prove the convergence of XN and to characterize their limit. To this aim, let us denote by γp the product of standard Gaussians in the countable product R∞ , and notice that the consistency assumption provides us with a unique vector field b : [0, T ] × R∞ → R∞ such that, denoting by EN the conditional expectation with respect to x 1 , . . . , x N and by πN : R∞ → RN the canonical projections, the identities EN πN bt = (bN )t hold. In order to recover a Wiener space we fix a sequence (λi ) ∈ 2 and define
∞ i 2 i 2 E := x : λi x < ∞ . i=1
The space E can be endowed with the canonical scalar product, and obviously γp (E) = 1, so that b can be also viewed as a vector field in E and the induced measure γ in E is Gaussian. According to Remark 2.1, its Cameron–Martin space H can be identified with 2 . Then, we can apply the stability Theorem 4.10 (viewing, with a slight abuse, bN as vector fields in E and, consequently, their flows XN as flows in E which leave x N +1 , x N +2 , . . . fixed) to obtain that XN converge to the flow X relative to b in L1 (γ ; E). It follows that
lim
∞
N →∞ R∞
2 λ2i XiN (t, x) − Xi (t, x) dγp (x) = 0
∀t ∈ [0, T ], ∀(λi ) ∈ 2 .
(55)
i=1
Finally, notice that also X could be defined without an explicit mention to E, working in (R∞ , γp ) in place of (E, γ ). According to this viewpoint, E plays just the role of an auxiliary space, and deliberately we wrote (55) without an explicit mention to it.
L. Ambrosio, A. Figalli / Journal of Functional Analysis 256 (2009) 179–214
205
5. An extension to non-H-valued vector fields In [7,22], the authors consider the following equation: ˜ tx + X(t, x) = Q
t
Qt−s bs X(s, x) ds.
(56)
0
˜t :E → E Here (Qt )t∈R is a strongly continuous group of orthogonal operator on H, and Q denotes the measurable linear extension of Qt to E (which always exists and preserves the measure γ , see for instance [18]). Observe that, thanks to the Duhamel formula, (56) formally corresponds to the equation ˙ x) = LX(t, x) + bt X(t, x) , X(t, ˙ t = LQt ). where L denotes the generator of the group (i.e. Q The definition of Lr -regular flow can be extended in the obvious way to (56). Let us now see how our results allow to prove existence and uniqueness of Lr -regular flows under the assumptions of Theorem 1.2 (observe that this forces in particular r > 1). ˜ −t X(t, x). Then we have Let X(t, x) be a solution of (56), and define Y (t, x) := Q t Y (t, x) = x +
Q−s bs X(s, x) ds = x +
0
t
˜ s Y (s, x) ds. Q−s bs Q
0
˜ t x). Moreover Y is still Therefore Y is a flow associated to the vector field ct (x) := Q−t bt (Q r r a L -regular flow. Indeed, if ut ∈ L (γ ) denotes the density of the law of X(t, ·), then, for all φ ∈ Cyl(E, γ ), we have
φ Y (t, x) dγ (x) =
˜ −t X(t, x) dγ (x) = φ Q
φ(Q˜ −t x)ut (x) dγ (x)
ut Lr (γ ) φ ◦ Q˜ t Lr (γ ) = ut Lr (γ ) φLr (γ ) . Since r > 1, this implies that Y is Lr -regular. On the other hand we remark that, using the same ˜ t Y (t, x) argument, one obtains that, if Y is a Lr -regular flow associated to c, then X(t, x) := Q r is a L -regular flow for (56). We have therefore shown that there is a one-to-one correspondence between Lr -regular flows for (56) and Lr -regular flows associated to c. To conclude the existence and uniqueness of Lr -regular flows for (56), it suffices to observe that, thanks to the orthogonality ˜ t , if b satisfies all the assumptions in Theoof Qt and the measure-preserving property of Q rem 1.2, then so does c thanks to the identities ct (x)H = bt (Q˜ t x)H , (∇ct )sym (x)HS = (∇bt )sym (Q˜ t x)HS , and divγ ct (x) = divγ bt (Q˜ t x). Indeed, let us check the formula for the symmetric part of the derivative, the proof of the one concerning the divergence being similar and even simpler. Let h = j (e∗ ) ∈ H and notice that ˜ −t (y) . Using Remark 2.9 and the fact that φ → φ ◦ Q ˜t Qt h = j (f ∗ ), where f ∗ , y = e∗ , Q 1 maps Cyl(E, γ ) into Cb (E, γ ), for φ ∈ Cyl(E, γ ) we get:
206
L. Ambrosio, A. Figalli / Journal of Functional Analysis 256 (2009) 179–214
˜ t x), Qt h ∂h φ(x) dγ (x) bt (Q H
ct , h H ∂h φ dγ = E
E
˜ −t (y) dγ (y) bt (y), Qt h H (∂h φ) ◦ Q
= E
˜ −t )(y) dγ (y) bt (y), Qt h H ∂Qt h (φ ◦ Q
= E
=−
∂Qt h bt , Qt h H φ ◦ Q˜ −t dγ (y)
E
+
bt (y), Qt h H φ ◦ Q˜ −t f ∗ , y dγ (y)
E
=−
∂Qt h bt , Qt h H ◦ Q˜ t φ dγ (x) +
E
ct (x), h H φ e∗ , x dγ (x).
E
This proves that ∂h ct , h H = ∂Qt h bt , Qt h H ◦ Q˜ t , and using the fact that Qt maps orthonormal ˜ t. bases of H in orthonormal bases of H we get (∇ct )sym HS = (∇bt )sym HS ◦ Q 6. Finite-dimensional estimates This section is devoted to the proof of the crucial a priori bounds (28) and (37) in finitedimensional Wiener spaces. So, we shall assume that E = H = RN and, only in this section, denote by x · y the scalar product in RN , and by |x| the Euclidean norm (corresponding to the norm of the Cameron–Martin space). Also, only in this section we shall denote by γ the standard N , product of N standard Gaussians in R, and by integrals on the whole of RN . Gaussian in R The sums i (respectively i,j ) will always be understood with i (respectively i and j ) running from 1 to N . 6.1. Upper bounds on the flow density In this subsection we show the existence part of Theorem 3.1 in finite-dimensional Wiener spaces E = H = RN . Theorem 6.1. Let b : (0, T ) × RN → RN be satisfying the assumptions of the existence part of Theorem 3.1. Then, for any r ∈ [1, c/T ] there exists a generalized Lr -regular b-flow η. Its density ut satisfies also − exp T r[div (ut )r dγ dγ b ] ∀t ∈ [0, T ]. (57) γ t L∞ (0,T )
Proof. T Step 1. Here we consider first the case when bt are smooth, with 0 ∇bt L∞ (B) dt finite for all bounded open sets B ⊂ RN . Under this assumption, for all x ∈ RN the unique solution X(·, x)
L. Ambrosio, A. Figalli / Journal of Functional Analysis 256 (2009) 179–214
207
˙ x) = bt (X(t, x)), with the initial condition X(0, x) = x, is defined until some to the ODE X(t, maximal time τ (x) ∈ (0, T ]. Obviously, by the maximality of τ (x), if lim sup X(t, x) < +∞ t↑τ (x)
then τ (x) = T and the solution is continuous in [0, T ]. Let us fix s ∈ [0, T ). We denote Es the set {τ > s} and notice that standard stability results for ODE’s with a locally Lipschitz vector field ensure that Es is open and that x → X(t, x) is smooth in Es for t ∈ [0, s]. Furthermore, from the identity ∇˙ x X(t, x) = ∇bt (X(t, x))∇x X(t, x), obtained by spatial differentiation of the ODE (see [2] for details), one obtains J˙X(t, x) = div bt X(t, x) J X(t, x),
x ∈ Es , t ∈ [0, s],
(58)
where J X(t, x) is the determinant of ∇x X(t, x). We first compute a pointwise expression for the measure X(t, ·)# (χEs γ ) for t ∈ [0, s]. By the change of variables formula, the density ρts of X(t, ·)# (χEs γ ) with respect to LN is linked to the initial density ρ¯ s by ρ¯ s (x) ρts X(t, x) = , J X(t, x) where ρ¯ s (y) := χEs (y)e−|y| we get
2 /2
. Denoting by ust the density of X(t, ·)# (χEs γ ) with respect to γ ,
ρ¯ s (x) |X(t,x)|2 /2 e ust X(t, x) = . J X(t, x)
(59)
So, taking the identity (58) into account, we obtain ρ¯ s (x) |X(t,x)|2 /2 d s ut X(t, x) = − divγ bt X(t, x) e = − divγ bt X(t, x) ust X(t, x) . dt J X(t, x) By integrating the ODE, for t ∈ [0, s] we get t ust X(t, x) = χEs (x) exp − divγ bτ X(τ, x) dτ 0
t χEs (x) exp
− divγ bτ X(τ, x) dτ .
0
We can now estimate ust Lr (γ ) as follows:
s r ut dγ =
s r−1 s ut ut dγ
t
exp (r − 1) 0
− divγ bτ X(τ, x) dτ χEs (x) dγ (x)
208
L. Ambrosio, A. Figalli / Journal of Functional Analysis 256 (2009) 179–214
1 t
t
−
dτ χEs (x) dγ (x) exp t (r − 1) divγ bτ X(τ, x)
0
=
1 t
t
−
χEs (x) dγ (x) dτ exp t (r − 1) divγ bτ X(τ, x)
0
1 t
t
−
exp T (r − 1) divγ bτ (y) usτ (y) dγ (y) dτ.
0
Now, set Λ(t) :=
t 0
usτ rLr (γ ) dτ and apply the Hölder inequality to get
1 Λ (t) t
t
− dγ (y) dτ exp T r divγ bτ (y)
1/r Λ1/r (t)
0
Kt
1/r −1
Λ1/r (t) = Kt −1/r Λ1/r (t),
(60)
1/r with K := exp(T r[divγ bt ]− ) dγ L∞ (0,T ) . An integration of this differential inequality yields
Λ(t) K r t, which inserted into (60) gives
s r − ut dγ exp T r[divγ bt ] dγ
L∞ (0,T )
∀t ∈ [0, s], ∀s ∈ [0, T ).
(61)
Now, let us prove that the flow is globally defined in [0, T ] for γ -a.e. x: we have indeed
sup X(t, x) − x dγ (x)
[0,τ (x))
τ (x) T bt X(t, x) dt dγ (x) = |bt X(t, x) | dγ (x) dt 0 Et
0
T |bt |utt dγ dt.
= 0
Using (61) with s = t, we obtain that sup[0,τ (x)) |X(t, x) − x| dγ (x) is finite, so that τ (x) = T and X(·, x) is continuous up to t = T for γ -a.e. x. Letting s ↑ T in (61) we obtain (57). Denoting as in (52) by Xs the flow starting at time s, we also notice (this is useful in the proof, by approximation, of the semigroup property in Proposition 4.11) that the pointwise uniqueness of the flow implies the semigroup property Xs t, X r (s, x) = Xr (t, x) for all x where Xr (·, x) is globally defined in [r, T ].
∀0 r s t T
(62)
L. Ambrosio, A. Figalli / Journal of Functional Analysis 256 (2009) 179–214
209
Step 2. In this step we remove the regularity assumptions made on b, considering the vector fields bε defined by biε (t, ·) := Tε bit . It is immediate to check that the fields bε satisfy the regularity assumptions made in Step 1, so the existence of a Lr -regular bε -flow ηε satisfying
− ε r exp T r div dγ (b ) ut dγ γ ε t
(63)
L∞ (0,T )
is ensured by Step 1. In (63) the functions uεt are, as usual, the densities of (et )# ηε with respect to γ . Now, since divγ ((bε )t ) = e−ε Tε (divγ bt ), we may apply Jensen’s inequality to get
ε r −ε − ut dγ exp e T r[divγ bt ] dγ
L∞ (0,T )
(64)
.
Since T
bε (t, x)p dγ H
0
T
1/p dt
b(t, x)p dγ H
1/p dt,
0
the same tightness argument used in the proof of Theorem 4.5 to pass from finitely many to infinitely many dimensions provides us with a b-flow η satisfying (57): any weak limit point η of ηε as ε ↓ 0. 2 6.2. Commutator estimate This subsection is entirely devoted to the proof of the commutator estimate (37) in finitedimensional Wiener spaces. We will often use the “Gaussian rotations” (65) (x, y) → (z, w) := e−ε x + 1 − e−2ε y, − 1 − e−2ε x + e−ε y , mapping the product measure γ (dx) × γ (dy) into γ (dz) × γ (dw). Indeed, the transformations above preserve the Lebesgue measure in RN × RN (being their Jacobian identically equal to 1) and |x|2 + |y|2 = |z|2 + |w|2 . We now state two elementary Gaussian estimates. The first one
1/p 1/p p = |l| = Λ(p)|l| |l · w| dγ (w) |w1 | dγ (w) p
∀l ∈ RN ,
(66)
with Λ depending only on p, is a simple consequence of the rotation invariance of γ . Lemma 6.2. Let A : RN → RN be a linear map and c ∈ R. Then, if q 2, we have
1/q √ Aw, w − c q dγ (w) 2Asym HS + |tr A − c|.
(67)
210
L. Ambrosio, A. Figalli / Journal of Functional Analysis 256 (2009) 179–214
Proof. Obviously we can assume that A is symmetric. By rotation invariance, we can also assume that A is diagonal, and denote by λ1 , . . . , λN its eigenvalues. We have then 2 2 i 2 i 2 j 2 i 2 dγ (w) w w w w λ − c dγ (w) = λ λ − 2c λ + c i i j i i
i,j
=3
λ2i +
=2
λ2i +
i
=2
i
λi λj − 2c
i=j
i
λi + c 2
i
λi λj − 2c
i,j
λ2i +
i
λi + c 2
i
2 λi − c
.
i
If q = 2 we take the square roots of both sides and we conclude; if q 2 we apply the Hölder inequality. 2 q
Henceforth, a vector field c ∈ Lp (γ ; RN ) ∩ LDH (γ ; RN ) and a function v ∈ Lr (γ ) will be fixed, with r = max{p , q } and p > 1, 1 q 2. Our goal is to prove the estimate
Λ(p)ε
r v cLp (γ ;RN ) + 21/q divγ cLq (γ ) L (γ ) √ L1 (γ ) −2ε 1−e √
+ 21/q 2(∇c)sym HS Lq (γ ) ,
ε r
(68)
where r ε := eε c · ∇vε − Tε divγ (vc) . √
Since 21/q 2, this yields the finite-dimensional version of (37). In this setup the Ornstein–Uhlenbeck operator vε := Tε v takes the explicit form vε (x) :=
v e−ε x + 1 − e−2ε y dγ (y) =
v(z)ρε (x, z) dγ (z)
with ρε (x, z) := = This implies that
1 (1 − e−2ε )N/2 1 (1 − e−2ε )N/2
2 |z| |e−ε x − z|2 exp exp − −2ε 2 2(1 − e ) |e−ε x|2 − 2ε −ε x · z + |e−ε z|2 . exp − 2(1 − e−2ε )
(69)
L. Ambrosio, A. Figalli / Journal of Functional Analysis 256 (2009) 179–214
211
−ε e x −z v(z)∇x ρε (x, z) dγ (z) = −e−ε f (z)ρε (x, z) dγ (z) 1 − e−2ε y = e−ε v e−ε x + 1 − e−2ε y √ dγ (y). 1 − e−2ε
∇vε (x) =
(70)
Let us look for a more explicit expression of the commutator in (69). To this aim, we show first that Tε (divγ (vc)) is a function, and y dγ (y) − Tε (z · vc)(x). (71) Tε divγ (vc) (x) = (vc) e−ε x + 1 − e−2ε y · √ 1 − e−2ε If c and v are smooth, this is immediate to check: indeed, thanks to (14), we need only to show that y Tε div(vc) (x) = (vc) e−ε x + 1 − e−2ε y · √ dγ (y). 1 − e−2ε The latter is a direct consequence of (70) (with v replaced by vci ) and of the relation ∂i Tε (vci ) = e−ε Tε (∂i (vci )). If v and c are not smooth, we argue by approximation. Therefore, taking (70) and (71) into account, we have that r ε (x) is given by
√ −ε c(x) − c(e−ε x + 1 − e−2ε y) −2ε v e x + 1−e y · y dγ (y) √ 1 − e−2ε + v e−ε x + 1 − e−2ε y c e−ε x + 1 − e−2ε y · e−ε x + 1 − e−2ε y dγ (y) √ v(e−ε x + 1 − e−2ε y) = √ 1 − e−2ε ! " × c(x) · y − c e−ε x + 1 − e−2ε y · e−2ε y − e−ε 1 − e−2ε x dγ (y).
Now, using the abbreviations αε (x, y) := v(e−ε x + polate and write −r ε (x) as √
1 1 − e−2ε
d αε (x, y) dt
1
√ √ 1 − e−2ε y), βε := ε/ 1 − e−2ε , we inter-
c e−tε x + 1 − e−2εt y
0
× e−2tε y − e−tε 1 − e−2tε x dt dγ (y) = βε αε (x, y), 1 ∂j ci e−tε x + 1 − e−2tε y e−tε 1 − e−2tε x i − e−2tε y i 0
i,j
e−2tε + ci e−tε x + 1 − e−2tε y × e−tε x j − √ yj −2tε 1−e i
(72)
212
L. Ambrosio, A. Figalli / Journal of Functional Analysis 256 (2009) 179–214
e−3tε x i − 2e−2tε y i dt dγ (y) e−tε 1 − e−2tε − √ 1 − e−2tε =: βε αε (x, y) Aε (x, y) + Bε (x, y) dγ (y),
×
(73)
where, adding and subtracting e−2tε −tε i e x + 1 − e−2tε y i , ci e−tε x + 1 − e−2tε y √ −2tε 1−e i we have set
Aε (x, y) :=
1
∂j ci e−tε x + 1 − e−2tε y e−tε 1 − e−2tε x i − e−2tε y i
i,j
0
e−2tε × e−tε x j − √ yj 1 − e−2tε e−2tε −tε i − e x + 1 − e−2tε y i dt, ci e−tε x + 1 − e−2tε y √ 1 − e−2tε i
Bε (x, y) :=
1 i
0
Let us estimate βε
i −tε
c e x + 1 − e−2tε y e−tε 1 − e−2tε x i − e−tε y i dt.
|αε Bε | dγ dγ first: the change of variables (65) and Fubini’s theorem give
1 |αε Bε | dγ (x) dγ (y) βε
βε
0
e−εt
i i v(z) c (z)w dγ (z) dγ (w) dt. i
Using (66) with f = c(z), we get |αε Bε | dγ (x) dγ (y)
βε
βε
i i v(z) c (z)w dγ (z) dγ (w) βε Λ(p)cLp (γ ;RN ) vLp (γ ) .
(74)
i
Now, we estimate βε
|αε Aε | dγ dγ ; again, we use the change of variables (65) to write
e−tε 1 − e−2tε x i − e−2tε y i = −e−tε w i , Therefore we get
e−2tε e−tε e−tε x j − √ yj = − √ wj . −2tε −2tε 1−e 1−e
L. Ambrosio, A. Figalli / Journal of Functional Analysis 256 (2009) 179–214
213
βε
|αε Aε | dγ (x) dγ (y) e−2tε e−2tε i i j i i v(z) ∂j c (z) √ ww − c (z) √ z dγ (z) dγ (w) dt βε −2tε −2tε 1−e 1−e i,j i 0 i i j i i v(z) = ∂ c (z)w w − c (z)z j dγ (z) dγ (w), 1
i,j
i
where we used the identity 1 √ 0
e−2tε 1 − e−2tε
√ 1 − e−2ε = βε−1 . dt = ε
Eventually we use (67) with A = ∇c(z) and c = c(z) · z to obtain βε |αε Aε | dγ (x) dγ (y) vLq (γ )
q 1/q i i j i i ∂ c (z)w w − c (z)z dγ (w) dγ (z) j i,j
2
1−1/q
vLq (γ )
21−1/q vLq (γ )
i
1/q √ q sym q q (∇c) 2 + |divγ c| dγ (z) HS
√ 2(∇c)sym HS Lq (γ ) + divγ cLq (γ ) .
(75)
Combining (72), (74) and (75), we have proved (68). References [1] L. Ambrosio, Transport equation and Cauchy problem for BV vector fields, Invent. Math. 158 (2004) 227–260. [2] L. Ambrosio, Transport equation and Cauchy problem for non-smooth vector fields, in: B. Dacorogna, P. Marcellini (Eds.), Calculus of Variations and Non-Linear Partial Differential Equations, CIME Series, Cetraro, 2005, in: Lecture Notes in Math., vol. 1927, Springer, Berlin, 2008, pp. 2–41. [3] L. Ambrosio, G. Crippa, Existence, uniqueness, stability and differentiability properties of the flow associated to weakly differentiable vector fields, in: F. Ancona, S. Bianchini, R.M. Colombo, C. De Lellis, A. Marson, A. Montanari (Eds.), Transport Equations and Multi-D Hyperbolic Conservation Laws, in: UMI Lecture Notes, vol. 5, Springer, Berlin, 2008, pp. 3–57. [4] L. Ambrosio, C. De Lellis, Existence of solutions for a class of hyperbolic systems of conservation laws in several space dimensions, Int. Math. Res. Not. 41 (2003) 2205–2220. [5] L. Ambrosio, F. Bouchut, C. De Lellis, Well-posedness for a class of hyperbolic systems of conservation laws in several space dimensions, Comm. Partial Differential Equations 29 (2004) 1635–1651. [6] V. Bogachev, Gaussian Measures, Math. Surv. Monogr., vol. 62, Amer. Math. Soc., Providence, RI, 1998. [7] V. Bogachev, E.M. Wolf, Absolutely continuous flows generated by Sobolev class vector fields in finite and infinite dimensions, J. Funct. Anal. 167 (1999) 1–68. [8] I. Capuzzo Dolcetta, B. Perthame, On some analogy between different approaches to first order PDE’s with nonsmooth coefficients, Adv. Math. Sci. Appl. 6 (1996) 689–703. [9] F. Cipriano, A.B. Cruzeiro, Flows associated with irregular Rd -vector fields, J. Differential Equations 219 (1) (2005) 183–201.
214
L. Ambrosio, A. Figalli / Journal of Functional Analysis 256 (2009) 179–214
[10] G. Crippa, C. De Lellis, Estimates and regularity results for the DiPerna–Lions flow, J. Reine Angew. Math. 616 (2008) 15–46. [11] A.B. Cruzeiro, Équations différentielles ordinaires: non explosion et mesures quasi-invariantes, J. Funct. Anal. 54 (1983) 193–205. [12] A.B. Cruzeiro, Équations différentielles sur l’espace de Wiener et formules de Cameron–Martin non linéaires, J. Funct. Anal. 54 (1983) 206–227. [13] A.B. Cruzeiro, Unicité de solutions d’équations différentielles sur l’espace de Wiener, J. Funct. Anal. 58 (1984) 335–347. [14] M. Cullen, M. Feldman, Lagrangian solutions of semigeostrophic equations in physical space, SIAM J. Math. Anal. 37 (2006) 1371–1395. [15] R.J. DiPerna, P.L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math. 98 (1989) 511–547. [16] A. Figalli, Existence and uniqueness of martingale solutions for SDEs with rough of degenerate coefficients, J. Funct. Anal. 254 (2008) 109–153. [17] Y. Hu, A.S. Ustunel, M. Zakai, Tangent processes on Wiener spaces, J. Funct. Anal. 192 (1) (2002) 234–270. [18] S. Kusuoka, Analysis on Wiener space. I. Nonlinear maps, J. Funct. Anal. 98 (1991) 122–168. [19] M. Ledoux, Isoperimetry and Gaussian analysis, in: Lectures on Probability Theory and Statistics, Saint-Flour, 1994, in: Lecture Notes in Math., vol. 1648, Springer, Berlin, 1996, pp. 165–294. [20] P.L. Lions, Mathematical Topics in Fluid Mechanics, vol. I: Incompressible Models, Oxford Lecture Ser. Math. Appl., vol. 3, Oxford Univ. Press, Oxford, 1996. [21] P.L. Lions, Mathematical Topics in Fluid Mechanics, vol. II: Compressible Models, Oxford Lecture Ser. Math. Appl., vol. 10, Oxford Univ. Press, Oxford, 1998. [22] G. Peters, Anticipating flows on the Wiener space generated by vector fields of low regularity, J. Funct. Anal. 142 (1996) 129–192. [23] D.W. Stroock, S.R.S. Varadhan, Multidimensional Diffusion Processes, Grundlehren Math. Wiss., vol. 233, Springer, Berlin, 1979. [24] R. Temam, Problémes mathématiques en plasticité, Gauthier–Villars, Paris, 1983. [25] A.S. Ustunel, M. Zakai, Transformation of Measure on Wiener Space, Springer, Berlin, 2000.
Journal of Functional Analysis 256 (2009) 215–257 www.elsevier.com/locate/jfa
The Cauchy problem of the Ward equation Derchyi Wu Institute of Mathematics, Academia Sinica, Taipei, Taiwan, ROC Received 12 March 2008; accepted 13 June 2008 Available online 17 July 2008 Communicated by C. Kenig
Abstract We generalize the results of [J. Villarroel, The inverse problem for Ward’s system, Stud. Appl. Math. 83 (1990) 211–222; A.S. Fokas, T.A. Ioannidou, The inverse spectral theory for the Ward equation and for the 2 + 1 chiral model, Comm. Appl. Anal. 5 (2001) 235–246; B. Dai, C.L. Terng, K. Uhlenbeck, On the space–time Monopole equation, arXiv:math.DG/0602607] to study the inverse scattering problem of the Ward equation with non-small data and solve the Cauchy problem of the Ward equation with a non-small purely continuous scattering data. © 2008 Elsevier Inc. All rights reserved. Keywords: Self-dual Yang–Mills equation; Lax pair; Inverse scattering problem; Riemann–Hilbert problem; Cauchy integral operator
Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Direct problem I: eigenfunctions with small data . . . . . . . . 3. Direct problem II: asymptotic analysis with small data . . . . 4. Direct problem III: eigenfunctions with non-small data . . . 5. Direct problem IV: asymptotic analysis with non-small data 6. Inverse problem: continuous scattering data . . . . . . . . . . . 7. The Cauchy problem: continuous scattering data . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
E-mail address:
[email protected]. 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.06.009
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
216 220 223 231 243 250 255 257 257
216
D. Wu / Journal of Functional Analysis 256 (2009) 215–257
1. Introduction The Ward equation (or the modified 2 + 1 chiral model) ∂t J −1 ∂t J − ∂x J −1 ∂x J − ∂y J −1 ∂y J − J −1 ∂t J, J −1 ∂y J = 0,
(1.1)
for J : R2,1 → SU(n), ∂w = ∂/∂w, is obtained from a dimension reduction and a gauge fixing of the self-dual Yang–Mills equation on R2,2 [8,19]. It is an integrable system which possesses the Lax pair [16,21,23] λ∂x − ∂ξ − J −1 ∂ξ J, λ∂η − ∂x − J −1 ∂x J = 0
(1.2)
t−y −1 ∂ J = −∂ Q, J −1 ∂ J = −∂ Q. Then by a with ξ = t+y ξ x x η 2 , η = 2 . Note (1.2) implies that J change of variables (η, x, ξ ) → (x, y, t), (1.2) is equivalent to
(∂y − λ∂x )Ψ (x, y, t, λ) = ∂x Q(x, y, t) Ψ (x, y, t, λ), ∂t − λ2 ∂x Ψ (x, y, t, λ) = (λ∂x Q + ∂y Q)Ψ (x, y, t, λ),
(1.3) (1.4)
see [11], and the Ward equation (1.1) turns into ∂x ∂t Q = ∂y2 Q + [∂y Q, ∂x Q].
(1.5)
The construction of solitons, the study of the scattering properties of solitons, and Darboux transformation of the Ward equation have been studied intensively by solving the degenerate Riemann–Hilbert problem and studying the limiting method [2,3,14,15,21,22,24]. In particular, Dai and Terng gave an explicit construction of all solitons of the Ward equation by establishing a theory of Backlund transformation [7]. For the investigation of the Cauchy problem of the Ward equation, Villarroel [20], Dai, Terng and Uhlenbeck [8] use Fourier analysis in the x, y-space to study the spectral theory of Lλ = ∂y − λ∂x in (1.3), while Fokas and Ioannidou [11] invert Lλ by interpreting it as a 1-dimensional spectral operator with coefficients being the x-Fourier transform of functions. In both cases, small data conditions of Q are required to ensure the invertibility of Lλ and the solvability of the inverse problem. Under the small data condition, the eigenfunctions Ψ possesses continuous scattering data only and therefore the solutions for the Ward equation do not include the solitons in previous study. Nontheless, the approach of Fokas and Ioannidou [11] shows that after taking the Fourier transform in the x-space, (1.3) looks similar to the spectral problem of the AKNS system (∂x − λJ )Ψ (x, t, λ) = q(x, t)Ψ (x, t, λ), where J is a constant diagonal matrix with distinct eigenvalues. The solution of the forward and inverse scattering problem of the AKNS system is fairly complete, due to the work of Beals, Coifman, Deift, Tomei, Zhou [4,6,9]. In particular, the inverse scattering problem for the AKNS system and its associated nonlinear evolution equations is rigorously solved for generic q ∈ L1 without small data condition [5].
D. Wu / Journal of Functional Analysis 256 (2009) 215–257
217
The purpose of the present paper is to remove the small data condition in solving the scattering and inverse scattering problem of (1.3) and the Cauchy problem of the Ward equation (1.5) with a purely continuous scattering data. We summarize principal results as follows. Theorem 1.1. Let Q ∈ P∞,2,0 . Then there is a bounded set Z ⊂ C such that • Z ∩ (C \ R) is discrete in C \ R; • For λ ∈ C \ (R ∪ Z), the problem (1.3) has a unique solution Ψ and Ψ − 1 ∈ DH2 ; • For (x, y) ∈ R × R, the eigenfunction Ψ (x, y, ·) is meromorphic in λ ∈ C \ R with poles precisely at the points of Z ∩ (C \ R); • Ψ (x, y, λ) satisfies: lim Ψ (·, y, λ) = 1,
lim Ψ (x, ·, λ) = 1,
|x|→∞
|y|→∞
for λ ∈ C \ (R ∪ Z),
Ψ (x, y, ·) tends to 1 uniformly as |λ| → ∞;
(1.6) (1.7)
• Ψ (x, 0, λ) satisfies: ∂xi (Ψ − 1), i = 0, 1, 2, are uniformly bounded in L2 (dx) for λ∈C\ R Dj (λj ) . For any zj ∈ C \ R, fixing k for ∀k = j λj ∈Z −hj
and letting j → 0, these L2 (dx)-norms increase as Cj j with uniform constants Cj , hj > 0. Ψ − 1, ∂x Ψ → 0
in L2 (dx) as λ → ∞,
(1.8) (1.9)
where j > 0 are any given constants, D (λj ) denotes the disk of radius centered at λj . Here the function spaces P∞,2,0 , and DH2 are defined as follows. Definition 1.
j
P∞,k1 ,k2 = qx (x, y): R × R → su(n) ξ i y s q L (dξ dy) ,
ξ h q (ξ, y) L (y) L (dξ ) , ∂x ∂yl q L , ∞ 1 1 2
j l
j l
sup ∂x ∂ q
, ∂x ∂ q
< ∞ for 1 i max{5, k1 }, y
y
L1 (dx)
y
L1 (dx dy)
0 j, l max{5, k1 }, 1 h k1 , 0 s k2 ,
DHk = f ∂xi f (x, y) are uniformly bounded in L2 (R, dx), 0 i k . To derive Theorem 1.1, we transform the existence problem of Ψ into a Riemann–Hilbert problem with a non-small continuous data by the translating invariant and the derivation properties of the spectral operator Lλ , and an induction scheme. Hence the scheme of [4, Section 10] can be adapted to solve the Riemann–Hilbert problem. That is, we first approximate the solution
218
D. Wu / Journal of Functional Analysis 256 (2009) 215–257
by a piecewise rational function. Then the correction is made by a solution of a Riemann–Hilbert problem with small data and a solution of a finite linear system. Since the eigenfunction obtained in each induction step consists the data of the Riemann–Hilbert problem in the next step, we need to obtain the H 2 -estimate (1.8) of the eigenfunction. Besides, the boundary estimate (1.9) and the meromorphic property are derived in each step to assure the solvability of the linear system. In general, the points in Z, i.e., poles of Ψ (x, y, λ), will occur or accumulate on the real line, or the limit points will accumulate themselves. Assuming higher regularities on the potential Q and Z = Z(Ψ ) = φ (there are no poles of Ψ (x, y, λ)), we can extract the continuous scattering data: Theorem 1.2. For Q ∈ P∞,k,1 , k 7, if Z = φ, then there exists uniquely a function v(x, y, λ) ∈ Sc,k which satisfies Ψ+ (x, y, λ) = Ψ− (x, y, λ)v(x, y, λ),
λ ∈ R.
Where the space Sc,k is defined by Definition 2. Let Sc,k , k 7, be the space consisting of continuous scattering data v(x, y, λ), λ ∈ R, such that v satisfies the algebraic constraints: det(v) ≡ 1,
(1.10)
v = v ∗ > 0,
(1.11)
and the analytic constraints: for i + j k − 4, Lλ v = 0,
v(x, y, λ) = v(x + λy, λ) for ∀x, y ∈ R,
j
∂xi ∂y (v − 1) are uniformly bounded in L∞ ∩ L2 (R, dλ) ∩ L1 (R, dλ),
(1.12) (1.13)
j
∂xi ∂y (v − 1) → 0 uniformly in L∞ ∩ L2 (R, dλ) ∩ L1 (R, dλ) as |x| or |y| → ∞, (1.14) ∂λ v are in L2 (R, dλ) and the norms depend continuously on x, y,
(1.15)
where Lλ = ∂y − λ∂x . The characterization of the scattering data v ∈ Sc,k is necessary. Since the Cauchy integral operator will play a key role in the inverse problem. The study of the asymptotic behavior of the scattering data v (hence the asymptotic behavior of the eigenfunctions Ψ ) is important. Because the Cauchy operator is bounded in L2 [18], in general, an L2 -estimate of Ψ and its derivatives will be good enough. However, a formal calculation will yield (1.19) if the inverse problem is solvable. Hence we provide the estimates (1.13)–(1.15). The derivation of (1.13)–(1.15) basically relies on the L2 -boundedness of the Cauchy operator and the estimates obtained in the small-data problem. In particular, both of the 1-dimensional (Fokas and Ioannidou [11] or (2.7)) and the 2-dimensional formulation (Villarroel [20] or (3.1)) of the spectral problem are crucial in the derivation of the estimates with small data condition. That is, using (2.7), boundedness or integrability in x-variable of the eigenfunctions Ψ comes first from the differentiability and integrability of the potentials Q via the Fourier transform. Then, strong asymptote in x, y or λ-variable of the eigenfunctions Ψ can be obtained by (3.1)
D. Wu / Journal of Functional Analysis 256 (2009) 215–257
219
and previous estimates. We lose some regularities in deriving strong asymptote. See the proof of Theorem 3.1 for example. For the inverse problem, our results are: Theorem 1.3. Given v(x, y, λ) ∈ Sc,k , k 7, there exists a unique solution Ψ (x, y, ·) for the Riemann–Hilbert problem (λ ∈ R, v(x, y, λ)) such that Ψ − 1, ∂x Ψ, ∂y Ψ are uniformly bounded in L2 (R, dλ).
(1.16)
Moreover, for each fixed λ ∈ / R, and i + j k − 4, j
∂xi ∂y Ψ ∈ L∞ (dx dy), j ∂xi ∂y (Ψ
− 1) → 0 in L∞ (dx dy), as x or y → ∞.
(1.17) (1.18)
Theorem 1.3 is proved by a Riemann–Hilbert problem with a non-small purely continuous scattering data. Without uniform boundedness of ∂λ v, we need to handle separately the Riemann– Hilbert problem for |λ| > M 1 and |λ| M. For |λ| > M 1, the Riemann–Hilbert problem is a small-data problem and hence can be solved. For |λ| M, the Riemann–Hilbert problem is again factorized into a diagonal problem, a Riemann–Hilbert problem with small data, and a finite linear system. Note we obtain the globally solvability by applying the Fredholm property and the reality condition (1.11). Moreover, good estimates for Ψ can be derived only for λ ∈ / R. However, it is enough to imply satisfactory analytical properties of the potentials. Theorem 1.4. Given v(x, y, λ) ∈ Sc,k , k 7, the eigenfunction Ψ obtained by Theorem 1.3 satisfies (1.3) with 1 Q(x, y) = 2πi
Ψ− (v − 1) dζ,
(1.19)
R j
and Ψ (x, ·, λ) → 1 as y → −∞, where ∂x Q(x, y) ∈ su(n), and for i + j k − 4, i > 0, ∂xi ∂y Q, j ∂y Q, Q ∈ L∞ , ∂xi ∂y Q, ∂y Q, Q → 0 as x or y → ∞. Applying Theorems 1.1–1.4, we extend the results of [8,11,20] as follows. Theorem 1.5. If Q0 ∈ P∞,k,1 , k 7, and there are no poles of the eigenfunction Ψ0 of Q0 , then the Cauchy problem of the Ward equation (1.5) with initial condition Q(x, y, 0) = Q0 (x, y) admits a global solution satisfying: for i + j + k − 4, i 2 + j 2 > 0, ∂x Q(x, y, t) ∈ su(n), j
∂xi ∂y ∂th Q, ∂t Q, Q ∈ L∞ , j
∂xi ∂y ∂th Q, ∂t Q, Q → 0, as x, y, t, → ∞.
220
D. Wu / Journal of Functional Analysis 256 (2009) 215–257
The paper is organized as follows. In Section 2, we review an existence theorem of Fokas and Ioannidou [11] by an analytical treatment. In Section 3, under the small-data constraint, we analyze the asymptotic behavior of the eigenfunctions. In Sections 4 and 5, we solve the direct problem by justifying Theorems 1.1 and 1.2. The inverse problem is complete in Section 6 by proving Theorems 1.3 and 1.4. Finally, Theorem 1.5 is proved in Section 7. 2. Direct problem I: eigenfunctions with small data Given a potential ∂x Q(x, y) : R × R → su(n), and a constant λ ∈ C, we consider the boundary value problem ∂y Ψ (x, y, λ) − λ∂x Ψ (x, y, λ) − (∂x Q)Ψ (x, y, λ) = 0,
(2.1)
Ψ (x, y, λ) → 1,
(2.2)
as y → −∞.
To investigate the problem, we denote throughout as follows. Definition 3.
q (ξ, y) L (dξ dy) < 1 , P1 = ∂x q(x, y) : R × R → su(n): ξ 1
(ξ, y) L (dξ ) < ∞ , X = w(x, y) : R × R → Mn (C): sup w y
1
y
1
X = f (ξ, y) : R × R → Mn (C): sup f (ξ, y) L
< ∞ , (dξ )
where is the Fourier transform with respect to the x-variable, Mn (C) is the space of n × n matrices, and for f ∈ Mn (C) 1 |f | = trace f ∗ f 2 , f ∗ = f T ,
f (ξ, y)
= f (ξ, y) dξ. L (dξ ) 1
R
Theorem 2.1. Suppose Q ∈ P1 . Then for all fixed λ ∈ C± , there is uniquely a solution Ψ of (2.1) and (2.2) such that Ψ − 1 ∈ X. Moreover, for λ ∈ C± , lim Ψ (·, y, λ) = I,
|x|→∞
lim Ψ (x, ·, λ) = I.
|y|→∞
(2.3)
Proof. Write Ψ = 1 + W . Then (2.1), (2.2) are transformed into ∂y W − λ∂x W = (∂x Q)W + ∂x Q, W (x, y, λ) → 0 as y → −∞. Taking the Fourier transform with respect to the x-variable (in distribution sense), we obtain (ξ, y, λ) − iξ λW (ξ, y, λ) = (∂ ∂y W x Q)W (ξ, y, λ) + ∂ x Q(ξ, y).
D. Wu / Journal of Functional Analysis 256 (2009) 215–257
221
Thus we are led to consider the following integral equations ⎧ y iλξ(y−y ) (∂
+ ∂ ⎪ xQ ∗ W x Q) dy , ⎪ −∞ e ⎪ ⎪ ⎪
⎨ − ∞ eiλξ(y−y ) (∂ + ∂ xQ ∗ W x Q) dy , y W (ξ, y, λ) = ∞
⎪ + ∂ − y eiλξ(y−y ) (∂ ⎪ xQ ∗ W x Q) dy , ⎪ ⎪ ⎪
⎩ y eiλξ(y−y ) (∂
+ ∂ xQ ∗ W x Q) dy , −∞
if λ ∈ C+ , ξ 0; if λ ∈ C+ , ξ 0; if λ ∈ C− , ξ 0;
(2.4)
if λ ∈ C− , ξ 0,
where ∗ is the convolution operator with respect to the ξ -variable. Define ⎧ y iλξ(y−y ) (∂
⎪ x Q ∗ f )(ξ, y , λ) dy , ⎪ −∞ e ⎪ ⎪ ⎪
⎨ − ∞ eiλξ(y−y ) (∂ x Q ∗ f )(ξ, y , λ) dy , y Kλ f (ξ, y, λ) = ∞
⎪ − y eiλξ(y−y ) (∂ ⎪ x Q ∗ f )(ξ, y , λ) dy , ⎪ ⎪ ⎪ ⎩ y eiλξ(y−y ) (∂
x Q ∗ f )(ξ, y , λ) dy , −∞
if λ ∈ C+ , ξ 0; if λ ∈ C+ , ξ 0; if λ ∈ C− , ξ 0; if λ ∈ C− , ξ 0.
Thus (2.4) turns into ⎧
+ y eiλξ(y−y ) ∂ ⎪ Kλ W x Q(ξ, y ) dy , ⎪ −∞ ⎪ ⎪ ⎪
⎨ Kλ W − ∞ eiλξ(y−y ) ∂ x Q(ξ, y ) dy , y W=
⎪ − ∞ eiλξ(y−y ) ∂ Kλ W ⎪ x Q(ξ, y ) dy , y ⎪ ⎪ ⎪ y ⎩K W iλξ(y−y ) ∂
λ + −∞ e x Q(ξ, y ) dy , where
y
−∞ e
if λ ∈ C+ , ξ 0; if λ ∈ C+ , ξ 0; if λ ∈ C− , ξ 0; if λ ∈ C− , ξ 0,
∞ iλξ(y−y ) iλξ(y−y ) ∂
∂ x Q(ξ, y ) dy , y e x Q(ξ, y ) dy
Kλ f (ξ, y)
L
1 (dξ )
∞
∂ x Q(ξ, y ) L
1 (dξ )
(2.5)
∈ X by Q ∈ P1 . Note that
f (ξ, y )
L1 (dξ )
dy
−∞
∂ x Q(ξ, y) L
1 (dξ dy)
sup |f |L1 (dξ ) . y
Hence X→ X, Kλ :
Kλ ∂ x Q(ξ, y) L
1 (dξ
dy)
< 1.
So ⎧ y
⎪ (1 − Kλ )−1 −∞ eiλξ(y−y ) ∂ x Q(ξ, y ) dy , ⎪ ⎪ ⎪
⎪ ∞ iλξ(y−y ) ∂
⎨ −(1 − Kλ )−1 x Q(ξ, y ) dy , y e = W
∞
⎪ −(1 − Kλ )−1 y eiλξ(y−y ) ∂ ⎪ x Q(ξ, y ) dy , ⎪ ⎪ ⎪ ⎩ (1 − K )−1 y eiλξ(y−y ) ∂
λ x Q(ξ, y ) dy , −∞
if λ ∈ C+ , ξ 0; if λ ∈ C+ , ξ 0; if λ ∈ C− , ξ 0; if λ ∈ C− , ξ 0.
(2.6)
222
D. Wu / Journal of Functional Analysis 256 (2009) 215–257
Hence (2.4) is solvable if Q ∈ P1 . Furthermore, the eigenfunction of (2.1), (2.2) is given by ⎧ ∞ y 0 ∞ 1 1 + 2π ( 0 dξ −∞ dy − −∞ dξ y dy ) ⎪ ⎪ ⎪ ⎪
⎨
(ξ, y , λ) + ∂ × eiξ(x+λ(y−y )) (∂ xQ ∗ W x Q(ξ, y )) Ψ (x, y, λ) = 0 y ∞ ∞ 1 ⎪ 1 + 2π ( −∞ dξ −∞ dy − 0 dξ y dy ) ⎪ ⎪ ⎪ ⎩
(ξ, y , λ) + ∂ × eiξ(x+λ(y−y )) (∂ xQ ∗ W x Q(ξ, y ))
if λ ∈ C+ ;
(2.7)
if λ ∈ C− .
The uniqueness follows from (2.1), (2.2), (2.6), the definition of X, and the contraction property of Kλ . The uniform boundedness of Ψ comes from Definition 3, (2.6) and Q ∈ P1 . By (2.7), , ∂ ∂ xQ ∗ W x Q ∈ L1 (dξ dy) and the Riemann–Lebesgue theorem, we obtain Ψ (·, y, λ) → 1 as , ∂ |x| → ∞. On the other hand, (2.7), ∂ xQ ∗ W x Q ∈ L1 (dξ dy) and the Lebesgue convergence theorem imply that Ψ (x, ·, λ) → 1 when |y| → ∞. 2 Lemma 2.1. Suppose Ψ satisfies (2.1), (2.2). Then for λ ∈ / R, det Ψ (x, y, λ) ≡ 1. Proof. Let e1 , . . . , en denote the standard basis for Cn , ψk the kth column vector of the matrix Ψ . Let Λk (Cn ) denote the space of alternating k forms on Cn . Hence ψ1 ∧ ψ2 ∧ · · · ∧ ψn = (det Ψ )(e1 ∧ e2 ∧ · · · ∧ en ). Taking derivatives of both sides, we derive
(∂y − λ∂x )(det Ψ ) (e1 ∧ e2 ∧ · · · ∧ en )
= (∂y − λ∂x ) (det Ψ )(e1 ∧ e2 ∧ · · · ∧ en )
= (∂y − λ∂x ) ψ1 ∧ ψ2 ∧ · · · ∧ ψn
= (∂y − λ∂x )ψ1 ∧ · · · ∧ ψn + · · · + ψ1 ∧ · · · ∧ (∂y − λ∂x )ψn = (∂x Q)ψ1 ∧ · · · ∧ ψn + · · · + ψ1 ∧ · · · ∧ (∂x Q)ψn = (trace ∂x Q)ψ1 ∧ ψ2 ∧ · · · ∧ ψn . So (∂y − λ∂x )(det Ψ ) = 0 by ∂x Q ∈ su(n). Moreover, for λ ∈ / R, the equation turns into the debar equation ∂z¯ (det Ψ ) = 0,
x, y ∈ R,
by the change of variables: x + λy = x˜ + i y˜ = z,
x, ˜ y˜ ∈ R.
Therefore the Liouville’s theorem and (2.3) imply that det Ψ ≡ 1, for λ ∈ / R.
(2.8) 2
D. Wu / Journal of Functional Analysis 256 (2009) 215–257
223
Lemma 2.2. Suppose that Q ∈ P1 . Then the reality condition Ψ (x, y, λ)Ψ (x, y, λ¯ )∗ = I holds for the eigenfunction Ψ . Proof. By Lemma 2.1, one derives (∂y − λ∂x )Ψ (x, y, λ)∗
−1
−1 (∂y − λ∂x )Ψ (x, y, λ)∗ Ψ (x, y, λ)∗ −1 −1 = −Ψ (x, y, λ)∗ (∂y − λ∂x )Ψ (x, y, λ)T Ψ (x, y, λ)∗ ∗ −1 −1 = −Ψ (x, y, λ)∗ (∂x Q)Ψ (x, y, λ) Ψ (x, y, λ)∗ −1 = − ∂x Q∗ Ψ (x, y, λ)∗ = −Ψ (x, y, λ)∗
−1
= (∂x Q)Ψ (x, y, λ)∗
−1
.
Besides, noting |fn |L1 (dξ ) |f |nL1 (dξ ) and the boundary condition of Ψ , we obtain Ψ −1 − 1 ∈ X. Hence the lemma follows from the uniqueness property in Theorem 2.1. 2 3. Direct problem II: asymptotic analysis with small data The results and arguments will be applied or adapted in Sections 4 and 5. Denote ∞ ∞ (f ∗x,y g)(x, y) =
f (x − x , y − y )g(x , y ) dx dy ,
−∞ −∞
(f ∗z,¯z g)(z, z¯ ) =
f (z − ζ, z¯ − ζ¯ )g(ζ, ζ¯ ) dζ d ζ¯ .
C
By the change of variables (2.8), we then have (∂y − λ∂x )−1 =
i −1 1 1 sgn(λI ) ∗z,¯z = − ∗x,y ∂ =− 2λI z¯ 4πλI z 2πi x + λy
with λ = λR + iλI . Now let S be the set of Schwartz functions. If Q ∈ P1 ∩ S, then the eigenfunction Ψ obtained by Theorem 2.1 satisfies Ψ = 1 + Gλ (∂x Q)Ψ ,
(3.1)
where 1 Gλ f (x, y, λ) = − 2πi
∞ ∞ −∞ −∞
The following lemma is due to R. Beals.
sgn(λI )f (x − x , y − y , λ) dx dy . x + λy
(3.2)
224
D. Wu / Journal of Functional Analysis 256 (2009) 215–257
Lemma 3.1. Suppose ϕ ∈ S. For |λ| = 0 and |λI | < 1, |Gλ ϕ|
C sup |∂y ϕ|L1 (dx) + sup |ϕ|L1 (dx) + |ϕ|L1 (dx dy) , |λ| y y
where C is a constant. Proof. Let
1 s
=
λR . λ2R +λ2I
So λI 1 1 1 = −i , λ s λR s
1
y +
1
. x |y + xs | λ
(3.3)
Write Gλ ϕ =
−1 2πiλ
sgn (λI )
ϕ(x − x , y − y ) − ϕ(x − x , y + y
|y + xs |<1
+
sgn (λI )
ϕ(x − x , y − y ) y +
|y + xs |>1
+
sgn (λI )
x λ
ϕ(x − x , y +
|y + xs |<1
y +
x λ
x s )
x λ
+
x s )
dx dy
dx dy
dx dy
= I1 + I2 + I 3 . In view of (3.3), it is easy to see that |I1 |
1 2π|λ|
∂y ϕ(x − x , y − z) dx
sup
z: |z+ xs |<1
C1 sup |∂y ϕ|L1 (dx) , |λ| y
ϕ(x − x , y − y ) C2 1
dx dy |ϕ|L1 (dx dy) . |I2 |
2π|λ| |λ| y + xλ
|y + xs |>1
Finally,
sgn (λI )
|y + xs |<1
1 y +
x λ
1 − i λλIRxs
dy =
log
−1 − i λλIRxs
λI x λI x
=
i arg 1 − i − arg −1 − i
π. λ s λ s R
R
(3.4) (3.5)
D. Wu / Journal of Functional Analysis 256 (2009) 215–257
225
This yields 1 I3 2|λ|
ϕ x − x , y + x dx
s
C3 sup |ϕ|L1 (dx) . |λ| y
Combining (3.4), (3.5), and (3.6), we prove the lemma.
(3.6) 2
Lemma 3.2. Suppose that Q ∈ P1 ∩ S. Then there exists a constant CN such that
N
∂ Ψ CN , x where CN is a constant depending on Q. Proof. Since = ξ N ∂ xQ ∗ W
N N (ξ − ξ )k ξ N −k ∂ xQ ∗ W k k=0
=
N N k N −k ξ ∂x Q ∗ ξ W , k k=0
∈ X for 0 k N . This can be proved by induction on k and using the it suffices to prove ξ i W same argument as in the proof of Theorem 2.1 if |ξ N ∂ x Q|L1 (dξ dy) < ∞. 2 Definition 4. Define
q (ξ, y) L (dξ dy) < 1, and P1,k = ∂x q(x, y) : R × R → su(n): ξ 1
j h
j
i
j h
ξ q L (dξ dy) , ∂x ∂y q L , sup ∂x ∂y q L (dx) , ∂x ∂yh q L (dx dy) < ∞ 1
∞
y
1
1
for 1 i max{5, k}, 0 j, h max{5, k} . j
j
Note that P1 ∈ P1,k . For simplicity we abuse the notation ∂xi ∂y Q, ∂xi ∂y Ψ by Qx · · · x y · · · y , i
and Ψx · · · x y · · · y in the remaining part of this section. i
j
Lemma 3.3. Suppose that Q ∈ P1,k , k 5. Then
N
∂ Ψ CN , 0 N 4. x Moreover, as |λ| → ∞,
C , |∂x Ψ |, ∂x2 Ψ , ∂x3 Ψ |λ|
j
226
D. Wu / Journal of Functional Analysis 256 (2009) 215–257
where CN , C is a constant depending on Q. Proof. The uniform boundedness of ∂xN Ψ , 0 N 4, in Lemma 3.2 will be used in the proof. A direct computation yields Q Q = Ψ 1 − Ψ −1 1 − , Ψ − 1− λ λ Q 1 −1 1− = − Ψ −1 (Qy − Qx Q). (∂y − λ∂x ) Ψ λ λ
(3.7) (3.8)
So Ψx +
Q Q Qx = Ψx 1 − Ψ −1 1 − − Ψ Ψ −1 1 − = I1 + I2 , λ λ λ x
by (3.7). Therefore, inverting the operator ∂y − λ∂x in (3.8) and applying Lemmas 3.1, 3.2, we have
1 |Ψx | Gλ Ψ −1 (Qy − Qx Q)
|λ|
C x |L1 (dξ dy) sup Ψ −1 (Qy − Qx Q)
2 |ξ Q y L1 (dx) |λ| y
+ sup Ψ −1 (Qy − Qx Q) L (dx) + Ψ −1 (Qy − Qx Q) L
|I1 | =
1 C
i
ξ |Ψy |L∞ + 1 sup |Qy − Qx Q|L1 (dx) Q x L (dξ dy) 2 1 |λ| y i=0
+ sup (Qy − Qx Q)y L (dx) + |Qy − Qx Q|L1 (dx dy) 1
y
1 C |λΨx + Qx Ψ |L∞ + 1 sup |Qy − Qx Q|L1 (dx) |ξ Q | x L (dξ dy) 1 2 |λ| y i=0
+ sup (Qy − Qx Q)y L (dx) + |Qy − Qx Q|L1 (dx dy) 1
y
1 C
i
2 ξ Qx L (dξ dy) 1 |λ| i=0
1 (dx dy)
1
y
!
2 " j,k=0
2
j sup ∂x ∂yk Q L y
1
j
2 + ∂x ∂yk Q L (dx)
C |λ|
as |λ| → ∞. Taking the x-derivatives of both the sides of (3.8), we derive |I2 | =
1
Ψ Gλ Ψ −1 (Qy − Qx Q) x
|λ|
1 (dx dy)
#
D. Wu / Journal of Functional Analysis 256 (2009) 215–257
C sup Ψ −1 (Qy − Qx Q) xy L (dx) 2 1 |λ| y
+ sup Ψ −1 (Qy − Qx Q) x L (dx) + Ψ −1 (Qy − Qx Q) x L
2 C
i
2 ξ Qx L (dξ dy) 1 |λ|
!
i=0
1 (dx dy)
1
y
227
3 "
2
j sup ∂x ∂yk Q L
1
y
j,k=0
#
j k 2
∂x ∂ Q
+ y L (dx)
1 (dx dy)
C . |λ|
Here we have used (2.1) and Lemma 3.2. By the same scheme as above and the following equalities: Q Q Qxx = Ψxx 1 − Ψ −1 1 − + 2Ψx 1 − Ψ −1 1 − λ λ λ x Q + Ψ 1 − Ψ −1 1 − , λ xx Q Q Qxxx Ψxxx + = Ψxxx 1 − Ψ −1 1 − + 3Ψxx 1 − Ψ −1 1 − λ λ λ x Q Q + 3Ψx 1 − Ψ −1 1 − + Ψ 1 − Ψ −1 1 − , λ λ xx xxx Ψxx +
one derives 3 C
i
2 ξ Qx L (dξ dy) |Ψxx | 1 |λ|
!
i=0
4 C
i
2 ξ Qx L (dξ dy) |Ψxxx | 1 |λ| i=0
4 "
2
j sup ∂x ∂yk Q L
!
1
y
j,k=0
5 "
2
j sup ∂x ∂yk Q L
Hence the estimates for Ψxx and Ψxxx follow.
1
y
j,k=0
j k 2
∂x ∂ Q
+ y L (dx)
#
j
2 + ∂x ∂yk Q L (dx)
#
,
1 (dx dy)
1 (dx dy)
.
2
Lemma 3.4. Suppose that Q ∈ P1,k , k 5. Then |∂y Ψ |
C , |λ|
(3.9)
|∂x ∂y Ψ |
C , |λ|
(3.10)
as |λ| → ∞. Here C is a constant depending on Q.
228
D. Wu / Journal of Functional Analysis 256 (2009) 215–257
Proof. Using the formula Qy Q Q −1 −1 1− 1− Ψy + = Ψy 1 − Ψ −Ψ Ψ = II 1 + II 2 , λ λ λ y and Lemma 3.3, one can derive
1
Ψy Gλ Ψ −1 (Qy − Qx Q)
|λ|
C 2 |Ψy | sup Ψ −1 (Qy − Qx Q) y L (dx) + sup Ψ −1 (Qy − Qx Q) L (dx) 1 1 |λ| y y
+ Ψ −1 (Qy − Qx Q)
|II 1 | =
L1 (dx dy)
C 2 |λ|
2
i 2
ξ Q x
!
3 "
L1 (dξ dy)
i=0
2
j sup ∂x ∂yk Q L
1
y
j,k=0
#
j
2 + ∂x ∂yk Q L (dx)
1 (dx dy)
(by estimates of I1 , and I2 in Lemma 3.3) C , |λ|2
1
Ψ Gλ Ψ −1 (Qy − Qx Q) y
|II 2 | = |λ|
C 2 sup Ψ −1 (Qy − Qx Q) yy L (dx) + sup Ψ −1 (Qy − Qx Q) y L (dx) 1 1 |λ| y y
+ Ψ −1 (Qy − Qx Q)y
L1 (dx dy)
3 C
i
2 ξ Qx L (dξ dy) 1 |λ| i=0
!
4 "
2
j sup ∂x ∂yk Q L j,k=0
y
1
j
2 + ∂x ∂yk Q L (dx)
#
1 (dx dy)
(by estimates of Ψxx in Lemma 3.3)
C , |λ|
where the estimate |Ψyy | = |λ2 Ψxx + λ(Qx Ψ )x + (Qx Ψ )y | has been used. Thus (3.9) is proved. On the other hand, we write Qxy Q Q −1 −1 1− 1− = Ψxy 1 − Ψ + Ψx 1 − Ψ Ψxy + λ λ λ y Q Q − Ψy 1 − Ψ −1 1 − − Ψ 1 − Ψ −1 1 − λ λ x xy = III 1 + III 2 + III 3 + III 4 . Similarly, one can verify
D. Wu / Journal of Functional Analysis 256 (2009) 215–257
|III 1 |
4 " 3 #
j k 2
j k 2 C
i
2
∂ ξ ∂ sup , ∂ Q + ∂ Q Q x L1 (dξ dy) x y x y L1 (dx) L1 (dx dy) |λ|2 y i=0
|III 2 |
j,k=0
4 " 3 #
j k 2
j k 2 C
i
2
∂ ξ ∂ sup , ∂ Q + ∂ Q Q x L1 (dξ dy) x y x y L1 (dx) L1 (dx dy) |λ|2 y i=0
|III 3 |
j,k=0
4 " 3 #
2
j
2
j C
i
2 ξ sup ∂x ∂yk Q L (dx) + ∂x ∂yk Q L (dx dy) , Q x L1 (dξ dy) 3 1 1 |λ| y i=0
|III 4 |
229
j,k=0
5 " 4 #
2
j
2
j C
i
2 ξ Qx L (dξ dy) sup ∂x ∂yk Q L (dx) + ∂x ∂yk Q L (dx dy) , 1 1 1 |λ| y i=0
j,k=0
by Lemma 3.3, (3.9). Hence we prove (3.10).
2
Theorem 3.1. If Q ∈ P1,k , k 5, then as |λ| → ∞,
Ψ (x, y, λ) − 1 − Q
λ
∂x Ψ (x, y, λ) + ∂x Q , ∂y Ψ (x, y, λ) + ∂y Q
λ λ
C , |λ|2
(3.11)
C , |λ|2
(3.12)
where C is a constant depending on Q. Proof. Applying (3.8), Lemmas 3.3 and 3.4, we obtain
Ψ − 1 − Q
λ
Q
= |Ψ |
1 − Ψ −1 1 − λ
|Ψ |
−1 = Gλ Ψ (Qy − Qx Q)
|λ|
C
−1 2 |Q (Qy − Qx Q) y
x |L1 (dξ dy) sup Ψ L1 (dx) |λ| y
+ sup Ψ −1 (Qy − Qx Q)
+ Ψ −1 (Qy − Qx Q) L y
L1 (dx)
1 (dx dy)
4 " 3 #
j k 2
j k 2 C
i
2
∂x ∂ Q
∂x ∂ Q
ξ sup + Q x y y L1 (dξ dy) L1 (dx) L1 (dx dy) |λ|2 y i=0
j,k=0
as |λ| → ∞. Therefore, (3.11) is proved. To proved (3.12), we used the results of Lemmas 3.3 and 3.4 to improve the estimates of I1 , I2 , II 1 , and II 2 in the proof of Lemmas 3.3, 3.4. More precisely,
230
D. Wu / Journal of Functional Analysis 256 (2009) 215–257
1 |Ψx | Gλ Ψ −1 (Qy − Qx Q)
|λ|
C 2 |Ψx | sup Ψ −1 (Qy − Qx Q) y L (dx) 1 |λ| y
−1
−1 + sup Ψ (Qy − Qx Q) L (dx) + Ψ (Qy − Qx Q) L
|I1 | =
1
y
1 (dx dy)
4 " 3 #
2
j
2
j C
i
2 ξ Qx L (dξ dy) sup ∂x ∂yk Q L (dx) + ∂x ∂yk Q L (dx dy) , 3 1 1 1 |λ| y i=0
j,k=0
1
Ψ Gλ Ψ −1 (Qy − Qx Q) x
|I2 | = |λ|
C 2 sup Ψ −1 (Qy − Qx Q) xy L (dx) 1 |λ| y
+ sup Ψ −1 (Qy − Qx Q) x L (dx) + Ψ −1 (Qy − Qx Q) x L
1 (dx dy)
1
y
4 " 3 #
2
j
2
j C
i
2 ξ sup ∂x ∂yk Q L (dx) + ∂x ∂yk Q L (dx dy) , Q x L1 (dξ dy) 2 1 1 |λ| y i=0
j,k=0
1
Ψy Gλ Ψ −1 (Qy − Qx Q)
|λ|
C 2 |Ψy | sup Ψ −1 (Qy − Qx Q) y L (dx) 1 |λ| y
+ sup Ψ −1 (Qy − Qx Q) L (dx) + Ψ −1 (Qy − Qx Q) L
|II 1 | =
1 (dx dy)
1
y
4 " 3 #
j k 2
j k 2 C
i
2
∂x ∂ Q
∂x ∂ Q
ξ sup , + Q x y y L1 (dξ dy) L1 (dx) L1 (dx dy) |λ|3 y i=0
j,k=0
1
Ψ Gλ Ψ −1 (Qy − Qx Q) y
|λ|
C 2 sup Ψ −1 (Qy − Qx Q) yy L (dx) 1 |λ| y
−1
+ sup Ψ (Qy − Qx Q) y L (dx) + Ψ −1 (Qy − Qx Q)y L
|II 2 | =
1
y
1 (dx dy)
5 " 4 #
2
j
2
j C
i
2 ξ Qx L (dξ dy) sup ∂x ∂yk Q L (dx) + ∂x ∂yk Q L (dx dy) . 2 1 1 1 |λ| y i=0
j,k=0
Here |Ψyy | = |λΨxy + Qxy Ψ + Qx Ψy | and (3.10) have been used in the estimation of II 2 . By induction, we can generalize the results of Lemmas 3.2–3.4 and Theorem 3.1 to
2
D. Wu / Journal of Functional Analysis 256 (2009) 215–257
231
Corollary 3.1. Suppose that Q ∈ P1,k . Then for i + h max{k, 5} − 4 and as |λ| → ∞,
i h
∂ ∂ Ψ (x, y, λ) − ∂ i ∂ h 1 − Q C . x y
x y λ |λ|2 Remark 1. In general, the scattering transformation is a generalized Fourier transform. That is, it maps smooth potentials to decaying scattering data, and decaying potentials to smooth scattering data. As is known, the asymptotic expansion of eigenfunctions is related to the decayness of the scattering data. However, in the case of Ward equation, even for the Schwartz potentials, the second order asymptotic expansion of Theorem 3.1 seems difficult to be improved. To see it, the second-order coefficient of the asymptotic expansion Ψ , and an analogue of (3.8) need to be introduced. That is x Ψ2 (x, y) =
(−Qy + Qx Q)(x , y) dx ,
−∞
∞ c(y) =
(−Qy + Qx Q)(x , y) dx ,
−∞
x Φ(x) =
∞
φ(x ) dx ,
−∞
φ(x ) dx = 1,
−∞
f (x, y) = Ψ2 (x, y) − c(y)Φ(x), and Q Ψ2 1 = 2 Ψ −1 (∂y Ψ2 − Qx Ψ2 ), (∂y − λ∂x ) Ψ −1 1 − + 2 λ λ λ
(3.13)
where φ is a Schwartz function. Then f (x, y), c(y) are Schwartz. It can be checked that Ψ2 does Ψ2 not possess integrability in the x-variable. This causes troubles in estimating |Ψ − (1 − Q λ + λ2 )| while inverting (3.13) to derive a higher order asymptotic expansion of Ψ . 4. Direct problem III: eigenfunctions with non-small data First we introduce Definition 5. The Cauchy operator C and its limits C± are defined as follows: 1 Cf (λ) = 2πi
∞ −∞
f (ζ ) dζ, ζ −λ
1 C± f (λ) = lim + →0 2πi
∞ −∞
λ ∈ C \ R,
f (ζ ) dζ, λ ∈ R. ζ − (λ ± i)
232
D. Wu / Journal of Functional Analysis 256 (2009) 215–257
It is well known that C± are bounded operators on Lp (R) for 1 < p < ∞, and C± f (λ) = ˜ λ ∈ R, λ˜ ∈ C± [18]. limλ˜ →λ Cf (λ), Definition 6. Suppose v(λ) is defined on R. A function Ψ (λ) is called a solution of the Riemann– Hilbert problem (λ ∈ R, v) if Ψ (λ) = 1 +
1 2πi
R
Ψ− (t)(v(t) − 1) dt t −λ
= 1 + C Ψ− (v − 1) , where Ψ± (λ) = limλ˜ →λ Ψ (λ˜ ), λ ∈ R, λ˜ ∈ C± . Moreover, the function v(λ) is called the data of the Riemann–Hilbert problem (λ ∈ R, v). Suppose the data v(λ), λ ∈ R satisfies ∂λi (Ψ − 1) ∈ L2 (R, dλ), for i = 0, 1, 2. It can be seen that Ψ is a solution of the Riemann–Hilbert problem (λ ∈ R, v) if and only if ∂λ¯ Ψ = 0,
λ ∈ C± ,
Ψ+ = Ψ− v,
λ ∈ R,
Ψ → 1,
as |λ| → ∞.
Lemma 4.1. Suppose the data v(λ), λ ∈ R, satisfies: v − 1 ∈ L2 (dλ), |v − 1|L∞ (dλ) C± 2 < 1. Then the Riemann–Hilbert problem (λ ∈ R, v) has a unique solution Ψ such that Ψ − 1 ∈ j L∞ (dλ) ∩ L2 (dλ). Moreover, if H k = {f | ∂λ f ∈ L2 (dλ), 0 j k} and |v − 1|H k (dλ) 1, then |Ψ± − 1|H k (dλ) C|v − 1|H k (dλ) for some constant C. Proof. The proof can be derived by an adaptation of the proof of Theorems 8.9 and 9.20 in [4]. 2 Lemma 4.2. Suppose the data v(λ), λ ∈ R, is a scalar function satisfying: • v(λ) ∞ = 0, ∀λ; • −∞ d arg v(λ) = 0; • v − 1, ∂λ v ∈ L2 (dλ).
D. Wu / Journal of Functional Analysis 256 (2009) 215–257
233
Then the Riemann–Hilbert problem (λ ∈ R, v) has a unique solution Ψ . Moreover, if v − 1 ∈ H k (dλ) then |Ψ± − 1|H k (dλ) C|v − 1|H k (dλ) , where H k (dλ) = {f | ∂λi f ∈ L2 (dλ), 0 i k}, and C is a constant depending on |v|L∞ , |1/v|L∞ . Proof. Note that by the Sobolev’s theorem, v − 1 ∈ C0 by condition v − 1, ∂λ v ∈ L2 (dλ). Here C0 denotes continuous functions with limit 0 at ∞. Hence the proof can be found in [4, Appendix]. 2 Lemma 4.3. Suppose Q ∈ P∞,2,0 ∩ P1 . Then the eigenfunction obtained in Theorem 2.1 satisfies: (1) ∂xi (Ψ (·, y, λ) − 1), i = 0, 1, 2, are uniformly bounded in L2 (dx); (2) Ψ (·, y, λ) − 1, ∂x Ψ (·, y, λ) → 0 uniformly in L2 (dx) as λ → ∞. Proof. By noting that the Fourier transform is an isometry on the L2 spaces, to prove (1), it , i = 0, 1, 2, are uniformly bounded in L2 (dξ ). We will only treat the suffices to show that ξ i W case of λ ∈ C+ and ξ 0 for simplicity. Other cases can be handled similarly. Note that
Kλ f (ξ, y, λ)
L2 (dξ )
∂ x Q(ξ, y) L
1 (dξ
dy)
sup |f |L2 (dξ ) . y,λ
Denote X2 = {f (ξ, y, λ) : R × R × C → Mn (C): supy,λ |f (ξ, y, λ)|L2 (dξ ) < ∞}. So X∩ X2 → X∩ X2 , Kλ :
Kλ ∂ x Q(ξ, y) L
1 (dξ
dy)
.
By the assumption Q ∈ P∞,2,0 , we have y
eiλξ(y−y ) ∂ X∩ X2 . x Q(ξ, y ) dy ∈
−∞
of (2.5) is in Therefore the solution W X∩ X2 . Moreover, one can derive = ξW
y
−∞
e
iλξ(y−y )
x ) ∗ W dy + (ξ Q
y
−∞
x eiλξ(y−y ) Q
) dy + ∗ (ξ W
y
x dy eiλξ(y−y ) ξ Q
−∞
∈ from (2.4). As a result, we have ξ W X∩ X2 , if Q ∈ P∞,2,0 ∩ P1 . The same argument can prove 2 ξ W ∈ X ∩ X2 , if Q ∈ P∞,2,0 ∩ P1 . Hence (1) is justified.
234
D. Wu / Journal of Functional Analysis 256 (2009) 215–257
(ξ, y, λ) can be approxiTo prove (2), by the definition of X and result of (1), the function W mated uniformly by g where − g|L2 (dξ )∩L1 (dξ ) < , |W and g is a linear combination of step functions in ξ with uniformly bounded coefficients in y, λ. Hence ! ∞ eiξ x g(ξ, y, λ) dξ χ|x|>N → 0 uniformly in L2 (dx) as N → ∞, −∞
where χ|x|>N is the characteristic function of the set {|x| > N }. The above two inequalities imply that (Ψ (x, y, λ) − 1)χ|x|>N → 0 uniformly in L2 (dx) as N → ∞. We can prove the case of (∂x Ψ (x, y, λ))χ|x|>N by the similar method. Combining with Theorem 3.1 and the Lebesgue convergence theorem, one can prove (2). 2 Lemma 4.4. Let x + λy = z, ∂z¯ = 12 (∂x + i∂y ), and f±,z (x, λ) = lim|y|→0± f (x, y, λ). If f (x, y, λ) is the solution of the Riemann–Hilbert problem (x ∈ R, F (x, λ)) and F (·, λ) − 1, ∂x F (·, λ), f±,z (·, λ) − 1, ∂x f±,z (·, λ) → 0 in L2 (dx) as |λ| → ∞, then f (x, y, ·) tends to 1 uniformly as |λ| → ∞. Proof. For y = 0, the lemma follows from the Sobolev’s theorem, Lemma 4.1 and the assumption on f±,z , F . For simplicity, we omit the words “for |λ| 1” in the following proof. Decompose f (x, y, λ) into 1 f− (F (t, λ) − 1) f− (F (t, λ) − 1) 1 dt + dt f (x, y, λ) = 1 + 2πi t −z 2πi t −z |t−x|<1
|t−x|1
= 1 + I (x, y, λ) + II(x, y, λ). Note that f− (F − 1)(·, λ) is uniformly Hölder continuous by the assumption on F , f± and the imbedding theorem of Morrey [13]. Hence one has I (x, y, λ) → I±,z (x, λ) uniformly as y → 0± [12]. The uniform convergence of II(x, y, λ) → II ±,z (x, λ) as y → 0± can be justified by the Hölder inequality. Moreover, one can check that this convergence is independent of x. As a result, f (x, y, λ) → f±,z (x, λ) uniformly as y → 0± . Since the lemma holds on the x-axis the uniform convergence provided above implies: for any > 0, one can find N1 , δ such that |f (x, y, λ) − 1| < for ∀|λ| N1 , ∀|y| δ . Besides, by the Hölder inequality, we can find N2 such that |f (x, y, λ) − 1| < for ∀|λ| > N2 , |y| δ . Hence for any > 0, we obtain
f (x, y, λ) − 1 < , ∀|λ| > max{N , N }. 2 1 2
D. Wu / Journal of Functional Analysis 256 (2009) 215–257
235
We can start to prove Theorem 1.1. Proof of Theorem 1.1. We will prove Theorem 1.1 by induction on the norm of
∂ x Q(ξ, y) L (dξ dy) . 1
3 0 Step 1 (The case of n = 0). If |∂ x Q(ξ, y)|L1 (dξ dy) < ( 2 ) , the existence and (1.6) are proved by Theorem 2.1. The conditions (1.7), (1.8) and (1.9) are shown by Theorem 3.1 and Lemma 4.3. The holomorphic property comes from (2.7).
Step 2 (Transforming to a Riemann–Hilbert problem). Suppose Theorem 1.1 holds for 3 n |∂ x Q(ξ, y)|L1 (dξ dy) < ( 2 ) . Note the eigenfunction corresponding to a y-translate of Q is the y-translate of the eigenfunction. Thus after translation we may have 0
∂ x Q(ξ, y) dy dξ =
R −∞
∞
∂ x Q(ξ, y) dy dξ <
n+1 n 3 3 1 < , 2 2 2
R 0
3 n+1 for a potential ∂x Q(x, y) with |∂ . Let χ ± = χ ± (y) 1 be smooth x Q(ξ, y)|L1 (dξ dy) < ( 2 ) real-valued functions such that $ n
1, for y 0, 3 − − − −
∂x Q L (dξ dy) < ∂x Q = ∂x Q(x, y)χ (y), , χ = 1 0, for y 1, 2 $ n
1, for y 0, 3
∂x Q+
χ+ = ∂x Q+ = ∂x Q(x, y)χ + (y), < . L1 (dξ dy) 0, for y −1, 2 3 n ± So Q± ∈ P∞,4,0 and |∂ x Q (ξ, y)|L1 (dξ dy) < ( 2 ) . By the induction hypothesis there exist bounded sets Z ± such that Z ± ∩ (C \ R) are discrete in C \ R and for all λ ∈ C \ Z ± , Q± have eigenfunctions Ψ ± which fulfill the statements of Theorem 1.1. Here we remark that the meaning of the notation Ψ + is different from that of Ψ+ . The former is a function defined in the half plane y 0, the latter means limλI →0+ Ψ (x, y, λ). Hence any eigenfunction Ψ for Q, whenever it exists, must be of the form $ − Ψ (x, y, λ)a − (x + λy, λ), y 0, (4.1) Ψ (x, y, λ) = Ψ + (x, y, λ)a + (x + λy, λ), y 0,
where for y ∈ R± , ⎧ ± ⎨ a (x + λy, λ) is meromorphic in λ ∈ C \ R with discrete poles, a ± (x, y, λ) satisfies (1.6), (1.7), ⎩ ± a±,z (x, 0, λ) satisfies (1.8), (1.9).
(4.2)
Conversely, if we can find a ± such that a ± satisfies (4.2) for y ∈ R± and a + (a − )−1 (x, 0, λ) = (Ψ + )−1 Ψ − (x, 0, λ) (the invertibility of a ± , Ψ ± is implied by Lemma 2.1). Then we can define 3 n+1 . Therefore, Ψ (x, y, λ) by (4.1) and prove Theorem 1.1 in case of |∂ x Q(ξ, y)|L1 (dξ dy) < ( 2 ) we conclude this step by
236
D. Wu / Journal of Functional Analysis 256 (2009) 215–257
Lemma 4.5 (Transforming into a Riemann–Hilbert problem). To prove Theorem 1.1, it is equivalent to solving the problem: find a bounded set Z, f (x, ˜ y, ˜ λ), and f˜(x, ˜ y, ˜ λ) such that Z ± ⊂ Z and • Z ∩ (C \ R) is discrete in C \ R; • For λ ∈ C+ \ (R ∪ Z), f is the unique solution of the Riemann–Hilbert problem (x˜ ∈ R, F (x, ˜ λ)); • For λ ∈ C− \ (R ∪ Z), f˜ is the unique solution of the Riemann–Hilbert problem (x˜ ∈ ˜ λ)); R, F −1 (x, • f, f˜ are meromorphic in λ ∈ C \ R with poles at the points of Z ∩ (C \ R); • f±,z , f˜±,z satisfy (1.8), (1.9), where x + λy = x˜ + i y˜ = z,
x, ˜ y˜ ∈ R,
(4.3)
and F (x, ˜ λ) = Ψ − (x, ˜ 0, λ)−1 Ψ + (x, ˜ 0, λ).
(4.4)
Proof. Note that if f, f˜ exist for Lemma 4.5, then by Lemma 4.4 f, f˜ satisfy (1.6), (1.7) as well. Therefore, the lemma can be proved by the change of variables (4.3) (or (2.8)) and setting a − (x + λy, λ) = A− (x, ˜ y, ˜ λ), a + (x + λy, λ) = A+ (x, ˜ y, ˜ λ), with x, ˜ y˜ ∈ R where $ f (x, ˜ y, ˜ λ) =
˜ y, ˜ λ), (A+ )−1 (x, (A− )−1 (x, ˜ y, ˜ λ),
$ − −1 ˜ y, ˜ λ), (A ) (x, f˜(x, ˜ y, ˜ λ) = + −1 (A ) (x, ˜ y, ˜ λ), in the above discussion.
for y˜ 0, λ ∈ C+ , for y˜ 0, λ ∈ C+ . for y˜ 0, λ ∈ C− , for y˜ 0, λ ∈ C− ,
2
Step 3 (Factorization: a diagonal problem, a Riemann–Hilbert problem with small data and a rational function). For any square matrix A we let dk+ (A) denote the upper (k × k)-principal minors. Also let βik , i k be the minor of A formed of the first i rows, the first i − 1 columns, and the kth column, and γki be the minor of A formed of the first i columns, the first i − 1 rows, and the kth row. The following factorization theorem can be found in [10]. Lemma 4.6. Suppose the principal minors dk+ (A) = 0, for 1 k n. Then the matrix A can be represented as A = CSB,
D. Wu / Journal of Functional Analysis 256 (2009) 215–257
237
where ⎛
1
⎜ γγ21 ⎜ 11 C=⎜ . ⎝ ..
γn1 γ11
0 1 .. .
γn2 γ22
..
⎛
⎜ ⎜ ⎜ S=⎜ ⎜ ⎝
. ···
⎞
⎛1
⎟ ⎟ ⎟, ⎠
⎜ ⎜ B =⎜ ⎝
1
d1+ (A)
β12 β11
···
1
··· .. .
0 0
d2+ (A) d1+ (A)
.. 0
.
dn+ (A) + dn−1 (A)
⎞
β1n β11 β2n β22
⎞
⎟ ⎟ , .. ⎟ ⎠ . 1
⎟ ⎟ ⎟ ⎟. ⎟ ⎠
From now on, we only deal with the case of λ ∈ C+ for simplicity. The other case can be proved in an analogous argument. Lemma 4.7. For λ ∈ C+ \ [Z + ∪ Z − ], we have a factorization F (x, ˜ λ) = (1 + gl )−1 δ(1 + gu ), where δ is diagonal and gu (gl ) is strictly upper (lower) triangular;
(4.5)
δ, gu , gl are λ-meromorphic in C+ with poles at [Z + ∪ Z − ];
(4.6)
˜ for ∂xi (δ − 1), ∂xi gu , ∂xi gl , i = 0, 1, 2, are uniformly bounded in L2 (d x) + λ ∈ C+ \ λj ∈[Z + ∪Z − ] D (λj ). For any zj ∈ C \ R, fixing k for ∀k = j −hj
and letting j → 0, these L2 (dx)-norms increase as Cj j
with
uniform constants Cj , hj > 0;
(4.7)
δ − 1, gu , gl , ∂x δ, ∂x gu , ∂x gl → 0 in L2 (dx) as λ → ∞.
(4.8)
Proof. By the same technique of the proof of Lemma 2.1, one proves det Ψ ± = 1 for λ ∈ / R. So det F ≡ 1. As a result, if di+ (F )(x˜0 , λ0 ) = 0 for some 1 i < n, then F must have a pole at (x˜0 , λ0 ). By det Ψ ± = 1 and (4.4), we obtain λ0 ∈ [Z + ∪ Z − ]. Therefore for λ ∈ C+ \ [Z + ∪ Z − ], we obtain a factorization by Lemma 4.6. The properties (4.5)–(4.8) are implied by F (x, ˜ λ) is meromorphic in λ ∈ C+ with poles at [Z + ∪ Z − ] at most; F (x, ˜ λ) satisfies (1.8), (1.9) which come from the induction hypothesis.
(4.9) (4.10)
2
238
D. Wu / Journal of Functional Analysis 256 (2009) 215–257
Lemma 4.8 (A diagonal Riemann–Hilbert problem). For λ ∈ C+ \ [Z + ∪ Z − ], the Riemann– Hilbert problem (x˜ ∈ R, δ(x, ˜ λ)) has a solution Δ(z, λ). Moreover, • Δ is λ-meromorphic in C+ with poles at [Z + ∪ Z − ] ∩ C+ ; • Δ±,z satisfies (1.8), (1.9). Proof. For λ ∈ C+ \ [Z + ∪ Z − ], the matrix δ is a diagonal matrix nonvanishing entries. So with 1 d the winding number of δ(x, ˜ λ) is well defined by N (λ) = − 2πi arg δ(t, λ) dt. By (4.6) and dt (4.7), N(λ) is a continuous integer-valued function for x ∈ C+ \ [Z + ∪ Z − ]. Thus N (λ) ≡ 0 by (4.8). Combining with (4.7), and (4.8), Lemma 4.2 implies the existence of Δ which satisfies the Riemann–Hilbert problem (x˜ ∈ R, δ(x, ˜ λ)), (1.8), and (1.9). The meromorphic property of Ψ (x, y, ·) is proved by (4.6), and [17] $ , 1 log δ(t, λ) Δ(z, λ) = exp dt . 2πi t −z
2
R
Lemma 4.9. For λ ∈ C+ \
+
λj ∈[Z + ∪Z − ] D (λj ),
$ R =
there exists
˜ y, ˜ λ), R,u (x, R,l (x, ˜ y, ˜ λ),
for y˜ 0, for y˜ 0,
such that
Δ−,z 1 + (R )−,z F 1 + (R )+,z −1 Δ−1 (x, ˜ λ) − 1
1;
(4.11)
C± < 1;
(4.12)
+,z
H 2 (R,d x) ˜
+,z
L∞
Δ−,z 1 + (R )−,z F 1 + (R )+,z −1 Δ−1 (x, ˜ λ) − 1
(R )u (R )l is strictly upper (lower) triangular;
R can be meromorphically extended in λ ∈ C+ with poles at Z + ∪ Z − ; R
∈ H 2 (R, d x) ˜
and is rational in
z ∈ C± ,
(4.13) (4.14)
with finite simple poles
(independent of λ) and each corresponding residue is an off-diagonal matrix with only one non-zero entry. Moreover, the non-zero entry tends to 0 as |λ| → ∞.
(4.15)
Proof. By the condition (4.8), there exists δ such that |gu χ|λ|>δ |H 2 (d x) ˜ < . Moreover, by (4.7), + + for each λ0 ∈ C \ λj ∈[Z + ∪Z − ] D (λj ), |λ0 | δ , there exists N = N (, λ0 ) such that |gu − p,u |H 2 (d x) ˜ < where
for λ in a small neighborhood of λ0 ,
D. Wu / Journal of Functional Analysis 256 (2009) 215–257
2
p,u (z, λ) =
N
gu
j =−N 2
P (t) =
239
j j , λ P z − , N N
1 1 − is the Poisson kernel [4, Appendix A.2]. t − i t + i
One can check that p,u ∈ H 2 (R, d x) ˜ satisfies (4.13), (4.14). Hence choosing a bigger N or δ , there exists a z-rational function, denoted as p˜ ,u , |gu − p˜ ,u |H 2 (d x) ˜ <
for ∀λ ∈ C+ \
+
λj ∈[Z + ∪Z − ] D (λj ),
and p˜ ,u satisfies (4.13), (4.14). Consequently, using (4.9), (4.10), Lemmas 4.7, 4.8, and the off-diagonal form of gu , one can find a z-rational function Ru (z, λ) which is an approximation of gu on z ∈ R and satisfies (4.11)–(4.15). The case of gl can be done in analogy. 2 With Lemma 4.9, one can find a solution to the small-data Riemann–Hilbert problem (x˜ ∈ R, Δ−,z (1 + (R )−,z )F (1 + (R )+,z )−1 Δ−1 +,z ). However, it is difficult to analyze the meromorphic property of the solution in a neighborhood of points in [Z + ∪ Z − ]. Hence we need to improve + + − + + Lemma 4.9. First of all, let us denote C+ = {λ ∈ C | λI }, and [Z ∪ Z ] = {λ ∈ [Z ∪ − Z ] | λI } for simplicity. Lemma 4.10. For λ ∈ C+ , there exist $ ˜ y, ˜ λ), R˜ ,u (x, ˜ R = ˜ ˜ y, ˜ λ), R,l (x,
for y˜ 0, for y˜ 0
such that
Δ−,z 1 + (R˜ )−,z F 1 + (R˜ )+,z −1 Δ−1 − 1
+,z
1; H 2 (R,d x) ˜
−1 −1
Δ−,z 1 + (R˜ )−,z F 1 + (R˜ )+,z Δ − 1 C± < 1; +,z L∞ (R˜ )u (R˜ )l is strictly upper (lower) triangular;
(4.17)
+ − + R˜ is meromorphic in λ ∈ C+ with poles at [Z ∪ Z ] ;
(4.19)
R˜
∈ H 2 (R, d x) ˜
and is rational in
z ∈ C± ,
(4.16)
(4.18)
with finite simple poles
(independent of λ) and each corresponding residue is an off-diagonal matrix with only one non-zero entry. Moreover, the non-zero entry tends to 0 as |λ| → ∞.
(4.20)
Proof. One can multiply gu (gl respectively) by product P,u =
λj ∈[Z + ∪Z − ]+
λ − λj λ+i
hj
240
D. Wu / Journal of Functional Analysis 256 (2009) 215–257
so that G,u = P,u gu is holomorphic in λ ∈ C+ . Then using (4.7) and the same argument as
. Let the proof of Lemma 4.9, one can approximate G,u by a piecewise z-rational function R,u −1 R . R˜ ,u = P,u ,u . λ−λ Next, choose kj sufficiently large in U (λ) = λj ∈[Z + ∪Z − ]+ ( λ+ij )kj to make U δ, U Δ holomorphic in λ ∈ C+ . Hence the lemma can be proved by an adaptation of the proof of Lemma 4.9. (Note the factors U , P,u , P,l are cancelled out.) 2 Lemma 4.11 (A Riemann–Hilbert problem with small data). The Riemann–Hilbert problem + (x˜ ∈ R, Δ−,z (1 + (R˜ ,u )−,z )F (1 + (R˜ ,u )+,z )−1 Δ−1 +,z ) admits a solution f,s (z, λ) for λ ∈ C \ + − + [Z ∪ Z ] . Moreover, + − + • f,s is meromorphic in λ ∈ C+ with poles at [Z ∪ Z ] ; • (f,s )±,z satisfies (1.8), (1.9).
Proof. By the assumption (4.16), (4.17), one can apply Lemma 4.1 to find f,s which satisfies (1.8) and the Riemann–Hilbert problem (x˜ ∈ R, Δ−,z (1 + (R˜ ,u )−,z )F (1 + (R˜ ,u )+,z )−1 Δ−1 +,z ). Moreover, f,s satisfies (1.9) by Lemma 4.1, (4.10), Lemma 4.8, and (4.20). Finally, f,s is + − + meromorphic in λ ∈ C+ with poles at [Z ∪ Z ] by (4.9), Lemma 4.8, and (4.19). 2 We conclude this step by a characterization of Lemma 4.5. Lemma 4.12 (Factorization of the Riemann–Hilbert problem). Suppose f (z, λ) fulfills the statement in Lemma 4.5. Then there exist a unique function r (z, λ) and a set Z , such that r (z, λ) = 1 +
N (z − zk )−1 ck, (λ),
(4.21)
k=1
for some integer N , Z ⊂ Z, and for λ ∈ C+ \ Z, ck, is meromorphic in λ ∈ C+ with poles at Z ;
(4.22)
ck, (λ) → 0
(4.23)
as |λ| → ∞;
f = r f,s Δ(1 + R˜ ).
(4.24)
Conversely, suppose there are uniformly bounded sets Z , and functions {r } which are λmeromorphic in C+ with poles at Z , satisfy (4.21)–(4.23), and r f,s Δ(1 + R˜ ) is holomorphic in z ∈ C±
(4.25)
+ − + + ˜ for λ ∈ C+ \ (Z ∪ [Z ∪ Z ] ). Define f = r f,s Δ(1 + R ) for λ ∈ C . Then we have + − + f is meromorphic in λ ∈ C+ with poles at Z ∪ [Z ∪ Z ] ;
f 1 = f 2
for λ ∈ C+ 1 if 1 > 2 .
(4.26) (4.27)
+ Hence f = f is well defined, and f satisfies the statements in Lemma 4.5 with Z = Z(f ) ∪ {λj ∈ R | lim sup→0 |f (D2 (λj ) ∩ C+ )| = ∞}. Here Z(f ) denotes the poles of f .
D. Wu / Journal of Functional Analysis 256 (2009) 215–257
241
Proof. First of all, by Lemma 2.1, det f,s (z, λ) = det (1 + R˜ (z, λ)) = det Δ(z, λ) = 1. So they are invertible at regular λ. Besides, f (z, λ) and f,s (z, λ)Δ(z, λ)(1 + R˜ (z, λ)) are zmeromorphic, possess the same jump singularity across z ∈ R, and tend to 1 at infinity. Therefore −1 f f,s (z, λ)Δ(z, λ) 1 + R˜ (z, λ) is z-rational and (4.21)–(4.23) are satisfied by Lemmas 4.8–4.11 and the assumption on f . For the converse part, (4.26) comes immediately from the definition of f and the meromorphic properties of r , Δ, R˜ , f,s implied by assumption and Lemmas 4.8–4.11. Besides, by assumption, f1 , f2 satisfy the same Riemann–Hilbert problem in Lemma 4.5 for λ ∈ C+ 1 \ Z1 . Thus (4.27) follows from the Liouville’s theorem and the meromorphic properties. As a result, the well-defined property follows from (4.26) and (4.27). The conditions (1.8), + (1.9) can be proved by Lemmas 4.8–4.11, and (4.21)–(4.23), f = f (i.e., (4.24)), and Z = Z ∪ {λj ∈ R: lim sup→0 |f (D2 (λj ) ∩ C+ )| = ∞}. 2 Step 4 (Solving the Riemann–Hilbert problem). We complete the proof of Theorem 1.1 by finding a rational function r in Lemma 4.12. Lemma 4.13 (Existence of the rational function r ). There exist a function r and a uniformly bounded set Z such that r is λ-meromorphic in C+ with poles at the points of Z and satisfies + − + (4.21)–(4.23), (4.25) for λ ∈ C+ \ (Z ∪ [Z ∪ Z ] ). Proof. For simplicity, we drop in the notation r , f,s , R , . . . in the following proof. (a) A linear system for r(z, λ). Let {zk = x˜k + i y˜k }, k = 1, . . . , N be the simple poles of R in C± by (4.15). Denote 1 + R(z, λ) = (z − zj )−1 dj + nj + O |z − zj | , fs Δ(z, λ) = αj + βj (z − zj ) + O |z − zj |2
(4.28) (4.29)
at zj . Thus fs Δ(1 + R)(z, λ) = (z − zj )−1 αj dj + (βj dj + αj nj ) + O |z − zj | . Now let r(z, λ) = 1 +
N (z − zk )−1 ck .
(4.30)
k=1
Hence at zj , r(z, λ) = (z − zj )−1 cj + bj + O |z − zj | , where bj = 1 +
(zj − zk )−1 ck . k =j
(4.31)
242
D. Wu / Journal of Functional Analysis 256 (2009) 215–257
We then try to find cj , such that r(z, λ)fs (z, λ)Δ(z, λ)(1 + R(z, λ)) is holomorphic at zj . This yields the linear system for cj : cj αj dj = 0,
1 j N,
bj αj dj + cj (βj dj + αj nj ) = 0,
(4.32)
1 j N.
(4.33)
(b) Solving the linear system (4.32), (4.33). The properties (4.13), (4.15) imply that nj are in2 vertible and (dj n−1 j ) = 0. Therefore, it can be justified that (4.32) are consequences of (4.33). Inserting (4.31) into (4.33), we obtain a system of N n2 linear equations in N n2 unknowns (the entries of ck ) with coefficients in entries of dj (λ), nj (λ), αj (λ), βj (λ). Observing that as |λ| → ∞, dj → 0,
nj → 1,
αj → 1,
βj → 0
by Lemmas 4.8–4.11 we have (4.33) are solvable as |λ| → ∞. Precisely, ck can be written in + − rational forms of dj , nj , αj , βj which are all holomorphic in λ ∈ C+ \ [Z ∪ Z ]. Therefore, + (4.33) are solvable for λ ∈ C \ Z where Z are uniformly bounded sets. Consequently, (4.21), (4.22), (4.23), and (4.25) are fulfilled. 2 By the same argument as the proof of Theorem 1.1, we have Corollary 4.1. Suppose that Q ∈ P∞,k,0 , k 2, and Ψ (x, y, λ) is the associated eigenfunction. Then Ψ − 1 is uniformly bounded in DH for λ ∈ C \ R ∪ D (λj ) . k
λj ∈Z
In particular, if λ0 is a removable singularity of Ψ (x, y, λ), then Ψ − 1 is uniformly bounded in DHk in a neighborhood of λ0 . By a similar argument as that in Lemmas 2.1 and 2.2 and using the uniqueness property in Theorem 1.1, we can derive the same algebraic characterization of the eigenfunctions: Lemma 4.14. Suppose that Q ∈ P∞,k,0 , k 2. Then the eigenfunction Ψ satisfies det Ψ (x, y, λ) ≡ 1,
(4.34)
¯ =I Ψ (x, y, λ)Ψ (x, y, λ) ∗
for λ ∈ C \ R.
(4.35)
D. Wu / Journal of Functional Analysis 256 (2009) 215–257
243
5. Direct problem IV: asymptotic analysis with non-small data We define the continuous scattering data and study its algebraic and analytic characteristics in this section. We first show that the existence of continuous scattering data for Q ∈ P1 is automatic. Lemma 5.1. If Q ∈ P1 , then the eigenfunction Ψ (x, y, ·) obtained by Theorem 2.1 has limits Ψ± on R. k instead of W (ξ, y, λk ) Proof. Suppose {λk } ⊂ C+ , and λk converge to a point of R. Write W and / y iλ ξ(y−y )
k when ξ 0, ∂ x Q(ξ, y ) dy , −∞ e fk = ∞ iλ ξ(y−y )
− e k ∂ x Q(ξ, y ) dy , when ξ 0. y
Then (2.4) and (2.5) imply k − W h = (1 − Kλk )−1 (Kλk − Kλh )W h + (1 − Kλk )−1 (fk − fh ) W = I1 + I 2 .
(5.1)
Now write h I1 = (1 − Kλk )−1 (Kλk − Kλh )W =
N
Kλi k (Kλk
h + K N +1 − Kλh )W λk
i=0 = I1 + I1
.
∞
h Kλi k (Kλk − Kλh )W
i=0
(5.2)
−1 imply h |L1 (dξ ) (1 − |∂ Note that (2.6) and supy |W x Q(ξ, y)|L1 (dξ dy) )
∞
i h
sup
Kλk (Kλk − Kλh )W < C
y
i=0
and |I1
|L1 (dξ )
L1 (dξ )
∞
N +1 i
= sup Kλk Kλk (Kλk − Kλh )Wh
y
i=0
→ 0,
as N → ∞.
L1 (dξ )
On the other hand,
(Kλ − Kλ )W h
k h L
1 (dξ )
y
iλ ξ(y−y )
e k h |L1 (dξ ) dy − eiλh ξ(y−y ) |∂ x Q|L1 (dξ ) |W
−∞
∞ +
iλ ξ(y−y )
e k − eiλh ξ(y−y ) |∂ x Q|L
1 (dξ )
y
→ 0,
as k, h → ∞
h |L1 (dξ ) dy |W
(5.3)
244
D. Wu / Journal of Functional Analysis 256 (2009) 215–257
by the Lebesgue convergence theorem and Q ∈ P1 . So
N
h
Kλi k (Kλk − Kλh )W |I1 |L1 (dξ ) =
i=0
→ 0,
as k, h → ∞.
(5.4)
L1 (dξ )
Hence |I1 |L1 (dξ ) → 0 as k, h → ∞ by (5.2)–(5.4). A similar argument will induce |I2 |L1 (dξ ) = k − W h |L1 (dξ ) → 0 as k, h → |(1 − Kλk )−1 (fk − fh )|L1 (dξ ) → 0 as well. Therefore, we have |W ∞ by (5.1). Taking the Fourier transform, we prove the lemma when λ ∈ C+ . The case of λ ∈ C− can be proved by analogy. 2 Lemma 5.2. Suppose that Q ∈ P1 and
2
ξ Q
L
1 (dξ
dy)
< ∞.
(5.5)
Then Ψ+ and Ψ− are continuously differentiable with respect to x and y. Proof. If λk → λ± and I1 , I2 are closed intervals on R, • ∂x Ψ (x, y, λk ), and ∂y Ψ (x, y, λk ) are Cauchy for each (x, y) ∈ I1 × I2 ; • ∂x Ψ (x, y, λk ), and ∂y Ψ (x, y, λk ) are uniformly bounded on I1 × I2 , then Ψ± is differentiable and ∂x Ψ± = (∂x Ψ )± , and ∂y Ψ± = (∂y Ψ )± by the Lebesgue convergence theorem. Therefore, the continuous differentiability will be implied by proving the uniform Cauchy property of ∂x Ψ (x, y, λk ), and ∂y Ψ (x, y, λk ) with respect to x, y in compact subsets. Lemma 5.1 and (2.1) imply that the uniform convergence of ∂y Ψ (x, y, λk ) comes from that of ∂x Ψ (x, y, λk ). So it is sufficient to show
ξ W (ξ, y, λk ) − ξ W (ξ, y, λh )
→ 0. L (dξ ) 1
By replacing ∂ x Q(ξ, y ) with ξ ∂ x Q(ξ, y ) in the representation of fk in (5.1), it can be shown by adopting a similar argument as that in the proof of Lemma 5.1. 2
Lemma 5.3. For Q ∈ P1 and Q satisfies (5.5), the eigenfunction Ψ (x, y, ·) is holomorphic in C± and has limits Ψ± on R. Moreover, there exists a continuously differentiable function v(x +λy, λ) such that Ψ+ (x, y, λ) = Ψ− (x, y, λ)v(x + λy, λ),
Lλ v = 0,
λ ∈ R,
where Lλ = ∂y − λ∂x . Proof. The holomorphicity has been proved in Theorem 1.1. By assumption, Lemmas 5.1 and 2.1, Ψ± is invertible. Hence Lemma 5.2 implies
(∂y − λ∂x ) Ψ−−1 Ψ+ = (∂y − λ∂x )Ψ−−1 Ψ+ + Ψ−−1 (∂y − λ∂x )Ψ+ = −Ψ−−1 (∂x Q)Ψ+ + Ψ−−1 (∂x Q)Ψ+ = 0.
2
D. Wu / Journal of Functional Analysis 256 (2009) 215–257
245
We denote Z = Z(Ψ ) = φ if there are no poles of Ψ (x, y, λ). Lemma 5.4. For Q ∈ P∞,k,0 , k 2, if Z = φ, then there exists a continuously differentiable function v(x + λy, λ) such that Ψ+ (x, y, λ) = Ψ− (x, y, λ)v(x + λy, λ),
Lλ v = 0,
λ ∈ R.
Proof. Since Z = φ, the eigenfunction Ψ (x, y, ·) has limits Ψ± by Corollary 4.1 and the Sobolev’s theorem. Moreover, we have the uniform convergence of ∂x Ψ (x, y, λk ), λk → λ0 . Hence the lemma can be proved by using the same argument as the proof of Lemma 5.3. 2 Since we are going to solve the inverse problem by the Riemann–Hilbert problem (λ ∈ R, v). By the scheme of Section 4, we need to investigate L2 (R, dλ) condition on v and ∂λ v. Hence the λ-asymptote of v and ∂λ v will be investigated in the remaining part of this section. We extend Theorem 3.1, and Corollary 3.1 as follows. Lemma 5.5. If Q ∈ P∞,k,0 , k 5 and Z = φ, then for i + j k − 4,
i j
∂ ∂y Ψ± − 1 − ∂x Q C ,
x
|λ|2 λ as |λ| → ∞. Where C is a constant depending on Q. Proof. We follow the scheme in Section 3 to prove this lemma. Note that all of the arguments there can be repeated except the proof of Lemma 3.2, where the small data condition has been used to assure the uniform boundedness of ∂xN Ψ , 0 N k − 1. Hence to prove this lemma, one needs only to show the uniform boundedness of ∂xN Ψ± , 0 N k − 1 as |λ| → ∞.
(5.6)
However, since Q ∈ P∞,k,0 , k 5, Ψ± exists, the property (5.6) can be justified by Corollary 4.1 and the Sobolev’s theorem. 2 We improve the boundary properties (1.6), (1.7) of Theorem 1.1 as follows. Lemma 5.6. If Q ∈ P∞,k,0 , k 5, and Z = φ, then for i + j k − 4, j
∂xi ∂y (Ψ± − 1) → 0 uniformly in L∞ as |x| or |y| → ∞. Proof. By the results of Lemma 5.5, it is sufficient to prove this lemma for |λ| < c where c is any fixed constant. However, for |λ| < c, i + j k − 4, j ∂xi ∂y Ψ± (x, 0, λ) − 1 → 0 uniformly in L∞ as |x| → ∞
246
D. Wu / Journal of Functional Analysis 256 (2009) 215–257
follow from (2.1), Corollary 4.1, and the Sobolev’s theorem. For y = 0, one can follow the j j argument of Lemma 4.4 to show the uniform convergence of ∂xi ∂y Ψ → ∂xi ∂y Ψ±,z . Then the lemma is proved by the uniform convergence and applying Hölder inequality to 1 Ψ (x, y, λ) = 1 + 2πi
∞ −∞
Ψ+ (t, 0, λ) − Ψ− (t, 0, λ) dt. t − (x + λy)
2
Lemma 5.7. For Q ∈ P∞,k,1 ∩ P1 , k 7, we have |∂λ Ψ± |, |∂λ ∂x Ψ± | <
C , |λ|
as |λ| → ∞,
and C depends continuously on x, y. Proof. By formula (2.7), we have 1 Ψ (x, y, λ) = 1 + 2π
∞
(ξ, y, λ) dξ. eiξ x W
(5.7)
−∞
Write (ξ, y, λ) = 1 A(ξ, y, λ). W λ ∈ Note that W X with X defined by Definition 3. Therefore Theorem 3.1 implies A is uniformly bounded in X.
(5.8)
Now we define B1 (ξ, y, λ) = λ
y
eiλξ(y−y ) ∂ x Q(ξ, y ) dy ,
if λ ∈ C+ , ξ 0;
−∞
B2 (ξ, y, λ) =− λ
∞
if λ ∈ C+ , ξ 0;
if λ ∈ C− , ξ 0;
eiλξ(y−y ) ∂ x Q(ξ, y ) dy ,
y
B3 (ξ, y, λ) =− λ
∞
eiλξ(y−y ) ∂ x Q(ξ, y ) dy ,
y
B4 (ξ, y, λ) = λ
y
−∞
eiλξ(y−y ) ∂ x Q(ξ, y ) dy ,
if λ ∈ C− , ξ 0.
(5.9)
D. Wu / Journal of Functional Analysis 256 (2009) 215–257
247
By (2.5), (2.6), (5.8), (5.9), and Theorem 3.1, we obtain X. B1 , B2 , B3 , B4 are uniformly bounded in
(5.10)
Differentiating both the sides of (2.5), we obtain
= iy (1 − Kλ )∂λ W
0 y e
iλξ(y−y )
) dy + ξ(∂ xQ ∗ W
−∞
1
y e
iλξ(y−y )
ξ ∂ x Q dy
−∞
0 y −i
e
iλξ(y−y )
) dy + y ξ(∂ xQ ∗ W
−∞
e
iλξ(y−y )
y ξ ∂ x Q dy
−∞
0 y = iy
1
y
e
iλξ(y−y )
dy + ∂x2 Q ∗ W
−∞
1
y
2 eiλξ(y−y ) ∂ x Q dy
−∞
0 y −i
e
2 y ∂ x Q ∗ W dy +
iλξ(y−y )
−∞
1
2 eiλξ(y−y ) y ∂ x Q dy
−∞
y + iy
y
e
iλξ(y−y )
) dy − i (∂ xQ ∗ ξW
−∞
y
) dy (5.11) eiλξ(y−y ) y (∂ xQ ∗ ξW
−∞
for λ ∈ C+ , ξ 0 (other cases can be done similarly). Define C1 (ξ, y, λ) = λ
y
2 eiλξ(y−y ) ∂ x Q(ξ, y ) dy ,
−∞
C2 (ξ, y, λ) = λ
y
∂ 2 Q(ξ, y ) dy , eiλξ(y−y ) y x
−∞
C3 (ξ, y, λ) (ξ, y, λ). = ξW λ Using the definition of P∞,k,1 , and following the way to prove (5.10), one can show that X C1 , C2 , C3 are uniformly bounded in
(5.12)
C if Q ∈ P∞,k,1 and k 6. Combining (5.7), (5.8), (5.11), (5.12), and (2.6), we prove |∂λ Ψ± | < |λ| as |λ| → ∞ and C depends continuously on x, y. ∞ iξ x i (ξ, y, λ) dξ , modifying the above argument and letting Since ∂x Ψ (x, y, λ) = 2π ξW −∞ e k 7 in P∞,k,1 , one can obtain the estimate for |∂λ ∂x Ψ± | as well. 2
248
D. Wu / Journal of Functional Analysis 256 (2009) 215–257
Lemma 5.8. If Q ∈ P∞,k,1 , k 7, and Z = φ, then |∂λ Ψ± | <
C , |λ|
as |λ| → ∞,
and C depends continuously on x, y. Proof. Since the property we wish to justify is a local property. Without loss of generality, we need only to show
χ(x, y)∂λ Ψ± < C , |λ|
as |λ| → ∞,
(5.13)
where C depends continuously on x, y, and χ(x, y) is any fixed smooth function with compact support. Now by the induction scheme as the proof of Theorem 1.1, we have $ − Ψ (x, y, λ)a − (x, y, λ), y 0, (5.14) Ψ (x, y, λ) = Ψ + (x, y, λ)a + (x, y, λ), y 0, and ∂λ Ψ = ∂λ Ψ ± a ± + Ψ ± ∂λ a ± . By induction and applying Lemmas 5.6, and 5.7, it reduces to showing
χ(x, y)∂λ a < C |λ|
as |λ| → ∞,
where $ a(x, y, λ) =
a − (x, y, λ), a + (x, y, λ),
y 0, y 0.
By (5.14), one can derive the inhomogeneous Riemann–Hilbert problem −1 (χ∂λ a)+,z (x, 0, λ) = g(x, λ) + Ψ + Ψ − (χ∂λ a)−,z (x, 0, λ), with −1 g = ∂λ Ψ + Ψ − χa−,z . Hence [1] χ∂λ a(x, y, λ) = Ψ˜ (x, y, λ)−1
1 2πi
R
(Ψ + )(t, 0, λ)g(t, λ) dt t −z
with x + λy = z, and $ − Ψ (x, y, λ), Ψ˜ (x, y, λ) = Ψ + (x, y, λ),
y 0, y 0.
(5.15)
D. Wu / Journal of Functional Analysis 256 (2009) 215–257
249
Therefore by Lemma 5.7 and (5.15),
−1
(Ψ + )(t, 0, λ)g(t, λ)
˜ dt
|χ∂λ a|L2 (R,dx) < C Ψ
t −z R
−1
χ(t, 0)(Ψ + )(t, 0, λ)g(t, λ)
˜
t −z R
χ(t, 0)
< C|g|
dt
t −z L2 (R,dx) R
<
C |λ|
(5.16)
as |λ| → ∞. Furthermore, differentiating both the sides of (5.15) and using Corollary 4.1, Lemma 5.7, we obtain
∂x (χ∂λ a)
(Ψ + )(t, 0, λ)g(t, λ)
∂x Ψ˜ −1
1 dt <
2πi
L2 (R,dx) t −z L2 (R,dx) R
−1
(Ψ + )(t, 0, λ)g(t, λ)
∂x 1
dt + Ψ˜
2πi t −z R
<
L2 (R,dx)
C . |λ|
(5.17)
Hence the lemma follows from (5.16), (5.17), and Sobolev’s theorem.
2
We conclude this section by the proof of Theorem 1.2 and the definition of continuous scattering transformation. Proof of Theorem 1.2. The condition (1.12) follows from Lemma 5.4. The identity (1.10) comes from (4.34) and Lemma 5.4. Besides, (4.35) and Lemma 5.4 imply that for λ ∈ R v(x + λy, λ) = Ψ− (x + λy, λ)−1 Ψ+ (x + λy, λ) = Ψ+ (x + λy, λ)∗ Ψ+ (x + λy, λ). Therefore (1.11) follows. Next note that Lemma 5.5 implies that j
∂xi ∂y (Ψ± − 1) are uniformly bounded in L∞ ∩ L2 (R, dλ) ∩ L1 (R, dλ). j
(5.18)
So (1.13) follows. Combining Lemma 5.6, (5.18), one obtains ∂xi ∂y (v −1) → 0 uniformly in L∞ . So condition (1.14) follows from (1.13), and the Lebesgue convergence theorem. Finally, condition (1.15) is derived by applying Lemma 5.8. 2
250
D. Wu / Journal of Functional Analysis 256 (2009) 215–257
Definition 7. For Q ∈ P∞,k,1 , k 7, if the eigenfunction Ψ (x, y, ·) has limits Ψ± on R, then we define the continuous scattering data of Q to be v ∈ Sc,k obtained by Theorem 1.2. Moreover, the continuous scattering transformation Sc on Q is defined by Sc (Q) = v. 6. Inverse problem: continuous scattering data We first prove Theorem 1.3 by solving the Riemann–Hilbert problem via a modified scheme of Section 4. Proof of Theorem 1.3. First of all, (1.13), (1.14) and Lemma 4.1 imply that there exists a constant M > 0 such that, as |x| or |y| > M − 1, the Riemann–Hilbert problem (λ ∈ R, v(x, y, λ)) can be solved and
i j
∂ ∂y (Ψ± − 1)
x
L2 (dλ)
C|v − 1|L2 (dλ)
(6.1)
for a constant C. Hence (1.16) holds as |x| or |y| > M − 1. Applying Hölder inequality, (1.13), (1.14), and (6.1), we then derive: For each fixed λ ∈ / R, ∀|x| or |y| > M − 1, j
∂xi ∂y (Ψ − 1) ∈ L∞ (dx dy), j
∂xi ∂y (Ψ − 1) → 0 in L∞ (dx dy), as x or y → ∞. Hence, to prove Theorem 1.3, it is sufficient to solve the Riemann–Hilbert problem (λ ∈ R, v(x, y, λ)) and establish (1.16), (1.17) for max(|x|, |y|) < M. The scheme in Section 4, in particular Lemmas 4.7–4.13, can be adapted to the solving of this problem. More precisely, Lemma 6.1. For λ, x, y ∈ R, we have a factorization v(x, y, λ) = (1 + hl )−1 χ(1 + hu )(x, y, λ), and for i + j k − 4, χ is diagonal and hu (hl ) is strictly upper (lower) triangular; j ∂xi ∂y (χ
− 1),
j j ∂xi ∂y hu , ∂xi ∂y hl , ∂λ χ, ∂λ hu , ∂λ hl
(6.2)
are in L∞ ∩ L2 (R, dλ)
and the norms depend continuously on x, y;
(6.3)
χ − 1, hu , hl → 0 uniformly in L∞ ∩ L2 (R, dλ) as |x| or |y| → ∞.
(6.4)
Proof. We use the positivity condition (1.11) to prove that di+ , 1 i n vanishes nowhere for λ ∈ R. Hence the statements can be proved by the same method as that in the proof of Lemma 4.7. 2 Lemma 6.2 (A diagonal Riemann–Hilbert problem). For max(|x|, |y|) < M, there exists a uniquely solution Ξ (x, y, λ) to the Riemann–Hilbert problem (λ ∈ R, χ) such that
D. Wu / Journal of Functional Analysis 256 (2009) 215–257
251
Ξ − 1 are uniformly bounded in H 1 (R, dλ);
(6.5)
Ξ − 1, ∂x Ξ, ∂y Ξ are uniformly bounded in L2 (R, dλ),
(6.6)
and for each fixed λ ∈ / R, j ∂xi ∂y Ξ ∈ L∞ (dx dy) for max |x|, |y| < M.
(6.7)
Proof. Applying (6.3), and (6.4), one obtains that ∂λi (χ − 1) are uniformly bounded in L∞ ∩ L2 (R, dλ), i = 0, 1. d arg χ 1 Hence the winding number N (x, y) = − 2πi dζ (x, y, ζ ) dζ is integer-valued. Moreover, the condition (6.4) implies that N (x, y) ≡ 0. Thus for max(|x|, |y|) < M, the existence of Ξ , and (6.5) can be implied by (6.3), the Sobolev’s theorem, and Lemma 4.2. By (6.3), (6.5), and the formulas , $ log χ(x, y, ζ ) 1 dζ , Ξ (x, y, λ) = exp 2πi ζ −λ R
, $ 1 log χ(x, y, ζ ) ∂x χ(x, y, ζ ) 1 ∂x Ξ (x, y, λ) = exp dζ dζ , 2πi ζ −λ 2πi χ(x, y, ζ )(ζ − λ) R
R
R
R
R
R
$ , ∂y χ(x, y, ζ ) log χ(x, y, ζ ) 1 1 ∂y Ξ (x, y, λ) = exp dζ dζ , 2πi ζ −λ 2πi χ(x, y, ζ )(ζ − λ) , 2 $ 1 log χ(x, y, ζ ) ∂x χ(x, y, ζ ) 1 dζ dζ ∂x2 Ξ (x, y, λ) = exp 2πi ζ −λ 2πi χ(x, y, ζ )(ζ − λ) , $ 1 log χ(x, y, ζ ) (∂x χ(x, y, ζ ))2 1 dζ dζ − exp 2πi ζ −λ 2πi χ 2 (x, y, ζ )(ζ − λ) R
R
R
R
, $ 1 1 log χ(x, y, ζ ) ∂xx χ(x, y, ζ ) dζ dζ , + exp 2πi ζ −λ 2πi χ(x, y, ζ )(ζ − λ) ... ,
we derive (6.6). Finally, we obtain (6.7) by Hölder inequality.
2
Lemma 6.3. For max(|x|, |y|) < M, there exists a function H (x, y, λ) satisfying $ H=
Hu (x, y, λ), Hl (x, y, λ),
for λ ∈ C+ , for λ ∈ C− ,
and j
• H (x, y, λ) ∈ L∞ ∩ H 1 (R, dλ), and ∂xi ∂y H (x, y, λ) ∈ L∞ ∩ L2 (R, dλ); • |Ξ− (1 + H− )v(1 + H+ )−1 Ξ+−1 (x, y, λ) − 1|H 1 (R,dλ) < ∞;
252
D. Wu / Journal of Functional Analysis 256 (2009) 215–257
• |Ξ− (1 + H− )v(1 + H+ )−1 Ξ+−1 (x, y, λ) − 1|L∞ C± < 1; • Hu (Hl ) is strictly upper (lower) triangular; • H is rational in λ ∈ C± , with only simple poles and each corresponding residue is off diagj onal, with only one non-zero entry κ and ∂xi ∂y κ ∈ L∞ (dx dy). Proof. Combining (6.5) with the results of Lemmas 6.1, 6.2, and the same method as in the proof of Lemma 4.9, the lemma can be proved. 2 Lemma 6.4 (A Riemann–Hilbert problem with small data). For max(|x|, |y|) < M, the Riemann– Hilbert problem (λ ∈ R, Ξ− (1 + H− )v(1 + H+ )−1 Ξ+−1 ) admits a solution ϕs (x, y, λ). Moreover, ϕs − 1, ∂x ϕs , ∂y ϕs are uniformly bounded in L2 (R, dλ), and for each fixed λ ∈ / R, j
∂xi ∂y (ϕs − 1) ∈ L∞ (dx dy). Proof. The existence of the solution and its properties can be proved by Lemmas 4.1, 6.1–6.3, the property of the Cauchy operator C and Hölder inequality. 2 Lemma 6.5 (Factorization of the Riemann–Hilbert problem). Suppose Ψ (x, y, λ) satisfies Theorem 1.3. Then for max(|x|, |y|) < M, there exists a unique function u, u(x, y, λ) = 1 +
N (λ − λk )−1 ak (x, y),
(6.8)
k=1
and j
∂xi ∂y ak ∈ L∞ (dx dy),
(6.9)
Ψ (x, y, λ) = uϕs Ξ (1 + H ).
(6.10)
Conversely, if for max(|x|, |y|) < M, ∃u(x, y, λ) satisfying (6.8), (6.9) and uϕs Ξ (1 + H ) is holomorphic for λ ∈ C± .
(6.11)
Define Ψ = uϕs Ξ (1 + H ) for max(|x|, |y|) < M. Hence Ψ satisfies Theorem 1.3. We then use Lemma 6.5 to prove Theorem 1.3. (a) A linear system for u(x, y, λ). Let u(x, y, λ) = 1 +
p (λ − λk )−1 ak . k=1
(6.12)
D. Wu / Journal of Functional Analysis 256 (2009) 215–257
253
u(x, y, λ) = (λ − λj )−1 aj + bj + O |λ − λj | ,
(6.13)
Then at λj
with bj = 1 +
(λj − λk )−1 ak .
(6.14)
k =j
Since λj is a simple pole of H and ϕs Ξ is regular at λj , we can write 1 + H (x, y, λ) = (λ − λj )−1 hj + nj + O |λ − λj | , ϕs Ξ (x, y, λ) = αj + βj (λ − λj ) + O |λ − λj |2 .
(6.15) (6.16)
We then try to find ak , such that u(x, y, λ)ϕs (x, y, λ)Ξ (x, y, λ)(1 + H (x, y, λ)) is holomorphic at λj . This yields the linear system for ak : aj αj hj = 0,
1 j p,
(6.17)
bj αj hj + aj (βj hj + αj nj ) = 0,
1 j p.
(6.18)
(b) Solving the linear system (6.17)–(6.18). Note by Lemma 6.3, one can conclude 2 = 0. hj n−1 j
(6.19)
Therefore, it can be justified that (6.17) is a consequence of (6.18). Note the off-diagonal form of hl (hu ) in Lemma 6.1 is crucial here. Inserting (6.14) into (6.18), we obtain a system of pn2 linear equations in pn2 unknowns (the entries of ak with coefficients in entries of hj (x, y), nj (x, y), αj (x, y), βj (x, y)). Therefore, we conclude the existence problem of Ψ is Fredholm. (c) Solving the Riemann–Hilbert problem. Using the Fredholm alternative, we need only to show that for any fixed x, y the homogeneous problem (with limit 0 rather than 1 as λ → ∞) has only the trivial solution. Suppose f (x, y, λ) solves this homogeneous problem. Consider ¯ ∗ . Since f (x, y, ·) ∈ L2 (R, dλ), we have g(λ) ∈ L1 (R, dλ) and g(x, y, λ) = f (x, y, λ)f (x, y, λ) ± is holomorphic in C . Thus the Cauchy’s theorem implies
g+ (s) ds =
0= R
R
f+ (s)f− (s)∗ ds =
f− (s)v(s)f− (s)∗ ds.
R
Because of (1.11) we conclude f− ≡ 0 on R, so also f+ ≡ 0 and f ≡ 0. Hence we prove the solvability of the Riemann–Hilbert problem in Theorem 1.3.
2
Lemma 6.6. For the solution Ψ of the Riemann–Hilbert problem obtained in Theorem 1.3, we have
254
D. Wu / Journal of Functional Analysis 256 (2009) 215–257
det Ψ (x, y, λ) ≡ 1,
(6.20)
¯ ∗ ≡ 1. Ψ (x, y, t, λ)Ψ (x, y, t, λ)
(6.21)
Proof. By (1.10), det Ψ (x, y, ·) has no jump across the real line. So applying the Liouville’s theorem, (6.20) follows from the holomorphic property in C± and Ψ → 1 as |λ| → ∞. Hence Ψ (x, y, λ) is invertible for all λ ∈ C, limits (Ψ (x, y, z, λ¯ )∗ )−1 ± for λ ∈ R exist, and ¯ ∗ )−1 fulfills the boundary condition as |λ| → ∞. (Ψ (x, y, z, λ) Secondly, by (1.11) and Ψ+ = Ψ− v, we obtain ¯ ∗ = Ψ− (x, y, λ) ¯ ∗ = Ψ+ (x, y, λ)v ¯ −1 ∗ Ψ (x, y, λ) + ¯ ∗ . ¯ ∗ = v −1 Ψ (x, y, λ) = v −1 Ψ+ (x, y, λ) −
(6.22)
So ¯ ∗ −1 v. ¯ ∗ −1 = Ψ (x, y, λ) Ψ (x, y, λ) + − ¯ ∗ )−1 satisfies the same Riemann–Hilbert problem in Theorem 1.3. ConseTherefore (Ψ (x, y, λ) ¯ ∗ )−1 by the uniqueness property of Theorem 1.3 (the Liouville’s quently Ψ (x, y, λ) = (Ψ (x, y, λ) theorem) and (6.21) is established. 2 We conclude this section by the proof of Theorem 1.4 and the definition of inverse scattering transformation. Proof of Theorem 1.4. By (1.18), the boundary condition (2.2) is satisfied. Besides, the Cauchy integral formula, and Theorem 1.3 imply Ψ (x, y, λ) = I + CΨ− (v − 1).
(6.23)
For fixed x, y ∈ R, applying Lλ = ∂y − λ∂x to (6.23) and using (1.16), (1.13), we obtain Lλ Ψ = Lλ CΨ− (v − 1) = C(Lζ Ψ− )(v − 1) + [Lλ , C]Ψ− (v − 1) 1 Ψ− (x, y, ζ ) v(x + ζy, ζ ) − 1 dζ = ∂x 2πi R
+ C(Lζ Ψ− )(v − 1) = ∂x Q(x, y) + C [Lζ Ψ ]− (v − 1)
(6.24)
with Q(x, y) given by (1.19). Hence comparing (6.23) and (6.24) and using the uniqueness result of Theorem 1.3, we obtain (2.1). Besides, (1.13), (1.16), (1.19), and Hölder inequality show that Q, ∂x Q, and ∂y Q ∈ L∞ . j Furthermore, by (2.1), (1.17), (6.20), and the λ-independence of Q, we derive ∂xi ∂y Q ∈ L∞ and j ∂xi ∂y Q, ∂y Q, Q → 0 as x or y → ∞, for i + j k − 4, i > 0. Finally, by (6.21) and (2.1), we have
D. Wu / Journal of Functional Analysis 256 (2009) 215–257
(∂x Q)Ψ (x, y, t, λ)∗
255
−1 −1
= (∂y − λ∂x )Ψ (x, y, t, λ)∗ −1 −1 = −Ψ (x, y, t, λ)∗ (∂y − λ∂x )Ψ (x, y, t, λ)∗ Ψ (x, y, t, λ)∗ T −1 −1 = −Ψ (x, y, t, λ)∗ (∂y − λ∂x )Ψ (x, y, t, λ) Ψ (x, y, t, λ)∗ ∗ −1 −1 = −Ψ (x, y, t, λ)∗ (∂x Q)Ψ (x, y, t, λ) Ψ (x, y, t, λ)∗ = −(∂x Q)∗ Ψ (x, y, t, λ)∗ Thus ∂x Q(x, y) ∈ su(n).
−1
.
2
Definition 8. For a function v ∈ Sc , we define the inverse scattering transformation Sc−1 on v by Sc−1 (v) = Q, where Q is obtained by Theorems 1.3 and 1.4. 7. The Cauchy problem: continuous scattering data We prove Theorem 1.5 in this section. Proof of Theorem 1.5. We can apply Theorem 1.1 to find the eigenfunction Ψ (x, y, 0, λ). By assumption, and Theorem 1.2, Sc (Q0 ) ∈ Sc,k . Now let us define v(t) by
v(t) = v(x, y, t, λ) = v x + λy + λ2 t, λ . (7.1) For each t ∈ R, rewriting x + λy + λ2 t = x + λ(y + λt) = x + λ2 (t + λ1 y) and modifying the approach in proving lemmas in Sections 3–5, one can justify that v(t) ∈ Sc,k (see Definition 2). So v satisfies the algebraic constraints: • det (v) ≡ 1; • v = v ∗ > 0, and the analytic constraints: for i + j + h k − 4, • • • •
Lλ v = 0, Mλ v = 0; j ∂xi ∂y ∂th (v − 1) are uniformly bounded in L∞ ∩ L2 (R, dλ) ∩ L1 (R, dλ); j ∂xi ∂y ∂th (v − 1) → 0 uniformly in L∞ ∩ L2 (R, dλ) ∩ L1 (R, dλ) as |x| or |y| or t → ∞; ∂λ v ∈ L2 (R, dλ) and the norms depend continuously on x, y,
where Lλ = ∂y − λ∂x , and Mλ = ∂t − λ∂y . Now we apply Theorems 1.3 and 1.4 to show the existence of Ψ (x, y, t, λ) and Q(x, y, t) satisfying (2.1) and (2.2). More precisely, Ψ (x, y, t, λ) = I + CΨ− (v − 1) Ψ− (x, y, ζ )(v(x + ζy + ζ 2 t, ζ ) − 1) 1 =I + dζ, 2πi λ−ζ R
Ψ± − 1, ∂x Ψ± , ∂y Ψ± , ∂t Ψ± are uniformly bounded in L2 (dλ),
(7.2)
256
D. Wu / Journal of Functional Analysis 256 (2009) 215–257
and for each fixed λ ∈ / R, i + j + h k − 4, j
∂xi ∂y ∂th Ψ ∈ L∞ (dx dy dt).
(7.3)
In addition, Q(x, y, t) =
1 2πi
Ψ− (x, y, t, ζ ) v x + ζy + ζ 2 t, ζ − 1 dζ,
R
and for i + j + h k − 4, i 2 + j 2 > 0, j
∂xi ∂y ∂th Q, ∂t Q, Q ∈ L∞ ,
(7.4)
j
(7.5)
∂xi ∂y ∂th Q, ∂t Q, Q → 0
in L∞ .
To prove (1.4), we note it is equivalent to prove Mλ Ψ = (∂y Q)(x, y, t)Ψ (x, y, t, λ).
(7.6)
Applying Mλ to both sides of (7.2) and using similar approach as that in the proof of Theorem 1.4, we obtain Mλ Ψ = (∂y Q)(x, y, t) + C(Mζ Ψ )− (v − 1).
(7.7)
Comparing (7.2) and (7.7) and using the uniqueness result of Theorem 1.3, we obtain (7.6). The smooth and decay properties of Q can be derived by an argument similar to the proof of Theorem 1.4 and conditions (7.3)–(7.5). Since we have obtain the differentiability of Ψ (x, y, t, λ) and Q(x, y, t). The compatibility condition of (2.1) and (7.6) yields (1.5). 2 We conclude this report by a brief remark on examples of Q0 ∈ P∞,k,1 , k 7, and the corresponding eigenfunction Ψ0 has no poles. The first class of examples is P1 ∩ S (S is the set of Schwartz functions and P1 is defined by Definition 3). To construct an example with large norm, we let v(x, y, λ) = v(x + λy, λ) satisfy det(v) = 1,
v = v ∗ > 0,
v − 1 ∈ S,
and for ∀i, j, h 0, j
∂xi ∂y ∂λh (v − 1) ∈ L2 (R, dλ) ∩ L1 (R, dλ) uniformly, j
∂xi ∂y ∂λh (v − 1) → 0 in L2 (R, dλ) uniformly, as |x|, |y| → ∞. We can solve the inverse problem and obtain Ψ0 ∈ S by the argument in proving Theorem 1.1. ∗ Note here we need to use the reality condition v = v > 0 to show the global solvability. More1 over, by using the fomula Q0 (x, y) = 2πi R ψ0,− (v − 1) dξ , one obtains that Q0 is Schwartz and possesses purely continuous scattering data.
D. Wu / Journal of Functional Analysis 256 (2009) 215–257
257
Acknowledgments I would like to express my gratitude to Richard Beals for the hospitality during my visit at Yale in the summer of 2006 and many helpful discussions during the preparation of this report. I would like to thank Chuu-Lian Terng and Karen Uhlenbeck for pointing out the problem of Ward equation and for the generous encouragement during this work. References [1] M.J. Ablowitz, A.S. Fokas, Complex Variables: Introduction and Applications, Cambridge Texts Appl. Math., Cambridge Univ. Press, Cambridge, 1997. [2] C.K. Anand, Ward’s solitons, Geom. Topol. 1 (1997) 9–20. [3] C.K. Anand, Ward’s solitons II, exact solutions, Canad. J. Math. 50 (1998) 1119–1137. [4] R. Beals, R.R. Coifman, Scattering and inverse scattering for first order systems, Comm. Pure Appl. Math. 37 (1) (1984) 39–90. [5] R. Beals, R.R. Coifman, Inverse scattering and evolution equations, Comm. Pure Appl. Math. 88 (1985) 29–42. [6] R. Beals, P. Deift, C. Tomei, Direct and Inverse Scattering on the Line, Math. Surveys Monogr., vol. 28, Amer. Math. Soc., Providence, RI, 1988. [7] B. Dai, C.L. Terng, Backlund transformations Ward solitons, and unitons, J. Differential Geom. 75 (1) (2007) 57– 108. [8] B. Dai, C.L. Terng, K. Uhlenbeck, On the space–time Monopole equation, arXiv:math.DG/0602607. [9] P. Deift, X. Zhou, Direct and inverse scattering on the line with arbitrary singularities, Comm. Pure Appl. Math. 44 (5) (1991) 485–533. [10] D.K. Faddeev, V.N. Faddeeva, Computational Methods of Linear Algebra, Freeman, New York, 1963. [11] A.S. Fokas, T.A. Ioannidou, The inverse spectral theory for the Ward equation and for the 2 + 1 chiral model, Comm. Appl. Anal. 5 (2001) 235–246. [12] F.D. Gakhov, Boundary Value Problems, Addison–Wesley, Reading, MA, 1966. [13] D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, second ed., Grundlehren Math. Wiss., vol. 224, Springer, Berlin, 1983. [14] T. Ioannidou, Soliton solutions and nontrivial scattering in an integrable chiral model in dimensions, J. Math. Phys. 37 (1996) 3422–3441. [15] T. Ioannidou, W. Zakrzewski, Solutions of the modified chiral model in dimensions, J. Math. Phys. 39 (1998) 2693– 2701. [16] K. Pohlmeyer, Integrable Hamiltonian systems and interactions through quadratic constraints, Comm. Math. Phys. 46 (1976) 207–221. [17] D.H. Sattinger, Flat connection and scattering theory on the line, SIAM J. Math. Anal. 21 (1990) 729–756. [18] E.M. Stein, G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, Princeton, NJ, 1971. [19] C.L. Terng, Applications of loop group factorization to geometric soliton equations, in: Proceedings of the International Congress of Mathematicians, Madrid, 2006, vol. II, ICM, Madrid, 2006, pp. 927–950. [20] J. Villarroel, The inverse problem for Ward’s system, Stud. Appl. Math. 83 (1990) 211–222. [21] R.S. Ward, Soliton solutions in an integrable chiral model in 2 + 1 dimensions, J. Math. Phys. 29 (1988) 386–389. [22] R.S. Ward, Nontrivial scattering of localized solutions in a 2 + 1-dimensional integrable system, Phys. Lett. A 208 (1995) 203–208. [23] V.E. Zakharov, A.V. Mikhailov, Relativistically invariant two-dimensional models of field theory which are integrable by means of the inverse scattering problem method, Soviet Phys. JETP 47 (6) (1978) 1017–1027. [24] Z. Zhou, Constructions of explicit solutions of modified principal chiral field in dimensions via Darboux transformations, in: C.H. Gu, et al. (Eds.), Differential Geometry, World Scientific, Singapore, 1993, pp. 325–332.
Journal of Functional Analysis 256 (2009) 258–274 www.elsevier.com/locate/jfa
Weak-type inequalities for higher order Riesz–Laguerre transforms Liliana Forzani a , Emanuela Sasso b,∗ , Roberto Scotto a a Departamento de Matemática and IMAL, Universidad Nacional del Litoral and CONICET, Santa Fe 3000, Argentina b Dipartimento di Matematica, Universitá di Genova. Genova, Via Dodecaneso 35, Genova 16146, Italy
Received 2 April 2008; accepted 5 September 2008 Available online 25 September 2008 Communicated by C. Kenig
Abstract In this paper we study the weak-type (1, 1) boundedness of the higher order Riesz–Laguerre transforms associated with the Laguerre polynomials. In particular, we obtain the boundedness for the Riesz–Laguerre transforms of order 2 and we find also the sharp polynomial weight ω that makes the Riesz–Laguerre transforms of order greater than two continuous from L1 (ω dμα ) into L1,∞ (dμα ), being μα the Laguerre measure. © 2008 Elsevier Inc. All rights reserved. Keywords: Riesz transforms; Laguerre; Polynomial expansion; Weak-type
1. Introduction: Riesz–Laguerre transforms of order |m| The aim of this paper is to study the weak type (1, 1) boundedness of Rm α , the mth Riesz– Laguerre transform, with m ∈ Zd0 , associated with the multidimensional Laguerre operator Lα , where α = (α1 , . . . , αd ) is a multi-index with αi 0, i = 1, . . . , d. The Laguerre operator Lα is a self-adjoint “Laplacian” on L2 ( dμα ), where μα is the Laguerre measure of type α = (α1 , . . . , αd ) with αi > −1, i = 1, . . . , d; defined on Rd+ = {x ∈ Rd : xi > 0, * Corresponding author.
E-mail addresses:
[email protected] (L. Forzani),
[email protected] (E. Sasso),
[email protected] (R. Scotto). 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.09.004
L. Forzani et al. / Journal of Functional Analysis 256 (2009) 258–274
259
for each i = 1, . . . , d}, by dμα (x) =
d xiαi e−xi dx. (αi + 1) i=1
It is well known that the spectral resolution of Lα is Lα =
∞
nPnα ,
n=0
where Pnα is the orthogonal projection on the space spanned by Laguerre polynomials of total degree n and type α in d variables, cf. [4,10]. The operator Lα is the infinitesimal generator of a “heat” semigroup, called the Laguerre semigroup, {e−t Lα : t 0}, defined in the spectral sense as e−t Lα =
∞
e−nt Pnα .
n=0
For any multi-index m = (a1 , . . . , ad ) ∈ Zd0 , the Riesz–Laguerre transforms Rm α of order |m| = a1 + · · · + ad are defined by m −|m|/2 α ⊥ P0 , Rm α = ∇α (Lα )
√ √ where ∇α is the gradient associated to Lα defined as ∇α = ( x1 ∂x1 , . . . , xd ∂xd ), and P0α ⊥ denotes the orthogonal projection onto the orthogonal complement of the eigenspace corresponding to the eigenvalue 0 of Lα . For every multi-index α such that αi 0, for all i = 1, . . . , d, we have the following main result. Theorem 1. The second order Riesz–Laguerre transforms map L1 ( dμα ) continuously into L1,∞ ( dμα ). This result was proved recently by Graczyk et al. [4] only for half-integer α that includes αi = − 12 . The corresponding proof is based on the technique of transference to the Hermite setting. This technique firstly appears for the Riesz–Laguerre transform of order one in Gutiérrez et al. [5]. The method seems to be inapplicable for any other value of α. It is known that the first order Riesz–Laguerre transforms are weak-type (1, 1), cf. [9]. Furthermore, we also know from that same paper that the Riesz–Laguerre transforms of order higher than 2 need not be weak-type (1, 1) with respect to μα . However, we can prove the following result that has to do with certain kind of weights we can add on the domain of these transforms to make them satisfy a weak-type inequality. Let us mention that in the Gaussian context something quite similar occur with the higher order Riesz–Gauss transforms. Pérez proved that for |m| > 2, the Riesz–Gauss transforms of order |m|, associated to the Ornstein–Uhlenbeck semigroup, map 2 L1 ((1 + |y||m|−2 ) dγ ) continuously into L1,∞ ( dγ ), with dγ (x) = e−|x| dx (see [6]). Recently Forzani, Harboure and Scotto in [1] extend this result for a family of singular integral operators
260
L. Forzani et al. / Journal of Functional Analysis 256 (2009) 258–274
related with the Gaussian measure γ . Regarding the weights for the Riesz–Laguerre transforms of order higher than 2 we have Theorem 2. The Riesz–Laguerre transforms √ of order |m|, with |m| > 2, map L1 (w dμα ) contin1,∞ uously into L (dμα ), where w(y) = (1 + |y|)|m|−2 . Moreover, Theorem 3. The weight w is the optimal polynomial weight needed to get the weak type (1, 1) inequality for the Riesz–Laguerre transforms of order |m|. It should be noted that there is another proof of Theorem 2 for multi-indices of half-integer type (even for the negative one). Indeed, take f w as the function f in Lemma 2.1 of the paper by Gutiérrez et al. [5] after using Corollary 4.8 of the reference [4]. The paper is organized as follows. Section 2 contains basic facts and notation needed in the sequel and the proof of Theorem 1. In particular, in order to exploiting the well-known relationship with the Ornstein–Uhlenbeck context, we introduce the “modified” Riesz–Laguerre transforms related to the “modified” Laguerre measure. In Section 3, we prove auxiliary propositions and Theorems 2 and 3. 2. Some preliminaries and proof of Theorem 1 In order to use the well-known relationship with the Ornstein-Uhlenbeck context, but not too much exploited in the weak-type inequalities, we are going to perform a change of coordinates in Rd+ . If x = (x1 , . . . , xd ) is a vector in Rd+ , then x 2 will denote the vector (x12 , . . . , xd2 ). Let Ψ : Rd+ → Rd+ be defined as Ψ (x) = x 2 and let dμ˜ α = dμα ◦ Ψ −1 be the pull-back measure from dμα . Then the modified Laguerre measure dμ˜ α is the probability measure dμ˜ α (x) = 2d
2 d 2α +1 x i e−xi
i
i=1
(αi + 1)
dx = 2d
d 2α +1 xi i 2 e−|x| dx, (αi + 1)
(2.1)
i=1
on Rd+ . The map f → UΨ f = f ◦ Ψ is an isometry from Lq (dμα ) onto Lq (dμ˜ α ) and from Lq,∞ (dμα ) onto Lq,∞ (dμ˜ α ), for every q in [1, ∞]. So we may reduce the problem of studying the weak-type boundedness of Rm α to the study of the same boundedness for the modified −1 m ˜ Riesz–Laguerre transforms Rα = UΨ Rm ˜ α. α UΨ with respect to the measure dμ m ˜ α coincides, up to a multiplicative constant, with ∇ m (L˜ α )−|m| P α ⊥ , being Observe that R 0 α ⊥ = UΨ P α ⊥ U −1 and ∇ the gradient on Rd associated to the Laplacian L˜ α = UΨ Lα UΨ−1 , P Ψ 0 0 operator (for more details, see [8, Section 2]). ˜m For the sequel it is convenient to express the kernel of R α with respect to the polynomial d measure mα defined on R+ as dmα (x) = e|x| dμ˜ α (x). 2
(2.2)
L. Forzani et al. / Journal of Functional Analysis 256 (2009) 258–274
261
According to [6,9], for αi > −1/2, i = 1, . . . , d, the kernel of the modified Riesz–Laguerre transforms of order |m| with respect to the polynomial measure mα is defined, off the diagonal, as Km (x, y) = Km (x, y, s)Πα (s) ds [−1,1]d
with 1 K (x, y, s) = m
0
q− (rx 2 ,y 2 ,s) √ |m|−2 d 2 √ |m|−2 log r rxi − yi si e− 1−r − ( r) Hai dr √ 1−r 1−r (1 − r)|α|+d+1
(2.3)
i=1
where Hai is the Hermite polynomial of degree ai and q± (x, y, s) =
d xi + yi ± 2(xi yi )1/2 si , i=1
d
α −1/2 (αi + 1) , √ 1 − si2 i (αi + 1/2) π i=1
d xi yi si , cos θ = cos θ (x, y, s) = i=1 |x||y| 2 1/2 d i=1 xi yi si 2 1/2 sin θ = sin θ (x, y, s) = 1 − cos θ = 1− . |x||y| Πα (s) =
Observe that cos θ (and also sin θ ) does not depend on |x| nor on |y|. Throughout this paper the symbol a b means a C b where C is a constant that may be different on each occurrence. And we write a ∼ b whenever a b and b a. The proof of Theorem 1 follows from the three propositions below whose proofs can be found in Section 3. Proposition 4. For |m| = 2, m K (x, y, s) e|x|2 K∗ (x, y, s)e−|y|2 on the global region G = Rd+ × [−1, 1]d \ R0 = R1 ∪ R2 ∪ R3 ∪ R4 , with 1/2 R0 = (y, s) ∈ Rd+ × [−1, 1]d : q− x 2 , y 2 , s R1 = (y, s) ∈ / R0 : cos θ < 0 ,
C , 1 + |x|
262
L. Forzani et al. / Journal of Functional Analysis 256 (2009) 258–274
R2 = (y, s) ∈ / R0 : cos θ 0, |y| |x| , R3 = (y, s) ∈ / R0 : cos θ 0, |x| |y| 2|x| , R4 = (y, s) ∈ / R0 : cos θ 0, |y| 2|x| , and ⎧ 2 ⎪ e−|x| , ⎪ ⎪ ⎪ 1/2 2 2 ⎪ ⎪ ⎪ |x|2|α|+2d e−c|x|q− (x ,y ,s) , ⎪ ⎨ 4 2 − 2 c|x| 2 sin θ 2 K∗ (x, y, s) = |y| −|x| +sin θ|x| e ⎪ 2|α|+2d ⎪ 1 ∧ , |x| ⎪ 2|α|+2d−1 ⎪ ⎪ 2 − |x|2 + sin θ |x|2 ) 2 ⎪ (|y| ⎪ ⎪ ⎩ 2 2 (1 + |x|)e− sin θ|x| ,
(y, s) ∈ R1 ,
(1)
(y, s) ∈ R2 ,
(2)
(y, s) ∈ R3 ,
(3)
(y, s) ∈ R4 .
(4)
Proposition 5. The operator K∗ defined as
K∗ f (x) = e|x|
2
χG (y, s)K∗ (x, y, s)Πα (s) ds f (y) dμ˜ α (y),
Rd+ [−1,1]d
is of weak type (1, 1) with respect to the measure μ˜ α . Proposition 6. For all m, the operator
T0m f (x) = p.v.
χR0 (y, s)Km (x, y, s)Πα (s) ds f (y) dmα (y),
Rd+ [−1,1]d
which is the modified Riesz–Laguerre transform restricted to the local region R0 , is of weak type (1, 1) with respect to the measure μ˜ α . Proof of Theorem 1. The result follows by splitting the modified Riesz–Laguerre transforms of second order into a local operator and a global one. Let us observe that for a simple covering lemma, we may pass from estimates with respect to the measure mα on the local part R0 to estimates with respect to the modified Laguerre measure μ˜ α . Therefore the local operator is equivalent to T0m defined in Proposition 6 for |m| = 2. On the other hand, the global operator, taking into account Proposition 4, is bounded by the operator defined in Proposition 5. Both are weak type (1,1) and therefore so are the second order modified Riesz–Laguerre transforms. 2 3. Proofs of propositions and Theorems 2 and 3 From [6] and [9], it is known that an upper bound for |Km (x, y, s)| on G is K˜ m (x, y, s) =
⎧ 2 ⎨ (|x|2 + |y|2 ) |m|−2 2 e −|y|
if cos θ < 0,
⎩ (q+ q− )
if cos θ 0,
|m|−2 4
|α|+d −u0 ( qq+− ) 2 (1 + (q+ q− )1/4 |x||x||y| 2 +|y|2 )e
(3.1)
L. Forzani et al. / Journal of Functional Analysis 256 (2009) 258–274
263
with u0 =
|y|2 − |x|2 (q+ (x 2 , y 2 , s)q− (x 2 , y 2 , s))1/2 + . 2 2
Proof of Proposition 4. In this proposition, |m| = 2. If cos θ < 0, it is immediate that m K (x, y, s) e|x|2 K∗ (x, y, s)e−|y|2 . Let us then assume that cos θ 0. (1) First let us consider |x| > |y|. 1/2 1/2 1/2 Since cos θ 0, q+ |x| and since |x| > |y|, then q+ 2|x|. Therefore q+ ∼ |x|. On 1/2 C the other hand, since q− 1+|x| then |x| c. Thus m K (x, y, s)
q+ q−
|α|+d 2
q+ + q−
|α|+d 2
1/4
(q+ q− )
e−u0
1/2 |α|+d+1/2 |α|+d (q+ ) e−u0 |x||α|+d 1 + |x| + 1/2 |α|+d−1/2 (q− ) |x|2|α|+2d e−u0 = e|x| |x|2|α|+2d e− 2
e|x| |x|2|α|+2d e− 2
1/2 |x|q− (x 2 ,y 2 ,s) 2
(q+ q− )1/2 2
e
|y|2 −|x|2 2
e−|y|
2
e−|y| = e|x| K∗ (x, y, s)e−|y| . 2
2
2
(2) Now let us assume |y| |x| and rewrite u0 in the following way: u0 =
|y|2 − |x|2 (q+ (x 2 , y 2 , s)q− (x 2 , y 2 , s))1/2 + 2 2
= |y|2 − |x|2 +
(q+ (x 2 , y 2 , s)q− (x 2 , y 2 , s))1/2 − (|y|2 − |x|2 ) 2
= |y|2 − |x|2 +
q+ q− − (|y|2 − |x|2 )2 2(|y|2 − |x|2 + (q+ q− )1/2 )
= |y|2 − |x|2 +
2 sin2 θ |x|2 |y|2 . |y|2 − |x|2 + (q+ q− )1/2
(3.2)
Since 2 2 q+ q− = |x|2 + |y|2 − 4|x|2 |y|2 cos2 θ = |y|2 − |x|2 + 4|x|2 |y|2 sin2 θ, and taking into account that sin θ is non-negative, we obtain that (q+ q− )1/2 ∼ |y|2 − |x|2 + |x||y| sin θ |y|2 − |x|2 + |x|2 sin θ.
(3.3)
264
L. Forzani et al. / Journal of Functional Analysis 256 (2009) 258–274
Thus, from (3.2) together with (3.3) we get u0 |y|2 − |x|2 +
c|x|4 sin2 θ . |y|2 − |x|2 + |x||y| sin θ
(3.4)
Claim. max(|y|2 − |x|2 , |x|2 sin θ ) 1. Proof. If sin θ
1 , |x|2
the inequality is immediate.
If sin θ then |y|2 |x|2 + 1. This inequality is immediate when |x| 1 by adjusting conveniently the constant C in the definition of the global zone and it is also immediate for d = 1 and |x| > 1. Now let us assume that d 2 and |x| > 1. 1 , |x|2
2 2 (C/2)2 C2 2 2 x q , y , s = |x| + |y| − 2|x||y| 1 − sin2 θ − |x|2 (1 + |x|)2 1 2 2 |x| + |y| − 2|x||y| 1 − 4 . |x| Hence |y| − 2|x| 1 − 2
1 (C/2)2 |y| + |x|2 − 0 4 |x| |x|2
for all |y| |x|, then 1 |y| |x| 1 − 4 + |x|
(C/2)2 − 1 |x|
which implies that 1 (C/2)2 − 2 |y|2 |x|2 + 2 (C/2)2 − 1 1 − 4 + |x|2 + 1. |x| |x|2
2
Therefore by applying this claim to inequality (3.3) we obtain that q+ q− c in this context. If |x| |y| 2|x|, by taking into account (3.4), we get u0 |y|2 − |x|2 +
c|x|4 sin2 θ , |y|2 − |x|2 + |x|2 sin θ
then m K (x, y, s)
q+ q−
|α|+d 2
(q+ q− )1/4 e−u0
|α|+d
q+
(q+ q− )
2|α|+2d 4
(q+ q− )1/4 e−u0
(3.5)
L. Forzani et al. / Journal of Functional Analysis 256 (2009) 258–274
|x|2|α|+2d [(q+ q−
e|x|
)1/2 ]
2|α|+2d−1 2
e−u0
|x|2|α|+2d
2
265
(|y|2 − |x|2 + sin θ |x|2 )
e 2|α|+2d−1
−
c|x|4 sin2 θ |y|2 −|x|2 +|x|2 sin θ
e−|y| . 2
2
To get the last inequality we have used (3.3) and (3.5). On the other hand, since q+ q− c, it is immediate the following inequality m K (x, y, s) |x|2|α|+2d e|x|2 −|y|2 . Thus m K (x, y, s) e|x|2 K∗ (x, y, s)e−|y|2 . Now if |y| 2|x|, then (q+ q− )1/4 (|x|2 + |y|2 )1/2 |y|, and thus 1 + (q+ q− )
1/4
|x||y| |x|2 + |y|2
1 + |x| .
Besides q− (|y| − |x|)2 c|y|2 and q+ C|y|2 therefore qq+− C. On the other hand, from (3.2) together with (q+ q− )1/2 |y|2 − |x|2 + 2|x||y| sin θ and |y|2 − |x|2 + |x||y| sin θ 2|y|2 we get u0 |y|2 − |x|2 +
sin2 θ |x|2 |y|2 sin2 θ 2 2 2 |x| . |y| − |x| + 2 |y|2 − |x|2 + |x||y| sin θ
Therefore m K (x, y, s) e|x|2 K∗ (x, y, s)e−|y|2 .
2
Proof of Proposition 5. The method of proof we use here is an adaptation to our context of the techniques developed in [1–3], which allows us to get rid of the classical one called “forbidden regions technique.” The kernels (1) and (2) define strong type (1, 1) operators. Indeed, 2 2 e|x| χR1 (y, s)e−|x| Πα (s) ds f (y) dμ˜ α (y) C f 1 . Rd+ [−1,1]d
Moreover, for semi-integer values of the parameter α, by [6], 1/2
|x|2|α|+2d e−c|x|q−
(x 2 ,y 2 ,s)
(3.6)
is in L1 (dmα ) uniformly in y and s and so the operator is of strong type with respect to μ˜ α on R2 . Finally the result for the other values of α is obtained via the multidimensional Stein’s complex interpolation theorem (the same argument is used at the end of this proof).
266
L. Forzani et al. / Journal of Functional Analysis 256 (2009) 258–274
So to get the weak-type (1, 1) inequality for the operator K∗ it suffices to prove that the operators 2 χRi+3 (y, s)K∗ (x, y, s)Πα (s) ds f (y) dμ˜ α (y), i = 0, 1, Si f (x) = e|x| Rd+ [−1,1]d
map L1 (dμ˜ α ) continuously into L1,∞ (dμ˜ α ). Without loss of generality, we may assume that f 0. Fix λ > 0 and let Ei = {x ∈ Rd+ : Si f (x) > λ}, for i = 0, 1. We must prove that μ˜ α (Ei ) C fλ 1 . Let r0 and r1 be the positive roots of the equations 2|α|+2d r02
2
e f 1 = λ and r1 er1 f 1 = λ.
r0
We may observe that Ei ∩ {x ∈ Rd+ : |x| < ri } = ∅: indeed, if |x| < ri , we have S0 f (x) |x|2|α|+2d e|x| f 1 < λ, 2
S1 f (x) |x|e|x| f 1 < λ. 2
On the other hand, we may take λ > K f 1 and by choosing K large enough we may assume that both r0 and r1 are larger that one. Hence
μ˜ α x ∈ Rd+ : |x| > 2ri
d
2αj +1 −|x|2
xj
e
dx
|x|>2ri j =1 2|α| −4ri2
ri
e
C
f 1 . λ
Thus we only need to estimate μ˜ α {x ∈ Rd+ : ri |x| 2ri }. We let Ei denote the set of x ∈ S d−1 for which there exists a ρ ∈ [ri , 2ri ] with ρx ∈ E. For each x ∈ Ei we let ρi (x ) be the smallest such ρ. Observe that sin θ (x, y, s) = sin θ (x , y, s) = sin θ . Then Si f (ρi (x )x ) = λ, by continuity. This implies for i = 0 and x ∈ E0 , λ = S0 f ρ0 (x )x
Rd+
[−1,1]d
=
χR3 e
|x|2
2|α|+2d 1∧ |x|
e
−
c|x|4 sin2 θ |y|2 −|x|2 +sin θ|x|2
(|y|2 − |x|2 + sin θ |x|2 )
2|α|+2d−1 2
× Πα (s) ds f (y) dμ˜ α (y) 2
2
2|α|+2d
Rd+
[−1,1]d
eρ0 (x ) r0
χ{|y|r0 } (y) 1 ∧
× Πα (s) ds f (y) dμ˜ α (y),
e
−
cr04 sin2 θ 2 |y| −r02 +sin θr02
(|y|2 − r02 + sin θ r02 )
2|α|+2d−1 2
(3.7)
L. Forzani et al. / Journal of Functional Analysis 256 (2009) 258–274
267
and for i = 1 and x ∈ E1 , λ = S1 f ρi (x )x 2 2 2 = χR5 (y, s)e|x| 1 + |x| e− sin θ|x| Πα (s) ds f (y) dμ˜ α (y) Rd+ [−1,1]d 2
2
eρ1 (x ) r1
χ{|y|>r1 } (y)e−c sin
2 θr 2 1
Πα (s) ds f (y) dμ˜ α (y).
(3.8)
Rd+ [−1,1]d
Clearly, since r0 and r1 are greater than one, we have
2ri
μ˜ α x ∈ Ei : ri |x| 2ri
dσ (x ) Ei
e−ρ ρ 2|α|+2d−1 dρ 2
ρi (x )
2
e−ρi (x ) ri
2|α|+2d−2
dσ (x ).
Ei
Combining this estimate for i = 0 with (3.7), we get C 4|α|+4d−2 r0 dσ (x )(I0 + II 0 ), μ˜ α x ∈ Ei : ri |x| 2ri λ
(3.9)
E0
with
e
I0 = [−1,1]d {|y|r0 ,sin θr02 c}
and
(|y|2
−
cr04 sin2 θ 2 |y| −r02 +sin θr02
− r02 + sin θ r02 )
2|α|+2d−1 2
f (y) dμ˜ α (y) Πα (s) ds,
II 0 =
f (y) dμ˜ α (y) Πα (s) ds.
[−1,1]d {sin θr02 c}
Similarly for i = 1 with (3.8), we obtain C μ˜ α x ∈ E1 : r1 |x| 2r1 λ with
I1 = [−1,1]d {|y|r1 ,sin θr12 c}
2|α|+2d−1
r1
dσ (x )(I1 + II 1 ),
E1
e−c sin
2 θr 2 1
f (y) dμ˜ α (y) Πα (s) ds,
(3.10)
268
L. Forzani et al. / Journal of Functional Analysis 256 (2009) 258–274
and
II 1 =
f (y) dμ˜ α (y) Πα (s) ds.
[−1,1]d {sin θr12 c}
It is immediate to verify that 4|α|+4d−2 r0
dσ (x ) Πα (s) ds C
[−1,1]d {x : sin θr02 c}
and 2|α|+2d−1
dσ (x ) Πα (s) ds C
r1
[−1,1]d {x : sin θr02 c}
(for more details, see [8, p. 254]), which give, after changing the order of integration in (3.9) and (3.10), the desired estimate for the terms involving II 0 and II 1 , respectively. Now let us prove that for |y| r0
4|α|+4d−2
e
r0
[−1,1]d {x : sin θr02 c}
(|y|2
−
− r02
cr04 sin2 θ |y|2 −r02 +sin θr02 2|α|+2d−1 + sin θ r02 ) 2
dσ (x ) Πα (s) ds C
and for |y| r1 2|α|+2d−1 r1
e−c sin
2 θr 2 1
dσ (x ) Πα (s) ds C.
[−1,1]d {x : sin θr12 c}
Firstly, one considers the case where α = ( n21 − 1, . . . , n2d − 1), with ni ∈ N and ni > 1 for each i = 1, . . . , d. In this case the inner integrals can be interpreted as integrals over S |n|−1 with respect to the Lebesgue measure, expressed in polyradial coordinates. For these cases, the desired d estimates can be found in [2, p. 13]. The same estimates are obtained also for α ∈ N2 − 1 + iRd . Finally the result for the other values of α are obtained via the multidimensional Stein’s complex interpolation theorem. Indeed, let F : Cd → C the function defined by
4ξ +4d−2
F (ξ ) = r0
{sin θr02 c}
e
−
cr04 sin2 θ |y|2 −r02 +sin θr02
(|y|2 − r02 + sin θ r02 )
2|ξ |+2d−1 2
Πξ (s) ds.
We have seen that |F ( n2 − 1)| C and it is easy to prove that |F ( n2 − 1 + iζ )| |F ( n2 − 1)|, whenever n is a integer vector and ζ ∈ Rd . 2
L. Forzani et al. / Journal of Functional Analysis 256 (2009) 258–274
269
Proof of Proposition 6. The proof of this result follows the same steps like the proof of the weak-type boundedness on the local zone of the first order Riesz–Laguerre transforms done in [9]. Indeed the only two things left to prove are Lemma 3.3 of [9] for the kernel of a modified Riesz–Laguerre transform of any order and the L2 (dμ˜ α )-boundedness of this transform over the global region G. For the former we have the Calderón–Zygmund-type estimates for the kernel Km . 2 Lemma 7. There exists a constant C such that m K (x, y, s) ϕ(x, y, s) Cq− x 2 , y 2 , s −|α|−d , ∇(x,y) Km (x, y, s)ϕ(x, y, s) Cq− x 2 , y 2 , s −|α|−d−1/2 being ϕ(x, y, s) a cut-off function defined in [9] and (y, s) ∈ R0 . For the latter we have the following theorem regarding the Lp (dμ˜ α )-boundedness for 1 < p < ∞ of the modified Riesz–Laguerre transform of any order on G. Theorem 8. The operator
Rm g f (x) =
χG (y, s)Km (x, y, s) Πα (s) ds f (y) dmα (y)
Rd+ [−1,1]d
is strong-type (p, p) for 1 < p < ∞ with respect to the measure μ˜ α . √ 1/2 Proof of Lemma 7. Since | rxi − yi si | q− (rx 2 , y 2 , s), then d √ |m| rxi − yi si − q− (rx 2 ,y 2 ,s) q− (rx 2 , y 2 , s) k/2 − q− (rx 2 ,y 2 ,s) 1−r 1−r Hai e √ e 1−r 1−r i=1
k=0
e−
q− (rx 2 ,y 2 ,s) 2(1−r)
e−c
q− (x 2 ,y 2 ,s) 1−r
,
where last inequality follows from this one: q− rx 2 , y 2 , s q− x 2 , y 2 , s − 2C 1 − r 1/2 when (y, s) ∈ R0 (see [7]). Thus on R0 , m K (x, y, s) ϕ(x, y, s)
1 0
|m|−2 −c q− (x 2 ,y 2 ,s) 2 1−r √ |m|−2 log r e − ( r) dr |α|+d+1 1−r (1 − r)
2 ,y 2 ,s) 1/2 1 −c q− (x1−r √ |m|−2 |m|−2 e ( r) (− log r) 2 dr + dr (1 − r)|α|+d+1
0
1/2
1 + q− x 2 , y 2 , s
−|α|−d
.
270
L. Forzani et al. / Journal of Functional Analysis 256 (2009) 258–274
In computing the gradient of the kernel with respect to y we are going to have integrals such as Km (x, y, s)∂xj ϕ(x, y, s), 1
√ |m|−2 2 √ log r rxi − yi si ( r)|m|−1 − Hai √ 1−r 1−r i=j
0
√ × Haj −1
rxj − yj sj √ 1−r
e−
q− (rx 2 ,y 2 ,s) 1−r
(1 − r)|α|+d+3/2
dr,
with H−1 ≡ 0 and 1
√ |m|−2 d 2 √ |m|−1 log r rxi − yi si − ( r) Hai √ 1−r 1−r i=1
0
√ rxj − yj sj e × √ dr. 1 − r (1 − r)|α|+d+3/2 q (rx 2 ,y 2 ,s) − − 1−r
In order to estimate these three integrals we use the same estimates described at the beginning of this proof. For the first one we have to use also that ∇x ϕ(x, y, s) + ∇y ϕ(x, y, s) The gradient with respect to x is treated similarly.
1 1/2 q− (x 2 , y 2 , s)
.
2
Proof of Theorem 8. The proof of this result is an adaptation to our context of the same result for the higher order Riesz–Gauss transforms done in [6]. Taking into account that on G, q+ (x 2 , y 2 , s)q− (x 2 , y 2 , s) c when cos θ 0, by (3.1) an upper bound for |Km (x, y, s)| is ⎧ 2 ⎨ (|x|2 + |y|2 ) |m|−2 2 e −|y| if cos θ < 0, ˜ m K˜ (x, y, s) = 2 −|x|2 (q+ q− )1/2 |y| |m|−1 ⎩ q |α|+d (q q ) 4 e−( 2 + 2 ) if cos θ 0. +
+ −
Thus,
p K (x, y, s)Πα (s) ds f (y) dmα (y) dμ˜ α (x)
m
Rd+ G∩{cos θ<0}
Rd+
G∩{cos θ<0}
Rd+
p ˜˜ m (x, y, s)Π (s) ds f (y) dm (y) dμ˜ (x) K α α α
Rd+
p−1 2 p (|m|−2) p 2 |x| + |y|2 dμ˜ α (y) dμ˜ α (x) f Lp ( dμ˜ α ) .
L. Forzani et al. / Journal of Functional Analysis 256 (2009) 258–274
271
For the region G ∩ {cos θ 0} we are going to use the following estimates: q− |x − y|, (q+ q− )1/2 |y|2 − |x|2 , |x|2 + |y|2 q+ |x + y|2 , 1 1 1 1 2 (q+ q− )1/2 2 − |y| − |x| − (q+ q− )1/2 < since p > 1, p 2 p 2 2 p Km (x, y, s)Πα (s) ds f (y) dmα (y) dμ˜ α (x)
Rd+ G∩{cos θ0}
p m ˜ ˜ K (x, y, s)Πα (s) ds f (y) dmα (y) dμ˜ α (x)
Rd+
G∩{cos θ0}
|x + y|2|α|+2d (q+ q− )
Rd+
m−1 4
e
m−1 4
e
− |y|
2 −|x|2 |y|2 −|x|2 (q+ q− )1/2 + − 2 p 2
Πα (s) ds
G∩{cos θ0}
p − |y|2 p × f (y) e dmα (y) dmα (x)
|x + y|2|α|+2d (q+ q− )
Rd+
−( 12 −| p1 − 12 |)(q+ q− )1/2
Πα (s) ds
G∩{cos θ0}
p − |y|2 p × f (y) e dmα (y) dmα (x)
p − |y|2 1/2 |x + y|2|α|+2d e−cp (q+ q− ) Πα (s) ds f (y) e p dmα (y) dmα (x).
Rd+
G∩{cos θ0}
To finish the proof we just need to check that the kernel H (x, y, s) := |x + y|2|α|+2d e−cp (q+ q− ) χG∩{cos θ0} , 1/2
1/2
C for G = {(x, y, s): q− (x 2 , y 2 , s) 1+|x|+|y| }, is in L1 ( dmα (y)) and L1 (dmα (x)) independently of the remaining variables. Due to the symmetry of the kernel we are going to check only the first claim:
1/2
|y|2|α|+2d e−cp |y|q− dmα (y)
H (x, y, s) dmα (y) Rd+
|x−y|1
+
2|α|+2d |x| + |y|2|α|+2d e−c˜p (|x|+|y|) dmα (y).
|x−y|>1
It is clear that the second integral is bounded independently of x and s, for the first one see (3.6) with x substituted by y. 2
272
L. Forzani et al. / Journal of Functional Analysis 256 (2009) 258–274
Proof of Theorem 2. As we mention in the preliminaries to prove this theorem is equivalent to prove that the modified Riesz–Laguerre transforms of order higher than 2 map L1 (w˜ dμ˜ α ) continuously into L1,∞ (dμ˜ α ), with w(y) ˜ = (1 + |y|)|m|−2 . For each x ∈ Rd+ , let us write Rd+ × [−1, 1]d =
4
Ri .
i=0
Therefore, in order to get the result, it will be enough to prove that each of the following operators
Tim f (x) =
χRi (y, s)Km (x, y, s)Πα (s) ds f (y) d(mα ),
Rd+ [−1,1]d
for i = 0, . . . , 4, maps L1 (w˜ dμ˜ α ) continuously into L1,∞ (dμ˜ α ). Observe that according to Proposition 6, for all m the operator T0m is weak-type (1, 1) with respect to μ˜ α . On the other hand, for the ‘global parts’: R1 , R2 , R3 , and R4 , we have the following estimate for the kernel Km : 2 2 |m|−2 |x|2 ∗ −|y|2 , cos θ < 0, m K (x, y, s) (|x| + |y| ) 2 e K (x, y, s)e |m|−2 2 2 cos θ 0 (q+ q− ) 4 e|x| K∗ (x, y, s)e−|y| , (for more details see kernel (3.1) together with proof of Proposition 4 or papers [6,9]). 2 If (y, s) ∈ R1 , |Km (x, y, s)| is controlled by C(1 + max{|x|, |y|})|m|−2 e−|y| and therefore it m 1 1 is immediate to prove that T1 maps L (w˜ dμ˜ α ) into L (dμ˜ α ). Now if (y, s) ∈ Ri , with i = 2, 3, 4, we claim that m 2 |x|2 ∗ K (x, y, s) w(y)e ˜ K (x, y, s)e−|y| . Assuming the claim, the conclusion follows from Proposition 5. We must finally prove the claim. If (y, s) ∈ R2 , since q+ (|x| + |y|)2 |x|2 , then m 1/2 |m|−2 K (x, y, s) (q+ q− ) 4 e|x|2 e−c|x|q− e−|y|2 |m|−2 1/2 2 2 1/2 2 |x|q− e|x| e−c|x|q− e−|y| 1/2
|x| −c|x|q− −|y| w(y)e ˜ e e . 2
2
If (y, s) ∈ R3 ∪ R4 , 2 q+ q− = |x|2 + |y|2 − 4|x|2 |y|2 sin2 θ 2 |x|2 + |y|2 |y|4 .
L. Forzani et al. / Journal of Functional Analysis 256 (2009) 258–274
273
Thus m |m|−2 K (x, y, s) (q+ q− ) 4 e|x|2 K∗ (x, y, s)e−|y|2 |x| w(y)e ˜ K∗ (x, y, s)e−|y| , 2
and this concludes the proof of the theorem.
2
2
Proof of Theorem 3. This proof follows essentially from the proof of Theorem 4.1 in [9]. With the notation of that theorem, one takes η ∈ Rd+ with |η| sufficiently large, away from the axis and obtains the following lower bound for Km (x, η) K (x, η) = C m
Km (x, η, s)Πα (s) ds C|η||m|−2|α|−d−1 eξ
2 −|η|2
(3.11)
[−1,1]d η + v: v ⊥ η, |v| < 1, 12 |η| < ξ < 32 |η|}. for x ∈ J = {ξ |η| Now if we assume that the Riesz–Laguerre transforms of order |m| > 2 map L1 (w˜ dμ˜ α ) continuously into L1,∞ (dμ˜ α ) with w˜ (y) = (1 + |y|) and 0 < < |m| − 2 then by taking f 0 in L1 (w˜ dμ˜ α ) close to an approximation of a point mass at η, with f L1 (w˜ dμ˜ α ) = 1 we
|η| Km (x, η)|η|− and by applying inequality (3.11) we get that have that Rm α f (x) is close to e 2
e|η| Km (x, η)|η|− |η||m|−2|α|−d−1− e( we obtain 2
e−(
|η| 2 2 )
|η| 2 2 )
. Therefore by setting λ = c|η||m|−2|α|−d−1− e(
|η| 2 2 )
|η|2|α|+d−1 μ˜ α (J ) μ˜ α x ∈ Rd+ : Rm α f (x) > λ
|η| 2 1 = C|η|2|α|+d−|m|+1+ e−( 2 ) . λ
Hence |η||m|−2− must be bounded which is a contradiction. Therefore the conclusion of Theorem 3 holds. 2 Acknowledgment We thank the referee on pointing us out Gutiérrez et al. [5] and the alternative proof of Theorem 2 for half-integer multi-indices. References [1] L. Forzani, E. Harboure, R. Scotto, Weak type inequality for a family of singular integral operators related with the Gaussian measure, preprint. [2] L. Forzani, E. Harboure, R. Scotto, Harmonic analysis related to Hermite expansions, preprint. [3] J. García-Cuerva, G. Mauceri, S. Meda, P. Sjögren, J.L. Torrea, Maximal operators for the holomorphic Ornstein– Uhlenbeck semigroup, J. London Math. Soc. (2) 67 (1) (2003) 219–234. [4] P. Graczyk, J. Loeb, I. López, A. Nowak, W. Urbina, Higher order Riesz transforms, fractional derivatives, and Sobolev spaces for Laguerre expansions, J. Math. Pures Appl. (9) 84 (3) (2005) 375–405.
274
L. Forzani et al. / Journal of Functional Analysis 256 (2009) 258–274
[5] C. Gutiérrez, A. Incognito, J.L. Torrea, Riesz transforms, g-functions, and multipliers for the Laguerre semigroup, Houston J. Math. 27 (3) (2001) 579–592. [6] S. Pérez, F. Soria, Operators associated with the Ornstein–Uhlenbeck semigroup, J. London Math. Soc. (2) 61 (3) (2000) 857–871. [7] E. Sasso, Spectral multipliers of Laplace transform type for the Laguerre operator, Bull. Austral. Math. Soc. 69 (2) (2004) 255–266. [8] E. Sasso, Maximal operators for the holomorphic Laguerre semigroup, Math. Scand. 97 (2) (2005) 235–265. [9] E. Sasso, Weak type estimates for the Riesz–Laguerre transforms, Bull. Austral. Math. Soc. 75 (3) (2007) 397–408. [10] G. Szegö, Orthogonal Polynomials, Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, RI, 1959.
Journal of Functional Analysis 256 (2009) 275–306 www.elsevier.com/locate/jfa
Multipeak solutions to the Bahri–Coron problem in domains with a shrinking hole ✩ Mónica Clapp a , Monica Musso b,c , Angela Pistoia d,∗ a Instituto de Matemáticas, Universidad Nacional Autónoma de México, Circuito Exterior, C.U.,
04510 México DF, Mexico b Departamento de Matemática, Pontificia Universidad Católica de Chile, Avenida Vicuna Mackenna 4860, Macul,
Santiago, Chile c Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy d Dipartimento di Metodi e Modelli Matematici, Università di Roma “La Sapienza”, via A. Scarpa 16,
00161 Roma, Italy Received 14 April 2008; accepted 13 June 2008 Available online 28 August 2008 Communicated by J.-M. Coron
Abstract We construct positive and sign changing multipeak solutions to the pure critical exponent problem in a bounded domain with a shrinking hole, having a peak which concentrates at some point inside the shrinking hole (i.e. outside the domain) and one or more peaks which concentrate at interior points of the domain. These are, to our knowledge, the first multipeak solutions in a domain with a single small hole. © 2008 Elsevier Inc. All rights reserved. Keywords: Nonlinear elliptic boundary value problem; Pure critical exponent; Multipeak solutions
✩
The second and the third authors are supported by the M.I.U.R. National Project “Metodi variazionali e topologici nello studio di fenomeni non lineari” (Italy). The second author is supported by Fondecyt 1040936 (Chile). The first author is supported by CONACYT 58049 and PAPIIT IN105106 (Mexico). * Corresponding author. E-mail addresses:
[email protected] (M. Clapp),
[email protected] (M. Musso),
[email protected] (A. Pistoia). 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.06.034
276
M. Clapp et al. / Journal of Functional Analysis 256 (2009) 275–306
1. Introduction In this paper we investigate the existence of solutions, both positive and sign changing, to the problem
4
−u = |u| N−2 u u=0
in Ω\ ε(ω + ξ0 ), on ∂ Ω \ ε(ω + ξ0 ) ,
(1)
where Ω is a connected bounded smooth domain in RN , ξ0 ∈ Ω, ω is a closed bounded neighborhood of 0 in RN with smooth boundary, N 3, and ε > 0 is small enough. The exponent of the nonlinearity is 2∗ − 1 where 2∗ := N2N −2 is the so-called critical Sobolev exponent. This problem has a rich geometric structure: it is invariant under the group of Möbius transformations; in particular, it is invariant under dilations. This fact is responsible for the lack of ∗ compactness of the Sobolev embedding H01 (D) → L2 (D) even when D is a bounded domain, and produces a dramatic change in the behavior of this problem with respect to the subcritical one. Indeed, whereas for q ∈ (2, 2∗ ) problem −u = |u|q−2 u in D,
u=0
on ∂D,
(2)
has infinitely many solutions in every bounded smooth domain D of RN , for q = 2∗ Pohožaev [20] showed that it has only the trivial solution if D is strictly starshaped. Moreover, for q = 2∗ this problem does not have a nontrivial least energy solution unless D = RN . Solvability for q = 2∗ is, thus, a difficult issue. There are some well known existence results for q = 2∗ . The first one was given by Kazdan and Warner [13] who showed that, if D is an annulus, then (2) has infinitely many radial solutions. Later, without any symmetry assumption, Coron [10] proved the existence of a positive solution to (2) if D is annular shaped, i.e. x ∈ RN : 0 < R1 < |x| < R2 ⊂ D
and 0 ∈ / D,
(3)
and R2 /R1 is small enough. Substantial improvement was obtained by Bahri and Coron [3] (see also [2]) who showed that, if the reduced homology of D with coefficients in Z2 is nontrivial, then problem (2) has at least one positive solution. Concerning Coron’s result, an interesting issue is the study of the asymptotic behavior of Coron’s solution for R2 fixed and R1 → 0, in other words, when D has a small hole whose diameter tends to zero. If the hole is the ball of radius R1 , then the solution found by Coron concentrates around the hole and it converges, in the sense of measure, to a Dirac delta centered at the center of the hole as R1 → 0. We refer the reader to [14,15,21] where the study of existence of positive multipeak solutions to (2) in domains with several small circular holes and their asymptotic behavior as the size of the holes goes to zero has been carried out. Recently, Clapp and Weth [9] extended Coron’s result. They showed that, if D has a small enough hole, then (2) has at least two solutions. But nothing can be said about the sign of the second one. Existence of sign changing solutions for symmetric domains with a small hole was first shown by Clapp and Weth [8]. Musso and Pistoia [16] proved that, if the domain has certain symmetries and a small spherical hole, then the number of sign changing solutions becomes arbitrarily large as the radius of the sphere goes to zero. Recently, Clapp and Pacella [7] considered
M. Clapp et al. / Journal of Functional Analysis 256 (2009) 275–306
277
Fig. 1.
annular shaped domains D which are invariant under a finite group Γ of orthogonal transformations of RN and established the existence of multiple sign changing solutions even if the hole is large provided the cardinality of the minimal Γ -orbit of D is also large. Finally, if the domain D has two small holes, then Musso and Pistoia [18] proved that problem (2) has at least one pair of sign changing solutions. Results obtained so far suggest that solutions to problem (1) should concentrate at points outside the domain. In this paper we shall construct positive and sign changing multipeak solutions to (1) having a peak which concentrates at some point inside the shrinking hole ε(ω + ξ0 ) (i.e. outside the domain) and one or more peaks which concentrate at interior points of the domain Ω \ ε(ω + ξ0 ), for certain points ξ0 ∈ Ω. These are, to our knowledge, the first known solutions to problem (1) exhibiting this kind of concentration behavior, and the first multipeak solutions in a domain with a single small hole. Our first three results concern existence of positive multipeak solutions. Set
N AN := N (N − 2) 2
−(N +2)/2 1 + |y|2 dy.
RN
Theorem 1. Assume that ∂Ω is not connected. There exists ρ0 > 0, depending only on Ω, such that, for each point ξ0 ∈ Ω with dist(ξ0 , ∂Ω) ρ0 , there exist ζ ∗ ∈ Ω \ {ξ0 } and ε0 > 0 with the following property: for every ε ∈ (0, ε0 ) there is a positive solution uε to problem (1) satisfying |∇uε |2 dx AN δξ0 + δζ ∗
in the sense of measures, as ε → 0.
Under some symmetry assumptions on the domain, we obtain multiplicity of positive multipeak solutions. The domains considered in Theorems 2 and 3 are illustrated by Figs. 1 and 2, respectively. Theorem 2. Assume that, for some n N , (x1 , . . . , xn , xn+1 , . . . , xN ) ∈ Ω
⇔
(x1 , . . . , xn , −xn+1 , . . . , −xN ) ∈ Ω,
(4)
(x1 , . . . , xn , xn+1 , . . . , xN ) ∈ ω
⇔
(x1 , . . . , xn , −xn+1 , . . . , −xN ) ∈ ω.
(5)
There exists ρ0 > 0, depending only on Ω, such that, for each ξ0 ∈ Ω ∩ (Rn × {0}) with dist(ξ0 , ∂Ω) ρ0 and each connected component C of Ω ∩(Rn ×{0}) with nonconnected bound-
278
M. Clapp et al. / Journal of Functional Analysis 256 (2009) 275–306
Fig. 2.
ary, such that ξ0 ∈ / C if n = 1, there exist ζC∗ ∈ C \ {ξ0 } and ε0 > 0 with the following property: for every ε ∈ (0, ε0 ) and every C there is a positive solution uC,ε to problem (1) satisfying |∇uC,ε |2 dx AN (δξ0 + δζC∗ )
in the sense of measures, as ε → 0.
Theorem 3. Let Ω := B(0, 1) \ T (r, ρ), where B(0, 1) := x ∈ RN : |x| < 1 , T (r, ρ) := x ∈ RN : dist x, S(r) < ρ , S(r) := (x1 , x2 , 0, . . . , 0) ∈ RN : x12 + x22 = r 2 ,
r∈
1 ,1 . 2
Let ξ0 = 0 and assume that ω satisfies (x1 , x2 , x3 , . . . , xN ) ∈ ω
⇔
(x1 , x2 , −x3 , . . . , −xN ) ∈ ω.
Then, for each integer k 1 there exists ρ0 ∈ ( 12 , 1) such that, if r + ρ ∈ (ρ0 , 1), there exist r∗ ∈ (r + ρ, 1) and ε0 > 0 with the following property: for every ε ∈ (0, ε0 ) there is a positive solution uε to problem (1) satisfying |∇uε | dx AN δ0 + 2
k−1
δζ j
in the sense of measures, as ε → 0,
j =0 2πj where ζj := r∗ (cos 2πj k , sin k , 0, . . . , 0).
Concerning existence of sign changing multipeak solutions to (1), we prove the following two results.
M. Clapp et al. / Journal of Functional Analysis 256 (2009) 275–306
279
Theorem 4. Assume that x ∈ Ω iff −x ∈ Ω and let ξ0 = 0. Then there exist ζ ∗ ∈ Ω \ {0} and ε0 > 0 such that, for each ε ∈ (0, ε0 ), there is a pair ±uε of sign changing solutions to problem (1) satisfying |∇uε |2 AN (δ0 − δζ ∗ − δ−ζ ∗ )
in the sense of measures, as ε → 0.
Theorem 5. Let Ω := B(0, 1) and ξ0 = 0. If N is odd, assume that ω satisfies (x1 , x2 , x3 , . . . , xN ) ∈ ω
iff (x1 , x2 , −x3 , . . . , −xN ) ∈ ω.
Then, for every integer k 1 there exist r∗ ∈ (0, 1) and ε0 > 0 such that, for each ε ∈ (0, ε0 ), there exists a pair ±uε of sign changing solutions to problem (1) satisfying |∇uε | dx AN δ0 − 2
k−1
in the sense of measures, as ε → 0,
δζ j
j =0 2πj where ζj := r∗ (cos 2πj k , sin k , 0, . . . , 0).
One may ask whether the solutions given by the above results are solely created by the topology of Ω or whether there is really an effect of the hole. In other words, do these solutions persist for ε = 0? The answer is that, in general, they do not persist. In fact, Ben Ayed, El Mehdi and Hammami [4] showed that for thin annuli the least energy of a positive solution goes to infinity as the width of the annulus goes to zero. In particular, a thin annulus does not have 2-peak solutions, so the solutions provided by Theorem 1 for small ε blow up as ε → 0. This paper is organized as follows. In Section 2, we describe the construction of a first approximation for a solution to problem (1) and we give the scheme of the proof of our results, which is based in a finite-dimensional reduction. Section 3 is devoted to the proof of our main results. In particular, we state and prove a general existence result for solutions to problem (1) under some general symmetry assumptions. This result, together with a topological lemma, are the tools to construct positive and sign changing solutions to (1), as asserted in the previous theorems. In Section 4 we give the expansion of the energy functional associated to the problem at the ansatz. Finally, Section 5 is devoted to the study of the associated nonlinear problem which provides the finite-dimensional reduction. 2. An approximate solution and scheme of the proof In this section we describe a first approximation of the solution to problem (1). To simplify notation, we shall assume from now on that ξ0 = 0. Let δ be a positive real number and z be a point in RN . The basic element to construct a solution to problem (1) is the so called standard bubble Uδ,z defined by Uδ,z (x) = αN N−2
δ (δ 2
+ |x
N−2 2
− z|2 )
N−2 2
,
δ > 0, z ∈ RN ,
with αN := [N(N − 2)] 4 . It is well known (see [1,6,24]) that these functions are the positive +2 solutions of the equation −u = up in RN , where p := N N −2 . These are the basic cells to build
280
M. Clapp et al. / Journal of Functional Analysis 256 (2009) 275–306
an actual solution of (1) after we perform a suitable correction to fit in the boundary condition. To this purpose, we replace Uδ,z by its projection Pε Uδ,z onto H01 (Ω \ εω), defined by
p
−Pε Uδ,z = Uδ,z Pε Uδ,z = 0
in Ω \ εω, on ∂(Ω \ εω).
We will look for a solution to (1) of the form u = Vλ,ζ + φ,
Vλ,ζ := Pε Uμ,ξ +
k
(6)
νj Pε Uλj ,ζj ,
j =1
where Vλ,ζ represents the leading term and φ is a lower order term. Here νj = ±1, λ = (μ, λ1 , . . . , λk ) ∈ (0, ∞)k+1 and ζ = (ξ, ζ1 , . . . , ζk ) ∈ Ω k+1 . We will choose points ξ, ζj ∈ Ω and parameters μ, λj ∈ (0, ∞), j = 1, . . . , k, as follows: √ μ := d ε,
η < d < η−1
and ξ := μτ,
τ ∈ RN , |τ | < η,
(7)
and for j = 1, . . . , k, √ λj := Λj ε, |ζj | > 2η,
η < Λj < η−1 , |ζj − ζs | > 2η
dist(ζj , ∂Ω) > 2η,
(8) if j = s,
(9)
for some positive small fixed η. Set Λ¯ := (Λ1 , . . . , Λk ) ∈ (0, ∞)k and ζ¯ := (ζ1 , . . . , ζk ) ∈ Ω k . In terms of φ, problem (1) becomes
Lλ,ζ (φ) = Nλ,ζ (φ) + Rλ,ζ φ=0
in Ω \ εω, on ∂(Ω \ εω),
(10)
where Lλ,ζ (φ) := −φ − f (Vλ,ζ )φ, Nλ,ζ (φ) := f (Vλ,ζ + φ) − f (Vλ,ζ ) − f (Vλ,ζ )φ, Rλ,ζ := f (Vλ,ζ ) + Vλ,ζ . 2 N λj ,ζj the kernel of the operator −−pU Here f (u) := |u| N−2 u. We denote by K λj ,ζj on L (R ), and consider the spaces 4
p−1
Kλ,ζ := span f (Vλ,ζ )Pε θ : θ ∈ ⊥ Kλ,ζ
:= φ
∈ H01 (Ω
\ εω): Ω\εω
k j =0
Kλj ,ζj ,
φψ = 0 for all ψ ∈ Kλ,ζ ,
M. Clapp et al. / Journal of Functional Analysis 256 (2009) 275–306
281
where Pε θ denotes the orthogonal projection of θ onto H01 (Ω \ εω), i.e. Pε θ = θ in Ω \ εω, Pε θ = 0 on ∂(Ω \ εω). To prove the existence of a solution to (10), we first solve the problem ⎧ ⎨ Lλ,ζ (φ) = Nλ,ζ (φ) + Rλ,ζ + ψ, ψ ∈ Kλ,ζ , ⎩ φ ∈ K⊥ . λ,ζ
(℘λ,ζ )
(11)
δ,z has dimension N + 1 and is spanned Now, in order to solve this problem we recall [5] that K by the functions
0 (x) := Zδ,z
N − 2 (N −4)/2 |x − z|2 − δ 2 ∂Uδ,z (x) = αN δ , ∂δ 2 (δ 2 + |x − z|2 )N/2
i Zδ,z (x) :=
xi − zi ∂Uδ,z (x) = −αN (N − 2)δ (N −2)/2 2 , ∂zi (δ + |x − z|2 )N/2
x ∈ RN , x ∈ RN ,
for i = 1, . . . , N . So solving problem (℘λ,ζ ) in (11) is equivalent to finding φ and coefficients cji , i = 0, . . . , N , j = 0, . . . , k, such that
⎧ ⎪ Lλ,ζ (φ) = Nλ,ζ (φ) + Rλ,ζ + cji f (Vλ,ζ )Pε Zλi j ,ζj ⎪ ⎪ ⎪ ⎪ i,j ⎨ φ = 0 ⎪ ⎪ ⎪ φf (Vλ,ζ )Pε Zλi j ,ζj = 0 ⎪ ⎪ ⎩
in Ω \ εω, on ∂(Ω \ εω),
(12)
i = 0, . . . , N, j = 0, . . . , k.
Ω\εω
For technical reasons, it is useful to scale the problem. Let Ωε :=
Ω \ εω √ ε
x and y = √ ∈ Ωε . ε
1 √ Then u is a solution to (1) if and only if the function u(y) ˆ := ε p−1 u( εy) solves the problem
4
−v = |v| N−2 v v=0
in Ωε , on ∂Ωε .
(13)
ˆ In this expanded variables, the solution we are looking for looks like u(y) ˆ = Vˆλ,ζ + φ(y), where 1
Vˆλ,ζ (y) := ε p−1 V √λ
, √ζε ε
√ ( εy)
1 √ ˆ and φ(y) := ε p−1 φ( εy),
y ∈ Ωε .
282
M. Clapp et al. / Journal of Functional Analysis 256 (2009) 275–306
ˆ problem (12) becomes Now, in terms of φ,
⎧ ˆ = Nˆ λ,ζ (φ) ˆ + Rˆ λ,ζ + Lˆ λ,ζ (φ) cji f (Vˆλ,ζ )Zˆ ji ⎪ ⎪ ⎪ ⎪ i,j ⎪ ⎨ ˆ φ = 0 ⎪ ⎪ ⎪ ˆ (Vˆλ,ζ )Zˆ ji = 0 φf ⎪ ⎪ ⎩
in Ωε , on ∂Ωε ,
(14)
i = 0, . . . , N, j = 0, . . . , k,
Ωε 1
where Zˆ ji (y) := ε p−1 Pε Z iλj √
ζ
, √jε ε
√ ( εy) and
ˆ := −φˆ − f (Vˆλ,ζ )φ, ˆ Lˆ λ,ζ (φ) ˆ := f (Vˆλ,ζ + φ) ˆ − f (Vˆλ,ζ ) − f (Vˆλ,ζ )φ, ˆ Nˆ λ,ζ (φ) Rˆ λ,ζ := f (Vˆλ,ζ ) −
k
j =0
f (U √λj
ε
ζ
, √jε
).
We point out that φˆ solves (14) if and only if φ solves (12). The solution to problem (14) will be obtained as a fixed point of a certain contraction map, which will be defined thanks to the solvability of the following linear problem. Fix points and parameters as in (7)–(9). Given a function h, we consider the problem of finding φˆ such that for certain real numbers cji the following is satisfied
⎧ ˆ λ,ζ (φ) ˆ =h+ L cji f (Vˆλ,ζ )Zˆ ji in Ωε , ⎪ ⎪ ⎪ ⎪ i,j ⎪ ⎨ ˆ φ = 0 on ∂Ωε , (15) ⎪ ⎪ i ⎪ ˆ (Vˆλ,ζ )Zˆ j = 0 φf i = 0, . . . , N, j = 0, . . . , k. ⎪ ⎪ ⎩ Ωε
In order to perform an invertibility theory for Lˆ λ,ζ subject to the above orthogonality conditions, ∞ we introduce L∞ ∗ (Ωε ) and L∗∗ (Ωε ) to be, respectively, the spaces of functions defined on Ωε with finite · ∗ -norm (respectively · ∗∗ -norm), where 1 ψ∗ = sup w −β (x)ψ(x) + w −(β+ N−2 ) (x)Dψ(x) , x∈Ωε
with 2 − N−2 − N−2 2 + 2 , w(x) = 1 + |x − ξ |2 1 + x − ζj j
β = 1 if N = 3 and β =
2 N −2
if N 4. Similarly we define, for any dimension N 3, 4 ψ∗∗ = sup w − N−2 (x)ψ(x). x∈Ωε
M. Clapp et al. / Journal of Functional Analysis 256 (2009) 275–306
283
The operator Lˆ λ,ζ is indeed uniformly invertible with respect to the above weighted L∞ -norm, for all ε small enough. This fact is established in the next proposition. We refer the reader to [11,17] for the proof. Proposition 6. Let η > 0 be fixed. There are numbers ε0 > 0, C > 0, such that, for points and parameters satisfying (7)–(9), problem (15) admits a unique solution φˆ =: Tλ,ζ (h) for all 0 < ε < ε0 and all h ∈ C α (Ω¯ ε ). Moreover, ∂ ¯ ¯ Tλ,ζ (h) Ch∗∗ Tλ,ζ (h) Ch∗∗ , (16) d,Λ,τ,ζ ∗ ∗ and |ci | Ch∗∗ .
(17)
The solvability of problem (14) is established in the following proposition. Proposition 7. Let η > 0 be fixed. There are numbers ε0 > 0, C > 0, such that, for points and ˆ ¯ τ, ζ¯ ) to problem (14), parameters satisfying (7)–(9) there exists a unique solution φˆ = φ(d, Λ, 1 ¯ ¯ ˆ ¯ ¯ such that the map (d, Λ, τ, ζ ) → φ(d, Λ, τ, ζ ) is of class C for the · ∗ -norm and ˆ ∗ Cε φ
N−2 2
(18)
,
ˆ ∗ Cε ∇(d,Λ,τ, ¯ ζ¯ ) φ
N−2 2
.
(19)
The proof of the previous proposition will be postponed to Section 5. Here we just mention that the size of φˆ and its derivatives, given in (18) and (19), is strictly related to the size of N−2 Rˆ λ,ζ ∗∗ , which turns out to be of order ε 2 in all the different existence results we obtain, as shown in the proof of Proposition 7. Looking back at (14), we conclude that, in the expanded variable, the function Vˆλ,ζ + φˆ is an actual solution to (13), or equivalently that the function Vλ,ζ + φ in (6) is an actual solution to ¯ τ, ζ¯ ), the constants ci our original problem (1), if we show that, for a proper election of (d, Λ, j are all zero. This reduces our problem to a finite-dimensional one. Let Jε : H01 (Ω \ εω) → R be the energy functional given by 1 1 Jε (u) = |Du|2 − |u|p+1 . (20) 2 p+1 Ω\εω
Ω\εω
It is well known that critical points of Jε are solutions to (1). We introduce the function Jε∗ : (0, ∞)k+1 × RN × Ω k → R given by ¯ τ, ζ¯ ) := Jε (Vλ,ζ + φ) Jε∗ (d, Λ, where φ is the unique solution to problem (℘λ,ζ ) in (11) given by Proposition 7. Using standard tools one can prove the following results. Lemma 8. uε = Vλ,ζ + φ is a solution of problem (1), i.e. cji = 0 in (12) for all i, j , if and only ¯ τ, ζ¯ ) is a critical point of Jε∗ . if (d, Λ, A direct consequence of estimates (18) and (19) is the following expansion.
284
M. Clapp et al. / Journal of Functional Analysis 256 (2009) 275–306
Proposition 9. Let η > 0 be fixed and assume (7)–(9) hold true. Then we have the following expansion N−2 ¯ τ, ζ¯ ) = Jε (Vλ,ζ ) + o ε 2 , Jε∗ (d, Λ, where, as ε goes to zero, the term o(ε
N−2 2
(21)
¯ τ, ζ¯ )’s satisfying (7)–(9). ) is C 1 -uniform over all (d, Λ,
Finally, we conclude this section with the asymptotic expansion of the main part of the energy Jε (Vλ,ζ ), which will be obtained in Section 4. The expansion of Jε (Vλ,ζ ) is given in terms of the Green function of the Laplace operator vanishing at the boundary ∂Ω, defined by
1 G(x, y) = κN − H (x, y) , (22) |x − y|N −2 1 with κN := (N −2)|∂B| , where |∂B| denotes the surface area of the unit sphere in RN . The function H denotes the regular part of the Green function, which for all y ∈ Ω satisfies
H (x, y) = 0
H (x, y) = κN
in Ω,
1 , |x − y|N −2
x ∈ ∂Ω.
(23)
The function H (x, x) is called the Robin function of Ω at x. It is useful to point out the following properties of G and H : 1 for any x, y ∈ Ω, |x − y|N −2 lim H (x, x) = +∞
0 G(x, y) κN
(24) (25)
x→∂Ω
and H (x, x) min H (x, x) =: HΩ > 0.
(26)
x∈Ω
Proposition 10. Let η > 0 be fixed and assume that (7)–(9) hold. Then we have the following expansion Jε (Vλ,ζ ) = (k + 1)a1 − ε
N−2 2
N−2 ¯ τ, ζ¯ ) + o ε 2 , Ψ (d, Λ,
(27)
where Ψ is defined by ¯ τ, ζ¯ ) := F (τ ) Ψ (d, Λ,
1 d N −2
+ a2
k
+ a2 H (0, 0)d N −2
−2 H (ζj , ζj )ΛN j
−
j =1
− 2a2
k
j =1
k
N−2 2
νj νs G(ζj , ζs )Λj
N−2 2
Λs
j,s=1 s =j N−2
νj G(0, ζj )Λj 2 d
N−2 2
,
(28)
M. Clapp et al. / Journal of Functional Analysis 256 (2009) 275–306
285
where
1
p+1
F (τ ) := αN cω
(1 + |τ |2 )
N−2 2
RN
1 1 dy, N+2 N −2 |y + τ | (1 + |y|2 ) 2
a1 and a2 are positive constants and, as ε goes to zero, the term o(ε ¯ τ, ζ¯ )’s satisfying (7)–(9). (d, Λ,
N−2 2
) is C 1 -uniform over all
Roughly speaking, we may say that any critical point of Ψ stable with respect C 1 -perturbation generates a solution to (1) which has a positive blow-up point at the origin and k positive (if νj = +1) or negative (if νj = −1) blow-up points ζj ∈ Ω \ {0}. 3. Multipeak solutions Let Γ be a closed subgroup of the group O(N) of orthogonal transformations of RN . We denote by Γ x := {γ x: γ ∈ Γ } the Γ -orbit of x ∈ RN . A subset X of RN is said to be Γ -invariant if Γ x ⊂ X for every x ∈ X, and a function u : X → R is Γ -invariant if it is constant on every Γ -orbit of X. The Green function satisfies the following. Lemma 11. If Ω is Γ -invariant then G(γ x, γ y) = G(x, y)
and H (γ x, γ y) = H (x, y),
for all x, y ∈ Ω, γ ∈ Γ . Proof. Fix x ∈ Ω, γ ∈ Γ . The map y → H (x, γ −1 y) is harmonic and, since γ y ∈ ∂Ω for every y ∈ ∂Ω, it satisfies H x, γ −1 y =
1 1 = |x − γ −1 y|N −2 |γ x − y|N −2
∀y ∈ ∂Ω.
Therefore, H (γ x, y) = H x, γ −1 y This proves our claim.
∀x, y ∈ Ω, γ ∈ Γ.
2
Let Γ be a group of the form Γ := Γ1 × Γ2 , where Γ1 is a closed subgroup of O(n) and Γ2 is a closed subgroup of O(m), n + m = N , acting on RN = Rn × Rm by (γ1 , γ2 )(y, z) := (γ1 y, γ2 z)
∀γ1 ∈ Γ1 , γ2 ∈ Γ2 , y ∈ Rn , z ∈ Rm .
From now on, we assume that these groups have the following properties:
286
M. Clapp et al. / Journal of Functional Analysis 256 (2009) 275–306
(I) Γ1 is a finite group which acts freely on Rn \ {0}, that is, γ y = y for every γ ∈ Γ1 , y ∈ Rn . (II) Γ2 acts without fixed points on Rm \ {0}, that is, for every z ∈ Rm \ {0} there exists γ ∈ Γ2 such that γ z = z. To simplify notation we write Γ1 for the subgroup Γ1 × {1} of Γ and Γ2 for the subgroup {1} × Γ2 of Γ . Property (II) implies that the fixed point space of the Γ2 -action on RN is
x ∈ RN : γ x = x ∀γ ∈ Γ2 = Rn × {0},
(29)
thus Γ y = Γ1 y
∀y ∈ Rn × {0},
and, since Γ1 acts freely on Rn \ {0}, its cardinality #Γ y is the order |Γ1 | of the group Γ1 . For ζ ∈ (Ω \ {0}) ∩ (Rn × {0}) we define
α(ζ ) := H (ζ, ζ ) −
G(ζ, γ ζ ).
γ ∈Γ1 \{1}
Set Ω1 := ζ ∈ Ω \ {0} ∩ Rn × {0} : α(ζ ) = 0 , and let ϕ : Ω1 → R be defined by ϕ(ζ ) := H (0, 0) −
|Γ1 |G2 (0, ζ ) . α(ζ )
By Lemma 11, both α and ϕ are Γ1 -invariant, that is, α(γ ζ ) = α(ζ )
for all γ ∈ Γ1 , ζ ∈ Ω \ {0} ∩ Rn × {0} ,
ϕ(γ ζ ) = ϕ(ζ )
for all γ ∈ Γ1 , ζ ∈ Ω1 .
(30)
The following holds. Theorem 12. Assume that Ω is Γ -invariant and ω is Γ2 -invariant, and let ζ ∗ ∈ Ω1 be a C 1 stable critical point of ϕ. (i) If α(ζ ∗ ) > 0 and ϕ(ζ ∗ ) > 0, then there exists ε0 > 0 such that, for each ε ∈ (0, ε0 ), there is a positive Γ2 -invariant solution uε to problem (1) which satisfies |∇uε | dx AN δ0 + 2
γ ∈Γ1
δγ ζ ∗
in the sense of measures, as ε → 0.
M. Clapp et al. / Journal of Functional Analysis 256 (2009) 275–306
287
(ii) If α(ζ ∗ ) < 0, then there exists ε0 > 0 such that, for each ε ∈ (0, ε0 ), there is a sign changing Γ2 -invariant solution uε to problem (1) which satisfies
2 δγ ζ ∗ |∇uε | dx AN δ0 − in the sense of measures, as ε → 0. γ ∈Γ1
Proof. We look for a Γ2 -invariant solution to problem (1) of the form u = V + φ,
V := Pε Uμ,ξ +
νPε Uλ,γ ζ
(31)
γ ∈Γ1
with ν ∈ {1, −1}, and μ, λ ∈ (0, ∞), ξ, ζ ∈ Ω ∩ (Rn × {0}), such that conditions (7)–(9) hold with λj = λ and ζj = γj ζ , that is, √ μ := d ε,
√ λ := Λ ε,
ξ := μτ, |ζ | > 2η,
dist(ζ, ∂Ω) > 2η,
η < d, Λ < η−1 ,
τ ∈ RN , |τ | < η,
(32) (33)
|ζ − γ ζ | > 2η
∀γ ∈ Γ1 , γ = 1,
(34)
for some η > 0. In this case, by Lemma 11, the function Ψ defined in (28) reduces to 1 + a2 H (0, 0)d N −2 d N −2
+ a2 kH (ζ, ζ ) − k G(ζ, γ ζ ) ΛN −2
Ψ (d, Λ, τ, ζ ) := F (τ )
γ ∈Γ1 \{1}
− 2a2 νkG(0, ζ )Λ
N−2 2
d
N−2 2
,
where k := |Γ1 | and, abusing notation, we have set Λ := (Λ, . . . , Λ) and ζ := (ζ, γ2 ζ, . . . , γk ζ ) for some chosen ordering of the elements of Γ1 = {γ1 := 1, γ2 , . . . , γk }. We will now show that, for some η > 0, the restriction of Ψ to the set ¯ τ, ζ¯ ) ∈ (0, ∞)k+1 × Rn × {0} × Ω ∩ Rn × {0} k : (32)–(34) hold Sη := (d, Λ, has a critical point which is stable with respect to C 1 -perturbation. The claim will then follow from Propositions 10, 9, and Lemma 13 below. It is easy to check that, if νG(0, ζ ) kG2 (0, ζ ) > 0 and ϕ(ζ ) := H (0, 0) − > 0, α(ζ ) α(ζ )
(35)
there exist unique d(ζ ), Λ(ζ ) > 0, τ (ζ ) ∈ Rn , such that ∇(d,Λ,τ ¯ ) Ψ d(ζ ), Λ(ζ ), τ (ζ ), ζ = 0. In fact, τ (ζ ) = 0,
(36)
288
M. Clapp et al. / Journal of Functional Analysis 256 (2009) 275–306
α(ζ ) F (0) d(ζ ) = a2 H (0, 0)α(ζ ) − kG2 (0, ζ ) 2 νG(0, ζ ) N−2 d(ζ ). Λ(ζ ) = α(ζ )
1 2(N−2)
,
If follows from (24) and (26) that conditions (35) hold if, either ν = 1, α(ζ ) > 0 and ϕ(ζ ) > 0, or if ν = −1 and α(ζ ) < 0. An easy computation shows that 1/2 Ψ d(ζ ), Λ(ζ ), τ (ζ ), ζ = 2 a2 F (0)k ϕ(ζ )1/2 . Therefore, if ζ ∗ is a C 1 -stable critical point of ϕ satisfying (35) then, by (30), γ ζ ∗ is a C 1 -stable critical point of ϕ for all γ ∈ Γ1 and, by (36), (d(ζ ∗ ), Λ(ζ ∗ ), 0, ζ ∗ ) is a critical point of the 2 restriction of Ψ to the set Sη for some η > 0. Moreover, since D(d, ¯ ) Ψ (d(ζ ), Λ(ζ ), τ (ζ ), ζ ) is Λ,τ invertible, the critical point (d(ζ ∗ ), Λ(ζ ∗ ), 0, ζ ∗ ) is C 1 -stable. This concludes the proof.
2
¯ τ, ζ¯ ) is a critical point of the restriction Lemma 13. If (d, Λ, Jε∗ |(0,∞)k+1 ×(Rn ×{0})×(Ω∩(Rn ×{0}))k then u = Vλ,ζ + φ is a Γ2 -invariant solution to problem (1). Proof. It suffices to show that Jε∗ is Γ2 -invariant with respect to the Γ2 -action on (0, ∞)k+1 × ¯ τ, ζ¯ ) := (d, Λ, ¯ γ τ, γ ζ¯ ), where γ ζ¯ := (γ ζ1 , . . . , γ ζk ). Indeed, propRN × Ω k given by γ (d, Λ, erty (II) implies that the fixed point set of this action is (0, ∞)k+1 × (Rn × {0}) × (Ω ∩ (Rn × {0}))k . Therefore, by the principle of symmetric criticality [19,25], we conclude that, ¯ τ, ζ¯ ) is a critical point of the restriction if (d, Λ, Jε∗ |(0,∞)k+1 ×(Rn ×{0})×(Ω∩(Rn ×{0}))k , then it is a critical point of Jε∗ . The claim now follows from Lemma 8. To prove that Jε∗ is Γ2 -invariant first observe that, since Ω and ω are Γ2 -invariant, the domain Ω \ εω is Γ2 -invariant for every ε > 0, and one has an action of Γ2 on H01 (Ω \ εω) given by (γ u)(x) := u(γ −1 x). This action preserves the Sobolev and the Lp+1 norms, i.e. ∇(γ u)∇(γ v) = ∇u∇v and |γ u|p+1 = |u|p+1 Ω\εω
Ω\εω
Ω\εω
Ω\εω
for all γ ∈ Γ2 , u, v ∈ H01 (Ω \ εω). Therefore, the functional Jε defined in (20) is Γ2 -invariant with respect to this action, i.e. Jε (γ u) = Jε (u)
for all γ ∈ Γ2 , u ∈ H01 (Ω \ εω).
(37)
Secondly, we claim that for any γ ∈ Γ2 (φ, ψ) solves (℘λ,ζ )
⇔
(γ φ, γ ψ) solves (℘λ,γ ζ ),
(38)
M. Clapp et al. / Journal of Functional Analysis 256 (2009) 275–306
289
where problems (℘λ,ζ ) and (℘λ,γ ζ ) are defined in (11), and γ ζ := (γ ξ, γ ζ1 , . . . , γ ζk ). Indeed, first notice that Uλj ,γ ζj (x) = Uλj ,ζj γ −1 x =: γ Uλj ,ζj (x)
for all γ ∈ Γ2 ,
(39)
j = 0, . . . , k, ζ0 := ξ . Since
∇(γ θ )∇(γ v) = RN
∇(θ )∇(v) = p
RN
=p
p−1
Uλj ,ζj θ v
RN
(γ Uλj ,ζj )p−1 (γ θ )(γ v)
for all v ∈ Cc∞ RN ,
RN
λj ,ζj iff γ θ ∈ K λj ,γ ζj . Similar arguments show that ψ ∈ Kλ,ζ iff γ ψ ∈ Kλ,γ ζ , we have that θ ∈ K ⊥ ⊥ that φ ∈ Kλ,ζ iff γ φ ∈ Kλ,γ ζ , and that Lλ,ζ (φ) = Nλ,ζ (φ) + Rλ,ζ + ψ holds iff Lλ,γ ζ (γ φ) = Nλ,γ ζ (γ φ) + Rλ,γ ζ + γ ψ holds. Therefore (38) follows. This allows us to conclude Jε∗ is Γ2 -invariant. Indeed, since the solution (φ, ψ) to problem (℘λ,ζ ) in (11) is unique, (38) guarantees that (γ φ, γ ψ) is the unique solution to problem (℘λ,γ ζ ). It follows from (37) and (39) that ¯ γ τ, γ ζ¯ ) = Jε (Vλ,γ ζ + γ φ) = Jε γ (Vλ,ζ + φ) Jε∗ (d, Λ, ¯ τ, ζ¯ ) = Jε (Vλ,ζ + φ) = Jε∗ (d, Λ, as claimed.
for all γ ∈ Γ2 ,
2
To prove Theorem 2 we need the following topological lemma. Lemma 14. Let D be a connected bounded smooth domain in Rn , n 2, with nonconnected boundary. Then there exists a point x0 ∈ Rn \ D with the following property: for every v ∈ Rn , v = 0, there exist t2 > t1 > 0 such that x0 + t1 v and x0 + t2 v are in different components of ∂D, and x0 + tv ∈ D for every t ∈ (t1 , t2 ). Proof. Let K1 , . . . , Kk be the connected components of ∂D, k 2. Then Kj is an (n − 1)dimensional compact connected submanifold of Rn . By Alexander’s and Poincaré’s duality theorems [23, Chapter 6, Section 2, Theorems 16 and 18], 0 Rn \ Kj ; Z ∼ H = H n−1 (Kj ; Z) ∼ = H0 (Kj ; Z) ∼ = Z,
290
M. Clapp et al. / Journal of Functional Analysis 256 (2009) 275–306
where H∗ (·; Z) and H ∗ (·; Z) denote singular homology and cohomology with integer coefficients. Hence, Rn \ Kj has precisely two connected components Dj and Uj , with Dj bounded and Uj unbounded. Note that, if Dj ∩ Ds = ∅ and j = s then, either Dj ⊂ Ds , or Ds ⊂ Dj . Now, since D is bounded, it must be contained in one of the Dj ’s and, since D is connected, such Dj is unique. So, after reordering, we conclude that D = D1 \ (D2 ∪ · · · ∪ Dk ), Dj ⊂ D1
for all j = 2, . . . , k,
Dj ∩ Ds = ∅ for all j, s = 2, . . . , k, j = s. Let x0 ∈ D2 and let v ∈ Rn , v = 0. Define t1 := max{t > 0: x0 + tv ∈ D2 } and t2 := min{t > t1 : x0 + tv ∈ ∂D}. It is easy to check that they have the desired properties.
2
Proof of Theorem 2. Let Γ1 := {1}, and Γ2 := {1, −1} acting by multiplication on RN −n . We will prove that the function ϕ : (Ω \ {0}) ∩ (Rn × {0}) → R defined by ϕ(ζ ) := H (0, 0) −
G2 (0, ζ ) H (ζ, ζ )
has a critical point of mountain pass type ζ ∗ ∈ C, which is stable with respect to C 1 -perturbations, such that ϕ(ζ ∗ ) > 0 if 0 is close enough to ∂Ω. Note that, in this case, α(ζ ) := H (ζ, ζ ) > 0 for all ζ . The claim then follows from Theorem 12. First note that, since Ω is Γ -invariant and Rn × {0} is the fixed point set of the Γ -action on RN , the normal to ∂Ω at each point x ∈ ∂Ω ∩ (Rn × {0}) lies in Rn × {0}. Hence, Ω ∩ (Rn × {0}) is a bounded smooth domain in Rn × {0}. Consider the function f : (Ω \ {0}) ∩ (Rn × {0}) → R defined by f (ζ ) :=
G2 (0,ζ ) H (ζ,ζ )
0
if ζ ∈ (Ω \ {0}) ∩ (Rn × {0}), if ζ ∈ ∂Ω ∩ (Rn × {0}).
Note that f (ζ ) → 0 as dist(ζ, ∂Ω) → 0. Let C0 be the connected component of Ω ∩ (Rn × {0}) containing 0. We consider two cases. Case 1: C = C 0 . Fix two points ξ1 , ξ2 ∈ ∂C in different connected components of ∂C, and consider the set Θ := σ ∈ C 0 [0, 1], C : σ (0) = ξ1 , σ (1) = ξ2 . It is not difficult to check that there exists ζ ∗ ∈ C such that f (ζ ∗ ) = inf max f σ (t) σ ∈Θ t∈[0,1]
M. Clapp et al. / Journal of Functional Analysis 256 (2009) 275–306
291
Fig. 3.
and ζ ∗ is a critical point of mountain pass type of the function f which is stable with respect to C 1 -perturbation (see [12]). Therefore, ζ ∗ is a C 1 -stable critical point of the function ϕ. Now, let us estimate f (ζ ∗ ). Since Ω ∩ (Rn × {0}) is a bounded smooth domain in Rn × {0}, we have that C ∩ C0 = ∅. Hence, rC := dist(C, C0 ) > 0. By (24) and (26), there is a constant a := f (ζ ) =
2 κN HΩ
> 0 such that, for any ζ ∈ C,
G2 (0, ζ ) −2(N −2) . a|ζ |−2(N −2) arC H (ζ, ζ )
In particular, −2(N −2)
f (ζ ∗ ) arC
.
Therefore, by (25), there exists ρ0 > 0, depending only on Ω, such that −2(N −2)
ϕ(ζ ∗ ) H (0, 0) − arC
>0
if dist(0, ∂Ω) < ρ0 . Case 2: C = C 0 . Let x0 ∈ (Rn × {0}) \ C0 be as in Lemma 14, choose v ∈ Rn × {0}, v = 0, orthogonal to x0 , and let t2 > t1 > 0 be such that ξ1 := x0 + t1 v and ξ2 := x0 + t2 v lie in different components of ∂C0 and x0 + tv ∈ C0 for every t ∈ (t1 , t2 ), see Fig. 3. Consider the set Θ := σ ∈ C 0 [0, 1], C0 \ {0} : σ (0) = ξ1 , σ (1) = ξ2 . As in the previous case, there exists ζ ∗ ∈ C such that f (ζ ∗ ) = inf max f σ (t) , σ ∈Θ t∈[0,1]
ζ ∗ is a C 1 -stable critical point of the function f and, hence, also of ϕ.
292
M. Clapp et al. / Journal of Functional Analysis 256 (2009) 275–306
To estimate f (ζ ∗ ), set r0 := dist(x0 , C0 ) > 0, and consider the path τ ∈ Θ given by τ (t) := (1 − t)ξ1 + tξ2 , t ∈ [0, 1]. From (24) and (26), since x0 is orthogonal to v, we obtain that −2(N −2) G2 (0, τ (t)) −2(N −2) f τ (t) = a|x0 |−2(N −2) ar0 a τ (t) H (τ (t), τ (t)) with a :=
2 κN HΩ
> 0 and, consequently, f (ζ ∗ ) max f τ (t) ar0−2(N −2) . t∈[0,1]
So, by (25), there exists ρ0 > 0, depending only on Ω, such that −2(N −2)
ϕ(ζ ∗ ) H (0, 0) − ar0 if dist(0, ∂Ω) < ρ0 . This concludes the proof.
>0
2
Remark 15. Observe that Theorem 2 remains true if instead of (4) and (5) we assume that Ω and ω are Γ2 -invariant for some closed subgroup Γ2 of O(N − n) satisfying property (II) above. Proof of Theorem 1. Since Ω is assumed to be connected, Theorem 1 follows from Theorem 2 taking n = N . 2 Proof of Theorem 3. Let Γ1 := {e2πij/k ∈ C: j = 0, . . . , k − 1}, acting on R2 ≡ C by complex multiplication, and let Γ2 := {1, −1}, acting by multiplication on RN −2 . For every ζ ∈ C with |ζ | ∈ ( 12 , 1) using (24) we obtain G(0, ζ ) κN
1 2κN =: c1 , |ζ |
and, for j = 1, . . . , k − 1, G ζ, e2πij/k ζ κN
2κN 1 =: c2 . |ζ − e2πij/k ζ | |1 − e2πi/k |
The Robin function H depends on r and ρ. Nevertheless it is not difficult to check that lim H (0, 0)
r+ρ→1
min
|ζ |∈(r+ρ,1)
H (ζ, ζ ) − (k − 1)c2 = +∞.
Consequently, there exists ρ0 ∈ ( 12 , 1) such that, if r + ρ ∈ (ρ0 , 1) then
M. Clapp et al. / Journal of Functional Analysis 256 (2009) 275–306
α(ζ ) = H (ζ, ζ ) −
293
k−1
G ζ, e2πij/k ζ j =1
min
|ζ |∈(r+ρ,1)
H (ζ, ζ ) − (k − 1)c2 > 0 for all |ζ | ∈ (r + ρ, 1),
and ϕ(ζ ) = H (0, 0) − H (0, 0) −
kG2 (0, ζ ) α(ζ ) kc12 min|ζ |∈(r+ρ,1) H (ζ, ζ ) − (k − 1)c2
> 0 for all |ζ | ∈ (r + ρ, 1). Let C := {ζ ∈ Ω ∩ (R2 × {0}): |ζ | ∈ (r + ρ, 1)}. Arguing as in the proof of Theorem 2, we prove that the function 2 kG (0,ζ ) if ζ ∈ C, α(ζ ) f (ζ ) := 0 if ζ ∈ ∂C has a critical point ζ ∗ ∈ C which is stable with respect to C 1 -perturbation. Therefore, ζ ∗ is a C 1 -stable critical point of ϕ with α(ζ ∗ ) > 0 and ϕ(ζ ∗ ) > 0. Since Ω is O(2)-invariant we may take ζ ∗ := ρ∗ (1, 0, . . . , 0). The result now follows from Theorem 12. 2 Theorems 4 and 5 are special cases of the following result. Theorem 16. Assume that Ω is Γ -invariant and ω is Γ2 -invariant, and that |Γ1 | 2. Then there exists ε0 > 0 such that, for each ε ∈ (0, ε0 ), there exists a pair ±uε of Γ2 -invariant sign changing solutions to problem (1) satisfying
in the sense of measures, as ε → 0, δγ ζ ∗ |∇uε |2 dx AN δ0 − γ ∈Γ1
for some ζ ∗ ∈ (Ω \ {0}) ∩ (Rn × {0}). Proof. Since Γ1 acts without fixed points on Sn−1 := {x ∈ Rn : |ζ | = 1}, one has that min |ζ − γ ζ | = a0 > 0.
min
ζ ∈Sn−1 γ ∈Γ1 \{1}
Hence, for every γ ∈ Γ1 \ {1} and every ζ ∈ (Ω \ {0}) ∩ (Rn × {0}), we obtain that G(ζ, γ ζ )
κN κN N −2 N −2 |ζ − γ ζ | a0 |ζ |N −2
and, therefore, α(ζ ) := H (ζ, ζ ) −
γ ∈Γ1 \{1}
G(ζ, γ ζ ) H (ζ, ζ ) −
κN . N −2 a0 |ζ |N −2
294
M. Clapp et al. / Journal of Functional Analysis 256 (2009) 275–306
This, together with (25), implies that lim α(ζ ) = −∞
|ζ |→0
and
lim α(ζ ) = +∞.
ζ →∂Ω
Let O := ζ ∈ Ω \ {0} ∩ Rn × {0} : α(ζ ) < 0 . Then O is open in Rn and |Γ1 |G2 (0, ζ ) H (0, 0). inf ϕ(ζ ) = inf H (0, 0) − α(ζ ) ζ ∈O ζ ∈O Since ϕ(ζ ) → +∞ as dist(ζ, ∂O) → 0, there exists ζ ∗ ∈ O such that ϕ(ζ ∗ ) = inf ϕ(ζ ). ζ ∈O
ζ ∗ is a C 1 -stable critical point of ϕ with ϕ(ζ ∗ ) < 0. The result now follows from Theorem 12. 2 Proof of Theorem 4. Apply Theorem 16 with n = N , Γ1 = {−1, 1} acting by multiplication on RN , and Γ2 = {1}. 2 Proof of Theorem 5. If N is odd, apply Theorem 16 with n = 2, Γ1 := {e2πij/k ∈ C: j = 0, . . . , k − 1} acting on R2 ≡ C by complex multiplication, and Γ2 := {1, −1} acting by multiplication on RN −2 . If N is even, apply Theorem 16 with n = N , Γ1 := {e2πij/k ∈ C: j = 0, . . . , k − 1} acting on RN ≡ CN/2 by complex multiplication, and Γ2 := {1}. 2 4. The expansion of the energy This section is devoted to prove Proposition 10. First, we describe the asymptotic expansion of the projection of the standard bubble centered at a point which is inside the hole of our domain. The following result holds (see [17, Lemma 2.1]). Lemma 17. Problem ⎧ ⎨ −u = 0 in RN \ ω, u = 1 on ∂ω, ⎩ u ∈ D1,2 RN \ ω
(40)
M. Clapp et al. / Journal of Functional Analysis 256 (2009) 275–306
295
has a unique solution ϕω . Moreover, c2 c1 ϕω (x) N −2 N −2 |x| |x|
∀x ∈ RN \ ω
for some positive constants c1 , c2 . Furthermore, lim
|x|→+∞
|x|N −2 ϕω (x) = cω
with cω =
1 (N − 2)|S N −1 |
Observe that, if ω = B(0, 1) then ϕω (x) = Lemma 2.2]).
∇ϕω (x)2 dx.
RN \ω
1 . |x|N−2
The following expansion holds (see [17,
Lemma 18. Let N−2 ε Rd,τ (x) := Pε Uμ,ξ (x) − Uμ,ξ (x) + αN μ 2 H (x, ξ ) + αN
x . ϕ N−2 N−2 ω ε μ 2 (1 + |τ |2 ) 2 1
Then there exists a positive constant c such that for any x ∈ Ω \ εω
N−1 ε 2 +ε if N 4, |x|N −2
ε √ R (x) cε 14 ε + ε if N = 3, d,τ |x|
N−1 ε 2 ∂d R ε (x) cε N−2 4 +ε if N 4, d,τ |x|N −2
√ ∂d R ε (x) cε 14 ε + ε if N = 3 d,τ |x|
N−2 2 N−3 ∂τ R ε (x) cε N4 ε 2 if N 3, +ε i d,τ |x|N −2
ε R (x) cε N−2 4 d,τ
(41) (42) (43) (44) (45)
ε solves −R = 0 in Ω \ εω with Proof. The function R := Rd,τ
R(x) = αN −
μ
N−2 2
(μ2 + |x − ξ |2 )
N−2 2
N−2 x μ 2 1 , + + N−2 ϕ N−2 ω N −2 ε |x − ξ | μ 2 (1 + |τ |2 ) 2
x ∈ ∂Ω,
R(x) = αN −
μ
N−2 2
(μ2 + |x − ξ |2 )
N−2 2
+μ
N−2 2
H (x, ξ ) +
1 μ
N−2 2
(1 + |τ |2 )
, N−2 2
x ∈ ∂εω.
296
M. Clapp et al. / Journal of Functional Analysis 256 (2009) 275–306
Therefore (41) and (42) follow, because ε−
N−2 4
N−2 R(x) = O ε + ε 2 ,
and ε −
x ∈ ∂Ω
N−2 4
N−3 R(x) = O ε − 2 ,
x ∈ ∂εω.
ε (x) solves −R = 0 in Ω \ εω with The function Rd (x) = ∂d Rd,τ d
μ2 − |x − ξ |2 N − 2 N−4 1 (x − ξ, τ ) 2 2 μ Rd (x) = αN ε − 2μ N N 2 (μ2 + |x − ξ |2 ) 2 (μ2 + |x − ξ |2 ) 2
x 1 (x − ξ, τ ) 1 , x ∈ ∂Ω, + + 2μ − ϕ ω N−2 N N −2 |x − ξ | ε |x − ξ | μN −2 (1 + |τ |2 ) 2 μ2 − |x − ξ |2 N − 2 N−4 1 (x − ξ, τ ) 2 2 Rd (x) = αN μ ε − 2μ N N 2 2 2 2 (μ + |x − ξ | ) 2 (μ + |x − ξ |2 ) 2 2μ 1 , x ∈ ∂εω. ∇y H (x, ξ ), τ − + H (x, ξ ) + N−2 N −2 μN −2 (1 + |τ |2 ) 2 Therefore (43) and (44) follow, because ε−
N−2 4
N−2 Rd (x) = O ε + ε 2 ,
and ε −
x ∈ ∂Ω
N−2 4
N−3 Rd (x) = O ε − 2 ,
x ∈ ∂εω.
ε (x) solves −R = 0 in Ω \ εω with The function Ri (x) = ∂τi Rd,τ i
Ri (x) = αN (N − 2)μ
N 2
(x − ξ )i N
(μ2 + |x − ξ |2 ) 2
x ∈ ∂Ω, Ri (x) = αN (N − 2)μ
N 2
(x − ξ )i N
(μ2 + |x − ξ |2 ) 2
−
(x − ξ )i N
|x − ξ | 2
x , − N −1 ϕω N ε μ (1 + |τ |2 ) 2 1
τi
∂yi H (x, ξ ) 1 τi , − N −1 + N N −2 μ (1 + |τ |2 ) 2
x ∈ ∂εω. Therefore (45) follows, because N N−3 ε − 4 Ri (x) = O ε + ε 2 , This finishes the proof.
x ∈ ∂Ω
N−2 N and ε − 4 Ri (x) = O ε − 2 ,
x ∈ ∂εω.
2
The asymptotic expansion of the projection of the standard bubble centered at a point inside the domain is, by now, a standard fact. We refer the reader to [22]. We state the result in the following. Lemma 19. Let η > 0 be fixed. If (8) and (9) hold, then the following facts hold true. Let ε (x) := Pε Uλ,ζ (x) − Uλ,ζ (x) + αN λ rΛ,ζ
N−2 2
H (x, ζ ).
M. Clapp et al. / Journal of Functional Analysis 256 (2009) 275–306
297
Then, for any x ∈ Ω \ εω, ε (x) cλ 0 rΛ,ζ
N+2 2
(46)
,
for some positive and fixed constant c. Furthermore, for any x ∈ Ω \ εω ∂Λ r ε (x) cε N+2 4 Λ,ζ
if N 4,
∂Λ r ε (x) cε 34 Λ,ζ
if N = 3
(47)
and for i = 1, . . . , N ∂ζ r ε (x) cε N+2 4 , i Λ,ζ
(48)
for some positive and fixed constant c. We have now all the elements needed to perform the expansion (27). Proof of Proposition 10. For the sake of simplicity, we will prove estimate (27) when k = 1. Let η > 0 be fixed and assume (7)–(9) hold with λ, ζ instead of λ1 , ζ1 . We will prove that Jε (Pε Uμ,ξ ± Pε Uλ,ζ )
N −2 ε = 2a1 − F (τ ) 1 + o(1) μ N−2 N−2 − a2 H (0, 0)μN −2 + H (ζ, ζ )λN −2 ∓ 2G(0, ζ )λ 2 μ 2 1 + o(1) ,
(49)
uniformly in the C 1 -sense for (τ, ζ, d, Λ) satisfying (7)–(9). The constants that appear in (49) are given by p+1
1 dy, (1 + |y|2 )N
a1 := αN
RN
1 p+1 a2 := αN 2
1
RN
(1 + |y|2 )
N+2 2
(50)
(51)
dy,
We have that Jε (Pε Uμ,ξ ± Pε Uλ,ζ ) 1 ∇(Pε Uμ,ξ ± Pε Uλ,ζ )2 − 1 = 2 p+1 Ω\εω
=
1 2
p
Pε Uμ,ξ Uμ,ξ + Ω\εω
1 2
|Pε Uμ,ξ ± Pε Uλ,ζ |p+1
Ω\εω
p
Pε Uλ,ζ Uλ,ζ ± Ω\εω
Ω\εω
p
Pε Uλ,ζ Uμ,ξ
298
M. Clapp et al. / Journal of Functional Analysis 256 (2009) 275–306
1 − p+1 1 = N
|Pε Uμ,ξ ± Pε Uλ,ζ |p+1 Ω\εω
p+1 Uμ,ξ
Ω\εω
+
p+1
Uλ,ζ + Ω\εω
−
p
(Pε Uμ,ξ − Uμ,ξ )Uμ,ξ Ω\εω
1 N
1 + 2
1 p+1
1 2
p
(Pε Uλ,ζ − Uλ,ζ )Uλ,ζ ± Ω\εω
p
Pε Uλ,ζ Uμ,ξ
Ω\εω
p+1 p+1 |Pε Uμ,ξ ± Pε Uλ,ζ |p+1 − Uμ,ξ − Uλ,ζ .
(52)
Ω\εω
Now, setting x = μy we obtain
p+1
p+1
Uμ,ξ = αN Ω\εω
(μ2 Ω\εω
p+1
= αN
Ω\εω μ
p+1
= αN
RN
μN dx + |x − ξ |2 )N
1 dy (1 + |y − τ |2 )N
1 dy + O (1 + |y|2 )N
N ε + μN . μ
(53)
By Lemma 18 we have that
p
(Pε Uμ,ξ − Uμ,ξ )Uμ,ξ dx Ω\εω p+1 = −αN
μ
N−2 2
Ω\εω
+
N+2 x μ 2 H (x, ξ ) + N−2 ϕ dx N−2 ω N+2 ε μ 2 (1 + |τ |2 ) 2 (μ2 + |x − ξ |2 ) 2 1
p
ε Rd,τ Uμ,ξ dx.
(54)
Ω\εω
Now, setting x − ξ = μy we have μ Ω\εω
N−2 2
= Ω\εω−ξ μ
H (x, ξ )
μ
N+2 2
(μ2 + |x − ξ |2 )
μN −2 H (μy + ξ, ξ )
N+2 2
dx
1 (1 + |y|2 )
N+2 2
dy
M. Clapp et al. / Journal of Functional Analysis 256 (2009) 275–306
= μN −2 H (0, 0)
1
RN
(1 + |y|2 )
N+2 2
dy + o(1) .
299
(55)
Moreover, we get
1 μ
N−2 2
=
(1 + |τ |2 )
N−2 2
Ω\εω
1 (1 + |τ |2 )
N+2 x μ 2 ϕω dx ε (μ2 + |x − ξ |2 ) N+2 2
N−2 2
ϕω Ω\εω−ξ μ
N −2 1 ε = N−2 μ (1 + |τ |2 ) 2 =
μ 1 (y + τ ) dy N+2 ε (1 + |y|2 ) 2 fε (y)
Ω\εω−ξ μ
1 1 dy N+2 N −2 |y + τ | (1 + |y|2 ) 2
N −2
1 1 1 ε cω dy + o(1) . N−2 μ |y + τ |N −2 (1 + |y|2 ) N+2 2 (1 + |τ |2 ) 2
(56)
RN
Here we have set fε (y) := ( με )N −2 |y + τ |N −2 ϕω ( με (y + τ )) and applied Lebesgue’s dominated convergence theorem and Lemma 17. Therefore
p
(Pε Uμ,ξ − Uμ,ξ )Uμ,ξ dx Ω\εω
N −2 ε p+1 1 + o(1) , = −αN c3 H (0, 0)μN −2 1 + o(1) − F (τ ) μ
(57)
where c3 := RN
1 (1 + |y|2 )
N+2 2
dy
and p+1 F (τ ) := αN cω
1 (1 + |τ |2 )
N−2 2
RN
1 1 dy. |y + τ |N −2 (1 + |y|2 ) N+2 2
A standard computation proves
p+1
p+1
Uλ,ζ = αN Ω\εω
and also
RN
1 p+1 dy + O λN = αN c1 + O λN 2 N (1 + |y| )
(58)
300
M. Clapp et al. / Journal of Functional Analysis 256 (2009) 275–306
p
(Pε Uλ,ζ − Uλ,ζ )Uλ,ζ dx Ω\εω
p+1
= −αN
λ
N−2 2
H (x, ζ )
Ω\εω
p+1 = −αN λN −2 H (ζ, ζ )
λ (λ2
+ |x
N+2 2
− ζ |2 )
(1 + |y|2 )
p
rε,λ Uλ,ζ dx
Ω\εω
dy + o(1)
1
RN
N+2 2
dx +
N+2 2
p+1 = −αN c3 λN −2 H (ζ, ζ ) 1 + o(1) .
(59)
Now we have to estimate the interaction. Setting μy = x − ξ we obtain
p
Uμ,ξ Pε Uλ,ζ dx Ω\εω
p+1 = αN
μ
N+2 2
(μ2 + |x − ξ |2 )
Ω\εω
λ
×
N−2 2
−λ
N−2 2
(λ2 + |x − ζ |2 ) p+1 N−2 N−2 = αN λ 2 μ 2
N−2 2
(1 + |y|2 )
1
×
H (x, ζ ) + rε,λ (x) dx
1
{Ω\εω}−ξ μ
N+2 2
N−2
N+2 2
− H (μy + ξ, ζ ) dy
(λ2 + |μy + ξ − ζ |2 ) 2 1 p+1 N−2 r (μy + ξ ) dy + αN μ 2 N+2 ε,μ 2) 2 (1 + |y| {Ω\εω}−ξ μ
p+1
= αN λ
N−2 2
μ
N−2 2
G(ζ, ξ ) RN
p+1
= αN c3 λ
N−2 2
μ
N−2 2
1 (1 + |y|2 )
N+2 2
N−2 N−2 +o λ 2 μ 2
G(ζ, ξ ) 1 + o(1) .
(60)
It remains to estimate the term 1 p+1 p+1 |Pε Uμ,ξ ± Pε Uλ,ζ |p+1 − Uμ,ξ − Uλ,ζ . p+1 Ω\εω
Let η > 0 be fixed such that B(0, η) ∩ B(ζ, η) = ∅. If ε is small enough then εω ⊂ B(0, η) and we can write
M. Clapp et al. / Journal of Functional Analysis 256 (2009) 275–306
301
p+1 p+1 |Pε Uμ,ξ ± Pε Uλ,ζ |p+1 − Uμ,ξ − Uλ,ζ
Ω\εω
=
... +
B(0,η)\εω
... +
B(ζ,η)
(61)
....
Ω\{B(0,η)∪B(ζ,η)}
It is easy to check that
p+1 p+1 Uμ,ξ + Uλ,ζ
... = O Ω\{B(0,η)∪B(ζ,η)}
Ω\{B(0,η)∪B(ζ,η)}
= O μN + λN .
(62)
Via a Taylor expansion we have, for some t ∈ [0, 1],
p+1 p+1 |Pε Uμ,ξ ± Pε Uλ,ζ |p+1 − Uμ,ξ − Uλ,ζ dx
B(0,η)\εω
Uμ,ξ + (Pε Uμ,ξ − Uμ,ξ ± Pε Uλ,ζ )p+1 − U p+1 dx − μ,ξ
= B(0,η)\εω
p+1
Uλ,ζ dx
B(0,η)\εω
p
Uμ,ξ (Pε Uμ,ξ − Uμ,ξ ± Pε Uλ,ζ ) dx
= (p + 1) B(0,η)\εω
+
p(p + 1) 2
Uμ,ξ + t (Pε Uμ,ξ − Uμ,ξ ± Pε Uλ,ζ )p
B(0,η)\εω
p+1
× (Pε Uμ,ξ − Uμ,ξ ± Pε Uλ,ζ ) dx −
Uλ,ζ dx.
B(0,η)\εω
Setting again μy = x − ξ , we have that
p
Uμ,ξ Pε Uλ,ζ dx B(0,η)\εω p+1
μ
= αN
B(0,η)\εω
×
λ
N+2 2
(μ2 + |x − ξ |2 )
N−2 2
(λ2 + |x − ζ |2 )
p+1 N−2 N−2 = αN λ 2 μ 2
N−2 2
−λ
{B(0,η)\εω}−ξ μ
N+2 2
N−2 2
H (x, ζ ) + rε,λ (x) dx 1
(1 + |y|2 )
N+2 2
(63)
302
M. Clapp et al. / Journal of Functional Analysis 256 (2009) 275–306
1
×
N−2
− H (μy + ξ, ζ ) dy
(λ2 + |μy + ξ − ζ |2 ) 2 1 p+1 N−2 + αN μ 2 r (μy + ξ ) dy N+2 ε,μ 2) 2 (1 + |y| {B(0,η)\εω}−ξ μ
p+1
= αN λ
N−2 2
μ
N−2 2
G(ζ, ξ ) RN
p+1
= αN c3 λ
N−2 2
μ
N−2 2
1 (1 + |y|2 )
N+2 2
N−2 N−2 +o λ 2 μ 2
G(ζ, ξ ) 1 + o(1) .
(64)
! p The term B(0,η)\εω Uμ,ξ (Pε Uμ,ξ − Uμ,ξ ) dx was estimated in (57). The remaining terms are of lower order. In a similar way, via a Taylor expansion we have, for some t ∈ [0, 1],
p+1 p+1 |Pε Uμ,ξ ± Pε Uλ,ζ |p+1 − Uμ,ξ − Uλ,ζ dx
B(ζ,η)
Uλ,ζ + (Pε Uλ,ζ − Uλ,ζ ± Pε Uμ,ξ )p+1 − U p+1 dx − λ,ζ
= B(ζ,η)
p+1
Uμ,ξ dx
B(ζ,η)
p
= (p + 1)
Uλ,ζ (Pε Uλ,ζ − Uλ,ζ ± Pε Uμ,ξ ) dx
B(ζ,η)
+
p(p + 1) 2
Uλ,ζ + t (Pε Uλ,ζ − Uλ,ζ ± Pε Uμ,ξ )p
B(ζ,η)
×(Pε Uλ,ζ − Uλ,ζ ± Pε Uμ,ξ ) dx −
p+1
Uμ,ξ dx.
B(ζ,η)
Setting now λy = x − ζ , we have
p
Uλ,ζ Pε Uμ,ξ dx B(ζ,η) p+1
= αN
B(ζ,η)
×
λ
N+2 2
(λ2 + |x − ζ |2 ) μ
N−2 2
(μ2 + |x − ξ |2 ) N−2 p+1 N−2 2 2 μ = αN λ
N−2 2
B(0,η/λ)
N+2 2
−μ
N−2 2
x + Rε,μ (x) dx H (x, ξ ) − N−2 ϕω ε μ 2 1
1 (1 + |y|2 )
N+2 2
(65)
M. Clapp et al. / Journal of Functional Analysis 256 (2009) 275–306
×
1 N−2
303
− H (λy + ζ, ξ ) dy
(μ2 + |λy + ζ − ξ |2 ) 2
1 λy + ζ 1 p+1 N−2 + R ϕ (λy + ζ ) dy − αN λ 2 ε,μ N+2 N−2 ω ε (1 + |y|2 ) 2 μ 2 B(0,η/λ) N−2 N−2 1 p+1 N−2 N−2 = αN λ 2 μ 2 G(ζ, ξ ) +o λ 2 μ 2 N+2 (1 + |y|2 ) 2 N R
N−2 N−2 p+1 = αN c3 λ 2 μ 2 G(ζ, ξ ) 1 + o(1) .
(66)
! p The term B(ζ,η) Uλ,ζ (Pε Uλ,ζ − Uλ,ζ ) dx was estimated in (59). Collecting all the previous estimates, we get that expansion (49) holds true uniformly for (d, Λ, τ, ζ ) satisfying (7)–(9). Arguing in a similar way and using estimates (44), (47) and (48), we prove that the expansion holds true also uniformly in the C 1 -sense. This proves our claim. 2 5. The associated nonlinear problem This section is devoted to prove Proposition 7. First, we estimate the ∗∗ -norm of Nˆ λ,ζ (ϑ). It is convenient, and sufficient for our purposes, to assume ϑ∗ < 1. In order to estimate Nˆ λ,ζ (ϑ)∗∗ we need to distinguish two cases: N 6 and N > 6. If N 6, then p 2 and we can estimate − 4 4 w N−2 Nˆ λ,ζ (ϑ) Cw (p−2)β+2β− N−2 ϑ2 , ∗
hence Nˆ λ,ζ (ϑ)
∗∗
Cϑ2∗ .
Assume now that N > 6. If |ϑ| 12 |Vˆλ,ζ |, we see directly that |Nˆ λ,ζ (ϑ)| C|ϑ|p and hence − 4 p w N−2 Nˆ λ,ζ (ϑ) Cw p−2 ϑp∗ Cε − N−6 2 ϑ . ∗ 1 Let us consider now the case |ϑ| 12 |Vˆλ,ζ |. In the region where dist(y, ∂Ωε ) δε − 2 for some δ > 0, one has that Vˆλ,ζ (y) αδ w(y) for some αδ > 0; hence in this region, we have
− 4 w N−2 Nˆ λ,ζ (ϑ) Cw 2β−1 ϑ2 Cε (2β−1) N−2 2 ϑ2 . ∗ ∗ 1 On the other hand, when dist(y, ∂Ωε ) δε − 2 , the following facts occur: w(y), Vˆλ,ζ (y) = N−2 O(ε 2 ), and
Vˆλ,ζ (y) = Cε
N−1 2
dist(y, ∂Ωε ) 1 + o(1) as y → ∂Ωε .
304
M. Clapp et al. / Journal of Functional Analysis 256 (2009) 275–306
This second assertion is a consequence of the fact that the Green function of the domain Ω vanishes linearly with respect to dist(x, ∂Ω) as x → ∂Ω. These two facts imply that, if 1 dist(y, ∂Ωε ) δε − 2 , then − 4 4 w N−2 Nˆ λ,ζ (ϑ) w − N−2 |Vˆλ,ζ |p−2 |ϑ|2 2 p−2 4 N−1 Cw − N−2 ε 2 dist(y, ∂Ωε ) dist(y, ∂Ωε )2Dϑ(y) 4
2
Cw − N−2 +2β+ N−2 ε
p N−1 2 (p−2)− 2
ϑ2∗ Cε −
N−4 2
ϑ2∗ .
Combining these relations we get 2 ∗ Nˆ λ,ζ (ϑ) CϑN−4 p ∗∗ C(ε − 2 ϑ2∗ + ε p−2 ϑ∗ ) Next we estimate the term Rˆ λ,ζ . In the region |y − √ζiε | >
if N 6, if N > 6.
√δ , for any i ε
= 0, 1, . . . , k and some
. Assume now that |y − √ζiε | positive small δ, direct computations show that |Rˆ λ,ζ | Cε for some i = 0, 1, . . . , k. Then, in this region, using either Lemma 18 or Lemma 19, we get N+2 2
|Rˆ λ,ζ | Cε Using the boundedness of
λi √ , ε
N−2 2
p−1
U λi √
ε
ζ
, √iε
√δ ε
.
we conclude that Rˆ λ,ζ ∗∗ Cε
N−2 2
(67)
.
˜ with Now, we are in position to prove that problem (14) has a unique solution φˆ = φ˜ + ψ, ˆ ˜ ψ := Tλ,ζ (Rλ,ζ ) (see Proposition 6), having the required properties. Problem (14) is equivalent to solving a fixed point problem. Indeed, φˆ = φ˜ + ψ˜ is a solution of (14) if and only if ˜ =: Aλ,ζ (φ), ˜ φ˜ = Tλ,ζ Nˆ λ,ζ (φ˜ + ψ) because ψ˜ = Tλ,ζ (R). We shall prove that the operator Aλ,ζ defined above is a contraction inside a properly chosen region. First observe that, from the definition of ψ˜ , from (67) and from Proposition 6, we infer that N−2 ˜ ∗∗ C |λ log ε| + ε 2 ψ and for ϑ∗ 1, Nˆ λ,ζ (ψ˜ + ϑ)
∗∗
C(ϑ2∗ + ε N −2 ) N−4 N p C(ε − 2 ϑ2∗ + ε p−2 ϑ∗ + ε 2 )
if N 6, if N > 6.
(68)
M. Clapp et al. / Journal of Functional Analysis 256 (2009) 275–306
305
Let us set N−2 Fε := ϑ ∈ H01 (Ωε ): ϑ∗ δε 2 . From Proposition 6 and (68) we conclude that, for ε sufficiently small and any ϑ ∈ Fε we have Aλ,ζ (ϑ) ε N−2 2 . ∗ Now we will show that the map Aλ,ζ is a contraction for any ε small enough. That will imply that Aλ,ζ has a unique fixed point in Fε and, hence, that problem (14) has a unique solution. For any ϑ1 , ϑ2 in Fε we have Aλ,ζ (ϑ1 ) − Aλ,ζ (ϑ2 ) C Nˆ λ,ζ (ψ˜ + ϑ1 ) − Nˆ λ,ζ (ψ˜ + ϑ2 ) , ∗ ∗∗ hence we just need to check that Nˆ λ,ζ is a contraction in its corresponding norms. By definition of Nˆ λ,ζ Dϑ Nˆ λ,ζ (ϑ) = p f (Vˆλ,ζ + ϑ) − f (Vˆλ,ζ ) . Hence we get Nˆ λ,ζ (ψ˜ + ϑ1 ) − Nˆ λ,ζ (ψ˜ + ϑ2 ) C Vˆ p−2 |ϑ||ϑ ¯ 1 − ϑ2 | λ,ζ
¯ ∗, for some ϑ¯ in the segment joining ψ˜ + ϑ1 and ψ˜ + ϑ2 . Hence, we get for small enough ϑ 4 ¯ ∗ ϑ1 − ϑ2 ∗ . ω− N−2 Nˆ λ,ζ (ψ˜ + ϑ1 ) − Nˆ λ,ζ (ψ˜ + ϑ2 ) Cε p−2+2β ϑ We conclude that there exists c ∈ (0, 1) such that Nˆ λ,ζ (ψ˜ + ϑ1 ) − Nˆ λ,ζ (ψ˜ + ϑ2 ) cϑ1 − ϑ2 ∗ . ∗∗ Arguing like in [11], we obtain the estimate (19). This concludes the proof. References [1] T. Aubin, Problemes isoperimetriques et espaces de Sobolev, J. Differential Geom. 11 (1976) 573–598. [2] A. Bahri, Critical Points at Infinity in Some Variational Problems, Pitman Res. Notes Math. Ser., vol. 182, Longman, Harlow, 1989. [3] A. Bahri, J.M. Coron, On a nonlinear elliptic equation involving the critical Sobolev exponent: The effect of the topology of the domain, Comm. Pure Appl. Math. 41 (1988) 253–294. [4] M. Ben Ayed, K. El Mehdi, M. Hammami, A nonexistence result for Yamabe type problems on thin annuli, Ann. Inst. H. Poincaré Anal. Non Linéaire 19 (2002) 715–744. [5] G. Bianchi, H. Egnell, A note on the Sobolev inequality, J. Funct. Anal. 100 (1991) 18–24. [6] L. Caffarelli, B. Gidas, J. Spruck, Asymptotic symmetry and local behaviour of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math. 42 (1989) 271–297. [7] M. Clapp, F. Pacella, Multiple solutions to the pure critical exponent problem in domains with a hole of arbitrary size, Math. Z. 259 (2008) 575–589. [8] M. Clapp, T. Weth, Minimal nodal solutions of the pure critical exponent problem on a symmetric domain, Calc. Var. Partial Differential Equations 21 (2004) 1–14.
306
M. Clapp et al. / Journal of Functional Analysis 256 (2009) 275–306
[9] M. Clapp, T. Weth, Two solutions of the Bahri–Coron problems in punctured domains via the fixed point transfer, Commun. Contemp. Math. 10 (2008) 81–101. [10] J.M. Coron, Topologie et cas limite des injections de Sobolev, C. R. Acad. Sci. Paris Ser. I Math. 299 (1984) 209–212. [11] M. del Pino, P. Felmer, M. Musso, Two-bubble solutions in the super-critical Bahri–Coron’s problem, Calc. Var. Partial Differential Equations 16 (2) (2003) 113–145. [12] H. Hofer, The topological degree at a critical point of mountain-pass type, in: Nonlinear Functional Analysis and Its Applications, part 1, Berkeley, CA, 1983, in: Proc. Sympos. Pure Math., vol. 45, Amer. Math. Soc., Providence, RI, 1986, pp. 501–509. [13] J. Kazdan, F.W. Warner, Remarks on some quasilinear elliptic equations, Comm. Pure Appl. Math. 28 (1975) 567– 597. [14] R. Lewandowski, Little holes and convergence of solutions of −u = u(N +2)/(N −2) , Nonlinear Anal. 14 (1990) 873–888. [15] G. Li, S. Yan, J. Yang, An elliptic problem with critical growth in domains with shrinking holes, J. Differential Equations 198 (2004) 275–300. [16] M. Musso, A. Pistoia, Sign changing solutions to a nonlinear elliptic problem involving the critical Sobolev exponent in pierced domains, J. Math. Pures Appl. 86 (2006) 510–528. [17] M. Musso, A. Pistoia, Persistence of Coron’s solutions in nearly critical problems, Ann. Sc. Norm. Super. Pisa Cl. Sci. 6 (2) (2007) 331–357. [18] M. Musso, A. Pistoia, Sign changing solutions to a Bahri–Coron’s problems in pierced domains, Adv. Differential Equations 21 (1) (2008) 295–306. [19] R. Palais, The principle of symmetric criticality, Comm. Math. Phys. 69 (1979) 19–30. [20] S.I. Pohožaev, On the eigenfunctions of the equation u + λf (u) = 0, Dokl. Akad. Nauk SSSR 165 (1965) 36–39 (in Russian). [21] O. Rey, Sur un probléme variationnel non compact: l’effect de petits trous dans le domain, C. R. Acad. Sci. Paris Sér. I Math. 308 (1989) 349–352. [22] O. Rey, The role of the Green’s function in a nonlinear elliptic equation involving the critical Sobolev exponent, J. Funct. Anal. 89 (1990) 1–52. [23] E.H. Spanier, Algebraic Topology, McGraw–Hill, New York, 1966. [24] G. Talenti, Best constants in Sobolev inequality, Ann. Mat. Pura Appl. 110 (1976) 353–372. [25] M. Willem, Minimax Theorems, Progr. Nonlin. Differential Equations Appl., vol. 24, Birkhäuser Boston, Boston, MA, 1996.
Journal of Functional Analysis 256 (2009) 307–322 www.elsevier.com/locate/jfa
A class of simple C∗-algebras with stable rank one ✩ George A. Elliott a , Toan M. Ho b , Andrew S. Toms b,∗ a Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 2E4 b Department of Mathematics and Statistics, York University, 4700 Keele St., Toronto, Ontario, Canada M3J 1P3
Received 13 March 2007; accepted 4 August 2008 Available online 8 November 2008 Communicated by Alain Connes
Abstract We study the limits of inductive sequences (Ai , φi ) where each Ai is a direct sum of full matrix algebras over compact metric spaces and each partial map of φi is diagonal. We give a new characterisation of simplicity for such algebras, and apply it to prove that the said algebras have stable rank one whenever they are simple and unital. Significantly, our results do not require any dimension growth assumption. © 2008 Elsevier Inc. All rights reserved. Keywords: AH algebras; Stable rank; Dimension growth
1. Introduction Let X and Y be compact Hausdorff spaces. A ∗-homomorphism φ : Mm C(X) → Mnm C(Y ) is called diagonal if there are n continuous maps λi : Y → X such that ✩
The authors were supported by the Natural Science and Engineering Council of Canada.
* Corresponding author. Fax: +1 416 736 5757.
E-mail addresses:
[email protected] (G.A. Elliott),
[email protected] (T.M. Ho),
[email protected] (A.S. Toms). 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.08.001
308
G.A. Elliott et al. / Journal of Functional Analysis 256 (2009) 307–322
⎛
f ◦ λ1 ⎜ 0 ⎜ φ(f ) = ⎜ ⎜ .. ⎝ . 0
0 f ◦ λ2 .. . 0
... ... .. . ...
⎞ 0 0 ⎟ ⎟ ⎟ .. ⎟ . . ⎠ f ◦ λn
The λi are called the eigenvalue maps or simply eigenvalues of φ. The multiset {λ1 , λ2 , . . . , λn } is called the eigenvalue pattern of φ and is denoted by ep(φ). This definition can be extended to ∗-homomorphisms φ:
n
m
Mni C(Xi ) → Mmj C(Yj ) j =1
i=1
by requiring, roughly, that each partial map φ ij : Mni C(Xi ) → Mmj C(Yj ) induced by φ be diagonal. (A precise definition can be found in Section 2.) C∗ -algebras obtained as limits of inductive systems (Ai , φi ) where Ai =
ni
Mni,j C(Xi,j )
j =1
and each φi is diagonal form a rich class. They include AF algebras, simple unital AT algebras (and hence the irrational rotation algebras) [6], Goodearl algebras [10], and some interesting examples of Villadsen and the third named author connected to Elliott’s program to classify amenable C∗ -algebras via K-theory [15,17]. The structure of these algebras is only well understood when they satisfy some additional conditions such as (very) slow dimension growth or the combination of real rank zero, stable rank one, and weak unperforation of the K0 -group— situations in which the strong form of Elliott’s classification conjecture can be verified [3,5,7–9]. In this paper we give a new characterisation of simplicity for AH algebras with diagonal connecting maps. As a consequence we are able to prove that such algebras have stable rank one whenever they are unital and simple. The significance of our result derives from the fact that we make no assumptions on the dimension growth of the algebras; we obtain a general theorem on the structure of algebras heretofore considered “wild.” As suggested by M. Rørdam in his recent ICM address, it is high time we became friends with such algebras, as opposed to treating them simply as a source of pathological examples. 2. Preliminaries 2.1. Basic notation We use Mn to denote the set of n × n complex matrices. Given a closed subset E of a compact metric space (X, d) and δ > 0 we set
Bδ (E) = x ∈ X d(E, x) < δ , and make the convention that Bδ (∅) = ∅.
G.A. Elliott et al. / Journal of Functional Analysis 256 (2009) 307–322
309
2.2. AH systems with diagonal maps Definition 2.1. We will say that a unital ∗-homomorphism φ:
n
Mni C(Xi ) → Mk C(Y ) ∼ = Mk ⊗ C(Y )
i=1
is diagonal if there exist natural numbers k1 , . . . , kn such that ι:
n
i ki
= k and ni |ki , an embedding
Mki → Mk ,
i=1
and diagonal maps φi : Mni C(Xi ) → Mki ⊗ C(Y ) such that φ=
n
φi .
i=1
(Notice that ki = 0 is allowed.) We will say that a unital ∗-homomorphism φ:
n
m
Mni C(Xi ) → Mmj C(Yj ) j =1
i=1
is diagonal if each restriction φj :
n
Mni C(Xi ) → Mmj C(Yj )
i=1
is diagonal. Let A be the limit of the inductive sequence (Ai , φi ), where Ai =
ki
Mni,t C(Xi,t ) ,
t=1
Xi,t is a connected compact metric space, and ni,t and ki are natural numbers. Define Ai,t := Mni,t C(Xi,t ) , Xi := Xi,1 Xi,2 · · · Xi,ki , and
(1)
310
G.A. Elliott et al. / Journal of Functional Analysis 256 (2009) 307–322
φi,j := φj −1 ◦ · · · ◦ φi . Let t,l φi,j : Mni,t C(Xi,t ) → Mnj,l C(Xj,l ) and l φi,j :
ki
Mni,t C(Xi,t ) → Mnj,l C(Yj,l )
t=1
denote the appropriate restrictions of φ. If each φi is unital and diagonal, then we refer to (Ai , φi ) as an AH system with diagonal maps. The limit algebra A will be called a diagonal AH algebra, and we will refer to (Ai , φi ) as a decomposition of A. t,l as a diagonal map from Mni,t (C(Xi,t )) Assume that A as above is diagonal. We will view φi,j t,l into the cut-down of Mnj,l (C(Xj,l )) by φi,j (1). For fixed i and j , we will denote by epij the t,l multiset which is the union, counting multiplicity, of the eigenvalue patterns of each φi,j ; epij is the eigenvalue pattern of φi,j ; an element of epij is an eigenvalue map of φi,j . For fixed i, j , t,l . and l, we will denote by eplij the multiset which is the union of the eigenvalue patters of each φi,j Let us now show that the bonding maps φi may be assumed to be injective. Let (Ai , φi ) be a decomposition for a diagonal AH algebra A as above. Fix i ∈ N and 1 t ki . For each j > i j,l and 1 l kj , Let Xi,t denote the closed subset of Xi,t which is the union of the images of the j,l j j j j +1 eigenvalue maps of φ t,l . Put X = l X , and X˜ i,t = j X . Since X ⊇ X , we have i,j
i,t
that X˜ i,t is closed subset of Xi,t . Define
i,t
i,t
i,t
i,t
A˜ i,t = Mni,t C(X˜ i,t ) and A˜ i =
ki
A˜ i,t .
t=1 t,l t,l Define diagonal maps φ˜ i,i+1 : A˜ i,t → A˜ i+1,l by replacing the eigenvalue maps of φi,i+1 with ˜ ˜ ˜ ˜ their restrictions to Xi+1,l . Define φi : Ai → Ai+1 in a manner analogous to the definition of φi . It follows that (A˜ i , φ˜ i ) is a diagonal AH system with limit A, and φ˜ i is injective by construction. We assume from here on that all bonding maps in diagonal AH systems are injective. One way to construct a simple diagonal AH algebra is to ensure that for each i ∈ N and x in a specified dense subset of Xi there is some j i such that for each l ∈ {1, . . . , kj } the diagonal t,l contains the eigenvalue map evx : Xj,l → Xi,t given by evx (y) = x. The next definition map φi,j gives and approximate version of this situation.
Definition 2.2. Say that a diagonal AH algebra A with decomposition (Ai , φi ) has the property P if for any i ∈ N, element f in Ai , > 0, and x ∈ Xi there exist j i and unitaries ul ∈ Aj,l , l ∈ {1, . . . , kj } such that
G.A. Elliott et al. / Journal of Functional Analysis 256 (2009) 307–322
t,l ul φ (f )u∗ − f (x) l i,j 0
311
0 < bl
for some appropriately sized bl . (Note that diag(f (x0 ), bl ) ∈ Aj,l .) We will prove in the sequel that A as in Definition 2.2 is simple if and only if it has property P. 2.3. A characterisation of simplicity Proposition 2.1 of [4] gives some necessary and sufficient conditions for the simplicity of an AH algebra. We will have occasion to apply these in the proof of our main result, and so restate the said proposition in the particular case of an AH system with diagonal injective maps. Proposition 2.3. Let (Ai , φi ) be an AH system with diagonal injective maps, and set A = limi→∞ (Ai , φi ). The following conditions are equivalent: (i) A is simple; (ii) For any positive integer i and any nonempty open subset U of Xi , there is a j0 i such that for every j j0 and l ∈ {1, . . . , kj } we have l −1 epij (U ) = Xj,l , where (eplij )−1 (U ) denotes the union of the sets λ−1 (U ), λ ∈ eplij ; (iii) For any nonzero element a in Ai , there is a j0 i such that for every j j0 , φij (a)(x) is not zero, for every x in Xj . 2.4. Paths between permutation matrices Given any permutation π ∈ Sn , let U [π] denote the permutation matrix in Mn corresponding to π , that is, U [π] is obtained from the identity of Mn by moving the ith row to the π(i)th row, for i ∈ {1, 2, . . . , n}. Any two permutation matrices are homotopic inside the unitary group U(Mn ) of Mn , but we want to define some particular homotopies for use in the sequel. Let π and σ be elements of Sn , viewed as permutations of the canonical basis vectors e1 , . . . , en of Cn . Let R = {eW,1 , . . . , eW,dim(W ) } be the set of basis vectors upon which π and σ agree, and choose γ ∈ Sn be such that γ σ (v) = γ π(v) = v,
∀v ∈ R.
Then U [γ ]U [σ ] and U [γ ]U [π] fix e1 , . . . , e|R| . Put W = span{e1 , . . . , e|R| }, and let V be the orthogonal complement of W . There are a canonical unital embedding of Mdim(W ) ⊕ Mdim(V ) into Mn and unitaries u, v ∈ U(Mdim(V ) ) such that U [γ ]U [π] = 1Mdim(W ) ⊕ v;
U [γ ]U [σ ] = 1Mdim(W ) ⊕ u.
Choose a homotopy g(t) between u and v inside U(Mdim(V ) )—g(0) = v and g(1) = u—and put u(t) = U [γ ]−1 1Mdim(V ) ⊕ g(t) .
312
G.A. Elliott et al. / Journal of Functional Analysis 256 (2009) 307–322
Then u(t) is a homotopy of unitaries between U [π] and U [σ ] such that u(t)(v) = U [π](v) = U [σ ](v),
∀v ∈ R, ∀t ∈ [0, 1].
2.5. Applications of Urysohn’s Lemma Lemma 2.4. Let σ ∈ Sn be given. There is a homotopy u : [0, 1] → U(Mn ) between u(0) = 1n and u(1) = U [σ ] which moreover has the following property: for any complex numbers λ1 , λ2 , . . . , λn such that λi = λσ (i) for every i = 1, 2, . . . , n, we have ⎛
λ1 ⎜0 ⎜ u(t) ⎜ ⎜ .. ⎝ . 0
0 λ2 .. . 0
... ... .. . ...
⎛ ⎞ λ1 0 ⎜0 0⎟ ⎜ ⎟ ∗ ⎜ .. ⎟ ⎟ u (t) = ⎜ .. ⎝ . ⎠ . λn 0
0 λ2 .. . 0
... ... .. . ...
⎞ 0 0⎟ ⎟ .. ⎟ ⎟, . ⎠ λn
∀t ∈ [0, 1].
Proof. Let us first consider the case that σ is a k-cycle. The hypotheses of the lemma guarantee that the desired conclusion holds already for t ∈ {0, 1}. Choose the homotopy u(t) as in Section 2.4 by using our given value of σ and setting π equal to the identity element of Sn . The hypothesis λi = λσ (i) , i ∈ {1, . . . , n}, implies that there is a λ ∈ C such that for each i ∈ {1, . . . , n} which is not fixed by σ we have λi = λ. In other words, if one decomposes diag(λ1 , . . . , λn ) into a direct sum of two diagonal matrices using the decomposition Cn = W ⊕ V —V and W as in Section 2.4—then the direct summand corresponding to V is scalar k × k matrix. By construction, u(t) = v(t) ⊕ 1n−k , with v(t) ∈ U(Mk ). It follows that u(t) commutes with diag(λ1 , . . . , λn ) for each t ∈ (0, 1). Now suppose that σ is any permutation on n letters, and write σ as a product of disjoint cycles: σ = σ1 σ2 . . . σl . For each j ∈ {1, . . . , l}, let ui (j ) denote the unitary path between U [id] and U [σj ], constructed as in Section 2.4. Now u(t) := u1 (t)u2 (t) · · · ul (t) is a path of unitaries with u(0) = U [id] and u(1) = U [σ ], and u(t) commutes with diag(λ1 , . . . , λn ) since each uj (t) does. 2 Lemma 2.5. Let σ be any permutation in Sn , and A, B disjoint nonempty closed subsets of a metric space X. Let λ1 , . . . , λn : X → C be continuous. Then, there exists a unitary v ∈ Mn (C(X)) such that (i) v(x) = 1n , ∀x ∈ A, (ii) v(x) = U [σ ], ∀x ∈ B, and (iii) v(x) commutes with diag(λ1 (x), . . . , λn (x)) whenever λi (x) = λσ (i) (x) for each i ∈ {1, . . . , n}. Proof. Find a unitary path u(t) connecting U [id] to U [σ ] using Lemma 2.4, so that u(t) commutes with diag(λ1 (x), . . . , λn (x)) for each t ∈ (0, 1) and each x ∈ X for which λi (x) = λσ (i) (x), i ∈ {1, . . . , n}. By Urysohn’s Lemma, there is a continuous map f : X → [0, 1] which is equal to
G.A. Elliott et al. / Journal of Functional Analysis 256 (2009) 307–322
313
zero on A and equal to one on B. It is straightforward to check that v(x) := u(f (x)) satisfies the conclusion of the lemma. 2 Lemma 2.6. Let Y be a closed subset of a normal space X, and let f : Y → Sn be continuous. Then, there is a continuous map f˜ from X to the n + 1 disk D n+1 which extends f . Proof. View D n+1 as the cube [0, 1]n+1 , with S n as its boundary. Then f (y) = f1 (y), . . . , fn+1 (y) , where each fi : Y → [0, 1] is continuous. Extend each fi to a continuous map f˜i : X → [0, 1], and put f˜(y) = f˜1 (y), . . . , f˜n+1 (y) .
2
3. The main theorem Let A be a diagonal AH algebra with decomposition (Ai , φi ), and assume that the φi are injective. In this section we will prove that if A is simple then it has the property P (cf. Section 2). (The converse also holds, but is easier by far.) Let us begin with an outline of our strategy, before plunging headlong into the proof. Assume first that A is simple and Ai = Mni (C(Xi )), so that there are no partial maps to contend with. Let there be given a natural number i, an element f of Ai , a point x0 ∈ Xi , and some > 0. Put U = B (x0 ). By Proposition 2.3 there is a j0 with j0 i such that for any j j0 , −1 −1 Xj = λ−1 1 (U ) ∪ λ2 (U ) ∪ · · · ∪ λn (U ),
where ⎛
f ◦ λ1 ⎜ .. φi,j (f ) = ⎝ . 0
0 .. . 0
... .. . ...
⎞ 0 .. ⎟ . . ⎠ f ◦ λn
On each closed subset λ−1 t (U ), the range of the eigenvalue map λt is within of x0 . To show that A has the property P, we require a unitary u in Aj and an element bf ∈ Mnj −ni (C(Xj )) such that uφi,j (f )u∗ − f (x0 ) 0 < . 0 bf We would like u(y) to exchange the first and tth diagonal entries of φi,j (f ) whenever y ∈ −1 −1 λ−1 t (U ), but this operation is unlikely to be well-defined—the sets λ1 (U ), . . . , λn (U ) need not be mutually disjoint. The remainder of this section is devoted to overcoming this complication.
314
G.A. Elliott et al. / Journal of Functional Analysis 256 (2009) 307–322
Given positive integers m n, let us denote by Mm (C(Y )) ⊕ 1n−m the set of all n × n matrices of the form a 0 , 0 1 where a ∈ Mm (C(Y )) and 1n−m ∈ Mn−m . Theorem 3.1. Let there be given a diagonal ∗-homomorphism φ : C(X) → Mn (C(Y )) with the eigenvalue pattern {λ1 , λ2 , . . . , λn }, a point x0 in X, an element f of C(X), and a tolerance > 0. Choose η > 0 such that |f (x) − f (y)| < whenever d(x, y) < 2η (d is the metric on X). Suppose that F1 , . . . , Fm are nonempty closed subsets of Y (m n) such that d(λi (y), x0 ) < η whenever y ∈ Fi . Then, there is a unitary u in Mm (C(Y )) ⊕ 1n−m and an element b ∈ Mn−m−1 (C(Y )) such that for each y ∈ m i=1 Fi we have ⎛ f (x0 ) ⎜ 0 ⎜ ⎜ u(y)φ(f )(y)u∗ (y) − ⎜ 0 ⎜ ⎜ .. ⎝ . 0
0 b(y) 0 .. . 0
0 0 λm+1 (y) .. . 0
... ... ... .. . ...
⎞ 0 ⎟ 0 ⎟ ⎟ 0 ⎟ < 2. ⎟ .. ⎟ . ⎠ λn (y)
(2)
(Note that if n − m − 1 = 0, then there is no b in the matrix in (2).) Proof. Let ρ denote the metric on Y . Choose δ > 0 such that d(λi (x), λi (y)) < η whenever ρ(x, y) δ, i ∈ {1, . . . , n}. For each 1 i m, f ◦ λi (y) − f (x0 ) < ,
for all y ∈ Bδ (Fi ).
Set εi (y) = f ◦ λi (y) − f (x0 ) for all y in Bδ (Fi ). Then, εi is a continuous map from Bδ (Fi ) to the disk of radius in the complex plane. By Lemma 2.6, εi can be extended to a continuous function from Y to the complex plane such that εi (let us also denote this extension map by εi ). For m < i n, set εi = 0. For each i ∈ {1, . . . , n}, put gi = f ◦ λi − i , so that gi ∈ C(X). Set g = diag(g1 , g2 , . . . , gn ). Then, for each i ∈ {1, . . . , m} and y ∈ Bδ (Fi ), we have gi (y) = f (x0 ); if i ∈ {m + 1, . . . , n}, then gi = f ◦ λi . For any unitary u ∈ Mn (C(Y )) we have uφ(f )u∗ − ugu∗ = diag(ε1 , . . . , εn ) < 2. We have therefore reduced our problem to proving the following claim.
G.A. Elliott et al. / Journal of Functional Analysis 256 (2009) 307–322
315
Claim. There is a unitary u in Mm (C(Y )) ⊕ 1n−m such that ⎛ ⎜ ⎜ ⎜ ∗ ugu = ⎜ ⎜ ⎜ ⎝
f (x0 )
0
0
...
0
⎞
0 0 .. .
b(x) 0 .. .
0 gm+1 (x) .. .
... ... .. .
0 0 .. .
⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎠
0
0
0
...
gn (x)
∀x ∈
m
Fi ,
i=1
where b ∈ Mn−m−1 (C(Y )). Proof. We will assume that for some 1 k < m there is a unitary uk ∈ Mk (C(Y )) ⊕ 1n−k such that ⎛
f (x0 ) ⎜ 0 ⎜ ⎜ 0 uk gu∗k = ⎜ ⎜ ⎜ .. ⎝ . 0
0 b(x) 0 .. . 0
0 0 gk+1 (x) .. . 0
... ... ... .. . ...
⎞ 0 0 ⎟ ⎟ ⎟ 0 ⎟, ⎟ .. ⎟ . ⎠ gn (x)
∀x ∈
k
Fi ,
(3)
i=1
and then prove that the same statement holds with k replaced by k + 1. Since (3) clearly holds when k = 1—just take u1 to be the identity matrix of Mn (C(Y ))—this recursive argument will prove our claim. 2 Assume that (3) holds for some k < m. Put B = Fk+1 and A = Y \Bδ (B). Apply Lemma 2.5 with these choices of A and B and with σ = (1 k + 1) to obtain a unitary v ∈ Mn (C(Y )). We then have that v = 1n on A and v = U [(1 k + 1)] on B. Inspecting the construction of v, we find that it has the following form: v (y) v(y) = U (2 k + 1) 0
0 U (2 k + 1) , 1
∀y ∈ Y,
(4)
where v (y) is a unitary matrix in M2 (C(Y )) equal to 12 on A and equal to U [(12)] on B. Define uk+1 := vuk . Let us show that uk+1 satisfies the requirements of the claim. It is clear that uk+1 is an element of Mk+1 (C(Y )) ⊕ 1n−k−1 . First suppose that y ∈ B, so that gk+1 (y) = f (x0 ). Since uk+1 = vuk we have ⎛
c(y) ⎜ 0 ⎜ uk+1 (y)g(y)u∗k+1 (y) = U (1 k + 1) ⎜ ⎜ .. ⎝ . 0
0 f (x0 ) .. .
... ... .. .
0 0 .. .
0
...
gn (y)
⎞ ⎟ ⎟ ⎟ U (1 k + 1) ⎟ ⎠
316
G.A. Elliott et al. / Journal of Functional Analysis 256 (2009) 307–322
⎛
f (x0 ) ⎜ 0 ⎜ ⎜ 0 =⎜ ⎜ ⎜ .. ⎝ . 0
0 b(y) 0 .. . 0
0 0 gk+2 (y) .. . 0
... ... ... .. . ...
⎞ 0 0 ⎟ ⎟ ⎟ 0 ⎟ ⎟ .. ⎟ . ⎠ gn (y)
for some c(y), b(y) ∈ Mk . Now suppose that y ∈ m i=1 Fi \Bδ (B) ⊆ Y \Bδ (B). In this case v(y) = 1n and uk+1 (y) = uk (y) and there is nothing to prove. Finally, suppose that y ∈ (Bδ (B)\B) ∩ ( m i=1 Fi ). As in the case y ∈ B, we have gk+1 (y) = f (x0 ). From (3) and this last fact we have ⎛
f (x0 ) ⎜ 0 ⎜ ⎜ ⎜ 0 ⎜ ∗ U (2 k + 1) uk (y)g(y)uk (y)U (2 k + 1) = ⎜ 0 ⎜ ⎜ ⎜ .. ⎝ . 0
0 f (x0 ) 0
0 0 d(y)
0 0 0
... ... ...
0 0 0
0 .. . 0
0 .. . 0
gk+2 (y) .. . 0
... .. .
0 .. . gn (y)
...
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
for some d(y) ∈ Mk−1 . Since the upper left 2 × 2 corner of the matrix above is scalar, the entire matrix commutes with v (y) ⊕ 1n−2 . It follows that uk+1 (y)g(y)u∗k+1 (y) is equal to ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ U (2 k + 1) ⎜ ⎜ ⎜ ⎜ ⎝
0 0
0 f (x0 ) 0
0 0 b1 (y)
0 0 0
... ... ...
0 0 0
0 .. . 0
0 .. . 0
0 .. . 0
gk+2 (y) .. . 0
... .. .
0 .. . gn (y)
f (x0 )
...
Computing this product yields a matrix of the form ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
f (x0 ) 0 0 .. . 0
as required.
0
0
b(y) 0 0 gk+2 (y) .. .. . . 0
0
...
0
⎞
... ... .. .
0 0 .. .
⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎠
...
gn (y)
2
For any C∗ -algebra B there is an isomorphism π : B ⊗ Mn → Mn (B)
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ U (2 k + 1) . ⎟ ⎟ ⎟ ⎠
G.A. Elliott et al. / Journal of Functional Analysis 256 (2009) 307–322
317
given by ⎛ ba11 ⎜ . π b ⊗ (aij ) = ⎝ .. ban1
··· .. .
⎞ ba1n .. ⎟ . ⎠.
···
bann
(5)
Proposition 3.2. Suppose that φ : Mm (C(X)) → Mnm (C(Y )) is a diagonal ∗-homomorphism with ep(φ) = {λ1 , λ2 , . . . , λn }. Let φ˜ : C(X) → Mn (C(Y )) be the diagonal ∗-homomorphism given by ⎛ ⎜ ⎜ ˜ φ(f ) = ⎜ ⎜ ⎝
f ◦ λ1
0
...
0 .. . 0
f ◦ λ2 .. . 0
... .. . ...
⎞
0
0 ⎟ ⎟ ⎟ .. ⎟ . . ⎠ f ◦ λn
Then, φ˜ ⊗ idMm : C(X) ⊗ Mm → Mn (C(Y )) ⊗ Mm is unitarily equivalent to φ. Proof. On the one hand we have ⎛
f ◦ λ1 ⎜ 0 ⎜ φ˜ ⊗ idMm f ⊗ (cij ) = ⎜ ⎜ .. ⎝ . 0 while on the other we have ⎛ (f ◦ λ1 ) ⊗ (cij ) ⎜ 0 ⎜ φ f ⊗ (cij ) = ⎜ .. ⎜ ⎝ . 0
0 f ◦ λ2 .. . 0
... ... .. . ...
0 (f ◦ λ2 ) ⊗ (cij ) .. . 0
⎞ 0 0 ⎟ ⎟ ⎟ .. ⎟ ⊗ (cij ), . ⎠ f ◦ λn
... ... ..
.
...
⎞ 0 ⎟ 0 ⎟ ⎟. .. ⎟ ⎠ . (f ◦ λn ) ⊗ (cij )
With the identifications C(X) ⊗ Mn ∼ = Mn (C(X)) and Mn (C(Y )) ⊗ Mm ∼ = Mnm (C(Y )) given by (5) in mind, one sees that Ad U [π] ◦ (φ˜ ⊗ idMm ) = φ, where π is the permutation in Snm and given by π(kn + i) = (i − 1)m + k + 1 for k = 0, 1, 2, . . . , m − 1 and i = 1, 2, . . . , n. 2 Corollary 3.3. Let φ : Mm (C(X)) → Mnm (C(Y )) be a diagonal ∗-homomorphism with eigenvalue pattern {λ1 , λ2 , . . . , λn }, and let > 0 be given. Let f be any element of Mm (C(X)) and choose η > 0 such that f (x) − f (y) <
whenever d(x, y) < 2η
318
G.A. Elliott et al. / Journal of Functional Analysis 256 (2009) 307–322
(d is the metric on X). Let U be the open ball centered at x0 ∈ X with radius η, and suppose that −1 −1 Y = λ−1 1 (U ) ∪ λ2 (U ) ∪ · · · ∪ λn (U ).
Then there is a unitary u in Mnm (C(Y )) and element b ∈ Mnm−m (C(Y )) such that uφ(f )u∗ − f (x0 ) 0
0 < . b
Proof. By Proposition 3.2, we can assume that m = 1. Let Fi be the closure of λ−1 i (U ) for each i. Now, apply Theorem 3.1. Since Y = ni=1 Fi , we are done. 2 Now, we are ready to prove the main theorem of this section. (A , φi ) be a unital diagonal AH algebra. Then, A is simple if and Theorem 3.4. Let A = lim −→ i only if A has the property P of Definition 2.2. Proof. Suppose that A has property P. Let f ∈ Ai be nonzero, so that there is a point x0 in Xi such that f (x0 ) = 0. By the definition of property P, there are an integer j > i and unitaries ul ∈ Aj,l , l ∈ {1, . . . , kj } such that t,l ul φ (f )u∗ − f (x0 ) 0 < l i,j 0 bl for some appropriately sized bl . We may assume that < f (x0 ), so that φij (f ) is nowhere zero. This implies that the ideal of Aj generated by φij (f ) is all of Aj , and that the ideal of A generated by the image of f is all of A. Since f was arbitrary, A is simple. Now assume that A is simple, and let f ∈ Ai,t be nonzero. Recall that the φi may be taken to be injective. By Proposition 2.3 there exists, for each x0 ∈ Xi,t and > 0, a j0 > i with the following property: for every j j0 and every l ∈ {1, . . . , kj } we have t,l Bδ (x0 ) = Xj,l , ep−1 φi,j where δ is some positive number such that d f (x), f (y) <
whenever d(x, y) < 2δ.
t,l As pointed out in Section 2, the map φi,j may be viewed as a diagonal map from Ai,t into the cutt,l down of Aj,l by the projection φi,j (1); any unitary u in this corner of Aj,l gives rise to a unitary u˜ t,l in Aj,l by setting u˜ = u ⊕ (1Aj,l − φi,j (1). Combining this observation with Corollary 3.3 we see that there exists, for each l ∈ {1, . . . , kj }, a unitary ul ∈ Aj,l such that
t,l ul φ (f )u∗ − f (x0 ) l i,j 0 Thus, A has property P, as desired.
2
0 < . bl
G.A. Elliott et al. / Journal of Functional Analysis 256 (2009) 307–322
319
4. Stable rank (A , φi ) be a simple unital diagonal AH algebra. Then, A has stable Theorem 4.1. Let A = lim −→ i rank one. Before proving Theorem 4.1, we situate it relative to other results on the stable rank of general approximately homogeneous (AH) algebras. Recall that an AH algebra is an inductive limit C∗ algebra A = limi→∞ (Ai , φi ), where Ai =
ni
pi,l C(Xi,l ) ⊗ K pi,l
(6)
l=1
for compact connected Hausdorff spaces Xi,l , projections pi,l ∈ C(Xi,l ) ⊗ K, and natural numbers ni . If A is separable, then one may assume that the Xi,l are finite CW-complexes [1,11]. The inductive system (Ai , φi ) is referred to as a decomposition for A. All AH algebras in this paper are assumed to be separable. If an AH algebra A admits a decomposition as in (6) for which max
1lni
dim(Xi,1 ) dim(Xi,ni ) ,..., rank(pi,1 ) rank(pi,ni )
i→∞
−→ 0,
then we say that A has slow dimension growth. Theorem 1 of [2] states that every simple unital AH algebra with slow dimension growth has stable rank one. Villadsen in [17] constructed simple diagonal AH algebras which do not have slow dimension growth, but which do have stable rank one; the converse of [2, Theorem 1] does not hold. There are in fact a wealth of simple diagonal AH algebras without slow dimension growth which exhibit all sorts of interesting behaviour (cf. [14–16]), whence Theorem 4.1 is widely applicable. Simple AH algebras may have stable rank strictly greater than one, and there is reason to believe that Theorem 4.1 is quite close to being best possible. One might be able to generalise our result to the setting of AH algebras where the projections φi,j (pi,l ) appearing in (6) can be decomposed into a direct sum of a trivial projection θj and a second projection qj such that τ (qj ) → 0 as j → ∞ for any trace τ . Otherwise, one finds oneself in a situation very similar to the construction of [18], where the stable rank is always strictly greater than one. Let us now prepare for the proof of Theorem 4.1. Lemma 4.2. Let a ∈ Mm (C(X)) be block diagonal, i.e., ⎛
0
⎜0 ⎜ a=⎜ ⎜ .. ⎝. 0
0
...
0
a1 .. .
... .. . ...
an
0
⎞
0⎟ ⎟ ⎟, ⎟ ⎠
where ai ∈ Mki (C(X)) for natural numbers k1 , . . . , kn . If the size of the matrix 0 in the upper lefthand corner of a is strictly greater than max1in ki , then a can be approximated arbitrarily closely by invertible elements in Mm (C(X)).
320
G.A. Elliott et al. / Journal of Functional Analysis 256 (2009) 307–322
Proof. Let > 0 be given, and let k denote the size of the matrix 0 in the upper left-hand corner of a. Since k > ki , i ∈ {1, . . . , n}, there is a permutation matrix U such that ⎛0 ⎜0 ⎜ Ua = ⎜ . ⎝ .. 0
∗ 0 .. .
∗⎞ ∗⎟ ⎟ ⎟ ⎠
... ... .. . ...
0
0
is nilpotent. As was proved in [13], every nilpotent element in a unital C∗ -algebra can be approximated arbitrarily closely by invertible elements. We may thus find an invertible element b ∈ Mm (C(X)) such that U a − b < , and a − U −1 b = U −1 U a − U −1 b U −1 · U a − b < . The lemma now follows from the fact that U −1 b is invertible.
2
It easy to prove that a ∈ Mn (C(X)) is invertible if and only if a(x) is invertible for each x ∈ X. The proof of the next lemma is also straightforward. Lemma 4.3. Let p, q be orthogonal projections in a C∗ -algebra A, and let > 0 be given. If elements a and b in A can be approximated to within by invertible elements in pAp and qAq, respectively, then a + b can be approximated to within by an invertible element in (p + q)A(p + q). Now, we are ready to prove Theorem 4.1. Proof of Theorem 4.1. Since every element in A can be approximated arbitrarily closely by elements in ∞ i=1 Ai , it will suffice to prove for any > 0 and any a ∈ Ai , there is an invertible element in A whose distance to a is less than . (Note that we are using the injectivity of the φi to identify Ai with its image in A.) By Lemma 4.3, we may assume that Ai = Mni (C(Xi )). We also assume that a is not invertible. By the comment preceding Lemma 4.3, there is a point x0 ∈ Xi such that det(a(x0 )) = 0. There are permutation matrices u, v ∈ Mni and a matrix c ∈ Mni −1 such that ua(x0 )v =
0 0 0
c
.
Let b denote the element uav. Following the lines of the proof of Lemma 4.2, it will suffice to prove that b can be approximated to within by an invertible element of A. For each j > i, φi,j (b) is a tuple of kj elements. If each coordinate of φi,j (b) can be approximated to within by an invertible element in the corner of A generated by the unit of Aj,l , then φi,j (b) can be approximated to within by an invertible element of A. We may therefore assume that Aj = Mnj (C(Yj ), and concern ourselves with proving that φi,j (b) is approximated to within by an invertible element in A.
G.A. Elliott et al. / Journal of Functional Analysis 256 (2009) 307–322
321
By Theorem 3.4, there exist an integer j > i, a unitary w ∈ Aj and an element b such that wφij (b)w ∗ − b(x0 ) 0
0 < /2. b
We have
b(x0 )
0
0
b
=
0
0
0 b
,
where b = diag(c, b ). Put d = wφi,j (b)w ∗ , and note that it will suffice to prove that φj,m (d) is approximated to within /2 by an invertible element in Am for some m > j . Since A is simple, there is an integer m > j large enough so that, for each t ∈ {1, . . . , km }, 1,t counted with multiplicity is strictly larger either the number of the eigenvalue maps of φj,m 1,t is finite-dimensional. In the latter case, than the size of the matrix b , or the image of φj,m 1,t 1,t (d) is approximated to within by an invertible element in the image of φj,m since finiteφj,m ∗ dimensional C -algebras have stable rank one, so we may assume that the number of eigenvalue 1,t , counted with multiplicity, is strictly larger than the size of the matrix b . Then, maps of φj,m 1,t (d) is unitarily equivalent to φj,m
⎛
0
0
⎜0 b 1 ⎜ ⎜. . ⎜. . ⎝. . 0 0
...
0
⎞
... .. .
0⎟ ⎟ .. ⎟ ⎟ . ⎠
...
bl
1,t inside Am,t , where bk is the composition of b and the kth eigenvalue map of φj,m , and the size of the matrix 0 at the upper left-hand corner of the above matrix is strictly bigger than the size of the matrix bk for every k. By Lemma 4.2, the matrix above can be approximated to within /2 by an invertible in Am,t , as required. 2
Corollary 4.4. Let A be a simple unital diagonal AH algebra. If A has real rank zero and weakly unperforated K0 -group, then A is tracially AF. Proof. By Theorem 4.1, A has stable rank one. The corollary then follows from a result of Lin [12, Theorem 2.1]. 2 Corollary 4.4 applies, for instance, to simple unital diagonal AH algebras for which the spaces Xi in some diagonal decomposition for A are all contractible. This contractibility hypothesis may seem strong, but it does not substantially restrict the complexity of A; if one wants to classify all such A via K-theory and traces, then the additional assumption of very slow dimension growth and the full force of [8] and [9] are required; the collection of all such cannot be classified by topological K-theory and traces alone [15].
322
G.A. Elliott et al. / Journal of Functional Analysis 256 (2009) 307–322
Acknowledgments The third named author thanks Hanfeng Li, Cristian Ivanescu, and Zhuang Niu for several helpful discussions and comments. References [1] B. Blackadar, Matricial and ultramatricial topology, in: Operator Algebras, Mathematical Physics, and LowDimensional Topology, Istanbul, 1991, in: Res. Notes Math., vol. 5, pp. 11–38. [2] B. Blackadar, M. Dadarlat, M. Rordam, The real rank of inductive limit C∗ -algebras, Math. Scand. 69 (1991) 211– 216. [3] M. Dadarlat, Reduction to dimension three of local spectra of real rank zero C∗ -algebras, J. Reine Angew. Math. 460 (1995) 189–212. [4] M. Dadarlat, G. Nagy, A. Nemethi, C. Pasnicu, Reduction of topological stable rank in inductive limits of C∗ algebras, Pacific J. Math. 153 (1992) 267–276. [5] G.A. Elliott, On the classification of C∗ -algebras of real rank zero, J. Reine Angew. Math. 443 (1993) 179–219. [6] G.A. Elliott, D.E. Evans, The structure of irrational rotation C∗ -algebras, Ann. of Math. (2) 138 (1993) 477–501. [7] G.A. Elliott, G. Gong, On the classification of C∗ -algebras of real rank zero. II, Ann. of Math. (2) 144 (3) (1996) 497–610. [8] G.A. Elliott, G. Gong, L. Li, On the classification of simple inductive limit C∗ -algebras, II. The isomorphism theorem, Invent. Math., in press. [9] G. Gong, On the classification of simple inductive limit C∗ -algebras I. The reduction theorem, Doc. Math. 7 (2002) 255–461. [10] K.R. Goodearl, Notes on a class of simple C∗ -algebras with real rank zero, Publ. Mat. 36 (2A) (1992) 637–654. [11] K.R. Goodearl, Riesz decomposition in inductive limit C∗ -algebras, Rocky Mountain J. Math. 24 (1994) 1405– 1430. [12] H. Lin, Simple AH algebras of real rank zero, Proc. Amer. Math. Soc. 131 (12) (2003) 3813–3819. [13] M. Rordam, On the structure of simple C∗ -algebras tensored with a UHF-algebra, J. Funct. Anal. 100 (1991) 1–17. [14] A. Toms, Flat dimension growth for C∗ -algebras, J. Funct. Anal. 238 (2006) 678–708. [15] A. Toms, On the classification problem for nuclear C∗ -algebras, Ann. of Math. (2), in press. [16] A. Toms, An infinite family of non-isomorphic C∗ -algebras with identical K-theory, preprint, arXiv: math.OA/ 0609214, 2006. [17] J. Villadsen, Simple C∗ -algebras with perforation, J. Funct. Anal. 154 (1998) 110–116. [18] J. Villadsen, On the stable rank of simple C∗ -algebras, J. Amer. Math. Soc. 12 (1999) 1091–1102.
Journal of Functional Analysis 256 (2009) 323–351 www.elsevier.com/locate/jfa
Banach Lie algebras with Lie subalgebras of finite codimension: Their invariant subspaces and Lie ideals Edward Kissin a,∗ , Victor S. Shulman a , Yurii V. Turovskii b,1 a Department of Computing, Communications Technology and Mathematics, London Metropolitan University,
166-220 Holloway Road, London N7 8DB, Great Britain b Institute of Mathematics and Mechanics, National Academy of Sciences of Azerbaijan, 9 F. Agayev Street,
Baku AZ1141, Azerbaijan Received 26 February 2008; accepted 7 October 2008 Available online 7 November 2008 Communicated by K. Ball
Abstract The paper studies the existence of closed invariant subspaces for a Lie algebra L of bounded operators on an infinite-dimensional Banach space X. It is assumed that L contains a Lie subalgebra L0 that has a non-trivial closed invariant subspace in X of finite codimension or dimension. It is proved that L itself has a non-trivial closed invariant subspace in the following two cases: (1) L0 has finite codimension in L and there are Lie subalgebras L0 = L0 ⊂ L1 ⊂ · · · ⊂ Lp = L such that Li+1 = Li + [Li , Li+1 ] for all i; (2) L0 is a Lie ideal of L and dim(L0 ) = ∞. These results are applied to the problem of the existence of non-trivial closed Lie ideals and closed characteristic Lie ideals in an infinite-dimensional Banach Lie algebra L that contains a non-trivial closed Lie subalgebra of finite codimension. © 2008 Elsevier Inc. All rights reserved. Keywords: Invariant subspaces; Lie algebras of bounded operators
* Corresponding author.
E-mail addresses:
[email protected] (E. Kissin),
[email protected] (V.S. Shulman),
[email protected] (Y.V. Turovskii). 1 The support received from INTAS project No. 06-1000017-8609 is gratefully acknowledged by the third author. 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.10.012
324
E. Kissin et al. / Journal of Functional Analysis 256 (2009) 323–351
1. Introduction The link between the structure of Lie subalgebras and Lie ideals of Lie algebras attracted attention of researchers both in the classical situation of finite-dimensional Lie algebras and in the general case. Barnes [2] and Towers [20] studied finite-dimensional Lie algebras possessing the property that all their maximal Lie subalgebras have codimension 1. In [9] the first author considered finite-dimensional Lie algebras L over C that have “sufficiently” many Lie subalgebras of codimension 1. Amayo in [1] studied Lie subalgebras L0 of codimension 1 in finite-dimensional and infinite-dimensional Lie algebras L over a field F . He showed that there is the largest Lie ideal I (L0 ) of L contained in L0 and that, for char(F ) = 0, either dim(L/I (L0 )) 3, or dim(L/I (L0 )) = ∞. He also constructed a simple infinite-dimensional Lie algebra L with a Lie subalgebra L0 of codimension 1, so that I (L0 ) = {0} and therefore dim(L/I (L0 )) = ∞. The situation changes if L is a complex Banach Lie algebra. The first author in [10] investigated the structure of Banach Lie algebras L with “sufficiently” many Lie subalgebras of codimension 1 and showed that dim(L/I (L0 )) 3 for each Lie subalgebra L0 of codimension 1. The question was also raised as to whether dim(L/I (L0 )) < ∞ for each closed Lie subalgebra L0 of L of finite codimension. In the last section of this paper we show that already when codim(L0 ) = 2 the answer to the above question is negative, but nevertheless L in this case has a closed proper Lie ideal of finite codimension. For comparison it should be noted that associative algebras A with associative subalgebras A0 of finite codimension always have two-sided ideals of finite codimension that lie in A0 (Laffey [12]). Lie subalgebras and Lie ideals of finite codimension of associative Banach algebras and, in particular, of the algebra C(H ) of all compact operators on a separable Hilbert space H, of Schatten classes Cp , 1 p < ∞, and of uniformly hyperfinite C*-algebras were studied in [5,13,14]. Note that Lie ideals of Banach Lie algebras L are just invariant subspaces with respect to the adjoint representation: ad(a)(x) = [a, x]
for a, x ∈ L.
Using this link, we will study the question about the existence of Lie ideals in the much broader context of Invariant Subspace theory. Throughout the paper X denotes an infinite-dimensional complex Banach space and B(X) the algebra of all bounded operators on X. A Lie subalgebra of B(X) is called reducible if it has a non-trivial closed invariant subspace; otherwise, it is irreducible. We consider a Lie subalgebra L of B(X) and assume that it contains a Lie subalgebra L0 that has a closed invariant subspace X0 in X. We will always assume that X0 is a proper subspace of X and that L0 is a proper Lie subalgebra of L closed in L: {0} = X0 = X,
{0} = L0 = L
and L0 ∩ L = L0 ,
where L0 is the closure of L0 in B(X). We investigate the problem whether L itself has a nontrivial closed invariant subspace. For finite-dimensional X, the problem has a negative solution. Indeed, take L = B(X) and L0 = C1. If L and L0 are associative algebras and L0 has finite codimension in L, then this problem has a simple positive solution (see Section 8) based on the already mentioned result of [12]. If, however, both of them are Lie algebras, then the problem is much more delicate and requires a more careful investigation. Belti¸ta˘ and Sabac ¸ [3] proved that L is reducible in the case when it
E. Kissin et al. / Journal of Functional Analysis 256 (2009) 323–351
325
contains a finite-dimensional Lie ideal of nilpotent operators; the second and the third authors [19] proved this in the case when L contains a finite-dimensional Lie ideal of compact operators. Another sufficient condition of reducibility of L considered in [19, Theorem 1.1] is the assumption that L consists of compact operators and L0 is a Lie ideal of L of some special type: for example, it is E-solvable, or consists of quasinilpotent operators (a Volterra Lie ideal). In this paper we neither assume that a Lie algebra L consists of compact operators, nor that L0 is a Lie ideal of a special type. Instead, we assume that L0 has a closed invariant subspace X0 in X of finite codimension (sometimes these assumptions can be weakened). Our approach is based on the study of the natural representation θ of L0 on the quotient space L/L0 . To describe the results we need the following definition. Definition 1.1. (i) A Lie subalgebra L0 of a Lie algebra L is called non-degenerate in L, if L0 has finite codimension in L and L = L0 + [L0 , L]. (ii) A Lie subalgebra L0 is related to L, if there are Lie subalgebras L0 = L0 ⊆ L1 ⊆ · · · ⊆ p L = L such that Li is non-degenerate in Li+1 . The term “non-degenerate” in (i) is due to the fact that L0 is non-degenerate if and only if the representation θ is non-degenerate: θ (L0 )(L/L0 ) = L/L0 . Note also that if L0 lies in a Lie ideal of L, then L0 is not related to L. We will prove that L is reducible in two “opposite extreme” cases: Case 1. L0 is related to L; Case 2. L0 is a Lie ideal of L and dim(L0 ) = ∞. Moreover, in Case 1 (see Section 5) a non-trivial closed invariant subspace Y of L can be chosen so that codim(Y ) < ∞ and Y ⊆ X0 . In particular, if L0 is a Lie subalgebra (and not a Lie ideal) of codimension 1 and codim(X0 ) < ∞, then L always has a non-trivial closed invariant subspace of finite codimension contained in X0 . In Case 2 (see Section 6) L may have no invariant subspaces of finite codimension. However, if X0 is a maximal subspace invariant for L0 , then the subspace Y can be chosen so that it lies in X0 (if L0 has a non-trivial closed invariant subspace of finite codimension, it has a maximal one). We will construct examples which show that if the conditions in Cases 1 and 2 are weakened in some way, then L may become irreducible; or even if it has invariant subspaces, they do not necessarily lie in X0 . For example, it follows from Example 9.6 that if codim(X0 ) < ∞ and codim(L0 ) < ∞, but [L0 , L] + L0 = L (cf. Case 1), then non-trivial closed subspaces invariant for L (even if they exist) may not lie in X0 . We also show in Example 6.6 that if L0 is a Lie ideal of L but either dim(L0 ) < ∞ or codim(X0 ) = ∞ (cf. Case 2), then L may be irreducible. In Section 7 we consider the case when X0 is a finite-dimensional subspace invariant for L0 . Using duality, we establish that L has a non-trivial closed invariant subspace that contains X0 , if either L0 is related to L, or L0 is an infinite-dimensional Lie ideal of L. In Section 8 we combine the results of the previous sections and study the case when L0 ⊆ L ⊆ L are Lie subalgebras of B(X), dim(L0 ) = ∞ and L0 has a closed invariant subspace of finite codimension or dimension. We show that if L0 is related to L while L is a Lie ideal of L, or if L0 is a Lie ideal of L while L is related to L, then L is reducible. We also consider the
326
E. Kissin et al. / Journal of Functional Analysis 256 (2009) 323–351
situation when L and L0 are associative algebras, L0 is reducible and L is finitely generated as a module over L0 . In the last section we apply the results of the previous sections to the problem of the existence of Lie ideals in infinite-dimensional Banach Lie algebras L with closed Lie subalgebras L0 of finite codimension. We show that if L0 is related to L, then L has a closed Lie ideal of finite codimension contained in L0 . From this we infer that if codim(L0 ) 2, then L always has a closed Lie ideal K of finite codimension and, if codim(L0 ) = 1 then K ⊆ L0 and codim(K) 3. The question whether L always has a closed ideal if codim(L0 ) > 2 is still open. Moreover, we show in Corollary 9.5 that if codim(L0 ) 2 then even if L has a non-trivial closed Lie ideal, it does not necessarily lie in L0 . At the end of Section 9 we prove that if an infinite-dimensional, non-commutative Banach Lie algebra L has a proper closed Lie subalgebra related to L, or a non-trivial closed Lie ideal of finite codimension or dimension, then it has a closed characteristic Lie ideal—the Lie ideal invariant for all bounded derivations of L. Sections 2 and 3 contain some results about finite-dimensional representations of Lie algebras that we use in later sections (in spite of the general character of these results we could not find them in the literature). In Section 4 we introduce and study properties of (L, X0 )-filtrations of Banach spaces X with respect to closed subspaces X0 of X and Lie subalgebras L of B(X). These filtrations provide one of the main tools for our study of invariant subspaces of L. 2. Eigen-representations and non-degenerate representations of Lie algebras on finite-dimensional spaces Throughout this section V is a finite-dimensional linear space. Let L0 be a Lie algebra. A representation θ of L0 on V is a Lie homomorphism from L0 into the algebra L(V ) of all operators on V . It is irreducible, if the Lie subalgebra θ (L0 ) of L(V ) has no non-zero invariant proper subspaces in V . It is cyclic if there is u ∈ V such that V = θ (L0 )u. A representation of a Lie algebra can be irreducible but not cyclic; it can also be cyclic but not irreducible. Example 2.1. Consider the algebra M3 (C) of all 3 × 3 matrices as the algebra of all operators on a 3-dimensional space V . Then L1 = {A = (aij ) ∈ M3 (C): ai1 = ai2 = 0 for all i} and L2 = {A ∈ M3 (C): At = −A}, where At is the transposed matrix, are Lie subalgebras of M3 (C). Clearly, the identity representation of L1 is cyclic but not irreducible. On the other hand, the identity representation of L2 is irreducible but not cyclic. Indeed, the enveloping algebra of L2 coincides with M3 (C). Thus the identity representation of L2 is irreducible. Let us show that it is not cyclic. Consider u ∈ V as a 3 × 1 matrix and denote by ut its transposed matrix. Then ut Au ∈ C for A ∈ L2 . As (ut Au)t = ut At u = −ut Au, we have ut Au = 0 for all A ∈ L2 and u ∈ V . Fix u. As ut Au = 0, for all A ∈ L2 , we have that dim(L2 u) 2, so that the identity representation of L2 is not cyclic. For 0 = λ ∈ C, set Eλ = {x ∈ V : θ (h)x = λx for some h ∈ L0 }. Denote by Vλ the linear span in V of all elements in Eλ and by Ve the linear span of all elements from all Eλ , 0 = λ ∈ C. For a subset U of V , denote by θ (L0 )U the linear span of all θ (h)u, where h ∈ L0 , u ∈ U. Lemma 2.2. For all 0 = λ ∈ C, we have θ (L0 )Vλ = Vλ and θ (L0 )Ve = Ve .
E. Kissin et al. / Journal of Functional Analysis 256 (2009) 323–351
327
Proof. Set L = θ (L0 ). For a, b ∈ L and t ∈ C, exp(tb)a exp(−tb) = exp(t ad(b))(a) ∈ L, where ad(b)(a) = [b, a]. Let x ∈ Eλ and ax = λx for some a ∈ L. Then exp(tb)a exp(−tb) exp(tb)x = exp(tb)ax = λ exp(tb)x. Hence exp(tb)x ∈ Eλ , so that bx = limt→0 1t (exp(tb)x − x) ∈ Vλ . Thus Vλ is invariant for L. As Eλ ⊆ LEλ , we have LVλ = Vλ . Therefore LVe = Ve . 2 Definition 2.3. Let θ be a representation of a Lie algebra L0 on V . (i) θ is nilpotent if θ (h) is nilpotent for each h ∈ L0 , that is, θ (h)n = 0 for some n that depends on h. (ii) θ is an eigen-representation if V = Ve . (iii) θ is non-degenerate if θ (L0 )V = V . Clearly, θ is nilpotent if and only if all operators θ (h), h ∈ L0 , have only zero eigenvalues. Proposition 2.4. Every representation of a Lie algebra on a finite-dimensional space that decomposes into the direct sum of non-zero irreducible or cyclic representations is an eigenrepresentation. Proof. Let θ be a representation of L0 on V and dim(V ) = n. Set L = θ (L0 ). (1) If θ is irreducible then, by Lemma 2.2, either Ve = {0} or Ve = V . In the first case all eigenvalues of all a ∈ L are zero. Hence all a are nilpotent, that is, a n = 0. By Engel’s theorem, there is 0 = x ∈ V such that ax = 0 for all a ∈ L. As L has no invariant subspaces in V , V = Cx and L = {0}, so θ is the zero representation. Thus Ve = V , so θ is an eigen-representation. (2) If θ is cyclic then Lu = V for some u ∈ V . Then u ∈ Ve . By Lemma 2.2, V = Lu ⊆ LVe ⊆ Ve . Hence Ve = V , so θ is an eigen-representation. The general case follows immediately from (1) and (2). 2 Proposition 2.5. A representation θ of L0 on V is nilpotent if and only if V has no non-zero subspaces M invariant for θ (L0 ) such that the restrictions of θ to M are non-degenerate. Proof. By Lemma 2.2, the restriction of θ to each Vλ is non-degenerate. If V has no non-zero subspaces invariant for θ (L0 ) such that the restrictions of θ to them are non-degenerate, all Vλ = {0}. Thus all θ (h), h ∈ L0 , have only zero eigenvalues, so that θ is nilpotent. Conversely, if θ is nilpotent, its restriction θM to any invariant subspace M of V is nilpotent. It follows from the Engel theorem (see [8, Theorem II.2.1 ]) that there are subspaces {0} = V0 ⊂ V1 ⊂ · · · ⊂ Vp = M invariant for θ such that each quotient representation of L0 on Vi /Vi−1 is a one dimensional zero representation. Hence θ (L0 )M = M. 2 Let subspaces K and M of V be invariant for a representation θ of L0 and K ⊂ M. We will call the corresponding representation of L0 on the quotient space M/K a quotient representation. Theorem 2.6. Let θ be a representation of L0 on V . The following conditions are equivalent: (i) V has no subspaces M = V invariant for θ (L0 ) such that the quotient representations of L0 on V /M are nilpotent;
328
E. Kissin et al. / Journal of Functional Analysis 256 (2009) 323–351
(ii) θ is non-degenerate; (iii) there are subspaces {0} = V0 ⊂ V1 ⊂ · · · ⊂ Vp = V invariant for θ such that each quotient representation θi of L0 on Vi /Vi−1 is an eigen-representation. Proof. (i) ⇒ (iii). As θ is not nilpotent, there is h ∈ L0 with a non-zero eigenvalue. Hence Ve = {0}. Set V1 = Ve . By Lemma 2.2, V1 is invariant for θ and θ1 = θ |V1 is an eigenrepresentation. The quotient representation ψ of L0 on V /V1 is not nilpotent, so (V /V1 )e = {0}. By Lemma 2.2, (V /V1 )e is invariant for ψ and ψ|(V /V1 )e is an eigen-representation. Let V2 be the subspace of V such that V1 ⊂ V2 and V2 /V1 is isomorphic to (V /V1 )e . Then V2 is invariant for θ and the quotient representation θ2 on V2 /V1 is equivalent to ψ|(V /V1 )e and, therefore, is an eigen-representation. Continuing this process, we conclude the proof. (iii) ⇒ (ii). As θi are eigen-representations, θ (L0 )V1 = θ1 (L0 )V1 = V1 and θi (L0 )(Vi /Vi−1 ) = Vi /Vi−1 , for i > 1. Therefore θ (L0 )Vi = Vi , for all i. So θ (L0 )V = θ (L0 )Vp = Vp = V . (ii) ⇒ (i). Suppose that there is a subspace M in V invariant for θ such that the quotient θ is not non-degenerate. Hence representation θ of L0 on V /M is nilpotent. By Proposition 2.5, θ (L0 )(V /M) = V /M. Therefore θ (L0 )V = V . 2 3. Nilpotent part of spaces of commuting operators Let X be a Banach space and let M be a linear subspace of B(X) that consists of mutually commuting operators. Denote by NM the set of all nilpotent operators in M: NM = A ∈ M: An = 0 for some n that depends on A .
(3.1)
As operators in M commute, NM is a linear subspace of M. Let M be the closure of M. For n ∈ N, let M n be the linear subspace of B(X) generated by all products of n elements from M. Proposition 3.1. Let B ∈ B(X) be such that [B, A] ∈ M for all A ∈ M. Then (i) [B, Y ] ∈ NM for all Y ∈ NM . (ii) If [B, A] − μA ∈ NM , for some A ∈ M and μ = 0, then An ∈ NM n for some n. If dim(M) < ∞ then A ∈ NM . Proof. For S ∈ B(X), denote by δS the operator on B(X) that acts by the formula δS (C) = [S, C] for C ∈ B(X). Let Y ∈ NM . As δY (B) = [Y, B] ∈ M, we have δY2 (B) = [Y, [Y, B]] = 0. The fact that [Y, B] is nilpotent, if [Y, [Y, B]] = 0 and Y is nilpotent is well known (see [7]); we will give the proof of it for the sake of completeness. Since Y k = 0, for some k, and all terms in δY2k−1 (C), for each C ∈ B(X), contain Y m with m k, we have δY2k−1 (C) = 0. Then (see [6, p. 335]), n!(δY (B))n = δYn (B n ) for all n. Hence (δY (B))2k−1 = 0. Thus [Y, B] ∈ NM . Part (i) is proved. Assume now that δB (A) = μA + Y , where Y ∈ NM and μ = 0. As δB is a derivation, δB (CD) = CδB (D) + δB (C)D
for C, D ∈ B(X).
(3.2)
E. Kissin et al. / Journal of Functional Analysis 256 (2009) 323–351
329
Hence, since all operators in M commute and δB (A) ∈ M, δB (A2 ) = 2AδB (A). Continuing this, we obtain that δB (An ) = nAn−1 δB (A). Hence δB An = nμAn + nAn−1 Y.
(3.3)
As δB 2 B and μ = 0, the operator Tn = δB − nμ1 on B(X) is invertible for n > 2 B μ . n Fix such an n. All operators of M mutually commute. As δB is bounded and maps M into itself, it follows from (3.2) that δB also maps M n into itself. k k Applying (i) to M n , we have that δB maps NM n into itself. As Tn−1 = −nμ ∞ k=0 δB /(nμ) , it maps NM n into itself. By (3.3), Tn (An ) = nAn−1 Y. Since Y ∈ NM , there are Yp ∈ NM that converge to Y . As A and Yp commute, all nAn−1 Yp ∈ NM n . Hence nAn−1 Y ∈ NM n . Therefore An ∈ NM n . If dim(M) < ∞, then NM n = NM n . Hence A is nilpotent, so that A ∈ NM . 2 Let V be a finite-dimensional space and assume that there is a linear map u → E u from V into B(X) such that Ev Eu = EuEv
for all u, v ∈ V .
Set V nil = u ∈ V : E u is nilpotent .
(3.4)
Then V nil is a linear subspace of V . Let θ be a representation of a Lie algebra L0 on V and let ρ be a representation of L0 on X that satisfy ρ(h), E u = E θ(h)u
for h ∈ L0 and u ∈ V .
(3.5)
Proposition 3.2. (i) The subspace V nil is invariant for all θ (h), h ∈ L0 . (ii) Let u ∈ V and let there exist h ∈ L0 such that θ (h)u − μu ∈ V nil , for some μ = 0. Then u ∈ V nil . Proof. M = {E u : u ∈ V } is a finite-dimensional linear subspace of B(X), all its elements commute and NM = {E u : u ∈ V nil }. Hence part (i) follows from (3.5) and Proposition 3.1(i). If θ (h)u − μu ∈ V nil , for some μ = 0, then [ρ(h), E u ] = E θ(h)u = μE u + E v , where v ∈ nil V . Hence part (ii) follows from Proposition 3.1(ii). 2 Corollary 3.3. If θ is non-degenerate then V = V nil . θ of L0 on Proof. If V = V nil , it follows from Proposition 3.2 that the quotient representation θ (h), h ∈ L0 , has only zero eigenvalues. Hence θ is nilpotent. By V /V nil is such that each Theorem 2.6, θ is not non-degenerate. This contradiction shows that V = V nil . 2
330
E. Kissin et al. / Journal of Functional Analysis 256 (2009) 323–351
4. Filtrations of spaces and Lie algebras Let L be a Lie algebra of bounded operators on a Banach space X, that is, a linear subspace of B(X) closed under the Lie multiplication [a, b] = ab − ba. For linear subspaces K and M of L, set [K, M] = span [b, a]: b ∈ K, a ∈ M and KM = span{ba: b ∈ K, a ∈ M}. If [K, L] ⊆ K then K is a Lie ideal of L; in this case we will write K L. Let Y be a subspace of X and x ∈ X. Set Lx = {ax: a ∈ L}
and LY = span{ax: a ∈ L, x ∈ Y }.
The subspace Y is called invariant for L if LY ⊆ Y. Throughout the paper X0 will be a proper subspace of X. Set X−1 = X and Xn = {x ∈ Xn−1 : Lx ⊆ Xn−1 } for n 1.
(4.1)
Then Xn ⊆ Xn−1 and, for n 1, Xn is the largest linear subspace of Xn−1 such that LXn ⊆ Xn−1 .
(4.2)
We denote by N the set of all non-negative integers. Definition 4.1. The decreasing sequence {Xn }∞ n=−1 of subspaces of X constructed in (4.1) is called (L, X0 )-filtration of X. A filtration is non-trivial if Xn = {0} for all n ∈ N. For a non-trivial filtration {Xn } of X, the following three cases are possible: (1)
Xn = Xn+1 , for some n; (2) Xn = {0} and all Xn are distinct; (3) Xn = {0}.
In Cases 1 and 2 the subspace Xn is non-zero and invariant for L. Throughout the paper L0 will be a proper Lie subalgebra of L that leaves X0 invariant and we assume that L0 is closed in L: L0 ∩ L = L0 . Set L1 = b ∈ L0 : [b, L] ⊆ L0 and Z(L, L0 , X0 ) = {b ∈ L1 : bX ⊆ X0 }. (4.3) We will often write Z(L) instead of Z(L, L0 , X0 ). Lemma 4.2. Let {Xn } be (L, X0 )-filtration of X. Then (i) L0 Xn ⊆ Xn and Z(L)Xn ⊆ Xn+1 for n ∈ N ∪ {−1}. (ii) Z(L) L0 . (iii) Let q = codim(L0 ) < ∞ and p = codim(X0 ) < ∞. Then dim(L0 /Z(L)) < ∞ and dim(Xn−1 /Xn ) pq n . (iv) If X0 is a closed subspace of X, then all Xn are closed subspaces of X.
E. Kissin et al. / Journal of Functional Analysis 256 (2009) 323–351
331
Proof. We have L0 X0 ⊆ X0 . It follows from (4.2) that, for n 1, LL0 Xn ⊆ L0 LXn + [L, L0 ]Xn ⊆ L0 Xn−1 + LXn ⊆ L0 Xn−1 + Xn−1 .
(4.4)
Assume, by induction, that L0 Xi ⊆ Xi for all i n − 1. Then it follows from (4.4) that LL0 Xn ⊆ Xn−1 . Hence, by (4.2), L0 Xn ⊆ Xn . Set Z = Z(L). As Z ⊆ L1 and [L1 , L] ⊆ L0 , we have [Z, L] ⊆ L0 . Hence LZXn ⊆ ZLXn + [L,Z]Xn ⊆ ZXn−1 + L0 Xn ⊆ ZXn−1 + Xn .
(4.5)
Then LZX0 ⊆ ZX + X0 ⊆ X0 . Using this and (4.5), we obtain by induction that LZXn ⊆ Xn . Therefore, by (4.2), ZXn ⊆ Xn+1 . Part (i) is proved. As Z ⊆ L1 and L1 L0 , [L0 , Z] ⊆ [L0 , L1 ] ⊆ L1 . Also [L0 , Z]X ⊆ L0 ZX + ZL0 X ⊆ L0 X0 + ZX ⊆ X0 + X0 = X0 . Thus [L0 , Z] ⊆ Z, so that Z L0 . Part (ii) is proved. q There are {ei }i=1 in L \ L0 such that L = L0 Ce1 · · · Ceq . Set pn = dim(Xn−1 /Xn ). Fix i and assume that pn < ∞. By (4.2), ei (Xn ) ⊆ Xn−1 . Hence the subspace Li = {x ∈ Xn : ei x ∈ Xn } has codimension in Xn less or equal to pn . As Xn+1 ⊆ Xn and L0 Xn ⊆ Xn , Xn+1 = {x ∈ Xn : Lx ⊆ Xn } = {x ∈ Xn : ei x ∈ Xn for i = 1, . . . , q} =
q
Li
i=1
has codimension in Xn less or equal to qpn . Hence pn+1 qpn , so pn pq n . Let Y be a subspace of X such that X = X0 Y. Then dim(Y ) = p. As L1 X0 ⊆ L0 X0 ⊆ X0 , we have that Z = {b ∈ L1 : bY ⊆ X0 }. Therefore there is an injective linear map from L1 /Z into the algebra of all operators on Y. Hence dim(L1 /Z) p 2 . Let us show now that dim(L0 /L1 ) < ∞. Fix i. Then [L0 , ei ] ⊆ L and the subspace {b ∈ L0 : [b, ei ] ∈ L0 } has a finite codimension in L0 . Hence L1 has finite codimension in L0 , as L1 = b ∈ L0 : [b, L] ⊆ L0 = b ∈ L0 : [b, ei ] ⊆ L0 for i = 1, . . . , q =
q b ∈ L0 : [b, ei ] ⊆ L0 . i=1
Thus dim(L0 /Z) = dim(L0 /L1 )+ dim(L1 /Z) < ∞. Part (iii) is proved. Finally, let X0 be closed. Let xk ∈ Xn and xk → x. Then x ∈ X0 and, for each a ∈ L, ax − axk a x − xk → 0. Assume, by induction, that Xn−1 is closed. As axk ∈ Xn−1 , we have that ax ∈ Xn−1 . Hence, by (4.2), x ∈ Xn . Thus Xn is closed. 2 For n ∈ N (cf. (4.3)), set Ln = b ∈ L0 : [b, L] ⊆ Ln−1 ,
so Ln ⊆ Ln−1 and [L, Ln ] ⊆ Ln−1 .
(4.6)
The sequence {Ln } of Lie subalgebras of L is called L0 -filtration of L (see [11]). We will consider now the action of Lie subalgebras Lp on subspaces Xn .
332
E. Kissin et al. / Journal of Functional Analysis 256 (2009) 323–351
Lemma 4.3. Let X0 be invariant for L0 , let {Xn } be (L,X0 )-filtration of X and {Ln } be L0 filtration of L. If Lm X ⊆ X0 , for some m 1, then Lp Xn ⊆ Xp−m+n+1
when p + n m − 1.
(4.7)
Proof. We have from (4.6) that LLm ⊆ Lm L + [L, Lm ] ⊆ Lm L + L0 , for all n. Therefore LLm X0 ⊆ Lm X + L0 X0 ⊆ X0 , so (4.2) implies Lm X0 ⊆ X1 . Assume that Lm Xk ⊆ Xk+1 , for some k 0. Then, by Lemma 4.2(i), LLm Xk+1 ⊆ Lm LXk+1 + L0 Xk+1 ⊆ Lm Xk + L0 Xk+1 ⊆ Xk+1 . Hence Lm Xk+1 ⊆ Xk+2 . By induction, Lm Xn ⊆ Xn+1 for all n, so (4.7) holds for p = m. Assume now that (4.7) holds for some p = k m and all n ∈ N. By (4.6), LLk+1 ⊆ Lk+1 L + [L, Lk+1 ] ⊆ Lk+1 L + Lk .
(4.8)
Hence LLk+1 X−1 ⊆ Lk X−1 ⊆ Xk−m , so, by (4.2), Lk+1 X ⊆ Xk−m+1 . Using (4.8), we have LLk+1 X0 ⊆ Lk+1 X + Lk X0 ⊆ Xk−m+1 . Hence, by (4.2), Lk+1 X0 ⊆ X(k+1)−m+1 . Assume that Lk+1 Xl ⊆ X(k+1)−m+l+1 , for some l 0. Then, by (4.8), LLk+1 Xl+1 ⊆ Lk+1 Xl + Lk Xl+1 ⊆ X(k+1)−m+l+1 , whence Lk+1 Xl+1 ⊆ X(k+1)−m+l+1 . By induction, Lk+1 Xn ⊆ X(k+1)−m+n+1 , for all n −1, so (4.7) holds for p = k + 1. Thus, by induction, (4.7) holds for all p m and n −1. Let 0 p < m. Then Lp Xn ⊆ L0 Xn ⊆ Xn ⊆ Xp−m+n+1 , as p − m + n + 1 n. If p = −1 then L−1 Xn ⊆ LXn ⊆ Xn−1 ⊆ Xp−m+n+1 , as p − m + n + 1 = n − m n − 1. 2 Given a Lie algebra L and its subalgebra L0 , we can consider L as a space X and L0 as its subspace X0 . Let ad(L) = {ad(a): a ∈ L} be the Lie algebra of operators on L where each ad(a) acts by ad(a)x = [a, x] for x ∈ L. Lemmas 4.2 and 4.3 yield (cf. [11]): Corollary 4.4. Let L0 be a Lie subalgebra of L and {Ln } be L0 -filtration of L. Then (i) {Ln } coincides with (ad(L), L0 )-filtration of L considered as a space and Z(ad(L), ad(L0 ), L0 ) = ad(L1 ); (ii) {ad(Ln )} is ad(L0 )-filtration of ad(L); (iii) Ln are Lie ideals of L0 and [Lp , Ln ] ⊆ Lp+n for −1 p + n; (iv) If q = codim(L0 ) < ∞ then dim(Ln−1 /Ln ) q n+1 . Let L be a subspace of B(X) and let X0 be invariant for L. Denote by Lk , k 1, the linear subspace of B(X) generated by all products a1 · · · · · ak , ai ∈ L. We say that L is operatornilpotent (to distinguish it from Lie nilpotency) on X0 if Lk X0 = {0} for some k 1. We will consider now some cases when (L,X0 )-filtration of X is non-trivial. Corollary 4.5. Let X0 be invariant for L0 , let {Xn } be (L,X0 )-filtration of X and {Ln } be L0 filtration of L. The filtration {Xn } is non-trivial if one of the following conditions holds: (i) codim(L0 ) < ∞ and codim(X0 ) < ∞; (ii) Lm ⊆ Z(L, L0 , X0 ), for some m 1, and Lm is not operator-nilpotent on X0 ; (iii) Lm ⊆ Z(L, L0 , X0 ), for some m 1, and Lp |X0 = {0} for all p m. Proof. Part (i) follows from Lemma 4.2(iv).
E. Kissin et al. / Journal of Functional Analysis 256 (2009) 323–351
333
Let Lm X ⊆ X0 , for some m. If Lnm X0 = {0} for all n, repeatedly applying (4.7), we have n−1 n−2 n−2 {0} = Lnm X0 = Ln−1 m (Lm X0 ) ⊆ Lm X1 = Lm (Lm X1 ) ⊆ Lm X2 = · · · ⊆ Lm Xn−1 ⊆ Xn .
If Lp |X0 = {0} for all p m, it follows from (4.7) that {0} = Lp X0 ⊆ Xp−m+1 . Hence in both cases Xn = {0} for all n. 2 Let, for example, X = L and X0 = L0 . By Corollary 4.4(i), Z(ad(L)) = ad(L1 ). If ad(L1 ) is not operator-nilpotent on L0 , it follows from Corollary 4.5(ii) that the filtration {Ln } is nontrivial. 5. Invariant subspaces of operator Lie algebras: the case when L0 is related to L Recall that a Lie subalgebra of B(X) is reducible if it has a non-trivial closed invariant subspace. In this section L is a Lie subalgebra (not necessarily infinite-dimensional) of B(X) and L0 is a non-trivial reducible Lie subalgebra of L of finite codimension. As L0 and L0 have the same closed invariant subspaces, we assume without loss of generality that L0 ∩ L = L0 (otherwise, we replace L0 by L0 ∩ L). Let φ be the canonical linear map from L onto L/L0 . For each a ∈ L0 , denote by θ (a) the operator on L/L0 defined by θ (a)φ(e) = φ [a, e] , for e ∈ L. (5.1) Then θ : a → θ (a) is a representation of L0 on L/L0 and Ker(θ ) = L1 . It is easy to see that θ is non-degenerate (Definition 2.3)
⇔
L0 is non-degenerate in L (Definition 1.1). (5.2)
Let X0 be a closed subspace of X invariant for L0 . Set X−1 = X and let {Xn }∞ n=−1 be (L,X0 ) filtration of X. For each n ∈ N ∪ {−1}, Xn = Xn /Xn+1 is a Banach space. Denote by τn the n . By Lemma 4.2, there is a representation ρn of L0 on X n canonical map from Xn onto X defined by ρn (h)τn (x) = τn (hx),
for h ∈ L0 , x ∈ Xn .
(5.3)
n into X n−1 by the formula Each e ∈ L defines operators Ene , for all n ∈ N, from X Ene τn (x) = τn−1 (ex),
where x ∈ Xn .
(5.4)
The maps e ∈ L → Ene are linear. By (4.1), for each 0 = x ∈ Xn \ Xn+1 , n ∈ N, there is e ∈ L such that τn−1 (ex) = 0. From this, from (5.4) and Lemma 4.2 it follows that Ene = 0 if e ∈ L0 , and Ker Ene : e ∈ L = {0} for all n ∈ N. (5.5) For each u = φ(e) ∈ L/L0 and n ∈ N, set Enu = Ene . As Ene = 0 if e ∈ L0 , the linear maps u → Enu are well defined on L/L0 for all n.
(5.6)
334
E. Kissin et al. / Journal of Functional Analysis 256 (2009) 323–351
Let h ∈ L0 and e ∈ L. Then, by (5.3) and (5.4), for x ∈ Xn , ρn−1 (h)Ene τn (x) = ρn−1 (h)τn−1 (ex) = τn−1 (hex) = τn−1 [h, e]x + τn−1 (ehx) = En[h,e] τn (x) + Ene ρn (h)τn (x). From this and from (5.5) we obtain that ρn−1 (h)Enu = Enu ρn (h) + Enθ(h)u for h ∈ L0 and u ∈ L/L0 ; Ker Enu : u ∈ L/L0 = {0} for all n ∈ N.
(5.7) (5.8)
Lemma 5.1. (i) Z(L) = L1 ∩ Ker(ρ−1 ) ⊆ Ker(ρn ) and ρn (h) h for all n ∈ N and h ∈ L0 . n , X n−1 ) and Enu e if u = φ(e). (ii) Enu ∈ B(X u v u (iii) En En+1 = Env En+1 for all n ∈ N and u, v ∈ L/L0 . Proof. Part (i) follows from Lemma 4.2(i) and the fact that, for all x ∈ Xn ,
ρn (h)τn (x) = τn (hx) = inf hx + y X inf h(x + y) X X X n
n
y∈Xn+1
y∈Xn+1
inf h x + y X = h τn (x) X . n
y∈Xn+1
We have that Enu e , where u = φ(e) and e ∈ L, since for x ∈ Xn ,
φ(e)
E n τn (x) X
n−1
= τn−1 (ex) X
n−1
= inf ex + y X inf e(x + z) X y∈Xn
inf e x + z X = e τn (x) X .
z∈Xn+1
n
z∈Xn+1
Let u = φ(e) and v = φ(e ), for some e, e ∈ L. By (4.2), τn−1 (ax) = 0 for all x ∈ Xn+1 , a ∈ L and n ∈ N. As L is a Lie algebra, [e, e ] ∈ L, so e e
e τn+1 (x) = τn−1 [e, e ]x = 0. En En+1 − Ene En+1 u . v This proves that Enu En+1 = Env En+1
2
Note that the condition that L is a Lie algebra and not just a module over a Lie algebra L0 is important and used in the proof of part (iii) of the above lemma. Let M = {0} be a subspace of L/L0 . For all n ∈ N, set n Kern {0} = X
and
Kern (M) =
Ker Enu : u ∈ M .
n . If M ⊆ M then Kern (M ) ⊆ Kern (M) for all n. Then Kern (M) ⊆ X
(5.9)
E. Kissin et al. / Journal of Functional Analysis 256 (2009) 323–351
335
Lemma 5.2. Let M be a subspace of L/L0 and h ∈ L0 . n invariant for ρn (h). (i) If M is invariant for θ (h) then Kern (M) is a closed subspace of X (ii) Each Env , v ∈ L/L0 , maps Kern (M) into Kern−1 (M). (iii) If Kern (M) = {0}, for some n 0, then Kern+1 (M) = {0}. n , Ker(Enu ) is a closed subspace Proof. As each Enu , u ∈ L/L0 , is a bounded operator on X of Xn . Hence Kern (M) is a closed subspace of Xn . Let u ∈ M. Since M is invariant for θ (h), θ(h)u θ (h)u ∈ M. If ξ ∈ Kern (M), then Enu ξ = 0 and En ξ = 0. Hence it follows from (5.7) that Enu ρn (h)ξ = ρn−1 (h)Enu ξ − Enθ(h)u ξ = 0. Therefore ρn (h)ξ ∈ Kern (M), so Kern (M) is invariant for ρn (h). Part (i) is proved. u E v ξ = E v E u ξ = 0, for ξ ∈ Ker (M), u ∈ M and v ∈ L/L . By Lemma 5.1(iii), En−1 n 0 n n−1 n v Hence En maps Kern (M) into Kern−1 (M). n+1 , there is v ∈ L/L0 such that E v ξ = 0. If {0} = ξ ∈ By (5.8), for each 0 = ξ ∈ X n+1 v Kern+1 (M) then, by (ii), 0 = En+1 ξ ∈ Kern (M). 2 Proposition 5.3. Let M ⊂ K be subspaces of L/L0 invariant for θ, q = dim(K/M) < ∞. Let the quotient representation θ |K/M of L0 on K/M be an eigen-representation. If Kerm (K) = {0} for some m, then there is r ∈ N such that Kern (M) = {0} for n r. q
Proof. As θ |K/M is an eigen-representation, there are {ui }i=1 in K \ M such that together with q M they span K, and fi = θ (hi )ui − μi ui ∈ M, for some {hi }i=1 in L0 and 0 = μi ∈ C. Set n invariant for ρn . Un = Kern (M). By Lemma 5.2, Un are closed subspaces of X q f Set ri = [2 hi /|μi |] and r = m+ i=1 ri . Assume that Ur = {0}. Then Er i ξ = 0 for ξ ∈ Un . fi θ(hi )ui ui ui Thus Er ξ = μi Er ξ + Er ξ = μi Er ξ. By (5.7), ρr−1 (hi )Erui ξ = Erui ρr (hi )ξ + Erθ(hi )ui ξ = Erui ρr (hi )ξ + μi Erui ξ. ui By Lemma 5.2(ii), Erui ξ ∈ Ur−1 . Using (5.7), we obtain similarly that Er−k . . . Erui ξ ∈ Ur−k−1 and u
u
u
i i i ρr−k−1 (hi )Er−k . . . Erui ξ = Er−k . . . Erui ρr (hi )ξ + (k + 1)μi Er−k . . . Erui ξ.
(5.10)
Fix i. Set A = ρr−k−1 (hi )|Ur−k−1 and B = ρr (hi )|Ur . Consider the map T : C → AC − CB on the Banach space of all bounded operators C from Ur to Ur−k−1 . By Rosenblum’s theorem (see [15, Theorem 0.12]), the spectrum Sp(T ) of T is contained in the set {α − β: α ∈ Sp(A), β ∈ Sp(B)}. By Lemma 5.1(i), max( A , B ) hi . Thus Sp(T ) lies in the circle of radius 2 hi . ui u From this and from (5.10) it follows that, if Er−k . . . Er i |Ur = 0, then (k + 1)μi ∈ Sp(T ), so that (k + 1)|μi | 2 hi . Hence ui Er−k . . . Erui U = 0, r
if k = ri .
(5.11)
336
E. Kissin et al. / Journal of Functional Analysis 256 (2009) 323–351
u1 Let k1 be the smallest number such that Er−k . . . Eru1 Ur = {0}. By (5.11), 0 k1 r1 . 1 u1 Set Y1 = Ur , if k1 = 0, and Y1 = Er−k . . . Eru1 Ur , otherwise. Then {0} = Y1 ⊆ Ur−k1 and 1 +1 u1 Er−k1 Y1 = {0}. By Lemma 5.1(iii), u2 u2 u2 u2 u1 u1 u2 Er−k . . . Er−k Y1 = Er−k . . . Er−k E u1 . . . Eru1 Ur = Er−k . . . Er−k−1 Er−k . . . Eru2 Ur . 1 −k 1 1 r−k1 +1 1 −k u2 u2 . . . Er−k Y1 = {0}. Set By (5.11), there is the smallest k2 , 0 k2 r2 , such that Er−k 1 −k2 1 u2 u2 Y2 = Y1 , if k2 = 0, and Y2 = Er−k1 −k2 +1 . . . Er−k1 Y1 , otherwise. Then {0} = Y2 ⊆ Ur−k1 −k2 u2 and Er−k Y2 = {0}. Moreover, by Lemma 5.1(iii), 1 −k2 u1 u1 u2 u2 u2 u1 Er−k Y2 = Er−k E u2 . . . Er−k Y1 = Er−k . . . Er−k . . . Er−k Y1 = {0}. 1 −k2 1 −k2 r−k1 −k2 +1 1 1 −k2 1 1 −1
Repeating this procedure, we obtain integers ki , 0 ki ri , for 1 i q, and the subspace ui Yq = {0} for all i. Set p = r − k1 − k2 − {0} = Yq ⊆ Ur−k1 −k2 −···−kq such that Er−k 1 −k2 −···−kq · · · − kq . Then p m and {0} = Yq ⊆ Up ∩ Ker Epui : 1 i q = Kerp (K). On the other hand, as Kerm (K) = {0} and p m, we have from Lemma 5.2(iii) that Kerp (K) = {0}. This contradiction shows that Ur = {0}. 2 We shall now prove the main result of this section. Theorem 5.4. Let X be an infinite-dimensional Banach space and L be a finite- or infinitedimensional Lie subalgebra of B(X). Let X0 be a closed subspace of X invariant for a Lie subalgebra L0 of L and let (L, X0 )-filtration {Xn } of X be non-trivial. If L0 is non-degenerate in L then, for some m, {0} = Xm ⊆ X0 is a closed subspace of X invariant for L. Proof. By (5.2), θ is non-degenerate. Theorem 2.6 implies that there are subspaces {0} = V0 ⊂ quotient representations θj of L0 on Vj /Vj −1 V1 ⊂ · · · ⊂ Vk = L/L0 invariant for θ such that all
are eigen-representations. By (5.8), Ker0 (Vk ) = {Ker(E0u ): u ∈ L/L0 } = {0}. As the representation θk of L0 on Vk /Vk−1 is an eigen-representation, Proposition 5.3 implies that there is nk−1 0 such that Kernk−1 (Vk−1 ) = {0}. Repeating this process, we obtain n0 such that n0 . Hence X n0 = {0}. Kern0 (V0 ) = {0}. As V0 = {0}, we have from (5.9) that Kern0 (V0 ) = X Thus Xn0 is invariant for L. 2 Corollary 5.5. Let X and L be as in Theorem 5.4. Let X0 be a closed subspace of X invariant for a Lie subalgebra L0 of L ⊆ B(X) and let L0 be related to L. (i) If (L,X0 )-filtration {Xn } of X is non-trivial, then L has a closed non-trivial invariant subspace contained in X0 . (ii) If codim(X0 ) < ∞ then L has a closed non-trivial invariant subspace W contained in X0 and codim(W ) < ∞. Proof. Consider Lie subalgebras L0 = L0 ⊆ L1 ⊆ · · · ⊆ Lp = L of L such that each Li is nondegenerate in Li+1 . Set Y0 = X0 . As L1 ⊆ L, we have from (4.2) that (L1 , Y0 )-filtration {Yn }
E. Kissin et al. / Journal of Functional Analysis 256 (2009) 323–351
337
of X satisfies Xn ⊆ Yn ⊆ X0 , so it is non-trivial. Hence, by Theorem 5.4, for some m1 , the closed subspace Ym1 ⊆ X0 is invariant for L1 . Set Z0 = Ym1 . Then Xm1 ⊆ Z0 . It follows from (4.2) that (L2 , Z0 )-filtration {Zn } of X satisfies Xn+m1 ⊆ Zn ⊆ Z0 , for n ∈ N, so that it is nontrivial. If p = 2 then L = L2 and, by Theorem 5.4, for some m2 , the closed subspace Zm2 ⊆ X0 is invariant for L. If p > 2, repeating this, we obtain that L has a closed non-trivial invariant subspace contained in X0 . Part (i) is proved. If X0 has finite codimension, then (L,X0 )-filtration {Xn } of X is non-trivial and Ym1 has finite codimension, so that Zm2 has finite codimension. Repeating this process, we complete the proof. 2 Remark 5.6. (1) Let L ⊆ B(X), let L0 be a Lie subalgebra of L with codim(L0 ) < ∞, let X0 be a non-trivial closed subspace of X invariant for L0 and let (L, X0 )-filtration of X be nontrivial. Suppose that a one-dimensional subspace Cφ(e), e ∈ L, of L/L0 is invariant for all θ (h), h ∈ L0 . Then L = Ce L0 is a Lie subalgebra of L. If L0 is not a Lie ideal of L and L is related to L, then L0 is related to L. Hence L0 is related to L and, by Corollary 5.5, L is reducible. The case when L0 L and L is related to L will be considered in Theorem 8.1(ii). Corollary 5.7. Let L0 be a Lie subalgebra of L ⊆ B(X) and let L0 be related to L. If L0 is operator-nilpotent (Ln0 = {0} for some n), then L is reducible. Proof. Any operator-nilpotent Lie subalgebra L0 of B(X) has a closed invariant subspace of codimension 1. Indeed, L0 X is a proper subspace of X and each subspace containing it is invariant for L0 . Applying now Corollary 5.5, we complete the proof. 2 In the above corollary one can consider more general condition L0 X = X instead of the condition that L0 is operator-nilpotent. It should be noted that if a Lie subalgebra L0 of L is operator-nilpotent and has finite codimension but not related to L, then L may be irreducible as the following example shows. Example 5.8. Let Y be a Banach space, dim(Y ) = ∞. Then X = Y ⊕ Y is a Banach space with norm x ⊕ y = sup( x , y ). Let a bounded operator S on Y have no non-trivial closed a a12 invariant subspaces. Set S = S ⊕S. For a = a11 ∈ M2 (C), let Aa be the operator on X acting 21 a22 by the formula Aa (x ⊕ y) = (a11 x + a12 y) ⊕ (a21 x + a22 y) and let A = {Aa : a ∈ M2 (C)}. Then L = C S + A is an irreducible Lie subalgebra of B(X). Set L0 = {Aa : a ∈ M2 (C), a11 = a21 = a22 = 0}. Then L0 is a Lie subalgebra of L, it is operator-nilpotent and codim(L0 ) = 4. However, since A is a Lie ideal of L, L0 is not related to L. We shall now develop a new approach to the problem of the existence of invariant subspaces for Lie subalgebras of B(X). It will allow us to give a different proof of Theorem 5.4 and to treat in the next section the case when L0 is a Lie ideal of L. Let {Xn } be (L,X0 )-filtration of X. Then n = Xn /Xn+1 are Banach spaces with norms · n . Consider the graded Banach space X = X
∞ n=−1
n with n = x = x−1 ⊕ · · · ⊕ xn ⊕ · · · : xn ∈ X x = sup xn n < ∞ . X
338
E. Kissin et al. / Journal of Functional Analysis 256 (2009) 323–351
Set V = L/L0 . Let u → Enu , n ∈ N, be the linear maps from V into the Banach spaces of all n−1 defined in (5.6). Denote by E u the operators on X that act n into X bounded operators from X by x = E0u x0 ⊕ E1u x1 ⊕ · · · ⊕ Enu xn ⊕ · · · . E u
(5.12)
E u e , for It follows from Lemma 5.1 that u → E u is a linear map from V into B(X), each e ∈ L satisfying u = φ(e), and Ev Eu = EuEv
for all u, v ∈ V .
(5.13)
by Ψ (e) = E φ(e) . Note that Ψ This map extends to a linear bounded map Ψ from L into B(X) is not, generally speaking, a Lie homomorphism. Set V nil = {u ∈ V : E u is nilpotent} (see (3.4)). Then M = {E u : u ∈ V } is a linear subspace of mutually commuting bounded operators and NM = {E u : u ∈ V nil }. Set of B(X) Yn =
n . Ker E u : u ∈ V nil ∩ X
(5.14)
p
Let V nil = {0}, dim(V nil ) = p < ∞ and let {ui }i=1 be a basis in V nil . Then (E ui )ki = 0 for n−1 and E k = 0 for all i and n. n ⊆ X some ki . Set Ei = E ui and k = max(ki ). Then Ei X i n = Proposition 5.9. If X {0} for some n pk, then Ym = {0} for some m satisfying n − pk m n. n = {0}. Then 0 m1 < k and E1 K1 = {0}. Proof. Take the largest m1 such that K1 = E1m1 X Take the largest m2 such that K2 = E2m2 K1 = {0}. Then 0 m2 < k and E2 K2 = {0}. As E1 and E2 commute (see (5.13)), E1 K2 = E1 E2m2 K1 = E2m2 E1 K1 = {0}. Repeating this process, we obtain p−1 n−m1 −···−mp , n ⊆ X . . . E2m2 E1m1 X {0} = Kp = Ep p Kp−1 = Ep p Ep−1
m
m
m
for some 0 mp < k, such that E1 Kp = E2 Kp = · · · = Ep Kp = {0}. As NM is the linear span of all Ei , {0} = Kp ⊆ Yn−m1 −···−mp . Setting m = n − m1 − · · · − mp , we complete the proof. 2 n defined in (5.3), satisfying Let ρn , n ∈ N ∪ {−1}, be the representations of L0 on X that acts by the formula ρn (h) h for h ∈ L0 . Let ρ(h) be the operator on X x−1 ⊕ · · · ⊕ ρn (h) xn ⊕ · · · . ρ(h) x = ρ−1 (h)
(5.15)
and ρ(h) h . Then ρ is a representation of L0 on X Let θ be the representation of L0 on V defined in (5.1). It follows from (5.7) that ρ(h)E u = E u ρ(h) + E θ(h)u
for h ∈ L0 and u ∈ V .
Using this approach, we will now give another proof of Theorem 5.4.
(5.16)
E. Kissin et al. / Journal of Functional Analysis 256 (2009) 323–351
339
Proof. We have from (5.14) that Yn = {Ker Enu : u ∈ V nil }. Comparing (5.16) and (3.5), and n = {0} then, using that θ is non-degenerate, we obtain from Corollary 3.3 that V = V nil . If all X u : u ∈ V } for some m 0, which contradicts (5.8). by Proposition 5.9, {0} = Ym = {KerEm n = {0} for some n, so Xn is invariant for L. 2 Thus X 6. Invariant subspaces of operator Lie algebras: the case when L0 is a Lie ideal of L In this section we will turn to the case when L0 is a Lie ideal of L. The restriction codim(L0 ) < ∞ will be dropped; instead we will only assume that dim(L0 ) = ∞. We will show that if L0 has a closed invariant subspace X0 of finite codimension, then L has a non-trivial closed invariant subspace. Recall (see (4.3)) that Z(L) ⊆ L1 and Z(L)X ⊆ X0 . Proposition 6.1. Let L0 be a Lie subalgebra of L ⊆ B(X), let X0 be a closed subspace
of X invariant for L0 and {Xn } be (L,X0 )-filtration of X. Assume that Z(L) L. Set X = ∞ n=0 Xn . Then (i) Z(L)X ⊆ X. (ii) If dim(L0 ) = ∞, codim(L0 ) < ∞ and codim(X0 ) < ∞, then X is a non-trivial closed subspace invariant for L. (iii) If dim(L0 ) = ∞, L0 L and codim(X0 ) < ∞, then X is a non-trivial closed subspace invariant for L and dim(L0 /Z(L)) < ∞. Proof. Set Z = Z(L). We have ZX ⊆ X0 . If ZX ⊆ Xk , for some k 0, then, as Z L, LZX ⊆ [L, Z]X + ZLX ⊆ ZX + ZX ⊆ Xk . Hence it follows from (4.2) that ZX ⊆ Xk+1 . By induction, ZX ⊆ Xn for all n ∈ N. Part (i) is proved. Clearly, X is invariant for L. By Lemma 4.2(iv), X is closed. Since Z L, we have from (i) that ZX ⊆ X. Let us show that X = {0}. (ii) Since codim(L0 ) < ∞ and codim(X0 ) < ∞, it follows from Lemma 4.2(iii) that dim(L0 /Z) < ∞. As dim(L0 ) = ∞, Z = {0}. Hence X = {0}. (iii) Since codim(X0 ) < ∞, the representation ρ−1 of L0 on X/X0 is finite-dimensional. As L0 L, L0 = L1 . Hence, by (4.3), Ker(ρ−1 ) = Z. Thus dim(L0 /Z) < ∞. If X = {0} then Z = {0}, so that dim(L0 ) < ∞ which contradicts our assumption. 2 We will now show that the conditions codim(X0 ) < ∞ and L0 L automatically imply Z(L) L. By Proposition 6.1(iii), this will guarantee ∞ n=0 Xn = {0}. By Lemma 4.2(i), zXn ⊆ Xn+1 for z ∈ Z(L). Hence, for all n ∈ N, z defines linear operators n−1 into X n by the formula Fnz from X Fnz τn−1 (x) = τn (zx),
for x ∈ Xn−1 .
(6.1)
that acts on by the formula Denote by F z the operator on X x = x−1 ⊕ · · · ⊕ xn ⊕ · · · ∈ X z F z x = 0 ⊕ F0z x−1 ⊕ F1z x0 ⊕ · · · ⊕ Fn+1 xn ⊕ · · · .
(6.2)
340
E. Kissin et al. / Journal of Functional Analysis 256 (2009) 323–351
Proposition 6.2. If Z(L) ⊆ L2 , the operators ρ([z, e]) are quasinilpotent, for all e ∈ L, z ∈ Z(L). Proof. Let e ∈ L. For u = φ(e) and z ∈ Z(L), we have from (5.12) and (6.2) that z u z u F ,E x = −E0u F0z x−1 ⊕ F0z E0u − E1u F1z x0 ⊕ · · · ⊕ Fnz Enu − En+1 xn ⊕ · · · . Fn+1 As zX ⊆ X0 , τ−1 (zX) = 0. Since [z, e] ∈ L0 , it follows from (5.4) and (6.1) that −E0u F0z τ−1 (x) = −E0u τ0 (zx) = τ−1 (−ezx) = τ−1 (ze − ez)x = ρ−1 [z, e] τ−1 (x),
(6.3)
for x ∈ X. For x ∈ Xn , we have z u z u u Fn En − En+1 τn (x) = Fnz τn−1 (ex) − En+1 Fn+1 τn+1 (zx) = τn (zex) − τn (ezx) = τn [z, e]x = ρn [z, e] τn (x). Therefore [F z , E u ] x = ρ([z, e]) x , so [F z , E u ] = ρ([z, e]). Since Z(L) ⊆ L2 , we have [z, e] ∈ L1 = Ker(θ ). Hence, by (5.16), [ρ([z, e]), E u ] = θ([z,e])u E = 0. Therefore it follows from the Kleinecke–Shirokov theorem (see [6]) that ρ([z, e]) is quasinilpotent. 2 Let X0 be a closed subspace of X invariant for L0 . We will say that X0 is maximal, if it is not contained in a larger closed proper subspace of X invariant for L0 . If X0 is not maximal and codim(X0 ) < ∞, it can always be extended to a maximal invariant subspace. In the proof of our next result we use the following extension of Engel Theorem proved in [8]: if K is a set of nilpotent operators on a finite-dimensional space such that [k1 , k2 ] ∈ K, for all k1 , k2 ∈ K, then Ker(K) = 0. This is why we have to impose the condition codim(X0 ) < ∞. Proposition 6.3. Let X0 be a maximal closed subspace of X invariant for L0 and let {Xn } be (L,X0 )-filtration of X. If codim(X0 ) < ∞ and L0 L then Z(L) L. Proof. Set Z = Z(L). As L0 L, L0 = L1 . Hence, by Lemma 5.1(i), Z = Ker(ρ−1 ).
(6.4)
−1 is irreducible. Hence the Lie alSince X0 is maximal, the representation ρ−1 of L0 on X gebra L = ρ−1 (L0 ) of operators on X−1 has no non-trivial invariant subspaces. Its subset K = {ρ−1 ([z, e]): e ∈ L, z ∈ Z} consists of nilpotent operators. Indeed, since Z ⊆ L2 , we have for e ∈ L and z ∈ Z. As X −1 is finitefrom Proposition 6.2 that ρ([z, e]) is quasinilpotent in X dimensional, each ρ−1 ([z, e]) is nilpotent. Let us show that all ρ−1 ([z, e]) = 0. As ρ−1 is a Lie homomorphism, we have for h ∈ L0 that ρ−1 (h), ρ−1 [z, e] = ρ−1 h, [z, e] = ρ−1 [h, z], e + ρ−1 z, [h, e] .
E. Kissin et al. / Journal of Functional Analysis 256 (2009) 323–351
341
As L0 L, we have [h, e] ∈ L0 . By Lemma 4.2(ii), Z L0 . Therefore [z, [h, e]] ∈ Z and [h, z] ∈ Z. Hence, by (6.4), ρ−1 ([z, [h, e]]) = 0, so that [ρ−1 (h), ρ−1 ([z, e])] = ρ−1 ([[h, z], e]) ∈ K. Thus [a, k] ∈ K
for all a ∈ L and k ∈ K.
(6.5)
In particular, [k1 , k2 ] ∈ K for all k1 , k2 ∈ K. As K consists of nilpotent operators, it follows from −1 such that kξ0 = 0 for all k ∈ K. [8, Theorem II.2.1 ] that there is 0 = ξ0 ∈ X −1 : Kξ = {0}}. Then R = {0} and, by (6.5), R is invariant for L. As L has no Let R = {ξ ∈ X −1 . Hence K X −1 = {0}, so K = {0}. Thus ρ−1 ([z, e]) = 0 non-trivial invariant subspaces, R = X for all e ∈ L and z ∈ Z. Therefore, by (6.4), [Z, L] ⊆ Z, so that Z L. 2 The following theorem is the central result of this section. Theorem 6.4. Let L0 L ⊆ B(X), let dim(L0 ) = ∞ and let L0 have a closed invariant subspace X0 of finite codimension. Then (i) L has a non-trivial closed invariant subspace Y such that Y + X0 = X. (ii) L has a Lie ideal C ⊆ L0 such that CX
⊆ Y and dim(L0 /C) < ∞. (iii) If X0 is maximal, Y can be taken as Xn (so Y ⊆ X0 ) and C = Z(L, L0 , X0 ). Proof. If X0 is not maximal, replace it by a larger maximal closed subspace Y0 invariant for L0 . Let {Yn } be (L,Y0 )-filtration of X. The closed subspace Y = Yn is invariant for L and Y + X0 ⊆ Y0 = X. Set C = Z(L, L0 , Y0 ). By Proposition 6.3, C L. Applying Proposition 6.1(iii), we complete the proof. 2 Remark 6.5. The closed subspace Y in Theorem 6.4 invariant for L is not, generally speaking, finite codimensional. Indeed, let X be an infinite-dimensional Banach space such that there exists a bounded operator e on X that has no non-trivial closed invariant subspaces (for example, X = l1 (see [17])). Let Z be a closed subspace of codimension 1 in X. Set X = X ⊕ X, X0 = Z ⊕ X. Set L = Ce ⊕ B(X) and L0 = {0} ⊕ B(X). Then L0 L and codim(L0 ) = 1, the subspace X0 is invariant for L0 and codim(X0 ) = 1, while the Lie algebra L has only two non-trivial closed invariant subspaces {0} ⊕ X and X ⊕ {0}; both of them have infinite codimension. As the following example shows, the conditions dim(L0 ) = ∞ and codim(X0 ) < ∞ in Theorem 6.4 are crucial. Example 6.6. Let L0 L ⊆ B(X) and let L0 have a closed invariant subspace X0 . If either dim(L0 ) < ∞, or codim(X0 ) = ∞, then L may be irreducible. Indeed, let H and K be Hilbert spaces, dim(H ) = ∞ and dim(K) < ∞. Let X = H ⊗ K and L = B(H ) ⊗ C1K + C1H ⊗ B(K). Let H be a subspace of H of codimension 1. Consider the Lie ideals B(H ) ⊗ C1K and C1H ⊗ B(K) of L. (i) Let L0 = B(H ) ⊗ C1K . Then dim(L0 ) = ∞, the closed subspace X0 = H ⊗ Ck, for each k ∈ K, is invariant for L0 and codim(X0 ) = ∞; (ii) Let L0 = C1H ⊗ B(K). Then dim(L0 ) < ∞, the closed subspace X0 = Ch ⊗ K, for each h ∈ H, is invariant for L0 and codim(X0 ) = ∞;
342
E. Kissin et al. / Journal of Functional Analysis 256 (2009) 323–351
(iii) Let L0 = C1H ⊗ B(K). Then dim(L0 ) < ∞, the closed subspace X0 = H ⊗ K is invariant for L0 and codim(X0 ) < ∞. However, as the commutant of L coincides with C1X , L has no non-trivial closed invariant subspaces. It follows from Proposition 6.3 that if a Lie ideal of an irreducible Lie algebra has an invariant subspace of finite codimension then it has a special simple structure. Proposition 6.7. Let L0 L ⊆ B(X). Suppose
that L0 has a maximal closed invariant subspace X0 such that k = codim(X0 ) < ∞ and Xn = {0}, where {Xn } is (L,X0 )-filtration of X. (In particular, this holds if L is irreducible.) Then (i) L0 is isomorphic to an irreducible Lie algebra of operators on k-dimensional space. (ii) If k = 1 then L0 = C1.
Proof. By Proposition 6.3, Z(L) L. If Xn = {0} then, by Proposition 6.1(i), Z(L) = {0}. By (6.4), Z(L) = Ker(ρ−1 ). Hence ρ−1 is a faithful representation of L0 on the k-dimensional −1 . Thus L0 is isomorphic to the Lie algebra ρ−1 (L0 ) and, since X0 is maximal, ρ−1 (L0 ) space X has no invariant subspaces. part (i) is proved. If k = 1, dim(L0 ) = 1. Hence L0 = Ca for some 0 = a ∈ L. Suppose that a = λ1 and consider the Lie algebra C = L + C1. Then C0 = L0 + C1 C, X0 is invariant for C0 and {Xn } is also (C,X0 )-filtration of X. By the above argument, dim(C0 ) = 1. This contradiction shows that L0 = C1. 2 7. Finite-dimensional invariant subspaces In this section we consider the case when L0 has an invariant subspace X0 of finite dimension. Let X ∗ be the dual space of X and X0⊥ = {f ∈ X ∗ : f (x) = 0 for all x ∈ X0 } be the annihilator of X0 in X ∗ . Then X0⊥ is a closed subspace of X ∗ and codim(X0⊥ ) = dim(X0 ) < ∞. For a ∈ B(X), denote by a ∗ the conjugate operator on X ∗ defined by (a ∗ f )(x) = f (ax) for f ∈ X ∗ and x ∈ X. Set L∗ = {a ∗ : a ∈ L} and L∗0 = {a ∗ : a ∈ L0 }. As [a, b]∗ = −[a ∗ , b∗ ],
(7.1)
L∗ is a Lie subalgebra of B(X ∗ ) and L∗0 is its Lie subalgebra. The subspace X0⊥ is invariant for L∗0 . Theorem 7.1. Let L be a finite- or infinite-dimensional Lie subalgebra of B(X). Let a Lie subalgebra L0 of L have a non-trivial finite-dimensional invariant subspace X0 . (i) If codim(L0 ) < ∞ and the representation θ of L0 on L/L0 is non-degenerate, then L has a finite-dimensional invariant subspace that contains X0 . (ii) If dim(L0 ) = ∞ and L0 L, then L has a non-trivial closed invariant subspace W . If X0 is a minimal subspace invariant for L0 , then there is a non-trivial closed invariant subspace W such that X0 ⊆ W.
E. Kissin et al. / Journal of Functional Analysis 256 (2009) 323–351
343
Proof. Since codim(X0⊥ ) < ∞, (L∗ , X0⊥ )-filtration {Xn∗ } of X ∗ is non-trivial. (i) Let codim(L0 ) < ∞ and θ be non-degenerate. By (5.2), L = L0 +[L0 , L]. Hence it follows from (7.1) that L∗ = L∗0 + [L∗0 , L∗ ]. Therefore, by (5.2), θ ∗ is non-degenerate. ∗ is a closed subSince codim(L∗0 ) < ∞, it follows from Theorem 5.4 that, for some m, Xm ∗ ∗ ∗ space of X invariant for L and codim(Xm ) < ∞. ∗ in X ∗∗ . Then Y is a subspace Let X ∗∗ be the dual space of X ∗ and Y be the annihilator of Xm ∗∗ ∗∗ ∗∗ of X invariant for the Lie subalgebra L = {a : a ∈ L} of B(X ∗∗ ) and dim(Y ) < ∞. The space X can be considered as a closed subspace of X ∗∗ invariant for L∗∗ and a ∗∗ |X = a for a ∈ L. ∗ ⊆ X ∗ = X ⊥ . Therefore the finite-dimensional subspace W = Y ∩ X of X Hence X0 ⊆ Y , as Xm 0 0 is invariant for L and X0 ⊆ W. Part (i) is proved. (ii) Let dim(L0 ) = ∞ and L0 L. Then dim(L∗0 ) = ∞ and, by (7.1), L∗0 L∗ . We may assume that X0 is the minimal subspace invariant for L0 . Then X0⊥ is a maximal subspace invariant for L∗0 and codim(X0⊥ ) < ∞. Hence it follows from Theorem 6.4 that L∗ has a non-trivial closed
invariant subspace Y = Xn∗ . Then the annihilator Y ⊥ of Y in X ∗∗ is a closed subspace invariant for the Lie algebra L∗∗ . Consider X as a closed subspace of X ∗∗ invariant for L∗∗ . Then a ∗∗ |X = a, for a ∈ L, and X0 ⊆ Y ⊥ , as Y ⊆ X0∗ = X0⊥ . Hence the subspace W = Y ⊥ ∩ X of X is invariant for L and X0 ⊆ W. Thus W = {0}. We have that X is dense in X ∗∗ in the σ (X ∗∗ , X ∗ ) topology. On the other hand, Y ⊥ is closed in ∗∗ X in the σ (X ∗∗ , X ∗ ) topology and {0} = Y ⊥ = X ∗∗ . Therefore X Y ⊥ . Hence W = X. 2 Remark 7.2. As Example 6.6(ii) shows, if dim(L0 ) < ∞ in Theorem 7.1(ii), then L can be irreducible. Corollary 7.3. Let L be a finite- or infinite-dimensional Lie subalgebra of B(X). Let a Lie subalgebra L0 of L have a non-trivial finite-dimensional invariant subspace X0 . If L0 is related to L, then L has a finite-dimensional invariant subspace that contains X0 . Proof. As L0 is related to L, there are Lie subalgebras L0 = L0 ⊆ L1 ⊆ · · · ⊆ Lp = L of L such that the representations θi , 0 i p − 1, of Li on Li+1 /Li are non-degenerate. Apply now Theorem 7.1(i) to each pair (Li , Li+1 ). 2 8. Combined cases. Invariant subspaces of associative operator algebras and semigroups 8.1. Combined cases In this subsection we shall combine the results of Sections 5–7. Theorem 8.1. Let L0 ⊆ L ⊆ L be Lie subalgebras of B(X) and let dim(L0 ) = ∞. Suppose that L0 has a closed invariant subspace X0 . (i) Let L0 be related to L and L L. If codim(X0 ) < ∞ or dim(X0 ) < ∞ then L is reducible. (ii) Let L0 L, let codim(L0 ) < ∞ and L be related to L. If codim(X0 ) < ∞ and L0 has no operator-nilpotent Lie ideals of finite codimension (see Section 4), then L is reducible. Proof. (i) As L0 and L are related, it follows from Corollaries 5.5 and 7.3 that L has a non-trivial closed invariant subspace of finite codimension (respectively, of finite dimension). As L L and dim(L) = ∞, we have from Theorems 6.4 and 7.1(ii) that L is reducible.
344
E. Kissin et al. / Journal of Functional Analysis 256 (2009) 323–351
(ii) As L0 L, dim(L0 ) = ∞ and codim(X0 ) < ∞, it follows from Theorem 6.4(i) and (ii) that L has a non-trivial closed invariant subspace Y0 and a Lie ideal C contained in L0 such that CX ⊆ Y0 and dim(L0 /C) < ∞. If dim(Y0 ) < ∞ then, as L is related to L, it follows from Corollary 7.3 that L has a finitedimensional invariant subspace. Let dim(Y0 ) = ∞. As codim(L0 ) < ∞ and dim(L0 /C) < ∞, we have codim(C) < ∞. Set C0 = C and consider C0 -filtration {Cn } of L. By Corollary 4.4(iii), codim(C1 ) < ∞, so that C1 = {0}. As C1 X ⊆ CX ⊆ Y0 , we have C1 ⊆ Z(L, C 0 , Y0 ). We also have [C1 , L0 ] ⊆ [C0 , L0 ] ⊆ C0 and [C1 , L0 ], L ⊆ C1 , [L0 , L] + [C1 , L], L0 ⊆ [C1 , L] + [C0 , L0 ] ⊆ C0 . Hence [C1 , L0 ] ⊆ C1 , so that C1 L0 . As the Lie ideal C1 is not operator-nilpotent on X, we have that, for any n 1, {0} = C1n+1 X = C1n (C1 X) ⊆ C1n Y0 . Therefore, by Corollary 4.5(ii), (L,Y0 )filtration {Yn } of X is non-trivial. As Y0 is invariant for L and L is related to L, we obtain from Corollary 5.5 that L is reducible. 2 Problem 1. Let L0 be a Lie subalgebra of L ⊆ B(X) of finite codimension and dim(L0 ) = ∞. Assume that there is L such that L0 L and L L. If L0 has a closed invariant subspace of finite codimension or dimension, is L always reducible? Note that it follows from Theorems 6.4 and 7.1(ii) that in the conditions of Problem 1 L has a closed non-trivial invariant subspace W. However, W may have infinite codimension and dimension. Let L0 be a Lie subalgebra of B(X). Set Nor(L0 ) = {A ∈ B(X): [A, L0 ] ⊆ L0 }. Then Nor(L0 ) is a Lie subalgebra of B(X) and L0 Nor(L0 ). A closed subspace Y of X is superinvariant for L0 if it is invariant for Nor(L0 ). Corollary 6.4 and Theorem 7.1(ii) yield Corollary 8.2. Let L0 be a Lie subalgebra of B(X) and dim(L0 ) = ∞. If L0 has a closed invariant subspace X0 of finite codimension (respectively, finite dimension), then it has a nontrivial superinvariant subspace W such that W ⊆ X0 (respectively, W ⊇ X0 ). The example below shows that the conditions codim(X0 ) < ∞ and, respectively, dim(X0 ) < ∞ in Corollary 8.2 cannot be dropped. Example 8.3. (Theorems 4.2 and 4.3(ii) of [9].) Let H be an infinite-dimensional Hilbert space and S be a closed unbounded self-adjoint operator on H with domain D(S) dense in H. Let A(S) be the associative subalgebra of B(H ) of all operators A such that AD(S) ⊆ D(S) and that the operator [S, A] on D(S) extends to a bounded operator [S, A] on H. Set X = H ⊕ H, E=
0 0
A i[S, A] : A ∈ A(S) , 0 A iSx + tx Xt = : x ∈ D(S) x
1H 0
,
L0 =
for t ∈ C. Then L0 is an associative subalgebra of B(X), Nor(L0 ) = L0 CE and L0 is a Lie ideal of codimension 1 in Nor(L0 ). Non-trivial invariant subspaces of L0 are H ⊕ {0} and all Xt ,
E. Kissin et al. / Journal of Functional Analysis 256 (2009) 323–351
345
for t ∈ C. On the other hand, Nor(L0 ) has a unique non-trivial invariant subspace H ⊕ {0}. It neither contains, nor is contained in any subspace Xt . 8.2. Operator Lie algebras with Lie subalgebras of small codimension In order to illustrate the situation we will consider Lie algebras of operators with Lie subalgebras L0 satisfying codim(L0 ) 3. Corollary 8.4. Let L be a finite- or infinite-dimensional Lie subalgebra of B(X). Let L0 be a Lie subalgebra of codimension 1 in L and let L0 have a non-trivial closed invariant subspace of finite codimension (respectively, dimension). (i) If L0 L, then L has a non-trivial closed invariant subspace of finite codimension (respectively, dimension). (ii) If L0 L and dim(L0 ) = ∞, then L is reducible. Proof. If L0 L, the representation θ of L0 on L/L0 is non-degenerate and (i) follows from Theorems 5.4 and 7.1(i). Part (ii) follows from Theorems 6.4 and 7.1(ii). 2 Set L = L0 + [L0 , L]. Then L0 ⊆ L ⊆ L. If dim(L/L0 ) = 1 then L is a Lie subalgebra of L. Corollary 8.5. Let codim(L0 ) = 2 and let L0 have a non-trivial closed invariant subspace of finite codimension (respectively, dimension). (i) If L0 = L and dim(L0 ) = ∞, then L is reducible. (ii) Let dim(L/L0 ) = 1. (1) If L0 L and L L, then L has a non-trivial closed invariant subspace of finite codimension (respectively, dimension). (2) If L0 L, L L and dim(L0 ) = ∞, then L is reducible. (3) If L0 L, L L, dim(L0 ) = ∞ and L0 has no operator-nilpotent Lie ideals of finite codimension (see Section 4), then L is reducible. (iii) If L = L then L has a non-trivial closed invariant subspace of finite codimension (respectively, dimension). Proof. Part (i) follows from Theorems 6.4 and 7.1(ii). In (ii)(1) and (iii) L0 is related to L, so that their proofs follow from Corollaries 5.5 and 7.3. In (ii)(2) L0 is related to L and in (ii)(3) L is related to L. Hence their proofs follow from Theorem 8.1. 2 If dim(L0 ) < ∞ in Corollary 8.5(ii)(2), then L may be irreducible. Indeed, if in Example 5.8 L = A and L0 = {Aa : a ∈ M2 (C), a21 = 0}, then codim(L0 ) = 2, L0 has a non-trivial closed invariant subspace of finite codimension, L0 is related to L and L L. However, dim(L0 ) = 3 < ∞ and L is irreducible. We will now briefly consider the case when codim(L0 ) = 3 and L0 has a non-trivial closed invariant subspace of finite codimension or dimension. 1. Let L0 = L. Then L0 L. If dim(L0 ) = ∞, it follows from Theorems 6.4 and 7.1(ii) that L is reducible.
346
E. Kissin et al. / Journal of Functional Analysis 256 (2009) 323–351
2. Let dim(L/L0 ) = 1. Then L is a Lie subalgebra of L of codimension 1. (i) If L0 L, then, by Corollary 8.4(i), L has a non-trivial closed invariant subspace of finite codimension or dimension. As codim(L) = 2, the problem is reduced to the case considered in Corollary 8.5. (ii) If L0 L and dim(L0 ) = ∞ then, by Corollary 8.4(ii), L has a closed non-trivial invariant subspace Y . If Y has finite codimension or dimension, the problem is reduced to the case considered in Corollary 8.5. Otherwise, the problem is open. 3. Let dim(L/L0 ) = 2. If L is a Lie subalgebra of L, then in Corollary 8.5 we have various conditions when L has a closed non-trivial invariant subspace Y. If Y has finite codimension or dimension then, by Corollary 8.4, L is reducible. 4. If L = L then L0 and L are related, so that L has a non-trivial closed invariant subspace of finite codimension or dimension. 8.3. Invariant subspaces of associative algebras and semigroups of operators The situation turns out to be much simpler if instead of a Lie algebra with a reducible Lie subalgebra of finite codimension one considers an associative algebra A of operators with a reducible subalgebra A0 of finite codimension or such that A is finitely generated as a module over A0 . In this subsection we will obtain several (possibly known) results about invariant subspaces of A. If I = {0} is a reducible two-sided ideal of A, then (see [22, Lemma 5]) A is reducible. Indeed, if Y is a non-trivial subspace invariant for I, the closed subspaces I Y and Ker(I ) are invariant for A. At least one of them is non-trivial, since Ker(I ) = X and, if Ker(I ) = {0} then {0} = I Y ⊆ Y = X. Proposition 8.6. Let A be an associative subalgebra of B(X) and let A0 be a reducible subalgebra of A. If dim(A/A0 ) < ∞ then A is reducible. Proof. Let dim(A) < ∞. For each x ∈ / Ker(A), Ax is a non-trivial, finite-dimensional invariant subspace of A. Let dim(A) = ∞. As dim(A/A0 ) < ∞, it follows from [12, Lemma 2.1] that A0 contains a two-sided ideal I of A such that dim(A/I ) < ∞. Thus I = {0}. Since A0 is reducible, I is reducible. By the argument before the proposition, A is reducible. 2 We will now consider the case when an associative algebra A is finitely generated as a module over a reducible subalgebra. Proposition 8.7. Let A0 be a subalgebra of an associative algebra A ⊆ B(X) and let X0 be a non-trivial closed subspace invariant for A0 . (i) Suppose that there are e1 , . . . , en in B(X) such that A ⊆ ni=1 A0 ei . If codim(X0 ) < ∞, then A has a closed invariant subspace X1 ⊆ X0 with codim(X 1 ) < ∞. (ii) Suppose that there are e1 , . . . , en in B(X) such that A ⊆ ni=1 ei A0 . If dim(X0 ) < ∞, then A has a closed invariant subspace X1 with dim(X1 ) < ∞ and X0 ⊆ X1 .
E. Kissin et al. / Journal of Functional Analysis 256 (2009) 323–351
347
Proof. (i) As in (4.1), set X1 = {x ∈ X0 : ax ∈ X0 , a ∈ A}. By Lemma 4.2(iv), X1 is a closed subspace of X0 . For x ∈ X1 and b ∈ A, a(bx) = (ab)x ∈ X0 for all a ∈ A. Hence bx ∈ X1 , so that X1 is invariant for A. The closed subspaces
Li = {x ∈ X0 : ei x ∈ X0 } have finite codimenhence in X. Therefore Y = Li has finite codimension in X. As ei Y ⊆ X0 , we sion in X0 and, have AY ⊆ i A0 ei Y ⊆ A0 X0 ⊆ X0 whence Y ⊆ X1 . Hence codim(X1 ) < ∞. = X0 + AX0 is invariant for A and contains X0 . Since X1 ⊆ X0 + (ii) The subspace X1 ( ni=1 ei A0 )X0 ⊆ X0 + ni=1 (ei X0 ), we have that X1 is finite-dimensional. 2 We will apply now Proposition 8.7 to the Invariant Subspace problem for semigroups of bounded operators (see the excellent discussion of this subject in [16]). For semigroups containing compact operators, we refer the reader to [18,21]. Corollary 8.8. Let H be a subsemigroup of a semigroup G ⊂ B(X). Let H have a non-trivial closed invariant subspace X0 . (i) If codim(X0 ) < ∞ and there are e1 , . . . , en ∈ B(X) such that G ⊆ H ∪ He1 ∪ · · · ∪ Hen , then G has a closed invariant subspace X1 ⊆ X0 with codim(X1 ) < ∞. (ii) If dim(X0 ) < ∞ and there are e1 , . . . , en ∈ B(X) such that G ⊆ H ∪ e1 H ∪ · · · ∪ en H, then G has a closed invariant subspace X1 with dim(X1 ) < ∞ and X0 ⊆ X1 . Proof. Let A = span(G), A0 = span(H). Then A ⊆ in (ii). It only remains to apply Proposition 8.7. 2
n
i=1 A0 ei
in (i) and A ⊆
n
i=1 ei A0
9. Lie ideals in Banach Lie algebras In this section we consider infinite-dimensional complex Banach Lie algebras. Recall that a complex Lie algebra L is called a Banach Lie algebra, if it is a Banach space in some norm · that satisfies
[a, b] D a b for some D > 0 and all a, b ∈ L.
(9.1)
For example, closed Lie subalgebras of B(X) are Banach Lie algebras. Throughout this section we assume that L has a proper closed Lie subalgebra L0 of finite codimension and study the question when L has Lie ideals. We will write L0 L when L0 is a closed Lie ideal of L. Set M = ad(L)
and M0 = ad(L0 ).
Then M0 ⊆ M ⊆ B(L) and dim(M/M0 ) dim(L/L0 ); whence either M0 = M, or M0 has finite non-zero codimension in M. If the representation θ of L0 on L/L0 is non-degenerate then, by (5.2), L = L0 + [L0 , L]. As ad([a, b]) = [ad(a), ad(b)], we have M = M0 + [M0 , M]. Hence, if M0 = M then the representation of M0 on the quotient space M/M0 is nondegenerate. It implies that if L0 is related to L, then M0 is related to M. The next result gives a partial answer to the question raised in [10].
348
E. Kissin et al. / Journal of Functional Analysis 256 (2009) 323–351
Corollary 9.1. Let L0 be related to L. Then L has a closed Lie ideal of finite codimension contained in L0 . Proof. Set X = L and X0 = L0 . If M0 = M, L0 is the required Lie ideal. If M0 = M, then M0 is related to M ⊆ B(X), X0 is invariant for M0 and codim(X0 ) < ∞. By Corollary 5.5(ii), there is a non-trivial closed subspace W of X0 invariant for M and codim(W ) < ∞. Hence W is the required Lie ideal. 2 Amayo showed in Lemma 2.2(c) of [1] that, if L is a finite-dimensional Lie algebra over field of characteristic 0 and codim(L0 ) = 1, then L0 contains a Lie ideal K of L such that dim(L/K) 3. The extension of this result to Banach Lie algebras obtained in [10] is an easy consequence of Corollary 9.1. Corollary 9.2. Let L0 be a closed Lie subalgebra of L and let codim(L0 ) = 1. Then L0 contains a closed Lie ideal K of L such that codim(K) 3. Proof. If L0 is not a Lie ideal of L then L0 is non-degenerate in L. By Corollary 9.1, L has a closed Lie ideal W of finite codimension contained in L0 . Applying the result of Amayo stated above to L/W , we obtain that L0 contains a closed Lie ideal K of L such that codim(K) 3. 2 Let us show now that the above result extends to the case when codim(L0 ) = 2. Corollary 9.3. Let L0 be a closed Lie subalgebra of L and codim(L0 ) = 2. Then L has a non-trivial closed Lie ideal K of finite codimension. Moreover, if L0 + [L0 , L] = L then codim(K) 3. Proof. If L0 L, the result holds. Let L0 L. Set L = L0 + [L0 , L]. Then L = L0 . If L = L then L0 and L are related. By Corollary 9.1, L has a closed Lie ideal of finite codimension contained in L0 . Let L = L. Since codimension of L0 in L is 1, [L, L] = [L0 , L] ⊆ [L0 , L] ⊆ L. Thus L is a Lie subalgebra of L. Since codim(L) = 1, the result follows from Corollary 9.2. 2 Let codim(L0 ) = 3 and set L = L0 + [L0 , L]. If L0 is not a Lie ideal of L, then either L = L, or codim(L) = 2, or codim(L) = 1. If L = L then L0 is related to L and, by Corollary 9.1, L0 contains a Lie ideal of L of finite codimension. Let codim(L) = 2. As L0 is closed and has codimension 1 in L, L is a closed Lie subalgebra. By Corollary 9.3, L has a non-trivial closed Lie ideal of finite codimension. Let codim(L) = 1. If L is a Lie algebra, then, by Corollary 9.2, L has a non-trivial closed Lie ideal of finite codimension. However, if L is not a Lie algebra, the question is open. Problem 2. Let 3 codim(L0 ) < ∞. Does L always have a Lie ideal of finite codimension? We will consider now examples of Banach Lie algebras L with closed Lie subalgebras L0 of finite codimension such that L has non-trivial closed Lie ideals but none of them lies in L0 . Let L be a closed Lie subalgebra of the algebra B(X) of all bounded operators on a Banach space X. Denote by L the direct sum L = L ⊕ X and endow it with the following binary operation: (a; x), (b; y) = [a, b]; ay − bx , for a, b ∈ L and x, y ∈ X. (9.2)
E. Kissin et al. / Journal of Functional Analysis 256 (2009) 323–351
349
Direct calculations (see [4, Chapter I, 1, 8, Example 2 ]) show that L is a Lie algebra. It is a Banach space with respect to the norm (a; x) = max{ a , x }. Note that L is a Banach Lie algebra, as
(a; x), (b; y) = [a, b]; ay − bx max 2 a b , a y + b x
2 max a , x max b , y = 2 (a; x)
(b; y) . Lemma 9.4. Let a closed Lie subalgebra L ⊆ B(X) be irreducible. Then each closed non-zero Lie ideal of the Banach Lie algebra L = L ⊕ X is of the form K = L0 ⊕ X, where L0 is a Lie ideal of L. Proof. Set Y = {y ∈ X: (0; y) ∈ K}. Then Y is a linear subspace of X. Let 0 = (a; x) ∈ K. If a = 0 then 0 = x ∈ Y. If a = 0 then, by (9.2), [(a; x), (0; z)] = (0; az) ∈ K for all z ∈ X. Thus Y = {0}. As K is closed, Y is closed. For all y ∈ Y and a ∈ L, [(a; 0), (0; y)] = (0; ay) ∈ K. Hence ay ∈ Y, so Y is invariant for L. Thus Y = X. Hence {0} ⊕ X ⊆ K which concludes the proof. 2 Corollary 9.5. Let a closed Lie subalgebra L of B(X) be irreducible (for example, let X = l1 and L = Ce, where e is a bounded operator on X that has no non-trivial closed invariant subspaces (see [17])). Then, for each closed subspace Y X, L0 = {(0; y): y ∈ Y } is a closed Lie subalgebra of the Banach Lie algebra L = L ⊕ X and 2 codim(L0 ) = codim(Y ) + dim(L). However, L0 contains no non-trivial closed Lie ideals of L. Suppose that L0 ⊆ L ⊆ B(X), L0 has an invariant subspace X0 of finite codimension and codim(L0 ) < ∞, but L0 is not related to L. In Example 5.8 we considered the case when dim(L0 ) < ∞ and L is irreducible. Below we consider an example when dim(L0 ) = ∞ and L is reducible, but its invariant subspaces do not lie in X0 (cf. Corollary 5.5). Example 9.6. Let X = l1 , let e be a bounded irreducible operator on X and let Y be a proper closed subspace of X of finite codimension. As above, consider the Banach Lie algebra L = Ce ⊕ X. Then L0 = {0} ⊕ Y is a closed Lie subalgebra of L of finite codimension and dim(L0 ) = ∞. Since L0 is contained in the Lie ideal {0} ⊕ X of L, it is not related to L. As Ker(ad) = {0}, L is isomorphic to the Lie subalgebra ad(L) of the algebra of all bounded operators on the Banach space X = L. Then the Lie subalgebra ad(L0 ) of finite codimension in ad(L) has a closed invariant subspace X0 = L0 of finite codimension in X. By Lemma 9.4, ad(L) has only one non-trivial closed invariant subspace {0} ⊕ X and it does not lie in X0 . We will now consider various cases when L0 contains a closed Lie subalgebra K0 : K0 ⊆ L0 ⊆ L. Corollary 9.7. Let L0 be related to L and let K0 L0 . (i) If dim(L/K0 ) < ∞, then L has a closed Lie ideal of finite codimension in K0 . (ii) If dim(K0 ) < ∞, then L has a finite-dimensional Lie ideal that contains K0 . Proof. Set X = L and X0 = K0 . Then the Lie subalgebra M0 = ad(L0 ) is related to M = ad(L) or M0 = M; and X0 is invariant for M0 . The case M0 = M is evident.
350
E. Kissin et al. / Journal of Functional Analysis 256 (2009) 323–351
(i) If dim(L/K0 ) < ∞ then codim(X0 ) < ∞. It follows from Corollary 5.5(ii) that there is a closed subspace W of X0 invariant for M and codim(W ) < ∞. Hence W is the required Lie ideal. (ii) If dim(K0 ) < ∞ then dim(X0 ) < ∞. We have from Corollary 7.3 that M has an invariant subspace W, X0 ⊆ W and dim(W ) < ∞. Hence W is the required Lie ideal. 2 Corollary 9.8. Let L0 be a closed, non-commutative Lie ideal of L and dim(L0 ) = ∞. Then L has a non-trivial closed Lie ideal K L0 in the following cases: (i) L0 contains a proper closed Lie subalgebra K0 related to L0 ; (ii) L0 contains a proper closed Lie ideal K0 and dim(L0 /K0 ) < ∞; (iii) L0 contains a proper closed Lie ideal K0 and dim(K0 ) < ∞. Proof. Let C be the center of L0 . As L0 is non-commutative, C = L0 . If C = {0}, then K = C is the required Lie ideal of L. Indeed, as L0 is a Lie ideal of L, [C, L], L0 ⊆ C, [L, L0 ] + [C, L0 ], L = {0}. Assume now that C = {0}. Set X = L0 , N = ad(K0 )|X , N0 = ad(L0 )|X and N = ad(L)|X ⊆ B(X). Then dim(X) = dim(N0 ) = ∞, N0 N and X0 = K0 = X is a closed subspace. (i) The subspace X0 is invariant for N and codim(X0 ) < ∞. As N is related to N0 , it follows from Corollary 5.5(ii) that there is a closed subspace W of X0 invariant for N0 and codim(W ) < ∞. Applying now Theorem 6.4(i), we obtain that N has a non-trivial closed invariant subspace K. Hence K is a Lie ideal of L and K L0 . (ii) The subspace X0 is invariant for N0 and codim(X0 ) < ∞. As N0 N and dim(N0 ) = ∞, it follows from Theorem 6.4(i) that N has a non-trivial closed invariant subspace K of X. Hence K is a closed Lie ideal of L and {0} = K L0 . (iii) The subspace X0 is invariant for N0 and dim(X0 ) < ∞. Replacing the pair L0 L in Theorem 7.1(ii) by N0 N , we obtain that there is a non-trivial closed subspace K invariant for N . Hence K is a Lie ideal of L. 2 A linear map δ on L is a bounded derivation if there is C > 0 such that δ(a) C a , for all a ∈ L, and δ([a, b]) = [δ(a), b] + [a, δ(b)], for a, b ∈ L. Denote by D(L) the set of all bounded derivations on L. A Lie ideal of L is called characteristic if it is invariant for all δ ∈ D(L). Theorem 9.9. Let L be an infinite-dimensional, non-commutative Banach Lie algebra and L0 be a non-trivial closed Lie subalgebra of L. Then L has a non-trivial closed characteristic Lie ideal W if one of the following conditions holds: (i) L0 is related to L. (ii) L0 L and codim(L0 ) < ∞. (iii) L0 L and dim(L0 ) < ∞. Proof. Replacing (K0 , L0 , L) in Corollary 9.8 by (L0 , L, D(L)), we obtain the proof of the theorem. 2
E. Kissin et al. / Journal of Functional Analysis 256 (2009) 323–351
351
In many cases a Banach Lie algebra can have a big variety of closed Lie ideals of finite codimension but no characteristic ideals. For example, if L is commutative then each closed subspace of L is a Lie ideal and each bounded operator on L is a derivation. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
R.K. Amayo, Quasi-ideals of Lie algebras II, Proc. London Math. Soc. (3) 33 (1976) 37–64. D.W. Barnes, On the cohomology of soluble algebras, Math. Z. 101 (1967) 343–349. D. Belti¸ta˘ , M. Sabac, ¸ Lie Algebras of Bounded Operators, Birkhäuser, Basel, 2001. N. Bourbaki, Groupes et algébres de Lie, Hermann, Paris, 1971. M. Brešar, E. Kissin, V.S. Shulman, Lie ideals: From pure algebra to C*-algebra, J. Reine Angew. Math. (Crelle) 623 (2008) 73–121. P.R. Halmos, A Hilbert Space Problem Book, Van Nostrand, Princeton, NJ, 1967. N. Jacobson, Rational methods in the theory of Lie algebras, Ann. of Math. (2) 36 (4) (1935) 875–881. N. Jacobson, Lie Algebras, Interscience, New York, 1962. E. Kissin, On some reflexive algebras of operators and the operator Lie algebras of their derivations, Proc. London Math. Soc. 49 (1984) 1–35. E. Kissin, On normed Lie algebras with sufficiently many subalgebras of codimension 1, Proc. Edinburgh Math. Soc. 29 (1986) 199–220. A.I. Kostrikin, Vokrug Bernsajda (Around Burnside), Nauka, Moscow, 1986. T. Laffey, On the structure of algebraic algebras, Pacific J. Math. 62 (1976) 461–471. L.W. Marcoux, On the closed Lie ideals of certain C*-algebras, Integral Equation Operator Theory 22 (1995) 463– 475. G.J. Murphy, H. Radjavi, Associative and Lie subalgebras of finite codimension, Studia Math. 76 (1983) 81–85. H. Radjavi, P. Rosenthal, Invariant Subspaces, Springer, Berlin, 1973. H. Radjavi, P. Rosenthal, Simultaneous Triangularization, Springer, Berlin, 2000. C. Read, A solution of the invariant subspace problem on the space l1 , Bull. London Math. Soc. 17 (1985) 305–317. V.S. Shulman, Yu.V. Turovskii, Joint spectra radius, operator semigroups and a problem of W. Wojty´nski, J. Funct. Anal. 177 (2000) 383–441. V.S. Shulman, Yu.V. Turovskii, Invariant subspaces of operator Lie algebras and Lie algebras with compact adjoint action, J. Funct. Anal. 223 (2005) 425–508. D. Towers, Lie algebras all of whose maximal subalgebras have codimension one, Proc. Edinburgh Math. Soc. 24 (1981) 217–219. Yu.V. Turovskii, Volterra semigroups have invariant subspaces, J. Funct. Anal. (2) 162 (1999) 313–323. W. Wojty´nski, On the existence of closed two-sided ideals in radical Banach algebras with compact elements, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astr. Phys. 26 (2) (1978) 109–113.
Journal of Functional Analysis 256 (2009) 352–384 www.elsevier.com/locate/jfa
Operators L1 (R+) → X and the norm continuity problem for semigroups ✩ Ralph Chill a , Yuri Tomilov b a Université Paul Verlaine, Metz, Laboratoire de Mathématiques et Applications de Metz, CNRS, UMR 7122,
Bât. A, Ile du Saulcy, 57045 Metz Cedex 1, France b Faculty of Mathematics and Computer Science, Nicolas Copernicus University, ul. Chopina 12/18,
87-100 Torun, Poland Received 17 March 2008; accepted 24 May 2008 Available online 26 June 2008 Communicated by N. Kalton
Abstract We present a new method for constructing C0 -semigroups for which properties of the resolvent of the generator and continuity properties of the semigroup in the operator-norm topology are controlled simultaneously. It allows us to show that (a) there exists a C0 -semigroup which is continuous in the operator-norm topology for no t ∈ [0, 1] such that the resolvent of its generator has a logarithmic decay at infinity along vertical lines; (b) there exists a C0 -semigroup which is continuous in the operator-norm topology for no t ∈ R+ such that the resolvent of its generator has a decay along vertical lines arbitrarily close to a logarithmic one. These examples rule out any possibility of characterizing norm-continuity of semigroups on arbitrary Banach spaces in terms of resolvent-norm decay on vertical lines. © 2008 Elsevier Inc. All rights reserved. Keywords: C0 -semigroup; Norm continuity; Resolvent; Banach algebra homomorphism; Laplace transform
✩ Supported by the Marie Curie “Transfer of Knowledge” programme, project “TODEQ”. This research was started during a Research in Pair stay at the Mathematisches Forschungsinstitut Oberwolfach. Supports are gratefully acknowledged. E-mail addresses:
[email protected] (R. Chill),
[email protected] (Y. Tomilov).
0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.05.019
R. Chill, Y. Tomilov / Journal of Functional Analysis 256 (2009) 352–384
353
1. Introduction The study of continuity properties of C0 -semigroups (T (t))t0 on a Banach space X in the uniform operator topology of L(X) (norm-continuity) has been initiated in [28] and attracted considerable attention over the last decades; see in particular [2–4,15,16,27,30,37,43,44]. The classes of immediately norm-continuous semigroups, of eventually norm-continuous semigroups, and of asymptotically norm-continuous semigroups (or, equivalently, semigroups norm continuous at infinity) emerged and were studied in depth during this period. The interest in these classes comes mainly from the fact that a condition of norm continuity of a semigroup implies a variant of the spectral mapping theorem, and thus asymptotic properties of a semigroup are essentially determined by the spectrum of the generator. One of the main issues in the study of norm-continuity is to characterize these classes in terms of the resolvent of the semigroup generator (or in other a priori terms). In particular, the so called norm-continuity problem for C0 -semigroups attributed to A. Pazy was a focus for relevant research during the last two decades. Given a C0 -semigroup (T (t))t0 on a Banach space X, with generator A, the problem is to determine whether the resolvent decay condition lim R(ω + iβ, A) = 0 for some ω ∈ R
|β|→∞
(1.1)
implies that the semigroup is immediately norm-continuous, that is, norm-continuous for t > 0. The decay condition (1.1) is certainly necessary for immediate norm-continuity, by the fact that the resolvent of the generator is the Laplace transform of the semigroup, and by a simple application of the lemma of Riemann–Lebesgue. Hence, the question is whether condition (1.1) characterizes immediate norm-continuity. The resolvent decay condition (1.1) does characterize immediate norm continuity if the underlying Banach space is a Hilbert space [15,43,44], [1, Theorem 3.13.2], or if it is an Lp space and the semigroup is positive, [27]. Only very recently, T. Matrai [37] constructed a counterexample showing that the answer to the norm continuity problem is negative in general. The generator in his example is an infinite direct sum of Jordan blocks on finite dimensional spaces. The infinite sum is equipped with an appropriate norm and the resulting Banach space is reflexive. This kind of counterexample going back to [45] has been used in the spectral theory of semigroups to show the failure of the spectral mapping theorems or certain relationships between semigroup growth bounds, see for example [1,18]. We point out that the resolvent decay condition (1.1) implies that the resolvent exists and is uniformly bounded in a domain of the form Σϕ := λ ∈ C: Re λ > −ϕ |Im λ| , where ϕ ∈ C(R+ ) satisfies limβ→∞ ϕ(β) = ∞. It is known that the existence of the resolvent and its uniform boundedness in such a domain can imply regularity properties of the semigroup if the function ϕ is growing sufficiently fast: we recall corresponding results for analytic, immediately differentiable and eventually differentiable semigroups, [1, Theorem 3.7.11], [39, Theorems 4.7, 5.2]. It follows from the proofs of these results (which use the complex inversion formula for Laplace transforms) that there are similar results in the more general context of Laplace transforms of vector-valued functions; see, for example, [1, Theorem 2.6.1], [14, I.4.7, II.7.4], [41,42]. One could therefore think of the following Laplace transform version of
354
R. Chill, Y. Tomilov / Journal of Functional Analysis 256 (2009) 352–384
the norm-continuity problem: if the Laplace transform of a bounded scalar (or vector-valued) function extends analytically to a bounded function in some domain Σϕ , where ϕ ∈ C(R+ ) satisfies limβ→∞ ϕ(β) = ∞, is the function immediately or eventually continuous? It is relatively easy to give counterexamples to this Laplace transform version of the norm-continuity problem. It follows from the main result in this article (Theorem 4.2) that every counterexample to the norm-continuity problem for scalar functions yields a counterexample to the norm-continuity problem for semigroups. As indicated in the title of this article, we approach the problem of norm-continuity via Banach algebra homomorphisms L1 (R+ ) → A. The connection between semigroups and such homomorphisms is well known. We recall that to every bounded C0 -semigroup (T (t))t0 on a Banach space X one can associate an algebra homomorphism T : L1 (R+ ) → L(X) given by ∞ Tg =
T (t)g(t) dt,
g ∈ L1 (R+ )
(integral in the strong sense).
(1.2)
0
Conversely, every algebra homomorphism L1 (R+ ) → A is, after passing to an equivalent homomorphism, of this form; cf. Lemma 3.1 below. It is therefore natural to ask how regularity properties of the semigroup or the resolvent of its generator are encoded in the corresponding algebra homomorphism or its adjoint. We will discuss some of the connections in the first part of this article, partly in the context of general operators L1 (R+ ) → X. Then, given a function f ∈ L∞ (R+ ) such that its Laplace transform extends to a bounded analytic function on some domain Σϕ , we will show how to construct an algebra homomorphism T : L1 (R+ ) → L(X) which is represented (in the strong sense) by a C0 -semigroup such that f ∈ range T ∗ and such that the resolvent of the generator satisfies a precise decay estimate. In fact, the space X will be continuously embedded into L∞ (R+ ) and left-shift invariant, and the operator T will be represented by the left-shift semigroup on X. In this way, we will be able to show that the norm-continuity problem has a negative solution and at the same time we will be able to estimate the resolvent decay along vertical lines. It turns out that the decay R(ω + iβ, A) = O(1/ log |β|) implies eventual differentiability but not immediate norm-continuity, and that any slower decay does not imply eventual norm-continuity. The paper is organized as follows. In Section 2, we remind some basic properties and definitions from the theory of operators L1 → X needed in the sequel, and introduce the notion of a Riemann–Lebesgue operator. In Section 3, we set up a framework of homomorphisms L1 → A and establish the relation to the norm continuity problem for semigroups. The main, Section 4, is devoted to the construction of Riemann–Lebesgue homomorphisms. Finally, in Section 5, we apply the main result from Section 4 to give counterexamples to the norm-continuity problem. R+ ) → X 2. Operators L1 (R Operators L1 → X and their representations is a classical subject of both operator theory and geometric theory of Banach spaces. For a more or less complete account of basic properties of these operators one may consult [12], and a selection of more recent advances pertinent to our studies include [6,7,11,22–26,29,31,33]. The following representation of operators L1 (R+ ) → X by vector-valued Lipschitz continuous functions on R+ will be used in the sequel. We denote by Lip0 (R+ ; X) the Banach
R. Chill, Y. Tomilov / Journal of Functional Analysis 256 (2009) 352–384
355
space of all Lipschitz continuous functions F : R+ → X satisfying F (0) = 0. Then, for every F ∈ Lip0 (R+ ; X) the operator TF : L1 (R+ ) → X given by the Stieltjes integral ∞ TF g :=
g(t) dF (t),
g ∈ L1 (R+ ),
(2.1)
0
is well defined and bounded, and it turns out that every bounded operator T : L1 (R+ ) → X is of this form. In fact, by the Riesz–Stieltjes representation theorem [1, Theorem 2.1.1], the operator F → TF is an (isometric) isomorphism from Lip0 (R+ ; X) onto L(L1 (R+ ), X). There are several analytic properties of operators L1 → X which have been defined and studied in the literature. Among them, we will recall Riesz representability and the (local) Dunford–Pettis property, and we introduce the Riemann–Lebesgue property. The first and the last will be relevant for this article while the (local) Dunford–Pettis property is mentioned for reasons of comparison. Throughout the following, for every λ ∈ C and every t ∈ R+ , we define eλ (t) := e−λt . If λ belongs to the open right half-plane C+ , then eλ ∈ L1 (R+ ). Recall that an operator between two Banach spaces is called Dunford–Pettis or completely continuous if it maps weakly convergent sequences into norm convergent sequences. Definition 2.1. Let T : L1 (R+ ) → X be a bounded operator. (a) We call T Riesz representable, or simply representable, if there exists a function f ∈ L∞ (R+ ; X) such that Tg =
g(s)f (s) ds
for every g ∈ L1 (R+ ).
(2.2)
R+
(b) We call T locally Dunford–Pettis if for every measurable K ⊂ R+ of finite measure the restriction of T to L1 (K) is Dunford–Pettis. (c) We call T Riemann–Lebesgue if lim|β|→∞ T (eiβ g) = 0 for every g ∈ L1 (R+ ). The definitions of representable and local Dunford–Pettis operators clearly make sense on general L1 spaces. For some operator theoretical questions it may be more natural to consider operators on L1 (0, 1) or a similar L1 space. In the context of (bounded) C0 -semigroups, the space L1 (R+ ) is appropriate. We point out that if F ∈ Lip0 (R+ ; X) and if T = TF : L1 (R+ ) → X is representable by some t function f ∈ L∞ (R+ ; X), then necessarily F (t) = 0 f (s) ds. In fact, T = TF is representable if and only if the function F admits a Radon–Nikodym derivative in L∞ (R+ ; X). Proposition 2.2. Let T : L1 (R+ ) → X be a bounded operator. The following implications are true: T is weakly compact ⇓
356
R. Chill, Y. Tomilov / Journal of Functional Analysis 256 (2009) 352–384
T is representable ⇓ T is locally Dunford–Pettis ⇓ T is Riemann–Lebesgue. Proof. The first implication follows from [12, Theorem 12, p. 75], while the second implication is a consequence of [12, Lemma 11, p. 74, and Theorem 15, p. 76]. Note that these results only deal with finite measure spaces whence the necessity to consider local Dunford–Pettis operators. In order to prove the last implication, one has to remark that for general T : L1 (R+ ) → X the space
E := g ∈ L1 (R+ ): lim T (eiβ g) = 0 is closed in L1 (R+ ). β→∞
(2.3)
Next, by the lemma of Riemann–Lebesgue, for every g ∈ L1 (R+ ) one has w-limβ→∞ eiβ g = 0 in L1 (R+ ) and L1 (K), where K is any compact subset of R+ . Hence, if T is locally Dunford– Pettis, then the space E contains all compactly supported functions in L1 (R+ ), and since this space is dense in L1 (R+ ), the operator T must be Riemann–Lebesgue. 2 The properties from Definition 2.1 have also been defined for Banach spaces instead of single operators. For example, a Banach space X has the Radon–Nikodym property if every operator T : L1 (R+ ) → X is representable, [12], or, equivalently, if every function in Lip0 (R+ ; X) admits a Radon–Nikodym derivative in L∞ (R+ ; X). Similarly, a Banach space X has the complete continuity property if every operator T : L1 (R+ ) → X is locally Dunford–Pettis. Note that the Dunford–Pettis property for Banach spaces has also been defined in the literature, but is different from the complete continuity property, [38, Definition 3.7.6]. Finally, a Banach space X has the Riemann–Lebesgue property if every operator T : L1 (R+ ) → X is Riemann–Lebesgue. The Riemann–Lebesgue property for Banach spaces has been defined only recently, [8], and Definition 2.1 is perhaps the first instance where the Riemann–Lebesgue property is defined for a single operator. It has been recently shown that the complete continuity property and the Riemann–Lebesgue property for Banach spaces are equivalent, [31]. It is therefore natural to ask whether a similar result holds for single operators. Problem 2.3. Is every Riemann–Lebesgue operator T : L1 (R+ ) → X a local Dunford–Pettis operator? The following theorem gives a characterization of Riemann–Lebesgue operators using only exponential functions. Theorem 2.4. An operator T : L1 (R+ ) → X is a Riemann–Lebesgue operator if and only if lim|β|→∞ T eω+iβ = 0 for some/all ω > 0.
R. Chill, Y. Tomilov / Journal of Functional Analysis 256 (2009) 352–384
357
Proof. Assume first that T : L1 (R+ ) → X is a bounded operator satisfying lim|β|→∞ T eω+iβ = 0 for some ω > 0. Let 0 < a < ω < b < ∞, and define the closed strip S := {λ ∈ C+ : a Re λ b}. The function f : S → X, λ → T eλ , is bounded, continuous on S, and analytic in the interior of S. By a standard argument from complex function theory (involving Vitali’s theorem) and the assumption we obtain lim T (eα+iβ ) = 0,
|β|→∞
for all α ∈ (a, b). Since a ∈ (0, ω) and b ∈ (ω, ∞) are arbitrary, the above equation is true for every α ∈ (0, ∞). Next, recall from (2.3) that the space of all g ∈ L1 (R+ ) such that lim|β|→∞ T (eiβ g) = 0 is closed in L1 (R+ ). By the preceding argument, this space contains the set {eα : α > 0}. Since this set is total in L1 (R+ ), by the Hahn–Banach theorem and by uniqueness of the Laplace transform, it therefore follows that T is a Riemann–Lebesgue operator. The other implication is trivial. 2 Corollary 2.5. Let F ∈ Lip0 (R+ ; X), and let TF : L1 (R+ ) → X be the corresponding bounded the Laplace–Stieltjes transform of F , that is, operator given by (2.1). Denote by dF (λ) := dF
∞
e−λt dF (t),
λ ∈ C+ .
0
Then TF is a Riemann–Lebesgue operator if and only if (ω + iβ) = 0 for some/all ω > 0. lim dF
|β|→∞
Proof. This is a direct consequence from Theorem 2.4 and the definition of the representing function. 2 R+ ) → A and the norm-continuity problem 3. Algebra homomorphisms L1 (R In the following, we will equip L1 (R+ ) with the usual convolution product given by t (f ∗ g)(t) =
f (t − s)g(s) ds,
f, g ∈ L1 (R+ ).
0
Then L1 (R+ ) is a commutative Banach algebra with bounded approximate identity; for example, the net (λeλ )λ ∞ is an approximate identity bounded by 1.
358
R. Chill, Y. Tomilov / Journal of Functional Analysis 256 (2009) 352–384
Let A be a Banach algebra. If (a(t))t>0 ⊂ A is a uniformly bounded and continuous semigroup, then the operator T : L1 (R+ ) → A given by ∞ Tg =
a(t)g(t) dt
(3.1)
0
is an algebra homomorphism as one easily verifies. Conversely, if T : L1 (R+ ) → A is an algebra homomorphism, then T is represented as above, but (a(t))t>0 is a semigroup of multipliers on A and the integral is to be understood in the sense of the strong topology of the multiplier algebra M(A); see, for example, [10, Theorems 3.3 and 4.1], [40, Proposition 1.1]. We will state this result in a slightly different form, more convenient to us, using the notion of equivalent operators which we introduce here. We call two operators T : L1 (R+ ) → X and S : L1 (R+ ) → Y equivalent if there exist constants c1 , c2 0 such that T gX c1 SgY c2 T gX
for every g ∈ L1 (R+ ).
It is easy to check that properties like weak compactness, representability, the local Dunford– Pettis property and the Riemann–Lebesgue property are invariant under equivalence, that is, for example, if T and S are equivalent, then T is representable if and only if S is representable; one may prove that if FT and FS are the representing Lipschitz functions, then FT is differentiable almost everywhere if and only if FS is differentiable almost everywhere (use that difference quotients are images of multiples of characteristic functions). We point out that two operators T and S are equivalent if and only if range T ∗ = range S ∗ ; compare with [17, Theorem 1]. The proof of the following lemma is similar to the proof of [9, Theorem 1] (see also [34, Corollary 4.3], [35, Theorem 10.1], [5, Theorem 1.1]). Lemma 3.1. For every algebra homomorphism T : L1 (R+ ) → A there exists a Banach space X0 , an equivalent algebra homomorphism S : L1 (R+ ) → L(X0 ) and a uniformly bounded C0 semigroup (S(t))t0 ⊂ L(X0 ) such that for every g ∈ L1 (R+ ) ∞ Sg =
S(t)g(t) dt
(integral in the strong sense).
0
If A ⊂ L(X) as a closed subspace, then X0 can be chosen to be a closed subspace of X. Proof. We first assume that A ⊂ L(X) as a closed subspace, and we put R(λ) := T eλ ∈ L(X) (λ ∈ C+ ). Since T is an algebra homomorphism, the function R is a pseudoresolvent, that is, R(λ) − R(μ) = (μ − λ)R(λ)R(μ)
for every λ, μ ∈ C+ .
This resolvent identity implies that range R(λ) is independent of λ ∈ C+ .
R. Chill, Y. Tomilov / Journal of Functional Analysis 256 (2009) 352–384
359
We put X0 := range R(λ)·X ⊂ X. Clearly, X0 is invariant under R(λ), and since λeλ L1 = 1 for every λ > 0, we obtain the estimate n λR(λ) n L(X ) λR(λ) L(X) T for every λ > 0, n 1. 0
(3.2)
By using this estimate (with n = 1), for every x ∈ X and every λ ∈ C+ one obtains lim μR(μ)R(λ)x = lim
μ→∞
μ→∞
μR(λ)x − μR(μ)x = R(λ)x, μ−λ
which implies lim μR(μ)x = x
μ→∞
for every x ∈ X0 .
This relation and the resolvent identity imply that R(λ) is injective on X0 and the range of R(λ) is dense in X0 . As a consequence, there exists a densely defined, closed operator A on X0 such that for every λ ∈ C+ , x ∈ X0 .
R(λ)x = R(λ, A)x
(3.3)
By (3.2) and the Hille–Yosida theorem, A is the generator of a uniformly bounded C0 -semigroup ∞ (S(t))t0 ⊂ L(X0 ). Let S : L1 (R+ ) → L(X0 ) be the operator defined by Sg = 0 S(t)g(t) dt, where the integral is understood in the strong sense. Let F ∈ Lip0 (R+ ; L(X)) be the function representing T (Riesz–Stieltjes representation). Then the equality (3.3) and the definition of R imply ∞ e
−λt
∞ dF (t)x =
0
e−λt S(t)x dt
for every λ ∈ C+ , x ∈ X0 .
0
By the uniqueness of the Laplace–Stieltjes transform, we obtain t S(s)x ds = F (t)x
for every t 0, x ∈ X0 ,
0
and hence (Sg)(x) = (T g)(x)
for every g ∈ L1 (R+ ), x ∈ X0 .
Clearly, this implies SgL(X0 ) T gL(X)
for every g ∈ L1 (R+ ).
360
R. Chill, Y. Tomilov / Journal of Functional Analysis 256 (2009) 352–384
On the other hand, for every x ∈ X one has (T g)(x) = lim μR(μ)(T g)(x) μ→∞ = lim (Sg) μR(μ)x μ→∞ SgL(X0 ) sup μR(μ)x μ>0
SgL(X0 ) T x. The last two inequalities imply that T and S are equivalent. The general case can be reduced to the case A ⊂ L(X) in the following way. First of all, we may assume without loss of generality that range T is dense in A. Since L1 (R+ ) admits a bounded approximate identity, it is then easy to verify that also A admits a bounded approximate identity. From this one deduces that the natural embedding A → L(A), which to every element a ∈ A associates the multiplier Ma ∈ L(A) given by Ma b = ab, is an isomorphism onto its range. 2 Remark 3.2. It is in general an open problem to give conditions on an algebra homomorphism T : L1 (R+ ) → A which imply that there exists an equivalent algebra homomorphism S : L1 (R+ ) → L(X) on a Banach space X having additional properties, for example, being reflexive, being an Lp space, etc. If T has dense range, this is essentially the problem of representing A as a closed subalgebra of L(X). The next lemma relates the norm-continuity problem with the problem of representability of homomorphisms L1 (R+ ) → A. Lemma 3.3. An algebra homomorphism T : L1 (R+ ) → A is representable if and only if there exists a uniformly bounded and continuous semigroup (a(t))t>0 ⊂ A (no continuity condition at zero) such that T is represented by (3.1). Proof. The sufficiency part is trivial. So assume that T is representable by some a ∈ L∞ (R+ , A). Without loss of generality we may assume that T has dense range in A. By the proof of Lemma 3.1, there exists an equivalent algebra homomorphism S : L1 (R+ ) → L(A) which is (strongly) represented by a uniformly bounded C0 -semigroup (S(t))t0 ⊂ L(A). Moreover, ∞
∞ a(t)bg(t) dt =
0
S(t)bg(t) dt
for every b ∈ A, g ∈ L1 (R+ ),
0
which in turn implies a(t)b = S(t)b
for every b ∈ A and almost every t > 0.
R. Chill, Y. Tomilov / Journal of Functional Analysis 256 (2009) 352–384
361
As a consequence, after changing a on a set of measure zero, (a(t))t>0 is a semigroup. Since S is also representable, the semigroup (S(t))t0 is measurable in L(A), and hence immediately norm continuous by [28, Theorem 9.3.1]. Since A admits a bounded approximate identity, one thus obtains that also (a(t))t>0 is norm continuous. 2 Remark 3.4. One can also prove that an algebra homomorphism T : L1 (R+ ) → A is weakly compact if and only if it is represented by a uniformly bounded and continuous semigroup (a(t))t0 ⊂ A (continuity at 0 included!); compare with [20,21]. Let T : L1 (R+ ) → L(X) be an algebra homomorphism which is represented in the strong sense by a uniformly bounded C0 -semigroup (T (t))t0 with generator A. By Lemma 3.3 above, (T (t))t0 is immediately norm continuous if and only if T is representable. By Theorem 2.4 and Corollary 2.5, the resolvent of A satisfies the resolvent decay condition (1.1) if and only if T is Riemann–Lebesgue. Hence, by Lemma 3.1, the norm-continuity problem can be reformulated in the following way. Problem 3.5 (Norm-continuity problem reformulated). If A is a Banach algebra and if T : L1 (R+ ) → A is a Riemann–Lebesgue algebra homomorphism, is T representable? We recall from Section 1, that the norm-continuity problem has a negative answer in general, but that there are some positive answers in special cases. For example, by the representation theorem for C ∗ algebras as subalgebras of L(H ) (H a Hilbert space), and by Lemma 3.1, the answer to Problem 3.5 is positive if A is a C ∗ algebra. This follows from the result in [44]. The fact that the answer to Problem 3.5 is in general negative follows from Matrai’s example [37]. The aim of the following section is to construct suitable Riemann–Lebesgue homomorphisms and to deduce from this different counterexamples to Problem 3.5 for which it is possible to control the resolvent decay along vertical lines. At the same time, we are not able to answer the following variant of the norm-continuity problem. Observe that since Problem 3.5 has in general a negative answer, this variant and Problem 2.3 are not independent of each other. Problem 3.6 (Variant of the norm-continuity problem). If A is a Banach algebra and if T : L1 (R+ ) → A is a local Dunford–Pettis algebra homomorphism, is T representable? We finish this section by collecting some basic properties of algebra homomorphisms L1 (R+ ) → A and their adjoints which are needed in the sequel. For every g ∈ L1 (R+ ), h ∈ L∞ (R+ ) we define the adjoint convolution g h ∈ L∞ (R+ ) by ∞ g(s)h(t + s) ds.
(g h)(t) = 0
With this definition, for every f, g ∈ L1 (R+ ) and every h ∈ L∞ (R+ ) we have the identities f (g h) = (f ∗ g) h
362
R. Chill, Y. Tomilov / Journal of Functional Analysis 256 (2009) 352–384
and f ∗ g, h L1 ,L∞ = f, g h L1 ,L∞ , which will be frequently used in the following. The second identity explains the name of the product . From this identity one can also deduce that is separately continuous on (L1 , weak) × (L∞ , weak∗ ) with values in (L∞ , weak∗ ). Whenever X is some Banach space, we denote by BX the closed unit ball in X. Lemma 3.7. Let T : L1 (R+ ) → A be an algebra homomorphism. Then the following are true: (a) The set T ∗ BA∗ ⊂ L∞ (R+ ) is non-empty, convex, weak∗ compact and T ∗ BA∗ = −T ∗ BA∗ . (b) If T = 1, then for every g ∈ BL1 and every h ∈ T ∗ BA∗ ⊂ L∞ (R+ ) one has g h ∈ T ∗ BA∗ . In particular, range T ∗ is invariant under adjoint convolution. (c) If T is representable, then range T ∗ ⊂ C(0, ∞). (d) If T is represented (in the strong sense) by a bounded C0 -semigroup which is normcontinuous for t > t0 , then every function in range T ∗ is continuous on (t0 , ∞). Proof. The properties in (a) are actually true for general bounded linear operators T and do not depend on the spaces L1 (R+ ) and A. The weak∗ compactness follows from Banach–Alaoglu and the other properties are true for any unit ball in a Banach space. In order to prove (b), let g ∈ BL1 and h = T ∗ a ∗ ∈ T ∗ BA∗ for some a ∗ ∈ BA∗ . Since T is an algebra homomorphism, for every f ∈ L1 (R+ ), f, g T ∗ a ∗ L1 ,L∞ = f ∗ g, T ∗ a ∗ L1 ,L∞ = Tf T g, a ∗ A,A∗ =: Tf, T ga ∗ A,A∗ , so that g T ∗ a ∗ = T ∗ (T ga ∗ ). However, T ga ∗ A∗ T gA a ∗ A∗ 1, so that (b) is proved. If T is representable, then, by Lemma 3.3, there exists a bounded norm-continuous semigroup (a(t))t>0 ⊂ A such that ∞ Tg =
a(t)g(t) dt,
g ∈ L1 (R+ ).
0
Hence, for every a ∗ ∈ A∗ and every g ∈ L1 (R+ ), g, T ∗ a ∗ L1 ,L∞ = T g, a ∗ A,A∗ ∞ =
g(t) a(t), a ∗ A,A∗ dt.
0
This implies T ∗ a ∗ = a(·), a ∗ A,A∗ ∈ C(0, ∞) so that (c) is proved. The last assertion is very similar to (c), if we use in addition that L1 (R+ ) is the direct sum of 1 L (0, t0 ) and L1 (t0 , ∞) (and L∞ (R+ ) is the direct sum of the corresponding duals). 2
R. Chill, Y. Tomilov / Journal of Functional Analysis 256 (2009) 352–384
363
4. Construction of Riemann–Lebesgue homomorphisms This section is devoted to the main result of this article: we will describe a procedure how to construct Riemann–Lebesgue algebra homomorphisms T : L1 (R+ ) → A for which one can estimate the norm decay of the pseudoresolvent (T eλ )λ∈C+ along vertical lines. In the following, for every function ϕ ∈ C(R+ ) we define the domain Σϕ := λ ∈ C: Re λ > −ϕ |Im λ| . The domains Σϕ are symmetric with respect to the real axis. Domains of the form Σϕ with limβ→∞ ϕ(β) = ∞ play an important role in connection with Riemann–Lebesgue algebra homomorphisms. The following proposition contains a necessary condition for algebra homomorphisms to be Riemann–Lebesgue. Proposition 4.1. If A is a Banach algebra and if T : L1 (R+ ) → A is a Riemann–Lebesgue algebra homomorphism, then there exists a function ϕ ∈ C(R+ ) satisfying limβ→∞ ϕ(β) = ∞ such that for every f ∈ range T ∗ the Laplace transform fˆ extends to a bounded analytic function on Σϕ . Proof. Assume that T : L1 (R+ ) → A is a Riemann–Lebesgue homomorphism. Then lim|β|→∞ T e2+iβ = 0. Expanding the pseudoresolvent λ → T eλ in a power series at the points 2 + iβ with β ∈ R, one easily verifies that this pseudoresolvent extends to a bounded analytic function in some domain Σϕ , where ϕ is as in the statement. If f = T ∗ a ∗ ∈ range T ∗ , then for every λ ∈ C+ one has fˆ(λ) = eλ , f L1 ,L∞ = T eλ , a ∗ A,A∗ , and therefore the Laplace transform fˆ extends to a bounded analytic function on Σϕ .
2
The main result in this section goes in the opposite direction to Proposition 4.1. Throughout the following, we put for every β ∈ R λβ := 2 + iβ, and if ϕ ∈ C(R+ ) is a given nonnegative function, then we also put dβ := dist(λβ , ∂Σϕ ). It will not be necessary to make the dependence of dβ on ϕ explicit in the notation since the function ϕ will always be clear from the context. Theorem 4.2. Let f ∈ L∞ (R+ ) be a function such that its Laplace transform fˆ extends to a bounded analytic function in some domain Σϕ , where ϕ ∈ C(R+ ) satisfies inf ϕ > 0. Then there exists a Banach space X which embeds continuously into L∞ (R+ ) and which is left-shift invariant such that
364
R. Chill, Y. Tomilov / Journal of Functional Analysis 256 (2009) 352–384
(i) the left-shift semigroup (T (t))t0 on X is bounded and strongly continuous, (ii) the resolvent of the generator A satisfies the decay estimate R(λβ , A)
L(X)
C
log dβ dβ
for every β ∈ R,
(4.1)
(iii) if T : L1 (R+ ) → L(X) is the algebra homomorphism which is represented (in a strong sense) by (T (t))t0 , then f ∈ range T ∗ , and (iv) the following inclusion holds: ∞ ,weak∗ )
X ⊂ L1 (R+ ) f (L
.
If, in addition, the function C+ → L∞ (R+ ), λ → eλ f, extends analytically to Σϕ and if there exists some r ∈ (0, 1) such that sup λ∈B(λβ ,rdβ )
eλ f ∞ C
1 dβ
for every β ∈ R,
(4.2)
then the space X can be chosen in such a way that the resolvent satisfies the stronger estimate 1 R(λβ , A) L(X) C d β
for every β ∈ R.
Remark 4.3. The condition inf ϕ > 0 in the above theorem simplifies the proof in some places but is not essential. Moreover, it can always be achieved by rescaling the function f or the semigroup (T (t))t0 . The important points in the above theorem are the statements that the resolvent decay condition (1.1) is satisfied as soon as limβ→∞ ϕ(β) = ∞, and that at the same time f ∈ range T ∗ . Thus, if we are able to find a function f ∈ L∞ (R+ ) such that its Laplace transform fˆ extends to a bounded analytic function on Σϕ , where ϕ ∈ C(R+ ) satisfies limβ→∞ ϕ(β) = ∞, and such that f is not continuous on (0, ∞), then the Riemann–Lebesgue operator from Theorem 4.2 is not representable by Lemma 3.7(c), that is, the semigroup (T (t))t0 is not immediately norm continuous (Lemma 3.3). In other words, the existence of such a function f solves the normcontinuity problem. It is straightforward to check that the characteristic function f = 1[0,1] provides such an example. This and another example will be discussed in Section 5. For these examples it will be of substantial interest that Theorem 4.2 also gives an estimate of the resolvent R(·, A) along vertical lines, in terms of the Laplace transform fˆ, the decay of the function λ → eλ f and the growth of the function ϕ. We point out that a decay condition weaker than (4.2) is always true, as we will prove in Lemma 4.14 below. We do not know whether the decay condition (4.2) is always satisfied.
R. Chill, Y. Tomilov / Journal of Functional Analysis 256 (2009) 352–384
365
The rest of this section will be devoted to the proof of Theorem 4.2, that is, to the construction of the Banach space X and the algebra homomorphism T : L1 (R+ ) → L(X). The space X will be a closed subspace of an appropriate Banach space M which is continuously embedded into L∞ (R+ ); we will first construct M by constructing its unit ball. Lemma 4.4. Let (fn ) ⊂ L∞ (R+ ) be a bounded sequence and define the set BM :=
gn fn : (gn ) ∈ Bl 1 (L1 (R+ ))
(L∞ ,weak∗ )
⊂ L∞ (R+ ).
(4.3)
n
Then: (a) The set BM is non-empty, convex, weak∗ compact and BM = −BM . (b) For every g ∈ BL1 and every h ∈ BM one has g h ∈ BM . (c) For every n one has fn ∈ BM . Proof. The properties in (a) are either trivial or easy to check. Next, let g ∈ BL1 and h ∈ BM . Assume first that h = n gn fn for some sequence (gn ) ∈ Bl 1 (L1 (R+ )) . Then gh=
g (gn fn ) =
(g ∗ gn ) fn .
n
n
h ∈ BM . For general h ∈ BM , by the definition Since (g ∗ gn ) ∈ Bl 1 (L1 (R+ )) , this implies g of BM , there exists a net (hα ) ⊂ BM , hα = n gnα fn for some (gnα ) ∈ Bl 1 (L1 (R+ )) , such that w∗ - limα hα = h. However, then g h = w∗ - limα g hα as one easily verifies. Since BM is weak∗ closed, we have proved (b). By definition of BM , one has g fn ∈ BM for every g ∈ BL1 (R+ ) . Taking an approximate unit (gj ) in BL1 (R+ ) , one easily shows w∗ - limj gj fn = fn . Since BM is weak∗ closed, this proves (c). 2 For a bounded sequence (fn ) ⊂ L∞ (R+ ) we define the set BM ⊂ L∞ (R+ ) as in (4.3), and then we put M := R+ BM .
(4.4)
Then M is a (in general nonclosed) subspace of L∞ (R+ ) and becomes a normed space when it is equipped with the Minkowski norm hM := inf{λ > 0: h ∈ λBM }. When M is equipped with this Minkowski norm, then BM is the unit ball of M, and there is no ambiguity with our previously introduced notation. Moreover, M embeds continuously into L∞ (R+ ). By a result by Dixmier, M is a dual space, and in particular M is a Banach space [13]. To be more precise, consider the natural embedding S : L1 (R+ ) → M ∗ given by
366
R. Chill, Y. Tomilov / Journal of Functional Analysis 256 (2009) 352–384
Sg, m M ∗ ,M := g, m L1 ,L∞ ,
(4.5)
and let ∗
M∗ := range S M .
(4.6)
Then we have the following result; the short proof follows [32, proof of Theorem 1]. Lemma 4.5. The space M is isometrically isomorphic to M∗ ∗ , that is, to the dual of M∗ . Proof. The key point is the fact that, by construction, BM is weak∗ compact in L∞ (R+ ). By the definition of the operator S and by the definition of the space M∗ , this implies that the unit ball BM is compact with respect to the σ (M, M∗ ) topology. Consider the contraction J : M → M∗ ∗ which maps every m ∈ M to the functional J m ∈ M∗ ∗ given by J m, m∗ M∗ ∗ ,M∗ := m, m∗ M,M ∗ . The space M∗ separates the points in M because the space L1 (R+ ) separates the points in M ⊂ L∞ (R+ ). Therefore, the operator J is injective. Next, let m∗∗ ∈ M∗ ∗ and assume for simplicity that m∗∗ M∗ ∗ = 1. By Hahn–Banach, we may consider m∗∗ also as an element in BM ∗∗ . By Goldstine’s theorem, there exists a net (mα ) ⊂ BM which converges to m∗∗ in σ (M ∗∗ , M ∗ ). Since BM is compact with respect to the σ (M, M∗ ) topology, there exists m ∈ BM such that
m, m∗ M,M ∗ = lim mα , m∗ M,M ∗ = m∗∗ , m∗ M∗ ∗ ,M∗ for every m∗ ∈ M∗ . α
Hence, J m = m∗∗ and mM 1 = J mM∗ ∗ . We have thus proved that J is also surjective and isometric. 2 Remark 4.6. Using only the definitions of the operators J and S, it is straightforward to verify that S ∗ J is the natural embedding of M into L∞ (R+ ) and that range S ∗ = M.
(4.7)
Lemma 4.7. The space M is an L1 (R+ ) module in a natural way: for every g ∈ L1 (R+ ) and every m ∈ M (⊂ L∞ (R+ )) the adjoint convolution g m belongs to M and g mM gL1 mM . Proof. By Lemma 4.4(b), for every nonzero g ∈ L1 (R+ ) and m ∈ M one has BM . The claim follows immediately. 2
g gL1
m mM
∈
In the following, we will always consider M as an L1 (R+ ) module via the adjoint convolution. Note that together with M also the dual space M ∗ is an L1 (R+ ) module if for every g ∈ L1 (R+ ) and every m∗ ∈ M ∗ we define the product g ∗ m∗ ∈ M ∗ by g ∗ m∗ , m M ∗ ,M := m∗ , g m M ∗ ,M ,
m ∈ M.
We use again the notation ∗ for the adjoint of the adjoint convolution. If M = L∞ (R+ ), then the product ∗ coincides with the usual convolution in L1 (R+ ) ⊂ L∞ (R+ )∗ and there is no ambiguity in the notation.
R. Chill, Y. Tomilov / Journal of Functional Analysis 256 (2009) 352–384
367
Lemma 4.8. The natural embedding S : L1 (R+ ) → M ∗ defined in (4.5) is an L1 (R+ ) module homomorphism. The space M∗ is an L1 (R+ ) submodule of M ∗ . Proof. For every g, h ∈ L1 (R+ ) and every m ∈ M one has
S(g ∗ h), m M ∗ ,M = g ∗ h, m L1 ,L∞ = h, g m L1 ,L∞ = Sh, g m M ∗ ,M = g ∗ Sh, m M ∗ ,M .
Since this equality holds for every m ∈ M, this proves S(g ∗ h) = g ∗ Sh for every g, h ∈ L1 (R+ ), and therefore S is an L1 (R+ ) module homomorphism. At the same time, this equality proves that the closure of the range is a submodule of M ∗ . 2 We omit the proof of the following lemma which is straightforward. Lemma 4.9. Let M∗ be defined as in (4.6). The natural embedding T∗ : L1 (R+ ) → L(M∗ ) given by T∗ g(m∗ ) := g ∗ m∗ , m∗ ∈ M∗ , is an algebra homomorphism. The following lemma allows us to calculate T∗ gL(M∗ ) in terms of the sequence (fn ). Lemma 4.10. Let (fn ) ⊂ L∞ (R+ ) be a bounded sequence, let M∗ be defined as in (4.6), and let T∗ : L1 (R+ ) → L(M∗ ) be the induced algebra homomorphism from Lemma 4.9. Then, for every g ∈ L1 (R+ ), one has T∗ gL(M∗ ) = sup g fn M . n
Proof. Let g ∈ L1 (R+ ). Then, by the definition of T∗ , S, and by the definition of M∗ , T∗ gL(M∗ ) = sup g ∗ m∗ M ∗ m∗ ∈BM∗
= sup
sup g ∗ m∗ , m M ∗ ,M
m∗ ∈BM∗ m∈BM
=
sup
sup g ∗ Sh, m M ∗ ,M .
h∈L1 (R+ ) m∈BM ShM ∗ 1
Since S is an L1 (R+ ) module homomorphism, and by the definition of S,
(4.8)
368
R. Chill, Y. Tomilov / Journal of Functional Analysis 256 (2009) 352–384
T∗ gL(M∗ ) =
sup S(g ∗ h), m M ∗ ,M
sup
h∈L1 (R+ ) m∈BM ShM ∗ 1
=
sup g ∗ h, m L1 ,L∞ .
sup
h∈L1 (R+ ) m∈BM ShM ∗ 1
Since, by definition, { n gn fn : (gn ) ∈ Bl 1 (L1 (R+ )) } is weak∗ dense in BM (with respect to the weak∗ topology in L∞ (R+ )), and since M∗ is norming for M by Lemma 4.5, we can continue to compute T∗ gL(M∗ ) =
=
sup
sup
h∈L1 (R+ ) (gn )∈Bl 1 (L1 (R+ )) ShM ∗ 1
sup
sup
h∈L1 (R+ ) (gn )∈Bl 1 (L1 (R+ )) ShM ∗ 1
=
sup
sup
h∈L1 (R+ ) (gn )∈Bl 1 (L1 (R+ )) ShM ∗ 1
=
n
∞
L1 ,L
h, g (g f ) n n n
∞
L1 ,L
Sh, gn (g fn ) n
M ∗ ,M
g (g f ) n n .
sup (gn )∈Bl 1 (L1 (R
g ∗ h, gn f n
M
n
+ ))
This immediately implies T∗ gL(M∗ )
h (g fn )
sup h∈BL1 (R
=
+)
sup h∈BL1 (R
(h ∗ g) fn
M
M
for every n.
+)
By putting h = λeλ , letting λ → ∞, and using Lemma 4.7, we obtain T∗ gL(M∗ ) g fn M
for every n.
On the other hand, by Lemma 4.7 again, T∗ gL(M∗ )
sup (gn )∈Bl 1 (L1 (R
+ ))
gn L1 g fn M
n
sup g fn M . n
The preceding two estimates imply the claim.
2
The operator T∗ from Lemma 4.9 will be equivalent to the operator we are looking for in Theorem 4.2. However, so far we have not said anything about the sequence (fn ) ⊂ L∞ (R+ )
R. Chill, Y. Tomilov / Journal of Functional Analysis 256 (2009) 352–384
369
which served for the construction of M, and which allows us by Lemma 4.10 to obtain the desired resolvent estimate in Theorem 4.2. It remains to explain how the sequence (fn ) is constructed in order to prove Theorem 4.2. For the time being, let f ∈ L∞ (R+ ) be a fixed function, and suppose that the Laplace transform fˆ extends analytically to a bounded function on Σϕ , where ϕ ∈ C(R+ ) satisfies inf ϕ > 0. In order to simplify the notation, we define for every k ∈ Z λk := 2 + ik, dk := dist(λk , ∂Σϕ ),
and
e˜k = eλk , and we will choose numbers ck > 0 depending on the functions λ → eλ f and ϕ; see Proposition 4.11 below for the precise definition of ck . We define inductively for n 1 and k ∈ Zn+1 e˜k := e˜k¯ ∗ e˜kn+1 = e˜k1 ∗ · · · ∗ e˜kn+1 and ck := ck¯ · ckn+1 = ck1 · · · · · ckn+1 , ¯ kn+1 ). where k¯ ∈ Zn is such that k = (k, Then, for every n 1 and every k ∈ Zn we put fk :=
e˜k f e˜k ∗ · · · ∗ e˜kn = 1 f. ck ck1 · · · · · ckn
Finally, we set f∞ := f, and I := {∞} ∪
Zn ,
n1
and we will define the unit ball BM , the space M and the space X starting from the family (fk )k∈I .
(4.9)
370
R. Chill, Y. Tomilov / Journal of Functional Analysis 256 (2009) 352–384
Proposition 4.11. Let f ∈ L∞ (R+ ) be such that the Laplace transform fˆ extends to a bounded analytic function in Σϕ , where ϕ ∈ C(R+ ) satisfies inf ϕ > 0. Let r ∈ (0, 14 ) be arbitrary. For every k ∈ Z we put ck =
4 log dk . r dk
(4.10)
Then the family (fk )k∈I given by (4.9) is bounded in L∞ (R+ ). The same is true if the condition (4.2) is satisfied and if we then put, for every k ∈ Z, ck =
4 1 . r dk
The proof of Proposition 4.11 is based on the following series of four lemmas. The statement and the proof of the following lemma should be compared to [1, Lemmas 4.6.6, 4.7.9]. Lemma 4.12. Let f ∈ L∞ (R+ ) be such that the Laplace transform fˆ extends to a bounded analytic function in Σϕ , where ϕ ∈ C(R+ ) satisfies inf ϕ > 0. Then also the function λ → eλ f , C+ → BUC(R+ ) extends to a bounded analytic function in Σϕ . Proof. For every t ∈ R+ and every λ ∈ C+ one has ∞ (eλ f )(t) =
e−λs f (t + s) ds
0
= e fˆ(λ) −
t
λt
eλ(t−s) f (s) ds. 0
From this identity we obtain first that for every fixed t ∈ R+ the function λ → (eλ f )(t) extends to an analytic function on Σϕ , and we obtain second for every t ∈ R+ and every λ ∈ Σϕ the estimate (eλ f )(t)
f ∞
| Re λ| f ∞ | Re λ|
if Re λ > 0, + fˆ∞
if Re λ < 0.
(4.11)
By assumption, there exists 0 < α 1 such that inf ϕ > α. The above estimate immediately yields sup λ∈Σϕ |Re λ| α2
2f ∞ + fˆ∞ . sup (eλ f )(t) α t∈R+
In order to show that the function (eλ f )(t) is bounded in the strip {λ ∈ C: |Re λ| α2 } (with a bound independent of t ∈ R+ ) we can argue as follows. For every β ∈ R, by the maximum principle and by the estimate (4.11),
R. Chill, Y. Tomilov / Journal of Functional Analysis 256 (2009) 352–384
371
(λ − iβ)2 sup (eλ f )(t) 1 + α2 |λ−iβ|α (λ − iβ)2 = sup (eλ f )(t) 1 + α2 |λ−iβ|=α
4f ∞ + 4fˆ∞ . α
Hence, for every t ∈ R+ and every β ∈ R sup |λ−iβ| α2
(eλ f )(t) 6f ∞ + 6fˆ∞ , α
which yields the desired estimate in the strip {λ ∈ C: |Re λ| < α2 }. So we finally obtain sup sup (eλ f )(t) < ∞, λ∈Σϕ t∈R+
and in particular the function λ → eλ f is bounded on Σϕ with values in BC(R+ ). Now one may argue as in the proof of [1, Corollary A.4]. Pointwise analyticity and uniform boundedness imply, by [1, Proposition A.3], that the function λ → eλ f is bounded and analytic on Σϕ with values in BC(R+ ). Since eλ f ∈ BUC(R+ ) for every λ ∈ C+ , by the identity theorem for analytic functions (see, for example, the version in [1, Proposition A.2]), we finally obtain the claim. 2 The main argument in the proof of the following lemma (the two constants theorem) is also used in [30, proof of Theorem 5.3], but the following lemma gives a better estimate. Recall that λβ = 2 + iβ and dβ = dist(λβ , ∂Σϕ ). Lemma 4.13. Let X be some Banach space and let ϕ ∈ C(R+ ) be a nonnegative function. Let h : Σϕ → X be a bounded analytic function satisfying the estimate h(λ) C Re λ
for every λ ∈ C+ and some C 0.
Then for every r ∈ (0, 14 ) there exists Cr 0 such that for every β ∈ R sup d
λ∈B(λβ ,r logβd )
h(λ) Cr log dβ . dβ
β
Proof. We may assume that the constant C from the hypothesis satisfies C h∞ . Fix r ∈ (0, 14 ). We may in the following consider only those β ∈ R for which 4 < log dβ . For the other β, the estimate in the claim becomes trivial if the constant Cr is chosen sufficiently large. Let Ω := λ ∈ C: |Re λ|, |Im λ| < 1 ,
372
R. Chill, Y. Tomilov / Journal of Functional Analysis 256 (2009) 352–384
and let Γ0 := {λ ∈ ∂Ω: Re λ = 1} and Γ1 := ∂Ω \ Γ0 . By the two constants theorem, for every analytic function g : Ω → X having a continuous extension to Ω¯ and satisfying the boundary estimate g(λ) Ci if λ ∈ Γi (i = 0, 1), one has the estimate g(λ) C w(λ) C 1−w(λ) 0
for every λ ∈ Ω,
1
where w = wΩ (·, Γ0 ) is the harmonic measure of Γ0 with respect to Ω, that is, w : Ω → [0, 1] is the harmonic function satisfying w = 1 on Γ0 and w = 0 on Γ1 . For every β ∈ R with log dβ > 4 we apply this two constants theorem to the function given by dβ 1 , λ−1+ g(λ) = h λβ + 4 log dβ which satisfies by assumption the estimates log dβ g(λ) C dβ C
if λ ∈ Γ0 ,
¯ λ ∈ Ω,
and
if λ ∈ Γ1 .
We then obtain w(λ) g(λ) C log dβ dβ
for every λ ∈ Ω.
By the Schwarz reflection principle, the function 1 − w extends to a harmonic function in the rectangle {λ ∈ C: −1 < Re λ < 3, |Im λ| < 1}, and in particular the function w is continuously differentiable there. We can therefore find a constant c > 0 such that inf
λ∈B(1− log1d β
,4r
1 log dβ
w(λ) 1 − )
c(1 + 4r) . log dβ
Combining the preceding two estimates, we obtain h(λ) =
sup d
λ∈B(λβ ,r logβd ) β
g(λ)
sup λ∈B(1− log1d ,4r log1d ) β
β
C
sup λ∈B(1− log1d ,4r log1d ) β
C
log dβ dβ
β
1− c(1+4r) log dβ
log dβ dβ
w(λ)
R. Chill, Y. Tomilov / Journal of Functional Analysis 256 (2009) 352–384
= Ce Cr The claim is proved.
c(1+4r)(1−
log log dβ log dβ
373
) log dβ
dβ log dβ . dβ
2
Lemma 4.14. Let f ∈ L∞ (R+ ) be such that the Laplace transform fˆ extends to a bounded analytic function in Σϕ , where ϕ ∈ C(R)+ satisfies inf ϕ > 0. Then for every r ∈ (0, 14 ) there exists Cr 0 such that eλ f ∞ Cr
sup d
λ∈B(λk ,r logkd )
log dk dk
for every k ∈ Z.
k
Proof. Since eλ f ∞ mas 4.12 and 4.13. 2
C Re λ
for every λ ∈ C+ , this lemma is a direct consequence of Lem-
The following is a consequence of the resolvent identity and should probably be known. We will give the easy proof here. Lemma 4.15. For every n 1, every λ1 , . . . , λn ∈ C+ , and every closed path Γ ⊂ C+ such that λ1 , . . . , λn are in the interior of Γ one has eλ1 ∗ · · · ∗ eλn
(−1)n+1 = 2πi
Γ
eλ dλ. (λ − λ1 ) · · · · · (λ − λn )
(4.12)
Proof. The proof goes by induction on n. If n = 1, then the formula (4.12) is just Cauchy’s integral formula. So assume that the formula (4.12) is true for some n 1. Let λ1 , . . . , λn , λn+1 ∈ C+ , and let Γ ⊂ C+ be a closed path such that λ1 , . . . , λn , λn+1 are in the interior of Γ . Then, by the resolvent identity and the induction hypothesis, eλ1 ∗ · · · ∗ eλn ∗ eλn+1 =
(−1)n+1 2πi
Γ
=
(−1)n+2
2πi Γ
eλ ∗ eλn+1 dλ (λ − λ1 ) · · · · · (λ − λn ) eλ dλ (λ − λ1 ) · · · · · (λ − λn )(λ − λn+1 )
(−1)n+1 + eλn+1 2πi
Γ
1 dλ. (λ − λ1 ) · · · · · (λ − λn )(λ − λn+1 )
For the induction step it suffices to show that the second integral on the right-hand side of this equality vanishes. In order to see that this integral vanishes, we replace the path Γ by a circle
374
R. Chill, Y. Tomilov / Journal of Functional Analysis 256 (2009) 352–384
centered in 0 and having radius R > 0 large enough, without changing the value of the integral. A simple estimate then shows that 1 dλ = O R −n as R → ∞. (λ − λ1 ) · · · · · (λ − λn )(λ − λn+1 ) |λ|=R
Since the left-hand side is independent of R and since n 1, by letting R → ∞, we obtain that the integral above is zero. 2 Proof of Proposition 4.11. It will be convenient in this proof to define the function h(s) := s 2. Then ck =
4 1 r h(dk )
s log s ,
for every k ∈ Z,
where r ∈ (0, 14 ) is fixed as in the assumption. Let Cr 0 be as in Lemma 4.14. We will show that sup
k∈I, k=∞
fk ∞ Cr .
The proof goes by induction on n. By Lemma 4.14, for every k ∈ Z, e˜k f
∞
Cr Cr c k h(dk )
by the definition of ck and since r 1, and therefore fk ∞ Cr
for every k ∈ Z.
Next, we assume that there exists n 1 such that fk ∞ Cr
for every k ∈ Zn .
Let k = (kν )1νn+1 ∈ Zn+1 . Assume first that there exist 1 ν, μ n + 1 such that |kν − kμ | >
r h(dkν ) + h(dkμ ) . 4
(4.13)
There exists k˜ ∈ I such that fk =
e˜kν ∗ e˜kμ fk˜ , ckν · ckμ
and therefore, by the resolvent identity, by the induction hypothesis, by the definition of ck , and by (4.13),
R. Chill, Y. Tomilov / Journal of Functional Analysis 256 (2009) 352–384
375
e˜kν fk˜ − e˜kμ fk˜ fk ∞ = (k − k )c · c μ ν kν kμ ∞ 1 1 1 Cr + |kν − kμ | ckν ckμ = Cr
r 1 h(dkν ) + h(dkμ ) |kν − kμ | 4
Cr . Hence, we may suppose that |kν − kμ |
r h(dkν ) + h(dkμ ) 4
for every 1 ν, μ n + 1.
(4.14)
After a permutation of the indices, we may assume in addition that h(dk1 ) =
max
1νn+1
h(dkν ).
From the estimate |λkν − λk1 | = |kν − k1 | 2r h(dk1 ) we obtain
3r λkν ∈ B λ1 , h(dk1 ) for every 1 ν n + 1. 4 As a consequence, by Lemma 4.15, and since λ → eλ f extends to a bounded analytic function on B(λk1 , h(dk1 )), e˜k f = (eλk1
1 ∗ · · · ∗ eλkn+1 ) f = 2πi
∂B(λk1 , 3r 4 h(dk1 ))
eλ f dλ. (λ − λk1 ) · · · · · (λ − λkn+1 ) (4.15)
Note that for every λ ∈ ∂B(λ1 , 3r4 h(dk1 )) and every 1 ν n + 1 one has |λ − λkν | |λ − λk1 | − |λk1 − λkν | 3r r h(dk1 ) − h(dk1 ) + h(dkν ) 4 4 r h(dk1 ) 4 r 1 h(dkν ) = 4 ckν or 1 ckν . |λ − λkν | This inequality, the equality (4.15), and the decay condition from Lemma 4.14 yield
376
R. Chill, Y. Tomilov / Journal of Functional Analysis 256 (2009) 352–384
e˜k f ∞
3r h(dk1 ) sup 4 λ∈∂B(λ , 3r h(d k1
4
eλ f ∞ ck1 · · · · · ckn+1
k1 ))
Cr ck1 · · · · · ckn+1 = Cr c k . This implies fk ∞ Cr
for every k ∈ Zn+1 ,
and by induction, the first claim is proved. If the estimate (4.2) holds and if ck = 4r d1k , then one may repeat the above proof replacing the function h by the function h(s) := s. 2 We are ready to prove Theorem 4.2. Proof of Theorem 4.2. Let f ∈ L∞ (R+ ) and ϕ ∈ C(R+ ) be as in the hypothesis. Define the numbers ck > 0 as in Proposition 4.11 (depending on whether the condition (4.2) holds or not), and let the family (fk )k∈I be defined as in (4.9). By Proposition 4.11, the family (fk )k∈I is uniformly bounded in L∞ (R+ ). Define the unit ball BM , and the spaces M and M∗ as above. We recall that the space M embeds continuously into L∞ (R+ ), and that by construction ∞ ,weak∗ )
M ⊂ L1 (R+ ) f (L
.
Let T∗ : L1 (R+ ) → L(M∗ ) be the algebra homomorphism defined in Lemma 4.9, and let T : L1 (R+ ) → L(M) be the algebra homomorphism given by T g(m) := g m, m ∈ M. Clearly, T gL(M) = T∗ gL(M∗ ) for every g ∈ L1 (R+ ). By Lemma 4.10, and since supk∈I fk M 1 by Lemma 4.4 (c), for every k ∈ Z, T eλk L(M) = T∗ eλk L(M∗ ) = sup eλk fk¯ M ¯ k∈I
e˜k¯ f = supe˜k ck¯ M ¯ k∈I f e˜(k,k) ¯ = sup c k c¯ ¯ M (k,k) k∈I = sup f(k,k) ¯ ck M ¯ k∈I
ck . By the definition of ck (see Proposition 4.11), this leads to the estimate
R. Chill, Y. Tomilov / Journal of Functional Analysis 256 (2009) 352–384
T eλk L(M)
C logdkdk ,
or
for every k ∈ Z,
C d1k ,
377
(4.16)
depending on whether the condition (4.2) holds or not. By Lemma 3.1, after replacing the space M by a closed subspace X, if necessary, we can assume that the homomorphism T is represented (in the strong sense) by a bounded C0 -semigroup (T (t))t0 ∈ L(X). Since T was defined by adjoint convolution, it follows that the semigroup (T (t))t0 is the left-shift semigroup on X. If A is the generator of this semigroup, then the estimate (4.16) implies
R(λk , A)
L(X)
C logdkdk ,
or
C d1k ,
for every k ∈ Z,
depending on whether the condition (4.2) holds or not. Now let β ∈ R be arbitrary, and let k ∈ Z be such that |β − k| 1. By the resolvent identity and boundedness of the semigroup (T (t))t0 , R(λβ , A) L(X) = R(λk , A) + i(k − β)R(λβ , A)R(λk , A) L(X) (1 + C)R(λk , A)L(X) log dk C dk , or C d1k , depending on whether the condition (4.2) holds or not. By contractivity of the distance function we have |dβ − dk | |β − k| 1, so that dβ dk + 1 2dk ; recall that dk 2. This estimate for dβ implies 1 1 2 dk dβ
and
log dβ log dk 2 , dk dβ
and therefore R(λβ , A)
L(X)
⎧ ⎨ C log dβ , or dβ ⎩C
1 dβ ,
for every β ∈ R,
depending on whether the condition (4.2) holds or not. It remains to show that f ∈ range T ∗ . For every g ∈ L1 (R+ ) we can estimate
378
R. Chill, Y. Tomilov / Journal of Functional Analysis 256 (2009) 352–384
SgM ∗ = sup S(g ∗ h)M ∗ h∈BL1
= sup g ∗ ShM ∗ h∈BL1
Sh g ∗ = S sup S M ∗ h∈BL1 S
sup
g ∗ ShM ∗
h∈L1 ShM ∗ 1
= S sup g ∗ m∗ M ∗ m∗ ∈BM∗
= ST∗ gL(M∗ ) = ST gL(M) . In other words, there is a bounded operator R : range T ·L(M) → M∗ ⊂ M ∗ such that S = RT . Hence, T ∗ R ∗ = S ∗ : M ∗∗ → L∞ (R+ ), which implies range S ∗ ⊂ range T ∗ .
(4.17)
On the other hand, we recall from (4.7) that range S ∗ = M. Since f = f∞ ∈ BM by Lemma 4.4(c), we thus obtain f ∈ range T ∗ . Theorem 4.2 is completely proved. 2 Remark 4.16. It would be interesting to understand the geometric structure of the spaces M and X, for example, whether they might be UMD spaces or spaces having nontrivial Fourier type. 5. The norm continuity problem In this section we present two examples showing that the norm continuity problem has a negative answer. In these two examples emphasis will be put on precise decay estimates for the resolvent along vertical lines. Before stating the two examples, we recall the following known result; see [39, Theorem 4.9], [19, Theorem 4.1.3]. Proposition 5.1. Let A be the generator of a bounded C0 -semigroup (T (t))t0 . Then the following are true: (i) If R(2 + iβ, A) = o
1 log |β|
as |β| → ∞,
then the semigroup (T (t))t0 is immediately differentiable.
R. Chill, Y. Tomilov / Journal of Functional Analysis 256 (2009) 352–384
379
(ii) If R(2 + iβ, A) = O
1 log |β|
as |β| → ∞,
then the semigroup (T (t))t0 is eventually differentiable. The following is our first counterexample to the norm continuity problem. Theorem 5.2. There exists a Banach space X and a uniformly bounded C0 -semigroup (T (t))t0 ⊂ L(X) with generator A such that: (i) the resolvent satisfies the estimate R(2 + iβ, A) = O
1 log |β|
as |β| → ∞,
and in particular the resolvent satisfies the resolvent decay condition (1.1), (ii) T (1) = 0, that is, the semigroup (T (t))t0 is nilpotent, and (iii) whenever t0 ∈ [0, 1), then the semigroup (T (t))t0 is not norm-continuous for t > t0 . Proof. Let f = 1[0,1] be the characteristic function of the interval [0, 1]. Since f has compact support, the Laplace transform fˆ and also the function λ → eλ f extend to entire functions, and for every λ ∈ C \ {0} and every t ∈ R+ , (eλ f )(t) =
1 −λ(1−t) ) λ (1 − e
if 0 t 1, if t > 1.
0
Hence, for every λ ∈ C \ {0}, eλ f ∞
−Re λ
2 e |λ|
if Re λ < 0,
1 2 |λ|
if Re λ 0, λ = 0.
Let ϕ ∈ C(R+ ) be the function given by ϕ(β) = 1 + log+ (β),
β 0,
where log+ is the positive part of the logarithm. Clearly limβ→∞ ϕ(β) = +∞. It follows from (5.1) that sup fˆ(λ) < ∞, λ∈Σϕ
so that f satisfies the hypothesis of Theorem 4.2. By the definition of ϕ, 3 + log+ |β| dβ 3 + log+ |β| 2
for every β ∈ R,
(5.1)
380
R. Chill, Y. Tomilov / Journal of Functional Analysis 256 (2009) 352–384
where, as before, we put dβ := dist(λβ , ∂Σϕ ) and λβ = 2 + iβ. From (5.1) one therefore also obtains for every r ∈ (0, 1) the estimate sup λ∈B(λβ ,rdβ )
eλ f ∞ 2
max{e−Re λ , 1} |λ| λ∈B(λβ ,rdβ )
C
sup
er log |β| |β| − r log+ |β|
C|β|−(1−r) , and in particular sup λ∈B(λβ ,rdβ )
eλ f ∞ Cr
1 dβ
for every β ∈ R.
This means that the function f satisfies the decay condition (4.2) from Theorem 4.2. By Theorem 4.2, there exists a left-shift invariant Banach space X → L∞ (R+ ) such that the resolvent of the generator A of the left-shift semigroup (which is strongly continuous on X) satisfies the decay estimate R(2 + iβ, A)
L(X)
C
1 1 + log+ |β|
for every β ∈ R,
so that the resolvent satisfies the resolvent decay condition (1.1). Moreover, if T : L1 (R+ ) → L(X) is the algebra homomorphism which is represented (in the strong sense) by the left-shift semigroup, then f = 1[0,1] ∈ range T ∗ . In particular, by Lemma 3.7(d), the semigroup cannot be continuous for t > t0 whenever t0 ∈ [0, 1). On the other hand, it follows from Theorem 4.2(iv) that every function in X is supported in the interval [0, 1] so that the left-shift semigroup on X vanishes for t 1. 2 In the second example we show that there are also C0 -semigroups which are never normcontinuous, whose generator satisfies the resolvent decay condition (1.1), and the decay of the resolvent along vertical lines is even arbitrarily close to a logarithmic decay. Note that, by Proposition 5.1, a logarithmic decay as in Theorem 5.2(i) is not possible for semigroups which are not eventually norm-continuous, that is, norm-continuous for t > t0 . Theorem 5.3. Let h ∈ C(R+ ) be a positive, increasing and unbounded function such that also the function log+ / h is increasing and unbounded. Then there exists a Banach space X and a uniformly bounded C0 -semigroup (T (t))t0 ⊂ L(X) with generator A such that: (i) the resolvent satisfies the estimate R(2 + iβ, A) = O h(|β|) as |β| → ∞, log |β| and in particular the resolvent satisfies the resolvent decay condition (1.1), and (ii) the semigroup (T (t))t0 is not eventually norm-continuous.
R. Chill, Y. Tomilov / Journal of Functional Analysis 256 (2009) 352–384
381
Proof. Since the function log+ / h is increasing and unbounded, then also the function s → s 1/ h(s) = elog s/ h(s) is increasing and unbounded for s 1. We may assume that this function is strictly increasing for s 1. In particular, the function ψ given by 1 ψ es 1/ h(s) := h(s), 4
s 1,
(5.2)
is well defined, increasing and unbounded. Choose coefficients an > 0 such that 1 an an+1 and such that ∞
an r n+1 r ψ(r)
for every r 1;
(5.3)
n=0
it is an exercise to show that such coefficients exist (see also [36, Problem 2, p. 1]). We put f=
∞
an 1[n,n+1] .
n=0
Then clearly f ∈ L∞ (R+ ) and it follows from (5.3) that the function λ → eλ f extends to an entire function. It is straightforward to show that for every λ ∈ C \ {0} and every t ∈ R+ one has (eλ f )(t) =
∞ e−λ([t]−t) − 1 eλt 1 − e−λ an e−λn − a[t] λ λ n=[t]
∞ eλt 1 − e−λ([t]+1−t) −λ 1−e . = an e−λn + a[t] λ λ n=[t]+1
If Re λ < 0, this yields the estimate ∞ 2 (eλ f )(t) 2 an+[t] e−Re λ(n+1) + |λ| |λ| n=0 ∞ 2 an e−Re λ(n+1) + 1 |λ| n=0
2 −Re λψ(e−Re λ ) e +2 , |λ| where in the second line we have used the fact that the sequence (an ) is decreasing, and in the third line we have used the estimate (5.3). If Re λ 0, then we obtain the estimate (eλ f )(t) C |λ|
382
R. Chill, Y. Tomilov / Journal of Functional Analysis 256 (2009) 352–384
for some C 1, so that eλ f ∞
C −Re λψ(e−Re λ ) e +2 |λ|
for every λ ∈ C \ {0}.
+
β Let ϕ(β) := 1+ log h(β) , β 0. By assumption, limβ→∞ ϕ(β) = ∞. Moreover, for every β ∈ R large enough,
sup λ∈B(2+iβ,2+ϕ(β))
eλ f ∞
C −Re λψ(e−Re λ ) e +2 λ∈B(2+iβ,2+ϕ(β)) |λ| sup
ϕ(|β|)
C
)+2 eϕ(|β|)ψ(e . |β| − ϕ(|β|)
For all β large enough we have ϕ(|β|) 12 |β|. Moreover, if |β| 2, then eϕ(|β|)ψ(e
ϕ(|β|) )
1/ h(|β|) )
= eψ(e|β|
1 h(|β|)
|β|
ψ(e|β|1/ h(|β|) ) h(|β|)
1
= |β| 4 log |β| |β| 4 1
C|β| 2 , by definition of the functions ϕ and ψ . Hence, if β is large enough, then sup λ∈B(2+iβ,2+ϕ(β))
1
eλ f ∞ C|β|− 2 .
In particular, sup eλ f ∞ < ∞.
λ∈Σϕ
Moreover, if we let, as before, λβ = 2 + iβ and dβ = dist(λβ , ∂Σϕ ), then 2 + ϕ(|β|) dβ 2 + ϕ |β| 2
for every β ∈ R large enough
and therefore sup λ∈B(λβ ,dβ )
eλ f ∞ C
1 dβ
for every β ∈ R.
By Theorem 4.2, there exists a left-shift invariant Banach space X → L∞ (R+ ) such that the left-shift semigroup (T (t))t0 ⊂ L(X) is strongly continuous and such that the resolvent of the generator A satisfies the estimate R(2 + iβ, A) C h(|β|) log |β|
for every β ∈ R, |β| > 1,
R. Chill, Y. Tomilov / Journal of Functional Analysis 256 (2009) 352–384
383
so that the resolvent decay condition (1.1) is satisfied. Moreover, still by Theorem 4.2, if T : L1 (R+ ) → L(X) is the algebra homomorphism which is represented (in the strong sense) by the semigroup (T (t))t0 , then f ∈ range T ∗ . Since the function is not continuous on any interval of the form (t0 , ∞), by Lemma 3.7(d), the semigroup (T (t))t0 cannot be eventually norm-continuous. 2 Acknowledgments The authors would like to thank the referee for his/her careful reading and helpful remarks and suggestions to improve on the original version. References [1] W. Arendt, C.J.K. Batty, M. Hieber, F. Neubrander, Vector-valued Laplace Transforms and Cauchy Problems, Monogr. Math., vol. 96, Birkhäuser, Basel, 2001. [2] C.J.K. Batty, Differentiability of perturbed semigroups and delay semigroups, in: W. Arendt, C.J.K. Batty, M. Mbekhta, Y. Tomilov, J. Zemánek (Eds.), Perspectives in Operator Theory, in: Banach Center Publications, vol. 75, Polish Acad. Sci., Warsaw, 2007, pp. 39–53. [3] M. Blake, A spectral bound for asymptotically norm-continuous semigroups, J. Operator Theory 45 (2001) 111– 130. [4] O. Blasco, J. Martinez, Norm continuity and related notions for semigroups on Banach spaces, Arch. Math. 66 (1996) 470–478. [5] A. Bobrowski, Notes on the algebraic version of the Hille–Yosida–Feller–Phillips–Miyadera theorem, Houston J. Math. 27 (2001) 75–95. [6] J. Bourgain, Dunford–Pettis operators on L1 and the Radon–Nikodym property, Israel J. Math. 37 (1980) 34–47. [7] J. Bourgain, A characterization of non-Dunford–Pettis operators on L1 , Israel J. Math. 37 (1980) 48–53. [8] S. Bu, R. Chill, Banach spaces with the Riemann–Lebesgue or the analytic Riemann–Lebesgue property, Bull. London Math. Soc. 34 (2002) 569–581. [9] W. Chojnacki, On the equivalence of a theorem of Kisynski and the Hille–Yosida generation theorem, Proc. Amer. Math. Soc. 126 (1998) 491–497. [10] W. Chojnacki, Multiplier algebras, Banach bundles, and one-parameter semigroups, Ann. Sc. Norm. Super. Pisa 28 (1999) 287–322. [11] G. Crombez, W. Govaerts, Completely continuous multipliers from L1 (G) into L∞ (G), Ann. Inst. Fourier (Grenoble) 34 (1984) 137–154. [12] J. Diestel, J.J. Uhl, Vector Measures, Math. Surveys Monogr., vol. 15, Amer. Math. Soc., Providence, RI, 1977. [13] J. Dixmier, Sur un théorème de Banach, Duke Math. J. 15 (1948) 1057–1071. [14] G. Doetsch, Handbuch der Laplace-Transformation, vols. I–III, Birkhäuser, Basel, 1950. [15] O. ElMennaoui, K.J. Engel, On the characterization of eventually norm continuous semigroups in Hilbert space, Arch. Math. 63 (1994) 437–440. [16] O. ElMennaoui, K.J. Engel, Towards a characterization of eventually norm continuous semigroups on Banach spaces, Quaest. Math. 19 (1996) 183–190. [17] M.R. Embry, Factorization of operators on Banach space, Proc. Amer. Math. Soc. 38 (1973) 587–590. [18] K.J. Engel, R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Grad. Texts in Math., vol. 194, Springer, Berlin, 1999. [19] H.O. Fattorini, The Cauchy Problem, Addison–Wesley, London, 1983. [20] J.E. Galé, Weakly compact homomorphisms and semigroups in Banach algebras, J. London Math. Soc. 45 (1992) 113–125. [21] J.E. Galé, T.J. Ransford, M.C. White, Weakly compact homomorphisms, Trans. Amer. Math. Soc. 331 (1992) 815– 824. [22] N. Ghoussoub, M. Talagrand, A noncompletely continuous operator in L1 (G) whose random Fourier transform is ˆ Proc. Amer. Math. Soc. 92 (1984) 229–232. in c0 (G), [23] N. Ghoussoub, G. Godefroy, B. Maurey, W. Schachermayer, Some topological and geometrical structures of Banach spaces, Mem. Amer. Math. Soc. 378 (1987).
384
R. Chill, Y. Tomilov / Journal of Functional Analysis 256 (2009) 352–384
[24] M. Girardi, Dunford–Pettis operators on L1 , and the complete continuity property, PhD thesis, University of Illinois at Urbana-Champaign, 1990. [25] M. Girardi, Compactness in L1 , Dunford–Pettis operators, geometry of Banach spaces, Proc. Amer. Math. Soc. 111 (1991) 767–777. [26] M. Girardi, W. Johnson, Universal non-completely-continuous operators, Israel J. Math. 99 (1997) 207–219. [27] V. Goersmeyer, L. Weis, Norm continuity of c0 -semigroups, Studia Math. 134 (1999) 169–178. [28] E. Hille, R.S. Phillips, Functional Analysis and Semi-Groups, Amer. Math. Soc., Providence, RI, 1957. [29] J. Holub, Dunford–Pettis and strongly Dunford–Pettis operators on L1 (μ), Glasgow Math. J. 31 (1989) 49–57. [30] P.S. Iley, Perturbations of differentiable semigroups, J. Evol. Equ. 7 (2007) 765–781. [31] Ye. Ivakhno, The Riemann–Lebesgue property is equivalent to the Complete Continuity property, Bull. London Math. Soc. 39 (2007) 583–585. [32] S. Kaijser, A note on dual Banach spaces, Math. Scand. 41 (1977) 325–330. [33] R. Kaufman, M. Petrakis, L.H. Riddle, J.J. Uhl Jr., Nearly representable operators, Trans. Amer. Math. Soc. 312 (1989) 315–333. [34] J. Kisynski, The Widder spaces, representations of the convolution algebra L1 (R+ ), and one-parameter semigroups of operators, preprint no. 588, Institute of Mathematics, Polish Acad. Sci., Warsaw, 1998. [35] J. Kisynski, Around Widder’s characterization of the Laplace transform of an element of L∞ (R+ ), Ann. Polon. Math. 74 (2000) 161–200. [36] B.Ya. Levin, Lectures on Entire Functions, Transl. Math. Monogr., vol. 150, Amer. Math. Soc., Providence, RI, 1996. [37] T. Matrai, Resolvent norm decay does not characterize norm continuity, Israel J. Math. (2008), in press. [38] P. Meyer-Nieberg, Banach Lattices, Springer, Berlin, 1991. [39] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Appl. Math. Sci., vol. 44, Springer, Berlin, 1983. [40] T.V. Pedersen, Sinclair homomorphisms and semigroups of analytic functions, J. Funct. Anal. 145 (1997) 527–554. [41] R.L. Rebarber, A Laplace transform relevant to holomorphic semigroups, Proc. Roy. Soc. Edinburgh Sect. A 105 (1987) 243–258. [42] M. Sova, The Laplace transform of analytic vector-valued functions (complex conditions), Cas. Pest. Mat. 104 (1979) 267–280. [43] Peng Fei Yao, On the inversion of the Laplace transform of C0 semigroups and its applications, SIAM J. Math. Anal. 26 (1995) 1331–1341. [44] Puhong You, Characteristic conditions for a C0 -semigroup with continuity in the uniform operator topology for t > 0 in Hilbert space, Proc. Amer. Math. Soc. 116 (1992) 991–997. [45] J. Zabczyk, A note on C0 -semigroups, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 23 (8) (1975) 895– 898.
Journal of Functional Analysis 256 (2009) 385–408 www.elsevier.com/locate/jfa
Stochastic Calculus of Variations on complex line bundle and construction of unitarizing measures for the Poincaré disk Ana-Bela Cruzeiro a,∗ , Paul Malliavin b a Dep. Matemática I.S.T, T.U.L.-Lisbon and Grupo de Física-Matemática U.L.,
Av. Rovisco Pais, 1049-001 Lisboa, Portugal b 10 rue S. Louis-en-l’Ile, 75004 Paris, France
Received 17 March 2008; accepted 17 March 2008 Available online 28 April 2008 Communicated by Paul Malliavin
Abstract Holomorphic representation of Lie algebra can be realized through Kählerian symplectic formalism; underlying holomorphic convexity requires then the introduction of elliptic operators with complex coefficients. We construct the Stochastic Calculus of Variations for those elliptic operators; remote past vanishing of projections of the underlying process implies convergence in law; then limit laws lead to the unitarizing measure of the given representation; this general approach is developed in full details on Poincaré disk. © 2008 Elsevier Inc. All rights reserved. Keywords: Unitarizing measure; Malliavin Calculus in complex geometry
Contents 1. Transfer principle, projected transfer principle . . . . . . . . . . . . . . . 2. Stochastic Calculus of Variations on complex line bundle . . . . . . . . 3. Commutation of differential operators on Poincaré disk . . . . . . . . . 4. Remote past vanishing, convergence in law and unitarizing measure References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
* Corresponding author.
E-mail addresses:
[email protected] (A.-B. Cruzeiro),
[email protected] (P. Malliavin). 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.03.011
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
386 387 397 404 408
386
A.-B. Cruzeiro, P. Malliavin / Journal of Functional Analysis 256 (2009) 385–408
1. Transfer principle, projected transfer principle In this paper we develop a new appearance of the following general mathematical principle: remote past vanishing implies ergodicity. This principle has already given rise to theorems in the context of Zakai filtering [5]. In [4] the same principle gives rise to a general criterium, confinement by curvature positivity, for the existence of an invariant measure of a diffusion. Let us recall below shortly this last result. Let M be a Riemannian manifold; let its Laplace–Beltrami operator; let ρ be a fixed 1differential form on M: consider the elliptic operator defined by 1 L0 f := f − (ρ | df ). 2
(1.1)a
Let x the Brownian motion driving the diffusion associated to L0 : set Φx,t the stochastic flow associated to L0 ; set m0 be a well-chosen point of M such that ρ0 = 0; in the normal chart at 0 the L0 -process is asymptotic to an Euclidean Brownian motion y. Consider the semi-martingale ζx,t () = Φx,t (y()); by Itô Calculus its stochastic differential is given, for small enough, by 1 (m0 )(dy) + Φx,t (m0 )(dy, dy). dζx,t = Φx,t 2
(1.1)b
We have made this calculus in the normal chart at the end point Φx,t (m0 ): then define (m0 ) , Q(t, m) := E Φx,t (m0 )=m Φx,t 1 (m0 )(∗, ∗) . q(t, m) := trace E Φx,t (m0 )=m Φx,t 2
(1.1)c
Consider the process driven by the following Itô SDE: dly,t = Q(t, ly,t ) dy(t) + q(t, ly,t ) dt.
(1.1)d
Then in [4] the following results are proved. 1.2. Theorem (Transfer from the initial to the final value). Set μt the heat measure which is the law of Φx,t (m0 ); start the process (1.1)d at time t0 taking for starting measure μt0 ; then the law of lt,y is equal to μt , t > t0 . 1.3. Theorem (Remote past vanishing). Assume that Qt , qt decay exponentially in time, then limt→∞ μt exists =: μ∞ and is an invariant measure for mx,t . 1.4. Projected transfer Suppose that a map H : M → R d is given; set μ˜ t the law of H ◦ Φx,t that is the image measure
A.-B. Cruzeiro, P. Malliavin / Journal of Functional Analysis 256 (2009) 385–408
387
μ˜ t := H∗ (μt ). Set Φ˜ x,t = H ◦ Φx,t . Using Itô Calculus compute the stochastic differential of ζ˜x,t := H (ζx,t ): 1 (m0 )(dy) + Φ˜ x,t (m0 )(dy, dy). d ζ˜x,t = Φ˜ x,t 2
(1.4)a
Then define ˜ h) := E Φ˜ x,t (m0 )=h Φ˜ x,t Q(t, (m0 ) , ˜ 1 (m0 )(∗, ∗) . q(t, ˜ h) := trace E Φx,t (m0 )=h Φ˜ x,t 2
(1.4)b
Consider the process driven by the following Itô SDE: ˜ l˜y,t ) dy(t) + q( d l˜y,t = Q(t, ˜ l˜y,t , t) dt.
(1.4)c
Then 1.5. Theorem (Projected transfer). Start the process (1.4)c at time t0 taking for starting measure μ˜ t0 ; then the law of l˜t,y is equal to μ˜ t , t > t0 . ˜ t , q˜t decay exponentially in time, then the random variable (H ◦ Φx,t )(m0 ) Assume that Q converges in law when t → +∞. Proof. By Itô Calculus the image ξy,t := H (ly,t ) is a semi-martingale dξy,t = α dy + β dt, ˜ h), E H (ly,t )=h (α) = Q(t,
where
(1.5)a
E H (ly,t )=h (β) = q(t, ˜ h),
(1.5)b
following [1], to ϕ any C 2 -function defined on the range of H associate the semi-martingale F (y, t) = (ϕ ◦ H )(l˜y,t ). Then, similarly to Theorem 1.3. the following limit exists lim E F (y, t) = lim
t→∞
t→∞
ϕ(h)μ˜ t (dh).
2
2. Stochastic Calculus of Variations on complex line bundle In the geometry of Kähler manifolds the de Rham cohomology must be replaced by the Hodge cohomology; the corresponding Weitzenbök formula involves derivation along complex vector fields. Probabilistic representation of the corresponding semi-group, together with its associated Stochastic Calculus of Variations, is the object of this section.
388
A.-B. Cruzeiro, P. Malliavin / Journal of Functional Analysis 256 (2009) 385–408
2.1. Complex derivatives on a Kähler manifold Following [7], we shall set well-known facts into the framework of orthonormal frame bundle. Denote by M a Kähler manifold, that is a Riemannian manifold of dimension 2n such that there exists an atlas where change of local coordinates are given by holomorphic diffeomorphisms and such that in each local chart the Riemannian metric can be expressed as the (1, 1) Hessian of a real function. An orthonormal frame {eq } of M is compatible with the complex structure J if J (e2s ) = −e2s−1 . Denote by U (M) the set of all orthonormal frames which are compatible with the complex structure; set p the natural projection p : U (M) → M. Then U (M) is a principal bundle having for structural group the n-dimensional unitary group. Set O(M) the bundle of all orthonormal frames; then U (M) ⊂ O(M). Given r0 ∈ U (M) denote by Ak (r0 ), k ∈ [1, 2n], the canonical horizontal vector fields of O(M) at the points r0 then Ak (r0 ) are tangent at the submanifold U (n) ⊂ O(M); therefore by parallel transport a frame which is compatible with the complex structure stays compatible with the complex structure: the Levi-Civita parallel transport preserves U (M). Introduce on U (M) the following differential operators with complex coefficients: 1 ∂q = (∂A2q−1 − i∂A2q ), 2
1 ∂¯q = (∂A2q−1 + i∂A2q ), 2
q ∈ [1, n].
The horizontal Laplacian is defined as = ∂q ∂¯q + ∂¯q ∂q 2 ∂q ∂¯q , q
(2.1)a
(2.1)b
q
where the sign means that the two operators take the same values on functions F of the form F = f ◦ p (basic functions): in fact [∂q , ∂¯q ] are vertical vector fields. If we suppose that M is compact we have a natural volume measure dr of finite total mass on U (M). Given a real function K denote L2K the Hilbert space of complex-valued functions which are square integrable for the measure exp(−K) dr: (h1 | h2 )L2 = h1 h¯ 2 exp(−K) dr. K
U (M)
¯ The adjoint in L2K of ∂q is ∂q∗ = −∂¯q + ∂K. In fact,
(∂q f1 )f¯2 exp(−K) dr = −
f1 (∂q f¯2 ) exp(−K) dr +
(2.1)c
f1 f¯2 (∂q K) exp(−K) dr.
The operator L :=
n q=1
This results from the equality
∂¯q ∂q − ∂¯q K × ∂q is autoadjoint in L2K .
(2.1)d
A.-B. Cruzeiro, P. Malliavin / Journal of Functional Analysis 256 (2009) 385–408
(Lf1 |f2 )L2 = K
389
n (∂¯q − ∂¯q K)(∂q f1 )f¯2 exp(−K) dr q=1
=−
n (∂q f1 |∂q f2 )L2 . K
q=1
If F = f ◦ p is basic then Lf is basic and exp(tL) defines a semi-group on M. (2.1)e In fact the expression q ∂¯q K × ∂q is invariant under the action of an unitary operator on U (M). Consider xs , s ∈ [1, 2n], independent R-valued Brownian motions; then the horizontal lift to U (M) of the Brownian motion on the Riemannian manifold M is given by the following Stratonovitch SDE: drx,t =
2n
As (rx,t ) ◦ dxs (t).
(2.1)f
s=1
Theorem. Assume 2 t n ¯ E exp ∂q K rx (s) ds < ∞ ∀t.
0
q=1
Then the semi-group associated to L has the following Girsanov type expression: t n
exp(tL)F (r0 ) = Er0 exp − ∂¯q K(dx2q−1 + i dx2q ) × F rx (t) ,
(2.1)g
0 q=1
where the integral appearing inside the exponential is an Itô integral and where F = f ◦ p is basic. Proof. Set ∗ the Itô contraction between stochastic differentials, then we have: (dx2q−1 + i dx2q ) ∗ (dx2q−1 + i dx2q ) = 0, (dx2q−1 + i dx2q ) ∗ (dx2q−1 − i dx2q ) = 2 dt.
(2.1)h 2
2.2. Weitzenböck formula Our objective is to construct a Stochastic Calculus of Variations (SCV) associated to the semigroup exp(tL). The infinitesimal form of the SCV will correspond to Weitzenböck–Kodaira formula associated to L; we rewrite below these classical results translated into frame bundle terminology. Set ξk the real valued 1-differential form on U (M) which vanish on the fibers p −1 (m) and which constitute a dual basis of the horizontal vector fields As . Introduce the following complexvalued differentials forms ηq = ξ2q−1 + iξ2q ,
η¯ q = ξ2q−1 − iξ2q .
(2.2)a
390
A.-B. Cruzeiro, P. Malliavin / Journal of Functional Analysis 256 (2009) 385–408
Then ηq , η¯ q , q ∈ [1, n], constitute a C-basis of complex coefficients basic differential forms. Introduce the operators going from functions to differential forms defined by ∂F :=
n
¯ := ∂F
∂q F × ηq ,
q=1
n
∂¯q F × η¯ q .
(2.2)b
q=1
Then if F is basic, ∂F = p ∗ where is a 1-differential form on M. Theorem. For all basic smooth function F = f ◦ p we have 1 ηq × (∂q ∂¯j K) × ∂j F =: L1 (∂F ), ∂(LF ) = L(∂F ) − Ricci(∂F ) − 2 q
(2.2)c
j
where the operator L does not operate in the first term of the right-hand side on the ηq . Proof. See for instance [8].
2
2.3. Line bundle formalism The line bundle machinery will consist in adding two auxiliary dimensions to M; on the extended space all complex coefficients first order differential operators on M will appear as real coefficients second order differential operators and therefore could be integrated trough suitable SDE. The line bundle above M will be the product L = M × C.
(2.3)a
√ As we want to make √ −1 disappear from our formalism, we denote the R-basis of C by the two letters 0 = 1, 1 = −1. Then the line bundle appear as a fiber bundle above M the fibers, being two-dimensional Euclidean spaces. We associate to a complex-valued function f = u + iv the complex-valued function σf defined on L by σf is linear on each fiber, σf (0 ) := u + iv, σf (1 ) := σi×f (0 ) = −v + iu.
(2.3)b
On the Euclidean space Lm0 we have two natural vector fields: the vector field of infinitesimal homotheties and the vector field of infinitesimal rotations; they prolongate into two vertical vector fields on L denoted ∂α , ∂β . Then we have ∂α σf = σf ,
∂β σf = σi×f .
(2.3)c
The multiplicative group C ∗ of complex numbers acts on C as the group of similitudes constituted by homotheties and rotations. We denote by S this group and by S its Lie algebra; set ∂ˆα , ∂ˆβ the natural basis of S. Then [∂α ]l = exp( ∂ˆα )l,
[∂β ]l = exp( ∂ˆβ )l,
→ 0.
A.-B. Cruzeiro, P. Malliavin / Journal of Functional Analysis 256 (2009) 385–408
391
Given two vector fields B, C on M, consider the complex coefficients first order operator Z = ∂B + i∂C . We have σZf = ∂α ∂B σf + ∂β ∂C σf ,
(2.3)d
where B , C are the lift of B, C to L through the trivial connection. A complex coefficients first order operator on M can be looked upon as a real coefficients second order operator on L. From this fact it is clear that the operator L can be written as a real second order operator on L; we shall explicit in (2.6)∗ this fact. 2.4. A connection on the line bundle Introduce on U (M) the following real operators sending a real function into differential forms with real coefficients: ¯ d = ∂ + ∂,
¯ d c = −i × (∂ − ∂).
(2.4)a
Then 2∂ = d + id c , 2∂¯ = d − id c . The operators d and d c have the following explicit expressions: 2n dF = (∂As F ) × ξs , s=1
n d F= (∂A2q−1 F ) × ξ2q − (∂A2q F ) × ξ2q−1 . c
(2.4)b
q=1
We have ¯ = 2(∂ + ∂)F
n (∂A2q−1 − i∂A2q )F × (ξ2q−1 + iξ2q ) + (∂A2q−1 + i∂A2q )F × (ξ2q−1 − iξ2q ) q=1
=2
2n (∂As F ) × ξs . s=1
We deduce the expression for d c in an analogous way. The first formula in (2.4)b is the usual expression of the differential of a function on frame bundle, fact which justifies a posteriori our notation. Remark that if F is basic function then dF and d c F are basic differential forms. Define a differential form ω on M with value in the Lie algebra S of the group of similitudes S of R 2 by the formula 2ω = −∂α × dK + ∂β × d c K.
(2.4)c
Then ω defines a connection on the principal bundle S × M =: P . On each fiber Lm of the line bundle L define the Euclidean metric (ζ1 | ζ2 )m := exp −K(m) ζ1 ζ¯2 .
(2.4)d
392
A.-B. Cruzeiro, P. Malliavin / Journal of Functional Analysis 256 (2009) 385–408
Set σ a section of L and set ∇ ω the covariant derivative on sections associated to the connection ω; then for any vector field z on M the connection preserves the metric: d
(σ1 | σ2 )m+zm = ∇zω σ1 σ2 + σ1 ∇zω σ2 . d=0
(2.4)e
In fact, d
(σ1 | σ2 )m+zm = (∂z σ1 | σ2 ) + (σ1 | ∂z σ2 ) − z, dK × (σ1 | σ2 ). d=0 On the other hand, 2z, dK × (σ1 | σ2 )
= z, dK − i z, d c K × σ1 σ2 + σ1 z, dK − i z, d c K × σ2 . 2.5. An hypoelliptic operator Set A˜ k the lift to U (L) of the horizontal vector field Ak defined in (2.1) through the connection ω defined in (2.4)c , A˜ k = Ak , ω(Ak ) .
(2.5)a
Let L be the infinitesimal generator associated to the lift to L of the horizontal Brownian motion on U (M): 1 2 ∂A˜ . k 2 2n
L =
(2.5)b
s=1
Theorem. For every complex-valued function f defined on M we have L σ f = σ L 2 f ,
where L2 f = Lf − (K) × f.
(2.6)a
Proof. We have 2
1 1 ∂A2˜ = ∂As − As , dK∂α + As , d c K ∂β . s 2 2
(2.6)b
Using the commutation [∂As , ∂α ] = 0, [∂As , ∂β ] = 0,
1 1 ∂A2˜ = ∂A2 s − As , dK∂α ∂As + As , d c K ∂β ∂As + Qs + Rs , s 2 4 where s
Qs =
−∂As As , dK ∂α + ∂As As , d c K ∂β = 2K∂α ,
(2.6)c
A.-B. Cruzeiro, P. Malliavin / Journal of Functional Analysis 256 (2009) 385–408
393
granted that the second term vanishes. On the other hand,
2 Rs := As , dK∂α − As , d c K ∂β . Using the fact that ∂β σf = iσf , ∂α σf = σf we get Rs σ f =
2
2
− 2iAs , dK As , d c K σf . As , dK − As , d c K
Lemma. For every real function K, we have 2 dK2 = d c K ,
dK d c K = 0.
(2.6)d
Proof. 1 1 ∂q = (∂A2q−1 − i∂A2q ), ∂¯q = (∂A2q−1 + i∂A2q ), 2 2 ηq = ξ2q−1 + iξ2q , ∂K =
n
¯ = ∂K
∂K × ηq ,
q=1
n
q ∈ [1, n],
¯ × η¯ q , ∂K
q=1
dK = 2∂K; d K = 2∂K, ∂A2q−1 K × ξ2q−1 + ∂A2q K × ξ2q , dK = c
q
dcK =
∂A2q−1 K × ξ2q − ∂A2q K × ξ2q−1 ,
q
dK2 =
[∂A2q−1 K]2 + [∂A2q K]2 , q
c 2 d K = [∂A
2q−1
K]2 + [∂A2q K]2 ,
q
[−∂A2q−1 K × ∂A2q K + ∂A2q K × ∂A2q−1 K] = 0. dK d c K =
2
q
Applying the lemma we get
2n
Rs σ f =
2 dK2 − d c K − 2i dK | d c K σf = 0.
s=1
Using (2.3)d ,
2n
c As , dK∂α ∂As − As , d K ∂β ∂As σf − s=1
(2.6)e
394
A.-B. Cruzeiro, P. Malliavin / Journal of Functional Analysis 256 (2009) 385–408
=−
2n
As , dK∂As − i × As , d K ∂As σf . c
s=1
We use now (2.4)a ,
2n ¯ = −2 As , ∂K∂As σf = −σ∂K∂f ¯ s=1
granted the following identity, n
n A2s−1 , η¯ s ∂A2s−1 f + A2s , η¯ s ∂A2s f (∂A2s−1 K + i∂A2s K) = ∂s f × ∂¯s K.
s=1
s=1
Corollary. Assume
t
(K)(rx,s ) ds < ∞
E exp
∀t.
0
Then the process associated to L is given by
(rx,t , γx,t )
where γx,t
t n ¯ = exp − ∂q K(dx2q−1 + i dx2q ) − K dt .
(2.6)f
0 q=1
The lift to L of the Brownian motion mx,t on M is given by (mx,t , ζx,t ),
ζx,t := γx,t (ζ0 ), ζ0 ∈ Lmx,0 .
(2.6)g
The process ζx,t lives on the circle bundle which means that (ζx,t | ζx,t )mx,t = const.
(2.6)h
Proof. The result follows from the stochastic representation for the semigroup associated to L given by (2.1)g together with a Feynman–Kac formula (cf. [7]) for the zero order term −(K) × f in L2 . Finally (2.6)h results from (2.4))e . 2 Corollary. Consider the subspace W of functions on L which are linear on each fiber. Then L (W) ⊂ W,
exp(tL )(W) ⊂ W.
(2.6)i
Proof. Any element of W is of the form σf . According to (2.6)a we have σexp(t L2 )f (m0 ) = E(m0 ,0 ) f (mx,t )ζx,t .
2
(2.6)j
A.-B. Cruzeiro, P. Malliavin / Journal of Functional Analysis 256 (2009) 385–408
395
2.7. Stochastic Calculus of Variation for L Set Φx,t the flow of diffemorphisms on U (M) × C generated by the SDE associated to L : dΦx,t (l0 ) =
2n
A˜ k Φx,t (l0 ) ◦ dxk ,
Φx,0 (l0 ) = l0 .
(2.7)a
k=1
Then Φx,t {r0 , ζ } = Ψx (t), γx,t × ζ , t n γx,t := exp − ∂¯q K (dx2q−1 + i dx2q ) − K dt ,
(2.7)b
0 q=1
where Ψx,t is the horizontal Brownian flow on U (M). The construction of a SCV for Φx,t will be done by lifting the SCV for Ψx,t through (2.7)b . We recall the Bismut formula [2] governing the SCV on Ψx,t following the notation of [6] and [3]. A tangent process is the infinitesimal transformation of the Brownian motion x defined, as → 0, by t x(t) → x := x +
ρ(s) dx(s) + h(t),
(2.7)c
0
where ρ(s) is an adapted process taking its values in the antisymmetric matrices. Given a smooth functional Θ on the Wiener space, its derivative will be defined as (D(ρ,h) Θ)(x) :=
d Θ(x ). d=0
We propagate a variation of the initial condition through the variation of the path on U (M) given by 1 h˙ + Ricci h = 0, 2
dρ = −Ω(◦dx, h),
(2.7)d
where ρ(0) = 0 and h(0) is given and where Ω denotes the curvature tensor. We have t Dρ,h (γx,t ) = −γx,t 0
d ¯ ∂q K x(∗) + h(∗) × (dx2q−1 + i dx2q ) d=0
t
d K x(∗) + h(∗) × dt + {∂¯q K} × ρ ∗ (dx2∗−1 + i dx2∗ ) q . + d=0
0
(2.7)e
396
A.-B. Cruzeiro, P. Malliavin / Journal of Functional Analysis 256 (2009) 385–408
2.8. Stochastic Calculus of Variations for L We have interpreted the semi-group associated to the complex coefficients differential operator L2 defined in (2.6)a , (2.1)d in terms of the stochastic flow Φx,t defined on the line bundle. In this section we make the basic hypothesis K = const =: c.
(2.8)
Then L = L2 + c. We want to recapture the stochastic theory corresponding to the formula of differential geometry (2.2)c . The tangent space T∗ (U (M)) at a generic point of U (M) is identified through the horizontal parallelism to C n × U(n) where U(n) is the Lie algebra of the unitary group of dimension n. The tangent space T∗ (L(M)) at a generic point ∗ of L(M) is identified to T∗ (U (M)) × S then :T Φx,t Φx,0 (L(M)) → TΦx,t is given by (h0 , ρ0 , ζ0 ) = (ht , ρt , ζt ), Φx,t
(2.8)a
where (ht , ρt ) are given in (2.7)d and where t ζt = ζ0 −
{Dh ∂¯q K} × (dx2q−1 + i dx2q ) −
0
t
{∂¯q K} × ρ(dx2∗−1 + i dx2∗ ) q . (2.8)b
0
We say that a first order differential operator is of type (1.0) if it is a linear combination with complex coefficients of the differential operators ∂q , q ∈ [1, n]. If h(0) is of type (1, 0)
then h(t) will be of type (1, 0).
(2.8)c
In fact, as Ricci tensor preserves the complex structure, this results from (2.7)d . t If h is of type (1, 0) we have D0,h (γx,t ) = −γx,t
∂h ∂¯q K × (dx2q−1 + i dx2q ).
(2.8)d
0
The stochastic flow Φx,t preserves the vector fields ∂α , ∂β .
(2.8)e
In fact a variation of ζ0 in (2.8)b leads to the same variation of ζt . Operator L1 defined in (2.2)c can be extended as operating on (1, 0) differential forms and generates a semi-group exp(tL1 ) operating on (1, 0) differential forms on M. Theorem. For every (1, 0) differential form ω defined on M we have
∗ exp(tL1 )(ω), ηq∗ = E ω, Φx,t ηq , where ηq∗ denotes the basis of (1, 0) vector fields which is dual of the basis ηq .
(2.9)a
A.-B. Cruzeiro, P. Malliavin / Journal of Functional Analysis 256 (2009) 385–408
397
Proof. Remark that the right-hand side of (2.9)a is a basic (1, 0) differential form; it defines a semi-group on (1, 0) differential forms. We shall prove identity (2.9)a by showing the equality of the two infinitesimal generators. Let us compute ∗
1 Q := lim E ω, Φx,t ηq − ηq∗ , t→0 t ∗
ω, Φx,t ηq m = ωΦx,t (m) , (ht , ρt , ζt ) ,
(2.9)b
where (ht , ρt , ζt ) has been defined in (2.8)a . As we shall take in (2.9)a an expectation we can forget the contribution of Dρ,0 . Then we get
1 t Q = lim E ωΦx,t (m) , h0 − ωm , Ricci(h0 ) − ωΦx,t (m) , h0 × G , t→0 t 2
(2.9)c
where G is the horizontal derivative of the Girsanov factor which has been defined in (2.8)d ; by inspection of the last formula we get Q = L1 (ω). 2 3. Commutation of differential operators on Poincaré disk Let D be the Poincaré disk D = {z ∈ C: |z| < 1} with the Kähler metric ds 2 = Consider the transformation D → D defined by z →
dzd z¯ . (1−z¯z)2
where |a|2 − |b|2 = 1. Denote by G the group SU(1, 1) constituted by matrices g = ab¯ ab¯ with |a|2 − |b|2 = 1. Its iα β Lie algebra su(1, 1) is identified with β¯ −iα , where α is real. The Poincaré disk D appears as D = SU(1, 1)/S 1 , where S 1 is the subgroup of G defined by the equation b = 0. The Kähler potential az+b ¯ a¯ bz+
K(z) := −log 1 − |z|2 satisfies ¯ + a) K(gz) = 2 log(bz ¯ + K(z),
(3.1)a
¯ is invariant under the action of G. Indeed, identity which implies that the Kählerian metric ∂ ∂K we have az + b 2 + log 1 − |z|2 K(gz) − K(z) = −log 1 − ¯bz + a¯ ¯ + a| |bz ¯ 2 (1 − |z|2 ) ¯ + a| = log = log|bz ¯ 2. ¯ + a| |bz ¯ 2 − |az + b|2 Theorem. Let H(D) be the space of holomorphic functions on the Poincaré disk D. Let Hγ (D) be the subspace of holomorphic functions for which the following Hilbertian norm is finite 2 (3.1)b f 2γ := f (z) exp −γ K(z) dv(z), D
398
A.-B. Cruzeiro, P. Malliavin / Journal of Functional Analysis 256 (2009) 385–408
where dv is the hyperbolic volume measure on D: dv := (1 − |z|2 )−2 dz ∧ d z¯ . Define an action of G on H(D) by γ Ta,b f (z) =
1 az + b . f ¯ + a) ¯ + a¯ (bz ¯ γ bz
(3.1)c
γ
Then Ta,b induces on Hγ (D) an unitary action. Proof. We have γ T (f )2 = a,b γ
1 f g(z) 2 exp −γ K(z) dv(z) 2γ ¯ + a| |bz ¯
D
=
f g(z) 2 exp −γ K g(z) dv(z),
D
equality resulting from (3.1)a . Make the change of variable z = g −1 (z ); we get by the invariance of the hyperbolic measure: γ T (f )2 = a,b γ
2 exp −γ K(z ) × f (z ) dv(z ).
2
D
Definition. We shall call the finite mass measure γ μγ = 1 − |z|2 dv(z),
γ > 1,
(3.1)d
the unitarizing measure associated to the representation T . Let G be the Lie algebra of G; take for orthonormal basis of G: 1 e1 = 2
i 0
0 −i
,
1 e2 = 2
0 1
1 , 0
1 e3 = 2
0 −i
i . 0
Then [e1 , e2 ] = e3 ,
[e1 , e3 ] = −e2 ,
[e2 , e3 ] = −e1 .
(3.1)e
We consider G as a real Lie algebra defined by the above bracket relations. The vector e1 generates a group which is isomorphic to the circle S 1 . The quotient π : G → G/S 1 is the Poincaré disk D, which is a Riemannian manifold, homogeneous under the left action of G. The transform of the base point 0 ∈ D by g ∈ G is g(0) = b(a) ¯ −1 . The Kähler potential is defined on D 2 by −log(1 − |z| ); it lifts to G as 2 ¯ −1 = log 1 + |b|2 . K(g) := −log 1 − b(a)
(3.1)f
A.-B. Cruzeiro, P. Malliavin / Journal of Functional Analysis 256 (2009) 385–408
399
Remark that if g=
b , a¯
a b¯
then g
−1
=
−b ; a
a¯ −b¯
therefore K g −1 = K(g).
(3.1)g
We define the following left (respectively right) derivatives on G, g = l ∂ei F (g) :=
d F exp(ei )g , d=0
r ∂ei F (g) :=
a b¯ , b¯ a¯
d F g exp(ei ) d=0
then Proposition. Consider the functions a, b defined as the first line of the matrix g ∈ SU(1, 1). Then 2∂er2 (a) = b,
2∂er2 (b) = a,
¯ 2∂el 2 (a) = b,
2∂er3 (a) = −ib,
2∂el 2 (b) = a, ¯
2∂er3 (b) = ia,
¯ 2∂el 3 (a) = i b,
2∂el 3 (b) = i a. ¯
(3.1)h
Proof. 2∂er2
a b¯
b a¯
d = d=0 d = d=0
a b¯ a b¯
d b a exp(e2 ) = a¯ d=0 b¯ b b a + . a¯ a¯ b¯
b a¯
1
1
The derivative with respect to e3 gives d d=0
a b¯
b a¯
1 −i
i 1
=
d d=0
a b¯
b a¯
+ i
−b −a¯
a b¯
.
2
The computation of the left derivatives is analogous. Corollary. Let ϑ(g) := g −1 (0) = −b(a)−1 . Then r ∂e2 K (g) = − ϑ(g) ,
∂er3 K (g) = − ϑ(g) .
(3.1)i
400
A.-B. Cruzeiro, P. Malliavin / Journal of Functional Analysis 256 (2009) 385–408
3.2. Horizontal Brownian motion Consider the G-valued process defined by the Stratonovitch SDE, dgx,t = gx,t e2 ◦ dx2 (t) + e3 ◦ dx3 (t) ,
(3.2)a
where x is an R 2 -valued Brownian motion. The infinitesimal generator of this process is =
1 r 2 r 2 ∂ + ∂e3 . 2 e2
(3.2)b
Write (3.2)a in Itô formalism, then appears the Itô contraction 1 1 [e2 ]2 + [e3 ]2 = Identity 2 4
(3.2)c
and (3.2)a becomes 1 dgx,t = gx,t e2 dx2 (t) + e3 dx3 (t) + dt . 4
(3.2)d
3.3. Theorem. The projection of gx,t on the Poincaré disk D is the Brownian motion of the Riemannian manifold D. The bundle of orthonormal frame O(D) of D can be identified to G, the canonical horizontal vector fields being ∂er2 , ∂er3 . The process t → gx,t is the horizontal lift of the Brownian motion on D to O(D). Proof. Set p the projection G → G/S 1 = D defined by b p(g) = g(0) = . a¯ By homogeneity it is sufficient to verify the statement nearby zero; there, for t small, and writing the first line of the matrix g(t), we obtain b(t)
1 1 x2 (t) + ix3 (t) + o(t), a(t) = 1 + t + o(t), 2 4 b(t) 1 = x2 (t) + ix3 (t) + o(t). a(t) ¯ 2
(3.3)a
The normal chart at zero coincides with the embedding of D into C. Therefore straight lines τ → τ ∈ C and τ → iτ are osculatrix to geodesics at 0. Compare these straight lines to the exponentials 1 τ → exp(τ e2 ) = 1 + τ e2 + τ 2 + o τ 2 . 2
A.-B. Cruzeiro, P. Malliavin / Journal of Functional Analysis 256 (2009) 385–408
401
Then b τ = = τ + o τ2 2 τ a¯ 1+ 2 and up to the second order the exponential map on G coincides to the exponential map of the normal chart. 2 3.4. First order complex differential operators By (3.3)a the complex structure on the Poincaré disk sends e2 → e3 and e3 → −e2 and is therefore realized by ad(e1 ), the adjoint action of e1 . Introduce the operators ∂r =
1 r ∂e2 − i∂er3 , 2
1 ∂¯ r = ∂er2 + i∂er3 , 2
∂l =
1 l ∂e2 − i∂el 3 , 2
1 ∂¯ l = ∂el 2 + i∂el 3 . 2 (3.4)a
A function f on G is said right-G-holomorphic if it satisfies the equation ∂¯ r f = 0.
(3.4)b
A function f on G is said basic-holomorphic if there exists an holomorphic function f˜ on D such that f = f˜ ◦ p. A function f on G is basic-holomorphic if and only if it is invariant under the right action of S 1 and satisfies Eq. (3.4)b . Indeed the operator ∂¯ r is the horizontal lift of the Cauchy–Riemann operator. 3.5. Proposition. Define ϕ(g) = g(0). Then ϕ is right-holomorphic. Define ϑ(g) = g −1 (0) = ϕ(g −1 ). Then ϑ is left-holomorphic. Proof. As ϑ(g) = ϕ(g −1 ) it is sufficient to prove proposition for ϕ. We have ϕ = b(a) ¯ −1 . Using (3.1)h , 2∂er2 (ϕ) =
|a|2 − |b|2 1 = 2, a¯ 2 a¯
i|a|2 − i|b|2 i = 2, 2 a¯ a¯ 1 r ∂ + i∂er3 (ϕ) = 0. 2 2 e2
2∂er3 (ϕ) =
3.6. Corollary. The function ∂¯ r K is left holomorphic.
(3.4)c
402
A.-B. Cruzeiro, P. Malliavin / Journal of Functional Analysis 256 (2009) 385–408
Proof. Reading (3.1)i we have (∂er2 K)(g) = −2(ϑ(g)), (∂er3 K)(g) = −2(ϑ(g)); therefore ∂¯ r K = −2ϑ then Proposition 3.5 proves the corollary, since l 1 1 ∂e2 + i∂el 3 b(a)−1 = 2 − 2 = 0. a a
2
Remark. We have b2 ∂er2 ∂er2 K(g) = 1 − 2 , a 2 b ∂er2 ∂er3 K(g) = − 2 , a
b2 ∂er3 ∂er2 K(g) = − 2 , a b2 ∂er3 ∂er3 K(g) = 1 + 2 . a
We can check that the Hessian is always a positive matrix, but has an arbitrary small eigenvalue for some parameters a, b. The degeneracy of this Hessian forbids the direct use of [4]. 3.7. A complex coefficient elliptic operator Define on G the following differential operator: Lf = − γ ∂¯ r K × ∂ r f,
(3.7)a
where γ is a positive constant. Introduce the Girsanov-type functional
t
Ax,g0 (t) := exp −γ
2 3 ∂ K gx (t) × dx (s) + i dx (s) ,
¯r
(3.7)b
0
where the integral appearing in this formula is an Itô stochastic integral. The process Ax,g0 (∗) is a martingale.
(3.7)c
In fact the following Itô contraction vanishes: 2 dx (s) + i dx 3 (s) ∗ dx 2 (s) + i dx 3 (s) = 1 + i 2 ds = 0. Theorem. The semi-group associated to the elliptic operator L has the following representation (3.7)d exp(tL)f (g0 ) = E Ax,g0 (t)f g0 gx (t) . Proof. Itô Calculus.
2
Theorem (Algebraic commutations). We have ∂ r L − L∂ r = (1 − γ ), ¯l
¯l
(3.7)e
∂ L = L∂ ,
(3.7)f
∂el 1 L = L∂el 1 .
(3.7)g
A.-B. Cruzeiro, P. Malliavin / Journal of Functional Analysis 256 (2009) 385–408
403
Proof. We have ∂ = ∂ −
1 Ricci ∂ 2
and, from [8, Chapter 3, Proposition 6.4], we have Ricci = −2 Id since 2 1 = , 2 2 (1 − |z| ) (1 − |z|2 )2 ∂ r ∂¯ r K × ∂ r f = ∂ r ∂¯ r K × ∂ r f + ∂¯ r K × ∂ r ∂ r f = ∂ r f + ∂¯ r K × ∂ r ∂ r f ¯ log ∂∂
since, from (3.1)a , r r ∂ ∂¯ K (g) = ∂ r ∂¯ r K (0) = − ∂ r ∂¯ r log(1 − z¯z) z=0 = 1. We have ∂¯ l = ∂el 2 + i∂el 3 ,
l r ∂ej , ∂eq = 0.
Therefore
l ∂¯ , L = −γ × ∂¯ l ∂¯ r K × ∂ r . Using Corollary 3.6, ∂¯ l ∂¯ r K = 0. On the other hand, ∂el 1 ∂¯ r K = ∂¯ r ∂el 1 K. Using (3.1)g we get ∂¯ r ∂el 1 K = ∂¯ r ∂er1 K = 0.
2
Corollary. ∂ r exp(tL) = exp (1 − γ )t exp(tL)∂ r .
(3.7)h
3.8. The space H Consider the vector space H of functions on G which are of the form f (g) = φ g −1 (0) , where φ is a bounded holomorphic function on D.
(3.8)a
Obviously, H is an algebra for the pointwise multiplication.
(3.8)b
Theorem. A function f defined on G belongs to H if and only if it satisfies the following two first order differential equations ∂el 1 f = 0,
∂¯ l f = 0.
(3.8)c
404
A.-B. Cruzeiro, P. Malliavin / Journal of Functional Analysis 256 (2009) 385–408
Proof. Define F by F (g) = f (g −1 ). Then ∂er1 F = 0,
∂¯ r F = 0.
The first equation means that there exists a function ϕ defined on D such that F = ϕ ◦ p; the second equation (3.8)c means that ϕ is holomorphic. 2 3.9. Theorem. The semi-group exp(tL) conserves the space H. Proof. The commutation formulae (3.7)f , (3.7)g imply that ∂¯ l exp(tL) = exp(tL)∂¯ l ,
∂el 1 exp(tL) = exp(tL)∂el 1 ,
relations which imply that the kernels of the two differential operators ∂¯ l , ∂e1 are preserved by exp(tL). 2 4. Remote past vanishing, convergence in law and unitarizing measure Remark that in the case of the Poincaré disk, we can apply (2.8) according the fact that K = 1,
therefore L = L2 + 1,
and the Stochastic Calculus of Variations for L and L2 are equivalent. We have already in (2.9)∗ transfered the commutation formula (3.7)e in terms of semi-group: it remains to transfer (3.7)f . Set t Ax,t = exp
r ¯∂ K (gx,t ) × (dx2 + idx3 ) .
(4.1)a
0
Theorem.
∂¯ l E σf , Φx,t (l) = E σ∂¯ l f , Φx,t (l) .
(4.1)b
Proof. The operator defined in (2.1)b commutes with the left translation. On the other hand, from (3.6), ¯l
t
∂ Ax,t = −Ax,t ×
∂¯ l ∂¯ r K(dx2 + i dx3 ) = 0.
2
0
Let H∞ (D) be the Banach space of bounded holomorphic functions on the disk D. To ϕ ∈ let us associate
H∞ (D)
−1 b fϕ (g) := ϕ g (0) = ϕ − . a
(4.2)a
A.-B. Cruzeiro, P. Malliavin / Journal of Functional Analysis 256 (2009) 385–408
405
Set δ is the initial point of L defined as δ = (Identity, 0 , 0)
(4.2)b
and define g
Bt (g) = Eδ x,t
=g
(4.2)c
(Ax,t ).
The notion of measure with values in a line bundle it is given by a probability measure on M together with a section of L; set pt (dg) the probability measure describing the law of the Brownian motion starting from the identity; then the couple (Bt (g), pt (dg)) defines a measure with values in L which can be considered as the heat measure associated to L. Definition. The heat measure defined on G will be the complex-valued measure νt := Bt (g) × pt (dg).
(4.2)d
Theorem. Assume γ > 1. When t → ∞, the heat measures νt converge weakly to a finite mass measure ν∞ . (4.2)e Proof. Firstly recall the notion of image measure of νt : given Ψ a R d -valued function on G, define the image measure Ψ∗ (νt ) as the complex-valued measure on R d , associating to a Borelian A ⊂ R d the complex number (Ψ∗ νt )(A) := νt Ψ −1 (A) .
(4.2)f
Main lemma. Given ϕ := ϕ1 , . . . , ϕd ∈ H∞ (D), associate Ψ ϕ : G → C d defined by Ψ (g) := (ϕ1 (g), . . . , ϕd (g)); then, when t → ∞, ϕ
ϕ
Ψ∗ νt converges weakly to ν∞ , ϕ ν∞
has all its moments finite.
(4.3)a (4.3)b
Proof. The proof is written for d = 2. Set Φx,t the stochastic flow on L(M) defined in (2.7)a . Define a C 2 -valued function H on L(M) by H (l) = (lσfϕ1 , lσfϕ2 ). Set Φ˜ x,t := H ◦ Φx,t . In order to apply the projected transfer principle (1.5) we must compute ˜ (δ) , E Φx,t (δ)=z Φ˜ x,t
˜ E Φx,t (δ)=z Φ˜ x,t (δ) ,
(4.3)c
406
A.-B. Cruzeiro, P. Malliavin / Journal of Functional Analysis 256 (2009) 385–408
where δ has been defined in (4.2)b . Set y() = e2 y1 () + e3 y2 () the Brownian motion on D around 0 read in the normal chart at 0. We have the following expression of a stochastic differential of a function u defined in a neighborhood of 0: u = ∂ r u y2 () + iy3 () + ∂¯ l u y2 () − iy3 () + u + o(). 2
(4.3)d
We apply this identity to u := Φ˜ x,t y() .
(4.3)e
Then according to (4.1)b , ∂¯ l Φ˜ x,t = σ∂¯ l fϕ ◦ Φx,t (δ) = 0, j
j = 1, 2.
(4.3)f
The Laplacian at the origin of the local chart has the following expression: = ∂¯ l ∂ r = ∂ r ∂¯ l . By the last equality and by (4.3)d , (Φ˜ x,t )(δ) = 0, Φ˜ x,t y() = H ◦ (∂ r Φx,t )(δ) × y2 () + iy3 () + o().
(4.3)g (4.3)h
Using (2.9)a and (2.2)c we get Φ˜ x,t y() = exp t (1 − γ ) σ∂ r fϕj Φx,t (δ) × y2 () + iy3 () j =1,2 + o().
(4.3)i
We conclude by proving r d g ∂ fϕ (g) = dz ϕ (0) ϕH∞ ,
(4.3)j
where ϕ g (z) := ϕ(g −1 (z)). Indeed ϕ g is a bounded holomorphic function in the unit disk and its derivative at 0, by the Schwarz lemma, is bounded by ϕ g H∞ . 2 Proof of (4.2)e . Denote by V the complex vector space generated by the fϕ1 , f¯ϕ2 , ϕ1 , ϕ2 ∈ H ∞ (D). It separates points; it constitutes an algebra; it is stable by conjugation. By Stone– Weierstrass it is dense in the space of continuous functions. Applying (4.3)a , with d = 2, we obtain that lim fϕ1 f¯ϕ2 dνt exists. (4.4)a t→∞
G
A.-B. Cruzeiro, P. Malliavin / Journal of Functional Analysis 256 (2009) 385–408
407
To conclude we need to have a uniform estimate of the total variation νt of νt . We pick a function ϕ0 which is injective on the unit disk, then ϕ (4.4)b νt = νt 0 , the right-hand side being uniformly bounded in virtue of the Main lemma.
2
Theorem. Denote by νt1 the heat measure for the real elliptic operator L1 := − γ ∇K ∗ ∇ corresponding to the source at time 0 being the identity. Then νt = νt1
∀t > 0.
(4.5)a
Proof. Set Rθ the action of θ ∈ S 1 and Rθ∗ the action of Rθ by inverse image on functions; then we have
L1 , Rθ∗ = 0. (4.5)b L, Rθ∗ = 0, In fact we have that Rθ∗ K = K and Rθ preserves the Riemannian structure and the complex structure. As the source identity is preserved by Rθ we get (Rθ )∗ νt = νt , (L − L1 )f = iγ d c K df , c d d K df dνt . f dνt = Lf dνt = L1 f dνt + i dt
(4.5)c
Make the holomorphic change of coordinates log z = ζ = ξ + iη. As K(z) = −log(1 − exp(2ξ )) we have c d K df = 2
1 × ∂er1 f, sinh(ξ ) c 1 r ∂ . d K df dνt = −2 f × divνt sinh(ξ ) e1 We have, divνt
1 1 1 ∂er1 = −∂er1 + divνt ∂er1 = 0 sinh(ξ ) sinh(ξ ) sinh(ξ )
granted (4.5)b . Therefore, d dt
f dνt =
L1 f dνt .
(4.5)d
408
A.-B. Cruzeiro, P. Malliavin / Journal of Functional Analysis 256 (2009) 385–408
As by definition d dt
f dνt1 =
we deduce the theorem by transposition.
L1 f dνt1 ,
2
4.6. Corollary. When t → ∞ the measure νt1 converges towards a probability measure. Remark. The result (4.6) can be obtained by much shorter approaches as, for instance, by Lyapounov function methodology. Our purpose was to prove it from infinitesimal considerations, in the spirit of [4], managing to take in account the lack of strong positivity for the full Hessian of K. References [1] H. Airault, P. Malliavin, Functorial analysis in geometrical probability theory, in: A.B. Cruzeiro, J.C. Zambrini (Eds.), Stochastic Analysis and Mathematical Physics, Birkhäuser, Basel, 2001. [2] J.-M. Bismut, Large Deviations and the Malliavin Calculus, Birkhäuser, Basel, 1984. [3] A.B. Cruzeiro, P. Malliavin, Renormalized differential geometry on path space: Structural equation, curvature, J. Funct. Anal. 139 (1) (1996) 119–181. [4] A.B. Cruzeiro, P. Malliavin, Nonperturbative construction of invariant measure through confinement by curvature, J. Math. Pures Appl. 9 77 (6) (1998) 527–537. [5] G. Da Prato, M. Fuhrman, P. Malliavin, Asymptotic ergodicity of the process of conditional law in non-linear filtering, J. Funct. Anal. 164 (1999) 356–377. [6] S. Fang, P. Malliavin, Stochastic calculus on the path space of a Riemannian manifold. I. Markovian stochastic calculus, J. Funct. Anal. 118 (1993) 249–274. [7] P. Malliavin, Formule de la moyenne, calcul de perturbations et théoreme d’annulation pour les formes harmoniques, J. Funct. Anal. 17 (3) (1974) 274–291. [8] J. Morrow, K. Kodaira, Complex Manifolds, Holt, Rinehart & Winston, 1971.
Journal of Functional Analysis 256 (2009) 409–431 www.elsevier.com/locate/jfa
Property A and affine buildings ✩ S.J. Campbell School of Mathematics, University of Southampton, Highfield, Southampton, SO17 1BJ, UK Received 18 March 2008; accepted 13 October 2008 Available online 31 October 2008 Communicated by K. Ball
Abstract Yu’s Property A is a non-equivariant generalisation of amenability introduced in his study of the coarse ˜ 2 have Baum Connes conjecture. In this paper we show that all affine buildings of type A˜ 2 , B˜ 2 and G Property A. Together with results of Guentner, Higson and Weinberger, this completes a programme to show that all affine building have Property A. In passing we use our technique to obtain a new proof for groups acting on A˜ n buildings. © 2008 Elsevier Inc. All rights reserved. Keywords: Property A; Affine buildings; Exotic buildings; Baum Connes conjecture
0. Introduction Property A was introduced by Yu in [20] as a non-equivariant generalisation of the group theoretic notion of amenability. He used it in his seminal proof that the property implies uniform embedding into a Hilbert space before appealing to a theorem by Higson and Roe to obtain the result that such groups satisfy the coarse Baum Connes conjecture. It is defined as follows Definition 1. A discrete metric space X has Property A if for all R, > 0 there exists S > 0 and a family of finite non-empty subsets Ax of X × N, indexed by x in X, such that: ✩
The author was supported by EPSRC postdoctoral fellowship EP/C53171X/1. E-mail address:
[email protected].
0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.10.014
410
S.J. Campbell / Journal of Functional Analysis 256 (2009) 409–431 |A ΔA |
• for all x, x with d(x, x ) < R we have x|Ax | x < ; • for all (x , n) in Ax we have d(x, x ) S. This property is a coarse invariant. We will review the definitions of coarse functions and coarse invariance later on in Section 2. There are several generalisations of Property A for nondiscrete spaces however, as remarked in [3]. The following is the most general: Definition 2. An arbitrary metric space X is said to have Property A if it contains a discrete coarsely dense subset with Property A. Property A has been extensively studied [1,9,12–14,18,20]. Any group with Property A (for example any group acting properly on a metric space with Property A), is exact or in other words its reduced C ∗ -algebra is exact. It is uniformly embeddable in a Hilbert space. Its action on its ˘ Stone–Cech compactification is amenable. Furthermore it satisfies both the coarse Baum Connes and the strong Novikov conjectures. Examples of groups with Property A include free groups and amenable groups [4], word hyperbolic groups [20], discrete subgroups of connected Lie groups [20], and groups acting properly on CAT(0) cube complexes of finite dimension [5]. Examples of metric spaces with Property A include trees [9] and finite dimensional CAT(0) cube complexes [3]. In particular it is known that linear groups and affine buildings arising as buildings of linear groups have Property A [11]. However that leaves the cases of so-called exotic affine buildings which are only found in dimension 2 and consist of those buildings which are not associated with a linear ˜ 2 [16]. group. Such buildings are of type A˜ 2 , B˜ 2 and G In this paper we will prove the following results: ˜ 2 have Property A. Theorem 3. Affine buildings of type A˜ 2 , B˜ 2 or G Together with results of [11] we obtain: Corollary 4. All affine buildings have Property A. All classical affine buildings admit a natural transitive action. However some exotic buildings also admit interesting actions, including a vertex transitive automorphism group [7,19] and others with an orbit on two vertices but no transitive action on vertices and apartments [2]. Remark 5. As commented in [8], a non-linear group with Property T admitting a proper action with compact quotient could only possibly be found as an exotic group in a thick affine building of dimension 2. As a further result of this paper, any such group would also have Property A. We will use the method developed in [14] and [4] to give explicit constructions for groups acting on buildings of dimension greater than two, which are partially based on those used in the first part of the paper, thus giving a new proof of the following theorem: Theorem 6. A˜ n groups have Property A.
S.J. Campbell / Journal of Functional Analysis 256 (2009) 409–431
411
Fig. 1.
1. Background We will first introduce some necessary background on affine buildings and Property A and the main theorems on which our method is based. A building Δ is affine when each apartment is an affine Coxeter complex. The Coxeter diagram must therefore be connected and the Coxeter matrix must be positive semi-definite [10]. Each apartment arises from a tiling of Euclidean space [16]. The simplest example of an affine building is a tree. ˜ 2 and are obtained When the affine buildings are two-dimensional, they are of type A˜ 2 , B˜ 2 or G from the appropriate Coxeter diagrams as illustrated in Fig. 1. The vertices labelled s are the special vertices which will be defined in a moment. In all 3 cases the apartments are isometric to E2 tessellated by triangles. When considering the tessellation of an apartment as a Euclidean space, A˜ 2 is a tessellation by equilateral triangles, ˜ 2 a tessellation by right angled triangles B˜ 2 a tessellation by right angled isoceles triangles and G with angles π/6 and π/3. A chamber is a maximal simplex. A wall is associated to a reflection s and consists of the simplices which are fixed pointwise by s [10]. In the buildings we consider, the chambers are triangles and walls are lines. Some vertices have particular properties and are called special. These are identified as follows. Fix a chamber and let S be the set of reflections in its walls and W the Coxeter group generated by it. Let W¯ be the group of linear parts of W . A point x such that Wx → W¯ is an isomorphism is special [10]. A wall divides any apartment into two roots. Consider a special vertex s and a base chamber c containing it. The intersection of the roots determined by the panels of c (i.e. the roots containing c) which contain s is a simplicial cone. We call this a sector. We define a sector face to mean a face of a sector treated as a simplicial cone and say that they are parallel if the distance between them is bounded. All sectors in a given affine building are isometric. It is easy to see that parallelism is an equivalence relation. Two sectors are parallel if and only if their intersection contains a sector [16]. By using equivalence classes of sectors, we can construct the “building at infinity” Δ∞ . Its chambers are defined to be parallel classes of sectors. Two chambers are adjacent if there are representative sectors having sector panels which are parallel. Our method will rely on the following lemma: Lemma 7. (See [16].) If s is a special vertex, then each parallel class of sectors contains a unique sector having vertex s. This means that if we fix a chamber in the building at infinity, then each special vertex has a unique sector associated to it. We will use this induced structure to construct our functions and prove that the metric space has Property A.
412
S.J. Campbell / Journal of Functional Analysis 256 (2009) 409–431
Fig. 2.
Our method is inspired by a proof by Dranishnikov and Januszkiewicz that trees have Property A [9]. Their proof is as follows. Proof. Let T be the Cayley graph of a free group (a tree) and V its set of vertices. Fix a point at infinity and let γ0 : R → T be a fixed geodesic ray in T . For any vertex z, let γz be the unique geodesic ray issuing from z and intersecting γ0 along a geodesic ray. Let γzn be the initial ray of γz of length n as represented in the diagram in Fig. 2. Dranishnikov and Januszkiewicz then use the following characterisation of Property A for metric spaces of bounded geometry: Definition 8. A discrete metric space Z has the Property A if and only if there is a sequence of maps a n : Z → P (Z) such that: (1) for every n there is some R > 0 with the property that for every z ∈ Z; supp azn ⊂ {z ∈ Z | d(z, z ) < R}; n = 0. (2) for every K > 0, limn→∞ < K supd(z,w)
0, such that for each n, x the function fn,x is supported in BSn (x), and for any R > 0 fn,x − fn,x 1 →0 fn,x 1 uniformly on the set {(x, x ): d(x, x ) R} as n → ∞.
S.J. Campbell / Journal of Functional Analysis 256 (2009) 409–431
413
2. Property A for affine buildings Let S, Δ be metric spaces. Recall that two maps i, j : S → Δ are said to be “close” if there is D < ∞ such that d(i(s), j (s)) < D for all s ∈ S. A map i : S → Δ is called coarse if it is both coarsely proper (the inverse image of any bounded set is bounded) and bornologous (∀R > 0, ∃S > 0 such that dΔ (i(x), i(x )) < S whenever dS (x, x ) < R). A map i is called a coarse equivalence if it is a coarse map S → Δ and there exists another coarse map j : Δ → S such that i ◦ j is close to IdS and j ◦ i is close to IdΔ . In this case S is coarsely equivalent to Δ. Let Δ be an affine building. We equip the 1-skeleton with the path metric where each edge has length 1. Note that this metric is quasi-isometric to the geodesic metric. In Δ every chamber is a simplex and contains a special vertex. Thus every vertex is either a special vertex or is adjacent to a special vertex. Let S be the metric space of the set of all special vertices in Δ equipped with the subspace metric. Now consider the two maps i from S to Δ which maps every special vertex to itself and j from Δ to S which maps every special vertex to itself and every other vertex to a special vertex distance 1 away. Lemma 10. The maps i, j are coarse maps between Δ and S. Δ and S are coarsely equivalent. Proof. The maps i and j are both coarsely proper since the inverse image of any bounded set is bounded. For any r > 0, d(j (x), j (x )) r + 2 for all pairs x, x such that d(x, x ) r. The maps i and j are both bornologous and hence are coarse maps. Consider the map i ◦ j . Every vertex is mapped to itself and then back to itself. This map is exactly the identity map IdS . Consider the map j ◦ i. Every special vertex is mapped to itself and then back to itself. Every non-special vertex is mapped to a vertex distance one away and then that vertex is mapped back to itself, ending up distance one away from the original point. The map j ◦ i is close to IdB , with D = 1. 2 Note that when Δ has type A˜ 2 then S = Δ. Let Δ be any affine building of dimension 2 equipped with the 1-skeleton metric. Let Δ∞ be its building at infinity and fix a chamber ω ∈ Δ∞ . Now let S be the metric space of special vertices of Δ equipped with the subspace metric. Given any vertex x ∈ S we consider the sector based at x defined by ω and denote it S x (ω). Let B(n, x) denote the ball or radius n around x. For any vertex y ∈ S, let fn,x denote the characteristic function of the intersection S x (ω) ∩ B(n, x), so fn,x (y) =
1 if y ∈ S x (ω) and d(x, y) n, 0 if d(x, y) > n and/or y ∈ / S x (ω).
Proposition 11. Let Δ be an affine building and S the metric space of its special vertices equipped with the 1-skeleton path metric. If Δ is of type A˜ 2 , then S has Property A. Proof. Let Δ be an affine building of type A˜ 2 and consider fn,x (y). The diagram in Fig. 3 represents the contributing vertices in the case n = 5.
414
S.J. Campbell / Journal of Functional Analysis 256 (2009) 409–431
Fig. 3.
Fig. 4.
Note that fn,x 1 is the number of vertices in the enclosed black area. The following observation is elementary. Lemma 12. If Δ is an A˜ 2 building and x ∈ Δ is a vertex, then fn,x 1 =
n2 +3n+2 . 2
Now consider two adjacent vertices x and x . Since they are adjacent, x and x lie on a wall which separates the building into half spaces. By definition ω must belong to at least one of these and x, x and ω must all lie in a common plane. Given a vertex x, there are only 6 adjacent vertices. By symmetry we need only consider 3 of them. The sector based at the vertex x intersects two of the adjacent vertices symmetrically. Thus there are only two possible generic situations. An adjacent vertex x can either belong to the sector based at x or not. This is illustrated in the diagrams shown in Fig. 4 (here for n = 5). The squares represent vertices belonging to S x but not S x and the triangles represent vertices belonging to S x but not Sx . Since we are working in an isometrically embedded Euclidean plane, no distortion or scaling occurs irrespective of the size of n. Note that this will be the case throughout this paper, whichever affine building we are considering. And so in the more general case, the number of square vertices or the number of triangle vertices is always n + 1, giving a total of 2(n + 1). Thus fn,x − fn,x 1 = 2(n + 1) for adjacent vertices. Now consider x, x distance i apart and a sequence of vertices {x = x0 , x1 , . . . , xi = x }. We have
S.J. Campbell / Journal of Functional Analysis 256 (2009) 409–431
415
S x S x = S x0 S x1 S x1 S x2 · · · S xi−1 S xi .
The size of S x S x is thus bounded above by the size of x S 0 S x1 ∪ S x1 S x2 ∪ · · · ∪ S xi−1 S xi and in general for any two vertices x, x , using the calculation fn,x − fn,x 1 4d(x, x )(n + 1) we get fn,x − fn,x 1 8d(x, x )(n + 1) . fn,x 1 n2 + 3n + 2 On the set d(x, x ) R, this tends uniformly to 0 as n tends to infinity as required. The conditions of Proposition 9 are satisfied and S has Property A. 2 Since a building is coarsely equivalent to the space of its special vertices S we obtain: Corollary 13. Affine buildings of type A˜ 2 have Property A. We now turn to buildings of type B˜ 2 . Proposition 14. Let Δ be an affine building and S the metric space of its special vertices equipped with the 1-skeleton path metric. If Δ is of type B˜ 2 , then S has Property A. Proof. Let Δ be an affine building of type B˜ 2 . Let Δ∞ be its building at infinity and fix a chamber ω ∈ Δ∞ . Now let S be the subspace of Δ consisting of all the special vertices and equipped with the subspace metric. A diagram of an apartment and a sector in S is shown in Fig. 5. The thick black lines delimit a sector. The squares represent vertices belonging to it, while the circles are vertices outside it. Comment: since we are using the subspace metric the distances between vertices remain the same as in B˜ 2 . For example the distance between two adjacent square vertices on the thick vertical line delimiting the sector is 2. We consider the function fn,x defined by the sector S x (ω). Note that fn,x 1 is the number of vertices in the sector truncated at n (i.e. all vertices in S x at distance at most n away from the base point x). Consider the diagram in Fig. 6 of a sector in S, delimited by the thick black lines. The horizontal thick and dashed lines represent the points belonging to the sector and distance exactly n away from the base point x, where thick is the case when n is odd and dashed is the case when n is even. The numbers on the left are the distance n. The numbers on the right are the number of such special vertices. The number of special vertices in the sector at distance exactly n is (n + 1)/2 if n is odd and (n/2 + 1) if n is even. Thus if n is odd, then
416
S.J. Campbell / Journal of Functional Analysis 256 (2009) 409–431
Fig. 5.
Fig. 6.
S.J. Campbell / Journal of Functional Analysis 256 (2009) 409–431
417
Fig. 7. n+1
2 n2 special vertices y ∈ S x (ω), d(y, x) n = 2 ∗ i= + n + 3. 4
i=1
On the other hand, if n is even, then n
+1
n
2 2 n2 special vertices y ∈ S x (ω), d(y, x) n = + n + 1. i+ i= 4
i=1
2
2
i=1
Hence fn,x 1 is either n4 + n + 3 or n4 + n + 1. Now consider the case of two adjacent vertices x, x in Δ. Again we see that they must lie in a common apartment. Given a vertex x, an adjacent vertex x can lie in four different positions. By symmetry we need only consider two of them. There are two general possibilities, either x belongs to the sector based at x or it does not. This is illustrated in the diagrams shown in Figs. 7 and 8 where we have taken the specific examples with n = 8 and n = 9. The triangles represent vertices which belong to S x but not S x and the squares are vertices which belong to S x but not S x . In the first case, in general the number of special vertices which belong to S x (ω) or S x (ω) but do not lie in the intersection is either n + 1 or n + 2. It is the sum of the number of triangle and square vertices. The number of triangle or square vertices is n2 + 1 when n is even and n+1 2 when n is odd. Hence when n is odd, the result is n + 1 and when n is even it is n + 2. In the second case the number of triangle or square vertices is always n + 1 regardless of whether n is odd or even and so the general total is 2(n + 1). Thus the only possibilities for fn,x − fn,x 1 are n + 1, n + 2 and 2(n + 1).
418
S.J. Campbell / Journal of Functional Analysis 256 (2009) 409–431
Fig. 8.
Looking at the worst case scenario, the largest fn,x − fn,x 1 can be is 2(n + 1), and the 2 smallest fn,x 1 can be is n4 + n + 1. Arguing as in the A˜ 2 case, fn,x − fn,x 1 2d(x, x )2(n + 1). Hence fn,x − fn,x 1 4d(x, x )(n + 1) . n2 fn,x 1 4 +n+1 On the set d(x, x ) R, this tends uniformly to 0 as n tends to infinity as required. By Proposition 9, S has Property A. 2 By coarse equivalence we deduce: Corollary 15. Affine buildings of type B˜ 2 have Property A. ˜ 2. Now we consider buildings of type G Proposition 16. Let Δ be an affine building and S the metric space of its special vertices ˜ 2 , then S has Property A. equipped with the 1-skeleton path metric. If Δ is of type G ˜ 2 . Let Δ∞ be its building at infinity and fix a Proof. Let Δ be an affine building of type G ∞ chamber ω ∈ Δ . Now let S be the subspace of Δ consisting of all the special vertices and equipped with the 1-skeleton path subspace metric. A diagram of an apartment and sector in S is as is shown in Fig. 9.
S.J. Campbell / Journal of Functional Analysis 256 (2009) 409–431
419
Fig. 9.
As before, since we are using the subspace metric the distances between special vertices ˜ 2 . For example the distances between two adjacent vertices on the top remain the same as in G and horizontal thick lines delimiting the sector are 4 and 2 respectively. As before we consider fn,x , the function determined by the sector S x (ω). As earlier, fn,x 1 is the number of vertices in the sector truncated at n. We need to calculate the number of vertices contained within a sector x, distance n or less away from the base point x. The diagram in Fig. 10 represents this. These are all the vertices in the sector from the base point up to and belonging to the appropriate line. ˜ 2 , note that when n is odd, the number Since adjacent special vertices are distance 2 apart in G of vertices is exactly the same as that for n − 1. In view of this, let us concentrate on the case when n is even. Consider the diagram given in Fig. 11. Both dashed and thick lines follow a pattern of form i=k i=1 i for some k. If n is a multiple of 4 (i.e. n = 4 ∗ j ) for some integer j , or if n = 4 ∗ j + 1 (since we saw that for n odd, the number of vertices is the same as that of n − 1), then the number of vertices is the appropriate sum of the dashed vertices and that of the thick vertices. Specifically: If n is of the form 4 ∗ j or 4 ∗ j + 1, then j +1 j vertices y ∈ S x (ω), d(y, x) n = j+ j = (j + 1)2 . j =1
j =1
If n is of the form 4 ∗ j + 2 or 4 ∗ j + 3, then j +1 vertices y ∈ S x (ω), d(y, x) n = 2 ∗ j = (j + 1)(j + 2). j =1
420
S.J. Campbell / Journal of Functional Analysis 256 (2009) 409–431
Fig. 10.
Fig. 11.
In particular, |vertices y ∈ S x (ω), d(y, x) n| is at least ( n4 + 1)2 and so we have fn,x 1
2 n +1 . 4
S.J. Campbell / Journal of Functional Analysis 256 (2009) 409–431
421
Fig. 12.
Now consider the case of two vertices x, x distance 2 apart, and their associated sectors. Assume that ω lies in the same plane as x and x . Given a vertex x, there are six possible positions for x , three of which we can ignore by symmetry. The other three are all distinct and we must consider each one in turn. x could lie on the sector based at x. In the other two cases it does not and the intersection point of its sector with the sector based at x is either distance 2 or 4 away from x . We have illustrated the three possibilities in the diagrams in Figs. 12–14 where we have taken the specific examples with n = 12, 13. In the first case, in general the number of vertices which belong to S x (ω) or S x (ω) but do not lie in the intersection is the number of triangle vertices and since the distance between special vertices is 4, is equal to n4 + 1. In the other cases it is the sum of the triangle and square vertices. The number of triangle or square vertices is n2 + 1 when n is even and n+1 2 when n is odd. Hence when n is odd, the result is n + 1 and when n is even it is n + 2. Unlike the B˜ 2 case x, x and ω need not lie in the same plane, since the distance between x and x is 2. However the short distance imposes restrictions on the amount of branching which can occur. The plane on which ω lies has to intersect the wall between x and x . Otherwise there exists a plane on which x, x and ω all lie, which contradicts our original assumption. By symmetry we only need to consider two possible outcomes, depending upon whether the delimiting sector lines issuing from x and x intersect the ω-plane once or twice. We first look at the case where they intersect the ω-plane twice. Since the ω-plane constitutes a plane of symmetry between x and x , the points of intersection from x and those from x are the same. We will call them y and y . In order to obtain two intersection points, y and y must lie distance 1 or 2 away. Otherwise one of the delimiting sector lines is parallel to the intersection of the ω-plane with the (x, x )-plane and there is only one intersection point. An example is illustrated in Fig. 15.
422
S.J. Campbell / Journal of Functional Analysis 256 (2009) 409–431
Fig. 13.
Fig. 14.
The intersection of the two sectors lies entirely within the ω-plane. Thus the only special vertices which do not belong to the intersection but do belong to the sector are x and x themselves and the number of such vertices is 2. In the second case, if only one delimiting sector line from each of x and x intersects the ω-plane, then we get the following generic situation (see Fig. 16).
S.J. Campbell / Journal of Functional Analysis 256 (2009) 409–431
423
Fig. 15.
Fig. 16.
The intersection of the sectors based at x and x is a sector lying entirely in the ω-plane based at the intersection point y. However the union of these sectors includes in addition a strip lying
424
S.J. Campbell / Journal of Functional Analysis 256 (2009) 409–431
in the (x, x )-plane, which is delimited by the two parallel lines to the intersection of the ω-plane with the (x, x )-plane as shown in the diagram. The special vertices lie distance 4 apart along these lines. Hence the total number of such vertices is equal to 2 n4 + 2. By definition, the sectors based at x and x must intersect the ω-plane. This plane intersects the (x, x )-plane along a wall which divides this plane into two roots. In order for the sector to intersect the ω-plane, the lines emanating from x or x respectively must not lie entirely in just one root. However this is the situation in all other cases, so there are no more cases to consider. So the possibilities for fn,x − fn,x 1 are 2, n4 + 1, 2 n4 + 2, n + 1 or n + 2. Considering the worst case scenario, when d(x, x ) = 2, fn,x − fn,x 1 n + 2. Now Δ is connected and the distance between two special vertices is always 2. This implies that S is 2-connected, i.e. any 2 vertices can be connected by a path where the distance at each step is 2. So in general, fn,x − fn,x 1 2d(x, x )(n + 2). Hence fn,x − fn,x 1 2d(x, x )(n + 2) . fn,x 1 ( n4 + 1)2 On the set d(x, x ) R, this tends uniformly to 0 as n tends to infinity as required. Thus by Proposition 9 S has Property A. 2 By coarse equivalence we obtain: ˜ 2 have Property A. Corollary 17. Affine buildings of type G Putting together Propositions 11, 14 and 16 we obtain our main theorem: ˜ 2 have Property A. Theorem 3. Affine buildings of type A˜ 2 , B˜ 2 or G Since affine buildings in higher dimensions arise from linear groups and have Property A [11], we also deduce our main corollary: Corollary 4. All affine buildings have Property A. 3. A direct construction for buildings of type B˜ 2 The use of the coarse equivalence obscures the nature of the functions except in the case of the buildings of type A˜ 2 where every vertex is special. Here we show that in the case of affine buildings of type B˜ 2 the functions can also be constructed explicitly. This may be of value in attempting to generalise these methods to other spaces of non-positive curvature. Theorem 18. Affine buildings of type B˜ 2 have Property A. Proof. Let Δ be an affine building of type B˜ 2 . Let Δ∞ be its building at infinity and fix a chamber ω ∈ Δ∞ . The situation here is slightly tricky since not every vertex is a special vertex. Since non-special vertices do not admit sectors at infinity, we cannot proceed directly as we did in the A˜ 2 case where every vertex is a special vertex.
S.J. Campbell / Journal of Functional Analysis 256 (2009) 409–431
425
Fig. 17.
This is illustrated in the diagram in Fig. 17 of an apartment in Δ. The thick black lines delimit a sector. The square vertices represent the special vertices and the small black round ones the nonspecial vertices within this sector. The emphasized black vertices represent the special vertices which do not belong to the sector. For every vertex x ∈ Δ, choose sx to be one of the special vertices closest to it. This means that if x is a special vertex, then sx is x itself. If on the other hand x is not a special vertex, then sx is one of the special vertices distance 1 away. The actual choice of special vertex is not important for the purposes of this paper. 3.1. The function Given any vertex x ∈ Δ we will consider the sector based at sx defined by ω and denote it S sx (ω). We now define our function as follows: fn,x (y) =
1 if y is a special vertex ∈ S sx (ω) and d(sx , y) n, 0 otherwise.
All values of fn,x are positive and fn,x 1 is the number of special vertices in the sector 2 truncated at n, which as seen earlier in this paper in the proof of Proposition 14 is either n4 +n+3 n2 4
+ n + 1. Now consider the case of two adjacent vertices x, x . There are only two possibilities. Either both are special vertices, or one is special and one is not. Let us first assume that both are special and thus sx and sx are the points x, x themselves. or
426
S.J. Campbell / Journal of Functional Analysis 256 (2009) 409–431
Fig. 18.
As seen previously in the paper during the proof of Theorem 4, the only possibilities for fn,x − fn,x 1 in this situation are n + 1, n + 2 and 2(n + 1). Now let us assume that one of x, x is special but the other is not. Without loss of generality, assume that x is special and thus sx is x itself. x can be mapped to several special vertices. sx could be sx itself. In this case fn,x − fn,x 1 = 0. x could be mapped to a special vertex distance 1 away from sx . In this case sx and sx are distance 1 apart and as we have just seen, fn,x − fn,x 1 is either n + 1, n + 2 or 2(n + 1). Finally, x could also be mapped to a special vertex distance 2 away from sx . Assume the special vertex remains in the same apartment as the sector based at x. There are four positions it could go, two of which we can ignore by symmetry. We need to consider the remaining two cases. We illustrate them in the diagrams in Figs. 18 and 19 with the special cases of n = 9, 10, 11. In fact these are essentially the same and the number of special vertices which belong to S sx (ω) or S sx (ω) but do not lie in the intersection remains identical. The number of square vertices is once again n + 1, as is the number of triangle vertices. In both cases fn,x − fn,x 1 = 2(n + 1). It is feasible that sx not lie in the same apartment as the sector based at x, in which case the sector based at sx intersects that based at x = sx at the first two vertices along the sector as illustrated in the diagram in Fig. 20 and fn,x − fn,x 1 = 2. Looking at the worst case scenario, the largest fn,x − fn,x 1 can be is 2(n + 1), and the 2 smallest fn,x 1 can be is n4 + n + 1. In general, fn,x − fn,x 1 2d(x, x )2(n + 1).
S.J. Campbell / Journal of Functional Analysis 256 (2009) 409–431
427
Fig. 19.
Fig. 20.
Hence fn,x − fn,x 1 4d(x, x )(n + 1) . n2 fn,x 1 4 +n+1 On the set d(x, x ) R, this tends uniformly to 0 as n tends to infinity as required. Thus by Proposition 9, buildings of type B˜ 2 have Property A. 2 4. Groups acting on A˜ n buildings We turn our attention more specifically to groups acting on affine buildings and look at an example for higher dimensions. The result for A˜ n groups had previously been deduced to be true by a different method. If Γ is a closed subgroup of Aut(Δ) where Δ is a Euclidean building of dimension > 2, then Γ acts amenably on its boundary and so as previously discussed has Property A [13,15].
428
S.J. Campbell / Journal of Functional Analysis 256 (2009) 409–431
Fig. 21.
We will now use the ideas from [14] and [12] to obtain a shorter direct geometric construction proof for A˜ n . In practise this simply involves slightly modifying the general method used in the previous sections of our paper of weighting the vertices in the intersection of sectors. To do this we use the following group characterisation of Property A [12]: Proposition 19. Let Γ be a finitely generated discrete group equipped with word length and metric associated to a finite symmetric set of generators. Then Γ is exact and has Property A iff there exists a sequence of positive functions uk : Γ × Γ → R satisfying: (1) For all C > 0, uk → 1 uniformly on the strip x, y: d(x, y) < C. (2) For all k there exists R such that uk (s, t) = 0 if d(x, y) R. We first introduce a labelling commonly used in buildings of type A˜ 2 . This labelling is already known and used in [6] and [15]. During our proof we will extend it to A˜ n buildings. Given any vertex x ∈ Δ we consider the sector based at x defined by ω and denote it S x (ω). x is a vertex ∈ S x (ω) such that its distance from the left side of the sector is i and A vertex si,j similarly its distance from the right side of the sector is j . We define Skx (ω) to be the part of the sector S x (ω) such that max(i + j ) = k. These are all the vertices distance at most k away from the base point x. This is illustrated in the diagram in Fig. 21 for k = 6. Let Δ be an affine building of type A˜ n . Let Ω be its building at infinity and fix a chamber ω ∈ Ω. Given any vertex x ∈ Δ we consider the sector based at x defined by ω and denote it S x (ω). A vertex skx1 ,k2 ,...,kn is a vertex ∈ S x (ω) such that its distances from the sides of the sector are k 1 , k 2 , . . . , kn . We define Skx (ω) to be the part of the sector S x (ω) such that max(k1 + k2 + · · · + kn ) = k. We now need to prove the following technical theorem:
S.J. Campbell / Journal of Functional Analysis 256 (2009) 409–431
429
Fig. 22.
Theorem 20. Let Δ be an affine building of type A˜ n . Then given any vertex x ∈ Δ, the number of vertices in Skx (ω) is a polynomial in k of degree n. Proof. Let An (k) = |Sk (ω)| in an A˜ n building. For example, when n = 1 (a tree), An (k) = k. k+1 1 2 When n = 2 (the case A˜ 2 seen previously), An (k) = k+1 i=1 A1 (k) = i=1 i = 2 (k + 3k + 2). More generally since we are considering simplices, then by construction each sector is a stack of sectors of degree one below as illustrated in Fig. 22 for n = 3 and k = 5, and An (k) =
k+1
An−1 (i).
i=1
By [17], ki=1 i p = cp+1 k p+1 + · · · + c1 k, where c1 , . . . , cp+1 can be found by solving the p+1 system of p + 1 equations i=j +1 (−1)i−j +1 ji ci = δj,p , where δj,p is the Kronecker delta. Note that these are independent of k. The sum of some polynomial of degree n: ki=1 (an i n + an−1 i n−1 + · · · + a0 ) can be rewritten k n k n−1 k as an i=1 i + an−1 i=1 i + · · · + i=1 a0 . This is another polynomial in k of degree n + 1: bn+1 k n+1 + bn k n + · · · + b0 , with new constants b0 , . . . , bn+1 dependent on a0 , . . . , an and n via the above Schultz equality. Since An (k) is obtained by the repeated n − 1 summing of polynomials, the first of which is of degree 1, An (k) is in fact ultimately a polynomial of degree n with coefficients dependent on n and obtained using the Schultz equalities. So we have |Sk (ω)| = bn k n + · · · + b1 k + b0 . 2 Theorem 21. Given an A˜ n group, there exists a sequence of positive functions uk : A˜ n × A˜ n → R satisfying: (1) For all C > 0, uk → 1 uniformly on the strip x, y: d(x, y) < C. (2) For all k there exists R such that uk (s, t) = 0 if d(x, y) R.
430
S.J. Campbell / Journal of Functional Analysis 256 (2009) 409–431
Proof. Given two vertices x, y ∈ Δ and some k ∈ N we consider the intersection of the sectors y Skx (ω) and Sk (ω). We define y
uk (x, y) =
|Skx (ω) ∩ Sk (ω)| , |Sk (ω)|
where |Sk (ω)| is the number of vertices in the defined sector at any base point. uk (x, y) is a positive function ∀k as required. This could be shown explicitly by calculation y by rewriting |Skx (ω) ∩ Sk (ω)| as a double sum in terms of x and y and obtaining a sum of squares as seen in [4]. Alternatively note that this intersection is in fact the inner product of two characteristic functions. (1) By definition, the vertex sd(x,y),...,d(x,y) (with n coordinates) relative to the sector based at x and that based at y lies in the intersection of these two sectors. Choose the vertex which maximizes the distance between it and one of those base points and call it z. The distance between z and x or y is at most (n + 1) ∗ d(x, y). To simplify notation, we will call this value m. Furthermore, the sector S z is a common subsector to S x (ω) and S y (ω) [6]. Fix some C > 0. For all pairs of points (x, y) distance less than C apart, m < (n + 1) ∗ C, which is a fixed number. We thus have the following: x S (ω) ∩ S y (ω) Sk (ω).
z S
k−m (ω)
k
k
By Theorem 20, we have |Sk (ω)| = bn k n + · · · + b1 k + b0 . Similarly |Sk−m (ω)| = bn (k − m)n + · · · + b1 (k − m) + b0 , where b0 , . . . , bn are constants dependent on n, obtained using the Schultz equalities, z |Sk−m (ω)|
|Sk (ω)|
=
bn (k − m)n + · · · + b1 (k − m) + b0 . bn k n + · · · + b1 k + b0
Hence lim
z |Sk−m (ω)|
|Sk (ω)|
k→∞
= 1.
Since we had z |Sk−m (ω)|
|Sk (ω)|
uk (x, y)
|Sk (ω)| |Sk (ω)|
limk→∞ uk (x, y) = 1 for all pairs of points (x, y) such that d(x, y) < C. (2) Given some fixed k ∈ N, define R to be 2k. In that case, when d(x, y) > R = 2k, there is y no intersection between the sectors Skx (ω) and Sk (ω) and so uk (x, y) = 0. 2 By Proposition 19, we get the following corollary to Theorem 21: Corollary 22. A˜ n groups have Property A.
S.J. Campbell / Journal of Functional Analysis 256 (2009) 409–431
431
Acknowledgment The author would like to thank Guyan Robertson for his guidance and useful talks during the course of this research. References [1] G.N. Arzhantseva, V.S. Guba, M.V. Sapir, Metrics on diagram groups and uniform embeddings in a Hilbert space, Comment. Math. Helv. 81 (4) (2006) 911–929. [2] Sylvain Barre, Immeubles de Tits Triangulaires Exotiques, Ann. Fac. Sci. Toulouse 9 (2000) 575–603. [3] J. Brodzki, S.J. Campbell, E. Guentner, G.A. Niblo, N.J. Wright, Property A and CAT(0) cube complexes, J. Funct. Anal. (2008), in press. [4] S.J. Campbell, Exactness for free and amenable groups by construction of Ozawa kernels, Bull. London Math. Soc. 38 (2006) 441–446. [5] Sarah Campbell, Graham Niblo, Hilbert space compression and exactness of discrete groups, J. Funct. Anal. 222 (2005) 292–305. [6] Donald I. Cartwright, Harmonic functions on buildings of type A˜ n , in: Massimo Picardello, Wolfgang Woess (Eds.), Random Walks and Discrete Potential Theory, in: Symposia Mathematica Volume, vol. 39, Cambridge University Press, 1999, pp. 104–138. [7] Donald I. Cartwright, A.M. Mantero, T. Steger, A. Zappa, Groups acting simply transitively on the vertices of a building of type A˜ 2 , I, II, Geom. Dedicata 49 (1993) 143–223. [8] Donald I. Cartwright, Wojciech Mlotkowski, Tim Steger, Property (T) and A˜ 2 groups, Ann. Inst. Fourier 44 (1) (1994) 231–248. [9] A. Dranishnikov, T. Januszkiewicz, Every Coxeter group acts amenably on a compact space, in: Proceedings of the 1999 Topology and Dynamics Conference, Salt Lake City, UT, in: Topology Proc., vol. 24, Spring 1999, pp. 135– 141. [10] Paul Garrett, Buildings and Classical Groups, Chapman & Hall, 1997, April 1. [11] E. Guentner, N. Higson, S. Weinberger, The Novikov conjecture for linear groups, Publ. Math. Inst. Hautes Études Sci. 101 (2005) 243–268. [12] Erik Guentner, Jerome Kaminker, Exactness and uniform embeddability of discrete groups, J. Lond. Math. Soc. 70 (3) (2004) 703–718. [13] Nigel Higson, John Roe, Amenable group actions and the Novikov conjecture, J. Reine Angew. Math. 519 (2000) 143–153. [14] N. Ozawa, Amenable actions and exactness for discrete groups, C. R. Acad. Sci. Paris Sér. I Math. 330 (2000) 691–695. [15] Guyan Robertson, Tim Steger, C ∗ -algebras arising from group actions on the boundary of a triangle building, Proc. London Math. Soc. 72 (1996) 613–637. [16] Mark Ronan, Lectures on Buildings, Perspect. Math., vol. 7, 1989. [17] H.J. Schultz, The sums of the th powers of the first integers, Amer. Math. Monthly 87 (1980) 478–481. [18] Jean-Louis Tu, Remarks on Yu’s “Property A” for discrete metric spaces and groups, Bull. Soc. Math. France 129 (1) (2001) 115–139. [19] Van Maldeghem, Automorphisms of nonclassical triangle buildings, Bull. Soc. Math. Belg. Ser. B 42 (1990) 201– 237. [20] G. Yu, The Coarse Baum Connes conjecture for spaces, which admit a uniform embedding into a Hilbert space, Invent. Math. 139 (2000) 201–240.
Journal of Functional Analysis 256 (2009) 432–478 www.elsevier.com/locate/jfa
Poisson cluster measures: Quasi-invariance, integration by parts and equilibrium stochastic dynamics Leonid Bogachev a,∗ , Alexei Daletskii b a Department of Statistics, University of Leeds, Leeds LS2 9JT, UK b Department of Mathematics, University of York, York YO10 5DD, UK
Received 28 March 2008; accepted 7 October 2008 Available online 28 October 2008 Communicated by Paul Malliavin
Abstract (with i.i.d. clusters) is studied via an auxThe distribution μcl of a Poisson cluster process in X = Rd iliary Poisson measure on the space of configurations in X = n X n , with intensity measure defined as a convolution of the background intensity of cluster centres and the probability distribution of a generic cluster. We show that the measure μcl is quasi-invariant with respect to the group of compactly supported diffeomorphisms of X and prove an integration-by-parts formula for μcl . The corresponding equilibrium stochastic dynamics is then constructed using the method of Dirichlet forms. © 2008 Elsevier Inc. All rights reserved. Keywords: Cluster point process; Poisson measure; Configuration space; Quasi-invariance; Integration by parts; Dirichlet form; Stochastic dynamics
1. Introduction In the mathematical modelling of multi-component stochastic systems, it is conventional to describe their behaviour in terms of random configurations of “particles” whose spatio-temporal dynamics is driven by interaction of particles with each other and the environment. Examples are ubiquitous and include various models in statistical mechanics, quantum physics, astrophysics, chemical physics, biology, computer science, economics, finance, etc. (see [16] and the extensive bibliography therein). * Corresponding author.
E-mail addresses: [email protected] (L. Bogachev), [email protected] (A. Daletskii). 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.10.009
L. Bogachev, A. Daletskii / Journal of Functional Analysis 256 (2009) 432–478
433
Initiated in statistical physics and theory of point processes, the development of a general mathematical framework for suitable classes of configurations was over decades a recurrent research theme fostered by widespread applications. More recently, there has been a boost of more specific interest in the analysis and geometry of configuration spaces. In the seminal papers [5,6], an approach was proposed to configuration spaces as infinite-dimensional manifolds. This is far from straightforward, since configuration spaces are not vector spaces and do not possess any natural structure of Hilbert or Banach manifolds. However, many “manifold-like” structures can be introduced, which appear to be nontrivial even in the Euclidean case. We refer the reader to papers [3,6,7,25,29] and references therein for further discussion of various aspects of analysis on configuration spaces and applications. Historically, the approach in [5,6] was motivated by the theory of representations of diffeomorphism groups (see [17,20,33]). To introduce some notation, let ΓX be the space of countable subsets (configurations) without accumulation points in a topological space X (e.g., Euclidean space Rd ). Any probability measure μ on ΓX , quasi-invariant with respect to the action of the group Diff0 (X) of compactly supported diffeomorphisms of X (lifted pointwise to transformations of ΓX ), generates a canonical unitary representation of Diff0 (X) in L2 (ΓX , μ). It has been proved in [33] that this representation is irreducible if and only if μ is Diff0 (X)-ergodic. Representations of such type are instrumental in the general theory of representations of diffeomorphism groups [33] and in quantum field theory [17,18]. According to a general paradigm described in [5,6], configuration space analysis is determined by the choice of a suitable probability measure μ on ΓX (quasi-invariant with respect to Diff0 (X)). It can be shown that such a measure μ satisfies a certain integration-by-parts formula, which enables one to construct, via the theory of Dirichlet forms, the associated equilibrium dynamics (stochastic process) on ΓX such that μ is its invariant measure [5,6,27]. In turn, the equilibrium process plays an important role in the asymptotic analysis of statistical-mechanical systems whose spatial distribution is controlled by the measure μ; for instance, this process is a natural candidate for being an asymptotic “attractor” for motions started from a perturbed (nonequilibrium) configuration. This programme has been successfully implemented in [5] for the Poisson measure, which is the simplest and most well-studied example of a Diff0 (X)-quasi-invariant measure on ΓX , and in [6] for a wider class of Gibbs measures, which appear in statistical mechanics of classical continuous gases. In particular, it has been shown that in the Poisson case, the equilibrium dynamics amounts to the well-known independent particle process, that is, an infinite family of independent (distorted) Brownian motions started at the points of a random Poisson configuration. In the Gibbsian case, the dynamics is much more complex due to interaction between the particles. The Gibbsian class (containing the Poisson measure as a simple “interaction-free” case) is essentially the sole example so far that has been fully amenable to such analysis. In the present paper, our aim is to develop a similar framework for a different class of random spatial structures, namely the well-known cluster point processes (see, e.g., [15,16]). Cluster process is a simple model to describe effects of grouping (“clustering”) in a sample configuration. The intuitive idea is to assume that the random configuration has a hierarchical structure, whereby independent clusters of points are distributed around a certain (random) configuration of invisible “centres.” The simplest model of such a kind is the Poisson cluster process, obtained by choosing a Poisson point process as the background configuration of the cluster centres. Cluster models have been very popular in numerous practical applications ranging from neurophysiology (nerve impulses) and ecology (spatial distribution of offspring around the parents) to seismology (statistics of earthquakes) and cosmology (formation of constellations and galax-
434
L. Bogachev, A. Daletskii / Journal of Functional Analysis 256 (2009) 432–478
ies). More recent examples include applications to trapping models of diffusion-limited reactions in chemical kinetics [1,9,12], where clusterization may arise due to binding of traps to a substrate (e.g., a polymer chain) or trap generation (e.g., by radiation damage). An exciting range of new applications in physics and biology is related to the dynamics of clusters consisting of a few to hundreds of atoms or molecules. Investigation of such “mesoscopic” structures, intermediate between bulk matter and individual atoms or molecules, is of paramount importance in the modern nanoscience and nanotechnology (for an authoritative account of the state of the art in this area, see a recent review [14] and further references therein). In the present work, we consider Poisson cluster processes in X = Rd . We prove the Diff0 (X)quasi-invariance of the Poisson cluster measure μcl and establish the integration-by-parts formula. We then construct an associated Dirichlet form, which implies in a standard way the existence of equilibrium stochastic dynamics on the configuration space ΓX . Our technique is based on the representation of μcl as a natural “projection” imageof a certain Poisson measure on an auxiliary configuration space ΓX over a disjoint union X = n X n , comprising configurations of “droplets” representing individual clusters of variable size. A suitable intensity measure on X is obtained as a convolution of the background intensity λ(dx) (of cluster centres) with the probability distribution η(dy) ¯ of a generic cluster. This approach enables one to apply the well-developed apparatus of Poisson measures to the study of the Poisson cluster measure μcl . Let us point out that the projection construction of the Poisson cluster measure is very general, and in particular it works even in the case when “generalized” configurations (with possible accumulation or multiple points) are allowed. However, to be able to construct a well-defined differentiable structure on cluster configurations, we need to restrict ourselves to the space ΓX of “proper” (i.e., locally finite and simple) configurations. Using the technique of Laplace functionals, we obtain necessary and sufficient conditions of almost sure (a.s.) properness for Poisson cluster configurations, set out in terms of the background intensity λ(dx) of cluster centres and the in-cluster distribution η(dy). ¯ To the best of our knowledge, these conditions appear to be new (cf., e.g., [16, §6.3]) and may be of interest for the general theory of cluster point processes. Some of the results of this paper have been sketched in [11] (in the case of clusters of fixed size). We anticipate that the projection approach developed in the present paper can be applied to the study of more general cluster measures on configurations spaces, especially Gibbs cluster measure (see [10] for the case of fixed-size clusters). Such models, and related functional-analytic issues, will be addressed in our future work. The paper is organized as follows. In Section 2.1, we set out a general framework of probabil ity measures in the space of generalized configurations ΓX . In Section 2.2, we recall the definition and discuss the construction and some basic properties of the Poisson measure on the space ΓX , while Section 2.3 goes on to describe the Poisson cluster measure. In Section 2.4, we discuss criteria for Poisson cluster configurations to be a.s. locally finite and simple (Theorem 2.7, theproof of which is deferred to Appendix A). An auxiliary intensity measure λ on the space X = n X n is introduced and discussed in Section 3.1, which allows us to define the corresponding Poisson measure πλ on the configuration space ΓX (Section 3.2). Theorem 3.6 of Section 3.3 shows that the Poisson cluster measure μcl can be obtained as a push-forward of the Poisson measure πλ on ¯ := xi ∈x¯ {xi } ∈ ΓX . In Section 3.4, we describe a ΓX under the “unpacking” map X x¯ → p(x)
more general construction of μcl using another Poisson measure defined on the space ΓX×X of configurations of pairs (x, y) ¯ (x = cluster centre, y¯ = in-cluster configuration), with the product intensity measure λ(dx) ⊗ η(dy). ¯ Following a brief compendium on differentiable functions in configuration spaces (Section 4.1), Section 4.2 deals with the property of quasi-invariance of the
L. Bogachev, A. Daletskii / Journal of Functional Analysis 256 (2009) 432–478
435
measure μcl with respect to the diffeomorphism group Diff0 (X) (Theorem 4.3). Further on, an integration-by-parts formula for μcl is established in Section 4.3 (Theorem 4.5). The Dirichlet form Eμcl associated with μcl is defined and studied in Section 5.1, which enables us to construct in Section 5.2 the canonical equilibrium dynamics (i.e., diffusion on the configuration space with invariant measure μcl ). In addition, we show that the form Eμcl is irreducible (Theorem 5.4, Section 5.3). Finally, Appendix A includes the proof of Theorem 2.7 (Section A.1) and the proof of a well-known general result on quasi-invariance of Poisson measures, adapted to our purposes (Section A.2). 2. Poisson and Poisson cluster measures in configuration spaces In this section, we fix some notations and describe the setting of configuration spaces that we shall use. As compared to a standard exposition (see, e.g., [15,16]), we adopt a more general standpoint by allowing configurations with multiple points and/or accumulation points. With this modification in mind, we recall the definition and some properties of Poisson point process (as a probability measure in the generalized configuration space ΓX ). We then proceed to introduce the main object of the paper, the cluster Poisson point process and the corresponding measure μcl in ΓX . The central result of this section is the projection constriction showing that μcl can be obtained as a push-forward of a suitable Poisson measure in the auxiliary “vector” configuration space ΓX , where X = n X n . 2.1. Generalized configurations Let X be a Polish space (i.e., separable completely metrizable topological space), equipped with the Borel σ -algebra B(X) generated by the open sets. Denote Z+ := Z+ ∪ {∞}, where 2, . . .}, and consider the space X built from Cartesian powers of X, that is, a disjoint Z+ = {0, 1, union X := n∈Z+ X n including X 0 = {∅} and the space X ∞ of infinite sequences (x1 , x2 , . . .).
That is to say, x¯ = (x1 , x2 , . . .) ∈ X if and only if x¯ ∈ X n for some n ∈ Z+ . For simplicity of ¯ notation, we take the liberty to write xi ∈ x¯ if xi is a coordinate of the vector x. Each space X n is equipped with the product topology induced by X, that is, the coarsest topology in which all coordinate projections (x1 , . . . , xn ) → xi are continuous (i = 1, . . . , n). Hence, the space X is endowed with the natural disjoint union topology, that is, the finest topology in which the canonical injections jn : X n →X are continuous (n ∈ Z+ ). In other words, a set U ⊂ X is open in this topology whenever U = n∈Z+ Un , where each Un is an open subset in X n (n ∈ Z+ ). Hence, the Borel σ -algebra on X is given by B(X) = n∈Z+ B(X n ), that is, consists of sets of the form B = n∈Z+ Bn , where Bn ∈ B(X n ), n ∈ Z+ . Remark 2.1. Note that a set K ⊂ X is compact if and only if K = N n=0 Kn , where N < ∞ and Kn are compact subsets of X n , respectively. This becomes clear by considering an open cover of K by the sets Un = X n , n ∈ Z+ . Denote by N (X) the space of Z+ -valued measures N (·) on B(X) with countable (i.e., finite or countably infinite) support supp N := {x ∈ X: N {x} > 0} (here and below, we use N {x} as a
436
L. Bogachev, A. Daletskii / Journal of Functional Analysis 256 (2009) 432–478
shorthand for a more accurate N ({x}); the same convention applies to other measures). Consider the natural projection
X x¯ → p(x) ¯ :=
δxi ∈ N (X),
(2.1)
xi ∈x¯
whereδx is Dirac measure at point x ∈ X. Gathering any ¯ the measure coinciding points xi ∈ x, N = xi ∈x¯ δxi in (2.1) can be written down as N = x ∗ ∈supp N ki δxi∗ , where ki = N {xi∗ } > 0 i is the “multiplicity” (possibly infinite) of the point xi∗ ∈ supp N . Any such measure N can be conveniently associated with a generalized configuration γ of points in X,
N ↔ γ :=
xi∗ ∈supp N
xi∗ · · · xi∗ ,
ki
where the disjoint union {x ∗ } · · · {x ∗ } signifies the inclusion of several distinct copies of point x ∗ ∈ supp N . Thus, the mapping (2.1) can be symbolically rewritten as
p(x) ¯ = γ :=
{xi },
x¯ = (x1 , x2 , . . .) ∈ X.
(2.2)
xi ∈x¯
That is to say, under the projection mapping p each vector from X is “unpacked” into distinct components, resulting in a countable aggregate of points in X (with possible multiple points), which we interpret as a generalized configuration γ . Note that, formally, x¯ may be from the “trivial” component X 0 = {∅}, in which case the union in (2.2) (as well as the sum in (2.1)) is vacuous and hence corresponds to the empty configuration, γ = ∅. Even though generalized configurations are not, strictly speaking, subsets of X (due to possible multiple points), it is convenient to keep using set-theoretic notations, which should not cause any confusion. For instance, we write γB := γ ∩ B for the restriction of configuration γ to a subset B ∈ B(X). Similarly, for a function f : X → R we denote f, γ :=
xi ∈γ
f (xi ) ≡
xi∗ ∈supp N
N xi∗ f xi∗ =
f (x) N (dx).
(2.3)
X
This formula motivates the following convention that will be used throughout: if γ = ∅ then x∈γ f (x) := 0. In what follows, we shall identify generalized configurations γ with the corresponding mea sures N = xi ∈γ δxi , and we shall opt to interpret the notation γ either as an aggregate of (multiple) points in X or as a Z+ -valued measure or both, depending on the context. For example, if 1B (x) is the indicator function of a set B ∈ B(X) then 1B , γ = γ (B) is the total number of points (counted with their multiplicities) in the restriction γB of the configuration γ to B.
Definition 2.1. Configuration space ΓX is the set of generalized configurations γ in X, en dowed with the cylinder σ -algebra B(ΓX ) generated by the class of cylinder sets CBn := {γ ∈ ΓX : γ (B) = n}, B ∈ B(X), n ∈ Z+ .
L. Bogachev, A. Daletskii / Journal of Functional Analysis 256 (2009) 432–478
437
Remark 2.2. Note that the set CB∞ = {γ ∈ ΓX : γ (B) = ∞} is measurable:
CB∞ =
∞ ∞ ∞
γ ∈ ΓX : γ (B) n = CBk ∈ B ΓX . n=0 k=n
n=0
The mapping p : X → ΓX defined by formula (2.2) is measurable, since for any cylinder set ∈ B(ΓX ) we have
CBn
p
−1
n n CB = DB := x¯ ∈ X: 1B (xi ) = n ∈ B(X).
(2.4)
xi ∈x¯
As already mentioned, conventional theory of point processes (and their distributions as probability measures on configuration spaces) usually rules out the possibility of accumulation points or multiple points (see, e.g., [16]).
Definition 2.2. Configuration γ ∈ ΓX is said to be locally finite if γ (K) < ∞ for any compact set K ⊂ X. Configuration γ ∈ ΓX is called simple if γ {x} 1 for each x ∈ X. Configuration γ ∈ ΓX is called proper if it is both locally finite and simple. The set of proper configurations will be denoted by ΓX and called the proper configuration space over X. The corresponding σ -algebra B(ΓX ) is generated by the cylinder sets {γ ∈ ΓX : γ (B) = n} (B ∈ B(X), n ∈ Z+ ). Like in the standard theory for proper configuration spaces (see, e.g., [16, §6.1]), every mea sure μ on the generalized configuration space ΓX can be characterized by its Laplace functional Lμ [f ] :=
e−f,γ μ(dγ ),
f ∈ M+ (X),
(2.5)
ΓX
where M+ (X) is the set of measurable non-negative functions on X (so that the integral in (2.5) is well defined since 0 e−f,γ 1). To see why Lμ [·] completely determines the measure μ on B(ΓX ), note that if B ∈ B(X) then Lμ [s1B ] as a function of s > 0 gives the Laplace–Stieltjes transform of the distribution of the random variable γ (B) and as such determines the values of the measure μ on the cylinder sets CBn ∈ B(ΓX ) (n ∈ Z+ ). In particular, Lμ [s1B ] = 0 if and only if γ (B) = ∞ (μ-a.s.). Similarly, using linear combinations ki=1 si 1Bi we can recover the values of μ on the cylinder sets k CBn11,...,n ,...,Bk :=
k
CBnii = γ ∈ ΓX : γ (Bi ) = ni , i = 1, . . . , k
i=1
and hence on the ring C(X) of finite disjoint unions of such sets. Since the ring C(X) generates the cylinder σ -algebra B(ΓX ), the extension theorem (see, e.g., [19, §13, Theorem A] or [16, Theorem A1.3.III]) ensures that the measure μ on B(ΓX ) is determined uniquely.
438
L. Bogachev, A. Daletskii / Journal of Functional Analysis 256 (2009) 432–478
2.2. Poisson measure We recall here some basic facts about Poisson measures in configuration spaces. As compared to the customary treatment, another difference, apart from working in the space of generalized configurations ΓX , is that we use a σ -finite intensity measure rather than a locally finite one. Poisson measure on the configuration space ΓX is defined descriptively as follows (cf. [16, §2.4]). Definition 2.3. Let λ be a σ -finite measure in (X, B(X)) (not necessarily infinite, i.e., λ(X) ∞). The Poisson measure πλ with intensity λ is a probability measure on B(ΓX ) satisfying the following condition: for any disjoint sets B1 , . . . , Bk ∈ B(X) (i.e., Bi ∩ Bj = ∅ for i = j ), such that λ(Bi ) < ∞ (i = 1, . . . , k), and any n1 , . . . , nk ∈ Z+ , the value of πλ on the cylinder set k CBn11,...,n ,...,Bk is given by k
n ,...,n λ(Bi )ni e−λ(Bi ) πλ CB11 ,...,Bkk = ni !
(2.6)
i=1
(with the convention 00 := 1). That is, for disjoint sets Bi the values γ (Bi ) are mutually independent Poisson random variables with parameters λ(Bi ), respectively. A well-known “explicit” construction of the Poisson measure πλ is as follows (cf. [5,31]). For a fixed set Λ ∈ B(X) such that λ(Λ) < ∞, consider the restriction mapping pΛ ,
ΓX γ → pΛ γ = γ ∩ Λ ≡ γΛ ∈ ΓΛ .
n ) = {γ˜ ∈ Γ : γ˜ (Λ) = n}. For A ∈ B(Γ ) and n ∈ Z , let A n Clearly, pΛ (CΛ + Λ,n := A∩pΛ (CΛ ) ∈ Λ Λ B(ΓΛ ) and define the measure
πλΛ (A) := e−λ(Λ)
∞ 1 ⊗n −1 λ ◦ p (AΛ,n ), n!
A ∈ B ΓΛ ,
(2.7)
n=0
where λ⊗n = λ · · ⊗ λ is the product measure in (X n , B(X n )) (we formally set λ⊗0 := δ{∅} ) ⊗ · n
and p is the projection operator defined in (2.2). In particular, (2.7) implies that πλΛ is a proba bility measure on ΓΛ . It is easy to check that the “cylindrical” measure πλΛ ◦ pΛ in ΓX (in fact, ∞ n supported on n=0 CΛ ) satisfies Eq. (2.6) for any disjoint Borel sets Bi ⊂ Λ. It is also clear that the family {πλΛ , Λ ⊂ X} is consistent, that is, the restriction of the measure πλΛ to a smaller −1 Λ configuration space ΓΛ (with Λ ⊂ Λ) coincides with πλΛ , that is, πλΛ ◦ (pΛ pΛ ) = πλ . Existence (and uniqueness) of a measure πλ in (ΓX , B(ΓX )) such that, for any Λ ∈ B(X), −1 ∗ the push-forward measure pΛ πλ ≡ πλ ◦ pΛ coincides with πλΛ (which implies that πλ sat isfies Definition 2.3 and is therefore a Poisson measure on the configuration space ΓX ), now follows by a projective version of the fundamental Kolmogorov extension theorem (see, e.g., [16, §A1.5] or [28, Chapter 5]). More precisely, recall that the measure λ on X is σ -finite, hence there is a countable family of sets Bk ∈ B(X) such that λ(Bk ) < ∞ and ∞ k=1 Bk = X.
L. Bogachev, A. Daletskii / Journal of Functional Analysis 256 (2009) 432–478
439
Then Λm := m k=1 Bk ∈ B(X) (m ∈ N) is a monotone increasing sequence of sets such that λ(Λm ) < ∞ and ∞ m=1 Λm = X. By the construction (2.7), we obtain a consistent family of probability measures πλΛm on the configuration spaces ΓΛm , respectively. Using the metric in X (which is assumed to be a Polish space, see Section 2.1), one can define a suitable distance between finite configurations in each space ΓΛm and thus convert ΓΛm into a Polish space (see [31]), which ensures that the Kolmogorov extension theorem is applicable. Remark 2.3. Even though the paper [31] deals with simple configurations only, its methods may be easily extended to a more general case of configurations with multiple points. However, finiteness of configurations in each Λm is essential. Remark 2.4. The requirement that X is a Polish space (see Section 2.1) is only needed in order to equip the spaces of finite configurations in the sets Λm with the structure of a Polish space and thus to be able to apply the Kolmogorov extension theorem as explained above (see [31]). This assumption may be replaced by a more general condition that (X, B(X)) is a standard Borel space (i.e., Borel isomorphic to a Borel subset of a Polish space, see [21,28]). Remark 2.5. Formula (2.7), rewritten in the form πλΛ (A) =
∞ λ(Λ)n e−λ(Λ) λ⊗n ◦ p−1 (AΛ,n ) · , n! λ(Λ)n n=0
gives an explicit way of sampling a Poisson configuration γΛ in the set Λ: first, a random value of γ (Λ) is sampled as a Poisson random variable with parameter λ(Λ) < ∞, and then, conditioned on the event {γ (Λ) = n} (n ∈ Z+ ), the n points are distributed over Λ independently of each other, with probability distribution λ(dx)/λ(Λ) each (cf. [22, §2.4]). Decomposition (2.7) implies that if F (γ ) ≡ F (γΛ ) for some set Λ ∈ B(X) such that λ(Λ) < ∞, then F (γ ) πλ (dγ ) = F (pΛ γ ) πλ (dγ ) = F (γ ) πλΛ (dγ )
ΓX
ΓX
ΓΛ
= e−λ(Λ)
∞
1 F {x1 , . . . , xn } λ(dx1 ) . . . λ(dxn ). n! n=0
(2.8)
Λn
A well-known formula for the Laplace functional of a Poisson point process without accumulation points (see, e.g., [5,16]]) is easily verified in the case of generalized configurations. Proposition 2.1. The Laplace functional Lπλ [f ] :=
ΓX
e−f,γ πλ (dγ ) of the Poisson measure
πλ on the configuration space ΓX is given by
−f (x) Lπλ [f ] = exp − 1−e λ(dx) , X
f ∈ M+ (X).
(2.9)
440
L. Bogachev, A. Daletskii / Journal of Functional Analysis 256 (2009) 432–478
Proof. Repeating a standard derivation, suppose that λ(Λ) < ∞ and set fΛ := f · 1Λ . Applying formula (2.8) we have e
−fΛ ,γ
πλ (dγ ) = e
−λ(Λ)
n ∞ 1 exp − fΛ (xi ) λ(dx1 ) . . . λ(dxn ) n! n=0
ΓX
=e
−λ(Λ)
i=1
Λn
n ∞ 1 −fΛ (x) e λ(dx) n! n=0
Λ
1 − e−fΛ (x) λ(dx) . = exp −
(2.10)
X
Since fΛ (x) ↑ f (x) as Λ ↑ X (more precisely, setting Λ = Λm as in the above construction of πλ and passing to the limit as m → ∞), by applying the monotone convergence theorem to both sides of (2.10) we obtain (2.9). 2 Formula (2.6) implies that if B1 ∩ B2 = ∅ then the restricted configurations γB1 and γB2 are independent under the Poisson measure πλ . That is, if B := B1 ∪ B2 then the distribution πλB = pB∗ πλ of composite configurations γB = γB1 γB2 coincides with the product measure B πλB1 ⊗ πλB2 (πλ i = pB∗ i πλ ). Building on this observation, we obtain the following useful result. Proposition 2.2. Suppose that (Xn , B(Xn )) (n ∈ N) is a family of disjoint measurable spaces (i.e., Xi ∩ Xj = ∅, i = j ), with measures λn , respectively, and let πλn be the corresponding Poisson measures on the configuration spaces ΓXn (n ∈ N). Consider the disjoint-union space ∞ ∞ X = n=1 Xn endowed with theσ -algebra B(X) = ∞ n=1 B(Xn ) and measure λ = n=1 λn . Then the product measure πλ = ∞ π exists and is a Poisson measure on the configuration n=1 λn space ΓX with intensity measure λ. Proof. Note that ΓX is a Cartesian product space, ΓX = X∞ n=1 ΓXn , endowed with the product ∞ σ -algebra B(ΓX ) = n=1 B(ΓXn ). The existence of the product measure πλ := ∞ n=1 πλn on
(ΓX , B(ΓX )) now follows by a standard result for infinite products of probability measures (see, e.g., [19, §38, Theorem B] or [21, Corollary 5.17]). Let us point out that this theorem is valid without any regularity conditions on the spaces Xn . To show that πλ is a Poisson measure, one could check the cylinder condition (2.7), but it is easier to compute its Laplace functional. Note that each function f ∈ M+ (X) is decomposed as f = ∞ the restriction of f to Xn ; similarly, each n=1 fXn · 1Xn , where fXn ∈ M+ (Xn ) is ∞ configuration γ ∈ ΓX may be represented as γ = n=1 γXn , where γXn = pXn γ ∈ ΓXn . Hence, ∞ f, γ = n=1 fXn , γXn and, using Proposition 2.1 for each πλn , we obtain
ΓX
∞ ∞ exp − fXn , γn πλn (dγn )
e−f,γ πλ (dγ ) = ∞
Xn=1 ΓXn
n=1
n=1
L. Bogachev, A. Daletskii / Journal of Functional Analysis 256 (2009) 432–478
=
∞ n=1
441
e−fXn ,γn πλn (dγn )
ΓXn
∞
−fXn (xn ) 1−e λn (dxn ) = exp −
n=1 X
n
−f (x) = exp − 1−e λ(dx) , X
and it follows, according to formula (2.9), that πλ is a Poisson measure.
2
Remark 2.6. Using Proposition 2.2, one can give a construction of a Poisson measure πλ on the configuration space ΓX avoiding any additional topological conditions upon the space X (e.g., that X is a Polish space) that are needed for the sake of the Kolmogorov extension theorem (similar ideas are developed in [22,23] in the context of proper configuration spaces). To do so, recall that the measure λ is σ -finite and define Xn := Λn \Λn−1 (n ∈ N), where the sets ∅ = Λ0 ⊂ Λ1 ⊂ · · · ⊂ Λn ⊂ · · · ⊂ X, such that λ(Λn ) < ∞ and ∞ n=1 Λn = X, were considered ) is a disjoint partition of X (i.e., Xi ∩ Xj = ∅ for i = j above. Then the family of sets (X n and ∞ n=1 Xn = X), such that λ(Xn ) < ∞ for all n ∈ N. Using formula (2.6), we construct the Poisson measures πλn ≡ pXn πλ on each ΓXn , where λn = λXn is the restriction of the measure λ to the set Xn . Now, it follows by Proposition 2.2 that the product measure πλ = ∞ n=1 πλn is the required Poisson measure on ΓX . Remark 2.7. Although not necessary for the existence of the Poisson measure, in order to develop a sensible theory one needs to ensure that there are enough measurable sets and in particular any singleton set {x} is measurable. To this end, it is suitable to assume (see [22, §2.1]) that the diagonal set {x = y} is measurable in the product space X 2 = X × X, that is, (2.11) D := (x, y) ∈ X 2 : x = y ∈ B(X 2 ). This condition readily implies that {x} ∈ B(X) for each x ∈ X. Note that if X is a Polish space, condition (2.11) is automatically satisfied because then the diagonal D is a closed set in X 2 . Let us also record one useful general result known as the Mapping Theorem (see [22, §2.3], where configurations are assumed proper and the mapping is one-to-one). Let ϕ : X → Y be a measurable mapping (not necessarily one-to-one) of X to another (or the same) measurable space Y endowed with Borel σ -algebra B(Y ). The mapping ϕ can be lifted to a measurable “diagonal” mapping (denoted by the same letter) between the configuration spaces ΓX and ΓY :
ΓX γ → ϕ(γ ) :=
ϕ(x) ∈ ΓY .
(2.12)
x∈γ
Proposition 2.3 (Mapping Theorem). If πλ is a Poisson measure on ΓX with intensity measure λ, then under the mapping (2.12) the push-forward measure ϕ ∗ πλ ≡ πλ ◦ ϕ −1 is a Poisson measure on ΓY with intensity measure ϕ ∗ λ ≡ λ ◦ ϕ −1 .
442
L. Bogachev, A. Daletskii / Journal of Functional Analysis 256 (2009) 432–478
Proof. It suffices to compute the Laplace functional of ϕ ∗ πλ . Using Proposition 2.1, for any f ∈ M+ (Y ) we have Lϕ ∗ πλ [f ] =
e−f,γ ϕ ∗ πλ (dγ ) =
e−f,ϕ(γ ) πλ (dγ )
ΓY
ΓX
1 − e−f (ϕ(x)) λ(dx) = exp − X
∗ −f (y) 1−e ϕ λ (dy) = Lπϕ ∗ λ [f ], = exp − Y
and the proof is complete.
2
We conclude this section with necessary and sufficient conditions in order that πλ -almost all (a.a.) configurations γ ∈ ΓX be proper (see Definition 2.2). Although being apparently wellknown folklore, these criteria are not always proved or even stated explicitly in the literature, most often being mixed up with various sufficient conditions, e.g., using the property of orderliness etc. (see, e.g., [15,16,22]). We do not include the proof here, as the result follows from a more general statement for the Poisson cluster measure (see Theorem 2.7 below). Proposition 2.4. (a) If B ∈ B(X) then γ (B) < ∞ (πλ -a.s.) if and only if λ(B) < ∞. In particular, in order that πλ -a.a. configurations γ ∈ ΓX be locally finite, it is necessary and sufficient that λ(K) < ∞ for any compact set K ∈ B(X). (b) In order that πλ -a.a. configurations γ ∈ ΓX be simple, it is necessary and sufficient that the measure λ be non-atomic, that is, λ{x} = 0 for each x ∈ X. 2.3. Poisson cluster measure Let us first recall the notion of a general cluster point process (CPP). The intuitive idea is to construct its realizations in two steps: (i) take a background random configuration of (invisible) “centres” obtained as a realization of some point process γc governed by a probability measure μc on ΓX , and (ii) relative to each centre x ∈ γc , generate a set of observable secondary points (referred to as a cluster centred at x) according to a point process γx with probability measure μx on ΓX (x ∈ X). The resulting (countable) assembly of random points, called the cluster point process, can be symbolically expressed as γ=
x∈γc
γx ∈ ΓX ,
L. Bogachev, A. Daletskii / Journal of Functional Analysis 256 (2009) 432–478
443
where the disjoint union signifies that multiplicities of points should be taken into account. More precisely, assuming that the family of secondary processes γx (·) is measurable as a function of x ∈ X, the integer-valued measure corresponding to a CPP realization γ is given by γ (B) = X
γx (B) γc (dx) =
γx (B) =
x∈γc
δy (B),
B ∈ B(X).
(2.13)
x∈γc y∈γx
A tractable model of such a kind is obtained when (i) X is a linear space so that translations X y → y + x ∈ X are defined, and (ii) random clusters are independent and identically distributed (i.i.d.), being governed by the same probability law translated to the cluster centres, μx (A) = μ0 (A − x),
A ∈ B(ΓX ).
(2.14)
From now on, we make both of these assumptions. Remark 2.8. Unlike the standard theory of CPPs whose sample configurations are presumed to be a.s. locally finite (see, e.g., [16, Definition 6.3.I]), the description of the CPP given above only implies that its configurations γ are countable aggregates in X, but possibly with multiple and/or accumulation points, even if the background point process γc is proper. Therefore, the distribution μ of the CPP (2.13) is a probability measure defined on the space ΓX of generalized configurations. It is a matter of interest to obtain conditions in order that μ be actually supported on the proper configuration space ΓX , and we shall address this issue in Section 2.4 below in the case of Poisson CPPs. Let νx := γx (X) be the total (random) number of points in a cluster γx centred at point x ∈ X (referred to as the cluster size). According to our assumptions, the random variables νx are i.i.d. for different x, with common distribution pn := μ0 {ν0 = n} (n ∈ Z+ )
(2.15)
(so in principle the event {ν0 = ∞} may have a positive probability, p∞ 0). Remark 2.9. One might argue that allowing for vacuous clusters (i.e., with νx = 0) is superfluous since these are not visible in a sample configuration, and in particular the probability p0 cannot be estimated statistically [16, Corollary 6.3.VI]. In fact, the possibility of vacuous cluster may be ruled out without loss of generality, at the expense of rescaling the background intensity measure, λ → (1 − p0 )λ. However, we keep this possibility in our model in order to provide a suitable framework for evolutionary cluster point processes with annihilation and creation of particles, which we intend to study elsewhere. The following fact is well known in the case of CPPs without accumulation points (see, e.g., [16, §6.3]).
Proposition 2.5. The Laplace functional Lμ [·] of the probability measure μ on ΓX corresponding to the CPP (2.13) is given, for all functions f ∈ M+ (X), by
444
L. Bogachev, A. Daletskii / Journal of Functional Analysis 256 (2009) 432–478
Lμ [f ] = Lμc − ln Lμx [f ] = Lμc − ln Lμ0 f (· + x) ,
(2.16)
where Lμc acts in variable x. Proof. The representation (2.13) of cluster configurations γ implies that f, γ =
f (z) =
f (y).
x∈γc y∈γx
z∈γ
Conditioning on the background configuration γc and using the independence of the clusters γx for different x, we obtain e ΓX
−f,γ
μ(dγ ) =
ΓX
=
x∈γc
e
−
y∈γx
f (y)
μx dγx
μc (dγc )
ΓX
exp ln Lμx [f ] μc (dγc ) = Lμc − ln Lμx [f ] , x∈γc
ΓX
which proves the first formula in (2.16). The second one easily follows by shifting the measure μx to the origin using (2.14). 2 In this paper, we are mostly concerned with the Poisson CPPs, which are specified by assuming that μc is a Poisson measure on configurations, with some intensity measure λ. The corresponding probability measure on the configuration space ΓX will be denoted by μcl and called the Poisson cluster measure. The combination of (2.9) and (2.16) yields a formula for the Laplace functional of the measure μcl .
Proposition 2.6. The Laplace functional Lμcl [f ] of the Poisson cluster measure μcl on ΓX is given, for all f ∈ M+ (X), by
− f (y+x) 1 − e y∈γ0 μ0 dγ0 λ(dx) . (2.17) Lμcl [f ] = exp − X
ΓX
According to the convention made in Section 2.1 (see after Eq. (2.3)), if γ0 = ∅ then the function under the internal integral in (2.17) vanishes, so the integral over ΓX is reduced to that over the subset {γ0 ∈ ΓX : γ0 = ∅}. 2.4. Criteria of local finiteness and simplicity In this section, we give criteria for the Poisson CPP to be locally finite and simple. As mentioned in Section 1, these results appear to be new (e.g., a general criterion of local finiteness in [16, Lemma 6.3.II and Proposition 6.3.III] is merely a more formal rewording of the finiteness condition).
L. Bogachev, A. Daletskii / Journal of Functional Analysis 256 (2009) 432–478
445
For a given set B ∈ B(X) and each in-cluster configuration γ0 centred at the origin, consider the set (referred to as droplet cluster)
DB γ0 := (B − y), (2.18) y∈γ0
which is a set-theoretic union of “droplets” of shape B shifted to the centrally reflected points of γ0 .
Theorem 2.7. Let μcl be a Poisson cluster measure on the generalized configuration space ΓX .
(a) In order that μcl -a.a. configurations γ ∈ ΓX be locally finite, it is necessary and sufficient that the following two conditions hold: (a-i) in-cluster configurations γ0 are a.s. locally finite, that is, for any compact set K ∈ B(X), γ0 (K) < ∞ (μ0 -a.s.),
(2.19)
(a-ii) for any compact set K ∈ B(X), the mean λ-measure of the droplet cluster DK (γ0 ) is finite,
(2.20) λ DK γ0 μ0 dγ0 < ∞.
ΓX
(b) In order that μcl -a.a. configurations γ ∈ ΓX be simple, it is necessary and sufficient that the following two conditions hold: (b-i) in-cluster configurations γ0 are a.s. simple, sup γ0 {x} 1 (μ0 -a.s.),
(2.21)
x∈X
(b-ii) for any x ∈ X, the “point” droplet cluster D{x} (γ0 ) has a.s. zero λ-measure,
λ D{x} γ0 = 0 (μ0 -a.s.).
(2.22)
The proof of Theorem 2.7 is deferred to Appendix A (Section A.1). Let us discuss the conditions of properness. First of all, the interesting question is whether the local finiteness of the Poisson CPP is compatible with the possibility that the number of points in a cluster, ν0 = γ0 (X), is infinite (see (2.15)). The next proposition describes a simple situation where this is not the case. Proposition 2.8. Let both conditions (a-i) and (a-ii) be satisfied, and suppose that for any compact set K ∈ B(X), the λ-measure of its translations is uniformly bounded from below, cK := inf λ(K + x) > 0. x∈X
Then ν0 < ∞ (μ0 -a.s.).
(2.23)
446
L. Bogachev, A. Daletskii / Journal of Functional Analysis 256 (2009) 432–478
Proof. Suppose that γ0 is an infinite configuration. Due to (a-i), γ0 must be locally finite (μ0 a.s.), which implies that there is an infinite subset of points yj ∈ γ0 such that the sets K − yj are disjoint (j ∈ N). Hence, using (2.23) we get ∞
λ DK γ0 λ(K − yj ) = ∞, j =1
which, according to condition (a-ii), may occur only with zero probability.
2
On the other hand, it is easy to construct examples of locally finite Poisson CPPs with a.s.infinite clusters. Example 2.1. Let X = Rd and choose a measure λ such that, for any compact set K ⊂ Rd , λ(K − x) ∼ Cd λ(K)|x|−α as x → ∞, where α > 0 (e.g., take λ(dx) = (1 + |x|)−α−d+1 dx). Suppose now that the in-cluster configurations γ0 = {xn , n ∈ N} are such that n2/α < |xn | (n + 1)2/α , n ∈ N (μ0 -a.s.). Then for any compact set K
λ(K − xn ) < ∞, λ DK γ0 xn ∈γ0
because λ(K − xn ) ∼ Cd λ(K)|xn |−α = O(n−2 ) as n → ∞. It is easy to give conditions sufficient for (a-ii). The first set of conditions below is expressed in terms of the intensity measure λ and the mean number of points in a cluster, while the second condition focuses on the location of in-cluster points. Proposition 2.9. Suppose that ν0 < ∞ (μ0 -a.s.). Then either of the following conditions is sufficient for condition (a-ii) in Theorem 2.7. (a-ii ) For any compact set K ∈ B(X), the λ-measure of its translations is uniformly bounded from above, CK := sup λ(K + x) < ∞,
(2.24)
x∈X
and, moreover, the mean number of in-cluster points is finite,
ΓX
γ0 (X) μ0 dγ0 = npn < ∞
(2.25)
n∈Z+
(this necessarily implies that p∞ = 0). (a-ii ) In-cluster configuration γ0 as a set in X is μ0 -a.s. bounded, that is, there exists a compact set K0 ∈ B(X) such that γ0 ⊂ K0 (μ0 -a.s.).
L. Bogachev, A. Daletskii / Journal of Functional Analysis 256 (2009) 432–478
447
Proof. From (2.18) and (2.24) we obtain
λ(K − y) CK γ0 (X) = CK ν0 , λ DK γ0 y∈γ0
and condition (a-ii) follows by (2.25),
λ DK γ0 μ0 dγ0 CK γ0 (X) μ0 dγ0 < ∞.
ΓX
ΓX
If condition (a-ii ) holds then
(K − y) =: K − K0 , DK γ0 ⊂ y∈K0
where the set K − K0 is compact. Therefore,
λ DK γ0 μ0 dγ0 λ(K − K0 ) μ0 dγ0 = λ(K − K0 ) < ∞,
ΓX
ΓX
and condition (a-ii) follows.
2
The impact of conditions (a-ii ) and (a-ii ) is clear: (a-ii ) imposes a bound on the number of points which can be contributed from remote clusters, while (a-ii ) restricts the range of such contribution. Similarly, one can work out simple sufficient conditions for (b-ii). The first condition below is set in terms of the measure λ, whereas the second one exploits the in-cluster distribution μ0 . Proposition 2.10. Suppose that ν0 < ∞ (μ0 -a.s.). Then either of the following conditions is sufficient for condition (b-ii) of Theorem 2.7. (b-ii ) The measure λ is non-atomic, that is, λ{x} = 0 for each x ∈ X. (b-ii ) In-cluster configurations γ0 have no fixed points (μ0 -a.s.), that is, μ0 {γ0 ∈ ΓX : x ∈ γ0 } = 0 for each x ∈ X. Proof. Condition (b-ii ) readily implies (b-ii):
0 λ D{x} γ0 λ{x − y} = 0. y∈γ0
Further, if condition (b-ii ) holds then
ΓX
λ D{x} γ0 μ0 dγ0 =
1∪y∈γ {x−y} (z) μ0 dγ0 λ(dz) 0
X
ΓX
448
L. Bogachev, A. Daletskii / Journal of Functional Analysis 256 (2009) 432–478
=
1γ (z − x) μ0 dγ0 λ(dz) 0
X
=
ΓX
μ0 γ0 ∈ ΓX : z − x ∈ γ0 λ(dz) = 0,
(2.26)
X
and condition (b-ii) follows.
2
3. Poisson cluster processes via Poisson measures In this section, we construct an auxiliary Poisson measure πλ on the “vector” configuration space X and prove that the Poisson cluster measure μcl coincides with the projection of πλ onto the configuration space ΓX (Theorem 3.6). This furnishes a useful description of Poisson cluster measures that will enable us to apply to their study the well-developed calculus on Poisson configuration spaces. 3.1. An auxiliary intensity measure λ Recall that the space X = n∈Z+ X n of finite or infinite vectors x¯ = (x1 , x2 , . . .) was introduced in Section 2.1 The probability distribution μ0 of a generic cluster γ0 centred at the origin (see Section 2.3) determines a probability measure η in X which is symmetric with respect to permutations of coordinates. Conversely, μ0 is a push-forward of the measure η under the projection mapping p : X → ΓX defined by (2.2), that is, μ0 = p∗ η ≡ η ◦ p−1 .
(3.1)
Conditional measure induced by η on the space X n via the condition γ0 (X) = n will be denoted ηn (n ∈ Z+ ); in particular, η0 = δ{∅} . Hence (recall (2.15)), η(B) =
pn η n B ∩ X n ,
B ∈ B(X).
(3.2)
n∈Z+
Note that if pn = η{γ0 (X) = n} = 0 then ηn is not well defined; however, this is immaterial since the corresponding term vanishes from the sum (3.2) (cf. also the decomposition (3.5) below). The following definition is fundamental for our construction. Definition 3.1. We introduce the measure λ on X as a special “convolution” of the measures η and λ: λ (B) :=
η(B − x) λ(dx), X
B ∈ B(X);
(3.3)
L. Bogachev, A. Daletskii / Journal of Functional Analysis 256 (2009) 432–478
449
equivalently, if M+ (X) is the set of all non-negative measurable functions on X then, for any f ∈ M+ (X),
f (y) ¯ λ (dy) ¯ =
X
X
f (y¯ + x) η(dy) ¯ λ(dx).
(3.4)
X
Here and below, we use the shift notation y¯ + x := (y1 + x, y2 + x, . . .),
y¯ = (y1 , y2 , . . .) ∈ X, x ∈ X.
Using the decomposition (3.2), the measure λ on X can be represented as a weighted sum of contributions from the constituent spaces X n : λ (B) =
pn λn B ∩ X n ,
B ∈ B(X),
(3.5)
Bn ∈ B(X n ).
(3.6)
n∈Z+
where, for each n ∈ Z+ , λn (Bn ) :=
ηn (Bn − x) λ(dx), X
Remark 3.1 (Case n = 0). Recall that X 0 = {∅} and B(X 0 ) = {∅, X 0 } = {∅, {∅}}. Since ∅ − x = ∅, {∅} − x = {∅} (x ∈ X) and η0 = δ{∅} , formula (3.6) for n = 0 must be interpreted as follows: λ0 (∅) =
η0 (∅) λ(dx) = 0, X
λ0 {∅} =
η0 {∅} λ(dx) =
X
λ(dx) = λ(X) = ∞.
(3.7)
X
If p∞ = 0 (i.e., clusters are a.s. finite) and X = Rd , then in order that the measure ∞η be ab(d y) ¯ ⊕ solutely continuous (a.c.) with respect to the “Lebesgue measure” d y ¯ = δ {∅} n=1 dy1 ⊗ n · · · ⊗ dyn on X = ∞ n=0 X , with some density h, η(dy) ¯ = h(y) ¯ dy, ¯
y¯ ∈ X,
(3.8)
it is necessary and sufficient that each measure ηn is a.c. with respect to Lebesgue measure on ¯ = hn (y) ¯ dy, ¯ y¯ ∈ X n (n ∈ Z+ ); in this case, the density h is decomposed as X n , that is, ηn (dy) h(y) ¯ =
∞
pn hn (y)1 ¯ Xn (y), ¯
y¯ ∈ X.
(3.9)
n=0
Moreover, it follows that the measures λ and λn (n ∈ Z+ ) are also a.c., with the corresponding densities
450
L. Bogachev, A. Daletskii / Journal of Functional Analysis 256 (2009) 432–478
s(y) ¯ = sn (y) ¯ =
¯ λ (dy) = dy¯ ¯ λn (dy) dy¯
h(y¯ − x) λ(dx),
y¯ ∈ X,
X
=
hn (y¯ − x) λ(dx),
y¯ ∈ X n ,
(3.10)
X
related by the equation (cf. (3.5), (3.9)) s(y) ¯ =
∞
pn sn (y)1 ¯ Xn (y), ¯
y¯ ∈ X.
(3.11)
n=0
Remark 3.2. In the case n = 1, the definition (3.6) is reduced to λ1 (B1 ) = η1 (B1 − x) λ(dx) = λ(B1 − x) η1 (dx), X
B1 ∈ B(X).
(3.12)
X
In particular, if λ is translation invariant (i.e., λ(B1 − x) = λ(B1 ) for each B1 ∈ B(X) and any x ∈ X), then λ1 coincides with λ. Remark 3.3. There is a possibility that the measure λn defined by (3.6) is not σ -finite (even if λ is), and moreover, λn may appear to be locally infinite, in that λn (B) = ∞ for any compact set B ⊂ Rn with non-empty interior, as in the following example. Example 3.1. Let X = R, and for n 1 set λ(dx) := e|x| dx,
η1 (dx) :=
|x| dx (x 2 + 1)2
(x ∈ R),
and ηn (dx) ¯ := η1 (dx1 ) ⊗ · · · ⊗ η1 (dxn ), x¯ = (x1 , . . . , xn ) ∈ Rn . Note that for a < b and any x∈ / [a, b], η1 [a − x, b − x] =
b−a (b − a)|a + b − 2x| ∼ 2 2 2((a − x) + 1)((b − x) + 1) |x|3
(x → ∞),
so, for any rectangle B = Xni=1 [ai , bi ] ⊂ Rn (ai < bi ), by (3.12) we obtain λ1 (B) =
∞ n
η1 [ai − x, bi − x] e|x| dx = ∞.
−∞ i=1
The next example illustrates a non-pathological situation. Example 3.2. Let X = R, and for n 1 set ¯ = hn (y)
1 ¯ 2 /2 e−y , n/2 (2π)
y¯ = (y1 , . . . , yn ) ∈ Rn ,
L. Bogachev, A. Daletskii / Journal of Functional Analysis 256 (2009) 432–478
451
where · is the usual Euclidean norm in Rn . Thus, ηn is a standard Gaussian measure on Rn . Assume that λ is the Lebesgue measure on R, λ(dx) = dx. For n = 1, from Eq. (3.10) we obtain 1 s1 (y) = √ 2π
∞
e−(y−x)
2 /2
dx = 1,
−∞
hence λ1 = λ, in accord with Remark 3.2. If n = 2 then from (3.10) we get 1 s2 (y1 , y2 ) = 2π
∞
e−((y1 −x)
2 +(y −x)2 )/2 2
−∞
1 2 dx = √ e−(y1 −y2 ) /4 . 2 π
√ √ Via the orthogonal transformation z1 = (y1 + y2 )/ 2, z2 = (y1 − y2 )/ 2, the measure λ2 is reduced to 1 2 λ2 (dz1 , dz2 ) = √ e−z2 /2 dz1 dz2 , 2 π which is a product of the√ standard Gaussian measure (along the coordinate axis z1 ) and the scaled Lebesgue measure dz2 / 2. Note that λ2 (R2 ) = ∞, but any vertical or horizontal strip of finite width (in coordinates y) ¯ has finite λ2 -measure. In general (n 2), integration in (3.10) yields 1 1 2 −1 2 y ¯ , ¯ = √ − n |y + · · · + y | sn (y) exp − √ 1 n 2 ( 2π )n−1 n
y¯ ∈ Rn .
It is easy to check that after an orthogonal transformation z¯ = yU ¯ such that z1 = n−1/2 (y1 + · · · + yn ), the measure λn takes the form dz1 1 2 2 e−(z2 +···+zn )/2 dz2 . . . dzn , λn (d¯z) = √ · √ n−1 n ( 2π )
z¯ = (z1 , . . . , zn ).
√ That is, λn (d¯z) is a product of the scaled Lebesgue measure dz1 / n and the standard Gaussian measure in coordinates z2 , . . . , zn . Hence λn (Rn ) = ∞, but for any coordinate strip Ci = {y¯ ∈ Rn : |yi | c} we have λn (Ci ) < ∞. Example 3.2 can be generalized as follows. Proposition 3.1. Suppose that p∞ = 0 and X = Rd . For each n 1, consider an orthogonal linear transformation z¯ = yU ¯ n of the space X n such that z1 =
y1 + · · · + yn , √ n
z¯ = (z1 , . . . , zn ),
Set z¯ := (z2 , . . . , zn ) and consider the measures
y¯ = (y1 , . . . , yn ).
(3.13)
452
L. Bogachev, A. Daletskii / Journal of Functional Analysis 256 (2009) 432–478
ηn (B ) :=
ηn dz1 , B = ηn (X × B ),
B ∈ B X n−1 ,
(3.14)
X
λ˜ n (B1 |¯z ) :=
B1 − z1 ηn dz1 |¯z , λ √ n
B1 ∈ B(X),
(3.15)
X
where ηn (dz1 |¯z ) is the measure on X obtained from ηn via conditioning on z¯ . Then the measure λ can be decomposed as λ (d¯z) = p0 λ0 (d¯z) +
∞
pn λ˜ n (dz1 |¯z ) ηn d¯z ,
(3.16)
n=1
where λ0 is defined in (3.7). In particular, if the measure λ on X = Rd is translation invariant then λ
(d¯z) = p0 λ0 (d¯z) +
∞ n=1
pn
λ(dz1 ) η (d¯z ). nd/2 n
(3.17)
Proof. For a fixed n 1, let z¯ = yU ¯ n and consider a Borel set in X n of the form Bn = {y¯ ∈ X n : z1 ∈ B1√ , z¯ ∈ Bn }. By Eq. (3.13) and orthogonality of Un , we have Bn − x = {¯z ∈ X n : z1 ∈ B1 − x n, z¯ ∈ Bn }. Therefore, from (3.6) we obtain λn (Bn ) =
1(B1 −x √n )×Bn (¯z) ηn (d¯z) λ(dx)
Xn
X
=
1B1 −x
Xn
X
= X×X n−1
= Bn
=
√
z ) ηn (d¯z) n (z1 ) λ(dx) 1Bn (¯
1(B1 −z1 )/√n (x) λ(dx) 1Bn (¯z ) ηn dz1 |¯z ηn (d¯z )
X
√ λ (B1 − z1 )/ n ηn dz1 |¯z ηn (d¯z )
X
λ˜ n B1 |¯z ηn (d¯z ),
Bn
and by inserting this √ into Eq. (3.5) we get (3.16). Finally, the translation invariance of λ implies that λ((B1 − z1 )/ n ) = n−d/2 λ(B1 ). Formula (3.15) then gives λ˜ n (B1 |¯z ) = n−d/2 λ(B1 ), and (3.17) readily follows from (3.16). 2 Using decomposition (3.16), it is easy to obtain the following criterion of absolute continuity of the measure λ .
L. Bogachev, A. Daletskii / Journal of Functional Analysis 256 (2009) 432–478
453
d . Then the measure λ (dx) Corollary 3.2. Suppose that p∞ = 0 and X = R ¯ on X is a.c. with respect to the Lebesgue measure dx¯ = δ{∅} (dx) ¯ ⊕ ∞ dx ⊗ · · · ⊗ dx if and only if the following 1 n n=1 two conditions hold:
(i) for each n 1, the measure ηn (d¯z ) is a.c. with respect to the Lebesgue measure d¯z on X n−1 ; (ii) for a.a. z¯ , the measure λ˜ n (dz1 |¯z ) is a.c. with respect to the Lebesgue measure dz1 on X. In particular, if λ is translation invariant then condition (ii) is automatically fulfilled and hence condition (i) alone is necessary and sufficient for the absolute continuity of λ . Remark 3.4. The absolute continuity of η is sufficient (cf. (3.8), (3.10)), but not necessary, for condition (i). This is illustrated by the following example: 1 1 η(dy1 , dy2 ) = δ{1} (dy1 ) f (y2 ) dy2 + δ{1} (dy2 ) f (y1 ) dy1 , 2 2
(y1 , y2 ) ∈ R2 ,
where f (y) (y ∈ R) is some probability density function. Then the projection measure η on R (see (3.14)) is given by √ √ √ 2 f (1 − 2z ) + f (1 + 2z ) dz , η (dz ) = 2
z =
y1 − y2 √ , 2
and so η (dz ) is absolutely continuous. The next result shows that the absolute continuity of λ implies that the Poisson cluster process a.s. has no multiple points (see Definition 2.2). Proposition 3.3. Suppose that p∞ = 0, X = Rd , and the measure λ (dx) ¯ on X is a.c. with respect to the Lebesgue measure dx. ¯ Then μcl -a.a. configurations γ ∈ ΓX are simple. Proof. By Theorem 2.7, it suffices to check conditions (b-i) and (b-ii). First, note that if condition (b-i) is not satisfied (i.e., if the set of points y¯ ∈ X with two or more coinciding coordinates has positive η-measure), than the projected measure η (d¯z ) charges a hyperplane (of codimension 1) in the space X spanned over the coordinates z¯ . But this contradicts the absolute continuity of λ , since such hyperplanes have zero Lebesgue measure. Furthermore, similarly to (2.26) and using the definition (3.3), for each x ∈ X we obtain λ X
yi ∈y¯
{x − yi } η(dy) ¯ = η{y¯ ∈ X: z − x ∈ y} ¯ λ(dz) X
= λ y¯ ∈ X: −x ∈ p(y) ¯ = 0,
by the absolute continuity of λ . Hence, λ( yi ∈y¯ {x − yi }) = 0 (η-a.s.) and condition (b-ii) follows. 2
454
L. Bogachev, A. Daletskii / Journal of Functional Analysis 256 (2009) 432–478
3.2. An auxiliary Poisson measure πλ
Recall that the “unpacking” map p : X → ΓX is defined in (2.2). For any Borel subset B ∈ B(X), denote
¯ ∩ B = ∅ ∈ B ΓX . XB := x¯ ∈ X: p(x)
(3.18)
The following result is crucial for our purposes (cf. Example 3.2). Proposition 3.4. Let B ∈ B(X) be a set such that λ(B) < ∞. Then condition (2.20) of Theorem 2.7(a) (i.e., that the mean λ-measure of the droplet cluster DB is finite) is necessary and sufficient in order that λ (XB ) < ∞, or equivalently, γ¯ (XB ) < ∞ for πλ -a.a. γ¯ ∈ ΓX . Proof. Using (3.3) we obtain
λ (XB ) =
η(XB − x) λ(dx) = X
λ (XB ) =
X
= X
(3.19)
X
X
By definition (3.18), y¯ + x ∈ XB if and only if x ∈ (3.19) can be rewritten as
1XB (y¯ + x) λ(dx) η(dy). ¯
yi ∈y¯ (B
− yi ) ≡ DB (y) ¯ (see (2.18)). Hence,
1DB (y) ¯ ¯ (x) λ(dx) η(dy)
X
λ DB (y) ¯ η(dy) ¯ =
λ DB γ0 μ0 dγ0 ,
ΓX
by the change of measure (3.1). Thus, the bound λ (XB ) < ∞ is nothing else but condition (2.20) applied to B. The second part follows by Proposition 2.4(a). 2
Let us consider the cluster configuration space ΓX over the space X with generic elements γ¯ ∈ ΓX . Our next goal is to define a Poisson measure πλ on ΓX with intensity λ . However, as Remark 3.3 and Example 3.1 indicate, the measure λ may not be σ -finite, in which case a general construction of the Poisson measure as developed in Section 2.2 would not be applicable. It turns out that Proposition 3.4 provides a suitable basis for a good theory. Proposition 3.5. Suppose that condition (2.20) of Theorem 2.7(a) is fulfilled for any set B ∈ B(X) such that λ(B) < ∞. Then the measure λ on X is σ -finite. Proof. Since the measure λ on X is σ -finite, there is a sequence of sets Bk ∈ B(X) (k ∈ N) such that λ(Bk ) < ∞ and ∞ k=1 Bk = X. Hence, by Proposition 3.4, λ (XBk ) < ∞ for each Bk , and from the definition (3.18) it is clear that ∞ k=1 XBk = X. 2 By virtue of Proposition 3.5 and according to the discussion in Section 2.2, the Poisson mea sure πλ on the configuration space ΓX does exist. Moreover, due to Remark 2.6, this is true
L. Bogachev, A. Daletskii / Journal of Functional Analysis 256 (2009) 432–478
455
even without any extra topological assumptions, except that of σ -finiteness of the basic intensity measure of πλ may be elaborated further by applying Proposition 2.2 to λ. The construction X = n∈Z+ X n and λ = n∈Z+ pn λn ; namely, one first defines the Poisson measures πpn λn on
the constituent configuration spaces ΓXn (of course, the measures λn are σ -finite together with λ ) and then constructs the Poisson measure πλ on ΓX = Xn∈Z+ ΓXn as a product measure, πλ = n∈Z+ πpn λn .
Remark 3.5. A degenerate Poisson measure πp0 λ0 on ΓX0 is defined as πp0 λ0 := δ{γ¯∞ } , where γ¯∞ = ({∅}, {∅}, . . .), i.e., γ¯∞ (X 0 ) = ∞. The component πp0 λ0 is actually irrelevant in the projection construction described in the next section. 3.3. Poisson cluster measure via the Poisson measure πλ
We can lift the projection mapping (2.2) to the configuration space ΓX by setting
ΓX γ¯ → p(γ¯ ) :=
p(x) ¯ ∈ ΓX .
(3.20)
x∈ ¯ γ¯
Disjoint union in (3.20) highlights the fact that p(γ¯ ) may have multiple points, even if γ¯ is proper. It is not difficult to see that (3.20) is a measurable mapping. Indeed, using the sets DBn introduced in (2.4), for any cylinder set CBn ⊂ ΓX (B ∈ B(X), n ∈ Z+ ) we have p−1 (CBn ) = AnB ∈ B(ΓX ), where, for instance,
A0B = γ¯ ∈ ΓX : γ¯ X \ DB0 = 0 ,
A1B = γ¯ ∈ ΓX : γ¯ DB1 = 1 ,
A2B = γ¯ ∈ ΓX : γ¯ DB2 = 1 or γ¯ DB1 = 2 , k and, more generally, AnB = (nk ) ∞ k=1 {γ¯ ∈ ΓX : γ¯ (DB ) = nk }, where the union is taken over integer arrays (nk ) = (n1 , n2 , . . .) such that nk > 0 and k knk = n. Finally, we introduce the measure μ on ΓX as a push-forward of the Poisson measure πλ under the mapping p,
μ(A) := (p∗ πλ )(A) ≡ πλ p−1 (A) ,
A ∈ B ΓX .
(3.21)
The next theorem is the main result of this section. Theorem 3.6. The measure μ = p∗ πλ on ΓX defined by (3.21) coincides with the Poisson cluster measure μcl .
Proof. According to Section 2.1, it is sufficient to compute the Laplace functional of the measure μ. For any f ∈ M+ (X), by the change of measure (3.21) we have ˜ e−f,γ μ(dγ ) = e−f,p(γ¯ ) πλ (dγ¯ ) = e−f ,γ¯ πλ (dγ¯ ), (3.22)
ΓX
ΓX
ΓX
456
L. Bogachev, A. Daletskii / Journal of Functional Analysis 256 (2009) 432–478
where f˜(y) ¯ := takes the form
yi ∈y¯
f (yi ) ∈ M+ (X). According to (2.9) and (3.4), the right-hand side of (3.22)
−f˜(y) ¯ −f˜(y+x) ¯ 1−e λ (dy) 1−e η(dy) ¯ λ(dx) ¯ = exp − exp − XX
X
− y ∈y¯ f (yi +x) i 1−e η(dy) ¯ λ(dx) , = exp − X
X
which, after the change of measure (3.1), coincides with the expression (2.17) for the Laplace functional of the Poisson cluster measure μcl . 2 Remark 3.6. As an elegant application of the technique developed here, let us give a transparent proof of Theorem 2.7(a) (cf. Appendix A.1). Indeed, in order that a given compact set K ⊂ X contain finitely many points of configuration γ = p(γ¯ ), it is necessary and sufficient that (i) each cluster “point” x¯ ∈ γ¯ is locally finite, which is equivalent to the condition (a-i), and (ii) there are finitely many points x¯ ∈ γ¯ which contribute to the set K under the mapping p, the latter being equivalent to condition (a-ii) by Proposition 3.4. 3.4. An alternative construction of the measures πλ and μcl The measure πλ was introduced in the previous section as a Poisson measure on the configuration space ΓX with a certain intensity measure λ prescribed ad hoc by Eq. (3.3). In this section, we show that πλ can be obtained in a more natural way as a suitable skew projection of a canonical Poisson measure πˆ defined on a bigger configuration space ΓX×X , with the product intensity measure λ ⊗ η. More specifically, given a Poisson measure πλ in ΓX , let us construct a new measure μˆ in ΓX×X as the probability distribution of random configurations γˆ ∈ ΓX×X obtained from Poisson
configurations γ ∈ ΓX by the rule γ → γˆ := (x, y¯x ): x ∈ γ , y¯x ∈ X ,
(3.23)
where the random vectors {y¯x } are i.i.d., with common distribution η(dy). ¯ Geometrically, such a construction may be viewed as pointwise i.i.d. translations of the Poisson configuration γ ∈ X into the space X × X, X x ↔ (x, 0) → (x, y¯x ) ∈ X × X. Remark 3.7. Vector y¯x in each pair (x, y¯x ) ∈ X × X can be interpreted as a mark attached to the point x ∈ X, so that γˆ becomes a marked configuration, with the mark space X (see [16,24]).
Theorem 3.7. The probability distribution μˆ of random configurations γˆ ∈ ΓX×X constructed
in (3.23) is given by the Poisson measure πλˆ on the configuration space ΓX×X , with the product intensity measure λˆ := λ ⊗ η.
L. Bogachev, A. Daletskii / Journal of Functional Analysis 256 (2009) 432–478
457
Proof. Let us check that, for any non-negative measurable function f (x, y) ¯ on X × X, the Laplace functional of the measure μˆ is given by formula (2.9). Using independence of the vectors y¯x corresponding to different x, we obtain
ΓX×X
e−f,γˆ μ(d ˆ γˆ ) =
ΓX
x∈γ
¯ e−f (x,y) η(dy) ¯ πλ (dγ )
X
¯ 1 − e−f (x,y) = exp − η(dy) ¯ λ(dx) X
X
¯ 1 − e−f (x,y) λ(dx) η(dy) ¯ = exp − X X
−f (x,y) ¯ ˆ 1−e λ(dx, dy) ¯ = exp − =
X×X
e−f,γˆ πλˆ (dγˆ ),
ΓX×X
where we have applied formula (2.9) for the Laplace functional of the Poisson measure πλ with ¯ η(dy)) the function f˜(x) = − ln( X e−f (x,y) ¯ ∈ M+ (X). 2 Remark 3.8. The measure μ, ˆ originally defined on configurations γˆ of the form (3.23), naturally extends to a probability measure on the entire space ΓX×X . Remark 3.9. Theorem 3.7 can be regarded as a generalization of the well-known invariance property of Poisson measures under random i.i.d. translations (see, e.g., [15,16,22]). A novel element here is that starting from a Poisson point process in X, random translations create a new (Poisson) point process in a bigger space, X × X, with the product intensity measure. On the other hand, note that the pointwise coordinate projection X × X (x, y¯x ) → x ∈ X recovers the original Poisson measure πλ , in accord with the Mapping Theorem (see Proposition 2.3). Therefore, Theorem 3.7 provides a converse counterpart to the Mapping Theorem. To the best of our knowledge, these interesting properties of Poisson measures have not been pointed out in the literature so far. Theorem 3.7 can be easily extended to more general (skew) translations.
Theorem 3.8. Suppose that random configurations γˆ+ ∈ ΓX×X are obtained from Poisson con ΓX
by pointwise translations x → (x, y¯x + x), where y¯x ∈ X (x ∈ X) are i.i.d. figurations γ ∈ with common distribution η(dy). ¯ Then the corresponding probability measure μˆ + on ΓX×X coincides with the Poisson measure of intensity λˆ + (dx, dy) ¯ := λ(dx) η(dy¯ − x).
(3.24)
458
L. Bogachev, A. Daletskii / Journal of Functional Analysis 256 (2009) 432–478
Corollary 3.9. Under the pointwise projection (x, y) ¯ → y¯ applied to configurations γˆ+ ∈ ΓX×X ,
the Poisson measure μˆ + of Theorem 3.8 is pushed forward to the Poisson measure πλ on ΓX with intensity measure λ defined in (3.3). Proof. By the Mapping Theorem (see Proposition 2.3), the image of the measure μˆ + under the projection (x, y¯ + x) → y¯ + x is a Poisson measure with intensity given by the push-forward of the measure (3.24), that is, λˆ + (dx, B) = η(B − x) λ(dx) = λ (B), B ∈ B(X), X
X
according to the definition (3.3).
2
Remark 3.10. According to Corollary 3.9, σ -finiteness of the intensity measure λ (see Proposition 3.5) is not necessary for the existence of the Poisson measure πλ . Finally, combining Theorems 3.7, 3.8 and Corollary 3.9 with Theorem 3.6, we arrive at the following result. Theorem 3.10. Suppose that all the conditions of Theorems 3.7 and 3.8 are fulfilled. Then, under the composition mapping p˜ : (x, y) ¯ → (x, y¯ + x) → y¯ + x → p(y¯ + x),
the Poisson measure πλˆ constructed in Theorem 3.7 is pushed forward from the space ΓX×X
directly to the space ΓX where it coincides with the prescribed Poisson cluster measure μcl ,
∗ p˜ πλˆ (A) ≡ πλˆ p˜ −1 (A) = μcl (A),
A ∈ B ΓX .
Remark 3.11. The construction used in Theorem 3.10 may prove instrumental for more complex (e.g., Gibbs) cluster processes, as it enables one to avoid the intermediate space ΓX where the push-forward measure (analogous to πλ ) may have no explicit description. 4. Quasi-invariance and integration by parts From now on, we restrict ourselves to the case where X = Rd . We shall assume throughout that conditions (a-i) and (a-ii) of Theorem 2.7 are fulfilled, so that μcl -a.a. configurations γ ∈ ΓX are locally finite. Furthermore, all clusters are assumed to be a.s. finite, hencep∞ ≡ μ0 {ν0 = ∞} = 0 and the component X ∞ may be dropped from the disjoint union X = n X n . We shall also require the absolute continuity of the measure λ (see the corresponding necessary and sufficient conditions in Corollary 3.2). By Proposition 3.3, this implies that configurations γ are μcl -a.s. simple (i.e., have no multiple points). In particular, these assumptions ensure that μcl -a.a. configurations γ belong to the proper configuration space ΓX . Under these conditions, in this section we prove the quasi-invariance of the measure μcl with respect to the action of compactly supported diffeomorphisms of X and establish an integrationby-parts formula. We begin with a brief description of some convenient “manifold-like” concepts
L. Bogachev, A. Daletskii / Journal of Functional Analysis 256 (2009) 432–478
459
and notations first introduced in [5], which provide the suitable framework for analysis on configuration spaces. 4.1. Differentiable functions on configuration spaces Let Tx X be the tangent space of X = Rd at point x ∈ X. It can be identified in the natural way with Rd , with the corresponding (canonical) inner product denoted by a “fat” dot ·. The gradient on X is denoted by ∇. Following [5], we define the “tangent space” of the configuration space ΓX at γ ∈ ΓX as the Hilbert space Tγ ΓX := L2 (X → T X; dγ ), or equivalently Tγ ΓX = x∈γ Tx X. The scalar product in Tγ ΓX is denoted by ·,·γ . A vector field V over ΓX is a mapping ΓX γ → V (γ ) = (V (γ )x )x∈γ ∈ Tγ ΓX . Thus, for vector fields V1 , V2 over ΓX we have ! V1 (γ ), V2 (γ ) γ = V1 (γ )x · V2 (γ )x ,
γ ∈ ΓX .
x∈γ
For γ ∈ ΓX and x ∈ γ , denote by Oγ ,x an arbitrary open neighbourhood of x in X such that Oγ ,x ∩ γ = {x}. For any measurable function F : ΓX → R, define the function Fx (γ , ·) : Oγ ,x → R by Fx (γ , y) := F ((γ \ {x}) ∪ {y}), and set ∇x F (γ ) := ∇Fx (γ , y)|y=x ,
x ∈ X,
provided Fx (γ , ·) is differentiable at x. Denote by F C(ΓX ) the class of functions on ΓX of the form
F (γ ) = f φ1 , γ , . . . , φk , γ ,
γ ∈ ΓX ,
(4.1)
where k ∈ N, f ∈ Cb∞ (Rk ) (:= the set of C ∞ -functions on Rk bounded together with all their derivatives), and φ1 , . . . , φk ∈ C0∞ (X) (:= the set of C ∞ -functions on X with compact support). Each F ∈ F C(ΓX ) is local, that is, there is a compact set K ⊂ X (which may depend on F ) such that F (γ ) = F (γK ) for all γ ∈ ΓX . Thus, for a fixed γ there are only finitely many non-zero derivatives ∇x F (γ ). For a function F ∈ F C(ΓX ), its Γ -gradient ∇ Γ F is defined as follows:
∇ Γ F (γ ) := ∇x F (γ ) x∈γ ∈ Tγ ΓX ,
γ ∈ ΓX ,
(4.2)
so the directional derivative of F along a vector field V is given by ! ∇VΓ F (γ ) := ∇ Γ F (γ ), V (γ ) γ = ∇x F (γ ) · V (γ )x ,
γ ∈ ΓX .
x∈γ
Note that the sum on the right-hand side contains only finitely many non-zero terms. Further, let F V(ΓX ) be the class of cylinder vector fields V on ΓX of the form V (γ )x =
k i=1
Ai (γ )vi (x) ∈ Tx X,
x ∈ X,
(4.3)
460
L. Bogachev, A. Daletskii / Journal of Functional Analysis 256 (2009) 432–478
where Ai ∈ F C(ΓX ) and vi ∈ Vect0 (X) (:= the space of compactly supported C ∞ -smooth vector fields on X), i = 1, . . . , k (k ∈ N). Any vector filed v ∈ Vect0 (X) generates a constant vector field V on ΓX defined by V (γ )x := v(x). We shall preserve the notation v for it. Thus, ∇vΓ F (γ ) =
∇x F (γ ) · v(x),
γ ∈ ΓX .
(4.4)
x∈γ
Recall (see Proposition 2.4(a)) that if λ(Λ) < ∞ then γ (Λ) < ∞ for πλ -a.a. γ ∈ ΓX . This motivates the definition of the class F C λ (ΓX ) of functions on ΓX of the form (4.1), where φ1 , . . . , φk are C ∞ -functions with λ(supp φi ) < ∞, i = 1, . . . , k. Any function F ∈ F C λ (ΓX ) is local in the sense that there exists a set B ∈ B(X) (depending on F ) such that λ(B) < ∞ and F (γ ) = F (γB ) for all γ ∈ ΓX . As in the case of functions from F C(ΓX ), for a fixed γ there are only finitely many non-zero derivatives ∇x F (γ ). The approach based on “lifting” of the differential structure from the underlying space X ton the configuration space ΓX as described above can also be applied to the spaces X = ∞ n=0 X and ΓX . First of all, the space X is endowed with the natural differential structure inherited from the constituent spaces X n . Namely, the tangent space of X at point x¯ ∈ X is defined piecewise as Tx¯ X := Tx¯ X n for x¯ ∈ X n (n ∈ Z+ ), with the scalar product in Tx¯ X induced from the tangent spaces Tx¯ X n and again denoted by the dot ·; furthermore, for a function f : X → R its ¯ = (∇x1 f (x), ¯ . . . , ∇xn f (x)) ¯ ∈ Tx¯ X n , where ∇xi gradient ∇f acts on each space X n as ∇f (x) n is the “partial” gradient with respect to the component xi ∈ x¯ ∈ X . A vector field on X is a map X x¯ → V (x) ¯ ∈ Tx¯ X; in other words, the restriction of V to X n is a vector field on X n (n ∈ Z+ ). The derivative of a function f : X → R along a vector field V on X is then defined by ¯ := ∇f (x) ¯ · V (x) ¯ (x¯ ∈ X). ∇V f (x) The functional class C ∞ (X) is defined, as usual, as the set of C ∞ -functions f : X → R; similarly, C0∞ (X) is the subclass of C ∞ (X) consisting of functions with compact support. Since differentiability is a local property, C ∞ (X) admits a component-wise description: f ∈ C ∞ (X) if and only if for each n ∈ Z+ the restriction of f to X n is in C ∞ (X n ). However, this is not true ¯ ≡ x¯ for the class C0k (X) which, according to Remark 2.1, involves a stronger condition that f (x) (x¯ ∈ X n ) for all large enough n. Lifting of this differentiable structure from the space X to the configuration space ΓX can be done by repeating the same constructions as before with only obvious modifications, so we do not dwell on details. This way, we introduce the tangent space Tγ¯ ΓX = x∈ ¯ γ¯ Tx¯ X, vector fields V over ΓX , and differentiable functions Φ : ΓX → R. Similarly to (4.1) and (4.3) one can define the spaces F C(ΓX ), F C λ (ΓX ) and F V(ΓX ) of C ∞ -smooth local functions and vector fields on X, and we shall use these notations without further explanation. 4.2. Diff0 -quasi-invariance In this section, we discuss the property of quasi-invariance of the measure μcl with respect to diffeomorphisms of X. Let us start by describing how diffeomorphisms of X act on configuration spaces. For a measurable mapping ϕ : X → X, its support supp ϕ is defined as the smallest closed set containing all x ∈ X such that ϕ(x) = x. Let Diff0 (X) be the group of diffeomorphisms of X with compact support. For any ϕ ∈ Diff0 (X), we define the “diagonal” diffeomorphism ϕ¯ : X → X acting on each space X n (n ∈ Z+ ) as follows:
¯ x) ¯ := ϕ(x1 ), . . . , ϕ(xn ) ∈ X n . X n x¯ = (x1 , . . . , xn ) → ϕ(
L. Bogachev, A. Daletskii / Journal of Functional Analysis 256 (2009) 432–478
461
Remark 4.1. Although K := supp ϕ is compact in X, note that the support of the diffeomorphism ϕ¯ (again defined as the closure of the set {x¯ ∈ X: ϕ(x) ¯ = x}) ¯ is given by supp ϕ¯ = XK (see (3.18)) and hence is not compact in the topology of X (see Remark 2.1). However, λ (XK ) < ∞ (by Proposition 3.4), which is sufficient for our purposes. The mappings ϕ and ϕ¯ can be lifted to measurable “diagonal” transformations (denoted by the same letters) of the configuration spaces ΓX and ΓX , respectively: ΓX γ → ϕ(γ ) := ϕ(x), x ∈ γ ∈ ΓX , ΓX γ¯ → ϕ( ¯ γ¯ ) := ϕ( ¯ x), ¯ x¯ ∈ γ¯ ∈ ΓX .
(4.5a) (4.5b)
Let I : L2 (ΓX , μcl ) → L2 (ΓX , πλ ) be the isometry defined by the projection p,
(IF )(γ¯ ) := F p(γ¯ ) ,
γ¯ ∈ ΓX ,
(4.6)
and let I ∗ : L2 (ΓX , πλ ) → L2 (ΓX , μcl ) be the adjoint operator. Remark 4.2. The definition implies that I ∗ I is the identity operator in L2 (ΓX , μcl ). However, the operator II ∗ acting in the space L2 (ΓX , πλ ) is a non-trivial orthogonal projection, which plays the role of an infinite particle symmetrization operator. Unfortunately, general explicit form of the operators I ∗ and II ∗ is not known, and may be hard to obtain. By the next lemma, the action of Diff0 (X) commutes with the operators p and I. Lemma 4.1. For any ϕ ∈ Diff0 (X), we have ϕ ◦ p = p ◦ ϕ¯ and furthermore, I(F ◦ ϕ) = (IF ) ◦ ϕ¯ for any F ∈ L2 (ΓX , μcl ). Proof. The first statement follows from the definition (3.20) of the mapping p and the diagonal form of ϕ¯ (see (4.5b)). The second statement then readily follows by the definition (4.6) of the operator I. 2 Let us now consider the configuration space ΓX equipped with the Poisson measure πλ introduced in Section 3.2. As already mentioned, we assume that the intensity measure λ is a.c. with respect to the Lebesgue measure on X and, moreover, s(x) ¯ :=
¯ λ (dx) > 0 for a.a. x¯ ∈ X. dx¯
(4.7)
This implies that the measure λ is quasi-invariant with respect to the action of diagonal transformations ϕ¯ : X → X (ϕ ∈ Diff0 (X)) and the corresponding Radon–Nikodym derivative is given by ϕ¯
¯ = ρλ (x)
¯ s(ϕ¯ −1 (x)) Jϕ¯ (x) ¯ −1 s(x) ¯ ϕ¯
for a.a. x¯ ∈ X,
(4.8)
¯ = 1 if s(x) ¯ = 0 or s(ϕ¯ −1 (x)) ¯ = 0). where Jϕ¯ is the Jacobian determinant of ϕ¯ (we set ρλ (x)
462
L. Bogachev, A. Daletskii / Journal of Functional Analysis 256 (2009) 432–478
Proposition 4.2. The Poisson measure πλ is quasi-invariant with respect to the action of diagonal diffeomorphisms ϕ¯ : ΓX → ΓX (ϕ ∈ Diff0 (X)). The corresponding Radon–Nikodym density ϕ¯ Rπλ := d(ϕ¯ ∗ πλ )/dπλ is given by
ϕ¯ ϕ¯ Rπϕ¯λ (γ¯ ) = exp 1 − ρλ (x) ¯ λ (dx) ¯ · ρλ (x), ¯
γ¯ ∈ ΓX ,
(4.9)
x∈ ¯ γ¯
X ϕ¯
where ρλ is defined in (4.8). Proof. The result follows from Remark 4.1 and Proposition A.1 in Appendix A (applied to the space X with measure λ and mapping ϕ). ¯ 2 ϕ¯
ϕ¯
Remark 4.3. The function Rπλ is local in the sense that, for πλ -a.a. γ¯ ∈ ΓX , we have Rπλ (γ¯ ) = ϕ¯ Rπλ (γ¯ ∩ XK ), where K := supp ϕ. ϕ¯
Remark 4.4 (Explicit form of Rπλ ). Let the measure η(dy) ¯ be a.c. with respect to Lebesgue measure dy¯ on X, with density h(y) ¯ (see (3.8)). According to (4.8), ϕ¯ ρλ (y) ¯ =
n −1 −1 X h(ϕ (y1 ) − x, . . . , ϕ (yn ) − x) λ(dx) X h(y1
− x, . . . , yn − x) λ(dx)
Jϕ (yi )−1 ,
y¯ ∈ X n ,
i=1
" where Jϕ (y) ¯ = det(∂ϕi /∂yj ) is the Jacobian determinant of ϕ (note that Jϕ¯ (y) ¯ = ni=1 Jϕ (yi ) ϕ¯ for y¯ ∈ X n ). Then Rπλ (γ¯ )" can be calculated using formula (4.9). In particular, if clusters have i.i.d. points, so that h(y) ¯ = ni=1 h0 (yi ), then ϕ¯ ρλ (y) ¯ =
"n X
Jϕ (yi )−1 h0 (ϕ −1 (yi ) − x) λ(dx) i=1 "n , i=1 h0 (yi − x) λ(dx) X
y¯ = (y1 , . . . , yn ) ∈ X n ,
and Rπϕ¯λ (γ¯ ) = C where C := exp{
y∈ ¯ γ¯
" X
−1 −1 y∈y¯ Jϕ (y) h0 (ϕ (y) − x) λ(dx)
" X
ϕ¯ ¯ λ (dy)} ¯ X (1 − ρλ (y))
y∈y¯
h0 (y − x) λ(dx)
,
γ¯ ∈ ΓX ,
is a normalizing constant.
Now we can prove the main result of this section. Theorem 4.3. Under condition (4.7), the Poisson cluster measure μcl on ΓX is quasiϕ invariant with respect to the action of Diff0 (X) on ΓX . The Radon–Nikodym density Rμcl := ϕ ϕ ¯ ϕ ¯ d(ϕ ∗ μcl )/dμcl is given by Rμcl = I ∗ Rπλ , where the density Rπλ = d(ϕ¯ ∗ πλ )/dπλ is defined in (4.9). Proof. According to Theorem 3.6 (see (3.21)) and Lemma 4.1,
L. Bogachev, A. Daletskii / Journal of Functional Analysis 256 (2009) 432–478
463
ϕ ∗ μcl = p∗ πλ ◦ ϕ −1 = πλ ◦ (ϕ ◦ p)−1
= πλ ◦ (p ◦ ϕ) ¯ −1 = ϕ¯ ∗ πλ ◦ p−1 = p∗ ϕ¯ ∗ πλ . Hence, by the change of variables γ = p(γ¯ ), for any non-negative measurable function F on ΓX we obtain
∗
F (γ ) ϕ μcl (dγ ) = ΓX
F (γ ) p ϕ¯ ∗ πλ (dγ ) = ∗
ΓX
=
IF (γ¯ ) ϕ¯ ∗ πλ (dγ¯ )
ΓX
IF (γ¯ )Rπϕ¯λ (γ¯ ) πλ (dγ¯ ) =
ΓX
F (γ ) I ∗ Rπϕ¯λ (γ ) μcl (dγ ),
ΓX
where we have also used formula (4.6) and Proposition 4.2. Thus, the measure ϕ ∗ μcl is a.c. with ϕ ϕ¯ respect to the measure μcl , with the Radon–Nikodym density Rμcl = I ∗ Rπλ , and the theorem is proved. 2 ϕ
Remark 4.5. We do not know an explicit form of the density Rμcl (cf. Remark 4.2). Remark 4.6. The Poisson cluster measure μcl on the configuration space ΓX can be used to construct the canonical unitary representation U of the diffeomorphism group Diff0 (X) by operators in L2 (ΓX , μcl ), given by the formula Uϕ F (γ ) =
#
ϕ Rμcl (γ ) F ϕ −1 (γ ) ,
F ∈ L2 (ΓX , μcl ).
Such representations, which can be defined for arbitrary quasi-invariant measures on ΓX , play a significant role in the representation theory of the diffeomorphism group Diff0 (X) [20,33] and quantum field theory [17,18]. An important question is whether the representation U is irreducible. According to [33], this is equivalent to the Diff0 (X)-ergodicity of the measure μcl , which in our case is equivalent to the ergodicity of the measure πλ with respect to the group of transformations ϕ, ¯ where ϕ ∈ Diff0 (X). The latter is an open question. 4.3. Integration-by-parts formula The main objective of this section is to establish an integration-by-parts (IBP) formula for the Poisson cluster measure μcl , in the spirit of the IBP formula for Poisson measures proved in [5]. To this end, we shall use the projection operator p and the properties of the auxiliary Poisson measure πλ . Since our framework is somewhat different from that in [5], we give a proof of the IBP formula for πλ . First, recall that the classical IBP formula for a Borel measure on a Euclidean space Rm (see, e.g., [13, Chapter 5]) is expressed by the following identity that should hold for any vector field v ∈ Vect0 (Rm ) and all functions f, g ∈ C0∞ (Rm ):
f (y)∇v g(y) (dy) = − Rm
Rm
g(y)∇v f (y) (dy)
464
L. Bogachev, A. Daletskii / Journal of Functional Analysis 256 (2009) 432–478
−
v f (y)g(y)β (y) (dy),
(4.10)
Rm v ∈ L1 (Rm , ) is a measurwhere ∇v φ(y) is the derivative of φ along v at point y ∈ Y and β loc able function called the logarithmic derivative of along the vector field v. It is easy to see that v can be represented in the form β v β (y) = β (y) · v(y) + div v(y),
where the corresponding mapping β : Rm → Rm is called vector logarithmic derivative of . Suppose that the measure is a.c. with respect to the Lebesgue measure dy, with density w 1,2 (Rm ) (:= the local Sobolev space of order 1 in L2 (Rm ; dy), i.e., the such that w 1/2 ∈ Hloc space of functions on Rm whose first-order partial derivatives are locally square integrable). Then the measure satisfies the IBP formula (4.10) with the vector logarithmic derivative β (y) = w(y)−1 ∇w(y) (note that w(y) = 0 for -a.a. y ∈ Rm ). 1,2 Assume that the density s(x) ¯ = λ (dx)/d ¯ x¯ (x¯ ∈ X) satisfies the condition s 1/2 ∈ Hloc (X) 2 (:= the local Sobolev space of order 1 in L (X; dx)). ¯ By formula (3.10) and decompositions (3.5) and (3.11), the latter condition is equivalent to the set of analogous conditions for the restrictions of s(x) ¯ to the spaces X n . That is, assuming without loss of generality that pn = 0, 1/2 1,2 ¯ = λn (dx)/d ¯ x¯ (x¯ ∈ X n ) we have sn ∈ Hloc (X n ). By the general result alluded to for each sn (x) above, this ensures that the IBP formula holds for each measure λn , with the vector logarithmic ¯ = (β1 (x), ¯ . . . , βn (x)) ¯ (x¯ ∈ X n ), where derivative βλn (x) ∇i hn (x1 − x, . . . , xn − x) λ(dx) ¯ ∇i sn (x) = X ¯ := βi (x) sn (x) ¯ X hn (x1 − x, . . . , xn − x) λ(dx)
(4.11)
if sn (x) ¯ = 0 and βi (x) ¯ := 0 if sn (x) ¯ = 0. For any v ∈ Vect0 (X), let us define the vector field v¯ on X by setting
v( ¯ x) ¯ := v(x1 ), . . . , v(xn ) ,
x¯ = (x1 , . . . , xn ) ∈ X n
(n ∈ Z+ ).
(4.12)
The logarithmic derivative of the measure λn along the vector field v¯ is given by ¯ = βλv¯n (x)
βi (x) ¯ · v(xi ) + div v(xi ) ,
x¯ ∈ X n .
(4.13)
xi ∈x¯
Proposition 4.4. The measure λ satisfies the following IBP formula:
f (x)∇ ¯ v¯ g(x) ¯ λ (dx) ¯ =−
X
g(x)∇ ¯ v¯ f (x) ¯ λ (dx) ¯ −
X
f (x)g( ¯ x)β ¯ λv¯ (x) ¯ λ (dx), ¯ (4.14)
X
where f, g ∈ C0∞ (X) and βλv¯ (x) ¯ = βλv¯ (x) ¯ if x¯ ∈ X n (n ∈ Z+ ). n
Proof. The result easily follows from the decomposition (3.5) of the measure λ and the IBP formula for each measure λn such that pn = 0 (n ∈ Z+ ). 2
L. Bogachev, A. Daletskii / Journal of Functional Analysis 256 (2009) 432–478
465
Remark 4.7. Formula (4.14) can be rewritten in the form ∇x g(x) ∇x f (x) ¯ · v(x) λ (dx) ¯ = − g(x) ¯ ¯ · v(x) λ (dx) ¯
f (x) ¯
x∈p(x) ¯
X
x∈p(x) ¯
X
−
f (x)g( ¯ x)β ¯ λv¯ (x) ¯ λ (dx). ¯
X
Recall that the functional classes F C(ΓX ), F C(ΓX ), and F C λ (ΓX ) of local functions on the configuration spaces ΓX and ΓX are defined in Section 4.1. Theorem 4.5. For each v ∈ Vect0 (X) and any F, G ∈ F C(ΓX ), the following IBP formula holds:
F (γ )∇vΓ G(γ ) μcl (dγ ) = − ΓX
G(γ )∇vΓ F (γ ) μcl (dγ )
ΓX
−
F (γ )G(γ )Bμv cl (γ ) μcl (dγ ),
(4.15)
ΓX
where ∇vΓ is the Γ -gradient along the vector field v defined by (4.4), Bμv cl (γ ) := I ∗ βλv¯ , γ¯ , and βλv¯ is the logarithmic derivative of λ along the corresponding vector field v¯ (see (4.12)). Proof. Denote Q(γ ) := F (γ )∇vΓ G(γ ) = F (γ )
∇x G(γ ) · v(x),
x∈γ
then (IQ)(γ¯ ) = (IF )(γ¯ )
∇x G p(γ¯ ) · v(x).
(4.16)
x∈p(γ¯ )
Note that IQ ∈ F C λ (ΓX ), so we can use (2.8) in order to integrate IQ with respect to πλ . Using Theorem 3.6 (see (3.21)) and formula (4.16), we obtain F (γ )∇vΓ G(γ ) μcl (dγ ) ΓX
=
(IF )(γ¯ )
∇x G p(γ¯ ) · v(x) πλ (dγ¯ )
x∈p(γ¯ )
ΓX
= e−λ
(X
K)
∞ 1 m!
m=0
(XK )m
F
p(x¯1 ), . . . , p(x¯m )
466
L. Bogachev, A. Daletskii / Journal of Functional Analysis 256 (2009) 432–478 m
×
m
∇x G p(x¯1 ), . . . , p(x¯m ) · v(x) λ (dx¯i )
i=1 x∈p(x¯i )
= e−λ
(X
K)
∞ m=0
×
i=1
1 m!
i=1
m
F (XK )m−1
p(x¯1 ), . . . , p(x¯m )
XK
∇x G p(x¯1 ), . . . , p(x¯m ) · v(x) λ (dx¯i ) λ (dx¯j ).
(4.17)
j =i
x∈p(x¯i )
By the IBP formula for λ , the inner integral in (4.17) can be rewritten as − XK
×
G p(x¯1 ), . . . , p(x¯m )
∇x F
v¯ p(x¯1 ), . . . , p(x¯m ) · v(x) + F p(x¯1 ), . . . , p(x¯m ) βλ (x¯i ) λ (dx¯i ).
x∈p(x¯i )
Hence, the right-hand side of (4.17) is reduced to −e−λ
(X
K)
∞ 1 m!
m=0
×
G p(x¯1 ), . . . , p(x¯m )
(XK )m
∇x F
p(x¯1 ), . . . , p(x¯m ) · v(x)
x∈p({x¯1 ,...,x¯m })
m v¯
λ (dx¯i ) + F p(x¯1 ), . . . , p(x¯m ) Bπλ {x¯1 , . . . , x¯m } =− ΓX
G p(γ¯ )
x∈p(γ¯ )
i=1
∇x F p(γ¯ ) · v(x) + F p(γ¯ ) Bπv¯ λ (γ¯ ) πλ (dγ¯ )
=−
G(γ )∇vΓ F (γ )μcl (dγ ) −
ΓX
F (γ )G(γ )Bμv cl (γ ) μcl (dγ ),
ΓX
where Bπv¯ λ (γ¯ ) :=
! βλv¯ (x) ¯ = βλv¯ , γ¯ ,
γ¯ ∈ ΓX ,
(4.18)
x∈ ¯ γ¯
¯ < ∞, so there are only and Bμv cl := I ∗ Bπv¯ λ . Note that Bπv¯ λ is well defined since λ (supp v) finitely many non-zero terms in the sum (4.18). Moreover, finiteness of the first and second moments of πλ implies that Bπv¯ λ ∈ L2 (ΓX , πλ ). 2
L. Bogachev, A. Daletskii / Journal of Functional Analysis 256 (2009) 432–478
467
Remark 4.8. The logarithmic derivative Bπv¯ λ can be written in the form (cf. (4.11)) Bπv¯ λ (γ¯ ) =
βi (x) ¯ · v(xi ) + div v(xi ) x∈ ¯ γ¯ xi ∈x¯
βλ (x) ¯ · v( ¯ x) ¯ + div v( ¯ x) ¯ , =
γ¯ ∈ ΓX .
x∈ ¯ γ¯
Formula (4.15) can be extended to more general vector fields on ΓX . For any vector field V ∈ F V(ΓX ) of the form (4.3), we set BμVcl (γ ) :=
k
Ai (γ )Bμvi (γ ) +
∇x Ai (γ ) · vi (x) ,
γ ∈ ΓX .
x∈γ
i=1
Theorem 4.6. For any V ∈ F V(ΓX ) and all F, G ∈ F C(ΓX ), we have
F (γ )∇VΓ G(γ ) μcl (dγ ) = − ΓX
G(γ )∇VΓ F (γ ) μcl (dγ )
ΓX
−
F (γ )G(γ )BμVcl (γ ) μcl (dγ ).
(4.19)
ΓX
Proof. The result readily follows from Theorem 4.5 and linearity of the right-hand side of (4.13) with respect to v. 2 Remark 4.9. An explicit form of BμVcl is not known (cf. Remarks 4.2 and 4.5). Remark 4.10. The logarithmic derivative BμVcl can be represented in the form BμVcl = I ∗ BπIλV , where BπIλV is the logarithmic derivative of πλ along the vector field IV (γ¯ ) := V (p(γ¯ )). Note that the equality Tγ¯ ΓX =
$ x∈ ¯ γ¯
Tx¯ X =
$$ x∈ ¯ γ¯ xi ∈x¯
Txi X =
$
Tx X = Tp(γ¯ ) ΓX
x∈p(γ¯ )
implies that V (p(γ¯ )) ∈ Tγ¯ ΓX , and thus IV (γ¯ ) is a vector field on ΓX . 5. Dirichlet forms and equilibrium stochastic dynamics In this section, we construct a Dirichlet form Eμcl associated with the Poisson cluster measure μcl and prove the existence of the corresponding equilibrium stochastic dynamics on the configuration space. We also show that the Dirichlet form Eμcl is irreducible. We assume throughout that the measure λ satisfies all the conditions set out at the beginning of Section 4 and in Section 4.3.
468
L. Bogachev, A. Daletskii / Journal of Functional Analysis 256 (2009) 432–478
5.1. The Dirichlet form associated with μcl Let us introduce the pre-Dirichlet form Eμcl associated with the Poisson cluster measure μcl , defined on F C(ΓX ) ⊂ L2 (ΓX , μcl ) by Eμcl (F, G) :=
! ∇ Γ F (γ ), ∇ Γ G(γ ) γ μcl (dγ ),
F, G ∈ F C(ΓX ),
(5.1)
ΓX
where ∇ Γ is the Γ -gradient on the configuration space ΓX (see (4.2)). The next proposition shows that the form Eμcl is well defined. Proposition 5.1. For any F, G ∈ F C(ΓX ), we have Eμcl (F, G) < ∞. Proof. The statement follows from the existence of the first moments of μcl . Indeed, let F, G ∈ F C(ΓX ) have representations
F (γ ) = f φ1 , γ , . . . , φk , γ ,
G(γ ) = g ψ1 , γ , . . . , ψ , γ
(see (4.1)), then a direct calculation shows that ! ∇x F (γ ) · ∇x G(γ ) = Qij (γ )qij , γ , ∇ Γ F (γ ), ∇ Γ G(γ ) γ = x∈γ
i,j
where qij (x) := ∇φi (x) · ∇ψj (x) ∈ C0 (X) and
Qij (γ ) := ∇i f φ1 , γ , . . . , φk , γ ∇j g ψ1 , γ , . . . , ψ , γ ∈ F C(ΓX ). ˜ x) ¯ := Denoting for brevity q(x) := qij (x) and setting q(
! q, p(γ¯ ) πλ (dγ¯ ) =
q, γ μcl (dγ ) = ΓX
ΓX
x∈x¯
q(x), by Theorem 3.6 we have
q, ˜ γ¯ πλ (dγ¯ ) =
ΓX
q( ˜ y) ¯ λ (dy) ¯ < ∞, X
because λ (supp q) ˜ = λ (Xsupp q ) < ∞ by Proposition 3.4. Therefore, q, γ ∈ L1 (ΓX , μcl ) and the required result follows. 2 Let us also consider the pre-Dirichlet form Eπλ associated with the Poisson measure πλ , defined on the space F C(ΓX ) ⊂ L2 (ΓX , πλ ) by Eπλ (Φ, Ψ ) :=
! ∇ Γ Φ(γ¯ ), ∇ Γ Ψ (γ¯ ) γ¯ πλ (dγ¯ ),
Φ, Ψ ∈ F C(ΓX )
ΓX
(here ∇ Γ is the Γ -gradient on the configuration space ΓX , cf. (4.2)). Pre-Dirichlet forms of such type associated with general Poisson measures were introduced and studied in [5]. Finiteness of
L. Bogachev, A. Daletskii / Journal of Functional Analysis 256 (2009) 432–478
469
the first moments of the Poisson measure πλ implies that Eπλ is well defined. It follows from the IBP formula for πλ that Eπλ (Φ, Ψ ) = Hπλ Φ(γ¯ )Ψ (γ¯ ) πλ (dγ¯ ), Φ, Ψ ∈ F C(ΓX ), (5.2) ΓX
where Hπλ is a symmetric non-negative operator in L2 (ΓX , πλ ) (called the Dirichlet operator of the Poisson measure πλ , see [5]) defined on the domain F C(ΓX ) by (Hπλ Φ)(γ¯ ) := −
x¯ Φ(γ¯ ) + ∇x¯ Φ(γ¯ ) · βλ (x) ¯
(γ¯ ∈ ΓX ).
(5.3)
x∈ ¯ γ¯
Since function Φ ∈ F C(ΓX ) is local (see Section 4.1), there are only finitely many non-zero terms in the sum (5.3). Remark 5.1. Note that the operator Hπλ is well defined by formula (5.3) on the bigger space F C λ (ΓX ). Similar arguments as before show that the pre-Dirichlet form Eπλ (Φ, Ψ ) is well defined on F C λ (ΓX ) and formula (5.2) holds for any Φ, Ψ ∈ F C λ (ΓX ). Consider a symmetric operator in L2 (ΓX , μcl ) defined on F C(ΓX ) by the formula Hμcl := I ∗ Hπλ I.
(5.4)
Note that the domain F C(ΓX ) is dense in L2 (ΓX , μcl ). Theorem 5.2. For any F, G ∈ F C(ΓX ), the form (5.1) satisfies the equality Eμcl (F, G) = Hμcl F (γ )G(γ ) μcl (dγ ).
(5.5)
ΓX
In particular, this implies that Hμcl is a non-negative operator on F C(ΓX ). Proof. Let us fix F, G ∈ F C(ΓX ) and set Q(γ ) := ∇ Γ F (γ ), ∇ Γ G(γ )γ . From the definition (4.6) of the operator I, it readily follows that (IQ)(γ¯ ) =
∇x IF (γ¯ ) · ∇x IG(γ¯ ) =
∇x¯ IF (γ¯ ) · ∇x¯ IG(γ¯ ),
(5.6)
x∈ ¯ γ¯
x∈p(γ¯ )
where ∇x¯ := (∇x1 , . . . , ∇xn ) when x¯ = (x1 , . . . , xn ) ∈ X n (n ∈ N). Thus, by Theorem 3.6 and formulas (4.6) and (5.6) we obtain Eμcl (F, G) = Q(γ ) μcl (dγ ) = (IQ)(γ¯ ) πλ (dγ¯ ) ΓX
=
ΓX
x∈ ¯ γ¯
ΓX
∇x¯ IF (γ¯ ) · ∇x¯ IG(γ¯ ) πλ (dγ¯ ) = Eπλ (IF, IG)
(5.7)
470
L. Bogachev, A. Daletskii / Journal of Functional Analysis 256 (2009) 432–478
(note that IF, IG ∈ F C λ (ΓX ) ⊂ D(Eπλ )). Finally, combining (5.7) with formula (5.2) we get (5.5). 2 Remark 5.2. The operator Hμcl defined in (5.4) can be represented in the following form separating its diffusive and drift parts: (Hμcl F )(γ ) = −
x F (γ ) − I ∗ ΨF (γ ),
F ∈ F C(ΓX ),
(5.8)
x∈γ
where ΨF (γ¯ ) :=
x∈ ¯ γ¯
∇x¯ IF (γ¯ ) · βλ (x) ¯ (γ¯ ∈ ΓX ).
Remark 5.3. Formulas (5.5) and (5.8) can also be obtained directly from the IBP formula (4.19). 5.2. The associated equilibrium stochastic dynamics Formula (5.5) implies that the form Eμcl is closable on L2 (ΓX , μcl ), and we preserve the same notation for its closure. Its domain D(Eμcl ) is obtained as a completion of F C(ΓX ) with respect to the norm
1/2
F Eμcl := Eμcl (F, F ) +
2
F dμcl
.
ΓX
In the canonical way, the Dirichlet form (Eμcl , D(Eμcl )) defines a non-negative self-adjoint operator in L2 (ΓX , μcl ) (i.e., the Friedrichs extension of Hμcl = I ∗ Hπλ I from the domain F C(ΓX )), for which we keep the same notation Hμcl . In turn, this operator generates the semigroup exp(−tHμcl ) in L2 (ΓX , μcl ). According to a general result (see [27, §4]), it follows that Eμcl is a quasi-regular local Dirichlet form on a bigger space L2 (Γ¨X , μcl ), where Γ¨X is the space of all locally finite configurations γ with possible multiple points (note that Γ¨X can be identified in the standard way with the space of Z+ -valued Radon measures on X, cf. [5,27,30]). Then, by the general theory of Dirichlet forms (see [26]), we obtain the following result. Theorem 5.3. There exists a conservative diffusion process X = (Xt , t 0) on Γ¨X , properly associated with the Dirichlet form Eμcl ; that is, for any function F ∈ L2 (Γ¨X , μcl ) and all t 0, the mapping Γ¨X γ → pt F (γ ) :=
F (Xt ) dPγ Ω
is an Eμcl -quasi-continuous version of exp(−tHμcl )F . Here Ω is the canonical sample space (of Γ¨X -valued continuous functions on R+ ) and (Pγ , γ ∈ Γ¨X ) is the family of probability distributions of the process X conditioned on the initial value γ = X0 . The process X is unique up to μcl -equivalence. In particular, X is μcl -symmetric (i.e., Fpt G dμcl = Gpt F dμcl for all measurable functions F, G : Γ¨X → R+ ) and μcl is its invariant measure.
L. Bogachev, A. Daletskii / Journal of Functional Analysis 256 (2009) 432–478
471
Remark 5.4. It can be proved that in the case of Poisson and Gibbs measures, under certain technical conditions the diffusion process X actually lives on the proper configuration space ΓX (see [30]). It is plausible that a similar result should be valid for the Poisson cluster measure, but this is an open problem. Remark 5.5. Formula (5.2) implies that the “pre-projection” form Eπλ is closable. According to the general theory of Dirichlet forms [26,27], its closure is a quasi-regular local Dirichlet ¯ on Γ¨X . This process coincides with the form on Γ¨X and as such generates a diffusion process X independent infinite particle process, which amounts to independent distorted Brownian motions in X with drift given by the vector logarithmic derivative of λ (see [5]). However, it is not clear in what sense the process X constructed in Theorem 5.3 can be obtained directly via the projection ¯ from Γ¨X onto Γ¨X . of X 5.3. Irreducibility of the Dirichlet form Eμcl Let us recall that a Dirichlet form E is called irreducible if the condition E(F, F ) = 0 implies that F = const. Theorem 5.4. The Dirichlet form (Eμcl , D(Eμcl )) is irreducible. Proof. For any F ∈ D(Eμcl ), we have 2 2 F Eμ = Eμcl (F, F ) + F dμcl = Eπλ (IF, IF ) + (IF )2 dπλ = IF 2Eπ , cl
λ
ΓX
ΓX
which implies that ID(Eμcl ) ⊂ D(Eπλ ). It is obvious that if IF = const (πλ -a.s.) then F = const (μcl -a.s.). Therefore, according to formula (5.7), it suffices to prove that the Dirichlet form (Eπλ , D(Eπλ )) is irreducible, which is established in Lemma 5.6 below. 2 We first need the following general result (see [4, Lemma 3.3]). Lemma 5.5. Let A and B be self-adjoint, non-negative operators in separable Hilbert spaces H and K, respectively. Then Ker(A B) = Ker A ⊗ Ker B, where A B is the closure of the operator A ⊗ I + I ⊗ B from the algebraic tensor product of the domains of A and B. Proof. Ker A and Ker B are closed subspaces of H and K, respectively, and so their tensor product Ker A ⊗ Ker B is a closed subspace of the space H ⊗ K. The inclusion Ker A ⊗ Ker B ⊂ Ker(A B) is trivial. Let f ∈ Ker(A B). Using the theory of operators admitting separation of variables (see, e.g., [8, Chapter 6]), we have
0 = (A Bf, f ) = (x1 + x2 ) d E(x1 , x2 )f, f R2+
= R2+
x1 d E(x1 , x2 )f, f +
x2 d E(x1 , x2 )f, f
R2+
= (A ⊗ If, f ) + (I ⊗ Bf, f ),
(5.9)
472
L. Bogachev, A. Daletskii / Journal of Functional Analysis 256 (2009) 432–478
where E is a joint resolution of the identity of the commuting operators A ⊗ I and I ⊗ B. Since both operators A ⊗ I and I ⊗ B are non-negative, we conclude from (5.9) that f ∈ Ker(A ⊗ I ) ∩ Ker(I ⊗ B) = Ker A ⊗ Ker B, which completes the proof of the lemma.
2
Lemma 5.6. The Dirichlet form (Eπλ , D(Eπλ )) is irreducible. Remark 5.6. Irreducibility of Dirichlet forms associated with Poisson measures on configuration spaces of connected Riemannian manifolds was shown in [5,6]. However, the space X consists of countably many disjoint connected components X n , so we need to adapt the result to this situation. Proof of Lemma 5.6. Let us recall that, according to the general theory (see, e.g., [2]), irreducibility of a Dirichlet form is equivalent to the condition that the kernel of its generator consists of constants (uniqueness of the ground state). Thus, it suffices to prove that Ker Hπλ = {const}. ˜ n := ∞ X k , n ∈ Z+ , endowed with the measures Let us consider the “residual” spaces X k=n 0 1 n ˜ λ˜ n := ∞ k=n pk λk . Hence, X = X X · · · X Xn+1 , which implies that ΓX = ΓX 0 × ΓX1 × · · · × ΓXn × ΓX˜ n+1 and, according to Proposition 2.2, πλ = π0 ⊗ π1 ⊗ · · · ⊗ πn ⊗ π˜ n+1 , where we use a shorthand notation πn := πpn λn , π˜ n := πλ˜ . Therefore, there is an isomorphism n of Hilbert spaces L2 (ΓX , πλ ) ∼ = L2 (ΓX , π1 ) ⊗ · · · ⊗ L2 (ΓXn , πn ) ⊗ L2 (ΓXn+1 , π˜ n+1 ). Consequently, the Dirichlet operator Hπλ can be decomposed as Hπλ = Hπ1 · · · Hπn Hπ˜ n+1 .
(5.10)
Since all operators on the right-hand side of (5.10) are self-adjoint and non-negative, it follows by Lemma 5.5 that Ker Hπλ = Ker Hπ1 ⊗ · · · ⊗ Ker Hπn ⊗ Ker Hπ˜ n+1 .
(5.11)
The Dirichlet forms of all measures πk are irreducible (as Dirichlet forms of Poisson measures on connected manifolds), hence Ker Hπk = R and (5.11) implies that Ker Hπλ = Ker Hπ˜ n+1 . Since n is arbitrary, it follows that every function F ∈ Ker Hπλ does not depend on any finite number of variables, and thus F = const (πλ -a.s.). 2 Remark 5.7. The result of Lemma 5.6 (and the idea of its proof) can be viewed as a functionalanalytic analogue of Kolmogorov’s zero–one law (see, e.g., [21, Chapter 2]), stating that for a sequence of independent random variables (Xn ), the corresponding tail σ -algebra F∞ := n Fn is trivial (where Fn := σ {Xk : k n}), and in particular, all F∞ -measurable random variables are a.s. constants. Remark 5.8. According to the general theory of Dirichlet forms (see, e.g., [2]), the irreducibility of Eμcl is equivalent to each of the following properties:
L. Bogachev, A. Daletskii / Journal of Functional Analysis 256 (2009) 432–478
473
(i) The semigroup e−tHμcl is L2 -ergodic, that is, as t → ∞, 2 e−tHμcl F (γ ) − F (γ ) μcl (dγ ) μcl (dγ ) → 0. ΓX
ΓX
(ii) If F ∈ D(Hμcl ) and Hμcl F = 0 then F = const. Acknowledgments Part of this research was done during the authors’ visits to the Institute of Applied Mathematics of the University of Bonn supported by SFB 611. Financial support through DFG Grant 436 RUS 113/722 is gratefully acknowledged. The authors would like to thank Sergio Albeverio, Yuri Kondratiev and Eugene Lytvynov for useful discussions. Thanks are also due to the anonymous referee for the careful reading of the manuscript and valuable comments. Appendix A A.1. Proof of Theorem 2.7 Note that the droplet cluster DB (γ0 ) = y∈γ (B − y) (see (2.18)) can be decomposed into 0 disjoint components according to the number of constituent “layers” (including infinitely many):
DB γ0 =
DB γ0 ,
1∞
where
DB γ0 := x ∈ X: γ0 (B − x) = ,
∈ Z+ .
(a) Set fq := − ln q · 1K ∈ M+ (X) (0 < q < 1), then Lμcl [fq ] =
q γ (K) μcl (dγ ) =
∞
q n μcl γ ∈ ΓX : γ (K) = n
(A.1)
n=0
ΓX
→ μcl γ ∈ ΓX : γ (K) < ∞
(q ↑ 1).
Therefore, γ (K) < ∞ (μcl -a.s.) if and only if limq↑1 ln Lμcl [fq ] = 0. Clearly, condition (2.19) is necessary for local finiteness of μcl -a.a. configurations γ ∈ ΓX . Furthermore, (2.19) implies that, for any compact set K ⊂ X and any x ∈ X, we have γ0 (K − x) < ∞ (μ0 -a.s.). Hence, according to (2.17), − ln Lμcl [fq ] = X
ΓX
γ0 (K−x) 1−q μ0 dγ0 λ(dx)
474
L. Bogachev, A. Daletskii / Journal of Functional Analysis 256 (2009) 432–478
∞
1 − q 1D (γ ) (x) λ(dx) μ0 dγ0 = K
ΓX
=
X =0
0
∞
1 − q λ DK γ0 μ0 dγ0 .
ΓX
(A.2)
=1
Note that, for 0 < q < 1, 0
∞ ∞
1 − q λ DK γ0 γ0 = λ DK γ0 , λ DK =1
=1
so if condition (2.20) is satisfied then we can apply Lebesgue’s dominated convergence theorem and pass termwise to the limit on the right-hand side of (A.2) as q ↑ 1, which gives limq↑1 ln Lμcl [fq ] = 0, as required. Conversely, since ∞ ∞
1 − q λ DK γ0 (1 − q) γ0 = (1 − q)λ DK γ0 0, λ DK =1
=1
from (A.2) we must have (1 − q)
λ DK γ0 μ0 dγ0 → 0 (q ↑ 1),
ΓX
which implies (2.20). (b) Let us first prove the “only if” part. Clearly, condition (2.21) is necessary in order to avoid any in-cluster ties. Furthermore, each fixed x0 ∈ X cannot belong to more than one cluster; in particular, for any 2 ∞,
γ0 = 0 (μ0 -a.s.). λ D{x (A.3) 0} Let fq := − ln q · 1{x0 } (0 < q < 1). The expansion (A.1) then implies that in order for x0 to be simple (μcl -a.s.), Lμcl [fq ] must be a linear function of q. But from (A.2) and (A.3) we have
=1 Lμcl [fq ] = exp −(1 − q) λ D{x0 } γ0 μ0 dγ0 ,
ΓX
=1 (γ0 ) = 0 (μ0 -a.s.). Together with (A.3), this gives and it follows that λ D{x 0}
γ0 = 0 λ D{x λ D{x0 } γ0 = } 0 1∞
and condition (2.22) follows.
(μ0 -a.s.),
L. Bogachev, A. Daletskii / Journal of Functional Analysis 256 (2009) 432–478
475
To prove the “if” part, it suffices to show that, under conditions (2.21) and (2.22), with probability one there are no cross-ties between the clusters whose centres belong to a set Λ ⊂ X, λ(Λ) < ∞. Conditionally on the total number of cluster centres in Λ (which are then i.i.d. and have the distribution λ(·)/λ(Λ)), the probability of a tie between a given pair of (independent) clusters is given by
1 λ(Λ)2
λ⊗2 BΛ (γ1 , γ2 ) μ0 (dγ1 ) μ0 (dγ2 ),
ΓX ×ΓX
where BΛ (γ1 , γ2 ) := (x1 , x2 ) ∈ Λ2 : x1 + y1 = x2 + y2 for some y1 ∈ γ1 , y2 ∈ γ2 . But
λ⊗2 BΛ (γ1 , γ2 ) =
λ
Λ
=
y1 ∈γ1 y2 ∈γ2
y1 ∈γ1
λ
{x1 + y1 − y2 } λ(dx1 )
y2 ∈γ2
Λ
y1 ∈γ1
{x1 + y1 − y2 } λ(dx1 )
λ D{x1 +y1 } (γ2 ) λ(dx1 ) = 0 (μ0 -a.s.),
Λ
since, by assumption (2.22), λ(D{x1 +y1 } (γ2 )) = 0 (μ0 -a.s.) and γ1 is a countable set. Thus, the proof is complete. A.2. Quasi-invariance of Poisson measures The next general result is a direct consequence of Skorokhod’s theorem [32] on the absolute continuity of Poisson measures (see also [5]). Although essentially well known, we give its simple proof adapted to our slightly more general setting, whereby transformations ϕ have support of finite measure rather than compact. Suppose that πλ is a Poisson measure on the configuration space ΓX with intensity measure λ. Let ϕ : X → X be a measurable mapping; as explained earlier (see (4.5a)), it can be lifted to a (measurable) transformation of ΓX : ΓX γ → ϕ(γ ) := ϕ(x), x ∈ γ ∈ ΓX .
(A.4)
Proposition A.1. Let ϕ : X → X be a measurable bijection such that λ(supp ϕ) < ∞. Assume that the measure λ is quasi-invariant with respect to ϕ, that is, the push-forward measure ϕ ∗ λ ≡ λ ◦ ϕ −1 is a.c. with respect to λ, with density ϕ
ρλ (x) :=
ϕ ∗ λ(dx) , λ(dx)
x ∈ X.
(A.5)
476
L. Bogachev, A. Daletskii / Journal of Functional Analysis 256 (2009) 432–478
Then the measure πλ is quasi-invariant with respect to the action (A.4), that is, ϕ ∗ πλ (dγ ) = Rπϕλ (γ ) πλ (dγ ),
γ ∈ ΓX ,
(A.6)
ϕ
where the density Rπλ is given by
ϕ ϕ Rπϕλ (γ ) = exp 1 − ρλ (x) λ(dx) · ρλ (x),
γ ∈ ΓX ,
(A.7)
x∈γ
X ϕ
and moreover, Rπλ ∈ L2 (ΓX , πλ ). ϕ
Proof. Note that ρλ ≡ 1 outside the set K := supp ϕ. By Proposition 2.4(a), the condition λ(K) " <ϕ∞ implies that, for πλ -a.a. γ ∈ ΓX , there are only finitely many terms in the product x∈γ ρλ (x) not equal to 1, thus the right-hand side of Eq. (A.7) is well defined. Using formulas ϕ ϕ (A.5), (A.7) and Proposition 2.1, the Laplace functional of the measure πλ := Rπλ πλ is obtained as follows: ϕ
ϕ Lπ ϕ [f ] = exp 1 − ρλ (x) λ(dx) · e−f,γ ρλ (x) πλ (dγ ) λ
x∈γ
ΓX
X
ϕ
ϕ = exp 1 − ρλ (x) λ(dx) · exp − 1 − e−f (x)+ln ρλ (x) λ(dx) X
ϕ 1 − e−f (x) ρλ (x) λ(dx) = exp −
X
X
∗ −f (x) 1−e ϕ λ(dx) = Lπϕ ∗ λ [f ], = exp − X ϕ
and so πλ = πϕ ∗ λ . But, according to the Mapping Theorem (see Proposition 2.3), we have πϕ ∗ λ = ϕ ∗ πλ , and formula (A.6) follows. ϕ To check that Rπλ ∈ L2 (ΓX , πλ ), let us compute its L2 -norm:
% % ϕ ϕ
ϕ %R (γ )%2 πλ (dγ ) = exp 1 − ρ (x) λ(dx) · e2 ln ρλ ,γ πλ (dγ ) πλ λ
ΓX
X
ΓX
ϕ
ϕ 2 ln ρλ (x) = exp 1 − ρλ (x) λ(dx) · exp − 1−e λ(dx) X
X
% ϕ %2 ϕ % % ρλ (x) − ρλ (x) λ(dx) < ∞, = exp X ϕ
ϕ
because |ρλ (x)|2 − ρλ (x) = 0 outside the set K = supp ϕ.
2
L. Bogachev, A. Daletskii / Journal of Functional Analysis 256 (2009) 432–478
477
References [1] S. Albeverio, L. Bogachev, Brownian survival in a clusterized trapping medium, Rev. Math. Phys. 10 (1998) 147– 189. [2] S. Albeverio, Yu. Kondratiev, M. Röckner, Ergodicity of L2 -semigroups and extremality of Gibbs states, J. Funct. Anal. 144 (1997) 394–423. [3] S. Albeverio, A. Daletskii, A. Kalyuzhnyj, Traces of semigroups associated with interacting particle systems, J. Funct. Anal. 246 (2007) 196–216. [4] S. Albeverio, A. Daletskii, E. Lytvynov, De Rham cohomology of configuration spaces with Poisson measure, J. Funct. Anal. 185 (2001) 240–273. [5] S. Albeverio, Yu. Kondratiev, M. Röckner, Analysis and geometry on configuration spaces, J. Funct. Anal. 154 (1998) 444–500. [6] S. Albeverio, Yu. Kondratiev, M. Röckner, Analysis and geometry on configuration spaces: The Gibbsian case, J. Funct. Anal. 157 (1998) 242–291. [7] S. Albeverio, Yu. Kondratiev, M. Röckner, Diffeomorphism groups and current algebras: Configuration spaces analysis in quantum theory, Rev. Math. Phys. 11 (1999) 1–23. [8] Yu.M. Berezanski˘ı, Selfadjoint Operators in Spaces of Functions of Infinitely Many Variables, Transl. Math. Monogr., vol. 63, Amer. Math. Soc., Providence, RI, 1986. [9] A.M. Berezhkovskii, Yu.A. Makhnovskii, L.V. Bogachev, S.A. Molchanov, Brownian-particle trapping by clusters of traps, Phys. Rev. E 47 (1993) 4564–4567. [10] L. Bogachev, A. Daletskii, Analysis on configuration spaces and Gibbs cluster ensembles, Russian J. Math. Phys. 14 (2007) 397–408. [11] L. Bogachev, A. Daletskii, Equilibrium stochastic dynamics of Poisson cluster ensembles, Condens. Matter Phys. 11 (2008) 261–273. [12] L.V. Bogachev, S.A. Molchanov, Yu.A. Makhnovskii, A.M. Berezhkovskii, Correlation effects in the trapping problem: General approach and rigorous results, in: F. Gesztesy, et al. (Eds.), Stochastic Processes, Physics and Geometry: New Interplays II, Proc. Conf. on Infinite Dimensional (Stochastic) Analysis and Quantum Physics, Leipzig, 18–22 January 1999, in: CMS Conf. Proc. Ser., vol. 29, Amer. Math. Soc., Providence, RI, 2000, pp. 29– 42. [13] V.I. Bogachev, Gaussian Measures, Math. Surveys Monogr., vol. 62, Amer. Math. Soc., Providence, RI, 1998. [14] P.Y. Cheng, J.S. Baskin, A.H. Zewail, Dynamics of clusters: From elementary to biological structures, Proc. Natl. Acad. Sci. USA 103 (2006) 10570–10576. [15] D.R. Cox, V. Isham, Point Processes, Monogr. Appl. Probab. Statist., Chapman & Hall, London, 1980. [16] D.J. Daley, D. Vere-Jones, An Introduction to the Theory of Point Processes, vol. I: Elementary Theory and Methods, second ed., Probab. Appl. (N.Y.), Springer, New York, 2003. [17] G.A. Goldin, J. Grodnik, R.T. Powers, D.H. Sharp, Nonrelativistic current algebra in the N/V limit, J. Math. Phys. 15 (1974) 88–100. [18] G.A. Goldin, U. Moschella, Generalized configuration spaces for quantum systems, in: F. Gesztesy, et al. (Eds.), Stochastic Processes, Physics and Geometry: New Interplays II, Proc. Conf. on Infinite Dimensional (Stochastic) Analysis and Quantum Physics, Leipzig, 18–22 January 1999, in: CMS Conf. Proc. Ser., vol. 29, Amer. Math. Soc., Providence, RI, 2000, pp. 243–251. [19] P.R. Halmos, Measure Theory, Grad. Texts in Math., vol. 18, Springer, New York, 1974. [20] R.S. Ismagilov, Representations of Infinite-Dimensional Groups, Transl. Math. Monogr., vol. 152, Amer. Math. Soc., Providence, RI, 1996. [21] O. Kallenberg, Foundations of Modern Probability, Probab. Appl. (N.Y.), Springer, New York, 1997. [22] J.F.C. Kingman, Poisson Processes, Oxford Stud. Probab., vol. 3, Clarendon, Oxford, 1993. [23] J.F.C. Kingman, Poisson processes revisited, Probab. Math. Statist. 26 (2006) 77–95. [24] T. Kuna, Studies in Configuration Space Analysis and Applications, Bonner Math. Schrift., vol. 324, Univ. Bonn, Bonn, 1999, PhD dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, 1999, 187 pp. [25] V. Liebscher, Integration by parts formulae for point processes, in: F. Gesztesy, et al. (Eds.), Stochastic Processes, Physics and Geometry: New Interplays II, Proc. Conf. on Infinite Dimensional (Stochastic) Analysis and Quantum Physics, Leipzig, 18–22 January 1999, in: CMS Conf. Proc. Ser., vol. 29, Amer. Math. Soc., Providence, RI, 2000, pp. 447–454. [26] Z.-M. Ma, M. Röckner, Introduction to the Theory of (Non-Symmetric) Dirichlet Forms, Universitext, Springer, Berlin, 1992. [27] Z.-M. Ma, M. Röckner, Construction of diffusions on configuration spaces, Osaka J. Math. 37 (2000) 273–314.
478
L. Bogachev, A. Daletskii / Journal of Functional Analysis 256 (2009) 432–478
[28] K.R. Parthasarathy, Probability Measures on Metric Spaces, Probab. Math. Statist., Academic Press, New York, 1967. [29] M. Röckner, Stochastic analysis on configuration spaces: Basic ideas and recent results, in: J. Jost, et al. (Eds.), New Directions in Dirichlet Forms, in: Stud. Adv. Math., vol. 8, Amer. Math. Soc., Providence, RI, 1998, pp. 157–232. [30] M. Röckner, B. Schmuland, A support property for infinite-dimensional interacting diffusion processes, C. R. Acad. Sci. Paris Sér. I Math. 326 (1998) 359–364. [31] H. Shimomura, Poisson measures on the configuration space and unitary representations of the group of diffeomorphisms, J. Math. Kyoto Univ. 34 (1994) 599–614. [32] A.V. Skorokhod, On the differentiability of measures which correspond to stochastic processes. I. Processes with independent increments, Theory Probab. Appl. 2 (1957) 407–432. [33] A.M. Vershik, I.M. Gel’fand, M.I. Graev, Representations of the group of diffeomorphisms, Russian Math. Surveys 30 (6) (1975) 1–50.
Journal of Functional Analysis 256 (2009) 479–508 www.elsevier.com/locate/jfa
Initial boundary value problems for nonlinear dispersive wave equations Joachim Escher a , Zhaoyang Yin b,∗ a Institute for Applied Mathematics, Leibniz University of Hanover, D-30167 Hanover, Germany b Department of Mathematics, Sun Yat-sen University, 510275 Guangzhou, China
Received 1 April 2008; accepted 2 July 2008 Available online 31 July 2008 Communicated by C. Kenig
Abstract In this paper we study initial value boundary problems of two types of nonlinear dispersive wave equations on the half-line and on a finite interval subject to homogeneous Dirichlet boundary conditions. We first prove local well-posedness of the rod equation and of the b-equation for general initial data. Furthermore, we are able to specify conditions on the initial data which on the one hand guarantee global existence and on the other hand produce solutions with a finite life span. In the case of finite time singularities we are able to describe the precise blow-up scenario of breaking waves. Our approach is based on sharp extension results for functions on the half-line or on a finite interval and several symmetry preserving properties of the equations under discussion. © 2008 Elsevier Inc. All rights reserved. Keywords: The Camassa–Holm equation and the rod equation; The Degasperis–Procesi equation and the b-equation; Initial boundary value problems; Local well-posedness; Blow-up; Global existence
1. Introduction In this paper we present a thorough study on initial boundary value problems of two type of nonlinear dispersive wave equations on the half-line and on a finite interval. One type is the so called rod equation: ut − utxx + 3uux = γ (2ux uxx + uuxxx ), * Corresponding author.
E-mail addresses: [email protected] (J. Escher), [email protected] (Z. Yin). 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.07.010
(1.1)
480
J. Escher, Z. Yin / Journal of Functional Analysis 256 (2009) 479–508
where γ is a real constant. Eq. (1.1) was obtained recently by Dai in [20,21] as a model for nonlinear waves in cylindrical hyperelastic rods with u(t, x) representing the radial stretch relative to a pre-stressed state. Among Eq. (1.1), there are two other important equations. With γ = 0 in Eq. (1.1) we find the BBM equation which is a well-known model for surface waves in a channel [3]. All solutions are global and solitary waves are smooth and orbitally stable [54]. Although having a Hamiltonian structure, the equation is not integrable and its solitary waves are not solitons [26]. For γ = 1, Eq. (1.1) becomes the Camassa–Holm equation modeling unidirectional propagation of shallow water waves over a flat bottom. Here u(t, x) stands for the fluid velocity at time t and position x, cf. [5,27,40]. It has a bi-Hamiltonian structure [36] and is completely integrable [5,9]. Its solitary waves are peaked [6]. The peaked solitons are orbitally stable [19]. The explicit interaction of the peaked solitons is given in [2]. The peakons capture a characteristic of the travelling waves of greatest height—exact travelling solutions of the governing equations for water waves with a peak at their crest, cf. [10,15,55]. Simpler approximate shallow water models (like KdV) do not present travelling wave solutions with this feature (see the discussion in [59]). The Cauchy problems of the rod equation (1.1) on the line and on the circle had been investigated in [18] and [62], respectively. Several blow-up results and global existence results of strong solutions to the rod equation (1.1) on the line and on the circle had been obtained in [18] and [62], respectively. The stability of a class of solitary waves for the rod equation (1.1) on the line were also investigated in [18]. The other type of nonlinear dispersive wave equations is the b-equation: ut − α 2 utxx + (b + 1)uux = α 2 (bux uxx + uuxxx ),
(1.2)
where b and α are arbitrary real constants. The b-equation (1.2) can be derived as the family of asymptotically equivalent shallow water wave equations that emerges at quadratic order accuracy for any b = −1 by an appropriate Kodama transformation, cf. [28,29]. For the case b = −1, the corresponding Kodama transformation is singular and the asymptotic ordering is violated, cf. [28,29]. The solutions of the b-equation (1.2) were studied numerically for various values of b in [38,39], where b was taken as a bifurcation parameter. The KdV equation, the Camassa– Holm equation and the Degasperis–Procesi equation are the only three integrable equations in the b-equation (1.2), which was shown in [23,25] by using Painlevé analysis. The b-equation admits peakon solutions for any b ∈ R, cf. [25,38,39]. If α = 0 and b = 2, then Eq. (1.2) becomes the well-known KdV equation which describes unidirectional propagation of waves at the free surface of shallow water under the influence of gravity, cf. [27]. In this model u(t, x) represents the wave’s height above a flat bottom, x is proportional to distance in the direction of propagation and t is proportional to the elapsed time. The KdV equation is completely integrable and its solitary waves are solitons [50]. The Cauchy problem of the KdV equation has been the subject of a number of studies, and a satisfactory local or global (in time) existence theory is now in hand (for example, see [41,42]). It is shown that the KdV equation is globally well-posed for u0 ∈ H −1 , cf. [41]. It is observed that the KdV equation does not accommodate wave breaking (by wave breaking we understand that the wave remains bounded while its slope becomes unbounded in finite time [59]). For α = 1 and b = 2, Eq. (1.2) becomes the Camassa–Holm equation. The Cauchy problems of the Camassa–Holm equation on the line and on the circle have been studied extensively. It has been shown that this equation is locally well-posed [11,14,22,45,52] for initial data u0 ∈ H s (I ) with s > 32 , where I = R or I = S = R/Z. More interestingly, it has global strong solutions
J. Escher, Z. Yin / Journal of Functional Analysis 256 (2009) 479–508
481
[8,11,12,22] and also finite time blow-up solutions [8,11–13,16,22,63]. On the other hand, it has global weak solutions in H 1 (I ), cf. [4,17,22,51,58,60]. The advantage of the Camassa–Holm equation in comparison with the KdV equation lies in the fact that the Camassa–Holm equation has peaked solitons and models wave breaking [6,13]. If α = 1 and b = 3 in Eq. (1.2), then we find the Degasperis–Procesi equation [23]. The formal integrability of the Degasperis–Procesi equation was obtained in [24] by constructing a Lax pair. It has a bi-Hamiltonian structure with an infinite sequence of conserved quantities and admits exact peakon solutions which are analogous to the Camassa–Holm peakons [24]. The Degasperis–Procesi equation can be regarded as a model for nonlinear shallow water dynamics and its asymptotic accuracy is the same as for the Camassa–Holm shallow water equation [28,29]. An inverse scattering approach for computing n-peakon solutions to the Degasperis– Procesi equation was presented in [48]. Its traveling wave solutions was investigated in [44,57]. The Cauchy problems for the Degasperis–Procesi equation on the line and on the circle have been studied recently. Local well-posedness of this equation has been established in [63,64] for initial data u0 ∈ H s (I ) with s > 32 , where I = R or I = S = R/Z. Similar to the Camassa–Holm equation, the Degasperis–Procesi equation has also global strong solutions [34,35,46,63,65] and as well as finite time blow-up solutions [30,34,35,46,63–65]. On the other hand, it has global weak solutions in H 1 (I ) [34,65,66] and global entropy weak solutions belonging to the class L1 (R) ∩ BV (R) and to the class L2 (R) ∩ L4 (R), cf. [7]. Although the Degasperis–Procesi equation is similar to the Camassa–Holm equation in several aspects, these two equations are truly different. One of the novel features of the Degasperis– Procesi different from the Camassa–Holm equation is that it has not only peakon solutions [24] and periodic peakon solutions [66], but also shock peakons [47] and periodic shock waves [35]. The Cauchy problem for Eq. (1.2) with α = 0 on the line has been discussed recently in [33]. The local well-posedness for the b-equation, a precise blowup scenario, several blowup results and global existence results of strong solutions, and the uniqueness and existence of global weak solutions to the b-equation on the line have been proved recently in [33]. Furthermore, the initial boundary value problem for the Camassa–Holm equation and the Degasperis–Procesi equation on the half-line and on a finite interval were studied in [31,32,43,61]. However, the results in these papers are not sharp. There are still unsolved problems needed to be investigated further. Initial boundary value problems for the rod equation and the b-equation on the half-line and on a finite interval have not been investigated so far. The aim of this paper is to find a general approach to investigate these two problems. Our approach heavily depends on sharp results on the odd extension of functions and the conservation of symmetry of the equation. By using a density argument and Hardy’s inequality, we first establish sharp a priori estimates. By using these estimates and the intrinsic norm of fractional Sobolev spaces, we then establish sharp results on the odd extensions. Note that both the rod equation and the b-equation enjoy the conservation of symmetry. Thus, by this general approach, we can convert initial boundary value problems of the rod equation and the b-equation on the half-line and on the finite interval into Cauchy problems on the whole line and on the circle, respectively. Applying known results for the rod equation and the b-equation, we obtain local well-posedness results, blow-up and global existence results for strong solutions on the half-line and on the finite interval, respectively. Our new results for the rod equation and the b-equation on the half-line and on the finite interval are sharp. In particular, these results cover and improve considerably previous results for the Camassa–Holm equation and the Degasperis–Procesi equation on the half-line and on the finite interval.
482
J. Escher, Z. Yin / Journal of Functional Analysis 256 (2009) 479–508
Our paper is organized as follows. In Section 2, we derive sharp extension results, which are crucial for our approach. In Section 3, by using the conservation of symmetry enjoyed by the rod equation, we study initial boundary value problems of the rod equation on the half-line and on the finite interval. These results improve considerably the previous results for the Camassa–Holm equation obtained in [32,43,49]. In Section 4, we investigate initial boundary value problems of the b-equation on the half-line and on the finite interval by this same method. Here our results improve the previous results for the Degasperis–Procesi equation in [31]. Notation. In the following, we denote by ∗ the spatial convolution. Given a Banach space X, we denote its norm by · X . If s is a real number then we use the notation s = [s]− + {s}+ , where [s]− is an integer and 0 < {s}+ 1. 2. Several key lemmas In the section we present several technical lemmas which will be crucial for our purpose. We first consider the case of the half-line (0, ∞). Lemma 2.1. (a) If s ∈ [0, 12 ], then C0∞ (R+ ) is dense in H s (R+ ). (b) If s ∈ ( 12 , ∞), then C0∞ (R+ ) is dense in H0s (R+ ), where − H0s (R+ ) = f (x) f (x) ∈ H s (R+ ), f (r) (0) = 0, r = 0, . . . , s − 12 . The proof of Lemma 2.1 is given in [56, pp. 219, 220]. Lemma 2.2. Let s ∈ (0, 1) with s = ∞ 0
1 2
be given. If f (x) ∈ C0∞ (R+ ), then
∞∞ |f (x)|2 |f (x) − f (y)|2 4 dx 2 1 + dx dy. x 2s (2s − 1)2 |x − y|1+2s 0 0
Remark 2.1. Lemma 2.2 can be deduced from the conclusion [56, p. 261] and Hardy’s inequality, see [56, p. 262]. However, in order to get the exact constant in the above inequality and to show that s = 12 is a critical value for our problems, we would like to write down the complete proof of Lemma 2.2. Proof. For 1 < x < ∞, we set 1 g(x) := f (x) − x
x
1 f (y) dy = x
0
x
f (x) − f (y) dy.
0
Since f (x) ∈ C0∞ (R+ ), it follows from (2.1) that lim g(x) = lim g(x) = 0.
x→0
x→∞
(2.1)
J. Escher, Z. Yin / Journal of Functional Analysis 256 (2009) 479–508
483
A straightforward computation yields ∞ g(y) g(x) − dy − f (x) y x
1 f (x) + 2 = f (x) − x x
x f (y) dy +
g(x) − f (x) = 0. x
(2.2)
0
By (2.2) and limx→∞ g(x) = 0, we have ∞ f (x) = g(x) −
g(y) dy. y
(2.3)
x
In the case 0 < 2s < 1, we take −2r + 2 = Schwarz inequality, we have
1+2s 2
so that 2s < −2r + 2 < 1. By the Cauchy–
∞ ∞ g(y) 2 ∞ −2r dy y dy g 2 (y)y 2r−2 dy y x
x
x
1 x −2r+1 2r − 1
∞ g 2 (y)y 2r−2 dy. x
The above inequality and Fubini’s theorem lead to ∞ x 0
∞
∞ ∞ g(y) 2 1 −2s−2r+1 2 2r−2 dy dx x g (y)y dy dx y 1 − 2r
−2s
x
x
0
1 = 1 − 2r =
=
2
g (y)y
2r−2
0
x
−2s−2r+1
dx dy
0
1 1 1 − 2r 2s + 2r − 2 4 (2s − 1)2
y
∞
∞
∞
y −2s g 2 (y) dy
0
x −2s g 2 (x) dx.
(2.4)
0
In the case 2s > 1, we take −2r + 2 = 1+2s 2 so that 2s > −2r + 2 > 1. In view of (2.1) and (2.3), ∞ g(y) we have that g(0) = 0 y dy = 0. Thus, an application of the Cauchy–Schwarz inequality yields
484
J. Escher, Z. Yin / Journal of Functional Analysis 256 (2009) 479–508
∞ g(y) 2 x g(y) 2 dy = dy y y x
0
x
y
−2r
x g 2 (y)y 2r−2 dy
dy
0
0
1 x −2r+1 2r − 1
x g 2 (y)y 2r−2 dy. 0
Consequently, ∞ 0
∞
x ∞ g(y) 2 1 −2s −2s−2r+1 2 2r−2 dy dx x x g (y)y dy dx y 1 − 2r x
0
1 = 1 − 2r =
=
0
∞
∞ g 2 (y)y 2r−2
x −2s−2r+1 dx dy
y
0
1 1 1 − 2r 2s + 2r − 2 4 (2s − 1)2
∞
∞
y −2s g 2 (y) dy
0
x −2s g 2 (x) dx.
0
Using (2.4)–(2.5) and Fubini’s theorem, we get ∞ x 0
−2s
∞ f (x) dx 2 2
x
∞
−2s 2
g (x) dx + 2
0
2 1+ 2 1+ =2 1+ =2 1+
x
x
0
4 (2s − 1)2 4 (2s − 1)2 4 (2s − 1)2 4 (2s − 1)2
∞
x −2s g 2 (x) dx
0
∞
∞ g(y) 2 dy dx y
−2s
x −2s−1
0
∞∞
x
2 f (x) − f (y) dy dx
0
2 x −2s−1 f (x) − f (y) dx dy
0 y
∞∞ 2 (x + y)−2s−1 f (x + y) − f (y) dx dy 0 0
(2.5)
J. Escher, Z. Yin / Journal of Functional Analysis 256 (2009) 479–508
4 2 1+ (2s − 1)2
4 2 1+ (2s − 1)2 This completes the proof of the lemma.
∞∞
2 x −2s−1 f (x + y) − f (y) dx dy
0 0
∞∞
2 |x − y|−2s−1 f (x) − f (y) dx dy.
0 0
2
Lemma 2.3. Given s ∈ [0, 12 ) ∪ ( 12 , 1]. Assume that v ∈ H s (R+ ) if 0 s < if 12 < s 1 with v(0) = 0. Let furthermore v(x) ˜ =
485
1 2
or that v ∈ H s (R+ )
v(x), if x 0, −v(−x), if x < 0.
Then v(x) ˜ ∈ H s (R). Proof. It is obvious that the lemma holds true for s = 0. For s = 1, our assumption implies that v(x) ∈ H01 (R+ ). If v ∈ C 1 (R+ ) ∩ H01 (R+ ), then one can readily obtain lim v˜ (x) = lim
x→0−
x→0−
v(x) ˜ − v(0) ˜ −v(−x) = lim = lim v (−x) = v (0) x −0 x x→0− x→0−
= lim v (x) = lim x→0+
x→0+
v(x) v(x) ˜ − v(0) ˜ = lim = lim v˜ (x). x x −0 x→0+ x→0+
This shows that v˜ ∈ C 1 (R) ∩ H 1 (R). Since C 1 (R+ ) ∩ H01 (R+ ) is dense in H01 (R+ ), the above relation shows the conclusion of the lemma is true for s = 1. Next, we prove that the lemma holds true for 0 < s < 1 with s = 12 . Consider first v ∈ C0∞ (R+ ). Applying Lemma 2.2 and the relation (v(x) + v(y))2 2(v 2 (x) + v 2 (y)), we deduce R R
2 |v(x) ˜ − v(y)| ˜ dx dy |x − y|1+2s
∞∞ = 0 0
2 |v(x) ˜ − v(y)| ˜ dx dy + |x − y|1+2s
0 ∞ + −∞ 0
∞∞ = 0 0
∞ 0 0 −∞
2 |v(x) ˜ − v(y)| ˜ dx dy + |x − y|1+2s
|v(x) − v(y)|2 dx dy + |x − y|1+2s
2 |v(x) ˜ − v(y)| ˜ dx dy |x − y|1+2s
0 0
−∞ −∞
∞ 0 0 −∞
2 |v(x) ˜ − v(y)| ˜ dx dy |x − y|1+2s
|v(−x) + v(y)|2 dx dy |x − y|1+2s
486
J. Escher, Z. Yin / Journal of Functional Analysis 256 (2009) 479–508
0 ∞ + −∞ 0
∞∞ =2 0 0
∞∞ 2 0 0
∞∞ 2 0 0
|v(x) + v(−y)|2 dx dy + |x − y|1+2s
0 0
−∞ −∞
∞∞
|v(x) − v(y)|2 dx dy + 2 |x − y|1+2s
0 0
∞∞
|v(x) − v(y)|2 dx dy + 4 |x − y|1+2s
0 0
|v(x) − v(y)|2 4 dx dy + s |x − y|1+2s
2 + 8s −1 + 32s −1 (2s − 1)−2
∞ 0
|v(x) + v(y)|2 dx dy |x + y|1+2s (v 2 (x) + v 2 (y)) dx dy |x + y|1+2s
v 2 (x) dx dy |x|2s
∞∞ 0 0
|v(−x) − v(−y)|2 dx dy |x − y|1+2s
|v(x) − v(y)|2 dx dy. |x − y|1+2s
(2.6)
Note that the intrinsic norm of the fractional Sobolev space H n+σ (I ), where n = 0, 1, 2, . . . , 0 < σ < 1, and I is an open interval of R, is defined as follows, cf. [1]: 2 = f (x)H n (I ) + H n+σ (I )
f (x)2
I I
|f (n) (x) − f (n) (y)|2 dx dy. |x − y|1+2σ
(2.7)
Thus, (2.6) and (2.7) imply that v˜ ∈ H s (R) as v ∈ C0∞ (R+ ). By Lemma 2.1, we know that C0∞ (R+ ) is dense in H s (R+ ) if 0 < s < 12 and in v ∈ H s (R+ ) 1 if 2 < s < 1 with v(0) = 0. This also leads to v˜ ∈ H s (R) for v ∈ H s (R+ ) with 0 < s < 12 or for v ∈ H s (R+ ) with 12 < s < 1 and v(0) = 0. 2 Lemma 2.4. Given s ∈ [0, 1]. Assume that v ∈ H s (R+ ). Let furthermore v(x) ˆ =
v(x), v(−x),
if x 0, if x < 0.
Then v(x) ˆ ∈ H s (R). Proof. It is obvious that the lemma holds true for s = 0. 0,1 (R+ ). In For s = 1, we first consider v ∈ C 1 (R+ ) ∩ H 1 (R+ ). Thus, we have that u ∈ Cloc 0,1 view of the definition of v, ˆ we also have that vˆ ∈ Cloc (R). Note that a locally uniformly Lipschitz 1,1 (R) and continuous function is weakly differentiable, cf. [37], that is vˆ ∈ Wloc v (x), (v) ˆ (x) = v (x) = −v (−x),
A straightforward calculation shows that
if x 0, if x < 0.
J. Escher, Z. Yin / Journal of Functional Analysis 256 (2009) 479–508
2 v(x) ˆ
H 1 (R)
487
2 2 2 2 ˆ L2 (R) + vˆ (x)L2 (R) = 2v(x)L2 (R ) + 2v (x)L2 (R ) = v(x) + + 2 = 2v(x)H 1 (R ) . +
Since C 1 (R+ ) ∩ H 1 (R+ ) is dense in H 1 (R+ ), the above relation shows that v(x) ˆ ∈ H s (R). Next, we prove that the lemma holds true for 0 < s < 1. By the definition of vˆ and the relation |x + y| |x − y| for x, y 0, we can obtain R R
2 |v(x) ˆ − v(y)| ˆ dx dy |x − y|1+2s
∞∞ = 0 0
2 |v(x) ˆ − v(y)| ˆ dx dy + |x − y|1+2s
0 ∞ + −∞ 0
∞∞ = 0 0
+ −∞ 0
∞∞ =2 0 0
∞∞ 4 0 0
0 −∞
2 |v(x) ˆ − v(y)| ˆ dx dy + |x − y|1+2s
2 |v(x) ˆ − v(y)| ˆ dx dy |x − y|1+2s
0 0
−∞ −∞
|v(x) − v(y)|2 dx dy + |x − y|1+2s
0 ∞
∞ 0
∞ 0 0 −∞
|v(x) − v(−y)|2 dx dy + |x − y|1+2s
2 |v(x) ˆ − v(y)| ˆ dx dy |x − y|1+2s
|v(−x) − v(y)|2 dx dy |x − y|1+2s 0 0
−∞ −∞
|v(x) − v(y)|2 dx dy + 2 |x − y|1+2s
∞∞ 0 0
|v(−x) − v(−y)|2 dx dy |x − y|1+2s
|v(x) − v(y)|2 dx dy |x + y|1+2s
|v(x) − v(y)|2 dx dy. |x − y|1+2s
(2.8)
In view of the definition of the norm of H s (R) with 0 < s < 1 given in (2.7), we deduce that vˆ ∈ H s (R). This completes the proof. 2 Lemma 2.5. Given s ∈ ( 12 , 52 ). Assume that v ∈ H s (R+ ) with v(0) = 0. Let furthermore v(x) ˜ =
v(x), if x 0, −v(−x), if x < 0.
Then v(x) ˜ ∈ H s (R). Proof. By Lemma 2.3 and the assumption of the lemma, we see that the lemma holds true for 1 2 < s 1.
488
J. Escher, Z. Yin / Journal of Functional Analysis 256 (2009) 479–508
For 1 s 2, we have v˜ (x) ∈ L2 (R) and v (x), v˜ (x) = v (x) = v (−x),
if x 0, if x < 0.
Note that if u ∈ H n (I ), n = 1, 2, . . . , and σ ∈ [0, 1], then u ∈ H n+σ (I ) iff u(n) ∈ H σ (I ). Set σ = s − 1, we have v (x) ∈ H σ (R+ ), 0 σ 1, it follows from Lemma 2.4 that v˜ (x) = v (x) ∈ H σ (R), 0 σ 1. By v(x) ˜ ∈ H 1 (R), we therefore get v(x) ˜ ∈ H s (R) for 1 s 2. 5 2 For 2 s < 2 , we have that v˜ (x) ∈ L (R) and v (x), v˜ (x) = v (x) = −v (−x),
if x 0, if x < 0.
Setting σ = s − 2, we have v (x) ∈ H σ (R+ ), 0 σ < 12 . Thus it follows from Lemma 2.3 that v˜ (x) = v (x) ∈ H σ (R), 0 σ < 12 . By v(x) ˜ ∈ H 2 (R), we therefore get v(x) ˜ ∈ H s (R) for 5 2 s < 2 . This completes the proof of the lemma. 2 Remark 2.2. For s 52 , under the same assumption of Lemma 2.5, one can not deduce v˜ ∈ H s (R) generally. In order to obtain v˜ ∈ H s (R), one has to add additional conditions. For this we let k = 0, 1, 2, . . . , and for 2k + 12 < s < 2k + 52 we set Dks (R+ ) = v ∈ H s (R+ ) v (2k) (0) = v (2k−2) (0) = · · · = v(0) = 0 . We now have the following lemma. Lemma 2.6. Assume that v ∈ Dks (R+ ), where k = 0, 1, 2, . . . , and 2k + furthermore v(x) ˜ =
v(x), −v(−x),
1 2
< s < 2k + 52 . Let
if x 0, if x < 0.
Then v(x) ˜ ∈ H s (R). Proof. We prove the lemma by induction. It is obvious that the assertion holds true for k = 0 and 1 5 2 < s < 2. Assume that the conclusion holds true for some k ∈ N. To proceed from k to k + 1, pick s ∈ (2(k + 1) + 12 , 2(k + 1) + 52 ). s (R ), it then follows that v (2k+2) ∈ H σ (R ), where σ = s − 2(k + 1). By Due to v ∈ Dk+1 + + (2k+2) ∈ H σ (R). Lemma 2.5, we have v By the assumption, we get v(x) ˜ ∈ H s−2 (R). Thus, we have v˜ (2k) (x) is even and (2k) v˜ (2k) (x) = v (2k) (x), v (−x),
if x 0, if x < 0.
J. Escher, Z. Yin / Journal of Functional Analysis 256 (2009) 479–508
489
Since v (2k+1) (0) = 0 and v ∈ H s (R+ ), it follows from the above relation and Lemma 2.3 that v˜ ( 2k + 1)(x) is odd and
v˜
(2k+1)
(2k+1) (x), (x) = v (2k+1) (−x), −v
if x 0, if x < 0.
By Lemma 2.4, we may obtain that v(x) ˜ ∈ H 2k+2 (R) and
v˜
(2k+2)
(2k+1) (x), (x) = v (2k+1) (−x), −v
if x 0, if x < 0.
(2k+2) ∈ H σ (R), in view of (2.7), we deduce that v(x) ˜ ∈ By the above relations together with v H s (R). This completes the proof. 2
We now consider the case of the finite interval (0, l) with l > 0. Lemma 2.7. (a) If s ∈ [0, 12 ], then C0∞ (0, l) is dense in H s (0, l). (b) If s ∈ ( 12 , ∞), then C0∞ (0, l) is dense in H0s (0, l), where − . H0s (0, l) = f (x) f (x) ∈ H s (0, l), f (r) (0) = f (r) (l) = 0, r = 0, . . . , s − 12 The proof of Lemma 2.7 is given in [56, p. 264]. Lemma 2.8. Let functions f1 (x) and f2 (x) be given on [a, c] and [c, b], respectively, −∞ < a < c < b < ∞ and let f1 (x), if a x c, f (x) = f2 (x), if c < x b. If f1 (x) ∈ H s (a, c) and f1 (x) ∈ H s (c, b), 0 < s < 12 , then f (x) ∈ H s (a, b). Proof. In view of Theorem 18.3, [53, p. 337], we know that if 0 < s < 12 , then H s (a, b) = I s L2 (a, b) . By Theorem 13.11 [53, p. 237], we see that if f1 (x) ∈ I s [L2 (a, c)] and f1 (x) ∈ I s [L2 (c, b)], 0 < s < 12 , then f (x) ∈ I s [L2 (a, b)]. Thus, we deduce that f (x) ∈ H s (a, b). 2 Lemma 2.9. Assume that l 0
1 2
< s < 1. If f (x) ∈ C0∞ (0, l), then
11 |f (x)|2 |f (x) − f (y)|2 4 dx 2 1 + dx dy. x 2s (2s − 1)2 |x − y|1+2s 0 0
490
J. Escher, Z. Yin / Journal of Functional Analysis 256 (2009) 479–508
x Proof. For 0 < x 1, we use the transformation t := l−x to map (0, l) onto (0, ∞) and its lt lt inverse x := t+1 to map (0, ∞) onto (0, l). Therefore, if f (x) ∈ C0∞ (0, l), then g(t) = f ( t+1 )∈ 1 ∞ C0 (0, ∞). By Lemma 2.2, in view of the assumption 2 < s < 1, we obtain
l
|f (x)|2 dx = l 1−2s x 2s
0
∞ 0
2l
1−2s
2 1+ 2 1+
|g(t)|2 dt l 1−2s t 2s (t + 1)(2−2s)
4 1+ (2s − 1)2
4 (2s − 1)2 4 (2s − 1)2
∞∞ 0 0
l l 0 0
l l 0 0
∞
|g(t)|2 dt t 2s
0
|g(t) − g(τ )|2 dt dτ |t − τ |1+2s
x 2s−1 y 2s−1 |f (x) − f (y)|2 1 − 1 − dx dy l l |x − y|1+2s |f (x) − f (y)|2 dx dy. |x − y|1+2s
2
This completes the proof of the lemma.
Lemma 2.10. Given s ∈ [0, 12 ) ∪ ( 12 , 1]. Assume that v ∈ H s (0, l) if 0 s < H s (0, l) if 12 < s 1 with v(0) = v(l) = 0. Let furthermore v(x) ˜ =
1 2
or that v ∈
v(x), if x ∈ [0, l), −v(−x), if x ∈ (−l, 0).
Then v(x) ˜ ∈ H s (−l, l). Proof. Following a similar argument as in the proof of Lemma 2.3, we see that the lemma holds true for s = 0 and s = 1. For 0 < s < 12 , we can show that the conclusion of the lemma is true by using Lemma 2.8 with f1 (x) = v(x) and f2 (x) = −v(−x). Next, we prove that the assertion for 12 < s < 1. Consider first v ∈ C0∞ (0, l). Applying Lemma 2.9 and following a similar argument as in the proof of Lemma 2.3, we can obtain l l 0 0
2 |v(x) ˜ − v(y)| ˜ dx dy |x − y|1+2s
l l 2 0 0
|v(x) − v(y)|2 dx dy + 4 |x − y|1+2s
l l 0 0
(v 2 (x) + v 2 (y)) dx dy |x + y|1+2s
J. Escher, Z. Yin / Journal of Functional Analysis 256 (2009) 479–508
l l 2 0 0
|v(x) − v(y)|2 4 dx dy + 1+2s s |x − y|
2 + 8s −1 + 32s −1 (2s − 1)−2
l 0
l l 0 0
491
v 2 (x) dx dy |x|2s |v(x) − v(y)|2 dx dy. |x − y|1+2s
(2.9)
Thus, the above inequality and (2.7) imply that v˜ ∈ H s (−l, l) as v ∈ C0∞ (0, l). By Lemma 2.7(b), we know that C0∞ (0, l) is dense in H s (0, l) as 12 < s < 1 with v(0) = 0. This leads to v˜ ∈ H s (−l, l) if v ∈ H s (0, l) as 12 < s < 1 with v(0) = 0 and completes the proof of the lemma. 2 Arguing as in the proof of Lemma 2.4, we obtain Lemma 2.11. Given s ∈ [0, 1], assume that v ∈ H s (0, l). Let furthermore v(x), if x ∈ [0, l), v(x) ˆ = v(−x), if x ∈ (−l, 0). Then v(x) ˆ ∈ H s (−l, l). Applying Lemmas 2.10, 2.11, we get the following analogue of Lemma 2.5 on the interval (0, l). Lemma 2.12. Given s ∈ ( 12 , 52 ). Assume that v ∈ H s (0, l) with v(0) = 0. Let furthermore v(x) ˜ =
v(x), −v(−x),
if x ∈ [0, l), if x ∈ (−l, 0).
Then v(x) ˜ ∈ H s (−l, l). Remark 2.3. For s 52 , under the same assumption of Lemma 2.12, one can not deduce v˜ ∈ H s (−l, l) generally. In order to obtain v˜ ∈ H s (−l, l), one has to add additional conditions. For this pick k = 0, 1, 2, . . . . For 2k + 12 < s < 2k + 52 we set
Dks (0, l) = v ∈ H s (0, l) v (2k) (0) = v (2k) (l) = v (2k−2) (0) = v (2k−2) (l) = · · · = v(0) = v(l) = 0 . Applying an induction argument and following the lines of the proof of Lemma 2.6, we can get Lemma 2.13. Assume that v ∈ Dks (0, l), where k = 0, 1, 2, . . . , and 2k + furthermore v(x), if x 0, v(x) ˜ = −v(−x), if x < 0.
1 2
< s < 2k + 52 . Let
492
J. Escher, Z. Yin / Journal of Functional Analysis 256 (2009) 479–508
Then v(x) ˜ belongs to Dks (−l, l) = v ∈ H s (−l, l) v (2k) (−l) = v (2k) (l) = v (2k−2) (−l) = v (2k−2) (l) = · · · = v(−l) = v(l) = 0 . 3. The rod equation and Camassa–Holm equation In this section, we investigate initial boundary value problems of the rod equation on the half-line and on a finite interval. As a special case of the rod equation, our results improve considerably the recent results in [32,49,61] on the Camassa–Holm equation on the half-line and on the finite interval. 3.1. The case of the half-line Let us now consider the following initial boundary value problem of Eq. (1.1) on the half-line:
ut − utxx + 3uux = γ (2ux uxx + uuxxx ), u(0, x) = u0 (x), u(t, 0) = 0,
t > 0, x ∈ R+ , x ∈ R+ , t 0.
(3.1)
Here γ is an arbitrary real constant. In the case γ = 1, Eq. (3.1) becomes the Camassa–Holm equation. We first present the following local well-posedness result for Eq. (3.1). Theorem 3.1. Assume that u0 ∈ H s (R+ ) ∩ H01 (R+ ) with 32 < s < 52 . Then there exists a maximal T = T (u0 ) > 0 and a unique solution u(t, x) to Eq. (3.1) such that u = u(·, u0 ) belongs to
C [0, T ); H s (R+ ) ∩ H01 (R+ ) ∩ C 1 [0, T ); H s−1 (R+ ) ∩ H01 (R+ ) .
(3.2)
Moreover, the solution depends continuously on the initial data, i.e. the mapping u0 → u(·, u0 ) : H s (R+ ) ∩ H01 (R+ ) → C([0, T ); H s (R+ ) ∩ H01 (R+ )) ∩ C 1 ([0, T ); H s−1 (R+ ) ∩ H01 (R+ )) is continuous. Furthermore, the maximal T is independent of s in the following sense: if the solution u = u(·, u0 ) to Eq. (3.1) satisfies (3.2) and if u0 ∈ H s (R+ ) ∩ H01 (R+ ) with 32 < s s < 52 , then
u ∈ C [0, T ); H s (R+ ) ∩ H01 (R+ ) ∩ C 1 [0, T ); H s −1 (R+ ) ∩ H01 (R+ ) with the same T . Proof. We first convert the initial boundary value problem of Eq. (3.1) into the Cauchy problem of the rod equation on the line. In order to do so, we extend the initial data u0 (x) defined on the half-line into an odd function defined on the line: x 0, u0 (x), u˜ 0 (x) = (3.3) −u0 (−x), x < 0. Note that u0 (x) ∈ H s (R+ ) ∩ H01 (R+ ) with 32 < s < 52 . The relation (3.3) and Lemma 2.5 show that u˜ 0 (x) ∈ H s (R) with 32 < s < 52 is an odd function.
J. Escher, Z. Yin / Journal of Functional Analysis 256 (2009) 479–508
We now can convert Eq. (3.1) into the rod equation on the whole line. u˜ t − u˜ txx + 3u˜ u˜ x = γ (2u˜ x u˜ xx + u˜ u˜ xxx ), t > 0, x ∈ R, x ∈ R. u(0, ˜ x) = u˜ 0 (x) (odd),
493
(3.4)
Note that if p(x) := 12 e−|x| , x ∈ R, then (1 − ∂x2 )−1 f = p ∗ f for all f ∈ L2 (R) and p ∗ (u˜ − ˜ Using this identity, we can rewrite Eq. (3.4) as a quasi-linear hyperbolic evolution u˜ xx ) = u. equation of the following type: ⎧ 3−γ 2 γ 2 ⎨ u˜ + u˜ x = 0, t > 0, x ∈ R, u˜ t + γ u˜ u˜ x + ∂x p ∗ (3.5) 2 2 ⎩ u(0, ˜ x) = u˜ 0 (x) (odd), x ∈ R. Applying the previous local well-posedness result of the Cauchy problem for the rod equation on the line [18], we conclude that there exists a maximal T = T (u˜ 0 ) > 0, and a unique solution u(t, ˜ x) to Eq. (3.5) such that
u˜ = u(·, ˜ u˜ 0 ) ∈ C [0, T ); H s (R) ∩ C 1 [0, T ); H s−1 (R) . Moreover, the solution depends continuously on the initial data, i.e. the mapping u˜ 0 → u(·, ˜ u˜ 0 ) : H s (R) → C([0, T ); H s (R)) ∩ C 1 ([0, T ); H s−1 (R)) is continuous. Furthermore, the maximal T is independent of s in the following sense: if the solution u˜ = u(·, ˜ u˜ 0 ) to Eq. (3.5) in C([0, T ); H s (R)) ∩ C 1 ([0, T ); H s−1 (R)) and if u˜ 0 ∈ H s (R) with 32 < s s < 52 , then u˜ ∈ C([0, T ); H s (R)) ∩ C 1 ([0, T ); H s −1 (R)) with the same T . In addition, if u(t, ˜ x) is a solution to Eq. (3.5), then it is not difficult to check that the function ˜ −x), (t, x) ∈ [0, T ) × R is also a solution of Eq. (3.5) in C([0, T ); H s (R)) ∩ u˜ 1 (t, x) := −u(t, ˜ Thus C 1 ([0, T ); H s−1 (R)) with the same initial data u˜ 0 . By uniqueness we conclude that u˜ 1 ≡ u. u(t, ˜ x) is odd for any t ∈ [0, T ). In particular, we have u(t, ˜ 0) ≡ 0 for all t ∈ [0, T ). Set u(t, x) = u(t, ˜ x) for all (t, x) ∈ [0, T ) × R+ . Then we see that u(t, x) ∈ C([0, T ); H s (R+ ) ∩ H01 (R+ )) ∩ C 1 ([0, T ); H s−1 (R+ ) ∩ H01 (R+ )) is a solution to Eq. (3.1). On the other hand, if v(t, x) is also a solution to Eq. (3.1) with the same initial data u0 (x), then v(t, x), x 0, v(t, ˜ x) = (3.6) −v(t, −x), x < 0, is the unique solution in (3.2) to Eq. (3.5) with the initial data u˜ 0 (x). By the uniqueness, we conclude that u(t, x) = v(t, x) is the unique solution to Eq. (3.1) with the initial data u0 (x). ˜ u˜ 0 ) implies that of u0 → u(·, u0 ) as well. 2 Obviously, the continuity of u˜ 0 → u(·, Remark 3.1. Assume that u0 ∈ H s (R+ ) ∩ H01 (R+ ) with s 52 . Thus, given r ∈ ( 32 , 52 ), it follows from Theorem 3.1 that there exists a maximal T = T (u0 ) > 0 and a unique solution u(t, x) to Eq. (3.1) in the class
C [0, T ); H r (R+ ) ∩ H01 (R+ ) ∩ C 1 [0, T ); H r−1 (R+ ) ∩ H01 (R+ ) . However, the arguments of the proof of Theorem 3.1 cannot be used to conclude that
u ∈ C [0, T ); H s (R+ ) ∩ H01 (R+ ) ∩ C 1 [0, T ); H s−1 (R+ ) ∩ H01 (R+ ) .
494
J. Escher, Z. Yin / Journal of Functional Analysis 256 (2009) 479–508
In order to study more regular solution, we may consider the following initial boundary value problem: ⎧ t > 0, x ∈ R+ , ⎨ ut − utxx + 3uux = γ (2ux uxx + uuxxx ), (3.7) x ∈ R+ , u(0, x) = u0 (x), ⎩ (2k) u (t, 0) = u(2k−2) (t, 0) = · · · = u(t, 0) = 0, t 0. We next present the following local well-posedness result. Theorem 3.2. Assume that u0 ∈ Dks (R+ ), where k = 1, 2, . . . , and 2k + 12 < s < 2k + 52 . Then there exists a maximal T = T (u0 ) > 0, and a unique solution u(t, x) to Eq. (3.7) such that u = u(·, u0 ) ∈ C([0, T ); Dks (R+ )) ∩ C 1 ([0, T ); Dks−1 (R+ )). Moreover, the solution depends continuously on the initial data, i.e. the mapping u0 → u(·, u0 ) : Dks (R) → C([0, T ); Dks (R+ )) ∩ C 1 ([0, T ); Dks−1 (R+ )) is continuous. Furthermore, the maximal T is independent of s in the following sense. If the solution
u = u(·, u0 ) ∈ C [0, T ); Dks (R+ ) ∩ C 1 [0, T ); Dks−1 (R+ )
is the solution to (3.7) and if u0 ∈ Dks (R+ ) with 2k + 12 < s s < 2k + 52 , then u ∈ C([0, T ); Dks (R+ )) ∩ C 1 ([0, T ); Dks −1 (R+ )) with the same T . Proof. Following a similar argument as in Theorem 3.1, we first extend the initial data u0 (x) defined on the half-line into an odd function u˜ 0 (x) defined in (3.3) on the line. Since u0 ∈ Dks (R+ ), Lemma 2.6 implies that u˜ 0 (x) ∈ H s (R) is an odd function. The conclusions follow now as in Theorem 3.1. 2 Remark 3.2. From Lemmas 2.5, 2.6 and Theorems 3.1, 3.2, we see that s = 2k + 12 , where k = 1, 2, . . . , are the critical points for the problem (3.7). Moreover, for γ = 1, Theorems 3.1, 3.2 improve considerably the well-posedness results in [33,49]. We now present a precise blow-up scenario of strong solutions to Eq. (3.1). Theorem 3.3. Given u0 ∈ H s (R+ ) ∩ H01 (R+ ) with 32 < s < 52 , blow up of the solution u = u(·, u0 ) to Eq. (3.1) in finite time T < +∞ occurs if and only if lim inf inf γ ux (t, x) = −∞. t↑T
x∈R+
Proof. As before, we first extend the initial data u0 (x) defined on the half-line into an odd func˜ x) tion u˜ 0 (x) defined in (3.3) on the line. By Theorem 3.1, we can obtain the odd solution u(t, which is the corresponding strong solution to Eq. (3.5) with the initial data u˜ 0 (x). Moreover, u(t, x) = u(t, ˜ x) confined on [0, T ) × R+ is the unique strong solution to Eq. (3.1) with the initial data u0 (x). For the rod equation on the line [18], we know that blow up of the solution u˜ = u(·, ˜ u˜ 0 ) to Eq. (3.1) in finite time T < +∞ occurs if and only if lim inf inf γ u˜ x (t, x) = −∞. t↑T
x∈R
J. Escher, Z. Yin / Journal of Functional Analysis 256 (2009) 479–508
495
Since u(t, ˜ ·) is odd, it follows that u˜ x (t, ·) is even. Thus, we have that lim inf inf γ u˜ x (t, x) = lim inf inf γ ux (t, x) . t↑T
x∈R
t↑T
(3.8)
x∈R+
The above two relations imply the desired result of the theorem.
2
Next, we present two blow-up results and one global existence result for Eq. (3.1). Theorem 3.4. Let u0 ∈ H s (R+ ) ∩ H01 (R+ ) with 32 < s < 52 . Assume that there exists x0 ∈ R+ | 3+γ 2 2 2 such that [u 0 (x0 )]2 > 3+|γ 2|γ | u0 H 1 (R+ ) if γ < 0 or [u0 (x0 )] > γ u0 H 1 (R+ ) if γ > 0. Then the corresponding solution to Eq. (3.1) blows up in finite time. Proof. Again, we may extend the initial data u0 (x) defined on the half-line into an odd function ˜ x) which is the u˜ 0 (x) defined in (3.3) on the line. By Theorem 3.1, we obtain the odd solution u(t, corresponding strong solution to Eq. (3.5) with the initial data u˜ 0 (x). Moreover, u(t, x) = u(t, ˜ x) confined on [0, T ) × R+ is the unique strong solution to Eq. (3.1) with the initial data u0 (x). For the rod equation on the line, we know that if there exists x1 ∈ R such that [u˜ 0 (x1 )]2 > 3+|γ | ˜ 0 2H 1 (R ) as γ < 0 or [u˜ 0 (x1 )]2 > 3+γ ˜ 0 2H 1 (R ) as γ > 0, then the corresponding 2|γ | u γ u + + solution to Eq. (3.5) blows up in finite time, see [18]. Since u(t, ˜ ·) is odd, in view of Theorem 3.3 and (3.8), it follows that there exists x0 ∈ R+ such that 2 2 3 + |γ | 3 + |γ | u0 2H 1 (R ) = u˜ 0 2H 1 (R ) u˜ 0 (x0 ) = u 0 (x0 ) > + + |γ | 2|γ |
if γ < 0,
or 2 2 3 + γ 3+γ u0 2H 1 (R ) = u˜ 0 2H 1 (R ) u˜ 0 (x1 ) = u 0 (x0 ) > + + 2γ γ Thus the corresponding solution u(t, x) to Eq. (3.1) blows up in finite time.
if γ > 0. 2
Theorem 3.5. Let u0 ∈ H s (R+ ) ∩ H01 (R+ ) with 32 < s < 52 . If γ = 3 and u0 ≡ 0, then the corresponding solution to Eq. (3.1) blows up in finite time. Proof. Let u˜ 0 (x) be defined in (3.3). Then the assumption of the theorem ensure u˜ 0 (x) ≡ 0. For the rod equation on the line, we know that if γ = 3 and u˜ 0 (x) ≡ 0, then the corresponding solution u(t, ˜ x) to Eq. (3.5) blows up in finite time, see [18]. Since u(t, ˜ ·) is odd, in view of Theorem 3.3 and (3.8), it follows that the corresponding solution u(t, x) to Eq. (3.1) blows up in finite time. 2 Theorem 3.6. Let u0 ∈ H s (R+ ) ∩ H01 (R+ ) with solution to Eq. (3.1) exists globally in time.
3 2
< s < 52 . If γ = 0, then the corresponding
Proof. Let u˜ 0 (x) be defined in (3.3). In the case of the rod equation on the line, we know that if γ = 0, then the corresponding solution u(t, ˜ x) to Eq. (3.5) exists globally in time, see [18]. Thus,
496
J. Escher, Z. Yin / Journal of Functional Analysis 256 (2009) 479–508
u(t, x) = u(t, ˜ x) confined on [0, T ) × R+ is the unique global strong solution to Eq. (3.1) with the initial data u0 (x). 2 Note that if γ = 1, then the rod equation becomes the Camassa–Holm equation. In view of Lemma 2.5 and of [32, Theorems 2.4–2.6], we obtain the following blow-up results and global existence results for the Camassa–Holm equation, which improve earlier results. Theorem 3.7. Assume that u0 ∈ H s (R+ ) ∩ H01 (R+ ) with 32 < s < 52 and that there exists x0 ∈ R+ such that u 0 (x0 ) < −u0 H 1 (R+ ) . Then the corresponding solution to Eq. (3.1) with γ = 1 blows up in finite time. Theorem 3.8. Let u0 ∈ H s (R+ ) ∩ H01 (R+ ) with 32 < s < 52 and y0 (x) := u0 (x) − u0,xx (x). Assume that y0 (x) ≡ 0 and y0 (x) 0 for all x ∈ R+ . Then the corresponding solution to Eq. (3.1) with γ = 1 blows up in finite time. Theorem 3.9. Let u0 ∈ H s (R+ ) ∩ H01 (R+ ) with 32 < s < 52 and y0 (x) := u0 (x) − u0,xx (x). Assume that y0 (x) 0 for all x ∈ R+ . Then the corresponding solution to Eq. (3.1) with γ = 1 exists globally in time. Remark 3.3. Let u0 ∈ Dks (R+ ), where k = 1, 2, . . . , and 2k + 12 < s < 2k + 52 . Then Theorems 3.3–3.9 hold true for the corresponding solution u(t, x) to Eq. (3.7). 3.2. The case of the interval (0, 12 ) In this subsection, we study initial boundary value problems of the rod equation on the interval (0, 12 ). More precisely, let us consider the following problem: ⎧
⎪ x ∈ 0, 12 , ⎨ ut − utxx + 3uux = γ (2ux uxx + uuxxx ), t > 0,
(3.9) u(0, x) = u0 (x), x ∈ 0, 12 , ⎪
1 ⎩ u(t, 0) = u t, 2 = 0, t 0. We first present the local well-posedness result for Eq. (3.9). Theorem 3.10. Assume that u0 ∈ H s (0, 12 ) ∩ H01 (0, 12 ) with 32 < s < 52 . Then there exists a maximal T = T (u0 ) > 0, and a unique solution u(t, x) to Eq. (3.9) such that u = u(·, u0 ) belongs to
C [0, T ); H s 0, 12 ∩ H01 0, 12 ∩ C 1 [0, T ); H s−1 0, 12 ∩ H01 0, 12 . (3.10) Moreover, the solution depends continuously on the initial data, i.e. the mapping u0 → u(·, u0 ) : H s (0, 12 ) ∩ H01 (0, 12 ) → C([0, T ); H s (0, 12 ) ∩ H01 (0, 12 )) ∩ C 1 ([0, T ); H s−1 (0, 12 ) ∩ H01 (0, 12 )) is continuous. Furthermore, the maximal T is independent of s in the following sense: if the solution u = u(·, u0 ) to Eq. (3.9) satisfies (3.10) and if u0 ∈ H s (0, 12 ) ∩ H01 (0, 12 ) with 32 < s s < 52 , then
u ∈ C [0, T ); H s 0, 12 ∩ H01 0, 12 ∩ C 1 [0, T ); H s −1 0, 12 ∩ H01 0, 12 with the same T .
J. Escher, Z. Yin / Journal of Functional Analysis 256 (2009) 479–508
497
Proof. We first convert the initial boundary value problem of the rod equation on the interval (0, 12 ) into the Cauchy problem of the periodic rod equation with period 1. In order to do so, we extend the initial data u0 (x) defined on the interval (0, 12 ) into a periodic odd function defined on the line: u˜ 0 (x) =
Note that u0 (x) ∈ H s (0, 12 ) ∩ H01 (0, 12 ) with lim x→ 12
−
x ∈ [n, 12 + n], n ∈ Z, x ∈ [n − 12 , n], n ∈ Z.
u0 (x), −u0 (−x), 3 2
(3.11)
< s < 52 . Then we have
u˜ 0 (x) − u˜ 0 ( 12 ) d u˜ 0 (x) u0 (x) −u0 (−x) = lim = lim = lim 1 1 1 − − dx x−2 x→(− 12 )+ x + 2 x→ 12 x→ 12 x − 2 =
lim
x→(− 12 )+
u˜ 0 (x) − u˜ 0 (− 12 ) x
− (− 12 )
=
lim
x→(− 12 )+
d u˜ 0 (x) d u˜ 0 (x) = lim . 1+ dx dx x→ 2
Combining the above relation with (3.11) and Lemma 2.12 with l = 12 , we have that u˜ 0 (x) ∈ H s (− 12 , 12 ) ∩ H01 (− 12 , 12 ) with 32 < s < 52 is a periodic odd function. Thus, we may convert the rod equation on the interval (0, 12 ) into the following periodic problem: ⎧ u˜ t − u˜ txx + 3u˜ u˜ x = γ (2u˜ x u˜ xx + u˜ u˜ xxx ), ⎪ ⎪ ⎨ u(0, ˜ x) = u˜ 0 (x) (odd), 1 ⎪ u˜ 0 (0) = u˜ 0 2 = 0, ⎪ ⎩ u(t, ˜ x) = u˜ 0 (t, x + 1),
t > 0, x ∈ R, x ∈ R,
(3.12)
t 0, x ∈ R.
Applying the local well-posedness result of the periodic rod equation [62], we have that there exists a maximal T = T (u˜ 0 ) > 0 and a unique solution u(t, ˜ x) to Eq. (3.12) such that u˜ = u(·, ˜ u˜ 0 ) belongs to
C [0, T ); H s (0, 1) ∩ H01 (0, 1) ∩ C 1 [0, T ); H s−1 (0, 1) ∩ H01 (0, 1) .
(3.13)
Moreover, the solution depends continuously on the initial data, i.e. the mapping u˜ 0 → u(·, ˜ u˜ 0 ) : H s (0, 1) ∩ H01 (0, 1) → C([0, T ); H s (0, 1) ∩ H01 (0, 1)) ∩ C 1 ([0, T ); H s−1 (0, 1) ∩ H01 (0, 1)) is continuous. Furthermore, the maximal T is independent of s in the following sense: if the solu tion u˜ = u(·, ˜ u˜ 0 ) to Eq. (3.4) satisfies (3.5) and if u˜ 0 ∈ H s (0, 1) ∩ H01 (0, 1) with 32 < s s < 52 , then u˜ belongs to
C [0, T ); H s (0, 1) ∩ H01 (0, 1) ∩ C 1 [0, T ); H s −1 (0, 1) ∩ H01 (0, 1) with the same T . Note that if u(t, ˜ x) is a solution to Eq. (3.12), then one can check that the function v(t, x) := −u(t, ˜ −x), (t, x) ∈ [0, T ) × R is also a solution of Eq. (3.12) satisfying (3.13) with the initial data u˜ 0 . By uniqueness we conclude that v ≡ u˜ and u(t, ˜ ·) is odd for any t ∈ [0, T ). Therefore, we have u(t, ˜ 0) ≡ 0 for all t ∈ [0, T ).
498
J. Escher, Z. Yin / Journal of Functional Analysis 256 (2009) 479–508
Set u(t, x) = u(t, ˜ x) for all (t, x) ∈ [0, T ) × [0, 12 ]. Then we know that u(t, x) ∈ C([0, T ); H s (0, 12 ) ∩ H01 (0, 12 )) ∩ C 1 ([0, T ); H s−1 (0, 12 ) ∩ H01 (0, 12 )) is a solution to Eq. (3.9). On the other hand, if v(t, x) is a solution to Eq. (3.9) with the initial data u0 (x), then v(t, ˜ x) =
v(t, x), x ∈ [n, 12 + n], n ∈ Z, −v(t, −x), x ∈ [n − 12 , n], n ∈ Z,
(3.14)
is the unique solution to Eq. (3.12) with the initial data u˜ 0 (x) satisfying (3.13). By the uniqueness of u(t, ˜ x), we conclude that u(t, x) = v(t, x) is the unique solution to Eq. (3.9) with the initial data u0 (x). Moreover, the solution u(t, x) depends continuously on the initial data u0 (x) and the maximal T is independent of s. This completes the proof of the theorem. 2 As in the case on the line, our method of proof is not suitable to study more regular solution in the class
with s > 52 . C [0, T ); H s 0, 12 ∩ H01 0, 12 ∩ C 1 [0, T ); H01 0, 12 But, we may consider the following initial boundary value problem: ⎧ ut − utxx + 3uux = γ (2ux uxx + uuxxx ), ⎪ ⎪ ⎪ ⎨ u(0, x) = u (x), 0 ⎪ u(2k) (t, 0) = u(2k−2) (t, 0) = · · · = u(t, 0) = 0, ⎪ ⎪
⎩ (2k) 1 u t, 2 = u(2k−2) t, 12 = · · · = u t, 12 = 0,
t > 0, x ∈ 0, 12 , x ∈ 0, 12 , t 0, t 0.
(3.15)
Then we have the following local well-posedness result. Theorem 3.11. Assume that u0 ∈ Dks (0, 12 ), where k = 1, 2, . . . , and 2k + 12 < s < 2k + 52 . Then there exists a maximal T = T (u0 ) > 0, and a unique solution u(t, x) to Eq. (3.15) such that u = u(·, u0 ) ∈ C([0, T ); Dks (0, 12 )) ∩ C 1 ([0, T ); Dks−1 (0, 12 )). Moreover, the solution depends continuously on the initial data, i.e. the mapping
u0 → u(·, u0 ) : Dks 0, 12 → C [0, T ); Dks 0, 12 ∩ C 1 [0, T ); Dks−1 0, 12 is continuous. Furthermore, the maximal T is independent of s in the following sense. If the solution
u = u(·, u0 ) ∈ C [0, T ); Dks 0, 12 ∩ C 1 [0, T ); Dks−1 0, 12
is the solution to Eq. (3.15) and if u0 ∈ Dks (0, 12 ) with 2k + 12 < s s < 2k + 52 , then u ∈ C([0, T ); Dks (0, 12 )) ∩ C 1 ([0, T ); Dks −1 (0, 12 )) with the same T . Proof. Following a similar argument of as in the proof of Theorem 3.10, we first extend the initial data u0 (x) defined on the interval (0, 12 ) into an periodic odd function u˜ 0 (x) defined in (3.11) on the line. Since u0 ∈ Dks (0, 12 ), Lemma 2.13 and (3.11) show that u˜ 0 (x) ∈ D s (0, 1) is a periodic odd function. Thus the result follows from Theorem 3.10. 2
J. Escher, Z. Yin / Journal of Functional Analysis 256 (2009) 479–508
499
Remark 3.4. For s = 4 and k = 1, Theorem 3.4 with γ = 1 covers and improves considerably the well-posedness results of the Camassa–Holm equation in [32,43]. Next, we present a precise blow-up scenario of strong solutions to Eq. (3.9). Theorem 3.12. Given u0 ∈ H s (0, 12 ) ∩ H01 (0, 12 ) with 32 < s < 52 , blow up of the solution u = u(·, u0 ) to Eq. (3.9) in finite time T < +∞ occurs if and only if lim inf inf γ ux (t, x) = −∞. t↑T
x∈[0, 12 ]
Proof. As before, we first extend the initial data u0 (x) defined on the interval [0, 12 ] into an periodic odd function u˜ 0 (x) defined in (3.11) on the line. By Theorem 3.10, we obtain the odd function u(t, ˜ x) which is the corresponding strong solution to Eq. (3.12) with the initial data ˜ x) confined on [0, T ) × [0, 12 ] is the unique strong solution to u˜ 0 (x). Moreover, u(t, x) = u(t, Eq. (3.9) with the initial data u0 (x). For the periodic rod equation [62], blow up of the solution u˜ = u(·, ˜ u˜ 0 ) to Eq. (3.12) in finite time T < +∞ occurs if and only if lim inf inf γ u˜ x (t, x) = −∞. t↑T
x∈[0,1]
Since u(t, ˜ ·) is odd, it follows that u˜ x (t, ·) is even. Thus, we have that lim inf inf γ u˜ x (t, x) = lim inf inf γ ux (t, x) . t↑T
t↑T
x∈[0,1]
The above two relation imply the desired result.
x∈[0, 12 ]
(3.16)
2
Next, we present a global existence result and two blow-up results for Eq. (3.9). Theorem 3.13. Let u0 ∈ H s (0, 12 ) ∩ H01 (0, 12 ) with solution to Eq. (3.9) exists globally in time.
3 2
< s < 52 . If γ = 0, then the corresponding
Proof. Let u˜ 0 (x) be defined in (3.11). For the rod equation on the line, we know that if γ = 0, then the corresponding solution u(t, ˜ x) to Eq. (3.12) exists globally in time, see [63]. Thus, u(t, x) = u(t, ˜ x) confined on [0, T ) × [0, 12 ] is the unique global strong solution to Eq. (3.9) with the initial data u0 (x). 2 Theorem 3.14. Let u0 ∈ H s (0, 12 ) ∩ H01 (0, 12 ) with (−∞, −
3 cosh( 12 ) 2−cosh( 12 )
) ∪ (0, ∞), or if γ ∈ [−
γ u0 (0) < −
3 cosh( 12 ) 2−cosh( 12 )
3 2
<s<
5 2
be given and u0 (x) ≡ 0. If γ ∈
, 0) and
γ (γ − 3) cosh( 12 ) − 2γ 2 2 sinh( 12 )
1 2
u0 H 1 (0, 1 ) , 2
then the corresponding strong solution to Eq. (3.9) blows up in finite time.
500
J. Escher, Z. Yin / Journal of Functional Analysis 256 (2009) 479–508
Proof. Again, we may extend the initial data u0 (x) defined on the interval [0, 12 ] into an periodic odd function u˜ 0 (x) defined in (3.11) on the line. By Theorem 3.10, we obtain the periodic odd function u(t, ˜ x) which is the corresponding strong solution to Eq. (3.12) with the initial data ˜ x) confined on [0, T ) × [0, 12 ] is the unique strong solution to u˜ 0 (x). Moreover, u(t, x) = u(t, Eq. (3.9) with the initial data u0 (x). For the periodic rod equation, see [62, Theorem 3.4], we know that if u˜ 0 (x) is odd and u˜ 0 (x) ≡ 0 and if γ ∈ (−∞, − −(
3 cosh( 12 ) ) 2−cosh( 12 )
γ (γ −3) cosh( 12 )−2γ 2 1 ) 2 u˜ 0 H 1 (0,1) , 4 sinh( 12 )
∪ (0, ∞), or if γ ∈ [−
3 cosh( 12 ) , 0) 2−cosh( 12 )
and γ u˜ 0 (0) <
then the corresponding strong solution to Eq. (3.12) blows
up in finite time. Note that u(t, ˜ ·) is odd and
γ u˜ 0 (0) = γ u0 (0) < − =−
γ (γ − 3) cosh( 12 ) − 2γ 2
1
2 sinh( 12 )
γ (γ − 3) cosh( 12 ) − 2γ 2 4 sinh( 12 )
2
u0 H 1 (0, 1 ) 2
1 2
u˜ 0 H 1 (0,1) .
By the assumptions of the theorem and Theorem 3.11, we deduce that u(t, x) = u(t, ˜ x) confined on [0, T ) × [0, 12 ] to Eq. (3.9) blows up in finite time. 2 As a generalization of Theorem 3.14, we have the following result. Theorem 3.15. Assume that u0 ∈ Dks (0, 12 ), where k = 1, 2, . . . , and 2k + and u0 ≡ 0. If γ ∈ (−∞, − −(
3 cosh( 12 ) 2−cosh( 12 )
γ (γ −3) cosh( 12 )−2γ 2 1 ) 2 u0 H 1 (0, 1 ) , 2 sinh( 12 ) 2
) ∪ (0, ∞), or if γ ∈ [−
3 cosh( 12 ) 2−cosh( 12 )
1 2
< s < 2k + 52 ,
, 0) and γ u0 (0) <
then the corresponding solution to Eq. (3.15) blows up in
finite time.
Remark 3.5. For s = 4 and k = 1, Theorem 3.15 with γ = 1 covers and improves considerably the blow-up results of the Camassa–Holm equation in [32,43]. 4. The b-equation and the Degasperis–Procesi equation In this section, we use the same method in Sections 2, 3 to deal with initial boundary value problems of the b-equation on the half-line (0, ∞) and on the interval (0, 12 ). As a special case of the b-equation, our results improve considerably the recent results in [32] on the Degasperis– Procesi equation on the half-line and on the finite interval. 4.1. The case of the half-line Consider the following initial boundary value problem of the b-equation on the half-line: ⎧ ⎨ ut − α 2 utxx + (b + 1)uux = α 2 (bux uxx + uuxxx ), t > 0, x ∈ R+ , (4.1) x ∈ R+ , u(0, x) = u0 (x), ⎩ u(t, 0) = 0, t 0.
J. Escher, Z. Yin / Journal of Functional Analysis 256 (2009) 479–508
501
Here b and α 2 > 0 are real constants. For the values b = 2 and b = 3, Eq. (4.1) with α = 1 becomes the Camassa–Holm equation and the Degasperis–Procesi equation. We first present the local well-posedness result for Eq. (4.1). Theorem 4.1. Assume that u0 ∈ H s (R+ ) ∩ H01 (R+ ) with 32 < s < 52 . Then there exists a maximal T = T (u0 ) > 0 and a unique solution u(t, x) to Eq. (4.1) such that u = u(·, u0 ) belongs to
C [0, T ); H s (R+ ) ∩ H01 (R+ ) ∩ C 1 [0, T ); H s−1 (R+ ) ∩ H01 (R+ ) .
(4.2)
Moreover, the solution depends continuously on the initial data, i.e. the mapping u0 → u(·, u0 ) : H s (R+ ) ∩ H01 (R+ ) → C([0, T ); H s (R+ ) ∩ H01 (R+ )) ∩ C 1 ([0, T ); H s−1 (R+ ) ∩ H01 (R+ )) is continuous. Furthermore, the maximal T is independent of s in the following sense: if the solution u = u(·, u0 ) to Eq. (4.1) satisfies (4.2) and if u0 ∈ H s (R+ ) ∩ H01 (R+ ) with 32 < s s < 52 , then
u ∈ C [0, T ); H s (R+ ) ∩ H01 (R+ ) ∩ C 1 [0, T ); H s −1 (R+ ) ∩ H01 (R+ ) with the same T . Proof. The outline of the proof of Theorem 4.1 is the same as that of Theorem 3.1. We first extend the initial data u0 (x) defined on the half-line into an odd function u˜ 0 (x) defined in (3.3) on the line. Then, we know that u˜ 0 (x) ∈ H s (R) is an odd function by using Lemma 2.5 and (3.3). Note that if the strong solution to Eq. (4.1) is odd initially, then it will be odd as long as it exists. We thus can convert Eq. (4.1) into the b-equation on the whole line.
u˜ t − α 2 u˜ txx + (b + 1)u˜ u˜ x = α 2 (bu˜ x u˜ xx + u˜ u˜ xxx ), u(0, ˜ x) = u˜ 0 (x) (odd),
t > 0, x ∈ R, x ∈ R.
(4.3)
Applying the previous local well-posedness result of the Cauchy problem for Eq. (4.3) in [33] and following the similar proof in Theorem 3.1, we finally obtain local well-posedness result of the Cauchy problem for (4.1). 2 Odd solutions of higher regularity satisfy the following initial boundary value problem: ⎧ ⎨ ut − α 2 utxx + (b + 1)uux = α 2 (bux uxx + uuxxx ), u(0, x) = u0 (x), ⎩ (2k) u (t, 0) = u(2k−2) (t, 0) = · · · = u(t, 0) = 0,
t > 0, x ∈ R+ , x ∈ R+ , t 0.
(4.4)
Similar to Theorem 3.2, we have the following result. Theorem 4.2. Assume that u0 ∈ Dks (R+ ), where k = 1, 2, . . . , and 2k + 12 < s < 2k + 52 . Then there exists a maximal T = T (u0 ) > 0 and a unique solution u(t, x) to Eq. (4.4) such that u = u(·, u0 ) ∈ C([0, T ); Dks (R+ )) ∩ C 1 ([0, T ); Dks−1 (R+ )). Moreover, the solution depends continuously on the initial data, i.e. the mapping u0 → u(·, u0 ) : Dks (R) → C([0, T ); Dks (R+ )) ∩ C 1 ([0, T ); Dks−1 (R+ )) is continuous. Furthermore, the maximal T is independent of s in the following sense: if the solution
u = u(·, u0 ) ∈ C [0, T ); Dks (R+ ) ∩ C 1 [0, T ); Dks−1 (R+ )
502
J. Escher, Z. Yin / Journal of Functional Analysis 256 (2009) 479–508
is the solution to (4.4) and if u0 ∈ Dks (R+ ) with 2k + 12 < s s < 2k + 52 , then u ∈ C([0, T ); Dks (R+ )) ∩ C 1 ([0, T ); Dks −1 (R+ )) with the same T . Remark 4.1. From Lemmas 2.5, 2.6 and Theorems 4.1, 4.2, we see that s = 2k + 12 , where k = 1, 2, . . . , are the critical values for the problem (4.4). If b = 3 and α = 1, then Theorems 4.1, 4.2 improve considerably the well-posedness result of the Degasperis–Procesi equation obtained in [31]. We now present a precise blow-up scenario of strong solutions to Eq. (4.1). Theorem 4.3. Given u0 ∈ H s (R+ ) ∩ H01 (R+ ) with 32 < s < 52 , blow up of the solution u = u(·, u0 ) to Eq. (4.1) in finite time T < +∞ occurs if and only if lim inf inf (2b − 1)ux (t, x) = −∞. t↑T
x∈R+
Proof. As we did before, we first extend the initial data u0 (x) defined on the half-line into an odd function u˜ 0 (x) defined in (3.3) on the line. By Theorem 4.1, we can obtain the odd solution u(t, ˜ x) which is the corresponding strong solution to Eq. (4.3) with the initial data u˜ 0 (x). Moreover, u(t, x) = u(t, ˜ x) confined on [0, T ) × R+ is the unique strong solution to Eq. (4.1) with the initial data u0 (x). For the b-equation on the line, see [33, Theorem 3.2], we know that blow up of the solution u˜ = u(·, ˜ u˜ 0 ) to Eq. (4.3) in finite time T < +∞ occurs if and only if lim inf inf (2b − 1)u˜ x (t, x) = −∞. t↑T
x∈R
Since u(t, ˜ ·) is odd, then it follows that u˜ x (t, ·) is even. Thus, we have that lim inf inf (2b − 1)u˜ x (t, x) = lim inf inf (2b − 1)ux (t, x) . t↑T
x∈R
t↑T
x∈R+
The above two relations imply the desired result of the theorem.
(4.5)
2
Next, we present two blow-up results and two global existence results for Eq. (4.1). Theorem 4.4. Let u0 ∈ H s (R+ ) ∩ H01 (R+ ) with 32 < s < 52 be given and b 3. If u0 ≡ 0 such that y0 (x) = u0 (x) − α 2 u0,xx (x) 0 for all x 0, then the corresponding solution to Eq. (4.1) blows up in finite time. Moreover, the maximal existence time of the corresponding solution is strictly less than − u0,x1(0) . Proof. Let us extend u0 (x) into an odd function u˜ 0 (x) defined in (3.3). Then the assumption ˜ x) to of the theorem ensures that u˜ 0 (x) ≡ 0. By Theorem 4.1, we obtain the odd solution u(t, ˜ x) confined on [0, T ) × R+ is the Eq. (4.3) with the initial data u˜ 0 (x). Moreover, u(t, x) = u(t, unique strong solution to Eq. (4.1) with the initial data u0 (x). Assume now that u˜ 0 ≡ 0 is odd and satisfies y˜0 (x) = u˜ 0 (x) − α 2 u˜ 0,xx (x) 0 for x 0, y˜0 (x) = u˜ 0 (x) − α 2 u˜ 0,xx (x) 0 for x 0.
J. Escher, Z. Yin / Journal of Functional Analysis 256 (2009) 479–508
503
Then the corresponding solution to the b-equation on the line blows up in finite time, provided b 3. Moreover, the maximal existence time of the corresponding solution is strictly less than − u˜ 0,x1(0) , cf. [33]. Since u˜ 0 (x) and y˜0 (x) are odd, it follows from the assumption of the theorem that the solution u(t, ˜ x) to Eq. (4.3) blows up in finite time. Moreover, the maximal existence time is strictly less than − u˜ 0,x1(0) . By Theorem 4.3 and (4.5), we obtain the desired results of the theorem. 2 Theorem 4.5. Let u0 ∈ H s (R+ ) ∩ H01 (R+ ) with 32 < s < 52 be given and 1 < b 3. If u0 (x) ≡ 0 and u0,x (0) 0, then the corresponding solution to Eq. (4.1) blows up in finite time. Moreover, if u0,x (0) < 0, then the maximal existence time of the corresponding solution is strictly less than − (b−1)u2 0,x (0) . Proof. Let u˜ 0 (x) be defined in (3.3). Then the assumption of the theorem ensures that u˜ 0 (x) ≡ 0. For the b-equation on the line, we know that if 1 < b 3, u˜ 0 (x) ≡ 0 is odd and u˜ 0,x (0) 0, it is known that the corresponding solution to the b-equation on the line blows up in finite time. Moreover, if u˜ 0,x (0) < 0, then the maximal existence time of the corresponding solution is strictly less than − (b−1)2u˜ 0,x (0) , see [33, Theorem]. Since u(t, ˜ ·) is odd, in view of Theorems 4.3 and (4.5), it follows that the conclusion of the theorem holds true. 2 By applying Lemma 2.6 and two global existence results of the b-equation on the line, see [33, Theorems 4.3 and 4.4], and following a similar argument as in the proof of Theorem 3.6, we obtain the following results. 1
Theorem 4.6. Let u0 ∈ Dks (R+ ) ∩ W 3,− b (R+ ), where k = 2, . . . , and 2k + 12 < s < 2k + 52 1 be given and b = − 2n , for some n = 1, 2, . . . . Then the corresponding solution to Eq. (4.4) is defined globally in time. 1
Theorem 4.7. Let u0 ∈ Dks (R+ ) ∩ W 2, b (R+ ), where k = 1, 2, . . . , and 2k + 12 < s < 2k + 52 be given and 0 b 1. Then the corresponding solution to Eq. (4.4) is defined globally in time. Note that if α = 1 and b = 3, then the b-equation becomes the Degasperis–Procesi equation. In view of Lemma 2.5 and of [31, Theorems 3.4–3.6], we obtain the following blow-up results and global existence result for the Degasperis–Procesi equation on the half-line, which improve earlier results. Theorem 4.8. Let ε > 0 and u0 ∈ H s (R+ ) ∩ H01 (R+ ) with x0 ∈ R+ such that
3 2
< s < 52 . Assume that there exists
√ 1 √ 2 6 2 . u0 L∞ + 4 6u0 2L2 ln 1 + + u0 L2∞ 4 ε
(1 + ε) u 0 (x0 ) < −
Then the corresponding solution to Eq. (4.1) with α = 1 and b = 3 blows up in finite time.
504
J. Escher, Z. Yin / Journal of Functional Analysis 256 (2009) 479–508
Theorem 4.9. Let u0 ∈ H s (R+ ) ∩ H01 (R+ ) with 32 < s < 52 and y0 (x) := u0 (x) − u0,xx (x). Assume that y0 (x) ≡ 0 and y0 (x) 0 for all x ∈ R+ . Then the corresponding solution to Eq. (4.1) with α = 1 and b = 3 blows up in finite time. Theorem 4.10. Let u0 ∈ H s (R+ ) ∩ H01 (R+ ) with 32 < s < 52 and y0 (x) := u0 (x) − u0,xx (x). Assume that y0 (x) 0 for all x ∈ R+ . Then the corresponding solution to Eq. (4.1) with α = 1 and b = 3 exists globally in time. Remark 4.2. Let u0 ∈ Dks (R+ ), where k = 1, 2, . . . , and 2k + 12 < s < 2k + 52 . Then Theorems 4.3–4.5 and 4.8–4.10 hold true for the corresponding solution u(t, x) to Eq. (4.4). 4.2. The case of the interval (0, 12 ) In the subsection, we study initial boundary value problems of the b-equation on the interval (0, 12 ). ⎧ 2 2 ⎪ ⎨ ut − α utxx + (b + 1)uux = α (bux uxx + uuxxx ), u(0, x) = u0 (x), ⎪
⎩ u(t, 0) = u t, 12 = 0,
t > 0, x ∈ 0, 12 ,
1 x ∈ 0, 2 , t 0.
(4.6)
Here b and α 2 > 0 are real constants. Applying the local well-posedness result of the periodic b-equation which is similar to the case of the whole line [33] and following the proof of Theorem 3.10, we can obtain the local well-posedness result for Eq. (4.6). Theorem 4.11. Assume that u0 ∈ H s (0, 12 ) ∩ H01 (0, 12 ) with 32 < s < 52 . Then there exists a maximal T = T (u0 ) > 0 and a unique solution u(t, x) to Eq. (4.6) such that u = u(·, u0 ) belongs to
C [0, T ); H s 0, 12 ∩ H01 0, 12 ∩ C 1 [0, T ); H s−1 0, 12 ∩ H01 0, 12 .
(4.7)
Moreover, the solution depends continuously on the initial data, i.e. the mapping u0 → u(·, u0 ) : H s (0, 12 ) ∩ H01 (0, 12 ) → C([0, T ); H s (0, 12 ) ∩ H01 (0, 12 )) ∩ C 1 ([0, T ); H s−1 (0, 12 ) ∩ H01 (0, 12 )) is continuous. Furthermore, the maximal T is independent of s in the following sense: if the solution u = u(·, u0 ) to Eq. (4.7) satisfies (4.8) and if u0 ∈ H s (0, 12 ) ∩ H01 (0, 12 ) with 32 < s s < 52 , then
u ∈ C [0, T ); H s 0, 12 ∩ H01 0, 12 ∩ C 1 [0, T ); H s −1 0, 12 ∩ H01 0, 12 with the same T . Odd solutions of higher regularity satisfy the following initial boundary value problem: ⎧ ut − α 2 utxx + (b + 1)uux = α 2 (bux uxx + uuxxx ), ⎪ ⎪ ⎪ ⎨ u(0, x) = u0 (x), ⎪ u(2k) (t, 0) = u(2k−2) (t, 0) = · · · = u(t, 0) = 0, ⎪ ⎪
⎩ (2k) 1 t, 2 = u(2k−2) t, 12 = · · · = u t, 12 = 0, u
t > 0, x ∈ 0, 12 , x ∈ 0, 12 , t 0, t 0.
(4.8)
J. Escher, Z. Yin / Journal of Functional Analysis 256 (2009) 479–508
505
Here b and α 2 > 0 are real constants. Similar to Theorem 3.11, we get the following result. Theorem 4.12. Assume that u0 ∈ Dks (0, 12 ), where k = 1, 2, . . . , and 2k + 12 < s < 2k + 52 . Then there exists a maximal T = T (u0 ) > 0, and a unique solution u(t, x) to Eq. (4.8) such that u = u(·, u0 ) ∈ C([0, T ); Dks (0, 12 )) ∩ C 1 ([0, T ); Dks−1 (0, 12 )). Moreover, the solution depends continuously on the initial data, i.e. the mapping
u0 → u(·, u0 ) : Dks 0, 12 → C [0, T ); Dks 0, 12 ∩ C 1 [0, T ); Dks−1 0, 12 is continuous. Furthermore, the maximal T is independent of s in the following sense: if the solution
u = u(·, u0 ) ∈ C [0, T ); Dks 0, 12 ∩ C 1 [0, T ); Dks−1 0, 12
is the solution to Eq. (4.8) and if u0 ∈ Dks (0, 12 ) with 2k + 12 < s s < 2k + 52 , then u ∈ C([0, T ); Dks (0, 12 )) ∩ C 1 ([0, T ); Dks −1 (0, 12 )) with the same T . Remark 4.3. Theorems 4.11, 4.12 with α = 1 and b = 2 or b = 3 improve considerably the wellposedness results of the Camassa–Holm equation on the finite interval [32] or the Degasperis– Procesi equation on the finite interval [31]. Similar to the argument on the case of on the half-line, we obtain a precise blow-up scenario, a blow-up result and two global existence results of strong solutions to Eq. (4.6) as follows. Theorem 4.13. Given u0 ∈ H s (0, 12 ) ∩ H01 (0, 12 ) with 32 < s < 52 , blow up of the solution u = u(·, u0 ) to Eq. (4.6) in finite time T < +∞ occurs if and only if lim inf inf (2b − 1)ux (t, x) = −∞. t↑T
x∈[0, 12 ]
Theorem 4.14. Let u0 (x) ∈ H s (0, 12 ) ∩ H01 (0, 12 ) with 32 < s < 52 be given and 1 < b 3. If u0 (x) ≡ 0 and u0,x (0) 0, then the corresponding solution to Eq. (4.6) blows up in finite time. Moreover, if u0,x (0) < 0, then the maximal existence time of the corresponding solution is strictly less than − (b−1)u2 0,x (0) . 1
Theorem 4.15. Let u0 ∈ Dks (0, 12 ) ∩ W 3,− b (0, 12 ), where k = 2, . . . , and 2k + 12 < s < 2k + 52 1 be given and b = − 2n , for some n = 1, 2, . . . . Then the corresponding solution to Eq. (4.8) is defined globally in time. 1
Theorem 4.16. Let u0 ∈ Dks (0, 12 ) ∩ W 2, b (0, 12 ), where k = 1, 2, . . . , and 2k + 12 < s < 2k + 52 be given and 0 b 1. Then the corresponding solution to Eq. (4.8) is defined globally in time. For the Degasperis–Procesi equation on the finite interval (0, 12 ), in view of Theorem 4.4 in [31], we have the following sharp blow-up result.
506
J. Escher, Z. Yin / Journal of Functional Analysis 256 (2009) 479–508
Theorem 4.17. Assume that u0 ∈ H s (0, 12 ) ∩ H01 (0, 12 ) with 32 < s < 52 and u0 ≡ 0. Then the corresponding solution to Eq. (4.6) with α = 1 and b = 3 blows up in finite time. Remark 4.4. Let u0 ∈ Dks (0, 12 ), where k = 1, 2, . . . , and 2k + 12 < s < 2k + 52 . Then Theorems 4.13, 4.14 and 4.17 hold true for the corresponding solution u(t, x) to Eq. (4.8). Acknowledgments The second author was partially supported by the Alexander von Humboldt Foundation, the NNSF of China (No. 10531040), the SRF for ROCS, SEM and the NSF of Guangdong Province. The authors thank the referee for valuable comments and suggestions. References [1] R.A. Adams, Sobolev Spaces, Academic Press, New York, 1975. [2] R. Beals, D. Sattinger, J. Szmigielski, Acoustic scattering and the extended Korteweg–de Vries hierarchy, Adv. Math. 140 (1998) 190–206. [3] T.B. Benjamin, J.L. Bona, J.J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. Roy. Soc. London 272 (1972) 47–78. [4] A. Bressan, A. Constantin, Global conservative solutions of the Camassa–Holm equation, Arch. Ration. Mech. Anal. 183 (2007) 215–239. [5] R. Camassa, D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett. 71 (1993) 1661– 1664. [6] R. Camassa, D. Holm, J. Hyman, A new integrable shallow water equation, Adv. Appl. Mech. 31 (1994) 1–33. [7] G.M. Coclite, K.H. Karlsen, On the well-posedness of the Degasperis–Procesi equation, J. Funct. Anal. 233 (2006) 60–91. [8] A. Constantin, Global existence of solutions and breaking waves for a shallow water equation: A geometric approach, Ann. Inst. Fourier (Grenoble) 50 (2000) 321–362. [9] A. Constantin, On the scattering problem for the Camassa–Holm equation, Proc. Roy. Soc. London Ser. A 457 (2001) 953–970. [10] A. Constantin, The trajectories of particles in Stokes waves, Invent. Math. 166 (2006) 523–535. [11] A. Constantin, J. Escher, Global existence and blow-up for a shallow water equation, Ann. Sc. Norm. Sup. Pisa Cl. Sci. (4) 26 (1998) 303–328. [12] A. Constantin, J. Escher, Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation, Comm. Pure Appl. Math. 51 (1998) 475–504. [13] A. Constantin, J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math. 181 (1998) 229–243. [14] A. Constantin, J. Escher, Global weak solutions for a shallow water equation, Indiana Univ. Math. J. 47 (1998) 1527–1545. [15] A. Constantin, J. Escher, Particle trajectories in solitary water waves, Bull. Amer. Math. Soc. 44 (2007) 423–431. [16] A. Constantin, H.P. McKean, A shallow water equation on the circle, Comm. Pure Appl. Math. 52 (1999) 949–982. [17] A. Constantin, L. Molinet, Global weak solutions for a shallow water equation, Comm. Math. Phys. 211 (2000) 45–61. [18] A. Constantin, W.A. Strauss, Stability of a class of solitary waves in compressible elastic rods, Phys. Lett. A 270 (2000) 140–148. [19] A. Constantin, W.A. Strauss, Stability of peakons, Comm. Pure Appl. Math. 53 (2000) 603–610. [20] H.H. Dai, Model equations for nonlinear dispersive waves in a compressible Mooney–Rivlin rod, Acta Mech. 127 (1998) 193–207. [21] H.H. Dai, Y. Huo, Solitary shock waves and other travelling waves in a general compressible hyperelastic rod, Proc. Roy. Soc. London Ser. A 456 (2000) 331–363. [22] R. Danchin, A few remarks on the Camassa–Holm equation, Differential Integral Equations 14 (2001) 953–988. [23] A. Degasperis, M. Procesi, Asymptotic integrability, in: A. Degasperis, G. Gaeta (Eds.), Symmetry and Perturbation Theory, World Scientific, Singapore, 1999, pp. 23–37.
J. Escher, Z. Yin / Journal of Functional Analysis 256 (2009) 479–508
507
[24] A. Degasperis, D.D. Holm, A.N.W. Hone, A new integral equation with peakon solutions, Theoret. Math. Phys. 133 (2002) 1463–1474. [25] A. Degasperis, D.D. Holm, A.N.W. Hone, Integral and non-integrable equations with peakons, in: Nonlinear Physics: Theory and Experiment, II, Gallipoli, 2002, World Sci. Publ., River Edge, NJ, 2003, pp. 37–43. [26] R.K. Dodd, J.C. Eilbeck, J.D. Gibbon, H.C. Morris, Solitons and Nonlinear Wave Equations, Academic Press, New York, 1984. [27] H.R. Dullin, G.A. Gottwald, D.D. Holm, An integrable shallow water equation with linear and nonlinear dispersion, Phys. Rev. Lett. 87 (2001) 4501–4504. [28] H.R. Dullin, G.A. Gottwald, D.D. Holm, Camassa–Holm, Korteweg–de Vries-5 and other asymptotically equivalent equations for shallow water waves, Fluid Dynam. Res. 33 (2003) 73–79. [29] H.R. Dullin, G.A. Gottwald, D.D. Holm, On asymptotically equivalent shallow water wave equations, Phys. D 190 (2004) 1–14. [30] J. Escher, Wave breaking and shock waves for a periodic shallow water equation, Philos. Trans. Roy. Soc. Ser. A 365 (2007) 2281–2289. [31] J. Escher, Z. Yin, Initial boundary value problems of the Degasperis–Procesi equation, Phys. Lett. A 368 (2007) 69–76. [32] J. Escher, Z. Yin, Initial boundary value problems of the Camassa–Holm equation, Comm. Partial Differential Equations 33 (2008) 377–395. [33] J. Escher, Z. Yin, Well-posedness, blow-up phenomena, and global solutions for the b-equation, J. Reine Angew. Math., in press. [34] J. Escher, Y. Liu, Z. Yin, Global weak solutions and blow-up structure for the Degasperis–Procesi equation, J. Funct. Anal. 241 (2006) 457–485. [35] J. Escher, Y. Liu, Z. Yin, Shock waves and blow-up phenomena for the periodic Degasperis–Procesi equation, Indiana Univ. Math. J. 56 (2007) 87–117. [36] A. Fokas, B. Fuchssteiner, Symplectic structures, their Bäcklund transformation and hereditary symmetries, Phys. D 4 (1981) 47–66. [37] D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, New York, 1977. [38] D.D. Holm, M.F. Staley, Wave structure and nonlinear balances in a family of evolutionary PDEs, SIAM J. Appl. Dyn. Syst. 2 (2003) 323–380 (electronic). [39] D.D. Holm, M.F. Staley, Nonlinear balance and exchange of stability in dynamics of solitons, peakons, ramps/cliffs and leftons in a 1–1 nonlinear evolutionary PDE, Phys. Lett. A 308 (2003) 437–444. [40] R.S. Johnson, Camassa–Holm, Korteweg–de Vries and related models for water waves, J. Fluid. Mech. 457 (2002) 63–82. [41] T. Kappeler, P. Topalov, Global wellposedness of KdV in H −1 (T, R), Duke Math. J. 135 (2006) 327–360. [42] C. Kenig, G. Ponce, L. Vega, Well-posedness and scattering results for the generalized Korteweg–de Vries equation via the contraction principle, Comm. Pure Appl. Math. 46 (1993) 527–620. [43] K.H. Kwek, H. Gao, W. Zhang, C. Qu, An initial boundary value problem of Camassa–Holm equation, J. Math. Phys. 41 (2000) 8279–8285. [44] J. Lenells, Traveling wave solutions of the Degasperis–Procesi equation, J. Math. Anal. Appl. 306 (2005) 72–82. [45] Y.A. Li, P.J. Olver, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation, J. Differential Equations 162 (2000) 27–63. [46] Y. Liu, Z. Yin, Global existence and blow-up phenomena for the Degasperis–Procesi equation, Comm. Math. Phys. 267 (2006) 801–820. [47] H. Lundmark, Formation and dynamics of shock waves in the Degasperis–Procesi equation, J. Nonlinear Sci. 17 (2007) 169–198. [48] H. Lundmark, J. Szmigielski, Multi-peakon solutions of the Degasperis–Procesi equation, Inverse Problems 19 (2003) 1241–1245. [49] S. Ma, S. Ding, On the initial boundary value problem for a shallow water equation, J. Math. Phys. 45 (2004) 3479–3497. [50] H.P. McKean, Integrable systems and algebraic curves, in: Global Analysis, in: Lecture Notes in Math., vol. 755, Springer-Verlag, Berlin, 1979, pp. 83–200. [51] L. Molinet, On well-posedness results for Camassa–Holm equation on the line: A survey, J. Nonlin. Math. Phys. 11 (2004) 521–533. [52] G. Rodriguez-Blanco, On the Cauchy problem for the Camassa–Holm equation, Nonlinear Anal. 46 (2001) 309– 327.
508
J. Escher, Z. Yin / Journal of Functional Analysis 256 (2009) 479–508
[53] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives Theory and Applications, Gordon&Breach, Amsterdam, 1993. [54] P. Souganidis, W.A. Strauss, Instability of a class of dispersive solitary waves, Proc. Roy. Soc. Edinburgh Sect. A 114 (1990) 195–212. [55] J.F. Toland, Stokes waves, Topol. Methods Nonlinear Anal. 7 (1996) 1–48. [56] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam, New York, 1978. [57] V.O. Vakhnenko, E.J. Parkes, Periodic and solitary-wave solutions of the Degasperis–Procesi equation, Chaos Solitons Fractals 20 (2004) 1059–1073. [58] E. Wahlén, Global existence of weak solutions to the Camassa–Holm equation, Int. Math. Res. Not. (2006), Art. ID 28976, 12 pp. [59] G.B. Whitham, Linear and Nonlinear Waves, Wiley, New York, 1980. [60] Z. Xin, P. Zhang, On the weak solutions to a shallow water equation, Comm. Pure Appl. Math. 53 (2000) 1411– 1433. [61] L. Yang, B. Guo, Initial boundary value problem for a shallow water equation, Math. Theory Appl. (Changsha) 23 (2003) 1–10 (in Chinese). [62] Z. Yin, On the blow-up of solutions of a periodic nonlinear dispersive wave equation in compressible elastic rods, J. Math. Anal. Appl. 288 (2003) 232–245. [63] Z. Yin, Global existence for a new periodic integrable equation, J. Math. Anal. Appl. 283 (2003) 129–139. [64] Z. Yin, On the Cauchy problem for an integrable equation with peakon solutions, Illinois J. Math. 47 (2003) 649– 666. [65] Z. Yin, Global solutions to a new integrable equation with peakons, Indiana Univ. Math. J. 53 (2004) 1189–1210. [66] Z. Yin, Global weak solutions to a new periodic integrable equation with peakon solutions, J. Funct. Anal. 212 (2004) 182–194.
Journal of Functional Analysis 256 (2009) 509–593 www.elsevier.com/locate/jfa
Pseudo-localization of singular integrals and noncommutative Calderón–Zygmund theory ✩ Javier Parcet a Instituto de Ciencias Matemáticas, CSIC-UAM-UC3M-UCM, Consejo Superior de Investigaciones Científicas,
C/Serrano 121, 28006 Madrid, Spain Received 2 April 2008; accepted 8 April 2008 Available online 9 May 2008 Communicated by J. Bourgain
Abstract The weak type (1, 1) boundedness of singular integrals acting on matrix-valued functions has remained open since the 1980s, mainly because the methods provided by the vector-valued theory are not strong enough. In fact, we can also consider the action of generalized Calderón–Zygmund operators on functions taking values in any other von Neumann algebra. Our main tools for its solution are two. First, the lack of some classical inequalities in the noncommutative setting forces to have a deeper knowledge of how fast a singular integral decreases—L2 sense—outside of the support of the function on which it acts. This gives rise to a pseudo-localization principle which is of independent interest, even in the classical theory. Second, we construct a noncommutative form of Calderón–Zygmund decomposition by means of the recent theory of noncommutative martingales. This is a corner stone in the theory. As application, we obtain the sharp asymptotic behavior of the constants for the strong Lp inequalities, also unknown up to now. Our methods settle some basics for a systematic study of a noncommutative Calderón–Zygmund theory. © 2008 Elsevier Inc. All rights reserved. Keywords: Calderón–Zygmund operator; Almost orthogonality; Noncommutative martingale
Contents 0. 1.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510 Noncommutative integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520
✩ Partially supported by ‘Programa Ramón y Cajal, 2005’ and also by grants MTM2007-60952 and CCG06-UAM/ESP0286, Spain. E-mail address: [email protected].
0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.04.007
510
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
2. A pseudo-localization principle . . . . . . . . . . . 3. Calderón–Zygmund decomposition . . . . . . . . . 4. Weak type estimates for diagonal terms . . . . . . 5. Weak type estimates for off-diagonal terms . . . 6. Operator-valued kernels . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A. On pseudo-localization . . . . . . . . . . . Appendix B. On Calderón–Zygmund decomposition References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
523 555 560 565 575 582 583 589 592
0. Introduction After the pioneer work of Calderón and Zygmund in the 1950s, the systematic study of singular integrals has become a corner stone in harmonic analysis with deep implications in mathematical physics, partial differential equations and other mathematical disciplines. Subsequent generalizations of Calderón–Zygmund theory have essentially pursued two lines. We may either consider more general domains or ranges for the functions considered. In the first case, the Euclidean space is replaced by metric spaces equipped with a doubling or non-doubling measure of polynomial growth. In the second case, the real or complex fields are replaced by a Banach space in which martingale differences are unconditional. Historically, the study of singular integrals acting on matrix or operator-valued functions has been considered part of the vector-valued theory. This is however a limited approach in the noncommutative setting and we propose to regard these functions as operators in a suitable von Neumann algebra, generalizing so the domain and not the range of classical functions. A far √ reaching aspect of our approach is the stability of the product f g and the absolute value |f | = f ∗ f for operator-valued functions, a fundamental property not exploited in the vector theory. In this paper we follow the original Calderón–Zygmund program and present a noncommutative scalar-valued Calderón–Zygmund theory, emancipated from the vector theory. Noncommutative harmonic analysis (understood in a wide sense) has received much attention in recent years. The functional analytic approach given by operator space theory and the new methods from quantum/free probability have allowed to study a great variety of topics. We find in the recent literature noncommutative analogs of Khintchine and Rosenthal inequalities, a settled noncommutative theory of martingale inequalities, new results on Fourier/Schur multipliers, matrix Ap weights and a sharpened Carleson embedding theorem, see [20,30,31,43,47,57] and the references therein. However, no essential progress has been made in the context of singular integral operators. Our original motivation was the weak type boundedness of Calderón–Zygmund operators acting on operator-valued functions, a well-known problem which has remained open since the beginning of the vector-valued theory in the 1980s. This fits in the context of Mei’s recent paper [37]. Our main tools for its solution are two. On one hand, the failure of some classical estimates in the noncommutative setting forces us to have a deep understanding of how the L2 mass of a singular integral is concentrated around the support of the function on which it acts. To that aim, we have developed a pseudo-localization principle for singular integrals which is of independent interest, even in the classical theory. This is used in conjunction with a noncommutative form of Calderón–Zygmund decomposition which we have constructed using the theory of noncommutative martingales. As a byproduct of our weak type inequality, we obtain the sharp
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
511
asymptotic behavior of the constants for the strong Lp inequalities as p → 1 and p → ∞, which are not known. At the end of the paper we generalize our results to certain singular integrals including operator-valued kernels and functions at the same time. A deep knowledge of this kind of fully noncommutative operators is a central aim in noncommutative harmonic analysis. Our methods in this paper open a door to work in the future with more general classes of operators. I. Terminology. Let us set some notation that will remain fixed all through out the paper. Let M be a semifinite von Neumann algebra equipped with a normal semifinite faithful (n.s.f.) trace τ . Let us consider the algebra AB of essentially bounded M-valued functions AB = f : Rn → M f strongly measurable such that ess supf (x)M < ∞ , x∈Rn
equipped with the n.s.f. trace ϕ(f ) =
τ f (x) dx.
Rn
The weak-operator closure A of AB is a von Neumann algebra. If 1 p ∞, we write Lp (M) and Lp (A) for the noncommutative Lp spaces associated to the pairs (M, τ ) and (A, ϕ). The lattices of projections are written Mπ and Aπ , while 1M and 1A stand for the unit elements. The set of dyadic cubes in Rn is denoted by Q. The size of any cube Q in Rn is defined as the length (Q) of one of its edges. Given an integer k ∈ Z, we use Qk for the subset of Q formed by cubes Q of the kth generation, i.e. those of size 1/2k . If Q is a dyadic cube and f : Rn → M is integrable on Q, we set the average fQ =
1 |Q|
f (y) dy. Q
Let us write (Ek )k∈Z for the family of conditional expectations associated to the classical dyadic filtration on Rn . Ek will also stand for the tensor product Ek ⊗ idM acting on A. If 1 p < ∞ and f ∈ Lp (A) Ek (f ) = fk =
fQ 1Q .
Q∈Qk
We shall denote by (Ak )k∈Z the corresponding filtration Ak = Ek (A).
is the only dyadic cube containing Q with double size. Given If Q ∈ Q, its dyadic father Q δ > 1, the δ-concentric father of Q is the only cube δQ concentric with the cube Q and such that (δQ) = δ(Q). In this paper we will mainly work with dyadic and 9-concentric fathers. Note that in the classical theory 2-concentric fathers are typically enough. We shall write just Lp to refer to the commutative Lp space on Rn equipped with the Lebesgue measure dx. II. Statement of the problem. Just to motivate our problem and for the sake of simplicity, the reader may think for the moment that (M, τ ) is given by the pair (Mm , tr) formed by the algebra
512
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
of m × m square matrices equipped with the standard trace. In this particular case, the von Neumann algebra A = AB becomes the space of essentially bounded matrix-valued functions. Let us consider a Calderón–Zygmund operator formally given by Tf (x) =
k(x, y)f (y) dy. Rn
As above, let Lp (M) be the noncommutative Lp space associated to (M, τ ). If M is the algebra of m × m matrices we recover the Schatten p-class over Mm , for a general definition see below. The first question which arises is whether or not the singular integral T is bounded on Lp (A) for 1 < p < ∞. The space Lp (A) is defined as the closure of AB with respect to the norm f p =
τ |f (x)|p dx
1
p
.
Rn
In other words, Lp (A) is isometric to the Bochner Lp space with values in Lp (M). In particular, when dealing with the Hilbert transform and by a well-known result of Burkholder [5,6], the boundedness on Lp (A) reduces to the fact that Lp (M) is a UMD Banach space for 1 < p < ∞, see also [2,4]. After some partial results of Bourgain [3], it was finally Figiel [15] who showed in 1989 (using an ingenious martingale approach) that the UMD property implies the Lp boundedness of the corresponding vector-valued singular integrals associated to generalized kernels. The second natural question has to do with a suitable weak type inequality for p = 1. Namely, such inequality is typically combined in the classical theory with the real interpolation method to produce extrapolation results on the Lp boundedness of Calderón–Zygmund and other related operators. The problem of finding the right weak type inequality is subtler since arguments from the vector-valued theory are no longer at our disposal. Indeed, in terms of Bochner spaces we may generalize the previous situation by considering the mapping T from L1 (Rn ; X) to L1,∞ (Rn ; X) with X = L1 (M). However, L1 (M) is not UMD and the resulting operator is not bounded. On the contrary, using operators rather than vectors (i.e. working directly on the algebra A) we may consider the operator T : L1 (A) → L1,∞ (A) where L1,∞ (A) denotes the corresponding noncommutative Lorentz space, to be defined below. The only result on this line is the weak type (1, 1) boundedness of the Hilbert transform for operator-valued functions, proved by Randrianantoanina in [49]. He followed Kolmogorov’s approach, exploiting the conjugation nature of the Hilbert transform (defined in a very wide setting via Arveson’s [1] maximal subdiagonal algebras) and applying complex variable methods. As is well known this is no longer valid for other Calderón–Zygmund operators and new real variable methods are needed. In the classical case, these methods live around the celebrated Calderón–Zygmund decomposition. One of the main purposes of this paper is to supply the right real variable methods in the noncommutative context. As we will see, there are significant differences. Using real interpolation, our main result gives an extrapolation method which produces the Lp boundedness results discussed in the paragraph above and provides the sharp asymptotic behavior of the constants, for which the UMD approach is inefficient. Moreover, when working with operator-valued kernels we obtain new strong Lp inequalities. We should warn the reader not to confuse this setting with that of Rubio de Francia, Ruiz and Torrea [53], Hytönen [21] and Hytönen, Weis [22,23], where the mentioned limitations of the vector-valued theory appear.
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
513
III. Calderón–Zygmund decomposition. Let us recall the formulation of the classical decomposition for scalar-valued integrable functions. If f ∈ L1 is positive and λ ∈ R+ , we consider the level set Eλ = x ∈ Rn Md f (x) > λ , where the dyadic Hardy–Littlewood maximal function Md f is greater than λ. If we write Eλ =
j Qj as a disjoint union of maximal dyadic cubes, we may decompose f = g + b where the good and bad parts are given by g = f 1Ecλ +
fQj 1Qj
and b =
j
(f − fQj )1Qj .
j
Letting bj = (f − fQj )1Qj , we have n (i) g1 f 1 and g∞ 2 λ. (ii) supp bj ⊂ Qj , Qj bj = 0 and j bj 1 2f 1 .
These properties are crucial for the analysis of singular integral operators. In this paper we use the so-called Cuculescu’s construction [9] to produce a sequence (pk )k∈Z of disjoint projections in A which constitute the noncommutative counterpart of the characteristic functions supported by the sets Eλ (k) =
Qj .
Qj ⊂Eλ (Qj )=1/2k
Cuculescu’s construction will be properly introduced in the text. It has proved to be the right tool from the theory of noncommutative martingales to deal with inequalities of weak type. Indeed, Cuculescu proved in [9] the noncommutative Doob’s maximal weak type inequality. Moreover, these techniques were used by Randrianantoanina to prove several weak type inequalities for noncommutative martingales [50–52] and by Junge and Xu in their remarkable paper [30]. In fact, a strong motivation for this paper relies on [44], where similar methods were applied to obtain Gundy’s decomposition for noncommutative martingales. It is well known that the probabilistic analog of Calderón–Zygmund decomposition is precisely Gundy’s decomposition. However, in contrast to the classical theory, the noncommutative analogue of Calderón–Zygmund decomposition turns out to be much harder than Gundy’s decomposition. Although we shall justify this below in further detail, the main reason is that singular integral operators do not localize the support of the function on which it acts, something that happens for instance with martingale transforms or martingale square functions. Let us now formulate the noncommutative Calderón–Zygmund decomposition. If f ∈ L1 (A)+ and λ ∈ R+ , we consider the disjoint projections (pk )k∈Z given by Cuculescu’s construction. Let p∞ denote the projection onto the ortho-complement of the range of k pk . In particular, using the terminology
Z = Z ∪ {∞} we find the relation k∈
Z
pk = 1 A .
514
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
Then, the good and bad parts are given by
g=
pi fi∨j pj
and b =
i,j ∈
Z
pi (f − fi∨j )pj ,
i,j ∈
Z
with i ∨ j = max(i, j ). We will show how this generalizes the classical decomposition. IV. Main weak type inequality. Let Δ denote the diagonal of Rn × Rn . We will write in what follows T to denote a linear map S → S from test functions to distributions which is associated to a given kernel k : R2n \ Δ → C. This means that for any smooth test function f with compact support, we have Tf (x) =
for all x ∈ / supp f.
k(x, y)f (y) dy Rn
Given two points x, y ∈ Rn , the distance |x − y| between x and y will be taken for convenience with respect to the ∞ (n) metric. As usual, we impose size and smoothness conditions on the kernel: (a) If x, y ∈ Rn , we have k(x, y)
1 . |x − y|n
(b) There exists 0 < γ 1 such that γ k(x, y) − k(x , y) |x − x | |x − y|n+γ γ k(x, y) − k(x, y ) |y − y | n+γ |x − y|
1 if |x − x | |x − y|, 2 1 if |y − y | |x − y|. 2
We will refer to this γ as the Lipschitz smoothness parameter of the kernel. Theorem A. Let T be a generalized Calderón–Zygmund operator associated to a kernel satisfying the size and smoothness estimates above. Assume that T is bounded on Lq for some 1 < q < ∞. Then, given any f ∈ L1 (A), the estimate below holds for some constant cn,γ depending only on the dimension n and the Lipschitz smoothness parameter γ sup λϕ |Tf | > λ cn,γ f 1 . λ>0
In particular, given 1 < p < ∞ and f ∈ Lp (A), we find Tf p cn,γ
p2 f p . p−1
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
515
The expression supλ>0 λϕ{|Tf | > λ} is just a slight abuse of notation to denote the noncommutative weak L1 norm, to be rigorously defined below. We find it though more intuitive, since it is reminiscent of the classical terminology. Theorem A provides a positive answer to our problem for any singular integral associated to a generalized Calderón–Zygmund kernel satisfying the size/smoothness conditions imposed above. Moreover, the asymptotic behavior of the constants as p → 1 and p → ∞ is optimal. Independently, Tao Mei has recently obtained another argument for this which does not include the weak type inequality [38]. We shall present it at the end of the paper, since we shall use it indirectly to obtain weak type inequalities for singular integrals associated to operator-valued kernels. In the language of operator space theory and following Pisier’s characterization [45,46] of complete boundedness we immediately obtain: Corollary. Let T be a generalized Lq -bounded and γ -Lipschitz Calderón–Zygmund operator. Let us equip Lp with its natural operator space structure. Then, the cb-norm of T : Lp → Lp is controlled by
cn,γ
p2 . p−1
Thus, the growth rate as p → 1 or p → ∞ coincides with the Banach space case. Before going on, a few remarks are in order. (a) It is standard to reduce the proof of Theorem A to the case q = 2. (b) The reader might think that our hypothesis on Lipschitz smoothness for the first variable is unnecessary to obtain the weak type inequality and that only smoothness with respect to the second variable is needed. Namely, this is the case in the classical theory. It is however not the case in this paper because the use of certain almost orthogonality arguments (see below) forces us to apply both kinds of smoothness. We refer to Remark 2.11 for the specific point where the x-Lipschitz smoothness is applied and to Remark 5.5 for more in depth discussion on the conditions imposed on the kernel. (c) In the classical case Eλ is a perfectly delimited region of Rn . In particular, we may construct the dilation 9Eλ = j 9Qj . This set is useful to estimate the bad part b since it has two crucial properties. First, it is small because |9Eλ | ∼ |Eλ | and Eλ satisfies the Hardy–Littlewood weak maximal inequality. Second, its complement is far away from Eλ (the support of b) so that T b restricted to Rn \ 9Eλ avoids the singularity of the kernel. The problem that we find in the noncommutative case is that Eλ is no longer a region in Rn . Indeed, given a dyadic cube Q and a positive f ∈ L1 , we have either fQ > λ or not and this dichotomy completely determines the set Eλ . However, for f ∈ L1 (A)+ the average fQ is a positive operator (not a positive number) and the dichotomy disappears since the condition fQ > λ is only satisfied in part of the spectrum of fQ . This difficulty is inherent to the noncommutativity and is motivated by the lack of a total order in the positive cone of M. It also produces difficulties to define noncommutative maximal functions [9,24], a problem that required the recent theory of operator spaces for its solution and is in the heart of the matter. Our construction of the right-noncommutative analog ζ of Rn \ 9Eλ is a key step in this paper, see Lemma 4.2 below. Here it is relevant to recall that, quite unexpectedly (in contrast with the classical case) we shall need the projection ζ to deal with both the good and the bad parts.
516
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
(d) Another crucial difference with the classical setting and maybe the hardest point to overcome is the lack of estimates (i) and (ii) above in the noncommutative framework. Indeed, given f ∈ L1 (A)+ we only have such estimates for the diagonal terms
pk f k pk
and
k
pk (f − fk )pk .
k
A more detailed discussion on this topic is given in Appendix B below. Let us now explain how we face the lack of the classical inequalities. Since 1A − ζ is the noncommutative analog of 9Eλ which is small as explained above, we can use the noncommutative Hardy–Littlewood weak maximal inequality to reduce our problem to estimate the terms ζ T (g)ζ and ζ T (b)ζ . A very naive and formally incorrect way to explain what to do here is the following. Given a fixed positive integer s, we find something like −γ s ζ T pi fi∨j pj ζ s2 pk f k pk , 2
|i−j |=s
k
2
−γ s ζ T pi (f − fi∨j )pj ζ s2 pk (f − fk )pk , 1
|i−j |=s
k
1
where γ is the Lipschitz smoothness parameter of the kernel. In other words, we may estimate the action of ζ T (·)ζ on the terms in the sth upper and lower diagonals by s2−γ s times the corresponding size of the main diagonal. Then, recalling that (i) and (ii) hold on the diagonal, it is standard to complete the argument. We urge however the reader to understand this just as a motivation (not as a claim) since the argument is quite more involved than this. For instance, we will need to replace the off-diagonal terms of g by other gk,s ’s satisfying k,s
gk,s =
pi fi∨j pj .
i=j
A rough way of rephrasing this phenomenon is to say that Calderón–Zygmund operators are almost diagonal when acting on operator-valued functions. In other contexts, this almost diagonal nature has already appeared in the literature. The wavelet proof of the T 1 theorem [39] exhibits this property of singular integrals with respect to the Haar system in Rn . This also applies in the context of Clifford analysis [40]. Moreover, some deep results in [7,13] (which we will comment below) use this almost diagonal nature as a key idea. It is also worthy of mention that these difficulties do not appear in [44]. The reason is that the operators for which Gundy’s decomposition is typically applied (martingale transforms or martingale square functions) do not move the support of the original function/operator. This means that the action of T over the offdiagonal terms is essentially supported by 1A − ζ . Consequently, these terms are controlled by means of the noncommutative analog of Doob’s maximal weak type inequality, see [44] for further details. As a byproduct, we observe that the pseudo-localization principle which we present below is not needed in [44]. V. Pseudo-localization. A key point in our argument is the behavior of singular integrals acting on the off-diagonal terms pi fi∨j pj and pi (f − fi∨j )pj , as a function of the parameter s = |i − j | in a region ζ ≈ Rn \ 9Eλ which is in some sense far away from their (left and right)
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
517
support. The idea we need to exploit relies on the following principle: more regularity of the kernel of T implies a faster decay of Tf far away from the support of f . That is why the bterms pi (f − fi∨j )pj are better than the g-terms pi fi∨j pj . Indeed, the cancellation of f − fi∨j allows to subtract a piecewise constant function from the kernel (in the standard way) to apply the smoothness properties of it and obtain suitable L1 estimates. However, the off-diagonal g-terms are not mean-zero (at least at first sight) with respect to Rn and we need more involved tools to prove this pseudo-localization property in the L2 metric. For the sake of clarity and since our result might be of independent interest even in the classical theory, we state it for scalar-valued functions. The way we apply it in our noncommutative setting will be clarified along the text, see Theorem 5.2 for the noncommutative form of this principle. Since we are assuming that T is bounded on L2 , we may further assume by homogeneity that it is of norm 1. In the sequel, we will only consider L2 -normalized Calderón–Zygmund operators. Our result is related to the following problem. An L2 -localization problem. Given f : Rn → C in L2 and 0 < δ < 1, find the sets Σf,δ such that the inequality below holds for all normalized Calderón–Zygmund operator satisfying the imposed size/smoothness conditions
Rn \Σ
Tf (x)2 dx
1 2
δ
f (x)2 dx
1 2
.
Rn
f,δ
Given f : Rn → C in L2 , let fk and dfk denote the kth condition expectation of f with respect to the standard dyadic filtration and its corresponding kth martingale difference. That is, we have n dfk = Q∈Qk (fQ − fQ
)1Q . Let Rk be the class of sets in R being the union of a family of
cubes in Qk . Given such an Rk -set Ω = j Qj , we shall work with the dilations 9Ω = j 9Qj , where 9Q denotes the 9-concentric father of Q. We shall prove the following result. A pseudo-localization principle. Let us fix a positive integer s. Given a function f in L2 and any integer k, we define Ωk to be the smallest Rk -set containing the support of dfk+s . If we further consider the set 9Ωk , Σf,s = k∈Z
then we have the localization estimate
Tf (x)2 dx
1 2
cn,γ s2−γ s/4
Rn \Σf,s
f (x)2 dx
1 2
,
Rn
for any L2 -normalized Calderón–Zygmund operator with Lipschitz parameter γ . Given any integer k ∈ Z, we are considering the smallest set Ωk containing supp dfk+s and belonging to an s times coarser topology. This procedure gives rise to an apparently artificial shift condition supp dfk+s ⊂ Ωk
518
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
which gives a measure of how much we should enlarge Σf = k supp dfk at every scale. However, this condition (or its noncommutative analog) is quite natural in our setting since it is satisfied by the off-diagonal terms of g, precisely those for which our previous tools did not work. In the classical/commutative setting there are some natural situations for which our result applies and some others which limit the applicability of it. For instance, at first sight our result is only applicable for functions f satisfying fm = 0 for some integer m. There are also some other natural questions such as an Lp analog of our result or an equivalent formulation using a Littlewood–Paley decomposition, instead of martingale differences. For the sake of clarity in our exposition, we prove the result in the body of the paper and we postpone these further comments to Appendix A below. The proof of this result reduces to a shifted form of the T 1 theorem in a sense to be explained below. In particular, almost orthogonality methods are essential in our approach. Compared to the standard proofs of the T 1 theorem, with wavelets [39] or more generally with approximations of the identity [54], we need to work in a dyadic/martingale setting forced by the role of Cuculescu’s construction in this paper. This produces a lack of smoothness in the functions we work with, requiring quite involved estimates to obtain almost orthogonality results. An apparently new aspect of our estimates is the asymmetry of our bounds when applying Schur lemma, see Remark 2.1 for more details. Let us briefly comment the relation of our result with two papers by Christ [7] and Duoandikoetxea, Rubio de Francia [13]. Although both papers already exploited the almost diagonal nature of Calderón–Zygmund operators, only convolution-type singular integrals are considered and no localization result is pursued there. Being more specific, a factor 2−γ s is obtained in [7] for the bad part of Calderón–Zygmund decomposition. As explained above, we need to produce this factor for the good part. This is very unusual (or even new) in the literature. Nevertheless, the way we have stated our pseudo-localization result shows that the key property is the shift condition supp dfk+s ⊂ Ωk , regardless we work with good or bad parts. On the other hand, in [13] Littlewood–Paley theory and the commutativity produced by the use of convolution operators is used to obtain related estimates in Lp with p = 2. In particular, almost orthogonality does not play any role there. The lack of a suitable noncommutative Littlewood–Paley theory and our use of generalized Calderón–Zygmund operators make their argument not applicable here. VI. Operator-valued kernels. At the end of the paper we extend our main results to certain Calderón–Zygmund operators associated to kernels k : R2n \ Δ → M satisfying the canonical size/smoothness conditions. In other words, we replace the absolute value by the norm in M: (a) If x, y ∈ Rn , we have k(x, y)
M
1 . |x − y|n
(b) There exists 0 < γ 1 such that k(x, y) − k(x , y)
M
|x − x |γ |x − y|n+γ
1 if |x − x | |x − y|, 2
k(x, y) − k(x, y )
M
|y − y |γ |x − y|n+γ
1 if |y − y | |x − y|. 2
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
519
Unfortunately, not every such kernel satisfies the analog of Theorem A. Namely, we shall construct (using classical Littlewood–Paley methods) a simple kernel satisfying the size and smoothness conditions above and giving rise to a Calderón–Zygmund operator bounded on L2 (A) but not on Lp (A) for 1 < p < 2. However, a detailed inspection of our proof of Theorem A and a few auxiliary results will show that the key condition (together with the size/smoothness hypotheses on the kernel) for the operator T is to be an M-bimodule map. Of course, this always holds in the context of Theorem A. When dealing with operator-valued kernels this is false in general, but it holds for instance when dealing with standard Calderón–Zygmund operators Tf (x) = ξf (x) + lim
ε→0 |x−y|>ε
k(x, y)f (y) dy
associated to a commuting kernel k : R2n \ Δ → ZM , with ZM = M ∩ M standing for the center of M. Note that we are only requiring the M-bimodule property to hold on the singular integral part, since the multiplier part is always well behaved as far as ξ ∈ A. Note also that when M is a factor, any commuting kernel must be scalar-valued and we go back to Theorem A. Theorem B. Let T be a generalized Calderón–Zygmund operator associated to an operatorvalued kernel k : R2n \ Δ → M satisfying the imposed size/smoothness conditions. Assume that T is an M-bimodule map bounded on Lq (A) for some 1 < q < ∞. Then, the following weak type inequality holds for some constant cn,γ depending only on the dimension n and the Lipschitz smoothness parameter γ sup λϕ |Tf | > λ cn,γ f 1 . λ>0
In particular, given 1 < p < ∞ and f ∈ Lp (A), we find Tf p cn,γ
p2 f p . p−1
The strong Lp inequalities stated in Theorem B do not follow from a UMD-type argument as it happened with Theorem A. In particular, these Lp estimates seem to be new and independently obtained by Tao Mei as pointed above. VII. Appendices. We conclude the paper with two appendices. A further analysis on pseudolocalization is given in Appendix A. This mainly includes remarks related to our result, some conjectures on possible generalizations and a corollary on the rate of decreasing of the L2 mass of a singular integral far away from the support of the function on which it acts. In Appendix B we study the noncommutative form of Calderón–Zygmund decomposition in further detail. In particular, we give some weighted inequalities for the good and bad parts which generalize the classical L1 and L2 estimates satisfied by these functions. The sharpness of our estimates remains as an open interesting question. Remark. The value of the constant cn,γ will change from one instance to another.
520
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
1. Noncommutative integration We begin with a quick survey of definitions and results on noncommutative Lp spaces and related topics that will be used along the paper. All or most of it will be well known to experts in the field. The right framework for a noncommutative analog of measure theory and integration is von Neumann algebra theory. We refer to [32,55] for a systematic study of von Neumann algebras and to the recent survey by Pisier, Xu [48] for a detailed exposition of noncommutative Lp spaces. 1.1. Noncommutative Lp A von Neumann algebra is a weak-operator closed C∗ -algebra. By the Gelfand–Naimark– Segal theorem, any von Neumann algebra M can be embedded in the algebra B(H) of bounded linear operators on some Hilbert space H. In what follows we will identify M with a subalgebra of B(H). The positive cone M+ is the set of positive operators in M. A trace τ : M+ → [0, ∞] on M is a linear map satisfying the tracial property τ (a ∗ a) = τ (aa ∗ ). It is said to be normal if supα τ (aα ) = τ (supα aα ) for any bounded increasing net (aα ) in M+ ; it is semifinite if for any non-zero a ∈ M+ , there exists 0 < a a such that τ (a ) < ∞ and it is faithful if τ (a) = 0 implies a = 0. Taking into account that τ plays the role of the integral in measure theory, all these properties are quite familiar. A von Neumann algebra M is called semifinite whenever it admits a normal semifinite faithful (n.s.f. in short) trace τ . Except for a brief comment in Remark 5.4 below we shall always work with semifinite von Neumann algebras. Recalling that any operator a can be written as a linear combination a1 − a2 + ia3 − ia4 of four positive operators, we can extend τ to the whole algebra M. Then, the tracial property can be restated in the familiar way τ (ab) = τ (ba) for all a, b ∈ M. According to the GNS construction, it is easily seen that the noncommutative analogs of measurable sets (or equivalently characteristic functions of those sets) are orthogonal projections. Given a ∈ M+ , the support projection of a is defined as the least projection q in M such that qa = a = aq and will be denoted by supp a. Let S+ be the set of all a ∈ M+ such that τ (supp a) < ∞ and set S to be the linear span of S+ . If we write |x| for the operator (x ∗ x)1/2 , we can use the spectral measure γ|x| : R+ → B(H) of the operator |x| to define |x|p = s p dγ|x| (s) for 0 < p < ∞. R+ 1
We have x ∈ S ⇒ |x|p ∈ S+ ⇒ τ (|x|p ) < ∞. If we set xp = τ (|x|p ) p , it turns out that p is a norm in S for 1 p < ∞ and a p-norm for 0 < p < 1. Using that S is a w ∗ -dense ∗-subalgebra of M, we define the noncommutative Lp space Lp (M) associated to the pair (M, τ ) as the completion of (S, p ). On the other hand, we set L∞ (M) = M equipped with the operator norm. Many of the fundamental properties of classical Lp spaces like duality, real and complex interpolation . . . can be transferred to this setting. The most important properties for our purposes are the following: • Hölder inequality. If 1/r = 1/p + 1/q, we have abr ap bq . • The trace τ extends to a continuous functional on L1 (M): |τ (x)| x1 . We refer to [48] for a definition of Lp over non-semifinite von Neumann algebras.
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
521
1.2. Noncommutative symmetric spaces Let
M = b ∈ B(H) ab = ba for all a ∈ M
be the commutant of M. A closed densely-defined operator on H is affiliated with M when it commutes with every unitary u in the commutant M . Recall that M = M and this implies that every a ∈ M is affiliated with M. The converse fails in general since we may find unbounded operators. If a is a densely-defined self-adjoint operator on H and a = R s dγa (s) is its spec tral decomposition, the spectral projection R dγa (s) will be denoted by χR (a). An operator a affiliated with M is τ -measurable if there exists s > 0 such that τ χ(s,∞) |a| = τ |a| > s < ∞. The generalized singular-value μ(a) : R+ → R+ is defined by μt (a) = inf s > 0 τ |x| > s t . This provides us with a noncommutative analogue of the so-called non-increasing rearrangement of a given function. We refer to [14] for a detailed exposition of the function μ(a) and the corresponding notion of convergence in measure. If L0 (M) denotes the ∗-algebra of τ -measurable operators, we have the following equivalent definition of Lp 1 p Lp (M) = a ∈ L0 (M) μt (a)p dt <∞ . R+
The same procedure applies to symmetric spaces. Given the pair (M, τ ), let X be a rearrangement invariant quasi-Banach function space on the interval (0, τ (1M )). The noncommutative symmetric space X(M) is defined by X(M) = a ∈ L0 (M) μ(a) ∈ X with aX(M) = μ(a)X . It is known that X(M) is a Banach (respectively quasi-Banach) space whenever X is a Banach (respectively quasi-Banach) function space. We refer the reader to [11,58] for more in depth discussion of this construction. Our interest in this paper is restricted to noncommutative Lp spaces and noncommutative weak L1 -spaces. Following the construction of symmetric spaces of measurable operators, the noncommutative weak L1 -space L1,∞ (M), is defined as the set of all a in L0 (M) for which the quasi-norm a1,∞ = sup tμt (x) = sup λτ |x| > λ t>0
λ>0
is finite. As in the commutative case, the noncommutative weak L1 -space satisfies a quasitriangle inequality that will be used below with no further reference. Indeed, the following inequality holds for a1 , a2 ∈ L1,∞ (M) λτ |a1 + a2 | > λ λτ |a1 | > λ/2 + λτ |a2 | > λ/2 .
522
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
1.3. Noncommutative martingales Consider a von Neumann subalgebra (a weak∗ closed ∗-subalgebra) N of M. A conditional expectation E : M → N from M onto N is a positive contractive projection. The conditional expectation E is called normal if the adjoint map E ∗ satisfies E ∗ (M∗ ) ⊂ N∗ . In this case, there is a map E∗ : M∗ → N∗ whose adjoint is E. Note that such normal conditional expectation exists if and only if the restriction of τ to the von Neumann subalgebra N remains semifinite, see e.g. [55, Theorem 3.4]. Any such conditional expectation is trace preserving (i.e. τ ◦ E = τ ) and satisfies the bimodule property E(a1 ba2 ) = a1 E(b)a2
for all a1 , a2 ∈ N and b ∈ M.
Let (Mk )k1 be an increasing sequence of von Neumann subalgebras of M such that the union of the Mk ’s is weak∗ dense in M. Assume that for every k 1, there is a normal conditional expectation Ek : M → Mk . Note that for every 1 p < ∞ and k 1, Ek extends to a positive contraction Ek : Lp (M) → Lp (Mk ). A noncommutative martingale with respect to the filtration (Mk )k1 is a sequence a = (ak )k1 in L1 (M) such that Ej (ak ) = aj
for all 1 j k < ∞.
If additionally a ⊂ Lp (M) for some 1 p ∞ and ap = supk1 ak p < ∞, then a is called an Lp -bounded martingale. Given a martingale a = (ak )k1 , we assume the convention that a0 = 0. Then, the martingale difference sequence da = (dak )k1 associated to x is defined by dak = ak − ak−1 . The next result due to Cuculescu [9] was the first known result in the theory and will be crucial in this paper. It can be viewed as a noncommutative analogue of the classical weak type (1, 1) boundedness of Doob’s maximal function. Cuculescu’s construction. Suppose a = (a1 , a2 , . . .) is a positive L1 martingale relative to the filtration (Mk )k1 and let λ be a positive number. Then there exists a decreasing sequence of projections q(λ)1 , q(λ)2 , q(λ)3 , . . . in M satisfying the following properties: (i) q(λ)k commutes with q(λ)k−1 ak q(λ)k−1 for each k 1. (ii) q(λ)k belongs to Mk for each k 1 and q(λ)k ak q(λ)k λq(λ)k . (iii) The following estimate holds 1 τ 1M − q(λ)k sup ak 1 . λ k1 k1
Explicitly, we set q(λ)0 = 1M and define q(λ)k = χ(0,λ] (q(λ)k−1 ak q(λ)k−1 ).
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
523
The theory of noncommutative martingales has achieved considerable progress in recent years. The renewed interest on this topic started from the fundamental paper of Pisier and Xu [47], where they introduced a new functional analytic approach to study Hardy spaces and the Burkholder–Gundy inequalities for noncommutative martingales. Shortly after, many classical inequalities have been transferred to the noncommutative setting. A noncommutative analogue of Doob’s maximal function [24], the noncommutative John–Nirenberg theorem [26], extensions of Burkholder inequalities for conditioned square functions [29] and related weak type inequalities [50–52]; see [44] for a simpler approach to some of them. 2. A pseudo-localization principle Let us now proceed with the proof of the pseudo-localization principle stated in the introduction. In the course of it we will see the link with a shifted form of the T 1 theorem, which is formulated in a dyadic martingale setting. Since we are concerned with its applications to our noncommutative problem, we leave a more in depth analysis of our result to Appendix A below. 2.1. Three auxiliary results We need some well-known results that live around David–Journé’s T 1 theorem. Cotlar lemma is very well known and its proof can be found in [12,54]. We include the proof of Schur lemma, since our statement and proof is non-standard, see Remark 2.1 below for details. The localization estimate at the end follows from [39]. We give the proof for completeness. Cotlar lemma. Let H be a Hilbert space and let us consider a family (Tk )k∈Z of bounded operators on H with finitely many non-zero Tk ’s. Assume that there exists a summable sequence (αk )k∈Z such that 2 max Ti∗ Tj B(H) , Ti Tj∗ B(H) αi−j for all i, j ∈ Z. Then we automatically have Tk
B (H )
k
αk .
k
Schur lemma. Let T be given by Tf (x) =
k(x, y)f (y) dy. Rn
Let us define the Schur integrals associated to k S2 (y) = k(x, y) dx. S1 (x) = k(x, y) dy, Rn
Rn
Assume that both S1 and S2 belong to L∞ . Then, T is bounded on L2 and T B(L2 ) S1 ∞ S2 ∞ .
524
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
Proof. By the Cauchy–Schwarz inequality we obtain 2 1 2 1 2 2 k(x, y)f (y) dy dx k(x, y)f (y) dy dx Rn Rn
Rn
Rn
Rn
k(x, y) dy
Rn
1 2 k(x, y)f (y)2 dy dx
Rn
S1 ∞
k(x, y)f (y)2 dy dx
1 2
Rn Rn
S1 ∞ S2 ∞
f (y)2 dy
1 2
.
2
Rn
Remark 2.1. Typically, Schur lemma is formulated as T B(L2 )
1 S1 ∞ + S2 ∞ , 2
see e.g. [39,54]. This might happen because we usually have S1 ∞ ∼ S2 ∞ , by certain symmetry in the estimates. In particular, the cases for which the arithmetic mean does not help but the geometric mean does are very rare in the literature, or even (as far as we know) not existent! However, motivated by a lack of symmetry in our estimates, this is exactly the case in this paper. A localization estimate. Assume that k(x, y)
1 |x − y|n
for all x, y ∈ Rn .
Let T be a Calderón–Zygmund operator associated to the kernel k and assume that T is L2 normalized. Then, given x0 ∈ Rn and r1 , r2 ∈ R+ with r2 > 2r1 , the estimate below holds for any pair f, g of bounded scalar-valued functions respectively supported by Br1 (x0 ) and Br2 (x0 ) Tf, g cn r n log(r2 /r1 )f ∞ g∞ . 1
Proof. Let us write B for the ball B3r1 /2 (x0 ) and let us consider a smooth function ρ which is identically 1 on B and identically 0 outside B2r1 (x0 ). Set η = 1 − ρ so that we may decompose Tf, g = Tf, ρg + Tf, ηg. For the first term we have Tf, ρg Tf 2 ρg2 f 2 ρg2 f ∞ g∞ |supp f |supp(ρg) cn r1n f ∞ g∞ .
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
525
On the other hand, for the second term we have . Tf, ηg = k(x, y)f (y) dy ηg(x) dx Br2 (x0 )\B
Br1 (x0 )
The latter integral is clearly bounded by f ∞ g∞ Ω
dx dy |x − y|n
with Ω = (Br2 (x0 ) \ B) × Br1 (x0 ). However, it is easily checked that an upper bound for the double integral given above is provided by cn r1n log(r2 /r1 ), where cn is a constant depending only on n. This completes the proof. 2 2.2. Shifted T 1 theorem By the conditions imposed on T in the introduction, it is clear that its adjoint T ∗ is an L2 normalized Calderón–Zygmund operator with kernel k ∗ (x, y) = k(y, x) satisfying the same size and smoothness estimates. This implies that T ∗ 1 (understood in a weak sense, see e.g. [54] for details) belongs to BMO, the space of functions with bounded mean oscillation. In addition, if Δj = Ej − Ej −1 denotes the dyadic martingale difference operator, it is also well known that for any ρ ∈ BMO the dyadic paraproduct against ρ Πρ (f ) =
∞
Δj (ρ)Ej −1 (f )
j =−∞
is bounded on L2 . Here it is necessary to know how BMO is related to its dyadic version BMOd , see [16] and [35] for details. It is clear that Πρ (1) = ρ and the adjoint of Πρ is given by the operator Πρ∗ (f ) =
∞
Ej −1 Δj (ρ)f .
j =−∞
Thus, since T ∗ 1 ∈ BMO we may write T = T0 + ΠT∗ ∗ 1 . According to our previous considerations, both T0 and ΠT∗ ∗ 1 are Calderón–Zygmund operators bounded on L2 and their kernels satisfy the standard size and smoothness conditions imposed on T with the same Lipschitz smoothness parameter γ , see [54] for the latter assertion. Moreover, the operator T0 now satisfies T0∗ 1 = 0. Now we use that T0∗ 1 is the weak∗ limit of a sequence (T0∗ ρk )k1 in BMO, where the ρk ’s are increasing bump functions which converge to 1. In particular, the relation below holds for any f ∈ H1 T0 f (x) dx = 0. Rn
(2.1)
526
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
Indeed, we have T0 f, 1 = f, T0∗ 1 = 0. The use of paraproducts is exploited in the T 1 theorem to produce the cancellation condition (2.1), which is a key assumption to make Cotlar lemma effective in this setting. The paraproduct term is typically estimated using Carleson’s lemma, although we will not need it here. What we shall do is to prove that our theorem for T0 and ΠT∗ ∗ 1 reduces to prove a shifted form of the T 1 theorem. In this paragraph we only deal with T0 . Let T be a generalized Calderón–Zygmund operator as in the statement of our result and assume that T satisfies the cancellation condition (2.1), so that there is no need to use the notation T0 in what follows. Let us write Rn \ Σf,s =
Θk
with Rn \ Θk = 9Ωk .
k∈Z
Denote by Ek the kth dyadic conditional expectation and by Δk the martingale difference operator Ek − Ek−1 , so that Ek (f ) = fk and Δk (f ) = dfk . Recall that Ωk and Θk are Rk -sets. In particular, the action of multiplying by the characteristic functions 1Ωk or 1Θk commutes with Ej for all j k. Then we consider the following decomposition 1Rn \Σf,s Tf = 1Rn \Σf,s
Ek T Δk+s 1Ωk
+ (id − Ek )1Θk T 1Ωk Δk+s (f ).
k
k
Note that we have used here the shift condition supp dfk+s ⊂ Ωk as well as the commutation relations mentioned above in conjunction with Rn \ Σf,s ⊂ Θk . Next we observe that 1Θk T 1Ωk = 1Θk T4·2−k 1Ωk , where Tε denotes the truncated singular integral formally given by Tε f (x) =
k(x, y)f (y) dy.
|x−y|>ε
Indeed, we have 1Θk T 1Ωk f (x) = 1Θk (x)
1Rn \9Q (x)
Q∈Qk Q∩Ωk =∅
k(x, y)f (y) dy, Q
from where the claimed identity follows, since we have dist Q, Rn \ 9Q = 4 · 2−k for all Q ∈ Qk . Taking all these considerations into account, we deduce 1Rn \Σf,s Tf = 1Rn \Σf,s
Ek T Δk+s +
(id − Ek )T4·2−k Δk+s (f ).
k
k
In particular, our problem reduces to estimate the norm in B(L2 ) of Φs =
k
Ek T Δk+s
and Ψs =
(id − Ek )T4·2−k Δk+s . k
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
527
Both Φs and Ψs are reminiscent of well-known operators (in a sense Ek T and T4·2−k behave here in the same way) appearing in the proof of the T 1 theorem by David and Journé [10]. Indeed, what we find (in the context of dyadic martingales) is exactly the s-shifted analogs meaning that we replace Δk by Δk+s . In summary, we have proved that under the assumption that cancellation condition (2.1) holds our main result reduces to the proof of the theorem below. Shifted T 1 theorem. Let T be an L2 -normalized Calderón–Zygmund operator with Lipschitz parameter γ . Assume that T ∗ 1 = 0 or, in other words, that we have Rn Tf (x) dx = 0 for any f ∈ H1 . Then, we have Ek T Δk+s Φs B(L2 ) =
B(L2 )
k
cn,γ s2−γ s/4 .
Moreover, regardless the value of T ∗ 1 we also have (id − Ek )T4·2−k Δk+s Ψs B(L2 ) =
B(L2 )
k
cn,γ 2−γ s/2 .
Remark 2.2. For some time, our hope was to estimate T4·2−k Δk+s
B(L2 )
k
since we believed that the truncation of order 2−k in conjunction with the action of Δk+s was enough to produce the right decay. Note that our pseudo-localization result could also be deduced from this estimate. However, the cancellation produced by the paraproduct decomposition in Φs and by the presence of the term id − Ek in Ψs play an essential role in the argument. 2.3. Paraproduct argument Now we show how the estimate of the paraproduct term also reduces to the shifted T 1 theorem stated above. Indeed, let us write Π instead of ΠT∗ ∗ 1 to simplify the notation. Then, as we did above, it is straightforward to see that 1
Rn \Σ
f,s
Πf = 1
Rn \Σ
f,s
+ (id − Ek )Π4·2−k Δk+s (f ).
Ek ΠΔk+s 1Ωk
k
k
Recalling one more time that Π is an L2 -bounded generalized Calderón–Zygmund operator satisfying the same size and smoothness conditions as T , the estimate for the second operator (id − Ek )Π4·2−k Δk+s k
B(L2 )
cn,γ 2−γ s/2
follows from the second assertion of the shifted T 1 theorem. Here it is essential to note that the hypothesis T ∗ 1 = 0 is not needed for Ψs . Therefore, it only remains to estimate the first
528
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
operator. However, we claim that 1Rn \Σf,s k Ek ΠΔk+s 1Ωk f is identically zero. Let us prove this assertion. We have Ek ΠΔk+s 1Ωk f = 1Rn \Σf,s Ek Ej −1 Δj T ∗ 1 1Ωk dfk+s . 1Rn \Σf,s k
k
j
If we fix the integer k, all the j -terms on the second sum above vanish except for the term associated to j = k + s. Indeed, if j < k + s we use Ej −1 = Ej −1 Ek+s−1 and obtain
Ej −1 Δj T ∗ 1 1Ωk dfk+s = Ej −1 Ek+s−1 Δj T ∗ 1 1Ωk dfk+s
= Ej −1 Δj T ∗ 1 Ek+s−1 (1Ωk dfk+s ) = 0.
If j > k + s we have
Ej −1 Δj T ∗ 1 1Ωk dfk+s = Ej −1 Δj T ∗ 1 1Ωk dfk+s = 0.
In particular, we obtain the following identity 1Rn \Σf,s
Ek ΠΔk+s 1Ωk f = 1Rn \Σf,s
k
k
= 1Rn \Σf,s
Ek Δk+s T ∗ 1 1Ωk dfk+s
1Ωk Ek Δk+s T ∗ 1 dfk+s = 0.
k
The last identity follows from the fact that Ωk ⊂ Σf,s and Rn \ Σf,s are disjoint. 2.4. Estimating the norm of Φs Now we estimate the operator norm of the sum Φs under the assumption that the cancellation condition (2.1) holds for T . We begin by identifying the kernel of the operators appearing in Φs . Let us denote by ke,k and kδ,k+s the kernels of Ek and Δk+s respectively. The kernel of the operator Ek T Δk+s is then given by ks,k (x, y) =
ke,k (x, w)k(w, z)kδ,k+s (z, y) dw dz.
Rn ×Rn
It is straightforward to verify that ke,k (x, w) = 2nk
R∈Qk
kδ,k+s (z, y) = 2n(k+s)
1R×R (x, w),
Q∈Qk+s
1Q×Q (z, y) −
1 1 (z, y) .
2n Q×Q
Given x, y ∈ Rn , define Rx to be the only cube in Qk containing x, while Qy will stand for the only cube in Qk+s containing y. Moreover, let Q2 , Q3 , . . . , Q2n be the remaining cubes in Qk+s sharing dyadic father with Qy . Let us introduce the following functions
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
φRx (w) =
529
1 1R (w), |Rx | x n
2 1 ψQ 1Qy (z) − 1Qj (z).
y (z) =
y | |Q j =2
Then the kernel ks,k (x, y) can be written as follows ks,k (x, y) = T (ψQ
y ), φRx .
(2.2)
Notice that ψQ
y ∈ H1 since it is a linear combination of atoms. 2.4.1. Schur type estimates In this paragraph we give pointwise estimates for the kernels ks,k and use them to obtain upper bounds of the Schur integrals associated to them. Both will be used below to produce Cotlar type estimates. Lemma 2.3. The following estimates hold: (a) If y ∈ Rn \ 3Rx , we have ks,k (x, y) cn 2−γ (k+s)
1 . |x − y|n+γ
(b) If y ∈ 3Rx \ Rx , we have ks,k (x, y) cn,γ 2−γ (k+s) 2nk min Rx
dw γ (k+s) . , s2 |w − cy |n+γ
(c) Similarly, if y ∈ Rx we have ks,k (x, y) cn,γ 2−γ (k+s) 2nk min
Rn \Rx
dw γ (k+s) . , s2 |w − cy |n+γ
y . The constant cn,γ only depends on n and γ ; cy denotes the center of the cube Q Proof. We proceed in several steps. The first estimate. Using ψQ
y (z) dz = 0, Rn
530
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
y we obtain the following identity where cy denotes the center of Q ks,k (x, y) = φRx (w) k(w, z) − k(w, cy ) ψQ
y (z) dw dz.
y Rx × Q
y , Lipschitz smoothness gives Since |z − cy | 12 |w − cy | for (w, z) ∈ Rx × Q
ks,k (x, y)
φRx (w)
y Rx × Q
|z − cy |γ ψQ
y (z) dw dz. n+γ |w − cy |
y Then, we use |w − cy | 13 |x − y| and |z − cy | 2−(k+s) for (w, z) ∈ Rx × Q −γ (k+s) ks,k (x, y) cn 2 |x − y|n+γ
y Rx × Q
2−γ (k+s) φRx (w)ψQ .
y (z) dw dz cn |x − y|n+γ
The second estimate. By (2.1), we have T (ψQ
y )(w) dw = 0. Rn
Using this cancellation, we shall use the following relations:
y ⊂ • If y ∈ / Rx ⇒ Q Rx and |ks,k (x, y)| =
y ⊂ Rx and |ks,k (x, y)| = • If y ∈ Rx ⇒ Q
1
y )(w) dw|. |Rx | | Rx T (ψQ 1 | T
y )(w) dw|. |Rx | Rn \Rx (ψQ
In the first case, we may have
y ∩ Rx = ∅, (b1) 3Q
y ∩ Rx = ∅. (b2) 3Q
y ∩ Rx = ∅, we may use Lipschitz smoothness as above to obtain If 3Q ks,k (x, y) 1 |Rx | cn 2
y Rx × Q
|z − cy |γ dw dz ψ (z)
Q y |w − cy |n+γ
−γ (k+s) nk
2
Rx
dw . |w − cy |n+γ
y ∩ Rx = ∅ we use the latter estimate on Rx \ 3Q
y On the other hand, if 3Q ks,k (x, y) cn 2−γ (k+s) 2nk
y Rx \3Q
dw cn +
y | |w − cy |n+γ |Rx ||Q
y )×Q
y (Rx ∩3Q
k(w, z) dw dz.
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
531
√ Fig. 1. We have α = δz and β 2 n2−(k+s−1) .
We claim that the second term on the right-hand side is dominated by the first one, up to a constant
y ) with ∂Ω denoting the cn,γ depending only on n and γ . Indeed, let us write δz = dist(z, ∂ Q boundary of Ω. The size estimate for the kernel gives 1
y | |Rx ||Q
k(w, z) dw dz 2nk 2n(k+s)
y Rx ∩3Q
y Q
y )×Q
y (Rx ∩3Q
dw dz. |w − z|n
According to Fig. 1, we easily see that
nk n(k+s)
2 2
y Rx ∩3Q
y Q
dw dz cn 2nk 2n(k+s) |w − z|n
y Q
cn 2nk 2n(k+s)
y Q √
√ 2 n2−(k+s−1)
dr r
√ −(k+s−1) 2 n2 dz · σ (Sn−1 ) log δz √ 4 n2−(k+s) r n−1 dr log √ −(k+s) n2 −r
k+s n/2
∼ cn 2 2
0
cn 2 . nk
This gives rise to Rx
dσ dz
δz
Sn−1
nk n(k+s)
ks,k (x, y) cn 2−γ (k+s) 2nk
dw x (y), + cn 2nk 1Us,k |w − cy |n+γ
x is defined by where the set Us,k
x = y ∈ Rn \ Rx dist(y, ∂Rx ) < 2−(k+s−1) . Us,k
532
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
x we have However, it is easily seen that for y ∈ Us,k
2−γ (k+s)
Rx
dw cn 2−γ (k+s) |w − cy |n+γ
2−(k+s) +2−k 2−(k+s)
Sn−1
dr
dσ cn,γ .
r 1+γ
In particular, we deduce our claim and so ks,k (x, y) cn,γ 2−γ (k+s) 2nk
Rx
dw . |w − cy |n+γ
y ⊂ Rx , we may have In the second case Q
y ∩ (Rn \ Rx ) = ∅, (c1) 3Q
y ∩ (Rn \ Rx ) = ∅. (c2) 3Q The argument in this case is entirely similar. Indeed, if the intersection is empty we use Lipschitz smoothness one more time and the same argument as above gives
ks,k (x, y) 2−γ (k+s) 2nk
Rn \Rx
dw . |w − cy |n+γ
If the intersection is not empty, the inequality 1
y | |Rx ||Q
k(w, z) dw dz cn 2nk
y )×Q
y ((Rn \Rx )∩3Q
can be proved as above. This gives rise to the estimate ks,k (x, y) cn 2−γ (k+s) 2nk
Rn \Rx
dw x (y), + cn 2nk 1Vs,k |w − cy |n+γ
x is defined by where the set Vs,k
x Vs,k = y ∈ Rx dist(y, ∂Rx ) < 2−(k+s−1) . x we have Now we use that for y ∈ Vs,k
2
−γ (k+s)
Rn \Rx
dw cn 2−γ (k+s) |w − cy |n+γ
Sn−1
∞ 2−(k+s)
dr r 1+γ
Our estimates prove the first halves of inequalities (b) and (c) above.
dσ cn,γ .
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
533
The third estimate. It remains to prove that ks,k (x, y) = T (ψ ), φR cn s2nk . x Qy Since y ∈ 3Rx , the localization estimate in Section 2.1 gives ks,k (x, y) cn (Q
y )n log (3Rx ) φRx ∞ ψQ
y ∞
y ) (Q n
y | log 32s−1 1 2 − 1 cn s2nk . = cn |Q
y | |Rx | |Q
We have used that T is assumed to be L2 -normalized. The proof is complete.
2
Lemma 2.4. Let us define 1 Ss,k (x) =
ks,k (x, y) dy,
Rn
2 Ss,k (y) =
ks,k (x, y) dx.
Rn
Then, there exists a constant cn,γ depending only on n, γ such that 1 (x) Ss,k
cn,γ s 2γ s
2 Ss,k (y) cn,γ s
for all (x, k) ∈ Rn × Z, for all (y, k) ∈ Rn × Z.
1 and S 2 in turn. Proof. We estimate Ss,k s,k
Estimate of S 1s,k (x). Given x ∈ Rn , define the cube Rx as above. Then we decompose the 1 (x) into three regions according to Lemma 2.3 and estimate each one indeintegral defining Ss,k pendently. Using Lemma 2.3(a) we find dy ks,k (x, y) dy cn 2−γ (k+s) cn 2−γ s . (2.3) |x − y|n+γ Rn \3Rx
Rn \3Rx
On the other hand, the first estimate in Lemma 2.3(b) gives ks,k (x, y) dy cn,γ 2−γ (k+s) 2nk 3Rx \Rx
3Rx \Rx Rx
1 dw dy. |w − cy |n+γ
Now we set δw = dist(w, ∂Rx ) for w ∈ Rx . Then we clearly have
δw ≡ δw + 2−(k+s) δw + dist(cy , ∂Rx ) |w − cy |.
534
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
Fig. 2. Even if x and y are close, we have a (
δw − δw )-margin.
In particular, we find (see Fig. 2) 3Rx \Rx
|B δw (w)| dy + n+γ n+γ
|w − cy | δw
dy ∼ 1/
δwγ . |w − y|n+γ
Rn \B
δw (w)
This provides us with the estimate
3Rx \Rx Rx
1 dw dy cn |w − cy |n+γ
2−k
Sn−1 0
r n−1 dr dσ. (2−(k+s) + 2−k − r)γ
Using t = 2−k + 2−(k+s) − r and the bound r 2−k
3Rx \Rx Rx
1 dw dy cn 2−(n−1)k |w − cy |n+γ cn
2−k
2−(k+s)
s2−(n−1)k cγ 2−nk 2γ k
dt tγ if γ = 1, if 0 < γ < 1.
In summary, combining our estimates we have obtained ks,k (x, y) dy cn,γ s2−γ s . 3Rx \Rx
It remains to control the integral over Rx . By Lemma 2.3(c) ks,k (x, y) dy cn,γ 2−γ (k+s) 2nk Rx
Rx Rn \Rx
1 dw dy. |w − cy |n+γ
(2.4)
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
535
For any given y ∈ Rx , we set again δcy = dist(cy , ∂Rx ) 2−(k+s) . Arguing as above, we may use polar coordinates to obtain
Rx Rn \Rx
1 dw dy |w − cy |n+γ
∞ Sn−1 δcy
Rx
r n−1 dy dr dσ dy ∼ γ n+γ r δc y Rx
2−k −2 −(k+s)
∼
r n−1 dr dσ − r)γ
(2−k
Sn−1
0
2−k
r n−1
+ Sn−1 2−k −2−(k+s)
2−γ (k+s)
dr dσ.
The first integral is estimated as above Sn−1
2−k −2 −(k+s)
0
−(n−1)k r n−1 s2 dr dσ c n cγ 2−nk 2γ k (2−k − r)γ
if γ = 1, if 0 < γ < 1,
as for the second we obtain an even better bound. Indeed, we have
2−k
Sn−1 2−k −2−(k+s)
r n−1 2−γ (k+s)
n dr dσ ∼ 2γ (k+s) 2−nk − 2−k − 2−(k+s) n = 2γ (k+s) 2−nk 1 − 1 − 2−s n n −sj 2 2γ (k+s) 2−nk j j =1
cn 2−nk 2γ k . Writing all together we finally get
ks,k (x, y) dy cn,γ s2−γ s .
(2.5)
Rx 1 (x) c −γ s . According to (2.3), (2.4) and (2.5) we obtain the upper bound Ss,k n,γ s2
Estimate of S 2s,k (y). Given a fixed point y, we consider a partition Rn = Ω1 ∪ Ω2 where Ω1 is the set of points x such that y ∈ / 3Rx and Ω2 = Rn \ Ω1 . In the region Ω1 we may proceed as
536
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
in (2.3). On the other hand, inside Ω2 and according to Lemma 2.3 we know that |ks,k (x, y)| cn,γ s2nk . This means that we have 2 (y) cn,γ 2−γ s + |Ω2 |s2nk = cn,γ 2−γ s + |3Ry |s2nk cn,γ s. Ss,k This upper bound holds for all (y, k) ∈ Rn × Z. Hence, the proof is complete.
2
2.4.2. Cotlar type estimates Let us write Λs,k for Ek T Δk+s . According to the pairwise orthogonality of martingale differences, we have Λs,i Λ∗s,j = 0 whenever i = j . In particular, it follows from Cotlar lemma that it suffices to control the norm of the operators Λ∗s,i Λs,j . Explicitly, our estimate for Φs stated in the shifted T 1 theorem will be deduced from ∗ 2 −γ s/2 2 Λ Λ αi−j s,i s,j B (L ) cn,γ s 2 2
for some summable sequence (αk )k∈Z . The kernel of Λ∗s,i Λs,j is given by s ki,j (x, y) =
ks,i (z, x)ks,j (z, y) dz. Rn
Before proceeding with our estimates we need to point out another cancellation property which easily follows from (2.1). Given r > 0 and a point y ∈ Rn , let f (z) = 1Br (y) (z)/|Br (y)|. Then it is clear that Δk+s f = dfk+s is in H1 since it can be written as a linear combination of atoms. According to our cancellation condition (2.1) we find
Ek T Δk+s f (x) dx = Rn
T dfk+s (x) dx = 0. Rn
In terms of the kernels, this identity is written as
ks,k (x, z)f (z) dz dx = 0.
Rn
Rn
Using Fubini theorem (our estimates in Lemma 2.3 ensure the integrability) and taking the limit as r → 0, the Lebesgue differentiation theorem implies the following identity, which holds for almost every point y: ks,k (x, y) dx = 0. Rn
This holds for all k ∈ Z and we deduce s (x, y) = ks,i (z, x) ks,j (z, y) − ks,j (x, y) dz ki,j Rn
(2.6)
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
537
ks,i (z, x) − ks,i (y, x) ks,j (z, y) dz.
= Rn
s , we use the first or the second expression above according In order to estimate the kernels ki,j to whether i j or not. Since the estimates are entirely similar we shall assume in what follows that i j and work in the sequel with the first expression above. Moreover, given w ∈ Rn we shall write all through out this paragraph Rw for the only cube in Qj containing w. Then, since Rz = Rx whenever z ∈ Rx , it follows from (2.2) that s ki,j (x, y) = ks,i (z, x) ks,j (z, y) − ks,j (x, y) dz Rn \3Rx
+
ks,i (z, x) ks,j (z, y) − ks,j (x, y) dz.
3Rx \Rx s (x, y) and β s (x, y) are the first and second terms above, let If αi,j i,j
1,α Si,j,s (x) =
s α (x, y) dy, i,j
Rn
2,α Si,j,s (y) =
s α (x, y) dx, i,j
Rn 1,β
Si,j,s (x) =
s β (x, y) dy, i,j
Rn 2,β
Si,j,s (y) =
s β (x, y) dx. i,j
Rn
According to Schur lemma from Section 2.1, we obtain the upper bound ∗ 2,β 2,α Λ Λ S 1,α + S 1,β S s,i s,j B (L ) i,j,s ∞ i,j,s ∞ + Si,j,s ∞ . i,j,s ∞ 2
Lemma 2.5. We have −γ s 1,α 1,β , S . max Si,j,s i,j,s ∞ cn,γ s s + |i − j | 2 ∞ Proof. According to Lemma 2.3, we know that ks,i (z, x) cn 2−γ (i+s)
1 |x − z|n+γ
whenever z ∈ / 3Rx . Moreover, Lemma 2.4 gives ks,j (z, y) − ks,j (x, y) dy cn,γ s2−γ s . Rn
(2.7)
538
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
If we combine the two estimates above, we obtain
ks,i (z, x)ks,j (z, y) − ks,j (x, y) dz dy
1,α Si,j,s (x) Rn
Rn \3Rx
ks,i (z, x)
=
Rn \3Rx
cn,γ s2
Rn −γ s −γ (i+s)
ks,j (z, y) − ks,j (x, y) dy dz
2
Rn \3Rx
dz cn,γ s2−2γ s 2−γ |i−j | . |x − z|n+γ
The last inequality uses the assumption i j , so that i − j = |i − j |. The estimate above holds for all x ∈ Rn . Hence, since cn,γ s2−2γ s 2−γ |i−j | is much smaller than cn,γ s(s + |i − j |)2−γ s , it is clear that the first function satisfies the thesis. Let us now proceed with the second function. To that aim we observe that ks,j (z, y) is j -measurable as a function in z, meaning that Ej (ks,j (·, y))(z) = ks,j (z, y). This follows from (2.2). In particular, the same holds for the function 13Rx \Rx (z) ks,j (z, y) − ks,j (x, y) . Therefore, using the integral invariance of conditional expectations
s βi,j (x, y) =
Ej ks,i (·, x) (z) ks,j (z, y) − ks,j (x, y) dz
3Rx \Rx
=
R∼Rx R
1 |R|
ks,i (w, x) dw ks,j (z, y) − ks,j (x, y) dz,
R
where R ∼ Rx is used to denote that R is a neighbor of Rx in Qj . That is, the neighbors of Rx form a partition of 3Rx \ Rx formed by 3n − 1 cubes in Qj . If cR denotes the center of R, we use that ks,j (z, y) = ks,j (cR , y) for z ∈ R and obtain the estimate s ks,i (w, x) dw ks,j (cR , y) − ks,j (x, y). β (x, y) i,j
(2.8)
R∼Rx R
This, combined with Lemma 2.4, produces Si,j,s (x) cn,γ s2−γ s 1,β
ks,i (w, x) dw . R∼Rx R
Let us now estimate the integral. If w ∈ Sw ∈ Qi and x ∈ Ox ∈ Qi+s
(2.9)
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
539
ks,i (w, x) dw = R
T ψO x , φSw dw R
=
S⊂R, S∈Qi S
T ψO x (z) dz = T ψO x , 1R .
Now we use the localization estimate from Section 2.1 to obtain (3R) n T ψ , 1R cn (O
ψO x ∞ 1R ∞ cn s + |i − j | . x ) log Ox
(Ox ) Since there are 3n − 1 neighbors, this estimate completes the proof with (2.9).
2
Lemma 2.6. We have 2,α 2,β −γ |i−j | 2 , S max Si,j,s . i,j,s ∞ cn s 1 + |i − j | 2 ∞ Proof. Once again, Lemma 2.3 gives ks,i (z, x) cn 2−γ (i+s)
1 |x − z|n+γ
for z ∈ / 3Rx .
This, together with Fubini theorem produces
2,α Si,j,s (y) cn Rn
Rn \3Rx
2−γ (i+s) ks,j (z, y) − ks,j (x, y) dz dx |x − z|n+γ
cn Rn
Rn \B2−j (z)
2−γ (i+s) ks,j (z, y) dz dx n+γ |x − z|
+ cn Rn
Rn \B2−j (x)
= cn 2−γ s 2−γ |i−j |
2−γ (i+s) dz ks,j (x, y) dx n+γ |x − z|
ks,j (w, y) dw.
Rn
Now, according to Lemma 2.4 we now that the integral on the right-hand side is bounded by cn,γ s for all y in Rn . Therefore, the L∞ norm of the first function is much smaller than our upper bound. Let us now estimate the second function. If we proceed as in Lemma 2.5 and use (2.8), we find 2,β
Si,j,s (y)
ks,i (w, x) dw ks,j (cR , y) − ks,j (x, y) dx. Rn R∼Rx R
540
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
Now we need a different estimate for the integral of ks,i (·, x) over the neighbor cubes R of Rx . Indeed, combining the pointwise estimates obtained in Lemma 2.3 it easily follows that ks,i (w, x) cn,γ
s2ni (1 + 2i |x − w|)n+γ
for all (w, x) ∈ Rn × Rn .
(2.10)
If we set δx = dist(x, ∂Rx ) dist(x, ∂R), we get 2ni ks,i (w, x) dw cn,γ s dw i (1 + 2 |x − w|)n+γ R
R
∞
cn,γ s δx
Sn−1
∞ = cn,γ s 2i δx
2ni r n−1 dr dσ (1 + 2i r)n+γ
zn−1 1 dz cn,γ s . (1 + z)n+γ (1 + 2i δx )γ
Using (2.10) for ks,j , we have 2,β
Si,j,s (y) cn,γ s 2 2nj Υ (i, j, γ ) where the term Υ (i, j, γ ) is given by 1 1 1 + dx. (1 + 2i δx )γ (1 + 2j |cR − y|)n+γ (1 + 2j |x − y|)n+γ Rn R∼Rx
It is straightforward to see that it suffices to estimate the integral 1 1 dx. (1 + 2i δx )γ (1 + 2j |x − y|)n+γ
(2.11)
Rn
Indeed, both functions inside the big bracket above are comparable and the sum R∼Rx can be deleted since it only provides an extra factor of 3n − 1. Now, the main idea to estimate (2.11) is to observe that the two functions in the integrand are nearly independent inside any dyadic cube of Qj . Let us be more explicit, we have Rn
1 1 dx i γ j (1 + 2 δx ) (1 + 2 |x − y|)n+γ
R∈Qj R
1 1 dx i γ j (1 + 2 δx ) (1 + 2 dist(R, Ry ))n+γ
sup
R∈Qj
R
1 dx (1 + 2i δx )γ
R∈Qj
1 (1 + 2j dist(R, Ry ))n+γ
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
∼ sup
R∈Qj
R
1 dx (1 + 2i δx )γ
Rn
541
2nj dx. − y|)n+γ
(1 + 2j |x
The integral on the right-hand side is majorized by an absolute constant. Moreover, recalling that δx stands for dist(x, ∂Rx ) and that Rx = R for any x ∈ R ∈ Qj , it is clear that the integral on the left-hand side does not depend on the chosen cube R, so that the supremum is unnecessary. To estimate this integral we set λ = 1 − 2j −i R
2−j
dx ∼ (1 + 2i δx )γ
Sn−1 λ2−j
∼ 0
0
r n−1 dr dσ (1 + 2i (2−j − r))γ
r n−1 dr + (1 + 2i (2−j − r))γ
2−j
λ2−j
r n−1 dr. (1 + 2i (2−j − r))γ
The first integral is majorized by
2−γ i
λ2 −j
0
r n−1 dr 2−γ i λn−1 2−nj 2j (2−j − r)γ 2−nj
λ2 −j
dr − r)γ
(2−j 0
− j |2−|i−j |
|i cγ 2−γ |i−j |
if γ = 1, if 0 < γ < 1.
The second integral is majorized by 2−j
r n−1 dr 2−nj 2j 2−j − λ2−j = 2−nj 2−|i−j | .
λ2−j
Combining our estimates we finally get 2,β Si,j,s (y) cn,γ s 2 1 + |i − j | 2−γ |i−j | . Since the last estimate holds for all y ∈ Rn , the proof is complete.
2
Conclusion. According to (2.7), Lemmas 2.5 and 2.6 give ∗ 2 2 −γ s/2 2 3 −γ s 2−γ |i−j | c Λ Λ c αi−j , n,γ s s + |i − j | 2 n,γ s 2 s,i s,j B (L ) 2
1
where αk = (1 + |k|) 2 2−γ |k|/4 . In particular, Cotlar lemma provides the estimate Φs B(L2 ) = Ek T Δk+s cn,γ s2−γ s/4 αk = cn,γ s2−γ s/4 . k
B(L2 )
k
542
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
2.5. Estimating the norm of Ψs We finally estimate the operator norm of Ψs . This will complete the proof of our pseudolocalization principle. We shall adapt some of the notation introduced in the previous paragraph. Namely, we shall now write Λs,k when referring to the operator (id − Ek )T4·2−k Δk+s and ks,k (x, y) will be reserved for its kernel. Arguing as above it is simple to check that we have ks,k (x, y) = T4·2−k ψQ
y (x) − T4·2−k ψQ
y , φRx . We shall use the terminology 1 ks,k (x, y) = T4·2−k ψQ
y (x),
2 ks,k (x, y) = T4·2−k ψQ
y , φRx .
2.5.1. Schur type estimates Lemma 2.7. Let us consider the sets x = w ∈ Rn 4 · 2−k − 2−(k+s−1) |x − w| < 4 · 2−k + 2−(k+s−1) . Ws,k Then, the following pointwise estimate holds ks,k (x, y) cn 1Rn \B Proof. We have
−γ (k+s) 2 nk x (y) + 2 1 (y) . Ws,k 2·2−k (x) |x − y|n+γ
1 ks,k (x, y) =
1Rn \B4·2−k (x) (z)k(x, z)ψQ
y (z) dz.
y Q
If |x − y| 3 · 2−k we have |x − z| |x − y| + |y − z| |x − y| + 2−(k+s−1) 4 · 2−k
y . In particular, we obtain since z ∈ Q 1 ks,k (x, y) = 0
whenever |x − y| 3 · 2−k .
y If |x − y| > 5 · 2−k , then we have for z ∈ Q |x − z| |x − y| − |z − y| |x − y| − 2−(k+s−1) > 4 · 2−k . Thus, we can argue in the usual way and obtain 1 k (x, y) = k(x, z) − k(x, cy ) ψ (z) dz s,k Qy
y Q
cn
2−γ (k+s) |x − y|n+γ
y Q
−γ (k+s) ψ (z) dz cn 2 . Qy |x − y|n+γ
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
543
1 (x, y) as a sum of two integrals If 3 · 2−k < |x − y| 5 · 2−k , we write ks,k
1Rn \B4·2−k (x) (z) k(x, z) − k(x, cy ) ψQ
y (z) dz
1 ks,k (x, y) = Rn
+
k(x, cy ) 1Rn \B4·2−k (x) (z) − 1Rn \B4·2−k (x) (cy ) ψQ
y (z) dz
Rn
= A1 + B1 . Here we have used that ψQ
y is mean-zero. Lipschitz smoothness gives once more |A1 | cn
2−γ (k+s) . |x − y|n+γ
To estimate B1 we use the size condition on the kernel 1 cn |B1 |
y | |x − cy |n |Q
1Rn \B
4·2−k (x)
(z) − 1Rn \B4·2−k (x) (cy ) dz.
y Q
x = B (x) \ B (x), we have Q
y ∩ ∂ B −k (x) = ∅. Fig. 3. If y ∈ / Ws,k α β 4·2
Since 3 · 2−k < |x − y| 5 · 2−k , we have cn |x − cy |−n ∼ cn 2nk . Moreover, the only z’s for which the integrand above is not zero are those with (z, cy ) lying at different sides of ∂ B4·2−k (x). x (see Fig. 3) and we get This can only happen when y ∈ Ws,k |B1 | cn 2nk
x (y) 1Ws,k 1Rn \B x (y) . (z) − 1Rn \B4·2−k (x) (cy ) dz cn 2nk 1Ws,k 4·2−k (x)
|Qy |
y Q
Combining our estimates obtained so far we get −γ (k+s) 1 2 nk k (x, y) cn 1Rn \B x (y) + 2 1 (y) . (x) Ws,k s,k 3·2−k |x − y|n+γ
(2.12)
544
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
Let us now study pointwise estimates for the kernel 2 (x, y) = ks,k
1 |Rx |
1Rn \B4·2−k (w) (z)k(w, z)ψQ
y (z) dz dw.
y Q
Rx
If |x − y| 2 · 2−k we have |w − z| |w − x| + |x − y| + |y − z| 4 · 2−k
y . This gives for all (w, z) ∈ Rx × Q whenever |x − y| 2 · 2−k .
2 ks,k (x, y) = 0
y If |x − y| > 6 · 2−k , then we have for (w, z) ∈ Rx × Q |w − z| |x − y| − |x − w| − |z − y| > 4 · 2−k . Therefore, we obtain as usual the estimate 2 k (x, y) 1 s,k |Rx |
y Q
Rx
2−γ (k+s) |Rx | cn
k(w, z) − k(w, cy )ψ (z) dz dw Qy
1 |w − cy |n+γ
Rx
2−γ (k+s) |Rx |
Rx
ψ (z) dz dw Qy
y Q
dw 2−γ (k+s) = c . n |w − cy |n+γ |x − y|n+γ
When 2 · 2−k < |x − y| 6 · 2−k we have the two integrals 2 ks,k (x, y) =
1 |Rx | +
1Rn \B4·2−k (w) (z) k(w, z) − k(w, cy ) ψQ
y (z) dw dz
y Rx × Q
1 |Rx |
k(w, cy ) 1Rn \B4·2−k (w) (z) − 1Rn \B4·2−k (w) (cy ) ψQ
y (z) dw dz
y Rx × Q
= A2 + B2 . By Lipschitz smoothness, we may estimate A2 by |A2 | cn
2−γ (k+s)
y | |Rx ||Q
y Rx × Q
1Rn \B4·2−k (w) (z) |w − z|n+γ
dw dz cn 2nk 2−γ s
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
545
since 4 · 2−k |w − z| |w − x| + |x − y| + |y − z| 8 · 2−k . That is |A2 | cn
2−γ (k+s) |x − y|n+γ
2 · 2−k < |x − y| 6 · 2−k .
To estimate B2 we first observe that 1 −k 1 −k 2 = 2 . |w − cy | |x − y| − |x − w| − |cy − y| 2 − 1 − 2 2 Then we apply the size estimate for the kernel and Fubini theorem |B2 |
1
y | |Rx ||Q
cn 2nk
y | |Rx ||Q
|1Rn \B4·2−k (w) (z) − 1Rn \B4·2−k (w) (cy )| |w − cy |n
y Rx × Q
y Q
1Rn \B
dw dz
(z) − 1Rn \B4·2−k (w) (cy ) dw dz. 4·2−k (w)
Rx
y we know In the integral inside the brackets, the points z and cy are fixed. Moreover, since z ∈ Q −(k+s) that |z − cy | 2 . Therefore, we find that the only w’s for which the integrand of the inner integral is not zero live in cy Ws+1,k = w ∈ Rn 4 · 2−k − 2−(k+s) |w − cy | < 4 · 2−k + 2−(k+s) . This automatically gives the estimate |B2 |
2−γ (k+s) cn 2nk cy Ws+1,k cn 2nk 2−s cn . |Rx | |x − y|n+γ
Our partial estimates so far produce the global estimate 2 2−γ (k+s) k (x, y) cn 1Rn \B (y) . (x) −k s,k 2·2 |x − y|n+γ The assertion then follows from a combination of inequalities (2.12) and (2.13). Lemma 2.8. Let us define 1 Ss,k (x) =
ks,k (x, y) dy,
Rn
2 Ss,k (y) =
ks,k (x, y) dx.
Rn
Then there exists a constant cn such that 1 2 max Ss,k (x), Ss,k (y) cn 2−γ s .
(2.13) 2
546
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
x | c 2−nk 2−s Proof. According to Lemma 2.7 and |Ws,k n
1 Ss,k (x) cn Rn \B2·2−k (x)
2−γ (k+s) nk x + 2 1 (y) dy cn 2−γ s . Ws,k |x − y|n+γ
2 (y), since we have 1 x (y) = 1 y (x). The same argument applies for Ss,k Ws,k W s,k
2
2.5.2. Cotlar type estimates We have again Λs,i Λ∗s,j = 0 for i = j , so that we are reduced (by Cotlar lemma) to estimate the norms of Λ∗s,i Λs,j in B(L2 ). The kernel of Λ∗s,i Λs,j is given by s ki,j (x, y) =
ks,i (z, x)ks,j (z, y) dz. Rn
Taking f (z) = 1Br (y) (z)/|Br (y)|, we note 1 |Br (y)|
ks,k (x, z) dx dz = Br (y)
Rn
(id − Ek )T4·2−k Δk+s f (x) dx = 0, Rn
due to the integral invariance of conditional expectations. Taking the limit as r → 0, we deduce from Lebesgue differentiation theorem that the cancellation condition (2.6) also holds for our new kernels ks,k (x, y) and for a.e. y ∈ Rn . In particular, the same discussion as above leads us to use (2.6) in one way or another according to i j or vice versa. Both cases can be estimated in the same way. Thus we assume in what follows that i j and use the expression
ks,i (z, x) ks,j (z, y) − ks,j (x, y) dz
s (x, y) = ki,j Rn
ks,i (z, x) ks,j (z, y) − ks,j (x, y) dz
= Rn \B2·2−j (x)
+
ks,i (z, x) ks,j (z, y) − ks,j (x, y) dz.
B2·2−j (x)\B2·2−i (x)
Observe that the integrand vanishes for z in B2·2−i (x) since ks,i (z, x) does, according to s (x, y) and β s (x, y) for the first and second terms on the rightLemma 2.7. Let us write αi,j i,j hand side. Then (as before) we need to estimate the quantity S 1,α
i,j,s ∞
where the S functions are given by
1,β 2,α + S 2,β , + Si,j,s ∞ Si,j,s i,j,s ∞ ∞
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
547
s α (x, y) dy, i,j
1,α Si,j,s (x) = Rn
s α (x, y) dx, i,j
2,α Si,j,s (y) = Rn
1,β
s β (x, y) dy, i,j
Si,j,s (x) = Rn
2,β
s β (x, y) dx. i,j
Si,j,s (y) = Rn
Lemma 2.9. We have 1,α 1,β −2γ s , S max Si,j,s . i,j,s ∞ cn 2 ∞ Proof. According to Lemma 2.7, we know that ks,i (z, x) cn
2−γ (i+s) ni x + 2 1 (z) Ws,i |x − z|n+γ
for all z ∈ Rn \ B2·2−j (x). Moreover, Lemma 2.8 gives
ks,j (z, y) − ks,j (x, y) dy cn 2−γ s .
Rn 1,α Combining these estimates we find an L∞ bound for Si,j,s 1,α Si,j,s (x) cn 2−γ s
Rn \B2·2−j (x)
2−γ (i+s) ni x + 2 1 (z) dz Ws,i |x − z|n+γ
x . cn 2−2γ s 2−γ |i−j | + cn 2−γ s 2ni Rn \ B2·2−j (x) ∩ Ws,i 1,α We claim that Si,j,s (x) cn 2−2γ s 2−γ |i−j | . Indeed, if the intersection above is empty there is nothing to prove. If it is not empty, the following inequality must hold
4 · 2−i + 2−(i+s−1) > 2 · 2−j . This implies that we can only have i = j or i = j + 1 and hence x x 2−2γ s ∼ 2−2γ s 2−γ |i−j | . 2−γ s 2ni Rn \ B2·2−j (x) ∩ Ws,i 2−γ s 2ni Ws,i Therefore, the first function clearly satisfies the thesis. Let us now analyze the second function. To that aim we proceed exactly as above in B2·2−j (x) \ B2·2−i (x) and obtain
548
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
Si,j,s (x) cn 2−γ s 1,β
B2·2−j (x)\B2·2−i (x)
2−γ (i+s) ni x + 2 1 (z) dz Ws,i |x − z|n+γ
x cn 2−2γ s . cn 2−2γ s + cn 2−γ s 2ni Rn \ B2·2−j (x) ∩ Ws,i
2
Lemma 2.10. We have 2,α 2,β −γ |i−j | , S max Si,j,s . i,j,s ∞ cn,γ 1 + |i − j | 2 ∞ Proof. For the first function we have −γ (i+s) 2 2,α ni x + 2 1Ws,i (z) ks,j (z, y) dz dx Si,j,s (y) cn |x − z|n+γ Rn
Rn \B2·2−j (x)
+ cn Rn
Rn \B2·2−j (x)
= cn Rn
Rn \B2·2−j (z)
2−γ (i+s) ni z (x) dx ks,j (z, y) dz + 2 1 Ws,i n+γ |x − z|
+ cn Rn
2−γ (i+s) ni x (z) + 2 1Ws,i ks,j (x, y) dz dx |x − z|n+γ
Rn \B2·2−j (x)
2−γ (i+s) ni x (z) dz ks,j (x, y) dx, + 2 1Ws,i n+γ |x − z|
cn 2−3γ s 2−γ |i−j | . x (z) = 1 z (x). The last inequality Here we have used Lemma 2.7, Fubini theorem and 1Ws,i Ws,i follows arguing as in Lemma 2.9. Let us now estimate the second S function. We may assume 2,β i = j because otherwise Si,j,s = 0. Let us decompose
ks,j (z, y) − ks,j (x, y) A + B, where these terms are given by
1 1 A = ks,j (z, y) − ks,j (x, y), 2 2 B = ks,j (z, y) − ks,j (x, y).
Moreover, we further decompose the A-term into A 1Rn \B4·2−j (z) (w)k(z, w) − k(x, w)ψQ
y (w) dw
y Q
+
y Q
k(x, w)1Rn \B
(z) 4·2−j
(w) − 1Rn \B4·2−j (x) (w)ψQ
y (w) dw = A1 + A2 .
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
This gives rise to 2,β Si,j,s (y)
Rn
ks,i (z, x)(A1 + A2 + B) dz dx
B2·2−j (x)\B2·2−i (x)
cn Rn
B2·2−j (x)\B2·2−i (x)
2−γ (i+s) ni x + 2 1 (z) ( A + A + B ) dz dx 1 2 Ws,i |x − z|n+γ
= A1 + A2 + B. The A1 -term. We have A1
1Rn \B4·2−j (z) (w)
y Q
cn 2nj 2γj
y | |Q
y Q
|x − z|γ ψQ
y (w) dw n+γ |z − w|
|x − z|γ 2nj 2γj |x − z|γ dw ∼ c . n (1 + 2j |z − w|)n+γ (1 + 2j |z − y|)n+γ
Lipschitz smoothness is applicable since z ∈ B2·2−j (x) \ B2·2−i (x). We then have
A1 cn Rn
B2·2−j (x)\B2·2−i (x)
2−γ (i+s) 2nj 2γj |x − z|γ dz dx |x − z|n+γ (1 + 2j |z − y|)n+γ
+ cn
2nj 2γj |x − z|γ x (z) 2 1Ws,i dz dx (1 + 2j |z − y|)n+γ ni
Rn
B2·2−j (x)\B2·2−i (x)
= A11 + A12 . The estimate of A11 is standard A11 = cn Rn
2−γ (i+s) 2nj 2γj (1 + 2j |z − y|)n+γ
= cn |i − j | Rn
B2·2−j (z)\B2·2−i (z)
dx dz |x − z|n
2−γ (i+s) 2nj 2γj dz ∼ cn |i − j |2−γ s 2−γ |i−j | . (1 + 2j |z − y|)n+γ
The term A12 can be written as follows A12 = cn Rn
2ni 2nj 2γj (1 + 2j |z − y|)n+γ
B2·2−j (z)\B2·2−i (z)
z (x) dx dz. |x − z|γ 1Ws,i
549
550
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
z (x) implies that Now, the presence of 1Ws,i
|x − z| 4 · 2−i + 2−(i+s−1) 5 · 2−i . Therefore we find A12 cn 2
−γ |i−j |
Rn
z 2ni 2nj |Ws,i |
(1 + 2j |z − y|)n+γ
dz cn 2−s 2−γ |i−j | .
This means that A11 dominates A12 and we conclude A1 cn |i − j |2−γ s 2−γ |i−j | . The A2 -term. Consider the symmetric difference j Zx,z = B4·2−j (x) B4·2−j (z) = B4·2−j (x) \ B4·2−j (z) ∪ B4·2−j (z) \ B4·2−j (x) . Then we clearly have A2 = j
y ∩Zx,z Q
j
k(x, w)ψ (w) dw cn 2nj |Qy ∩ Zx,z | , Qy
y | |Q
where the 2nj comes from the size condition on the kernel and the inequality j |x − w| dist x, ∂Zx,z 4 · 2−j − |x − z| 2 · 2−j , j
which holds for any w ∈ Zx,z and z ∈ B2·2−j (x) \ B2·2−i (x). This allows us to write
A2 cn 2nj Rn
B2·2−j (x)\B2·2−i (x)
j
y ∩ Zx,z | 2−γ (i+s) |Q dz dx
y | |x − z|n+γ |Q
+ cn 2nj
x (z) 2ni 1Ws,i
Rn
B2·2−j (x)\B2·2−i (x)
j
y ∩ Zx,z |Q | dz dx
y | |Q
= A21 + A22 . Before proceeding with the argument, we note • If |x − y| > 7 · 2−j |z − w| |x − y| − |x − z| − |w − y| > 4 · 2−j
y × (B2·2−j (x) \ B2·2−i (x)). Similarly, we have for all (w, z) ∈ Q |x − w| |x − y| − |w − y| > 6 · 2−j .
(2.14)
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
551
y , so that Q
y ∩ Zx,z = ∅. This implies that w ∈ / Zx,z for any w ∈ Q • If |x − y| < 2−j j
j
|z − w| |z − x| + |x − y| + |y − w| < 4 · 2−j
y × (B2·2−j (x) \ B2·2−i (x)). Similarly, we have for all (w, z) ∈ Q |x − w| |x − y| + |y − w| < 2 · 2−j . j j
y , so that Q
y ∩ Zx,z This implies that w ∈ / Zx,z for any w ∈ Q = ∅.
In particular, we conclude that
A21 + A22 = cn 2
nj B7·2−j (y)\B2−j (y)
B2·2−j (x)\B2·2−i (x)
+ cn 2
j
y ∩ Zx,z | 2−γ (i+s) |Q dz dx
y | |x − z|n+γ |Q
nj
ni
2 1 B7·2−j (y)\B2−j (y)
x Ws,i
B2·2−j (x)\B2·2−i (x)
j
y ∩ Zx,z |Q | (z) dz dx.
y | |Q
y behaves as a ball of radius 2−(j +s) while Zx,z behaves like an annulus of Observe now that Q −j radius 4 · 2 and width |x − z|. Therefore, the measure of the intersection can be estimated by j
j Q
y ∩ Zx,z cn min 2−(n−1)(j +s) |x − z|, 2−n(j +s) . This provides us with the estimate j
y ∩ Zx,z |Q | cn min 2j +s |x − z|, 1 .
|Qy |
If 2−(j +s) 2 · 2−i ,
nj −γ (i+s)
A21 cn 2 2
|x−y|<7·2−j
|z−x|>2−(j +s)
dz |x − z|n+γ
dx cn 2−γ |i−j | .
If 2−(j +s) > 2 · 2−i ,
nj −γ (i+s)
dz |x − z|n+γ
A21 cn 2 2
|x−y|<7·2−j nj −γ (i+s)
|z−x|>2−(j +s)
+ cn 2 2
|x−y|<7·2−j
cn 2−γ |i−j | + cn
B2−(j +s) (x)\B2·2−i (x)
(i − j − s)2−|i−j | cγ 2−γ |i−j |
dx
2j +s |x − z| dz dx |x − z|n+γ
if γ = 1, if 0 < γ < 1.
552
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
This gives A21 cn,γ |i − j |2−γ |i−j | . On the other hand, we also have
ni nj j +s
A22 cn 2 2 2
|x − z|1 B7·2−j (y)\B2−j (y)
x Ws,i
(z) dz dx.
B2·2−j (x)\B2·2−i (x)
x and |W x | c 2−ni 2−s , we get Since we have |x − z| < 5 · 2−i for z ∈ Ws,i n s,i
A22 cn 2−|i−j | . Therefore, A21 dominates A22 and we conclude A2 cn,γ |i − j |2−γ |i−j | .
(2.15)
The B-term. As usual, we decompose
B cn Rn
B2·2−j (x)\B2·2−i (x)
2−γ (i+s) B dz dx |x − z|n+γ
x (z)B dz dx = B1 + B2 , 2 1Ws,i
+ cn
ni
Rn
B2·2−j (x)\B2·2−i (x)
2 (z, y) − k 2 (x, y)|. We have with B = |ks,j s,j
B 1 cn Rn
B2·2−j (x)\B2·2−i (x)
2−γ s 2ni B dz dx, (1 + 2i |x − z|)n+γ
since for z ∈ B2·2−j (x) \ B2·2−i (x) both integrands are comparable. Recalling that 2 (x, y) = T4·2−j ψQ ks,j
y , φRx ,
we observe (as in our analysis of Φs ) that B = Ej (B) when regarded as a function of z. This means that B = 0 for any z ∈ Rx . This, together with the fact that B2·2−j (x) ⊂ 5Rx , implies
B 1 cn 5Rx \Rx
Rn
2 2−γ s 2ni k (z, y) − k 2 (x, y) dz dx. s,j (1 + 2i |x − z|)n+γ s,j
Moreover, arguing as in Lemma 2.5
B 1 cn
Ej
Rn
cn
5Rx \Rx
Rn R≈Rx
R
2 2−γ s 2ni k (z, y) − k 2 (x, y) dz dx (z) s,j s,j (1 + 2i |x − ·|)n+γ
2 2−γ s 2ni k (cR , y) + k 2 (x, y) dx. dw s,j s,j i n+γ (1 + 2 |x − w|)
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
553
Here we write R ≈ Rx to denote that R is a dyadic cube in Qj contained in 5Rx \ Rx . Then we apply the argument in Lemma 2.6 B 1 cn
Rn R≈Rx
2 2−γ s k (cR , y) + k 2 (x, y) dx. s,j s,j i γ (1 + 2 δx )
Now, it is clear from (2.13) that we have 2 k (x, y) cn s,j
2−γ s 2nj . (1 + 2j |x − y|)n+γ
These estimates in conjunction with the argument in Lemma 2.6 give B1 cn,γ 2−2γ s |i − j |2−γ |i−j | . To estimate B2 we use that B = 0 for any z ∈ Rx and B2·2−j (x) ⊂ 5Rx
2 k (z, y) + k 2 (x, y) dz dx. s,j s,j
B2 cn 2ni Rn
x (5Rx \Rx )∩Ws,i
2 (·,·)| given above for (z, y) and (x, y) Then we apply our estimate of |ks,j
B2 cn 2ni 2−γ s 2nj
Rn
ni −γ s nj
+ cn 2 2
x (5Rx \Rx )∩Ws,i
2
Rn
ni −γ s nj
∼ cn 2 2
dz (1 + 2j |z − y|)n+γ
x (5Rx \Rx )∩Ws,i
2
Rn
x (5Rx \Rx )∩Ws,i
dx
dz (1 + 2j |x − y|)n+γ
dz j (1 + 2 |x − y|)n+γ
dx
dx.
x . Next, the set Last equivalence follows from the presence of Ws,i x (5Rx \ Rx ) ∩ Ws,i
forces z to be outside Rx but at a distance of x controlled by 4 · 2−i + 2−(i+s−1) . Thus, the only x ∈ Rn for which the inner integral does not vanish are those x for which dist(x, ∂Rx ) 4 · 2−i + 2−(i+s−1) . Notice that for |i − j | 3 this suppose no restriction but for |i − j | large does. Given R ∈ Qj we set Rs,i = w ∈ R dist(w, ∂R) 4 · 2−i + 2−(i+s−1) . Our considerations allows us to complete our estimate as follows
554
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
Fig. 4. The factor 2−|i−j | comes from |Rs,i | cn 2−|i−j | |R|.
B2 cn 2ni 2−γ s 2nj
R∈Qj R
= cn 2ni 2−γ s 2nj
x | |Ws,i
(1 + 2j |x − y|)n+γ
dx
s,i
x | |Rs,i | |Ws,i 1 dx |R| |R| |Rs,i | (1 + 2j |x − y|)n+γ
R∈Qj
Rs,i
cn 2−(1+γ )s 2nj 2−|i−j |
R∈Qj
∼ cn 2−(1+γ )s 2nj 2−|i−j |
Rn
|R|
1 |Rs,i |
Rs,i
1 dx − y|)n+γ
(1 + 2j |x
dx ∼ cn 2−(1+γ )s 2−|i−j | − y|)n+γ
(1 + 2j |x
(see Fig. 4). Combining our estimates for B1 and B2 we get B cn,γ 2−2γ s |i − j |2−γ |i−j | .
(2.16)
Finally, the sum of (2.14), (2.15) and (2.16) produces Si,j,s (y) cn,γ |i − j |2−γ |i−j | . 2,β
2,α satisfies a better estimate, the proof is complete. As we have proved that Si,j,s
2
Remark 2.11. The estimate given for A1 in the proof of Lemma 2.10 above is the only point in the whole argument for our pseudo-localization principle where the Lipschitz smoothness with respect to the x variable is used. Conclusion. Now we have all the necessary estimates to complete the argument. Namely, a direct application of Lemmas 2.9 and 2.10 in conjunction with Schur lemma give us the following estimate ∗ −γ s 2 −2γ s 1 + |i − j | 2−γ |i−j | c Λ Λ c αi−j n,γ 2 n,γ 2 s,i s,j B (L ) 2
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
555
1
where αk = (1 + |k|) 4 2−γ |k|/4 . Therefore, Cotlar lemma provides Ψs B(L2 ) = (id − Ek )T Δk+s
B(L2 )
k
cn,γ 2−γ s/2
αk = cn,γ 2−γ s/2 .
k
3. Calderón–Zygmund decomposition We now go back to the noncommutative setting and present a noncommutative form of Calderón–Zygmund decomposition. Let us recall from the introduction that, for a given semifinite von Neumann algebra M equipped with a n.s.f. trace τ , we shall work on the weak-operator closure A of the algebra AB of essentially bounded functions f : Rn → M. Recall also the dyadic filtration (Ak )k∈Z in A. 3.1. Cuculescu revisited A difficulty inherent to the noncommutativity is the absence of maximal functions. It is however possible to obtain noncommutative maximal weak and strong inequalities. The strong inequalities follow by recalling that the Lp norm of a maximal function is an Lp (∞ ) norm. As observed by Pisier [45] and further studied by Junge [24], the theory of operator spaces is the right tool to define noncommutative Lp (∞ ) spaces; see [30] for a nice exposition. We shall be interested on weak maximal inequalities, which already appeared in Cuculescu’s construction above and are simpler to describe. Indeed, given a sequence (fk )k∈Z of positive functions in L1 and any λ ∈ R+ , we are interested in describing the noncommutative form of the Lebesgue measure of ! " sup fk > λ . k∈Z
If fk ∈ L1 (A)+ for k ∈ Z, this is given by inf ϕ(1A − q) q ∈ Aπ , qfk q λq for all k ∈ Z . Given a positive dyadic martingale f = (f1 , f2 , . . .) in L1 (A) and looking one more time at Cuculescu’s construction, it is apparent that the projection q(λ)k represents the following set ! " q(λ)k ∼ sup fj λ . 1j k
Therefore, we find 1A −
! " q(λ)k ∼ sup fk > λ .
k1
k1
However, in this paper we shall be interested in the projection representing the set where supk∈Z fk > λ since we will work with the full dyadic filtration (Ak )k∈Z , where Ak stands for Ek (A). We shall clarify below why it is not enough to work with the truncated filtration (Ak )k1 . The construction of the right projection for supk∈Z fk > λ does not follow automatically from Cuculescu’s construction, see Proposition 3.2 below. Moreover, given a general
556
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
function f ∈ L1 (A)+ , we are not able at the time of this writing to construct the right projections qλ (f, k) which represent the sets qλ (f, k) ∼
! sup
j ∈Z, j k
" fj λ .
Indeed, the weak∗ limit procedure used in the proof of Proposition 3.2 below does not preserve the commutation relation (i) of Cuculescu’s construction and we are forced to work in the following dense class of L1 (A) −−→ Ac,+ = f : Rn → M f ∈ A+ , − suppf is compact ⊂ L1 (A).
(3.1)
−−→ Here − supp means the support of f as a vector-valued function in Rn . In other words, we have suppf = suppf M . We employ this terminology to distinguish from supp f (the support of −−→ f as an operator in A) defined in Section 1. Note that − suppf is a measurable subset of Rn , while supp f is a projection in A. In the rest of the paper we shall work with functions f in Ac,+ . This impose no restriction due to the density of span Ac,+ in L1 (A). The following result is an adaptation of Cuculescu’s construction which will be the one to be used in the sequel. −−−→
Lemma 3.1. Let f ∈ Ac,+ and fk = Ek (f ) for k ∈ Z. The sequence (fk )k∈Z is a (positive) dyadic martingale in L1 (A). Given any positive number λ, there exists a decreasing sequence (qλ (f, k))k∈Z of projections in A satisfying (i) qλ (f, k) commutes with qλ (f, k − 1)fk qλ (f, k − 1) for each k ∈ Z. (ii) qλ (f, k) belongs to Ak for each k ∈ Z and qλ (f, k)fk qλ (f, k) λqλ (f, k). (iii) The following estimate holds ϕ 1A −
k∈Z
1 qλ (f, k) f 1 . λ
Proof. Since f ∈ Ac,+ we have for all Q ∈ Qj 1 fQ = |Q|
−−→ f (x) dx 2j f A |− suppf |1A −→ 0 as j → −∞.
Q
In particular, given any λ ∈ R+ , we have fj λ1A for all j < mλ < 0 and certain mλ ∈ Z \ N with |mλ | large enough. Then we define the desired projections by the following relations ⎧ ⎨ 1A qλ (f, k) = χ(0,λ] (fk ) ⎩ χ(0,λ] (qλ (f, k − 1)fk qλ (f, k − 1))
if k < mλ , if k = mλ , if k > mλ .
To prove (iii) we observe that our projections are exactly the ones obtained when applying Cuculescu’s construction over the truncated filtration (Ak )kmλ . Thus we get
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
557
1 1 ϕ 1A − qλ (f, k) = ϕ 1A − q(λ)k sup fk 1 = f 1 . λ kmλ λ k∈Z
kmλ
The rest of the properties of the sequence (qλ (f, k))k∈Z are easily verifiable.
2
3.2. The maximal function We now recall the Hardy–Littlewood weak maximal inequality. In what follows it will be quite useful to have another expression for the qλ (f, k)’s constructed in Lemma 3.1. It is not difficult to check that ξλ (f, Q)1Q qλ (f, k) = Q∈Qk
for k ∈ Z, with ξλ (f, Q) projections in M defined by ⎧ ⎨ 1M ξλ (f, Q) = χ(0,λ] (fQ ) ⎩
Q ξλ (f, Q))
χ(0,λ] (ξλ (f, Q)f
if k < mλ , if k = mλ , if k > mλ .
As for Cuculescu’s construction, we have • • • •
ξλ (f, Q) ∈ Mπ .
ξλ (f, Q) ξλ (f, Q).
Q ξλ (f, Q).
ξλ (f, Q) commutes with ξλ (f, Q)f ξλ (f, Q)fQ ξλ (f, Q) λξλ (f, Q).
The noncommutative weak type (1, 1) inequality for the Hardy–Littlewood dyadic maximal function [37] follows as a consequence of this. We give a proof including some details (reported by Quanhua Xu to the author) not appearing in [37]. Proposition 3.2. If (f, λ) ∈ L1 (A) × R+ , there exists qλ (f ) ∈ Aπ with 8 supqλ (f )fk qλ (f )A 16λ and ϕ 1A − qλ (f ) f 1 . λ k∈Z Proof. Let us fix an integer m ∈ Z \ N. Assume f ∈ L1 (A)+ and consider the sequence (q(λ)m,k )km provided by Cuculescu’s construction applied over the filtration (Ak )km . Define qm (λ) =
q(λ)m,k
for each m ∈ Z \ N.
km
Let us look at the family (qm (λ))m∈Z\N . By the weak* compactness of the unit ball BA and the positivity of our family, there must exist a cluster point a ∈ BA+ . In particular, we may find a subsequence with qmj (λ) → a as j → ∞ (note that mj → −∞ as j → ∞) in the weak∗ topology. Then we set qλ (f ) = χ[1/2,1] (a) and define positive operators δ(a) and β(a) bounded by 21A and determined by
558
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
qλ (f ) = aδ(a) = δ(a)a, 1A − qλ (f ) = χ(1/2,1] (1A − a) = (1A − a)β(a) = β(a)(1A − a). In order to prove the first inequality stated above, we note that qλ (f )fk qλ (f ) = sup ϕ qλ (f )fk qλ (f )b . A bL1 (A) 1
However, we have ϕ qλ (f )fk qλ (f )b = ϕ afk aδ(a)bδ(a) = lim ϕ qmj (λ)fk qmj (λ)δ(a)bδ(a) j →∞
lim qmj (λ)fk qmj (λ)∞ δ(a)bδ(a)1 . j →∞
Therefore we conclude 2 ϕ qλ (f )fk qλ (f )b b1 δ(a)∞ lim q(λ)mj ,k fk q(λ)mj ,k ∞ 4λ. j →∞
This proves the first inequality, as for the second 2 ϕ 1A − qλ (f ) = ϕ (1A − a)β(a) 2ϕ(1A − a) = 2 lim ϕ 1A − qmj (λ) f 1 . j →∞ λ Finally, for a general f ∈ L1 (A) we decompose f = (f1 − f2 ) + i(f3 − f4 ) with fj ∈ L1 (A)+ and define qλ (f ) =
qλ (fj ).
1j 4
Then, the estimate follows easily with constants 16λ and 8/λ respectively.
2
3.3. The good and bad parts If f ∈ L1 positive and λ ∈ R+ , define 1 f (y) dy Md f (x) = sup x∈Q∈Q |Q|
and Eλ = x ∈ Rn Md f (x) > λ .
Q
Writing Eλ = j Qj as a disjoint union of maximal dyadic cubes with fQ λ < fQj for all dyadic Q ⊃ Qj , we may decompose f = g + b where the good and bad parts are given by g = f 1Ecλ +
j
fQj 1Qj
and b =
(f − fQj )1Qj . j
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
559
If bj = (f − fQj )1Qj , we have n (i) g1 f 1 and g∞ 2 λ. (ii) supp bj ⊂ Qj , Qj bj = 0 and j bj 1 2f 1 .
Before proceeding with the noncommutative Calderón–Zygmund decomposition, we simplify our notation for the projections ξλ (f, Q) and qλ (f, k). Namely, (f, λ) will remain fixed in Ac,+ × R+ , see (3.1). These choices lead us to set (ξQ , qk , q) = ξλ (f, Q), qλ (f, k),
qλ (f, k) .
k∈Z
Moreover, we shall write (pk )k∈Z for the projections pk = qk−1 − qk =
(ξQ
− ξQ )1Q =
Q∈Qk
πQ 1Q .
(3.2)
Q∈Qk
The terminology πQ = ξQ
− ξQ will be frequently used below. Recall that the pk ’s are pairwise disjoint and (according to our new terminology) we have qj = 1A for all j < mλ . In particular, we find
pk = 1A − q.
k
Z stands for Z ∪ {∞}, our noncommutative analogue for the Calderón– If we write p∞ for q and
Zygmund decomposition can be stated as follows. If f ∈ Ac,+ and λ ∈ R+ , we consider the decomposition f = g + b with g=
pi fi∨j pj
and b =
i,j ∈
Z
pi (f − fi∨j )pj ,
(3.3)
i,j ∈
Z
where i ∨ j = max(i, j ). Note that i ∨ j = ∞ whenever i or j is infinite. In particular, since f = f∞ by definition, the extended sum defining b is just an ordinary sum over Z × Z. Note also that our expressions are natural generalizations of the classical good and bad parts stated in the classical decomposition. Indeed, recalling the orthogonality of the pk ’s, all the off-diagonal terms vanish in the commutative setting and we find something like gd = qf q +
pk f k pk
and bd =
k
pk (f − fk )pk .
k
In this form, and recalling that for M = C we have q ∼ Rn \ Eλ
and pk ∼ {Qj ⊂ Eλ | Qj ∈ Qk },
it is not difficult to see that we recover the classical decomposition.
(3.4)
560
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
Fig. 5. Commutative and noncommutative decompositions.
Remark 3.3. In the following we shall use the square-diagram in Fig. 5 to think of our decomposition. Namely, we first observe that for any f ∈ Ac,+ and for any λ ∈ R+ there will be an mλ ∈ Z such that fj λ1A for all j < mλ , see the proof of Lemma 3.1 above. In particular, since f and λ will remain fixed, by a simple relabelling we may assume with no loss of generality that mλ = 1. This will simplify very much the notation, since now we have pk = 0 for all non-positive k. Therefore, the terms pi fi∨j pj and pi (f − fi∨j )pj in our decomposition may be located in the (i, j )th position of an ∞ × ∞ matrix where the ‘last’ row and column are devoted to the projection q = p∞ . 4. Weak type estimates for diagonal terms In this section we start with the proof of Theorem A. Before that, a couple of remarks are in order. First, according to the classical theory it is clearly no restriction to assume that q = 2. In particular, since L2 (A) is a Hilbert space valued L2 -space, boundedness in L2 (A) will hold. Second, we may assume the function f ∈ L1 (A) belongs to Ac,+ . Indeed, this follows by decomposing f as a linear combination (f1 − f2 ) + i(f3 − f4 ) of positive functions fj ∈ L1 (A)+ and using the quasi-triangle inequality on L1,∞ (A) stated in Section 1. Then we approximate each fj ∈ L1 (A)+ by functions in Ac,+ . Third, since f 0 by assumption, we may break it for any fixed λ ∈ R+ following our Calderón–Zygmund decomposition. In this section we prove our main result for the diagonal terms in (3.4). According to the quasi-triangle inequality for L1,∞ (A), this will reduce the problem to estimate the off-diagonal terms. 4.1. Classical estimates The standard estimates (i) and (ii) satisfied by the good and bad parts of Calderón–Zygmund decomposition are satisfied by the diagonal terms (3.4). Indeed, since f is positive so is gd and gd 1 = ϕ(qf q) +
ϕ(pk fk pk ) = ϕ f q + f (1A − q) = f 1 .
k1
On the other hand, by orthogonality we have " ! gd ∞ = max qf q∞ , sup pk fk pk ∞ . k1
To estimate the first term, take a ∈ L1 (A) of norm 1 with qf q∞ ϕ(qf qa) + δ.
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
561
Since fk → f as k → ∞ in the weak* topology, we deduce that qf q∞ ϕ(qf qa) + δ = lim ϕ(qfk qa) + δ k→∞
lim qfk q∞ a1 + δ λ + δ, k→∞
where the last inequality follows from qfk q = qqk fk qk q λq. Therefore, taking δ → 0+ we deduce that qf q∞ λ. Let us now estimate the second terms. To that aim, we observe that
fk =
Q∈Qk
1 |Q|
f (y) dy1Q = 2n
Q∈Qk
Q
1
|Q|
f (y) dy1Q 2n fk−1 . Q
Therefore, we obtain pk fk pk ∞ 2n qk−1 fk−1 qk−1 ∞ 2n λ.
(4.1)
This completes the proof of our assertions for gd . Let us now prove the assertions for bd . If we ∗ and also that supp b take bd,k to be pk (f − fk )pk , it is clear that bd,k = bd,k d,k pk . Moreover, recalling that bd,k =
(ξQ
− ξQ )(f − fQ )(ξQ
− ξQ )1Q ,
Q∈Qk
the following identity holds for any Q0 ∈ Qk ,
bd,k (y) dy = (ξQ f (y) − fQ0 (y) dy (ξQ
0 − ξQ0 )
0 − ξQ0 ) = 0.
Q0
Q0
Finally, we observe that k1
bd,k 1
ϕ pk (f + fk )pk = 2ϕ f (1A − q) 2f 1 . k1
This completes the proof of our assertions for the function bd . As we shall see in the following section, the estimates for the off-diagonal terms require more involved arguments which do not appear in the classical (scalar-valued) theory. Remark 4.1. It is important to note that the doubling estimate (4.1) is crucial for our further analysis and also that such inequality is the one which imposes to work with the full filtration (Ak )k∈Z instead with the truncated one (Ak )k1 . Indeed, if we truncate at k 1 (not at k mλ as we have done), then condition (4.1) fails in general for k = 1. This is another difference with the approach in [44], where the doubling condition above was not needed.
562
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
4.2. An Rn -dilated projection As above, given a positive function f ∈ L1 , let Eλ be the set in Rn where the dyadic Hardy– Littlewood maximal function Md f is greater than λ. If we decompose Eλ = j Qj as a disjoint union of maximal dyadic cubes, let us write 9Eλ for the dilation 9Eλ =
9Qj .
j
As we pointed out in the introduction, this is a key set to give a weak type estimate for the bad part in Calderón–Zygmund decomposition. On the other hand, we know from Cuculescu’s construction that 1A − q represents the noncommutative analog of Eλ , so that the noncommutative analog of 9Eλ should look like ‘9(1A − q)’ in the sense that we dilate on Rn but not on M. In the following result we construct the right noncommutative analog of 9Eλ . Lemma 4.2. There exists ζ ∈ Aπ such that (i) λϕ(1A − ζ ) 9n f 1 . (ii) If Q0 ∈ Q and x ∈ 9Q0 , then ζ (x) 1M − ξQ
0 + ξQ0 . In particular, in this case we immediately find ζ (x) ξQ0 . Proof. Given k ∈ Z+ , we define ψk =
k
(ξQ
− ξQ )19Q
and ζk = 1A − supp ψk .
s=1 Q∈Qs
Since we have ξQ ξQ
for all dyadic cube Q, it turns out that (ψk )k1 is an increasing sequence of positive operators. However, enlarging Q by its concentric father 9Q generates overlapping and the ψk ’s are not projections. This forces us to consider the associated support projections and define ζ1 , ζ2 , . . . as above. The sequence of projections (ζk )k1 is clearly decreasing and we may define ζ=
ζk .
k1
Now we are ready to prove the first estimate λϕ(1A − ζ ) = λ lim ϕ(1A − ζk ) λ k→∞
= 9n λ
∞
∞
ϕ (ξQ
− ξQ )19Q
s=1 Q∈Qs
s=1 Q∈Qs
n n ϕ (ξQ
− ξQ )1Q = 9 λϕ(1A − q) 9 f 1 .
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
563
Now fix Q0 ∈ Q, say Q0 ∈ Qk0 for some k0 ∈ Z. If k0 0, the assertion is trivial since we know from Remark 3.3 that ξQ0 = ξQ
0 = 1M . Thus, we assume that k0 1. Then we have (ξQ
0 − ξQ0 )19Q0 ψk0
⇒
ζk0 1A − (ξQ
0 − ξQ0 )19Q0
⇒
ζ (x) ζk0 (x) 1M − ξQ
0 + ξQ0
for any x ∈ 9Q0 . It remains to prove that in fact ζ (x) ξQ0 . Let us write Qj for the j th dyadic antecessor of Q0 . In other words, Q1 is the dyadic father of Q0 , Q2 is the dyadic father of Q1 and so on until Qk0 −1 ∈ Q1 . Since the family Q0 , Q1 , . . . is increasing, the same happens for their concentric fathers and we find x∈
k 0 −1
9Qj .
j =0
In particular, applying the estimate proved so far ζ (x)
k 0 −1 j =0
(1M − ξQ
j + ξQj ) = ξQ0 .
The last identity easily follows from ξQ
k
0 −1
= 1M .
Indeed, we have agreed in Remark 3.3 to assume qk = 1A for all k 0.
2
4.3. Chebychev’s inequalities By Section 4.1, we have 1 1 gd 22 = ϕ gd2 gd gd2 gd 1 gd ∞ 2n λf 1 .
In particular, the estimate below follows from Chebychev’s inequality 1 1 λϕ |T gd | > λ T gd 22 gd 22 2n f 1 . λ λ As it is to be expected, here we have used our assumption on the L2 -boundedness of T . Now we are interested on a similar estimate with bd in place of gd . Using the projection ζ introduced in Lemma 4.2, we may consider the following decomposition T bd = (1A − ζ )T (bd )(1A − ζ ) + ζ T (bd )(1A − ζ ) + (1A − ζ )T (bd )ζ + ζ T (bd )ζ. In particular, we find λϕ |T bd | > λ λϕ(1A − ζ ) + λϕ ζ T (bd )ζ > λ .
564
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
Indeed, according to our decomposition of T bd and the quasi-triangle inequality on L1,∞ (A), the estimate above reduces to observe that the first three terms in such decomposition are left or right supported by 1A − ζ . Hence, since the quasi-norm in L1,∞ (A) is adjoint-invariant [14], we easily deduce it. On the other hand, according to the first estimate in Lemma 4.2, it suffices to study the last term above. Let us analyze the operator ζ T (bd )ζ . In what follows we shall freely manipulate infinite sums with no worries of convergence. This is admissible because we may assume from the beginning (by a simple approximation argument) that f ∈ An for some finite n 1. In particular, we could even think that all our sums are in fact finite sums. We may write ζ T (bd )ζ =
ζ T (bd,k )ζ
k1
with bd,k = pk (f − fk )pk for all k 1. Then, Chebychev’s inequality gives ∞ ζ T (bd,k )ζ . λϕ ζ T (bd )ζ > λ 1 k=1
According to Lemma 4.2 and using ξQ πQ = πQ ξQ = 0 (recall the definition of πQ from (3.2) above), we have ζ (x)bd,k (y)ζ (x) = 0 whenever x lies in the concentric father 9Q of the cube Q ∈ Qk for which y ∈ Q. In other words, we know that x lives far away from the singularity of the kernel k and ζ T (bd,k )ζ (x) =
k(x, y) ζ (x)bd,k (y)ζ (x) dy
Rn
=
Q∈Qk
k(x, y) ζ (x)bd,k (y)ζ (x) dy 1(9Q)c (x)
Q
k(x, y)bd,k (y) dy 1(9Q)c (x) ζ (x). = ζ (x) Q∈Qk
Q
Now we use the mean-zero condition of bd,k from Section 4.1 k(x, y) − k(x, cQ ) bd,k (y) dy 1(9Q)c (x) ζ (x), ζ T (bd,k )ζ (x) = ζ (x) Q∈Qk
Q
where cQ is the center of Q. Then we use the Lipschitz γ -smoothness to obtain ∞ ∞ ζ T (bd,k )ζ k(·, y) − k(·, cQ ) bd,k (y)1(9Q)c (·) dy 1 1 k=1 Q∈Qk Q
k=1
=
∞ k=1 Q∈Qk Q
τ (9Q)c
k(x, y) − k(x, cQ ) dx bd,k (y) dy
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
∞ k=1 Q∈Qk Q
∞
(9Q)c
565
|y − cQ |γ dx τ bd,k (y) dy n+γ |x − cQ |
τ bd,k (y) dy
k=1 Q∈Qk Q
=
∞
τ bd,k (y) dy
k=1 Rn
=
∞
bd,k 1 2f 1 ,
k=1
where the last inequality follows once more from Section 4.1. This completes the argument for the diagonal part. Indeed, for any fixed λ ∈ R+ we have seen that the diagonal parts of g and b (which depend on the chosen λ) satisfy λϕ |T gd | > λ + λϕ |T bd | > λ cn f 1 .
(4.2)
5. Weak type estimates for off-diagonal terms Given λ ∈ R+ , we have broken f with our Calderón–Zygmund decomposition for such λ. In the last section, we have estimated the diagonal terms gd and bd . Let us now consider the offdiagonal terms goff and boff determined by g = gd + goff and b = bd + boff . As we pointed out in the introduction, it is paradoxical that the bad part behaves (when dealing with off-diagonal terms) better than the good one! 5.1. An expression for goff We have goff =
i=j i,j ∈
Z
pi fi∨j pj = qf (1A − q) + (1A − q)f q +
∞ ∞
pk fk+s pk+s + pk+s fk+s pk .
s=1 k=1
Here we have restricted the sum k∈Z to k1 according to Remark 3.3. Applying property (i) of Cuculescu’s construction, we know that the projection qj commutes with qj −1 fj qj −1 for all j 1. Taking i ∧ j = min(i, j ), this immediately gives that pi fi∧j pj = 0 for i = j . Indeed, we have pi fi∧j pj = pi qi−1 fi qi−1 pj = 0
if i < j,
pi fi∧j pj = pi qj −1 fj qj −1 pj = 0
if i > j.
Using this property and inverting the order of summation, we deduce
566
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593 ∞ ∞
pk fk+s pk+s + pk+s fk+s pk
s=1 k=1
=
∞
pk (fk+s − fk )pk+s + pk+s (fk+s − fk )pk
s,k=1
=
∞ s
pk dfk+j pk+s + pk+s dfk+j pk =
s,k=1 j =1
∞ ∞
pk dfk+j pk+s + pk+s dfk+j pk .
j,k=1 s=j
Recall that we may use Fubini theorem since, as we observed in Section 4.3, we may even assume that all our sums are finite sums. Now we can sum in s and apply the commutation property above to obtain ∞ ∞
pk fk+s pk+s + pk+s fk+s pk
s=1 k=1
=
∞
pk dfk+j (qk+j −1 − q) + (qk+j −1 − q) dfk+j pk
j,k=1
=
∞
pk dfk+j qk+j −1 + qk+j −1 dfk+j pk −
j,k=1
=
∞
∞
pk (f − fk )q + q(f − fk )pk
k=1
pk dfk+j qk+j −1 + qk+j −1 dfk+j pk −
j,k=1
=
∞
∞
pk f q + qfpk
k=1
pk dfk+j qk+j −1 + qk+j −1 dfk+j pk − (1A − q)f q + qf (1A − q).
j,k=1
Indeed, we have used pk fk q = pk qk−1 fk qk−1 q = 0 = qqk−1 fk qk−1 pk = qfk pk . Combined the identities obtained so far, we get goff =
∞ ∞
pk dfk+s qk+s−1 + qk+s−1 dfk+s pk =
s=1 k=1
∞ ∞
gk,s .
s=1 k=1
We shall use through out this expression for goff in terms of the functions gk,s . 5.2. Noncommutative pseudo-localization Now we formulate and prove the noncommutative extension of our pseudo-localization principle. We need a weak notion of support from [44] which is quite useful when dealing with weak type inequalities. For a non-necessarily self-adjoint f ∈ A, the two-sided null projection of f is
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
567
the greatest projection q in Aπ satisfying qf q = 0. Then we define the weak support projection of f as supp∗ f = 1A − q. It is clear that supp∗ f = supp f when A is abelian. Moreover, this notion is weaker than the usual support projection in the sense that we have supp∗ f supp f for any self-adjoint f ∈ A and supp∗ f is a subprojection of both the left and right supports in the non-self-adjoint case. Remark 5.1. Below we shall use the following characterization of the weak support projection. The projection supp∗ f is the smallest projection p in Aπ satisfying the identity f = pf + fp − pfp. Indeed, let q be the two-sided null projection of f and let p = 1A − q. Then we have (1A − p)f (1A − p) = 0 by definition. In other words, f = pf + fp − pfp and p is the smallest projection with this property because q is the greatest projection satisfying the identity qf q = 0. The following constitutes a noncommutative analog of the pseudo-localization principle that we have stated in the introduction. The terminology has been chosen to fit with that of the noncommutative Calderón–Zygmund decomposition. This will make the exposition more transparent. Theorem 5.2. Let us fix a positive integer s. Given a function f ∈ L2 (A) and any integer k, let us consider any projection qk in Aπ ∩ Ak satisfying that 1A − qk contains supp∗ dfk+s as a subprojection. If we write ξQ 1Q qk = Q∈Qk
with ξQ ∈ Mπ , we may further consider the projection & ζf,s = 1A − (1M − ξQ )19Q . Q∈Qk
k∈Z
Then we have the following localization estimate in L2 (A) Rn
2 τ [ζf,s Tf ζf,s ](x) dx
1 2
cn,γ s2
−γ s/4
2 τ f (x) dx
1 2
,
Rn
for any L2 -normalized Calderón–Zygmund operator with Lipschitz parameter γ . Proof. We shall reduce this result to its commutative counterpart. More precisely to the shifted form of the T 1 theorem proved above. According to Remark 5.1 and the shift condition supp∗ dfk+s ≺ 1A − qk , we have dfk+s = qk⊥ dfk+s + dfk+s qk⊥ − qk⊥ dfk+s qk⊥
568
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
where we write qk⊥ = 1A − qk for convenience. On the other hand, let &
ζk = 1A −
(1M − ξQ )19Q ,
Q∈Qk
' so that ζf,s = k ζk . Following Lemma 4.2, it is easily seen that 1A − ζk represents the Rn dilated projection associated to 1A − qk with a factor 9. Let La and Ra denote the left and right multiplication maps by the operator a. Let also LRa stand for La + Ra − La Ra Then our considerations so far and the fact that Lζk , Rζk and LRq ⊥ commute with Ej for j k give k
ζf,s Tf ζf,s = Lζf,s Rζf,s
Ek T Δk+s LRq ⊥ + (id − Ek )Lζk Rζk T LRq ⊥ Δk+s (f ). k
k
k
k
Now we claim that Lζk Rζk T LRq ⊥ = Lζk Rζk T4·2−k LRq ⊥ . k
k
Indeed, this clearly reduces to see Lζk T Lq ⊥ = Lζk T4·2−k Lq ⊥ k
k
and Rζk T Rq ⊥ = Rζk T4·2−k Rq ⊥ . k
k
By symmetry, we just prove the first identity Lζk T Lq ⊥ f (x) = k
ζk (x)(1M − ξQ )
Q∈Qk
k(x, y)f (y) dy. Q
Assume that x ∈ 9Q for some Q ∈ Qk , then it easily follows from the definition of the projection ζk that ζk (x) ξQ . Note that this is simpler than the argument in Lemma 4.2 because we do not need to prove here a property like (i) there. In particular, we deduce from the expression above that for each y ∈ Q we must have x ∈ Rn \ 9Q. This implies that |x − y| 4 · 2−k as desired. Finally, since the operators L and R were created from properties of f and ζf,s , we can eliminate them and obtain ζf,s Tf ζf,s = Lζf,s Rζf,s Ek T Δk+s + (id − Ek )T4·2−k Δk+s (f ). k
k
Assume that T ∗ 1 = 0. According to our shifted form of the T 1 theorem, we know that the operator inside the brackets has norm in B(L2 ) controlled by cn,γ s2−γ s/4 . In particular, the same happens when we tensor with the identity on L2 (M), which is the case. This proves the assertion for convolution-type operators. When T ∗ 1 is a non-zero element of BMO, we may follow verbatim the paraproduct argument given above since LRq ⊥ commutes with Ek and ζf,s qk⊥ = qk⊥ ζf,s = 0.
2
k
Remark 5.3. It is apparent that 1A − qk represents in the noncommutative setting the set Ωk in the commutative formulation. Moreover, ζf,s and ζk represent Rn \ Σf,s and Rn \ 9Ωk respectively. The only significant difference is that in the commutative statement we take Ωk to be the
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
569
smallest Rk -set containing supp dfk+s . This is done to optimize the corresponding localization estimate. Indeed, the smaller are the Ωk ’s the larger is 1Rn \Σf,s Tf . However, it is in general false that the smaller are the 1A − qk ’s the larger is ζf,s T ζf,s . That is why we consider any sequence of qk ’s satisfying the shift condition. 5.3. Estimation of T goff Our aim is to estimate λϕ |T goff | > λ . As usual, we decompose the term T goff in the following way (1A − ζ )T (goff )(1A − ζ ) + ζ T (goff )(1A − ζ ) + (1A − ζ )T (goff )ζ + ζ T (goff )ζ, where ζ denotes the projection constructed in Lemma 4.2. According to this lemma and the argument in Section 4.3, we are reduced to estimate the last term above. This will be done in several steps. 5.3.1. Orthogonality It is not difficult to check that the terms gk,s in Section 5.1 are pairwise orthogonal. It follows from the trace-invariance of conditional expectations and the mutual orthogonality of the pk ’s. We first prove the following implication ϕ gk,s gk∗ ,s = 0
⇒
k + s = k + s .
Indeed, if we assume without loss of generality that k + s > k + s , we get ϕ gk,s gk∗ ,s = ϕ Ek+s−1 gk,s gk∗ ,s = ϕ Ek+s−1 (gk,s )gk∗ ,s = 0. Now, assume that k = k and k + s = k + s . By the orthogonality of the pk ’s ϕ gk,s gk∗ ,s = ϕ(pk dfk+s qk+s−1 pk dfk+s qk+s−1 ) + ϕ(qk+s−1 dfk+s pk qk+s−1 dfk+s pk ) = ϕ(pk dfk+s qk+s−1 pk dfk+s qk+s−1 ) + ϕ(qk+s−1 dfk+s pk qk+s−1 dfk+s pk ) =0 since pk qk+s−1 = qk+s−1 pk = 0. This means that ϕ(gk,s gk∗ ,s ) = 0 unless k = k and k + s = k + s or, equivalently, (k, s) = (k , s ). Therefore, the gk,s ’s are pairwise orthogonal and goff 22
=
∞ ∞ s=1 k=1
gk,s 22 .
570
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
5.3.2. An ∞ (2 ) estimate Following the classical argument in Calderón–Zygmund decomposition or our estimate for the diagonal terms in Section 4.1, it would suffice to prove that goff 22 λf 1 . According to the pairwise orthogonality of the gk,s ’s, that is to say ∞ ∞
gk,s 22 λf 1 .
s=1 k=1
However, we just have the weaker inequality sup
∞
s1 k=1
gk,s 22 λf 1 .
(5.1)
Let us prove this estimate before going on with the proof gk,s 22 = 2ϕ(pk dfk+s qk+s−1 dfk+s pk ) = 2ϕ(pk fk+s qk+s−1 fk+s pk ) − 2ϕ(pk fk+s qk+s−1 fk+s−1 pk ) − 2ϕ(pk fk+s−1 qk+s−1 fk+s pk ) + 2ϕ(pk fk+s−1 qk+s−1 fk+s−1 pk ). By Cuculescu’s construction (ii) and fj 2n fj −1 (see Section 4.1), we find 1 12 2 f q k+s k+s−1 fk+s ∞ = qk+s−1 fk+s qk+s−1 ∞ λ, 1 12 2 f k+s−1 qk+s−1 fk+s−1 ∞ = qk+s−1 fk+s−1 qk+s−1 ∞ λ.
The crossed terms require Hölder’s inequality 1
1
ϕ(pk fk+s qk+s−1 fk+s−1 pk ) ϕ(pk fk+s qk+s−1 fk+s pk ) 2 ϕ(pk fk+s−1 qk+s−1 fk+s−1 pk ) 2 1
1
λϕ(pk fk+s pk ) 2 ϕ(pk fk+s−1 pk ) 2 = λϕ(pk fpk ), where the last identity uses the trace-invariance of the conditional expectations Ek+s and Ek+s−1 respectively. The same estimate holds for the remaining crossed term. This proves that sup
∞
s1 k=1
gk,s 22
λ sup
∞
ϕ(pk fpk ) λf 1 .
s1 k=1
5.3.3. The use of pseudo-localization Consider the function g(s) =
gk,s .
k
It is straightforward to see that dg(s) k+s = gk,s . In particular, we have supp∗ dg(s) k+s pk = qk−1 − qk 1A − qk .
(5.2)
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
571
According to the terminology of Theorem 5.2, we consider the projection
ζg(s) ,s =
&
1A −
(1M − ξQ )19Q .
Q∈Qk
k1
Notice that we are just taking k 1 and not k ∈ Z as in Theorem 5.2. This is justified by the fact that the qk ’s are now given by Cuculescu’s construction applied to our f ∈ Ac,+ and our assumption in Remark 3.3 implies that 1M − ξQ = 0 for all Q ∈ Qk with k < 1. Now, if we compare this projection with the one provided by Lemma 4.2 ζ=
& 1A − (ξQ − ξ )1
Q 9Q , 1j k Q∈Qj
k1
it becomes apparent that ζ ζg(s) ,s . On the other hand, Chebychev’s inequality gives ( ∞ ) ∞ 1 ζ T (g(s) )ζ λϕ ζ T (goff )ζ > λ = λϕ ζ T (g(s) )ζ > λ 2 λ s=1
s=1
This automatically implies ∞ 2 1 λϕ ζ T (goff )ζ > λ ζg(s) ,s T (g(s) )ζg(s) ,s 2 . λ s=1
Now, combining (5.1) and (5.2), we may use pseudo-localization and deduce c2n,γ λϕ ζ T (goff )ζ > λ λ =
c2n,γ λ
∞
2
s2−γ s/4 g(s) 2
s=1 ∞
s2
−γ s/4
s=1
1 gk,s 22
2
2
k
cn,γ f 1 . This completes the argument for the off-diagonal terms of g. 5.4. Estimation of T boff As above, it suffices to estimate ζ T (boff )ζ =
∞ ∞ s=1 k=1
ζ T pk (f − fk+s )pk+s + pk+s (f − fk+s )pk ζ.
2
.
572
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
In the sequel we use the following notation. For any dyadic cube Q ∈ Qk+s , we shall write Qk to denote the sth antecessor of Q. That is, Qk is the only dyadic cube in Qk containing Q. If we set bk,s = pk (f − fk+s )pk+s + pk+s (f − fk+s )pk , the identity below follows from ξQk πQk = πQk ξQk = 0 and Lemma 4.2 ζ T (bk,s )ζ (x) =
k(x, y) ζ (x)bk,s (y)ζ (x) dy
Rn
k(x, y)πQk f (y) − fQ πQ dy 1(9Qk )c (x) ζ (x) = ζ (x) Q∈Qk+s Q
+ ζ (x) k(x, y)πQ f (y) − fQ πQk dy 1(9Qk )c (x) ζ (x) Q∈Qk+s Q
= ζ (x) k(x, y)bk,s (y) dy 1(9Qk )c (x) ζ (x). Q∈Qk+s Q
Fig. 6. Decomposition into disjoint diagonal boxes for s = 2.
Before going on with the proof, let us explain a bit our next argument. Our terms bk,s are located in the (s + 1)th upper and lower diagonals and we want to compare their size with that of the main diagonal. To do so we write each bk,s , located in the entries (k, k + s) and (k + s, k), as a linear combination of four diagonal boxes in a standard way. However, this procedure generates overlapping and we are forced to consider only those integers k congruent to a fixed 1 j s +1 at a time. Fig. 6 will serve as a model (s = 2) for our forthcoming estimates. According to Chebychev’s inequality we obtain ∞ ) ( ∞ ∞ ∞ λϕ ζ T (bk,s )ζ > λ ζ T (bk,s )ζ s=1 k=1
s=1 k=1
∞ ∞
1
k(·, y)bk,s (y) dy 1(9Q )c (·) k
s=1 k=1 Q∈Qk+s Q
1
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593 ∞ s
573
k(·, y)bk,s (y) dy 1(9Q )c (·) . k
1
k≡j Q∈Qk+s Q mod s+1
s=1 j =0
We now use the decomposition bk,s =
s
pk+r (f − fk+s )
r=0
−
s
s
pk+r −
r=0
pk+r (f − fk+s )
r=1
s
s−1
pk+r +
r=1
pk+r (f − fk+s )
r=0
s−1
s−1
pk+r
r=0
pk+r (f − fk+s )
r=1
s−1
pk+r
r=1
1 2 3 4 − bk,s − bk,s + bk,s , = bk,s
of bk,s as a linear combination of four diagonal terms. Let us recall that the four projections r pk+r above (with 0 r s and meaning either < or ) belong to Ak+s . In particular, since Ek+s (f − fk+s ) = 0, the following identity holds for any Q ∈ Qk+s and any 1 i 4 i bk,s (y) dy = 0. Q
Therefore, we find ) ( ∞ ∞ ζ T (bk,s )ζ > λ λϕ s=1 k=1
∞ 4 s s=1 i=1 j =0
k≡j Q∈Qk+s Q mod s+1
k(·, y) − k(·, cQ ) bi (y)1(9Q )c (·) dy. k k,s 1
However, by Lipschitz γ -smoothness we have
k(·, y) − k(·, cQ ) bi (y)1(9Q )c (·) dy k k,s 1
Q
=
k(x, y) − k(x, cQ ) dx bi (y) dy k,s
τ
(9Qk )c
Q
Q
(9Qk )c
i |y − cQ |γ dx τ bk,s (y) dy n+γ |x − cQ |
(Q) /(Qk ) ϕ γ
γ
s r=0
pk+r (f + fk+s )
s r=0
pk+r 1Q
574
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
2
−γ s
ϕ
s
pk+r f
r=0
s
pk+r 1Q .
r=0
Finally, summing over (s, i, j, k, Q) we get ) ( ∞ ∞ ∞ 4 s λϕ ζ T (bk,s )ζ > λ s=1 i=1 j =0
s=1 k=1
s=1 i=1 j =0
=4
∞ s+1 s=1
2γ s
2
−γ s
k≡j mod s+1
∞ 4 s
ϕ
s r=0
pk+r f
s
pk+r
r=0
2
−γ s
f 1
f 1
= 4cγ f 1 . This completes the argument for the off-diagonal terms of b. 5.5. Conclusion Combining the results obtained so far in Sections 4 and 5, we obtain the weak type inequality announced in Theorem A. The strong Lp estimates follow for 1 < p < 2 from the real interpolation method, see e.g. [48] for more information on the real interpolation of noncommutative Lp spaces. In the case 2 < p < ∞, our estimates follow from duality since our size/smoothness conditions on the kernel are symmetric in x and y. Remark 5.4. Recent results in noncommutative harmonic analysis [25,27–29] show the relevance of non-semifinite von Neumann algebras in the theory. The definition of the corresponding Lp spaces (so called Haagerup Lp spaces) is more involved, see [19,56]. A well-known reduction argument due to Haagerup [18] allows us to extend our strong Lp estimates in Theorems A and B to functions f : Rn → M with M a type III von Neumann algebra M. Indeed, if σ denotes the one-parameter unimodular
group associated to (A, ϕ), we take the crossed product R = A σ G with the group G = n∈N 2−n Z. According to [18], R is the closure of a union of finite von Neumann algebras k1 Ak directed by inclusion. We know that our result holds on Lp (Ak ) for 1 < p < ∞ and with constants independent of k. Therefore, the same will hold on Lp (R). Then, using that Lp (A) is a (complemented) subspace of Lp (R), the assertion follows. Remark 5.5. According to the classical theory, it seems that some hypotheses of Theorem A could be weakened. For instance, the size condition on the kernel is not needed for scalar-valued functions. Moreover, it is well known that the classical theory only uses Lipschitz smoothness on the second variable to produce weak type (1, 1) estimates. Going even further, it is unclear whether or not we can use weaker smoothness conditions, like Hörmander type conditions. Nevertheless, all these apparently extra assumptions become quite natural if we notice that all of them where used to produce our pseudo-localization principle, a key point in the whole argument. Under this point of view, we have just imposed the natural hypotheses which appear around the T 1 theorem. This leads us to pose the following problem.
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
575
Problem. Can we weaken the hypotheses on the kernel as pointed above? Remark 5.6. We believe that our methods should generalize if we replace Rn by any other space of homogeneous type. In other words, a metric space equipped with a non-negative Borel measure which is doubling with respect to the given metric. More general notions can be found in [8,33,34]. Of course, following recent results by Nazarov, Treil, Volberg and Tolsa, it is also possible to study extensions of non-doubling Calderón–Zygmund theory in our setting. It is not so clear that the methods of this paper can be easily adapted to this case. 6. Operator-valued kernels We now consider Calderón–Zygmund operators associated to operator-valued kernels k : R2n \ Δ → M satisfying the canonical size/smoothness conditions. In other words, we replace the absolute value by the M-norm, see the introduction for details. We begin by constructing certain bad kernels which show that there is no hope to extend Theorem A in full generality to this context. Then we obtain positive results assuming some extra hypotheses. 6.1. Negative results The origin of the counterexample we are constructing goes back to a lack (well known to experts in the field) of noncommutative martingale transforms
dfk →
k
ξk−1 dfk .
k
Indeed, the boundedness of this operator on Lp might fail when the predictable sequence of ξk ’s is operator-valued. Here is a simple example. Let A be the algebra of m × m matrices equipped with the standard trace tr and consider the filtration A1 , A2 , . . . , Am , where As denotes the subalgebra spanned by the matrix units eij with 1 i, j s and the matrix units ekk with k > s. • If 1 < p < 2, we take f =
m
k=2 e1k
and ξk = ek1 , so that
√ 1/p ξk−1 dfk = (m − 1) m − 1 = dfk . p
k
• If 2 < p < ∞, we take f =
m
k=2 ek−1,k
k
p
and ξk = e1k , so that
√ 1/p dfk = (m − 1) m − 1 = ξk−1 dfk . k
p
k
p
Letting m → ∞, we see that Lp boundedness might fail for any p = 2 even having L2 boundedness. Our aim is to prove that the same phenomenon happens in the context of singular integrals with operator-valued kernels. The examples above show us the right way to proceed. Namely, we shall construct a similar operator using Littlewood–Paley type arguments. Note that a dyadic martingale approach is also possible here, but this would give rise to certain operators having non-smooth kernels and we want to show that smoothness does not help in this particular case.
576
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
Let SR be the Schwartz class in R and consider a non-negative function ψ in SR bounded above by 1, supported in 1 |ξ | 2 and identically 1 in 5/4 |ξ | 7/4. Define ψk (ξ ) = ψ 2−k ξ .
= ψ . If we construct the functions Let Ψ denote the inverse Fourier transform of ψ, so that Ψ
k = ψk and we may define the following convolution-type operaΨk (x) = 2k Ψ (2k x), we have Ψ tors T1 f (x) =
ek1 Ψk ∗ f,
k1
T2 f (x) =
e1k Ψk ∗ f.
k1
In this case we are taking M = B(2 ) and both T1 and T2 become contractive operators in L2 (A). Indeed, let FA = FR ⊗ idL2 (M) denote the Fourier transform on L2 (A). According to Plancherel’s theorem, FA is an isometry and the following inequality holds 1 2 2
T1 f 2 = ek1 Ψk ∗ f = ek1 ψk f 2 2
k1
k1
k1
f (ξ )2 dξ
1 2
f 2 .
2k |ξ |2k+1
The same argument works for T2 . Now we show that the kernels of T1 and T2 also satisfy the expected size and smoothness conditions. These are convolution-type kernels given by k1 (x, y) =
ek1 Ψk (x − y) and k2 (x, y) =
k1
e1k Ψk (x − y).
k1
We clearly have k1 (x, y)
ek1 Ψk (x − y) M=
M
k1
k2 (x, y)
= e Ψ (x − y) 1k k M k1
M
=
Ψk (x − y)2
1 2
,
k1
=
Ψk (x − y)2
1 2
.
k1
Therefore, for the size condition it suffices to see that 1 2 1 Ψk (x)2 . |x| k∈Z
(6.1)
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
577
Similarly, using the mean value theorem in the usual way, the condition 1 2 2 1 Ψ (x) 2 k |x|
(6.2)
k∈Z
implies Lipschitz smoothness for any 0 < γ 1. The proof of (6.1) and (6.2) is standard. Namely, since Ψ and Ψ belong to the Schwartz class SR , there exist absolute constants c1 and c2 such that Ψ (x) c2 min 1, 1 . Ψ (x) c1 min 1, 1 and |x|2 |x|3 If 2−j |x| < 2−j +1 , we find the estimate 1 1 2 2 2k −4 −2k Ψk (x)2 c1 2 + c1 |x| 2 22j + k∈Z
k>j
kj
1 22j |x|4
1 2
1 . |x|
1 . |x|2
Similarly, using that Ψk (x) = 22k Ψ (2k x), we have 1 1 2 2 2 4k −6 −2k Ψ (x) c2 2 + c2 |x| 2 24j + k k∈Z
k>j
kj
1 2j 2 |x|6
1 2
Thus, T1 and T2 are bounded on L2 (A) with operator-valued kernels satisfying the standard size and smoothness conditions. Now we shall see how the boundedness on Lp (A) fails for p = 2. By definition, we know that • ψk is supported by 2k |ξ | 2k+1 , • ψk is identically 1 in 5 · 2k /4 |ξ | 7 · 2k /4. If I0 = [5/4, 7/4] and Ik = I0 + 32 (2k − 1), it is easily seen that ψk 1Ik = 1Ik
(6.3)
for all non-negative integer k. Now we are ready to show the behavior of T1 and T2 on Lp . Indeed, let us fix an integer m 1 and let gk be the inverse Fourier transform of 1Ik for 1 k m. Then we set f1 =
m
e1k gk
and f2 =
k=1
m
ekk gk .
k=1
By (6.3) we have Ψj ∗ gk = δj k gk for 1 k m. Moreover, 3 k j
gj (ξ ) =
gk ξ + 2 − 2 2
⇒
gj (x) = gk (x).
These observations allow us to obtain the following identities
578
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593 1 p p 1 1 m ( m T1 f1 p − k=1 ekk gk p k=1 |gk | ) p = m = = m p 2 g1 p , 1 m f1 p k=1 e1k gk p 2 2 ( k=1 |gk | ) p m 2 12 1 1 k=1 e1k gk p ( m T2 f2 p − k=1 |gk | ) p = m = = m 2 p g1 p . 1 m f2 p k=1 ekk gk p ( k=1 |gk |p ) p p
Therefore, letting m → ∞ we see that T1 and T2 are not bounded on Lp (A) for 1 < p < 2 and 2 < p < ∞ respectively. Since we have seen that both are bounded on L2 (A) and are equipped with good kernels, we deduce that Theorem A does not hold for T1 and T2 . This is a consequence of the matrix units we have included in the kernels of our operators. Remark 6.1. We refer to [36] and [43] for a study of paraproducts associated to operator-valued kernels. There it is shown that certain classical estimates also fail when dealing with noncommuting operator-valued kernels. The results in [43] give new light to Carleson embedding theorem. 6.2. The L∞ → BMO boundedness In what follows we shall work under the hypotheses of Theorem B. In other words, with Calderón–Zygmund operators which are M-bimodule maps bounded on Lq (A) and are associated to operator-valued kernels satisfying the standard size/smoothness conditions, see the introduction for further details. Let us define the noncommutative form of dyadic BMO associated to our von Neumann algebra A. According to [37,47], we may define the space BMOA as the closure of functions f in L1,loc (Rn ; M) with f BMOA = max f BMOrA , f BMOcA < ∞, where the row and column BMO norms are given by f BMOrA
1 2 1 ∗ f (x) − fQ f (x) − fQ dx = sup |Q|
,
f BMOcA
1 2 1 ∗ f (x) − fQ f (x) − fQ dx = sup |Q|
.
Q∈Q
Q
Q∈Q
Q
M
M
In order to extend our pseudo-localization result to the framework of Theorem B, we shall need to work with the identity 1A and show that T ∗ 1A belongs to the noncommutative form of BMO. In fact, the (still unpublished) result below due to Tao Mei [38] gives much more. Theorem 6.2. If T is as above, then Tf BMOA cn,γ f A . Mei’s argument for Theorem 6.2 is short and nice for q = 2. The case q = 2 requires the noncommutative analog of John–Nirenberg theorem obtained by Junge and Musat in [26].
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
579
Remark 6.3. Let us fix an index q < p < ∞. By a recent result of Musat [41] adapted to our setting by Mei [37], we know that Lq (A) and BMOA form an interpolation couple. Moreover, both the real and complex methods give the isomorphism BMOA , Lq (A) q/p Lp (A) with constant cp ∼ p for p large. The proof of the latter assertion was achieved in [26], refining the argument of [41]. In particular, the Lp estimates announced in Theorems A and B automatically follow from Theorem 6.2 combined with Musat’s interpolation. Although this approach might look much simpler, the proof of the necessary interpolation results from [41] and of the noncommutative John–Nirenberg theorem (used in Mei’s argument) are also quite technical. Remark 6.4. It also follows from Theorem 6.2 that the problem posed in Remark 5.5 is only interesting for weak type inequalities. Indeed, if we are given a kernel with no size condition and only satisfying the Hörmander smoothness condition in the second variable, then we may obtain the strong Lp estimates provided by Theorems A and B for 1 < p 2. We just need to apply Mei’s argument for Theorem 6.2 (which works under these weaker assumptions) to the adjoint mapping and dualize backwards. A similar argument holds for Hörmander smooth kernels in the first variable and 2 p < ∞. 6.3. Proof of Theorem B Before proceeding with the argument, we set some preliminary results. According to Theorem 6.2 and the symmetry of the conditions on the kernel, we know that T ∗ 1A belongs to BMOA . In the following result, we shall write H1 for the Hardy space associated to the dyadic filtration on Rn . That is, the predual of dyadic BMO, see [17]. Lemma 6.5. If T is as above and T ∗ 1A = 0, then Tf (x) dx = 0 for any f ∈ H1 . Rn
Proof. Since T ∗ 1A = 0 vanishes as an element of BMOA , we will have τ T φ(x) dx = T φ, 1A = φ, T ∗ 1A = 0
(6.4)
Rn
for any φ ∈ H1 (A), the Hardy space associated to the dyadic filtration (Ak )k∈Z , see [47] for details and for the noncommutative analogue of Fefferman’s duality theorem H1 (A)∗ = BMOA . Given any projection q ∈ Mπ of finite trace and f ∈ H1 , it is clear that φ = f q ∈ H1 (A). In particular, using M-modularity again T φ(x) dx = τ q Tf (x) dx = 0 τ Rn
Rn
for any such projection q. Clearly, this immediately implies the assertion.
2
580
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
Lemma 6.6. Let T be as above for q = 2 and L2 (A)-normalized. Then, given x0 ∈ Rn and r1 , r2 > 0 with r2 > 2r1 , the following estimate holds for any pair f, g of bounded scalar-valued functions respectively supported by Br1 (x0 ) and Br2 (x0 ) Tf (x)g(x) dx Rn
M
cn r1n log(r2 /r1 )f ∞ g∞ .
Proof. We proceed as in the proof of the localization estimate given in Section 2.1. Let B denote the ball B3r1 /2 (x0 ) and consider a smooth function ρ identically 1 on B and 0 outside B2r1 (x0 ). Taking η = 1 − ρ, we may decompose Tf (x)g(x) dx Rn
M
= Tf (x)ρg(x) dx
M
Rn
+ Tf (x)ηg(x) dx Rn
. M
For the first term we adapt the commutative argument using the convexity of the function a → |a|2 . Indeed, if M embeds isometrically in B(H), it suffices to see that a → a ∗ ah, hH is a convex function for any h ∈ H. However, this follows from the identity a ∗ ah, hH = ah2H . As an immediate consequence of this, we find the inequality 1 |B (x )| 2r1 0
B2r1 (x0 )
2 1 Tf (x)ρg(x) dx |B2r1 (x0 )|
Tf (x)ρg(x)2 dx.
B2r1 (x0 )
This combined with M-modularity gives Tf (x)ρg(x) dx Rn
M
= B2r1 (x0 ) B2r1 (x0 )
n/2 = cn r1
n/2 cn r1
n/2
cn r1
1 |B2r1 (x0 )| 1 |B2r1 (x0 )|
sup
sup
cn r1n f ∞ g∞ ,
M
2 τ 1B2r1 (x0 ) (x)Tf (x)ρg(x)a dx
Rn
aL2 (M) 1
1 2 Tf (x)ρg(x)2 dx
B2r1 (x0 )
sup
aL2 (M) 1
M
B2r1 (x0 )
aL2 (M) 1
Tf (x)ρg(x) dx
2 τ T (f a)(x) dx
Rn
f 2 a2 g∞
1 2
g∞
1 2
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
581
since supp f ⊂ Br1 (x0 ). On the other hand, the second term equals Tf (x)ηg(x) dx Rn
M
=
Br2 (x0 )\B
k(x, y)f (y) dy ηg(x) dx
Br1 (x0 )
This term is estimated exactly in the same way as in Section 2.1.
. M
2
Sketch of the proof of Theorem B. As in the proof of Theorem A, we first observe that there is no restriction by assuming that q = 2. Indeed, according to Theorem 6.2 and Remark 6.3, it is easily seen that boundedness on Lq (A) is equivalent to boundedness on L2 (A). Moreover, we may assume that f ∈ Ac,+ and decompose it for fixed λ ∈ R+ applying the noncommutative Calderón–Zygmund decomposition. This gives rise to f = g + b. The diagonal parts are estimated in the same way. Indeed, since we have gd 22 2n λf 1 , the L2 -boundedness of T suffices for the good part. On the other hand, we use Lemma 4.2 for the bad part bd in the usual way. This reduces the problem to estimate ζ T (bd )ζ . By M-bimodularity, we can proceed verbatim with the argument given for this term in the proof of Theorem A. Moreover, exactly the same reasoning leads to control the off-diagonal part boff . It remains to estimate the term associated to goff . By Lemma 4.2 one more time, it suffices to study the quantity λϕ ζ T (goff )ζ > λ . As in the proof of Theorem A, we write goff = k,s gk,s as a sum of martingale differences and use pseudo-localization. To justify our use of pseudo-localization we follow the argument in Theorem 5.2 using M-bimodularity. This reduces the problem to study the validity of the paraproduct argument and of the shifted form of the T 1 theorem for our new class of Calderón– Zygmund operators. The paraproduct argument is simple. Indeed, since T is M-bimodular, the same holds for T ∗ so that T ∗ 1A becomes an element of BMOZA where ZA denotes the center of A. According to [38], the dyadic paraproduct Πξ associated to the term ξ = T ∗ 1A defines a bounded map on L2 (A). Moreover, since it is clear that Πξ is M-bimodular, this allows us to consider the usual decomposition T = T0 + Πξ∗ . Now following the argument in Section 2.3, with the characteristic functions 1Rn \Σf,s and 1Ωk replaced by the corresponding projections provided by Theorem 5.2, we see that the estimate of the paraproduct also reduces here to the shifted T 1 theorem. At this point we make crucial use of the fact that ξ = T ∗ 1A is commuting, so that the same holds for Δj (ξ ) for all j ∈ Z. Let us now sketch the main (slight) differences that appear when reproving the shifted T 1 theorem for operator-valued kernels. Lemma 6.5 will play the role of the cancellation condition (2.1). On the other hand, we also have at our disposal the three auxiliary results (suitably modified) in Section 2.1. Namely, Cotlar lemma as it was stated there will be used below with the only difference that we apply it over the Hilbert space H = L2 (A) instead of the classical L2 . Regarding Schur lemma, it is evident how to adapt it to the present setting. We just need to replace the Schur integrals by S1 (x) = Rn
k(x, y)
M dy
and S2 (y) = Rn
k(x, y)
M dx.
582
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
We leave the reader to complete the straightforward modifications in the original argument. Finally, Lemma 6.6 given above is the counterpart in our context of the localization estimate that we use several times in the proof of Theorem A. Once these tools are settled, the proof follows verbatim just replacing the absolute value | · | by the norm · M when corresponds. Maybe it is also worthy of mention that the two instances in the proof of the shifted T 1 theorem where the Lebesgue differentiation theorem is mentioned, we should apply its noncommutative analog from [37]. This completes the proof. 2 Remark 6.7. The following question related to Theorem B was communicated to me by Tao Mei. It is clear that when the kernel takes values in the center ZM of M, the corresponding Calderón– Zygmund operator is M-bimodular and Theorem B applies. Assume now that the kernel k takes values in the commutant M . This gives rise to an M-bimodular Calderón–Zygmund operator T : L2 (A) → L2 Rn ; L2 B(H) where both M and M embed in B(H). Assume further that such operator is bounded on L2 (A). The question is whether our arguments in this paper can be suitably modified to produce the corresponding weak type inequality. We observe that some difficulties appear in Mei’s argument for Theorem 6.2 and also in Lemmas 6.5 and 6.6. Note however that a positive answer to this question would produce, by real interpolation and duality, new strong Lp inequalities for a much wider class of operators. Remark 6.8. After Theorem B, it is also natural to wonder about a vector-valued noncommutative Calderón–Zygmund theory. Let us be more precise, if the von Neumann algebra M is hyperfinite, Pisier’s theory [45] allows us to consider the spaces Lp (A; X) with values in the operator space X. Here it is important to recall that we must impose on X an operator space structure since a Banach space structure is not rich enough. Then, we can consider vector-valued noncommutative singular integrals and study for which operator spaces we obtain weak type (1, 1) and/or strong type (p, p) inequalities. Of course, this is closely related to the geometry of the operator space in question and in particular to the notion of UMDp operator spaces, also defined by Pisier. In this context a great variety of problems come into scene, like the independence of the UMDp condition with respect to p (see [42] for some advances) or the operator space analog of Burkholder’s geometric characterization of the UMD property in terms of ζ -convexity [5]. Remark 6.9. Another problem is the existence of T 1 type theorems. This follows however from results by Hytönen [21] and Hytönen, Weis [22]. Namely, given a pair of Banach spaces (X, Y), they consider X-valued functions and B(X, Y)-valued kernels. In this general context, they need to impose R-boundedness conditions on the kernel. However, in our setting X = L2 (M) = Y and R-boundedness coincides with classical boundedness. Moreover, since our operators act by left or right multiplication, their norms in B(L2 (M)) coincide with the norm of the corresponding multiplier k(x, y) in M. Therefore, up to some extra conditions imposed in [21,22], their results are applicable here. Acknowledgments I would like to thank J.M. Martell, F. Soria and Q. Xu for discussions related to the content of this paper and specially to Tao Mei for keeping me up to date on his related work.
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
583
Appendix A. On pseudo-localization A.1. Applicability We begin by analyzing how the pseudo-localization principle is applied to a given L2 function. At first sight, it is only applicable to functions f in L2 satisfying that Em (f ) = fm = 0 for some integer m. Indeed, according to the statement of the pseudo-localization principle we have supp f ⊂ supp dfk+s ⊂ Ωk . k∈Z
k∈Z
Given ε = (ε1 , ε2 , . . . , εn ) with εj = ±1 for 1 j n, let Rn(ε) = x ∈ Rn sgn xj = εj for 1 j n be the n-dimensional quadrant associated to ε and define f(ε) to be the restriction of f to such quadrant. If fm = 0 for all m ∈ Z, the same will happen to f(ε) for some index ε. Assume (with no loss of generality) that ε = (1, 1, . . . , 1) or, in other words, that f itself is supported by the first quadrant. Let Λf be the set of negative k’s satisfying supp dfk+s = ∅. Our hypothesis fm = 0 for all m ∈ Z implies that Λf has infinitely many elements. According to the shift condition, we know that Ωk = ∅ for each k ∈ Λf and therefore contains at least a cube in Qk , since Ωk is an Rk -set. In fact, for k small enough the Qk -cube in the first quadrant closest to the origin will be large enough to intersect the support of f . A moment of thought gives rise to the conclusion that Ωk contains such cube for infinitely many negative k’s and Σf,s =
9Ωk = Rn .
k∈Z
Therefore, our result does not provide any information in this case. It is convenient to explain how to apply our result for an arbitrary function f in L2 not satisfying the condition fm = 0. By homogeneity, we may assume that f 2 = 1. On the other hand, if supp f is not compact we approximate f by a compactly supported function f0 such that f − f0 2 cn,γ s2−γ s/4 . This clearly reduces our problem to find the set Σf,s around the support of f0 . Next we decompose f0 = 1j 2n fj , with fj being the restriction of f0 to the j th quadrant and work independently with each of these functions. In other words our localization problem reduces to study functions f in L2 with compact support contained in the first n-dimensional quadrant. Let f be such a function and take Q to be the smallest dyadic cube containing the support of f . We have Q ∈ Qm for some integer m. Then we find fm = λ1Q with 1 λ = |Q| Rn f (x) dx and thus we decompose f = f − λ2−γ s/2 1Qs + λ2−γ s/2 1Qs = f 1 + f 2 where Qs is a cube satisfying:
584
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
• Qs contains Q; • Qs is contained in a dyadic antecessor of Q; • The Lebesgue measure of Qs is |Qs | = 2γ s/2 |Q|. It is clear that we have
2 2 f (x) dx
1 2
= λ2−γ s/2 2γ s/4 |Q| = 2−γ s/4 fm 2 2−γ s/4 .
Rn
s be the dyadic Q-antecessor Therefore, f 2 is small enough for our aims. On the other hand, let Q of generation m − j0 with j0 being the smallest positive integer such that j0 γ s/2n. In other words, this cube is the smallest dyadic Q-antecessor containing Qs . If we set m0 = m − j0 , we 1 = 0 and supp df 1 = ∅ so that there is clearly have fm1 0 = 0. When k m0 − s we have dfk+s k+s no set to control. When k + s > m0 we use 1
s . ⊂Q supp dfk+s
s . In the worst case k = Hence, we may choose Ωk to be the smallest Rk -set containing Q
s . That is, the m0 − s + 1 we are forced to take Ωk as the (s − 1)th dyadic antecessor of Q
(j0 + s − 1)-dyadic antecessor Q(j0 + s − 1) of Q. This gives rise to the set Σf,s =
γ
0 + s − 1) ⊂ 9 · 2j0 +s Q ∼ 9 · 2(1+ 2n )s supp f 9Ωk = 9Q(j
k∈Z
and completes the argument for arbitrary L2 functions. To conclude, we should mention that the dependence on the n-dimensional quadrants, due to the geometry imposed by the standard dyadic filtration, is fictitious. Indeed, we can always translate the dyadic filtration, so that the role of the origin is played by another point which leaves the support of f in the new first quadrant. Remark A.1. Given a function f in L2 and a parameter δ ∈ R+ , we have analyzed so far how to find appropriate sets Σf,δ satisfying the localization estimate which motivated our pseudolocalization principle
Tf (x)2 dx
1 2
Rn \Σf,δ
δ
f (x)2 dx
1 2
.
Rn
Reciprocally, given a set Σ in Rn and δ ∈ R+ , it is quite simple to find functions fΣ,δ satisfying such estimate on Rn \ Σ . Indeed, let s 1 be the smallest possible integer satisfying cn,γ s2−γ s/4 δ and write Σ = k∈Z 9Ωk as a disjoint union of 9-dilations of maximal Rk sets. In this case, any function of the form fΣ,δ =
k∈Z
1Ωk dgk+s
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
585
with g ∈ L2 satisfies the hypotheses of our pseudo-localization principle with Σ as the final localization set. Indeed, we have d(fΣ,δ )k+s = 1Ωk dgk+s because 1Ωk is (k + s)-predictable and we deduce that fΣ,δ satisfies the shift condition. A.2. Decreasing rate of singular integrals in the L2 metric As an immediate consequence of the pseudo-localization principle, we can give a lower estimate of how fast decreases a singular integral far away from a set Σf associated to f . To be more specific, the following result holds. Corollary A.2. Let f be in L2 and define Σf =
with Γk = supp dfk ∈ Rk .
9Γk
k∈Z
Then, the following holds for any ξ > 4,
Tf (x)2 dx
1 2
cn,γ ξ −γ /4 log ξ
Rn \ξ Σf
f (x)2 dx
1 2
Rn
and any L2 -normalized Calderón–Zygmund operator with Lipschitz parameter γ . Proof. Let
Ωk be the smallest Rk -set containing Γk+s . In the worst case, Γk+s can be written as
α (s) to be the sth dyadic antecessor of Qα , we observe a union α Qα of Qk+s -cubes. Taking Q that Ωk ⊂
α (s) ⊂ 2s+1 Γk+s . Q
α
Then we construct Σf,s =
9Ωk ⊂ 2s+1 Σf ,
k∈Z
and the theorem above automatically gives
Rn \2s+1 Σf
Tf (x)2 dx
1 2
cn,γ s2
−γ s/4
f (x)2 dx
Rn
Since this holds for every positive integer s, the assertion follows.
2
Remark A.3. All the considerations in Section A.1 apply to this result. Remark A.4. This result might be quite far from being optimal, see below.
1 2
.
586
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
A.3. Atomic pseudo-localization in L1 Maybe the oldest localization result was already implicit in the Calderón–Zygmund decomposition. Indeed, let b denote the bad part of f associated to a fixed λ > 0 and let Σλ be the level set where the dyadic Hardy–Littlewood maximal function Md f is bigger than λ. Note that b is supported by Σλ . Then, we have T b(x) dx cn bj 1 cn f 1 , j
Rn \2Σλ
where the bj ’s are the atoms in which we decompose b. In fact, this reduces to a well-known localization result for dyadic atoms in L1 . Namely, let a denote an atom supported by a dyadic cube Qa . Then, the mean-zero of a gives the following estimate for any ξ > 2 T a(x) dx = k(x, y) − k(x, cQa ) a(y) dy dx Rn \ξ Qa
Rn \ξ Qa Rn
Rn \ξ Q
Rn
a
|y − cQa |γ a(y) dy dx cn ξ −γ a1 . n+γ |x − y|
(A.1)
Note that the only condition on T that we use is the γ -Lipschitz smoothness on the second variable, not even an a priori boundedness condition. Under these mild assumptions, we may generalize (A.1) in the language of our pseudo-localization principle. Namely, the following result (maybe known to experts) holds. Theorem A.5. Let us fix a positive integer s. Given a function f in L1 and any integer k, we define Ωk to be the smallest Rk -set containing the support of dfk+s and consider the set Σf,s = 3Ωk . k∈Z
Then, we have for any Calderón–Zygmund operator as above Tf (x) dx cn 2−γ s f (x) dx. Rn \Σf,s
Rn
Proof. We may clearly assume that fm = 0 for some integer m. Namely, otherwise we can argue as in the previous paragraph to deduce that Σf,s = Rn and the assertion is vacuous. Define inductively A1 = supp dfm+1 , Aj = supp dfm+j \
Aw .
w<j
Use that supp f ⊂
j Aj
and pairwise disjointness of Aj ’s to obtain
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
Tf (x) dx
=
j
T (f 1Q )(x) dx
Q∈Qm+j Rn \Σ f,s Q⊂Aj
j
Rn \Σf,s
587
Q∈Qm+j Rn \Σ f,s Q⊂Aj
∞ dfk (x) dx. T 1 Q k=m+j
s be the sth dyadic antecessor of Q. Since Let Q Q ⊂ Aj ⊂ supp dfm+j ⊂ Ωm+j −s
s ⊂ Σf,s ⇒ Rn \ Σf,s ⊂ Rn \ 2s Q and and Q ∈ Qm+j , we deduce 2s Q ⊂ 3Q ∞ Tf (x) dx dfk (x) dx. T 1 Q j
Rn \Σf,s
Q∈Qm+j Rn \2s Q Q⊂Aj
k=m+j
On the other hand, by (A.1)
Tf (x) dx cn 2−γ s j
Rn \Σf,s
∞ dfk 1Q
Q∈Qm+j Q⊂Aj
k=m+j
1
∞ = cn 2−γ s dfk 1Aj j
= cn 2
−γ s
k=m+j
1
1Aj f 1 .
j
Using once more the pairwise disjointness of the Aj ’s we deduce the assertion.
2
Remark A.6. Here we should notice that the condition fm = 0 cannot be removed as we did in the L2 case and its applicability is limited to this atomic setting. On the other hand, if we try to use the argument of Theorem A.5 for p = 2, we will find a nice illustration of why the ideas around almost orthogonality that we have used in the paper come into play. In the L1 framework, almost orthogonality is replaced by the triangle inequality. A.4. Other forms of pseudo-localization Once we have obtained results in L1 and L2 , it is quite natural to wonder about Lp pseudolocalization for other values of p. If we only deal with atoms, it easily seen that (A.1) generalizes to any p > 1 in the following way
Rn \ξ Q
T a(x)p dx a
1
p
cn ξ
−(γ +n/p )
Rn
a(x)p dx
1
p
.
(A.2)
588
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
This gives rise to two interesting problems: (i) In Theorem A.5 we showed that (A.1) generalizes to more general functions in L1 , those satisfying fm = 0 for some integer m. On the other hand, as we have seen in Section A.1, the condition fm = 0 is not a serious restriction for p = 2, or any p > 1. Therefore, inequality (A.2) suggests that our pseudo-localization principle might hold with s2−γ s/4 replaced by the better constant 2−(γ +n/2)s . However, this result and its natural Lp generalization are out of the scope of this paper. (ii) Although the constant that we have obtained in our pseudo-localization principle on L2 might be far from being optimal, it still makes a lot of sense to wonder whether or not the corresponding interpolated inequality holds for 1 < p < 2. Below we give some guidelines which might lead to such a result. We have not checked details, since the necessary estimates might be quite technical, as those in the proof for p = 2. All our ideas below can be thought as problems for the interested reader. The interpolated inequality that comes to mind is
Tf (x)p dx
1
p
Rn \Σf,s
cn,γ
s2−γ s/4 (s23γ s/4 )
2 p −1
f (x)p dx
1
p
.
Rn
However, by the presence of Σf,s , a direct interpolation argument does not apply and we need a more elaborated approach. Namely, following the proof of our result in L2 verbatim, it suffices to find suitable upper bounds for Φs and Ψs in B(Lp ). Here we might use Rubio de Francia’s idea of extrapolation and content ourselves with a rough estimate (i.e. independent of s) for the norm of these operators from L1 to L1,∞ . Of course, by real interpolation this would give rise to the weaker inequality
Rn \Σ
Tf (x)p dx f,s
1
p
2− p2
(s2−γ s/4 ) cn,γ p−1
f (x)p dx
1
p
.
(A.3)
Rn
However, this would be good enough for many applications. The Calderón–Zygmund method will be applicable to both Φs and Ψs if we know that their kernels satisfy a suitable smoothness estimate. The lack of regularity of Ek and Δk+s appears again as the main difficulty to overcome. In this case, it is natural to wonder if the Hörmander condition k(x, y) − k(x, 0) dx cn,γ , |x|>2|y|
holds for the kernels of Φs and Ψs . We believe this should be true. Anyway, a more in depth application of Rubio’s extrapolation method (which we have not pursued so far) might be quite interesting here. Remark A.7. According to the classical theory [54], it is maybe more natural to replace (in the shifted form of the T 1 theorem) the dyadic martingale differences Δk+s by a Littlewood– Paley decomposition and the conditional expectations Ek by their partial sums. This result will
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
589
be surely easier to prove since there is no lack of regularity as in the dyadic martingale setting. This alternative approach to the shifted T 1 theorem might give rise to some sort of pseudolocalization result in terms of Littlewood–Paley decompositions. Although this is not helpful in the noncommutative setting (by our dependence on Cuculescu’s construction), it makes the problem on the smoothness of the kernels of Φs and Ψs more accessible. Remark A.8. If the argument sketched above for inequality (A.3) works, another natural question is whether results for p > 2 can be deduced by duality. On one hand, the operator Φs behaves well with respect to duality. In fact, the analysis of k Δk+s T Ek just requires (in analogy with the T 1 theorem) to assume first that we have T 1 = 0. As pointed above, this kind of cancellation conditions are only necessary for Φs , since the presence of the terms id − Ek in Ψs produce suitable cancellations. However, this is exactly why the adjoint Ψs∗ =
∗ Δk+s T4·2 −k (id − Ek )
k
does not behave as expected. This leaves open the problem for p > 2. Appendix B. On Calderón–Zygmund decomposition B.1. Weighted inequalities Given a positive function f in L1 and λ ∈ R+ , let us consider the Calderón–Zygmund decomposition f = g + b associated to λ. As pointed out and well known, the most significant inequalities satisfied by these functions are
g(x)2 dx 2n λf 1
and
bj (x) dx 2f 1 , j Rn
Rn
where the bj ’s are the atoms in which b is decomposed. We already saw in Section 4 that these inequalities remain true for the diagonal terms of the noncommutative Calderón–Zygmund decomposition. However, we do not have at our disposal (see Section B.2) such inequalities for the off-diagonal terms. As we have explained in the introduction, our way to solve this lack has been to prove the off-diagonal estimates • ζ T ( k bk,s )ζ 1 αs f 1 , • ζ T ( k gk,s )ζ 22 βs λf 1 , for some fast decreasing sequences αs , βs . The proof of these estimates has exploited the properties of the projection ζ in conjunction with our localization results. We have therefore hidden the actual inequalities satisfied by the off-diagonal terms which are independent of the behavior of ζ T (·)ζ . Namely, we have (a) Considering the atoms bk,s = pk (f − fk+s )pk+s + pk+s (f − fk+s )pk
590
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
in boff =
k,s bk,s ,
we have for any positive sequence (αs )s1 s
αs bk,s 1
sαs f 1 .
s
k
(b) Considering the martingale differences gk,s = pk dfk+s qk+s−1 + qk+s−1 dfk+s pk in goff =
k,s gk,s ,
we have for any positive sequence (βs )s1
2 2 2 β g = β g βs2 λf 1 . s k,s k,s 2 s s
2
k
s
s
k
As the careful reader might have noticed, the proof of these estimates is implicit in our proof of Theorem A. It is still to be determined whether these estimates for the weights αs and βs are sharp. On the other hand, it is also possible to study weighted Lp estimates for the off-diagonal terms of the good part and p > 1. We have not pursued any of these lines. B.2. On the lack of a classical L2 estimate The pseudo-localization approach of this paper has been motivated by the lack of the key estimate g22 λf 1 in the noncommutative setting. Although we have not disproved such inequality so far, we end this paper by giving some evidences that it must fail. Recalling from Section 4 that the diagonal terms of g satisfy the estimate gd 22 λf 1 , it suffices disprove the inequality goff 22 λf 1 . By the original expression for goff , we have goff =
i
pi f i pj +
j
j
pi f j pj =
i<j
pk fk (1A − qk−1 ) + (1A − qk−1 )fk pk .
k
By orthogonality of the pk ’s and the tracial property, it is easily seen that 1 2 goff 22 = ϕ (1A − qk−1 )fk pk fk (1A − qk−1 ) λ λ k qk−1 fk pk fk qk−1 f k pk f k +2 = A + B. ϕ ϕ =2 λ λ k
k
By the tracial property B=
2 ϕ(pk fk qk−1 fk pk ). λ k
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
591
Moreover, we also have 1 12 f qk−1 f 2
k
∞
k
= qk−1 fk qk−1 ∞ 2n qk−1 fk−1 qk−1 ∞ 2n λ.
Thus, we find the inequality B cn
ϕ(pk fk ) cn f 1
k
and our problem reduces to disprove 2 f k pk λf 1 .
(B.1)
2
k
As in the argument given in Section 6 to find a bad-behaved noncommuting kernel, our motivation comes from a matrix construction. Namely, let A be the algebra of 2m × 2m matrices equipped with the standard trace tr and consider the filtration A1 , A2 , . . . , A2m , where As denotes the subalgebra spanned by the matrix units eij with 1 i, j s and the matrix units ekk with k > s. Let us set λ = 1 and define f=
2m
eij .
i,j =1
It is easily checked that q1 = χ(0,1] (f1 ) = 1A , q2 = χ(0,1] (q1 f2 q1 ) =
k>2 ekk ,
q3 = χ(0,1] (q2 f3 q2 ) = χ(0,1] (q2 ) = q2 , q4 = χ(0,1] (q3 f4 q3 ) = k>4 ekk , q5 = χ(0,1] (q4 f5 q4 ) = χ(0,1] (q4 ) = q4 , q6 = · · · . Hence, p2k−1 = 0 and p2k = e2k−1,2k−1 + e2k,2k . This gives 2m
f k pk =
k=1
m
f2k p2k =
m 2k
ej,2k−1 + ej,2k .
k=1 j =1
k=1
We have λ = 1 and it is clear that f 1 = 2m, while the L2 norm is 2m 2 m f k pk = 4k = 2m(m + 1). k=1
2
k=1
Therefore, if we let m → ∞ we see that (B.1) fails in this particular setting.
592
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
Problem. Adapt the construction given above to the usual von Neumann algebra A of operatorvalued functions f : Rn → M, equipped with the standard dyadic filtration. This would disprove (B.1) and thereby the inequality g22 λf 1 in the noncommutative setting. References [1] W.B. Arveson, Analyticity in operator algebras, Amer. J. Math. 89 (1967) 578–642. [2] J. Bourgain, Some remarks on Banach spaces in which martingale difference sequences are unconditional, Ark. Mat. 21 (1983) 163–168. [3] J. Bourgain, Extension of a result of Benedek, Calderón and Panzone, Ark. Mat. 22 (1984) 91–95. [4] J. Bourgain, Vector valued singular integrals and the H 1 –BMO duality, in: J.-A. Chao, W.A. Woyczynski (Eds.), Probability Theory and Harmonic Analysis, Dekker, 1986, pp. 1–19. [5] D.L. Burkholder, A geometrical characterization of Banach spaces in which martingale difference sequences are unconditional, Ann. Probab. 9 (1981) 997–1011. [6] D.L. Burkholder, A geometric condition that implies the existence of certain singular integrals of Banach space valued functions, in: Conference on Harmonic Analysis in Honor of Antoni Zygmund, Wadsworth, 1983, pp. 270– 286. [7] M. Christ, Weak type (1, 1) bounds for rough operators, Ann. of Math. 128 (1988) 19–42. [8] R. Coifman, G. Weiss, Analyse harmonique non-commutative sur certains espaces homogènes, Lecture Notes in Math., vol. 242, Springer, 1971. [9] I. Cuculescu, Martingales on von Neumann algebras, J. Multivariate Anal. 1 (1971) 17–27. [10] G. David, J.L. Journé, A boundedness criterion for generalized Calderón–Zygmund operators, Ann. of Math. 120 (1984) 371–397. [11] P.G. Dodds, T.K. Dodds, B. de Pagter, Noncommutative Banach function spaces, Math. Z. 201 (1989) 583–597. [12] J. Duoandikoetxea, Fourier Analysis, translated and revised from the 1995 Spanish original by D. Cruz-Uribe, Grad. Stud. Math., vol. 29, Amer. Math. Soc., 2001. [13] J. Duoandikoetxea, J.L. Rubio de Francia, Maximal and singular integral operators via Fourier transform estimates, Invent. Math. 84 (1986) 541–561. [14] T. Fack, H. Kosaki, Generalized s-numbers of τ -measurable operators, Pacific J. Math. 123 (1986) 269–300. [15] T. Figiel, Singular integral operators: A martingale approach, in: P.F.X. Müller, W. Schachermayer (Eds.), Geometry of Banach Spaces, in: London Math. Soc. Lecture Note Ser., vol. 158, Cambridge Univ. Press, 1989, pp. 95–110. [16] J. Garnett, P. Jones, BMO from dyadic BMO, Pacific J. Math. 99 (1982) 351–371. [17] A.M. Garsia, Martingale Inequalities: Seminar Notes on Recent Progress, Math. Lecture Notes Ser., Benjamin, 1973. [18] U. Haagerup, Non-commutative integration theory, unpublished manuscript, 1978, see also Haagerup’s lecture given at the Symposium in Pure Mathematics of the Amer. Math. Soc., Queens University, Kingston, Ontario, 1980. [19] U. Haagerup, Lp spaces associated with an arbitrary von Neumann algebra, in: Algèbres d’opérateurs et leurs applications en physique mathématique, CNRS, 1979, pp. 175–184. [20] A. Harcharras, Fourier analysis, Schur multipliers on S p and non-commutative Λ(p)-sets, Studia Math. 137 (1999) 203–260. [21] T. Hytönen, An operator-valued T b theorem, J. Funct. Anal. 234 (2006) 420–463. [22] T. Hytönen, L. Weis, A T 1 theorem for integral transformations with operator-valued kernel, J. Reine Angew. Math. 599 (2006) 155–200. [23] T. Hytönen, L. Weis, Singular convolution integrals with operator-valued kernels, Math. Z. 255 (2007) 393–425. [24] M. Junge, Doob’s inequality for non-commutative martingales, J. Reine Angew. Math. 549 (2002) 149–190. [25] M. Junge, Embedding of the operator space OH and the logarithmic ‘little Grothendieck inequality’, Invent. Math. 161 (2005) 225–286. [26] M. Junge, M. Musat, A non-commutative version of the John–Nirenberg theorem, Trans. Amer. Math. Soc. 359 (2007) 115–142. [27] M. Junge, J. Parcet, Rosenthal’s theorem for subspaces of noncommutative Lp , Duke Math. J. 141 (2008) 75–122. [28] M. Junge, J. Parcet, Operator space embedding of Schatten p-classes into von Neumann algebra preduals, Geom. Funct. Anal. (2008), doi: 10.1007/s00039-008-0660-0. [29] M. Junge, Q. Xu, Noncommutative Burkholder–Rosenthal inequalities, Ann. Probab. 31 (2003) 948–995. [30] M. Junge, Q. Xu, Noncomutative maximal ergodic theorems, J. Amer. Math. Soc. 20 (2007) 385–439. [31] M. Junge, J. Parcet, Q. Xu, Rosenthal type inequalities for free chaos, Ann. Probab. 35 (2007) 1374–1437.
J. Parcet / Journal of Functional Analysis 256 (2009) 509–593
593
[32] R.V. Kadison, J.R. Ringrose, Fundamentals of the Theory of Operator Algebras I, Grad. Stud. Math., vol. 15, Amer. Math. Soc., 1997; II, Grad. Stud. Math., vol. 16, Amer. Math. Soc., 1997. [33] R.A. Macías, C. Segovia, Lipschitz functions on spaces of homogeneous type, Adv. Math. 33 (1979) 257–270. [34] R.A. Macías, C. Segovia, A decomposition into atoms of distributions on spaces of homogeneous type, Adv. Math. 33 (1979) 271–309. [35] T. Mei, BMO is the intersection of two translates of dyadic BMO, C. R. Math. Acad. Sci. Paris 336 (2003) 1003– 1006. [36] T. Mei, Notes on matrix valued paraproducts, Indiana Univ. Math. J. 55 (2006) 747–760. [37] T. Mei, Operator valued Hardy spaces, Mem. Amer. Math. Soc. 188 (2007). [38] T. Mei, personal communication. [39] Y. Meyer, Ondelettes et Opérateurs II: Opérateurs de Calderón–Zygmund, Hermann, 1990. [40] M. Mitrea, Clifford Wavelets, Singular Integrals and Hardy Spaces, Lecture Notes in Math., vol. 1575, Springer, 1994. [41] M. Musat, Interpolation between non-commutative BMO and non-commutative Lp -spaces, J. Funct. Anal. 202 (2003) 195–225. [42] M. Musat, On the operator space UMD property for noncommutative Lp -spaces, Indiana Math. J. 55 (2006) 1857– 1891. [43] F. Nazarov, G. Pisier, S. Treil, A. Volberg, Sharp estimates in vector Carleson imbedding theorem and for vector paraproducts, J. Reine Angew. Math. 542 (2002) 147–171. [44] J. Parcet, N. Randrianantoanina, Gundy’s decomposition for non-commutative martingales and applications, Proc. London Math. Soc. 93 (2006) 227–252. [45] G. Pisier, Non-commutative vector valued Lp -spaces and completely p-summing maps, Astérisque 247 (1998). [46] G. Pisier, Introduction to Operator Space Theory, Cambridge Univ. Press, 2003. [47] G. Pisier, Q. Xu, Non-commutative martingale inequalities, Comm. Math. Phys. 189 (1997) 667–698. [48] G. Pisier, Q. Xu, Non-commutative Lp -spaces, in: W.B. Johnson, J. Lindenstrauss (Eds.), Handbook of the Geometry of Banach Spaces II, North-Holland, 2003, pp. 1459–1517. [49] N. Randrianantoanina, Hilbert transform associated with finite maximal subdiagonal algebras, J. Austral. Math. Soc. Ser. A 65 (1998) 388–404. [50] N. Randrianantoanina, Non-commutative martingale transforms, J. Funct. Anal. 194 (2002) 181–212. [51] N. Randrianantoanina, A weak type inequality for non-commutative martingales and applications, Proc. London Math. Soc. 91 (2005) 509–544. [52] N. Randrianantoanina, Conditioned square functions for non-commutative martingales, Ann. Probab. 35 (2007) 1039–1070. [53] J.L. Rubio de Francia, F.J. Ruiz, J.L. Torrea, Calderón–Zygmund theory for operator-valued kernels, Adv. Math. 62 (1986) 7–48. [54] E.M. Stein, Harmonic Analysis: Real Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Math. Ser., vol. 43, Princeton Univ. Press, 1993. [55] M. Takesaki, Theory of Operator Algebras I, Springer-Verlag, New York, 1979. [56] M. Terp, Lp spaces associated with von Neumann algebras, Math. Institute Copenhagen University, 1981. [57] A. Volberg, Matrix Ap weights via S-functions, J. Amer. Math. Soc. 10 (1997) 223–246. [58] Q. Xu, Analytic functions with values in lattices and symmetric spaces of measurable operators, Math. Proc. Cambridge Philos. Soc. 109 (1991) 541–563.
Journal of Functional Analysis 256 (2009) 594–602 www.elsevier.com/locate/jfa
Generalized Widder Theorem via fractional moments Ami Viselter 1 Department of Mathematics, Bar Ilan University, 52900 Ramat-Gan, Israel Received 4 April 2008; accepted 4 August 2008 Available online 3 September 2008 Communicated by D. Voiculescu
Abstract We provide a necessary and sufficient condition for the representability of a function as the classical multidimensional Laplace transform, when the support of the representing measure is contained in some generalized semi-algebraic set. This is done by employing a method of Putinar and Vasilescu [M. Putinar, F.-H. Vasilescu, Solving moment problems by dimensional extension, Ann. of Math. (2) 149 (3) (1999) 1087–1107] for the corresponding multidimensional moment problem. © 2008 Elsevier Inc. All rights reserved. Keywords: Multidimensional fractional moment problem; Widder Theorem; Laplace transform
0. Introduction A well-known theorem of Widder states that a necessary and sufficient condition for a function f : (0, ∞) → R to be representable in the form ∞ (∀x > 0)
f (x) =
e−xt dμ(t),
−∞
where μ is a positive measure over R, is that f (x) be continuous and of positive type (cf. [13, Ch. VI, §21]). This theorem has been generalized to the multidimensional case in several works, E-mail address: [email protected]. 1 This paper is a part of the author’s PhD thesis, written under the direction of Prof. Shmuel Kantorovitz in the
Department of Mathematics and Statistics, Bar-Ilan University, Israel. 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.08.002
A. Viselter / Journal of Functional Analysis 256 (2009) 594–602
595
using various methods (see Akhiezer [1], Devinatz [3], Shucker [10] and their references, to mention a few), and has many applications. The same is true for closely related representation theorems, such as Bernstein’s Theorem [2, Ch. 4] and the Paley–Wiener–Schwartz Theorem on the Fourier–Laplace transform [6, Ch. 7]. If, however, we wished to characterize those functions with a representing measure whose support is contained in some rather general, not necessarily convex, fixed set, these results would not be helpful. On the other hand, Putinar and Vasilescu used in [9] a method of dimensional extension to solve the multidimensional moment problem. In their paper, the moment problem is translated to the problem of representation of a certain linear functional, which is obtained by means of the spectral theory of selfadjoint operators, over some special Hilbert space. The bonus in their method is that it enables them to characterize moment sequences, whose representing measure’s support lies in a given semi-algebraic set. The connection between a moment problem and the corresponding Laplace transform representation problem has been successfully established in the past (e.g. in [12], and the references therein). In this note we modify Putinar and Vasilescu’s method of dimensional extension to obtain a generalized version of Widder’s Theorem, which characterizes the functions that can be represented by the multidimensional Laplace transform of a measure with support in a given (generalized) semi-algebraic set. Essentially, non-negative integral powers of the variables are replaced by non-negative rational powers. 1. Preliminaries Let R be an algebra of complex functions, such that f ∈ R for all f ∈ R (that is, R is selfadjoint). We say that a linear functional Λ over R is positive semi-definite if Λ(|f |2 ) 0 for each f ∈ R. When this is the case, one can define the semi-inner product (f, g) := Λ(f g). Thus, if N = {f ∈ R: Λ(|f |2 ) = 0}, then R/N is an inner-product space. Hence, its completion, H, is a complex Hilbert space. For simplicity, we often write r instead of r + N for elements r ∈ R/N . The standard notations R+ = [0, ∞), Q+ = R+ ∩ Q, etc. are used. Fix an n ∈ N. For t = (t1 , . . . , tn ), α = (α1 , . . . , αn ) ∈ Rn+ , we write t α for t1α1 · · · tnαn . We let Pn denote the set of all complex polynomials with n real variables. By Qn we shall denote the complex algebra of all “fractional polynomials” of positive rational exponents and n variables. That is, Qn is the set of all of the functions in the form Rn+ t → α∈Qn+ aα t α , where the aα ’s are complex, and differ from zero only for a finite number of indices α. Let A be a subsemigroup of Qn+ . A family of complex numbers δ = (δα )A induces the linear functional Lδ over the subalgebra of Qn generated by {t α : α ∈ A}, defined by Lδ (t α ) = δα for all α ∈ A. We say that δ is positive semi-definite if the functional Lδ is positive semi-definite. For the rest of the section, H denotes an arbitrary complex Hilbert space. Lemma 1.1. Let A be a positive selfadjoint operator over H, and let q1 , q2 be positive real numbers. Then there exists a unique positive selfadjoint operator B, so that B q2 = Aq1 , namely B = Aq1 /q2 . Proof. Let E(·; A) denote the resolution of the identity of the selfadjoint operator A. By [4, Theorem XII.2.9], a positive operator B satisfies the theorem’s statement if and only if for every Borel set δ ⊆ R+ , E δ 1/q2 ; B = E δ; B q2 = E δ; Aq1 = E δ 1/q1 ; A .
(1.1)
596
A. Viselter / Journal of Functional Analysis 256 (2009) 594–602
Since the mapping δ → δ q2 is bijective from the set of all Borel subsets of R+ into itself, (1.1) is equivalent to that E(δ; B) = E δ q2 /q1 ; A
(1.2)
for all Borel sets δ ⊆ R+ . But by the same theorem from [4], there exists a unique positive selfadjoint operator B that satisfies (1.2), which is B = Aq1 /q2 . 2 We now state two results from [9]. Proposition 1.2. (See [9, Proposition 2.1], originally from [7,8].) Let T1 , . . . , Tn be symmetric operators in H. Assume that there exists a dense linear space D ⊆ nj,k=1 D(Tj Tk ) such that Tj Tk x = Tk Tj x for all x ∈ D, j = k, j, k = 1, . . . , n. If the operator (T12 + · · · + Tn2 )|D is essentially selfadjoint, then the operators T1 , . . . , Tn are essentially selfadjoint, and their canonical closures T1 , . . . , Tn commute. Lemma 1.3. (See [9, Lemma 2.2].) Let A be a positive densely defined operator in H, such that AD(A) ⊆ D(A). Suppose that I + A is bijective on D(A). Then A is essentially selfadjoint. 2. Generalized Widder Theorem Let p = (p1 , . . . , pm ), where pk are real fractional polynomials in Qn . For this fixed set of polynomials, let θp : Rn+ → C be defined as −1 θp (t) := 1 + t12 + · · · + tn2 + p1 (t)2 + · · · + pm (t)2 . We denote by R the complex algebra generated by Qn and the function θp . The following is the main operator-theoretic result, leading to the moments theorem to follow. Theorem 2.1. Let Λ be a positive semi-definite functional over R. Then there exists a unique representing measure for Λ. The support of that measure is contained in Rn+ . Moreover, if 2 Λ(pk |r| m) 0−1for all r ∈ R, 1 k m, then the support of that (unique) measure is a subset of k=1 pk (R+ ). Proof. Let H be the Hilbert space generated by Λ, as explained in Section 1. For 1 i n, 1 j m, we define the operators Ti , Pj over R/N by Ti : r + N → ti r + N ,
Pj : r + N → pj r + N .
Let B be the operator B := T12 + · · · + Tn2 + P12 + · · · + Pm2 . Then B : R/N → R/N is a positive 2 operator, since for all r ∈ R, (Br, r) = ni=1 Λ(|ti r|2 ) + m j =1 Λ(|pj r| ) 0, by the positivity of Λ. Moreover, I + B is bijective, since for all r ∈ R, (I + B)u = r for some u ∈ R if and only if u = θp r. Therefore, by Lemma 1.3, B is essentially selfadjoint. Thus, by Proposition 1.2, the operators Ti and Pj are essentially selfadjoint for all 1 i n, 1 j m. Moreover, the selfadjoint operators A1 := T1 , . . . , An := Tn commute, and thus have a common resolution of the identity, E (cf. [11, Ch. IV, Theorem 10.3]). Set A := (A1 , . . . , An ). For r ∈ R, r(A) will denote
A. Viselter / Journal of Functional Analysis 256 (2009) 594–602
597
the normal operator Rn r(t)E(dt), and for q ∈ Qn+ , Aq will stand for f (A) where f (t) = t q . + We define the operators Ti (qi ) : R/N → R/N , 1 i n, by q
Ti (qi ) : r + N → ti i r + N , and set T (q) := T1 (q1 ) · · · Tn (qn ). Claim 1. For all q ∈ Qn+ , T (q) ⊆ Aq . To prove the claim, we first notice that T (q) is positive for q ∈ Qn+ , as Λ(t q |r|2 ) = Λ(|t q/2 r|2 ) 0 for all r ∈ R. Let q = ( k11 , . . . , knn ), where k1 , . . . , kn ∈ N ∪ {0}, 1 , . . . , n ∈ N.
Fix an 1 i n. Since the operator Ti ( 1i ) is positive, it has an (a priori, not necessarily unique)
positive selfadjoint extension, Ai ( 1i ). Now, observe that Ti = Ti ( 1i )i ⊆ Ai ( 1i )i . But Ti is
essentially selfadjoint and Ai ( 1i )i is selfadjoint, which implies that Ai = Ti = Ai ( 1i )i . There1/i
fore, by the uniqueness part of Lemma 1.1, Ti ( 1i ) ⊆ Ai ( 1i ) = Ai Lemma 1.1, and the fact that (∀r1 , r2 ∈ R)
. Hence, once again by
r1 (A)r2 (A) ⊆ (r1 r2 )(A)
(2.1)
(which follows readily from [11, Ch. IV, Theorem 10.3]), T (q) = T1
1 1
k1
· · · Tn
1 n
kn
1/ k 1/ k k / k / ⊆ A1 1 1 · · · An n n = A11 1 · · · Ann n ⊆ Aq ,
(2.2)
and the claim is proved. Claim 2. For all r ∈ R, Λ(r) =
r(t) E(dt)(1 + N ), 1 + N .
(2.3)
Rn+
In order to prove the claim, fix an r ∈ R. Let the operator r(T ) : R/N → R/N be the operator of multiplication by r. We shall show that r(T ) ⊆ r(A). By linearity and (2.1), it is sufficient to prove this for r(t) = t q , q ∈ Qn+ , and for r = θp . The first case is exactly Claim 1, since r(T ) = T (q) and r(A) = Aq . As for the case r = θp , it follows from the fact that θp−1 (T ) ⊆ θp−1 (A), and so for all f ∈ R/N , θp (A)f = θp (A)[θp−1 (T )θp (T )]f = θp (A)θp−1 (A)θp (T )f = θp (T )f (by (2.1)). Finally, to prove (2.3), we note that by the Spectral Theorem, Λ(r) = (r + N , 1 + N ) = r(T )(1 + N ), 1 + N = r(A)(1 + N ), 1 + N = r(t) E(dt)(1 + N ), 1 + N Rn+
(the domain of integration is Rn+ since the operators A1 , . . . , An are positive), and the claim is proved.
598
A. Viselter / Journal of Functional Analysis 256 (2009) 594–602
Consequently, the (positive) Borel measure μ over Rn defined by μ(·) = (E(·)(1 + N ), 1 + N ) is a representing measure for Λ, whose support lies in Rn+ . We have thus proved the existence part of the theorem. The uniqueness of μ is proved as in [9], using an argument taken from [5]. Let us assume that there exists another positive measure ν over Rn+ , that represents the functional Λ. It is clear that when this is the case, R/N can be identified as a subspace of the Hilbert space L2 (ν). Hence, H can be identified as the closure of R/N in L2 (ν). For all 1 j n, let us now define the selfadjoint operators Hj over L2 (ν) by Hj f := tj f . Denote the spectral measure of Hj by Ej . Since the operators H1 , . . . , Hn commute, they have a joint spectral measure, EH . Obviously, Tj ⊆ Hj for all j . Since the operators Hj are closed, Aj ⊆ Hj for all j . Therefore, R(ζ ; Aj ) ⊆ R(ζ ; Hj ) for all ζ ∈ C\R, and so R(ζ ; Hj ) leaves H invariant, whence we conclude (cf. [4, Theorem XII.2.10]) that Ej also leaves H invariant for each 1 j n. Thus, as EH (B1 × · · · × Bn ) = E1 (B1 ) · · · En (Bn ) for all Borel sets B1 , . . . , Bn in R, EH leaves H invariant as well. In particular, for each Borel set B in Rn , IB = EH (B)1 ∈ H (where IB is the indicator function of B over Rn ). Since the simple functions are dense in L2 (ν), we infer that H = L2 (ν), and so Aj = Hj for each 1 j n. In particular, E = EH , and so for each Borel set B in Rn , by the definition of μ, μ(B) = E(B)(1 + N ), 1 + N = EH (B)1, 1 =
IB dν = ν(B), Rn
and the proof of the uniqueness of μ is completed. Assume now that Λ(pk |r|2 ) 0 for all r ∈ R and 1 k m. This condition is equivalent to the operators P1 , . . . , Pm being positive. We recall that these operators are essentially selfadjoint. But for all such k, Pk ⊆ pk (A) by Claim 1, and pk (A) is selfadjoint; thus, Pk = pk (A) is a positive selfadjoint operator. Equivalently, its spectral measure is supported by R+ . But the spectral measure of pk (A) is Fk (δ) = E(pk−1 (δ)). Hence, E itself is supported by pk−1 (R+ ). Since that is −1 true for all 1 k m, the support of E is therefore a subset of m k=1 pk (R+ ). 2 Lemma 2.2. Let ϑ ∈ Pn be such that ϑ(t) > 0 for all t ∈ Rn , and let p(t, s) ∈ Pn+1 (t ∈ Rn , s ∈ R) be such that p(t, ϑ −1 (t)) ≡ 0. Then there exists a complex polynomial q ∈ Pn+1 such that (∀t, s)
p(t, s) = q(t, s) · sϑ(t) − 1 .
Proof. This is a simple generalization of [9, Lemma 2.3]; simply replace their θp by ϑ −1 . We omit the details. 2 n the complex algebra generated by Qn and the algebra of all comDefinition 2.3. Denote by Q plex polynomials with one positive real variable. Its elements will take the form p(t, s), t ∈ Rn+ , s ∈ R+ . n → R be the mapping defined by p(t, s) → p(t, θp (t)). Then ρ is a Proposition 2.4. Let ρ : Q surjective algebras homomorphism, whose kernel is the ideal generated by the function σ (t, s) = sθp (t)−1 − 1.
A. Viselter / Journal of Functional Analysis 256 (2009) 594–602
599
Proof. We first note that ρ is indeed well defined, since θp (t) ∈ R+ for all t ∈ Rn+ . It is clearly a surjective algebras homomorphism. Assume p ∈ ker(ρ), that is, p(t, θp (t)) = 0 for all t ∈ Rn+ . For 1 j n, we let cj denote the l.c.m. of all of the denominators of the exponents of tj in the 1/cj
polynomial p. The mappings uj = tj tj by
c ujj
are bijective mappings from R+ onto itself. Replacing
in the above equality yields ∀u ∈ Rn+
p uc , θp uc = 0.
(2.4)
The expression on the left side of (2.4), after the reduction of the fractions in the exponents of the uj ’s, becomes a (not fractional) polynomial in u = (u1 , . . . , un ). Hence, (2.4) is true (as equality of polynomials) for all u ∈ Rn , and by Lemma 2.2, there exists a q ∈ Pn+1 such that
−1 −1 . ∀u ∈ Rn+ , s ∈ R+ p uc , s = q(u, s) sθp uc q (t, s) = q(t 1/c , s), we conclude that We can now replace u by t 1/c , and by defining
p(t, s) = q (t, s) sθp (t)−1 − 1 n , as wanted. q ∈Q for all t ∈ Rn+ , s ∈ R+ , where
2
Definition 2.5. Let γ = (γα )α∈Rn+ be a family of non-negative numbers. (1) We say that γ is continuous if the function α → γα is continuous (as a function from Rn+ to R+ ). (2) We say that γ is an (n-dimensional) fractional moments family if there exists a positive Borel measure μ over Rn+ , such that ∀α ∈ Rn+
γα =
t α dμ.
(2.5)
Rn+
Note that (2.5) is equivalent to the (multidimensional) Laplace representation ∀α ∈ Rn+ γα = e−α·s dν(s) Rn
obtained by the change of variable ti = e−si . The following is the main theorem, whose proof is almost identical to that of Theorem 2.7 in [9], basing on our Theorem 2.1 and Proposition 2.4 instead of the parallel ones in [9], and using Lebesgue’s Dominated Convergence Theorem to derive (2.5) for all of Rn+ . For the sake of completeness, we include the details. Theorem 2.6. Let γ = (γα )α∈Rn+ be a continuous family of non-negative numbers. Let p1 , . . . , pm ∈ Qn , pk (t) = ξ ∈Ik akξ t ξ (Ik ⊆ Qn+ is finite) for k = 1, 2, . . . , m. Then γ is a frac −1 tional moments family with a representing measure whose support is a subset of m k=1 pk (R+ ) if and only if there exists a positive semi-definite family
600
A. Viselter / Journal of Functional Analysis 256 (2009) 594–602
δ = (δ(α,β) )(α,β)∈Qn+ ×Z+ that satisfies: ∈ Qn+ . (1) δ(α,0) = γα for all α (2) δ(α,β) = δ(α,β+1) + nj=1 δ(α+2ej ,β+1) + m k=1 ξ,η∈Ik akξ akη δ(α+ξ +η,β+1) for all (α, β) ∈ n Q+ × Z+ . (3) The families ( ξ ∈Ik akξ δ(α+ξ,β) )(α,β)∈Qn+ ×Z+ are positive semi-definite for all k = 1, . . . , m. Moreover, the representing measure of γ (with the properties mentioned above) is unique if and only if the family δ is unique. Proof. Necessity. Assume that γ isa fractional moments family with a representing measure μ, −1 whose support is a subset of E := m k=1 pk (R+ ). We define the family δ by ∀(α, β) ∈ Qn+ × Z+ δ(α,β) :=
t α θp (t)β dμ.
(2.6)
E
Then δ is a positive semi-definite family, that satisfies (1). (2) is a result of the obvious equality
θp (t) 1 + t12 + · · · + tn2 + p1 (t)2 + · · · + pm (t)2 − 1 t α θp (t)β dμ = 0,
E
which is true for all α ∈ Qn+ , β ∈ Z+ . Finally, (3) is true since
2 pk (t) p t, θp (t) dμ 0
E
n , 1 k m. for all p ∈ Q Sufficiency. Let δ be as in the theorem’s statement, and the algebra R be defined as in the beginning of this section. We define the linear functional Λ over R by Λ(r) = Lδ (p) n , and p ∈ Q n is such that for all r ∈ R, where Lδ is the linear functional induced by δ over Q n ∼ n /I, where r(t) = p(t, θp (t)) for all t ∈ R+ . Λ is well defined, since by Proposition 2.4, R = Q −1 n , generated by the element sθp (t) − 1; and indeed, by (2), (Lδ )|I = 0. I is the ideal in Q Thus, Λ is a well-defined positive semi-definite mapping on R. From (3) we deduce that n , 1 k m, hence Λ(pk |r|2 ) 0 for all r ∈ R, 1 k m. Lδ (pk |p|2 ) 0 for all p ∈ Q By virtue of Theorem 2.1, there exists a unique representing measure μ for Λ, whose support is a subset of E. Particularly, by (1), γα = δ(α,0) =
t α dμ E
(2.7)
A. Viselter / Journal of Functional Analysis 256 (2009) 594–602
601
for all α ∈ Qn+ . But since the family γ is continuous, Lebesgue’s Dominated Convergence Theorem implies that γα = E t α dμ for all α ∈ Rn+ , that is, γ is a fractional moments sequence, as wanted. Assume that the family δ, that satisfies the conditions in the theorem’s statement, is unique. Let μ1 , μ2 be two representing measures for γ . By the uniqueness of δ, Eq. (2.6) and the dis α n β dμ = α β cussion that follows, t θ (t) p 1 E E t θp (t) dμ2 for each α ∈ Q+ , β ∈ Z+ . Therefore, E r dμ1 = E r dμ2 for all r ∈ R, and by the uniqueness part of Theorem 2.1, it follows that μ1 = μ2 . Conversely, assume that μ is unique. Suppose that both δ1 , δ2 satisfy the conditions in the theorem’s statement. As explained above, δ1 , δ2 induce the positive semi-definite linear functionals Λ1 , Λ2 , respectively, over R, which, in turn, have the unique representing measures μ1 , μ2 , respectively (by Theorem 2.1). Both measures represent γ as a fractional moments family, and so, by the uniqueness of μ, μ1 = μ2 , hence Λ1 = Λ2 . Finally, for each α ∈ Qn+ , β ∈ Z+ , (δ1 )(α,β) = Λ1 (t α θp (t)β ) = Λ2 (t α θp (t)β ) = (δ2 )(α,β) , that is, δ1 = δ2 . 2 Remark 2.7. Throughout this section, Qn+ might have been replaced, e.g., by
k A := l : k, l ∈ N ∪ {0} 2
n .
We are limited by the mere requirements that A be a subsemigroup of Qn+ which contains (N ∪ {0})n , and that a2 ∈ A for all a ∈ A (the latter is used in the proof of Claim 1 of Theorem 2.1). Such A is, of course, dense in Rn+ . As a concrete demonstration, we have the following immediate corollary of Theorem 2.6. Corollary 2.8. Denote F := {t ∈ R2+ : t12 t2 }. In order for a continuous 2-dimensional family (γα )α∈R2 of non-negative numbers to be representable in the form +
γα =
t α dμ F
where μ is a non-negative measure over F , it is necessary and sufficient that there exist a positive semi-definite family (δ(α,β) )(α,β)∈Q2 ×Z+ , such that the following conditions hold: +
(1) δ(α,0) = γα for all α ∈ Q2+ . (2) δ(α,β) = δ(α,β+1) + δ(α+2e1 ,β+1) + 2δ(α+2e2 ,β+1) + δ(α+4e1 ,β+1) − 2δ(α+2e1 +e2 ,β+1) for all (α, β) ∈ Q2+ × Z+ . (3) The family (δ(α+e2 ,β) − δ(α+2e1 ,β) )(α,β)∈Q2 ×Z+ is positive semi-definite. +
Acknowledgments The author would like to thank his thesis advisor, Prof. Shmuel Kantorovitz, for his encouragement, many valuable discussions, and for having suggested the topic of this paper. He would also like to thank the referee for his/her useful remarks.
602
A. Viselter / Journal of Functional Analysis 256 (2009) 594–602
References [1] N.I. Akhiezer, The Classical Moment Problem and Some Related Questions in Analysis, Hafner Publishing Co., New York, 1965. [2] S. Bochner, Harmonic Analysis and the Theory of Probability, Univ. of California Press, Berkeley and Los Angeles, 1955. [3] A. Devinatz, The representation of functions as Laplace–Stieltjes integrals, Duke Math. J. 22 (1955) 185–191. [4] N. Dunford, J.T. Schwartz, Linear Operators, Part II, Interscience Publishers, New York, 1968. [5] B. Fuglede, The multidimensional moment problem, Expo. Math. 1 (1) (1983) 47–65. [6] L. Hörmander, The Analysis of Linear Partial Differential Operators I, Grundlehren Math. Wiss., Springer-Verlag, Berlin, Heidelberg, 1983. [7] E. Nelson, Analytic vectors, Ann. of Math. (2) 70 (3) (1959) 572–615. [8] A.E. Nussbaum, Commutativity theorem for unbounded operators in Hilbert space, Trans. Amer. Math. Soc. 140 (1969) 485–491. [9] M. Putinar, F.-H. Vasilescu, Solving moment problems by dimensional extension, Ann. of Math. (2) 149 (3) (1999) 1087–1107. [10] D.S. Shucker, Extensions and generalizations of a theorem of Widder and of the theory of symmetric local semigroups, J. Funct. Anal. 58 (3) (1984) 291–309. [11] F.-H. Vasilescu, Analytic Functional Calculus and Spectral Decompositions, Editura Academiei/D. Reidel Publ. Co., Bucharest/Dordrecht, 1982. [12] D.V. Widder, Necessary and sufficient conditions for the representation of a function by a doubly infinite Laplace integral, Bull. Amer. Math. Soc. 40 (1934) 321–326. [13] D.V. Widder, The Laplace Transform, Princeton Univ. Press, Princeton, 1941.
Journal of Functional Analysis 256 (2009) 603–634 www.elsevier.com/locate/jfa
A C ∗ -algebra of geometric operators on self-similar CW-complexes. Novikov–Shubin and L2-Betti numbers ✩ Fabio Cipriani a , Daniele Guido b,∗ , Tommaso Isola b a Dipartimento di Matematica, Politecnico di Milano, piazza Leonardo da Vinci 32, 20133 Milano, Italy b Dipartimento di Matematica, Università di Roma “Tor Vergata”, I-00133 Roma, Italy
Received 2 March 2007; accepted 9 October 2008 Available online 8 November 2008 Communicated by Alain Connes
Abstract A class of CW-complexes, called self-similar complexes, is introduced, together with C ∗ -algebras Aj of operators, endowed with a finite trace, acting on square-summable cellular j -chains. Since the Laplacian j belongs to Aj , L2 -Betti numbers and Novikov–Shubin numbers are defined for such complexes in terms of the trace. In particular a relation involving the Euler–Poincaré characteristic is proved. L2 -Betti and Novikov–Shubin numbers are computed for some self-similar complexes arising from self-similar fractals. © 2008 Elsevier Inc. All rights reserved. Keywords: Self-similar CW-complexes; Fractal graphs; Homological Laplacians; Geometric operators; Traces on amenable spaces; L2 -invariants
✩
This work has been partially supported by GNAMPA, MIUR and by the European Networks “Quantum Spaces—Noncommutative Geometry” HPRN-CT-2002-00280, and “Quantum Probability and Applications to Physics, Information and Biology”. * Corresponding author. E-mail addresses: [email protected] (F. Cipriani), [email protected] (D. Guido), [email protected] (T. Isola). 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.10.013
604
F. Cipriani et al. / Journal of Functional Analysis 256 (2009) 603–634
1. Introduction In this paper we study the possibility of extending the definition of some L2 -invariants, like the L2 -Betti numbers and Novikov–Shubin numbers, to geometric structures which are not coverings of compact spaces. The first attempt in this sense is due to John Roe [29], who defined a trace on finite propagation operators on amenable manifolds, allowing the definition of L2 -Betti numbers on these spaces. However such trace was defined in terms of a suitable generalised limit, hence the corresponding L2 -Betti and Novikov–Shubin numbers depend, in principle, on this generalised limit procedure. Here we show that, on spaces possessing a suitable self-similarity, it is possible to select a natural C ∗ -algebra of operators, generated by operators with finite propagation and locally commuting with the transformations giving the self-similar structure, on which a Roe-type trace is well defined, independently of any generalised limit. The theory of L2 -invariants was started by M.F. Atiyah in a celebrated paper [1], where he observed that on covering manifolds Γ → M → X, a trace on Γ -periodic operators may be defined, called Γ -trace, with respect to which the Laplace operator has compact resolvent. Replacing the usual trace with the Γ -trace, he defined the L2 -Betti numbers and proved an index theorem for covering manifolds. Based on this paper, Novikov and Shubin [28] observed that, since for noncompact manifolds the spectrum of the Laplacian is not discrete, new global spectral invariants can be defined, which necessarily involve the density near zero of the spectrum. L2 -Betti numbers were proved to be Γ -homotopy invariants by Dodziuk [7], whereas Novikov–Shubin numbers were proved to be Γ -homotopy invariants by Gromov–Shubin [12]. L2 -Betti numbers (depending on a generalised limit procedure) were subsequently defined for open manifolds by Roe, and were proved to be invariant under quasi-isometries [30]. The invariance of Novikov–Shubin numbers was proved in [13]. The basic idea of the present analysis is the notion of self-similar CW-complex, which is defined as a complex endowed with a natural exhaustion {Kn } in such a way that Kn+1 is a union (with small intersections) of a finite number of copies of Kn . The identification of the different copies of Kn in Kn+1 gives rise to many local isomorphisms on such complexes. Then we consider finite propagation operators commuting with these local isomorphisms up to boundary terms, and call them geometric operators. Geometric operators generate a C ∗ -algebra Aj , containing the j -Laplace operator, on the space of 2 -chains of j -cells, for any j ranging from 0 to the dimension of the complex. For any operator T in this C ∗ -algebra, we consider the sequence of the traces of T En , renormalised with the volume of the j -cells of Kn , where En denotes the projection onto the space generated by the j -cells of Kn . Such a sequence is convergent, and the corresponding functional is indeed a finite trace on Aj . By means of these traces, L2 -Betti numbers and Novikov–Shubin numbers are defined. In the Γ -covering case, L2 -Betti numbers are defined as Γ -dimensions of the kernels of Laplace operators, namely as Γ -traces of the corresponding projections. This is not allowed in our framework. Indeed, our traces being finite, and the C ∗ -algebras being weakly dense in the algebra of all bounded linear operators, our traces cannot extend to the generated von Neumann algebras. In particular they are not defined on the spectral projections of the Laplace operators. Therefore we define L2 -Betti numbers as the infimum of the traces of all continuous functional calculi of the Laplacian, with functions taking value 1 at 0, namely L2 -Betti numbers are defined as the “external measure” of the spectral projections of the Laplace operators.
F. Cipriani et al. / Journal of Functional Analysis 256 (2009) 603–634
605
Since we are in an infinite setting, the Euler–Poincaré characteristic is naturally defined as a renormalised limit of the Euler–Poincaré characteristic of the truncations Kn of the complex. We prove that such characteristic coincides with the alternating sum of the L2 -Betti numbers. An analogous result, though obtained with a different proof, for amenable simplicial complexes is contained in [9]. Concerning the stability of the L2 -Betti and the Novikov–Shubin numbers, here we prove that as far as 1-dimensional CW-complexes are considered, and in particular prefractal graphs determined by nested fractals, a result by Hambly and Kumagai applies [20], implying that Novikov–Shubin numbers are invariant under rough isometries. We then show that in some cases L2 -Betti and Novikov–Shubin numbers can be computed, relying on results of several authors concerning random walks on graphs. In particular, it turns out that the Novikov–Shubin numbers of some prefractal complexes coincide with the spectral dimensions of the corresponding fractals, thus strengthening the interpretation of such numbers as (asymptotic) spectral dimensions given in [13]. Our framework was strongly influenced by the approach of Lott and Lück [25], in particular we also consider invariants relative to the boundary, however we are not able to prove the Poincaré duality shown in [25]. Approximation results in terms of finite subgraphs are contained in [8,31]. When this article was already completed, a paper of Elek [10] appeared on the arXive, containing some ideas and results similar to ours. Starting from results of Lenz and Stollmann [24] on Delone sets, Elek introduces the notion of abstract quasi-crystal graph, and proves that there is a well-defined trace on the algebra of pattern invariant operators. It is remarkable that such different starting points produced two notions with such a vast intersection (indeed Elek notion is more general when graphs are concerned, while our self-similar CW-complexes describe also higher dimensional objects). The paper is organised as follows. In Section 2 we recall some notions from the theory of CW-complexes and introduce the basic operators. Section 3 introduces the notion of local isomorphisms of CW-complexes and the algebra of geometric operators. The notion of self-similar CW-complex is given in Section 4, and a finite trace on geometric operators is constructed. In Section 5 we introduce L2 -Betti and Novikov–Shubin numbers for the above setting, and prove the mentioned result on the Euler–Poincaré characteristic. Section 6 focuses on the subclass of self-similar CW-complexes given by prefractal complexes, and on some properties of the associated Laplacians. Computations of the Novikov–Shubin numbers for fractal graphs in terms of transition probabilities, together with an invariance result under rough isometries are discussed in Sections 7 and 8, and the top-dimensional relative Novikov–Shubin number is computed for two examples of 2-dimensional CW-complexes. In closing this introduction, we note that the C ∗ -algebra and the trace for self-similar graphs constructed in this paper, are used in [19] to study the Ihara zeta function for fractal graphs and in [6] to study BEC on inhomogeneous amenable networks. The results contained in this paper were announced in the conferences C ∗ -Algebras and Elliptic Theory, Bedlewo, 2006, and 21st International Conference on Operator Theory, Timisoara, 2006. 2. CW-complexes and basic operators In this paper we shall consider a particular class of infinite CW-complexes, therefore we start by recalling some notions from algebraic topology, general references being [26,27]. A CW-
606
F. Cipriani et al. / Journal of Functional Analysis 256 (2009) 603–634
complex M of dimension p ∈ N is a Hausdorff space consisting of a disjoint union of (open) j cells of dimension j ∈ {0, 1, . . . , p} such that: (i) for each j -cell σα , there is a continuous map j j fα : {x ∈ Rj : x 1} → M that is a homeomorphism of {x ∈ Rj : x < 1} onto σα , and maps j {x ∈ R : x = 1} into a finite union of cells of dimension < j ; (ii) a set A ⊂ M is closed in M j j j j iff A ∩ σ α is closed in σ α , for all j , α, where σ α denotes the closure of σα in M. Let us denote j j j by σ˙ α = fα ({x ∈ Rj : x = 1}) the boundary of σα , for all j , α. A CW-complex is regular if j fα is a homeomorphism, for all j , α. j We denote by Ej (M) := {σα : α ∈ Aj }, j = 0, 1, . . . , p, the family of j -cells, and by M j := j j j −1 , Z) is the (abelian) group of k=0 Ek (M), the j -skeleton of M. Then Cj (M) := Hj (M , M j j -dimensional cellular chains, and is generated by the class of σα , α ∈ Aj . Let ∂j : Cj (M) → Cj −1 (M) be the boundary operator, which is the connecting homomorphism of the homology sequence of the triple (M j , M j −1 , M j −2 ). Let us choose an orientation of M, that is, a basis j j { σα : α ∈ Aj } of Cj (M), j ∈ {0, . . . , p}, where each σα is (up to sign) the class of one (open) j j j -cell. We will usually identify the algebraic cell σα with the geometric cell σα , and denote by j j −σα the cell σα with the opposite orientation. Then the action of ∂j on the chosen basis is given j j j −1 j −1 j j −1 by ∂j σα = β∈Aj −1 [σα : σβ ]σβ , where [σα : σβ ] ∈ Z depends on the chosen orientation j
j −1
and is called incidence number. If M is regular, [σα : σβ
j −1 σβ
j
j −1
] ∈ {−1, 0, 1}, and [σα : σβ
]=0⇔
j ∩ σα
= ∅. Let us recall that the orientation of the zero-cells is chosen in such a way that, for any 1-cell σ 1 , α [σ 1 , σα0 ] = 0. In the following we will consider only regular CW-complexes,unless otherwise stated. A Hilbert norm on Cj (M) ⊗Z C is then defined as c2 := i |ci |2 when c = i ci · σi ∈ (2) Cj (M) ⊗Z C. The Hilbert space Cj (M) ≡ 2 (Ej M) is the completion of Cj (M) ⊗Z C under this norm. (2) (2) We can extend ∂j to a densely defined linear operator Cj (M) → Cj −1 (M). Then the halfLaplace operators j ± are j + := ∂j +1 ∂j∗+1 , j − := ∂j∗ ∂j and the Laplace operators are j := j + + j − . These are operators on 2 (Ej M) densely defined on Cj (M) ⊗Z C. Let us observe that ∂j is a bounded operator under some condition. Definition 2.1 (Bounded complex). Let M be a regular CW-complex, denote by Vj+ := Vj−
:=
sup τ ∈ Ej +1 (M): τ˙ ⊃ σ ,
σ ∈Ej (M)
sup ρ ∈ Ej −1 (M): ρ ⊂ σ˙ ,
σ ∈Ej (M)
where | · | denotes the cardinality. We say that M is a bounded complex if Vj± < ∞, for all j . Let us note that for a 1-dimensional complex, i.e. a graph, bounded means with bounded degree.
F. Cipriani et al. / Journal of Functional Analysis 256 (2009) 603–634
607
Lemma 2.2. Let M be a bounded regular CW-complex. Then ∂j : 2 (Ej M) → 2 (Ej −1 M) is bounded. Proof. If c =
i ci
· σi , setting α ∼ β if there is ρ ∈ Ej −1 (M) such that ρ ⊂ σ˙ α ∩ σ˙ β , we have ∂j c2 =
cα · cβ · (∂j σα , ∂j σβ )
α∼β
|cα | · |cβ | · (∂j σα , ∂j σβ )
α∼β
1 Vj− |cα |2 + |cβ |2 2 α∼β
2 Vj− Vj+−1 c2 . Indeed (∂j σα , ∂j σβ )
[σα : ρ] · [σβ : ρ] ρ⊂σ˙ α ∩σ˙ β
ρ ∈ Ej −1 : ρ ⊂ σ˙ α ∩ σ˙ β Vj− , while, for any α ∈ Aj , |{β ∈ Aj : β ∼ α}| Vj− Vj+−1 .
2
Lemma 2.3. ∂j∗+1 σ =
[τ : σ ]τ.
τ ∈Ej +1 (M)
Proof. Indeed, with τ ∈ Ej +1 (M),
τ, ∂j∗+1 σ = (∂j +1 τ, σ ) =
[τ : σ ](σ , σ ) = [τ : σ ].
2
σ ∈Ej (M)
Proposition 2.4. Let M be a bounded regular CW-complex. Then, for σ, σ ∈ Ej (M), we have (σ, j + σ ) =
[τ : σ ][τ : σ ],
τ ∈Ej +1 (M)
and (σ, j − σ ) =
[σ : τ ][σ : τ ].
τ ∈Ej −1 (M)
In particular,
608
F. Cipriani et al. / Journal of Functional Analysis 256 (2009) 603–634
(σ, j + σ ) = τ ∈ Ej +1 (M): τ˙ ⊃ σ , (σ, j − σ ) = τ ∈ Ej −1 (M): τ ⊂ σ˙ . Proof. Straightforward computation.
2
Remark 2.5. It follows that j ± does not depend on the orientation of the (j ± 1)-cells, but only on the orientation of the j -cells. 3. Local isomorphisms and geometric operators In this section, we define geometric operators and prove that the Laplacians (absolute or relative to the boundary subcomplex) are geometric. Definition 3.1 (Combinatorial distance). Let M be a connected, regular, bounded CW-complex. Let σ, σ be distinct cells in Ej (M). We set (i) d− (σ, σ ) = 1, if there is ρ ∈ Ej −1 (M) such that ρ ⊂ σ˙ ∩ σ˙ , (ii) d+ (σ, σ ) = 1, if there exists τ ∈ Ej +1 (M) such that σ ∪ σ ⊂ τ˙ , (iii) d(σ, σ ) = 1, if either d− (σ, σ ) = 1 or d+ (σ, σ ) = 1. The distances d, d− , d+ between two general distinct cells σ and σ are then defined as the minimum number of steps of length one needed to pass from σ to σ , and as +∞ if such a path does not exist. We say that Ej (M) is d± -connected if d± (σ, σ ) < +∞ for any σ, σ ∈ Ej (M). Proposition 3.2. Let M be a p-dimensional, regular, bounded CW-complex. (i) If Ej (M) is d+ -connected, then it is d− -connected. (ii) Assume any j -cell is contained in the boundary of some (j + 1)-cell, j + 1 p. Then, if Ej +1 (M) is d− -connected, then Ej (M) is d+ -connected. Proof. (i) Let us show that if d+ (σ0 , σ1 ) = 1, σ0 , σ1 ∈ Ej (M), then d− (σ0 , σ1 ) Vj−+1 − 1. Let oriented according to some τ ∈ Ej +1 (M) be such that σ0 , σ1 ⊂ τ˙ . Let {σi } be a basis of j -cells unique j -cycle orientation on τ˙ , which is homeomorphic to the j -sphere. Then i σi is the (up to constant multiples) representing the non-trivial homology class, hence ∂j i σi = 0. This corresponds to the fact that any (j − 1)-cell has non-trivial incidence number with exactly two j -cells, one incidence number being 1 and the other −1. Assume now there is a d− -connected in the boundary of τ . Since component k σik which is properly contained k σik is not a cycle, there exists a (j − 1)-cell ρ such that (ρ, ∂j k σik ) = 0. Then there is exactly one j -cell, not belonging to k σik , having non-trivial incidence number with ρ. But this is impossible, since k σik is a d− -connected component. Since the maximum number of j -faces of τ ∈ Ej +1 (M) is Vj−+1 , the thesis follows. (ii) Let ρ1 = ρ2 ∈ Ej (M), σ1 , σ2 ∈ Ej +1 such that ρi ⊂ σ˙ i . Then, since a d− -path from σ1 to σ2 gives rise to a d+ -path from ρ1 to ρ2 , we have d+ (ρ1 , ρ2 ) d− (σ1 , σ2 ) + 1.
2
F. Cipriani et al. / Journal of Functional Analysis 256 (2009) 603–634
609
If σ ∈ Ej (M), r ∈ N, we write Br (σ ) := {σ ∈ Ej (M): d(σ , σ ) r}, namely Br denotes the ball with respect to the metric d above. Definition 3.3 (Finite propagation operators). A bounded linear operator A on 2 (Ej M) has finite propagation r = r(A) 0 if, for all σ ∈ Ej (M), supp(Aσ ) ⊂ Br (σ ) and supp(A∗ σ ) ⊂ Br (σ ). Lemma 3.4. Finite propagation operators form a ∗ -algebra. Proof. The set of finite propagation operators is ∗ -closed by definition. To prove that it is also an algebra, one can choose, for example, r(λA + B) = r(A) ∨ r(B),
r(AB) = r(A) + r(B).
2
Given two CW-complexes M, N , a continuous map f : M → N is called cellular if f (M j ) ⊂ j ; it induces linear maps fj : Cj (M) ⊗Z C → Cj (N ) ⊗Z C intertwining the boundary maps. The cellular map f is called regular if, for all j , σ ∈ Ej (M), there are k, τ ∈ Ek (N ) such that f (σ ) = τ , f (σ˙ ) = τ˙ ; then, necessarily, k j . We call f an isomorphism if it is a j j −1 j j −1 bijective regular map such that [fj σα : fj −1 σβ ] = [σα : σβ ], for all j, α, β. Then f is a homeomorphism and fj is a linear isomorphism. A subcomplex N of M is a closed subspace of M which is a union of (open) cells. We call N a full subcomplex if, for all j , σ ∈ Ej (M), σ˙ ⊂ N imply σ ⊂ N . To prove that a cell belongs to a full subcomplex, we will find it convenient in the sequel to refer to the following N j , for all
Lemma 3.5. Let N be a full subcomplex of the regular CW-complex M. Let τ ∈ Ej (M) be such that, for all ρ ∈ Ej −1 (τ˙ ), one has ρ ∈ N . Then τ ∈ N . Proof. As N is a subcomplex, it follows that τ˙ ⊂ N ; therefore τ ∈ N , because N is full.
2
Definition 3.6 (Local isomorphisms and geometric operators). A local isomorphism1 of the CWcomplex M is a triple
s(γ ), r(γ ), γ where s(γ ), r(γ ) are full subcomplexes of M and γ : s(γ ) → r(γ ) is an isomorphism. For any j = 0, . . . , dim(M), the local isomorphism γ defines a partial isometry Vj (γ ): 2 (Ej M) → 2 (Ej M), by setting Vj (γ )(σ ) :=
γj (σ ), 0,
σ ∈ Ej (s(γ )), σ∈ / Ej (s(γ )),
1 Compare with the notion of pseudogroup (of local homeomorphisms), cf. e.g. [32].
610
F. Cipriani et al. / Journal of Functional Analysis 256 (2009) 603–634
and extending by linearity. An operator T ∈ B(2 (Ej M)) is called geometric if there exists r such that T has finite propagation r and, for any local isomorphism γ , any σ ∈ Ej (M) such that Br (σ ) ⊂ s(γ ) and Br (γ σ ) ⊂ r(γ ), one has T ∗ Vj (γ )σ = Vj (γ )T ∗ σ.
TV j (γ )σ = Vj (γ )T σ,
Proposition 3.7. Let M be a regular, bounded CW-complex. Then, for any j , geometric operators on 2 (Ej M) form a ∗ -algebra. The half-Laplacians j ± belong to it. Proof. The first statement is obvious. Concerning the second, let us note that, since the complex is bounded, half-Laplacians j ± are bounded (cf. Lemma 2.2). / r(γ ), because Let σ, σ ∈ Ej (M), with B1 (σ ) ⊂ s(γ ) and B1 (γ σ ) ⊂ r(γ ). Then, if σ ∈ supp(j ± σ ) ⊂ B1 (σ ) ⊂ s(γ ) and supp(j ± (γj σ )) ⊂ B1 (γ σ ) ⊂ r(γ ), we get
σ , j ± Vj (γ )σ = 0 = σ , Vj (γ )j ± σ . So, let us suppose that σ ∈ r(γ ), so that σ = γj σ , for σ ∈ s(γ ) and
σ , j − Vj (γ )σ =
[σ : τ ][γj σ : τ ]
τ ∈Ej −1 (M)
=
τ ∈E
=
[σ : γj −1 τ ][γj σ : γj −1 τ ]
j −1 (M)
[σ : τ ][σ : τ ]
τ ∈Ej −1 (M)
= Vj (γ )∗ σ , j − σ = σ , Vj (γ )j − σ , where the third equality comes from the incidence-preserving property of γ , and in the second equality we used the fact that the non-zero terms in the sum come from τ ’s which are “components” of the chain ∂j γj σ = γj −1 ∂j σ = ci γj −1 ρi , if ∂j σ = ci ρi , so that τ = γj −1 ρi , for some i. By linearity we get that j − is geometric. As for j + ,
σ , j + Vj (γ )σ =
[τ : σ ][τ : γj σ ]
τ ∈Ej +1 (M)
=
[γj +1 τ : σ ][γj +1 τ : γj σ ]
τ ∈Ej +1 (M)
=
[τ : σ ][τ : σ ]
τ ∈Ej +1 (M)
= Vj (γ )∗ σ , j + σ = σ , Vj (γ )j + σ , where the third equality comes from the incidence-preserving property of γ , and in the second equality we used the fact that the non-zero terms in the sum come from τ ’s such that [τ : γj σ ] = 0, so that, for all ρ ∈ Ej (τ˙ ), we get d(ρ, γj σ ) = 1, hence ρ ∈ r(γ ); from Lemma 3.5, τ ∈ r(γ ), so there is τ ∈ s(γ ) such that τ = γj +1 τ . By linearity we get that j + is geometric. 2
F. Cipriani et al. / Journal of Functional Analysis 256 (2009) 603–634
611
We now consider a version of the boundary operators relative to the boundary subcomplex. This idea is due to Lott and Lück [25], who introduced relative invariants for covering CWcomplexes. In this way, other non-trivial L2 -Betti numbers are available, as shown in Section 8. Let M be a p-dimensional, regular, bounded CW-complex. We shall consider the (p − 1)dimensional boundary subcomplex ∂M, defined as follows: (i) a (p − 1)-cell of M is in ∂M if it is contained in at most one p-cell. (ii) a j -cell of M is in ∂M if it is contained in a (p − 1)-cell in ∂M. Then ∂M is a regular bounded CW-complex. Lemma 3.8. Let N be a full subcomplex of M, and σ0 ∈ Ej (N ) be such that Bk (σ0 ) ⊂ N . Then, for any τ0 ∈ Ej +1 (N ) such that σ0 ⊂ τ˙0 , one has B (τ0 ) ⊂ N , for V −k −1 . j +1
Proof. Let τ1 ∈ Ej +1 (M) be such that d(τ1 , τ0 ) . Then, for any σ1 ∈ Ej (M), σ1 ⊂ τ˙1 , one has, from the proof of Proposition 3.2, d(σ1 , σ0 ) (Vj−+1 − 1) k. Therefore σ1 ⊂ N . As N is full, τ1 ⊂ N , and the thesis follows. 2 Lemma 3.9. Let γ be a local isomorphism, σ ∈ Ej (M) be such that Bk (σ ) ⊂ s(γ ), Bk (γ σ ) ⊂ − − − 1)(Vp−2 − 1) · · · (Vj−+1 − 1). Then σ ∈ ∂M iff γ σ ∈ ∂M. r(γ ), where k (Vp−1 Proof. (⇒) Let σ ∈ Ep−1 (∂M). If there were τ = τ ∈ Ep (M) such that γ σ ⊂ τ˙ ∩ τ˙ , then for all ρ ∈ Ep−1 (M), ρ ⊂ τ˙ we would get d(ρ, γ σ ) = 1, hence ρ ∈ r(γ ); from Lemma 3.5, τ ∈ r(γ ); analogously τ ∈ r(γ ). As γp preserves incidences and boundaries, σ ⊂ γ −1 τ˙ ∩ γ −1 τ˙ , which implies σ ∈ / ∂M, and we have reached a contradiction. Therefore, there is a unique τ ∈ Ep (M) such that γ σ ⊂ τ˙ , which means that γ σ ∈ ∂M. If σ ∈ Ej (∂M), there is τ ∈ Ep−1 (∂M) such that σ ⊂ τ˙ , and γ σ ⊂ (γ τ )˙= γ (τ˙ ). Then, from Lemma 3.8, B1 (τ ) ⊂ s(γ ), and B1 (γ τ ) ⊂ r(γ ). From what has already been proved, γ τ ∈ ∂M. Therefore γ σ ∈ ∂M, because ∂M is a subcomplex. (⇐) follows from the above applied to γ −1 . 2 Let ∂ j ≡ ∂jM,∂M be the boundary operator of the relative cellular complex Cj (M, ∂M) := Hj (M j ∪ ∂M, M j −1 ∪ ∂M, Z). As Cj (M, ∂M) ∼ = σ ∈E j (M) Zσ , where E j (M) := {σ ∈ (2)
Ej (M): σ ∩ ∂M = ∅}, we can identify Cj (M, ∂M), the 2 -completion of Cj (M, ∂M) ⊗Z C, (2)
with 2 (E j (M)), a closed subspace of 2 (Ej (M)). Moreover we can consider ∂ j : Cj (M) → (2) (2) (2) ∗ ⊥ or Cj(2) −1 (M), ∂ j : Cj −1 (M) → Cj (M), by extending them to 0 on Cj (M, ∂M)
Cj −1 (M, ∂M)⊥ , respectively. Define j + := ∂ j +1 ∂ j∗+1 , j − := ∂ j∗ ∂ j . Then (2)
Lemma 3.10. (i) j ± σ = 0, for σ ∈ Cj (M, ∂M)⊥ , (2)
(ii) for σ, σ ∈ Cj (M, ∂M), (2)
612
F. Cipriani et al. / Journal of Functional Analysis 256 (2009) 603–634
(σ , j + σ ) =
[τ : σ ][τ : σ ],
(σ , j − σ ) =
τ ∈E j +1 (M)
[σ : τ ][σ : τ ].
τ ∈E j −1 (M)
Proposition 3.11. Let γ be a local isomorphism, σ ∈ Ej (M) be such that Bk (σ ) ⊂ s(γ ),
p−1 Bk (γ σ ) ⊂ r(γ ), for some k 1 + i=j +1 (Vi− − 1). Then j − Vj (γ )σ = Vj (γ ) j − σ,
j + Vj (γ )σ = Vj (γ ) j + σ.
Proof. Let us prove that, for any σ ∈ E j (M), we have
s , j ± Vj (γ )σ = σ , Vj (γ ) j ± σ . If σ ∈ / B1 (γ σ ), the thesis is true. Indeed, from supp(j ± σ ) ⊂ B1 (σ ) ⊂ s(γ ), and supp(j ± γ σ ) ⊂ B1 (γ σ ) ⊂ r(γ ), it follows (σ , j ± Vj (γ )σ ) = 0, whereas, if σ ∈ / r(γ ) we get (σ , Vj (γ )j ± σ ) = 0, while, if σ ∈ r(γ ) \ B1 (γ σ ), we get (σ , Vj (γ )j ± σ ) = (γj−1 σ , j ± σ ) = 0, as d(γj−1 σ , σ ) = d(σ , γj σ ) > 1. Therefore, we can assume σ ∈ B1 (γ σ ). Moreover, if σ ∈ ∂M, so that γ σ ∈ ∂M (by Lemma 3.9), we get j ± Vj (γ )σ = 0 = Vj (γ )j ± σ . Therefore, we now assume σ ∈ / ∂M, σ ∈ B1 (γ σ ). Then
σ , j − Vj (γ )σ =
[σ : τ ][γj σ : τ ].
τ ∈E j −1 (M)
Let τ ∈ E j −1 (M), τ ⊂ (γ σ )˙ ∩ σ˙ . Then, as in the first part of the proof of Proposition 3.7, there is τ ∈ Ej −1 (s(γ )) such that τ = γ τ ; moreover τ ⊂ σ˙ , as [σ : τ ] = [γ σ : γ τ ] = 0. Let us now show that τ ∈ ∂M ⇔ τ ∈ ∂M; indeed, if τ ∈ ∂M, then there is ρ ∈ Ej (∂M) such that τ ⊂ ρ; ˙ therefore d(ρ, γ σ ) 1, and Bk−1 (ρ) ⊂ r(γ ), so, from Lemma 3.9, it follows that ρ := γj−1 ρ ∈ ∂M; then [ρ : τ ] = [ρ : τ ] = 0, hence τ ⊂ ρ˙ , and τ ∈ ∂M. The other implication follows similarly. Therefore
σ , j − Vj (γ )σ =
[σ : τ ][γj σ : τ ]
τ ∈E j −1 (M)
=
[σ : γj −1 τ ][γj σ : γj −1 τ ]
τ ∈E j −1 (M)
=
−1 γj σ : τ [σ : τ ]
τ ∈E j −1 (M)
= Vj (γ )∗ σ , j − σ = σ , Vj (γ ) j − σ . As for j + , we get
σ , j + Vj (γ )σ =
[τ : σ ][τ : γj σ ].
τ ∈E j +1 (M)
F. Cipriani et al. / Journal of Functional Analysis 256 (2009) 603–634
613
Let τ ∈ Ej +1 (M) be such that γ σ ∪ σ ⊂ τ˙ ; as for any ρ ∈ Ej (M), ρ ⊂ τ˙ , it holds d(ρ, γ σ ) 1, so ρ ∈ B1 (γ σ ) ⊂ r(γ ), from Lemma 3.5 we get τ ∈ r(γ ); therefore there is τ ∈ s(γ ) such that τ = γ τ . From Lemma 3.8 it follows B (τ ) ⊂ r(γ ), for k/(Vj−+1 − 1), and Lemma 3.9 gives us τ ∈ ∂M ⇔ τ ∈ ∂M. Therefore
[τ : σ ][τ : γj σ ] σ , j + Vj (γ )σ = τ ∈E j +1 (M)
=
τ ∈E j +1 (M)
=
[γj +1 τ : σ ][γj +1 τ : γj σ ] −1 τ : γj σ [τ : σ ]
τ ∈E j +1 (M)
= Vj (γ )∗ σ , j + σ = σ , Vj (γ ) j + σ .
2
We have proved the following. Proposition 3.12. Let M be a p-dimensional, regular, bounded CW-complex. The relative halfLaplacians j ± are geometric operators. 4. Self-similar CW-complexes In this section we introduce self-similar complexes, and show that there is a natural trace state on the algebra of geometric operators. If K is a subcomplex of M, we call j -frontier of K, and denote it by F (Ej K), the family of cells in Ej K having distance 1 from the complement of Ej K in Ej (M). Definition 4.1 (Amenable CW-complexes). A countably infinite CW-complex M is amenable if it is regular and bounded, and has an amenable exhaustion, namely, an increasing family of finite subcomplexes {Kn : n ∈ N} such that Kn = M and for all j = 0, . . . , dim(M), |F (Ej Kn )| →0 |Ej Kn |
as n → ∞.
Definition 4.2 (Self-similar CW-complexes). A countably infinite CW-complex M is self-similar if it is regular and bounded, and it has an amenable exhaustion by full subcomplexes {Kn : n ∈ N} such that the following conditions (i) and (ii) hold: (i) for all n there is a finite set of local isomorphisms G(n, n + 1) such that, for all γ ∈ G(n, n + 1), one has s(γ ) = Kn ,
γj Ej (Kn ) = Ej (Kn+1 ), j = 0, . . . , dim(M), γ ∈G (n,n+1)
and moreover if γ , γ ∈ G(n, n + 1) with γ = γ
Ej γ (Kn ) ∩ Ej γ (Kn ) = F Ej γ (Kn ) ∩ F Ej γ (Kn ) ,
j = 0, . . . , dim(M). (4.1)
614
F. Cipriani et al. / Journal of Functional Analysis 256 (2009) 603–634
Fig. 2. Lindstrom graph.
Fig. 1. Gasket graph.
(ii) We then define G(n, m), with n < m, as the set of all admissible products γm−1 · · · · · γn , γi ∈ G(i, i + 1), where admissible means that the range of γj is contained in the source of γj +1 . We let G(n, n) consist of the identity isomorphism on Kn , and G(n) = mn G(n, m). We now define the G-invariant j -frontier of Kn :
FG (Ej Kn ) = γj−1 F Ej γ (Kn ) , γ ∈G (n)
and we ask that |FG (Ej Kn )| → 0 as n → ∞. |Ej Kn | Remark 4.3. We may replace the condition in (4.1) with the following:
Ej γ (Kn ) ∩ Ej γ (Kn ) ⊆ Br F Ej γ (Kn ) ∩ Br F Ej γ (Kn ) , j = 0, . . . , dim(M), for a suitable r > 0. It is easy to see that all the theory developed below will remain valid. Some examples of self-similar CW-complexes are given below, cf. Section 6 for more details on the construction. Example 4.4. The Gasket graph in Fig. 1, the Lindstrom graph in Fig. 2, the Vicsek graph in Fig. 3 are examples of 1-dimensional self-similar complexes. The Carpet 2-complex in Fig. 4 is an example of a 2-dimensional self-similar CW-complex. Theorem 4.5. Let M be a self-similar CW-complex, A(Ej M) the C ∗ -algebra given by the closure of the ∗ -algebra of geometric operators. Then, on A(Ej M) there is a well defined trace state Φj given by Φj (T ) = lim n
Tr(E(Ej Kn )T ) Tr(E(Ej Kn ))
where E(Ej Kn ) is the orthogonal projection of 2 (Ej M) onto 2 (Ej Kn ).
F. Cipriani et al. / Journal of Functional Analysis 256 (2009) 603–634
615
Fig. 4. Carpet 2-complex.
Fig. 3. Vicsek graph.
Proof. Fix j ∈ {0, . . . , p}, and for a finite subset N ⊂ Ej M denote by E(N ) ∈ B(2 (Ej M)) the projection onto span N . Let us observe that, since N is an orthonormal basis for 2 (N ), then Tr(E(N)) = |N|. First step. (Some combinatorial results.) (a) Let μ ≡ μj = supσ ∈Ej M |B1 (σ )|. First observe that μ is finite, since μ Vj+ + Vj− . Then, since Br+1 (σ ) = B1 (σ ), σ ∈Br (σ )
we get |Br+1 (σ )| |Br (σ )|μ, giving |Br (σ)| μr , ∀σ ∈ Ej M, r 0. As a consequence, for any finite set Ω ⊂ Ej M, we have Br (Ω) = σ ∈Ω Br (σ ), giving Br (Ω) |Ω|μr , ∀r 0. (4.2) (b) Let us set Ω(n, r) = Ej Kn \ Br (FG (Ej Kn )). Then, for any γ ∈ G(n), we have
γj Ω(n, r) ⊂ γj Ej Kn ⊂ γj Ω(n, r) ∪ Br FG (γj Ej Kn ) . Now assume r 1. Then, the γj Ω(n, r)’s are disjoint, for different γ ’s in G(n, m). Therefore, |Ej Kn | Ω(n, r) + FG (Ej Kn )μr , Ej Km \ γj Ω(n, r) G(n, m) FG (Ej Kn )μr ,
(4.3) (4.4)
γ ∈G (n,m)
G(n, m) Ω(n, r) |Ej Km | G(n, m) |Ej Kn |. Indeed, (4.3) and (4.5) are easily verified, whereas
(4.5)
616
F. Cipriani et al. / Journal of Functional Analysis 256 (2009) 603–634
Ej Km \
γ ∈G (n,m)
γj Ω(n, r) =
γj Ej Kn \
γ ∈G (n,m)
γ ∈G (n,m)
γj Ω(n, r)
γj Ej Kn \ Ω(n, r)
γ ∈G (n,m)
G(n, m)Br FG (Ej Kn ) G(n, m)FG (Ej Kn )μr . (c) Let εn =
|FG (Ej Kn )| |Ej Kn | ,
and recall that εn → 0. Putting together (4.3) and (4.5) we get
G(n, m) |Ej Kn | − G(n, m) FG (Ej Kn )μr |Ej Km | G(n, m) |Ej Kn |, which implies 1 − εn μr
|Ej Km | 1. |G(n, m)| |Ej Kn |
Choosing n0 such that, for n > n0 , εn μr 1/2, we obtain 0
|G(n, m)| |Ej Kn | − 1 2εn μr 1. |Ej Km |
(4.6)
Therefore, from (4.4), we obtain Ej Km \
γ ∈G (n,m)
γj Ω(n, r) G(n, m)FG (Ej Kn )μr = G(n, m)|Ej Kn |εn μr 2|Ej Km |εn μr .
(4.7)
Second step. (The existence of the limit for geometric operators.) (a) By definition of Vj (γ ), we have, for γ ∈ G(n, m), n < m, Vj∗ (γ )Vj (γ ) = E(Ej Kn ),
Vj (γ )Vj∗ (γ ) = E γj Ej Kn .
Assume now T ∈ B(2 (Ej M)) is a geometric operator with finite propagation r. Then,
TV j (γ )E Ω(n, r) = Vj (γ )TE Ω(n, r) ,
E γj Ω(n, r) = Vj (γ )E Ω(n, r) Vj (γ )∗ .
As a consequence,
Tr TE γj Ω(n, r) = Tr T Vj (γ )E Ω(n, r) Vj (γ )∗ = Tr Vj (γ )TE Ω(n, r) Vj (γ )∗
= Tr TE Ω(n, r) Vj (γ )∗ Vj (γ ) = Tr TE Ω(n, r) E(Ej Kn )
= Tr TE Ω(n, r) . (4.8) (b) Let us show that the sequence is Cauchy:
F. Cipriani et al. / Journal of Functional Analysis 256 (2009) 603–634
617
Tr TE(Ej Kn ) Tr TE(Ej Km ) Tr E(E K ) − Tr E(E K ) j n j m
| Tr T (E(Ej Kn ) − E(Ω(n, r)))| | Tr T (E(Ej Km ) − E( γ ∈G (n,m) γj Ω(n, r)))| + |Ej Kn | |Ej Km | Tr TE(Ω(n, r)) |G(n, m)| |Ej Kn | Tr TE(Ω(n, r)) − + |Ej Kn | |Ej Km | |Ej Kn | |G(n, m)| |Ej Kn | |Ej Kn \ Ω(n, r)| |Ej Km \ γ ∈G (n,m) γj Ω(n, r)| T + + 1 − |Ej Kn | |Ej Km | |Ej Km | 5T εn μr ,
where we used (4.8), in the first inequality, and (4.7), (4.6), in the third inequality. Third step. (Φj is a state on A(Ej (M)).) (a) Let T ∈ A(Ej (M)), ε > 0. Now find a geometric operator T such that T − T ε/3, Tr AE(E K )
and set φn (A) := Tr E(EjjKnn) . Then choose n such that, for m > n, |φm (T ) − φn (T )| ε/3. We get φm (T ) − φn (T ) φm (T − T ) + φm (T ) − φn (T ) + φn (T − T ) ε namely lim φn (T ) exists. (b) The functional Φj is clearly linear, positive and takes value 1 at the identity, hence it is a state on A(Ej (M)). Fourth step. (Φj is a trace on A(Ej (M)).) Let A be a geometric operator with propagation r. Then
AE(Ej Kn ) = E Br (Ej Kn ) AE(Ej Kn ),
E Ω(n, r) A = E Ω(n, r) AE(Ej Kn ). Indeed,
Ω(n, r) ⊂ Ej Kn \ Br F (Ej Kn ) = σ ∈ Ej Kn : d(σ, M \ Ej Kn ) r + 2 , so that
Br Ω(n, r) ⊂ σ ∈ Ej Kn : d(σ, M \ Ej Kn ) 2 ⊂ Ej Kn . Since A∗ has propagation r, we get
A∗ E Ω(n, r) = E Br Ω(n, r) A∗ E Ω(n, r) = E(Ej Kn )A∗ E Ω(n, r) , which proves the claim. Therefore,
AE(Ej Kn ) = E Br (Ej Kn ) \ Ω(n, r) AE(Ej Kn ) + E Ω(n, r) A
= E Br (Ej Kn ) \ Ω(n, r) AE(Ej Kn ) − E Ej Kn \ Ω(n, r) A + E(Ej Kn )A.
618
F. Cipriani et al. / Journal of Functional Analysis 256 (2009) 603–634
Therefore, if B ∈ A(Ej (M)),
|Br (Ej Kn ) \ Ω(n, r)| + |Ej Kn \ Ω(n, r)| φn [B, A] A B |Ej Kn | 2A Bεn μr , as Br (Ej Kn ) \ Ω(n, r) ⊂ Br (FG (Ej Kn )). Taking the limit as n → ∞ we get Φj ([B, A]) = 0. By continuity, the result holds for any A, B ∈ A(Ej (M)). 2 In the following we use a different normalisation for the traces and, by giving up the state property, we obtain that the trace of the identity operator in Aj measures the relative volume of Ej (M). This simplifies the relations in Corollaries 5.6 and 5.8. Lemma 4.6. Let M be a p-dimensional self-similar complex. The following limits exist and are finite: lim n
|Ej (Kn )| , |Ep (Kn )|
0 j p.
Proof. We show that the sequences are Cauchy. Indeed, by inequalities (4.6) in the proof of Theorem 4.5, we have, for m > n and j = 0, . . . , p, (1 + 2εn μ)−1 G(n, m)|Ej Kn | |Ej Km | G(n, m)|Ej Kn |, where the sequence εn = supj =0,...,p supj supσ ∈Ej M |B1 (σ )|. Therefore (1 − 2εn μ)
|FG (Ej Kn )| |Ej Kn |
is infinitesimal and less than 1, and μ =
|Ej Km | |Ej Kn | |Ej Kn | (1 + 2εn μ) . |Ep Kn | |Ep Km | |Ep Kn |
|E K |
Hence, the sequence |Epj Knn | is bounded by some constant M > 0, and |Ej Km | |Ej Kn | |E K | − |E K | 2Mμεn . p m p n The thesis follows.
2
Definition 4.7. Let M be a p-dimensional self-similar complex. On the C ∗ -algebras Aj we shall consider the traces TrG j (T ) = lim n
|Ej (Kn )| Tr(E(Ej Kn )T ) Φj (T ) = lim . n Tr(E(Ep Kn )) |Ep (Kn )|
In this way, TrG j (I ) measures the relative volume of Ej (M) with respect to Ep (M).
F. Cipriani et al. / Journal of Functional Analysis 256 (2009) 603–634
619
5. L2 -Betti numbers and Novikov–Shubin numbers for self-similar CW-complexes In this section, we define L2 -Betti numbers and Novikov–Shubin numbers for self-similar CW-complexes, prove various relations among them, and give a result on the Euler–Poincaré characteristic of a complex. Let M be a self-similar CW-complex, let be one of the operators j ± , j , j ± , j , and define Definition 5.1 (L2 -Betti and Novikov–Shubin numbers). (i) β() := limt→∞ TrG (e−t ), the L2 -Betti number of , G −t )−β()) , the Novikov–Shubin number of , and the lower and (ii) α() := 2 limt→∞ log(Tr (e − log t upper versions, if the above limits do not exist. Then set βj± (M) := β(j ± ), βj (M) := β(j ), βj± (M, ∂M) := β(j ± ), βj (M, ∂M) := β(j ), and analogously for the Novikov–Shubin numbers. Remark 5.2. (i) The L2 -Betti numbers and Novikov–Shubin numbers could have been defined also in terms of the spectral density function Nλ . This is usually defined in terms of spectral projections, which belong to the generated von Neumann algebra, hence, in our case, are not necessarily in the domain of the trace. However we may consider the spectral measure μj ± associated, via Riesz theorem, to the functional ϕ ∈ C 0 [0, ∞) → TrG j (ϕ(j ± )) ∈ C, and then λ 2 define Nλ (j ± ) := 0 dμj ± . The two definitions for the L -Betti numbers clearly coincide, while Novikov–Shubin numbers can be related via a Tauberian theorem, as in [12]. (ii) We followed [25] for the definition of the relative L2 -invariants, even though we considered only the two cases of no boundary and of full boundary. It would be interesting to prove their Poincaré duality result in our context. (iii) We have used the same normalizing sequence for each trace TrG j , see Definition 4.7, in 2 order to compare L -Betti numbers. This will imply the relation in Corollary 5.8. Lemma 5.3 (Hodge decomposition). The following decomposition holds true: 2 (Ej M) = Im j − ⊕ Im j + ⊕ ker j . Proof.
⊥
⊥ Im j + = ker ∂j +1 ∂j∗+1 = ker ∂j∗+1 = Im ∂j +1 ⊆ ker ∂j = ker ∂j∗ ∂j = (Im j − )⊥ .
620
F. Cipriani et al. / Journal of Functional Analysis 256 (2009) 603–634
Then since (Im j + )⊥ ∩ (Im j − )⊥ = ker ∂j ∩ ker ∂j∗+1 = ker j the thesis follows.
2
Theorem 5.4. With the notation above, we have the relations: βj (M) = βj+ (M) + βj− (M) − TrG j (I ), βj (M, ∂M) = βj + (M, ∂M) + βj − (M, ∂M) − TrG j (I ), αj (M) = min αj + (M), αj − (M) , αj (M, ∂M) = min αj + (M, ∂M), αj − (M, ∂M) . Proof. By the orthogonality of the ranges we have j + j − = 0. Hence (j + + j − )n = nj+ + nj− from which we get e−tj = e−tj + + e−tj − − I . Now the thesis easily follows. The proof for the relative invariants is analogous. 2 G ∗ k ∗ k Proposition 5.5. TrG j −1 ((∂j ∂j ) ) = Trj ((∂j ∂j ) ), k ∈ N.
Proof. Let us set Ωn := Ej −1 Kn \ FG (Ej −1 Kn ). First we note that E(Ωn )∂j = E(Ωn )∂j E(Ej Kn ). Indeed they coincide on the range of E(Ej Kn ), and both vanish on its kernel. Analogously ∂j E(Ej Kn ) = E(B1 (Ej −1 Kn ))∂j E(Ej Kn ). Let us note that if ∂j = Vj |∂j | denotes the polar decomposition, and A ∈ B(2 (Ej (M))), then Tr(∂j A∂j∗ ) = Tr(|∂j |A|∂j |). Then
k
k−1
Tr E(Ωn ) ∂j ∂j∗ = Tr E(Ωn )∂j E(Ej Kn )∂j∗ ∂j ∂j∗ , and
k
k−1
Tr E(Ej Kn ) ∂j∗ ∂j = Tr |∂j |E(Ej Kn ) ∂j∗ ∂j |∂j |
k−1 ∗
∂j = Tr ∂j E(Ej Kn ) ∂j∗ ∂j
k−1
. = Tr E B1 (Ej −1 Kn ) ∂j E(Ej Kn )∂j∗ ∂j ∂j∗ Since B1 (Ej −1 Kn ) \ Ωn ⊂ B1 (FG (Ej −1 Kn )), we obtain
Tr E(Ej Kn ) ∂ ∗ ∂j k − Tr E(Ωn ) ∂j ∂ ∗ k j
j
k−1
Tr E B1 FG (Ej −1 Kn ) ∂j E(Ej Kn )∂j∗ ∂j ∂j∗ k μ∂j ∂j∗ Ej −1 (Kn )εn ,
F. Cipriani et al. / Journal of Functional Analysis 256 (2009) 603–634
where εn =
|FG (Ej −1 Kn )| |Ej −1 Kn |
621
and |B1 (F (Ej −1 Kn ))| μ|Ej −1 Kn |εn . Finally,
Tr(E(Ej −1 Kn )(∂j ∂j∗ )k ) Tr(E(Ej Kn )(∂j∗ ∂j )k ) − |Ep Kn | |Ep Kn |
k
|Ep Kn |−1 Tr E FG (Ej −1 Kn ) ∂j ∂j∗
k
k
+ Tr E(Ej Kn ) ∂j∗ ∂j − Tr E(Ωn ) ∂j ∂j∗ by Lemma 4.6.
k |Ej −1 Kn | (μ + 1)∂j ∂j∗ εn → 0, |Ep Kn |
as n → ∞,
2
Corollary 5.6. For any continuous bounded function f : [0, ∞) → C vanishing at zero, one has G ∗ ∗ TrG j −1 (f (∂j ∂j )) = Trj (f (∂j ∂j )). In particular −t∂j ∂ ∗
G −t∂j∗ ∂j
j − TrG TrG − TrG j (I ). j −1 e j −1 (I ) = Trj e Therefore G − βj+−1 (M) − TrG j −1 (I ) = βj (M) − Trj (I ), G − βj+−1 (M, ∂M) − TrG j −1 (I ) = βj (M, ∂M) − Trj (I ),
αj+−1 (M) = αj− (M), αj+−1 (M, ∂M) = αj− (M, ∂M).
Proof. The proof for α(j ± ) and β(j ± ) follows directly by the previous results. All the arguments above may be rephrased for the relative invariants, giving the corresponding equality. 2 Remark 5.7. Let us recall that in [25] Novikov–Shubin numbers have been associated to the boundary operator ∂, namely depend on an index varying from 1 to the dimension p of the complex. As a consequence of Corollary 5.6, there are only p independent Novikov–Shubin numbers in our framework too. Concerning L2 -Betti numbers, they have been defined in [25] as Γ -dimensions of the L2 homology, hence coincide with the trace of the kernel of the full Laplacians. The relations proved in this section show that the βj± ’s are completely determined by the βj ’s. Moreover such relations imply a further identity which is the basis of the following theorem. Theorem 5.8 (Euler–Poincaré characteristic). Let M be a p-dimensional self-similar CWcomplex, with exhaustion {Kn }. Then χ G (M) :=
p χ(Kn ) . (−1)j βj (M) = lim n |Ep (Kn )| j =0
Proof. Let us first observe that, by using Theorem 5.4 and Corollary 5.6, we get
622
F. Cipriani et al. / Journal of Functional Analysis 256 (2009) 603–634
χ G (M) = β0+ (M) +
p−1
p − (−1)j βj+ (M) + βj− (M) − TrG j (I ) + (−1) βp (M)
j =1
=
=
p−1
p
j =0
j =1
(−1)j βj+ (M) + (−1)p TrG p (I ) +
(−1)j βj+−1 (M) −
p (−1)j TrG j −1 (I ) j =1
p (−1)j TrG j (I ). j =0
On the other hand, p p |Ej (Kn )| χ(Kn ) (−1)j TrG (I ) = lim (−1)j = lim . j n n |Ep (Kn )| |Ep (Kn )| j =0
The thesis follows.
j =0
2
Example 5.9. (i) For the Gasket graph of Fig. 1 we get |VK n | = 12 3n + 32 and |EK n | = 3n , so that χ G (X) = − 12 . (ii) For the Vicsek graph of Fig. 3 we get |VK n | = 3 · 5n + 1 and |EK n | = 4 · 5n , so that χ G (X) = − 14 . (iii) For the Lindstrom graph of Fig. 2 we get |VK n | = 4 · 7n + 2 and |EK n | = 6 · 7n , so that χ G (X) = − 13 . 6. Prefractals as CW-complexes We say that a j -dimensional polyhedron in some Euclidean space Rm is strictly convex if it is convex and any (j − 1)-hyperplane contains at most one of its (j − 1)-dimensional faces. Definition 6.1. A polyhedral complex is a regular CW complex whose topology is that of a closed subset of some Euclidean space Rm and whose j -cells are flat strictly convex j -polyhedra in Rm . The following proposition motivates the name of the boundary subcomplex. Proposition 6.2. Let M be a p-dimensional polyhedral complex in Rp . Then the boundary subcomplex ∂M gives a CW structure on the boundary of M, seen as a subspace of Rp . Proof. Clearly, given a (p − 1)-cell σ of M and a point x ∈ σ , we may find ε > 0 such that the ball B(x, ε) in Rp is contained in σ ∪ τ1 ∪ τ2 , if σ is contained in the two distinct p-cells τ1 , τ2 , and is not contained in M (indeed half of it is in the complement of M with respect to Rp ), if σ is contained in only one p-cell τ . This proves the thesis. 2 Proposition 6.3. Let M be a p-dimensional polyhedral complex, j = 1, . . . , p. If σ, σ are distinct cells in Ej −1 (M), there exists at most one polyhedron τ ∈ Ej (M) such that (σ, ∂j τ )(σ , ∂j τ ) = 0. If σ, σ ∈ Ej (M), there exists at most one polyhedron ρ ∈ Ej −1 (M) such that (∂j σ, ρ)(∂j σ , ρ) = 0.
F. Cipriani et al. / Journal of Functional Analysis 256 (2009) 603–634
623
Proof. Let σ = σ ∈ Ej −1 (M) and C denote the convex hull of σ ∪ σ . If C has dimension j − 1, a τ as above would have two faces in the same (j − 1)-plane, against the strict convexity. If C has dimension greater or equal to j , two different τ, τ ∈ Ej (M) containing both σ and σ in their boundaries would contain C also, implying τ = τ . Finally, if σ, σ ∈ Ej (M), and there were ρ, ρ ∈ Ej −1 (M) as above, they would belong to the (j − 1)-plane separating σ and σ , namely σ and σ would have two faces in the same (j − 1)plane, against strict convexity. 2 Our main examples of self-similar CW-complexes will be a special class of polyhedral complexes, namely prefractal complexes. Let us recall that a self-similar fractal K in Rp is determined by contraction similarities w1 , . . . , wq as the unique (compact) solution of the fixed point equation K = W K, where W q is a map on subsets defined as W A = j =1 wj A. The fractal K satisfies the open set condition with open set U if wj U ⊂ U,
wj U ∩ wi U = ∅,
i = j.
Assume now we are given a self-similar fractal in Rp determined by similarities w1 , . . . , wq , with the same similarity parameter. Assume Open Set Condition holds for a bounded open set whose closure is a strictly convex p-dimensional polyhedron P. If σ is a multiindex of length n, we set wσ := wσn ◦ · · · ◦ wσ1 . If σ is an infinite multiindex, we denote by σ |n its nth truncation. Assume that wσ P ∩ wσ P is a (facial) subpolyhedron of both wσ P and wσ P, |σ | = |σ |. Finally we choose an infinite multiindex I . We construct a polyhedral CW-complex as follows. First set n Kn := wI−1 |n W P =
|σ |=n
wI−1 |n wσ P.
Lemma 6.4. Kn is a finite polyhedral complex satisfying the following properties: ∀j < p, σ ∈ Ej (Kn ),
∃τ ∈ Ep (Kn ): σ ⊂ τ˙ ,
(6.1)
Kn ⊂ Kn+1 .
(6.2)
−1 −1 Proof. Observe that wI−1 |n wσ P ∩ wI |n wσ P = wI |n (wσ P ∩ wσ P), hence is a common facial subpolyhedron, namely Kn has a natural structure of polyhedral complex. Property (6.1) is obvious. Let us now prove (6.2).
−1 Kn+1 = wI−1 |n wIn+1
q j =1
n wj W n P ⊃ wI−1 |n W P = Kn .
2
Corollary 6.5 (Prefractal complexes). M = n∈N Kn is a polyhedral complex satisfying property (6.1). {Kn } is an exhaustion for M. M is called a prefractal complex. Proof. Obvious.
2
Proposition 6.6. A prefractal complex is a regular bounded CW-complex.
624
F. Cipriani et al. / Journal of Functional Analysis 256 (2009) 603–634
Proof. Regularity is obvious by construction. Let us estimate Vj+ . Any σ ∈ Ej is contained in some copy of the fundamental polyhedron P, therefore any τ ∈ Ej +1 such that τ˙ ⊃ σ will be contained in the same copy of P or in some neighboring copy. As a consequence, we may estimate the number of such τ ’s with the product of |Ej +1 (P)| times the maximum number of disjoint copies of P having a j -cell in common. Since such copies are contained in a ball of radius diam(P), their number may be estimated e.g. by the ratio of the volume of the ball of radius diam(P) and the volume of P. As for Vj− , again any σ ∈ Ej is contained in some copy of the fundamental polyhedron P, therefore any ρ ∈ Ej −1 such that ρ ⊂ σ˙ will be contained in the same copy of P. The number of such ρ’s is majorised by |Ej −1 (P)|. 2 In order to show that M is a self-similar complex, we shall prove that Kn is a regular exhaustion satisfying Definition 4.2. Lemma 6.7. Assume K is a p-dimensional polyhedral complex in Rp satisfying property (6.1), and we have σi ∈ Ei (K), i = j0 , . . . , j1 , j1 < p, with σi ⊂ σ˙ i+1 . Then there exist σi ∈ Ei (K), i = j0 , . . . , j1 + 1, such that , i = j ,...,j , (α) σi ⊂ σ˙ i+1 0 1 (β) σi−1 ⊂ σ˙ i , i = j0 + 1, . . . , j1 + 1, (γ ) ph(σi ∪ σj1 ) = σj1 +1 , i = j0 , . . . , j1 , where ph(E) denotes the polyhedral hull, namely the minimal polyhedron in K (if it exists) containing E ⊂ K.
Proof. The proof will be done by descending induction on i, starting from j1 + 1. Take any σj1 +1 ⊃ σj1 , which exists, because of (6.1). Assume the statement for k + 1. If , there is (one and only one) σ ⊂ σ , such that σ ˙ k+1 ˙ k , and k > j0 , as σk−1 ⊂ σ˙ k , σk ⊂ σ˙ k+1 k−1 ⊂ σ k σk = σk , by regularity of the CW-complex. If k = j0 , simply take σk as any k-dimensional face in , distinct from σ . Let us observe that, for i = j , property (β) is empty. This the boundary of σk+1 k 0 is strictly convex, and σk , σk are two distinct k-dimensional faces, gives (α) and (β). Since σk+1 . Then ph(σ ∪ σ ) = ph(σ ∪ σ ∪ σ ) = ph(σ ∪ σ ) = σ we have ph(σk ∪ σk ) = σk+1 j1 j1 k j1 k k k+1 j1 +1 , which is property (γ ). 2 Theorem 6.8. Any prefractal complex M is a self-similar polyhedral complex. Proof. We already proved regularity and boundedness. Let us now observe that Kn+1 =
q j =1
n wI−1 |n+1 wj W P
=
q
γn Kn ,
=1
where γn = wI−1 |n+1 w wI |n are local isomorphisms. Moreover, by OSC, −1 n γkn Kn = wI−1 |n+1 wk W P ⊂ wI |n+1 wk P.
Therefore, γkn Kn ∩ γn Kn ⊂ wI−1 |n+1 (wk P ∩ w P), which is contained in some affine hyperplane π ; hence γkn Kn ∩ γn Kn has no p-dimensional polyhedra. Let σj0 be a j0 -dimensional polyhedron in γkn Kn ∩ γn Kn , and σi , i = j0 , . . . , j1 , be a maximal family of polyhedra in
F. Cipriani et al. / Journal of Functional Analysis 256 (2009) 603–634
625
γkn Kn ∩ π such that σi ⊂ σ˙ i+1 , i = j0 , . . . , j1 − 1. Now apply the lemma with K = γkn Kn , and get σi ∈ Ei (γkn Kn ), i = j0 , . . . , j1 + 1, with ph(σi ∪ σj1 ) = σj1 +1 . Then σj0 ⊂ π , otherwise / Ej0 (γn Kn ). Moreover, since π ⊃ ph(σj0 ∪ σj1 ) = σj1 +1 , against the maximality. Therefore σj0 ∈ σj0 , σj0 ⊂ σ˙ j0 +1 , we have d+ (σj0 , σj0 ) = 1, namely σj0 ∈ F (Ej0 (γn Kn )). Similarly, one proves σj0 ∈ F (Ej0 (γkn Kn )). Therefore, γkn Kn ∩ γn Kn = F (γkn Kn ) ∩ F (γn Kn ). 2 Remark 6.9. The construction above can be easily generalised to the case of translation limit fractals [14–18]. We now study some properties of polyhedral complexes, which are valid in particular for the prefractal complexes. Definition 6.10. We say that j ± is a graph-like Laplacian if there exists a suitable orientation of M such that the off-diagonal entries of the matrix associated with j ± , in the corresponding orthonormal basis, belong to {0, −1}. Theorem 6.11. Assume M is a p-dimensional polyhedral complex in Rm , m p. Then j + is graph-like if and only if j = 0. Proof. If σ, σ ∈ E0 (M), then (σ, 0+ σ ) = α [τα1 , σ ][τα1 , σ ]. By Proposition 6.3, the sum consists of at most one non-vanishing summand, corresponding to some 1-cell τ . By the choice we made for the orientation on the 0-cells, the sum of [τα1 , σ ] and [τα1 , σ ] is 0, hence the product is −1. Let j > 0 and choose τ ∈ Ej +1 (M). Since j + 1 2, τ has at least three distinct faces σ1 , σ2 , σ3 ∈ Ej (M). Setting [τ, σi ] = λi = ±1, we obtain, for i = k, (σi , j + σk ) = λi · λk . Therefore the product (σ1 , j + σ2 ) · (σ2 , j + σ3 ) · (σ3 , j + σ1 ) = 1 so that the three off-diagonal matrix elements will never be all equal to −1.
2
Lemma 6.12. Assume (j +1)− is not diagonal. Then j − is not graph-like. Proof. By assumption, there exist τ1 , τ2 ∈ Ej +1 (M) such that (∂j +1 τ1 , ∂j +1 τ2 ) = (τ1 , (j +1)− τ2 ) = 0, namely there exists σ3 ⊂ τ˙1 ∩ τ˙2 . If ρ ⊂ σ˙ 3 , ∂j (∂j +1 τi ) = 0 implies the existence of σi ⊂ τ˙i for which ρ ∈ σ˙ i . Setting λi = [σi , ρ], i = 1, 2, 3, (σi , j − σk ) = λi λk , the proof goes on as in Theorem 6.11. 2 In order to prove a general result on the possibility of j − to be graph-like, we shall exclude some trivial cases. Definition 6.13. We say that an operator A acting on 2 (Ej (M)) is irreducible if the only self-adjoint idempotent multiplication operators commuting with A are 0 and I . We say that a (connected) polyhedral complex is q-irreducible if q is the maximum number such that j + is irreducible for any j < q, and j − is irreducible for any j q.
626
F. Cipriani et al. / Journal of Functional Analysis 256 (2009) 603–634
Theorem 6.14. Let M be a q-irreducible p-dimensional polyhedral complex in Rp , j q. Then j − graph-like implies j = q. If q = p the above implication is indeed an equivalence. Proof. If j < q, j − is not graph-like by Lemma 6.12. If q = p we may choose the orientation for p-cells according to a given orientation of Rp . Then, if τ, τ ∈ Ep (M) have a face in common, i.e. (τ, p− τ ) = 0, such face receives opposite orientations from the embeddings in τ˙ , respectively τ˙ , i.e. (τ, p− τ ) = (∂p τ, ∂p τ ) = −1. 2 Definition 6.15. If M is a p-irreducible polyhedral complex in Rp , by the previous theorem, a graph G is associated to p− , and is constructed as follows. The set of vertices of G is Ep (M), while σ, σ ∈ Ep (M) are adjacent iff there is ρ ∈ Ep−1 (M) such that ρ ⊂ σ˙ , σ˙ . We call G the dual graph of M. In this way, clearly p− is unitary equivalent to the graph Laplacian of G. 7. Computation of the Novikov–Shubin numbers for fractal graphs Let us observe that a 1-dimensional regular CW-complex is the same as a simple graph, and boundedness means bounded degree. Recall that a simple graph G = (VG, EG) is a collection VG of objects, called vertices, and a collection EG of unordered pairs of distinct vertices, called edges. We call a self-similar 1-dimensional CW-complex simply a self-similar graph. The results in this section will allow us to calculate α0 = α1 of some self-similar graphs, and also αp of a p-irreducible prefractal complex in Rp , in the sense of Definition 6.13. In the rest of this section, G is a countably infinite graph with bounded degree. We denote by the Laplacian on 0-cells (points), hence = C − A, where C is a diagonal operator, with C(x, x) = c(x) the number of edges starting from the point x, and A is the adjacency matrix. Let P be the transition operator, i.e. p(x, y) is the transition probability from x to y of the simple random walk on G. Let A0 be a C ∗ -algebra of operators, acting on 2 (G) = 2 (E0 (G)), which contains , and possesses a finite trace τ0 . We can also consider the Hilbert space 2 (G, c) with scalar product (v, w)c = x∈G c(x)v x wx . On this space the transition operator P is selfadjoint, and the Laplace operator is defined as c = I − P . Since G has bounded degree, C is bounded from above by a multiple of the identity, and, since G is connected, it is bounded from below by the identity. Also, the two spaces 2 (G) and 2 (G, c) coincide as topological vector spaces, with the obvious identification. With this identification we have (u, v) = (u, c v)c . We may also identify operators on the two spaces, hence the C ∗ -algebras A0 , respectively A0,c , acting on 2 (G), respectively 2 (G, c), can be identified as topological algebras. We use this identification to carry the trace τ0 onto A0,c . Theorem 7.1. Let μ be the maximal degree of G. Then τ0
1 1 + μtc
τ0
1 1 + t
τ0
1 , 1 + tc
t 0.
Proof. Let us consider the positive self-adjoint operator Q = C −1/2 C −1/2 on 2 (G). Since C and belong to A0 , Q ∈ A0 too. Since A = CP , we have Q = C 1/2 (I − P )C −1/2 . Clearly Q Q1/2 CQ1/2 μQ, hence, by operator monotonicity, τ0
1 1 + μtQ
τ0
1 1 + tQ1/2 CQ1/2
τ0
1 , 1 + tQ
t 0.
F. Cipriani et al. / Journal of Functional Analysis 256 (2009) 603–634
627
Now observe that, for sufficiently small t, τ0
1 1 + tQ
=
∞ ∞
(−t)n τ0 Qn = (−t)n τ0 nc = τ0 n=0
n=0
1 . 1 + tc
Since both the left- and the right-hand side are analytic functions for t 0, they coincide for any t 0. Analogously, τ0
1 1 + tQ1/2 CQ1/2
The result follows.
=
∞ ∞
n
= (−t)n τ0 Q1/2 CQ1/2 (−t)n τ0 n = τ0 n=0
n=0
1 . 1 + t
2
In order to prove the main result of this section, we need a Tauberian theorem. It is a quite simple modification of a theorem of de Haan and Stadtmüller, cf. [5, Theorem 2.10.2], and, on the same book, also Theorem 1.7.6, by Karamata, showing that the bound α < 1 below is a natural one. Definition 7.2. (i) Let us denote by OR(1) the space of positive, non increasing functions f on [0, ∞) such that ∃T > 0, c > 0, α ∈ (0, 1) such that f (λt) cλ−α , f (t)
∀λ > 0, t T .
(7.1)
(ii) If f, g are functions on [a, +∞) we write f g, t → ∞ if ∃T a, k > 1 such that k −1 g(t) f (t) kg(t),
t T.
(7.2)
Remark 7.3. If the functions are bounded and defined on [0, ∞), we may equivalently assume T = 0 both in Eqs. (7.1) and (7.2), possibly changing the constants c and k. Let us now denote by fˆ the Laplace transform of f , fˆ(t) = t
∞
e−ts f (s) ds.
0
Lemma 7.4. Let f be a positive bounded function. Then f ∈ OR(1) iff fˆ(1/·) ∈ OR(1). In this case f fˆ(1/·),
t → ∞.
Proof. Let us notice that since f is bounded also fˆ is bounded, hence, according to Remark 7.3, the properties above should hold for all t 0. Now observe that fˆ(s) =
∞ 0
e−y f (y/s) dy,
628
F. Cipriani et al. / Journal of Functional Analysis 256 (2009) 603–634
hence fˆ(1/s) =
∞ e
−y
x f (sy) dy f (xs)
0
e−y dy,
(7.3)
0
giving f (t)
ex ˆ f (x/t), ex − 1
∀x, t.
(7.4)
Now assume f ∈ OR(1). By (7.3) we get fˆ(1/t) f (λt) = e−λ dλ. f (t) f (t) ∞
0
Therefore, splitting the domain of integration and using property OR(1), we get fˆ(1/t) c λ−α e−λ dλ + e−λ dλ cΓ (1 − α) + 1. f (t) ∞
1
0
1
This, together with (7.4) for x = 1, implies f fˆ(1/·), t → ∞. Moreover, f (λt) f (λt) fˆ(1/(λt)) c λ−α , c c ˆ ˆ f (t) f (1/t) f (1/t) showing that fˆ(1/·) ∈ OR(1). Now assume fˆ(1/·) ∈ OR(1). Then, by (7.4) with x = 1,
fˆ(s) =
a e
−y
∞ f (y/s) dy + a
0
e e−1 c
e−y f (y/s) dy
a e
−y
fˆ(s/y) dy + f (a/s)
e−y dy
a
0
e ˆ f (s) e−1
∞
a 0
y −α e−y dy + f (a/s).
F. Cipriani et al. / Journal of Functional Analysis 256 (2009) 603–634 e Choosing a sufficiently small, we get c e−1 using (7.4) with x = a, we get
a 0
629
y −α e−y dy 1/2, hence fˆ(s) 2f (a/s). Now,
f (λt) 2c ea −α 2ea fˆ λta a λ , a a f (t) e − 1 fˆ e −1 t
namely f ∈ OR(1). The proof is complete.
2
Corollary 7.5. Let (A, τ ) be a C ∗ -algebra with a finite trace, A be a positive element of A. Then τ (e−tA ) ∈ OR(1) iff τ ((1 + tA)−1 ) ∈ OR(1). In this case τ (e−tA ) τ ((1 + tA)−1 ), t → ∞. Proof. Let us consider, in the von Neumann algebra ∞ of the GNS representation of τ , the function N(t) = τ (e[0,t) (A)). Then f (t) := τ (e−tA ) = 0 e−ts dN (s), hence its Laplace transform is fˆ(x) = x
∞
∞ dN(s)
0
dt e
−t (s+x)
∞ =
0
The result now follows from the lemma above.
0
1 x dN (s) = τ . s +x 1 + x −1 A
2
Now we come back to the Laplacians on G. Corollary 7.6. Let G be a countably infinite connected graph with bounded degree. Let , respectively c , be the homological, respectively probabilistic Laplacian. Let A0 be a C ∗ -algebra of operators, acting on 2 (G) = 2 (E0 (G)), which contains (hence c ), and possesses a finite trace τ0 . Consider the functions ϑ(t) = τ0 (e−t ), ϑc (t) = τ0 (e−tc ). Then ϑ ∈ OR(1) iff ϑc ∈ OR(1). In this case ϑ ϑc for t → ∞. Proof. Assume ϑ ∈ OR(1). Then, by Corollary 7.5, ϑ(t) τ0 ((1 + t)−1 ), t → ∞ and τ0 ((1 + t)−1 ) ∈ OR(1). Therefore, recalling Theorem 7.1, and denoting by μ the maximum degree of G, we get 1
τ0 ((1 + tc )−1 ) τ0 ((1 + μ−1 t)−1 ) cμα , τ0 ((1 + t)−1 ) τ0 ((1 + t)−1 )
namely τ0 ((1 + tc )−1 ) τ0 ((1 + t)−1 ). Therefore τ0 ((1 + tc )−1 ) ∈ OR(1). Applying Corollary 7.5 again we get ϑc ∈ OR(1) and ϑ ϑc . The converse implication is proved analogously. 2 Now we relate the large n asymptotics of the probability of returning to a point in n steps with the large time heat kernel asymptotics. Since for bipartite graphs the probability is zero for odd n, the estimates are generally given in terms of the sum of the n-step plus the (n + 1)-step return probability. In order to match the above treatment we shall use a suitable mean for the return probability, namely the trace of the nth power of the transition operator P . First we need the auxiliary function described in the following
630
F. Cipriani et al. / Journal of Functional Analysis 256 (2009) 603–634
Lemma 7.7. Let us denote by ϕγ , γ > 0, the function x −γ
x
ϕγ (x) := e x
e−t d t γ ,
x 0.
0
Then ϕγ extends to the entire function −1 ∞ n x n+γ , n n!
x ∈ C.
(7.5)
n=0
Proof. Let us observe that the power series in (7.5) is an entire function ϕ satisfying ⎧ γ γ ⎨ ϕ+ , ϕ = 1− x x ⎩ ϕ(0) = 1. It is easy to see that ϕγ is the unique solution of the differential equation in (0, +∞) which tends to 1 when x → 0+ . 2 Theorem 7.8. Let G be a countably infinite connected graph with bounded degree. Let A0 be a C ∗ -algebra of operators, acting on 2 (G) = 2 (E0 (G)), which contains the homological Laplacian and possesses a finite trace τ0 . If τ0 (P n + P n+1 ) n−γ , then ϑc (t) t −γ , t → ∞. Proof. Let us observe that e−x x γ ϕγ (x) → γ (γ ),
x → +∞,
and that
n+γ n
n−γ → 1,
n → +∞.
On the one hand, we have ∞ n ∞ n
t t −γ τ0 P n Ke−t n ϑc (t) = e−t τ0 etP = e−t n! n!
K e−t
∞ n t n=0
n!
On the other hand, setting ψ(t) = ∞ n=0 e−t (ψ(t) + ψ (t)). Let us note that ψ (t) =
n=0
n+γ n
−1
n=0
= K e−t ϕγ (t) K t −γ .
tn n −t n! τ0 (P ), we get ϑc (t) = e ψ(t), and 2ϑc (t)+ϑc (t) =
∞ ∞ n
t n−1 t τ0 P n = τ0 P n+1 , (n − 1)! n! n=1
n=0
F. Cipriani et al. / Journal of Functional Analysis 256 (2009) 603–634
631
therefore ψ(t) + ψ (t) =
∞ n t n=0
∞ n
t −γ τ0 P n + P n+1 c n c et t −γ . n! n! n=0
Finally, since ϑc is negative,
2ϑc 2ϑc (t) + ϑc (t) = e−t ψ(t) + ψ (t) c t −γ . The thesis follows.
2
Corollary 7.9. Let G be a countably infinite connected graph with bounded degree. Let A0 be a C ∗ -algebra of operators, acting on 2 (G) = 2 (E0 (G)), which contains , and possesses a finite trace τ0 . Assume τ0 (P n + P n+1 ) n−γ , for γ > 0. (i) If γ ∈ (0, 1), then the Novikov–Shubin number α(G) = 2γ . (ii) If γ > 0 and G has constant degree, then α(G) = 2γ . Proof. (i) It follows from Corollary 7.6 and Theorem 7.8. (ii) It follows from Theorem 7.8 and the observation that if the degree is constantly equal to μ, then τ0 (e−t ) = τ0 (e−μtc ). 2 Corollary 7.10. Let G be a countably infinite connected graph with bounded degree. Let A0 be a C ∗ -algebra of operators, acting on 2 (G) = 2 (E0 (G)), which contains , and possesses a finite trace τ0 . Denote by pn (x, y) the (x, y)-element of the matrix P n , which is the probability that a simple random walk started at x reaches y in n steps. Assume that there are γ ∈ (0, 1), c1 , c2 > 0 such that, for all x ∈ G, n ∈ N, pn (x, x) c2 n−γ , pn (x, x) + pn+1 (x, x) c1 n−γ .
(7.6)
Then the Novikov–Shubin number α(G) = 2γ . Proof. It is just a restatement of the previous corollary.
2
As Novikov–Shubin numbers of covering manifolds are large scale invariants, one expects that graphs which are asymptotically close should have the same Novikov–Shubin number. We show that this happens in case of roughly isometric graphs. Definition 7.11. Let G1 , G2 be infinite graphs with bounded degree. A map ϕ : G1 → G2 is called a rough isometry if there are a, b, M > 0 such that (i) a −1 d1 (x, y) − b d2 (ϕ(x), ϕ(y)) ad1 (x, y) + b, for x, y ∈ G1 , (ii) d2 (ϕ(G1 ), y) M, for y ∈ G2 . Then G1 and G2 are said to be roughly isometric.
632
F. Cipriani et al. / Journal of Functional Analysis 256 (2009) 603–634
Observe that being roughly isometric is an equivalence relation. Theorem 7.12. Let G1 , G2 be roughly isometric, infinite graphs with bounded degree. For j = 1, 2, let Aj be a C ∗ -algebra of operators, acting on 2 (Gj ) = 2 (E0 (Gj )), which contains the Laplace operator j of the graph Gj , and possesses a finite trace τj . Assume G1 satisfies (7.6) with γ < 1, then G2 does as well. As a consequence, α0 (G1 ) = α0 (G2 ). Proof. It is a consequence of Corollary 7.10, [11, Theorem 3.1], [3, Lemma 1.1], and [20, Theorem 5.11]. 2 8. Examples In this section, we compute the Novikov–Shubin numbers of some explicit examples. Our first class of examples is that of nested fractal graphs, namely prefractal graphs as in Section 6, based on a particular class of self-similar fractals, called nested fractals, see [20] for a definition.2 Assume we are given a nested fractal K in Rp determined by similarities w1 , . . . , wq , with the same similarity parameter, and let S be the Hausdorff dimension of K in the resistance metric [21]. Let M be the nested fractal graph based on K. Theorem 8.1. Let K, S, and M be as above. Then (7.6) hold for M, with γ = Therefore, α0 (M) = 2γ . Proof. The thesis follows from [20, Corollary 4.13], and Corollary 7.10 above.
S S+1
∈ (0, 1).
2
Example 8.2. Using the previous theorem we can compute α0 for some self-similar graphs coming from fractal sets as in Section 6. Moreover, since β0 = 0 by the estimate in Theorem 7.8, we get, by Corollary 5.8, β1 = −χ , the latter being explicitly computed in Example 5.9. log 3 1 For the Gasket graph in Fig. 1 we obtain α0 = 2log 5 , see [2], β0 = 0, and β1 = 2 . For the Vicsek graph in Fig. 3 we obtain α0 =
2 log 5 log 15 ,
For the Lindstrom graph in Fig. 2 we obtain α0 = β0 = 0, and β1 =
see [22], β0 = 0, and β1 = 14 .
2 log 7 log 12.89027
computed numerically, see [23],
1 3.
Our second class of examples is given by the following Proposition 8.3. Let M be a p-irreducible prefractal complex in Rp , let G be the dual graph of M, as in Definition 6.15, and assume that (7.6) hold on G. Then the Novikov–Shubin number αp (M, ∂M) = 2γ . Proof. It is a consequence of Corollary 7.10.
2
Example 8.4. Let us consider the 2-dimensional complex M in Fig. 5. We want to compute its second relative Novikov–Shubin number α2 (M, ∂M). If we extend the definition of self-similar 2 The constructions of the graph from a fractal described here and in [20] are slightly different, however the corresponding graphs are roughly isometric.
F. Cipriani et al. / Journal of Functional Analysis 256 (2009) 603–634
633
Fig. 5. Dodecagon 2-complex.
Fig. 6. Carpet graph (following Barlow).
Fig. 7. Dual Carpet graph.
CW-complex as explained in Remark 4.3, it is easy to see that the complex in Fig. 5 is selfsimilar and its dual graph, defined in Definition 6.15, coincides with the Gasket graph considered log 3 in Fig. 1. Therefore, by Proposition 8.3, α2 (M, ∂M) = 2log 5 . Example 8.5. The Carpet 2-complex M in Fig. 4 is an example of a 2-dimensional self-similar CW-complex. Barlow [2] associates to M the graph G1 in Fig. 6, which, by [2] Theorem 3.4, log 8 satisfies the estimates in Corollary 7.10, with γ = log(8ρ) , where ρ ∈ [ 76 , 32 ], while computer calculations suggest that ρ ∼ = 1.251. The dual graph of M, as in Definition 6.15, is the graph G2 in Fig. 7, also associated to M by Barlow and Bass in [4]. It is easy to see that the graphs G1 and G2 are roughly-isometric. Therefore, by Theorem 7.12, α2 (M, ∂M) = α(2− ) = 2γ (so that α2 (M, ∂M) ∈ [1.67, 1.87]).
634
F. Cipriani et al. / Journal of Functional Analysis 256 (2009) 603–634
References [1] M.F. Atiyah, Elliptic operators, discrete groups and von Neumann algebras, Astérisque 32–33 (1976) 43–72. [2] M.T. Barlow, Heat kernels and sets with fractal structure, in: Heat Kernels and Analysis on Manifolds, Graphs, and Metric Spaces, Paris, 2002, in: Contemp. Math., vol. 338, Amer. Math. Soc., Providence, RI, 2003, pp. 11–40. [3] M.T. Barlow, Which values of the volume growth and escape time exponents are possible for a graph? Rev. Math. Iberoamericana 20 (2004) 1–31. [4] M.T. Barlow, R.F. Bass, Random walks on graphical Sierpinski carpets, in: Random Walks and Discrete Potential Theory, Cortona, 1997, in: Sympos. Math., vol. XXXIX, Cambridge Univ. Press, Cambridge, 1999, pp. 26–55. [5] N.H. Bingham, C.M. Goldie, J.L. Teugels, Regular Variation, Cambridge Univ. Press, Cambridge, 1987. [6] F. Fidaleo, D. Guido, T. Isola, Bose Einstein condensation on inhomogeneous amenable graphs, preprint, 2008. [7] J. Dodziuk, De Rham–Hodge theory for L2 -cohomology of infinite coverings, Topology 16 (1977) 157–165. [8] J. Dodziuk, V. Mathai, Approximating L2 invariants of amenable covering spaces: A combinatorial approach, J. Funct. Anal. 154 (1998) 359–378. [9] G. Elek, Combinatorial heat kernels and index theorems, J. Funct. Anal. 129 (1995) 64–79. [10] G. Elek, Aperiodic order, integrated density of states and the continuous algebras of John von Neumann, preprint, arXiv: math-ph/0606061, 2006. [11] A. Grigor’yan, A. Telcs, Harnack inequalities and sub-Gaussian estimates for random walks, Math. Ann. 324 (2002) 521–556. [12] M. Gromov, M. Shubin, Von Neumann spectra near zero, Geom. Funct. Anal. 1 (1991) 375–404. [13] D. Guido, T. Isola, Noncommutative Riemann integration and singular traces for C ∗ -algebras, J. Funct. Anal. 176 (2000) 115–152. [14] D. Guido, T. Isola, Fractals in Noncommutative Geometry, in: R. Longo (Ed.), Proceedings of the Conference Mathematical Physics in Mathematics and Physics, Siena 2000, in: Fields Inst. Commun., vol. 30, Amer. Math. Soc., Providence, RI, 2001. [15] D. Guido, T. Isola, Dimensions and singular traces for spectral triples, with applications to fractals, J. Funct. Anal. 203 (2003) 362–400. [16] D. Guido, T. Isola, Dimensions and spectral triples for fractals in RN , in: F. Boca, O. Bratteli, R. Longo, H. Siedentop (Eds.), Advances in Operator Algebras and Mathematical Physics, Proceedings of the Conference, Sinaia, Romania, June 2003, in: Theta Ser. Adv. Math., Theta, Bucharest, 2005. [17] D. Guido, T. Isola, Tangential dimensions I. Metric spaces, Houston J. Math. 31 (2005) 1023–1045. [18] D. Guido, T. Isola, Tangential dimensions II. Measures, Houston J. Math. 32 (2006) 423–444. [19] D. Guido, T. Isola, M.L. Lapidus, A trace on fractal graphs and the Ihara zeta function, Trans. Amer. Math. Soc., in press. [20] B.M. Hambly, T. Kumagai, Heat kernel estimates for symmetric random walks on a class of fractal graphs and stability under rough isometries, in: Proc. Sympos. Pure Math., vol. 72, part 2, Amer. Math. Soc., Providence, RI, 2004, pp. 233–259. [21] J. Kigami, Effective resistances for harmonic structures on p.c.f. self-similar sets, Math. Proc. Cambridge Philos. Soc. 115 (1994) 291–303. [22] J. Kigami, M.L. Lapidus, Self-similarity of volume measures for Laplacians on p.c.f. self-similar fractals, Comm. Math. Phys. 217 (2001) 165–180. [23] T. Kumagai, Estimates of transition densities for Brownian motion on nested fractals, Probab. Theory Related Fields 96 (1993) 205–224. [24] D. Lenz, P. Stollmann, Aperiodic order and quasicrystals: Spectral properties, Ann. Henri Poincaré 4 (2) (2003) 933–942. [25] J. Lott, W. Lück, L2 -topological invariants of 3-manifolds, Invent. Math. 120 (1995) 15–60. [26] A.T. Lundell, S. Weingram, The Topology of CW-Complexes, Van Nostrand–Reinhold, New York, 1969. [27] J.R. Munkres, Elements of Algebraic Topology, Addison–Wesley, Menlo Park, CA, 1984. [28] S.P. Novikov, M.A. Shubin, Morse Theory and von Neumann II1 Factors, Dokl. Akad. Nauk SSSR 289 (1986) 289–292. [29] J. Roe, An index theorem on open manifolds. I, II, J. Differential Geom. 27 (1988) 87–113, 115–136. [30] J. Roe, On the quasi-isometry invariance of L2 -Betti numbers, Duke Math. J. 59 (1989) 765–783. [31] G. Elek, L2 -spectral invariants and convergent sequences of finite graphs, J. Funct. Anal. 254 (2008) 2667–2689. [32] S. Kobayashi, K. Nomizu, Foundations of Differential Geometry, vol. I, Interscience, New York, 1963.
Journal of Functional Analysis 256 (2009) 635–672 www.elsevier.com/locate/jfa
Weighted irregular Gabor tight frames and dual systems using windows in the Schwartz class Jean-Pierre Gabardo 1 Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario L8S 4K1, Canada Received 21 June 2007; accepted 31 October 2008 Available online 2 December 2008 Communicated by L. Gross
Abstract We give a characterization for the weighted irregular Gabor tight frames or dual systems in L2 (Rn ) in terms of the distributional symplectic Fourier transform of a positive Borel measure on R2n naturally associated with the system and the short-time Fourier transform of the windows in the case where the window (or at least one of the windows in the case of dual systems) belongs to S(Rn ). This result implies that, for certain classes of windows such as generalized Gaussians or “extreme-value” windows, the only weighted irregular Gabor tight frames (or even dual systems with both windows in the same class) that can be constructed with these windows are the trivial ones, corresponding to the measure μ = 1 on R2n . Furthermore, we show that, if a such Gabor system admits a dual which is of Gabor type, then the Beurling density of the associated measure exists and is equal to one. © 2008 Elsevier Inc. All rights reserved. Keywords: Irregular Gabor systems; Translation-bounded measures; Parseval frames; Gabor duality
1. Introduction Let g ∈ L2 (Rn ) be a window function with g2 = 1. The corresponding short-time Fourier transform is the mapping Vg : L2 (Rn ) → L2 (R2n ) defined, for f ∈ L2 (Rn ), by Vg f (x, ν) = f (t)e−2πiν·t g(t − x) dt, (x, ν) ∈ R2n . Rn
E-mail address: [email protected]. 1 Supported by an NSERC grant.
0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.10.025
636
J.-P. Gabardo / Journal of Functional Analysis 256 (2009) 635–672
(Note that in the following, R2n will be identified with Rn × Rn , and thus, the notation (x, ν) ∈ R2n means, in particular, that both x and ν belong to Rn .) The following result is central in the theory of the short-time Fourier transform (see [9, Corollary 3.2.2]). Theorem 1.1. Given g ∈ L2 (Rn ) with g2 = 1, we have Vg f (x, ν)2 dx dν = f (t)2 dt,
f ∈ L2 R n .
(1.1)
Rn
R2n
The short-time Fourier transform is thus, for a fixed g as above, an isometric linear transformation from L2 (Rn ) onto a closed proper subspace of L2 (R2n ) consisting of continuous functions. The identity (1.1) immediately yields an integral representation formula for functions in L2 (Rn ) in terms of the continuous Gabor system {e2πiν·t g(t − x)}(x,ν)∈R2n : f (t) =
Vg f (x, ν)e2πiν·t g(t − x) dx dν,
f ∈ L2 R n .
(1.2)
R2n
This formula represents f as a continuous superposition of elementary signals given by modulations and translations of the window function g. One major goal of Gabor analysis is to obtain discrete representation for functions in L2 (Rn ) analogous to the one given in (1.2). In fact, most of the work done in Gabor analysis concerns systems of the form {e2πibl·t g(t − ak)}(k,l)∈Zn ×Zn , where a, b > 0 are two parameters. More recently, several authors have begun to investigate irregular Gabor systems {e2πiν·t g(t − x)}(x,ν)∈Λ where Λ is a discrete set in the time–frequency space [1–5,7,18,19,21,27–29] as well as weighted irregular Gabor systems, i.e. systems of the form {w(x, ν)1/2 e2πiν·t g(t − x)}(x,ν)∈Λ where Λ is a discrete set in the time–frequency space and w is a positive function defined on Λ [16]. (The term “irregular” in this context might be somewhat misleading as it also includes the “regular” case.) If such a system forms a Parseval tight frame for L2 (Rn ), we have, by definition, 2 2 (1.3) w(x, ν)Vg f (x, ν) = f (t) dt, f ∈ L2 Rn . (x,ν)∈Λ
Rn
Introducing the positive measure μ on R2n defined by μ= w(x, ν)δ(x,ν) , (x,ν)∈Λ
where δ(x,ν) denotes the Dirac mass at the point (x, ν), we can rewrite Eq. (1.3) as Vg f (x, ν)2 dμ(x, ν) = f (t)2 dt, f ∈ L2 Rn .
(1.4)
Rn
R2n
The analogue of the reconstruction formula (1.2) now reads f (t) = Vg f (x, ν)e2πiν·t g(t − x) dμ(x, ν), R2n
f ∈ L2 R n .
(1.5)
J.-P. Gabardo / Journal of Functional Analysis 256 (2009) 635–672
637
The discrete and continuous Gabor expansions considered above lead naturally to the following general question: given a window g ∈ L2 (Rn ), which positive Borel measure μ on R2n satisfy the identity (1.4)? It is also clear that the problem of constructing discrete, weighted irregular Gabor tight frames with a given window g is equivalent to finding discrete measures μ which satisfy the identity (1.4). Another question that we will address is the following. We know that, if g has norm 1, the Lebesgue measure on R2n , which we identify with the constant function 1, is always a solution of our problem (1.4). Are there windows g ∈ L2 (R) for which this positive measure is the only solution of our problem? In such cases, it is clear that no (discrete) weighted irregular Gabor system constructed with the given windows will yield a tight frame for L2 (Rn ). For technical reasons, we will restrict our analysis to windows belonging to the Schwartz class (or, with at least one of the windows in the Schwartz class in the case of dual systems). Indeed, these windows are well suited for Gabor analysis as they are very well localized in the time– frequency space. More importantly, it is not even clear how to state our main results in general, if this assumption is not made, as they would involve products of distributions which are not well defined. The paper is organized as follows. One of our main goals will be to obtain a characterization of the positive measures μ on R2n which satisfy the identity (1.4) in the case where the window function g belongs to the Schwartz space S(Rn ). This characterization involves the symplectic Fourier transform of the measure and, in order to make sense of it, we first need to show that any positive measure μ satisfying the identity (1.4) for a given window g must be a tempered measure. In fact, we show in Section 2 that if a window g ∈ L2 (R) satisfies the Bessel condition with respect to the measure μ, which means that there exists a constant C1 0, such that R2n
Vg f (x, ν)2 dμ(x, ν) C1
f (t)2 dt,
f ∈ L2 R n ,
(1.6)
Rn
then the measure μ must be translation-bounded and thus tempered (Proposition 2.2). Conversely, we also show that the Bessel condition (1.6) always holds if μ is translation-bounded and the window g belongs to the Schwartz class (Proposition 2.4). In Section 3, we state our characterization of the identity (1.4) (Theorem 3.2) and use it to show that the uniqueness of the measure μ appearing in (1.4) for a fixed window g is equivalent to the non-vanishing of the short-time Fourier transform Vg g (Corollary 3.3). We show, in particular that this last property is satisfied by the class of generalized Gaussians windows (Theorem 3.10) and also by the so-called “extreme-value” windows (Proposition 3.11). This implies, of course, that no discrete, weighted irregular Gabor system can yield a tight frame for L2 (Rn ) for such windows. In the case of even windows, we prove, using a result of Hudson [14], that the generalized Gaussians are the only ones with the property that Vg g does not vanish and thus for which the uniqueness of the measure in (1.4) holds (Proposition 3.8). In the case of odd windows, non-uniqueness always occurs (Proposition 3.9). We also show that, if the window is a generalized Gaussian multiplied by a polynomial, then, although there might be more than one measure μ satisfying (1.4), none of these can be discrete, thus preventing again the existence of discrete irregular Gabor tight frame constructed with such windows (Theorem 3.13). In Section 4, we consider the problem of characterizing the “duals of Gabor type” associated with a window g in the Schwartz class and a positive Borel measure μ defined on the time– frequency space (see Definition 4.1). Although it is known that, a standard dual can always be associated with a given frame (even in the case of frames associated with measures; see [8] for
638
J.-P. Gabardo / Journal of Functional Analysis 256 (2009) 635–672
more details), in the case of irregular Gabor systems, it is possible that no dual of Gabor type exists, contrary to the situation for “regular Gabor systems.” We obtain a characterization for the possible duals of Gabor type, h, associated with a pair (g, μ), where g is in the Schwartz class in terms of the symplectic Fourier transform of the measure μ and of the short-time Fourier transform Vg h (Theorem 4.6). The characterization for Gabor duality obtained is also valid for a weaker definition of Gabor duality in which the dual h is allowed to be a tempered distribution and the expansion associated with it takes place in S (Rn ) instead of L2 (Rn ) (Theorem 4.4). In both situations, the characterization obtained implies, in particular, that the Fourier transform of the measure μ must be equal to a positive multiple of the Dirac mass at the origin on a neighborhood of the origin (Proposition 4.8), a property which is always destroyed by applying a non-zero local perturbation to the measure μ. Since the frame property itself is not so sensitive to local perturbations, it follows that any such small non-zero perturbation of a Gabor system admitting a dual of Gabor type will result in a Gabor frame admitting no such dual (even if the window is replaced by another window in the Schwartz class). As in Section 3, we again obtain a characterization for the uniqueness of a measure μ with the property that h is a dual of Gabor type for the pair (g, μ): the short-time Fourier transform Vg h cannot vanish simultaneously at any points (x, ν) and (−x, −ν) in the time–frequency space (Corollary 4.7). Using this characterization, we extend the results of Section 3 to dual systems and show that if g and h are both generalized Gaussian (associated with possibly different parameters) and (g, h)2 = 1, then the only measure μ such that h is a dual of Gabor type for the pair (g, μ) is the trivial one, μ = 1 (Theorem 4.12). A similar result holds for the extreme-value windows (Proposition 4.15). We also show that if g and h are both generalized Gaussian multiplied by polynomials, the possible measures μ allowing for Gabor duality between these windows have to be non-discrete, preventing thus discrete expansions in terms of such dual Gabor windows to exist (Theorem 4.14). Theorem 4.6 is used 2 again, in the case where the window g is the one-dimensional Gaussian g(t) = 21/4 e−πt , to show that, if a dual of Gabor type exists for the pair (g, μ), then the symplectic Fourier transform of μ must be supported on a discrete set and that the discrete sets obtained in this way can be characterized as certain subsets of the zero sets of the entire functions F (z) in the Bargmann– Fock space F 2 (C) satisfying F (0) = 1 (Proposition 4.9). In Section 5, we point out that, when applied to unweighted regular Gabor systems, where the sampling set Λ in the time–frequency space is a full-rank lattice, our main results translate into some well-known results of Gabor analysis such as the Ron–Shen or the Wexler–Raz identity (Theorems 5.1 and 5.2) and their generalizations (of course, with the restriction that at least one of the windows involved belong to the Schwartz class). In the last section of this paper, we define the Beurling density of a positive Borel measure μ on R2n and show that the existence of a dual of Gabor type for a Gabor system (g, μ), with g in the Schwartz class, implies that the Beurling density of μ exists and is equal to one (Theorem 6.3). As an immediate consequence, it follows that, if a weighted, discrete irregular Gabor system associated with such a window g form a Parseval frame for L2 (Rn ), then the associate sampling set Λ in the time–frequency space must have a lower Beurling density at least equal to one (Corollary 6.4). Finally, we conclude this paper by displaying examples of discrete sampling sets Λ having arbitrary large Beurling density and having the property that the associated irregular Gabor system G := {e2πiνt g(t − x)}(x,ν)∈Λ , with g being any window in S(Rn ), does not admit a dual of Gabor type (Proposition 6.5). We point out that the bracket ·,·, linear in both variables, will be used to denote the duality between S (Rn ) and S(Rn ). We will use the notation (·,·)2 for the usual inner product in L2 (Rn ), which is thus linear in the first variable and anti-linear in the second. C0∞ (U ) will denote the space
J.-P. Gabardo / Journal of Functional Analysis 256 (2009) 635–672
639
of infinitely differentiable complex-valued functions with compact support in the open set U . If α, β are multi-indices in Nn , we will write α β (respectively α < β) when αi βi , for each i = 1, . . . , n (respectively αi β i , for each i = 1, . . . , n with strict inequality for at least one i). Also, |α| = i αi , βα = i αβii if α β, and, if ξ ∈ Rn , ξ α = i ξiαi . 2. Translation-bounded measures and the Bessel condition The notion of translation bounded measure is a useful tool in the theory of quasicrystals (see e.g. [17]). It will also play an important role in our following discussion as it characterizes the measures for which the Bessel condition holds for Gabor systems with a window in the Schwartz class as we will prove in this section. Definition 2.1. We say that a positive Borel measure μ on R2n is translation bounded if for every compact K ⊂ R2n , there exists a constant C > 0 such that μ(K + x) C,
∀x ∈ R2n .
(2.1)
Clearly, a measure will be translation bounded if the condition above holds for a fixed compact set with non-empty interior. Proposition 2.2. Let g ∈ L2 (Rn ) be a window function with g2 = 1 and suppose that μ is a positive Borel measure on R2n which satisfies the Bessel condition (1.6). Then, μ is translation bounded. In particular, μ is a tempered measure on R2n , i.e. there exists an integer M 1 such that 1 dμ(x, ν) < ∞. (1 + |x|2 + |ν|2 )M R2n
Proof. Note first that Vg g(0, 0) = 1 and thus, using the continuity of Vg g, there exists r > 0 such that |Vg g(x, ν)|2 12 if |x|2 + |ν|2 r 2 . Given any point (x0 , ν0 ) ∈ R2n , choose f ∈ L2 (Rn ) of the form f (t) = e2πit·ν0 g(t − x0 ). With this choice of f , we have Vg f (x, ν) = e−2πit·(ν−ν0 ) g(t − x0 )g(t − x) dt Rn
= e−2πix0 ·(ν−ν0 )
e−2πis·(ν−ν0 ) g(s)g s − (x − x0 ) ds
Rn
=e
−2πix0 ·(ν−ν0 )
Vg g(x − x0 , ν − ν0 ).
If |x − x0 |2 + |ν − ν0 |2 r 2 , we have thus Vg f (x, ν)2 = Vg g(x0 − x, ν − ν0 )2 1 2
640
J.-P. Gabardo / Journal of Functional Analysis 256 (2009) 635–672
and, letting Br (x0 , ν0 ) = {(x, ν) ∈ R2n , |x − x0 |2 + |ν − ν0 |2 r 2 }, we deduce that 1 2
1 dμ(x, ν) Br (x0 ,ν0 )
Vg f (x, ν)2 dμ(x, ν) C1 .
R2n
This shows that μ Br (x0 , ν0 ) 2C1 ,
∀(x0 , ν0 ) ∈ R2n ,
from which the conclusion of the lemma easily follows.
2
The condition (2.1) is, in general, not sufficient to guarantee that the Bessel condition (1.6) holds since it does not involve the window g. (See the paper [18] where sufficient conditions for the Bessel condition to hold are given in the case of unweighted discrete irregular Gabor frame.) 2πilt g(t − k)} For example, in the one-dimensional (k,l)∈Z2 case, the classical Gabor system {e corresponds to the measure μ = (k,l)∈Z2 δ(k,l) which is certainly translation bounded, but the system does not generate a Bessel collection for certain windows g ∈ L2 (R), since it is well known that a necessary and sufficient condition for this to happen is that the Zak transform of g be bounded a.e., while the Zak transform of an arbitrary function in g ∈ L2 (R) could be, when restricted to the set I 2 , any function in L2 (I 2 ), where I = [0, 1]. On the other hand, the following proposition shows that the condition (2.1) guarantees that the Bessel condition (1.6) holds for windows in the Schwartz class. Before stating it, we will need the following lemma [9, Lemma 11.3.3]. Lemma 2.3. Let g0 , g, γ ∈ S(Rn ) be such that (g, γ )2 = 0 and let f ∈ S (Rn ). Then, we have Vg f (x, ν) 0
1 |Vg f | ∗ |Vg0 γ | (x, ν), |(g, γ )2 |
(x, ν) ∈ R2n ,
(2.2)
where ∗ denotes the convolution product on R2n . Proposition 2.4. Let μ be a positive, translation bounded Borel measure on R2n and let g ∈ S(Rn ). Then, there exists a constant C1 0 such that (1.6) holds. Proof. We can assume that g = 0. Applying the inequality (2.2) in Lemma 2.3 with the functions g0 = γ = g ∈ S(Rn ) and f ∈ L2 (Rn ), we obtain that Vg f (x, ν) C |Vg f | ∗ |Vg g| (x, ν),
(x, ν) ∈ R2n ,
with C = 1/g22 . Using the Cauchy–Schwarz inequality, we have 2 |Vg f | ∗ |Vg g|(x, ν)
2 Vg f (y, ω) Vg g(x − y, ν − ω) dy dω = R2n
(2.3)
J.-P. Gabardo / Journal of Functional Analysis 256 (2009) 635–672
Vg f (y, ω)2 Vg g(x − y, ν − ω) dy dω
R2n
= C
641
Vg g(x − y, ν − ω) dy dω
R2n
Vg f (y, ω)2 Vg g(x − y, ν − ω) dy dω,
R2n
where C = Vg gL1 < ∞ since Vg g ∈ S(R2n ) by a result in [12] (see also [9, Theorem 11.2.5]). We can thus find, using (2.3), a constant C depending only on g such that
Vg f (x, ν)2 dμ(x, ν)
R2n
C
R2n
=C
Vg f (y, ω)2 Vg g(x − y, ν − ω) dy dω dμ(x, ν)
R2n
Vg f (y, ω)2
R2n
Vg g(x − y, ν − ω) dμ(x, ν) dy dω.
R2n
The translation boundedness of μ, i.e. (2.1), easily shows the existence of a constant D > 0 such that
Vg g(x − y, ν − ω) dμ(x, ν) D,
(y, ω) ∈ R2n .
R2n
We deduce thus the existence of a constant C1 > 0 such that
Vg f (x, ν)2 dμ(x, ν) C1
R2n
Vg f (y, ω)2 dy dω,
f ∈ L2 R n ,
R2n
and the conclusion follows from Theorem 1.1.
2
3. A characterization for tight Gabor systems Our next goal is to give a characterization for the positive Borel measures on R2n satisfying the identity (1.4) and to derive some immediate consequences of this theorem. Note that many of the results in this section are given without proofs as they are particular cases of analogous results valid for dual systems, which are proved in Section 4. We first need the following definition. Definition 3.1. If h ∈ L1 (R2n ), we define its Fourier transform, F h, by the formula (F h)(ξ, η) = R2n
e−2πi(ξ ·x+η·y) h(x, y) dx dy,
(ξ, η) ∈ R2n ,
642
J.-P. Gabardo / Journal of Functional Analysis 256 (2009) 635–672
while the symplectic Fourier transform of h, F S h, is given by the formula S F h (ξ, η) = e−2πi(η·x−ξ ·y) h(x, y) dx dy, (ξ, η) ∈ R2n . R2n
Note that (F S h)(ξ, η) = (F h)(η, −ξ ). The definition of the Fourier transform and that of the symplectic Fourier transform are extended in the usual way to the elements of S (R2n ): if σ ∈ S (R2n ) and ϕ ∈ S(R2n ), we have S F σ, ϕ = σ, F ϕ, F σ, ϕ = σ (ξ, η), F ϕ(−η, ξ ) . The following characterization for our generalized tight Gabor frames associated with a measure is actually a particular case of a more general result valid for dual systems which will be proved in Section 4 (Theorem 4.6). Theorem 3.2. Let μ be a positive, translation bounded Borel measure on R2n and let g ∈ S(Rn ). Then, the identity (1.4) holds for all f ∈ L2 (Rn ) if and only if S F μ (Vg g) = δ(0,0) on R2n . (3.1) Remark. The product on the left-hand side of (3.1) has to be understood as the product of a distribution in S (Rn ) with a function in S(Rn ) and is thus a well-defined tempered distribution. Note that this product might no longer make sense if we assumed that the window was just in L2 (Rn ) since the short-time Fourier transform Vg g might not necessarily be infinitely differentiable, but just continuous, and the symplectic Fourier transform of a positive, translation bounded Borel measure need not be a measure. For example, there exist real-valued functions in L∞ (R2n ) whose symplectic Fourier transforms are not measures (locally), and adding a small multiple of one of these to the function 1 will yield a density f (t) for a measure μ = f (t) dt which is positive, translation bounded, and such that F S (μ) is not a measure. Remark. We note that, for windows in S(Rn ), Theorem 1.1 follows immediately from the preceding theorem since F S (1) = δ(0,0) . The following corollary provides a very simple criterion to determine whether or not the Lebesgue measure on R2n (which we identify with the function 1) is the only measure satisfying (1.4) for all f ∈ L2 (Rn ). Corollary 3.3. Let g ∈ S(Rn ) satisfy g2 = 1. Then, the measure μ = 1 is the only translationbounded positive Borel measure on R2n satisfying the identity (1.4) for all f ∈ L2 (Rn ) if and only if Vg g(x, ν) = 0,
for all (x, ν) ∈ R2n .
(3.2)
Proof. Let g ∈ S(Rn ) satisfy g2 = 1. If (3.2) holds and μ is a translation-bounded measure satisfying (1.4), we have using (3.1) that S F (μ − 1) Vg g = 0 on R2n ,
J.-P. Gabardo / Journal of Functional Analysis 256 (2009) 635–672
643
and thus μ = 1. On the other hand, if, at some point (x0 , ν0 ) ∈ R2n , we have Vg g(x0 , ν0 ) = 0, it follows that Vg g(−x0 , −ν0 ) = 0 also, since Vg g(−x, −ν) = e−2πix·ν Vg g(x, ν),
(x, ν) ∈ R2n .
Therefore, using the identity
1 F S cos 2π(xν0 − νx0 ) = (δ(x0 ,ν0 ) + δ(−x0 ,−ν0 ) ), 2 we find that μ = 1 + cos(2π(xν0 − νx0 )) is a translation-bounded measure different from 1 which satisfies (3.1) and the result follows from Theorem 3.2. 2 Perhaps surprisingly, the even window functions g in the Schwartz class satisfying the uniqueness condition in the previous theorem which is equivalent to the non-vanishing of the function Vg g can be exactly characterized using a result of Hudson ([14]; see also [9, Theorem 4.4.1]): they must be “generalized Gaussians.” Before stating this result, we need the following definitions. Definition 3.4. A function g ∈ L2 (Rn ) of the form g(x) = e−Ax·x+2πb·x+c ,
x ∈ Rn ,
(3.3)
where A is an n × n invertible matrix with complex entries and positive-definite real part (A + A∗ )/2, where b ∈ Cn and c ∈ C, is called a generalized Gaussian. We can assume that A is symmetric in this definition since the function g is unchanged when A is replaced by (A + At )/2, where At denotes the transpose of A. The Wigner distribution, which was first introduced by Wigner, plays an important role in quantum mechanics. Definition 3.5. The Wigner distribution of a function f ∈ L2 (Rn ) is defined by Wf (x, ν) = f (x + t/2)f (x − t/2)e−2πiν·t dt, (x, ν) ∈ R2n .
(3.4)
Rn
Note that Wf is real-valued. Definition 3.6. The ambiguity function of a function f ∈ L2 (Rn ) is defined by Af (x, ν) = f (t + x/2)f (t − x/2)e−2πiν·t dt, (x, ν) ∈ R2n .
(3.5)
Rn
It satisfies Af (−x, −ν) = Af (x, ν) and is related to the short-time Fourier transform via the formula Ag(x, ν) = eiπx·ν Vg g(x, ν),
(x, ν) ∈ R2n .
(3.6)
644
J.-P. Gabardo / Journal of Functional Analysis 256 (2009) 635–672
We should point out that, although the formulas defining the Wigner distribution and the ambiguity function are very similar, these two functions can be very different. In particular, the sets where they vanish might be completely unrelated. (See the comments after Theorem 3.10.) The following result of Hudson completely characterizes as generalized Gaussians the functions in L2 (Rn ) with a non-vanishing Wigner distribution. Theorem 3.7. (See [14].) Let f ∈ L2 (Rn ). Then, Wf (x, ν) > 0 for all (x, ν) ∈ R2n if and only if f is a generalized Gaussian of the form (3.3). As an immediate consequence of this last result and of Corollary 3.3, we have the following. Proposition 3.8. Let g ∈ S(Rn ) be even (i.e. g(−x) = g(x), for all x ∈ Rn ) and satisfy g2 = 1. Then, the following are equivalent: (a) The measure μ = 1 is the only translation-bounded positive Borel measure on R2n satisfying the identity (1.4) for all f ∈ L2 (Rn ). (b) Vg g(x, ν) = 0, for all (x, ν) ∈ R2n . (c) g(x) = Ce−Ax·x , where C = 0 is a constant and A ∈ GL(n, C) is an n × n invertible matrix with positive-definite real part. Proof. The equivalence of (a) and (b) is the statement of Corollary 3.3. Furthermore, note that for any g ∈ L2 (Rn ) and g is even, we have Wg(x/2, ν/2)2−n = Ag(x, ν),
(x, ν) ∈ R2n ,
(3.7)
and, in particular, Wg(0, 0) = Ag(0, 0)2n = 2n > 0. Since Wg is real, the fact that Wg does not vanish means that Wg > 0 on R2n . By Hudson’s result [14] and the identity (3.6), the fact that Vg g is never zero is then equivalent to g being a generalized Gaussian of the form (c), using the evenness of g. 2 In the case of an odd window, we always have the non-uniqueness of the measure μ. Proposition 3.9. Let g ∈ S(Rn ) be odd (i.e. g(−x) = −g(x), for all x ∈ Rn ) and satisfy g2 = 1. Then, there is more then one translation-bounded Borel measure μ on R2n satisfying the identity (1.4) for all f ∈ L2 (Rn ). Proof. If g is odd, we have Wg(x/2, ν/2)2−n = −Ag(x, ν) = −eiπx·ν Vg g(x, ν),
(x, ν) ∈ R2n ,
(3.8)
and, in particular, Wg(0, 0) = −Ag(0, 0)2n = −2n < 0. On the other hand, it is known (see, e.g. [9, Lemma 4.3.6]) that, for any f ∈ L2 (Rn ), we have Wf (x, ν) dx dν = f 22 . R2n
J.-P. Gabardo / Journal of Functional Analysis 256 (2009) 635–672
645
Since Wg is real-valued, Wg must thus take on some positive values. The continuity of Wg together with the intermediate value theorem shows then that Wg(x0 , ν0 ) = 0 for some (x0 , ν0 ) ∈ R2n and the conclusion follows from Corollary 3.3. 2 Our next result shows that, whether they are even or not, the functions in L2 (Rn ) with a positive Wigner distribution, which by Hudson’s theorem are exactly the generalized Gaussians of the form (3.3), have a non-vanishing ambiguity function. Theorem 3.10. Let g be a generalized Gaussian of the form (3.3). Then, Vg g(x, ν) = 0 for all (x, ν) ∈ R2n and if, in addition g2 = 1, the measure μ = 1 is the only translation-bounded Borel measure for which (1.4) holds. In particular, no weighted irregular tight frame for L2 (Rn ) can be constructed using a generalized Gaussian as a window function. Proof. The fact that Vg g does not vanish follows from Lemma 4.11 (with h = g) which will be proved later. The remainder of the statement follows from Corollary 3.3 and the identity (1.3). 2 For a general window g ∈ S(Rn ), the positivity of the associated Wigner distribution implies thus the non-vanishing of the ambiguity function or, equivalently, that of Vg g. For windows that are either even or odd, the relations (3.7) and (3.8) between the ambiguity function and the Wigner distribution show that the non-vanishing of one is equivalent to that of the other. This suggests that, perhaps, these two properties are equivalent in general and that the generalized Gaussians are the only windows for which the ambiguity function does not vanish. However, for windows without (even or odd) symmetry this is not necessarily the case. For example, in one dimension, if g(x) = e−πx χ[0,∞) (x), we have A(g)(x, ν) =
e−π|x| e−iπ|x|ν = 0, 2π(1 + iν)
(x, ν) ∈ R2 ,
while W(g)(x, ν) = e−2πx
sin(4πxν) χ[0,∞) (x), πν
(x, ν) ∈ R2 ,
vanishes for all x 0. Of course, the previous window belongs to L2 (R) but not to the Schwartz class. Nevertheless, as we will show next, counterexamples in the Schwartz class can also be constructed. The so-called “extreme value” window function appears as a density function in probability theory and is defined by t
ψ(t) = et−e ,
t ∈ R.
More generally, we consider the functions ψk,m defined by t
ψk,m = ekt−me ,
t ∈ R,
646
J.-P. Gabardo / Journal of Functional Analysis 256 (2009) 635–672
where k, m > 0 are parameters. Clearly, ψk,m decays exponentially at −∞ and superexponentially at ∞. Furthermore, since t t t ψk,m (t) = k − met ekt−me = kekt−me − me(k+1)t−me , it follows, by induction, that the rth derivative of ψk,m is a linear combination of ψk+j,m , j = 0, . . . , r, and thus has also exponential decay at ±∞. Hence, ψk,m ∈ S(R). It is known that the Fourier transform of ψk,m can be expressed in terms of the Gamma function: F ψk,m (ξ ) = m−k+2πiξ (k − 2πiξ ),
ξ ∈ R.
The corresponding ambiguity function can also be expressed in terms of the Gamma function which yields the analogue of Theorem 3.10 for the normalized windows ψk,m . This result is a particular case of a more general result for dual systems (Proposition 4.15) which will be proved later. Proposition 3.11. We have −2k+2πiν
A(ψk,m )(x, ν) = m ex/2 + e−x/2 (2k − 2πiν). In particular, (a) A(ψk,m ) does not vanish anywhere. (b) If g(t) = (2m)k ((2k))−1/2 ψk,m , we have g2 = 1 and the measure μ = 1 is the only translation-bounded Borel measure for which (1.4) holds. Note that the Wigner distribution of ψk,m has to vanish somewhere in the time–frequency plane by Hudson’s theorem. Even in the case where the measure μ in (1.4) is not unique, it is possible that no discrete measure is solution of the problem which then again prevents the existence of discrete, weighted tight irregular Gabor frames associated with the given window. This will be the case if, for example, the zero set of Vg g is compact as we prove next. Proposition 3.12. Let g ∈ S(Rn ) satisfy g2 = 1 and suppose that the set (x, ν) ∈ R2n , Vg g(x, ν) = 0 is a compact subset of R2n . Then, any positive Borel measure μ satisfying the identity (1.4) must be absolutely continuous with respect to the Lebesgue measure, i.e. dμ = f (t) dt. Furthermore, f (t) is the restriction to Rn of an entire function of exponential type on Cn . Proof. Using (3.1) and our assumption, we deduce that F S μ, and thus also F μ, has compact support. The result follows then immediately from the Paley–Wiener–Schwartz theorem, see [24]. 2
J.-P. Gabardo / Journal of Functional Analysis 256 (2009) 635–672
Example. If we let g(t) = te−πt , for t ∈ R, we have then √
2 − π (x 2 +ν 2 ) 2 1 2 Ag(x, ν) = − x +ν − e 2 , 8 π
647
2
(x, ν) ∈ R2 ,
and the previous result applies, since if μ is any positive measure satisfying the identity (1.4), the support of F S μ will then be contained in the circle centered at the origin and of radius √1π in the time–frequency plane. The compactness of the zero set of Vg g implies the absolute continuity of the measure μ in the previous proposition but, in fact, much less is required to prevent that measure from being discrete as the following particular case of Theorem 4.14 shows. Theorem 3.13. Let g(x) = p(x)e−πAx·x+πa·x where A is a symmetric matrix in GL(n, C) with positive definite real part, a ∈ Cn and p(x) is a polynomial in n variables chosen so that g2 = 1. Then, no discrete, irregular weighted tight Gabor frame can be constructed using g(x) as a window function. Note that if the “tightness” restriction is removed, regular Gabor frames can be constructed with windows such as those appearing in the previous theorem. For example, Gröchenig and Lyubarski˘ı [11] have recently showed that the regular Gabor system associated with the Hermite 2 d n −πt 2 ) (e ) and a lattice A(Z2 ), where A is an invertible 2×2 real matrix, function Hn (t) = eπt ( dt 2 forms a frame for L (R) as long as |det(A)| < (n + 1)−1 . Our next result shows that the property of a translation-bounded measure μ to satisfy the identity (1.4) for a certain window function is destroyed by any local perturbation of that measure. Proposition 3.14. Let μ be a translation-bounded measure on R2n which satisfies the identity (1.4) for a certain window g0 ∈ S(Rn ) with g0 2 = 1. Then, there exists r > 0, such that F μ = δ(0,0)
on Br ,
(3.9)
where Br denotes the open ball of radius r centered at the origin in R2n . In particular, if K is a compact subset of R2n and ρ is a translation-bounded measure on R2n different from μ such that μ = ρ on R2n \ K, then ρ fails to satisfy the identity (1.4) for any window g ∈ S(Rn ) with g2 = 1. Proof. If μ satisfies (1.4) with the window g0 , then (3.9) follows immediately from the identity (3.1) in Theorem 3.2 since Vg0 g0 (0, 0) = 1. If ρ is as above and satisfies (1.4), then the measure μ − ρ is translation-bounded, compactly supported and has a Fourier transform that vanishes on Bs , for some s > 0. By the Paley–Wiener–Schwartz theorem [24], F (μ − ρ) is the restriction to R2n of an entire function of exponential type on C2n and has thus to vanish identically if it vanishes on Bs . This contradicts the fact that ρ = μ and proves our claim. 2 4. Dual windows of Gabor type In this section we consider the problem of constructing dual Gabor windows associated with a Gabor window belonging to the Schwartz class. The possibility of constructing a dual Gabor
648
J.-P. Gabardo / Journal of Functional Analysis 256 (2009) 635–672
window different than the original one is quite important since, as was seen in the previous section, the generalized Gabor expansions that can associated with certain windows (such as the generalized Gaussians) are quite limited if one insists on self-duality. In the following, we do not require the dual window to belong to the Schwartz class but we allow it to be a priori any function in L2 (Rn ) as in our next definition or even any tempered distribution as in Theorem 4.4. Definition 4.1. Given g ∈ S(Rn ) and a positive, translation-bounded measure μ on R2n , we say that the function h ∈ L2 (Rn ) is a dual window of Gabor type for the pair (g, μ) if h satisfies the Bessel condition (1.6) (with g replaced with h) with respect to μ and if we have (4.1) Vg f1 (x, ν)Vh f2 (x, ν) dμ(x, ν) = f1 (t)f2 (t) dt, f1 , f2 ∈ L2 Rn . Rn
R2n
Note that, under the conditions of the previous definition, the left-hand side of Eq. (4.1) is well defined since g satisfies the Bessel condition with respect to μ by Proposition 2.4. It is also worth mentioning that if a dual of Gabor type h exists for the pair (g, μ), there exists a constant C > 0 such that f 2 CVg f 2,μ holds for all f ∈ L2 (Rn ), by applying the Cauchy–Schwarz inequality (with f1 = f2 ) to the left-hand side of (4.1). The system (g, μ) is thus a “generalized frame” in the sense of the theory developed in [8]. Formula (4.1) leads directly to the following generalized Gabor expansion formulas in the case where h is a dual window for the pair (g, μ): Vg f (x, ν)e2πiν·t h(t − x) dμ(x, ν)
f (t) = R2n
f ∈ L2 R n .
Vh f (x, ν)e2πiν·t g(t − x) dμ(x, ν),
= R2n
Lemma 4.2. Let g ∈ S(Rn ) and h ∈ S (Rn ). Define ϕ ∈ S R2n .
(Kϕ)(x, ν) = g(t) ⊗ h(s), e−2πiν·(t−s) ϕ(t + x, s + x) , Then, K is a continuous mapping from S(R2n ) to itself. Proof. Note that if ϕ ∈ S(R2n ), we have
g(t) ⊗ h(s), ϕ(t, s) = h(s), g(t), ϕ(t, s)
where k(s) = g(t), ϕ(t, s) belongs to S(Rn ). For any integer m 0, let m ψSm = sup D α ψ(x) 1 + |x|2 ∞ , |α|m
ψ ∈ S Rn .
Define R[ϕ](x, ν, s) = g(t), e−2πiν·(t−s) ϕ(t + x, s + x) ,
ϕ ∈ S R2n .
J.-P. Gabardo / Journal of Functional Analysis 256 (2009) 635–672
649
If β1 , β2 and γ are multi-indices in Nn , we have γ
Dxβ1 Dνβ2 Ds R[ϕ](x, ν, s) γ
(2πiν)γ −γ1 g(t), e−2πiν·(t−s) D (0,γ1 ) ϕ(t + x, s + x) = Dxβ1 Dνβ2 γ1 γ1 γ
γ (2πi)γ −γ1 = Dxβ1 γ1 γ1 γ β2
β −δ (ν)γ −γ1 −δ g(t), e−2πiν·(t−s) 2πi(s − t) 2 D (0,γ1 ) ϕ(t + x, s + x) . × δ δβ2 δγ −γ1
Moreover, ν−δ −2πiν·(t−s) (0,γ ) e D 1 ϕ(t + x, s + x) Dxβ1 g(t), 2πi(s − t) ν−δ −2πiν·(t−s) (β ,γ ) = g(t), 2πi(s − t) e D 1 1 ϕ(t + x, s + x) ν−δ −2πiν·(t−s) (0,β +γ ) + g(t), 2πi(s − t) e D 1 1 ϕ(t + x, s + x) = g(t), e−2πiν·(t−s) ψ(t + x, s + x) , where ν−δ (β ,γ ) D 1 1 ϕ(t, s) + D (0,β1 +γ1 ) ϕ(t, s) ψ(t, s) = 2πi(s − t) and the mapping ϕ → ψ is continuous from S(R2n ) to itself. It follows that γ Dxβ1 Dνβ2 Ds R[ϕ](x, ν, s) = cσ (ν)σ g(t), e−2πiν·(t−s) ψσ (t + x, s + x)
(4.2)
σ γ
where, for each multi-index σ , cσ is a complex constant and ψσ ∈ S(R2n ) with the linear mapping ϕ → ψσ from S(R2n ) to itself being continuous. Note that (ν)σ g(t), e−2πiν·(t−s) ψσ (t + x, s + x) 1 = g(t)ψσ (t + x, s + x)Dtσ e−2πiν·(t−s) dt. |σ | (−2πi) Rn
Since, for fixed x and s, the function t → g(t)ψσ (t + x, s + x) belongs to S(Rn ), integration by parts shows that the last expression can be written as 1 Dtσ g(t)ψσ (t + x, s + x) e−2πiν·(t−s) dt |σ | (2πi) Rn
=
σ σ 1 Dt 1 g(t)Dtσ −σ1 ψσ (t + x, s + x)e−2πiν·(t−s) dt. σ1 (2πi)|σ | σ1 σ
Rn
650
J.-P. Gabardo / Journal of Functional Analysis 256 (2009) 635–672
Since g and each ψσ belong to the Schwartz class, we can find, for any integer m > n/2, a constant C (depending on g) such that the modulus of the previous expression is bounded by 1 1 dtϕSk C (1 + |t|2 )m (1 + |t + x|2 + |s + x|2 )m Rn
where k = k(β1 , β2 , γ , m, g). Using Peetre’s inequality (see [23, Lemma 1.18]), this last expression is itself bounded by C1 Rn
1 1 1 dtϕSk (1 + |t|2 )m (1 + |t|2 + |s|2 )m (1 + |x|2 )m
C2
1 1 ϕSk . (1 + |s|2 )m (1 + |x|2 )m
Using (4.2) and the previous estimate, it is then easy to see that, for each integer m 0, there is a constant Cm such that, for ϕ ∈ S(R2n ), m −m γ 1 + |x|2 + |ν|2 Dxβ1 Dνβ2 Ds R[ϕ](x, ν, s) Cm 1 + |s|2 ϕSk , (4.3) if max{|β1 |, |β2 |, |γ |} m, where k = k(m, g). Since h ∈ S (Rn ), there exists C0 > 0 and an integer m0 0 such that h(s), ψ(s) C0 ψS , ψ ∈ S Rn . (4.4) m0 Therefore, for any multi-indices β1 , β2 in Nn , we have Dxβ1 Dνβ2 (Kϕ)(x, ν) = Dxβ1 Dνβ2 h(s), R[ϕ](x, ν, s) = h(s), Dxβ1 Dνβ2 R[ϕ](x, ν, s) and, using the inequalities (4.3) and (4.4), we have thus, if m m0 and max{|β1 |, |β2 |, |γ |} m, that β β D 1 D 2 (Kϕ)(x, ν) C0 sup D β1 D β2 Dsγ R[ϕ](x, ν, s) 1 + |s|2 m0 x ν x ν s∈Rn |γ |m0
−m Cm 1 + |x|2 + |ν|2 ϕSk , for all ϕ ∈ S(R2n ), where k = k(β1 , β2 , m, g, h). This proves our claim.
2
The following lemma will be needed. It offers a slight improvement to [9, Theorem 11.2.3]. Lemma 4.3. Let g ∈ S(Rn ). Then, if h ∈ S (Rn ), the function Vg h(x, ν) belongs to OM (R2n ), i.e. for any multi-indices α, β ∈ R2n , there is a constant C(α, β) > 0 and an integer m(α, β) such that α β D D Vg h(x, ν) C(α, β) 1 + |x|2 + |ν|2 m(α,β) , (x, ν) ∈ R2n . (4.5) x ν
J.-P. Gabardo / Journal of Functional Analysis 256 (2009) 635–672
651
Furthermore, if B is a bounded set in S (Rn ), the constants C(α, β) and m(α, β) can be chosen such that the estimate (4.5) holds uniformly for all h ∈ B. Proof. Note that if B is a bounded set in S (Rn ), there exists a constant C 0 and an integer m 0 such that h, ϕ CϕS , ϕ ∈ S Rn , h ∈ B. m In the case where α = β = 0 and B consists of a single element, the estimate (4.5) is exactly the statement of Theorem 11.2.3 in [9], but the proof is the same if B is a bounded set in S (Rn ). In general, we have Dxα Dνβ Vg h(x, ν) = (−1)|α| h(t), (−2πit)β e−2πiν·t D α g(t − x) β
|α| |β| x β−γ h(t), (t − x)γ e−2πiν·t D α g(t − x) = (−1) (−2πi) γ γ β β
|α| |β| = (−1) (−2πi) x β−γ Vψα,γ h(x, ν), γ γ β
where ψα,γ (t) = t γ D α g(t), and the result follows immediately from the previous case.
2
We prove next a version of our dual window characterization in which the functions to be expanded are in the Schwartz class instead of being square-integrable but where we allow the dual window to be a tempered distribution and the expansion takes place in the space of tempered distributions. This situation occurs, for example, in the case of the one-dimensional regular Gabor system associated with the lattice Z2 , where the Balian–Low theorem prevents any function nicely localized in the time–frequency plane, and in particular, any Schwartz function, to form a frame for L2 (R) and thus to admit a Gabor dual in L2 (R) in the sense of Definition 4.1. However, the weaker duality defined by (4.6) can still occur. An example of such duality in the distributional sense in the case where the window is a Gaussian can be found in the paper [15] by Janssen. Theorem 4.4. Let μ be a tempered positive Borel measure on R2n . Let g ∈ S(Rn ) and let h ∈ S (Rn ). Then, the identity (4.6) Vg ψ1 (x, ν)Vh ψ2 (x, ν) dμ(x, ν) = ψ1 (t)ψ2 (t) dt, ψ1 , ψ2 ∈ S Rn , Rn
R2n
holds if and only if S F μ (Vg h) = δ(0,0)
on R2n .
Proof. We first define a tempered distribution T on R2n by the formula T (t, s), ϕ(t, s) = g(t) ⊗ h(s), e−2πiν·(t−s) ϕ(t + x, s + x) dμ(x, ν), R2n
(4.7)
652
J.-P. Gabardo / Journal of Functional Analysis 256 (2009) 635–672
for each ϕ ∈ S(R2n ). Note that T is a well-defined tempered distribution by Lemma 4.2. It is also clear that, if ψ1 , ψ2 ∈ S(Rn ), we have
Vg ψ1 (x, ν)Vh ψ2 (x, ν) dμ(x, ν) = T (t, s), ψ1 (t) ⊗ ψ2 (s) .
R2n
If (4.6) holds, we have thus
T (t, s), ψ1 (t) ⊗ ψ2 (s) =
ψ1 (t)ψ2 (t) dt,
ψ1 , ψ2 ∈ S R n .
Rn
Defining the tempered distribution ρ on R2n by the formula
ρ(t, s), ϕ(t, s) =
ϕ(t, t) dt,
ϕ ∈ S R2n ,
Rn
and using the density in S(R2n ) of the span of the functions of the form ψ1 (t) ⊗ ψ2 (s), where ψ1 , ψ2 ∈ S(Rn ), we deduce that T = ρ. The change of variable u = s − t, w = s induces a transformation Φ from S (R2n ) to itself defined by
Φ(σ )(u, w), ϕ(u, w) = σ (t, s), ϕ(s − t, s) ,
σ ∈ S R2n , ϕ ∈ S R2n .
With this change of variable, we have
Φ(T )(u, w), ϕ(u, w) =
g(t) ⊗ h(s), e−2πiν·(t−s) ϕ(s − t, s + x) dμ(x, ν),
R2n
for each ϕ ∈ S(R2n ), while
Φ(ρ)(u, w), ϕ(u, w) =
ϕ(0, w) dw, Rn
for each such ϕ, showing that Φ(ρ)(u, w) = δ0 (u) ⊗ 1(w). Denoting by F2 the Fourier transform with respect to the second variable w and defined by the formula F2 ϕ(u, ξ ) =
e−2πiξ ·w ϕ(u, w) dw,
ϕ ∈ S R2n ,
Rn
for functions in the Schwartz class and, by duality, F2 σ, ϕ = σ, F2 ϕ,
ϕ ∈ S R2n , σ ∈ S R2n ,
for tempered distributions on R2n , we have, that
J.-P. Gabardo / Journal of Functional Analysis 256 (2009) 635–672
653
g(t) ⊗ h(s), e−2πiν·(t−s) F2 ϕ(s − t, s + x) = h(s), g(t)e−2πiν·(t−s) F2 ϕ(s − t, s + x) = h(s), g(t)e−2πiν·(t−s) e−2πi(s+x)·ξ ϕ(s − t, ξ ) dξ dt . R2n
After the change of variables s − t = u, this last expression can be written as −2πi(−ν)·u −2πi(s+x)·ξ h(s), g(s − u)e e ϕ(u, ξ ) du dξ
(4.8)
R2n
or as
e−2πi[(−ν)·u+x·ξ ] h(s), g(s − u)e−2πis·ξ ϕ(u, ξ ) du dξ
R2n
= F (Vg h)ϕ (−ν, x).
(4.9)
To justify the equality between (4.8) and (4.9), consider a sequence (hk ) in S(Rn ) which converges to h in S (Rn ) as k → ∞. For fixed x and ν, let ζ (s) =
g(s − u)e−2πi(−ν)·u e−2πi(s+x)·ξ ϕ(u, ξ ) du dξ.
R2n
Since ζ ∈ S(Rn ), we have hk (s), ζ (s) → h(s), ζ (s) as k → ∞. Furthermore, since, for each k, hk (s)g(s − u)ϕ(u, ξ ) du dξ ds < ∞, R3n
Fubini’s theorem shows that hk (s), ζ (s) = F {(Vg hk )ϕ}(−ν, x). Since the sequence {hk } is convergent in S (Rn ), it must be weakly bounded and thus strongly bounded in S (Rn ). Using Lemma 4.3 with α = β = 0, we can find a constant C and an integer m, such that, for all k, (Vg hk )(x, ν) C 1 + |x|2 + |ν|2 m ,
(x, ν) ∈ R2n ,
and thus (Vg hk )(x, ν)ϕ(x, ν)
C , (1 + |x|2 + |ν|2 )n+1
(x, ν) ∈ R2n ,
for some constant C independent of k. Since the right-hand side of the previous inequality is integrable on R2n and Vg hk converges to Vg h pointwise as k → ∞, our claim follows from the Lebesgue dominated convergence theorem. Noting that, by Lemma 4.3, the function (Vg h)ϕ belongs to S(R2n ) and letting S = Φ(T ), we have thus
654
J.-P. Gabardo / Journal of Functional Analysis 256 (2009) 635–672
F2 S, ϕ =
F (Vg h)ϕ (−ν, x) dμ(x, ν) = μ(x, ν), F (Vg h)ϕ (−ν, x)
R2n
= F S μ(x, ν), (Vg h)(x, ν)ϕ(x, ν) = F S μ(x, ν)(Vg h)(x, ν), ϕ(x, ν)
while F2 Φ(ρ) = δ(0,0) (x, ν), from which the identity (4.7) follows immediately. Conversely, if the identity (4.7) holds, the preceding arguments can easily be reversed to obtain (4.6), which concludes the proof. 2 Note that the identity (4.6) means that every function ϕ ∈ S(Rn ) admits the expansion Vg ϕ(x, ν)e2πit·ν h(t − x) dμ(x, ν) ϕ(t) = R2n
in the space S (Rn ). Remark. The product appearing on the left-hand side formula (4.7) is well defined in S (R2n ) as the product between a distribution in S (R2n ) and a function in OM (R2n ). By taking the inverse Fourier transform of both sides of (4.7) and using the fact that the so-called cross Rihaczek distribution of g and h, defined by −2πix·ν ˆ , e R(g, h)(x, ν) = g(x) ⊗ h(ν)
(x, ν) ∈ R2n ,
satisfies the identity (see [10, Lemma 8.9] for more details) F R(g, h) (u, ξ ) = Vg h(−ξ, u), (u, ξ ) ∈ R2n , we deduce that (4.7) is equivalent to μ ∗ R(g, h) = 1 on R2n . The following lemma is needed before stating the L2 -version of the previous theorem. Lemma 4.5. Let μ be a complex Borel measure on Rm with the property that its total variation, |μ|, is translation-bounded. Suppose that for some r > 0 and some τ ∈ Rm , supp(F μ) ∩ Br (τ ) = {τ }, where Br (τ ) = {ξ ∈ Rm , |ξ − τ | < r}. Then, there exists a ∈ C such that F μ = aδτ
on Br (τ ).
Proof. Choose a function ϕ which is infinitely differentiable with compact support in the ball B = {ξ ∈ Rm , |ξ | < 1}, and define
ξ −τ ψ (ξ ) = ϕ , for 0 < < r.
J.-P. Gabardo / Journal of Functional Analysis 256 (2009) 635–672
655
Then, F ψ (x) = e−2πix·τ m F ϕ(x) and, thus, μ, F ψ = e−2πix·τ m F ϕ(x) dμ(x) m F ϕ(x) d|μ|(x). Rm
Rm
Since F ϕ ∈ S(Rm ), there exists, for each integer N 1, a constant BN such that F ϕ(x)
BN 1 + |x|2N
and, since |μ| is translation-bounded, there exists C > 0 such that |μ| [0, 1)m + x C,
∀x ∈ Rm .
Hence μ, F ψ CBN m
sup
k∈Zm
y∈[0,1)m
1 . 1 + 2N |k + y|2N
(4.10)
√ Now, if |k| 6 m and y1 , y2 ∈ [0, 1)m , we have 2 √ 2 √ 2 |k + y1 |2 |k| + |y1 | |k| + m 2 |k| − m 2|k + y2 |2 . √ This shows that, if |k| 6 m, sup y∈[0,1)m
1 2N 1 + |k + y|2N
[0,1)m
1 1 + 2N 2−2N |k
+ y|2N
dy.
Therefore, using this last √estimate together with (4.10), we obtain, letting A be the cardinality of the set {k ∈ Zm , |k| < 6 m}, that μ, F ψ CBN m A +
1
m k∈Z√ [0,1)m |k|6 m
CBN m A +
Rm
1 + 2N 2−2N |k + y|2N
1 dx 1 + |x/2|2N
m −m m = CBN A + 2 Rm
dy
1 dx . 1 + |x|2N
If N > m/2, it follows that the integral above is finite and we obtain the existence of a constant M > 0 such that μ, F ψ M,
0 < < r.
656
J.-P. Gabardo / Journal of Functional Analysis 256 (2009) 635–672
On the other hand, our assumption on the support of F μ implies that Fμ =
c α D α δτ
on Br (τ ),
α∈F
where F ⊂ Nm is a finite set of multi-indices, by a well-known characterization of distributions supported on a single point (see [24]). We have thus, if 0 < < r, |α| −|α| α μ, F ψ = F μ, ψ = cα (−1) D ϕ(0). α∈F
Since the values of the partial derivatives of ϕ at 0 can be arbitrary, this last expression will not remain bounded for all ϕ ∈ C0∞ (Br ) as approaches 0 unless cα = 0 whenever α = 0. This proves our claim. 2 Theorem 4.6. Let μ be a translation-bounded, positive Borel measure on R2n . Let g ∈ S(Rn ) and let h ∈ L2 (Rn ) satisfies the Bessel condition (1.6) (with g replaced with h) with respect to μ. Then, the function h is a dual window for the pair (g, μ) if and only if S F μ (Vg h) = δ(0,0)
on R2n .
(4.11)
In particular, if h is a dual window for the pair (g, μ), we must have that (g, h)2 = 0. Proof. If h is a dual window for the pair (g, μ), the identity (4.1) has to hold, in particular, for all functions in S(Rn ). This implies (4.11) using Theorem 4.4. Conversely, using that same result, if (4.11) holds, the identity (4.6) holds for all functions in S(Rn ). Since h satisfies (1.6) by assumption and g as well by Proposition 2.4, it follows that the identity (4.6) can be extended to all functions in L2 (Rn ) by continuity, showing that h is a dual window for the pair (g, μ). Finally, if h is a dual window for the pair (g, μ), the identity (4.7) implies that the support of F S μ is the set {(0, 0)} on a neighborhood of the origin. Using Lemma 4.5, it follows that F S μ = Cδ0,0 on that neighborhood. Thus CVg h(0, 0) = C(g, h)2 = 1, showing that (g, h)2 = 0. 2 Note that the well-known orthogonality conditions for the short-time Fourier transform (see [9, Theorem 3.2.1]) can be obtained from the previous theorem in the case where one of the windows belongs to the Schwartz class, since, under the condition that (g, h)2 = 1, the identity (4.11) holds when μ = 1. We now consider the analogue of Corollary 3.3 for Gabor dual systems. Corollary 4.7. Let g ∈ S(Rn ) and h ∈ L2 (Rn ) satisfy (g, h)2 = 1. Then, the measure μ = 1 is the only translation-bounded positive Borel measure on R2n such that h is a dual window for the pair (g, μ) if and only if Vg h(x, ν)2 + Vg h(−x, −ν)2 = 0,
for all (x, ν) ∈ R2n .
(4.12)
Proof. If the condition (4.12) holds and h is a dual window for the pair (g, μ), we must have F S (μ − 1)(x, ν)Vg h(x, ν) = 0
J.-P. Gabardo / Journal of Functional Analysis 256 (2009) 635–672
657
which implies that 2 F S (μ − 1)(x, ν)Vg h(x, ν) = 0. Since μ − 1 is a real measure, we have F S (μ − 1)(x, ν) = F S (μ − 1)(−x, −ν). This together with the previous identity yields 2 F S (μ − 1)(x, ν)Vg h(−x, −ν) = 0 and thus 2 2 F S (μ − 1)(x, ν) Vg h(x, ν) + Vg h(−x, −ν) = 0. Hence, F S (μ − 1) = 0 and μ = 1. If the condition (4.12) is not satisfied at some point (x0 , ν0 ), the construction of a measure μ = 1 such that h is a dual window for the pair (g, μ) is the same as in Corollary 3.3. Note that, since the measure μ constructed is given by a function in L∞ (R2n ), any function in L2 (Rn ) satisfies the Bessel condition (1.6) with respect to μ as can be seen using Theorem 1.1. 2 Example. Consider the pair of windows g(t) = e−πt and h(t) = − C \ {0}, in one dimension. Then, 2
Vg h(x, ν) = −
1 2 2 (x − iν − 2ζ )e−π(x +ν +2ixν)/2 , 2ζ
√ ζ
2
(t − ζ )e−πt , where ζ ∈ 2
(x, ν) ∈ R2 .
Since, Vg h has a single zero located at the point (x, ν) = (2 Re(ζ ), −2 Im(ζ )), the condition (4.12) in the previous corollary is satisfied and, since (g, h)2 = 1, the only translation-bounded positive Borel measure μ on R2n such that h is a dual window for the pair (g, μ) is the trivial one, μ = 1. We also have the analogue of Proposition 3.14 for dual windows, with a similar proof which uses the identity (4.7). Proposition 4.8. Let μ be a positive tempered measure on R2n and assume that g0 ∈ S(Rn ) and h0 ∈ S (Rn ) satisfy h0 , g0 = 1 as well as (4.6) with g = g0 and h = h0 . Then, there exists r > 0, such that F μ = δ(0,0)
on Br ,
(4.13)
where Br denotes the open ball of radius r centered at the origin in R2n . In particular, if K is a compact subset of R2n and ρ is tempered measure on R2n different from μ such that μ = ρ on R2n \ K, then ρ fails to satisfy the identity (4.6) for any window g ∈ S(Rn ) and any distribution h ∈ S (Rn ). Note that the previous result has the following somewhat surprising consequence. Suppose for example that the collection {e2πibl·t g(t − ak)}(k,l)∈Zn ×Zn , where a, b > 0 are two parameters and g ∈ S(Rn ) forms a frame for L2 (Rn ). Then, the standard dual of the frame is again a Gabor
658
J.-P. Gabardo / Journal of Functional Analysis 256 (2009) 635–672
system {e2πibl·t h(t − ak)}(k,l)∈Zn ×Zn with h ∈ L2 (Rn ). However, if a small non-zero perturbation is applied to the frame as in Proposition 4.8, the resulting system will still be a frame but it can no longer admit a dual which is of “Gabor type.” We will now use Theorem 4.6 to obtain more information about the measures μ having the property that the pair (ϕ, μ) admits a dual window in the case where ϕ is the Gaussian ϕ(x) = 2 2n/4 e−π|x| . Recall that the Bargmann–Fock space F 2 (Cn ) is the Hilbert space of entire functions F on Cn for which 2 2 F 2F = F (z) e−π|z| dz Cn
is finite. It is known that the Bargmann transform B : L2 (Rn ) → F 2 (Cn ), defined by Bf (z) = 2
n/4
f (t)e2πt·z−πt
2 − π z2 2
dt,
f ∈ L2 R n , z ∈ C n ,
Rn
is an isometry from L2 (Rn ) onto F 2 (Cn ). Furthermore, if we write z = x + iν ∈ Cn , then we have 2 Vϕ f (x, −ν) = eπix·ν e−π|z| /2 Bf (z), f ∈ L2 Rn . (See [9, Section 3.4] for more details on the Bargmann transform.) It was proved by Lyubarski˘ı [20] and, independently by Seip and Wallstén [25,26], that, in dimension n = 1, the regular Gabor system generated by the Gaussian {e2πiblt ϕ(t − ak)}(k,l)∈Z2 , where a, b > 0 are two parameters, forms a frame for L2 (R) if ab < 1. In particular, in that case, the standard dual provides a dual of Gabor type for the system. Note that this situation corresponds to the case where the measure μ associated with the system is the counting measure of a lattice and its symplectic Fourier transform, F S μ, is thus a measure supported on the adjoint lattice (see the discussion at the beginning of Section 5) and it is in particular discrete. The following result, which is only valid in dimension n = 1, shows, in particular, that, if μ is any measure such that the pair (ϕ, μ) admits a dual window h ∈ L2 (R), then the support of F S μ must necessarily be discrete. Proposition 4.9. Let ϕ(x) = 21/4 e−πx and let μ be a translation-bounded measure on R2 . Suppose that the pair (ϕ, μ) admits a dual window h ∈ L2 (R) with (ϕ, h)2 = 1, Then, there exists a discrete set Λ ⊂ R2 with −Λ = Λ and a function F ∈ F 2 (C) satisfying F (0) = 1 such that F S μ = δ(0,0) + cx,ν δ(x,ν) (4.14) 2
(x,ν)∈Λ
where Λ ⊂ ZF := (x, ν) ∈ R2 , F (x − iν) = F (−x + iν) = 0 and cx,ν ∈ C \ {0} for each (x, ν) ∈ Λ.
J.-P. Gabardo / Journal of Functional Analysis 256 (2009) 635–672
659
Conversely, given any function F ∈ F 2 (C) with F (0) = 1 and any subset Λ of R2 satisfying −Λ = Λ and Λ ⊂ ZF , there exists a translation-bounded measure μ on R2 such that F S μ is of the form (4.14) and a function h ∈ L2 (R) such that the pair (ϕ, μ) admits h as a dual window. Proof. Suppose first that h ∈ L2 (R) is a dual window for the pair (ϕ, μ). Since μ is a positive measure, we have F S μ(x, ν) = F S μ(−x, −ν). Hence, if F S μ has the form (4.14), we must have −Λ = Λ (and c−x,−ν = cx,ν for (x, ν) ∈ Λ). To prove that (4.14) holds, we use Theorem 4.6 to obtain the identity S F μ (x, ν)Vϕ h(x, ν) = δ(0,0) (x, ν) which is equivalent to S F μ (x, ν)Bh(x − iν) = δ(0,0) (x, ν). This last identity combined with the fact that the support of F S is invariant under the transformation (x, ν) → (−x, −ν) implies that the support of F S μ is contained in the union of the set {(0, 0)} with ZF , where F := Bh ∈ F 2 (C). Since F is an entire function of one complex variable, its zeros are isolated and the representation (4.14) follows from Lemma 4.5. Conversely, if F ∈ F 2 (C) satisfies F (0) = 1 and Λ is a subset of ZF with −Λ = Λ, let L = {(xk , νk ), k ∈ K} be a subset of ZF such that Λ = L ∪ (−L) and L ∩ (−L) = ∅ where K = ∅ if Λ = ∅, K = {1, 2, . . . , m} if card(L) = m and K = N = {1, 2, . . .} if L is infinite. Defining μ by letting F S μ = δ(0,0) + 4−k (δ(xk ,νk ) + δ(−xk ,−νk ) ) k∈K
we have μ = g ∈ L∞ (R2 ) with 1/3 g 5/3 and the result follows immediately from Theorem 4.6 letting h = B −1 F . 2 Remark. It is clear that in dimension n 2, the proposition just proved is not true. For example, if h ∈ L2 (R2 ) is of the form h = h1 ⊗ h2 where h1 , h2 ∈ L2 (R) and ϕ = ϕ1 ⊗ ϕ1 , where ϕ1 (x) = 2 21/4 e−π|x| , we have Vϕ h(x1 , x2 , ν1 , ν2 ) = Vϕ1 h1 (x1 , ν1 )Vϕ1 h1 (x2 , ν2 ) and if, for example, Vϕ1 h1 (x0 , ν0 ) = Vϕ1 h1 (−x0 , −ν0 ) = 0, then Vϕ h(x0 , x2 , ν0 , ν2 ) = Vϕ h(−x0 , −x2 , −ν0 , −ν2 ) = 0 for all (x2 , ν2 ) ∈ R2 and the zeros of Vϕ h are not isolated. It is then easy to use Theorem 4.6 to construct a counterexample. Definition 4.10. The cross-ambiguity function of two functions f and g in L2 (Rn ) is defined by A(f, g)(x, ν) = f (t + x/2)g(t − x/2)e−2πiν·t dt, (x, ν) ∈ R2n . (4.15) Rn
660
J.-P. Gabardo / Journal of Functional Analysis 256 (2009) 635–672
Lemma 4.11. Let g and h be two generalized Gaussian on Rn of the form g(x) = e−πAx·x+πa·x ,
h(x) = e−πBx·x+πb·x
where both A and B are symmetric matrices in GL(n, C) with positive definite real part and a, b ∈ Cn . Then, A(g, h)(x, ν) = eR(x,ν) ,
(x, ν) ∈ R2n ,
where R(x, ν) is a polynomial of degree 2 (in 2n variables). Proof. First note that, if A satisfies the conditions of the lemma, then −πAx·x −1 (ξ ) = e−πAx·x e−2πiξ ·x dx = (det A)−1/2 e−πA ξ ·ξ , F e
ξ ∈ Rn .
Rn
In the previous formula, the term (det A)1/2 is well defined (and positive) if A is positive definite and can be extended by analyticity to the set of symmetric matrices with positive definite real part, since the range of the mapping A → det A, where A varies over such matrices, is a simply connected open subset of C \ {0}. (See [9,13] for more details.) Since the integral defining the Fourier transform is absolutely convergent even when ξ ∈ Cn and the terms in the previous identity are all well defined and analytic when viewed as functions of ξ ∈ Cn , it follows, by analyticity, that the previous identity also holds for ξ ∈ Cn . Therefore, we have −πAx·x+2πa·x (ξ ) = e−πAx·x+2πa·x e−2πiξ ·x dx F e Rn
=
e−πAx·x e−2πi(ξ +ia)·x dx
Rn −1 (ξ +ia)·(ξ +ia)
= (det A)−1/2 e−πA
,
ξ ∈ Rn .
Hence, A(g, h)(x, ν) ∗ = e−πA(t+x/2)·(t+x/2)+2πa·(t+x/2) e−πB (t−x/2)·(t−x/2)+2π b·(t−x/2) e−2πiν·x dt Rn
= e−π(A+B
∗ )x·x/4+π(a−b)·x
e−π(A+B
∗ )t·t+2π(a−b)·t
e−π(A−B
∗ )x·t
e−2πiν·t dt
Rn
=e
−π(A+B ∗ )x·x/4+π(a−b)·x
(det C)−1/2 e−πC
−1 ξ ·ξ
,
where C = A + B ∗ and ξ = ν + i(a − b + (B ∗ − A)x/2). The conclusion of the lemma follows. 2 As an immediate consequence, we have the following result.
J.-P. Gabardo / Journal of Functional Analysis 256 (2009) 635–672
661
Theorem 4.12. Let g(x) = c1 e−πAx·x+πa·x and h(x) = c2 e−πBx·x+πb·x , where both A and B are symmetric matrices in GL(n, C) with positive definite real part, a, b ∈ Cn and c1 , c2 are constants chosen so that (g, h)2 = 1. Then, the measure μ = 1 is the only positive tempered measure on R2n such that the identity of (4.6) holds for this choice of g and h. In particular, the measure μ = 1 is the only positive, translation-bounded measure on R2n such that h is a dual for the pair (g, μ). Proof. The result follows immediately from Lemma 4.11 and Theorems 4.4 and 4.6, using the identity (3.6). 2 If the generalized Gaussians in the previous theorem are multiplied by polynomials, the uniqueness of the measure μ is no longer true, but the impossibility to construct discrete measure solution of the problem still remains as our next result will show. Before stating it, we need first the following lemma whose proof is almost obvious in the case where the set Λ supporting the measure μ is not dense in Rm . A bit more work is needed if we do not make this assumption. Lemma 4.13. Let μ = a∈Λ ca δa ∈ S (Rm ) be a discrete measure on Rm , where Λ ⊂ Rm is at most countable, and assume that, for some N > 0, Rm
1 d|μ| < ∞, (1 + |x|2 )N
where |μ| denotes the total variation of μ, so that, in particular, μ is a tempered measure on Rm . Then, there exists no polynomial of m variable P (ξ ) such that F μ(ξ )P (ξ ) = δ0
on Rm .
(4.16)
Proof. The inverse Fourier transform of P (ξ ) has the form F −1 P (ξ ) = b α D α δ0 α∈F
where F is a finite subset of Nm and bα = 0 for each α ∈ F . Taking inverse Fourier transform, Eq. (4.16) becomes μ∗
bα D α δ0 = 1.
(4.17)
α∈F
Choose a function ϕ which is infinitely differentiable and also compactly supported in the ball {x ∈ Rm , |x| < 1} and satisfies ϕ(0) = 1. Choose γ ∈ F such that |α| |γ | for all α ∈ F . Fix d ∈ Λ with μ({d}) = cd = 0 and define ψ (x) = (x − d)γ ϕ We have
x −d ,
for > 0.
662
J.-P. Gabardo / Journal of Functional Analysis 256 (2009) 635–672
γ x−d D ψ (x) = γ !ϕ
γ γ ! x − d γ −β x −d + D γ −β ϕ β (γ − β)! 0β<γ
= γ !ϕ
x−d
+ζ
x−d
(4.18)
where ζ ∈ C0∞ (Rm ) is supported in the ball {x ∈ Rm , |x| < 1} and satisfies ζ (0) = 0. Similarly, if α ∈ F and α = γ , we have
α γ ! α 1 x −d (x − d)γ −β |α|−|β| (D α−β ϕ) D ψ (x) = β (γ − β)! βγ βα
α γ ! x − d γ −β α−β x − d |γ |−|α| . D ϕ = β (γ − β)! βγ βα
This shows that
|α| x −d D α ψ (x) = , k ρk
(4.19)
k=0
where, for each k, ρk ∈ C0∞ (Rm ) is supported in the ball {x ∈ Rm , |x| < 1} and satisfies ρk (0) = 0. Clearly, lim 1, ψ = lim
→0+
→0+ Rm
ψ (x) dx = 0.
Also, letting Λ0 = Λ ∩ B(d, 1), we have
|cα | < ∞
α∈Λ0
and, if 0 < < 1, α∈F
=
bα D α μ, ψ α∈F
=
α∈F
bα
ca (−1)α D α ψ (a)
a∈Λ
bα cd (−1)α D α ψ (d) +
α∈F
bα
a∈Λ0 \{d}
ca (−1)α D α ψ (a).
(4.20)
J.-P. Gabardo / Journal of Functional Analysis 256 (2009) 635–672
663
Using (4.18) and (4.19), we see that D γ ψ (d) = γ ! and, as → 0+ , D α ψ (d) approaches 0 if α = γ . Furthermore, for any α ∈ F ,
lim
→0+
|ca |D α ψ (a) = 0.
a∈Λ0 \{d}
Therefore, lim
→0+
bα D α μ, ψ = bγ cd (−1)γ γ ! = 0.
(4.21)
α∈F
The identity (4.17) is contradicted by (4.20) and (4.21), which proves the lemma.
2
Theorem 4.14. Let g(x) = p(x)e−πAx·x+πa·x and h(x) = q(x)e−πBx·x+πb·x , where both A and B are symmetric matrices in GL(n, C) with positive definite real part, a, b ∈ Cn and p(x), q(x) are polynomials in n variables chosen so that (g, h)2 = 0. Let μ be a positive translationbounded measure on R2n . Then, h cannot be a dual window for the pair (g, μ) if the measure μ is discrete. In particular, no weighted irregular tight Gabor frame can be constructed using g as a window function. Proof. Let ψ1 , ψ2 ∈ S(Rn ) and let α, β ∈ Nn be multi-indices. We have A t α ψ1 (t), t β ψ2 (t) (x, ν) = (t + x/2)α ψ1 (t + x/2)(t − x/2)β ψ2 (t − x/2)e−2πiν·t dt Rn
=
α β
(x/2)α+β−γ −δ (−1)β−δ t γ +δ ψ1 (t + x/2)ψ2 (t − x/2)e−2πiν·t dt γ δ
0γ α 0δβ
=
Rn
α β
(−1)β−δ ∂ γ +δ (x/2)α+β−γ −δ A(ψ1 , ψ2 )(x, ν) . |γ +δ| γ +δ γ δ (−2πi) ∂ν
0γ α 0δβ
Therefore, if g and h are as above with p(x) = A g(t), h(t) (x, ν) =
α∈F cα x
α
and q(x) =
β∈G dβ x
β,
we have,
cα dβ A t α e−πAt·t+πa·t , t β e−πBt·t+πb·t (x, ν).
α∈F,β∈G
By the previous computations and Lemma 4.11, this last expression can be written in the form A g(t), h(t) (x, ν) = R1 (x, ν)eR2 (x,ν) where R1 and R2 are both polynomials in 2n variables and with R2 of degree at most 2. We can also assume that R2 (0, 0) = 0 (by replacing R2 (x, ν) by R2 (x, ν) − R2 (0, 0) if necessary). If μ
664
J.-P. Gabardo / Journal of Functional Analysis 256 (2009) 635–672
is a positive, translation-bounded measure on R2n which satisfies the identity (4.1) with g and h as above, we have, by Theorem 4.6, that S F μ (x, ν)R1 (x, ν)eR2 (x,ν) = δ(0,0) (x, ν)
on R2n
or, equivalently, multiplying both sides of the previous identity by e−R2 (x,ν) , S F μ (x, ν)R1 (x, ν) = δ(0,0) (x, ν)
on R2n .
After a change of variables, we obtain that (F μ)(x, ν)R(x, ν) = δ(0,0) (x, ν)
on R2n
for the polynomial R(x, ν) = R1 (−ν, x). Using Lemma 4.13, it follows that μ cannot be a discrete measure which proves our claim. 2 We now prove the analogue of Theorem 4.12 for the family of extreme value windows t ψk,m (t) = ekt−me , k, m > 0. Proposition 4.15. Let k1 , k2 , m1 , m2 be positive numbers. Then, for (x, ν) ∈ R2 , we have A(ψk1 ,m1 , ψk2 ,m2 )(x, ν)
−(k1 +k2 −2πiν) (k1 + k2 − 2πiν). = e(k1 −k2 )x/2 m1 ex/2 + m2 e−x/2 In particular, (a) A(ψk1 ,m1 , ψk2 ,m2 ) does not vanish anywhere on R2 . (b) Let g(t) = c1 ψk1 ,m1 (t) and h(t) = c2 ψk2 ,m2 (t), where the constants c1 and c2 are chosen so that (g, h)2 = 1. Then, the measure μ = 1 is the only positive tempered measure on R2 such that the identity of (4.6) holds for this choice of g and h. In particular, the measure μ = 1 is the only positive, translation-bounded measure on R2 such that h is a dual for the pair (g, μ). Proof. We have, for any (x, ν) ∈ R2 , A(ψk1 ,m1 , ψk2 ,m2 )(x, ν) t+x/2 k (t−x/2)−m et−x/2 −2πiνt 2 = ek1 (t+x/2)−m1 e e2 e dt R
=e
(k1 −k2 )x/2
e(k1 +k2 )t−(m1 e
x/2 +m
2e
−x/2 )et
e−2πiνt dt
R
= e(k1 −k2 )x/2 F ψk1 +k2 ,m1 ex/2 +m2 e−x/2 (ν)
−(k1 +k2 −2πiν) (k1 + k2 − 2πiν), = e(k1 −k2 )x/2 m1 ex/2 + m2 e−x/2
J.-P. Gabardo / Journal of Functional Analysis 256 (2009) 635–672
665
and the statement in (a) follows from the fact that the gamma function does not vanish anywhere while (b) is again a consequence of (a) together with Theorems 4.4 and 4.6. 2 5. Regular Gabor systems In this section, we briefly point out how some of the results obtained in the previous sections relate to known ones in the case of (unweighted) regular Gabor systems. Although these results are known, it is worthwhile to mention them as they are immediate consequences of our main theorems. Let Λ be a 2n-dimensional lattice in R2n . Then Λ can be described as the set CZ2n , for some 2n × 2n invertible real matrix C. Consider the measure μ = det(C) δCk . k∈Z2n
Then, Fμ =
δDk ,
k∈Z2n
where D = (C t )−1 . Denoting by J the linear mapping from R2n to itself defined by J (x, y) = (−y, x) for (x, y) ∈ Rn × Rn , we have also FSμ =
δJ Dk .
k∈Z2n
The lattice J DZ2n appearing in the previous formula is called the adjoint lattice and will be denoted by Λ◦ . Applying Theorem 3.2 to this particular translation-bounded measure μ, we obtain thus the following result. 2n Theorem 5.1. Let g ∈ S(Rn ) with g2 = 1 and consider the lattice √ Λ = CZ , where C is a ◦ 2n × 2n invertible real matrix and the adjoint lattice Λ . Let r = |det(C)|. Then, the following are equivalent:
(1) {re2πiν·t g(t − x)}(x,ν)∈Λ is a Parseval tight frame for L2 (Rn ). (2) Vg g(x, ν) = 0, (x, ν) ∈ Λ◦ \ {(0, 0)}. (3) The collection {e2πit·ν g(t − x)}(x,ν)∈Λ◦ is orthonormal. When restricted to the case of separable lattices, this result can be seen as a particular case of the Ron–Shen duality [22] or the Wexler–Raz identity [30] with the window being in the Schwartz class. A similar result holds for dual systems, using Theorem 4.6. Theorem 5.2. Let Λ, Λ◦ and r be as in the previous theorem. Let g ∈ S(Rn ) and let h ∈ L2 (Rn ) be such that the collection {e2πiν·t h(t − x)}(x,ν)∈Λ has the Bessel property and assume that (g, h)2 = 1. Then, the following are equivalent:
666
J.-P. Gabardo / Journal of Functional Analysis 256 (2009) 635–672
(1) The collections {re2πiν·t g(t − x)}(x,ν)∈Λ and {re2πiν·t h(t − x)}(x,ν)∈Λ are Gabor duals in L2 (Rn ). (2) Vg h(x, ν) = 0, (x, ν) ∈ Λ◦ \ {(0, 0)}. (3) The collections {e2πit·ν g(t − x)}(x,ν)∈Λ◦ and {e2πit·ν h(t − x)}(x,ν)∈Λ◦ are biorthogonal. Again, this last result, when restricted to the case of separable lattices, yields the Wexler–Raz identity [30], of course with the restriction that one of the windows belongs to the Schwartz class. These results were obtained in full generality for windows in L2 (Rn ) in [6]. (See also [9, Section 9.4] for the case of symplectic lattices.) 6. Beurling density and Gabor duality In this last section, we study the relationship between our characterization formula (4.11) for Gabor duality and the Beurling density of the corresponding measure μ. Several authors have obtained density results for discrete irregular Gabor systems showing that certain properties of the system such as being a frame or a Riesz basis have certain implications on the upper and lower Beurling densities associated with the corresponding sampling points in the time– frequency space (see [1,3,21] for the case of unweighted systems and [16] for weighted ones). We will define analogously the upper and lower Beurling density of a positive Borel measure and show that if a Gabor system associated with a measure admits a dual in the sense of Definition 4.1, then necessarily the associated measure has a well-defined Beurling density which must be equal to one. As we will see, this property is a direct consequence of the fact that the measure is translation-bounded and that its Fourier transform is equal to the Dirac mass at 0 in a neighborhood of the origin. Given a point z ∈ Rm and r > 0, we denote by Qr (z) the cube in Rm centered at z with side length r. Given a positive Borel measure μ on Rm , we define its upper and lower Beurling density, D + (μ) and D − (μ), by the formulas D + (μ) = lim sup sup
r→∞ z∈Rm
μ(Qr (z)) , rm
D − (μ) = lim inf infm r→∞ z∈R
μ(Qr (z)) , rm
and, if these two densities are equal, we define the Beurling density of μ to be D(μ) = D + (μ) = D − (μ). Lemma 6.1. Let μ be a positive translation-bounded Borel measure on Rm . Then, (a) There exists constant C such that lim sup sup r→∞
z∈Rm
μ(Qr (z)) C. rm
(b) We have lim r −m μ Qr (z) − μ Qr (z − u) = 0,
r→∞
uniformly for z ∈ Rm and u ∈ Q1 (0).
J.-P. Gabardo / Journal of Functional Analysis 256 (2009) 635–672
667
Proof. To prove the statement in (a), note that, since μ is translation-bounded, there is a constant C > 0 such that supy∈Rm μ(Q1 (y)) C. Denoting by [r] the integer part of r, we have the inclusion Qr (z) ⊂ Q[r]+1 (z) and this last set can be written as the union of ([r] + 1)m cubes of side length 1. Hence, lim sup sup r→∞ z∈Rm
μ(Qr (z)) μ(Q[r]+1 (z)) lim sup sup m r rm r→∞ z∈Rm lim sup r→∞
C([r] + 1)m = C. rm
Since μ Qr (z) − μ Qr (z − u) μ Qr (z) \ Qr (z − u) + μ Qr (z − u) \ Qr (z) , in order the prove (b), it is clearly enough to show that lim r −m μ Qr (z) \ Qr (z − u) = 0
r→∞
uniformly for z ∈ Rm and u ∈ Q1 (0). We have Qr (z) \ Qr (z − u) =
m j =1
m r Qr (z) ∩ x ∈ Rm , |xj + uj − zj | > Aj = 2 j =1
where, for j = 1, . . . , m, r r m Aj = x ∈ R , |xj + uj − zj | > , |xi − zi | , i = 1, . . . , m . 2 2 Note that we have the inclusion Aj ⊂ Bj ∪ Cj , with Bj being the set 1 r r [r] + 1 m x ∈ R , zj − xj zj − + , |xi − zi | , 1 i m, i = j 2 2 2 2 and Cj the set 1 r r [r] + 1 x ∈ Rm , zj + − xj zj + , |xi − zi | , 1 i m, i = j , 2 2 2 2 where [r] denotes the integer part of r. It is clear that Bj is contained in the union of at most ([r] + 1)m−1 cubes of side length 1 and the same is true for Cj . Hence, max{μ(Bj ), μ(Cj )} is bounded by C([r] + 1)m−1 . It follows immediately that m−1 −m r −m μ Qr (z) \ Qr (z − u) 2mC [r] + 1 r → 0,
r → ∞,
where the convergence is obviously uniform for z ∈ Rm and u ∈ Q1 (0), which proves our claim. 2
668
J.-P. Gabardo / Journal of Functional Analysis 256 (2009) 635–672
Proposition 6.2. Let μ be a positive translation-bounded measure on Rm such that, for some ρ > 0, F μ = δ0
on Bρ ,
where Bρ denotes the open ball centered at 0 with radius ρ in Rm . Then, the Beurling density of μ, D(μ), exists and is equal to 1. Proof. The statement that D(μ) exists and is equal to 1 is clearly equivalent to the fact that
z−y dμ(y) = 1 lim r −m χQ r→∞ r Rm
uniformly for z ∈ Rm , where Q = Q1 (0). Let ϕ ∈ S(Rm ) have the properties that ϕ 0 on ˆ ⊂ Bρ . Define, for η > 0, ϕη (x) = η−m ϕ(x/η), for Rm , that Rm ϕ(x) dx = 1 and that supp(ϕ) m x ∈ R . Note that the function ˆ ), F r −m (χQ ∗ ϕη )(·/r) (ξ ) = χˆ Q (rξ )ϕ(ηrξ
ξ ∈ Rm ,
has its support contained in Bρ if r > η−1 and, in that case, we have thus ˆ )χˆ Q (rξ )ϕ(ηrξ ˆ ) = δ0 (ξ )χˆ Q (rξ )ϕ(ηrξ ˆ ) F μ ∗ r −m (χQ ∗ ϕη )(·/r) (ξ ) = μ(ξ ˆ = χˆ Q (0)ϕ(0)δ 0 (ξ ) = δ0 (ξ ). It follows thus, by taking the inverse Fourier transform in the previous equality, that, if r > η−1 , we have
z−y dμ(y) = 1, for all z ∈ Rm . r −m (χQ ∗ ϕη ) r Rm
Hence, for r > η−1 , we have r −m
χQ
Rm
= r −m
z−y dμ(y) − 1 r
χQ
Rm
=r
−m
χQ
Rm
= r −m
z−y r z−y r
z−y − (χQ ∗ ϕη ) dμ(y) r
−
χQ
Rm
z−y −u ϕη (u) du dμ(y) r
z−y z−y −u χQ − χQ ϕη (u) du dμ(y). r r
Rm Rm
Using the Fubini–Tonelli theorem, we can write this last expression as
J.-P. Gabardo / Journal of Functional Analysis 256 (2009) 635–672
r −m
χQ
Rm
Rm
z−y r
− χQ
669
z−y −u dμ(y) ϕη (u) du r
= I1 (η, r, z) + I2 (η, r, z) where we define, letting B = {x ∈ Rm , |x| 1}, I1 (η, r, z) = r
−m
χQ
Rm \B
Rm
z−y r
− χQ
z−y−u dμ(y) ϕη (u) du r
and I2 (η, r, z) = r −m
χQ
B
Rm
z−y r
− χQ
z−y −u dμ(y) ϕη (u) du. r
By part (a) of Lemma 6.1, there exist r0 > 0 and C > 0 depending only on μ such that, if r r0 I1 (η, r, z) 3C
ϕη (u) du → 0,
as η → 0.
Rm \B
Thus, given > 0, we can choose η > 0 small enough so that 3C Rm \B
ϕη (u) du < . 2
For this fixed η, we have, using part (b) of Lemma 6.1, that
I2 (η, r, z) sup r −m χQ z − y − χQ z − y − u dμ(y) → 0, m r r y∈R u∈Q
Rm
uniformly for z ∈ Rm . It follows that, if r is large enough, I1 (η, r, z) + I2 (η, r, z) < , which proves our claim.
for all z ∈ Rm ,
2
An immediate consequence of the previous lemma and Theorem 4.4 is the following density result. A similar conclusion can be found in a paper by Kutyniok [16] in which the case of weighted irregular Gabor tight frames with a window in L2 (Rn ) is considered. Theorem 6.3. Let μ be a translation-bounded, positive Borel measure on R2n and let g ∈ S(Rn ). If there exists h ∈ L2 (Rn ) such that (h, g)2 = 1 and with h being a dual of Gabor type for the pair (g, μ), then necessarily D(μ) exists and is equal to 1.
670
J.-P. Gabardo / Journal of Functional Analysis 256 (2009) 635–672
We note that the same result holds, again assuming that μ is translation-bounded, if we allow the dual h to be a distribution in S (Rn ) as long as the duality is understood in the sense of the identity (4.6). We can use the previous result to obtain a generalization to weighted irregular Gabor systems of the so-called density theorem for regular Gabor systems in the particular case where the window is a Schwartz function and the generated system form a tight frame for L2 (Rn ). Theorem 6.4. Let g ∈ S(Rn ) be such that g2 = 1 and consider the discrete irregular system G := w(x, ν)1/2 e2πiν·t g(t − x) (x,ν)∈Λ , where Λ is a discrete set in the time–frequency space and w is a positive function defined on Λ. Let D − (Λ) = D − (ρ), as defined above with ρ being the (unweighted) measure ρ = λ∈Λ δλ . Then, if G forms a tight frame for L2 (Rn ), we must have that D − (Λ) 1. Proof. Multiplying the weight function w by an appropriate constant if necessary, we can assume that G forms a Parseval tight frame for L2 (Rn ). Using the fact that g2 = 1, it then follows by using the definition of a Parseval tight frame with f (t) = e2πit·ν0 g(t − x0 ) and (x0 , ν0 ) ∈ Λ, that 2 w(x, ν)Vg f (x, ν) w(x0 , ν0 )g22 = w(x0 , ν0 ), f 22 = g22 = 1 = (x,ν)∈Λ
which shows that w 1 on Λ. Defining the positive Borel measure μ as w(x, ν)δ(x,ν) , μ= (x,ν)∈Λ
we have thus that F μ = δ(0,0) on a neighborhood of the origin by Theorem 3.2. Hence, using the previous lemma and the fact that ρ μ, we have, D − (ρ) D − (μ) = 1, which proves our claim.
2
We conclude this paper by showing that there exist sets Λ ⊂ R2 which can be expressed as finite union of translates of a separable lattice and can have an arbitrarily large Beurling density while, at the same time, have the property that, for any window g in the Schwartz class, the system G := {e2πiνt g(t − x)}(x,ν)∈Λ does not admit a dual of Gabor type in L2 (R) (in the sense of Definition 4.1). This example illustrates the sharp contrast with known results [5,7] about irregular Gabor frames stating that if the window is sufficiently nice and the density is sufficiently high, the corresponding system does form a frame for L2 (R). It emphasizes again the fundamental difference which exists between general irregular Gabor systems and those admitting a dual which is also of Gabor type. Proposition 6.5. Let a > 1 and choose b1 , b2 > 0 such that b1 /a ∈ / Q and b2 ∈ / Q. Given positive integers K and L, define the set Λ ⊂ R2 by Λ = (kb1 + ma, lb2 + n), 0 k K − 1, 0 l L − 1, m, n ∈ Z .
J.-P. Gabardo / Journal of Functional Analysis 256 (2009) 635–672
671
Then, for any g ∈ S(R), the system G := {e2πiνt g(t − x)}(x,ν)∈Λ does not admit a dual of Gabor type in L2 (R). Proof. Let Λ0 = aZ ⊕ Z. Consider the measures μ0 = a
δ(x,ν)
and μ =
(x,ν)∈Λ0
=
a δ(x,ν) KL (x,ν)∈Λ
K−1 L−1 1 δ(kb1 ,lb2 ) ∗ μ0 . KL k=0 l=0
Note that, since the density of the lattice Λ0 is a −1 < 1, the system G := {e2πiνt g(t − x)}(x,ν)∈Λ0 is not dense in L2 (R) by the standard density theorem for regular Gabor systems (see [9]). In particular, there cannot exist a function h ∈ L2 (R) satisfying the identity (4.11) with μ replaced by μ0 . On the other hand, if the system G admitted a dual h ∈ L2 (R) with (g, h)2 = 1, the identity (4.11) would hold for μ. Letting P (ξ ) =
K−1 1 −2πiξ kb1 e K
and Q(ξ ) =
k=0
L−1 1 −2πiξ lb2 e , L l=0
we have F {μ}(ξ1 , ξ2 ) = P (ξ1 )Q(ξ2 )F {μ0 }(ξ1 , ξ2 ) and thus F S {μ}(ξ1 , ξ2 ) = P (ξ2 )Q(−ξ1 )F {μ0 }(ξ2 , −ξ1 ) δ(m, na ) (ξ1 , ξ2 ) = P (ξ2 )Q(−ξ1 ) (m,n)∈Z2
=
n Q(−m)δ(m, na ) (ξ1 , ξ2 ). P a 2
(m,n)∈Z
Using our hypothesis on b1 and b2 , it follows that P ( na )Q(−m) = 0 whenever (m, n) ∈ Z2 . Thus, the support of F S {μ} is the same as that of F S {μ0 } and, since P (0)Q(0) = 1, we deduce that the identity (4.11) also holds if μ is replaced by μ0 , contradicting our earlier statement. Hence, G cannot admit a Gabor dual. 2 Note that the Beurling density of the set Λ constructed in the previous theorem is KL/a and can thus be arbitrarily large. References [1] R. Balan, P. Casazza, C. Heil, Z. Landau, Density, overcompleteness, and localization of frames. II. Gabor systems, J. Fourier Anal. Appl. 12 (2006) 309–344. [2] P.G. Casazza, O. Christensen, Gabor frames over irregular lattices, Adv. Comput. Math. 18 (2003) 329–344. [3] O. Christensen, B. Deng, C. Heil, Density of Gabor frames, Appl. Comput. Harmon. Anal. 7 (1999) 292–304. [4] O. Christensen, S. Favier, F. Zó, Irregular wavelet frames and Gabor frames, Approx. Theory Appl. (N.S.) 17 (2001) 90–101. [5] H.G. Feichtinger, K. Gröchenig, Banach spaces related to integrable group representations and their atomic decompositions. I, J. Funct. Anal. 86 (1989) 307–340.
672
J.-P. Gabardo / Journal of Functional Analysis 256 (2009) 635–672
[6] H.G. Feichtinger, W. Kozek, Quantization of TF lattice-invariant operators on elementary LCS groups, in: Gabor Analysis and Algorithms, Birkhäuser Boston, Boston, MA, 1998, pp. 233–266. [7] H.G. Feichtinger, Wenchang Sun, Sufficient conditions for irregular Gabor frames, Adv. Comput. Math. 26 (2007) 403–430. [8] J.-P. Gabardo, D. Han, Frames associated with measurable spaces, Adv. Comput. Math. 18 (2003) 127–147. [9] K. Gröchenig, Foundations of Time–Frequency Analysis, Birkhäuser, Basel, 2001. [10] K. Gröchenig, A pedestrian approach to pseudodifferential operators, in: Harmonic Analysis and Application, in Honor of John J. Benedetto, Birkhäuser Boston, Boston, MA, 2006, pp. 139–169. [11] K. Gröchenig, Y.I. Lyubarski˘ı, Gabor frames with Hermite functions, C. R. Math. Acad. Sci. Paris 344 (2007) 157–162. [12] K. Gröchenig, G. Zimmermann, Hardy’s theorem and the short-time Fourier transform of Schwartz functions, J. London Math. Soc. (2) 63 (2001) 205–214. [13] L. Hörmander, The Analysis of Linear Partial Differential Operators I. Distribution Theory and Fourier Analysis, Springer-Verlag, Berlin, 1983. [14] R.L. Hudson, When is the Wigner quasi-probability density non-negative?, Rep. Math. Phys. 6 (1974) 249–252. [15] A.J.E.M. Janssen, Gabor representation of generalized functions, J. Math. Anal. Appl. 83 (1981) 377–394. [16] G. Kutyniok, Beurling density and shift-invariant weighted irregular Gabor systems, Sampl. Theory Signal Image Process. 5 (2006) 163–181. [17] J.C. Lagarias, Mathematical quasicrystals and the problem of diffraction, in: Directions in Mathematical Quasicrystals, in: CRM Monogr. Ser., vol. 13, Amer. Math. Soc., Providence, RI, 2000, pp. 61–93. [18] J.D. Lakey, Y. Wang, On perturbations of irregular Gabor frames, in: Approximation Theory, Wavelets and Numerical Analysis, Chattanooga, TN, 2001, J. Comput. Appl. Math. 155 (1) (2003) 111–129. [19] Y. Liu, Y. Wang, The uniformity of non-uniform Gabor bases, Adv. Comput. Math. 18 (2003) 345–355. [20] Y.I. Lyubarski˘ı, Frames in the Bargmann space of entire functions, in: Entire and Subharmonic Functions, in: Adv. Soviet Math., vol. 11, Amer. Math. Soc., Providence, RI, 1992, pp. 167–180. [21] J. Ramanathan, T. Steger, Incompleteness of sparse coherent states, Appl. Comput. Harmon. Anal. 2 (1995) 148– 153. [22] A. Ron, Z. Shen, Frames and stable bases for shift-invariant subspaces of L2 (Rd ), Canad. J. Math. 47 (1995) 1051–1094. [23] X. Saint Raymond, An Elementary Introduction to the Theory of Pseudodifferential Operators, Stud. Adv. Math., CRC Press, Boca Raton, FL, 1991. [24] L. Schwartz, Théorie des distributions, vols. 1 and 2, Hermann, Paris, 1956. [25] K. Seip, Density theorems for sampling and interpolation in the Bargmann–Fock space. I, J. Reine Angew. Math. 429 (1992) 91–106. [26] K. Seip, R. Wallstén, Density theorems for sampling and interpolation in the Bargmann–Fock space. II, J. Reine Angew. Math. 429 (1992) 107–113. [27] W. Sun, X. Zhou, Irregular wavelet/Gabor frames, Appl. Comput. Harmon. Anal. 13 (2002) 63–76. [28] W. Sun, X. Zhou, Irregular Gabor frames and their stability, Proc. Amer. Math. Soc. 131 (9) (2003) 2883–2893. [29] Y. Wang, Sparse complete Gabor systems on a lattice, Appl. Comput. Harmon. Anal. 16 (2004) 60–67. [30] J. Wexler, S. Raz, Discrete Gabor expansions, Signal. Process. 21 (1990) 207–220.
Journal of Functional Analysis 256 (2009) 673–699 www.elsevier.com/locate/jfa
Measurable selectors and set-valued Pettis integral in non-separable Banach spaces ✩ B. Cascales a , V. Kadets b , J. Rodríguez c,d,∗ a Departamento de Matemáticas, Universidad de Murcia, 30100 Espinardo (Murcia), Spain b Department of Mechanics and Mathematics, Kharkov National University, pl. Svobody 4, 61077 Kharkov, Ukraine c Departamento de Análisis Matemático, Universidad de Valencia, Avda. Doctor Moliner 50,
46100 Burjassot (Valencia), Spain d Departamento de Matematica Aplicada, Facultad de Informatica, Universidad de Murcia,
30100 Espinardo (Murcia), Spain Received 16 July 2007; accepted 29 October 2008 Available online 25 November 2008 Communicated by K. Ball
Abstract Kuratowski and Ryll-Nardzewski’s theorem about the existence of measurable selectors for multifunctions is one of the keystones for the study of set-valued integration; one of the drawbacks of this result is that separability is always required for the range space. In this paper we study Pettis integrability for multi-functions and we obtain a Kuratowski and Ryll-Nardzewski’s type selection theorem without the requirement of separability for the range space. Being more precise, we show that any Pettis integrable multi-function F : Ω → cwk(X) defined in a complete finite measure space (Ω, Σ, μ) with values in the family cwk(X) of all non-empty convex weakly compact subsets of a general (non-necessarily separable) Banach space X always admits Pettis integrable selectors and that, moreover, for each A ∈ Σ the Pettis integral A F dμ coincides with the closure of the set of integrals over A of all Pettis integrable selectors of F . As a consequence we prove that if X is reflexive then every scalarly measurable multi-function F : Ω → cwk(X) admits scalarly measurable selectors; the latter is also proved when (X ∗ , w∗ ) is angelic and has density character at most ω1 . In each of these two situations the Pettis integrability of a multi-function F : Ω → cwk(X) is equivalent to the uniform integrability of the family {sup x ∗ (F (·)): x ∗ ∈ BX∗ } ⊂ RΩ . Results about norm-Borel measurable selectors for multi-functions sat-
✩ B. Cascales and J. Rodríguez were supported by MEC and FEDER (project MTM2005-08379) and Fundación Séneca (project 00690/PI/04). J. Rodríguez was also supported by the “Juan de la Cierva” Programme (MEC and FSE). * Corresponding author at: Departamento de Matematica Aplicada, Facultad de Informatica, Universidad de Murcia, 30100 Espinardo (Murcia), Spain. E-mail addresses: [email protected] (B. Cascales), [email protected] (V. Kadets), [email protected] (J. Rodríguez).
0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.10.022
674
B. Cascales et al. / Journal of Functional Analysis 256 (2009) 673–699
isfying stronger measurability properties but without the classical requirement of the range Banach space being separable are also obtained. © 2008 Elsevier Inc. All rights reserved. Keywords: Pettis integral; Multi-function; Measurable selector; Multi-measure
1. Introduction Set-valued integration has its origin in the seminal papers by Aumann [2] and Debreu [9] and has been a very useful tool in areas like optimization and mathematical economics. The setvalued Pettis integral theory, which goes back to the monograph by Castaing and Valadier [7], has attracted recently the attention of several authors, see for instance [1,5,6,11,12,15,24,44,45]. All these studies deal with multi-functions whose values are subsets of a Banach space X that is always assumed to be separable. The main reason for this limitation on X relies on the fact that an integrable multifunction should have integrable (measurable) selectors and the tool to find these measurable selectors has always been the well-known selection theorem of Kuratowski and Ryll-Nardzewski [29] that only works when the range space is separable. For a detailed account on measurable selection results and set-valued integration we refer the reader to the monographs [7,27] and the survey [23]. Our main goal here is to show that most of the Pettis integral theory for multi-functions can be done without the restriction of separability on the range space. The extension from the separable case to the non-separable one is not so obvious and to do so we have to obtain a number of new measurable selection results for multi-functions in the non-separable case. Throughout this paper (Ω, Σ, μ) is a complete finite measure space, X is a Banach space and cwk(X) (respectively ck(X)) denotes the family of all convex weakly compact (respectively norm compact) non-empty subsets of X. We write δ ∗ (x ∗ , C) := sup{x ∗ (x): x ∈ C} for any bounded set C ⊂ X and x ∗ ∈ X ∗ . A multi-function F : Ω → cwk(X) is said to be Pettis integrable if ∗ ∗ • δ ∗ (x ∗ , F ) is integrable for each x ∈ X ; • for each A ∈ Σ, there is A F dμ ∈ cwk(X) such that
∗ δ x , F dμ = δ ∗ (x ∗ , F ) dμ ∗
A
for every x ∗ ∈ X ∗ .
A
Here the function δ ∗ (x ∗ , F ) : Ω → R is defined by δ ∗ (x ∗ , F )(ω) = δ ∗ (x ∗ , F (ω)). The paper is organized as follows. In Section 2 we study Pettis integrable multi-functions via their selectors. Our Theorem 2.5 states that every Pettis integrable multi-function F : Ω → cwk(X) admits indeed Pettis integrable selectors. Moreover, in this case, for each A ∈ Σ the integral A F dμ coincides with the closure of the set of integrals over A of all Pettis integrable selectors of F , Theorem 2.6. In the previous statement, the “closure” can be dropped provided that X ∗ is w ∗ -separable, Corollary 2.7. These results are the non-trivial extension of part of Theorem A below that is considered as the milestone result in the set-valued Pettis integral theory for separable Banach spaces. Theorem A. (See [15,44,45] and [7, Chapter V, §4].) Let X be a separable Banach space and F : Ω → cwk(X) a multi-function. The following conditions are equivalent:
B. Cascales et al. / Journal of Functional Analysis 256 (2009) 673–699
675
(i) F is Pettis integrable. (ii) The family WF = {δ ∗ (x ∗ , F ): x ∗ ∈ BX∗ } is uniformly integrable. (iii) The family WF is made up of measurable functions and any scalarly measurable selector of F is Pettis integrable. In this case, for each A ∈ Σ the integral Pettis integrable selectors of F .
AF
dμ coincides with the set of integrals over A of all
To get ready for the proof of a full counterpart to Theorem A for non-separable Banach spaces we quote in Section 3 some known facts about the existence of countably additive selectors and the Orlicz–Pettis theorem for multi-measures which are due to Godet-Thobie [20], Costé [8] and Pallu de la Barrière [33]: new proofs for these results are included. In Section 4 we discuss the possible extensions of Theorem A to the non-separable setting. The implications (i) ⇒ (ii) and (i) ⇒ (iii) hold without any assumption on X, Theorem 4.1 and Corollary 2.3. We show in Theorem 4.2 that the equivalence (i) ⇔ (iii) holds true if X has the following property: every scalarly measurable multi-function F : Ω → cwk(X) (meaning that δ ∗ (x ∗ , F ) is measurable for all x ∗ ∈ X ∗ ) admits a scalarly measurable selector. This condition, which we call Scalarly Measurable Selector Property with respect to μ, shortly μ-SMSP, is shared by many Banach spaces besides the separable ones, as explained a few lines below. On the other hand, to prove (ii) ⇒ (i) we have to require, in addition to the μ-SMSP, that X has the so-called Pettis Integral Property with respect to μ (shortly μ-PIP), Corollary 4.3. The last part of Section 4 is devoted to characterize Pettis integrability of multi-valued functions via singlevalued ones and we pay particular attention to the case of multi-functions with norm compact values. In Section 5 we are concerned with the existence of “measurable” selectors for multi-functions F : Ω → cwk(X) which satisfy one of the following measurability properties: (α) {ω ∈ Ω: F (ω) ∩ M = ∅} ∈ Σ for every closed half-space M ⊂ X (equivalently, F is scalarly measurable). (β) {ω ∈ Ω: F (ω) ∩ M = ∅} ∈ Σ for every convex closed set M ⊂ X. (γ ) {ω ∈ Ω: F (ω) ∩ M = ∅} ∈ Σ for every norm closed set M ⊂ X. When X is separable, it is known that (α), (β) and (γ ) are equivalent to the Effros measurability of F (i.e. the same property than (γ ) but replacing “closed” by “open,” cf. [7, Theorem III.37]). In this case, the selection theorem of Kuratowski and Ryll-Nardzewski, cf. [7, Theorem III.30], ensures that such an F admits a Borel(X, norm)-measurable (hence strongly measurable) selector. In the non-separable case these measurability notions are not equivalent in general and the situation becomes more complicated. Section 5.1 is started with Theorem 5.1 by proving that reflexive Banach spaces have μ-SMSP. Beyond that, our Theorem 5.4 shows that many other Banach spaces have μ-SMSP: for instance, this happens if the dual space is w ∗ -angelic and has w ∗ density character less than or equal to the uncountable cardinal number κ(μ), Example 5.5—we recall that the class of Banach spaces having w ∗ -angelic dual is very large and contains all weakly Lindelöf determined spaces and, in particular, all weakly compactly generated ones. Amongst other things we provide in Theorem 5.15 a different proof of Valadier’s result [43] saying that spaces with w ∗ -separable dual also have μ-SMSP. To this end we prove that a multi-function F : Ω → cwk(X) is scalarly measurable if and only if {ω ∈ Ω: F (ω) ∩ M = ∅} ∈ Σ for every set M ⊂ X which can be written as a finite intersection of closed half-spaces, Theorem 5.10.
676
B. Cascales et al. / Journal of Functional Analysis 256 (2009) 673–699
The paper is closed by Section 5.2 where we study the existence of Borel(X, norm)-measurable selectors for multi-functions F : Ω → cwk(X) satisfying (β). We prove, for instance, that such selectors always exist provided that X admits an equivalent locally uniformly rotund norm, Corollary 5.19: this improves a result by Leese [30] who obtained the same conclusion for multifunctions satisfying (γ ) when X admits an equivalent uniformly rotund norm. 1.1. Terminology Our unexplained terminology can be found in our standard references for multi-functions [7,27], Banach spaces [16] and vector integration [13,42]. The cardinality of a set Γ is denoted by card(Γ ). The cardinality of N (respectively R) is denoted by ℵ0 (respectively c). The symbol ω1 stands for the first uncountable ordinal. Our topological spaces (T , T) are always assumed to be Hausdorff. The density character of (T , T), denoted by dens(T , T) or simply by dens(T ), is the minimal cardinality of a dense set in T . All vector spaces here are assumed to be real. Given a subset S of a vector space, we write co(S) and span(S) to denote, respectively, the convex and linear hull of S. By letters X and Y we always denote Banach spaces. BY is the closed unit ball of Y and Y ∗ stands for the topological dual of Y . Given y ∗ ∈ Y ∗ and y ∈ Y , we write either y ∗ , y or y ∗ (y) to denote the evaluation of y ∗ at y. The weak (respectively weak∗ ) topology on Y (respectively Y ∗ ) is denoted by w (respectively w ∗ ). Given a non-empty set Γ (respectively a compact topological space K), we write ∞ (Γ ) (respectively C(K)) to denote the Banach space of all bounded (respectively continuous) real-valued functions on Γ (respectively K), equipped with the supremum norm. A function f : Ω → Y is said to be scalarly measurable if, for each y ∗ ∈ Y ∗ , the composition ∗ y , f := y ∗ ◦ f : Ω → R is measurable. By a result of Edgar [14], f is scalarly measurable if and only if it is Baire(Y, w)-measurable. Recall also that f is said to be Pettis integrable if (i) y ∗ ◦ f is integrable for every y ∗ ∈ Y∗ ; (ii) for each A ∈ Σ, there is an element A f dμ ∈ Y such that ∗ y , f dμ = y ∗ ◦ f dμ for every y ∗ ∈ Y ∗ . A
A
A function f : Ω → Y is strongly measurable if it is the μ-a.e. limit of a sequence of simple functions or, equivalently, if it is Borel(Y, norm)-measurable (or just scalarly measurable) and there is E ∈ Σ with μ(Ω \ E) = 0 such that f (E) is separable, cf. [13, Theorem 2, p. 42]. 2. Set-valued Pettis integral and selectors In order to prove our main result in this section stating that any Pettis integrable multi-function admits Pettis integrable selectors, Theorem 2.5, we need some previous work. Recall first that a function ϕ : X ∗ → R is said to be positively homogeneous if ϕ(αx ∗ ) = αϕ(x ∗ ) for every α > 0 and x ∗ ∈ X ∗ . ϕ is said to be subadditive if ϕ(x ∗ + y ∗ ) ϕ(x ∗ ) + ϕ(y ∗ ) for all pairs (x ∗ , y ∗ ) ∈ X ∗ × X ∗ . ϕ is said to be sublinear if it is both positively homogeneous and subadditive. We note that if C ∈ cwk(X) then the map x ∗ → δ ∗ (x ∗ , C) is a sublinear functional in X ∗ that is τ (X ∗ , X)-continuous. Here τ (X ∗ , X) stands for the Mackey topology on X ∗ , that is, the topology of uniform convergence on weakly compact subsets of X, cf. [28, §21.4]. Recall
B. Cascales et al. / Journal of Functional Analysis 256 (2009) 673–699
677
that, by the Mackey–Arens theorem, τ (X ∗ , X) is the finest locally convex topology on X ∗ whose topological dual is X, hence the w ∗ -closure and the τ (X ∗ , X)-closure of any convex set C ⊂ X ∗ coincide, cf. [28, §21.4(2) and §20.8(6)]. Lemma 2.1. Let F : Ω → cwk(X) be a multi-function such that δ ∗ (x ∗ , F ) is integrable for every x ∗ ∈ X ∗ . The following statements are equivalent: (i) F is Pettis integrable. (ii) For each A ∈ Σ , the mapping F ϕA : X ∗ → R,
x ∗ →
δ ∗ (x ∗ , F ) dμ,
A
is τ (X ∗ , X)-continuous. Proof. The implication (i) ⇒ (ii) follows from the fact that δ ∗ x ∗ , F dμ = δ ∗ (x ∗ , F ) dμ A
for every x ∗ ∈ X ∗ ,
A
and the τ (X ∗ , X)-continuity of the map x ∗ → δ ∗ (x ∗ , A F dμ). Conversely, assume that (ii) F is a sublinear function, it is convex. This fact and the τ (X ∗ , X)holds and fix A ∈ Σ. Since ϕA F allow us to deduce that for every t ∈ R the set {x ∗ ∈ X ∗ : ϕ F (x ∗ ) t} continuity of ϕA A F is w ∗ -lower semicontinuous is convex and τ (X ∗ , X)-closed, hence w ∗ -closed. Therefore ϕA and [7, Theorem II-16] applies to provide us with a non-empty convex, closed and bounded F (x ∗ ) = δ ∗ (x ∗ , C) for every x ∗ ∈ X ∗ . Finally, the fact that ϕ F is set C ⊂ X such that ϕA A ∗ τ (X , X)-continuous can be applied again to conclude that C is weakly compact. Indeed, the F (x ∗ ) < 1} ∩ {x ∗ ∈ X ∗ : ϕ F (−x ∗ ) < 1} is a τ (X ∗ , X)-neighborhood of 0 set U := {x ∗ ∈ X ∗ : ϕA A ◦ and thus its polar U = {x ∈ X: |x ∗ (x)| 1 for all x ∗ ∈ U } is weakly compact, [28, §21.4.1]. Since C is weakly closed and contained in U ◦ , C is weakly compact as well. 2 Observe that for every bounded set C ⊂ X and every x ∗ ∈ X ∗ we have inf x ∗ (x): x ∈ C = −δ ∗ (−x ∗ , C). Lemma 2.2. Let F, G : Ω → cwk(X) be two multi-functions such that F is Pettis integrable, ∗ ∗ ∗ ∗ ∗ ∗ G is scalarly measurable and, for each x ∈ X , we have δ (x , G) δ (x , F ) μ-a.e. Then G is Pettis integrable and A G dμ ⊂ A F dμ for every A ∈ Σ . Proof. Given x ∗ ∈ X ∗ , we have −δ ∗ (−x ∗ , F ) δ ∗ (x ∗ , G) δ ∗ (x ∗ , F ) μ-a.e. and so δ ∗ (x ∗ , G) G is subadditive and satisfies ϕ G (x ∗ ) ϕ F (x ∗ ) for all is integrable. Fix A ∈ Σ. The mapping ϕA A A x ∗ ∈ X ∗ , hence
G ∗
ϕ (x ) − ϕ G (y ∗ ) ϕ F (x ∗ − y ∗ ) + ϕ F (y ∗ − x ∗ )
A
A
A
A
678
B. Cascales et al. / Journal of Functional Analysis 256 (2009) 673–699
F is τ (X ∗ , X)-continuous and the previous for every x ∗ , y ∗ ∈ X ∗ . Since F is Pettis integrable, ϕA G ∗ inequality implies that ϕA is also τ (X , X)-continuous. Since A ∈ Σ is arbitrary, an appeal to Lemma 2.1 ensures that G is Pettis integrable. Moreover, for each A ∈ Σ we have A G dμ ⊂ F dμ, by the Hahn–Banach separation theorem and the fact that A
δ ∗ x ∗ , G dμ = δ ∗ (x ∗ , G) dμ δ ∗ (x ∗ , F ) dμ = δ ∗ x ∗ , F dμ A
A
A
for every x ∗ ∈ X ∗ . The proof is over.
A
2
Given a multi-function F : Ω → cwk(X) and A ∈ Σ we write f dμ: f is a Pettis integrable selector of F . ISF (A) := A
Note that ISF (A) might be empty in general and that otherwise it is a convex subset of X. Next corollary says, in particular, that ISF (A) ⊂ A F dμ whenever F is Pettis integrable. Corollary 2.3. Let F : Ω → cwk(X) be a Pettis integrable multi-function. If f : Ω → X is a scalarly measurable selector of F , then f is Pettis integrable and f dμ ∈ F dμ for every A ∈ Σ. A
A
Proof. Apply Lemma 2.2 to the multi-function G(ω) := {f (ω)}.
2
To prove the main result of this section we also need the following lemma: Lemma 2.4. (See [43, Lemme 3].) Let F : Ω → cwk(X) be a scalarly measurable multi-function. Fix x0∗ ∈ X ∗ and consider the multi-function G : Ω → cwk(X),
G(ω) := x ∈ F (ω): x0∗ (x) = δ ∗ x0∗ , F (ω) .
Then G is scalarly measurable. Theorem 2.5. Let F : Ω → cwk(X) be a Pettis integrable multi-function. Then F admits a Pettis integrable selector. Proof. Since A F dμ ∈ cwk(X), we can find an exposed point x0 ∈ A F dμ(cf. [4, Theorem 3.6.1]), that is, there is some x0∗ ∈ X ∗ such that x0∗ (x0 ) > x0∗ (x) for every x ∈ A F dμ \ {x0 }. Let us consider the multi-function G : Ω → cwk(X),
G(ω) := x ∈ F (ω): x0∗ (x) = δ ∗ x0∗ , F (ω) .
By Lemma 2.4, G is scalarly measurable. Since G(ω) ⊂ F (ω) for every ω ∈ Ω andF is Pettis integrable, an appeal to Lemma 2.2 ensures that G is Pettis integrable too, with Ω G dμ ⊂
B. Cascales et al. / Journal of Functional Analysis 256 (2009) 673–699
679
Ω F dμ. Let g : Ω → X be any selector of G. Clearly, g is also a selector of F . We will prove that g is scalarly measurable. Observe that
∗ δ x0 , G dμ = δ ∗ (x0∗ , G) dμ ∗
Ω
Ω
δ ∗ (x0∗ , F ) dμ = δ ∗ x0∗ , F dμ = x0∗ (x0 )
Ω
∗ −δ (−x0∗ , G) dμ = −δ ∗ −x0∗ , G dμ .
=
=
Ω
Ω
It follows that
Ω
Ω
G dμ = {x0 }. Given x ∗ ∈ X ∗ , we have −δ ∗ (−x ∗ , G) δ ∗ (x ∗ , G) and
∗ −δ (−x ∗ , G) dμ = x ∗ (x0 ) =
Ω
δ ∗ (x ∗ , G) dμ,
Ω
hence −δ ∗ (−x ∗ , G) = δ ∗ (x ∗ , G) μ-a.e. Therefore, x ∗ ◦ g = δ ∗ (x ∗ , G) μ-a.e. and, in particular, x ∗ ◦ g is measurable. Since x ∗ ∈ X ∗ is arbitrary, g is scalarly measurable. Finally, an appeal to Corollary 2.3 allows us to conclude that g is Pettis integrable. 2 In our next result we establish that in fact any Pettis integrable multi-function admits a collection of Pettis integrable selectors which are dense in it (a kind of “generalized” Castaing representation). Theorem 2.6. Let F : Ω → cwk(X) be a Pettis integrable multi-function. Then F admits a collection {fα }α<dens(X∗ ,w∗ ) of Pettis integrable selectors such that F (ω) = fα (ω): α < dens(X ∗ , w ∗ ) Moreover,
AF
for every ω ∈ Ω.
dμ = ISF (A) for every A ∈ Σ .
Proof. Notice first that κ := dens(X ∗ , w ∗ ) = dens(X ∗ , τ (X ∗ , X)). Fix a τ (X ∗ , X)-dense set {xα∗ : α < κ} ⊂ X ∗ . For each α < κ, the multi-function
Lα : Ω → cwk(X), Lα (ω) := x ∈ F (ω): xα∗ (x) = δ ∗ xα∗ , F (ω) , is scalarly measurable by Lemma 2.4 and so Pettis integrable by Lemma 2.2. Then Theorem 2.5 applied to Lα ensures that there is a Pettis integrable selector sα : Ω → X of Lα . Clearly, each sα is also a selector of F . We claim that
F (ω) = co sα (ω): α < κ
for every ω ∈ Ω.
Indeed, fix ω ∈ Ω and set C := co({sα (ω): α < κ}) ⊂ F (ω). Then C ∈ cwk(X) and
δ ∗ xα∗ , F (ω) δ ∗ xα∗ , C xα∗ sα (ω) = δ ∗ xα∗ , F (ω)
680
B. Cascales et al. / Journal of Functional Analysis 256 (2009) 673–699
for every α < κ. Since the set {xα∗ : α < κ} is τ (X ∗ , X)-dense in X ∗ and the maps x ∗ → δ ∗ (x ∗ , C) and x ∗ → δ ∗ (x ∗ , F (ω)) are τ (X ∗ , X)-continuous we obtain the equality δ ∗ (x ∗ , F (ω)) = δ ∗ (x ∗ , C) for every x ∗ ∈ X ∗ and, therefore, F (ω) = C as asserted. Observe that the collection {fα }α<κ made up of all convex combinations of the sα ’s with rational coefficients fulfills the required properties. In order to prove the last assertion, fix A ∈ Σ . Using Corollary 2.3, we obtain that ISF (A) ⊂ A F dμ. On the other hand, for each α < κ, the following holds: xα∗
sα dμ = A
xα∗
◦ sα dμ =
A
∗ ∗ ∗ δ xα , F dμ = δ xα , F dμ , ∗
A
A
and so δ ∗ (xα∗ , ISF (A)) δ ∗ (xα∗ , A F dμ). Since {xα∗ : α < κ} is τ (X ∗ , X)-dense in X ∗ , the inequality δ ∗ (x ∗ , ISF (A)) δ ∗ (x ∗ , A F dμ) holds true for every x ∗ ∈ X ∗ and we infer that A F dμ ⊂ ISF (A). Therefore ISF (A) = A F dμ and the proof is finished. 2 It turns out that, when X ∗ is w ∗ -separable, the sets ISF (A) are closed for any Pettis integrable multi-function F : Ω → cwk(X). The proof imitates that given in [15, Proposition 5.2] for a separable X and so we omit the details. Combining this fact with Theorem 2.6 we get the following result. Corollary 2.7. Suppose X ∗ is w ∗ -separable. Let F : Ω → cwk(X) be a Pettis integrable multifunction. Then A F dμ = ISF (A) for every A ∈ Σ. 3. Multi-measures and countably additive selectors Given a sequence (Cn ) in cwk(X), the series n Cn is said to be unconditionally convergent provided that for every choice xn ∈ Cn , n ∈ N, the series n xn is unconditionally convergent in X. In this case, the set Cn := xn : xn ∈ Cn for all n ∈ N n
n
also belongs to cwk(X), see [5, Lemma 2.2]. Recall that the family cwk(X), equipped with the Hausdorff metric h, is a complete metric space that can be isometrically embedded into the Banach space ∞ (BX∗ ) by means of the mapping j : cwk(X) → ∞ (BX∗ ),
j (C)(x ∗ ) := δ ∗ (x ∗ , C),
see e.g. [7, Chapter II]. It is known that a series n Cn as above is unconditionally convergent if ∗ and only if theseries n j (Cn ) is unconditionally convergent in ∞ (BX ) (in this case, we have j ( n Cn ) = n j (Cn )), cf. [5, Lemma 2.3]. Definition 3.1. A mapping M : Σ → cwk(X) is said to be a finitely additive (respectively countably additive) multi-measure if M(A ∪ B) = M(A) + M(B) whenever A, B ∈ Σ are disjoint (respectively if for every disjoint sequence (E ) in Σ the series M(E n n ) is unconditionally n convergent and M( n En ) = n M(En )).
B. Cascales et al. / Journal of Functional Analysis 256 (2009) 673–699
681
Note that M : Σ → cwk(X) is a finitely (respectively countably) additive multi-measure if and only if the composition j ◦ M : Σ → ∞ (BX∗ ) is a finitely (respectively countably) additive measure. Therefore, if for x ∗ ∈ X ∗ we define δ ∗ (x ∗ , M) : Σ → R by A → δ ∗ (x ∗ , M(A)), then M is a finitely additive multi-measure if and only if δ ∗ (x ∗ , M) is finitely additive for every x ∗ ∈ X ∗ . For countably additive multi-measures the analogue characterization is also true, see Theorem 3.4, but requires some work that we present in this section: this result, due to Costé [8] and Pallu de la Barrière [33], can be seen as the set-valued version of the well-known fact that weakly countably additive vector measures are norm countably additive (Orlicz–Pettis theorem, cf. [13, Corollary 4, p. 22]). From a technical point of view, the novelty of our approach to Theorem 3.4 relies mostly in the way of finding “finitely additive selectors” for finitely additive multi-measures, see Theorem 3.3, via a method of “linearization” of Lipschitz functions on Banach spaces that goes back to Pelczynski [34, p. 61]. Let Lip0 (X ∗ ) be the Banach space of all Lipschitz functions h : X ∗ → R satisfying h(0) = 0, equipped with the norm |h(x1∗ ) − h(x2∗ )| ∗ ∗ ∗ ∗ ∗ : x hLip0 (X∗ ) := sup , x ∈ X , x = x 1 2 1 2 . x1∗ − x2∗ Fix an invariant mean on X ∗ (considered as additive abelian group), that is, a linear mapping I : ∞ (X ∗ ) → R such that I(g) 0 whenever g 0, I(1) = 1 and I(g) = I(g(· + x ∗ )) for every g ∈ ∞ (X ∗ ) and every x ∗ ∈ X ∗ , cf. [25, Theorem 17.5]. It is known that we can define an operator P : Lip0 (X ∗ ) → X ∗∗ by the formula
P (h), x ∗ := I h(· + x ∗ ) − h(·) , h ∈ Lip0 (X ∗ ), x ∗ ∈ X ∗ , cf. [3, Proposition 7.5]. Lemma 3.2. Let C ∈ cwk(X). Then δ ∗ (·, C) ∈ Lip0 (X ∗ ) and P (δ ∗ (·, C)) ∈ C. Proof. The first assertion is clear, since
∗ ∗
δ x , C − δ ∗ x ∗ , C x ∗ − x ∗ · sup x: x ∈ C for every x1∗ , x2∗ ∈ X ∗ . 1 2 1 2 The proof of the second assertion is by contradiction. Suppose that P (δ ∗ (·, C)) ∈ / C. Since C is a convex w ∗ -closed subset of X ∗∗ , the Hahn–Banach separation theorem guarantees the existence of some x ∗ ∈ X ∗ such that ∗ P δ (·, C) , x ∗ > sup x ∗ (x): x ∈ C = δ ∗ (x ∗ , C). (1) On the other hand, we have δ ∗ (y ∗ + x ∗ , C) − δ ∗ (y ∗ , C) δ ∗ (x ∗ , C) for every y ∗ ∈ X ∗ , and the properties of I yield
P δ ∗ (·, C) = I δ ∗ (· + x ∗ , C) − δ ∗ (·, C) I δ ∗ (x ∗ , C) = δ ∗ (x ∗ , C), which contradicts (1). The proof is over.
2
We are now ready to deal with the aforementioned results about multi-measures.
682
B. Cascales et al. / Journal of Functional Analysis 256 (2009) 673–699
Theorem 3.3. (See [8,20,33].) Let M : Σ → cwk(X) be a finitely additive multi-measure. Then there is a finitely additive measure m : Σ → X such that m(A) ∈ M(A) for every A ∈ Σ . Proof. Lemma 3.2 ensures that δ ∗ (·, M(A)) ∈ Lip0 (X ∗ ) and
m(A) := P δ ∗ ·, M(A) ∈ M(A)
for every A ∈ Σ.
Since M is a finitely additive multi-measure and P is linear, m is finitely additive.
2
For a given x ∗ ∈ BX∗ , let ex ∗ denote the element of B ∞ (BX∗ )∗ defined by the formula ex ∗ (ϕ) := ϕ(x ∗ ). Theorem 3.4 (Costé-Pallu de la Barrière). Let M : Σ → cwk(X) be a mapping. The following statements are equivalent: (i) M is a countably additive multi-measure. (ii) δ ∗ (x ∗ , M) is countably additive for every x ∗ ∈ X ∗ . (iii) δ ∗ (x ∗ , M) is countably additive for every x ∗ ∈ X ∗ and there is a countably additive measure m : Σ → X such that m(A) ∈ M(A) for every A ∈ Σ . Proof. The implication (i) ⇒ (ii) follows from the fact that δ ∗ (x ∗ , M) = ex ∗ , j ◦ M for every x ∗ ∈ BX∗ . Let us prove (ii) ⇒ (iii). By Theorem 3.3 there is a finitely additive measure m : Σ → X such that m(A) ∈ M(A) for every A ∈ Σ . We claim that m is countably additive. To prove that it suffices to show that the composition x ∗ ◦ m is countably additive for every x ∗ ∈ X ∗ and then appeal to the Orlicz–Pettis theorem, see [13, Corollary 4, p. 22]. Given x ∗ ∈ X ∗ , we have −δ ∗ (−x ∗ , M(A)) (x ∗ ◦ m)(A) δ ∗ (x ∗ , M(A)) for every A ∈ Σ . Since both −δ ∗ (−x ∗ , M) and δ ∗ (x ∗ , M) are countably additive and x ∗ ◦ m is finitely additive, it follows that x ∗ ◦ m is countably additive, as claimed. To finish we prove (iii) ⇒ (i). We will prove that the finitely additive measure ν := j ◦ M : Σ → ∞ (BX∗ ) is countably additive. The proof is divided into two cases. Particular case. Suppose m(A) = 0 for every A ∈ Σ . Take a disjoint sequence (An ) in Σ. We will show first that the series n ν(An ) is unconditionally convergent. This is equivalent to saying that the series of sets n M(An ) is unconditionally convergent. Fix xn ∈ M(An ) for every n ∈ N, and take a sequence n1 < n2 < · · · in N. Define sk = ki=1 xni for every k ∈ N. Note that sk = sk + 0 ∈
k
M(Ani ) + M Ω \
i=1
k
Ani
= M(Ω)
for every k ∈ N.
i=1
On the other hand, for each x ∗ ∈ X ∗ the series bear in mind that
∞
i=1 x
∗ (x
ni )
is convergent. Indeed, it suffices to
∞ ∞ ∞
∗
∗ ∗
∗ ∗
x (xn )
δ x , M(An ) +
δ −x , M(An ) < +∞. i i i i=1
i=1
i=1
B. Cascales et al. / Journal of Functional Analysis 256 (2009) 673–699
683
This ensures that the sequence (sk ) has at most one weak clusterpoint in X. Since (sk ) is contained in the weakly compact set M(Ω), it follows that the series ∞ i=1 xni is weakly convergent. As the sequence n1 < n < · · · is arbitrary, the Orlicz–Pettis theorem (cf. [13, Corollary 4, p. 22]) 2 ensures that the series n xn is unconditionally convergent. This proves that the series n ν(An ) ∗ converges unconditionally ∞in ∞ (BX ). ∞ We claim now that n=1 ν(An ) = ν( n=1 An ). Indeed, for each x ∗ ∈ BX∗ we have
∞ n=1
ν(An ) (x ∗ ) = lim
N
N →∞
=δ
∗
ν(An )(x ∗ ) = lim
n=1
∗
x ,M
N
N →∞
∞ n=1
An
=ν
n=1 ∞
δ ∗ x ∗ , M(An )
An (x ∗ ).
n=1
The proof of Particular case is finished. General case. Define the mapping M : Σ → cwk(X),
M (A) = −m(A) + M(A).
It is clear that δ ∗ (x ∗ , M ) = −x ∗ ◦ m + δ ∗ (x ∗ , M) for every x ∗ ∈ X ∗ . Note also that 0 ∈ M (A) for every A ∈ Σ . Particular case already proved ensures that the mapping ν := j ◦ M : Σ → ∞ (BX∗ ) is a countably additive measure. On the other hand, the mapping ν : Σ → ∞ (BX∗ ) given by ν (A)(x ∗ ) := x ∗ (m(A)) is obviously a countably additive measure. It follows that ν = ν + ν is countably additive, as required. 2 For further information on the theory of multi-measures, we refer the reader to [23, Section 7], [27, Chapter 19] and the references therein. 4. Characterization of Pettis integrability for multi-functions The aim of this section is to discuss the validity of Theorem A in Section 1 within the setting of non-separable Banach spaces. Note that Corollary 2.3 gives us the extension to the non-separable case of (i) ⇒ (iii) in Theorem A. With the help of the results about multi-measures isolated in Section 3 we start by proving Theorem 4.1 below that extends to the non-separable case the implication (i) ⇒ (ii) in Theorem A, see (d) ⇒ (e) in [15, Theorem 5.4]. Given F : Ω → cwk(X) we write WF := δ ∗ (x ∗ , F ): x ∗ ∈ BX∗ ⊂ RΩ . Recall that a family H of real-valued integrable functions defined on Ω is said to beuniformly integrable if it is bounded for · 1 and for each ε > 0 there is δ > 0 such that suph∈H E |h| dμ ε whenever μ(E) δ.
684
B. Cascales et al. / Journal of Functional Analysis 256 (2009) 673–699
Theorem 4.1. Let F : Ω → cwk(X) be a Pettis integrable multi-function. Define the indefinite Pettis integral of F by IF : Σ → cwk(X), IF (A) := F dμ. A
Then: (i) IF is a countably additive multi-measure. (ii) WF is uniformly integrable. Proof. Clearly, δ ∗ (x ∗ , IF ) is countably additive for every x ∗ ∈ X ∗ and we can apply Theorem 3.4 to conclude that IF is a countably additive multi-measure. This proves (i). We prove now statement (ii). The composition ν := j ◦ IF : Σ → ∞ (BX∗ ) is a countably additive vector measure that vanishes on all μ-null sets. Hence ν is μ-continuous, that is, ν(A) = 0 (cf. [13, Theorem 1, p. 10]). On the other hand, observe that ex ∗ , ν (A) = limμ(A)→0 ∗ (x ∗ , F ) dμ for every x ∗ ∈ B ∗ and every A ∈ Σ . In view of the above, the uniform integraδ X A bility of WF now follows from the fact that
∗ ∗
δ (x , F ) dμ ν(A) sup ex ∗ , ν (A) = sup x ∗ ∈BX∗
for every A ∈ Σ.
x ∗ ∈BX∗
A
2
We turn our attention now to the implication (iii) ⇒ (i) in Theorem A for the non-separable case: the proof below is inspired by some of the ideas in [15, Theorems 3.9 and 5.4]. We say that a Banach space X has the Scalarly Measurable Selector Property with respect to μ, shortly μSMSP, if every scalarly measurable multi-function F : Ω → cwk(X) has a scalarly measurable selector. Theorem 4.2. Suppose X has the μ-SMSP. Let F : Ω → cwk(X) be a scalarly measurable multifunction such that every scalarly measurable selector of F is Pettis integrable. Then F is Pettis integrable. Proof. For any fixed A ∈ Σ the set ISF (A) is closed and convex. We prove now that ISF (A) ∈ cwk(X). By James’ theorem (cf. [17, §5]) we only have to prove that every x ∗ ∈ X ∗ attains its supremum on ISF (A). Fix x ∗ ∈ X ∗ and consider the multi-function
Gx ∗ : Ω → cwk(X), Gx ∗ (ω) := x ∈ F (ω): x ∗ (x) = δ ∗ x ∗ , F (ω) . Since Gx ∗ is scalarly measurable (by Lemma 2.4) and X has the μ-SMSP, there is a scalarly measurable selector gx ∗ of Gx ∗ . In particular, gx ∗ is a selector of F and so it is Pettis integrable. Hence δ ∗ (x ∗ , F ) = x ∗ ◦ gx ∗ is integrable. By the very definition, we have A gx ∗ dμ ∈ ISF (A). We claim that gx ∗ dμ . sup x ∗ (x): x ∈ ISF (A) = x ∗ A
B. Cascales et al. / Journal of Functional Analysis 256 (2009) 673–699
685
Indeed, notice that for each Pettis integrable selector f of F we have x
∗
gx ∗ dμ = A
∗
x ◦ gx ∗ dμ = A
A
δ ∗ (x ∗ , F ) dμ
A
x ∗ ◦ f dμ = x ∗
f dμ ,
A
hence sup x ∗ (x): x ∈ ISF (A) = sup x ∗ (x): x ∈ ISF (A) = x ∗
g dμ . x∗
A
This proves that IS F (A) is weakly compact. Moreover, the previous equality can be read as δ ∗ (x ∗ , ISF (A)) = A δ ∗ (x ∗ , F ) dμ. It follows that F is Pettis integrable. 2 Recall that the Banach space X is said to have the μ-Pettis Integral Property (shortly μ-PIP) if every scalarly measurable and scalarly bounded function f : Ω → X is Pettis integrable. Here f : Ω → X is said to be scalarly bounded if there is M > 0 such that for each x ∗ ∈ BX∗ we have |x ∗ ◦ f | M μ-a.e. (the exceptional set depending on x ∗ ). Equivalently, X has the μ-PIP if and only if the Pettis integrability of any function f : Ω → X is equivalent to the fact that the family Zf = {x ∗ ◦ f : x ∗ ∈ BX∗ } ⊂ RΩ is uniformly integrable. Corollary 4.3. Suppose X has the μ-SMSP and the μ-PIP. Let F : Ω → cwk(X) be a multifunction. Then F is Pettis integrable if and only if WF is uniformly integrable. Proof. It only remains to prove the “if” part. Observe that F is scalarly measurable. Each scalarly measurable selector f of F satisfies −δ ∗ (−x ∗ , F ) x ∗ ◦ f δ ∗ (x ∗ , F ) for all x ∗ ∈ BX∗ . Since WF is uniformly integrable, the same holds for Zf and thus f is Pettis integrable (because X has the μ-PIP). The result now follows from Theorem 4.2. 2 The Banach space X has the PIP if it has the μ-PIP for any complete probability measure μ. The class of Banach spaces with the PIP is very large and contains, for instance, all spaces having Corson’s property (C), see [42, Theorem 5-2-4], hence all weakly Lindelöf Banach spaces and all Banach spaces with w ∗ -angelic dual [35]. Recall that a topological space T is said to be angelic if each relatively countably compact set C ⊂ T is relatively compact and, moreover, each point in the closure of C is the limit of a sequence in C. The following cardinal number will be used in several examples that follow: E >0 , κ(μ) = min card(E): E ⊂ Σ, μ(E) = 0 for every E ∈ E, μ∗ defined if there exist such infinite families E (this happens, for instance, if μ is not purely atomic). Here μ∗ denotes the outer measure induced by μ. Notice that κ(μ) ω1 . We point out that the
686
B. Cascales et al. / Journal of Functional Analysis 256 (2009) 673–699
intersection of less than κ(μ) elements of Σ also belongs to Σ , cf. [37, Lemma 4.4]. When κ(μ) cannot be defined, the intersection of any family of measurable sets is measurable and all our results involving κ(μ) are true without the restrictions on the cardinalities or density characters appearing in their statement. It is well known (cf. [40]) that Martin’s axiom implies the statement “κ(Lebesgue measure on [0, 1]) = c” (Axiom M). The Banach space X has both the μ-SMSP and the PIP in each of the following cases: • • • •
X is separable. X is reflexive, Theorem 5.1. (X ∗ , w ∗ ) is angelic and dens(X ∗ , w ∗ ) κ(μ), Example 5.5. X = Y ∗ has property (C) and dens(Y ) κ(μ), Example 5.6.
On the other hand, we will also see that X has the μ-SMSP whenever X ∗ is w ∗ -separable, Theorem 5.15. However, such an X does not have the μ-PIP in general. Indeed, Fremlin and Talagrand [18] showed that ∞ (N) fails the μ-PIP for certain pathological measure μ. They also proved that, at least under Axiom M, if BX∗ is w ∗ -separable for some equivalent norm on X (equivalently, X is isomorphic to a subspace of ∞ (N)), then X has the PIP with respect to any perfect measure (for instance, a Radon finite measure on a topological space), cf. [42, Theorems 6-1-2 and 6-1-3]. We end up this section turning our attention to the following question, thoroughly studied in [5,6] within the setting of separable Banach spaces: What is the relationship between the Pettis integrability of the multi-function F : Ω → cwk(X) and that of the single-valued composition j ◦ F : Ω → ∞ (BX∗ )? As in the separable case, see [5, Proposition 3.5], F is Pettis integrable whenever j ◦ F is. The proof of this fact given here is more direct. Proposition 4.4. Let F : Ω → cwk(X) be a multi-function such that j ◦ F is Pettis integrable. Then F is Pettis integrable and
j IF (A) =
j ◦ F dμ for every A ∈ Σ. A
Proof. Since j ◦ F is Pettis integrable, the composition ex ∗ , j ◦ F = δ ∗ (x ∗ , F ) is integrable for every x ∗ ∈ BX∗ . Fix A ∈ Σ. The Pettis integrability of j ◦ F and the Hahn–Banach separation theorem ensure that
j ◦ F dμ ∈ μ(A) · co (j ◦ F )(A) ,
A
cf. [13, proof of Corollary 8, p. 48]. Since j (cwk(X)) is a closed convex cone, we conclude that A j ◦ F dμ = j (CA ) for some CA ∈ cwk(X). Then
B. Cascales et al. / Journal of Functional Analysis 256 (2009) 673–699
δ ∗ (x ∗ , F ) dμ =
A
687
ex ∗ , j ◦ F dμ = ex ∗ , j ◦ F dμ = δ ∗ (x ∗ , CA )
A
A
for every x ∗ ∈ BX∗ . This shows that F is Pettis integrable, with j (IF (A)) = every A ∈ Σ. 2
Aj
◦ F dμ for
It is known that the converse of Proposition 4.4 does not hold in general even for separable Banach spaces, see [6, Theorem 2.1]. However, it is valid under some additional assumptions on the given multi-function. Proposition 4.5. Let F : Ω → cwk(X) be a multi-function such that (j ◦ F )(Ω) is contained in a subspace of ∞ (BX∗ ) having w ∗ -angelic dual (this happens, for instance, if F (Ω) is separable for the Hausdorff distance). The following statements are equivalent: (i) F is Pettis integrable; (ii) WF is uniformly integrable; (iii) j ◦ F is Pettis integrable. Proof. The implication (i) ⇒ (ii) follows from Theorem 4.1 and (iii) ⇒ (i) from Proposition 4.4. Let us prove (ii) ⇒ (iii). Let Y ⊂ ∞ (BX∗ ) be a subspace containing (j ◦ F )(Ω) such that Y ∗ is w ∗ -angelic. Notice that the set B := {ex ∗ |Y : x ∗ ∈ BX∗ } ⊂ BY ∗ is norming. The desired conclusion now follows by applying [5, Lemma 3.3] to the Y -valued function j ◦ F , see the comments in [5, p. 552]. 2 Recall that a convex, closed, bounded, non-empty set C ⊂ X is norm compact if and only if the real-valued mapping given by x ∗ → δ ∗ (x ∗ , C) is w ∗ -continuous on BX∗ , cf. [31, Section 7]. Thus j (ck(X)) ⊂ C(BX∗ ) = C(BX∗ , w ∗ ). Proposition 4.6. Suppose X ∗ is w ∗ -angelic. Let F : Ω → cwk(X) be a multi-function with norm compact values such that WF is uniformly integrable. Then F is Pettis integrable and IF (A) is norm compact for every A ∈ Σ. ∗ ∗ F : X ∗ → R given by ϕ F (x ∗ ) = Proof. Fix A ∈ Σ. We claim that the mapping ϕA A A δ (x ∗ , F ) dμ ∗ ∗ ∗ ∗ is w -continuous when restricted to BX . Indeed, fix B ⊂ BX and take x ∈ B w . Since (X ∗ , w ∗ ) is angelic, there is a sequence (xn∗ ) in B converging to x ∗ in the w ∗ -topology. Given ω ∈ Ω, the set F (ω) is norm compact and so the mapping δ ∗ (·, F (ω)) is w ∗ -continuous on BX∗ , hence δ ∗ (xn∗ , F (ω)) → δ ∗ (x ∗ , F (ω)) as n → ∞. Since WF is uniformly integrable, an appeal to Vitali’s convergence theorem ensures that
F ∗ F ∗ xn = δ ∗ xn∗ , F dμ → δ ∗ (x ∗ , F ) dμ = ϕA (x ) as n → ∞. ϕA A ∗
A ∗
F (B w ) ⊂ ϕ F (B). Since this inclusion holds for As x ∗ ∈ B w is arbitrary, we conclude that ϕA A F| F ∗ any set B ⊂ BX∗ , the restriction ϕA BX∗ is w -continuous, as claimed. Similarly, ϕA |nBX∗ is F ∗ w -continuous for every n ∈ N. Bearing in mind that ϕA is convex, an appeal to the Banach– F is w ∗ -lower semicontinuous. By Dieudonné theorem (cf. [16, Theorem 4.44]) ensures that ϕA
688
B. Cascales et al. / Journal of Functional Analysis 256 (2009) 673–699
F (x ∗ ) = [7, Theorem II-16], there is a convex, closed, bounded, non-empty set C ⊂ X such that ϕA F ∗ ∗ ∗ ∗ ∗ δ (x , C) for every x ∈ X . The w -continuity of ϕA |BX∗ guarantees that C is norm compact and the proof is over. 2
5. Measurable selectors 5.1. Scalarly measurable selectors The first measurable selection results of this subsection follow from the existence of scalarly measurable selectors for Pettis integrable cwk(X)-valued functions, Theorem 2.5 above. Theorem 5.1. If X is reflexive, then it has the μ-SMSP. Proof. Let F : Ω → cwk(X) be a scalarly measurable multi-function. Since ∗ ∗ δ (x , F ): x ∗ ∈ X ∗ , x ∗ = 1 is a pointwise bounded family of measurable functions, we can find a countable partition E1 , E2 , . . . of Ω in Σ and a sequence (Mn ) of positive real numbers such that, for each n ∈ N and each x ∗ ∈ X ∗ with x ∗ = 1, we have |δ ∗ (x ∗ , F )|En | Mn μ-a.e. (cf. [32, Proposition 3.1]). Fix n ∈ N and consider the (constant) Pettis integrable multi-function Hn : En → cwk(X) given by Hn (ω) := Mn BX . Observe that for each x ∗ ∈ X ∗ we have δ ∗ (x ∗ , F |En ) δ ∗ (x ∗ , Hn ) μ-a.e. From Lemma 2.2 it follows that F |En is Pettis integrable. By Theorem 2.5, we know that F |En admits a scalarly measurable selector fn : En → X. Define f : Ω → X by f (ω) := fn (ω) if ω ∈ En , n ∈ N. Clearly, f is a scalarly measurable selector of F . 2 Theorem 5.2. Suppose X ∗ is w ∗ -angelic. Then every scalarly measurable multi-function F : Ω → ck(X) admits a scalarly measurable selector. Proof. Again, since WF is a pointwise bounded family of measurable functions, there is a countable partition E1 , E2 , . . . of Ω in Σ and a sequence (Mn ) of positive real numbers such that, for each n ∈ N and each x ∗ ∈ BX∗ , we have |δ ∗ (x ∗ , F )|En | Mn μ-a.e. Given n ∈ N, the previous inequality ensures that the family WF |En is uniformly integrable and Proposition 4.6 can be applied to conclude that F |En is Pettis integrable. The proof finishes as in Theorem 5.1. 2 At this point it is convenient to introduce the following terminology. Given a topological space T , we denote by k(T ) the collection of all compact non-empty subsets of T . Let M be a non-empty family of closed subsets of T . We say that a multi-function F : Ω → k(T ) is Mmeasurable if {ω ∈ Ω: F (ω) ∩ M = ∅} ∈ Σ for every M ∈ M. Clearly, with this terminology, a multi-function F : Ω → cwk(X) is scalarly measurable if and only if it is M-measurable for M = collection of all closed half-spaces of X. Lemma 5.3. Let T be a topological space and M a non-empty family of closed subsets of T . Let γ < κ(μ) and, for each α < γ , let Fα : Ω → k(T ) be a M-measurable multi-function. Suppose Fβ (ω) ⊃ Fα (ω) for every β < α < γ and every ω ∈ Ω. Then:
B. Cascales et al. / Journal of Functional Analysis 256 (2009) 673–699
689
(i) For each ω ∈ Ω, the set F (ω) := α<γ Fα (ω) is compact and non-empty. (ii) The multi-function F : Ω → k(T ) is M-measurable. Proof. Given ω ∈ Ω, the net of compact non-empty sets (Fα (ω))α<γ is decreasing and so it has compact non-empty intersection. In order to prove the second assertion, take M ∈ M and observe that, since (Fα (ω) ∩ M)α<γ is a decreasing net of compact sets, we have
ω ∈ Ω: Fα (ω) ∩ M = ∅ . ω ∈ Ω: F (ω) ∩ M = ∅ = α<γ
The M-measurability of each Fα ensures that {ω ∈ Ω: Fα (ω) ∩ M = ∅} ∈ Σ . Since card(γ ) < κ(μ), it follows that {ω ∈ Ω: F (ω) ∩ M = ∅} ∈ Σ . 2 Our approach to the next theorem is inspired somehow by some of the ideas in the original proof of Valadier’s result [43] saying that Banach spaces with w ∗ -separable dual always have the μ-SMSP (Theorem 5.15 below). Theorem 5.4. Suppose there is a set Γ ⊂ X ∗ satisfying the following properties: (i) card(Γ ) κ(μ). (ii) Γ separates the points of X. (iii) A function f : Ω → X is scalarly measurable if and only if x ∗ ◦ f is measurable for every x∗ ∈ Γ . Then X has the μ-SMSP. Proof. Enumerate Γ = {xα∗ : α < card(Γ )}. Fix a scalarly measurable multi-function F : Ω → cwk(X). We divide the proof of the existence of a scalarly measurable selector of F into several steps. Step 1. Define F0 := F . We will construct by transfinite induction a family of scalarly measurable multi-functions Fα : Ω → cwk(X), with α < card(Γ ), such that Fα (ω) =
x ∈ Fβ (ω): xβ∗ (x) = δ ∗ xβ∗ , Fβ (ω)
for all ω ∈ Ω,
(2)
β<α
for every 0 < α < card(Γ ). To this end, assume that 0 < γ < card(Γ ) and that we have already constructed a family (Fα )α<γ of scalarly measurable multi-functions satisfying (2) for every 0 < α < γ . Given α < γ , Lemma 2.4 applies to conclude that the multi-function Gα : Ω → cwk(X) given by
Gα (ω) := x ∈ Fα (ω): xα∗ (x) = δ ∗ xα∗ , Fα (ω) is scalarly measurable. Observe that Gβ (ω) ⊃ Gα (ω) for every β < α < γ and every ω ∈ Ω. Since γ < card(Γ ) κ(μ), Lemma 5.3 allows us to define a scalarly measurable multi-function Fγ : Ω → cwk(X) by the formula Fγ (ω) := α<γ Gα (ω). Obviously, Fγ satisfies (2) by construction.
690
B. Cascales et al. / Journal of Functional Analysis 256 (2009) 673–699
Step 2. Given ω ∈ Ω, the net of weakly compact non-empty sets (Fα (ω))α
xβ∗ (x1 ) = δ ∗ xβ∗ , Fβ (ω) = xβ∗ (x2 ) for every β < card(Γ ), and the fact that Γ separates the points of X implies x1 = x2 . Therefore, there is a function f : Ω → X such that
Fα (ω) = f (ω)
for every ω ∈ Ω.
α
Step 3. Clearly, f is a selector of F . By assumption, in order to prove that f is scalarly measurable we only have to check that xβ∗ ◦ f is measurable for every β < card(Γ ). Indeed, take β < α < card(Γ ). Then f (ω) ∈ Fα (ω) and therefore xβ∗ (f (ω)) = δ ∗ (xβ∗ , Fβ (ω)) for every ω ∈ Ω. Since Fβ is scalarly measurable, we conclude that xβ∗ ◦ f is measurable. The proof is over. 2 A well-known result of Edgar, see [14, Theorem 2.3], states that the Baire σ -algebra of a locally convex space endowed with its weak topology is exactly the σ -algebra generated by all the elements of the topological dual. In particular, if Γ ⊂ X ∗ is a set separating the points of X and σ (X, Γ ) denotes the topology on X of pointwise convergence on Γ , then Baire(X, σ (X, Γ )) is just the σ -algebra on X generated by Γ . Thus, condition (iii) in Theorem 5.4 is equivalent to “f is Baire(X, σ (X, Γ ))-measurable.” Bearing this in mind, observe that Theorem 5.4 ensures that X has the μ-SMSP in the following two cases: Example 5.5. (X ∗ , w ∗ ) is angelic and dens(X ∗ , w ∗ ) κ(μ). By a result of Gulisashvili [21], when (X ∗ , w ∗ ) is angelic, the equality Baire(X, σ (X, Γ )) = Baire(X, w) holds for any set Γ ⊂ X ∗ separating the points of X. A wide class of spaces having w ∗ -angelic dual is that of weakly Lindelöf determined (WLD) Banach spaces. This class contains all weakly compactly generated spaces (cf. [16, Chapters 11 and 12]) and for every WLD space X the equality dens(X ∗ , w ∗ ) = dens(X) holds. In particular, any weakly compactly generated Banach space with density character less than or equal to ω1 has the μ-SMSP. For instance, this applies to c0 (ω1 ), separable Banach spaces, etc. Example 5.6. X = Y ∗ has property (C) and dens(Y ) κ(μ). Indeed, any norm dense set Γ ⊂ Y separates the points of X and satisfies Baire(X, σ (X, Γ )) = Baire(X, w ∗ ). On the other hand, since X is a dual space having property (C), the equality Baire(X, w ∗ ) = Baire(X, w) holds, see [39, Corollary 3.10]. Next three lemmas are needed to prove Theorem 5.10. Lemma 5.7. Let A ∈ cwk(X) and x0∗ ∈ X ∗ satisfying inf x0∗ (A) < b < sup x0∗ (A) for some b ∈ R. Let x ∈ A such that x0∗ (x) b. Then for every ε > 0 there is y ∈ A such that x − y ε and x0∗ (y) ∈ [b, sup x0∗ (A)] ∩ Q. Proof. Since A ∈ cwk(X), we have x0∗ (A) = [inf x0∗ (A), sup x0∗ (A)]. There are two possibilities:
B. Cascales et al. / Journal of Functional Analysis 256 (2009) 673–699
691
Case 1. Suppose x0∗ (x) < sup x0∗ (A). Fix z ∈ A such that x0∗ (z) = sup x0∗ (A) and consider the mapping ϕ : [0, 1] → [x0∗ (x), sup x0∗ (A)] given by ϕ(θ ) := x0∗ (θ z + (1 − θ )x). We can choose 0 < θ < min{ε/x − z, 1} such that ϕ(θ ) ∈ Q. Then the vector y := θ z + (1 − θ )x satisfies the required properties. Case 2. Suppose x0∗ (x) = sup x0∗ (A). Take z ∈ A such that x0∗ (z) = b and consider now the mapping ϕ : [0, 1] → [b, sup x0∗ (A)] given by ϕ(θ ) := x0∗ (θ z + (1 − θ )x). Choose 0 < θ < min{ε/x − z, 1} such that ϕ(θ ) ∈ Q. Then y := θ z + (1 − θ )x works. 2 Lemma 5.8. (See [43, Lemme 3] or [7, Proposition I-24].) Let C ∈ cwk(X), x0∗ ∈ X ∗ and α ∈ R. Suppose H := {x ∈ X: x0∗ (x) = α} intersects C. Then C ∩ H ∈ cwk(X) and δ ∗ (x ∗ , C ∩ H ) = inf δ ∗ x ∗ − λx0∗ , C + λα: λ ∈ Q for every x ∗ ∈ X ∗ . Lemma 5.9. Let F : Ω → cwk(X) be a scalarly measurable multi-function and consider a measurable function h : Ω → R. Fix x0∗ ∈ X ∗ and write L(ω) := x ∈ X: x0∗ (x) h(ω) for every ω ∈ Ω. Then E := {ω ∈ Ω: F (ω) ∩ L(ω) = ∅} ∈ Σ and the multi-function G : E → cwk(X),
G(ω) := F (ω) ∩ L(ω),
is scalarly measurable. Proof. Clearly, the set E = {ω ∈ Ω: δ ∗ (x0∗ , F (ω)) h(ω)} belongs to Σ . Note that −δ ∗ (−x0∗ , F (ω)) = inf x0∗ (F (ω)) for every ω ∈ Ω. The sets
E1 := ω ∈ E: inf x0∗ F (ω) h(ω) ,
E2 := ω ∈ E: sup x0∗ F (ω) = h(ω) ,
E3 := ω ∈ E: inf x0∗ F (ω) < h(ω) < sup x0∗ F (ω) belong to Σ and E = E1 ∪ E2 ∪ E3 . We have G(ω) = F (ω) whenever ω ∈ E1 , thus the restriction G|E1 is scalarly measurable. On the other hand, we also have
for every ω ∈ E2 , G(ω) = x ∈ F (ω): x0∗ (x) = δ ∗ x0∗ , F (ω) hence Lemma 2.4 can be applied to conclude that G|E2 is scalarly measurable. In order to finish the proof it only remains to show that G|E3 is scalarly measurable as well. By Lemma 5.7, for each ω ∈ E3 we have G(ω) =
norm F (ω) ∩ x ∈ X: x0∗ (x) = q ,
q∈I (ω)
where I (ω) := {q ∈ Q: h(ω) q δ ∗ (x0∗ , F (ω))}. Define
J (q) := ω ∈ E3 : h(ω) q δ ∗ x0∗ , F (ω) ∈ Σ
(3)
692
B. Cascales et al. / Journal of Functional Analysis 256 (2009) 673–699
for every q ∈ Q. Fix x ∗ ∈ X ∗ and a ∈ R, and write W := {x ∈ X: x ∗ (x) > a}. Given q ∈ Q, Lemma 5.8 ensures that the multi-function J (q) → cwk(X) given by ω → F (ω) ∩ {x ∈ X: x0∗ (x) = q} is scalarly measurable, so the set ω ∈ J (q): F (ω) ∩ x ∈ X: x0∗ (x) = q ∩ W = ∅ belongs to Σ. Since W is open, equality (3) yields ∗ ω ∈ E3 : G(ω) ∩ W = ∅ = ω ∈ E3 : F (ω) ∩ x ∈ X: x0 (x) = q ∩ W = ∅ q∈I (ω)
ω ∈ J (q): F (ω) ∩ x ∈ X: x0∗ (x) = q ∩ W = ∅ ∈ Σ. = q∈Q
This shows that G is scalarly measurable.
2
Let Mw be the collection of all finite intersections of closed half-spaces of X. Theorem 5.10. Let F : Ω → cwk(X) be a multi-function. Then F is scalarly measurable if and only if F is Mw -measurable. Proof. It only remains to check the “only if.” We prove the following statement by induction on n ∈ N: (∗) For each scalarly measurable multi-function G : E → cwk(X), where E ∈ Σ , the set {ω ∈ E: G(ω) ∩ C = ∅} belongs to Σ whenever C is the intersection of n closed half-spaces of X. The case n = 1 follows directly from the scalar measurability. Assume n > 1 and the induction hypothesis. Fix a scalarly measurable multi-function G : E → cwk(X), where E ∈ Σ. Take C := ni=1 {x ∈ X: xi∗ (x) ai }, where x1∗ , . . . , xn∗ ∈ X ∗ and a1 , . . . , an ∈ R. Define E := {ω ∈ E: δ ∗ (xn∗ , G(ω)) an } ∈ Σ and consider the multi-function G : E → cwk(X),
G (ω) := G(ω) ∩ x ∈ X: xn∗ (x) an ,
which is scalarly measurable by Lemma 5.9. Define C := induction hypothesis, the set
n−1
i=1 {x
∈ X: xi∗ (x) ai }. Now, by
ω ∈ E : G (ω) ∩ C = ∅ = ω ∈ E: G(ω) ∩ C = ∅ belongs to Σ. The proof is over.
2
The following lemma is a nice tool to get measurable selectors that will also be applied in the next subsection. Lemma 5.11. Let T be a topological space and M a non-empty family of closed subsets of T . Suppose M is closed under finite intersections. Let g : T → [0, ∞) be a function such that g −1 ([0, a]) ∈ M for every a 0. Let F : Ω → k(T ) be a M-measurable multi-function. Then:
B. Cascales et al. / Journal of Functional Analysis 256 (2009) 673–699
693
(i) For each ω ∈ Ω, the set G(ω) := t ∈ F (ω): g(t) = inf g(t ): t ∈ F (ω) is compact and non-empty. (ii) The multi-function G : Ω → k(T ) is M-measurable. Proof. Since M is made up of closed sets, g is lower semicontinuous and (i) follows straightforwardly bearing in mind that each F (ω) is compact and non-empty. We divide the proof of (ii) into several steps. Step 1. Fix n ∈ N. For each m ∈ N we define An,m := g −1 ([0, m/2n ]) ∈ M and B n,m := {ω ∈ Ω: F (ω) ∩ An,m = ∅} ∈ Σ . Clearly, Bn,m ⊂ Bn,m+1 for every m ∈ N and Ω = ∞ m=1 Bn,m . Define Cn,1 := Bn,1 and Cn,m := Bn,m \ Bn,m−1 for every m 2, so that Cn,1 , Cn,2 , . . . is a countable partition of Ω in Σ. Consider the multi-function Fn : Ω → k(T ) defined by Fn (ω) := F (ω) ∩ An,m whenever ω ∈ Cn,m . Then Fn is M-measurable. Indeed, given M ∈ M, note that An,m ∩ M ∈ M for every m ∈ N and we have
∞
ω ∈ Ω: Fn (ω) ∩ M = ∅ = Cn,m ∩ ω ∈ Ω: F (ω) ∩ (An,m ∩ M) = ∅ ∈ Σ m=1
since F is M-measurable. Step 2. Clearly, Cn,m = Cn+1,2m−1 ∪ Cn+1,2m and An+1,2m−1 ⊂ An+1,2m = An,m for every n, m ∈ N, by the very definitions. It follows that Fn+1 (ω) ⊂ Fn (ω) for every ω ∈ Ω and every n ∈ N. In view of Lemma 5.3, we can define a M-measurable multi-function H : Ω → k(T ) by H (ω) := ∞ n=1 Fn (ω). Step 3. Given ω ∈ Ω, note that a point t ∈ F (ω) does not belong to G(ω) if and only if g(t ) < m/2n < g(t) for some t ∈ F (ω) and some n, m ∈ N, which is equivalent to saying that ω ∈ Cn,m for some 1 m m and t ∈ / An,m . It follows that G(ω) = H (ω) for every ω ∈ Ω and the proof is over. 2 Lemma 5.12. Let T be a topological space and M a non-empty family of closed subsets of T . Suppose M is closed under finite intersections. Let κ < κ(μ) be a cardinal and write M(κ) to denote the collection of all intersections of at most κ elements of M. Then a multi-function F : Ω → k(T ) is M-measurable if and only if it is M(κ)-measurable. Proof. It only remains to prove the “only if.” We will check that F is M(κ)-measurable for every cardinal κ < κ(μ) by transfinite induction. Fix such a cardinal and assume that F is M(κ )measurable for every cardinal κ < κ. Clearly, the conclusion follows automatically if κ is finite, since M is closed under finite intersections. So assume that κ is infinite. Take a family {Mα : α < κ} ⊂ M and define, for each ordinal β < κ, the set Nβ :=
α<β
Mα ∈ M card(β) ,
694
B. Cascales et al. / Journal of Functional Analysis 256 (2009) 673–699
so that {ω ∈ Ω: F (ω) ∩ Nβ = ∅} ∈ Σ by induction hypothesis. Given ω ∈ Ω, the net of compact sets (F (ω) ∩ Nβ )β<κ is decreasing and, therefore, we have ω ∈ Ω: F (ω) ∩ Nβ = ∅ = ω ∈ Ω: F (ω) ∩ Nβ = ∅ . β<κ
β<κ
Observe that β<κ Nβ = α<κ Mα . Since the intersection of less than κ(μ) elements of Σ also belongs to Σ and κ < κ(μ), we conclude that ω ∈ Ω: F (ω) ∩ Mα = ∅ ∈ Σ. α<κ
This shows that F is M(κ)-measurable, as required.
2
In the next two theorems we apply the previous work to present sufficient conditions on X to have the μ-SMSP. Recall that a norm · on X is said to be strictly convex if x = x whenever x, x ∈ X are such that x = x = 1 and x + x = 2. Theorem 5.13. If X admits an equivalent strictly convex norm with the property that dens(BX∗ , w ∗ ) < κ(μ), then X has the μ-SMSP. Proof. Write κ := dens(BX∗ , w ∗ ). Let F : Ω → cwk(X) be a scalarly measurable multifunction. By Theorem 5.10 and Lemma 5.12, F is Mw (κ)-measurable. Let · be an equivalent strictly convex norm with dens(BX∗ , w ∗ ) < κ(μ) and define g : X → [0, ∞) by g(x) := x. Observe that
g −1 [0, a] = x ∈ X: x ∗ (x) a ∈ Mw (κ)
for every a 0,
x ∗ ∈D
where D ⊂ BX∗ is any w ∗ -dense set with card(D) = κ. Given ω ∈ Ω, the set G(ω) := x ∈ F (ω): x = inf x : x ∈ F (ω) contains only one point, say f (ω), because F (ω) ∈ cwk(X) and · is w-lower semicontinuous and strictly convex. Note that the function f : Ω → X is a selector of F . We can now apply Lemma 5.11 (working with the topological space (X, w) and considering the family M = Mw (κ)) to conclude that f −1 (C) ∈ Σ for every C ∈ Mw (κ), so that f is scalarly measurable. 2 A norm · on X is called locally uniformly rotund (shortly LUR) if xn − x → 0 whenever the sequence (xn ) in X and x ∈ X satisfy xn → x and xn + x → 2x. Clearly, this property implies strict convexity. Many Banach spaces admit an equivalent LUR norm, for instance, the WLD ones, cf. [10, Corollary 1.10, p. 286]. For complete information about renormings in Banach spaces we refer the reader to [10,19,46]. As an application of the previous theorem we obtain:
B. Cascales et al. / Journal of Functional Analysis 256 (2009) 673–699
695
Example 5.14. C([0, ω1 ]) has the μ-SMSP whenever κ(μ) > ω1 . Indeed, it is known that C([0, ω1 ]) admits an equivalent LUR (in particular, strictly convex) norm, because [0, ω1 ] is a Valdivia compactum, cf. [10, Corollary 1.10, p. 286]. On the other hand, the dual unit ball of any equivalent norm on C([0, ω1 ]) has w ∗ -density character ω1 (bear in mind that this space contains a subspace isomorphic to c0 (ω1 )). A similar argument allows us to give an alternative proof of the previously announced result of Valadier, see [43, Proposition 6]. Theorem 5.15 (Valadier). If X ∗ is w ∗ -separable, then X has the μ-SMSP. Proof. Let F : Ω → cwk(X) be a scalarly measurable multi-function. By Theorem 5.10 and Lemma 5.12, we know that F is Mw (ℵ0 )-measurable. Fix a countable w ∗ -dense set {xn∗ : n ∈ N} ⊂ X ∗ and consider the operator T : X → 2 (N),
T (x) :=
xn∗ (x) . 2n
Define g : X → [0, ∞) by g(x) := T (x) 2 (N) . Since B 2 (N)∗ is w ∗ -separable, we have g −1 ([0, a]) ∈ Mw (ℵ0 ) for every a 0. Since g is a w-lower semicontinuous strictly convex norm on X (non-necessarily equivalent to the original one!), the arguments in the proof of Theorem 5.13 (dealing now with the family of weakly closed sets Mw (ℵ0 )) ensure that F admits a scalarly measurable selector. 2 It is well known that X admits an equivalent strictly convex norm whenever X ∗ is w ∗ separable, cf. [10, Theorem 2.4, p. 46]. However, the fact that such an X has the μ-SMSP cannot be deduced, in general, from Theorem 5.13 above. Indeed, the Johnson–Lindenstrauss space JL2 has w ∗ -separable dual but, for any equivalent norm on JL2 , the corresponding dual unit ball is not w ∗ -separable, see [26, Example 1]. The technique used in the proof of Theorem 2.6 can be used to prove Theorem 5.16 below: the particular case of Banach spaces having w ∗ -separable dual was first proved by Valadier in [43, Proposition 7]. Theorem 5.16. Suppose X has the μ-SMSP. Let F : Ω → cwk(X) be a scalarly measurable multi-function. Then there is a collection {fα }α<dens(X∗ ,w∗ ) of scalarly measurable selectors of F such that F (ω) = fα (ω): α < dens(X ∗ , w ∗ )
for every ω ∈ Ω.
5.2. Borel measurable selectors In this subsection we exploit Lemma 5.11 in order to find nice selectors for multi-functions with stronger measurability properties. It is convenient to recall first some facts concerning measurability in Banach spaces.
696
B. Cascales et al. / Journal of Functional Analysis 256 (2009) 673–699
Let Mnc (respectively Mcc ) be the collection of all norm closed (respectively convex closed) subsets of X. Write σ (Mcc ) to denote the smallest σ -algebra on X containing Mcc . In general, we have
Baire(X, w) ⊂ σ Mcc ⊂ Borel(X, w) ⊂ Borel(X, norm). All these σ -algebras coincide for separable X but some inclusions may be strict beyond the separable case. Talagrand [41] showed that Borel( ∞ (N), w) = Borel( ∞ (N), norm) and Edgar [14] proved that the equality Borel(X, w) = Borel(X, norm) holds whenever X admits an equivalent Kadec norm (i.e. a norm for which the weak and norm topologies coincide on the unit sphere; clearly, every LUR norm is Kadec). A result of Raja [36, Theorem 1.2] states that X admits an equivalent LUR norm if and only if every norm open set U ⊂ X can be writcc for every n ∈ N; in this case, we have ten as U = ∞ n=1 (Cn \ Dn ), where Cn , Dn ∈ M cc σ (M ) = Borel(X, norm). On the other hand, it is known that Baire(X, w) = σ (Mcc ) whenever X ∗ is not w ∗ -separable, cf. [22, Theorem 1.5.3], but also for ∞ (N) and the Johnson– Lindenstrauss spaces [26], see [38, Theorem 2.3]. Theorem 5.17. Suppose X admits an equivalent strictly convex norm. Then every Mcc measurable multi-function F : Ω → cwk(X) admits a σ (Mcc )-measurable selector. Proof. Fix an equivalent strictly convex norm · on X. Given ω ∈ Ω, the set G(ω) := x ∈ F (ω): x = inf x : x ∈ F (ω) contains only one point f (ω) because F (ω) ∈ cwk(X) and · is w-lower semicontinuous and strictly convex. The function f : Ω → X is a selector of F . Obviously, the mapping g : X → [0, ∞) given by g(x) := x satisfies g −1 ([0, a]) ∈ Mcc for every a 0. We can apply Lemma 5.11 (working with the topological space (X, w) and taking M = Mcc ) to conclude that f is σ (Mcc )-measurable. 2 In fact, under the same assumption we can say more: Theorem 5.18. Suppose X admits an equivalent strictly convex norm. Then every Mcc measurable multi-function F : Ω → cwk(X) admits a collection {fα }α<dens(X) of σ (Mcc )measurable selectors such that F (ω) = fα (ω): α < dens(X)
for every ω ∈ Ω.
Proof. Fix a dense set {xα : α < κ} ⊂ X, where κ := dens(X), and take an equivalent strictly convex norm · on X. Fix α < κ. Since the multi-function Fα : Ω → cwk(X) given by Fα (ω) := −xα + F (ω) is Mcc -measurable, a glance at the proof of Theorem 5.17 reveals that Fα admits a σ (Mcc )-measurable selector gα : Ω → X with the property that gα (ω) = inf x − xα : x ∈ F (ω)
for every ω ∈ Ω.
(4)
Let us consider the σ (Mcc )-measurable selector fα : Ω → X of F defined by the formula fα (ω) := gα (ω) + xα . We claim that the collection {fα }α<κ fulfills the required property.
B. Cascales et al. / Journal of Functional Analysis 256 (2009) 673–699
697
Indeed, fix ω ∈ Ω and x ∈ F (ω). Given ε > 0, there is α < κ such that x − xα ε, hence (4) yields fα (ω) − x gα (ω) + x − xα 2x − xα 2ε. As x ∈ F (ω) and ε > 0 are arbitrary, we get F (ω) = {fα (ω): α < κ}.
2
As we have mentioned at the beginning of the subsection, if X admits an equivalent LUR norm then σ (Mcc ) = Borel(X, norm). Bearing in mind that every LUR norm is strictly convex, from Theorem 5.18 we deduce the following corollary. Corollary 5.19. Suppose X admits an equivalent LUR norm. Let F : Ω → cwk(X) be a Mcc measurable multi-function. Then F admits a collection of Borel(X, norm)-measurable selectors {fα }α<dens(X) such that F (ω) = fα (ω): α < dens(X)
for every ω ∈ Ω.
We stress that the previous corollary improves a result of Leese [30, Theorem 2], who proved the existence of Borel(X, norm)-measurable selectors for Mnc -measurable multi-functions when X admits an equivalent uniformly rotund norm. Similar arguments to those of Theorems 5.17 and 5.18, now dealing with the norm topology of X, allow us to deduce the following result. Theorem 5.20. Suppose X admits an equivalent strictly convex norm. Let F : Ω → ck(X) be a Mnc -measurable multi-function. Then F admits a collection {fα }α<dens(X) of Borel(X, norm)measurable selectors such that F (ω) = fα (ω): α < dens(X) for every ω ∈ Ω. Under such assumptions, the existence of at least one Borel(X, norm)-measurable selector was first proved by Leese [30, Theorem 1]. To the best of our knowledge, the question below remains unanswered in full generality: Open problem. Does every Banach space have the μ-SMSP for any μ? References [1] A. Amrani, Lemme de Fatou pour l’intégrale de Pettis, Publ. Mat. 42 (1) (1998) 67–79, MR 1628138 (99h:28027). [2] R.J. Aumann, Integrals of set-valued functions, J. Math. Anal. Appl. 12 (1965) 1–12, MR 0185073 (32 #2543). [3] Y. Benyamini, J. Lindenstrauss, Geometric Nonlinear Functional Analysis, vol. 1, Amer. Math. Soc. Colloq. Publ., vol. 48, Amer. Math. Soc., Providence, RI, 2000, MR 1727673 (2001b:46001). [4] R.D. Bourgin, Geometric Aspects of Convex Sets with the Radon–Nikodým Property, Lecture Notes in Math., vol. 993, Springer, Berlin, 1983, MR 704815 (85d:46023). [5] B. Cascales, J. Rodríguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2) (2004) 540– 560, special issue dedicated to John Horváth, MR 2088679 (2005f:26021). [6] B. Cascales, V. Kadets, J. Rodríguez, The Pettis integral for multi-valued functions via single-valued ones, J. Math. Anal. Appl. 332 (1) (2007) 1–10, MR 2319640 (2008e:28025). [7] C. Castaing, M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Math., vol. 580, Springer, Berlin, 1977, MR 0467310 (57 #7169).
698
B. Cascales et al. / Journal of Functional Analysis 256 (2009) 673–699
[8] A. Costé, Set-valued measures, in: Proceedings of the Conference Topology and Measure, I, part 1 (Greifswald), Zinnowitz, 1974, Ernst-Moritz-Arndt Univ., 1978, pp. 55–74, MR 540563 (82b:28019). [9] G. Debreu, Integration of correspondences, in: Proc. Fifth Berkeley Sympos. Math. Statist. and Probability, vol. II: Contributions to Probability Theory, part 1, Berkeley, CA, 1965/1966, Univ. California Press, Berkeley, CA, 1967, pp. 351–372, MR 0228252 (37 #3835). [10] R. Deville, G. Godefroy, V. Zizler, Smoothness and Renormings in Banach Spaces, Pitman Monographs Surveys Pure Appl. Math., vol. 64, Longman Sci. Tech., Harlow, 1993, MR 1211634 (94d:46012). [11] L. Di Piazza, K. Musiał, Set-valued Kurzweil–Henstock–Pettis integral, Set-Valued Anal. 13 (2) (2005) 167–179, MR 2148134 (2006a:28013). [12] L. Di Piazza, K. Musiał, A decomposition theorem for compact-valued Henstock integral, Monatsh. Math. 148 (2) (2006) 119–126, MR 2235359. [13] J. Diestel, J.J. Uhl Jr., Vector Measures, Math. Surveys, vol. 15, Amer. Math. Soc., Providence, RI, 1977, with a foreword by B.J. Pettis, MR 0453964 (56 #12216). [14] G.A. Edgar, Measurability in a Banach space, Indiana Univ. Math. J. 26 (4) (1977) 663–677, MR 0487448 (58 #7081). [15] K. El Amri, C. Hess, On the Pettis integral of closed valued multifunctions, Set-Valued Anal. 8 (4) (2000) 329–360, MR 1802239 (2002e:26025). [16] M. Fabian, P. Habala, P. Hájek, V. Montesinos Santalucía, J. Pelant, V. Zizler, Functional Analysis and InfiniteDimensional Geometry, CMS Books Math./Ouvrages Math. SMC, vol. 8, Springer, New York, 2001, MR 1831176 (2002f:46001). [17] K. Floret, Weakly Compact Sets, Lecture Notes in Math., vol. 801, Springer, Berlin, 1980, lectures held at SUNY, Buffalo, in Spring 1978, MR 576235 (82b:46001). [18] D.H. Fremlin, M. Talagrand, A decomposition theorem for additive set-functions, with applications to Pettis integrals and ergodic means, Math. Z. 168 (2) (1979) 117–142, MR 544700 (80k:28004). [19] G. Godefroy, Renormings of Banach Spaces, Handbook of the Geometry of Banach Spaces, vol. I, North-Holland, Amsterdam, 2001, pp. 781–835, MR 1863706 (2003f:46023). [20] C. Godet-Thobie, Some results about multimeasures and their selectors, in: Proc of the Conf. Measure Theory, Oberwolfach, 1979, in: Lecture Notes in Math., vol. 794, Springer, Berlin, 1980, pp. 112–116, MR 577965 (81g:28011). [21] A.B. Gulisashvili, Estimates for the Pettis integral in interpolation spaces, and a generalization of some imbedding theorems, Soviet Math. Dokl. 25 (1982) 428–432, MR 0651235 (83g:46068). [22] M. Heiliö, Weakly summable measures in Banach spaces, Ann. Acad. Sci. Fenn. Ser. A I Math. Diss. 66 (1988), MR 938371 (89e:46048). [23] C. Hess, Set-valued integration and set-valued probability theory: An overview, in: Handbook of Measure Theory, vols. I, II, North-Holland, Amsterdam, 2002, pp. 617–673, MR 1954624 (2004g:60020). [24] C. Hess, H. Ziat, Théorème de Komlós pour des multifonctions intégrables au sens de Pettis et applications, Ann. Sci. Math. Québec 26 (2) (2002) 181–198, MR 1980843 (2004i:28023). [25] E. Hewitt, K.A. Ross, Abstract Harmonic Analysis, vol. I: Structure of Topological Groups. Integration Theory, Group Representations, Grundlehren Math. Wiss., vol. 115, Academic Press, New York, 1963, MR 0156915 (28 #158). [26] W.B. Johnson, J. Lindenstrauss, Some remarks on weakly compactly generated Banach spaces, Israel J. Math. 17 (1974) 219–230, MR 0417760 (54 #5808). [27] E. Klein, A.C. Thompson, Theory of Correspondences, Canad. Math. Soc. Ser. Monogr. Adv. Texts, Wiley– Interscience, New York, 1984, includes applications to mathematical economics, MR 752692 (86a:90012). [28] G. Köthe, Topological Vector Spaces. I, Grundlehren Math. Wiss., vol. 159, Springer, New York, 1969, MR 40 #1750, translated from the German by D.J.H. Garling. [29] K. Kuratowski, C. Ryll-Nardzewski, A general theorem on selectors, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 13 (1965) 397–403, MR 0188994 (32 #6421). [30] S.J. Leese, Measurable selections in normed spaces, Proc. Edinburgh Math. Soc. (2) 19 (2) (1974/1975) 147–150, MR 0437710 (55 #10634). [31] J.J. Moreau, Fonctionnelles convexes, in: Séminaire sur les Équations aux Dérivées Partielles (1966–1967). II, Collège de France, Paris, 1967, MR 0390443 (52 #11269). [32] K. Musiał, Topics in the theory of Pettis integration, in: School on Measure Theory and Real Analysis, Grado, 1991, Rend. Istit. Mat. Univ. Trieste 23 (1) (1991) 177–262, (1993), MR 1248654 (94k:46084). [33] R. Pallu de la Barrière, Introduction à l’étude des multi-mesures, in: G. Choquet, M. Rogalski, J. Saint-Raymond (Eds.), Initiation Seminar on Analysis, 19th Year: 1979/1980, in: Publ. Math. Univ. Pierre et Marie Curie, vol. 41, Univ. Paris VI, Paris, 1980, Exp. No. 7, 23, MR 670777 (84e:28013).
B. Cascales et al. / Journal of Functional Analysis 256 (2009) 673–699
699
[34] A. Pełczy´nski, Linear extensions, linear averagings, and their applications to linear topological classification of spaces of continuous functions, Dissertationes Math. (Rozprawy Mat.) 58 (1968) 92, MR 0227751 (37 #3335). [35] R. Pol, On a question of H.H. Corson and some related problems, Fund. Math. 109 (2) (1980) 143–154, MR 597061 (82a:46022). [36] M. Raja, First Borel class sets in Banach spaces and the asymptotic-norming property, Israel J. Math. 138 (2003) 253–270, MR 2031959 (2005f:46031). [37] J. Rodríguez, The Bourgain property and convex hulls, Math. Nachr. 280 (11) (2007) 1302–1309, MR 2337347. [38] J. Rodríguez, Weak Baire measurability of the balls in a Banach space, Studia Math. 185 (2) (2008) 169–176, MR 2379966. [39] J. Rodríguez, G. Vera, Uniqueness of measure extensions in Banach spaces, Studia Math. 175 (2) (2006) 139–155, MR 2261730. [40] J.R. Shoenfield, Martin’s axiom, Amer. Math. Monthly 82 (1975) 610–617, MR 0373887 (51 #10087). [41] M. Talagrand, Comparaison des boreliens d’un espace de Banach pour les topologies fortes et faibles, Indiana Univ. Math. J. 27 (6) (1978) 1001–1004, MR 511253 (80d:46033). [42] M. Talagrand, Pettis integral and measure theory, Mem. Amer. Math. Soc. 51 (307) (1984), ix+224, MR 756174 (86j:46042). [43] M. Valadier, Multi-applications mesurables à valeurs convexes compactes, J. Math. Pures Appl. (9) 50 (1971) 265– 297, MR 0299752 (45 #8800). [44] H. Ziat, Convergence theorems for Pettis integrable multifunctions, Bull. Polish Acad. Sci. Math. 45 (2) (1997) 123–137, MR 1466838 (98i:46035). [45] H. Ziat, On a characterization of Pettis integrable multifunctions, Bull. Polish Acad. Sci. Math. 48 (3) (2000) 227– 230, MR 1779005 (2001d:46063). [46] V. Zizler, Nonseparable Banach spaces, in: Handbook of the Geometry of Banach Spaces, vol. 2, North-Holland, Amsterdam, 2003, pp. 1743–1816, MR 1999608 (2004g:46030).
Journal of Functional Analysis 256 (2009) 700–717 www.elsevier.com/locate/jfa
Functional calculi for convolution operators on a discrete, periodic, solvable group ✩ Andrzej Hulanicki a,b , Małgorzata Letachowicz b,∗ a Opole University, Institute of Mathematics and Informatics, Oleska 48, 45 052 Opole, Poland b 50-384 Wroclaw University, pl. Grunwaldzki 2/4, Poland
Received 3 August 2007; accepted 12 November 2008 Available online 13 December 2008 Communicated by L. Gross
Abstract Suppose T is a bounded self-adjoint operator on the Hilbert space L2 (X, μ) and let T=
λ dE(λ)
SpL2 T
be its spectral resolution. Let F be a Borel bounded function on [−a, a], SpL2 T ⊂ [−a, a]. We say that F is a spectral Lp -multiplier for T , if F (T ) =
F (λ) dE(λ)
SpL2 T
✩ This research project has been partially supported by European Commission via Marie Curie Transfer of Knowledge Fellowship Harmonic Analysis, Nonlinear Analysis and Probability (contract number MTKD-CT-2004-013389). The authors were also supported by KBN grants 1 P03A 018 26 and N201 012 31/1020. The results of this paper were presented at Workshop on Algebraic, Geometric and Probabilistic Aspects of Amenability at The Erwin Schrödinger International Institute for Math. Phys. Vienna, July 1–July 13, 2007. The authors would like to express their gratitude to the Organizers: Anna Erschler, Vadim Kaimanovich and Klaus Schmidt, for their hospitality. * Corresponding author. E-mail addresses: [email protected] (A. Hulanicki), [email protected] (M. Letachowicz).
0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.11.011
A. Hulanicki, M. Letachowicz / Journal of Functional Analysis 256 (2009) 700–717
701
is a bounded operator on Lp (X, μ). The paper deals with l 1 -multipliers, where X = G is a discrete (countable) solvable group with ∀x∈G , x 4 = 1, μ is the counting measure and TΦ : l 2 (G) ξ → ξ ∗ Φ ∈ l 2 (G), where Φ = Φ ∗ is a l 1 (G) function, supp Φ generates G. The main result of the paper states that there exists a Ψ on G such that all l 1 -multipliers for TΨ are real analytic at every interior point of Spl 2 (G) TΨ . We also exhibit self-adjoint Φ s in l 1 (G) such that supp Φ generates G and F ∈ Cc2 are l 1 -multipliers for TΦ . © 2008 Elsevier Inc. All rights reserved. Keywords: Discrete group; l 1 -multipliers; Convolution operator; Functional calculi
1. Introduction Suppose T is a bounded self-adjoint operator on the Hilbert space L2 (X, μ) and let T=
λ dE(λ)
SpL2 T
be its spectral resolution. Let F be a Borel bounded function on [−a, a], SpL2 T ⊂ [−a, a]. We say that F is a spectral Lp -multiplier for T , if F (T ) =
F (λ) dE(λ)
SpL2 T
is a bounded operator on Lp (X, μ). If G is a locally compact group, Lp (G) the space of functions p-integrable with respect to the right-invariant Haar measure, then for every Φ = Φ ∗ ∈ L1 (G) TΦ : L2 (G) ξ → ξ ∗ Φ ∈ L2 (G) is a self-adjoint bounded operator. Let TΦ =
λ dEΦ (λ)
SpL2 T
be its spectral resolution. Let F be a Borel bounded function on [−a, a], SpL2 T ⊂ [−a, a]. We say that F is a spectral Lp -multiplier for T , if F (T ) = SpL2 T
F (λ) dEΦ (λ)
702
A. Hulanicki, M. Letachowicz / Journal of Functional Analysis 256 (2009) 700–717
is a bounded operator on Lp (G). If G is a Lie group considerable attention has been paid to spectral Lp multipliers for TΦ , where Φ = p1 , {pt }t>0 being the heat kernel associated to a subLaplacian L on G. This of course can be rephrased as a property of the spectral multipliers of L, since Tp1 =
eλ dEL (λ), R
EL (λ) being the spectral resolution of L. Depending on the group and L it may happen that there exist L1 -spectral multipliers F of finite smoothness or, just the opposite, F must be real analytic at every interior point of SpL2 L. There is a host of results about the subject, some of the references being [1–3,6]. The existence of this alternative, G being an infinite discrete group has been also noticed and studied. To wit, if G is a non-commutative free group, S the set of the free generators and Φ=
1 1 −1 , |S ∪ S −1 | S∪S
then all spectral l p -multipliers for TΦ , p = 2 must be real analytic. As is well known [7], if G is commutative there is no Φ = Φ ∗ ∈ l 1 (G) such that all the spectral multipliers for TΦ are real analytic. If G is finitely generated and is of polynomial growth, then for every Φ = Φ ∗ of finite support TΦ admits spectral multipliers of finite smoothness. Also a combination of theorems by M. Gromov [5] and J. Ludwig [9] yields that for a finitely generated group of polynomial growth for every Φ = Φ ∗ ∈ l 1 (G) there is a non-analytic spectral multiplier F for TΦ . For solvable infinite discrete groups the question has not been settled. In this paper we construct a discrete infinite group G for which, on one hand side, there exists a Ψ = Ψ ∗ ∈ l 1 (G) such that all l 1 -multipliers for TΨ must be real analytic at every interior point of Spl 2 (G) Ψ and on the other there are self-adjoint Ψ s ∈ l 1 (G) such that supp Ψ generates G and for some k all functions F ∈ Cck are l 1 -multipliers for TΨ . Let us remark that we know only about theorems that state that Lp -multipliers F are real analytic, in all the cases of semi-simple, solvable Lie groups or the free group, the proofs of which are based on representation theory. In the present paper, we show exponential decay of the Fourier coefficients of the multiplier F and we do not appeal to representation theory. Our proofs are of the spirit of the classical theorems of Y. Katznelson [8], cf. also [7]. 2. The group The group operation will be written as multiplication. Let C, D, E be three copies of the direct product of infinitely many copies of Z2 = {1, −1} with multiplication. C = ε = (ε1 , ε2 , . . .): εj ∈ {1, −1}, εj = 1, except a finite number of j , D = d = (d1 , d2 , . . .): dj ∈ {1, −1}, dj = 1, except a finite number of j , E = x = (x1 , x2 , . . .): xj ∈ {1, −1}, xj = 1, except a finite number of j .
A. Hulanicki, M. Letachowicz / Journal of Functional Analysis 256 (2009) 700–717
703
We denote by E the direct product of countably many copies of E numbered by elements of D. Thus E is isomorphic with the set of functions η : D → E such that η(d) = 1 for finitely many d s only. We introduce an action of D on E by writing ηd (d ) = η(dd ). The wreath product E D (with fiber E) we denote by L = (η, d): η ∈ E, d ∈ D , the multiplication being given by (η1 , d1 )(η2 , d2 ) = η1 η2d1 , d1 d2 . The group that is going to be the object of our investigations is G = C × L. The following theorem is obvious. Theorem 2.1. The group G is locally finite, meta-abelian, for each element x of G we have x 4 = 1. We define an increasing sequence of finite subgroups Gn of G such that Let
∞
n=1 Gn
Cn = (ε1 , ε2 , . . .): εj ∈ {1, −1}, 1 = εn+1 = εn+2 = · · · , Dn = (d1 , d2 , . . .): dj ∈ {1, −1}, 1 = dn+1 = dn+2 = · · · , En = (x1 , x2 , . . .): xj ∈ {1, −1}, 1 = xn+1 = xn+2 = · · · .
= G.
(2.2)
For the abelian group D we define the set of free generators by di = (1, . . . , 1, −1, 1, . . .): i = 1, 2, . . . ,
d0 = (1, 1, 1, . . .),
(2.3)
similarly for C ci = (1, . . . , 1, −1, 1, . . .): i = 1, 2, . . .
(2.4)
xi = (1, . . . , 1, −1, 1, . . .): i = 1, 2, . . . .
(2.5)
and for E
We put Gn = (c, η, d): c ∈ Cn , supp η ⊂ Dn , range η ⊂ En , d ∈ Dn .
704
A. Hulanicki, M. Letachowicz / Journal of Functional Analysis 256 (2009) 700–717
Clearly n−1 ]
n
|Gn | = 2n 2n2 2n = 22n[1+2
.
(2.6)
Remark 2.7. In virtue of (2.6), G is a residually nilpotent group. This property, however, is not relevant for us here. 3. Functions on E Let E =
∞
n=1 Z2
and ϕ be a function in l 1 (E). We define E(·, ϕ) : R u → E(u, ϕ) ∈ l 1 (E)
by E(u, ϕ)(x) =
∞ (iu)k ϕ ∗ k (x) k=0
k!
.
Of course, E(u, ϕ) ∈ l 1 (E), E(u + v, ϕ) = E(u, ϕ) ∗ E(v, ϕ), E(u, αϕ) = E(αu, ϕ), ϕ∗ψ =ψ ∗ϕ ϕ=ϕ
∗
⇒
⇒
E(u, ϕ + ψ) = E(u, ϕ) ∗ E(v, ψ),
The operator of convolution by E(u, ϕ) is a unitary.
Now, we are going to find a function f0 ∈ l 1 (E), such that
∀N E(u, f0 ) l ∞ cN u−N . The proof is a modification of the classical argument that goes back to Malliavin [10], cf. also [8]. Let us set 1 fm = (δ1 + δx2m x2m−1 − δx2m − δx2m−1 ), 4 where xj is defined in (2.5). The function f0 is defined for an appropriate sequence of real numbers {am } (see (3.2)) by f0 =
∞
am fm ,
m=1
where ∞ m=1
|am | < ∞.
A. Hulanicki, M. Letachowicz / Journal of Functional Analysis 256 (2009) 700–717
705
Since the group E is commutative, so is l 1 (E) with convolution ∗. Since the functions fm commute and supp fm ∩ supp fn = {1} for n = m, we have
∗
∞
E(u, f0 ) ∞ =
E(uam , fm ) ∞ , E(ua , f ) m m
l (E) l (E)
∞ m∈N
here and subsequently the symbols shows that
l (E)
∗
m∈N
m=1
denotes the convolution product. A simple verification fm∗ 2 = fm ,
whence E(α, fm )(x) =
∞ (iα)k f ∗ k (x) m
k!
k=0
= fm (x)
∞ (iα)k fm (x) k=1
∞ (iα)k k=0
=
k!
k!
+ δ1 (x)
− fm (x) + δ1 (x)
= fm (x) eiα − 1 + δ1 (x), hence
1 1
E(α, fm ) ∞ = max 1 2 − 2 cos(α) 2 , 1 10 + 6 cos(α) 2 1. l (E) 4 4 If
2π 3
α
4π 3 ,
then
E(α, fm ) ∞
l (E)
3 . 4
(3.1)
Let {Nk } be an increasing sequence of positive integers such that
Nk 2−k < ∞.
We set am =
π 3 · 2k−1
for
k−1 j =1
Nj < m
k j =1
We have ∞ m=1
so f0 ∈ l 1 (E).
am =
∞ 2π Nk 2−k < ∞, 3 k=1
Nj .
(3.2)
706
A. Hulanicki, M. Letachowicz / Journal of Functional Analysis 256 (2009) 700–717
We have we obtain
2π 3
uam
4π 3
for 2k u 2k+1 and for Nk values am =
C(u) :=
∞
E(u, am fm ) ∞ l
(E)
m=1
π . 3·2k−1
Hence by (3.1)
Nk 3 . 4
If we take Nk = 2k k −2 we obtain C(u) = C(−u) c
u log−2 u 3 , 4
E(u, f0 ) ∞
< cN |u|−N ,
as u → ∞.
Thus ∀N
l (E)
as u → ±∞.
(3.3)
For λ ∈ R we define a function ∞ Tλ = c λ
iu E u, (f0 − λδ1 ) du.
(3.4)
−∞
By (3.3), Tλ ∈ l ∞ (E). For f ∈ l 1 (E), g ∈ l ∞ (E) we write f, g =
f (x)g(x).
x∈E
Now we look for a number λ ∈ R and a function S ∈ l ∞ (E) such that f0 − λδ1 , Tλ = 0,
(3.5)
δ1 , Tλ = 0,
(3.6)
f0 − λδ1 , S = 0,
(3.7)
δ1 , S = 1.
(3.8)
We define M : R u −→ E(u, f0 )(1) ∈ C. takes Since f0 is function with real values, M(u) = M(−u), whence the Fourier transform M real values. Since by (3.3) M is integrable, M vanishes at infinity. Therefore exists a ι such that d dλ M(λ)|λ=ι = 0 and M(ι) = 0. We have
A. Hulanicki, M. Letachowicz / Journal of Functional Analysis 256 (2009) 700–717
dM d (λ) = dλ dλ
∞ E(u, f0 )(1)e
−∞
∞ du = −
−∞
∞ =
−iuλ
707
E(u, f0 )(1)iue−iuλ du
−∞
1 iuE u, (f0 − λδ1 ) (1) du = − δ1 , Tλ cλ
and 1 f0 − λδ1 , Tλ = (f0 − λδ1 )(x) cλ x
∞ ∞
iu
−∞ k=0
∞ = uE(u, f0 − λδ1 )(1)
u=−∞
∞ =−
(iu)k (f0 − λδ1 )∗ k (x) du k! ∞
−
E(u, f0 − λδ1 )(1) du
−∞
E(u, f0 )(1)e−iuλ du = −M(λ).
∞
So (3.5) and (3.6) hold with λ = ı. Let fι = f0 − ιδ1 . What is more, ∀g∈l 1 (E)
g ∗ fι∗ 2 , Tι = 0.
Indeed, ∞ ∗ 2 ∗ 2 g ∗ f , Tι = g ∗ fι (x) iuE(u, fι )(x) du ι x
−∞
∞ ∞ (iu)k fι∗ k (x) ∗2 = du g ∗ fι (x)iu k! −∞
x
k=0
∞ ∞ k ∗k ∗ 2 (iu) fι = (1) du iug ∗ fι ∗ k! −∞ k=0
∞ ∞ ∗ (k+1) uk i k+1 fι = ug ∗ fι ∗ (1) du k! −∞ k=0
(3.9)
708
A. Hulanicki, M. Letachowicz / Journal of Functional Analysis 256 (2009) 700–717
u=∞ ∞ = g ∗ fι ∗ E(u, fι )(1)u − g ∗ fι ∗ E(u, fι )(1) du u=−∞ −∞
u=∞
u=∞
g ∗ fι l 1 |u| · E(u, fι ) l ∞ + gl 1 E(u, fι ) l ∞ = 0. u=−∞
u=−∞
From (3.5), (3.6) and (3.9) we conclude that fι is not invertible, hence, by the Wiener theorem, such that fι (χ0 ) = 0. We easily see that (3.7) and (3.8) hold if we set S = χ0 . there exists χ0 ∈ E Hence (3.7) and (3.8) are proved. Moreover ∀g∈l 1 (E)
g ∗ fι , S = 0.
(3.10)
To summarize, putting cλ = fι , Tι −1 in (3.4), in virtue of (3.9), (3.10) and (3.5)–(3.8), we have fι , Tι = 1,
(3.11)
δ1 , Tι = 0, ∀g∈l 1 (E) g ∗ fι∗ 2 , Tι = 0,
(3.12)
δ1 , S = 1,
(3.14)
g ∗ fι , S = 0.
∀g∈l 1 (E)
(3.13)
(3.15)
4. Functions on E and on L Now we are going to consider l 1 (E). Since E is abelian, l 1 (E) is commutative. Let
f(η) =
fι (η(1)), 0,
if η(d) = 1 for all d = 1, otherwise,
where fι is as in Section 3. Let d ∈ D. We write fd (η) = f ηd . Consequently,
f (η) = d
fι (η(d)), 0,
if η(d ) = 1 for all d = d, otherwise.
We define functions in l ∞ (E) by TW =
d∈W
Tι ⊗
d ∈W /
S,
(4.1)
A. Hulanicki, M. Letachowicz / Journal of Functional Analysis 256 (2009) 700–717
where W ⊂ D is finite. In virtue of (3.11)–(3.15) we have ∗ ∗ j 1, if W = K and jd = 1 for all d ∈ K, d d f , TW = 0, otherwise.
709
(4.2)
d∈K
The space l 1 (L) is isometrically isomorphic to the space of functions Φ : D → l 1 (E), with Φ =
Φ(d) 1 l
Φ ∗ Ψ (d) =
, (E)
d Φ(d ) ∗ Ψ (d d) .
(4.3)
d ∈D
d∈D
To simplify the notation we write Φ ∗ Ψ = ΦΨ,
δ(1,d) ∗ Φ = dΦ,
Φ ∗ δ(1,d) = Φd.
Also we imbed l 1 (E) g → δ1 g ∈ l 1 (L), that is g ∈ l 1 (E) is identified with the function on D (with the value of g in l 1 (E) supported by the identity of D). Consequently, for g ∈ l 1 (E) and d ∈ D, by (4.3) dg = gd d.
(4.4)
Let dk be the basis of D as in (2.3). We define a function Φι ∈ l 1 (L) by Φι =
∞ 1 (fdk + dk f). k+1 2
(4.5)
k=0
To estimate the nth convolution power of Φι at 1, 1 ∈ D, we write Φι∗ n (1) =
α(di1 )α(di2 ) · . . . · α(din )(fdi1 + di1 f) . . . (fdin + din f)(1),
di1 ·...· din =1
where α(dk ) =
1 . 2k+1
The product (fdi1 + di1 f) . . . (fdin + din f) is the sum of following summands
(4.6)
710
A. Hulanicki, M. Letachowicz / Journal of Functional Analysis 256 (2009) 700–717
(. . . dij ffdij +1 . . .),
(4.7)
(. . . fdij dij +1 f . . .),
(4.8)
(fdi1 fdi2 · · · fdin ),
(4.9)
(di1 fdi2 f · · · din f),
(4.10)
where (4.9) and (4.10) appear exactly once. Clearly, some terms of the form (4.7) can be written also in the form (4.8). Since (4.4) holds, we can rewrite (4.7) as d d ·...· dij di1 di2 ·...· dij f . . . (di1 · . . . · din ). . . . f i1 i2 Assume now that (4.8) cannot be written also in the form (4.7). Then the factor fim dim dim+1 fim+1 appears exactly once in (4.8). Moreover, by (4.4) we can rewrite (4.8) as ∗2 d d ···d d ···d f f i1 · · · f i1 ij −1 f i1 ij +1 · · · f d1 d2 ...din−1 (di1 · · · din ), because di1 di2 · · · din = 1 and the algebra l 1 (E) is commutative. In virtue of (4.2) summands (4.7) and (4.8) are annihilated by TW for every finite W ⊂ D. Thus we have
Φι∗ n (1), TW =
α(di1 )α(di2 ) · . . . · α(din ) fdi1 fdi2 · · · fdin (1), TW
di1 ·...· din =1
+ di1 fdi2 f · · · din f(1), TW α(di1 )α(di2 ) · . . . · α(din )2 f ∗ f di1 ∗ · · · ∗ f di1 ·...·din−1 , TW , (4.11) = di1 ·...· din =1
where, by (4.2) and (4.6), each of the summands is nonnegative. Observe that for a finite subset W of D and the function TW in l ∞ (E) defined in (4.1), we have
∗n ∗n Φι (1), TW , n = |W |, (4.12) Φι (1), TW = 0, otherwise. Indeed, if n < |W |, then in each of the summands (4.11) there is an element d in W , which does not appear as the exponent in the f ∗ f di1 ∗ · · · ∗ f di1 ·...·din−1 . On the other hand, if n > |W |, then each summand in (4.11) is annihilated by TW , because the product f ∗ f di1 ∗ · · · ∗ f di1 ·...·din−1
contains at least one factor of the form f d , d ∈ / W, or (f d )∗ 2 , d ∈ W. Therefore in both cases ∗n Φι (1), TW = 0. Now we consider the function TDk ∈ l ∞ (E), Dk being as in (2.2). For each nk = 2k = |Dk |, nk by induction. k ∈ N, we define a sequence {dik }i=1 d11 = d21 = d1 .
A. Hulanicki, M. Letachowicz / Journal of Functional Analysis 256 (2009) 700–717
711
Then we put k k−1 k−1 d1 , . . . , dnkk = d1k−1 , . . . , dnk−1 . , d , d , . . . , d , d k k 1 n −1 −1 k−1 k−1 It is clear that for a fixed k all elements of the sequence d1k , . . . , dnkk
(4.13)
belong to the set {d1 , . . . , dk } and that for j < k dj appears 2k−j times and dk appears 2 times in (4.13). It is also easy to see that all nk elements in the set
d1k . . . dnkk , d2k . . . dnkk , d3k . . . dnkk , . . . , dnkk = Dk
are different and d1k . . . dnkk = 1. It suffices to show that ∀i<j
dik . . . djk = 1.
(4.14)
We argue by induction on k: if i nk−1 j < nk , then dk appears only once in the product (4.14), so it is; if i < j < nk−1 , then dik . . . djk = dik−1 . . . djk−1 = 1; k−1 if nk−1 < i < j < nk , then dik . . . djk = di−n . . . djk−1 −nk−1 = 1 and (4.14) follows by inductive k−1 hypothesis. We observe that, by (4.11)
Φι∗ nk (1), TDk =
di1 ... din =1
α(di1 )α(di2 ) · . . . · α(dink ) fdi1 fdi2 · · · fdink (1), TDk
k
+ di1 fdi2 f · · · dink−1 f(1), TDk
k k k d k ·...·dnk k , TD α d1k α d2k · . . . · α dnkk 2 f ∗ f d1 ∗ f d1 d2 ∗ · · · ∗ f 1 k = 2 · 2−2·2
k−1
2−3·2
k−2
· . . . · 2−k·2 2−(k+1)2
= 2 · 2−2(2·2
k−2 +3·2k−3 +...+k)−2(k+1)
= 2 · 2−2(2
k +2k−1 −k−2+k+1)
= 2−3·2
k +2
= 8 · 8−nk .
We then have
∗n
−nk
Φ k 1 Φ ∗ nk (1), TD TD −1 . ι ι k k l ∞ (E) 8 · Tι l (L) Now, using our last estimate and (4.12), we are going to present the lower bound for
712
A. Hulanicki, M. Letachowicz / Journal of Functional Analysis 256 (2009) 700–717
E(n, Φι ) 1
l (L)
E(n, Φι )(1) l 1 (E)
∞
(in)l Φ ∗ l (1)
ι =
1
l! l=0 l (E) ∞ (in)l Φ ∗ l (1) ι , TDk TDk −1 l ∞ (E) l! l=0
n2 Φι∗ 2 (1), TDk TDk −1 = l ∞ (E) 2k ! n 2k k
k
8Tι 2k !
.
Whence for n 16Tι , there exists k ∈ N such that 2k
E(n, Φι ) 1 l
n 8Tι
2k+1 and
k
(L)
(2k )2 Ceε|n| . 2k !
Thus we arrive to the main theorem of this section. Theorem 4.16. If Φι ∈ l 1 (L) is defined by (4.5), then for a ε > 0 and C > 0 we have
E(n, Φι ) 1
l (L)
Ceε|n| .
5. Functions on G Now we consider the group G = C × L. Let φ ∗ = φ ∈ l 1 (C) be such that Spl 2 (C ) φ = [−π, π], e.g. φ(c) = π
∞ δcj (c) j =1
2j
,
where {cj : j ∈ N} is the basis of C as in (2.4). Let Ψ0 = φ ⊗ δ1 + δ1 ⊗ Φι . Consider the Hilbert space l 2 (G) = l 2 (C) ⊗ l 2 (L) and the operator TΨ0 = Tφ ⊗ idl 2 (L) + idl 2 (C ) ⊗ TΦι
(4.15)
A. Hulanicki, M. Letachowicz / Journal of Functional Analysis 256 (2009) 700–717
713
on l 2 (G). Let Ω = Spl 2 (G) TΨ0 . Clearly, Ω = [−π; π] + Spl 2 (L) TΦι . Suppose that for a bounded Borel function G we have
G(λ1 , λ2 ) dEΦι (λ2 ) dEφ (λ1 )
l 1 (G)→l 1 (G)
[−π,π]×Spl 2 (L) TΦι
C0 .
Then ess
sup
λ1 ∈[−π,π]
G(λ1 , λ2 ) dEΦι (λ2 )
l 1 (L)→l 1 (L)
Spl 2 (L) TΦι
C0 ,
(5.1)
with the same constant C0 . Let H be 2π -periodic bounded Borel function. Suppose H is an l 1 -spectral multiplier for TΨ0 , that is,
H (λ) dEΨ0 (λ)
Spl 2 (G) TΨ0
l 1 (G)
C0 .
Set G(λ1 , λ2 ) = H (λ1 + λ2 ). Then (5.1) gives ess
sup
s∈[−π,π]
H (s + λ2 ) dEΦι (λ2 )
Spl 2 (L) TΦι
l 1 (L)→l 1 (L)
C0 .
(5.2)
(n) of the 2π -periodic function H . Now we are going to compute the Fourier coefficients H We write π
H (λ + t) dEΦι (λ)e
−π Spl 2 (L) TΦι
int
π
dt =
H (λ + t)eint dt dEΦι (λ)
Spl 2 (L) TΦι −π
π
=
H (s)eins e−inλ ds dEΦι (λ)
Spl 2 (L) TΦι −π
= Spl 2 (L) TΦι
(n)e−inλ dEΦι (λ) 2π H
714
A. Hulanicki, M. Letachowicz / Journal of Functional Analysis 256 (2009) 700–717
(n) = 2π H
e−inλ dEΦι (λ)
Spl 2 (L) TΦι
(n)E(n, Φι ). = 2π H Thus, by (5.2) we have
H (n) · E(n, Φι ) 1
l (L)→l 1 (L)
C0 ,
hence from (4.15) it may be conclude that H (n) Ce−ε|n| .
(5.3)
We are now in a position to define the function Ψ on G = C × L by putting
Ψ = sin Ψ0 =
∗ (2j +1) ∞ (−1)j Ψ 0
j =0
(2j + 1)!
∈ l 1 (G).
Of course Tsin Ψ0 =
sin(λ) dEΨ0 (λ),
Spl 2 (G) TΨ0
where EΨ0 is the spectral measure of TΨ0 : l 2 (G) ξ → ξ ∗ Ψ0 ∈ l 2 (G). Theorem 5.4. Suppose a function F is an l 1 -multiplier for Ψ . By dilating F we may assume that F is defined on [−1, 1]. Then for every z0 ∈ (−1, 1) the function F (z) has a holomorphic extension in a neighborhood z0 ⊂ C. Indeed, let H (λ) = F (sin λ), then
1 F (z) dEΨ = −1
H (λ) dEΨ0 (λ), Ω
(n)ein(λ+iη) is absolutely and uniformly converH is periodic and by (5.3) H (λ) = n∈Z H gent for |η| < ε. Consequently, for every x ∈ (−1, 1) ⊂ C the function F is real analytic in a neighborhood of x.
A. Hulanicki, M. Letachowicz / Journal of Functional Analysis 256 (2009) 700–717
715
6. Functional calculai The following argument goes back to J. Dixmier [4] and has been used many times since then. Let H be a Hilbert space, B be a Banach space contained in H. Suppose that T : H → H is a bounded hermitian operator on H. Let λ dE(λ) T= SpH T
be its spectral resolution. Assume that SpH T ⊂ [−a, a]. Let BT be the self-adjoint algebra of Borel bounded functions F on [−a, a] such that F (λ) dE(λ) F (T ) = SpH T
is a bounded operator on B. Since F (T )G(T ) = (F G)(T ), BT is indeed a ∗-algebra. A functional calculus for an operator T as above is, by definition, the correspondence between functions F on R and the operators F (T ) that are bounded on B. For a hermitian operator T we define the operator eiT by eiT =
∞ (iT )n n=0
n!
(6.1)
.
Of course eiT is a unitary operator. Theorem 6.2. Since T is hermitian (multiplying T by a constant) we may assume that SpH (T ) ⊂ (n) be the Fourier coefficients of F . Since [−π, π]. Let F T B→B T H→H ,
(6.3)
then ∞
F (n) einT
B→B
<∞
⇒
F (T )
B→B
< ∞.
n=−∞
Indeed, since the series (6.1) is absolutely convergent, in virtue of (6.3), the operator ∞
(n)einT = F (T ) F
n=−∞
is bounded on B and on H. On the other hand, for every λ in SpH T we have F (λ) =
∞
(n)einλ , F
n=−∞
which implies that F (T ) is a bounded operator B → B.
716
A. Hulanicki, M. Letachowicz / Journal of Functional Analysis 256 (2009) 700–717
Let F ∈ C k+2 [−π, π], F (j ) (−π) = F (j ) (π), j = 0, . . . , k + 2. Then F (n) C|n|−(k+2) . This implies that if
inT
e
B→B
= O |n|k
for n → ∞,
then F (T ) is a bounded operator on B for F ∈ C k+2 [−π, π]. 7. Locally finite groups Let us consider G=
∞
Gm ⊂ Gm+1 ,
Gm ,
|Gm | < ∞,
m=1
where Gm is an increasing family of group. We define a length function τ(αn ) on G by τ(αn ) (x) = 1 +
∞
αn 1Gn \Gn−1 (x),
n=1
where 1 α1 α2 · · · . Set w(αn ) (x) = eτ(αn ) (x) . We have τ(αn ) (xy) τ(αn ) (x) + τ(αn ) (y),
whence ω(αn ) (xy) ω(αn ) (x)ω(αn ) (y).
(7.1)
Now we put αn = e|Gn | and we write τ = τ(αn )
and ω = eτ .
Then x: τ (x) < α ⊂ Gn ,
where |Gn | < log α.
Let lω1 (G) =
f (x) ω(x) = f w < ∞ . f: x∈G
(7.2)
A. Hulanicki, M. Letachowicz / Journal of Functional Analysis 256 (2009) 700–717
717
By (7.1), lω1 (G) is a Banach ∗-algebra with the norm · ω . Let us note that eif =
∞ (if )∗ k
k!
k=0
has the following estimate
if
e ef ω . ω Theorem 7.3. Let f ∗ = f ∈ lω1 (G). Then
inf
e 1
l (G)
1
c(log n) 2 .
Indeed, in virtue of (7.2) and the Schwarz inequality, for n 1 we have
inf
e 1 l
= (G)
inf e (x) +
τ (x)<f ω n
inf e (x)ω(x)ω(x)−1
τ (x)f ω n
1 1
x: τ (x) < f ω n 2 einf l 2 (G)→l 2 (G) + enf ω −nf ω c log nf ω 2 . Acknowledgments The second author wishes to express her gratitude to Jacek Dziuba´nski for his thorough reading of the manuscript and many helpful comments. References [1] J.P. Anker, Lp Fourier multipliers on Riemannian symmetric spaces of the noncompact type, Ann. of Math. (2) 132 (3) (1990) 597–628. [2] M. Christ, D. Müller, On Lp spectral multipliers for a solvable Lie group, Geom. Funct. Anal. 6 (5) (1996) 860–876. [3] M. Cowling, S. Giulini, A. Hulanicki, G. Mauceri, Spectral multipliers for a distinguished Laplacian on certain groups of exponential growth, Studia Math. 111 (2) (1994) 103–121. [4] J. Dixmier, Opérateur de rang fini dans les représentation unitaires, Publ. Math. Inst. Hautes Études Sci. (1959) 85–92 (in French). [5] M. Gromov, Groups of polynomial growth and expanding maps, Publ. Math. Inst. Hautes Etudes Sci. 53 (1981) 53–78. [6] W. Hebisch, J. Ludwig, D. Müller, Sub-Laplacians of holomorphic Lp -type on exponential solvable groups, J. London Math. Soc. 72 (2) (2005) 364–390. [7] J.P. Kahane, Séries de Fourier Absolument Convergentes, Ergeb. Math. Grenzgeb., vol. 50, Springer-Verlag, Berlin– New York, 1970 (in French). [8] Y. Katznelson, An Introduction to Harmonic Analysis, John Wiley & Sons, Inc., New York–London–Sydney, 1968. [9] J. Ludwig, A class of symmetric and a class of Wiener group algebras, J. Funct. Anal. 31 (1979) 187–194. [10] P. Malliavin, Impossibilité de la synthese spectrale sur les groupes abéliens non compacts, Publ. Math. Inst. Hautes Études Sci. 6 (1960) 305–317 (in French).
Journal of Functional Analysis 256 (2009) 718–746 www.elsevier.com/locate/jfa
Strichartz estimates for the Schrödinger equation with time-periodic Ln/2 potentials Michael Goldberg ∗,1 Department of Mathematics, Johns Hopkins University, 3400 N. Charles St., Baltimore, MD 21218, USA Received 28 August 2007; accepted 11 November 2008 Available online 28 November 2008 Communicated by I. Rodnianski
Abstract We prove Strichartz estimates for the Schrödinger operator H = −Δ + V (t, x) with time-periodic comn/2 plex potentials V belonging to the scaling-critical space Lx L∞ t in dimensions n 3. This is done directly from estimates on the resolvent rather than using dispersive bounds, as the latter generally require a stronger regularity condition than what is stated above. In typical fashion, we project onto the continuous spectrum of the operator and must assume an absence of resonances. Eigenvalues are permissible at any location in the spectrum, including at threshold energies, provided that the associated eigenfunction decays sufficiently rapidly. © 2008 Elsevier Inc. All rights reserved. Keywords: Schrödinger equation; Strichartz inequalities; Periodic potential; Threshold eigenvalue; Stein–Tomas restriction
1. Introduction and definitions The past decade has seen considerable progress in identifying classes of Schrödinger operators which retain the same dispersive properties as the Laplacian. In many cases these operators are described by a simple perturbation of the Laplacian, taking the form H = −Δ + L(t, x). Typically L is a self-adjoint differential operator of degree d = 0, 1, 2 representing electrostatic, * Fax: +1 410 516 5549.
E-mail address: [email protected]. 1 The author received partial support from NSF grant DMS-0600925 during the preparation of this work.
0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.11.005
M. Goldberg / Journal of Functional Analysis 256 (2009) 718–746
719
magnetic, and/or geometric perturbations, respectively. In this paper we consider the Floquettype potential L(t, x) = V (t, x) satisfying V (t + 2π, x) = V (t, x) for all t ∈ R and x ∈ Rn . We do not assume any self-adjointness in our main theorem, instead allowing V to be a complex-valued function. Further improvements for real and/or time-independent potentials are examined as corollaries and applications of the first result. The propagator e−itΔ of the free Schrödinger equation in Rn may be represented as a con2 volution operator with kernel (4πit)−n/2 e−i(|x| /4t) . From this formula it is clear that the free evolution satisfies the dispersive bound itΔ e
1→∞
−n/2 4π|t|
at all times t = 0. A T T ∗ argument combined with fractional integration bounds for the t variable then leads to the family of Strichartz inequalities −itΔ e u0 Lp Lq Cp u0 2 , t
x
2 n n + = , p q 2
p, q ∈ [2, ∞]
(1)
for all u0 ∈ L2 (Rn ). To be precise, the p = 2 endpoint requires a more detailed computation [10] and is false when n = 2. We will focus on dimensions n 3 in order to take advantage of the full range of exponents p ∈ [2, ∞] in (1). The Schrödinger propagator of H generally fails to satisfy estimates like (1) due to the possible existence of bound states, quasiperiodic solutions obeying u(t + 2π, x) = e2πiλ u(t, x) for all t, x ∈ R1+n and possessing moderate spatial decay. These are best understood in terms of the Floquet Hamiltonian K = i∂t − Δx + V (t, x)
(2)
acting on 2π -periodic functions with domain T × Rn . Each bound state φ(t, x) solves the distributional equation (K − λ)e−iλt φ = 0. If e−iλt φ is time-periodic and belongs to the space L2 (T × Rn ) then it is an eigenfunction of K with eigenvalue λ. We say that K has a resonance at λ if the resolvent (K − (λ ± i0))−1 is singular but the associated “eigenfunction” is not squareintegrable. The precise technical definition is postponed until Section 5, where we attempt to estimate the resolvent of K in the neighborhood of singularities. The spectrum of K is invariant under integer shifts, as (K − n) = e−int Keint for any n ∈ Z. In this paper we prove that the Schrödinger evolution of H = −Δ + V (t, x) observes a space– time estimate identical to (1) once a finite-dimensional space of bound states are projected away. For time-independent potentials, our conclusion takes the form itH e I − Pac (H ) u0 Lp Lq u0 2 t
x
(3)
over the entire range of Strichartz-admissible exponents in (1). The primary assumptions are that n/2 V (t, x) be periodic and belong to the scaling-invariant space Lx L∞ t and that each of the bound states is an eigenfunction of sufficient decay and/or regularity. If one further assumes that V is real-valued with polynomial pointwise decay and some smoothness with respect to t, then only the bound states at λ ∈ Z are a concern, and only in dimensions n 6. Improvements of this type are discussed immediately following our statement of the main theorem.
720
M. Goldberg / Journal of Functional Analysis 256 (2009) 718–746
Our assumptions do not imply that K is self-adjoint, so the familiar L2 conservation law for solutions of the Schrödinger equation cannot be taken for granted. This is best illustrated whenever K contains point spectrum at some λ ∈ / R. Since |e2πiλ | = e−2π Im(λ) = 1, the L2 norm of each eigenfunction φλ decreases exponentially in one time direction and grows in the other. Without a well-articulated conservation law, even the (p, q) = (∞, 2) case of (3) appears to be nontrivial. For each λ ∈ C define Nλ to be the solution space Nλ = φ: (K − λ)e−iλt φ = 0, e−iλt φ ∈ L2 T × Rn .
(4)
Local regularity theory dictates that every true eigenfunction also satisfies e−iλt φ ∈ C(T; L2 (Rn )). It is then permissible to discuss the initial value of an eigenfunction Φ = φ(0, ·). The projection of Nλ onto the space of initial data has as its image Xλ = {Φ: φ ∈ Nλ } ⊂ L2 Rn .
(5)
We will show via a compactness argument that both Nλ and Xλ are always finite-dimensional. Similarly define N˜ λ and X˜ λ to represent the eigenfunctions of K¯ (the Floquet operator with ¯ x)) that have eigenvalue λ. ¯ These spaces are all invariant if λ is replaced by any potential V (t, λ + m, m ∈ Z. We will assume that K and K¯ have no resonances along the real axis, in order to apply the above treatment of eigenfunctions to each one of the bound states. In addition, we require that the behavior at each eigenvalue λ ∈ C satisfy these conditions. ¯ (C1) e−iλt Nλ and e−i λt N˜ λ are both contained in the space
2n L n+2 Rn ; L2 (T) + x −1 L2 T × Rn . 2n
(C2) Xλ and X˜ λ are subspaces of x −1 L2 (Rn ) + W 1, n+2 (Rn ). (C3) The L2 -orthogonal projection of Xλ onto X˜ λ is bijective. The first two conditions are concerned primarily with the decay of eigenfunctions as |x| → ∞, and correspond to a homogeneous rate of x −β , β > n2 + 1. The last one describes a desired ¯ as explained in Proposition 5. algebraic/spectral property of the Floquet operators K and K, Remark 1. The unweighted portion of condition (C2) is not sharp in terms of the number of derivatives required. Lemma 16 and its supporting propositions construct a family of lowerregularity spaces which may be used in place of W 1,2n/(n+2) (Rn ). 2. Statement of results Theorem 1. Let V (t, x) be a time-periodic function on R1+n , n 3, satisfying V (t + 2π, x) = n/2 ¯ V (t, x) at almost every t, x and belonging to the class Lx L∞ t . Suppose that K and K have no resonances along the real axis, and that their behavior at each eigenvalue λ ∈ C satisfies conditions (C1)–(C3). Under these assumptions, there exist at most finitely many eigenvalues of
M. Goldberg / Journal of Functional Analysis 256 (2009) 718–746
721
K, K¯ in the strip C/Z, counted with multiplicity. Furthermore, the initial value problem for the Schrödinger equation i∂t − Δx + V (t, x) u(t, x) = 0, u(0, x) = f (x),
x ∈ Rn , t ∈ R, x ∈ Rn 2n/(n−2)
possesses a unique weak solution in the Strichartz space L2t Lx
, satisfying
uL2 L2n/(n−2) + uCb (R:L2 (Rn )) f 2 t
x
for all initial data f in the L2 -orthogonal complement of X˜ =
(6)
(7)
˜
λ Xλ .
Remark 2. If V is real-valued, then each eigenvalue λ is also real. Since K¯ = K, it also follows that N˜ λ = Nλ and X˜ λ = Xλ , making condition (C3) unnecessary. Corollary 2. Suppose that the time-periodic potential V (t, x) is real valued and satisfies the bound sup x β V (·, x)H s (T) < ∞
(8)
x∈Rn
for some β > 2 and s > 12 . The Strichartz estimates in Theorem 1 are valid provided that λ ∈ Z is not a resonance, and any eigenvectors at λ ∈ Z belong to x −1 L2 . In dimensions n 7, Theorem 1 is valid for all real-valued potentials satisfying (8). No further conditions are necessary. Proof. Due to the self-adjointness of K, there are no eigenvalues off of the real axis. Following the proof of Lemma 2.8 in [3], resonances can only exist at λ ∈ Z, and if λ is not an integer then the eigenfunctions additionally satisfy φλ ∈ x −N H s (T; L2 (Rn )). The main ingredients are an Agmon-type bootstrapping argument (based on [1]) and the fact that multiplication by a function in H s (T) preserves the H s−1/2 (T) norm. When λ ∈ Z, the bootstrapping process produces only as much spatial decay for φλ as is present in the Green’s function of the Laplacian. In general, the Green’s function belongs to x σ L2 (aside from the local singularity) for all σ > 4−n 2 . For n 7, the desired value σ = −1 is part of this range. 2 n
Corollary 3. Let V (x) ∈ L 2 (Rn ) be a complex valued time-independent potential. The Strichartz estimates in Theorem 1 are valid provided the equation (−Δ + V − λ)φ = 0 has no solutions φ ∈ L2n/(n−2) (Rn ) for any λ ∈ [0, ∞) ⊂ C, and condition (C3) is satisfied at every eigenvalue. Proof. Similar to the preceding corollary, the point is that all of the permitted bound states φλ = eiλt Φ(x) are necessarily eigenfunctions that decay rapidly enough to satisfy condition (C2).
722
M. Goldberg / Journal of Functional Analysis 256 (2009) 718–746
In this case the bootstrapping is based on the relation Φ = −(I + (−Δ − λ)−1 V )Φ. Since λ ∈ / [0, ∞), the resolvent of the Laplacian is bounded from every Lp (Rn ) to itself, 1 p ∞. Starting with Φ ∈ L2n/(n−2) , one iteration decreases the exponent so that Φ ∈ L2n/(n+2) . Furthermore it is quite easy to take two derivatives: ΔΦ = V Φ − λΦ ∈ L2n/(n+2) . Thus φ ∈ W 1,2n/(n+2) as is required by (C2). 2 n
Corollary 4. If V ∈ L 2 (Rn ) is a real-valued potential, then (7) holds provided the Schrödinger operator H = −Δ + V does not have a resonance or an eigenvalue at zero energy. Proof. In this case the spectrum of H is purely absolutely continuous on the interval (0, ∞) due to the combined results of [4] and [6]. According to the previous corollary, the only remaining spectral point of concern is the behavior of H at λ = 0. The additional assumption ensures that zero is a regular point of the spectrum as well. 2 Returning for a moment to the statement of Theorem 1, there are three conditions placed on ¯ Two of them deal with the spatial decay properties of the eigenspaces of K − λ and K¯ − λ. individual eigenfunctions. The third is presented as a non-orthogonality condition, and it is used this way during the proof. However it serves equally well as an assumption regarding the absence of generalized eigenfunctions. ¯ 2= Proposition 5. Condition (C3) implies that ker(K − λ)2 = ker(K − λ) and ker(K¯ − λ) ¯ ¯ ker(K − λ). ˜ = 0. Proof. If Condition (C3) holds, then for each Φ ∈ Xλ there exists Φ˜ ∈ X˜ λ so that Φ, Φ
Based on the identity (24), this property extends to elements of Nλ as well. Given any φ ∈ Nλ there exists φ˜ ∈ N˜ λ so that
¯
e−iλt φ, e−i λt φ˜
L2 (T×Rn )
˜ L2 = 0. = 2π Φ, Φ
x
The kernel of K − λ consists of functions e−λt φ, φ ∈ Nλ . If it were possible to solve (K − λ)ψ = e−iλt φ with any ψ ∈ L2 (T × Rn ), it would lead to the contradiction
¯
¯
¯
e−iλt φ, e−i λt φ˜ = (K − λ)ψ, e−i λt φ˜ = ψ, (K¯ − λ¯ )e−i λt φ˜ = 0
¯ ¯ because e−i λt N˜ λ is the kernel of K¯ − λ.
2
A formal argument along these lines suggests that the converse statement should also be true. There are some technical issues regarding the domain of K and K¯ which stand in the way. We state and prove one possible converse as an appendix to Section 5. Combined with Proposition 5 this gives a condition on the algebraic structure of K − λ that is equivalent to (C3). 3. Summary of methods Although Theorem 1 is presented as a perturbation of the Strichartz inequality (1), which in turn is based on dispersive estimates for the free Schrödinger evolution, we do not attempt to prove comparable dispersive estimates for H . This is partly a matter of convenience, as the
M. Goldberg / Journal of Functional Analysis 256 (2009) 718–746
723
study of time-asymptotics for Floquet operators (as in [3]) presents its own set of technical challenges. More importantly, the conditions for Theorem 1 include numerous potentials for which the corresponding dispersive estimate are known to fail. The discrepancy is especially apparent in dimensions n 4. No pointwise or Lp condition on the potential is sufficient by itself to imply an L1 → L∞ dispersive bound [5]. Either some extra regularity of V is needed, as in [8], or one must expect to suffer a loss of derivatives in the solution [18]. On the other hand, Strichartz estimates were proven in [14] for time-independent potentials satisfying |V (x)| x −2−ε . In this work the authors used the L2 theory of Kato smoothing estimates [9] as an intermediary step in place of the nonexistent dispersive bounds. Corollary 4 represents a modest extension. We wish to emphasize one additional feature of Theorem 1 that appears to be unique in the literature: the treatment of eigenvalues depends only on the nature of the associated eigenfunction, not on its location relative to the spectrum of K. While it may be true in certain applications that threshold eigenvalues and/or resonances enjoy distinct properties from those embedded in the continuous spectrum or from isolated points, the criteria (C1)–(C3) apply equally in all these cases. The proof of Theorem 1 is based on a direct application of Duhamel’s formula. We consider the behavior of solutions when t 0; the reasoning for t 0 is identical. Let U + denote the forward propagator of the free Schrödinger equation, that is U + g(t, x) :=
e−i(t−s)Δ g(s, x) ds.
s
We will use U0+ to indicate the free forward propagation of initial data from time zero, U0+ g(t, x) := χ[0,∞) (t)e−itΔ g(x). The adjoint of U + is the backward propagator U − . The full range of mapping properties of U + and U0+ are established in [10]; of particular concern are the bounds ∩ C R; L2x , 2n/(n−2) U + : L1t L2x → L2t Lx ∩ C R; L2x , 2n/(n−2) U0+ : L2x → L2t Lx ∩ C [0, ∞); L2x .
U + : L2t Lx
2n/(n+2)
2n/(n−2)
→ L2t Lx
(9)
Every weak solution of (6) on the time interval [0, ∞) must solve the functional equation u(t, x) = U0+ f (t, x) + iU + V u(t, x). This leads to the formal solution −1 + u = I − iU + V U0 f 2n/(n−2)
where the inverse is taken among bounded operators on L2t Lx n setting of L2t L2x , factorize V = ZW , with Z, W ∈ L∞ t Lx and write
−1 u = U0+ f + iU + Z I − iW U + Z W U0+ f.
. In order to work in the
(10)
724
M. Goldberg / Journal of Functional Analysis 256 (2009) 718–746
In the event that I − iW U + Z is invertible as an operator on L2 ([0, ∞); L2 (Rn )), one concludes that (7) holds for all f ∈ L2 which implies an absence of bound states. This occurs for all V ∈ n/2 L∞ t Lx of sufficiently small norm. In every other case, the challenge is to find a condition on f so that W U0+ f belongs to the domain of the unbounded operator (I − iW U + Z)−1 . Much of our analysis is done with respect to the Fourier transform of the time variable, in deference to the fact that U + and V preserve the space of functions satisfying g(t + 2π, x) = e2πiλ g(t, x) for each λ ∈ [0, 1]. We show that I − iW U + Z is a compact perturbation of the identity on each of these spaces. The Fredholm Alternative then equates invertibility with the absence of eigenvalues or resonances at λ. Common sense suggests that the singularities caused by a particular bound state φ can be avoided by requiring the initial data f to be orthogonal to Φ. Even in the time-independent case, however, eigenvalues at zero energy are known to disturb dispersive estimates after such a projection. This phenomenon is first identified in [7] and described in more detail in [2]. A full asymptotic expansion for Floquet solutions has recently been computed in three dimensions in [3]. We note that the intuitive suggestion above is also incorrect when the Schrödinger propagation is not unitary (i.e. when K has complex values). The projection employed in Theorem 1 is actually orthogonal to a function Φ˜ ∈ X˜ rather than Φ. In order to determine the success of a projection, we closely examine the behavior of (I − iW U + Z)−1 for all λ in the neighborhood of an eigenvalue and assess whether it is compatible with the input W U0+ f . The resulting eigenvalue condition appears in the form of a discrete-time Kato smoothing bound. This last computation, parts of which are adapted from [11] and [16], may be of independent interest. 4. Resolvents, compactness, and continuity We cannot in general expect I − iW U + Z to possess a bounded inverse on L2 ([0, ∞); If it instead belongs to the Fredholm class, however, then the inverse still exists as a map between two closed subspaces of finite codimension. Our next step is to decompose L2t L2x into a “Fourier basis” of invariant subspaces, and to show that the restriction of I − iW U + Z to each of these is a compact perturbation of the identity. For each λ ∈ C, define L2 (Rn )).
Yλ = g ∈ Lt2,loc L2x : g(t + 2π, x) = e2πiλ g(t, x) . Each g ∈ Yλ is naturally associated with the time-periodic function e−itλ g ∈ L2 (T × Rn ), and we use this identification to give Yλ the structure of a Hilbert space. For each λ ∈ R/Z, there exists a “projection” Pλ from L2t L2x onto Yλ given by Pλ g(t, x) =
e−2πiλm g(t + 2πm, x).
m∈Z
Clearly Pλ commutes with pointwise multiplication (in (t, x)) by any 2π -periodic function. The family of operators Pλ can be understood as a discrete Fourier transform in the time direction, acting on the space 2m (L2 ([2πm, 2π(m + 1)] × Rn )) ∼ = L2t L2x and setting λ ∈ [0, 1] as the Fourier dual variable to m ∈ Z. There is a corresponding Plancherel identity which takes the form
M. Goldberg / Journal of Functional Analysis 256 (2009) 718–746
1 Pλ g2Yλ dλ =
g2L2 ([2πm,2π(m+1)]×Rn ) = g2L2 L2 . t
m∈Z
0
725
(11)
x
For functions g with support in the halfline t ∈ [0, ∞), the definition of Pλ g extends to the strip λ = λ + iμ, μ 0, λ ∈ R/Z with the value e−μt Pλ (eμt g). The Plancherel identity in this case becomes 1 Pλ +iμ g2Yλ +iμ
1
dλ =
0
Pλ eμt g 2 dλ = eμt g 2 2 2 . Y L L λ
t
x
0
ˆ x) be the On the Fourier side with respect to time, Pλ has a very clear interpretation. Let g(τ, partial Fourier transform of g. By definition P0 g is the periodization (in t) of g, so that (P0 g)ˆ restricts gˆ to the cross-sections τ ∈ Z. For every other value of λ, there is the relation Pλ = eiλt P0 e−iλt . Consequently, (Pλ g)ˆ is the restriction of gˆ to the cross-sections {τ ∈ λ + Z}. If g is supported on {t 0} then gˆ even has a holomorphic extension to all τ in the lower halfplane, making the restrictions to {τ ∈ λ + iμ + Z} well-defined. The action of U + in this setting is also easy to characterize. Since U + convolves functions in the time variable with the integral kernel K(t) = limε↓0 e−itΔ−εt χt0 , on the Fourier side it ˆ ) = limε↓0 i(−Δ − (τ − iε))−1 . Using the performs pointwise (in τ ) “multiplication” by K(τ notation of resolvents + U g ˆ(τ, x) = iR − (τ )g(τ, ˆ x)
(12)
where R − (τ ) represents the branch of the resolvent of −Δ which continues analytically to {Im(τ ) 0}. Similarly, + U0 g ˆ(τ, ·) = iR − (τ )g. This shows that U + commutes with each of the projections Pλ , as both operators act pointwise in τ on the Fourier side (and the actions commute with one another). Once again, if suppt g ⊂ [0, ∞), the identity (12) remains valid for all τ in the lower halfplane, with the understanding that g(τ, ˆ x) = eIm(τ )t g ˆ Re(τ ), x . Therefore the operator I − iW U + Z admits a restriction to each Yλ , Im(λ) < 0, and most importantly, μt e I − iW U + Z −1 W U + f 2 2 2 = 0 L L t
1
I − iW U + Z −1 Pλ +iμ W U + f 2 0 Y
λ +iμ
x
0
dλ .
(13)
726
M. Goldberg / Journal of Functional Analysis 256 (2009) 718–746
The proof of Theorem 1 will be complete once we bound this quantity in terms of the L2 (Rn ) norm of f , uniformly over μ 0. The particular factorization we choose for V (t, x) is to let 1 W (t, x) = w(x) = V (·, x)∞ 2 .
(14)
By our assumptions, w ∈ Ln (Rn ). The remaining factor can be decomposed as w(x)z(t, x), with w the same function as above and z(t, x) periodic in time and bounded almost everywhere by 1. Multiplication by z is a bounded operator of unit norm on Yλ , so compactness of the operator wU + wz follows directly from compactness of wU + w. Proposition 6. Given any function w ∈ Ln (Rn ), the collection {wR − (τ )w: Im(τ ) 0} forms a uniformly continuous family of compact operators on L2 (Rn ) with norm decreasing to zero as |τ | → ∞. Proof. This is a compilation of well-known resolvent estimates, primarily the fact (proved 2n 2n in [11]) that R − (τ ) are uniformly bounded as operators from L n+2 to L n−2 . All of the desired properties—compactness, continuity, and norm decay—are preserved if w is approximated in Ln by a sequence of bounded compactly supported functions w ε . For compactness, observe that (−Δ + 1)R − (τ )w ε g = w ε g + (τ + 1)R − (τ )w ε g ∈ L2 Rn . 2n
Within any ball of finite radius R, the Sobolev space H 2 embeds compactly inside L n−2 . If this ball is much larger than the support of w ε , then there is a pointwise bound
− 1−n
R (τ )w ε g(x) |τ | n−3 4 g2 w ε |x| 2 2 in the complementary region {|x| R}. Increasing R → ∞ allows w ε R − (τ )w ε to be expressed as a norm-limit of compact operators on L2 . 1 For continuity, recall that the integration kernel of R − (τ ) is |x − y|2−n F (τ 2 |x|), where F can be expressed explicitly in terms of Hankel functions. In dimensions n 3 it satisfies the pointwise bounds
F (z) , F (z) z (n−3)/2 . Using the mean value theorem, if |τ − σ | < 12 |τ | then
−
R (τ ) − R − (σ )(x, y) 1 1 |τ 2 − σ 2 | |x − y|3−n , n−3 1 1 3−n |τ | 4 |τ 2 − σ 2 ||x − y| 2 ,
1
if |x − y| < |τ |− 2 , 1 1 1 if |τ |− 2 < |x − y| < |τ 2 − σ 2 |−1 .
The case where |x − y| is large is unimportant because w ε has compact support. The Schur test then shows that w ε R − (τ )w ε is continuous with respect to τ .
M. Goldberg / Journal of Functional Analysis 256 (2009) 718–746
727
Finally, decay as |τ | → ∞ is an immediate consequence of another resolvent bound from [11], 1 2n+2 2n+2 namely that |τ | n+1 R − (τ ) is a uniformly bounded family of maps from L n+3 to L n−1 . The combination of continuity and decay at infinity immediately implies uniform continuity. 2 Corollary 7. Given any w ∈ Ln (Rn ), the set {e−iλt wU + weiλt : Im(λ) 0} is a continuous family (with respect to λ) of compact operators on L2 (T × Rn ), with norm decreasing to zero as Im(λ) → −∞. The same is also true for the family of operators e−iλt wU + wzeiλt for any bounded 2π periodic function z. Proof. For every choice of λ in the lower halfplane, the Fourier series coefficients of e−iλt wU + we+iλt g are precisely {wR − (λ + k)w g(k, ˆ x): k ∈ Z}. At each k this is a compact operator on Rn , and the norms decrease as |k| → ∞. It follows that their collective action on
2 (k; L2 (Rn )) is a compact operator with norm supk wR − (λ + k)w. As Im(λ) → −∞, the norm is bounded by sup
|τ |>| Im(λ)|
wR − (τ )w
which decreases to zero. Given two numbers λ1 and λ2 , the norm difference of their associated operators is supw R − (λ1 + k) − R − (λ2 + k) w . k
The uniform continuity assertion in Proposition 6 takes this to zero in the limit λ2 → λ1 . Neither the compactness nor continuity properties of e−iλt wU + weiλt are affected by composition with the bounded operator e−iλt zeiλt . 2 5. Estimates for inverse operators There are two main elements in the expression (13), the typically unbounded operator (I − iwU + wz)−1 and a family of functions Pλ wU0+ f ∈ Yλ . In this section we prove uniform bounds for (I − iwU + wz)−1 on Yλ where possible, and describe the singularities that occur as λ approaches the spectrum of K. The spaces Yλ are a natural setting for working with bound states, especially those bound states that grow exponentially over time. When we wish to vary λ as a parameter, however, a unified approach based on L2 (T × Rn ) is preferred. Define the family of operators T (λ) = I − ie−iλt wU + wzeiλt = I − iw e−iλt U + eiλt wz acting on L2 (T × Rn ). The kernel of T (λ) provides valuable information about the spectrum of K, thanks to the intertwining relations (K − λ) e−iλt U + eiλt wz = iwzT (λ), −iλt + iλt we U e (K − λ) = iT (λ)w.
(15)
728
M. Goldberg / Journal of Functional Analysis 256 (2009) 718–746
Each element g ∈ ker T (λ) corresponds to a bound state φ = U + wzeiλt g. Proposition 8 below shows that e−iλt φ is an eigenfunction of K in L2 (T × Rn ) if Im(λ) < 0. Additional tools are available [3,19] if V is real-valued and λ ∈ / Z. In any of the remaining cases it is possible that the spatial decay of φ fails to be square-integrable. We say that K has a resonance at λ when this occurs; that is, when there exists some g ∈ ker T (λ) for which φ = U + wzeiλt g does not belong to L2 (T × Rn ). The defining property T (λ)g = 0 has a corresponding expression in terms of the associated eigenfunction (or resonance) φ, namely U + V φ = −iφ. ¯ will be used repeatedly to This, and the analogous identity U − V¯ φ˜ = i φ˜ for eigenfunctions of K, simply calculations during the next two sections. Note that T (λ + 1) is a unitary conjugate of T (λ), so one only needs to check the invertibility of T (λ) inside the strip Ω − = λ ∈ C: Re(λ) ∈ [0, 1), Im(λ) 0 . The set Ω − ⊂ C is a fundamental domain for the lower halfplane modulo the integers, and will always be given the quotient topology. We make some remarks about the size and differentiability properties of e−λt U + eiλ for future reference. Proposition 8. For each λ with Im(λ) < 0, the operator e−iλt U + eiλt is subject to the following estimates. −iλt + iλt e U e g
−1
Im(λ) gL2 (T×Rn )
(16)
−iλt + iλt e U e g
Im(λ)
(17)
L2 (T×Rn ) L2 (T×Rn )
− 12
gL2 (T;L2n/(n+2) (Rn )) .
The difference between its evaluation at the points λ1 , λ2 can be expressed as e−iλ1 t U + eiλ1 t − e−iλ2 t U + eiλ2 t = −i(λ1 − λ2 ) e−iλ1 t U + eiλ1 t e−iλ2 t U + eiλ2 t .
(18)
Therefore the family of operators e−iλt U + eiλt possesses the holomorphic derivative 2 d −iλt + iλt e = −ie−iλt U + eiλt U e dλ
(19)
over the domain Im(λ) < 0. Proof. The estimates (16) and (17) both exploit the facts that U + g(t, x) depends only on χs
1
L2 (T; L n+2 (Rn )) then the L2t Lx norm of χ(−∞,t) eiλs g is bounded by | Im(λ)|− 2 e− Im(λ)t . In either case the propagator estimates (9) complete the argument. 2n/(n+2)
M. Goldberg / Journal of Functional Analysis 256 (2009) 718–746
729
The difference and derivative formulas can be verified directly or by expressing U + according to its Fourier representation (12). The equivalent identities for resolvents are R − (λ1 ) − R − (λ2 ) = d − (λ1 − λ2 )R − (λ1 )R − (λ2 ) and dλ R (λ) = (R − (λ))2 . 2 Corollary 7 shows that each T (λ), Im(λ) 0, is a compact perturbation of the identity. Furthermore, T (λ)−1 varies continuously over its domain of definition, is periodic with respect to translation by Z, and is bounded by 2 once the imaginary part of λ is sufficiently negative. If T (λ)−1 existed everywhere, this would suffice to bound its norm uniformly in λ. By the Fredholm Alternative, only an eigenvalue or resonance at λ can prevent T (λ) from being invertible. We examine the structure of these singularities in Lemmas 9 and 11. Lemma 9. Let w ∈ Ln (Rn ) and z ∈ L∞ (T × Rn ). Suppose the operator T (λ0 ) fails to be invertible for some λ0 ∈ C with Im(λ0 ) < 0. Then the solution spaces Nλ0 ⊂ Yλ0 and N˜ λ0 ⊂ Yλ¯ 0 are both nontrivial and finite-dimensional. The set of their initial values, Xλ0 and X˜ λ0 , are well defined finite-dimensional subspaces of L2 (Rn ). If the orthogonal projection from Xλ0 onto X˜ λ0 is bijective, then T (λ) is invertible for every other λ in a neighborhood of λ0 . More precisely, T (λ)−1 (h1 + h2 )
L2 (T×Rn )
C(w, z, λ0 ) |λ − λ0 |−1 h1 + h2
(20)
¯ ˜ φ˜ ∈ N˜ λ0 , and h2 belongs to the L2 -orthogonal complement of ¯ φ, where h1 = e−i λ0 t zw ¯ ¯ N˜ λ0 . e−i λ0 t zw
Proof. The operator T (λ0 ) is a compact perturbation of the identity, and by assumption it is not invertible. The Fredholm Alternative asserts that T (λ0 ) has a finite-dimensional kernel, a cokernel of the same dimension, and that it is an invertible map between their respective orthogonal complements. Every element g ∈ L2 (T × Rn ) in the kernel of T (λ0 ) is associated to a prospective eigenfunction e−iλ0 t φ by the relations φ = U + wzeiλ0 t g and g = ie−iλ0 t wφ. Note that wzg ∈ 2n L2 (T; L n+2 (Rn )), so the mapping estimate (17) implies that e−λ0 t φ belongs to L2 (T × Rn ). That makes e−iλ0 t φ an eigenfunction of K, and φ ∈ Nλ0 by definition. It follows immediately that ker T (λ0 ) = e−iλ0 t wNλ0 . In general, a function φ ∈ L2 (T ×Rn ) should not have a meaningful initial value Φ(x) = φ(0, x). On the other hand, φ solves the inhomogeneous Schrödinger equation 2n/(n+2)
(i∂t − Δ)φ = −V φ ∈ Lt2,loc Lx
from which Duhamel’s formula (averaged over all starting times s ∈ [−2π, 0]) yields
730
M. Goldberg / Journal of Functional Analysis 256 (2009) 718–746
−1
0
φ(0, x) = (2π)
e
φ(s, x) + i
−2π −1
0 iΔs
e
V φ(r, x) dr ds
s
0
= (2π)
iΔr
0 e
iΔs
φ(s, x) ds + i
−2π
e
iΔr
(r + 2π)V ψ(r, x) dr .
(21)
−2π
The first integral evaluates to a function in L2 (Rn ) because eiΔs is unitary and φ ∈ Lt1,loc L2x . The second integral does likewise, via the dual statement of (9). Remark 3. Because ker T (λ0 ) is a finite-dimensional space, the norms of g (as an element of ker T (λ0 ) ⊂ L2 (T × Rn )) and φ (in Nλ0 ⊂ Yλ0 ) are equivalent. These norms are also equivalent to the norm of Φ ∈ Xλ0 ⊂ L2 (Rn ) for the same reason. The image of T (λ0 ) consists of all functions orthogonal to the kernel of its adjoint, namely ¯
¯
¯ − we ¯ i λ0 t . T (λ0 )∗ = I + ie−i λ0 t zwU ¯ Every element g˜ in the kernel of T (λ0 )∗ is associated to an eigenfunction e−i λ0 t φ˜ ∈ N˜ λ0 of K˜ by ¯ ¯ ˜ The argument which places φ˜ in N˜ λ0 and ¯ i λ0 t g˜ and g˜ = −ie−i λ0 t zw ¯ φ. the relations φ˜ = U − we establishes the existence of Φ˜ is the same as the one for φ and Φ above. We can now express the image of T (λ0 ) as
¯ ¯ φ˜ = 0, φ˜ ∈ N˜ λ0 , image T (λ0 ) = g ∈ L2 (T × Rn ): g, e−i λ0 t zw
(22)
¯ and the cokernel of T (λ0 ) as the subspace e−i λ0 t zw ¯ N˜ λ0 . Our next goal is to find an inverse image for each h1 ∈ coker T (λ0 ) with respect to the map T (λ), λ = λ0 . At first, let g and h be any two functions in L2 (T×Rn ). By Proposition 8, the scalar restriction of T (λ) described by
ag,h (λ) = T (λ)g, h is a holomorphic function in the lower halfplane, with derivative
a (λ) = we−iλt U + 2 eiλt wzg, h Im(λ) −1 gh. g,h
(23)
¯ Now fix a particular h1 = e−i λ0 t zw ¯ φ˜ 1 with φ˜ 1 ∈ N˜ λ0 of approximately unit norm, and suppose −iλ t 0 that g = e wφ, φ ∈ Nλ0 . By construction ag,h1 (λ0 ) = 0 and
ag,h (λ0 ) = −i 1
+ U V φ, U − V¯ φ˜ 1 = i φ, φ˜ 1 = i
2π
φ(t, ·), φ˜ 1 (t, ·) L2 dt = 2πi Φ, Φ˜ 1 L2x . x
0
M. Goldberg / Journal of Functional Analysis 256 (2009) 718–746
731
The last line in this chain of equations is a non-selfadjoint version of the unitarity of propagation. More generally, if u(t, x) is any solution of (6) and v(t, x) satisfies (−i∂t − Δx + ¯ x))v(t, x) = 0, then V (t,
d u(t, ·), v(t, ·) L2 = 0. x dt
(24)
If the orthogonal projection of Xλ0 onto X˜ λ0 is bijective, then there exists a unique unit vector Φ1 ∈ Xλ0 such that
Φ1 , Φ˜ 1
∼ Φ˜ 1 2 1 while Φ1 , Φ˜ = 0 for all Φ˜ ∈ X˜ λ0 orthogonal to Φ˜ 1 . For the associated function g1 = e−iλ0 t wφ1 , this provides the lower bound
a g
1 ,h1
(λ) |λ − λ0 |
while at the same time
a g
1 ,h
(λ) |λ − λ0 |2
for all unit vectors h ∈ coker T (λ0 ) orthogonal to h1 . Returning to the derivative estimate (23), we observe that
T (λ)g1
image T (λ0 )
|λ − λ0 |.
Switching the roles of g and h gives the bound
T (λ)g, h1 + h |λ − λ0 |g for every g ∈ L2 (T × Rn ) and any unit vector h1 + h ∈ coker T (λ0 ). Recall that T (λ0 ) is an invertible map between its co-image and image. By continuity, the restrictions of T (λ) to these spaces are uniformly invertible within a small neighborhood of λ0 . Therefore, given g1 as constructed above there exists a unique element g (λ) ∈ coimage T (λ0 ) so that T (λ)(g1 + g (λ)) ∈ coker T (λ0 ). The norm of g (λ) is of order |λ − λ0 |. Let gh1 (λ) = g1 + g (λ). This is a vector of approximately unit norm that satisfies both T (λ)gh1 (λ) = Ch1 (λ − λ0 )h1 + O |λ − λ0 |2 and also T (λ)gh1 (λ) ∈ coker T (λ0 ). Choose any basis {hj } for coker T (λ0 ). The desired inverse image T (λ)−1 h1 will be a linear combination (with bounded coefficients) of the functions (λ − λ0 )−1 ghj (λ). For any unit vector h2 ∈ image T (λ0 ), there exists a unique gh2 (λ) in the co-image of T (λ0 ) so that T (λ)gh2 (λ) − h2 = h ∈ coker T (λ0 ).
732
M. Goldberg / Journal of Functional Analysis 256 (2009) 718–746
The norms of gh2 and h will be of order 1 and |λ − λ0 |, respectively. Thus T (λ)−1 h , and finally T (λ)−1 h2 = gh2 + T (λ)−1 h will both be of bounded norm. 2 The fact that Im(λ0 ) < 0 only played a role to the extent that we relied upon the propagator estimates of Proposition 8. If λ0 ∈ R instead, these can be replaced with a weaker set of bounds based on the mapping properties of R − (λ) along the real axis. Proposition 10. For each λ ∈ C, Im(λ) 0, the operator e−iλt U + eiλt is subject to the following estimates. −iλt + iλt e U e g
g
2n
(25)
−1 −iλt + iλt x e U e g
x g L2 (Rn ×T)
(25 )
−iλt + iλt e U e g
x g L2 (Rn ×T)
(25
)
2n
L n−2 (Rn ;L2 (T)) L2 (Rn ×T)
2n L n−2 (Rn ;L2 (T))
L n+2 (Rn ;L2 (T))
The difference between its evaluation at any two points λ1 , λ2 can still be expressed formally as e−iλ1 t U + eiλ1 t − e−iλ2 t U + eiλ2 t = −i(λ1 − λ2 ) e−iλ1 t U + eiλ1 t e−iλ2 t U + eiλ2 t .
(18)
Proof. The order of variables is interchanged from Proposition 8 so that we may work entirely on the Fourier side with respect to t. By Minkowski’s inequality for mixed norms [12] and Plancherel’s identity, g ˆ 2 L2n/(n+2) g ˆ L2n/(n+2) 2 = g n x
x
n
2n
L n+2 (Rn ;L2 (T))
Following the Fourier characterization of U + given in (12) leads to the statement of (25), −iλt + iλt e U e g
2n L n−2 (Rn ;L2 (T))
= e−iλt U + eiλt g ˆ L2n/(n−2) 2 e−iλt U + eiλt g ˆ
x
n
2n/(n−2)
2n Lx
g ˆ 2 L2n/(n+2) n x
g
2n
L n+2 (Rn ;L2 (T)) 2n
2n
where the second to last inequality is the uniform L n+2 → L n−2 bound for R − (λ + n), n ∈ Z proved in [11]. A proof of (25 ) which captures the sharp constant is given in [16]. The basic argument is the same as the one above, however the Hilbert space structure of x L2 (Rn ) and the Plancherel identity permit precise computation of the various norms. Finally, the statement (25
) is equivalent to the resolvent bound − R (τ )ψ
2n n−2
x ψ 2
(26)
M. Goldberg / Journal of Functional Analysis 256 (2009) 718–746
733
uniformly over all Im(τ ) 0. It is conceivable that (26) can be derived directly from the resolvent estimates in [16] and [11] by factorizing R − (τ ) through unweighted L2 . Theorem 3.1 of [15] is another closely related statement, differing only in the weights and regularity of the domain 1 1 ( x − 2 −ε H˙ − 2 (Rn ) versus x −1 L2 (Rn )). We present a complete proof as Lemma 14, in the section devoted to Fourier analysis. 2 Lemma 11. Let w ∈ L2 (Rn ) and z ∈ L∞ (T × Rn ). Suppose the operator T (λ0 ) fails to be invertible at λ0 ∈ R and that neither K nor K¯ has a resonance at λ0 . The solution spaces Nλ0 ⊂ Yλ0 and N˜ λ0 ⊂ Yλ0 are nontrivial and finite-dimensional, and their initial values form finite-dimensional subspaces Xλ0 , X˜ λ0 ⊂ L2 (Rn ). If the orthogonal projection from Xλ0 onto X˜ λ0 is bijective, and if the spaces e−iλ0 t Nλ0 and −iλ e 0 t N˜ λ0 are both contained inside 2n L n+2 Rn ; L2 (T) + x −1 L2 Rn × T then T (λ) is invertible for every other λ in the lower halfplane sufficiently close to λ0 , with the norm estimate T (λ)−1 (h1 + h2 ) 2 C(w, z, λ0 ) |λ − λ0 |−1 h1 + h2 . (27) L (T×Rn ) ¯ N˜ λ0 , and h2 belongs to the L2 -orthogonal complement of In this expression h1 ∈ e−iλ0 t zw −iλ t 0 e zw ¯ N˜ λ0 . Proof. As in Lemma 9, one determines that each g ∈ ker T (λ0 ) is associated with an eigenfunction φ ∈ Nλ0 by the relations φ = U + eiλ0 t wg and g = ie−iλ0 t zwφ. Because the available estimate (25) for U + does not map into L2 (Rn × T), the extra assumption that λ0 is not a resonance is required in order to place φ ∈ Nλ0 . It then follows that ker T (λ0 ) = eiλ0 t wNλ0 and coker T (λ0 ) = e−iλ0 t zw ¯ N˜ λ0 . The next step is again to evaluate T (λ)−1 h1 for h1 ∈ coker T (λ0 ) using the function ag,h (λ) = T (λ)g, h as a guide. While ag,h (λ) is holomorphic inside the lower halfplane, in general one expects it to be merely continuous at the boundary, based on Corollary 7. Better behavior occurs locally if h ∈ coker T (λ0 ). Choose any h1 = e−iλ0 t zw ¯ φ˜ 1 , φ˜ 1 ∈ N˜ λ0 . By construction, ag,h1 (λ0 ) = 0, and the statements in Proposition 10 imply the local Lipschitz bound
ag,h (λ) = |λ − λ0 | wzg, e−iλt U − ei(λ−λ0 )t φ˜ 1 1 |λ − λ0 |gL2 (Rn ×T) e−λ0 t φ˜ 1 2n n 2 −1 2 n L n+2 (R ;L (T))+ x
L (R ×T)
for all λ in the lower halfplane. A similar bound holds for ag1 ,h (λ), where g1 ∈ ker T (λ0 ) and h is any vector in L2 (Rn × T). We do not claim any differentiability unless both g = e−iλ0 wφ ∈ ker T (λ0 ) and h1 ∈ coker T (λ0 ). In that case,
ag,h1 (λ) = (λ − λ0 ) e−iλ0 t φ, e−iλt U − eiλt wh ¯ 1
= i(λ − λ0 ) φ, φ˜ 1 + (λ − λ0 )2 e−iλ0 t φ, e−iλt U − ei(λ−λ0 )t φ˜ 1 = 2πi(λ − λ0 ) Φ, Φ˜ 1 L2x + O |λ − λ0 |2 e−iλ0 t φ e−iλ0 t φ˜ 1 .
734
M. Goldberg / Journal of Functional Analysis 256 (2009) 718–746 2n
The norms in the last line can be taken with respect to L n+2 (Rn ; L2 (T)) + x −1 L2 (Rn × T), since e−iλt U − eiλt maps this space to its dual (see Proposition 10). Once again the finitedimensionality of Nλ0 and N˜ λ0 makes every norm space for e−iλ0 t φ equivalent to gL2 (Rn ×T) and similarly for φ˜ 1 and h1 . If the projection of Xλ0 onto X˜ λ0 is bijective, then for a fixed unit vector h1 ∈ coker T (λ0 ) there exists a unique unit vector g1 ∈ ker T (λ0 ) with the properties
T (λ)g1 , h1 |λ − λ0 |,
|λ − λ0 |, T (λ)g1 image T (λ0 )
T (λ)g1 , h |λ − λ0 |2 for all λ in a small neighborhood of λ0 in the lower halfplane, and all unit vectors h ∈ coker T (λ0 ) orthogonal to h1 . From this point onward one can follow the proof of Lemma 9 exactly. By continuity, T (λ) is an invertible map between the co-image and image of T (λ0 ). Given g1 with the properties above there exists a unique g (λ) ∈ coimage T (λ0 ) with g (λ) |λ − λ0 | so that T (λ)(g1 + g (λ)) ∈ coker T (λ0 ). The combined vector gh1 (λ) = g1 + g (λ) is still of approximately unit norm and satisfies T (λ)gh1 (λ) = Ch1 (λ − λ0 )h1 + O |λ − λ0 |2 with the error lying entirely in coker T (λ0 ). After choosing a (finite) basis for coker T (λ0 ), the true inverse T (λ)−1 h1 can be expressed as a linear combination of (λ − λ0 )−1 ghj (λ). The inverse image of h2 ∈ image T (λ0 ) is first approximated by considering the restricted operator T (λ) : coimage T (λ0 ) → image T (λ0 ). This may produce an error h ∈ coker T (λ0 ) which can be removed via a correction of size proportional to that of h2 . 2 Corollary 12. Let V = w 2 z be a complex potential in Lx L∞ t . Suppose the associated Floquet operators K and K¯ have no resonances on the real axis, that condition (C1) is satisfied at every real eigenvalue, and condition (C3) at every eigenvalue. Then K has finitely many eigenvalues λj , counted with multiplicity, inside the strip λ ∈ Ω − . Similarly, K¯ has only the eigenvalues λ¯ j in the reflected strip Ω + = {λ¯ : λ ∈ Ω − }. For all λ ∈ Ω − , the action of T (λ)−1 is governed by the bound n/2
T (λ)−1 g
L2 (T×Rn )
2π
−i λ¯ t
1 + i cot π(λ − λj )
g, e gL2 (T×Rn ) + zw ¯ φ˜ j L2 dt x
j
= gL2 (T×Rn )
0
1 + i cot π(λ − λj )
g, e−i λ¯ t zw + ¯ φ˜ j j
t∈[0,2π] L2t L2x
(28)
where φ˜ j ∈ N˜ λj enumerate the linearly independent eigenvectors of K¯ with eigenvalues in Ω + .
M. Goldberg / Journal of Functional Analysis 256 (2009) 718–746
735
Proof. The continuity and norm-decay properties of Corollary 7 imply that T (λ)−1 is invertible for all λ in an open subset of Ω − , with uniform bounds once Im(λ) is sufficiently large. Its complement is therefore compact in Ω − . If conditions (C1) and (C3) are satisfied, then Lemmas 9 and 11 show that the complement is discrete as well, making it a finite set. At each point where T (λ)−1 fails to exist, the corresponding eigenvalues of K and K¯ have finite multiplicity as a consequence of the Fredholm Alternative. For the quantitative statement, first recall that T (λ + 1) = e−it T (λ)eit . This makes T (λ)−1 periodic with respect to integer translations. A finite number of local statements such as (20) and (27) is sufficient to completely categorize the singularities of T (λ)−1 in the entire lower halfplane. The conclusion (28) rewrites these local bounds to make them periodic and gathers them into a finite sum. For example, the single pole (λ − λj )−1 is replaced with a cotangent function. The alterations to the inner product are designed to express projection onto the cokernel of T (λ) as a periodic operation. Note that coker T (λ + 1) = e−it coker T (λ) for every λ, and Nλ+1 = Nλ exactly. In the neighborhood of λj we have the estimate 2π
¯ ¯ g, e−i(λ−λj )t h L2 dt = g, h L2 (T×Rn ) + O |λ − λj | g2 h2 x
0
and it is bounded everywhere by (1 + e2π Im(λj −λ) )g2 h2 . Choosing a specific unit vector hj gives us
2π
−i(λ¯ −λ¯ )t
j h (t, ·)
1 + i cot π(λ − λj )
g, e j 2 dt L x
0
−imt
1
g, e = sup hj L2 (T×Rn ) + O g m∈Z π|λ − (λj + m)| in each neighborhood of λj + Z and it is bounded by g over the remainder of Ω − . To construct the global bound we have also used the fact that |1 + i cot π(λ)| ∼ e2π Im(λ) as Im(λ) → −∞. ¯ Taking hj = e−i λj t zw ¯ φ˜ j , the expression in (28) is seen to possess the same poles as (20) and (27) near each point λj + Z and the appropriate global bound away from these singularities. 2 5.1. Remarks on condition (C3) In the introduction, we observed a relation between the non-orthogonality condition (C3) and the ability to diagonalize K and K¯ over their respective eigenspaces. This can be phrased more precisely as a mapping property of the bounded operator T (λ). As a starting point, the kernel of (K − λ)2 should be strictly larger than the kernel of K − λ if there exists a solution of (K − λ)e−iλt ψ = e−iλt φ for some φ ∈ Nλ . Applying the operator we−iλt U + eiλt to both sides yields the equation iT (λ)we−iλt ψ = we−iλt U + φ thanks to the intertwining identity (15).
(29)
736
M. Goldberg / Journal of Functional Analysis 256 (2009) 718–746
Proposition 13. Suppose that Nλ and N˜ λ both satisfy (C1). Condition (C3) is then equivalent to the following statement. (C3 ) The image of T (λ) does not contain we−iλt U + φ ∈ L2 (T × Rn ) for any nonzero φ ∈ Nλ . Proof. The fact that we−iλt U + φ always belongs to L2 (T × Rn ) is a consequence of (C1) and ¯ Proposition 10. It belongs to the image of T (λ) if it is orthogonal to zwe ¯ −i λt N˜ λ = ker T (λ)∗ . This would occur if the inner products
¯
¯ −i λt φ˜ we−iλt U + φ, zwe
L2 (T×Rn )
˜ = −2πi Φ, Φ
˜ L2 = φ, U − V¯ φ˜ = −i φ, φ
x
vanish for every φ˜ ∈ N˜ λ . The last identity is due to the conservation law (24). In other words, we−iλt U + φ is in the image of T (λ) precisely if φ belongs to the kernel of an orthogonal projection from Nλ onto N˜ λ . Because this is a linear map between vector spaces of the same finite dimension, there is a nontrivial kernel whenever condition (C3) fails. 2 6. Proof of Theorem 1 Based on the solution formula (10), it suffices to show that (I − iwU + wz)−1 wU0+ f belongs to L2t L2x , with support on the time halfline t ∈ [0, ∞). The method of choice is suggested by (13), namely to demonstrate the finiteness of 2 −1 sup eμt I − iwU + wz wU0+ f L2 L2 t
μ0
1 = sup μ0
T (λ + iμ)−1 e−i(λ +iμ)t Pλ +iμ wU + f 2 2 0
L (T×Rn )
dλ
0
1 = sup μ0
x
T (λ + iμ)−1 e−iλ t Pλ eμt wU + f 2 2 0
L (T×Rn )
dλ .
0
Using the inequality (28) to control the behavior of T (λ + iμ)−1 , we are left to show that 1
Pλ eμt wU + f 2 dλ 0
2
0
2
1 + i cot π λ − λ + i(μ − μj ) 2 eμt U + f, Pλ e−μt V¯ φ˜ j
dλ + j 0 t∈[0,2π] L2 L2 1
t
j
f 22
x
0
(30)
uniformly in μ 0. To write things in this form we have taken advantage of the facts that Pλ is self-adjoint on L2t L2x and commutes with pointwise multiplication by w(x).
M. Goldberg / Journal of Functional Analysis 256 (2009) 718–746
737
The first integral above is exactly eμt wU0+ f 2 2 2 f 2 as a result of the Plancherel Lt Lx identity (11) and the free Strichartz inequality (9). The second integral appears more complicated, but it is also evaluated (separately for each j ) using Plancherel’s identity in the λ variable. Designate by bj,μ (λ ) the function
bj,μ (λ ) = 1 + i cot π λ − λ j + i(μ − μj ) eμt U0+ f, Pλ e−μt V¯ φ˜ j t∈[0,2π] L2 L2 . (31) x
t
The desired bound (30) is achieved by showing that bj,μ L2 ([0,1]) Cj f 2 for each j and all μ 0. Let k ∈ Z be the Fourier variable dual to λ . Given any function g ∈ L2t L2x and a multiplier M(λ ), the inverse Fourier transform of M(λ )Pλ g has the form (MPλ )ˇg(k, t, x) =
ˇ − m)g(t + 2πm, x). M(k
m∈Z
Integration inside the infinite sum is justified in the same manner as the Fourier inversion formula. The fact that Pλ resides in the conjugate-linear half of an inner product creates some minor bookkeeping issues. When we wish to find the inverse Fourier transform of a function B(λ ) = M(λ ) F, Pλ g , the end result is instead ˇ B(k) =
ˇ ¯+ m)g(t + 2πm, x) . F, M(k
m∈Z
The multiplier of interest, M(λ ) = 1 + i cot π(λ − λ j + i(μ − μj )), has as its inverse Fourier transform 2πiλ 2π(μ−μ ) k −2|k1 if μ μj , j j ˇ e × (32) M(k) = e if μ > μj . 2|k0 We have chosen to handle the case μ = μj by analytic continuation from μ < μj rather than as a principal value. For our purposes the distinction is irrelevant, as the inner product in (30) will be made to vanish wherever there is a singularity of the cotangent function. We are now prepared to evaluate bj,μ 2 . First consider the case μ μj . Applying the top line from (32) to the function g(t, x) = e−μt V¯ φ˜ j χt∈[0,2π] and recalling the periodicity relation for φ˜ j yields
k
ˇ B(k) = −2 e−2πiλj e2π(μ−μj ) F, e−μt V¯ φ˜ j t2πk . After substituting F (t, x) = eμt U0+ f into this expression, Plancherel’s identity tells us that bj,μ 2L2 ([0,1]) =
k∈Z
=
k∈Z
∗
2
4e4π(μ−μj )k f, U0+ (V¯ φ˜ j |t2πk ) L2 (Rn )
2
4e4π(μ−μj )k f, (U − (V¯ φ˜ j |t2πk )(0, ·) L2 (Rn ) .
(33)
738
M. Goldberg / Journal of Functional Analysis 256 (2009) 718–746
The support of U − (V¯ φ˜ j |t2πk ) is contained in the time interval t ∈ (−∞, 2πk], therefore the inner product vanishes for each k 0 (it vanishes when k = 0 because of local L2 continuity). ¯ For each k 1 we use the eigenvector property φ˜ j = U − V¯ φ˜ j and the periodicity of e−i λj t φ˜ j to assert that U − (V¯ φ˜ j |t2πk )(0, ·) = U − (V¯ φ˜ j ) − U − (V¯ φ˜ j |t>2πk ) (0, ·) ¯ = Φ˜ j − e2πi λj k e2πikΔ Φ˜ j
with the conclusion bj,μ 2L2 [0,1] 8
2
2 e4π(μ−μj )k f, Φ˜ j
+ 8 e4πμk f, e2πikΔ Φ˜ j
k1
k1
2
|μ − μj |−1 f, Φ˜ j
+ |μ|−1 f 22 Φ˜ j 22 . If μj < 0, then we have shown that bj,μ |μj |−1/2 f for all f ∈ L2 (Rn ) orthogonal to Φ˜ j and all μ μj . The extra assumption (C2) is unnecessary in this case. The calculations are more delicate when μj = 0 because the unitarity of e2πikΔ on L2 does not provide a satisfactory estimate of the inner product. In its place we use the bound
e−2πikΔ f, ψ 2 f 2 ψ2 2
(34)
x −1 L2 +W 1,2n/(n+2)
k∈Z
which is proved as Lemma 16 in the last section. This is essentially a discrete-time version of more familiar Kato smoothing estimates
−itΔ
2
e f, ψ dt f 2 ψ2 2
x −1 L2 +L2n/(n+2)
R
gathered from [16] and [15]. It is worth re-iterating that Φ˜ j has approximately unit norm in any space that contains the finite-dimensional subspace X˜ λj . The remaining case μj < μ 0 is treated similarly. The same sequence of computations using the appropriate case of (32) leads to the identity bj,μ 2L2 ([0,1]) =
2
4e4π(μ−μj )k f, U − (V¯ φ˜ j |t2πk )(0, ·) L2 (Rn ) .
k∈Z
This time the properties of φ˜ j simplify the inner product so that bj,μ 2L2 ([0,1]) = 4
k0
2
2 e4π(μ−μj )k f, Φ˜ j
+ 4 e4πμk f, e2πikΔ Φ˜ j k1
2
|μ − μj |−1 f, Φ˜ j
+ f 22 Φ˜ j 2
x −1 L2 +W 1,2n/(n+2)
.
This concludes the proof of Theorem 1, with the exception of the technical lemmas whose proofs are presented below.
M. Goldberg / Journal of Functional Analysis 256 (2009) 718–746
739
7. Fourier analysis Our remaining task is to justify some of the technical estimates employed during the proofs of Lemma 11 and Theorem 1. The recurring theme here will be the use of Fourier restriction theorems, with particular emphasis on whether the restriction to a sphere varies smoothly with respect to changes in radius. Lemma 14. The resolvents R − (τ ) observe the following inequality − R (τ )ψ
2n n−2
x ψ 2
(26)
with a constant that is uniform over the closed halfplane Im(τ ) 0. ˆ ω) indicate the restriction of ψˆ to the sphere with radius r. Since Proof. Let ψˆ r (ω) = ψ(r, x ˆ we have assumed that x ψ ∈ L2 (Rn ), the radial derivative ∂r ψˆ r (ω) = ∇ ψ(x) · |x| is squareintegrable with respect to spherical coordinates. Combined with the convexity of norms, this means
∞ r
n−1
∞ 2 2 2 d ˆ ψr L2 (S n−1 ) dr r n−1 ∂r ψˆ r (ω)L2 (S n−1 ) dr x ψ 2 . dr
0
(35)
0
The left-hand side is a weighted L2 norm of the derivative of ψˆ r . Hardy’s inequality (or the ˆ itself, Schur test when n 4) then gives a weighted L2 estimate for ψ ∞
2 r n−3 ψˆ r 2L2 (S n−1 ) dr x ψ 2 ,
0 1
which is in effect a bound on (−Δ)− 2 ψ2 . Applying the Lp fractional integration bound for 1 (−Δ)− 2 on top of this leads to the conclusion − R (0)ψ
2n n−2
= Cn ψ ∗ |x|2−n
2n n−2
x ψ 2 .
(36)
Applying the Cauchy–Schwartz inequality to (35) gives a pointwise bound for ψˆ r instead. n ψˆ r L2 (S n−1 ) r 1− 2 x ψ 2 .
(37)
The resolvent R − (λ) multiplies Fourier transforms by (|ξ |2 − λ)−1 . If Re(λ) < | Im(λ)| then standard estimates show that the convolution kernel of R − (λ) is bounded pointwise by |x|2−n , uniformly in λ over this range. The conclusion of the lemma is verified by taking absolute values and applying (36). The case Re(λ) > | Im(λ)| requires more care. Let χ be a smooth function identically equal to 1 on [ 12 , 2] and supported on [ 14 , 4]. Decompose the resolvent into two pieces,
740
M. Goldberg / Journal of Functional Analysis 256 (2009) 718–746
− 1 −1 ˆ ), (R1 ψ)ˆ(ξ ) = 1 − χ Re(λ) 2 |ξ | |ξ |2 − λ ψ(ξ −1 1 ˆ ). (R2 ψ)ˆ(ξ ) = χ Re(λ)− 2 |ξ | |ξ |2 − λ ψ(ξ The convolution kernel associated to R1 is again controlled pointwise by |x|2−n , making it subject to the same bound as in (36). Each restriction of φˆ to the sphere radius r makes the contribution (ψˆ r )ˇ(x) = (2π)−n r n−1
eirx·ω ψˆ r (ω) dω
S n−1
toward the original function ψ . Once the normalization is taken into account, the Stein–Tomas theorem [17] indicates that (ψˆ r )ˇ
r 2 ψˆ r L2 (S n−1 ) . n
2n n−2
(38)
Set r0 = | Re(λ)| 2 and write out ψˆ r = (ψˆ r − ψˆ r0 ) + ψˆ r0 . This splits R2 ψ into the sum of two pieces. 1
4r0 −1 r 2 r − λ (ψˆ r − ψˆ r0 )ˇ(x) dr R2 ψ(x) = χ r0 r0 4
4r0 + r0 4
r r0
n−1 −1 r 2 r0 x dr χ r − λ (ψˆ r0 )ˇ r0 r
= I1 + I2 . For the first integral, (35) shows that r (n−1)/2 ψˆ r , viewed as a L2 (S n−1 )-valued function of r, has a square-integrable weak derivative. Therefore ψˆ r is Hölder-continuous of order 1/2 in the (1−n)/2 x ψ2 . Combined with (38) this interval r ∈ [ r40 , 4r0 ], with constant no greater than r0 shows 4r0 I1
2n n−2
n 2
1−n 2
r r0 r0 4
1/2 r0
4r0 r0 4
x ψ 2 . The primary estimate for I2 is that
|r − r0 |1/2 x ψ dr 2 |r 2 − λ|
|r − r0 |1/2 dr x ψ 2 2 2 |r − r0 |
M. Goldberg / Journal of Functional Analysis 256 (2009) 718–746
(ψˆ r )ˇ 0
2n n−2
741
r0 x ψ 2
by virtue of (37) and (38). After a suitable change of variables, this function can be transformed into I2 via a singular integral operator that preserves Lp norms. The proposition below completes the proof. 2 Proposition 15. Given the cutoff χ as defined above and any λ = r02 + iμ with |μ| r02 , the operator 4r0 Sg = r0 4
r r0
n−1 −1 r0 r 2 r −λ g x dr χ r0 r
satisfies the bounds Sgp Cp r0−1 gp for every 1 < p < ∞. Proof. Consider the logarithmic spherical coordinates (s, ω) ∈ R × S n−1 defined by s = log |x| x . The Jacobian factor transforms the Lp norms according to the rule and ω = |x| p gp
= S n−1
g(s, ω) p ens ds dω =
R
g(·, ω)p p
L (ens ds)
dω.
S n−1
In these coordinates the action of S takes place entirely along the s variable. Let ρ = log( rr0 ). Then log 4
Sg(s, ω) = r0−1
enρ χ eρ
− log 4
= r0−1 g
e2ρ
g(s − ρ, ω) dρ − (1 + iμ/r02 )
enρ χ(eρ ) (s, ω), ∗ e2ρ − (1 + μ/r02 )
where the convolution takes place in the s variable only. This is a Calderón–Zygmund singular integral which can be controlled by the Hilbert transform independently of the value of μ. The unweighted bounds for the Hilbert transform apply here (despite the fact that ens belongs to no Ap class) because the convolution kernel is supported in [−2, 2] and the exponential function is essentially constant over any interval of similar length. 2 Lemma 16. The propagator of the free Schrödinger equation obeys the sampling estimates
e−2πikΔ f, ψ 2 f 2 ψ2 2
x −1 L2
,
k∈Z
e−2πikΔ f, ψ 2 f 2 ψ2 2 2
k∈Z
provided γ ∈ [ n+1 2 , n + 1] and α +
1 γ
> 1.
L ∩L2n/(n+2) ∩W˙ α,2γ /(γ +2)
,
(39)
742
M. Goldberg / Journal of Functional Analysis 256 (2009) 718–746
Proof. For each k the inner product e−2πikΔ f, ψ represents the integral
− n2
ˆ ¯ )e2πik|ξ |2 dξ fˆ(ξ )ψ(ξ
(2π)
Rn − n2
∞
= (2π)
s
n−2 2
0
ˆ ¯ ω)e2πiks dω ds, fˆ(s, ω)ψ(s,
S n−1
x where (s, ω) are the spherical coordinates s = |ξ |2 , ω = |x| . This in turn describes (up to con stants) the kth Fourier coefficient of the periodic function m F (s + m), where
F (s) = (s)
n−2 2
ˆ ¯ ω) dω. fˆ(s, ω)ψ(s,
S n−1
We finding conditions on ψ that lead to the periodization are therefore concerned with 2 ([0, 1]). It would be sufficient to show instead that F ∈ F (s + m) belonging to L m
1m (L2 ([m, m + 1])). Plancherel’s identity dictates that s (n−2)/4 fˆ(s, ω) is precisely an element of 2m L2 ([m, m + 1]; L2 (S n−1 )). Bounds of the type (39) will follow provided that s (n−2)/4 ψˆ belongs to
2m L∞ ([m, m + 1]; L2 (S n−1 )). Taking ψˆ s to be the restriction of ψˆ to the sphere |ξ |2 = s, we wish to show that
sup
s
n−2 2
m0 s∈[m,m+1]
ψˆ s 2L2 (S n−1 )
is controlled by the norm of ψ in a space of our choosing. Suppose ψ ∈ x −1 L2 . Changing variables from r to s in (35) leads to the derivative estimates ∞
n
s2
2 2 d ψˆ s L2 (S n−1 ) ds x ψ 2 . ds
(40)
0
Local differences in the value of ψˆ s are estimated by the mean value theorem and Cauchy– Schwartz. For any pair of points s1 , s2 ∈ [m, m + 1],
(n−2)/4 (n−2)/4 2
s ψˆ s2 − s2 ψˆ s1 1
m+1
2 d (n−2)/4 s ψˆ s L2 (S n−1 ) ds ds
m m+1
2
s
n−6 2
ψˆ s 2 + s
n−2 2
d ψˆ s ds
2 ds.
m
The L∞ norm of a positive function over a unit interval is controlled by its integral and the variation of its values, hence
M. Goldberg / Journal of Functional Analysis 256 (2009) 718–746
sup
s
n−2 2
ψˆ s 2
m1 s∈[m,m+1]
743
m+1 2 n−2 n−6 n−2 d 2 ˆ ˆ s 2 + 2s 2 ψs + 2s 2 ψs ds ds
m1 m
∞
s
n−2 2
ψˆ s 2 ds +
1
2 ψ22 + x ψ 2
∞ s
n 2
2 d ˆ ψs ds ds
1
(41)
by Plancherel and (40), respectively. The supremum over the interval s ∈ [0, 1] is controlled separately by the estimate
s
n−2 4
ψˆ s s
n−2 4
∞ 1/2
∞
d 2 n d
ˆ ψˆ s
ds s 2
x ψ 2
ds ψs ds ds s
s
which is a combination of Cauchy–Schwartz and (40). 2n For the second statement, the condition ψ ∈ L n+2 is most important in the interval s ∈ [0, 1] and the Sobolev regularity condition plays a major role as s → ∞. It is clearly necessary to have ψ ∈ L2 , otherwise the inner product in (39) could be undefined for one or more values of k. The dual statement to (38), when normalized with the correct factor of r n−1 indicates that (n−2)/4 s ψˆ s ψ 2n for all s > 0. In particular, the supremum over s ∈ [0, 1] is bounded in n+2 this manner. The fact that ψ ∈ L2 implies that s (n−2)/4 ψˆ s 2 is integrable. Controlling its L∞ norm on a unit interval in terms of its L1 norm generally requires some degree of continuity. In the previous case we were able to infer differentiability of ψˆ s from the polynomial weighted decay of ψ. With ψ merely belonging to an Lp space, it may still be true that ψˆ is continuous, but the modulus of continuity is not determined by ψ alone. We exploit the observation (also used in [4]), that the norm of ψˆ s varies smoothly even when the restrictions themselves do not. 2γ
γ +2 (Rn ) satisfy the Proposition 17. Let γ ∈ [ n+1 2 , n + 1]. The Fourier restrictions of ψ ∈ L continuity bound
m
n−2 2
ψˆ s1 2 − ψˆ s2 2
|s1 − s2 |n+1−γ m
1/γ ψ22γ
(42)
γ +2
for every pair s1 , s2 ∈ [m, m + 1], m 1. The power of |s1 − s2 | does not matter much so long as it is nonnegative. Of considerably greater interest is the factor of m−1/γ , as it contributes meaningfully to the bound
(n−2)/4 2 (n−2)/4 2 −(α+ 1 ) n
s γ + m−(α+2− γ ) ψ2 ψˆ s1 − s ψˆ s2 m ˙ α,2γ /(γ +2) 1
2
W
for each pair s1 , s2 ∈ [m, m + 1], m 1. The first term is derived from (42), and the second (which is dominated by the first) from the Stein–Tomas theorem. As before, the L∞ norm of
744
M. Goldberg / Journal of Functional Analysis 256 (2009) 718–746
a function on a unit interval is controlled by the its L1 norm and the diameter of its image. Consequently,
(n−2)/4 2 s ψˆ s
sup
m0 s∈[m,m+1]
ψ
2 2n n+2
+
m+1 −(α+ γ1 ) (n−2)/2 ˆ 2 2 s ψs ds + m ψW˙ α,2γ /(γ +2)
m1
m
∞ ψ
2 2n n+2
+ ψ2W˙ α,2γ /(γ +2)
+
s (n−2)/2 ψˆ s 2 ds
1
= ψ22n + ψ2W˙ α,2γ /(γ +2) + ψ22 n+2
provided the sum of m−(α+1/γ ) is convergent.
2
Proof of Proposition 17. On each interval [m, m + 1] the function s (n−2)/4 can be replaced by the constant m(n−2)/4 . Recalling the proof of the Stein–Tomas theorem, Fourier restriction to the sphere is described by a convolution operator, with the T T ∗ estimate ψˆ s 2 = Cn
1 ¯ dx dy. f (x)K s 2 (x − y) f (y)
(43)
R2n
The kernel is an oscillatory function bounded pointwise by |K(z)| z −(n−1)/2 . The related ˜ function K(z) = zK (z) is also oscillatory, and bounded pointwise by z −(n−3)/2 . If f is a Schwartz function it is permissible to differentiate (43) with respect to s, obtaining d ψˆ s 2 = Cn s −1 ds
1 ¯ dx dy. f (x)K˜ s 2 (x − y) f (y)
R2n
The same interpolation argument that proves the Stein–Tomas theorem also suffices to show 2n+2 2n+2 that convolution with K˜ is a bounded operator from L n+5 (Rn ) to its dual space L n−3 (Rn ). Combining this with the usual restriction estimate and scaling appropriately,
m(n−2)/2 ψˆ s1 2 − ψˆ s2 2 max m1/(n+1) f 22n+2 , m2/(n+1) |s1 − s2 |f 22n+2 . n+3
n+5
These represent the cases γ = n + 1 and γ = n+1 2 , respectively. The intermediate cases follow from Riesz–Thorin interpolation, noting that the norm of a self-adjoint linear operator agrees with the extremal value of its quadratic form. 2 Remark 4. The proof of Lemma 16 hinges on placing the spherical restrictions of ψˆ inside a mixed-norm space 2 (L∞ ) with respect to the radial variable. This consists of three essentially independent requirements.
M. Goldberg / Journal of Functional Analysis 256 (2009) 718–746
745
1. Because of the embedding 2 (L∞ ) ⊂ 2 (L2 ) and the Plancherel identity, we must have ψ ∈ L2 . This is the only way to produce 2 decay as m → ∞. 2. Since 2 (L∞ ) also embeds into ∞ (L∞ ), the normalized restrictions s (n−2)/4 ψˆ s must be uniformly bounded, in particular as s → 0. This is achieved so long as ψ belongs to either of the spaces x −1 L2 or L2n/(n+2) . 3. The norm of the restrictions must also be sufficiently continuous so that the 2 (L2 ) bound implied by the first item can be improved into 2 (L∞ ). Proposition 17 provides one estimate for the modulus of continuity of ψˆ s 2 based on the Stein–Tomas restriction theorem. Another estimate, based on L2 trace properties, is available when x β ψ ∈ L2 for some β > 12 . The latter bounds are well known from the proof of the limiting absorption principle [13] and spectral theory of Schrödinger operators. The norm spaces in the statement of Proposition 17 were chosen to meet these requirements entirely with weights, or entirely with homogeneous Lp conditions, respectively. To create a more comprehensive list, one can mix and match the two approaches in any combination. A precise but unwieldy formulation is presented below. Proposition 18. The propagator of the free Schrödinger equation obeys the sampling estimates
e−2πikΔ f, ψ 2 f 2 ψ, k∈Z
where the norm of ψ is taken in the interpolation space 2n n 2n+2 n−1 2n+2 1 ψ ∈ L2 ∩ x −1 L2 + L n+2 ∩ x − 2 −ε L2 + W˙ n+1 +ε, n+3 + W˙ n+1 +ε, n+5 . References [1] S. Agmon, Spectral properties of Schrödinger operators and scattering theory, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 2 (2) (1975) 151–218. [2] M.B. Erdo˘gan, W. Schlag, Dispersive estimates for Schrödinger operators in the presence of a resonance and/or an eigenvalue at zero energy in dimension three: I, Dyn. Partial Differ. Equ. 1 (4) (2004) 359–379. [3] A. Galtbayar, A. Jensen, K. Yajima, Local time-decay of solutions to Schrödinger equations with time-periodic potentials, J. Stat. Phys. 116 (2004) 231–282. [4] M. Goldberg, W. Schlag, A limiting absorption principle for the three-dimensional Schrödinger equation with Lp potentials, Int. Math. Res. Not. 2004:75 (2004) 4049–4071. [5] M. Goldberg, M. Visan, A counterexample to dispersive estimates for Schrödinger operators in higher dimensions, Comm. Math. Phys. 266 (2006) 211–238. [6] A. Ionescu, D. Jerison, On the absence of positive eigenvalues of Schrödinger operators with rough potentials, Geom. Funct. Anal. 13 (2003) 1029–1081. [7] A. Jensen, T. Kato, Spectral properties of Schrödinger operators and time-decay of the wave functions, Duke Math. J. 46 (3) (1979) 583–611. [8] J.-L. Journé, A. Soffer, C.D. Sogge, Decay estimates for Schrödinger operators, Comm. Pure Appl. Math. 44 (5) (1991) 573–604. [9] T. Kato, Wave operators and similarity for some non-selfadjoint operators, Math. Ann. 162 (1965/1966) 258–279. [10] M. Keel, T. Tao, Endpoint Strichartz inequalities, Amer. J. Math. 120 (1998) 955–980. [11] C.E. Kenig, A. Ruiz, C.D. Sogge, Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators, Duke Math. J. 55 (2) (1987) 329–347. [12] E. Lieb, M. Loss, Analysis, second ed., Grad. Stud. Math., vol. 14, Amer. Math. Soc., Providence, RI, 2001. [13] M. Reed, B. Simon, Methods of Modern Mathematical Physics. IV. Analysis of Operators, Academic Press, New York, London, 1978.
746
M. Goldberg / Journal of Functional Analysis 256 (2009) 718–746
[14] I. Rodnianski, W. Schlag, Time decay for solutions of Schrödinger equations with rough and time-dependent potentials, Invent. Math. 155 (3) (2004) 451–513. [15] A. Ruiz, L. Vega, On local regularity of Schrödinger equations, Int. Math. Res. Not. 1 (1993) 13–27. [16] B. Simon, Best constants in some operator smoothness estimates, J. Funct. Anal. 107 (1) (1992) 66–71. [17] P. Tomas, Restriction theorems for the Fourier transform, Proc. Sympos. Pure Math. 35 (1979) 111–114. [18] G. Vodev, Dispersive estimates of solutions to the Schrödinger equation in dimensions n 4, Asymptot. Anal. 49 (2006) 61–86. [19] K. Yajima, Exponential decay of quasi-stationary states of time periodic Schrödinger equations with short range potentials, Sci. Papers Collega Arts Sci. Univ. Tokyo 40 (1990) 27–36.
Journal of Functional Analysis 256 (2009) 747–776 www.elsevier.com/locate/jfa
Global branches of non-radially symmetric solutions to a semilinear Neumann problem in a disk ✩ Yasuhito Miyamoto Department of Mathematics, Tokyo Institute of Technology, Meguro-ku, Tokyo 152-8551, Japan Received 16 November 2007; accepted 20 November 2008 Available online 10 December 2008 Communicated by H. Brezis
Abstract Let D ⊂ R2 be a disk, and let f ∈ C 3 . We assume that there is a ∈ R such that f (a) = 0 and f (a) > 0. In this article, we are concerned with the Neumann problem u + λf (u) = 0 in D,
∂ν u = 0 on ∂D.
We show the following: There are unbounded continuums consisting of non-radially symmetric solutions emanating from the second and third eigenvalues. If f (u) = −u + u|u|p−1 (a = 1) or if f is of bistable type, then the unbounded branches emanating from non-principal eigenvalues are unbounded in the positive direction of λ. The branch emanating from the second eigenvalue is unique near the bifurcation point up to rotation. The main tool of this article is the zero level set (the nodal curve) of uθ and ux . © 2008 Elsevier Inc. All rights reserved. Keywords: Global branch; Bifurcation; Nodal curve; Nodal domain
1. Introduction and main results Let D ⊂ R2 be a disk centered at the origin with radius 1, and let f ∈ C 3 . Throughout the present article, we assume that there is a ∈ R such that f (a) = 0 and f (a) > 0. ✩
This work was partially supported by JSPS Research Fellowships for Young Scientists. E-mail address: [email protected].
0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.11.023
(A0)
748
Y. Miyamoto / Journal of Functional Analysis 256 (2009) 747–776
We are concerned with the Neumann problem in a domain Ω ⊂ RN u + λf (u) = 0 in Ω,
∂ν u = 0 on ∂Ω,
(BPΩ )
where λ > 0. In order to specify the dependence of the problem on the domain we put the index Ω as above. Because f (a) > 0 and the change of the variable f (a)λ → λ does not affect the form of the equation in (BPΩ ), we can assume without loss of generality that f (a) = 1. The following are examples of f : There are a− , a+ ∈ R such that a− < a < a+ , f (a− ) = f (a+ ) = 0, f < 0 in (a− , a), and f > 0 in (a, a+ ), f (u) = −u + u|u|p−1 /(p − 1) (p > 1), and a = 1.
(A1) (A2)
The problem (BPΩ ) with (A2) is equivalent to the problem ε 2 u − u + u|u|p−1 = 0 in Ω,
∂ν u = 0 on ∂Ω,
(1.1)
√ where ε = (p − 1)/λ. Since, for any λ > 0, u ≡ a is a solution of (BPΩ ), we call u ≡ a a trivial solution (or a trivial branch). In this article, we study branches of non-radially symmetric solutions to (BPD ) bifurcating from the trivial branch. Let X be a functional space to which the solution u of (BPΩ ) belongs. We call (λ∗ , a) ∈ R × X is a bifurcation point if for any neighborhood U ⊂ R × X of (λ∗ , a) there is a non-trivial solution (λ, u) in U . A brief statement of our main results is Theorem A. Let N denote the Laplacian with the Neumann boundary condition, and let μ1 (D), μ2 (D) be the second and third eigenvalue of N on D without counting multiplicities, respectively. Then (BPD ) has bifurcation points (μ1 (D), a) and (μ2 (D), a) from which unbounded branches consisting of non-radially symmetric solutions emanate. When (A1) or (A2) holds, the unbounded branches are unbounded in the positive direction of λ. The branch emanating from (μ1 (D), a) is unique near the bifurcation point up to rotation if f (a) = 0. A rotation of a solution to (BPD ) is a solution. Hence, a branch is actually a sheet provided that the branch consists of non-radially symmetric solutions. The precise statements are in Theorems 3.1, 3.5, 3.6, and 4.1 and Corollaries 3.8 and 4.2. In the proof of the main results, we use a celebrated work of Rabinowitz [33] called the Rabinowitz alternative. This theorem says that, for a large class of nonlinear eigenvalue problems, a continuum C bifurcates from a characteristic value (eigenvalue) of odd (algebraic) multiplicity, and the continuum either (1) is unbounded or (2) meets another characteristic value (eigenvalue). For a precise statement of this theorem, see Proposition 2.1 of the present article. In order to prove the existence of unbounded branches, one must exclude the case (2). Several methods were developed. For methods that are not used in this article, see the introduction of [14]. In our proof the main tools to show that (2) does not occur are the zero level set (the zero curve or the nodal curve) and the non-zero level set (the nodal domain), which are intensively used in [16,17].
Y. Miyamoto / Journal of Functional Analysis 256 (2009) 747–776
749
Before going to the details of our proof, we consider the one-dimensional problem uxx + λf (u) = 0 in I := (0, π),
ux = 0 at x = 0 and π,
(BPI )
where f satisfies f (0) = 0 and f (0) = 1. This problem is a simplified problem of an example given in [33, (2.1)] as an application of the abstract theory in [33]. We will show a sketch of the proof of the existence of unbounded branches emanating from a trivial solution u ≡ 0. The Sturm–Liouville theory says that every eigenvalue of the associated eigenvalue problem ϕxx + λf (0)ϕ = κϕ
in I,
ϕx = 0 at x = 0 and π
(1.2)
is simple hence of odd (algebraic) multiplicity. It follows from the Rabinowitz alternative that if 0 is an eigenvalue of (1.2), i.e., λ = n2 (n 1), then (λn , 0) is a bifurcation point, where λn := n2 . There is a continuum Cλn consisting of non-trivial solutions of (BPI ) and emanating from (λ, u) = (λn , 0). If (λ, u) ∈ Cλn is near the bifurcation point (λn , 0), then u is close to the nth eigenfunction, C0 cos((n − 1)x) (C0 is small), in the C 1 -sense. Therefore u has exactly n − 1 simple zero(s) in I . Moreover the number of the zeros of u does not change along the branch. If not, then u must have a degenerate zero for some λ in order to change the zero number, because u continuously depends on λ. By the uniqueness of solutions of ODEs u should be 0, which indicates that Cλn does not meet other bifurcation points and that Cλn and Cλm (n = m) do not meet each other. Thus (2) does not occur and the branch Cλn is unbounded. The key of this argument is the uniqueness of solutions to ODEs. Let us consider the Dirichlet problem in a ball B ⊂ RN centered at the origin with radius 1 u + λf (u) = 0
in B,
u = 0 on ∂B.
(1.3)
If we restrict ourselves to consider only the positive solutions, then it follows from Gidas, Ni and Nirenberg [13] that all the positive solutions of (1.3) are radially symmetric. Hence (1.3) can be reduced to the ODE urr +
ur + λf (u) = 0 in (0, 1), r
ur (0) = 0,
u(1) = 0.
(1.4)
Although one has to modify the method of the 1D case mentioned above to overcome the technical difficulty coming from the singularity at r = 0, a similar argument is applicable to (1.4), and we can show the existence of unbounded branches. In this setting, detailed analysis can be done. Among others, see [25] for the bistable type nonlinearity, [30,31] for a class of nonlinearities including f (u) = u(u − b)(c − u) (0 < 2b < c) and f (u) = up − uq (1 < p < q), [11,19,28,39] for f (u) = eu , and [10] for a nonlinearity having an S-shaped bifurcation curve. The Neumann problem tends to have more solutions than the Dirichlet problem, as pointed out by Shi [37]. In fact, the Neumann problem can have a positive non-radially symmetric solution when the domain is a ball. We cannot reduce (BPD ) into an ODE. When the spatial dimension is 2 or larger, the multiplicity of the eigenvalues may not be 1. Therefore the theory of Crandall and Rabinowitz [8], which studies the bifurcations from simple eigenvalues, cannot be directly applied. Even if one can show that u near a bifurcation point is close to an eigenfunction in some sense, we cannot argue a high-dimensional case similarly in the 1D case, because (I) Courant’s nodal domain theorem does not give the one-to-one correspondence, which is given by the Sturm–Liouville theory in 1D cases, between the index of an eigenvalue and the number
750
Y. Miyamoto / Journal of Functional Analysis 256 (2009) 747–776
of the nodal domains of the associated eigenfunction, (II) the topology of the nodal domains can change along the branch in contrast to 1D cases, and (III) the analysis of the changes of the topology seems to be difficult. Then to avoid these difficulties we restrict the domain of the problem (BPD ) to (1.5) below. In this article, we also consider the Neumann problem (BPDn ), where Dn :=
{(r, θ ); 0 < r < 1, 0 < θ < π/n} if n ∈ {1, 2, 3, . . .}, D if n = 0.
(1.5)
Here (r, θ ) is a polar coordinate of R2 centered at O. In Sections 3 and 4, we show that (BPDn ) (n = 1, 2) has a branch consisting of non-trivial solutions such that −∂θ u 0 in Dn (n = 1, 2), where ∂θ := −y∂x + x∂y . Then proving the non-negativity (or the non-positivity) of uθ and ux along the branch is much easier than the analysis of the changes of the topology. The difficulties (II) and (III) can be avoided. Since the sign of uθ does not change and uθ satisfies uθ + λf (u)uθ = 0 with mixed boundary condition, uθ is the first eigenfunction of the problem ϕ + λf (u)ϕ = μϕ
in Dn ,
ϕ = 0 on ∂Dn \∂D,
∂ν ϕ = 0 on ∂Dn ∩ ∂D.
Assume that the branch meets another eigenvalue. Although Dn (n = 1, 2) has corner points where the interior sphere condition is not satisfied, we can show that uθ has a zero in Dn by analyzing the local behavior of the solution near a corner point (Remark 2.5). If uθ ≡ 0 in Dn , then we obtain a contradiction, which means that the branch does not meet another eigenvalue. When uθ ≡ 0 in Dn , this method cannot exclude the possibility where the branch meets an eigenvalue having a radially symmetric eigenfunction. In this case we use the zero level set of ux , and argue similarly. Then we obtain a contradiction. See the proofs of Theorems 3.1 and 4.1 for the detail of this argument. Even if the one-to-one correspondence stated in (I) does not exist, we can exclude the possibility (2) as far as the branch emanating from the first eigenvalue of the restricted problem is concerned. There are a vast amount of literature on the Neumann problem of semilinear elliptic equations in a high-dimensional domain. However, there are few articles proving the existence of unbounded branches from a viewpoint of the bifurcation theory. Ni and Takagi in [29] prove the existence of unbounded branches in a domain of the type RN := [0, a1 ] × [0, a2 ] × · · · × [0, aN ], 2 where 1/a12 , 1/a22 , . . . , 1/aN are independent over Q.
(1.6)
It follows from this condition that every eigenvalue of N is simple. Shi in [37] considers (BPΩ ) in a rectangle, proves the existence of unbounded branches consisting of non-one-dimensional solutions, and studies the shape of solutions when λ is large. When the domain is not a rectangle, few results are known about the existence of unbounded branches consisting of non-trivial solutions of the Neumann problem.
Y. Miyamoto / Journal of Functional Analysis 256 (2009) 747–776
751
Let us mention a boundary one-spike layer. In 1D cases, when f (u) = −u + up , (BPI ) has an unbounded branch emanating from the second eigenvalue, and the branch goes toward a boundary spike as λ → +∞. Refs. [27,40–42] study this phenomenon for several classes of nonlinearities. In the case of high-dimensional domains, it is expected that the same phenomenon occurs. However, there is a technical difficulty. We discuss this point later (Remark 3.9). In spite that the changes of the zero number (or the lap-number in the sense of [22]) for solutions to 1D parabolic equations have been studied much [1,2,22,26], a little is known about the zero curve and the changes of the topology of the nodal domains except the theory of Carleman, Hartman and Wintner [5,18]. The author hopes that this article sheds light on the zero curve and that techniques analyzing the changes of the topology are developed in future. Watanabe [43] studies the changes of the topology of the zero level set of the solutions to the heat equation in a planar domain. Specifically, he gives necessary conditions when the topology does not change. Although [43] is not directly related to our problem, [43] may give hints of analysis of the zero curve. This article consists of four sections. In Section 2, we state known results about the Rabinowitz alternative and a theory of Crandall and Rabinowitz (Subsection 2.1), the eigenvalues and the eigenfunctions of the Neumann Laplacian in a disk (Subsection 2.2), and the zero level set of eigenfunctions (Subsection 2.3). We give some extension of known results about the shape of the solution near a bifurcation point (Subsection 2.4). In Subsection 2.4 we prove the existence of an unbounded branch for an arbitrary domain with smooth boundary. In Section 3, we study the existence of unbounded branch emanating from the second eigenvalue (Theorem 3.1), the shape of solutions near the bifurcation point (Theorem 3.5), and the direction of the unbounded branches (Theorem 3.6). In Section 4, we establish the existence of an unbounded branch emanating from the third eigenvalue (Theorem 4.1). 2. Preliminaries 2.1. Bifurcation from an eigenvalue with odd multiplicity Let X be a real Banach space with norm · , and let F : R × X → X be a compact and continuous mapping. We consider the equation u = F (λ, u)
(λ ∈ R, u ∈ X).
(2.1)
We assume that F (λ, u) = λLu + F0 (λ, u), where F0 (λ, u) = o( u ) ( u → 0) uniformly on −1 2 bounded λ intervals and L is a compact linear map on X. In our case, L = and X = {u ∈ C ; (u − a) dx = 0}. Since {(λ, 0); λ ∈ R} is a solution of (2.1), we call the solution a trivial solution. Let σ (L) be the set of the characteristic values (eigenvalues) of L, i.e., for λ∗ ∈ σ (L), there exists 0 = v ∈ X such that v = λ∗ Lv. The next proposition is one of the main results of [33]. Proposition 2.1. (See [33, Theorem 1.3].) If λ∗ ∈ σ (L) is of odd (algebraic) multiplicity, then there exists a continuum C consisting of non-trivial solutions of (2.1) and emanating from (λ∗ , 0) such that C either (1) is unbounded in R × X, or ˆ 0), where λ∗ = λˆ ∈ σ (L). (2) meets (λ, Here the algebraic multiplicity of λ∗ ∈ σ (L) is the dimension of
∞
j =1 ker((I
− λ∗ L)j ).
752
Y. Miyamoto / Journal of Functional Analysis 256 (2009) 747–776
Crandall and Rabinowitz in [8] study the bifurcation from the simple eigenvalue. We use their theory to study the shape of the solutions near a bifurcation point. Proposition 2.2. (See [8, Theorem 1].) Let F be as (2.1). Suppose F ∈ C 2 (R × X, X). Let λ∗ be a characteristic value of L. If the following hold: (a) (b) (c) (d)
there is u∗ ∈ X such that ker(I − Fu (λ∗ , 0)) = spanu∗ , hence dim ker(I − Fu (λ∗ , 0)) = 1, Ran(I − Fu (λ∗ , 0)) is closed, codim Ran(I − Fu (λ∗ , 0)) = 1, / Ran(I − Fu (λ∗ , 0)), where u∗ is defined in (a), Fu,λ (λ∗ , 0)u∗ ∈
then λ∗ is a bifurcation point. In addition the closure of the set of non-trivial solutions of (2.1) is, near (λ∗ , 0), a unique C 1 -curve. 2.2. Eigenvalues and eigenfunctions of N on Dn We recall known results on the eigenvalues and the eigenfunctions of N on Dn (n 0). Let us consider the eigenvalue problem u + μu = 0 in Dn ,
∂ν u = 0 on ∂Dn .
(EVPLDn )
To begin with, we consider the case where the domain is D0 (= D). We use a separation of variables. Let u(x, y) = ξ(r)η(θ ). Here (r, θ ) is a polar coordinate of D. Then it is well known that if ξ(r)η(θ ) is in C 2 (D), then, for some integer m 0, ξ and η satisfy ξrr +
m2 ξr + μ − 2 ξ = 0 and ηθθ = −m2 η. r r
Since u(x, y) = ξ(r)η(θ ) is of class C 2 at the origin, the solution of these equation should be √ ξ(r) = C0 Jm ( μr)
(0 r < 1),
η(θ ) = C1 cos(mθ ) + C2 sin(mθ )
(0 θ < 2π).
Here Jm (r) (m 0) is the Bessel function of the first kind of order m. Specifically, m ∞ r (−1)j (r/2)2j Jm (r) = . 2 j !(m + j )! j =0
See [6,38] for more details of calculations of eigenvalues of the Laplacian on a disk. Since ξ √ 2 , where p satisfies the Neumann boundary condition at r = 1, Jm ( μ ) = 0. Hence μ = pm,l m,l (l 1, m 0) is the lth positive zero of Jm (·). Therefore, Jm (pm,l r) cos(mθ )
and Jm (pm,l r) sin(mθ )
(l 1, m 0)
2 is the eigenvalue corresponding to these consist of all the eigenfunctions of (EVPLD ) and pm,l eigenfunctions. (0) Let {μj }j 0 denote the set of the eigenvalues of (EVPLD ) without counting multiplicities, (0)
(0)
(0)
(0)
2 } i.e., 0 = μ0 < μ1 < μ2 < · · · . Then {μj }j 0 becomes the set {0} ∪ {pm,l l1, m0 sorted
Y. Miyamoto / Journal of Functional Analysis 256 (2009) 747–776
753
in increasing order. (However, we remove the same values if there are.) In particular, (0)
(0)
μ0 = 0,
2 μ1 = p1,1 ≈ 3.390,
(0)
(0)
2 μ2 = p2,1 ≈ 9.328,
(0)
2 ≈ 14.681, μ3 = p0,1
2 μ4 = p3,1 ≈ 17.650.
Next, we consider the case where the domain is Dn (n 1). In this case, in order to satisfy the Neumann boundary condition on ∂Dn , u should be (n) φm,l (r, θ ) := Jmn (pmn,l r) cos(mnθ )
(0 < r < 1, 0 < θ < π/n) (l 1, m 0).
2 Therefore pmn,l (l 1, m 0) is an eigenvalue corresponding to this eigenfunction. Let
{μ(n) j }j 0 denote the set of the eigenvalues of (EVPLDn ) without counting multiplicities. Then (n)
2 }l1, m0 sorted in increasing order. (We remove the same {μj }j 0 becomes the set {0} ∪ {pmm,l values if there are.) The set D can be considered as
D=
2n−1
Rπj /n Dn ,
j =0
where Rθ is the counterclockwise rotation with center O and angle θ.
(2.2)
For example, Rπj/n Dn := {(r, θ ); 0 < r < 1, πj/n < θ < π(j + 1)/n}. Let u be a solution to (BPDn ), and let u˜ be the extended function of u with even reflections. Specifically, for r ∈ [0, 1) and θ ∈ [0, 2π), u(r, ˜ θ ) :=
u(r, θ − πn [ nθ π ])
if θ ∈
u(r, πn − θ + πn [ nθ π ])
if θ ∈
n−1
2πj (2j +1)π ], j =1 [ n , n (2j +1)π (2j +2)π , n ]. j =1 [ n
n−1
(2.3)
2 , i.e., Then u˜ is in C 2 (D) and it satisfies (BPD ). φ˜ m,l (r, θ ) satisfies (EVPLD ) with μ = pmn,l (n) φ˜ m,l is an eigenfunction. Hence each set of the eigenvalues of (EVPLDn ) (n 1) is a subset of the eigenvalues of (EVPLD ). We state facts in this subsection as Proposition 2.3 in order that we refer easily. (n)
(n)
Proposition 2.3. Let {μj }j 0 (n 0) be the eigenvalues of (EVPLDn ) without counting (0)
(n)
2 } multiplicities. Then {μj }j 0 (respectively {μj }j 0 (n 1)) is the set {0} ∪ {pm,l l1, m0 2 (respectively {0} ∪ {pmn,l }l1, m0 ) sorted in increasing order.
754
Y. Miyamoto / Journal of Functional Analysis 256 (2009) 747–776
2.3. The zero level set Let Ω ⊂ R2 be an open set with piecewise smooth boundary, and let V ∈ C 1 (Ω). In this subsection, we study the local behavior of the zero level set of the solution to linear elliptic equations near zeros. Let p0 = (x0 , y0 ) ∈ Ω and let α = (α1 , α2 ) (α1 0, α2 0) be a multiindex with length |α| = α1 + α2 . We say that p0 is a degenerate zero of order (at least) n (n 2), if (∂α u)(p0 ) = 0 for all |α| n − 1, where ∂α := ∂xα1 ∂yα2 . When u(p0 ) = 0, we say that p0 is not a degenerate zero of u if ux (p0 ) = 0 or uy (p0 ) = 0. The following is the main proposition in this subsection: Proposition 2.4. Let V (x, y) ∈ C 1 (Ω), and let u(x, y) be a function such that u + V u = 0 in Ω. Then u ∈ C 2 (Ω). Furthermore, u has the following properties: (i) If u has a zero of any order at p0 in Ω, then u ≡ 0 in Ω. (ii) If u has a zero of order (exactly) l at p0 in Ω, then the Taylor expansion of u is u(p) = Hl (p − p0 ) + O |p − p0 |l+1 , where Hl is a real valued, non-zero, harmonic, homogeneous polynomial of degree l. Therefore, {u = 0} has exactly 2l branches at p0 . (iii) If u has a zero of order (exactly) l at p0 on ∂Ω and if u satisfies the Neumann (respectively Dirichlet) boundary condition, then u(p) = C0 r l cos(lθ ) + O r l+1 for some non-zero C0 ∈ R, where (r, θ ) is a polar coordinate of p = (x, y) around p0 . The angle θ is chosen so that the tangent to the boundary at p0 is given by the equation sin θ = 0 (respectively cos θ = 0). (i) and (ii) are first proved by Carleman [5] and generalized by Hartman and Wintner [18]. The statement of this proposition is taken from [15]. Remark 2.5. Applying Proposition 2.4(iii) to the solution extended by the reflection, we can analyze the local behavior of the solution near a corner point more in detail than Serrin’s corner point lemma [36, Lemma 1]. However this method is applicable for a planar domain. Next, we study the zero level set of u in the whole domain. Let p0 ∈ Ω be a zero of u. If p0 is not a degenerate zero, then (ux (p0 ))2 + (uy (p0 ))2 = 0, hence the equation u(x, y) = 0 can be solved locally with respect to x or y by the implicit function theorem. Thus the zero level set of u near p0 is a C 1 -curve, and the zero curve can be extended locally at p0 . If the curve cannot / ∂Ω, then p0 should be a degenerate zero of order at least 1. In be extended at p0 and if p0 ∈ that case, it follows from Proposition 2.4(ii) that p0 should be an intersection point among zero curves of u. We have obtained the following: Proposition 2.6. Let u( ≡ 0) be as in Proposition 2.4. Then the zero level set of u consists of C 1 -curves and intersection points among those curves. Moreover if several curves meet at an interior point, then the number of the curves that meet at the point should be even, say 2l (l 1),
Y. Miyamoto / Journal of Functional Analysis 256 (2009) 747–776
755
and all the angles between two adjacent curves are equal, hence π/ l. The same assertion as Proposition 2.4(iii) holds when the intersection point is on the boundary. Several authors use the zero curve to prove interesting properties of solutions of elliptic equations in a planar domain. For example, Ref. [32] shows that the first eigenfunction of the Dirichlet Laplacian in a planar convex domain has exactly one critical point. The author [23] shows that if the solution of the Neumann problem u + f (u) = 0 in a disk has a critical point inside the disk, then the Morse index of u is 2 or larger. See also [4] for other applications of the zero curve. In this paper, we consider (BPΩ ). The nonlinearity f in (BPΩ ) is not necessarily of the type f (u) = f0 (u)u. Hence we cannot directly apply Proposition 2.4 to u. However, we can apply the proposition to uθ (and ux ), because uθ (respectively ux ) satisfies the linear equation uθ + λf (u)uθ = 0 (respectively ux + λf (u)ux = 0). In Sections 3 and 4, we use the zero level set of uθ and ux . 2.4. The shape of solutions near the bifurcation points In this subsection, we study the shape of non-trivial solutions near the bifurcation points. Let Ω ⊂ RN be a bounded domain with piecewise smooth boundary. We consider the Neumann problem u + g(λ, u) = 0
in Ω,
∂ν u = 0 on ∂Ω,
(2.4)
where g is of class C 2 . We assume that there is a ∈ R such that, for any λ ∈ R, g(λ, a) = 0. Since, for any λ ∈ R, u ≡ a is a solution to (2.4), we call this solution a trivial solution (or a trivial branch). We consider the bifurcation problem (2.4). In this article, we restrict ourselves to bifurcations from constant solutions. This restriction reduces technical complexities, for example gu (λ, a) and gλu (λ, a) are constants. Let Lλ := + gu (λ, a). It is well known that if the operator Lλ with the Neumann boundary condition is an isomorphism between certain functional spaces, then (λ, a) ∈ R × C 2 (Ω) is not a bifurcation point. Therefore, candidates of the bifurcation points are points (λ, a) satisfying that 0 is an eigenvalue of Lλ . Let λ∗ be one of them. If 0 is a simple eigenvalue of Lλ∗ , and if certain other conditions are satisfied, then [8, Theorem 1.7] (Proposition 2.2 of this article) says that there are exactly two branches near (λ∗ , a). However, in our case, dim ker Lλ∗ may be 2 or larger. This makes the next lemma less trivial than it looks. Lemma 2.7. Let Ω ⊂ RN be a bounded domain. Suppose that (λ∗ , a) ∈ R × C 2 (Ω) is a bifurcation point of (2.4), i.e., there exists a set C ⊂ R × C 2 (Ω) consisting of non-trivial solutions of (2.4) such that C has at least one sequence converging to (λ∗ , a). Let {(λj , uj )}j 1 ⊂ C be such a sequence. If gλu (λ∗ , a) = 0, then there are a subsequence {(λjk , ujk )}k1 , {tk }k1 ⊂ R converging to 0, a sequence {vˆk }k1 ⊂ C 2 (Ω), and v∗ ∈ ker Lλ∗ \{0} such that ujk = a + tk v∗ + vˆk ,
where vˆk C 2 = o(tk ) as k → ∞.
In particular, ujk − a k→∞ v∗ −−−→
ujk − a
v∗
in C 2 .
(2.5)
756
Y. Miyamoto / Journal of Functional Analysis 256 (2009) 747–776
This lemma is the main result of this subsection. We postpone the proof of Lemma 2.7. Remark 2.8. Refs. [3] and [21] study the local bifurcation of (BPΩ ) in an abstract setting, where Ω ⊂ RN is an arbitrary bounded domain with smooth boundary. See [34, Theorem 11.4] or [35] for a precise statement of this result. Let {μj (Ω)}j 0 be the set of eigenvalues of N on Ω without counting multiplicities. Using this result, one can show that, for every non-principal eigenvalue μn (Ω) (n 1), there are at least two (local) continuums consisting of non-constant solutions of (BPΩ ) and emanating from (μn (Ω), 1). As far as (BPΩ ) is concerned, every eigenvalue is a bifurcation point. An immediate consequence of this lemma is the following: Corollary 2.9. Let {ujk }k1 , {tk }k1 , {vˆk }k1 , and v∗ be as in Lemma 2.7. The following hold: (i) Let Ω ⊂ R2 be a bounded domain with piecewise smooth boundary. Then ∂θ ujk k→∞ ∂θ v∗ − −−→
ujk − a
v∗
in C 1 ,
and
∂x ujk k→∞ ∂x v∗ − −−→
ujk − a
v∗
in C 1 .
(ii) Let Ω ⊂ RN be a bounded domain with smooth boundary. If any v ∈ ker Lλ∗ \{0} has a zero in Ω, then, for large k, ujk − a has a zero in Ω. Roughly speaking, Corollary 2.9 indicates that if u is near the bifurcation point, then the zero level set of ∂θ u is close to that of ∂θ v∗ , where v∗ is an eigenfunction of L. We will later use the nodal properties of ∂θ u and ∂x u to show that Proposition 2.1(2) does not occur for certain bifurcation problems. Proof of Corollary 2.9. (i) is a direct consequence of Lemma 2.7. We will prove (ii). Since v( ≡ 0) has a zero in Ω and v + gv (λ∗ , a)v = 0, it follows from the strong maximum principle that 0 is not the maximum or minimum value. Hence the sign of v changes in Ω. By the C 2 -convergence (2.5) we see that ujk − a = 0 has a zero in Ω provided that k is large. 2 We give a simple example (Corollary 2.10) for which Corollary 2.9 is applied. We use Corollary 2.9(ii) in order to show that the branch emanating from the first eigenvalue does not meet any other eigenvalue. Corollary 2.10. Let Ω ⊂ RN be a bounded domain with smooth boundary. Assume that guλ (λ∗ , a) = 0, gu (λ∗ , a) = 0, and gu (λ, a) = 0 for all λ ∈ R\{λ∗ }. Then there exists an unbounded continuum of non-trivial solutions to (2.4) in R × C 2 which meets (λ∗ , a). Proof. Let Lλ∗ := +gu (λ∗ , a). Since gu (λ∗ , a) = 0, 0 is the principal eigenvalue of Lλ∗ , hence the eigenvalue 0 is simple which is of odd (algebraic) multiplicity. Applying the Rabinowitz alternative (Proposition 2.1), we see that there is a continuum C, which consists of non-trivial solutions, emanating from (λ∗ , a). We will exclude (2) of Proposition 2.1 by contradiction. ˜ a) be a bifurcation point different from (λ∗ , a). Because of the assumption, 0 is not Let (λ, ˜ a). Therefore, for any v ∈ ker L ˜ \{0}, the sign of v the first eigenvalue of Lλ˜ := + gu (λ, λ
Y. Miyamoto / Journal of Functional Analysis 256 (2009) 747–776
757
˜ a). Then there is a subconchanges in Ω, hence v has a zero in Ω. Assume that C meets (λ, ˆ a)} such that (λ(0), u(λ(0))) = (λ∗ , a), tinuum B := {(λ(s), u(λ(s)))}0s1 of C ∪ {(λ∗ , a), (λ, (λ(1), u(λ(1))) = {(λˆ , a)}, and {u(λ(s))}0<s<1 does not contain the trivial solution. It follows from Corollary 2.9(ii) that, for some small s > 0, u(λ(s)) − a does not have a zero in Ω. On the other hand, u(λ(s)) − a has a zero in Ω for some s near 1, because of Corollary 2.9(ii). Since (λ, u(λ)) ∈ B is continuous in λ, there is (λ0 , u0 ) ∈ B such that u0 − a does not change its sign, u0 ≡ a in Ω, and u0 = a for some point on Ω. Let (g(λ0 , u0 ) − g(λ0 , a))/(u0 − a) if u0 = a, V := if u0 = a, gu (u0 , a) and let w := u0 − a. Then w satisfies w + V w = 0 in Ω, ∂ν w = 0 on ∂Ω. 0 is the maximum or minimum value of u0 . It follows from the strong maximum principal that w ≡ 0 in Ω, since Ω satisfies the interior sphere condition. Since u0 ≡ a, we obtain a contradiction. Hence C does not ˜ a), and C is unbounded. 2 meet (λ, Example 2.11. Let us consider u + λu − u2
in Ω,
∂ν u = 0 on ∂Ω.
Let g(λ, u) = λu − u2 . Then gλu (0, 0) = 1, gu (0, 0) = 0, and gu (λ, 0) = λ. Therefore the assumptions in Corollary 2.10 are satisfied. Hence there is an unbounded branch emanating from (λ, u) = (0, 0), which is u ≡ λ. A similar result holds for a Dirichlet problem. See [33, Theorem 2.12] for a problem on a domain satisfying the interior sphere condition. Even if the interior sphere condition is not satisfied, there is a similar result for a Dirichlet problem [17, Corollary 1.5] under certain symmetry assumptions on the nonlinearity and the domain. Proof of Lemma 2.7. The proof is almost the same as one of [8, Lemma 1.12]. However, we have to modify that proof, because the multiplicity of the eigenvalue may not be 1. Without loss of generality, we can assume that a = 0. Then g(λ, 0) = 0 for all λ ∈ R. Let {vj }j 1 ⊂ ker L and {wj }j 1 ⊂ (ker L)⊥ be sequences of functions such that uj = vj + wj . We consider the sequence vj / vj . Since dim ker L < ∞, there are a subsequence {vjk }k1 and v∗ ∈ ker L\{0} such that vjk / vjk → v∗ (k → ∞) and v∗ = 1. Note that v∗ ≡ 0. Let tk := vjk , v˜jk := (vjk / vjk − v∗ ). Then v˜jk → 0 (k → ∞), and we can rewrite u as u = tk v∗ + tk v˜jk + wjk . It is enough to show that
tk v˜jk + wjk C 2+γ = o |tk | as k → ∞.
(2.6)
Hereafter, we omit the suffixes jk and k for simplicity. Let G(η, u) = u + g(λ∗ + η, u) : R × C 2+γ → C γ . Then G(η, 0) = 0 for all η ∈ R. Here we assume that there is c > 0 such that
(2.7) c |tη| + w Gu (0, 0)[t v˜ + w] + ηGηu (0, 0)[tv∗ ] .
758
Y. Miyamoto / Journal of Functional Analysis 256 (2009) 747–776
We prove (2.7) later. Using (2.7), we prove (2.6). We have 0 = G(η, tv∗ + t v˜ + w) = G(η, tv∗ + t v˜ + w) − G(η, tv∗ ) + G(η, tv∗ ) − G(η, 0) = G(η, tv∗ + t v˜ + w) − G(η, tv∗ ) − Gu (η, tv∗ )[t v˜ + w] + Gu (η, tv∗ )[t v˜ + w] − Gu (0, 0)[t v˜ + w] + G(η, tv∗ ) − G(η, 0) − Gu (η, 0)[tv∗ ] + Gu (η, 0)[tv∗ ] − ηGηu (0, 0)[tv∗ ] + Gu (0, 0)[t v˜ + w] + ηGηu (0, 0)[tv∗ ] =: I1 + I2 + I3 + I4 + I5 . On the other hand, we have
I1 = o t v˜ + w ,
I2 = o t v˜ + w ,
I3 = o |t| ,
and I4 = o |tη| .
Using these estimate and (2.7), we have
c |tη| + w Gu (0, 0)[t v˜ + w] + ηGηu (0, 0)[tv∗ ] = I5 I1 + I2 + I3 + I4 = o t v˜ + w + o |t| + o |tη| . When |tη| is small, there is c > 0 such that c (|tη| + w ) o( t v˜ + w ) + o(|t|). When |t|, |η|, w , and v ˜ are small, we have c t v˜ + w + c |tη| c t v ˜ + c w + c |tη| ˜ + o t v˜ + w + o |t| . c t v ˜ = o(|t|), Since c t v c + o(1) t v˜ + w c + o(1) t v˜ + w + c |tη| o |t| , which proves (2.6). We will prove (2.7) which is left. As used in [8, (1.9)], we also use the mapping H (t, η, v, ˜ z) :=
t −1 G(η, t (v∗ + v˜ + z)) Gu (η, 0)[v∗ + v˜ + z]
if t = 0, if t = 0.
˜ 0) = Gu (0, 0)[v∗ + v] ˜ = 0. Note that Hη and Hz are continuous in (t, η, z) and that H (0, 0, v, (η, z) := H (0, η, 0, z) at (η, z) = (0, 0) is The Fréchet derivative of the map H |(η,z)=(0,0) )[η∗ , z∗ ] = η∗ Gηu (0, 0)[v∗ ] + Gu (0, 0)[z∗ ] (∂(η,z) H = αη∗ v∗ + Lz∗ , where α(= Gηu (0, 0)) is non-zero constant. Since v∗ ∈ ker L and L is invertible as an operator
Y. Miyamoto / Journal of Functional Analysis 256 (2009) 747–776
759
(ker L)⊥ ∩ C 2+γ → Ran L ∩ C γ , we see that the operator |(η,z)=(0,0) : R × (ker L)⊥ ∩ C 2+γ → spanv∗ × Ran L ∩ C γ ∂(η,z) H is invertible, hence the inverse is bounded. Therefore there is c > 0 such that
c |η| + z C 2+γ ηGηu (0, 0)[v∗ ] + Gu (0, 0)[z] .
(2.8)
Since Gu (0, 0)[t v] ˜ = L[t v] ˜ = 0, Gu (0, 0)[z] = Gu (0, 0)[t v˜ + z]. Substituting this equality into (2.8) and replacing η by tη, we obtain (2.7). We prove (2.5). First, we will show that 1 o(1) 1 (2.9) u − a − |t| v |t| v . ∗ ∗ Since u − a = tv∗ + v, ˆ we have
u − a = tv∗ + v ˆ |t| v∗ + v ˆ = |t| v∗ 1 + o(1) .
On the other hand, since tv∗ = u − a − v ˆ u − a + v , ˆ we have
u − a |t| v∗ − v ˆ = |t| v∗ 1 − o(1) . Therefore, 1 1 1 . (1 + o(1))|t| v∗ u − a (1 − o(1))|t| v∗ Subtracting these inequalities from 1/(|t| v∗ ), we have o(1) 1 1 o(1) − . (1 + o(1))|t| v∗ u − a |t| v∗ (1 − o(1))|t| v∗ We obtain (2.9). Second, we will show that
u−a v∗
|t| v − v = o(1). ∗ ∗
Dividing u − a = tv∗ + vˆ by |t| v∗ , we have (2.10). Using (2.9) and (2.10), we have
u−a
u−a v∗ u−a u−a v∗
u − a − v = u − a − |t| v + |t| v − v ∗ ∗ ∗ ∗
1
1
u − a − v∗ u − a + −
u − a |t| v∗ |t| v∗ v∗
|t|(1 + o(1))o(1) + o(1) = o(1). |t| v∗
The proof of Lemma 2.7 is complete.
2
(2.10)
760
Y. Miyamoto / Journal of Functional Analysis 256 (2009) 747–776
3. Bifurcation from the second eigenvalue Let Xn := C 2+γ (Dn ) (0 < γ < 1, n 0). As stated in Subsection 2.2, if u ∈ Xn (n 1) is a solution to (BPDn ), then u, ˜ which is defined by (2.3), becomes a solution to (BPD ). We work on X1 mainly. We consider the bifurcation problem (BPDn ) (n 1). The candidate of the bifurcation points of (BPDn ) is a pair (λ, a) such that 0 is an eigenvalue of ϕ + λf (a)ϕ = κϕ
in Dn ,
∂ν ϕ = 0
on ∂Dn .
(EVPDn )
Since f (a) = 1, λ = μj (j 1) is a candidate, where μj is defined in Proposition 2.3. (n)
(n)
(1)
First, we prove the existence of an unbounded branch emanating from (μ1 , a). In this section, let Γ1 := {(cos θ, sin θ ); 0 < θ < π}, Γ2 := {(x, 0); −1 < x < 1}, O = (0, 0), P := (1, 0), and Q := (−1, 0). (0) Theorem 3.1. There is an unbounded continuum of (BPD ), C1 , emanating from (μ1 , a) and consisting of non-radially symmetric solutions such that, for any (λ, u) ∈ C1 , u is symmetric with respect to {y = 0},
−uθ > 0 in D1 ∪ Γ1 ,
and ux > 0 in D1 \{P , Q}.
(3.1)
Hence P and Q are the maximum and minimum points of u in D, respectively. (1)
(0)
(1)
2 = μ . Since μ We consider (BPD1 ). Then μ1 = p1,1 1 1 is a simple eigenvalue of (EVPLD1 ), (1)
0 is a simple eigenvalue of (EVPDn ) when λ = μ1 . Using the Rabinowitz alternative (Propo(1) sition 2.1), we see that there exists a continuum C1 ⊂ R × X1 emanating from (μ1 , a) and consisting of non-trivial solutions of (BPD1 ). Next, we will show by contradiction that C1 is unbounded. Assume that C1 is not unbounded. (1) Proposition 2.1(2) should occur, i.e., C1 meets (μn , a) for n 2. Thus it is enough to show (1) that C1 does not meet (μn , a) for n 2. Let B := {(λ(s), u(s))}0s1 be a subcontinuum (1) (1) (1) (1) of C1 ∪ {(μ1 , a), (μn , a)} such that (λ(0), u(0)) = (μ1 , a), (λ(1), u(1)) = (μn , a), and {u(s)}0<s<1 does not contain the trivial solution. Moreover, by Remark 3.7 below we see that the branch emanating from a non-principal eigenvalue cannot touch a constant solution except the trivial one. We can assume that {u(s)}0<s<1 does not contain a constant solution. Before proving Theorem 3.1, we need two lemmas. Lemma 3.2. Let B := {(λ(s), u(s))}0s1 be as above. Then the following hold: (i) There is s0 > 0 such that −∂θ u(s0 ) > 0
in D1 ∪ Γ1 ,
and ∂θ u(s0 ) = 0 on Γ2 ∪ {P , Q}.
(ii) If, for some s1 ∈ (s0 , 1), ∂θ u(s1 ) has a zero in D1 , then there is s2 ∈ (s0 , s1 ] such that ∂θ u(s2 ) ≡ 0 in D1 .
Y. Miyamoto / Journal of Functional Analysis 256 (2009) 747–776
761
(1)
Proof. We will prove (i). Let ϕ(:= φ1,1 ) := J1 (p1,1 r) cos θ be an eigenfunction corresponding (1)
to μ1 , and let ψ := ∂ϕ. We have ψ = −J1 (p1,1 r) sin θ < 0, 1 ∂y ψ = −p1,1 J1 (p1,1 r) sin2 θ − J1 (p1,1 r) cos2 θ. r
(3.2)
We have 1 ∂y ψ = − J1 (p1,1 r) r
on Γ2 ∪ {P , Q}.
Since J1 (r) = r/2 + O(r 3 ), we see that −J1 (p1,1 r)/r < −δ < 0 on r ∈ [0, 1]. Hence ∂y ψ < −δ < 0
on Γ2 ∪ {P , Q}.
(3.3)
We also have ψ = 0 on Γ2 ∪ {P , Q}
and ∂θ u = 0 on Γ2 ∪ {P , Q}.
(3.4) (1)
Using (3.2)–(3.4), and the C 1 -convergence in Corollary 2.9(i) with simplicity of μ1 , we see that (i) holds. We will prove (ii). Let s0 ∈ (0, 1) be as in (i). Suppose that there is s1 ∈ (s0 , 1) such that ∂θ u(s1 ) has a zero in D1 . Because of (i) and the continuous dependence of u(s) on s, there is s2 ∈ (s0 , s1 ] such that −∂θ u(s2 ) 0, and one of the following occurs: (1) (2) (3) (4)
∂θ u(s2 ) = 0 at p0 ∈ D1 , ∂θ u(s2 ) = 0 at p0 ∈ Γ1 , ∂y ∂θ u(s2 ) = 0 at p0 ∈ Γ2 , ∂y ∂θ u(s2 ) = 0 at P or Q.
Assume that (1) occurs. Since p0 is a maximum point of ∂θ u in D1 and ∂θ u satisfies ∂θ u + f (u)∂θ u = 0, it follows from the maximum principle that ∂θ u ≡ 0 in D1 . If (2) occurs, then p0 is a maximum point. Since the interior sphere condition is satisfied at p0 , it follows from Hopf’s boundary point lemma that ∂ν ∂θ u = 0 at p0 provided that ∂θ u ≡ 0. It contradicts that ∂θ u satisfies the Neumann boundary condition at p0 . Thus ∂θ u ≡ 0. If (3) occurs, then ∂θ u satisfies the Neumann boundary condition at p0 . Since p0 is a maximum point of ∂θ u, it follows from Hopf’s boundary point lemma that ∂y ∂θ u = 0 provided that ∂θ u ≡ 0. It means that ∂θ u ≡ 0. Assume that (4) occurs. Then ∂θ u˜ is a solution to ∂θ u˜ + λf (u)∂ ˜ θ u˜ = 0 in D,
∂ν ∂θ u˜ = 0 on ∂D,
where u˜ is defined by (2.3). Since ∂θ u = ∂x ∂θ u = 0 at P (or Q), P (respectively Q) is a degenerate zero of order at least 2, because ∂y ∂θ u = 0 at P (respectively Q). Since {y = 0} ⊂ {∂θ u˜ = 0}, the order is 3 or larger (Proposition 2.4(iii)). Therefore the sign of ∂θ u changes in D1 . We obtain a contradiction. It follows that ∂θ u ≡ 0. All the cases are verified. The proof is complete. 2
762
Y. Miyamoto / Journal of Functional Analysis 256 (2009) 747–776
Lemma 3.3. Let B := {(λ(s), u(s))}0s1 be as above. Suppose that, for λ ∈ (0, 1), u(s) is not radially symmetric. Then the following hold: (i) There is s0 ∈ (0, 1) such that ∂x u(s0 ) > 0 on D1 \{P , Q}. (ii) If, for some s1 ∈ (s0 , 1), ∂x u(s1 ) has a zero in D1 , then there is s2 ∈ (s0 , s1 ] such that ∂x u(s2 ) ≡ 0 in D1 . (1)
Proof. We will prove (i). Let ϕ(:= φ1,1 ) := J1 (p1,1 r) cos θ be an eigenfunction corresponding (1)
to μ1 , and let ψ := ∂x ϕ. We have 1 ψ = p1,1 J1 (p1,1 r) cos2 θ + J1 (p1,1 r) sin2 θ > 0 r
in D1 \{P , Q}.
(3.5)
We easily see that and ∂x u = 0 at P and Q, 2 2 ∂x ψ = p1,1 J1 (p1,1 ) < 0 at P . J1 (p1,1 ) = 1 − p1,1 ψ =0
at P and Q
(3.6) (3.7)
Here we use J1 (r) = −J1 (r)/r − (1 − 1/r 2 )J1 (r). By a similar calculation we see that ∂x ψ > 0 at Q. Using (3.5)–(3.7), and the C 1 -convergence in Corollary 2.9(i), we see that (i) holds. We will prove (ii). Let s0 ∈ (0, 1) be as in (i). Suppose that there is s1 ∈ (s0 , 1) such that ∂x u(s1 ) has a zero in D1 . Because of (i) and the continuous dependence of u(s) on s, there is s2 ∈ (s0 , s1 ] such that ∂x u(s2 ) 0, ∂x u(s2 ) ≡ 0 in D1 , and one of the following occurs: (1) (2) (3) (4)
∂x u(s2 ) = 0 at p0 ∈ D1 , ∂x u(s2 ) = 0 at p0 ∈ Γ1 , ∂x u(s2 ) = 0 at p0 ∈ Γ2 , ∂x ∂x u(s2 ) = 0 at P or Q.
If (1) occurs, then we can obtain a contradiction by an argument similar to one of (1) in the proof of Lemma 3.2. We omit the details. If (2) occurs, then ∂y u = 0 at p0 , because ∂x u = ∂ν u = 0 at p0 . Hence ∂θ u = 0 at p0 . We see by Proposition 2.4 that there is a nodal curve of ∂θ u emanating from p0 and that the sign of ∂θ u changes in D1 . This is a contradiction. If (3) occurs, then ∂x u = 0 at p0 ∈ Γ2 . By Proposition 2.4 we see that there is a nodal curve of ∂x u emanating from p0 and that the sign of ∂x u changes in D1 , which indicates that ∂x u ≡ 0 in D1 . If (4) occurs, then P (or Q) is a minimum point of ∂x u˜ which satisfies ˜ x u˜ = 0 in D. ∂x u˜ + λf (u)∂ Since ∂D satisfies the interior sphere condition at P (respectively Q), it follows from Hopf’s boundary point lemma that ∂ν ∂x u˜ = 0 at P (respectively Q) which contradicts that ∂x ∂x u = 0 at P (respectively Q). We have checked all the cases. The proof is complete. 2 (1)
Proof of Theorem 3.1. Let B = {(λ(s), u(s))}0s1 be as above, and let A := B\{(μ1 , a), (1) (μn , a)}. We will show by contradiction that B does not meet any other eigenvalue. We divide the possibilities into two cases.
Y. Miyamoto / Journal of Functional Analysis 256 (2009) 747–776
763
Case 1: A does not have a radially symmetric solution. Because of (i) of Lemma 3.2, for some s0 > 0, ∂θ u(s0 ) = 0
on Γ2 ∪ {P , Q}
and −∂θ u(s0 ) > 0 in D1 ∪ Γ1 .
(3.8)
Let ϕ be an eigenfunction corresponding to μ1 . Let L1 := + μn f (a) (n 2). Since ψ(∈ ker L1 ) satisfies (1)
(1)
ψ + μ(1) n f (a)ψ = 0 in D1 ,
∂ν ψ = 0 on ∂D1 ,
ψθ satisfies ∂θ ψ + μ(1) n f (a)∂θ ψ = 0 in D1 ,
∂ν ∂θ ψ = 0
on Γ1 ,
and ∂θ ψ = 0 on Γ2 .
Multiplying this equation by ∂θ ϕ and integrating it over D1 , we have μ(1) n f (a)∂θ ψ, ∂θ ϕ = −
∂θ ψ∂θ ϕ dx
D1
∂ν ∂θ ψ∂θ ϕ dS +
=− ∂D1
∇∂θ ψ · ∇∂θ ϕ dx
D1
∂θ ∂θ ψ∂θ ϕ dS −
=− ∂D1
∂θ ψ∂ν ∂θ ϕ dS −
∂D1
∂ν ∂θ ψ∂θ ϕ dS −
=−
∂D1
∂θ ψ∂θ ϕ dx
D1
∂θ ψ∂ν ∂θ ϕ dS + μ1 f (a)∂θ ψ, ∂θ ϕ, (1)
∂D1
where we use Green’s formula. Because ∂ν ∂θ ϕ = ∂ν ∂θ ψ = 0 on Γ1 and ∂θ ϕ = ∂θ ψ = 0 on Γ2 , (1) (1) ∂D1 (∂ν ∂θ ψ∂θ ϕ + ∂θ ψ∂ν ∂θ ϕ) dS = 0. Since μ1 = μn (n 2), we have ∂θ ψ, ∂θ ϕ = 0.
(3.9)
From the assumption of the contradiction we assume that B meets (μ(1) n , a) for n 2. It follows from Lemma 2.7 that there are ψ ∈ ker L1 \{0} and a sequence {uj }j 0 such that uj − a j →∞ ψ −−−→
uj − a
ψ
in C 2 .
We divide Case 1 into two cases. Case 1-1: ψ is not radially symmetric. Then ∂θ ψ ≡ 0. Because of (3.9), ∂θ ψ has a zero in D1 . Moreover by Proposition 2.4(ii) we see that the sign of ∂θ ψ changes in D1 . Therefore, for large j , ∂θ uj has a zero in D1 . It follows from Lemma 3.2(ii) that ∂θ uj ≡ 0 in D1 , which is a contradiction. Case 1-2: ψ is radially symmetric.
764
Y. Miyamoto / Journal of Functional Analysis 256 (2009) 747–776
Because of Lemma 3.3(i), for some s0 > 0, ∂x u(s0 ) > 0 in D1 \{P , Q}
and ∂x u(s0 ) = 0 at P and Q.
(3.10)
Since ψ ≡ 0 is radially symmetric, ∂x ψ ≡ 0 and ∂x ψ = 0 on {x = 0} ∩ D1 . Proposition 2.4 says that the sign of ∂x ψ changes in D1 . It follows from Corollary 2.9(i) that there is (λ(s), u) ∈ B near (μ(1) n , a) such that ∂x u has a zero in D1 . By Lemma 3.3(ii) we see that ∂x u(s2 ) ≡ 0 in D1 for some s2 ∈ (0, 1) near 1. u(s2 ) satisfies the Neumann boundary condition, hence u(s2 ) is constant. This is a contradiction. Case 2: A contains a radially symmetric solution. Since the set {s ∈ (0, 1); u(s) is not radially symmetric} is open, there is s0 ∈ (0, 1) such that u(s) (0 < s < s0 ) is not radially symmetric and u(s0 ) is radially symmetric. Let v := u(s0 ). Then u → v in C 2 (s ↑ s0 ). Since v is radially symmetric, we can argue similarly in Case 1-2. Specifically, we assume that B meets a radially symmetric solution. Since ∂x v ≡ 0 and ∂x v has a zero in D1 , if u is close to v, u should be constant (Lemma 3.3). We obtain a contradiction. Case 2 also does not occur. We have checked all the cases. Then the branch C1 is unbounded. It is clear from the proof that the property (3.1) holds. 2 (0)
We study the shape of solutions to (BPD ) near the bifurcation point (μ1 , a). 2 ) in Lemma 3.4. (See [24, Lemma 4.1].) Let u be a solution of (BPD ). If λf (u) < μ2 (= p2,1 the range of u, then u is symmetric with respect to a line containing the origin. (0)
This lemma is already known. However we prove the lemma for the readers’ convenience. Proof. Let u be a solution to (BPD ), and let S be the reflection with respect to the x-axis, i.e., (x, y) → (x, −y). We define u(θ) (x, y), u¯ (θ) (x, y) by u(θ) (x, y) := (Rθ u)(x, y),
u¯ (θ) (x, y) = (SRθ u)(x, y),
namely u(θ) (x, y) := u(x cos θ + y sin θ, −x sin θ + y cos θ ) and u¯ (θ) := u(x cos θ − y sin θ, −x sin θ − y cos θ ). We define w (θ) by w (θ) := u(θ) − u¯ (θ) .
(3.11)
Then w (θ) (x, −y) = −w (θ) (x, y), hence the set {w (θ) = 0} is symmetric with respect to the x-axis. Moreover w (θ) satisfies that w (θ) + V w (θ) = 0 in D,
∂ν w (θ) = 0 on ∂D,
where V (x, y) :=
λf (u(θ) (x,y))−λf (u¯ (θ) (x,y)) u(θ) (x,y)−u¯ (θ) (x,y) λf (u(θ) (x, y))
if u(θ) (x, y) = u¯ (θ) (x, y), if u(θ) (x, y) = u¯ (θ) (x, y).
Y. Miyamoto / Journal of Functional Analysis 256 (2009) 747–776
765
Here (0, w (θ) ) is an eigenpair of the eigenvalue problem ϕ + V ϕ = κϕ
in D,
∂ν ϕ = 0 on ∂D,
(3.12)
provided that w (θ) ≡ 0 in D. It follows from (3.12) that, for any θ ∈ [0, 2π), w (θ) = 0 on D ∩ {y = 0}. In particular, w (θ) (0, 0) = wx(θ) (0, 0) = 0. Since wy(θ) (0, 0) = 2(ux (0, 0) sin θ + (θ ) uy (0, 0) cos θ ), there exists θ0 ∈ [0, 2π) such that wy 0 (0, 0) = 0. It follows from Proposition 2.4(ii) that either w (θ0 ) ≡ 0 in D or {w (θ) = 0} has at least four branches at (0, 0). We will show by contradiction that w (θ0 ) ≡ 0 in D. If we prove this, then u(θ0 ) (x, y) ≡ (θ 0 u¯ ) (x, y) in D. Therefore Rθ u is symmetric with respect to the x-axis, hence the proof is complete. Suppose the contrary. Since {w (θ0 ) = 0} is symmetric with respect to the x-axis, {w (θ) = 0} has at least one branch in each side of D\{y = 0} and the branch should divide each side into at least two subdomains. Thus w (θ0 ) has at least four nodal domains. By Courant’s nodal domain theorem we see that there is an integer n0 3 such that κn0 = 0, where {κj }∞ j =0 is the set of the eigenvalues of (3.12) counting multiplicities. In particular, the forth eigenvalue is 0 is larger. On the other hand, the forth eigenvalue of the eigenvalue problem (0)
ψ + μ2 ψ = ζ ψ
in D,
∂ν ψ = 0 on ∂D
(3.13)
(0)
is 0. Because of the assumption of the lemma, V < μ2 . It follows from the comparison principle of the eigenvalues of linear elliptic partial differential operators that the forth eigenvalue of (3.12) should be strictly less than the forth eigenvalue of (3.13). Let {ζn }∞ n=0 be the set of the eigenvalues of (3.13). When we count multiplicities of the eigenvalues of (3.13), we have that (0) (0) (0) ζ0 = μ2 , ζ1 = μ2 − μ1 , and ζ2 = ζ3 = 0. Hence the forth eigenvalue of (3.13) is 0. This is a contradiction, because the forth eigenvalue of (3.12) is 0 or larger. 2 The second result in this section is the symmetry of solutions near the bifurcation point (0) (μ1 , a). Theorem 3.5. Let C be a continuum consisting of non-trivial solutions to (BPD ) and emanating (0) from (μ(0) 1 , a). Then there is a neighborhood U0 ⊂ R × X of (μ1 , a) such that if (λ, u) ∈ C ∩ U0 , then u is symmetric with respect to a line containing the origin. Moreover if f (a) = 0, then C (0) is unique up to rotation near (μ1 , a). Specifically, there is a neighborhood U1 ⊂ R × X of (0) (μ1 , a) such that if (λ0 , u), (λ0 , v) ∈ C ∩ U1 , then u = Rθ v for some θ ∈ [0, 2π). Proof. We can choose U0 such that, for any (λ, u) ∈ U0 , λf (u) < μ2 , since μ1 f (a) = (0) μ(0) 1 < μ2 . The assertion of the first part follows from Lemma 3.4. We will prove the uniqueness of the branch near the bifurcation point. (0) Let (λ, u) ∈ C be a pair near the bifurcation point (μ1 , a). Then u is symmetric, hence u (1) can be considered as a solution to (BPD1 ). We study the solutions of (BPD1 ) near (μ1 , a). (1) (0) Note that μ1 (= μ1 ) is a simple eigenvalue. We can check the assumptions of Proposition 2.2 [8, Theorem 1] from the assumptions of this theorem. Applying Proposition 2.2, we see that there (0)
(0)
766
Y. Miyamoto / Journal of Functional Analysis 256 (2009) 747–776 (1)
is exactly one curve C1 ⊂ R × X1 consisting of non-trivial solutions to (BPD1 ) near (μ1 , a). Moreover it is well known that when ϕ, f (a)[ϕ, ϕ] = 0, the curve can be described as u=±
(1) 1/2
λ − μ1 b
(1) ϕ + O λ − μ1 ,
(3.14)
(1)
where ϕ is a normalized eigenfunction corresponding to μ1 , i.e.,
ϕ = J1 (p1,1 r) cos θ/ J1 (p1,1 r) cos θ L2 and b = −μ1 ϕ, f (a)[ϕ, ϕ, ϕ]/ϕ, f (a)[ϕ]( = 0). Hence, for each λ = μ1 near μ1 , there are exactly two solutions near u ≡ a. Moreover, u(−x, y) = u(x, y), because of (3.14). Therefore u(x, y) and u(−x, y) are different solutions, Thus the two solutions obtained by Proposition 2.2 should be u(x, y) and u(−x, y), which indicates that a solution to (BPD1 ) near (μ(1) , a) is unique (0) up to the reflection (x, y) → (−x, y). Hence a solution of (BPD ) near (μ1 , a) should be unique up to rotation. 2 (1)
(1)
(1)
The third result is the direction of unbounded branches. The next result holds for general domains. Theorem 3.6. Let Ω ⊂ RN be a bounded domain with smooth boundary, and let {μj (Ω)}j 0 denote the set of the eigenvalues of N . Suppose that (A1) holds. If (BPΩ ) has an unbounded continuum C emanating from (μn (Ω), a) (n 1), then C is unbounded in the positive direction of λ. Hence the branch of (BPD ) obtained in Theorem 3.1 is unbounded in the positive direction of λ. Proof. First, we recall the well-known fact that (0) the solution u of (BPD ) is trivial if sup λf u(p) < μ1 .
(3.15)
p∈Ω
Since f (a) = 1, if λ > 0 is small, (BPΩ ) has no non-trivial solution, hence there is no bifurcation point for small λ > 0. For this property, see [7] and [12]. Second, we will show by contradiction that, for any (λ, u) ∈ C, there is p ∈ Ω such that u(p) = a.
(3.16)
Suppose the contrary. Then there is (λ1 , u1 ) ∈ C such that minΩ u1 > a or maxΩ u1 < a. Moreover there are an integer n 1 and a subcontinuum B := {(λ(s), u(s))}0s1 of C ∪ {(μ1 (Ω), a)} such that (λ(0), u(0)) = (μ1 (Ω), a), (λ(1), u(1)) = (λ1 , u1 ), and B\{(μ1 (Ω), a), (λ1 , u1 )} does not have the trivial solution. Note that B cannot meet (μ0 (Ω), a). Since μ1 (Ω) is not the principal eigenvalue, we see by Corollary 2.9(ii) that there is small s > 0 such that u(s) − a changes its sign. Because of the continuous dependence of u(s) on s, there is s1 ∈ (0, 1) such that minΩ u(s1 ) = a or maxΩ u(s1 ) = a. Since Ω satisfies the interior sphere condition and 0
Y. Miyamoto / Journal of Functional Analysis 256 (2009) 747–776
767
is the maximum or minimum value of w := u(s1 ) − a, we apply the strong maximum principle to w which satisfies w + V w = 0
in Ω,
∂ν w = 0
on ∂Ω,
where V :=
λf (u(s1 ))−λf (a) u(s1 )−a λf (a)
if u(s1 ) = a, if u(s1 ) = a.
The solution w should be identically equal to 0, hence u(λ(s1 )) ≡ a, which is a contradiction. Third, we show by the contradiction that, for any (λ, u) ∈ C, (3.17)
a− < u < a+ .
Suppose the contrary. There is (λ2 , u2 ) ∈ C such that u does not satisfy (3.17). Using the same argument as above, we see that if there is subcontinuum B0 meeting (λ2 , u2 ), then B0 should contain u ≡ a+ or u ≡ a− which contradicts to (3.16). Combining (3.15) and (3.17), we see that the branch C should be unbounded in the positive direction of λ. 2 Remark 3.7. The bound (3.17) does not immediately follow from the existence of an upper solution u ≡ a+ and a lower solution u ≡ a− . Let us consider the equation λ−1 u + u u − (1 − u) = 0 in Ω, 2−λ
∂ν u = 0
on ∂Ω,
where Ω ⊂ RN is a bounded domain with smooth boundary. We easily see that u ≡ 0, (λ − 1)/(2 − λ), 1 are solutions. The solution uλ := (λ − 1)/(2 − λ) bifurcates from u ≡ 0 at λ = 1, and 0 < uλ < 1 if 0 < λ < 3/2. However, uλ → +∞ as λ ↑ 2. This example and the proof of Theorem 3.6 indicate that the branches emanating from a non-principal eigenvalue cannot touch solutions above or below the trivial solution and that the branches emanating from the principal eigenvalue can go through those solutions. We consider the case where (A2) holds. Ni and Takagi [29] obtain a priori estimates of solutions to (BPΩ ) with (A2). Let (λ, u) be a non-negative solution to (BPΩ ) with (A2). Then there is a positive increasing function β(p) such that u satisfies
u C γ (Ω) C max 1, λβ(p)
(3.18)
for some γ ∈ (0, 1) and C > 0 independent of u and λ. They show that if λ > 0 is small, then (BPΩ ) with (A2) has no non-trivial solution.
(3.19)
Using (3.18) and (3.19), they show that (BPRN ) with (A2) has branches emanating from the trivial branch {(λ, 1)} and that these branches are unbounded in the positive direction of λ. Here RN is defined in (1.6).
768
Y. Miyamoto / Journal of Functional Analysis 256 (2009) 747–776
When the domain is D, (BPD ) has an unbounded branch C, because of Theorem 3.1. Since C emanates from a non-principal eigenvalue, we see by the same argument in the proof of Theorem 3.6 that, for any (λ, u) ∈ C, u > 0 in D. Using (3.18) and (3.19), we obtain Corollary 3.8. Suppose that (A2) holds. Then the branch of (BPD ) obtained in Theorem 3.1 is unbounded in the positive direction of λ. Remark 3.9. Let us consider (1.1). Let uε be a positive solution to (1.1) such that uε satisfies (3.1). Moreover, we suppose that uε satisfies lim sup ε −1 Iε [uε ] < ∞,
(3.20)
ε↓0
where Iε [ϕ] := D
ε2 ϕ p+1 ϕ2 |∇ϕ|2 + − dx. 2 2 p+1
Then we can show that uε (ε ↓ 0) is the boundary one-spike layer, using the so-called blowup argument. However, proving (3.20) seems to be difficult, as Dancer [9] pointed out. Other methods to show that the branch C1 consists of boundary one-spike layers seem to be unknown when the spatial dimension is 2 or larger. 4. Bifurcation from the third eigenvalue We establish the existence of an unbounded branch emanating from the third eigenvalue. Our goal of this section is to prove (0) Theorem 4.1. There is an unbounded continuum of (BPD ), C2 , emanating from (μ2 , a) and consisting of non-radially symmetric solutions such that if (λ, u) ∈ C2 , then u is symmetric with respect to {x = 0} and {y = 0},
uθ > 0 in Rπ/2 D2 ∪ R3π/2 D2 ,
and uθ < 0 in D2 ∪ Rπ D2 ,
where Rθ is defined by (2.2). Combining Theorems 4.1 and 3.6, (3.18), and (3.19), we immediately obtain the following: Corollary 4.2. Suppose that (A1) or (A2) holds. Then the branch C2 of (BPD ) obtained in Theorem 4.1 is unbounded in the positive direction of λ. Hereafter we are devoted to prove Theorem 4.1. We consider (BPD2 ). We easily see that 2 = μ(2) and that μ(2) is a simple eigenvalue of (EVPL ). All the assumptions in = p2,1 D2 1 1 Proposition 2.1 are satisfied. Thus (BPD2 ) has a continuum C2 of non-trivial solutions emanat(2) ing from (μ1 , a). We show by contradiction that Proposition 2.1(2) does not occur. Suppose (2) the contrary. C2 meets (μn , a) (n 2). There is a subcontinuum B := {(λ(s), u(s))}0s1 (0) μ2
Y. Miyamoto / Journal of Functional Analysis 256 (2009) 747–776 (2)
(2)
(2)
769 (2)
of C2 ∪ {(μ1 , a), (μn , a)} such that (λ(0), u(0)) = (μ1 , a), (λ(1), u(1)) = (μn , a), and {u(s)}0<s<1 does not contain a trivial solution. In the proof of Theorem 4.1 we see that ∂θ u and ∂x u are not enough to exclude the possibility (2) (2) of the connection to any other eigenvalues. Specifically, ∂θ φ0,1 ≡ 0, and ∂x φ0,1 does not have 2 a zero in D2 . Hence we cannot exclude the possibility where C2 meets (p0,1 , a). Then we use ∂α u,
√ where ∂α := (−∂x + ∂y )/ 2.
(4.1)
(2)
∂α φ0,1 has a zero in D2 . Before proving Theorem 4.1, we need three lemmas. From now on, let Γ1 := {(cos θ, sin θ ); 0 < θ < π/2}, Γ2 := {(x, 0); 0 < x < 1}, Γ3 := {(0, y); 0 < y < 1}, O = (0, 0), P = (1, 0), and R = (0, 1). Lemma 4.3. Let B := {(λ(s), u(s))}0s1 be as above. Then the following hold: (i) There is s0 > 0 such that −∂θ u(s0 ) > 0 in D2 ∪ Γ1 ,
and ∂θ u(s0 ) = 0 on Γ2 ∪ Γ3 ∪ {O, P , R}.
(ii) If, for some s1 ∈ (s0 , 1), ∂θ u(s1 ) has a zero in D2 , then there is s2 ∈ (s0 , s1 ] such that ∂θ u(s2 ) ≡ 0 in D2 . ˜ Then ϕ satisfies Proof. We will prove (i). Let (λ, u) ∈ B, and let ϕ := ∂θ u. ϕ + λfu (u)ϕ = 0 in D2 ,
∂ν ϕ = 0 on ∂D2 , (2)
and {x = 0} ∪ {y = 0} ⊂ {ϕ = 0}. Suppose that λ(s) is close to μ1 . It follows from Corollary 2.9(i) that ϕ ≡ 0. 0 is an eigenvalue and ϕ is a corresponding eigenfunction provided that (2) 2 ) is the fourth eigenvalue of on D counting multiplicity and the ϕ ≡ 0. Since μ1 (= p2,1 N multiplicity of this eigenvalue is 2, the number of the nodal domains of ϕ should be 5 or smaller provided that λ(s) is near μ(2) 1 . If ϕ has a zero in D2 , then ϕ has at least 2 nodal domains in D2 , hence ϕ has at least 8 nodal domains in D. This is a contradiction. Thus when s0 > 0 is small, −∂θ u(s0 ) > 0 in D2 . (i) is proven. We will prove (ii). Let s0 be as in (i). Suppose that there is s1 ∈ (s0 , 1) such that ∂θ u(s1 ) has a zero in D2 . Because of (i), there is s2 ∈ (s0 , s1 ] such that −∂θ u(s2 ) 0 and one of the following occurs: (1) (2) (3) (4) (5)
∂θ u(s2 ) = 0 at p0 ∈ D2 , ∂θ u(s2 ) = 0 at p0 ∈ Γ1 , ∂ν ∂θ u(s2 ) = 0 at p0 ∈ Γ2 ∪ Γ3 , ∂θ ∂θ u(s2 ) = 0 at R or P , ∂θ u(s2 ) = o(r 2 ) near O. (Otherwise ∂θ u(s) = A(λ(s))xy + o(r 2 ) and A(λ(s)) < 0 if s is close to s2 . Thus ∂θ u(s) < 0 near O if s is close to s2 .)
If (1), (2), or (3) occurs, then we can see by a way similar to one used in the proof of Lemma 3.2 that ∂θ u(s2 ) ≡ 0 in D2 .
770
Y. Miyamoto / Journal of Functional Analysis 256 (2009) 747–776
We consider the case (4). Suppose that ∂θ ∂θ u(s2 ) = 0 at R. Let ϕ := ∂θ u(s ˜ 2 ). Then ϕ = ∂y ϕ = 0 at R. Since ∂θ ϕ = −∂x ϕ = 0 at R, R is a degenerate zero of ϕ order at least 2. Since ϕ satisfies the Neumann boundary condition and {x = 0} ⊂ {ϕ = 0}, we see by Proposition 2.4(iii) that the order of the degeneracy at R should be 3 or larger. Therefore there is a nodal curve of ϕ in D2 emanating from R and ϕ has a zero in D2 . Hence ∂θ u also has a zero in D2 . Since ∂θ u(λ(s2 )) 0, ∂θ u(s2 ) ≡ 0 in D2 . Otherwise the sign of ∂θ u(s2 ) changes in D2 , which is a contradiction. We consider the case (5). Since {x = 0} ∪ {y = 0} ⊂ {ϕ = 0} and ∂θ u = o(r 2 ) near O, the order of the degeneracy of ϕ at O should be 4 or larger. Therefore ϕ has a zero in D2 (Proposition 2.4(iii)). Since ∂θ u(s2 ) 0, ∂θ u(s2 ) ≡ 0 in D2 . We have checked all the cases. The proof is complete. 2 Lemma 4.4. Let B := {(λ(s), u(s))}0s1 be as above. Suppose that, for λ ∈ (0, 1), u(s) is not radially symmetric. Then the following hold: (i) There is s0 ∈ (0, 1) such that ∂x u(s0 ) > 0 in D2 ∪ Γ1 and ∂x u(s0 ) = 0 on Γ3 ∪ {O, P , R}. (ii) If, for some s1 ∈ (s0 , 1), ∂x u(s1 ) has a zero in D2 , then there is s2 ∈ (s0 , s1 ] such that ∂x u(s2 ) ≡ 0 in D2 . (2) Proof. Let ϕ(:= φ˜ 1,1 ) = J2 (p2,1 r) cos 2θ . We study the shape of ψ := ∂x ϕ near {x = 0}. We have
4 2 ∂x ψ = p2,1 J2 (p2,1 r) cos2 θ cos 2θ + J2 (p2,1 r) sin θ cos θ sin 2θ r p2,1 4 + J2 (p2,1 r) sin2 θ cos 2θ − 2 J (p2,1 r) sin θ cos θ sin 2θ r r 4 2 − 2 J2 (p2,1 r) sin θ cos 2θ. r Using −J2 (r) = −J1 (r) + 2J2 (r)/r, we have ∂x ψ|θ= π2 = −
p2,1 4 p2,1 2 J (p2,1 r) + 2 J2 (p2,1 r) = J1 (p2,1 r) + 2 J2 (p2,1 r). r r r r
Using J2 (r) =
r2 + O r4 , 8
J2 (r) =
r + O r3 , 4
J2 (r) =
1 + O r2 , 4
(4.2)
we see that ∂x ψ > δ > 0 on Γ3 ∪ {O, R}, where we use the fact that J1 (p2,1 r) > 0 for r ∈ [0, 1]. We have 2 2 ∂x ψ = p2,1 J2 (p2,1 ) < 0 at P , J (p2,1 ) = 4 − p2,1 where we use J2 (r) = −J2 (r)/r − (1 − 4/r 2 )J2 (r).
(4.3)
(4.4)
Y. Miyamoto / Journal of Functional Analysis 256 (2009) 747–776
771
On the other hand, we easily see that ∂x u = ψ = 0 on Γ3 ∪ {O, P , R}.
(4.5)
Next we will show that ψ >0
on D2 .
(4.6)
Suppose that ψ has a zero in D2 . Because of (4.3) and (4.5), there is no degenerate zero of u on Γ3 ∪ {O, P , R}. We easily see that ψ > 0 on Γ1 ∪ Γ2 . Therefore {ψ = 0} does not connect to the boundary and should have a loop. However, {ψ = 0} does not have a loop, because ψ + (0) μ2 ψ = 0 (see [20]). Thus (4.6) holds. Using (4.3)–(4.6) and the C 1 -convergence in Corollary 2.9(i), we see that (i) holds. We will prove (ii). Let s0 be as in (i). Suppose that there is s1 ∈ (s0 , 1) such that ∂x u(s1 ) has a zero in D2 . Because of (i), there is s2 ∈ (s0 , s1 ] such that ∂x u(s2 ) 0 in D2 and one of the following occurs: (1) (2) (3) (4) (5) (6)
∂x u(s2 ) = 0 at p0 ∈ D2 , ∂x u(s2 ) = 0 at p0 ∈ Γ1 , ∂x u(s2 ) = 0 at p0 ∈ Γ2 , ∂ν ∂x u(s2 ) = 0 at p0 ∈ Γ3 , ∂x ∂x u(s2 ) = 0 at R, ∂x ∂x u(s2 ) = 0 at O or P .
If (1) occurs, then it follows from the maximum principle that ∂x u ≡ 0 in D2 . Suppose that (2) occurs. Since ∂ν u = 0 at p0 , ∂x u = ∂y u = 0 at p0 . Hence ∂θ u = 0 at p0 . By (2) in the proof of Lemma 4.3 we see that ∂θ u ≡ 0 in D2 . It contradicts that u(s2 ) is not radially symmetric. If (3) occurs, then p0 is a (degenerate) zero of order at least 1. Since ∂x u satisfies the Neumann boundary condition on Γ2 , we see by Proposition 2.4(iii) that ∂x u has a zero in D2 . Since ∂x u 0, ∂x u ≡ 0 in D2 . If (4) occurs, then by Hopf’s boundary point lemma we see that ∂x u ≡ 0 in D2 . If (5) occurs, then by (4) in the proof of Lemma 3.2 we see that ∂θ u ≡ 0, which is a contradiction. Suppose that (6) occurs. The proofs of the cases for O and P are similar. We consider only the case for O. ϕ(:= ∂x u(s ˜ 2 )) is a solution of ϕ + λf (u)ϕ = 0 in D2 ∪ R−π/2 D2 ∪ Γ2 . Since ϕ = 0 at O and O is a minimum value of ϕ, we see by Hopf’s boundary point lemma that ϕ ≡ 0 in D2 . We have checked all the cases. The proof is complete. 2 Lemma 4.5. Let ∂α be as defined by (4.1). Let B := {(λ(s), u(s))}0s1 be as above. Suppose that {u(s)}0<s<1 does not contain a radially symmetric. Then the following hold: (i) There is s0 > 0 such that ∂α u(s0 ) > 0 in D2 \{O, P , R}
and ∂α u(s0 ) = 0 at O, P , and R.
772
Y. Miyamoto / Journal of Functional Analysis 256 (2009) 747–776
(ii) If, for some s1 ∈ (s0 , 1), ∂α u(s1 ) has a zero in D2 , then there is s2 ∈ (s0 , s1 ] such that ∂α u(s2 ) ≡ 0 in D2 . Proof. We change the coordinates to make notation simple. Specifically, we rotate (EPD2 ) by −π/4 and consider 2 , uˆ + λf (u) ˆ = 0 in D
2 , ∂ν uˆ = 0 on D
2 := R−π/4 D2 and uˆ = R−π/4 u. The second eigenfunction of N on D 2 is where D := −J2 (p2,1 r) sin 2θ . ∂α is transformed to ∂y . Let ϕ := −J2 (p2,1 r) sin 2θ , P := R−π/4 P , R R−π/4 R. We will prove (i). To begin with, we study the shape of ψ := ∂y ϕ. We have 2 ψ = −p2,1 J2 (p2,1 r) sin θ sin 2θ − J2 (p2,1 r) cos θ cos 2θ. r We see by direct calculation that ψ <0
, R} and ψ = 0 at O, P , and R. 2 \{O, P in D
(4.7)
, and R. We have We will obtain the derivative of ∂y ϕ at O, P 2 J2 (p2,1 r) sin θ sin 2θ − ∂r ψ = −p2,1
2p2,1 J2 (p2,1 r) cos θ cos 2θ r
2 J2 (p2,1 r) cos θ cos 2θ, r2 ∂θ ψ = −p2,1 J2 (p2,1 r)(cos θ sin 2θ + 2 sin θ cos 2θ ) +
2 + J2 (p2,1 r)(sin θ cos 2θ + 2 cos θ sin 2θ ). r Using the asymptotics (4.2), we have ⎫ 2 p2,1 ⎪ ⎪ ⎪ ∂r ψ|(r,θ)=(0, π4 ) = ∂r ψ|(r,θ)=(0,− π4 ) = − √ , ⎪ ⎪ ⎪ 4 2 ⎪ ⎪ 2 ⎬ p2,1 4 , , ∂r ψ = − √ J2 (p2,1 )(> 0) at P ∂θ ψ = − √ J2 (p2,1 )(< 0) at P ⎪ 2 2 ⎪ ⎪ ⎪ ⎪ 2 ⎪ p2,1 4 ⎪ ⎪ ⎭ ∂θ ψ = √ J2 (p2,1 )(> 0) at R. ∂r ψ = − √ J2 (p2,1 )(> 0) at R, 2 2
(4.8)
2 )J (r ) > 0. We see Here we use J (p2,1 ) = (1 − 4/p2,1 2 2,1
, and R. ∂y uˆ = 0 at O, P Using (4.7)–(4.9), and the C 1 -convergence in Corollary 2.9(i), we see that (i) holds.
(4.9)
Y. Miyamoto / Journal of Functional Analysis 256 (2009) 747–776
773
We will prove (ii). Let s0 be as in (i). Suppose that there is s1 ∈ (s0 , 1) such that ∂u u(s ˆ 1 ) has ˆ 2 ) 0 in D2 and one of the following a zero in D2 . Then there is s2 ∈ (s0 , s1 ] such that ∂y u(s holds: (1) (2) (3) (4-1) (4-2) (4-3) (5-1) (5-2) (5-3) (6-1) (6-2) (6-3)
2 , ∂y u(s ˆ 2 ) = 0 at pˆ 0 ∈ D ∂y u(s ˆ 2 ) = 0 at pˆ 0 ∈ Γ1 , where Γ1 := R−π/4 Γ , ∂y u(s ˆ 2 ) = 0 at pˆ 0 ∈ Γ2 ∪ Γ3 , where Γ2 := R−π/4 Γ2 and Γ3 := R−π/4 Γ3 , , ˆ 2 ) = 0 at P ∂r ∂y u(s , ∂θ ∂y u(s ˆ 2 ) = 0 at P ∂β ∂α u(s2 ) = 0 at P , where ∂β denotes a unit inward derivative at P , ∂r ∂y u(s ˆ 2 ) = 0 at R, ˆ 2 ) = 0 at R, ∂θ ∂y u(s ∂β ∂α u(s2 ) = 0 at R, where ∂β denotes a unit inward derivative at R, ∂x ∂α u(s2 ) = 0 at O, ∂y ∂α u(s2 ) = 0 at O, ˆ 2 ) = 0 at O, where ∂β denotes a unit inward derivative at O. ∂β ∂y u(s
2 . If (1) occurs, then it follows from the strong maximum principle that ∂y u(s ˆ 2 ) ≡ 0 in D Suppose that (2) occurs. Since ∂ν uˆ = 0 at pˆ 0 , ∂x uˆ = ∂y uˆ = 0 at pˆ 0 . Therefore ux = uy = 0 at p0 , where p0 := Rπ/4 pˆ 0 . We see that ∂θ u = 0 at p0 . (2) in the proof of Lemma 4.3 occurs. Thus ∂θ u ≡ 0 in D2 , which is a contradiction. We consider the case (3). Suppose that ∂y uˆ = 0 at pˆ ∈ Γ2 . Then ∂α u = 0 at p0 . Therefore ∂x u = ∂y u = 0 at p0 . (3) in the proof of Lemma 4.4 occurs, and u(s2 ) is constant, which is a contradiction. The proof of the case where √ ∂y uˆ = 0 at pˆ 0 ∈ Γ3 is similar. We consider the case (4). If (4-1) occurs, then ∂x ∂α u(s2 ) = 1/ 2∂x (−∂x + ∂y )u(s2 ) = 0 at P . On the other hand, ∂r ∂θ u(s2 ) = ∂x ∂y u(s2 ) = 0 at P . Hence ∂x ∂x u(s2 ) = 0 at P . (6) in the proof of Lemma 4.4 holds. Thus u(s2 ) is constant, which is √ a contradiction. If (4-2) occurs, then ∂θ ∂α u(s2 ) = 1/ 2∂θ (−∂r + ∂θ )u(s2 ) = 0 at P . Since ∂θ ∂r u(s2 ) = 0 at P , ∂θ ∂θ u(s2 ) = 0 at P . (4) in the proof of Lemma 4.3 holds. Thus u(s2 ) is radially symmetric, which is a contradiction. Suppose that (4-3) holds. Since −∂r u(s0 ) < 0 and ∂θ u(s0 ) < 0 at P , by the continuous dependence of u(s) on s we can assume that −∂r ∂y u(s2 ) < 0 at P
and ∂θ ∂y u(s2 ) < 0 at P .
(4.10)
Otherwise, there is s3 ∈ (s0 , s1 ] such that −∂r ∂y u(s3 ) = 0 at P or ∂θ ∂y u(s3 ) = 0 at P . Thus (4-1) or (4-2) occurs. Because of (4.10), we see that ∂β ∂y u < 0 at P , which is a contradiction. The proof of the case (5) is similar to one of the√case (4). We omit the proof. We consider the case (6). If (6-1) holds, then 1/ 2∂x (−∂x + ∂y )u = 0 at O. Since ∂u ∂x u = 0 at O, ∂x ∂x u = 0 at O. We see by (6) in√the proof of Lemma 4.4 that u is constant, which is a contradiction. If (6-2) occurs, then 1/ 2∂y (−∂x + ∂y )u(s2 ) = 0 at O. Since ∂y ∂x u(s2 ) = 0 at O, ∂y ∂y u(s2 ) = 0 at O. We can see by the same argument as the case of ∂x u(s2 ) that u(s2 ) is constant, which is a contradiction. If (6-3) holds, then we can obtain a contradiction by the same argument as the case (4-3). We omit the details. We have checked all the cases. The proof is complete. 2 Proof of Theorem 4.1. It is enough to show that C2 , which is the branch of (BPD2 ) defined above, is unbounded. If we can prove this, then we easily see that, for any (λ, u) ∈ C2 , u˜ satisfies all the properties stated in Theorem 4.1. Hence C2 := {(λ, u); ˜ (λ, u) ∈ C2 } is a desired branch.
774
Y. Miyamoto / Journal of Functional Analysis 256 (2009) 747–776
We show by contradiction that C2 does not meet other eigenvalues. Let B := {(λ(s), (2) (2) (2) u(s))}0s1 be as defines above, and let A := B\{(μ1 , a), (μn , a)} and L := + μn f (a) (n 2). By Proposition 2.7 we see that there are ψ ∈ ker L\{0} and a sequence {tj }j 1 converging to 0 such that u(tj ) − a j →∞ ψ −−−→
u(tj ) − a
ψ
in C 2 .
We divide the possibilities into two cases. Case 1: A does not contain a radially symmetric solution. We divide this case into two cases. Case 1-1: ψ is not radially symmetric. (2) Let ϕ be an eigenfunction corresponding to μ1 . Then by a calculation similar to one used in the proof of (3.9) we see that ∂θ ψ, ∂θ ϕ = 0. Since ∂θ ϕ does not change sign in D2 , ∂θ ψ changes the sign in D2 and ∂θ ψ has a zero in D2 . Therefore ∂θ u(s) has a zero in D2 if s is close to 1. It follows from Lemma 4.3(ii) that there is s1 ∈ (0, 1) such that ∂θ u(s1 ) ≡ 0 in D2 . It contradicts that A does not contain a radially symmetric solution. Case 1-2: ψ is radially symmetric. (2) 2 for some l 1. First, we will show that B does not Since ψ is radially symmetric, μn = p0,l 2 2 , a) (l 2) and that there is a sequence {t } meet (p0,l , a) (l 2). Suppose that B meets (p0,l j j 1 converging to 1 such that ∂x u(tj ) j →∞ ∂x ψ −−−→
u(tj ) − a
ψ
in C 2 .
(0)
Since ψ = φ0,l (l 2), ψx has a zero in D2 . Therefore, for large j > 0, ∂x u(tj ) has a zero in D2 . It follows from Lemma 4.4(ii) that, for some s1 ∈ (0, 1), ∂x u(s1 ) ≡ 0 in D2 . Since u(s1 ) satisfies the Neumann boundary condition, u(s1 ) should be constant. This contradicts that A does not contain a radially symmetric solution. 2 , a). Suppose that B meets (p 2 , a). Let Second, we will show that B does not meet (p0,1 0,1 (2)
ψ := φ0,1 . There is a sequence {tj }j 1 converging to 1 such that ∂α u(tj ) j →∞ ∂α ψ −−−→
u(tj ) − a
ψ
in C 2 .
Here ∂α ψ has a zero in D2 , hence, for large j 1, ∂α u(tj ) has a zero in D2 . It follows from Lemma 4.5(ii) that there is s1 ∈ (0, 1) such that ∂α u(λ(s1 )) is constant. This is a contradiction. Case 2: A contains a radially symmetric solution. The proof of Case 2 is similar to the one of Case 1-2. Let s0 := inf{s ∈ (0, 1); u(s) is radially symmetric}. ∂α u(s0 ) has a zero in D2 , hence, for some s1 ∈ (0, s0 ), ∂α u(s1 ) has a zero in D2 . Thus there is s2 ∈ (0, s1 ) such that u(s2 ) is constant in D2 . This contradicts the definition of s1 . We have checked all the cases. B does not meet other eigenvalues. Thus C2 is unbounded. 2
Y. Miyamoto / Journal of Functional Analysis 256 (2009) 747–776
775
References [1] S. Angenent, The zero set of a solution of a parabolic equation, J. Reine Angew. Math. 390 (1988) 79–96. [2] S. Angenent, B. Fiedler, The dynamics of rotating waves in scalar reaction diffusion equations, Trans. Amer. Math. Soc. 307 (1988) 545–568. [3] R. Böhme, Die Lösung der Verzweigungsgleichungen für nichtlineare Eigenwertprobleme, Math. Z. 127 (1972) 105–126 (in German). [4] P. Bauman, N. Carlson, D. Phillips, On the zeros of solutions to Ginzburg–Landau type systems, SIAM J. Math. Anal. 24 (1993) 1283–1293. [5] T. Carleman, Sur les systèmes linéaires aux derivées partielles du premier ordre à deux variables, C. R. Math. Acad. Sci. Paris 197 (1933) 471–474. [6] R. Courant, D. Hilbert, Methods of Mathematical Physics, vol. I, Interscience Publishers Inc., New York, NY, 1953. [7] E. Conway, D. Hoff, J. Smoller, Large time behavior of solutions of systems of nonlinear reaction–diffusion equations, SIAM J. Appl. Math. 35 (1978) 1–16. [8] M. Crandall, P. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal. 8 (1971) 321–340. [9] E. Dancer, Stable and finite Morse index solutions on Rn or bounded domains with small diffusion, Trans. Amer. Math. Soc. 357 (2005) 1225–1243. [10] Y. Du, Exact multiplicity and S-shaped bifurcation curve for some semilinear elliptic problems from combustion theory, SIAM J. Math. Anal. 32 (2000) 707–733. [11] I. Gelfand, Some problems in the theory of quasilinear equations, Trans. Amer. Math. Soc. 29 (1963) 295–381. [12] M. Gurtin, H. Matano, On the structure of equilibrium phase transitions within the gradient theory of fluids, Quart. Appl. Math. 46 (1988) 301–317. [13] B. Gidas, W.-M. Ni, L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979) 209–243. [14] J. Gawrycka, S. Rybicki, Solutions of systems of elliptic differential equations on circular domains, Nonlinear Anal. 59 (2004) 1347–1367. [15] B. Helffer, M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, M.P. Owen, Nodal sets for groundstates of Schrödinger operators with zero magnetic field in non-simply connected domains, Comm. Math. Phys. 202 (1999) 629–649. [16] T. Healey, H. Kielhöfer, Symmetry and nodal properties in the global bifurcation analysis of quasi-linear elliptic equations, Arch. Ration. Mech. Anal. 113 (1991) 299–311. [17] T. Healey, H. Kielhöfer, Preservation of nodal structure on global bifurcating solution branches of elliptic equations with symmetry, J. Differential Equations 106 (1993) 70–89. [18] P. Hartman, A. Wintner, On the local behavior of solutions of non-parabolic partial differential equations, Amer. J. Math. 75 (1953) 449–476. [19] D. Joseph, T. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Ration. Mech. Anal. 49 (1973) 241–269. [20] B. Kawohl, Rearrangements and Convexity of Level Sets in PDE, Lecture Notes in Math., vol. 1150, SpringerVerlag, Berlin, 1985. [21] A. Marino, La biforcazione nel caso variazionale, Confer. Sem. Mat. Univ. Bari 132 (1973) (in Italian). [22] H. Matano, Nonincrease of the lap-number of a solution for a one-dimensional semilinear parabolic equation, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 29 (1982) 401–441. [23] Y. Miyamoto, An instability criterion for activator–inhibitor systems in a two-dimensional ball II, J. Differential Equations 239 (2007) 61–71. [24] Y. Miyamoto, On the shape of the stable patterns for activator–inhibitor systems in two-dimensional domains, Quart. Appl. Math. 65 (2007) 357–374. [25] A.S. do Nascimento, Bifurcation of radially symmetric equilibria of a parabolic equation, Nonlinear Anal. 7 (1983) 1061–1070. [26] K. Nickel, Gestaltaussagen über Lösungen parabolischer Differentialgleichungen, J. Reine Angew. Math. 211 (1962) 78–94. [27] Y. Nishiura, Global structure of bifurcating solutions of some reaction–diffusion systems, SIAM J. Math. Anal. 13 (1982) 555–593. [28] K. Nagasaki, T. Suzuki, Spectral and related properties about the Emden–Fowler equation −u = λeu on circular domains, Math. Ann. 299 (1994) 1–15. [29] W.-M. Ni, I. Takagi, On the Neumann problem for some semilinear elliptic equations and systems of activator– inhibitor type, Trans. Amer. Math. Soc. 297 (1986) 351–368.
776
Y. Miyamoto / Journal of Functional Analysis 256 (2009) 747–776
[30] T. Ouyang, J. Shi, Exact multiplicity of positive solutions for a class of semilinear problems, J. Differential Equations 146 (1998) 121–156. [31] T. Ouyang, J. Shi, Exact multiplicity of positive solutions for a class of semilinear problem. II, J. Differential Equations 158 (1999) 94–151. [32] L. Payne, On two conjectures in the fixed membrane eigenvalue problem, J. Appl. Math. Phys. 24 (1973) 721–729. [33] P. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal. 7 (1971) 487–513. [34] P. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. Math., vol. 65, Amer. Math. Soc., Providence, RI, 1986, Published for the Conference Board of the Mathematical Sciences, Washington, DC. [35] T. Senba, Bifurcation from constant solutions to an elliptic equation, Studies on structure of solutions of nonlinear PDEs and its analytical methods (Kyoto, 1999), S¯urikaisekikenky¯usho K¯oky¯uroku 1204 (2001) 21–33 (in Japanese). [36] J. Serrin, A symmetry problem in potential theory, Arch. Ration. Mech. Anal. 43 (1971) 304–318. [37] J. Shi, Semilinear Neumann boundary value problems on a rectangle, Trans. Amer. Math. Soc. 354 (2002) 3117– 3154. [38] W.A. Strauss, Partial Differential Equations: An Introduction, John Wiley and Sons Inc., New York, 1992. [39] T. Suzuki, Semilinear Elliptic Equations, GAKUTO Internat. Ser. Math. Sci. Appl., vol. 3, Gakk¯otosho Co. Ltd., Tokyo, 1994. [40] J. Smoller, A. Wasserman, Global bifurcation of steady-state solutions, J. Differential Equations 39 (1981) 269–290. [41] I. Takagi, Point-condensation for a reaction–diffusion system, J. Differential Equations 61 (1986) 208–249. [42] S.-H. Wang, A correction for a paper by J. Smoller and A.G. Wasserman, J. Differential Equations 77 (1989) 199– 202. [43] K. Watanabe, Zero sets of solutions of second order parabolic equations with two spatial variables, J. Math. Soc. Japan 57 (2005) 127–136.
Journal of Functional Analysis 256 (2009) 777–809 www.elsevier.com/locate/jfa
On symmetries in the theory of finite rank singular perturbations Seppo Hassi a , Sergii Kuzhel b,∗ a Department of Mathematics and Statistics, University of Vaasa, PO Box 700, 65101 Vaasa, Finland b Institute of Mathematics of the National Academy of Sciences of Ukraine,
3 Tereshchenkivska Street, 01601, Kiev-4, Ukraine Received 29 November 2007; accepted 30 October 2008 Available online 28 November 2008 Communicated by L. Gross
Abstract For a nonnegative self-adjoint operator A0 acting on a Hilbert space H singular perturbations of the form A0 + V , V = n1 bij ψj , ·ψi are studied under some additional requirements of symmetry imposed on the initial operator A0 and the singular elements ψj . A concept of symmetry is defined by means of a one-parameter family of unitary operators U that is motivated by results due to R.S. Phillips. The abstract framework to study singular perturbations with symmetries developed in the paper allows one to incorporate physically meaningful connections between singular potentials V and the corresponding self-adjoint realizations of A0 + V . The results are applied for the investigation of singular perturbations of the Schrödinger operator in L2 (R3 ) and for the study of a (fractional) p-adic Schrödinger type operator with point interactions. © 2008 Elsevier Inc. All rights reserved. Keywords: Self-adjoint operator; Singular perturbation with symmetries; Friedrichs and Krein–von Neumann extensions; Scaling transformation; p-Adic analysis
* Corresponding author.
E-mail addresses: [email protected] (S. Hassi), [email protected] (S. Kuzhel). 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.10.023
778
S. Hassi, S. Kuzhel / Journal of Functional Analysis 256 (2009) 777–809
1. Introduction Let A0 be an unbounded nonnegative self-adjoint operator acting on a Hilbert space H and let H2 (A0 ) ⊂ H1 (A0 ) ⊂ H ⊂ H−1 (A0 ) ⊂ H−2 (A0 ) be the standard scale of Hilbert spaces associated with A0 . More precisely, k/2 Hk (A0 ) = D A0 ,
k = 1, 2,
(1.1)
equipped with the norm uk = (A0 + I )k/2 u. The dual spaces H−k (A0 ) can be defined as the completions of H with respect to the norms u−k = (A0 + I )−k/2 u (u ∈ H). The resolvent operator (A0 + I )−1 can be continuously extended to an isometric mapping (A0 + I )−1 from H−2 (A0 ) onto H and the relation ψ, u = (A0 + I )u, (A0 + I )−1 ψ ,
u ∈ H2 (A0 ),
(1.2)
enables one to identify the elements ψ ∈ H−2 (A0 ) as linear functionals on H2 (A0 ). Consider the heuristic expression n
A0 +
bij ψj , ·ψi ,
bij ∈ C, n ∈ N,
(1.3)
i,j =1
where elements ψj (1 j n) form a linearly independent system in H−2 (A0 ). In what follows it is supposed that the linear span X of {ψj }nj=1 satisfies the condition X ∩ H = {0}, i.e., elements ψj are H-independent. In this case, the perturbation V = ni,j =1 bij ψj , ·ψi is said to be singular and the formula D(Asym ) = u ∈ D(A0 ): ψj , u = 0, 1 j n
Asym = A0 D(Asym ),
(1.4)
determines a closed densely defined symmetric operator in H. In the theory of singular perturbations, cf. e.g. [3,5,23], each intermediate extension A of Asym , i.e., Asym ⊂ A ⊂ A∗sym , can be viewed to be singularly perturbed with respect to A0 and, in general, such an extension can be regarded as an operator-realization of (1.3) in H. In this context, the natural question arises whether and how one could establish a physically meaningful correspondence between the parameters bij of the singular potential V and the intermediate extensions of Asym . The investigation of this problem is one of goals of the present paper. In the approach developed by S. Albeverio and P. Kurasov in [4,5] one considers an operator realization A of (1.3) by setting A = AR D(A),
D(A) = f ∈ D A∗sym : AR f ∈ H ,
(1.5)
where AR = A0 +
n i,j =1
is seen as a regularization of (1.3).
bij ψjex , · ψi
(1.6)
S. Hassi, S. Kuzhel / Journal of Functional Analysis 256 (2009) 777–809
779
Formula (1.6) involves a construction of the extended functionals ψjex , · defined on D(A∗sym ). These functionals are uniquely determined by the choice of a Hermitian matrix R = (rjp )nj,p=1 . Since for elements ψ ∈ X ∩ H−1 (A0 ) the functionals ψ, · admit extensions by continuity onto H1 (A0 ) ∩ D(A∗sym ), a lot of natural restrictions appears in the choice of R. For their preservation the concept of admissible matrices R for the regularization of (1.3) has been introduced in [4, Definition 3.1.2]. However, this definition involves certain spectral measures and, in what follows, their calculation will be avoided. In fact, an equivalent operator concept of admissible large coupling limits of (1.3) is introduced in the form convenient for the further studies in the present paper. If the singular potential V in (1.3) is not form-bounded (i.e., X ⊂ H−1 (A0 )), then an admissible large coupling limit A∞ cannot be determined uniquely and one needs to impose some extra assumptions to achieve the uniqueness. For instance, in many applications, the condition of extremality [9,10] allows one to select a unique operator A∞ (see Theorem 3.12). It should be noted that the concept of extremality is physically reasonable. For example, extremal operators determine free evolutions in the Lax–Phillips scattering theory [31]. Another approach inspired by [4,5,30] deals with the preservation of initially existing symmetries of singular elements ψj in the definition of the extended functionals ψjex . To study this problem in an abstract framework, one needs to define the notion of symmetry for the unperturbed operator A0 and for the singular elements ψj in (1.3). Generalizing the ideas suggested in [5,26,37], the required definitions will be formulated here as follows: Let T be a subset of the real line R and let U = {Ut }t∈T be a one-parameter family of unitary operators acting on H with the following property: Ut ∈ U
⇔
Ut∗ ∈ U.
(1.7)
Definition 1.1. (See [20].) A linear operator A ( = 0) acting in H is said to be p(t)-homogeneous with respect to U if there exists a real function p(t) defined on T such that Ut A = p(t)AUt ,
∀t ∈ T.
(1.8)
In other words, the set U determines the structure of a symmetry and the property of A to be p(t)-homogeneous with respect to U means that A possesses a certain symmetry with respect to U. Definition 1.2. (See [20].) A singular element ψ ∈ H−2 (A0 ) \ H is said to be ξ(t)-invariant with respect to U if there exists a real function ξ(t) defined on T such that Ut ψ = ξ(t)ψ,
∀t ∈ T,
(1.9)
where Ut is the continuation of Ut onto H−2 (A0 ) (see Section 4 for details). The main aim of the paper is to study (1.3) assuming that the initial operator A0 is p(t)homogeneous and the singular elements ψj are ξj (t)-invariant with respect to U. It appears that the preservation of ξj (t)-invariance for the extended functionals ψjex , · is equivalent to the p(t)-homogeneity of the operator A∞ which is used for the regularization of (1.3) (Theorem 4.8). Combining this result with the complete description of admissible large coupling limits (Theorem 3.6) allows one to select a unique admissible large coupling limit A∞ by imposing the
780
S. Hassi, S. Kuzhel / Journal of Functional Analysis 256 (2009) 777–809
condition of p(t)-homogeneity (Theorems 4.13, 4.14). One of interesting properties discovered here is the possibility to get the Friedrichs and the Krein–von Neumann extension (and more generally, all p(t)-homogeneous self-adjoint extensions transversal to A0 ) as solutions of a system of equations involving the functions p(t) and ξj (t) (Corollary 4.10, Proposition 4.16). The choice of a p(t)-homogeneous admissible large coupling limit A∞ for the regularization of (1.3) immediately gives a new specific relation for the corresponding Weyl function M(z) (Theorem 5.5) and enables one to establish simple relations involving the functions p(t) and ξj (t), and the properties of operator realizations of (1.3) (Theorem 5.1, Proposition 5.3). It is well known, see e.g. [2,13,25,30] that the Schrödinger operators perturbed by potentials homogeneous with respect to a certain set U of unitary operators might possess a lot of interesting properties. Obviously, such properties become even more meaningful if, in addition to (1.7), the set U has further algebraic group properties. In particular, if U is the set of scaling transformations, then the additional multiplicative property Ut1 Ut2 = Ut2 Ut1 = Ut1 t2 of it elements enables one to get simple solutions of many problems (like description of nonnegative operator realizations, spectral properties, completeness of the wave operators, explicit form of the scattering matrix) for Schrödinger operators with singular potentials ξ(t)-invariant with respect to scaling transformations in R3 (Section 6). The abstract approach to the notion of symmetry developed in the paper can be also useful for the study of supersingular perturbations [30], for applications in the non-Archimedean analysis (Example 5.6), and for the investigation of Weyl families of boundary relations [15]. In a very recent paper [36], K.A. Makarov and E. Tsekanovskii considered the so-called μscale invariant operators, which can be seen as a special case of p(t)-homogeneous operators in the present paper. The main result of [36] is intimately related to [20, Lemma 4.5], see also Section 4 below. Throughout the paper D(A), R(A), and ker A denote the domain, the range, and the nullspace of a linear operator A, respectively, while A D stands for the restriction of A to the set D. 2. Preliminaries on operator realizations Following [4,5] an operator realization A of (1.3) in H are defined by (1.5), (1.6). To clarify the meaning of A0 and ψjex in (1.6), observe that A0 stands for the continuation of A0 as a bounded linear operator acting from H into H−2 (A0 ). Using the extended resolvent (A0 + I )−1 this continuation can be determined also by the formula −1
A0 f = (A0 + I )−1 f − f,
∀f ∈ H.
(2.1)
The linear functionals ψjex , · are extensions of ψj , · onto D(A∗sym ). Using the well-known relation ˙ H, D A∗sym = D(A0 ) +
where H = ker A∗sym + I ,
(2.2)
one concludes that ψj , · can be extended onto D(A∗sym ) by fixing their values on H. It follows from (1.2) and (1.4) that the vectors hj = (A0 + I )−1 ψj ,
j = 1, . . . , n,
(2.3)
S. Hassi, S. Kuzhel / Journal of Functional Analysis 256 (2009) 777–809
781
form a basis of the defect subspace H = ker(A∗sym + I ) of Asym . Hence, the functionals ψjex , · are well defined by the formula ψjex , f = ψj , u +
n
(2.4)
αp rjp
p=1
for all elements f = u + np=1 αp hp ∈ D(A∗sym ) (u ∈ D(A0 ), αp ∈ C) if the entries rjp = ψj , (A0 + I )−1 ψp = ψj , hp of the matrix R = (rjp )nj,p=1 are known. If all ψj ∈ H−1 (A0 ), then rjp are well defined and R is a Hermitian matrix [5]. Otherwise, the matrix R is not uniquely determined. In what follows, it is assumed that R is already chosen as a Hermitian matrix. The problem of an appropriate choice of R will be discussed in Section 3. In order to describe an operator realization A of (1.3) in terms of parameters bij of the singular perturbation V , the method of boundary triplets (see [16,18] and the references therein) is now incorporated. Definition 2.1. (See [18].) A triplet (N, Γ0 , Γ1 ), where N is an auxiliary Hilbert space and Γ0 , Γ1 are linear mappings of D(A∗sym ) into N , is called a boundary triplet of A∗sym if (A∗sym f, g) − (f, A∗sym g) = (Γ1 f, Γ0 g)N − (Γ0 f, Γ1 g)N for all f, g ∈ D(A∗sym ) and the mapping (Γ0 , Γ1 ) : D(A∗sym ) → N ⊕ N is surjective. The next two results (Lemma 2.2 and Theorem 2.3) are known (see e.g. [6,14]). For the convenience of the reader some principal steps of their proofs are repeated. Lemma 2.2. The triplet (Cn , Γ0 , Γ1 ), where the linear operators Γi : D(A∗sym ) → Cn are defined by the formulas ⎛ ψ ex , f ⎞ Γ0 f = ⎝
1
.. .
⎠,
ψnex , f
⎛
⎞ α1 . Γ1 f = − ⎝ .. ⎠ , αn
(2.5)
where f = u + j =1 αj hj ∈ D(A∗sym ) (u ∈ D(A0 ), αj ∈ C) and ψjex , f is defined by (2.4), forms a boundary triplet for A∗sym . Proof. Using (1.2), (2.2), and (2.3) it is easy to verify that the mappings ⎛
⎞ α1 . Γ0 f = ⎝ .. ⎠ , αn
⎛
⎞ ψ1 , u Γ1 f = ⎝ ... ⎠ , ψn , u
f =u+
αj hj
(2.6)
j =1
satisfy the conditions of Definition 2.1. Thus (Cn , Γ0 , Γ1 ) is a boundary triplet for A∗sym . It follows from (2.4), (2.5), and (2.6) that (2.7) Γ1 f = −Γ0 f, f ∈ D A∗sym . Γ0 f = Γ1 f + RΓ0 f, These relations between Γi and Γi and the fact that (Cn , Γ0 , Γ1 ) is a boundary triplet for A∗sym imply that (Cn , Γ0 , Γ1 ) also is a boundary triplet for A∗sym . 2
782
S. Hassi, S. Kuzhel / Journal of Functional Analysis 256 (2009) 777–809
Theorem 2.3. The operator realization A of (1.3) is an intermediate extension of Asym which coincides with the operator AB = A∗sym D(AB ), D(AB ) = f ∈ D A∗sym : BΓ0 f = Γ1 f , (2.8) where Γi are defined by (2.5) and B = (bij )ni,j =1 is the coefficient matrix of the singular pertur bation V = ni,j =1 bij ψj , ·ψi in (1.3). If V is symmetric, i.e., V u, v = u, V v (u, v ∈ H2 (A0 )), then the corresponding operator realization AB becomes self-adjoint. Proof. It follows from (2.1) that A0 hj = ψj − hj for all hj defined by (2.3). Rewriting f ∈ D(A∗sym ) in the form f = u + ni=1 αi hi , where u ∈ D(A0 ), hi ∈ H, αi ∈ C, and using (1.6) and (2.5) leads to AR f = A0 u −
n
αi hi +
n bij ψjex , f ψi + αi ψi
i,j =1
i=1
= A∗sym f
n
i=1
+ (ψ1 , . . . , ψn )[BΓ0 f − Γ1 f ].
This equality and (1.5) show that f ∈ D(A) if and only if BΓ0 f − Γ1 f = 0. Therefore, the operator realization A of (1.3) is an intermediate extension of Asym and A coincides with the operator AB defined by (2.8). To complete the proof it suffices to finally observe that V is symmetric if and only if the corresponding matrix of coefficients B = (bij )ni,j =1 is Hermitian. In this case (2.8) immediately implies the self-adjointness of AB . 2 Corollary 2.4. The operator realization AB of (1.3) in Theorem 2.3 determined by the boundary condition BΓ0 f = Γ1 f in (2.8) takes the form AB f = A0 f +
n
bij ψjex , f ψi ,
f ∈ D(AB ),
(2.9)
i,j =1
where the extended functionals ψjex , ·, j = 1, . . . , n, are determined by (2.4). Proof. Since the vectors hj in (2.3) span the defect subspace H = ker(A∗sym + I ) of Asym , one has A0 hj = ψj − hj = ψj + A∗sym hj and hence A∗sym f = A0 u +
n i=1
αi (A0 hi − ψi ) = A0 f −
n
αi ψi
(2.10)
i=1
for f = u + ni=1 αi hi ∈ D(A∗sym ). By substituting the boundary condition BΓ0 f = Γ1 f in (2.10) yields the desired perturbation formula for AB in (2.9). 2 Remark 2.5. Another approach, also involving the use of boundary triplets, to determine selfadjoint operator realizations of finite rank singular perturbations of the form A0 + GαG∗ , where G is an injective linear mapping from Cn to H−k (A0 ) was presented in [14, Section 4].
S. Hassi, S. Kuzhel / Journal of Functional Analysis 256 (2009) 777–809
783
3. Admissible matrices and admissible large coupling limits There are certain natural requirements for the determination of the entries rjp of the matrix R in (2.4). Indeed, if the linear span X of {ψj }nj=1 has a nonzero intersection with H−1 (A0 ), then for any ψ ∈ X ∩ H−1 (A0 ), the corresponding element h = (A0 + I )−1 ψ belongs to H1 (A0 ) and, hence, the functional ψ, · defined by (1.2) admits the following extension by continuity onto H1 (A0 ): (3.1) ψ, f = (A0 + I )1/2 f, (A0 + I )1/2 h , ∀f ∈ H1 (A0 ). To preserve such natural extensions of ψ, · onto D(A∗sym ) ∩ H1 (A0 ) in the definition (2.4), the concept of admissible matrices R as introduced in [4] is used. Definition 3.1. A Hermitian matrix R = (rjp )nj,p=1 is called admissible for the regularization AR of (1.3) if its entries rjp are chosen in such a way that if a singular element ψ = c1 ψ1 +· · ·+cn ψn belongs to H−1 (A0 ), then for all f ∈ D(A∗sym ) ∩ H1 (A0 ) n ψ ex , f = (A0 + I )1/2 f, (A0 + I )1/2 h = cj ψjex , f ,
(3.2)
j =1
where ψjex , f are defined by (2.4) and h = (A0 + I )−1 ψ . It is convenient to describe the set of admissible matrices in terms of a certain associated operators. It follows from the relations in (2.7) that the choice of a matrix R in (2.4) is equivalent to the choice of an operator A∞ defined by (3.3) A∞ = A∗sym D(A∞ ), D(A∞ ) = ker Γ0 = f ∈ D A∗sym : −RΓ0 f = Γ1 f . Since R is Hermitian, the general theory of boundary triplets [16] implies that A∞ is a selfadjoint extension of Asym . By the construction, A∞ and A0 , D(A0 ) = ker Γ1 (= ker Γ0 ), are transversal extensions of Asym , i.e., D(A0 ) + D(A∞ ) = D(A∗sym ). Furthermore, it follows from Theorem 2.3 that A∞ and the operator realization AB of (1.3) determined by the boundary condition BΓ0 f = Γ1 f , f ∈ D(A∗sym ) are also transversal extensions of Asym for every coefficient matrix B in (1.3), i.e., the operator A∞ determined by (3.3) is always transversal to the singular perturbations AB in (2.9). The operator A∞ corresponds formally to the matrix B with infinite entries in (2.9) (such an extension of Asym need not be unique). In this sense, A∞ can be considered as a large coupling limit of operator realizations AB of (1.3) with finite entries of B. Definition 3.2. An operator A∞ is called admissible large coupling limit of (1.3) if A∞ is defined by (3.3) with an admissible matrix R. So, the choice of an admissible large coupling limit A∞ of (1.3) is equivalent to the choice of an admissible matrix R for the regularization AR of (1.3). The next lemma contains some useful facts concerning the (unperturbed) nonnegative selfadjoint operator A0 and its relation to the Friedrichs extension AF of Asym . They can be considered to be well known from the extension theory of nonnegative operators, therefore details for the present formulations with their proofs are left to the reader; see e.g. [8,17,21,22,29,32].
784
S. Hassi, S. Kuzhel / Journal of Functional Analysis 256 (2009) 777–809
Lemma 3.3. Let C = (A0 + I )−1 − (AF + I )−1 and let S0 = A0 ∩ AF . Moreover, denote H = ker(A∗sym + I ) and H = ker(S0∗ + I ). Then: (i) (ii) (iii) (iv)
R(C) = H ; ker C = R(S0 + I ) = R(Asym + I ) ⊕ H , where H = H H ; 1/2 R(C 1/2 ) = D(A0 ) ∩ H = H ; 1/2 1/2 ˙ R(C 1/2 ). D(A0 ) = D(AF ) +
Using the spaces introduced in (1.1) and (iii) in Lemma 3.3 one can rewrite the decomposition in part (iv) of Lemma 3.3 as follows: H1 (A0 ) = D ⊕1 H ,
H = H ∩ H1 (A0 ) = (A0 + I )−1 X ∩ H−1 (A0 ) ,
(3.4)
1/2
where D (= D(AF )) stands for the completion of D(Asym ) in H1 (A0 ), ⊕1 denotes the orthogonal sum in H1 (A0 ), and X is the linear span of {ψj }nj=1 . The set of admissible large coupling limits of (1.3) can now be characterized in ‘coordinate free’ manner as follows. of Asym is an admissible large coupling limit of (1.3) if Theorem 3.4. A self-adjoint extension A is transversal to A0 (i.e., D(A0 ) + D(A) = D(A∗sym )) and and only if A ∩ H1 (A0 ) ⊂ D(AF ), D(A)
(3.5)
where AF is the Friedrichs extension of Asym . of Asym is transversal to A0 and it satisfies the Proof. Assume that the self-adjoint extension A and A0 is condition (3.5). In view of (2.6), D(A0 ) = ker Γ0 . Therefore, the transversality of A equivalent to the representation of D(A) in the form (3.3) with an n × n Hermitian matrix R (here Asym has finite defect numbers (n, n)), cf. [17, Proposition 1.4]. Since D(AF ) = D ∩ D A∗sym ,
(3.6)
the decomposition (3.4) shows that the condition (3.5) is equivalent to the relation (A0 + I )1/2 f˜, (A0 + I )1/2 h = 0,
∩ H1 (A0 ), ∀h ∈ H . ∀f˜ ∈ D(A)
(3.7)
Now it is shown that R is an admissible matrix in the sense of Definition 3.1 by verifying (3.2) for all ψ ∈ X ∩ H−1 (A0 ). Observe, that the mapping Γ0 defined in Lemma 2.2, see also (2.7), determines the extended functionals ψjex , f in (2.4). and A0 yields the following decomposition for the elements f ∈ The transversality of A ∗ D(Asym ): f = f˜ + u,
(3.8)
S. Hassi, S. Kuzhel / Journal of Functional Analysis 256 (2009) 777–809
785
and u ∈ D(A0 ) are uniquely determined modulo D(Asym ). If ψ = nj=1 cj ψj ∈ where f˜ ∈ D(A) H−1 (A0 ), then by (3.4) h = (A0 + I )−1 ψ ∈ H . Now with f ∈ D(A∗sym ) ∩ H1 (A0 ) decomposed as in (3.8) one obtains:
ψ ex , f
=
n
(3.8) cj ψjex , f = cΓ0 f = cΓ0 (f˜ + u)
j =1
(2.6) (1.2) = c(Γ1 + RΓ0 )u = cΓ1 u = ψ, u = (A0 + I )u, h
(2.7)
(3.9)
where c := (c1 , . . . , cn ). On the other hand, it follows from (3.7) that (A0 + I )1/2 f, (A0 + I )1/2 h = (A0 + I )1/2 (f˜ + u), (A0 + I )1/2 h = (A0 + I )u, h , (= A∞ ) is an which combined with (3.9) proves (3.2). Thus, R is an admissible matrix and A admissible large coupling limit of (1.3). = A∞ satisfies the condition of Definition 3.2. Then (3.3) ensures Conversely, assume that A and A0 and R determines the extended functionals ψ ex , · via (2.4). the transversality of A j Reasoning as in (3.9) it is seen that (3.2) implies 0 = (A0 + I )1/2 f, (A0 + I )1/2 h − ψ ex , f = (A0 + I )1/2 f˜, (A0 + I )1/2 h for all f ∈ D(A∗sym ) ∩ H1 (A0 ) and h ∈ H . Thus, the relation (3.7) and, equivalently, the relation (3.5) is satisfied. Theorem 3.4 is proved. 2 For some further study of admissible large coupling limits the following lemma is needed. be a subspace of H = ker(A∗sym + I ). Then the symmetric operator Lemma 3.5. Let H S = AF D(S) ,
D(S) = (AF + I )−1 R(Asym + I ) ⊕ H
(3.10)
satisfies the relations D(S) ∩ D(A0 ) = D(Asym )
˙ H and D(S) + D(A0 ) = D(AF ) +
(3.11)
if and only if = dim H dim H
∩ H = {0}, and H
(3.12)
where H = H ∩ H1 (A0 ) and H = H H . In this case, the domain of S admits the description ˙ h + u: h ∈ H , u = u(h ) , D(S) = D(Asym ) +
(3.13)
where u = u(h ) ∈ D(A0 ) is (uniquely) determined by h ∈ H and satisfies the relation
(A0 + I )u, h⊥ = ψ, u = 0,
ψ = (A0 + I ) ∀ h⊥ ∈ H H, h⊥ .
(3.14)
786
S. Hassi, S. Kuzhel / Journal of Functional Analysis 256 (2009) 777–809
Proof. Denote S0 = AF ∩ A0 . By Lemma 3.3
D(S0 ) = (A0 + I )−1 R(Asym + I ) ⊕ H = (AF + I )−1 R(Asym + I ) ⊕ H ,
(3.15)
where H = H H . Comparing (3.10) and (3.15), one concludes that
∩ H ) . D(S) ∩ D(A0 ) = D(S) ∩ D(S0 ) = (AF + I )−1 R(Asym + I ) ⊕ (H Thus, D(S) ∩ D(A0 ) = D(Asym )
⇔
∩ H = {0}. H
The relations (3.10) and (3.15) also show that
+ ˙ H ) + (A0 + I )−1 H . D(S) + D(A0 ) = (AF + I )−1 R(Asym + I ) ⊕ (H
(3.16)
Here (A0 + I )−1 H can be represented as (A0 + I )−1 H = (AF + I )−1 h + Ch : h ∈ H ,
(3.17)
where C = (A0 + I )−1 − (AF + I )−1 . It follows from Lemma 3.3 that R(C) = H ,
ker C = ran(Asym + I ) ⊕ H .
(3.18)
+ ˙ H = H. Relations (3.16)–(3.18) show that the second identity in (3.11) holds if and only if H Obviously, this representation is possible only in the case where dim H = dim H . where ˙ (AF + I )−1 H, The definition (3.10) shows that D(S) = D(Asym ) + = (A0 + I )−1 . (AF + I )−1 H h − C h: h∈H satisfies (3.12), it follows from (3.18) that C H = H . Now, setting u = (A0 + I )−1 Since H h and and therefore h = C −1 h ∈ H, h = −C h, one obtains (3.13) and (3.14). Note that the preimage also u, is uniquely determined by h ∈ H . 2 The next theorem gives a description of all admissible large coupling limits. be a self-adjoint extension of Asym and let the symmetric operator S = Theorem 3.6. Let A of H. Then the following statements ∩ AF be represented as in (3.10) with some subspace H A are equivalent: (= A∞ ) is an admissible large coupling limit of (1.3); (i) A is a self-adjoint extension of S transversal to the Friedrichs extension SF of S and the (ii) A satisfies the conditions in (3.12). subspace H be an admissible large coupling limit. Since A and A0 are transversal, one has Proof. Let A ∩ D(A0 ) = D(Asym ), D(A)
+ D(A0 ) = D(AF ) + ˙ H = D A∗sym . D(A)
(3.19)
S. Hassi, S. Kuzhel / Journal of Functional Analysis 256 (2009) 777–809
787
The condition (3.5) is equivalent to ∩ D(AF ) = D(A ∩ AF ). ∩ H1 (A0 ) = D(A) D(A) Thus, intersecting all parts of (3.19) with H1 (A0 ) one concludes that the relations (3.11) are satisfies (3.12). Furthermore, since the ∩ AF . By Lemma 3.5, the subspace H true for S = A = D(AF ) ∩ D(A) = Friedrichs extension SF of S coincides with AF , one gets D(SF ) ∩ D(A) D(S). This implies the transversality of SF and A. The implication (i) ⇒ (ii) is proved. is a self-adjoint extenNow, assume that (ii) is satisfied. Since S ⊃ Asym , the operator A ∗ sion of Asym . It follows from (3.10) that ker(S + I ) = H H and hence, D(S ∗ ) = D(SF ) + On the other hand, the transversality of SF and A gives ˙ (H H). ker(S ∗ + I ) = D(AF ) + ∗ ˙ (H H). This equality and D(S ) = D(AF ) + D(A). Therefore, D(AF ) + D(A) = D(AF ) + the second relation in (3.11) yield = D(S) + D(A0 ) + D(A) D(A0 ) + D(A) = D(AF ) + ˙ H + D(A) ˙ H + ˙ (H H). = D(AF ) +
(3.20)
= H. Hence, (3.20) shows that D(A0 ) + D(A) = ˙ (H H) The conditions (3.12) imply that H + ∗ ˙ D(AF ) + H = D(Asym ), i.e., A and A0 are transversal. Furthermore, by Lemma 3.3, see also ˙ H = H1 (A0 ) ∩ D(A∗sym ). Now, employing the second relation in (3.11) one (3.6), D(AF ) + obtains ∩ D(S) + D(A0 ) = D(S)+D(Asym ) = D(S) ⊂ D(AF ). ∩ H1 (A0 ) = D(A) D(A) is an admissible large coupling limit of (1.3). The According to Theorem 3.4 this means that A implication (ii) ⇒ (i) is proved. 2 It follows from Theorem 3.6 that there is at least one admissible large coupling limit of (1.3). Some further specifications are given in the following two corollaries. Corollary 3.7. If all the elements ψj in (1.3) belong to H−1 (A0 ), then there exists a unique admissible large coupling limit A∞ and it coincides with the Friedrichs extension AF of Asym . Proof. Assume that ψj ∈ H−1 (A0 ) for all j = 1, . . . , n. Then D(A∗sym ) ⊂ H1 (A0 ) and H = H. = A∞ be an admissible large coupling limit of (1.3) and let S = A ∩ AF . By Theorem 3.6 Let A satisfies (3.12) in Lemma 3.5, so that H = H. Now (3.10) gives the corresponding subspace H ∩ AF , one concludes that A = AF . This completes the proof. 2 S = AF and since S = A Corollary 3.8. If all the elements ψj in (1.3) are H−1 (A0 )-independent (i.e. X ∩H−1 (A0 ) = {0}), of Asym transversal to A0 is an admissible large coupling then every self-adjoint extension A limit of (1.3). The Friedrichs extension of Asym coincides with A0 . Proof. The condition of H−1 (A0 )-independency means that H = {0}. In this case, only the = {0} can satisfy (3.12). The corresponding operator S coincides with Asym . zero subspace H is Moreover, since H = {0}, Lemma 3.3 shows that SF = AF = A0 . Thus, by Theorem 3.6, A an admissible large coupling limit if and only if A is transversal to A0 . 2
788
S. Hassi, S. Kuzhel / Journal of Functional Analysis 256 (2009) 777–809
Observe, that the condition X ⊂ H−1 (A0 ) in Corollary 3.7 is equivalent to D(A∗sym ) ⊂ H1 (A0 ); see (2.2). Since in this case all the elements ψj ∈ H−1 (A0 ) admit their natural extension by continuity onto H1 (A0 ) via (3.1), the matrix R = (rjp )nj,p=1 in (2.4) is uniquely determined and the extended functionals ψjex , · in (2.4) are obtained by restricting their natural continuations to the subset D(A∗sym ) of H1 (A0 ). It follows that if X ⊂ H−1 (A0 ), then the operator realizations of (1.3) described in Theorem 2.3 reduce to the so-called form bounded perturbations of A0 : Corollary 3.9. If all the elements ψj in (1.3) belong to H−1 (A0 ), then the operator realization AB of (1.3) in Theorem 2.3 determined by the boundary condition BΓ0 f = Γ1 f in (2.8) takes the form AB f = A0 f +
n
bij ψjex , f ψi ,
f ∈ D(AB ),
i,j =1
where the extended functionals ψjex , ·, j = 1, . . . , n, are determined by their continuations onto H1 (A0 ) via (3.1) and A0 as defined by (2.1) can be considered as a bounded operator acting from H−1 (A0 ) into H1 (A0 ). Proof. The statement is immediate from Corollary 2.4 and the fact that in this case D(A∗sym ) ⊂ H1 (A0 ). Note that A0 defined by (2.1) satisfies A0 (H1 (A0 )) ⊂ H−1 (A0 ) and its restriction to H1 (A0 ) coincides with the continuation of A0 as a bounded operator from H−1 (A0 ) into H1 (A0 ). 2 The properties of admissible large coupling limits are closely related to the transversality of the Friedrichs and the Krein–von Neumann extensions of Asym . Theorem 3.10. There exists a nonnegative admissible large coupling limit of (1.3) if and only if the Friedrichs extension AF and the Krein–von Neumann extension AN of Asym are transversal. be a nonnegative admissible large coupling limit. Then A is a nonnegative extenProof. Let A sion of Asym and therefore + I )−1 (AN + I )−1 , (AF + I )−1 (A
(3.21)
where AF is the Friedrichs extension and AN is the Krein–von Neumann extension of Asym (see e.g. [21] and the references therein). 2 of Asym is equivalent to 1 and A Recall that transversality of self-adjoint extensions A
2 + I )−1 H = H 1 + I )−1 − (A (A
(3.22)
(see e.g. [16]). Hence, if AF and AN are not transversal then (AF + I )−1 h = (AN + I )−1 h for + I )−1 h = (A0 + I )−1 h due to and A0 yields (A some nonzero h ∈ H. Then nonnegativity of A (3.21) (with similar inequalities for A0 ), so that
+ I )−1 − (A0 + I )−1 H ⊂ H h (A
S. Hassi, S. Kuzhel / Journal of Functional Analysis 256 (2009) 777–809
789
and A0 cannot be transversal. This is a contradiction to the admissibility of A. and by (3.22) A Thus AF and AN are transversal. be a subspace To prove the converse statement assume that AF and AN are transversal. Let H of H, which satisfies (3.12) and let the symmetric operator S be defined by (3.10) in Lemma 3.5. be the Krein–von Neumann extension of S. Clearly, A is a nonnegative selfMoreover, let A adjoint extension of Asym . It remains to prove that A is an admissible large coupling limit of (1.3). To see this, observe that the Friedrichs extension of S coincides with AF . Then it follows from [10, Proposition 7.2] that the Friedrichs extension SF = AF and the Krein–von Neumann of S are transversal with respect to S. Therefore, by Theorem 3.6, A is an admissible extension A large coupling limit. 2 Observe that S in Theorem 3.10 is a restriction of the Friedrichs extension AF of Asym . Since constructed in Theorem 3.10 is the Krein–von Neumann the admissible large coupling limit A is an extremal extension of Asym extension of S it is a consequence of [10, Theorem 6.4] that A in the sense of the following definition. of Asym is called extremal if it is nonDefinition 3.11. (See [9,10].) A self-adjoint extension A negative and satisfies the condition
inf
u∈D (Asym )
− u), f − u = 0 for all f ∈ D(A). A(f
Theorem 3.12. Let the Friedrichs extension AF and the Krein–von Neumann extension AN of Asym be transversal, and let S be defined by (3.10) and (3.12). Then among all self-adjoint of (1.3). extensions of S there exists a unique extremal admissible large coupling limit A of S is the Proof. By Theorem 3.10, it suffices to show that the Krein–von Neumann extension A only extremal extension of Asym which coincides with admissible large coupling limit of (1.3). is extremal and admissible in the sense of Definition 3.2. Then To prove this assume that A as an extremal extension of Asym is the Krein–von Neumann extension by [10, Theorem 6.4] A means ∩ AF . Moreover, by Theorem 3.6 the admissibility of A of the symmetric operator S=A satisfies (3.12). that S is determined via (3.10) where the corresponding subspace H ⊆ H, where the subspaces H is an extension of S, one has S ⊆ Since A S or, equivalently, H and H correspond to S and S in (3.10). Now the first equality in (3.12) forces that H = H and = A and this completes the proof. 2 hence S = S. Therefore, A transversal to the initial one A0 (but Remark 3.13. The selection of a self-adjoint operator A without the condition (3.5)) is also a key point of the approach used in [11] to the determination of self-adjoint realizations of a formal expression A0 + V , where a singular perturbation V is assumed to be (in general) an unbounded self-adjoint operator V : H2 (A0 ) → H−2 (A0 ) such that ker V is dense in H. In this case, the regularization of A0 + V takes the form AP ,V = A0 + V P and it is well defined on the domain D(AP ,V ) = {f ∈ D(A∗sym ): Pf ∈ D(V )}, where P is the skew projection onto H2 (A0 ) in D(A∗sym ) that is uniquely determined by A.
790
S. Hassi, S. Kuzhel / Journal of Functional Analysis 256 (2009) 777–809
4. Singular perturbations with symmetries and uniqueness of admissible large coupling limits According to (2.4) and (3.3) the regularization AR of (1.3) depends on the choice of an admissible large coupling limit A∞ . Apart from the case of form bounded singular perturbations, admissible large coupling limits are not determined uniquely, cf. Theorem 3.6. However, in many cases (see e.g. [4,5]), the uniqueness can be attained by imposing extra assumptions of symmetry motivated by the specific nature of the underlying physical problem. In this section, we study this problem in an abstract framework. 4.1. Preliminaries First some general facts concerning p(t)-homogeneous operators are given. Let an operator A in H be p(t)-homogeneous with respect to a one-parameter family U = {Ut }t∈T of unitary operators acting on H, cf. Definition 1.1. It follows from (1.7) and (1.8) that p(t)p(g(t)) = 1,
∀t ∈ T,
(4.1)
where the function of conjugation g(t) : T → T is determined by the formula Ug(t) = Ut∗ ,
∀t ∈ T.
(4.2)
Lemma 4.1. Let A be a p(t)-homogeneous operator with respect to a family U = {Ut }t∈T . Then for all t ∈ T and all z ∈ C, Ut ker(A − zI ) = ker p(t)A − zI .
(4.3)
In particular, ker A is a reducing subspace for every Ut , t ∈ T. Furthermore, z ∈ σa (A) ⇔ zp(t)n ∈ σa (A), n ∈ Z, t ∈ T, a ∈ {p, r, c}. If p(t) = 1 at least for one point t ∈ T, then the essential spectrum of A contains the point z = 0. Proof. In view of (4.1), p(t) = 0 for all t ∈ T. Using (1.8) one gets z I Ut Ut (A − zI ) = p(t)A − zI Ut = p(t) A − p(t)
(4.4)
that gives Ut (ker(A − zI )) ⊂ ker(p(t)A − zI ). The reverse inclusion is obtained by using (4.1). The property of ker A to be a reducing subspace for every Ut follows from (4.3) with z = 0 if one takes into account that p(t) = 0. The remaining assertions of the lemma immediately follow from (4.4). 2 Lemma 4.2. Let A be a closed densely defined p(t)-homogeneous operator with respect to a family U = {Ut }t∈T . Then also its adjoint A∗ is p(t)-homogeneous with respect to U. Proof. Since A is p(t)-homogeneous one has Ut A = p(t)AUt for all t ∈ T. As a unitary operator Ut is bounded with bounded inverse, and therefore, the previous equality is equivalent to
S. Hassi, S. Kuzhel / Journal of Functional Analysis 256 (2009) 777–809
791
A∗ Ut∗ = p(t)Ut∗ A∗ ⇔ Ut A∗ = p(t)A∗ Ut , ∀t ∈ T, which means that A∗ is p(t)-homogeneous with respect to U. 2 In the case that A is symmetric the formula (4.3) in Lemma 4.1 shows how the unitary operators Ut , t ∈ T, transform the defect subspaces ker(A∗ − zI ) of A. Corollary 4.3. Let A in Lemma 4.2 be nonnegative and p(t)-homogeneous with respect to U = {Ut }t∈T and let A0 be a nonnegative selfadjoint extension of A. Then (p(t)A0 + I )(A0 + I )−1 Ut (ker(A∗ + I )) = ker(A∗ + I ). Proof. By Lemma 4.2 the adjoint A∗ of A is also p(t)-homogeneous and (4.3) implies that Ut (ker(A∗ + I )) = ker(A∗ + 1/p(t)I ). Moreover, the equality 1 I = ker(A∗ + I ) p(t)A0 + I (A0 + I )−1 ker A∗ + p(t) is always satisfied for a nonnegative self-adjoint extension A0 of A.
2
For the next result recall that if A is a nonnegative operator (or in general a nonnegative relation) in a Hilbert space H, then the Friedrichs extension AF and the Krein–von Neumann extension AN of A can be characterized as follows (see [8] for the densely defined case and [19,21,22] for the general case): If {f, f } ∈ A∗ , then {f, f } ∈ AF if and only if inf f − h2 + (f − h , f − h): {h, h } ∈ A = 0. (4.5) If {f, f } ∈ A∗ , then {f, f } ∈ AN if and only if inf f − h 2 + (f − h , f − h): {h, h } ∈ A = 0.
(4.6)
Lemma 4.4. Let A be a nonnegative densely defined p(t)-homogeneous operator with respect to U. Then the Friedrichs extension AF and the Krein–von Neumann extension AN of A are also 1/2 1/2 1/2 p(t)-homogeneous with respect to U. Moreover, Ut (D(AF )) ⊂ D(AF ) and Ut (R(AN )) ⊂ 1/2 R(AN ) for all t ∈ T. Proof. By Lemma 4.2 A∗ is p(t)-homogeneous with respect to U. Hence, in view of (1.7) and of A is p(t)-homogeneous with respect to U if and only if (1.8), an intermediate extension A → D(A), Ut : D(A)
∀t ∈ T.
(4.7)
To prove that AF is p(t)-homogeneous with respect to U, assume that f ∈ D(AF ). Then g = Ut f ∈ D(A∗ ) and there is a sequence hn ∈ D(A) attaining the infimum in (4.5). Then Ut hn ∈ D(A), Ut hn → Ut f = g, and (4.8) (A∗ Ut f − AUt hn , Ut f − Ut hn ) = p g(t) (A∗ f − Ahn , f − hn ) → 0, so that g ∈ D(AF ) by (4.5). Therefore, Ut (D(AF )) ⊂ D(AF ) and AF is p(t)-homogeneous with respect to U.
792
S. Hassi, S. Kuzhel / Journal of Functional Analysis 256 (2009) 777–809
To prove the p(t)-homogeneity of AN assume that f ∈ D(AN ). Then again g = Ut f ∈ D(A∗ ) and there is a sequence hn ∈ D(A) attaining the infimum in (4.6). In particular, Ahn → A∗ f , Ut hn ∈ D(A), and AUt hn = p g(t) Ut Ahn → p g(t) Ut A∗ f = A∗ Ut f = A∗ g. Moreover, (4.8) is satisfied. Therefore, (4.6) shows that g ∈ D(AN ). This proves that Ut (D(AN )) ⊂ D(AN ) and thus AN is p(t)-homogeneous with respect to U. 1/2 Finally, recall that the domain D = D(AF ), see (3.4), can be characterized as the set of vectors f ∈ H satisfying A(hn − hm ), hn − hm → 0,
hn → f,
m, n → ∞,
1/2
and the range R(AN ) as the set of vectors g ∈ H satisfying Ahn → g,
A(hn − hm ), hn − hm → 0,
m, n → ∞,
with hn ∈ D(A), see (4.5) and (4.6). The last statement is clear from these characterizations using similar arguments as above with the sequence hn . This completes the proof. 2 Let the operator A0 in (1.3) be p(t)-homogeneous with respect to U = {Ut }t∈T . Define a family of self-adjoint operators on H by Gt = p(t)A0 + I (A0 + I )−1 ,
t ∈ T.
(4.9)
Clearly, Gt is positive and bounded with bounded inverse for all t ∈ T. Moreover, it follows from (1.8) and (4.1) that (A0 + I )−1 Ut = Ut (p(g(t))A0 + I )−1 and −1 Gt Ut = Ut G−1 g(t) = (Gg(t) Ug(t) ) .
(4.10)
Since u−2 = (A0 + I )−1 u, the identity (A0 + I )−1 Ut = Gt Ut (A0 + I )−1 implies that Ut u−2 Gt u−2 for all u ∈ H. Hence, the operators Ut can be continuously extended to bounded operators Ut in H−2 (A0 ) and, furthermore, (A0 + I )−1 Ut ψ = Gt Ut (A0 + I )−1 ψ
(4.11)
for all ψ ∈ H−2 (A0 ) and t ∈ T. The equality (4.2) shows that Ut has a bounded inverse which satisfies U−1 t = Ug(t) . The operator Ut can be characterized also as the dual mapping (adjoint) of Ug(t) with respect to the form defined in (1.2). In fact, using (1.2), (1.8), (4.2), and (4.11), it is seen that the action of the functional Ut ψ, · on the elements u ∈ H2 (A0 ) is determined by the formula Ut ψ, u = (A0 + I )u, Gt Ut h = Ug(t) p(t)A0 + I u, h = (A0 + I )Ug(t) u, h = ψ, Ug(t) u, where h = (A0 + I )−1 ψ .
(4.12)
S. Hassi, S. Kuzhel / Journal of Functional Analysis 256 (2009) 777–809
793
Now consider a singular element ψ ∈ H−2 (A0 ), cf. (1.3). The assumption that ψ is ξ(t)invariant with respect to U, i.e. Ut ψ = ξ(t)ψ for all t ∈ T (see Definition 1.2), implies some relations between ξ(t), p(t), and g(t). Proposition 4.5. Let the operator A0 in (1.3) be p(t)-homogeneous with respect to the family U and let ψ ∈ H−2 (A0 ) \ H be ξ(t)-invariant with respect to U. Then for all t ∈ T one has ξ(t)ξ g(t) = 1
(4.13)
and, moreover, |ξ(t)| = 1 if p(t) = 1 and min{1, p(t)} < |ξ(t)| < max{1, p(t)} if p(t) = 1. Proof. It follows from (1.9) and (4.11) that ψ ∈ H−2 (A0 ) \ H is ξ(t)-invariant with respect to U if and only if Gt Ut h = ξ(t)h,
∀t ∈ T,
(4.14)
where h = (A0 + I )−1 ψ . This together with (4.10) implies that h = (Gg(t) Ug(t) )(Gt Ut )h = ξ(t)Gg(t) Ug(t) h = ξ(t)ξ g(t) h, which proves (4.13). Moreover, (4.14) shows that |ξ(t)|h = Gt Ut h. In particular, if p(t) = 1, then Gt = I and |ξ(t)|h = Ut h = h that gives |ξ(t)| = 1. In the case where p(t) = 1 the formula for Gt in (4.9) with an evident reasoning leads to the estimates α(t)h = α(t)Ut h < Gt Ut h < β(t)Ut h = β(t)h, where α(t) = min{1, p(t)} and β(t) = max{1, p(t)}. This completes the proof. 2 4.2. p(t)-homogeneous self-adjoint extensions of Asym Let Asym be defined by (1.4). This means that Asym is a nonnegative symmetric operator with finite defect numbers. Lemma 4.6. If p(t) = 1 at least for one point t ∈ T, then an arbitrary p(t)-homogeneous selfadjoint extension of the symmetric operator Asym is nonnegative. Proof. Assume that z is a negative eigenvalue of a p(t)-homogeneous self-adjoint extension A of Asym and that p(t) = 1 for t ∈ T. Then, according to Lemma 4.1, there exists infinite series of negative eigenvalues zp(t)n (n ∈ Z) of A that contradicts to the assumption of finite defect numbers of Asym . Hence, A is a nonnegative extension of Asym . 2 Lemma 4.7. Let A0 be p(t)-homogeneous and let ψj be ξj (t)-invariant with respect to U, j = 1, . . . , n. Then the symmetric operator Asym defined by (1.4) and its adjoint A∗sym are also p(t)homogeneous with respect to U. Proof. It follows from (1.4) and (4.12) that ψj , Ut u = Ug(t) ψj , u = ξj g(t) ψj , u = 0
794
S. Hassi, S. Kuzhel / Journal of Functional Analysis 256 (2009) 777–809
for every u ∈ D(Asym ). Thus Ut : D(Asym ) → D(Asym ) and hence by (1.8) Asym is p(t)homogeneous: Ut Asym = p(t)Asym Ut . By Lemma 4.2 also the adjoint A∗sym is p(t)-homogeneous with respect to U. 2 In view of (1.9) and (4.12) the ξj (t)-invariance of ψj is equivalent to the relation ξj (t)ψj , u = ψj , Ug(t) u,
∀u ∈ H2 (A0 ), ∀t ∈ T,
(4.15)
where the linear functionals ψj , · are defined by (1.2). The next theorem shows that the preservation of (4.15) for the extended functionals ψjex , · is closely related to the existence of p(t)-homogeneous self-adjoint extensions of Asym transversal to A0 . Theorem 4.8. Let A0 be p(t)-homogeneous, let ψ1 , . . . , ψn be ξj (t)-invariant with respect to U, and let ψjex , f be defined by (2.4). Then the relations ξj (t) ψjex , f = ψjex , Ug(t) f ,
1 j n, ∀t ∈ T,
(4.16)
are satisfied for all f ∈ D(A∗sym ) if and only if the corresponding self-adjoint operator A∞ defined by (3.3) is p(t)-homogeneous with respect to U. Proof. Denote ⎛
ξ1 (t) ⎜ 0 Ξ (t) = ⎜ ⎝ .. . 0
0 ξ2 (t) .. .
... ... .. .
0
...
⎞ 0 0 ⎟ .. ⎟ ⎠. .
(4.17)
ξn (t)
Then det Ξ (t) = 0, t ∈ T, by Proposition 4.5, since ψi is ξj (t)-invariant with respect to U. By using (2.5) in Lemma 2.2 the relations (4.16) can be rewritten as follows: Ξ (t)Γ0 f = Γ0 Ug(t) f,
∀f ∈ D A∗sym , ∀t ∈ T.
(4.18)
Since D(A∞ ) = ker Γ0 , (4.18) immediately implies that Ut (D(A∞ )) ⊂ D(A∞ ), cf. (4.2). Thus the equalities (4.16) ensure p(t)-homogeneity of A∞ with respect to U. Conversely, assume that A∞ is p(t)-homogeneous with respect to U. According to (3.3), (4.2), and (4.7) this is equivalent to −RΓ0 Ug(t) f = Γ1 Ug(t) f,
∀f ∈ D(A∞ ), ∀t ∈ T.
(4.19)
Using (4.9), (4.13), and (4.14) it is seen that Ug(t) hj = p(t)Gg(t) Ug(t) hj + I − p(t)Gg(t) Ug(t) hj =
p(t) hj + 1 − p(t) (A0 + I )−1 Ug(t) hj , ξj (t)
(4.20)
S. Hassi, S. Kuzhel / Journal of Functional Analysis 256 (2009) 777–809
795
where hj = (A0 + I )−1 ψj , j = 1, . . . , n. This expression and relations (2.6), (4.12) yield the following equalities for all f = u + j =1 αj hj ∈ D(A∗sym ) and t ∈ T: Γ0 Ug(t) f = p(t)Ξ −1 (t)Γ0 f,
Γ1 Ug(t) f = Ξ (t)Γ1 f + 1 − p(t) G (t)Γ0 f,
(4.21)
where G (t) is the transpose of the matrix G(t) = ((hi , Ut hj ))ni,j =1 . Now with f ∈ D(A∞ ) substituting these expressions into (4.19), using (3.3), and taking into account that Γ0 (D(A∞ )) = Cn , one concludes that the p(t)-homogeneity of A∞ is equivalent to the matrix equality Ξ (t)R − p(t)RΞ −1 (t) = 1 − p(t) G (t),
∀t ∈ T.
(4.22)
Finally, employing (2.7) and (4.21) it is easy to see that equality (4.22) is equivalent to (4.18). Therefore, the extended functionals ψjex , · satisfy the relations (4.16). Theorem 4.8 is proved. 2 Remark 4.9. In the particular case where p(t) = t β and ξ(t) = t θ with β, θ ∈ R, another condition for the preservation of ξ(t)-invariance for ψjex , · has been obtained in [5, Lemma 1.3.2]. of Asym transversal to A0 is p(t)-homogeneous if Corollary 4.10. A self-adjoint extension A is defined by (3.3) and the entries rij of R in (3.3) satisfy the following system of and only if A equations for all t ∈ T: βij (t)rij = 1 − p(t) (hj , Ut hi ),
p(t) , βij (t) = ξi (t) − ξj (t)
1 i, j n.
(4.23)
Proof. Since ker Γ0 = D(A0 ), formula (3.3) describe all self-adjoint extensions of Asym transversal to A0 when the parameter R = (rij )ni,j =1 runs the set of all Hermitian matrices. = A∞ for some choice of R in (3.3). The proof of Theorem 4.8 shows that A∞ is Hence, A p(t)-homogeneous if and only if R is a solution of (4.22) that does not depend on t ∈ T. Rewriting (4.22) componentwise one gets (4.23). 2 Remark 4.11. In the case that p(x) ≡ 1, the right-hand side of (4.23) vanishes and (4.23) reduces to βij (t)rij = 0, 1 i, j n. Moreover, by Proposition 4.5 βii (t) ≡ 0 and, therefore, the entries rii cannot be uniquely determined from (4.23). This implies the existence of infinitely many 1-homogeneous self-adjoint extensions of Asym transversal to A0 . be defined by Example 4.12. Let α > 0 and let A α ), α = A∗sym D(A A
α ) = D(Asym ) + ˙ ker A∗sym + αI . D(A
α is a 1-homogeneous self-adjoint extensions of Asym transversal to A0 . Then for all α > 0, A 4.3. Uniqueness of p(t)-homogeneous admissible large coupling limits of (1.3) Let the operator A0 be p(t)-homogeneous and let the singular elements ψj appearing in (1.3) be ξj (t)-invariant with respect to U.
796
S. Hassi, S. Kuzhel / Journal of Functional Analysis 256 (2009) 777–809
If all ψj belong to H−1 (A0 ), then the extended functionals ψjex , · are determined by continuity onto D(A∗sym ) and they automatically possess the property of ξj (t)-invariance (4.16), since Ut D(A0 ) can be extended by continuity onto H1 (A0 ). In this case, the set of admissible large coupling limits consists of a unique element (the Friedrichs extension AF , see Corollary 3.7) and this operator is p(t)-homogeneous. If H−1 (A0 ) does not contain all ψj , then admissible large coupling limits A∞ of (1.3) are not determined uniquely. In this case, the natural assumption of ξj (t)-invariance for the extended functionals ψjex , · can be used to select a unique operator A∞ . By Theorem 4.8 the ξj (t)invariance of ψjex , · is equivalent to the p(t)-homogeneity of the corresponding operator A∞ defined by (3.3). Therefore, instead of assumption of ξj (t)-invariance one can use the requirement of p(t)-homogeneity imposed on the set of admissible large coupling limits A∞ of (1.3) to achieve their uniqueness. Theorem 4.13. Assume that the singular elements ψj in (1.3) are H−1 (A0 )-independent and the system of equations (4.23) has a unique solution R = (rij )ni,j =1 that does not depend on t ∈ T. Then there exists a unique p(t)-homogeneous admissible large coupling limit A∞ of (1.3) and it coincides with the Krein–von Neumann extension AN of Asym . Proof. Let R = (rij )ni,j =1 be a unique solution of (4.23) and let A∞ be the corresponding selfadjoint extension of Asym determined by (3.3). Since (4.23) has a unique solution, p(t) = 1 for at least one point t ∈ T (see Remark 4.11). In this case, Lemma 4.6 and relation (3.3) imply that A∞ is a nonnegative extension of Asym transversal to A0 . Then also AF and AN are transversal extensions of Asym ; cf. the proof of Theorem 3.10. These extensions are also p(t)-homogeneous (see Lemmas 4.7, 4.4). Since elements ψj in (1.3) form an H−1 (A0 )-independent system, Corollary 3.8 gives that any self-adjoint extension of Asym transversal to A0 is an admissible large coupling limit of (1.3) and A0 = AF . The unique solution of (4.23) allows one to select a unique p(t)-homogeneous self-adjoint extension A∞ of Asym transversal to A0 = AF . Obviously, it coincides with the Krein–von Neumann extension AN . 2 The next statement concerns to the general case. Theorem 4.14. Let AF and AN be transversal, let the operator S defined in (3.10) be p(t) satisfying conditions (3.12), and assume that for every βij (t) homogeneous for some choice of H in (4.23) there exists at least one point tij ∈ T such that βij (tij ) = 0. Then there exists a unique p(t)-homogeneous admissible large coupling limit of (1.3). be the Krein–von Neumann extension of S. The second part of the proof of TheProof. Let A is an admissible large coupling limit of (1.3). By Lemma 4.4, A is orem 3.10 shows that A p(t)-homogeneous. Its uniqueness follows from the fact that condition βij (tij ) = 0 ensures in view of (4.23) the uniqueness of p(t)-homogeneous self-adjoint extensions of Asym transversal to A0 . 2 The next statement contains conditions for the p(t)-homogeneity of the symmetric operator S defined by (3.10) in Lemma 3.5 which appear to be useful in applications.
S. Hassi, S. Kuzhel / Journal of Functional Analysis 256 (2009) 777–809
797
Proposition 4.15. Let A0 be p(t)-homogeneous, let the singular elements ψj in (1.3) be ξj (t) Then: invariant with respect to U, and let Y = (A0 + I )(H H). (i) S is p(t)-homogeneous if and only if Y is invariant under Ut , t ∈ T, and ∀t ∈ T0 = t ∈ T: p(t) = 1 . h , Ut h⊥ ∈ H H, h⊥ = 0, ∀h ∈ H , ∀
(4.24)
= H is p(t)-homogeneous if and only if (4.24) (ii) If Gt Ut , t ∈ T, is self-adjoint, then S with H holds. (iii) If Y is a linear span of some singular elements ψj in (1.3), then S is p(t)-homogeneous if and only if (4.24) holds. Hence, if S is p(t)Proof. (i) The definition (3.10) shows that ker(S ∗ + I ) = H H. =HH by Corollary 4.3. According to homogeneous with respect to U then Gt Ut (H H) is in is invariant under Gt Ut if and only if Y = (A0 + I )(H H) (4.11) the subspace H H variant under the operator Ut , t ∈ T. Thus, if S is p(t)-homogeneous with respect to U then Y is invariant under Ut , t ∈ T. By Lemma 4.7, A∗sym is p(t)-homogeneous with respect to U. Since S is an intermediate extension of Asym its p(t)-homogeneity is equivalent to the relation Ug(t) (D(S)) ⊂ D(S), t ∈ T, see (4.7). The definition of S in (3.10) implies that h⊥ ∈ H H. (AF + I )Ug(t) f, h⊥ = 0, ∀ (4.25) Ug(t) f ∈ D(S) ⇔ Now let f = h + u ∈ D(S) be decomposed as in Lemma 3.5, see (3.13), (3.14). It follows from (4.20) that (AF + I )Ug(t) f = A∗sym + I Ug(t) f = 1 − p(t) Ug(t) h + (A0 + I )Ug(t) u. By taking (4.12) into account one obtains (AF + I )Ug(t) f, h⊥ = 1 − p(t) Ug(t) h , h⊥ + (A0 + I )Ug(t) u, h⊥ = 1 − p(t) h , Ut h⊥ + Ut ψ, u.
(4.26)
If Y is invariant under Ut , t ∈ T, then Ut ψ, u = 0 for all f = h + u ∈ D(S). Now (4.25) and (4.26) show that S is p(t)-homogeneous if and only if Y is invariant under Ut and (4.24) holds. (ii) Since A0 and AF are p(t)-homogeneous, the symmetric restriction S0 := AF ∩ A0 and its adjoint S0∗ are also p(t)-homogeneous, see Lemma 4.2. It follows from (3.15) that f ∈ D(S0 ) if and only if f ∈ D(A0 ) and (A0 + I )f, h = 0, ∀h ∈ H = H ∩ H1 (A0 ). Hence, ker(S0∗ + I ) = H and Gt Ut H = H for all t ∈ T by Corollary 4.3. Similarly Gt Ut H = H for all t ∈ T, since Asym is p(t)-homogeneous. Therefore, if Gt Ut is self-adjoint, then H and H are reducing subspaces for the operators Gt Ut and consequently Gt Ut H ⊂ H is satisfied for all t ∈ T. Then, according to (4.11), Y = (A0 + I )H is invariant under Ut . Now the claim = H . = H and H H follows from part (i) with H
798
S. Hassi, S. Kuzhel / Journal of Functional Analysis 256 (2009) 777–809
(iii) If Y has a basis formed by some ξj (t)-invariant singular elements ψj , then Y is invariant under Ut , see (1.9). So, the statement is reduced to (i). 2 4.4. The case of rank one singular perturbations In the case of rank one singular perturbations A0 + bψ, ·ψ , where A0 is p(t)-homogeneous and ψ is ξ(t)-invariant, the system (4.23) takes the form 2 ξ (t) − p(t) r = ξ(t) 1 − p(t) (h, Ut h)
h = (A0 + I )−1 ψ , ∀t ∈ T.
(4.27)
Proposition 4.16. (1) If (4.27) has no solutions r ∈ R, then there is only one p(t)-homogeneous extension A0 = AF = AN and any self-adjoint extension of Asym different from A0 has a negative eigenvalue. (2) If (4.27) has at least two solutions r1 , r2 ∈ R, then all self-adjoint extensions of Asym are p(t)-homogeneous. (3) If (4.27) has a unique solution r ∈ R, then the symmetric operator Asym associated with A0 + bψ, ·ψ possesses exactly two p(t)-homogeneous extensions: the Friedrichs AF and the Krein–von Neumann AN extensions. One of them coincides with A0 , another one is the unique p(t)-homogeneous admissible large coupling limit A∞ of A0 + bψ, ·ψ. More precisely, A0 = AF and A∞ = AN if ψ ∈ H−2 (A0 ) \ H−1 (A0 ); A0 = AN and A∞ = AF if ψ ∈ H−1 (A0 ). Proof. In the case of rank one perturbations, an arbitrary self-adjoint extension A( = A0 ) of the symmetric operator Asym = A0 {u ∈ D(A0 ): ψ, u = 0} is transversal to A0 . This means that there is a one-to-one correspondence between the set of solutions r ∈ R of (4.27) and the set of p(t)-homogeneous self-adjoint extensions A( = A0 ) of Asym . By Lemmas 4.4, 4.7 the symmetric operator Asym and its Friedrichs AF and Krein–von Neumann AN extensions are p(t)-homogeneous. Therefore, if (4.27) has no solutions, then AN = AF = A0 that justifies assertion (1). Two different solutions of (4.27) may appear only in the case where ξ 2 (t) = p(t) and (1 − p(t))(h, Ut h) = 0 for all t ∈ T. But these equalities are equivalent to the fact that any r ∈ R is a solution of (4.27). Therefore, an arbitrary self-adjoint extension of Asym is p(t)-homogeneous. Assertion (2) is proved. Finally, assume that (4.27) has a unique solution. It follows from Corollary 4.10 that the set of all p(t)-homogeneous extensions of Asym is exhausted by the Friedrichs AF and the Krein–von Neumann AN extensions. One of them coincides with A0 , another one is the unique p(t)-homogeneous admissible large coupling limit A∞ . To complete the proof it suffices to use Theorem 4.13 for ψ ∈ H−2 (A0 ) \ H−1 (A0 ) and Corollary 3.7 for ψ ∈ H−1 (A0 ). 2 Example 4.17. One point interaction in Rn (n = 1, 2, 3). Consider the singular rank one perturbation − + bδ, ·δ(x), where A0 = − (D(A0 ) = W22 (Rn )) is the Laplace operator in H = L2 (Rn ) and the associated symmetric operator Asym = − {u(x) ∈ W22 (Rn ): u(0) = 0}. The operator A0 is t −2 -homogeneous with respect to the set of scaling transformations U = {Ut }t∈(0,∞) in L2 (Rn ), where Ut f (x) = t n/2 f (tx). Furthermore, the singular element ψ = δ is t −n/2 -invariant (cf. [5]).
S. Hassi, S. Kuzhel / Journal of Functional Analysis 256 (2009) 777–809
799
If n = 1, then δ(x) ∈ H−1 (A0 ) = W2−1 (R), Eq. (4.27) has a unique solution and by Proposition 4.16 the free Laplace operator − coincides with the Krein–von Neumann extension AN of Asym . The Friedrichs extension AF has the form AF = −d 2 /dx 2 {u(x) ∈ W22 (R \ {0}) ∩ W21 (R): u(0) = 0}. If n = 2, then (4.27) has no solutions and there exists the unique nonnegative self-adjoint extension − = AN = AF of Asym . If n = 3, then δ(x) ∈ W2−2 (R3 ) \ W2−1 (R3 ), Eq. (4.27) has a unique solution and − = AF . The Krein–von Neumann extension AN has the form AN f (x) = − u(x) − u(0)
e−|x| , |x|
e−|x| D(AN ) = f = u(x) + u(0) : u ∈ W22 R3 . |x|
Another description of the Krein–von Neumann extension of Asym obtained with the aid of the Fourier transformation can be founded in [12]. 5. Operator realizations in the case of singular perturbations with symmetries In this section, operator realizations AB of (1.3) given by formulas (2.8), (2.9) are studied under the condition that the unperturbed operator A0 and the singular elements ψj in (1.3) are, respectively, p(t)-homogeneous and ξj (t)-invariant with respect to U. 5.1. p(t)-Homogeneous operator realizations Theorem 5.1. Let an admissible large coupling limit A∞ of (1.3) be chosen to be p(t)homogeneous. Then the operator AB defined by (2.8) is p(t)-homogeneous if and only if the relations ξi (t)ξj (t) = p(t),
∀t ∈ T,
hold for all indices 1 i, j n corresponding to non-zero entries bij of B. Proof. By Lemma 4.7, the operator A∗sym is p(t)-homogeneous. Hence, in view of (4.7), AB is p(t)-homogeneous if and only if Ug(t) : D(AB ) → D(AB ), ∀t ∈ T. By (2.8), this relation can be rewritten as BΓ0 Ug(t) f = Γ1 Ug(t) f,
∀t ∈ T, ∀f ∈ D(AB ).
(5.1)
Since the admissible large coupling limit A∞ is p(t)-homogeneous, the boundary operator Γ0 satisfies (4.18). Therefore, BΓ0 Ug(t) f = BΞ (t)Γ0 f . On the other hand, relations (2.7) and (4.21) lead to the equality Γ1 Ug(t) f = p(t)Ξ −1 (t)Γ1 f,
∀f ∈ D A∗sym .
(5.2)
The last two equalities and (2.8) show that the relation (5.1) is equivalent to the matrix equality Ξ (t)BΞ (t) = p(t)B, t ∈ T. Rewriting this componentwise, one obtains the equalities ξi (t)ξj (t)bij = p(t)bij , 1 i, j n. 2
800
S. Hassi, S. Kuzhel / Journal of Functional Analysis 256 (2009) 777–809
Corollary 5.2. If there exists a point t0 ∈ T such that p(t0 ) = 1 and relations ξi (t0 )ξj (t0 ) = p(t0 ) hold for all indices 1 i, j n corresponding to non-zero entries bij of B, then: (i) the point λ = 0 belongs to the essential spectrum of AB and λ ∈ σ (AB ) ⇔ λp(t0 )n ∈ σ (AB ), n ∈ Z; (ii) the operator AB is nonnegative if and only if the matrix B is Hermitian. Proof. If the matrix B satisfies the conditions above, then AB is p(t)-homogeneous with respect to the family U0 := {Ut ∈ U: t ∈ {t0 , g(t0 )}}. Now, to establish (i), it suffices to use Lemma 4.1 with A = AB . Obviously, the matrix B is Hermitian if and only if the operator AB defined by (2.8) is selfadjoint. Using Lemma 4.6 and Theorem 5.1 one derives (ii). 2 Proposition 5.3. Assume that the singular elements ψj in (1.3) form a H−1 (A0 )-independent orthonormal system in H−2 (A0 ), the system (4.23) has a unique solution R, and a p(t)homogeneous admissible large coupling limit A∞ of (1.3) is chosen. Then a self-adjoint operator realization AB of (1.3) is nonnegative if and only if det(BR + E) = 0 and 0 −(BR + E)−1 B −R−1 , where E stands for the identity matrix. Proof. By Theorem 4.13, the Krein–von Neumann extension AN of Asym coincides with a p(t)homogeneous admissible large coupling limit A∞ and it is defined by (3.3), where R is the solution of (4.23). Furthermore, the Friedrichs extension AF coincides with A0 . Combining these observations with [33, Theorem 3] the statement follows. For completeness some of the details are repeated here. By (3.21) a self-adjoint operator AB is nonnegative if and only if −1 ∈ ρ(AB ) and 0 CB C N ,
(5.3)
where CB = (AB + I )−1 − (A0 + I )−1 and CN = (AN + I )−1 − (A0 + I )−1 are self-adjoint operators in H = ker(A∗sym + I ). It follows from (2.7) and (2.8) that D(AB ) = f ∈ D A∗sym : BΓ1 f = −(BR + E)Γ0 f .
(5.4)
Relations (2.5) and (5.4) imply −1 ∈ ρ(AB ) ⇔ D(AB ) ∩ H = {0} ⇔ det(BR + E) = 0. Since the elements ψj are orthonormal in H−2 (A0 ), the corresponding vectors hj in (2.3) form an orthonormal basis of H. In that case, the domain D(AB ) can be also presented as D(AB ) = {f ∈ D(− ∗sym ): CB Γ1 f = Γ0 f }, where CB is the matrix representation of CB with respect to the basis {hj }n1 . Comparing this with (5.4) one gets CB = −(BR + E)−1 B. Similar reasonings for the operator AN defined by (3.3) give det R = 0 (since −1 ∈ ρ(AN )) and CN = −R−1 . By substituting the obtained expressions for CB and CN into (5.3) one completes the proof. 2 Remark 5.4. A description of nonnegative self-adjoint operator realizations of (1.3) given above is based on the specific form of boundary operators Γi . A general approach to the description of nonnegative self-adjoint extensions of a symmetric operator has been proposed recently in [12].
S. Hassi, S. Kuzhel / Journal of Functional Analysis 256 (2009) 777–809
801
5.2. The Weyl function and the resolvent formula Let (Cn , Γ0 , Γ1 ) be the boundary triplet of A∗sym constructed in Lemma 2.2 and let A∞ be a self-adjoint extension of Asym defined by (3.3). The γ -field γ (z) and the Weyl function M(z) associated with the boundary triplet (Cn , Γ0 , Γ1 ) are defined by γ (z) = (Γ0 Hz )−1 ,
M(z) = Γ1 γ (z),
z ∈ ρ(A∞ ),
(5.5)
see [16,17]. Here Hz = ker(A∗sym − zI ), z ∈ C, denote the defect subspaces of Asym . The mappings Γi are defined by (2.5) and M(z) is an n × n-matrix function. Theorem 5.5. The operator A∞ is p(t)-homogeneous if and only if for at least one point z = z0 ∈ C \ R (and then for all non-real points z) the Weyl function M(z) satisfies the relation p(t)M(z) = Ξ (t)M p(t)z Ξ (t),
∀t ∈ T,
(5.6)
where Ξ (t) is defined by (4.17). Proof. Let fz ∈ Hz , z ∈ C. Then Lemma 4.1 and relation (4.1) imply Ug(t) fz ∈ ker A∗sym −
z I p(g(t))
= ker A∗sym − p(t)zI = Hp(t)z .
(5.7)
Putting f = fz ∈ Hz in (5.2), using (5.7), and observing that M(z)Γ0 fz = Γ1 fz , z ∈ C (see (5.5)), one can rewrite (5.2) as follows: M p(t)z Γ0 Ug(t) fz = p(t)Ξ −1 (t)M(z)Γ0 fz .
(5.8)
If the identity (5.6) holds for some non-real z = z0 , then (5.8) implies that Γ0 Ug(t) f = Ξ (t)Γ0 f
(5.9)
for all f = fz0 ∈ Hz0 . Since M∗ (z) = M(z) [16] and hence, (5.6) holds for z0 , the relation (5.9) is also true for f = fz0 ∈ Hz0 . Moreover, (5.9) holds for all f ∈ D(Asym ) since Γ0 f = Γ0 Ug(t) f = 0 by (1.4). Consequently, (5.9) is true on the domain D(A∗sym ) = ˙ Hz0 + ˙ Hz0 . By Theorem 4.8 this provides the p(t)-homogeneity of A∞ . D(Asym ) + Conversely, assume that A∞ is p(t)-homogeneous. In this case, (5.9) holds for all f ∈ D(A∗sym ) (see (4.18)). But then, for all non-real z and all fz ∈ Hz , (5.9) (5.7) M p(t)z Ξ (t)Γ0 fz = M p(t)z Γ0 Ug(t) fz = Γ1 Ug(t) fz (5.2)
= p(t)Ξ −1 (t)Γ1 fz =p(t)Ξ −1 (t)M(z)Γ0 fz
that justifies (5.6). Theorem 5.5 is proved.
2
802
S. Hassi, S. Kuzhel / Journal of Functional Analysis 256 (2009) 777–809
Let AB be a self-adjoint realization of (1.3) defined by (2.8). Then the resolvents of AB and A∞ are connected via Krein’s formula −1 (AB − zI )−1 = (A∞ − zI )−1 + γ (z) B − M(z) γ (z)∗ ,
z ∈ ρ(AB ) ∩ ρ(A∞ ).
(5.10)
The explicit form of M(z) can be found as follows. By (2.7) it is easy to see that the Weyl functions M(z) and M(z) associated with the boundary triplets (2.5) and (2.6), respectively, are connected via the linear fractional transform −1 M(z) = − R + M(z) ,
z ∈ C \ R.
(5.11)
The boundary triplet (2.6) is one of the most used boundary triplets and the corresponding Weyl function M(z) is studied well. In particular, if the singular elements ψj in (1.3) form an orthonormal system in H−2 , then (see [16, Remark 4])
M(z) = (z + 1)PH I + (z + 1)(A0 − zI )−1 PH . By combining this relation with (5.11) one gets an explicit form for M(z). Example 5.6. A point interaction for p-adic Schrödinger type operator. Let p be a fixed prime number and let Qp be the field of p-adic numbers. The operation of differentiation is not defined in the p-adic analysis of complex-valued functions defined on Qp and the Vladimirov operator of the fractional p-adic differentiation D α f (x) =
pα − 1
1 − p−1−α
Qp
f (x) − f (y) |x − y|1+α p
dμ(y),
α > 0,
is used as an analog of it (see [27] for details). Here | · |p and dμ(y) are, respectively, the p-adic norm and the Haar measure on Qp . The operator D α is positive and self-adjoint in the Hilbert space L2 (Qp ) of complex-valued square integrable functions on Qp . p-Adic Schrödinger-type operators with potentials V (x) : Qp → C are defined as D α + V (x). Denote T = {t = pn : n ∈ Z} and consider a family U = {Ut }t∈T of unitary operators Ut f (x) = t −1/2 f (tx) acting in L2 (Qp ). Obviously, Ut satisfies (1.7) with the function of conjugation g(t) = 1/t, cf. (4.2). It follows from [28] that Ut D α = t α D α Ut , t ∈ T. Hence, D α is t α -homogeneous with respect to U. Since D α is a p-adic pseudo-differential operator its domain of definition D(D α ) need not contain functions continuous on Qp and, in general, it may happen that the formal expression D α + bδ, ·δ(x),
b ∈ R,
(5.12)
and the associated symmetric operator Asym = D α {u(x) ∈ D(D α ): u(0) = 0} are not defined on D(D α ). It is known [35] that the domain D(D α ) consists of continuous functions on Qp and the Dirac delta function δ(x) is well defined on H2 (D α ) = D(D α ) if and only if α > 1/2. √ Furthermore, δ(x) is t-invariant with respect to U and δ(x) ∈ H−2 (D α ) \ H−1 (D α ) if 1/2 < α 1, while δ(x) ∈ H−1 (D α ) if α > 1.
S. Hassi, S. Kuzhel / Journal of Functional Analysis 256 (2009) 777–809
803
It follows from [27, Lemma 3.7] and [35, Lemma 2.1] that p−1 ∞ α
−1 −1 h(x) = D + I δ= p−N/2 pα(1−N ) + 1 ψNj 0 (x),
N =−∞ j =1
where the functions ψNj 0 (x) (N ∈ Z, j = 1, . . . , p − 1) form a part of the p-adic wavelet basis {ψNj (x)} recently constructed in [28]. Eq. (4.27) takes the form √ t − t α r = t 1 − t α (h, Ut h), ∀t ∈ T. (5.13) A simple analysis shows that (5.13) has no solutions r ∈ R for α = 1. In that case the initial operator D 1 is a unique nonnegative self-adjoint extension of Asym , see Proposition 4.16. If α = 1 (α > 1/2), then (5.13) has a unique solution r ∈ R that determines a unique t α -homogeneous admissible large coupling limit A∞ of (5.12) by the formula (cf. (3.3)) α u(0) u(0) α h(x), D(A∞ ) = f = u(x) − h(x): u ∈ D D . A∞ f (x) = D u(x) + r r In view of Proposition 4.16, the operator A∞ coincides with the Krein–von Neumann (Friedrichs) extension of Asym for 1/2 < α < 1 (respectively for α > 1). Let (Cn , Γ0 , Γ1 ) be the boundary triplet of A∗sym constructed in Lemma 2.2 so that ker Γ0 = D(A∞ ). By Theorem 2.3, self-adjoint operator realizations of (5.12) in L2 (Qp ) have the form Ab f = Ab (u + ch) = D α u − ch, ∀u ∈ D(D α ), where the parameter c = c(u, b) ∈ C is uniquely determined by the relation bu(0) = −c[1 + br]. Since ξ 2 (t) = t = t α = p(t) (α = 1), Theorem 5.1 shows that Ab is t α -homogeneous if and only if b = 0 or b = ∞. Let α > 1. It follows from [7] that the Weyl function associated with (Cn , Γ0 , Γ1 ) has the form M(z) = −
(p − 1)
∞
1 p−N
.
N =−∞ pα(1−N) −z
By virtue of Theorem 5.5, M(z) satisfies the relation t α−1 M(z) = M(t α z), ∀t ∈ T. This simplifies the spectral analysis of Ab , see [7] for details. Example 5.7. A general zero-range potential in R. A one-dimensional Schrödinger operator corresponding to a general zero-range potential at the point x = 0 can be given by the expression A0 + b11 δ, ·δ(x) + b12 δ , ·δ(x) + b21 δ, ·δ (x) + b22 δ , ·δ (x), where A0 = −d 2 /dx 2 (D(A0 ) = W22 (R)) acts in H = L2 (R), δ (x) is the derivative of the Dirac δ-function (with support at 0). In this case, Asym = −d 2 /dx 2 {u(x) ∈ W22 (R): u(0) = u (0) = 0} and the corresponding Friedrichs and Krein–von Neumann extensions are transversal (see, e.g., [10]). The functions 1 e−x , x > 0, h (x) = (A0 + I )−1 ψ2 = −(sign x)h (x), h (x) = (A0 + I )−1 ψ1 = x < 0, 2 ex ,
804
S. Hassi, S. Kuzhel / Journal of Functional Analysis 256 (2009) 777–809
where ψ1 = δ(x) and ψ2 = δ (x), form an orthogonal basis of H = ker(A∗sym + I ) such that H = H ∩ H1 (A0 ) = h (x) and H = H H = h (x). Define U = {Ut }t∈[0,∞) as a collection of the space parity √ operator U0 f (x) = f (−x) (f (x) ∈ L2 (R)) and the set of scaling transformations Ut f (x) = tf (tx), t > 0. In this case, A0 is p(t)homogeneous with respect to U, where p(0) = 1 and p(t) = t −2 if t > 0. The elements ψj (j = 1, 2) are ξj (t)-invariant, where ξ1 (0) = 1, ξ1 (t) = t −1/2 (t > 0) and ξ2 (0) = −1, ξ2 (t) = t −3/2 (t > 0). Furthermore, for such a choice of U, T0 = {t ∈ [0, ∞): p(t) = 1} = (0, ∞) and
(h , Ut h ) = t
∞ 1/2
h (x)h (tx) dx = 0,
∀t ∈ T0 .
−∞
= H . Then Y = (A0 + I )H = ψ2 and part (iii) of Proposition 4.15 implies Let us put H that the corresponding operator S defined by (3.10) is p(t)-homogeneous. Calculating βij (t) in (4.23) for ξ1 (t), ξ2 (t), and p(t) as given above, it is easy to see that βij (0) = 0 if i = j and βii (t) = 0 for all t > 0. In this case, by Theorem 4.14 there exists a unique p(t)-homogeneous admissible large coupling limit A∞ . To identify A∞ it suffices to determine the entries rij of R in (3.3) with the aid of (4.23): for t = 0, (4.23) takes the form −2r0 21 2r012 = 0 and, hence, r12 = r21 = 0; on the other hand, for t > 0 calculating both sides of (4.23) leads to √ t 2(1+t) 0 0 r11 −3/2 −2 √ = 1−t t (t − 1) t 0 −r22 0 2(1+t)
and thus r11 = 1/2, r22 = −1/2. Substituting the coefficients rij in (2.4) results in the wellknown extensions of δ(x) and δ (x) onto D(A∗sym ) = W22 (R \ {0}) (see [5]): δex , f =
f (+0) + f (−0) , 2
f (+0) + f (−0) δex . ,f = − 2
The corresponding operator A∞ is the restriction of −d 2 /dx 2 to D(A∞ ) = {f (x) ∈ −f (−0) = f (+0), −f (−0) = f (+0)} and A∞ is transversal to the singular perturbations AB of A0 that are determined by (2.9). It follows from Theorem 5.1 that AB is t −2 -homogeneous with respect to the scaling transfor mations Ut (t > 0) if and only if B = b021 b012 . In that case AB = A∗B (i.e., b21 = b12 ) ⇔ AB 0 (by Corollary 5.2).
W22 (R \ {0}):
6. Schrödinger operators with singular perturbations ξ(t)-invariant with respect to scaling transformations in R 3 It is well known (see, e.g., [5,13]) that the Schrödinger operator A0 = − (D( ) = W22 (R3 )), is t −2 -homogeneous with respect to the set of scaling transformations U = {Ut }t∈(0,∞) (Ut f (x) = t 3/2 f (tx)) in L2 (R3 ). It is clear that Ut satisfies (1.7) with the function of conjugation g(t) = 1/t. The elements Ut of U possess the additional multiplicative property Ut1 Ut2 = Ut2 Ut1 = Ut1 t2 that enables one to describe all measurable functions ξ(t) for which there exist ξ(t)-invariant singular elements ψ ∈ W2−2 (R3 ).
S. Hassi, S. Kuzhel / Journal of Functional Analysis 256 (2009) 777–809
805
Theorem 6.1. Let ξ(t) be a real measurable function defined on (0, ∞). Then ξ(t)-invariant singular elements ψ ∈ W2−2 (R3 ) \ L2 (R3 ) exist if and only if ξ(t) = t −α , where 0 < α < 2. Proof. Let ψ ∈ W2−2 (R3 ) \ L2 (R3 ) be ξ(t)-invariant with respect to U. Since Ut1 Ut2 = Ut2 Ut1 = Ut1 t2 , equality (1.9) gives ξ(t1 )ξ(t2 ) = ξ(t1 t2 ) (ti > 0) that is possible only if ξ(t) = 0 or ξ(t) = t −α (α ∈ R) [24, Chapter IV]. Furthermore, Proposition 4.5 enables one to restrict the set of possible functions ξ(t) as follows: ξ(t) = t −α , where 0 < α < 2. To complete the proof of Theorem 6.1 it suffices to construct t −α -invariant singular elements for 0 < α < 2. Fix m(w) ∈ L2 (S 2 ), where L2 (S 2 ) is the Hilbert space of square-integrable functions on the unit sphere S 2 in R3 , and determine the functional ψ(m, α) ∈ W2−2 (R3 ) by the formula 2 m(w) ψ(m, α), u = |y| + 1 (6.1) u(y) dy y = |y|w ∈ R3 , |y|3/2−α (|y|2 + 1) R3
where u(y) = (2π)1 3/2 R3 eix·y u(x) dx is the Fourier transformation of u(·) ∈ W22 (R3 ). It is easy to verify that 1 (U u)(y) = (U u)(y) = eiy·x u(x/t) dx = Ut u(y) = t 3/2 u(ty). 1/t g(t) (2πt)3/2
(6.2)
R3
Using (6.1) and (6.2), one obtains ψ(m, α), Ug(t) u = t −α ψ(m, α), u for all u ∈ W22 (R3 ). By (4.15) this means that the functional ψ(m, α) is t −α -invariant with respect to U. Theorem 6.1 is proved. 2 A more detailed study of functionals that are t −α -invariant with respect to scaling transformations and the results of [38] lead to the conclusion that the collection Lα of all t −α invariant singular elements ψ ∈ W2−2 (R3 ) \ L2 (R3 ) can be described as follows: Lα = {ψ = ψ(m, α) : m(w) ∈ L2 (S 2 ), m(w) = 0}. Let us consider the formal expression − +
n
bij ψj , ·ψi ,
bij ∈ C, n ∈ N,
(6.3)
i,j =1
where all singular elements ψj are assumed to be t −α -invariant with respect to scaling transformations for a fixed α, i.e., ψj = ψ(mj , α). The symmetric operator Asym = − sym associated with (6.3) takes the form − sym = − D( sym ) , D( sym ) = u(x) ∈ W22 R3 : ψj , u = 0, 1 j n , (6.4) where ψj , u are defined by (6.1). Comparing (1.2) and (6.1), one sees that the functions hj = (A0 + I )−1 ψ(mj , α) in (2.3) have the form hj (x) =
mj (w) 3/2−α |y| (|y|2 + 1)
∨
(x) =
mj (w) 3/2−α |y| (|y|2 + 1)
∧ (x),
(6.5)
806
S. Hassi, S. Kuzhel / Journal of Functional Analysis 256 (2009) 777–809
where the symbol ∨ denotes the inverse Fourier transformation. A simple analysis of (6.5) shows that hj ∈ L2 (R3 ) \ W21 (R3 ) for 1 α < 2 and hj ∈ W21 (R3 ) for 0 < α < 1. In the latter case, Corollary 3.7 and Lemma 4.4 imply that the Friedrichs extension − F is a unique t −2 -homogeneous admissible large coupling limit of (6.3). Proposition 6.2. Let 1 < α < 2. Then the Krein–von Neumann extension − N of − sym is a unique t −2 -homogeneous admissible large coupling limit of (6.3). Proof. If 1 < α < 2, then all the elements ψj in (6.3) are W2−1 (R3 )-independent. Let us show that the system (4.23) has a unique solution R = (rij )ni,j =1 that does not depend on t > 0. Since the both parts of (4.23) are equal to zero for t = 1, one can suppose that t > 0 and t = 1. It follows from (6.2) and (6.5) that
∧ ∧ mi (w) mi (w) (x) = U (x) 1/t |y|3/2−α (|y|2 + 1) |y|3/2−α (|y|2 + 1) ∧ mi (w) 2−α =t (x). |y|3/2−α (|y|2 + t 2 )
Ut hi (x) = Ut
Hence, (hj , Ut hi ) = t
2−α R3
mi (w)mj (w) dy |y|3−2α (|y|2 + t 2 )(|y|2 + 1) ∞
= (mi , mj )L2 0
= cα where cα =
∞ 0
|y|3−2α |y|2 +1
t 2−α d|y| |y|1−2α (|y|2 + t 2 )(|y|2 + 1)
t α − t 2−α (mi , mj )L2 , t2 − 1
d|y| and (mi , mj )L2 =
S2
mi (w)mj (w) dw is the scalar product in
L2 (S 2 ). Substituting the expression for (hj , Ut hi ) into (4.23) one gets a unique solution R = (rij )ni,j =1 , where rij = −cα (mi , mj )L2 . By Theorem 4.13, the obtained solution determines a unique t −2 -homogeneous admissible large coupling limit A∞ of (6.3) that coincides with − N . 2 Remark 6.3. If α = 1, then (4.23) has no solution, there are no t −2 -homogeneous admissible large coupling limits of (6.3), and the Friedrichs − = − F and the Krein–von Neumann − N extensions of − sym are not transversal. Corollary 6.4. For a fixed 1 < α < 2 assume that ψj = ψ(mj , α) in (6.3) form an orthonormal system in W2−2 (R3 ) and self-adjoint operator realizations AB = − B of (6.3) are defined by
S. Hassi, S. Kuzhel / Journal of Functional Analysis 256 (2009) 777–809
807
(2.8) with ker Γ0 = D(− N ). Then − B is nonnegative if and only if det(βα B − E) = 0 and 0 βα B[βα B − E]−1 E, where ∞ βα = 0
∞ −1 |y|3−2α 1 d|y| d|y| . |y|2 + 1 |y|1−2α (|y|2 + 1)2
(6.6)
0
Proof. Since ψ(mj , α) are orthonormal in W2−2 (R3 ) the functions hj (x) determined by (6.5) are orthonormal in L2 (R3 ). This means that ∞ (mi , mj )L2 = 0 (i = j )
and (mi , mi )L2 0
1 |y|1−2α (|y|2
+ 1)2
d|y| = 1.
The obtained relations allow one to rewrite the unique solution R = −cα ((mi , mj )L2 )ni,j =1 of (4.23) in a more explicit form: R = −βα E, where βα is defined by (6.6). Using Proposition 5.3 one completes the proof. 2 Note that the delta function δ(·) belongs to L3/2 . For this reason, the expression (6.3) where all ψj ∈ L3/2 can be considered as a generalization of the classical one-point interaction − + bδ, ·δ. In that case the parameter βα in Corollary 6.4 can be easily calculated: β3/2 = 2. Theorem 6.5. Let α = 3/2. Then for any self-adjoint operator realization AB = − B of (6.3) defined by (2.8), the following statements are true: (i) if − B is nonnegative, then the wave operators W± = limt→±∞ e−it B ei t exist and are unitary operators in L2 (R3 ); (ii) if − B is nonnegative and the singular elements ψj = ψ(mj , 3/2) in (6.3) form an orthonormal system in W2−2 (R3 ), then the S-matrix S(− B ,− ) = F W+∗ W− F −1 (F is the Fourier transformation in L2 (R3 )) of the Schrödinger equation iut = − B u coincides with the boundary value S(− B ,− ) (δ) (δ ∈ R) of the contractive operator-valued function S(− B ,− ) (z) = (E − 2izB)(E + 2izB)−1 ,
z ∈ C+ ,
(6.7)
analytic in the upper half-plane C+ . Proof. The statements follow from [34, Theorem 3.3] and [33, Section 4]. Remark 6.6. In [33] the expression (6.7) was obtained by using the Lax–Phillips scattering scheme. Another description of S(− B ,− ) (z) in terms of the Krein’s resolvent formula was obtained in [1]. In that paper, the stationary scattering theory approach has been used.
808
S. Hassi, S. Kuzhel / Journal of Functional Analysis 256 (2009) 777–809
Acknowledgments The authors thank S. Albeverio and Yu. Arlinskii for useful discussions and the referee for remarks which led to improvements in the paper. The first author (S.H.) is grateful for the support from the Research Institute for Technology of the University of Vaasa. The second author (S.K.) expresses his gratitude to the Academy of Finland (Projects 208056, 117656) for the support and the Department of Mathematics and Statistics of the University of Vaasa for the warm hospitality. References [1] V. Adamyan, B. Pavlov, Zero-radius potentials and M.G. Krein’s formula for generalized resolvents, Zap. Nauchn. Sem. LOMI 149 (1986) 7–23. [2] S. Albeverio, L. Dabrowski, P. Kurasov, Symmetries of Schrödinger operators with point interactions, Lett. Math. Phys. 45 (1998) 33–47. [3] S. Albeverio, F. Gesztesy, R. Høegh-Krohn, H. Holden, Solvable Models in Quantum Mechanics, Springer, Berlin, 1988. [4] S. Albeverio, P. Kurasov, Rank one perturbations, approximations and self-adjoint extensions, J. Funct. Anal. 148 (1997) 152–169. [5] S. Albeverio, P. Kurasov, Singular perturbations of differential operators, in: Solvable Schrödinger Type Operators, in: London Math. Soc. Lecture Note Ser., vol. 271, Cambridge Univ. Press, Cambridge, 2000. [6] S. Albeverio, S. Kuzhel, L. Nizhnik, Singularly perturbed self-adjoint operators in scales of Hilbert spaces, Ukrainian Math. J. 59 (2007) 723–744. [7] S. Albeverio, S. Kuzhel, S. Torba, p-Adic Schrödinger-type operator with point interactions, J. Math. Anal. Appl. 338 (2008) 1267–1281. [8] T. Ando, K. Nishio, Positive selfadjoint extensions of positive symmetric operators, Tôhoku Math. J. 22 (1970) 65–75. [9] Yu.M. Arlinskii, Positive spaces of boundary values and sectorial extensions of nonnegative symmetric operators, Ukrainian Math. J. 40 (1988) 8–15. [10] Yu.M. Arlinskii, S. Hassi, Z. Sebestyen, H.S.V. De Snoo, On the class of extremal extensions of a nonnegative operator, in: Oper. Theory Adv. Appl., vol. 127, Birkhäuser, Basel, 2001, pp. 41–81. [11] Yu.M. Arlinskii, E.R. Tsekanovskii, Some remarks of singular perturbations of self-adjoint operators, Methods Funct. Anal. Topology 9 (2003) 287–308. [12] Yu.M. Arlinskii, E.R. Tsekanovskii, On von Neumann’s problem in extension theory of nonnegative operators, Proc. Amer. Math. Soc. 131 (2003) 3143–3154. [13] H.L. Cycon, R.G. Froese, W. Kirsch, B. Simon, Schrödinger Operators with Applications to Quantum Mechanics and Global Geometry, Springer, Berlin, 1987. [14] V. Derkach, S. Hassi, H. de Snoo, Singular perturbations of self-adjoint operators, Math. Phys. Anal. Geometry 6 (2003) 349–384. [15] V. Derkach, S. Hassi, M. Malamud, H. de Snoo, Boundary relations and their Weyl families, Trans. Amer. Math. Soc. 358 (2006) 5351–5400. [16] V.A. Derkach, M.M. Malamud, Generalized resolvents and the boundary value problems for Hermitian operators with gaps, J. Funct. Anal. 95 (1991) 1–95. [17] V.A. Derkach, M.M. Malamud, The extension theory of Hermitian operators and the moment problem, J. Math. Sci. 73 (1995) 141–242. [18] M.L. Gorbachuk, V.I. Gorbachuk, Boundary-Value Problems for Operator-Differential Equations, Kluwer, Dordrecht, 1991. [19] S. Hassi, On the Friedrichs and the Kre˘ın–von Neumann extension of nonnegative relations, Acta Was. 122 (2004) 37–54. [20] S. Hassi, S. Kuzhel, On symmetries in the theory of singular perturbations, working papers of the University of Vaasa, 2006, 29 pp., http://lipas.uwasa.fi/julkaisu/sis.html. [21] S. Hassi, M. Malamud, H. de Snoo, On Krein’s extension theory of nonnegative operators, Math. Nachr. 274–275 (2004) 40–73. [22] S. Hassi, A. Sandovici, H.S.V. de Snoo, H. Winkler, A general factorization approach to the extension theory of nonnegative operators and relations, J. Operator Theory 58 (2007) 351–386.
S. Hassi, S. Kuzhel / Journal of Functional Analysis 256 (2009) 777–809
809
[23] S. Hassi, H. de Snoo, One-dimensional graph perturbations of self-adjoint relations, Ann. Acad. Sci. Fenn. Ser. A I Math. 22 (1997) 123–164. [24] E. Hille, R.S. Phillips, Functional Analysis and Semi-Groups, Amer. Math. Soc., Providence, RI, 1957. [25] A.A. Kiselev, B.S. Pavlov, N.N. Penkina, M.G. Suturin, Interaction symmetry in the theory of extensions technique, Teor. Mat. Phys. 91 (1992) 179–191. [26] A.N. Kochubei, About symmetric operators commuting with a family of unitary operators, Funktsional. Anal. i Prilozhen. 13 (1979) 77–78. [27] A.N. Kochubei, Pseudodifferential Equations and Stochastics over Non-Archimedian Fields, Dekker, New York, 2001. [28] S.V. Kozyrev, Wavelet analysis as a p-adic spectral analysis, Izv. Ross. Akad. Nauk Ser. Mat. 66 (2002) 149–158. [29] M.G. Krein, The theory of self-adjoint extensions of semibounded Hermitian operators and its applications, I, Mat. Sb. 20 (1947) 431–495. [30] P. Kurasov, Yu.V. Pavlov, On field theory methods in singular perturbation theory, Lett. Math. Phys. 64 (2003) 171–184. [31] S. Kuzhel, On the determination of free evolution in the Lax–Phillips scattering scheme for second-order operatordifferential equations, Math. Notes 68 (2000) 724–729. [32] A. Kuzhel, S. Kuzhel, Regular Extensions of Hermitian Operators, VSP, Utrecht, 1998. [33] S. Kuzhel, L. Matsyuk, On an application of the Lax–Phillips scattering approach in the theory of singular perturbations, Ukrainian Math. J. 80 (2005) 232–241. [34] S. Kuzhel, Ul. Moskalyova, The Lax–Phillips scattering approach and singular perturbations of Schrödinger operator homogeneous with respect to scaling transformations, J. Math. Kyoto Univ. 45 (2005) 265–286. [35] S. Kuzhel, S. Torba, p-Adic fractional differentiation operator with point interactions, Methods Funct. Anal. Topology 13 (2007) 169–180. [36] K.A. Makarov, E. Tsekanovskii, On μ-scale invariant operators, Methods Funct. Anal. Topology 13 (2007) 181– 186. [37] R.S. Phillips, The extension of dual subspaces invariant under an algebra, in: Proceedings of the International Symposium on Linear Spaces, Jerusalem, 1960, Jerusalem Academic Press, 1961, pp. 366–398. [38] E.M. Stein, G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, Princeton, NJ, 1971.
Journal of Functional Analysis 256 (2009) 810–864 www.elsevier.com/locate/jfa
Ricci curvature of Markov chains on metric spaces Yann Ollivier ENS Lyon, UMPA, 46 allée d’Italie, Lyon, France Received 20 December 2007; accepted 3 November 2008 Available online 28 November 2008 Communicated by Cédric Villani
Abstract We define the coarse Ricci curvature of metric spaces in terms of how much small balls are closer (in Wasserstein transportation distance) than their centers are. This definition naturally extends to any Markov chain on a metric space. For a Riemannian manifold this gives back, after scaling, the value of Ricci curvature of a tangent vector. Examples of positively curved spaces for this definition include the discrete cube and discrete versions of the Ornstein–Uhlenbeck process. Moreover this generalization is consistent with the Bakry–Émery Ricci curvature for Brownian motion with a drift on a Riemannian manifold. Positive Ricci curvature is shown to imply a spectral gap, a Lévy–Gromov–like Gaussian concentration theorem and a kind of modified logarithmic Sobolev inequality. The bounds obtained are sharp in a variety of examples. © 2008 Elsevier Inc. All rights reserved. Keywords: Ricci curvature; Markov chains; Metric geometry; Concentration of measure
0. Introduction In Riemannian geometry, positively curved spaces in the sense of Ricci curvature enjoy numerous properties, some of them with very natural probabilistic interpretations. A basic result involving positive Ricci curvature is the Bonnet–Myers theorem bounding the diameter of the space via curvature; let us also mention Lichnerowicz’s theorem for the spectral gap of the Laplacian (Theorem 181 in [7]), hence a control on mixing properties of Brownian motion; and the Lévy–Gromov theorem for isoperimetric inequalities and concentration of measure [27]. The scope of these theorems has been noticeably extended by Bakry–Émery theory [5,6], which E-mail address: [email protected]. 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.11.001
Y. Ollivier / Journal of Functional Analysis 256 (2009) 810–864
811
highlights the analytic and probabilistic significance of Ricci curvature; in particular, they show that in positive Ricci curvature, a logarithmic Sobolev inequality holds. We refer to the nice survey [30] and the references therein for a discussion of the geometric interest of lower bounds on Ricci curvature and the need for a generalized notion of positive Ricci curvature for metric measure spaces. Here we define a notion of Ricci curvature which makes sense for a metric space equipped with a Markov chain (or with a measure), and allows to extend the results above. Namely, we compare the transportation distance between the measures issuing from two given points to the distance between these points (Definition 3), so that Ricci curvature is positive if and only if the random walk operator is contracting on the space of probability measures equipped with this transportation distance (Proposition 20). Thus, the techniques presented here are a metric version of the usual coupling method; namely, Ricci curvature appears as a refined version of Dobrushin’s classical ergodic coefficient ([16,17], or e.g. Section 6.7.1 in [9]) using the metric structure of the underlying space. Our definition is very easy to implement on concrete examples. Especially, in ε-geodesic spaces, positive curvature is a local property (Proposition 19), as can be expected of a notion of curvature. As a result, we can test our notion in discrete spaces such as graphs. An example is the discrete cube {0, 1}N , which from the point of view of concentration of measure or convex geometry [29,35] behaves very much like the sphere S N , and is thus expected to somehow have positive curvature. Our notion enjoys the following properties: when applied to a Riemannian manifold equipped with (a discrete-time approximation of) Brownian motion, it gives back the usual value of the Ricci curvature of a tangent vector. It is consistent with the Bakry–Émery extension, and provides a visual explanation for the curvature contribution −∇ sym b of the drift term b in this theory. We are able to prove generalizations of the Bonnet–Myers theorem, of the Lichnerowicz spectral gap theorem and of the Lévy–Gromov isoperimetry theorem, as well as a kind of modified logarithmic Sobolev inequality. As a by-product, we get a new proof for Gaussian concentration and the logarithmic Sobolev inequality in the Lévy–Gromov or Bakry–Émery context (although with some loss in the numerical constants). We refer to Section 1.3 for an overview of the results. Some of the results of this text have been announced in a short note [40]. Historical remarks and related work. In the respective context of Riemannian manifolds or of discrete Markov chains, our techniques reduce, respectively, to Bakry–Émery theory or to a metric version of the coupling method. As far as I know, it had not been observed that these can actually be viewed as the same phenomenon. From the discrete Markov chain point of view, the techniques presented here are just a version of the usual coupling method using the metric structure of the underlying space. Usually the coupling method involves total variation distance (see e.g. Section 6.7.1 in [9]), which can be seen as a transportation distance with respect to the trivial metric. The coupling method is especially powerful in product or product-like spaces, such as spin systems. The work of Marton [32,33] emphasized the relationship between couplings and concentration of measure in product-like situations, so it is not surprising that we are able to get the same kind of results. The relationship between couplings and spectral gap is thoroughly explored in the works of Chen (e.g. [11,13,14]). The contraction property of Markov chains in transportation distance seems to make its appearance in Dobrushin’s paper [18] (in which the current wide interest in transportation distances originates), and is implicit in the widely used “Dobrushin criterion” for spin systems [18,20].
812
Y. Ollivier / Journal of Functional Analysis 256 (2009) 810–864
It later appears sporadically in the literature, as in Chen and Wang [14] (Theorem 1.9, as a tool for spectral gap estimates, using the coupling by reflection); at the very end of Dobrushin’s notes [19] (Dobrushin’s study of the topic was stopped by his death); in Bubley and Dyer [4] for the particular case of product spaces, after Dobrushin; in the second edition of [12, Section 5.3]; in Djellout, Guillin and Wu [15] in the context of dependent sequences of random variables to get Gaussian concentration results; in lecture notes by Peres [42] and in [44, p. 94]. See also the related work mentioned below. However, the theorems exposed in our work are new. From the Riemannian point of view, our approach boils down to contraction of the Lipschitz norm by the heat equation, which is one of the results of Bakry and Émery ([5,6], see also [1] and [43]). This latter property was suggested in [43] as a possible definition of a lower bound on Ricci curvature for diffusion operators in general spaces, though it does not provide an explicit value for Ricci curvature at a given point. Another notion of lower bound on Ricci curvature, valid for length spaces equipped with a measure, has been simultaneously introduced by Sturm [46], Lott and Villani [31], and Ohta [37] (see also [43] and [41]). It relies on ideas from optimal transportation theory and analysis of paths in the space of probability measures. Their definition keeps a lot of the properties traditionally associated with positive Ricci curvature, and is compatible with the Bakry–Émery extension. However, it has two main drawbacks. First, the definition is rather involved and difficult to check on concrete examples. Second, it is infinitesimal, and difficult to adapt to discrete settings [10]. Related work. After having written a first version of this text, we learned that related ideas appear in some recent papers. Joulin [28] uses contraction of the Lipschitz constant (under the name “Wasserstein curvature”) to get a Poisson-type concentration result for continuous-time Markov chains on a countable space, at least in the bounded, one-dimensional case. Oliveira [38] considers Kac’s random walk on SO(n); in our language, his result is that this random walk has positive coarse Ricci curvature, which allows him to improve mixing time estimates significantly. Notation. We use the symbol ≈ to denote equality up to a multiplicative universal constant (typically 2 or 4); the symbol ∼ denotes usual asymptotic equivalence. The word “distribution” is used as a synonym for “probability measure”. Here for simplicity we will mainly consider discrete-time processes. Similar definitions and results can be given for continuous time (see e.g. Section 3.3.4). 1. Definitions and statements 1.1. Coarse Ricci curvature In Riemannian geometry, positive Ricci curvature is characterized [43] by the fact that “small spheres are closer (in transportation distance) than their centers are.” More precisely, consider two very close points x, y in a Riemannian manifold, defining a tangent vector (xy). Let w be another tangent vector at x; let w be the tangent vector at y obtained by parallel transport of w from x to y. Now if we follow the two geodesics issuing from x, w and y, w , in positive curvature the geodesics will get closer, and will part away in negative curvature. Ricci curvature along (xy) is this phenomenon, averaged on all directions w at x. If we think of a direction w at x as a point on a small sphere Sx centered at x, this shows that, on average, Ricci curvature controls whether the distance between a point of Sx and the corresponding point of Sy is smaller or larger than the distance d(x, y).
Y. Ollivier / Journal of Functional Analysis 256 (2009) 810–864
813
In a more general context, we will use a probability measure mx depending on x as an analogue for the sphere (or ball) Sx centered at x.
Definition 1. Let (X, d) be a Polish metric space, equipped with its Borel σ -algebra. A random walk m on X is a family of probability measures mx (·) on X for each x ∈ X, satisfying the following two technical assumptions: (i) the measure mx depends measurably on the point x ∈ X; (ii) each measure mx has finite first moment, i.e. for some (hence any) o ∈ X, for any x ∈ X one has d(o, y) dmx (y) < ∞. Instead of “corresponding points” between two close spheres Sx and Sy , we will use transportation distances between measures. We refer to [47] for an introduction to the topic. This distance is usually associated with the names of Kantorovich, Rubinstein, Wasserstein, Ornstein, Monge, and others (see [36] for a historical account); we stick to the simpler and more descriptive “transportation distance.” Definition 2. Let (X, d) be a metric space and let ν1 , ν2 be two probability measures on X. The L1 transportation distance between ν1 and ν2 is W1 (ν1 , ν2 ) :=
inf
ξ ∈Π(ν1 ,ν2 ) (x,y)∈X×X
d(x, y) dξ(x, y)
where Π(ν1 , ν2 ) is the set of measures on X × X projecting to ν1 and ν2 . Intuitively, dξ(x, y) represents the mass that travels from x to y, hence the constraint on the projections of ξ , ensuring that the initial measure is ν1 and the final measure is ν2 . The infimum is actually attained (Theorem 1.3 in [47]), but the optimal coupling is generally not unique. In what follows, it is enough to choose one such coupling. The data (mx )x∈X allow to define a notion of curvature as follows: as in the Riemannian case, we will ask whether the measures mx and my are closer or further apart than the points x and y are, in which case Ricci curvature will be, respectively, positive or negative. Definition 3 (Coarse Ricci curvature). Let (X, d) be a metric space with a random walk m. Let x, y ∈ X be two distinct points. The coarse Ricci curvature of (X, d, m) along (xy) is κ(x, y) := 1 −
W1 (mx , my ) . d(x, y)
814
Y. Ollivier / Journal of Functional Analysis 256 (2009) 810–864
We will see below (Proposition 19) that in geodesic spaces, it is enough to know κ(x, y) for close points x, y. Geometers will think of mx as a replacement for the notion of ball around x. Probabilists will rather think of this data as defining a Markov chain whose transition probability from x to y in n steps is dm∗n x (y) :=
dmx∗(n−1) (z) dmz (y)
z∈X
where of course m∗1 x := mx . Recall that a measure ν on X is invariant for this random walk if dν(x) = y dν(y) dmy (x). It is reversible if moreover, the detailed balance condition dν(x) dmx (y) = dν(y) dmy (x) holds. Other generalizations of Ricci curvature start with a metric measure space [31,46]. Here, as in Bakry–Émery theory, the measure appears as the invariant distribution of some process on the space (e.g. Brownian motion on a Riemannian manifold), which can be chosen in any convenient way. The following remark produces a random walk from a metric measure space, and allows to define the “Ricci curvature at scale ε” for any metric space. Example 4 (ε-step random walk). Let (X, d, μ) be a metric measure space, and assume that balls in X have finite measure and that Supp μ = X. Choose some ε > 0. The ε-step random walk on X, starting at a point x, consists in randomly jumping in the ball of radius ε around x, with probability proportional to μ; namely, mx = μ|B(x,ε) /μ(B(x, ε)). (One can also use other functions of the distance, such as Gaussian kernels.) As explained above, when (X, d) is a Riemannian manifold and mx is the ε-step random walk with small ε, for close enough x, y this definition captures the Ricci curvature in the direction xy (up to some scaling factor depending on ε, see Example 7). In general there is no need for ε to be small: for example if X is a graph, ε = 1 is a natural choice. If a continuous-time Markov kernel is given, one can also define a continuous-time version of coarse Ricci curvature by setting κ(x, y) := −
d W1 (mtx , mty ) dt d(x, y)
when this derivative exists (or take a lim inf), but for simplicity we will mainly work with the discrete-time version here. Indeed, for continuous-time Markov chains, existence of the process is already a non-trivial issue, even in the case of jump processes [12]. We will sometimes use our results on concrete continuous-time examples (e.g. M/M/∞ queues in Section 3.3.4), but only when they appear as an obvious limit of a discrete-time approximation. One could use the Lp transportation distance instead of the L1 one in the definition; however, this will make positive curvature a stronger assumption, and is never needed in our theorems. Notation. In analogy with the Riemannian case, when computing the transportation distance between measures mx and my , we will think of X × X equipped with the coupling measure as a
Y. Ollivier / Journal of Functional Analysis 256 (2009) 810–864
815
tangent space, and for z ∈ X × X we will write x + z and y + z for the two projections to X. So in this notation we have 1 κ(x, y) = d(x, y) − d(x + z, y + z) dz d(x, y) where implicitly dz is the optimal coupling between mx and my . 1.2. Examples Example 5 (ZN and RN ). Let m be the simple random walk on the graph of the grid ZN equipped with its graph metric. Then for any two points x, y ∈ Zd , the coarse Ricci curvature along (xy) is 0. Indeed, we can transport the measure mx around x to the measure my by a translation of vector y − x (and this is optimal), so that the distance between mx and my is exactly that between x and y. This example generalizes to the case of ZN or RN equipped with any distance and random walk which are translation-invariant (consistently with [31]). For example, the triangular tiling of the plane has 0 curvature. We now justify the terminology by showing that, in the case of the ε-step random walk on a Riemannian manifold, we get back the usual Ricci curvature (up to some scaling factor). Proposition 6. Let (X, d) be a smooth complete Riemannian manifold. Let v, w be unit tangent vectors at x ∈ X. Let ε, δ > 0. Let y = expx δv and let w be the tangent vector at y obtained by parallel transport of w along the geodesic expx tv. Then ε2 d(expx εw, expy εw ) = δ 1 − K(v, w) + O ε 3 + ε 2 δ 2 as (ε, δ) → 0. Here K(v, w) is the sectional curvature in the tangent plane (v, w).
Example 7 (Riemannian manifold). Let (X, d) be a smooth complete N -dimensional Riemannian manifold. For some ε > 0, let the Markov chain mε be defined by dmεx (y) := if y ∈ B(x, ε), and 0 otherwise.
1 d vol(y) vol(B(x, ε))
816
Y. Ollivier / Journal of Functional Analysis 256 (2009) 810–864
Let x ∈ X and let v be a unit tangent vector at x. Let y be a point on the geodesic issuing from v, with d(x, y) small enough. Then κ(x, y) =
ε 2 Ric(v, v) + O ε 3 + ε 2 d(x, y) . 2(N + 2)
The proof is postponed to Section 8; it is a refinement of Theorem 1.5 (condition (xii)) in [43], except that therein, the infimum of Ricci curvature is used instead of its value along a tangent vector. Basically, the value of κ(x, y) is obtained by averaging Proposition 6 for w in the unit ball of the tangent space at x, which provides an upper bound for κ. The lower bound requires use of the dual characterization of transportation distance (Theorem 1.14 in [47]). Example 8 (Discrete cube). Let X = {0, 1}N be the discrete cube equipped with the Hamming metric (each edge is of length 1). Let m be the lazy random walk on the graph X, i.e. mx (x) = 1/2 and mx (y) = 1/2N if y is a neighbor of x. Let x, y ∈ X be neighbors. Then κ(x, y) = 1/N . This examples generalizes to arbitrary binomial distributions (see Section 3.3.3). Here laziness is necessary to avoid parity problems: if no laziness is introduced, points at odd distance never meet under the random walk; in this case one would have to consider coarse Ricci curvature for points at even distance only. Actually, since the discrete cube is a 1-geodesic space, one has κ(x, y) 1/N for any pair x, y ∈ X, not only neighbors (see Proposition 19). Proof. We can suppose that x = 00 . . . 0 and y = 10 . . . 0. For z ∈ X and 1 i N , let us denote by zi the neighbor of z in which the ith bit is switched. An optimal coupling between mx and my is as follows: for i 2, move x i to y i (both have mass 1/2N under mx and my respectively). Now mx (x) = 1/2 and my (x) = 1/2N , and likewise for y. So it is enough to move a mass 1/2 − 1/2N from x to y. All points are moved over a distance 1 by this coupling, except for a mass 1/2N which remains at x and a mass 1/2N which remains at y, and so the coarse Ricci curvature is at least 1/N . Optimality of this coupling is obtained as follows: consider the function f : X → {0, 1} which sends a point of X to its first bit. This is a 1-Lipschitz function, with f (x) = 0 and f (y) = 1. The expectations of f under mx and my are 1/2N and 1 − 1/2N respectively, so that 1 − 1/N is a lower bound on W1 (mx , my ). A very short but less visual proof can be obtained with the L1 tensorization property (Proposition 27). 2 Example 9 (Ornstein–Uhlenbeck process). Let s 0, α > 0 and consider the Ornstein– Uhlenbeck process in RN given by the stochastic differential equation dXt = −αXt dt + s dBt where Bt is a standard N -dimensional Brownian motion. The invariant distribution is Gaussian, of variance s 2 /2α.
Y. Ollivier / Journal of Functional Analysis 256 (2009) 810–864
817
Let δt > 0 and let the random walk m be the flow at time δt of the process. Explicitly, mx is a Gaussian probability measure centered at e−αδt x, of variance s 2 (1 − e−2αδt )/2α ∼ s 2 δt for small δt. Then the coarse Ricci curvature κ(x, y) of this random walk is 1 − e−αδt , for any two x, y ∈ RN . Proof. The transportation distance between two Gaussian distributions with the same variance −αδt x−e−αδt y| . 2 is the distance between their centers, so that κ(x, y) = 1 − |e |x−y| Example 10 (Discrete Ornstein–Uhlenbeck). Let X = {−N, −N + 1, . . . , N − 1, N} and let m be the random walk on X given by mk (k) = 1/2,
mk (k + 1) = 1/4 − k/4N,
mk (k − 1) = 1/4 + k/4N
which is a lazy random walk with linear drift towards 0. The binomial distribution reversible for this random walk. Then, for any two neighbors x, y in X, one has κ(x, y) = 1/2N . Proof. Exercise.
2N 1 22N N +k
is
2
Example 11 (Bakry–Émery). Let X be an N -dimensional Riemannian manifold and F be a tangent vector field. Consider the differential operator 1 L := + F.∇ 2 associated with the stochastic differential equation dXt = F dt + dBt where Bt is the Brownian motion in X. The Ricci curvature (in the Bakry–Émery sense) of this operator is 12 Ric −∇ sym F where ∇ sym F ij := 12 (∇ i F j + ∇ j F i ) is the symmetrized of ∇F . Consider the Euler approximation scheme at time δt for this stochastic equation, which consists √ in following the flow of F for a time δt and then randomly jumping in a ball of radius (N + 2)δt. Let x ∈ X and let v be a unit tangent vector at x. Let y be a point on the geodesic issuing from v, with d(x, y) small enough. Then √ 1 sym Ric(v, v) − ∇ F (v, v) + O d(x, y) + O( δt) . κ(x, y) = δt 2
818
Y. Ollivier / Journal of Functional Analysis 256 (2009) 810–864
Proof. First let us explain the normalization: jumping in a ball of radius ε generates a variance ε 2 N 1+2 in a given direction. On the other hand, the N -dimensional Brownian motion has, by definition, a variance dt per unit of time dt in a given direction, so√a proper discretization of Brownian motion at time δt requires jumping in a ball of radius ε = (N + 2)δt . Also, as noted in [6], the generator of Brownian motion is 12 instead of , hence the 12 factor for the Ricci part. Now the discrete-time process begins by following the flow F for some time δt. Starting at points x and y, using elementary Euclidean geometry, it is easy to see that after this, the distance between the endpoints behaves like d(x, y)(1 + δt v.∇v F + O(δt 2 )). Note that v.∇v F = ∇ sym F (v, v). Now, just as in Example 7, randomly jumping in a ball of radius ε results in a gain of 2 d(x, y) 2(Nε +2) Ric(v, v) on transportation distances. Here ε 2 = (N + 2)δt. So after the two steps of the process, the distance between the endpoints is δt d(x, y) 1 − Ric(v, v) + δt ∇ sym F (v, v) 2 as needed, up to higher-order terms.
2
Maybe the reason for the additional −∇ sym F in Ricci curvature à la Bakry–Émery is made clearer in this context: it is simply the quantity by which the flow of X modifies distances between two starting points. It is clear on this example why reversibility is not fundamental in this theory: the antisymmetric part of the force F generates an infinitesimal isometric displacement. With our definition, combining the Markov chain with an isometry of the space has no effect whatsoever on curvature. Example 12 (Multinomial distribution). Consider the set X = {(x0 , x1 , . . . , xd ), xi ∈ N, xi = N } viewed as the configuration set of N balls in d + 1 boxes. Consider the process which consists in taking a ball at random among the N balls, removing it from its box, and putting it back at random in one of the d + 1 boxes. More precisely, the transition probability from (x0 , . . . , xd ) to (x0 , . . . , xi − 1, . . . , xj + 1, . . . , xd ) (with maybe i = j ) is xi /N (d + 1). The multinomial distribution (d+1)NN! x ! is reversible for this Markov chain. i
Y. Ollivier / Journal of Functional Analysis 256 (2009) 810–864
819
Equip this configuration space with the metric d((xi ), (xi )) := 12 |xi − xi | which is the graph distance w.r.t. the moves above. The coarse Ricci curvature of this Markov chain is 1/N . Proof. Exercise (see also the discussion after Proposition 27).
2
Example 13 (Geometric distribution). Let the random walk on N be defined by the transition probabilities pn,n+1 = 1/3, pn+1,n = 2/3 and p0,0 = 2/3. This random walk is reversible with respect to the geometric distribution 2−(n+1) . Then for n 1 one has κ(n, n + 1) = 0. Proof. The transition kernel is translation-invariant except at 0.
2
Section 5 contains more material about this latter example and how non-negative coarse Ricci curvature sometimes implies exponential concentration. Example 14 (Geometric distribution, 2). Let the random walk on N be defined by the transition probabilities pn,0 = α and pn,n+1 = 1 − α for some 0 < α < 1. The geometric distribution α(1 − α)n is invariant (but not reversible) for this random walk. The coarse Ricci curvature of this random walk is α. Proof. Exercise.
2
Example 15 (δ-hyperbolic groups). Let X be the Cayley graph of a non-elementary δ-hyperbolic group with respect to some finite generating set. Let k be a large enough integer (depending on the group) and consider the random walk on X which consists in performing k steps of the simple random walk. Let x, y ∈ X. Then κ(x, y) = −2k/d(x, y) (1 + o(1)) when d(x, y) and k tend to infinity. Note that −2k/d(x, y) is the smallest possible value for κ(x, y), knowing that the steps of the random walk are bounded by k. Proof. For z in the ball of radius k around x, and z in the ball of radius k around y, elementary δ-hyperbolic geometry yields d(z, z ) = d(x, y) + d(x, z) + d(y, z ) − (y, z)x − (x, z )y up to some multiple of δ, where (·,·) denotes the Gromov product with respect to some basepoint [24]. Since this decomposes as the sum of a term depending on z only and a term depending on z only, to compute the transportation distance it is enough to know the expectation of (y, z)x for z in the ball around x, and likewise for (x, z )y . Using that balls have exponential growth, it is not difficult (see Proposition 21 in [39]) to see that the expectation of (y, z)x is bounded by a constant, whatever k, hence the conclusion. The same argument applies to trees or discrete δ-hyperbolic spaces with a uniform lower bound on the exponential growth rate of balls. 2 Example 16 (Kac’s random walk on orthogonal matrices, after [38]). Consider the following random walk on the set of N × N orthogonal matrices: at each step, a pair of indices 1 i < j N is selected at random, an angle θ ∈ [0; 2π) is picked at random, and a rotation of angle θ is performed in the coordinate plane i, j . Equip SO(N ) with the Riemannian metric induced by It is proven in a preprint by the Hilbert–Schmidt inner product Tr(a ∗ b) on its tangent space. √ Oliveira [38] that this random walk has coarse Ricci curvature 1 − 1 − 2/N (N − 1) ∼ 1/N 2 .
820
Y. Ollivier / Journal of Functional Analysis 256 (2009) 810–864
This is consistent with the fact that SO(N ) has, as a Riemannian manifold, a positive Ricci curvature in the usual sense. However, from the computational point of view, Kac’s random walk above is much nicer than either the Brownian motion or the ε-step random walk of Example 7. Oliveira uses his result to prove a new estimate O(N 2 ln N ) for the mixing time of this random walk, neatly improving on previous estimates O(N 4 ln N ) by Diaconis–Saloff-Coste and O(N 2.5 ln N) by Pak–Sidenko; Ω(N 2 ) is an easy lower bound, see [38]. Example 17 (Glauber dynamics for the Ising model). Let G be a finite graph. Consider the configuration space X := {−1, 1}G together with the energy function U (S) := −
x∼y∈G
S(x)S(y) − h
S(x)
for S ∈ X,
x
where h ∈ R is the external magnetic field. For some β 0, equip X with the Gibbs distribution μ := e−βU /Z where as usual Z := S e−βU (S) . The distance between two states is defined as the number of vertices of G at which their values differ. For S ∈ X and x ∈ G, denote by Sx+ and Sx− the states obtained from S by setting Sx+ (x) = +1 and Sx− (x) = −1, respectively. Consider the following random walk on X (known as the Glauber dynamics): at each step, a vertex x ∈ G is chosen at random, and a new value for S(x) is picked according to local equilibrium, i.e. S(x) is set to 1 or −1 with probabilities proportional to e−βU (Sx+ ) and e−βU (Sx− ) respectively (note that only the neighbors of x influence the ratio of these probabilities). The Gibbs distribution is reversible for this Markov chain. Then the coarse Ricci curvature of this Markov chain is at least eβ − e−β 1 1 − vmax β |G| e + e−β where vmax is the maximal valency of a vertex of G. In particular, if vmax + 1 1 β < ln 2 vmax − 1 then curvature is positive. Consequently, the critical β is at least this quantity. This estimate for the critical temperature coincides with the one derived in [26]. Actually, our argument generalizes to different settings (such as non-constant/negative values of the coupling Jxy between spins, or continuous spin spaces), and the positive curvature condition for the Glauber dynamics exactly amounts to the well-known one-site Dobrushin criterion [18] (or to G(β) < 1 in the notation of [26, Eq. (19]). By comparison, the exact value of the critical β for v ), which shows asymptotic the Ising model on the regular infinite tree of valency v is 12 ln( v−2 optimality of this criterion. When block dynamics (see [34]) are used instead of single-site updates, positive coarse Ricci curvature of the block dynamics Markov chain is equivalent to the Dobrushin–Shlosman criterion [20]. As shown in the rest of this paper, positive curvature implies several properties, especially, exponential convergence to equilibrium, concentration inequalities and a modified logarithmic Sobolev inequality. For the Glauber dynamics, the constants we get in these inequalities are essentially the same as in the infinite-temperature (independent) case, up to some factor depending
Y. Ollivier / Journal of Functional Analysis 256 (2009) 810–864
821
on temperature which diverges when positive curvature ceases to hold. This is more or less equivalent to the main results of the literature under the Dobrushin–Shlosman criterion (see e.g. the review [34]). Note however that in our setting we do not need the underlying graph to be ZN . Proof. Using Proposition 19, it is enough to bound coarse Ricci curvature for pairs of states at distance 1. Let S, S be two states differing only at x ∈ G. We can suppose that S(x) = −1 and S (x) = 1. Let mS and mS be the law of the step of the random walk issuing from S and S respectively. We have to prove that the transportation distance between mS and mS is at most β −β 1 1 − |G| (1 − vmax eeβ −e ). +e−β y 1 The measure mS decomposes as mS = |G| y∈G mS , according to the vertex y ∈ G which is modified by the random walk, and likewise for mS . To evaluate the transportation distance, we y y will compare mS to mS . If the step of the random walk consists in modifying the value of S at x (which occurs with 1 probability |G| ), then the resulting state has the same law for S and S , i.e. mxS = mxS . Thus in
1 . this case the transportation distance is 0 and the contribution to coarse Ricci curvature is 1 × |G| If the step consists in modifying the value of S at some point y in G not adjacent to x, then y y the value at x does not influence local equilibrium at y, and so mS and mS are identical except at x. So in this case the distance is 1 and the contribution to coarse Ricci curvature is 0. Now if the step consists in modifying the value of S at some point y ∈ G adjacent to x (which occurs with probability vx /|G| where vx is the valency of x), then the value at x does influence the law of the new value at y, by some amount which we now evaluate. The final distance between the two laws will be this amount plus 1 (1 accounts for the difference at x), and the contribution to coarse Ricci curvature will be negative. Let us now evaluate this amount more precisely. Let y ∈ G be adjacent to x. Set a = e−βU (Sy+ ) /e−βU (Sy− ) . The step of the random walk consists in setting S(y) to 1 with proba a 1 bility a+1 , and to −1 with probability a+1 . Setting likewise a = e−βU (Sy+ ) /e−βU (Sy− ) for S , a 1 ; a+1 ) we are left to evaluate the distance between the distributions on {−1, 1} given by ( a+1
and ( a a+1 ; a 1+1 ). It is immediate to check, using the definition of the energy U , that a = e4β a. Then, a simple computation shows that the distance between these two distributions is at most eβ −e−β . This value is actually achieved when y has odd valency, h = 0 and switching the value at eβ +e−β x changes the majority of spin signs around y. (Our argument is suboptimal here when valency is even—a more precise estimation yields the absence of a phase transition on Z.) Combining these different cases yields the desired curvature evaluation. To convert this into 1 an evaluation of the critical β, reason as follows: magnetization, defined as |G| x∈G S(x), is a 1 |G| -Lipschitz μh the Gibbs
function of the state. Now let μ0 be the Gibbs measure without magnetic field, and measure with external magnetic field h. Use the Glauber dynamics with magnetic field h, but starting with an initial state picked under μ0 ; Corollary 22 yields that the magneti1 zation under μh is controlled by |G| W1 (μ0 , μ0 ∗ m)/κ where κ is the coarse Ricci curvature, and W1 (μ0 , μ0 ∗ m) is the transportation distance between the Gibbs measure μ0 and the measure obtained from it after one step of the Glauber dynamics with magnetic field h; reasoning 1 eβh −e−βh as above this transportation distance is easily bounded by |G| , so that the derivative of eβh +e−βh magnetization w.r.t. h stays bounded when |G| → ∞, which is the classical criterion used to define critical temperature. (Compare Eq. (22) in [26].) 2
822
Y. Ollivier / Journal of Functional Analysis 256 (2009) 810–864
Further examples. More examples can be found in Sections 3.3.3 (binomial and Poisson distributions), 3.3.4 (M/M/∞ queues and generalizations), 3.3.5 (exponential tails), 3.3.6 (heavy tails) and 5 (geometric distributions on N, exponential distributions on RN ). 1.3. Overview of the results Notation for random walks. Before we present the main results, we need to define some quantities related to the local behavior of the random walk: the jump, which will help control the diameter of the space, and the coarse diffusion constant, which controls concentration properties. Moreover, we define a notion of local dimension. The larger the dimension, the better for concentration of measure. Definition 18 (Jump, diffusion constant, dimension). Let the jump of the random walk at x be J (x) := Emx d(x, ·) = W1 (δx , mx ). Let the (coarse) diffusion constant of the random walk at x be 1/2 1 2 σ (x) := d(y, z) dmx (y) dmx (z) 2 and, if ν is an invariant distribution, let σ := σ (x) L2 (X,ν) be the average diffusion constant. Let also σ∞ (x) := 12 diam Supp mx and σ∞ := sup σ∞ (x). Let the local dimension at x be nx :=
σ (x)2 sup{Varmx f, f : Supp mx → R 1-Lipschitz}
and finally n := infx nx . About this definition of dimension. Obviously nx 1. For the discrete-time Brownian motion on a N -dimensional Riemannian manifold, one has nx ≈ N (see the end of Section 8). For the simple random walk on a graph, nx ≈ 1. This definition of dimension amounts to saying that√in a space of dimension n, the typical variations of a (1-dimensional) Lipschitz function are 1/ n times the typical distance between two points. This is the case in the sphere S n , in the Gaussian measure on Rn , and in the discrete cube {0, 1}n . So generally one could define the “statistical dimension” of a metric measure space (X, d, μ) by this formula i.e. 1 d(x, y)2 dμ(x) dμ(y) 2 StatDim(X, d, μ) := sup{Varμ f, f 1-Lipschitz} so that for each x ∈ X the local dimension of X at x is nx = StatDim(X, d, mx ). With this definition, RN equipped with a Gaussian measure has statistical dimension N and local dimension ≈ N , whereas the discrete cube {0, 1}N has statistical dimension ≈ N and local dimension ≈ 1. We now turn to the description of the main results of the paper.
Y. Ollivier / Journal of Functional Analysis 256 (2009) 810–864
823
Elementary properties. In Section 2 are gathered some straightforward results. First, we prove (Proposition 19) that in an ε-geodesic space, a lower bound on κ(x, y) for points x, y with d(x, y) ε implies the same lower bound for all pairs of points. This is simple yet very useful: indeed in the various graphs given above as examples, it was enough to compute the coarse Ricci curvature for neighbors. Second, we prove equivalent characterizations of having coarse Ricci curvature uniformly bounded below: a space satisfies κ(x, y) κ if and only if the random walk operator is (1 − κ)-contracting on the space of probability measures equipped with the transportation distance (Proposition 20), and if and only if the random walk operator acting on Lipschitz functions contracts the Lipschitz norm by (1 − κ) (Proposition 29). An immediate corollary is the existence of a unique invariant distribution when κ > 0. The property of contraction of the Lipschitz norm easily implies, in the reversible case, that the spectral gap of the Laplacian operator associated with the random walk is at least κ (Proposition 30); this can be seen as a generalization of Lichnerowicz’s theorem, and provides sharp estimates of the spectral gap in several examples. (A similar result appears in [14].) In analogy with the Bonnet–Myers theorem, we prove that if coarse Ricci curvature is bounded below by κ > 0, then the diameter of the space is at most 2 supx J (x)/κ (Proposition 23). In case J is unbounded, we can evaluate instead the average distance to a given point x0 under the invariant distribution ν (Proposition 24); namely, d(x , y) dν(y) J (x0 )/κ. In particular we 0 have d(x, y) dν(x) dν(y) 2 inf J /κ. These are L1 versions of the Bonnet–Myers theorem √ rather than generalizations: from the case of manifolds one would expect 1/ κ instead of 1/κ. Actually this L1 version is sharp in all our examples except Riemannian manifolds; in Section 6 we investigate additional conditions for an L2 version of the Bonnet–Myers theorem to hold. Let us also mention some elementary operations preserving positive curvature: composition, superposition and L1 tensorization (Propositions 25–27). Concentration results. Basically, if coarse Ricci curvature is bounded below by κ > 0, then the invariant distribution satisfies concentration results with variance σ 2 /nκ (up to some constant factor). This estimate is often sharp, as discussed in Section 3.3 where we revisit some of the examples. However, the type of concentration (Gaussian, exponential, or 1/t 2 ) depends on further local assumptions: indeed, the tail behavior of the invariant measure cannot be better than that of the local measures mx . Without further assumptions, one only gets that the variance of a 1-Lipschitz function is at most σ 2 /nκ, hence concentration like σ 2 /nκt 2 (Proposition 32). If we make the further assumption that the support of the measures mx is uniformly bounded (i.e. σ∞ < ∞), then we get mixed Gaussian-then-exponential concentration, with variance σ 2 /nκ (Theorem 33). The width of the Gaussian window depends on σ∞ , and on the rate of variation of the diffusion constant σ (x)2 . For the case of Riemannian manifolds, simply considering smaller and smaller steps for the random walks makes the width of the Gaussian window tend to infinity, so that we recover full Gaussian concentration as in the Lévy–Gromov or Bakry–Émery context. However, for lots of discrete examples, the Gaussian-then-exponential behavior is genuine. Examples where tails are Poisson-like (binomial distribution, M/M/∞ queues) or exponential are given in Sections 3.3.3 to 3.3.5. Examples of heavy tails (when σ∞ = ∞) are given in 3.3.6. We also get concentration results for the finite-time distributions m∗k x (Remark 35).
824
Y. Ollivier / Journal of Functional Analysis 256 (2009) 810–864
Log-Sobolev inequality. Using a suitable non-local notion of norm of the gradient, we are able to adapt the proof by Bakry and Émery of a logarithmic Sobolev inequality for the invariant dis(z)| tribution. The gradient we use (Definition 41) is (Df )(x) := supy,z |f (y)−f exp(−λd(x, y) − d(y,z) λd(y, z)). This is a kind of “semi-local” Lipschitz constant for f . Typically the value of λ can be taken large at the “macroscopic” level; for Riemannian manifolds, taking smaller and smaller steps for the random walk allows to take λ → ∞ so that we recover the usual gradient for smooth functions. The inequality takes the form Ent f C (Df )2 /f dν (Theorem 45). The main tool of the proof is the gradient contraction relation D(Mf ) (1 − κ/2)M(Df ) where M is the random walk operator (Theorem 44). That the gradient is non-local, with a maximal possible value of λ, is consistent with the possible occurrence of non-Gaussian tails. Exponential concentration and non-negative curvature. The simplest example of a Markov chain with zero coarse Ricci curvature is the simple random walk on N or Z, for which there is no invariant distribution. However, we show that if furthermore there is a “locally attracting” point, then non-negative coarse Ricci curvature implies exponential concentration. Examples are the geometric distribution on N or the exponential distribution e−|x| on RN associated with the Xt stochastic differential equation dXt = dBt − |X dt. In both cases we recover correct orders of t| magnitude. Gromov–Hausdorff topology. One advantage of our definition is that it involves only combinations of the distance function, and no derivatives, so that it is more or less impervious to deformations of the space. In Section 7 we show that coarse Ricci curvature is continuous for Gromov– Hausdorff convergence of metric spaces (suitably reinforced, of course, so that the random walk converges as well), so that having non-negative curvature is a closed property. We also suggest a loosened definition of coarse Ricci curvature, requiring that W1 (mx , my ) (1 − κ)d(x, y) + δ instead of W1 (mx , my ) (1 − κ)d(x, y). With this definition, positive curvature becomes an open property, so that a space close to one with positive curvature has positive curvature. 2. Elementary properties 2.1. Geodesic spaces The idea behind curvature is to use local properties to derive global ones. We give here a simple proposition expressing that in near-geodesic spaces such as graphs (with ε = 1) or manifolds (for any ε), it is enough to check positivity of coarse Ricci curvature for nearby points. Proposition 19 (Geodesic spaces). Suppose that (X, d) is ε-geodesic in the sense that for any two points x, y ∈ X, there exists an integer n and a sequence x0 = x, x1 , . . . , xn = y such that d(x, y) = d(xi , xi+1 ) and d(xi , xi+1 ) ε. Then, if κ(x, y) κ for any pair of points with d(x, y) ε, then κ(x, y) κ for any pair of points x, y ∈ X. Proof. Let (xi ) be as above.Using the triangle inequality for W1 , one has W1 (mx , my ) W1 (mxi , mxi+1 ) (1 − κ) d(xi , xi+1 ) = (1 − κ)d(x, y). 2
Y. Ollivier / Journal of Functional Analysis 256 (2009) 810–864
825
2.2. Contraction on the space of probability measures Let P(X) by the space of all probability measures μ on X with finite first moment, i.e. for some (hence any) o ∈ X, d(o, x) dμ(x) < ∞. On P(X), the transportation distance W1 is finite, so that it is actually a distance. Let μ be a probability measure on X and define the measure μ ∗ m :=
dμ(x) mx
x∈X
which is the image of μ by the random walk. A priori, it may or may not belong to P(X). The following proposition and its corollary can be seen as a particular case of Theorem 3 in [18] (viewing a Markov chain as a Markov field on N). Equivalent statements also appear in [19, Proposition 14.3], in the second edition of [12, Theorem 5.22], in [15] (in the proof of Proposition 2.10), in [42] and in [38]. Proposition 20 (W1 contraction). Let (X, d, m) be a metric space with a random walk. Let κ ∈ R. Then we have κ(x, y) κ for all x, y ∈ X, if and only if for any two probability distributions μ, μ ∈ P(X) one has W1 (μ ∗ m, μ ∗ m) (1 − κ)W1 (μ, μ ). Moreover in this case, if μ ∈ P(X) then μ ∗ m ∈ P(X). Proof. First, suppose that convolution with m is contracting in W1 distance. For some x, y ∈ X, let μ = δx and μ = δy be the Dirac measures at x and y. Then by definition δx ∗ m = mx and likewise for y, so that W1 (mx , my ) (1 − κ)W1 (δx , δy ) = (1 − κ)d(x, y) as required. The converse is more difficult to write than to understand. For each pair (x, y) let ξxy be a coupling (i.e. a measure on X × X) between mx and my witnessing for κ(x, y) κ. According to Corollary 5.22 in [48], we can choose ξxy to depend measurably on the pair (x, y). Let Ξ be a coupling between μ and μ witnessing for W1 (μ, μ ). Then X×X dΞ (x, y) ξxy is a coupling between μ ∗ m and μ ∗ m and so W1 (μ ∗ m, μ ∗ m)
d(x, y) d
x,y
dΞ (x , y ) ξx y (x, y)
x ,y
dΞ (x , y ) dξx y (x, y) d(x, y)
= x,y,x ,y
dΞ (x , y ) d(x , y ) 1 − κ(x , y )
x ,y
(1 − κ)W1 (μ, μ ) by the Fubini theorem applied to d(x, y) dΞ (x , y ) dξx ,y (x, y).
826
Y. Ollivier / Journal of Functional Analysis 256 (2009) 810–864
To see that in this situation P(X) is preserved by the random walk, fix some origin o ∈ X and note that for any μ ∈ P(X), the first moment of μ ∗ m is W1 (δo , μ ∗ m) W1 (δo , mo ) + W1 (mo , μ ∗ m) W1 (δo , mo ) + (1 − κ)W1 (δo , μ). Now W1 (δo , μ) < ∞ by assumption, and W1 (δo , mo ) < ∞ by Definition 1. 2 As an immediate consequence of this contracting property we get: Corollary 21 (W1 convergence). Suppose that κ(x, y) κ > 0 for any two distinct x, y ∈ X. Then the random walk has a unique invariant distribution ν ∈ P(X). Moreover, for any probability measure μ ∈ P(X), the sequence μ ∗ m∗n tends exponentially fast to ν in W1 distance. Namely W1 μ ∗ m∗n , ν (1 − κ)n W1 (μ, ν) and in particular n W1 m∗n x , ν (1 − κ) J (x)/κ. The last assertion follows by taking μ = δx and noting that J (x) = W1 (δx , mx ) so that W1 (δx , ν) W1 (δx , mx ) + W1 (mx , ν) J (x) + (1 − κ)W1 (δx , ν), hence W1 (δx , ν) J (x)/κ. This is useful to provide bounds on mixing time. For example, suppose that X is a graph; since the total variation distance between two measures μ, μ is the transportation distance with respect to the trivial metric instead of the graph metric, we obviously have |μ − μ |TV W1 (μ, μ ), t hence the corollary above yields the estimate |m∗t x − ν|TV (diam X)(1 − κ) for any x ∈ X. N Applied for example to the discrete cube {0, 1} , with κ = 1/N and diameter N , this gives the correct estimate O(N ln N ) for mixing time in total variation distance, whereas the traditional estimate based on spectral gap and passage from L2 to L1 norm gives O(N 2 ). Also note that t the pointwise bound |m∗t x − ν|TV (1 − κ) J (x)/κ depends on local data only and requires no knowledge of the invariant measure (compare [21]) or diameter; in particular it applies to infinite graphs. Another immediate interesting corollary is the following, which allows to estimate the average of a Lipschitz function under the invariant measure, knowing some of its values. This is useful in concentration theorems, to get bounds not only on the deviations from the average, but on what the average actually is. Corollary 22. Suppose that κ(x, y) κ > 0 for any two distinct x, y ∈ X. Let ν be the invariant distribution. Let f be a 1-Lipschitz function. Then, for any distribution μ, one has |Eν f − Eμ f | W1 (μ, μ ∗ m)/κ. In particular, for any x ∈ X one has |f (x) − Eν f | J (x)/κ. Proof. One has W1 (μ ∗ m, ν) (1 − κ)W1 (μ, ν). Since by the triangle inequality, W1 (μ ∗ m, ν) W1 (μ, ν) − W1 (μ, μ ∗ m), one gets W1 (μ, ν) W1 (μ, μ ∗ m)/κ. Now if f is a 1-Lipschitz function, for any two distributions μ, μ one has |Eμ f − Eμ f | W1 (μ, μ ) hence the result. The last assertion is simply the case when μ is the Dirac measure at x. 2
Y. Ollivier / Journal of Functional Analysis 256 (2009) 810–864
827
2.3. L1 Bonnet–Myers theorems We now give a weak analogue of the Bonnet–Myers theorem. This result shows in particular that positivity of coarse Ricci curvature is a much stronger property than a spectral gap bound: there is no coarse Ricci curvature analogue of a family of expanders. Proposition 23 (L1 Bonnet–Myers). Suppose that κ(x, y) κ > 0 for all x, y ∈ X. Then for any x, y ∈ X one has d(x, y)
J (x) + J (y) κ(x, y)
diam X
2 supx J (x) . κ
and in particular
Proof. We have d(x, y) = W1 (δx , δy ) W1 (δx , mx ) + W1 (mx , my ) + W1 (my , δy ) J (x) + (1 − κ)d(x, y) + J (y) hence the result. 2 This estimate is not sharp at all for Brownian motion in Riemannian manifolds (since J ≈ ε and κ ≈ ε 2 Ric /N , it fails by a factor 1/ε compared to the Bonnet–Myers theorem!), but is sharp in many other examples. For the discrete cube X = {0, 1}N (Example 8 above), one has J = 1/2 and κ = 1/N , so we get diam X N which is the exact value. For the discrete Ornstein–Uhlenbeck process (Example 10 above) one has J = 1/2 and κ = 1/2N , so we get diam X 2N which once more is the exact value. For the continuous Ornstein–Uhlenbeck process on R (Example 9 with N = 1), the diameter is infinite, consistently with the fact that J√is unbounded. If we consider points x, y lying in some large interval [−R; R] with R s/ α, then sup J ∼ αRδt on this interval, and κ = (1 − eαδt ) ∼ αδt so that the diameter bound is 2R, which is correct. √ These examples show that one cannot replace J /κ with J / κ in this result (as could be expected from the example of Riemannian manifolds). In fact, Riemannian manifolds seem to be √ the only simple example where there is a diameter bound behaving like 1/ κ. In Section 6 we investigate conditions under which an L2 version of the Bonnet–Myers theorem holds. In case J is not bounded, we can estimate instead the “average” diameter d(x, y) dν(x) dν(y) under the invariant distribution ν. This estimate will prove very useful in several examples, to get bounds on the average of σ (x) in cases where σ (x) is unbounded but controlled by the distance to some “origin” (see e.g. Sections 3.3.4 and 3.3.5). Proposition 24 (Average L1 Bonnet–Myers). Suppose that κ(x, y) κ > 0 for any two distinct x, y ∈ X. Then for any x ∈ X, d(x, y) dν(y) X
J (x) κ
828
Y. Ollivier / Journal of Functional Analysis 256 (2009) 810–864
and so d(x, y) dν(x) dν(y)
2 infx J (x) . κ
X×X
Proof. The first assertion follows from Corollary 22 with f = d(x, ·). For the second assertion, choose an x0 with J (x0 ) arbitrarily close to inf J , and write
d(y, z) dν(y) dν(z)
X×X
d(y, x0 ) + d(x0 , z) dν(y) dν(z)
X×X
= 2W1 (δx0 , ν) 2J (x0 )/κ which ends the proof.
2
2.4. Three constructions Here we describe three very simple operations which trivially preserve positive curvature, namely, composition, superposition and L1 tensorization. Proposition 25 (Composition). Let X be a metric space equipped with two random walks m = (mx )x∈X , m = (mx )x∈X . Suppose that the coarse Ricci curvature of m (resp. m ) is at least κ (resp. κ ). Let m be the composition of m and m , i.e. the random walk which sends a probability measure μ to μ ∗ m ∗ m . Then the coarse Ricci curvature of m is at least κ + κ − κκ . Proof. Trivial when (1 − κ) is seen as a contraction coefficient.
2
Superposition states that if we are given two random walks on the same space and construct a new one by, at each step, tossing a coin and deciding to follow either one random walk or the other, then the coarse Ricci curvatures mix nicely. Proposition 26 (Superposition). Let X be a metric space equipped with a family (m(i) ) of random walks. Suppose that for each i, the coarse Ricci curvature of m(i) is at least κi . Let (αi ) be a family of non-negative real numbers with αi = 1. Define a random walk m on X by mx := αi m(i) . Then the coarse Ricci curvature of m is at least αi κi . x (i) Proof. Let x, y ∈ X and for each i let ξi be a coupling between m(i) x and my . Then (i) (i) coupling between αi mx and αi my , so that
W1 (mx , my )
(i) αi W1 m(i) x , my
αi (1 − κi )d(x, y)
= 1− αi κi d(x, y).
αi ξi is a
Y. Ollivier / Journal of Functional Analysis 256 (2009) 810–864 (i)
829
(i)
Note that the coupling above, which consists in sending each mx to my , has no reason to be optimal, so that in general equality does not hold. 2 Tensorization states that if we perform a random walk in a product space by deciding at random, at each step, to move in one or the other component, then positive curvature is preserved. Proposition 27 (L1 tensorization). Let ((Xi , di ))i∈I be a finite family of metric spaces and suppose that Xi is equipped with a random walk m(i) . Let X be the product of the spaces Xi , equipped with the distance d := di . Let (αi ) be a family of non-negative real numbers with αi = 1. Consider the random walk m on X defined by m(x1 ,...,xk ) :=
αi δx1 ⊗ · · · ⊗ mxi ⊗ · · · ⊗ δxk .
Suppose that for each i, the coarse Ricci curvature of m(i) is at least κi . Then the coarse Ricci curvature of m is at least inf αi κi . For example, this allows for a very short proof that the curvature of the lazy random walk on the discrete cube {0, 1}N is 1/N (Example 8). Indeed, it is the N -fold product of the random walk on {0, 1} which sends each point to the equilibrium distribution (1/2, 1/2), hence is of curvature 1. Likewise, we can recover the coarse Ricci curvature for multinomial distributions (Example 12) as follows: consider a finite set S of cardinal d + 1, representing the boxes of Example 12, endowed with an arbitrary probability distribution ν. Equip it with the trivial distance and the Markov chain sending each point of S to ν, so that coarse Ricci curvature is 1. Now consider the N -fold product of this random walk on S N . Each component represents a ball of Example 12, and the product random walk consists in selecting a ball and putting it in a random box according to ν, as in the example. By the proposition above, the coarse Ricci curvature of this N -fold product is (at least) 1/N . This evaluation of curvature carries down to the “quotient” Markov chain of Example 12, in which only the number of balls in each box is considered instead of the full configuration space. The case when some αi is equal to 0 shows why coarse Ricci curvature is given by an infimum: indeed, if αi = 0 then the corresponding component never gets mixed, hence curvature cannot be positive (unless this component is reduced to a single point). This is similar to what happens for the spectral gap. The statement above is restricted to a finite product for the following technical reasons: first, to define the L1 product of an infinite family, a basepoint has to be chosen. Second, in order for the formula above to define a random walk with finite first moment (see Definition 1), some uniform assumption on the first moments of the m(i) is needed. (i)
Proof. For x ∈ X let m ˜ x stand for δx1 ⊗ · · · ⊗ mxi ⊗ · · · ⊗ δxk . Let x = (xi ) and y = (yi ) be two points in X. Then
(i) (i) ˜ x ,m αi W1 m ˜y
(i) (i)
dj (xj , yj ) αi W1 mx , my +
W1 (mx , my )
j =i
830
Y. Ollivier / Journal of Functional Analysis 256 (2009) 810–864
=
αi (1 − κi )di (xi , yi ) + dj (xj , yj ) j =i
αi −κi di (xi , yi ) +
dj (xj , yj )
αi κi di (xi , yi ) di (xi , yi ) −
(1 − inf αi κi ) di (xi , yi )
=
= (1 − inf αi κi )d(x, y).
2
2.5. Lipschitz functions and spectral gap Definition 28 (Averaging operator, Laplacian). For f ∈ L2 (X, ν) let the averaging operator M be Mf (x) := f (y) dmx (y) y
and let := M − Id. 2
d (This is the layman’s convention for the sign of the Laplacian, i.e. = dx 2 on R, so that on a Riemannian manifold is a negative operator.) The following proposition also appears in [15] (in the proof of Proposition 2.10). For the classical case of Riemannian manifolds, contraction of the norm of the gradient is one of the main results of Bakry–Émery theory.
Proposition 29 (Lipschitz contraction). Let (X, d, m) be a random walk on a metric space. Let κ ∈ R. Then the coarse Ricci curvature of X is at least κ, if and only if, for every k-Lipschitz function f : X → R, the function Mf is k(1 − κ)-Lipschitz. Proof. First, suppose that the coarse Ricci curvature of X is at least κ. Then, using the notation presented at the end of Section 1.1, we have Mf (y) − Mf (x) =
f (y + z) − f (x + z) z
k
d(x + z, y + z) z
= kd(x, y) 1 − κ(x, y) . Conversely, suppose that whenever f is 1-Lipschitz, Mf is (1 − κ)-Lipschitz. The duality theorem for transportation distance (Theorem 1.14 in [47]) states that
Y. Ollivier / Journal of Functional Analysis 256 (2009) 810–864
831
W1 (mx , my ) = =
sup
f d(mx − my )
sup
Mf (x) − Mf (y)
f 1-Lipschitz f 1-Lipschitz
(1 − κ)d(x, y).
2
Let ν be an invariant distribution of the random walk. Consider the space L2 (X, ν)/{const} equipped with the norm f 2L2 (X,ν)/{const}
:= f
− Eν f 2L2 (X,ν)
1 = Varν f = 2
2 f (x) − f (y) dν(x) dν(y).
X×X
The operators M and are self-adjoint in L2 (X, ν) if and only if ν is reversible for the random walk. It is easy to check, using associativity of variances, that Varν f = Varmx f dν(x) + Varν Mf so that Mf 2 f 2 . It is also clear that Mf ∞ f ∞ . Usually, spectral gap properties for are expressed in the space L2 . The proposition above only implies that the spectral radius of the operator M acting on Lip(X)/{const} is at most (1 − κ). In general it is not true that a bound for the spectral radius of an operator on a dense subspace of a Hilbert space implies a bound for the spectral radius on the whole space. This holds, however, when the operator is self-adjoint or when the Hilbert space is finite-dimensional. Proposition 30 (Spectral gap). Let (X, d, m) be a metric space with random walk, with invariant distribution ν. Suppose that the coarse Ricci curvature of X is at least κ > 0 and that σ < ∞. Suppose that ν is reversible, or that X is finite. Then the spectral radius of the averaging operator acting on L2 (X, ν)/{const} is at most 1 − κ. Compare Theorem 1.9 in [14] (Theorem 9.18 in [12]). Proof. First, if X is finite then Lipschitz functions coincide with L2 functions, and the norms are equivalent, so that there is nothing to prove. So we suppose that ν is reversible, i.e. M is self-adjoint. Let f be a k-Lipschitz function. Proposition 32 below implies that Lipschitz functions belong to L2 (X, ν)/{const} and that the Lipschitz norm controls the L2 norm (this is where we use k(1 − κ)t -Lipschitz one gets Var Mt f Ck 2 (1 − κ)2t for some that σ < ∞). Since Mt f is constant C so that limt→∞ ( Var Mt f )1/t (1 − κ). So the spectral radius of M is at most 1 − κ on the subspace of Lipschitz functions. Now Lipschitz functions are dense in L2 (X, ν) (indeed, a probability measure on a metric space is regular, so that indicator functions of measurable sets can be approximated by Lipschitz functions). Since M is bounded and self-adjoint, its spectral radius is controlled by its value on a dense subspace using the spectral decomposition. 2
832
Y. Ollivier / Journal of Functional Analysis 256 (2009) 810–864
Corollary 31 (Poincaré inequality). Let (X, d, m) be an ergodic random walk on a metric space, with invariant distribution ν. Suppose that the coarse Ricci curvature of X is at least κ > 0 and that σ < ∞. Suppose that ν is reversible. Then the spectrum of − acting on L2 (X, ν)/{const} is contained in [κ; ∞). Moreover the following discrete Poincaré inequalities are satisfied for f ∈ L2 (X, ν): 1 Varν f κ(2 − κ)
Varmx f dν(x)
and Varν f
1 2κ
2 f (y) − f (x) dν(x) dmx (y).
Proof. These are rewritings of the inequalities Varν Mf (1 − κ)2 Varν f f, Mf L2 (X,ν)/{const} (1 − κ) Varν f , respectively. 2
and
The quantities Varmx f and 12 (f (y) − f (x))2 dmx (y) are two possible analogues of ∇f (x)2 in a discrete setting. Though the latter is more common, the former is preferable when the support of mx can be far away from x because it cancels out the “drift.” Moreover one always has Varmx f (f (y) − f (x))2 dmx (y), so that the first form is generally sharper. Reversibility is really needed here to turn an estimate of the spectral radius of M into an inequality between the norms of Mf and f , using that M is self-adjoint. When the random walk is not reversible, applying the above to MM∗ does not work since the coarse Ricci curvature of the latter is unknown. However, a version of the Poincaré inequality with a non-local gradient still holds (Theorem 45). As proven by Gromov and Milman ([25], or Corollary 3.1 and Theorem 3.3 in [29]), in quite a general setting a Poincaré inequality implies exponential concentration. Their argument adapts √ well here, and provides a concentration bound of roughly exp(−t κ σ∞ ). We do not include the details, however, since Theorem 33 below is always more precise and covers the non-reversible case as well. Let us compare this result to Lichnerowicz’s theorem in the case of the ε-step random walk on an N -dimensional Riemannian manifold with positive Ricci curvature. This theorem states that the smallest eigenvalue of the usual Laplacian is NN−1 inf Ric, where inf Ric is the largest K such that Ric(v, v) K for all unit tangent vectors v. On the other hand, the operator
associated with the random walk is the difference between the mean value of a function on a ball 2 of radius ε, and its value at the center of the ball: when ε → 0 this behaves like 2(Nε +2) times the usual Laplacian (take the average on the ball of the Taylor expansion of f ). We saw (Example 7) 2 that in this case κ ∼ 2(Nε +2) inf Ric. Note that both scaling factors are the same. So we miss the N N −1 factor, but otherwise get the correct order of magnitude. Second, let us test this corollary for the discrete cube of Example 8. In this case the eigenbase of the discrete Laplacian is well-known (characters, or Fourier/Walsh transform), and the spectral gap of the discrete Laplacian associated with the lazy random walk is 1/N . Since the coarse Ricci curvature κ is 1/N too, the value given in the proposition is sharp. Third, consider the Ornstein–Uhlenbeck process on R, as in Example 9. Its infinitesimal gens 2 d2 d erator is L = 2 dx 2 − αx dx , and the eigenfunctions are known to be Hk (x α/s 2 ) where Hk is
Y. Ollivier / Journal of Functional Analysis 256 (2009) 810–864 k
833
d −x the Hermite polynomial Hk (x) := (−1)k ex dx . The associated eigenvalue of L is −kα, so ke that the spectral gap of L is α. Now the random walk we consider is the flow eδtL at time δt of the process (with small δt), whose eigenvalues are e−kαδt . So the spectral gap of the discrete Laplacian eδtL − Id is 1 − e−αδt . Since coarse Ricci curvature is 1 − e−αδt too, the corollary is sharp again. 2
2
3. Concentration results 3.1. Variance of Lipschitz functions We begin with the simplest kind of concentration, namely, an estimation of the variance of Lipschitz functions. Contrary to Gaussian or exponential concentration, the only assumption needed here is that the average diffusion constant σ is finite. Since our Gaussian concentration result will yield basically the same variance σ 2 /nκ, we discuss sharpness of this estimate in various examples in Section 3.3. Proposition 32. Let (X, d, m) be a random walk on a metric space, with coarse Ricci curvature at least κ > 0. Let ν be the unique invariant distribution. Suppose that σ < ∞. Then the variance of a 1-Lipschitz function is at most 2
σ2 nκ(2−κ) .
2
σ Note that since κ 1 one has nκ(2−κ) σnκ . this implies that all Lipschitz functions are in L2 /{const}; especially, In particular, 2 d(x, y) dν(x) dν(y) is finite. The fact that the Lipschitz norm controls the L2 norm was used above in the discussion of spectral properties of the random walk operator. The assumption σ < ∞ is necessary. As a counterexample, consider a random walk on N that sends every x ∈ N to some fixed distribution ν on N with infinite second moment: coarse Ricci curvature is 1, yet the identity function is not in L2 .
Proof. Suppose for now that |f | is bounded by A ∈ R, so that Var f < ∞. We first prove that Var Mt f tends to 0. Let Br be the ball of radius r in X centered at some basepoint. Ust t ing on Br and bounded by A on X \ Br , we get Var Mt f = thatt M f is (1t − κ) -Lipschitz 1 2 (M f (x) − M f (y)) dν(x) dν(y) 2(1 − κ)2t r 2 + 2A2 ν(X \ Br ). Taking for example 2 r = 1/(1 − κ)t/2 shows that Var Mt f → 0. As already mentioned, one has Var f = Var Mf + Varmx f dν(x). Since Var Mt f → 0, by induction we get
Var f =
∞
Varmx Mt f dν(x).
t=0
Now since f is 1-Lipschitz, by definition Varmx f σ (x)2 /nx . Since Mt f is (1 − κ)t -Lipschitz, σ2 . The case of we have Varmx Mt f (1 − κ)2t σ (x)2 /nx so that the sum above is at most nκ(2−κ) unbounded f is treated by a simple limiting argument. 2
834
Y. Ollivier / Journal of Functional Analysis 256 (2009) 810–864
3.2. Gaussian concentration As mentioned above, positive coarse Ricci curvature implies a Gaussian-then-exponential concentration theorem. The estimated variance is σ 2 /nκ as above, so that this is essentially a more precise version of Proposition 32, with some loss in the constants. We will see in the discussion below (Section 3.3) that in the main examples, the order of magnitude is correct. The fact that concentration is not always Gaussian far away from the mean is genuine, as exemplified by binomial distributions on the cube (Section 3.3.3) or M/M/∞ queues (Section 3.3.4). The width of the Gaussian window is controlled by two factors. First, variations of the diffusion constant σ (x)2 can result in purely exponential behavior (Section 3.3.5); this leads to the assumption that σ (x)2 is bounded by a Lipschitz function. Second, as Gaussian phenomena only emerge as the result of a large number of small events, the “granularity” of the process must be bounded, which leads to the (comfortable) assumption that σ∞ < ∞. Otherwise, a Markov chain which sends every point x ∈ X to some fixed measure ν has coarse Ricci curvature 1 and can have arbitrary bad concentration properties depending on ν. In the case of Riemannian manifolds, simply letting the step of the random walk tend to 0 makes the width of the Gaussian window tend to infinity, so that we recover Gaussian concentration as in the Lévy–Gromov or Bakry–Émery theorems. For the uniform measure on the discrete cube, the Gaussian width is equal to the diameter of the cube, so that we get full Gaussian concentration as well. In a series of other examples (such as Poisson measures), the transition from Gaussian to non-Gaussian regime occurs roughly as predicted by the theorem. Theorem 33 (Gaussian concentration). Let (X, d, m) be a random walk on a metric space, with coarse Ricci curvature at least κ > 0. Let ν be the unique invariant distribution. Let Dx2 :=
σ (x)2 nx κ
and D 2 := Eν Dx2 . Suppose that the function x → Dx2 is C-Lipschitz. Set tmax :=
D2 . max(σ∞ , 2C/3)
Then for any 1-Lipschitz function f , for any t tmax we have t2 ν x, f (x) t + Eν f exp − 6D 2 and for t tmax 2 tmax t − tmax . ν x, f (x) t + Eν f exp − − 6D 2 max(3σ∞ , 2C)
Y. Ollivier / Journal of Functional Analysis 256 (2009) 810–864
835
Remark 34. Proposition 24 or Corollary 22 often provide very sharp a priori bounds for Eν Dx2 even when no information on ν is available, as we shall see in the examples. Remark 35. It is clear from the proof below that σ (x)2 /nx κ itself need not be Lipschitz, only bounded by some Lipschitz function. In particular, if σ (x)2 is bounded one can always set D 2 = 2 supx σn(x) and C = 0. xκ Remark 36 (Continuous-time situations). If we replace the random walk m = (mx )x∈X with the lazy random walk m whose transition probabilities are mx := (1 − α)δx + αmx , when α tends to 0 this approximates the law at time α of the continuous-time random walk with transition rates mx , so that the continuous-time random walk is obtained by taking the lazy random walk m and speeding up time by 1/α when α → 0. Of course this does not change the invariant distribution. The point is that when α → 0, both σx2 and κ scale like α (and nx tends to 1), so that D 2 has a finite limit. This means that we can apply Theorem 33 to continuous-time examples that naturally appear as limits of a discrete-time, finite-space Markov chain, as illustrated in Sections 3.3.4 to 3.3.6. Remark 37. The condition that σ∞ is uniformly bounded can be replaced with a Gaussian-type assumption, namely that for each measure mx there exists a number sx such that Emx eλf 2 2 eλ sx /2 eλEmx f for any 1-Lipschitz function f . Then a similar theorem holds, with σ (x)2 /nx replaced with sx2 . (When sx2 is constant this is Proposition 2.10 in [15].) However, this is generally not well-suited to discrete settings, because when transition probabilities are small, the best sx2 for which such an inequality is satisfied is usually much larger than the actual variance σ (x)2 : for example, if two points x and y are at distance 1 and mx (y) = ε, sx must satisfy sx2 1/2 ln(1/ε) ε. Thus making this assumption will provide extremely poor estimates of the variance D 2 when some transition probabilities are small (e.g. for binomial distributions on the discrete cube), and in particular, this cannot extend to the continuous-time limit. In Section 3.3.5, we give a simple example where the Lipschitz constant of σ (x)2 is large, resulting in exponential rather than Gaussian behavior. In Section 3.3.6 we give two examples of positively curved process with heavy tails: one in which σ∞ = 1 but with non-Lipschitz growth of σ (x)2 , and one with σ (x)2 1 but with unbounded σ∞ (x). These show that the assumptions cannot be relaxed. Proof. This proof is a variation on standard martingale methods for concentration (see e.g. Lemma 4.1 in [29], or [45]). Lemma 38. Let ϕ : X → R be an α-Lipschitz function with α 1. Assume λ 1/3σ∞ . Then for x ∈ X we have λϕ 2 2 σ (x)2 Me (x) eλMϕ(x)+λ α nx . 2 α2 σ 2 ∞
Note that the classical Proposition 1.16 in [29] would yield (Meλϕ )(x) eλMϕ(x)+2λ which is too weak to provide reasonable variance estimates.
,
836
Y. Ollivier / Journal of Functional Analysis 256 (2009) 810–864
Proof of Lemma 38. For any smooth function g and any real-valued random variable Y , a Taylor expansion with Lagrange remainder gives Eg(Y ) g(EY ) + 12 (sup g ) Var Y . Applying this with g(Y ) = eλY we get λϕ λ2 sup eλϕ Varmx ϕ Me (x) = Emx eλϕ eλMϕ(x) + 2 Supp mx and note that since diam Supp mx 2σ∞ and ϕ is 1-Lipschitz we have supSupp mx ϕ Emx ϕ + 2σ∞ , so that λϕ λ2 λMϕ(x)+2λσ∞ e Me (x) eλMϕ(x) + Varmx ϕ. 2 Now, by definition we have Varmx ϕ α 2 hence the result. 2
σ (x)2 nx . Moreover for λ 1/3σ∞
we have e2λσ∞ 2,
Back to the proof of the theorem, let f be a 1-Lipschitz function and λ 0. Define by induc2 2k tion f0 := f and fk+1 (x) := Mfk (x) + λ σ (x) nx (1 − κ/2) . 2
Suppose that λ 1/2C. Then λ σ (x) nx is κ/2-Lipschitz. Using Proposition 29, we can show by induction that fk is (1 − κ/2)k -Lipschitz. Consequently, the lemma yields λf 2 σ (x)2 2k Me k (x) eλMfk (x)+λ nx (1−κ/2) = eλfk+1 (x) so that by induction k λf M e (x) eλfk (x) . Now setting g(x) :=
σ (x)2 nx
and unwinding the definition of fk yields
k
k−i fk (x) = Mk f (x) + λ M g (x) (1 − κ/2)2(i−1) i=1
so that lim fk (x) = Eν f + λ
k→∞
∞
Eν g (1 − κ/2)2(i−1) Eν f + λEν g
i=1
4 . 3κ
Meanwhile, (Mk eλf )(x) tends to Eν eλf , so that 4λ2
Eν eλf lim eλfk eλEν f + 3κ k→∞
Eν σ (x) nx
2
.
We can conclude by a standard Chebyshev inequality argument. The restrictions λ 1/2C and λ 1/3σ∞ give the value of tmax . 2
Y. Ollivier / Journal of Functional Analysis 256 (2009) 810–864
837
Remark 39 (Finite-time concentration). The proof provides a similar concentration result for the finite-time measures m∗k x as well, with variance 2 Dx,k
k 2
2(i−1) k−i σ (y) = (1 − κ/2) M (x) ny i=1
2 instead of D 2 in the expression for t and Dx,k max .
3.3. Examples revisited Let us test the sharpness of these estimates in some examples, beginning with the simplest ones. In each case, we gather the relevant quantities in a table. Recall that ≈ denotes an equality up to a multiplicative universal constant (typically 4), while symbol ∼ denotes usual asymptotic equivalence (with sharp constant). 3.3.1. Riemannian manifolds First, let X be a compact N -dimensional Riemannian manifold with positive Ricci curvature. Equip this manifold with the ε-step random walk as in Example 7. The measure vol B(x,ε) vol BEucl (ε) d vol(x) is reversible for this random walk. In particular, when ε → 0, the density of this measure with respect to the Riemannian volume is 1 + O(ε 2 ). Let inf Ric denote the largest K > 0 such that Ric(v, v) K for any unit tangent vector v. The relevant quantities for this random walk are as follows (see Section 8 for the proofs). Coarse Ricci curvature Coarse diffusion constant
2
κ ∼ 2(Nε +2) inf Ric σ (x)2 ∼ ε2 NN+2 ∀x
Dimension
n≈N
Variance (Lévy–Gromov thm.)
≈ 1/ inf Ric
Gaussian variance (Theorem 33)
D 2 ≈ 1/ inf Ric
Gaussian range
tmax ≈ 1/(ε inf Ric) → ∞
So, up to some (small) numerical constants, we recover Gaussian concentration as in the Lévy–Gromov theorem. The same applies to diffusions with a drift on a Riemannian manifold, as considered by Bakry and Émery. To√be consistent with the notation of Example 11, in the table above ε has to be replaced with (N + 2)δt, and inf Ric with inf(Ric(v, v) − 2∇ sym F (v, v)) for v a unit tangent vector. (In the non-compact case, care has to be taken since the solution of the stochastic differential equation of Example 11 on the manifold may not exist, and even if it does its Euler scheme approximation at time δt may not converge uniformly on the manifold. In explicit examples such as the Ornstein–Uhlenbeck process, however, this is not a problem.) 3.3.2. Discrete cube Consider now the discrete cube {0, 1}N equipped with its graph distance (Hamming metric) and lazy random walk (Example 8). For a random walk on a graph one always has σ ≈ 1, and n 1 in full generality. The following remark allows for more precise constants.
838
Y. Ollivier / Journal of Functional Analysis 256 (2009) 810–864
Remark 40. Let m be a random walk on a graph. Then, for any vertex x one has σ (x)2 /nx 1 − mx ({x}). Proof. By definition σ (x)2 /nx is the maximal variance, under mx , of a 1-Lipschitz function. So let f be a 1-Lipschitz function on the graph. Since variance is invariant by adding a constant, we can assume that f (x) = 0. Then |f (y)| 1 for any neighbor y of x. The mass, under mx , of all neighbors of x is 1 − mx ({x}). Hence Varmx f = Emx f 2 − (Emx f )2 Emx f 2 1 − mx ({x}). This value is achieved, for example, with a lazy simple random walk when x has an even number of neighbors and when no two distinct neighbors of x are mutual neighbors; in this case one can take f (x) = 0, f = 1 on half the neighbors of x and f = −1 on the remaining neighbors of x. 2 Applying this to the lazy random walk on the discrete cube, one gets: Coarse Ricci curvature
κ = 1/N
Coarse diffusion constant & dimension σ (x)2 /nx 1/2 Estimated variance (Proposition 32)
σ 2 /nκ(2 − κ) ∼ N/4
Actual variance
N/4
Gaussian variance (Theorem 33)
D 2 N/2
Gaussian range
tmax = N/2
In particular, since N/2 is the maximal possible value for the deviation from average of a 1-Lipschitz function on the cube, we see that tmax has the largest possible value. 3.3.3. Binomial distributions The occurrence of a finite range tmax for the Gaussian behavior of tails is genuine, as the following example shows. Let again X = {0, 1}N equipped with its Hamming metric (each edge is of length 1). Consider the following Markov chain on X: for some 0 < p < 1, at each step, choose a bit at random among the N bits; if it is equal to 0, flip it to 1 with probability p; if it is equal to 1, flip it to 0 with probability 1 − p. The binomial distribution ν((x1 , . . . , xN )) = p xi (1 − p)1−xi is reversible for this Markov chain. The coarse Ricci curvature of this Markov chain is 1/N , as can easily be seen directly or using the tensorization property (Proposition 27). Let k be the number of bits of x ∈ X which are equal to 1. Then k follows a Markov chain on {0, 1, . . . , N}, whose transition probabilities are: pk,k+1 = p(1 − k/N ), pk,k−1 = (1 − p)k/N, pk,k = pk/N + (1 − p)(1 − k/N ). The binomial distribution with parameters N and p, namely Nk p k (1 − p)N −k , is reversible for this Markov chain. Moreover, the coarse Ricci curvature of this “quotient” Markov chain is still 1/N .
Y. Ollivier / Journal of Functional Analysis 256 (2009) 810–864
839
Now, fix some λ > 0 and consider the case p = λ/N . Let N → ∞. It is well known that the invariant distribution tends to the Poisson distribution e−λ λk /k! on N. Let us see how Theorem 33 performs on this example. The table below applies either to the full space {0, 1}N , with k the function “number of 1’s,” or to its projection on {0, 1, . . . , N }. Note the use of Proposition 24 to estimate σ 2 a priori, without having to resort to explicit knowledge of the invariant distribution. All constants implied in the O(1/N ) notation are small and completely explicit. Coarse Ricci curvature
κ = 1/N
Coarse diffusion constant
σ (k)2 = (λ + k)/N + O(1/N 2 )
Estimated Ek (Proposition 24)
Ek J (0)/κ = λ
Actual Ek
Ek = λ
Average diffusion constant
σ 2 = Eσ (k)2 = 2λ/N + O(1/N 2 )
Dimension
n1
Estimated variance (Proposition 32) Actual variance
σ 2 /nκ(2 − κ) λ + O(1/N ) λ
Gaussian variance (Theorem 33)
D 2 2λ + O(1/N )
Lipschitz constant of Dx2
C = 1 + O(1/N )
Gaussian range
tmax = 4λ/3
The Poisson distribution has a roughly Gaussian behavior (with variance λ) in a range of size approximately λ around the mean; further away, it decreases like e−k ln k which is not Gaussian. This is in good accordance with tmax the table above, and shows that the Gaussian range cannot be extended. 3.3.4. A continuous-time example: M/M/∞ queues Here we show how to apply Theorem 33 to a continuous-time example, the M/M/∞ queue. These queues were brought to my attention by D. Chafaï. The M/M/∞ queue consists of an infinite number of “servers.” Each server can be free (0) or busy (1). The state space consists of all sequences in {0, 1}N with a finite number of 1’s. The dynamics is at follows: fix two numbers λ > 0 and μ > 0. At a rate λ per unit of time, a client arrives and the first free server becomes busy. At a rate μ per unit of time, each busy server finishes its job (independently of the others) and becomes free. The number k ∈ N of busy servers is a continuous-time Markov chain, whose transition probabilities at small times t are given by t pk,k+1 = λt + O t 2 , t pk,k−1 = kμt + O t 2 ,
t = 1 − (λ + kμ)t + O t 2 . pk,k
This system is often presented as a discrete analogue of an Ornstein–Uhlenbeck process, since asymptotically the drift is linear towards the origin. However, it is not symmetric around the mean, and moreover the invariant (actually reversible) distribution ν is a Poisson distribution with parameter λ/μ, rather than a Gaussian.
840
Y. Ollivier / Journal of Functional Analysis 256 (2009) 810–864
This continuous-time Markov chain can be seen as a limit of the binomial Markov chain above as follows: first, replace the binomial Markov chain with its continuous-time equivalent (Remark 36); Then, set p = λ/N and let N → ∞, while speeding up time by a factor N . The analogy is especially clear in the table below: if we replace λ with λ/N and μ with 1/N , we get essentially the same table as for the binomial distribution. It is easy to check that Proposition 32 (with σ 2 /2nκ instead of σ 2 /nκ(2 − κ)) and Theorem 33 pass to the limit. In this continuous-time setting, the definitions become the following: κ(x, y) := − dtd W1 (mtx , mty )/d(x, y) (as mentioned in the introduction) and σ (x)2 := 1 d d(y, z) dmtx (y) dmtx (z), where mtx is the law at time t of the process starting at x. 2 dt Then the relevant quantities are as follows. Coarse Ricci curvature
κ =μ
Coarse diffusion constant
σ (k)2 = kμ + λ
Estimated Ek (Proposition 24)
Ek J (0)/κ = λ/μ
Actual Ek
Ek = λ/μ
Average diffusion constant
σ 2 = Eσ (k)2 = 2λ
Dimension
n1
Estimated variance (Proposition 32)
σ 2 /2nκ = λ/μ
Actual variance
λ/μ
Gaussian variance (Theorem 33)
D 2 2λ/μ
Lipschitz constant of Dx2
C=1
Gaussian range
tmax = 4λ/3μ
So once more Theorem 33 is in good accordance with the behavior of the random walk, whose invariant distribution is Poisson with mean and variance λ/μ, thus Gaussian-like only in some interval around this value. An advantage of this approach is that is can be generalized to situations where the rates of the servers are not constant, but, say, bounded between μ0 /10 and 10μ0 , and clients go to the first free server according to some predetermined scheme, e.g. the fastest free server. Indeed, the M/M/∞ queue above can be seen as a Markov chain in the full configuration space of the servers, namely the space of all sequences over the alphabet {free, busy} containing a finite number of “busy.” It is easy to check that the coarse Ricci curvature is still equal to μ in this configuration space. Now in the case of variable rates, the number of busy servers is generally not Markovian, so one has to work in the configuration space. If the rate of the ith server is μi , the coarse Ricci curvature is inf μi in the configuration space, whereas the diffusion constant is controlled by sup μi . So if the rates vary in a bounded range, coarse Ricci curvature still provides a Gaussian-type control, though an explicit description of the invariant distribution is not available. Let us consider more realistic queue models, such as the M/M/k queue, i.e. the number of servers is equal to k (with constant or variable rates). Then, on the part of the space where some servers are free, coarse Ricci curvature is at least equal to the rate of the slowest server; whereas on the part of the space where all servers are busy, coarse Ricci curvature is 0. If, as often, an abandon rate for waiting clients is added to the model, then coarse Ricci curvature is equal to this abandon rate on the part of the space where all servers are busy (and in particular, coarse Ricci curvature is positive on the whole space).
Y. Ollivier / Journal of Functional Analysis 256 (2009) 810–864
841
3.3.5. An example of exponential concentration We give here a very simple example of a Markov chain which has positive curvature but for which concentration is not Gaussian but exponential, due to large variations of the diffusion constant, resulting in a large value of C. Compare Example 14 above where exponential concentration was due to unbounded σ∞ . This is a continuous-time random walk on N defined as follows. Take 0 < α < β. For k ∈ N, the transition rate from k to k + 1 is (k + 1)α, whereas the transition rate from k + 1 to k is (k + 1)β. It is immediate to check that the geometric distribution with decay α/β is reversible for this Markov chain. The coarse Ricci curvature of this Markov chain is easily seen to be β − α. We have σ (k)2 = (k + 1)α + kβ, so that σ (k)2 is (α + β)-Lipschitz and C = (α + β)/(β − α). The expectation of k under the invariant distribution can be bounded by J (0)/κ = α/(β − α) by Proposition 24, which is actually the exact value. So the expression above for σ (k)2 yields σ 2 = 2αβ/(β − α). Consequently, the estimated variance σ 2 /2nκ (obtained by the continuoustime version of Proposition 32) is at most αβ/(β − α)2 , which is the actual value. Now consider the case when β − α is small. If the C factor in Theorem 33 is not taken into account, we get blatantly false results since the invariant distribution is not Gaussian at all. Indeed, in the regime where β − α → 0, the width of the Gaussian window in Theorem 33 is D 2 /C ≈ α/(β − α). This is fine, as this is the decay distance of the invariant distribution, and in this interval both the Gaussian and geometric estimates are close to 1 anyway. But without the C factor, we would get D 2 /σ∞ = αβ/(β − α)2 , which is much larger; the invariant distribution is clearly not Gaussian on this interval. Moreover, Theorem 33 predicts, in the exponential regime, a exp(−t/2C) behavior for concentration. Here the asymptotic behavior of the invariant distribution is (α/β)t ∼ (1 − 2/C)t ∼ e−2t/C when β − α is small. So we see that (up to a constant 4) the exponential decay rate predicted by Theorem 33 is genuine. 3.3.6. Heavy tails It is clear that a variance control alone does not imply any concentration bound beyond the Bienaymé–Chebyshev inequality. We now show that this is still the case even under a positive curvature assumption. Namely, in Theorem 33, neither the assumption that σ (x)2 is Lipschitz, nor the assumption that σ∞ is bounded, can be removed (but see Remark 37). Heavy tails with non-Lipschitz σ (x)2 . Our next example shows that if the diffusion constant σ (x)2 is not Lipschitz, then non-exponential tails may occur in spite of positive curvature. Consider the continuous-time random walk on N defined as follows: the transition rate from k to k + 1 is a(k + 1)2 , whereas the transition rate from k to k − 1 is a(k + 1)2 + bk for k 1. Here a, b > 0 are fixed. We have κ = b and σ (k)2 = 2a(k + 1)2 + bk, which is obviously not Lipschitz. This Markov chain has a reversible measure ν, which satisfies ν(k)/ν(k − 1) = ak 2 /(a(k + 1)2 + bk) = 1 − k1 (2 + ab ) + O(1/k 2 ). Consequently, asymptotically ν(k) behaves like k k 1 1 b 1− 2+ ≈ e−(2+b/a) i=1 i ≈ k −(2+b/a) i a i=1
thus exhibiting heavy, non-exponential tails.
842
Y. Ollivier / Journal of Functional Analysis 256 (2009) 810–864
This shows that the Lipschitz assumption for σ (x)2 cannot be removed, even though σ∞ = 1. It would seem reasonable to expect a systematic correspondance between the asymptotic behavior of σ (x)2 and the behavior of tails. Heavy tails with unbounded σ ∞ . Consider the following random walk on N∗ : a number k goes to 1 with probability 1 − 1/4k 2 and to 2k with probability 1/4k 2 . One can check that κ 1/2. These probabilities are chosen so that σ (k)2 = (2k − 1)2 × 1/4k 2 × (1 − 1/4k 2 ) 1, so that the variance of the invariant distribution is small. However, let us evaluate the probability that, starting at 1, the first all consist in doing a multiplication by 2, so that we end i steps 1 −1−i(i−1)/2 . Setting i = log k, we see that the invariant at 2i ; this probability is i−1 2 j =0 4·(2j )2 = 4 distribution ν satisfies ν(k)
ν(1) − log2 k(log2 k−1) 2 4
for k a power of 2. This is clearly not Gaussian or exponential, though σ (k)2 is bounded. 4. Local control and logarithmic Sobolev inequality We now turn to control of the gradient of Mf at some point, in terms of the gradient of f at neighboring points. This is closer to classical Bakry–Émery theory, and allows to get a kind of logarithmic Sobolev inequality. Definition 41 (Norm of the gradient). Choose λ > 0 and, for any function f : X → R, define the λ-range gradient of f by |f (y) − f (y )| −λd(x,y)−λd(y,y ) e . d(y, y ) y,y ∈X
(Df )(x) := sup
This is a kind of “mesoscopic” Lipschitz constant of f around x, since pairs of points y, y far away from x will not contribute much to Df (x). If f is a smooth function on a compact Riemannian manifold, when λ → ∞ this quantity tends to |∇f (x)|. It is important to note that log Df is λ-Lipschitz. We will also need a control on negative curvature: in a Riemannian manifold, Ricci curvature might be 1 because there is a direction of curvature 1000 and a direction of curvature −999. The next definition captures these variations. Definition 42 (Unstability). Let 1 κ+ (x, y) := d(x, y)
d(x, y) − d(x + z, y + z) +
z
and κ− (x, y) :=
1 d(x, y)
z
d(x, y) − d(x + z, y + z) −
Y. Ollivier / Journal of Functional Analysis 256 (2009) 810–864
843
where a+ and a− are the positive and negative part of a ∈ R, so that κ(x, y) = κ+ (x, y) − κ− (x, y). (The integration over z is under a coupling realizing the value of κ(x, y).) The unstability U (x, y) is defined as U (x, y) :=
κ− (x, y) κ(x, y)
and U :=
sup
x,y∈X, x =y
U (x, y).
Remark 43. If X is ε-geodesic, then an upper bound for U (x, y) with d(x, y) ε implies the same upper bound for U . In most discrete examples given in the introduction (Examples 8, 10, 12, 13, 14), unstability is actually 0, meaning that the coupling between mx and my never increases distances. (This could be a possible definition of non-negative sectional curvature for Markov chains.) In Riemannian manifolds, unstability is controlled by the largest negative sectional curvature. Interestingly, in Example 17 (Glauber dynamics), unstability depends on temperature. Due to the use of the gradient D, the theorems below are interesting only if a reasonable estimate for Df can be obtained depending on “local” data. This is not the case when f is not λ-log-Lipschitz (compare the similar phenomenon in [8]). This is consistent with the fact mentioned above, that Gaussian concentration of measure only occurs in a finite range, with exponential concentration afterwards, which implies that no true logarithmic Sobolev inequality can hold in general. Theorem 44 (Gradient contraction). Suppose that coarse Ricci curvature is at least κ > 0. Let λ 20σ∞1(1+U ) and consider the λ-range gradient D. Then for any function f : X → R with Df < ∞ we have D(Mf )(x) (1 − κ/2)M(Df )(x) for all x ∈ X. Theorem 45 (Log-Sobolev inequality). Suppose that coarse Ricci curvature is at least κ > 0. Let λ 20σ∞1(1+U ) and consider the λ-range gradient D. Then for any function f : x → R with Df < ∞, we have 4σ (x)2 Varν f sup (Df )2 dν κnx x and for positive f ,
4σ (x)2 Entν f sup κnx x
(Df )2 dν f
where ν is the invariant distribution. If moreover the random walk is reversible with respect to ν, then Varν f
V (x) Df (x)2 dν(x)
844
Y. Ollivier / Journal of Functional Analysis 256 (2009) 810–864
and Entν f
V (x)
Df (x)2 dν(x) f (x)
where V (x) = 2
∞
σ (x)2 . (1 − κ/2)2t Mt+1 nx t=0
The form involving V (x) is motivated by the fact that, for reversible diffusions in RN with non-constant diffusion coefficients, the coefficients naturally appear in the formulation of functional inequalities (see e.g. [2]). The quantity V (x) Df (x)2 is to be thought of as a crude version of the Dirichlet form associated with the random walk. It would be more satisfying to obtain inequalities involving the latter (compare Corollary 31), but I could not get a version of the commutation property DM (1 − κ/2)MD involving the Dirichlet form. Remark 46. If
σ (x)2 nx κ
is C-Lipschitz (as in Theorem 33), then V (x)
4σ 2 κn
+ 2C
J (x) κ .
Examples. Let us compare this theorem to classical results. In the case of a Riemannian manifold, for any smooth function f we can choose a random walk with small enough steps, so that λ can be arbitrarily large and Df arbitrarily close to |∇f |. Since moreover σ (x)2 does not depend on x for the Brownian motion, this theorem allows to recover the logarithmic Sobolev inequality in the Bakry–Émery framework, with the correct constant up to a factor 4. Next, consider the two-point space {0, 1}, equipped with the measure ν(0) = 1 − p and ν(1) = p. This is the space on which modified logarithmic Sobolev inequalities were introduced [8]. We endow this space with the Markov chain sending each point to the invariant distribution. Here we have σ (x)2 = p(1 − p), nx = 1 and κ = 1, so that we get the inequality )2 Entν f 4p(1 − p) (Df f dν, comparable to the known inequality [8] except for the factor 4. The modified logarithmic Sobolev inequality for Bernoulli and Poisson measures is traditionally obtained by tensorizing this result [8]. If, instead, we directly apply the theorem above to the Bernoulli measure on {0, 1}N or the Poisson measure on N (see Sections 3.3.3 and 3.3.4), we get slightly worse results. Indeed, consider the M/M/∞ queue on N, which is the limit when N → ∞ of the projection on N of the Markov chains on {0, 1}N associated with Bernoulli measures. Keeping the notation of Section 3.3.4, we get, in the continuous-time version, σ (x)2 = xμ + λ, which is not bounded. So we have to use the version with V (x); Remark 46 and the formulas in Section 3.3.4 yield V (x) 8λ/μ + 2(λ + xμ)/μ so that we get the inequality Entν f
λ μ
=
λ μ
Df (x)2 (10 + 2xμ/λ) dν(x) f (x) Df (x)2 2 dν(x − 1) + 10 dν(x) f (x)
Y. Ollivier / Journal of Functional Analysis 256 (2009) 810–864
845
which is to be compared to the inequality λ Entν f μ
D+ f (x)2 dν(x) f (x)
obtained in [8], with D+ f (x) = f (x + 1) − f (x). So asymptotically our version is worse by a factor dν(x − 1)/dν(x) ≈ x. One could say that our general, non-local notion of gradient fails to distinguish between a point and an immediate neighbor, and does not take advantage of the particular directional structure of a random walk on N as the use of D+ does. Yet being able to handle the configuration space directly rather than as a product of the twopoint space allows us to deal with more general, non-product situations. Consider for example the queuing process with heterogeneous server rates mentioned at the end of Section 3.3.4, where newly arrived clients go to the fastest free server (in which case the number of busy servers is not Markovian). Then coarse Ricci curvature is equal to the infimum of the server rates, and Theorem 45 still holds, though the constants are probably not optimal when the rates are very different. I do not know if this result is new. We now turn to the proof of Theorems 44 and 45. The proof of the former is specific to our setting, but the passage from the former to the latter is essentially a copy of the Bakry–Émery argument. Lemma 47. Let x, y ∈ X with κ(x, y) > 0. Let (Z, μ) be a probability space equipped with a map π : Z → Supp mx × Supp my such that π sends μ to an optimal coupling between mx 1 and my . Let A be a positive function on Z such that sup A/ inf A eρ with ρ 2(1+U ) . Then
d(x + z, y + z) 1 − κ(x, y)/2 A(z) d(x, y)
A(z) z
z∈Z
and in particular
A(z) d(x + z, y + z) − d(x, y) 0
z∈Z
where, as usual, x + z and y + z denote the two projections from Z to Supp mx and Supp my respectively. Proof.The idea is the following: when A is constant, the result obviously holds since by definition d(x + z, y + z)/d(x, y) = 1 − κ(x, y). Now when A is close enough to a constant, the same holds with some numerical loss. Set F = supz A(z). Then A(z) z
d(x + z, y + z) = d(x, y)
A(z) + F
z
z
A(z) d(x + z, y + z) −1 . F d(x, y)
846
Y. Ollivier / Journal of Functional Analysis 256 (2009) 810–864
Let Z− = {z ∈ Z, d(x, y) < d(x + z, y + z)} and Z+ = Z \ Z− . Recall that by definition, κ− (x, y) = Z− (d(x + z, y + z)/d(x, y) − 1) and κ+ (x, y) = Z+ (1 − d(x + z, y + z)/d(x, y)), so that κ = κ+ − κ− . Using that A(z) F on Z− and A(z) e−ρ F on Z+ , we get d(x + z, y + z) A(z) + F κ− (x, y) − e−ρ κ+ (x, y) . A(z) d(x, y) z
z
Now by definition of U we have κ− (x, y) U κ(x, y). It is not difficult to check that ρ is enough to ensure that e−ρ κ+ (x, y) − κ− (x, y) κ(x, y)/2, hence
1 2(1+U )
A(z)
d(x + z, y + z) d(x, y)
z
A(z) − F κ(x, y)/2 z
1 − κ(x, y)/2
A(z) z
as needed.
2
Proof of Theorem 44. Let y, y ∈ X. Let ξxy and ξyy be optimal couplings between mx and my , my and my respectively. Apply the gluing lemma for couplings (Lemma 7.6 in [47]) to obtain a measure μ on Z = Supp mx × Supp my × Supp my whose projections on Supp mx × Supp my and Supp my × Supp my are ξxy and ξyy respectively. We have |Mf (y) − Mf (y )| −λ(d(x,y)+d(y,y )) e d(y, y ) −λ(d(x,y)+d(y,y )) e = f (y + z) − f (y + z) d(y, y ) z∈Z
−λ(d(x,y)+d(y,y )) f (y + z) − f (y + z) e d(y, y )
z∈Z
Df (x + z)
d(y + z, y + z)
e−λ(d(x+z,y+z)+d(y+z,y +z))
e−λ(d(x,y)+d(y,y )) d(y, y )
z∈Z
=
A(z)B(z)
d(y + z, y + z) d(y, y )
z∈Z
where A(z) = Df (x + z) and B(z) = eλ(d(x+z,y+z)−d(x,y)+d(y+z,y +z)−d(y,y )) . Since diam Supp mx 2σ∞ and likewise for y, for any z, z we have d(x + z, y + z) − d(x + z , y + z ) 4σ∞ , d(y + z, y + z) − d(y + z , y + z ) 4σ∞
Y. Ollivier / Journal of Functional Analysis 256 (2009) 810–864
847
so that B varies by a factor at most e8λσ∞ on Z. Likewise, since Df is λ-log-Lipschitz, A varies by a factor at most e2λσ∞ . So the quantity A(z)B(z) varies by at most e10λσ∞ . So if λ 20σ∞1(1+U ) , we can apply Lemma 47 and get A(z)B(z)
d(y + z, y + z) (1 − κ/2) d(y, y )
z∈Z
A(z)B(z).
z∈Z
Now we have z A(z)B(z) = z A(z) + z A(z)(B(z) − 1). Unwinding B(z) and using that ea − 1 aea for any a ∈ R, we get
A(z)(B(z) − 1) λ z
A(z)B(z) d(x + z, y + z) − d(x, y) + d(y + z, y + z) − d(y, y )
z
which decomposes as a sum of two terms λ z A(z)B(z)(d(x + z, y + z) − d(x, y)) and λ A(z)B(z)(d(y + z, y + z) − d(y, y )), each of which is non-positive by Lemma 47. Hence z z A(z)(B(z) − 1) 0 and z A(z)B(z) z A(z) = z Df (x + z) = M(Df )(x). So we have shown that for any y, y in X we have |Mf (y) − Mf (y )| −λd(x,y)+d(y,y ) e (1 − κ/2)M(Df )(x) d(y, y ) as needed.
2
Lemma 48. Let f be a function with Df < ∞. Let x ∈ X. Then f is e4λσ∞ M(Df )(x)-Lipschitz on Supp mx . Proof. For any y, z ∈ Supp mx , by definition of D we have |f (y) − f (z)| Df (y)d(y, z)eλd(y,z) Df (y)d(y, z)e2λσ∞ . Since moreover Df is λ-log-Lipschitz, we have Df (y) e2λσ∞ infSupp mx Df e2λσ∞ M(Df )(x), so that finally f (y) − f (z) d(y, z) M(Df )(x) e4λσ∞ as announced.
2
Proof of Theorem 45. Let ν be the invariant distribution. Let f be a positive measurable function. Associativity of entropy (e.g. Theorem D.13 in [22] applied to the measure f (y) dν(x)dmx (y) on X × X) states that Ent f =
Entmx f dν(x) + Ent Mf x
=
t0 x
Entmx Mt f dν(x)
848
Y. Ollivier / Journal of Functional Analysis 256 (2009) 810–864
by induction, and similarly Var f =
Varmx Mt f dν(x).
t0 x
Since by the lemma above, f is M(Df )(x) e4λσ∞ -Lipschitz on Supp mx and moreover < 2, we have
e8λσ∞
Varmx f
2(M(Df )(x))2 σ (x)2 nx
and, using that a log a a 2 − a, we get that Entmx f Entmx f
1 Mf (x)
Varmx f so
2(M(Df )(x))2 σ (x)2 . nx Mf (x)
Thus Var f 2
σ (x)2 2 M DMt f (x) dν(x) nx t0 x
and Ent f 2
σ (x)2 (M(DMt f )(x))2 dν(x). nx Mt+1 f (x) t0 x
By Theorem 44, we have (DMt f )(y) (1 − κ/2)t Mt (Df )(y), so that Var f 2
σ (x)2 2 Mt+1 Df (x) (1 − κ/2)2t dν(x) nx t0 x
and Ent f 2
σ (x)2 (Mt+1 Df (x))2 (1 − κ/2)2t dν(x). nx Mt+1 f (x) t0 x
Now, for variance, convexity of a → a 2 yields t+1 2 M Df Mt+1 (Df )2 and for entropy, convexity of (a, b) → a 2 /b for a, b > 0 yields 2 (Mt+1 Df (x))2 t+1 (Df ) M (x). f Mt+1 f (x)
Y. Ollivier / Journal of Functional Analysis 256 (2009) 810–864
849
Finally we get
σ (x)2 t+1 2t Var f 2 (Df )2 (x) dν(x) (1 − κ/2) M nx t0
x
and Ent f 2
σ (x)2 t+1 (Df )2 (x) dν(x). (1 − κ/2)2t M nx f t0
x
Now, in the non-reversible case, simply apply the identity t+1 t+1 g(x) M h(x) dν(x) (sup g) M h(x) dν(x) = (sup g) h dν 2
2 2 to the functions g(x) = σ (x) nx and h(x) = (Df )(x) (for variance) or h(x) = (Df )(x) /f (x) (for entropy). For the reversible case, use the identity t+1 g(x) M h(x) dν(x) = h(x) Mt+1 g(x) dν(x)
instead.
2
5. Exponential concentration in non-negative curvature We have seen that positive coarse Ricci curvature implies a kind of Gaussian concentration. We now show that non-negative coarse Ricci curvature and the existence of an “attracting point” imply exponential concentration. The basic example to keep in mind is the following. Let N be the set of non-negative integers equipped with its standard distance. Let 0 < p < 1 and consider the nearest-neighbor random walk on N that goes to the left with probability p and to the right with probability 1−p; explicitly mk = pδk−1 + (1 − p)δk+1 for k 1, and m0 = pδ0 + (1 − p)δ1 . Since for k 1 the transition kernel is translation-invariant, it is immediate to check that κ(k, k + 1) = 0; besides, κ(0, 1) = p. There exists an invariant distribution if and only if p > 1/2, and it satisfies exponential concentration with decay distance 1/ log(p/(1 − p)). For p = 1/2 + ε with small ε this behaves like 1/4ε. Of course, when p 1/2, there is no invariant distribution so that non-negative curvature alone does not imply concentration of measure. Geometrically, what entails exponential concentration in this example is the fact that, for p > 1/2, the point 0 “pulls” its neighbor, and the pulling is transmitted by non-negative curvature. We now formalize this situation in the following theorem. Theorem 49. Let (X, d, (mx )) be a metric space with random walk. Suppose that for some o ∈ X and r > 0 one has: • κ(x, y) 0 for all x, y ∈ X, • for all x ∈ X with r d(o, x) < 2r, one has W1 (mx , δo ) < d(x, o), • X is r-geodesic,
850
Y. Ollivier / Journal of Functional Analysis 256 (2009) 810–864
• there exists s > 0 such that each measure mx satisfies the Gaussian-type Laplace transform inequality 2 s 2 /2
Emx eλf eλ
eλEmx f
for any λ > 0 and any 1-Lipschitz function f : Supp mx → R. Set ρ = inf{d(x, o) − W1 (mx , δo ), r d(o, x) < 2r} and assume ρ > 0. Then there exists an invariant distribution for the random walk. Moreover, setting D = s 2 /ρ and m = r + 2s 2 /ρ + ρ(1 + J (o)2 /4s 2 ), for any invariant distribution ν we have
ed(x,o)/D dν(x) 4 + J (o)2 /s 2 em/D
and so for any 1-Lipschitz function f : X → R and t 0 we have Pr f − f (o) t + m 8 + 2J (o)2 /s 2 e−t/D . So we get exponential concentration with characteristic decay distance s 2 /ρ. The last assumption is always satisfied with s = 2σ∞ (Proposition 1.16 in [29]). Examples. Before proceeding to the proof, let us show how this applies to the geometric distribution above on N. We take of course o = 0 and r = 1. We can take s = 2σ∞ = 2. Now there is only one point x with r d(o, x) < 2r, which is x = 1. It satisfies m1 = pδ0 + (1 − p)δ2 , so that W1 (m1 , δ0 ) = 2(1 − p), which is smaller than d(0, 1) = 1 if and only if p > 1/2 as was to be expected. So we can take ρ = 1 − 2(1 − p) = 2p − 1. Then we get exponential concentration with characteristic distance 4/(2p − 1). When p is very close to 1 this is not so good (because the discretization is too coarse), but when p is close to 1/2 this is within a factor 2 of the optimal value. Xt Another example is the stochastic differential equation dXt = S dBt − α |X dt on Rn , for t| which exp(−2|x|α/S 2 ) is a reversible measure. Take as a Markov chain the Euler approximation scheme at time δt for this stochastic differential equation, as in Example 11. Taking r = nS 2 /α yields that ρ α δt/2 after some simple computation. Since we have s 2 = S 2 δt for Gaussian measures at time δt, we get exponential concentration with decay distance 2S 2 /α, which is correct up to a factor 4. The additive constant in the deviation inequality is m = r + ρ(1 + J (o)2 /4s 2 ) + 2s 2 /ρ which is equal to (n + 4)S 2 /α + O(δt) (note that J (o)2 ≈ ns 2 ). For comparison, the actual value for the average distance to the origin under the exponential 2 distribution e−2|x|α/S is nS 2 /2α, so that up to a constant the dependency on dimension is recovered. In general, the invariant distribution is not unique under the assumptions of the theorem. For example, start with the random walk on N above with geometric invariant distribution; now consider the disjoint union N ∪ (N + 12 ) where on N + 12 we use the same random walk translated by 12 : the assumptions are satisfied with r = 1 and o = 0, but clearly there are two disjoint invariant distributions. However, if κ > 0 in some large enough ball around o, then the invariant distribution will be unique.
Y. Ollivier / Journal of Functional Analysis 256 (2009) 810–864
851
Proof of Theorem 49. Let us first prove a lemma which shows how non-negative curvature transmits the “pulling.” Lemma 50. Let x ∈ X with d(x, o) r. Then W1 (mx , o) d(x, o) − ρ. Proof. If d(o, x) < 2r then this is one of the assumptions. So we suppose that d(o, x) 2r. Since X is r-geodesic, let o = y0 , y1 , y2 , . . . , yn = x be a sequence of points with d(yi , yi+1 ) = d(o, x). We can assume that d(o, y2 ) > r (otherwise, red(yi , yi+1 ) r and move y1 ). Set z = y1 if d(o, y1 ) = r and z = y2 if d(o, y1 ) < r, so that r d(o, z) < 2r. Now W1 (δo , mx ) W1 (δo , mz ) + W1 (mz , mx ) d(o, z) − ρ + d(z, x) since κ(z, x) 0. But d(o, z) + d(z, x) = d(o, x) by construction, hence the conclusion.
2
We are now ready to prove the theorem. The idea is to consider the function eλd(x,o) . For points far away from the origin, since under the random walk the average distance to the origin decreases by ρ by the previous lemma, we expect the function to be multiplied by e−λρ under the random walk operator. Close to the origin, the evolution of the function is controlled by the variance s 2 and the jump J (o) of the origin. Since the integral of the function is preserved by the random walk operator, and it is multiplied by a quantity < 1 far away, this shows that the weight of faraway points cannot be too large. More precisely, we need to tamper a little bit with what happens around the origin. Let ϕ : R+ → R+ be defined by ϕ(x) = 0 if x < r; ϕ(x) = (x − r)2 /kr if r x < r( k2 + 1) and ϕ(x) = x − r − kr/4 if x r( k2 + 1), for some k > 0 to be chosen later. Note that ϕ is a 1-Lipschitz function and that ϕ 2/kr. If Y is any random variable with values in R+ , we have Eϕ(Y ) ϕ(EY ) +
1 1 Var Y sup ϕ ϕ(EY ) + Var Y. 2 kr
Now choose some λ > 0 and consider the function f : X → R defined by f (x) = eλϕ(d(o,x)) . Note that ϕ(d(o, x)) is 1-Lipschitz, so that by the Laplace transform assumption we have 2 s 2 /2
Mf (x) eλ
eλMϕ(d(o,x)) .
The Laplace transform assumption implies that the variance under mx of any 1-Lipschitz function is at most s 2 . So by the remark above, we have s2 s2 Mϕ d(o, x) ϕ Md(o, x) + = ϕ W1 (mx , δo ) + kr kr so that finally 2 s 2 /2+λs 2 /kr
Mf (x) eλ for any x ∈ X.
eλϕ(W1 (mx ,δo ))
852
Y. Ollivier / Journal of Functional Analysis 256 (2009) 810–864
We will use different bounds on ϕ(W1 (mx , δo )) according to d(o, x). First, if d(x, o) < r, then use non-negative curvature to write W1 (mx , δo ) W1 (mx , mo ) + J (o) d(x, o) + J (o) so that ϕ(W1 (mx , δo )) ϕ(r + J (o)) J (o)2 /kr so that 2 s 2 /2+λs 2 /kr+λJ (o)2 /kr
Mf (x) eλ
2 s 2 /2+λs 2 /kr+λJ (o)2 /kr
= eλ
f (x)
since f (x) = 1. Second, for any x with d(x, o) r, Lemma 50 yields 2 s 2 /2+λs 2 /kr
Mf (x) eλ
eλϕ(d(x,o)−ρ) .
If d(x, o) r( k2 + 1) + ρ then ϕ(d(x, o) − ρ) = ϕ(d(x, o)) − ρ so that 2 s 2 /2+λs 2 /kr−λρ
Mf (x) eλ
f (x).
If r d(x, o) < r( k2 + 1) + ρ, then ϕ(d(x, o) − ρ) ϕ(d(x, o)) so that 2 2
2
Mf (x) eλ s /2+λs /kr f (x). Let ν be any probability measure such that f dν < ∞. Let X = {x ∈ X, d(x, o) < r( k2 + 1)} and X = X \ X . Set A(ν) = X f dν and B(ν) = X f dν. Combining the cases above, we have shown that A(ν ∗ m) + B(ν ∗ m) = f d(ν ∗ m) = Mf dν = Mf dν + Mf dν X
e
X
λ2 s 2 /2+λs 2 /kr+λJ (o)2 /kr
f dν + e X
λ2 s 2 /2+λs 2 /kr−λρ
f dν X
= αA(ν) + βB(ν) 2 2
2
2
2 2
2
with α = eλ s /2+λs /kr+λJ (o) /kr and β = eλ s /2+λs /kr−λρ . Choose λ small enough and k large enough (see below) so that β < 1. Using that A(ν) eλkr/4 for any probability measure ν, we get αA(ν) + βB(ν) (α − β)eλkr/4 + β(A(ν) + B(ν)). λkr/4 λkr/4 In particular, if A(ν) + B(ν) (α−β)e , we get αA(ν) + βB(ν) (α−β)e again. So setting 1−β 1−β (α−β)eλkr/4 , we have just shown that the set C of probability measures ν such that f dν R R= 1−β is invariant under the random walk. Moreover, if A(ν) + B(ν) > R then αA(ν) + βB(ν) < A(ν) + B(ν). Hence, if ν is an invariant distribution, necessarily ν ∈ C. This, together with an evaluation of R given below, will provide the bound for f dν stated in the theorem. We now turn to existence of an invariant distribution. First, C is obviously closed and convex. Moreover, C is tight: indeed if K is a compact, say included in a ball of radius a around o, then for any ν ∈ C we have ν(X \ K) Re−λa . So by Prokhorov’s theorem, C is compact in the weak convergence topology. So C is compact convex in the topological vector space of all
Y. Ollivier / Journal of Functional Analysis 256 (2009) 810–864
853
(signed) Borel measures on X, and is invariant by the random walk operator, which is an affine map. By the Markov–Kakutani theorem (Theorem I.3.3.1 in [23]), it has a fixed point. Let us finally evaluate R. We have 2
α/β − 1 λkr/4 eλJ (o) /kr+λρ − 1 eλkr/4 R= = e 2 2 2 1/β − 1 eλρ−λs /kr−λ s /2 − 1
ρ + J (o)2 /kr 2 eλJ (o) /kr+λρ+λkr/4 ρ − s 2 /kr − λs 2 /2
using ea − 1 aea and ea − 1 a. Now take λ = ρ/s 2 and k = 4s 2 /rρ. This yields 2 2 2 R 4 + J (o)2 /s 2 eλ(s /ρ+ρ(1+J (o) /4s )) . Let ν be some invariant distribution: it satisfies f dν R. Since d(x, o) ϕ(d(x, o)) + r(1 + k/4) we have eλd(x,o) dν eλr(1+k/4) f dν Reλr(1+k/4) , hence the result in the theorem. 2 6. L2 Bonnet–Myers theorems As seen in Section 2.3, it is generally not possible to give a bound for the diameter of a positively curved space similar to the usual Bonnet–Myers theorem involving the square root of curvature, the simplest counterexample being the discrete cube. Here we describe additional conditions which provide such a bound in two different kinds of situation. We first give a bound on the average distance between two points rather than the diameter; it holds when there is an “attractive point” and is relevant for examples such as the Ornstein– Uhlenbeck process (Example 9) or its discrete analogue (Example 10). Next, we give a direct generalization of the genuine Bonnet–Myers theorem for Riemannian manifolds. Despite lack of further examples, we found it interesting to provide an axiomatization of the Bonnet–Myers theorem in our language. This is done by reinforcing the positive curvature assumption, which compares the transportation distance between the measures issuing from two points x and y at a given time, by requiring a transportation distance inequality between the measures issuing from two given points at different times. 6.1. Average L2 Bonnet–Myers We now describe a Bonnet–Myers–like estimate on the average distance between two points, provided there is some “attractive point.” The proof is somewhat similar to that of Theorem 49. Proposition 51 (Average L2 Bonnet–Myers). Let (X, d, (mx )) be a metric space with random walk, with coarse Ricci curvature at least κ > 0. Suppose that for some o ∈ X and r 0, one has W1 (δo , mx ) d(o, x) for any x ∈ X with r d(o, x) < 2r, and that moreover X is r-geodesic.
854
Y. Ollivier / Journal of Functional Analysis 256 (2009) 810–864
Then
d(o, x) dν(x)
1 κ
σ (x)2 dν(x) + 5r nx
where as usual ν is the invariant distribution. Note that the assumption d(o, y) dmx (y) d(o, x) cannot hold for x in some ball around o unless o is a fixed point. This is why the assumption is restricted to an annulus. As in the Gaussian concentration theorem (Theorem 33), in case σ (x)2 is Lipschitz, Corol 2 lary 22 may provide a useful bound on σ (x) nx dν(x) in terms of its value at some point. As a first example, consider the discrete Ornstein–Uhlenbeck process of Example 10, which is the Markov chain on {−N, . . . , N} given by the transition probabilities pk,k = 1/2, pk,k+1 = 1/4 − k/4N and pk,k−1 = 1/4 + k/4N ; the coarse Ricci curvature is κ = 1/2N , and the invariant 2N 1 distribution is the binomial 22N N +k . This example is interesting because the diameter is 2N (which is the bound provided by Proposition 23), whereas the average distance between two √ points is ≈ N . It is immediate to check that 0 is attractive, namely that o = 0 and r = 1 fulfill the assumptions. Since σ (x)2 ≈ 1 and κ ≈ 1/N , the proposition recovers the correct order of magnitude for distance to the origin. Our next example is the Ornstein–Uhlenbeck process dXt = −αXt dt + s dBt on RN (Example 9). Here it is clear that 0 is attractive in some sense, so o = 0 is a natural choice. The invariant distribution is Gaussian of variance s 2 /α; under this distribution the average distance to 0 is ≈ Ns 2 /α. At small time τ , a point x ∈ RN is sent to a Gaussian centered at (1 − ατ )x, of variance τ s 2 . The average quadratic distance to the origin under this Gaussian is (1 − ατ )2 d(0, x)2 + N s 2 τ + o(τ ) by asimple computation. If d(0, x)2 > N s 2 /2α this is less than d(0, x)2 , so that we can take r = N s 2 /2α. Considering the random walk discretized at time τ we have we have κ ∼ ατ , ≈ N . So in the proposition above, the first term is ≈ s 2 /α, whereas the σ (x)2 ∼ N s 2 τ and nx second term is 5r ≈ N s 2 /α, which is thus dominant. So the proposition gives the correct order of magnitude; in this precise case, the first term in the proposition reflects concentration of measure (which is dimension-independent for Gaussians), whereas it is the second term 5r which carries the correct dependency on dimension for the average distance to the origin. Proof. Let ϕ : R → R be the function defined by ϕ(x) = 0 if x 2r, and ϕ(x) = (x − 2r)2 otherwise. For any real-valued random variable Y , we have Eϕ(Y ) ϕ(EY ) +
1 Var Y sup ϕ = ϕ(EY ) + Var Y. 2
Now let f : X → R be defined by f (x) = ϕ(d(o, x)). We are going to show that Mf (x) (1 − κ)2 f (x) +
σ (x)2 + 9r 2 nx
for all x ∈ X. Since f dν = Mf dν, we will get f dν (1 − κ)2 f dν + which easily implies the result.
σ (x)2 nx
dν + 9r 2
Y. Ollivier / Journal of Functional Analysis 256 (2009) 810–864
855
First, suppose that r d(o, x) < 2r. We have f (x) = 0. Now d(o, y) dmx (y) is at most d(o, y) by assumption. Using the bound above for ϕ, together with the definition of σ (x)2 and nx , we get Mf (x) =
ϕ d(o, y) dmx (y) ϕ
σ (x)2 σ (x)2 = d(o, y) dmx (y) + nx nx
since d(o, y) dmx (y) 2r by assumption. Second, suppose that d(x, o) 2r. Using that X is r-geodesic, we can find a point x such that d(o, x) = d(o, x ) + d(x , x) and r d(o, x ) < 2r (take the second point in a sequence joining o to x). Now we have d(o, y) dmx (y) = W1 (δo , mx ) W1 (δo , mx ) + W1 (mx , mx ) W1 (δo , mx ) + (1 − κ)d(x , x) d(o, x ) + (1 − κ)d(x , x) (1 − κ)d(o, x) + 2κr and as above, this implies Mf (x) ϕ
σ (x)2 d(o, y) dmx (y) + nx
2 σ (x)2 (1 − κ)d(o, x) + 2κr − 2r + nx σ (x)2 = (1 − κ)2 ϕ d(o, x) + nx as needed. The last case to consider is d(o, x) < r. In this case we have d(o, y) dmx (y) = W1 (δo , mx ) W1 (δo , mo ) + W1 (mo , mx ) = J (o) + W1 (mo , mx ) J (o) + (1 − κ)d(o, x) J (o) + r. So we need to bound J (o). If X is included in the ball of radius r around o, the result trivially holds, so that we can assume that there exists a point x with d(o, x) r. Since X is r-geodesic we can assume that d(o, x) < 2r as well. Now J (o) = W1 (mo , δo ) W1 (mo , mx ) + W1 (mx , δo ) (1 − κ)d(o, x) + W1 (mx , δo ) (1 − κ)d(o, x) + d(o, x) by assumption, so that J (o) 4r.
856
Y. Ollivier / Journal of Functional Analysis 256 (2009) 810–864
Plugging this into the above, for d(o, x) < r we get 2 ϕ( d(o, y) dmx (y)) 9r 2 hence Mf (x) 9r 2 + σ (x) nx . Combining the results, we get that whatever x ∈ X
Mf (x) (1 − κ)2 f (x) +
as needed.
d(o, y) dmx (y) 5r so that
σ (x)2 + 9r 2 nx
2
6.2. Strong L2 Bonnet–Myers √ As mentioned above, positive coarse Ricci curvature alone does not imply a 1/ κ-like diameter control, because of such simple counterexamples as the discrete cube or the Ornstein– Uhlenbeck process. We now extract a property satisfied by the ordinary Brownian motion on Riemannian manifolds (without drift), which guarantees a genuine Bonnet–Myers theorem. Of course, this is of limited interest since the only available example is Riemannian manifolds, but nevertheless we found it interesting to find a sufficient condition expressed in our present language. Our definition of coarse Ricci curvature controls the transportation distance between the measures issuing from two points x and x at a given time t. The condition we will now use controls the transportation distance between the measures issuing from two points at two different times. It is based on what holds for Gaussian measures in RN . For any x, x ∈ RN and t, t > 0, let m∗t x at and m∗t be the laws of the standard Brownian motion issuing from x at time t and from x x time t , respectively. It is easy to check that the L2 transportation distance between these two measures is √ √ ∗t 2 W2 m∗t = d(x, x )2 + N ( t − t )2 x , mx hence √ √ ∗t ∗t N ( t − t )2 W1 mx , mx d(x, x ) + . 2d(x, x ) The important feature here is that, when t tends to t, the second term is of second order in t − t. This is no more the case if we add a drift term to the diffusion. We now take this inequality as an assumption and use it to copy the traditional proof of the Bonnet–Myers theorem. Here, for simplicity of notation we suppose that we are given a continuous-time Markov chain; however, the proof uses only a finite number of different values of t, so that discretization is possible (this is important in Riemannian manifolds, because the heat kernel is positive on the whole manifold at any positive time, and there is no simple control on it far away from the initial point; taking a discrete approximation with bounded steps solves this problem).
Y. Ollivier / Journal of Functional Analysis 256 (2009) 810–864
857
Proposition 52 (Strong L2 Bonnet–Myers). Let X be a metric space equipped with a continuoustime random walk m∗t . Assume that X is ε-geodesic, and that there exist constants κ > 0, C 0 such that for any two small enough t, t , for any x, x ∈ X with ε d(x, x ) 2ε one has √ √ ∗t ∗t C( t − t )2 −κ inf(t,t ) W1 mx , mx e d(x, x ) + 2d(x, x ) with κ > 0. Assume moreover that ε Then
1√ 2 C/2κ.
diam X π
4ε C 1+ √ . 2κ C/2κ
∗t −κt d(x, x ), which is just the When t = t , the assumption reduces to W1 (m∗t x , mx ) e continuous-time version of the positive curvature assumption. The constant C plays the role of a diffusion constant, and is equal to N for (a discrete approximation of) Brownian motion on a Riemannian manifold. We restrict the assumption to d(x, x ) ε to avoid divergence problems √ √ C( t− t )2 for 2d(x,x ) when x → x.
For Brownian motion on an N -dimensional Riemannian manifold, we can take κ = 12 inf Ric by Bakry–Émery theory (the 12 is due to the fact that the infinitesimal generator of Brownian motion is 12 ), and C = N as in RN . So we get the usual Bonnet–Myers theorem, up to a √ √ factor N instead of N − 1 (similarly to our spectral gap estimate in comparison with the Lichnerowicz theorem), but with the correct constant π . Proof. Let x, x ∈ X. Since X is ε-geodesic, wecan find a sequence x = x0 , x1 , . . . , xk−1 , d(xi , xi+1 ) = d(x0 , xk ). By taking a subxk = x of points in X with d(xi , xi+1 ) ε and sequence (denoted xi again), we can assume that ε d(xi , xi+1 ) 2ε instead. i) 2 Set ti = η sin( πd(x,x d(x,x ) ) for some (small) value of η to be chosen later. Now, since t0 = tk = 0 we have
∗ti+1 i W1 m∗t xi , mxi+1 √ √
C( ti+1 − ti )2 e−κ inf(ti ,ti+1 ) d(xi , xi+1 ) + 2d(xi , xi+1 )
d(x, x ) = W1 (δx , δx )
a+b a+b by assumption. Now we have | sin b − sin a| = |2 sin b−a 2 cos 2 | |b − a|| cos 2 | so that
√ √ C( ti+1 − ti )2 Cηπ 2 d(xi , xi+1 ) d(x, xi ) + d(x, xi+1 ) 2 π . cos 2d(xi , xi+1 ) 2d(x, x ) 2d(x, x )2 Besides, if η is small enough, one has e−κ inf(ti ,ti+1 ) = 1 − κ inf(ti , ti+1 ) + O(η2 ). So we get d(x, x )
d(xi , xi+1 ) − κ inf(ti , ti+1 )d(xi , xi+1 ) d(x, xi ) + d(x, xi+1 ) Cηπ 2 d(xi , xi+1 ) 2 cos π + + O η2 . 2 2d(x, x ) 2d(x, x )
858
Y. Ollivier / Journal of Functional Analysis 256 (2009) 810–864
i )+d(x,xi+1 ) d(xi , xi+1 ) cos2 (π d(x,x2d(x,x ) and inf(ti , ti+1 )d(xi , xi+1 ) are close ) 1 1 to the integrals d(x, x ) 0 cos2 (πu) du and d(x, x )η 0 sin2 (πu) du respectively; the relative error in the Riemann sum is easily bounded by πε/d(x, x ) so that πε 1 − d(x, x ) d(x, x ) − κηd(x, x ) 2 d(x, x ) Cηπ 2 πε 1 + + + O η2 d(x, x ) 2 2 d(x, x ) 2d(x, x ) Now the terms
hence, taking η small enough, Cπ 2 1 + 2πε/d(x, x ) 2κ 1 − 2πε/d(x, x ) √ √ so that either d(x, x ) π C/2κ, or 2πε/d(x, x ) 2πε/π C/2κ 1/2 by the assumption that ε is small, in which case we use (1 + a)/(1 − a) 1 + 4a for a 1/2, hence the conclusion. 2 d(x, x )2
7. Coarse Ricci curvature and Gromov–Hausdorff topology One of our goals was to define a robust notion of curvature, not relying on differential calculus or the small-scale structure of a space. Here we first give two remarks about how changes to the metric and the random walk affect curvature. Next, in order to be able to change the underlying space as well, we introduce a Gromov–Hausdorff–like topology for metric spaces equipped with a random walk. First, since coarse Ricci curvature is defined as a ratio between a transportation distance and a distance, we get the following remark. Remark 53 (Change of metric). Let (X, d, m = (mx )) be a metric space with random walk, and let d be a metric on X which is bi-Lipschitz equivalent to d, with constant C 1. Suppose that the coarse Ricci curvature of m on (X, d) is at least κ. Then the coarse Ricci curvature of m on (X, d ) is at least κ where 1 − κ = C 2 (1 − κ). As an example, consider the ε-step random walk on a Riemannian manifold with positive Ricci curvature; κ behaves like ε 2 times the usual Ricci curvature, so that small bi-Lipschitz deformations of the metric, smaller than O(ε 2 ), will preserve positivity of curvature of the εstep random walk. The next remark states that we can deform the random walk m = (mx ) if the deformation depends on x in a Lipschitz way. Given a metric space (X, d), consider the space of 0-mass signed measures P0 (X) = {μ+ − μ− } where μ+ , μ− are measures on X with finite first moment and withthe same total mass. Equip this space with the norm (it is one) μ+ − μ− := supf 1-Lipschitz f d(μ+ − μ− ) = W1 (μ+ , μ− ). Then the following trivially holds. Remark 54 (Change of random walk). Let (X, d) be a metric space and let m = (mx )x∈X , m = (mx )x∈X be two random walks on X. Suppose that the coarse Ricci curvature of m is at least κ, and that the map x → mx − mx ∈ P0 (X) is C-Lipschitz. Then the coarse Ricci curvature of m is at least κ − 2C.
Y. Ollivier / Journal of Functional Analysis 256 (2009) 810–864
859
We now turn to changes in the space itself, for which we need to give a generalization of Gromov–Hausdorff topology taking the random walk data into account. Two spaces are close in this topology if they are close in the Gromov–Hausdorff topology and if moreover, the measures issuing from each point x are (uniformly) close in the L1 transportation distance. Recall [3] that two metric spaces (X, dX ) and (Y, dY ) are at Gromov–Hausdorff distance at most e ∈ [0; ∞] if there exists a semi-metric space (Z, dZ ) and isometries fX : X → Z, fY : Y → Z, such that for any x ∈ X, there exists y ∈ Y with dZ (fX (x), fY (y)) e, and likewise for any y ∈ Y (i.e. the Hausdorff distance between fX (X) and fY (Y ) is at most e). We extend this definition as follows to incorporate the random walk. Definition 55. Let (X, (mx )x∈X ) and (Y, (my )y∈Y ) be two metric spaces equipped with a random walk. For e ∈ [0; ∞], we say that these spaces are e-close if there exists a metric space Z and two isometric embeddings fX : X → Z, fY : Y → Z such that for any x ∈ X, there exists y ∈ Y such that dZ (fX (x), fY (y)) e and the L1 transportation distance between the pushforward measures fX (mx ) and fY (my ) is at most 2e, and likewise for any y ∈ Y . It is easy to see that this is defines a semi-metric on the class of metric spaces equipped with a random walk. We say that a sequence of spaces with random walks (X N , (mN x )x∈X N ) converges )) and (X, m ) tends to 0. We say, moreover, to (X, (mx )) if the semi-distance between (X N , (mN x x that a sequence of points x N ∈ X N tends to x ∈ X if we can take x N and x to be corresponding points in the definition above. We give a similar definition for convergence of tuples of points in X N . Coarse Ricci curvature is a continuous function in this topology. Namely, a limit of spaces with coarse Ricci curvature at least κ has coarse Ricci curvature at least κ, as expressed in the following proposition. Proposition 56 (Gromov–Hausdorff continuity). Let (X N , (mN x )x∈X N ) be a sequence of metric spaces with random walk, converging to a metric space with random walk (X, (mx )x∈X ). Let x, y be two distinct points in X and let (x N , y N ) ∈ X N × X N be a sequence of pairs of points converging to (x, y). Then κ(x N , y N ) → κ(x, y). In particular, if all spaces X N have coarse Ricci curvature at least κ, then so does X. Thus, having coarse Ricci curvature at least κ is a closed property. W (m ,m )
x y and likewise for κ(x N , y N ). The definition ensures that Proof. We have κ(x, y) = 1 − 1d(x,y) N N N N d(x , y ) and W1 (mx , my ) tend to d(x, y) and W1 (mx , my ) respectively, hence the result. 2
Note however, that the coarse Ricci curvature of (X, (mx )) may be larger than the limsup of N the coarse Ricci curvatures of (X N , (mN x )), because pairs of points in X , contributing to the N curvature of X , may tend to the same point in X; for example, X may consist of a single point. This collapsing phenomenon prevents positive curvature from being an open property. Yet it is possible to relax the definition of coarse Ricci curvature so as to allow any variation at small scales; with this perturbed definition, having coarse Ricci curvature greater than κ will become an open property. This is achieved as follows (compare the passage from trees to δ-hyperbolic spaces).
860
Y. Ollivier / Journal of Functional Analysis 256 (2009) 810–864
Definition 57. Let (X, d) be a metric space equipped with a random walk m. Let δ 0. The coarse Ricci curvature up to δ along x, y ∈ X is the largest κ 1 for which W1 (mx , my ) (1 − κ)d(x, y) + δ. With this definition, the following is easy. Proposition 58. Let (X, (mx )) be a metric space with random walk with coarse Ricci curvature at least κ up to δ 0. Let δ > 0. Then there exists a neighborhood VX of X such that any space Y ∈ VX has coarse Ricci curvature at least κ up to δ + δ . Consequently, the property “having curvature at least κ for some δ ∈ [0; δ0 )” is open. It would be interesting to study which properties of positive coarse Ricci curvature carry to this more general setting. 8. Transportation distance in Riemannian manifolds Here we give the proofs of Proposition 6 and of the statements of Example 7 and Section 3.3.1. We begin with Proposition 6 and evaluation of the coarse Ricci curvature of the ε-step random walk. The argument is close to the one in [43, Theorem 1.5(xii)], except that we use the value of Ricci curvature at a given point instead of its infimum on the manifold. Let X be a smooth N -dimensional Riemannian manifold and let x ∈ X. Let v, w be unit tangent vectors at x. Let δ, ε > 0 small enough. Let y = expx (δv). Let x = expx (εw) and y = expy (εw ) where w is the tangent vector at y obtained by parallel transport of w along the 2
geodesic t → expx (tv). The first claim is that d(x , y ) = δ(1 − ε2 K(v, w) + O(δε 2 + ε 3 )). We suppose for simplicity that w and w are orthogonal to v. We will work in cylindrical coordinates along the geodesic t → expx (tv). Let vt = d exp x (tv) be the speed of this geodesic. Let Et be the orthogonal of vt in the tangent dt space at expx (tv). Each point z in some neighborhood of x can be uniquely written as expexpx (τ (z)v) (εζ (z)) for some τ (z) ∈ R and ζ (z) ∈ Eτ (z) . Consider the set expx (E0 ) (restricted to some neighborhood of x to avoid topological problems), which contains x . Let γ be a geodesic starting at some point of expx (E0 ) and ending at y , which realizes the distance from expx (E0 ) to y . The distance from x to y is at least the length of γ . If δ and ε are small enough, the geodesic γ is arbitrarily close to the Euclidean one so that the coordinate τ is strictly increasing along γ . Let us parametrize γ using the coordinate τ , so that τ (γ (t)) = t. Let also wt = ζ (γ (t)) ∈ Et . In particular wδ = w . Consider, for each t, the geodesic ct : s → expexpx (tv) (swt ). We have γ (t) = ct (ε). For each D given t, the vector field dt ct (s) is a Jacobi field along the geodesic s → ct (s). The initial conD D d D ct (s)|s=0 = vt and dt ditions of this Jacobi field for s = 0 are given by dt ds ct (s)|s=0 = dt wt . Applying the Jacobi equation yields dγ (t) 2 dct (ε) 2 3 2 2 2 2 = dt = |vt | + 2εvt , w˙ t + ε |w˙ t | − ε R(wt , vt )wt , vt + O ε dt
Y. Ollivier / Journal of Functional Analysis 256 (2009) 810–864
where w˙ t = we get
D dt wt . But since by definition wt
861
∈ Et , we have vt , w˙ t = 0. Since moreover |vt | = 1
dγ (t) 3 ε2 ε2 2 dt = 1 + 2 |w˙ t | − 2 R(wt , vt )wt , vt + O ε . Integrating from t = 0 to t = δ and using that R(wt , vt )wt , vt = K(w, v) + O(δ) yields that the length of the geodesic γ is δ 3 2 ε2 ε2 δ 1 − K(v, w) + O ε + O ε δ + |w˙ t |2 2 2 t=0
so that the minimal value is achieved for w˙ t = 0. But by definition w˙ t = 0 means that the geodesic γ starts at x . So first, we have estimated d(x , y ), which proves Proposition 6, and second, we have proven that the distance from y to expx (E0 ) is realized by x up to the higher-order terms, which we will use below. Let us now prove the statement of Example 7. Let μ0 , μ1 be the uniform probability measures on the balls of radius ε centered at x and y respectively. We have to prove that W1 (μ0 , μ1 ) = d(x, y) 1 −
ε2 Ric(v, v) 2(N + 2)
up to higher-order terms. Let μ0 , μ1 be the images under the exponential map, of the uniform probability measures on the balls of radius ε in the tangent spaces at x and y respectively. So μ0 is a measure having density 1 + O(ε 2 ) w.r.t. μ0 , and likewise for μ1 . If we average Proposition 6 over w in the ball of radius ε in the tangent space at x, we get that W1 μ0 , μ1 d(x, y) 1 −
ε2 Ric(v, v) 2(N + 2)
up to higher-order terms, since the coupling by parallel transport realizes this value. Indeed, the average of K(v, w) on the unit sphere of the tangent plane at x is N1 Ric(v, v). Averaging on the ball instead of the sphere yields an N 1+2 factor instead. Now the density of μ0 , μ1 with respect to μ0 , μ1 is 1 + O(ε 2 ). More precisely write dμ1 dμ1
dμ0 dμ0
=
= 1 + ε 2 f1 (where f0 and f1 can be written very explicitly in terms of the 1 + ε 2 f0 and metric and its derivatives). Note that f1 = f0 + O(d(x, y)), and that moreover f0 integrates to 0 since both μ0 and μ0 are probability measures. Plugging all this in the estimate above, we get the inequality for W1 (μ0 , μ1 ) up to the desired higher-order terms. The converse inequality is proven as follows: if f is any 1-Lipschitz function, the L1 transportation distance between measures μ0 and μ1 is at least the difference of the integrals of f under μ0 and μ1 . Consider the function f equal to the distance of a point to expx (E0 ) (taken in some small enough neighborhood of x), equipped with a − sign if the point is not on the same side of E0 as y. Clearly f is 1-Lipschitz. We computed above a lower bound for f in cylindrical
862
Y. Ollivier / Journal of Functional Analysis 256 (2009) 810–864
coordinates, which after integrating yields a lower bound for W1 (μ0 , μ1 ). Arguments similar to the above turns this into the desired lower bound for W1 (μ0 , μ1 ). Finally, let us briefly sketch the proofs of the other statements of Section 3.3.1, namely, evaluation of the diffusion constant and local dimension (Definition 18). Up to a multiplicative factor 1 + O(ε), these can be computed in the Euclidean space. A simple computation shows that the expectation of the square distance of two points taken 2 N at random in a ball of radius ε in RN is ε 2 N2N +2 , hence the value ε N +2 for the diffusion constant σ (x)2 . (x)2 To evaluate the local dimension nx = sup Varm σf,f 1-Lipschitz (Definition 18), we have to bound x
the maximal variance of a 1-Lipschitz function on a ball of radius ε in RN . We will prove that the local dimension nx is comprised between N − 1 and N . A projection to a coordinate axis 2 provides a function with variance Nε+2 , so that local dimension is at most N . For the other bound, let f be a 1-Lipschitz function on the ball and let us compute an upper bound for its variance. Take ε = 1 for simplicity. Write the ball of radius 1 as the union of the spheres Sr of radii r 1. Let v(r) be the variance of f restricted to the sphere Sr , and let a(r) be the average of f on Sr . Then associativity of variances gives 1 v(r) dμ(r) + Varμ a(r)
Var f = r=0
1 N−1 where μ is the measure on the interval [0; 1] given by r Z dr with Z = r=0 r N −1 dr = N1 . Since the variance of a 1-Lipschitz function on the (N − 1)-dimensional unit sphere is at 1 2 most N1 , we have v(r) rN so that r=0 v(r) dμ(r) N 1+2 . To evaluate the second term, note that a(r) is again 1-Lipschitz as a function of r, so that Varμ a(r) = 12 (a(r) − a(r ))2 dμ(r)dμ(r ) is at most 12 (r − r )2 dμ(r)dμ(r ) = (N +1)N2 (N +2) . So finally Var f
N 1 + N + 2 (N + 1)2 (N + 2)
so that the local dimension nx is bounded below by
N (N +1)2 N 2 +3N +1
N − 1.
Acknowledgments I would like to thank Vincent Beffara, Fabrice Debbasch, Alessio Figalli, Pierre Pansu, Bruno Sévennec, Romain Tessera and Cédric Villani for numerous inspiring conversations about coarse geometry and Ricci curvature, as well as Thierry Bodineau, Djalil Chafaï, Aldéric Joulin, Shinichi Ohta and Roberto Imbuzeiro Oliveira for useful remarks on the manuscript and bibliographical references. Special thanks to Pierre Py for the two points x and y. References [1] C. Ané, S. Blachère, D. Chafaï, P. Fougères, I. Gentil, F. Malrieu, C. Roberto, G. Scheffer, Sur les Inégalités de Sobolev Logarithmiques, Panor. Synthèses, vol. 10, Société Mathématique de France, 2000. [2] A. Arnold, P. Markowich, G. Toscani, A. Unterreiter, On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker–Planck type equations, Comm. Partial Differential Equations 26 (1–2) (2001) 43–100.
Y. Ollivier / Journal of Functional Analysis 256 (2009) 810–864
863
[3] D. Burago, Y. Burago, S. Ivanov, A Course in Metric Geometry, Grad. Stud. Math., vol. 33, Amer. Math. Soc., 2001. [4] R. Bubley, M.E. Dyer, Path coupling: A technique for proving rapid mixing in Markov chains, Found. Comput. Sci. (1997) 223–231. [5] D. Bakry, M. Émery, Hypercontractivité de semi-groupes de diffusion, C. R. Math. Acad. Sci. Paris Sér. I 299 (15) (1984) 775–778. [6] D. Bakry, M. Émery, Diffusions hypercontractives, in: Séminaire de Probabilités, XIX, 1983/1984, in: Lecture Notes in Math., vol. 1123, Springer, Berlin, 1985, pp. 177–206. [7] M. Berger, A Panoramic View of Riemannian Geometry, Springer, Berlin, 2003. [8] S. Bobkov, M. Ledoux, On modified logarithmic Sobolev inequalities for Bernoulli and Poisson measures, J. Funct. Anal. 156 (2) (1998) 347–365. [9] P. Brémaud, Markov Chains, Texts Appl. Math., vol. 31, Springer, New York, 1999. [10] A.-I. Bonciocat, K.-T. Sturm, Mass transportation and rough curvature bounds for discrete spaces, preprint. [11] M.-F. Chen, Trilogy of couplings and general formulas for lower bound of spectral gap, in: Probability Towards 2000, New York, 1995, in: Lecture Notes in Statist., vol. 128, Springer, New York, 1998, pp. 123–136. [12] M.-F. Chen, From Markov Chains to Non-Equilibrium Particle Systems, second ed., World Scientific, 2004. [13] M.-F. Chen, S.-F. Li, Coupling methods for multidimensional diffusion processes, Ann. Probab. 17 (1) (1989) 151– 177. [14] M.-F. Chen, F.-Y. Wang, Application of coupling method to the first eigenvalue on manifold, Sci. China Ser. A 37 (1) (1994) 1–14. [15] H. Djellout, A. Guillin, L. Wu, Transportation cost-information inequalities and applications to random dynamical systems and diffusions, Ann. Probab. 32 (3B) (2004) 2702–2732. [16] R.L. Dobrušin, On the condition of the central limit theorem for inhomogeneous Markov chains, Dokl. Akad. Nauk SSSR (N.S.) 108 (1956) 1004–1006 (in Russian). [17] R. Dobrushin, Central limit theorem for non-stationary Markov chains I, Teor. Veroyatnost. i Primenen. 1 (1956) 72–89 (in Russian). [18] R.L. Dobrušin, Definition of a system of random variables by means of conditional distributions, Teor. Verojatnost. i Primenen. 15 (1970) 469–497 (in Russian). [19] R. Dobrushin, Perturbation methods of the theory of Gibbsian fields, in: R. Dobrushin, P. Groeneboom, M. Ledoux (Eds.), Lectures on Probability Theory and Statistics, Lectures from the 24th Saint-Flour Summer School held July 7–23, 1994, in: P. Bernard (Ed.), Lecture Notes in Math., vol. 1648, Springer, Berlin, 1996, pp. 1–66. [20] R.L. Dobrushin, S.B. Shlosman, Constructive criterion for the uniqueness of Gibbs field, in: J. Fritz, A. Jaffe, D. Szász (Eds.), Statistical Physics and Dynamical Systems, papers from the Second Colloquium and Workshop on Random Fields: Rigorous Results in Statistical Mechanics, held in Köszeg, August 26–September 1, 1984, in: Progress in Phys., vol. 10, Birkhäuser, Boston, 1985, pp. 347–370. [21] P. Diaconis, L. Saloff-Coste, Logarithmic Sobolev inequalities for finite Markov chains, Ann. Appl. Probab. 6 (3) (1996) 695–750. [22] A. Dembo, O. Zeitouni, Large Deviations Techniques and Applications, second ed., Appl. Math., vol. 38, Springer, New York, 1998. [23] A. Granas, J. Dugundji, Fixed Point Theory, Springer Monogr. Math., Springer, New York, 2003. [24] É. Ghys, P. de la Harpe, Sur les Groupes Hyperboliques d’Après Mikhael Gromov, Progr. Math., vol. 83, Birkhäuser, 1990. [25] M. Gromov, V. Milman, A topological application of the isoperimetric inequality, Amer. J. Math. 105 (1983) 843– 854. [26] R.B. Griffiths, Correlations in Ising ferromagnets III, Commun. Math. Phys. 6 (1967) 121–127. [27] M. Gromov, Isoperimetric inequalities in Riemannian manifolds, in: V. Milman, G. Schechtman (Eds.), Asymptotic Theory of Finite Dimensional Normed Spaces, in: Lecture Notes in Math., vol. 1200, Springer, Berlin, 1986, pp. 114–129. [28] A. Joulin, Poisson-type deviation inequalities for curved continuous time Markov chains, Bernoulli 13 (3) (2007) 782–798. [29] M. Ledoux, The Concentration of Measure Phenomenon, Math. Surveys Monogr., vol. 89, Amer. Math. Soc., 2001. [30] J. Lott, Optimal transport and Ricci curvature for metric-measure spaces, expository manuscript. [31] J. Lott, C. Villani, Ricci curvature for metric-measure spaces via optimal transport, preprint. [32] K. Marton, Bounding d-distance by informational divergence: A method to prove measure concentration, Ann. Probab. 24 (2) (1996) 857–866. [33] K. Marton, A measure concentration inequality for contracting Markov chains, Geom. Funct. Anal. 6 (3) (1996) 556–571, Geom. Funct. Anal. 7 (3) (1997) 609–613, Erratum.
864
Y. Ollivier / Journal of Functional Analysis 256 (2009) 810–864
[34] F. Martinelli, Relaxation times of Markov chains in statistical mechanics and combinatorial structures, in: H. Kesten (Ed.), Probability on Discrete Structures, in: Encyclopaedia Math. Sci., vol. 110, Springer, Berlin, 2004, pp. 175– 262. [35] V. Milman, G. Schechtman, Asymptotic Theory of Finite Dimensional Normed Spaces, Lecture Notes in Math., vol. 1200, Springer, Berlin, 1986. [36] M. Hazewinkel, The Online Encyclopaedia of Mathematics, article Wasserstein metric, http://eom.springer.de/W/ w120020.htm. [37] S.-i. Ohta, On the measure contraction property of metric measure spaces, Comment. Math. Helv. 82 (4) (2007) 805–828. [38] R. Imbuzeiro Oliveira, On the convergence to equilibrium of Kac’s random walk on matrices, preprint, arXiv: 0705.2253. [39] Y. Ollivier, Sharp phase transition theorems for hyperbolicity of random groups, Geom. Funct. Anal. 14 (3) (2004) 595–679. [40] Y. Ollivier, Ricci curvature of metric spaces, C. R. Math. Acad. Sci. Paris 345 (11) (2007) 643–646. [41] F. Otto, C. Villani, Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality, J. Funct. Anal. 173 (2000) 361–400. [42] Y. Peres, Mixing for Markov chains and spin systems, Lecture Notes, 2005. [43] M.-K. von Renesse, K.-T. Sturm, Transport inequalities, gradient estimates, and Ricci curvature, Comm. Pure Appl. Math. 68 (2005) 923–940. [44] M.D. Sammer, Aspects of mass transportation in discrete concentration inequalities, PhD thesis, Georgia institute of technology, 2005, etd.gatech.edu/theses/available/etd-04112005-163457/unrestricted/sammer_marcus_ d_200505_phd.pdf. [45] G. Schechtman, Concentration, results and applications, in: Handbook of the Geometry of Banach Spaces, vol. 2, North-Holland, Amsterdam, 2003, pp. 1603–1634. [46] K.-T. Sturm, On the geometry of metric measure spaces, Acta Math. 196 (1) (2006) 65–177. [47] C. Villani, Topics in Optimal Transportation, Grad. Stud. Math., vol. 58, Amer. Math. Soc., Providence, RI, 2003. [48] C. Villani, Optimal Transport, Old and New, Grundlehren Math. Wiss., vol. 338, Springer, 2008.
Journal of Functional Analysis 256 (2009) 865–880 www.elsevier.com/locate/jfa
Boundaries of compact convex sets and fragmentability ✩ Ondˇrej F.K. Kalenda ∗ , Jiˇrí Spurný Charles University, Faculty of Mathematics and Physics, Department of Mathematical Analysis, Sokolovská 83, 186 75 Praha 8, Czech Republic Received 28 February 2008; accepted 11 November 2008 Available online 28 November 2008 Communicated by N. Kalton
Abstract If X is a compact convex set in a real locally convex space, B ⊂ X is said to be its boundary if every affine continuous function on X attains its maximum at some point of B. We study relations between fragmentability of B and the whole set X. As a byproduct we obtain a characterization of separable Asplund spaces. We also study the possibility of finding the Haar system in a boundary of a metrizable compact convex set. © 2008 Elsevier Inc. All rights reserved. Keywords: Compact convex sets; Boundary; Extreme points; Fragmentability; Asplund spaces; Haar system
1. Introduction If X is a convex compact set (i.e., a convex compact subset of a real Hausdorff locally convex space), then a subset B ⊂ X is called a boundary of X if each real-valued affine continuous function on X attains its maximum on X at some point of B. We investigate which properties of a boundary are transferred to the whole X. Some results of this type are contained in [21,24,30]. In [30] it is proved that the “norm-separability” can be transferred (see Section 5 for an explanation and [16] for an alternative proof). The transfer of the existence of a countable network is ✩
The work is a part of the research project MSM 0021620839 financed by MSMT and partly supported by the research ˇ 201/06/0018 and GA CR ˇ 201/07/0388. grants GA CR * Corresponding author. E-mail addresses: [email protected] (O.F.K. Kalenda), [email protected] (J. Spurný). 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.11.004
866
O.F.K. Kalenda, J. Spurný / Journal of Functional Analysis 256 (2009) 865–880
proved in [24], the case B = ext X is contained already in [21]. We focus namely on the transfer of fragmentability. Let us briefly describe the content of the paper. Section 2 contains some notation and definitions used throughout the paper. Section 3 is devoted to a study of fragmentability of compact convex sets. Its main theme is the question under what conditions we can deduce fragmentability of a compact convex set from the fragmentability of a boundary. Positive results are summed up in Theorem 3.1. Counterexamples showing limits of the positive results are given at the end of the section. We also obtain another characterization of separable Asplund spaces (see Corollary 3.2). Section 4 contains a general result on the relationship of fragmentability and the existence of the Haar system (see Theorem 4.1). As a consequence we get an improvement of a result of M. López-Pellicer and V. Montesinos by finding the Haar system in any analytic boundary of a separable non-Asplund space. In Section 5 we collect some results on transferring other properties from a boundary to the whole compact convex set, namely the norm density and the network weight. Remark 1.1. Our original motivation for the investigation of fragmentability was an attempt to solve the so-called boundary problem asked by G. Godefroy [19, Question V.2], which in our setting reads as follows: Let X be a compact convex set and B ⊂ X a boundary. Suppose that A is a uniformly bounded subset of A(X), the space of all affine continuous functions on X, which is compact in the topology τB of pointwise convergence on B. Is then A compact also in the topology τX of pointwise convergence on X? Until recently there were only partial positive solutions, some of them quite involved, see [3–5,8–10,32,33]. We were inspired by the observation that (in the above situation) A is τX -compact if and only if X is fragmented by ρA (using the notation introduced in the next section). Indeed, the only if part follows from the Namioka theorem [25, Theorem 2.3]; the if part can be proved using [9, Theorem B(iii) ⇒ (i)] and a slight generalization of [15, Lemma 2.1.1]. However, Examples 3.6 and 3.5 show that this method could not easily yield the solution. Moreover, after the first version of our paper was submitted, the boundary problem was solved in the positive by H. Pfitzner by a surprisingly elementary method (see [28]). 2. Notation and definitions In this section we collect basic notation and definitions used in the paper. If E is a Banach space, by BE we denote the closed unit ball of E. If X is a compact convex set, A(X) is the space of all real-valued affine continuous functions on X equipped with the supremum norm. An important notion is the pseudometric defined by a family of continuous affine functions. More precisely, let X be a compact convex set and A a bounded subset of A(X). We define a pseudometric ρA on X by the formula ρA (x, y) = supa(x) − a(y), x, y ∈ X. a∈A
It is clear that it is a lower semicontinuous pseudometric on X. Moreover, if we consider X canonically embedded into A(X)∗ (recall that this canonical embedding is defined by assigning
O.F.K. Kalenda, J. Spurný / Journal of Functional Analysis 256 (2009) 865–880
867
to each x ∈ X the evaluation map εx ∈ A(X)∗ defined by εx (f ) = f (x), f ∈ A(X)), then ρA is generated by a weak∗ lower semicontinuous seminorm on A(X)∗ . Further, ρBA(X) is the metric defined by the dual norm of A(X)∗ . Note that the definition of ρA has a good meaning if X is just a compact space and A a bounded subset of C(X). In a compact convex set X we denote by ext X the set of extreme points of X. If (X, ρ) is a metric space, we write ρ-diam(A) for the diameter of a set A ⊂ X and ρ- dist(A1 , A2 ) for the distance between sets A1 , A2 in X. If X is a set and B a subset of X, we write τB for the topology of pointwise convergence on B for the space RX of all functions from X to R. Recall also that a topological space is K-analytic if it is an upper semicontinuous compact valued image of a separable completely metrizable space. If the domain is allowed to be non-complete, we get K-countably determined spaces. For a detailed study of K-analytic spaces we refer to [31], basic facts on K-countably determined spaces can be found for example in [15, Chapter 7]. ˇ We further recall that a space T is almost Cech-analytic if there is some H ⊂ T × NN such that the projection of H onto the first coordinate is whole T and any nonempty closed subset of H ˇ ˇ contains a dense Cech-complete subset (i.e., H is hereditarily almost Cech-complete). This is a large class of spaces containing all K-analytic spaces, all scattered spaces and, more generally, the class of spaces which are called scattered-K-analytic spaces in [20], almost K-descriptive in [22] and cover-analytic in [26]. Finally we recall definitions related to fragmentability. Let (T , τ ) be a topological space and ρ a pseudometric on T . We say that (T , τ ) is fragmented by ρ down to ε (where ε > 0) if any nonempty subset of T has a nonempty relatively τ -open subset of ρ-diameter less that ε. The space (T , τ ) is said to be fragmented by ρ if it is fragmented by ρ down to ε for every ε > 0. Further, (T , τ ) is σ -fragmented by ρ if for each ε > 0 there is a countable cover T = n∈N Tn such that each Tn is fragmented by ρ down to ε. 3. Transferring fragmentability In this section we collect results on transferring fragmentability from a boundary to the whole set X. Positive results form the content of the following theorem. Some limits of these results are witnessed by Examples 3.5 and 3.6. Theorem 3.1. Let X be a compact convex set, A a bounded set of affine continuous functions and B be a boundary of X. Suppose, moreover, that at least one of the following conditions is satisfied. (a) B is K-analytic. (b) X is metrizable. (c) B is hereditarily Lindelöf. Then the following assertions are equivalent. (i) (ii) (iii) (iv) (v)
B is fragmented by ρA . B is σ -fragmented by ρA . For every countable C ⊂ A the space (B, ρC ) is separable. For every countable C ⊂ A the space (X, ρC ) is separable. X is fragmented by ρA .
868
O.F.K. Kalenda, J. Spurný / Journal of Functional Analysis 256 (2009) 865–880
First note that many implications are true even if X is any compact space, A a bounded subset of C(X) and B any subset of X. Namely, the implications (v) ⇒ (i) ⇒ (ii) and (iv) ⇒ (iii) are obviously true. Moreover, (iv) ⇔ (v) by [7, Theorem 2.1]. We are interested namely in the validity of (i) ⇒ (v). In general, i.e., if B is a boundary of X but does not satisfy any of the conditions (a)–(c), this implication is false as witnessed by Examples 3.5 and 3.6 below. As a corollary we get the following characterization of separable Asplund Banach spaces. Corollary 3.2. Let E be a separable Banach space. Then the following assertions are equivalent: (i) (ii) (iii) (iv)
E is Asplund. There exists a boundary B ⊂ BE ∗ that is σ -fragmented by the norm. Every boundary B ⊂ BE ∗ is σ -fragmented by the norm. BE ∗ is fragmented by the norm.
Proof. The equivalence (i) ⇔ (iv) is well known and holds also in the nonseparable case, see e.g. [11, Theorem I.5.2]. The implications (iv) ⇒ (iii) ⇒ (ii) are trivial. The implication (ii) ⇒ (iv) follows from Theorem 3.1 (take X = (BE ∗ , w ∗ ) which is a metrizable compact convex set). 2 Remark that this characterization is false in the nonseparable case, see Examples 3.5 and 3.6 below. We proceed by a proof of Theorem 3.1. We will use the following lemmata. The first lemma proves (iii) ⇒ (iv). Lemma 3.3. Let X be a compact convex set, C a bounded set of affine continuous functions and B be a boundary of X. If B is ρC -separable, then so is X. In case C = BA(X) this lemma is an immediate consequence of Rodé’s theorem [30] (see also [16, Corollary 2.4] or [17, Theorem 5.7]). Proof. We will use [16, Theorem 2.3] and the idea of the proof of [16, Proposition 2.2]. Set D = BA(X) . Without loss of generality we can suppose that C ⊂ D and hence ρC ρD . Suppose that B is ρC -separable. Let {bn : n ∈ N} be a countable ρC -dense subset of B. We will prove that the convex hull of this set is ρC -dense in B. If we succeed, we are done, as the convex hull is clearly ρC -separable. (Note that ρC is induced by a seminorm.) For a fixed ε > 0, we set Bn = x ∈ X: ρC (x, bn ) ε ,
n ∈ N.
Then each Bn is convex and closed (by lower semicontinuity of ρC ). Moreover, the sets Bn cover B and hence by the Fonf–Lindenstrauss theorem [16, Theorem 2.3] the convex hull of n∈N Bn is ρD -dense in X. As ρC is weaker, it is also ρC -dense.
O.F.K. Kalenda, J. Spurný / Journal of Functional Analysis 256 (2009) 865–880
869
nSo, if x ∈ X is arbitrary, we can find n ∈ N, yi ∈ Bi and ti ∈ [0, 1] for i = 1, . . . , n, such that i=1 ti = 1 and ρC x,
n
ti yi < ε.
i=1
As ρC (yi , bi ) ε for every i = 1, . . . , n, we get n ρC x, ti bi < 2ε. i=1
As ε > 0 is arbitrary, we get that conv{bn : n ∈ N} is ρC -dense in X and we are done.
2
The following lemma shows how the case (b) of metrizable X can be reduced to the case (a) of a K-analytic boundary. Lemma 3.4. Let X be a metrizable space, ρ a lower semicontinuous pseudometric on X, B ⊂ X and ε > 0. If B is fragmented by ρ down to ε, then there is a set B˜ ⊂ X containing B which is simultaneously Fσ and Gδ and is also fragmented by ρ down to ε. Proof. Suppose that B is fragmented by ρ down to ε. It is easy to check (using transfinite induction) that there is a continuous increasing well-ordered family (Uα : α < κ) of open sets covering B such that U0 = ∅ and for each α < κ we have
(1) ρ- diam B ∩ (Uα+1 \ Uα ) < ε. We set Bα = B ∩ (Uα+1 \ Uα ),
α < κ.
By (1) and by lower semicontinuity of ρ we have ρ-diam Bα < ε. Therefore B˜ = Bα ∩ (Uα+1 \ Uα ) α<κ
contains B and is fragmented by ρ down to ε. Moreover, B˜ is simultaneously Fσ and Gδ by Montgomery’s lemma [27, Lemma 16.2]. This completes the proof. 2 Proof of Theorem 3.1. Note that Lemma 3.3 proves the implication (iii) ⇒ (iv). Further, if B is K-analytic, then (ii) ⇒ (iii) by [6, Theorem 2.1], Therefore the proof of (a) is completed (the remaining implication being valid by the above discussion). Let us prove (b). The only missing implication is (ii) ⇒ (iii). So suppose that B is σ -fragmented by ρA . It follows easily from Lemma 3.4 that there is B˜ ⊃ B which is Fσ δ in X and is σ -fragmented by ρA . As this B˜ is analytic, we may use [6, Theorem 2.1] to conclude the proof. For the proof of (c), we have to verify (ii) ⇒ (iii) under the assumption that B is hereditarily Lindelöf. In fact, we will prove that B is ρA -separable. Suppose it is not.
870
O.F.K. Kalenda, J. Spurný / Journal of Functional Analysis 256 (2009) 865–880
Since (B, ρA ) is a nonseparable pseudometric space, there exists δ > 0 and an uncountable set D ⊂ B that is δ-discrete in ρA . By (ii), the set D is σ -fragmented by ρA . Therefore we can write D = n∈N Dn where each Dn is fragmented by ρA down to 2δ . We choose n ∈ N such that Dn is uncountable and set U=
{V : V ⊂ B open, V ∩ Dn is countable}.
Since B is hereditarily Lindelöf, U ∩ Dn is countable. Then Dn = Dn \ U is an uncountable set such that V ∩ Dn is uncountable for each open set V ⊂ B intersecting Dn . Hence every nonempty relatively open subset of Dn has ρA -diameter at least δ. Hence Dn is not fragmented by ρA down to 2δ . This contradiction concludes the proof. 2 We continue with the examples announced above. (We refer the reader to [1] and [17] for information on simplices.) Example 3.5. There is a Choquet simplex X with the following properties: (a) The set B = ext X is relatively discrete and hence fragmented by any pseudometric. (b) Each element of B is a Gδ point in X. (c) If A is the unit ball of A(X), then X is not fragmented by ρA . In particular, A(X) is not Asplund but the set of extreme points of BA(X)∗ is weak∗ relatively discrete and hence fragmented by the norm metric. Proof. We take the well-known example of a Choquet simplex from [2, Section VII] (see also [1, Proposition II.3.17]). We briefly describe its construction. Let K = [0, 1] × {−1, 0, 1} be equipped with the porcupine topology, i.e., the points of [0, 1] × {−1, 1} are isolated and a basis of neighborhoods of a point (t, 0) is formed by sets (U × {−1, 0, 1}) \ {(t, −1), (t, 1)}, where U is a neighborhood of t in [0, 1]. Further, set
1
H = f ∈ C(K): f (t, 0) = f (t, −1) + f (t, 1) for t ∈ [0, 1] 2 and let X be the state space of H, i.e., X = ξ ∈ H∗ : ξ 1 and ξ(1) = 1 . Then X is a Choquet simplex and H is canonically identified with the space of all continuous affine functions on X. Moreover, K canonically homeomorphically embeds into X and using this embedding we have ext X = [0, 1] × {−1, 1} ⊂ K. Now it is clear that each extreme point is isolated in ext X and hence (a) is proved. Further, fix t ∈ [0, 1] and define ft ∈ H such that ft (t, 1) = 1, ft (t, −1) = −1 and ft = 0 elsewhere. This ft witnesses that both (t, 1) and (t, −1) are Gδ points of X (in fact, they are exposed points of X). This proves (b).
O.F.K. Kalenda, J. Spurný / Journal of Functional Analysis 256 (2009) 865–880
871
Finally, let us prove (c). We will show that [0, 1] × {0} is discrete in ρA . As it carries the euclidean topology as a subset of X, we get that X is not fragmented by ρA . Take s, t ∈ [0, 1] with s < t. Define f ∈ C([0, 1]) by the formula
= −1, f (u) is linear, = 1,
u ∈ [0, s], on [s, t], u ∈ [t, 1].
Further define h ∈ H by h(u, ) = f (u) for (u, ) ∈ K. Then h ∈ A and so ρA ((s, 0), (t, 0)) 2. To prove the ‘in particular’ part note that A(X) contains the space C[0, 1] as a closed subspace and hence it is not Asplund while the set of extreme points of BA(X)∗ is identified with ext(X) ∪ (−ext(X)) and hence it is relatively discrete and thus fragmented by the norm. 2 Example 3.6. There is a compact space K and B ⊂ K such that (a) B is scattered (and hence fragmented by any pseudometric). (b) B is a boundary for C(K). (c) K is not scattered, and hence the space of probability measures M1 (K) ⊂ C(K)∗ is not fragmented by the norm. In particular, C(K) is not Asplund space but admits a boundary fragmented by the norm. Note that we consider K canonically embedded into the dual space C(K)∗ , identifying any point of K with the respective Dirac measure. Using this identification we understand the assertion (b). Therefore, (b) means that any f ∈ C(K) attains its maximum on K at some point of B. Proof. It is enough to take a scattered pseudocompact space B which admits a non-scattered compactification. We describe an easy example of such a space. Let A denotes the set of all rational numbers from [0, 1]. For each x ∈ [0, 1] let Ux be a maximal family of infinite subsets of A which is almost disjoint (i.e., any two distinct members have finite intersection) such that each member of Ux is the set of points of a one-to-one sequence converging to x. Let U = x∈[0,1] Ux . Then U is clearly a maximal almost disjoint family of subsets of A. Let B be the Mrówka space defined by U , i.e., B = A ∪ U such that the points of A are isolated and neighborhoods of U ∈ U are formed by sets {U } ∪ (U \ F ), F ⊂ A finite. As U is maximal, it is easy to see that B is pseudocompact (see e.g. [13, Proposition 11.6]). Finally, define g : B → [0, 1] by g(t) = t,
t ∈ A,
g(U ) = lim U,
U ∈ U.
Then g is continuous and g(B) = [0, 1]. It follows that K = βB is not scattered. Indeed, the ˇ Cech–Stone extension of g maps continuously K onto [0, 1] and compact scattered spaces are preserved by continuous images. 2
872
O.F.K. Kalenda, J. Spurný / Journal of Functional Analysis 256 (2009) 865–880
We finish this section by asking the following question. Question 3.7. Let X be a compact convex set, A ⊂ A(X) a uniformly bounded set and B ⊂ X a boundary which is K-countably determined and σ -fragmented by ρA . Is then X fragmented by ρA ? Note that K-countably determined spaces are a common generalization of K-analytic spaces and separable metrizable spaces. In both these special cases the answer is positive by Theorem 3.1 (note that separable metrizable spaces are hereditarily Lindelöf). 4. Fragmentability and the Haar system In this section we will prove a theorem on embedding the Cantor set together with the Haar system to certain non-σ -fragmented spaces. We will also give applications to the study of boundaries. Let us start with fixing some notation. By 2N we denote the Cantor set, i.e., the countable Cartesian power of the two point discrete set {0, 1} for which we use the set-theoretic shortcut 2 = {0, 1}. So, the elements of 2N are infinite sequences of elements of {0, 1}. The set of all finite sequences of elements of {0, 1}, including the empty sequence, is denoted by 2 0 such that for every ε > 0 there exist a compact set K ⊂ B, a family {ft : t ∈ 2
ft − χΔt ◦ ϕ C (K) < ε.
t∈2
Further, if all compact subsets of B are metrizable, then the mapping ϕ in condition (vi ) can be chosen to be moreover injective. The conditions are denoted by (ii ) and (vi ) as this theorem can be viewed as a complement of Theorem 3.1—the condition (ii ) is the negation of the condition (ii) from Theorem 3.1. In particular, the negation of (vi ) can be added to Theorem 3.1 as another equivalent condition if B is K-analytic. Corollary 4.2. Let E be a separable Banach space with nonseparable dual and B ⊂ BE ∗ be a weak∗ -analytic boundary.
O.F.K. Kalenda, J. Spurný / Journal of Functional Analysis 256 (2009) 865–880
873
Then there exists M > 0 such that for each ε > 0 there is a weak∗ compact subset K ⊂ B, a family {xt : t ∈ 2
xt − χΔt ◦ ϕ C (K) < ε,
t∈2
where each xt is canonically identified with a continuous function on K. Proof. We apply Theorem 4.1 to X = BE ∗ and A = BE . As E ∗ is nonseparable, we get by Corollary 3.2 that B is not σ -fragmented by ρA (which is the norm metric). Therefore (ii ) is satisfied. The assertion now follows from (vi ). 2 Corollary 4.2 in case B = ext BE ∗ follows from [34, Corollary 1 on p. 218] (see also [23, Theorem 3.3]). Indeed, ext BE ∗ is a weak∗ Gδ set which is norm nonseparable, which are the assumptions of the quoted results. The following question remains open. Question 4.3. Is the assertion of Corollary 4.2 true if we omit the assumption of analyticity of B? The rest of this section is devoted to the proof of Theorem 4.1. We prove first the easy implication. Proof of Theorem 4.1 (vi ) ⇒ (ii ). Assume (vi ) holds. Let C ⊂ A and M > 0 be the objects obtained by (vi ). We set ε = 14 and obtain K, ft : t ∈ 2
1 ρM conv(C∪−C) - dist ϕ −1 (τ1 ), ϕ −1 (τ2 ) , 2
τ1 , τ2 ∈ 2N distinct.
Since ρM conv(C∪−C) = Mρconv(C∪−C) = MρC , K is not ρC -separable. By [6, Theorem 2.1], K is not fragmented by ρA . (In fact, it is easy to deduce the non-fragmentability of K. Indeed, let K0 ⊂ K be a minimal closed subset such that ϕ(K0 ) = 2N . Then for each nonempty relatively open U ⊂ K0 its image ϕ(U ) contains at least 1 two points, and hence ρC - diam U MC . So, K is not fragmented by ρC and a fortiori it is not fragmented by ρA .) As K is compact and ρA is lower semicontinuous, we get that K is not σ -fragmented by ρA . Therefore neither B is σ -fragmented by ρA . 2 We start the proof of the converse implication by the following lemma that is originated in the technique of [34, Lemma 1] (see also [23, Lemma 3.2]).
874
O.F.K. Kalenda, J. Spurný / Journal of Functional Analysis 256 (2009) 865–880
Lemma 4.4. Let B be a compact space and A ⊂ C(B) be a closed convex symmetric subset of BC (B) . If B is ρA -nonseparable, then there exists M > 0 and a transfinite sequence (xα , gα ) ∈ B × M · B∞ (B) : α < ω1 , such that (a) gβ (xα ) = 0, α < β < ω1 , (b) gα (xα ) = 1, α < ω1 , (c) gα ∈ M · AτB , α < ω1 . Proof. Given B and A as in the premise, let E stand for the linear span of A. We consider on E the norm · A that has A for the unit ball. We write · ∗A for the dual norm on the dual space (E, · A )∗ . Then the space B can be mapped onto a set ε(B) ⊂ (E, · A )∗ via the mapping x → εx , where εx is the respective evaluation functional. Moreover, ρA (x1 , x2 ) = εx1 − εx2 ∗A , x1 , x2 ∈ B, and hence ε : (B, ρA ) → (ε(B), · ∗A ) is an isometry. Since B is not ρA -separable, the set ε(B) is not · ∗A separable. Hence there exists 0 < M1 < M2 such that ε(B) ∩ (M2 · BE ∗ \ M1 · BE ∗ ) is not separable. Then the set B =
b ∗ ∗ : b ∈ ε(B) ∩ (M · B \ M · B ) 2 E 1 E b ∗A
is contained in the unit sphere SE ∗ and is not separable. By the italicized claim in the proof of [23, Lemma 3.2], there exists η > 0 such that B is not contained in ∗
span · A S + ηBE ∗ for any S ⊂ E ∗ countable. Now we can proceed with the construction as in the proof of the mentioned [23, Lemma 3.2]. We select b1 ∈ B and x1∗∗ ∈ SE ∗∗ such that x1∗∗ (b1 ) = 1. Assume that α < ω1 and bβ ∈ B and xβ∗∗ have been already chosen for all β < α. Then there exists a point bα ∈ B such that
∗ dα = · ∗A - dist bα , span · A {bβ : β < α} η. By the Hahn–Banach theorem, there exists a functional xα∗∗ ∈ SE ∗∗ such that xα∗∗ = 0 on all points bβ , β < α, and xα∗∗ (bα ) = dα . Having all the points bα ∈ B and functionals xα∗∗ ∈ SE ∗∗ , α < ω1 , constructed, we finish the proof by defining the required objects. ε First we find points xα ∈ B such that εxα ∈ (M1 , M2 ) and bα = εxxα ∗ for α < ω1 . (It is possible by the definition of B .) Next we define functions gα as gα (x) =
xα∗∗ (εx ) , xα∗∗ (εxα )
x ∈ B, α < ω1 .
α A
O.F.K. Kalenda, J. Spurný / Journal of Functional Analysis 256 (2009) 865–880
875
Then (xα , gα ), α < ω1 , satisfy conditions (a) and (b). Further we notice that xα∗∗ (εxα ) = εxα ∗A xα∗∗ (bα ) M1 η. As moreover, εxα ∗A M2 , we get gα ∞ (B)
M2 , ηM1
so the family (gα ) is uniformly bounded. Further, given α < ω1 , by the Goldstine theorem there is a net aν ∈ A weak∗ converging to xα∗∗ . Thus, x ∗∗a(εν x ) τB -converges to gα . If we set now α
α
1 M2 , , M = max ηM1 ηM1 we conclude the proof.
2
Lemma 4.5. Let B be a closed metrizable subset of a compact space X and A ⊂ C(X) be a convex closed symmetric subset of BC (X) such that B is ρA -nonseparable. Then there exists M > 0 such that for every ε > 0 there exist a compact set K ⊂ B, a family {ft : t ∈ 2
Proof. We start the proof by using Lemma 4.4 to find M > 0 and (xα , gα ) ∈ B × M · B∞ (B) : α < ω1 with the properties (a), (b) and (c) of Lemma 4.4. We fix on B a compatible metric σ with σ -diam(B) < 1. Since (B, σ ) is a compact metric space, by discarding countably many points and functions if necessary we may assume that U ∩ {xα : α < ω1 } is uncountable whenever it is nonempty and U ⊂ B is relatively open. Set J = {xα : α < ω1 }. Let ε > 0 be given. We find strictly positive numbers (εn ) such that n∈N 2n εn < ε. Claim 4.5.1. For each n ∈ N, there exist functions ft ∈ M · A and relatively open sets Ut ⊂ B, t ∈ 2
Ut ∩ J = ∅, t ∈ 2
x∈
t ∈2
Ut .
876
O.F.K. Kalenda, J. Spurný / Journal of Functional Analysis 256 (2009) 865–880
Proof. Set U∅ = B. Then conditions (a) and (c) are obviously satisfied and condition (d) is vacuous. We assume now that n ∈ N ∪ {0}, the construction has been completed for each t ∈ 2 max{α2 , . . . , α2n+1 } such that xα1 ∈ Vt1 (as Vt1 ∩ J is uncountable). Then
gα1 (xαl ) =
1, l = 1, 0, l ∈ {1, . . . , 2n+1 } \ {1}.
Since gα1 ∈ M · AτB , there exists a function f1 ∈ M · A such that f1 (xα ) − gα (xα ) < εn+1 , l l 1
l = 1, . . . , 2n+1 .
As f1 is continuous, we can find open sets Vt,1 ⊂ Vt , t ∈ 2n+1 , such that xαl ∈ Vtl ,1 and (g) holds for j = 1. This finishes the first step of the construction. Assume now that the objects have been defined for some j ∈ {1, . . . , 2n+1 − 1}. As above we first find indices αl < ω1 , l ∈ {1, . . . , 2n+1 } \ {j + 1}, such that xαl ∈ Vtl ,j for l ∈ {1, . . . , 2n+1 } \ {j + 1}. Further we find αj +1 > max{α1 , . . . , αj , αj +2 , . . . , α2n+1 } such that xαj +1 ∈ Vtj +1 ,j . It is possible as Vtj +1 ,j ∩ J = ∅. Then
1, l = j + 1, gαj +1 (xαl ) = 0, l ∈ {1, . . . , 2n+1 } \ {j + 1}. We use the pointwise approximation to find a function fj +1 ∈ M · A such that fj +1 (xα ) − gα l
j +1
(xαl ) < εn+1 ,
l = 1, . . . , 2n+1 .
O.F.K. Kalenda, J. Spurný / Journal of Functional Analysis 256 (2009) 865–880
877
Again using continuity of fj +1 we find suitable open sets Vt,j +1 ⊂ Vt,j , t ∈ 2n+1 , such that xαl ∈ Vtl ,j +1 and (g) holds for j + 1. After 2n+1 steps we finish the construction. Step 3. For each t ∈ 2n+1 , we set • Ut = Vt,2n+1 , • ft = fj , where j is the index of t in the natural enumeration of 2n+1 . Then all conditions (a)–(d) are satisfied, which finishes the proof of the claim.
2
Now we continue with the proof of Lemma 4.5. Let K=
∞
∞
=
Ut
n=1 |t|=2n
Uτ |n .
τ ∈2N n=1
By (b) and (c), ∞ n=1 Uτ |n is a singleton contained in B. Hence K is a subset of B homeomorphic with 2N via the mapping ψ : 2N → K, τ →
∞
Uτ |n ,
τ ∈ 2N .
n=1
Let ϕ = ψ −1 . We claim that {ft : t ∈ 2
t = t,
ft (x) ε|t| ,
t = t,
hence ft − χΔt ◦ ϕ C (K) ε|t| . By adding all these estimates together we get
ft − χΔt ◦ ϕ C (K) < ε.
t∈2
This finishes the proof.
2
ˇ Proof of Theorem 4.1 (ii ) ⇒ (vi ). Assume that B is an almost Cech-analytic subset of a compact convex set X that is not σ -fragmented by ρA , where A ⊂ BA(X) . According to [26, Theorem 5.2], there exists a compact set L ⊂ B that is not fragmented by ρA . Using [6, Theorem 2.1] we find a countable set C ⊂ A such that L is not ρC -separable. We enumerate C = {hn : n ∈ N} and define ψ : X → RN by
ψ(x) = h1 (x), h2 (x), . . . ,
x ∈ X.
878
O.F.K. Kalenda, J. Spurný / Journal of Functional Analysis 256 (2009) 865–880
Then X = ψ(X) is a compact metrizable space. Denote by h n the projection to nth coordinate restricted to X . (Note that hn = h n ◦ ψ , n ∈ N.) We write C = {h n : n ∈ N} and notice that L = ψ(L) is ρC -nonseparable. Hence L is not ρconv(C ∪−C ) -separable as well. We use Lemma 4.5 to get M > 0. We claim that M is the sought number. To this end, let ε > 0 be given. According to Lemma 4.5, there exist a homeomorphism ϕ : K → 2N of a compact set K ⊂ L onto 2N and a family {ft : t ∈ 2
By setting K = L ∩ ψ −1 (K )
and ft = ft ◦ ψ,
t ∈ 2
we finish the proof. To verify the particular case, we assume that every compact subset of B is metrizable. We follow the above proof until finding L. Since L is metrizable by the assumptions, we use Lemma 4.5 directly to conclude the proof. 2 5. Transferring other properties It is natural to ask whether some other properties can be transferred from a boundary to the whole space. We will briefly address two such properties—the network weight (i.e., the minimal cardinality of a network, see [14, p. 127]) and the norm density (the minimal cardinality of a ρBA(X) -dense subset). In both cases the countable case can be transferred. In case of the norm density it follows from the Rodé theorem [30] (see also Lemma 3.3 above). The case of network weight follows from [24, Theorem 2.6] (the particular case of extreme points is proved in [21, Theorem 4.6]). We collect some examples showing that uncountable values of the norm density and the network weight need not transfer from a boundary to the whole space. We start by two examples concerning the norm density. Example 5.1. Under Martin’s axiom there is a compact convex set X (even a Bauer simplex) such that ext X has cardinality 2ω and the ρBA(X) -density of X is strictly greater than 2ω . Proof. By a result of D. Fremlin and G. Plebanek [18, Theorem 3A], under Martin’s axiom there ω exists a compact space K of cardinality 2ω such that K admits 22 mutually orthogonal Radon probability measures. From this fact the proof easily follows. Indeed, extreme points of X = M1 (K) are just Dirac measures and hence are weak∗ homeomorphic to K. Moreover, they are norm discrete, so the density of this set is equal to the cardinality of K. On the other hand, the norm density of M1 (K) is equal to the norm density of C(K)∗ and this is greater than the density of extreme points. 2 We note that under continuum hypothesis the cardinality of this K is ℵ1 , hence we get a counterexample of the smallest possible cardinality.
O.F.K. Kalenda, J. Spurný / Journal of Functional Analysis 256 (2009) 865–880
879
Example 5.2. Let E = 1 (Γ ) and X = (BE ∗ , w ∗ ). Then the norm density of X is 2|Γ | and there is a boundary of X with cardinality |Γ |ω . In particular, if |Γ | = 2ω , the norm density of X is strictly larger than the norm density of a boundary. Proof. As E ∗ = ∞ (Γ ), it is well known and easy to see that the norm density of X is 2|Γ | . For a boundary we can take
B = (xγ ) ∈ X: (∀γ ∈ Γ ) xγ ∈ {−1, 0, 1} and {γ ∈ Γ : xγ = 0} is countable . Then B is clearly a boundary and |B| = |Γ |ω .
2
We continue by an example concerning the network weight. Recall that for a compact space the network weight coincides with the weight and that the network weight of a space is not greater than its cardinality. ω
Example 5.3. There is a compact space K of weight 22 such that there is a boundary B ⊂ K for C(K) of cardinality 2ω . 2ω
ω
Proof. Take K = {0, 1}2 . Then obviously the weight of K is 22 . By [12, §3] there is a dense countably compact subset B ⊂ K of cardinality 2ω . (It is easy to construct B starting from an arbitrary dense subset of cardinality 2ω .) As B is countably compact and dense, it is a boundary for C(K). This completes the proof. 2 In case B is the set of all extreme points, there are some more positive results. By [29, Theorem 3.2] for the set of extreme points the network weight and the weight coincide. Moreover, the weight of the whole set X equals to the (network) weight of ext X if either X is a Choquet simplex [29, Theorem 3.4] of ext X is Lindelöf [29, Theorem 3.3]. It is an open question whether this equality holds in general: Question 5.4. If X is a compact convex set, is it true that weight of X is equal to the (network) weight of ext X? As remarked above, for ext X the weight and the network weight coincide. For a general ω boundary it is not the case. Note that in Example 5.3 the weight of B is equal to 22 (it is easy to check that any dense subset of {0, 1}Γ has weight equal to |Γ |). Hence, the network weight of B is strictly smaller than its weight. Further, the weight of B is equal to the weight of K. Therefore it is natural to ask whether the weight of a compact convex set is equal to the weight of any boundary. References [1] E.M. Alfsen, Compact Convex Sets and Boundary Integrals, Springer-Verlag, 1971. [2] E. Bishop, K. de Leeuw, The representations of linear functionals by measures on sets of extreme points, Ann. Inst. Fourier (Grenoble) 9 (1959) 305–331. [3] J. Bourgain, M. Talagrand, Compacité extrémale, Proc. Amer. Math. Soc. 80 (1) (1980) 68–70. [4] B. Cascales, G. Godefroy, Angelicity and the boundary problem, Mathematika 45 (1) (1998) 105–112. [5] B. Cascales, G. Manjabacas, G. Vera, A Krein–Šmulian type result in Banach spaces, Quart. J. Math. Oxford Ser. (2) 48 (190) (1997) 161–167.
880
O.F.K. Kalenda, J. Spurný / Journal of Functional Analysis 256 (2009) 865–880
[6] B. Cascales, I. Namioka, The Lindelöf property and σ -fragmentability, Fund. Math. 180 (2) (2003) 161–183. [7] B. Cascales, I. Namioka, J. Orihuela, The Lindelöf property in Banach spaces, Studia Math. 154 (2) (2003) 165–192. [8] B. Cascales, R. Shvydkoy, On the Krein–Šmulian theorem for weaker topologies, Illinois J. Math. 47 (4) (2003) 957–976. [9] B. Cascales, G. Vera, Topologies weaker than the weak topology of a Banach space, J. Math. Anal. Appl. 182 (1) (1994) 41–68. [10] M. De Wilde, Pointwise compactness in spaces of functions and R.C. James theorem, Math. Ann. 208 (1974) 33–47. [11] R. Deville, G. Godefroy, V. Zizler, Smoothness and Renormings of Banach Spaces, Pitman Monographs, 1993. [12] E.K. van Douwen, The weight of a pseudocompact (homogeneous) space whose cardinality has countable cofinality, Proc. Amer. Math. Soc. 80 (4) (1980) 678–682. [13] E.K. van Douwen, The integers and topology, in: K. Kunen, J.E. Vaughan (Eds.), Handbook of Set-Theoretical Topology, North-Holland, 1988, pp. 111–167. [14] R. Engelking, General Topology, Heldermann, Berlin, 1989. [15] M. Fabian, Gâteaux Differentiability of Convex Functions and Topology: Weak Asplund Spaces, Wiley– Interscience, New York, 1997. [16] V.P. Fonf, J. Lindenstrauss, Boundaries and generation of convex sets, Israel J. Math. 136 (2003) 157–172. [17] V.P. Fonf, J. Lindenstrauss, R.R. Phelps, Infinite dimensional convexity, in: W.B. Johnson, J. Lindenstrauss (Eds.), Handbook of the Geometry of Banach Spaces, vol. 1, North-Holland, 2001, pp. 599–670. [18] D. Fremlin, D. Plebanek, Large families of mutually singular Radon measures, Bull. Pol. Acad. Sci. Math. 51 (2) (2003) 169–174. [19] G. Godefroy, Boundaries of a convex set and interpolation sets, Math. Ann. 277 (2) (1987) 173–184. [20] R.W. Hansell, Descriptive topology, in: M. Hušek, J. van Mill (Eds.), Recent Progress in General Topology, NorthHolland, 1992, pp. 275–315. [21] R. Haydon, An extreme point criterion for separability of a dual Banach space, and a new proof of a theorem of Corson, Quart. J. Math. Oxford Ser. (2) 27 (107) (1976) 379–385. ˇ [22] P. Holický, Cech-analytic and almost-K-descriptive spaces, Czechoslovak Math. J. 43 (118) (3) (1993) 451–466. [23] M. López-Pellicer, V. Montesinos, Cantor sets in the dual of a separable Banach space. Applications, in: T. Banakh (Ed.), General Topology in Banach Spaces, Nova Sci. Publ., Huntington, 2001, pp. 35–59. [24] W.B. Moors, E.A. Reznichenko, Separable subspaces of affine function spaces on convex compact sets, Topology Appl. 155 (12) (2008) 1306–1322. [25] I. Namioka, Separate continuity and joint continuity, Pacific J. Math. 51 (1974) 515–531. [26] I. Namioka, R. Pol, σ -fragmentability and analyticity, Mathematika 43 (1) (1996) 172–181. [27] J.C. Oxtoby, Measure and Category, second ed., Springer-Verlag Inc., New York, 1980. [28] H. Pfitzner, Boundaries for Banach spaces determine weak compactness, http://arxiv.org/abs/0807.2810. [29] E.A. Reznichenko, Convex compact spaces and their maps, Topology Appl. 36 (2) (1990) 117–141. [30] G. Rodé, Superkonvexität und schwache Kompaktheit, Arch. Math. (Basel) 36 (1) (1981) 62–72 (in German). [31] C.E. Rogers, J.E. Jayne, K-analytic sets, in: C.E. Rogers, J.E. Jayne (Eds.), Analytic Sets, Academic Press, 1980, pp. 1–181. [32] S. Simons, A convergence theorem with boundary, Pacific J. Math. 40 (1972) 703–708. [33] J. Spurný, Boundary problem for L1 preduals, Illinois J. Math., http://www.karlin.mff.cuni.cz/~rokyta/preprint/ index.php. [34] C. Stegall, The Radon–Nikodym property in conjugate Banach spaces, Trans. Amer. Math. Soc. 206 (1975) 213– 223.
Journal of Functional Analysis 256 (2009) 881–927 www.elsevier.com/locate/jfa
A critical functional framework for the inhomogeneous Navier–Stokes equations in the half-space Raphaël Danchin a,∗ , Piotr Bogusław Mucha b a Université Paris-Est, Laboratoire d’Analyse et de Math. Appliquées, UMR 8050, 61 avenue du Général de Gaulle,
94010 Créteil Cedex, France b Instytut Matematyki Stosowanej i Mechaniki, Uniwersytet Warszawski, ul. Banacha 2, 02-097 Warszawa, Poland
Received 27 March 2008; accepted 24 November 2008 Available online 4 December 2008 Communicated by C. Villani
Abstract This paper is devoted to solving globally the boundary value problem for the incompressible inhomogeneous Navier–Stokes equations in the half-space in the case of small data with critical regularity. In dimension n 3, we state that if the initial density ρ0 is close to a positive constant in L∞ ∩ W˙ n1 (Rn+ ) and 0 (Rn ) then the initial velocity u0 is small with respect to the viscosity in the homogeneous Besov space B˙ n,1 + the equations have a unique global solution. The proof strongly relies on new maximal regularity estimates 0 (Rn )), interesting for their own sake. for the Stokes system in the half-space in L1 (0, T ; B˙ p,1 + © 2008 Elsevier Inc. All rights reserved. Keywords: Critical regularity; Inhomogeneous viscous fluids; Stokes system; Homogeneous Besov spaces; Half-space
1. Introduction We want to investigate the global well-posedness for the incompressible inhomogeneous Navier–Stokes equations in the half-space Rn+ := {(x1 , . . . , xn ) ∈ Rn : xn > 0}. The corresponding system reads * Corresponding author.
E-mail addresses: [email protected] (R. Danchin), [email protected] (P.B. Mucha). 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.11.019
882
R. Danchin, P.B. Mucha / Journal of Functional Analysis 256 (2009) 881–927
∂t ρ + u · ∇ρ = 0
in (0, T ) × Rn+ ,
ρ(∂t u + u · ∇u) − μu + ∇Π = ρf div u = 0
in (0, T ) × Rn+ ,
in (0, T ) × Rn+ ,
u|xn =0 = 0 on (0, T ) × Rn−1 , u0 |t=0 = u0 ,
ρ|t=0 = ρ0
on Rn+ ,
(INS)
where ρ, u = (u1 , . . . , un ) and Π stand respectively for the unknown density, velocity and pressure of the fluid. The given positive real number μ is the viscosity coefficient and f represents external body forces. Due to the compatibility conditions, we assume that the initial velocity u0 is divergence free and that its normal component u0n is zero on ∂Rn+ in the distributional meaning; the initial density ρ0 is required to be strictly positive in the half-space and we restrict our attention to solutions such that the density tends (weakly) to a positive constant (say 1), and the velocity tends to 0 at infinity. The homogeneous case ρ0 ≡ 1 — the classical incompressible Navier–Stokes equations — has been extensively studied from a mathematical viewpoint. It is well established that as far as one is interested by global existence results with uniqueness, it is important to work with critical norms for the initial data u0 and for the solution u, that is with norms invariant for all λ > 0 by the rescaling1 (u, Π)(t, x) → λu, λ2 Π λ2 t, λx ,
u0 (x) → λu0 (λx).
(1.1)
This is due to the fact that the homogeneous Navier–Stokes equations are invariant by (1.1) so that any proof based on contracting mapping arguments in a Banach space requires norms with the above scaling invariance. Solving the (homogeneous) Navier–Stokes equations in critical spaces goes back to the pioneering work by H. Fujita and T. Kato in [17,18]. There, in the case of a bounded domain of Rn , it is stated that any small enough initial velocity with n/2 − 1 derivative(s) in L2 generates a global (unique) solution. In the whole space case, Fujita and Kato’s approach has been adapted to a plethora of critical functional frameworks (see e.g. [8,25]). Let us mention in particular that, in the whole space case, the classical incompressible Navier–Stokes equations are globally wellposed if taking u0 small with respect to the viscosity in the Lebesgue space Ln (Rn ) (see [19,23]). This latter result has been adapted to the case of bounded domains by Y. Giga and T. Miyakawa in [20] and to the case of exterior domains by Y. Giga and H. Sohr in [21], and H. Iwashita in [22]. In the half-space case, the well-posedness issue has been studied by H. Kozono in [24] (see also [9] for results related to critical Besov spaces with negative index of regularity). Motivated by the fact that in real life, a fluid is hardly homogeneous, we want to study whether the aforementioned approach is relevant for the inhomogeneous incompressible Navier–Stokes equations. Now, the scaling invariance for (INS) reads (ρ, u, Π)(t, x) → ρ, λu, λ2 Π λ2 t, λx ,
(ρ0 , u0 )(x) → (ρ0 , λu0 )(λx)
(1.2)
which, roughly, means that the critical spaces for the velocity (and pressure) are the same as in the homogeneous case, and that one has to take one more derivative for the density. 1 Here we take f ≡ 0 for simplicity.
R. Danchin, P.B. Mucha / Journal of Functional Analysis 256 (2009) 881–927
883
The first example of solving (INS) in a critical functional framework has been given by the first author in [11], in the whole space case. There, global well-posedness is shown whenever n n 2 2 −1 (ρ0 − 1, u0 ) belongs to the homogeneous Besov space B˙ 2,1 × B˙ 2,1 which is critical in the sense of (1.2). Data in more general critical Besov spaces related to Lp spaces have been considered in [1,2]. All those results strongly rely on the use of the Fourier transform on Rn so that their extension to more general domains is far from being obvious. A first attempt in this direction (for bounded domains) has been done in [13]. There, Fourier analysis has been replaced by standard maximal regularity estimates for the Stokes system (after the pioneering work by O. Ladyzhenskaya and V. Solonnikov in [26]). However, it is not clear that the critical index may be attained by this method. In the present paper, we aim at proving global in time existence (and uniqueness) for (INS) in the half-space for small data with critical regularity. In fact, we strive for a statement as close as possible to the one given by [24] in the homogeneous case. Based on (1.2), it is thus natural to take the initial data (ρ0 , u0 ) so that ∇ρ0 Ln (Rn+ ) and u0 Ln (Rn+ ) be small. However, in order to get a control on the ellipticity of the velocity equation, assuming in addition that ρ0 is bounded away from zero, and in L∞ (Rn+ ) (a space which has the desired scaling) seems unavoidable. We shall also rather take u0 in a subspace of Ln (Rn+ ), namely the homoge0 (Rn ) which still has the right scaling. This assumption will ensure that neous Besov space B˙ n,1 + ∇u ∈ L1,loc (R+ ; L∞ (Rn+ )), a property which is needed to propagate the regularity of the initial density. In fact, in the framework of critical Besov spaces, having 1 as a third index is the only possibility to get a control over ∇u in L1,loc (R+ ; L∞ (Rn+ )). More explanations (together with 0 (Rn )) will be given in the next section. Here we touch an open question, the definition of B˙ n,1 + whether one may investigate general parabolic-type systems in L1 (0, T ; X), where X stands for a Banach space determining the regularity of solutions with respect to space directions. Positive answers [14,21], obtained by techniques of the theory of semigroups, are known only for spaces Lq (0, T ; X) with q ∈ (1, ∞), but the case q = 1 is beyond this approach. Thus, our paper is devoted to this critical case, however only for our particular system. Let us now introduce the functional spaces that we shall use in our global existence statement. For p ∈ [1, +∞] and T ∈ [0, +∞] we define Ep (T ) as the set of functions (ρ, u, Π) such that2 (ρ − 1) ∈ L∞ (0, T ) × Rn+ ∩ Cb [0, T ); Wp1 Rn+ , n n n 0 0 R+ and ∂t u, ∇ 2 u, ∇Π ∈ L1 0, T ; B˙ p,1 R+ u ∈ Cb [0, T ); B˙ p,1 . If T = +∞ then we simply denote the above space by Ep . We shall also use the notation Ep,loc = T >0 Ep (T ). Theorem 1. Let n 2. Let ρ0 be a bounded positive function such that (ρ0 − 1) ∈ Wn1 (Rn+ ). Let u0 be a divergence free vector field on Rn+ with u0,n = 0 at the boundary and coefficients 0 (Rn ). Assume that f ∈ L (R ; B 0 n in B˙ n,1 1 + ˙ n,1 (R+ )). There exist two positive constants c and M + depending only on n, and such that if
ρ0 − 1 L∞ (Rn+ ) + ∇ρ0 Ln (Rn+ ) c
and u0 B˙ 0
n n,1 (R+ )
+ f L1 (R+ ;B˙ 0
n n,1 (R+ ))
cμ
(1.3)
2 It is understood that W 1 (Rn ) stands for the set of L functions over Rn with (weak) first-order derivatives in p p + + Lp (Rn+ ).
884
R. Danchin, P.B. Mucha / Journal of Functional Analysis 256 (2009) 881–927
then system (INS) has a global solution (ρ, u, Π) in the space En such that for all t ∈ R+ , ρ(t) − 1
ρ(t) − 1 1 n 2 ρ0 − 1 1 n , (1.4) = ρ0 − 1 L∞ (Rn+ ) , Wn (R+ ) Wn (R+ ) ∇Π, ∂t u, μD 2 u n 0 0 (Rn )) + u L (R ;B L1 (R+ ;B˙ n,1 ∞ + ˙ n,1 (R+ )) + (1.5) M u0 B˙ 0 (Rn ) + f L1 (R+ ;B˙ 0 (Rn )) .
L∞ (Rn+ )
n,1
+
n,1
+
If n 3 then uniqueness holds true in the space En . A few comments are in order. • Because the space L∞ ∩ Wn1 (Rn+ ) fails to be embedded in the set of continuous functions over Rn+ , the initial density need not be continuous. The initial velocity need not be continuous either. In particular it may have a jump across a smooth interface (if compactly 1 0 (Rn )). n (Rn+ ), hence also in B˙ n,1 supported, such data are in B˙ n,∞ + 0 n • In the case of a large initial velocity in B˙ n,1 (R+ ), an easy variation over our method would provide a local solution. • In dimension n 3, one may weaken slightly the assumptions on the density: it is possible 1 (Rn ) as done in [1] in the whole space case. However, to replace the space Wn1 (Rn+ ) by Bn,∞ + 1 (Rn ), a study this improvement requires estimates for the transport equation in spaces Bn,∞ + that we decided to omit in the present paper, for simplicity. 0 (R2 ) • Uniqueness in dimension two may be obtained if assuming that ρ0 − 1 is small in B˙ 2,1 + (see the whole space case in [11]). However proving this result also requires estimates for the transport equation in Besov spaces. • We expect to have global well-posedness for large data in dimension two (this fact is well known for smooth data in the case of a bounded domain, see [26,3,28] and has been extended to the R2 case with critical data in [12]). This study would require a rather different approach in the treatment of the nonlinear terms, though. Remark 1. Even though taking the initial velocity in Ln (Rn+ ) (as in the homogeneous case) may seem more natural, it is very unlikely that one may prove a global well-posedness result under 0 (Rn ) may be replaced by the larger this assumption. It is not clear either that the space B˙ n,1 + n 0 space B˙ p,q (R+ ) with q > 1. The reason why is that at the level of the linearized equations, having q = 1 is the only possibility to get a control of ∇u in L1,loc (R+ ; L∞ (Rn+ )), a property which is needed to propagate the Wn1 (Rn+ ) regularity of the density for all time. Proving Theorem 1 requires three main ingredients: • time-independent maximal estimates for the linearized velocity equation, namely the evolutionary Stokes system: ∂t v − μv + ∇P = F
in (0, T ) × Rn+ ,
div v = 0 in (0, T ) × Rn+ , v|xn =0 = 0 on (0, T ) × Rn−1 , v|t=0 = v0
on Rn+ ;
(1.6)
R. Danchin, P.B. Mucha / Journal of Functional Analysis 256 (2009) 881–927
885
• bilinear estimates in functional spaces related to homogeneous Besov spaces; • a compactness argument. In contrast with the homogeneous Navier–Stokes equations and owing to the hyperbolicity of the equation for the density, our existence theorem does not come up as a consequence of a contracting mapping argument in a suitable Banach space. Therefore, we shall rather make use of a compactness method, proving first uniform estimates for a sequence of smooth solutions pertaining to smoothed out data. This may be achieved thanks to standard estimates in Lebesgue spaces for the transport equation satisfied by the density and estimates in homogeneous Besov spaces for (1.6) taking for F all the nonlinear terms of the velocity equations. Bilinear estimates in Besov spaces will be needed for bounding those nonlinear terms. Uniqueness will be obtained afterward, proving a stability estimate in low norm and taking advantage of a logarithmic interpolation argument. Note that at the formal level, the general method and the main ingredients are the same as in the whole space case treated in [11,1,2]. However, in the half-space case, one has to face several additional difficulties. First, in contrast with the Rn case, the Stokes system cannot be reduced to the basic heat equation after suitable projection. Second, the adaptation of the bilinear estimates to this new framework requires some care. Third, there is no explicit definition of homogeneous 0 (Rn ). In fact, the only reasonable definition is given by restriction of functions Besov spaces B˙ p,1 + defined on the whole space. Furthermore, because we did not make any additional assumption on the potential part of the source term f (in contrast with what has been done in [1,2,11]), recovering the full L1 -in-time regularity for ∇ 2 u, ∂t u and ∇Π in (INS) turns out to be not so straightforward and requires a novel approach for this problem. Of course this ultimate difficulty would also occur in the whole space case. Let us now state the estimates that we have obtained for the Stokes system in the half-space: s (Rn )) and Theorem 2. Let p ∈ (1, ∞), s ∈ ( p1 − 1, p1 ) and T ∈ (0, ∞]. Let F ∈ L1 (0, T ; B˙ p,1 + s (Rn ) with div v = 0 and v | v0 ∈ B˙ p,1 = 0 in the meaning of the trace. Then there exists a 0 0n x =0 + n unique solution to problem (1.6) such that
n s v ∈ Cb [0, T ); B˙ p,1 R+ ,
n n s s ∇ 2 v ∈ L1 0, T ; B˙ p,1 R+ and ∇P ∈ L1 0, T ; B˙ p,1 R+
and the following estimate is valid:
v L∞ (0,T ;B˙ s
n p,1 (R+ ))
+ ∂t v, μ∇ 2 v, ∇P L
C F L1 (0,T ;B˙ s
n p,1 (R+ ))
˙ s (Rn+ )) 1 (0,T ;B p,1
+ v0 B˙ s
n p,1 (R+ )
,
(1.7)
where C is a constant depending only on s, p and n. Remark 2. Proving the existence part of Theorem 1 requires only the case s = 0. However, combining the general statement with real interpolation will enable us to get another family of estimates (see Lemma 10 below) which turns out to be the key to the uniqueness for (INS). The s (Rn ), see Proposition 3. choice of s is restricted by properties of B˙ p,1 +
886
R. Danchin, P.B. Mucha / Journal of Functional Analysis 256 (2009) 881–927
Remark 3. Choosing homogeneous semi-norms enables us to get time-independent estimates for the Stokes system. This is of course the key to proving a global result for (INS). Indeed using the more standard inhomogeneous framework would introduce a linear time dependency in the estimates. Here also, having Besov spaces with third index 1 is fundamental: inequality (1.7) is gener0 (Rn ) is changed into ically false (even in the whole space or for the heat equation), if B˙ p,1 + n n 0 Lp (R+ ), or into B˙ p,q (R+ ) for some q > 1. Difficulties are carried by the L1 -regularity with respect to time, precluding us from using the standard Marcinkiewicz theorem for the Fourier multipliers or general Calderon–Zygmund theory for singular operators [16,35]. This follows that even the advanced theory of semigroups [14, 21] cannot deliver us such results. This technique deals with the regularity of type Lq (0, T ; X) with q ∈ (1, ∞) and q = 1 cannot be reached. Hence Theorem 2 should not be viewed as an element of the standard theory. It is worthwhile to underline that, in contrast with the heat equation, one cannot adapt directly the results for the Stokes system in the whole space to our case by a suitable method of symmetry. Even though the solution may be explicitly computed in terms of the data (see in particular [36,27,34]) it is not clear that the above theorem may be obtained by a direct use of the corresponding formula. We also think that any approach based on the characterization of the Besov spaces in terms of the heat flow (such that the one used in [9] for instance) is bound to fail for nonnegative index of regularity. The main tool for proving Theorem 2 is the Fourier transform. After a suitable preparation (reducing the study to the case where div F = 0 and Fn |xn =0 = 0 then extending the Stokes problem to the whole real line with nonhomogeneous boundary data on xn = 0) we may perform a Fourier transform with respect to the time t and tangential variables x := (x1 , . . . , xn−1 ). We then obtain a system of ordinary differential equations with respect to the normal variable xn , which may be explicitly solved. By taking advantage of harmonic analysis techniques and of the theory of Besov spaces, it is then possible to get the maximal regularity estimate (1.7). At this point, handling the trace of the gradient of the velocity is the main problem. Surprisingly, the explicit representation does not give any exploitable information for standard methods in the half-space with homogeneous equations and inhomogeneous boundary conditions are not allowed here. A way to overcome this ultimate difficulty is to obtain the “explicit” form of the velocity in the half-space with the homogeneous boundary data as in [15]. Then the difficulty related to the traces can be omitted by construction of explicit extensions. To finish with, let us emphasize that the maximal regularity estimates — called sometimes Schauder’s estimates — are irreplaceable in the analysis of quasi-linear systems [21,30–33]. They allow us to treat the nonlinear problem as a perturbation of a linear one, since we loose no regularity. This feature is particularly important here as the functions we work with have critical regularity so that the nonlinearities can be controlled only by the highest/whole norms of sought solutions. We thus expect our study to be an important step to understand more advanced boundary problems in (possibly) more general domains. The paper unfolds as follows. In the next section we recall definitions of the Besov spaces and some auxiliary results from this field. In Section 3 we analyze the Stokes system and prove Theorem 2. Then we return to the nonlinear system (INS) and prove Theorem 1. In Appendix A we give the proofs of some technical results needed in the paper.
R. Danchin, P.B. Mucha / Journal of Functional Analysis 256 (2009) 881–927
887
2. Besov spaces In this section we introduce the homogeneous Besov spaces required in our analysis, and give a few basic results. Throughout we fix a smooth function φ : R+ → [0, 1] supported in {1/2 r 2} and such that φ 2−k r = 1 for all r > 0.
(2.1)
k∈Z
Then we introduce the homogeneous Littlewood–Paley decomposition (k )k∈Z over Rn by setting k u := ϕ 2−k D u = F −1 ϕ 2−k · F u with ϕ(ξ ) := φ |ξ | . Above F stands for the Fourier transform on Rn . Let us first define the homogeneous Besov spaces on Rn . For that, we introduce the following homogeneous Besov semi-norms (for all s ∈ R and p, q ∈ [1, ∞]):
u B˙ s
n p,q (R )
:= 2sk k u Lp (Rn ) q (Z) .
(2.2)
Owing to the lack of control of low frequencies (in particular when s is large), there is no cons (Rn ). When one deals with nonlinear PDEs, sensus for defining homogeneous Besov spaces B˙ p,q the following definition turns out to be convenient: n s R = u ∈ Sh Rn : u B˙ s B˙ p,q
p,q (R
n)
<∞ ,
where Sh (Rn ) stands for the set of tempered distributions u over Rn such that for all smooth compactly supported function θ over Rn , we have lim θ (λD)u = 0
λ→+∞
in L∞ Rn .
Note that the above condition implies that any distribution in Sh (Rn ) tends weakly to 0 at infinity. In particular, Sh (Rn ) contains no nonzero polynomial. Note also that if u ∈ Sh (Rn ) then one may write u=
k u
in Sh Rn ,
(2.3)
k∈Z
and that, conversely, if (2.3) is satisfied and u B˙ s (Rn ) < ∞ for some index s such that s < n/p p,q s (Rn ). (or s n/p if q = 1) then u is in B˙ p,q One can prove the following two fundamental properties (see [4]):
888
R. Danchin, P.B. Mucha / Journal of Functional Analysis 256 (2009) 881–927
Proposition 1. s (Rn ) is complete whenever s n/p if q = 1, and s < n/p if q > 1. 1. The space B˙ p,q 2. The set S0 (Rn ) of Schwartz functions with Fourier transform supported away from the origin s (Rn ) if and only if p and q are finite. is dense in B˙ p,q
Remark 4. For the special case s ∈ (0, 1) the Besov semi-norms may be defined in terms of finite differences of order 1 according to [35, Chapter 2.5]. More precisely, the quantity defined in (2.2) is equivalent to
n
∞
k=1 0 n
1
dh u(x1 , . . . , xk + h, . . . , xn ) − u(x1 , . . . , xk , . . . , xn )q
q
Lp (Rn )
h1+sq
sup h−s u(x1 , . . . , xk + h, . . . , xn ) − u(x1 , . . . , xk , . . . , xn )L
p (R
k=1 h>0
n)
if q < ∞,
if q = ∞.
Let us now consider the Poisson equation −u = f
in Rn .
(2.4)
It is obvious that if f ∈ S0 (Rn ) then the function u defined by F u(ξ ) =
F f (ξ ) |ξ |2
(2.5)
is the unique solution of (2.4) in S0 (Rn ) and that, in addition, u B˙ s+2 (Rn ) C f B˙ s (Rn ) . Since p,q p,q s (Rn ) if p and q are finite, one may deduce the following result: S0 (Rn ) is dense in B˙ p,q Proposition 2. If p and q are finite then the map f → u defined by (2.5) has a unique continuous s (Rn ) to B s+2 (Rn ). ˙ p,q extension from B˙ p,q Let us now define the homogeneous Besov spaces on the half-space. s (Rn ) Definition 1. For s ∈ R and 1 p, q ∞, we define the homogeneous Besov space B˙ p,q + n s n ˙ over the half-space as the restriction (in the distributional sense) of Bp,q (R ) on R+ , that is
n s φ ∈ B˙ p,q R+
⇔
φ = ψ|Rn+
n s R . for some ψ ∈ B˙ p,q
We then set
φ B˙ s
n p,q (R+ )
:=
inf
ψ|Rn =φ +
ψ B˙ s
p,q (R
n)
.
s (Rn ) for small |s|. It will enable us to consider the symThe result below characterizes B˙ p,q + 0 (Rn ) on the whole space, preserving metric and antisymmetric extension of functions from B˙ p,q + the class of regularity.
R. Danchin, P.B. Mucha / Journal of Functional Analysis 256 (2009) 881–927
Proposition 3. Let 1 p, q < ∞. Then for
1 p
889
− 1 < s < p1 , we have
n
· ˙ s n s s R+ = f ∈ B˙ p,q Rn : supp f ⊂ Rn+ Bp,q (R ) . B˙ p,q s (Rn ) then one can find a sequence (ul ) s (Rn ) functions, ˙ p,q In other words, if u ∈ B˙ p,q l∈N of B + n n supported in R+ and such that, denoting by u˜ the extension of u by 0 on R− , we have
lim ul − uB˙ s
l→+∞
p,q (R
n)
= 0.
The proof is based on a result from [35] for nonhomogeneous spaces and the fact that for 0 < s (R) is embedded in some Lebesgue space with finite index. Since it is s < p1 , the space B˙ p,1 fundamental for our analysis, a sketch of it is given in Appendix A. Remark 5. From Proposition 3, it is not difficult to prove that if 1 p, q < ∞ and 1/p − 1 < s (Rn ). This result fails to be true for B s ˙ p,∞ (Rn+ ). s < 1/p then the space C0∞ (Rn+ ) is dense in B˙ p,q +
· ˙ s s (Rn+ ). In other words, the space C ∞ (Rn ) Bp,∞ is a strict subspace of B˙ p,∞ 0
The following embedding results will be of constant use in the study of (INS). Proposition 4. Let 1 p p ∞, 1 q ∞ and s ∈ R. We have: 0 (Rn ) → L (Rn ) → B 0 (Rn ). ˙ p,∞ • B˙ p,1 p + + + 1 s−n( p1 − p )
s (Rn ) → B ˙ • B˙ p,q + p ,q
(Rn+ ).
Proof. Those properties are well known for the Besov spaces defined on Rn (see e.g. [4,5]). The result for the spaces on Rn+ thus follows readily from the definition by restriction. 2 Let us now consider the Poisson equation in the half-space. s (Rn ) with 1 < p < ∞, 1 q ∞ and Lemma 1. For any H ∈ B˙ p,q +
z = div H
in Rn+ ,
z|xn =0 = 0 on Rn−1 ,
1 p
− 1 < s < p1 , system
z → 0 as |x| → ∞,
(2.6)
s (Rn ), and we have has a unique solution z such that ∇z ∈ B˙ p,q +
∇z B˙ s
n p,q (R+ )
C H B˙ s
n p,q (R+ )
.
(2.7)
Proof. The uniqueness stems from the standard theory for the Laplace equation. For proving |Rn = H and, on Rn by H existence, one may extend H by antisymmetry as in [29]: we define H + τ := (H 1 , . . . , H n−1 ), denoting Hτ := (H1 , . . . , Hn−1 ) and H ∀xn ∈ (0, +∞),
τ (x , −xn ) = −Hτ (x , xn ) H
n (x , −xn ) = Hn (x , xn ). and H
(2.8)
890
R. Danchin, P.B. Mucha / Journal of Functional Analysis 256 (2009) 881–927
s (Rn ) and ∈ B˙ p,q Note that as s ∈ ( p1 − 1, p1 ), Proposition 3 guarantees that H
˙ s
H B
p,q (R
n)
C H B˙ s
n p,q (R+ )
,
s+1 (Rn ) satisfying has a solution so that Proposition 2 ensures that equation z = div H z ∈ B˙ p,q
∇ z B˙ s
p,q (R
˙ s C H B
n)
p,q (R
n)
(2.9)
.
n has no “jump” on ∂Rn+ , we have Because the normal component H ∀xn ∈ (0, +∞),
)(x , −xn ) = − div H (x , xn ) (div H
in the sense of distributions
z|∂Rn+ = 0. In so that z is antisymmetric with respect to the hyperplane ∂Rn+ . This implies that on Rn in the sense of distribution, one may write addition, as z = div H (2.10) zπ dx = ∇ z · ∇π dx = H · ∇π dx for all π ∈ C0∞ Rn+ . − Rn
Rn+
Rn+
Therefore, z := z|Rn+ is a weak solution of (2.6). Inequality (2.7) follows from (2.9).
2
Our analysis also requires some estimates for the linear heat equation in the half-space: ∂t v − v = F
in (0, T ) × Rn+ ,
v|xn =0 = 0 on (0, T ) × Rn−1 , v|t=0 = v0 on Rn+ ,
v → 0 as |x| → ∞.
(2.11)
Let us first recall the following result pertaining to the heat equation in the whole space (see the proof in [10]). s (Rn )) and v ∈ B ˙ s (Rn ) Proposition 5. Let p ∈ (1, ∞) and s ∈ R. For any F ∈ L1 (0, T ; B˙ p,1 0 p,1 there exists a unique solution v to
∂t v − v = F in (0, T ) × Rn , v|t=0 = v0 on Rn ,
v → 0 as |x| → ∞,
such that n s v ∈ C [0, T ]; B˙ p,1 R
n s and ∂t v, ∇ 2 v ∈ L1 0, T ; B˙ p,1 R .
Besides, the following estimate is valid
v L∞ (0,T ;B˙ s (Rn )) + ∂t v, ∇ 2 v L (0,T ;B˙ s (Rn )) 1 p,1 p,1 C F L1 (0,T ;B˙ s (Rn )) + v0 B˙ s (Rn ) p,1
where C is a constant depending only on n.
p,1
(2.12)
R. Danchin, P.B. Mucha / Journal of Functional Analysis 256 (2009) 881–927
891
This entails the following result for the heat equation on the half-space: s (Rn )) and Proposition 6. Let p ∈ (1, ∞) and s ∈ (−1 + 1/p, 1/p). For any F ∈ L1 (0, T ; B˙ p,1 + s n v0 ∈ B˙ p,1 (R+ ), system (2.11) has a unique solution v such that
n s R+ v ∈ C [0, T ]; B˙ p,1
n s and ∂t v, ∇ 2 v ∈ L1 0, T ; B˙ p,1 R+
and inequality (2.12) holds true (with Rn replaced by Rn+ ). Proof. The proof is based again on the use of an antisymmetric extension of the data v0 and F. the extended data, Proposition 3 ensures that Denoting by v0 and F
v0 B˙ s
p,1 (R
n)
C v0 B˙ s
n p,1 (R+ )
and F L1 (0,T ;B˙ s
p,1 (R
n ))
C v0 L1 (0,T ;B˙ s
n p,1 (R+ ))
.
Now, the previous proposition provides a unique solution n s+2 n s v ∈ C [0, T ]; B˙ p,1 R ∩ L1 0, T ; B˙ p,1 R . Owing to the antisymmetry of v0 to the heat equation in the whole space with data v0 and F and F , the solution v is antisymmetric hence vanishes on xn = 0. It is thus clear that the restriction v to v on [0, T ] × Rn+ is a solution to (2.11) and satisfies the required properties. 2 Let us state another two basic results the proof of which may be found in Appendix A. The first one will enable us to solve the Laplace equation in the half-space for the Dirichlet problem with nonzero boundary conditions. Lemma 2. Let s > 0, 1 < p < ∞ and 1 q ∞. Then there exists a constant C such that for s−1/p all h ∈ B˙ p,q (Rn−1 ), we have −1 −|ξ |x n F [h] F e x ˙s
Bp,q (Rn+ )
x
C h B˙ s−1/p (Rn−1 ) p,q
(2.13)
where Fx stands for the Fourier transform with respect to x := (x1 , . . . , xn−1 ) and ξ denotes the corresponding Fourier variable. The second result combined with Proposition 3 will allow us to generalize the standard trace theorem. Lemma 3. Let 1 < p < ∞, 1 q ∞ and s ∈ (−1 + 1/p, 1/p). For any vector field F with cos−1/p s (Rn ) and div F = 0 in D (Rn ), we have F | ˙ p,q efficients in B˙ p,q (Rn−1 ). In addition, n xn =0 ∈ B + + there exists a constant C depending only on n and such that
Fn |xn =0 B˙ s−1/p (Rn−1 ) C F B˙ s p,q
n p,q (R+ )
.
(2.14)
892
R. Danchin, P.B. Mucha / Journal of Functional Analysis 256 (2009) 881–927
The proof of uniqueness in Theorem 1 requires our introducing modified Lebesgue–Besov s (Ω)) (where I is any interval of R, and Ω = Rn or Rn ). Those spaces related to L1 (I ; B˙ p,q + s (Ω)), have been introduced in [10]. In the case 1 (I ; B˙ p,q spaces, which will be denoted by L Ω = Rn they may be seen as the set of distributions u in S (I × Rn ) such that
u L 1 (I ;B˙ s
p,q (R
n ))
:=
2ks k u L1 (I ;Lp (Rn )) < ∞
lim θ (λD)u = 0
and
λ→+∞
k∈Z
(2.15)
for all smooth compactly supported functions θ over Rn . The above definition together with a result by H. Triebel [35, 1.18.2] implies the following important fact. Proposition 7. Let p, q ∈ [1, ∞] and s1 , s2 ∈ R with s1 = s2 . Then for any interval I and any θ ∈ (0, 1) we have n n n s s1 s2 1 I ; B˙ p,q 1 I ; B˙ p,q 1 I ; B˙ p,q L R = L R ,L R θ,q 1 2
with s := θ s2 + (1 − θ )s1 .
In other words, Proposition 7 gives us the possibility to omit direct analysis on this type of spaces, just by interpolation, taking q1 = q2 = 1 above. Indeed, by (2.15) we have 1 I ; B˙ s Rn = L1 I ; B˙ s Rn . L p,1 p,1
(2.16)
To extend the above definition to the case Ω = Rn+ , one may follow the same approach s (Rn )) is defined as the restriction 1 (I ; B˙ p,q as for the standard Besov spaces: the space L + s (Rn )) on I × Rn , that is of L1 (I ; B˙ p,q + n s 1 I ; B˙ p,q R+ φ∈L
⇔
φ = ψ|I ×Rn+
n s 1 I ; B˙ p,q R . for some ψ ∈ L
We then set
φ L 1 (I ;B˙ s
n p,q (R+ ))
:=
inf
ψ|I ×Rn =φ
ψ L 1 (I ;B˙ s
p,q (R
+
n ))
.
It is easy to prove embeddings similar to those of Proposition 4. The following two lemmas will be used in the proof of the existence part of Theorems 1 and 3. Proving them requires paradifferential calculus (see Appendix A). Lemma 4. Let q0 > 1 and q ∈ [q0 , ∞]. There exists a constant C = Cq0 ,n such that for all 0 (Rn ) ∩ B 1 (Rn ) we have ˙ 0 (Rn+ ) and G ∈ L∞ (Rn+ ) ∩ B˙ q,∞ F ∈ B˙ n,1 + + q,1
F G B˙ 0
n q,1 (R+ )
C F B˙ 0
n n,1 (R+ )
∇G B˙ q,∞ 0 (Rn ) + F B ˙0 +
n q,1 (R+ )
G L∞ (Rn+ ) .
Lemma 5. Let 1 < q < ∞ and I be an interval of R. There exists a constant C = Cn,q depending continuously on q such that for all (r, r1 , r2 ) ∈ [1, +∞]3 such that 1/r = 1/r1 + 1/r2 and all 0 (Rn )) × (L (I ; L∞ ) ∩ L 0 (Rn ))), we have r1 (I ; B˙ q,∞ r2 (I ; B˙ n,∞ (F, G) in L r2 +
F G L 0 ) C F L r (I ;B˙ q,∞ r
˙0 1 (I ;Bq,∞ )
G Lr2 (I ;L∞ ) + ∇G L r
˙0 2 (I ;Bn,∞ )
.
R. Danchin, P.B. Mucha / Journal of Functional Analysis 256 (2009) 881–927
893
3. The Stokes system This section is devoted to the study of the Stokes system (1.6) in the half-space. In the first three subsections, we shall focus on the proof of Theorem 2, while the last subsection is de0 (Rn )), which will be needed for proving 1 (0, T ; B˙ p,∞ voted to estimates in the (larger) spaces L + the uniqueness for (INS). Our technique follows from standard approaches to the subject [15,30, 31,33]. For notational simplicity, we shall assume that μ = 1. Of course a convenient change of variables gives the general case. 3.1. Reduction to a model problem on R × Rn+ . The first step is to extend the problem on the whole real line for the time direction. Without loss of generality, one may assume that T = +∞, extending the source term f by zero if need be. Next, we want to “eliminate” the initial datum v0 . As 1/p − 1 < s < 1/p, Proposition 3 v0 |Rn+ = v0 and guarantees that the function v0 defined on Rn by v0τ (x , xn ) = − v0τ (x , −xn )
and v0n (x , xn ) = v0n (x , −xn )
for xn < 0,
(3.1)
s (Rn ), is divergence free and satisfies v0 B˙ s (Rn ) C v0 B˙ s (Rn ) . belongs to B˙ p,1 p,1 p,1 + Now, according to Proposition 5, the heat equation
∂t v − v = 0 in R+ × Rn , v|t=0 = v0
on Rn ,
(3.2)
s+2 s (Rn )) ∩ L (R ; B n has a unique solution E v0 in Cb (R+ ; B˙ p,1 1 + ˙ p,1 (R )) and we have
E v0 L∞ (R;B˙ s
p,1 (R
n ))
+ E v0 L1 (R+ ;B˙ s+2 (Rn )) C v0 B˙ s p,1
n p,1 (R+ )
.
(3.3)
Because div v0 = 0, uniqueness for the heat equation guarantees that div E v0 = 0 and, since the symmetry of v0 is preserved during the evolution, we have (E v0 )τ |xn =0 = 0,
where (E v0 )τ := (E v0 )1 , . . . , (E v0 )n−1 .
(3.4)
Note however that the normal (nth) component (E v0 )n may be nonzero at xn = 0. Now, introducing the new unknown function vnew = vold − E v0
for (t, x) ∈ R+ × Rn+ ,
(3.5)
reduces our study of (1.6) to the case where the initial data is zero and the boundary data is v0 |xn =0 . v|xn =0 = vb := −E Extending the problem on the whole time line by setting v = 0 for t < 0 is the next step. The properties of (3.5) allow us to do it. However, we also have to get a suitable control over vb for proving Lemma 8 below. In our case, as vbτ = 0, we only have to worry about the nth component. So let us consider the solution w to the auxiliary problem
894
R. Danchin, P.B. Mucha / Journal of Functional Analysis 256 (2009) 881–927
∂t w − w = 0 in R+ × Rn+ , w|xn =0 = 0
on R+ × Rn−1 ,
w|t=0 = v0n
on Rn+ .
(3.6)
s+2 s (Rn )) ∩ L (R ; B n From Proposition 6, we get w ∈ Cb (R+ ; B˙ p,1 1 + ˙ p,1 (R+ )) and +
w L∞ (R;B˙ s
n p,1 (R+ ))
+ ∂t w, ∇ 2 w L
˙ s (Rn+ )) 1 (R;B p,1
C v0n B˙ s
n p,1 (R+ )
.
Then we set Evbn :=
w − (E v0 )n 0
for (t, x) ∈ R+ × Rn+ , for (t, x) ∈ R− × Rn+ .
(3.7)
As its trace on t = 0 is zero, function Evbn satisfies ∂t Evbn − Evbn = 0 in R × Rn+ , −E v0 |xn =0 for t > 0, on R × Rn−1 . Evbn |xn =0 = 0 for t 0,
(3.8)
In addition, as Evbn = 0 for negative times, Proposition 6 and inequality (3.3) guarantee that s (Rn )), ∂ Ev , ∇ 2 Ev ∈ L (R; B ˙ s (Rn+ )) and Evbn ∈ Cb (R; B˙ p,1 t bn bn 1 + p,1
Evbn L∞ (R;B˙ s
n p,1 (R+ ))
+ ∂t Evbn , ∇ 2 Evbn L
˙ s (Rn+ )) 1 (R;B p,1
C v0n B˙ s
n p,1 (R+ )
.
(3.9)
Note that the above inequality provides us with an information on the regularity of Evbn at the boundary, thus also on vb . Therefore one can now consider the following boundary value problem ∂t v − v + ∇P = F
in R × Rn+ ,
div v = 0 in R × Rn+ , v|xn =0 = vb
on R × Rn−1 ,
(3.10)
with boundary data vb given by vbτ ≡ 0,
vbn = Evbn |xn =0
on R × Rn−1 ,
(3.11)
and Evbn : R × Rn+ → R satisfying (3.9). Removing the potential part of F and the trace of its normal component at the boundary will be our next task. For that, formally, it suffices to solve the elliptic equation P = div F
in R × Rn+ ,
(∂xn P − Fn )|xn =0 = 0 on R × Rn−1 .
(3.12)
R. Danchin, P.B. Mucha / Journal of Functional Analysis 256 (2009) 881–927
895
Note that, by virtue of Lemma 3, the second line makes sense as soon as ∇P − F is in s (Rn )), since the first line ensures that div(∇P − F ) = 0. However, because ∂ P L1 (R; B˙ p,1 xn + and Fn need not have a trace on ∂Rn+ , two steps are required for solving system (3.12). So let us first consider the following Poisson equation: P1 = div F
in R × Rn+ ,
P1 |xn =0 = 0 on R × Rn−1 .
(3.13)
Applying Lemma 1 for all fixed t ∈ R then integrating over R yields
∇P1 L1 (R;B˙ s
n p,1 (R+ ))
C F L1 (R;B˙ s
n p,1 (R+ ))
(3.14)
.
s (Rn )) to that the nth compoBy construction, div(F − ∇P1 ) = 0 and (F − ∇P1 ) ∈ L1 (R; B˙ p,1 + n n := Fn −∂xn P1 has a trace on the boundary ∂R+ , which, according to Lemma 3 and (3.14) nent F satisfies
n |x =0
F n
s− 1
p L1 (R;B˙ p,1 (Rn−1 ))
C F − ∇P1 L1 (R;B˙ s
n p,1 (R+ ))
C F L1 (R;B˙ s
n p,1 (R+ ))
. (3.15)
As a second step for solving (3.12), we thus consider the following Neumann problem: P2 = 0 in R × Rn+ , n |xn =0 ∂xn P2 |xn =0 = F
on R × Rn−1 .
(3.16)
System (3.16) can be solved explicitly by applying the Fourier transform Fx with respect to the tangential space variables x := (x1 , . . . , xn−1 ). Denoting ξ := (ξ1 , . . . , ξn−1 ) the corresponding Fourier variables, we get P2 = Fx−1
−|ξ |xn 1 −e Fx [Fn |xn =0 ] . |ξ |
(3.17)
Taking advantage of Lemma 2 and of inequality (3.15), one can conclude that ∇P2 belongs to L1 (R; B˙ s (Rn+ )) and that p,1
∇P2 L1 (R;B˙ s
n p,1 (R+ ))
n |x =0
C F n
1 s− p
L1 (R;B˙ p,1 (Rn−1 ))
C F L1 (R;B˙ s
n p,1 (R+ ))
.
(3.18)
So finally, changing the divergence free source term F into F − ∇P1 − ∇P2 reduces the study of system (3.10) to the case where div F = 0 in Rn+
and Fn = 0 on ∂Rn+ .
(3.19)
896
R. Danchin, P.B. Mucha / Journal of Functional Analysis 256 (2009) 881–927
3.2. The boundary problem This step, which is the cornerstone of the proof of Theorem 2, is devoted to the study of system (3.10) under hypothesis (3.19). Our main result reads: s (Rn )) satisfy div F = 0 Lemma 6. Let 1 < p < ∞ and 1/p − 1 < s < 1/p. Let F ∈ L1 (R; B˙ p,1 + and Fn |xn =0 = 0 in the meaning of the trace, and let vb be given by (3.11). Then system (3.10) has a solution (v, P ) verifying
n s v ∈ Cb R; B˙ p,1 R+
n s and ∂t v, ∇ 2 v, ∇P ∈ L1 R; B˙ p,1 R+ .
(3.20)
In addition, we have for some constant C depending only on n, p and s,
v L∞ (R;B˙ s (Rn )) + ∂t v, ∇ 2 v, ∇P L (R;B˙ s (Rn )) 1 p,1 + p,1 + C F L1 (R;B˙ s (Rn )) + v0 B˙ s (Rn ) . p,1
+
p,1
+
(3.21)
Proof. Denoting (ξ0 , ξ ) ∈ R × Rn−1 the Fourier variables pertaining to (t, x ) and
e−itξ0 −ix ·ξ v(t, x , xn ) dx dt,
u(ξ0 , ξ, xn ) = Ft,x [v] = R Rn−1
e−itξ0 −ix ·ξ P (t, x , xn ) dx dt,
q(ξ0 , ξ, xn ) = Ft,x [P ] = R Rn−1
e−itξ0 −ix ·ξ F (t, x , xn ) dx dt,
f (ξ0 , ξ, xn ) = Ft,x [F ] = R Rn−1
system (3.10) reduces to the following ordinary differential system: r 2 uτ − ∂x2n uτ = fτ − iξ q
in R × Rn−1 × (0, ∞),
r 2 un − ∂x2n un = fn − ∂xn q
in R × Rn−1 × (0, ∞),
iξ · uτ + ∂xn un = 0
in R × Rn−1 × (0, ∞),
u|xn =0 = ub
on R × Rn−1 ,
(3.22)
with r 2 = iξ0 + |ξ |2 chosen so that3 arg r ∈ [− π4 , π4 ] and u = (uτ , un ),
f = (fτ , fn ),
where uτ = (u1 , . . . , un−1 ) and fτ = (f1 , . . . , fn−1 ).
System (3.22) may be solved explicitly. Indeed, taking the divergence of (3.10)1 and knowing that div F = 0, we discover that 3 In other words, r := (
ξ02 +|ξ |4 +|ξ |2 1 ) 2 + i sgn(ξ0 )( 2
ξ02 +|ξ |4 −|ξ |2 1 )2 . 2
R. Danchin, P.B. Mucha / Journal of Functional Analysis 256 (2009) 881–927
|ξ |2 q − ∂x2n q = 0,
897
(3.23)
so that, because we want q to tend to 0 for xn → ∞, we get q(ξ0 , ξ, xn ) = q0 (ξ0 , ξ )e−|ξ |xn
for an unknown function q0 .
(3.24)
The first step is to determine q0 and its properties. The theory of ordinary differential equations gives us the following explicit formulae for the solutions to (3.22) — see [15]: uτ (ξ0 , ξ, xn ) = ubτ e
−rxn
1 + 2r
∞ −r|x −s | n n − e −r(xn +sn ) f (ξ , ξ, s ) − iξ e −|ξ |sn q e τ 0 n 0 dsn , 0
∞ 1 −r|xn −sn | −rxn un (ξ0 , ξ, xn ) = ubn e e + − e−r(xn +sn ) fn (ξ0 , ξ, sn ) + |ξ |e−|ξ |sn q0 dsn . 2r 0
In our case ubτ is zero and only ubn may be nontrivial — see (3.11). Therefore, taking the derivative of the second equality with respect to xn , we get ∂xn un = −rubn e
−rxn
1 + 2
∞ −r|x −s | n n sgn(s − x ) + e −r(xn +sn ) f + |ξ |e −|ξ |sn q e n n n 0 dsn . 0
Letting xn → 0, we find that ∞ ∂xn un |xn =0 = −rubn +
e−rsn fn (sn ) + |ξ |e−|ξ |sn q0 dsn ,
0
whence ∞ ∂xn un |xn =0 = −rubn +
e−rsn fn (sn ) dsn +
0
|ξ | q0 . r + |ξ |
(3.25)
Note that our assumption on Fn implies that fn |xn =0 = 0 in the meaning of the trace. Therefore, owing to div F = 0, one may write ∞ −r
e
−rsn
∞ fn dsn =
0
∂sn e−rsn fn dsn = −
0
∞ e
−rsn
0
∞ ∂sn fn dsn =
e−rsn iξ · fτ dsn .
0
As ∂xn un = −iξ · uτ is zero at the boundary, we thus get q0 =
∞ iξ0 iξ + r + |ξ | ubn + e−rsn · fτ − fn dsn . |ξ | |ξ | 0
(3.26)
898
R. Danchin, P.B. Mucha / Journal of Functional Analysis 256 (2009) 881–927
Now we want to investigate the properties of regularity of q0 . Together with (3.24) this will enable us to compute the pressure, and to prove that its gradient satisfies (3.21). For this purpose we need two results. The first one will enable us to handle the second term in (3.26): s (Rn )) and h := Ft,x h. Then Lemma 7. Let h ∈ L1 (R; B˙ p,1 +
−1 := Ft,x
H
∞
e
−rsn
(3.27)
h(ξ0 , ξ, sn ) dsn
0
∈ L1 (R; B˙ s+1 (Rn+ )) on R × Rn+ such that admits an extension H p,1 |xn =0 = H H
and H L1 (R;B˙ s+1 (Rn )) C h L1 (R;B˙ s p,1
n p,1 (R+ ))
+
.
(3.28)
, let us consider the following heat equation: Proof. In order to construct H ∂t v − v = h v|xn =0 = 0
in R × Rn+ ,
on R × Rn−1 .
(3.29)
Arguing as for solving (3.10), we obtain the explicit formula: v
−1 = Ft,x
1 2r
∞ −r|x −s | −r(x +s ) n n −e n n e h(ξ0 , ξ, sn ) dsn . 0
Differentiating the above formulation with respect to xn , and taking xn = 0, we get
∂xn v|xn =0 = H
−1 := Ft,x
∞
e
−rsn
h dsn ,
0
:= ∂xn v has the required properties.4 so that Proposition 6 ensures that H
2
The second result concerns the terms of (3.26) related to the boundary data: −1 iξ0 ˙ Lemma 8. Let vb be given by (3.11). Then Ft,x [ |ξ | ubn ] ∈ L1 (R; Bp,1
s+1−1/p
−1 iξ0 F u t,x |ξ | bn
s+1−1/p L1 (R;B˙ p,1 (Rn−1 ))
C v0 B˙ s
n p,1 (R+ )
(Rn−1 )) and
.
4 In fact, we have to consider first the case where h ≡ 0 on (−∞, T ), and next have T tend to −∞.
(3.30)
R. Danchin, P.B. Mucha / Journal of Functional Analysis 256 (2009) 881–927
899
−1 ˙ s+1 n Furthermore, if g = Ft,x [(r + |ξ |)ubn ] then there exists an extension G ∈ L1 (R; Bp,1 (R+ )) such that G|xn =0 = g and
G L1 (R;B˙ s+1 (Rn )) C v0 B˙ s p,1
n p,1 (R+ )
+
(3.31)
.
Proof. Let us first prove (3.31). Remind that v := Evbn (see (3.8)) satisfies ∂t v − v = 0 in R × Rn+ , v|xn =0 = vbn
in R × Rn−1 .
(3.32)
s+1 Therefore we have Ft,x [v] = ubn e−rxn and, according to (3.9), ∇v ∈ L1 (R; B˙ p,1 (Rn+ )). Hence it follows that F −1 [|ξ |ubn e−rxn ] ∈ L1 (R; B˙ s+1 (Rn+ )) and that p,1
t,x
−1 F |ξ |ubn e−rxn
s+1 n L1 (R;B˙ p,1 (R+ ))
t,x
C v0n B˙ s
n p,1 (R+ )
.
(3.33)
−1 −rxn u ]. Hence F −1 [re−rxn u ] ∈ L (R; B ˙ s+1 (Rn+ )) Next, we notice that ∂xn v = −Ft,x [re bn bn 1 p,1 t,x and satisfies (3.33) too. Therefore, one may construct an extension G satisfying (3.31). −1 iξ0 Formally, the most difficult term, Ft,x [ |ξ | ubn ], may be viewed as the trace of ∂t v at xn = 0 s (Rn )) together divided by |ξ |. Since (3.32) is satisfied, we already know that ∂t v ∈ L1 (R; B˙ p,1 + with a suitable estimate. In order to show that the trace at xn = 0 of ∂t v is well defined, we plan to express ∂t v as the nth component of a convenient divergence free vector field with coefficients s (Rn )). Then applying Lemma 3 will enable us to get (3.31). in L1 (R; B˙ p,1 + So, for k = 1, . . . , n − 1, let us consider the following system:
∂t w − w = 0 ∂xn w|xn =0 = 0
in R+ × Rn+ , on R+ × Rn−1 ,
w|t=0 = v0k
on Rn+ .
By combining symmetric extension and Proposition 5, it is not difficult to see that the above system has a solution wk with the usual properties of regularity and satisfying in particular 2 ∇ w k
s (Rn )) L1 (R+ ;B˙ p,1 +
C v0 B˙ s
n p,1 (R+ )
.
Now, if we denote by wn the solution to (3.6), the vector field V := (w1 , . . . , wn ) is divergence free — because V |t=0 = v0 on Rn+ — and ∂t V , ∇ 2 V
s (Rn )) L1 (R+ ;B˙ p,1 +
C v0 B˙ s
n p,1 (R+ )
.
s (Rn )), Lemma 3 guarOf course, we also have div ∂t V = 0. Hence, as ∂t V is in L1 (R+ ; B˙ p,1 + s−1/p n ˙ antees that ∂t wn |xn =0 ∈ L1 (R+ ; Bp,1 (R+ )) with a suitable inequality. Finally, we notice that v0 )n , that div(∂t E v0 ) = 0 and that ∂t E v0 ∈ L1 (R+ ; B˙ s (Rn+ )). So applying ∂t v = ∂t wn − ∂t (E p,1
again Lemma 3 yields the desired bound for ((∂t E v0 )n )|xn =0 and thus for (∂t v)|xn =0 . Now, it
900
R. Danchin, P.B. Mucha / Journal of Functional Analysis 256 (2009) 881–927
−1 iξ0 ˙ is easy to conclude that Ft,x [ |ξ | ub ] ∈ L1 (R; Bp,1 proved. 2
s+1−1/p
(Rn+ )) and satisfies (3.30). Lemma 8 is
3.3. The estimate for the pressure Here we complete the proof of Lemma 6 and Theorem 2. s+1 Lemma 9. Under the assumptions of Lemma 6 we have P ∈ L1 (R; B˙ p,1 (Rn+ )) and
∇P L1 (R;B˙ s
n p,1 (R+ ))
C F L1 (R;B˙ s
n p,1 (R+ ))
+ v0 B˙ s
n p,1 (R+ )
.
(3.34)
Proof. According to (3.24), we have Ft,x [P ] = q0 e−|ξ |xn , where q0 is given by (3.26). Split q0 into q0 = q1 + q2 with q2 = r + |ξ | ubn +
iξ0 ubn , q1 = |ξ |
∞
e−rs
0
iξ · fτ − fn ds. |ξ |
(3.35)
By virtue of Lemma 7 and of the first part of Lemma 8, one can find some function Q2 in s+1 −1 (Rn+ )) so that Q2 |xn =0 = Ft,x L1 (R; B˙ p,1 [q2 ] and
Q2 L1 (R;B˙ s+1 (Rn )) C F L1 (R;B˙ s p,1
n p,1 (R+ ))
+
+ v0 B˙ s
n p,1 (R+ )
.
(3.36)
−1 −|ξ |xn Next, combining Lemmas 2 and 8, we gather that the function Q1 := Ft,x [e q1 ] is in s+1 −1 n L1 (R; B˙ p,1 (R+ )), satisfies Q1 |xn =0 = Ft,x [q1 ] and Q1 = 0, and
Q1 L1 (R;B˙ s+1 (Rn )) C v0 B˙ s +
p,1
n p,1 (R+ )
(3.37)
.
So finally, because we want to have P = 0 (remind that div F = 0) the pressure can be sought s (Rn )) and satisfies in the form P = P1 + P2 where P1 := Q1 + Q2 (so that ∇P1 is in L1 (R; B˙ p,1 + the desired estimate), and P2 fulfills the system P2 = −Q2
in Rn+ ,
P2 |xn =0 = 0 on Rn−1 .
(3.38)
s (Rn )), hence the assumptions of According to (3.36) and (3.37), we have ∇Q2 ∈ L1 (R; B˙ p,1 + s (Rn )) Lemma 1 are fulfilled. Therefore Eq. (3.38) has a solution P2 such that ∇P2 ∈ L1 (R; B˙ p,1 + and
∇P2 L1 (R;B˙ s
n p,1 (R+ ))
C ∇Q2 L1 (R;B˙ s
n p,1 (R+ ))
C F L1 (R;B˙ s
Combining this with (3.36) completes the proof of estimate (3.34).
n p,1 (R+ ))
+ v0 B˙ s
n p,1 (R+ )
.
2
Armed with Lemma 9, one may now look at the original system (1.6) as a heat equation, namely
R. Danchin, P.B. Mucha / Journal of Functional Analysis 256 (2009) 881–927
∂t v − v = F − ∇P
901
in (0, T ) × Rn+ ,
v|xn =0 = 0 on (0, T ) × Rn−1 , v|t=0 = v0
on Rn+ .
(3.39)
Let us emphasize that the incompressibility condition div v = 0 is hidden in the construction of the pressure. In addition, Proposition 6 ensures that the solution v to the above system satisfies
v L∞ (0,T ;B˙ s (Rn )) + ∂t v, ∇ 2 v L (0,T ;B˙ s (Rn )) 1 p,1 + p,1 + C F − ∇P L1 (0,T ;B˙ s (Rn )) + v0 B˙ s (Rn ) . p,1
+
p,1
+
Since ∇P is bounded according to Lemma 9, Theorem 2 is proved. 0 (Rn )) 1 (R; B˙ p,∞ 3.4. The estimate in L +
As a consequence of the above analysis we readily get the following estimates which turn out to be the key to the proof of uniqueness in Theorem 1. Lemma 10. Let 1 < p < ∞ and s ∈ (−1 + 1/p, 1/p). Assume that the time-dependent vector s 1 (R; B˙ p,∞ field F has coefficients in L (Rn+ )). Then system (3.10) with null boundary data has a unique solution (v, P ) such that n n s s 1 R; B˙ p,∞ v ∈ L∞ R; B˙ p,∞ R+ and ∂t v, ∇ 2 v, ∇P ∈ L R+ .
(3.40)
In addition, 2
v L∞ (R;B˙ p,∞ s (Rn )) + ∂t v, ∇ v, ∇P L +
s ˙ p,∞ (Rn+ )) 1 (R;B
C F L s 1 (R;B˙ p,∞ (Rn )) . +
(3.41)
Proof. The proof follows from a direct application of the interpolation theory (see Proposition 7) to Theorem 2. 2 4. The nonlinear problem The previous section will enable us to solve the nonlinear system (INS) with the initial velocity in a critical Besov space of index 0. In order to prove our main existence result, Theorem 1, we shall proceed as follows: • first, we show the existence of solutions for data with more integrability; • second, we prove the uniqueness part of Theorem 1; • last, we tackle the proof of existence in the case of data with critical regularity. For that, we shall use the first step to construct smoother solutions pertaining to smoothed out data, then resort to compactness arguments. Notation. In this section, we shall only consider functions or distributions defined on the halfs instead of B s (Rn ). ˙ p,r space Rn+ so that, for notational simplicity, we shall write B˙ p,r +
902
R. Danchin, P.B. Mucha / Journal of Functional Analysis 256 (2009) 881–927
4.1. Smoother solutions This subsection is dedicated to the proof of existence of “smooth” global solutions. Our main result reads: Theorem 3. Let (ρ0 , u0 , f ) satisfy the assumptions of Theorem 1 for a small enough con0 and f ∈ L ˙ 0 ) for some stant c. Assume in addition that (ρ0 − 1) ∈ Wp1 , u0 ∈ B˙ p,1 1,loc (R+ ; B p,1 p ∈ (n, ∞). Then system (INS) has a unique global solution (ρ, u, ∇Π) ∈ En ∩ Ep,loc satisfying the inequalities of Theorem 1 and, for all t ∈ R+ , ∇ρ(t)
Lq
2 ∇ρ0 Lq
for q = n, p,
(4.1)
∇u L1 (0,t;L∞ ) log 2,
u L∞ (0,t;B˙ 0 ) + ∂t u, μ∇ 2 u, ∇Π L (0,t;B˙ 0 ) 1 p,1 p,1 C u0 B˙ 0 + f L1 (0,t;B˙ 0 ) + μ ∇ρ0 Lp p,1
(4.2)
(4.3)
p,1
with C depending only on n. Proof. As a first step, let us state a priori estimates in En ∩ Ep,loc for system (INS). So we assume that we are given a solution (ρ, u) ∈ En (T ) ∩ Ep (T ). We claim that if condition (1.3) is satisfied for some small enough constant c then estimates (1.4), (1.5), (4.1), (4.2) and (4.3) are true for all t ∈ [0, T ]. Let us first consider the density. Owing to the incompressibility condition and to the fact that u · n = 0 on ∂Rn+ , all the Lp norms of ρ are time independent and we obtain by a Gronwall type argument the following inequalities: ∇ρ(t)
Lq
t
e 0 ∇u L∞ dτ ∇ρ0 Lq for q ∈ {n, p}, 1 − ρ(t) = 1 − ρ0 L . ∞ L
(4.4)
∞
Therefore, in particular, 1 − ρ(t)
L∞ ∩W˙ n1
e
t 0
∇u L∞ dτ
1 − ρ0 L∞ ∩W˙ 1 .
(4.5)
n
In order to bound the velocity, we may apply Theorem 2 to the system ∂t u − μu + ∇Π = (1 − ρ)∂t u + ρ(f − u · ∇u), u|xn =0 = 0,
div u = 0,
u|t=0 = u0 .
(4.6)
We get for q ∈ {n, p}, Uq (t) C Uq (0) + (1 − ρ)∂t uL
˙0 ) 1 (0,t;B q,1
+ ρ(f − u · ∇u)L
˙0 ) 1 (0,t;B q,1
R. Danchin, P.B. Mucha / Journal of Functional Analysis 256 (2009) 881–927
903
with Uq (0) := u0 B˙ 0 and q,1
Uq (t) := u L∞ (0,t;B˙ 0
q,1 )
+ ∂t u L1 (0,t;B˙ 0
q,1 )
+ μ ∇ 2 u L1 (0,t;B˙ 0
q,1 )
+ ∇Π L1 (0,t;B˙ 0 ) . q,1
For bounding the right-hand side of Un (t) one may use Proposition 4 from Section 2, which gives 1 1 B˙ n,1
→ L∞ ∩ W˙ n1 → L∞ ∩ B˙ n,∞ ,
and Lemma 4. We find that (1 − ρ)∂t u 1 ∂t u L (0,t;B 0 ) C ρ − 1 L (0,t;L ∩W ˙ 0 ), L1 (0,t;B˙ n,1 ∞ ∞ ˙n) 1 n,1 ρ(f − u · ∇u) C 1 + ρ − 1 L∞ (0,t;L∞ ∩W˙ 1 ) f − u · ∇u L1 (0,t;B˙ 0 ) . L (0,t;B˙ 0 ) 1
n
n,1
n,1
Since Lemma 4 also ensures that
u · ∇u L1 (0,t;B˙ 0
n,1 )
C u L∞ (0,t;B˙ 0 ) ∇u L1 (0,t;B˙ 1
n,1 )
n,1
(4.7)
we end up with Un (t) C Un (0) + ρ − 1 L∞ (0,t;L∞ ∩W˙ 1 ) Un (t) n + 1 + ρ − 1 L∞ (0,t;L∞ ∩W˙ 1 ) f L1 (0,t;B˙ 0
n,1 )
n
+ μ−1 Un2 (t) .
Therefore, there exist two positive constants c and M depending only on n such that if
ρ − 1 L∞ (0,T ;L∞ ∩W˙ 1 ) c n
and Un (T ) cμ
then the above inequality implies that Un (t) M Un (0) + f L1 (0,t;B˙ 0 ) n,1
for all t ∈ [0, T ].
(4.8)
Obviously, inequality (4.5) implies that the smallness condition for ρ − 1 is satisfied on [0, T ] provided
ρ0 − 1 L∞ ∩W˙ 1 c/2 and ∇u L1 (0,T ;L∞ ) log 2. n
(4.9)
1 (Rn ) → L (Rn ) (see Proposition 4), the latter condition is satisfied provided Because B˙ n,1 ∞ + + Un (T ) cμ with c small enough. As the function t → Un (t) is continuous, combining inequality (4.8) with a standard bootstrap argument enables us to conclude that (4.8) and the second part of (4.9) are true for all t ∈ [0, T ] provided u0 B˙ 0 + f L1 (R+ ;B˙ 0 ) < cμ for a small enough n,1 n,1 positive constant c.
904
R. Danchin, P.B. Mucha / Journal of Functional Analysis 256 (2009) 881–927
Let us now bound Up (t). By virtue of Lemma 4 and inequalities (4.4), (4.9), one can write (1 − ρ)∂t u L
˙0 ) 1 (0,t;B p,1
C ∇ρ0 Lp ∂t u L1 (0,t;B˙ 0
n,1 )
ρ(f − u · ∇u) L
˙0 ) 1 (0,t;B p,1
+ 1 − ρ0 L∞ ∂t u L1 (0,t;B˙ 0
C 1 + ρ0 − 1 L∞ f − u · ∇u L1 (0,t;B˙ 0 ) p,1 + ∇ρ0 Lp f − u · ∇u L1 (0,t;B˙ 0 ) .
p,1 )
,
n,1
1 (Rn ) → L ∩ W ˙ n1 (Rn+ ) and B˙ 1 (Rn+ ) → W˙ p1 (Rn+ ), Lemma 4 also yields Now, since B˙ n,1 ∞ + p,1
u · ∇u L1 (0,t;B˙ 0
p,1 )
C u L∞ (0,t;B˙ 0 ) ∇u L1 (0,t;B˙ 1
p,1 )
n,1
Cμ
−1
+ u L∞ (0,t;B˙ 0 ) ∇u L1 (0,t;B˙ 1
n,1 )
p,1
Un (t)Up (t).
Using also (4.7), we finally find that Up (t) C Up (0) + Un (t) ∇ρ0 Lp + ρ0 − 1 L∞ Up (t) + μ−1 Un (t)Up (t) 1 + 1 − ρ0 L∞ + μ−1 Un2 (t) ∇ρ0 Lp + 1 + 1 − ρ0 L∞ f L1 (0,t;B˙ 0 ) + ∇ρ0 Lp f L1 (0,t;B˙ 0 ) . p,1
n,1
Because 1 − ρ0 L∞ + μ−1 Un (t) is small, one can deduce (use (4.8)) that Up (t) C Up (0) + f L1 (0,t;B˙ 0
p,1 )
+ ∇ρ0 Lp u0 B˙ 0 + f L1 (0,t;B˙ 0 ) , n,1
n,1
(4.10)
which implies inequality (4.3). Remark 6. Let us stress the fact that since the constant c may be computed from the constant involved in Lemma 4, it is independent of p ∈ (n, ∞). The proof of the existence part of Theorem 3 unfolds as follows: • • • •
first, we construct a sequence of approximate solutions by solving a linear system; second, we prove uniform bounds for those approximate solutions; third, we prove convergence in low norm; last, we check that the limit is indeed a solution to (INS), and satisfies the required properties of regularity.
Throughout, we assume that condition (1.3) is satisfied and that, in addition, ρ0 − 1 belongs 0 and f ∈ L ˙ 0 ) for some p ∈ (n, ∞). to Wp1 , u0 is in B˙ p,1 1,loc (R+ ; B p,1 1. Construction of a sequence of approximate solutions. Starting from (ρ 0 , u0 ) := (1, 0), one can solve inductively the following system of linear PDEs:
R. Danchin, P.B. Mucha / Journal of Functional Analysis 256 (2009) 881–927
∂t ρ l+1 + ul · ∇ρ l+1 = 0 in (0, T ) × Rn+ , ∂t ul+1 − μul+1 + ∇Π l+1 = ρ l+1 f − ul · ∇ul + 1 − ρ l+1 ∂t ul
905
in (0, T ) × Rn+ ,
div ul+1 = 0 in (0, T ) × Rn+ , ul+1 |xn =0 = 0 ul+1 |t=0 = u0 ,
on (0, T ) × Rn−1 , ρ l+1 |t=0 = ρ0
on Rn+ .
(4.11)
Having ul in En implies that ∇ul ∈ L1 (R+ ; L∞ ) (see Proposition 4). Therefore the classical theory for transport equations provides a solution ρ l+1 to (4.11)1 . Next, Theorem 2 enables us to solve the Stokes system (4.11)2,3,4,5 . Then an easy induction ensures that for all l ∈ N, the above system has a global solution (ρ l+1 , ul+1 , ∇Π l+1 ) with 0 0 ul+1 ∈ C R+ ; B˙ n,1 ∩ B˙ p,1
l+1 ρ − 1 ∈ C R+ ; Wn1 ∩ Wp1 ,
0 0 and ∂t ul+1 , ∇ 2 ul+1 , ∇Π l+1 ∈ L1,loc R+ ; B˙ n,1 . ∩ B˙ p,1
2. Uniform bounds in En ∩ Ep,loc . Arguing exactly as in the first part of the proof, it is easy to see that if the smallness condition (1.3) is satisfied then (ρ l , ul , ∇Π l )l∈N is bounded in En ∩ Ep,loc . Besides inequality (4.4) is satisfied and l u
C u0 B˙ 0 + f L1 (R+ ;B˙ 0 ) , n,1 n,1 l u for all T > 0. C u0 B˙ 0 + f L1 (0,T ;B˙ 0 ) + μ ∇ρ0 Lp E (T )
(4.12)
En
p
p,1
(4.13)
p,1
3. Convergence in low norm. In order to complete the proof of existence, one has to show that (ρ l , ul , ∇Π l )l∈N converges to some function (ρ, u, ∇Π) with the required properties of regularity. Owing to the hyperbolic nature of the equation for the density, it is not clear that convergence may be proved in the space En ∩ Ep,loc . Therefore, we shall show that convergence holds true in a larger space. More precisely, we claim that for all T0 > 0, l ρ − 1 l1
is a Cauchy sequence in C [0, T0 ]; Lp , l 1 0, T0 ; B˙ 2p u − u1 l1 is a Cauchy sequence in L∞ 0, T0 ; B˙ 0p ,∞ ∩ L , ,∞ 2 2 l 1 0, T0 ; B˙ 1p Π − Π 1 l1 is a Cauchy sequence in L . ,∞
and
2
In all that follows, we fix some positive time T0 . Let us first consider the density. Denoting l , δul ) := (ρ l+m − ρ l , ul+m − ul ), we see that δρ l satisfies (δρm m m l+1 l+1 + ul+m · ∇δρm = −δulm · ∇ρ l , ∂t δρm
whence, because div ul+m = 0, l+1 δρ (t) m
Lp
t
l δu
m L∞
0
l ∇ρ
Lp
dτ.
906
R. Danchin, P.B. Mucha / Journal of Functional Analysis 256 (2009) 881–927
Because (∇ρ l )l∈N is bounded according to (4.1), we thus have for all t ∈ R+ , l+1 δρ (t) m
t Lp
2 ∇ρ0 Lp
l δu (τ ) dτ. m L
(4.14)
∞
0
Next, we notice that l+1 l+1 l ∂t δul+1 m − μδum + ∇δΠm = Rm
with δΠml+1 := Π l+1+m − Π l+1
and l l+1 Rm f − ∂t ul − ul · ∇ul − ρ l+1+m δulm · ∇ul + ul+m · ∇δulm := −δρm + 1 − ρ l+1+m ∂t δulm . By virtue of Lemma 10 from Section 3, for proving that, for all l 1, we have ⎧ l n 1 0, T0 ; B˙ 2p ⎨ u − u1 ∈ L∞ 0, T0 ; B˙ 0p ,∞ Rn+ ∩ L R+ , 2 ,∞ 2 l ⎩ ∇ Π − Π1 ∈ L 1 0, T0 ; B˙ 0p Rn+ , ,∞
(4.15)
2
1 belongs to L 1 (0, T0 ; B˙ 0p ). Actually, combining inequalities (4.12) it suffices to show that Rl−1 2 ,∞ and (4.13), Proposition 4 and interpolation, we discover that
• • • •
(1 − ρ l )l∈N is bounded in L∞ (0, T0 ; Lp ), (ul )l∈N is bounded in L∞ (0, T0 ; Lp ) ∩ L2 (0, T0 ; L∞ ), (∇ul )l∈N is bounded in L2 (0, T0 ; Lp ), (∂t ul )l∈N is bounded in L1 (0, T0 ; Lp ).
1 1 (0, T0 ; B˙ 0p ) by virtue of the Hence Rl−1 belongs to L1 (0, T0 ; L p ) which is a subspace of L 2 2 ,∞ following chain of embedding:
1 0, t; B˙ 0p L1 (0, t; L p ) → L1 0, t; B˙ 0p ,∞ → L . ,∞ 2
2
Let δUml (t) := ∂t δulm , μ∇ 2 δulm , ∇δΠml L 1 (0,t;B˙ 0p
2 ,∞
)
tion 10 to the equation satisfied by δul+1 m , we see that
+ δulm L∞ (0,t;B˙ 0p
2 ,∞
l l l (t) + Cm (t) + Dm (t) δUml+1 (t) C Alm (t) + Bm with
(4.16)
2
).
Applying Proposi-
R. Danchin, P.B. Mucha / Journal of Functional Analysis 256 (2009) 881–927
l+1 f − ∂t ul − ul · ∇ul L Alm (t) := δρm
˙ 0p ) 1 (0,t;B 2 ,∞
l Bm (t) := ρ l+1+m δulm · ∇ul L
˙ 0p ) 1 (0,t;B 2 ,∞
907
,
,
l Cm (t) := ρ l+1+m ul+m · ∇δulm L
˙ 0p ) 1 (0,t;B 2 ,∞
,
˙ 0p ) 1 (0,t;B 2 ,∞
.
l Dm (t) := 1 − ρ l+1+m ∂t δulm L
Bounding Alm is easy. Indeed, thanks to (4.16) and Hölder’s inequality, we have t Alm (t) C
l+1 δρ m
Lp
f − ∂t ul − ul · ∇ul dτ. L
(4.17)
p
0
Next, by virtue of Lemma 5 and Proposition 4, there exists a constant C depending continuously on (p, n) such that t l Bm (t) C
l+1+m l ρ ∇u ˙ 1
Wn ∩L∞
l δu ˙ 0
m Bp
dτ.
2 ,∞
0
The uniform bounds of the previous step and the smallness condition (1.3) guarantee that l Bm (t) Ccδulm L
˙ 0p ) ∞ (0,t;B 2 ,∞
(4.18)
.
Similarly, Lemma 5 and Proposition 4 yield l Cm (t) C ρ l+1+m ul+m L
˙ n1 ∩L∞ ) 2 (0,t;W
∇δul
m L2 (0,t;B˙ 0p
2 ,∞
)
.
Lemma 4 combined with an obvious embedding yields l+1+m l+m ρ u
L2 (0,t;W˙ n1 ∩L∞ )
C ρ l+1+m L
˙ n1 ) ∞ (0,t;L∞ ∩W
l+m u
1 ). L2 (0,t;B˙ n,1
Hence, using condition (1.3), inequality (4.12) and interpolation, we gather that 1 l Cm (t) Ccμ 2 δulm L
˙ 1p ) 2 (0,t;B 2 ,∞
Cc δUml (t).
(4.19)
Similar arguments lead to Dlt (t) 1 − ρ l+1+m L
˙ n1 ) ∞ (0,t;L∞ ∩W
∂t δul
m L1 (0,t;B˙ 0p
2 ,∞
)
Cc∂t δulm L
˙ 0p ) 1 (0,t;B 2 ,∞
. (4.20)
So finally, putting together inequalities (4.17)–(4.20) and taking c smaller if needed, we get
908
R. Danchin, P.B. Mucha / Journal of Functional Analysis 256 (2009) 881–927
1 l δU (t) + C 2 m
δUml+1 (t)
t
f − ∂t ul − ul · ∇ul (τ ) δρ l+1 (τ ) dτ m L L p
p
0
for all t ∈ [0, T0 ]. Plugging inequality (4.14) in the above inequality, we end up with 1 δUml+1 (t) δUml (t) 2 t + C ∇ρ0 Lp
f − ∂t ul − ul · ∇ul (τ ) δul m L L
1 (0,τ ;L∞ )
p
dτ.
(4.21)
0
Let us admit for a while that there exists some constant C such that l δu
m Lp/n (0,T ;L∞ )
CδUml (T )
for all T ∈ [0, T0 ].
(4.22)
Then inserting inequality (4.22) in inequality (4.21), we see that for all T ∈ [0, T0 ], 1 1− n δUml+1 (T ) δUml (T ) + CT p ∇ρ0 Lp f − ∂t ul − ul · ∇ul L (0,T ;L ) δUml (T ). p 1 2 Now, because f − ∂t ul − ul · ∇ul is bounded in L1 (0, T0 ; Lp ), we see that if T has been chosen small enough then 3 δUml+1 (T ) δUml (T ), 4 so that (ul − u1 )l∈N and (Π l − Π 1 )l∈N are Cauchy sequences in the desired space restricted to interval [0, T ]. Let us emphasize that, according to inequality (4.10), the smallness of T depends only on the magnitude of the data and on T0 . Therefore, starting from time T , the above arguments can be used again to show that (ul − u1 )l∈N and (Π l − Π 1 )l∈N are also Cauchy sequences in the desired space restricted to interval [T , 2T ], and so on, until the whole interval [0, T0 ] is exhausted. l In order to justify inequality (4.22), one may use the fact that z := δum satisfies z = k<0 k z + k0 k z. Hence
z Lp/n (0,T ;L∞ )
k z Lp/n (0,T ;L∞ ) +
k<0
k z Lp/n (0,T ;L∞ )
k0
2
− 2kn p
k z Lp/n (0,T ;L∞ ) +
k<0
T
n p
n
1− n
k z Lp 1 (0,T ;L∞ ) k z L∞p(0,T ;L∞ )
k0
k<0
+
2
− 2kn p
k z L∞ (0,T ;L∞ )
k(2− 2n ) n − 2nk 1− n p 2 p p z
p z
2 k L1 (0,T ;L∞ ) k L∞ (0,T ;L∞ ) k0
R. Danchin, P.B. Mucha / Journal of Functional Analysis 256 (2009) 881–927 n
T p z
− 2n p L∞ (0,T ;B˙ ∞,∞ )
+ z
1− pn
909
n
− 2n p L∞ (0,T ;B˙ ∞,∞ )
z p
2− 2n p
L1 (0,T ;B˙ ∞,∞ )
.
Now, Proposition 4 ensures that 2− 2n − 2n p p ˙ 0p ˙ ∞,∞ 1 0, T ; B˙ 2p 1 0, T ; B˙ ∞,∞ L
→ L and L 0, T ; B
→ L 0, T ; B . ∞ ∞ ,∞ ,∞ 2
2
So we get inequality (4.22). 4. Regularity of the solution. Let ρ be the limit of (ρ l )l∈N , and u := u + u1 (resp. Π := Π + Π 1 ) where u (resp. Π ) stands for the limit of (ul − u1 )l∈N (resp. (Π l − Π 1 )l∈N ). Interpolating the bounds of step 2 with the results of convergence of step 3, it is easy to check that (ρ, u, ∇Π) is a global solution to (INS). In addition, the compactness properties for the weak ∗ 0 ∩B ˙ 0 )) topology guarantee that ρ − 1 (resp. u) is in L∞ (R+ ; Wn1 ∩ Wp1 ) (resp. L∞ (R+ ; B˙ n,1 p,1 and satisfies the desired inequalities. 0 (Rn )), one can proceed as follows. First, we extend In order to show that ∇ 2 u ∈ L1 (R+ ; B˙ n,1 + 2 l 2 v l (resp. v ), we the terms ∇ u (resp. ∇ u) by 0 on the whole space. Denoting this extension by thus have according to Propositions 3 and 7, v1 → v − v1 vl −
n 1 R+ ; B˙ 0p R . in L ,∞
(4.23)
2
Introduce the spectral cut-off operator Ek := |j |k j . It is easy to check that Ek v l satisfies exactly the same inequalities as ∇ 2 ul (up to an irrelevant constant). In addition, owing to the spectral localization of Ek v l , the following Bernstein inequality5 l j v − v
Ln
C2
j ( 2n p −1)
l j v − v L p 2
associated with (4.23) implies that for fixed k ∈ N, n 0 R . v l → Ek v in L1 R+ ; B˙ n,1 Ek
(4.24)
On the other hand for all k, l ∈ N μ E k v l L
˙ 0 (Rn )) 1 (R+ ;B n,1
C u0 B˙ 0
n n,1 (R+ )
+ f L1 (R+ ;B˙ 0
n n,1 (R+ ))
with C independent of k and l. Therefore by (4.24) we have for all k ∈ N, μ Ek v L1 (R+ ;B˙ 0
n,1 (R
n ))
C u0 B˙ 0
n n,1 (R+ )
+ f L1 (R+ ;B˙ 0
n n,1 (R+ ))
.
0 (Rn ), and (4.25), one may write So finally, using the definition of the norm in B˙ n,1
5 Note that one can assume here that p/2 n taking p smaller in step 3 if need be.
(4.25)
910
R. Danchin, P.B. Mucha / Journal of Functional Analysis 256 (2009) 881–927
v L1 (R+ ;B˙ 0
n n,1 (R ))
= lim
k→+∞
l v L1 (R+ ;Lp ) C u0 B˙ 0
n n,1 (R+ )
|l|k
+ f L1 (R+ ;B˙ 0
n n,1 (R+ ))
.
0 (Rn )), and thus ∇ 2 u ∈ L (R ; B 0 n Hence, one can conclude that v ∈ L1 (R+ ; B˙ n,1 1 + ˙ n,1 (R+ )). 0 2 Of course, the same method works for proving that ∇ u ∈ L1,loc (R+ ; B˙ p,1 ) and that ∂t u and ∇Π have the desired regularity. Finally, because we have ∇u ∈ L1 (R+ ; L∞ ), continuity with respect to time for the density stems from standard properties for the transport equation. Continuity for the velocity follows from Theorem 2. Uniqueness may be justified from arguments similar to those which have been used for showing that (ρ l , ul , ∇Π l ) is a Cauchy sequence. We do not give any details since a more general result will be proved below in Proposition 8. 2
4.2. The proof of uniqueness The following statement implies uniqueness in Theorem 1. Proposition 8. Let (ρ1 , u1 , ∇Π1 ) and (ρ2 , u2 , ∇Π2 ) solve system (INS) on [0, T ] × Rn+ with the same data. Assume that n 3 and that for i = 1, 2 we have (ρi − 1) ∈ L∞ 0, T ; L∞ ∩ Wn1
2 0 and ui ∈ L1 0, T ; B˙ n,1 ∩ C [0, T ]; B˙ n,1 .
There exists a constant c = c(n) such that if in addition
ρ1 − 1 L∞ (0,T ×Rn+ ) + ∇ρ1 L∞ (0,T ;Ln ) c,
(4.26)
then (ρ2 , u2 , ∇Π2 ) ≡ (ρ1 , u1 , ∇Π1 ) on [0, T ] × Rn+ . Proof. The system satisfied by the difference (δρ, δu, δΠ) := (ρ2 − ρ1 , u2 − u1 , Π2 − Π1 ) between the two solutions reads ∂t δρ + u1 · ∇δρ = −δu · ∇ρ2 , ∂t δu − μδu + ∇δΠ = f − (∂t + u2 · ∇)u2 δρ − ρ1 u1 · ∇δu − δu · (ρ1 ∇u2 ) + (1 − ρ1 )∂t δu, div δu = 0.
(4.27)
Note that the right-hand side of (4.27)1 is at most in L∞ (0, T ; Ln ) no matter how smooth δu is. Therefore, we shall estimate δρ in L∞ (0, T ; Ln ). Owing to the coupling between the two equations, this will induce also a loss in the stability estimates for the velocity. For instance, by using 0 ), u ∈ L (0, T ; B ˙ 0 ), ∇u2 ∈ L1 (0, T ; B˙ 1 ) and that the fact that ∂t u2 ∈ L1 (0, T ; B˙ n,1 2 ∞ n,1 n,1 0
→ Ln B˙ n,1
1 and B˙ n,1
→ L∞ ,
R. Danchin, P.B. Mucha / Journal of Functional Analysis 256 (2009) 881–927
911
we readily see that (f − (∂t + u2 · ∇)u2 ) ∈ L1 (0, T ; Ln )6 so that the first term of (4.27)2 may be estimated in L1 (0, T ; L n2 ) if a bound on δρ L1 (0,t;Ln ) is available. However, the solution to the evolutionary Stokes equations with the right-hand side in L1 (0, T ; L n2 ) fails to have its first-order time derivative in L1 (0, T ; L n2 ). Actually, according to Lemma 10, it belongs to the 1 (0, T ; B˙ 0n ), that we shall use for bounding δu. slightly larger space L 2 ,∞ Let us now tackle the proof of uniqueness. Bounding δρ in L∞ (0, T ; Ln ) is straightforward. Indeed, since div u1 = 0, one can write for all t ∈ [0, T ]: δρ(t)
t
Ln
∇ρ2 Ln δu L∞ dτ ∇ρ2 L∞ (0,t;Ln ) δu L1 (0,t;L∞ ) .
(4.28)
0
1 (0, T ; B˙ 0n ). Let us now check that δu is in L∞ (0, T ; B˙ 0n ,∞ ) and that ∂t δu, ∇ 2 δu, ∇δΠ ∈ L 2 2 ,∞ We claim that the right-hand side of (4.27)2 belongs to L1 (0, T ; L n2 ) which is a subspace 1 (0, T ; B˙ 0n ) according to the following chain of embeddings: of L ,∞ 2
1 0, t; B˙ 0n L1 (0, t; L n2 ) → L1 0, t; B˙ 0n ,∞ → L ,∞ . 2
2
Since δu|t=0 = 0, Lemma 10 will entail that δu and ∇δΠ have the required regularity. We have already seen that (f −(∂t +u2 ·∇)u2 )δρ is in L1 ([0, T ]; L n2 ). Next, because ∇u1 and 0 ) (argue by interpolation) we gather that ∇δu is in L (0, T ; L ). Since ∇u2 are in L2 (0, T ; B˙ n,1 2 n ρ1 u1 ∈ L∞ (0, T ; Ln ) this implies that ρ1 u1 · ∇δu is in L2 (0, T ; L n2 ). Similar arguments yield ρ1 δu · ∇u2 ∈ L2 (0, T ; L n2 ). Finally, because (1 − ρ1 ) ∈ L∞ (0, T ; Ln ) and ∂t u1 ∈ L1 (0, T ; Ln ), the last term in the right-hand side of the equation for δu is also in L1 (0, T ; L n2 ). Now, bounding δu relies on Lemma 10. Denoting δU (t) := ∂t δu L 1 (0,t;B˙ 0n
2 ,∞
)
+ μ δu L 1 (0,t;B˙ 2n
2 ,∞
)
+ δu L∞ (0,t;B˙ 0n
2 ,∞
)
+ ∇δΠ L 1 (0,t;B˙ 0n
2 ,∞
),
we get for some constant C depending only on n, δU (t) C δU1 (t) + δU2 (t) + δU3 (t) + δU4 (t)
(4.29)
with δU1 (t) := δρ f − (∂t + u2 · ∇)u2 L δU3 (t) := δu · (ρ1 ∇u2 )L
˙ 0n ) 1 (0,t;B 2 ,∞
˙ 0n ) 1 (0,t;B 2 ,∞
,
,
δU2 (t) := ρ1 u1 · ∇δu L 1 (0,t;B˙ 0n
δU4 (t) := (1 − ρ1 )∂t δuL
˙ 0n ) 1 (0,t;B 2 ,∞
For bounding δU1 (t), we proceed as explained above. We get for all t ∈ [0, T ], 6 In fact, this term belongs to L (0, T ; B˙ 0 ) but this does not help in what follows. 1 n,1
2 ,∞
.
),
912
R. Danchin, P.B. Mucha / Journal of Functional Analysis 256 (2009) 881–927
t
δU1 (t)
L1 (0,t;B˙ 0n ,∞ )
C
2
f − (∂t + u2 · ∇)u2 δρ L dτ. n L n
0
Therefore, there exists some integrable function V over [0, T ] such that t δU1 (t)
V (τ )δρ(τ )L dτ.
(4.30)
n
0
In order to bound the other terms, one may resort to Lemma 5. Indeed, for δU2 (t), it suffices to apply this lemma with F = ∇δu, G = ρ1 u1 and r1 = r2 = 2. This is indeed possible since hav2 ) ∩ L (0, T ; B ˙ 0 ) implies (by interpolation) that u1 ∈ L2 (0, T ; B˙ 1 ), ing u1 ∈ L1 (0, T ; B˙ n,1 ∞ n,1 n,1 whence u1 ∈ L2 (0, T ; L∞ ∩ W˙ n1 ) by embedding. Now, ρ1 is in L∞ (0, T ; W˙ n1 ∩ L∞ ) and W˙ n1 ∩ L∞ is an algebra, so ρ1 u1 does belong to L2 (0, T ; W˙ n1 ∩ L∞ ). One can thus write that δU2 (t) A(t) ∇δu L 2 (0,t;B˙ 0n
2 ,∞
)
for some bounded function A : [0, T ] → R+ such that limt→0 A(t) = 0. From straightforward −1 interpolation arguments, we see that ∇δu L 2 (0,t;B˙ 0n ) μ 2 δU (t). Hence, up to a change of 2 ,∞
the function A,
δU2 (t) A(t)δU (t).
(4.31)
Similar arguments lead to δU3 (t) C ρ1 ∇u2 L1 (0,t;L∞ ∩W˙ 1 ) δu L∞ (0,t;B˙ 0n n
2 ,∞
),
whence, as ρ1 ∇u2 ∈ L1 (0, T ; L∞ ∩ Wn1 ), δU3 (t) A(t)δU (t).
(4.32)
Finally, we have δU4 (t) C 1 − ρ1 L∞ (0,t;L∞ ∩W˙ 1 ) ∂t δu L 1 (0,t;B˙ 0n n
2 ,∞
).
(4.33)
So plugging inequalities (4.30) to (4.33) in (4.29) and using the smallness condition (4.26) and the fact that limt→0 A(t) = 0, we get t δU (t)
V (τ )δρ(τ )L dτ n
0
for all t in a small enough interval [0, T0 ].
(4.34)
R. Danchin, P.B. Mucha / Journal of Functional Analysis 256 (2009) 881–927
913
At this stage, we are in trouble for bounding δρ through inequality (4.28) requires an L1 (0, t; L∞ ) control of δu which is not given by δU (t). In order to overcome this, one may use the following logarithmic interpolation estimate, proved in Appendix A (see Proposition 9):
δu L1 (0,t;Ln ) + ∇δu L1 (0,t;L∞ ) .
δu L1 (0,t;L∞ ) C δu L 0 1 (0,t;B˙ ∞,∞ ) log e +
δu L 0 1 (0,t;B˙ ∞,∞ )
(4.35)
0 ) ∩ L (0, T ; B ˙ 2 ), and to the embedding Let us notice that, due to ui ∈ L∞ (0, T ; B˙ n,1 1 n,1 0
→ Ln B˙ n,1
2 1 and B˙ n,1
→ W˙ ∞ ,
1 (0, t; B˙ 2n ) → L 1 (0, t; the numerator is a bounded function over [0, T ]. Remark also that L 2 ,∞ 0 ) (see Proposition 4). So finally, inserting inequality (4.35) in (4.28), using that for all B˙ ∞,∞ α > 0, the map r → r log(e + α/r) is nondecreasing, then coming back to (4.34), we conclude that for some positive constant C and integrable function V we have t δU (t)
V (τ )δU (τ ) log e +
C dτ. δU (τ )
0
Osgood’s lemma thus guarantees that δU ≡ 0 on [0, T0 ]. Note that the above proof works if we start from any time t0 ∈ [0, T ] such that u2 (t0 ) = u1 (t0 ). Therefore {t ∈ [0, T ]: u2 (t) = u1 (t)} is a nonempty open set of [0, T ]. Let us also notice that if (tn )n∈N is a sequence of [0, T ] such that u2 (tn ) = u1 (tn ) for all n ∈ N then, due to the fact that 1 , we also have u (t) = u (t). Hence the above set u1 and u2 are continuous with values in B˙ n,1 2 1 is also closed, and one can conclude that it equals [0, T ]. 2 4.3. The proof of existence in the critical case In order to prove the existence part of Theorem 1, we shall proceed as follows: • first, we solve system (INS) for mollified data (taking advantage of Theorem 3) and state uniform bounds; • second, we resort to compactness arguments in order to pass to the limit; • third, we check that the limit is indeed a solution; • last, we prove that it has the desired properties of regularity. 1. Uniform bounds for the solution with smoothed out data. Fix some p > n. From Proposition 3 and Remark 5, one can construct a sequence (ρ0l , ul0 , f l ) in7 1 0 n 0 n 0 0 Wn ∩ Wp1 × B˙ n,1 ∩ B˙ p,1 ∩ L1 R+ ; B˙ n,1 ∩ B˙ p,1 tending strongly to (ρ0 , u0 , f ) in 7 For the density, consider the symmetric extension and use the fact that the set of smooth compactly supported functions in Rn is dense in Wn1 (Rn ).
914
R. Danchin, P.B. Mucha / Journal of Functional Analysis 256 (2009) 881–927
n 0 n 0 Wn1 × B˙ n,1 ∩ L1 R+ ; B˙ n,1 . According to Theorem 3, system (INS) has a unique global solution (ρ l , ul , ∇Π l ) in En ∩ Ep,loc satisfying in addition for all t 0, l ρ (t) − 1
l ∇ρ (t) 2∇ρ l , = ρ0l − 1L , 0 L Ln ∞ n ul ul ˙ 0 + f l ∇Π l , ∂t ul , μ∇ 2 ul + M 0 0 ˙ ˙ 0 B L (0,t;B ) L (0,t;B ) L L∞
1
∞
n,1
n,1
n,1
˙0 ) 1 (0,t;B n,1
.
(4.36)
Note that all the terms of the right-hand sides may be bounded independently of l. Therefore (ρ l , ul , ∇Π l ) is bounded in the space En . 2. Compactness. We claim that sequence (ρ l , ul , ∇Π l )l∈N converges weakly (up to an omitted extraction) to some distribution (ρ, u, ∇Π). Compactness for the density stems from the fact that (∂t ρ l )l∈N is bounded in the space L2 (R+ ; Ln ). Indeed, we have ∂t ρ l = −ul · ∇ρ l and we know that (∇ρ l )l∈N is bounded in L∞ (R+ ; Ln ) and that (by interpolation and embedding) (ul )l∈N is bounded in L2 (R+ ; L∞ ). 1 Therefore (ρ l − 1)l∈N is bounded in C 2 ([0, T ]; Ln ) for all T > 0. Now, because (ρ l − 1)l∈N is 1 bounded in C(R+ ; Wn ) and the embedding of Wn1 in Ln is locally compact, Ascoli’s theorem combined with Cantor diagonal process ensures that, up to extraction, sequence (ρ l )l∈N tends to some function ρ in C(R+ ; Ln,loc ). From (4.36) and interpolation, one can thus conclude that (ρ − 1) ∈ L∞ R+ ; L∞ ∩ Wn1
and satisfies (1.4)
(4.37)
and that, up to an omitted extraction, lim ρ l = ρ
l→+∞
s for all s ∈ [0, 1). in L∞,loc R+ ; Wn,loc
(4.38)
Let us now turn to the study of the velocity. The important fact is that (4.36) implies that (ul )l∈N is bounded in L1,loc (R+ ; Wn2 ) and that (∂t ul )l∈N is bounded in L1,loc (R+ ; Ln ). This implies that 1 (R × Rn ) which is compactly embedded in L (ul )l∈N is bounded in the space W1,loc + p,loc (R+ × + n R+ ) for any p < (n + 1)/n. Hence we gather that, up to extraction, lim ul = u in Lp,loc R+ × Rn+ for any p < (n + 1)/n.
l→+∞
(4.39)
Of course, the divergence free condition is preserved. Finally, because (∇Π l )l∈N is bounded in L1 (R+ ; Ln ), there exists some distribution Q such that, up to extraction, (∇Π l )l∈N tends weakly to Q. Of course, Q is the gradient of some distribution Π. 3. Passing to the limit in (INS). Let us first show that (ρ, u, ∇Π) satisfies the conservative formulation of (INS).8 Passing to the limit in the linear terms is straightforward. In order to pass to the limit in the nonlinear terms, we shall first state some properties of strong convergence. 8 That is u · ∇ρ is replaced by div(ρu), and ρ(∂ u + u · ∇u), by ∂ (ρu) + div(ρu ⊗ u). t t
R. Danchin, P.B. Mucha / Journal of Functional Analysis 256 (2009) 881–927
915
From step 1 and interpolation, we know that (ul )l∈N is bounded in L∞ (R+ ; Ln ) ∩ L2 (R+ ; L∞ ), hence, using the weak ∗ compactness properties of those spaces, we gather that ul u weak ∗
in L∞ R+ ; Ln ∩ L2 R+ ; L∞ .
(4.40)
Interpolating with (4.39), we thus get strongly in Lp,loc R+ × Rn+ for some p > 2.
ul → u
(4.41)
Since the density converges strongly in L∞ (0, T ; Ln,loc ), this is enough to pass to the weak limit in the terms div(ρ l ul ), ∂t (ρ l ul ) and div(ρ l ul ⊗ ul ). Finally, the properties of regularity stated hitherto for (ρ, u) are enough to show that (ρ, u, ∇Π) satisfies (INS). For instance, having (ρ − 1) ∈ C(R+ ; Lp,loc ) for all p < ∞, ∂t ρ ∈ L2 (R+ ; Ln ) and (∂t ul )l∈N bounded in L1 (R+ ; Ln ) ensures that ∂t (ρu) = ρ∂t u + u∂t ρ in the sense of distributions. Similar computations may be done for the other nonlinear terms. 4. Regularity. The bounds of step 1 combined with interpolation guarantee that (ul )l∈N is 2 bounded in Lr (R+ ; B˙ r ) for all r 1. Using the Fatou properties for those spaces in the case n,1
2
r ). We now want to show that r > 1 ensures that u ∈ Lr (R+ ; B˙ n,1
0 ∇ 2 u is in L1 R+ ; B˙ n,1 and satisfies the bounds of step 1.
(4.42)
Starting from the fact that ∇ 2 ul tends weakly to ∇ 2 u and that functions ∇ 2 ul and ∇ 2 u are −2 in (say) L5/4 (R+ ; B˙ n,15 (Rn+ )), we extend ∇ 2 ul and ∇ 2 u by 0 on the whole space. Since v 1/n − 1 < −2/5 < 1/n, Proposition 3 guarantees that the corresponding extensions v l and 2 −5 n l v v in the weak sense. are in L5/4 (R+ ; B˙ n,1 (R )) and, obviously, we still have Now, using the spectral truncation operator Ek defined just above (4.23) and Bernstein inequality, one can assert that for any fixed k ∈ N, sequence (Ek v l )l∈N is bounded in 0 0 n n L5/4 (R+ ; B˙ n,1 (R )), thus also in L1,loc (R+ ; B˙ n,1 (R )), and tends to Ek v in the weak sense. 0 n ˙ v is actually in L1,loc (R+ ; Bn,1 (R )), one may write for all T > 0 and k ∈ N, As Ek T
T
Ek v B˙ 0
n dt n,1 (R )
lim inf
Bn,1 (Rn )
l→∞
0
Ek vl ˙ 0
dt,
0
whence, according to (4.36), T
Ek v (t) ˙ 0
Bn,1 (Rn )
dt C u0 B˙ 0
n n,1 (R+ )
+ f L1 (0,T ;B˙ 0
n n,1 (R+ ))
.
0 0 ), whence ∇ 2 u ∈ Arguing exactly as in the proof of Theorem 3, we then get v ∈ L1 (R+ ; B˙ n,1 0 ) (with the desired bound). L1 (R+ ; B˙ n,1
916
R. Danchin, P.B. Mucha / Journal of Functional Analysis 256 (2009) 881–927
Getting regularity for ∇Π and ∂t u is more involved. As a first step, let us fix some bounded interval I of R+ and let us prove that ∂t u
belongs to L1 (I ; Ln ).
(4.43)
In what follows we denote by Lp,σ the completion of the set of smooth compactly supported divergence free vector fields over Rn+ , for the Lp (Rn+ ) norm. Let us notice that (4.42) implies that ∇u ∈ L1 (I ; L∞ ). Because we also know that u ∈ L∞ (I ; Ln ) and ρ ∈ L∞ (I × Rn+ ), we deduce that ρu · ∇u ∈ L1 (I ; Ln ). Taking advantage of the momentum equation and of the fact that the Helmholtz projector PH onto divergence free vector fields maps Ln onto Ln,σ — see [21], we conclude that h := PH [ρ∂t u] ∈ L1 (I ; Ln ).
(4.44)
We claim that this implies that ∂t u itself is in L1 (I ; Ln ). For proving that, let us introduce the operator Φ : w → PH [ρw].
(4.45)
Lemma 11. Let p ∈ (1, ∞). There exists a constant c such that if
1 − ρ L∞ (I ×Rn+ ) c
(4.46)
then Φ is an invertible self-map on L1 (I ; Lp,σ ) and on Cb (I ; Lp,σ ). Proof. Obviously it suffices to consider the stationary case (viz. proving that Φ is an invertible self-map of Lp,σ if 1 − ρ L∞ (Rn+ ) c for a small enough c). First, it is clear that Φ is a linear bounded self-map on Lp,σ and that (4.46) implies that it is one-to-one provided c is sufficiently small. Next, a simple implementation of the Banach fixed point theorem guarantees the existence of solutions to the equation PH [ρw] = g
for arbitrary g ∈ Lp,σ ,
with the estimate w Lp 2 g Lp . Indeed for solving this equation, one may consider the following iterative scheme: w 0 := 0, Lemma 11 is proved.
w n+1 := g + PH (1 − ρ)w n .
2
We now plan to use the above lemma for showing that ∂t u ∈ L1 (I ; Ln ). We have already seen that PH (ρ∂t u) = h ∈ L1 (I ; Ln ). Hence it suffices to state that the distribution ∂t u coincides with the unique solution ∂t u in L1 (I ; Ln ) to the equation Φ(w) = h. For that, we are going to show that (∂t ul )l∈N tends weakly to ∂t u, or, in other words, that for all φ ∈ Cb (I ; Ln ,σ ) (with n = n/(n − 1)) we have
R. Danchin, P.B. Mucha / Journal of Functional Analysis 256 (2009) 881–927
lim
l→∞ I ×Rn+
l ∂t u − ∂t u · φ dx dt = 0.
917
(4.47)
Fix some φ ∈ Cb (I ; Ln ,σ ) and denote ψ := Φ −1 (φ). According to Lemma 11, we have ψ ∈ Cb (I ; Ln ,σ ). Now, using the definition of Φ and the fact that ∂t ul − ∂t u and ψ are divergence free, and that PH is symmetric, one may write
l ∂t u − ∂t u · φ dx dt =
I ×Rn+
l ∂t u − ∂t u · (ρψ) dx dt
I ×Rn+
l
∂t u − ∂t u · ψ + (ρ − 1) ∂t ul − ∂t u · PH ψ dx dt
= I ×Rn+
l
∂t u − ∂t u + PH (ρ − 1) ∂t ul − ∂t u · ψ dx dt.
= I ×Rn+
Because ul satisfies the momentum equation of (INS), we have ∂t ul − ∂t u + PH (ρ − 1) ∂t ul − ∂t u = hl − h + PH ρ − ρ l ∂t ul with hl = PH μul − ρ l ul · ∇ul . Hence
l ∂t u − ∂t u · φ dx dt =
I ×Rn+
I ×Rn+
l h − h · ψ dx dt +
PH ρ − ρ l ∂t ul · ψ dx dt.
I ×Rn+
The boundedness and convergence properties stated so far ensure that (hl )l∈N is bounded in L1 (I ; Ln ) and tends to h in the weak sense. As h ∈ L1 (I ; Ln ), this ensures that the first term of the right-hand side of the above equality tends to 0 as l goes to infinity. Next, combining (4.38) and the fact that (∂t ul )l∈N is bounded in L1 (I ; Ln ), we readily get that PH [(ρ − ρ l )∂t ul ] tends to zero in the sense of distributions. Because PH [(ρ − ρ l )∂t ul ] is also bounded in L1 (I ; Ln ) one can thus conclude that the second term of the above equality also tends to 0. This completes the proof of (4.47). Hence (4.43) is true for all bounded interval I of R+ . Knowing that ∂t u is in L1,loc (R+ ; Ln ), that ∂t ul tends weakly to ∂t u and that (4.36) is ful0 ). It suffices to use the spectral truncation filled, it is not difficult to show that ∂t u ∈ L1 (R+ ; B˙ n,1 operator Ek and to follow the proof of (4.42). Finally, from the momentum equation combined 0 ) and that u ∈ C (R ; B 0 with Lemma 4 and Theorem 2, we deduce that ∇Π ∈ L1 (R+ ; B˙ n,1 b + ˙ n,1 ). Theorem 1 is proved. Acknowledgments The second author has been supported by Polish KBN grant No. 1 P03A 021 30 and by ECFP6 M. Curie ToK program SPADE2, MTKD-CT-2004-014508 and SPB-M. The second author thanks Laboratoire Jacques-Louis Lions, the research was performed, for its hospitality.
918
R. Danchin, P.B. Mucha / Journal of Functional Analysis 256 (2009) 881–927
Appendix A Proof of Proposition 3. Let us denote for ε > 0 and t ∈ R,
ηε (t) :=
⎧ ⎨0 ⎩
2 εt
for t < ε, − 2 for ε t 3ε,
1
for t > 3ε.
We agree that η0 stands for the characteristic function of R+ and for all function u over Rn , we denote Φε (u) : x → ηε (xn )u(x). Let us admit for a while the following lemma: Lemma 12. For all 1 p < ∞, 1 q ∞ and −1 + 1/p < σ < 1/p the operator Φε maps σ (Rn ) in B σ (Rn ) uniformly with respect to ε 0. ˙ p,q B˙ p,q s (Rn ), we have Moreover, if q is finite then for all u ∈ B˙ p,q Φε (u) →ε→0 Φ0 (u)
n s R . in B˙ p,q
s (Rn ) (with 1 p, q < ∞ and 1/p − 1 < s < 1/p). Then u is the restriction to Let u ∈ B˙ p,q + s (Rn ). Now, the above lemma ensures that Φ ( Rn+ of some function u ∈ B˙ p,q ε u) is supported in n s R+ and tends to 1Rn+ u in B˙ p,q (Rn ). As 1Rn+ u coincides with the extension of u by 0, Proposition 3 is proved. 2
For the sake of completeness, let us now prove Lemma 12. σ (Rn ) in B σ (Rn ) with ˙ p,∞ Proof of Lemma 12. As a first step, let us state that Φε maps B˙ p,1 uniform bounds with respect to ε 0, whenever 0 < σ < 1/p and 1 p < ∞. Because σ ∈ (0, 1), one can use the definition of Besov norms in terms of finite differences (see Remark 4) to write that
Φε (u) ˙ σ
Bp,∞
(Rn )
n−1
ηε (xn ) u(x1 , . . . , xi + hi , . . . , xn ) − u(x1 , . . . , xn ) sup h−σ i L
p (R
i=1 hi >0
ηε (xn + hn )u(x , xn + hn ) − ηε (xn )u(x , xn ) + sup h−σ n L
p (R
hn >0
n)
n)
.
Because |ηε (xn )| 1 for all xn ∈ R, we have Φε (u) ˙ σ B
p,∞ (R
n)
u B˙ σ
p,∞ (R
n)
ηε (xn + hn ) u(x , xn + hn ) − u(x , xn ) + sup h−σ n L
p (R
hn >0
ηε (xn + hn ) − ηε (xn ) u(x , xn ) + sup h−σ n L hn >0
whence, according to Hölder inequality,
p (R
n)
n)
R. Danchin, P.B. Mucha / Journal of Functional Analysis 256 (2009) 881–927
Φε (u) ˙ σ
2 u B˙ σ
Bp,∞ (Rn )
p,∞ (R
919
n)
ηε (· + hn ) − ηε + sup h−σ n L
q ∗ (R)
hn >0
u Lp∗ (R;Lp (Rn−1 ))
(A.1)
with q ∗ = 1/σ and 1/p ∗ = 1/p − 1/q ∗ (a choice which is in accordance with the assumption that 0 < σ < 1/p). Let us bound the last term in (A.1). On the one hand, for hn 2ε one may write owing to the fact that ηε (· + hn ) − ηε is supported in [ε − hn , 3ε] and valued in [0, 1], 3ε
ηε (· + hn ) − ηε
Lq ∗ (R)
=
∗ ηε (t + hn ) − ηε (t) q dt
1/q ∗
1/q ∗
Chn
.
ε−hn
On the other hand, if hn < 2ε then one may split the integral over [ε − hn , 3ε] into integrals over [ε − hn , ε], [ε, 3ε − hn ] and [3ε − hn , 3ε]. As the first and last intervals have length hn , the 1/q ∗ corresponding integrals may be bounded by hn . As for the second integral, one may write
1/q ∗ 3ε−h 3ε−h q ∗ 1/q ∗ n n q ∗ 2h n ηε (t + hn ) − ηε (t) dt = dt ε ε
ε 1/q ∗
Chn
hn ε
1−
1 q∗
1/q ∗
Chn
.
Finally, let us notice that according to e.g. [35, Chapter 2.8] or [5, Chapter 18], we have n σ B˙ p,1 R → Lp∗ Rxn ; Lp Rn−1 . x Hence, putting together the above two inequalities and (A.1), one ends up with Φε (u) ˙ σ
Bp,∞ (Rn )
C u B˙ σ
p,1 (R
n)
uniformly in ε 0.
(A.2)
One can now deduce that for all 0 < σ < 1/p and 1 p, q ∞, the map Φε is (uniformly) σ (Rn ) to B σ (Rn ). Indeed, one may find δ > 0 so small as to satisfy 0 < δ < ˙ p,q bounded from B˙ p,q min{σ, p1 − σ }. Hence we have, uniformly with respect to ε 0, n σ +δ n σ +δ Φε : B˙ p,1 R → B˙ p,∞ R
n σ −δ n σ −δ and Φε : B˙ p,1 R → B˙ p,∞ R .
So we get the result by interpolation. Let us now state the properties of convergence of Φε (u) in the case 0 < σ < 1/p and 1 σ (Rn ) → B σ (Rn ) is ˙ p,q q < ∞. We already know that for all σ ∈ (σ, 1/p), operator Φε : B˙ p,q uniformly bounded. In addition, as p is finite, Lebesgue theorem ensures that for all function u in Lp (Rn ) then Φε (u) tends to Φ0 (u) in Lp (Rn ). These two properties combined with an σ (Rn ) we have Φ (u) → Φ (u) interpolation argument will enable us to prove that for all u ∈ B˙ p,q ε 0 σ n ˙ in Bp,q (R ). Indeed, fix σ ∈ (σ, 1/p) and set θ = σ/σ . For m ∈ N, denote
920
R. Danchin, P.B. Mucha / Journal of Functional Analysis 256 (2009) 881–927
u1m :=
u2m :=
j u,
j u and u3m :=
|j |m
j <−m
(A.3)
j u.
j >m
σ . Therefore, owing to the uniform bounds for Φ stated previNote that u2m belongs to Lp ∩ B˙ p,r ε ously and to an obvious interpolation inequality, one may write that Iε := Φε (u) − Φ0 (u) B˙ σ p,q satisfies
Iε Φε u1m − Φ0 u1m B˙ σ + Φε u2m − Φ0 u2m B˙ σ + Φε u3m − Φ0 u3m B˙ σ p,q
p,q
1−θ θ C u1m B˙ σ + Φε u2m − Φ0 u2m Lp Φε u2m − Φ0 u2m B˙ σ + u3m B˙ σ p,q
p,q
1−θ θ C u1m B˙ σ + Φε u2m − Φ0 u2m Lp u2m B˙ σ + u3m B˙ σ . p,q
p,q
Because q < ∞ the terms u1m B˙ σ
p,q
fixed m we have
u2m
∈ Lp
and u3m B˙ σ
p,q
p,q
p,q
p,q
tend to 0 when m goes to +∞. Next, for
so that 2 Φε u − Φ0 u2 m
m
→ 0.
Lp ε→0
σ . Putting those two results together, it is now easy to conclude that Φε (u) tends to Φ0 (u) in B˙ p,q
Remark 7. Note that the above convergence result holds true for q = ∞ if we assume in addition σ (Rn ) norm. Indeed, in this that u belongs to the completion of the Schwartz class of the Bp,∞ case the terms u1m B˙ σ and u3m B˙ σ defined in (A.3) tend to 0 when m goes to −∞ and +∞ p,∞ p,∞ respectively. In order to complete the proof, let us now focus on the case of negative index of regularity σ (Rn ) → B σ (Rn ) uniformly. The ˙ p,q (that is p1 − 1 < σ < 0). First, we want to prove that Φε : B˙ p,q σ n basic idea is that the negative space B˙ p,q (R ) can be represented as the dual (or pre-dual) of the n positive space B˙ p−σ ,q (R ) where p and q are the conjugate exponents of p and q (see e.g. [7] for the homogeneous framework). More precisely, one may write Φε (u) ˙ σ
Bp,q (Rn )
= sup
n ηε uf dx: f ∈ S0 R and f B˙ −σ (Rn ) 1 .
(A.4)
p ,q
Rn
However, as 0 < −σ < 1 −
1 p
=
1 p ,
−σ (Rn ) and that the previous steps ensure that ηε f ∈ B˙ p,q
ηε f B˙ −σ (Rn ) C f B˙ −σ (Rn ) . p ,q
p ,q
Therefore, one can conclude that Φε (u) ˙ σ
Bp,q (Rn )
C u B˙ σ
p,q (R
n)
.
(A.5)
R. Danchin, P.B. Mucha / Journal of Functional Analysis 256 (2009) 881–927
921
σ (Rn ). The case q = 1 A similar argument works for proving that Φε (u) tends to Φ0 (u) in B˙ p,q stems from Remark 7. Finally, the remaining case σ = 0 follows by interpolation. This completes the proof of the lemma. 2
Proof of Lemma 2. Let us first consider the case q = 1 and s ∈ (0, 1). We have to show that if s− 1
−|ξ |xn F h] satisfies h ∈ B˙ p,1p (Rn−1 ) then H := Fx−1 [e x
H B˙ s
n p,1 (R+ )
C h B˙ s−1/p (Rn−1 ) . p,1
Taking advantage of an interpolation argument (see [35, Section 2.5]), we can write n n−1 s s s R+ = B˙ p,1 R ; Lp (R+ ) ∩ B˙ p,1 R+ ; Lp Rn−1 . B˙ p,1
(A.6)
s (Rn−1 ; L (R )). For ξ ∈ Rn−1 , denote ϕ (ξ ) = φ(2−k |ξ |) and So let us first bound H in B˙ p,1 p + k −k (2 |ξ |) where φ is a smooth function on R+ , with support in {1/3 r 3} and ϕk (ξ ) = φ value 1 on a neighborhood of supp φ, and φ defined in (2.1). Denote by ( k )k∈Z the corresponding Littlewood–Paley decomposition on Rn−1 . By definition of the Besov norm, one can write that
H B˙ s
n−1 ;L (R )) p + p,1 (R
=
−|ξ |x nF h ϕk e 2sk Fx−1 x L
n p (R+ )
k∈Z
.
Now, it is not difficult to check that ξ → ϕk (ξ )e(2
k−2 −|ξ |)x n
(A.7)
is a multiplier with bounds independent of k ∈ Z and xn ∈ R+ . Therefore, the Marcinkiewicz theorem (see [16]) ensures that
H B˙ s
p,1 (R
n−1 ;L (R )) p +
C
k−2 2ks e−2 xn L
p (R+ )
k∈Z
C
2ks 2
k hL
− p1 k
p (R
h k
Lp (Rn−1 )
n−1 )
k∈Z
C h B˙ s−1/p (Rn−1 ) . p,1
s (R ; L (Rn−1 )). Because s ∈ (0, 1), the norm Next, we want to show that H ∈ B˙ p,1 + p
· B˙ s (R+ ;Lp (Rn−1 )) may be expressed in terms of finite differences of order one (see [35]): p,1
∞
H B˙ s
p,1 (R+ ;Lp (R
n−1 ))
=
dw H (·,·) − H (·, · +w) =: I. Lp (Rn+ ) 1+s w
0
To estimate the right-hand side of (A.8), we use the obvious inequality
(A.8)
922
R. Danchin, P.B. Mucha / Journal of Functional Analysis 256 (2009) 881–927
z Lp (Rn+ )
F −1 ϕ Fx z k
x
k∈Z
Lp (Rn+ )
.
Hence, using again the Marcinkiewicz theorem for the multiplier defined in (A.7), we discover that H (·,·) − H (·, · +w) L
n p (R+ )
C
−|ξ |x n − e −|ξ |(xn +w) F h F −1 ϕ e x
Lp (Rn+ )
k
x
k∈Z
C
2
− pk
Fx−1 ϕk 1 − e−|ξ |w Fx h L
p (R
n−1 )
.
k∈Z
Now, returning to (A.8) we get
∞
I C
k∈Z 0
dw − pk k 1 − e−2 w Fx−1 ψk ϕk Fx h L (Rn−1 ) 2 1+s p w
k (ξ ) with ψk (ξ ) = φ
1 − e−|ξ |w 1 − e−2
kw
.
Again, it turns out that ψk is a multiplier with bounds independent of k and w. So combining the Marcinkiewicz theorem and the change of variables u = 2k w, we get
∞
I C
k∈Z 0
du k(s− p1 ) 1 − e−u k hL (Rn−1 ) , 2 1+s p u
whence ∞ I C 0
1 − e−u du h s− p1 . u1+s B˙ p,1 (Rn−1 )
This completes the proof in the case s ∈ (0, 1) and q = 1. To extend the result for any s ∈ R+ \ N, it suffices to differentiate [s] times the expression of H, and to repeat the above proof with exponent s − [s]. Finally, because the Besov spaces are an interpolation family, namely s1 s2 s = B˙ p,1 , B˙ p,1 B˙ p,q θ,q
with s = θ s1 + (1 − θ )s2 ,
one gets the desired result for all s > 0 and q ∈ [1, ∞]. Lemma 2 is proved.
2
Proof of Lemma 3. Take p ∈ (1, ∞) and s ∈ (−1 + 1/p, 1/p). We consider the case q = p, the general case will follow from interpolation. Using the properties of duality of Besov spaces (see [4]), one can write (with p conjugate exponent of p)
R. Danchin, P.B. Mucha / Journal of Functional Analysis 256 (2009) 881–927
923
n−1 and φ B˙ −s+1/p (Rn−1 ) 1 . Fn φ dx : φ ∈ S0 R
Fn |xn =0 B˙ s−1/p (Rn−1 ) = sup
p ,p
p,p
Rn−1
Because div F = 0, we have
Fn φ dx =
F · ∇(Eφ) dx,
Rn+
Rn−1
n where Eφ is the extension of φ in B˙ p−s+1 ,p (R+ ) given by Lemma 2 — the assumptions guarantee that −s + 1 > 0. Next, ∇(Eφ) ∈ B˙ p−s ,p (Rn+ ) with −s ∈ (−1 + 1/p , 1/p ). Then, thanks to Proposition 3, both functions ∇(Eφ) and F can be extended by zero for xn < 0. We thus get
Fn φ dx C ∇Eφ B˙ −s
p ,p
s (Rn ) (Rn+ ) F B˙ p,p +
C φ B˙ −s+1/p (Rn−1 ) F B˙ s
n p,p (R+ )
p ,p
.
Rn−1
This completes the proof of Lemma 3.
2
Proof of Lemma 4. Let us first prove the result in the case when F and G are defined on 1 (Rn ) and F ∈ B ˙ 0 (Rn ), the the whole space Rn . Because G belongs to the Besov space B˙ q,∞ n,1 equalities F=
˙ jF
and G =
j ∈Z
˙ jG
j ∈Z
make sense in the set of tempered distributions. Therefore, one can decompose the product F G according to the following homogeneous Bony decomposition (see the original paper [6] by J.-M. Bony, and [4] for the homogeneous framework): ˙ F G = T˙F G + R(F, G) + T˙G F. Above, the paraproduct operator T˙ is defined by T˙F G :=
Sk Fk G with Sk :=
j ,
j k−3
k∈Z
and the remainder operator R˙ is defined by ˙ R(F, G) :=
k∈Z
We claim that
k G with k := k F
|i|2
k+i .
924
R. Danchin, P.B. Mucha / Journal of Functional Analysis 256 (2009) 881–927
T˙F G B˙ 0
q,1 (R
n)
C F B˙ 0
n,1 (R
n)
G B˙ 1
q,∞ (R
n)
(A.9)
,
T˙G F B˙ 0 (Rn ) C G L∞ (Rn ) F B˙ 0 (Rn ) , q,1 q,1 R(F, ˙ G) B˙ 0 (Rn ) C F B˙ 0 (Rn ) G B˙ 1 (Rn ) ,
(A.11)
q,∞
n,1
q,1
(A.10)
where the constant C in the first two inequalities depends only on the dimension, and depends continuously on q > 1 in the last inequality. 0 (Rn ) and of the As an example, let us prove inequality (A.9). By definition of the norm in B˙ q,1 paraproduct, we have l (k Fj G) n = q,1 (R )
T˙F G B˙ 0
l∈Z
j ∈Z kj −3
. Lq (Rn )
Note that, by virtue of the support properties of the function φ defined in (2.1), the second sum may be restricted to those j such that |j − l| 3. Hence we have, for some constant C depending only on φ and n,
T˙F G B˙ 0
n q,1 (R )
C
k F L∞ (Rn ) j G Lq (Rn ) .
l∈Z |j −l|3 kj −3
From Bernstein inequalities, we get
k F L∞ (Rn ) C2k k F Ln (Rn )
and j G Lq (Rn ) C2−j ∇j G Lq (Rn )
so that the above inequality becomes
T˙F G B˙ 0
n q,1 (R )
C
2k−j k F Ln (Rn ) ∇j G Lq (Rn ) .
l∈Z |j −l|3 kj −3
Applying convolution inequalities for series thus yields (A.9). The proof of inequalities (A.10) and (A.11) goes along the same lines (for more details, refer to e.g. [4, Chapter 2]). Finally, putting inequalities (A.9), (A.10) and (A.11) together, one can conclude that
F G B˙ 0
n q,1 (R )
C F B˙ 0
0 (Rn ) n ∇G B ˙ q,∞ n,1 (R )
+ F B˙ 0
n G L∞ (Rn ) q,1 (R )
,
(A.12)
with C depending only on n and on q0 (if q q0 > 1). Next, assume that F and G are defined only on the half-space Rn+ . Then, according to and G of F and G are in B˙ 0 (Rn ) ∩ B˙ 0 (Rn ) and Proposition 3 the symmetric extensions F n,1 q,1 ∞ n 1 n L (R ) ∩ B˙ q,∞ (R ) respectively and satisfy ˙ 0
F B
n,1 (R
n)
2 F B˙ 0
n n,1 (R+ )
L∞ (Rn ) = G L (Rn ) ,
G
∞ +
,
˙ 0
F B
q,1 (R
n)
2 F B˙ 0
n q,1 (R+ )
,
˙0
∇ G
0 (Rn ) . Bq,∞ (Rn ) 2 ∇G B˙ q,∞ +
R. Danchin, P.B. Mucha / Journal of Functional Analysis 256 (2009) 881–927
925
G
˙ 0 n , plugging the above inequalities in (A.12) completes the As F G B˙ 0 (Rn ) F Bq,1 (R ) q,1 + proof. 2 Proof of Lemma 5. As for the previous lemma, it suffices to prove the result for functions defined on I × Rn so that one may resort to Bony’s decomposition. Note that the continuity s ) spaces are the same as for B s . ˙ p,q r (I ; B˙ p,q results for the paraproduct and the remainder in L The time Lebesgue exponents just behave according to Hölder inequality (see e.g. [4]). So we have
TG F L 0 ) C F L r (I ;B˙ q,∞ r
˙0 1 (I ;Bq,∞ )
G Lr2 (I ;L∞ ) ,
and, because q < ∞,
TF G L 0 ) C F L r (I ;B˙ q,∞ r
˙0 1 (I ;Bq,∞ )
∇G L r
˙0 2 (I ;Bn,∞ )
.
Finally, since q > 1, we have R(F, G)
0 ) Lr (I ;B˙ q,∞
This gives the result.
C F L r
˙0 1 (I ;Bq,∞ )
∇G L r
˙0 2 (I ;Bn,∞ )
.
2
Remark 8. In the above two statements, the assumption q > 1 comes into play only for bounding the remainder term. Actually, even if the time is not involved the lemma is false for q = 1 because if it were true, we could for instance multiply a Wn1 ∩ L∞ function by a Dirac mass 0 ). That the case q = 1 is false is the main reason why one cannot prove (which belongs to B1,∞ uniqueness in dimension two if no additional regularity assumptions on the density. Proposition 9. There exists a constant C such that for any z ∈ L1 (0, T ; Ln ∩ L∞ (Rn+ )) with ∇z ∈ L1 (0, T ; L∞ (Rn+ )) the following inequality holds true:
z
L1 (0,T ;L∞ (Rn+ ))
z L1 (0,T ;Ln (Rn+ )) + ∇z L1 (0,T ;L∞ (Rn+ )) . C z L 0 1 (0,T ;B˙ ∞,∞ (Rn+ )) log e +
z L 0 1 (0,T ;B˙ ∞,∞ (Rn )) +
0 1 (0, T ; B˙ ∞,∞ Proof. Let us first point out that L1 ((0, T ); L∞ (Rn+ )) ⊂ L (Rn+ )) so that the above inequality makes sense. Let z be the symmetric extension to z. Obviously, z ∈ L1 ((0, T ); Ln ∩ L∞ (Rn )), ∇ z ∈ L1 ((0, T ); L∞ (Rn )) and we have 1
z L1 (0,T ;Lp (Rn )) = 2 p z L1 (0,T ;Lp (Rn+ ))
for p = n, ∞,
∇ z L1 (0,T ;L∞ (Rn )) = ∇z L1 (0,T ;L∞ (Rn+ )) ,
z L 0 0 1 (0,T ;B˙ ∞,∞ 1 (0,T ;B˙ ∞,∞ (Rn )) 2 z L (Rn )) , +
+
so it suffices to state the desired inequality for z in the whole space.
926
R. Danchin, P.B. Mucha / Journal of Functional Analysis 256 (2009) 881–927
In order to do so, one may split z into low, medium and high frequencies according to Littlewood–Paley decomposition. More precisely, for any nonnegative integer m, we have9
z L 1 (0,T ;L∞ )
k z L1 (0,T ;L∞ ) +
k z L1 (0,T ;L∞ ) +
|k|<m
k−m
k z L1 (0,T ;L∞ ) .
km
Therefore, taking advantage of Bernstein inequality and of the definition of · L 0 1 (0,T ;B˙ ∞,∞ ),
z L 1 (0,T ;L∞ ) C
2k k z L1 (0,T ;Ln )
k−m
+ (2m − 1) z L 0 1 (0,T ;B˙ ∞,∞ )+
2
−k
k ∇ z L1 (0,T ;L∞ ) .
km
Because
k z L1 (0,T ;Ln ) C z L1 (0,T ;Ln )
L1 (0,T ;L∞ ) , L1 (0,T ;L∞ ) C ∇z
and k ∇z
one can thus conclude that −m −m
z L1 (0,T ;Ln ) + (2m − 1) z L
∇z L1 (0,T ;L∞ ) .
z L 0 1 (0,T ;L∞ ) C 2 1 (0,T ;B˙ ∞,∞ )+2 Choosing for m the closest positive integer to log2 ( inequality.
2
z L1 (0,T ;Ln ) + ∇ z L1 (0,T ;L∞ ) )
z L (0,T ;B˙ 0 ) 1
yields the desired
∞,∞
References [1] H. Abidi, Équations de Navier–Stokes avec densité et viscosité variables dans l’espace critique, Rev. Mat. Iberoamericana 23 (2) (2007) 537–586. [2] H. Abidi, M. Paicu, Existence globale pour un fluide inhomogène, Ann. Inst. Fourier 57 (3) (2007) 883–917. [3] S. Antontsev, A. Kazhikhov, V. Monakhov, Boundary Value Problems in Mechanics of Nonhomogeneous Fluids, Stud. Math. Appl., vol. 22, North-Holland Publishing Co., Amsterdam, 1990. [4] H. Bahouri, J.-Y. Chemin, R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Springer, in press. [5] O.V. Besov, V.P. Ilin, S.M. Nikolskij, Integral Function Representation and Imbedding Theorem, Moscow, 1975. [6] J.-M. Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Sci. École Norm. Sup. 14 (1981) 209–246. [7] G. Bourdaud, Réalisations des espaces de Besov homogènes, Ark. Mat. 26 (1988) 41–54. [8] M. Cannone, Ondelettes, paraproduits et Navier–Stokes. With a preface by Yves Meyer, Diderot Editeur, Paris, 1995. [9] M. Cannone, F. Planchon, M. Schonbek, Strong solutions to the incompressible Navier–Stokes equations in the half-space, Comm. Partial Differential Equations 25 (5–6) (2000) 903–924. [10] J.-Y. Chemin, Théorèmes d’unicité pour le système de Navier–Stokes tridimensionnel, J. Anal. Math. 77 (1999) 27–50. [11] R. Danchin, Density-dependent incompressible fluids in critical spaces, Proc. Roy. Soc. Edinburgh 133 (6) (2003) 1311–1334. 9 Here it is understood that all the norms are taken on Rn .
R. Danchin, P.B. Mucha / Journal of Functional Analysis 256 (2009) 881–927
927
[12] R. Danchin, Local and global well-posedness results for flows of inhomogeneous viscous fluids, Adv. Differential Equations 9 (2004) 353–386. [13] R. Danchin, Density-dependent incompressible fluids in bounded domains, J. Math. Fluid Mech. 8 (3) (2006) 333– 381. [14] R. Denk, M. Hieber, J. Pruss, R-boundedness, Fourier multipliers and problems of elliptic and parabolic type, Mem. Amer. Math. Soc. 166 (788) (2003), viii+114 pp. [15] W. Desch, M. Hieber, J. Prüss, Lp -theory of the Stokes equation in a half-space, J. Evol. Equ. 1 (1) (2001) 115–142. [16] J. Duoandikoetxea, Fourier Analysis, Amer. Math. Soc., Providence, RI, 2001. [17] H. Fujita, T. Kato, On the nonstationary Navier–Stokes system, Rend. Sem. Mat. Univ. Padova 32 (1962) 243–260. [18] H. Fujita, T. Kato, On the Navier–Stokes initial value problem I, Arch. Ration. Mech. Anal. 16 (1964) 269–315. [19] Y. Giga, Solutions for semilinear parabolic equations in Lp and regularity of weak solutions of the Navier–Stokes system, J. Differential Equations 62 (2) (1986) 186–212. [20] Y. Giga, T. Miyakawa, Solutions in Lr of the Navier–Stokes initial value problem, Arch. Ration. Mech. Anal. 89 (1985) 267–281. [21] Y. Giga, H. Sohr, Abstract Lp estimates for the Cauchy problem with applications to the Navier–Stokes equations in exterior domains, J. Funct. Anal. 102 (1991) 72–94. [22] H. Iwashita, Lq –Lr estimates for solutions of the nonstationary Stokes equations in an exterior domain and the Navier–Stokes initial value problems in Lq spaces, Math. Ann. 285 (2) (1989) 265–288. [23] T. Kato, Strong Lp -solutions of the Navier–Stokes equation in Rm , with applications to weak solutions, Math. Z. 187 (4) (1984) 471–480. [24] H. Kozono, Global Ln -solution and its decay property for the Navier–Stokes equations in half-space Rn+ , J. Differential Equations 79 (1) (1989) 79–88. [25] H. Kozono, M. Yamazaki, Semilinear heat equations and the Navier–Stokes equation with distributions in new function spaces as initial data, Comm. Partial Differential Equations 19 (5–6) (1994) 959–1014. [26] O. Ladyzhenskaya, V. Solonnikov, The unique solvability of an initial-boundary value problem for viscous incompressible inhomogeneous fluids, J. Soviet Math. 9 (1978) 697–749. [27] P.-G. Lemarié-Rieusset, A. Zhioua, Weakly singular initial values for the Stokes equation on a half space, J. Math. Anal. Appl. 320 (1) (2006) 205–229. [28] P.-L. Lions, Mathematical Topics in Fluid Dynamics, vol. 1. Incompressible Models, Oxford Univ. Press, 1996. [29] P.B. Mucha, On an estimate for the linearized compressible Navier–Stokes equations in the Lp -framework, Colloq. Math. 87 (2) (2001) 159–169. [30] P.B. Mucha, On weak solutions to the Stefan problem with Gibbs–Thomson correction, Differential Integral Equations 20 (7) (2007) 769–792. [31] P.B. Mucha, W. Zaj¸aczkowski, On the existence for the Cauchy–Neumann problem for the Stokes system in the Lp -framework, Studia Math. 143 (2000) 75–101. [32] Y. Shibata, S. Shimizu, Lp –Lq maximal regularity and viscous incompressible flows with free surface, Proc. Japan Acad. Ser. A Math. Sci. 81 (2005) 151–155. [33] V.A. Solonnikov, Unsteady motion of an isolated volume of a viscous incompressible fluid, translation in: Math. USSR Izv. 31 (2) (1988) 381–405. [34] V.A. Solonnikov, On nonstationary Stokes problem and Navier–Stokes problem in a half-space with initial data nondecreasing at infinity, J. Math. Sci. 114 (5) (2003) 1726–1740. [35] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland Math. Library, vol. 18, North-Holland Publishing Co., Amsterdam, 1978. [36] S. Ukai, A solution formula for the Stokes equation in Rn+ , Comm. Pure Appl. Math. 40 (5) (1987) 611–621.
Journal of Functional Analysis 256 (2009) 928–939 www.elsevier.com/locate/jfa
A property of Bp (G). Applications to convolution operators Antoine Derighetti EPFL SB SMA-GE (Bâtiment MA), Station 8, CH-1015 Lausanne, Switzerland Received 18 April 2008; accepted 25 June 2008 Available online 16 July 2008 Communicated by D. Voiculescu
Abstract A new property of Bp (G), permits to obtain an approximation theorem for p-convolution operators and a non-commutative version of the Lohoué’s monomorphism theorem concerning the norm closure of the set of all p-convolution operators with compact support. © 2008 Elsevier Inc. All rights reserved. Keywords: Convolution operators; Multipliers; Herz Figà-Talamanca algebra; Amenable groups
1. Introduction Let G be a locally compact group and H a dense subgroup. Let 1 < p < ∞ and T ∈ CV p (G). Suppose that Gd is amenable. We prove the existence of a net (μα )α∈I of finitely supported measures such that: (1) supp μα ⊂ H , (2) the norm of μα , considered as a convolution operator of Lp (G), is not larger than the norm of T , (3) the net of operators (μα )α∈I converges strongly to T .
E-mail address: [email protected]. 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.06.029
A. Derighetti / Journal of Functional Analysis 256 (2009) 928–939
929
In 1971 N. Lohoué obtained this assertion assuming that the group G is abelian. Even for G = R, H = Q and the convolution operator of Lp (R) T ϕ = f ∗ ϕ, with f a continuous function with compact support, this result is not trivial. Let G be a locally compact amenable group, H a locally compact group and ω a continuous injective homomorphism of G into H . We show that for every bounded measure μ on G, the norm of μ, considered as a convolution operator of Lp (G), is equal to the norm of ω(μ), as a convolution operator of Lp (H ). The above approximation theorem and the preceding monomorphism’s theorem, are derived from the following property of the famous Banach algebra Bp (G) introduced by C. Herz [7,8]. Theorem. Let G be an arbitrary locally compact group, H a dense subgroup, 1 < p < ∞ and u a complex valued continuous function on G with ResH u ∈ Bp (Hd ). Then u ∈ Bp (G) and uBp (G) = ResH uBp (Hd ) . We briefly recall that Bp (G) is the set of all pointwise multipliers of the Herz Figà-Talamanca algebra Ap (G) if G is amenable. The Banach algebra Bp (G) has been until recently intensively investigated by many authors (see for instance [13]). Even the case p = 2 is highly interesting. One of the reasons for this interest is that the Fourier algebra of many non-amenable groups G admits approximate units bounded in the norm of B2 (G). In this paper, Bp (G) appears as a tool for the investigation of the Banach algebra CV p (G) of all p-convolution operators of a locally compact group G. Our approach is new even for G abelian. We say that a continuous operator T of Lp (G) is a p-convolution operator if T (a ϕ) = a T ϕ for a ∈ G and ϕ ∈ Lp (G) where a ϕ(x) = ϕ(ax). In Section 2 we recall the definition of the Banach algebra Bp (G). The three main results of this paper are Theorem 7 (Section 2), Theorem 9 (Section 3) and Theorem 10 (Section 4). 2. A property of Bp (G) Let X be a locally compact T2 space, μ a positive Radon measure on X and 1 < p < ∞. Suppose that μ(U ) > 0 for every non-empty relatively compact open subset U of X. Let k ∈ CX×X , μ ⊗ μ-measurable and bounded. Suppose the existence of A, μ-integrable subset of X, with {(x, y) | k(x, y) = 0} ⊂ A × A. For f , p-integrable the relation Tk [f ] = [g] with g(x) = k(x, y)f (y) dμ(y) X p
for every x ∈ X, defines a linear operator of LC (X, μ), we have |||Tk |||p k∞ μ(A) where |||Tk |||p is the bound of the operator Tk . For f ∈ CX we denote by [f ] the set {g ∈ CX | f (x) = g(x) almost everywhere} and for f , p p-integrable (f ∈ LC (X, μ)) we put
930
A. Derighetti / Journal of Functional Analysis 256 (2009) 928–939
Np (f ) =
1/p f (x)p dμ(x) .
X p
p
L(LC (X, μ)) is the Banach space of all continuous operators of LC (X, μ). For the definitions of PM p (G) (p-pseudomeasures) and Ap (G) (Figà-Talamanca Herz algebra) for a locally compact group G, we refer to [5, pp. 232–233]. If k ∈ C00 (X; C) ⊗ C00 (X; C) then Tk has finite range. For k ∈ C00 (X × X; C), Tk is p in the norm ||| |||p closure of {Tl | l ∈ C00 (X; C) ⊗ C00 (X; C)} in L(LC (X, μ)). We precise that C(X; C) is the set of all continuous maps of X into C and that C00 (X; C) is the set {f ∈ C(X; C) | supp f is compact}. Lemma 1. Let k ∈ C00 (X × X; C), ϕ ∈ C(X × X; C) and ε > 0. Then there are U1 , . . . , UN open relatively compact subsets of X such that: (i) pr1 (supp k) ∪ pr2 (supp k) ⊂ U1 ∪ · · · ∪ UN ,1 (ii) for 1 p, q N , for x, x ∈ Up and for y, y ∈ Uq we have ϕ(x, y)k(x, y) − ϕ(x , y )k(x, y) < ε. Proof. Let K = pr1 (supp k) ∪ pr2 (supp k), U an open relatively compact neighborhood of K and r1 , . . . , rm , s1 , . . . , sm , t1 , . . . , tn , u1 , . . . , un ∈ C00 (X; C) vanishing on X \ K with m ε , ri ⊗ si < k − 16C i=1
∞
where C = 1 + sup ϕ(x, y) (x, y) ∈ U × U and with n tj ⊗ uj ϕk − i=1
<
∞
ε . 16
For every x ∈ K there is U(x) , open neighborhood of x with U(x) ⊂ U , such that for every x ∈ U(x) , for every 1 i m and for every 1 j n ri (x) − ri (x ) < si (x) − si (x ) < uj (x) − uj (x ) <
1 pr (x, y) = x and pr (x, y) = y. 1 2
16C(1 + 16C(1 + 16(1 +
ε
m
,
ε
m
,
l=1 sl ∞ ) l=1 rl ∞ )
ε
n
l=1 tl ∞ )
,
A. Derighetti / Journal of Functional Analysis 256 (2009) 928–939
tj (x) − tj (x ) < There are x1 , . . . , xN in K with K ⊂ required properties. 2
16(1 +
N
p=1 U(xp ) .
ε
n
l=1 ul ∞ )
931
.
Then U1 = U(x1 ) , . . . , UN = U(xN ) satisfy the
Let E(X, μ) be the set {(E1 , . . . , En ) | E1 , . . . , En are disjoint Borel μ-integrable subsets of X with μ(Ei ) > 0 for 1 i n}. We define on E(X, μ) a non-filtering partial order δ δ with δ = (E1 , . . . , En ) and δ =
(E1 , . . . , En ) by: (1) for every 1 j n there is 1 i n with Ej ⊂ Ei , (2) for every i n there are P , μ-negligible and J a subset of {1, . . . , n }, with Ei = 1
P r∈J Er . Lemma 2. Let k ∈ C00 (X × X; C), ϕ ∈ C(X × X; C), ε > 0 and δ ∈ E(X, μ). Then there is δ1 ∈ E(X, μ) such that: (i) δ δ1 , (ii) for every δ = (E1 , . . . , En ) ∈ E(X, μ) with δ1 δ and for every x1 ∈ E1 , . . . , xn ∈ En , we have
ϕ(x, y)k(x, y) − ϕ x , y k(x, y) < ε i
j
for 1 i, j n , x ∈ Ei and y ∈ Ej . Proof. Let δ = (E1 , . . . , Em ), K = pr1 (supp k) ∪ pr2 (supp k), I = {i | 1 i m, μ(Ei ∩ K) > 0} and J = {j | 1 j m, μ(Ej \ K) > 0}. By Lemma 1 there are U1 , . . . , UN open relatively compact subsets of X with K ⊂ U1 ∪ · · · ∪ UN and ϕ(x, y)k(x, y) − ϕ(x , y )k(x, y) < ε for 1 p, q N , x, x ∈ Up and y, y ∈ Uq . Let F = {K ∩ Ei ∩ Up | i ∈ I, 1 p N, μ(K ∩ Ei ∩ Up ) > 0} ∪ {Ej \ K | j ∈ J } and (1) (1) δ1 = (E1 , . . . , En ) ∈ E(X, μ) such that: (1)
(1) for every 1 i n there is F ∈ F with Ei ⊂ F , (1) (1) (2) for every F ∈ F there are N , μ-negligible and G ⊂ {E1 , . . . , En } such that F = N ∪ G∈G G. It is straightforward to verify that δ1 satisfies the desired properties.
2
Let δ = (E1 , . . . , En ) and δ = (E1 , . . . , En ) elements of E(X, μ). We say that δ δ if for every 1 i n there are P , μ-negligible and J a subset of {1, . . . , n }, with Ei = P r∈J Er . The relation is a filtering partial order on E(X, μ). For δ = (E1 , . . . , En ) ∈ E(X; μ) and f ∈ p LC (X, μ) we put
932
A. Derighetti / Journal of Functional Analysis 256 (2009) 928–939
Pδ [f ] =
n
X f (x)1Ej (x) dμ(x)
μ(Ej )
j =1
[1Ej ].
p
Then |||Pδ |||p = 1 and Pδ is a projection of LC (X, μ) onto p Lδ (X, μ) =
n
cj [1Ej ] c1 , . . . , cn ∈ C .
j =1
Proposition 3. For every compact operator T of Lp (G), the net (Pδ T Pδ )δ∈E (X,μ), converges strongly to T . p
Proof. Let f ∈ LC (X, μ) and ε > 0. There are B1 , . . . , Bs , μ-integrable Borel subsets of X with μ(Bi ) > 0 for every 1 i s, c1 , . . . , cs ∈ C such that Np (f − h) <
ε 4(1 + |||T |||p )
where h = si=1 ci 1Bi . There is also T0 operator of finite-dimensional range with |||T − T0 |||p < p ε 4(1+Np (h)) . For every ϕ ∈ LC (X, μ) we have T0 [ϕ] =
m j =1
ϕ(x)ϕj (x) dμ(x) [gj ],
X p
p
where g1 , . . . , gm ∈ LC (X, μ) and ϕ1 , . . . , ϕm ∈ LC (X, μ). For every 1 i m there are: (1) ni ∈ N, (2) Borel sets {Eij | 1 j ni } with μ(Eij ) ∈ (0, ∞), (3) {dij | 1 j ni } ⊂ C, such that Np (gi − hi ) <
ε , 8m(1 + M)
where hi =
ni
dij 1Eij
j =1
and M = max h(x)ϕk (x) dμ(x) 1 k m . X
There is δ = (E1 , . . . , En ) ∈ E(X; μ) such that:
A. Derighetti / Journal of Functional Analysis 256 (2009) 928–939
933
(1) for every F ∈ F (= {B1 , . . . , Bs , E11 , . . . , E1n1 , . . . , Emn1 , . . . , Emnm }) there are N , μnegligible and E subset of {E1 , . . . , En } with F = N E∈E E, (2) for every 1 i n there is F ∈ F with Ei ⊂ F . Let δ ∈ E(X; μ) with δ δ . We have T [f ] − Pδ T Pδ [f ] < 3ε + T0 [h] − Pδ T0 Pδ [h] , p p 4 but m h(x)ϕj (x) dμ(x)[gj ] − Pδ [gj ] . T0 [h] − Pδ T0 Pδ [h] p p j =1 X
Taking in account that [gj ] − Pδ [gj ] < p
ε 4m(1 + M)
for every 1 j m, we indeed obtain T [f ] − Pδ T Pδ [f ] < ε. p
2
For x1 , . . . , xn ∈ X and ϕ ∈ CX×X we denote by l(x1 , . . . , xn ; ϕ) the matrix of Mn (C) (ϕ(xi , xj )). Let δ = (E1 , . . . , En ) be an element of E(X, μ). The map Sδ : (a1 , . . . , an ) →
n j =1
p
p
aj [1Ej ] μ(Ej )1/p p
is a linear isometry of ln onto Lδ (X, μ). For T ∈ L(LC (X, μ)), we denote by aδ (T ) the matrix p of the map Sδ−1 Pδ T Pδ Sδ with respect to the canonical basis of ln . Lemma 4. Let k ∈ C00 (X × X; C), ϕ ∈ C(X × X; C), ε > 0 and δ0 ∈ E(X, μ). Then there is δ1 ∈ E(X, μ) such that: (i) δ0 δ1 , (ii) for every δ = (E1 , . . . , En ) ∈ E(X, μ) with δ1 δ and for every x1 ∈ E1 , . . . , for every xn ∈ En , we have
aδ (Tϕk ) − l x , . . . , x ; ϕ ∗ aδ (Tk ) < ε, n 1 p where
l x1 , . . . , xn ; ϕ ∗ aδ (Tk ) denotes the Hadamard product of the matrices l(x1 , . . . , xn ; ϕ) and aδ (Tk ).
934
A. Derighetti / Journal of Functional Analysis 256 (2009) 928–939 (0)
(0)
Proof. Let δ0 = (E1 , . . . , Em ). According to Lemma 2 there is δ1 ∈ E(X, μ) such that: (i) δ0 δ1 , (ii) for every δ = (E1 , . . . , En ) ∈ E(X, μ) with δ1 δ , for every x1 ∈ E1 , . . . , xn ∈ En , for every 1 i, j n , for every x ∈ Ei and for every y ∈ Ej we have
k(x, y)ϕ(x, y) − k(x, y)ϕ x , x < ε , i j 2A where A = xn
m
(0) r=1 μ(Er ).
Let therefore δ = (E1 , . . . , En ) be an element of E(X, μ) with δ1 δ and x1 ∈ E1 , . . . , ∈ En . We put
d = (dij ) = aδ (Tϕk ) − l x1 , . . . , xn ; ϕ ∗ aδ (Tk ).
For 1 i, j n we have dij =
X X 1Ei (x)1Ej (y)(ϕ(x, y)k(x, y) − ϕ(xi , xj )k(x, y)) dμ(x) dμ(y)
μ(Ei )1/p μ(Ej )1/p
and consequently |dij |
εμ(Ei )1/p μ(Ej )1/p 2A
.
The inequality p |||d|||p
n
n
i=1
p/p
p
|dij |
j =1
implies n
ε
μ Ej |||d|||p 2A j =1
and finally |||d|||p < ε.
2
Lemma 5. Let k ∈ C00 (X × X; C), ϕ ∈ C(X × X; C) and ε > 0. Then there is δ0 ∈ E(X, μ) such that for every δ ∈ E(X, μ) with δ0 δ and for every x1 ∈ E1 , . . . , xn ∈ En , where δ = (E1 , . . . , En ), we have |||Tϕk |||p < ε + l(x1 , . . . , xn ; ϕ) ∗ aδ (Tk )p .
A. Derighetti / Journal of Functional Analysis 256 (2009) 928–939
935
Proof. By Proposition 3 there is δ0 ∈ E(X, μ) such that for every δ ∈ E(X, μ) with δ0 δ we have |||Tϕk |||p −
ε < |||Pδ Tϕk Pδ |||p . 2
By Lemma 4 there is δ1 ∈ E(X, μ) such that: (i) δ0 δ1 , (ii) for every δ = (E1 , . . . , En ) ∈ E(X, μ) with δ1 δ and for every x1 ∈ E1 , . . . , xn ∈ En we have aδ (Tϕk ) − l(x1 , . . . , xn ; ϕ) ∗ aδ (Tk ) < ε . p 2 Let therefore δ = (E1 , . . . , En ) ∈ E(X, μ) with δ1 δ. Let also x1 ∈ E1 , . . . , xn ∈ En . Then |||Tϕk |||p −
ε ε < aδ (Tϕk )p < + l(x1 , . . . , xn ; ϕ) ∗ aδ (Tk )p . 2 2
2
We recall now an important notion due to C. Herz [8, p. 150]. Definition 1. We say that ϕ ∈ C(X × X; C) belongs to Vp (X × X, μ ⊗ μ), where 1 < p < ∞, if there exists C ∈ [0, ∞) such that |||Tϕk |||p C|||Tk |||p for every k ∈ C00 (X; C) ⊗ C00 (X; C). The smallest possible C is denoted ϕVp (X×X,μ⊗μ) . Lemma 6. Let Y be a dense subset of X, 1 < p < ∞ and ϕ ∈ C(X × X; C). Suppose that ResY ×Y ϕ ∈ Vp (Yd × Yd , m ⊗ m), where m is the counting measure of the topological space Yd . Then ϕ ∈ Vp (X × X, μ ⊗ μ) and ϕVp (X×X,μ⊗μ) = ResY ×Y ϕVp (Yd ×Yd ,m⊗m) . Proof. Let k ∈ C00 (X; C) ⊗ C00 (X; C) and ε > 0. By Lemma 5 there is δ = (E1 , . . . , En ) ∈ E(X, μ) such that for every x1 ∈ E1 , . . . , xn ∈ En |||Tϕk |||p < Let 0 < η <
ε 2(1+A1/p )
ε + l(x1 , . . . , xn ; ϕ) ∗ aδ (Tk )p . 2
where
A=
n n i=1
and (aij ) = aδ (Tk ).
j =1
p/p
p
|aij |
936
A. Derighetti / Journal of Functional Analysis 256 (2009) 928–939
For every 1 i, j n there is Wij open neighborhood of (xi , xj ) in X × X with ϕ(xi , xj ) − ϕ(x, y) < η for every (x, y) ∈ Wij . Let Wi , open neighborhood of xi , Zj , open neighborhood of xj , with Wi × Zj ⊂ Wij and finally Ui = Wi ∩ Zi . For every 1 i n there is yi ∈ Y ∩ Ui . Consequently for 1 i, j n we have ϕ(xi , xj ) − ϕ(yi , yj ) < η. We have therefore
l(x1 , . . . , xn ; ϕ) − l(y1 , . . . , yn ; ϕ) ∗ aδ (Tk ) < ε , p 2 this implies l(x1 , . . . , xn ; ϕ) ∗ aδ (Tk ) < ε + l(y1 , . . . , yn ; ϕ) ∗ aδ (Tk ) p p 2 and consequently |||Tϕk |||p < ε + ResY ×Y ϕVp (Yd ×Yd ,m⊗m) |||Tk |||p . We obtain that ϕ ∈ Vp (X × X, μ ⊗ μ) and ϕVp (X×X,μ⊗μ) ResY ×Y ϕVp (Yd ×Yd ,m⊗m) . Finally Lemma 2 of [8, p. 150] permits to conclude.
2
Let us finally recall the definition of Bp (G) proposed by C. Herz [8, p. 146]. Definition 2. Let G be an arbitrary locally compact group, 1 < p < ∞ and u ∈ C(G; C). We say that u ∈ Bp (G) if MG u ∈ Vp (G × G, mG ⊗ mG ) where mG is a left-invariant measure of G and where MG f (x, y) = f (y −1 x) for f ∈ CG . For u ∈ Bp (G) we put uBp (G) = MG uVp (G×G,mG ⊗mG ) . In [8], C. Herz proved, for G arbitrary locally compact group, that Bp (G) = Bp (Gd ) ∩ C(G; C) and that uBp (G) = uBp (Gd ) . One of the main results of this paper is the following theorem. Theorem 7. Let G be an arbitrary locally compact group, H a dense subgroup, 1 < p < ∞ and u ∈ C(G; C). Suppose that ResH u ∈ Bp (Hd ). Then u ∈ Bp (G) and uBp (G) = ResH uBp (Hd ) .
A. Derighetti / Journal of Functional Analysis 256 (2009) 928–939
937
Proof. We have ResH ×H ◦MG = MH ◦ ResH . This implies ResH ×H MG u ∈ Vp (Hd × Hd , m ⊗ m). By Lemma 6 MG u ∈ Vp (G × G, mG ⊗ mG ) and MG uVp (G×G,mG ⊗mG ) = ResH ×H MG uVp (Hd ×Hd ,m⊗m) .
2
Remark. Even for G compact, the result is new. 3. Application to spectral synthesis p
Let μ be a bounded complex measure on G (μ ∈ MC1 (G)) then λG (μ) denotes the element of CV p (G) defined by p 1/p λG (μ)[ϕ] = ϕ ∗ ΔG μˇ for ϕ ∈ C00 (G; C), where ΔG is the modular function of G. We precise that μˇ is the measure p defined by μ(ϕ) ˇ = μ(ϕ) ˇ where ϕ(x) ˇ = ϕ(x −1 ). The map λG is a faithful representation of the p Banach algebra MC1 (G) into Lp (G). Therefore, via λG , every bounded measure on G can be considered as a p-convolution operator. Using the commutation theorem and the Kaplansky’s density theorem, we easily get the following result. Proposition 8. Let G be an arbitrary locally compact group and H a dense subgroup. For every T ∈ CV 2 (G) there is a net (μα ) of complex measures on G such that: (i) (ii) (iii) (iv)
supp μα is finite, supp μα ⊂ H , |||λ2G (μα )|||2 |||T |||2 , the net (λ2G (μα )) converges strongly to T .
We intend to generalize this proposition to p = 2. In [11, Théorème 1] N. Lohoué succeeded assuming G abelian. Theorem 9. Let G be a locally compact group and H a dense subgroup. Suppose that Gd is amenable. Let 1 < p < ∞ and T ∈ CV p (G). Then there is a net (μα ) of complex measures on G such that: (i) (ii) (iii) (iv)
supp μα is finite, supp μα ⊂ H , p |||λG (μα )|||p |||T |||p , p the net (λG (μα )) converges strongly to T .
Proof. It suffices to consider the case where |||T |||p = 1. Let D be the closure in CV p (G), with p respect to the strong operator topology, of the set of all convolution operators λG (μ), where μ is p a complex measure with finite support contained in H with |||λG (μ)|||CV p (G) 1. Suppose that T ∈ / D. Then there is u ∈ Ap (G) with u, T Ap (G),PMp (G) > 1 and |u, SAp (G),PMp (G) | 1 for
938
A. Derighetti / Journal of Functional Analysis 256 (2009) 928–939
every S ∈ D. For the definitions of PM p (G) and of the pairing , Ap (G),PMp (G) , we again refer to [5, pp. 232, 233]. Theorem 2 of [8, p. 147] and Theorem 7 above imply p ˜ supp μ ⊂ H, supp μ is finite, λHd (μ)CV uAp (G) = sup μ(u)
p (Hd )
1 .
We precise that for every measure μ, μ(ϕ) ˜ = μ(ϕ) ˜ where ϕ(x) ˜ = ϕ(x −1 ) for ϕ ∈ C00 (G; C). For every measure μ with finite support in H we have p λ (μ) Hd CV
p (Hd )
p = λGd (μ)CV
p (Gd )
by [4, Théorème 1, p. 72]. But, by a recent result of Delmonico [1,3], p λ (μ) Gd CV
p (Gd )
p = λG (μ)CV
p (G)
.
Consequently uAp (G) 1 and therefore |u, T Ap (G),PM p (G) | 1. This contradiction implies that T ∈ D. 2 Remarks. (1) See [1,3] for the case H = G. For G = H = Rn see [9]. (2) It is possible to have more informations on the net (μα ) for certain classes of locally compact groups. See [10, Théorème II, p. 82] and [2]. 4. The Lohoué’s monomorphism theorem Theorem 10. Let G be a locally compact amenable group, H a locally compact group and ω a continuous injective homomorphism of G into H . Then for 1 < p < ∞ and every bounded complex measure μ on G we have p λ (μ) G
CV p (G)
p
= λH ω(μ) CV
p (H )
.
Proof. (1) Suppose at first that ω(G) is dense in H . Let ω∗ be the map u → u ◦ ω. According to Theorem 7 ω∗ is a linear isometry of Bp (H ) into Bp (G). Let ω∗∗ be the dual of ω∗ . p ˜ for u ∈ Bp (G). We have F Bp (G)∗ = |||λG (μ)|||p = For μ ∈ MC1 (G) we put F (u) = μ(u) p ω∗∗ (F )Bp (H )∗ = |||λH (ω(μ))|||p . (2) Suppose now that ω(G) is not dense in H . Let I be the closure of ω(G) in H . We denote by ω the map ω considered as a continuous homomorphism of G into I . Let i be the p inclusion of I into H . We have i ◦ ω = ω. For μ ∈ MC1 (G) we have |||λH (ω(μ))|||CV p (H ) = p p p |||λI (ω (μ))|||CV p (I ) by [4]. The part (1) implies that |||λI (ω (μ))|||CV p (I ) = |||λG (μ)|||CV p (G) . 2 Remarks. (1) C. Delmonico showed in [1], that the map ω extends canonically to the norm closure in p L(LC (G)) of the space of all p-convolution operators with compact support. Replacing in
A. Derighetti / Journal of Functional Analysis 256 (2009) 928–939
939
p
the preceding proof, the operator λG (μ) by any T ∈ CV p (G) with compact support, we obtain the isometry for this larger class of operators. (2) This theorem was first obtained by N. Lohoué in [12] for G and H both abelian. (3) Theorem 10 does not require the amenability of Gd . The following density property is also due to Lohoué [9] for G commutative. Corollary 11. Let G be a locally compact amenable group, H a locally compact group and ω a continuous injective homomorphism of G into H with ω(G) dense in H . Then for 1 < p < ∞ the set {u ◦ ω | u ∈ Ap (H ) with uAp (H ) 1} is dense in {v | vBp (G) 1} for the topology of uniform convergence on compact subsets of G. Proof. Lohoué’s proof [12, pp. 146, 147], taking into account Theorem 10, adapts without change! 2 Final remarks. (1) This paper solves Problems 9.5, 9.6, 9.10 and 9.12 of [6]. (2) For G abelian, our approach avoids the use of the structure theory of locally compact abelian groups in Theorems 7, 9 and 10. (3) Even for p = 2, Theorems 7 and 10 are new. References [1] C. Delmonico, Opérateurs de convolution et homomorphismes de groupes localement compacts, Thèse de doctorat, Ecole Polytechnique Fédérale de Lausanne, 2003. [2] C. Delmonico, Atomization process for convolution operators on locally compact groups, Proc. Amer. Math. Soc. 134 (2006) 3231–3241. [3] C. Delmonico, Convolution operators and homomorphisms of locally compact groups, J. Aust. Math. Soc., in press. [4] A. Derighetti, Relations entre les convoluteurs d’un groupe localement compact et ceux d’un sous-groupe fermé, Bull. Sci. Math. 106 (1982) 69–84. [5] A. Derighetti, Conditional expectations on CV p (G). Applications, J. Funct. Anal. 247 (2007) 231–251. [6] P. Eymard, Algèbres Ap et convoluteurs, Sém. Bourbaki 22 (1969/1970) 1–15. [7] C. Herz, Remarque sur la note précédente de M. Varopoulos, C. R. Acad. Sci. Paris Sér. A 260 (1965) 6001–6004. [8] C. Herz, Une généralisation de la notion de transformée de Fourier–Stieltjes, Ann. Inst. Fourier (Grenoble) 24 (1974) 145–157. [9] N. Lohoué, Sur le critère de S. Bochner dans les algèbres Bp (Rn ) et l’approximation bornée des convoluteurs, C. R. Acad. Sci. Paris Sér. A (1970) 247–250. [10] N. Lohoué, Algèbres Ap (G) et convoluteurs, Doctorat d’Etat, Paris-Sud, Orsay, 1971. [11] N. Lohoué, La synthèse des convoluteurs sur un groupe abélien localement compact, C. R. Acad. Sci. Paris Sér. A 272 (1971) 27–29. [12] N. Lohoué, Approximation et transfert d’opérateurs de convolution, Ann. Inst. Fourier 26 (1976) 133–150. [13] N. Spronk, Measurable Schur multipliers and completely bounded multipliers of the Fourier algebras, Proc. London Math. Soc. (3) 89 (2004) 161–192.
Journal of Functional Analysis 256 (2009) 940–957 www.elsevier.com/locate/jfa
Mass transport generated by a flow of Gauss maps Vladimir I. Bogachev ∗ , Alexander V. Kolesnikov Moscow State University, Department of Mechanics and Mathematics, Moscow, Russian Federation Received 20 April 2008; accepted 9 May 2008 Available online 20 June 2008 Communicated by Paul Malliavin
Abstract Let A ⊂ Rd , d 2, be a compact convex set and let μ = 0 dx be a probability measure on A equivalent to the restriction of Lebesgue measure. Let ν = 1 dx be a probability measure on Br := {x: |x| r} equivalent to the restriction of Lebesgue measure. We prove that there exists a mapping T such that ν = μ ◦ T −1 and T = ϕ · n, where ϕ : A → [0, r] is a continuous potential with convex sub-level sets and n is the Gauss map of the corresponding level sets of ϕ. Moreover, T is invertible and essentially unique. Our proof employs the optimal transportation techniques. We show that in the case of smooth ϕ the level sets of ϕ are K(x) · n(x), where K is the Gauss curvature. As governed by the Gauss curvature flow x(s) ˙ = −s d−1 1 (sn) 0 (x) a by-product one can reprove the existence of weak solutions to the classical Gauss curvature flow starting from a convex hypersurface. © 2008 Elsevier Inc. All rights reserved. Keywords: Optimal transportation; Monge–Kantorovich problem; Monge–Ampère equation; Gauss curvature flow; Gauss map
1. Introduction The goal of this paper is to introduce a new class of transformations of measures on Rd which (heuristically) have the form T = ϕ · ∇ϕ/|∇ϕ| with some function ϕ. Our work is motivated by two intensively developing areas: optimal transportation and curvature flows, and establishes an interesting link between these areas. Optimal transportation can be described as a problem of optimization of a certain functional associated with a pair of measures. The quadratic transportation * Corresponding author.
E-mail address: [email protected] (V.I. Bogachev). 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.05.006
V.I. Bogachev, A.V. Kolesnikov / Journal of Functional Analysis 256 (2009) 940–957
941
cost W22 (μ, ν) between two probability measures μ, ν on Rd is defined as the minimum of the following functional (the Kantorovich functional): m →
|x1 − x2 |2 dm(x1 , x2 ),
m ∈ P(μ, ν),
(1.1)
Rd ×Rd
where P(μ, ν) is the set of all probability measures on Rd × Rd with the marginals μ and ν; here |v| denotes the Euclidean norm of v ∈ Rd . The problem of minimizing (1.1) is called the mass transportation problem. This formulation is due to Kantorovich [14]. A detailed discussion of the mass transportation problem in this setting can be found in [19]. In many cases there exists a mapping T : Rd → Rd , called the optimal transport between μ and ν (or a solution to the Monge problem), such that ν = μ ◦ T −1 and W22 (μ, ν) =
x − T (x)2 μ(dx).
Rd
The minimization of the latter integral in the class of measurable mappings T such that μ ◦ T −1 = ν is called the Monge problem. If T is a solution to the Monge problem, then the image of μ under the mapping x → (x, T (x)) to Rd × Rd minimizes the Kantorovich functional. However, it may happen that the Monge problem has no solution, while the Kantorovich problem is always solvable. It is worth mentioning that the first rigorous results related to existence of optimal mappings were obtained in the classical work of A.D. Alexandroff [1] on convex surfaces with prescribed curvature! If μ and ν are absolutely continuous, then, as show by Brenier [6] and McCann [16], there exists an optimal transportation T which takes μ to ν. Moreover, this mapping is μ-unique and has the form T = ∇W , where W is convex. Under broad assumptions, W solves the following non-linear PDE (the Monge–Ampère equation): ν (∇W ) det Da2 W = μ , where μ and ν are densities of μ and ν and Da2 W is the absolutely continuous part of the distributional derivative of D 2 W . At present the optimal transportation theory attracts attention of researchers from the most diverse fields, including probability, partial differential equations, geometry, and infinite-dimensional analysis (see Villani’s book [24] and papers [2] and [21]). The study of curvature flows is a very popular subject in geometry. The theory of Ricci flows attracted particular interest after the famous works of G. Perelman on the Poincaré conjecture. The theory of geometrical flows began, however, with flows of embedded manifolds. Let F0 : M d−1 → Rd be a smooth embedding of a smooth compact Riemannian manifold M d−1 (without boundary). Denote by A the enclosed body: ∂A = F0 (M). We say that an evolution F (·,·) : M × [0, T ) → Rd is a geometrical flow if F0 = F (·, 0) and ∂ F (x, t) = −g F (x, t) · n F (x, t) , ∂t
(1.2)
942
V.I. Bogachev, A.V. Kolesnikov / Journal of Functional Analysis 256 (2009) 940–957
where g : M → R is some curvature function and n is the outer unit normal vector. If g = H is the mean curvature, then F is called the mean curvature flow. If g = K is the Gauss curvature, then F is called the Gauss curvature flow. The simplest case of the curvature flow is given by the following planar flow: ∂x(t, s) = −k(x) · n(x), ∂t
x ∈ R2 , M = S 1 .
Consider the flow of closed curves t → x(t, ·). Under this flow the enclosed volume decreases with the constant speed −2π . In addition, any non-convex curve becomes convex in finite time and then remains convex. Finally, any curve shrinks to a point in finite time; the shape of any curve becomes more and more rotund (see [10] and [12]). Any multi-dimensional mean curvature flow or Gauss curvature flow starting from a convex surface preserves convexity and shrinks the surface to a point (see [13] and [23]). In [9] and [7], Eq. (1.2) in the case of the mean curvature was investigated from the PDEs point of view. It turns out that the surfaces driven by (1.2) can be obtained as level sets of a function u(t, x) which satisfies a non-linear degenerate second order parabolic equation of the Monge–Ampère type. A solution of this equation is in general understood in some weak sense (viscosity solutions). For the PDE approach and viscosity solutions, see the recent book [11]. Concerning Gauss flows, see [3]. In this paper, we establish the existence and uniqueness of a special measure transportation mapping between two probability measures μ and ν. It has the following heuristic expression: T =ϕ
∇ϕ . |∇ϕ|
The potential ϕ has convex sub-level sets. Note that this transportation mapping may not be a gradient. Nevertheless, T can be obtained as a degenerate limit of some transportation mappings which are constructed by means of the optimal transportation techniques. The limiting potential satisfies a degenerate Monge–Ampère equation (see the proof of the main theorem). In addition, we show that the resulting limit is naturally connected with the Gauss flow. The level sets of the potential ϕ can be associated to a special Gauss flow associated to the measures μ and ν according to x(s) ˙ = −s d−1 In the case 1 (x) =
Cd,r |x|d−1
1 (sn) K(x) · n(x). 0 (x)
and 0 (x) = Hd1(A) we obtain a weak solution to x(s) ˙ = −cK(x) · n(x),
which is the classical Gauss flow starting from some initial convex hypersurface. Finally, we note that in [15,17,22] the reader can find other interesting links between mass transportation and geometrical flows (in particular, the Ricci flows). Some analogs of the presented results in the case of a manifold will be considered in our forthcoming joint paper with F.-Y. Wang.
V.I. Bogachev, A.V. Kolesnikov / Journal of Functional Analysis 256 (2009) 940–957
943
2. Main result Throughout we assume that d 2 and denote by Hn the n-dimensional Hausdorff measure. For Lebesgue measure we often use another common notation dx. Let Int A denote the interior of a set A. Recall that, given a compact smooth orientable (d − 1)-dimensional surface M in Rd , one has the Gauss map n : M → S d−1 , where n(x) is the global unit outer normal vector field. Let Dn : TM x → TSd−1 n(x) be the differential of n. Choose an orthonormal basis {e2 , . . . , ed } ⊂ TM x . Then the matrix Dn can be written as ∂ei n, ej , where ∂ei n are the usual partial derivatives of n. The determinant of Dn(x) is called the Gauss curvature and is denoted throughout by K(x). Below we deal with the case where M is a surface (possibly, non-smooth) of the form M = ∂V , where V is a convex compact set. In this case the normal n(x) is well-defined almost everywhere on M. More precisely, for an arbitrary point x ∈ M, let us set NM,x := η ∈ S d−1 ; ∀z ∈ V , η, z − x 0 . If NM,x contains a single element n(x), then n(x) is the unit normal in the usual sense. We shall use the fact that one has Hd−1 (S) = 0, where S = {x: NM,x contains more than one element}. Hence the Gauss map n(x) is well-defined Hd−1 -almost everywhere on M. Moreover, one can show that K(x) is well-defined Hd−1 -almost everywhere on M (but this fact is not used below). We shall consider the following Hausdorff distance between non-empty compact sets: dist(B1 , B2 ) = max sup dist(x, B2 ), sup dist(x, B1 ) . x∈B1
x∈B2
Theorem 2.1. Let A ⊂ Rd be a compact convex set and let μ = 0 dx be a probability measure on A equivalent to the restriction of Lebesgue measure. Let ν = 1 dx be a probability measure on Br = {x: |x| r} equivalent to the restriction of Lebesgue measure. Then, there exist a Borel mapping T : A → Br and a continuous function ϕ : A → [0, r] with convex sub-level sets As = {ϕ s} such that ν = μ ◦ T −1 and T = ϕ · n Hd -almost everywhere, where n = n(x) is a unit outer normal vector to the level set {y: ϕ(y) = ϕ(x)} at the point x. If ϕ is smooth, the level sets of ϕ are moving according to the following Gauss curvature flow equation: x(s) ˙ = −s d−1
1 (sn) K(x) · n(x), 0 (x)
where x(s) ∈ ∂Ar−s , 0 s r, x(0) ∈ ∂A is any initial point.
(2.1)
944
V.I. Bogachev, A.V. Kolesnikov / Journal of Functional Analysis 256 (2009) 940–957
To prove this theorem we develop an approach based on the optimal transportation techniques. For every t 0, we consider a mapping Tt that takes μ to ν and maximizes the functional F →
t x, F (x) F (x) μ(dx)
(2.2)
in the class of mappings F with μ ◦ F −1 = ν. Equivalently, it minimizes the functional F →
x − F (x)F (x)t 2 μ(dx)
in the class of mappings F with μ ◦ F −1 = ν. For t = 0 (2.2) becomes the classical Monge– Kantorovich problem. For t = 0 standard arguments from the Monge–Kantorovich theory show that the set t x, Tt (x)Tt (x) , x ∈ A is cyclically monotone, hence belongs to the graph of the gradient of some convex function Wt (see [24, Chapter 2]). This can be shown, for instance, by a cyclical permutation of small balls (see [24]). More formally, this can be obtained by variation of the corresponding Lagrange functional (see [8]). If the reader does not want to be concerned with the cyclical monotonicity or calculus of variations, we note that ∇Wt is just the optimal transportation of μ to ν ◦ St−1 , where St (x) = x|x|t . This can be taken for a definition of Wt . One has the following relations: Tt =
∇Wt |∇Wt |
t 1+t
,
t ∇Wt (x) = Tt (x)Tt (x) .
Clearly, |Tt (x)| r since Tt transforms μ into ν. Throughout the paper we choose Wt in such a way that minx∈A Wt (x) = 0. Define a new potential function ϕt by Wt =
1 ϕ t+2 . t +2 t
One has Tt = ϕt
∇ϕt t
|∇ϕt | t+1
.
We show below that the limits lim ϕt = ϕ,
t→∞
lim Tt = T
t→∞
exist almost everywhere (for a suitable sequence tn → ∞) and then we prove that T is the desired mapping.
V.I. Bogachev, A.V. Kolesnikov / Journal of Functional Analysis 256 (2009) 940–957
945
Lemma 2.2. One has 1 1+t 1 ϕt (2 + t) 2+t diam(A) 2+t r 2+t , 1 1+t 1 ∇ϕt (x) dx ϕt dHd−1 (2 + t) 2+t diam(A) 2+t r 2+t Hd−1 (∂A). A
∂A
Proof. By the convexity of Wt we have
Wt (x) − Wt (y) x − y, ∇Wt (x) . Choosing y0 in such a way that Wt (y0 ) = 0, we find Wt (x) diam(A)|∇Wt (x)| for every x ∈ A. 1 ϕtt+2 , we obtain Since Wt = 2+t ϕt (t + 2) diam(A)|∇ϕt |. Let α =
1+t 2+t .
Note that
1−α α
=
1 1+t ,
hence ϕt |∇ϕt |
1−α α
= |Tt | r. Therefore, one has
1−α ϕt = ϕt1−α ϕtα (2 + t) diam(A) |∇ϕt |1−α ϕtα 1−α 1−α α ϕt |∇ϕt | α (2 + t) diam(A) 1−α α 1 1+t (2 + t) diam(A) r = (2 + t) diam(A) 2+t r 2+t . By using the convexity of Wt again, we get ∇Wt ∇ϕt 0 div = div , |∇Wt | |∇ϕt | ∇Wt where under div( |∇W ) we understand the distributional derivative of the vector field t| Integrating with respect to ϕt dx we obtain
0 A
∇Wt |∇Wt | .
∇ϕt ∇ϕt ϕt dx = − |∇ϕt | dx + ϕt nA , dHd−1 . div |∇ϕt | |∇ϕt | A
∂A
Hence
|∇ϕt | dx A
ϕt dHd−1 .
∂A
Applying the above uniform estimate for ϕt we complete the proof. In fact, one could do these calculations in the case of smooth densities, where ϕt has a better regularity, and then approximate our densities by smooth ones (the corresponding optimal transports converge to ∇Wt ). 2 Corollary 2.3. There exists a sequence {tn } → ∞ such that {ϕtn } converges almost everywhere to a finite function ϕ.
946
V.I. Bogachev, A.V. Kolesnikov / Journal of Functional Analysis 256 (2009) 940–957
Proof. By Lemma 2.2, every sequence {ϕtn } is bounded in W 1,1 (A). By the compactness of the embedding W 1,1 (B) ⊂ L1 (B) for any ball B ⊂ A and the diagonal argument (in fact, since in the present situation A is convex, the embedding of the whole space W 1,1 (A) is compact), we obtain the claim. 2 Lemma 2.4. There exists a sequence {tn } → ∞ such that 1
lim |∇ϕn | 1+tn = 1
tn →∞
almost everywhere, where ϕn := ϕtn . Proof. By Lemma 2.2 one has ϕt (t + 2) diam(A)|∇ϕt |. Hence 1
1
Ct |Tt | 2+t |∇ϕt | 1+t , 1
1
where Ct−1 = (2 + t) 2+t (diam(A)) 2+t . Changing variables one gets the following estimate for any δ > 0: 1 1 1 |∇ϕn | 1+tn 1 − δ μ |Ttn | 2+tn 1 − δ = ν |x| 2+tn 1 − δ . μ Ct−1 n Hence 1 μ 1 − Ct−1 |∇ϕn | 1+tn δ → 0. n 1
|∇ϕn | 1+tn )+ tends to zero in μ-measure as tn → ∞. Passing to an This implies that (1 − Ct−1 n almost everywhere convergent subsequence one can assume additionally that 1
limtn →∞ |∇ϕn | 1+tn 1
(2.3)
almost everywhere. Since supt ∇ϕt L1 (dx) < ∞ by Lemma 2.2, we see that the sequence 1
{|∇ϕn | 1+tn } is bounded in Lp (A) for any p < ∞. Moreover, by Hölder’s inequality p limtn →∞ |∇ϕn | 1+tn dx Hd (A). A 1
Hence, choosing an Lp (A)-weakly convergent subsequence |∇ϕnm | 1+tnm → f , one has f dx Hd (A). A
On the other hand, (2.3) and Fatou’s lemma show that f 1 a.e., which yields p lim |∇ϕnm | 1+tnm dx = 1. tnm →∞
A
V.I. Bogachev, A.V. Kolesnikov / Journal of Functional Analysis 256 (2009) 940–957
947
1
Hence |∇ϕnm | 1+tnm → 1 in the norm of any Lp (A), p < ∞. Extracting again an almost everywhere convergent subsequence we get the claim. 2 1
In what follows we set ϕn := ϕtn and assume that ϕn → ϕ and |∇ϕn | 1+tn → 1 almost everywhere. Lemma 2.5. Let Cn ⊂ Br be convex sets such that ICn → IC almost everywhere. If C is of positive measure, then dist(∂Cn , ∂C) → 0. Proof. The set C can be taken convex by letting IC := limn ICn . We may assume that r = 1. It is known and readily verified by induction that every convex set U ⊂ Br with Hd (U ) δ contains a ball of volume at least κ1 (d)δ, where κ1 (d) depends only on d. Let B be a ball of radius ε > 0 centered at some point x0 ∈ ∂U . Then Hd (U ∩ B) κ2 (d)ε d δ, where κ2 (d) depends only on d. It follows that, whenever Hd (Cn ) Hd (C)/2, one has IC − ICn L1 = Hd (C Cn ) 2−1 κ2 (d)Hd (C) dist(∂Cn , ∂C)d . 2 Note that due to convexity one has dist(∂Cn , ∂C) = dist(Cn , C). Lemma 2.6. The sequence of potentials ϕtn converges to ϕ uniformly on A. In particular, ϕ is continuous and has convex sub-level sets As = {y: ϕ(y) s}. Proof. Clearly, it is sufficient to prove the claim for a subsequence. As noted above, one can 1
assume, in addition, that |∇ϕn | 1+tn → 1 almost everywhere. Let us redefine ϕ as follows: ϕ := limn ϕn . Then the sub-level sets As of ϕ are convex since the corresponding sub-level sets As,n = 1
{x: ϕn (x) s} of ϕn are convex. Since |Tn | = ϕn |∇ϕn | 1+tn , we have shown that |Tn | → ϕ almost everywhere. Then the equality ν = μ ◦ Tn−1 yields that the image of μ under the mapping x → ϕ(x) ∈ R+ , denoted by μϕ ∈ P(R+ ), coincides with ν|x| ∈ P(R+ ), where ν|x| is the image of ν under the mapping x → |x|. Due to our assumptions on ν, this implies that μϕ has a strictly increasing continuous distribution function, i.e., (1) μ(As1 ) < μ(As2 ) whenever s1 < s2 , (2) μ({ϕ = t}) = 0 for all t ∈ [0, r]. Note that (2) implies that IAs,n → IAs almost everywhere for each s > 0. By Lemma 2.5 we have dist(∂As,n , ∂As ) → 0,
s > 0.
Now, given ε > 0, we divide [0, r] by points s1 , . . . , sN with |si+1 − si | < ε and take δ = maxiN dist(∂Asi , ∂Asi+1 ). There exists M such that dist(∂Asi ,n , ∂Asi ) < δ/2
948
V.I. Bogachev, A.V. Kolesnikov / Journal of Functional Analysis 256 (2009) 940–957
for every i = 1, . . . , N and every n > M. This implies that supx∈A |ϕn (x) − ϕ(x)| 2ε for all n M. Hence ϕn → ϕ uniformly. Since ϕn are continuous as powers of convex functions, ϕ is continuous as well. The proof is complete. 2 Lemma 2.7. Let Nx := N∂Aϕ(x) ,x , where Aϕ(x) = {y: ϕ(y) ϕ(x)} and S := {x ∈ A: Nx contains more than one element}, i.e. S is the set of all the points x such that the boundary of the sub-level set containing x is not differentiable at x. Then Hd (S) = 0. Proof. First we consider the case d = 2. Fix an orthonormal basis {e1 , e2 } and identify every unit vector n with α ∈ [0, 2π), where α is the angle between e1 and n. We write n := nα := cos α · e1 + sin α · e2 . The set S is a countable union of the sets Sp,q := x: [p − q, p + q] ⊂ Nx ,
p, q ∈ Q ∩ [0, 2π).
If S has a positive measure, then Hd (Sp,q ) > 0 for some p, q. If x ∈ Sp,q , then we have Aϕ(x) ⊂ {z: z − x, np 0}. Note that the line lx,p (z) = z: z − x, np = 0 intersects Sp,q exactly at x. Indeed, otherwise we get two points x, y such that the sub-level sets Aϕ(x) and Aϕ(y) both intersect lx,p at two different points and belong to the same half-plane P with ∂P = lx,p . Hence neither Aϕ(x) ⊂ Aϕ(y) nor Aϕ(y) ⊂ Aϕ(x) hold, which is impossible. Thus we obtain that lx,p ∩ Sp,q = {x}. Finally, applying Fubini’s theorem and disintegrating Lebesgue measure along the lines parallel to l0,p , we obtain H2 (Sp,q ) = 0, which is a contradiction. So the lemma is proved for d = 2. The multi-dimensional case follows by induction and Fubini’s theorem. Indeed, fix an orthonormal basis {e1 , . . . , ed }. Note that all sections of a convex body are convex. Disintegrating S along ei and applying the result for d − 1, we obtain that Si = {projection of Nx on xi = 0 has more than one element} has measure zero. Since S =
d
i=1 Si ,
the proof is complete.
2
Proof of Theorem 2.1. According to Lemmas 2.6 and 2.4, we have ϕn → ϕ uniformly and 1
|∇ϕn | 1+tn → 1 almost everywhere. It remains to prove that ∇ϕn /|∇ϕn | → n almost everywhere. Let us fix x ∈ A. Since ϕn (x) → ϕ(x), one has IAϕn (x),n → IAϕ(x) almost everywhere, where Aϕn (x),n = y: ϕn (y) ϕn (x) ,
Aϕ(x) = y: ϕ(y) ϕ(x) .
V.I. Bogachev, A.V. Kolesnikov / Journal of Functional Analysis 256 (2009) 940–957
949
According to Lemma 2.7, n(x) is well-defined for almost all x. The same holds for every ∇ϕn (x)/|∇ϕn (x)|. So, without loss of generality we can fix x in the interior of A such that n(x) and ∇ϕn (x)/|∇ϕn (x)| are well-defined. If the vectors ∇ϕn (x)/|∇ϕn (x)| do not converge to n(x), then, extracting a convergent subsequence from a sequence of unit vectors {∇ϕn (x)/|∇ϕn (x)|}, we obtain a unit vector η = n(x). By using convergence IAϕn (x),n → IAϕ(x) , one can show that η, z − x 0 for all z ∈ A, i.e., η ∈ Nx , which contradicts the choice of x. It remains to verify the evolution equation for a smooth potential ϕ. Indeed, let us choose an orthonormal basis {ei } at x such that e1 = n and every vector ei , 2 i d, belongs to the tangent space of ∂At at x. Let us write the change of variables formula for T = ϕ · n. Differentiating along n we find ∂n T = ∂n ϕ · n + ϕ · ∂n n. Differentiating the identity n, n = 1, we see that ∂n n belongs to the tangent space of ∂At at x. In addition, ∂n ϕ = |∇ϕ|. Next we note that ∂ei T = ϕ · ∂ei n,
∂ei n, n = 0,
1 i d.
Hence det DT = |∇ϕ|ϕ d−1 det ∂ei n, ej . Since K = det( ∂ei n, ej ), we have det DT = |∇ϕ|ϕ d−1 K. Thus one obtains the following change of variables formula (the Monge–Ampère equation): 0 = 1 (ϕ · n)|∇ϕ|ϕ d−1 K. It remains to note that the level sets ∂As are shrinking with the velocity 1/|∇ϕ| in the direction of −n. Hence (2.1) follows from the change of variables formula. The proof is complete. Example 2.8. Let A be a strictly convex compact set. Set 1 (x) :=
Cd,r , |x|d−1
0 (x) :=
1 , Hd (A)
−1 . Varying r we can show the existence of a weak solution (in the where Cd,r = ( Br |x|dx d−1 ) “transportation sense”) to the classical Gauss curvature flow which starts from ∂A and satisfies the equation x(s) ˙ = −cK(x) · n(x), where c can be chosen arbitrarily. Certainly, a rigorous justification of this formula requires some additional work, since we have not proved that ϕ is differentiable.
950
V.I. Bogachev, A.V. Kolesnikov / Journal of Functional Analysis 256 (2009) 940–957
3. Injectivity and uniqueness In this section, we prove that T is invertible and essentially unique. Recall that the Legendre transform of a convex function W on a convex set A is defined by W ∗ (y) = sup x, y − W (x) . x∈A
Let ∂W (x) denote the subdifferential of W at x. Recall also the following known fact from the theory of convex functions (see, e.g., [20, Theorem 23.5]). Lemma 3.1. Let v ∈ ∂W (x) for every x ∈ [x1 , x2 ], where x1 = x2 and [x1 , x2 ] = {tx1 + (1 − t)x2 , t ∈ [0, 1]}. Then [x1 , x2 ] ⊂ ∂W ∗ (v). In particular, W ∗ is not differentiable at v. In addition to the singular set S ⊂ A of all points x such that N∂At ,x , where t = ϕ(x), contains more than one element, we introduce another set of degeneracy of n defined by U = x ∈ A \ S: there is x ∈ ∂At , t = ϕ(x), such that x = x and n(x) ∈ N∂At ,x . Proposition 3.2. (i) Consider the set C = ∂At for some fixed t. Then the set n(U ∩ C) in S d−1 has Hd−1 -measure zero. (ii) The sets T (U ) and ϕ(x) · N∂Aϕ(x) ,x T(S) := x∈S
have ν-measure zero. Proof. (i) It is sufficient to prove our claim locally on C in a small neighborhood O of a point x0 where n(x0 ) is unique. We may assume that n(x0 ) = −ed , the surface C ∩ O is the graph of a convex function W : B ⊂ Rd−1 → R, where B is an open ball containing 0, and that W attains minimum at 0. In addition, we may assume that ∂W (B) is a bounded set. We parameterize C ∩ O in the following way: B (x1 , . . . , xd−1 ) → x1 , . . . , xd−1 , W (x) . Since W is Lipschitzian on B, the surface measure Hd−1 on C ∩ O corresponds to the measure 1 (1 + |∇W |2 ) 2 Hd−1 on Rd−1 . The Gauss map n is given by n=
1 1 + |∇W |2
(−∂x1 W, . . . , −∂xd−1 W, 1).
/ S ∩ C ∩ O. The proThis holds for every (x1 , . . . , xd−1 ) ∈ B such that (x1 , . . . , xd−1 , W (x)) ∈ jection of S ∩ C ∩ O on B coincides with the points of non-differentiability of W . It is convenient to identify the half-sphere S d−1 ∩ {yd 0} with its projection Π d−1 on Rd−1 and n with the mapping n: − √ ∇W 2 taking values in Π d−1 . Note that the surface measure 1+|∇W |
V.I. Bogachev, A.V. Kolesnikov / Journal of Functional Analysis 256 (2009) 940–957
on S d−1 has the form md−1 := √
1 Hd−1 1−|y|2
951
in the local chart (on the set where y12 + · · · +
2 yd−1 < 1)
2 (y1 , . . . , yd−1 ) → y1 , . . . , yd−1 , 1 − y12 − · · · − yd−1 . Hence we have to show that md−1 F ◦ ∇W (U ) = 0,
(3.1)
where U is the corresponding projection of U ∩ C and F (x) = −
x 1 + |x|2
.
The mapping F is smooth and non-degenerate everywhere. Hence in order to prove (3.1) it suffices to show that Hd−1 (∇W (U )) = 0. Let us set W := +∞ outside of B. The Legendre transform W ∗ is finite everywhere. By Lemma 3.1, the set ∇W (U ) is contained in the set of non-differentiability of W ∗ , hence has Hd−1 -measure zero. (ii) By Fubini’s theorem, it suffices to show that for each t the intersection of the set T (U ) with the sphere of radius t has zero Hd−1 -measure. By construction, these intersection coincide with the sets T (∂At ∩ U ) defined similarly. Therefore, the claim for T (U ) follows by assertion (i). In order to see that the set T(S) has ν-measure zero, we observe that its intersection with the set T (A \ S) of full ν-measure belongs to T (U ), which is clear from the definition of U . 2 Now we can show that T is invertible. Corollary 3.3. The mapping T is injective on a set of full μ-measure. Hence there exists a measurable mapping T −1 : Br → A such that T (T −1 (y)) = y for ν-almost all y and T −1 (T (x)) = x for μ-almost all x. Proof. Since the equality T (x1 ) = T (x2 ) may only happen if ϕ(x1 ) = ϕ(x2 ), i.e., x1 and x2 belong to the same level set ∂At , it follows from our previous considerations that T is injective outside the set T −1 (T(S) ∪ T (U )). This set has μ-measure zero because the set T(S) ∪ T (U ) has ν-measure zero by the above proposition. 2 Theorem 3.4. The mapping T constructed above is unique in the following sense: if a measurable mapping T0 : A → Br is such that ν = μ ◦ T0−1 and T0 = ϕ0 · n0 , where ϕ0 : A → [0, r] is a continuous function with convex sub-level sets At,0 := {ϕ0 t} and n0 is the corresponding Gauss map, then T = T0 μ-a.e. Proof. Let us show that ϕ0 (x) ϕ(x) for all x ∈ A. This will yield the equality ϕ0 = ϕ because otherwise there is t such that μ({ϕ0 t}) > μ({ϕ t}), which is impossible since both sides equal ν(Bt ). Set Ct := {x ∈ A: x ∈ ∂At,0 ∩ At }, Ct , Uτ := tτ
Dt := x ∈ A: x ∈ ∂At \ Int(At,0 ) , Vτ := Dt . tτ
952
V.I. Bogachev, A.V. Kolesnikov / Journal of Functional Analysis 256 (2009) 940–957
We observe that for every x ∈ ∂At \ Int(At,0 ) there exists t t depending on x such that x ∈ Ct . Indeed, if x ∈ ∂At,0 ∩ ∂At , then x ∈ Ct . Otherwise one has x ∈ ∂At ,0 for some t = t (x) > t. Since x ∈ ∂At , we have x ∈ Int(At ). Hence Vτ ⊂ Uτ .
(3.2)
For every Borel set C ⊂ A, set T(C) :=
ϕ(x) · N∂Aϕ(x) ,x .
x∈C
Let us show that T0 (Ct ) ⊂ T(Dt ).
(3.3)
Suppose that x0 ∈ ∂At,0 ∩ At , t = ϕ0 (x0 ), v = nx0 ,0 ∈ N∂At,0 ,x0 . We show that tv ∈ T (∂At \ Int(At,0 )). Let us consider the support hyperplane Lx0 ,v ⊥ v to At,0 at x0 . If x0 ∈ ∂At and v ∈ N∂At ,x0 , the claim is obvious. Otherwise Lx0 ,v splits At in two convex parts At and At . Since Lx0 ,v is a support hyperplane to At,0 , one of these parts, say, At , and At,0 are separated by Lx0 ,v . There exists a hyperplane L parallel to Lx0 ,v that is supporting to At and passes through a point x1 ∈ ∂At . Then v ∈ N∂At ,x1 . Note that one could also take x2 such that T (x2 ) = T0 (x0 ) (such x2 exists for a.e. x0 ); then x2 ∈ ∂At \ Int(At,0 ). This proves (3.3). Hence we have T0 (Ut ) ⊂ T(Vt ),
0 t r.
(3.4)
Suppose now that there exists x0 such that ϕ0 (x0 ) > ϕ(x0 ). Then, by the continuity of ϕ and ϕ0 , there is τ > 0 for which the inclusion in (3.2) is strict and there is a neighborhood in Uτ not intersecting Vτ . Therefore, μ(Uτ ) > μ(Vτ ). Taking into account that T is injective on a full measure set, we obtain ν T0 (Uτ ) = μ T0−1 T0 (Uτ ) μ(Uτ ) > μ(Vτ ) = ν T (Vτ ) , which contradicts (3.4) because ν(T (Vτ )) = ν(T(Vτ )) according to Corollary 3.2.
2
4. Duality Now we consider certain duality properties of the potential ϕ. The duality principle of Kantorovich is a powerful tool for investigating the Monge–Kantorovich problem. In our case we also have a kind of the duality formula which relates the potential ϕ to some function ψ that can be considered as the support function of the family of level sets At . Note that some interesting duality results for the solution of the Monge–Kantorovich problem on a sphere with applications to the prescribed Gauss curvature problem have been obtained in [18]. For every y ∈ Br we set ψ(y) =
sup x: ϕ(x)|y|
x, y .
(4.1)
V.I. Bogachev, A.V. Kolesnikov / Journal of Functional Analysis 256 (2009) 940–957
953
Note that the restriction of ψ to ∂B|y| coincides with the support function SA|y| of A|y| = {x: ϕ(x) |y|}, where the support function is defined by SA|y| (v) := sup v, x . x∈A|y|
Lemma 4.1. For ν-almost all y one has
ψ(y) = T −1 (y), y .
(4.2)
Proof. It is clear that the supremum on the right-hand side of (4.1) is attained at a point p such that y ∈ N∂Aϕ(p) ,p . This implies that p coincides with T −1 (y) for ν-almost all y, hence is ν-almost everywhere well-defined, which yields our claim. 2 Now we show how to describe ψ as a limit of certain functions depending on pre-limit potentials ϕt . Recall that the Legendre transform Wt∗ satisfies the inequality Wt (x) + Wt∗ (y) x, y .
(4.3)
An equality holds if and only if y ∈ ∂Wt (x) and x ∈ ∂Wt∗ (y). Moreover, Wt and Wt∗ satisfy the identities ∇Wt∗ ◦ ∇Wt (x) = x,
∇Wt ◦ ∇Wt∗ (y) = y
almost everywhere on the sets A and ∇Wt (A). Since ∇Wt = |Tt |t Tt , one has Tt−1 (y) = ∇Wt∗ |y|t y . In what follows we denote by I the identity matrix and by Iz the orthogonal projector on the one-dimensional vector subspace generated by z, i.e., Iz v =
v, z z . |z| |z|
We have found a sequence tn → +∞ for which the mappings Ttn converge to T almost everywhere on A, hence converges in measure μ. For this sequence, the following holds. converge to T −1 in measure ν. Hence there exists a subsequence Lemma 4.2. The mappings Tt−1 n −1 −1 tn → ∞ such that Tt → T ν-almost everywhere. n
= ν, for any ν-measurable function f , the functions f ◦ Ttn Proof. Since μ ◦ T −1 = μ ◦ Tt−1 n converge to f ◦ T in measure μ (see, e.g., [4, Corollary 9.9.11]). Hence the mappings T −1 ◦ Ttn converge to T −1 ◦ T = I in measure μ. Therefore, for every c > 0 one has ν y: Tt−1 (y) − T −1 (y) c = μ x ∈ A: x − T −1 Ttn (x) c → 0 n
954
V.I. Bogachev, A.V. Kolesnikov / Journal of Functional Analysis 256 (2009) 940–957
as n → ∞, which completes the proof.
2
Theorem 4.3. Let a function ψt be defined by the relation t t Wt∗ (z) = |z| 1+t ψt z|z|− 1+t . Equivalently, ψt (y) =
Wt∗ (y|y|t ) . |y|t
Then one has ψ = limtn →∞ ψtn almost everywhere for some sequence {tn }. Proof. Note that it is consistent with our previous choice of Wt to assume that Wt∗ (0) = 0. Indeed, Wt (x) x, y − Wt∗ (y), hence taking y = 0 we find Wt (x) 0. Taking any x0 ∈ ∂Wt∗ (0) we easily obtain Wt (x0 ) = 0. Indeed, for (x0 , 0) inequality (4.3) becomes an equality, hence Wt (x0 ) + Wt∗ (0) = x0 , 0 = 0. The inequality Wt∗ (a) − Wt∗ (b) a − b, ∇W ∗ (a) yields, by substituting b = 0 and a = y|y|t , that
ψt (y) ∇Wt∗ y|y|t , y = y, Tt−1 (y) . Similarly, if a = 0 and b = y|y|t , one has ψt (y) v, y for any v ∈ ∂Wt∗ (0). In particular, ψt (y) diam(A)|y|. One has ∇Wt∗ (z) =
− t 1 t z t t − 1+t |z| · ψt z|z| 1+t + I − Iz ∇ψt z|z|− 1+t . 1+t |z| 1+t
Substituting z = |y|t y we obtain Tt−1 (y) =
t y t ψt (y) 2 + I − Iy ∇ψt (y). 1+t 1+t |y|
(4.4)
Taking the inner product with y we find Tt−1 (y), y =
t 1
ψt (y) + ∇ψt (y), y . 1+t 1+t
In view of Lemma 4.2 and equality (4.2) it suffices to show that everywhere for some {tn }. Indeed, since
Tt−1 (y), y
ψt Tt−1 (y), y ,
1 1+tn ∇ψtn (y), y
(4.5) → 0 almost
we obtain from (4.5) that
t −1 1
Tt (y), y + ∇ψt (y), y . 1+t 1+t
V.I. Bogachev, A.V. Kolesnikov / Journal of Functional Analysis 256 (2009) 940–957
955
Hence
ψt Tt−1 (y), y ∇ψt (y), y . Taking into account that ψt −diam(A)|y|, we see that { ∇ψt (y), y } is uniformly bounded from below. The integration by parts formula yields
∇ψt (y), y dx = −d ψt dx + |y|ψt dHd−1 . Br
Br
∂Br
Applying again the estimate |ψt | diam(A)|y| we obtain
sup ∇ψt (y), y dx < ∞, t
Br
hence supt ∇ψt (y), y L1 (Br ) < ∞. Therefore,
1 ∇ψt (y), y 1 = 0. L (Br ) t→∞ 1 + t lim
2
Extracting a subsequence we complete the proof.
Remark 4.4. Taking a scalar product of (4.4) with any vector v⊥y we obtain the equality ∂v ψn (y) = Tn−1 (y), v . Let us set ∂v ψ(y) := lim ∂v ψn (y). tn →∞
In view of convergence Tn → T this definition makes sense. Moreover, we have
∂v ψ(y) = T −1 (y), v ,
(4.6)
for any v⊥y. Taking into account (4.2) we obtain the following remarkable relation: T −1 (y) =
ψ(y) ∂ei (y) ψ(y)ei (y), e1 (y) + |y| d
i=2
where {ei (y)} is an orthonormal system of unit vectors chosen in such a way that e1 (y) = y/|y| and ei (y)⊥y, 2 i d. Remark 4.5. Let us see what happens in the limit with the duality formula Wt (x) + Wt∗ (z) x, z . It can be rewritten as 1 t+2 ϕ (x) + |y|t ψt (y) x, y |y|t t +2 t
956
V.I. Bogachev, A.V. Kolesnikov / Journal of Functional Analysis 256 (2009) 940–957
∇ϕt by letting z := y|y|t . If y = Tt (x) = ϕt |∇ϕ |∇ϕt | 1+t , then an equality holds. It is known that t| this is possible only if the pair (x, y) belongs to the graph of Tt . Hence we obtain the following duality relation: 1
− t
1 2 ϕt (x) ∇ϕt (x) t+1 + ψt Tt (x) = x, Tt (x) . t +2 In the limit t → ∞ we find
ψ T (x) = x, T (x) . It is worth noting that for the constructed transformation T one can establish a change of variables formula involving certain analogs of Alexandroff’s determinants. Such formulas which neglect singular components are known for optimal transformations and triangular transformations (concerning the latter, see [4, Chapter 10] and [5]). Acknowledgments This work was supported by the RFBR projects 07-01-00536, 08-01-91205-JF, GFEN-06-0139003, RF President Grant MD-764.2008.1, DFG Grant 436 RUS 113/343/0(R), the Russian– Ukrainian RFBR Grant, ARC Discovery Grant DP0663153, and the SFB 701 at the University of Bielefeld. The second-named author thanks Franck Barthe for his hospitality and fruitful discussions during the author’s visit to the University of Paul Sabatier in Toulouse, where this work was partially done. References [1] A.D. Alexandroff, Existence and uniqueness of a convex surface with a given integral curvature, Dokl. Akad. Nauk USSR 35 (1942) 131–134 (in Russian). [2] L. Ambrosio, Lecture notes on optimal transport problems, in: Mathematical Aspects of Evolving Interfaces, Funchal, 2000, in: Lecture Notes in Math., vol. 1812, Springer, Berlin, 2003, pp. 1–52. [3] B. Andrews, Motion of hypersurfaces by Gauss curvature, Pacific J. Math. 195 (1) (2000) 1–34. [4] V.I. Bogachev, Measure Theory, vols. 1, 2, Springer, Berlin, 2007. [5] V.I. Bogachev, A.V. Kolesnikov, K.V. Medvedev, Triangular transformations of measures, Mat. Sb. 196 (3) (2005) 309–335 (in Russian); English transl.: Sb. Math. 196 (3) (2005) 3–30. [6] Y. Brenier, Polar factorization and monotone rearrangement of vector valued functions, Comm. Pure Appl. Math. 44 (1991) 375–417. [7] Y.-G. Chen, Yo. Giga, S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flows, J. Differential Geom. 33 (1991) 749–786. [8] L.C. Evans, Partial differential equations and Monge–Kantorovich mass transfer, in: Current Developments in Mathematics, Internat. Press, Boston, MA, 1999, pp. 65–126. [9] L.C. Evans, J. Spruck, Motion of level-sets by mean curvature. I, J. Differential Geom. 33 (1991) 635–681. [10] M.E. Gage, R.S. Hamilton, The heat equation shrinking convex plane curves, J. Differential Geom. 23 (1986) 69–96. [11] Y. Giga, Surface Evolution Equations. A Level Sets Approach, Birkhäuser, Basel, 2006. [12] M. Grayson, The heat equation shrinks embedded closed curves to round points, J. Differential Geom. 26 (1987) 285–314. [13] G. Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differential Geom. 20 (1984) 237–266. [14] L.V. Kantorovich, On the translocation of masses, C. R. (Dokl.) Acad. Sci. Nauk URSS 37 (7–8) (1942) 199–201. [15] J. Lott, C. Villani, Ricci curvature for metric-measure spaces via optimal transport, preprint, http://www.math.lsa. umich.edu/~lott/. [16] R.J. McCann, Existence and uniqueness of monotone measure-preserving maps, Duke Math. J. 80 (1995) 309–323. [17] R.J. McCann, P. Topping, Ricci flows, entropy and optimal transportation, 2007, preprint.
V.I. Bogachev, A.V. Kolesnikov / Journal of Functional Analysis 256 (2009) 940–957
957
[18] V. Oliker, Embeddings of S n into Rn+1 with given integral Gauss curvature and optimal mass transport on S n , Adv. Math. 213 (2007) 600–620. [19] S.T. Rachev, L. Rüschendorf, Mass Transportation Problems, vols. I, II, Springer, New York, 1998. [20] R.T. Rockafellar, Convex Analysis, Princeton Univ. Press, Princeton, NJ, 1970. [21] K.-T. Sturm, On the geometry of metric measure spaces. I, II, Acta Math. 196 (1) (2006) 65–131, 133–177. [22] P. Topping, L-optimal transportation for Ricci flows, preprint, 2007. [23] K. Tso, Deforming a hypersurfaces by Gauss–Kronecker curvature, Comm. Pure Appl. Math. 38 (1985) 876–882. [24] C. Villani, Topics in Optimal Transportation, Amer. Math. Soc., Providence, RI, 2003.
Journal of Functional Analysis 256 (2009) 959–991 www.elsevier.com/locate/jfa
Indecomposable representations of quivers on infinite-dimensional Hilbert spaces Masatoshi Enomoto a , Yasuo Watatani b,∗ a College of Business Administration and Information Science, Koshien University, Takarazuka, Hyogo 665, Japan b Department of Mathematical Sciences, Kyushu University, Hakozaki, Fukuoka, 812-8581, Japan
Received 29 December 2007; accepted 5 December 2008
Communicated by D. Voiculescu
Abstract We study indecomposable representations of quivers on separable infinite-dimensional Hilbert spaces by bounded operators. We exhibit several concrete examples and investigate duality theorem between reflection functors. We also show a complement of Gabriel’s theorem. Let Γ be a finite, connected quiver. If its underlying undirected graph contains one of extended Dynkin diagrams A˜ n (n 0), D˜ n (n 4), E˜ 6 , E˜ 7 and E˜ 8 , then there exists an indecomposable representation of Γ on separable infinite-dimensional Hilbert spaces. © 2008 Elsevier Inc. All rights reserved. Keywords: Quiver; Indecomposable representation; Dynkin diagram; Reflection functor; Hilbert space
1. Introduction We studied the relative position of several subspaces in a separable infinite-dimensional Hilbert space in [6] after Gelfand and Ponomarev [11]. In this paper we extend it to the relative position of several subspaces along quivers. More generally we study representations of quivers on infinite-dimensional Hilbert spaces by bounded operators. We call them Hilbert representations for short. Gabriel’s theorem says that a finite, connected quiver has only finitely many indecomposable representations if and only if the underlying undirected graph is one of Dynkin diagrams * Corresponding author.
E-mail address: [email protected] (Y. Watatani). 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.12.011
960
M. Enomoto, Y. Watatani / Journal of Functional Analysis 256 (2009) 959–991
An , Dn , E6 , E7 , E8 [8]. The theory of representations of quivers on finite-dimensional vector spaces has been developed by Bernstein, Gelfand and Ponomarev [2], Brenner [3], Donovan and Freislish [5], Dlab and Ringel [4], Gabriel and Roiter [10], Kac [18], Nazarova [25], . . . . Infinitedimensional representatios of quivers have also been investigated in purely algebraic setting. See Krause and Ringel [19] and Reiten and Ringel [27]. Furthermore locally scalar representations of quivers in the category of Hilbert spaces were introduced by Kruglyak and Roiter [21]. They associate operators and their adjoint operators with arrows and classify them up to the unitary equivalence. They proved an analog of Gabriel’s theorem. Their study is connected with representations of *-algebras generated by linearly related orthogonal projections, see for example, S. Kruglyak, V. Rabanovich and Y. Samoilenko [22]. In this paper we study duality theorem between reflection functors and the existence of indecomposable representations of quivers on infinite-dimensional Hilbert spaces. We associate bounded operators with arrows but we do not associate their adjoint operators simultaneously as in [21]. In particular if we consider a certain quiver Γ whose underlying undirected graph is the extended Dynkin diagram D˜ 4 , then indecomposability of Hilbert representations of Γ is reduced to indecomposability of systems of four subspaces studied in [11] and [6]. We consider a complement of Gabriel’s theorem for Hilbert representations and prove one direction: If the underlying undirected graph of a finite, connected quiver Γ contains one of extended Dynkin diagrams A˜ n (n 0), D˜ n (n 4), E˜ 6 , E˜ 7 and E˜ 8 , then there exists an indecomposable representation of Γ on separable infinite-dimensional Hilbert spaces. The result does not depend on the choice of orientation. But we cannot prove the converse. In fact if the converse were true, then a long standing problem in [13] on transitive lattices of subspaces of Hilbert spaces would be settled. Recall that we study relative position of n subspaces in a separable infinite-dimensional Hilbert space in [6]. See Y.P. Moskaleva and Y.S. Samoilenko [23] on a connection with *algebras generated by projections. Let H be a Hilbert space and E1 , . . . En be n subspaces in H . Then we say that S = (H ; E1 , . . . , En ) is a system of n subspaces in H or an n-subspace system in H . A system S is called indecomposable if S cannot be decomposed into a non-trivial direct sum. For any bounded linear operator A on a Hilbert space K, we can associate a system SA of four subspaces in H = K ⊕ K by SA = H ; K ⊕ 0, 0 ⊕ K, graph A, (x, x); x ∈ K . In particular on a finite-dimensional space, Jordan blocks correspond to indecomposable systems. Moreover on an infinite-dimensional Hilbert space, the above system SA is indecomposable if and only if A is strongly irreducible, which is an infinite-dimensional analog of a Jordan block, see books by Jiang and Wang [15,16]. For example, a unilateral shift operator is a typical example of strongly irreducible operator. Such a system of four subspaces give an indecomposable Hilbert representation of a quiver with underlying undirected graph D˜ 4 . We transform these representations and make up indecomposable Hilbert representations of other quivers in this paper. In purely algebraic case many such functors are introduced, see [5,10] and [28], for example. We follow some of their constructions. But we have not yet checked all such functors preserve indecomposability in infinite-dimensional Hilbert setting in general. We need to prove the indecomposability of the Hilbert representations in our concrete examples directly. One of our main theorems of the paper is the following: Let Γ be a finite, connected quiver. If its underlying undirected graph contains one of extended Dynkin diagrams A˜ n (n 0), D˜ n (n 4), E˜ 6 , E˜ 7 and E˜ 8 , then there exists an indecomposable representation of Γ on sep-
M. Enomoto, Y. Watatani / Journal of Functional Analysis 256 (2009) 959–991
961
arable infinite-dimensional Hilbert spaces. There were two difficulties which did not appear in finite-dimensional case. Firstly we need to find indecomposable, infinite-dimensional representations of a certain class of Γ . We constructed them by studying the relative position of several subspaces along quivers, where vertices and arrows are represented by subspaces and natural inclusion maps. Secondly we need to change the orientation of the quiver preserving indecomposability. Here comes reflection functors. Being different from finite-dimensional case, we need to check the co-closedness condition at sources to show that indecomposability is preserved under reflection functors. We introduce a certain nice class, called positive-unitary diagonal Hilbert representations, such that co-closedness is easily checked and preserved under reflection functors at any source. We believe that there exists an analogy between study of Hilbert representations of quivers and subfactor theory invented by V. Jones [17]. In fact Dynkin diagrams also appear in the classification of subfactors, see, for example, Goodman, de la Harpe and Jones [9], Evans and Kawahigashi [7]. But we have not yet understood the full relations between them. There exists a close interplay between finite-dimensional representations of quivers and finitedimensional representations of path algebras in purely algebraic sense. Any Hilbert representation of a quiver gives an operator algebra representation of the corresponding path algebra. Therefore we expect some relation between Hilbert representations of quivers and certain operator algebras associated with quivers. There exist some related works. See, for example, P. Muhly [24], D.W. Kribs and S.C. Power [20] and B. Solel [29]. But the relation is not so clear for us. Throughout the paper a projection means an operator e with e2 = e = e∗ and an idempotent means an operator p with p 2 = p. By a subspace we mean a closed subspace unless otherwise stated. In purely algebraic setting, it is known that if a finite-dimensional algebra R is not of representation-finite type, then there exist indecomposable R-modules of infinite length as in M. Auslander [1]. Since we consider bounded operator representations on Hilbert spaces, the result in [1] cannot be applied directly. 2. Representations of quivers A quiver Γ = (V , E, s, r) is a quadruple consisting of the set V of vertices, the set E of arrows, and two maps s, r : E → V , which associate with each arrow α ∈ E its support s(α) and range r(α). We sometimes denote by α : x → y an arrow with x = s(α) and y = r(α). Thus a quiver is just a directed graph. We denote by |Γ | the underlying undirected graph of a quiver Γ . A quiver Γ is said to be connected if |Γ | is a connected graph. A quiver Γ is said to be finite if both V and E are finite sets. Definition. Let Γ = (V , E, s, r) be a finite quiver. We say that (H, f ) is a Hilbert representation of Γ if H = (Hv )v∈V is a family of Hilbert spaces and f = (fα )α∈E is a family of bounded linear operators fα : Hs(α) → Hr(α) . Definition. Let Γ = (V , E, s, r) be a finite quiver. Let (H, f ) and (K, g) be Hilbert representations of Γ. A homomorphism T : (H, f ) → (K, g) is a family T = (Tv )v∈V of bounded operators Tv : Hv → Kv satisfying, for any arrow α ∈ E Tr(α) fα = gα Ts(α) .
962
M. Enomoto, Y. Watatani / Journal of Functional Analysis 256 (2009) 959–991
The composition T ◦ S of homomorphisms T and S is defined by (T ◦ S)v = Tv ◦ Sv for v ∈ V . Thus we have obtained a category H Rep(Γ ) of Hilbert representations of Γ . We denote by Hom((H, f ), (K, g)) the set of homomorphisms T : (H, f ) → (K, g). We denote by End(H, f ) := Hom((H, f ), (H, f )) the set of endomorphisms. We denote by Idem(H, f ) := T ∈ End(H, f ) T 2 = T the set of idempotents in End(H, f ). Let 0 = (0v )v∈V be the family of zero endomorphisms 0v and I = (Iv )v∈V be the family of identity endomorphisms Iv . The both 0 and I are in Idem(H, f ). Let Γ = (V , E, s, r) be a finite quiver and (H, f ), (K, g) be Hilbert representations of Γ. We say that (H, f ) and (K, g) are isomorphic, denoted by (H, f ) (K, g), if there exists an isomorphism ϕ : (H, f ) → (K, g), that is, there exists a family ϕ = (ϕv )v∈V of bounded invertible operators ϕv ∈ B(Hv , Kv ) such that, for any arrow α ∈ E, ϕr(α) fα = gα ϕs(α) . We say that (H, f ) is a finite-dimensional representation if Hv is finite-dimensional for all v ∈ V . And (H, f ) is an infinite-dimensional representation if Hv is infinite-dimensional for some v ∈ V . 3. Indecomposable representations of quivers In this section we shall introduce a notion of indecomposable representation, that is, a representation which cannot be decomposed into a direct sum of smaller representations anymore. Definition (Direct sum). Let Γ = (V , E, s, r) be a finite quiver. Let (K, g) and (K , g ) be Hilbert representations of Γ. Define the direct sum (H, f ) = (K, g) ⊕ (K , g ) by Hv = Kv ⊕ Kv
(for v ∈ V )
and fα = gα ⊕ gα
(for α ∈ E).
We say that a Hilbert representation (H, f ) is zero, denoted by (H, f ) = 0, if Hv = 0 for any v ∈V. Definition (Indecomposable representation). A Hilbert representation (H, f ) of Γ is called decomposable if (H, f ) is isomorphic to a direct sum of two non-zero Hilbert representations. A non-zero Hilbert representation (H, f ) of Γ is said to be indecomposable if it is not decomposable, that is, if (H, f ) ∼ = (K, g) ⊕ (K , g ) then (K, g) ∼ = 0 or (K , g ) ∼ = 0. We start with an easy fact. Let H be a Hilbert space and K1 , K2 be closed subspaces of H . Assume that K1 ∩ K2 = 0 and H = K1 + K2 . But we do not assume that K1 and K2 are orthogonal. Let T : H → H be a bounded operator with T Ki ⊂ Ki for i = 1, 2. Define Si = T |Ki : Ki → Ki . Consider the (orthogonal) direct sum K1 ⊕ K2 and the bounded operator S1 ⊕ S2 on K1 ⊕ K2 . Define a bounded invertible operator ϕ : H → K1 ⊕ K2 by ϕ(h) = (h1 , h2 ) for h = h1 + h2 with hi ∈ Ki , as in the proof of [6, Lemma 2.1.] Then we have T = ϕ −1 ◦ (S1 ⊕ S2 ) ◦ ϕ. The following proposition is used frequently to show the indecomposability in concrete examples.
M. Enomoto, Y. Watatani / Journal of Functional Analysis 256 (2009) 959–991
963
Proposition 3.1. Let (H, f ) be a Hilbert representation of a quiver Γ . Then the following conditions are equivalent: (1) (H, f ) is indecomposable. (2) Idem(H, f ) = {0, I }. Proof. ¬(1) ⇒ ¬(2): Assume that (H, f ) is not indecomposable. Then there exist non-zero representations (K, g) and (K , g ) of Γ , such that (H, f ) ∼ = (K, g) ⊕ (K , g ). For any x ∈ V , define the projection Qx ∈ B(Kx ⊕ Kx ) of Kx ⊕ Kx onto Kx . Then Q := (Qx )x∈V is in End(K ⊕ K , g ⊕ g ), because Qr(α) gα , gα = (gα , 0) = gα , gα Qs(α) for any α ∈ E. Since there exist v, w ∈ V such that Kv = 0 and Kw = 0, we have Qv = 0 and Qw = I . Thus Q = 0 and Q = I . Let ϕ = (ϕx )x∈V : (H, f ) → (K, g) ⊕ (K , g ) be an isomorphism. Put Px = (ϕx )−1 Qx ϕx for x ∈ V and P := (Px )x∈V ∈ Idem(H, f ). Then P = 0 and P = I. ¬(2) ⇒ ¬(1): Assume that there exists P ∈ Idem(H, f ) with P = 0 and P = I . Thus there exist v ∈ V and w ∈ V such that Pv = 0v , Pw = Iw . For any x ∈ V , define closed subspaces Kx = Px (Hx )
and Kx = (I − Px )(Hx ).
Then K := (Kx )x = 0, K := (Kx )x = 0 and H ∼ = K ⊕ K . For any α ∈ E, let x = s(α) and y = r(α). Since fα Px = Py fα , we have fα Kx ⊂ Ky . Similarly, fα (I − Px ) = (I − Py )fα implies that fα Kx ⊂ Ky . We can define gα = fα |Kx : Kx → Ky and gα = fα |Kx : Kx → Ky . Put g = (gα )α and g = (gα )α . Then (K, g) and (K , g ) are representations of Γ . Define ϕx : Hx → Kx ⊕ Kx by ϕx (ξ ) = (Px ξ, (I − Px )ξ ) for ξ ∈ Hx . Then ϕ := (ϕx )x∈V : (H, f ) → (K, g) ⊕ (K , g ) is an isomorphism. Since K := (Kx )x = 0 and K := (Kx )x = 0, (H, f ) is decomposable. 2 Remark. (1) The proof of the above Proposition 3.1 shows that (H, f ) is decomposable if and only if there exist non-zero families K = (Kx )x∈V and K = (Kx )x∈V of closed subspaces Kx and Kx of Hx with Kx ∩ Kx = 0 and Kx + Kx = Hx such that fα Kx ⊂ Ky and fα Kx ⊂ Ky for any arrow α : x → y. (2) In the statement of the above Proposition 3.1, we cannot replace the set Idem(H, f ) of idempotents of endomorphisms by the set of projections of endomorphisms. For example, let H0 = C2 . Fix an angle θ with 0 < θ < π/2. Put H1 = C(1, 0) and H2 = C(cos θ, sin θ ). Then the system (H0 ; H1 , H2 ) of two subspaces is isomorphic to 2 C ; C ⊕ 0, 0 ⊕ C ∼ = (C; C, 0) ⊕ (C; 0, C). Hence (H0 ; H1 , H2 ) is decomposable. See Example 2 in [6] and Remark after it. Now consider the following quiver Γ : α1
α2
◦1 −→ ◦0 ←− ◦2 . Define a Hilbert representation (H, f ) of Γ by H = (Hi )i=0,1,2 and canonical inclusion maps fi = fαi : Hi → H0 for i = 1, 2. Then the Hilbert representation (H, f ) is also decomposable,
964
M. Enomoto, Y. Watatani / Journal of Functional Analysis 256 (2009) 959–991
see Example 3 below in this paper. But for any P = (Pi )i=0,1,2 ∈ End(H, f ), if Pi ∈ B(Hi ) is a projection for i = 0, 1, 2, then P = 0 or P = I . In fact P0 (Hi ) ⊂ Hi for i = 1, 2. Let e1 ∈ B(H0 ) and e2 ∈ B(H0 ) be the projections of H0 onto H1 and H2 . Then the C ∗ -algebra C ∗ ({e1 , e2 }) generated by e1 and e2 is exactly B(H0 ) ∼ = M2 (C). Since P0 commutes with e1 and e2 , P0 = 0 or P0 = I . Because Pi = P0 |Hi , Pi = 0 or Pi = I simultaneously. Example 1. Let Γ be a loop with one vertex 1 and one arrow α : 1 → 1, that is, the underlying undirected graph is an extended Dynkin diagram A˜ 0 . Let H1 = 2 (N) and fα = S : H1 → H1 be a unilateral shift. Then the Hilbert representation (H, f ) is infinite-dimensional and indecomposable. In fact, any T ∈ Idem(H, f ) can be identified with T ∈ B(2 (N)) with T 2 = T and T S = ST . Since T commutes with a unilateral shift S, the operator T is a lower triangular Toeplitz matrix. Since T is an idempotent, T = 0 or T = I . Thus (H, f ) is indecomposable. Replacing S by S + λI for λ ∈ C, we obtain a family of infinite-dimensional, indecomposable Hilbert representations (H λ , f λ ) of Γ . Since (H λ , f λ ) and (H μ , f μ ) are isomorphic if and only if S + λI and S + μI is similar, we have uncountably many infinite-dimensional, indecomposable Hilbert representations of Γ . Example 2. Let Γ = (V , E, s, r) be a quiver whose underlying undirected graph is an extended Dynkin diagram A˜ n , (n 1). Then there exist uncountably many infinite-dimensional, indecomposable Hilbert representations of Γ . For example, consider
Define a Hilbert representation (H, f ) of Γ by H1 = H2 = · · · = Hn+1 = 2 (N), fα2 = fα3 = · · · = fαn+1 = I and fα1 = S, the unilateral shift. Let P = (Pk )k∈V ∈ Idem(H, f ). Then P1 = P2 = · · · = Pn+1
and SP1 = P2 S.
Since P1 is an idempotent and SP1 = P1 S, we have P1 = 0 or P1 = I . This implies P = 0 or P = I . Therefore (H, f ) is indecomposable. Replacing S by S + λI for λ ∈ C, we obtain uncountably many infinite-dimensional, indecomposable Hilbert representations of Γ . Example 3. Let L be a Hilbert space and E1 , . . . En be n subspaces in L. Then we say that S = (L; E1 , . . . , En ) is a system of n subspaces in L. A system S is called indecomposable if S cannot be decomposed into a non-trivial direct sum, see [6]. Consider the following quiver Γn = (V , E, s, r)
Define a Hilbert representation (H, f ) of Γn by Hk := Ek (k = 1, . . . , n), H0 := L and fk = fαk : Hk = Ek → H0 = L be the inclusion map. Then the system S of n subspaces is indecom-
M. Enomoto, Y. Watatani / Journal of Functional Analysis 256 (2009) 959–991
965
posable if and only if the corresponding Hilbert representation (H, f ) of Γ is indecomposable. In fact, assume that S is indecomposable. Let P = (Pk )k∈V ∈ Idem(H, f ). Then fk Pk = P0 fk . This implies P0 (Hk ) ⊂ Hk for k = 1, . . . , n. Since P0 is an idempotent and S is indecomposable, P0 = 0 or P0 = I by [6, Lemma 3.2]. Since fk Pk = P0 fk , Pk = 0 or Pk = I simultaneously. Thus P = 0 or P = I , that is, (H, f ) is indecomposable. Conversely assume that (H, f ) is indecomposable. Let R ∈ B(L) be an idempotent with R(Ek ) ⊂ Ek for k = 1, . . . , n. Define P = (Pk )k∈V by P0 = R and Pk = P0 |Hk . Then P ∈ Idem(H, f ). Therefore P = 0 or P = I . Thus R = O or R = I . Hence S is indecomposable. We can also show that two systems S and S of n subspaces are isomorphic if and only if the corresponding Hilbert representations (H, f ) and (H , f ) of Γ are isomorphic. Since there exist uncountably many, indecomposable systems of fours subspaces in an infinitedimensional Hilbert space as in [6], there exist uncountably many infinite-dimensional, indecomposable Hilbert representations of Γ4 whose underlying undirected graph is the extended Dynkin diagram D˜ 4 . In particular, let K = 2 (N) and A ∈ B(K) be a strongly irreducible operator studied in [15,16] for example, a unilateral shift. Define H1 = K ⊕ 0, H2 = 0 ⊕ K, H4 = (x, x) ∈ K ⊕ K x ∈ K . H3 = (x, Ax) ∈ K ⊕ K x ∈ K ,
H0 = K ⊕ K,
Let fk = fαk : Hk → H0 be the inclusion map for k = 1, 2, 3, 4. Put H (A) = (Hv )v∈V and f (A) = (fα )α∈E . Then (H (A) , f (A) ) is an infinite-dimensional, indecomposable Hilbert representation of D˜ 4 . Moreover let A and B be strongly irreducible operators on 2 (N). Then two indecomposable Hilbert representations (H (A) , f (A) ) and (H (B) , f (B) ) of D˜ 4 are isomorphic if and only if two operators A and B are similar. Example 4. Consider the following quiver Γ = (V , E, s, r)
Then underlying undirected graph is an extended Dynkin diagram E˜ 6 . Let K = 2 (N) and S a unilateral shift on K. We define a Hilbert representation (H, f ) := ((Hv )v∈V , (fα )α∈E ) of Γ as follows: Put H0 = K ⊕ K ⊕ K,
H1
H1 = K ⊕ 0 ⊕ K,
H2 = 0 ⊕ 0 ⊕ K,
H2 = 0 ⊕ K ⊕ 0, H1 = K ⊕ K ⊕ 0, and H2 = (x, x, x) ∈ K 3 x ∈ K . = (x, x, x) + (y, Sy, 0) ∈ K 3 x, y ∈ K
Then H1 is a closed subspace of H0 . In fact, let
966
M. Enomoto, Y. Watatani / Journal of Functional Analysis 256 (2009) 959–991
(xn , xn , xn ) + (yn , Syn , 0) = (xn + yn , xn + Syn , xn ) ∈ H1 converges to (a, b, c) ∈ H0 . Then xn → c, yn → a −c and c +S(a −c) = b. Define x = c and y = a − c. Then (a, b, c) = (x, x, x) + (y, Sy, 0) ∈ H1 . For any arrow α ∈ E, let fα : Hs(α) → Hr(α) be the canonical inclusion map. We shall show that the Hilbert representation (H, f ) is indecomposable. Take T = (Tv )v∈V ∈ Idem(H, f ). Since T ∈ End(H, f ), for any v ∈ {1, 2, 1 , 2 , 1 , 2 } and any x ∈ Hv , we have T0 x = Tv x. In particular, T0 Hv ⊂ Hv . Since H1 ∩ H1 = K ⊕ 0 ⊕ 0, H2 = 0 ⊕ K ⊕ 0 and H2 = 0 ⊕ 0 ⊕ K, T0 preserves these subspaces. Hence T0 is a block diagonal operator with T0 = P ⊕ Q ⊕ R ∈ B(K ⊕ K ⊕ K). Since T0 (H2 ) ⊂ H2 , for any x ∈ K, T0 (x, x, x) = (y, y, y) for some y ∈ K. Therefore P = Q = R and T0 = P ⊕ P ⊕ P . Moreover P is an idempotent, because so is T0 . Since T0 preserves H1 ∩ H1 = {(y, Sy, 0) ∈ K 3 | y ∈ K}, for any y ∈ K, there exists z ∈ K such that T0
y Sy 0
=
Py P Sy 0
=
z Sz . 0
Therefore P Sy = Sz = SP y for any y ∈ K, i.e., P S = SP . Since P is an idempotent, P = 0 or P = I. This means that T0 = 0 or T0 = I. Because T0 x = Tv x for any x ∈ Hv for v ∈ {1, 2, 1 , 2 , 1 , 2 }, we have Tv = 0 or Tv = I simultaneously. Thus T = 0 or T = I , that is Idem(H, f ) = {0, I }. Therefore (H, f ) is indecomposable. Example 5. We have a different kind of infinite-dimensional, indecomposable Hilbert representation (L, g) = ((Lv )v∈V , (gα )α∈E ) of the same Γ in Example 4 as follows: Let K = 2 (N) and S a unilateral shift on K. Define L0 = K ⊕ K ⊕ K,
L2 = 0 ⊕ (y, Sy) ∈ K 2 y ∈ K , L2 = (x, x) ∈ K 2 x ∈ K ⊕ 0, L2 = (x, 0, x) ∈ K 3 x ∈ K .
L1 = 0 ⊕ K ⊕ K,
L1 = K ⊕ K ⊕ 0, L1 = K ⊕ 0 ⊕ K,
For any arrow α ∈ E, let gα : Ls(α) → Lr(α) be the canonical inclusion map. We can similarly prove that the Hilbert representation (L, g) is indecomposable. We shall show that two Hilbert representations in Examples 4 and 5 are not isomorphic. In fact, on the contrary, suppose that there were an isomorphism ϕ = (ϕv )v∈V : (H, f ) → (L, g). Since any arrow is represented by the canonical inclusion, ϕ0 : H0 → L0 satisfies that ϕv = ϕ0 |Hv : Hv → Lv . This implies that ϕ0 (Hv ) ⊂ Lv for any v ∈ V . Since ϕ0 (H1 ) ⊂ L1 and ϕ0 (H1 ) ⊂ L1 , ϕ0 has a form such that ϕ0 =
0 B 0
A C 0
0 D E
.
M. Enomoto, Y. Watatani / Journal of Functional Analysis 256 (2009) 959–991
967
Since ϕ0 (H2 ) ⊂ L2 , for any z ∈ K there exists y ∈ K such that (0, Dz, Ez) = (0, y, Sy). Hence Ez = Sy = SDz, so that E = SD. Then Im ϕ0 ⊂ K ⊕ K ⊕ Im S = L0 . This contradicts the assumption that ϕ0 : H0 → L0 is onto. Therefore Hilbert representations (H, f ) and (L, g) of Γ are not isomorphic. 4. Reflection functors Reflection functors are crucially used in the proof of the classification of finite-dimensional, indecomposable representations of tame quivers. In fact many indecomposable representations of tame quivers can be reconstructed by iterating reflection functors on simple indecomposable representations. We cannot expect such a best situation in infinite-dimensional Hilbert representations. But reflection functors are still useful to show that some property of representations of quivers on infinite-dimensional Hilbert spaces does not depend on the choice of orientations and does depend on the fact underlying undirected graphs are (extended) Dynkin diagrams or not. Let Γ = (V , E, s, r) be a finite quiver. A vertex v ∈ V is called a sink if v = s(α) for any α ∈ E. Put E v = {α ∈ E | r(α) = v}. We denote by E the set of all formally reversed new arrows α for α ∈ E. Thus if α : x → y is an arrow, then α : x ← y. Definition. Let Γ = (V , E, s, r) be a finite quiver. For a sink v ∈ V , we construct a new quiver σv+ (Γ ) = (σv+ (V ), σv+ (E), s, r) as follows: All the arrows of Γ having v as range are reversed and all the other arrows remain unchanged. More precisely, σv+ (V ) = V ,
σv+ (E) = E \ E v ∪ E v ,
where E v = {α | α ∈ E v }. Definition (Reflection functor Φv+ ). Let Γ = (V , E, s, r) be a finite quiver. For a sink v ∈ V , we define a reflection functor at v Φv+ : H Rep(Γ ) → H Rep σv+ (Γ ) between the categories of Hilbert representations of Γ and σv+ (Γ ) as follows: For a Hilbert representation (H, f ) of Γ , we shall define a Hilbert representation (K, g) = Φv+ (H, f ) of σv+ (Γ ). Let hv :
Hs(α) → Hv
α∈E v
be a bounded linear operator defined by
hv (xα )α∈E v = fα (xα ). α∈E v
Define
Kv := Ker hv = (xα )α∈E v ∈ Hs(α) fα (xα ) = 0 . α∈E v
α∈E v
968
M. Enomoto, Y. Watatani / Journal of Functional Analysis 256 (2009) 959–991
Consider also the canonical inclusion map iv : Kv → Ku = Hu . For β ∈ E v , let
Pβ :
α∈E v
Hs(α) . For u ∈ V with u = v, put
Hs(α) → Hs(β)
α∈E v
be the canonical projection. Then define gβ : Ks(β) = Kv → Kr(β) = Hs(β)
by gβ = Pβ ◦ iv ,
that is gβ ((xα )α∈E v ) = xβ . For β ∈ / E v , let gβ = fβ . For a homomorphism T : (H, f ) → (H , f ), we shall define a homomorphism S = (Su )u∈V = Φv+ (T ) : (K, g) = Φv+ (H, f ) → (K , g ) = Φv+ (H , f ). If u = v, a bounded operator Sv : Kv → Kv is given by Sv (xα )α∈E v = Ts(α) (xα ) α∈E v . It is easy to see that Sv is well-defined and we have the following commutative diagram:
hv iv −−−→ Hv 0 −−−−→ Kv −−−−→ α∈E v Hs(α) − ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ (Ts(α) )α∈E v Sv Tv iv
0 −−−−→ K v −−−−→
α∈E v
hv
H s(α) −−−−→ H v
For other u ∈ V with u = v, we put Su = Tu : Ku = Hu → Ku = Hu . We shall consider a dual of the above construction. A vertex v ∈ V is called a source if v = r(α) for any α ∈ E. Put Ev = {α ∈ E | s(α) = v}. Definition. Let Γ = (V , E, s, r) be a finite quiver. For a source v ∈ V , we construct a new quiver σv− (Γ ) = (σv− (V ), σv− (E), s, r) as follows: All the arrows of Γ having v as source are reversed and all the other arrows remain unchanged. More precisely, σv− (V ) = V ,
σv− (E) = (E \ Ev ) ∪ Ev ,
where Ev = {α | α ∈ Ev }. In order to define a reflection functor at a source, it is convenient to consider the orthogonal complement M ⊥ of a closed subspace M of a Hilbert space H instead of the quotient H /M. Define an isomorphism f : M ⊥ → H /M by f (y) = [y] = y + M for y ∈ M ⊥ ⊂ H . Then the ⊥ (x) for x ∈ H , where P ⊥ is the projection inverse f −1 : H /M → M ⊥ is given by f −1 ([x]) = PM M ⊥ of H onto M . We shall use the following elementary fact frequently:
M. Enomoto, Y. Watatani / Journal of Functional Analysis 256 (2009) 959–991
969
Lemma 4.1. Let K and L be Hilbert spaces, M ⊂ K and N ⊂ L be closed subspaces. Let A : K → L be a bounded operator. Assume that A(M) ⊂ N . Let A˜ : K/M → L/N be the in˜ duced map such that A([x]) = [Ax] for x ∈ K. Identifying K/M and L/N with M ⊥ and N ⊥ , A˜ is identified with the bounded operator S : M ⊥ → N ⊥ such that S(x) = PN⊥ (Ax). Then S = (A∗ |N ⊥ )∗ . Proof. Consider A∗ : L → K. Since A(M) ⊂ N , we have A∗ (N ⊥ ) ⊂ M ⊥ . Hence the restriction A∗ |N ⊥ : N ⊥ → M ⊥ has the adjoint (A∗ |N ⊥ )∗ : M ⊥ → N ⊥ . For any m ∈ M ⊥ and n ∈ N ⊥ ∗ (A |N ⊥ )∗ m | n = (m | A∗ |N ⊥ n) = (m | A∗ n) = (Am | n) = PN⊥ (Am) | n .
2
Definition (Reflection functor Φv− ). Let Γ = (V , E, s, r) be a finite quiver. For a source v ∈ V , we define a reflection functor at v Φv− : H Rep(Γ ) → H Rep σv− (Γ ) between the categories of Hilbert representations of Γ and σv− (Γ ) as follows: For a Hilbert representation (H, f ) of Γ , we shall define a Hilbert representation (K, g) = Φv− (H, f ) of σv− (Γ ). Let hˆ v : Hv →
Hr(α)
α∈Ev
be a bounded linear operator defined by hˆ v (x) = fα (x) α∈E
v
for x ∈ Hv .
Define Kv := (Im hˆ v )⊥ = Ker hˆ ∗v ⊂
Hr(α) ,
α∈Ev
where hˆ ∗v : α∈Ev Hr(α) → Hv is given hˆ ∗v ((xα )α∈Ev ) = fα∗ (xα ). For u ∈ V with u = v, put Ku = Hu . Let Qv : α∈Ev Hr(α) → Kv be the canonical projection. For β ∈ Ev , let jβ : Hr(β) →
Hr(α)
α∈Ev
be the canonical inclusion. Define gβ : Ks(β) = Hr(β) → Kr(β) = Kv
by gβ = Qv ◦ jβ .
970
M. Enomoto, Y. Watatani / Journal of Functional Analysis 256 (2009) 959–991
For β ∈ / Ev , let gβ = fβ . For a homomorphism T : (H, f ) → (H , f ), we shall define a homomorphism S = (Su )u∈V = Φv− (T ) : (K, g) = Φv− (H, f ) → (K , g ) = Φv− (H , f ), recalling the above Lemma 4.1. For u = v, a bounded operator Sv : Kv → Kv is given by Sv (xα )α∈Ev = Qv Tr(α) (xα ) α∈E , v
Qv
α∈Ev Hr(α)
→ Kv
where : be the canonical projection. We have the following commutative diagram:
Qv hˆ v Hv −−−−→ −−−→ Kv −−−−→ 0 α∈Ev Hr(α) − ⏐ ⏐ ⏐
⏐ ⏐ ⏐ Tv Sv α∈Ev Tr(α) hˆ v
H v −−−−→
Q v
α∈Ev
H r(α) −−−−→ K v −−−−→ 0
For other u ∈ V with u = v, we put Su = Tu : Ku = Hu → Ku = Hu . We shall explain a relation between two (covariant) functors Φv+ and Φv− . We need to introduce another (contravariant) functor Φ ∗ in the first place. Let Γ = (V , E, s, r) be a finite quiver. We define the opposite quiver Γ = (V , E, s, r) by reversing all the arrows, that is, V =V
and E = {α | α ∈ E}.
Definition. Let Γ = (V , E, s, r) be a finite quiver and Γ = (V , E, s, r) its opposite quiver. We introduce a contravariant functor Φ ∗ : H Rep(Γ ) → H Rep(Γ ) between the categories of Hilbert representations of Γ and Γ as follows: For a Hilbert representation (H, f ) of Γ , we shall define a Hilbert representation (K, g) = Φ ∗ (H, f ) of Γ by Ku = Hu for u ∈ V
and gα = fα∗ for α ∈ E.
For a homomorphism T : (H, f ) → (H , f ), we shall define a homomorphism S = (Su )u∈V = Φ ∗ (T ) : (K , g ) = Φ ∗ (H , f ) → (K, g) = Φ ∗ (H, f ), by bounded operators Su : Ku = Hu → Ku = Hu given by Su = Tu∗ . Proposition 4.2. Let Γ = (V , E, s, r) be a finite quiver. If v ∈ V is a source of Γ , then v is a sink of Γ , σv− (Γ ) = σv+ (Γ ) and we have the following:
M. Enomoto, Y. Watatani / Journal of Functional Analysis 256 (2009) 959–991
971
(1) For a Hilbert representation (H, f ) of Γ , Φv− (H, f ) = Φ ∗ Φv+ Φ ∗ (H, f ) . (2) For a homomorphism T : (H, f ) → (H , f ), Φv− (T ) = Φ ∗ Φv+ Φ ∗ (T ) . Proof. (1) It is enough to consider around a source v. For each α ∈ Ev with α : v → u = r(α), a bounded operator fα : Hv → Hu is assigned in (H, f ). Taking Φ ∗ , we have Φ ∗ (Hu ) = Hu and Φ ∗ (fα ) = fα∗ : Hu → Hv in Φ ∗ (H, f ). Let hv :
Hr(α) → Hv
α∈Ev
be a bounded operator given by ∗ hv (xα )α∈Ev = fα (xα ). α∈Ev
Define
∗ Wv := (xα )α∈Ev ∈ Hr(α) fα (xα ) = 0 . α∈Ev
α∈Ev
Then Φv+ (Φ ∗ (Hv )) = Wv and Φv+ (Φ ∗ (Hu )) = Hu in Φ + (Φ ∗ (H, f )). Consider the canonical inclusion map iv : Wv → α∈Ev Hr(α) . For β ∈ Ev , let Pβ :
Hr(α) → Hr(β)
α∈Ev
be the canonical projection. Then Φv+ (Φ ∗ (fβ )) = Pβ ◦ iv . Finally take Φ ∗ again. Since
∗ hv : Hv → α∈Ev Hr(α) is given by
h∗v (y) = fα (y) α∈E = hˆ v (y) v
for y ∈ Hv .
we have ⊥ Φ ∗ Φv+ Φ ∗ (Hv ) = Wv = Ker hv = Im h∗v = (Im hˆ v )⊥ = Φv− (Hv ). Moreover iv∗ = Qv :
α∈Ev
Hr(α) → Wv is the canonical projection. For β ∈ Ev , we have Pβ∗ = jβ : Hr(β) →
α∈Ev
Hr(α) .
972
M. Enomoto, Y. Watatani / Journal of Functional Analysis 256 (2009) 959–991
Therefore Φ ∗ Φv+ Φ ∗ (fβ ) = (Pβ ◦ iv )∗ = iv∗ ◦ Pβ∗ = Qv ◦ jβ = Φv− (fβ ). (2) If u = v, then
Φ ∗ Φv+ Φ ∗ (T ) u = Tu∗∗ = Tu = Φv− (T ) u .
, M If u = v, then, apply Lemma 4.1 by putting that K = α∈Ev Hr(α) , L = α∈Ev Hr(α) is the closure of {(fα (x))α∈Ev ∈ K | x ∈ Hv } in K, N is the closure of {(fα (x))α∈Ev ∈ L | x ∈ Hv } in L and A : K → L with A((yα )α∈Ev ) = (Tr(α) yα )α∈Ev . Then (Φ ∗ (Φv+ (Φ ∗ (T ))))v = (Φv− (T ))v . 2 Proposition 4.3. Let Γ = (V , E, s, r) be a finite quiver. If v ∈ V is a sink of Γ , then v is a source of Γ , σv+ (Γ ) = σv− (Γ ) and we have the following: (1) For a Hilbert representation (H, f ) of Γ , Φv+ (H, f ) = Φ ∗ Φv− Φ ∗ (H, f ) . (2) For a homomorphism T : (H, f ) → (H , f ), Φv+ (T ) = Φ ∗ Φv− Φ ∗ (T ) . Proof. It follows immediately from Proposition 4.2 and the fact that (Φ ∗ )2 = Id.
2
5. Duality theorem We shall show a certain duality between reflection functors, which is analogous to Takesaki duality in operator algebras. Bernstein, Gelfand and Ponomarev [2] introduced reflection functors and Coxeter functors and clarify a relation with the Coxeter–Weyl group and Dynkin diagrams in the case of finite-dimensional representations of quivers. In the case of infinite-dimensional Hilbert representations, duality theorem between reflection functors does not hold as in the purely algebraic setting. We need to modify and assume a certain closedness condition at a sink or a source. v Definition. Let Γ = (V , E, s, r) be a finite quiver and v ∈ V a sink. Recall that E = {α | r(α) = v}. We say that a Hilbert representation (H, f ) of Γ is closed at v if α∈E v Im fα ⊂ Hv is a closed subspace. We say that (H, f ) is full at v if α∈E v Im fα = Hv .
Remark. Recall that a bounded operator hv : α∈E v Hs(α) → Hv is given by hv ((xα )α∈E v ) = α∈E v fα (xα ). Then a Hilbert representation (H, f ) of Γ is closed at v if and only if Im hv is closed. A Hilbert representation (H, f ) is full at v if and only if hv is onto. Definition. Let Γ = (V , E, s, r) be a finite quiver and v ∈ V a source. Recall that Ev = {α | s(α) = ∗v}. We say that a Hilbert representation (H, f ) of Γ isco-closed ∗at v if α∈Ev Im fα ⊂ Hv is a closed subspace. We say that (H, f ) is co-full at v if α∈Ev Im fα = Hv .
M. Enomoto, Y. Watatani / Journal of Functional Analysis 256 (2009) 959–991
973
Remark. Recall that a bounded operator hˆ v : Hv → α∈Ev Hr(α) is given by hˆ v (x) = (fα (x))α∈Ev for x ∈ Hv . Then a Hilbert representation (H, f ) of Γ is co-closed at v if and only if Im hˆ ∗v is closed. A Hilbert representation (H, f ) is co-full at v if and only if hˆ ∗v is onto if and only if Im hˆ v is closed and α∈Ev Ker fα = 0. In fact the latter condition is equivalent to (Im hˆ ∗v )⊥ = Ker hˆ v = 0. We also see that (H, f ) is co-closed at v if and only if Φv∗ (H, f ) is closed at v. And (H, f ) is co-full at v if and only if Φv∗ (H, f ) is full at v. In order to prove a duality theorem, we need to prepare a lemma. Lemma 5.1. Let H and K be Hilbert spaces and T : H → K be a bounded operator. Let T = U |T | be its polar decomposition and U a partial isometry with supp U = Im |T | and Im U = Im T . Suppose that Im T is closed. Then we have the following: (1) Im |T | = Im T ∗ is a closed subspace of H . (2) Under the orthogonal decomposition H = Ker |T |⊥ ⊕ Ker |T | = Im |T | ⊕ Ker |T |, the restriction |T ||Im |T | : Im |T | → Im |T | is a bounded invertible operator. (3) Let S = (|T ||Im |T | )−1 be its inverse. Define a bounded operator B : K → Im T ∗ by Bx = SU ∗ x for x ∈ K. Let Q : H → Im T ∗ be the canonical projection. Then BT = Q. Moreover B|Im T : Im T → Im T ∗ is a bounded invertible operator. Proof. (1) Since Im T is closed, Im T ∗ is also closed. Since U (|T |x) = T x by definition of U and Im T is closed, Im |T | is closed. (2) Since Ker |T |⊥ = Im |T |, |T ||Im |T | is one to one. Since |T |(H ) = |T |(Im |T |) is closed, |T ||Im |T | is onto. Hence |T ||Im |T | is bounded invertible. (3) For any x = x1 + x2 ∈ H with x1 ∈ Im |T | = Im T ∗ and x2 ∈ Ker |T |, BT x = SU ∗ U |T |x = S|T |x = S|T |x1 = x1 = Qx. It is clear that B|Im T is a bounded invertible operator.
2
Theorem 5.2. Let Γ = (V , E, s, r) be a finite quiver
and v ∈ V a sink. Assume that a Hilbert representation (H, f ) of Γ is closed at v. Let hv : α∈E v Hs(α) → Hv be a bounded operator defined by hv ((xα )α∈E v ) = α∈E v fα (xα ). Define a Hilbert representation (H˜ , f˜) of Γ by H˜ v = (Im hv )⊥ ⊂ Hv , H˜ u = 0 for u = v and f˜ = 0. Then we have (H, f ) ∼ = Φv− Φv+ (H, f ) ⊕ (H˜ , f˜). + , f + ) = Φ + (H, f ) and (H +− , f +− ) = Φ − (Φ + (H, f )). Then H + = Ker h = Proof. Let (H v v v v v + {(xα )α∈E v ∈ α∈E v Hs(α) | α∈E v fα (xα ) = 0}, and Hu = Hu for u = v. We have / Ev . fβ+ ((xα )α∈E v ) = xβ for β ∈ E v , and fβ+ = fβ for β ∈
Let hˆ v : Hv+ → α∈E v Hs(α) be a bounded operator given by
hˆ v (xα )α∈E v = fβ+ (xα )α∈E v β∈E v = (xβ )β∈E v = (xα )α∈E v .
974
M. Enomoto, Y. Watatani / Journal of Functional Analysis 256 (2009) 959–991
Hence hˆ v is the canonical embedding. Since (H, f ) is closed at v, Im hv and Im h∗v are closed subspaces. Therefore ⊥ Hv+− = (Im hˆ v )⊥ = Hv+ = (Ker hv )⊥ = Im h∗v . For any other u ∈ V with u = v, Hu+− = Hu . Let Qv : projection. For β ∈ E v , let jβ : Hs(β) →
α∈E v
Hs(α) → Hv+− be the canonical
Hs(α)
α∈E v
be the canonical inclusion. Then fβ+− : Hs(β) → Hv+− is given by fβ+− = Qv ◦ jβ . For other β∈ / E v , we have fβ+− = fβ . We shall define an isomorphism ϕ : (H, f ) → Φv− Φv+ (H, f ) ⊕ (H˜ , f˜).
Apply Lemma 5.1 by putting T = hv , H = α∈E v Hs(α) and K = Hv . Consider the polar decomposition hv = U |hv |. Put S = (|hv ||Im |hv | )−1 . Define a bounded operator B : Hv → Im h∗v by B = SU ∗ . Then Bhv is the canonical projection Qv of Hv onto Im h∗v . We define ϕv : Hv = Im hv ⊕ (Im hv )⊥ → Hv+− ⊕ H˜ v = Im h∗v ⊕ (Im hv )⊥ by ϕv (x, y) = (B|Im hv x, y) for x ∈ Im hv and y ∈ (Im hv )⊥ . By Lemma 5.1 (2), ϕv is a bounded invertible operator. For u ∈ V with u = v, put ϕu : Hu → Hu ⊕ 0 by ϕu (x) = (x, 0) for x ∈ Hu . For any β ∈ E v and x ∈ Hs(β) , ϕv ◦ fβ (x) = ϕv hv jβ (x) = B hv jβ (x) , 0 = Qv jβ (x) , 0 . On the other hand, +− fβ ⊕ 0 ◦ ϕs(β) (x) = fβ+− ⊕ 0 (x, 0) = fβ+− (x), 0 = Qv ◦ jβ (x), 0 . For other β ∈ / E v , we have ϕr(β) ◦ fβ+− = ϕr(β) ◦ fβ = (fβ ⊕ 0) ◦ ϕs(β) = fβ+− ⊕ 0 ◦ ϕs(β) . Hence ϕ : (H, f ) → Φv− (Φv+ (H, f )) ⊕ (H˜ , f˜) is an isomorphism.
2
Counter example. If we do not assume that a Hilbert representation (H, f ) of Γ is closed at v, then the above Theorem 5.2 does not hold in general. In fact, consider the following quiver Γ = (V , E, s, r): α1
α2
◦1 −→ ◦0 ←− ◦2 .
M. Enomoto, Y. Watatani / Journal of Functional Analysis 256 (2009) 959–991
975
Let K = 2 (N) with the canonical basis (en )n∈N . Define a Hilbert representation (H, f ) of Γ by H0 = K ⊕ K, H1 = K ⊕ 0 and H2 is the closed subspace of H0 spanned by π π {(cos n+2 en , sin n+2 en ) ∈ K ⊕ K | n ∈ N}. Then H1 ∩ H2 = 0 and H1 + H2 is a dense subspace of H0 but not closed in H0 . Let fk = fαk : Hk → H0 be the inclusion map for k = 1, 2. Then (H, f ) is not closed at a sink v = 0. It is easy to see that H0+ = Ker h0 = 0, f1+ = 0 and f2+ = 0. Therefore H0+− = H1 ⊕ H2 and H1+− = H1 , H2+− = H2 . We have fk+− : Hk → H1 ⊕ H2 is a canonical inclusion for k = 1, 2. Since H˜ 0 = (Im h0 )⊥ = 0, we have (H˜ , f˜) = (0, 0). Therefore Φ0− Φ0+ (H, f ) ⊕ (H˜ , f˜) = Φ0− Φ0+ (H, f ) = H +− , f +− is closed at a sink v = 0. But (H, f ) is not closed at a sink v = 0. Therefore there exists no isomorphism between (H, f ) and Φ0− (Φ0+ (H, f )) ⊕ (H˜ , f˜). Note that (H, f ) is not full at a sink v = 0 and Φ0− (Φ0+ (H, f )) is full at a sink v = 0. Therefore this example also shows that, if we do not assume that a Hilbert representation (H, f ) of Γ is full at v, then the following duality theorem (Corollary 5.3) does not hold in general. Corollary 5.3 (Duality theorem). Let Γ = (V , E, s, r) be a finite quiver and v ∈ V a sink. If a Hilbert representation (H, f ) of Γ is full at v, then (H, f ) ∼ = Φv− Φv+ (H, f ) . Proof. Since (H, f ) is full at v, H˜ v = (Im hv )⊥ = Hv⊥ = 0. Hence (H˜ , f˜) = (0, 0) in Theorem 5.2. 2 Remark. (1) If we regard reflection functors Φv+ and Φv− as crossed products by an action and its dual action, then the above formula (H, f ) ∼ = Φv− (Φv+ (H, f )) is analogous to Takesaki duality theorem in operator algebras. (2) It is also necessary that (H, f ) is full at the sink v in order that the above duality theorem holds. It follows from Lemma 5.8 below. We have a dual version. Theorem 5.4. Let Γ = (V , E, s, r) be a finite quiver and v ∈ V a source. Assume that a Hilbert representation (H, f ) of Γ is co-closed at v. Let hˆ v : Hv → α∈Ev Hr(α) is a bounded operator defined by hˆ v (x) = (fα (x))α∈Ev for x ∈ Hv . Define a Hilbert representation (Hˇ , fˇ) of Γ by ∗ ⊥ ˆ ˇ ˆ = Ker hv = Hv = Im hv Ker fα ⊂ Hv , α∈Ev
Hˇ u = 0 for u = v and fˇ = 0. Then (H, f ) ∼ = Φv+ Φv− (H, f ) ⊕ (Hˇ , fˇ).
976
M. Enomoto, Y. Watatani / Journal of Functional Analysis 256 (2009) 959–991
Proof. We see that v is a sink in Γ , because v is a source in Γ . Since a Hilbert representation (H, f ) of Γ is co-closed at v, a Hilbert representation Φ ∗ (H, f ) is closed at v. By Theorem 5.2, there exists a Hilbert representation (H˜ , f˜) of Γ such that Φ ∗ (H, f ) ∼ = Φv− Φv+ Φ ∗ (H, f ) ⊕ (H˜ , f˜). Put (Hˇ , fˇ) = Φ ∗ (H˜ , f˜). Then (H, f ) ∼ = Φ ∗ Φv− Φv+ Φ ∗ (H, f ) ⊕ Φ ∗ (H˜ , f˜) = Φ ∗ Φ ∗ (H, f ) ∼ ∼ = Φ ∗ Φv− Φ ∗ Φ ∗ Φv+ Φ ∗ (H, f ) ⊕ Φ ∗ (H˜ , f˜) ∼ = Φv+ Φv− (H, f ) ⊕ (Hˇ , fˇ). Moreover it is easy to see that Hˇ v =
Im fα∗
α∈Ev
⊥
=
Ker fα .
2
α∈Ev
Counter example. If we do not assume that a Hilbert representation (H, f ) of Γ is co-closed at the source v, then the above Theorem 5.4 does not hold in general. In fact, consider the following quiver Γ = (V , E, s, r): α1
α2
◦1 ←− ◦0 −→ ◦2 . Let K = 2 (N) with the canonical basis (en )n∈N . Define a Hilbert representation (H, f ) of Γ by H0 = K ⊕ K, H1 = K ⊕ 0 and H2 is the closed subspace H0 spanned by π π en , sin n+2 en ) ∈ K ⊕ K | n ∈ N}. Let fk = fαk : H0 → Hk be the canonical pro{(cos n+2 jection for k = 1, 2. Then (H, f ) is not co-closed at a source v = 0. It is easy to see that H0− = (Im hˆ 0 )⊥ = 0, f1− = 0 and f2− = 0. Therefore H0−+ = H1 ⊕ H2 and H1−+ = H1 , H2−+ = H2 . We have that fk−+ : H1 ⊕ H2 → Hk is the canonical projection for k = 1, 2. Since Hˇ0 = Ker hˆ 0 = 0, we have (Hˇ , fˇ) = (0, 0). Therefore Φ0+ Φ0− (H, f ) ⊕ (Hˇ , fˇ) = Φ0+ Φ0− (H, f ) = H −+ , f −+ is co-closed at a source v = 0. But (H, f ) is not co-closed at a source v = 0. Therefore there exists no isomorphism between (H, f ) and Φ0+ (Φ0− (H, f )) ⊕ (Hˇ , fˇ). Note that (H, f ) is not co-full at a source v = 0 and Φ0+ (Φv− (H, f )) is co-full at a source v = 0. Therefore this example also shows that, if we do not assume that a Hilbert representation (H, f ) of Γ is co-full at v, then the following duality theorem (Corollary 5.5) does not hold in general. Corollary 5.5 (Duality theorem). Let Γ = (V , E, s, r) be a finite quiver and v ∈ V a source. If a Hilbert representation (H, f ) of Γ is co-full at v, then (H, f ) ∼ = Φv+ Φv− (H, f ) .
M. Enomoto, Y. Watatani / Journal of Functional Analysis 256 (2009) 959–991
Proof. Since (H, f ) is co-full at v, Hˇ v = rem 5.4. 2
α∈Ev
977
Ker fα = 0. Hence (Hˇ , fˇ) = (0, 0) in Theo-
Remark. It is also necessary that (H, f ) is co-full at the source v in order that the above duality theorem holds. It follows from Lemma 5.6 below. Lemma 5.6. Let Γ = (V , E, s, r) be a finite quiver and v ∈ V a sink. Then for any Hilbert representation (H, f ) of Γ , Φv+ (H, f ) is co-full at v.
, f + ) = Φv+ (H, f ). Recall that hv : is given by Proof. Put (H + α∈E v Hs(α) → Hv Hv+ = Ker hv . And for β ∈ E v , let iv : Hv+ → α∈E v Hs(α) hv ((xα )α∈E v ) = α∈E v fα (xα ), and be the canonical inclusion and Pβ : α∈E v Hs(α) → Hs(β) the canonical projection. We define + + = Hv+ → Hr(β) = Hs(β) fβ+ : Hs(β)
by gβ = Pβ ◦ iv .
∗ ∗ Therefore fβ+ : Hs(β) → Hv+ is given by fβ+ = iv∗ ◦ Pβ ∗ . Since Pβ∗ : Hs(β) → α∈E v Hs(α) is
the canonical inclusion and iv∗ : α∈E v Hs(α) → Hv+ is the canonical projection, we have
∗
Im fβ+ =
β∈Ev
Therefore (H + , f + ) is co-full at v.
Im iv∗ ◦ Pβ ∗ = Hv+ .
β∈E v
2
Proposition 5.7. Let Γ = (V , E, s, r) be a finite quiver and v ∈ V a sink. If (H, f ) is a Hilbert representation of Γ , then Φv+ Φv− Φv+ (H, f ) ∼ = Φv+ (H, f ). Proof. Since Φv+ (H, f ) is co-full at the source v in σv+ (Γ ) by the above Lemma 5.6, duality theorem (Corollary 5.5) yields the conclusion. 2 Lemma 5.8. Let Γ = (V , E, s, r) be a finite quiver and v ∈ V a source. Then for any Hilbert representation (H, f ) of Γ , Φv− (H, f ) is full at v.
Proof. Put (H − , f − ) = Φv− (H, f ). Recall that hˆ v : Hv → α∈Ev Hr(α) is given by hˆ v (x) =
Hr(α) . Let Qv : α∈Ev Hr(α) → Hv− be (fα (x))α∈Ev for x ∈ Hv and Hv− = (Im hˆ v )⊥ ⊂ α∈Ev the canonical projection. For β ∈ Ev , let jβ : Hr(β) → α∈Ev Hr(α) be the canonical inclusion. Then − − fβ− : Hs(β) = Hr(β) → Hr(β) = Hv−
Therefore
Im fβ− = Qv
α∈Ev
β∈E v
Thus (H − , f − ) is full at v.
2
by fβ− = Qv ◦ jβ .
Hr(α) = Hv− .
978
M. Enomoto, Y. Watatani / Journal of Functional Analysis 256 (2009) 959–991
Proposition 5.9. Let Γ = (V , E, s, r) be a finite quiver and v ∈ V a source. If (H, f ) is a Hilbert representation of Γ , then Φv− Φv+ Φv− (H, f ) ∼ = Φv− (H, f ). Proof. Since Φv− (H, f ) is full at the source in σv− (Γ ) by the above Lemma 5.8, duality theorem (Corollary 5.3) yields the conclusion. 2 We examine on which representation a reflection functor vanishes. Lemma 5.10. Let Γ = (V , E, s, r) be a finite quiver and v ∈ V a sink. Then, for any Hilbert representation (H, f ) of Γ , the following are equivalent: (1) Φv+ (H, f ) ∼ = (0, 0), (2) Hu = 0 for any u ∈ V with u = v. Furthermore if the above conditions are satisfied and (H, f ) is indecomposable, then Hv ∼ = C. If the above conditions are satisfied and (H, f ) is full at the sink v, then (H, f ) ∼ = (0, 0).
, f + ) = Φv+ (H, f ). Recall that hv : Proof. Put (H + α∈E v Hs(α) → Hv is given by hv ((xα )α∈E v ) = α∈E v fα (xα ), and Hv+ = Ker hv . For other u ∈ V with u = v, Hu+ = Hu . (1) ⇒ (2): Assume that Φv+ (H, f ) = 0. Then, for any u ∈ V with u = v we have Hu = Hu+ = 0. (2) ⇒ (1): Assume that Hu = 0 for any u ∈ V with u = v. Then Hv+ = 0, because Hv+ = Ker hv ⊂ α∈E v Hs(α) = 0. For other u ∈ V with u = v, Hu+ = Hu = 0. Furthermore assume that the above conditions are satisfied and (H, f ) is indecomposable. Then f = 0. Suppose that dim Hv 2. Then a non-trivial decomposition Hv = K ⊕ L gives a non-trivial decomposition of (H, f ). This contradicts that (H, f ) is indecomposable. Hence that the above conditions are satisfied and (H, f ) is full at v. Then f = 0, so Hv ∼ = C. Assume that Hv = α∈E v Im fα = 0. Hence (H, f ) ∼ = (0, 0). 2 Lemma 5.11. Let Γ = (V , E, s, r) be a finite quiver and v ∈ V a source. Then, for any Hilbert representation (H, f ) of Γ , the following condition are equivalent: (1) Φv− (H, f ) ∼ = (0, 0), (2) Hu = 0 for any u ∈ V with u = v. Furthermore if the above conditions are satisfied and (H, f ) is indecomposable, then Hv ∼ = C. If the above conditions are satisfied and (H, f ) is co-full at the source v, then (H, f ) ∼ = (0, 0).
Proof. Put (H − , f − ) = Φv− (H, f ). Recall that hˆ v : Hv → α∈Ev Hr(α) is given by hˆ v (x) =
(fα (x))α∈Ev for x ∈ Hv , and Hv− = (Im hˆ v )⊥ ⊂ α∈Ev Hr(α) . For other u ∈ V with u = v, Hu− = Hu . (1) ⇒ (2): Assume that Φv− (H, f ) = 0. Then, for any u ∈ V with u = v we have Hu = − Hu = 0. (2) ⇒ (1): Assume that Hu = 0 for any u ∈ V with u = v. Then Hv− = 0, because Hv− =
(Im hˆ v )⊥ ⊂ α∈Ev Hr(α) = 0. For other u ∈ V with u = v, Hu− = Hu = 0.
M. Enomoto, Y. Watatani / Journal of Functional Analysis 256 (2009) 959–991
979
∗ Assume that the above conditions are satisfied and (H, f ) is co-full at v. Since fα = 0 for any α ∈ E, Hv = α∈Ev Im fα∗ = 0. Hence (H, f ) ∼ (0, 0). The rest is clear. 2 =
We shall show that a reflection functor preserves indecomposability of a Hilbert representation unless vanishing on it, under the assumption that the Hilbert representation is closed (resp. coclosed) at a sink (resp. source). Theorem 5.12. Let Γ = (V , E, s, r) be a finite quiver and v ∈ V a sink. Suppose that a Hilbert representation (H, f ) of Γ is indecomposable and closed at v. Then we have the following: (1) If Φv+ (H, f ) = 0, then Hv = C, Hu = 0 for any u ∈ V with u = v and fα = 0 for any α ∈ E. (2) If Φv+ (H, f ) = 0, then Φv+ (H, f ) is also indecomposable and (H, f ) ∼ = Φv− (Φv+ (H, f )).
Proof. Recall an operator hv : α∈E v Hs(α) → Hv defined by hv ((xα )α∈E v ) = α∈E v fα (xα ). Since (H, f ) is closed at a sink v, we have a decomposition such that (H, f ) ∼ = Φv− Φv+ (H, f ) ⊕ (H˜ , f˜) by Theorem 5.2, where H˜ v = (Im hv )⊥ ⊂ Hv , H˜ u = 0 for u = v and f˜ = 0. Since (H, f ) is indecomposable, Φv− (Φv+ (H, f )) ∼ = (0, 0) or (H˜ , f˜) ∼ = (0, 0). − + ∼ ∼ ˜ ˜ Case 1. Suppose that Φv (Φv (H, f )) = (0, 0). Then (H, f ) = (H , f ). Hence Hu ∼ = H˜ u = 0 + ∼ for u = v. This implies that Φv (H, f ) = (0, 0) by Lemma 5.10. Since (H, f ) is indecomposable, Hv ∼ = C. Case 2. Suppose that (H˜ , f˜) ∼ = (0, 0). Then (H, f ) ∼ = Φv− (Φv+ (H, f )). Since (H, f ) is non+ + zero, Φv (H, f ) is non-zero. We shall show that Φv (H, f ) is indecomposable. Assume that Φv+ (H, f ) ∼ = (K, g) ⊕ (K , g ). Then (H, f ) ∼ = Φv− (K, g) ⊕ Φv− (K , g ). = Φv− Φv+ (H, f ) ∼ ∼ (0, 0) or Φ − (K , g ) ∼ Since (H, f ) is indecomposable, Φv− (K, g) = = (0, 0). By Lemma 5.6, v + Φv (H, f ) is co-full at v, so are its direct summands (K, g) and (K , g ). Then (K, g) ∼ = (0, 0) or (K , g ) ∼ = (0, 0) by Lemma 5.11. Thus Φv+ (H, f ) is indecomposable. Since Cases 1 and 2 are mutually exclusive and either of them occurs, we get the conclusion. 2 We have a dual version. Theorem 5.13. Let Γ = (V , E, s, r) be a finite quiver and v ∈ V a source. Suppose that a Hilbert representation (H, f ) of Γ is indecomposable and co-closed at v. Then we have the following: (1) If Φv− (H, f ) = 0, then Hv = C, Hu = 0 for any u ∈ V with u = v and fα = 0 for any α ∈ E. (2) If Φv− (H, f ) = 0, then Φv− (H, f ) is also indecomposable and (H, f ) ∼ = Φv+ Φv− (H, f )). Proof. A dual argument of the proof in Theorem 5.12 works.
2
980
M. Enomoto, Y. Watatani / Journal of Functional Analysis 256 (2009) 959–991
6. Extended Dynkin diagrams Gabriel’s theorem says that a finite, connected quiver has only finitely many indecomposable representations if and only if the underlying undirected graph is one of Dynkin diagrams An , Dn , E6 , E7 , E8 . In this section, we consider a complement of Gabriel’s theorem for Hilbert representations. We need to construct some examples of indecomposable, infinite-dimensional representations of quivers with the underlying undirected graphs extended Dynkin diagrams D˜ n (n 4), E˜ 7 and E˜ 8 . We consider the relative position of several subspaces along the quivers, where vertices are represented by a family of subspaces and arrows are represented by natural inclusion maps. Lemma 6.1. Let Γ = (V , E, s, r) be the following quiver with the underlying undirected graph an extended Dynkin diagram D˜ n for n 4:
Then there exists an infinite-dimensional, indecomposable Hilbert representation (H, f ) of Γ . Proof. Let K = 2 (N) and S a unilateral shift on K. We define a Hilbert representation (H, f ) := ((Hv )v∈V , (fα )α∈E ) of Γ as follows: Define H1 = K ⊕ 0, H2 = 0 ⊕ K, H4 = (x, x) ∈ K ⊕ K x ∈ K ,
H3 = (x, Sx) ∈ K ⊕ K x ∈ K , H5 = H6 = · · · = Hn+1 = K ⊕ K.
Let fαk : Hs(αk ) → Hr(αk ) be the inclusion map for any αk ∈ E for k = 1, 2, 3, 4, and fβ = id for other arrows β ∈ E. Then we can show that (H, f ) is indecomposable as in Example 3 in Section 3. 2 Let Γ = (V , E, s, r) be the quiver of Example 4 in Section 3 with the underlying undirected graph an extended Dynkin diagram E˜ 6 . We have already shown that there exists an infinitedimensional, indecomposable Hilbert representation (H, f ) of Γ . Lemma 6.2. Let Γ = (V , E, s, r) be the following quiver with the underlying undirected graph an extended Dynkin diagram E˜ 7 :
Then there exists an infinite-dimensional, indecomposable Hilbert representation (H, f ) of Γ .
M. Enomoto, Y. Watatani / Journal of Functional Analysis 256 (2009) 959–991
981
Proof. Let K = 2 (N) and S a unilateral shift on K. We define a Hilbert representation (H, f ) := ((Hv )v∈V , (fα )α∈E ) of Γ as follows: Let H0 = K ⊕ K ⊕ K ⊕ K, H1 = K ⊕ 0 ⊕ K ⊕ K, H2 = K ⊕ 0 ⊕ (x, x); x ∈ K , H3 = K ⊕ 0 ⊕ 0 ⊕ 0, H1 = 0 ⊕ K ⊕ K ⊕ K, H2 = 0 ⊕ K ⊕ (y, Sy) ∈ K 2 y ∈ K , H3 = 0 ⊕ K ⊕ 0 ⊕ 0 and H1 = (x, y, x, y) ∈ K 4 x, y ∈ K . For any arrow α ∈ E, let fα : Hs(α) → Hr(α) be the canonical inclusion map. We shall show that the Hilbert representation (H, f ) is indecomposable. Take T = (Tv )v∈V ∈ Idem(H, f ). Since T ∈ End(H, f ) and any arrow is represented by the inclusion map, we have T0 x = Tv x for any v ∈ {1, 2, 3, 1 , 2 , 3 , 1 } and any x ∈ Hv . In particular, T0 Hv ⊂ Hv . Since T0 preserves H3 = K ⊕ 0 ⊕ 0 ⊕ 0, H3 = 0 ⊕ K ⊕ 0 ⊕ 0, and H1 ∩ H1 = 0 ⊕ 0 ⊕ K ⊕ K, T0 is written ⎛
A 0 ⎜0 B T0 = ⎝ 0 0 0 0
0 0 X Z
⎞ 0 0⎟ ⎠, Y W
for some A, B, X, Y, Z, W ∈ B(K). Because H1 = {(x, y, x, y) ∈ K 4 | x, y ∈ K} is also invariant under T0 , for any x, y ∈ K, there exist x , y ∈ K such that ⎛
A 0 ⎜0 B ⎝ 0 0 0 0
0 0 X Z
⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 0 x Ax x 0 ⎟ ⎜ y ⎟ ⎜ By ⎟ ⎜ y ⎟ ⎠⎝ ⎠ = ⎝ ⎠ = ⎝ ⎠. x Y x Xx + Yy y W y Zx + Wy
Putting y = 0, we have Ax = Xx and 0 = Zx for any x ∈ K. Hence A = X and Z = 0. Similarly, letting x = 0, we have Y = 0 and W = B. Therefore T0 has a block diagonal form such that ⎛
A ⎜0 T0 = ⎝ 0 0
0 B 0 0
0 0 A 0
⎞ 0 0⎟ ⎠ = A ⊕ B ⊕ A ⊕ B. 0 B
Furthermore, as T0 preserves H1 ∩ H2 = {(0, 0, x, x) ∈ K 4 | x ∈ K}, for any x ∈ K there exists y ∈ K such that (0, 0, Ax, Bx) = (0, 0, y, y). Hence A = B. Therefore T0 = A ⊕ A ⊕ A ⊕ A. Moreover H1 ∩ H2 = {(0, 0, x, Sx) ∈ K 4 | x ∈ K} is also invariant under T0 . Hence for any x ∈ K, there exists y ∈ K such that (0, 0, Ax, ASx) = (0, 0, y, Sy). Thus AS = SA. Since T ∈ Idem(H, f ), T0 is an idempotent, so that A is also an idempotent. Because AS = SA and A2 = A, we have A = 0 or A = I . Thus T0 = 0 or T0 = I . Since for any v ∈ V and any x ∈ Hv T0 x = Tv x, we have Tv = 0 or Tv = I simultaneously. Thus T = (Tv )v∈V = 0 or T = I , that is, Idem(H, f ) = {0, I }. Therefore (H, f ) is indecomposable. 2
982
M. Enomoto, Y. Watatani / Journal of Functional Analysis 256 (2009) 959–991
Remark. Replacing S by S + λI for λ ∈ C, we have uncountably many infinite-dimensional, indecomposable Hilbert representations of Γ . Lemma 6.3. Let Γ = (V , E, s, r) be the following quiver with the underlying undirected graph an extended Dynkin diagram E˜ 8 :
Then there exists an infinite-dimensional, indecomposable Hilbert representation (H, f ) of Γ . Proof. Let K = 2 (N) and S a unilateral shift on K. We define a Hilbert representation (H, f ) := ((Hv )v∈V , (fα )α∈E ) of Γ as follows: Let H0 = K ⊕ K ⊕ K ⊕ K ⊕ K ⊕ K, H1 = (x, x) ∈ K 2 x ∈ K ⊕ K ⊕ K ⊕ K ⊕ K,
H2 = 0 ⊕ 0 ⊕ K ⊕ K ⊕ K ⊕ K, H3 = 0 ⊕ 0 ⊕ 0 ⊕ K ⊕ K ⊕ K, H5 = 0 ⊕ 0 ⊕ 0 ⊕ K ⊕ 0 ⊕ 0, H4 = 0 ⊕ 0 ⊕ 0 ⊕ K ⊕ (y, Sy) ∈ K 2 y ∈ K , H2 = K ⊕ K ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0, H1 = K ⊕ K ⊕ (x, y, x, y) ∈ K 4 x, y ∈ K , H1 = (y, z, x, 0, y, z) ∈ K 6 x, y, z ∈ K . For any arrow α ∈ E, let fα : Hs(α) → Hr(α) be the canonical inclusion map. We shall show that the Hilbert representation (H, f ) is indecomposable. Take T = (Tv )v∈V ∈ Idem(H, f ). Since T ∈ End(H, f ) and any arrow is represented by the inclusion map, we have T0 x = Tv x for any v ∈ V and any x ∈ Hv . In particular, T0 Hv ⊂ Hv . Since T0 preserves subspaces H2 = K ⊕ K ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0, H2 = 0 ⊕ 0 ⊕ K ⊕ K ⊕ K ⊕ K, T0 has a form such that ⎛
∗ ⎜∗ ⎜ ⎜0 T0 = ⎜ ⎜0 ⎝ 0 0
∗ ∗ 0 0 0 0
0 0 ∗ ∗ ∗ ∗
0 0 ∗ ∗ ∗ ∗
0 0 ∗ ∗ ∗ ∗
⎞ 0 0⎟ ⎟ ∗⎟ A ⎟= ∗⎟ 0 ⎠ ∗ ∗
0 B
,
for some A ∈ B(K ⊕ K) and B ∈ B(K ⊕ K ⊕ K ⊕ K). Moreover H1 ∩ H2 = 0 ⊕ 0 ⊕ K ⊕ 0 ⊕ 0 ⊕ 0 and H3 = 0 ⊕ 0 ⊕ 0 ⊕ K ⊕ K ⊕ K are invariant under T0 . Furthermore H5 = 0 ⊕ 0 ⊕ 0 ⊕ K ⊕ 0 ⊕ 0 and T0 (H5 ) ⊂ H5 . Therefore T0 is written as
M. Enomoto, Y. Watatani / Journal of Functional Analysis 256 (2009) 959–991
⎛
T0 =
a ⎜ ⎜c 0 ⎜0 =⎜ B ⎜0 ⎝ 0 0
A 0
b d 0 0 0 0
0 0 0 0 e 0 0 f 0 0 0 0
0 0 0 g i k
983
⎞ 0 0⎟ ⎟ 0⎟ ⎟ h⎟ ⎠ j l
for some a, b, c, d, e, f, g, h, i, j, k, l ∈ B(K). Since H1 ∩ H3 = 0 ⊕ 0 ⊕ 0 ⊕ {(y, 0, y) ∈ K 3 | y ∈ K} is invariant under T0 , for any y ∈ K, there exists y ∈ K such that ⎛ ⎞ ⎛ 0 e 0 ⎜y ⎟ ⎜0 f B⎝ ⎠=⎝ 0 0 0 y 0 0
0 g i k
⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 0 0 0 0 h ⎟ ⎜ y ⎟ ⎜ fy + hy ⎟ ⎜ y ⎟ ⎠⎝ ⎠ = ⎝ ⎠ = ⎝ ⎠. j 0 jy 0 l y ly y
Therefore f + h = l and j = 0. Next consider H1 ∩ H2 = 0 ⊕ 0 ⊕ {(x, y, x, y); x, y ∈ K}. Since H1 ∩ H2 is invariant under T0 , for any x, y ∈ K there exist x , y ∈ K such that ⎛ ⎞ ⎛ e x ⎜y ⎟ ⎜0 B⎝ ⎠=⎝ 0 x 0 y
0 f 0 0
0 g i k
⎞ ⎛ ⎞ ⎞⎛ ⎞ ⎛ x ex x 0 h ⎟ ⎜ y ⎟ ⎜ fy + gx + hy ⎟ ⎜ y ⎟ ⎠ = ⎝ ⎠. ⎠⎝ ⎠ = ⎝ ix x 0 x kx + ly y l y
Putting y = 0, we have ex = x = ix,
gx = y = kx
for any x ∈ K.
Hence e = i and g = k. Letting x = 0, we have fy + hy = y = ly for any y ∈ K. Hence f + h = l. Since T0 preserves H2 ∩ H1 = {(x, x) ∈ K 2 | x ∈ K} ⊕ 0 ⊕ 0 ⊕ 0 ⊕ 0, for any x ∈ K, there exists x ∈ K such that x a b x ax + bx x . A = = = x x c d x cx + dx Hence ax + bx = cx + dx, for any x ∈ K, so that a + b = c + d. Furthermore H1 = {(y, z, x, 0, y, z) ∈ K 6 | x, y, z ∈ K} is invariant under T0 . Therefore for any x, y, z ∈ K there exist x , y , z ∈ K satisfying ⎛
a ⎜c ⎜ ⎜0 ⎜ ⎜0 ⎝ 0 0
b d 0 0 0 0
0 0 e 0 0 0
0 0 0 f 0 0
0 0 0 g e g
⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 0 y ay + bz y 0 ⎟ ⎜ z ⎟ ⎜ cy + dz ⎟ ⎜ z ⎟ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 0 ⎟ ⎜ x ⎟ ⎜ ex ⎟ ⎜ x ⎟ ⎟⎜ ⎟ = ⎜ ⎟ = ⎜ ⎟. h ⎟ ⎜ 0 ⎟ ⎜ gy + hz ⎟ ⎜ 0 ⎟ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠ y 0 y ey z l z gy + lz
Put x = z = 0. Then for any y ∈ K, we have ay = y = ey, cy = z = gy and gy = 0. Hence we have a = e and c = g = 0.
984
M. Enomoto, Y. Watatani / Journal of Functional Analysis 256 (2009) 959–991
Letting x = y = 0, for any z ∈ K we have bz = y = 0, dz = z = lz and hz = 0. Therefore b = 0, d = l and h = 0. Combining these with f + h = l and a + b = c + d, we have a = d and f = l = d. Thus T0 is a block diagonal such that T0 = a ⊕ a ⊕ a ⊕ a ⊕ a ⊕ a ⊕ a ⊕ a. Since T0 is an idempotent, a is also an idempotent. Finally consider that H4 = 0 ⊕ 0 ⊕ 0 ⊕ K ⊕ {(y, Sy) ∈ K 2 | y ∈ K} is invariant under T0 . Then for any x, y ∈ K, there exist x , y ∈ K such that T0 (0, 0, 0, x, y, Sy) = (0, 0, 0, ax, ay, aSy) = (0, 0, 0, x , y , Sy ). Hence aSy = Sy = Say, so that aS = Sa. Since S is a unilateral shift and a is an idempotent, we have a = 0 or a = I. This implies that T0 = 0 or T0 = I. Since for any v ∈ V and any x ∈ Hv T0 x = Tv x, we have Tv = 0 or Tv = I simultaneously. Thus T = (Tv )v∈V = 0 or T = I , that is, Idem(H, f ) = {0, I }. Therefore (H, f ) is indecomposable. 2 Remark. In many cases of our construction of indecomposable, infinite-dimensional representations, we can replace a unilateral shift S by any strongly irreducible operator. We shall show that the existence of indecomposable, infinite-dimensional representations does not depend on the choice of the orientation of quivers. Suppose that two finite, connected quivers Γ and Γ have the same underlying undirected graph and one of them, say Γ , has an infinitedimensional, indecomposable, Hilbert representation. We need to prove that another quiver Γ also has an infinite-dimensional, indecomposable, Hilbert representation. Reflection functors are useful to show it. But we need to check the co-closedness at a source. We introduce a certain nice class of Hilbert representations such that co-closedness is easily checked and preserved under reflection functors at any source. Definition. Let Γ be a quiver whose underlying undirected graph is Dynkin diagram An . We count the arrows from the left as αk : s(αk ) → r(αk ) (k = 1, . . . , n − 1). Let (H, f ) be a Hilbert representation of Γ . We denote fαk by fk for short. For example, f1
f2
f3
f4
f5
◦H1 ←− ◦H2 −→ ◦H3 ←− ◦H4 −→ ◦H5 −→ ◦H6 . We say that (H, f ) is positive-unitary diagonal if there exist m ∈ N and orthogonal decompositions (admitting zero components) of Hilbert spaces Hk =
m
Hk,i
(k = 1, . . . , n)
i=1
and decompositions of operators fk =
m i=1
fk,i :
m i=1
Hs(αk ),i →
m i=1
Hr(αk ),i
(k = 1, . . . , n),
M. Enomoto, Y. Watatani / Journal of Functional Analysis 256 (2009) 959–991
985
such that each fk,i : Hs(αk ),i → Hr(αk ),i is written as fk,i = 0 or fk,i = λk,i uk,i for some positive scalar λk,i and onto unitary uk,i ∈ B(Hs(αk ),i , Hr(αk ),i ). It is easy to see that if (H, f ) is positive-unitary diagonal, then Φ ∗ (H, f ) is also positiveunitary diagonal. Example (Inclusions of subspaces). Consider the following quiver Γ : α1
α2
◦1 −→ ◦2 −→ ◦3 . Let H3 be a Hilbert space and H1 ⊂ H2 ⊂ H3 inclusions of subspaces. Define a Hilbert representation (H, f ) of Γ by H = (Hi )i=1,2,3 and canonical inclusion maps fi = fαi : Hi → Hi+1 for i = 1, 2. Then (H, f ) is positive-unitary diagonal. In fact, define K1 = H1 ,
K2 = H2 ∩ H1⊥ ,
Consider orthogonal decompositions Hk =
3
i=1 Hk,i
K3 = H3 ∩ H2⊥ . (k = 1, 2, 3) by
H1 = K1 ⊕ 0 ⊕ 0, H2 = K1 ⊕ K2 ⊕ 0 and H3 = K1 ⊕ K2 ⊕ K3 . Then f1 = I ⊕ 0 ⊕ 0 and f2 = I ⊕ I ⊕ 0. Hence (H, f ) is positive-unitary diagonal. It is trivial that the example can be extended to the case of inclusion of n subspaces. Lemma 6.4. Let Γ be a quiver whose underlying undirected graph is Dynkin diagram An and (H, f ) be a Hilbert representation of Γ . Assume that (H, f ) is positive-unitary diagonal. Then (H, f ) is closed at any sink of Γ and co-closed at any source of Γ . Proof. Let v be a sink of Γ . Then α∈E v Im fα is a finite sum of some of orthogonal subspaces {Hv,i | i} of Hv which correspond to the images of positive times unitaries in the direct component of fα . Hence it is a closed subspace of Hv . Therefore (H, f ) is closed at v. Similarly (H, f ) co-closed at any source of Γ . 2 Proposition 6.5. Let Γ be a quiver whose underlying undirected graph is Dynkin diagram An and (H, f ) be a Hilbert representation of Γ . Let v be a source of Γ . Assume that (H, f ) is positive-unitary diagonal. Then Φv− (H, f ) is also positive-unitary diagonal. ∼ , f ) ⊕ (H , f ), then Φ − (H, f ) = ∼ Φ − (H , f ) ⊕ Φ − (H , f ). ThereProof. If (H, v v v
fm) = (H − − fore Hk = i=1 Hk,i . Hence it is enough to consider orthogonal components. We may and do examine locally the following cases: Case 1. A Hilbert representation (H, f ) is given by T1
T2
◦H1 ←− ◦H0 −→ ◦H2 with T1 = λ1 U1 and T2 = λ2 U2 for some positive scalars λ1 , λ2 and onto unitaries U1 , U2 . Put (H − , f − ) = Φ0− (H, f ): T1−
T2−
◦H1 −→ ◦H − ←− ◦H2 . 0
986
M. Enomoto, Y. Watatani / Journal of Functional Analysis 256 (2009) 959–991
Then (a, b) ∈ H1 ⊕ H2 is in H0− = (Im hˆ 0 )⊥ if and only if ((a, b) | (T1 z, T2 z)) = 0 for any z ∈ H0 , so that T1∗ a + T2∗ b = 0. Hence ∗ a, −λ1 λ−1 2 U2 U1 a ∈ H1 ⊕ H2 a ∈ H1 ∗ = −λ−1 1 λ2 U1 U2 b, b ∈ H1 ⊕ H2 b ∈ H2 .
H0− =
Solving − ∗ ˆ (x, 0) = a, −λ1 λ−1 2 U2 U1 a + (λ1 U1 z, λ2 U2 z) ∈ H0 ⊕ Im h0 , we have T1− x =
λ22
x, − 2
λ21 + λ2
λ1 λ2 U2 U1∗ x 2 λ1 + λ22
for x ∈ H1 .
Similarly we have λ22 λ1 λ2 ∗ T2− y = − 2 U U y, y for y ∈ H2 . 1 2 λ1 + λ22 λ21 + λ22 Let − λ := 1
λ22 λ21 + λ22
2
+
λ1 λ2 2 λ1 + λ22
2
−1 − > 0 and U1− := (λ− 1 ) T1 .
− − Then U1− is an onto unitary and T1− = λ− 1 U1 . Similarly T2 is a positive scalar times unitary. Case 2. A Hilbert representation (H, f ) is given by T1
T2
◦H1 ←− ◦H0 −→ ◦H2 with T1 = 0 and T2 = 0. Then it is easy to see that H0− = H1 ⊕ H2 , T1− and T2− are canonical inclusions: T1− x = (x, 0) ∈ H1 ⊕ H2 for x ∈ H1 and T2− y = (0, y) ∈ H1 ⊕ H2 for y ∈ H2 . We may write that T1− = I ⊕ 0 : H1 ⊕ 0 → H1 ⊕ H2 and T2− = 0 ⊕ I : 0 ⊕ H2 → H1 ⊕ H2 . Hence (H − , f − ) is positive-unitary diagonal. Case 3. A Hilbert representation (H, f ) is given by T1
T2
◦H1 ←− ◦H0 −→ ◦H2 with T1 = λ1 U1 and T2 = 0 for some positive scalar λ1 and onto unitary U1 . Then we see that H0− = 0 ⊕ H2 , T1− = 0 and T2− y = (0, y) ∈ 0 ⊕ H2 for y ∈ H2 . Hence (H − , f − ) is positive-unitary diagonal. Case 4. A Hilbert representation (H, f ) is given by T1
◦H0 −→ ◦H1
M. Enomoto, Y. Watatani / Journal of Functional Analysis 256 (2009) 959–991
987
with T1 = λ1 U1 for some positive scalar λ1 and onto unitary U1 . Put (H − , f − ) = Φ0− (H, f ): T1−
◦H − ←− ◦H1 . 0
Then we see that H0− = 0 and T1− = 0. Case 5. A Hilbert representation (H, f ) is given by T1
◦H0 −→ ◦H1 with T1 = 0. Then we have that H0− = H1 and T1− = I : H1 → H1 = H0− .
2
We shall show that we can change the orientation of Dynkin diagram An using only the iteration of σv− at sources v except the right end. Lemma 6.6. Let Γ0 and Γ be quivers whose underlying undirected graphs are the same Dynkin diagram An for n 2. We assume that Γ0 is the following: ◦1 −→ ◦2 −→ ◦3 . . . ◦n−1 −→ ◦n . Then there exists a sequence v1 , . . . , vm of vertices in Γ0 such that (1) for each k = 1, . . . , m, vk is a source in σv−k−1 . . . σv−2 σv−1 (Γ0 ), (2) σv−m . . . σv−2 σv−1 (Γ0 ) = Γ , (3) for each k = 1, . . . , m, vk = n. Proof. The proof is by induction on the number n of vertices. Let n = 2. Since σ1− (◦1 → ◦2 ) = ◦1 ← ◦2 , the statement holds. Assume that the statement holds for n − 1. If Γ has an arrow ◦n−1 → ◦n , then we can directly apply the assumption of the induction. If Γ has an arrow ◦n−2 → ◦n−1 ← ◦n , replace only this part by ◦n−2 ← ◦n−1 → ◦n to get Γ . Then n − 1 is a − (Γ ) = Γ . Applying the induction assumption for Γ , we can construct source of Γ , and σn−1 the desired iteration. Consider the case that Γ has an arrow ◦n−2 ← ◦n−1 ← ◦n . If there exists a vertex u such that ◦u−1 → ◦u and ◦k ← ◦k+1 for k = u, . . . , n − 1, then define a new quiver Γ by putting ◦u−1 ← ◦u , ◦n−1 → ◦n and other arrows unchanged with Γ . By the induction assumption, there exists a sequence v1 , . . . , vm of vertices in Γ0 such that σv−m . . . σv−2 σv−1 (Γ0 ) = Γ and, for each k = 1, . . . , m, vk = n and vk = n − 1. Then − − − . . . σn−2 σn−1 σv−m . . . σv−2 σv−1 (Γ0 ) = Γ. σu− σu+1
If all the arrows between 1 and n are of the form ◦k ← ◦k+1 for k = 1, . . . , n − 1, then − . . . σ2− σ1− (Γ0 ) = Γ . 2 σn−1 Lemma 6.7. Let Γ = (V , E, s, r) and Γ = (V , E , s , r ) be finite, connected quivers and Γ contains Γ as a subgraph, that is, V ⊂ V , E ⊂ E , s = s |E and r = r |E . If there exists an infinite-dimensional, indecomposable, Hilbert representation of Γ , then there exists an infinitedimensional, indecomposable, Hilbert representation of Γ .
988
M. Enomoto, Y. Watatani / Journal of Functional Analysis 256 (2009) 959–991
Proof. Let (H, f ) be an infinite-dimensional, indecomposable, Hilbert representation of Γ . Define Hv = Hv for v ∈ V and Hv = 0 for v ∈ V \ V . We put fα = fα for α ∈ E and fα = 0 for α ∈ E \ E. Then it is clear that (H , f ) is an infinite-dimensional, indecomposable, Hilbert representation of Γ . 2 We prove one of our main theorems. Theorem 6.8. Let Γ be a finite, connected quiver. If the underlying undirected graph |Γ | contains one of the extended Dynkin diagrams A˜ n (n 0), D˜ n (n 4), E˜ 6 , E˜ 7 and E˜ 8 , then there exists an infinite-dimensional, indecomposable, Hilbert representation of Γ . Proof. By Lemma 6.7, we may assume that the underlying undirected graph |Γ | is exactly one of the extended Dynkin diagrams A˜ n (n 0), D˜ n (n 4), E˜ 6 , E˜ 7 and E˜ 8 . The case of extended Dynkin diagrams A˜ n (n 0) was already verified in Examples 1 and 2 in Section 3. Next suppose that |Γ | is E˜ 6 . Let Γ0 be the quiver of Example 4 in Section 3 and we denote here by (H (0) , f (0) ) the Hilbert representation constructed there. Then |Γ0 | = |Γ | = E˜ 6 , but their orientations are different in general. Three “wings” of |Γ0 | 2 − 1 − 0, 2 − 1 − 0, 2 − 1 − 0 can be regarded as Dynkin diagrams A3 . Applying Lemma 6.6 for these wings locally, we can find a sequence v1 , . . . , vm of vertices in Γ0 such that (1) for each k = 1, . . . , m, vk is a source in σv−k−1 . . . σv−2 σv−1 (Γ0 ), (2) σv−m . . . σv−2 σv−1 (Γ0 ) = Γ , (3) for each k = 1, . . . , m, vk = 0. We note that co-closedness of Hilbert representations at a source can be checked locally around the source. Since the restriction of the representation (H (0) , f (0) ) to each “wing” is positiveunitary diagonal and the iteration of reflection functors does not move the vertex 0, we can apply Lemma 6.4 and Proposition 6.5 locally that Φv−k−1 . . . Φv−2 Φv−1 (H (0) , f (0) ) is co-closed at vk for k = 1, . . . , m. Therefore Theorem 5.13 implies that (H, f ) := Φv−m . . . Φv−2 Φv−1 (H (0) , f (0) ) is the
desired indecomposable, Hilbert representation of Γ . Since the particular Hilbert space H0(0) associated with the vertex 0 is infinite-dimensional and remains unchanged under the iteration of the reflection functors above, (H, f ) is infinite-dimensional. The case that the |Γ | is E˜ 7 or E˜ 8 is shown similarly if we apply iteration of reflection functors on the representations in Lemma 6.2 or Lemma 6.3. Finally consider the case that the |Γ | is D˜ n . Let Γ0 be the quiver of Lemma 6.1 and (H (0) , f (0) ) the Hilbert representation constructed there. Then |Γ0 | = |Γ | = D˜ n , but their orientations are different in general. Let Γ1 be a quiver such that |Γ1 | = D˜ n and the orientation is as same as Γ on the path between 5 and n + 1 and as same as Γ0 on the rest four “wings.” Define a Hilbert representation (H (1) , f (1) ) of Γ1 similarly as (H (0) , f (0) ). For any arrow β in (1) the path between 5 and n + 1, fβ = I . Hence the same proof as for (H (0) , f (0) ) shows that (H (1) , f (1) ) is indecomposable. By a certain iteration of reflection functors at a source 1, 2, 3 or 4 on (H (1) , f (1) ) yields an infinite-dimensional, indecomposable, Hilbert representation of Γ . Here the co-closedness at a source 1, 2, 3 or 4 on (H (1) , f (1) ) is easily checked, because the map is the canonical inclusion. Thus we can apply Theorem 5.13 in this case too. 2
M. Enomoto, Y. Watatani / Journal of Functional Analysis 256 (2009) 959–991
989
Corollary 6.9. Let Γ be a finite, connected quiver. If there exists no infinite-dimensional, indecomposable, Hilbert representation of Γ , then the underlying undirected graph |Γ | is one of the Dynkin diagrams An (n 1), Dn (n 4), E6 , E7 and E8 . Proof. It directly follows from a well-known fact that if the underlying undirected graph |Γ | contains no extended Dynkin diagrams, then |Γ | is one of the Dynkin diagrams. 2 Remark. We have not yet proved the converse. In fact if the converse were true, then a long standing problem on transitive lattices of subspaces of Hilbert spaces would be settled. Recall that Halmos initiated the study of transitive lattices and gave an example of transitive lattice consisting of seven subspaces in [13]. Harrison, Radjavi and Rosenthal [14] constructed a transitive lattice consisting of six subspaces using the graph of an unbounded closed operator. Hadwin, Longstaff and Rosenthal found a transitive lattice of five non-closed linear subspaces in [12]. Any finite transitive lattice which consists of n subspaces of a Hilbert space H gives an indecomposable system of n − 2 subspaces by withdrawing 0 and H . It is still unknown whether or not there exists a transitive lattice consisting of five subspaces. See also [26]. Therefore it is also an interesting problem to know whether there exists an indecomposable system of three subspaces in an infinite-dimensional Hilbert space. The problem can be rephrased as whether there exists an indecomposable representation of a certain quiver whose underlying undirected graph is D4 in an infinite-dimensional Hilbert space. We have a partial evidence for a certain quiver whose underlying undirected graph is An . We prepare an elementary lemma. Let H be a Hilbert space. For a, b ∈ H we denote by θa,b a 2 =θ rank one operator on H such that θa,b (x) = (x | b)a for x ∈ H . Then θa,b a,b if and only if (a | b) = 1 or a = 0 or b = 0. Moreover if dim H 2 and (a | b) = 1, then θa,b is an idempotent such that θa,b = 0 and θa,b = I . Lemma 6.10. Let H1 and H2 be Hilbert spaces and T : H1 → H2 a bounded operator. Take a, b ∈ H1 and c, d ∈ H2 . Suppose that there exists a scalar λ such that T a = λc and T ∗ d = λb. Then T θa,b = θc,d T . Proof. T θa,b = θT a,b = θλc,b = θc,λb = θc,T ∗ d = θc,d T .
2
Proposition 6.11. Let Γ be the following quiver whose underlying undirected graph is An for n 1: α1
α2
αn−1
◦1 −→ ◦2 −→ ◦3 . . . ◦n−1 −→ ◦n . Then there exists no infinite-dimensional, indecomposable, Hilbert representation of Γ . Proof. The case n = 1 is clear by a non-trivial decomposition H1 = L1 ⊕ K1 . We may assume that n 2. Suppose that there were an infinite-dimensional, indecomposable, Hilbert representation (H, f ) of Γ . Put Tk = fαk : Hk → Hk+1 for k = 1, . . . , n − 1.
990
M. Enomoto, Y. Watatani / Journal of Functional Analysis 256 (2009) 959–991
Case 1. Suppose that Tn−1 Tn−2 . . . T1 = 0. Then there exists a1 ∈ H1 such that Tn−1 Tn−2 . . . T1 a1 = 0. Consider non-zero vectors ak = Tk−1 Tk−2 . . . T1 a1 ∈ Hk for k = 1, . . . , n. Put bn = ∗ · · · T ∗ b ∈ H for i = 1, 2, . . . , n − 1. Then an −2 an ∈ Hn . Define bi = Ti∗ Ti+1 i n−1 n ∗ ∗ (ai | bi ) = ai | Ti∗ Ti+1 · · · Tn−1 bn = (Tn−1 Tn−2 . . . Ti ai | bn ) = (an | bn ) = 1. Since Tk ak = ak+1 and Tk∗ bk+1 = bk , the above Lemma 6.10 implies that Tk θak ,bk = θak+1 ,bk+1 Tk for k = 1, . . . , n − 1. Define the non-zero idempotents Pk = θak ,bk . Since (H, f ) is infinitedimensional, there exists some vertex m such that Hm is infinite-dimensional. Then Pm = I . Define P = (Pk )k , then P ∈ Idem(H, f ) and P = O and P = I . This contradicts the assumption that (H, f ) is indecomposable. Case 2. Suppose that there exists r such that Tr−1 Tr−2 . . . T1 = 0 and Tr Tr−1 . . . T1 = 0 for some r = 1, . . . , n − 1 and dim Hm 2 for some m = 1, . . . , r. Then there exists a1 ∈ H1 such that Tr−1 Tr−2 . . . T1 a1 = 0. Consider non-zero vectors ak = Tk−1 Tk−2 . . . T1 a1 ∈ Hk for k = ∗ · · · T ∗ b ∈ H for i = 1, 2, . . . , r − 1. 1, . . . , r. Put br = ar −2 ar ∈ Hr . Define bi = Ti∗ Ti+1 i r−1 r Then we have Tk θak ,bk = θak+1 ,bk+1 Tk for k = 1, . . . , r −1 as Case 1. Define non-zero idempotents Pk = θak ,bk for k = 1, . . . , r. Put Pk = 0 for k = r + 1, . . . , n. Then Tr θar ,br = θTr ar ,br = θ0,br = 0 and Tk Pk = Pk+1 Tk = 0 for k = r, . . . , n − 1. Since dim Hm 2, the non-zero idempotent Pm = I . Define P = (Pk )k , then P ∈ Idem(H, f ) and P = O and P = I . This is a contradiction. Case 3. Suppose that there exists r such that Tr−1 Tr−2 . . . T1 = 0 and Tr Tr−1 . . . T1 = 0 for some r = 1, . . . , n and dim Hk = 1 for k = 1, . . . , r. Therefore Tr = 0. We may put Pk = 0 for k = 1, . . . , r. Then for any a, b ∈ Hr+1 and Pr+1 = θa,b , we have Tk Pk = Pk+1 Tk = 0 for k = 1, . . . , r. Hence we may choose freely Pk for k = r + 1, . . . , n. Starting form Hr+1 , we can repeat the argument from the beginning. After finite steps, we can reduce to the situation of Case 1 or Case 2. And finally we obtain a contradiction. 2 Acknowledgment The authors are supported by the Grant-in-Aid for Scientific Research of JSPS. References [1] M. Auslander, Large modules over artin algebras, in: Algebra, Topology and Category Theory, Academic Press, New York, 1976, pp. 1–17. [2] I.N. Bernstein, I.M. Gelfand, V.A. Ponomarev, Coxeter functors and Gabriel’s theorem, Russian Math. Surveys 28 (1973) 17–32. [3] S. Brenner, Endomorphism algebras of vector spaces with distinguished sets of subspaces, J. Algebra 6 (1967) 100–114. [4] V. Dlab, C.M. Ringel, Indecomposable representations of graphs and algebras, Mem. Amer. Math. Soc. 6 (173) (1976). [5] P. Donovan, M.R. Freislish, The representation theory of finite graphs and associated algebras, Carleton Math. Lect. Notes 5 (1973) 1–119. [6] M. Enomoto, Y. Watatani, Relative position of four subspaces in a Hilbert space, Adv. Math. 201 (2006) 263–317. [7] D. Evans, Y. Kawahigashi, Quantum Symmetries on Operator Algebras, Oxford Univ. Press, 2000. [8] P. Gabriel, Unzerlegbare Darstellungen I, Manuscripta Math. 6 (1972) 71–103. [9] F. Goodman, P. de la Harpe, V. Jones, Coxeter Graphs and Towers of Algebras, Math. Sci. Res. Inst. Publ., vol. 14, Springer-Verlag, Berlin, 1989. [10] P. Gabriel, A.V. Roiter, Representations of Finite-Dimensional Algebras, Springer-Verlag, 1997. [11] I.M. Gelfand, V.A. Ponomarev, Problems of linear algebra and classification of quadruples of subspaces in a finitedimensional vector space, in: Proc. Internat. Conf., Tihany, 1970, in: Colloq. Math. Soc. Janos Bolyai, vol. 5, North-Holland, Amsterdam, 1972, pp. 163–237.
M. Enomoto, Y. Watatani / Journal of Functional Analysis 256 (2009) 959–991
991
[12] D.W. Hadwin, W.E. Longstaff, P. Rosenthal, Small transitive lattices, Proc. Amer. Math. Soc. 87 (1983) 121–124. [13] P.R. Halmos, Ten problems in Hilbert space, Bull. Amer. Math. Soc. 76 (1970) 887–933. [14] K.J. Harrison, H. Radjavi, P. Rosenthal, A transitive medial subspace lattice, Proc. Amer. Math. Soc. 28 (1971) 119–121. [15] C. Jiang, Z. Wang, Strongly Irreducible Operators on Hilbert Space, Longman, 1998. [16] C. Jiang, Z. Wang, Structure of Hilbert Space Operators, World Scientific, 2006. [17] V. Jones, Index for subfactors, Invent. Math. 72 (1983) 1–25. [18] V.G. Kac, Infinite root systems, representations of graphs and invariant theory, Invent. Math. 56 (1980) 57–92. [19] H. Krause, C.M. Ringel (Eds.), Infinite Length Modules, Birkhäuser, 2000. [20] D.W. Kribs, S.C. Power, Free semigroupoid algebras, J. Ramanujan Math. Soc. 19 (2004) 117–159. [21] S.A. Kruglyak, A.V. Roiter, Locally scalar representations of graphs in the category of Hilbert spaces, Funct. Anal. Appl. 39 (2005) 91–105. [22] S. Kruglyak, V. Rabanovich, Y. Samoilenko, On sums of projections, Funct. Anal. Appl. 36 (2002) 182–195. [23] Y.P. Moskaleva, Y.S. Samoilenko, Systems of n subspaces and representations of *-algebras generated by projections, preprint, arXiv:math.OA/0603503. [24] P. Muhly, A finite-dimensional introduction to operator algebras, in: A. Katavolos (Ed.), Operator Algebras and Applications, Kluwer Academic, 1997, pp. 313–354. [25] L.A. Nazarova, Representation of quivers of infinite type, Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973) 752–791. [26] H. Radjavi, P. Rosenthal, Invariant Subspaces, Springer-Verlag, 1973. [27] I. Reiten, C.M. Ringel, Infinite dimensional representations of canonical algebras, Canad. J. Math. 58 (2006) 180– 224. [28] C.M. Ringel, The rational invariants of the tame quivers, Invent. Math. 58 (1980) 217–239. [29] B. Solel, You can see the arrows in a quiver operator algebra, J. Aust. Math. Soc. 77 (2004) 111–122.
Journal of Functional Analysis 256 (2009) 992–1064 www.elsevier.com/locate/jfa
Classification of type I and type II behaviors for a supercritical nonlinear heat equation ✩ Hiroshi Matano a,∗ , Frank Merle b a Graduate School of Mathematical Sciences, University of Tokyo, Tokyo, Japan b Université de Cergy-Pontoise, CNRS and IHES, France
Received 6 April 2008; accepted 22 May 2008 Available online 11 July 2008 Communicated by J. Coron
Abstract We study blow-up of radially symmetric solutions of the nonlinear heat equation ut = u + |u|p−1 u ei+2 ther on RN or on a finite ball under the Dirichlet boundary conditions. We assume p > pS := N N −2 and that the initial data is bounded, possibly sign-changing. Our first goal is to establish various characterizations of type I and type II blow-ups. Among many other things we show that the following conditions are equivalent: (a) the blow-up is of type II; (b) the rescaled solution w(y, s) converges to either ϕ ∗ (y) or −ϕ ∗ (y) as s → ∞, where ϕ ∗ denotes the singular stationary solution; (c) u(x, T )/ϕ ∗ (x) tends to ±1 as x → 0, where T is the blow-up time. Our second goal is to study continuation beyond blow-up. Among other things we show that if a blow-up is of type I and incomplete, then its limit L1 continuation becomes smooth immediately after blow-up, and that type I blow-up implies “type I regularization,” that is, (t − T )1/(p−1) u(·, t)L∞ is bounded as t T . We also give various criteria for complete and incomplete blow-ups. © 2008 Elsevier Inc. All rights reserved. Keywords: Blow-up; Nonlinear heat equation; Supercritical power; Self-similar; Asymptotics; Continuation
✩
This work has been partially supported by the program ONDENONLIN.
* Corresponding author.
E-mail addresses: [email protected] (H. Matano), [email protected] (F. Merle). 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.05.021
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
993
1. Introduction In this paper we discuss blow-up phenomena for the nonlinear heat equation
ut = u + |u|p−1 u u(x, 0) = u0 (x)
(x ∈ Ω, t > 0), (x ∈ Ω),
(1.1)
where either Ω = RN or Ω = BR := {x ∈ RN | |x| < R}. In the latter case, we impose the Dirichlet boundary condition u(x, t) = 0 (x ∈ ∂Ω, t > 0).
(1.2)
The exponent p is supercritical in the Sobolev sense, that is, p > pS :=
N +2 , N −2
N 3,
and we assume u0 ∈ L∞ (Ω) ∩ C(Ω). Throughout this paper we deal with radially symmetric solutions. In other words, u is expressed in the form u(x, t) = U |x|, t ,
(1.3)
where the function U (r, t) satisfies the equation Ut = Urr +
N −1 Ur + |U |p−1 U. r
(1.4)
It is well known that for each initial data u0 Eq. (1.1) has a unique solution u ∈ C([0, T ), L∞ (Ω)) for some 0 < T +∞, and that either T = +∞ or T < +∞ and
lim u(·, t)L∞ = +∞.
t→T
In the latter case we say that the solution blows up in finite time, and T is called the blow-up time. The solution is smooth on the time interval 0 < t < T . The simplest example of blow-up solution is a spatially uniform solution, which is nothing but a solution of the following ordinary differential equation: du = |u|p−1 u. dt More precisely, 1 − p−1
1 − p−1
u(t) = κ(T − t)
where κ = (p − 1)
.
(1.5)
Another typical example is the so-called self-similar solution, which is given in the form 1 − p−1
u(x, t) = (T − t)
x −a ψ √ , T −t
(1.6)
994
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
where a is any point in RN and ψ(y) is a bounded solution of the equation 1 1 ψ − y · ∇ψ − ψ + |ψ|p−1 ψ = 0 for y ∈ RN . 2 p−1 1 − p−1
In both cases the blow-up solutions satisfy u(·, t)L∞ = O((T − t)
(1.7)
).
Definition 1.1. We say that the blow-up is of type I if u satisfies 1 lim sup(T − t) p−1 u(·, t)L∞ < ∞,
(1.8)
1 lim sup(T − t) p−1 u(·, t)L∞ = ∞.
(1.9)
t→T
while it is of type II if
t→T
−
1
Since a simple comparison argument deduces that κ(T − t) p−1 u(·, t)L∞ , the above condition (1.8) is equivalent to the existence of some constant C 1 such that 1 κ (T − t) p−1 u(·, t)L∞ Cκ
for 0 t < T .
(1.10)
+2 In the subcritical range 1 < p < pS := N N −2 , it is known that any blow-up is of type I. This is true even for non-radial solutions, at least in convex regions (see [13–16,33,34], also [39] for an earlier work). It is also known, for any p > 1, that the blow-up is of type I if the solution satisfies ut > 0 near the blow-up point (see [11]). Furthermore, type I blow-up solutions are known to behave like self-similar solutions near the blow-up point. More precisely, at any point a ∈ Ω where |u(a, t)| → ∞ as t → T , one can find a bounded solution ψ(y) of (1.7) such that u behaves like a self-similar solution (1.6) in a certain “local” sense:
√ − 1 u a + T − ty, t ∼ (T − t) p−1 ψ(y)
as t → T .
(1.11)
The asymptotic self-similarity of type I blow-up can be better explained by using the rescaled solution 1
wa,T (y, s) := (T − t) p−1 u(x, t) √ 1 = (T − t) p−1 u a + T − ty, t s − s = e p−1 u a + e− 2 y, T − e−s where x−a y=√ , T −t
s = −log(T − t).
(1.12)
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
995
The function wa,T (y, s) satisfies the rescaled equation ∂w 1 1 = w − y · ∇w − w + |w|p−1 w, ∂s 2 p−1
(1.13)
and any solution of Eq. (1.7) is a stationary solution of (1.13). In other words, a solution u(x, t) is self-similar with respect to the point x = a if the corresponding rescaled solution wa,T (y, s) is independent of s. It is also clear that the blow-up is of type I if and only if wa,T (y, s) remains bounded as s → ∞. Once we have the boundedness of wa,T , then by an energy argument and parabolic estimates, we can show that w(y, s) approaches a nonzero stationary solution—or a set of stationary solutions—as s → ∞. This is a more accurate interpretation of the asymptotic self-similarity (1.11). Remark 1.2. Whether w converges to precisely one stationary solution or simply approaches a set of stationary solutions is a subtle question. This amounts to asking whether the ω limit set of w is a singleton or not. In the subcritical range 1 < p < pS , the answer is easy, since there are only two nonzero bounded stationary solutions ψ = ±κ, where κ is as in (1.5) (see [12]). The connectedness of the ω limit set then implies that this set is a singleton. On the other hand, in the supercritical range p > pS , the structure of stationary solutions can be far more complex, therefore a more elaborate argument is needed to prove the convergence. See the remark after (1.16). Let us briefly review known results on the existence of type II blow-up. It is Herrero and Velázquez [18,19] who first discovered the existence of type II blow-ups for Eq. (1.1). More precisely, it is shown in [18,19] that if Ω = RN and if N 11 and p > pJL := 1 +
4
√ N −4−2 N −1
(1.14)
then there exists a radially symmetric solution u(x, t) that satisfies 1 lim (T − t) p−1 u(·, t)L∞ = ∞.
t→T
They constructed such blow-up solutions by using a matched asymptotic expansion and a fixed point argument. Later [36] showed that type II blow-up can also occur for Ω = BR , if N 12 7 . A formal analysis of [9] suggests that type II blow-up may occur for the and p > 1 + N −11 critical power p = pS if the solution changes sign. On the other hand, our previous paper [28] shows that no type II blow-up can occur in the low supercritical range pS < p < pJL . Note that all these studies are done in the framework of radially symmetric solutions. One of the main objectives of the present paper is to give complete characterization of type II and type I blow-up behaviors in terms of the local and global blow-up profiles. Here, by a “local profile” we mean w ∗ (y) := lim w0,T (y, s) s→∞
1 √ = lim (T − t) p−1 u T − t y, t
t→T
(1.15)
and by a “global profile” we mean u(x, T ) := lim u(x, t). t→T
(1.16)
996
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
Note that, since the estimate (2.27) implies that type II blow-up phenomena can occur only at x = 0, the rescaled solution wa,T with a = 0 will play a central role when we study type II blow-up. As we have mentioned in Remark 1.2, the existence of the limit (1.15) is a subtle question if p > pS . As far as type I blow-ups with radial symmetry are concerned, the convergence has been known when Ω = BR , and partially known when Ω = RN (see [31], also [28]). As we will see later in Theorem 3.1, the limit w ∗ always exists no matter whether Ω = BR or Ω = RN , and regardless of the type of blow-up. The existence of the limit (1.16), on the other hand, is much easier to show (see Proposition 3.14). Before proceeding further, let us introduce some notations. Given a solution u that blows up at, say, t = T , we define its blow-up set by
B(u0 ) := x ∈ Ω ∃xn → x, tn → T such that u(xn , tn ) → ∞ ,
(1.17)
where u0 denotes the initial data of solution u. Any element of B(u0 ) is called a blow-up point of u. Let us also introduce the following notation: T (u0 ) := the blow-up time of the solution with initial data u0 .
(1.18)
Here we understand that T (u0 ) = ∞ if the solution does not blow-up. Next, let us recall the structure of radially symmetric stationary solutions of (1.1), that is, solutions of Urr +
N −1 Ur + |U |p−1 U = 0. r
There are two kinds of nonzero solutions of the above equation: the singular one and the regular ones. The singular one, denoted by ϕ ∗ (x) = Φ ∗ (|x|), is given by Φ ∗ (r) = c∗ r
2 − p−1
,
where (c∗ )p−1 =
2 2 N −2− . p−1 p−1
(1.19)
The regular ones, denoted by ϕa (x) = Φa (|x|) with a = 0, are defined as the solution of Urr +
N −1 Ur + |U |p−1 U = 0 r
with U (0) = a, U (0) = 0.
(1.20)
By the self-similar structure of Eq. (1.20), we have p−1 Φa (r) = aΦ1 a 2 r .
(1.21)
Note that ϕ ∗ is also a singular stationary solution of the rescaled Eq. (1.13). This solution ϕ ∗ will play a key role throughout the present paper.
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
997
1.1. Characterization of blow-up profiles Our first main results are the following (see Theorems 3.1 and 3.2 for details). Existence of the local blow-up profile. Let pS < p < ∞. Then the limit (1.15) exists locally uniformly in y ∈ RN \ {0} and it is either a bounded solution of (1.7) or a singular stationary solution ±ϕ ∗ . Various characterizations of type II blow-up. Let p, u, w ∗ be as above. Then type II blow-up at t = T
⇔ ⇔ ⇔
1 lim (T − t) p−1 u(·, t)L∞ = ∞
t→T ∗
w = ϕ ∗ or −ϕ ∗ u(x, T ) = 1 or −1. lim ∗ x→0 ϕ (x)
(1.22)
By using the above characterization results, we will give an alternative proof to our earlier result in [28] on the nonexistence of type II blow-up in the range pS < p < pJL (Theorems 3.7– 3.9). This alternative proof is simpler and works under a slightly milder assumption in the case where Ω = RN . Our next result gives classification of “focused” blow-up (that is, a blow-up that occurs at x = 0) in terms of its global profile. Previously the nature of global profiles has not been well understood compared with the local profile w ∗ . Classification of focused blow-up (pS < p < ∞). ⎧ ⇔ ⎪ ∞ or −∞ u(x, T ) ⎨ finite but = ±1, 0 ⇔ lim = ⇔ ⎪ x→0 ϕ ∗ (x) ⎩ 1 or −1 0 ⇔
type I with w ∗ = κ or −κ, type I with nonconstant w ∗ , type II, no blow-up at x = 0.
(1.23)
See Theorem 4.1 for details. Note that the third statement of (1.23) is included in (1.22). One of the immediate consequences of the above classification result is that lim sup u(x, t) M for some M 0 and every x = 0 t→T
implies no blow-up at t = T . In other words, a δ-function type singularity never occurs for Eq. (1.1), provided p = pS (Corollary 4.2). This fact is in marked contrast with the critical case p = pS , for which the formal analysis of [9] suggests the existence of such a thin needle-like singularity. The result (1.23) is partly a consequence of the following identity (Proposition 4.4): u(x, T ) w ∗ (y) = lim . |y|→∞ ϕ ∗ (y) x→0 ϕ ∗ (x) lim
This identity implies that the asymptotics in the rescaled coordinates are well reflected in the original coordinates. We prove this identity by using our general estimates on the derivatives of u near the blow-up point.
998
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
The following result characterizes the type of blow-up in terms of “the number of vanishing intersections” m0 (U ). Here m0 (U ) roughly denotes the number of the zeros of |U (r, t)| − Φ ∗ (r) (where ϕ ∗ (x) = Φ ∗ (|x|)) that approach r = 0 as t → T ; see (3.15) for a more precise definition. Characterization by the intersection number (pJL < p < ∞). Let u(x, t) = U (|x|, t) be a solution of (1.1) that blows up at t = T . Then type II blow-up
⇒
m0 (U ) 2.
See Theorem 3.11 for details. 1.2. Continuation beyond blow-up We next discuss the behavior of solutions after the blow-up time. Following [3], we introduce the notion of complete and incomplete blow-ups. Given a positive solution u of (1.1) that blows up at t = T , one can define its “minimal extension” u¯ of u for all t T by using a certain approximation procedure. A blow-up is called “complete” if u(x, ¯ t) = ∞ (a.e. x ∈ Ω) for every t > T . The blow-up is “incomplete” if u(x, ¯ t) ≡ ∞ (a.e. x ∈ Ω) on some time interval T < t < T + δ. There is another notion of extension called a “limit L1 continuation,” which is defined by using a different kind of approximation procedure and can apply to sign-changin solutions. As far as positive solutions are concerned, the minimal extension u¯ and the minimal L1 continuation u˜ are equal until the former becomes ∞ everywhere; see Section 5.1 for details. We first recall the following useful estimate for limit L1 continuation u˜ established in our earlier paper [28] (see Lemma 5.12 of the present paper): − 2 u(x, ˜ t) C 1 + |x| p−1 . This estimate holds even after the blow-up time T so far as u˜ is defined. From this estimate we easily see that any incomplete blow-up can occur only at x = 0 (Proposition 5.13). Also, by refining the above estimate, we can derive the following results. Eventual regularity (pS < p < ∞). Suppose that the solution u can be continued globally for 0 t < ∞ as a limit L1 solution. In the case where Ω = RN , assume further that u0 ∈ H 1 (Ω) ∩ L∞ (Ω). Then there exists t0 0 such that u is smooth on Ω × [t0 , ∞) and u(·, t)
L∞
→ 0 as t → ∞.
Threshold behavior (pS < p < ∞). Let v ∈ H 1 (Ω) ∩ L∞ (Ω) satisfy v 0, ≡ 0. Denote by uλ the solution of (1.1) with initial data u0 = λv. Then there exists λ∗ > 0 such that 0 λ < λ∗
⇒
λ u (·, t)
∗
⇒
blow-up in finite time;
λ = λ∗
⇒
λ>λ
L∞
→ 0 as t → ∞;
∗ blow-up in finite time and uλ (·, t)L∞ → 0 as t → ∞.
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
999
See Theorems 5.14 and 5.15 for details. The former result implies, in particular, that no positive stationary solution—singular or regular—can be reached from an H 1 initial data. This is in some sense remarkable, as there are infinitely many positive stationary solutions if Ω = RN . The latter result on the threshold behavior has been mostly known in the literature, particularly when Ω is bounded or when Ω = RN and u > 0 is radially decreasing. Our emphasis here is on the fact that the result follows as an immediate consequence of our general blow-up estimates. We ∗ remark that Chou, Du and Zheng [7] prove the eventual regularity of the threshold solution uλ without assuming radial symmetry, provided that Ω is a bounded convex domain. In a forthcoming paper [29] we will study further properties of the threshold behavior. Among other things we will show in [29] that the blow-up of uλ (λ > λ∗ ) is of type I and complete except for at most finitely many exceptional values of λ. Our next result, which is the main result of Section 5, is concerned with the regularity of solutions after the blow-up time (see Theorem 5.19). Immediate regularization for type I blow-up (pS < p < ∞). Suppose that solution u blows up at t = T and that the blow-up is of type I. Suppose also that there exists a limit L1 continuation u˜ of u on some interval 0 t < T ∗ with T ∗ > T . Then u˜ is smooth in some interval T < t < T + δ and satisfies 1 ˜ t)L∞ < ∞. lim sup(t − T ) p−1 u(·, tT
We may call the above estimate type I regularization. Applying this result to solutions with u0 0, we see that either of the following always holds for any nonnegative solution that blows up at t = T (Corollary 5.20): (a) the blow-up is complete; (b) the minimal extension u¯ is smooth in some interval T < t < T + δ. The above result improves that of [8], which studies the range pS < p < pJL and shows immediate regularization only for minimal continuation. Our result, on the other hand, can apply to sign-changing solutions and possibly non-minimal continuations. More importantly, our result reveals how fast the regularization occurs. (Note, however, that [8] also deals with the equation ut = u + eu , which is outside the scope of the present paper.) For a type II blow-up, we focus on the range p > pJL since no type II blow-up occurs if pS < p < pJL , as shown in [28] and in Theorems 3.7–3.9 of the present paper. Immediate regularization for type II blow-up (pJL < p < ∞, u0 0). Suppose that solution u blows up at t = T and that the blow-up is of type II. Assume further that u(x, T ) ≡ ϕ ∗ . Then either of the following holds: (a) the blow-up is complete; (b) the minimal continuation of u is smooth in some interval T < t < T + δ. Next we summarize our results on the speed of regularization, which we have partly stated above (see Theorems 5.19 and 5.22 for details).
1000
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
Rate of regularization. Suppose that the solution u blows up at t = T and that the blow-up is incomplete. Then type I blow-up, p > pS
⇒
type II blow-up, p > pJL , u0 0
⇒
type I regularization, 1 lim sup(t − T ) p−1 u(·, t)L∞ = ∞. tT
The next results are concerned with the relation between the completeness of the blow-up and the intersection number. Before stating the results, we introduce some notation. Given a function v(r) on an interval J ⊂ R, we define ZJ [v] := the number of zeros of v(r) in J.
(1.24)
The symbol m0 (U ) denotes the number of vanishing intersections defined in (3.15). In the statement below, w ∗ and ϕ ∗ are regarded as functions of r = |y| (see Theorems 5.27 and 5.28). Zero-number criterion (pJL < p < ∞, u0 0). Suppose that the blow-up occurs only at r = 0. Then type I and Z(0,∞) [w ∗ − ϕ ∗ ] is odd ∗
∗
type I and Z(0,∞) [w − ϕ ] is even
⇒
complete blow-up,
⇒
immediate regularization.
Single-intersection blow-up (pS < p < ∞, p = pJL ). Suppose that u blows up at t = T and that m0 (U ) = 1. Then the blow-up is of type I and complete. Furthermore, w ∗ = ±κ. 1.3. Organization of the paper This paper is organized as follows. In Section 2, which is a preliminary section, we present various fundamental estimates for blow-up solutions. Among other things we prove useful pointwise estimates for blow-up in Sections 2.2 to 2.4, which include the general blow-up estimate (see Proposition 2.5) of the form 1 2 u(x, t) CT (T − t)− p−1 + |x|− p−1 and the following estimate for a focused blow-up (see Proposition 2.7): 2 u(x, t) C 1 + |x|− p−1 . The former was introduced in our earlier paper [28], but the latter is new. In Section 2.5, we derive an L∞ estimate for solutions with small H 1 initial data. More precisely, u(·, t0 ) 1 : small ⇒ u(·, t0 + δ) ∞ : small, H L u-loc
where ·H 1 denotes a uniform local H 1 norm to be defined in Section 2.5. Such an estimate is u-loc well known for the subcritical range of p, but was not known in the supercritical range p > pS .
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
1001
In Section 2.6, we prove what we call the “no-needle lemma,” which states that convergence outside the origin automatically implies convergence at the origin. In other words, a thin needlelike singularity does not appear in (1.1), provided that p = pS . This lemma will be of central importance in later sections. In Section 2.7, we show that if u0 ∈ H 1 (RN ) ∩ L∞ (RN ), then there exists R0 > 0 such that lim sup∇u(·, t)L2 (RN \B t→T
R0 )
< ∞,
lim supu(·, t)Lp+1 (RN \B t→T
R0 )
< ∞,
where RN \ BR0 = {x ∈ RN | |x| R0 }. This estimate gives a uniform bound on the solution near |x| = ∞ and will be used in the proof of Theorems 3.9 and 4.9. In Section 3, we prove the existence of the local profile w ∗ (y) := lims→∞ w(y, s) (Theorem 3.1) and give various equivalent characterizations of type II blow-up (Theorem 3.2). The proof of the convergence w(y, s) → ϕ ∗ (y) will be done by combining the weaker convergence result in our earlier paper [28] and the above-mentioned no-needle lemma in Section 2.6. We then use the above characterization to prove nonexistence of type II blow-up in the range pS < p < pJL (Theorems 3.7–3.9), thereby giving an alternative proof to the result we obtained in [28]. We also study the vanishing intersections of type II blow-ups. In Section 4, we classify focused blow-up in terms of their local and global profiles. This classification turns out to be exceedingly useful in studying complete and incomplete blow-ups in Section 5. In Section 5, we present various results on continuation beyond blow-up. One of the highlights is Theorem 5.19, which states that type I blow-up implies type I regularization. In proving this result, the above-mentioned no-needle lemma again plays a central role. In Appendix A, we prove a lemma concerning the asymptotics of solutions of (1.7) as |x| → ∞. Finally, in Appendix B, we consider a more general equation of the form ut = u + f (u),
x ∈ RN , t > 0,
and show that the blow-up set is compact if the initial data u0 has compact support, or if u0 is radially symmetric and u0 (x) → 0 as |x| → ∞. This result will be used in Section 5.1 to study basic properties of L1 continuation beyond blow-up when Ω = RN . 2. Fundamental estimates In this section we will establish various fundamental estimates that will be used later. Section 2.1 is concerned with basic energy estimates that are well known in the literature. Section 2.2 is concerned with general blow-up estimates that were introduced in our earlier paper [28]. The estimates in other subsections are new. 2.1. Basic estimates In this subsection we derive some basic pointwise estimates away from the origin by using energy methods that are well known in the literature for the subcritical range of p (see [13,33]). We modify these estimates slightly so that they apply to the supercritical range pS < p < ∞ under radial symmetry.
1002
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
Note that, unlike the subcritical case, L∞ estimates at the origin would not follow automatically from these basic estimates. We need further ideas, which we will discuss in Sections 2.4– 2.6. In what follows, given a ∈ Ω and T1 > 0, wa,T1 (y, s) will denote the rescaled solution 1
wa,T1 (y, s) := (T1 − t) p−1 u(x, t) = e
1 − p−1 s
u a + e−s/2 y, T1 − e−s ,
(2.1)
where x−a y=√ , T1 − t
s = − log(T1 − t).
(2.2)
It is important to note that we do not assume that solution u blows up exactly at t = T1 , but we simply assume that u is defined as a classical solution at least for 0 t < T1 . In particular, if u is a global classical solution, then T1 can be any positive number. Just as in the case where T1 = T , the function wa,T1 (y, s) satisfies the same rescaled equation as (1.13), that is, ∂w 1 w = w − y · ∇w − + |w|p−1 w ∂s 2 p−1
for y ∈ Ωa,s , s > − log T1 ,
along with the boundary condition w = 0 on ∂Ωa,s , where N s/2 s/2 < R}, Ωa,s := {yN∈ R | |y + e a| < Re } if Ω = {|x| if Ω = RN . R Note that Ωa,s → RN as s → ∞ for any a ∈ Ω. We associate with this equation the following energy functional: 1 1 1 |∇w|2 + |w|2 − |w|p+1 ρ(y) dy, E(w) = 2 2(p − 1) p+1
(2.3)
(2.4)
RN
where N |y|2 ρ(y) := (4π)− 2 exp − . 4
(2.5)
In the case where Ω = BR := {|x| < R}, it will be understood that wa,T1 is defined for all y ∈ RN by setting wa,T1 = 0 outside Ωa,s . For any solution w(y, s) of (1.13), we have d E w(·, s) = − ds
∂w ∂s
2 ρ(y) dy,
(2.6)
ρ(y) dy − β(s),
(2.7)
RN
in the case of Ω = RN , and d E w(·, s) = − ds
Ωa,s
∂w ∂s
2
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
1003
in the case of Ω = {|x| < R}, where β(s) =
1 4
|y||∇wa,T1 |2 ρ(y) dSy . ∂Ωa,s
In either case, E(w(·, s)) is monotonically decreasing. Lemma 2.1. Let a ∈ Ω and let u be a solution of (1.1) defined for (at least) 0 t < T1 . Set s0 := − log T1 and E(a, s) := E(wa,T1 (·, s)). Then for any δ > 0 there exists a monotone increasing function h : [0, ∞) → [0, ∞) dependent only on p, N and δ such that wa,T (0, s + δ) h E(a, s) 1
(2.8)
for any s s0 satisfying es/2 |a| 1. Furthermore, h satisfies 1 h(z) = O z p+1
as z → 0.
(2.9)
Proof. In what follows, for notational simplicity, we write w instead of wa,T1 . As we mentioned before, if Ω = {|x| < R}, we extend w by setting w = 0 outside Ωa,s , hence w is defined for all y ∈ RN . By Green’s formula, 1 d 2 ds
w 2 ρ dy = RN
−|∇w|2 −
RN
1 |w|2 + |w|p+1 ρ dy p−1
p−1 = −2E(w) + p+1
|w|p+1 ρ dy.
(2.10)
RN
Now choose s ∗ ∈ [s0 , ∞) arbitrarily. Then the above identity and Hölder’s inequality, along with the monotone decreasing property of E(w(·, s)), yield 1 d 2 ds
RN
p−1 w 2 ρ dy −2E w(·, s ∗ ) + p+1
p+1
w 2 ρ dy
2
.
(2.11)
RN
for any s s ∗ . Here we have used the fact that ρ(y) dy = 1. RN
Since w(y, s) is defined for all s s ∗ , the quantity w 2 ρ dy cannot blow up in finite time. In view of this and the differential inequality (2.11), we see that p−1 −2E w(·, s ∗ ) + p+1
2
∗
w (y, s )ρ(y) dy RN
p+1 2
0
(2.12)
1004
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
(cf. [34], also [28]). This inequality in particular implies that E(w(·, s ∗ )) 0. Since s ∗ can be chosen arbitrarily in [s0 , ∞), we have E w(·, s) 0 for s s0 .
(2.13)
The inequality (2.12) also implies that
2
w 2 (y, s)ρ(y) dy Cp E0p+1
for s s0 ,
(2.14)
RN
where Cp :=
2(p + 1) p−1
2 p+1
.
Integrating (2.10) from s = s ∗ to s ∗ + δ, and using (2.14), we obtain p−1 p+1
s∗ +δ
|w|p+1 ρ dy ds s ∗ RN
1 2
w 2 (y, s ∗ + δ)ρ dy + 2δE w(·, s ∗ )
RN 2 1 Cp E0p+1 + 2δE0 . 2
(2.15)
Now we return to the original notation of wa,T1 instead of w and recall the relation wa,T1 (y, s) = w0,T1 y + es/2 a, s .
(2.16)
Since w0,T1 (y, s) is radially symmetric because of the radial symmetry of u, we can write w0,T1 (y, s) = W |y|, s , where W (r, s) is defined for r 0, s s0 and satisfies the equation ∂W ∂ 2W r ∂W 1 N − 1 ∂W = − − W + |W |p−1 W. + ∂s r ∂r 2 ∂r p−1 ∂r 2
(2.17)
The estimates (2.14) and (2.15) then imply
2
W 2 (r, s) dr Cp,N E0p+1
for any s s0 ,
(2.18)
|r−r0 (s)| 34 s∗ +δ
2
|W |p+1 (r, s) dr
s∗
|r−r0 (s)| 34
ds C0
for any s ∗ s0 ,
(2.19)
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
1005
where r0 (s) := es/2 |a|, and Cp,N is a constant depending only on p, N , while C0 is a constant depending only on p, N and E0 . Next we convert the above equation for W (r, s) into an equation (z, s) = W (r, s) with z = r − r0 (s): for W ∂ 2W z ∂W N − 1 ∂W ∂W , = − + ξW + 2 s/2 ∂s 2 ∂z ∂z z + e |a| ∂z
(2.20)
where ξ(z, s) = −
1 |p−1 . + |W p−1
In these variables, the estimates (2.18) and (2.19) imply that
2
2 (z, s) dz Cp,N E p+1 W 0
for any s ∈ [s ∗ , s ∗ + δ],
|z| 12 s∗ +δ
|ξ |
s∗
p+1 p−1
2 ds C0 .
dz
|z| 12
Other coefficients of (2.20) are uniformly bounded in the region |z| 12 , s ∗ s s ∗ + δ under ∗ the assumption that es /2 |a| 1. Combining these bounds and parabolic a priori estimates for the one-dimensional linear parabolic equation (2.20) (see, for instance, [23, Section III, Theorem 8.1]), we obtain 1 W (0, s ∗ + δ) C1 E p+1 0
for any s ∗ ∈ [s0 , ∞)
∗
provided that es /2 |a| 1, where C1 is a constant depending only on Cp,N and C0 (hence only (0, s) = wa,T1 (0, s), we obtain the desired estimates (2.8) and (2.9). on p, N , δ and E0 ). Since W The lemma is proven. 2 Lemma 2.2. Let T1 , δ, h(E) be as in Lemma 2.1 and κ be as in (1.5). Set hmax (s; a, R) :=
sup |b−a|Re−s/2
h E wb,T1 (·, s) .
Then for any R0 0 there exists a constant M0 1, dependent only on p, N , δ and R0 , such that if hmax (s1 ; a, R0 )
κ 2
for some s1 − log T1 satisfying es1 /2 |a| 1 + R0 , then 1 wa,T (0, s) M0 hmax (s1 ; a, R0 )e− p−1 (s−s1 ) 1
for s s1 + δ.
(2.21)
1006
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
The following lemma states that a solution cannot blow up too slowly. We will use this lemma to prove Lemma 2.2. Since this result holds without the assumption of radial symmetry, we state it in full generality. Note that this lemma generalizes the earlier result [14, Theorem 2.1], which was for the subcritical case 1 < p < pS . Our proof is much simpler and gives more explicit estimates. Lemma 2.3. Let u(x, t) be a solution of (1.1) which is not necessarily radially symmetric and is defined on a cylindrical domain D × [0, T1 ). Suppose that there exist t0 ∈ [0, T1 ), a ∈ D, a real number r0 > 0 with {|x − a| < r0 } ⊂ D and a constant θ ∈ (0, 1) such that 1 u(x, t) θ κ(T1 − t)− p−1
for |x − a| < r0 , t0 t < T1 .
Then there exists a constant M > 0 depending only on p, N , r0 , T1 − t0 such that 1 u(x, t) Mθ κ 1 − θ p−1 − p−1
for |x − a|
r0 , t 0 t < T1 . 2
(2.22)
In particular, u cannot blow up in a neighborhood of a as t → T1 . Remark 2.4. The constant M that appears in (2.22) can be estimated as M=
2 2B + 2 T1 − t0 r0
1 p−1
− 2 − 1 = O (T1 − t0 ) p−1 + r0 p−1 ,
(2.23)
where B is some constant. Proof of Lemma 2.3. Let h be a C 2 function on [0, 1] satisfying 1 1 h = , 2 2
h(0) = 1, h (r) < 0 for 0 < r < 1,
h(1) = 0,
h (0) = h (1) = 0,
h (1) > 0.
(For example, h(r) = sin2 πr.) Then, since h , r −1 h , (h )2 / h are all bounded, we have −B := inf
0
h (r) +
N −1 p (h (r))2 h (r) − r p − 1 h(r)
> −∞.
This constant B depends on p, N and the choice of the function h. For example, B 12 π 2 (N + 2 2 p−1 ) if we set h(r) = sin πr, but its precise value is not important in the following argument. Now we construct a supersolution in the form − 1 u(x, ˜ t) = θ κ T1 − t + g(r, t) p−1 , where r := |x − a| and r g(r, t) = μ(t − t0 )h , r0
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
1007
with μ being a constant to be specified later. Then ˜ p−1 u˜ = u˜ t − u˜ − |u|
θκ − p (T − t + g) p−1 F, p−1
where F = 1 − θ p−1 − gt + grr + 1 − θ p−1 − gt + grr +
gr2 N −1 p gr − r p − 1 T1 − t + g N −1 p gr2 gr − r p−1 g
1 − θ p−1 − μ − μ(T1 − t0 )r0−2 B. Now we set 1 − θ p−1
μ=
1 + B(T1 − t0 )r0−2
(< 1).
Then F 0, hence ˜ p−1 u˜ 0 for |x − a| < r0 , t0 < t < T1 , u˜ t − u˜ − |u| which implies that u˜ is a supersolution of (1.1). Furthermore, since g(r, t0 ) = g(r0 , t) = 0, 1 − p−1
u(x, ˜ t0 ) = θ κ(T1 − t0 )
1 − p−1
u(x, ˜ t) = θ κ(T1 − t)
u(x, t0 ) u(x, t)
for |x − a| < r0 , for |x − a| = r0 , t0 t < T1 .
Therefore, by the comparison principle, we have u u. ˜ Since 0 μh(r) 1, the function T1 − t + μ(t − t0 )h(r) is monotone decreasing in t, hence u(x, ˜ t) u(x, ˜ T1 ). Consequently − 1 p−1 . u(x, t) u(x, ˜ T1 ) = θ κ μ(T1 − t0 )h r0−1 |x − a| Recall also that h(r) is monotone decreasing and that h( 12 ) = 12 . Thus
μ(T1 − t0 ) u(x, t) θ κ 2
−
1 p−1
− 1 = Mθ κ 1 − θ p−1 p−1
for |x − a| < r20 , where M is the constant given in (2.23). Applying the same estimate to −u(x, t), we obtain an upper bound for −u, which proves the lemma. 2 Proof of Lemma 2.2. By Lemma 2.1 we have wb,T (0, s) hmax := hmax (s1 ; a, R0 ) 1
for s s1 + δ
(2.24)
1008
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
for any b ∈ Ω satisfying |b − a| R0 e−s1 /2 . In the u variable, this is equivalent to 1 u(b, t) hmax (T1 − t)− p−1
for |b − a| R0 e−s1 /2 , t0 t < T1 ,
where t0 = T1 − e−(s1 +δ) . Thus the assumptions of Lemma 2.3 are fulfilled, with r0 = R0 e−s1 /2 ,
θ = κ −1 hmax ,
t0 = T1 − e−(s1 +δ) .
Consequently 1 u(b, t) M 1 − θ p−1 − p−1 hmax
for |b − a|
R0 −s1 /2 , t 0 t < T1 . e 2
In the w variable, this implies that 1 1 wb,T (0, s) M 1 − θ p−1 − p−1 e− p−1 s hmax 1
for s s1 + δ.
Recalling that r0 = R0 e−s1 /2 , we obtain 1 wb,T (0, s) M0 e− p−1 (s−s1 ) hmax 1
for s s1 + δ, 2+2BR −2
1
0 where b is any point satisfying |b − a| 12 R0 e−s1 /2 and M0 = ( 1−2−(p−1) ) p−1 . The lemma is proven. 2
2.2. General blow-up estimates In this subsection we present estimates that hold for any blow-up with radial symmetry. These estimates are established in our earlier paper [28], and they play an important role again in the present paper. Before stating the results let us define the following space: H˙ 1 (Ω) = the closure of compactly supported smooth functions 1/2 2 ∇v(x) dx in the seminorm .
(2.25)
Ω
Note that H˙ 1 (Ω) = H01 (Ω) if Ω is bounded. Proposition 2.5 (General blow-up estimates). (See [28, Corollary 3.2].) Let 1 < p < ∞ and let u0 ∈ L∞ (Ω). Suppose that the solution u is defined for (at least) 0 t < T and let w0,T be the rescaled solution introduced in (2.1) with a = 0. Then, given any δ > 0, there exists a constant CT , dependent only on N , p, δ, u0 L∞ and T , such that 2 w0,T (y, s) CT 1 + |y|− p−1 for |y| > 0, s − log T + δ, 1 2 u(x, t) CT (T − t)− p−1 + |x|− p−1 for x ∈ Ω \ {0}, δ1 T t < T ,
(2.26) (2.27)
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
1009
where δ1 = 1 − e−δ . If ∇u0 is also bounded, the constant CT is given, for instance, by CT =
sup
a∈Ω, T1 ∈[T /2,T ]
h E wa,T1 (·, − log T1 ) .
(2.28)
If u0 belongs to H˙ 1 (Ω) ∩ L∞ (Ω), then CT is estimated as 1 − 1 ( N−2 − 2 ) . CT = O T p+1 2 p−1 ∇u0 Lp+1 2
(2.29)
Note that the estimate (2.27) implies, among other things, that type II blow-up does not occur 1
outside the origin. More precisely, (T − t) p−1 u(x, t) remains bounded as t → T outside any small neighborhood of x = 0. Corollary 2.6 (Derivative estimates). (See [28, Corollary 3.2].) Let the assumptions of Proposition 2.5 hold. Then, for j = 1, 2, 3, j 2 ∇ w0,T (y, s) CT 1 + |y|− p−1 −j for |y| > 0, s − log T + δ, (2.30) j j 1 2 ∇ u(x, t) CT (T − t)− p−1 − 2 + |x|− p−1 −j for x ∈ Ω \ {0}, δ1 T t < T , (2.31) where the constant CT is the same as in Proposition 2.5. Note that we do not assume any sign conditions on u nor ∂u/∂r in Proposition 2.5 and Corollary 2.6. Proof of Proposition 2.5. For simplicity we only consider the case where Ω = RN . The case Ω = BR can be treated virtually the same way; see [28, Theorem 3.1] for details. Let us first assume that ∇u0 is bounded. For each T1 ∈ [T /2, T ], we have, by Lemma 2.1, that wa,T (0, s + δ) h E wa,T (·, − log T1 ) , 1 1 provided that s − log T1 ,
es/2 |a| 1.
This and (2.16) imply w0,T1 (y, s + δ) CT
for |y| 1, s − log T1 ,
where CT is as in (2.28). By the definition of wa,T1 , we have x 1 w0,T √ , log T −t T −t x 1 − 1 . , log = (T1 − t) p−1 w0,T1 √ T1 − t T1 − t 1 − p−1
u(x, t) = (T − t)
(2.32)
1010
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
By setting x y=√ , T −t
s = log
1 , T −t
s1 = log
1 , T1 − t
we get w0,T (y, s) = Putting λ =
T −t T1 −t
T −t T1 − t
1 p−1
w0,T1
T −t y, s1 . T1 − t
and observing s1 = s + log λ, we obtain 1
w0,T (y, s) = λ p−1 w0,T1
√ λy, s + log λ .
(2.33)
Now we let t vary over [T /2, T ) and T1 over (t, T ]. Then we see that (2.33) holds for any s log(2/T ), λ ∈ [1, ∞). In particular, for each y with 0 < |y| 1, we can choose λ = 1/|y|2 and apply (2.32), to obtain 2 w0,T (y, s) CT |y|− p−1
for 0 < |y| 1, s − log(T /2).
Combining this and (2.32), we get (2.26). Next let us consider the case where ∇u0 is not necessarily bounded. Then by parabolic estimates, ∇u(x, t0 ) is bounded for any 0 < t0 < T . Choosing sufficiently small t0 and arguing as above, we obtain the desired estimate (2.26), where CT depends only on N , p, u0 L∞ and T . Finally we consider the case where u0 ∈ H˙ 1 ∩ L∞ . Since 1 wa,T1 (y, − log T1 ) = T1p−1 u0 a + T1 y ,
we have E wa,T1 (·, − log T1 ) 1 1 1 |∇wa,T1 |2 + |wa,T1 |2 − |wa,T1 |p+1 ρ(y) dy = 2 2(p − 1) p+1 RN
2 N−2 p−1 − 2
= T1
RN 2 N−2 p−1 − 2
T1
RN 2 N−2 p−1 − 2
T1
RN
1 x −a 1 1 2 2 p+1 ρ √ dx |∇u0 | + |u0 | |u0 | − 2 2(p − 1)T1 p+1 T1 1 |u0 |2 |x − a|2 2 x−a 2 dx ρ √ |∇u0 | + 2 p − 1 |x − a|2 4T1 T1 1 |u0 |2 2 dx. |∇u0 | + C 2 |x − a|2
(2.34)
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
1011
Here we have used the boundedness of the function y 2 e−y . Combining this, (2.28), (2.9) and Hardy inequality, we obtain (2.29). The estimate (2.27) follows immediately from (2.26). Indeed, 2
1 u(x, t) = (T − t)− p−1 w0,T √ x , log 1 T −t T −t 2 x − p−1 − 1 CT (T − t) p−1 1 + √ T −t − 1 − 2 CT (T − t) p−1 + |x| p−1 . The proposition is proven.
2
Proof of Corollary 2.6. By (2.26) and parabolic estimates, we have, for any T1 ∈ [T /2, T ], j ∇ w0,T (y, s) C0 1
for any |y| 1, s − log
T , j = 1, 2, 3. 2
On the other hand, differentiation of (2.33) gives 1
∇ j w0,T (y, s) = λ p−1
+ j2
∇ j w0,T1
√ λy, s + log λ .
The estimate (2.30) now follows by setting λ = 1/|y|2 and from the above estimate. The estimate (2.31) is just a restatement of (2.30) in different variables. The proof of the corollary is complete. 2 2.3. Refined estimates for highly focused blow-up In this subsection we establish pointwise estimates for blow-up that occurs at the origin x = 0 with a nonconstant local profile. Proposition 2.7. Let pS < p < ∞ and suppose that w ∗ (y) := lim w0,T (y, s) ≡ ±κ.
(2.35)
s→∞
Then there exist r0 > 0, t0 ∈ [0, T ) and C > 0 such that 2 u(x, t) C|x|− p−1 ,
j 2 ∇ u(x, t) C|x|− p−1 −j ,
2 ut (x, t) C|x|− p−1 −2 (2.36)
for any 0 < |x| r0 , t0 t < T and j = 1, 2, 3. Proof. It is well known that any nonconstant bounded solution of (1.7) decays as r → ∞ with the order r
2 − p−1
(see, for example, [5,25,28]). Thus
− 2 w ∗ (y) = O |y| p−1 ,
p+1 ∇w ∗ (y) = O |y|− p−1 as r → ∞.
1012
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
Consequently, E w ∗ (· + a) =
RN
1 1 1 ∗ 2 ∗ 2 ∗ p+1 ρ(y − a) dy → 0 |∇w | + |w | − |w | 2 2(p − 1) p+1
as |a| → ∞. Therefore if we choose R1 1 sufficiently large, then κ h E w ∗ (· + a) 4
for |a| R1 ,
where h is as in Lemma 2.1. Since w0,T (y, s) converges to w ∗ (y) as s → ∞ in C 1 (RN \ {0}), there exists s ∗ such that κ h E w0,T (· + a, s) 2
for R1 |a| R1 + 2R0 , s s ∗ ,
where R0 > 0 is the constant that appears in Lemma 2.2. By virtue of (2.16), the above estimate implies κ h E wa,T (·, s) 2
for R1 e−s/2 |a| (R1 + 2R0 )e−s/2 , s s ∗ .
Therefore, for any s1 ∈ [s ∗ , ∞) and any a satisfying |a| = (R0 + R1 )e−s1 /2 , we have hmax (s1 ; a, R0 )
κ , 2
where hmax is as in Lemma 2.2. Consequently, 1 wa,T (0, s) M0 κ e− p−1 (s−s1 ) 2
for s s1 + δ.
This means that 1 u(a, t) M0 κ (T − t1 )− p−1 2
for t1 + δ1 t < T ,
(2.37)
where t1 = T − e−s1 and t1 + δ1 = T − e−s1 −δ . Since s1 ∈ [s ∗ , ∞) can be chosen arbitrarily, the ∗ inequality (2.37) holds for any t1 ∈ [T − e−s , T ) and a ∈ Ω satisfying |a| = (R0 + R1 )e−s1 /2 = (R0 + R1 ) T − t1 . Thus t1 + δ1 = T − e−δ (R0 + R1 )−2 |a|2 , hence (2.37) can be rewritten as 2
p−1 − 2 u(a, t) M0 κ(R0 + R1 ) |a| p−1 2
for T −
eδ (R
|a|2 t < T. 2 0 + R1 )
This inequality holds for any a ∈ Ω satisfying |a| (R0 + R1 )e−s interval 0 t T − e−δ (R0 + R1 )−2 |a|2 , we have
∗ /2
(2.38)
. On the other hand, in the
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
1013
a w0,T √ ,s T −t − 2 2 δ (R0 + R1 ) p−1 e p−1 supw0,T (·, s)L∞ |a| p−1 . 1 − p−1
u(a, t) = (T − t)
s
Combining this and (2.38), we obtain the desired estimate. The proposition is proven.
2
Remark 2.8. Note that the above proposition implies, in particular, that the origin is an isolated blow-up point. This, in fact, is always true even if the local blow-up profile at the origin is ±κ. This follows from the fact the infinitely many peaks cannot accumulate to the origin as t → T . See the proof of Theorem 1.11 in [28] for details. (In Theorem 1.11, we have assumed that u0 belongs to H 1 , but we have not used this assumption to show that the blow-up points do not accumulate at the origin.) 2.4. L∞ estimates below the singular states In this subsection we derive L∞ estimates for solutions whose initial data lies strictly below the singular states near the origin. A combination of these estimates and the general blow-up estimates in Section 2.2 will yield the “no-needle lemma” (Lemmas 2.14 and 2.13) in Section 2.6, which play a crucial role in proving the convergence result in Theorem 3.2. The estimate will also be used later in Theorem 5.14 to show that any global weak solution will become a classical solution in finite time and decay to 0 as t → ∞. In the following two lemmas, u(x, t) will denote a solution of (1.1) and w(y, s) will denote a solution of the rescaled Eq. (1.13). Throughout this section, we assume NN−2 < p < ∞. This is the range of p for which the singular stationary solution ϕ ∗ defined in (1.19) exists. Lemma 2.9. Let 0 < μ < 1. Then for any sufficiently small δ > 0 there exists a constant Mδ > 1 such that, given any C > 0, u(x, t0 ) μϕ ∗ (x) + C for x ∈ Ω \ {0} (2.39) implies u x, t0 + C −(p−1) δ Mδ C
for x ∈ Ω.
(2.40)
Lemma 2.10. Let 0 < μ < 1. Then for any C > 0 and any sufficiently small σ > 0 there exists a constant Mσ > 0 such that the inequality w(y, s0 ) μϕ ∗ (y) + C for y ∈ RN \ {0} (2.41) implies w(y, s0 + σ ) Mσ C
for y ∈ RN .
Proof of Lemma 2.9. Fix μ˜ ∈ (μ, 1) and let u(x, ˜ t) be the minimal solution of the problem ut = u + |u|p−1 u x ∈ RN , t > 0 , (2.42) x ∈ RN \ {0} . u(x, 0) = μϕ ˜ ∗ (x)
1014
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
More precisely, u˜ is characterized as the limit function of an increasing sequence of classical solutions u1 u2 u3 · · · whose initial data satisfies un (x, 0) → μϕ ˜ ∗ (x)
as n → ∞
for every x ∈ RN \ {0}. (See Section 5.1 for details.) As is shown in [11, Lemma 10.4], such a solution u˜ exists for every 0 < μ˜ < 1, is smooth for all t > 0 and is a forward self-similar solution of the form |x| − 1 (2.43) u(x, ˜ t) = t p−1 Ψf √ , t where Ψf (r) is a bounded solution of Ψ +
N −1 r 1 Ψ + Ψ + Ψ + |Ψ |p−1 Ψ = 0 for 0 < r < ∞ r 2 p−1
(2.44)
satisfying Ψ (0) > 0, Ψ (0) = 0 and lim
r→∞
Ψ (r) = μ. ˜ Φ ∗ (r)
(2.45)
In other words, ψ(y) := Ψ (|y|) is a bounded radially symmetric solution of the equation ψ 1 + |ψ|p−1 ψ = 0 in RN . ψ + y · ∇ψ + 2 p−1
(2.46)
Next set θ := μ/μ˜ and let g(t) be the solution of ⎧ dg ⎪ ⎨ = g p (t > 0), dt ⎪ ⎩ g(0) = C . 1−θ
(2.47)
The function g(t) can be written explicitly as g(t) =
1−θ C
p−1 − (p − 1)t
−
1 p−1
.
Now, by the convexity of the function u → up (u > 0), the linear interpolation of the two solutions u˜ and g of (1.1), namely u(x, ˆ t) := θ u(x, ˜ t) + (1 − θ )g(t), is a supersolution. More precisely, uˆ satisfies
uˆ t uˆ + uˆ p x ∈ RN , t > t0 , u(x, ˆ 0) = θ u(x, ˜ 0) + (1 − θ )g(0) = μϕ ∗ (x) + C.
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
1015
Similarly, −uˆ is a subsolution with initial data −μϕ ∗ (x) − C. Thus, by the comparison principle, we have u(x, t) u(x, ˆ t − t0 )
for x ∈ Ω, t0 t < T ∗ ,
where T ∗ denotes the time when the right-hand side blows up, namely, T ∗ = t0 + C −(p−1) δ ∗
with δ ∗ =
(1 − θ )p−1 . (p − 1)
Fix any δ ∈ (0, δ ∗ ), where δ ∗ is as above. Then u x, t0 + C −(p−1) δ uˆ x, C −(p−1) δ = θ u˜ 0, C −(p−1) δ + (1 − θ )g C −(p−1) δ − 1 − 1 = θ Cδ p−1 Ψf (0) + (1 − θ )C (1 − θ )p−1 − (p − 1)δ p−1 1 δ − p−1 − 1 . = C θ δ p−1 Ψf (0) + 1 − ∗ δ Thus (2.40) holds with 1 δ − p−1 − 1 M = θ δ p−1 Ψf (0) + 1 − ∗ . δ 2
This completes the proof of the lemma.
Proof of Lemma 2.10. If we define a function u(x, t) by 1 − p−1
u(x, t) = (T − t)
1 , w √ , log T −t T −t
x
then u is a solution of (1.1). The assumption (2.41) implies u(x, T − λ) = λ
1 − p−1
x x − 1 + C = μ ϕ ∗ (x) + C1 , w √ , s0 λ p−1 μϕ ∗ √ λ λ
where λ = e−s0 and C1 = Cλ
1 − p−1
. Applying Lemma 2.9, we obtain
−(p−1) δ Mδ C 1 . u x, T − λ + C1 This is equivalent to 1
w(y, s0 + σ ) λ p−1 Mδ C1 = Mδ C,
1016
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
where σ is a real number defined by the relation −(p−1)
−e−(s0 +σ ) = −λ + C1
δ
= −λ 1 − C −(p−1) δ ,
or, equivalently, σ = log
1 . 1 − C −(p−1) δ
By choosing δ > 0 small, we can make σ arbitrarily small. The lemma is proven.
2
2.5. L∞ estimates for small-energy initial data In this subsection we show that small local energy implies small L∞ bounds. Such an estimate is standard in the Sobolev subcritical range 1 < p < pS , but is new in the supercritical range pS < p < ∞. We start with some notation. The symbol H˙ 1 stands for the space introduced in (2.25). We define vH 1
u-loc
:=
∇v(x) 2 + v 2 (x) dx
sup
1/2 .
a∈Ω |x−a|1
Proposition 2.11. Let pS < p < ∞ and let u(x, t) be a radially symmetric solution of (1.1) defined on some interval t0 t < t0 + T1 and satisfying u(·, t0 ) ∈ H˙ 1 (Ω) ∩ L∞ (Ω). Then for any δ ∈ (0, T1 ) there exist positive constants γ0 , M depending only on p, N , T1 and δ such that if u(·, t0 )
Hu1-loc
γ0 ,
(2.48)
then 1 u(·, t0 + δ) ∞ M u0 1 p+1 . H L u-loc
(2.49)
γ0 , M Proposition 2.12. Let pS < p < ∞. Then for any δ > 0 there exist positive constants depending only on p, N and δ such that if w(y, s) is a radially symmetric solution of (1.13) defined on some interval s0 s < ∞ and if γ0 , Emax w(·, s0 ) := sup E w(· + a, s0 ) a∈RN
then w(·, s0 + δ)
L∞
1 MEmax w(·, s0 ) p+1 .
(2.50)
Proof of Proposition 2.11. Without loss of generality we may assume t0 = 0. Fix δ > 0 arbitrarily. Let wa,T1 be as defined in (2.1). Then the same kind of computation as in (2.34) shows
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064 2 − N−2 E wa,T1 (·, s0 ) T1p−1 2
RN
Cu0 H 1
u-loc
1017
1 x−a 1 dx |∇u0 |2 + |u0 |2 ρ √ 2 2(p − 1)T1 T1
,
where C is a constant dependent only on p, N, T1 . Therefore, the constant CT that appears in Proposition 2.5 satisfies the estimate CT = O u0 H 1
1 p+1
u-loc
.
(2.51)
It follows that there exists a constant C0 > 0 dependent only on p, N , δ such that wa,T (y, s) C0 u0 1 1 H
u-loc
provided that u0 H 1
u-loc
1 p+1
∗ ϕ (y) + 1 for s s0 + δ,
is not too large. Now choose γ0 such that 1
μ := C0 γ0p+1 < 1. Then by Lemma 2.10, we have, for any δ1 ∈ (0, δ ∗ ), wa,T (y, s + δ + δ1 ) M1 1
for s s0 ,
where M1 is a constant dependent only on p, N , γ0 , δ and δ1 . In the original u variable, this 1
implies (2.49) with M = e p−1
(δ+δ1 )
M1 . The proposition is proven.
2
Proof of Proposition 2.12. If we define a function u(x, t) by 1 x − 1 , u(x, t) = (T1 − t) p−1 w √ , log T1 − t T1 − t then w(y, s) coincides with the rescaled solution w0,T1 defined in (2.1), and wa,T1 coincides with w(y + es/2 a, s). Thus the conclusion follows directly from the proof of the previous proposition. 2 2.6. Nonexistence of needle-like singularity In this subsection we prove what we call the “no-needle lemma,” which states that convergence outside the origin automatically implies convergence at the origin. In other words, δ function type thin singularity will not appear in our problem. The results will play an important role in the characterization of type II blow-up (Theorems 3.1 and 3.2) and in the study of regularization speed after blow-up (Theorem 5.19). Lemma 2.13 (No-needle lemma for u). Let pS < p < ∞ and let un (x, t) (n = 1, 2, 3, . . .) be a family of radially symmetric classical solutions of (1.1) that are defined for x ∈ Ω, t0 t T1 and satisfy supn un (·, t0 )L∞ < ∞, un (x, t ∗ ) → v(x)
(n → ∞), for every x ∈ Ω \ {0}
1018
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
for some t ∗ ∈ (t0 , T1 ) and some bounded function v on Ω \ {0} with bounded gradient ∇v. Then for any sufficiently small δ > 0 there exists a constant M1 > 0 such that sup∇ j un (·, t ∗ + δ)L∞ (Ω) M1 for j = 0, 1, 2, 3. (2.52) n
Lemma 2.14 (No-needle lemma for w). Let pS < p < ∞ and let wn (y, s) (n = 1, 2, 3, . . .) be a family of radially symmetric classical solutions of (1.13) that are defined for y ∈ RN , s0 s < ∞ and satisfy supn E(wn (·, s0 )) < ∞. Suppose that wn (y, s ∗ ) → ψ(y)
(n → ∞), for every y ∈ RN \ {0}
for some s ∗ > s0 and some bounded function ψ(y) on RN \ {0} with bounded gradient ∇w. Then for any sufficiently small δ > 0 there exists a constant M1 > 0 such that (2.53) sup∇ j wn (·, s ∗ + δ)L∞ M1 for j = 0, 1, 2, 3. n
Remark 2.15. In Lemma 2.13, the case where t ∗ = T1 is not included. This case will be treated in Corollary 4.2. Proof of Lemma 2.13. Without loss of generality we may set t0 = 0. Define 1
wn (y, s) := (T1 − t) p−1 un (x, t) = e
1 − p−1 s
u e−s/2 y, T1 − e−s ,
where x , y=√ T1 − t
s = − log(T1 − t).
Then wn satisfies the rescaled equation (1.13) and wn (y, s ∗ ) → ψ(y)
(n → ∞), for every y ∈ RN \ {0}, −
1
s∗
(2.54)
∗
where s ∗ = − log(T1 − t ∗ ) and ψ(y) = e p−1 v(e−s /2 y). By the assumption supun (·, t0 )L∞ < ∞ n
along with Proposition 2.5 and Corollary 2.6, we have 2 wn (y, s ∗ ) C 1 + |y|− p−1 , for |y| > 0, s s ∗ ,
2p 2 ∇ wn (y, s ∗ ) C 1 + |y|− p−1
(2.55)
where the constant C is independent of n. Thus the convergence (2.54) takes place in C 1 (RN \ {0}). Hence there exists a constant M > 0 such that for any small ε > 0 and any large R0 > 0 we have wn (y, s ∗ ) + ∇wn (y, s ∗ ) M for ε |y| R0
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
1019
for sufficiently large n. Combining this and (2.55), we obtain wn (y, s ∗ ) + ∇wn (y, s ∗ ) M for ε |y| < ∞
(2.56)
with an appropriate choice of constant M > 0. Now, given a constant 0 < θ 1, we set T2 := (1 − θ )t ∗ + θ T1 , and define 1
w˜ n (y, s) := (T2 − t) p−1 un (x, t) = e
1 − p−1 s
u e−s/2 y, T2 − e−s ,
where x−a y=√ , T2 − t
s = − log(T2 − t).
Then w˜ n satisfies the same rescaled Eq. (1.13), and it holds that 1
w˜ n (y, s) = λ p−1 (s)wn
λ(s)y, s + log λ(s) ,
where λ(s) =
1 T2 − t = ∗. T1 − t 1 + (1 − θ )es−s
In particular, we have 1
w˜ n (y, s ∗ − log θ ) = θ p−1 wn
√ θ y, s ∗ .
Let us compute the energy of w˜ n (y + a, s ∗ − log θ ) for each a ∈ RN : p+1
θ p−1 E w˜ n (· + a, s ∗ − log θ ) 2
√ ∇wn θ y 2 ρa (y) dy
RN
2
+
θ p−1 2(p − 1)
wn2
√ θ y ρa (y) dy,
RN
where wn (z) stands for wn (z, s ∗ ) and ρa (y) = ρ(y − a). Using (2.30), the first integral on the right-hand side is estimated as √ √ ∇wn θ y 2 ρa (y) dy = ∇wn θ y 2 ρa (y) dy + RN
|y|ε
C
|y|ε
2
√ − 2(p+1) θ y p−1 ρa (y) dy + M 2
|y|ε
C2 (4π)N/2
θ
− p+1 p−1
|y|ε − 2(p+1) p−1
|y| |y|ε
dy + M 2 .
ρa (y) dy
1020
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
Here we have used the fact that
0 ρa (y) (4π)−N/2 ,
ρa (y) dy = 1. RN
Similarly, the second integral is estimated as wn2
√ θ y ρa (y) dy =
wn2
√
θ y ρa (y) dy +
|y|ε
RN
C2 − 2 θ p−1 N/2 (4π)
wn2
√ θ y ρa (y) dy
|y|ε
4 − p−1
|y|
dy + M 2 .
|y|ε
Combining these, we obtain E w˜ n (· + a, s ∗ − log θ )
C2 2(4π)N/2 +M
− 2(p+1) p−1
Note that |y| 4 − p−1
2
θ
− 2(p+1) p−1
|y| |y|ε p+1 p−1
2
4 1 − p−1 dy |y| + (p − 1)
2 θ p−1 + . 2(p − 1)
(2.57)
is locally integrable around the origin if (and only if) p > pS . Then the
term |y| becomes automatically locally integrable. Consequently, choosing ε and θ small enough, we can make the right-hand side of the above inequality smaller than the constant γ0 that appears in Proposition 2.12. More precisely, γ0 E w˜ n (· + a, s ∗ − log θ )
for any a ∈ RN ,
hence, by Proposition 2.12, w˜ n (·, s ∗ − log θ + δ)
1
L∞
M γ0p+1 .
Consequently, wn (·, s ∗ + δ)
L∞
− 1 = λ(s ∗ + δ) p−1 w˜ n (·, s ∗ − log θ + δ)L∞ 1 1 1 + (1 − θ )eδ p−1 M γ0p+1 .
Parabolic regularization then implies that j ∇ wn (·, s ∗ + 2δ)
L∞
M0
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
1021
for j = 1, 2, 3, where M0 is a constant independent of n. Going back to the original function x , − log(T1 − t) , un (x, t) = wn √ T1 − t we obtain the estimate (2.52) after an appropriate redefinition of M1 and δ. The lemma is proven. 2 Proof of Lemma 2.14. As in the proof of Proposition 2.12, if we define u(x, t) by 1 − p−1
u(x, t) = (T1 − t)
1 , , log w √ T1 − t T1 − t
x
then w(y, s) coincides with w0,T1 defined in (2.1), and wa,T1 coincides with w(y + es/2 a, s). Thus the conclusion follows directly from the proof of Lemma 2.13. 2 2.7. Estimates near |x| = ∞ for H 1 initial data The following proposition will be used in the proof of Theorems 3.9 and 4.9. Proposition 2.16. Let Ω = RN , pS < p < ∞ and let u be a solution of (1.1) that blows up at t = T . Assume that u0 ∈ H 1 (RN ) ∩ L∞ (RN ). Then there exists R0 > 0 such that lim supu(·, t)H 1 (RN \B
R0 )
t→T
< ∞,
lim supu(·, t)Lp+1 (RN \B t→T
R0 )
< ∞,
where RN \ BR0 = {x ∈ RN | |x| R0 }. Proof. As we have shown in the proof of Proposition 2.5, we have 2 − N−2 E wa,T (·, − log T ) T1p−1 2
RN
1 x −a 1 dx. |∇u0 |2 + |u0 |2 ρ √ 2 2(p − 1)T1 T1
Since u0 ∈ H 1 (RN ), the right-hand side tends to 0 as |a| → ∞, hence so does E(wa,T ). Consequently, by Lemma 2.2 and (2.9), there exists some constant C > 0 such that u(a, t) p+1 CE wa,T (·, − log T ) for all large |a|, say |a| R0 , and for all t close to the blow-up time T . Integrating the above inequality by a and applying the previous estimate, we obtain |a|R0
u(a, t) p+1 da C1 u0 2 1 . H
(2.58)
1022
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
This proves the latter part of the desired estimates. Next we prove the former part. The above observation shows that u(a, t) is uniformly bounded in the region |a| R0 , hence by parabolic estimates there exists a constant C2 > 0 and t2 ∈ [0, T ) such that ∇u(·, t) ∞ + ut (·, t)L∞ (|x|R ) C2 for t1 t < T . (2.59) L (|x|R ) 0
0
Now, integration by parts yields d dt
|x|R0
1 1 2 p+1 dx = |∇u| − |u| 2 p+1
∇ut · ∇u − |u|p−1 uut dx
|x|R0
=−
ut
|x|=R0
u2t dx
|x|R0
−
∂u dSx − ∂r
ut
|x|=R0
∂u dSx . ∂r
The right-hand side of the above inequality is bounded for t1 t < T by virtue of (2.59). Con 1 |u|p+1 ) dx is bounded from above as t → T . Combining this sequently, |x|R0 ( 12 |∇u|2 − p+1 and (2.58), we obtain lim sup |∇u|2 dx < ∞. t→T
|x|R0
2
This completes the proof of the proposition. 3. Properties of type II blow-up
In this section we study various properties of type II blow-up. Throughout this section, Ω is either a finite ball BR := {|x| < R} or the entire space RN . 3.1. Various characterizations of type II blow-up The following two theorems are fundamental in this section. Theorem 3.1 (Existence of the local profile). Let pS < p < ∞ and let u0 ∈ L∞ (Ω). Suppose the solution u of (1.1) blows up at t = T . Then the limit w ∗ (y) := lim w0,T (y, s) s→∞
1 √ = lim (T − t) p−1 u T − t y, t
t→T
(3.1)
exists locally uniformly in y ∈ RN \ {0} (hence in C 2 (RN \ {0}), and w ∗ is either a bounded radially symmetric solution of (1.7) or the singular stationary solution ϕ ∗ (y) or −ϕ ∗ (y). Theorem 3.2 (Characterizations of type II blow-up). Let pS < p < ∞ and let u and w ∗ be as in Theorem 3.1. Then the following conditions are equivalent:
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
1023
(a) the blow-up is of type II; 1
(b) limt→T (T − t) p−1 u(·, t)L∞ = ∞; (c) w ∗ (y) = ϕ ∗ (y) or −ϕ ∗ (y); ) (d) limx→0 u(x,T ϕ ∗ (x) = 1 or −1. Remark 3.3. The main novelty of Theorem 3.1 is the existence of the limit (3.1) for a type II blow-up. This result is exceedingly useful in studying general properties of type II blow-up. As for type I blow-up, the existence of the limit (3.1) has been established by [31] when Ω = BR (see also [28, Theorem 1.13(i)]), while only partial results have been known when Ω = RN . Remark 3.4. Condition (b) in Theorem 3.2 asserts that the “lim sup” in Definition 1.1 can be replaced by “lim.” Incidentally, the general estimate (2.27) implies 1 1 − 2 (T − t) p−1 u(x, t) C (T − t) p−1 |x| p−1 + 1 , 1
therefore the quantity (T − t) p−1 |u(x, t)| remains bounded as t → T outside any small neighborhood of x = 0. In other words, a type II behavior can occur only at x = 0. Remark 3.5. The term u(x, T ) in condition (d) denotes limt→T u(x, t), which we call the global blow-up profile. The existence of this limit will be proven in Proposition 3.14. Before starting the proof of the above theorem, let us recall the well known zero-number properties for parabolic equations (a Sturm type theorem). The following is a radial version of this property whose proof is found in [6]. Lemma 3.6 (Zero number properties). Let v(x, t) := V (r, t) be a smooth radially symmetric solution of the linear parabolic equation vt = v + b |x|, t (x · ∇v) + a |x|, t v,
|x| < R0 , t ∈ (t1 , t2 ),
(3.2)
where 0 < R0 < +∞, −∞ t1 < t2 +∞ and a(r, t), b(r, t) are bounded continuous functions on [0, R0 ] × (t1 , t2 ). Assume that V (r, t) is not identically equal to zero and satisfies either of the following boundary conditions: t ∈ (t1 , t2 ) , V (R0 , t) = 0 t ∈ (t1 , t2 ) .
V (R0 , t) ≡ 0
Then the following hold: (i) Z[0,R0 ] [V (·, t)] is finite for any t ∈ (t1 , t2 ); (ii) t → Z[0,R0 ] [V (·, t)] is monotone non-increasing; (iii) if Vr (r ∗ , t ∗ ) = V (r ∗ , t ∗ ) = 0 for some r ∗ ∈ [0, R0 ], t ∗ ∈ (t1 , t2 ), then Z[0,R0 ] V (·, t) > Z[0,R0 ] V (·, s)
(t1 < t < t ∗ < s < t2 ).
(3.3a) (3.3b)
1024
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
Typical situations to which the above lemma applies is: V = U1 − U2 , where U1 , U2 are solutions of (1.4); V = Ut , where U is a solutions of (1.4); V = W1 − W2 , where W1 , W2 are radial solutions of (1.13); V = Wt , where W is a radial solution of (1.13); V = U − Φ ∗ or V = W − Φ ∗ , where Φ is the singular stationary solutions. In the last case, we have replace [0, R0 ] by (0, R0 ]. It is also easily that the lemma remains true if R0 is replaced by a smooth function R0 (t) > 0 (see [28, Remark 2.8]). Note also that the statements (ii), (iii) are still valid if [0, R0 ] (or (0, R0 ]) is replaced by [0, ∞) (or (0, R0 ]). Proof of Theorem 3.2. First we prove the existence of the limit (3.1). Denote by ω(w0,T ) the ωlimit set of w0,T in the topology of C 1 (RN \ {0}). In other words, ω(w0,T ) consists of functions obtained as a limit of any convergent sequence w0,T (y, sn ) with sn → ∞. Then as is shown in [28, Theorem 1.13], we have ω(w0,T ) ⊂ E ∪ {ϕ ∗ , −ϕ ∗ }.
(3.4)
Here E denotes the set of radially symmetric bounded solutions of (1.7). Furthermore ω(w0,T ) is a compact connected set in C 1 (RN \ {0}). What we have to show is that ω(w0,T ) contains precisely one element. If 0 is not a blow-up point of u, then clearly w0,T (y, s) → 0 as s → ∞, so this case is trivial. Next, if 0 is a blow-up point and if the blow-up is of type I, then w0,T (y, s) remains bounded as s → ∞. In this case, ω(w0,T ) ⊂ E, and there is no distinction between the topology of C 1 (RN \ {0}) and that of C 1 (RN ). As we have mentioned in Remark 3.3, the existence of the limit (3.1) for this special case is established in [31] provided that Ω = BR . In order to deal with the case Ω = RN , we have to modify the argument. For the clarity of the present paper, let us recall how the argument of [31] (also that of [28]) goes for the case Ω = BR . Thus we start with the case where Ω = BR and the blow-up is of type I. The argument below is based largely on the ideas of [26]. Suppose that ω(w0,T ) contains more than one element. Then, since ω(w0,T ) is connected, it must contain infinitely many elements. Let ψ1 (y) = Ψ1 (|y|), ψ2 (y) = Ψ2 (|y|) and ψ3 (y) = Ψ3 (|y|) be elements of ω(w0,T ) ∩ E satisfying Ψ1 (0) < Ψ3 (0) < Ψ2 (0). These functions are ω-limit points of w0,T in the topology of C 1 (RN ). Consequently, we can choose a sequence 0 < s1 < s2 < s3 < · · · → ∞ such that W (r, s2k−1 ) → Ψ1 (r),
W (r, s2k ) → Ψ2 (r)
as k → ∞
in C 1 ([0, ∞)), where W (r, s) := w0,T (y, s) with r = |y|. Then W (0, s2k−1 ) − Ψ3 (0) < 0,
W (0, s2k ) − Ψ3 (0) > 0
for k sufficiently large. Hence for each large n there exists s¯n ∈ (sn , sn+1 ) such that W (0, s¯n ) − Ψ3 (0) = 0
for n = 1, 2, 3, . . . .
(3.5)
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
1025
Since Wr (0, s) = Ψ (0) = 0, the function W (r, s¯n ) − Ψ3 (r) has a degenerate zero at the origin. Applying Lemma 3.6(iii) to W (r, s) − Ψ3 (r), we see that Z[0,∞) [W (·, s) − Ψ3 ] must drop at least by 1 each time s crosses s¯n . Now recall that any non-trivial bounded radial solution of (1.7) does not vanish for r large (see, for example, the proof of [5, Theorem 3]). On the other hand, in the case where Ω = BR , we have W (Res/2 , s) = 0 for s s0 . Therefore the function W (Res/2 , s) − Ψ3 (Res/2 ) has a constant sign for all large s. By Lemma 3.6 and its subsequent remark, Z[0,Res/2 ] W (·, s) − Ψ3 is finite and monotonically decreasing for s sufficiently large. Therefore, this quantity can drop only finitely many times, contradicting the previous observation. This contradiction proves that ω(w0,T ) cannot contain more than one element, hence the convergence (3.1) follows. Next we assume that the blow-up is of type I and that Ω = RN . By exactly the same argument 3 the self-similar solution of (1.1) corresponding as above, we see that (3.5) holds. Denote by U to Ψ3 ; namely 1 r − p−1 . U3 (r, t) := (T − t) Ψ3 √ T −t Then (3.5) implies 3 (0, t¯n ) for n = 1, 2, 3, . . . , U (0, t¯n ) = U where t¯n = T − e−¯sn . Now fix r0 > 0 arbitrarily. Then by Lemma 3.6, 3 (·, t) < ∞ Z[0,r0 ] U (·, t) − U for every 0 < t < T , while this quantity drops at least by one each time t crosses t¯n . In order 3 (r, t) have to enter the interval from the for this to be possible, new zeros of r → U (r, t) − U 3 (r0 , t) changes sign infinitely many times as t approaches T . endpoint r0 , therefore U (r0 , t) − U Consequently, 1
3 (r0 , t) = lim λ p−1 Ψ3 U (r0 , T ) = lim U t→T
√
λ→∞
− 2 λr0 = μ∞ (Ψ3 )r0 p−1 ,
where μ∞ (·) is as in (A.2). On the other hand, since ω(w0,T ) is connected, it contains another element Ψ4 satisfying Ψ (1) < Ψ3 (0) < Ψ4 (0) < Ψ2 (0). Then the same argument as above shows 2 − p−1
U (r0 , T ) = μ∞ (Ψ4 )r0
,
hence μ∞ (Ψ4 ) = μ∞ (Ψ3 ). This, however, contradicts Lemma A.1 in Appendix A, which states ). This contradiction shows that ω(w0,T ) is a singleton. implies μ∞ (Ψ ) = μ∞ (Ψ that Ψ ≡ Ψ Next we consider the case where the blow-up is of type II. In view of (3.4), all we have to show is that ω(w0,T ) ∩ E = ∅.
(3.6)
1026
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
Indeed, once this is shown, then the connectedness of ω(w0,T ) implies either ω(w0,T ) = {ϕ ∗ } or ω(w0,T ) = {−ϕ ∗ }, which means the convergence w(·, s) → ±ϕ ∗ . Assuming the contrary, we will derive a contradiction. Suppose the blow-up is of type II and that ω(w0,T ) ∩ E = ∅. Then there exists a sequence sn → ∞ and ψ ∈ E such that w0,T (y, sn ) → ψ(y)
as n → ∞
in C 1 (RN \{0}). Then by Lemma 2.14, w0,T (y, sn +δ) remains bounded in C 3 (RN ), where δ > 0 is some fixed small number. On the other hand, since ψ is a stationary solution, w0,T (y, sn + δ) converges to ψ in C 1 (RN \ {0}). It follows that w0,T (y, sn + δ) → ψ(y)
as n → ∞
in C 2 (RN ). If ψ is the only element of ω(w0,T ), we have w0,T (y, s) → ψ(y) as s → ∞ in C 1 (RN \ {0}). Hence, by the same argument as above, w0,T (y, s + δ) → ψ(y) as s → ∞ in C 2 (RN ), hence locally uniformly in RN as s → ∞. This, together with the general estimate (2.26), implies that w0,T remains bounded in L∞ as s → ∞, contradicting the assumption that the blow-up is of type II. Therefore ω(w0,T ) must contain more than one element. By the connectedness of ω(w0,T ), it contains infinitely many elements, hence the same is true of ω(w0,T ) ∩ E. We can then derive a contradiction by the same argument as in the case of type I blow-up, since any point in ω(w0,T ) ∩ E is also an ω limit point with respect to the C 2 (RN ) topology, as we have seen above. This completes the proof of the convergence (3.1). Next we prove the equivalence of the conditions (a)–(d). The equivalence (c) ⇔ (d) is included in Theorem 4.1 (and follows immediately from Proposition 4.4), so we postpone the proof of this assertion and simply prove the equivalence of (a)–(c). As we have shown above, (a) implies (c). The relation (c) ⇒ (b) obviously holds since w0,T (·, − log(T − t))L∞ = 1
(T − t) p−1 u(·, t)L∞ . The relation (b) ⇒ (a) is clear from the definition of type II blow-up. The theorem is proven. 2 3.2. Nonexistence of type II blow-up for pS < p < pJL In this subsection, as an application of Theorem 3.2 above, we give an alternative proof to the result of our earlier paper [28], which states that type II blow-up does not occur if pS < p < pJL . The idea in [28] was first to show that any type II blow-up solution converges to a stationary solution of (1.1) after an appropriate rescaling. Our proof here, on the other hand, is based on the fact that type II blow-up implies w ∗ = ϕ ∗ . Theorem 3.7 below for the case Ω = BR is identical to Theorem 1.5 in [28], while Theorem 3.8 for the case Ω = RN slightly improves Theorem 1.6 in [28], as we do not impose any condition on the zero number of Ut (r, 0). Theorem 3.9 for H 1 initial data is new. Theorem 3.7. Let pS < p < pJL and let Ω = BR . Then no type II blow-up occurs. Theorem 3.8. Let pS < p < pJL and let Ω = RN . Assume that Z(0,∞) [|U0 | − Φ ∗ ] < ∞, where U0 (r) := u0 (x) with r = |x|. Then no type II blow-up occurs.
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
1027
Theorem 3.9. Let pS < p < pJL and let Ω = RN . Assume that u0 ∈ L∞ (RN ) ∩ H 1 (RN ). Then no type II blow-up occurs. Remark 3.10. We do not know if the conclusion of Theorem 3.8 holds true without the assumption Z(0,∞) [|U0 | − Φ ∗ ] < ∞. However, we can at least say that a type II blow-up is an extremely rare phenomenon in the range pS < p < pJL . More precisely, if a type II blow-up ever occurs in the range pS < p < pJL , then we must have u(x, T ) := lim u(x, t) = ±ϕ ∗ (x) t→T
for every x ∈ RN \ {0}.
(3.7)
We will prove this assertion later in this subsection. Proof of Theorem 3.7. Let ϕ1 (x) = Φ1 (|x|) be the stationary solution defined in (1.20) with a = 1. It is known that, in the range pS < p < pJL , the graph of Φ1 (r) and that of Φ ∗ (r) intersects infinitely many times. In other words, Z(0,∞) [Φ1 − Φ ∗ ] = ∞, see [21]. The proof of this well known fact can also be found in [28, Lemma 2.2]. Now suppose that the blow-up is of type II. Then by Theorem 3.2, w0,T (y, s) → ϕ ∗ (y) as s → ∞. In the variable r := |y|, this convergence is expressed as W0,T (r, s) → Φ ∗ (r)
(s → ∞) locally uniformly in 0 < r < ∞.
Combining this and the above estimate, and recalling that W0,T (r, s) is defined for 0 r Res/2 , we obtain lim Z(0,Res/2 ) Φ1 − W0,T (·, s) = ∞.
s→∞
Choose any sequence 0 < t1 < t2 < t3 < · · · → T and set λn = T − tn . Since 1
W0,T (r, sn ) = λnp−1 U
λn r, tn ,
where sn = − log (T − tn ), the estimate (3.8) implies 1 lim Z(0,Rn ) Φ1 − λnp−1 U λn r, tn = ∞,
n→∞
where Rn =
√R . λn
Note also that, by (1.21), 1
Φ1 (r) − λnp−1 U −
1 λn r, tn = λnp−1 Φan λn r − U λn r, tn ,
1
where an = λn p−1 . Consequently, 1 Z(0,Rn ) Φ1 − λnp−1 U λn r, tn = Z(0,R) Φan − U (·, tn ) ,
(3.8)
1028
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
hence lim Z(0,R) Φan − U (·, tn ) = ∞.
n→∞
Since Z(0,R) [Φan − U (·, t)] is monotone non-increasing in t, and since Φa is a stationary solution, we have Z(0,R) Φan − U (·, t0 ) → ∞
as n → ∞
(3.9)
for any fixed t0 ∈ (0, T ). By Lemma 3.6, we can choose t0 such that the function r → Φ ∗ (r) − U (r, t0 ) has only simple zeros in (0, R] and that m0 := Z(0,R) Φ ∗ − U (·, t0 ) < ∞. Then, since Φa (r) → Φ ∗ (r) as a → ∞ locally uniformly in 0 < r < ∞, the simplicity of the zeros of Φ ∗ (r) − U (r, t0 ) implies that Z(0,R) Φa − U (·, t0 ) = m0 for all large a, which contradicts (3.9). This contradiction proves the theorem.
2
Proof of Theorem 3.8. By the assumption of the theorem, we have m0 := Z(0,∞) [Φ ∗ − U0 ] < ∞. Therefore, by Lemma 3.6, we can choose t0 such that the function Φ ∗ (r) − U (r, t0 ) has only simple zeros in (0, ∞). Furthermore, Z(0,∞) Φ ∗ − U (·, t0 ) m0 . The convergence Φa → Φ ∗ as a → ∞ and the simplicity of the zeros of Φ ∗ (r) − U (r, t0 ) imply Z(0,∞) Φa − U (·, t0 ) m0 for all large a > 0. On the other hand, by the same argument as we have used to derive (3.9), we obtain Z(0,∞) Φan − U (·, t0 ) → ∞ This contradiction proves the theorem.
as n → ∞.
2
Before proving Theorem 3.9, let us first show the claim (3.7). Proof of (3.7). By the same argument as we used to derive (3.8), we obtain lim Z(0,R0 es/2 ) Φ1 − W0,T (·, s) = ∞
s→∞
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
1029
for any R0 > 0, if the blow-up is of type II. Thus, arguing as in the proof of Theorem 3.7, we see that lim Z(0,R0 ) Φan − U (·, tn ) = ∞
(3.10)
n→∞
for any R0 > 0. Now suppose that U (R0 , T ) = Φ ∗ (R0 ). Then there exist ε0 > 0 and δ > 0 such that ∗ Φ (R0 ) − U (R0 , t) ε0
for t ∈ [T − δ, T ).
(3.11)
Then by Lemma 3.6, Z(0,R0 ) [Φ ∗ − U (·, t)] is finite for every t ∈ (T − δ, T ) and is monotonically non-increasing. Furthermore, for possibly finitely many exceptional values of t, the zeros of the function r → Φ ∗ (r) − U (r, t) lying in the interval (0, R0 ) are all simple. Fix any such t0 ∈ (T − δ, T ). Then by exactly the same argument as in the proof of Theorem 3.7, we have Z(0,R) Φa − U (·, t0 ) = m0 := Z(0,R) Φ ∗ − U (·, t0 ) for all large a. Furthermore, by (3.11) and the fact that Φa (R0 ) → Φ ∗ (R0 ) as a → ∞, we have Φa (R0 ) − U (R0 , t) = 0 for t ∈ [T − δ, T ) for all large a. Consequently, Z(0,R0 ) [Φa − U (·, t)] is monotone non-increasing in t for every large a > 0. In particular, Z(0,R0 ) Φan − U (·, tn ) Z(0,R) Φan − U (·, t0 ) m0 for every large a > 0, contradicting (3.10). This contradiction shows that U (R0 , T ) = Φ ∗ (R0 ). Since R0 > 0 is arbitrary, (3.7) is proven. 2 Proof of Theorem 3.9. Since u0 ∈ H 1 (RN ) ∩ L∞ (RN ), we see from Proposition 2.16 and Fatou’s lemma that u(x, T ) p+1 dx < ∞ for all large R0 > 0. |x|R0
This, however, is impossible since, by (3.7), the above integral is equal to |x|R0
∗ p+1 ϕ (x) dx = (c∗ )p+1
|x|−2(p+1)/(p−1) dx,
|x|R0
which is infinite if p pS . This contradiction proves the theorem.
2
1030
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
3.3. Type II blow-up and the intersection number In this subsection we prove that if the blow-up is of type II, then at least two intersection points between the graph of U (r, t) and that of the singular stationary solution Φ ∗ (r) escape to ∞ as t → T . This is equivalent to saying that at least two zeros of the function r → U (r, t) − Φ ∗ (r) approaches r = 0 as t → T . In order to state our result more clearly, let u(x, t) be a solution of (1.1) that blows up at t = T , and let U (r, t) := u(x, t) with r = |x|. Then a comparison argument (Lemma 3.13 below) shows that if u satisfies |u(x, t)| ϕ ∗ (x) (x ∈ Ω) for some t ∈ [0, T ), then u cannot blow up in finite time. Therefore
Z(t) := r > 0 U (r, t) − Φ ∗ (r) = 0
(3.12)
is not empty for any t ∈ [0, T ). Now we define rmin (t) := min Z(t).
(3.13)
Again by Lemma 3.13, in order for the solution u to blow up, it must hold that lim inf rmin (t) = 0. t→T
(3.14)
Now, given a solution U (r, t) that blows up at t = T , we define the number of vanishing intersections m0 (U ) as follows:
m0 (U ) := lim lim sup max Z(0,r0 ] U (·, t) − Φ ∗ , Z(0,r0 ] U (·, t) + Φ ∗ . r0 →0 t→T
(3.15)
By (3.14), we always have m0 (U ) 1 for any blow-up that occurs at r = 0. As is easily seen, m0 (U ) = 1 if and only if there exist r0 > 0 and t0 ∈ [0, T ) such that Z(0,r0 ] U (·, t) − Φ ∗ 1,
Z(0,r0 ] U (·, t) + Φ ∗ 1 for t0 < t < T .
(3.16)
We call such a blow-up a “single-intersection blow-up.” Our main result in this section is as follows. Theorem 3.11. Let pJL < p < ∞ and let u be a solution that blows up at t = T . Suppose that the blow-up is of type II. Then m0 (U ) 2. In other words, any single-intersection blow-up is of type I. Remark 3.12. See Theorem 5.28 and Remark 5.31 for further properties of a single-intersection blow-up. Proof of Theorem 3.11. Let w0,T (y, s) = W (|y|, s) be the rescaled solution. By Theorem 3.2, W (r, s) converges to either Φ ∗ (r) or −Φ ∗ (r) as s → ∞ locally uniformly in r > 0. Without loss of generality we may assume the former, since otherwise we can consider −u instead of u. By assuming (3.16), we will derive a contradiction.
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
1031
We linearize the right-hand side of Eq. (2.17) (namely the rescaled equation (1.13) under the radial symmetry) around Φ ∗ and denote this operator by L; namely LΨ := Ψ +
N −1 r 1 p(c∗ )p−1 Ψ. Ψ − Ψ − Ψ+ r 2 p−1 r2
Let Hρ1 be the space of radially symmetric functions on RN with the norm ψ2H 1 ρ
:=
|∇ψ|2 + ψ 2 ρ dy,
RN
where ρ is as in (2.5). Then, as is shown in [18,19], the eigenvalues of L in the space Hρ1 are given by the form λj =
α 1 + + j − 1, p−1 2
(3.17)
where α=
−(N − 2) +
(N − 2)2 − 4pB , 2
B=
2 2 N −2− . p−1 p−1
One can easily check that λ1 < λ2 < 0
(3.18)
for any pJL < p < ∞. (Furthermore, it can further be shown that λ3 > 0 if and only if p > pL := 6 1 + N −10 , though we do not need this fact in the present paper.) Since the second eigenvalue is strictly negative, one can find a constant M > 0 such that the second eigenvalue of L restricted to the interval M −1 < r < M under the Dirichlet boundary conditions at r = M −1 , M is 0. We denote this eigenfunction by ψ2 . By the Sturm–Liouville theory, ψ2 (r) changes sign exactly once at some point r1 ∈ (M −1 , M). Therefore we can assume ψ2 (r) < 0 for M −1 < r < r1 ,
ψ2 (r) > 0
for r1 < r < M.
Denote by r(s) the smallest zero of W (r, s) − Φ ∗ (r) for each fixed s. By the assumption (3.16), we can choose s1 ∈ R such that, for every s s1 , the function W (r, s) − Φ ∗ (r) has no zero other than r(s) in the range 0 < r M + 1. Whenever the zero r(s) appears in the range 0 < r M, it is a simple zero, since, by the result of [1], a degenerate zero can occur only when two zeros merge and disappear. Consequently, r(s) is a smooth function of s whenever 0 < r(s) M. Choose ε > 0 small enough so that there exists a stationary solution v ε (r) of (2.17) that approximates the function Φ ∗ (r) + εψ2 (r). More precisely, by a bifurcation argument, one can easily find a value δε = M −1 + O(ε) and a solution v ε of the equation V +
1 r 1 V − V − V + |V |p−1 V = 0 N −1 2 p−1
1032
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
on the interval [δε , M] such that v ε (r) = Φ ∗ (r) + εψ2 (r) + O ε 2 for r ∈ [δε , M] in the C 1 sense and such that v ε (δε ) = Φ ∗ (δε ), < Φ ∗ (r) v ε (r) > Φ ∗ (r)
v ε (M) = Φ ∗ (M), for r ∈ (δε , rε∗ ), for r ∈ (rε∗ , M),
(3.19)
where rε∗ is a point in (δε , M) satisfying rε∗ = r1 + O(ε). Furthermore, if ε is chosen small enough, then Z[δε ,M] W (·, s1 ) − v ε = Z[δε ,M] W (·, s1 ) − Φ ∗ 1. This is because W (r, s1 ) − Φ ∗ (r) has no zero other than r(s1 ) in the interval (0, M], and because r(s1 ) is a simple zero if it lies in (0, M]. As s varies over the interval [s1 , ∞), the number of the zeros of W (r, s) − v ε (r) in the interval [δε , M] may vary, but it can vary only when r(s) crosses either δε or M. Since W (r, s)
< Φ ∗ (r) > Φ ∗ (r)
if 0 < r < r(s), if r(s) < r M,
(3.20)
and since v ε satisfies (3.19), we easily see that there is at least one zero of W (r, s) − v ε (r) between δε and r(s) if r(s) ∈ [δε , rε∗ ), and there is at least one zero between r(s) and M if r(s) ∈ (rε∗ M]. Consequently, each time r(s) leaves the interval [δε , M], at least one zero of W (r, s) − v ε (r) is lost. On the other hand, it is easily seen that each time r(s) enters the interval [δε , M], precisely one zero of W (r, s) − v ε (r) is created. It follows that Z[δε ,M] W (·, s) − v ε 1 for s1 s < ∞. However, since W (r, s) converges to Φ ∗ as s → ∞ uniformly in [δε , M], and since W satisfies (3.20) while v ε satisfies (3.19), we must have Z[δε ,M] W (·, s) − v ε 2 for s sufficiently large, regardless of the position of r(s). This contradiction proves the theorem.
2
To end this subsection, let us prove the following well-known lemma which we have used in defining m0 (U ) in (3.15). Lemma 3.13. Suppose that U (r, 0) is bounded and that there exists a constant r0 > 0 such that U (r, t) Φ ∗ (r)
for 0 < r < r0 , 0 t < T .
Then U remains bounded in 0 r < r0 , 0 t < T .
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
1033
Proof. The proof is found in [11], but since it is short, we prove it here for the convenience of the reader. By a simple comparison argument, one can show that there exist a constant δ > 0 and r1 ∈ (0, r0 ) such that U (r1 , t) Φ ∗ (r1 ) − δ
for 0 t < T .
Now choose ε > 0 sufficiently small so that Φ ∗ (r1 + ε) > Φ ∗ (r1 ) − δ,
U (r, 0) Φ ∗ (r + ε)
for r ∈ [0, r1 ].
Then it is easily seen that the function Φ ∗ (|x| + ε) is a supersolution for (1.1) in the region |x| r1 and that U (|x|, t) Φ ∗ (|x| + ε) on the parabolic boundary of Br1 (0) × [0, T ). Thus by the comparison principle, we get U |x|, t Φ ∗ |x| + ε Φ ∗ (ε)
for |x| r1 , 0 t < T .
The boundedness of U in the region r1 |x| r0 follows from the assumption. The proof of the lemma is complete. 2 3.4. Existence of the global profile In this subsection, partly as an application of Theorems 3.1 and 3.2, we show that the global blow-up profile u(x, T ) is well defined. Proposition 3.14. Let pS < p < ∞ and let u be a radially symmetric solution of (1.1) that blows up at t = T . Then the following pointwise limit exists for every x ∈ Ω \ {0}: lim u(x, t) =
t→T
a finite value if x ∈ / B(u0 ), ∞ or −∞ if x ∈ B(u0 ).
(3.21)
Furthermore, the limit also exists at x = 0 if the blow-up is of type I, or if the blow-up is of type II and u0 0. Proof. If x0 ∈ / B(u0 ), then u is bounded in a neighborhood of x0 as t → T , hence standard parabolic estimates imply that ut (x0 , t) remains bounded as t → T , from which the convergence (3.21) at x = x0 follows. Next let x0 ∈ B(u0 ). If x0 = 0, then by [28, Theorem 1.13] the local profile at x = x0 is either κ or −κ and the convergence wx0 ,T → ±κ takes place locally uniformly in RN , hence the limit (3.21) equals either ∞ or −∞. Similarly, if x0 = 0 ∈ B(u0 ) and if the blow-up is of type I, then the limit is either ∞ or −∞. Finally we consider the case where x0 = 0 and the blow-up is of type II and u0 0. In this case, by Theorems 3.1 and 3.2, w0,T (y, s) = W (|y|, s) converges to Φ ∗ (|y|) as s → ∞ locally uniformly in RN \ {0}. Then, for any small δ > 0, there exists s0 such that 1 W (δ, s) Φ ∗ (δ) 2
for s s0 .
(3.22)
1034
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
Now choose δ > 0 sufficiently small so that Φ ∗ (δ) κ. Then, since W > 0, (3.22) and a simple comparison argument show that min W (r, s) > κ
0rδ
for sufficiently large s,
which implies U (0, t) → ∞ as t → T . The proposition is proven.
(3.23)
2
Remark 3.15. If the blow-up is of type II and if u0 0, then one can easily show that U (·, t)
L∞ (Ω)
= U (0, t) for t ∈ [t1 , T )
(3.24)
for some t1 ∈ [0, T ), or, equivalently, W (·, s)
L∞
= W (0, s)
for all large s.
(3.25)
Proof of (3.25)). Let δ > 0 be as in (3.23) and choose M > 0 such that Φ ∗ (M) < κ. Since W (r, s) converges to Φ ∗ as s → ∞ in C 1 ([δ, M]), and since the graph of Φ ∗ has precisely one intersection with κ in the interval [δ, M], which is transverse, we see that Z[δ,M] W (·, s) − κ = 1 for all large s. Combining this and (3.23), we obtain Z[0,M] W (·, s1 ) − κ = 1
for some s1 .
Since the intersection is transverse, we see that, for some small ε ∗ > 0, Z[0,M] W (·, s1 ) − (κ + ε) = 1 for 0 < ε ε ∗ . Denote by ηε (s) the solution of (2.17) with initial data κ + ε at s = s1 . Then Z[0,M] W (·, s) − ηε (s) 1 for s s1 , 0 < ε ε ∗ .
(3.26)
Note that ηε (s) blows up in finite time. Let s ∗ be the blow-up time of ηε∗ (s). Then (3.26) implies Z[0,M] W (·, s) − C 1
for s s ∗ and any constant C > κ.
This means that W (r, s) is monotone decreasing in r ∈ [0, M] so far as W > κ, hence its maximum in [0, M] is attained at r = 0. This and (2.26), along with the fact that W (·, s)L∞ → ∞ imply (3.25). 2
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
1035
4. Properties of focused blow-up In this section we study focused blow-up. This is a blow-up that occurs at x = 0. Unlike the subcritical case 1 < p < pS , the constants ±κ are no longer the only nonzero bounded solutions of (1.7), but there are also nonconstant solutions in some part of the supercritical range p > pS (see [5]). Therefore even type I blow-up behaviors can be much more complex than in the subcritical case. In what follows we study blow-up behaviors around x = 0, but for generality we will not assume that 0 is the only blow-up point. However, since any blow-up that occurs outside x = 0 always has a constant local blow-up profile (see [28, Theorem 1.13]; also [30]), a truly complex behavior can be observed only around the origin. 4.1. Classification of focused blow-up The following theorem gives classification of focused blow-up in terms of the global profile u(x, T ). It shows that the local structure of blow-up is well reflected in the global profile. Previously most studies of blow-up were concerned mainly with the local profile w ∗ , therefore the role of the global profile was not well understood. Theorem 4.1 (Classification of focused blow-up). Let pS < p < ∞ and let u be a solution of (1.1) that blows up at t = T . Then limx→0 u(x, T )/ϕ ∗ (x) exits and satisfies ⎧ ⎪ ∞ or −∞ u(x, T ) ⎨ finite but = ±1, 0 = lim ⎪ 1 or −1 x→0 ϕ ∗ (x) ⎩ 0
⇔ ⇔ ⇔ ⇔
type I with w ∗ = κ or −κ, type I with nonconstant w ∗ , type II, no blow-up occurs at x = 0.
(4.1)
The following corollary implies that a thin needle-like singularity does not occur at the blowup time. This result is similar to Lemma 2.13 in its spirit, but does not follow from it directly. Corollary 4.2 (Formation of no needle). Let pS < p < ∞ and let u be a solution of (1.1) defined (at least) for 0 t < T . Suppose there exists a constant M > 0 such that lim sup u(x, t) M for every x ∈ Ω \ {0}. t→T
Then u is bounded on Ω × [0, T ). Proof. By the assumption, u(x, T ) is a bounded function for x = 0. Therefore u(x, T )/ϕ ∗ (x) → 0 as x → 0. Thus, by Theorem 4.1, x = 0 is not a blow-up point, hence u remains bounded in a neighborhood of x = 0 as t → T . To complete the proof, we need to show that u does not blow up outside x = 0. This follows from the above assumption and Proposition 3.14. 2 Remark 4.3. Note that the ‘lim sup’ in the above corollary is taken pointwise at each x, not uniformly in Ω \ {0}. The same result as the above corollary can easily be shown for the subcritical case 1 < p < pS since every blow-up is of type I. However, the situation looks different in the critical case p = pS , where the formal analysis of [9] suggests the occurrence of a needle-like singularity for sign-changing solutions.
1036
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
Theorem 4.1 is partly a consequence of the following two propositions. Proposition 4.4. Let pS < p < ∞ and let w ∗ be as in (3.1). Then u(x, T ) w ∗ (y) = lim . |y|→∞ ϕ ∗ (y) x→0 ϕ ∗ (x) lim
(4.2)
Proposition 4.5. Let pS < p < ∞ and let ψ be a radially symmetric solution of (1.7). Then lim|y|→∞ ψ(y)/ϕ ∗ (y) exists and ⎧ ⎪ ∞ or −∞ ψ(y) ⎨ finite but = ±1, 0 = lim ⎪ |y|→∞ ϕ ∗ (y) ⎩ 1 or −1 0
⇔ ⇔ ⇔ ⇔
ψ = ±κ, ψ = ±κ, ±ϕ ∗ , 0, ψ = ±ϕ ∗ , ψ = 0.
(4.3)
The proof of Proposition 4.5 will be given in Appendix A. Proposition 4.4 follows partly from the next lemma. Lemma 4.6. Let pS < p < ∞ and let w ∗ be as in (3.1). Suppose that w ∗ = ±κ. Then for any ε, δ > 0 there exist r0 > 0 and t0 ∈ [0, T ) such that 2 ˆ t) < ε |x| p−1 u(x, t) − u(x, 1 − p−1
where u(x, ˆ t) := (T − t) local profile w ∗ .
√ for δ T − t < |x| r0 , t0 t < T ,
(4.4)
√ w ∗ (x/ T − t ) is the self-similar solution corresponding to the
√ Proof. Using the rescaled variables y = x/ T − t, s = − log(T − t), we can write 2 2 ˆ t) = |y| p−1 w(y, s) − w ∗ (y) . |x| p−1 u(x, t) − u(x, The right-hand side converges to 0 as s → ∞ locally uniformly in y ∈ RN \ {0}. Therefore, for any ε > 0 and any 0 < δ < M, there exist s ∗ > − log T such that ε 2 |y| p−1 w(y, s) − w ∗ (y) < 2
for δ < |y| M, s > s ∗ ,
hence ε 2 ˆ t) < |x| p−1 u(x, t) − u(x, 2
√ √ for δ T − t < |x| M T − t, t ∗ < t < T ,
∗
where t ∗ = T − e−s . By Proposition 2.7 we have 2p ut (x, t) C|x|− p−1
for 0 < |x| r0 , t0 t < T ,
(4.5)
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
1037
for some r0 > 0, t0 ∈ [0, T ). It follows that, for 0 < |x| r0 and t0 t1 < t2 < T , u(x, t2 ) − u(x, t1 )
t2
2p ut (x, τ ) dτ C|x|− p−1 (T − t1 ),
t1
and the same estimate naturally holds for u. ˆ Consequently 2 2 ˆ t) |x| p−1 u(x, t1 ) − u(x, |x| p−1 u(x, t) − u(x, ˆ t1 ) + C|x|−2 (T − t1 )
(4.6)
for 0 < |x| r0 and t0 t1 < t < T . Now choose M > 0 sufficiently large so that CM −2 < ε/2, and, for each r > 0 define the number t1 (r) by r = M T − t1 (r)
or t1 (r) = T − r 2 /M 2 .
Replacing r0 > 0 by a smaller constant if necessary, we may assume that t1 (r0 ) max{t0 , t ∗ }. Hereafter we put t˜0 := t1 (r0 ). Then substituting t1 = t1 (|x|) in (4.6), we obtain 2 2 ˆ t) |x| p−1 u x, t1 |x| − uˆ x, t1 |x| + C|x|−2 T − t1 |x| |x| p−1 u(x, t) − u(x, 2 ε < |x| p−1 u x, t1 |x| − uˆ x, t1 |x| + 2 √ for 0 < |x| r0 , t1 (|x|) < t < T , or, equivalently, for t˜0 < t < T , M T − t < |x| r0 . Combining this and (4.5), we obtain (4.4). The lemma is proven. 2 Proof of Proposition 4.4. By Theorem 3.2, w ∗ is either a bounded solution of (1.7) or ±ϕ ∗ . Let us first consider the case where w ∗ ≡ ±κ. Then by Lemma 4.6, we have √ u(x, t) u(x, ˆ t) u(x, t) w ∗ (x/ T − t) ε = − ∗ − > √ c∗ Φ ∗ (x) Φ (x) Φ ∗ (x) ϕ ∗ (x/ T − t) for any 0 < |x| r0 and t0 < t < T . Letting t → T , we obtain u(x, T ) ε w ∗ (x) ∗ for 0 < |x| r0 = r0 (ε). − lim Φ ∗ (x) |y|→∞ ϕ ∗ (x) c Since ε is arbitrary, we obtain (4.2). Next we consider the case where w ∗ = κ or −κ. Since the latter case can be argued the same way, we assume that w ∗ = κ. In this case, it is shown in [32, Theorem 5], as a generalization of the result of [17] to higher dimensions, that u |x|, T ≈
8p (p − 1)2
1 p−1
1 log |x| p−1
2 − p−1
|x|
as |x| → 0,
1038
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
or − 2m u |x|, T ≈ C|x| p−1
as |x| → 0
for some constant C > 0 and an integer m 2. (Though [32, Theorem 5] is stated under the assumption that u 0, it is clear from its proof that the same estimate holds without this assumption.) This implies lim
|x|→0
u(x, T ) = ∞, ϕ ∗ (x)
which establishes (4.2) for the case w ∗ = κ. The proposition is proven.
2
Proof of Theorem 4.1. By virtue of Propositions 4.4 and 4.5, we have ⎧ ⎪ ∞ or −∞ u(x, T ) ⎨ finite but = ±1, 0 = lim ⎪ x→0 ϕ ∗ (x) ⎩ 1 or −1 0
⇔ ⇔ ⇔ ⇔
w ∗ = ±κ, w ∗ = ±κ, ±ϕ ∗ , 0, w ∗ = ±ϕ ∗ , w ∗ = 0.
By Theorem 3.2, the third condition is equivalent to a type II blow-up. Therefore, the other three cases are either type I blow-up or no blow-up. Consequently, the first two cases are type I blow-up. It remains to show that w ∗ = 0 implies that x = 0 is not a blow-up point. Suppose that x = 0 is a blow-up point. Then, as mentioned above, the blow-up is of type I. Therefore, w ∗ (y, s) converges to 0 in C 1 (RN ). It then follows from Lemma 2.2 that x = 0 is not a blow-up point, a contradiction. The theorem is proven. 2 4.2. Asymptotic self-similarity In this subsection we show that any highly focused type I blow-up is well approximated by a self-similar solution near x = 0. The result follows easily from Lemma 4.6. Proposition 4.7 (Asymptotic self-similarity). Let pS < p < ∞ and let u be a solution of (1.1) that blows up at t = T . Suppose that the blow-up is of type I and that 0 is a blow-up point. Furthermore, assume that the local blow-up profile w ∗ (y) := lims→∞ w0,T (y, s) does not coincide with ±κ. Then for any ε > 0 there exists r0 > 0 and t0 ∈ [0, T ) such that 2 ˆ t) < ε |x| p−1 u(x, t) − u(x, 2p |x| p−1 ut (x, t) − uˆ t (x, t) < ε 1 − p−1
where u(x, ˆ t) := (T − t) local profile w ∗ .
for 0 < |x| r0 , t0 t < T , for 0 < |x| r0 , t0 t < T ,
√ w ∗ (x/ T − t) is the self-similar solution corresponding to the
Proof. We only prove the former estimate since the latter follows easily from the former by standard parabolic estimates and a rescaling argument.
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
1039
Since w converges to w ∗ as s → ∞ locally uniformly in RN , the estimate (4.5) holds for √ 0 < |x| M T − t. The conclusion of the proposition then follows by arguing as in the proof of Lemma 4.6. 2 As an immediate corollary to the above proposition is the following. Corollary 4.8 (Behavior of intersection points). Let pS < p < ∞ and let u be a solution of (1.1) that blows up at t = T . Suppose that the blow-up is of type I and that 0 is a blow-up point. Put m := Z(0,∞) |W ∗ | − Φ ∗ . Then there exists t0 ∈ [0, T ) and r0 > 0 such that Z(0,r0 ) U (·, t) − Φ ∗ = m
for t0 t < T .
Furthermore, the zeros of |U (r, t)| − Φ ∗ (r) in the interval (0, r0 ), say r1 (t), . . . , rm (t), satisfy rj (t) → 0
as t → T for j = 1, 2, . . . , m.
Proof. Since the blow-up is of type I, we have u(x, T ) w ∗ (y) = lim
= ±1. |y|→∞ ϕ ∗ (y) x→0 ϕ ∗ (x) lim
Choose ε small enough so that u(x, T ) 0 < ε < lim ∗ − 1 . x→0 ϕ (x) Using the same argument√as in the proof of Proposition 4.7, we see that U (r, t) − Φ ∗ (r) has no zero in the region Rε T − t < |x| < r0 (ε). On the other hand, since W (y, s) converges to W ∗ (y) in the C 1 sense uniformly in |y| Rε and since W ∗ (r) − Φ ∗ (r) has precisely m simple zeros in this region, we obtain the desired conclusion. 2 The above corollary implies that each intersection point between U (r, t) and Φ ∗ (r) either converges to 0 as t → T or remains away from 0, provided that the blow-up is of type I. 4.3. Energy blow-up The energy E(w) of the rescaled solution w has so far played an important role in analyzing the blow-up behavior of solutions. In this subsection we focus on the energy of the original solution u, which is defined by J (u) := Ω
1 1 |∇u|2 − |u|p+1 dx. 2 p+1
(4.7)
1040
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
This functional is well defined if u ∈ H 1 (Ω) ∩ Lp+1 (Ω). Furthermore, if u is a classical solution of (1.1), a straightforward computation shows d J u(·, t) = − (ut )2 dx 0, dt Ω
therefore J (u(·, t)) is monotone non-increasing in t. It is well known that if J (u(·, t0 )) < 0 for some t0 0, then the solution u blows up in finite time (in the usual L∞ sense). This result holds without the assumption of radial symmetry. The proof differs between the case where Ω is a bounded domain of any shape (cf. [2,25]) and the case where Ω is a (possibly unbounded) convex domain such as BR and RN (cf. [34]). The proof for the former case is rather standard and better known. Indeed, by integrating (1.1) by parts and using Jensen’s inequality, one easily sees that u(·, t)L2 (Ω) tends to ∞ in finite time, which implies blow-up in the L∞ sense. On the other hand, if Ω is unbounded, the L2 blow-up does not automatically imply L∞ blow-up, therefore one needs a different argument. The key idea in [34] is to use the energy for the rescaled solution w. Using the first two equalities in (2.34), one obtains 2 N−2 1 1 p−1 − 2 2 p+1 dx |∇u0 | − |u0 | E w0,T1 (·, − log T1 ) ≈ T1 2 p+1 RN
for u0 ∈ H 1 (Ω) ∩ L∞ (Ω) and T1 sufficiently large. Thus, if J (u0 ) < 0, then E(w0,T1 ) < 0 for sufficiently large T1 , hence, by (2.13), the solution u must blow up before t = T1 . The same argument works if u0 is replaced by u(·, t0 ). In the subcritical case 1 < p < pS , it is known that every blow-up solution satisfies J (u(·, t)) → −∞ as t → T . This, however, is not always the case if p > pS , as shown in the following theorem. Theorem 4.9. Let pS < p < ∞ and u0 ∈ L∞ (Ω). If Ω = RN , assume further that u0 ∈ H 1 (Ω)∩ Lp+1 (Ω). Suppose that u blows up at t = T and that the blow-up is highly focused, that is, B(u0 ) = {0} and w ∗ = ±κ, where w ∗ is as in (2.35). Then lim sup |∇u|2 dx < ∞, lim sup |u|p+1 dx < ∞. t→T
t→T
Ω
Ω
Consequently, lim J u(·, t) > −∞.
t→T
Proof. By Proposition 2.7 there exists a constant C > 0 and t0 ∈ [0, T ) such that 2 u(x, t) C 1 + |x|− p−1 ,
2 ∇u(x, t) C 1 + |x|− p−1 −1
for x ∈ Ω \ {0} and t0 t < T . If Ω isa finite ball, then the above estimates and the fact that p > pS imply that both Ω |∇u|2 dx and Ω |u|p+1 dx remain bounded as t → T , hence J (u(·, t))
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
1041
is bounded. The same conclusion also holds for the case Ω = RN , by virtue of Proposition 2.16 and the above estimates. 2 5. Behavior after blow-up In this section we study continuation beyond the blow-up time. What we mean by continuation is the so-called proper extension (minimal solution), or, more generally, the limit L1 continuation, whose precise definition will be given below. 5.1. Basic concepts We first recall some basic notions for continuation of a solution beyond the blow-up time. Since the relation between different notions of continuation is not well explained in the literature, we will explain them in some details for the convenience of the reader. In this subsection we write the equation in (1.1) as ut = u + f (u)
(x ∈ Ω, t > 0).
(5.1)
Though our main focus is on the case where f (u) = |u|p−1 u, much of the argument in this subsection applies to a large class of nonlinearities; see Remark 5.6. We begin with the following concept of continuation, which is due to [3]. As in [11], we will call it a “proper solution” or a “proper extension” of a given solution. Definition 5.1 (Proper extension). Let u be a solution of (1.1) or (5.1) with u0 0. The “proper extension” of u is defined by u(x, ¯ t) = lim um (x, t) m→∞
for x ∈ Ω, 0 t < ∞,
(5.2)
where each um (x, t) is a classical solution of the approximating equation ut = u + fm (u),
u(x, 0) = u0 (x)
(5.3)
(under the Dirichlet boundary condition if Ω = RN ), and 0 f1 (u) f2 (u) f3 (u) · · · are globally Lipschitz, monotone non-decreasing functions converging to f (u). The monotonicity of the sequence u1 u2 u3 · · · follows from the standard comparison principle, therefore the pointwise limit in (5.2) is well defined. Note that, by the global Lipschitz continuity of each fm (u), the function um (x, t) is globally defined for x ∈ Ω and t 0; hence so is u. ¯ A typical example of fm is:
fm (u) = min f (u), m
for m = 1, 2, 3, . . . .
Since each um is a classical solution of the approximating equation, it satisfies t um (·, t) = e u0 + t
0
e(t−τ ) fm um (·, τ ) dτ
for 0 t < ∞.
(5.4)
1042
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
The integral on the right-hand side can be expressed as t
G(x, y, t − τ )fm um (y, τ ) dy dτ,
0 Ω
where G(x, y, t) is the fundamental solution of the heat equation on Ω. Considering that 0 fm um (x, t) f u(x, ¯ t) as m → ∞ and that G(x, y, t) is positive, we can apply the monotone convergence theorem, to obtain t u(·, ¯ t) = e u0 + t
e(t−τ ) f u(·, ¯ τ ) dτ
for 0 t < ∞,
(5.5)
0
where the integral identity is understood in a generalized sense, allowing u¯ and f (u) ¯ to take values in [0, +∞]. Furthermore, u¯ is the smallest element among all nonnegative functions that satisfy (5.5) in this generalized sense (see [3]). In particular, u¯ is independent of the choice of the sequence {fm } and is thus uniquely determined by the initial data u0 . The minimality of u¯ can be shown as follows. Define a sequence of functions {u¯ k } by the following iteration scheme: ⎧ t ⎪ ⎪ ⎨ t u¯ k+1 (·, t) = e u0 + e(t−τ ) f u¯ k (·, τ ) dτ (k = 0, 1, 2, . . .), (5.6) ⎪ ⎪ 0 ⎩ u¯ 0 (x, t) ≡ 0. Then clearly 0 = u¯ 0 u¯ 1 and u¯ 0 v, where v is any nonnegative solution of (5.5) in the generalized sense. Recalling that f is monotone increasing, we obtain, by induction, 0 = u¯ 0 u¯ 1 u¯ 2 · · · and u¯ k v (k = 1, 2, 3, . . .). Therefore limk→∞ u¯ k (x, t) exists pointwise and satisfies 0 lim u¯ k (x, t) v(x, t). k→∞
In particular, lim u¯ k (x, t) u(x, ¯ t).
k→∞
Next, for each positive integer m, we define the sequence u¯ k,m by ⎧ t ⎪ ⎪ ⎨ t u¯ k+1,m (·, t) = e u0 + e(t−τ ) fm u¯ k,m (·, τ ) dτ (k = 0, 1, 2, . . .), ⎪ 0 ⎪ ⎩ u¯ 0,m (x, t) ≡ 0. Since fm f we have u¯ k,m u¯ k for k = 0, 1, 2, . . . . Letting k → ∞, we obtain um (x, t) = lim u¯ k,m (x, t) lim u¯ k (x, t) k→∞
k→∞
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
1043
for each m. Thus u(x, ¯ t) = limm→∞ um (x, t) limk→∞ u¯ k (x, t). Consequently, u(x, ¯ t) = lim u¯ k (x, t) v(x).
(5.7)
k→∞
This proves the minimality of u. ¯ Clearly u(x, t) = u(x, ¯ t) for 0 t < T (u0 ), where T (u0 ) is as defined in (1.18). See [11] for a more general treatment of the notion of proper extension. Next we set
¯ t1 ) = ∞ for a.e. x ∈ Ω , Tc (u0 ) := sup t1 > 0 u(x,
(5.8)
where u¯ denotes the proper extension of solution u. Clearly we have T (u0 ) Tc (u0 ) ∞. As one can easily see from the identity (5.5), u(x, ¯ t) = ∞ for every x ∈ Ω and every t > Tc (u0 ). In what follows we often abbreviate Tc (u0 ) as Tc . Definition 5.2 (Complete blow-up). Let u be a solution of (1.1) with u0 0 that blows up at t = T < ∞. We say that the blow-up is “complete” if T = Tc and “incomplete” if T < Tc . It is known that if 1 < p < pS , every blow-up is complete; see [3] for the case of a bounded domain and [11] for Ω = RN . However, in the supercritical range pS < p < ∞, there are cases of incomplete blow-up, which is the main focus of this section. The notion of incomplete blow-up can be extended to sign-changing solutions by introducing the concept of limit L1 continuation. The following definition is a slightly modified version of what is found in [8], which deals with only the case where Ω is bounded. Definition 5.3 (Limit L1 solution). By a limit L1 -solution of (5.1) on the interval 0 t < T ∗ we mean a function u(x, ˜ t) that can be approximated by a sequence of classical solutions in the following way: There is a sequence {u˜ 0,n } in C(Ω) such that u˜ 0,n → u0
in C(Ω)
(5.9)
and that the solution u˜ n of (5.1) with u˜ n (·, 0) = u˜ 0,n exists for 0 t < T ∗ and satisfies u˜ n (·, t) → u(·, ˜ t) f (u˜ n ) → f (u) ˜
in L1loc (Ω) for every t ∈ [0, T ∗ ), in L1loc Ω × (0, t) for every t ∈ [0, T ∗ ).
(5.10)
Definition 5.4 (Minimal L1 solution). Suppose u0 0. A limit L1 -solution u˜ of (5.1) is called a minimal L1 -solution if the approximating sequence in (5.9) satisfies 0 u˜ 0,1 u˜ 0,2 u˜ 0,3 · · · → u0 ,
u˜ 0,n ≡ u0
(n = 1, 2, 3, . . .).
If Ω is unbounded, we further require that each u˜ 0,n has compact support.
(5.11)
1044
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
Given a solution u that blows up at t = T , we call u˜ a limit L1 continuation of u if u˜ is a ˜ t) = u(x, t) for limit L1 solution defined on some interval 0 t < T ∗ with T ∗ > T and if u(x, 0 t < T . If, in addition, u˜ is a minimal L1 solution, then we call u˜ a minimal L1 continuation of u. From the definition it is clear that a solution u has a limit L1 continuation if and only if there exists a sequence of initial data u˜ 0,k such that limk→∞ T (u˜ 0,k ) > T (u0 ). In other words, a limit L1 continuation exists if and only if u0 is a discontinuous point of the mapping u0 → T (u0 ). As far as Eq. (1.1) in the subcritical range 1 < p < pS is concerned, this mapping u0 → T (u0 ) is always continuous as shown in [3] for positive solutions, and in [38] for more general signchanging solutions. If u˜ 0,k (k = 1, 2, 3, . . .) is a sequence satisfying (5.11), then the monotone convergence theorem automatically guarantees the convergence (5.10), therefore the limit function u˜ is a minimal L1 solution. Furthermore, since each u˜ n is a classical solution, it holds that t u˜ n (·, t) = e u˜ 0,n + t
e(t−τ ) f u˜ n (·, τ ) dτ
for 0 t < T ∗ .
(5.12)
0
Letting n → ∞ and using again the monotone convergence theorem, we see that any minimal L1 solution u˜ satisfies t u(·, ˜ t) = e u0 + t
e(t−τ ) f u(·, ˜ τ ) dτ
for 0 t < T ∗ .
(5.13)
0
The definition and properties of minimal extension and L1 continuation we have stated so far apply to a large class of nonlinearities f that are monotone increasing and Lipschitz continuous. In the special case where u is radial and f (u) = |u|p−1 u with pS < p < ∞, the identity (5.13) holds for any limit L1 solutions as we will see in Proposition 5.10 below. The following proposition asserts that the minimal L1 continuation u˜ coincides with the proper extension u. ¯ In particular, u˜ does not depend on the choice of the approximating sequence u˜ n . Thus the two concepts of continuation are equivalent as far as positive solutions are concerned. Note that no radial symmetry is assumed here. Proposition 5.5. Let Ω be either a bounded smooth convex domain or the entire space RN , and let f be a function satisfying (B.2) and (B.3) or (B.4) in Appendix B. Let u0 be a bounded nonnegative function. Suppose that the solution u of (5.1) blows up at t = T . Let u˜ n (n = 1, 2, 3, . . .) be a sequence of classical solutions of (5.1) satisfying 0 u˜ n (x, 0) u0 (x),
u˜ n (x, 0) ≡ u0 (x)
u˜ n (x, 0) u0 (x)
for n = 1, 2, 3, . . . ,
as n → ∞, for x ∈ Ω.
If Ω = RN , assume also that the support of each u˜ n (·, 0) is compact. Then T (u˜ n (·, 0)) > Tc (u0 ) and lim u˜ n (x, t) = u(x, ¯ t)
n→∞
for x ∈ Ω, t ∈ [0, T ∗ ),
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
1045
where u¯ denotes the proper extension of u and T ∗ := limn→∞ T (u˜ n (·, 0)) Tc (u0 ). In the special case where f (u) = |u|p−1 u and 1 < p < ∞, it holds that T ∗ = Tc (u0 ). Remark 5.6. Examples of f (u) satisfying the conditions (B.2), (B.3) include up and λeu and many other nonlinearities. See the remark after Lemma B.1 in Appendix B. We will prove this proposition by using the following general result of [22]. Lemma 5.7. (See [22, Theorem 2].) Let Ω and f be as in Proposition 5.5 and let u1 , u2 be two (possibly non-radial) solutions of (5.1). Assume 0 u1 (x, 0) u2 (x, 0), u1 (x, 0) ≡ u2 (x, 0). If Ω = RN , assume also that sup
0
u1 (x, t) < ∞
(5.14)
for some R > 0. Then either T (u1 ) > Tc (u2 ) or T (u1 ) = T (u2 ) = ∞. Remark 5.8. In Theorem 2 of [22], (5.14) is not treated as an assumption but as a ‘fact’ proven in [10]. However, if Ω is unbounded, (5.14) does not hold in general without assuming some conditions on the initial data of u1 . For this reason we have slightly modified the statement of the result of [22] by putting (5.14) as an assumption. Proof of Proposition 5.5. The equality (5.13) and the minimality of u¯ imply u(x, ˜ t) u(x, ¯ t)
for x ∈ Ω, 0 t < T ∗ .
(5.15)
Therefore it suffices to prove the opposite inequality. We first consider the case where Ω is a bounded convex domain. By Lemma 5.7, we have T (u˜ n (·, 0)) > Tc (u0 ) or T (u˜ n (·, 0)) = Tc (u0 ) = ∞. The monotonicity of the sequence {u˜ n } implies T (u˜ 1 ) T (u˜ 2 ) T (u˜ 3 ) · · ·, hence T u˜ n (·, 0) → ∃ T ∗ Tc (u0 )
as n → ∞.
Consequently, given any T∗ ∈ (0, T ∗ ), each u˜ n is bounded on the interval 0 t T∗ . Therefore, for each n ∈ N, there exists mn ∈ N such that f (u˜ n ) mn in Ω × [0, T∗ ]. This implies that u˜ n is a solution of the equation ut = u + fm (u) for m mn , where fm (u) is as in (5.4). This fact and the inequality u˜ n (x, 0) u0 (x) = um (x, 0) yield u˜ n (x, t) um (x, t)
for x ∈ Ω, 0 t T∗
for all m mn , where um is as in Definition 5.1. Letting m → ∞, and considering that T∗ ∈ (0, T ∗ ) can be chosen arbitrarily close to T ∗ , we obtain ¯ t) u˜ n (x, t) u(x,
for x ∈ Ω, 0 t < T ∗ .
Since u˜ n is increasing in n, the following pointwise limit exits: u(x, ˜ t) := lim u˜ n (x, t). n→∞
(5.16)
1046
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
Letting n → ∞ in (5.16), we get u(x, ˜ t) u(x, ¯ t)
for x ∈ Ω, 0 t < T ∗ .
(5.17)
This and (5.15) yield u(x, ˜ t) = u(x, ¯ t) for 0 t < T ∗ . Next we consider the case where Ω = RN . The proof is the same as above, except that, in order to apply [22, Theorem 2] to deduce T (u˜ n (·, 0)) > Tc (u0 ), we have to verify the condition (5.14). This follows from Lemma B.1 in Appendix B. Finally, we consider the case where f (u) = |u|p−1 u with 1 < p < ∞. Let wn (y, s) and w(y, ˜ s) be the rescaled solutions of (1.13) as defined in (2.1) with a = 0, T1 = T ∗ and u replaced by un and u, ˜ respectively. Then by (2.14) we have wn2 (y, s)ρ(y) dy M (n = 1, 2, 3, . . .) RN
for some constant M > 0 and for all large s. Using Fatou’s lemma, we obtain w˜ 2 (y, s)ρ(y) dy M for all large s. RN
Consequently u(x, ˜ t) ≡ ∞ for any t ∈ [0, T ∗ ) sufficiently close to T ∗ , which implies T ∗ Tc (u0 ); hence Tc (u0 ) = T ∗ . The proof of the proposition is complete. 2 The following simple proposition is also worth noting. Proposition 5.9. The comparison principle holds for proper extensions of solutions of (1.1) or (5.1). More precisely, if u¯ 1 (x, t0 ) u¯ 2 (x, t0 ) (a.e. x ∈ Ω) for some t0 0, then u¯ 1 (x, t) u¯ 2 (x, t) (a.e. x ∈ Ω) for all t t0 . Proof. Let u1,0 and u2,0 be the initial data of the proper solutions u¯ 1 , u¯ 2 , respectively. Then by (5.6) we have u¯ i = limk→∞ u¯ i,k (i = 1, 2), where u¯ i,k (k = 0, 1, 2, . . .) is a monotone increasing sequence defined by ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
t u¯ i,k+1 (·, t) = e ui,0 + t
e(t−τ ) f u¯ i,k (·, τ ) dτ
(k = 0, 1, 2, . . .),
0
u¯ i,0 (x, t) ≡ 0.
From this one can easily deduce that ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
t u¯ i,k+1 (·, t0 + t) = e u¯ i,k+1 (·, t0 ) + t
u¯ i,0 (x, t) ≡ 0
for t 0. Setting i = 2 and letting k → ∞ yield
0
e(t−τ ) f u¯ i,k (·, t0 + τ ) dτ,
(5.18)
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
t u¯ 2 (·, t0 + t) = e u¯ 2 (·, t0 ) + t
e(t−τ ) f u¯ 2 (·, t0 + τ ) dτ
1047
(5.19)
0
for t 0. Comparing (5.19) and (5.18) with i = 1 and recalling that u¯ 1,k (x, t0 ) u¯ 1 (x, t0 ) u¯ 2 (x, t0 ), we obtain the following inequalities by induction: u¯ 1,k (x, t0 + t) u¯ 2 (x, t0 + t)
for x ∈ Ω, t 0, k = 0, 1, 2, 3, . . . .
The conclusion of the proposition now follows by letting k → ∞.
2
In what follows we focus on the case where f (u) = |u|p−1 u and u is radial. Proposition 5.10. Let pS < p < ∞. Then any radially symmetric limit L1 solution u˜ of (1.1) satisfies (5.13). Proof. Let u˜ n (n = 1, 2, 3, . . .) be the approximating sequence for u. ˜ By Proposition 2.5 and Corollary 2.6, there exists a constant C > 0 such that j j 2 1 ∇ u˜ n (x, t) C |x|− p−1 −j + (T ∗ − t)− p−1 − 2 for x ∈ Ω \ {0}, t ∈ [δ, T ∗ ), (5.20)
for j = 0, 1, 2, 3 and n = 1, 2, 3, . . . . In particular, the limit function u˜ satisfies − 2 − 1 u(x, ˜ t) C |x| p−1 + (T ∗ − t) p−1
for x ∈ Ω \ {0}, t ∈ [δ, T ∗ ).
(5.21)
Since each u˜ n is a classical solution, it satisfies (5.12). Letting n → ∞ and using the estimate (5.20) with j = 0, we obtain (5.13). The proposition is proven. 2 The identity (5.13) and the estimate (5.21) show that any limit L1 solution u˜ satisfies u˜ ∈ 1 (Ω) ∩ Lp+1 (Ω)); see [8, Proposition 2.15]. C((0, T ∗ ); Hu-loc loc Another immediate consequence of the estimate (5.20) is that the approximating sequence u˜ n of any limit L1 solution u˜ satisfies ˜ t) in C 2 Ω \ {0} for every t ∈ (0, T ∗ ). (5.22) u˜ n (·, t) → u(·, It is also clear from the standard parabolic estimates that u˜ n (·, t) → u(·, ˜ t)
in C 2 (Ω) for every t ∈ (0, T ).
The following corollary extends Proposition 2.11 of [8] by allowing sign-changing solutions. Corollary 5.11. (See [8].) Let u and u¯ be as in Proposition 5.5. Then for any δ ∈ (0, T ) and any T1 ∈ [T , Tc ) there exists a constant C > 0 such that − 2 u(x, ¯ t) C 1 + |x| p−1 for x ∈ Ω \ {0}, t ∈ [δ, T1 ]. p+1
Furthermore, u¯ ∈ C((0, Tc ); Hu1-loc (Ω) ∩ Lloc (Ω) ∩ C 2 (Ω \ {0}).
1048
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
Proof. Since the proper extension u¯ coincides with the minimal L1 continuation u˜ in the range ¯ t) 0 t < Tc , we obtain the desired estimate from (5.21). The continuous dependence t → u(·, follows from the integral expression (5.13). 2 p+1
1 (Ω) ∩ L Note that (1.1) is not well posed in the space Hu-loc loc (Ω), and the standard bootstrap argument does not yield much better regularity. To study further regularity of a solution after its blow-up time, we need more in-depth analysis, which is the main focus of Section 5.3.
5.2. Decay estimates of limit L1 solutions In this subsection we present various useful estimates for limit L1 solutions. Among other things we show that any time-global limit L1 solution decays as t → ∞. We start with a basic lemma which is for the most part a restatement of (5.21). Lemma 5.12. (See [28, Corollary 3.3 and Remark 3.4].) Let pS < p < ∞ and let u˜ be a limit L1 continuation of u defined on some time interval 0 t < T ∗ . Then for any T1 ∈ (0, T ∗ ) there exists a constant CT1 such that − 2 u(x, ˜ t) CT1 1 + |x| p−1 for x ∈ Ω \ {0}, t ∈ [T1 /2, T1 ].
(5.23)
Furthermore, if u0 ∈ H 1 (Ω) ∩ L∞ (Ω), then 1 − 1 ( N−2 − 2 ) . CT1 = O T1 p+1 2 p−1 ∇u0 Lp+1 2
Proof. The estimates follow immediately by applying Proposition 2.5 to the approximating sequence u˜ n and letting n → ∞. 2 The following proposition and theorem follow immediately from Lemma 5.12. Proposition 5.13. Let pS < p < ∞. Suppose either that the local blow-up profile is ±κ, or that the blow-up occurs at some point x = 0. Then the blow-up is complete. Proof. If the blow-up is incomplete, then u has a limit L1 continuation beyond the blow-up time t = T . Then, choosing T1 larger than T (but smaller than 2T ) and putting t = T in (5.23), we see that the blow-up can occur only at x = 0. It also follows from (5.23) that ∗ 2 w (y) CT |y|− p−1 , 1 which implies w ∗ = ±κ. The proof is complete.
2
Theorem 5.14. Let pS < p < ∞ and let u˜ be a limit L1 solution defined for all t 0 and assume that u(·, ˜ 0) ∈ H 1 (Ω) ∩ L∞ (Ω). Then there exists t0 0 such that u˜ is smooth on Ω × [t0 , ∞) and u(·, ˜ t)
L∞
→ 0 as t → ∞.
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
1049
Proof. By Lemma 5.12, u˜ satisfies (5.23) for any choice of T1 > 0. Since p > pS , we have CT1 → 0 as T1 → ∞. This yields an estimate of the form − 2 u(x, ˜ t) C(t) 1 + |x| p−1
for x ∈ Ω \ {0},
where C(t) → 0 as t → ∞. This and Lemma 2.9 prove the theorem.
(5.24)
2
Using the above theorem, we can prove the following threshold result, which has been mostly known if Ω = BR but only partly known if Ω = RN (see Remark 5.17 below). Our proof is different from any previously known ones. Theorem 5.15 (Threshold behavior). Let pS < p < ∞ and let v ∈ H 1 (Ω) ∩ L∞ (Ω) satisfy v 0, v ≡ 0. Then there exists λ∗ > 0 such that the solution uλ of (1.1) with initial data uλ (·, 0) = λv satisfies the following: (i) if 0 λ < λ∗ , then uλ is globally smooth and uλ (·, t)L∞ (Ω) → 0 as t → ∞; (ii) if λ > λ∗ , then uλ (x, t) blows up in finite time; (iii) if λ = λ∗ , then uλ (x, t) blows up in finite time, but its minimal L1 continuation exists for 0 t < ∞, eventually becomes smooth and uλ (·, t)L∞ (Ω) → 0 as t → ∞. Proof. If λ is sufficiently small, then the coefficient C(t) in (5.24) is very small for all t > 0. Indeed, the smallness of C(t) for small t follows from a simple comparison argument, while the smallness for large and middle-range t follows from the last estimate in Lemma 5.12. Consequently, by Lemma 2.9, uλ is smooth for all t > 0. The convergence uλ → 0 follows from Theorem 5.14. On the other hand, if λ is sufficiently large, uλ blows up in finite time. To see this, consider the energy J introduced in 4.3. Then, for λ large, J (λv) := Ω
λ2 λp+1 p+1 2 dx < 0, |∇v| − v 2 p+1
since p + 1 > 2. As seen in Section 4.3, the negativity of J implies a finite time blow-up. Thus we have
λ∗ := sup λ > 0 T (λv) = ∞ < ∞, where T (·) is as defined in (1.18). From the definition of λ∗ and the comparison principle, it is clear that T (λv) = ∞
for 0 λ < λ∗ ,
T (λv) < ∞
for λ > λ∗ ,
hence, by Theorem 5.14, λ u (·, t) ∞ → 0 as t → ∞ for 0 λ < λ∗ . L (Ω) Next we show that T (λ∗ v) < ∞.
(5.25)
1050
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
If Ω is a bounded domain, then a standard approach would be to use the stability of the stationary solution u = 0 and the nonexistence of a positive stationary solution (by Pohozaev) to derive supt>0 u∗ (·, t)L∞ = ∞ (see [37]), then to show that this implies blow-up in finite time. However, since this stability argument does not work if Ω = RN , we use a more direct argument. Suppose that T (λ∗ v) = ∞. Choose a sequence λ1 > λ2 > λ3 > · · · → λ∗ . Then the wellposedness of (1.1) in the space L∞ (Ω) ∩ C(Ω) implies Tn := T (λn v) → ∞
as n → ∞.
Furthermore Tn < ∞ for n = 1, 2, 3, . . . , since λn > λ∗ . By Proposition 2.5, λn 2 w (y, s) CT 1 + |y|− p−1 for |y| > 0, s − log Tn + δ. n 0,Tn Here we have CTn → 0 as n → ∞ by (2.29) and the uniform boundedness of ∇(λn v)L2 . This and Lemma 2.10 yield, for all sufficiently large n, λn w (·, s) ∞ N κ 0,Tn L (R ) 2 −
for s − log Tn + 2δ,
1
which implies uλn (·, t)L∞ κ2 (T − t) p−1 . Hence, by Lemma 2.3, uλn cannot blow-up at t = Tn . This contradiction establishes (5.25). Define u∗ (x, t) := lim∗ uλ (x, t). λλ
By the monotone dependence of uλ on λ, the above limit exists, and u∗ is a limit L1 continuation ∗ ∗ of uλ and is defined for all t 0. Consequently, the minimal L1 continuation of uλ exists for all t 0, and, by Theorem 5.14, it becomes smooth in finite time and decays to 0 as t → ∞. This completes the proof of the theorem. 2 Remark 5.16. Among the remarkable implications of Theorem 5.15 are the following: ∗
(1) uλ can blow up only at x = 0 even if all the mass of v(x) is accumulated far away from the origin. (To see this, simply apply Proposition 5.13.) (2) If Ω = RN , there is a family of positive stationary solutions (1.21), but none of them can be reached by solutions with initial data in H 1 ∩ L∞ . Remark 5.17. If Ω = BR , part of the assertion (iii) of Theorem 5.15 can be proven more simply. Indeed by using the fact that no positive stationary solution exists (by Pohozaev) and that u = 0 ∗ is stable, [37] showed that supx∈Ω,t>0 uλ (x, t) = ∞, thereby constructing the first example of ∗ an unbounded global weak solution of (1.1). For a long time it was not known whether uλ ∗ blows up in finite time or remains smooth for all t 0 and uλ L∞ → ∞ as t → ∞. Later [11] 6 confirmed that a finite time blow-up occurs if pS < p < 1 + N −10 . The upper restriction on p was removed in [35] for radially decreasing solutions. The results were extended to non-radial ∗ solutions by [7], which shows the blow-up and eventual regularity of uλ in a bounded convex ∗ domain. If Ω = RN , a different argument is needed to show that uλ blows up in finite time. The
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
1051
advantage of our proof of Theorem 5.15 is that the argument is much more direct and can handle both the case Ω = BR and the case RN simultaneously. Remark 5.18. In the forthcoming paper [29], we study further properties of uλ . Among other things we show that, for any λ > λ∗ except for at most finitely many exceptional values, the blow-up is complete and is a “single-intersection blow-up,” hence the blow-up is of type I and w ∗ = ±κ (see Theorems 3.11 and 5.28). 5.3. Immediate regularization after blow-up In [8], it is proven for Ω = BR that any positive minimal L1 solution of (1.1) that can be continued beyond the blow-up time becomes classical immediately after blow-up, provided pS < p < pJL . The following theorem extends the above result to all p > pS and all—possibly nonminimal and sign-changing—L1 solutions under the assumption that the blow-up is of type I. (Incidentally, this last assumption is automatically fulfilled if pS < p < pJL .) Furthermore, our proof reveals how fast the regularization occurs. Theorem 5.19 (Immediate regularization for type I). Let pS < p < ∞ and let u be a solution of (1.1) that blows up at t = T . Assume that the blow-up is of type I and that there exists a limit L1 continuation u˜ of u on some interval 0 t < T ∗ with T ∗ > T . Then u˜ is smooth in some interval T < t < T + δ and satisfies 1 ˜ t)L∞ < ∞. lim sup(t − T ) p−1 u(·,
(5.26)
tT
The estimate (5.26) shows that the speed of regularization is roughly the same as that of forward self-similar solutions. One may call this a “type I regularization.” Thus Theorem 5.19 states that an incomplete type I blow-up always leads to type I regularization. The following corollary is an immediate consequence of the above theorem and the fact that the proper extension coincides with the minimal L1 continuation (Proposition 5.5). Corollary 5.20. Let pS < p < ∞ and let u be a nonnegative solution of (1.1) that blows up at t = T . Assume that the blow-up is of type I and incomplete. Then the proper extension u¯ of u is smooth in some interval T < t < T + δ and satisfies (5.26). For a type II blow-up, we have the following result, which is somewhat weaker. We assume pJL < p < ∞ since otherwise type II blow-up does not occur by virtue of Theorems 3.7 and 3.8. Theorem 5.21 (Immediate regularization for type II). Let pJL < p < ∞ and let u be a nonnegative solution of (1.1) that blows up at t = T . Assume that the blow-up is of type II. If Ω = R N , assume also that u(x, T ) ≡ ϕ ∗ (x). Then either of the following holds: (a) the blow-up is complete; (b) the minimal continuation of u is smooth in some interval T < t < T + δ. The same is true for any limit L1 continuation of u. As for the speed of regularization, we have the following result.
1052
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
Theorem 5.22 (Speed of regularization for type II). Let pJL < p < ∞ and let u 0 be a solution of (1.1) that blows up at t = T . Assume that the blow-up is of type II and incomplete. Denote by u˜ the minimal L1 continuation of u on T t < Tc (u0 ). Then 1 ˜ t)L∞ = ∞. lim (t − T ) p−1 u(·,
tT
(5.27)
The proof of Theorem 5.19 is much more involved than that of Theorems 5.21 and 5.22. For the proof of the former, we need the following lemmas. Lemma 5.23. The no-needle lemma (Lemma 2.13) remains true for limit L1 solutions. More precisely, let pS < p < ∞ and let un (x, t) (n = 1, 2, 3, . . .) be a family of limit L1 solutions of (1.1) that are defined for x ∈ Ω \ {0}, t0 t < T1 and given as a limit of classical solutions un,k → un (k → ∞) satisfying supn,k un,k (·, t0 )L∞ < ∞. Suppose that un (x, t ∗ ) → v(x)
(n → ∞) for every x ∈ Ω \ {0},
for some t ∗ ∈ (t0 , T1 ) and some bounded function v on Ω \ {0} with bounded gradient ∇v. Then there exists δ0 > 0 such that un is a classical solution in the region Ω × (t ∗ , t ∗ + δ0 ) and that for any δ ∈ (0, δ0 ) there exists a constant M1 > 0 such that sup∇ j un (·, t ∗ + δ)L∞ (Ω) M1 for j = 0, 1, 2, 3. (5.28) n
Similarly, the conclusion of Lemma 2.14 remains true if wn (·, s) (n = 1, 2, 3, . . .) are limit L1 solutions of (1.13). Proof. Since un,k (x, t) converges to un (x, t) locally uniformly in x ∈ Ω \ {0}, t0 t < T1 , we can find a sequence kn → ∞ (n → ∞) such that un,kn (x, t) converges to v(x) as n → ∞ for every x ∈ Ω \ {0}, t0 t < T1 , so long as kn kn (n = 1, 2, 3, . . .). Applying Lemma 2.13, we see that there exists δ0 > 0 such that, for any δ ∈ (0, δ0 ), there exists M1 > 0 satisfying sup∇ j un,kn (·, t ∗ + δ)L∞ (Ω) M1 for j = 0, 1, 2, 3. n
Letting kn → ∞, we obtain sup∇ j un (·, t ∗ + δ)L∞ (Ω\{0}) M1 n
for j = 0, 1, 2, 3.
It follows, in particular, that un (x, t) remains bounded in the region x ∈ Ω \ {0}, t ∈ (t ∗ + ε, t ∗ + δ0 ) for any sufficiently small ε > 0. Since un satisfies Eq. (1.1) for x ∈ Ω \ {0}, its boundedness implies that x = 0 is a removable singularity. Hence un is a classical solution of (1.1) in the region x ∈ Ω, t ∈ (t ∗ , δ0 ), thus the above estimate in Ω \ {0} implies (5.28). The last part of the lemma can be shown by simply repeating the proof of Lemma 2.14. The lemma is proven. 2 Lemma 5.24. Let pS < p < ∞ and let u be a solution of (1.1) that blows up at t = T . Assume that the blow-up is of type I and that u has a limit L1 continuation u˜ on some interval 0 t < T ∗ with T ∗ > T . Denote by uk (k = 1, 2, 3, . . .) the sequence of classical solutions of (1.1) that
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
1053
defines u. ˜ Then there exist δ > 0 and integers m0 , k0 such that, for each k k0 , (uk )t (0, t) changes sign at most m0 times in the interval T < t < T + δ. Proof. By (5.22) we have lim uk (x, t) = u(x, ˜ t)
k→∞
in C 2 Ω \ {0} for every t ∈ (0, T ∗ )
(5.29)
and lim uk (x, t) = u(x, t)
k→∞
in C 2 (Ω) for every t ∈ (0, T ).
(5.30)
(r, t), respectively, where r = |x|. Now In what follows we write u, uk , u˜ as U (r, t), Uk (r, t), U choose r0 > 0 such that Ut (r0 , T ) = 0. Such an r0 exists, since otherwise Ut (r, T ) = 0 for every r > 0, which implies U (r, T ) is an unbounded stationary solution of (1.4), hence U (r, T ) ≡ ±Φ ∗ (r), but this contradicts Theorem 4.1 since the blow-up is of type I. Thus by the smoothness outside r = 0 and the convergence (5.29), we can find a small constant 0 < δ < T and an of U integer k0 > 0 such that Uk (r0 , t) = 0 for t ∈ [T − δ, T + δ], k k0 . By Lemma 3.6 we have Z[0,r0 ] [Ut (·, T − δ)] < ∞. Choosing a larger k0 if necessary, we see from (5.30) that Z[0,r0 ] (Uk )t (·, T − δ) = Z[0,r0 ] Ut (·, T − δ) =: m0
for k k0 .
Again by Lemma 3.6, Z[0,r0 ] [(Uk )t (·, t)] is non-increasing in t ∈ [T − δ, T + δ], and it drops strictly each time (Uk )t (0, t) changes sign. Consequently, for each k k0 , (Uk )t (0, t) can change sign at most m0 times in the interval [T − δ, T + δ]. The proof of the lemma is complete. 2 Lemma 5.25. Let u˜ be a limit L1 solution defined on some interval 0 < t < T ∗ and let w˜ be the corresponding rescaled solution as defined in (2.1) with a = 0, T1 ∈ (0, T ∗ ] and with u replaced by u. ˜ Then the energy E(w(·, ˜ s)) is well defined and non-increasing in s. Proof. Let uk (k = 1, 2, 3, . . .) be the approximating sequence of classical solutions converging to u˜ for x ∈ Ω \ {0}, t ∈ [0, T1 ), and let wk be the corresponding rescaled solutions as defined in (2.1) with a = 0. Here, we understand (as before) that each uk is defined for all x ∈ RN by setting uk = 0 outside Ω. Then, by applying the estimates (2.30) to wk , we see that the sequence p+1 p+1 {wk } is relatively compact in the space Hρ1 (RN ) ∩ Lρ (RN ), where Hρ1 and Lρ denote the weighted H 1 and Lp+1 spaces with ρ given in (2.5). Consequently E(w) ˜ is well defined and lim E wk (·, s) = E w(·, ˜ s) .
k→∞
Therefore E(w(·, ˜ s)) is monotone non-increasing in s. The lemma is proven.
2
Lemma 5.26. (See [8].) Let u˜ and w˜ be as in Lemma 5.25. Then any ω limit point of wˆ in the N ∗ topology of L∞ loc (R \ {0}) is either a bounded solution of (1.7) or ±ϕ .
1054
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
Proof. Applying (2.6), (2.7) and (2.13) to wk , one finds that ∞
∂wk ∂s
2 ρ(y) dy ds M
s0 RN
for some s0 ∈ R and some constant M that is independent of k. Letting k → ∞ and using Fatou’s lemma, one obtains ∞
∂ w˜ ∂s
2 ρ(y) dy ds < ∞.
s0 RN
The conclusion of the lemma now follows easily from this, the estimates (2.30) and (3.4). For more details, see also [8, Lemma 4.5 and Corollary 4.6]. 2 Proof of Theorem 5.19. All we have to show is the estimate (5.26). Since the blow-up is of type I, the local blow-up profile w ∗ (y) := lims→∞ w0,T (y, s) is a bounded solution of (1.7). By Propositions 5.13, we have w ∗ = ±κ; hence, by Theorem 4.1, we see that μ := lim w ∗ (y)/ϕ ∗ (y) = ±1, 0, ±∞. |y|→∞
Now suppose that (5.26) does not hold. Then there exists a sequence t1 > t2 > t3 > · · · → T such that 1 ˜ tn )L∞ → ∞ (tn − T ) p−1 u(·,
as n → ∞.
Define 1 uˆ n (x, t) := λnp−1 u˜ λn x, T + λn t ,
λ n = tn − T .
Since u˜ is a limit L1 solution, uˆ n is also a limit L1 solution, and it satisfies uˆ n (·, 1)
L∞
→∞
as n → ∞.
(5.31)
By (5.23), there exist constants C > 0, δ1 ∈ (0, T ) and T1 ∈ (T , T ∗ ) such that 2 uˆ n (x, t) C εn + |x|− p−1
T − δ1 T 1 − T , for x = 0, t ∈ − , λn λn
(5.32)
1
where εn = O(λnp−1 ). Thus, by parabolic estimates, {uˆ n } has a subsequence converging to some function uˆ locally uniformly in (x, t) ∈ (RN \ {0}) × R. This function uˆ is again a limit L1 solution of (1.1) and is defined for all x ∈ RN \ {0}, t ∈ R, since λn → 0. Clearly − 2 u(x, ˆ t) C|x| p−1
for (x, t) ∈ RN \ {0} × R.
(5.33)
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
1055
Furthermore, for each t < 0, we have 1 − p−1
uˆ n (x, t) = (−t)
w0,T
1 x x − p−1 ∗ w √ √ , − log(−t) − log λn → (−t) −t −t
as n → ∞. Considering this and that ε 1 − p−1
u(x, ˆ t) = (−t)
1 − p−1
√ w ∗ (y/ ε ) → μϕ ∗ (y) as ε → 0, we see that
x for t < 0, w∗ √ −t
u(x, ˆ 0) = μϕ ∗ (x).
(5.34)
Next we show that ˆ t)L∞ (RN \{0}) = ∞. lim supu(·,
(5.35)
t1
Suppose the contrary. Then there exists δ > 0 such that u(·, ˆ t)L∞ (RN ) remains bounded for N t ∈ [δ, 1). Since uˆ satisfies Eq. (1.1) for x ∈ R \ {0}, the boundedness of uˆ implies that x = 0 is a removable singularity, therefore uˆ can be extended to a smooth solution of (1.1) on RN × (1 − δ, 1). In particular, ∇ u(·, ˆ t)L∞ (RN ) is bounded in t ∈ [1 − δ1 , 1) for any δ1 ∈ (0, δ). This and Lemma 5.23 yield supn uˆ n (·, 1)L∞ < ∞, contradicting (5.31). (Here, note that the assumption of Lemma 5.23 is easily seen to hold by choosing t0 = −1.) This establishes (5.35). Next we show that ˆ t)L∞ (RN \{0}) = ∞ sup u(·,
for any 0 a < b 1.
(5.36)
a
Suppose that (5.36) does not hold. Then there exists an open interval I ⊂ (0, 1) such that ˆ t)L∞ (RN \{0}) < ∞. supu(·, t∈I
As before, uˆ can be extended to a smooth solution of Eq. (1.1) on RN × I . Denote by (τ− , τ+ ) the maximal subinterval of (0, ∞) containing I such that uˆ is a smooth solution on RN × (τ− , τ+ ). Then (5.35) and the well-posedness of (1.1) imply τ+ 1 and that ˆ t)L∞ (RN \{0}) = ∞. lim u(·,
tτ+
Consequently we can choose τ∗ , τ ∗ with τ− < τ∗ < τ ∗ < τ+ such that u(0, ˆ τ∗ ) < u(0, ˆ τ ∗ ). Now recall that uˆ n converges to uˆ as n → ∞ for every x ∈ RN \ {0}, t ∈ R. Considering that uˆ is a smooth solution for τ− < t < τ+ and using Lemma 5.23, we see that uˆ n (·, t) → u(·, ˆ t) in C 2 (RN ) as n → ∞ for every t ∈ (τ− , τ+ ). In particular, lim uˆ n (0, τ∗ ) = u(0, ˆ τ∗ ),
n→∞
lim uˆ n (0, τ ∗ ) = u(0, ˆ τ ∗ ),
n→∞
1056
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
or, equivalently, 1
˜ T + λn τ∗ ) = u(0, ˆ τ∗ ), lim λnp−1 u(0,
n→∞
1
lim λnp−1 u(0, ˜ T + λn τ ∗ ) = u(0, ˆ τ ∗ ),
n→∞
where λn = tn − T . Consequently, for all large n, 1
1
1
p−1 u(0, ˜ T + λn+1 τ ∗ ) > λnp−1 u(0, ˜ T + λn τ∗ ) < λnp−1 u(0, ˜ T + λn τ ∗ ). λn+1
Setting τ2m−1 = λm τ ∗ , τ2m = λm τ∗ (m = 1, 2, 3, . . .), and recalling that λ1 > λ2 > λ3 > · · · → 0, we see that τ1 > τ2 > τ3 > · · · → 0 and that ˜ T + τ2m ) < u(0, ˜ T + τ2m−1 ) u(0, ˜ T + τ2m+1 ) > u(0, for all large m. This means that, for any δ > 0, u(0, ˜ t) oscillates infinitely many times in the interval (T , T + δ). Thus, if we denote by uk (k = 1, 2, 3, . . .) the sequence of classical solutions of (1.1) that defines u, ˜ then (uk )t (0, t) changes sign arbitrarily many times in (T , T + δ) as k becomes larger, contradicting Lemma 5.24. This proves (5.36). ˆ s). Then (5.34) and Now we rescale uˆ as in (2.1) with a = 0, T1 = 1, and denote it by w(y, (5.35) imply w(y, ˆ 0) = μϕ ∗ (y),
ˆ s)L∞ (RN \{0}) = ∞ sup w(·,
a <s
(5.37)
for any 0 a < b < ∞.
(5.38)
N By Lemma 5.26, any ω limit point of wˆ in the topology of L∞ loc (R \ {0}) is either a bounded ∗ solution of (1.7) or ±ϕ . On the other hand, (5.38) and the last part of Lemma 5.23 show that the ω limit set of wˆ does not contain a bounded solution of (1.7). Hence
ˆ s) = ϕ ∗ (y) or −ϕ ∗ (y). lim w(y,
(5.39)
s→∞
p+1
Note that this convergence takes place in the topology of Hρ1 ∩ Lρ (see the proof of Lemma 5.25). Consequently, by (5.39), (5.37) and Lemma 5.25, we have E(μϕ ∗ ) E(ϕ ∗ ), but this is impossible since the quantity ∗
E(μϕ ) =
|μ|2 |μ|p+1 − 2 p+1
ρ(ϕ ∗ )p+1 dy
attains its strict maximum at μ = ±1. This contradiction establishes (5.26), and the proof of the theorem is complete. 2 Proof of Theorem 5.21. We work in the variable r = |x|. Since we are assuming U (r, T ) ≡ Φ ∗ (r), there exists r1 > 0 such that U (r1 , T ) = Φ ∗ (r1 ). (If Ω = BR , we simply set r1 = R.) Then since U (r1 , t) is continuous in t ∈ [0, T ], we can find ε > 0 such that U (r1 , t) = Φ ∗ (r1 , t) for t ∈ [T − ε, T ].
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
1057
By Lemma 3.6, we have Z(0,r1 ] U (·, t) − Φ ∗ < ∞
for t ∈ (T − ε < t < T ).
Consequently, U (r, T ) − Φ ∗ (r) changes sign at most finitely many times in the interval (0, r1 ]. Therefore, there exists r0 ∈ (0, r1 ] such that either (a) or (b) below holds: (a) U (r0 , T ) < Φ(r0 ), and U (r, t) Φ ∗ (r) for r ∈ (0, r0 ]; (b) U (r0 , T ) > Φ(r0 ), and U (r, t) Φ ∗ (r) for r ∈ (0, r0 ]. By Lemmas 5.29 and 5.30 below, (a) implies immediate regularization while (b) implies complete blow-up. The theorem is proven. 2 Proof of Theorem 5.22. We first observe that the limit function uˆ satisfies u(x, ˆ t) = ϕ ∗ (x)
for t 0
instead of (5.34). This is clear since w ∗ (y) = ϕ ∗ (y). For p > pJL , Theorem 10.1 of [11] states that the minimal solution of (1.1) with initial data ϕ ∗ is identically equal to ϕ ∗ , hence uˆ = ϕ ∗ for all t > 0. The assertion (5.27) follows immediately from this. 2 5.4. The intersection numbers Whether a blow-up is complete or not can be characterized by counting the number of intersections with the singular stationary solution Φ ∗ . Theorem 5.27 (Zero-number criterion). Let pJL < p < ∞ and u0 0. Suppose that the blow-up is of type I and occurs only at x = 0, and let w ∗ (y) = W ∗ (|y|) be the local blow-up profile. Then (i) Z(0,∞) [W ∗ − Φ ∗ ] being odd implies complete blow-up; (ii) Z(0,∞) [W ∗ − Φ ∗ ] being even implies immediate regularization. Theorem 5.28 (Single-intersection blow-up). Let pS < p < ∞ and p = pJL . Suppose that u blows up at t = T and that m0 (U ) = 1, where m0 (U ) is the number of vanishing intersections as defined in (3.15). Then the blow-up is complete, of type I and w ∗ = ±κ. The proofs of the above two theorems are entirely different. For the proof of the former, we use the following lemmas. Lemma 5.29. Let pJL < p < ∞ and let U be a nonnegative solution of (1.4) that blows up at t = T . Assume that, for some r0 > 0, U (r, T ) Φ ∗ (r)
(0 < r r0 ),
U (r0 , T ) > Φ ∗ (r0 ).
Then the proper extension U¯ (r, t) of U satisfies U¯ (r, t) = +∞ (a.e. r > 0, t > T ).
1058
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
Lemma 5.30. Let pJL < p < ∞ and let U be a nonnegative solution of (1.4) that blows up at t = T . Assume that, for some r0 > 0, 0 U (r, T ) Φ ∗ (r)
U (r0 , T ) < Φ ∗ (r0 ),
(0 < r r0 ),
and that supr>r0 U (r, T ) < ∞. Then there exists δ > 0 such that the proper extension U¯ (r, t) of U is smooth in the region Ω × (T , T + δ). Proof of Lemma 5.29. In [11, Theorem 10.4], it is shown for p > pJL that if U (r, T ) Φ ∗ (r) and U (r, T ) ≡ Φ ∗ (r), then U¯ (r, t) = +∞ (a.e. r > 0, t > T ). We will use this result below. Suppose that the conclusion of the lemma does not hold. Then T < Tc , hence, by Corollary 5.11, U¯ is smooth in r > 0, T t < Tc . Next let h(r, t) be the solution of the following problem: ⎧ N −1 ⎨ hr + hp (r0 < r < ∞, T < t < Tc ), ht = hrr + r ⎩ h(r0 , t) = U¯ (r0 , t) (T < t < Tc ). h(r, T ) = Φ ∗ (r) (r0 < r < ∞), Recalling that U¯ (r0 , T ) = U (r0 , T ) > Φ ∗ (r0 ) and that U¯ is smooth in r > 0, t ∈ [T , Tc ), we easily see that hr (r0 , t) → −∞ as t T . Consequently U¯ r (r0 , t) > hr (r0 + 0, t)
for t ∈ (T , T + δ1 )
provided that δ1 ∈ (0, Tc − T ) is chosen sufficiently small. It follows that the function ¯ V (r, t) := U (r, t) (0 < r r0 ), h(r, t) (r > r0 ) is a supersolution of (1.4) in the range r > 0, t ∈ (T , T + δ1 ). Therefore the minimal solution V¯ (r, t) of (1.1) for the initial data V¯ (r, T ) = V (r, T ) lies below V (r, t) for t ∈ (T , T + δ1 ), but this contradicts the above mentioned result of [11] since V (r, T ) Φ ∗ (r) and V (r, T ) ≡ Φ ∗ (r). The lemma is proven. 2 Proof of Lemma 5.30. Choose a function V (r) satisfying U (r, T ) V (r) Φ ∗ (r)
V (r) Φ ∗ (r)
for 0 < r < r0 ,
for r r0 ,
∗
U (r0 , T ) < V (r0 ) < Φ (r0 ), and let V¯ (x, t) (t T ) be the minimal solution of (1.4) for the initial data V¯ (r, T ) = V (r). Then, by Theorem 10.4 of [11], V¯ (·, t) ∈ L∞ (Ω) for t > 0. Next let h(r, t) be the solution of the following problem: ⎧ N −1 ⎨ hr + hp (r0 < r < ∞, T < t < ∞), ht = hrr + r ⎩ h(r0 , t) = V¯ (r0 , t) (T < t < ∞). h(r, T ) = U (r, T ) (r0 < r < ∞), Recalling that U (r0 , T ) < V (r0 ) = V¯ (r0 , T ), we easily see that V¯r (r0 , t) > hr (r0 + 0, t)
for t ∈ (T , T + δ1 ),
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
1059
provided that δ1 > 0 is sufficiently small. It follows that the function
¯ V (r, t) := V (r, t) h(r, t) ∗
(0 < r r0 ), (r > r0 )
is a supersolution in the range r > 0, t ∈ (T , T + δ1 ) and satisfies V ∗ (r, T ) U (r, T ). Consequently the proper extension U¯ of U satisfies U¯ (r, t) V ∗ (r, t) = V¯ (r, t)
for 0 < r < r0 , t ∈ [T , T + δ1 ).
Hence U¯ (r, t) ∈ L∞ (Ω) for 0 < t < δ1 . This completes the proof of the lemma.
2
Proof of Theorem 5.27. If Z(0,∞) [W ∗ − Φ ∗ ] is odd, then by Lemma A.1 lim W ∗ |y| /Φ ∗ |y| > 1.
|y|→∞
Hence, by Proposition 4.4, there exists r0 > 0 such that U (r, T ) > Φ ∗ (r)
for 0 < r r0 .
(5.40)
Thus, by Lemma 5.29, U¯ = +∞ for t > T , which proves the assertion (i). Similarly, if Z(0,∞) [w ∗ − ϕ ∗ ] is even, then by Proposition 4.4, u(x, T ) < ϕ ∗ (x)
for 0 < |x| r0 .
(5.41)
Thus, by Lemma 5.30, u(x, ˜ t) ∈ L∞ (Ω) for T < t < T + δ1 , which proves (ii). The proof of the theorem is complete. 2 Proof of Theorem 5.28. By Theorem 3.11, the blow-up is of type I if p > pJL . In the range pS < p < pJL , the blow-up is again of type I by virtue of Theorem 3.7 and Remark 3.10, provided that u(x, T ) ≡ ±ϕ ∗ (x). Let us show that u(x, T ) ≡ ±ϕ ∗ (x). We follow the notation in the proof of Theorem 3.8. Let r0 > 0, t0 ∈ [0, T ) be as in (3.16). Choose a > 0 sufficiently large so that there exist 0 < r1 < r2 < r3 < r0 such that Φa (ri ) = Φ ∗ (ri ) (i = 1, 2, 3), Φa (r) < Φ ∗ (r)
for r ∈ (r1 , r2 ),
Φa (r) > Φ ∗ (r)
for r ∈ (r2 , r3 )
and that U (r, t0 ) < Φa (r) for r ∈ (0, r3 ). Then, arguing as in the proof of Theorem 3.11, with v ε replaced by Φa , W by U and s → ∞ by t → T , we can derive a contradiction if we assume U (r, T ) ≡ Φ ∗ (r). Hence u(x, T ) ≡ ϕ ∗ (x). Similarly, u(x, T ) ≡ −ϕ ∗ (x). This proves type I blow-up for the case pS < p < pJL . Consequently the local profile w ∗ (y) = W ∗ (|y|) is a bounded solution of (1.7) and Z[W ∗ − ∗ Φ ] = 1. By Lemma 3.29 of [4], the only solutions of (1.7) satisfying this condition are ±κ; hence w ∗ = ±κ. Proposition 5.13 then implies that the blow-up is complete. 2 Remark 5.31. As we mentioned in Remark 5.18, we will show in our forthcoming paper [29] that a single-intersection blow-up is a generic phenomenon.
1060
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
Appendix A. The asymptotics of Ψ at |y| = ∞ The aim of this appendix is to prove Lemma A.1 below. The lemma is used in the proof of Proposition 4.5 and Theorem 3.2. As we have stated in Theorem 3.1, the rescaled solution W0,T (r, s) converges to a limit W ∗ (r) as s → ∞ and that W ∗ is either a bounded solution of the equation Ψ +
N −1 r 1 Ψ − Ψ − Ψ + |Ψ |p−1 Ψ = 0 r 2 p−1
for 0 < r < ∞,
(A.1)
or the singular solution Φ ∗ or −Φ ∗ . In the former case, Ψ can be extended to r = 0 smoothly and the following boundary condition is automatically satisfied: Ψ (0) = 0. As before, we denote by E all the bounded solutions of (A.1). It is well known that all elements of E ∪ {Φ ∗ , −Φ ∗ }, except the constant solutions ±κ, decay with the order O(r and that the following limit exists: 2
μ∞ (Ψ ) := lim r p−1 Ψ (r) = lim c∗ r→∞
r→∞
2 − p−1
) as r → ∞,
Ψ (r) ∈ R ∪ {−∞, ∞}, Φ ∗ (r)
(A.2)
where c∗ is the constant in (1.19). See [5,24] for details. The proof of this convergence is also found in [28, Lemma A.2]. This quantity μ∞ has the following property. Lemma A.1. Let Ψ1 , Ψ2 ∈ E ∪ {Φ ∗ , −Φ ∗ }. Then μ∞ (Ψ1 ) = μ∞ (Ψ2 ) if and only if Ψ1 = Ψ2 . Consequently μ∞ (Ψ ) = ±c∗ implies Ψ = ±Φ ∗ , μ∞ (Ψ ) = 0 implies Ψ = 0, and μ∞ (Ψ ) = ±∞ implies Ψ = ±κ. 2
Proof. Given a solution Ψ of (A.1), we put v(r) := r p−1 Ψ (r). Then μ∞ (Ψ ) = lim v(r), r→∞
and v satisfies the following equation: v + N −1−
r2 1 4 1 − v + 2 −(c∗ )p−1 v + v p = 0 for 0 < r < ∞. p−1 2 r r
Using the new space variable z = r 2 , we can rewrite this equation as vzz +
N −
4 p−1
2z
1 1 vz + 2 −(c∗ )p−1 v + v p = 0. − 4 4z
Now let Ψ1 , Ψ2 be as in the statement of the lemma, and put 2
vj (r) := r p−1 Ψj (r),
j = 1, 2.
(A.3)
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
1061
Then, by (A.3), the function h = v1 − v2 satisfies hzz +
N −
4 p−1
−
2z
1 hz = O z−2 |h| 4
as z → ∞,
hence hz +
N −
4 p−1
2z
1 h = O z−2 |h| − 4 z
as z → ∞.
(A.4)
Since h(z) → 0 as z → ∞ by the assumption of the lemma, there exists a sequence 0 < z1 < z2 < z3 < · · · → ∞ such that hz (zn ) → 0 as n → ∞. Integrating (A.4) from z to zn and letting n → ∞, we see that there exists a constant C > 0 such that hz − 1 h C max h(ζ ) . (A.5) 4 z ζ z Now choose z∗ > 0 such that z∗ 8C,
∗ h(z ) max h(ζ ) . ζ z∗
(A.6)
Such z∗ exists since h → 0 as z → ∞. This and (A.5) yield hz (z∗ ) − 1 h(z∗ ) 1 h(z∗ ) . 4 8 Thus, if h(z∗ ) = 0, then we would have hz (z∗ ) 1 1 − > 0. h(z∗ ) 4 8 This, however, contradicts the second inequality in (A.6), hence we have h(z∗ ) = 0, which implies, again by (A.6), that h(z) = 0 for any z z∗ . Consequently Ψ1 (r) = Ψ2 (r) for all large r > 0, hence Ψ1 ≡ Ψ2 since both are solutions of (A.1). The lemma is proven. 2 Appendix B. Boundedness of the blow-up set In this appendix we prove Lemma B.1, which asserts that the blow-up set of a given solution is contained in the convex hull of the support of initial data. We have used this lemma in the last part of the proof of Proposition 5.5. The lemma is stated in a rather general setting, without assuming radial symmetry. Let us consider the equation (B.1) ut = u + f (u) x ∈ RN , t > 0 , where f ∈ C 1 ([0, ∞)) ∩ C 2 ((0, ∞)) satisfies f (0) 0,
f (u) > 0,
f (u) > 0
for u > 0,
(B.2)
1062
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
and
f log f
∞ > 0,
log f (u) du < ∞. f (u)
(B.3)
M
Examples of functions f satisfying (B.3) include α u log(1 + u)
(α > 2),
up
(p > 1),
eβu
(β > 0).
Note that (B.3) is nearly equivalent to (but slightly stronger than) the so-called Friedman– McLeod condition (B.4) below. Lemma B.1. Let f satisfy the above assumptions (or (B.4) instead of (B.3)) and let u0 be a bounded nonnegative function whose support spt(u0 ) is compact. If the solution u of (B.1) with initial data u0 blows up in finite time, then its blow-up set B(u0 ) satisfies B(u0 ) ⊂ ch spt(u0 ) , where ch(A) denotes the convex hull of a set A. Proof. The assertion follows easily by combining the reflection argument found in [20,27] and an argument in [10]. More precisely, put K = ch(spt(u0 )). Without loss of generality, we may assume that 0 ∈ K. For each unit vector ξ ∈ RN , we define λ∗ (ξ ) 0 to be the smallest number such that
K ⊂ x ∈ RN x · ξ λ∗ (ξ ) . Fix such a unit vector ξ ∈ S N −1 arbitrarily and, for each λ λ∗ (ξ ), let uλ be the reflection of u with respect to the hyperplane P λ (ξ ) := {x · ξ = λ}. Then, since u = uλ on P λ (ξ ) and u(x, 0) uλ (x, 0) ≡ 0 in the halfspace D λ (ξ ) := {x · ξ λ}, the comparison principle yields u(x, t) uλ (x, t)
for x ∈ D λ (ξ ), 0 t < T (u0 ).
Furthermore u ≡ uλ since u0 ≡ uλ0 . Hence, by the Hopf boundary lemma, we have ξ · ∇u < ξ · ∇uλ
for x ∈ P λ (ξ ), 0 < t < T (u0 ),
which implies ξ · ∇u < 0 for x ∈ P λ (ξ ), 0 < t < T (u0 ) for all ξ ∈ S N −1 and λ > λ∗ (ξ ). Now applying the same argument as in the proof of Theorem 3.3 of [10], we see that no blow-up point appears on P λ (ξ ) if λ > λ∗ (ξ ). Since ξ ∈ S N −1 is arbitrary, this implies that B(u0 ) ⊂ K. 2
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
1063
Remark B.2. The condition (B.3) is nearly equivalent to (but slightly stronger than) the following well-known condition due to Friedman and McLeod [10]: there exist a function F and constants ε > 0, M > 0 such that ⎧ F (0) > 0, F (u) > 0, F (u) > 0 for u M, ⎪ ⎪ ⎪ ⎪ ⎨ f (u)F (u) − f (u)F (u) εF (u)F (u) for u M, ∞ (B.4) du ⎪ ⎪ < ∞. ⎪ ⎪ ⎩ F (u) M
That (B.3) implies (B.4) can be easily seen by setting F = f/ log f . References [1] S. Angenent, The zero set of a solution of a parabolic equation, J. Reine Angew. Math. 390 (1988) 79–96. [2] J. Ball, Remarks on blow-up and nonexistence theorems for nonlinear evolution equations, Quart. J. Math. Oxford Ser. (2) 28 (112) (1977) 473–486. [3] P. Baras, L. Cohen, Complete blow-up after Tmax for the solution of a semilinear heat equation, J. Funct. Anal. 71 (1987) 142–174. [4] J. Bebernes, D. Eberly, Mathematical Problems from Combustion Theory, Springer-Verlag, New York, 1989. [5] C.J. Budd, Y.-W. Qi, The existence of bounded solutions of a semilinear elliptic equations, J. Differential Equations 82 (1989) 207–218. [6] X.-Y. Chen, P. Poláˇcik, Asymptotic periodicity of positive solutions of reaction–diffusion equations on a ball, J. Reine Angew. Math. 472 (1996) 17–51. [7] K.-S. Chou, S.-Z. Du, G.-F. Zheng, On partial regularity of the borderline solution of semilinear parabolic problems, Calc. Var. Partial Differential Equations 30 (2007) 251–275. [8] M. Fila, H. Matano, P. Poláˇcik, Immediate regularization after blow-up, SIAM J. Math. Anal. 37 (2005) 752–776. [9] S. Filippas, M.A. Herrero, J.J.L. Velázquez, Fast blow-up mechanisms for sign-changing solutions of a semilinear parabolic equation with critical nonlinearity, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 456 (2004) (2000) 2957–2982. [10] A. Friedman, J.B. McLeod, Blow up of positive solutions of semilinear heat equations, Indiana Univ. Math. J. 34 (1985) 425–447. [11] V. Galaktionov, J.L. Vázquez, Continuation of blow-up solutions of nonlinear heat equations in several space dimensions, Comm. Pure Appl. Math. 50 (1997) 1–67. [12] Y. Giga, R.V. Kohn, Asymptotically self-similar blowup of semilinear heat equations, Comm. Pure Appl. Math. 38 (1985) 297–319. [13] Y. Giga, R.V. Kohn, Characterizing blowup using similarity variables, Indiana Univ. Math. J. 36 (1987) 1–40. [14] Y. Giga, R.V. Kohn, Nondegeneracy of blow up for semilinear heat equations, Comm. Pure Appl. Math. 42 (1989) 845–884. [15] Y. Giga, S. Matsui, S. Sasayama, Blow up rate for semilinear heat equation with subcritical nonlinearity, Indiana Univ. Math. J. 53 (2004) 483–514. [16] Y. Giga, S. Matsui, S. Sasayama, On blow up rate for sign-changing solutions in a convex domain, Math. Methods Appl. Sci. 27 (2004) 1771–1782. [17] M.A. Herrero, J.J.L. Velázquez, Blow-up behaviour of one-dimensional semilinear parabolic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 10 (2) (1993) 131–189. [18] M.A. Herrero, J.J.L. Velázquez, Explosion de solutions des équations paraboliques semilinéaires supercritiques, C. R. Acad. Sci. Paris Ser. A 319 (1994) 141–145. [19] M.A. Herrero, J.J.L. Velázquez, A blow up result for semilinear heat equations in the supercritical case, preprint. [20] C.K.R.T. Jones, Asymptotic behavior of a reaction–diffusion equation in higher space dimensions, Rocky Mountain J. Math. 13 (1983) 355–364. [21] D.D. Joseph, T.S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Ration. Mech. Anal. 49 (1973) 241–269. [22] A.A. Lacey, D. Tzanetis, Complete blowup for a semilinear diffusion equation with a sufficiently large initial data, IMA J. Appl. Math. 41 (1988) 207–215.
1064
H. Matano, F. Merle / Journal of Functional Analysis 256 (2009) 992–1064
[23] O.A. Ladyzhenskaya, V.A. Solonnikov, N.N. Ural’ceva, Linear and Quasilinear Equation of Parabolic Type, Transl. Math. Monogr., vol. 23, Amer. Math. Soc., Providence, RI, 1968. [24] L.A. Lepin, Self-similar solutions of a semilinear heat equation, Mat. Model. 2 (1990) 63–74. [25] H.A. Levine, Some nonexistence and instability theorems for solutions of formally parabolic equations of the form P ut = −Au + F (u), Arch. Ration. Mech. Anal. 51 (1973) 371–386. [26] H. Matano, Convergence of solutions one-dimensional semilinear parabolic equations, J. Math. Kyoto Univ. 18 (1978) 221–227. [27] H. Matano, Asymptotic behavior of the free boundaries arising in one phase Stefan problem in multi-dimensional spaces, in: North-Holland Math. Stud., vol. 81, North-Holland, Amsterdam, 1983, pp. 133–151. [28] H. Matano, F. Merle, On nonexistence of type II blowup for a supercritical nonlinear heat equation, Comm. Pure Appl. Math. 57 (2004) 1494–1541. [29] H. Matano, F. Merle, A threshold behavior for a supercritical nonlinear heat equation, in preparation. [30] J. Matos, Unfocused blow up solutions of semilinear parabolic equations, Discrete Contin. Dyn. Syst. 5 (1999) 905–928. [31] J. Matos, Convergence of blow-up solutions of nonlinear heat equations in the supercritical case, Proc. Roy. Soc. Edinburgh Ser. A 129 (1999) 1197–1227. [32] J. Matos, Self-similar blow up patterns in supercritical semilinear heat equations, Commun. Appl. Anal. 5 (2001) 455–483. [33] F. Merle, H. Zaag, Optimal estimates for blow-up rate and behavior for nonlinear heat equations, Comm. Pure Appl. Math. 51 (1998) 139–196. [34] F. Merle, H. Zaag, A Liouville theorem for vector-valued nonlinear heat equations and applications, Math. Ann. 316 (2000) 103–137. [35] N. Mizoguchi, Type II blowup for a semilinear heat equation, Adv. Differential Equations 9 (2004) 1279–1316. [36] N. Mizoguchi, Boundedness of global solutions for a supercritical heat equation and its application, Indiana Univ. Math. J. 54 (2005) 1047–1059. [37] W.-M. Ni, P.E. Sacks, J. Tavantzis, On the asymptotic behavior of solutions of certain quasilinear parabolic equations, J. Differential Equations 54 (1984) 97–120. [38] P. Quittner, Continuity of blowup time and a priori bounds for solutions of superlinear parabolic problems, Houston J. Math. 29 (3) (2003) 757–799. [39] F. Weissler, An L∞ blow-up estimate for a nonlinear heat equation, Comm. Pure Appl. Math. 38 (1985) 291–295.
Journal of Functional Analysis 256 (2009) 1065–1105 www.elsevier.com/locate/jfa
Construction and reconstruction of tight framelets and wavelets via matrix mask functions ✩ Marcin Bownik a,∗ , Ziemowit Rzeszotnik b a Department of Mathematics, University of Oregon, Fenton Hall, Eugene, OR 97403-1222, USA b Mathematical Institute, University of Wrocław, Pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland
Received 15 April 2008; accepted 10 December 2008
Communicated by N. Kalton
Abstract The paper develops construction procedures for tight framelets and wavelets using matrix mask functions in the setting of a generalized multiresolution analysis (GMRA). We show the existence of a scaling vector of a GMRA such that its first component exhausts the spectrum of the core space near the origin. The corresponding low-pass matrix mask has an especially advantageous form enabling an effective reconstruction procedure of the original scaling vector. We also prove a generalization of the Unitary Extension Principle for an infinite number of generators. This results in the construction scheme for tight framelets using lowpass and high-pass matrix masks generalizing the classical MRA constructions. We prove that our scheme is flexible enough to reconstruct all possible orthonormal wavelets. As an illustration we exhibit a pathwise connected class of non-MSF non-MRA wavelets sharing the same wavelet dimension function. © 2008 Elsevier Inc. All rights reserved. Keywords: Wavelet; Generalized multiresolution analysis; Matrix mask function
✩ The first author was partially supported by the NSF grants DMS-0441817 and DMS-0653881. The second author was supported under the EU-project MEXT-CT-2004-517154. The research for this paper was carried during a one-year visit of the second author to the University of Oregon. * Corresponding author. E-mail addresses: [email protected] (M. Bownik), [email protected] (Z. Rzeszotnik).
0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.12.006
1066
M. Bownik, Z. Rzeszotnik / Journal of Functional Analysis 256 (2009) 1065–1105
1. Introduction and preliminaries The main aim of this work is to develop a constructive procedure for constructing tight framelets and wavelets from more primitive objects given by low-pass and high-pass matrix mask functions. We should add that all of our results are shown in the general setting of expansive integer dilations in Rn . The novelty of our approach lies in its versatility allowing construction of all possible orthonormal wavelets without the customary restrictions on smoothness or decay in time or frequency domains. Hence, it applies to all sorts of exotic and little understood wavelets such as those with unbounded wavelet dimension function. In the case of multiresolution analysis (MRA) wavelets such procedure is well studied and understood. Usually, an MRA construction starts with a 1-periodic measurable function m, also called a low-pass mask and satisfying the quadrature-mirror equation m(ξ )2 + m(ξ + 1/2)2 = 1
for a.e. ξ ∈ R.
(1.1)
Under small regularity assumptions, such as m is Hölder continuous at 0 and m(0) = 1, one defines a refinable function ϕ ∈ L2 (R) by ϕ(ξ ˆ )=
∞ m 2−j ξ .
(1.2)
j =1
While ϕ might fail to be an orthogonal scaling function of an MRA, one can always obtain a tight frame wavelet ψ ∈ L2 (R) using a high-pass mask h by ˆ ) = h(ξ/2)ϕ(ξ/2), ψ(ξ ˆ
where h(ξ ) = e2πiξ m(ξ + 1/2),
(1.3)
see [20,26]. The fact that ψ is a tight framelet can be shown directly by employing the characterization equations [25, Section 7.1]. Alternatively, it is also a consequence of a Unitary Extension Principle of Ron and Shen [21,29]. While Hölder continuity of m at 0 is a relatively weak assumption, some MRA wavelets cannot be obtained by this scheme [18]. To circumvent this problem, Paluszy´nski, Šiki´c, Weiss, and Xiao [27] introduced the class of low-pass filters satisfying lim
n→∞
∞ −j m 2 ξ = 1 for a.e. ξ ∈ R. j =n
This is obviously a much weaker condition than Hölder continuity. Moreover, any low-pass mask m of an MRA scaling function must satisfy it by the characterization of scaling functions of MRAs [25]. While the infinite product (1.2) might not be convergent, the authors of [27] proved that one can always construct a refinable function ϕ satisfying ϕ(ξ ˆ ) = m(ξ/2)ϕ(ξ/2) ˆ
a.e. ξ ∈ R.
This is because the product (1.2) converges after taking absolute values and one must only recover the phase factor of ϕˆ by using a multiplier argument. As a consequence, the procedure of constructing MRA tight frame wavelets from [27] recovers all possible MRA wavelets.
M. Bownik, Z. Rzeszotnik / Journal of Functional Analysis 256 (2009) 1065–1105
1067
Similar ideas were used to prove the connectivity result for MRA wavelets by the Wutam Consortium [34]. Since MRA wavelets form only a special class among all orthonormal wavelets, one could ask whether similar construction and reconstruction procedures are possible for non-MRA wavelets. The most natural way of classifying non-MRA wavelets uses the wavelet dimension function Dψ (ξ ) =
∞ j ψˆ 2 (ξ + k) 2 . j =1 k∈Z
The wavelet dimension function has many interesting properties that were investigated by several authors [1,2,17,25,28,30]. One of its fundamental properties says that Dψ can be identified with the multiplicity function of the core space of a GMRA generated by the wavelet ψ. In particular, ψ is an MRA wavelet if and only if Dψ ≡ 1. Moreover, by the result of Speegle and the authors [17] all possible wavelet dimension functions D are characterized by the following 4 conditions: (D1) D : R → N ∪ {0} is a measurable 1-periodic function, (D2) D(ξ ) + D(ξ + 1/2) = D(2ξ ) + 1 for a.e. ξ ∈ R, (D3) k∈Z 1Δ (ξ + k) D(ξ ) for a.e. ξ ∈ R, where Δ = ξ ∈ R: D 2−j ξ 1 for j ∈ N ∪ {0} , (D4) lim infj →∞ D(2−j ξ ) 1 for a.e. ξ ∈ R. Note that we have intentionally omitted the integrability condition on D, since it is a consequence of (D1) and (D2) by Lemma 3.1 proved in this paper. The above characterization opens the possibility of constructing wavelets and framelets from more general low-pass matrix masks than the standard scalar masks satisfying (1.1). In general, one starts with a measurable matrix-valued 1-periodic function M satisfying M(ξ )M ∗ (ξ ) + M(ξ + 1/2)M ∗ (ξ + 1/2) = Ω(2ξ )
for a.e. ξ ∈ R,
(1.4)
where Ω(ξ ) = diag 1S1 (ξ ), 1S2 (ξ ), . . . ,
Sj = ξ ∈ R: D(ξ ) j ,
j ∈ Z.
More precisely, the values of M(ξ ) are infinite size matrices (doubly indexed by N) with only finitely many non-zero entries, which can be identified with bounded operators on 2 (N). Furthermore, in the case when the multiplicity function is bounded by N ∈ N, we can safely assume that the values of M(ξ ) are N × N matrices. Baggett, Jorgensen, Merrill, and Packer [4] showed that if the multiplicity function D is bounded and M satisfies some weak regularity assumptions then one can define a refinable vector function Φ = (ϕj )j ∈J ⊂ L2 (R), J = {1, . . . , N}, by
ˆ )= Φ(ξ
∞ j =1
−j M 2 ξ e,
(1.5)
1068
M. Bownik, Z. Rzeszotnik / Journal of Functional Analysis 256 (2009) 1065–1105
where e = (1, 0, . . . , 0). To make sure that the above product converges the authors of [4] assume that M is Lipschitz continuous at 0 and M(0) is a matrix having all zero entries except a single 1 in the upper-left corner. In general, Φ might not be a scaling vector of some GMRA in the sense that each ϕj is a quasi-orthogonal generator of S(ϕj ) = span ϕj (· − k): k ∈ Z and S(ϕi ) ⊥ S(ϕj ) for i = j . Nevertheless, the authors of [4] proved that by choosing an appropriate high-pass matrix mask H , one can always obtain a tight frame wavelet ψ ∈ L2 (R) by setting ˆ ) = H (ξ/2)Φ(ξ/2) ˆ ψ(ξ a.e. ξ ∈ R.
(1.6)
More precisely, H is 1-periodic measurable 1 × N matrix-valued function satisfying H (ξ )H ∗ (ξ ) + H (ξ + 1/2)H ∗ (ξ + 1/2) = 1
a.e. ξ ∈ T,
(1.7)
M(ξ )H ∗ (ξ ) + M(ξ + 1/2)H ∗ (ξ + 1/2) = 0 a.e. ξ ∈ T.
(1.8)
This naturally leads to a fundamental problem of the theory of non-MRA wavelets, which asks whether it is possible to use the above scheme of low-pass and high-pass matrix masks to construct all orthonormal wavelets. The goal of this paper is to give an affirmative answer to this problem. To give the idea of the level of difficulty behind this project one should realize that, a priori, no regularity assumption on the low-pass matrix mask functions can be assumed. Furthermore, an example in [17] demonstrates that the multiplicity function D could be unbounded which leads to a matrix-valued low-pass mask M of infinite size. Hence, the infinite product in (1.5) might not converge and special convergence procedures are needed to interpret such ill-defined expressions. The starting point of this paper is the investigation of the properties of scaling vectors corresponding to the core space of a GMRA. Unlike the case of an MRA, where the scaling function is unique (up to a unimodular 1-periodic phase factor in the Fourier domain), there are many possible choices for scaling vectors for a GMRA. This has been traditionally considered as an impediment of a successful theory, since different choices of a scaling vector Φ could lead to totally different low-pass masks M satisfying ˆ ) = M(ξ/2)Φ(ξ/2) Φ(ξ
a.e. ξ ∈ R.
(1.9)
It might seem that the only useful information extracted from (1.9) is a matrix analogue of the quadrature-mirror equation (1.4). Nevertheless, we show that abundance of choice is also a blessing if one carefully chooses generators of the scaling vector. The key idea is to choose the first generator ϕ1 such that it exhausts the entire spectrum of the core space near the origin. Consequently, the remaining generators ϕ2 , ϕ3 , . . . must be supported away from the origin in the Fourier domain. This leads to an especially advantageous form of the low-pass matrix mask M such that the first column of M(ξ ) has zeros in every entry, except the first where it has absolute value “almost equal” to 1 for ξ ≈ 0. For a precise statement see Theorem 2.2. In the case when a GMRA is associated with an orthonormal wavelet ψ, it is easy to verify that the high-pass mask H given by (1.6) must satisfy (1.7) and (1.8). The crux of our approach
M. Bownik, Z. Rzeszotnik / Journal of Functional Analysis 256 (2009) 1065–1105
1069
is the assertion claiming that one can reverse the above process. That is, given a low-pass mask in the above advantageous form and a high-pass matrix mask satisfying (1.7) and (1.8), we can construct an associated tight frame wavelet ψ . This leads to Theorem 4.3 which is the main construction result of our paper. The first key ingredient in the proof of this result is the existence of a refinable vector Φ, which is a result of a special convergence procedure making sense out of potentially divergent infinite product in (1.5). The second ingredient is a generalization of the Unitary Extension Principle to a situation when a refinable vector Φ has infinitely many components. The last part of the paper proves that the above scheme is flexible enough to reconstruct all possible wavelets ψ . A pivotal role in the reconstruction scheme is played by the concept of a multiplier. We say that a unimodular function ν is a multiplier associated to M = (mi,j ) if it satisfies ν(2ξ )ν(ξ )m1,1, (ξ ) = m1,1 (ξ )
for a.e. ξ ∈ R.
Then, our main reconstruction Theorem 5.4 says that the scaling vector Φ can be recovered by
N −N −j ˆ ) = lim ν 2 ξ M 2 ξ e Φ(ξ N →∞
for a.e. ξ,
(1.10)
j =1
and then the wavelet ψ can be recovered by (1.6). Finally, the paper ends with examples illustrating the inner workings of our construction and reconstruction procedures. In particular, we give an example of a class Wnik of non-MSF and non-MRA wavelets such that all of its members share the same Journé dimension function. Recall that a wavelet ψ is said to be minimally supported frequency (MSF), if the Lebesgue measure of the support of ψˆ is smallest possible, that is equal to 1. The classes of MSF wavelets and MRA wavelets are already well studied and understood. However, the class of non-MSF and non-MRA wavelets is the least understood and has inhibited the growth of L2 theory of wavelets. Nevertheless, we prove that our class Wnik is pathwise connected indicating that our techniques have a potential of attacking a recalcitrant problem of the connectivity of the set of all orthonormal wavelets. Despite the fact that all of our results are motivated by the classical case of dyadic dilations in R, we will adopt a more general setting of expansive integer-valued dilations in Rn . More specifically, we shall assume that we are given an n × n integer-valued matrix A that is expansive, i.e., all its eigenvalues have modulus greater than 1. For simplicity, its transpose will be denoted by B. We recall that a sequence {D j (V ): j ∈ Z} of closed subspaces of L2 (Rn ) is called a generalized multiresolution analysis (GMRA) if (M1) (M2) (M3) (M4)
Tk V = V for all k ∈ Zn , V ⊂ D(V ), D j (V ) = L2 (Rn ),
j ∈Z j j ∈Z D (V ) = {0}.
Here, the dilation operator D is given by Df (x) = |det A|1/2 f (Ax) for some n × n expansive integer-valued matrix A and the translation operator Tk f (x) = f (x − k) for some k ∈ Zn .
1070
M. Bownik, Z. Rzeszotnik / Journal of Functional Analysis 256 (2009) 1065–1105
As we can see, a GMRA is based on the core space V . Condition (M1) means that V is a shift-invariant (SI) space. If V satisfies (M2), then we call it refinable. Also, we shall often write Vj instead of D j (V ). We say that a finite family Ψ = {ψ 1 , . . . , ψ N } ⊂ L2 (Rn ) is a wavelet if its associated affine system ψj,k (x) = |det A|j/2 ψ Aj x − k ,
j ∈ Z, k ∈ Zn , ψ ∈ Ψ,
is an orthonormal basis of L2 (Rn ). In the more general case, when the affine system is a tight frame (with constant 1), we say that Ψ is a tight framelet. The latter are characterized by the wellknown equations that we list in (3.7) and (3.8). Moreover, a framelet Ψ is called semi-orthogonal if
D j (W ) = L2 Rn ,
where W = span ψ(· − k): k ∈ Zn , ψ ∈ Ψ .
j ∈Z
It turns out that every semi-orthogonal tight framelet comes from a GMRA. Indeed, for a finite family Ψ ⊂ L2 (Rn ) we define its space of negative dilates V by V = span ψj,k : j < 0, k ∈ Zn , ψ ∈ Ψ . We say that a framelet Ψ is associated with a GMRA (or that it generates a GMRA) if its space of negative dilates V satisfies (M1)–(M4). It is not hard to check that if Ψ is a semi-orthogonal tight framelet then conditions (M1)–(M4) hold and, therefore, V is a core space of a GMRA. 2. Scaling vectors for GMRA The main goal of this section is to provide a constructive procedure for selecting a suitable set of generators Φ for the core space V0 of a GMRA {Vj }j ∈Z . The result of this procedure is a collection of (mutually orthogonal) quasi-orthogonal generators Φ = (ϕj )j ∈J called a scaling vector for V0 . (In particular, if {Vj }j ∈Z is a usual MRA, we obtain a single orthogonal generator of V0 , usually called a scaling function.) Quasi-orthogonality means that integer shifts of the generator form a tight frame for the corresponding SI space. Such a space, that is generated by just one function, say ϕ, is called a principal shift-invariant (PSI) space and is denoted by S(ϕ). First, we shall review some basic results about SI spaces and the dimension and spectral functions. Every shift-invariant space V ⊂ L2 (Rn ) has a set of generators Φ, that is, a countable family of functions whose integer shifts form a tight frame (with constant 1) for V , see [10, Theorem 3.3]. Although this family is not unique, the function σV (ξ ) =
2 ϕ(ξ ˆ ) ϕ∈Φ
does not depend (except on a set of measure zero) on the choice of the family of generators, see [15, Lemma 2.3]. Here, the Fourier transform is defined by fˆ(ξ ) = f (x)e−2πi x,ξ dx. Rn
M. Bownik, Z. Rzeszotnik / Journal of Functional Analysis 256 (2009) 1065–1105
1071
We call σV the spectral function of V . This notion was introduced by the authors in [15]. The basic property of σ is that it is additive on countable orthogonal sums and that σL2 (Rn ) = 1. The spectral function also behaves nicely under dilations since σD(V ) (ξ ) = σV (B −1 ξ ). Moreover, if V is generated by a single function ϕ then
ˆ + k)|2 )−1 |ϕ(ξ ˆ )|2 ( k∈Zn |ϕ(ξ σV (ξ ) = 0
for ξ ∈ supp ϕ, ˆ otherwise.
There are several other equivalent ways of defining the spectral function. The original one involves the range function J , that is, a mapping from the torus Tn to the set of closed subspaces of 2 (Zn ). It turns that there is a 1–1 correspondence between SI spaces and measurable range functions J given by V = f ∈ L2 Rn : T f (ξ ) ∈ J (ξ ) for a.e. ξ ∈ Tn , see [10, Proposition 1.5]. Here, T : L2 (Rn ) → L2 (Tn , 2 (Zn )) is an isometric isomorphism given by T f (ξ ) = (fˆ(ξ + k))k∈Zn , where Tn is identified with [−1/2, 1/2)n . The spectral function σV can be equivalently defined by 2 σV (ξ + k) = PJ (ξ )ek
for ξ ∈ Tn and k ∈ Zn ,
where {ek }k∈Zn denotes the standard basis of 2 (Zn ) and PJ (ξ ) is an orthogonal projection of 2 (Zn ) onto J (ξ ). The spectral function also allows us to define the dimension function of V , dimV (ξ ) =
σV (ξ + k).
k∈Zn
The dimension function (also called the multiplicity function) is integer-valued and additive on countable orthogonal sums as well. Moreover, the minimal number of functions needed to generate V is equal to the L∞ norm of dimV . Again, we refer the reader to [10,15] for the proofs of all these facts. The main feature of our generator selecting procedure is that it distinguishes the first generator ϕ1 as having a dominating effect on all remaining generators. More precisely, the first generator ϕ1 is chosen so that it exhausts the entire spectrum of the core space V0 in some neighborhood of the origin. The fact that the space V0 is refinable and this exhaustion property of ϕ1 leads to a very special form of a matrix mask of Φ, whose first column has zeros in every, but the first entry, near the origin, see Theorem 2.2. Our procedure is somewhat reminiscent of the superfunction theory in the study of finitely generated shift-invariant (FSI) spaces by de Boor, DeVore, and Ron [22,23]. Among other things, the authors proved [23, Result 1.2] that an approximation order of an FSI space can be realized by some PSI space generated by a single function ψ , called a “superfunction.” Therefore, ψ has a dominating effect by providing the same approximation order as the whole FSI space generated by some finite collection of generators. This is analogous to our construction, where the first generator ϕ1 makes all other generators to be innocuous near the origin in the Fourier domain, thus producing a special form of a matrix mask.
1072
M. Bownik, Z. Rzeszotnik / Journal of Functional Analysis 256 (2009) 1065–1105
To achieve the above dominating effect we show the existence of a quasi-generator ϕ0 of an SI space V0 having the same spectral function as that of V0 in a pre-specified localized region of the Fourier domain. Therefore, the generator ϕ0 exhausts locally the space V0 in that region. Lemma 2.1. Assume that V0 ⊂ L2 (Rn ) is SI. Let K be a measurable subset of Rn such that K ∩ (K + l) = 0 for all l ∈ Zn \ {0}. Let ϕ = PV0 (1ˇ K ), where PV0 is the orthogonal projection onto V0 , and 1ˇ K is the inverse Fourier transform of the characteristic function 1K . Define ϕ0 ∈ L2 (Rn ) by ϕˆ0 (ξ ) =
ϕ(ξ ˆ )( k∈Zn |ϕ(ξ ˆ + k)|2 )−1/2 , ξ ∈ supp ϕ, ˆ 0, otherwise.
(2.1)
Then ϕ0 ∈ V0 is a quasi-orthogonal generator of S(ϕ0 ) and 2 σS (ϕ0 ) (ξ ) = ϕˆ 0 (ξ ) = σV0 (ξ )
for a.e. ξ ∈ K.
(2.2)
Proof. Let ϕK ∈ L2 (Rn ) be given by ϕˆK = 1K , and hence ϕ = PV0 ϕK . Clearly, ϕ0 is a quasiorthogonal generator of the PSI space S(ϕ0 ) = S(ϕ) ⊂ V0 . In particular, ϕˆ0 (ξ )2 = σS (ϕ ) (ξ ) = σS (ϕ) (ξ ). 0 Let J be the range function of V0 with the corresponding orthogonal projections PJ (ξ ). Then for any f ∈ L2 (Rn ) we have T (PV0 f )(ξ ) = PJ (ξ ) T f (ξ )
for a.e. ξ ∈ Tn .
Hence, for a.e. ξ ∈ Tn , PJ (ξ )ek , T ϕ(ξ ) = T (PV0 ϕK )(ξ ) = PJ (ξ ) T ϕK (ξ ) = 0,
ξ + k ∈ K, k ∈ Zn , otherwise.
Fix k ∈ Zn . If ξ + k ∈ K, ξ ∈ Tn , and T ϕ(ξ ) = 0, then we necessarily have σS (ϕ) (ξ + k) =
|ϕ(ξ ˆ + k)|2 | T ϕ(ξ ), ek |2 | PJ (ξ )ek , ek |2 | PJ (ξ )2 ek , ek |2 = = = T ϕ(ξ )2 T ϕ(ξ )2 PJ (ξ )ek 2 PJ (ξ )ek 2 2 | PJ (ξ )ek , PJ (ξ )ek |2 = = PJ (ξ )ek = σV0 (ξ + k). 2 PJ (ξ )ek
On the other hand, if ξ + k ∈ K, ξ ∈ Tn , and T ϕ(ξ ) = 0, then 2 σS (ϕ) (ξ + k) = 0 = PJ (ξ )ek = σV0 (ξ + k). Since k ∈ Zn is arbitrary, this proves (2.2).
2
M. Bownik, Z. Rzeszotnik / Journal of Functional Analysis 256 (2009) 1065–1105
1073
The next result provides a decomposition of any SI space V0 as an orthogonal sum of carefully chosen PSI spaces, such that the first PSI space exhausts the spectrum of V0 near the origin. Theorem 2.1. Assume that V0 ⊂ L2 (Rn ) is SI. Then there exists an orthogonal decomposition V0 =
∞
S(ϕj )
(2.3)
j =1
such that each ϕj is a quasi-orthogonal generator of S(ϕj ) and supp dimS (ϕj ) = ξ ∈ Rn : dimV0 (ξ ) j
for every j ∈ N.
(2.4)
Furthermore, the spectral function of S(ϕ1 ) coincides with that of V0 near the origin, i.e., 2 σS (ϕ1 ) (ξ ) = ϕˆ 1 (ξ ) = σV0 (ξ )
for a.e. ξ ∈ Tn ,
(2.5)
Proof. The existence of a decomposition satisfying (2.3) and (2.4) is already known, see [10]. The novelty of Theorem 2.1 lies in the fact that the first quasi-orthogonal generator ϕ1 can be chosen to satisfy (2.5). Let ϕ0 ∈ V0 be a quasi-orthogonal generator guaranteed by Lemma 2.1 with K = Tn . That is, 2 σS (ϕ0 ) (ξ ) = ϕˆ 0 (ξ ) = σV0 (ξ )
for a.e. ξ ∈ Tn .
Define E = supp dimV0 \ supp dimS (ϕ0 ) . Consider two possible cases. If |E| > 0, then define an SI space V = V0 ∩ Lˇ 2 (E). Here, Lˇ 2 (E) = f ∈ L2 Rn : supp fˆ ⊂ E . Let ϕ be a quasi-orthogonal generator of V such that supp dimS (ϕ) = ξ ∈ Rn : dimV (ξ ) 1 = ξ ∈ E: dimV0 (ξ ) 1 = E. Since ϕ0 ∈ Lˇ 2 (Rn \ E), ϕ ∈ Lˇ 2 (E), and the set E is invariant under translations by Zn , ϕ1 = ϕ0 + ϕ is a quasi-orthogonal generator of S(ϕ1 ). Moreover, ϕ1 ∈ V0 since ϕ0 , ϕ ∈ V0 , and supp dimS (ϕ1 ) = supp dimS (ϕ0 ) ∪ supp dimS (ϕ) = supp dimS (ϕ0 ) ∪E = supp dimV0 . Hence, (2.4) holds for j = 1. Since S(ϕ0 ) ⊂ S(ϕ1 ) ⊂ V0 , we have that for a.e. ξ ∈ Tn , σV0 (ξ ) σS (ϕ0 ) (ξ ) σS (ϕ1 ) (ξ ) σV0 (ξ ), which proves (2.5). In the case of |E| = 0, we let ϕ1 = ϕ0 . Trivially, (2.4) holds for j = 1 and (2.5) also holds. Finally, it suffices to consider an SI space V0 S(ϕ1 ) and its decomposition guaranteed by the first part of Theorem 2.1. That is we have
1074
M. Bownik, Z. Rzeszotnik / Journal of Functional Analysis 256 (2009) 1065–1105
V0 S(ϕ1 ) =
∞
S(ϕj ),
j =2
for j 2. supp dimS (ϕj ) = ξ ∈ Rn : dimV0 S (ϕ1 ) (ξ ) j − 1 = ξ ∈ Rn : dimV0 (ξ ) j Therefore, {ϕj }∞ j =1 is the sequence of quasi-orthogonal generators fulfilling (2.3)–(2.5).
2
Theorem 2.1 leads naturally to the definition of an exhausting quasi-orthogonal vector for general SI spaces and an exhausting scaling vector for refinable SI spaces. Definition 2.1. Suppose that V0 is SI and for j ∈ N let Sj = ξ ∈ Rn : dimV0 (ξ ) j .
(2.6)
Let J = {j ∈ N: |Sj | > 0}. Naturally, J=
{1, . . . , L} if L = ess supξ ∈Rn dimV0 (ξ ) < ∞, N
otherwise.
(2.7)
A quasi-orthogonal vector for V0 is defined as Φ = (ϕj )j ∈J , where {ϕj }j ∈J are quasi-orthogonal generators as in Theorem 2.1 satisfying (2.3) and (2.4) only. In addition, if (2.5) holds, then we say that Φ is an exhausting quasi-orthogonal vector for V0 . The Fourier transform of Φ, ˆ ) = ϕˆ j (ξ ) Φ(ξ j ∈J is treated as a column vector with values in 2 (J ), since Φ(ξ ˆ )22
(J )
= σV0 (ξ ) 1.
(2.8)
Also, define the diagonal matrix function of V0 as ⎡
1S1 (ξ ) ⎢ 0 ⎢ Ω(ξ ) = ⎢ ⎢ 0 ⎣ .. .
0 1S2 (ξ ) 0 .. .
0 0 1S3 (ξ ) .. .
⎤ ... ...⎥ ⎥ ⎥ ...⎥. ⎦ .. .
(2.9)
Suppose that Φ = (ϕj )j ∈J is a quasi-orthogonal vector for V0 . Since each ϕj is a quasiorthogonal generator of S(ϕj ) and S(ϕj ) ⊥ S(ϕj ) for j = j , we have that for a.e. ξ ∈ Rn ,
M. Bownik, Z. Rzeszotnik / Journal of Functional Analysis 256 (2009) 1065–1105
1075
ϕˆj (ξ + k)2 = 1S (ξ ), j
k∈Zn
ϕˆ j (ξ + k)ϕˆ j (ξ + k) = 0
for j = j .
k∈Zn
Hence, in short
ˆ + k)Φˆ ∗ (ξ + k) = Ω(ξ ) Φ(ξ
for a.e. ξ ∈ Rn .
(2.10)
k∈Zn
Definition 2.2. Suppose that an SI space V0 is refinable, that is V0 ⊂ D(V0 ). In this case a quasiorthogonal vector Φ = (ϕj )j ∈J for V0 is called a scaling vector for V0 . In addition, if (2.5) holds, then Φ is an exhausting scaling vector for V0 . The next result provides a characterization of elements of an SI space in terms of its quasiorthogonal vector. Proposition 2.1 is an immediate consequence of the corresponding wellknown result for PSI spaces; for example, see [27, Theorem 5.9]. Proposition 2.1. Suppose that an SI space V0 is decomposed as in (2.3) and each ϕj is a quasiorthogonal generator of S(ϕj ). Then f ∈ V0 if and only if fˆ(ξ ) =
rj (ξ )ϕˆj (ξ ),
(2.11)
j ∈N
where convergence is in L2 , each rj is Zn -periodic function in L2 (Sj ), and f 2 =
rj 2 .
j ∈J
Moreover, the sequence {rj }j ∈N of such functions is unique. Consequently, note that the series (2.11) converges a.e. after choosing a suitable subsequence. In particular, if the fibers of the SI space V0 are finitely dimensional, meaning that (2.12) holds, the convergence in (2.11) is also in the almost everywhere sense. This observation leads to a simple characterization of refinability of such SI spaces. Lemma 2.2. Suppose that V0 is an SI space such that dimV0 (ξ ) < ∞
for a.e. ξ.
(2.12)
Suppose that Φ is a quasi-orthogonal vector of V0 , and {Sj }j ∈J is given by (2.6) with J as in (2.7). Then the space V0 is refinable with respect to the dilation A if and only if ˆ ˆ ), Φ(Bξ ) = M(ξ )Φ(ξ
(2.13)
where B = AT and M is Zn -periodic matrix function with entries mi,j ∈ L2 (Sj ), i, j ∈ J . Moreover, if such an M exists, then it is unique.
1076
M. Bownik, Z. Rzeszotnik / Journal of Functional Analysis 256 (2009) 1065–1105
ˆ ) has finitely many non-zero entries and the Proof. Note that condition (2.12) guarantees that Φ(ξ matrix product in (2.13) is meaningful. By Proposition 2.1, (2.13) implies that each ϕj ∈ D(V0 ). Since D(V0 ) is SI, we must have V0 ⊂ D(V0 ) and V0 is refinable. Conversely, if V0 is refinable then the matrix M satisfying (2.13) is uniquely determined with the use of Proposition 2.1 for f = D −1 ϕj , j ∈ N. 2 A matrix function M satisfying (2.13) is often called a matrix mask function of Φ or a low-pass matrix mask. We are now ready to prove the main result of this section providing a description of a matrix mask corresponding to an exhausting scaling vector Φ for V0 . Theorem 2.2. Suppose that {Vj }j ∈Z is a GMRA such that (2.12) holds. Let Φ be an exhausting scaling vector for V0 and M be the matrix mask function as in Lemma 2.2. Then
M(ξ + d)M ∗ (ξ + d) = Ω(Bξ ),
(2.14)
d∈D
where D consists of representatives of distinct cosets of B −1 Zn /Zn . Moreover, the first column of M(ξ ) has zeros in every, but the first entry, near the origin in the sense that for a.e. ξ ∈ Rn , there exists N = N(ξ ) such that mi,1 B −j ξ = 0 for i 2, j > N.
(2.15)
Furthermore, the upper-left corner of M(ξ ) has absolute value “almost equal” to 1 near the origin, i.e., lim
k→∞
∞ m1,1 B −j ξ = 1 for a.e. ξ ∈ Rn .
(2.16)
j =k
Proof. The condition on the support of M implies that M(ξ )Ω(ξ ) = M(ξ ). Hence, by (2.10), Ω(Bξ ) =
k∈Zn
=
ˆ Φ(Bξ + k)Φˆ ∗ (Bξ + k) M ξ + B −1 k Φˆ ξ + B −1 k Φˆ ∗ ξ + B −1 k M ∗ ξ + B −1 k
k∈Zn
=
M(ξ + d)Ω(ξ + d)M ∗ (ξ + d) =
d∈D
M(ξ + d)M ∗ (ξ + d),
d∈D
which proves (2.14). To show (2.15) we will use the fact [15, Lemma 2.7] that 2 lim ϕˆ 1 B −j ξ = lim σV0 B −j ξ = lim σVj (ξ ) = 1 a.e. ξ ∈ Rn .
j →∞
j →∞
j →∞
(2.17)
Combining this with (2.8) yields that there exists N = N (ξ ) such that the first coordinate ˆ −j ξ ) is non-zero and all others are zero for all j N . By (2.13), of Φ(B
M. Bownik, Z. Rzeszotnik / Journal of Functional Analysis 256 (2009) 1065–1105
ϕˆi B −j +1 ξ = mi,1 B −j ξ ϕˆ1 B −j ξ
1077
for j N, i ∈ N.
Since ϕˆ i (B −j +1 ξ ) = 0 for i 2 and ϕˆ1 (B −j ξ ) = 0 for j > N , we have (2.15). To show (2.16) it suffices to observe that for every l > k N , l m1,1 B −j ξ . ϕˆ 1 B −k ξ = ϕˆ 1 B −l ξ j =k+1
By (2.17), l ∞ −k ϕˆ 1 B ξ = lim ϕˆ1 B −l ξ m1,1 B −j ξ = m1,1 B −j ξ , l→∞
j =k+1
which proves (2.16) by letting k → ∞.
j =k+1
2
The fact that M satisfies the condition (2.14), which is also called a generalized low-pass filter or generalized conjugate mirror filter, is due Baggett, Courter, and Merrill [3, Theorem 2.3]. This condition holds for all scaling vectors Φ (not necessarily exhausting) and it is an analogue of the usual quadrature-mirror equation (1.1). The additional assumption that Φ is an exhausting scaling vector implies that the first column of M must be of a special form (m1,1 (ξ ), 0, 0, . . .) with |m1,1 (ξ )| ≈ 1 for ξ near 0. Moreover, (2.14) implies that m1,1 (ξ + d)2 1S (Bξ ) 1 for a.e. ξ ∈ Tn . 1 d∈D
For these reasons m1,1 (ξ ) plays a role similar to that of a usual low-pass filter and it has a dominating effect on the entire matrix mask function M. These issues are further explored in Section 4, where the procedure for reconstructing a scaling vector from its matrix mask is presented. We should also emphasize that conditions (2.14)–(2.15) are only necessary and not sufficient for guaranteeing that M is a matrix mask of some scaling vector. This is a simple consequence of the usual MRA case, where (1.1) and (2.16) alone are not enough to produce the scaling function and some extra conditions, such as Lawton’s or Cohen’s conditions are needed [25,27]. Next, we will look at semi-orthogonal wavelets associated to a GMRA. In [16] we pointed out that one can always find a semi-orthogonal wavelet (possibly with infinite number of generators) associated to any GMRA. To be more precise, let us state the following Theorem 2.3. Suppose that {Vj }j ∈Z is a GMRA such that (2.12) holds. Then there exists a semiorthogonal wavelet (ψ j )j ∈J˜ ⊂ L2 (Rn ) such that W0 := V1 V0 =
j ∈J˜
S(ψj ),
supp dimS (ψj ) = S˜j := ξ ∈ Rn : dimW0 (ξ ) j . Here, J˜ is either {1, . . . , N} or N.
(2.18)
1078
M. Bownik, Z. Rzeszotnik / Journal of Functional Analysis 256 (2009) 1065–1105
˜ Conversely, suppose that we have a semi-orthogonal wavelet Ψ = (ψ j )j˜∈J˜ , where J˜ = {1, . . . , N} is finite, which is associated with a GMRA {Vj }j ∈Z ; that is (2.18) holds. Equivalently, a GMRA {Vj }j ∈Z associated to Ψ is given by
˜ Vj = span D i Tk ψ j : i < j, k ∈ Zn , j˜ ∈ J˜
for j ∈ Z.
Let Ψ be the column vector defined as Ψ = (ψ j )j ∈J˜ . By Proposition 2.1, there exists a matrix function H (ξ ) = (hi,j (ξ ))i∈J˜,j ∈J such that ˆ ), Ψˆ (Bξ ) = H (ξ )Φ(ξ and hi,j ∈ L2 (Sj ). Let Ω˜ be the diagonal matrix function corresponding to Ψ , i.e., ˜ ) = diag 1 ˜ (ξ ): j ∈ J˜ , Ω(ξ Sj
where S˜j = supp dimS (ψ j ) .
(2.19)
Then we have the following description of a matrix mask function H corresponding to a semiorthogonal wavelet Ψ , called a high-pass matrix mask or complementary conjugate mirror filter in [3]. Proposition 2.2. Suppose Ψ is a semi-orthogonal wavelet associated with a GMRA {Vj }j ∈Z . Let M and H be the low-pass and high-pass matrix mask functions as above. Then ˜ H (ξ + d)H ∗ (ξ + d) = Ω(Bξ ), (2.20)
d∈D
M(ξ + d)H ∗ (ξ + d) =
d∈D
H (ξ + d)M ∗ (ξ + d) = 0.
(2.21)
d∈D
Proof. The condition on the support of H implies that H (ξ )Ω(ξ ) = H (ξ ). Hence, by (2.10), ˜ Ω(Bξ )= Ψˆ (Bξ + k)Ψˆ ∗ (Bξ + k) k∈Zn
=
H ξ + B −1 k Φˆ ξ + B −1 k Φˆ ∗ ξ + B −1 k H ∗ ξ + B −1 k
k∈Zn
=
H (ξ + d)Ω(ξ + d)H ∗ (ξ + d) =
d∈D
H (ξ + d)H ∗ (ξ + d),
d∈D
which proves (2.20). Likewise, ˆ 0= Φ(Bξ + k)Ψˆ ∗ (Bξ + k) k∈Zn
=
M ξ + B −1 k Φˆ ξ + B −1 k Φˆ ∗ ξ + B −1 k H ∗ ξ + B −1 k
k∈Zn
=
M(ξ + d)Ω(ξ + d)H ∗ (ξ + d) =
d∈D
which proves (2.21).
d∈D
2
M(ξ + d)H ∗ (ξ + d),
M. Bownik, Z. Rzeszotnik / Journal of Functional Analysis 256 (2009) 1065–1105
1079
We can now summarize the procedure of generating low-pass M and high-pass H matrix masks as follows. Given a semi-orthogonal wavelet Ψ , we consider its associated GMRA {Vj }j ∈Z . Theorem 2.1 provides an exhausting scaling vector Φ for V0 . Then, Theorem 2.2 and Proposition 2.2 yield a low-pass matrix mask M and a high-pass matrix mask H satisfying (2.14)–(2.16) and (2.20)–(2.21), respectively. Since the decomposition of Theorem 2.1 is not unique, the above procedure yields a multitude of low-pass and high-pass matrix masks for a fixed semi-orthogonal wavelet. This fact makes the problem of reconstructing scaling vector and semi-orthogonal wavelet from their corresponding low-pass and high-pass matrix masks a highly non-trivial task. Fortunately, the exhausting property of Φ will counterbalance such inherent non-uniqueness. This will be explored in Sections 4 and 5, where the appropriate reconstruction procedure is provided, see Theorem 5.4. We end this section with two remarks involving orthogonality of columns versus rows for combined low-pass and high-pass matrix masks. Remark 2.1 is a slight refinement of [4, Theorem 2.5], where matrix mask functions M and H were assumed to have finite size. Remark 2.1. Let T be the combined matrix mask function of a scaling vector Φ and the associated semi-orthogonal wavelet Ψ given by
M(ξ + d1 ) T (ξ ) = H (ξ + d1 )
. . . M(ξ + dq ) , . . . H (ξ + dq )
where d1 , . . . , dq are representatives of distinct cosets of B −1 Zn /Zn . Note that for a.e. ξ ∈ Tn , T (ξ ) has only a finite number of non-zero entries. More precisely, there are only ˜ d∈D dimV0 (ξ +d) non-zero columns and dimV0 (Bξ )+dimW0 (Bξ ) non-zero rows. Let T (ξ ) be a finite sub-matrix of T (ξ ) consisting of only these columns and rows. The consistency equation of Baggett, see [6,7,15], says that dimV1 (Bξ ) =
dimV0 (ξ + d) = dimV0 (Bξ ) + dimW0 (Bξ )
for a.e. ξ ∈ Tn ,
(2.22)
d∈D
which implies that the matrix T˜ (ξ ) is square for a.e. ξ ∈ Tn . Moreover, Theorem 2.2 and Proposition 2.2 imply that the rows of the matrix T˜ (ξ ) are mutually orthogonal and normalized, that is T˜ (ξ ) is a unitary matrix. Consequently, the columns of T˜ (ξ ) are mutually orthogonal and normalized. Remark 2.2. One could consider the combined matrix mask function T (ξ ) corresponding to a more general situation when Ψ is a framelet obtained by a similar procedure. In this case, the rows of T (ξ ) do not have to be mutually orthogonal, anymore. In fact, the sub-matrix of non-zero rows and columns T˜ (ξ ) does not have to be square, since we can have many more generators in Ψ , and hence more rows in the matrix mask function H (ξ ). It turns out that unlike the situation of semi-orthogonal wavelets, where orthogonality of rows of T (ξ ) is necessary, orthogonality of columns plays a critical role for general framelets. This will be explored in the next section. 3. Unitary Extension Principle The Unitary Extension Principle, and its generalizations such as Oblique Extension Principle, are powerful tools in constructing tight framelets [8,9,21,29]. Since these techniques are used
1080
M. Bownik, Z. Rzeszotnik / Journal of Functional Analysis 256 (2009) 1065–1105
for constructing framelets with many desired properties such as smoothness, compact support, vanishing moments, etc., the Unitary Extension Principle is very often stated with some very mild and convenient regularity assumptions on a refinable function ϕ. Since the interest of our work lies mainly in L2 theory of framelets and wavelets, it is imperative to avoid any regularity assumptions, regardless of their mildness, limiting the applicability of our results. Furthermore, we are also forced to study situations where we are given a refinable vector consisting of infinite number of functions. Since these two problems were not adequately addressed yet, we provide an extension of Unitary Extension Principle, that is perfectly adapted to the L2 theory. We start with a definition of a refinable vector consisting of potentially infinitely many functions. Definition 3.1. We say that Φ = (ϕj )j ∈J ⊂ L2 (Rn ) is a refinable vector, where J = {1, . . . , N } or J = N, if ˆ ˆ ) Φ(Bξ ) = M(ξ )Φ(ξ
for a.e. ξ ∈ Rn ,
(3.1)
where B = AT and M = (mi,j )i,j ∈J is a matrix of Zn -periodic, measurable functions. In order to make sense of (3.1) in the case when J = N, we assume additionally that
1Rj (ξ ) < ∞
for a.e. ξ ∈ Tn ,
(3.2)
j ∈J
where Rj = supp dimS (ϕj ) . Note that we can always assume that supp mi,j ⊂ Rj , since the values of mi,j outside of Rj do not affect (3.1). Remark 3.1. Condition (3.2) is a technical matter that allows us to talk meaningfully about Eq. (3.1). However, if the matrix M(ξ ) has only finitely many non-zero entries for a.e. ξ ∈ Rn , then (3.1) makes sense right away. Moreover, in this simple case, (3.2) follows from (3.1). Theorem 3.1 is a generalization of the Unitary Extension Principle of Ron and Shen [29] to a situation when a refinable vector Φ is infinite. We note that the original result of Ron and Shen, in the case when Φ is finite, requires certain mild decay assumptions on Φ, see [21,29]. Nevertheless, Theorem 3.1 shows that these decay assumptions are unnecessary and they can be safely removed. Theorem 3.1. Suppose Φ = (ϕj )j ∈J is a refinable vector with a mask M such that j ∈J
ϕj 2 =
Φ(ξ ˆ )22
(J )
dξ < ∞
(3.3)
Rn
and lim Φˆ B −j ξ = 1 for a.e. ξ ∈ Rn .
j →∞
(3.4)
M. Bownik, Z. Rzeszotnik / Journal of Functional Analysis 256 (2009) 1065–1105
1081
Suppose also that Ψ = (ψ j )j ∈J˜ , where J˜ = {1, . . . , N} is finite, is given by ˆ ), Ψˆ (Bξ ) = H (ξ )Φ(ξ
(3.5)
where H = (hi,j )i∈J˜, j ∈J is Zn -periodic, measurable matrix function satisfying M ∗ (ξ )M(ξ + d) + H ∗ (ξ )H (ξ + d) = Ω(ξ )δ0,d
for a.e. ξ,
(3.6)
and for any d ∈ D. Then Ψ ⊂ L2 (Rn ) is a tight framelet. Proof. It suffices to verify that Ψ ⊂ L2 (Rn ) satisfies the characterization equations for tight framelets [11,24] Ψˆ B j ξ 2 = 1
for a.e. ξ,
(3.7)
j ∈Z ∞
Ψˆ ∗ B j ξ Ψˆ B j (ξ + q) = 0 for a.e. ξ, and all q ∈ Zn \ BZn .
(3.8)
j =0
Note that for any j ∈ Z, j 2 j 2 Ψˆ B ξ + Φˆ B ξ = Ψˆ ∗ B j ξ Ψˆ B j ξ + Φˆ ∗ B j ξ Φˆ B j ξ = Φˆ ∗ B j −1 ξ H ∗ B j −1 ξ H B j −1 ξ Φˆ B j −1 ξ + Φˆ ∗ B j −1 ξ M ∗ B j −1 ξ M B j −1 ξ Φˆ B j −1 ξ 2 = Φˆ ∗ B j −1 ξ Ω B j −1 ξ Φˆ B j −1 ξ = Φˆ B j −1 ξ ,
(3.9)
where in the last step we used that supp ϕˆi ⊂ Si . Therefore,
Ψˆ (ξ )2 dξ = |det A| − 1
Rn
Φ(ξ ˆ )2 dξ < ∞,
Rn
and the fact that Ψ ⊂ L2 (Rn ) is forced by (3.5) and (3.6). Next, we claim that lim Φˆ B j ξ = 0 for a.e. ξ ∈ Rn .
j →∞
(3.10)
ˆ j ξ ))j ∈Z , we could find δ > 0 such that Otherwise, due to monotonicity of the sequence (Φ(B E = ξ ∈ Rn : Φˆ B j ξ > δ for all j ∈ Z has a positive measure. Since BE = E, E must have infinite Lebesgue measure and consequently
1082
M. Bownik, Z. Rzeszotnik / Journal of Functional Analysis 256 (2009) 1065–1105
Φ(ξ ˆ )2 dξ
Rn
Φ(ξ ˆ )2 dξ = ∞,
E
which contradicts (3.3). Thus, (3.10) holds and together with (3.4), (3.9) it implies (3.7). Likewise for any j 1 and q ∈ Zn \ BZn , Ψˆ ∗ B j ξ Ψˆ B j (ξ + q) + Φˆ ∗ B j ξ Φˆ B j (ξ + q) = Φˆ ∗ B j −1 ξ H ∗ B j −1 ξ H B j −1 (ξ + q) Φˆ B j −1 (ξ + q) + Φˆ ∗ B j −1 ξ M ∗ B j −1 ξ M B j −1 (ξ + q) Φˆ B j −1 (ξ + q) = Φˆ ∗ B j −1 ξ Ω B j −1 ξ Φˆ B j −1 (ξ + q) = Φˆ ∗ B j −1 ξ Φˆ B j −1 (ξ + q) , where in the penultimate step we used Zn -periodicity of M and H . The same calculation for j = 0 together with the observation that H ∗ B −1 ξ H B −1 (ξ + q) + M ∗ B −1 ξ H B −1 (ξ + q) = 0 yields that ˆ + q) = 0. Ψˆ ∗ (ξ )Ψˆ (ξ + q) + Φˆ ∗ (ξ )Φ(ξ Combining these identities with lim Φˆ ∗ B j ξ Φˆ B j (ξ + q) lim Φˆ B j ξ Φˆ B j (ξ + q) = 0 for a.e. ξ ∈ Rn ,
j →∞
proves (3.8).
j →∞
2
Remark 3.2. The Unitary Extension Principle in the form of Theorem 3.1 yields not only a tight framelet Ψ but also two GMRAs. Indeed, every function Φ satisfying (3.1)–(3.4) generates a GMRA with a core space V0 generated by the integer shifts of the functions ϕj , j ∈ J . The other GMRA is the one, whose core space V˜0 is the space of negative dilates of the tight framelet Ψ , see Theorem 6.1. While we always have V˜0 ⊂ V0 , it is not clear whether we have the converse inclusion, i.e., whether these two GMRAs are the same. This open question was raised by Baggett, Jorgensen, Merrill, and Packer in [4]. In the next section we will use Theorem 3.1 to give a general construction procedure of tight framelets. To this end, it is convenient to prove the following fact about functions satisfying an inequality reminiscent of Baggett’s consistency equation. Lemma 3.1. Suppose that m : Rn → [0, ∞) is Zn -periodic, measurable function such that d∈D
m(ξ + d) m(Bξ ) + M
for a.e. ξ ∈ Tn ,
(3.11)
M. Bownik, Z. Rzeszotnik / Journal of Functional Analysis 256 (2009) 1065–1105
1083
for some M 0. Then m is integrable over its period and
m(ξ ) dξ M/ |det A| − 1 .
(3.12)
Tn
Heuristically, Lemma 3.1 seems to be trivial. Integrating (3.11) over Tn yields
|det A|
m(ξ ) dξ
Tn
m(ξ ) dξ + M. Tn
Unfortunately, we do not know a priori whether m is integrable and a much more complicated argument is necessary. Despite its simplicity, we could not find Lemma 3.1 in the existing literature and therefore we provide its proof. Proof. For an integer N 0, let RN (ξ ) be the “Riemann sum” of m of depth N given by 1 m RN (ξ ) = RN (ξ ) := |det A|N
0 ,...,N−1 ∈D
m ξ+
N −1
B −i i .
i=0
It is clear that RN (ξ ) is measurable and B −N Zn -periodic, since all the sums of the form N −1 −i −N Zn /Zn . i=0 B i , where 0 , . . . , N −1 ∈ D, are representatives of distinct cosets of B −1 n n Here, D consists as usual of representatives of distinct cosets of B Z /Z . By (3.11), |det A|RN (ξ ) RN −1 (Bξ ) + M
for any N 1.
Hence, by iteration, |det A|N − 1 . m B N ξ = R0 B N ξ |det A|N RN (ξ ) − M |det A| − 1 Take any C > M/(|det A| − 1) and let δ = C − M/(|det A| − 1). Then ξ ∈ Rn : RN (ξ ) C ⊂ ξ ∈ Rn : m B N ξ δ|det A|N .
(ξ ) = R m (ξ ), where m is a truncation of m at height K given by For a fixed K > 0, let RN N (ξ ) is B −N Zn -periodic, measurable and bounded m (ξ ) = min(m(ξ ), K). It is clear that each RN by K. Furthermore,
ξ ∈ Tn : R (ξ ) C ξ ∈ Tn : m B N ξ δ|det A|N N = ξ ∈ Tn : m(ξ ) δ|det A|N → 0 as N → ∞.
(3.13)
’s are bounded, there exists a subsequence {N } such that {R (ξ )} converges pointwise Since RN i Ni a.e. to some f (ξ ). Since f (ξ ) must be periodic with respect to every lattice B −N Zn ⊂ B −N +1 Zn ,
1084
M. Bownik, Z. Rzeszotnik / Journal of Functional Analysis 256 (2009) 1065–1105
−N Zn is dense, f (ξ ) must be a constant function, f (ξ ) = C . By (3.13), 0 C and ∞ 0 0 N =0 B M/(|det A| − 1). Moreover, C0 =
f (ξ ) dξ = lim
Tn
i→∞ Tn
RN (ξ ) dξ i
=
m (ξ ) dξ = Tn
min m(ξ ), K dξ.
Tn
Hence, letting K → ∞ allows to obtain (3.12) by the monotone convergence theorem.
2
4. Construction of tight framelets The main goal of this section is to provide a general reconstruction procedure for scaling vectors and semi-orthogonal wavelets from their corresponding low-pass and high-pass matrix masks. Hence, the goal is to reverse the flow of Section 2 by starting with a low-pass matrix mask function M satisfying conditions (2.14)–(2.16). Theorem 2.2 shows that this is a perfectly reasonable assumption, since any matrix mask function of an exhausting scaling vector must satisfy them. The key ingredient of our approach is a rather complicated procedure yielding a refinable vector Φ corresponding to the mask M, see Theorem 4.2. This, combined with the Unitary Extension Principle and appropriate conditions on a high-pass mask H , yields a tight framelet Ψ (see Theorem 4.3). In general, we can only expect that Ψ is a tight framelet. However, if we know a priori that our low-pass and high-pass matrix masks correspond to some semi-orthogonal wavelet Ψ , then we prove that our procedure is flexible enough to recover Ψ itself. In particular, every orthogonal wavelet Ψ can be obtained by our recovery procedure via low-pass and high-pass matrix masks manipulations. This will be shown in the following section. To start the construction of tight framelets we must recall a characterization of the dimension function associated to a GMRA proved in [15]. Theorem 4.1. Suppose {Vj }j ∈Z is a GMRA. Then the dimension function of the core space V0 , m(ξ ) = dimV0 (ξ ), satisfies the following conditions: n n (D1) m : R → N ∪ {0, ∞} is a measurable Z n-periodic function; (D2) d∈D m(ξ + d) m(Bξ ) for a.e. ξ ∈ R ; n (D3) k∈Zn 1Δ (ξ + k) m(ξ ) for a.e. ξ ∈ R , where
Δ = ξ ∈ Rn : m B −j ξ 1 for j ∈ N ∪ {0} ; (D4) lim infj →∞ m(B −j ξ ) 1 for a.e. ξ ∈ Rn . Conversely, if m satisfies (D1)–(D4), then there exists a GMRA {Vj }j ∈Z such that dimV0 (ξ ) = m(ξ ). Our construction is based on a function m that satisfies conditions (D1)–(D4) of the above theorem. However, to ensure the existence of a tight framelet we shall add two more assumptions. Namely, (D5) m ∈ L1 (Tn );
M. Bownik, Z. Rzeszotnik / Journal of Functional Analysis 256 (2009) 1065–1105
1085
and (D6) m is finite a.e. and there is N ∈ N such that for a.e. ξ ∈ Tn we have
m(ξ + d) m(Bξ ) + N.
d∈D
To motivate these final conditions we include the following Proposition 4.1. Let Ψ = {ψ 1 , . . . , ψ N } ⊂ L2 (Rn ) be a tight framelet and V0 its space of negative dilates. If {Vj }j ∈Z forms a GMRA, then m = dimV0 satisfies (D5) and (D6). Proof. First we conduct the standard orthogonalization procedure. That is, for j ∈ Z we define Wj := Vj +1 Vj and observe that
W j = L2 R n .
(4.1)
j ∈Z
Clearly, W0 is a shift-invariant space generated by {ψ − PV0 ψ}ψ∈Ψ , where PV0 is the orthogonal projection on V0 . By Theorem 2.1 we can find quasi-orthogonal generators Φ = {ϕ1 , ϕ2 , . . .} for W0 as in Theorem 2.1. Since our tight framelet Ψ consists of N of functions, Φ has N of non-zero elements as well. Condition 4.1 assures that Φ is a semi-orthogonal wavelet. This allows us to calculate the spectral function of V0 in terms of Φ. Indeed, a formula from [15] gives us σV0 (ξ ) =
ϕˆ B j ξ 2 . ϕ∈Φ j >0
After integrating the above formula and using the fact that ϕ 1 for all ϕ ∈ Φ, we obtain that σV0 = ϕ2 / |det A| − 1 N/ |det A| − 1 . Rn
ϕ∈Φ
Since Rn σV0 = Tn m, this shows that (D5) is satisfied. In order to justify (D6) we use basic properties of the dimension function that are given in [15]. Since V1 = V0 ⊕ W0 , we get that d∈D m(B −1 ξ + d) = m(ξ ) + dimW0 (ξ ). But W0 has N generators, therefore dimW0 N and (D6) follows. 2 Remark 4.1. In order to construct our GMRA only conditions (D1)–(D5) are going to be used. We shall also show that (D6) is necessary and sufficient to guarantee the existence of a “high pass filter” that will be used to define our framelet. We also want to point out, that (D5) follows from (D6), as was shown in Lemma 3.1. In short, our construction is guided by the standard procedure. We are going to consider a matrix mask function M that satisfies conditions (2.14)–(2.16). Then we will construct a corresponding refinable vector Φ and use Unitary Extension Principle to obtain an associated tight framelet Ψ .
1086
M. Bownik, Z. Rzeszotnik / Journal of Functional Analysis 256 (2009) 1065–1105
We start equipped with a function m that satisfies conditions (D1)–(D5). Then, we define the sets Sj , for j ∈ N, by a formula analogous to (2.6), that is Sj = ξ ∈ Rn : m(ξ ) j .
(4.2)
Let J = {i ∈ N: |Sj | > 0}. Hence, J = {1, 2, . . . , L} or J = N. The sets Sj , j ∈ J , are used to define the diagonal matrix function Ω as in (2.9). This allows us to consider a matrix mask function M with periodic entries mi,j ∈ L2 (Sj ), i, j ∈ J , that satisfies conditions (2.14)–(2.16). In order to find the corresponding refinable vector we shall use the ideas of [27]. First, we will modify M to assure that the product of the dilates of M is convergent. Then, we shall use multipliers to recover the solution to the original problem. To proceed in this direction we need the following basic lemma about multipliers. Lemma 4.1. Let μ be a unimodular measurable function on Rn (that is, μ : Rn → S 1 = {z ∈ C: |z| = 1}). If B is an expansive matrix, then there exists a unimodular measurable function ν such that ν(Bξ )ν(ξ ) = μ(ξ )
for a.e. ξ ∈ Rn .
(4.3)
Proof. It is well known, that for any expansive matrix B there is an ellipsoid E such that B(E), see e.g. [12, Lemma 2.2]. It follows, that for W = B(E) \ E we have E ⊂ j (W ) = Rn . Therefore, it is enough to define a unimodular function ν on W and then B j ∈Z extend it to Rn using Eq. (4.3). 2 The mentioned modification of the matrix mask function M is very simple. The most important entry of M is m1,1 . Let μ be a phase of m1,1 . That is, μ is a unimodular measurable function such that μ(ξ )m1,1 (ξ ) = m1,1 (ξ )
for a.e. ξ ∈ Rn .
(4.4)
A multiplier associated to the mask M is any unimodular measurable function ν satisfying (4.3) and (4.4). The modified mask is M := μM ¯
(4.5)
and the corresponding refinable vector is given by
ˆ
Φ (ξ ) := lim
N →∞
N
−j M B ξ e,
(4.6)
j =1
where e is the vector (1, 0, 0, . . .). Finally, the refinable vector Φ corresponding to the mask M will be given by Φˆ := ν Φˆ . In order to establish that Φ is refinable we use the following series of lemmas. Lemma 4.2. The vector function Φˆ in (4.6) is well defined.
M. Bownik, Z. Rzeszotnik / Journal of Functional Analysis 256 (2009) 1065–1105
1087
Proof. We need that the limit in (4.6) does exist for a.e. ξ ∈ Rn . We want to point out, to show, (B −j ξ ) may not exist, we are only interested in the first column of this M that although ∞ j =1 matrix. Later, we will prove that under some natural assumptions this product matrix exists and all of its columns but the first must be zero, see (5.1). By (2.15), for a.e. ξ ∈ Rn we can find N (ξ ) such that the first column of M (B −j ξ ) has only one non-zero entry (the first one) for all j > N(ξ ). Therefore, for N > N (ξ ) we have
N
N (ξ )
−j −j M B ξ e= M B ξ
N
j =1
j =1
m1,1 B −j ξ e.
j =N (ξ )+1
Thus, Φˆ (ξ ) := lim pN (ξ )v(ξ ), N →∞
N (ξ ) −j where v(ξ ) = [ j =1 M (B −j ξ )]e and pN (ξ ) = N j =N (ξ )+1 |m1,1 (B ξ )|. Since condition (2.14) guarantees that |m1,1 | 1, we see that {pN (ξ )} is a bounded decreasing sequence and our claim follows. 2 Lemma 4.3. The vector function Φˆ in (4.6) satisfies Φˆ (Bξ ) = M (ξ )Φˆ (ξ ). Proof. From (D5) it follows that our function m is finite a.e. Therefore, condition (2.14) implies that for a.e. ξ ∈ Tn the matrix M (ξ ) has only finitely many non-zero terms. This allows us to see that
Φˆ (Bξ ) = lim
N →∞
M (ξ )
N
N M B −j ξ e = M (ξ ) lim M B −j ξ e N →∞
j =1
ˆ
= M (ξ )Φ (ξ ).
j =1
2
Lemma 4.4. The vector function Φˆ in (4.6) satisfies limN →∞ Φˆ (B −N ξ ) = 1, for a.e. ξ ∈ Rn . Proof. By (2.15), for a.e. ξ ∈ Rn we can find N (ξ ) such that for all N > N (ξ ) we have Φˆ B −N ξ =
∞ −j m1,1 B ξ e. j =N +1
Therefore, our claim follows from (2.16).
2
Lemma 4.5. The vector function Φˆ in (4.6) satisfies
Rn
Φˆ (ξ )2 dξ < ∞.
Proof. For N ∈ N and a.e. ξ ∈ Rn let us consider the following matrix
MN (ξ ) =
N j =1
M B −j ξ 1B N (Tn ) (ξ ).
(4.7)
1088
M. Bownik, Z. Rzeszotnik / Journal of Functional Analysis 256 (2009) 1065–1105
We claim that for all N ∈ N and a.e. ξ ∈ Rn ,
MN (ξ + k)MN∗ (ξ + k) = Ω(ξ ).
(4.8)
k∈Zn
Indeed, for N = 1 we use (2.14) to obtain
M1 (ξ + k)M1∗ (ξ + k) =
k∈Zn
k∈Zn
=
M B −1 (ξ + k) M ∗ B −1 (ξ + k) 1B(Tn ) (ξ + k) M B −1 ξ + d M ∗ B −1 ξ + d = Ω(ξ ).
d∈D
To proceed with the induction we observe that MN +1 (ξ ) = M B −1 ξ MN B −1 ξ , for N ∈ N and a.e. ξ ∈ Rn . Therefore, k∈Zn
=
MN +1 (ξ + k)MN∗ +1 (ξ + k)
M B −1 (ξ + k) MN B −1 (ξ + k) MN∗ B −1 (ξ + k) M ∗ B −1 (ξ + k)
k∈Zn
=
M B −1 ξ + d MN B −1 ξ + d + l MN∗ B −1 ξ + d + l M ∗ B −1 ξ + d
d∈D l∈Zn
=
d∈D
=
M B −1 ξ + d Ω B −1 ξ + d M ∗ B −1 ξ + d M B −1 ξ + d M ∗ B −1 ξ + d = Ω(ξ ),
d∈D
what proves our claim (4.8). In order to use it, we observe that for all k ∈ Zn and a.e. ξ ∈ Rn , MN (ξ + k)e2 MN (ξ + k)2 MN (ξ + k)2 = tr MN (ξ + k)M ∗ (ξ + k) , N HS
(4.9)
where · H S denotes the Hilbert–Schmidt operator norm. The above estimate and (4.8) give us MN (ξ + k)e2 tr Ω(ξ ) = m(ξ ).
(4.10)
k∈Zn
Since limN →∞ (MN (ξ )e) = Φˆ (ξ ), we can use Fatou’s lemma to conclude that Φˆ (ξ + k)2 m(ξ ). k∈Zn
(4.11)
M. Bownik, Z. Rzeszotnik / Journal of Functional Analysis 256 (2009) 1065–1105
By (D5) the function m(ξ ) is integrable over Tn , thus 2 Φˆ (ξ ) dξ = Φˆ (ξ + k)2 dξ m(ξ ) dξ < ∞. Rn
Tn
k∈Zn
2
1089
(4.12)
Tn
Remark 4.2. We have that Φˆ = (ϕˆj )j ∈J . Since lemma shows that Φˆ ⊂ L2 (Rn ).
Rn
Φˆ (ξ )2 dξ =
j ∈J
ϕˆ j 2 , the above
To reverse the procedure given in (4.5) we use Lemma 4.1 to find a multiplier ν associated to M and define our refinable vector Φ by setting Φˆ := ν Φˆ .
(4.13)
The following result assures that such Φ has all of the properties that we need. Theorem 4.2. The vector function Φˆ given in (4.13) satisfies conditions (3.1)–(3.4). Proof. By Lemma 4.3 and (4.3) together with (4.5) we get that ˆ ) = M(ξ )Φ(ξ ˆ ), ˆ ¯ )M(ξ )¯ν (ξ )Φ(ξ Φ(Bξ ) = ν(Bξ )Φˆ (Bξ ) = ν(Bξ )M (ξ )Φˆ (ξ ) = ν(Bξ )μ(ξ therefore, (3.1) holds. As we mentioned before, (2.14) implies that the mask matrix M(ξ ) has only finitely many non-zero terms. Thus, condition (3.2) is satisfied, by Remark 3.1. ˆ ) = Φˆ (ξ ) a.e., properties (3.3) and (3.4) follow immediately from Lemmas 4.4 Since Φ(ξ and 4.5. 2 As an immediate consequence of Theorem 4.2 and the Unitary Extension Principle from the previous section, we obtain our framelet construction result. The precursor of Theorem 4.3 is a result of Baggett, Jorgensen, Merrill, and Packer [4, Theorem 3.4] where a low-pass matrix mask M is assumed to be finite and Lipschitz continuous near 0 (instead of satisfying our assumptions (2.15) and (2.16)). Theorem 4.3. Let m be a function that satisfies (D1)–(D5) with the sets Sj given by (4.2) and the corresponding matrix function Ω defined in (2.9). Let M = (mi,j )i,j ∈J be a matrix mask function with periodic entries mi,j ∈ L2 (Sj ), that satisfies (2.14)–(2.16). Then there is a refinable vector Φ such that ˆ ˆ ) Φ(Bξ ) = M(ξ )Φ(ξ
for a.e. ξ ∈ Rn .
(4.14)
Moreover, if a matrix function H = (hi,j )i∈J˜,j ∈J with J˜ = {1, . . . , N} finite and with periodic entries hi,j ∈ L2 (Sj ) satisfies M ∗ (ξ )M(ξ + d) + H ∗ (ξ )H (ξ + d) = Ω(ξ )δ0,d
for any d ∈ D and a.e. ξ ∈ Rn ,
(4.15)
then Ψ = (ψ j )j ∈J˜ given by ˆ ) Ψˆ (Bξ ) = H (ξ )Φ(ξ is a tight framelet for L2 (Rn ).
(4.16)
1090
M. Bownik, Z. Rzeszotnik / Journal of Functional Analysis 256 (2009) 1065–1105
Proof. The existence of a refinable vector Φ satisfying (4.14) is a consequence of Theorem 4.2. In addition, if a matrix function H satisfies (4.15), then by Theorem 3.1, Ψ given by (4.16) is a tight framelet for L2 (Rn ). 2 The next theorem gives the necessary and sufficient conditions for the existence of the highpass matrix mask H that satisfies (4.15). Some of the implications in Theorem 4.4 are already known. Indeed, (i) ⇒ (ii) is due to Baggett, Courter, and Merrill [3, Theorem 2.5], whereas (ii) ⇒ (iii) is due to Baggett, Jorgensen, Merrill, and Packer [4, Theorem 2.5] in the case of bounded m. We shall give the full proof of Theorem 4.4 for the sake of completeness. Theorem 4.4. Let m be any function satisfying (D1)–(D5). Let m ˜ : Rn → N∪{0} be a measurable n Z -periodic function satisfying
m(ξ + d) = m(Bξ ) + m(Bξ ˜ )
for a.e. ξ ∈ Rn .
(4.17)
d∈D
Define the sets Sj = ξ ∈ Rn : m(ξ ) j ,
S˜j = ξ ∈ Rn : m(ξ ˜ )j
and the corresponding matrix functions Ω and Ω˜ by (2.9). Assume that M = (mi,j )i,j ∈J is a matrix mask function with periodic entries mi,j ∈ L2 (Sj ) that satisfies (2.14). Then the following are equivalent: (i) m satisfies (D6). (ii) There exists a matrix function H = (hi,j )i∈J˜,j ∈J with J˜ finite and with periodic entries hi,j ∈ L2 (Sj ) satisfying
˜ H (ξ + d)H ∗ (ξ + d) = Ω(Bξ )
a.e. ξ ∈ Rn ,
(4.18)
d∈D
H (ξ + d)M ∗ (ξ + d) = 0 a.e. ξ ∈ Rn .
(4.19)
d∈D
(iii) There exists a matrix function H = (hi,j )i∈J˜,j ∈J with J˜ finite and with periodic entries hi,j ∈ L2 (Sj ) satisfying (4.15). Moreover, if a matrix function H satisfies (4.18) and (4.19) then it also satisfies (4.15). However, the converse is in general false. Remark 4.3. Note that m ˜ as in (4.17) always exists, since m(ξ ˜ )=
m B −1 ξ + d − m(ξ )
d∈D
is clearly Zn -periodic and non-negative by (D2). Moreover, let E be any measurable subset of Tn such that {E + d: d ∈ D} is a partition of Tn (modulo null sets). Then, it is easy to see using the
M. Bownik, Z. Rzeszotnik / Journal of Functional Analysis 256 (2009) 1065–1105
1091
periodicity of M and H that if (4.18) and (4.19) hold for a.e. ξ ∈ E, then they must hold for a.e. ξ ∈ Rn . Proof. First, suppose that a matrix function H = (hi,j )i∈J˜,j ∈J has periodic entries hi,j ∈ L2 (Sj ), satisfies (4.15) and the index set J˜ has N elements. Consider the combined matrix function M(ξ + d1 ) . . . M(ξ + dq ) . (4.20) T (ξ ) = H (ξ + d1 ) . . . H (ξ + dq ) By the support conditions and (4.15), the matrix T (ξ ) has precisely d∈D m(ξ + d) non-zero columns. Condition (4.15) says that these non-zero columns form an orthonormal system. On the other hand, (2.14) and the fact that J˜ has N elements imply that the matrix T (ξ ) has at most m(Bξ ) + N non-zero rows. Clearly, any collection of orthonormal vectors must be smaller than the dimension of the space where they live in. Consequently, (D6) must hold. This shows (iii) ⇒ (i). Conversely, suppose that (D6) holds and consider a matrix function T (ξ ) = [ M(ξ + d1 )
...
M(ξ + dq ) ] .
(4.21)
It is convenient to fix ξ ∈ E, where E is the same as in Remark 4.3. As before, by the support conditions, the matrix T (ξ ) has at most d∈D m(ξ + d) non-zero columns. On the other hand, system by (2.14). Therefore, the matrix T (ξ ) has m(Bξ ) non-zero rows forming an orthonormal for a fixed ξ , we have a finite submatrix T (ξ ) with c = d∈D m(ξ + d) columns and r = m(Bξ ) orthonormal rows. Now, it suffices to find an extension of this submatrix to a unitary c × c matrix. Since m(Bξ ˜ ) = c − r N by (D6), at most N extra rows must be added. Define [H (ξ + d1 ) . . . H (ξ + dq )] to be a matrix with rows indexed by J˜ = {1, . . . , N } such that the first m(Bξ ˜ ) = c − r rows of [H (ξ + d1 ) . . . H (ξ + dq )] are formed by inserting the extra rows from a finite submatrix T (ξ ) interspersed by zero columns, which were previously removed from the matrix T (ξ ). The remaining rows (if any) of [H (ξ + d1 ) . . . H (ξ + dq )] are defined to be zero. It is not hard to see that the these extra rows can be chosen in such a way that the resulting matrix function [H (ξ + d1 ) . . . H (ξ + dq )] has measurable entries as a function of ξ ∈ E. As a result, the combined matrix function T (ξ ) has the same number of non-zero ˜ ) columns equal to d∈D m(ξ + d) as the number of non-zero rows equal to m(Bξ ) + m(Bξ by (4.17). Furthermore, since the non-zero rows of T (ξ ) form an orthonormal sequence, the finite submatrix consisting of non-zero columns and rows must be unitary. Consequently, the constructed matrix H satisfies (4.18) and (4.19) for a.e. ξ ∈ E. By Remark 4.3 this shows that (i) ⇒ (ii). Next, if H is any matrix function as in (ii), then (2.14), (4.18), and (4.19) imply that the non-zero rows of the combined matrix function T (ξ ) form an orthonormal sequence. By (4.17) a finite submatrix consisting of non-zero columns and rows of T (ξ ) has the same number of non-zero columns as the number of non-zero rows and hence must be unitary. Since the rows of this finite submatrix are orthonormal, so are the columns, which implies that (4.15) holds. Therefore, (4.18) and (4.19) always imply (4.15). The converse implication is obviously false in general, since the combined matrix T (ξ ) may have a larger number of non-zero rows than non-zero columns and as a consequence the orthonormality of columns does not translate into orthonormality of rows. This proves (ii) ⇒ (iii) and completes the proof of Theorem 4.4. 2
1092
M. Bownik, Z. Rzeszotnik / Journal of Functional Analysis 256 (2009) 1065–1105
Remark 4.4. The origin of Eq. (4.17) is hidden in the consistency equation (2.22). Once we take m = dimV0 and m ˜ = dimW0 , the connection becomes clear. As a consequence of Theorems 4.3 and 4.4 we can deduce that any low-pass matrix mask function M satisfying (2.14)–(2.16) associated with the dimension function m satisfying (D1)– (D6) corresponds to some tight framelet Ψ via (4.14) and (4.16). To achieve this we must choose a high-pass mask matrix H such that the corresponding combined matrix function (4.20) has orthogonal columns, that is, (4.15) holds. Naturally, if we count on obtaining a semi-orthogonal wavelet Ψ , then the high-pass matrix mask H must satisfy more restrictive conditions (4.18)– (4.19) resulting in row orthogonality of the combined matrix (4.20), see Theorem 5.2. We would like to point out, that a refinable vector Φ in Theorem 4.3 is not unique. Indeed, the explicit formula for our choice of Φ is
ˆ ) = ν(ξ ) lim Φ(ξ
N →∞
N N −j −j −N −j μ¯ B ξ M B ξ e = lim ν B ξ M B ξ e, (4.22) N →∞
j =1
j =1
where e = (1, 0, 0, . . .), μ is the phase of m1,1 and ν is an arbitrary multiplier, i.e., a measurable unimodular function such that ν(Bξ )ν(ξ ) = μ(ξ ). Recall that Lemma 4.1 guarantees the existence of such multipliers. Equivalently, we can define a multiplier associated to the mask M = (mi,j ) as any function ν satisfying ν(Bξ )ν(ξ )m1,1, (ξ ) = m1,1 (ξ )
for a.e. ξ ∈ Rn .
(4.23)
Indeed, if ν satisfies (4.23), then μ(ξ ) = ν(Bξ )ν(ξ ) is a phase of m1,1 and ν is its corresponding multiplier. Note that a phase μ satisfying (4.4) might not be unique if m1,1 does not have a full support. Since there are many possibilities for multipliers ν satisfying (4.23) we obtain a lot of choices for Φ. Moreover, these different choices generate distinct GMRA’s. In general, if P is a matrix ˆ Nevreplaced by P Φ. function such that P (Bξ )−1 M(ξ )P (ξ ) = M(ξ ) a.e., then our Φˆ can be −j ξ ) that is ertheless, we can loosely think that Φ is given by the standard product ∞ M(B j =1 applied to the vector e. Even better, it turns out that if the product is convergent, then this standard choice of Φ is valid. Proposition 4.2. If M is as in Theorem 4.3 and the product a.e. ξ ∈ Rn , then Φ from Theorem 4.3 can be taken as
ˆ ) := Φ(ξ
∞
∞
−j M B ξ e.
j =1 M(B
−j ξ )
is convergent for
(4.24)
j =1
∞ −j −j Proof. Since ∞ j =1 M(B ξ ) is convergent a.e., we have that j =1 m1,1 (B ξ ) is convergent a.e. as well. In particular, lim
N →∞
∞ j =N
m1,1 B −j ξ = 1
a.e.
(4.25)
M. Bownik, Z. Rzeszotnik / Journal of Functional Analysis 256 (2009) 1065–1105
1093
By (4.3) this is equivalent to exists for a.e. ξ ∈ Rn . lim ν B −N ξ
N →∞
(4.26)
Let Φ be a refinable vector as in Theorem 4.3 and ν be any function as in Lemma 4.1. Define another function ν satisfying the conclusions of Lemma 4.1 by where α(ξ ) = lim ν B −N ξ .
ν (ξ ) = α(ξ )ν(ξ ),
N →∞
Indeed, ν (Bξ )ν (ξ ) = μ(ξ ),
lim ν B −N ξ = 1
N →∞
for a.e. ξ ∈ Rn .
Therefore, by (4.22),
Φˆ (ξ ) =
∞
−j M B ξ e
j =1
is a refinable vector function obtained by the procedure of Theorem 4.3 with the multiplier ν . This proves (4.24). 2 5. Reconstruction of wavelets The main goal of this section is to prove that every orthogonal wavelet can be reconstructed from its carefully chosen low-pass and high-pass matrix masks by the procedure described in the previous section. In order to achieve this, we will explore in more depth some subtle properties of the refinable vector Φ from Theorem 4.3. Recall that by starting from a dimension function m and an appropriate matrix mask M, we obtained a refinable vector Φ in Section 4 and, therefore, also the associated GMRA. The standard issue in this type of constructions is the problem of “vanishing mass.” In short, it may ˆ )2 dξ < n m(ξ ) dξ . In particular, Φ need not to be a scaling vector happen that Rn Φ(ξ R since quasi-orthogonality may fail. Also, the GMRA that results from such procedure can have a strictly smaller dimension function (of its core space) than the original one that was used to start the construction. This feature was already observed in the classical MRA case on R with dilation by 2. The familiar Cohen’s condition is one of the ways to assure that “no mass gets lost.” It is crucial if one hopes to obtain a wavelet. However, as pointed in [27] in the dyadic scalar case, even if “some of the mass does vanish” one can still construct corresponding tight framelet. In the general case, the problem gains on complexity. Below, we give a simple necessary condition that is needed for preserving the “mass.” In order to achieve this preservation, one has to impose that the matrix M given in (4.5) satisfies ⎡ ⎤ ∗ 0 0 ... N ⎢ ⎥ M B −j ξ = ⎣ ∗ 0 0 . . . ⎦ . lim (5.1) N →∞ .. .. .. . . j =1 . . . .
1094
M. Bownik, Z. Rzeszotnik / Journal of Functional Analysis 256 (2009) 1065–1105
More precisely, for a.e. ξ ∈ Rn the limit does exist and is equal to a matrix whose only non-zero entries are in the first column. This also sheds a new light on (4.6), where we defined Φˆ as the first column of such a product. We will show the necessity of the above condition in the following Proposition 5.1. Let m and Φ be as in Theorem 4.3. If
Φ(ξ ˆ )2 dξ =
Rn
m(ξ ) dξ,
(5.2)
Tn
then (5.1) holds. Proof. Let us show (5.1) by using calculations given in the proof of Lemma 4.5. First, we observe that (5.1) is equivalent to saying that limN →∞ MN (ξ )ei → 0 for a.e. ξ ∈ Rn , and every vector ei , i 2, of the standard basis. Also, we can use Φ instead of Φ in our considerations. By (4.12), the assumption (5.2) forces the inequality (4.11) to become an equality. Tracing back this fact through (4.10) and (4.9) we see that, eventually, one must have 2 lim tr MN (ξ )MN∗ (ξ ) − MN (ξ )e = 0,
N →∞
for a.e. ξ ∈ Rn . However, since tr[CD] = tr[DC], the above becomes lim
N →∞
MN (ξ )ei 2 = 0. i2
This shows the necessity of (5.1) and concludes the proof.
2
In the next result we present the full connection between the “mass preservation” and the properties of our refinable vector Φ. Theorem 5.1. Let m and Φ be as in Theorem 4.3. Then, Φ is a scaling vector that generates a GMRA with the same dimension function as m if and only if (5.2) holds. Proof. Again, we can consider Φ instead of Φ. If Φ is a scaling vector then the dimension function of the corresponding GMRA is equal to k∈Zn Φˆ (ξ + k)2 . Clearly, the assumption that this dimension function is the same as m implies that (5.2) holds. On the other hand, assume that (5.2) is satisfied. From (4.12) and (4.11) it follows that Φˆ (ξ + k)2 = m(ξ ),
(5.3)
k∈Zn
for a.e. ξ ∈ Tn . By Proposition 5.1, applying Fatou’s lemma to (4.8) yields k∈Zn
Φˆ (ξ + k)Φˆ ∗ (ξ + k) Ω(ξ ),
(5.4)
M. Bownik, Z. Rzeszotnik / Journal of Functional Analysis 256 (2009) 1065–1105
1095
in the operator sense, for a.e. ξ ∈ Tn . However, condition (5.3) simply says that tr
Φˆ (ξ + k)Φˆ ∗ (ξ + k) = tr Ω(ξ )
a.e.
k∈Zn
Therefore, we must have an equality in (5.4). This shows that Φ is a scaling vector. Moreover, (5.3) assures that the GMRA generated by Φ has the dimension function equal to m. 2 As a consequence of Theorem 5.1 we show that the procedure of Theorem 4.3 can result in a semi-orthogonal wavelet (with the expected size of generators) only if the combined matrix (4.20) has orthogonal rows. As a corollary, we conclude that the necessary condition for constructing orthogonal wavelets is that the high-pass filter H satisfies (4.18) and (4.19) with the diagonal matrix function Ω˜ constantly equal to the identity matrix. Theorem 5.2. Suppose that m is a function that satisfies (D1)–(D6) and m ˜ is given by (4.17). Suppose that M and H are low-pass and high-pass matrix masks as in Theorem 4.3. Let Ψ be the corresponding tight framelet. Then Rn
Ψˆ (ξ )2 dξ
m(ξ ˜ ) dξ.
(5.5)
Tn
Moreover, if Ψ is a semi-orthogonal wavelet such that the equality holds in (5.5), then the highpass filter H necessarily satisfies (4.18) and (4.19) with the diagonal matrix function Ω˜ given by (2.19). Proof. Let Φ be the refinable vector constructed in Theorem 4.3. Recall that the proof of Theorem 3.1 yields
Ψˆ (ξ )2 dξ = |det A| − 1
Rn
Φ(ξ ˆ )2 dξ < ∞.
(5.6)
Rn
On the other hand, since (D6) holds, condition (4.17) implies that m ˜ is bounded. Therefore, we can integrate (4.17) over Tn to obtain Tn
m(ξ ˜ ) dξ = |det A| − 1
m(ξ ) dξ < ∞.
(5.7)
Tn
Combining (4.12), (5.6), and (5.7) yields (5.5). In addition, suppose that Ψ is a semi-orthogonal wavelet such that the equality holds in (5.5). By Theorem 5.1, Φ is a scaling vector generating a GMRA {Vj }j ∈Z with the dimension function dimV0 = m. On the other hand, Ψ also generates a GMRA {Vj }j ∈Z given by ˜ Vj = span D i Tk ψ j : i < j, k ∈ Zn , j˜ ∈ J˜ for j ∈ Z.
1096
M. Bownik, Z. Rzeszotnik / Journal of Functional Analysis 256 (2009) 1065–1105
By [15, Corollary 4.3] the dimension function dimV0 of the core space V0 can be computed explicitly and equals the wavelet dimension function DΨ . Consequently, Tn
1 dimV0 (ξ ) dξ = |det A| − 1
Rn
Ψˆ (ξ )2 =
Φ(ξ ˆ )2 =
Rn
dimV0 (ξ ) dξ < ∞.
(5.8)
Tn
On the other hand, by (4.16) Ψ ⊂ V1 and hence V0 ⊂ V0 . Thus, dimV0 (ξ ) dimV0 (ξ ) for a.e. ξ and (5.8) implies that dimV0 (ξ ) = dimV0 (ξ ) < ∞
for a.e. ξ,
and hence V0 = V0 . Therefore, the semi-orthogonal wavelet ψ is associated with the GMRA {Vj }j ∈Z . By Proposition 2.2 the high-pass matrix mask H satisfies claimed properties. 2 As an immediate consequence of Theorem 5.2 we have Corollary 5.1. In addition to the assumptions of Theorem 5.2, assume that the equality holds in (D6). Hence, there is N ∈ N such that
m(ξ + d) = m(Bξ ) + N
for a.e. ξ ∈ Rn .
d∈D
Let Ψ be the tight framelet as in Theorem 4.3. If Ψ = {ψ 1 , . . . , ψ N } is an orthonormal wavelet, then the high-pass filter H necessarily satisfies (4.18) and (4.19) with the diagonal matrix func˜ ) ≡ IdN ×N . tion Ω(ξ Proof. Our hypotheses imply that m(ξ ˜ ) ≡ N a.e. ξ ∈ Tn . If Ψ = (ψ j )N j =1 is an orthonormal wavelet, then Rn
Ψˆ (ξ )2 dξ = N =
m(ξ ˜ ) dξ. Tn
By Theorem 5.2 and (2.19) H satisfies claimed properties since S˜j = supp dimS (ψ j ) = Rn for j = 1, . . . , N . 2 Finally, we will show that every exhausting scaling vector of a GMRA (that has an integrable dimension function of the core space) can be obtained by the procedure of Theorem 4.3 with an appropriate choice of a multiplier ν. Theorem 5.3. Suppose {Vj }j ∈Z is a GMRA such that dimV0 ∈ L1 (Tn ). Let M be the low-pass matrix mask function of an exhausting scaling vector Φ for the core space V0 . (i) Any multiplier ν associated to M corresponds by Theorem 4.3 to some scaling vector Φ with the same mask M (but generating not necessarily the same space V0 ).
M. Bownik, Z. Rzeszotnik / Journal of Functional Analysis 256 (2009) 1065–1105
1097
(ii) There exists a multiplier ν associated to M such that the scaling vector Φ is recovered via Theorem 4.3 by the product formula (4.22), that is
N −N −j ˆ ) = lim ν B ξ M B ξ e Φ(ξ N →∞
for a.e. ξ.
(5.9)
j =1
Proof. Note that the dimension function m = dimV0 of the core space V0 satisfies the assumptions (D1)–(D5). Moreover, by Theorem 2.2, the matrix mask function M of Φ satisfies conditions (2.14)–(2.16). Therefore, a fixed multiplier ν produces a refinable vector Φ by the procedure of Theorem 4.3. We are going to prove (i) and (ii) simultaneously. Observe that both Φ and Φ satisfy the same refinable equation, which takes the form Φˆ B −N ξ = m1,1 B −N −1 ξ Φˆ B −N −1 ξ , Φˆ B −N ξ = m1,1 B −N −1 ξ Φˆ B −N −1 ξ ,
(5.10)
for sufficiently large N > N(ξ ) dependent on the choice of ξ ∈ Rn . This is a simple consequence of the special form of the matrix mask M near the origin. Let ϕˆ1 and ϕˆ1 be the first entries of Φˆ and Φˆ , respectively. By (5.10), the sequence {ϕˆ1 (B −N ξ )/ϕˆ 1 (B −N ξ )}N >N (ξ ) must be constant whenever it is well defined. Let α(ξ ) be the constant value of this sequence. It is clear that α(Bξ ) = α(ξ ). Moreover, the fact that Φˆ and Φˆ have zeros in all but the first entry near the origin, (2.17), and Lemma 4.4, imply that |α(ξ )| = 1. Define another multiplier ν corresponding to the same matrix mask function M by ν(ξ ) = α(ξ )ν (ξ ). Finally, let Φ be the refinable vector obtained by the procedure of Theorem 4.3 with the multiplier ν. By (5.10) and the previously mentioned special form of Φ and Φ near the origin we have ε > 0 such that ˆ ) Φˆ (ξ ) = α(ξ )Φ(ξ
for a.e. |ξ | < ε.
On the other hand, by the product formula (4.22) we have that Φˆ (ξ ) = α(ξ )Φˆ (ξ )
for a.e. ξ ∈ Rn .
(5.11)
Since all functions Φ, Φ , and Φ satisfy the refinable equation with respect to the same matrix mask function M, we must necessarily have that Φ = Φ . This completes the proof of part (ii). To deduce part (i) observe that Φ = Φ together with (5.11) yields ˆ ) Φˆ (ξ ) = α(ξ )Φ(ξ
for a.e. ξ ∈ Rn ,
for some α such that |α(ξ )| = 1 and α(Bξ ) = α(ξ ) a.e. Therefore, we conclude that the refinable vector Φ must be necessarily a scaling vector. Obviously, there is no guarantee that the SI
1098
M. Bownik, Z. Rzeszotnik / Journal of Functional Analysis 256 (2009) 1065–1105
space V0 generated by Φ coincides with V0 unless function α is Zn -periodic. This completes the proof of part (i) of Theorem 5.3. 2 As an immediate consequence of Theorem 5.3 we have that every semi-orthogonal wavelet Ψ can be recovered by the procedure of Theorem 4.3. This is our main wavelet reconstruction result. Theorem 5.4. Suppose Ψ is a semi-orthogonal wavelet. Let {Vj }j ∈Z be the GMRA generated by Ψ , and let Φ be an exhausting scaling vector for V0 . Let M and H be the matrix mask functions of Φ and Ψ , respectively. Then there exists a multiplier ν associated to M such that the scaling vector Φ is recovered by the product formula (5.9) and Ψ is recovered by (4.16). Proof. Theorem 5.3 guarantees that we can recover Φ. To get back Ψ we use Proposition 2.2. It implies that the matrix mask functions M and H satisfy (2.20) and (2.21). As we pointed out in Theorem 4.4, these two conditions force H to satisfy (4.15). Thus, we can use such H to obtain Ψ via (4.16). 2 6. Examples and comments We shall construct examples of tight framelets and wavelets using the procedure that was described in Section 4. We remind the reader that the whole process starts from choosing a dimension function. In order to find a specific dimension function one can construct a wavelet set and calculate the associated dimension function. In this way all dimension functions can be obtained by the result of Speegle and the authors [17]. It is customary to test GMRA constructions on the original “non-MRA object,” that is, the 1 2 2 1 16 Journé wavelet ψ given by ψˆ = 1W , where W = [− 16 7 , −2] ∪ [− 2 , − 7 ] ∪ [ 7 , 2 ] ∪ [2, 7 ]. The associated dimension function, called here the Journé dimension function, is ⎧ 2 for ξ ∈ [− 17 , 17 ], ⎪ ⎨ m(ξ ) = 1 for ξ ∈ [− 12 , − 37 ] ∪ [− 27 , − 17 ] ∪ [ 17 , 27 ] ∪ [ 37 , 12 ], ⎪ ⎩ 0 for ξ ∈ [− 37 , − 27 ] ∪ [ 27 , 37 ].
(6.1)
Since m is Z-periodic we list only its values on the torus T identified with [−1/2, 1/2). The same convention shall be used for other periodic objects that appear in this section albeit with the identification T = [−3/7, 4/7). Clearly, m satisfies conditions (D1)–(D6) that are stated in Theorem 4.1 and thereafter. The corresponding sets Sj of (4.2) are S1 and S2 , where S1 is the periodization of [− 12 , − 37 ] ∪ [− 27 , 27 ] ∪ [ 37 , 12 ] and S2 is the periodization of [− 17 , 17 ]. The diagonal matrix function Ω of (2.9) is, therefore,
Ω(ξ ) =
1[− 1 ,− 3 ]∪[− 2 , 2 ]∪[ 3 , 1 ] (ξ )
0
0
1[− 1 , 1 ] (ξ )
2
7
7 7
7 2
,
7 7
for ξ ∈ T. All of this provides the ground for constructing tight framelets and wavelets by choosing appropriate low-pass and high-pass matrix mask functions. We should mention that there are already several interesting constructions of wavelets and tight framelets with the Journé dimension function in the literature. For example, Baggett, Courter, and Merrill [3] constructed an orthonormal wavelet ψ with the dimension function (6.1)
M. Bownik, Z. Rzeszotnik / Journal of Functional Analysis 256 (2009) 1065–1105
1099
such that ψˆ is C ∞ on an arbitrarily large interval. Later, Baggett, Jorgensen, Merrill, and Packer [4,5] gave an impressive construction of a tight framelet ψ with a prescribed smoothness ψ ∈ C r for r 0, and a global smoothness in the frequency ψˆ ∈ C ∞ . Our goal is to present another construction resulting in a large class of non-MSF wavelets sharing the same Journé dimension function. Unlike the previous constructions in [3–5], we shall insist that the first row of a low-pass matrix mask M remains unchanged compared with that of the Journé wavelet. Example 6.1. Consider the low-pass matrix mask function M given by
1F1 (ξ ) M(ξ ) = m1 (ξ )
0 , m2 (ξ )
(6.2)
where F1 is the periodization of [− 27 , − 14 ] ∪ [− 17 , 17 ] ∪ [ 14 , 27 ] and m1 , m2 are Z-periodic measurable functions. Condition (2.14) imposes certain restrictions on possible functions m1 and m2 . That is, we must stipulate that for ξ ∈ T, 1F1 (ξ )m1 (ξ ) + 1F1 (ξ + 1/2)m1 (ξ + 1/2) = 0, 2 m1 (ξ ) + m2 (ξ )2 + m1 (ξ + 1/2)2 + m2 (ξ + 1/2)2 = 1 1 1 3 4 (ξ ). [− , ]∪[ , ] 14 14
7 7
(6.3) (6.4)
Here and subsequently, we are using the identification T = [−3/7, 4/7). Since F1 and F1 + 1/2 are disjoint, m1 must vanish on F1 by (6.3) and consequently m1 must be supported on the 1 1 , 14 ] + 1/2 = [ 37 , 47 ] by (6.4). Consequently, we must have for periodization of the interval [− 14 ξ ∈ T, m1 (ξ ) = v(ξ )1[ 3 , 4 ] (ξ ),
m2 (ξ ) = v(ξ )1[− 1 , 1 ] (ξ ),
7 7
14 14
(6.5)
where v is an arbitrary Z-periodic measurable function satisfying v(ξ )2 + v(ξ + 1/2)2 = 1 1 1 (ξ ), [− , ]
for a.e. ξ ∈ (−1/4, 1/4).
14 14
(6.6)
It is easy to verify that as long as conditions (6.5) and (6.6) hold, the matrix mask function M satisfies (2.14)–(2.16). Thus, we can apply Theorem 4.3. The corresponding refinable vector Φˆ is the first column of the infinite product ∞ j =1
M 2
−j
ξ =
1E1 (ξ ) ∗
∞
0
j =1 m2 (2
−j ξ )
,
(6.7)
4 1 2 2 1 4 j where E1 = ∞ j =1 2 (F1 ) = [− 7 , − 2 ] ∪ [− 7 , 7 ] ∪ [ 2 , 7 ]. The lower left entry of the above matrix is represented by a more complicated infinite product which can be computed for some specific choices of the function v satisfying (6.6), see the next example. The corresponding framelet can be found by choosing a high-pass matrix mask H satisfying (3.6). In addition, if we hope on obtaining a wavelet we should apply Theorem 5.2. Since in our case m ˜ = 1 a.e., we see that H has to be 1 × 2 matrix-valued and must satisfy condi˜ ) = [1]. Then, a direct but tedious tions (4.18), (4.19), with the diagonal matrix function Ω(ξ
1100
M. Bownik, Z. Rzeszotnik / Journal of Functional Analysis 256 (2009) 1065–1105
calculation shows, that modulo a unimodular Z-periodic function, the high-pass matrix mask H is given by H (ξ ) = 1[− 1 ,− 1 ]∪[ 1 , 1 ] (ξ ) 4
7
7 4
1[− 1 ,− 1 ]∪[ 1 , 1 ] (ξ ) 7
14
+ e2πiξ v(ξ + 1/2) 1[ 3 , 4 ] (ξ ) 7 7
14 7
1[− 1 , 1 ] (ξ )
for ξ ∈ T.
14 14
ˆ ). Then, by Theorem 4.3, ψ is a tight framelet for any choice Define ψ ∈ L2 (R) by ψˆ = H (ξ )Φ(ξ of Z-periodic function v satisfying (6.6). Note that the choice of v = 1[− 1 , 1 ] , corresponds to the matrix mask 14 14
M(ξ ) =
1F1 (ξ ) 0
0 , 1F2 (ξ )
1 1 where F2 is the Z-periodization of [− 14 , 14 ]. A direct calculation shows that we obtain a tight ˆ framelet ψ given by |ψ| = 1[− 8 ,−1]∪[− 1 ,− 2 ]∪[ 2 , 1 ]∪[1, 8 ] . Thus, ψ is not a wavelet. This fact 7 2 7 7 2 7 can also be deduced as a consequence of Proposition 5.1. That is, the procedure of constructing refinable vector from low-pass matrix mask can only result in a scaling (with the same vector −j dimension function) if all but the first column of the product matrix ∞ M(2 ξ ) are zeros. j =1 Indeed, if ψ were a wavelet, then by (5.6), condition (5.2) would hold as well. Therefore, the mentioned proposition would imply that (5.1) must be satisfied. However, in our case (5.1) fails. Thus, ψ is a tight framelet, but not a wavelet. On the other hand, if we choose v = 1[ 3 , 4 ] , then 7 7
1F1 (ξ ) M(ξ ) = 1F3 (ξ )
0 , 0
where F3 is the periodization of [ 37 , 47 ]. A direct calculation shows that we obtain the usual ˆ = 1W , see also [3, Journé wavelet ψ modified by a negligible unimodular phase factor, i.e., |ψ| Example 4.3]. In the next example we construct a large class of non-MSF non-MRA wavelets by an appropriate choice of functions v satisfying (6.6). Naturally, each wavelet in this class must share the dimension function of the Journé wavelet given by (6.1). Example 6.2. Let w be an arbitrary Z-periodic measurable function satisfying w(ξ )2 + w(ξ + 1/2)2 = 1 1 1 [− ,− ]∪[ 1 , 1 ] (ξ ) 14
28
28 14
for a.e. ξ ∈ (−1/4, 1/4).
(6.8)
Then, v given for ξ ∈ T by v(ξ ) = w(ξ ) + 1[ 13 , 15 ] (ξ ) satisfies (6.6). Define Z-periodic functions 28 28
m1 (ξ ) = w(ξ )1[ 3 , 13 ]∪[ 15 , 4 ] (ξ ) + 1[ 13 , 15 ] (ξ ), 7 28
28 7
28 28
m2 (ξ ) = w(ξ )1[− 1 ,− 1 ]∪[ 1 , 1 ] (ξ ). 14
28
28 14
(6.9)
Finally, let M be given by (6.2). The same argument as in Example 6.1 shows that M satisfies (2.14)–(2.16) and hence, it is a low-pass matrix mask function. In fact, we obtain a proper
M. Bownik, Z. Rzeszotnik / Journal of Functional Analysis 256 (2009) 1065–1105
1101
subclass of low-pass matrix masks considered in the previous example. We can also choose a high-pass matrix mask H by emulating Example 6.1. That is, we define H (ξ ) = 1[− 1 ,− 1 ]∪[ 1 , 1 ] (ξ ) 4
7
1[− 1 ,− 1 ]∪[− 1 , 1 ]∪[ 1 , 1 ] (ξ )
7 4
7
14
28 28
+ e2πiξ w(ξ + 1/2) 1[ 3 , 13 ]∪[ 15 , 4 ] (ξ ) 7 28
28 7
14 7
1[− 1 ,− 1 ]∪[ 1 , 1 ] (ξ ) 14
28
for ξ ∈ T.
28 14
The advantage of our choice of low-pass and high-pass matrix masks is twofold. First, the corresponding refinable vector Φ can be easily computed. Second, it can be shown that Φ is a scaling vector and the resulting tight framelet ψ is a wavelet. Indeed, note that
1F1 (ξ/2)1F1 (ξ/4) M(ξ/2)M(ξ/4) = m1 (ξ/2)1F1 (ξ/4) + m2 (ξ/2)m1 (ξ/4)
0 , 0
(6.10)
since m2 (ξ )m2 (ξ/2) = 0 for a.e. ξ ∈ R. Moreover, 1E1 (ξ/4) 0 , M 2−j ξ = ∗ 0 j =3 ∞
(6.11)
where E1 = [− 47 , − 12 ] ∪ [− 27 , 27 ] ∪ [ 12 , 47 ]. Consequently, ˆ )= Φ(ξ
1E1 (ξ ) ϕˆ 1 (ξ ) = . ϕˆ 2 (ξ ) (m1 (ξ/2)1F1 (ξ/4) + m2 (ξ/2)m1 (ξ/4))1E1 (ξ/4)
Finally, a direct but tedious calculation shows that ϕˆ2 (ξ ) = 1[− 15 ,−1]∪[1, 15 ] + w(ξ/2)1[− 15 ,− 29 ]∪[− 8 ,− 15 ]∪[ 15 , 8 ]∪[ 29 , 15 ] . 14
14
7
14
7
14
14 7
14
(6.12)
7
To see that Φ is indeed a scaling vector it suffices to observe that ϕˆ2 (ξ + k)2 = 1 1 1 (ξ ) [− , ] 7 7
k∈Z
for a.e. ξ ∈ T,
and check that (2.10) holds. Finally, one can compute the formula for the corresponding wavelet ψ = ψw , ˆ ) = 1 29 ψ(ξ [− ,−2]∪[− 1 ,− 2 ]∪[ 2 , 1 ]∪[2, 29 ] (ξ ) + w(ξ/4)1[− 30 ,− 29 ]∪[− 16 ,− 15 ]∪[ 15 , 16 ]∪[ 29 , 30 ] (ξ ) 14
+e
πiξ
2
7
7 2
14
7
7
7
w(ξ/2 + 1/2)1[− 15 ,− 29 ]∪[− 8 ,− 15 ]∪[ 15 , 8 ]∪[ 29 , 15 ] (ξ ). 7
14
7
14
14 7
14
7
7
7
7
7
7
(6.13)
Figs. 1 and 2 show graphs of a typical scaling vector Φˆ and the corresponding wavelet ψˆ . We should also add that once the formula (6.13) is established, one can deduce that a wavelet ψw can be also obtained using interpolation pairs of wavelet sets [19,32]. Observe that the family of wavelets Wnik = ψw : w satisfies (6.8)
1102
M. Bownik, Z. Rzeszotnik / Journal of Functional Analysis 256 (2009) 1065–1105
Fig. 1. The graphs of ϕˆ 1 (solid line) and a typical ϕˆ 2 (dashed line).
Fig. 2. The graph of ψˆ (dashed line corresponds to the part that contains the phase factor eπ iξ ).
is pathwise connected in L2 (R). Indeed, given two Z-periodic measurable functions w0 and w1 both satisfying (6.8), it is not difficult to construct a family {wt }t∈[0,1] of functions satisfying (6.8) such that ws (ξ ) → wt (ξ )
for a.e. ξ ∈ T as s → t.
Then, by (6.13), we see that ψˆ ws (ξ ) → ψˆ wt (ξ )
for a.e. ξ ∈ R as s → t.
Since ψwt = 1 for all t ∈ [0, 1], the map t → ψwt is the required continuous path. Example 6.2 shows that a large class of wavelets can be constructed by the procedure of Theorem 4.3. Moreover, Theorem 5.4 shows that technically every imaginable wavelet can be obtained in that way. However, it is an open problem whether the same is true for all tight framelets. Two serious difficulties arise when one wants to design a constructive method for obtaining all tight framelets on Rn . The first problem is that it is not known if all such framelets are associated to a GMRA. This is often referred to as the “Baggett’s problem” [13]. Baggett observed that a
M. Bownik, Z. Rzeszotnik / Journal of Functional Analysis 256 (2009) 1065–1105
1103
tight framelet Ψ generates a GMRA if and only if its space of negative dilates V satisfies $
D j (V ) = {0}.
(6.14)
j ∈Z
We have treated this problem with detail in [16]. An earlier result of the second author [31] assures that if the spectral function of V is integrable, then the above condition is satisfied. It turns out that we can use Lemma 3.1 to improve on this result in the setting of the space of negative dilates. Theorem 6.1 can be also deduced as a consequence of a general result on the intersection of dilates of SI spaces due to the first author [14]. Theorem 6.1. Let Ψ be a tight framelet on Rn with its space of negative dilates V . If the set {ξ ∈ Rn : dimV (ξ ) < ∞} has a positive (Lebesgue) measure, then (6.14) holds and Ψ generates a GMRA. Proof. As in the proof of Proposition 4.1, let W = D(V ) V and observe that since Ψ consists of a finite number of functions, W has a finite number of generators. That is, we have dimW N for some N ∈ N. The equation D(V ) = V ⊕ W implies that
m B −1 ξ + d = m(ξ ) + dimW (ξ ) m(ξ ) + N,
(6.15)
d∈D
where m = dimV . Thus, condition (3.11) of Lemma 3.1 is satisfied for such m. However, to apply Lemma 3.1 we need to show that m is finite a.e. This can be done using a simple ergodic argument. Indeed, since the matrix B : Rn → Rn preserves the lattice Zn , it induces a measure preserving endomorphism B˜ : Tn → Tn . Moreover, B˜ is ergodic by [33, Corollary 1.10.1] because B is expansive. Define the set E = ξ ∈ Tn : m(ξ ) < ∞ . The condition (6.15) implies that B˜ −1 E ⊂ E. Since B˜ is measure preserving we must have ˜ we have either |E| = 0 or |E| = 1. B˜ −1 E = E (modulo null sets). Finally, by the ergodicity of B, Combining this with our hypothesis |E| > 0, proves that m(ξ ) < ∞ for a.e. ξ ∈ Rn . Since all the assumptions of Lemma 3.1 are satisfied for our m, we get that m ∈ L1 (Tn ). Equivalently, we have σV ∈ L1 (Rn ). As we mentioned before, the latter implies that (6.14) holds by the result of the second author [31]. Therefore, Ψ generates a GMRA. 2 If we consider an easier scenario and want to construct all tight framelets associated to a GMRA, we encounter the second difficulty. It is an open problem whether every tight framelet Ψ generating some GMRA {Vj }j ∈Z can be obtained by the procedure of Theorem 4.3 via the Unitary Extension Principle. In other words, is it possible to find appropriate matrix mask functions M and H resulting by the procedure of Theorem 4.3 in a tight framelet Ψ ? This problem remains open even for tight framelets Ψ associated to an MRA, i.e., when dimV0 ≡ 1.
1104
M. Bownik, Z. Rzeszotnik / Journal of Functional Analysis 256 (2009) 1065–1105
References [1] P. Auscher, Solution of two problems on wavelets, J. Geom. Anal. 5 (1995) 181–236. [2] L. Baggett, An abstract interpretation of the wavelet dimension function using group representations, J. Funct. Anal. 173 (2000) 1–20. [3] L. Baggett, J. Courter, K. Merrill, The construction of wavelets from generalized conjugate mirror filters in L2 (Rn ), Appl. Comput. Harmon. Anal. 13 (2002) 201–223. [4] L.W. Baggett, P.E.T. Jorgensen, K.D. Merrill, J.A. Packer, Construction of Parseval wavelets from redundant filter systems, J. Math. Phys. 46 (2005) 083502. [5] L.W. Baggett, P.E.T. Jorgensen, K.D. Merrill, J.A. Packer, A non-MRA C r frame wavelet with rapid decay, Acta Appl. Math. 89 (2005) 251–270. [6] L. Baggett, H. Medina, K. Merrill, Generalized multi-resolution analyses and a construction procedure for all wavelet sets in Rn , J. Fourier Anal. Appl. 5 (1999) 563–573. [7] L. Baggett, K. Merrill, Abstract harmonic analysis and wavelets in Rn , in: The Functional and Harmonic Analysis of Wavelets and Frames, San Antonio, TX, 1999, in: Contemp. Math., vol. 247, Amer. Math. Soc., Providence, RI, 1999, pp. 17–27. [8] J. Benedetto, S. Li, The theory of multiresolution analysis frames and applications to filter banks, Appl. Comput. Harmon. Anal. 5 (1998) 389–427. [9] J. Benedetto, O. Treiber, Wavelet frames: Multiresolution analysis and extension principles, in: Wavelet Transforms and Time-Frequency Signal Analysis, in: Appl. Numer. Harmon. Anal., Birkhäuser Boston, Boston, MA, 2001, pp. 3–36. [10] M. Bownik, The structure of shift-invariant subspaces of L2 (Rn ), J. Funct. Anal. 177 (2000) 282–309. [11] M. Bownik, A characterization of affine dual frames in L2 (Rn ), Appl. Comput. Harmon. Anal. 8 (2000) 203–221. [12] M. Bownik, Anisotropic Hardy spaces and wavelets, Mem. Amer. Math. Soc. 164 (2003), No. 781, 122 pp. [13] M. Bownik, Baggett’s problem for frame wavelets, in: Representations, Wavelets and Frames: A Celebration of the Mathematical Work of Lawrence Baggett, Birkhäuser, 2008, pp. 153–173. [14] M. Bownik, Intersection of dilates of shift-invariant spaces, Proc. Amer. Math. Soc. 137 (2009) 563–572. [15] M. Bownik, Z. Rzeszotnik, The spectral function of shift-invariant spaces, Michigan Math. J. 51 (2003) 387–414. [16] M. Bownik, Z. Rzeszotnik, On the existence of multiresolution analysis for framelets, Math. Ann. 332 (2005) 705– 720. [17] M. Bownik, Z. Rzeszotnik, D. Speegle, A characterization of dimension functions of wavelets, Appl. Comput. Harmon. Anal. 10 (2001) 71–92. [18] L. Brandolini, G. Garrigós, Z. Rzeszotnik, G. Weiss, The behaviour at the origin of a class of band-limited wavelets, in: The Functional and Harmonic Analysis of Wavelets and Frames, San Antonio, TX, 1999, in: Contemp. Math., vol. 247, Amer. Math. Soc., Providence, RI, 1999, pp. 75–91. [19] X. Dai, D. Larson, Wandering vectors for unitary systems and orthogonal wavelets, Mem. Amer. Math. Soc. 134 (1998), No. 640, viii+68 pp. [20] I. Daubechies, Ten Lectures on Wavelets, SIAM, Philadelphia, PA, 1992. [21] I. Daubechies, B. Han, A. Ron, Z. Shen, Framelets: MRA-based constructions of wavelet frames, Appl. Comput. Harmon. Anal. 14 (2003) 1–46. [22] C. de Boor, R.A. DeVore, A. Ron, The structure of finitely generated shift-invariant spaces in L2 (Rd ), J. Funct. Anal. 119 (1994) 37–78. [23] C. de Boor, R.A. DeVore, A. Ron, Approximation orders of FSI spaces in L2 (Rd ), Constr. Approx. 14 (1998) 631–652. [24] M. Frazier, G. Garrigós, K. Wang, G. Weiss, A characterization of functions that generate wavelet and related expansion, J. Fourier Anal. Appl. 3 (1997) 883–906. [25] E. Hernández, G. Weiss, A First Course on Wavelets, CRC Press, Boca Raton, FL, 1996. [26] W. Lawton, Tight frames of compactly supported affine wavelets, J. Math. Phys. 31 (1990) 1898–1901. [27] M. Paluszy´nski, H. Šiki´c, G. Weiss, S. Xiao, Generalized low pass filters and MRA frame wavelets, J. Geom. Anal. 11 (2001) 311–342. [28] M. Paluszy´nski, H. Šiki´c, G. Weiss, S. Xiao, Tight frame wavelets, their dimension functions, MRA tight frame wavelets and connectivity properties, Adv. Comput. Math. 18 (2003) 297–327. [29] A. Ron, Z. Shen, Affine systems in L2 (Rd ): The analysis of the analysis operator, J. Funct. Anal. 148 (1997) 408–447. [30] A. Ron, Z. Shen, The wavelet dimension function is the trace function of a shift-invariant system, Proc. Amer. Math. Soc. 131 (2003) 1385–1398.
M. Bownik, Z. Rzeszotnik / Journal of Functional Analysis 256 (2009) 1065–1105
1105
[31] Z. Rzeszotnik, Calderón’s condition and wavelets, Collect. Math. 52 (2001) 181–191. [32] Z. Rzeszotnik, D. Speegle, On wavelets interpolated from a pair of wavelet sets, Proc. Amer. Math. Soc. 130 (2002) 2921–2930. [33] P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, New York, 1982. [34] The Wutam Consortium, Basic properties of wavelets, J. Fourier Anal. Appl. 4 (1998) 575–594.
Journal of Functional Analysis 256 (2009) 1106–1117 www.elsevier.com/locate/jfa
On the separability problem for isometric shifts on C(X) ✩ Jesús Araujo Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria, E-39071 Santander, Spain Received 21 April 2008; accepted 18 November 2008 Available online 28 November 2008 Communicated by N. Kalton
Abstract We provide examples of nonseparable spaces X for which C(X) admits an isometric shift, which solves in the negative a problem proposed by Gutek et al. [A. Gutek, D. Hart, J. Jamison, M. Rajagopalan, Shift operators on Banach spaces, J. Funct. Anal. 101 (1991) 97–119]. © 2008 Elsevier Inc. All rights reserved. Keywords: Isometric shift; Weighted composition operator; Spaces of continuous functions
1. Introduction The usual concept of shift operator in the Hilbert space 2 has been introduced in the more general context of Banach spaces in the following way (see [4,15]): Given a Banach space E over K (the field of real or complex numbers), a linear operator T : E → E is said to be an isometric shift if (1) T is an isometry, (2) the codimension of T (E) in E is 1, n (3) ∞ n=1 T (E) = {0}.
✩
Research partially supported by the Spanish Ministry of Science and Education (Grant number MTM2006-14786). E-mail address: [email protected].
0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.11.013
J. Araujo / Journal of Functional Analysis 256 (2009) 1106–1117
1107
One of the main settings where isometric shifts have been studied is E = C(X), that is, the Banach space of all K-valued continuous functions defined on a compact and Hausdorff space X, equipped with its usual supremum norm. In this setting, major breakthroughs were made in [9] and [11]. On the one hand, in [9], Gutek, Hart, Jamison, and Rajagopalan studied in depth these operators. In particular, using the well-known Holszty´nski’s theorem [14], they classified them into two types, called type I and type II. On the other hand, in [11], Haydon showed a general method for providing isometric shifts of type II, as well as concrete examples. However, a very basic question has remained open since the publication in 1991 of the seminal paper [9]: If C(X) admits an isometric shift, must X be separable? This question is only meaningful for type I isometric shifts since it was already proved in [9, Corollary 2.2] that type II isometric shifts yield the separability of X. Let us recall the definitions. If T : C(X) → C(X) is an isometric shift, then there exist a closed subset Y ⊂ X, a continuous and surjective map φ : Y → X, and a function a ∈ C(Y ), |a| ≡ 1, such that (Tf )(x) = a(x) · f (φ(x)) for all x ∈ Y and all f ∈ C(X). T is said to be of type I if Y can be taken to be equal to X \ {p}, where p ∈ X is an isolated point, and is said to be of type II if Y can be taken equal to X. Moreover, if T is of type I, then the map φ : X \ {p} → X is indeed a homeomorphism. Not much is known about the possibility of finding a nonseparable space X such that C(X) admits an isometric shift since the problem was proposed. Interesting results in this direction say that such an X must have the countable chain condition (see [10, Theorem 1.4] or [20, Lemma 5.6]). In [10, Theorem 1.9], it is even proved that C0 (X \ clX {p, φ −1 (p), . . . , φ −n (p), . . .}) must have cardinality at most equal to c, that is, the cardinality of R (where, as usual, clX A denotes the closure of A in X and C0 (Z) is the space of K-valued continuous functions on Z vanishing at infinity). From this fact, we can easily deduce that if C(X) admits an isometric shift, then there exists a set S of cardinality at most c that is dense in X. To see it, we write C0 (X \ clX {p, φ −1 (p), . . . , φ −n (p), . . .}) = {fα : α ∈ I ⊂ R}. For each α ∈ I such that fα = 0, we pick a point xα ∈ X such that fα (xα ) = 0. Obviously, given any (nonempty) open set U ⊂ X \ clX {p, φ −1 (p), . . . , φ −n (p), . . .}, there exists fα = 0 whose support is contained in U . This implies that the set S consisting of the union of all points xα and {p, φ −1 (p), . . . , φ −n (p), . . .} is dense in X. In this paper, we will give an answer in the negative to the separability question: There are indeed examples of isometric shifts on C(X), with X not separable, and even having 2c infinite components (see Sections 3 and 4). The latter example can be connected with the question addressed in [15], where it was conjectured that the space X cannot have an infinite connected component (the only examples which appeared so far in the literature for type I isometric shifts, both of spaces containing exactly one infinite component, can be found in [9, Corollary 2.1] and [2]; for the case of type II isometric shifts in the complex setting, see [11]). Related to this, one of the main results in [9] states that C(X) does not admit any isometric shifts, whenever X has a countably infinite number of components, all of whom are infinite. Some other papers have recently studied questions related to isometric shifts (also defined on other spaces of functions). Among them, we will mention for instance [1,3,5,6,16,17,19–22] (see also references therein).
1108
J. Araujo / Journal of Functional Analysis 256 (2009) 1106–1117
2. Preliminaries and notation The unit circle in C will be denoted by T. L∞ (T) will be the space of all Lebesgue-measurable essentially bounded complex-valued functions on T, and M will be its maximal ideal space. m will denote the Lebesgue measure on T. It is well known that if ρ is an irrational number, then the rotation map [ρ] : T → T sending each z ∈ T to ze2πρi satisfies that {[ρ]n (z): n ∈ N} is dense in T for every z ∈ T (see [23, Proposition III.1.4]). Indeed, it is easy to see that this fact can be generalized to separable powers of T, that is, those of the form Tκ for κ c (similarly as it is mentioned for finite powers in [23, III.1.14]): Let Λ := {ρα : α ∈ R} be a set of irrational numbers linearly independent over Q; if P is any nonempty subset of R and [ρα ]α∈P : TP → TP is defined as [ρα ]α∈P ((zα )α∈P ) := (zα e2πρα i )α∈P , then the set {[ρα ]nα∈P ((zα )α∈P ): n ∈ N} is dense in TP for every (zα )α∈P ∈ TP . Given two topological spaces Z and W , we denote by Z + F their topological sum, that is, the union Z ∪ W endowed with the topology consisting of unions of open subsets of these spaces (see [24, p. 65]). ∞} will denote N := {1, 2, . . . , n, . . .} will be a discrete infinite countable space, and N ∪ {∞ its one-point compactification. In our examples, the point 1 will play the same role as p in the definition of isometric shift of type I. Throughout “homeomorphism” will be synonymous with “surjective homeomorphism.” We will usually write T = T [a, φ, Δ] to describe a codimension 1 linear isometry T : C(X) → C(X), where X is compact and contains N . It means that φ : X \ {1} → X is a homeomorphism, satisfying in particular φ(n + 1) = n for all n ∈ N. It also means that a ∈ C(X \ {1}), |a| ≡ 1, and that Δ is a continuous linear functional on C(X) with Δ 1. Finally, the description of T we have is (Tf )(x) = a(x)f (φ(x)), when x = 1, and (Tf )(1) = Δ(f ), for every f ∈ C(X). In general, given a continuous map f defined on a space X, we also denote by f its restrictions to subspaces of X and its extensions to other spaces containing X. All our results will be valid in the real and complex settings, unless otherwise stated. The only exceptions are the following: Results exclusively given for K = C appear just in Section 5. The only result valid just for the case K = R is given in Example 5.3. CC (X) and CR (X) will denote the Banach spaces of continuous functions on X taking complex and real values, respectively. 3. Nonseparable examples ∞}) admits an isometric shift. Theorem 3.1. C(M + N ∪ {∞ Once we have a first example, we can get more. For instance, the next result is essentially different in that it provides examples with 2c infinite connected components. ∞}) admits Theorem 3.2. Let κ be any cardinal such that 1 κ c. Then C(M × Tκ + N ∪ {∞ an isometric shift. Finally, we can also give examples with just one infinite component. ∞}) admits Theorem 3.3. Let κ be any cardinal such that 1 κ c. Then C(M + Tκ + N ∪ {∞ an isometric shift.
J. Araujo / Journal of Functional Analysis 256 (2009) 1106–1117
1109
Remark 3.1. Even if our space M is based on an algebra of complex-valued functions, Theorems 3.1–3.3 are valid both if K = R or C. Nevertheless, there are examples that can be constructed just in the complex setting (see Theorem 5.2 and Example 5.3). 4. The proofs It is well known that M is extremally disconnected, that is, the closure of each open subset is also open. In fact, each measurable subset A of T determines via the Gelfand transform an open and closed subset G(A) of M, and the sets obtained in this way form a basis for its topology (see [13, p. 170]). Now it is straightforward to see that M is not separable: Let (xn ) be a sequence in M, and consider a partition (a.e) of T by k arcs of equal length, k 3. This determines a partition of M into k closed and open subsets of T. Select the arc A1 such that G(A1 ) contains x1 . Next do the same process with k 2 arcs of equal length, and pick A2 with x2 ∈ G(A2 ). Repeat the n process infinitely many times, in such ∞a way that each time we take An of length 1/k such that xn ∈ G(An ). It is clear that if A := n=1 An , then m(A) < 2π , so G(T \ A) is a nonempty closed and open subset of M containing no point xn . Notice that, since M is not separable, every isometric shift on C(M) must be of type I. But there are none because M has no isolated points. Even more, in [9, Corollary 2.5], it is proved that no space L∞ (Z, Σ, μ) admits an isometric shift if μ is non-atomic. As usual, we consider T oriented counterclockwise, and denote by A(α, β) the (open) arc of T beginning at eiα and ending at eiβ . Proof of Theorem 3.1. We start by defining a linear and surjective isometry on L∞ (T). We first consider the rotation ψ(z) := zei for every z ∈ T, and then define the isometry S : L∞ (T) → L∞ (T) as Sf := −f ◦ ψ for every f ∈ L∞ (T). On the other hand, using the Gelfand transform we have that the Banach algebra L∞ (T) is isometrically isomorphic to C(M), so S determines a linear and surjective isometry TS : C(M) → C(M). Also, by the Banach-Stone theorem, there exists a homeomorphism φ : M → M such that TS f = −f ◦ φ for every f ∈ C(M). Notice that this is valid both in the real and complex cases (see for instance [7, p. 187]). ∞}. The definition of TS can be extended to a new isometry Let X := M + N ∪ {∞ T : C(X) → C(X) in three steps. First, for each f ∈ C(X), we put (Tf )(x) := (TS f )(x) if ∞} \ {1} (where φ : N \ {1} → N is x ∈ M. In the same way (Tf )(n) := (f ◦ φ)(n) if n ∈ N ∪ {∞ ∞) := ∞ ). the canonical map sending each n into n − 1, which obviously can be extended as φ(∞ Finally, we put (Tf )(1) :=
1 2π
f dm, A(0,2πΦ)
√ where Φ := ( 5 − 1)/2 is the golden ratio conjugate. It isi easy to verify that T is a codimension one linear isometry, so we just need to prove that ∞ i=1 T (C(X)) = {0}. i (C(X)). It is easy to check that Suppose then that f ∈ ∞ T i=1 f (n) = T −n+1 f (1) = T T −n f (1)
1110
J. Araujo / Journal of Functional Analysis 256 (2009) 1106–1117
=
(−1)n 2π
f ◦ ψ −n dm
A(0,2πΦ)
=
(−1)n
f dm.
2π A(n,n+2πΦ)
On the other hand, if we fix any α ∈ T, then there exist two increasing sequences (nk ) and (mk ) in 2N and 2N + 1, respectively, converging to α mod 2π . An easy application of the Dominated Convergence Theorem proves that A(α,α+2πΦ) f dm = 2π lim f (nk ), and A(α,α+2πΦ) f dm = −2π lim f (mk ). By continuity, we deduce that ∞) = − f dm = 2πf (∞ f dm. A(α,α+2πΦ)
A(α,α+2πΦ)
∞) = 0. In parObviously, this implies that A(α,α+2πΦ) f dm = 0 for every α ∈ T, and f (∞ ticular this proves that f (n) = 0 for every n ∈ N . As a consequence we can identify n (C(X)) with an element f ∈ L∞ (T) satisfying T f dm = 0 for every f∈ ∞ n=1 A(α,α+2πΦ) α ∈ T. On the other hand, it is clear that we may assume that f takes values just in R. Claim. A(α,α+2πΦ n ) f dm = (−1)n F (n − 1) T f dm for every α ∈ T and n ∈ N, where F (n) denotes the nth Fibonacci number. Let us prove the claim inductively on n. We know that it holds for n = 1. Also notice that Φ + Φ 2 = 1, so Φ n + Φ n+1 = Φ n−1 for every n ∈ N. The case n = 2 is immediate because, since T = A α, α + 2πΦ 2 ∪ A α + 2πΦ 2 , α + 2π Φ + Φ 2 a.e., then we have T f dm = A(α,α+2πΦ 2 ) f dm for every α ∈ T. Now assume that, given k 2, the claim is true for every n k. Then we see that, for any α ∈ T, A α, α + 2πΦ k−1 = A α, α + 2πΦ k+1 ∪ A α + 2πΦ k+1 , α + 2π Φ k + Φ k+1 a.e., so k−1
(−1)
F (k − 2)
f dm =
T
A(α,α+2πΦ k+1 )
f dm + (−1) F (k − 1) k
f dm, T
and the conclusion proves the claim. The claim, combined with the fact that f is essentially bounded, implies that T f dm = 0, and consequently A(α,α+2πΦ n ) f dm = 0 for every α ∈ T and every n ∈ N. Now, it is easy to see that if U is an open subset of T, then U is the union of countably many pairwise disjoint arcs whose lengths belong to theset {2πΦ n : n ∈ N}. Now, applying again the Dominated Convergence Theorem, we see that U f dm = 0. Obviously, this implies that K f dm = 0 whenever K ⊂ T is compact.
J. Araujo / Journal of Functional Analysis 256 (2009) 1106–1117
1111
Finally take C + := {z ∈ T: f (z) > 0}. We know that there exists a sequence of compact subsets Kn of C + , with Kn ⊂ Kn+1 for every n ∈ N, and such that limn→∞ m(C + \ Kn ) = 0. Clearly, the above fact and the Monotone Convergence Theorem imply that C + f dm = 0, and then m(C + ) = 0. Now we can easily conclude that f ≡ 0 a.e., and consequently T is a shift. 2 Next we prove Theorem 3.2. It provides nonseparable examples with 2c infinite connected components, each homeomorphic to a (finite or infinite dimensional) torus: It follows from the fact that M is homeomorphic to an infinite closed subset of βN \ N that its cardinality must be 2c (see [18] and [8, Corollary 9.2]). ∞}) → C(M + N ∪ {∞ ∞}) Proof of Theorem 3.2. Write the isometric shift T : C(M + N ∪ {∞ ∞}), and it given in the proof of Theorem 3.1 as T = T [a, φ, Δ]. Obviously, Δ ≡ 0 on C(N ∪ {∞ can be considered as an element of C(M) . Consider a subset P of R with cardinal equal to κ, and suppose that {1/2π} ∪ {ρα : α ∈ P} is a family of real numbers linearly independent over Q. Then put ρκ := [ρα ]α∈P . Define φκ : M × Tκ → M × Tκ as φκ (x, z) := (φ(x), ρκ (z)) for every x ∈ M, and z ∈ Tκ . Select now a point vκ in TP = Tκ , and consider the evaluation map Γvκ ∈ C(Tκ ) . Both Δ and Γvκ are positive linear functionals, and so is the product Δ × Γvκ ∈ C(M × Tκ ) , which also satisfies Δ × Γvκ 1 (see [12, §13] for details). Given f ∈ C(M × Tκ ) and z ∈ Tκ , we write fz : M → K meaning fz (x) := f (x, z) for every x ∈ M. Obviously fz belongs to C(M), and (Δ × Γvκ )(f ) = Γvκ (Δ(fz )) = Δ(fvκ ). ∞}, define aκ ∈ C(Xκ \ {1}) as aκ ≡ −1 on M × Tκ , and Now, for Xκ := M × Tκ + N ∪ {∞ aκ ≡ 1 everywhere else, and put Tκ := T [aκ , φκ , Δ × Γvκ ]. i Let ψ : T → T and Φ be as in the proof of Theorem 3.1. Given f ∈ ∞ i=1 Tκ (Xκ ), we have that for every k ∈ N, f (k) = Tκ−k+1 f (1) = (Δ × Γvκ ) Tκ−k f = (−1)k (Δ × Γvκ ) f ◦ φκ−k = (−1)k Δ f ◦ φκ−k v κ k = (−1) Δ fρκ−k (vκ ) ◦ φ −k (−1)k = fρκ−k (vκ ) ◦ ψ −k dm 2π A(0,2πΦ)
=
(−1)k
fρκ−k (vκ ) dm.
2π A(k,k+2πΦ)
To continue with the proof, we need an elementary result: Claim. Suppose that (zλ )λ∈D is a net in Tκ converging to z0 . Then limλ fzλ − fz0 = 0. Let us prove the claim. If it is not true, then there is an > 0 such that, for every λ ∈ D, there exists ν ∈ D, ν λ, such that fzν − fz0 . It is easy to see that the set E of all
1112
J. Araujo / Journal of Functional Analysis 256 (2009) 1106–1117
ν ∈ D satisfying the above inequality is a directed set, and that (zν )ν∈E is a subnet of (zλ )λ∈D . Moreover there is a net (xν )ν∈E in M such that |f (xν , zν )−f (xν , z0 )| for every ν ∈ E. Since M × Tκ is compact, there exist a point (x0 , z 0 ) ∈ M × Tκ and a subnet (xη , zη )η∈F of (xν , zν )ν∈E converging to (x0 , z 0 ). Obviously (zη )η∈F is a subnet of (zν )ν∈E , so z0 = z 0 . Consequently both (xη , zη )η∈F and (xη , z0 )η∈F converge to (x0 , z0 ). Taking limits, this implies |f (x0 , z0 ) − f (x0 , z0 )| , which is absurd. Now, fix (α, w) ∈ T × Tκ and > 0. We know that (α, w) belongs to the closure of both Nj := ein , ρκ−n (vκ ) : n ∈ 2N − j (α, w) , j = 0, 1. We first consider the case j = 0, and take a net (yλ )λ∈D = (einλ , ρκ−nλ (vκ ))λ∈D in N0 converging to (α, w). Since (einλ )λ∈D converges to α, there exists λ1 ∈ D such that f dm − f w w < 2 A(nλ ,nλ +2πΦ)
A(α,α+2πΦ)
for every λ λ1 . On the other hand, by the claim, there exists λ2 ∈ D such that, if λ λ2 , then fw − fρ −nλ (v ) < /4π , so κ
κ
fw dm −
A(nν ,nν +2πΦ)
A(nν ,nν +2πΦ)
for every ν ∈ D. We easily deduce that lim fρ −nλ (v ) dm = κ
λ
A(nλ ,nλ +2πΦ)
fρ −nλ (v ) dm < κ κ 2
κ
fw dm,
A(α,α+2πΦ)
∞) = A(α,α+2πΦ) fw dm. In a similar way, working with N1 , we see that and consequently 2πf (∞ ∞) = − A(α,α+2πΦ) fw dm. With the same arguments as in the proof of Theorem 3.1, we 2πf (∞ conclude that fw ≡ 0, and finally f ≡ 0, as we wanted to prove. 2 Proof of Theorem 3.3. Notice first that L∞ (T) is isometrically isomorphic to L∞ (T1 ∪ T2 ), where Ti , i = 1, 2, are disjoint copies of T endowed with the Lebesgue measure. It is not hard to see that this implies that C(M) and C(M + M) are isometrically isomorphic, so M and M + M are homeomorphic. Assume that T = T [a, φ, Δ] is the isometric shift given in the proof of Theorem 3.1. We first define a homeomorphism χ : M × {0, 1} → M × {0, 1} as χ(x, i) = (φ(x), i + 1 mod 2) for every (x, i). For i = 0, 1, and f ∈ C(M × {0, 1}), denote by f × {i} its restriction to M × {i}, and put Δi (f ) := Δ(f × {i}). Let ρκ : Tκ → Tκ , vκ , and Γvκ be as in the proof of Theorem 3.2. ∞}, and define Tκ : C(Xκ ) → C(Xκ ) to be Finally consider Xκ := M × {0, 1} + Tκ + N ∪ {∞ Tκ := T [aκ , φκ , Δκ ], where • aκ ≡ −1 on M × {0} ∪ Tκ , and aκ ≡ 1 everywhere else. • φκ = χ on M × {0, 1}, and φκ = ρκ on Tκ . • Δκ := (Δ0 + Δ1 + Γvκ )/3.
J. Araujo / Journal of Functional Analysis 256 (2009) 1106–1117
As above, if f ∈
∞
1113
k ∈ N, and
n n=1 Tκ (C(Xκ )),
τ (k) :=
k(k − 1) mod 4 , 2
then 3f (k) = 3 Tκ−k+1 f (1) = Δ0 Tκ−k f + Δ1 Tκ−k f + Γvκ Tκ−k f = Δ Tκ−k f × {0} + Δ Tκ−k f × {1} + Tκ−k f (vκ ) = (−1)τ (k) Δ f × {k mod 2} ◦ φ −k + (−1)τ (k+1) Δ f × {k + 1 mod 2} ◦ φ −k + (−1)k f ◦ ρκ−k (vκ ) =
(−1)τ (k) 2π
f × {k mod 2} ◦ ψ −k dm
A(0,2πΦ)
+
(−1)τ (k+1) 2π
f × {k + 1 mod 2} ◦ ψ −k dm
A(0,2πΦ)
+ (−1)k f ρκ−k (vκ ) =
(−1)τ (k) 2π
f × {k mod 2} dm A(k,k+2πΦ)
+
(−1)τ (k+1) 2π
f × {k + 1 mod 2} dm A(k,k+2πΦ)
+ (−1)k f ρκ−k (vκ ) . j
Next fix α ∈ T, w ∈ Tκ , and for j = 0, 1, 2, 3, take increasing sequences (nk ) in 4N + j such j
−nk
j
that limk→∞ nk = α mod 2π , and limk→∞ f (ρκ Xαi :=
1 2π
(vκ )) = f (w). Now put
f × {i} dm A(α,α+2πΦ)
1114
J. Araujo / Journal of Functional Analysis 256 (2009) 1106–1117 j
for i = 0, 1. Taking into account that τ (nk ) is constant for each j , and that τ (2) = 1 = τ (3), and τ (1) = 0 = τ (4), we have that the following equalities hold: ∞) = Xα1 − Xα0 − f (w) 3f (∞ = = =
−Xα0 − Xα1 + f (w) −Xα1 + Xα0 − f (w) Xα0 + Xα1 + f (w)
(case j = 1) (case j = 2) (case j = 3) (case j = 0).
We deduce that Xαi = 0 for every α ∈ T and i = 0, 1, and that f ≡ 0 on Tκ . As in the proof of Theorem 3.1, we easily conclude that f ≡ 0. 2 5. Some differences between the real and complex cases In this section we show that in the complex setting, it is possible to obtain nonseparable examples with arbitrary (finitely many) infinite connected components. For the different behavior in the real setting, see Example 5.3. Our first result in this section is indeed given for separable examples. The idea of the proof is used in Theorem 5.2 to obtain nonseparable examples. In both cases T0 denotes the set {0}. Definition 5.1. Let X be compact and Hausdorff, and suppose that T = T [a, φ, Δ]: C(X) → C(X) is an isometric shift of type I. For n ∈ N, we say that T is n-generated if n is the least number with the following property: There exist n points x1 , . . . , xn ∈ X \ clX N such that the set k φ (xj ): k ∈ Z, j ∈ {1, . . . , n} is dense in X \ clX N . Notice that isometries simultaneously of types I and II are always 1-generated (see [9, Theorem 2.5]), so the next theorem provides a way for constructing isometries that are not of type II. Theorem 5.1. Let K = C. Suppose that n ∈ N, and that (κj )nj=1 is a finite sequence of cardinals satisfying 0 κj c for every j . Then there exists an n-generated isometric shift ∞}. Tn : CC (Xn ) → CC (Xn ), where Xn = Tκ1 + · · · + Tκn + N ∪ {∞ Proof. Let P1 , . . . , Pn be any pairwise disjoint subsets of R of cardinalities κ1 , . . . , κn , respectively. Consider any family Λ := {ρα : α ∈ R} of real numbers linearly independent over Q, and put σj := [ρα ]α∈Pj for each j n (in the case when κj = 0, that is, Pj = ∅, σj is the identity). Also let vj be a point in TPj . ∞}, and define φn : Xn → Xn as σj on each TPj . Next write Xn := TPn + · · · + TP1 + N ∪ {∞ j −1 For j n, let zj ∈ C \ {0}, with |zj | 1/2j , and ζj := eiπ/2 . Define a codimension 1 linear Pj for each j n, and isometry Tn on CC (Xn ) as Tn := T [an , φ n , Δn ], where an ≡ ζj on T n ∞}, and where Δn (f ) := i=1 zj f (vj ) for every f . an ≡ 1 on N ∪ {∞ Of course, the construction of Tn depends on our choice of the sets Pj and Λ, the points vj , and the numbers zj . We will prove that for any choices, the operator Tn satisfies the theorem.
J. Araujo / Journal of Functional Analysis 256 (2009) 1106–1117
1115
We will do it inductively on n. We start at n = 1. It is easy to see that T1 : CC (X1 ) → CC (X1 ) is an isometric shift (both of type I and type II). Now let us show that if Tn is an n-generated isometric shift for n = Tl+1 is an (l + 1)-generated isometric shift. l ∈ N, then m (C (X Suppose that f ∈ ∞ T l+1 )). It is easy to check that m=1 l+1 C −k+1 f (k) = Tl+1 f (1) =
l+1
−k zj Tl+1 f (vj )
j =1 l
−k −k f ◦ σl+1 (vl+1 ) + zj ζj−k f ◦ σj−k (vj ), = zl+1 ζl+1 j =1
whenever k ∈ N. j Fix x1 ∈ TP1 , . . . , xl+1 ∈ TPl+1 . For j = 0, 1, we can take increasing sequences (nk ) in 2l+1 N l+1 l and 2 N + 2 , respectively, such that the sequences j j −n −n f ◦ σ1 k (v1 ), . . . , f ◦ σl+1k (vl+1 ) k∈N
converge to (f (x1 ), . . . , f (xl+1 )) ∈ Cl+1 for j = 0, 1. This means, on the one hand, that ∞) = lim f n0k f (∞ k→∞
l
−n0 −n0 −n0 −n0 zj ζj k f ◦ σj k (vj ) = lim zl+1 ζl+1k f ◦ σl+1k (vl+1 ) + k→∞
j =1
= zl+1 f (xl+1 ) +
l
zj f (xj ).
j =1
And, on the other hand, l
1 ∞) = lim f nk = −zl+1 f (xl+1 ) + f (∞ zj f (xj ). k→∞
j =1
We that f (xl+1 ) = 0, that is, f ≡ 0 on TPl+1 , and consequently ∞deduce m f ∈ m=1 Tl (CC (Xl )). Since Tl is a shift, we conclude that f ≡ 0 on Xl+1 . It is also easy to see that Tl+1 is (l + 1)-generated. 2 Theorem 5.2. Let K = C. Suppose that n ∈ N, and that (κj )nj=1 is a finite sequence of cardinals satisfying 0 κj c for every j . Then there exists an isometric shift TnM : ∞}. CC (XnM ) → CC (XnM ), where XnM = M + Tκ1 + · · · + Tκn + N ∪ {∞
1116
J. Araujo / Journal of Functional Analysis 256 (2009) 1106–1117
Proof. The proof is similar to that of Theorem 5.1. We consider the homeomorphism φ on M coming from the rotation ψ : T → T given in the proof of Theorem 3.1. Fix n ∈ N, and assume that Xn and Tn = T [an , φn , Δn ] are as in the proof of Theorem 5.1. n Take zn+1 ∈ C \ {0} such that |zn+1 | 1/2n+1 , and put ζn+1 := eiπ/2 . M We are going to define an isometric shift on Xn . First put zn+1 ΔM := f dm + Δn (f ). n 2π A(0,2πΦ)
Obviously we are assuming that 1/2π does not belong to the linear span (over Q) of {ρα : α ∈ P1 ∪ · · · ∪ Pn }. Let anM ∈ CC (XnM ) be equal to ζn+1 on M, and equal to an on Xn , and let φnM : XnM → XnM be defined as φn on Xn , and as φ on M. We consider TnM := T [anM , φnM , ΔM n ].Following the same process as in the proof of Theorem 5.1, we easily obtain that 0 = zn+1 A(α,2πΦ+α) f dm for every α ∈ T. As in the proof of Theorem 3.1, we see that f ≡ 0 on T, which is to say that f ≡ 0 on M. We deduce that m f∈ ∞ T m=1 n (CC (Xn )), and consequently f ≡ 0. 2 Remark 5.1. Notice that in both Theorems 5.1 and 5.2, we allow the possibility that κj = κk for some (or all) j = k. Our next example shows in fact that the procedure followed above is no longer valid when dealing with K = R. Example 5.3. Let K = R. Suppose that X = Y + X1 + X2 + X3 is compact, where each Xj is connected and nonempty, and N ⊂ Y . Let T = T [a, φ, Δ] be a codimension 1 linear isometry on CR (X), and assume that φ(Xj ) = Xj , j = 1, 2, 3. Let us see that T is not a shift. First, there are j, k, j = k, with a(Xj ) = a(Xk ) ∈ {−1, 1}. There are also αj , αk ∈ R such that |αj | + |αk | > 0 and Δ(αj ξXj + αk ξXk ) = 0, where ξA denotes the characteristic function on A. It is easy to check that αj ξXj + αk ξXk belongs to T n (CR (X)) for every n ∈ N, and consequently T is not a shift. ∞}) nor CR (M + T + T2 + T3 + In particular, we see that neither CR (T + T2 + T3 + N ∪ {∞ ∞ N ∪ {∞ }) admit an isometric shift. References [1] J. Araujo, J.J. Font, Codimension 1 linear isometries on function algebras, Proc. Amer. Math. Soc. 127 (1999) 2273–2281. [2] J. Araujo, J.J. Font, Isometric shifts and metric spaces, Monatsh. Math. 134 (2001) 1–8. [3] L.-S. Chen, J.-S. Jeang, N.-C. Wong, Disjointness preserving shifts on C0 (X), J. Math. Anal. Appl. 325 (2007) 400–421. [4] R.M. Crownover, Commutants of shifts on Banach spaces, Michigan Math. J. 19 (1972) 233–247. [5] F.O. Farid, K. Varadajaran, Isometric shift operators on C(X), Canad. J. Math. 46 (1994) 532–542. [6] J.J. Font, Isometries on function algebras with finite codimensional range, Manuscripta Math. 100 (1999) 13–21. [7] J.B. Garnett, Bounded Analytic Functions, Academic Press, New York, 1981. [8] L. Gillman, M. Jerison, Rings of Continuous Functions, Springer-Verlag, New York, 1976. [9] A. Gutek, D. Hart, J. Jamison, M. Rajagopalan, Shift operators on Banach spaces, J. Funct. Anal. 101 (1991) 97– 119. [10] A. Gutek, J. Norden, Type 1 shifts on C(X), Topology Appl. 114 (2001) 73–89. [11] R. Haydon, Isometric shifts on C(K), J. Funct. Anal. 135 (1996) 157–162.
J. Araujo / Journal of Functional Analysis 256 (2009) 1106–1117
1117
[12] E. Hewitt, K.A. Ross, Abstract Harmonic Analysis, vol. I, second ed., Springer-Verlag, Berlin, 1979. [13] K. Hoffman, Banach Spaces of Analytic Functions, Prentice Hall, Englewood Cliffs, NJ, 1962. [14] H. Holszty´nski, Continuous mappings induced by isometries of spaces of continuous functions, Studia Math. 26 (1966) 133–136. [15] J.R. Holub, On shift operators, Canad. Math. Bull. 31 (1988) 85–94. [16] K. Izuchi, Douglas algebras which admit codimension 1 linear isometries, Proc. Amer. Math. Soc. 129 (2001) 2069–2074. [17] J.-S. Jeang, N.-C. Wong, Isometric shifts on C0 (X), J. Math. Anal. Appl. 274 (2002) 772–787. [18] S. Negrepontis, On a theorem by Hoffman and Ramsay, Pacific J. Math. 20 (1967) 281–282. [19] M. Rajagopalan, K. Sundaresan, Backward shifts on Banach spaces C(X), J. Math. Anal. Appl. 202 (1996) 485– 491. [20] M. Rajagopalan, K. Sundaresan, An account of shift operators, J. Anal. 8 (2000) 1–18. [21] M. Rajagopalan, T.M. Rassias, K. Sundaresan, Generalized backward shifts on Banach spaces C(X, E), Bull. Sci. Math. 124 (2000) 685–693. [22] T.M. Rassias, K. Sundaresan, Generalized backward shifts on Banach spaces, J. Math. Anal. Appl. 260 (2001) 36–45. [23] J. de Vries, Elements of Topological Dynamics, Kluwer Academic, Dordrecht, 1993. [24] S. Willard, General Topology, Addison–Wesley, Reading, MA, 1970.
Journal of Functional Analysis 256 (2009) 1118–1136 www.elsevier.com/locate/jfa
On the structure of fractional degree vortices in a spinor Ginzburg–Landau model Stan Alama a,∗,1 , Lia Bronsard a,1 , Petru Mironescu b a Dept. of Mathematics and Statistics, McMaster Univ., Hamilton, Ontario, Canada L8S 4K1 b Université de Lyon, Université Lyon 1, Institut Camille Jordan CNRS UMR 5208, 43,
boulevard du 11 novembre 1918, F-69622 Villeurbanne Cedex, France Received 24 April 2008; accepted 23 October 2008 Available online 20 November 2008 Communicated by H. Brezis
Abstract We consider a Ginzburg–Landau functional for a complex vector order parameter Ψ = (ψ+ , ψ− ), whose minimizers exhibit vortices with half-integer degree. By studying the associated system of equations in R2 which describes the local structure of these vortices, we show some new and unconventional properties of these vortices. In particular, one component of the solution vanishes, but the other does not. We also prove the existence and uniqueness of equivariant entire solutions, and provide a second proof of uniqueness, valid for a large class of systems with variational structure. © 2008 Elsevier Inc. All rights reserved. Keywords: Partial differential equations; Calculus of variations; Ginzburg–Landau model; Vortices
1. Introduction Recent papers in the physics literature have introduced spin-coupled (or spinor) Ginzburg– Landau models for complex vector-valued order parameters in order to account for ferromagnetic (or antiferromagnetic) effects in high-temperature superconductors [10] and in optically confined Bose–Einstein condensates [9]. In [2] two of the authors studied one such model, for a complex * Corresponding author.
E-mail addresses: [email protected] (S. Alama), [email protected] (L. Bronsard), [email protected] (P. Mironescu). 1 Supported by an NSERC Research Grant. 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.10.021
S. Alama et al. / Journal of Functional Analysis 256 (2009) 1118–1136
1119
pair of order parameters and showed that minimizers exhibit new types of vortices, with fractional degrees. In this paper we consider the structure of these fractional degree vortices, and show that their cores are qualitatively different from Ginzburg–Landau vortices. Consider the following model problem, related to the superconductivity model introduced in [10]. Let Ω ⊂ R2 be a smooth, bounded domain, and Ψ ∈ H 1 (Ω; C2 ). We define an energy functional, 1 E (Ψ ) = 2
2 2γ 1 2 2 2 |∇Ψ | + 2 |Ψ | − 1 + 2 (ψ1 × ψ2 ) dx, 2 Ω
where Ψ = (ψ1 , ψ2 ), ψ1 × ψ2 = Im(ψ1 ψ2 ), γ > 0 and > 0 are parameters. The quantity S = ψ1 × ψ2 = Im{ψ 1 ψ2 } is interpreted as the z-component of a spin vector, which in this two-dimensional model is assumed to be orthogonal to the plane of Ω. As → 0, energy minimizers should converge pointwise to the manifold on which the potential term F (Ψ ) = (|Ψ |2 −1)2 + γ2 (ψ1 ×ψ2 )2 vanishes. Since γ > 0, we obtain a two-dimensional surface (a 2-torus) Σ ⊂ S 3 ⊂ C2 parametrized by two real phases, φ, ω: Σ:
Ψ = G(φ, ω) := eiφ cos ω, eiφ sin ω .
Notice that G is doubly-periodic with minimal period G(φ + π, ω ± π) = G(φ, ω), with each phase executing a half cycle. For a smooth function Ψ (x) taking values in Σ and a simple closed curve C contained in the domain of Ψ we may therefore define a pair of half-integer valued degrees (dφ , dω ) corresponding to the winding numbers of the two phases around Σ . From the above observation, these degrees satisfy dφ , dω ∈ 12 Z, and dφ + dω ∈ Z. When auxiliary conditions force one or the other of the two phases φ, ω to have nontrivial winding number the minimizer Ψ (x) cannot take values in Σ at every point in Ω and in the limit we observe vortices, just as in the classical Ginzburg–Landau model (see [6]). Each isolated vortex will carry a pair of half-integer degrees, (dφ , dω ) as above. The results of [2] describe the minimizers and their energies as → 0, with a given Dirichlet boundary condition Ψ |∂Ω = g, where g = (g1 , g2 ) is a given smooth function g : ∂Ω → Σ . The boundary condition admits degrees Dφ , Dω corresponding to its winding in each of the phases around ∂Ω. Assume for simplicity that Dφ |Dω |. The main theorem of [2] then states that the minimizers can exhibit vortices of three different topological types: two species of fractional degree vortices, (dφ , dω ) = ( 12 , 12 ) or ( 12 , − 12 ), and an integer-degree vortex, (dφ , dω ) = (1, 0). The integer-degree vortex can be seen as a superposition of the two different fractional-degree vortices at the same location in Ω, and indeed the energy expansion shows that there is a weak interaction which favors the combination of two nearby fractional-degree vortices into a single (1, 0)-vortex. We expect, however, that these distinct types of vortices are very different in their microscopic structure. In order to resolve the singularity at each vortex, the order parameter Ψ must deviate from the minimal manifold Σ ⊂ C2 . The surface Σ being of codimension two, there are two degrees of freedom for this to occur. The order parameter can choose to violate the condition |Ψ | = 1 and develop a zero at the core of the vortex, as is the case for the usual Ginzburg–Landau
1120
S. Alama et al. / Journal of Functional Analysis 256 (2009) 1118–1136
vortices. But there is another possibility: Ψ can rotate along the sphere |Ψ | = 1 and violate the condition S = 0, thus acquiring non-zero spin in its core and avoiding the vanishing of |Ψ | altogether. The integer-degree (dφ , dω ) = (1, 0) vortices will take the first option, and resemble usual Ginzburg–Landau vortices, but our results in this paper confirm that the two species of fractional-degree vortices will indeed prefer the second approach, and exhibit this new “coreless” structure. To do this, we blow up the minimizers around each vortex and study the associated limiting problem of entire solutions to the PDE system in the whole of R2 , using techniques introduced for the Ginzburg–Landau equation by Brezis, Merle, Rivière [8], Mironescu [12], and Shafrir [13]. In order to describe our results, we introduce a change of variable as in [1,2] which simplifies the accounting of degrees. Vortices are best described in terms of the (integer) indices [n+ , n− ], n+ = dφ + dω ,
n − = dφ − dω ,
(dφ , dω ) = n+ (1/2, 1/2) + n− (1/2, −1/2),
which count the number of these two species of fractional-degree vortices rather than their winding. Remarkably, this may be achieved via the linear transformation in the range, 1 ψ± := √ (ψ1 ± iψ2 ). 2 In the new coordinates we denote our order parameter as Ψ = [ψ+ , ψ− ]. Now the surface Σ is described more simply, Σ:
|ψ+ |2 =
1 = |ψ− |2 , 2
and is parametrized as
1 1 Ψ = √ eiα+ , √ eiα− 2 2
with phases α± carrying whole number degrees [n+ , n− ]. Note the following correspondences between the degrees (dφ , dω ) and the integer indices [n+ , n− ]: (dφ , dω ) =
1 1 , 2 2
↔ [n+ , n− ] = [1, 0],
(dφ , dω ) =
1 1 ,− 2 2
↔ [n+ , n− ] = [0, 1],
(dφ , dω ) = (1, 0) ↔ [n+ , n− ] = [1, 1]. In these coordinates, the spin is given by S = 12 (|ψ− |2 − |ψ+ |2 ). The equations for entire vortex solutions Ψ (x) = [ψ+ , ψ− ] then become, − ψ+ = 1 − |Ψ |2 ψ+ + γ |ψ− |2 − |ψ+ |2 ψ+ , − ψ− = 1 − |Ψ |2 ψ− − γ |ψ− |2 − |ψ+ |2 ψ− . Solutions to (1)–(2) obtained by blowing up will satisfy an integrability condition, 2 2 2 |Ψ | − 1 + γ |ψ− |2 − |ψ+ |2 dx < ∞, R2
(1) (2)
(3)
S. Alama et al. / Journal of Functional Analysis 256 (2009) 1118–1136
1121
analogous to the condition of [8] for the classical Ginzburg–Landau equation. It follows from arguments of [8] (see Lemma 2.3) that any solution satisfying (3) has a degree pair at infinity: ψ± ; SR ) (with SR the circle of radius R) for all sufficiently large radii R. As was the n± = deg( |ψ ±| case for the classical Ginzburg–Landau equations [8], the integral in (3) is quantized: 2 2 2 |Ψ | − 1 + γ |ψ− |2 − |ψ+ |2 dx = π n2+ + n2− . (4) R2
This fact, together with some asymptotic estimates of the behavior of solutions at infinity, will be proven in Proposition 2.1. We would like to relate solutions of (1)–(2) to energy minimization. If Ω ⊂ R2 is a bounded domain, we may define an energy locally by E(Ψ ; Ω) =
2 2 γ 1 1 dx. |∇Ψ |2 + |Ψ |2 − 1 + |ψ− |2 − |ψ+ |2 2 4 4
(5)
Ω
This energy diverges when either of ψ+ , ψ− has nontrivial winding number at infinity, so it is not well defined when Ω = R2 . Instead, we define locally minimizing solutions in R2 in the sense of De Giorgi: we say that Ψ is a locally minimizing solution of (1)–(2) if (3) holds and if for every bounded regular domain Ω ⊂ R2 , E(Ψ ; Ω) E(Φ; Ω) holds for every Φ = [ϕ+ , ϕ− ] ∈ H 1 (Ω; C2 ) with Φ|∂Ω = Ψ |∂Ω . We prove the following result concerning fractional-degree vortex solutions: Theorem 1.1. Suppose Ψ is a locally minimizing solution of (1)–(2) with degree pair [n+ , n− ] = [1, 0]. Then there exists a constant φ− ∈ [0, 2π) such that ψ− (x)eiφ− > 0 is real and positive in R2 . In particular, the ψ− -component of the order parameter is bounded away from zero in R2 , and the qualitative behavior is as we expected, with |Ψ | bounded away from zero and S = 1 2 2 2 (|ψ− | − |ψ+ | ) > 0 in the core. An analogous result is obtained for the [n+ , n− ] = [0, 1]vortex, except that now it is ψ+ which is bounded away from zero and the spin S < 0 in the core. By a straightforward argument, solutions obtained by blowing up minimizers of E around a vortex always yield local minimizers in the sense of De Giorgi, and hence we infer the following result. Theorem 1.2. Assume Ψ = [ψ+ , ψ− ] is a family of minimizers of E as → 0. If p ∈ Ω is such that Ψ converges in some deleted neighborhood Bδ (p) \ {p} to a canonical Σ -harmonic map with degrees [n+ , n− ] = [1, 0] at p, then for small , |ψ− (x)| is bounded away from zero in Bδ (p). Special solutions to (1)–(2) are obtained by an equivariant ansatz, ψ± (x) = f± (r)ein± θ , in polar coordinates (r, θ ) in R2 , with f± a pair of real-valued functions. In Section 3 we show that for each fixed choice of degrees n± at infinity, there exist unique equivariant entire solutions
1122
S. Alama et al. / Journal of Functional Analysis 256 (2009) 1118–1136
satisfying (3). Uniqueness is proven using the method of Brezis and Oswald [7]. In Section 4 we give an alternative proof of uniqueness of equivariant solutions by means of an extension of the Krasnoselskii theorem [11] to systems with variational structure. We also study the interesting role of the parameter γ in the monotonicity of the fractionaldegree vortex profiles. When 0 < γ < 1 the component of the order parameter which does not vanish (for example, f− for the [n+ , n− ] = [1, 0] vortex) is monotone decreasing, and approaches its limiting value at infinity from above. When γ 1, all vortices of any degree combination have density profiles which increase with r, just as in the Ginzburg–Landau case. Theorem 1.3. Assume Ψ (x) = [f+ (r)ein+ θ , f− (r)ein− θ ] is an equivariant solution satisfying (3). (i) If γ 1, then f± (r) 0 for all r > 0, for any degrees [n+ , n− ]. (ii) If 0 < γ < 1, n+ 1, and n− = 0, then f+ (r) 0 and f− (r) 0 for all r > 0. The methods used in Section 3 are derived from the work of Alama, Bronsard and Giorgi [3,4] on the SO(5)-model, which also featured a vector-valued order parameter and two different species of vortex profiles. A natural open question is whether all locally minimizing solutions to (1)–(2) must be radial. This fact was proven by Mironescu [12] for the classical Ginzburg–Landau equation, by dividing any given solution by an equivariant one (which must be of degree ±1) and calculating a sort of Pohozaev identity for the equation satisfied by the quotient. Our equations being a coupled system, the above procedure fails since the equation for the quotient is no longer clearly of gradient form. Although our system is in many ways very similar to the scalar Ginzburg–Landau equation, and many analytic results may be extended from one to the other, we have been careful in verifying which techniques derived for the scalar equation may be adapted to the system (1)–(2). 2. Locally minimizing solutions In this section we study solutions which are locally minimizing in the sense of De Giorgi, and prove Theorem 1.1. The proof is based on the following asymptotic description of solutions: Proposition 2.1. Let Ψ be a solution of (1)–(2) in R2 satisfying (3). There exist constants β+ , β− such that ψ± → √1 ei(n± θ+β± ) uniformly as |x| → ∞. Moreover: 2
(i) If [n+ , n− ] = [1, 0], then as r = |x| → ∞,
ψ+ (x) 2 = 1 − γ + 1 1 + o 1 , 2 4γ r 2 r2
ψ− (x) 2 = 1 − γ − 1 1 + o 1 . 2 4γ r 2 r2
(6)
(ii) If [n+ , n− ] = [1, 1], then as r = |x| → ∞,
ψ± (x) 2 = 1 − 1 + o 1 . 2 2r 2 r2
(7)
Remark 2.2. We observe that this asymptotic result already shows the qualitative difference between the case 0 < γ < 1 and the case γ 1, at least in the case of the fractional degree
S. Alama et al. / Journal of Functional Analysis 256 (2009) 1118–1136
1123
vortices. We will confirm this difference in monotonicity of the vortex profiles in our analysis of the equivariant (radially symmetric) vortex solutions in the next section. Proof of Theorem 1.1. By the first part of Proposition 2.1, we may assume without loss of 1 generality that ψ− (x) → √1 uniformly as |x| → ∞. In particular, if we fix any δ < √ , there 2
exists a radius R = R(δ) such that |ψ− (x) − x ∈ Ω define ψ˜ + (x) = ψ+ (x),
√1 | 2
2 2
< δ for all |x| R. Let Ω = BR (0), and for
ψ˜ − (x) = Re ψ− (x) + i Im ψ− (x).
Note that Ψ˜ := [ψ˜ + , ψ˜ − ] ∈ H 1 (Ω; C2 ), E(Ψ˜ ; Ω) = E(Ψ ; Ω), and (by the choice of R) Ψ˜ |∂Ω = Ψ |∂Ω . Therefore, Ψ˜ is also a local minimizer of E, in the sense described above. This implies that Ψ˜ also solves the Euler–Lagrange equations (1)–(2) in Ω. In particular, u = Re ψ˜ − is a non-negative solution of − u + (1 − γ )|ψ˜ + |2 + (1 + γ )|ψ˜ − |2 − 1 u = 0, which is strictly positive on ∂Ω = SR (again, by the choice of R). By the strong maximum principle, in fact u = Re ψ˜ − (x) > 0 in Ω. This implies Ψ˜ = Ψ , and Re ψ− (x) > 0 in R2 . Now let α be a constant with |α| < π2 , and consider ψˆ − (x) := ψ− (x)eiα . Note that Ψˆ := [ψ+ , ψˆ − ] is again a solution to (1)–(2), with the same energy in any domain Ω. By Proposition 2.1 and our definition of ψˆ − we now have ψˆ − (x) → √1 eiα uniformly as |x| → ∞. Choosing 2
δ = δ(α) > 0 such that Bδ ( √1 eiα ) is strictly contained inside the right half-plane {Re z > 0}, 2 there exists a radius R = R(α) such that |ψˆ − (x) − √1 eiα | < δ whenever |x| R. Repeating the 2
above argument, we conclude that Re ψˆ − (x) > 0 in R2 . Equivalently, Im ψ− (x) (cot α) Re ψ− (x)
when 0 < α < π/2,
Im ψ− (x) (cot α) Re ψ− (x)
when −π/2 < α < 0.
Letting α → ± π2 we conclude Im ψ− (x) ≡ 0.
2
To prove Proposition 2.1 we use the following modification of a similar result from [8]: Lemma 2.3. Let Ψ be an entire solution of (1)–(2) satisfying (3). (i) |Ψ (x)| 1 for all x ∈ R2 and |ψ± (x)|2 → 12 uniformly as |x| → ∞. (ii) There exist constants R0 > 0, n± ∈ Z, and smooth functions ρ± (x), φ± (x) for |x| R0 such that Ψ (x) = ψ+ (x), ψ− (x) = ρ+ (x)ei(n+ θ+φ+ (x)) , ρ− (x)ei(n− θ+φ− (x)) ,
1124
S. Alama et al. / Journal of Functional Analysis 256 (2009) 1118–1136
with
|∇ρ± |2 + |∇φ± |2 < ∞.
(8)
|x|>R0
Proof. Statement (i) follows as in Step 1 of the proof of Theorem 1 in [8]. Indeed, the quantity U (x) := |Ψ (x)|2 satisfies the equation 1 γ
U = (U − 1)u + S 2 + |∇Ψ |2 , 2 2 and hence the estimate U = |Ψ |2 < 1 follows from the strong maximum principle. The uniform convergence as |x| → ∞ then follows as in [8] from standard elliptic estimates, (3), and the following elementary inequality [2]: 2 min{1, γ }
1 |ψ+ | − 2 2
2
2 1 2 2 |Ψ |2 − 1 + 4γ S 2 . + |ψ− | − 2
The existence of R0 , ρ± , ψ± is an immediate consequence of (i). The first part of (ii) is an immediate consequence of the uniform limit |x| → ∞. To prove (8), we write the equations for ϕ± = n± θ + φ± and ρ± : 2 div ρ± ∇ϕ± = 0, 2 2 2 2 ρ± ∓ γ ρ− ρ± . − ρ− − ρ+ − ρ± + |∇ϕ± |2 ρ± = 1 − ρ+ The equations for ϕ± are identical to those in [8], and the analysis there applies with no modification. The equations for ρ± are of the same form, and the same approach as [8] leads easily to the same conclusion with only minor changes. We leave the details to the interested reader. 2 Proof of Proposition 2.1. The proof follows Shafrir [13]. Let Rm → ∞ be any increasing divergent sequence, m = 1/Rm , and let 0 < a < 1 < b be fixed. Denote by Ω = Bb (0) \ Ba (0) and Ωm = BbRm \ BaRm (0). Consider the rescaled solutions Ψm (x) = ψm+ (x), ψm− (x) = Ψ (Rm x). Then Ψm satisfies − ψm± +
1 (1 ± γ )|ψm+ |2 + (1 ∓ γ )|ψm− |2 − 1 ψm± = 0 in Ω. 2 m
(9)
We now apply Lemma 2.3 to obtain R0 > 0 and ρ± , φ± defined for |x| R0 . Since large |x| is equivalent to large m we may write, for large m, ψm± = ρm± exp(i(n± θ + φm± (x))). As in [13], we use (8) to calculate
S. Alama et al. / Journal of Functional Analysis 256 (2009) 1118–1136
|∇Ψm | = 2
Ω
2 |∇ρ± |2 + ρ± |∇Ψ | = |n± ∇θ + ∇φ± |2 2
Ωm
=
Ωm
=
1125
±
Ωm
±
2 |∇ρ± |2 + ρ±
n2± 2n± + 2 ∇φ± · (−y, x) + |∇φ± |2 r2 r
1 n2± + o(1) 2 r2 ±
Ωm
b = π n2+ + n2− ln + o(1). a
(10)
Up to a subsequence, we find Ψm Ψ˜ in H 1 (Ω; Σ). We claim that the convergence Ψm → Ψ˜ is strong in H 1 (Ω; Σ), and 1 Ψ˜ (x) = √ ei(n+ θ+β+ ) , ei(n+ θ+β− ) , 2
(11)
with β± real constants. Indeed, since Ψ˜ takes values in Σ we may represent it locally as Ψ˜ = √1 [exp(iϕ+ (x)), exp(iϕ− (x))], where ϕ± are possibly multivalued, real-valued functions. By 2 standard arguments we derive a lower bound which matches (10):
|∇ ψ˜ ± |2
b 2π a
Ω
1 2
1 (∇ϕ± · θˆ )2 r dθ dr 2
0
b [ ∂ϕ /∂s ds ]2 ± r Sr dr 2π 0 1r dθ a
b =
πn2± b dr = πn2± ln . r a
a
By lower semicontinuity, we conclude that this inequality is indeed an equality, Ω |∇ Ψ˜ |2 = π(n2+ + n2− ) ln ab . Hence, the convergence is strong in H 1 . In addition, we have the case of equality in the Cauchy–Schwarz inequality used in the second line of the lower bound above, which implies (11), and the claim is established. We now employ the main idea of [13]: to use the local convergence results away from vortices for the singularly perturbed problem (9), derived for the Ginzburg–Landau equation in [5] and k (Ω) for any k 0, extended to our spinor system in [2]. By Theorem 4.1 in [2], Ψm → Ψ˜ in Cloc and 1 ˜ m+ |2 + (1 ∓ γ )|ψ˜ m− |2 − 1 + 2|∇ ψ˜ m± |2 (1 ± γ )| ψ 2 m
k (Ω) Cloc
→ 0,
for all k 0.
1126
S. Alama et al. / Journal of Functional Analysis 256 (2009) 1118–1136
k convergence of Ψ to Ψ ˜ implies that we may replace 2|∇ ψ˜ m± |2 by Note that the Cloc m above estimate. Evaluating along ∂B1 (0) ⊂ Ω,
2 R (1 ± γ ) ψ+ (Rm x) 2 + (1 ∓ γ ) ψ− (Rm x) 2 − 1 + n2
± L∞ (∂B1 (0))
m
n2± r2
in the
→ 0.
Since Rm was an arbitrary divergent sequence we may conclude that the above holds for general r → ∞, that is,
2
2
n2± 1
(1 ± γ ) ψ+ (x) + (1 ∓ γ ) ψ− (x) − 1 + 2 = o 2 , r r
uniformly as |x| = r → ∞. This then implies that
1 n2+ (γ + 1) + n2− (γ − 1) 1 1 |ψ+ | = − +o 2 , 2 4γ r2 r
1 1 n2+ (γ − 1) + n2− (γ + 1) 1 2 |ψ− | = − +o 2 , 2 4γ r2 r 2
as r → ∞. The conclusions (6) and (7) then follow immediately. To obtain the uniform limit of φ± (x), we note that by taking the imaginary part of Eqs. (1)– (2) in polar form we arrive at the same equation (for conservation of current) as in the classical Ginzburg–Landau equation, 2 ∇(n± θ + φ± ) = 0. div ρ± Therefore the assertion that φ± (x) → β± uniformly as |x| → ∞ follows exactly as in [13].
2
We note the following further estimate which will be useful in our study of equivariant solutions in the next section: Corollary 2.4. Under the hypotheses above, with ρ± = |ψ± |, we have:
∂ρ± n2+ (γ ± 1) + n2− (γ ∓ 1) 1 1 = +o 3 , √ 3 ∂r r r 2 2γ √
2 1 ∂ ρ± 3 2 2 1 2 n+ (γ ± 1) + n− (γ ∓ 1) 4 + o 4 . =− 2 4γ ∂r r r k estimates above. These follow by differentiation in the Cloc Finally, we prove the quantization of the potential term for any entire solution satisfying (3):
Proposition 2.5. Let (for any choice of [n+ , n− ]) Ψ = [ψ+ , ψ− ] be a solution of (1)–(2) satisfying (3). Then 2 2 2 |Ψ | − 1 + γ |ψ− |2 − |ψ+ |2 dx = π n2+ + n2− . R2
S. Alama et al. / Journal of Functional Analysis 256 (2009) 1118–1136
1127
Proof. The proof continues from that of Lemma 2.3, following Step 3 in the proof of Theorem 1 in [8]. By the Pohozaev identity applied to our system, 1 r
∂ψ+ 2 ∂ψ− 2
G(Ψ ) dx +
∂r + ∂r ds
Br
∂Br
∂ψ+ 2 ∂ψ− 2
=
∂τ + ∂τ + G(Ψ ) ds,
(12)
∂Br
where τ indicates the unit tangent to ∂Br and 2 2 G(Ψ ) := |Ψ |2 − 1 + γ |ψ− |2 − |ψ+ |2 . Define
E(R) :=
G(Ψ ) dx,
E :=
BR
G(Ψ ) dx, R2
and note that E(R) → E as R → ∞, as well as 1 ln R
R
E(r) dr → E, r
R → ∞.
0
Integrating (12) over r ∈ (0, R),
R
∂ψ+ 2 ∂ψ− 2
∂ψ+ 2 ∂ψ− 2 1
+
+ E(r) dr =
+
∂r
∂τ
∂r
∂τ + 2 E(R). r BR
0
(13)
BR
+ 2 The radial derivatives | ∂ψ ∂r | are estimated as in (2.46) of [8], using (8) to obtain
∂ψ+ 2 ∂ψ− 2
+
∂r
∂r C BR
uniformly as R → ∞. The difference in our case is in the tangential derivative,
∂ψ± 2 n2±
−
|∇ρ± |2 + ρ 2 −
∂τ
2 2r
1
n2± 2 n± |∇φ± | + |∇φ± |2 , + 2ρ±
2 2 r r
proof of Lemma 2.3. Note the where we have decomposed ψ± (x) = ρ± ei(n± θ+φ± ) as in the √ extra factor 12 which appears in our case since ρ± = |ψ± | → 1/ 2 as |x| → ∞. Continuing as in (2.49), (2.50) of [8], we divide (13) by ln R and pass to the limit to obtain:
1128
S. Alama et al. / Journal of Functional Analysis 256 (2009) 1118–1136
1 G(Ψ ) dx = lim R→∞ ln R
∂ψ+ 2 ∂ψ− 2
+
∂τ
∂τ BR
R2
1 = lim R→∞ ln R
n2± dx = π n2+ + n2− . 2r 2
2
BR
3. Equivariant solutions In this section we consider special solutions of Eqs. (1)–(2) of the form ψ+ (x) = f+ (r)ein+ θ ,
ψ− (x) = f− (r)ein− θ ,
in polar coordinates (r, θ ) with given degree pair [n+ , n− ] ∈ Z2 . By taking complex conjugates if necessary, we may assume that n± 0. When f 0, the system (1)–(2) reduces to the following system of ODEs, ⎧ 1 n2± ⎪ ⎪ ⎪ − rf± + 2 f± + f+2 + f−2 − 1 f± ∓ γ f−2 − f+2 f± = 0, ⎪ ⎪ r r ⎪ ⎪ ⎨ f (r) 0 for all r ∈ [0, ∞),
for r ∈ (0, ∞),
±
(14)
1 ⎪ ⎪ f± (R) → √ as r → ∞, ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎩ f± (0) = 0 if n± = 0, f± (0) = 0 if n± = 0. We begin with their existence and uniqueness.
Lemma 3.1. Let n± ∈ Z be given. Then there exists a unique solution ∞ [f+ (r), f− (r)] to (14) for r ∈ [0, ∞) such that: f± ∈ C ∞ ((0, ∞)), f± (r) > 0 for all r > 0, 0 (1 − f+2 − f−2 )2 r dr < ∞, and f± (r) ∼ r n± for r ∼ 0. In particular, Ψ (x) = [f+ (r)ein+ θ , f− (r)ein− θ ] is an entire solution of (1)–(2) in R2 satisfying (3). Proof. To obtain existence we consider first the simpler problem defined in the ball BR , R > 0, ⎧ ⎪ 1 n2± ⎪ ⎪ − rf± + 2 f± + f+2 + f−2 − 1 f± ∓ γ f−2 − f+2 f± = 0, ⎪ ⎪ r ⎨ r 1 f± (R) = √ , ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎩ f± (0) = 0 if n± = 0, f± (0) = 0 if n± = 0.
for 0 < r < R, (15)
The existence of such a solution follows easily, for example, by minimization of the energy EnR+ ,n− (f+ , f− ) 1 = 2
R 2 2 n2 2 1 2 2 2 2 f f + f − 1 + γ f − f (fi )2 + ± + r dr, − − + 2 + r2 i 0
i=±
(16)
S. Alama et al. / Journal of Functional Analysis 256 (2009) 1118–1136
1129
over Sobolev functions satisfying the appropriate boundary conditions at r = 0 and r = R. Denote by [fR,+ (r), fR,− (r)] any solution of (15). As in the proof of (i) of Lemma 2.3 we have the simple a priori bound |Ψ |2 = (f+2 (r) + f−2 (r)) 1 for any solution to (16). By standard elliptic estimates, there exists a subsequence Rn → ∞ for which the solutions [fR,+ , fR,− ] → 1,α [f∞,+ , f∞,− ] in Cloc [0, ∞), and the limit functions [f+∞ (r), f−∞ (r)] give (weak) solutions to the ODE on (0, ∞) with the same boundary condition at r = 0. By standard estimates we obtain the behavior f∞,± ∼ r n± near r = 0, and therefore ψ± (x) = f∞,± (r)ein± θ is regular at x = 0 and solves (1)–(2) in R2 . On the other hand, by the strong maximum principle, we have either f± > 0 in (0, ∞), or f± ≡ 0 in (0, ∞). Therefore, in order to conclude, it suffices to establish (3). For this purpose
(r) and integrate we derive a Pohozaev identity: we multiply the equation of fR,± (r) by r 2 fR,± with respect to r ∈ (0, R). We obtain:
2 2 R fR,+ (R) + fR,− (R) +
R
2
2 2 2 2 2 2 1 − fR,+ r dr − fR,− + γ fR,− − fR,+
0
1 = n2+ + n2− . 2 By uniform convergence on [0, R0 ] for any R0 > 0 we have R0 2 2 2 1 2 2 2 1 − f∞,+ r dr n2+ + n2− , − f∞,− + γ f∞,− − f∞,+ 2 0
and so letting R0 → ∞ we recover the condition (3). This completes the existence part of Lemma 3.1. To prove uniqueness we use the basic approach of Brezis and Oswald [7]. Let [n+ , n− ] ∈ Z2 be given, and suppose [f+ , f− ] and [g+ , g− ] are two solutions of (14). Denote by r f := 1
r (rf (r)) the Laplacian for radial functions. Then we have:
r f+ r g+ 2 2 + (1 − γ ) f−2 − g− , + = − (1 + γ ) f+2 − g+ f+ g+
r f− r g− 2 2 − + (1 + γ ) f−2 − g− . + = − (1 − γ ) f+2 − g+ f− g−
−
(17) (18)
2 ) and (18) by (f 2 − g 2 ), and integrate over 0 < r < ∞. We then multiply (17) by (f+2 − g+ − − in θ Since ψ± (x) = f± (r)e ± defines a solution of the system (1)–(2) satisfying the condition (3), the estimates of Proposition 2.1 and Corollary 2.4 hold for f± , g± . In particular the integrals converge, and we may integrate by parts. As in [7] we obtain:
2
2
2
2 ∞
f − f+ g + g − g+ f + f − f− g + g − g− f r dr 0
+ g +
+ f +
− g −
− f − + + − − 0
1130
S. Alama et al. / Journal of Functional Analysis 256 (2009) 1118–1136
=−
∞
2 2 2 2 2 2 2 (1 + γ ) f+2 − g+ f − − g− + (1 + γ ) f−2 − g− r dr + 2(1 − γ ) f+2 − g+
0
∞ 2 2 2 2 2 −2 min{1, γ } f + − g+ r dr, + f−2 − g− 0
since the quadratic form (1 + γ )X 2 + 2(1 − γ )XY + (1 + γ )Y 2 2 min{1, γ }(X 2 + Y 2 ) is positive definite. Hence f± (r) = g ± (r) for all r ∈ (0, ∞), and we have proven uniqueness. 2 We next present the proof of Theorem 1.3 on the monotonicity of the radial profiles. First, we define the spaces X0 := H 1 (0, ∞); r dr ,
∞
Xn := u ∈ X0 :
u2 r dr < ∞ , r2
u2Xn
0
∞ n2 2
2 2 (u ) + u + 2 u r dr. = r 0
Of course the spaces Xn , n = 0, are all equivalent, but we define them this way for notational convenience. It is not difficult to show (see [3]) that for |n| 1, Xn is continuously embedded in the space of continuous functions on (0, ∞) which vanish at r = 0 and as r → ∞, and that C0∞ ((0, ∞)) is dense in X1 . It is possible to define a global variational framework for the equivariant problems in affine spaces based on Xn+ , Xn− to prove existence of solutions. The energy n2
is the same as in (16), except it must be “renormalized” to prevent divergence of the r±2 term at infinity. Here we are only interested in the (formal) second variation of this renormalized energy, D En+ ,n− (f+ , f− )[u+ , u− ] := 2
∞ 0
i=±
2 n2± 2 2 ui + 2 ui + f+ + f−2 − 1 u2i r
+ 2(f+ u+ + f− u− )2 + 2γ (f− u− − f+ u+ )2 2 2 2 2 + γ f− − f+ u− − u+ r dr, defined for [u+ , u− ] ∈ Xn+ × Xn− . We have the following remarkable fact about admissible radial solutions: Lemma 3.2. For any n± ∈ Z, if [f+ , f− ] is the (unique) admissible radial solution of (14), D 2 En+ ,n− (f+ , f− )[u+ , u− ] > 0 for all [u+ , u− ] ∈ Xn+ × Xn− \ [0, 0] . In other words, the radial solutions are non-degenerate local minimizers of the renormalized energy. An analogous statement for the Ginzburg–Landau equation with magnetic field was derived in [3], and this observation then became the main step in the proof of uniqueness of equivariant solutions proved there. The basic idea is that were there two admissible solutions to
S. Alama et al. / Journal of Functional Analysis 256 (2009) 1118–1136
1131
the equivariant vortex equations, each being a local minimizer of the energy there would be a third, non-minimizing solution via the Mountain-pass theorem. The argument was achieved by restriction to a convex constraint set (to eliminate the possibility of non-admissible solutions, which might not be local minimizers). The method works because the constraints play the role of a sub- and super-solution pair for the Ginzburg–Landau equations, and hence the mountain pass solutions obtained would lie in the interior of the constraint set. Unfortunately, in our vectorvalued case the sub-solution structure is not apparent and the argument does not seem to carry over. Proof of Lemma 3.2. We follow [3], and note the following calculus identity:
2
(f )2 u(r) 2 uu f
u + u2 2 = (u )2 − = (u )2 − 2 f . f 2 (r) f (r) f f f u2
u2
Let u± ∈ C0∞ ((0, ∞)) (if n± = 0, take u± ∈ C0∞ ([0, ∞)) instead). Then [ f++ , f−− ] gives an admissible test function in the weak form of the system (14), 2 u+ u2− , 0 = DEn+ ,n− (f+ , f− ) f+ f−
2
∞ 2 n2i 2 ui 2 ui + 2 ui − fi = fi r i=±
0
2 2 2 2 2 2 2 2 + f+ + f− − 1 u+ + u− + γ f− − f+ u− − u+ r dr.
Rearranging, we obtain the useful identity, ∞ 0
i=±
=
2 n2i 2 2 2 2 2 2 2 2 2 ui + 2 ui + f+ + f− − 1 u+ + u− + γ f− − f+ u− − u+ r dr r
∞
fi2
0 i=±
ui fi
2
0.
We then substitute into the expression for the second variation (19): D 2 En+ ,n− (f+ , f− )[u+ , u− ] =
∞ i=±
0
∞
0
fi2
ui fi
2
+ 2(f+ u+ + f− u− )2 + 2γ (f− u− − f+ u+ )2
2(f+ u+ + f− u− )2 + 2γ (f− u− − f+ u+ )2 ,
(19)
1132
S. Alama et al. / Journal of Functional Analysis 256 (2009) 1118–1136
valid for all u± ∈ C0∞ ((0, ∞)) (or, u± ∈ C0∞ ([0, ∞)) if the respective n± = 0). The case of general u± ∈ X1 (or X0 , in case one of n± = 0) then follows by density. Clearly, D 2 En+ ,n− (f+ , f− ) 0 (as a quadratic form). If it were zero for some [u+ , u− ], then we would have f+ u+ = f− u− = −f+ u+ almost everywhere. Since f± (r) > 0 for r > 0, we conclude that the second variation is strictly positive definite, as claimed. 2 Proof of Theorem 1.3. Let u± (r) := f± (r). Differentiating Eq. (14), we obtain n2 1 u± + f+2 + f−2 − 1 u± + 2(f+ u+ + f− u− )f± 0 = −u
± − u ± + ± 2 r r 2n2 1 ∓ 2γ (f− u− − f+ u= )f± − 3± f± + 2 u± . r r
(20)
We observe that all but the last two terms form part of the second variation of energy, (19). Define v± = min{0, u± } 0,
w± = max{0, u± } 0.
First, assume γ 1. We multiply the respective equation in (20) by v± (and use v+ w+ = 0 and v− w− = 0) and integrate by parts. Note that by conclusion (ii) of Lemma 3.1, f± (r) > 0 for all r and f± (r) ∼ r n± for r near zero. Therefore, if n± 1, u± (r) = f± (r) > 0 in some neighborhood r ∈ (0, δ). Thus, in case n± 1, v± is supported away from r = 0. By the asymptotic estimates of Corollary 2.4 we may then conclude that v± ∈ Xn± . Furthermore, ∞
v± u
±
∞ 2 1
v± r dr, + u± r dr = − r
0
0
with no boundary terms. In case n± = 0, then u± ∈ X0 by the regularity of solutions, and u± (0) = f± (0) = 0. The integration by parts formula above again holds with no boundary condition in this case as well. Combining terms and recognizing that many terms form part of the second variation of energy (19), we obtain: ∞ 0 = D En+ ,n− (f+ , f− )[v+ , v− ] + 2(1 − γ )
f+ f− (w− v+ + v− w+ )r dr
2
0
+
∞ i=± 0
1 2 2n2i v − 3 fi vi r dr. r2 i r
Each term above has a sign, and we obtain ∞ 0 D En+ ,n− (f+ , f− )[v+ , v− ] − 2
1 2 2 v+ + v− r dr < 0, 2 r
0
a contradiction to Lemma 3.2 unless both v± ≡ 0, that is unless f± (r) 0 for all r > 0. This proves (i).
S. Alama et al. / Journal of Functional Analysis 256 (2009) 1118–1136
1133
Now assume 0 < γ < 1, n− = 0 and n+ 1. This time we multiply the equation for u+ by v+ and the equation for u− by w− , and again integrate by parts. Just as in the previous case, w− ∈ X0 , and the boundary term in the integration will all vanish. This time we obtain: ∞ 0 = D 2 En+ ,n− (f+ , f− )[v+ , w− ] + 2(1 − γ )
f+ f− (v+ v− + w+ w− )r dr 0
∞ +
2n2+ 1 2 2 v + w− − 3 f+ v+ r dr. r2 + r
0
Since f+ > 0, v+ 0 and v+ v− , w+ w− 0, we conclude ∞ D En+ ,n− (f+ , f− )[v+ , w− ] − 2
1 2 2 v + + w− r dr < 0, 2 r
0
a contradiction with Lemma 3.2 unless v+ ≡ 0 and w− ≡ 0. That is, unless f+ (r) 0 and f− (r) 0 for all r > 0. 2 4. Another approach to uniqueness We give a second proof of the uniqueness of the equivariant solutions [ψ+ , ψ− ] = f+ (r)ein+ θ , f− (r)ein− θ which is based on an extension of Krasnoselskii’s method [11] to variational elliptic systems. For a vector u = (u1 , . . . , um ) ∈ Rm , we say u is positive, and write u > 0, if ui > 0 for all i = 1, . . . , m. We denote by u2 = (u21 , . . . , u2m ). Theorem 4.1. Suppose G : Ω × Rm → R, and G(x, u) is strictly convex in u > 0 for every fixed x ∈ Ω ⊂ Rn . Then, there is at most one positive solution u to ⎧ ⎨ − uj + ∂uj G x, u2 uj = 0, u > 0, x ∈ Ω, ⎩ u = 0, x ∈ ∂Ω.
x ∈ Ω, j = 1, . . . , m,
Proof. Define the energy associated to this problem, E(u) =
1 2
Ω
|∇u|2 + G x, u2
(21)
1134
S. Alama et al. / Journal of Functional Analysis 256 (2009) 1118–1136
for u ∈ H01 (Ω; Rm ). Suppose u, v are both positive solutions of (21), and define wj so that vj (x) = uj (x)wj (x) for all j = 1, . . . , m. By the Hopf boundary lemma, w ∈ C 1 (Ω). Multiplying the equation for uj by 12 uj (wj2 − 1) and integrating by parts, we arrive at the identity 1 2
1 |∇uj |2 wj2 − 1 + 2uj wj ∇uj · ∇wj = − 2
Ω
u2j wj2 − 1 ∂uj G x, u2 .
(22)
Ω
Next, we expand the energy of v, using the above identity: E(v) = E(u1 w1 , . . . , um wm ) m 1 2 = wj |∇uj |2 + 2uj wj ∇uj · ∇wj + u2j |∇wj |2 + G x, v 2 2 j =1 Ω
1 = 2 m
|∇uj |2 − u2j wj2 − 1 ∂uj G x, u2 + u2j |∇wj |2 + G x, v 2
j =1 Ω
1 = E(u) + 2
u2j |∇wj |2
1 + 2
Ω
m 2 2 2 2 2 vj − uj ∂uj G x, u . G x, v − G x, u − j =1
Ω
By the strict convexity of G, we have G(x, t) − G(x, s) −
m (tj − sj )∂sj G(x, s) 0 j =1
for all x ∈ Ω and for all s, t > 0 in Rm , with equality if and only if s = t. In particular, if u ≡ v, we have E(v) > E(u). Reversing the roles of the variables u and v we also see E(u) > E(v), a contradiction, unless u = v. 2 Remark 4.2. By the same proof, we obtain uniqueness for the more general semilinear variational system, ⎧ ⎨ − div Aj (x)∇uj + ∂uj G x, u2 uj = 0, x ∈ Ω, j = 1, . . . , m, u > 0, x ∈ Ω, ⎩ u = 0, x ∈ ∂Ω, with Aj (x) symmetric, n × n elliptic matrix-valued functions in Ω. The vortex profiles f± (r) being defined on the semi-infinite interval r ∈ [0, ∞), we must modify this basic uniqueness theorem to fit this setting. In particular, the energy associated to the equivariant vortices is infinite on the entire interval. However, by the basic estimates proven in Proposition 2.1, the difference between the energies of two solutions will converge. In this setting, we let u = (f+ (r), f− (r)) and G(r, s, t) =
n2+ n2− 1 γ s + t + (1 − s − t)2 + (s − t)2 . 2 2 r2 r2
S. Alama et al. / Journal of Functional Analysis 256 (2009) 1118–1136
1135
It is easy to verify that this G is strictly convex in (s, t) > 0 for each r ∈ [0, ∞); indeed, its Hessian is the constant matrix 1+γ 1−γ 2 . D(s,t) G(r, s, t) = 1−γ 1+γ Eqs. (15) for f± (r) take the form ⎧ ⎪ − r fj + ∂fj G r, f+2 , f−2 fj = 0, ⎪ ⎪ ⎨ f± (r) > 0, r ∈ [0, ∞), ⎪ 1 ⎪ ⎪ ⎩ f± (r) → √ , r → ∞. 2
r ∈ [0, ∞), j = +, −, (23)
Recall from the proof of Lemma 3.1 that f± (r) ∼ r |n± | as r → 0 when n = 0, and if n± = 0, we have f± (0) = 0. We recall also the localized energies in r ∈ [0, ∞) defined by (16), which take the form EnR+ ,n− (f+ , f− ) =
1 2
R 2 2 2 fj (r) + G r, f+ , f− r dr. 0
j =+,−
Now assume that there are two such solutions, u = (f+ , f− ) > 0 and v = (g+ , g− ) > 0, and as above we let w be chosen with vj = uj wj , j = +, −. By Lemma 3.1 we have w ∈ C 1 [0, ∞) and uniformly bounded. Because we no longer have a Dirichlet condition at r = R, the identity (22) takes the form: 1 2
R
2 2 fj wj − 1 + 2fj , wj fj wj r dr
0
1 =− 2
R
R fj2 wj2 − 1 ∂fj G r, f+2 , f−2 r dr + rfj (r)fj (r) wj2 (r) − 1 0
0
1 =− 2
R
fj2 wj2 − 1 ∂fj G r, f+2 , f−2 r dr + o(1),
0
for j = +, − and as R → ∞, where we have used Proposition 2.1 to estimate the boundary term at r = R → ∞ and Lemma 3.1 to eliminate the term at r = 0. As in the proof of Theorem 4.1 we compare the energies using the above identity, EnR+ ,n− (g+ , g− ) − EnR+ ,n− (f+ , f− ) = EnR+ ,n− (v) − EnR+ ,n− (u) 1 = 2
R 0 j =+,−
2 u2j wj r dr + o(1)
1136
S. Alama et al. / Journal of Functional Analysis 256 (2009) 1118–1136
1 + 2
R 2 2 2 2 2 vj − uj ∂uj G r, u r dr. G r, v − G r, u − j =+,−
0
By the estimates of Proposition 2.1, the term n2± 2 2 f ± − g± 2 r is integrable, as are all the other terms which appear in the energy density. Hence the left-hand side converges as R → ∞ and using Fatou’s lemma, we conclude lim EnR+ ,n− (g+ , g− ) − EnR+ ,n− (f+ , f− )
R→∞
1 2
∞ 2 2 2 2 2 G r, v − G r, u − vj − uj ∂uj G r, u r dr > 0, 0
j =+,−
unless u = v, by the strict convexity of G(r, s, t) in (s, t) > 0. Reversing the role of u and v, we arrive at a contradiction unless u = v. References [1] S. Alama, L. Bronsard, Des vortex fractionnaires pour un modèle Ginzburg–Landau spineur, C. R. Acad. Sci. Paris Sér. I Math. 337 (2003) 243–247. [2] S. Alama, L. Bronsard, Fractional degree vortices for a spinor Ginzburg–Landau model, Commun. Contemp. Math. 8 (3) (2006) 355–380. [3] S. Alama, L. Bronsard, T. Giorgi, Uniqueness of symmetric vortex solutions in the Ginzburg–Landau model of superconductivity, J. Funct. Anal. 167 (1999) 399–424. [4] S. Alama, L. Bronsard, T. Giorgi, Vortex structures for an SO(5) model of high-TC superconductivity and antiferromagnetism, Proc. Roy. Soc. Edinburgh Sect. A 130 (2000) 1183–1215. [5] F. Bethuel, H. Brezis, F. Heléin, Asymptotics for the minimization of a Ginzburg–Landau functional, Calc. Var. Partial Differential Equations 1 (1993) 123–148. [6] F. Bethuel, H. Brezis, F. Heléin, Ginzburg–Landau Vortices, Birkhäuser Boston, Boston, MA, 1994. [7] H. Brezis, L. Oswald, Remarks on sublinear elliptic equations, Nonlinear Anal. 10 (1986) 55–64. [8] H. Brezis, F. Merle, T. Rivière, Quantization effects for − u = u(1 − |u|2 ) in R2 , Arch. Ration. Mech. Anal. 126 (1994) 35–58. [9] T. Isoshima, K. Machida, Axisymmetric vortices in spinor Bose–Einstein condensates under rotation, Phys. Rev. A 66 (2002) 023602. [10] A. Knigavko, B. Rosenstein, Spontaneous vortex state and ferromagnetic behavior of type-II p-wave superconductors, Phys. Rev. B 58 (1998) 9354–9364. [11] M. Krasnoselskii, Positive Solutions of Operator Equations, Noordhoff, Groningen, 1964. [12] P. Mironescu, Les minimiseurs locaux de l’énergie de Ginzburg–Landau sont à symétrie radiale, C. R. Acad. Sci. Paris Sér. I Math. 323 (1996) 593–598. [13] I. Shafrir, Remarks on solutions of − u = (1 − |u|2 )u in R2 , C. R. Acad. Sci. Paris Sér. Math. I 318 (1994) 327–331.
Journal of Functional Analysis 256 (2009) 1137–1188 www.elsevier.com/locate/jfa
Decomposition of Triebel–Lizorkin and Besov spaces in the context of Laguerre expansions G. Kerkyacharian a , P. Petrushev b,c,∗,1 , D. Picard a , Yuan Xu d,2 a Laboratoire de Probabilités et Modèles Aléatoires, CNRS-UMR 7599, Université Paris VI et Université Paris VII,
rue de Clisson, F-75013 Paris, France b Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA c Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Bulgaria d Department of Mathematics, University of Oregon, Eugene, OR 97403-1222, USA
Received 28 April 2008; accepted 18 September 2008 Available online 14 October 2008 Communicated by N. Kalton
Abstract A pair of dual frames with almost exponentially localized elements (needlets) are constructed on Rd+ based on Laguerre functions. It is shown that the Triebel–Lizorkin and Besov spaces induced by Laguerre expansions can be characterized in terms of respective sequence spaces that involve the needlet coefficients. © 2008 Elsevier Inc. All rights reserved. Keywords: Laguerre functions; Localized kernels; Frames; Triebel–Lizorkin spaces; Besov spaces
1. Introduction The primary goal of this paper is to construct frames on Rd+ := (0, ∞)d with nearly exponentially localized elements, based on Laguerre functions and utilize them to the characterization of spaces of distribution on Rd+ . We are interested in extending the fundamental results of Frazier and Jawerth [4–6] on the ϕ-transform on Rd in the context of Laguerre expansions. * Corresponding author at Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA.
E-mail addresses: [email protected] (G. Kerkyacharian), [email protected] (P. Petrushev), [email protected] (D. Picard), [email protected] (Y. Xu). 1 Supported by NSF Grant DMS-0709046. 2 Supported by NSF Grant DMS-0604056. 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.09.015
1138
G. Kerkyacharian et al. / Journal of Functional Analysis 256 (2009) 1137–1188
From the three types of Laguerre functions available in the literature we focus our attention on the Laguerre functions {Fνα } (see (3.1)) which form an orthonormal basis for the space L2 (Rd+ , wα ) with weight wα (x) :=
d
2αj +1
xj
(1.1)
.
j =1
For various technical reasons we will assume that αj 0, while in general αj > −1. The other two classes of Laguerre functions {Lαν } and {Mαν } (see (3.4), (3.5)) form orthogonal bases for L2 (Rd+ ) (weight 1). The d-dimensional Laguerre functions Fνα are products of univariate Laguerre functions, namely, Fνα (x) := Fνα1 (x1 ) · · · Fναd (xd ) (see (3.1), (3.3)). Hence the kernel of the orthogonal projector onto Wn := span Fνα : |ν| = n is given by Fnα (x, y) := Fνα (x)Fνα (y).
(1.2)
|ν|=n
Denote Vn := nm=0 Wm . Evidently, Kn (x, y) := nm=0 Fmα (x, y) is the kernel of the orthogonal projector onto Vn . A main point in the present paper is that for compactly supported C ∞ cut-off functions a which are constant around zero the kernels Λn (x, y) :=
∞
j Fjα (x, y) a n
(1.3)
j =0
decay rapidly (almost exponentially) away from the main diagonal in Rd+ × Rd+ (Theorem 3.2). For the same kind of kernels associated with the Laguerre functions {Mαν } in dimension d = 1 this fact is established in [3]. We show that similar results are valid for {Mαν } and {Lαν } in dimension d > 1 as well. We utilize the kernels from (1.3) to the construction of a pair of dual frames {ϕξ }ξ ∈X and {ψξ }ξ ∈X with X a multilevel index set. As in other similar settings, the almost exponential localization of ϕξ and ψξ prompts us to call them “needlets.” The needlet systems from this paper can be regarded as analogues of the ϕ-transform of Frazier and Jawerth [4,5]. They are particularly well suited for characterization of the Triebel–Lizorkin and Besov spaces associated with Laguerre expansions. To be more precise, let a ∈ C ∞ , supp a ⊂ [1/4, 4], and | a | > c on [1/3, 3] and define Φ0 (x, y) := F0α (x, y)
and Φj (x, y) :=
∞ m a j −1 Fmα (x, y), 4
j 1.
m=0
sρ
Then for all appropriate indices (see Definition 6.1) the Laguerre–Triebel–Lizorkin space Fpq is defined as the set of all tempered distributions f on Rd+ such that sρ f Fpq
∞ 1/q −ρ/d q Φj ∗ f (·) 2sj Wα 4j ; · := < ∞. j =0
p
G. Kerkyacharian et al. / Journal of Functional Analysis 256 (2009) 1137–1188
1139
Here Φj ∗ f (x) := f, Φj (x, ·) (Definition 4.2) and the weight Wα (n; x) is define by Wα (n; x) :=
d 2α +1 xj + n−1/2 j .
(1.4)
j =1
Just for convenience we use dilations by factors of 4j on the frequency side as opposed to the traditional binary dilation. The Laguerre–Besov spaces are defined by the (quasi-)norm sρ := f Bpq
∞ q sj j −ρ/d 2 Wα 4 ; · Φj ∗ f (·) p
1/q .
j =0
Unlike in the classical case on Rd the weight wα creates some inhomogeneity which compels us to introduce the additional term Wα (4j ; ·)−ρ/d with parameter ρ ∈ R. This allows to consider different scales of Triebel–Lizorkin and Besov spaces. For instance, a “classical” choice would ss and B ss which embed “correctly” with be ρ = 0. However, more natural to us are the spaces Fpq pq respect to the smoothness parameter s. The main results in this article assert that the Laguerre Triebel–Lizorkin and Besov spaces can be characterized in terms of respective sequence spaces involving the needlet coefficients of the distributions (Theorems 6.7, 7.4). Along the same lines one can develop a similar theory on Rd+ with weight 1 using the Laguerre functions {Lαν } or {Mαν }. For such spaces induced by {Lαν }, see [2]. This paper is an integral part of a broader undertaking for needlet characterization of Triebel– Lizorkin and Besov spaces on nonstandard domains (and with weights) such as the sphere [11], interval [8], ball [9], and in the setting of Hermite expansions [13]. The outline of the paper is as follows. All the information we need about Laguerre polynomials and functions is given in Section 2. The localized kernels induced by Laguerre functions are given in Section 3. Some additional background material is collected in Section 4. The construction of needlets is given in Section 5. In Section 6 the Laguerre–Triebel–Lizorkin spaces are introduced and characterized in terms of needlet coefficients, while the characterization of the Laguerre–Besov spaces is given in Section 7. Some proofs for Sections 3, 4 are given in Section 8 and for Sections 5, 6 in Section 9. The following notation will be used throughout: x := maxi |xi |, |x| := di=1 |xi |, x2 := d ( i=1 |xi |2 )1/2 , f p := ( Rd |f (x)|p wα (x) dx)1/p ; |E| stands for the Lebesgue measure of + E ⊂ Rd+ , μ(E) := E wα (x) dx, 1E is the characteristic function of E, and 1˜ E := μ(E)−1/2 1E . Positive constants are denoted by c, c1 , c∗ , . . . and they may vary at every occurrence; A ∼ B means c1 A B c2 A. 2. Background: Laguerre polynomials and functions In this section we collect the information on Laguerre polynomials and functions that will be needed in this paper. The Laguerre polynomials Lαn (α > −1) can be defined by their generating function ∞ n=0
Lαn (x)r n = (1 − r)−α−1 e−xr/(1−r) ,
|r| < 1.
1140
G. Kerkyacharian et al. / Journal of Functional Analysis 256 (2009) 1137–1188
They are orthogonal on R+ = (0, ∞) with weight x α e−x , more precisely, ∞
Lαn (x)Lαm (x)e−x x α dx =
0
(n + α + 1) δn,m = (α + 1)Lαn (0)δn,m ,
(n + 1)
[15, (5.1.1)]. where we used that Lαn (0) = n+α n αd α1 α Let Lν (x) := Lν1 (x1 ) · · · Lνd (xd ) be the product Laguerre polynomials on Rd+ , where ν = (ν1 , . . . , νd ) ∈ Nd0 and α = (α1 , . . . , αd ). For δ > −1, define Pnα,δ (x; y) :=
n
Aδn−k
Lα (x)Lα (y) ν ν , Lαν (0)
Aδm
|ν|=k
k=0
m+δ := . m
(2.1)
This is a constant multiple of the nth Cesàro sum of the reproducing kernels for Laguerre polynomials in dimension d. Using the generating function of the Laguerre polynomials, it is shown in [18] that |x| . Pnα,δ (x, 0) = L|α|+δ+d n
(2.2)
The product formula for Laguerre polynomials (Hardy–Watson) [16, Proposition 6.1.1] asserts that for α > − 12 and x, y ∈ R+ ,
(n + 1) Lαn x 2 Lαn y 2
(n + α + 1) π 2α =√ Lαn x 2 + y 2 + 2xy cos θ e−xy cos θ jα−1/2 (xy sin θ ) sin2α θ dθ, 2π
(2.3)
0
where jα (x) := x −α Jα (x) with Jα (x) being the Bessel function. It will be convenient to denote x 2 := (x12 , . . . , xd2 ). Combining (2.1)–(2.3), we arrive at Pnα,δ x 2 , y 2 = cα
Pnα,δ z(x, y, θ ), 0 dμαx,y (θ )
[0,π]d
L|α|+δ+d n
= cα
x22
+ y22
+
d
xi yi cos θi dμαx,y (θ ),
(2.4)
i=1
[0,π]d
where cα = (2π)−d/2 2|α| di=1 (αi + 1), z(x, y, θ ) = (z1 (x, y, θ ), . . . , zd (x, y, θ )) with zi (x, y, θ ) = xi2 + yi2 + 2xi yi cos θi , and dμαx,y (θ ) := e−
d
i=1 xi yi
cos θi
d i=1
jαi −1/2 (xi yi sin θi ) sin2αi θi dθ.
(2.5)
G. Kerkyacharian et al. / Journal of Functional Analysis 256 (2009) 1137–1188
1141
Some standard asymptotic properties of Laguerre functions will be needed. The univariate Laguerre functions Lαn are defined by Lαn (x) :=
(n + 1)
(n + α + 1)
1/2
e−x/2 x α/2 Lαn (x).
(2.6)
Lemma 2.1. Set N := 4n + 2α + 2. The Laguerre functions Lαn satisfy ⎧ ⎪ (xN )α/2 , ⎪ ⎨ −1/4 , α L (x) c (xN ) n −1/4 ⎪ (N 1/3 + |N − x|)−1/4 , N ⎪ ⎩ −γ x , e
0 < x 1/N, 1/N x N/2, N/2 x 3N/2, x 3N/2,
(2.7)
where γ > 0 is an absolute constant. This lemma is contained in [15, Section 8.22] (see also [16, Lemma 1.5.3]). Using that (n + α + 1)/ (n + 1) ∼ nα one easily extracts from (2.7) the estimates e−x/2 Lαn (x) cnα/2−1/4 x −α/2−1/4 ,
x ∈ R+ \ (N/2, 3N/2),
(2.8)
−1/4 e−x/2 Lαn (x) cx −α/2 nα/2−1/4 n1/3 + |4n + 2α + 2 − x| .
(2.9)
and, for N/2 x 3N/2,
Also, from (2.7) e−x/2 Lαn (x) cnα ,
x ∈ R+ ,
(2.10)
and since Lαn ∞ c, again by (2.7), e−x/2 Lαn (x) c(n/x)α/2 ,
x ∈ R+ .
(2.11)
Let Knα (x, y) be the reproducing kernel of the Laguerre polynomials. Then Knα (x, y) = cα
n Lα (x)Lα (y) j j j =0
Lαj (0)
,
x, y ∈ R+ .
(2.12)
The Christoffel function is defined by
−1 λαn (x) := Knα (x, x) ,
x ∈ R+ .
(2.13)
For this function it is known that (see [10] and the references therein) c1 ϕn (x)
λαn (x) (x + n1 )α e−x
c2 ϕn (x),
0 x 4n,
(2.14)
1142
G. Kerkyacharian et al. / Journal of Functional Analysis 256 (2009) 1137–1188
where ϕn (x) :=
x + n1 . 4n − x + (4n)1/3
(2.15)
There are sharp estimates for Lαn (x) in terms of ϕn (x). For any x > 0, let tkx ,n denote the/a zero of Lαn (x) that is closest to x. Then (see e.g. [10])
α 2 1 α+1 −x (x − tkx ,n )2 e ∼ nα ϕn (x) , Ln (x) x + n (tkx ,n − tkx ±1,n )2
x ∈ [t1,n , tn,n ].
(2.16)
Here and in the following t1,n , . . . , tn,n denote the zeros of Lαn (x). They are known to satisfy [15, Section 6.31] cn−1 t1,n < t2,n < · · · < tn,n 4n + 2α + 2 − c(4n)1/3 .
(2.17)
Furthermore (see [15, (6.31.11)]), c∗
ν ν2 4ν 2 tν,n + c(α) n n n
and hence tν,n ∼
ν2 . n
(2.18)
In addition (see [10] and the references therein), tν+1,n − tν,n ∼ ϕn (tν,n ).
(2.19)
Therefore, if ν (1 − ε)n for some ε > 0, then by (2.18) tν,n (1 − ε)2 4n + c(α), and hence, using (2.19) and (2.15), tν+1,n − tν,n ∼
ν n
if ν (1 − ε)n.
(2.20)
On the other hand, by (2.19) and (2.15), in general, c tν+1,n − tν,n c n1/3 . n
(2.21)
We will need the Gaussian quadrature formula with weight t α e−t on (0, ∞) [15]: ∞ 0
f (t)t α e−t dt ∼
n
wν,n f (tν,n ),
wν,n := λαn (tν,n ),
(2.22)
ν=1
where tν,n are the zeros of Lαn (t) and λαn (x) is the Christoffel function, defined in (2.13). This quadrature is exact for all algebraic polynomials of degree 2n − 1.
G. Kerkyacharian et al. / Journal of Functional Analysis 256 (2009) 1137–1188
1143
3. Localized kernels associated with Laguerre functions 3.1. The setting There are three kinds of univariate Laguerre functions considered in the literature (see [16]), defined by Fnα (x) :=
2 (n + 1)
(n + α + 1)
1/2
e−x
2 /2
Lαn x 2 ,
(3.1)
Lαn (x) have already been defined in (2.6), and Mαn (x) := (2x)1/2 Lαn x 2 .
(3.2)
It is well known that {Fnα }n0 is an orthonormal basis for the weighed space L2 (R+ , x 2α+1 ), while {Lαn }n0 and {Mαn }n0 are orthonormal bases for L2 (R+ ). Throughout this paper we will use standard multi-index notation. Thus, for x ∈ Rd and α ∈ d R+ , we write x α := x1α1 . . . xdαd . We will use 1 to denote the vector 1 := (1, 1, . . . , 1). Then, for 1/2 1/2 instance, x 1/2 := x1 . . . xd . The d-dimensional Laguerre functions are defined by Fνα (x) := Fνα11 (x1 ) . . . Fναdd (xd ),
(3.3)
Lαν (x) := Lαν11 (x1 ) . . . Lανdd (xd ),
(3.4)
Mαν (x) := Mαν11 (x1 ) . . . Mανdd (xd ),
(3.5)
where ν = (ν1 , . . . , νd ) ∈ Nd0 and α = (α1 , . . . , αd ). Clearly, x −α e|x| Lαν (x) is a polynomial of degree n = |ν| = ν1 + . . . + νd and Fνα (x) = 2d/2 x −α Lαν x12 , . . . , xd2 .
(3.6)
Evidently, {Fνα } is an orthonormal basis for the weighed space L2 (Rd+ , wα ), wα (x) := x 2α+1 , while {Lαν } and {Mαν } are orthonormal bases for L2 (Rd+ ) (with weight 1). We will utilize the basis {Fνα } to the construction of frames for the space L2 (wα ) := L2 (Rd+ , wα ). The same scheme based on {Lαν } or {Mαν } can be used for the construction of frames in L2 (Rd+ ). As explained in the introduction, kernels of type (1.3) will play a critical role in the present paper. For our purposes we will be considering cut-off functions a that satisfy: Definition 3.1. A function a ∈ C ∞ [0, ∞) is said to be admissible of type (a) or type (b) if a satisfies one of the following conditions: (a) supp a ⊂ [0, 1 + v], a (t) = 1 on [0, 1], v > 0; or (b) supp a ⊂ [u, 1 + v], where 0 < u < 1 and v > 0. Here u, v are fixed constants.
1144
G. Kerkyacharian et al. / Journal of Functional Analysis 256 (2009) 1137–1188
For an admissible function a we introduce the kernels Λn (x, y) :=
∞
m Fmα (x, y) a n ∞
m α Lm (x, y) a n
with Lαm (x, y) :=
(3.7)
∞
m Mαm (x, y) a n
Lαν (x)Lαν (y),
(3.8)
|ν|=m
m=0
Λ∗n (x, y) :=
Fνα (x)Fνα (y),
|ν|=m
m=0
n (x, y) := Λ
with Fmα (x, y) :=
with Mαm (x, y) :=
Mαν (x)Mαν (y).
(3.9)
|ν|=m
m=0
n (x, y), and Λ∗n (x, y) and their partial derivatives The rapid decay of the kernels Λn (x, y), Λ d away from the main diagonal y = x in R+ × Rd+ will be vital for our further development. 3.2. The localization of Λn and its partial derivatives Recall the definition of the weight Wα (n; x) :=
d
i=1 (xi
+ n−1/2 )2αi +1 .
Theorem 3.2. Let a be admissible and let σ > 0. Then there is a constant cσ depending only on σ , α, and a such that for x, y ∈ Rd+ Λn (x, y) cσ √
nd/2 , √ Wα (n; x) Wα (n; y)(1 + n1/2 x − y)σ
(3.10)
and furthermore, for 1 r d, ∂ n(d+1)/2 cσ √ Λ (x, y) . √ ∂x n Wα (n; x) Wα (n; y)(1 + n1/2 x − y)σ r
(3.11)
a is of the form cσ = c(σ, α) max0lk a (l) L∞ , where k Here the dependence of cσ on σ + 2|α| + d/2. In addition to this, there exists a constant > 0 such that if x, y ∈ Rd+ and max{x, y} (6(1 + v)n + 3α + 3)1/2 , then Λn (x, y) cσ
e− max{x,y} (1 + n1/2 x − y)σ 2
(3.12)
and, for 1 r d, 2 ∂ e− max{x,y} ∂x Λn (x, y) cσ (1 + n1/2 x − y)σ .
(3.13)
r
To keep our exposition more fluid we relegate the proofs of these and the estimates to follow in this section to Section 8. p We next use estimate (3.10) to bound the L -integral of Λn (x, y), in particular, we show that Rd |Λn (x, y)|wα (y) dy c < ∞. +
G. Kerkyacharian et al. / Journal of Functional Analysis 256 (2009) 1137–1188
Proposition 3.3. For 0 < p < ∞, we have Λn (x, y)p wα (y) dy cn(d/2)(p−1) Wα (n; x)−(p−1) ,
1145
x ∈ Rd+ .
(3.14)
Rd+
Estimate (3.14) is immediate from (3.10) and the following lemma which will be instrumental in the subsequent development. Lemma 3.4. If s ∈ R and σ > d((2α + 1)(|s| + 1) + 1), then Rd+
Wα
wα (y) dy s (n; y) (1 + n1/2 x
− y)σ
cn−d/2 , Wα (n; x)s−1
x ∈ Rd+ .
(3.15)
We next give a lower bound estimate: Theorem 3.5. Let a be admissible in the sense of Definition 3.1 and | a | > c > 0 on [1, 1 + τ ], τ > 0. Then for any δ > 0
Λn (x, y)2 wα (y) dy c nd/2 Wα (n; x)−1 , x ∈ 0, (4 − δ)n d , (3.16) Rd+
where c > 0 depends only on α, d, τ , δ, and c . By the orthogonality of the Laguerre functions it readily follows that
∞ 2 Λn (x, y)2 wα (y) dy = a (m/n) F α (x, x), m
m=0
Rd+
and hence Theorem 3.5 is an immediate consequence of the following lemma. Lemma 3.6. For any ε > 0 and δ > 0 there exists a constant c > 0 such that n+ dεn
d x ∈ 0, (4 − δ)n .
Fmα (x, x) cnd/2 Wα (n; x)−1 ,
(3.17)
m=n
n and its partial derivatives 3.3. The localization of Λ n can be deduced from the localization of Λn given above. The localization of the kernels Λ Theorem 3.7. Let a be admissible. Then for any σ > 0 there is a constant cσ > 0 depending only on σ , α, and a such that for x, y ∈ Rd+ , Λ n (x, y) cσ
d
i=1 (xi
nd/2 1
1
+ n−1 ) 4 (yi + n−1 ) 4 (1 + n1/2 x 1/2 − y 1/2 )σ
,
(3.18)
1146
G. Kerkyacharian et al. / Journal of Functional Analysis 256 (2009) 1137–1188
and, for 1 r d, ∂ n (x, y) Λ ∂x d
cnd/2+1
i=1 (xi
r
1 4
1
+ n−1 ) (yi + n−1 ) 4 (1 + n1/2 x 1/2 − y 1/2 )σ
.
(3.19)
a is as in Theorem 3.7. Here the dependence of cσ on n like the ones of (3.12)–(3.16) can be extracted from (3.12)–(3.16). The Estimates for Λ results from this and the next subsections follow easily from Theorem 3.2, see Section 8.3. 3.4. The localization of Λ∗n and its partial derivatives The localization properties of Λ∗n (x, y) appear simpler: Theorem 3.8. Let a be admissible. Then for any σ > 0 there is a constant cσ such that for x, y ∈ Rd+ ∗ Λ (x, y) cσ n
nd/2 , (1 + n1/2 x − y)σ
(3.20)
and, for 1 r d, ∂ ∗ n(d+1)/2 cσ Λ (x, y) . ∂x n (1 + n1/2 x − y)σ
(3.21)
r
Estimates for Λ∗n similar to the ones of (3.12)–(3.16) can easily be obtained. 4. Additional background material 4.1. Norm equivalence Proposition 4.1. Let 0 < q p ∞ and g ∈ Vn (n 1). Then gp cn(d+|α|)(1/q−1/p) gq and, for any s ∈ R, Wα (n; ·)s g(·) cn(d/2)(1/q−1/p) Wα (n; ·)s+1/p−1/q g(·) . p q
(4.1)
(4.2)
Furthermore, for any s ∈ R gp cnM Wα (n; ·)s g(·) q ,
(4.3)
where M depends only on α, d, p, q, and s. The proof of this proposition employs the localized kernels from Section 3 and is rather standard. For completeness we give it in Section 8.
G. Kerkyacharian et al. / Journal of Functional Analysis 256 (2009) 1137–1188
1147
4.2. Maximal operator We define the “cube” centered at ξ ∈ Rd+ of “radius” r > 0 by Qξ (r) := {x ∈ Rd+ : x − ξ < r}. Let Mt be the maximal operator, defined by Mt f (x) := sup Q: x∈Q
1 μ(Q)
f (y)t wα (y) dy
1/t ,
x ∈ Rd+ ,
(4.4)
Q
where the sup is over all “cubes” Q in Rd+ with sides parallel to the coordinate axes which contain x. It is easy to see that d μ Qξ (r) ∼ r d (ξj + r)2αj +1 .
(4.5)
j =1
Hence μ(Qξ (2r)) cμ(Qξ (r)), i.e. μ(·) is a doubling measure. Therefore, the theory of maximal operators applies and the Fefferman–Stein vector-valued maximal inequality is valid (see [14]): if 0 < p < ∞, 0 < q ∞, and 0 < t < min{p, q}, then for any sequence of functions f1 , f2 , . . . on Rd+ ∞ ∞ 1/q 1/q q q fj (·) Mt fj (·) c , j =1
p
j =1
(4.6)
p
where c = c(p, q, t, d, α). 4.3. Distributions on Rd+ We will use as test functions the set S+ of all functions φ ∈ C ∞ ([0, ∞)d ) such that Pβ,γ (φ) := sup x γ ∂ β φ(x) < ∞
for all multi-indices γ and β,
(4.7)
x∈Rd+ of all temperate with the topology on S+ defined by the semi-norms Pβ,γ . Then the space S+ d distributions on R+ is defined as the set of all continuous linear functionals on S+ . The pairing of and φ ∈ S will be denoted by f, φ := f (φ) which is consistent with the inner product f ∈ S+ + f, g := Rd f (x)g(x)wα (x) dx in L2 (Rd+ , wα ). + It will be convenient for us to introduce the following “convolution.”
Definition 4.2. For functions Φ : Rd+ × Rd+ → C and f : Rd+ → C, we define Φ ∗ f (x) := Rd+
Φ(x, y)f (y)wα (y) dy.
(4.8)
1148
G. Kerkyacharian et al. / Journal of Functional Analysis 256 (2009) 1137–1188
and Φ : Rd × Rd → C is such that Φ(x, y) belongs to S as a function of In general, if f ∈ S+ + + + y (Φ(x, ·) ∈ S+ ), we define Φ ∗ f by
Φ ∗ f (x) := f, Φ(x, ·) ,
(4.9)
where on the right-hand side f acts on Φ(x, y) as a function of y. We now give some properties of the above convolution that can be proved in a standard way. Lemma 4.3. and Φ(·,·) ∈ S (Rd × Rd ), then Φ ∗ f ∈ S . Furthermore F α ∗ f ∈ V . (a) If f ∈ S+ + + n + + n , Φ(·,·) ∈ S (Rd × Rd ), and φ ∈ S , then Φ ∗ f, φ = f, Φ ∗ φ. (b) If f ∈ S+ + + + + , Φ(·,·), Ψ (·,·) ∈ S (Rd × Rd ), and Φ(y, x) = Φ(x, y), Ψ (y, x) = Ψ (x, y), then (c) If f ∈ S+ + + +
Ψ ∗ Φ ∗ f (x) = Ψ (x, ·), Φ(·,·) ∗ f.
(4.10)
Evidently the Laguerre functions {Fνα } belong to S+ . Moreover, the functions in S+ can be characterized by the coefficients in their Laguerre expansions. Denote Pr∗ (φ) :=
∞ ∞ 1/2 φ, F α 2 (n + 1)r Fnα ∗ φ 2 = (n + 1)r . ν n=0
n=0
(4.11)
|ν|=n
Lemma 4.4. A function φ ∈ S+ if and only if |φ, Fνα | ck (|ν| + 1)−k for all multi-indices ν and all k. Moreover, the topology in S+ can be equivalently defined by the semi-norms Pr∗ . The proof of this lemma is given in Section 8. 5. Construction of frame elements (needlets) In this section we construct frames utilizing the localized kernels from Section 3 and a cubature formula on Rd+ . As explained in the introduction, we will only use the Laguerre functions {Fνα } defined in (3.3). 5.1. Cubature formula We will utilize the Gaussian quadrature (2.22) for the construction of the needed cubature formula on Rd+ . Given n 1, we define, for ν = 1, . . . , n, ξν,n :=
tν,n
1 1 1 2 ξν,n 2 e , and cν,n := wν,n etν,n = λαn (tν,n )etν,n = λαn ξν,n 2 2 2
where {tν,n } are the zeros of Lαn (t) and {wν,n } are the weights from (2.22). It follows by (2.18) and (2.20), (2.21) that
(5.1)
G. Kerkyacharian et al. / Journal of Functional Analysis 256 (2009) 1137–1188
ν ξν,n ∼ √ , n ξν+1,n − ξν,n ∼ n−1/2
1149
(5.2)
if 1 ν (1 − ε)n,
(5.3)
and, in general, c1 n−1/2 ξν+1,n − ξν,n c2 n−1/6 .
(5.4)
Furthermore, using (2.14) and (2.19) we obtain α α 2α+1 cν,n ∼ ϕn (tν,n )tν,n ∼ (tν+1,n − tν,n )tν,n ∼ (ξν+1,n − ξν,n )ξν,n .
(5.5)
Now, for γ = (γ1 , . . . , γd ) ∈ Nd0 we set cγ ,n :=
d
cγj ,n
and ξγ ,n := (ξγ1 ,n , . . . , ξγd ,n ).
(5.6)
j =1
Proposition 5.1. The cubature formula f (x)g(x)wα (x) dx ∼
n γ1 =1
Rd+
...
n
cγ ,n f (ξγ ,n )g(ξγ ,n )
(5.7)
γd =1
is exact for all f ∈ V and g ∈ Vm provided + m 2n − 1. Proof. Evidently, it suffices to consider only the case d = 1. Suppose f ∈ V and g ∈ Vm with 2 2 + m 2n − 1. Let f (x) =: F (x 2 )e−x /2 and g(x) =: G(x 2 )e−x /2 , where F ∈ Π1 , G ∈ Πm1 with Πj1 being the set of all univariate polynomials of degree j . Then using the properties of quadrature formula (2.22), we get ∞
∞ f (x)g(x)wα (x) dx =
0
1 2 F x 2 G x 2 x 2α+1 e−x dx = 2
0
= =
1 2
F (t)G(t)t α e−t dt
0 n ν=1
n ν=1
which completes the proof.
∞
wν,n F (tν,n )G(tν,n ) =
n ν=1
2 2 1 wν,n F ξν,n G ξν,n 2
1 α 2 ξν,n 2 λ ξ e f (ξν,n )g(ξν,n ), 2 n ν,n
2
To construct our frame elements we need the cubature formula from (5.7) with √ n = nj := c∗−1 (1 + 11δ) 6 · 4j + 1 ∼ 4j ,
(5.8)
1150
G. Kerkyacharian et al. / Journal of Functional Analysis 256 (2009) 1137–1188
where 0 < c∗ 1 is the constant from (2.18) and 0 < δ < 1/26 is an arbitrary but fixed constant. For j 0, we define Xj := ξ ∈ Rd+ : ξ = ξγ ,nj , 1 γ nj , 1 d .
(5.9)
Note that #Xj = ndj ∼ 4j d . Now, if ξ ∈ Xj and ξ = ξγ ,nj , we set cξ := cγ ,nj . As an immediate consequence of Proposition 5.1 we get Corollary 5.2. The cubature formula f (x)g(x)wα (x) dx ∼
cξ f (ξ )g(ξ )
(5.10)
ξ ∈ Xj
Rd+
is exact for all f ∈ V and g ∈ Vm provided + m 2nj − 1. Tiling. We next introduce rectangular tiles {Rξ } with “centers” at the points ξ ∈ Xj . Set I1 := [0, (ξ1 + ξ2 )/2] and
Iν := (ξν−1 + ξν )/2, (ξν + ξν+1 )/2 ,
ν = 2, . . . , nj ,
where ξν := ξν,nj , ν = 1, . . . , nj , are from (5.1) and ξnj +1 := ξnj + 2j/3 . To every ξ = ξγ = (ξγ1 , . . . , ξγd ) in Xj we associate a tile Rξ defined by Rξ := Iγ1 × · · · × Iγd .
(5.11)
We also set Qj :=
!
(5.12)
Rξ .
ξ ∈ Xj
Evidently, different tiles Rξ do not overlap and Qj ∼ [0, 2j ]d . By (5.5) it readily follows that cξ ∼ μ(Rξ ) :=
wα (x) dx ∼ |Rξ |wα (ξ ) ∼ |Rξ |Wα 4j ; ξ .
(5.13)
Rξ
√ 1/2 −1/2 Assume ξ ∈ Xj , ξ := ξγ , and ξ (1 + 4δ) 6 · 2j . By (2.18) ξγ c∗ γ nj and hence √ −1/2 1/2 j γ c∗ (1 + 4δ) 6 · 2 nj (1 − δ)nj , where the last inequality follows by the selection of nj in (5.8). Therefore, for ξ ∈ Xj d
Rξ ∼ ξ + −2−j , 2−j
and μ(Rξ ) ∼ 2−j d wα (ξ )
while in general, for some positive constants c1 , c2 , c , c ,
√ if ξ (1 + 4δ) 6 · 2j , (5.14)
G. Kerkyacharian et al. / Journal of Functional Analysis 256 (2009) 1137–1188
d
ξ + −c1 2−j , c1 2−j ⊂ Rξ ⊂ ξ + [−c2 2−j/3 , c2 2−j/3 ]d
and
1151
(5.15)
c 2−j d wα (ξ ) μ(Rξ ) c 2−j d/3 wα (ξ ).
(5.16)
The following simple inequality is immediate from the definition of Wα (n; x) in (1.2) and will be useful in what follows: 2|α|+d Wα 4j ; y Wα 4j ; x 1 + 2j x − y ,
x, y ∈ Rd+ .
(5.17)
5.2. Definition of needlets Let a, b satisfy the conditions: a, b ∈ C ∞ (R), supp a , supp b ⊂ [1/4, 4], b(t) > c > 0 if t ∈ [1/3, 3], a (t),
(5.18)
a (t) b(t) + a (4t) b(4t) = 1
(5.20)
(5.19)
if t ∈ [1/4, 1].
Hence, ∞
a (4−m t) b 4−m t = 1,
t ∈ [1, ∞).
(5.21)
m=0
It is readily seen that (e.g. [5]) for any a satisfying (5.18), (5.19) there exists b satisfying (5.18), (5.19) such that (5.20) holds. Let a, b satisfy (5.18)–(5.20). Then we set Φ0 (x, y) := F0α (x, y),
Φj (x, y) :=
∞ m a j −1 Fmα (x, y), 4
and
(5.22)
j 1.
(5.23)
m=0
Ψ0 (x, y) := F0α (x, y),
Ψj (x, y) :=
∞ m b j −1 Fmα (x, y), 4
m=0
Let Xj be the set defined in (5.9) and let cξ be the coefficients of cubature formula (5.10). We define the j th level needlets by 1/2
ϕξ (x) := cξ Φj (x, ξ )
1/2
and ψξ (x) := cξ Ψj (x, ξ ),
ξ ∈ Xj .
(5.24)
" Set X := ∞ j =0 Xj . We will use X as an index set for our needlet systems Φ and Ψ . For this reason, (possibly) identical points from different levels Xj are considered as distinct elements of X . We define Φ := {ϕξ }ξ ∈X ,
Ψ := {ψξ }ξ ∈X .
We will term {ϕξ } analysis needlets and {ψξ } synthesis needlets.
(5.25)
1152
G. Kerkyacharian et al. / Journal of Functional Analysis 256 (2009) 1137–1188
Localization of needlets. An immediate consequence of Theorem 3.2 is the estimate: for any σ > 0 there exists a constant cσ > 0 such that for all x, y ∈ Rd+ Φj (x, y), Ψj (x, y)
cσ 2j d , Wα (4j , x) Wα (4j , y)(1 + 2j x − y)σ
(5.26)
√ while cσ 2j d can be replaced by c(σ, L)2−j L if max{x, y} (1 + δ) 6 · 2j , where L > 0 is an arbitrary constant but the constant c(σ, L) depends on L as well. We employ (5.26) and (5.13) to obtain for ξ ∈ Xj ϕξ (x), ψξ (x)
c2j d/2 Wα
(4j , x)(1 + 2j x
x ∈ Rd+ ,
(5.27)
√ if ξ (1 + δ) 6 · 2j .
(5.28)
− ξ )σ
,
and ϕξ (x), ψξ (x)
c2−j L Wα (4j , x)(1 + 2j x − ξ )σ
,
and Lp (Rd ) have discrete decompositions via needlets. We next show that S+ +
Proposition 5.3. , then (a) If f ∈ S+
f=
∞ j =0
f=
Ψj ∗ Φ j ∗ f ξ ∈X
f, ϕξ ψξ
in S+
and
(5.29)
in S+ .
(5.30)
(b) If f ∈ Lp (wα ), 1 p < ∞, then (5.29), (5.30) hold in Lp (wα ). Moreover, if 1 < p < ∞, then the convergence in (5.29), (5.30) is unconditional. Proof. (a) Note that Ψj ∗ Φ j (x, y) is well defined since Ψj (x, y) and Φj (x, y) are symmetric functions (e.g. Ψj (y, x) = Ψj (x, y)). By (5.22), (5.23) it follows that Ψ0 ∗ Φ 0 = P0 and
m m a j −1 b j −1 Fmα (x, y), 4 4 j −2
j
4
Ψj ∗ Φ j (x, y) =
j 1.
(5.31)
m=4
Hence, (5.21) and Lemma 4.4 imply (5.29). Evidently, Ψj (x, ·) and Φj (y, ·) belong to V4j and using the cubature formula from Corollary 5.2, we infer Ψj ∗ Φ j (x, y) = Rd+
Ψj (x, u)Φj (y, u) du
G. Kerkyacharian et al. / Journal of Functional Analysis 256 (2009) 1137–1188
=
cξ Ψj (x, ξ )Φj (y, ξ ) =
ξ ∈ Xj
1153
ψξ (x)ϕξ (y).
ξ ∈ Xj
Therefore, Ψj ∗ Φ j ∗ f = ξ ∈Xj f, ϕξ ψξ and combining this with (5.29) gives (5.30). (b) In Lp identity (5.29) follows easily by the rapid decay of the kernels of the nth partial sums. We skip the details. In Lp , identity (5.30) follows as above. The unconditional convergence in Lp (wα ), 1 < p < ∞, is a consequence of Proposition 6.3 and Theorem 6.7 below. 2 a and Remark 5.4. Suppose a 0. Then ϕξ = ψξ and that in the needlet construction b = (5.30) becomes f = ξ ∈X f, ψξ ψξ . It is easily seen that this representation holds in L2 and f L2 = ( ξ ∈X |f, ψξ |2 )1/2 , f ∈ L2 , i.e. {ψξ }ξ ∈X is a tight frame for L2 (Rd+ , wα ). 6. Laguerre–Triebel–Lizorkin spaces We follow the general idea of using spectral decompositions (see e.g. [12,17]) to introduce Triebel–Lizorkin spaces on Rd+ in the context of Laguerre expansions. Our main goal is to show that these spaces can be characterized via needlet representations. 6.1. Definition of Laguerre–Triebel–Lizorkin spaces Let a sequence of kernels {Φj } be defined by Φ0 (x, y) := F0α (x, y)
∞ m and Φj (x, y) := a j −1 Fmα (x, y), 4
j 1,
(6.1)
m=0
where {Fmα (x, y)} are from (3.7) and a obeys the conditions a ⊂ [1/4, 4], a ∈ C ∞ [0, ∞), supp a (t) > c > 0, if t ∈ [1/3, 3].
(6.2) (6.3)
Definition 6.1. Let s, ρ ∈ R, 0 < p < ∞, and 0 < q ∞. Then the Laguerre–Triebel–Lizorkin sρ sρ such that space Fpq := Fpq (F α ) is defined as the set of all distributions f ∈ S+ sρ f Fpq
∞ 1/q q j −ρ/d sj Φj ∗ f (·) 2 Wα 4 ; · := <∞ j =0
(6.4)
p
with the usual modification when q = ∞. As is shown in Theorem 6.7 below the above definition is independent of the choice of a as long as a satisfies (6.2), (6.3). sρ
Proposition 6.2. For all s, ρ ∈ R, 0 < p < ∞, and 0 < q ∞, Fpq is a (quasi-)Banach space . which is continuously embedded in S+
1154
G. Kerkyacharian et al. / Journal of Functional Analysis 256 (2009) 1137–1188 sρ
Proof. The completeness of the space Fpq follows easily (see e.g. [17, p. 49]) by the continuous sρ , which we establish next. embedding of Fpq in S+ sρ Let {Φj } be the kernels from the definition of Fpq with a obeying (6.2), (6.3) that are the same as (5.18), (5.19). As already indicated there exists a function b satisfying (5.18)–(5.20). sρ We use this function to define {Ψ } as in (5.23). Assume f ∈ F j pq . Then by Proposition 5.3 and hence f= ∞ Ψ ∗ Φ ∗ f in S j + j =0 j ∞
f, φ =
Ψj ∗ Φ j ∗ f, φ,
φ ∈ S+ .
j =0
We now employ (5.31) and the Cauchy–Schwarz inequality to obtain, for j 2, 2 Ψj ∗ Φ j ∗ f, φ =
j
4 m=4j −2 +1 j
4
2
m m α a j −1 b j −1 Fm ∗ f, Fmα ∗ φ 4 4
2 m 2 α a 4j −1 Fm ∗ f 2
m=4j −2 +1
j
Φj ∗ f 22
4
j
4 m=4j −2 +1
2 m 2 α b 4j −1 Fm ∗ φ 2
α F ∗ φ 2 . m 2
m=4j −2 +1
Using inequality (4.3) we get −ρ/d sρ , Φj ∗ f 2 c2j (M+|s|) 2sj Wα 2j ; · Φj ∗ f (·) p c2j (M+|s|) f Fpq where M depends on p, α, d, and ρ. From the above estimates we infer Ψj ∗ Φ j ∗ f, φ c2−j f sρ 2j k Fpq
α F ∗ f c2−j f sρ P ∗ (φ) m Fpq k 2
4j −2 <m4j
for k M + |s| + 1. A similar estimate trivially holds for j = 0, 1. Summing up we get f, φ cf which completes the proof.
∗ sρ Fpq Pk (φ),
2
Proposition 6.3. The following identification holds: 00 Fp2 ∼ Lp (wα ),
with equivalent norms.
1 < p < ∞,
(6.5)
G. Kerkyacharian et al. / Journal of Functional Analysis 256 (2009) 1137–1188
1155
The proof of this proposition is the same as the proof of Proposition 4.3 in [11] in the case of spherical harmonics. We omit it. Rough Lp multipliers for Laguerre expansions can be used for the proof. However, since we cannot find in the literature any multipliers for the Laguerre expansions we use in the present paper, we next give easy to prove but non-optimal multipliers. Proposition 6.4. Let k be a sufficiently large integer (k > (5/2)|α| + (7/4)d + 3 will do) and suppose m ∈ C k (R+ ) obeys sup t j m(j ) (t) c
for j = 0, 1, . . . , k.
(6.6)
t∈R+
Then the operator Tmα (f ) :=
∞
α j =0 m(j )Fj
∗ f is bounded on Lp (wα ), 1 < p < ∞.
The proof is given in Section 9. 6.2. Needlet decomposition of Laguerre–Triebel–Lizorkin spaces As a companion to Fpq we now introduce the sequence spaces fpq . Here {Xj }∞ j =0 is the sequence of points from (5.9) with associated " tiles {Rξ }ξ ∈Xj , defined in (5.11). Just as in the definition of needlets in Section 5, we set X := j 0 Xj . sρ
sρ
sρ
Definition 6.5. Suppose s, ρ ∈ R, 0 < p < ∞, and 0 < q ∞. Then fpq is defined as the space of all complex-valued sequences h := {hξ }ξ ∈X such that sρ hfpq
∞ 1/q j −ρ/d q sj q |hξ |Wα 4 ; ξ := 2 1˜ Rξ (·) <∞ j =0
ξ ∈ Xj
(6.7)
p
with the usual modification for q = ∞. Recall that 1˜ Rξ := μ(Rξ )−1/2 1Rξ . In analogy to the classical case on Rd we introduce “analysis” and “synthesis” operators by Sϕ : f → f, ϕξ ξ ∈X
and Tψ : {hξ }ξ ∈X →
(6.8)
hξ ψξ .
ξ ∈X sρ
We next show that the operator Tψ is well defined on fpq . sρ
Lemma 6.6. Let s, ρ ∈ R, 0 < p < ∞, and 0 < q ∞. Then for any h ∈ fpq , Tψ h := sρ ξ ∈X hξ ψξ converges in S+ . Moreover, the operator Tψ : fpq → S+ is continuous, i.e. there exist constants k > 0 and c > 0 such that Tψ h, φ | cP ∗ (φ)h k
sρ
sρ
fpq
sρ for h ∈ fpq and φ ∈ S+ .
sρ
Proof. Let h ∈ fpq . Using the definition of fpq we obtain −ρ/d 1˜ R (·) h sρ 2j s |hξ |Wα 4j ; ξ ξ fpq p
for ξ ∈ Xj , j 0.
(6.9)
1156
G. Kerkyacharian et al. / Journal of Functional Analysis 256 (2009) 1137–1188
But (5.16) gives 1˜ Rξ p = μ(Rξ )1/p−1/2 c[2−j d Wα (4j , ξ )]1/p−1/2 for ξ ∈ Xj and since 2−j (2|α|+d) Wα (4j , ξ ) c2j (2|α|+d) it follows that for ξ ∈ Xj with M := |s| + 2 |α| + d |ρ|/d + |1/p − 1/2| .
sρ |hξ | c2j M hfpq
By Lemma 4.4 φ = 1/2
∞
α n=0 Fn
(6.10)
∗ φ in S+ for φ ∈ S+ and hence for ξ ∈ Xj
m b j −1 Fmα (x, ξ ), 4 j
1/2
ψξ (x) := cξ Ψj (x, ξ ) = cξ
cξ ∼ |Rξ |Wα 4j , ξ .
4j −2 <m<4
Therefore,
m b j −1 Fmα ∗ φ 4 j
1/2 ψξ , φ = cξ
4j −2 <m<4
and hence
ψξ , φ c2−j (|α|+d)
4j −2 <m<4j
α F ∗ φ . m ∞
Since Fmα ∗ φ ∈ Vm , by Proposition 4.1 Fmα ∗ φ∞ cm(d+|α|)/2) Fmα ∗ φ2 and hence
ψξ , φ c2j (2|α|+2d)
α F ∗ φ . m 2
4j −2 <m<4j
This along with (6.10) and the fact that #Xj c4j d yields, for φ ∈ S+ , ξ ∈X
∞ |hξ |ψξ , φ |hξ |ψξ , φ j =0 ξ ∈Xj
ch
sρ fpq
sρ chfpq
∞ (#Xj )2j (M+2|α|+2d) j =0 ∞
α F ∗ φ m 2
4j −2 <m<4j ∞ (m + 1)k Fmα ∗ φ 2 2j (M+2|α|+4d+1−k)
m=0
j =0
∗ sρ P (φ), chfpq k
(6.11)
where k := M + 2|α| + 4d + 2 > M + 2|α| + 4d + 1. Therefore, the series ξ ∈X hξ ψξ con . We define T h by T h, φ := verges in S+ ψ ψ ξ ∈X hξ ψξ , φ for all φ ∈ S. Estimate (6.9) follows by (6.11). 2 We now present our main result on Laguerre–Triebel–Lizorkin spaces.
G. Kerkyacharian et al. / Journal of Functional Analysis 256 (2009) 1137–1188
1157 sρ
sρ
Theorem 6.7. Let s, ρ ∈ R, 0 < p < ∞ and 0 < q ∞. Then the operators Sϕ : Fpq → fpq and sρ sρ sρ sρ Tψ : fpq → Fpq are bounded and Tψ ◦ Sϕ = Id on Fpq . Consequently, f ∈ Fpq if and only if sρ {f, ϕξ }ξ ∈X ∈ fpq and sρ ∼ f, ϕξ sρ . f Fpq f
(6.12)
pq
sρ
In addition, the definition of Fpq is independent of the particular selection of a satisfying (6.2), (6.3). To prove this theorem we need several lemmas with proofs given in Section 9. Assume that {Φj } are the kernels from the definition of Laguerre–Triebel–Lizorkin spaces and {ϕξ }ξ ∈X and {ψξ }ξ ∈X are needlet systems defined as in (5.24) with no connection between the functions a ’s from (6.1) and (5.22). We also assume that p, q from the hypothesis of Theorem 6.7 are fixed and we choose 0 < t < min{p, q}. Lemma 6.8. For any σ > d there exists a constant cσ > 0 such that Φj ∗ ψξ (x)
cσ , μ(Rξ )1/2 (1 + 2m x − ξ )σ
ξ ∈ Xm , j − 1 m j + 1,
(6.13)
and Φj ∗ ψξ ≡ 0 for ξ ∈ Xm if |m − j | 2, where Xm := ∅ if m < 0. Definition 6.9. For any collection of complex numbers {hξ }ξ ∈Xj (j 0), we define h∗j (x) :=
η∈Xj
|hη | j (1 + 2 η − x)λ
(6.14)
and h∗ξ := h∗j (ξ ),
ξ ∈ Xj ,
(6.15)
where λ := 2d + 2(|α| + 3d)/t + 2(|α| + d)|ρ|/d. Lemma 6.10. For any set {hη }η∈Xj (j 0) of complex numbers h∗j (x) cMt
|hη |1Rη (x),
η∈Xj
x ∈ Rd+ .
(6.16)
Moreover, for ξ ∈ Xj −ρ/d ∗ W α 4j ; ξ hξ 1Rξ (x) cMt
−ρ/d |hη |Wα 4j ; η 1Rη (x),
η∈Xj
Here the constants depend only on d, α, ρ, δ, and t.
x ∈ Rd+ . (6.17)
1158
G. Kerkyacharian et al. / Journal of Functional Analysis 256 (2009) 1137–1188
Lemma 6.11. Suppose g ∈ V4j and denote Mξ := sup g(x),
and mη := inf g(x),
ξ ∈ Xj ,
x∈Rη
x∈Rξ
η ∈ Xj + .
Then there exists 1, depending only d, α, δ, and λ, such that for any ξ ∈ Xj Mξ∗ cm∗η
for all η ∈ Xj + , Rη ∩ Rξ = ∅,
(6.18)
and, therefore,
Mξ∗ 1Rξ (x) c
m∗η 1Rη (x),
x ∈ Rd ,
(6.19)
η∈Xj + ,Rη ∩Rξ =∅
where c > 0 depends only on d, α, δ, and t. Proof of Theorem 6.7. Choose σ so that σ λ + 2(|α| + d)|ρ|/d and recall that t has already been selected so that 0 < t < min{p, q}. Suppose {Φj } are from the definition of Laguerre–Triebel–Lizorkin spaces (see (6.1)–(6.3)). As already mentioned in Section 5.2, there exists a function b satisfying (5.18), (5.19) such that (5.20) holds as well. Using this function we define {Ψj } just as in (5.23). Then we use {Φj } and {Ψj } to define as in (5.24) a pair of dual needlet systems {ϕη } and {ψη }. η } is a second pair of needlet systems, defined as in (5.22)–(5.24) using Suppose { ϕη }, {ψ j }, {Ψ j }. another pair of kernels {Φ sρ sρ sρ We first show the boundedness of the operator Tψ : fpq → Fpq . Let h ∈ fpq and set f := ξ . Evidently Φj ∗ ψ ξ = 0 if ξ ∈ Xm and |j − m| 2, and hence Tψ h = ξ ∈X hξ ψ Φj ∗ f =
j +1
ξ hξ Φj ∗ ψ
(X−1 := ∅).
m=j −1 ξ ∈Xm
Denote Hξ := hξ Wα (4m ; ξ )−ρ/d μ(Rξ )−1/2 . Using Lemma 6.8 and (5.17) we get j +1 −ρ/d Φj ∗ f (x) W α 4j ; x
−ρ/d Φj ∗ ψ ξ (x) |hξ |Wα 4j ; x
m=j −1 ξ ∈Xm
c
j +1
|hξ |Wα (4m ; ξ )−ρ/d μ(Rξ )−1/2 (1 + 2m ξ − x)σ −2(|α|+d)|ρ|/d
m=j −1 ξ ∈Xm
c
j +1 m=j −1
Hm∗ (x)
∗ H−1 := 0 ,
(6.20)
sρ and apply where Hm∗ (x) is defined as in (6.14). We use this in the definition of f Fpq Lemma 6.10 and the maximal inequality (4.6) to obtain
G. Kerkyacharian et al. / Journal of Functional Analysis 256 (2009) 1137–1188
sρ f Fpq
1159
∞ 1/q js ∗ q 2 Hj (·) j =0
p
∞ #
$q 1/q −ρ/d M t 2j s c |hξ |Wα 4j ; ξ μ(Rξ )−1/2 1Rξ ξ ∈ Xj
j =0
c {hη }
sρ
fpq
p
.
sρ
sρ
Hence the operator Tψ : fpq → Fpq is bounded. sρ Let the space Fpq be defined using {Φ j } instead of {Φj }. We now prove the boundedness of sρ sρ sρ the operator Sϕ : Fpq → fpq . Let f ∈ Fpq and denote Mξ := sup Φ j ∗ f (x),
ξ ∈ Xj ,
and mη := inf Φ j ∗ f (x),
η ∈ Xj + ,
x∈Rη
x∈Rξ
where is the constant from Lemma 6.11. We have f, ϕξ c1/2 Φ j ∗ f (ξ ) cμ(Rξ )1/2 Mξ cμ(Rξ )1/2 M ∗ . ξ ξ
(6.21)
Evidently, Φ j ∗ f ∈ V4j , and applying Lemma 6.11 (see (6.19)), we get Mξ∗ 1Rξ (x) c
m∗η 1Rη (x),
x ∈ Rd .
(6.22)
η∈Xj + ,Rη ∩Rξ =∅
It is easy to see that Wα (4j + ; y) ∼ Wα (4j ; ξ ) for y ∈ Rξ . We use this, (6.21), (6.22), Lemma 6.10, and the maximal inequality (4.6) to obtain f, ϕξ
sρ
fpq
∞
q 1/q −ρ/d c 2sj q W α 4j ; ξ Mξ∗ 1Rξ j =0
ξ ∈ Xj
p
∞
q 1/q j + −ρ/d ∗ sj q c 2 Wα 4 ; η mη 1Rη j =0
η∈Xj +
p
∞
q 1/q j + −ρ/d sj c Mt 2 Wα 4 ; η mη 1Rη j =0
η∈Xj +
p
∞
q 1/q j + −ρ/d sj 2 c Wα 4 ; η mη 1Rη j =0
η∈Xj +
p
∞ 1/q −ρ/d q Φ j ∗ f (·) sρ . c 2sj q Wα 4j ; · = cf Fpq j =0
p
1160
G. Kerkyacharian et al. / Journal of Functional Analysis 256 (2009) 1137–1188
Here for the second inequality we used that each tile Rη , η ∈ Xj +l , intersects no more that finitely many (depending only on d) tiles Rη , η ∈ Xj . The above estimates prove the boundedness of the sρ sρ operator Sϕ : Fpq → fpq . The identity Tψ ◦ Sϕ = Id follows by Theorem 5.3. It remains to show the independence of the definition of Triebel–Lizorkin spaces from the j } are two sequences of kernels specific selection of a satisfying (6.2), (6.3). Suppose {Φj }, {Φ as in the definition of Triebel–Lizorkin spaces defined by two different functions a satisfying (6.2), (6.3). As above there exist two associated needlet systems {Φj }, {Ψj }, {ϕξ }, {ψξ } and j }, { ξ }. Denote by f F sρ (Φ) and f F sρ (Φ) j }, {Ψ ϕξ }, {ψ the F -norms defined via {Φj } {Φ pq pq and {Φj }. Then from above sρ ϕξ f sρ cf F sρ (Φ) f Fpq (Φ) c f, . pq
pq
sρ
The independence of the definition of Fpq of the specific choice of a in the definition of the j } and their complex conjufunctions {Φj } follows by interchanging the roles of {Φj } and {Φ gates. 2 sρ
ss are more natural than the spaces F To us the spaces Fpq pq with ρ = s since they embed correctly with respect to the smoothness index s.
Proposition 6.12. Let 0 < p < p1 < ∞, 0 < q, q1 ∞, and −∞ < s1 < s < ∞. Then we have the continuous embedding ss Fpq ⊂ Fps11sq11
if s/d − 1/p = s1 /d − 1/p1 .
(6.23)
The proof of this embedding result can be carried out similarly as the proof of Proposition 4.11 in [9], using the idea of the proof in the classical case on Rn (see e.g. [17, p. 129]). We omit it. 7. Laguerre–Besov spaces We introduce weighted Besov spaces on Rd+ in the context of Laguerre expansions using the kernels {Φj } from (6.1) with a satisfying (6.2), (6.3) (see [12,17] for the general idea of using orthogonal or spectral decompositions in defining Besov spaces). 7.1. Definition of Laguerre–Besov spaces sρ
sρ
Definition 7.1. Let s, ρ ∈ R and 0 < p, q ∞. The Laguerre–Besov space Bpq := Bpq (F α ) is such that defined as the set of all f ∈ S+ sρ := f Bpq
∞ j =0
2
q −ρ/d W α 4j ; · Φj ∗ f (·) p
sj
1/q < ∞,
(7.1)
where the q -norm is replaced by the sup-norm if q = ∞. Observe that as in the case of Laguerre–Triebel–Lizorkin spaces the above definition is indesρ pendent of the particular choice of a obeying (6.2), (6.3) (see Theorem 7.4). Also, as for Fpq the sρ Besov space Bpq is a quasi-Banach space which is continuously embedded in S+ . We skip the details.
G. Kerkyacharian et al. / Journal of Functional Analysis 256 (2009) 1137–1188
1161
7.2. Needlet decomposition of Laguerre–Besov spaces sρ
sρ
We next define the sequence spaces bpq associated to the Laguerre–Besov spaces Bpq . As in Section 6 we assume that {Xj }∞ j =0 are from (5.9) with associated tiles {Rξ }ξ ∈Xj from (5.11). As " before we set X := j 0 Xj . sρ
Definition 7.2. Let s, ρ ∈ R and 0 < p, q ∞. Then bpq is defined to be the space of all complex-valued sequences h := {hξ }ξ ∈X such that sρ := hbpq
∞ j =0
2
j sq
# $ 1/q j −ρ/d p q/p 1/p−1/2 Wα 4 ; ξ μ(Rξ ) |hξ |
(7.2)
ξ ∈ Xj
is finite, with the usual modification whenever p = ∞ or q = ∞. We shall utilize again the analysis and synthesis operators Sϕ and Tψ defined in (6.8). The sρ next lemma guarantees that the operator Tψ is well defined on bpq . sρ
Lemma 7.3. Let s, ρ ∈ R and 0 < p, q ∞. Then for any h ∈ bpq , Tψ h := . Moreover, the operator T : bsρ → S is continuous. verges in S+ ψ pq +
ξ ∈X
hξ ψξ con-
The proof of this lemma is quite similar to the proof of Lemma 6.6 and will be omitted. Our main result in this section is the following characterization of Laguerre–Besov spaces. sρ
sρ
Theorem 7.4. Let s, ρ ∈ R and 0 < p, q ∞. Then the operators Sϕ : Bpq → bpq and sρ sρ sρ we have Tψ : bpq → Bpq are bounded and Tψ ◦ Sϕ = Id on Bpq . Consequently, for f ∈ S+ sρ sρ that f ∈ Bpq if and only if {f, ϕξ }ξ ∈X ∈ bpq and sρ ∼ f, ϕξ sρ . f Bpq b pq
(7.3)
sρ
In addition, the definition of Bpq is independent of the particular selection of a satisfying (6.2), (6.3). The proof of this theorem relies on some lemmas from the proof of Theorem 6.7 as well as the next lemma with proof given in Section 9. Lemma 7.5. Let 0 < p ∞ and ρ ∈ R. Then for any g ∈ V4j , j 0, ξ ∈ Xj
1/p p j −ρp/d −ρ/d Wα 4 ; ξ max g(x) μ(Rξ ) c Wα 4j ; · g(·) p . x∈Rξ
(7.4)
Proof of Theorem 7.4. We will use some basic assumptions and notation from the proof of Theorem 6.7. Let 0 < t < p and σ λ + 2(|α| + d)|ρ|/d. Assume that {Φj }, {Ψj }, {ϕη }, {ψη } j }, {Ψ j }, { η } are two needlet systems, defined as in (5.22)–(5.24), that originate and {Φ ϕη }, {ψ from two completely different functions a satisfying (6.2), (6.3).
1162
G. Kerkyacharian et al. / Journal of Functional Analysis 256 (2009) 1137–1188 sρ
sρ
sρ
Let us first prove the boundedness of the operator Tψ : bpq → Bpq , assuming that Bpq is sρ ξ . defined by {Φj }. Suppose h ∈ bpq and set f := Tψ h = ξ ∈X hξ ψ Denote Hξ := hξ Wα (4m ; ξ )−ρ/d μ(Rξ )−1/2 , ξ ∈ Xm . Then by (6.20) and Lemma 6.10 j +1 j −ρ/d ∗ W α 4 ; · H Φj ∗ f (·) p c m p m=j −1
c
j +1 m −ρ/d −1/2 Mt 4 |h |W ; ξ μ(R ) 1 ξ α ξ Rξ ξ ∈ Xm
m=j −1
p
j +1 m −ρ/d 1/p 1/p−1/2 p |hξ |Wα 4 ; ξ c μ(Rξ ) , m=j −1
ξ ∈ Xm
sρ c{hη } sρ and hence the claimed boundedness of T . which yields f Bpq ψ bpq
sρ
sρ
We now prove the boundedness of the operator Sϕ : Bpq → bpq , where we assume that the sρ space Bpq is defined in terms of {Φ j } in place of {Φj }. Just as in (6.21) we have |f, ϕξ | cμ(Rξ )1/2 |Φ j ∗ f (ξ )|, ξ ∈ Xj . Since Φ j ∗ f ∈ V4j , Lemma 7.5 implies p −ρ/d W α 4j ; ξ μ(Rξ )1/p−1/2 f, ϕξ ξ ∈ Xj
c
ξ ∈ Xj
−ρp/d Φ j ∗ f (ξ )p μ(Rξ ) c Wα 4j ; · −ρ/d Φ j ∗ f (·) p , W α 4j ; ξ p
sρ cf sρ . which leads immediately to {f, ϕ}bpq Bpq
sρ
The identity Tψ ◦ Sϕ = Id follows by Proposition 5.3. The independence of Bpq of the specific selection of a in the definition of {Φj } follows from above exactly as in the Triebel–Lizorkin case (see the proof of Theorem 6.7). 2 sρ
The parameter ρ in the definition of the Besov spaces Bpq allows one to consider various scales of spaces. A “classical” choice of ρ would be ρ = 0. However, to us most natural are the ss (ρ = s) for they embed “correctly” with respect to the smoothness index s: spaces Bpq Proposition 7.6. Let 0 < p p1 ∞, 0 < q q1 ∞, and −∞ < s1 s < ∞. Then we have the continuous embedding ss Bpq ⊂ Bps11sq11
if s/d − 1/p = s1 /d − 1/p1 .
(7.5)
Proof. Assuming that Φj is from Definition 7.1 we have Φj ∗ f ∈ V4j +1 and applying estimate (4.2) from Proposition 4.1 we obtain j −s1 /d −s/d Wα 4 ; · Φj ∗ f (·) p c2j d(1/p−1/p1 ) Wα 4j ; · Φj ∗ f (·) p , 1
where we used that s/d − 1/p = s1 /d − 1/p1 . This implies (7.5) at once.
2
G. Kerkyacharian et al. / Journal of Functional Analysis 256 (2009) 1137–1188
1163
8. Proofs for Sections 3, 4 8.1. Proof of estimates (3.10) and (3.12) in Theorem 3.2 We may assume that n n0 , where n0 is a sufficiently large constant. Estimate (3.10) will be established by applying repeatedly summation by parts to the sum in the definition (3.7) of Λn (x, y). For a sequence of numbers {am } we denote by k am the kth forward differences, defined by am := am −am+1 and inductively k+1 am := (k am ). Choose k σ +2|α|+d/2 and denote
n m+k Akn−m Fmα (x, y), Akm := . (8.1) Ωnk (x, y) := k m=0
Using summation by parts k times, we obtain
∞
∞ j m α k+1 k Fj (x, y) = · Ωm Λn (x, y) := a a (x, y), n n j =0
(8.2)
m=0
k (x, y) = where k+1 is applied with respect to m. By (2.1) and (8.1), it easily follows that Ωm −(x22 +y22 )/2 α,k 2 2 ce Pm (x , y ) and combining this with (2.4) we get
k Ωm (x, y) = c
L|α|+k+d x22 + y22 + 2 m
d
xi yi cos θi
i=1
[0,π]d
×e
−(x22 +y22 +2
d
i=1 xi yi
cos θi )/2
d
jαi −1/2 (xi yi cos θi ) sin2αi θi dθ.
i=1
Using this in (8.2) we arrive at the identity
Kλn
Λn (x, y) = c
x22
+ y22
d
xi yi cos θi
i=1
[0,π]d
×
+2
d
jαi −1/2 (xi yi cos θi ) sin2αi θi dθ,
(8.3)
i=1
where λ := |α| + k + d and the kernel Kλn is defined by Kλn (t) :=
∞ m=0
m λ Lm (t)e−t/2 . k+1 a n
By a well-known property of finite differences we have k+1 m (k+1) (k+1) = n−k−1 ∞. a a a (ξ ) n−k−1 L n
(8.4)
(8.5)
1164
G. Kerkyacharian et al. / Journal of Functional Analysis 256 (2009) 1137–1188
Further, it is known that [1, p. 204] 1
jα− 1 (x) = x
−α+1/2
2
and j− 1 (x) = 2
%
2 π
2−α+ 2 Jα− 1 (x) = √ 2 π (α)
1 eixt (1 − t 2 )α−1 dt,
α > 0,
(8.6)
−1
cos x. Therefore, j
cα < ∞,
α− 12 (x)
x ∈ R+ , α 0.
(8.7)
By (8.5) and (2.11) (with α replaced by |α| + k + 1) we obtain for t > 0 λ K (t) c
(1+v)n
1
n
nk+1
m=max{un−k,1}
m t
(|α|+k+d)/2
cn(−k+|α|+d)/2 t −(|α|+k+d)/2 .
(8.8)
Using this in (8.3) we get d
Λn (x, y) cn(−k+|α|+d)/2
[0,π]d
2αi θi dθ i=1 sin . d 2 2 (x2 + y2 + 2 i=1 xi yi cos θi )(k+|α|+d)/2
Set τ := (k + |α| + d)/2. Substituting θi = π − ti in the above integral and using 1 − cos t = 2 sin2 2t ∼ t 2 we infer Λn (x, y) cn−k+τ
[0,π]d
cn−k+τ
d
(x
sin2 t2i )τ
d
[0,π]d
2αi ti dt i=1 sin d 2 − y2 + 4 i=1 xi yi
(x
2αi i=1 ti dt − y2 + di=1 xi yi ti2 )τ
=: cMnk,α (x, y).
(8.9)
We estimate the integral above in two ways. First, we trivially have −k+τ cn|α|+d Λn (x, y) cM k,α (x, y) cn 1/2 . n 2τ x − y (n x − y)k+|α|+d
(8.10)
The second estimate is really many estimates rolled into one. For a fixed 1 d we partition α into α = (α , α ) with α = (α1 , . . . , α ) and α = (α+1 , . . . , αd ). Since τ > |α| + d/2 and xi yi > 0 we have Mnk,α (x, y) cn−k+τ
[0,π]
(x
2αi i=1 ti dt − y2 + i=1 xi yi ti2 )τ
G. Kerkyacharian et al. / Journal of Functional Analysis 256 (2009) 1137–1188
cn−k+τ
i=1 (xi yi
)αi +1/2
π(xi yi )1/2
i=1
(x
0
− y2
+
1165
du
2 τ −|α | i=1 ui )
,
where we applied the substitutions ui = ti (xi yi )1/2 and used |α | power of the main term in the denominator to cancel the numerator. Enlarging the integral domain to R and using spherical coordinates, the above product of integrals is bounded by R
du = (x − y2 + u22 )τ −|α |
∞ 0
r −1 dr c . (x − y2 + r 2 )τ −|α | x − y2(τ −|α |)−
From above and a little algebra we obtain for 1 d Λn (x, y) cM k,α (x, y) n
αi + 12 i=1 (xi yi )
cnd/2
d
i=+1 (n
1 −1 )αi + 2 (n1/2 x
− y)k+|α|−2|α |+d−
. (8.11)
A third bound on |Λn (x, y)| will be obtained by estimating all terms in (3.7). By (2.10) and (3.6) it follows that α F i νi
α /2
∞
cνi i ,
1 i d,
and Fνα ∞ cν α/2 ,
(8.12)
and hence
α m + d − 1 |α| α F (x, y) c ν = c m cm|α|+d−1 , m m
yielding
|ν|=m
(1+v)n Λn (x, y) c m|α|+d−1 cn|α|+d .
(8.13)
m=0
We also need the estimate Λn (x, y)
cnd/2 , αi +1/2 d −1 αi +1/2 i=1 (xi yi ) i=+1 (n )
1 d.
(8.14)
By (2.8) it follows that α F (x) n
c x α+1/2 n1/4
,
if x 2 ∈ R+ \ (2n + 2α + 2, 6n + 3α + 3),
(8.15)
and if x 2 ∈ [2n + 2α + 2, 6n + 3α + 3), by (2.9) α F (x) n
x α n1/4 (n1/3
c . + |4n + 2α + 2 − x 2 |)1/4
(8.16)
1166
G. Kerkyacharian et al. / Journal of Functional Analysis 256 (2009) 1137–1188
From these two estimates one easily concludes that for x > 0 α F (x) n
c x α+1/2 n1/4
,
if n ∈ R+ \ x 2 /5, x 2 /3 ,
(8.17)
and α F (x) n
c , x α+1/2 (1 + |4n − x 2 |)1/4
if n ∈ x 2 /5, x 2 /3 .
(8.18)
Hence, |Fnα (x)| can be bounded by the sum of the right-hand side quantities in (8.17), (8.18). α /2 α Also, from (8.12) Fνii ∞ cνi i . From these along with (3.3), (3.4) we obtain Λn (x, y)
d (1+v)n F αi (xi )F αi (yi ) νi
i=1
νi
νi =0
αi +1/2 i=1 (xi yi )
×
(1+v)n i=1
d
(1+v)n
i=+1
νi =0
c
νi =0
(νi + 1)αi
1 1 + (1 + νi )1/4 (1 + |νi − ui |)1/4
1 1 , + (1 + νi )1/4 (1 + |νi − vi |)1/4
where ui , vi > 0 are some numbers. Clearly, each of the last sums can be bounded by four sums of the form
(1+v)n νi =0
1 cn1/2 . (1 + |νi − wi |)1/4 (1 + |νi − zi |)1/4
This last estimate apparently holds independently of wi and zi . Estimate (8.14) follows from above. We are now in a position to complete the proof of (3.10). Estimates (8.10) and (8.13) readily imply Λn (x, y)
cn|α|+d , (1 + n1/2 x − y)k+|α|+d
while by (8.11) and (8.14) we have for 1 i Λn (x, y)
cnd/2 . d αi +1/2 −1 αi +1/2 (1 + n1/2 x − y)k−|α| i=1 (xi yi ) i=+1 (n )
(8.19)
G. Kerkyacharian et al. / Journal of Functional Analysis 256 (2009) 1137–1188
1167
Clearly, this estimate holds for an arbitrary permutation i1 , i2 , . . . , id of the indices 1, 2, . . . , d. These estimates and (8.19) yield Λn (x, y) d
i=1 (xi yi
cnd/2 + n−1 )αi +1/2 (1 + n1/2 x − y)k−|α|
.
(8.20)
To complete the proof we need the following simple inequality: for x, y ∈ Rd+ xi + n−1/2 yi + n−1/2 3 xi yi + n−1 1 + n1/2 x − y ,
1 i d.
(8.21)
Combining these with (8.20) we get Λn (x, y) d
cnd/2
i=1 (xi
+ n−1/2 )α+1/2 (yi + n−1/2 )αi +1/2 (1 + n1/2 x − y)k−2|α|−d/2
,
which implies (3.10) since k was select so that k σ + 2|α| + d/2. The proof of (3.12) is trivial. Indeed, by Lemma 2.1 it follows that α F (x) cx −α e−γ x 2 n
for x (6n + 3α + 3)1/2 .
(8.22)
From this it easily follows that if max{x, y} (6(1 + v)n + 3α + 3)1/2 , then Λn (x, y) cnd e−γ max{x2 ,y2 } ,
γ > 0,
which readily implies (3.12). 8.2. Proof of estimates (3.11) and (3.13) in Theorem 3.2 Clearly, (3.13) implies (3.11) if max{x, y} (6(1 + v)n + 3α + 3)1/2 . Assume max{x, y} < (6(1 + v)n + 3α + 3)1/2 cn1/2 . We will prove (3.11) in this case by using the scheme of the proof of (3.10) with appropriate modifications. First, we need information about the derivative of Fnα . The Laguerre polynomials satisfy the relation [15, (5.1.14)]
d α Ln (x) = −Lα+1 (x) = x −1 nLαn (x) − (n + α)Lαn−1 (x) . n−1 dx
(8.23)
After taking the derivative of Fnα (see (3.1)), the first identity in (8.23) yields
√ α+1 d α (x) , Fn (x) = −x Fnα (x) + 2 nFn−1 dx
(8.24)
and from the second identity we similarly get x
d α α F (x) = −x 2 Fnα (x) + 2nFnα (x) − 2bn Fn−1 , dx n
bn :=
n(n + α).
(8.25)
1168
G. Kerkyacharian et al. / Journal of Functional Analysis 256 (2009) 1137–1188
Here and in what follows we assume Fkα (x) = 0 for k < 0. Also, from the recurrence relation for Laguerre polynomials [15, (5.1.10)] one readily derives the identity xLαn (x) = (2n + α + 1)Lαn (x) − (n + 1)Lαn+1 − (n + α)Lαn−1 (x),
n 1,
with Lα0 (x) = 1 and Lα1 (x) = −x + α + 1. From this with the definition of Fnα in (3.1), we get α α (x) + (2n + α + 1)Fnα (x) − bn Fn−1 (x), x 2 Fnα (x) = −bn+1 Fn+1
(8.26)
where bn is as above. Combining this with (8.25) gives
d α α α (x) − bn Fn−1 (x) . F (x) = x −1 −(α + 1)Fnα (x) + bn+1 Fn+1 dx n
(8.27)
We also need the relation [15, (5.1.13)] α+1 Lαn (x) = Lα+1 n (x) − Ln−1 (x).
(8.28)
From this and (3.1) we deduce Fnα (x) =
√ √ α+1 n + α + 1Fnα+1 (x) − nFn−1 (x).
(8.29)
Using this identity with α replaced by α − 1, (8.27), and the obvious fact that bn = n + O(1), we arrive at & d α ' α F (x) cx −1 F (x) + n1/2 max F α−1 (x) . (8.30) max m m dx n n−1mn+1 nmn+1 d By (8.24) and (8.12) we readily get the estimate | dx Fnα (x)| cxnα/2+1 , and by (8.30) and d Fnα (x)| cx −1 nα/2 . Therefore, (8.12), | dx
d α F (x) cnα/2 min x −1 , nx cn(α+1)/2 , dx n
x ∈ R+ .
(8.31)
We use this estimate to obtain (1+v)n ∂ ∂ α α F Λ (x, y) (y) F (x) ν ∂x ν ∂x n r
m=0
cn1/2
r
|ν|=m
(1+v)n
m=0
|ν|=m
ν α cn|α|+d+1/2 .
(8.32)
We next prove an analogue of (8.14). Let 0 < x cn1/2 . Assuming that m ∈ R \ (x 2 /5, x 2 /3) we derive as before from (8.15) and (8.24),
G. Kerkyacharian et al. / Journal of Functional Analysis 256 (2009) 1137–1188
d α F (x) x F α (x) + 2m1/2 F α+1 (x) m m−1 dx m
1 cn1/2 m1/2 cx α+1/2 1/4 + α+3/2 1/4 α+1/2 1/4 . x m x m x m
1169
(8.33)
From (8.16) and (8.24) we similarly obtain d α cn1/2 F (x) m dx x α+1/2 (1 + |4m − x 2 |)1/4
for m ∈ x 2 /5, x 2 /3 .
(8.34)
We further proceed exactly as in the proof of (8.14), with Fναrr (xr ) replaced by ∂x∂ r Fναrr (xr ) and for this term estimates (8.17), (8.18) are replaced by (8.33), (8.34), and we also use (8.31). As a result, we get ∂ ∂x Λn (x, y)
cn(d+1)/2 , αi +1/2 d −1 αi +1/2 i=1 (xi yi ) i=+1 (n )
r
1 d.
(8.35)
We now derive our main bound on |(∂/∂xr )Λn (x, y)|. It will be convenient to use the notation ∂f (t) := f (t). After differentiating the expression of Λn (x, y) in (8.3) we obtain for 1 r d, ∂ Λn (x, y) = Q1 (x, y) + Q2 (x, y), ∂tr
(8.36)
where
Q1 (x, y) :=
∂Kk+|α|+d n
x22
+ y22
+2
× (2xr − 2yr cos θr )
Kk+|α|+d n
d
xi yi cos θi
jαi − 1 (xi yi cos θi )(sin θi )2αi dθ,
i=1
x22
+ y22
2
+2
d
(8.37)
xi yi cos θi
i=1
[0,π]d
×
i=1
[0,π]d
Q2 (x, y) :=
d
d i=1, i=r
jαi − 1 (xi yi cos θi )∂jαr − 1 (xr yr cos θr )yr cos θr (sin θi )2αi dθ. (8.38) 2
2
We first estimate Q1 (x, y). By the left-hand side identity in (8.23) and (8.28) −t/2 d α α+1 Ln (t)e−t/2 = −(1/2) Lαn (t) + 2Lα+1 (t) e−t/2 = −(1/2) Lα+1 . n (t) + Ln−1 (t) e n−1 dt Hence, by the definition of Kλn in (8.4),
k+|α|+d+1 ∂Kk+|α|+d (t) = − Kk+|α|+d+1 (t) + K (t) /2, n n n
1170
G. Kerkyacharian et al. / Journal of Functional Analysis 256 (2009) 1137–1188
λn (t) is define as Kλn (t) but with Lλm in the sum in (8.4) replaced by Lλ . Evidently, where K m−1 λ Kn (t) has the same properties as Kλn (t). Substituting the above in (8.37) and taking into account (8.7), (8.8) we get |xr − yr cos tr | di=1 ti2αi dt . (x − y22 + di=1 xi yi ti2 )(k+|α|+d+1)/2
Q1 (x, y) cn
(−k+|α|+d+1)/2 [0,π]d
Now, using the fact that |xr − yr cos tr | |xr − yr | + 2xr yr sin2 (tr /2) |xr − yr | + xr−1 xr yr tr2 and noticing that |xr − yr | can be canceled by an 1/2 power of the main term in the denominator, whereas xr yr tr2 needs a square of that much, we conclude that d
Q1 (x, y) cn
(−k+|α|+d+1)/2 [0,π]d
(x
2αi i=1 ti dt − y22 + di=1 xi yi ti2 )(k+|α|+d)/2
d
+ cxr−1 n(−k+|α|+d+1)/2
[0,π]d
2αi i=1 ti
(x
− y22
+
d
dt
2 (k+|α|+d−1)/2 i=1 xi yi ti )
.
Both of the above integrals are of the form of Mnk,α defined in (8.9). In fact, we have Q1 (x, y) cn1/2 Mnk,α (x, y) + cxr−1 Mnk−1,α (x, y).
(8.39)
Furthermore, evidently |xr − yr cos tr | |xr − yr | + xr tr2 and inserting tr2 into the weight function of the integral, we obtain as above Q1 (x, y) cn1/2 Mnk,α (x, y) + cxr Mnk,α+er (x, y).
(8.40)
We next estimate Q2 . Using the integral representation (8.6) for jα− 1 (x) we get 2
1 ∂jα− 1 (x) = c 2
α−1 eixt t 1 − t 2 dt,
α > 0,
−1
while ∂j− 1 (x) = c sin x. Therefore, |∂jα− 1 (x)| c for α 0. Consequently, using also that 2
yr cn1/2 , we obtain as in (8.9)
2
Q2 (x, y) cn1/2 M k,α (x, y). n
Combining (8.39) and (8.41) gives ∂ −1 k−1,α (x, y) + cn1/2 Mnk,α (x, y), ∂x Λn (x, y) cxr Mn r
(8.41)
(8.42)
G. Kerkyacharian et al. / Journal of Functional Analysis 256 (2009) 1137–1188
1171
whereas combining (8.40) and (8.41) gives ∂ 1/2 k,α k,α+er (x, y). ∂x Λn (x, y) cn Mn (x, y) + cxr Mn r
(8.43)
We are now in a position to establish estimate (3.11). Using (8.10) in (8.42) and combining the result with (8.32), we conclude that for xr n−1/2 ∂ cn|α|+d+1/2 Λ (x, y) n ∂x (1 + n1/2 x − y)k+|α|+d−1 . r
(8.44)
On the other hand, using (8.10) in (8.43) and combining the result with (8.32) shows that estimate (8.44) holds for xr n−1/2 as well. Therefore, (8.44) holds for all x, y ∈ Rd+ . In going further, using (8.11) in (8.42) and combining the result with (8.35), we obtain for xr n−1/2 and 1 i ∂ Λ (x, y) ∂x n
αi + 12
i=1 (xi yi )
r
d
cn(d+1)/2 1
−1 αi + 2 (1 + n1/2 x − y)k−|α|−1 i=+1 (n )
. (8.45)
On the other hand, using (8.11) in (8.43) and combining the result with (8.35), we see that the same bound (8.45) holds for xr n−1/2 as well. Therefore, (8.45) holds in general. Moreover, (8.45) holds for all possible permutations of the indices and combining it with (8.44) leads to ∂ ∂x Λn (x, y) d
cn(d+1)/2
i=1 (xi yi
r
1
+ n−1 )αi + 2 (1 + n1/2 x − y)k−|α|−1
.
Now, estimate (3.11) follows using (8.21) as before. The proof of (3.13) is simple. By (8.22) and (8.24) it follows that d α F (x) cx −α+1 e−γ x 2 ce−γ x 2 for x 6(1 + v)n + 3α + 3 1/2 . dx n This and (8.22) imply that if max{x, y} (6(1 + v)n + 3α + 3)1/2 , then ∂ cnd e−γ max{x2 ,y2 } , γ > 0, Λ (x, y) ∂x n r which yields (3.13). 8.3. Proof of other localization estimates Proof of Lemma 3.4. We will derive estimate (3.15) from the following estimate: if s ∈ R, γ 0, σ > (2γ + 1)(|s| + 1) + 1, and z > 0, then ∞ I := 0
u2γ +1 du (1 + u)(2γ +1)s (1 + |u − z|)σ
c (1 + z)(2γ +1)(s−1)
.
(8.46)
1172
G. Kerkyacharian et al. / Journal of Functional Analysis 256 (2009) 1137–1188
Consider first the case when s 1. Then I = −σ
z
J1 (1 + z)
z/2 0
+
∞ z/2
=: J1 + J2 . Evidently,
1 du c(1 + z)−σ +1
0
and J2
∞
c (1 + z)(2γ +1)(s−1)
z/2
du c σ (2γ (1 + |u − y|) (1 + z) +1)(s−1)
(σ > 1).
Since σ > (2γ + 1)(s − 1) + 1 the above estimates for J1 and J2 yield (8.46). Let s < 1. Then we have ∞ I 0
(1 + u)(2γ +1)(1−s) du = (1 + |u − z|)σ
∞ c −∞
−∞
−z
(1 + v + z)(2γ +1)(1−s) du (1 + |v|)σ
(1 + |v|)(2γ +1)(1−s) + z(2γ +1)(1−s) du (1 + |v|)σ
∞ c
∞
du (1 + |v|)σ +(2γ +1)(s−1)
+ cz
(2γ +1)(1−s)
∞
−∞
du (1 + |v|)σ
c . (1 + z)(2γ +1)(1−s)
Here we used that σ > (2γ + 1)(1 − s) + 1. Therefore, (8.46) holds when s < 1 as well. We now proceed with the proof of (3.15). Denote by J the integral in (3.15). Using that |xj − yj | x − y, we get d
∞
J
i=1 0
2αi +1
yi
dyi (yi + n−1/2 )(2αi +1)s (1 + n1/2 |xi − yi |)σ/d d
∞
=n
(2|α|+d)s
i=1 0
yi2αi +1 dyi (1 + n1/2 yi )(2αi +1)s (1 + |n1/2 xi − n1/2 yi |)σ/d d
∞
= n(2|α|+d)(s−1)−d/2
i=1 0
cn(2|α|+d)(s−1)−d/2
d i=1
u2αi +1 du (1 + u)(2αi +1)s (1 + |u − n1/2 xi |)σ/d
1 cn−d/2 = . (1 + n1/2 xi )(2αi +1)(s−1) Wα (n; x)s−1
G. Kerkyacharian et al. / Journal of Functional Analysis 256 (2009) 1137–1188
Here for the last inequality we used (8.46).
1173
2
Proof of Theorems 3.7 and 3.8. By (3.6) we have Lαν (x) = 2−1/2 Fνα (x 1/2 )x α/2 and by (3.2), Mαν (x) = x α+1/2 Fνα (x). Hence n (x, y) = 2−1 Λn x 1/2 , y 1/2 x α/2 y α/2 Λ
and Λ∗n (x, y) = Λn (x, y)x α+1/2 y α+1/2 .
Now, it is easy to see that these relations and estimates (3.10) and (3.11) yield (3.18) and (3.19) as well as (3.20) and (3.21). 2 8.4. Proof of Lemma 3.6 The main step is to prove Lemma 3.6 for dimension d = 1. To this end we will need a lemma which goes back to van der Corput (see e.g. [19, vol. I, pp. 197, 198]). Lemma 8.1. If f (u) ρ > 0 or f (u) −ρ < 0 on [a, b], then 2πif (n) |f (b) − f (a)| + 2 4ρ −1/2 + c . e anb
Evidently, when d = 1 Lemma 3.6 is immediate from the following lemma. Lemma 8.2. For any ε > 0 and δ > 0 there exists a constant c > 0 such that for n 1/ε An (x) := e
−x
n+ εn m=n
1 −α−1/2 [Lαm (x)]2 1/2 x+ , cn Lαm (0) n
0 x (4 − δ)n.
(8.47)
Proof. We may assume that ε 1 and n n0 , where n0 is sufficiently large. The proof uses the asymptotic of Lαn (x) and is divided into several cases. Case 1. Let 0 x < c n−1 with c := (α + 1)(α + 3) (c n−1 is larger than the smallest zero of Lαn [15, (6.31.12)]). We need the asymptotic formula [15, (8.22.4), (8.22.5)] e−x/2 x α/2 Lαn (x) = N −α
(n + α + 1) Jα 2(N x)1/2 + x α/2+2 O nα , n!
where N = n + (α + 1)/2. Using also that Jα (z) =
zα 2α (α+1)
e−x/2 Lαn (x) ∼ nα + x 2 O nα cnα , Combining this with Lαn (0) =
n+α n
An (x) c
m=n
which proves (8.47) in this case.
+ O(zα+2 ), we obtain 0 x < c/n.
∼ nα we arrive at
n+ εn
mα ∼ nα+1 ,
0 < x c/n,
0 x < c n−1 ,
1174
G. Kerkyacharian et al. / Journal of Functional Analysis 256 (2009) 1137–1188
Case 2. Let c n−1 x c∗ n−1 , where the constant c∗ > 1 will be selected later on. In this case we use relations (2.16) and (2.19) to conclude that 2 e−x Lαn (x) ∼ n2α+2 (x − tkx ,n )2 . Furthermore, by a theorem of Tricomi (see [7] for the references), we know that for all the zeros j2
−2 of Lαn in the interval 0 < x < c/n we have tk,n = α,k n (1 + O(n )) as n → ∞, where jα,k , k = 1, 2, . . . , are the positive zeros, in increasing order, of the Bessel function Jα (x). Consequently,
An (x) cnα
n+ εn
2 2 mx − jα,k − cm−1 |mx − jα,kx | x
m=n
n+ εn 2 α 2 cn mx − jα,kx − c cnα+1 . m=n
Here for the last estimate we used that jα,k → ∞ as k → ∞ and hence there are only finitely 2 c n−1 (n + εn) c; the argument is the same as in the many zeros of Jα (x) such that jα,k ∗ analogous situation for Jacobi polynomials in [8]. Case 3. Let c∗ n−1 x c∗ , where c∗ is sufficiently large and its value is to be determined. In this case we use the asymptotic formula for Lαn (x) [15, (8.22.6)]: e−x/2 Lαn (x) = π −1/2 x −α/2−1/4 nα/2−1/4
× cos 2(nx)1/2 − απ/2 − π/4 + O(1)(nx)−1/2 , which holds for c n−1 x c and O(1) depends only on c , c . We denote γ := απ/2 + π/4 and deduce from above x α+1/2 An (x) cn−α e−x x α+1/2
n+ εn
α 2 Lm (x)
m=n
c
n+ εn
2 m−1/2 cos 2(mx)1/2 − γ + O(1)(nx)−1/2
m=n
cn−1/2
n+ εn
−1/2 cos2 2(mx)1/2 − γ + O(1)c∗ n1/2 .
m=n
Using the fact that 2 cos2 t = 1 + cos 2t and 2 cos 2t = e2it + e−2it , we see that Σ := 4
n+ εn
n+ εn √ √
1/2 e2πi(y m−γ ) + e−2πi(y m−γ ) , cos 2(mx) − γ 2 εn +
m=n
m=n
2
√ where y := (2/π) x and√γ := 2γ /π . The last sum can be estimated by making use of Lemma 8.1 with f (u) = y u, a = n and b = n + εn. We get
G. Kerkyacharian et al. / Journal of Functional Analysis 256 (2009) 1137–1188
1175
Σ 2 εn − 2 2 + x 1/2 n−1/2 c + 24x −1/4 n3/4 −1/4 2 εn − 2 2 + (c∗ )1/2 n−1/2 c + 24c∗ n . Putting the above estimates together, we arrive at −1/4 −1/2 x α+1/2 An (x) cn−1/2 2 εn − 2 2 + (c∗ )1/2 n−1/2 c + 24c∗ n + O(1)c∗ n1/2 . Choosing c∗ sufficiently large shows that the right-hand side of the above inequality is bounded below by cn1/2 for sufficiently large n. Thus (8.47) is proved in this case. Case 4. Let c∗ x (4 − δ)n. Here we apply another asymptotic formula of Laguerre polynomials [15, (8.22.9)]: for x = (4m + 2α + 2) cos2 φ with ε φ π/2 − εm−1/2 , x α/2+1/4 e−x/2 Lαm (x) = (−1)m (π sin φ)−1/2 mα/2−1/4 )
$ ( # α+1 −1/2 . (sin 2φ − 2φ) + 3π/4 + O(1)(mx) × sin m + 2 Note that the range of x above covers the range of this case. From above, as in Case 3, we obtain x α+1/2 An (x) cn−α e−x x α+1/2
n+ εn
α 2 Lm (x)
m=n
cn
−1/2
n+ εn
# sin
2
m=n
$ α+1 m+ (sin 2φ − 2φ) + 3π/4 + O(1)(c∗ )−1/2 . 2
The last sum is again bounded below by cn, which can be proved either by using Lemma 8.1 or by summing up using simple trigonometric identities. This shows again that (8.47) holds. 2 Proof of (3.16) in the case d 2. We may again assume ε 1. We will use induction on d. α := F α . Assume that (3.17) has been To indicate the dependence of Fmα on d we write Fm,d m established for dimensions up to d − 1. By definition α (x, x) = Fm,d
m
2 α Fkαd (xd ) Fm−k, d−1 (x , x ),
x = (x , xd ), α = (α , αd ),
k=0
and hence n+ dεn
α Fm,d (x, x)
m=n
n+ dεn εn
αd 2 α Fk (xd ) Fm−k,d−1 (x , x )
m=n
=
εn
k=0
n+ dεn
αd 2 α Fk (xd ) Fm−k,d−1 (x , x ) m=n
k=0
εn k=0
αd 2 Fk (xd )
n+ (d−1)εn j =n
α Fj, d−1 (x , x ).
(8.48)
1176
G. Kerkyacharian et al. / Journal of Functional Analysis 256 (2009) 1137–1188
It follows by (3.1) and (2.12)–(2.14) that for 0 x
√
(4 − δ)n
n
α 2 −2α−1 1 −α−1/2 2 Fk (x) = ce−x Knα x 2 , x 2 cn1/2 x 2 + cn1/2 x + n−1/2 . n k=0
Combining this estimate with (8.48) and the inductive assumption shows that (3.17) holds in dimension d. 2 Proof of Proposition 4.1. We first prove (4.2). Let g ∈ Vn . Assume 1 < q < ∞ and let Λn be the kernel from (3.7), with a admissible of type (a). Evidently g = Λn ∗ g and using Hölder’s inequality and Proposition 3.3 we obtain for x ∈ Rd+ 1 1 g(x) Wα (n; ·)s+ p − q g(·)
nd/2 c Wα (n; x)1/2 where β := q (s + to obtain
1 p
q
1 1 Λn (x, y)Wα (n; y)−s− p + q q wα (y) dy
Rd+
wα (y) dy
Rd+
Wα (n; y)
q 2 +β
1/q
(1 + n1/2 x − y)σ
1/q
1 1 Wα (n; ·)s+ p − q g(·) , q
− q1 ). To estimate the last integral we use estimate (3.15) from Lemma 3.4 g(x) c
1 1 nd/2q Wα (n; ·)s+ p − q g(·) q Wα (n; x)s+1/p
(8.49)
and hence 1 Wα (n; ·)s+ p g(·)
∞
s+ 1 − 1 cnd/2q Wα (n; ·) p q g(·) q ,
1 < q ∞.
(8.50)
If 0 < q 1, then the above estimate with q = 2 gives 1 Wα (n; ·)s+ p g(·)
∞
s+ 1 − 1 cnd/4 Wα (n; ·) p 2 g(·) 2 1−q/2 q/2 s+ 1 − 1 cnd/4 Wα (n; ·)s+1/p g(·) ∞ Wα (n; ·) p q g(·) q .
Consequently, (8.50) holds for 0 < q 1 as well. Let 0 < q < p < ∞. Using (8.50), we have Wα (n; ·)s g(·) = p
1 1 1 Wα (n; x)s+ p g(x)p−q Wα (n; x)s+ p − q g(x)q wα (x) dx
Rd+
1−q/p 1 1 s+ 1 Wα (n; ·)s+ p − q g(·) q/p Wα (n; ·) p g(·) ∞ q
s+ 1 − 1 = cn(d/2)(1/q−1/p) Wα (n; ·) p q g(·) q .
1/p
G. Kerkyacharian et al. / Journal of Functional Analysis 256 (2009) 1137–1188
1177
Hence (4.2) holds when p < ∞. In the case p = ∞ (4.2) follows from (8.50). To prove (4.1) we first assume that 1 < q < ∞. We use again that g = Λn ∗ g, Hölder’s inequality, Proposition 3.3, and that Wα (n; x) n−|α|−d/2 to obtain g(x) cgq nd/2 Wα (n; x)−1 1/q cn(d+|α|)/q gq ,
x ∈ Rd+ ,
and hence g∞ cn(d+|α|)/q gq . For the rest of the proof of (4.1) one proceeds similarly as in the proof of (4.2). We skip the details. To prove estimate (4.3) we first observe that (8.49) with s = γ + 1/p − 1/q yields g(x) c
nd/2q Wα (n; ·)s g(·) , q Wα (n; x)s+1/q
1 < q < ∞,
and, since Wα (n; x) n−|α|− 2 , we get g∞ cn(|α|+ 2 )s+(|α|+d)/q Wα (n; ·)s g(·)q . The remaining part of the proof is similar to the proof of (4.2). We omit it. 2 d
d
Proof of Lemma 4.4. (a) By (2.10) and the definition of Fnα , it follows that Fnα ∞ cnα/2 . Hence, using (8.27) if |x| 1 and (8.30) if |x| 1, we obtain d α F (x) cn(α+1)/2 , dx n
x ∈ R+ .
Furthermore, taking one more derivative of (8.24) and using (8.27) shows that
√ α+1 √ d2 α d d α+1 Fn (x) = − Fnα (x) + 2 nFn−1 (x) + x Fnα (x) + 2 n x Fn−1 (x) 2 dx dx dx
√ α+1 α α = − Fnα (x) + 2 nFn−1 (x) − (α + 1)Fnα (x) + bn+1 Fn+1 (x) − bn Fn−1 (x) √ α+1 α+1 (x) + bn Fnα+1 (x) − bn−1 Fn−2 (x) , + 2 n −(α + 1)Fn−1 k+1
k−1
d d α α which allows us to iterate and express dx k+1 Fn (x) in terms of dx k−1 Fn (x) and The recurrence relation (8.29) allows us to use induction to conclude that
k d α (α+k)/2 , dx k Fn (x) cn
d k−1 F α+1 (x). dx k−1 n
x ∈ R+ .
Therefore, for the product Laguerre functions, we have β α ∂ F (x) c |ν| + 1 (|α|+|β|)/2 , ν
|ν| = n, β ∈ Nd0 , x ∈ Rd+ .
Furthermore, together with the three term relation (8.26), the above inequality also shows that 2γ β α x ∂ F (x) c |ν| + 1 (|α|+|β|+2|γ |)/2 , ν
|ν| = n, β, γ ∈ Nd0 , x ∈ Rd+ .
1178
G. Kerkyacharian et al. / Journal of Functional Analysis 256 (2009) 1137–1188
Hence, if |φ, Fνα | ck (|ν| + 1)−k for all k, then
φ, Fνα x γ ∂ β Fνα (x),
x γ ∂ β φ(x) =
ν∈Nd0
where the series converges uniformly and hence γ β x ∂ φ(x) c φ, F α |ν| + 1 (|α|+|β|+2|γ |)/2 ck P ∗ (φ) ν k
(8.51)
ν∈Nd0
if k > d + |α| + |β| + 2|γ |)/2, which shows that φ ∈ S+ . (b) Assuming that φ ∈ S+ we next show that |φ, Fνα | has the claimed decay. From the wellknown second order differential equation satisfied by Lαn , a straightforward computation shows that Fnα (x) satisfies the equation y +
2α + 1 y − x 2 y + 2(2n + α + 1)y = 0. x
In particular, it follows that Fνα (x) satisfies, for each i = 1, 2, . . . , d, the equation Dxi u + xi2 u = 2(2νi + αi + 1)u, where Dxi := −∂i2 − (2αi + 1)xi−1 ∂i and ∂i =
∂ . ∂xi
(8.52)
Let k 1 and assume that the multi-index ν is fixed and ν = max1j d νj k. Choose xi = (x1 , . . . , xi−1 , 0, xi+1 , . . . , xd ). Denote briefly Ur (x) := i so that νi = ν and denote xi2 /2 r ∂i (φ(x)e ). Then by Taylor’s identity
φ(x)e
xi2 /2
−
2k−1
xir Ur ( x )/r! =
r=0
xi2k (2k − 1)!
1 (1 − t)2k−1 U2k ( x + txi ei ) dt, 0
which easily leads to φi (x) := φ(x) − e−xi /2 2
2k−1
xir Ur ( x )/r!
r=0
1 = xi2k
(1 − t)2k−1 0
2k
b2k−j (txi )∂i φ( x + txi ei )e−xi (1−t ) dt, j
2
2
(8.53)
j =0
where bj (·) (0 j 2k) is a polynomial of degree j and ei is the ith coordinate vector in Rd . α Then by the orthogonality of Fνii (recall that νi 2k) and (8.52) it follows that
φ, Fνα = φi , Fνα =
1 φi , (Dxi + xi2 )Fνα . 2(2νi + αi + 1)
G. Kerkyacharian et al. / Journal of Functional Analysis 256 (2009) 1137–1188
1179
* + The operator Dxi can be written in a self-adjoint form xi2αi +1 Dxi = ∂i xi2αi +1 ∂i . We use this and integration by parts to obtain
φi , Dxi Fνα
2α +1 φi (x)∂i xi i ∂i Fνα (x) dxi d x
= R+ Rd−1 +
∂i xi2αi +1 ∂i φi (x) Fνα (x) dxi d x = Dxi φi , Fνα .
= R+ Rd−1 +
Consequently,
φ, Fνα = =
1 Dxi + xi2 φi , Fνα 2(2νi + αi + 1) 2k (2νi
k 1 Dxi + xi2 φi , Fνα , k + αi + 1)
(8.54)
where we iterated k times. It is easy to see that there is a representation of the form 2k 2k−j 2 j −1 2 k 2 k Dxi + xi = −∂i − (2αi + 1)xi ∂i + xi = aj xi− ∂i j =0 =−2k
for some constants aj . On the other hand, by (8.53) it follows that if j + 2k j supxi− ∂i φi (x) c x
max
supx γ ∂ β φ(x) = c
|γ |4k,|β|2k+j x
max
|γ |4k,|β|2k+j
Pβ,γ (φ).
We use the above in (8.54) to obtain φ, F α ν
2k (2νi
1 + αi + 1)k
c|ν|−k+(|α|+d)/2
max
|γ |4k,|β|4k
max
|γ |4k,|β|4k
Pβ,γ (φ) Fνα 1
Pβ,γ (φ),
ν k.
(8.55)
Here we also used that Fνα 1 c|ν|(|α|+d)/2 which follows from Lemma 2.1. Estimate (8.55) shows that |φ, Fνα | ck (|ν| + 1)−k+(|α|+d)/2 for any k 1. Thus |φ, Fνα | has the claimed decay. The equivalence of the topologies on S+ induced by the semi-norms Pγ ,β from (4.7) and the norms Pk∗ from (4.11) follows readily by (8.51) and (8.55). 2
1180
G. Kerkyacharian et al. / Journal of Functional Analysis 256 (2009) 1137–1188
9. Proofs for Sections 6, 7 Proof of Proposition 6.4. We shall use a standard decomposition of unity argument. Suppose b ∈ C∞ satisfies the conditions: supp b ⊂ [1/4, 4], b 0, and b(t) + b(4t) = 1 on [1/4, 1]; (R) ∞ − hence =0 b(4 t) = 1, t ∈ [1, ∞). Now, define
Φ0 (x, y) := m(0)F0α (x, y)
and Φ (x, y) :=
4
m(j ) b j/4−1 Fjα (x, y),
1.
j =0
Then for the kernel K(x, y) of the operator Tmα we have K(x, y) = ∞ =0 Φ (x, y). By (6.6) it readily follows that (d/dt)k [m(t) b(t/4−1 )]∞ c4−k and just as in the proof of Theorem 3.2 (using also (5.17)) we get for x, y ∈ Rd+ Φ (x, y)
c2d , W (4 ; y)(1 + 2 x − y)σ
∂ c2(d+1) Φ (x, y) ∂y W (4 ; y)(1 + 2 x − y)σ , r
for 1 r d, where σ = k − (5/2)|α| − (3/4)d − 2. By a simple standard argument these two estimates (σ > d + 1) lead to K(x, y)
c wα (y)x − yd
∂ c and K(x, y) , ∂yr wα (y)x − yd+1
1 r d.
As in the weighted case on Rd (see [14]), these estimates show that Tmα is a Calderón–Zygmund type operator and hence Tmα is bounded on Lp (wα ), 1 < p < ∞. 2 Proof of Lemma 6.8. Using the orthogonality of Laguerre functions, we have Φj ∗ ψξ (x) = 0 for ξ ∈ Xm if |m − j | 2. √ Let ξ ∈ Xm , j − 1 m j + 1. Assume first that ξ (1 + δ) 6 · 2m . From (5.26), (5.27) it follows that m3d/2 Φj ∗ ψξ (x) cσ √ 2 Wα (4m ; x)
c2m3d/2 √ Wα (4m ; x)
Rd
Rd+
wα (y) dy Wα (4m ; y)(1 + 2m x − y)σ (1 + 2m y − ξ )σ
dy (1 + 2m x − y)σ (1 + 2m y − ξ )σ
(σ > d)
c2md/2 c2md/2 √ √ Wα (4m ; ξ )(1 + 2m x − ξ )σ −2|α|−2d Wα (4m ; x)(1 + 2m x − ξ )σ c , μ(Rξ )1/2 (1 + 2m x − ξ )σ −2|α|−2d where for the last two inequalities we used (5.14)–(5.17). Since σ can be arbitrarily large the claimed estimate (6.13) √ follows. Let ξ > (1 + δ) 6 · 2m . Just as above we use (5.26) and (5.28) to obtain
G. Kerkyacharian et al. / Journal of Functional Analysis 256 (2009) 1137–1188 m(d−L) Φj ∗ ψξ (x) cσ √ 2 Wα (4m ; x)
Rd+
1181
wα (y) dy Wα (4m ; y)(1 + 2m x − y)σ (1 + 2m y − ξ )σ
c2m(d−L) √ . Wα (4m ; ξ )(1 + 2m x − ξ )σ −2|α|−2d Since, in general, μ(Rξ ) c2−md/3 Wα (4m ; ξ ) and L can be arbitrarily large the above again leads to (6.13). 2 Proof of Lemma 6.10. Denote h j (x) :=
η∈Xj
|hη | (1 + 2j d(x, Rη ))κ
κ := λ − 2|α| + d |ρ|/d,
,
(9.1)
where d(x, E) := infy∈E x − y is the ∞ distance of x from E. We will show that h j (x) cMt
x ∈ Rd+ .
|hω |1Rω (x),
ω∈Xj
(9.2)
Evidently, h∗j (x) h j (x), x ∈ Rd+ , and hence (9.2) implies (6.16). On the other hand, using (5.17) we have for ξ ∈ Xj −ρ/d ∗ W α 4j ; ξ hξ η∈Xj
Wα (4j ; η)−ρ/d |hη | cHj (x) (1 + 2j ξ − η)λ−(2|α|+d)|ρ|/d
for x ∈ Rξ ,
where Hη := Wα (4j ; η)−ρ/d hη . Therefore, (9.2) yields (6.17) as well. By the definition of Qj in (5.12) it follows that there exists a constant c > 0 depending only on d such that ! d
Rξ ⊂ 0, c 2j . Qj := ξ ∈ Xj
Let x ∈ Rd . To prove (9.2) we consider two cases for x. Case 1. Let x > 2c 2j . Then d(x, Rη ) > x/2 for η ∈ Xj and hence h j (x) =
η∈Xj
|hη | (1 + 2j d(x, Rη ))κ
c4j d (2j x)κ
c (2j x)κ
|hη |
η∈Xj
1/t |hη |t
,
(9.3)
η∈Xj
where := 1 − min{1, 1/t} 1 and for the last estimate we use Hölder’s inequality if t > 1 and the t-triangle inequality if t < 1.
1182
G. Kerkyacharian et al. / Journal of Functional Analysis 256 (2009) 1137–1188
Denote Qx := [0, x]d . Evidently, μ(Qx ) ∼ x2(|α|+d) and combining this with (9.3) we arrive at h j (x)
c4j d x2(|α|+d)/t (2j x)κ
1 μ(Qx )
Qx
c2j (2d−κ) x2(|α|+d)/t−κ Mt
t
1/t |hη |1Rη (y) wα (y) dy
η∈Xj
|hη |1Rη (x) cMt |hη |1Rη (x)
η∈Xj
η∈Xj
as claimed. Here we used the fact that κ > max{2d, 2(|α| + d)/t}. Case 2. Let x 2c 2j . We first subdivide the tiles {Rη }η∈Xj into boxes of almost equal sides of length ∼ 2−j . By the construction of the tiles (see (5.11)) there exists a constant c˜ > 0 such ˜ −j . Now, evidently each tile Rη can be subdivided that the minimum side of each tile Rη is c2 into a disjoint union of boxes Rθ with centers θ such that
d d
θ + −c2 ˜ −j −1 , c2 ˜ −j −1 ⊂ Rθ ⊂ θ + −c2 ˜ −j , c2 ˜ −j . j the set of centers of all boxes obtained by subdividing the tiles from Xj . Also, set Denote by X hθ := hη if Rθ ⊂ Rη . Evidently, h j (x) :=
η∈Xj
|hη | |hθ | (1 + 2j d(x, Rη ))κ (1 + 2j d(x, Rθ ))κ
(9.4)
j θ∈X
and
|hη |1Rη =
|hθ |1Rθ .
(9.5)
j η∈X
η∈Xj
j : 2j θ − x c}, ˜ Denote Y0 := {θ ∈ X j : c2 Ym := θ ∈ X ˜ m−1 2j θ − x c2 ˜ m , and Qm := y ∈ Rd : y − x c(2 ˜ m + 1)2−j , m 1. Clearly, #Ym c2md , θ∈Ym
" θ∈Ym
= Rθ ⊂ Qm , and X
"
m0 Ym .
Similarly as in (9.3)
|hθ | (1 + 2j d(x, Rθ ))κ
c2−mκ
|hθ | c2−mκ 2md
θ∈Ym
c2−m(κ−d)
" θ∈Ym
1/t |hθ |t
θ∈Ym
Rθ θ∈Ym
μ(Rθ )−1 |hθ |t 1Rθ (y)wα (y) dy
1/t
G. Kerkyacharian et al. / Journal of Functional Analysis 256 (2009) 1137–1188
c2
−m(κ−d)
μ(Qm ) 1/t
1 μ(Qm )
θ∈Ym
Qm
Using (4.5) and that
" θ∈Ym
μ(Rθ )
1183
t
1/t |hθ |1Rθ (y) wα (y) dy .
Rθ ⊂ Qm we get
d 2(m−j )d xl + 2m−j 2αj +1 μ(Qm ) c −j d μ(Rθ ) 2 θl + 2−j l=1
c2md
d l=1
θl + 2 · 2m−j θl + 2−j
2αj +1
c2m(2|α|+3d) .
Therefore, θ∈Ym
|hθ | c2−m(κ−d−(2|α|+3d)/t) Mt j (1 + 2 d(x, Rθ ))κ
|hη |1Rη (x).
η∈Xj
Summing up over m 0, taking into account that κ > d + (2|α| + 3d)/t, and also using (9.4) we arrive at (9.2). 2 Proof of Lemma 6.11. For this proof we will need an additional lemma. Lemma 9.1. Let g ∈ V4j . For any σ > 0 and L > 0 we have for x , x ∈ 2Rξ , where ξ ∈ Xj , j 0, g(x ) − g(x ) c2j |x − x | η∈Xj
|g(η)| (1 + 2j ξ − η)σ
(9.6)
and g(x ) − g(x ) c∗ 2−j L |x − x | η∈Xj
|g(η)| , (1 + 2j ξ − η)σ
√ if ξ > (1 + 2δ) 6 · 2j . (9.7)
Here c and c∗ depend on α, d, δ, and σ and c∗ depends on L as well; 2Rξ ⊂ Rd is the set obtained by dilating Rξ by a factor of 2 and with the same center. Proof. Let Λ4j be the kernel from (3.7) with n = 4j , where a is admissible of type (a) with v := δ. Then Λ4j ∗ g = g and Λ4j (x, ·) ∈ V[(1+δ)4j ] . Note that [(1 + δ)4j ] + 4j 2nj − 1. Therefore, by Corollary 5.2 g(x) =
Λ4j (x, y)g(y)wα (y) dy = Rd
cη Λ4j (x, η)g(η),
η∈Xj
where cη ∼ |Rη |Wα (4j ; η). From this, we have for x , x ∈ 2Rξ , ξ ∈ Xj ,
1184
G. Kerkyacharian et al. / Journal of Functional Analysis 256 (2009) 1137–1188
g(x ) − g(x ) cη Λ4j (x , η) − Λ4j (x , η)g(η) η∈Xj
cx − x
cη sup ∇Λ4j (x, η) g(η).
(9.8)
x∈2Rξ
η∈Xj
√ Note that (6(1 + δ)4j + 3α + 3)1/2 (1 + δ) 6 · 2j for sufficiently large j (depending on α and δ). Therefore, using Theorem 3.2 we have for η ∈ Xj ∇Λ j (x, η) 4
c2j (d+1) , Wα (4j ; x) Wα (4j ; η)(1 + 2j x − η)σ
x ∈ Rd+ ,
(9.9)
and for any L > 0 ∇Λ j (x, η) 4
c2−j L , (1 + 2j x − η)σ
√ if min x, η > (1 + δ) 6 · 2j .
(9.10)
√ √ Suppose ξ (1 + 2δ) 6 · 2j and denote Xj := {η ∈ Xj : η (1 + δ) 6 · 2j } and X := Xj \ Xj . We split the sum in (9.8) over X and X to obtain g(x ) − g(x ) cx − x
... +
η∈Xj
. . . =: cx − x (Σ1 + Σ2 ).
η∈Xj
Using (9.9), (5.17), and that cη ∼ 2−j d Wα (4j ; η) for η ∈ Xj , we get Σ1 c2j
η∈Xj
c2j
sup x∈2Rξ
Wα (4j ; η) Wα (4j ; x)
η∈Xj
(1 + 2j ξ
1/2
|g(η)| (1 + 2j x − η)σ
|g(η)| . − η)σ −2(|α|+d)
(9.11)
To estimate Σ2 we use (9.10) and the rough estimate cη c2j d . We get Σ2 c2−j (L−d−2σ/3)
η∈Xj
|g(η)| . (1 + 2j ξ − η)σ
(9.12)
Here we also used that 1 + 2j ξ − η 1 + 2j c2−j/3 + x − η c22j/3 1 + 2j x − η for x ∈ 2Rξ . Estimates (9.11) (with sufficiently √ √ large σ ) and (9.12) (with L d + 2σ/3) imply (9.6). In the case ξ > (1 + 2δ) 6 · 2j , we have 2Rξ ⊂ {x ∈ Rd+ : x (1 + δ) 6 · 2j } for sufficiently large j and one proceeds just as above but uses only (9.10) as in the estimation of Σ2 . We skip the details. 2
G. Kerkyacharian et al. / Journal of Functional Analysis 256 (2009) 1137–1188
1185
We now proceed with the proof of Lemma 6.11. Let g ∈ V4j . Let 1 be sufficiently large (to be determined later on) and denote for ξ ∈ Xj Xj + (ξ ) := {η ∈ Xj + : Rη ∩ Rξ = ∅} and dξ := sup g(x ) − g(x ): x , x ∈ Rη for some η ∈ Xj + (ξ ) .
(9.13) (9.14)
Our first step is to estimate dξ , ξ ∈ Xj . Two cases are to be considered here. √ Case I. Let ξ (1 + 3δ) 6 · 2j . By (5.14) d
Rξ ∼ ξ + −2−j , 2−j
d
and Rη ∼ η + −2−j − , 2−j − ,
Hence, for sufficiently large ( = (d, δ)) we have
"
η∈Xj + (ξ ) Rη
η ∈ Xj + (ξ ).
(9.15)
⊂ 2Rξ . Now, using estimate
(9.6) of Lemma 9.1 with σ λ and the fact that diam(Rη ) ∼ 2−j − for η ∈ Xj + (ξ ), we get dξ c2−
η∈Xj
|g(η)| , (1 + 2j ξ − η)λ
(9.16)
where c > 0 is a constant independent of . √ √ j j Case " II. Let ξ > (1 + 3δ) 6 · 2 . By (5.14) it follows that x > (1 + 2δ) 6 · 2 for x ∈ η∈Xj + (ξ ) Rη if j is sufficiently large. We apply estimate (9.7) of Lemma 9.1 with σ λ and L = 1 to obtain dξ c2−j
η∈Xj
|g(η)| . (1 + 2j ξ − η)λ
(9.17)
We next estimate Mξ∗ , ξ ∈ Xj (see (6.14)). Two cases for ξ occur here. √ Case 1. Let ξ (1 + 4δ) 6 · 2j . Note that (9.15) is again valid. By the definition of dξ in (9.14) it follows that Mξ mω + dξ for some ω ∈ Xj + (ξ ) and hence, using (9.15), Mξ c
ω∈Xj +
mω + dξ =: m ξ + dξ , (1 + 2j + ξ − ω)λ
c = c(d, δ, λ, ).
Consequently, Mξ∗ m ∗ξ + dξ∗ .
(9.18)
√ Denote Xj := {η ∈ Xj : η (1 + 3δ) 6 · 2j } and Xj := Xj \ Xj . Now, we use (9.16), (9.17) to obtain
1186
G. Kerkyacharian et al. / Journal of Functional Analysis 256 (2009) 1137–1188
dξ∗ :=
dη j (1 + 2 ξ
η∈Xj
− η)λ
c2−
η∈Xj ω∈Xj
(1 + 2j ξ
|g(ω)| − η)λ (1 + 2j η − ω)λ
+ c2−j
η∈Xj ω∈Xj
(1 + 2j ξ
|g(ω)| . − η)λ (1 + 2j η − ω)λ
Replacing Xj and Xj by Xj above and shifting the order of summation we get g(ω) dξ∗ c 2− + 2−j ω∈Xj
c 2− + 2−j ω∈Xj
η∈Xj
1 (1 + 2j ξ
− η)λ (1 + 2j η − ω)λ
|g(ω)| c 2− + 2−j Mξ∗ . (1 + 2j ξ − ω)λ
(9.19)
Here the constant c is independent of and j , and we used that
1
η∈Xj
(1 + 2j ξ
− η)λ (1 + 2j η − ω)λ
c (1 + 2j ξ
− ω)λ
(λ > d).
(9.20)
This estimate easily follows from the fact that ξ − ξ c2−j for all ξ , ξ ∈ Xj . To estimate m ∗ξ we use again (5.14) and (9.20). We get m ∗ξ :=
η∈Xj
c
m η j (1 + 2 ξ − η)λ
ω∈Xj +
c
ω∈Xj +
mω
η∈Xj
c
η∈Xj ω∈Xj +
(1 + 2j ξ
mω − η)λ (1 + 2j η − ω)λ
1 (1 + 2j ξ − η)λ (1 + 2j η − ω)λ
mω mω λ c2 = cm∗θ (1 + 2j ξ − ω)λ (1 + 2j + θ − ω)λ ω∈Xj +
for each θ ∈ Xj + (ξ ). Combining this with (9.18), (9.19) we obtain Mξ∗ c1 m∗θ + c2 2− + 2−j Mξ∗
for θ ∈ Xj + (ξ ),
where c2 > 0 is independent of and j . Choosing and j sufficiently large (depending only on d, δ, and λ) this yields Mξ∗ cm∗θ for all θ ∈ Xj + (ξ ). For j c this relation follows as above but using only (9.6) and taking large enough. We skip the details. Thus we have shown (6.18) in Case 1.
G. Kerkyacharian et al. / Journal of Functional Analysis 256 (2009) 1137–1188
1187
√ Case 2. Let ξ > (1 + 4δ) 6 · 2j . Choose 1 the √ same as in Case " 1. Clearly, for sufficiently large j (depending only on d and δ) x (1 + 3δ) 6 · 2j for x ∈ η∈Xj + (ξ ) Rη . Hence, using (9.7) with L = 1, we have Mξ mω + c2−j
η∈Xj
|g(η)| mω + c2−j Mξ∗ (1 + 2j ξ − η)λ
for all ω ∈ Xj + (ξ ),
where c > 0 is independent of j . Fix θ ∈ Xj + (ξ ) and for each η ∈ Xj , η = ξ , choose ωη ∈ Xj + (η) so that θ − ωη = minω∈Xj + (η) θ − ω. Then from above Mξ∗
η∈Xj
mωη (1 + 2j ξ − η)λ
Mη∗
+ c2−j
η∈Xj
(1 + 2j ξ − η)λ
=: Σ1 + Σ2 .
(9.21)
From (2.19) it easily follows that ωη from above satisfies |θ − ωη | c|ξ − η| and hence Σ1 c
η∈Xj
mωη (1 + 2j θ − ωη )λ
c2λ
ω∈Xj +
mω c1 m∗θ . (1 + 2j + θ − ω)λ
(9.22)
On the other hand, using Definition 6.9 and (9.20), we have
Σ2 c2−j
η∈Xj ω∈Xj
c2−j
Mω
ω∈Xj
c2 2−j
ω∈Xj
Mω (1 + 2j ξ − η)λ (1 + 2j η − ω)λ
η∈Xj
1 (1 + 2j ξ
− η)λ (1 + 2j η − ω)λ
Mω = c2 2−j Mω∗ (1 + 2j ξ − ω)λ
with c2 > 0 independent of j . Combining this with (9.21), (9.22) we arrive at Mξ∗ c1 m∗θ + c2 2−j Mξ∗
for θ ∈ Xj + (ξ ).
Choosing j sufficiently large we get Mξ∗ c1 m∗θ for each θ ∈ Xj + (ξ ). For j c this estimate follows as in Case 1 but using only (9.6). This completes the proof of Lemma 6.11. 2 Proof of Lemma 7.5. Let g ∈ V4j and 0 < p < ∞. We will utilize Definition 6.9 and Lemmas 6.10, 6.11. To this end we select 0 < t < p and λ as in Definition 6.9. Set Mξ := supx∈Rξ |g(x)|, ξ ∈ Xj , and mη := infx∈Rη |g(x)|, η ∈ Xj + , where 1 is the constant from Lemma 6.11. By (1.2) and the properties of the tiles Rξ from (5.14)–(5.16) it readily follows that Wα (4j + ; y) ∼ Wα (4j , ξ ) for y ∈ Rξ . We now use this, Lemmas 6.10, 6.11 and the maximal inequality (4.6) to obtain
1188
G. Kerkyacharian et al. / Journal of Functional Analysis 256 (2009) 1137–1188
1/p p j −ρp/d Wα 4 ; ξ max g(x) μ(Rξ ) x∈Rξ
ξ ∈ Xj
j −ρ/d ∗ Wα 4 ; ξ Mξ 1Rξ ξ ∈ Xj
p
j + −ρ/d ∗ j + −ρ/d c Wα 4 ; η mη 1Rη c Mt Wα 4 ; η mη 1Rη η∈Xj +
p
η∈Xj +
p
η∈Xj +
j −ρ/d j + −ρ/d c Wα 4 ; η mη 1Rη g(·) p . c Wα 4 ; ·
p
2
References [1] G. Andrew, R. Askey, R. Roy, Special Functions, Cambridge Univ. Press, Cambridge, 1999. [2] J. Dziuba´nski, Triebel–Lizorkin spaces associated with Laguerre and Hermite expansions, Proc. Amer. Math. Soc. 125 (1997) 3547–3554. [3] J. Epperson, Hermite and Laguerre wave packet expansions, Studia Math. 126 (1997) 199–217. [4] M. Frazier, B. Jawerth, Decomposition of Besov spaces, Indiana Univ. Math. J. 34 (1985) 777–799. [5] M. Frazier, B. Jawerth, A discrete transform and decomposition of distribution spaces, J. Funct. Anal. 93 (1990) 34–170. [6] M. Frazier, B. Jawerth, G. Weiss, Littlewood–Paley Theory and the Study of Function Spaces, CBMS Reg. Conf. Ser. Math., vol. 79, Amer. Math. Soc., Providence, RI, 1991. [7] L. Gatteschi, Asymptotics and bounds for the zeros of Laguerre polynomials: A survey, J. Comp. Appl. Math. 144 (2002) 7–27. [8] G. Kyriazis, P. Petrushev, Yuan Xu, Jacobi decomposition of weighted Triebel–Lizorkin and Besov spaces, Studia Math. 186 (2008) 161–202. [9] G. Kyriazis, P. Petrushev, Yuan Xu, Decomposition of weighted Triebel–Lizorkin and Besov spaces on the ball, Proc. London Math. Soc. 97 (2008) 477–513. [10] G. Mastroianni, D. Occorsio, Lagrange interpolation at Laguerre zeros in some weighted uniform spaces, Acta Math. Hungar. 91 (2001) 27–52. [11] F.J. Narcowich, P. Petrushev, J.D. Ward, Decomposition of Besov and Triebel–Lizorkin spaces on the sphere, J. Funct. Anal. 238 (2006) 530–564. [12] J. Peetre, New Thoughts on Besov Spaces, Duke Univ. Math. Ser., Durham, NC, 1976. [13] P. Petrushev, Yuan Xu, Decomposition of spaces of distributions induced by Hermite expansion, J. Fourier Anal. Appl. 14 (2008) 372–414. [14] E. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Univ. Press, Princeton, NJ, 1993. [15] G. Szeg˝o, Orthogonal Polynomials, fourth ed., Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, RI, 1975. [16] S. Thangavelu, Lectures on Hermite and Laguerre Expansions, Princeton Univ. Press, Princeton, NJ, 1993. [17] H. Triebel, Theory of Function Spaces, Monogr. Math., vol. 78, Birkhäuser, Basel, 1983. [18] Yuan Xu, A note on summability of multiple Laguerre expansions, Proc. Amer. Math. Soc. 128 (2000) 3571–3578. [19] A. Zygmund, Trigonometric Series, Cambridge Univ. Press, Cambridge, 1959.
Journal of Functional Analysis 256 (2009) 1189–1237 www.elsevier.com/locate/jfa
On the Hardy–Littlewood majorant problem for random sets G. Mockenhaupt a , W. Schlag b,∗ a Fachgebiet Mathematik, Katholische Universität Eichstätt-Ingolstadt, Ostenstrasse 26, 85972 Eichstätt, Germany b Department of Mathematics, The University of Chicago, 5734 South University Avenue, Chicago, IL 60637, USA
Received 28 April 2008; accepted 10 June 2008 Available online 10 September 2008 Communicated by C. Kenig
Abstract We study the classical Hardy–Littlewood majorant problem for trigonometric polynomials. We show that the constant in the majorant inequality grows at most like an arbitrary small power of the degree provided the spectrum is chosen at random. We also give an example of a deterministic set where the majorant property fails, i.e., the constant grows like a fixed small power in the degree. © 2008 Published by Elsevier Inc. Keywords: Trigonometric polynomials; Classical harmonic analysis
1. The majorant property: Introduction This paper is concerned with versions of the majorant property of various randomly generated subsets of integers in [1, N]. More precisely, suppose A ⊂ [1, N] is a set of integers of size1 |A| N ρ for some fixed 0 < ρ < 1. For example, one can take A to be the squares, cubes, etc., or (multi-dimensional) arithmetic progressions in [1, N]. Given p 2 we ask for the smallest constant C such that uniformly for |an | majorized by 1 (we write e(nt) = e2πint ) * Corresponding author.
E-mail addresses: [email protected] (G. Mockenhaupt), [email protected] (W. Schlag). 1 Throughout this paper x y for x, y > 0 means that x Cy with a constant C. In general, constants in this paper are only allowed to depend on p (and they are referred to as “absolute” in that case). Similarly, x y means that x < C −1 y
with C large, whereas x y means that x y and y x.
0022-1236/$ – see front matter © 2008 Published by Elsevier Inc. doi:10.1016/j.jfa.2008.06.005
1190
G. Mockenhaupt, W. Schlag / Journal of Functional Analysis 256 (2009) 1189–1237
an e(n ·) C e(n ·) . p
n∈A
(1.1)
p
n∈A
If p is an even integer, then one can take C = 1, in particular C does not depend on A (respectively N ). In fact, Hardy and Littlewood [7] realized that whenever |an | bn and p even N N an e(n ·) C bn e(n ·) n=1
n=1
p
(1.2) p
holds with C = 1. On the other hand, it has been known for some time that if p is not an even integer the constant C in (1.2) does grow unboundedly with N (see, e.g. [10, p. 133]). A quantitative lower bound of order N c/ log log N , for some c > 0, is obtained in [9] for (1.1) with a particular sequence of integer sets AN in [1, N]. We will improve on this lower bound and show that for an appropriate sequence of integer sets AN ⊂ [1, N] the constants in (1.1) grow by a power in N (see Theorem 3.2). This result has also been obtained with a similar method by B. Green and I. Ruzsa [6] for the case p = 3. Unfortunately, both methods do not reveal a structural property of sets A which would guaranty a power growth in N of the constant C in (1.1). Inequality (1.1) for particular sets A plays an important role in analysis and number theory. For example, it is conjectured by H. Montgomery (see [10, p. 11]) that for 2 < p < 4 the frequency sets A=
p/2 N log n 1 n N ,
here [·] denotes the integer part, satisfy (1.1) with a slow growing bound C = Cε N ε , ε > 0. We may also interpret (1.1) for certain sets A as a reformulations of the restriction conjecture for the Fourier transform on Rd : f L1 (S d−1 ,dσ ) Cf Lp (Rd ) ,
p < 2d/(d + 1),
here σ is rotational invariant measure on the unit sphere S d−1 ⊂ Rd . This can be seen by localizing the above restriction inequality, i.e. assuming f is supported in a ball of radius N , and by using the uncertainty principle, which allows us to assume that fˆ is essentially constant on squares of size 1/N . The relevant sets A are of the form
n1 nd 0 < ni ∈ Z, N 2 < n21 + · · · + n2d (N + 1)2 , A= Q + ··· + Q1 Qd where Q = Q1 · · · Qd and the Qi ’s are relatively prime integers of order N (see [9] for those matters). The main objective of the paper is to show that random sets of integers A ⊂ [1, N] of size N ρ which are obtained by selecting each integer 1 n N with equal probability satisfy for all γ > 0 γ an e(n ·) Cγ N e(n ·) sup
|an |1 n∈A
p
n∈A
(1.3) p
G. Mockenhaupt, W. Schlag / Journal of Functional Analysis 256 (2009) 1189–1237
1191
with large probability (see Theorem 4.4). Theorem 4.4 relies on a nice argument developed by Bourgain in [3]. In Section 4.3 we first provide a method for proving a weaker variant of Theorem 4.4 (see Proposition 4.6) which will later allow us to extend this result for certain values of p by showing a that the N γ -term is not necessary (see Theorem 4.12). For this we rely on a probabilistic lemma from Bourgain’s work [1]. In addition to random subsets in the last section we also consider perturbations of arithmetic progressions. This means that each element of a given arithmetic progression is shifted independently and randomly by some small amount. We again show that most sets obtained in this fashion satisfy (1.3) for any γ > 0, see Theorem 5.6. As before, the method can be presented abstractly for perturbations of arbitrary sets A that satisfy condition (2.3). Given the fact that even a single explicit frequency set A ⊂ [1, N] satisfying |A ∩ [0, N]| ≈ N α , 0 < α < 1, as well as inequality (4.15) (without the expected value on the left-hand side) is not known, we think that this is worth mentioning. 2. Some generalities In order to justify the size restriction |A| N ρ , 0 < ρ < 1, on a frequency set A ⊂ [1, N] we remark that by Hausdorff–Young’s inequality one always has the bound 1 N p . a e(n ·) C e(n ·) sup n |A|
|an |1 n∈A
p
n∈A
p
p Together with the obvious lower bound n∈A e(n ·)p |A|p N −1 this settles the case of any large sets A, i.e. ρ = 1, as well as all arithmetic progressions. Another easy estimate can be obtained by interpolation. Indeed, if 2 < p < 4, say, then interpolating between 2 and 4 yields the bound C = O(N γ ), γ (1 − p4 )(1 − p2 ). It turns out that this interpolation can be done more carefully, which gives optimal results for sets A whose Dirichlet kernel satisfies a certain “reverse interpolation inequality.” To this end, consider the convex set of trigonometric polynomials given by PA := { n∈A an e(nθ ) | |an | 1}. Then for any odd integer p > 2, 1 p p−2 1 sup an e(nθ ) dθ = sup an e(nθ ) a¯ k e(−kθ ) a e( θ ) dθ
|an |1
0
|an |1 n∈A
n∈A
sup
g∈PA
k∈A
∈A
1 2 p−1 p−2 (n)2 g|g| |A| sup |A|g2(p−1) (2.1)
e(n ·) n∈A
0
g∈PA
n∈A
p−1 e(n ·) .
2 n∈A
(2.2)
2(p−1)
Here the first inequality sign in (2.1) follows by putting absolute values inside and Cauchy– Schwarz, the second is Plancherel, and (2.2) uses the majorant property on 2(p − 1) ∈ 2N. Now assume for all ε > 0 the following condition e(n ·) n∈A
p−1 e(n ·)
2 n∈A
2(p−1)
p Cε N e(n ·) , ε
n∈A
p
(2.3)
1192
G. Mockenhaupt, W. Schlag / Journal of Functional Analysis 256 (2009) 1189–1237
with Cε depending only on ε. In view of the preceding, one then has (1.3) for any γ > 0. This condition, which is of basic importance for most of our work, is basically the reverse of the usual interpolation inequality. One checks immediately that arithmetic progressions satisfy (2.3). Also, observe that any frequency set A for which (2.3) holds for all p satisfies (1.3) for all p with γ > 0. Indeed, this follows inductively from the argument leading up to (2.2) using the majorant property from the previous stage 2(p − 1) to pass to the next stage p. Finally, interpolation is required to obtain the desired bound for all p (at the cost of N ε ). Another case which is covered by this argument, but not the previous one based on Hausdorff–Young, are multi-dimensional arithmetic progressions. For example, one easily checks that A = {b + j1 a1 + j2 a2 | 0 j1 < L1 , 0 j2 < L2 }
(2.4)
with a1 L1 < a2 , satisfies p p−1 e(n ·) (L1 L2 ) p
n∈A
for p > 1. Another interesting case are the squares A = {n2 | 1 n known that the there is a “kink” at p = 4 (see e.g. [2]), Cε N ε+ 12 e(n ·) n∈A
p
n∈A
p
√ N }. In this case it is well
if 2 p 4,
2 Cε N 1− p +ε e(n ·)
if p 4,
so that (2.3) holds only for 2 p 3. In particular, the argument leading up to (2.2) gives the (trivial) statement that the majorant property holds at p = 3 for the squares. A nontrivial statement can be obtained by improving on the use of Plancherel in (2.1). Indeed, it is a wellknown fact that N N 1 2 2 a n e n θ Cε N ε |an |2 n=1
4
n=1
⇔
N 2 2 fˆ n
1 2
Cε N ε f
n=1
4
L 3 (T)
, (2.5)
the second statement being the dual of the first. This can be checked by reducing the L4 -norm to an L2 -norm by squaring, and then using Cauchy–Schwarz and the N ε -bound on the divisor function, see [2]. We now repeat the argument leading up to (2.2) to conclude the following. Let P :=
N
2 an e n θ |an | 1 .
n=1
If p = 3k + 1, then one can apply the majorant property at 43 (p − 1) so that
G. Mockenhaupt, W. Schlag / Journal of Functional Analysis 256 (2009) 1189–1237
1193
p N p−2 1 1 N N N 2 2 2 2 sup an e n θ dθ = sup an e n θ a¯ k e −k θ a e θ dθ |an |1 |an |1 0
n=1
n=1
sup
g∈P
k=1
0
=1
N p−2 n2 2 g|g| |A|
1 2
sup
n=1
N 2 e n · n=1
1
g∈P
p−1 3 (p−1)
|A|g 4
(2.6)
N p−1 2 e n · 4
2 n=1
3 (p−1)
5
Cε N ε N 2 N p− 2 Cε N ε N p−2 N p ε 2 Cε N e n · . n=1
p
Here we used (2.5) in (2.6). This implies that for the sequence of squares (1.3) holds with any γ > 0 at p = 7, 13, 19, etc. Another case of sets A that do not satisfy (2.3) are random subsets A ⊂ [1, N ]. Indeed, we show below that random sets A which are obtained by selecting each integer 1 n N with probability τ have the property that for p > 1 p p τ p N p−1 + (τ N ) 2 , E e(nθ ) n∈A
p
−1+ 2
p so that it is clear see Proposition 4.6. The two terms on the right balance at τcrit = N that (2.3) cannot hold in general. The main objective of the following section is to show that nevertheless, such random subsets do satisfy (1.3) with large probability. The method to some extent resembles the calculation from (2.2), but is of course more involved. We rely on a probabilistic lemma from Bourgain’s work [1]. It is possible to abstract the arguments below, and then verify that various examples satisfy the conditions of such an abstract theorem, the most important one being condition (2.3). More precisely, starting with a deterministic set A, define SN (ω) = {n ∈ A | ξn = 1} where ξn are i.i.d. selector variables satisfying P[ξn = 1] = τ = 1 − P[ξn = 0]. If, amongst other things, (2.3) holds for A, then much of what is done in the following section goes through. On the other hand, some improvements which we obtain below for the case of arithmetic progressions are not easily axiomatized. Moreover, since we do not have any examples apart from (multi-dimensional) arithmetic progressions, we have decided against casting this into a more general framework. Thus, we write out the main argument only for arithmetic progressions. If (2.3) is violated, then our method applies only to certain p or after suitable modifications. For example, one can check that the machinery which we develop below shows that with high probability random subset of the squares satisfy (1.3) at p = 7 for any γ > 0. This requires invoking the (almost) Λ(4) property of the squares as in (2.6). It seems difficult to obtain the desired bound for all p in case of the squares.
1194
G. Mockenhaupt, W. Schlag / Journal of Functional Analysis 256 (2009) 1189–1237
3. Failure of the majorant property In order to establish a frequency set A for which the constant in the majorant inequality (1.1) is growing by a power in N we will need the following lemma. Lemma 3.1. Suppose p > 2 is not an even integer, then there are trigonometric polynomials q ˆ and Q with coefficients in {0, 1, −1} such that |q(n)| ˆ = Q(n) and 2πit > (1 + δp )Q e2πit . q e p p Proof. For m, k ∈ N define polynomials q and Q as follows q(z) = 1 + zk 1 − zm
and Q(z) = 1 + zk 1 + zm ,
where z = e(t). Let cn be the Fourier coefficients of f (t) = | sin πt|p and define an (p) = 1 π p −int dt, which satisfies the following recurrence formulae: π 0 (sin t) e n an (p). 2ian (p) = an−1 (p − 1) − an+1 (p − 1) and an−1 (p − 1) = i 1 + p Since cn = a2n (p) a little algebra gives cn+1 =
n−α cn , n+1+α
where α = p/2,
(3.1)
(n) = (−1)n cn . By using and cn = c¯n = c−n . Note that for F (t) = | cos πt|p we have F Plancherel’s identity and by choosing m, k relatively prime we get 2πit p 2πit p − Q e = 4p q e p
1 F (kt)f (mt) − F (kt)F (mt) dt
p
0
= 4p (−1)kn cnm cnk − (−1)mn+kn cnm cnk n∈Z
=4 (−1)nk cnm cnk 1 − (−1)nk . p
n∈Z
We choose k even and m = k + 1. Hence only odd n contribute to the latter sum which evaluates to 4p+1
cnm cnk .
n1,n odd
By the recursion formula for cn we see that if k, m > p/2 both term in the sum have the same sign. The lemma follows. 2
G. Mockenhaupt, W. Schlag / Journal of Functional Analysis 256 (2009) 1189–1237
1195
Theorem 3.2. Suppose p > 2 is an not even integer and N a sufficiently large integer. Then there exist αp > 0, a frequency set E ⊂ [0, N] ∩ Z and a sequence j ∈ {−1, 1} such that i2πnx i2πnx αp . j e e N p
n∈E
(3.2)
p
n∈E
Proof. The previous lemma provides a trigonometric polynomial q of degree d > 1 with Fourier coefficients in {0, −1, 1} whose majorant Q satisfies | q (l)| = Q(l), for l ∈ Z, and qp (1 + δ)Qp
(3.3)
for some δ > 0. We will inductively construct a finite Riesz product qk (x) = kj =0 q(mj x) where mj ∈ Z are randomly chosen in the interval [M j , 2M j ]. Note that by choosing M > d p sufficiently large the Fourier coefficients of qk are again contained in {0, 1, −1}. We claim that for k (n) = | qk (n)| we have the majorant Qk with Q p
p
qk p (1 + δ)k Qk p .
(3.4)
This gives us (3.2) since qk is of degree at most N 2dM k . Inequality (3.4) will be shown inductively. Define p fk (x) = qk (x)
p and gk (x) = q(mk+1 x) .
Note that gk has frequencies in mk+1 Z. By Plancherel’s identity we obtain for T > 0 p
1
qk+1 p =
fk (x)gk (x) dx 0
= f k (0)g k (0) +
gk (−mk+1 l) + f k (mk+1 l)
gk (−mk+1 l) f k (mk+1 l)
|l|T
0<|l|
= Ak + Bk + Ck . We have by induction p
p
p
p
Ak = qk p qp (1 + δ)k+p Qk p Qp . (n)| c1 , To estimate Ck , note that F = |q|p is at least twice differentiable. Therefore |n2 F gk (−mk+1 l)| c1 / l 2 and by Cauchy–Schwarz and with c1 depending only on d and p, hence | Parseval’s identity we get
Ck2
1 2 2p c2 c2 c2 d 2pk qk (x) dx 3 , fk (mk+1 l) 3 T T T3 |l|>T
0
where we used qk ∞ d k . To estimate Bk we apply Cauchy–Schwarz
1196
G. Mockenhaupt, W. Schlag / Journal of Functional Analysis 256 (2009) 1189–1237
f k (mk+1 l)2 ,
Bk2 c32
0<|l|
where c3 is the L2 -norm of F , i.e. only dependent on d and p. We will need to specify n = mk+1 . To do this, let I be the set of integers in [M k+1 , 2M k+1 ]. Then with Bk = Bk (n) 2 1 1 Bk (n) c32 |I | |I | n∈I
n∈I, 0<|l|
f k (nl)2 c2 1 3 |I |
2 d(m)f k (m)
0<m<2T M k+1
with R = 2T M k+1 and d(m) is the number of divisors of m, which is at most of order ec log R/ log log R c R . By choosing T = M 100pk and since |I | = M k+1 we may bound the later term by c32 c k M fk 22 M k+1 for all > 0. Since 1 fk 22
=
qk (x)2p dx d 2kp ,
(3.5)
0
by pigeonholing we find n ∈ I such that Bk c˜
d 2p M
k
2
M k M −k/4
(3.6)
provided we chose M sufficiently large. By collecting the estimates for Ak , Bk and Ck by adjusting M (to absorb c2 ) we get: qk+1 p (1 + δ)k+p Qk p Qp − 2M −k/4 p p (1 + δ)k+p 1 − o(1) Qk p Qp , p
p
p
(3.7) (3.8)
where the o-term is refers to M → ∞. We can perform the same analysis for Qk+1 . We only need to possibly modify the choice of mk+1 . However, since (3.5) holds for q replaced by Q (in the definition of f and g) we can choose mk+1 such that, say, the sum of the moduli of the Bk -term’s for qk+1 and Qk+1 satisfy the above bound as well. Hence, p p p Qk+1 p 1 + o(1) Qk p Qp and therefore p p p p qk+1 p (1 + δ)k+p 1 − o(1) Qk p Qp (1 + δ)k Qk+1 p .
2
G. Mockenhaupt, W. Schlag / Journal of Functional Analysis 256 (2009) 1189–1237
1197
4. Random subsets have the majorant property 4.1. Random sums over asymmetric Bernoulli variables We first dispense with some simple technical statements about the behavior of random sums with asymmetric Bernoulli variables as summands. They are definitely standard, but lacking a precise reference we prefer to present them. Lemma 4.1. Let ηj be i.i.d. variables so that P[ηj = 1 − τ ] = τ , P[ηj = −τ ] = 1 − τ . Here N 2 2 0 < τ < 1 is arbitrary. Let N 1 and {aj }N j =1 |aj | . j =1 ∈ C be given. Define σ = τ (1 − τ ) Then for λ > 0, N λ2 aj ηj > λσ 4e− 8 P j =1
provided max λ|aj | 4σ.
(4.1)
1j N
Proof. Assume first that all aj ∈ R. Then for any t > 0 P
N
aj ηj > λσ e−tλσ E exp t
j =1
N
(4.2)
a j ηj
j =1
= e−tλσ
N (1−τ )ta j + (1 − τ )e −τ taj . τe
(4.3)
j =1
Next, we claim that τ e(1−τ )x + (1 − τ )e−τ x exp 2τ (1 − τ )x 2
for all |x| 1.
(4.4)
1
Observe that this property fails for x = τ − 2 . To prove this, set φτ (x) = exp 2τ (1 − τ )x 2 − τ e(1−τ )x − (1 − τ )e−τ x . By symmetry it suffices to consider the case 0 x 1 and to show that φτ 0 there. Clearly, φτ (x) = τ (1 − τ ) 4x exp 2τ (1 − τ )x 2 − e(1−τ )x + e−τ x τ (1 − τ ) 4x − e(1−τ )x + e−τ x . Differentiating the expression in brackets yields 4 − (1 − τ )e(1−τ )x − τ e−τ x 4 − (1 − τ )e(1−τ )x − τ e(1−τ )x 4 − e > 0
(4.5)
1198
G. Mockenhaupt, W. Schlag / Journal of Functional Analysis 256 (2009) 1189–1237
for all 0 x 1. It follows that φτ (x) 0 for 0 x 1 and since φτ (0) = 0 we also have φτ (x) 0 for 0 x 1, as desired. Inserting (4.4) into (4.3) gives P
N
λ2 aj ηj > λσ min e−tλσ exp 2t 2 σ 2 = e− 8 t>0
j =1
provided for the minimizing choice of t = t0 one has maxj |t0 aj | 1. But t0 = condition therefore reads max
1j N
λ 4σ
and this
|λ||aj | 1, 4σ
which is precisely (4.1). Evidently, the same bound also holds for deviations less than −λσ , 2 which gives 2e−λ /8 as an upper bound on the large deviation probability in the real case. Finally, if an ∈ C, then one splits into real and complex parts. 2 Lemma 4.1 immediately leads to the following version of the Salem–Zygmund inequality for asymmetric variables. Corollary 4.2. With ηn and σ as in the previous lemma N P sup an ηn e(nθ ) > 20σ log N 4N −8 θ∈T
n=1
for any an ∈ C provided the following conditions hold: sup 10|an |2 log N σ 2 = τ (1 − τ ) 1nN
N
|ak |2 ,
k=1
10 τ (1 − τ )N log N .
(4.6)
−2 Proof. Let {θj }N j =1 ⊂ T be a N -net. Denote 2
TN,ω (θ ) :=
N
an ηn (ω)e(nθ ).
n=1
(θ ) = T
By using TN,ω N,ω ∗ DN (θ ), where DN denotes the Dirichlet kernel, Cauchy–Schwarz and Parseval’s identity give
minTN,ω (θ ) − TN,ω (θj ) N −2 TN,ω ∞ j
N
−2
N 1 2 3 −2 2 TN,ω 2 DN 2 N |an | 2N 2 n=1
G. Mockenhaupt, W. Schlag / Journal of Functional Analysis 256 (2009) 1189–1237
1199
2σ N − 2 10σ log N . =√ τ (1 − τ ) 1
The final inequality here follows from our assumption (4.6). Therefore, by Lemma 4.1, N N 2 N P sup an ηn e(nθ ) > 20σ log N P an ηn e(nθj ) > 10σ log N θ∈T
j =1
n=1
n=1
4N 2 exp(−100 log N/8) 4N −8 , which is precisely the bound claimed in the lemma. The first condition in (4.6) ensures that (4.1) holds. 2 In the proof of Theorems 4.4 and 4.12 we shall need to know the typical size of the easier norm in (4.18). We determine this norm in the following lemma. Lemma 4.3. Let ξj be selector variables as above with τ = N −δ , 0 < δ < 1 fixed. Let p 2 and define p 1 N ξn (ω)e(nθ ) dθ. Ip,N (ω) = 0
n=1
Then for some constants Cp , p p Cp−1 τ p N p−1 + (τ N ) 2 EIp,N Cp τ p N p−1 + (τ N ) 2 . Moreover, there is some small constant cp such that p P Ip,N cp τ p N p−1 + (τ N ) 2 → 0 as N → ∞. Proof. Let ηn (ω) = ξn (ω) − τ , so that Eηn = 0 and Eηn2 = τ (1 − τ ). Then p p 1 1 N N τ e(nθ ) dθ + ηn e(nθ ) dθ Ip,N (ω) 0
n=1
0
n=1
p 1 N τ p N p−1 + ηn e(nθ ) dθ. 0
One now checks that
n=1
(4.7)
1200
G. Mockenhaupt, W. Schlag / Journal of Functional Analysis 256 (2009) 1189–1237
p 1 N p E ηn e(nθ ) dθ Cp N τ (1 − τ ) 2 . n=1
0
This can be verified by expanding the norm for even p and then interpolating. Indeed, 2k 1 N E ηn e(nθ ) dθ n=1
0
1 =E 0
=
N n1 ,...,nk =1
E n
Ck
Ck
2 ηn1 . . . ηnk e (n1 + · · · + nk )θ dθ
n1 +···+nk =n
2 ηn1 . . . ηnk =
E[ηn1 . . . ηnk ηm1 . . . ηmk ]
n1 +···+nk =m1 +···+mk
k
N
r=1
n1 ,...,nr =1 s1 +···+sr =2k, si 2
k
r k N r τ (1 − τ ) Ck N τ (1 − τ ) .
E|ηn1 |s1 · · · · · E|ηnr |sr
(4.8)
(4.9)
r=1
The constants in (4.8) and (4.9) are of a combinatorial nature and not necessarily the same. The relevant point in (4.8) is that si 2 which is due to independence and Eηj = 0. In particular, si 2 implies the important fact r k. Moreover, to pass to the last line we used that for every positive integer s 2 τ (1 − τ ) Eηjs = τ (1 − τ ) τ s−1 + (1 − τ )s−1 22−s τ (1 − τ ). To obtain the lower bound on the expectation, one splits the integral in θ into the region where the Dirichlet kernel dominates the mean zero random sum and vice versa. More precisely, with √ 1+δ h = τ N −1 = N − 2 , N p τ e(nθ ) dθ −
Ip,N |θ|< N1
n=1
N p ηn e(nθ ) dθ
|θ|< N1
n=1
p p 1−h N 1−h N + ηn e(nθ ) dθ − τ e(nθ ) dθ h
n=1
h
τ p N p−1 − C |θ|< N1
n=1
N p p N ηn e(nθ ) dθ + ηn e(nθ ) dθ − Cτ p h1−p . (4.10) n=1
|θ|>h n=1
G. Mockenhaupt, W. Schlag / Journal of Functional Analysis 256 (2009) 1189–1237
1201
According to Corollary 4.2, the first integral in (4.10) is p p N −1 (log N ) 2 τ (1 − τ )N 2
(4.11)
up to a negligible probability. For the second, one has because of p 2 p p p 1−h N 1 N N ηn e(nθ ) dθ ηn e(nθ ) dθ − ηn e(nθ ) dθ h
n=1
n=1
0
|θ|h n=1
2 p p 1 N N 2 ηn e(nθ ) dθ − ηn e(nθ ) dθ n=1
0
N
|θ|h n=1
p 2
ηn2
p
− Ch(N τ log N ) 2 ,
(4.12)
n=1
where the last term in (4.12) is obtained from Corollary 4.2. Using p 2 again, E
N
p 2
ηn2
E
n=1
N
p 2
ηn2
p N τ (1 − τ ) 2 .
n=1
In fact, Lemma 4.1 gives the following more precise estimate: N 2 2 2 2 2 P 4e−λ /8 ηn − Eηn λ N E η1 − Eη12
(4.13)
n=1
provided the conditions (4.1) hold. One checks that E(|η12 − Eη12 |2 ) τ (1 − τ ). Hence it follows from (4.13) that for large N
N 1 2 1 P E ηn = N τ (1 − τ ) 2 2 n=1 n=1 N N 1 2 2 P ηn − E ηn N τ (1 − τ ) 2 n=1 n=1 N N 2 P ηn2 − E ηn2 log N N τ (1 − τ ) 4e−(log N ) /8 , N
ηn2
n=1
n=1
since with our choice of parameters (4.1) hold for large N . Inserting this bound into (4.12) now √ 1+δ yields (recall that h = τ N −1 = N − 2 )
1202
G. Mockenhaupt, W. Schlag / Journal of Functional Analysis 256 (2009) 1189–1237
p 1−h N p 2 p p 1+δ 1 ηn e(nθ ) dθ − CN − 2 (N τ log N ) 2 (N τ ) 2 N τ (1 − τ ) 2 n=1
h
up to negligible probability. In view of this bound and (4.11), one obtains from (4.10) that N p p N ηn e(nθ ) dθ + ηn e(nθ ) dθ − Cτ p h1−p
Ip,N τ N p
p−1
−C |θ|< N1
n=1
|θ|>h n=1
p
τ p N p−1 + (N τ ) 2
p
up to negligible probability. To remove the final term in the first line we used that (N τ ) 2 τ p h1−p which follows from our choice of h provided N is big. 2 4.2. Random sets satisfy the majorant inequality (1.3) In this section we will show Theorem 4.4. Let 0 < δ < 1 be fixed. For every positive integer N we let ξj = ξj (ω) be i.i.d. variables with P[ξj = 1] = τ , P[ξj = 0] = 1 − τ where τ = N −δ . Define a random subset S(ω) = j ∈ [1, N] ξj (ω) = 1 . Then for every ε > 0 and p 2 one has an e(nθ ) P sup |an |1 n∈S(ω)
N e(nθ )
ε
Lp (T)
n∈S(ω)
→0
(4.14)
Lp (T)
as N → ∞. The proof of Theorem 4.4 relies on Slepian’s lemma and ideas in Bourgain’s paper [3]. In the next section we will present a variant of Theorem 4.4 which requires additional assumptions on the exponent δ as well as on p. However, this second method will lead us later to remove the N ε term in (4.14) in certain cases, for example when p = 3. This improvement (which we believe should hold in general, i.e. for 2 < p ∈ / 2N) relies on a method developed in Bourgain’s work on the solution of the Λ(p) problem, see [1] and [4]. In fact, in this situation we can avoid several complications that arose in Bourgain’s work. Notice that Theorem 4.4 is implied by Bourgain’s existence theorem of Λ(p) sets provided δ 1 − p2 , but not for δ < 1 − p2 . Indeed, in the former 2
case the random set S will typically have cardinality N p or smaller, and such sets were shown by Bourgain [1] to be Λ(p)-sets with large probability. Let ξj = ξj (ω), 1 j N , be i.i.d. variables with P[ξj = 1] = τ , P[ξj = 0] = 1 − τ where τ ∈ (0, 1). Define a random subset S(ω) = j ∈ [1, N] ξj (ω) = 1 .
G. Mockenhaupt, W. Schlag / Journal of Functional Analysis 256 (2009) 1189–1237
1203
We sometimes drop the argument ω and write simply S for S(ω). The following result is a discretized version of a result shown by Bourgain in [3]. Proposition 4.5. Let M ∈ N, M < N, and T (θ ) = n∈S(ω) an e(nθ ) be a trigonometric polynomial with frequencies in S(ω). Then there exists C > 0 independent of M and N such that
Eω sup
j 2 τ N + CM log N, sup ! T N
|an |1 |I |M
(4.15)
j ∈I
where the second supremum is over all integer sets I ⊂ [0, N ) with |I | M. In particular, we have for c1 sufficiently large Eω sup ! |an |1
2 T j Cτ N. N
|T ( Nj )|2 c1 τ N
(4.16)
log N
To see that (4.16) follows from (4.15), we choose M = τ N/ log N and note that for c1 > 0 sufficiently large the integer set X = {j ∈ [0, N ) | |T ( Nj )|2 c1 τ N log N } is of size at most M. Note that otherwise, X would contains a subset I of size |I | = M for which (4.15) implies: √ Mc1 τ N log N (C + 1)τ N , i.e. c1 (C + 1)2 . For convenience we will include below Bourgain’s proof of Proposition 4.5. With the bound on the expected size for Dirichlet kernels in Lp given by Lemma 4.3 we are prepared to the Proof of Theorem 4.4. By the Marcinkiewicz–Zygmund inequality, see [15, p. 28], the Lp norm (for p > 1) of a trigonometric polynomial of degree N is comparable with the Riemann sum over N equidistant points, i.e. for f (θ ) = n∈S(ω) an e(nθ ) with |an | 1 we have p an e(nθ ) p L
n∈S(ω)
N −1 p j 1 ≈ f N N (T) j =0
with hidden constants depending on p but independent of N . We divide the Riemann sum into I = {j | |f ( Nj )|2 c1 τ N log N } and its complement J in [0, N ) ∩ Z. Fix α, β > 0 with α(p − 2) + β = p/2. Since f ∞ |S| and the expected size of S is τ N we find ΩN ⊂ Ω with P(ΩN ) → 1, as N → ∞, such that f ∞ (log N )α τ N . Also, by (4.16) we find ΩN ⊂ ΩN with P(ΩN ) → 1 such that for ω ∈ ΩN we have j ∈I |f ( Nj )|2 (log N )β (τ N )2 . Hence, for ω ∈ ΩN , we get p−2 2 p N −1 p j j j j 1 1 1 f N = N f N f N + N f N N j =0
j ∈I
j ∈J
j 2 p α(p−2) p−2 1 + (c1 τ N log N ) 2 f (log N ) (τ N ) N N C(log N )
p 2
j ∈I
p τ p N p−1 + (τ N ) 2 .
1204
G. Mockenhaupt, W. Schlag / Journal of Functional Analysis 256 (2009) 1189–1237
Hence, by Lemma 4.3 we find p a e(nθ ) n p
p
C(log N ) 2 EIp,N
L (T)
n∈S(ω)
and the theorem follows with a possibly smaller subset of ΩN whose probability still approaches 1 as N → ∞. 2 Proof of Proposition 4.5. We need to show that the expectation L := Eω
sup
{|an |1,I }
N 2 1/2 aj ξj (ω)e(j m/N ) Cτ N.
m∈I j =1
Here the supremum is over all sets I with |I | τ N/ log N . By 2 (I )-duality we may express the left-hand side by N Eω sup sup aj ξj (ω) bm e(j m/N ) {|an |1,I } b 2 =1 (I )
= Eω
sup
j =1
N
I,b 2 (I ) =1 j =1
m∈I
ξj (ω) bm e(j m/N ). m∈I
Write ξj = ηj + τ , i.e. the ηj s have vanishing expectation. It follows that N N bm e(j m/N) + Eω sup ηj (ω) bm e(j m/N ) =: L1 + L2 . L τ sup I,b j =1 m∈I
I,b j =1
m∈I
By using the ( 1 , ∞ )-duality and Cauchy–Schwarz the term L1 is bounded by N 2 1/2 N N τ sup sup cj bm e(j m/N ) τ sup cj e(j m/N ) . b,I |cj |1 |cj |1 j =1
m∈I
m=1 j =1
2 2 So, Parseval’s identity gives L21 τ 2 N sup|cj |1 N k=1 |ck | (τ N ) . To bound the term L2 we first note that for each choice of εk = ±1 and each bounded sequence of complex-valued functions Ak (t) one has N N Eω sup ηk (ω)Ak (t) 2Eω sup εk ηk (ω)Ak (t). t t k=1
k=1
To see this, set X = {k | εk = 1} and Y = X c , the complement of X. Then
(4.17)
G. Mockenhaupt, W. Schlag / Journal of Functional Analysis 256 (2009) 1189–1237
1205
N Eω sup ηk (ω)Ak (t) Eω sup ηk (ω)Ak (t) + Eω sup ηk (ω)Ak (t). t t t k∈X
k=1
k∈Y
Since Eω ηk = 0 we may rewrite the first term as
Eω supEω εk ηk (ω) Ak (t) + εk ηk (ω ) Ak (t) , t
k∈X
k∈Y
which is bounded by N Eω Eω sup εk ηk (ω)Ak (t) + εk ηk (ω )Ak (t) = Eω sup εk ηk (ω)Ak (t), t t k∈X
k∈Y
k=1
where we used independence. Exchanging X and Y the second term is seen to be bounded by the same expression, i.e. (4.17) holds. Since (4.17) remains true if we average over εk we obtain for L2 the bound N εj ηj (ω) bm e(j m/N ). L2 2Eε Eω sup I,b j =1
m∈I
We may now employ the contraction principle (see [14, p. 222]) to majorize the above Rademacher sequence εj by Gaussian random variables gj , i.e. we have N L2 2Eω Eω sup gj (ω )ηj (ω) bm e(j m/N ). I,b j =1
m∈I
By Slepian’s lemma (see [14, p. 222]) for Gaussian processes we can bound right-hand side by N
CEω Eω sup gj (ω )ηj (ω) bm e(j m/N ). I,b j =1
m∈I
Hence, by evaluating the supremum over b 2 (I ) = 1 we find L2 CEω Eω
N 2 1/2 gj (ω )ηj (ω)e(j m/N )
m∈I j =1
C sup |I |Eω Eω I
N
sup gj (ω )ηj (ω)e(j m/N ). 1mN j =1
By the Salem–Zygmund’s inequality [13] for Gaussian Fourier series we finally get
1206
G. Mockenhaupt, W. Schlag / Journal of Functional Analysis 256 (2009) 1189–1237
L2 C(τ N/ log N )
1/2
1/2
(log N )
Eω
N
1/2 = Cτ N.
2
ηj (ω)
j =1
Hence L Cτ N .
2
4.3. Suprema of random processes In this section we will first derive a proof of the following somewhat weaker version of Theorem 4.4. Proposition 4.6. Let 0 < δ < 1 be fixed. For every positive integer N we let ξj = ξj (ω) be i.i.d. variables with P[ξj = 1] = τ , P[ξj = 0] = 1 − τ where τ = N −δ . Define a random subset S(ω) = j ∈ [1, N] ξj (ω) = 1 . Then for every ε > 0 and 4 p 2 one has P sup a e(nθ ) n |an |1 n∈S(ω)
Lp (T)
N ε e(nθ ) n∈S(ω)
→0
(4.18)
Lp (T)
as N → ∞. Moreover, under the additional restriction δ 12 , (4.18) holds for all p 4. For a proof of Theorem 4.6 as well as for its improvements we now collect the statements from Bourgain’s paper that we will need. The first is Lemma 1 from [1] with q0 = 1. In fact, Bourgain’s lemma is slightly stronger because of certain log τ1 -factors. While these factors are important for his purposes, they play no role in our argument. We present the proof for the reader’s convenience, following Bourgain’s original argument. Another proof was found by Ledoux and Talagrand [8] which is close to the ideology surrounding Dudley’s theorem on suprema of Gaussian processes. While their point of view is perhaps more conceptual, we have found it 2 12 advantageous to follow [1]. Throughout, if x ∈ RN , then |x| = |x| 2 = ( N j =1 xj ) is the EuN
clidean norm. Secondly, N2 (E, t) refers to the L2 -entropy of the set E at scale t. Recall that this is defined to be the minimal number of L2 -balls of radius t needed to cover E. Lemma 4.7. Let E ⊂ RN + , B = supx∈E |x|, and ξj be selector variables as above with P[ξj = 1] = τ , P[ξj = 0] = 1 − τ , and 0 < τ < 1 arbitrary. Let 1 m N . Then
E
sup
x∈E , |A|=m j ∈A
1 2
ξj xj (τ m + 1) B +
B
log N2 (E, t) dt,
0
where N2 refers to the L2 entropy. Proof. Let Ek be minimal 2−k -nets for E with 2−k B. Let B = 2−k0 . Then every x ∈ E can be written as
G. Mockenhaupt, W. Schlag / Journal of Functional Analysis 256 (2009) 1189–1237
x = xk0 +
∞
(xk+1 − xk ) = xk0 +
k=k0
∞
1207
2−k+1 yk ,
k=k0
where xk ∈ Ek for every k k0 . We can and do set xk0 = 0. Now, yk ∈ Fk where diam(Fk ) 1 and #(Fk ) #(Ek ) · #(Ek+1 ). Hence log #Fk C log #Ek+1 ,
(4.19)
and thus E
sup
x∈E , |A|=m j ∈A
ξj xj 2−k+1 E kk0
sup
ξi |yi |.
y∈Fk , |A|m i∈A
(4.20)
Now fix some k k0 and write F instead of Fk . Moreover, replacing every vector y = {yj }N j =1 ∈ N . Note that this changes neither the F with the vector {|yi |}N , we may assume that F ⊂ R + i=1 diameter nor the cardinality bound of F . With 0 < ρ1 < ρ2 to be determined, one has
ξi yi
yi +
yi ρ2
i∈A
yi +
i∈A,yi ρ1
ξi yi ρ2−1
yi ρ2
ρ1
yi2 + mρ1 +
ξi yi .
ρ1
Let q = 1 + log F . Since |y| 1, one concludes that E
sup
y∈F , |A|m i∈A
ξi yi ρ2−1 + mρ1 + E sup
y∈F ρ1
ρ2−1 + mρ1 + E ρ2−1 + mρ1 +
ξi yi
y∈F
ρ1
E y∈F
q
(4.21) q 1
q
(4.22)
ξi yi
ρ1
1 ρ2−1 + mρ1 + (#F ) q sup E ρ2−1 + mρ1 + sup
q 1 ξi yi
y∈F
|y|1 ρ1
q 1
q
(4.23)
ξi yi
ρ1
ξi (ω)yi
.
(4.24)
Lq (ω)
Here (4.21) follows from the embedding q (F ) → ∞ (F ), (4.22) follows from Hölder’s inequality, and to pass from (4.23) to (4.24) one uses that 1 (#F ) q = exp (log #F )/q e
by our choice of q = 1 + log F . To control the last term in (4.24), we need the following simple estimate, see [1, Lemma 2]. By the multinomial theorem (for any positive integer q),
1208
G. Mockenhaupt, W. Schlag / Journal of Functional Analysis 256 (2009) 1189–1237
E
n
q ξj
=
q1 +···+qn =q
j =1
=
q
q q q Eξ1 1 · · · · · Eξn n q 1 , . . . , qn
=1 1i1
q n =1
!
τ q
q q =1
q
q τ q i 1 , . . . , qi
q q q− q (nτ ) (eτ n) q!
q q−
=1
(q + eτ n) . q
(4.25)
It is perhaps more natural (and also more precise) to estimate qth moments by means of the Bernoulli law q n n n q τ (1 − τ )n− . ξj = E j =1
=0
But we have found the approach leading to (4.25) more flexible since it also applies to nonBernoulli cases. Continuing with the final term in (4.24) one concludes from (4.25) that sup
|y|1 ρ1
ξi (ω)yi
2
Lq (ω)
2
ρ2−2 <2j <ρ1−2
2
2j ξi (ω)
− j2
i=1
(4.26) Lq (ω)
j 2− 2 q + eτ 2j qρ2 + τρ1−1 .
(4.27)
ρ2−2 <2j <ρ1−2
Inserting this bound into (4.24) and setting ρ1 = E
sup
y∈F , |A|m i∈A
ξi yi
√
1
τ/m and ρ2 = q − 2 yields
√ √ √ mτ + q mτ + 1 + log #F.
The lemma now follows in view of (4.19) and (4.20).
2
4.4. Entropy bounds As in [1] we will need bounds on certain covering numbers, also called entropies. We recall those bounds starting with the so called “dual Sudakov inequality” for the reader’s convenience. More on this can be found in Pisier [12] and Bourgain, Lindenstrauss, Milman [5, Section 4]. Consider Rn with two norms, the Euclidean norm | · | and some other (semi)norm · . We set X = (Rn , · ) and denote the unit ball in this space by BX , whereas the Euclidean unit ball will be B n . As usual, for any set U ⊂ Rn and t > 0 one sets
N " (xj + tBX ) . E(U, BX , t) := inf N 1 ∃xj ∈ Rn , 1 j N, U ⊂ j =1
(4.28)
G. Mockenhaupt, W. Schlag / Journal of Functional Analysis 256 (2009) 1189–1237
1209
There are two closely related quantities, namely
N " ˜ (xj + tBX ) , E(U, BX , t) := inf N 1 ∃xj ∈ U, 1 j N, U ⊂ j =1
D(U, BX , t) := sup M 1 ∃yj ∈ U, 1 j M, yj − yk t, j = k .
(4.29)
There are the following comparisons between these quantities: ˜ D(U, BX , t) E(U, BX , t) E(U, BX , t) D(U, BX , 2t).
(4.30)
The final inequality holds because every covering of U by arbitrary t-balls gives rise to a covering by 2t-balls with centers in U . To see that E(U, BX , t) D(U, BX , 2t), let {yj }M j =1 ⊂ U be 2t#N separated and U ⊂ i=1 (xi + tBX ). Then every yj ∈ xi + tBX for some i = i(j ). Moreover, j = k ⇒ i(j ) = i(k). Hence N M. The “dual Sudakov inequality” Lemma 4.8 bounds E(B n , BX , t) in terms of the Levy mean MX :=
x dσ (x),
(4.31)
S n−1
where σ is the normalized measure on S n−1 . Alternatively, one has
− n2
MX = αn (2π)
e−
|x|2 2
x dx,
(4.32)
Rn
n gi (ω)ei dP(ω), MX = αn Ω
(4.33)
i=1
where αn =
( n2 ) 1 √ n− 2 ( n+1 ) 2 2
and gi are i.i.d. standard normal variables, and ei is an ONS. The probabilistic form (4.33) is of course just a restatement of (4.32), whereas the latter can be obtained from the definition (4.31) by means of polar coordinates. The following lemma is due to Pajor and Tomczak-Jaegermann [11] but the proof given below is due to Pajor and Talagrand, see [5]. Lemma 4.8. For any t > 0 n MX 2 log E B , BX , t Cn , t where C is an absolute constant.
(4.34)
1210
G. Mockenhaupt, W. Schlag / Journal of Functional Analysis 256 (2009) 1189–1237
n n Proof. Let {xi }N i=1 ⊂ B , xi − xj t for i = j and N maximal. Then E(B , BX , t) N . Let n
μ(dx) = (2π)− 2 e−
|x|2 2
dx. Then by definition (4.31),
1 μ x > 2MX αn−1 < 2
1 μ x 2MX αn−1 > . 2
⇒
(4.35)
N −1 Moreover, {xi + 12 tBX }N i=1 and therefore also {yi + 2MX αn BX }i=1 have mutually disjoint interiors, where we have set yi = 4MX (tαn )−1 xi . Now, by symmetry of BX and convexity of e−u ,
n μ yi + 2MX αn−1 BX = (2π)− 2
e−|y−yi |
2 /2
dy
2MX αn−1 BX
− n2
= (2π)
2MX αn−1 BX
− n2
(2π)
1 −|y−yi |2 /2 2 e + e−|y+yi | /2 dy 2 e−(|y−yi |
2 +|y+y |2 )/4 i
dy
2MX αn−1 BX
− n2
= (2π)
e−(|y|
2 +|y |2 )/2 i
2MX αn−1 BX
1 2 dy e−|yi | /2 , 2
where the last step follows from (4.35). Since |yi | 4MX (tαn )−1 , 1 1 μ yi + 2MX αn−1 BX exp − (4MX )2 (tαn )−2 . 2 2 Hence 1
N 1 μ yi + 2MX αn−1 BX N exp −(4MX )2 (tαn )−2 , 2 i=1
1
and the lemma follows since αn n− 2 .
2
Observe that (4.34) is a poor bound as t → 0. Indeed, rather than the exp(t −2 ) behavior exhibited by (4.34) the true asymptotics is t −n as t → 0. The point of Lemma 4.8 is to relate the size of t to both MX and n. This is best illustrated by some standard examples. • Firstly, take X = 1n . In that case, αn−1 MX
− n2
= (2π)
n Rn i=1
Therefore, MX
√ n. By (4.34),
|xi |e
− |x|2
2
n dx = √ 2π
∞ −∞
|x1 |e−
x12 2
2n dx1 = √ . 2π
G. Mockenhaupt, W. Schlag / Journal of Functional Analysis 256 (2009) 1189–1237
1211
sup E B n , B 1n , n C. n
This bound is somewhat wasteful. Indeed, since
√
nB 1n ⊃ B n , one actually has
√ sup E B n , B 1n , n C. n
The reason for this “overshoot” is that the major contribution to MX comes from the corners of B 1n . On the other hand, these corners do not determine the smallest r for which rBX ⊃ B n . • Secondly, consider X = ∞ n . Using (4.33), αn−1 MX = E sup |gi |
log n,
1in
where the latter bound is a rather obvious and well-known fact. Hence $ log n MX n which implies via (4.34) that , log n C. sup E B n , B ∞ n n
. In contrast to the previous This is the correct behavior up to the log n-factor since B n ⊂ B ∞ n that is also the most case, the bulk of the contribution to MX comes from that part of B ∞ n relevant for the covering of the Euclidean ball. • Finally, and most relevantly for our purposes, identify Rn with the space of trigonometric polynomials with real coefficients of degree n, i.e., n n R aj e(j θ ) aj ∈ R . (4.36) j =1
Furthermore, define · = · Lq (T) where q 2 is fixed. Then n MX = αn gj (ω)e(j θ ) j =1
Ω
dP(ω) Lq (T)
n ±gj (ω)e(j θ ) = αn E Ω
j =1
√
n
Cαn q
Cαn q
n
Ω j =1
(4.37)
1 2
gj2 (ω)
(4.38)
dP(ω)
j =1
Ω
√
dP(ω) Lq (T)
1 2
gj2 (ω) dP(ω)
√ √ √ = Cαn q n C q.
1212
G. Mockenhaupt, W. Schlag / Journal of Functional Analysis 256 (2009) 1189–1237
In (4.37) the expectation E refers to the random and symmetric choice of signs ±, whereas √ the q-factor in (4.38) is due to the fact that the constant in Khinchin’s inequality grows √ like q. Hence log E B n , BX , t Cqnt −2
(4.39)
in this case. The proof of Proposition 4.6 requires estimating Nq (PA , t) := E(PA , BLq (T) , t). Here
PA :=
an e(nθ ) |a| = |a|
2N
1 ,
n∈A
where A ⊂ [1, N]. Invoking (4.39) leads to log Nq (PA , t) Cq|A|t −2 .
(4.40)
This bound is basically optimal when t ∼ 1, but it can be improved for very small and very large t. Corollary 4.9. For q 2 and any A ⊂ [1, N] 1 1 log Nq (PA , t) Cq|A| 1 + log if 0 < t . t 2
(4.41)
Proof. Let m = |A|. Thus 1 m N . Notice firstly that log Nq
an e(nθ ) |a| 1 , t
n∈A
Cm log
1 + log Nq t
an e(nθ ) |a| 1 , 1 .
(4.42)
n∈A
This follows from the fact that for any norm · in Rm with unit-balls BX one has D(BX , BX , t) (4/t)m
for all 0 < t < 1
(4.43)
by scaling and volume counting, see (4.29) for the definition of D(BX , BX , t). Indeed, suppose M = D(BX , BX , t). Then there are M disjoint balls {xj + 12 tBX }M j =1 with centers xj ∈ BX . Since
xj + 12 tBX ⊂ 2BX if t < 1, it follows that
M 1 tBX |2BX | 2
⇒
M(t/2)m 2m ,
j =1
as claimed. Here | · | stands for Lebesgue measure. Thus (4.43) holds, and therefore also (4.42) in view of (4.30). Hence
G. Mockenhaupt, W. Schlag / Journal of Functional Analysis 256 (2009) 1189–1237
log Nq (PA , t) Cm log Cm log
1 + log Nq t
1213
an e(nθ ) |a| 1 , 1
n∈A
1 + Cqm, t
where the final term follows from (4.39).
2
We now turn to large t. The following corollary slightly improves on the rate of decay. Corollary 4.10. Let q 2 and A ⊂ [1, N]. With PA as above one has log Nq (PA , t) Cq|A|t −ν
1 if t > , 2
(4.44)
where ν = ν(q) > 2. Proof. Recall that Nq (PA , t) = E(PA , BLq , t). Using (4.30), one obtains from (4.40) that also ˜ A , BLq , t) Cq|A|t −2 . log E(P Let q < r,
1 q
=
1−θ 2
(4.45)
+ θr . Since for any f, g ∈ PA f − gq f − g21−θ f − gθr 2f − gθr ,
one concludes from (4.45) that ˜ A , BLq , t) log E˜ PA , BLr , (t/2)1/θ Cq|A|t −2/θ . log E(P Applying (4.30) again yields (4.44).
2
4.5. Decoupling lemma Lastly, we require a version of Bourgain’s decoupling technique, cf. [1, Lemma 4]. In contrast to his case we only need to decouple into two sets rather than three. Lemma 4.11. Let real-valued functions hα (u) on C be given for α = 1, 2, 3 that satisfy hα (u) 1 + |u| pα ,
hα (u) − hα (v) 1 + |u| + |v| pα −δ |u − v|δ
for all u, v ∈ C and some fixed choice of pα > 0, δ > 0. Let x, y, z ∈ 2N be sequences so that |x|, |y|, |z| 1 and suppose ζj = ζj (t) are i.i.d. random variables with P(ζj = 1) = P(ζj = 0) = 12 . We assume that P(dt) = dt on [0, 1], say. Set Rt1 = {1 j N | ζj (t) = 1}, Rt2 = {1 j N | ζj (t) = 0}. Then
1214
G. Mockenhaupt, W. Schlag / Journal of Functional Analysis 256 (2009) 1189–1237
1 1 1 h1 h h dt − h x y z xi h2 yi h3 zi i 2 i 3 i 1 2 2 2 i∈Rt1
i∈Rt2
i
i∈Rt2
i
i
p−δ C 1 + xi + yi + zi , i
i
(4.46)
i
where p = p1 + p2 + p3 and C is some absolute constant depending only on p and δ. Proof. By assumption, N N N pα −δ δ N 1 1 1 ζi − ζi − xi xi xi − hα xi 1 + xi + hα 2 2 2 1 i=1
i∈Rt
i=1
i=1
i=1
N pα −δ N pα 1 ζi − xi 1+ xi , 1+ 2 i=1 i=1 N N pα N pα 1 1 hα ζi − xi xi + hα xi 2 1 + xi 1+ 2 2 1
i=1
i∈Rt
i=1
i=1
for α = 1, 2, 3. Hence N N N 1 1 1 xi h2 yi h3 zi dt − h1 xi h2 yi h3 zi h1 2 2 2 1 2 2 i∈Rt
i∈Rt
i=1
i∈Rt
i=1
i=1
N N N p−δ C 1+ xi + yi + zi
×
i=1
i=1
i=1
N p N N 1 1 1 ζi − ζi − ζi − xi + yi + zi dt. 1+ 2 2 2 i=1
i=1
(4.47)
i=1
The lemma now follows from Khinchin’s inequality. Indeed, p N 1 ζi − xi dt Cp |x|p Cp , 2 i=1
by assumption.
2
4.6. The proof of Proposition 4.6 and its improvement for p = 3 We now start the proof of Proposition 4.6 for p = 3. In fact, we state a somewhat more precise form of this theorem for p = 3.
G. Mockenhaupt, W. Schlag / Journal of Functional Analysis 256 (2009) 1189–1237
1215
Theorem 4.12. Let 0 < δ < 1 be fixed. For every positive integer N we let ξj = ξj (ω) be i.i.d. variables with P[ξj = 1] = τ , P[ξj = 0] = 1 − τ where τ = N −δ . Define a random subset S(ω) = j ∈ [1, N] ξj (ω) = 1 . Then for every γ > 0 there is a constant Cγ so that sup P sup a e(nθ ) n
N 1
|an |1 n∈S(ω)
L3 (T)
Cγ e(nθ )
γ.
(4.48)
L3 (T)
n∈S(ω)
Proof. Firstly, note that for fixed 0 < δ < 1 and large N Lemma 4.1 implies that P
N
ξn 2τ N exp(−cτ N ).
n=1
Let E denote the restricted expectation N E sup ξn an e(nθ ) |an |1
L3 (T)
n=1
N := Eχ[ ξn 2τ N ] sup ξn an e(nθ ) |an |1
.
L3 (T)
n=1
Then N E sup ξn an e(nθ ) |an |1
L3 (T)
n=1
N N exp(−cτ N ) + E sup ξn an e(nθ ) |an |1
L3 (T)
n=1
N
O(1) + E sup ξn an e(nθ ) |an |1 n=1
. L3 (T)
From now on, we set m = 2τ N , and we will mostly work with E instead of E. Next, fix some {an }N n=1 with |an | 1. Then, rescaling Lemma 4.11 (with h1 (x) = h2 (x) = x and h3 (x) = |x|) one obtains that 1 8
3 1 1 N an ξn e(nθ ) dθ = an ξn e(nθ ) a¯ k ξk e(−kθ ) a ξ e( θ ) dθ dt 1 2 2 0
n=1
0 n∈Rt
3
k∈Rt
1
+ O m2 0 3
∈Rt
2 N a n 1+ √ ξn e(nθ ) dθ . m
(4.49)
n=1
N The O-term in (4.49) is O(m 2 ) by construction. Let {ξn (ω1 )}N n=1 and {ξn (ω2 )}n=1 denote two N 1 2 independent copies of {ξn (ω)}n=1 . Recall that Rt and Rt are disjoint for every t. Therefore, for fixed t
1216
G. Mockenhaupt, W. Schlag / Journal of Functional Analysis 256 (2009) 1189–1237
1 Eω sup an ξn (ω)e(nθ ) a¯ k ξk (ω)e(−kθ ) a ξ (ω)e( θ ) dθ |an |1 1 2 2 0 n∈Rt
k∈Rt
∈Rt
1 = Eω1 ,ω2 sup an ξn (ω1 )e(nθ ) a¯ k ξk (ω2 )e(−kθ ) a ξ (ω2 )e( θ ) dθ . |an |1 1 2 2 0 n∈Rt
k∈Rt
∈Rt
(4.50) This leads to 3 1 N Eω sup an ξn (ω)e(nθ ) dθ |an |1 n=1
0
3 2
m +
1 Eω1 ,ω2 sup an ξn (ω1 )e(nθ ) a¯ k ξk (ω2 )e(−kθ ) |an |1 1 2 0 n∈Rt
k∈Rt
× a ξ (ω2 )e( θ ) dθ dt ∈Rt2
3 2
m +
E ω1 E ω2
1 sup an ξn (ω1 )e(nθ ) b¯k ξk (ω2 )e(−kθ ) |an |1 1 2
|bn |1 0 n∈Rt
k∈Rt
× b ξ (ω2 )e( θ ) dθ dt ∈Rt2
3 2
m +
E ω1 E ω2
1 N N sup an ξn (ω1 )e(nθ ) b¯k ξk (ω2 )e(−kθ ) |an |1
n=1 |bn |1 0
k=1
N × b ξ (ω2 )e( θ ) dθ dt =1
3 2
m + E ω2 Eω1 sup
sup
x∈E (ω2 ) |A|=m n∈A
(4.51)
ξn (ω1 )xn .
Here N &N % N E(ω2 ) := b ξ (ω2 )e( ·) b¯k ξk (ω2 )e(−k ·) e(n ·), k=1
=1
n=1
sup |bn | 1 ⊂ RN +. 1nN
In the calculation leading up to (4.51) we firstly used (4.50), secondly the obvious fact that the N supremum only increases if we introduce {bn }N n=1 in addition to {an }n=1 , thirdly that one can
G. Mockenhaupt, W. Schlag / Journal of Functional Analysis 256 (2009) 1189–1237
1217
remove the restrictions to the sets Rt1 and Rt2 because they can be absorbed into the choice of the sequences an , bn , and lastly that n ξn m which allows us to introduce A ⊂ [1, N], |A| = m. If x ∈ E(ω2 ), then |x|2 2 N
4 4 2 sup ak ξk (ω2 )e(k ·) ξk (ω2 )e(k ·) =: B4 (ω2 ) |ak |1
4
k
(4.52)
4
k
by the L4 majorant property. By Lemma 4.3, 1
3
EB4 (EI4,N ) 2 τ 2 N 2 + τ N.
(4.53)
We now apply Lemma 4.7 to (4.51). This yields 3 1 N Eω sup an ξn (ω)e(nθ ) dθ |an |1 n=1
0
m
3 2
+ E ω2
√ ( τ m + 1)B4 (ω2 ) +
∞
log N2 E(ω2 ), t dt
0
1 3 (τ N ) + 1 + τ N 2 τ 2 N 2 + τ N + E ω2 3 2
∞
log N2 E(ω2 ), t dt.
(4.54)
0
It remains to deal with the entropy integral in (4.54). To this end, observe that the distance between any two elements in E(ω2 ) is of the form g|g| − h|h| g − h∞ g2 + h2 2 √ N ε g − hq g2 + h2 N ε mg − hq , ε where we chose √ q very large depending on ε (the factor N comes from Bernstein’s inequality). Here g, h ∈ mPA where A = A(ω2 ) = {n ∈ [1, N] | ξn (ω2 ) = 1} and
PA =
an e(n ·) |a|
2N
1 .
(4.55)
n∈A 2 Actually, our coefficients are in√the unit-ball of ∞ n , but we have embedded this into m in the obvious way, which leads to the m-factor in front of PA (at this point recall that we are working with E ω2 ). One concludes that, for ε > 0 small and q < ∞ large depending on ε,
log N2 E(ω2 ), t log Nq PA , N −ε m−1 t ε 1 + log mN 0 < t < mN ε , t , Cqm (m−1 N −ε t)−ν , t > N ε m,
(4.56)
1218
G. Mockenhaupt, W. Schlag / Journal of Functional Analysis 256 (2009) 1189–1237
where ν > 2, see Corollaries 4.9 and 4.10. It follows that the last term in (4.54) is at most ∞ 3 Eω2 log N2 E(ω2 ), t dt N ε m 2 . 0
Plugging this into (4.54) yields 3 1 N 3 1 3 3 an ξn (ω)e(nθ ) dθ (τ N ) 2 + 1 + τ N 2 τ 2 N 2 + τ N + N ε (τ N ) 2 Eω sup |an |1 0
n=1
3
τ 3 N 2 + N ε (τ N ) 2 .
(4.57)
3
Now suppose δ < 13 . Then τ 3 N 2 > N ε (τ N ) 2 provided ε > 0 is small and fixed, and provided N is large. Hence, combining (4.57) with Lemma 4.3 leads to Theorem 4.12 at least if δ < 13 . If one is willing to loose a N ε -factor, then (4.57) in combination with Lemma 4.3 leads to the desired 2 bounds in all cases. On the other hand, if δ 13 so that typically #(S(ω)) N 3 , then Bourgain showed that S(ω) is a Λ3 set with large probability. More precisely, he showed that the constant an e(n ·) K3 (ω) := sup |a| 2 1 n∈S(ω) N
3
satisfies EK33 C. Hence, in our case, N 3 3 an ξn (ω)e(n ·) (τ N ) 2 . E sup |an |1 n=1
3
Clearly, N N 1 ξn (ω)e(n ·) ξn (ω)e(n ·) = # S(ω) 2 , n=1
n=1
3
and we have thus proved (4.48) for δ
1 3
as well.
2
2
It is perhaps worth pointing out that interpolation of the L4 bound with the L2 bound gives 5
3
τ 2 N 2 + (τ N ) 2 , 3 so √ that the estimate we just obtained is better√by the initial τ -factor (note that this is due to the τ m-factor in Lemma 4.7 as compared to a τ N -factor).
G. Mockenhaupt, W. Schlag / Journal of Functional Analysis 256 (2009) 1189–1237
1219
4.7. The case of general p The strategy is to first generalize the previous argument to all odd integers using the fact that the majorant property holds for all even integers (for p = 3 we used this fact with p = 4). Then one runs the same argument again, using now that the (random) majorant property holds for all integers p and so on. For a given ε > 0 this yields that there is a set of p that is ε-dense in [2, ∞) and for which the majorant property holds. This is enough by interpolation, since we are allowing a loss of N ε in (4.18). Unfortunately, there are certain technical complications in carrying out this program having to do with the size of δ. In this section we finish a proof a Proposition 4.6 by employing the above method for p = 3. The next lemma formalizes the main probabilistic argument from the previous section. Let p 2. In this section, we say that the random majorant property (or RMP in short) holds at p if and only if for every ε > 0 there exists a constant Cε so that N p N p E sup an ξn e(nθ ) Cε N ε E ξn e(nθ ) |an |1 n=1
n=1
p
(4.58)
p
for all N 1. Note that (the proof of) Theorem 4.12 establishes that the random majorant property holds at p = 3. Moreover, if (4.58) holds for some p, then (4.18) also holds for that value of p, see Lemma 4.3. Lemma 4.13. Let 2 p 3. Suppose the random majorant property (4.58) holds at 2(p − 1). Then it also holds at p. Furthermore, suppose the RMP holds at p − 1, 2(p − 1) and 2(p − 2). If 1 4 p 3, then it also holds at p. If p > 4 and δ 12 (i.e., τ = N −δ N − 2 ), then it also holds at p. Proof. Assume first that p 3. Instead of (4.49), Lemma 4.11 implies in this case that p p−2 1 1 N −p an ξn e(nθ ) dθ = an ξn e(nθ ) a¯ k ξk e(−kθ ) a ξ e( θ ) dθ dt 2 1 2 2 0
n=1
0 n∈Rt
k∈Rt
∈Rt
p−1 N 1 a n +O m 1+ dθ . √ ξn e(nθ ) m
p 2
0
(4.59)
n=1
To bound the O-term in (4.59) note that by the RMP for p − 1 2, p−1 p−1 1 1 N N an ξn e(nθ ) dθ Cε N ε E ξn e(nθ ) dθ E sup |an |1 0
n=1
0
n=1
= Cε N ε EIp−1,N . A calculation analogous to that leading up to (4.51) therefore yields
(4.60)
1220
G. Mockenhaupt, W. Schlag / Journal of Functional Analysis 256 (2009) 1189–1237
p 1 N Eω sup an ξn (ω)e(nθ ) dθ |an |1 0
n=1
p
1
m 2 + Cε N ε m 2 EIp−1,N + E ω2 E ω1 sup
sup
x∈E (ω2 ) |A|=m n∈A
ξn (ω1 )xn ,
(4.61)
where now N p−2 &N % N E(ω2 ) = b ξ (ω2 )e( ·) b¯k ξk (ω2 )e(−k ·) e(n ·), k=1
=1
sup |bn | 1
n=1
1nN
⊂ RN +. If x ∈ E(ω2 ), then by Plancherel and the RMP at 2(p − 1),
E sup
x∈E (ω2 )
|x|2 2 N
Eω2
2(p−1) 1 sup ak ξk (ω2 )e(kθ ) dθ
|ak |1
0
(4.62)
k
2(p−1) 1 Cε N E ω 2 ξk (ω2 )e(kθ ) dθ Cε N ε EI2(p−1),N . ε
0
k
Thus, by (4.61) and Lemma 4.7, p 1 N Eω sup an ξn (ω)e(nθ ) dθ |an |1 0
Cε N
ε
n=1
p 2
1 2
m + m EIp−1,N
∞ √
+ (1 + mτ ) EI2(p−1),N + Eω2 log N2 E(ω2 ), t dt . 0
(4.63) To estimate the entropy term, let q be very large depending on ε. Then the distance between any two elements in E(ω2 ) is of the form g|g|p−2 − h|h|p−2 g − h∞ gp−2 + hp−2 2(p−2) 2(p−2) 2 p−2 p−2 Cε N ε g − hq g2(p−2) + h2(p−2) N p−2 ε Cε N sup an ξn (ω2 )e(n ·) g − hq , |an |1 n=1
1 2
=: Cε N ε J2(p−2),N (ω2 )g − hq ,
2(p−2)
(4.64)
G. Mockenhaupt, W. Schlag / Journal of Functional Analysis 256 (2009) 1189–1237
1221
where the N ε -term follows from Bernstein’s inequality and we have set N 2(p−2) sup an ξn (ω2 )e(n ·) =: J2(p−2),N (ω2 ). |an |1 n=1
2(p−2)
√ As before, g, h ∈ mPA , A = A(ω2 ) = {n ∈ [1, N] | ξn (ω2 ) = 1}, see (4.55). One concludes that, for ε > 0 small and q < ∞ large depending on ε, 1 −1 2 t log N2 E(ω2 ), t log Nq PA(ω2 ) , N −ε m− 2 J2(p−2),N ⎧ ⎨ 1 + log 1t Cq m 1 ⎩ (m− 12 J − 2 −ε −ν 2(p−2),N (ω2 )N t)
if 0 < t < N ε mJ2(p−2),N (ω2 ), if t > N ε mJ2(p−2),N (ω2 ),
where ν > 2, see Corollaries 4.9 and 4.10. Inserting this estimate into the last term of (4.63) yields by the random majorant property on 2(p − 2) 2, E ω2
∞
log N2 E(ω2 ), t dt Cε N ε m EI2(p−2),N
(4.65)
0
and therefore finally, by Lemma 4.3, p 1 N Eω sup an ξn (ω)e(nθ ) dθ |an |1 0
n=1
p √ 1 Cε N ε m 2 + m 2 EIp−1,N + (1 + mτ ) EI2(p−1),N + m EI2(p−2),N p p−1 1 Cε N ε (τ N ) 2 + (τ N ) 2 τ p−1 N p−2 + (τ N ) 2 √ 1 1 + (1 + τ N ) τ 2(p−1) N 2p−3 + (τ N )p−1 2 + τ N τ 2(p−2) N 2p−5 + (τ N )p−2 2 p 3 (4.66) Cε N ε τ p N p−1 + τ p−1 N p− 2 + (τ N ) 2 . 1
3−p
3
3
p
If τ N − 2 , then τ p N p−1 τ p−1 N p− 2 . Moreover, if τ N p−2 , then τ p−1 N p− 2 (τ N ) 2 . 3 In particular, if 3 p 4, then τ p−1 N p− 2 EIp,N , and the result follows. On the other hand, 1
3
if p 4, then τ N − 2 insures that τ p−1 N p− 2 τ p N p−1 EIp,N , as claimed. It remains to discuss 2 p 3. In that case, Lemma 4.11 implies that
2
−p
p p−2 1 1 N an ξn e(nθ ) dθ = an ξn e(nθ ) a¯ k ξk e(−kθ ) a ξ e( θ ) dθ dt 1 2 2 0
n=1
0 n∈Rt
k∈Rt
∈Rt
2 N 1 a n +O m 1+ √ ξn e(nθ ) dθ . m
p 2
0
n=1
(4.67)
1222
G. Mockenhaupt, W. Schlag / Journal of Functional Analysis 256 (2009) 1189–1237
The integral in (4.67) is O(1). Hence (4.61) changes to p 1 N p an ξn (ω)e(nθ ) dθ m 2 + E ω2 E ω1 sup sup ξn (ω1 )xn , (4.68) Eω sup |an |1 x∈E (ω2 ) |A|=m n∈A
n=1
0
with the same E(ω2 ), and (4.63) becomes p 1 N Eω sup an ξn (ω)e(nθ ) dθ |an |1 0
Cε N
ε
n=1
∞ √
log N2 E(ω2 ), t dt . m + (1 + mτ ) EI2(p−1),N + Eω2
p 2
(4.69)
0
Finally, the entropy estimate simplifies as 2(p − 2) 2 in this case: if g|g|p−2 , h|h|p−2 ∈ E(ω2 ), then g, h ∈ PA(ω2 ) and thus g|g|p−2 − h|h|p−2 g − h∞ gp−2 + hp−2 2(p−2) 2(p−2) 2 p−2 p−2 Cε N ε g − hq g2 + h2 Cε N ε m
p−2 2
g − hq ,
so that now E ω2
∞
p log N2 E(ω2 ), t dt Cε N ε m 2 .
0
We leave it to the reader to check that this again leads to (4.66). As already mentioned above, the p 3 term τ p−1 N p− 2 can be absorbed into (τ N ) 2 , since p 3. 2 This lemma quickly leads to a proof of Proposition 4.6 in case δ 0 < δ < 1 if 2 < p < 4.
1 2
for p > 4, and for all
Corollary 4.14. Suppose 0 < δ 12 and assume otherwise that the hypotheses of Proposition 4.6 are satisfied. Then (4.58) holds for all p 4. If 2 < p < 4, then (4.58) holds for all 0 < δ < 1. In particular, Proposition 4.6 is valid in these cases. Proof. As a first step, note that Lemma 4.13 immediately implies that all odd integers satisfy (4.58). Next, one checks that (4.58) holds at p = 52 since 2(p − 1) = 3 in that case. Now Lemma 4.13 implies that (4.58) holds at all other values p = 2 +1 2 , for all integers 3. Generally speaking, one checks by means of induction that (4.58) holds at all
p ∈ 2 + j ∈ Z+ =: Pj . 2
G. Mockenhaupt, W. Schlag / Journal of Functional Analysis 256 (2009) 1189–1237
1223
Indeed, we just verified that this holds for j = 0, 1. Now assume that it holds up to some integer j and we will prove it for j + 1. Thus take p = 2 + 2j +1 ∈ Pj +1 such that 2 < p < 3. Then 2(p − 1) = 2 + 2 j for which (4.58) holds by assumption. Hence Lemma 4.13 applies. Now suppose p ∈ Pj +1 is such that 3 < p < 4. Then (4.58) holds at p − 1 by what we just did, and at 2(p − 1), 2(p − 2) by assumption. Hence Lemma 4.13 applies again. One now continues with 4 < p < 5, etc., and we are done. Given any ε > 0 and p > 2 one can find p1 < p < p2 with p1 , p2 ∈ Pj where p2 − p1 < ε. Hence (4.58) holds for all p by interpolation, as desired. It remains to deal with δ > 12 if 2 < p < 4. Fix such a p. Then by Bourgain’s theorem on random Λ(p) sets, δ > 12 implies that the random set S(ω) is a Λ(p) set. More precisely, N p p E sup an ξn (ω)e(n ·) (τ N ) 2 . |an |1 n=1
p
Clearly, N N 1 ξn (ω)e(n ·) ξn (ω)e(n ·) = # S(ω) 2 , n=1
and we are done.
p
n=1
2
2
4.8. Choosing subsets by means of correlated selectors To conclude this section, we want to address the issue of obtaining a version of Proposition 4.6 for subsets which are obtained by means of selectors ξj that are allowed to have some degree of dependence. More precisely, we will work with the selectors from the following definition. Definition 4.15. Let 0 < τ < 1 be fixed. Define ξj (ω) = χ[0,τ ] (2j ω) for j 1. Here ω ∈ T = R/Z with probability measure P(dω) = dω equal to normalized Lebesgue measure. Since the doubling map ω → 2ω mod 1 is measure preserving, it follows that Eξj = τ and P[ξ = 1] = τ , P[ξj = 0] = 1 − τ , as in the random case. However, these selector variables are no longer independent. Nevertheless, they are close enough to being independent to make the following theorem accessible to the methods of the previous section. Theorem 4.16. Let 0 < δ < 1 be fixed. For every positive integer N we let ξj = χ[0,τ ] (2j ω) be as in Definition 4.15 with τ = N −δ . Define a subset S(ω) = j ∈ [1, N] ξj (ω) = 1
(4.70)
for every ω ∈ T. Then for every ε > 0 and 7 p 2 one has ε an e(nθ ) N e(nθ ) P sup |an |1 n∈S(ω)
Lp (T)
n∈S(ω)
→0
Lp (T)
as N → ∞. Moreover, under the additional restriction δ 12 , (4.71) holds for all p 7.
(4.71)
1224
G. Mockenhaupt, W. Schlag / Journal of Functional Analysis 256 (2009) 1189–1237
To prove this theorem we may of course assume that τ = 2−k for some positive integer k. Then ξj is measurable with respect to the dyadic intervals of length 2−k−j on the unit interval T, denoted by Dj +k . Moreover, it is easy to see that ξj and ξj +ak are independent variables. Lemma 4.17. Fix j 0 and k 1. Let τ = 2−k and ξi be as in Definition 4.15. Then the sequence {ξj +ak }∞ a=1 is a realization of a 0, 1-valued Bernoulli sequence with Eξi = τ . Proof. Fix a > 1 and note that the variable ξj +ak (ω) is 2−(j +ak) -periodic. On the other hand, each of the variables ξj +bk with b < a is constant on intervals from Dj +ak (which is the same as saying that these variables are all Dj +ak measurable). It follows that P[ξj +ak = 1 | ξj +bk = εb , 0 b a − 1] = τ = P[ξj +ak = 1], for any choice of εb = 0, 1, 0 b a − 1. This implies independence.
2
From now on, let τ = N −δ for some fixed 0 < δ < 1. In view of Lemma 4.17 we can decompose the sequence {ξj }N j =1 into about log N many subsequences, where the indices run along arithmetic progressions Pi of step-size equal to ∼ log N , and 1 i log N . Each of the subsequences consists of i.i.d. variables, but variables from different subsequences are not independent. This easily shows that Lemma 4.3 remains valid here, possibly with a logarithmic loss in the upper bound for EIp,N . Indeed, recall that the proof of that lemma is based upon splitting a random trigonometric polynomial into its expectation and a mean-zero part. Since
the Lp -norm of the Dirichlet kernel on an arithmetic progression of length K is about K 1/p , and here #Pi ∼ logNN , one sees immediately that the upper bound from (4.7) is the same up to logarithmic factors. As far as the lower bound of Lemma 4.3 is concerned, note that the proof relies on obtaining upper bounds on certain error terms, cf. (4.10)–(4.13). However, these upper bounds are again immediate corollaries of the random case by virtue of the splitting into the progressions Pi . The consequence of this is that basically all the main estimates from the previous section remain valid here, up to possibly an extra factor of log N . Clearly, such factors are irrelevant in this context. More precisely, with ξj as in Definition 4.15 and S(ω) as in (4.70), it is a corollary of the proof of Proposition 4.6 that p p (4.72) a e(nθ ) Cε N ε τ p N p−1 + (τ N ) 2 . E sup n |an |1 n∈S(ω)
Lp (T)
The proof of Theorem 4.16 is therefore completed as before by appealing to (the adapted version) of Lemma 4.3. Remark 4.18. Other examples of much more strongly correlated selectors are ξj (ω) = χ[0,τ ] (j s ω) where s is a fixed positive integer and ω ∈ T. It appears to be rather difficult to prove a version of Proposition 4.6 for these types of selectors. 5. Perturbing arithmetic progressions Let P ⊂ [1, N] be an arithmetic progression of length L, i.e., P = b + a 0 < b < a, 0 < L := N/a ⊂ [1, N ].
G. Mockenhaupt, W. Schlag / Journal of Functional Analysis 256 (2009) 1189–1237
1225
Fix some arbitrary ε0 > 0. Suppose N ε0 < s < a and let {ξj }j ∈P be i.i.d. variables, integer valued and uniformly distributed in [−s, s]. We define a random subset S(ω) := j + ξj (ω) j ∈ P .
(5.1)
For future reference, we set Ij := [j − s, j + s] for each j ∈ P. By construction, S(ω) ⊂ # j ∈P Ij , and the intervals Ij are congruent and pairwise disjoint. 5.1. Suprema of random processes The following lemma is related to Lemma 4.7. Lemma 5.1. Let E ⊂ RN + , B = supx∈E |x|, and S(ω) be as in (5.1). Then
Eω sup
x∈E j ∈S (ω)
xj B 1 + L/s +
B
log N2 (E, t) dt,
0
where N2 refers to the L2 entropy. Proof. As in the proof of Lemma 4.7, we introduce 2−k -nets Ek and Fk ⊂ RN so that diam(Fk ) 1, log #Fk C log #Ek+1 ,
(5.2)
and E sup
x∈E n∈S (ω)
xn
kk0
2−k+1 E sup
y∈Fk n∈S (ω)
|yn |.
(5.3)
Now fix some k k0 and write F instead of Fk . With 0 < ρ2 to be determined, one has for any |y| 1
yi
i∈S (ω)
yi +
yi ρ2
χS (ω) (i)yi ρ2−1 +
yi <ρ2
χS (ω) (i)yi .
yi <ρ2
Let q := 1 + log F . Then, as in (4.24), E sup
y∈F i∈S (ω)
yi ρ2−1
+ sup χS (ω) (i)yi |y|1 yi <ρ2
.
(5.4)
Lq (ω)
To control the last term in (5.4), we need the following analogue of (4.25). By the multinomial theorem (for any positive integer q),
1226
E
G. Mockenhaupt, W. Schlag / Journal of Functional Analysis 256 (2009) 1189–1237
q χA (n)
n∈S (ω)
=E
q χA j + ξj (ω) =
q1 +···+qL =q
j ∈P
=
q
ν=1 1i1
=
q
ν=1
1 ν ν! q
j ∈P
|A ∩ Ij | |Ij |
ν
q q E χA j + ξj (ω) j q 1 , . . . , qL j ∈P
ν q E χA it + ξit (ω) q i 1 , . . . , qi ν t=1
ν=1 1i1
q |A ∩ Ii1 | |A ∩ Ii2 | |A ∩ Iiν | · ··· · q i 1 , . . . , qi ν |Ii1 | |Ii2 | |Iiν |
# q ν q q! |A ∩ j ∈P Ij | ν q! ν! 2s + 1 ν=1
# q e|A ∩ j ∈P Ij | q q q−ν e|A ∩ j ∈P Ij | ν q+ . q ν 2s + 1 2s + 1 #
ν=1
Continuing with the final term in (5.4) one concludes that sup χS (ω) (i)yi
|y|1 yi <ρ2
Lq (ω)
ρ2−2 <2j
sup χA (n) j
|A|=2
ρ2−2 <2j
1
2
− j2
n∈S (ω)
Lq (ω)
j min(Ls, 2j ) qρ2 + L/s. 2− 2 q + s
1
Let ρ2 = q − 2 = (1 + log #F )− 2 . Inserting this bound into (5.4) therefore yields E sup
y∈F i∈S (ω)
yi
√ q + L/s L/s + 1 + log #F .
The lemma now follows in view of (5.2) and (5.3).
2
5.2. The Lp norm of the Dirichlet kernel over S(ω) The following lemma determines an upper bound on the typical size of the Dirichlet kernel over S(ω) in the Lp -norm, with 2 p 4. The lower bound, as well as the case p > 4 will be dealt with below.
G. Mockenhaupt, W. Schlag / Journal of Functional Analysis 256 (2009) 1189–1237
1227
Lemma 5.2. With S(ω) as in (5.1), there exists a constant Cp so that p p Lp−1 2 e(n ·) Cp L + E s p n∈S (ω)
for all 2 p 4. Proof. For every ∈ Z define A (ω) := # n, m ∈ S(ω) n − m = = χ[j −k+ξj −ξk = ] . j,k∈P
#
Clearly, P − P ⊂ i Ji where i ∈ aZ and Ji := [i − 2s, i + 2s]. These intervals are mutually disjoint since s a. This means that ∈ Ji
⇒
A (ω) =
χ[j −i∈P ] χ[ξj −ξj −i = −i] .
j ∈P
Let us denote the unique i for which ∈ Ji by i( ). For simplicity, we shall mostly write i. If i = 0, then A (ω) = Lδ0 ( ) (recall that #P = L). Otherwise, if i = 0, then one finds that EA =
j ∈P
=
| − i| | − i| 2 2 1− 1− χP (j − i) = L − |i|/a + (5.5) 2s + 1 s 2s + 1 s + +
2L * *L |i|/a Ks − i( ) K 2s + 1
*n (k) = (1 − |k|/n)+ denotes the Fejer kernel. Moreover, if i = 0, then where K
EA2 = E
χ[ξj −ξj −i = −i] χ[ξk −ξk−i = −i]
j,k∈P j −i∈P , k−i∈P
=
χ[j =k,j =k±i] Eχ[ξj −ξj −i = −i] Eχ[ξk −ξk−i = −i]
j,k∈P j −i∈P , k−i∈P
+
(χ[j =k,j =k±i] + χ[j =k,k+i,j =k−i] + χ[j =k,k−i,j =k+i] )
j,k∈P j −i∈P , k−i∈P
× Eχ[ξj −ξj −i = −i] χ[ξk −ξk−i = −i] . Hence EA2 =
j,k∈P j −i∈P , k−i∈P
Eχ[ξj −ξj −i = −i] Eχ[ξk −ξk−i = −i]
(5.6)
1228
G. Mockenhaupt, W. Schlag / Journal of Functional Analysis 256 (2009) 1189–1237
+
(χ[j =k,j =k±i] + χ[j =k,k+i,j =k−i] + χ[j =k,k−i,j =k+i] )
j,k∈P j −i∈P , k−i∈P
× E(χ[ξj −ξj −i = −i] χ[ξk −ξk−i = −i] ) (5.7) (χ[j =k,j =k±i] + χ[j =k,k+i,j =k−i] + χ[j =k,k−i,j =k+i] )Eχ[ξj −ξj −i = −i] − j,k∈P j −i∈P , k−i∈P
× Eχ[ξk −ξk−i = −i] | − i| L 2 . 1− = (EA ) + O s s +
(5.8) (5.9)
The O-term in (5.9) arises because the error terms in (5.7) and (5.8) basically reduce to the computation of a single expectation as in (5.5). Now consider
Vp,N
p 1 2 A (ω) − EA e( θ ) dθ. := ∈Z
0
p
Since p 4 by assumption, EVp,N (EV4,N ) 4 . Moreover, by (5.9), EV4,N = E
A (ω) − EA 2 = E A2 − (EA )2
∈Z
∈Z
| − i| L = E A20 − (EA0 )2 + (EA )2 + O − 1− (EA )2 s 2s + 1 + =0
L
=0
2
and therefore p
EVp,N L 2 . In view of (5.6),
EA e( θ ) =
∈Z
2L *L i( )/a e − i( ) θ e i( )θ *s − i( ) K K 2s + 1 ∈Z
=
2L * *L (j )e(j aθ ) = 2L Ks (θ )KL (aθ ). Ks (k)e(kθ ) K 2s + 1 2s + 1 j ∈Z
k∈Z
It follows that p p 1 p 1 2 2 p 2 −2 2 L 1 KL (aθ ) 2 dθ min s , θ EA e( θ ) dθ s s 0
∈Z
0
(5.10)
G. Mockenhaupt, W. Schlag / Journal of Functional Analysis 256 (2009) 1189–1237
1229
p p a 2 2 −1 p −1 L 2 p −1 p −1 1 −2 min s , (j/a) a L2 s 2 a L2 + s s j =1
Lp−1 s
(5.11)
.
Combining (5.10) with (5.11) one obtains for 2 p 4 p p 1 1 2 E e(nθ ) dθ = E A (ω)e( θ ) dθ 0
n∈S (ω)
∈Z
0
p p 1 1 2 2 A (ω) − EA e( θ ) dθ EA e( θ ) dθ + E 0
as claimed.
∈Z
0
∈Z
p Lp−1 +L2 , s
(5.12)
2
The following lemma is a special case of a well-known large deviation estimate for martingales with bounded increments. The norm · ∞ refers to the supremum norm with respect to the probability space. Lemma 5.3. Suppose {Xj }M j =1 are complex-valued independent variables with EXj = 0. Then for all λ > 0 M M 1 2 2 P Xj > λ Xj 2∞ < Ce−cλ j =1
j =1
with some absolute constants c, C. Lemma 5.3 implies the following simple generalization of the Salem–Zygmund bound. Corollary 5.4. Let s, L be positive integers. Suppose TL is a trigonometric polynomial with random coefficients that can be written in the form TL (θ ) =
L
aj (θ )e(j θ ),
j =−L
where aj (θ ) are trigonometric polynomials of degree at most s, and such that for fixed θ they are independent random variables with Eaj (θ ) = 0. Moreover, we assume that supθ∈T |aj (θ )| 1 for each j . Then for every A > 1 √ P TL ∞ > C log(s + L) L (s + L)−A ,
1230
G. Mockenhaupt, W. Schlag / Journal of Functional Analysis 256 (2009) 1189–1237
with some constant C = C(A). Proof. Fix θ ∈ T and apply Lemma 5.3 with Xj = aj (θ )e(j θ ). By assumption, these are complex-valued independent mean-zero variables with Xj ∞ 1. Therefore, L √ 2 aj (θ )e(j θ ) > λ L < Ce−cλ . sup P θ∈T
(5.13)
j =−L
If |θ − θ | < (s + L)−2 , then by Bernstein’s inequality TL (θ ) − TL (θ ) (s + L)TL ∞ |θ − θ | (s + L)L(s + L)−2 1. Now pick a (s + L)−2 -net on the circle. The corollary follows by setting λ = C log(s + L) with C large, and summing (5.13) over the elements of the net. 2 We can now state the general version of Lemma 5.2. It is possible to remove the log-term from the upper bound, but the bound given below suffices for our purposes. Lemma 5.5. For all p 2 there exists Cp so that p p−1 p L 2 + (L log N ) . E e(n ·) Cp s n∈S (ω)
(5.14)
p
Moreover, there is cp > 0 small so that p p Lp−1 2 →0 P e(n ·) < cp L + s p n∈S (ω)
as N → ∞. Proof. We work with the following splitting: n∈S (ω)
e(nθ ) =
EχS (ω) (n)e(nθ ) +
n∈Z
χS (ω) (n) − EχS (ω) (n) e(nθ ).
(5.15)
n∈Z
Clearly, n∈Z
and thus
EχS (ω) (n)e(nθ ) =
1 Ds (θ ) e(j θ ), 2s + 1 j ∈P
(5.16)
G. Mockenhaupt, W. Schlag / Journal of Functional Analysis 256 (2009) 1189–1237
1231
p p 1 L −1 −p min s, θ Eχ (n)e(nθ ) s e(j aθ ) dθ S (ω) p
n∈Z
j =1
0
s
−p
L
min(s, a/k) + s p
p
k=1
Lp−1 Lp−1 . a s
(5.17)
Conversely, p p 1/s L −p D Eχ (n)e(nθ ) s (θ ) e(j aθ ) dθ s S (ω) n∈Z
p
j =1
0
a Lp−1 Lp−1 = . s a s
(5.18)
Both (5.17) and (5.18) hold for all p > 1. The second sum in (5.15) can be written as χS (ω) (n) − EχS (ω) (n) e(nθ ) = aj (ω, θ )e(j θ ), j ∈P
n∈Z
1 Ds (θ ). Clearly, Eaj (ω, θ ) = 0, supθ |aj (ω, θ )| 2 where aj (ω, θ ) = χIj (ξj (ω))e(ξj (ω)) − 2s+1 and for fixed θ the random variables aj (ω, θ ) are independent. Thus Corollary 5.4 yields that
χ (n) − Eχ (n) e(nθ ) S (ω) S (ω)
∞
n∈Z
√ L log N
(5.19)
up to probability at most (s + L)−p . In particular, p p p p −p E χS (ω) (n) − EχS (ω) (n) e(nθ ) (L log N ) 2 + L (s + L) (L log N ) 2 . p
n∈Z
In conjunction with (5.17) this yields (5.14). For the lower bound, take N −ε0 /2 > h 1s . Then p 1 e(nθ ) dθ 0
n∈S (ω)
p p 1/s 1/s χS (ω) (n) − EχS (ω) (n) e(nθ ) dθ EχS (ω) (n)e(nθ ) dθ − 0
n∈Z
0
n∈Z
1−h 1−h p p χS (ω) (n) − EχS (ω) (n) e(nθ ) dθ − EχS (ω) (n)e(nθ ) dθ + h
n∈Z
=: I + II + III + IV.
h
n∈Z
(5.20)
1232
G. Mockenhaupt, W. Schlag / Journal of Functional Analysis 256 (2009) 1189–1237
By (5.18), I
Lp−1 s .
Secondly,
p 1−h L−1 1 e(j aθ ) dθ IV Ds (θ ) s j =0
h
s −p
(j/a)−p
j >ah
Lp−1 Lp−1 s −p h−p+1 Lp−1 , a s
(5.21)
where the final estimate follows from hs 1. Thirdly, in view of p 2 and (5.19), p 1 III χS (ω) (n) − EχS (ω) (n) e(nθ ) dθ n∈Z
0
h −
p 1 χS (ω) (n) − EχS (ω) (n) e(nθ ) dθ +
0
1−h
n∈Z
1 2 p2 p χS (ω) (n) − EχS (ω) (n) e(nθ ) dθ − Ch(L log N ) 2 0
n∈Z
p 2
p
L − Ch(L log N ) 2 ,
(5.22)
up to probability (s + L)−p = o(1) as N → ∞. Similarly, (5.19) implies that p
II s −1 (L log N ) 2
up to probability (s + L)−p . Combining this bound with (5.22), (5.21), and (5.20) implies that p 1 p−1 p p dθ L + L 2 − C h + s −1 (L log N ) 2 e(nθ ) s 0
n∈S (ω)
asymptotically with probability one. Since h < N −ε and s > N ε , the lemma follows.
2
5.3. The majorant property for randomly perturbed arithmetic progressions We are now ready to state our first result for perturbed arithmetic progressions as defined in (5.1). In this section, if S is the perturbation of an arithmetic progression of length L, then we write p . e(n ·) Ap,L (ω) := n∈S (ω)
p
Also, we say that the random majorant property (RMP) holds at p if
G. Mockenhaupt, W. Schlag / Journal of Functional Analysis 256 (2009) 1189–1237
p p ε . Eω sup a e(n ·) C N E e(nθ ) n ε ω |an |1 n∈S (ω)
p
1233
(5.23)
p
n∈S (ω)
Of course, this depends on the length L of the underlying arithmetic progression. Although L is arbitrary, it will be kept fixed in the course of any argument that uses (5.23). Theorem 5.6. Let S be as in (5.1). Then for every ε > 0 and 4 p 2 one has P sup a e(nθ ) n |an |1 n∈S (ω)
Lp (T)
N ε e(nθ )
→0
(5.24)
Lp (T)
n∈S (ω)
as N → ∞. Moreover, under the additional restriction L s, (5.24) holds for all p 4. Proof. The proof is similar to the random case of the previous section, so we shall be somewhat brief. We will show that the RMP holds at p provided either 2 p 3, or if the RMP holds at p − 1, 2(p − 1), and 2(p − 2). It is important to notice that the RMP at p implies (5.24). Firstly, recall that we can write S(ω) = {j + ξj | j ∈ P}. We apply the decoupling lemma, Lemma 4.11, to the progression P. I.e., in the notation of Lemma 4.11, Rt1 := {j ∈ P | ζj = 1}, and Rt2 := {j ∈ P | ζj = 0}. Set St1 (ω) := j + ξj (ω) j ∈ Rt1 ,
St2 (ω) := j + ξj (ω) j ∈ Rt2 .
Therefore, by Lemma 4.11, 1 8
p p−2 1 1 an e(nθ ) dθ = an e(nθ ) a¯ k e(−kθ ) a e( θ ) dθ dt 0
n∈S (ω)
1 0 n∈St (ω)
k∈St2 (ω)
∈St2 (ω)
max(p−1,2) 1 an +O L 1 + dθ . √ e(nθ ) L n∈S p 2
(5.25)
0
If either p 3, or if the RMP holds at p − 1, then the O-term in (5.25) is at most p
p
1
L 2 + Cε N ε L 2 EAp−1,L N ε L 2 ,
(5.26)
see Lemma 5.5. We therefore obtain as in (4.51),
Eω
p 1 sup an e(nθ ) dθ
|an |1
0
n∈S (ω)
p 2
Cε N L + ε
1 Eω1 ,ω2 sup |an |1 0
n∈St1 (ω1 )
an e(nθ )
k∈St2 (ω2 )
a¯ k e(−kθ )
1234
G. Mockenhaupt, W. Schlag / Journal of Functional Analysis 256 (2009) 1189–1237
×
∈St2 (ω2 )
p−2 a e( θ ) dθ dt
p 2
Cε N L + ε
Eω1 Eω2
1 sup |an |1
|bn |1 0
×
k∈St2 (ω2 )
b¯k e(−kθ )
p 2
∈St2 (ω2 )
Cε N L + ε
Eω1 Eω2
an e(nθ )
n∈St1 (ω1 )
p−2 b e( θ ) dθ dt
1 sup an e(nθ ) b¯k e(−kθ ) |an |1
n∈S (ω1 ) |bn |1 0
k∈S (ω2 )
p−2 × b e( θ ) dθ dt ∈S (ω2 )
p
Cε N ε L 2 + Eω2 Eω1 sup
x∈E (ω2 ) n∈S (ω ) 1
(5.27)
xn .
Here p−2 , N
+ E(ω2 ) := b e( ·) b¯k e(−k ·) e(n ·), k∈S (ω2 )
n=1
∈S (ω2 )
sup |bn | 1 ⊂ RN +. 1nN
By Lemma 5.1, it follows from (5.27) that
Eω
p 1 sup an e(nθ ) dθ
|an |1
0
(5.28)
n∈S (ω)
Cε N L + 1 + L/s Eω2 sup |x| + Eω2 ε
p 2
x∈E (ω2 )
∞
log N2 E(ω2 ), t dt.
0
Now suppose the RMP holds at 2(p − 1) (so this holds for sure if p is an odd integer). Then by Plancherel, Eω 2
p−1 sup |x| Cε N Eω e(n ·) ε
x∈E (ω2 )
n∈S (ω)
Cε N ε Eω A2(p−1),L (ω).
2(p−1)
As far as the entropy term in (5.28) is concerned, the same analysis as in the random case shows that if p 3, then
G. Mockenhaupt, W. Schlag / Journal of Functional Analysis 256 (2009) 1189–1237
Eω2
∞
1235
3 log N2 E(ω2 ), t dt Cε N ε L 2 ,
0
or if p > 3 and the RMP holds at 2(p − 2), then
Eω 2
∞
log N2 E(ω2 ), t dt Cε N ε L EA2(p−2),L ,
0
see (4.64) and (4.65) for the details. Inserting all of this into (5.28) yields, under the assumption that p > 3 and the RMP holds at p − 1, 2(p − 1), and 2(p − 2) (the case p 3 is similar),
Eω
p 1 sup an e(nθ ) dθ
|an |1
0
n∈S (ω)
p Cε N ε L 2 + (1 + L/s) Eω A2(p−1),L (ω) + L EA2(p−2),L
1 1 2p−3 2p−5 2 2 p L L p−1 p−2 2 +L +L Cε N L + (1 + L/s) +L s s p−1 3 p L Lp− 2 . Cε N ε +L2 + √ s s ε
p−1
(5.29)
p
Recall from Lemma 5.5 that the desired bound is L s + L 2 . If p = 3, then (5.29) does indeed agree with this bound. Since the hypotheses involving the RMP hold in case p = 3, we are done with that case, regardless of the relative size of L and s. Let us assume now that L s. Then (5.29) agrees with the desired bound for all p. This means that we can run the same type of inductive argument as in Corollary 4.14. We leave it to the reader to check that this√proves (5.24) for all p 2 provided L s. Finally, if L < s, then L < s a N L and thus L N . In partic√ ular, #S N in that case. In analogy with the random subset case, this suggests that S(ω) are Λ(p)-sets for 2 p 4 with high probability. Although perturbed arithmetic progressions are not covered by [1], it turns out that the strategy from [1] and [4] is still relevant. More precisely, suppose first that 2 p 3. Then (5.28) holds, even without the N ε -term. By Plancherel, but without appealing to any RMP, Eω2 sup |x| Eω2 x∈E (ω2 )
p−1 p p−2 Kp2 L 2 . sup an e(nθ )
|an |1
n∈S (ω)
2
Here
p Kp
:= Eω
p 1 dθ. sup a e(nθ ) n
|an |1
0
n∈S (ω)
(5.30)
1236
G. Mockenhaupt, W. Schlag / Journal of Functional Analysis 256 (2009) 1189–1237
To pass to (5.30), one writes 2(p − 1) = p + (p − 2) and then estimates the (p − 2)-power in L∞ . 2 Secondly, to bound the entropy term, set q = 3−p . Then by Plancherel the distance between any two elements in E(ω2 ) is at most g|g|p−2 − h|h|p−2 g − hq gp−2 + hp−2 2 2 2 L where g, h ∈
√
p−2 2
g − hq ,
LPS (ω2 ) , see (4.55). As before, the entropy estimate therefore reads
Eω2
∞
√ p−2 √ p log N2 E(ω2 ), t dt LL 2 L = L 2 .
0
Inserting these bounds into (5.28) yields p p−2 p p 1 p p Kp L 2 + 1 + L/s Kp2 L 2 CL 2 + Kp + C(1 + L/s)Lp−2 . 2 p
Since Lp−2 L 2 in view of p 4, one obtains the desired bound p
p
Kp L 2 +
Lp−1 s
if 2 p 3 and regardless of the relative size of L and s. If 3 p 4, then the previous argument needs to be modified in two places. Firstly, there is the issue of the O-term in (5.25). However, we just showed that the RMP holds at p − 1 3, and therefore (5.26) applies here as well (even without the N ε -term). Secondly, the entropy bounds need to be modified. In case 3 p 4, one has 2(p − 2) p. Hence g|g|p−2 − h|h|p−2 g − h∞ gp−2 + hp−2 2(p−2) 2(p−2) 2 p−2 p−2 Cε N ε g − hq gp + hp N p−2 ε an ξn (ω2 )e(n ·) g − hq , Cε N sup |an |1 n=1
p
with g, h as above. By the usual arguments, cf. (4.64), it follows that
Eω2
∞
p−2 p 1 p log N2 E(ω2 ), t dt LKp 2 Kp + L 2 . 2
0
Inserting these bounds into (5.28) implies the desired bound.
2
G. Mockenhaupt, W. Schlag / Journal of Functional Analysis 256 (2009) 1189–1237
1237
Remark 5.7. It is possible that one can make improvements on Theorem 5.6 similar to those in Proposition 4.6, thus removing the condition L s in some range of p 4. This would require working with Λ(p) type arguments as we just did in the end of the previous proof. But we do not pursue that issue here. Acknowledgments The authors would like to thank Jean Bourgain for several suggestions and comments. The first author was supported NSF grant DMS-0300416. The second author was partially supported by an NSF grant, DMS-0070538, and a Sloan fellowship. References [1] J. Bourgain, Bounded orthogonal systems and the Λ(p)-set problem, Acta Math. 162 (3–4) (1989) 227–245. [2] J. Bourgain, On Λ(p)-subsets of squares, Israel J. Math. 67 (3) (1989) 291–311. [3] J. Bourgain, Remarks on Halasz–Montgomery type inequalities, in: Geometric Aspects of Functional Analysis, Israel, 1992–1994, in: Oper. Theory Adv. Appl., vol. 77, Birkhäuser, Basel, 1995, pp. 25–39. [4] J. Bourgain, Λ(p)-sets in analysis: Results, problems and related aspects, in: Handbook of the Geometry of Banach Spaces, vol. I, North-Holland, Amsterdam, 2001, pp. 195–232. [5] J. Bourgain, J. Lindenstrauss, V. Milman, Approximation of zonoids by zonotopes, Acta Math. 162 (1–2) (1989) 73–141. [6] B. Green, I.Z. Ruzsa, On the Hardy–Littlewood majorant problem, Math. Proc. Cambridge Philos. Soc. 137 (3) (2004) 511–517. [7] G.H. Hardy, J.E. Littlewood, Notes on the theory of series (XIX): A problem concerning majorants of Fourier series, Q. J. Math. 6 (1935) 304–315. [8] M. Ledoux, M. Talagrand, Probability in Banach Spaces. Isoperimetry and Processes, Ergeb. Math. Grenzgeb. (3), vol. 23, Springer-Verlag, Berlin, 1991. [9] G. Mockenhaupt, Bounds in Lebesgue spaces of oscillatory integrals, Habilitationsschrift, Siegen, 1996, http:// www-math-analysis.ku-eichstaett.de/~gerdm/. [10] H. Montgomery, Ten Lectures on the Interface between Analytic Number Theory and Harmonic Analysis, CBMS Reg. Conf. Ser. Math., vol. 84, Amer. Math. Soc., Providence, RI, 1994. [11] A. Pajor, N. Tomczak-Jaegermann, Subspaces of small codimension of finite-dimensional Banach spaces, Proc. Amer. Math. Soc. 97 (4) (1986) 637–642. [12] G. Pisier, The Volume of Convex Bodies and Banach Space Geometry, Cambridge Tracts in Math., vol. 94, Cambridge Univ. Press, Cambridge, 1989. [13] R. Salem, A. Zygmund, Some properties of trigonometric series whose terms have random signs, Acta Math. 91 (1954) 245–301. [14] M. Talagrand, Sections of smooth convex bodies via majorizing measures, Acta Math. 175 (2) (1995) 273–300. [15] A. Zygmund, Trigonometric Series, Cambridge Univ. Press, Cambridge, 1959.
Journal of Functional Analysis 256 (2009) 1238–1257 www.elsevier.com/locate/jfa
Lp -uniqueness for elliptic operators with unbounded coefficients in RN ✩ Angela Albanese a , Luca Lorenzi b,∗ , Elisabetta Mangino a a Dipartimento di Matematica “E. De Giorgi”, Università del Salento, Via Per Arnesano, P.O. Box 193,
73100 Lecce, Italy b Dipartimento di Matematica, Università degli Studi di Parma, Viale G.P. Usberti, 53/A, 43100 Parma, Italy
Received 29 April 2008; accepted 21 July 2008 Available online 12 September 2008 Communicated by Paul Malliavin
Abstract Let A be an elliptic operator with unbounded and sufficiently smooth coefficients and let μ be a (sub)invariant measure of the operator A. In this paper we give sufficient conditions guaranteeing that the closure of the operator (A, Cc∞ (RN )) generates a sub-Markovian strongly continuous semigroup of contractions in Lp (RN , μ). Applications are given in the case when A is a generalized Schrödinger operator. © 2008 Elsevier Inc. All rights reserved. Keywords: Elliptic operators with unbounded coefficients; (Sub-)invariant measures; Cores
1. Introduction The qualitative properties of elliptic operators A with unbounded coefficients in RN have been investigated intensively in recent years, after the seminal papers [3,4,12,18], motivated by the important impact of these operators on stochastic processes, and their application to branches of applied sciences such as mathematical finance.
✩ Work partially supported by the research project “Equazioni di Kolmogorov” of the Ministero dell’Istruzione, dell’Università e della Ricerca (M.I.U.R.). * Corresponding author. E-mail addresses: [email protected] (A. Albanese), [email protected] (L. Lorenzi), [email protected] (E. Mangino).
0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.07.022
A. Albanese et al. / Journal of Functional Analysis 256 (2009) 1238–1257
1239
The natural settings where to study these operators are the space of bounded and continuous functions (say Cb (RN )) and Lp -spaces. In Cb (RN ) one can associate a semigroup of bounded operators with such operators under rather weak assumptions on their coefficients. Indeed, if
Aφ =
N i,j =1
qij Dij φ +
N
bj Dj φ + cφ,
j =1
on smooth functions φ, requiring that the matrix Q(x) = (qij (x)) is definite positive at any x ∈ RN , that the coefficients qij , bj (i, j = 1, . . . , N ) and c are locally α-Hölder continuous for some α ∈ (0, 1) and c is bounded from above, is all what one needs to show that, for any f ∈ Cb (RN ), the Cauchy problem
Dt u(t, x) = (Au)(t, x), u(0, x) = f,
t > 0, x ∈ RN , x ∈ RN ,
(1.1)
admits (at least) a classical solution, bounded with respect to the sup-norm in [0, T ] × RN , for any T > 0. Here, by classical solution we mean any function u which (i) is continuous up to t = 0, (ii) is continuously differentiable in (0, +∞) × RN , once with respect to time and twice with respect to the spatial variables, (iii) solves (1.1). In general, the Cauchy problem (1.1) admits more than a unique bounded classical solution (but it turns out to be uniquely solvable if an additional algebraic condition on its coefficients is prescribed). Nevertheless, it is possible to associate a semigroup of bounded operators with the operator A, setting T (t)f = u(t, ·) for any t 0, where for any positive f , u(t, ·) is the value at t of the minimal positive solution to (1.1). This semigroup is, in general, neither strongly continuous nor analytic in Cb (RN ). In fact, T (t)f tends to f in the weak topology of RN , i.e., the sequence {T (t)f } is bounded and converges to f locally uniformly in RN as t → 0+ . A rather complete analysis of the semigroup {T (t)} and its main smoothing properties is available nowadays and we refer the reader to [5]. The analysis of the operator A in the usual Lp -spaces related to the Lebesgue measure is much more difficult than in Cb (RN ). As a one-dimensional example in [30, Section 2] shows, whatever ε > 0 is fixed, the operator A defined by Aφ(x) = φ (x) − sign(x)|x|1+ε φ (x),
x ∈ R,
(1.2)
does not generate a strongly continuous semigroup in Lp (R) for any p ∈ [1, +∞). Hence, assumptions on the growth at infinity of the coefficients of the operator A more restrictive than in the Cb -case are to be prescribed. Typically, the diffusion coefficients are supposed to be bounded or to grow at most slightly more than quadratically at infinity, and some suitable compensation conditions on the coefficients are prescribed (see, e.g., the papers [9–11,16,17,23,25,26,28,30,31]). The suitable Lp -spaces where to analyze elliptic operators with unbounded coefficients are p L -spaces related to particular measures, the so-called invariant measures and infinitesimally invariant measures. Whenever existing, an invariant measure is any (probability) measures μ such that
1240
A. Albanese et al. / Journal of Functional Analysis 256 (2009) 1238–1257
T (t)f μ(dx) = RN
f μ(dx),
t > 0,
(1.3)
RN
for any f ∈ Cb (RN ). Here, {T (t)} is the semigroup introduced above. Condition (1.3) may be rephrased requiring that Aφ μ(dx) = 0,
(1.4)
RN
for any φ ∈ Cb (RN ) such that supt∈(0,1) t −1 T (t)φ − φ∞ < +∞ and t −1 (T (t)φ − φ) converges locally uniformly in RN to a bounded and continuous function. In particular, (1.4) should be satisfied by any smooth and compactly supported function φ. Any measure μ satisfying (1.4) for any φ ∈ Cc∞ (RN ) is usually called infinitesimally invariant. Note that whenever an infinitesimally invariant measure of A exists, the operator (A, Cc∞ (RN )) is dissipative in Lp (RN , μ) for any p ∈ [1, +∞). In the case when μ is an invariant measure of A, the semigroup {T (t)} extends in a rather straightforward way by a strongly continuous semigroup in Lp (RN , μ) for any p ∈ [1, +∞). Its infinitesimal generator Ap turns out to be an extension of the operator (A, Cc∞ (RN )). For 4 instance, in the case of the operator in (1.2) with ε = 2 the measure μ(dx) = Ke−|x| /4 dx (where K is a suitable normalizing constant) satisfies (1.3), so that the realization of A in Lp (RN , μ) generates a strongly continuous semigroup of contractions for any p ∈ [1, +∞). In general it is not known whether there exists only one strongly continuous semigroup on Lp (RN , μ) whose generator extends (A, Cc∞ (RN )). If the answer is positive, then (A, Cc∞ (RN )) is said to be Lp (RN , μ)-unique. Standard results in semigroup theory show that the Lp (RN , μ)uniqueness is equivalent to the condition that the closure of (A, Cc∞ (RN )) generates a strongly continuous semigroup in Lp (RN , μ). If this is the case, then Cc∞ (RN ) is a core for Ap and this is of great importance since, in general, characterizing the domain of Ap is a hard task. Indeed, it is rather easy to show that it contains the set of all compactly functions f ∈ Lp (RN , μ) such that their first- and second-order derivatives are in Lp (RN , μ) as well. Moreover, whenever pointwise gradient estimates for the function T (t)f are available for any f ∈ Cc∞ (RN ), one can partially characterize D(Ap ), showing that it is continuously embedded in the Sobolev space W 1,p (RN , μ) (the set of functions f ∈ Lp (RN , μ) such that the distributional gradient of f is in Lp (RN , μ)N ) for any p ∈ (1, +∞). Anyway, a complete characterization of D(Ap ) is known, to the best of our knowledge only in very few cases (see, e.g., [13,22,24,25]). In this paper, under suitable assumptions on the coefficients of the operator A and assuming that c ≡ 0, we prove the Lp (RN , μ)-uniqueness of (A, Cc∞ (RN )), thus generalizing the results in [1,2]. For the sake of generality, we assume that μ is just a sub-invariant measure of the operator A − α for some α ∈ R, in the sense that Aφ μ(dx) α φ μ(dx), RN
RN
for any positive function φ ∈ Cc∞ (RN ). Whenever it admits a sub-invariant measure, the operator (A − pα I, Cc∞ (RN )) turns out to be dissipative (see e.g. [14, Appendix B, Lemma 1.8]),
A. Albanese et al. / Journal of Functional Analysis 256 (2009) 1238–1257
1241
hence closable, in Lp (RN , μ) for any p ∈ [1, +∞). Under some integrability conditions involving a Lyapunov function and the coefficients of the operator A, we prove that the closure of (A, Cc∞ (RN )) generates a strongly continuous semigroup in Lp (RN , μ) for any p ∈ [1, +∞). Our results cover also some situations in which the coefficients of the operator A may grow at infinity with some exponential rate. In particular, we partly generalize some results in [14]. In the second part of the paper, we give another criterion ensuring that the closure of the operator (A, Cc∞ (RN )) generates a strongly continuous semigroup in Lp (RN , μ). In particular, this criterion is useful when the measure μ is symmetrizing for the operator A (i.e., in the case when RN φAψ μ(dx) = RN ψAφ μ(dx) for any φ, ψ ∈ Cc∞ (RN )). For instance, this is the case when A is a generalized Schrödinger operator. In this setting, our results generalize similar results obtained by Eberle [14], Liskevi˘c [20], Liskevi˘c and Semenov [21]. RN , μ)-uniqueness 2. A first criterion for Lp (R Let A be the second order elliptic partial differential operator defined on smooth functions by (Aψ)(x) =
N
qij (x)Dij ψ(x) +
i,j =1
N
bi (x)Di ψ(x),
x ∈ RN .
(2.1)
i=1
We make the following assumptions. Hypothesis 2.1. 1,r (i) The coefficients qij = qj i and bj (i, j = 1, . . . , N ) belong to Wloc (RN , dx) and Lrloc (RN , dx), respectively, for some r > N 2. (ii) The matrix Q := (qij )N i,j =1 satisfies the ellipticity condition
Q(x)ξ · ξ η(x)|ξ |2 ,
ξ ∈ RN , x ∈ RN ,
for some positive function η such that infK η > 0 for every compact K ⊂ RN . (iii) There exists a positive locally finite Borel measure μ on RN such that: (a) μ is absolutely continuous with respect to the Lebesgue measure and its density is everywhere positive in RN . Moreover, infK > 0 for any compact set K ⊂ RN . (b) μ is sub-invariant for the operator (A − α, Cc∞ (RN )) for some α 0, i.e.,
Af μ(dx) α
RN
f μ(dx)
RN
for all 0 f ∈ Cc∞ (RN ). We remark that condition (i) in Hypothesis 2.1 guarantees that the functions qij (i, j = 1, . . . , N ) are locally Hölder continuous and, hence, locally bounded in RN . In this setting we are able to prove the following result.
1242
A. Albanese et al. / Journal of Functional Analysis 256 (2009) 1238–1257
Theorem 2.2. Assume that Hypothesis 2.1 holds. Assume further that there exists a positive function V ∈ C 2 (RN ) such that lim|x|→+∞ V (x) = +∞ and AV ∈ Lp RN , μ and V log V
(Q∇V ) · ∇V ∈ Lp R N , μ , 2 V log V
(2.2)
p
for some 1 p < +∞ and bi ∈ Lloc (RN , μ) (i = 1, . . . , N ). Then, the closure of the operator (A, Cc∞ (RN )) on Lp (RN , μ) generates a sub-Markovian strongly continuous semigroup. In particular, (A, Cc∞ (RN )) is Lp (RN , μ) unique. Proof. To begin with we observe that, up to replacing V with V + e, we can assume, without any loss of generality, that V e in RN . Let us prove that the space (λI − A)(Cc∞ (RN )) is dense in Lp (RN , μ) for all λ > 0. For this purpose, let us denote by p the conjugate exponent of p and suppose that
(λI − A)ϕ, ψ
Lp (RN ,μ)
= 0,
(2.3)
for some ψ ∈ Lp (RN , μ), λ > 0, and all ϕ ∈ Cc∞ (RN ). Setting ν(dx) = ψ dx, condition (2.3) can be rewritten equivalently as (λ − A)ϕ ν(dx) = 0,
ϕ ∈ Cc∞ RN .
(2.4)
RN 1,r By [7, Corollary 2.10] the density ψ of the measure ν belongs to Wloc (RN , dx). Hence, it is N. continuous and locally bounded in R
r N ˆ ˆ For all i = 1, . . . , N , set bi = bi − N j =1 Dj qij . Since bi ∈ Lloc (R , dx) for any i, integrating by parts, from (2.4) we obtain that
λ
ϕψ dx =
RN
RN
Aϕψ dx =
−∇ϕ · Q∇(ψ) + ψbˆ · ∇ϕ dx,
(2.5)
RN
for every ϕ ∈ Cc∞ (RN ). By density, (2.5) can be extended to every ϕ ∈ Cc1 (RN ). Fix now v ∈ C 1 (RN ) and ζ ∈ Cc1 (RN ) with ζ 0. Replacing ϕ with vζ in (2.5), we get
λ RN
vζ ψ dx =
−v∇ζ · Q∇(ψ) − ζ ∇v · Q∇(ψ) + ψbˆ · ∇(vζ ) dx.
(2.6)
RN
Since ψ is locally Hölder continuous and (Q∇(ψ))i , bˆi ∈ Lrloc (RN , dx) for any i = 1, . . . , N ,
1,r (RN , dx), where r denotes the conjuwe can extend equality (2.6) by density to every v ∈ Wloc gate exponent of r. Let F : R → [−1, 1] be an increasing C 1 -function such that F (s) = 0 if |s| 1, F (s) = −1 if s −2 and F (s) = 1 if s 2. For every n ∈ N and x ∈ RN , set un (x) := F (nψ(x)(x)). Then, 1,r 1,r (RN , dx) ⊂ Wloc (RN , dx), since r < 2 < r. Moreover, |un | 1 and un pointwise un ∈ Wloc N tends to sign(ψ) in R , as n → +∞. Replacing v with un in (2.6) and observing that
A. Albanese et al. / Journal of Functional Analysis 256 (2009) 1238–1257
∇un · Q∇(ψ) = nF (nψ)∇(ψ) · Q∇(ψ) 0,
1243
n ∈ N,
we obtain −un ∇ζ · Q∇(ψ) − nζ F (nψ)∇(ψ) · Q∇(ψ) dx λ un ζ ψ dx = RN
RN
+
RN
−∇ζ · Q∇(un ψ) + ψ∇ζ · (Q∇un ) dx
RN
+
un ψbˆ · ∇ζ + nζ F (nψ)ψbˆ · ∇(ψ) dx
un ψbˆ · ∇ζ + nζ F (nψ)ψbˆ · ∇(ψ) dx
RN
un ψ div(Q∇ζ ) + ψnF (nψ)∇ζ · Q∇(ψ) dx
RN
+
un ψbˆ · ∇ζ + nζ F (nψ)ψbˆ · ∇(ψ) dx,
(2.7)
RN
for each n ∈ N. Since F (y) = 0 if |y| 1 or |y| 2, it holds that |nF (nψ)||ψ| 2F ∞ in RN for every n ∈ N. On the other hand, if ψ(x)(x) = 0, then there exists n0 ∈ N such that, for every n n0 , |ψ(x)(x)| > 2n−1 . Thus, nF (nψ(x)(x)) = 0 for all n n0 . It follows that nF (nψ)ψ tends to 0 in a dominated way as n → +∞. Hence, passing to the limit in the first and last sides of (2.7), and taking into account that the supports of all the involved functions are contained in the support of ζ , we get λ
ζ |ψ| dx
RN
RN
|ψ| div(Q∇ζ ) dx + RN
|ψ|bˆ · ∇ζ dx =
|ψ|Aζ dx.
(2.8)
RN
Next, let ζ : R → [0, 1] be a decreasing C 1 -function such that ζ (s) = 1 if s 1 and ζ (s) = 0 if s 2, and define ζn := H ( logn V ) for every n ∈ N. Since V blows up as |x| → +∞, each function ζn belongs to Cc1 (RN ). Moreover, ζn 1, limn→+∞ ζn = 1 pointwise in RN . A straightforward computation shows that Aζn =
AV log V log V (log V )2 log V (Q∇V ) · ∇V ζ ζ + n n V log V n n2 V 2 (log V )2
log V log V (Q∇V ) · ∇V ζ , − n n V 2 log V
for every n ∈ N. Hence, Aζn pointwise tends to 0 in RN as n → +∞. Moreover, observing that ζ ( Vn ), ζ ( Vn ) vanish if logn V ∈ / [1, 2], and V e, we get
1244
A. Albanese et al. / Journal of Functional Analysis 256 (2009) 1238–1257
AV + 4ζ ∞ + 2ζ ∞ (Q∇V ) · ∇V , |Aζn | 2ζ ∞ V log V V 2 log V
for every n ∈ N. Then, by replacing ζ with ζn in (2.8), recalling that the functions (V log V )−1 AV and (V 2 log V )−1 (Q∇V ) · ∇V belong to Lp (RN , μ) and applying the dominated convergence theorem, we obtain that λ
|ψ| dx 0,
RN
which, of course, implies that ψ = 0 μ-a.e. Since Hypothesis 2.1(iii)-(b) implies that the operator (A − pα , Cc∞ (RN )) is dissipative in Lp (RN , μ) (also in the case p = 1) (see, e.g., [14, Lemma 1.8 in Appendix B]), and the space ( pα + 1 − A)(Cc∞ (RN )) is dense in Lp (RN , μ), the closure of the operator (A − pα , Cc∞ (RN )) generates a strongly continuous semigroup of contractions in Lp (RN , μ) by the well-known Lumer–Phillips’ generation theorem (see, e.g., [15, Chapter 2, Theorem 3.15]). Therefore, the closure of the operator (A, Cc∞ (RN )) generates a strongly continuous semigroup on Lp (RN , μ). Finally, by [14, Lemma 1.9 in Appendix B] such a semigroup is also sub-Markov. This completes the proof. 2 Remark 2.3. Theorem 2.2 to be applied requires to prove the integrability of two suitable functions with respect to the measure μ. The fact that in most the cases the measure μ is not explicit makes things difficult. A strategy to prove the integrability of the functions in (2.2) consists in comparing them with functions which we know a priori that are in some Lp space related to the measure μ. For instance this is the case when the operator A admits a Lyapunov function, i.e., when there exists a positive smooth function ϕ diverging to +∞ as |x| → +∞, such that Aϕ tends to −∞ as |x| → +∞. Indeed, in this situation, the functions ϕ and Aϕ are integrable with respect to the measure μ. 2.1. An example In this subsection we provide the reader with a class of elliptic operators with unbounded coefficients to which Theorem 2.2 applies. We assume that the coefficients of the operator A satisfy conditions (i) and (ii) in Hypothesis 2.1. Let V ∈ C 1 (RN ) be any function such that V (x) 2 for any x ∈ RN and V (x) := 2 exp (δ|x|β ) for any x ∈ RN with |x| 1, where β and δ are positive constants. Further, assume that
b(x) · x < 0, (2.9) lim sup CΛ(x) + |x|β |x|→+∞ for some C > 0, where Λ(x) denotes the maximum eigenvalue of the matrix Q(x). By [27, Proposition 2.4], V is a Lyapunov function for the operator A defined in (2.1) if δ < β −1 C, i.e., 0 < V ∈ C 2 (RN ), V (x) → +∞ and AV (x) → −∞ as |x| → +∞. Therefore, by Khas’minskii theorem (see [19, Chapter 3, Theorem 5.1], see also [29, Theorem 6.3] or [5, Section 8.1.2]) A admits a unique invariant measure μ whose density is a positive and continuous function by
A. Albanese et al. / Journal of Functional Analysis 256 (2009) 1238–1257
1245
[8, Lemma 1.1], and the functions V and AV are integrable with respect to μ for δ < β −1 C by [27, Proposition 2.4]. Lemma 2.4. Let 1 p < +∞ and suppose that βδ < C. If lim sup |x|β−2 Λ(x)e
− pδ |x|β
|x|→+∞
< +∞,
(2.10)
then, the function (V 2 log V )−1 ((Q∇V ) · ∇V ) belongs to Lp (RN , μ). β
Proof. Since V (x) = 2eδ|x| for |x| 1, then ∇V (x) = δβ|x|β−2 xV (x) for such x’s. Hence, 2 Q(x)∇V (x) · ∇V (x) = δ 2 β 2 |x|2β−4 Q(x)x · x V (x) 2 δ 2 β 2 |x|2β−2 Λ(x) V (x) , for any |x| 1. Consequently, (Q(x)∇V (x)) · ∇V (x) δβ 2 |x|β−2 Λ(x), (V (x))2 log V (x)
|x| 1.
(2.11)
Using condition (2.10) it follows easily that the right-hand side of (2.11) can be estimated from above by K(V (x))1/p for some positive constant K. Since V ∈ L1 (RN , μ), the assertion follows. 2 Lemma 2.5. Suppose that βδ < C. Then, the function (V log V )−1 AV belongs to L1 (RN , μ). Moreover, let 1 < p < +∞ and assume that condition (2.10) is satisfied and lim sup
|x|→+∞
|b(x) · x| |x|2+β(p −1) exp (δ(p
− 1)|x|β )
< +∞,
(2.12)
where p is the conjugate exponent of p. Then the function (V log V )−1 AV belongs to Lp (RN , μ). Proof. Since, by assumptions, V 2, we can estimate (V log V )−1 AV log(4) −1 |AV |. Since AV ∈ L1 (RN , μ), it follows immediately that the function (V log V )−1 AV is in L1 (RN , μ) as well. Now, let us consider the case when p > 1. To prove that (V log V )−1 AV belongs to p L (RN , μ), we will show that such a function can be estimated from above by M|AV |1/p for some positive constant M or, equivalently, that the function (V log V )−p |AV | is bounded. For this purpose, we observe that
1246
A. Albanese et al. / Journal of Functional Analysis 256 (2009) 1238–1257
Tr(Q(x)) δβ(β − 2)(Q(x)x) · x + (AV )(x) = βδ|x|β−1 V (x) |x| |x|3 b(x) · x + δβ|x|β−3 Q(x)x · x , + |x| for any |x| 1. Since AV < 0 for large |x|, it follows
Tr(Q(x)) b(x) · x |(AV )(x)| β − − −1 −1) p p 1+β(p β |x| |x| (V (x) log V (x)) δ |x| exp(δ(p − 1)|x| ) (β − 2)(Q(x)x) · x − (2.13) − δβ|x|β−3 Q(x)x · x . 3 |x| We now consider the cases 0 < β < 2 and β 2 separately. Case β 2. Since all the terms in (2.13) are negative but the second one, we can estimate |(AV )(x)| |b(x) · x| β , p −1 (V (x) log V (x))p δ |x|2+β(p −1) exp(δ(p − 1)|x|β )
(2.14)
for |x| sufficiently large. From condition (2.12), it now follows immediately that the function (V log V )−p |AV | is bounded. Case 0 < β < 2. By (2.9), (2.10) and (2.13) we obtain, for a suitable κ > 0 and large |x|, |(AV )(x)| (V (x) log V (x))p
(Q(x)x) · x |b(x) · x| 1 + (2 − β) p −1 1+β(p −1) |x| |x|3 δ |x| exp(δ(p − 1)|x|β ) 1 β b(x) · x + (2 − β)Λ(x) p −1 2+β(p −1) β δ |x| exp(δ(p − 1)|x| )
exp( pδ |x|β ) β 1 p −1 2+β(p −1) . b(x) · x + (2 − β)κ |x|β−2 δ |x| exp(δ(p − 1)|x|β ) β
As p − 1 > p1 , it follows that the function (V log V )−p |AV | is bounded by using again condition (2.12). 2 In view of Theorem 2.2, we have proved the following result. Proposition 2.6. Let p 1 and let us assume that the diffusion and the drift coefficients of the opp erator A satisfy conditions (i) and (ii) in Hypothesis 2.1 and bi ∈ Lloc (RN , dx) (i = 1, . . . , N ). Further, assume that conditions (2.9), (2.10) and (2.12) are satisfied (the latter one only in the case when p > 1). Then, the closure of the operator (A, Cc∞ (RN )) generates a strongly continuous semigroup of contractions in Lp (RN , μ).
A. Albanese et al. / Journal of Functional Analysis 256 (2009) 1238–1257
1247
Remark 2.7. It is worth stressing that Proposition 2.6 covers also some situation in which the diffusion and the drift coefficients of A have some exponential growth at infinity. This is an improvement of [14, Theorem 2.3, p. 67] where it is required that (Q(x)x) · x < +∞. 4 2 |x|→+∞ |x| (log |x|)
(2.15)
lim sup
For instance, Theorem 2.2 applies to the operator A defined on smooth functions φ by 1
(Aφ)(x) = e 2 |x| φ − e|x| x · ∇φ(x), 2
2
x ∈ RN ,
which, of course, does not satisfy condition (2.15). RN , μ) uniqueness 3. A second criterion for Lp (R In this section we give a second criterion which guarantees that the operator (A, Cc∞ (RN )) uniquely extends to Lp (RN , μ) by a strongly continuous semigroup. As we will see in Section 3.1 this criterion is particularly useful in the case when the measure μ is symmetrizing for the operator A. Hypothesis 3.1. (i) Hypothesis 2.1 is satisfied. 1,r (ii) The density of μ belongs to Wloc (RN , dx), where r is the same exponent as in Hypothesis 2.1(i). Remark 3.2. Observe that if the measure μ is infinitesimally invariant for the operator A, i.e., Aφ μ(dx) = 0,
φ ∈ Cc∞ RN ,
RN
by [7, Corollary 2.10], the density of the measure μ with respect to the Lebesgue measure 1,r (RN , dx). Hence Hypothesis 3.1(ii) is satisfied. belongs to Wloc For our purposes, it is much more convenient to deal with operators whose principal part is in divergence form. Hence, we write A in the following equivalent way: Aψ =
N 1 1 Di (qij Dj ψ) + β · ∇ψ = div (Q∇ψ) + β · ∇ψ = A0 ψ + β · ∇ψ, i,j =1
r N for any smooth function ψ, where βj = bj − 1 N i=1 Di (qij ) ∈ Lloc (R , dx) (j = 1, . . . , n) as the functions qij (i, j = 1, . . . , N ) are locally Hölder continuous and is locally uniformly positive. The following lemma is essential in what follows.
1248
A. Albanese et al. / Journal of Functional Analysis 256 (2009) 1238–1257
Lemma 3.3. Let r be the conjugate exponent of r. Then, for any nonnegative function φ ∈ W 1,r (RN , dx) with compact support, we have
(β · ∇φ) μ(dx) α
RN
(3.1)
φ μ(dx).
RN
Proof. We begin the proof observing that an integration by parts shows that
f A0 φ μ(dx) =
RN
div(Q∇φ)f dx = −
RN
(Q∇φ) · ∇f dx = −
RN
(Q∇φ) · ∇f μ(dx),
RN
for every f ∈ C 1 (RN ) and every φ ∈ Cc∞ (RN ). In particular, taking f ≡ 1 we get A0 φ μ(dx) = 0.
(3.2)
RN
Taking Hypothesis 2.1(iii) into account and integrating both the sides of the equality β · ∇φ = Aφ − A0 φ, we get (3.1) for all nonnegative functions φ ∈ Cc∞ (RN ). To prove (3.1) in the general case, it suffices to observe that any nonnegative function φ ∈ W 1,r (RN ) with compact support is the limit in W 1,r (RN ) of a sequence {φn } ∈ Cc∞ (K) of nonnegative functions, where K is a suitable neighborhood of the support of φ, and use the dominated convergence theorem. 2 We can now prove the following result. Theorem 3.4. Let 1 < p < +∞. Assume that Hypothesis 3.1 holds and there exists a positive function V ∈ C 1 (RN ) such that lim|x|→+∞ V (x) = +∞ and β · ∇V −C V log V
and
(Q∇V ) · ∇V ∈ L∞ RN , dx , (V log V )2
(3.3)
for some positive constant C. Then, the closure of the operator (A, Cc∞ (RN )) on Lp (RN , μ) generates a sub-Markovian strongly continuous semigroup. In particular, (A, Cc∞ (RN )) is Lp (RN , μ) unique. Proof. Since V (x) tends to +∞ as |x| → +∞, up to replacing V with V + c for a suitable positive constant c, we can assume, without loss of generality, that V (x) > 1 for any x ∈ RN . Let p be the conjugate exponent of p. Fix λ > α + p1 and let g ∈ Lp (RN , μ) be such that (λφ − Aφ)g μ(dx) = 0, RN
for every φ ∈ Cc∞ (RN ). We claim that g ≡ 0.
A. Albanese et al. / Journal of Functional Analysis 256 (2009) 1238–1257
1249
1,r By [7, Corollary 2.10], the function F := g belongs to Wloc (RN , dx). In particular, since N r > N , F is continuous and locally bounded in R . Hence, g is locally bounded in RN as ∈ Lrloc (RN , dx). well, being everywhere positive and continuous in RN . Moreover ∇ 1 = − ∇ 2 1,r Therefore, g = F · 1 ∈ Wloc (RN , dx). Let us now fix φ ∈ Cc∞ (RN ) and observe that
φg μ(dx) =
λ RN
(Aφ)g μ(dx)
RN
=−
(Q∇φ) · ∇g μ(dx) +
RN
(β · ∇φ)g μ(dx).
(3.4)
RN
By density, the equality (3.4) extends to every φ ∈ W 1,r (RN , dx) ∩ L∞ (RN , dx) with 1,2 1,r (RN , dx) ⊂ Wloc (RN , dx), formula (3.4) holds true for every compact support. Since Wloc 1,2 N ∞ N φ ∈ W (R , dx) ∩ L (R , dx) with compact support. Next, let ζ : R → [0, 1] be a decreasing C 1 -function such that ζ (s) = 1 if s 1 and ζ (s) = 0 if s 2, and define ζn := ζ ( logn V ) for every n ∈ N. Since V (x) tends to +∞ as |x| → +∞, each function ζn belongs to Cc1 (RN ). Moreover, ζn 1 and ζn converges to 1 pointwise in RN as n → +∞. We now consider the cases p 2 and p > 2 separately.
Case 1 < p 2. Let us set φn = ζn2 sign(g)|g|p −1 for any n ∈ N. Let us observe that both the functions φn and φn g belong to W 1,2 (RN , dx) ∩ L∞ (RN , dx) by the local boundedness of g, and are compactly supported in RN . From (3.4) we obtain λ
ζn2 sign(g)|g|p −1 g μ(dx) + (p
RN
=−
2 n
− 1)
ζn2 |g|p −2 (Q∇g) · ∇g μ(dx)
RN
(Q∇V ) · ∇g μ(dx) + ζn ζ n−1 log V sign(g)|g|p −1 V
RN
=−
2 n
(Q∇V ) · ∇g μ(dx) ζn ζ n−1 log V sign(g)|g|p −1 V
p − 1 + p −
2 n
1 β · ∇(φn g) μ(dx) + p
RN
α p
β · ∇ζn2 |g|p μ(dx)
RN
(Q∇V ) · ∇g μ(dx) ζn ζ n−1 log V sign(g)|g|p −1 V
RN
+
(β · ∇φn )g μ(dx)
RN
RN
ζn2 |g|p μ(dx) + RN
2 p n
RN
β · ∇V μ(dx). ζn ζ n−1 log V |g|p V
1250
A. Albanese et al. / Journal of Functional Analysis 256 (2009) 1238–1257
In the second equality and in the last inequality, we have used, respectively, the formula p − 1 2 p 1 p 2 g∇ ζn2 sign(g)|g|p −1 = ∇ ζn |g| + |g| ∇ζn p p and Lemma 3.3, being φn g 0. Hence,
α λ− ζn2 sign(g)|g|p −1 g μ(dx) + (p − 1) ζn2 |g|p −2 (Q∇g) · ∇g μ(dx) p RN
−
2 n
(Q∇V ) · ∇g μ(dx) ζn ζ n−1 log V sign(g)|g|p −1 V
RN
2 p n
+
RN
β · ∇V μ(dx). ζn ζ n−1 log V |g|p V
(3.5)
RN
Note that both the two terms in the first side of (3.5) are positive and the last side of (3.5) is finite. It follows that the function ζn2 |g|p −2 (Q∇g) · ∇g belongs to L1 (RN , μ). Taking this fact into account, we can now estimate −1 2 ζn ζ n log V sign(g)|g|p −1 (Q∇V ) · ∇g μ(dx) n V RN
2 n
1
1 ((Q∇V ) · ∇V ) 2 (Q∇g) · ∇g 2 μ(dx) ζn ζ n−1 log V |g|p −1 V
RN
=2
ζn |g|
p −2 2
(Q∇g) · ∇g
1
2
1
p ((Q∇V ) · ∇V ) 2 1 −1 ζ n log V |g| 2 n V
μ(dx)
RN
ζn2 |g|p −2 (Q∇g) · ∇g μ(dx)
RN
1 + 2 n
−1 ζ n log V 2 |g|p (Q∇V ) · ∇V μ(dx). V2
(3.6)
RN
Hence, replacing (3.6) into (3.5), we get
α ζn2 sign(g)|g|p −1 g μ(dx) + (p − 2) ζn2 |g|p −2 (Q∇g) · ∇g μ(dx) λ− p RN
1 n2
−1 ζ n log V 2 |g|p (Q∇V ) · ∇V μ(dx) V2
RN
+
RN
2 p n
RN
β · ∇V μ(dx). ζn ζ n−1 log V |g|p V
(3.7)
A. Albanese et al. / Journal of Functional Analysis 256 (2009) 1238–1257
1251
Since ζ 0, ζn2 |g|p −2 (Q∇g) · ∇g 0 and p 2, from (3.7) and the conditions (3.3), we get
λ−
α p
ζn2 |g|p μ(dx)
1 n2
RN
−1 ζ n log V 2 |g|p (Q∇V ) · ∇V μ(dx) V2
RN
+
1 n2
2 p n
β · ∇V ζn ζ n−1 log V |g|p μ(dx) V
RN
−1 ζ n log V 2 |g|p (Q∇V ) · ∇V μ(dx) V2
RN
2C + pn
ζn ζ n−1 log V |g|p log V μ(dx).
(3.8)
RN
Observing that ζ (n−1 log V (x)) = 0 if V (x) ∈ / [en , e2n ] and taking conditions (3.3) into account, we can estimate 2C −1 4C ζn ζ n log V |g|p log V ζ ∞ |g|p pn p and 2 p (Q∇V ) · ∇V 1 −1 2 (Q∇V ) · ∇V ζ n log V |g| 4ζ ∞ |g|p . n2 V2 (V log V )2 ∞
Since |g|p ∈ L1 (RN , μ), we can apply Fatou’s lemma in (3.8) which yields
λ−
α p
|g|p μ(dx) 0, RN
thereby implying that g ≡ 0. p −2
Case p > 2. For all m ∈ N, let φm (x) = x(x 2 + m−1 ) 2 for any x ∈ R. Clearly, φm ∈ C ∞ (R). For all n, m ∈ N, set un,m = ζn2 φm (g), where ζn is as above. Then, un,m has compact support and is bounded by the local boundedness of g. Moreover, as 2 ∇V ∇un,m = ζn ζ n−1 log V φm (g) + ζn2 φm (g)∇g, n V
n, m ∈ N,
(3.9)
1,r 1,2 and g ∈ Wloc (RN , dx), the function un,m belongs to Wloc (RN , dx) for any n, m ∈ N. Hence, we can apply (3.4), replacing φ with un,m , which yields
1252
A. Albanese et al. / Journal of Functional Analysis 256 (2009) 1238–1257
λ
un,m g μ(dx) + (p − 1)
RN
p −4 2 1 ζn2 g 2 g 2 + (Q∇g) · ∇g μ(dx) m
RN
p −4 2 1 ζn2 g 2 + (Q∇g) · ∇g μ(dx) m
1 + m
RN
2 =− n
p −2
2 (Q∇V ) · ∇g −1 1 2 μ(dx) ζn ζ n log V g g + m V
RN
+
(β · ∇un,m )g μ(dx).
RN
Since ζn2 (g 2 +
1 p 2−4 (Q∇g) · ∇g m)
0 for any m, n ∈ N, we obtain
un,m g μ(dx) + (p − 1)
λ RN
ζn2 g 2
p −4 2 1 2 g + (Q∇g) · ∇g μ(dx) m
RN
2 − n
p −2
2 (Q∇V ) · ∇g −1 1 2 μ(dx) ζn ζ n log V g g + m V
RN
+
(β
· ∇g)ζn2
p −4 2 1 1 2 g + (p − 1)g 2 + g μ(dx) m m
RN
+
p −2 2 1 β · ∇ζn2 g 2 g 2 + μ(dx). m
(3.10)
RN
Observe that for every m, n ∈ N, we have p −2 ζn ζ n−1 log V |g| g 2 + m−1 2 (Q∇V ) · ∇g V −1 ζn |g|p −1 ζ ∞ |Q∇V |L∞ (supp(ζ )) |∇g|, n
(3.11)
and, since p < 2, p −4 p −2 (β · ∇g)g ζ 2 g 2 + m−1 2 (p − 1)g 2 + m−1 (β · ∇g)g ζ 2 g 2 + m−1 2 n
n
p −1 (β · ∇g) g 2 + m−1 2 . Moreover,
(3.12)
A. Albanese et al. / Journal of Functional Analysis 256 (2009) 1238–1257
p −2 p β · ∇ζ 2 g 2 g 2 + m−1 2 β · ∇ζ 2 g 2 + 1 2 . n n
1253
(3.13)
Note that the right-hand sides of (3.11)–(3.13) belong to L1 (RN , μ). Therefore, letting m tend to +∞ in both the sides of (3.10), by the Fatou’s lemma and dominated convergence we get λ
ζn2 |g|p
RN
2 n
μ(dx) + (p − 1)
ζn ζ n−1 log V |g|p −1 V −1 (Q∇V ) · ∇g μ(dx)
RN
2 n
ζn ζ n−1 log V |g|p V −1 (β · ∇V ) μ(dx)
RN
+ (p − 1) =
2 n
ζn2 |g|p −2 (Q∇g) · ∇g μ(dx)
RN
+
ζn2 sign(g)|g|p −1 (β · ∇g) μ(dx)
RN
ζn ζ n−1 log V |g|p −1 V −1 (Q∇V ) · ∇g μ(dx)
RN
+ (p − 1)
(β · ∇g)ζn2 |g|p −2 g μ(dx) +
RN
β · ∇ζn2 |g|p μ(dx).
RN
On the other hand, using Young inequality, we get 2 −1 ζn ζ n log V |g|p −1 V −1 (Q∇V ) · ∇g n 1 1 2 ζn ζ n−1 log V |g|p −1 V −1 (Q∇V ) · ∇V 2 (Q∇g) · ∇g 2 n
1 1 p −1 p −2 1 −1 2 2 2 2 ζ n log V |g| V (Q∇g) · ∇g (Q∇V ) · ∇V 2 ζn |g| n 2 1 (Q∇V ) · ∇V εζn2 |g|p −2 (Q∇g) · ∇g + 2 ζ n−1 log V |g|p , εn V2 for any ε > 0. Therefore, taking ε = p − 1 we get λ RN
ζn2 |g|p μ(dx)
1 (p − 1)n2
RN
+ (p − 1) + RN
−1 2 (Q∇V ) · ∇V ζ n log V |g|p μ(dx) V2
(β · ∇g)ζn2 |g|p −1 sign(g) μ(dx)
RN
β · ∇ζn2 |g|p μ(dx)
1254
A. Albanese et al. / Journal of Functional Analysis 256 (2009) 1238–1257
=
1 (p − 1)n2
−1 2 (Q∇V ) · ∇V ζ n log V |g|p μ(dx) V2
RN
1 β · ∇ |g|p ζn2 μ(dx) + 1− p RN
β · ∇ζn2 |g|p μ(dx).
+
(3.14)
RN
Arguing as in the case p 2, we can show that
β · ∇ |g|p ζn2 μ(dx) α
RN
|g|p ζn2 μ(dx) −
RN
β · ∇ζn2 |g|p μ(dx),
(3.15)
RN
for any n ∈ N. Hence, replacing (3.15) into (3.14), observing that ζ (n−1 log V ) pointwise vanishes as n → +∞ and applying Fatou’s lemma, we obtain that g ≡ 0. Thus (λ − A)(Cc∞ (RN )) is dense in Lp (RN , μ). The assertion now follows as in the proof of Theorem 2.2. 2 Proposition 3.5. Assume that Hypothesis 3.1 holds and there exists a strictly positive function V ∈ C 1 (RN ) such that lim|x|→+∞ V (x) = +∞ and β · ∇V ∈ L1 RN , μ and V log V
(Q∇V ) · ∇V ∈ L1 R N , μ . 2 (V log V )
Then, the closure of the operator (A, Cc∞ (RN )) on L1 (RN , μ) generates a sub-Markovian strongly continuous semigroup. In particular, (A, Cc∞ (RN )) is L1 (RN , μ) unique. Proof. Repeating verbatim the proof of Theorem 3.4 with p = 2, we can show that
α λ− 2
2 (log V )2 −1 (Q∇V ) · ∇V ζ n log V |g|2 μ(dx) 2 n (V log V )2
ζn2 |g|2 μ(dx) RN
RN
+
log V β · ∇V ζn ζ n−1 log V |g|2 μ(dx). n V log V
(3.16)
RN −2 The integrability assumptions on (V log V )−1 (β · V ) and (V log V ) (Q∇V ) · ∇V allow us to apply the Fatou’s lemma and still conclude from (3.8) that RN g 2 μ(dx) = 0, so that g ≡ 0. 2
3.1. The case of symmetrizing invariant measures In this subsection, we consider the case when the measure μ is symmetrizing for the operator A, i.e., the case when β ≡ 0 in RN . In this case, μ is an infinitesimally invariant measure for A (see (3.2)) and we can prove the following result.
A. Albanese et al. / Journal of Functional Analysis 256 (2009) 1238–1257
1255 p
Proposition 3.6. Fix p > 1. Assume that Hypothesis 2.1(i)–(iii) holds, bi ∈ Lloc (RN , dx) (i = 1, . . . , N) and that μ is a symmetrizing invariant measure for the operator A. Let Λ(x) denote the maximum eigenvalue of the matrix Q(x) for any x ∈ RN , and set λ(s) = max|x|=s Λ(x) for any s 0. Finally, assume that λ−1/2 is not integrable in a neighborhood of +∞. Then, the closure of the operator (A, Cc∞ (RN )) on Lp (RN , μ) generates a Markov strongly continuous semigroup. Proof. As it has been observed in Remark 3.2, the density of μ with respect to the Lebesgue 1,r (RN , dx). Thus, Hypothesis 3.1 is satisfied. Let V : RN → R be any measure belongs to Wloc positive C 1 -function such that V (x) = e
|x| 0
λ−1/2 (s) ds
for any x ∈ RN with |x| 1. Then,
(Q(x)x) · x 1 (Q(x)∇V (x)) · ∇V (x) = |x| , |x| 2 −1/2 2 −1/2 (V (x) log V (x))2 λ(|x|)|x| ( 0 λ (s) ds) ( 0 λ (s) ds)2 Hence, conditions (3.3) are satisfied and the assertion follows from Theorem 3.4.
|x| 1. 2
In the following corollary we specialize our result to the case when A is a generalized Schrödinger operator, i.e., in the case when Aφ = φ +
∇ · ∇φ,
(3.17)
on smooth functions φ. 1,p
Corollary 3.7. Let p > 1 and let ∈ Wloc (RN , dx) be locally uniformly positive. If there exists r > N such that ∇/ ∈ Lrloc (RN , dx), then the closure of the operator (A, Cc∞ (RN )) on Lp (RN , μ) generates a Markov strongly continuous semigroup, where μ(dx) = dx. If μ is finite, then the result holds also for p = 1. Proof. It suffices to apply Proposition 3.6 for p > 1, and Proposition 3.5 for p = 1, with V (x) = |x| for large |x|. 2 Remark 3.8. Some remarks are in order. (i) The one-dimensional case has been completely characterized by Eberle in [14]. (ii) Corollary 3.7 allows us to cover also some situations to which [14, Theorem 2.6] does not apply. Indeed, in the case when is not integrable with respect to the Lebesgue measure, the invariant measure μ(dx) = dx may not satisfy the condition lim sup r→+∞
1 rk
(x) dx < +∞,
(3.18)
Br
whatever k > 0 may be, which was one of the main requirement of [14, Theorem 2.6]. For instance, the condition (3.18) is not satisfied when is any smooth function such that 2 (x) = e|x| for large |x|.
1256
A. Albanese et al. / Journal of Functional Analysis 256 (2009) 1238–1257
Similarly, in the case when the measure μ is finite, our result generalizes the quoted theorem by Eberle for large p’s. Indeed, another important requirement in that theorem is that r N the function β = ∇ belongs to Lloc (R , dx) for some r > (1 + N/2)p. Since, by our r r N N assumptions Lloc (R , dx) = Lloc (R , dx), our result extends [14, Theorem 2.6] when (1 + N/2)p > N . (iii) In the case when p = 2 we recover the same result as in [6, Theorem 7]. (iv) A result similar to Corollary 3.7 has been proved by Liskevi˘c [20] and Liskevi˘c and Semenov [21] for p > 3/2. Our result generalizes the results by Liskevi˘c and Semenov in the sense that, differently from them, we do not assume any global integrability conditions and our result holds true for any p > 1. (v) Finally, we point out that a detailed discussion of L1 -uniqueness has been given by Stannat in [32]. Acknowledgment The authors thank G. Metafune for having suggested some improvements in our paper. References [1] A. Albanese, E. Mangino, Cores for Feller semigroups with an invariant measure, J. Differential Equations 225 (2006) 361–377. [2] A. Albanese, E. Mangino, Corrigendum to “Cores for Feller semigroups with an invariant measure”, J. Differential Equations 244 (11) (2008) 2980–2982. [3] D.G. Aronson, P. Besala, Uniqueness of the positive solutions of parabolic equations with unbounded coefficients, Colloq. Math. 18 (1967) 126–135. [4] D.G. Aronson, P. Besala, Parabolic equations with unbounded coefficients, J. Differential Equations 3 (1967) 1–14. [5] M. Bertoldi, L. Lorenzi, Analytical Methods for Markov Semigroups, Chapman & Hall/CRC, Boca Raton, FL, 2007. [6] V.I. Bogachev, N.V. Krylov, M. Röckner, Elliptic regularity and essential self-adjointness of Dirichlet operators on Rn , Ann. Sc. Norm. Super. Pisa Cl. Sci. 24 (1997) 451–461. [7] V.I. Bogachev, N.V. Krylov, M. Röckner, On regularity of transition probabilities and invariant measures of singular diffusions under minimal conditions, Comm. Partial Differential Equations 26 (2001) 2037–2080. [8] V.I. Bogachev, M. Röckner, A generalization of Haminskii’s theorem on existence of invariant measures for locally integrable drifts, Theory Probab. Appl. 45 (2001) 363–378. [9] P. Cannarsa, V. Vespri, Generation of analytic semigroups by elliptic operators with unbounded coefficients, SIAM J. Math. Anal. 18 (3) (1987) 857–872. [10] P. Cannarsa, V. Vespri, Generation of analytic semigroups in the Lp -topology by elliptic operators in RN , Israel J. Math. 61 (3) (1988) 235–255. [11] G. Cupini, S. Fornaro, Maximal regularity in Lp (RN ) for a class of elliptic operators with unbounded coefficients, Differential Integral Equations 17 (2004) 259–296. [12] G. Da Prato, A. Lunardi, On the Ornstein–Uhlenbeck operator in spaces of continuous functions, J. Funct. Anal. 131 (1995) 94–114. [13] G. Da Prato, A. Lunardi, Elliptic operators with unbounded drift coefficients and Neumann boundary condition, J. Differential Equations 198 (1) (2004) 35–52. [14] A. Eberle, Uniqueness and Non-uniqueness of Semigroups Generated by Singular Diffusion Operators, Lecture Notes in Math., vol. 1718, Springer-Verlag, Berlin, 1999. [15] K.J. Engel, R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, 2000. [16] S. Fornaro, L. Lorenzi, Generation results for elliptic operators with unbounded diffusion coefficient in Lp and Cb spaces, Discrete Contin. Dyn. Syst. Ser. A 18 (4) (2007) 747–772. [17] F. Gozzi, R. Monte, V. Vespri, Generation of analytic semigroups and domain characterization for degenerate elliptic operators with unbounded coefficients arising in financial mathematics. I, Differential Integral Equations 15 (9) (2002) 1085–1128.
A. Albanese et al. / Journal of Functional Analysis 256 (2009) 1238–1257
1257
[18] S. Ito, Fundamental solutions of parabolic differential equations and boundary value problems, Jpn. J. Math. 27 (1957) 55–102. [19] R.Z. Khas’minskii, Stochastic Stability of Differential Equations, Nauka, 1969 (in Russian); English transl.: Sijthoff Noordhoof, Alphen aan den Rijn, 1980. [20] V.A. Liskevi˘c, Smoothness estimates and uniqueness for the Dirichlet operators, in: Mathematical Results in Quantum Mechanics, Internat. Conference, Blossin, Germany, Birkhäuser, Basel, 1994. [21] V.A. Liskevi˘c, Y.A. Semenov, Dirichlet operators: A priori estimates and the uniqueness problem, J. Funct. Anal. 109 (1992) 199–213. [22] L. Lorenzi, A. Lunardi, Elliptic operators with unbounded diffusion coefficients in L2 spaces with respect to invariant measures, J. Evol. Equ. 6 (2006) 691–709. [23] A. Lunardi, V. Vespri, Generation of strongly continuous semigroups by elliptic operators with unbounded coefficients in Lp (RN ), Rend. Istit. Mat. Univ. Trieste 28 (1997) 251–279. [24] G. Metafune, J. Prüss, A. Rhandi, R. Schnaubelt, The domain of the Ornstein–Uhlenbeck operator on an Lp -space with an invariant measure, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) I (2002) 471–485. [25] G. Metafune, D. Pallara, J. Prüss, R. Schnaubelt, Lp -theory for elliptic operators on Rn with singular coefficients, Z. Anal. Anwend. 24 (2005) 497–521. [26] G. Metafune, J. Prüss, A. Rhandi, R. Schnaubelt, Lp -regularity for elliptic operators with unbounded coefficients, Adv. Differential Equations 10 (10) (2005) 1131–1164. [27] G. Metafune, D. Pallara, A. Rhandi, Global properties of invariant measures, J. Funct. Anal. 223 (2005) 396–424. [28] G. Metafune, D. Pallara, V. Vespri, Lp -estimates for a class of elliptic operators with unbounded coefficients in RN , Houston J. Math. 31 (2) (2005) 605–620. [29] G. Metafune, D. Pallara, M. Wacker, Feller semigroups on RN , Semigroup Forum 65 (2) (2002) 159–205. [30] J. Prüss, A. Rhandi, R. Schnaubelt, The domain of elliptic operators on Lp (Rd ) with unbounded drift coefficients, Houston J. Math. 32 (2006) 563–576. [31] P. Rabier, Elliptic problems in RN with unbounded coefficients in classical Sobolev spaces, Math. Z. 249 (2005) 1–30. [32] W. Stannat, (Nonsymmmetric) Dirichlet operators on L1 : Existence, uniqueness and associated Markov processes, Ann. Sc. Norm. Super. Pisa Cl. Sci. 28 (4) (1999) 99–140.
Journal of Functional Analysis 256 (2009) 1258–1268 www.elsevier.com/locate/jfa
Finitely strictly singular operators between James spaces Isabelle Chalendar a , Emmanuel Fricain a , Alexey I. Popov b , Dan Timotin c , Vladimir G. Troitsky b,∗ a Université de Lyon, Université Lyon 1, INSA de Lyon, Ecole Centrale de Lyon, CNRS, UMR5208,
Institut Camille Jordan, 43 bld. du 11 novembre 1918, F-69622 Villeurbanne cedex, France b Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB T6G 2G1, Canada c Institute of Mathematics of the Romanian Academy, PO box 1-764, Bucharest 014700, Romania
Received 2 May 2008; accepted 11 September 2008 Available online 2 October 2008 Communicated by N. Kalton
Abstract An operator T : X → Y between Banach spaces is said to be finitely strictly singular if for every ε > 0 there exists n such that every subspace E ⊆ X with dim E n contains a vector x such that T x < εx. We show that, for 1 p < q < ∞, the formal inclusion operator from Jp to Jq is finitely strictly singular. As a consequence, we obtain that the strictly singular operator with no invariant subspaces constructed by C. Read is actually finitely strictly singular. These results are deduced from the following fact: if k n then every k-dimensional subspace of Rn contains a vector x with x∞ = 1 such that xmi = (−1)i for some m1 < · · · < mk . © 2008 Elsevier Inc. All rights reserved. Keywords: Strictly singular operator; Zigzag vector; James space; Invariant subspace
* Corresponding author.
E-mail addresses: [email protected] (I. Chalendar), [email protected] (E. Fricain), [email protected] (A.I. Popov), [email protected] (D. Timotin), [email protected] (V.G. Troitsky). 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.09.010
I. Chalendar et al. / Journal of Functional Analysis 256 (2009) 1258–1268
1259
1. Introduction Recall that an operator T : X → Y between Banach spaces is said to be strictly singular if for every ε > 0 and every infinite-dimensional subspace E ⊆ X there is a vector x in the unit sphere of E such that T x < ε. Furthermore, T is said to be finitely strictly singular if for every ε > 0 there exists n ∈ N such that for every subspace E ⊆ X with dim E n there exists a vector x in the unit sphere of E such that T x < ε. Finitely strictly singular operators are also known in literature as superstrictly singular. Note that compact
⇒
finitely strictly singular
⇒
strictly singular,
and that each of these three properties defines a closed subspace in L(X, Y ). Actually, each property defines an operator ideal. We refer the reader to [1,7,9–11,13] for more information about strictly and finitely strictly singular operators. All the Banach spaces in this paper are assumed to be over real scalars. We say that a subspace E ⊆ X is invariant under an operator T : X → X if {0} = E = X and T (E) ⊆ E. Every compact operator has invariant subspaces by [2]. On the other hand, Read constructed in [12] an example of a strictly singular operator without nontrivial closed invariant subspaces (this answered a question of Pełczy´nski). Read’s operator acts on an infinite direct sum which involves James spaces. Recall that James’ p-space Jp is a sequence space consisting of all sequences x = (xn )∞ n=1 in c0 satisfying xJp < ∞ where xJp = sup
n−1
1/p |xki+1 − xki | : 1 k1 < · · · < kn , n ∈ N p
i=1
is the norm in Jp . For more information on James’ spaces we refer the reader to [3,6–8,14]. It was an open question whether every finitely strictly singular operator has invariant subspaces. Some partial results in this direction were obtained in [1,11]. We answer this question in the negative by showing that the operator in [12] is, in fact, finitely strictly singular. As an intermediate result, we prove that the formal inclusion operator from Jp to Jq with 1 p < q < ∞ is finitely strictly singular. The latter statement in a certain sense refines the result of Milman [9] that the formal inclusion operator from p to q with 1 p < q < ∞ is finitely strictly singular. Milman’s proof is based on the fact that every k-dimensional subspace E of Rn contains a vector “with a flat,” namely, a vector x with sup-norm one with (at least) k coordinates equal in modulus to 1. For such a vector, one has xq xp . The proofs of our results are based on the following refinement of this observation. We will show that x can be chosen so that these k coordinates have alternating signs. For such a “highly oscillating” vector x one has xJq
xJp . More precisely, a finite or infinite sequence of real numbers in [−1, 1] will be called a zigzag of order k if it has a subsequence of the form (−1, 1, −1, 1, . . .) of length k. Our results will be based on the following theorem; two different proofs of it will be presented in Sections 2 and 3. Theorem 1. For every k n, every k-dimensional subspace of Rn contains a zigzag of order k. Corollary 2. Let k ∈ N; then every k-dimensional subspace of c0 contains a zigzag of order k.
1260
I. Chalendar et al. / Journal of Functional Analysis 256 (2009) 1258–1268
Proof. Let F be a subspace of c0 with dim F = k. For every n ∈ N, define Pn : c0 → Rn via n Pn : (xi )∞ i=1 → (xi )i=1 . Let n1 be such that dim Pn1 (F ) = k. There exists n2 such that every vector in F attains its norm on the first n2 coordinates. Indeed, define g : F \ {0} → N via g(x) = max{i: |xi | = x∞ }. Then g is upper semi-continuous, hence bounded on the unit sphere of F , so that we put n2 = max{g(x): x ∈ F, x = 1}. Put n = max{n1 , n2 }. Since Pn (F ) is a k-dimensional subspace of Rn , by Theorem 1 there exists x ∈ F such that Pn x is a zigzag of order k. It follows from our definition of n that x is a zigzag of order k in F . 2 Suppose that 1 p < q. Since xJp is defined as the supremum of p -norms of certain sequences, · q · p implies · Jq · Jp . It follows that Jp ⊆ Jq and the formal inclusion operator ip,q : Jp → Jq has norm 1. We show next that it is finitely strictly singular. The proof is analogous to that of Proposition 3.3 in [13]. The main difference, though, is that we use Corollary 2 instead of the simpler lemma from [9,13]. Theorem 3. If 1 p < q < ∞ then the formal inclusion operator ip,q : Jp → Jq is finitely strictly singular. Proof. Given any x ∈ Jp , then |xi − xj |q (2x∞ )q−p |xi − xj |p for every i, j ∈ N, so that 1− p
p
1
−1
xJq (2x∞ ) q xJqp . Fix an arbitrary ε > 0. Let k ∈ N be such that (k − 1) p q > 1ε . Suppose that E is a subspace of Jp with dim E = k. By Corollary 2, there is a zigzag z ∈ E of 1
order k. By the definition of norm in Jp , we have zJp 2(k − 1) p . Put y =
z zJp
− p1
. Then y ∈ E with yJp = 1. Obviously, y∞ 12 (k − 1) ip,q (y)
1
Jq
= yJq (k − 1) q
Hence, ip,q is finitely strictly singular.
− p1
, so that
p
yJqp < ε.
2
We will now use Theorem 3 to show that the strictly singular operator T constructed by Read in [12] is finitely strictly singular. Let us briefly outline those properties of T that will be relevant for our investigation. The underlying space X for this operator is defined as the 2 -direct sum of 2 and Y , X = (2 ⊕ Y )2 , where Y itself is the 2 -direct sum of an infinite sequence of Jp -spaces Y = ( ∞ i=1 Jpi )2 , with (pi ) a certain strictly increasing sequence in (2, +∞). The operator T is a compact perturbation of 0 ⊕ W1 , where W1 : Y → Y acts as a weighted right shift, that is, W1 (x1 , x2 , x3 , . . .) = (0, β1 x1 , β2 x2 , β3 x3 , . . .),
xi ∈ Jpi ,
with βi → 0. Note that one should rather write βi ipi ,pi+1 xi instead of βi xi . Clearly, it suffices to show that W1 is finitely strictly singular. For n ∈ N, define Vn : Y → Y via Vn (x1 , x2 , x3 , . . .) = (0, β1 x1 , . . . , βn xn , 0, 0 . . .),
xi ∈ Jpi .
I. Chalendar et al. / Journal of Functional Analysis 256 (2009) 1258–1268
1261
It follows from βi → 0 that Vn − W1 → 0. Since finitely strictly singular operators from Y to Y form a closed subspace of L(Y ), it suffices to show that Vn is finitely strictly singular for every n. Given n ∈ N, one can write Vn =
n
βi ji+1 ipi ,pi+1 Pi ,
i=1
where Pi : Y → Jpi is the canonical projection and ji : Jpi → Y is the canonical inclusion. Thus, Vn is finitely strictly singular because finitely strictly singular operators form an operator ideal. This yields the following result. Theorem 4. Read’s operator T is finitely strictly singular. In the remaining two sections, we present two different proofs of Theorem 1, one based on combinatorial properties of polytopes and the other based on the geometry of the set of all zigzags and algebraic topology. 2. Proof of Theorem 1 via combinatorial properties of polytopes By a polytope in Rk we mean a convex set which is the convex hull of a finite set. A set is a polytope iff it is bounded and can be constructed as the intersection of finitely many closed half-spaces. A facet of P is a face of (affine) dimension k − 1. We refer the reader to [5,15] for more details on properties of polytopes. A polytope P is centrally symmetric iff it can be represented as the absolutely convex hull of its vertices, that is, P = conv{±u¯ 1 , . . . , ±u¯ n } where ±u¯ 1 , . . . , ±u¯ n are the vertices of P . Clearly, P is centrally symmetric iff it can be represented as the intersection of finitely many centrally symmetric “bands.” More precisely, there are vectors a¯ 1 , . . . , a¯ m ∈ Rk such that u¯ ∈ P iff −1 u, ¯ a¯ i 1 for all i = 1, . . . , m, and the facets of P are described by {u ∈ P : u, ¯ a¯ i = 1} or {u ∈ P : u, ¯ −a¯ i = 1} as i = 1, . . . , m. A simplex in Rk is the convex hull of k + 1 points with non-empty interior. A polytope P in k R is simplicial if all its faces are simplexes (equivalently, if all the facets of P are simplexes). Every polytope can be perturbed into a simplicial polytope by an iterated “pulling” procedure, see e.g., [5, Section 5.2] for details. We will outline a slight modification of the procedure such that it preserves the property of being centrally symmetric. Suppose that P is a centrally symmetric polytope with vertices, say ±u¯ 1 , . . . , ±u¯ n (see Fig. 1). Pull u¯ 1 “away from” the origin, but not too far, so that it does not reach any affine hyperplane spanned by the facets of P not containing u¯ 1 ; denote the resulting point u¯ 1 . Let Q = conv{u¯ 1 , −u¯ 1 , ±u¯ 2 , . . . , ±u¯ n }. By [5, 5.2.2, 5.2.3] this procedure does not affect the facets of P not containing u¯ 1 , while all the facets of Q containing u¯ 1 become pyramids having apex at u¯ 1 . Note that no facet of P contains both u¯ 1 and −u¯ 1 . Hence, if we put R = conv{±u¯ 1 , ±u¯ 2 , . . . , ±u¯ n }, then, by symmetry, all the facets of R containing −u¯ 1 become pyramids with apex at −u¯ 1 , while the rest of the facets (in particular, the facets containing u¯ 1 ) are not affected. Now iterate this procedure with every other pair of opposite vertices. Let P be the resulting polytope, P = conv{±u¯ 1 , . . . , ±u¯ n }. Clearly, P is centrally symmetric and simplicial as in [5, 5.2.4]. It also follows from the construction that if F is a facet of P then all the vertices of P corresponding to the vertices of F belong to the same facet of P .
1262
I. Chalendar et al. / Journal of Functional Analysis 256 (2009) 1258–1268
Fig. 1. Pulling out the first pair of vertices.
Fig. 2. Examples of marked polytopes in R2 and R3 .
We will call a polytope P marked if the following assumptions are satisfied: (i) P is simplicial, centrally symmetric, and has a non-empty interior. (ii) Every vertex is assigned a natural number, called its index, such that two vertices have the same index iff they are opposite to each other. (iii) All the vertices of P are painted in two colors, say, black and white, so that opposite vertices have opposite colors. See Fig. 2 for examples of marked polytopes. A face of a marked polytope is said to be happy if, when one lists its vertices in the order of increasing indices, the colors of the vertices alternate. For example, the front top facet of the marked polytope in the right-hand side of Fig. 2 is happy. See Fig. 3 for more examples of happy faces. We will reduce Theorem 1 to the claim that every marked polytope has a happy facet, which we will prove afterwards. Suppose that k n and E is a subspace of Rn with dim E = k. Let {b¯1 , . . . , b¯k } be a basis of E. We need to find a linear combination of these vectors x¯ := a1 b¯1 + · · · + ak b¯k such that x¯ is a zigzag. Let B be the n × k matrix with columns b¯1 , . . . , b¯k , and
I. Chalendar et al. / Journal of Functional Analysis 256 (2009) 1258–1268
1263
Fig. 3. Examples of happy simplexes in R2 and R3 .
let u¯ 1 , . . . , u¯ n be the rows of B. If a¯ = (a1 , . . . , ak ), then xi = u¯ i , a ¯ as i = 1, . . . , n. Thus, it suffices to find a¯ ∈ Rk such that the vector ( u¯ i , a) ¯ ni=1 is a zigzag of order k. Let P be the centrally symmetric convex polytope spanned by u¯ 1 , . . . , u¯ n , i.e., P = conv{±u¯ 1 , . . . , ±u¯ n }. Then some of the ±u¯ i ’s will be the vertices of P , while the others might end up inside P . Suppose that ±u¯ m1 , . . . , ±u¯ mr are the vertices of P , so that P = conv{±u¯ m1 , . . . , ±u¯ mr }. Following the “pulling” procedure that was described before, construct a simplicial centrally symmetric polytope P = conv{±u¯ m1 , . . . , ±u¯ mr }. Every vertex of P is either u¯ mi or −u¯ mi for some i. Paint the vertex white in the former case and black in the latter case; assign index i to this vertex. This way we make P into a marked polytope. We claim that happy facets of P correspond to zigzags. Indeed, suppose that P has a happy facet. Then this facet (or the facet opposite to it) is spanned by some −u¯ mi , u¯ mi , 1 2 −u¯ mi , u¯ mi , etc., for some 1 i1 < · · · < ik r. It follows that −u¯ mi1 , u¯ mi2 , −u¯ mi3 , u¯ mi4 , 3 4 etc., are all contained in the same facet of P . Hence, they are contained in an affine hyperplane, say L, such that P “sits” between L and −L. Let a¯ be the vector defining L, that is, L = {u: ¯ u, ¯ a ¯ = 1}. Since P is between L and −L, we have −1 u, ¯ a ¯ 1 for every u¯ in P . In particular, −1 xi = u¯ i , a ¯ 1 for i = 1, . . . , n. On the other hand, it follows from −u¯ mi1 , u¯ mi2 , −u¯ mi3 , u¯ mi4 , . . . ∈ L that xmi1 = −1, xmi2 = 1, xmi3 = −1, xmi4 = 1, etc. Hence, x¯ is a zigzag of order k. Thus, to complete the proof, it suffices to show that every marked polytope has a happy facet. Throughout the rest of this section, P will be a marked polytope in Rk ; Fj stands for the set of all j -dimensional faces of P for j = 0, . . . , k − 1. In particular, Fk−1 is the set of all facets of P , while F0 is the set of all vertices of P . By [5, 3.1.6], every (k − 2)-dimensional face E of P is contained in exactly two facets, say F and G; in this case E = F ∩ G. Suppose that R ⊆ Fk−1 . For E ∈ Fk−2 , we say that E is a boundary face of R if E = F ∩ G for some facets F and G such that F ∈ R and G ∈ / R. The set ˜ Clearly, of all boundary faces of R will be referred to as the face boundary of R and denoted ∂R. ˜ ⊂ Fk−2 . If F is a single facet, we put ∂F ˜ = ∂{F ˜ }. Clearly, ∂F ˜ is the set of all the facets of F . ∂R For a face F of P we define its color code to be the list of the colors of its vertices in the order of increasing indices. For example, the color codes of the simplexes in Fig. 3 are (wbw) and (bwbw). Here b and w correspond to “black” and “white” respectively. A face in P will be said to be a b-face if its color code starts with b and a w-face otherwise. Lemma 5. Suppose that F is a facet of P . The following are equivalent: (i) F is happy; ˜ contains exactly one happy b-face; (ii) ∂F ˜ has an odd number of happy b-faces. (iii) ∂F
1264
I. Chalendar et al. / Journal of Functional Analysis 256 (2009) 1258–1268
Proof. Note that since F is a simplex, every face of F can be obtained by dropping one vertex of F and taking the convex hull of the remaining vertices. Hence, the color code of the face is obtained by dropping one symbol from the color code of F . (i) ⇒ (ii). Suppose that F is happy, then its color code is either (bwbw . . .) or (wbwb . . .). In the former case, the only happy b-face of F is obtained by dropping the last vertex, while in the latter case the only happy b-face of F is obtained by dropping the first vertex. (ii) ⇒ (iii). Trivial. ˜ has an odd number of happy b-faces. Let E be a happy b-face (iii) ⇒ (i). Suppose that ∂F ˜ . Then the color code of E is the sequence (bwbw . . .) of length k − 1. The color code in ∂F of F is obtained by inserting one extra symbol into this sequence. Note that inserting the extra symbol should not result in two consecutive b’s or w’s, as in this case F would have exactly two happy b-faces (corresponding to removing each of the two consecutive symbols), which would contradict the assumption. Hence, the color code of F should be an alternating sequence, so that F is happy. 2 Lemma 6. For every R ⊆ Fk−1 , the number of happy facets in R and the number of happy ˜ have the same parity. b-faces in ∂R Proof. For R ⊆ Fk−1 , define the parity of R to be the parity of the number of happy b-faces in ˜ Observe that if R and S are two disjoints subsets of Fk−1 , then the parity of R ∪ S is the ∂R. sum of the parities of R and S (mod 2). It follows that the parity of R is the sum of the parities of all of the facets that make up R (mod 2). But this is exactly the parity of the number of happy facets in R by Lemma 5. 2 For every face F of P we write −F for the opposite face. If R is a set of facets, we write −R = {−F : F ∈ R}. Also, we write R for the set theoretic union of all the facets in R. Theorem 7. Every marked polytope has a happy facet. Proof. We will prove a stronger statement: every marked polytope in Rk has an odd number of happy b-facets. The proof is by induction on k. For k = 1, the statement is trivial. Let k > 1 and let P be a marked polytope in Rk . For every facet F , let n¯ F be the normal vector of F , directed outwards of P . Fix a vector v¯ of length one such that v¯ is not parallel to any of the facets of P (equivalently, not orthogonal to n¯ F for any facet F ); it is easy to see that such a vector exists. By rotating P we may assume without loss of generality that v¯ = (0, . . . , 0, 1). Let T be the projection from Rk to Rk−1 such that T : (x1 , . . . , xk−1 , xk ) → (x1 , . . . , xk−1 ). We can think of T as the orthogonal projection onto the “horizontal” hyperplane {x¯ ∈ Rk : xk = 0} in Rk . Let Q = T (P ). Since T is linear and surjective, Q is again a centrally symmetric convex polytope in Rk−1 with a non-empty interior (see Fig. 4). It follows from our choice of v¯ that the kth coordinate of n¯ F is non-zero for every facet F . Let R be the set of all the facets of P that “face upward,” that is, R = {F ∈ Fk−1 : the kth coordinate of n¯ F is positive}. Clearly, a facet F is in −R iff the kth coordinate of n¯ F is negative. Hence, −R ∩ R = ∅ and ˜ = ∂(−R); ˜ ˜ is centrally symmetric. Clearly, every hence ∂R −R ∪ R = Fk−1 . Observe that ∂R vertical line (i.e., a line parallel to v) ¯ that intersects the interior of P meets the boundary of P at
I. Chalendar et al. / Journal of Functional Analysis 256 (2009) 1258–1268
1265
Fig. 4. The images T (P ) of the polytopes in Fig. 2.
exactly two points and meets the interior of Q at exactly one point. It follows that the restriction of T to R is a bijection between R and Q. The same is also true for −R. Therefore, the ˜ is a face-preserving bijection between ∂R ˜ and the boundary of Q. restriction of T to ∂R ˜ Under this bijection, the faces in ∂R correspond to the facets of Q. Hence, this bijection induces a structure of a marked polytope on the boundary of Q, making Q into a marked polytope. It follows, by the induction hypothesis, that the boundary of Q has an odd number of happy b˜ has an odd number of happy b-faces. It follows from Lemma 6 that R has an facets. Hence, ∂R odd number of happy facets. Let m and be the numbers of all happy b-facets and w-facets in R, respectively. Then m + is odd. Observe that F is a happy b-facet iff −F is a happy w-facet. It follows that −R contains happy b-facets and m happy w-facets. Thus, the total number of happy b-facets of P is m + , which we proved to be odd. 2 3. Proof of Theorem 1 via algebraic topology n and S n−1 be, respectively, the unit ball and the unit sphere Fix a natural number n and let B∞ ∞ n n n n−1 = {x ∈ Rn : max |x | = 1}. For k 1 we of ∞ , i.e., B∞ = {x ∈ R : max |xi | 1} and S∞ i define
n Γk = x ∈ B∞ : x has at least k alternating coordinates ±1 ,
n A+ k = x ∈ B∞ : x has at least k alternating coordinates ±1, starting with 1 , + A− k = −Ak . + − n n Note that A− k is exactly the set of all zigzags of order k in R . Put also A0 = A0 = Γ0 = B∞ . ± n−1 For k 1, Γk , Ak ⊂ S∞ and we have − A+ k ∪ Ak = Γk , − A+ k ∩ Ak = Γk+1 .
Note that the first relation above is true also for k = 0. We start with a simple lemma. Lemma 8. Suppose p is a real polynomial of degree m, and there are m + 2 real numbers t1 < t2 < · · · < tm+2 , such that p(ti ) 0 for i odd and p(ti ) 0 for i even. Then p ≡ 0.
1266
I. Chalendar et al. / Journal of Functional Analysis 256 (2009) 1258–1268
Proof. We do induction with respect to m. If m = 0, the result is obvious. If the lemma has been proved up to m − 1, and p is a polynomial of degree m, then p has at least one real root s. We write p(t) = (t − s)q(t), and q (or −q) has a similar property, with respect to at least m − 1 values ti —so we can apply induction. 2 Lemma 9. There exists a sequence of subspaces πk ⊂ Rn , πk ⊃ πk+1 , dim πk = n − k, such that, if Pk is the orthogonal projection onto πk , then Pk |A+ k is injective. Proof. For 1 j n we define the vectors ζ j ∈ Rn by the formula ζi = i j −1 . One checks easily that the ζ j ’s are linearly independent. Define π0 = Rn , and, for k 1, πk = (span{ζ 1 , . . . , ζ k })⊥ . Suppose that x, y ∈ A+ k , and Pk x = Pk y. There exist scalars α1 , . . . , αk , such that x − y = k j j =1 αj ζ . We have indices 1 r1 < · · · < rk n and 1 s1 < · · · < sk n, such that xrl = ysl = (−1)l−1 . It follows that xrl − yrl 0 for l odd and 0 for l even, while xsl − ysl 0 for l odd and 0 for l even. Let the polynomial p of degree k − 1 be given by p(t) = kj =1 αj t j −1 . If rl = sl for all l, we obtain j αj ζrl = αj rl j −1 = 0 j
j
j
for all l = 1, . . . , k. Thus p has k distinct zeros; it must be identically 0, whence x = y. Suppose now that we have rl = sl for at least one index l. We claim then that among the union of the indices rl and sl we can find ι1 < ι2 < · · · < ιk+1 , such that xιl − yιl have alternating signs. This can be achieved by induction with respect to k. For k = 1 we must have r1 = s1 , so we may take ι1 = min{r1 , s1 }, ι2 = max{r1 , s1 }. For k > 1, there are two cases. If r1 = s1 , we take ι1 = r1 = s1 and apply the induction hypothesis to obtain the rest. If r1 = s1 , we take ι1 as the lesser of the two and ι2 as the other one, and then we continue “accordingly” to ι2 (that is, taking as ι’s the rest of r’s if ι2 = r1 and the rest of s’s if ι2 = s1 ). Now, the way ιl have been chosen implies that p(t) defined above satisfies the hypotheses of Lemma 8: it has degree k − 1 and the values it takes in ι1 , . . . , ιk+1 have alternating signs. It must then be identically 0, which implies x = y. 2 + − Since A− k = −Ak , it follows that Pk |Ak is also injective.
Lemma 10. If πk , Pk are obtained in Lemma 9, then Δk := Pk (Γk ) is a balanced, convex subset of πk , with 0 as an interior point (in πk ). Moreover, Δk = Pk (A− k )= ) and ∂Δ = P (Γ ) (the boundary in the relative topology of π ). Pk (A+ k k k+1 k k Proof. We will use induction with respect to k. The statement is immediately checked for k = 0 n−1 = Γ ). (note that P0 = IRn and ∂Δ0 = S∞ 1 Assume the statement true for k; we will prove its validity for k + 1. By the induction hypothesis, we have Δk+1 = Pk+1 Pk (Γk+1 ) = Pk+1 ∂Δk = Pk+1 Δk
I. Chalendar et al. / Journal of Functional Analysis 256 (2009) 1258–1268
1267
and is therefore a balanced, convex subset of πk+1 , with 0 as an interior point. −1 −1 (y) ∩ ∂Δk contains a single point. Then Pk+1 (y) ∩ Δk also Take then y ∈ Δ˚ k+1 . Suppose Pk+1 −1 contains a single point, and therefore Pk+1 (y) ∩ πk is a support line for the convex set Δk . This line is contained in a support hyperplane (in πk ); but then the whole of Δk projects onto πk+1 on one side of this hyperplane, and thus y belongs to the boundary of this projection. Therefore y cannot be in Δ˚ k+1 . −1 (y) ∩ ∂Δk contains at least two points. But The contradiction obtained shows that Pk+1
− ∂Δk = Pk (Γk+1 ) = Pk A+ k+1 ∪ Pk Ak+1 whence
− Pk+1 (∂Δk ) = Pk+1 A+ k+1 ∪ Pk+1 Ak+1 . Since Pk+1 restricted to each of the two terms in the right-hand side is injective by Lemma 9, − there exists a unique z+ ∈ A+ k+1 such that y = Pk+1 z+ and a unique z− ∈ Ak+1 such that y = Pk+1 z− . −1 − + (y) ∩ ∂Δk . Then either x ∈ Pk (A+ Take x ∈ Pk+1 k+1 ) or x ∈ Pk (Ak+1 ). If x ∈ Pk (Ak+1 ) then x = Pk z for some z ∈ A+ k+1 , so that y = Pk+1 x = Pk+1 z, which yields z = z+ ; hence x = Pk z+ . −1 Similarly, if x ∈ Pk (A− k+1 ) then x = Pk z− . It follows that Pk+1 (y) ∩ ∂Δk ⊆ {Pk z+ , Pk z− }. Since −1 −1 (y) ∩ ∂Δk contains at least two points, we conclude that Pk+1 (y) ∩ ∂Δk = {Pk z+ , Pk z− } Pk+1 and Pk z+ = Pk z− . It follows from y = Pk+1 z± that Δ˚ k+1 ⊂ Pk+1 (A± k+1 ). But, Δk+1 being a closed convex set with a nonempty interior, it is the closure of its interior Δ˚ k+1 ; since the two sets on the right are closed, we have actually Δk+1 = Pk+1 (A± k+1 ). We want to show now that ∂Δk+1 = Pk+1 (Γk+2 ). Suppose first that y ∈ Pk+1 (Γk+2 ) = − + − ˚ Pk+1 (A+ k+1 ∩ Ak+1 ); that is, y = Pk+1 z with z ∈ Ak+1 ∩ Ak+1 . Clearly, y ∈ Δk+1 . If y ∈ Δk+1 , ± then, defining z+ and z− as before, the injectivity of Pk+1 on Ak+1 implies z = z− = z+ . This contradicts Pk z+ = Pk z− ; consequently, y ∈ ∂Δk+1 . − Conversely, take y ∈ ∂Δk+1 = ∂(Pk+1 (Δk )). Again, take z+ ∈ A+ k+1 , z− ∈ Ak+1 , such that Pk+1 z+ = Pk+1 z− = y. We have then Pk z+ ∈ ∂Δk (if Pk z+ ∈ Δ˚ k , then Pk+1 z+ = Pk+1 Pk z+ must be in the interior of Pk+1 Δk , which is Δ˚ k+1 ). Similarly, Pk z− ∈ ∂Δk . If Pk z+ = Pk z− , then Pk+1 applied to the whole segment [Pk z+ , Pk z− ] is equal to y. − Therefore the segment belongs to ∂Δk . Since ∂Δk = Pk (A+ k+1 ∪ Ak+1 ), there exist two val+ − ues x1 , x2 either both in Ak+1 or both in Ak+1 , such that Pk x1 , Pk x2 ∈ [Pk z+ , Pk z− ], and thus Pk+1 x1 = Pk+1 x2 = y. This contradicts the injectivity of Pk+1 on A± k+1 . Therefore Pk z+ = Pk z− . But z+ and z− both belong to A+ , on which Pk is injective. It k − ∩ A = Γ , and P z = y. This ends the proof. 2 follows that z+ = z− ∈ A+ k+2 k+1 + k+1 k+1 The main consequence of Lemma 10, in combination with Lemma 9, is the fact that the linear map Pk−1 maps homeomorphically Γk into ∂Δk−1 , which is the boundary of a convex, balanced set, containing 0 in its interior. Proof of Theorem 1. As noted above, Pk−1 maps homeomorphically Γk onto the boundary of a x , we obtain convex, balanced set, containing 0 in its interior. Composing it with the map x → x n−k a homeomorphic map φ from Γk to S , which satisfies the relation φ(−x) = −φ(x).
1268
I. Chalendar et al. / Journal of Functional Analysis 256 (2009) 1258–1268
Suppose that E is a k-dimensional subspace of Rn with no zigzags. Then E ∩ Γk = ∅, so that the projection of Γk onto E ⊥ does not contain 0. Composing this projection with the map x , we obtain a continuous map from ψ : Γk → S n−k−1 , that satisfies ψ(−x) = −ψ(x). x → x Then the map Φ := ψ ◦ φ −1 : S n−k → S n−k−1 is continuous and satisfies Φ(−x) = −Φ(x). This is however impossible: it is known that such a map does not exist (see, for instance, [4]). 2 Remark 11. In Theorem 1, the alternating sequence (−1, 1, −1, 1, . . .) cannot generally be replaced with another “pattern,” i.e., another sequence of length k of ±1’s. Indeed, suppose that the pattern has two consecutive 1’s, say, in positions r and r + 1. Let E be the subspace of Rn defined by the relations xr + xr+1 + xr+2 = 0 and xi = 0 whenever r + 3 i n − k + r + 1. n has the required pattern. Then dim E = k and it is easy to see that no vector in E ∩ B∞ On the other hand, it follows easily from Theorem 1 that for every subspace E ⊆ Rn with n with any given pattern of length k. Generally, dim E = 2k − 1, one can find a vector in E ∩ B∞ 2k − 1 is a sharp estimate, as the following example shows. Consider the pattern (1, 1, . . . , 1) of length k. Consider the subspace E ⊂ Rn consisting of all the vectors whose first 2k − 1 coordinates add up to zero, and the remaining coordinates are zero. Then dim E = 2k − 2 and n contains no vectors conforming to the pattern. E ∩ B∞ Acknowledgments The authors would like to thank Charles J. Read and Jonathan R. Partington for useful discussions, and the anonymous referee for suggesting Remark 11 to us. References [1] G. Androulakis, P. Dodos, G. Sirotkin, V.G. Troitsky, Classes of strictly singular operators and their products, Israel J. Math., in press. [2] N. Aronszajn, K.T. Smith, Invariant subspaces of completely continuous operators, Ann. of Math. (2) 60 (1954) 345–350. [3] P. Casazza, R. Lohman, A general construction of spaces of the type of R.C. James, Canad. J. Math. 27 (6) (1975) 1263–1270. [4] W. Fulton, Algebraic Topology, Springer-Verlag, New York, 1995. [5] B. Grünbaum, Convex Polytopes, second ed., Grad. Texts in Math., vol. 221, Springer-Verlag, New York, 2003. [6] R.C. James, A non-reflexive Banach space isometric with its second conjugate, Proc. Natl. Acad. Sci. USA 37 (1951) 174–177. [7] J. Lindenstrauss, L. Tsafriri, Classical Banach Spaces. I. Sequence Spaces, Springer-Verlag, Berlin, 1977. [8] V. Maslyuchenko, A. Plichko, Quasireflexive locally convex spaces without Banach subspaces, Teor. Funktsiˇı Funktsional. Anal. i Prilozhen. 44 (1985) 78–84 (in Russian); translation in: J. Soviet Math. 48 (3) (1990) 307–312. [9] V.D. Milman, Operators of class C0 and C0∗ , Teor. Funktsiˇı Funktsional. Anal. i Prilozhen. 10 (1970) 15–26. [10] A. Plichko, Superstrictly singular and superstrictly cosingular operators, in: Functional Analysis and Its Applications, in: North-Holland Math. Stud., vol. 197, Elsevier, Amsterdam, 2004, pp. 239–255. [11] A. Popov, Schreier singular operators, Houston J. Math., in press. [12] C.J. Read, Strictly singular operators and the invariant subspace problem, Studia Math. 3 (132) (1999) 203–226. [13] B. Sari, Th. Schlumprecht, N. Tomczak-Jaegermann, V.G. Troitsky, On norm closed ideals in L(p ⊕ q ), Studia Math. 179 (3) (2007) 239–262. [14] I. Singer, Bases in Banach Spaces I, Springer-Verlag, New York, 1970. [15] G.M. Ziegler, Lectures on Polytopes, Grad. Texts in Math., vol. 152, Springer-Verlag, New York, 1994.
Journal of Functional Analysis 256 (2009) 1269–1298 www.elsevier.com/locate/jfa
Fokker–Planck equations and maximal dissipativity for Kolmogorov operators with time dependent singular drifts in Hilbert spaces V.I. Bogachev a,∗ , G. Da Prato b , M. Röckner c,d a Department of Mechanics and Mathematics, Moscow State University, 119991 Moscow, Russia b Scuola Normale Superiore di Pisa, Piazza dei Cavalieri 7, I-56125 Pisa, Italy c Fakultät für Mathematik, Universität Bielefeld, D-33501 Bielefeld, Germany d Department of Mathematics and Statistics, Purdue University, W. Lafayette, IN 47907, USA
Received 5 May 2008; accepted 9 May 2008 Available online 20 June 2008 Communicated by Paul Malliavin
Abstract We consider a Kolmogorov operator L0 in a Hilbert space H , related to a stochastic PDE with a timedependent singular quasi-dissipative drift F = F (t, ·) : H → H , defined on a suitable space of regular functions. We show that L0 is essentially m-dissipative in the space Lp ([0, T ] × H ; ν), p 1, where ν(dt, dx) = νt (dx) dt and the family (νt )t∈[0,T ] is a solution of the Fokker–Planck equation given by L0 . As a consequence, the closure of L0 generates a Markov C0 -semigroup. We also prove uniqueness of solutions to the Fokker–Planck equation for singular drifts F . Applications to reaction–diffusion equations with time-dependent reaction term are presented. This result is a generalization of the finite-dimensional case considered in [V. Bogachev, G. Da Prato, M. Röckner, Existence of solutions to weak parabolic equations for measures, Proc. London Math. Soc. (3) 88 (2004) 753–774], [V. Bogachev, G. Da Prato, M. Röckner, On parabolic equations for measures, Comm. Partial Differential Equations 33 (3) (2008) 397–418], and [V. Bogachev, G. Da Prato, M. Röckner, W. Stannat, Uniqueness of solutions to weak parabolic equations for measures, Bull. London Math. Soc. 39 (2007) 631–640] to infinite dimensions. © 2008 Elsevier Inc. All rights reserved. Keywords: Kolmogorov operators; Stochastic PDEs; Singular coefficients; Parabolic equations for measures; Fokker–Planck equations; Maximal dissipativity
* Corresponding author.
E-mail address: [email protected] (V.I. Bogachev). 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.05.005
1270
V.I. Bogachev et al. / Journal of Functional Analysis 256 (2009) 1269–1298
1. Introduction Given a separable Hilbert space H (with norm | · | and inner product ·,·), we denote the space of all linear bounded operators in H by L(H ) and the set of all Borel probability measures on H by P(H ). We study non-autonomous stochastic equations on H of the type √ dX(t) = AX(t) + F t, X(t) dt + C dW (t), (1.1) X(s) = x ∈ H, t s, where A : D(A) ⊂ H → H is the infinitesimal generator of a C0 -semigroup etA in H , C is a linear positive definite operator in H and F : D(F ) ⊂ [0, T ] × H → H is such that F (t, ·) is quasi-dissipative for all t ∈ [0, T ] (see Sections 2 and 3 for the precise assumptions). The case where no further regularity assumptions are made on F turns out to be very difficult because of the lack of parabolic regularity results in infinite dimensions. No existence (and uniqueness) results for solutions of (1.1) are known in this very general situation, in particular, when C is not of trace class. Therefore, in order to get a first grip on the dynamics described by (1.1), we study the corresponding Kolmogorov operator L on [0, T ] × H with the aim to prove that it generates a C0 -semigroup on a Banach space B of functions on [0, T ] × H . This semigroup is just the space–time homogenization of the family Ps,t , 0 s t T , of transition probabilities of the solution to (1.1) (if it exists), i.e. Ps,t solve the Chapman–Kolmogorov equation corresponding to L. The restriction L0 of the Kolmogorov operator L to an initial domain of nice functions, specified below, is given on [0, T ] × H , with T > 0 fixed, as follows: ∂ u(t, x) + N (t)u(t, x), ∂t
(1.2)
1 Tr CDx2 u(t, x) + x, A∗ Dx u(t, x) + F (t, x), Dx u(t, x) 2
(1.3)
(L0 u)(t, x) = where N(t)u(t, x) =
and A∗ is the adjoint of A. In order to define the initial domain of L0 we introduce some functional spaces. We denote the linear span of all real and imaginary parts of functions eix,h where h ∈ D(A∗ ) by EA (H ). Moreover, for any φ ∈ C 1 ([0, T ]) such that φ(T ) = 0 and any h ∈ C 1 ([0, T ]; D(A∗ )) we consider the function uφ,h (t, x) = φ(t)eix,h(t) ,
t ∈ R, x ∈ H,
and denote by EA ([0, T ] × H ) the linear span of all real and imaginary parts of such functions uφ,h . We shall define the operator L0 on the space D(L0 ) := EA ([0, T ] × H ). Our strategy to achieve the above described goal is the following: Step 1. Choose the Banach space B as B := Lp [0, T ] × H ; ν ,
V.I. Bogachev et al. / Journal of Functional Analysis 256 (2009) 1269–1298
1271
for p 1, with ν an appropriate measure on [0, T ] × H of the form ν(dt dx) = νt (dx) dt, where νt are probability measures in H . It turns out that appropriate are all measures ν of the above type such that for some α > 0
L0 u dν α
[0,T ]×H
u dν,
∀u ∈ D(L0 ), u 0.
(1.4)
[0,T ]×H
Then it follows that L0 is quasi-dissipative on Lp ([0, T ] × H ; ν), hence closable. Let Lp denote its closure. So, the first task is to find such measures. One way to do this is to solve the Fokker– Planck equation corresponding to L0 (i.e. the dual of the Kolmogorov equation). The resulting measure satisfies (1.4) with α = 0. Step 2. Prove that Lp is maximal-dissipative on Lp ([0, T ] × H ; ν). Hence it generates a C0 semigroup eτ Lp , τ 0, on Lp ([0, T ] × H ; ν) which turns out to be Markov. Then eτ Lp , τ 0, is the desired space–time homogenization of the transition probabilities Ps,t , 0 s t T , of the process that (if it exists) should solve (1.1). In this paper we realize both steps above, but emphasize that though this is already quite hard work, it constitutes only a partial result. It would be desirable to prove that eτ Lp is given by a probability kernel on [0, T ] × H and thus also get Ps,t as probability kernels on H . And furthermore one should prove the existence of a weak solution to (1.1) having Ps,t as transition probabilities. This second part of the programme is under study and will be the subject of forthcoming work. This paper consists of two parts, namely the case of regular F and non-regular F . In the first part of the paper (Section 2) we assume that F (t, x) is regular, see Hypothesis 2.1, and (extending [1,3] and [2] to infinite dimensions) prove that, for any ν0 ∈ P(H ), there exists a unique family of probability measures (νt )t∈[0,T ] ⊂ P(H ) with the same initial value ν0 such that they solve the Fokker–Planck equation for L0 , i.e., for each u ∈ D(L0 ) for almost all t ∈ [0, T ] one has d dt
u(t, x) νt (dx) = H
L0 u(t, x) νt (dx), H
or, equivalently, for each u ∈ D(L0 ) for almost all t ∈ [0, T ] one has
u(t, x) νt (dx) =
H
t u(0, x) ν0 (dx) +
H
L0 u(s, x) νs (dx).
(1.5)
0 H
Here we implicitly assume that the second integral on the right-hand side exists for all u ∈ D(L0 ), which is e.g. the case if |x| νt (dx) < +∞ H
1272
V.I. Bogachev et al. / Journal of Functional Analysis 256 (2009) 1269–1298
and F is Lipschitz, or if x, A∗ h , F, h ∈ L1 [0, T ] × H ; ν ,
∀h ∈ D A∗ ,
where ν(dt, dx) = νt (dx) dt. The following remark is crucial in this paper. Remark 1.1. (i) We note that even without F being regular, the relations νt (H ) = 1 for all t ∈ [0, T ] and limt→T u(t, x) = 0 for all x ∈ H along with (1.5) imply
T L0 u(t, x) νt (dx) dt = − 0 H
u(0, x) ν0 (dx),
∀u ∈ D(L0 ).
(1.6)
H
In particular, T L0 u(t, x) νt (dx) dt 0,
∀u ∈ D(L0 ), u 0.
(1.7)
0 H
(ii) If u ∈ D(L0 ), then u2 ∈ D(L0 ) and L0 u2 = 2uL0 u + |C 1/2 Dx u|2 . Hence by (1.6) we have T L0 u(t, x) u(t, x) νt (dx) dt 0 H
1 =− 2
T
1/2
C Dx u(t, x) 2 νt (dx) dt −
0 H
u2 (0, x) ν0 (dx).
(1.8)
H
If νt only satisfies (1.7) we still have T 0 H
1 L0 u(t, x) u(t, x) νt (dx) dt − 2
T
1/2
C Dx u(t, x) 2 νt (dx) dt.
(1.9)
0 H
After having established existence and uniqueness of (νt )t∈[0,T ] satisfying (1.5) in the regular case, we show that L0 is essentially m-dissipative in the space Lp ([0, T ] × H ; ν), i.e. (L0 , D(L0 )) is dissipative on Lp ([0, T ] × H ; ν) and (λ − L0 , D(L0 )) has dense range for all λ > 0. By the well-known Lumer–Phillips theorem [18] this means that the closure Lp of L0 generates a C0 -semigroup etLp , t 0, on Lp ([0, T ] × H ; ν), which in our case is even Markov. In the second part (Section 3), devoted to the case of irregular drifts, we prove (see Theorem 3.3 below) that L0 is essentially m-dissipative in Lp ([0, T ] × H ; ν) where ν(dt, dx) = νt (dx) dt and (νt )t∈[0,T ] is a suitable family of probability measures (see Hypothesis 3.1) as e.g. the solutions to the Fokker–Planck equation corresponding to L0 ; sufficient conditions for the existence of the latter have been obtained in [5], to which we refer for the proofs. However, in this paper, we prove uniqueness (see Theorem 3.6 below). Then, in Section 4, we apply the obtained
V.I. Bogachev et al. / Journal of Functional Analysis 256 (2009) 1269–1298
1273
results to reaction–diffusion equations with time-dependent coefficients. In this case existence and uniqueness for Eq. (1.1) is known. However, the m-dissipativity of its Kolmogorov operator and the uniqueness result for the Fokker–Planck equation are new. Finally, it would be interesting to prove existence and uniqueness for Eq. (1.5) when t varies on all R, generalizing results in [11,12]. This problem will be studied in a forthcoming paper. Some results of this work have been announced in our note [4]. We end this section by listing the assumptions on the linear operator A which we will assume throughout. Hypothesis 1.2. (i) There is ω ∈ R such that Ax, x ω|x|2 , ∀x ∈ D(A). (ii) C ∈ L(H ) is symmetric, nonnegative and such that the linear operator (α) Qt
t :=
∗
s −2α esA CesA ds
0
is of trace class for all t > 0 and some α ∈ (0, 1/2). t ∗ 1/2 (iii) Setting Qt := 0 esA CesA ds, one has etA (H ) ⊂ Qt (H ) for all t > 0 and there is Λt ∈ 1/2 L(H ) such that Qt Λt = etA and +∞ γλ := e−λt Λt dt < +∞, 0
where · denotes the operator norm in L(H ). We note that by our assumptions on F (see Hypothesis 2.1(ii) in the regular case and Hypothesis 3.1(ii) in the irregular case), by adding a constant times identity to F , we may assume without loss of generality that ω in Hypothesis 1.2(i) is strictly negative. We also note that Hypothesis 2.1(iii) implies that the Ornstein–Uhlenbeck operator associated to L0 (that is when F = 0) is strong Feller. This assumption is not essential but it allows to simplify several proofs below. In Appendix A we collect some results on the Ornstein–Uhlenbeck operator U ϕ(x) =
1 Tr CDx2 ϕ(x) + x, A∗ Dx ϕ(x) , 2
ϕ ∈ EA (H ), x ∈ D A∗ ,
needed throughout. In addition, we introduce the operator V0 u(t, x) = Dt u(t, x) + U u(t, x),
u ∈ EA [0, T ] × H ,
and its maximal monotone extension V (see (A.1)). Then we prove that the space EA ([0, T ] × H ) is a core (in a suitable sense) of V0 , generalizing a similar result for the operator U in [13].
1274
V.I. Bogachev et al. / Journal of Functional Analysis 256 (2009) 1269–1298
2. The case when F is regular In this section we assume that Hypothesis 2.1. (i) Hypothesis 1.2 is fulfilled. (ii) F : [0, T ] × H → H is continuous together with Dx F (t, ·) : H → L(H ) for all t ∈ [0, T ]. Moreover, there exists K > 0 such that
F (t, x) − F (t, y) K|x − y|,
x, y ∈ H, t ∈ [0, T ].
This clearly implies that x → F (t, x) − Kx is m-dissipative for any t ∈ [0, T ]. It is known (see, e.g., [14]) that, under Hypothesis 2.1, for any s 0, there exists a unique mild solution X(·, s, x) with P-a.s. H -continuous sample paths of the stochastic differential equation
√ dX(t) = AX(t) + F t, X(t) dt + C dW (t), X(s) = x ∈ H, t s,
(2.1)
where W (t), t ∈ R, is a cylindrical Wiener process in H defined on a filtered probability space (Ω, F , Ft , P). A mild solution X(t, s, x) of (2.1) is an adapted stochastic process X ∈ C([s, T ]; L2 (Ω, F , P)) such that t X(t, s, x) = e
(t−s)A
x+
e(t−r)A F r, X(r, s, x) dr + WA (t, s),
t s,
s
where WA (t, s) is the stochastic convolution: t WA (t, s) =
√ e(t−r)A C dW (r),
t s,
s
which is also P-a.s. H -continuous under our assumptions on A. In view of Hypothesis 1.2(ii), WA (t, s) is a Gaussian random variable in H with mean 0 and covariance operator Qs,t given by t Qs,t x =
∗
esA CesA x ds,
t s, x ∈ H.
s
The next result will be useful below. Lemma 2.2. For any m > 1/2 there is Cm > 0 such that for ω1 := ω − K
2m
E X(t, s, x)
Cm 1 + e−mω1 (t−s) |x|2m ,
t s.
(2.2)
V.I. Bogachev et al. / Journal of Functional Analysis 256 (2009) 1269–1298
1275
Proof. It is convenient to write Eq. (2.1) as a family of deterministic equations. Setting Y (t) = X(t, s, x) − WA (t, s), we see that Y (t) satisfies the equation Y (t) = AY (t) + F t, Y (t) + WA (t, s) , (2.3) Y (s) = x, t s, again in the mild sense. Also we set
M = sup F (t, 0) . t∈[0,T ]
Multiplying (2.3) by |Y (t)|2m−2 Y (t) and taking into account Hypothesis 2.1, yields for a suitable 1 that constant Cm
2m
2m−2
2m
1 d
Y (t) −ω Y (t) + F t, WA (t, s) , Y (t) Y (t)
2m dt
2m−2
+ F t, Y (t) + WA (t, s) − F t, WA (t, s) , Y (t) Y (t)
2m
2m−2
2m
−ω Y (t) + F t, WA (t, s) , Y (t) Y (t)
+ K Y (t)
2m
2m K − ω
1
F t, WA (t, s) . Y (t) + Cm − 2 This computation is formal, but can be made rigorous by approximation, cf. [14]. By a standard comparison result it follows that
Y (t) 2m e−mω1 (t−s) |x|2m + 2mC 1
t
m
2m e−mω1 (s−σ ) F σ, WA (σ, s) dσ,
s 2 one has and finally we find that, for some constant Cm
X(t, s, x) 2m C 2 e−mω(t−s) |x|2m m
t
2 + Cm
e
2m
2m
F σ, WA (σ, s) dσ + WA (t, s) .
−mω(σ −s)
(2.4)
s
Now the conclusion follows by taking the expectation and recalling that in view of Hypothesis 2.1 one has
F (t, x) F (t, 0) + F (t, x) − F (t, 0) M + K|x|, t ∈ [0, T ], x ∈ H, and using the fact that (see [15]) sup
t∈[0,T ],ts
The proof is complete.
2
2m E WA (t, s) < +∞.
1276
V.I. Bogachev et al. / Journal of Functional Analysis 256 (2009) 1269–1298
We define the transition evolution operator Ps,t ϕ(x) = E ϕ X(t, s, x) ,
t s, ϕ ∈ Cu (H ),
where Cu (H ) is the Banach space of all uniformly continuous and bounded functions ϕ : H → R endowed with the usual supremum norm
ϕ 0 = sup ϕ(x) . x∈H
For k ∈ N, Cuk (H ) is the subspace of Cu (H ) consisting of all functions with uniformly continuous and bounded derivatives of order l for all l k, equipped with its natural norm. ϕ(x) By Cu,2 (H ) we denote the set of all functions ϕ : H → R such that the function x → 1+|x| 2 is uniformly continuous and bounded. Endowed with the norm ϕ u,2 := sup x∈H
|ϕ(x)| , 1 + |x|2
Cu,2 (H ) is a Banach space. We notice that, in view of Lemma 2.2, the transition evolution operator Ps,t acts in Cu,2 (H ). We recall that, since F is Lipschitz, we have for some constant C > 0, Ps,t | · |(x) C 1 + |x| , ∀x ∈ H, 0 s t T , (2.5) and, by Lemma 2.2, for all m > 1/2 and some Cm > 0 one has Ps,t | · |2m (x) Cm 1 + |x|2m , ∀x ∈ H, 0 s t T .
(2.6)
The following result is well known (it follows from Itô’s formula, see [2]). Lemma 2.3. For each 0 s t T , Ps,t is Feller, and maps Cu,2 (H ) into itself. Moreover, for any u ∈ EA ([0, T ] × H ) we have ∂ Ps,t u(t, ·) = Ps,t L0 u(t, ·), ∂t
∀0 s t T .
It is useful to introduce an extension of the operator L0 in C([0, T ]; Cu,2 (H )). For any λ ∈ R, (s, x) ∈ [0, T ] × H set T Fλ f (s, x) :=
e−λ(r−s) Ps,r f (r, ·)(x) dr,
f ∈ C [0, T ]; Cu,2 (H ) .
s
Let us show that Fλ satisfies the resolvent identity Fλ − Fλ = (λ − λ)Fλ Fλ
(2.7)
for all real λ and λ , whence it follows that the range Fλ (C([0, T ]; Cu,2 (H ))) is independent of λ. Identity (2.7) is verified as follows:
V.I. Bogachev et al. / Journal of Functional Analysis 256 (2009) 1269–1298
T Fλ (Fλ f )(s, x) =
1277
e−λ(r−s) Ps,r Fλ (r, · ) (x) dr
s
T =
e
−λ(r−s)
Ps,r
e
s
−λ (u−r)
Pr,u f (u, ·)(x) du dr
r
T =
T
e
−(λ−λ )r λs
T
e
s
e−λ u Ps,u f (u, ·)(x) du dr.
r
Integrating by parts we obtain on the right-hand side, eλs − λ − λ =
T e
−λr
s
eλ s Ps,r f (r, ·)(x) dr + λ − λ
T
e−λ u Ps,u f (u, ·)(x) du
s
1 Fλ f (s, x) − Fλ f (s, x) . λ − λ
Furthermore, as λ → ∞, we have T −s
e−λr Ps,r+s f (r + s, ·)(x) dr
λFλ f (s, x) = λ 0
λ(T −s)
e−r Ps,rλ−1 +s f rλ−1 + s, · (x) dr → f (s, x).
= 0
Hence Fλ is one-to-one, continuous with D(Fλ ) := C([0, T ], Cu,2 (H )), so Fλ−1 exists and is closed on Fλ (D(Fλ )). Therefore, the operator L := λI − Fλ−1 is closed (as a densely defined operator on C([0, T ]; Cu,2 (H ))) and does not depend on λ (which follows by (2.7)). In addition, we have D(L) = Fλ C [0, T ]; Cu,2 (H ) , λ ∈ R.
Fλ = (λ − L)−1 ,
(2.8)
By Lemma 2.3 it follows that L is indeed an extension of L0 . Finally, it is easy to check that the semigroup Pτ , τ 0, in the space CT [0, T ]; Cu,2 (H ) = u ∈ C [0, T ]; Cu,2 (H ) : u(T , x) = 0, ∀x ∈ H , defined by Pτ f (t, x) =
Pt,t+τ f (t + τ, ·)(x) 0,
is generated by L in the sense of π -semigroups (cf. [19]).
if t + τ T , otherwise,
(2.9)
1278
V.I. Bogachev et al. / Journal of Functional Analysis 256 (2009) 1269–1298
Arguing as in [19] one can show that u ∈ D(L) and Lu = f if and only if ⎧ 1 ⎪ ⎪ ⎪ lim Ph u(t, x) − u(t, x) = f (t, x), ∀(t, x) ∈ [0, T ] × H, ⎨ h→0 h
(1 + |x|2 )−1
⎪ ⎪ ⎪ sup Ph u(t, x) − u(t, x) < +∞. ⎩ h h∈(0,1],(t,x)∈[0,T ]×H
(2.10i) (2.10ii)
2.1. Existence for problem (1.5) ∗ be the We denote the topological dual of Cu,2 (H ) by Cu,2 (H )∗ . If 0 s < t T , let Ps,t ∗ adjoint operator of Ps,t . It is easy to see that if ν0 ∈ P(H ) we have Ps,t ν0 ∈ P(H ) and ∗ ϕ(x) Ps,t ν0 (dx) = Ps,t ϕ(x) ν0 (dx), ∀ϕ ∈ Cu,2 (H ). H
H
∗ ν , Proposition 2.4. Let ν0 ∈ P(H ) be such that H |x| ν0 (dx) < +∞. Then, setting νt = P0,t 0 (νt ) is a solution of problem (1.5) for all (not just a.e.) t ∈ [0, T ] such that for any m 1/2 there exists Cm > 0 such that |x|2m νt (dx) Cm 1 + |x|2m ν0 (dx) . (2.11) H
In particular,
H
|x| νt (dx) < +∞,
∀t ∈ [0, T ].
(2.12)
H
Proof. Let u ∈ D(L0 ), i.e., u(t, x) = φ(t)eix,h(t) , where φ ∈ C 1 ([0, T ]), φ(T ) = 0 and h ∈ C 1 ([0, T ]; D(A∗ )). Then by definition u(t, x) νt (dx) = P0,t u(t, ·)(x) ν0 (dx). H
H
Hence by (2.5) and (2.6) we obtain (2.11) and (2.12). On the other hand, by Lemma 2.3 we have ∂ P0,t u(t, ·) = P0,t L0 u(t, ·). ∂t So, using (2.5) we obtain ∂ ∂ P0,t u(t, ·)(x) ν0 (dx) = P0,t L0 u(t, ·)(x) ν0 (dx) u(t, x) νt (dx) = ∂t ∂t H H H = L0 u(t, ·)(x) νt (dx). H
The proof is complete.
2
V.I. Bogachev et al. / Journal of Functional Analysis 256 (2009) 1269–1298
1279
2.2. Uniqueness for problem (1.5) Lemma 2.5. Let f ∈ C([0, T ]; Cu1 (H )), λ ∈ R, and let u = (λ − L)−1 f . Then (i) Dx u ∈ C([0, T ]; Cu (H ; H )), (ii) u ∈ D(V ), where V is the operator defined in Appendix A by (A.3), and we have λu − V u − F, Dx u = f.
(2.13)
Proof. By the definition of u we have that u ∈ D(L) and T u(t, x) =
e−λ(r−t) Pt,r f (r, ·)(x) dr,
t ∈ [0, T ], x ∈ H.
t
Let us prove (i). Since Pt,r f (r, ·)(x) = E[f (r, X(r, t, x))], and F is C 1 we have Dx Pt,r f (r, ·)(x) = E Dx X(r, t, x)∗ Dx f r, X(r, t, x) , which is also bounded in x since F is Lipschitz uniformly in t. Consequently, Dx u ∈ C([0, T ]; Cu (H ; H )) and T Dx u(t, x) =
e−λ(r−t) Dx Pt,r f (r, ·)(x) dr.
t
Let us now prove (ii). Fix t ∈ [0, T ] and h > 0 such that t + h T . Then t+h X(t + h, t, x) = Z(t + h, t, x) + e(t+h−s)A F s, X(s, t, x) ds, t
where Z(t + h, t, x) = e
hA
t+h √ x+ e(t+h−s)A C dW (s). t
Therefore, we have Rh u(t, x) = Rh u(t + h, ·)(x) = E u t + h, Z(h, 0, x) = E u t + h, Z(t + h, t, x) , where Rh is defined by (A.4). Set g(t + h, t, x) =
t+h e(t+h−s)A F s, X(s, t, x) ds. t
(2.14)
1280
V.I. Bogachev et al. / Journal of Functional Analysis 256 (2009) 1269–1298
Then, taking into account (2.14), we have for any h > 0 Rh u(t, x)
t+h (t+h−s)A e F s, X(s, t, x) ds = E u t + h, X(t + h, t, x) −
= E u t + h, X(t + h, t, x) 1 −
t
E Du t + h, X(t + h, t, x) − ξg(t + h, t, x) , g(t + h, t, x) dξ
0
1 = Ph u(t, x) −
E Du t + h, X(t + h, t, x) − ξg(t + h, t, x) , g(t + h, t, x) dξ.
0
It follows that 1 Rh u(t, x) − u(t, x) h 1 = Ph u(t, x) − u(t, x) h 1 1 E Du t + h, X(t + h, t, x) − ξg(t + h, t, x) , g(t + h, t, x) dξ. − h
(2.15)
0
Since u ∈ D(L) due to the equality u = (λ − L)−1 f , Lemma 2.2 yields 1 Rh u(t, x) − u(t, x) = Lu(t, x) − Dx u(t, x), F (t, x) , h→0 h lim
(t, x) ∈ [0, T ] × H.
To show that u ∈ D(V ) and V u = Lu − F, Du, it remains to prove (see (A.5)) that sup
h∈(0,1],(t,x)∈[0,T ]×H
(1 + |x|2 )−1
Rh u(t, x) − u(t, x) < +∞. h
By (2.5) and (2.15) we have
1
1
E Du t + h, X(t + h, t, x) − ξg(t + h, t, x) , g(t + h, t, x) dξ
h
0
c Du 0 1 + sup E X(t, s, x) c 1 + |x| . ts
The proof is complete.
2
V.I. Bogachev et al. / Journal of Functional Analysis 256 (2009) 1269–1298
1281
Corollary 2.6. Let f ∈ C([0, T ]; Cu1 (H )), λ ∈ R and u = (λ − L)−1 f . Then, for every bounded Borel measure μ on [0, T ] × H , there exists a sequence (un ) ⊂ D(L0 ) such that un → u, Dx un → Dx u, V0 un → V0 u, hence one has L0 un → Lu in measure μ and
un (t, x) + V0 un (t, x) + Dx un (t, x) c1 1 + |x|2 ,
∀(t, x) ∈ [0, T ] × H,
for some constant c1 . Proof. By Lemma 2.5 we know that u = (λ − L)−1 f belongs to D(V ) and Lu = V u + Dx u, F . Note that Dx u, F ∈ Cu,2 (H ) (this space is defined before Lemma 2.3) since F is Lipschitz continuous and consequently sub-linear. On the other hand, by Corollary A.3 there exists a sequence of elements un ∈ D(L0 ) and a constant c2 > 0 such that
un (t, x) + V0 un (t, x) + Dx un (t, x) c2 1 + |x|2 ,
∀(t, x) ∈ [0, T ] × H,
and un → u, V0 un → V u, Dx un → Dx u in measure μ. It follows that L0 un → Lu in μmeasure. 2 Proposition 2.7. Let (ζt )t∈[0,T ] be a solution of (1.5) such that |x|2 ζt (dx) < +∞.
sup
t∈[0,T ]
H
∗ ν for all t ∈ [0, T ]. Then ζt = P0,t 0 ∗ ν for all t ∈ [0, T ] and γ (dt, dx) = γ (dx) dt. Then Proof. Set γt = νt − ζt , where νt = P0,t 0 t for any u ∈ D(L0 ) by Remark 1.1(i) we have
T L0 u γt (dx) dt = 0.
(2.16)
0 H
Let now f ∈ C([0, T ]; Cu1 (H )) and set u = L−1 f . Then by Corollary 2.6 there exists a sequence (un ) ⊂ D(L0 ) such that un → u,
L0 un → L0 u in γ measure,
and
un (t, x) + L0 un (t, x) c 1 + |x|2 ,
∀(t, x) ∈ [0, T ] × H, ρ ∈ Γ,
1282
V.I. Bogachev et al. / Journal of Functional Analysis 256 (2009) 1269–1298
for a suitable constant c > 0. Then by (2.16) we find by the dominated convergence theorem T
T f (t, x) γt (dx) dt =
0 H
Lu(t, x) γt (dx) dt = 0. 0 H
This implies that γt dt = 0 since the set C([0, T ]; Cu1 (H )) is dense in the space L1 ([0, T ] × H ; γ ). 2 2.3. m-Dissipativity of L0 Theorem 2.8. Let p ∈ [1, ∞) and let ν be a positive bounded Borel measure on [0, T ] × H such that there exists a constant α > 0 such that T
T L0 u(t, x) ν(dt, dx) α
0 H
u(t, x) ν(dt, dx),
∀u ∈ D(L0 ), u 0,
0 H
and T
p
|x|2p 1 + F (t, x) ν(dt, dx) < ∞.
0 H
Then under Hypotheses 1.2 and 2.1, L0 − α/p is dissipative in the space Lp ([0, T ] × H ; ν). Consequently, L0 − α/p is closable. Its closure Lp − α/p is m-dissipative in the space Lp ([0, T ] × H ; ν). Hence Lp generates a C0 -semigroup eτ Lp , τ 0, on Lp ([0, T ] × H ; ν). Furthermore, this semigroup is Markov. In particular this holds for ν(dt, dx) = ν(dx) dt from Proposition 2.4 with α = 0, provided H |x|3p ν0 (dx) < ∞. Proof. By [17, Lemma 1.8 in Appendix B], the operator (L0 − α/p, D(L0 )) is dissipative in Lp ([0, T ] × H ; ν) for all p ∈ [1, ∞). Let f ∈ C([0, T ]; Cu1 (H )), λ ∈ R, and let u = (λ − L)−1 f . By Lemma 2.5 we know that u ∈ D(V ) ∩ C([0, T ]; Cu1 (H )) and λu − V u − Dx u, F = f. By Corollary A.3 there exists a sequence (un ) ⊂ EA ([0, T ] × H ) such that for some c1 > 0 one has
un (t, x) + Dx un (t, x) + V un (t, x) c1 1 + |x|2 , ∀n ∈ N, and un → u, V un → V u, Dx un → Dx u in measure ν. Set fn = λun − V un − Dx un , F = λun − L0 un . Then we have fn → f in measure ν and there exists c2 > 0 such that
fn (t, x) c1 1 + |x|2 + |x|2 F (t, x) , (t, x) ∈ [0, T ] × H.
V.I. Bogachev et al. / Journal of Functional Analysis 256 (2009) 1269–1298
1283
By assumption and the dominated convergence theorem it follows that fn → f in Lp ([0, T ] × H ; ν). So we have proved that the closure of the range of λ − L0 includes C([0, T ]; Cu1 (H )) which is dense in Lp ([0, T ]×H ; ν). The remaining part of the assertion is proved as Theorem 3.3 below. 2 3. General coefficients Suppose we are given a family {F (t, ·)}t∈[0,T ] of m-quasi-dissipative mappings F (t, ·) : D F (t, ·) ⊂ H → 2H . This means that D(F (t, ·)) is a Borel set in H and for some K > 0 u − v, x − y K|x − y|2 ,
∀x, y ∈ D F (t, ·) , u ∈ F (t, x), v ∈ F (t, y),
(3.1)
and Range (λ − F (t, ·)) := x∈D(F (t,·)) (x − F (t, x)) = H for any λ > K. We assume additionally that K is independent of t. For any x ∈ D(F (t, ·)) the set F (t, x) is closed, non-empty, and convex; we set F0 (t, x) := y0 (t), where y0 (t) ∈ F (t, x) is such that |y0 (t)| = miny∈F (t,x) |y|, x ∈ D(F (t, ·)). We are concerned with the Kolmogorov operator L0 u(t, x) := Dt u(t, x) + U u(t, x) + F0 (t, x), Dx u(t, x) ,
u ∈ D(L0 ),
where D(L0 ) = EA ([0, T ] × H ) and U is the Ornstein–Uhlenbeck operator defined by (A.1) in Appendix A. Our goal is to prove that the closure of L0 − α/p is m-dissipative in the space Lp ([0, T ] × H, ν), p ∈ [1, ∞), where ν(dt, dx) = νt (dx) dt and (νt )t∈[0,T ] is a given family of finite positive Borel measures on H such that for some α > 0 one has T
T L0 u(t, x) νt (dx) dt α
0 H
u(t, x) νt (dx) dt, 0 H
∀u ∈ D(L0 ), u 0.
(3.2)
We shall assume, in addition to Hypothesis 1.2, that Hypothesis 3.1. (i) There is a family {F (t, ·)}t∈[0,T ] of m-quasi-dissipative mappings in H such that 0 ∈ D(F (t, ·)) and F0 (t, 0) = 0 for all t ∈ R. (ii) There is a family (νt )t∈[0,T ] of Borel probability measures on H such that for some p ∈ [1, ∞), T
dt
0
H
p
p
2p
|x| + F0 (t, x) + |x|2p F0 (t, x) νt (dx) < +∞.
1284
V.I. Bogachev et al. / Journal of Functional Analysis 256 (2009) 1269–1298
(iii) For all u ∈ D(L0 ) we have L0 u ∈ Lp ([0, T ] × H, ν) and (3.2) is fulfilled. (iv) νt (D(F (t, ·)) = 1, ∀t ∈ [0, T ]. Remark 3.2. (i) For simplicity below we shall assume that K in (3.1) is zero. This is, however, no restriction since all our arguments below immediately extend to the case when we add a C ∞ -Lipschitz map to F and clearly F = F˜ + KId with F˜ satisfying (3.1) with K = 0. (ii) Obviously, in Hypothesis 3.1(iii) we have L0 u ∈ Lp ([0, T ] × H, ν) if and only if the maps x → x, A∗ h, (t, x) → F (t, x), h are in Lp ([0, T ] × H, ν) for all h ∈ D(A∗ ). (iii) In [5] a number of results have been proved that ensure the existence of measures ν(dt dx) = νt (dx) dt satisfying the required properties in Hypothesis 3.1. More precisely, it was proved that they even satisfy (1.5) which by Remark 1.1(i) is stronger than (3.2). Let us introduce the Yosida approximations of F (t, ·), t ∈ R. For any α > 0 we set Fα (t, x) :=
1 Jα (t, x) − x , α
x ∈ H,
where −1 Jα (t, x) := I − αF (t, ·) (x),
x ∈ H, t ∈ R, α > 0.
It is well known that lim Fα (t, x) = F0 (t, x),
∀x ∈ D F (t, ·) ,
Fα (t, x) F0 (t, x) ,
∀x ∈ D F (t, ·) .
α→0
and
Moreover, Fα (t, ·) is Lipschitzian with constant 2/α and Fα (t, 0) = 0. Since Fα (t, ·) is not differentiable in general, we introduce a further regularization, as in [10], by setting Fα,β (t, x) =
eβB Fα t, eβB x + y N 1 B −1 (e2βB −1) (dy), 2
α, β > 0,
H
where B : D(B) ⊂ H → H is a self-adjoint negative definite operator such that B −1 is of trace class. The mapping Fα,β (t, ·) is dissipative, of class C ∞ , possesses bounded derivatives of all orders, and Fα,β (t, ·) → Fα (t, ·) pointwise as β → 0, see [14, Theorem 9.19]. Moreover, Fα,β (t, ·) satisfies Hypothesis 2.1(ii) since it is Lipschitz continuous with Lipschitz constant 2/α and
Fα,β (t, 0)
H
Fα (t, y) N 1
2 −1 2βB −1) (dy) 2 B (e α
|y| N 1 B −1 (e2βB −1) (dy). 2
H
V.I. Bogachev et al. / Journal of Functional Analysis 256 (2009) 1269–1298
1285
3.1. m-Dissipativity of Lp − α/p We assume here that Hypotheses 1.2 and 3.1 hold for some p ∈ [1, ∞). As in the regular case, (3.2) implies that (Lp − α/p, D(L0 )) is dissipative, hence closable in Lp ([0, T ] × H, ν) for all p ∈ [1, ∞). We shall denote its closure with domain D(Lp ) by Lp − α/p. We are going to show that Lp is m-dissipative. Let us consider the approximating equation λuα,β − V uα,β − Fα,β , Dx uα,β = f,
α, β > 0,
(3.3)
where λ > 0 and f ∈ C([0, T ]; Cu1 (H )). In view of Lemma 2.5, Eq. (3.3) has a unique solution uα,β ∈ D(V ) ∩ C([0, T ]; Cu1 (H )) given by T uα,β (t, x) =
e−λ(s−t) E f s, Xα,β (s, t, x) ds,
t ∈ R, x ∈ H,
t
where Xα,β is the mild solution of the problem
√ dXα,β (s, t, x) = AXα,β (s, t, x) + Fα,β t, Xα,β (s, t, x) ds + C dW (s), Xα,β (t, t, x) = x ∈ H.
(3.4)
For all h ∈ H we have Dx uα,β (t, x), h =
T
h e−λ(s−t) E Dx f s, Xα,β (s, t, x) , ηα,β (s, t, x) ds,
(3.5)
t h (s, t, x) := D X where ηα,β x α,β (s, t, x), h is the mild solution of the problem
⎧ ⎨ d ηh (s, t, x) = Aηh (s, t, x) + Dx Fα,β s, Xα,β (s, t, x)ηh (s, t, x), α,β α,β ds α,β ⎩ h ηα,β (t, t, x) = h.
s t,
By a standard argument, based on approximations (see e.g. [9, Section 3.2]) and on the Gronwall lemma, we see that for some constant c > 0 one has
h
η (s, t, x) ec(s−t) |h|, α,β
T s t 0.
Consequently, by (3.5) it follows that for λ > c we have
Dx uα,β (t, x)
1 sup Dx f (t, x) , λ − c t∈[0,T ],x∈H
Now we can prove the main result of this section.
t ∈ [0, T ], x ∈ H.
(3.6)
1286
V.I. Bogachev et al. / Journal of Functional Analysis 256 (2009) 1269–1298
Theorem 3.3. Under Hypotheses 1.2 and 3.1, Lp − α/p is m-dissipative in the space Lp ([0, T ] × H, ν). Hence Lp generates a C0 -semigroup eτ Lp , τ 0, on the space Lp ([0, T ] × H, ν). Furthermore, this semigroup is Markov, i.e. positivity preserving and eτ Lp 1 = 1 for all τ 0. Finally, the resolvent set ρ(Lp ) of Lp coincides with R. Proof. Let f ∈ C([0, T ]; Cu1 (H )) and let uα,β be the solution to Eq. (3.3). Claim 1. One has lim lim sup Fα,β (t, ·) − F0 (t, ·), Dx uα,β = 0 in Lp [0, T ] × H, ν .
α→0 β→0
In fact, it follows by (3.6) that for λ > c
T Iα,β :=
dt
Fα,β (t, ·) − F0 , Dx uα,β (t, ·) p dν
H
0
1 (λ − c)p
sup
t∈[0,T ],x∈H
Dx f (t, x) p
T dt
Fα,β (t, ·) − F0 (t, ·) p dνt .
H
0
Now, since for fixed α > 0, Fα,β (t, ·) is Lipschitz continuous with Lipschitz constant 2/α, we see that for any α > 0 there is cα > 0 such that
Fα,β (t, x) cα 1 + |x| ,
x ∈ H,
and so lim sup Iα,β β→0
1 (λ − c)p
p
sup Dx f (t, x)
t0,x∈H
T dt 0
Fα (t, ·) − F0 (t, ·) p dνt .
H
Now the claim follows, in view of the dominated convergence theorem. Claim 2. One has uα,β ∈ D(Lp ) and for λ > c λuα,β − Lp uα,β = f + Fα,β − F0 , Dx uα,β .
(3.7)
Applying Corollary 2.6 with L being the Kolmogorov operator corresponding to (3.4) we can find un ∈ E([0, T ] × H ), n ∈ N, such that un → uα,β ,
Dx un → Dx uα,β ,
and V0 un → V uα,β
in ν-measure
and for all (t, x) ∈ [0, T ] × H one has
un (t, x) + V0 un (t, x) + Dx un (t, x) c1 1 + |x|2 .
V.I. Bogachev et al. / Journal of Functional Analysis 256 (2009) 1269–1298
1287
By Hypothesis 3.1(ii) it follows that the sequence {L0 un } is bounded in the space Lp ([0, T ] × H, ν). Hence Claim 2 follows by the closability of L0 on Lp ([0, T ] × H, ν). Claim 3. One has f ∈ R(λ − Lp ) for λ > c. Claim 3 immediately follows from Claim 1 and (3.7). Since C([0, T ]; Cu1 (H )) is dense in Lp ([0, T ] × H, ν) the first assertion of the theorem follows. The second one follows from the well-known Lumer–Phillips theorem. The final statement is now a consequence of [17, Lemma 1.9] and the fact that Lp 1 = 0. Claim 4. ρ(Lp ) = R. Let β ∈ R and λ > 0 such that λ + β > pα . Let f ∈ Lp ([0, T ] × H, ν). Then by what we proved above there exists v ∈ D(Lp ) such that (β + λ − Lp )v = eλ f, where eλ (t, x) := eλt , (t, x) ∈ [0, T ] × H. Define u := e−λ v. Then an easy approximation argument proves that u ∈ D(Lp ) and (β − Lp )u = e−λ (β + λ − Lp )v = f. So, (β − Lp , D(Lp )) is surjective. It is also injective because so is (β + λ − Lp , D(Lp )). Hence β ∈ ρ(Lp ), since (β − Lp , D(Lp )) is closed. 2 Remark 3.4. It immediately follows from Claim 1 above that lim uα,β = (λ − Lp )−1 f
α,β→0
in Lp [0, T ] × H, ν ,
that is, the space–time resolvent corresponding to (3.4) converges to the one of (1.1) in Lp ([0, T ] × H, ν) on functions in C([0, T ]; Cu1 (H )). 3.2. Uniqueness for problem (1.5) in the irregular case Let us fix a Borel probability measure ν0 on H . We introduce the set Mν0 of all Borel measures ν on [0, T ] × H having the following properties: (i) ν(dt dx) = νt (dx) dt, where for t ∈ [0, T ], νt is a Borel probability on H such that νt (D(F (t, ·)) = 1 for all t ∈ [0, T ]. (ii) L0 u ∈ L1 ([0, T ] × H, ν) for all u ∈ EA ([0, T ] × H ) and (νt )t∈[0,T ] satisfies (1.5). T (iii) 0 H (|x|2 + |F0 (t, x)| + |x|2 |F0 (t, x)|) νt (dx) dt < ∞. The aim of this subsection is to prove that under Hypotheses 1.2 and 3.1 Mν0 contains at most one element, i.e. #Mν0 1.
1288
V.I. Bogachev et al. / Journal of Functional Analysis 256 (2009) 1269–1298
Remark 3.5. As mentioned above, the existence of solutions of (1.5) under suitable conditions has been proved in [4]. There, however, (1.5) has been written equivalently as follows: T L0 u(t, x) νt (dx) dt = 0 0 H
for all u ∈ EA ([0, T ] × H ) such that u(t, x) = 0 if t or t T − for some > 0 and lim ζ (x) νt (dx) = ζ (x) ν0 (dx), ∀ζ ∈ EA (H ). t→0
H
H
The same proof as that of [2, Lemma 2.7] shows that this formulation is indeed equivalent to (1.5). Clearly, the above formulation is nothing but a generalization of the classical Fokker– Planck equation corresponding to the Kolmogorov operator L0 . So, as already mentioned in the introduction, our results can be summarized as follows: first solve the Fokker–Planck equation (for measures) corresponding to L0 and using its solution solve the Kolmogorov equation for L0 on Lp ([0, T ] × H, ν) (for functions) which is possible according to Theorem 3.3 above. Theorem 3.6. Let ν0 be a Borel probability measure on H . Under Hypotheses 1.2 and 3.1(i) we have #Mν0 1. Proof. Let ν (1) , ν (2) ∈ Mν0 and set 1 1 μ := ν (1) + ν (2) . 2 2 Then μ ∈ Mν0 and ν (i) = σi μ for some measurable functions σi : [0, T ] × H → [0, 2]. By (1.6) we have L0 u dν (1) = L0 u dν (2) , ∀u ∈ D(L0 ), [0,T ]×H
that is
[0,T ]×H
L0 u(σ1 − σ2 ) dμ = 0,
∀u ∈ D(L0 ).
[0,T ]×H
Since by the last statement of Theorem 3.3, the range of (L0 , D(L0 )) is dense in L1 ([0, T ]×H, μ) and (σ1 − σ2 ) is bounded, we conclude that σ1 = σ2 . 2 4. Application to reaction–diffusion equations We shall consider here a stochastic heat equation perturbed by a polynomial drift, with time dependent coefficients, of odd degree d > 1 of the form λξ − p(t, ξ ),
ξ ∈ R, t ∈ [0, T ],
where λ ∈ R is given, p(t, 0) = 0 and Dξ p(t, ξ ) 0 for all ξ ∈ R and t ∈ [0, T ].
V.I. Bogachev et al. / Journal of Functional Analysis 256 (2009) 1269–1298
1289
We set H = L2 (O) where O = (0, 1)n , n ∈ N, and denote by ∂O the boundary of O. We are concerned with the following stochastic PDE on O: ⎧ √ ⎪ ⎨ dX(t, s, ξ ) = ξ X(t, s, ξ ) + λX(t, s, ξ ) − p t, X(t, s, ξ ) dt + C dW (t, ξ ), (4.1) X(t, s, ξ ) = 0, t s, ξ ∈ ∂O, ⎪ ⎩ X(s, s, ξ ) = x(ξ ), ξ ∈ O, x ∈ H, where ξ is the Laplace operator, C ∈ L(H ) is positive, and W is a cylindrical Wiener process with respect to (Ft )t∈R in H defined on a filtered probability space (Ω, F , (Ft )t∈R , P). We choose W of the form W (t, ξ ) =
∞
ek (ξ )βk (t),
ξ ∈ O, t 0,
k=1
where (ek ) is a complete orthonormal system in H and (βk ) is a sequence of independent standard Brownian motions on a probability space (Ω, F , P). Then we extend W (t) to (−∞, 0) by symmetry. Let us write problem (4.1) as a stochastic differential equation in the Hilbert space H . For this we denote by A the realization of the Laplace operator with Dirichlet boundary conditions, i.e.,
Ax = ξ x, x ∈ D(A), D(A) = H 2 (O) ∩ H01 (O).
The operator A is self-adjoint and possesses a complete orthonormal system of eigenfunctions, namely ek (ξ ) = (2/π)n/2 sin(πk1 ξ1 ) · · · (sin πkn ξn ),
ξ = (ξ1 , . . . , ξn ) ∈ Rn ,
where k = (k1 , . . . , kn ), ki ∈ N. For any x ∈ H we set xk = x, ek , k ∈ Nn . Notice that Aek = −π 2 |k|2 ek ,
k ∈ Nn , |k|2 = k12 + · · · + kn2 .
Therefore, we have tA e e−π 2 t ,
t 0.
Concerning the operator C, we shall assume for simplicity that C = (−A)−γ with n/2 − 1 < γ < 1 (which implies n < 4). Now it is easy to check that Hypothesis 1.2 is fulfilled. In fact we have t Qt x =
t esA CesA∗ x ds =
0
(−A)−γ e2sA x ds
0
1 = (−A)−(1+γ ) 1 − e2tA x, 2
t 0, x ∈ H.
1290
V.I. Bogachev et al. / Journal of Functional Analysis 256 (2009) 1269–1298
Then −2(1+γ ) |k| < +∞, Tr (−A)−(1+γ ) = k∈Nn
since γ >
n 2
− 1. Similarly, one obtains that for any α ∈ (0, 1/2) 1
∗
s −2α esA CesA ds < +∞.
Tr 0
Hence parts (i) and (ii) of Hypothesis 1.2 hold. Part (iii) can also be derived easily. We refer to [7] for details. Now, setting X(t, s) = X(t, s, ·) and W (t) = W (t, ·), we shall write problem (4.1) as dX(t, s) = AX(t, s) + F t, X(t, s) dt + (−A)−γ /2 dW (t), (4.2) X(s, s) = x, where F is the mapping F : D(F ) = [0, T ] × L2d (O) ⊂ [0, T ] × H → H,
x(ξ ) → λξ − p t, x(ξ ) .
It is convenient, following [14], to introduce two different notions of solution of (4.2). For this purpose, for any s ∈ [0, T ), we consider the space CW [s, T ]; H := CW [s, T ]; L2 (Ω, F , P; H ) consisting of all continuous mappings F : [s, T ] → L2 (Ω, F , P; H ) adapted to the filtration (Ft )t∈R , endowed with the norm F CW ([s,T ];H ) =
2 1/2
sup E F (t)
;
t∈[s,T ]
the space CW ([s, T ]; H ) is complete. Definition 4.1. (i) Let x ∈ L2d (O). We say that X(·, s, x) ∈ CW ([s, T ]; H ) is a mild solution of problem (4.1) if X(t, s, x) ∈ L2d (O) for all t ∈ [s, T ] and the following integral equation holds: t X(t, s, x) = e(t−s)A x +
e(t−r)A F r, X(r, s, x) dr + WA (s, t),
s
where WA (s, t) is the stochastic convolution t WA (s, t) = s
e(t−r)A (−A)−γ /2 dW (s),
t s.
t s,
V.I. Bogachev et al. / Journal of Functional Analysis 256 (2009) 1269–1298
1291
(ii) Let x ∈ H and s ∈ [0, T ]. We say that X(·, s, x) ∈ CW ([s, T ]; H ) is a generalized solution of problem (4.2) if there exists a sequence (xn ) ⊂ L2d (O) such that lim xn = x
n→∞
in H
and the mappings X(·, s, xn ) from (i) satisfy in CW [s, T ]; H .
lim X(·, s, xn ) = X(·, s, x)
n→∞
One can show that Definition 4.1(ii) does not depend on (xn ), see [8, §4.2]. We shall denote both mild and generalized solutions of (4.1) by X(t, s, x). The following result can be proved arguing as in [14], see also [8, Theorem 4.8]. Theorem 4.2. The following statements are true. (i) If x ∈ L2d (O), problem (4.2) has a unique mild solution X(·, s, x). Moreover, for any m ∈ N there is cm,d,T > 0 such that
2m
E X(t, s, x) L2d (O) cm,d,T 1 + |x|2m , L2d (O )
0 s t T.
(ii) If x ∈ H, problem (4.1) has a unique generalized solution X(·, s, x). For any 0 s t T , let us consider the transition evolution operator Ps,t ϕ(x) = E ϕ X(t, s, x) ,
ϕ ∈ Cu (H ),
where X(t, s, x) is a generalized solution of (4.2). Then, given ν0 ∈ P(H ), as in Section 2 we set ∗ νt := P0,t ν0 ,
t ∈ [0, T ].
(4.3)
By Theorem 4.2 we find immediately the following result. Proposition 4.3. Let m ∈ N and assume that ν0 ∈ P(H ) satisfies |x|2m ν (dx) < +∞. L2d (O ) 0
(4.4)
H
Then we have
|x|2m ν (dx) cm,d,T L2d (O ) t
H
|x|2m ν (dx). L2d (O ) 0
(4.5)
H
Now let us consider the operator L0 defined by (1.2), (1.3) and associated with (4.2). Notice that if u ∈ EA ([0, T ] × H ) then L0 u does not belong to C([0, T ]; Cu (H )) in general. However,
1292
V.I. Bogachev et al. / Journal of Functional Analysis 256 (2009) 1269–1298
if ν0 ∈ P(H ) satisfies (4.4), then by (4.5) one has L0 u ∈ L2 ([0, T ] × H, ν), where ν(dt, dx) = νx (dt) dt. By Proposition 4.3 the family (νt )t∈[0,T ] obviously satisfies Hypothesis 3.1 for p = 2, provided ν0 satisfies (4.4) with m = 2d. Then by Theorems 3.3 and 3.6 we deduce the following result. Theorem 4.4. Assume that ν0 ∈ P(H ) satisfies (4.4) with m = 2d. Then the operator L0 with domain D(L0 ) = EA ([0, T ] × H ) associated with (4.2) is closable on L2 ([0, T ] × H ; ν), where ν(dt, dx) = νx (dt) dt and νt is defined by (4.3), and its closure L2 is m-dissipative. Furthermore, L2 generates a Markov C0 -semigroup of contractions on L2 ([0, T ] × H ; ν) and ν is the unique measure satisfying the Fokker–Planck equation (1.5) and having properties (i)–(iii) in Section 3.2. Acknowledgments This work has been supported in part by the RFBR project 07-01-00536, the Russian– Japanese Grant 08-01-91205-JF, the Russian–Chinese Grant 06-01-39003, the Russian–Ukrainian RFBR Grant, ARC Discovery Grant DP0663153, the DFG Grant 436 RUS 113/343/0(R), SFB 701 at the University of Bielefeld, the research programme “Equazioni di Kolmogorov” of the Italian “Ministero della Ricerca Scientifica e Tecnologica”. Most of the work was done during visits of the first and third authors to the Scuola Normale Superiore di Pisa and visits of the first and second authors to the University of Bielefeld. Appendix A. The Ornstein–Uhlenbeck semigroup In this section Hypothesis 1.2 is still in force. We denote by Rt the Ornstein–Uhlenbeck semigroup
ϕ etA x + y NQt (dy),
Rt ϕ(x) :=
ϕ ∈ Cu,2 (H ),
H
where t Qt x :=
∗
esA CesA x ds,
x ∈ H, t 0,
0
and NQt is the Gaussian measure in H with mean 0 and covariance operator Qt . Then (cf. [12]) we have Rt ϕ(x) = E ϕ Z(t, 0, x) . We shall consider Rt acting in the Banach space Cu,2 (H ) defined in Section 2. This will be needed in the proof of Proposition A.2 below.
V.I. Bogachev et al. / Journal of Functional Analysis 256 (2009) 1269–1298
1293
Let us define the infinitesimal generator U of Rt through its resolvent by setting, following λ (Cu,2 (H )), where λ −1 , D(U ) = G [6], U := λ − G λ f (x) = G
+∞ e−λt Rt f (x) dt,
x ∈ H, λ > 0, f ∈ Cu,2 (H ).
0
It is easy to see that for any h ∈ D(A∗ ) the function ϕh (x) = eix,h belongs to the domain of U in Cu,2 (H ) and we have U ϕh =
1 Tr CD 2 ϕh + x, A∗ Dϕh . 2
(A.1)
A.1. The strong Feller property The following identity for the derivative of Rt ϕ is well known, see [16]: −1/2 tA Dx Rt ϕ(x), h = Λt h, Qt y ϕ e x + y NQt (dy), ϕ ∈ Cu,2 (H ),
(A.2)
H −1/2 tA e .
where Λt = Qt
By Hölder’s inequality it follows that
Dx Rt ϕ(x), h 2 |Λt h|2
ϕ 2 etA x + y NQt (dy).
H
So, since h is arbitrary, one has
Dx Rt ϕ(x) 2 | Λt 2
ϕ 2 etA x + y NQt (dy).
H
It follows that 2 |Dx Rt ϕ(x)|2 |2 Λ(t) 2 2 (1 + |x| )
H
ϕ 2 (etA x + y) NQt (dy) (1 + |x|2 )2
2 Λ(t) ϕ 2
Cu,2 (H ) H
2 4 Λ(t) ϕ 2
Cu,2 (H )
(1 + |etA x + y|)2 NQt (dy) (1 + |x|2 )2 2 1 + |y|2 NQt (dy)
H
2 4c12 Λ(t) ϕ 2Cu,2 (H ) ,
where c1 is a positive constant. Now, recalling Hypothesis 1.2(iii) and using the Laplace transform we obtain the following result.
1294
V.I. Bogachev et al. / Journal of Functional Analysis 256 (2009) 1269–1298
Lemma A.1. Let ϕ ∈ D(U ). Then there exists c2 > 0 such that
Dx ϕ(x) c2 ϕ C (H ) + U ϕ C (H ) 1 + |x|2 , u,2 u,2
x ∈ H.
A.2. The Ornstein–Uhlenbeck semigroup in C([0, T ]; Cu,2 (H )) Let V0 u(t, x) = Dt u(t, x) + U u(t, x),
u ∈ EA [0, T ] × H .
It is clear that V0 u ∈ C([0, T ]; Cu,2 (H )) (note that U u(t, x) contains a term growing as |x|). Let us introduce an extension of the operator V0 . For any λ ∈ R set T Gλ f (t, x) =
e−λ(s−t) Rt−s f (s, x) ds,
f ∈ C [0, T ]; Cu,1 (H ) .
t
It is easy to see that Gλ satisfies the resolvent identity, so that there exists a unique linear closed operator V in C([0, T ]; Cu,1 (H )) such that D(V ) = Gλ C [0, T ]; Cu,2 (H ) , λ ∈ R. (A.3) Gλ = (λ − V )−1 , It is clear that V is an extension of V0 . Finally, it is easy to check that the semigroup Rτ , τ 0, generated by V in CT [0, T ]; Cu,1 (H ) := u ∈ CT [0, T ]; Cu,1 (H ) : u(T , x) = 0 is given by Rτ f (t, x) =
Rτ f (t + τ, ·)(x) 0,
if t + τ T , otherwise.
Arguing as in [19] one can show that u ∈ D(V ) and V u = f if and only if ⎧ 1 ⎪ ⎪ lim Rh u(t, x) − u(t, x) = f (t, x), ∀(t, x) ∈ [0, T ] × H, ⎪ ⎨ h→0 h
(1 + |x|2 )−1
⎪ ⎪ ⎪ sup Rh u(t, x) − u(t, x) < +∞. ⎩ h h∈(0,1],(t,x)∈[0,T ]×H
(A.4)
(A.5i) (A.5ii)
A.3. A core for V The following result is a generalization of [13]. Proposition A.2. Let u ∈ D(V ) and let ν be a finite nonnegative Borel measure on [0, t] × H . Then there exists a sequence (un ) ⊂ EA ([0, T ] × H ) such that for some c1 > 0 one has
un (t, x) + V0 un (t, x) c1 1 + |x|2 , ∀(t, x) ∈ [0, T ] × H, and un → u, V0 un → V0 u in measure ν.
V.I. Bogachev et al. / Journal of Functional Analysis 256 (2009) 1269–1298
1295
Proof. Let f ∈ C([0, T ; Cu (H )) and set u = V −1 f so that T u(t, x) = −
1 Rs−t f (s, x) ds = −
t
R(T −t)r f (T − t)r + t, x dr.
0
Arguing as in the proof of Proposition 1.2 in [8], it is easy to find a sequence (fn1 ,n2 ) ⊂ EA ([0, T ] × H ) such that lim fn1 ,n2 (t, x) = f (t, x),
lim
n1 →∞ n2 →∞
f n
(t, x) c1 ,
1 ,n2
(A.6)
where c1 is independent of n1 , n2 . Set T un1 ,n2 (t, x) = −
Rs−t fn1 ,n2 (s, x) ds t
1 =−
R(T −t)r fn1 ,n2 (T − t)r + t, x dr,
0
so that V un1 ,n2 = fn1 ,n2 . By (A.6) it follows that lim
lim un1 ,n2 (t, x) = u(t, x),
n1 →∞ n2 →∞
un ,n (t, x) c1 T . 1 2
(A.7)
Moreover, T V un1 ,n2 (t, x) = −
Rs−t Vfn1 ,n2 (s, x) ds t
1 =−
R(T −t)r Vfn1 ,n2 (T − t)r + t, x dr,
0
so that lim
lim V un1 ,n2 (t, x) = V u(t, x),
n1 →∞ n2 →∞
V un
1 ,n2
(t, x) c1 T .
(A.8)
Now we want to approximate un1 ,n2 by functions from EA ([0, T ] × H ). For this we consider the set Σ of all partitions σ = {t0 , t1 , . . . , tN } of [0, 1] with 0 = t0 < t1 < · · · < tN −1 < tN = T . We set |σ | = max |ti − ti−1 | 1in
1296
V.I. Bogachev et al. / Journal of Functional Analysis 256 (2009) 1269–1298
and endow Σ with the usual partial ordering σ1 < σ2
if and only if |σ1 | < |σ2 |.
Finally, for any σ = {t0 , t1 , . . . , tN } ∈ Σ we set un1 ,n2 ,σ (t, x) =
N
R(T −t)rk fn1 ,n2 (T − t)rk + t, x (rk − rk−1 ),
(A.9)
k=1
so that V un1 ,n2 ,σ (t, x) =
N
R(T −t)rk Vfn1 ,n2 (T − t)rk + t, x (rk − rk−1 ).
(A.10)
k=1
By (A.9) taking into account (A.7) it follows that lim
lim un1 ,n2 ,σ (t, x) = u(t, x),
lim
n1 →∞ n2 →∞ |σ |→0
un
1 ,n2 ,σ
(t, x) c1 T .
Similarly we see that lim
lim
lim V un1 ,n2 ,σ (t, x) = V u(t, x).
n1 →∞ n2 →∞ |σ |→0
However, (A.10) does not guarantee an estimate
V un
1 ,n2 ,σ
(t, x) c2 T ,
with c2 independent of n. Then we argue as follows. Note that if z ∈ EA ([0, T ] × H ), then the function F : [0, T ] × [0, T ] → H,
(t, s) → Rs z(t, x),
which is not continuous in the topology of Cu (H ), is continuous in that of Cu,2 (H ), consequently the integral 1
R(T −t)r f (T − t)r + t, x dr
0
is convergent in that topology. Therefore, for any ε > 0 there exists δε > 0 such that if |σ | < δε we have
N
R(T −t)rk Vfn1 ,n2 (T − t)rk + t, x (rk − rk−1 ) ε 1 + |x|2 .
V un1 ,n2 (t, x) −
k=1
V.I. Bogachev et al. / Journal of Functional Analysis 256 (2009) 1269–1298
1297
Consequently,
V un ,n ,σ (t, x) V un ,n (t, x) + ε 1 + |x|2 1 2 1 2 and, taking into account (A.8), we find
V un
1 ,n2 ,σ
(t, x) c1 T + ε 1 + |x|2 .
Let σn denote the partition formed by the points 0, 2−n T , 21−n T , . . . , T . We can find functions un1 ,n2 ,σn indexed by the triples (n1 , n2 , σn ) such that un1 ,n2 ,σn (t, x) → u(t, x) in the following sense: keeping n1 , n2 fixed, one has lim un1 ,n2 ,σn (t, x) = un1 ,n2 ,σn (t, x),
n→∞
next there is a limit un1 for any n1 fixed as n2 → ∞, and finally, un1 → u as n1 → ∞. Convergence V0 un1 ,n2 ,σn → V0 u takes place in the same sense. Clearly, we may assume that |x|2 is ν-integrable (just by multiplying ν by (|x|2 + 1)−1 ). By the dominated convergence theorem this yields L1 (ν)-convergence un1 ,n2 ,σn → u and V0 un1 ,n2 ,σn → V0 u in the same sense as above (first for any n1 , n2 fixed, etc.) and enables us to find a sequence of elements un in the net un1 ,n2 ,σn convergent in L1 (ν), hence in measure ν. 2 As in the proof of Lemma 2.5 one proves that if u ∈ D(V ) then u is differentiable in x. Hence the following result is a consequence of (A.2) and Lemma A.1. Corollary A.3. Let u ∈ D(V ) and let ν be a finite nonnegative Borel measure on [0, T ] × H . Then there exists a sequence (un ) ⊂ EA ([0, T ] × H ) such that for some c1 > 0 one has
un (t, x) + Dx un (t, x) + V0 un (t, x) c1 1 + |x|2 ,
∀(t, x) ∈ [0, T ] × H,
and un → u, Dx un → Dx u, V0 un → V u in measure ν. References [1] V. Bogachev, G. Da Prato, M. Röckner, Existence of solutions to weak parabolic equations for measures, Proc. London Math. Soc. (3) 88 (2004) 753–774. [2] V. Bogachev, G. Da Prato, M. Röckner, W. Stannat, Uniqueness of solutions to weak parabolic equations for measures, Bull. London Math. Soc. 39 (2007) 631–640. [3] V. Bogachev, G. Da Prato, M. Röckner, On parabolic equations for measures, Comm. Partial Differential Equations 33 (3) (2008) 397–418. [4] V. Bogachev, G. Da Prato, M. Röckner, Infinite dimensional Kolmogorov operators with time dependent drift coefficients, Dokl. Ross. Akad. Nauk 419 (5) (2008) 587–591 (in Russian); English transl.: in: Dokl. Akad. Nauk 77 (2) (2008) 276–280. [5] V. Bogachev, G. Da Prato, M. Röckner, Parabolic equations for measures on infinite-dimensional spaces, Dokl. Math. (2008), in press. [6] S. Cerrai, Weakly continuous semigroups in the space of functions with polynomial growth, Dynam. Syst. Appl. 4 (1995) 351–372.
1298
V.I. Bogachev et al. / Journal of Functional Analysis 256 (2009) 1269–1298
[7] S. Cerrai, Second Order PDE’s in Finite and Infinite Dimensions. A Probabilistic Approach, Lecture Notes in Math., vol. 1762, Springer-Verlag, 2001. [8] G. Da Prato, Kolmogorov Equations for Stochastic PDEs, Birkhäuser, 2004. [9] G. Da Prato, Introduction to Stochastic Analysis and Malliavin Calculus, Publ. Sc. Norm. Sup., vol. 6, Birkhäuser, 2007. [10] G. Da Prato, M. Röckner, Singular dissipative stochastic equations in Hilbert spaces, Probab. Theory Related Fields 124 (2002) 261–303. [11] G. Da Prato, M. Röckner, Dissipative stochastic equations in Hilbert space with time dependent coefficients, Atti Accad. Naz. Lincei 17 (2006) 397–403. [12] G. Da Prato, M. Röckner, A note on evolution systems of measures for time-dependent stochastic differential equations, in: Progr. Probab., vol. 59, Birkhäuser, 2007, pp. 115–122. [13] G. Da Prato, L. Tubaro, Some results about dissipativity of Kolmogorov operators, Czechoslovak Math. J. 51 (2001) 685–699. [14] G. Da Prato, J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge Univ. Press, 1992. [15] G. Da Prato, J. Zabczyk, Ergodicity for Infinite Dimensional Systems, London Math. Soc. Lecture Note Ser., vol. 229, Cambridge Univ. Press, 1996. [16] G. Da Prato, J. Zabczyk, Second Order Partial Differential Equations in Hilbert Spaces, London Math. Soc. Lecture Note Ser., vol. 293, Cambridge Univ. Press, 2002. [17] A. Eberle, Uniqueness and Non-uniqueness of Semigroups Generated by Singular Diffusion Operators, Lecture Notes in Math., vol. 1718, Springer-Verlag, 1999. [18] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, 1983. [19] E. Priola, On a class of Markov type semigroups in spaces of uniformly continuous and bounded functions, Studia Math. 136 (1999) 271–295.
Journal of Functional Analysis 256 (2009) 1299–1309 www.elsevier.com/locate/jfa
Wave front set for solutions to Schrödinger equations Shu Nakamura 1 Graduate School of Mathematical Science, University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8914, Japan Received 30 May 2008; accepted 10 June 2008 Available online 16 July 2008 Communicated by C. Kenig
Abstract We consider solutions to Schrödinger equation on Rd with variable coefficients. Let H be the Schrödinger operator and let u(t) = e−itH u0 be the solution to the Schrödinger equation with the initial condition u0 ∈ L2 (Rd ). We show that the wave front set of u(t) in the nontrapping region can be characterized by the wave front set of e−itH0 u0 , where H0 is the free Schrödinger operator. The characterization of the wave front set is given by the wave operator for the corresponding classical mechanical scattering (or equivalently, by the asymptotics of the geodesic flow). © 2008 Elsevier Inc. All rights reserved. Keywords: Schrödinger equation; Propagation of singularities; Wave front set
1. Introduction We consider a Schrödinger operator H =−
d 1 ∂xi aij (x)∂xj + V (x) 2 i,j =1
on Rd , where d 1. We suppose the coefficients {aij (x)} and the potential V (x) satisfy the following conditions: E-mail address: [email protected]. 1 Partially supported by JSPS Grant (C) 13640155.
0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.06.007
1300
S. Nakamura / Journal of Functional Analysis 256 (2009) 1299–1309
Assumption A. aij (x), V (x) ∈ C ∞ (Rd ; R) for i, j = 1, . . . , d, and there exist μ > 0, and Cα > 0 for each α ∈ Zd+ such that α ∂ aij (x) − δij Cα x−1−μ−|α| , x α ∂ V (x) Cα x1−μ−|α| , x ∈ Rd . x Moreover, H is elliptic, i.e., det(aij (x)) = 0 for each x ∈ Rd . Then it is well known that H is essentially self-adjoint on C0∞ (Rd ) (see e.g., [13, Chapter X]). We denote the self-adjoint extension by the same symbol H . We consider solutions to the Schrödinger equation ∂ u(t) = −iH u(t), ∂t
t ∈ R,
with initial condition u(0) = u0 ∈ L2 (Rd ). We study properties of the wave front set of the solution u(t) = e−itH u0 . In particular, we give a characterization of the wave front set of u(t) in the nontrapping region in terms of u0 , t = 0. We denote the kinetic energy part of the Hamiltonian by K, and the free Schrödinger operator by H0 , i.e., K =−
d 1 ∂xi aij (x)∂xj , 2 i,j =1
1 2 1 H0 = − = − ∂xj . 2 2 d
j =1
We denote the principal symbol of K by k(x, ξ ):
k(x, ξ ) =
d 1 aij (x)ξi ξj , 2
x, ξ ∈ Rd .
i,j =1
We denote y(t; x, ξ ), η(t; x, ξ ) = exp tHk (x, ξ ), i.e., (y(t), η(t)) is the solution to the Hamilton equation: d ∂k y(t) = y(t), η(t) , dt ∂ξ
d ∂k η(t) = − y(t), η(t) dt ∂x
(1)
with initial condition y(0) = x, η(0) = ξ . Definition 1. (x, ξ ) ∈ R2d is said to be forward nontrapping (backward nontrapping, respectively) if |y(t; x, ξ )| → +∞ as t → +∞ (t → −∞, respectively).
S. Nakamura / Journal of Functional Analysis 256 (2009) 1299–1309
1301
If (x, ξ ) is forward (backward, respectively) nontrapping, then it is well known that there exists (x± , ξ± ) ∈ R2d such that y(t; x, ξ ) − (x± + tξ± ) → 0 as t → ±∞, where “±” correspond to the forward and backward nontrapping cases, respectively. x± + tξ± is called the asymptotic trajectory, and we denote x± = x± (x, ξ ) and ξ± = ξ± (x, ξ ), respectively. We note (x, ξ ) → (x± , ξ± ) are local diffeomorphisms. We discuss them in detail in Section 2. We denote the wave front set of u ∈ D (Rd ) by WF(u) ⊂ R2d . Then our main result is stated as follows: Theorem 1. Suppose Assumption A, and suppose (x0 , ξ0 ) ∈ R2d is backward (forward, respectively) nontrapping. Let u(t) = e−itH u0 with u0 ∈ L2 (Rd ), and let t0 > 0 (t0 < 0, respectively). Then (x0 , ξ0 ) ∈ WF u(t0 )
⇐⇒
x∓ (x0 , ξ0 ), ξ∓ (x0 , ξ0 ) ∈ WF e−it0 H0 u0 .
Remark. By the discussion in Section 3, we see that WF(e−itH0 u0 ) is characterized as follows: 0 such that a(x, ξ ) is elliptic at (x , ξ ) ∈ / WF(e−itH0 u0 ) if and only if there exists a(x, ξ ) ∈ S1,0 (x , ξ ) and a(x + tDx , Dx )u0 ∈ C ∞ (Rd ). By this observation, we can show that the microlocal smoothing estimate of Craig, Kappeler, Strauss [1] follows from Theorem 1. It is also equivalent that there exists a(x, ξ ) ∈ C0∞ (R2d ) such that a(x , ξ ) = 0 and a(x + tDx , hDx )u0 = O(h∞ ) as h → 0. Our result may be considered as a refinement of the microlocal smoothing estimates, e.g., results by Craig, Kappler, Strauss [1], Robbiano, Zuilly [14]. They showed that decay of the initial condition u0 in a certain cone in Rd implies microlocal regularity of u(t) along classical flow starting from the cone. The theory is generalized to Schrödinger operators on manifolds with conic ends by Wunsch [17] and Robbiano, Zuily [15] using quadratic scattering wave front sets. On the other hand, the author studied it using different notion of wave front set, homogeneous wave front sets, to generalize results in [1] to Schrödinger equations with long-range type perturbation on Rd ([11]. See also a generalization to the analytic smoothing effect by Martinez, Nakamura and Sordoni [9].) Recently, Hassel and Wunsch obtained a characterization of the wave front set of the solution to Schrödinger equations on manifolds using the scattering wave front sets [5], and their results are closely related to our results. However, their formulation and assumptions are quite different from ours, and our proof is simpler. Doi proved smoothing effect using different settings [3,4]. In particular, this work is partially inspired by his work on the smoothing properties of perturbed harmonic oscillators [4]. After the original version of this work was completed, this result was extended to long-range perturbation cases by the author [12], and to Schrödinger operators on scattering manifolds by Ito and Nakamura [7]. It was also generalized to the characterization of the analytic singularities by Martinez, Nakamura and Sordoni [10]. We are investigating further extension and generalization of such characterization of microlocal singularities for Schrödinger equations. In Section 2, we prepare a few lemmas on the classical mechanical scattering, and we prove Theorem 1 in Section 3. The idea of the proof is to employ an Egorov-type theorem in the semiclassical limit. Here the semiclassical parameter is, essentially, the modulus of the momentum.
1302
S. Nakamura / Journal of Functional Analysis 256 (2009) 1299–1309
Notations. For a(x, ξ ) ∈ C ∞ (Rd × Rd ), we denote its Weyl quantization by a w (x, Dx ), namely, a w (x, Dx )u(x) = (2π)−d ei(x−y)·ξ a (x + y)/2, ξ u(y) dy dξ for u ∈ S(Rd ). We use the S(m, g) symbol class notation due to Hörmander [6, vol. 3]. In partic0 = S(1, dx 2 + dξ 2 /ξ 2 ). We ular, we write the standard Kohn–Nirenberg symbol class by S1,0 also use the standard notation x = 1 + |x|2 for a vector x, and R+ = [0, ∞); R− = (−∞, 0]. We denote the set of non-negative integers by Z+ , and the set of (d-dimensional) multi-indices by Zd+ . We write f (h) = O(h∞ ) if f (h) = O(hN ) as h → 0 for any N ∈ Z+ . L(H, K) denotes the Banach space of the bounded operators from H to K, and L(H) = L(H, H). C and C∗ denote generic constants, which may vary from line to line. 2. Classical trajectories Here we discuss several results on the scattering theory of classical mechanics. They seem to be well known, but we recall them for the completeness and also for reader’s convenience (see, e.g., [13, vol. 3, §11.2], [1, §2]). In the following, we only consider the case t0 > 0, and we fix a backward nontrapping point (x0 , ξ0 ) ∈ R2d . We study the asymptotic behavior of classical trajectories (i.e., the solutions to (1)) as t → −∞. Lemma 2. There exists Ω ⊂ R2d , a neighborhood of (x0 , ξ0 ) and constants c, C > 0 such that y(t; x, ξ ) c|t| − C
for t < 0, (x, ξ ) ∈ Ω.
Proof. Note k(x0 , ξ0 ) > 0, since otherwise (x0 , ξ0 ) is trapping. Hence k(x, ξ ) c0 > 0 in a small neighborhood of (x0 , ξ0 ). We note k(y(t), η(t)) = k(x, ξ ) for all t. By straightforward computations, we have 2 d 2 = 2 d y(t) · dy (t) = 2 d y(t) y(t) y a (t)η (t) ij i j dt dt dt dt 2 i,j = 4k y(t), η(t) + W y(t), η(t) , where W (x, ξ ) = O(|x|−1−μ ), locally uniformly in ξ ∈ Rd . Thus, if k(x, ξ ) > 0 and |y(t)| is sufficiently large, then |y(t)|2 is a convex function of t. We choose R > 0 so that |W (x, ξ )| 2c0 if |x| R and |ξ | sup{|η(t; x, ξ )| | t 0, (x, ξ ) ∈ Ω} < ∞. By the assumption, if Ω is sufficiently small neighborhood of (x0 , ξ0 ), there exists t1 < 0 such that |y(t1 ; x, ξ )| R and d dt |y(t1 ; x, ξ )| −c1 < 0 for all (x, ξ ) ∈ Ω. Then by the convexity, |y(t; x, ξ )| R + c1 |t − t1 | for all t t1 and (x, ξ ) ∈ Ω. The claim now follows immediately. 2 Lemma 3. Let Ω ⊂ R2d as in the previous lemma. Then there exist limits ξ− (x, ξ ) = lim η(t; x, ξ ), t→−∞ x− (x, ξ ) = lim y(t; x, ξ ) − tη(t; x, ξ ) t→−∞
S. Nakamura / Journal of Functional Analysis 256 (2009) 1299–1309
1303
for (x, ξ ) ∈ Ω, and there is C > 0 such that ξ− − η(t; x, ξ ) Ct−1−μ , x− − y(t; x, ξ ) + tη(t; x, ξ ) Ct−μ , for t 0, (x, ξ ) ∈ Ω. Moreover, the map S− : (x, ξ ) → (x− , ξ− ) is a diffeomorphism from Ω to S− (Ω). Remark. If (x0 , ξ0 ) is forward nontrapping, then the analogous results hold by replacing “t → −∞” by “t → +∞,” and “(∗)− ” by “(∗)+ .” Proof. We denote y(t) = y(t; x, ξ ), η(t) = η(t; x, ξ ) with (x, ξ ) ∈ Ω. Then we have d d 1 ηi (t) = − (∂xi aj k ) y(t) ηj (t)ηk (t) dt 2 j,k=1
−2−μ = O y(t) = O |t|−2−μ
as t → −∞
by Lemma 2. Similarly, if we set z(t) = y(t) − tη(t), we have d d d zi (t) = aij y(t) ηj (t) − ηi (t) − t ηi (t) dt dt j =1
=
d d t aij y(t) − δij ηj (t) + ∂xi aj k y(t) ηj (t)ηk (t) 2 j =1
= O |t|−1−μ as t → −∞.
j,k=1
The first claim follows immediately from these estimates. By the standard method, we can β also show (x, ξ ) → (x− , ξ− ) is a C ∞ -map. Namely, for each α, β ∈ Zd+ , ∂xα ∂ξ z(t; x, ξ ) and β
∂xα ∂ξ η(t; x, ξ ) satisfy similar first order ODEs with coefficients which are integrable in t. Hence β
β
β
β
∂xα ∂ξ z(t; x, ξ ) and ∂xα ∂ξ η(t; x, ξ ) converges uniformly to ∂xα ∂ξ x− (x, ξ ) and ∂xα ∂ξ ξ− (x, ξ ) as t → −∞. Moreover, if |t2 | is sufficiently large (and t2 < 0), then
∂(z, η)
1
(s) − E
∂(z , η )
2
for s 0,
where η(s) = η(s; z + t2 η , η ), z(s) = y(s; z + t2 η , y ) − (t2 + s)η(s) with (y , η ) ∈ Tt2 (Ω); E is the 2d × 2d unit matrix, and · denotes the operator norm in L(R2d ). Hence, by letting s → −∞, we have
1
∂(x− , ξ− )
∂(z , η ) − E 2 .
1304
S. Nakamura / Journal of Functional Analysis 256 (2009) 1299–1309
This implies (∂(x− , ξ− )/∂(z , η )) is invertible. By the inverse function theorem, we learn (z , η ) → (x− , ξ− ) is diffeomorphic. Since (x, ξ ) → (z , η ) is also diffeomorphic, we conclude the last assertion. 2 The lemma also implies y(t) − (x− + tξ− ) y(t) − x− + tη(t) + t η(t) − ξ− = O |t|−μ as t → −∞. Namely, y(t) asymptotically converges to the free motion x− + tξ− as t → −∞. We denote St (x, ξ ) = exp(−tHp0 ) ◦ exp(tHk )(x, ξ ) = y(t) − tη(t), η(t) , where p0 (ξ ) = 12 |ξ |2 is the free Hamiltonian, and exp(tHp0 ) is the free Hamilton flow, i.e., exp(tHp0 )(x, ξ ) = (x + tξ, ξ ). By the above lemma, we learn St converges to S− on Ω as t → β β −∞ in the sense of C ∞ -maps, i.e., for any α, β ∈ Zd+ , ∂xα ∂ξ St converges uniformly to ∂xα ∂ξ S− on Ω. By the proof of the above lemma, we also learn that the flow St is generated by a timedependent Hamiltonian 0 (t; x, ξ ) defined by 0 (t; x, ξ ) =
d 1 aj k (x + tξ ) − δj k ξj ξk , 2 j,k=1
i.e., (z(t), η(t)) = St (x, ξ ) satisfies the Hamilton equation d ∂ 0 zi (t) = t; z(t), η(t) , dt ∂ξi
d ∂ 0 ηi (t) = − t; z(t), η(t) dt ∂xi
with the initial condition (z(0), η(0)) = (x.ξ ). We also denote (t; x, ξ ) = 0 (t; x, ξ ) + V (x + tξ ), which generates the flow exp(−tHp0 ) ◦ exp(tHp ), where p(x, ξ ) = k(x, ξ ) + V (x) is our full Hamiltonian. 3. Proof of Theorem 1 In this section we suppose (x0 , ξ0 ) is backward nontrapping, and prove Theorem 1 for the case t0 > 0. We first consider the quantum evolution corresponding to the classical scattering evolution St . For v0 ∈ H 2 (Rd ), we consider v(t) = eitH0 e−itH v0 . If we differentiate v(t) in t, we have
S. Nakamura / Journal of Functional Analysis 256 (2009) 1299–1309
1305
d v(t) = i eitH0 (H0 − H )e−itH v0 dt = −i eitH0 (H − H0 )e−itH0 v(t) = −iL(t)v(t). We recall eitH0 a w (x, Dx )e−itH0 = a w (x + tDx , Dx ) for any symbol a(x, ξ ) without remainder terms. Hence we learn L(t) =
d 1 Dxi aijw (x + tDx ) − δij Dxj + V w (x + tDx ). 2 i,j =1
We note that the principal symbol of L(t) is given by (t; x, ξ ), and the remainder term is in S(ξ −1−μ , dx 2 + dξ 2 /ξ 2 ), locally in x. Thus the operator L(t) is a quantization of (t; x, ξ ), and it generates the corresponding quantum evolution. Let Ω ⊂ R2d be a small neighborhood of (x0 , ξ0 ) as in the last section, and let f0 ∈ C0∞ (R2d ) be a smooth cut-off function such that f0 (x0 , ξ0 ) = 0 and supported in Ω. For t 0, we set g0 (t; x, ξ ) = f0 ◦ St−1 (x, ξ ). Then g0 (t; ·,·) is supported in St (supp f0 ) ⊂ St (Ω), and by the discussion of the last section and the standard argument of the classical mechanics, we observe ∂g0 (t; x, ξ ) = −{ 0 , g0 }(t; x, ξ ), ∂t where {a, b} =
d
∂a j =1 ( ∂ξj
∂b ∂a ∂b · ∂x − ∂x · ) is the Poisson bracket. It is easy to see that the support j j ∂ξj
−1 of g0 (t; ·,·) is uniformly bounded and g(t; ·,·) converges to f0 ◦ S− in the C0∞ -topology as t → −∞ (cf. Lemma 3). Then we set
ψ0 (t; x, ξ ) = g0 h−1 t; x, hξ for x, ξ ∈ Rd and t 0 with a semiclassical parameter h > 0. By the scaling property of 0 (t; x, ξ ), it is easy to see that ψ0 satisfies the same Poisson equation as g0 , and satisfies the initial condition: ψ0 (0; x, ξ ) = f0 (x, hξ ). We consider G(t) = eitH0 e−itH f0w (x, hDx )eitH e−itH0 ∈ L L2 Rd for t 0, and study the behavior of G(−t0 ) as h → 0. By straightforward computation, we learn G(t) satisfies the Heisenberg equation:
d G(t) = −i L(t), G(t) , dt
G(0) = f0w (x, hDx ).
1306
S. Nakamura / Journal of Functional Analysis 256 (2009) 1299–1309
We now construct an asymptotic solution to the equation following the standard procedure of the pseudodifferential operator theory (cf., e.g., Taylor [16] or Hörmander [6]). Lemma 4. There exists ψ(t; x, ξ ) ∈ C0∞ (R2d ), t 0, such that: (i) ψ(0; x, ξ ) = f0 (x, hξ ). (ii) ψ(t; ·,·) is supported in {(x, ξ ) | (x, hξ ) ∈ Sh−1 t (supp[f0 ])}. (iii) For any α, β ∈ Zd+ , there is Cαβ > 0 such that α β ∂ ∂ ψ(t; x, ξ ) Cαβ h|β| , x ξ
t 0, x, ξ ∈ Rd .
(iv) The principal symbol of ψ(t; x, ξ ) is ψ0 (t; x, ξ ), i.e., for any α, β ∈ Zd+ , there is Cαβ > 0 such that α β ∂ ∂ ψ(t; x, ξ ) − ψ0 (t; x, ξ ) Cαβ h|β|+μ , x ξ
t 0, x, ξ ∈ Rd .
(v) We set G(t) = ψ w (t; x, Dx ). Then
d
∞
G(t) + i L(t), G(t)
2 d =O h
dt L(L (R ))
as h → 0.
The bound is locally uniform with respect to t 0. Proof. We note (t; ·,·), 0 (t; ·,·) ∈ S(ξ 2 + x + tξ 1−μ , dx 2 + dξ 2 ), locally uniformly in t. Moreover, for any compact sets Γ ⊂ Rd , I ⊂ R− , and for any α, β ∈ Zd+ , α β ∂ ∂ 0 (t; x, ξ ) Cαβ tξ −1−μ−|α| ξ 2 ξ −1 + ttξ −1 |β| x ξ
Cαβ tξ −1−μ−|α| ξ 2−|β| ,
α β ∂ ∂ V (x + tξ ) Cαβ tξ 1−μ−|α| ξ −|β| x ξ
for x ∈ Γ , ξ ∈ Rd and t ∈ I . We choose Γ so large that ψ0 (t; x, ξ ) = 0 for all t 0, ξ ∈ Rd if x∈ /Γ. We recall ∂ ψ0 (t; x, ξ ) = −{ 0 , ψ0 }(t; x, ξ ) ∂t thanks to the construction of ψ0 . If we set r0w (t; x, Dx ) =
∂ w ψ0 (t; x, Dx ) + i L(t), ψ0w (t; x, Dx ) , ∂t
then by the asymptotic expansion formula, we have α β ∂ ∂ r0 (t; x, ξ ) Cαβ h1+|β| + h h−1 t −μ h|β| + h−1 t 1−μ h1+|β| x ξ
Cαβ hμ+|β|
for x ∈ Γ, t ∈ I, ξ ∈ Rd ,
S. Nakamura / Journal of Functional Analysis 256 (2009) 1299–1309
1307
and it is supported in supp[ψ0 (t; ·,·)] essentially (i.e., modulo O(h∞ ) terms). Then we solve the transport equation ∂ ψ1 (t; x, ξ ) + { 0 , ψ1 }(t; x, ξ ) = −r0 (t; x, ξ ) ∂t with initial condition ψ1 (0; x, ξ ) = 0. Then we have α β ∂ ∂ ψ1 (t; x, ξ ) Cαβ hμ+|β| x ξ
for x ∈ Γ, t ∈ I, ξ ∈ Rd ,
with some Cαβ , and it is also supported essentially in supp[ψ0 (t; ·,·)]. If we set
∂ w ψ1 (t; x, Dx ) + i L(t), ψ1w (t; x, Dx ) + r0w (t; x, Dx ), ∂t
r1w (t; x, Dx ) = then r1 satisfies
α β ∂ ∂ r1 (t; x, ξ ) C h2μ+|β| x ξ αβ
for x ∈ Γ, t ∈ I, ξ ∈ Rd .
We iterate this procedure to obtain ψj , j = 2, 3, . . . , that satisfy α β ∂ ∂ ψj (t; x, ξ ) Cαβ hj μ+|β| x ξ
for x ∈ Γ, t ∈ I, ξ ∈ Rd ,
and rjw (t; x, Dx ) =
∂ w ψj (t; x, Dx ) + i L(t), ψjw (t; x, Dx ) + rjw−1 (t; x, Dx ) ∂t
is O(h(j +1)μ ). We then set ψ(t; x, ξ ) ∼
∞
ψj (t; x, ξ )
j =0
in the sense of an asymptotic sum as h → 0. We may choose ψ(t; x, ξ ) so that it is supported in supp[ψ0 (t; ·,·)], since the error is O(h∞ ). Now it is straightforward to check ψ(t; x, ξ ) satisfies the required properties. 2 Proof of Theorem 1. Let u0 ∈ L2 (Rd ) and t0 > 0 as in Theorem 1. By the construction of G(t) and Lemma 4, we have
d itH −itH
itH0 −itH
N 0
G(t)e e
CN h ,
dt e e and by integrating this in t, we learn
itH −itH
0 G(t)e itH0 e −itH v − G(0)v C |t|hN
e e N
1308
S. Nakamura / Journal of Functional Analysis 256 (2009) 1299–1309
for v ∈ L2 (Rd ), t 0. By setting t = −t0 and v = u(t0 ), we obtain
G(−t0 )e−it0 H0 u0 − G(0)u(t0 ) CN hN .
(2)
Recall that G(0) = f0w (x, hDx ) and G(−t0 ) = ψ w (−t0 ; x, Dx ) with ψ(t; x, ξ ) as in / WF(u(t0 )). Then by taking f0 supported in a sufficiently Lemma 4. Now we suppose (x0 , ξ0 ) ∈ small neighborhood of (x0 , ξ0 ), we may suppose G(0)u(t0 ) = O(h∞ ) as h → 0. By (2), this implies
G(−t0 )e−it0 H0 u0 = O h∞
as h → 0.
−1 −1 Since S−h −1 t (x− , ξ− ) converges to S− (x− , ξ− ) = (x0 , ξ0 ) as h → 0, we learn 0
−1 ψ0 −t0 ; x− , h−1 ξ− = f0 S−h −1 t (x− , ξ− ) → f0 (x0 , ξ0 ) = 0. 0
Hence there exist δ, ε > 0 and h0 > 0 such that ψ(−t0 ; x, ξ ) δ > 0 if |x − x− | < ε, |hξ − ξ− | < ε, and 0 < h h0 . In other words, G(−t0 ) is microlocally elliptic at (x− , ξ− ) in the semiclassical sense. Thus these / WF(e−it0 H0 u0 ) (cf. [2,8]). imply (x− , ξ− ) ∈ / WF(e−it0 H0 u0 ). We choose a neighborhood Λ of (x− , ξ− ) Conversely, we suppose (x− , ξ− ) ∈ 2d in R such that it is conic with respect to ξ , and that a w (x, Dx )(e−it0 H0 u0 ) ∈ S(Rd ) if a(x, ξ ) ∈ 0 is supported in Λ. If we choose the support of f (x, ξ ) sufficiently small, then ψ(−t ; ·,·) S1,0 0 0 is supported in Λ for small h, since supp ψ −t0 ; ·, h−1 (·) ⊂ S−h−1 t0 supp[f0 ] and the right-hand side converges to S− (supp[f0 ]) as h → 0. Then we have
w
ψ (−t0 ; x, Dx )e−it0 H0 u0 = O h∞ as h → 0. Again by (2), we learn f0w (x, Dx )u(t0 ) = O(h∞ ) and hence (x0 , ξ0 ) ∈ / WF(u(t0 )).
2
Acknowledgments The author wishes to thank Kenji Yajima, Shin-ichi Doi and André Martinez for valuable comments and discussions. References [1] W. Craig, T. Kappeler, W. Strauss, Microlocal dispersive smoothing for the Schrödinger equation, Comm. Pure Appl. Math. 48 (1996) 769–860. [2] M. Dimassi, J. Sjöstrand, Spectral Asymptotics in the Semi-Classical Limit, London Math. Soc. Lecture Note Ser., vol. 268, Cambridge Univ. Press, Cambridge, 1999. [3] S. Doi, Smoothing effects for Schrödinger evolution equation and global behavior of geodesic flow, Math. Ann. 318 (2000) 355–389. [4] S. Doi, Dispersion of singularities of solutions for Schrödinger equations, Comm. Math. Phys. 250 (2004) 473–505.
S. Nakamura / Journal of Functional Analysis 256 (2009) 1299–1309
1309
[5] A. Hassel, J. Wunsch, The Schrödinger propagator for scattering metrics, Ann. of Math. (2) 162 (2005) 487–523. [6] L. Hörmander, Analysis of Linear Partial Differential Operators, vols. I–IV, Springer-Verlag, Berlin, 1983–1985. [7] K. Ito, S. Nakamura, Singularities of solutions to Schrödinger equation on scattering manifold, preprint, November 2007, http://arxiv.org/abs/0711.3258. [8] A. Martinez, An Introduction to Semiclassical and Microlocal Analysis, Universitext, Springer-Verlag, New York, 2002. [9] A. Martinez, S. Nakamura, V. Sordoni, Analytic smoothing effect for the Schrödinger equation with long-range perturbation, Comm. Pure Appl. Math. 59 (2006) 1330–1351. [10] A. Martinez, S. Nakamura, V. Sordoni, Analytic wave front for solutions to Schrödinger equation, preprint, June 2007, http://arxiv.org/abs/0706.0415. [11] S. Nakamura, Propagation of the homogeneous wave front set for Schrödinger equations, Duke Math. J. 126 (2005) 349–367. [12] S. Nakamura, Semiclassical singularity propagation property for Schrödinger equations, J. Math. Soc. Japan, in press. [13] M. Reed, B. Simon, The Methods of Modern Mathematical Physics, vols. I–IV, Academic Press, New York, 1972– 1980. [14] L. Robbiano, C. Zuily, Microlocal analytic smoothing effect for the Schrödinger equation, Duke Math. J. 100 (1999) 93–129. [15] L. Robbiano, C. Zuily, Analytic theory for the quadratic scattering wave front set and application to the Schrödinger equation, Soc. Math. France, Astérisque 283 (2002) 1–128. [16] M. Taylor, Pseudodifferential Operators, Princeton Math. Ser., Princeton Univ. Press, Princeton, NJ, 1981. [17] J. Wunsch, Propagation of singularities and growth for Schrödinger operators, Duke Math. J. 98 (1999) 137–186.
