Phase Space Analysis of Partial Differential Equations

Progress in Nonlinear Differential Equations and Their Applications Volume 69 Editor Haim Brezis Universit´e Pierre et Marie Curie Paris and Rutgers University New Brunswick, N.J. Editorial Board Antonio Ambrosetti, Scuola Internationale Superiore di Studi Avanzati, Trieste A. Bahri, Rutgers University, New Brunswick Felix Browder, Rutgers University, New Brunswick Luis Caffarelli, The University of Texas, Austin Lawrence C. Evans, University of California, Berkeley Mariano Giaquinta, University of Pisa David Kinderlehrer, Carnegie-Mellon University, Pittsburgh Sergiu Klainerman, Princeton University Robert Kohn, New York University P. L. Lions, University of Paris IX Jean Mawhin, Universit´e Catholique de Louvain Louis Nirenberg, New York University Lambertus Peletier, University of Leiden Paul Rabinowitz, University of Wisconsin, Madison John Toland, University of Bath

Phase Space Analysis of Partial Differential Equations

Antonio Bove Ferruccio Colombini Daniele Del Santo Editors

Birkh¨auser Boston • Basel • Berlin

Ferruccio Colombini Universit`a di Pisa Dipartimento di Matematica I-56127 Pisa Italy

Antonio Bove Universit`a di Bologna Dipartimento di Matematica I-40126 Bologna Italy Daniele Del Santo Universit`a di Trieste Dipartimento di Scienze Matematiche I-34127 Trieste Italy

Mathematics Subject Classification (2000): 32H02, 32V20, 35L15, 26A15, 37L50, 35A07, 35S50, 78A60, 35H10, 35A17, 35K25, 35A05, 35L20, 58B20, 58D05, 35Q53, 35S05, 35P25, 81Q05, 35S10, 35H05, 93B07 Library of Congress Control Number: 2006931765 ISBN-10 0-8176-4511-X ISBN-13 978-0-8176-4511-3

e-ISBN-10: 0-8176-4521-7 e-ISBN-13: 978-0-8176-4521-2

Printed on acid-free paper. c 2006 Birkh¨auser Boston


All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Birkh¨auser Boston, c/o Springer Science+Business Media LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. (KeS) 987654321 www.birkhauser.com

Sunt, fateor, nuge; sed amant quoque seria nugas: Quis faciem Socratis semper habere queat? Æneas Silvius Piccolomini, Epigrammata

Preface

The present volume is a collection of papers mainly concerning Phase Space Analysis, or Microlocal Analysis, and its applications to the theory of Partial Differential Equations (PDEs). We would like to remark that major progress has been accomplished in the analysis of PDEs over the last twenty years, both theoretical and applied; many of these accomplishments are based on the development of powerful tools of microlocal analysis. The idea, at the crossing point of harmonic analysis, functional analysis, quantum mechanics and algebraic analysis, is that many phenomena depend jointly on position and frequency (or wave numbers, or momentum) and therefore must be understood and described in the phase space. Including time leads one to work in the space-time phase space. Various methods related to microlocal analysis constitute in fact a transversal theme of several articles in this volume. Nonetheless, the topics presented in the 15 research papers appearing in this volume span a great number of different subjects: unique continuation problems; uniqueness of solutions; Carleman estimates; inverse and ill-posed problems; hypoellipticity for systems or for “sums of squares” operators; Strichartz estimates for hyperbolic or Schr¨ odinger operators; estimates from below for systems and other problems related to the Fefferman–Phong inequality; hyperbolic operators with multiple characteristic; Benjamin–Ono equations; and traces on the Heisenberg group. Moreover the present volume includes a long review paper devoted to the study of some geometric evolution equations, including the Burgers’ and Korteweg–de Vries hierarchies. The editors wish to thank all the contributors whose effort has been essential for publication of the present volume. Many of these papers, all written by leading experts in their respective fields, are expanded versions of talks given at a meeting held in November 2005 in Pienza, an old town in the Tuscanian hills of Italy. The organizers, editors of this volume, wish to thank Enea Silvio Piccolomini, Pope Pius II, to whom we owe the existence of the town of Pienza as we know it nowadays,

viii

Preface

the town itself and, in particular, the Hotel “Il Chiostro di Pienza” for the warm welcome and the wonderful atmosphere. A number of institutions made the Pienza workshop possible through their financial support. We would like to list them here: the Italian Ministero dell’Istruzione, dell’Universit` a e della Ricerca, Gruppo Nazionale per l’Analisi Matematica, la Probabilit` a e le loro Applicazioni, Universit` a di Bologna, Universit`a di Pisa and the scientific cooperation agreement between the Universities of Pisa and Paris 6. We are indebted to all of them for their generosity.

Bologna, Pisa, Trieste March 2006

Antonio Bove Ferruccio Colombini Daniele Del Santo

Contents

Preface

vii

List of Contributors

xi

Trace theorem on the Heisenberg group on homogeneous hypersurfaces Hajer Bahouri, Jean-Yves Chemin and Chao-Jiang Xu . . . . . . . . . . . . . .

1

Strong unique continuation and finite jet determination for Cauchy–Riemann mappings M. Salah Baouendi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

On the Cauchy problem for some hyperbolic operator with double characteristics Enrico Bernardi and Antonio Bove . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

On the differentiability class of the admissible square roots of regular nonnegative functions Jean-Michel Bony, Ferruccio Colombini and Ludovico Pernazza . . . . .

45

The Benjamin–Ono equation in energy space Nicolas Burq and Fabrice Planchon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

Instabilities in Zakharov equations for laser propagation in a plasma Thierry Colin and Guy M´ etivier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63

Symplectic strata and analytic hypoellipticity Paulo D. Cordaro and Nicholas Hanges . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83

x

Contents

On the backward uniqueness property for a class of parabolic operators Daniele Del Santo and Martino Prizzi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95

Inverse problems for hyperbolic equations Gregory Eskin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

107

On the optimality of some observability inequalities for plate systems with potentials Xiaoyu Fu, Xu Zhang and Enrique Zuazua . . . . . . . . . . . . . . . . . . . . . . . . . .

117

Some geometric evolution equations arising as geodesic equations on groups of diffeomorphisms including the Hamiltonian approach Peter W. Michor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

133

Non-effectively hyperbolic operators and bicharacteristics Tatsuo Nishitani . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

217

On the Fefferman–Phong inequality for systems of PDEs Alberto Parmeggiani . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

247

Local energy decay and Strichartz estimates for the wave equation with time-periodic perturbations Vesselin Petkov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

267

An elementary proof of Fedi˘ı’s theorem and extensions David S. Tartakoff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

287

Outgoing parametrices and global Strichartz estimates for Schr¨ odinger equations with variable coefficients Daniel Tataru . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

291

On the analyticity of solutions of sums of squares of vector fields Fran¸cois Treves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

315

List of Contributors

Hajer Bahouri, D´epartement de Math´ematiques, Facult´e de Sciences de Tunis, 1060 Tunis, Tunisie [email protected] M. Salah Baouendi, Department of Mathematics, University of California, San Diego, La Jolla, CA 92093-0112, USA [email protected] Enrico Bernardi, Dipartimento di Matematica per le Scienze Economiche e Sociali, Universit` a di Bologna, Viale Filopanti 5 40126 Bologna, Italia [email protected] ´ Jean-Michel Bony, Centre de Math´ematiques, Laurent Schwartz, Ecole Polytechnique, 91128 Palaiseau, France [email protected] Antonio Bove, Dipartimento di Matematica, Universit` a di Bologna, Piazza di Porta San Donato 5, 40126 Bologna, Italia [email protected] Nicolas Burq, D´epartement de Math´ematiques, Universit´e Paris 11, 91405 Orsay, France [email protected] Jean-Yves Chemin, Laboratoire Jacques-Louis Lions, and CNRS UMR 7598, Universit´e Pierre et Marie Curie-Paris 6, 75005 Paris, France [email protected]

xii

List of Contributors

Thierry Colin, D´epartement de Math´ematiques Appliqu´ees, and CNRS UMR 5466, Universit´e Bordeaux 1, 351, Cours de la Lib´eration, 33405 Talence, France [email protected] Ferruccio Colombini, Dipartimento di Matematica Universit` a di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italia [email protected] Paulo D. Cordaro, Departamento de Matem´atica, Instituto de Mathem´ atica e Estatistica, University of S˜ ao Paulo, 05315 S˜ ao Paulo SP, Brazil [email protected] Daniele Del Santo, Dipartimento di Matematica e Informatica, Universit` a di Trieste, Via Valerio 12/1, 34127 Trieste, Italia [email protected] Gregory Eskin, Department of Mathematics, University of California, Los Angeles, Los Angeles, CA 90024, USA [email protected] Xiaoyu Fu, School of Mathematics, Sichuan University, Chengdu 610064, China rj [email protected] Nicholas Hanges, Department of Mathematics and Computer Science, Herbert H. Lehman College, CUNY, Bronx, NY 10468, USA [email protected] Guy M´etivier, D´epartement de Math´ematiques Appliqu´ees, and CNRS UMR 5466, Universit´e Bordeaux 1, 351, Cours de la Lib´eration, 33405 Talence, France [email protected] Peter W. Michor, Fakult¨ at f¨ ur Mathematik, Universit¨ at Wien, Nordbergstrasse 15, A-1090 Wien, Austria [email protected] and Erwin Schr¨ odinger International Institute of Mathematical Physics, Boltzmanngasse 9, A-1090 Wien, Austria [email protected]

List of Contributors

xiii

Tatsuo Nishitani, Department of Mathematics, Osaka University, Toyonaka, Osaka 560, Japan [email protected] Alberto Parmeggiani, Dipartimento di Matematica, Universit` a di Bologna, Piazza di Porta San Donato 5, 40126 Bologna, Italia [email protected] Ludovico Pernazza, Dipartimento di Matematica, Universit` a di Pavia, Via Ferrata 1, 27100 Pavia, Italia [email protected] Vesselin Petkov, D´epartement de Math´ematiques Appliqu´ees, Universit´e Bordeaux 1, 351, Cours de la Lib´eration, 33405 Talence, France [email protected] Fabrice Planchon, Laboratoire Analyse, G´eom´etrie & Applications, and CNRS UMR 7539, Institut Galil´ee, Universit´e Paris 13, 99 avenue J.B. Cl´ement, 93430 Villetaneuse, France [email protected] Martino Prizzi, Dipartimento di Matematica e Informatica, Universit` a di Trieste, Via Valerio 12/1, 34127 Trieste, Italia [email protected] David S. Tartakoff, Department of Mathematics, University of Illinois at Chicago, 851 So. Morgan St., Chicago, IL 60607, USA [email protected] Daniel Tataru, Department of Mathematics, University of California, Berkeley, Berkeley, CA 94720, USA [email protected] Fran¸cois Treves, Mathematics Department, Rutgers University, New Brunswick, NJ 08854, USA [email protected] Chao-Jiang Xu, Laboratoire de Math´ematiques, and CNRS UMR 6085, Universit´e de Rouen, Avenue de l’Universit´e, BP 12, ´ 76801 Saint-Etienne du Rouvray, France [email protected]

xiv

List of Contributors

Xu Zhang, Yangtze Center of Mathematics, Sichuan University, Chengdu 610064, China and Departamento de Matem´aticas, Facultad de Ciencias, Universidad Aut´ onoma de Madrid, 28049 Madrid, Spain [email protected] Enrique Zuazua, Departamento de Matem´aticas, Facultad de Ciencias, Universidad Aut´ onoma de Madrid, 28049 Madrid, Spain [email protected]

Trace theorem on the Heisenberg group on homogeneous hypersurfaces Hajer Bahouri1 , Jean-Yves Chemin2 and Chao-Jiang Xu3 1 2 3

D´epartement de Math´ematiques, Facult´e de Sciences de Tunis, Tunisie Laboratoire Jacques-Louis Lions and CNRS UMR 7598, Universit´e Pierre et Marie Curie-Paris 6, France Laboratoire de Math´ematiques, Universit´e de Rouen, France

Summary. We prove in this work the trace and trace lifting theorem for Sobolev spaces on the Heisenberg groups for homogeneous hypersurfaces.

2000 Mathematics Subject Classification: 35 A, 35 H, 35 S. Key words: Trace and trace lifting, Heisenberg group, H¨ ormander condition, Hardy’s inequality.

1 Introduction In this work, we continue the study of the problem of restriction of functions that belongs to Sobolev spaces associated to left invariant vector fields for the Heisenberg group Hd initiated in [3]. As observed in [3], the case when d = 1 is not very different from the case when d ≥ 2, but the statement in this particular case is less pleasant. Thus, for the sake of simplicity, we shall assume from now on that d ≥ 2. Let us recall that the Heisenberg group is the space R2d+1 of the (non-commutative) law of product w · w
= (x, y, s) · (x
, y
, s
) = (x + x
, y + y
, s + s
+ (y|x
) − (y
|x)). The left invariant vector fields are Xj = ∂xj + yj ∂s , Yj = ∂yj − xj ∂s , j = 1, . . . , d and S = ∂s =

1 [Yj , Xj ]. 2

In all that follows, we shall denote by Z this family and state Zj = Xj and Zj+d = Yj for j in {1, . . . , d}. Moreover, for any C 1 function f , we shall state def

∇H f = (Z1 · f, . . . , Z2d · f ).

2

Hajer Bahouri, Jean-Yves Chemin and Chao-Jiang Xu

The key point is that Z satisfies H¨ormander’s condition at order 2, which means that the family (Z1 , . . . , Z2d , [Z1 , Zd+1 ]) spans the whole tangent space T R2d+1 . For k ∈ N and V an open subset of Hd , we define the associated Sobolev space as H k (Hd , V ) = {f ∈ L2 (R2d+1 ) | Supp f ⊂ V def




and ∀α / |α| ≤ k , Z α f ∈ L2 (R2d+1 )}, def

where if α ∈ {1, . . . , 2d}k , |α| = k
and Z α = Zα1 · · · Zαk
. As in the classical case, when s is any real number, we can define the function space H s (Hd ) through duality and complex interpolation, Littlewood–Paley theory on the Heisenberg group (see [4] and [5]), or Weyl–H¨ ormander calculus (see [7], [8] and [9]). It turns out that these spaces have properties which look very much like the ones of usual Sobolev spaces, see [3] and their references. The purpose of this paper is to study the problems of trace and trace lifting on a smooth hypersurface of Hd in the frame of Sobolev spaces. Let us point out that the problem of existence of trace appears only when s is less than or equal to 1. Indeed, under the subellipicity of system Z, the space s H s (Hd ) is included locally in H 2 (R2d+1 ). So if s is strictly larger than 1, this implies that the trace on any smooth hypersurface exists and belongs locally s 1 to the usual Sobolev space H 2 − 2 of the hypersurface. Thus the case when s = 1 appears as the critical one. It is the case we study here. 1.1 Statement of the results Two very different cases then appear: first, when the hypersurface is noncharacteristic, which means that any point w0 of the hypersurface Σ is such that Z|w0 ⊂ Tw0 Σ, and second, when some point w0 of the hypersurface Σ is characteristic, which means that Z|w0 ⊂ Tw0 Σ. The non-characteristic case is now well understood. In [3], we give a full account of trace and trace lifting results on smooth non-characteristic hypersurfaces for s > 1/2. This result generalizes various previous results (see among others [6], [10] and [11]). Let us recall this theorem in the case of H 1 (see [3] for the details). If w0 is any non-characteristic point of Σ, then there exists at least one of the vector fields Z1 , . . . Z2d which is transverse to Σ at w0 . We denote by XΣ the subspace of T Σ defined, for w in Σ, by XΣ|w = Tw Σ ∩ X |w where X is the C ∞ -module of vector fields spanned by {Z1 , . . . , Z2d }. It is easily checked that, if g is a local defining function of Σ, the family def

Rj,k = (Zj · g)Zk − (Zk · g)Zj

Trace theorem on the Heisenberg group

3

generates XΣ and that it satisfies the H¨ ormander condition at order 2 (see for instance Lemma 4.1 of [3]). We define H k (Σ, ZΣ ) = {f ∈ L2 (Σ) | Supp f ⊂ V ∩ Σ

and ∀(j, k) , Rj,k u ∈ L2 (Σ)}.

We have proved the following trace and trace lifting theorem in [3]. Theorem 1.1 Let us suppose that Σ is non-characteristic on an open subset V of Hd . Then the trace operator on Σ denoted by γ Σ is an onto continuous def

1

map from H 1 (Hd , V ) onto [H 1 (Σ, ZΣ ), L2 (Σ)] 12 = H 2 (Σ, ZΣ ). Remark 1.1 As the system ZΣ satisfies the H¨ormander condition at order 2, Theorem 1.1 implies in particular that γ Σ maps H 1 (Hd , V ) into the classical Sobolev space H 1/4 (Σ, V ∩ Σ). We shall now consider the characteristic case. The set of characteristic points of Σ, Σc = {w ∈ Σ / Z|w ⊂ Tw Σ}, may have a complicated structure. For the sake of simplicity, we shall only consider here a particular case. By translation in the Heisenberg group, we work only in a neighborhood of w0 = (0, 0, 0). Near w0 , the hypersurface Σ can always be written as def

Σ = {w = (x, y, s) / g(w) = s − f (x, y) = 0} with f (0, 0) = 0 and Df (0, 0) = 0. From now on, we assume that f is a homogeneous polynomial of degree 2 on R2d . In this case, the equation is homogeneous of order 2 with respect to the dilation of Heisenberg group def

dλ (x, y, s) = (λx, λy, λ2 s). Then the set of characteristic points Σc is a submanifold defined by Σc = {w = (s, x, y) ∈ Hd / g(w) = 0 and Lj , (x, y) = 0} where Lj is the linear form (on R2d ) defined by Zj (g)(x, y). Let us denote by r the rank of the family (Lj )1≤j≤2d of linear forms on R2d . Let us observe that r is also the rank of the matrix (Zi · Zj · g)1≤i,j≤2d at w0 . Let us notice that if i ∈ {1, . . . , d} and j = i + d, (Zi · Zi+d · g)(w0 ) − (Zi+d · Zi · g)(w0 ) = −2∂s g(w0 ) = −2 and (Zi · Zj · g) = (Zj · Zi · g), then the rank of the matrix (Zi · Zj · g)1≤i,j≤2d and thus of (Lj )1≤j≤2d is at least d. From now on, we always consider this case for the sake of simplicity. Let us introduce some rings of functions adapted to our situation.

4

Hajer Bahouri, Jean-Yves Chemin and Chao-Jiang Xu

Definition 1.1 Let W be any open subset of Σ and F a closed subset of W . Let us denote by CF∞ (W ) the set of smooth functions a on W \ F such that for any multiindex α, a constant Cα exists such that |∂ α a(z)| ≤ Cα d(z, F )−|α| , where d denotes the distance on Σ induced by the euclidian distance on R2d+1 . Now let us define the vector fields on Σ which will describe the regularity on Σ. Definition 1.2 Let W be a neighborhood of w0 . We denote by ZΣ the ∞ (W ) modulus spanned by the set vector fields of Z ∩ T Σ|W that vanish CΣ c on Σc . As we shall see in Proposition 3.1, the modulus ZΣ is of finite type (of ∞ course as a CΣ (W ) modulus) if w0 is a regular characteristic point and W c is chosen small enough. If g is a local defining function of Σ, a generating system is given by def

Rj,k = (Zj · g)Zk − (Zk · g)Zj

for 1 ≤ j < k ≤ 2d.

(1.1)

Now we are ready to introduce the space of traces. Definition 1.3 Let W be a small enough neighborhood of w0 . We denote by H 1 (ZΣ , W ) the space of functions v of L2 (Σ) supported in W such that


def

v2H 1 (ZΣ ) = v2L2 (Σ) +

Rj,k v2L2 (Σ) < ∞

1≤j,k≤2d

where the family (Rj,k )1≤j,k≤2d is given by (1.1). If s ∈ [0, 1], we define H s (ZΣ , V ) by complex interpolation. Our theorem is the following. Theorem 1.2 Let V be a small enough neighborhood of w0 . Then the restric1 tion map γ Σ is an onto continuous map from H 1 (Hd , V ) onto H 2 (ZΣ , V ∩ Σ). Let us remark that, if r = 2d, this theorem is a particular case of Theorem 1.8 of [3]. 1.2 Structure of the proof In our paper [3], which corresponds to the case when r = 2d as thus Σc = {w0 }, we use a blowup of the point w0 . Here we blow up the submanifold Σc . In order to do it, let us introduce a function ϕ ∈ D(R \ {0}) such that ∀t ∈ [−1, 1] \ {0} ,

∞
p=0

ϕ(2p t) = 1.

(1.2)

Trace theorem on the Heisenberg group def

5

1

Let us define the function ρc by ρc = (g 2 + |∇H g|4 ) 4 . Now writing that for any function u in L2 (ρc ≤ 1), u=

∞


ϕp u with

def

ϕp (w) = ϕ(2p ρc (w)),

(1.3)

p=0

we apply Theorem 1.1 of trace and trace lifting to each piece ϕp u which is supported in a domain where Σ is non-characteristic because ρc ∼ 2−p in this domain. This decomposition leads immediately to the problem of estimating the norm H 1 (Hd ) of each piece ϕp u. Leibniz’s formula and the chain rule tell us that ∇H (ϕp u) = ϕp ∇H u + 2p ϕ
(2p ρc )u∇H ρc . Let us observe that, as Zj ρ4c = 2g(Zj · g) + 4|∇H g|2

2d


(Zk · g)Zj · (Zk · g) ,

k=1

we have, for any real number s, |∇H ρsc | ≤ Cs ρs−1 . As the support of ϕ
(2p ρc ) c
−p
p is included in ρc ∼ 2 , the supports of ϕ (2 ρc ) and ϕ
(2p ρc ) are disjoint if
|p − p | ≥ N0 for some N0 . Thus, we get that ∞





2 ϕ (2 2p

p=0

p

ρc )u∇H ρc 2L2

 2 u  ≤C ρ  2 . c L

This leads to the proof of the following Hardy type inequality. Theorem 1.3 A neighborhood V of w0 exists such that, for any u in the space H 1 (Hd , V ) of H 1 (Hd ) functions supported in V ,  1 u2 dw ≤ C∇H u2L2 , with ρc = (g 2 + |∇H g|4 ) 4 . 2 Hd ρc This theorem implies that, for any u in H 1 (Hd , V ), ∞


∇H (ϕp u)2L2 ≤ C∇H u2L2 .

(1.4)

p=0

The proof of this theorem, which is the core of this work, is the purpose of the second section. In the third section, after dilation, we apply Theorem 1.1. This gives a rather unpleasant description of the trace space. Then, we use an interpolation result which allows us to conclude the proof of Theorem 1.2.

6

Hajer Bahouri, Jean-Yves Chemin and Chao-Jiang Xu

2 A Hardy type inequality 2.1 The classical Hardy inequality As a warmup, let us recall briefly the usual proof of the classical Hardy inequality.1  u2 1 dw ≤ C∇H u2L2 with ρ(w) = (s2 + (|x|2 + |y|2 )2 ) 4 . (2.1) 2 ρ d H As D(Hd \ {0}) is dense in H 1 (Hd ), we restrict ourselves to functions u in D(Hd \ {0}). Then the proof mainly consists in an integration by parts with respect to the radial vector field RH adapted to the structure of Hd , namely def

RH = 2s∂s +

d


(xj ∂xj + yj ∂yj ) = s[Y1 , X1 ] +

j=1

d


(xj Xj + yj Yj ),

j=1

once we have noticed that RH · ρ−2 = −2ρ−2 and div RH = 2d + 2. More precisely, this gives    
 2   d u xj s yj u X Y dw = + −d udw − Y u(X1 u)dw j j 1 ρ2 ρ ρ ρ ρ2 j=1 +

   s X1 2 u(Y1 u)dw. ρ

   As we have Zj ρs2  ≤ Cρ−1 , the Cauchy–Schwarz inequality gives (2.1). 2.2 Construction of replacement for ρ and RH The classical case studied above corresponds to the case when r = 2d. Let us assume from now on that r < 2d and let us consider (Lj )1≤≤r a basis of the vector space generated by (Lj )1≤j≤2d . First, we have the following lemma. Lemma 2.1 A couple of vector fields (Z0 , Z 0 ) exists in (Z \ {Zj1 , . . . , Zjr })× (±Z) such that [Z0 , Z 0 ] = 2∂s

and

D(Z 0 · g)(w0 ) = 0.

Proof (Proof of Lemma 2.1). Let us consider Z0 ∈ Z \ {Zj1 , . . . , Zjr }, and Z 0 in ±Z such that [Z0 , Z 0 ] = 2∂s . If ±Z 0 belongs to {Zj1 , . . . , Zjr }, we infer from the definition of the family (Lj )1≤≤r that D(Z 0 · g)(w0 ) is different from 0 and then Z 0 = Z 0 fits. If ±Z 0 is not in {Zj1 , . . . , Zjr }, as (Z0 · (Z 0 · g))(w0 ) − (Z 0 · (Z0 · g))(w0 ) = 2, either D(Z0 ·g)(w0 ) or D(Z 0 ·g)(w0 ) is different from 0. Thus if D(Z 0 ·g)(w0 ) = 0, we get the lemma interchanging the roles of Z 0 and Z0 . 1

For a different approach based on Fourier analysis, see [2].

Trace theorem on the Heisenberg group

7

Let us state the following theorem, which immediately implies the Hardy type inequality stated in Theorem 1.3. Theorem 2.1 There exists a neighborhood V of w0 such that, for any u in H 1 (Hd , V ),  2 1 u def dw ≤ C∇H u2L2 with ρ0 = (g 2 + (Z 0 · g)4 ) 4 . ρ20 The above inequality is obviously better than that in Theorem 1.3 and it is surprisingly the one we are able to prove. Proof (Proof of Theorem 2.1). By definition of the family (Lj )1≤≤r , a family of real numbers (α )1≤≤r exists such that Z0 · g =

r


α (Zj · g).

(2.2)

=1

Let us, in our situation, define a field analogous to RH , 1 0 R1 = 2g∂s + (Z 0 · g)Z 2


0 def with Z = Z0 − α Zj . r

(2.3)

=1

In order to check that R1 is analogous to the radial field in the case of the classical Hardy inequality, let us prove that R1 · ρ40 = 4ρ40

and

div R1 = 3.

(2.4)

By definition of the function ρ0 , we have R1 · ρ40 = 2g(R1 · g) + 4(Z 0 · g)3 (R1 · (Z 0 · g)). 0 is tangent to Σ. Using that ∂s g ≡ 1, we get Equality (2.2) implies that Z that R1 · g = 2g. Let us compute R1 · (Z 0 · g). As ∂s (Z 0 · g) = 0, we have R1 · (Z 0 · g) =

1 0 · (Z 0 · g)). (Z 0 · g)(Z 2

Then we have R1 · ρ40 = 4ρ40 . Let us notice that Z0 does not belong to the family (Zj )1≤≤r . Thus Z 0 commutes with the vector fields Zj . By definition 0 , we infer that of Z 0 , Z 0 ] = [Z0 , Z 0 ] + [Z

r


α [Z , Z 0 ] = 2∂s .

(2.5)

=1

0 · g = 0. Thus we get 0 , we have Z By definition of Z 0 · g) + 2∂s g = 2. Z 0 · (Z 0 · g) = Z 0 · (Z

(2.6)

8

Hajer Bahouri, Jean-Yves Chemin and Chao-Jiang Xu

It turns out that R1 · ρ41 = 4g 2 + 4(Z 0 · g)4 = 4ρ40 . Now, let us compute div R1 . We have 1 div R1 = 2∂s g + Z 0 · (Z 0 · g) + (Z 0 · g) div Z0 . 2 Using that the vector fields Zj are divergence free, the fact that ∂s g ≡ 1 and (2.6), we get that div R1 = 3. Thus assertion (2.4) is proved. In order to continue the proof of Theorem 2.1, let us observe that, near d w0 , the set ρ−1 0 (0) is a submanifold of H of codimension 2. The following lemma will allow us to assume all along the proof that u belongs to D(V \ ρ−1 0 (0)). Lemma 2.2 Let V be a bounded domain of Hd and Γ a submanifold of codimension ≥ 2. Then D(V \ Γ ) is dense in the space H01 (Hd , V ) of functions of H01 (Hd ) supported in V equipped with the norm 1

(u2L2 + ∇H u2L2 ) 2 . Proof (Proof of Lemma 2.2). As H01 (Hd , V ) is a Hilbert space, it is enough to prove that the orthogonal of D(V \ Γ ) is {0}. Let u be in this space. For any v in D(V \ Γ ), we have (u|v)L2 + (∇H u|∇H v)L2 = 0. By integration by parts, this implies that ∀v ∈ D(V \ Γ ) , u − ∆H u, v = 0. Thus the support of u − ∆H u is included in Γ . As Zj u belongs to L2 , then Zj2 u belongs to H −1 (R2d+1 ) (the classical Sobolev space). And except for 0, no distribution of H −1 (R2d+1 ) can be supported in a submanifold of codimension greater than 1. Thus u−∆H u = 0. Taking the L2 scalar product with u implies that u ≡ 0. Thanks to equality (2.3), we have 1 −2 ρ−2 0 = − R1 · ρ0 . 2

(2.7)

Thus by integration by parts, we have, using equality (2.3),   2  2 3 u u u def dw = dw + I with I = (R1 · u)dw. ρ20 2 ρ20 ρ20 In order to estimate I, which contains terms of the type g∂s u, we have to compute the vector field R1 in term of elements of Z. Using (2.5), we infer that 0 , Z 0 ] + 1 (Z 0 · g)Z 0 . R1 = 2g[Z 2

Trace theorem on the Heisenberg group

9

We deduce that I = J1 + J2 with  u (Z 0 · g) def 1 J1 = (Z0 · u)dw 2 ρ0 ρ0  u def J2 = g[Z0 , Z 0 ] · udw. ρ20

and

By definition of ρ0 , the Cauchy–Schwarz inequality yields   u  |J1 | ≤ C   ρ  2 ∇H uL2 . 0 L

(2.8)

The estimate about J2 is a little bit more difficult to obtain. Let us write that J2 = K1 − K2 with   u u def def 0 · u)dw. g Z0 · (Z 0 · u)dw and K2 = gZ0 · (Z K1 = 2 ρ0 ρ20 By integration by parts, we have K1 = −K11 − K12 with  g def K11 = (Z0 · u)(Z 0 · u)dw ρ20    u g def def and K12 = f (Z 0 · u)dw with f = ρ0 Z0 · 2 · ρ0 ρ0 By definition of ρ0 , it is obvious that |K11 | ≤ C∇H u2L2 .

(2.9)

0 · g = 0, we get Using that Z     0 · g  = 2g |Z 0 · (Z 0 · g)||Z 0 · g|3 ≤ C g ≤ C · Z   2 6 ρ0 ρ0 ρ30 ρ0 This ensures that f is bounded on V and thus by the Cauchy–Schwarz inequality,   u  K12  ≤ C   ρ  2 ∇H uL2 . 0 L Together with (2.9), this proves that    u  + ∇H uL2 ∇H uL2 . |K1 | ≤ C  ρ  2 0 L

(2.10)

10

Hajer Bahouri, Jean-Yves Chemin and Chao-Jiang Xu

In order to estimate K2 , let us write that integrating by parts,     g g u (Z 0 · u)(Z0 · u)dw + ρ0 Z 0 · 2 (Z0 · u)dw. K2 = 2 ρ0 ρ0 ρ0 Using that Z 0 · ρ40 = 2g(Z 0 · g) + 4(Z 0 · (Z 0 · g))(Z 0 · g)3 ,

 we immediately get that the function ρ0 Z 0 · ρg2 is bounded on V and we 0 deduce that    u  |K2 | ≤ C   ρ  2 + ∇H uL2 ∇H uL2 . 0 L Together with (2.8) and (2.10), we infer that  u  ρ

0

2     ≤C u ρ  2

0

L

   



L2

+ ∇H uL2 ∇H uL2

which concludes the proof of Theorem 2.1.

3 The proof of the trace and trace lifting theorem 3.1 Some preliminary properties Proposition 3.1 A neighborhood W of w0 exists such that the CΣc (W ) modulus ZΣ spanned by the vector fields of Z ∩ T Σ|W which vanish on the characteristic submanifold Σc is of finite type and generated by def

Rj,k = (Zj · g)Zk − (Zk · g)Zj . Proof (Proof of Proposition 3.1). It is enough to prove that any element L of ∞ Z ∩ T Σ which vanishes on Σc is a combination (with coefficients in CΣ (W )) c of the Rj,k . By definition L=

2d
j=1

β j Zj

with

β j |Σ = 0 c

and

2d


β j (Zj · g) = 0.

j=1

)1≤j≤2d of the sphere S2d−1 such that Let us introduce a partition of unity (ψ j is included in the set of ζ of S2d−1 such that |ζ | ≥ (4d)−1 . the support of ψ j j Let us state that   ∇H g def ψj = ψ · j |∇H g| ∞ (W ). On It is an exercise left to the reader to check that ψ j belongs to CΣ c Σ \ Σc , we have, for any j in {1, . . . , 2d},

Trace theorem on the Heisenberg group

ψ j (L · g) =

2d


11

ψ j β k (Zk · g) = 0.

k=1

By definition of ψ j , (Zj · g) does not vanish on the support of ψ j . Thus we have
1 β j ψj = − ψ j β k (Zk · g). (Zj · g) k=j

From this, we deduce that ψj L =


k=j

=

  (Zk · g) Zj ψ j β k Zk − (Zj · g)


ψj β k ((Zj · g)Zk − (Zk · g)Zj ). (Zj · g) k=j

∞ and (Zj · g) do not vanish on the support of ψ j Now the facts that β k ∈ CΣ c ensure that def ϕj β k ∞ ∈ CΣ . β j,k = c (Zj · g)

So we have




L=

β j,k ((Zj · g)Zk − (Zk · g)Zj )

1≤j
and the proposition is proved. 3.2 The blowup procedure Using (1.3), Theorem 1.3 and its consequence (1.4), we have ∞


(∇H (ϕp u)2L2 + 22p ϕp u2L2 ) ≤ C∇H u2L2 .

(3.1)

p=0

Now let us use the Heisenberg dilation d22p and state that def

up (w) = ϕ0 (w)u(2p x, 2p y, 22p s). From (3.1), we immediately infer that ∞


2−2pd up 2H 1 (Hd ) ≤ C∇H u2L2 .

(3.2)

p=0

On the support of ϕ0 , the hypersurface defined by g(w) = s − f (x, y) = 0 (which is invariant under the action of Heisenberg dilation) is non-characteristic because on the support of ϕ0 , we have that ρc is between two positive constants. Thus, on g = 0, we have that |∇H g| is between two positive constants.

12

Hajer Bahouri, Jean-Yves Chemin and Chao-Jiang Xu

Thus Theorem 1.2 implies that a constant C exists (independent of p) such that γ(up )[L2 (Σ),H 1 (ZΣ ,Σ)] 1 ≤ Cup H 1 (Hd ) (3.3) 2

where [A, B]θ denotes the complex interpolation between A and B. We use the Heisenberg dilation dλ and the fact that ∀λ > 0 , ∀v , v ◦ dλ L2 (Σ) = λ−2d vL2 (Σ)

and

v ◦ dλ H 1 (ZΣ ,Σ) = λ−2d vH 1 (ZΣ ,Σ) .

(3.4)

Then, using (3.1)–(3.3), we get that ∞


γ(ϕp u)2[L2 (Σ),H 1 (ZΣ ,Σ)] 1 ≤ C∇H u2L2 . 2

p=0

Moreover, an integer N0 exists such that, if |p − p
| ≥ N0 , then the supports of ϕp and ϕp
are disjoint. Thus, we have in particular that γ(u)L2 (Σ) ≤ C∇H u2L2 . s Stating ϕΣ p = ϕp |Σ , let us define the following space T (Σ).

Definition 3.1 For s ∈ [0, 1], let us state


def def ϕΣ T s (Σ) = v ∈ L2 / v2T s (Σ) = p v[L2 (Σ),H 1 (ZΣ ,Σ)]s < ∞ . p

We have (almost) proved the following theorem. Theorem 3.1 The restriction map on the hypersurface Σ can be extended in 1 a continuous onto map from H 1 (Hd {ρc ≤ 1}) onto the space T 2 (Σ). Proof (Proof of Theorem 3.1). The only thing we still have to prove is the 1 fact that γ is onto. Let us consider a function v in T 2 (Σ). By definition of 1 T 2 (Σ), let us write ∞
ϕΣ v= p v. p=0

def

Stating vp = ϕΣ 0 v ◦ d2p , we infer from (3.4) that ∞


22pd vp 2[L2 (Σ),H 1 (ZΣ ,Σ)] 1 ≤ Cv2 1

p=0

2

T

2

(Σ)

.

As the support ϕΣ 0 , and thus the support of the functions vp , is included in the set |∇H g| ∼ 1, Theorem 1.1 tells us that a function up exists in H 1 (Hd ) such that γ(up ) = vp

and  up H 1 (Hd ) ≤ Cvp [L2 (Σ),H 1 (ZΣ ,Σ)] 1 . 2

Trace theorem on the Heisenberg group

13

Let us consider a function ϕ 0 such that the support of ϕ 0 is included in a set where ρc ∼ 1 and ϕ 0 has value 1 near the support of ϕ0 . We obviously have γ( ϕ0 up ) = vp

and  ϕ0 up H 1 (Hd ) ≤ C up H 1 (Hd ) . def

def

Then using (3.4) and stating ϕ p = ϕ 0 ◦ d2p , and u p = up ◦ d2p , we get after dilation that ∞


(∇H ( ϕp u p )2L2 + 22p  ϕp u p 2L2 ) ≤ Cv2

T

p=0

1 2

(Σ)

.

p As an integer N1 exists such that if |p − p
| ≥ N1 , then the supports of ϕ and ϕ p
are disjoint, we have that ∞


ϕ pu p ∈ H 1 (Hd ) and

p=0

 2 ∞ 
   ϕ pu p     p=0

≤ Cv2

T

1 2

(Σ)

.

H 1 (Hd )

This ends the proof of Theorem 3.1. Remark 3.1 The trace lifting theorem provides functions in H 1 (Hd ) the support of which is included in a set of the form s2 ≤ C(|x|2 + |y|2 )2 . Using this method obviously prevents us from proving the trace lifting theorem for very regular (for instance continuous) functions. The description given by Theorem 3.1 is not totally satisfactory. We want to describe this space of traces as an interpolation space to get Theorem 1.2. 3.3 Conclusion of the proof of Theorem 1.2 A theorem in interpolation theory asserts that T s (Σ) = [L2 (Σ), H 1 (ZΣ , Σ)]s . This is a consequence of the following two lemmas, the proofs of which are omitted (we refer to [1] for the details). Lemma 3.1 The space T 1 (Σ) is equal to H 1 (ZΣ , Σ) and the norms are equivalents. Lemma 3.2 Let us consider (Hj ,  · j )j∈{0,1} two Hilbert spaces such that H1 is densely included in H0 and a family (Hj,p )(j,p)∈{0,1}×N such that, for any p, Hj,p is a closed subset of Hj . Let us assume that a family of (Λp )p∈N of (unbounded) selfadjoint operators on H0,p exists such that H1,p equals the domain of Λp and ∀u ∈ H1,p , uH1 ∼ Λp uH0 .

(3.5)

14

Hajer Bahouri, Jean-Yves Chemin and Chao-Jiang Xu

Let us assume in addition that a family of operators (Ap )p∈N exists such that, for any (j, p) in {0, 1} × N, the operator Ap is continuous from Hj into Hj,p and   N  

  ∀v ∈ Hj , lim v − Ap v  = 0 and v2Hj ∼ Ap v2Hj . (3.6) p→∞   p p=0

Hj

Then, [H0 , H1 ]s =

v ∈ H0 /

∞


Ap v2Hs,p

<∞

with

def

Hs,p = [H0,p , H1,p ]s .

p=0

4 Concluding remarks Actually, Theorem 1.2 is valid in a more general situation than on a homogeneous hypersurface. But it remains necessary to make some assumptions about the nature of the characteristic set Σc . In fact, this characteristic must be a hypersurface, the codimension of which fits with the rank of the matrix (Zi ·Zj ·g(w0 ))1≤i,j≤2d . More precisely, we introduce in [1] the following notion of a regular characteristic point. Definition 4.1 A characteristic point w0 of a hypersurface Σ is a regular point of order r if and only if i) for any local defining function g of Σ, the rank of the matrix (Zi · Zj · g(w0 ))1≤i,j≤2d is equal to r; ii) near w0 , the characteristic set Σc is a submanifold of Σ of codimension r in Σ. Let us notice that, as w0 is a characteristic point, for any defining function g of Σ, Zj·g (w0 ) = 0 for all j. Thus the first condition of the definition does not depend on the choice of the function g. In this more general situation, the strategy of the proof is the same as in the present simplified case, but the proofs are more delicate because the functions Zj·g are of course no longer in linear form and the hypersurface Σ is not invariant under the action of the Heisenberg dilation dλ . We refer to [1] for complete proofs in this case.

References 1. H. Bahouri, J.-Y. Chemin and C.-J. Xu, Trace theorem on the Heisenberg group, preprint du Laboratoire J.-L. Lions, Universit´e de Paris VI, 2006. 2. H. Bahouri, J.-Y. Chemin and I. Gallagher, Precise Hardy inequality on ´ Polytechnique, Rd and on the Heisenberg group Hd , S´eminaire EDP de l’Ecole 2005.

Trace theorem on the Heisenberg group

15

3. H. Bahouri, J.-Y. Chemin and C.-J. Xu, Trace and trace lifting theorems in weighted Sobolev space, J. Inst. Math Jussieu 4(2005), 509–552. 4. H. Bahouri and I. Gallagher, Paraproduit sur le groupe de Heisenberg et applications, Rev. Math. IberoAmericana 17(2001), 69–105. 5. H. Bahouri, P. G´ erard and C.-J. Xu, Espaces de Besov et estimations de Strichartz g´en´eralis´ees sur le groupe de Heisenberg, J. Anal. Math. 82(2000), 93–118. 6. S. Berhanu and I. Pesenson, The trace problem for vector field satisfying H¨ ormander’s condition, Math. Z. 231(1999), 103–122. 7. J.-M. Bony and J.-Y. Chemin, Espaces fonctionnels associ´es au calcul de Weyl–H¨ ormander , Bull. Soc. Math. France 122(1994), 77–118. 8. C. E. Cancelier, J.-Y. Chemin and C.-J. Xu, Calcul de Weyl–H¨ ormander et op´erateurs sous-elliptiques, Ann. Inst. Fourier (Grenoble) 43(1993), 1157–1178. 9. J.-Y. Chemin and C.-J. Xu, Inclusions de Sobolev en calcul de Weyl– ´ H¨ ormander et champs de vecteurs sous-elliptiques, Ann. Sci. Ecole Norm. Sup. 30(1997), 719–751. 10. D. Danielli, N. Garofalo and D-M. Nhieu, Trace inequalities for Carnot– Carath´ edory spaces and applications, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 27(1998), 195–252. 11. I. Pesenson, The trace Problem and Hardy operator for non-isotropic function spaces on the Heisenberg group, Comm. Partial Differential Equations 19(1994), 655–676.

Strong unique continuation and finite jet determination for Cauchy–Riemann mappings M. Salah Baouendi Department of Mathematics, University of California, San Diego, USA

Summary. We give strong unique continuation and finite jet determination results for mappings between CR manifolds. Applications to the study of groups of local and global CR automorphisms are derived.

2000 Mathematics Subject Classification: 32H02, 32V20, 32V35, 32V40, 35F99. Key words: CR manifold, CR mappings, generic submanifolds, unique continuation, finite jet determination.

1 Introduction An (abstract) CR manifold is a smooth manifold M equipped with a subbundle V ⊂ CT M , the complexified tangent bundle of M , which satisfies the following two conditions: (a) V ∩ V = 0, (b) [V, V] ⊂ V. Here V denotes the complex conjugate of V, and condition (b) means that the Lie bracket of any two smooth sections of V is also a section of V. The subbundle V is called the CR bundle of M . If n is the fiber complex dimension of V and m := dim R M , then it follows from condition (a) that we necessarily have 0 ≤ 2n ≤ m. The integer n is called the CR dimension of M and the integer d := m−2n is called the CR codimension of M . The case n = 0 is trivial and will not be considered further in this paper. Making use of the well-known Newlander–Nirenberg theorem [NN57], it is easy to see that when m = 2n there is a unique complex structure on M such that V = T 0,1 M , the subbundle of antiholomorphic vectors on M . If (M, V) and (M
, V
) are two CR manifolds (not necessarily of the same dimension or the same CR codimension), a smooth mapping f : M → M
is called a CR mapping if for every p ∈ M , f∗ (Vp ) ⊂ Vf
(p) , that is the push forward by f of every tangent vector in the fiber Vp is in the fiber Vf
(p) . The strong unique continuation and the finite jet determination questions we

18

M. Salah Baouendi

will be concerned with in this survey paper are the following. Assume that f and g are two CR mappings as above and p ∈ M . Question (i): If f and g agree to infinite order at p, i.e., all jets at p of f and g coincide, do f and g agree in a neighborhood of p in M ? Question (ii): Is there a positive integer k such that if jpk f = jpk g, then f and g agree to infinite order at p? (Here jpk f denotes the jet of order k at p.) The most interesting examples of CR manifolds of positive CR codimension are the generic submanifolds of a complex manifold. Since the unique continuation questions asked here are of a local nature, we restrict ourselves to the case where the complex manifold is CN . An embedded smooth submanifold M ⊂ CN is called generic if for every p ∈ M , there are smooth real functions ρj (Z, Z), j = 1, . . . , d, defined in an open neighborhood U ⊂ CN of p, such that the complex-valued vectors ∂ρj /∂Z = (∂ρj /∂Z1 , . . . , ∂ρj /∂ZN ), j = 1, . . . , d, are linearly independent and M ∩ U = {Z ∈ U : ρj (Z, Z) = 0, j = 1, . . . , d}.

(1.1)

It is easy to see that in this case V := T 0,1 M , the set of antiholomorphic vectors in CN tangent to M , forms a subbundle of CT M and that (M, V) is a CR manifold of CR codimension d and CR dimension n := N − d. An abstract CR manifold (M, V) of CR dimension n and CR codimension d is called locally integrable if for every p ∈ M there is a smooth mapping f between Ω, an open neighborhood of p in M , and a generic submanifold  ⊂ CN with N = n + d, which is a diffeomorphism and a CR mapping M . (Here M  is considered as a CR manifold with the CR between Ω and M structure induced from the complex structure of CN as described above.) The Newlander–Nirenberg theorem mentioned above states that an abstract CR manifold whose CR codimension is zero is locally integrable. Smooth abstract CR manifolds with positive CR codimension need not be locally integrable, as shown by an example given by Nirenberg [Nir75] with n = 1 and d = 1. However an easy consequence of the classical complex Frobenius theorem is that a real-analytic abstract manifold of any CR codimension is always locally integrable (see e.g., [BER99b]). Here real-analytic means that both the manifold M and its CR subbundle V are real-analytic.

2 Local coordinates We shall now rephrase the local integrability of CR manifolds as well as the unique continuation questions (i) and (ii) discussed above in terms of local coordinates, and show how these problems can be formulated in terms of overdetermined systems of first order partial differential equations with complex coefficients.

Strong unique continuation for CR mappings

19

Let Ω ⊂ Rm be an open subset. We shall define a CR structure on Ω as follows. Assume that m = 2n + d with n and d both positive integers. Suppose that we are given n smooth complex vector fields in Ω, denoted by L1 , . . . , Ln , satisfying the following two conditions: (a) the collection of 2n vector fields {L1 , . . . , Ln , L1 , . . . , Ln } are linearly independent at every point in Ω, (b) for every 1 ≤ j, k ≤ n, the commutator [Lj , Lk ], is in the complex linear span of L1 , . . . , Ln at every point in Ω. Such data define a CR structure on Ω of CR dimension n and CR codimension d by considering the fiber of the CR bundle V ⊂ CT Ω at every point in Ω to be the linear span of L1 , . . . , Ln at that point. It is easy to see that such an abstract CR manifold (Ω, V) is locally integrable if and only if for every p ∈ Ω there is an open subset ω ⊂ Ω containing p, and N := n + d complex-valued smooth functions {Zj } defined in ω with linearly independent differentials {dZj } satisfying the system of equations Lj Zk = 0,

1 ≤ j ≤ n,

1 ≤ k ≤ N.

(2.1)

Observe that if ω is sufficiently small, it follows that the mapping Z := (Z1 , . . . , ZN ) is a CR diffeomorphism between ω and the generic submanifold Z(ω) ⊂ CN , (which is necessarily of CR dimension n and CR codimension d). If the vector fields Lj , 1 ≤ j ≤ n, are real-analytic, then the CR manifold (Ω, V) is real-analytic and hence locally integrable as mentioned above. In the sequel, we will not necessarily assume that (Ω, V) is locally integrable. For simplicity we will assume 0 ∈ Ω.
Let M
⊂ CN be a smooth generic submanifold of CR dimension n
and
CR codimension d
(N
= n
+ d
), and assume 0 ∈ M
. Let ρ
= (ρ
1 , . . . , ρ
d ) be defining equations for M
near 0 as in (1.1). We now reformulate the strong unique continuation and the finite jet determination problems, described in Questions (i) and (ii) in the introduction, for local CR mappings between Ω and M
. Let ω ⊂ Ω be an open neighborhood of 0. Then a CR mapping f sending (ω, V) into M
with f (0) = 0 is a complex vector-valued function f = (f1 , . . . , fN
) defined in ω and satisfying Lj fk = 0,

1 ≤ j ≤ n,

1 ≤ k ≤ N
,

f (0) = 0,

ρ
(f, f ) = 0.

(2.2)

One may now consider Questions (i) and (ii) with p = 0 in the context of two vector functions, f and g, both of which satisfy (2.2). Note that these unique continuation problems are nonlinear. Indeed in (2.2), although the components of f satisfy an overdetermined system of first order linear equations, the last equation in (2.2) is a nonlinear vector-valued relation that f must also satisfy. We shall now give some examples and state some positive results in the
case where M ⊂ CN and M
⊂ CN are generic submanifolds both passing
through points p in CN and p
in CN respectively. Note that in this case
if F : (CN , p) → (CN , p
) is a germ of a holomorphic mapping sending M into M
, then f := F |M , the restriction to M of F to M , is a germ of a CR mapping f : (M, p) → (M
, p
) for which the answer to Question (i) is always positive. In general this is not the case for Question (ii) even when N = N
, M = M
and F is a local biholomorphism, as shown by the following example.

20

M. Salah Baouendi

Example 2.1 Let M = M
⊂ C2 be the hypersurface given by Im Z2 = 0 and F (Z1 , Z2 ) := (Z1 , Z2 + h(Z2 )), where h(Z2 ) is a convergent power series in one variable with real coefficients, with h(0) = h
(0) = 0. Clearly F is a biholomorphism in a neighborhood of 0 in C2 sending (M, 0) into itself and there is no finite jet determination for F . Again, even when N = N
, M = M
, and f and g are local CR diffeomorphisms, the answer to Question (i) could be negative as can be seen by the following example. Example 2.2 Let M = M
⊂ C2 be the hypersurface given in Example 2.1. Let θ : R → R be a smooth function vanishing of infinite order at 0 such that θ(s) = 0 for any s = 0. It is easy to see that the mapping f defined by f (Z1 , Re Z2 ) = (Z1 , Re Z2 + θ(Re Z2 )) is a CR self mapping of (M, 0) that is a diffeomorphism in a sufficiently small neighborhood of 0 in M . The mapping f agrees with the identity map to infinite order at 0 and differs from the latter in any neighborhood of 0.

3 Nondegeneracy conditions The simple examples given above show that in order to obtain positive unique continuation and finite jet determination results, some nondegeneracy conditions on the manifolds need to be imposed. The first of these conditions is that of finite type. An abstract CR manifold (M, V) is said to be of finite type at a point p ∈ M if the Lie algebra (under commutation of vector fields) obtained by taking the CR vector fields, i.e., the smooth sections of V, and their complex conjugates span the complexified tangent space of M at p. This condition was first introduced by Kohn [Koh72] for hypersurfaces in complex space and then later generalized by Bloom and Graham [BG77] for CR manifolds of higher CR codimension. In the settings of Section 2, the property of being of finite type at a point p ∈ Ω means that the dimension of the span of the vector fields {L1 , . . . , Ln , L1 , . . . , Ln } and their commutators of any order, evaluated at p, is m = 2n + d. The second nondegeneracy condition we will now describe is that of finite nondegeneracy. For simplicity, we describe the latter only for generic submanifolds of complex space. For abstract CR manifolds we refer the reader to [BER99b]. Let M ⊂ CN be a smooth generic submanifold of codimension d, and hence of CR dimension n = N − d. Let p ∈ M , and ρ1 , . . . , ρd be smooth real-valued defining functions for M near p as in (1.1). If  is a nonnegative integer, we say that M is -nondegenerate at p if  is the smallest integer for which the set of vectors {(L1 . . . Ls ρjZ )(p, p) : 1 ≤ j ≤ d; 0 ≤ s ≤ }

(3.1)

spans CN , where L1 , . . . , Ls runs through arbitrary systems of CR vector fields on M , i.e., sections of V = T 0,1 M . Here ρjZ = (ρjZ1 , . . . , ρjZN ) denotes

Strong unique continuation for CR mappings

21

the complex gradient of ρj (with respect to the coordinates Z) and is regarded as a vector in CN . It can be shown, see e.g., [BER99b], that this definition is invariant, i.e., independent of the choice of the defining function ρ and the complex coordinates Z. If M is -nondegenerate at p for some finite integer , then M is called finitely nondegenerate at p. A real smooth hypersurface in CN is called Levi-nondegenerate at p if and only if it is 1-nondegenerate at that point. It is easy to see that a smooth hypersurface in CN that is finitely nondegenerate at some point is necessarily of finite type at that point. However the converse does not hold. For example, the hypersurface in C2 given by Im Z2 = |Z1 |4 is of finite type at every point but is not finitely nondegenerate at 0. The two notions are independent for generic submanifolds of higher codimension as can be shown by the following example. Example 3.1 Let M ⊂ C3 be the generic submanifold of codimension 2 given by Im Z3 = 0. Im Z2 = |Z1 |2 , Note that the CR dimension of M is 1 and that the CR vector field L=

∂ ∂ − 2iZ1 ∂Z 1 ∂Z 2

spans T 0,1 M at every point. The Lie algebra generated by L and L is of dimension 3 at every point, whereas M is of real dimension 4. Hence M is nowhere of finite type. However it is easy to see that M is 1-nondegenerate at every point. The mapping F (Z1 , Z2 , Z3 ) : = (Z1 , Z2 , Z3 + h(Z3 )), where h is as in Example 2.1, shows that a local biholomorphism sending (M, 0) into itself is not determined by a finite jet at 0. Similarly the restriction to M of the mapping f (Z1 , Z2 , Re Z3 ) := (Z1 , Z2 , Re Z3 + θ(Re Z2 )), where θ is as in Example 2.2, shows that the answer to Question (ii) is negative for germs of CR diffeomorphisms sending (M, 0) into itself. We conclude this section by a third nondegeneracy condition that is weaker than that of finite nondegeneracy. We restrict ourselves to the real analytic case since the definition is easier to state in that case. A real-analytic generic submanifold M ⊂ CN is called holomorphically degenerate at p ∈ M if there are germs at p of holomorphic functions a1 (Z),   . . . , aN (Z) such that 2 aj (Z)∂/∂Zj is tangent 1≤j≤N |aj (Z)| ≡ 0 and such that the vector field to M in a neighborhood of p. We say that M is holomorphically nondegenerate at p if M is not holomorphically degenerate at p. A wide class of simple examples of holomorphically degenerate generic submanifolds is obtained as follows. Let M1 ⊂ CN −1 be any real-analytic generic submanifold. Then the generic submanifold M := M1 × C ⊂ CN is holomorphically degenerate at every point. Indeed any defining functions of M are independent of one of the

22

M. Salah Baouendi

complex coordinates, say ZN , and hence the vector field ∂/∂ZN is everywhere tangent to M . The notions of holomorphic nondegeneracy and that of finite type are independent even in the hypersurface case. Indeed, the hypersurface in C2 given by Im Z2 = (Re Z2 )|Z1 |2 is everywhere holomorphically nondegenerate but is not of finite type at 0, whereas the hypersurface in C3 given by Im Z2 = |Z1 |2 is everywhere holomorphically degenerate, but is of finite type at every point. The connection between the condition of holomorphic nondegeneracy and that of finite nondegeneracy is more subtle (see [BER99b]). If M ⊂ CN is a real-analytic generic submanifold that is finitely nondegenerate at p, then M is also holomorphically nondegenerate at p. Although the converse is not true in general, the following holds: If M ⊂ CN is a connected real-analytic generic submanifold that is holomorphically nondegenerate at some point, then it is holomorphically nondegenerate at every point. Moreover, M is finitely nondegenerate at every point outside a possibly empty proper real-analytic subset of M . In particular, it follows from the above that if M is a connected holomorphically nondegenerate real-analytic generic submanifold of CN , then the set of points at which M is finitely nondegenerate is dense in M .

4 Necessary conditions and sufficient conditions for finite jet determination For a real-analytic germ (M, p) of a generic submanifold in CN , holomorphic nondegeneracy is a necessary condition for finite jet determination for local biholomorphisms mapping (M, p) into itself. Indeed if M is holomorphically degenerate at p, then for any positive integer k one can find two distinct germs of local biholomorphisms H 1 , H 2 : (CN , p) → (CN , p) mapping M into itself such that jpk H 1 = jpk H 2 . This can be seen as follows. Recall that since (M, p) is holomorphically degenerate, there exists a nontrivial germ at p of a (1,0)-vector field with holomorphic coefficients tangent to M . Multiplying this vector field by a germ at p of a nontrivial holomorphic function vanishing of order k if necessary, we may assume that the coefficients of the vector field vanish at least of order k at p. Clearly for any small time, the flow of this vector field yields a local biholomorphism of (CN , p) fixing (M, p). It is also easy to see that the jet at p of order k of such a biholomorphism coincides with that of the identity map. We now state a positive result giving finite jet determination for local biholomorphic mappings. The following result is an immediate consequence of a slightly more general result proved in [BMR02]. Theorem 4.1 Let (M, p) be a germ of a real-analytic generic submanifold of CN . If M is holomorphically nondegenerate and of finite type at p, then there

Strong unique continuation for CR mappings

23

exists a positive integer k such that if H 1 , H 2 : (CN , p) → (CN , p) are two germs of biholomorphisms sending M into itself with jpk H 1 = jpk H 2 , it follows that H 1 ≡ H 2 . As noted above, the condition of holomorphic nondegeneracy is necessary for the conclusion of Theorem 4.1 to hold. In Example 3.1 we have a germ of a real-analytic generic submanifold of C3 , which is holomorphically nondegenerate and nowhere of finite type, and for which the conclusion of Theorem 4.1 fails to hold. This leads us to state the following conjecture. Conjecture 4.1 Let M be a holomorphically nondegenerate connected realanalytic generic submanifold through a point q in CN . If M is of finite type at q, then the conclusion of Theorem 4.1 holds at every point p in M . It should be mentioned that Conjecture 4.1 has actually been proved in the case of a hypersurface in C2 by Ebenfelt, Lamel and Zaitsev [ELZ03] (see Theorem 4.3 below). Since the points of finite type on a holomorphically nondegenerate connected real-analytic hypersurface are dense (see Section 3), the assumption of the existence of a point of finite type in Conjecture 4.1 is redundant in the case of hypersurfaces. The proof of Theorem 4.1 does not give any control on the size of the integer k given in the conclusion of that theorem. It was shown by Poincar´e [Poi07] that germs at p of biholomorphisms in C2 , sending a piece of a real sphere through p in C2 into itself, are determined by their 2-jets at p. In fact, it follows from later work of Cartan [Car32], Tanaka [Tan62], and Chern–Moser [CM74] that the germs of biholomorphisms fixing a real-analytic Levi nondegenerate hypersurface in CN are also determined by their 2-jets. In the more restrictive case where M is assumed to be -nondegenerate at p, rather than merely holomorphically nondegenerate, the integer k given by Theorem 4.1 can be chosen not to exceed (d+1), where d is the CR codimension of M . (Note that for a Levi nondegenerate hypersurface we have d = 1 and  = 1 and hence (d + 1) = 2.) The latter is a special case of the following result, which gives positive answers to both Questions (i) and (ii) for smooth CR diffeomorphisms. Theorem 4.2 Let M be a smooth abstract CR manifold of CR codimension d and p ∈ M . If M is -nondegenerate and of finite type at p, and if f, g : (M, p) → (M, p) are two germs of smooth CR diffeomorphisms with (d+1) (d+1) f = jp g, then f ≡ g. jp As mentioned above, Theorem 4.2 has a long history, starting with Levi nondegenerate real-analytic hypersurfaces in CN and self-mappings that are restrictions of biholomorphisms. Under the additional assumption that M is a smooth generic submanifold of CN , it is proved in Baouendi–Ebenfelt– (d+1) f = Rothschild [BER99a] that if f, g are as in Theorem 4.2, with jp (d+1) r r jp g, then for any positive integer r it follows that jp f = jp g. In particular, the latter implies the result mentioned before the statement of Theorem 4.2,

24

M. Salah Baouendi

namely that the (d + 1)-jet determines a local biholomorphism sending (M, p) into itself when M is real-analytic. For smooth hypersurfaces of CN , Theorem 4.2 was first proved by Ebenfelt [Ebe01]. As stated above, it is due to Kim and Zaitsev [KZ05a]. We conclude this section by stating an optimal and somewhat unexpected result for hypersurfaces in C2 , due to Ebenfelt, Lamel, and Zaitsev [ELZ03]. Theorem 4.3 Let M ⊂ C2 be a connected real-analytic holomorphically nondegenerate hypersurface. For any p ∈ M there is a positive integer k such that if H 1 , H 2 : (CN , p) → (CN , p) are two germs of biholomorphisms sending M into itself with jpk H 1 = jpk H 2 , it follows that H 1 ≡ H 2 . Moreover, if p is a point of finite type, then the integer k can be chosen to be 2. It should be noted here that a connected real-analytic hypersurface in C2 is either holomorphically nondegenerate (at every point), or at every point it is locally biholomorphically equivalent to the flat hypersurface given in Example 2.1, for which there is no finite jet determination for local self biholomorphisms. Therefore Theorem 4.3 completely settles the question of finite jet determination for local self biholomorphisms in the case of a real-analytic hypersurface in C2 . If M is as in Theorem 4.3 and p ∈ M , then we know from Theorem 4.2 that 2-jets determine local biholomorphisms if M is Levi non-degenerate at p. It is noteworthy that only 2-jets are needed for any point of finite type, regardless of the length of commutators of CR vector fields and their conjugates needed to span the tangent space at that point. Unfortunately no such simple answer holds for hypersurfaces in CN with N > 2.

5 Lie group structures and jet parameterization It turns out that the question of finite jet determination discussed in Section 4 is closely related to the following one. Let M ⊂ CN be a smooth generic submanifold through a point p and denote by Aut(M, p), the stability group of (M, p), that is the group of germs at p of biholomorphisms sending (M, p) into itself. Endowed with the inductive limit topology on uniform convergence on compact neighborhoods of p, Aut(M, p) is a topological group. Is there a finitedimensional Lie group structure on Aut(M, p) (necessarily unique) compatible with its topology? Before we state a positive result when M is real analytic, we introduce some notation and preliminaries. For any positive integer k, denote by Gkp (CN ) the set of all k-jets of the form jpk H, with H a germ at p of a biholomorphism in CN fixing p. It is easy to see that Gkp (CN ) has a finite-dimensional complex Lie group structure with the multiplication defined by (j0k H 1 ) · (j0k H 2 ) := j0k (H 1 ◦ H 2 ). (Note that this multiplication is independent of the choice of representatives H 1 and H 2 .) Let (M, p) be a germ of a real-analytic generic submanifold of CN . If M is holomorphically nondegenerate and of finite type at p, then by Theorem 4.1,

Strong unique continuation for CR mappings

25

there exists a positive integer k such that the mapping Aut(M, p)  H → jpk H ∈ Gkp (CN ) is continuous and injective. Hence, in this case, Aut(M, p) may be identified with a subgroup of Gkp (CN ). To show that Aut(M, p) is a Lie group one must prove that its image is closed in Gkp (CN ). This still remains an open question. However the following is known. Theorem 5.1 Let (M, p) be a germ of a real-analytic generic submanifold of CN of codimension d. If M is -nondegenerate and of finite type at p, then the mapping Aut(M, p)  H → jp(d+1) H ∈ G(d+1) (CN ), p taking a germ of a local biholomorphism at p to its (d + 1)-jet, gives a diffeo(d+1) morphism of Aut(M, p) onto a real-algebraic Lie subgroup of Gp (CN ). For the case of a Levi-nondegenerate hypersurface (i.e.,  = d = 1), Theorem 5.1 follows from the work of Chern–Moser [CM74] and Burns– Shnider [BS77]. Recent works of the author with Ebenfelt and Rothschild [BER97], [BER99a], as well as that of Zaitsev [Zai97], are closely related to Theorem 5.1. The exact statement given here and its proof can be found in [BER99a]. Does the conclusion of Theorem 5.1 hold when the generic submanifold M is assumed to be merely smooth instead of real-analytic? The answer is negative even in the case of a hypersurface. Indeed recent work of Kim–Zaitsev [KZ05b] gives an ingenious construction of a smooth hypersurface M ⊂ CN , finitely nondegenerate at 0 (even Levi-nondegenerate) for which Aut(M, 0) is not a Lie group, although it is contained as a subgroup of a finite-dimensional Lie group. Note that in this case finite jet determination holds by Theorem 4.2. As mentioned above, in the context of Theorem 5.1, it is not known whether the finite nondegeneracy of M can be replaced by the weaker condition of holomorphic nondegeneracy to guarantee that Aut(M, p) has a Lie group structure. However, very recently Lamel and Mir [LM05] gave a positive answer in the case of a real-analytic generic submanifold that is of finite type and essentially finite at p. To avoid technicalities, we will not give here the exact definition of essential finiteness (see [BER99b] for the definition). It suffices to say that it is a weaker condition than finite nondegeneracy and stronger than holomorphic nondegeneracy. An interesting class of real-analytic essentially finite generic submanifolds of CN consists of those real-analytic generic submanifolds that do not contain germs of nontrivial complex varieties. For instance, it is known by a result of Diederich and Fornaess [DF78] that any real-analytic compact submanifold of CN does not contain nontrivial complex varieties. It follows that any compact real-analytic generic submanifold of CN (in particular any compact real-analytic boundary of a bounded domain in CN ) is essentially finite at every point. The result in [LM05] that we now state has other interesting applications that we will not address here.

26

M. Salah Baouendi

Theorem 5.2 Let (M, p) be a germ of a real-analytic generic submanifold of CN . If M is essentially finite and of finite type at p, then there exists a positive integer k such that the mapping Aut(M, p)  H → jpk H ∈ Gkp (CN ),

(5.1)

taking a germ of a local biholomorphism at p to its k-jet, gives a diffeomorphism of Aut(M, p) onto a real Lie subgroup of Gkp (CN ). Here is another intriguing related question. Let (M, p) be a germ of a real-analytic generic submanifold in CN , and assume that the stability group Aut(M, p) has a finite-dimensional Lie group structure compatible with its topology. Does it necessarily follow that finite jet determination at p holds for germs in Aut(M, p)? A partial answer is given in a joint work of the author with Rothschild, Winkelmann, and Zaitsev [BRWZ04] that we state here. Theorem 5.3 Let (M, p) be a germ of a real-analytic generic submanifold of CN . Assume that the stability group Aut(M, p) has a finite-dimensional Lie group structure compatible with its natural topology. If Aut(M, p) has finitely many connected components, then there exists a positive integer k such that the mapping Aut(M, p)  H → jpk H ∈ Gkp (CN ),

(5.2)

taking a germ of a local biholomorphism at p to its k-jet, is injective. The author does not know of any example of a germ of a real-analytic generic submanifold of CN whose stability group is a finite-dimensional Lie group with infinitely many components. It would be interesting to know if such examples exist. Also, under the assumptions of Theorem 5.3, it is not known if the image of the mapping (5.2) is a closed subgroup of Gkp (CN ). The conclusions of Theorems 5.1 and 5.2 are a consequence of a seemingly stronger property that we now define. Definition 5.1 Let (M, p) be a germ of a real-analytic generic submanifold of CN . We say that (M, p), or its stability group Aut(M, p), satisfies the jet parameterization property if and only if there exists a positive integer k such that for every λ0 ∈ Gp (CN ), there exist neighborhoods Ω
of p in CN , Ω

of λ0 in Gp (CN ), and a real-analytic mapping : Ψ : Ω
× Ω

→ CN , holomorphic in the first variable, such that f (·) = Ψ (·, jpk f ),

(5.3)

for any f ∈ Aut(M, p) with jpk f ∈ Ω

, where the equality in (5.3) holds in the sense of germs at p. It is clear that if (M, p) satisfies the jet parameterization property (5.3), then the germs in Aut(M, p) are determined by their k-jets at p. In fact the following holds (see e.g., [BER99a], [BRWZ04]).

Strong unique continuation for CR mappings

27

Proposition 5.1 If a germ (M, p) satisfies the jet parameterization property (5.3), then the mapping given by (5.1) is a diffeomorphism of Aut(M, p) onto a real Lie subgroup of Gkp (CN ). Hence Aut(M, p) has a Lie group structure. In the proofs of Theorems 5.1 and 5.2 it is shown that the stability group Aut(M, p) satisfies the jet parameterization property (5.3) (with k = (d + 1) in the case of Theorem 5.1); hence the conclusions of the theorems follow by making use of Proposition 5.1. There are no known cases of germs (M, p) of generic real-analytic submanifolds in CN for which Aut(M, p) has a Lie group structure but does not satisfy the jet parameterization property. It would be interesting to know if the two properties are equivalent. Although this paper deals only with local questions, I would like to briefly mention a related global result whose proof also uses Theorem 4.2. It is shown in [BRWZ04] that if M is a smooth abstract CR manifold that is finitely nondegenerate and of finite type at every point, then the group of global smooth CR automorphisms of M has the structure of a finite-dimensional Lie group. A similar result holds for the group of global real-analytic CR automorphisms of M if, in addition, M is assumed to be real-analytic. I would like to conclude by pointing out that the nondegeneracy conditions mentioned here hold for “most” generic submanifolds. Indeed, it is shown in [BRZ06] that any smooth generic submanifold in CN can be deformed into a generic submanifold that is everywhere finitely nondegenerate and of finite type. In addition to the references already mentioned, I would also like to refer to the excellent survey article by D. Zaitsev [Zai02], which addresses some of the questions discussed in this paper.

References [BER97]

M. S. Baouendi, P. Ebenfelt and L. P. Rothschild, Parametrization of local biholomorphisms of real analytic hypersurfaces, Asian J. Math. 1(1997), 1–16. [BER99a] M. S. Baouendi, P. Ebenfelt and L. P. Rothschild, Rational dependence of smooth and analytic CR mappings on their jets, Math. Ann. 315(1999), 205–249. [BER99b] M. S. Baouendi, P. Ebenfelt and L. P. Rothschild, Real submanifolds in complex space and their mappings, Princeton Mathematical Series, vol. 47, Princeton University Press, Princeton, NJ, 1999. [BG77] T. Bloom and I. Graham, On “type” conditions for generic real submanifolds of Cn , Invent. Math. 40(1977), 217–243. [BMR02] M. S. Baouendi, N. Mir and L. P. Rothschild, Reflection ideals and mappings between generic submanifolds in complex space, J. Geom. Anal. 12(2002), 543–580. [BRWZ04] M. S. Baouendi, L. P. Rothschild, J. Winkelmann and D. Zaitsev, Lie group structures on groups of diffeomorphisms and

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[BRZ06] [BS77]

[Car32] [CM74] [DF78] [Ebe01] [ELZ03]

[Koh72]

[KZ05a] [KZ05b] [LM05] [Nir75]

[NN57] [Poi07] [Tan62] [Zai97]

[Zai02]

applications to CR manifolds, Ann. Inst. Fourier (Grenoble) 54(2004), 1279–1303. M. S. Baouendi, L. P. Rothschild and D. Zaitsev, Deformation of generic submanifolds in a complex manifold, preprint. D. Burns, Jr. and S. Shnider, Real hypersurfaces in complex manifolds, Several complex variables (Proc. Sympos. Pure Math., Vol. XXX, Part 2, Williams Coll., Williamstown, Mass., 1975), Amer. Math. Soc., Providence, R.I., 1977, 141–168. E. Cartan, Sur la g´eom´etrie pseudo-conforme des hypersurfaces de deux variables complexes, I, Ann. Mat. Pura Appl. 11(1932), 17–90. S. S. Chern and J. K. Moser, Real hypersurfaces in complex manifolds, Acta Math. 133(1974), 219–271. K. Diederich and J. E. Fornaess, Pseudoconvex domains with realanalytic boundary, Ann. of Math.(2) 107(1978), 371–384. P. Ebenfelt, Finite jet determination of holomorphic mappings at the boundary, Asian J. Math. 5(2001), 637–662. P. Ebenfelt, B. Lamel and D. Zaitsev, Finite jet determination of local analytic CR automorphisms and their parametrization by 2-jets in the finite type case, Geom. Funct. Anal. 13(2003), 546–573. J. J. Kohn, Boundary behavior of δ on weakly pseudo-convex manifolds of dimension two, J. Differential Geometry 6(1972), 523–542, Collection of articles dedicated to S. S. Chern and D. C. Spencer on their sixtieth birthdays. S. -Y. Kim and D. Zaitsev, Equivalence and embedding problems for CR-structures of any codimension, Topology 44(2005), 557–584. S. -Y. Kim and D. Zaitsev, Remarks on the rigidity of CR-manifolds, preprint, http://arxiv.org/abs/math.CV/0501395. B. Lamel and N. Mir, Parametrization of local CR automorphisms by finite jets and applications, preprint, 2005. L. Nirenberg, On a problem of Hans Lewy, Fourier integral operators and partial differential equations (Colloq. Internat., Univ. Nice, Nice, 1974), Springer, Berlin, 1975, 224–234. Lecture Notes in Math., Vol. 459. A. Newlander and L. Nirenberg, Complex analytic coordinates in almost complex manifolds, Ann. of Math. (2) 65(1957), 391–404. H. Poincar´ e, Les fonctions analytiques de deux variables et la repr´esentation conforme, Rend. Circ. Mat. Palermo, II. 23(1907), 185–220. N. Tanaka, On the pseudo-conformal geometry of hypersurfaces of the space of n complex variables, J. Math. Soc. Japan 14(1962), 397–429. D. Zaitsev, Germs of local automorphisms of real-analytic CR structures and analytic dependence on k-jets, Math. Res. Lett. 4(1997), 823– 842. D. Zaitsev, Unique determination of local CR-maps by their jets: a survey, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 13(2002), 295–305.

On the Cauchy problem for some hyperbolic operator with double characteristics Enrico Bernardi1 and Antonio Bove2 1 2

Dipartimento di Matematica per le Scienze Economiche e Sociali, Universit` a di Bologna, Bologna, Italia Dipartimento di Matematica, Universit` a di Bologna, Bologna, Italia

Summary. We prove that the Cauchy problem for a class of hyperbolic operators with double characteristics and whose simple null bicharacteristics have limit points on the set of double points is not well posed in the C ∞ category, even though the usual Ivrii–Petkov conditions on the lower order terms are satisfied. According to the standard linear algebra classification these operators, at a double point, have fundamental matrices exhibiting a Jordan block of size 4 and cannot be brought into a canonical form known as “Ivrii decomposition”, due to higher order non-vanishing terms in the Taylor development of the principal symbol near the given double point.

2000 Mathematics Subject Classification: Primary: 35L15, Secondary: 37J05. Key words: Hyperbolic operators, double characteristics, Hamiltonian systems, Cauchy problem.

1 Introduction and statements In this paper we prove that the Cauchy problem in the C ∞ category for the operator: (1.1) P (x, D) = −D02 + 2x1 D0 Dn + D12 + bx31 Dn2 is not well posed if b = 0 (see Theorem 6.1 for a precise statement). ∂ . Here we adopt the notation Rn+1  x = (x0 , x1 , x

, xn ) and Dj = 1i ∂x j The operator P is hyperbolic w.r.t. x0 , at least if x1 is sufficiently small and has double characteristics at Σ2 = {ξ 0 = ξ 1 = x1 = 0} near ρ0 = (0, en ). Let us briefly explain the reason why we consider the operator P in (1.1) interesting. The standard linear symplectic classification (see e.g., [5]) readily yields that the spectrum of the fundamental matrix of P , FP (ρ0 ), is just 0 and FP (ρ0 ) has a Jordan block of size 4 in its canonical form.

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Enrico Bernardi and Antonio Bove

For operators whose fundamental matrices exhibit this kind of spectral properties, it is well known that one cannot, in general, exploit a remarkable result due to Ivrii, [7], which allows us to prove the well-posedness of the Cauchy problem in C ∞ , assuming a suitable factorization of the principal symbol holds. In fact in [3] and [4] we found necessary and sufficient conditions under which such factorization can be devised, if the fundamental matrix happens to have a Jordan block of size 4 for the 0 eigenvalue, at every double characteristic point of principal symbol. (For another proof of the result in [4] we refer also to [10].) In the case of the operator in (1.1), the fact that these conditions do not hold can be simply rephrased by saying that b = 0: on the other hand it can also be restated in a much more significant way by applying the results in [3] and remarking that b = 0 is equivalent to the existence of a simple bicharacteristic curve issued from a simple point and having a limit point onto Σ2 . This is easily seen for P in (1.1) solving the Hamilton equations:

x˙

= 0

ξ˙ 0 = 0 ξ˙ 1 = −2ξ 0 ξ n − 3bx21 ξ 2n ξ˙

= 0

x˙ n = 2x1 ξ 0 + 2bx31 ξ n

ξ˙ n = 0

x˙ 0 = −2ξ 0 + 2x1 ξ n x˙ 1 = 2ξ 1

(1.2)

and then finding that there is a curve γ landing onto Σ2 at ρ0 and having the form (1.3) γ(x0 ) ≡ (x0 , x20 , 0, x50 ; 0, x30 , 0, 1), when x0 → 0. It therefore follows from the results in [3] and [4] that P (x, D) in (1.1) is the microlocal model for a second order differential hyperbolic operator vanishing exactly of order 2 on its double characteristics and possessing the aforementioned behavior of the Hamilton flow. In [2], in an effort to prove a positive well-posedness result, we were able to show that the operator P in (1.1) is well posed in the Gs category, for 1 ≤ s ≤ 5, where Gs denotes, as usual, the Gevrey function space of order s. The technique used in the proof however breaks down beyond the threshold s = 5. This led us to believe that the operator P is indeed not well posed in C ∞ : we must however underline the strikingly singular coincidence of the Gevrey 5 threshold and the ratio 5 between the maximum and the minimum exponent in the powers of the parameter x0 in the curve γ(x0 ), actually shaping the curve near the double limit point. Even though the following pages will contain only an argument to show that (1.1) is not well posed in C ∞ , the solution explicity built in the proof

Hyperbolic operators with double characteristics

31

could be suitably modified in order to extend our result to a non-well-posedness one in Gs , for every s > 5. Furthermore a direct inspection of the solution we construct below allows us to see that we actually provide a solution whose singularities are carried by the particular null bicharacteristic landing onto Σ2 . We would like to end this short introduction by remarking that we believe that this strange interplay between the Hamiltonian flow and the lack of well-posedness for the Cauchy problem might not be accidental: we actually have reasons to believe that the existence of these tangent curves could actually represent a true obstruction to the C ∞ well-posedness of the initial value problem for a general hyperbolic differential operator, with arbitrary multiplicity characteristics and of any assigned order.

2 The model operator We will consider, as anticipated in the previous section, the following differential model operator: P (x, D) = −D02 + 2x1 D0 Dn + D12 + bx31 Dn2 .

(2.1)

Here b = 0, x ∈ Rn+1 , x = (x0 , x1 , . . . , xn ) = (x0 , x1 , x

, xn ), where x0 denotes the time variable. The operator (2.1) is a (microlocal) model for a weakly hyperbolic operator with double characteristics exhibiting a Jordan block in the canonical form of its Hamiltonian matrix F . We recall that F (ρ) = d(x,ξ) Hp (ρ), p denoting the symbol—principal symbol in this case—of (2.1) and ρ being a point in the double characteristic manifold Σ = {(x, ξ) | (x, ξ) ∈ T ∗ Rn+1 \ 0, x1 = 0, ξ 0 = ξ 1 = 0}. In (2.1) we write Dn2 instead of e.g., a Laplacian in the variables (x2 , . . . , xn ), since throughout the present paper it will be understood that we are in a microlocal (conical) neighborhood of the point (0, en ), where Dn is microlocally elliptic. It is well known that the Hamiltonian system of (2.1) is stable, i.e., there are no points on the simple characteristic manifold of (2.1) such that the bicharacteristic curves issued from them have limit points on Σ, if and only if b = 0. In [3] it was shown that condition b = 0 has an invariant meaning and actually amounts to asserting that HS3 p(ρ) = 0 for ρ ∈ Σ. Here S(x, ξ) is a smooth real function vanishing on Σ verifying (2.8) in [3]. In the present model we may assume S(x, ξ) = ξ 1 . Without loss of generality we may assume in the following that b = 1. It suffices to perform the symplectic dilation x0 → b−1 x0 , x1 → b−1 x1 , x

→ x

, xn → b−2 xn ,

(2.2)

where x

= (x2 , . . . , xn−1 ), to end up with the same operator as in (2.1) and b = 1.

32

Enrico Bernardi and Antonio Bove

Let us now consider the operator P (x, D) = −D02 + 2x1 D0 Dn + D12 + x31 Dn2

(2.3)

and perform a Fourier transform with respect to x0 and xn . The equation P u = 0 is now given by u

(x1 ) = (ξ 2n x31 + 2ξ 0 ξ n x1 − ξ 20 )u(x1 ).

(2.4)

Let us change the independent variable x1 ; set y = ξ 2/5 n x1 , where we once and for all assume that, in the microlocal region where we set ourselves, ξ n is positive. A completely symmetric argument holds for a negative ξ n . Equation (2.4) transforms into y − ξ 20 ξ −4/5 )v(y), v

(y) = (y 3 + 2ξ 0 ξ −1/5 n n

(2.5)

where we set v(y) = v(y, ξ 0 , ξ n ) = u(yξ −2/5 , ξ 0 , ξ n ). Finally putting ξ 0 = n z 1/5 ξ we arrive at the equation 2 n   z2 v

(y) = y 3 + zy − ξ −2/5 v(y). (2.6) 4 n We devote the next section to a detailed study of ordinary differential equations of the form (2.6).

3 Shibuya solutions In this section we shall be dealing with the following ordinary differential equation: w

(y) = (y 3 + ζy + ε)w(y), (3.1) where ζ, ε are complex numbers and ε will be thought of as small in the final arguments. We briefly recall, for this special situation, the general theory of subdominant solutions of equation (3.1), according to the exposition for instance in the book of Shibuya [11]. Theorem 6.1 in [11] states that the differential equation (3.1) has a solution w(y; ζ, ε) = Y3 (y; ζ, ε),

(3.2)

such that (i) Y3 (y; ζ, ε) is an entire function of (y; ζ, ε), (ii) Y3 (y; ζ, ε) admits an asymptotic representation   ∞
−3/4 −N/2 ∼ B3,N y 1+ exp[−E3 (y; ζ, ε)] Y3 (y; ζ, ε) = y N =1

(3.3)

Hyperbolic operators with double characteristics

33

uniformly on each compact set in the (ζ, ε) space as y tends to infinity in any closed subsector of the open sector, | arg y| < moreover E3 (y; ζ, ε) =

3π ; 5

2 5/2 y + ζy 1/2 5

(3.4)

(3.5)

and B3,N are polynomials in (ζ, ε). We note that, setting

and

  2 ω = exp i π 5

(3.6)

Y3,k (y; ζ, ε) = Y3 (ω −k y; ω −2k ζ, ω −3k ε),

(3.7)

where 0 ≤ k ≤ 4, all the five functions Y3,k (y; ζ, ε), k = 0, . . . , 4, solve (3.1); in particular Y3,0 = Y3 . Denoting by Y3 the r.h.s. of (3.3), an easy consequence of the above result is that: (i) Y3,k (y; ζ, ε) is an entire function of (y; ζ, ε). (ii) Y3,k (y; ζ, ε) ∼ = Y3 (ω −k y; ω−2k ζ, ω−3k ε), uniformly on each compact set in the (ζ, ε) space as y tends to infinity in any closed subsector of the open sector     arg y − 2k π  < 3π . (3.8)  5  5   π  Let Sk denote the open sector defined by  arg y − 2k 5 π < 5 , and let Sk be the closure of Sk . In the figure below the five sectors Sk are represented and we note that they cover the whole complex y plane. We say that a solution of (3.1) is subdominant in the sector Sk if it tends to 0 as y tends to infinity along any direction in the sector Sk . Analogously, a solution is called dominant in the sector Sk if this solution tends to ∞ as y tends to infinity along any direction in the sector Sk . Since (3.9) Re[y 5/2 ] > 0, for y ∈ S0 and Re[y 5/2 ] < 0, for y ∈ S−1 (= S4 ) and for y ∈ S1 , the solution Y3,0 (y; ζ, ε) is subdominant in S0 and dominant in S4 and S1 . Similarly Y3,k (y; ζ, ε) is subdominant in Sk and dominant in Sk−1 and Sk+1 .

34

Enrico Bernardi and Antonio Bove

4 Stokes multipliers From (3.9) we deduce that Y3,k+1 and Y3,k+2 are linearly independent. Therefore Y3,k is a linear combination of those two: Y3,k (y; ζ, ε) = Ck (ζ, ε)Y3,k+1 (y; ζ, ε) + C˜k (ζ, ε)Y3,k+2 (y; ζ, ε).

(4.1)

The above relation, (4.1), is known as a connection formula for Y3,k (y; ζ, ε) and the coefficients Ck , C˜k are called the Stokes multipliers for Y3,k (y; ζ, ε) with respect to Y3,k+1 (y; ζ, ε) and Y3,k+2 (y; ζ, ε). We summarize in the statement below some of the known and useful facts about the Stokes coefficients for our particular equation (3.1). Proofs can be found in Chapter 5 of [11]. Theorem 4.1 The following results hold true: (i) C˜k (ζ, ε) = −ω, ∀k, ε and ζ; (ii) Ck (ζ, ε) = C0 (ω −2k ζ, ω −3k ε), ∀k, ε, ζ and C0 (ζ, ε) is an entire function of (ζ, ε); (iii) for each fixed (ζ, ε) there exists k such that Ck (ζ, ε) = 0; (iv) Ck (0, 0) = 1 + ω, ∀k; (v) ∂ζ C0 (ζ, ε)|(ζ,ε)=(0,0) = 0; (vi) ∂ε C0 (ζ, ε)|(ζ,ε)=(0,0) = 0. A final and quite remarkable property comes from the following (see Theorem 21.3 in [11] pp. 85 and the following.)

Hyperbolic operators with double characteristics

35

Theorem 4.2 If we set 

 Ck (ζ, ε) 1 Sk (ζ, ε) = , −ω 0 then

k = 0, 1, 2, 3, 4,

(4.2)



 10 S4 (ζ, ε) · S3 (ζ, ε) · S2 (ζ, ε) · S1 (ζ, ε) · S0 (ζ, ε) = . 01

(4.3)

The proof of Theorem 4.2 is straightforward: a complete turn of 2π in the complex y plane brings us back to the same solution. We now state an interesting byproduct of Theorem 4.2 which will have far-reaching consequences. It is useful to state the next theorem in the case ε = 0, even though its conclusions would hold also for any non-zero ε. Theorem 4.3 Let us assume that ε = 0 and denote for the sake of brevity ck = Ck (ζ, 0). Then (4.3) is equivalent to ck + ω2 ck+2 ck+3 − ω3 = 0,

mod 5.

(4.4)

Or otherwise stated: c(ζ) + ω 2 c(ωζ)c(ω 4 ζ) − ω 3 = 0, ∀ζ ∈ C,

(4.5)

where we put c(ζ) = c0 (ζ) = Ck (ζ, 0). Proof. A straightforward computation from (4.3). We thus have c(ζ), an entire function verifying (4.5): we say that c(ζ) must vanish in some point ζ 0 ∈ C. Assume that c(ζ) = 0, ∀ζ ∈ C, then by (4.5) we would necessarily have that c(ζ) = ω3 and also ∀ζ ∈ C. Therefore by Picard’s little theorem c(ζ) would be an entire function not assuming two distinct values of the complex plane, thus c(ζ) should be constant, which contradicts (4.1), item (v). We have so proved the following Theorem 4.4 The Stokes coefficient C0 (ζ, 0) vanishes in at least one (nonzero) ζ 0 . Proof. ζ 0 = 0 because of Theorem 4.1, item (iv). Let us now consider the general case C0 (ζ, ε) with ε = 0. Let ζ 0 be a complex number where c(ζ 0 ) = 0. ζ 0 verifies C0 (ζ 0 , 0) = 0. Let µ be the multiplicity of this root. Since C0 is an entire function µ is finite and therefore, by the Weierstraß theorem we have, in a neighborhood of (ζ 0 , 0): ⎞ ⎛ µ
(4.6) aj (ε)(ζ − ζ 0 )µ−j ⎠ C0 (ζ, ε) = γ(ζ, ε) ⎝(ζ − ζ 0 )µ + j=1

where γ(ζ 0 , 0) = 0, aj (0) = 0 and aj (ε) is holomorphic in ε.

36

Enrico Bernardi and Antonio Bove

Let us now consider the above equation (4.6) in a neighborhood of (0, ζ 0 ). There exists a positive δ such that each root of the equation C0 (ζ, ε) = 0,

(4.7)

for some positive integer p, is a holomorphic function of ε1/p , for 0 < |ε| < δ, that is ∞
ζ(ε) = ζ 0 + cj (ε1/p )j = g(ε1/p ). j=0

This is actually a consequence of Theorem 3.2.6 in [6], observing that γ(ζ, ε) = 0 implies that the function g has no polar singularity at the origin. As a matter of fact the function g is holomorphic in a full neighborhood of the origin so that ζ(η p ) = g(η), which is a well-defined holomorphic function of η. In what follows we consider the equation   1 (4.8) w

(y) = y 3 + ζy − ζ 2 εp w(y), 4 ˜ ˜ ζ being holomorphic in a neighborso that (4.7) has a solution ζ(εp ) = ζ(ε), hood of ε = 0. Using this zero in (4.1) we see that ˜ Y3,0 (y; ˜ ζ(ε), εp ) = −ωY3,2 (y; ζ(ε), εp )

∀y ∈ C, |ε|  1.

(4.9)

Once this choice of root has been made, we denote by w(ζ) the function w(ζ) = Y3,0 (y; ˜ ζ(ε), εp ), which, by (4.9), is now subdominant both in S0 and S2 ; also note that the asymptotic development (3.3) holds for w(y) in the whole complex y-plane cut along the half-line y = rei7π/5 , r > 0. We also agree to use the complex plane cut along this line as the principal branch of y 1/2 , for y complex.

5 Asymptotic analysis 2

Consider now equation (2.6), of type (3.1) with ζ = z and ε = − z4 ξ −2/5 . n   −2/5 z 2 −2/5 ˜ Define C(z, ξ n ) = C0 z, − 4 ξ n . The arguments in the preceding section allow us to conclude that the Stokes multiplier connecting Y3,0 with ), with z(0) = ζ 0 . This is evident Y3,1 is zero along a curve z = z(ξ −2/5 n since

Hyperbolic operators with double characteristics

37

   2 ˜ ξ −2/5 ) = ∂z C0 z, − z ξ −2/5 ∂z C(z, n 4 n     z2 z −2/5 z 2 −2/5  = ∂z C0 z, − ξ −2/5 ξ ξ − z, − ∂ C ε 0  −2/5 4 n 2 n 4 n ε=ξ n = ∂z C0 (ζ 0 , 0) at ξ n = ∞. Since repeated derivatives behave in exactly the same way it is easy to see that we can reason as before, and conclude, via the ) for the Stokes Weierstraß theorem, the existence of a solution z = z(ξ −2/5 n multiplier. We can thus define the following function:    3/2 i ˜ −2/p −2/5p 1/5 5 10−θ 2 U (x, ρ) = exp iρ txn − ρ (t − 1) + ζ(ρ t )ρt x0 2 1/2 ˜ −2/p t−2/5p ))dt × w(x1 ρ2 t2/5 ; ζ(ρ

(5.1)

where εp = ρ−2 t−2/5 , 0 < θ < 5, ρ ∈ C, |ρ| large, and whose direction in C will be suitably chosen later. More precisely, in what follows we shall set 2 either ρ real, ρ > 0, or ρ complex, ρ = |ρ|ωk , where ω = ei 5 π and k = 1, 2, 3, 4. The integer k will then be chosen in a suitable way later. A final

2 e−ρ|x | factor for the mute tangent variables x

will be eventually added, see (5.13). It is easy to verify, using (2.4), (2.5) and (2.6), that P (x, D)U (x, ρ) = 0, ∀ρ, where P is the operator in (2.3). Next we are going to manipulate the expression in (5.1). We are interested in evaluating its asymptotic behavior in a region of the type δ ≤ |x| ≤ 2δ, δ > 0. Let us first consider the case x1 = 0. We start by changing the integration variable and set s = t − 1, so that s ∈] − 1/2, 1/2[. We have U (x, ρ) = Aρ (x)I(x, ρ),

(5.2)

where Aρ (x) = (x1 ρ2 )−3/4 

 ρθ 2 i ˜ −2/p 2 5 5/2 ˜ −2/p 1/2 × exp iρ xn + ρζ(ρ )x0 − ρ x1 − ζ(ρ )ρx1 e− 4 xn . 2 5 (5.3) 5

Defining ˜ −2/p ), Z(ρ, s) = ˜ ζ(ρ−2/p (1 + s)−2/5p )(1 + s)1/5 − ζ(ρ

(5.4)

38

Enrico Bernardi and Antonio Bove

we may write 

1/2

I(x, ρ) = −1/2

  exp −ρ10−θ (s − i/2ρθ−5 xn )2 (1 + s)−3/10

    i 2 5/2 5 1/2 x0 − x1 × exp ρ Z(ρ, s) − x1 ρ s 2 5    1 × 1+O ds. (x1 ρ2 (1 + s)2/5 )1/2

(5.5)

Here the asymptotic expansion (3.3) has been used. In order to asymptotically evaluate the above quantity we use a shift of the integration path. This is possible since the the function we are integrating is holomorphic. As an integration path we choose the path whose vertices are     1 1 1 − , − ρθ−5 xn , − ,0 , 2 2 2 

 1 1 θ−5 , − ρ xn , 2 2



 1 ,0 . 2

Let us show that the contribution coming from the “short” sides can be made exponentially small as the (absolute value of) parameter ρ becomes large.     Let us focus on the integral (5.5) along the segment 12 , − 21 ρθ−5 xn , 12 , 0 . Setting s = 12 +µA, A = − 2i ρθ−5 xn , µ ∈ [0, 1], we may write the integral along the short side as  2       1 i 1 1 1/2 + µA + ρ x0 − x1 exp −ρ10−θ A Z ρ, + µA 2 2 2 0 

  −3/10 3 1 + µA + µA 2 2  1 dµ.  3/5 (x1 ρ2 32 + µA )

2 5/2 × exp − x1 ρ5 5   ×

1+O



Since Z(ρ, 0) = 0 and |Z(ρ, s)| ≤ C, for a positive constant C, if |ρ|−1 + |s| ≤ C1 , C1 > 0, θ < 5, we have to estimate the quadratic part of the phase. Now because |A| = O(|ρ|θ−5 ), θ − 5 < 0, we have  2   1 10−θ + µA Re ρ µ ∈ [0, 1], (5.6) ≥ c|ρ|10−θ , 2

Hyperbolic operators with double characteristics

39

where, as before, it might be eventually necessary to set ρ = |ρ|ω k , k = 0, 1, . . . , 4, and the inequality (5.6) holds for every k, provided θ is sufficiently 5 small, e.g., 0 < θ < 16 . This proves our claim. Therefore, modulo errors that are exponentially decreasing as |ρ| → +∞ and uniformly for non-zero x’s in a compact set, I(x, ρ) ∼ = J(x, ρ) where  1/2 exp[−ρ10−θ s2 ](1 + s)−3/10 J(x, ρ) = −1/2

    i 2 5/2 5 1/2 x0 − x1 × exp ρ Z(ρ, s) − x1 ρ s 2 5    1 × 1+O ds. (x1 ρ2 (1 + s)2/5 )1/2

(5.7)

The latter integral can be estimated by use of the complex stationary phase. Set ˜ s), Z(ρ, s) = sZ(ρ, where Z has been defined in (5.4). The integral in (5.7) can then be rewritten as  1/2 v(s, ρ, x0 , x1 , xn ) −1/2



ρ10−θ × exp 4



 × exp −ρ

10−θ

 2  i 1/2 −9+θ ˜ −5+θ 2 5/2 Z(ρ, s) − ρ x0 − x1 x ρ 2 5 1

  2  i 1 1/2 −9+θ ˜ −5+θ 2 5/2 Z(ρ, s) − ρ x0 − x1 x ds, s+ ρ 2 2 5 1 (5.8)

where −3/10

v(s, ρ, x0 , x1 , xn ) = (1 + s)



 1+O

1 2 (x1 ρ (1 + s)2/5 )1/2

 ,

for x11 1= 0 fixed, as |ρ| tends to infinity and uniformly with respect to s ∈ − 2, 2 . The integral in (5.8) is    2   1/2 5 i x 1/2 θ 1 −8+θ ˜ s)2 Z(ρ, x0 − x1 exp ρ exp ρ 25 2 −1/2 −

4 5/2 ρ−4+θ x1 5



  i 1/2 ˜ Z(ρ, s) x0 − x1 2

40

Enrico Bernardi and Antonio Bove





× exp −ρ

10−θ

1 s+ 2



 2  i 1/2 −9+θ ˜ −5+θ 2 5/2 Z(ρ, s) − ρ ρ x0 − x1 x 2 5 1

× v(s, ρ, x0 , x1 , xn ) ds.

(5.9)

Define vˆ(s, ρ, x0 , x1 , xn ) = v(s, ρ, x0 , x1 , xn )  × exp ρ−8+θ



4 5/2 − ρ−4+θ x1 5 We point out that

i 1/2 x0 − x1 2



2 ˜ s)2 Z(ρ,

  i 1/2 ˜ s) . Z(ρ, x0 − x1 2

(5.10)

vˆ(s, ρ, x0 , x1 , xn ) = 1 + O(ρ−δ ),

uniformly in s for any fixed non-zero x. Next we change the integration variable in (5.9) according to    1 i 1/2 −9+θ ˜ −5+θ 2 5/2 σ =s+ Z(ρ, s) − ρ x0 − x1 x ρ 2 2 5 1 = s + ρ−5+θ G(s, ρ, x). Note that the above is a well-defined change of variables provided |ρ| is large  enough and that it maps the segment − 12 , 12 to a smooth curve, γ, in C. Moreover the projection from γ to the real axis is injective. The integral in (5.9) is then written as   5 10−θ 2 θ x1 σ e−ρ vˆ(σ, ρ, x)J(σ, ρ, x)dσ, (5.11) exp ρ 25 γ where J represents the term coming from the Jacobian of the change of variables. We may apply Cauchy’s theorem to (5.11), since all the functions of s involved are holomorphic in a neighborhood of the real axis and |G(s, ρ, x)| ≤ C, for a positive constant C. The stationary phase theorem then implies that the integral in (5.11) is bounded by a constant. Continuing to take into account (5.2) and (5.3) we may then write the asymptotics of U :

Hyperbolic operators with double characteristics

41

U (x, ρ) = α(x, ρ)ρM (x1 ρ2 )−3/4 

 ρθ 2 i ˜ −2/p 2 5 5/2 ˜ −2/p 1/2 ρ x1 − ζ(ρ × exp iρ xn + ρζ(ρ )x0 − )ρx1 e− 4 xn 2 5C   × vˆ(0, ρ, x) + O(ρ−δ 1 ) 5

= eiΦ(x,ρ) a(x, ρ),

(5.12)

where |α(x, ρ)| is bounded by a constant, M is a positive number and the 2/5 factor C in the denominator of the x1 term is due to the exponential in front of the integral in (5.11). Moreover at this stage we may replace vˆ(s, ρ, x0 , x1 , xn ) by vˆ(s, ρ, x0 , x1 , xn ) exp[−|ρ||x

|2 ], x

denoting the vector of tangent coordinates (x2 , . . . , xn−1 ). We agree to denote by vˆ(s, ρ, x) the latter function. Let us examine more closely the phase function Φ. We have Φ(x, ρ) = ρ5 xn + i

ρθ 2 1 ˜ −2/p x + ζ(ρ )ρx0 4 n 2

2 i 5/2 1/2 + i ρ5 x1 − ρθ x51 + i˜ ζ(ρ−2/p )ρx1 + i|ρ||x

|2 . 5 25

(5.13)

We are interested in computing the behavior of Φ for |ρ| large, x0 < 0 and 5/2 x1 < 0, since when x1 > 0 the leading term is clearly that involving ρ5 x1 , having a positive real part. We want a positive real part for −iΦ also when x1 < 0. 2 Recall that ρ = |ρ|ω k , ω = ei 5 π and k = 0, 1, . . . , 4. Clearly ρ5 = |ρ|5 , for every choice of the integer k. Hence    x i θ ρθ 0 − |x1 |1/2 + Im ρ |x1 |5 + |ρ||x

|2 . Im Φ(x, ρ) = Im i x2n + Im(ζ 0 ρ) 4 2 25 The above phase differs from Φ by error terms that are small when |ρ| is large, so that it suffices to discuss the sign of the imaginary part of the above function. Observe that the quantity x20 − |x1 |1/2 is negative. To obtain a positive imaginary part, it is enough to choose k in such a way that Im(ζ 0 ρ) < 0. Then if θ is small we obtain that Im Φ > 0. This completes the discussion of the asymptotics of U , as |ρ| tends to infinity, in the case x1 = 0. In the case x1 = 0 we make a similar, though simpler, argument obtaining basically the same formula as in (5.12) where x1 = 0. We explicitly remark in this case that the value vˆ(0, ρ, (x0 , 0, x
)), where
˜ −2/p )). x = (x2 , . . . , xn ), depends on w(0; ζ(ρ

42

Enrico Bernardi and Antonio Bove

6 Final steps in the proof of the necessary condition In this section we use the solution U in (5.1) to violate the standard a priori estimate which should hold, were we to assume the C ∞ Cauchy problem to be well posed. Let Ω be an open subset of Rn+1 containing the origin. We recall that if the Cauchy problem is well posed in the open set Ω ∩ {x0 < 0}, then, if K is a compact subset of Ω, there exists a positive integer N , such that − u− −N ≤ CP uN ,

(6.1)

s where  · − s denotes the standard H (Sobolev space of order s) norm defined on the restrictions to the half space x0 < 0. We shall proceed along the usual path: take a C0∞ function ϕ(x), ϕ(x) ≡ 1 if |x| ≤ δ, ϕ(x) ≡ 0 if |x| > 2δ, with δ > 0 suitably small. Now define (6.2) Uϕ (x, ρ) = ϕ(x)U (x, ρ).

We have that P (x, D)Uϕ (x, ρ) = P (x, D)[ϕ(x)U (x, ρ)] = [P, ϕ]U (x, ρ),

(6.3)

therefore, to be more precise, we will state our asymptotic behavior result for U (x, ρ) when δ ≤ |x| ≤ 2δ, x0 > 0. The right-hand side of (6.3) is identically zero in |x| ≤ δ and therefore, from (5.12), we deduce that |Uϕ (x, ρ)| ≤ C|ρ|M exp[−|ρ|θ |x|4 ],

(6.4)

for a suitable integer M > 0, a positive real θ and δ ≤ |x| ≤ 2δ. We recall that (6.4) holds once we choose ρ of the form ρ = |ρ|ω k , ω as defined in the preceding section, with a suitable k = 0, . . . , 4, and 0 < θ is sufficiently small. Let now χ ∈ C0∞ ({x | x0 < 0}) and let χλ (x) = λ8+2(n−2) χ(λx0 , λ2 x1 , λ2 x

, λ5 xn ), where λ denotes a positive large parameter. Then we have A(N ) − | Uϕ , χλ | ≤ χλ − Uϕ − N Uϕ −N ≤ Cλ −N ,

(6.5)

where A(N ) depends only on N . On the other hand, choosing λ = |ρ|, let us consider, when |ρ| → +∞, Uϕ , χ|ρ| .

(6.6)

Hyperbolic operators with double characteristics

43

By formula (5.1) we have that Uϕ , χ|ρ|

 =

χ|ρ| (x)ϕ(x) 



3/2

exp iρ xn t − ρ

×

5

10−θ

1/2

× w(x1 ρ t

2 2/5



3/2

=

i (t − 1) + ˜ζ(ρ−2/p t−2/5p )ρt1/5 x0 2



2



2 −2/p −2/5p ˜ ; ζ(ρ t ))dt e−|ρ||x | dx

10−θ

e−ρ

(t−1)

2

 χ(x)ϕ

1/2

1 |ρ|

(x)w(x1 ω 2k t2/5 ; ˜ζ(ρ−2/p t−2/5p ))

  i ˜ −2/p −2/5p k 1/5 −1

2 × exp ixn t + ζ(ρ t )ω t x0 − |ρ| |x | dx dt 2 



3/2

=

10−θ

e−ρ

(t−1)2

H(t, ρ)dt.

(6.7)

1/2

Here ϕ 1 has the same definition of χλ but without the Jacobian factor. |ρ| Let us examine the function H(t, ρ); it is real-analytic and H(t, ρ) = H0 (t) + o(1), where o(1) tends to zero uniformly in t when |ρ| → +∞. H0 is defined by the formula  k 1/5 i H0 (t) = ϕ(0) χ(x)eixn t+ 2 ζ 0 ω t x0 w(x1 ω2k t2/5 ; ζ 0 )dx. We apply the stationary phase method to the t-integral in (6.7) and obtain that   10−θ 2 1 Uϕ , χ|ρ| = (H0 (1) + O(|ρ|−δ )), (6.8) ρ with a suitable positive number δ, so that  k i H0 (1) = ϕ(0) χ(x)eixn + 2 ζ 0 ω x0 w(x1 ω2k ; ζ 0 )dx.

(6.9)

44

Enrico Bernardi and Antonio Bove

It is now possible to choose χ ∈ C0∞ ({x ∈ Rn+1 | x0 < 0}) in such a way that |H0 (1)| > 0, because of (6.9). Hence, replacing Uϕ by ρ(10−θ)/2 Uϕ —which has no effect on the exponential decay estimate (6.4)—from the relations (6.1), (6.4), (6.5) and the fact that as |ρ| → +∞ Uϕ , χ|ρ| −→ H0 (1) we obtain a contradiction. This achieves the proof of the following. Theorem 6.1 The Cauchy problem for the operator P in (1.1) is not C ∞ well posed.

References 1. E. Bernardi and A. Bove, Propagation of Gevrey singularities for hyperbolic operators with triple characteristics, I, Duke Math. J. 60(1990), 187–205. 2. E. Bernardi and A. Bove, A Remark on the Cauchy Problem for a Model Hyperbolic Operator, In V. Ancona, J. Gaveau (eds), Hyperbolic Differential Operators and Related Problems, 41–52, 2002. Marcel Dekker, New York. 3. E. Bernardi, A. Bove and C. Parenti, Geometric Results for a Class of Hyperbolic Operators with Double Characteristics, II, J. Funct. Anal. 116(1993), 62–82. 4. E. Bernardi and A. Bove, Geometric Results for a Class of Hyperbolic Operators with Double Characteristics, Comm. Partial Differential Equations 13(1988), 61–86. ¨ rmander, The Cauchy problem for differential equations with double char5. L.Ho acteristics, J. Anal. Math. 32(1977), 118–196. 6. S. G. Krantz and H. R. Parks, A primer of real analytic functions, Birkh¨ auser, Boston, 2002. 7. V.Ya.Ivrii, The Well-posednass of the Cauchy Problem for Nonstrictly Hyperbolic Operators. III. The Energy Integral, Trans. Moscow Math. Soc. 34(1978), 149–168. 8. T. Nishitani, Note on Some Non-Effectively Hyperbolic Operators, Sci. Rep. College Gen. Ed. Osaka Univ. 32(1983), 9–17. 9. T. Nishitani, The Hyperbolic Cauchy Problem, Lecture Notes in Mathematics 1505, 1991, Springer-Verlag, New York. 10. T. Nishitani, Non-effectively hyperbolic operators, Hamilton map and bicharacteristics,, J. Math. Kyoto Univ. 44(2004), 55–98. 11. Y. Sibuya, Global Theory of a Second Order Linear Ordinary Equation with a Polynomial Coefficient, North-Holland Mathematical Studies vol. 18, NorthHolland, Amsterdam–Oxford, 1975.

On the differentiability class of the admissible square roots of regular nonnegative functions Jean-Michel Bony1 , Ferruccio Colombini2 and Ludovico Pernazza3 1 2 3

´ Centre de Math´ematiques Laurent Schwartz, Ecole Polytechnique, Palaiseau, France Dipartimento di Matematica, Universit` a di Pisa, Pisa, Italia Dipartimento di Matematica, Universit` a di Pavia, Pavia, Italia

Summary. We investigate the possibility of writing f = g 2 when f is a C k nonnegative function with k ≥ 6. We prove that, assuming that f vanishes at all its local minima, it is possible to get g ∈ C 2 and three times differentiable at every point, but that one cannot ensure any additional regularity.

2000 Mathematics Subject Classification: 26A15, 26A27. Key words: Square roots, nonnegative functions, modulus of continuity, nondifferentiability.

1 Introduction We study the existence of a function g of a certain regularity satisfying g 2 = f for f a nonnegative real function of one variable (we will say that g is an admissible square root of f ). The starting point can be taken from the article by G. Glaeser ([4]), who proved that if f is C 2 and 2-flat on its zeros (i.e., f (x) = 0 implies f

(x) = 0) then f 1/2 is C 1 . Later, dropping the flatness assumption, Mandai (see [6]) showed that if f is C 2 , f has a C 1 admissible square root. More recently in [1] (see also [5] for an improved result) it was shown by D. Alekseevsky, A. Kriegl, P. W. Michor and M. Losik that if f is C 4 , a suitable admissible square root g can be chosen to be not only C 1 , but also twice differentiable at every point. In a joint work with F. Broglia (namely, [3]) we proved that this result is sharp, i.e., that in general it is not possible to find g in a fixed class C 1,ω for any modulus of continuity ω. In the same paper we also examined the case of nonnegative C 4 functions that vanish at all their minima; for these it is possible to find a C 2 admissible square root g. We again showed that no better regularity can be obtained, even for f ∈ C ∞ .

46

Jean-Michel Bony, Ferruccio Colombini and Ludovico Pernazza

In the present paper, we prove a natural extension of these results to nonnegative C 6 functions. If such a function f vanishes at all its (local) minima, then it has an admissible square root g which is three times differentiable at every point (Theorem 2.1). In general, g does not belong to C 3 and not even to C 2,ω (Example 2.1). We also give examples and counterexamples illustrating what happens when the conditions of Theorem 2.1 are not fulfilled: among other things, we find a necessary condition based on the characterization of functions admitting a C 2 admissible square root given in [3, Theorem 3.5].

2 Regularity of well-chosen admissible roots First of all, we would like to give a few examples of the relation between the regularity of f and that of its admissible square roots. Some of these are contained in [1] or in [3]. Example 2.1 Let k > 0 be an integer. The functions f1,k (t) = t2k sin2 (log |t|), f2,k (t) = t

2k

f3,k (t) = 2t

log |t|,

2k

+t

2k−1

for t = 0, f1,k (0) = 0, for t = 0, f1,k (0) = 0,

2

and

|t|

are all nonnegative and C 2k−1,α for every 0 < α < 1 (the f1,k and f3,k are even C 2k−1,1 ), but they have no C k admissible root (even though for f1,k and f3,k these can be chosen C k−1,1 ). In particular, there exist nonnegative C 1,1 functions without differentiable admissible square roots and C 3,1 nonnegative functions without twice differentiable admissible square roots. The function f (t) = t4 sin2 (1/t),

for t = 0, f (0) = 0

is twice differentiable, but has no C 1 admissible square root. Before giving our next example, we need to recall a few definitions. A modulus of continuity is a continuous increasing concave function ω, defined on an interval [0, t0 ], satisfying ω(0) = 0; if Ω is an open subset of R, a function f : Ω → R will be called ω-continuous (and ω is called a modulus of continuity for f ) on Ω if the quantity [f ]ω =

|f (y) − f (x)| 0<|x−y|<min(t0 ,d(x,
Ω)/2) ω(|y − x|) sup

is finite (we note that one can always suppose ω(s) ≥ s). We call C k,ω the functions whose kth derivative is ω-continuous; to avoid confusion, we will

Square roots of regular nonnegative functions

47

always use ω with this purpose, keeping the use of the letter α instead of ω (or even of a number between 0 and 1) for the special case ω(s) = sα , that is, the usual H¨ older continuity condition. From the analysis of the C 4 case we take the following result (see [3, Theorem 3.6]: Example 2.2 For any given modulus of continuity ω there is a C ∞ nonnegative function f on R, taking the value 0 at all its minima, which has no C 2,ω admissible square root. This is achieved by e.g., choosing an even C ∞ function χ vanishing outside (−2, 2), positive and logarithmically convex on (1, 2) and such that χ(t) = 1 for −1 ≤ t ≤ 1 and considering functions of the type f (t) =

∞
n=1

 χ

2

t − tn ρn

 (αn (t − tn )4 + β n (t − tn )2 )

for a careful choice of the sequences ρn , tn , αn and β n ; the logarithmic convexity here is requested because it ensures that f has no nonzero minima and all the calculations can be made explicitly. A similar counterexample was presented in [1] as a case of a C ∞ function, flat on its zeros, not having a C 2 admissible square root. We now prove the main result of this paper: Theorem 2.1 Let f be a nonnegative C 6 function of one variable such that it takes the value 0 at all its minima. Then f has an admissible square root g that is three times differentiable at every point. Proof. First, let us call F2k for k = 0, . . . , 3 the set of points where f and all its derivatives up to the 2kth vanish. We note that the points in F0 \ F6 are isolated, but can accumulate to points in F6 . We call Ai the connected components of R\F6 and choose a point xi ∈ Ai where f (xi ) = 0. If x ∈ Ai \F0 , call nx the number of points where f vanishes at order 2 or 6 between xi and x (they are isolated and thus nx is well defined); we set g(x) to be (−1)nx f (x) (in particular, g(xi ) = f (xi )). The remaining points are in F0 and there we set g|F0 = 0: g is then an admissible root of f . The function g is also three times differentiable at every point. It will be sufficient to show this on any given bounded interval I = [−2R, 2R]; moreover, this is trivial at the points where f = 0. Let then x0 be a point where f vanishes; up to a translation, we can assume that x0 = 0. We now argue on the order of f at 0. If f has a zero of order 2 at 0, we can write f (x) = x2 h(x) with h ∈ C 4 (see [7]) and h(0) = 0. By our choice of g, g(x) = ±x h(x), that is certainly three times differentiable at 0 (indeed, it is C 4 in a whole neighborhood of it). If the order of f at 0 is 4, writing f (x) = x4 h(x) as above (this time h ∈ C 2 ), again g(x) = x2 h(x) is three times differentiable at 0, since

48

Jean-Michel Bony, Ferruccio Colombini and Ludovico Pernazza

g

(x) = ±(2

h(x) + 4x( h(x) )
+ x2 ( h(x) )

)

and g

(0) = ±2 h(0), therefore g


(0) = ±6( h(x) )
(0). If the order is 6, writing f = x6 h(x) with h continuous and nonzero at 0 but C 6 outside 0, we see by recurrence that xk h(k) (x) for k = 0, . . . , 3 is continuous (since by [7] again xk h(x) is C k for k = 0, . . . , 6 and all the other terms of the kth derivative of xk h(x) apart from xk h(k) (x) are easily seen to be continuous) and therefore g(x) = ±x3 h(x) ∈ C 3 . We note that up to now we have relied on our “good” choice of the sign of the square roots. We are now left with the case where the order of f at 0 is bigger than 6 (that is, all the first six derivatives of f vanish at 0). There g

(0) = 0 and we expect the value of g


(0) to be also 0, because otherwise f should have order at most 6. We consider a small closed neighborhood J of the origin where f (6) (x) ≤ 1; let us call ω a modulus of continuity of f (6) (the sixth derivative of f ) in J, d the euclidean distance and α(x) = ω(d(x, F6 )). Since 0 ∈ F6 , the distance d(x, F6 ) ≤ |x| and up to dividing f by a constant M we can assume that |f (6) (x)| ≤ α(x). Thanks to this normalization, the various constants arising in the following estimates (that we will denote by c and C with the convention that c can be chosen small and C large enough for our aim) will not depend on the point and on the function, even though in principle they could do so without affecting our statement. To prove our assertion we will obtain an estimate of g

(x) at all the points x ∈ J. As a very first note, we observe that at the points of F6 ∩J, g

vanishes. We will then study the other points, where the function ⎧ 1/2 ⎫  (4) ⎬ ⎨ f (x) 1/6  f

(x) 1/4 f+ (x) + , , , ρ(x) = sup ⎭ ⎩ α(x) α(x) α(x) where a+ = sup{a, 0}, is well defined. We make use here and below of various results in [2], namely, Lemmas 3.4, 4.1 and 4.2 and Corollaries 3.5 and 3.6; we thus know that ρ(x) ≤ d(x, F6 ) ¯ and for a suitable positive constant C, 6−k ¯ |f (k) (x)| ≤ Cα(x)ρ(x) ,

We recall that ( f (x) )

=

for k = 0, . . . , 6.

f

(x) f
(x)2 − . 2f (x)1/2 4f (x)3/2

If now f (x)/α(x) ≥ c1 ρ(x)6 , where c1 < 1 is a small positive constant to be chosen below, it is easy to see that |g

(x)| = |(

f (x) )

| ≤

4 ¯ Cα(x)ρ(x) C¯ ≤ α(x)1/2 ρ(x). 1/2 3 c1 c1 α(x) ρ(x)

Square roots of regular nonnegative functions

49

Assume instead that f (x)/α(x) < c1 ρ(x)6 , in particular the supremum defining ρ is not attained by the first term, but by the second or the third one. In this case again from [2] we know that if c1 is smaller than a certain pos¯ independent of itive value ∆(1), then there exists a positive δ < 1/(12C), f and x (and c1 too, as far as c1 < ∆(1)), such that in a neighborhood [x − δρ(x)/3, x + δρ(x)/3] there is a minimum x0 (where now by our hypothesis f vanishes). Moreover, if k is the order of derivation of f in the dominating term in ρ, then 5 3 α(x)ρ(x)6−k ≤ f (k) (y) ≤ α(x)ρ(x)6−k 4 4 for y ∈ Ix = [x − 3δρ(x), x + 3δρ(x)]. ¯ 1/4 < Let us now choose the constant c1 such that 0 < c1 < ∆(1) and Cc 1 δ/6. We must separate it into two cases. The first occurs when (4)

f (x) f

(x) > c1 + 2 . α(x) α(x) In Ix then,

f

(x) 2 and so f

(y) > c1 α(x)ρ(x)4 /2. We can write f

(y) ≥

f (x) = (x − x0 )2 H(x) '1 where H(x) = 0 (1 − s)f

((1 − s)x0 + sx) ds is a C 4 nonnegative function that does not vanish at x0 . In this case we see easily that |H(x)| ≥ c1 α(x)ρ(x)4 /4,

|H
(x)| ≤ Cα(x)ρ(x)3 ,

|H

(x)| ≤ Cα(x)ρ(x)2

and since by our choice of g, g(x) = (x − x0 )H(x)1/2 (and |x − x0 | ≤ ρ(x)), then   Cα(x)ρ(x)2 Cα(x)2 ρ(x)6 + |g

(x)| ≤ C|x − x0 | α(x)1/2 ρ(x)2 α(x)3/2 ρ(x)6 +

Cα(x)ρ(x)3 ≤ Cα(x)1/2 ρ(x). α(x)1/2 ρ(x)2

In the second case, that is, when the term with the fourth derivative is dominating, if the function vanishes at order 4 or twice at order 2 in the same interval, we can write f (x) = (x − x0 )2 (x − x1 )2 K(x) (with maybe x0 = x1 ) and deduce the same inequalities as above from the formula  1  1 K(x) = (1 − s) t2 (1 − t)f (4) ((1 − t)x0 + t((1 − s)x1 + sx)) dt ds. 0

0

50

Jean-Michel Bony, Ferruccio Colombini and Ludovico Pernazza

If instead there is only one zero point of f of order 2, we write as above f (x) = (x − x0 )2 H(x) (we assume that x ≥ x0 ). We now claim that H attains a minimum in Ix . To prove this, first note that since f (4) (y) > 0 in Ix , also H

(y) > 0 in the same interval, so it is sufficient to prove that H
(y) vanishes. It is also clear that F − (y) ≤ f


(y) ≤ F + (y) where F − (y) = and F + (y) =

⎧ ⎨f


(x) − 5 |y − x| for y ≤ x 4 ⎩f


(x) + 3 |y − x| for y > x 4 ⎧ ⎨f


(x) − 3 |y − x| 4

for y ≤ x

⎩f


(x) + 5 |y − x| 4

for y > x

.

Let us look at the sign of f


(x0 ). If f


(x0 ) = 0 also H
(x0 ) = f


(x0 )/6 = 0 and we have nothing more to prove. If f


(x0 ) > 0, also H
(x0 ) = f


(x0 )/6 > 0; we estimate the integral defining H
(y) letting x0 = x and f


(y) = F + (y); but F + (y) is linear and so it is sufficient to prove that f


(x − 32 δρ(x)) < 0. We know again from [2], Lemma 4.1, that ¯ 1/4 α(x)ρ(x)3 , |f


(x)| ≤ Cc 1 ¯ 1/4 < 9 δ which is obvious by our so our condition is fulfilled as soon as Cc 1 8 choice of c1 . If instead f


(x0 ) < 0, we can assume that x0 = x − δρ(x)/3 and f


(y) = + F (y) and since now F + is a concave function, we only need to show that F + (x + 3δρ(x)) > −F + (x0 ); we use the same estimate as above for f


(x) ¯ 1/4 < δ/6, as provided and we have that the condition is fulfilled as soon as Cc 1 again by our choice of c1 . This ends the proof of our claim: H indeed attains a minimum in Ix . We can then write f (x) = (x − x0 )2 (a + (x − z)2 h(x)) where z is the minimum point of H(x) (we can assume that z < x0 ), a = H(z) and h(x) ≥ 0. Since H(x) can be written as a mean involving the values of f

in the interval between x0 and x, as seen above, h(x) is also a mean involving the values of H

, i.e., of f (4) in that interval and we have that cα(x)ρ(x)2 ≤ h(x) ≤ Cα(x)ρ(x)2 ,

|h
(x)| ≤ Cα(x)ρ(x),

|h

(x)| ≤ Cα(x).

Square roots of regular nonnegative functions

51

But if there are no more minima of f , f
must have constant (negative) sign in the interval (z, x0 ): let us compute its value in this interval at a general point x(s) = z + s(x0 − z), s ∈ (0, 1), f
(x(s)) = 2(s − 1)a(x0 − z) + ((s − 1)s2 + 2s(s − 1)2 )(x0 − z)3 h(x(s)) + (s − 1)2 s2 h
(x(s))(x0 − z)4 . We see that for s small (say s ≤ 1/4) the coefficient of the second term is positive and therefore the whole term is bigger than cs(x0 − z)3 α(x)ρ(x)2 ; the third term (whose coefficient depends on s2 ) is smaller than that quantity for sufficiently small s, therefore we must have (1 − s)(x0 − z)a > c(x0 − z)3 α(x)ρ(x)2 . We then write g(x) = (x − x0 )(a + (x − z)2 h(x))1/2 and g

(x) =

(x − z)h(x) + (x − z)2 h
(x) (a + (x − z)2 h(x))1/2 +

(x − x0 )(2h(x) + 4(x − z)h
(x) + (x − z)2 h

(x) 2(a + (x − z)2 h(x))1/2

+

(x − x0 )(2(x − z)h(x) + (x − z)2 h
(x))2 ; 4(a + (x − z)2 h(x))3/2

since (x − z)h(x) ≤ Cα(x)ρ(x)1/2 , (x0 − z)h(x) ≤ C(x0 − z)α(x)ρ(x)2 , a1/2 ≥ c(x0 − z)α(x)1/2 ρ(x) and (x − z)2 h(x)/(a + (x − z)2 h(x)) is bounded by 1, we get once again that |g

(x)| ≤ Cα(x)1/2 ρ(x). On the whole, since g

(0) = 0 and g

(x) ≤ Cα(x)1/2 ρ(x) we see that |g

(x) − g

(0)| |g

(x)| ≤ ≤ Cα(x)1/2 |x| d(x, F6 ) which tends to 0 as x tends to 0, as desired. Remark 2.1 It is clear that one cannot weaken the C 6 hypothesis: if we take as f the only continuous extension of t6 (log |t|)2 , we have a counterexample older continuous on every exponent, where f is C 5 and its fifth derivative is H¨ but there is no admissible square root of f even in C 2,1 ; but even for f C 5,1 , it was already pointed out in Example 2.1 that e.g., f = 2t6 + t5 |t| is a C 5,1 function such that the admissible square roots can be at most C 2,1 , but never three times differentiable. Moreover, the result does not directly extend to functions with nonzero minima that have C 2 admissible square roots. On this class of functions we recall the following result:

52

Jean-Michel Bony, Ferruccio Colombini and Ludovico Pernazza

Theorem 2.2 (see [3, Theorem 3.5]) A necessary and sufficient condition for a C 4 nonnegative function to admit C 2 square roots is that there be a continuous function γ vanishing on F4 such that at the nonzero minima x0 , f

(x0 ) ≤ γ(x0 )f (x0 )1/2 . One might think that this condition could be sufficient also for C 6 functions to admit three times differentiable square roots. But it is not difficult to find a function fulfilling the condition of Theorem 2.2 without any admissible square root of the desired regularity: indeed, the existence of such a function γ can only ensure that if a sequence xn of minima converges to a point x (in F6 ), then g

(xn ) converges to 0. Example 2.3 Let us define f (t) =

∞
n=1

 χ

2

t − tn ρn

 (αn (t − tn )2 + β n )

as already done above, with χ as in Example 2.2 and ρn =

1 , n2

∞


tn = 2ρn +

5ρj ,

j=n+1

In = [tn − 2ρn , tn + 2ρn ],

αn =

1 , 2n log n

βn =

1 . 4n

This function f is easily verified to be C ∞ . Here we can take γ(tn ) =

f

(tn ) 2 ; = 1/2 log n f (tn )

γ(tn ) tends to 0 and so the condition of Theorem 2.2 is fulfilled, but |g

(tn ) − g

(0)| n → ∞. ≥ tn 5 log n It is not difficult to find a necessary condition, analogous to that of Theorem 2.1 (though obviously slightly more restrictive) that a nonnegative C 6 function f has to fulfil to be eligible to have a three times differentiable admissible square root. Such a condition is contained in the following Proposition 2.1 For a nonnegative C 6 function to admit three times differentiable square roots, it is necessary that there exist a C 1 function γ vanishing along with its first derivative on F6 such that for every minimum x0 of f where f (x0 ) > 0, f

(x0 ) ≤ γ(x0 )f (x0 )1/2 . Proof. If such a function does not exist, there is a sequence xn of minima tending to a point x¯ ∈ F6 where f does not vanish and such that

Square roots of regular nonnegative functions

53





f (xn ) > 0. xn →¯ x f (xn )1/2 |xn − x ¯| lim

But then for every three times differentiable admissible square root g, |g

(xn ) − g

(¯ x)| f

(xn ) = lim > 0, xn →¯ x xn →¯ x 2f (xn )1/2 |xn − x |xn − x ¯| ¯| lim

which is impossible, because we have already observed that the only possible x) is 0, since x ¯ ∈ F6 . value for g


(¯ We do not know if this condition is also sufficient; this question and many other possible generalizations of the problems examined above should be further investigated.

References 1. D. Alekseevsky, A. Kriegl, P. W. Michor and M. Losik, Choosing Roots of Polynomials Smoothly, Israel J. Math. 105(1998), 203–233. 2. J. -M. Bony, Sommes de carr´es de fonctions d´erivables, to appear in Bull. Soc. Math. France 133(2005), 619–639. 3. J. -M. Bony, F. Broglia, F. Colombini and L. Pernazza, Nonnegative functions as squares or sums of squares, J. Funct. Anal. 232(2006), 137–147. 4. G. Glaeser, Racine carr´ ee d’une fonction diff´ erentiable, Ann. Inst. Fourier (Grenoble) 13(1963), 203–210. 5. A. Kriegl, M. Losik and P. W. Michor, Choosing Roots of Polynomials Smoothly. II, Israel J. Math. 139(2004), 183–188. 6. T. Mandai, Smoothness of roots of hyperbolic polynomials with respect to onedimensional parameter, Bull. Fac. Gen. Ed. Gifu Univ. 21(1985), 115–118. 7. J. -C. Tougeron, Id´eaux de fonctions diff´ erentiables, Ergeb. Math. 71, Springer-Verlag, Berlin, Heidelberg, New York 1972.

The Benjamin–Ono equation in energy space Nicolas Burq1 and Fabrice Planchon2 1 2

D´epartement de Math´ematiques, Universit´e Paris 11, Orsay, France Laboratoire Analyse, G´eom´etrie & Applications, Institut Galil´ee, Universit´e Paris 13, Villetaneuse, 55 France

Summary. We prove existence of solutions for the Benjamin–Ono equation with data in H s (R), s > 0. Thanks to conservation laws, this yields global solutions for 1 H 2 (R) data, which is the natural “finite energy” class. Moreover, unconditional 1 uniqueness is obtained in L∞ t (H 2 (R)), which includes weak solutions, while for 3 s > 20 , uniqueness holds in a suitable space.

2000 Mathematics Subject Classification: 35GXX, 37L50, 35L15. Key words: Dispersive equations, Schr¨odinger equation, gauge transform.

1 Introduction The purpose of this talk is to present some recent results on the Benjamin–Ono equation in the energy space H 1/2 . Let us consider ∂t u + H∂x2 u + u∂x u = 0, u(x, t = 0) = u0 (x), (t, x) ∈ R2 . Here and hereafter, H is the Hilbert transform, defined by  1 1 1 f (y) Hf (x) = dy = vp  u = F −1 (−isgn(ξ)fˆ(ξ)). π x−y π x

(1.1)

(1.2)

We will restrict ourselves to real-valued u0 . Equation (1.1) deals with wave propagation at the interface of layers of fluids with different densities (see Benjamin [1] and Ono [10]), and it belongs to a larger class of equation modeling for this type of phenomena, some of which are certainly more physically relevant. Mathematically, however, (1.1) presents several interesting and challenging properties; the exact balance between the degree of nonlinearity and the smoothing properties of the linear part preclude any hope to achieve results through a direct fixed point procedure, be it in Kato smoothing type of

56

Nicolas Burq and Fabrice Planchon

spaces or more elaborate conormal (“Bourgain”) spaces. In fact, the flow associated to (1.1) fails to be C ∞ (Molinet–Saut–Tzvetkov [9], and even uniformly continuous (Koch–Tzvetkov [8]). By standard energy methods (ignoring therefore the dispersive part), one may obtain local in time solutions for smooth data, e.g., u0 ∈ H s with s > 32 , and reach s = 32 by taking into account some form of dispersion (Ponce [11] and references therein). On the other hand, 1 (1.1) has global weak L2 and H 2 solutions (Ginibre–Velo [3]) and this result relies heavily on dispersive estimates for the nonlinear equation as well as the following two conservation laws:   2 u (x, t) dx = u20 (x) dx, 

R

1 √ 1 | −∆ 2 u(x, t)|2 dx + 3 R





R

u3 (x, t) dx = R

R

+

1 √ | −∆ 2 u0 (x)|2 dx

1 3

 R

u30 (x) dx.

(1.3)

Recently, progress has been achieved on the Cauchy problem for data in Sobolev spaces, by using more sophisticated methods: Koch and Tzvetkov [7] obtained s > 54 , and subsequently Kenig and Koenig [6] improved this result to s > 98 (both use Strichartz estimates which are tailored to the frequency, a procedure directly inspired by work on quasilinear wave equations and subsequently the semi-classical analysis developed in [2, Theorem 6]); Tao [12] obtained H 1 solutions, using a (complex) variant of the Hopf–Cole transform (which linearizes Burgers equation). These solutions can be immediately extended to global ones, thanks to another conservation law controlling the H˙ 1 norm (equation (1.1) which, being completely integrable, has an infinite hierarchy of conservation laws, a fact which at the moment cannot be connected with the Cauchy problem at low regularity and will not be used in this paper). Our main result reads as follows: Theorem 1.1 For any s > 14 , there exists a unique strong solution of the 0 Benjamin–Ono equation (1.1), which is Cloc (Rt ; H s (Rx )). Furthermore, if s ≥ ∞ 1/2, this solution is global and unique in Lt (Rt ; H s (Rx )), while for 14 < s < 1/2, uniqueness holds in suitable spaces. As a consequence, in the energy space, H 1/2 , weak solutions are strong. Remark 1.1 Remark that Benjamin–Ono is a Hamiltonian equation, with Hamiltonian given by (1.3) and H 1/2 is consequently the natural energy space. As such, existence and uniqueness of solutions in H 1/2 is of particular relevance. We shall use the fact that if u is a solution of Benjamin–Ono, so is uλ = λu(λx, λ2 t).

The Benjamin–Ono equation in energy space

57

As a consequence, a scaling invariant space is H −1/2 (Rx ) and since all spaces we will consider are above the scaling, local existence of solutions is equivalent with time T = 1 existence for small initial data. All the existence results are obtained by approximating the initial data and passing to the limit. As a consequence, for the existence part of our theorem, we have only to establish a priori estimates for smooth solutions. Remark 1.2 Recently, Ionescu and Koenig [5] improved existence all the way down to s = 0, which yields global L2 solutions. While this obviously supersedes our existence result, uniqueness is meant in the class of limits of smooth solutions.

2 Bourgain spaces We define the Bourgain spaces by XTs,0 = L2 ((−T, T ); H s (Rx )), XTs,1 = {U ∈ XTS,0 ; (∂t + H∂x2 )u ∈ X s,0 }, and for 0 ≤ b ≤ 1 and −1 ≤ b ≤ 0, XTs,b is defined by interpolation and duality. In fact we need a Besov version of these spaces, XTs,b,q . An alternative definition would be 2

X s,b = {u; etH∂x u ∈ H b ((−T, T ); Hxs )}. Projecting the equation on the positive (negative) part of the spectrum gives two linear Schr¨ odinger equations (∂t − i∂x2 )P + u = 0,

(∂t + i∂x2 )P − u = 0,

for which we can use the standard estimates. •

Strichartz. uL4t ;L∞
u0 L2x . x

(2.1)

•

Maximal function.

•

Bilinear smoothing. If Sj−1 is a spectral cut-off to frequencies smaller than 2j−1 and ∆j a spectral cut-off to frequencies of order 2j , we have

uL4x;L∞
u0 H 1/4 . t

j

Sj−1 u∆j vL2t,x
2− 2 u0 L2 v0 L2 . •

Smoothing. 2
uL2 . ∂x1/2 uL∞ x ;Lt

A most important feature of the spaces X s,b is that they inherit all properties of the linear flow.

58

Nicolas Burq and Fabrice Planchon

Proposition 2.1 For any b > 1/2, we have •

Strichartz.

•

Maximal function.

•

Bilinear smoothing. If Sj−1 is a spectral cut-off to frequencies smaller than 2j−1 and ∆j a spectral cut-off to frequencies of order 2j , we have


uX 0,b . uL4t ;L∞ x
uX 1/4,b . uL4x;L∞ t

j

Sj−1 u∆j vL2t,x
2− 2 uX 0,b vX 0,b . •

(2.2)

Smoothing 2
uX 0,b . ∂x1/2 uL∞ x ;Lt

Remark 2.1 This property is also true for the Besov version of X s,1/2 , X s,1/2,1 .

3 A priori estimate on weak solutions The first step concerning the uniqueness result for weak solutions is the following. Proposition 3.1 For any weak solution, 1/2

uL4t ;L∞ < CuL∞ ;H 1/2 uL∞ ;H 1/2 . x To prove this result, we localize the function on small time intervals whose size is adapted to the frequencies considered. The function χ(2j t)∆j (v) satisfies (∂t + H∂x2 )χ(2j t)∆j (v) ⎛ ⎞

1 = 2j χ
(2j t)∆j (v) + χ(2j t)∂x ∆j ⎝ vk vj + vk2 ⎠ . 2 k≤j

(3.1)

k>j

We can use Sobolev injection and take benefit from the fact that the integrals in times are taken on intervals of length 2−j to obtain ⎞ ⎛

j r.h.s.L2t,x
vL∞ 2−j + 2j 2−k ⎠ v2L∞ 1/2 ) + ⎝2 s t (H t (H ) k≤j


(1 + j)vL∞ 1/2 ) . t (H

k≥j

The Benjamin–Ono equation in energy space

59

As a consequence, we obtain χ(2j t)∆j (v)X 0,0
2−j vL∞ 1/2 , t ;H χ(2j t)∆j (v)X 0,1
(1 + j)vL∞ 1/2 ) t (H

(3.2)

1

⇒ χ(2j t)∆j (v)X 0,1/2+
2−j( 2 −) vL∞ 1/2 . t ;H We use Strichartz estimate (2.1) and obtain 1

χ(2j t)∆j (v)L4t ;L∞
2−j( 2 −) vL∞ 1/2 . t ;H Now we have to sum the time intervals (2j intervals) and we get 1

∆j (v)L4t ;L∞
2−j( 4 −) vL∞ 1/2 , t ;H which gives Proposition 3.1.

4 The gauge transformation Now let us give the main ingredients to prove existence and uniqueness at the level of regularity we obtained for the weak solutions. We first prove a priori estimates on smooth solutions. For this, we are going to paralinearize and project the solutions on the positive spectrum


uk ∂x P + uj + ∂x uk P + uj + ∂x uk uk . (∂t − i∂x2 )P + ∆j (u) = k j

k j

k≥j

Here and in the rest of the article we denote by uj any function obtained by truncating u spectrally to frequencies of order 2j . The essential point now is that a suitable gauge transform (Tao 04, see also Hayashi–Ozawa 94) eliminates the bad term. In order to eliminate the worst term in the nonlinear interaction, we would like to consider vj = e

−i 2

ÊxÈ kj

uk + uj

'x where g is a suitably defined primitive function of g. In fact, to keep the paralinearization properties, we consider vj = Sj−1 (e

−i 2

Êx

udx

)∆+ j (u).

Plugging this definition in the equation satisfied by u gives

(∂t − i∂x2 )vj+ = ∂x uk vj+ + ∂x uk vk+ + l.o.t., k j

k≥j

60

Nicolas Burq and Fabrice Planchon

and we see in the expression above that we eliminated the worst interaction term, namely
∂x uk vj+ j k

where the derivative falls on the high frequencies. The main problem is that we want to perform our estimates in the context of Bourgain spaces. Consequently, we have to understand the action in X s,b of the map (v, u) → w, wj = Sj−1 (e

−i 2

Êx

v

)∆j (u) = Fj ∆j (u) = T uj

(where v is a solution of the Benjamin–Ono equation). Let us fix v as such a 1 1 solution, bounded in X 4 , 2 and proceed by interpolation. X 0,0 = L2t,x ,

T : X 0,0 → X 0,0 ;

T  ∼ 1

X 0,1 = {u ∈ X 0,0 ; (∂t + H∂x2 )u ∈ L2t,x } and with F = e− 2 i

Êx

v

,

(∂t + H∂x2 )T u+ j = Fj (∂t − i∂x2 )uj + (∂t − i∂x2 )Fj uj − 2i∂x (Fj )∂x (uj )   x  Ê − 2i x v −i 2 = OK + Sj−1 e (∂t − i∂x ) v uj − 2i∂x (Fj )∂x (uj ), 2 (∂t + H∂x2 )T u+ j   i −i Ê x v = OK + Sj−1 − e 2 (∂t − i∂x2 ) 2

x

 v uj − 2i∂x (Fj )∂x (uj ).

Since the function v satisfies the Benjamin–Ono equation (1.1), we can exchange ∂t v for H∂x2 v + ∂x (v 2 ). The second term is better than the last one. Let us focus on this last term ∂x Fj ∼ Fj Sj−1 (u). At this point, we can use the bilinear smoothing (2.2): we lost 1 derivative, we regain 1/2. Finally, we have a 1/2 loss. To summarize, we have Proposition 4.1 Consider v a solution of the Benjamin–Ono equation. Then the gauge transform T : u → Sj−1 (e

−i 2

Êx

vdx

)∆+ j (u)

The Benjamin–Ono equation in energy space

61

satisfies T : X 0,0 → X 0,0 ;

T  ∼ 1,

1

T : X 2 ,1 → X 0,1 ; T :X

s, 12

→X

s− 14 , 12

T  ∼ 1, ;

T  ∼ 1.

Remark 4.1 At the level of Bourgain X s,b spaces the gauge transform loses p 1 q 4 of its derivatives. On the other hand, at the level of Strichartz Lt ; Lx or 4 ∞ maximal estimate, Lx ; Lt , since the solution is real, the exponential factor is transparent and we have no loss.

5 The existence and uniqueness result We prove a priori estimates for smooth solutions for 1 1

1. u ∈ X s− 4 , 2 and Strichartz at the level of regularity 14 , 1 2. w = T u ∈ X s, 2 . The tool to prove these a priori estimates is the use of classical bilinear estimates (for the Schr¨ odinger equation because we projected the equation on the positive part of the spectrum). Concerning the uniqueness part, we consider two solutions u, v of the Benjamin–Ono equation, with initial data in H s , s ≥ 1/2, one of which (u) we constructed, and we prove the estimate u − v

1 1

X− 2 , 2

 ≤ C(u0 , v0 )u0 − v0 

1

H− 2

.

The steps of the proof are the following. 1. We use the renormalization method using the bad solution (to eliminate the worst terms). 2. We use bilinear estimates in X s,b spaces as for the existence. 3. We use that the solution we constructed is better than merely L∞ ; H 1/2 . 4. We use the a priori estimate for weak solutions we proved above (Proposition 3.1). Remark 5.1 Let us remark that for existence, the method of proof (plus a careful study) goes down to s > 0 and for uniqueness, we can obtain unique3 3 . For 0 ≤ s ≤ 20 ness in (a smaller class than) the construction class for s > 20 the uniqueness status is not clear.

62

Nicolas Burq and Fabrice Planchon

References 1. T. Benjamin, Internal waves of permanent form in fluids of great depth, J. Fluid Mech. 29(1967), 559–592. 2. N. Burq, P. G´ erard, and N. Tzvetkov. Strichartz inequalities and the nonlinear Schr¨ odinger equation on compact manifolds, Amer. J. Math. 126(2004), 569–605. 3. J. Ginibre and G. Velo. Smoothing properties and existence of solutions for the generalized Benjamin–Ono equation, J. Differential Equations 93(1991), 150–212. 4. N. Hayashi and T. Ozawa. Remarks on nonlinear Schr¨ odinger equations in one space dimension, Differential Integral Equations 7(2): 453–461, 1994. 5. A. D. Ionescu and C. E. Kenig, Global well-posedness of the Benjamin–Ono equation in low regularity spaces, Preprint, arXiv: math. AP/0508632, 2005. 6. C. E. Kenig and K. D. Koenig, On the local well-posedness of the Benjamin– Ono and modified Benjamin–Ono equations, Math. Res. Lett. 10(2003), 879– 895. 7. H. Koch and N. Tzvetkov, On the local well-posedness of the Benjamin–Ono equation in H s (R), Int. Math. Res. Not. 26(2003), 1449–1464. 8. H. Koch and N. Tzvetkov, Nonlinear wave interactions for the Benjamin– Ono equation, Preprint, arXiv:math.AP/0411434, 2004. 9. L. Molinet, J. C. Saut, and N. Tzvetkov, Ill-posedness issues for the Benjamin–Ono and related equations, SIAM J. Math. Anal. 33(2001), 982–988 (electronic). 10. H. Ono, Algebraic solitary waves in stratified fluids, J. Phys. Soc. Japan 39(1975), 1082–1091. 11. G. Ponce, On the global well-posedness of the Benjamin–Ono equation, Differential Integral Equations 4(1991), 527–542. 12. T. Tao, Global well-posedness of the Benjamin–Ono equation in H 1 (R), J. Hyperbolic Differ. Equ. 1(2004), 27–49.

Instabilities in Zakharov equations for laser propagation in a plasma Thierry Colin and Guy M´etivier D´epartement de Math´ematiques Appliqu´ees and CNRS UMR 5466, Universit´e Bordeaux 1, Talence, France

Summary. In [LPS], F. Linares, G. Ponce and J.-C. Saut have proved that a nonfully dispersive Zakharov system arising in the study of laser-plasma interaction, is locally well posed in the whole space, for fields vanishing at infinity. Here we show that in the periodic case, seen as a model for fields non-vanishing at infinity, the system develops strong instabilities of Hadamard’s type, implying that the Cauchy problem is strongly ill posed.

2000 Mathematics Subject Classification: 35A07, 35Q53, 78A60. Key words: Ill-posedness, Zakharov, Langmuir turbulence, hyperbolicity.

1 Introduction 1.1 Physical context The construction of powerful lasers allows new experiments where hot plasmas are created in which laser beams can propagate. The main goal is to simulate, in a laboratory, nuclear fusion by inertial confinement. This requires precise and reliable models for laser-plasma interactions which can be used to produce numerical simulations that are usable to predict and illustrate the experiments. The kinetic-type models are the more precise ones but their cost in term of computations is exorbitant and, so far, no physically relevant situation for nuclear fusion can be simulated using these models. Another approach uses a bi-fluid model for the plasma, coupling two compressible Euler systems with Maxwell equations. Even in this form, it is not possible to perform direct computations because of the high frequency motions and of the small wavelength involved in the problem. At the beginning of the 1970s, Zakharov and his collaborators introduced the so-called Zakharov’s equations [ZMR] to describe electronic plasma waves. These systems couple the slowly varying

64

Thierry Colin and Guy M´etivier

envelope of the electric field and the low-frequency variation of the density of the ions. A typical non-dimensional form of such a system is: i∂t u + ∆u = nu, ∂t2 n − ∆n = ∆|u|2 . When considering the propagation of a laser beam in a plasma, several such systems have to be coupled in order to take into account the laser beam, the Raman component and the electronic plasma waves (see [CC1, CC2] for example). The laser beam and the Raman component correspond to transverse electromagnetic waves, while the electronic plasma waves are longitudinal waves. In the latter case, the electric field is a gradient E = ∇ψ which is linked to low-frequency variations of the density of the ions δn by the following equations: ⎧ 2 ⎨ i∂t ∇ψ + 3vth ∆(∇ψ) = ωpe ∇∆−1 div(δn∇ψ), 2ω pe 2n0 (1.1) ⎩ ∂ 2 δn − c2 ∆δn = 1 ∆(|∇ψ|2 ), t s 4πmi where vth is the thermal velocity of electrons, ω pe the plasma electronic frequency, n0 the mean density of the plasma, mi the mass of the ions and cs the sound velocity in the plasma. Typical values of vth is 0.1c where c is the speed of light and ω pe ∼ 1015 s−1 . The underlying wavelength is of the order of the micro-meter. For laser propagation or for the Raman component, one often uses the paraxial approximation, and the Zakharov system that couples the vector potential A of the electromagnetic field to the low-frequency variation of the density of the ions reads ⎧   ω 2pe c2 ⎪ ⎨ i ∂t + kc20 ω 0 ∂z A + 2ω ∆ A = nA, x 2n 0 0 ω0 (1.2) 2 ⎪ ⎩ (∂ 2 − c2 ∆ )n = ωpe ∆ |A|2 , t s x 4πmi c2 x where ω 0 is the frequency of the laser and k0 its wave number; they are linked by the dispersion relation ω 20 = ω2pe + k 2 c2 . The space variables are (z, x), z ∈ R and x ∈ R2 : z is the component in the direction of propagation of the laser beam and x denotes the components in directions that are transversal to the propagation. See [R] or [S] for a systematic use of this kind of model. In numerical simulations, systems (1.1) or (1.2) have to be used in various situations. Usually one considers that the unit of space is k10 while the relevant unit for time is ω1pe and the space and time steps have to be respectively of the order of magnitude of k10 and ω1pe . For experiments concerning fusion by inertial confinement, one has to consider domains with spatial dimension of order of the centimeter and over several millions of ω10 . In the 3-D configuration, this is often out of reach of computational capacities and one restricts attention to a small piece of the spatial domain. Moreover, in

Instabilities in Zakharov equations for laser propagation in a plasma

65

such scalings, it is not realistic to consider that the fields are localized and, to do numerics, one usually considers that the plasma as well as the laser have a locally periodic structure at least at the scales that are considered here. The systems are then endowed with periodic boundary conditions. On the contrary, for propagation of lasers in the air or in crystals, one uses propagation in the whole space Rn with functions tending to zero at infinity. In this paper, we will focus on the former case, that is periodic boundary conditions that are useful in the physical framework explained above. The propagation in the whole space studied in [LPS] would correspond to the latter case. 1.2 The mathematical framework and the main result The goal of this paper is to prove an ill-posedness result for a non-dimensional form of system (1.2): i(∂t + ∂z )E + ∆x E = nE, (1.3) (∂t2 − ∆x )n = ∆x |E|2 . We consider the Cauchy problem for (1.3) with initial data E|t=0 = E0 , n|t=0 = n0 ,

∂t n|t=0 = n1 .

(1.4)

The existence theorem (see [S, GTV, OT] and references therein) for the classical Zakharov system, that is when ∆x is replaced by ∆(z,x) , does not apply. In [LPS], it is proved that the Cauchy problem for (1.3) is well posed, locally in time, for data in suitable Sobolev spaces. The proof is based on dispersion estimates. For periodic data, these dispersion estimates are not valid. This is a well-known phenomenon, even in the simpler case of Schr¨ odinger equations. However, the new phenomenon here is that the consequences of this lack of dispersive effects are much more dramatic since it implies strong instabilities of Hadamard’s type, so that the Cauchy problem for periodic data is strongly ill posed in Sobolev spaces. For the applications that we have described in the preceding section, it is quite reasonable to consider that E ∼ E = 0 at infinity. Our result has therefore a practical application and means that the paraxial approximation is not a good model in this case: one should add the longitudinal dispersion, that is, replace ∆x E in the first equation by (α∂z2 + ∆x )E. A natural mathematical question will then be: how does the instability grow when one lets the longitudinal dispersion parameter α tend to zero? We will address this question in a future work. We look for solutions U = (E, n) of (1.3), which are periodic in x, with period 2π in x and periodic in z with period 2πZ, where Z is arbitrary. We denote by T the corresponding torus R/2πZ × (R/2π)2 .

66

Thierry Colin and Guy M´etivier

We consider the constant solution U = (E, 0),

E = 0,

(1.5)

which of course does not belong to the spaces used in [LPS], and we prove that this solution is strongly unstable. Theorem 1.1 For all s, there are families of solutions Uk = U + (ek , nk ), in C 1 ([0, Tk ]; H s (T)) such that ek (0), nk (0), ∂t nk (0)H s (T) → 0,

(1.6)

Tk → 0,

(1.7)

ek (Tk ), nk (Tk )L2 (T) → ∞.

(1.8)

This nonlinear instability result is pretty strong: not only the amplification u(T )0/u(0)s is arbitrarily large, in arbitrarily small time T , with arbitrary loss of derivatives s, but there is an effective blowup of the L2 norm.

2 The instability mechanism Our construction is based on the analysis of the dispersion relation for the Zakharov system. Consider the linearized equations around (E, 0) : i(∂t + ∂z )e + ∆x e − E n = f, (2.1) (∂t2 − ∆x )n − ∆x (2 Re E e) = g. With (e, e, n) as unknowns the system reads: ⎧ −i(∂t + ∂z )e − ∆x e + E n = −f, ⎪ ⎪ ⎨ i(∂t + ∂z )e − ∆e + E n = −f , ⎪ ⎪ ⎩ 2 (∂t − ∆x )n − E ∆x e − E ∆x e = g.

(2.2)

Denoting by (τ , ζ, ξ) the frequency variables dual to (t, z, x), its symbol is ⎞ ⎛ 0 E (τ + ζ) + |ξ|2 ⎟ ⎜ ⎟ ⎜ 0 −(τ + ζ) + |ξ|2 E (2.3) ⎠ ⎝ |ξ|2 E |ξ|2 − τ 2 |ξ|2 E and the relation dispersion is P = 0, where P is the determinant of the system, that is P = (|ξ|2 − τ 2 )(|ξ|4 − τ + ζ)2 ) − 2|E|2 |ξ|4 = P0 − 2|E|2 |ξ|4 .

(2.4)

Instabilities in Zakharov equations for laser propagation in a plasma

67

The remark is that for (ζ, ξ) real, P0 has four real roots in τ , −|ξ|,

−ζ − |ξ|2 ,

+|ξ|,

−ζ + |ξ|2 ,

(2.5)

with an intermediate double root when 0 < |ξ| = −ζ − |ξ|2 . Note that P0 is of degree 6 in ξ while the perturbation −|E|2 |ξ|4 is of degree 4 and negative. Therefore, for ξ large and ζ = −|ξ| − |ξ|2 , the double root of P0 is perturbed in two conjugated complex roots. More precisely, for |ξ|  0,

ζ = −|ξ| − |ξ|2

and τ = |ξ|(1 + σ),

(2.6)

the determinant P is P = −|ξ|5 (σ 2 (2 − σ/|ξ|)(2 + σ) + 2|E|2 /|ξ|).

(2.7)

The implicit function theorem shows that there are two non-real roots |E| 1 τ = ξ ± i √ |ξ| 2 + 0(1). 2

(2.8)

This means that waves at frequency (ζ, ξ) with ζ = −|ξ|−|ξ|2 are amplified by the exponential factor 1

eγt|ξ| 2 ,

|E| γ = √ > 0. 2

(2.9)

This implies that the Cauchy problem for the linearized equations (2.1) is ill posed in H ∞ : there are Cauchy data in H ∞ such that the homogeneous problem with f = g = 0 has no solution in C 0 ([0, T ]; H −∞ ). The goal of this paper is to translate this spectral instability into a nonlinear instability result for the Zakharov system (1.3). Remark 2.1 How is it that this spectral instability does not intervene in the analysis of [LPS]? The first answer is that the condition E = 0 is crucial for γ to be positive. In their case, where solutions vanish at infinity, linearizing the equation around non-vanishing constants has no real significance. However, the symbolic calculus above also makes sense in the case of variable coefficients and one expects that the dispersion relation P = 0, with E replaced by E(t, z, x), which still has non-real roots, should play an important role in the analysis. For instance, the symbolic analysis appears when one replaces the plane wave analysis used for constant coefficients, by geometric optics expansions associated to localized wave packets. In this case, for a wave packet with mean frequency (−|ξ| − |ξ|2 , ξ), an exponential amplification similar to (2.9) is expected. But the group velocity in x of this packet is of order 2ξ; therefore if E is confined (think of it as compactly supported) the time of amplification is short (typically O(|ξ|−1 )) so that the overall effect of the amplification is bounded. Of course, this is just a very rough explanation, but it is rather intuitive. The detailed balance between amplification and localization is indeed given by the dispersive estimates proved in [LPS].

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Thierry Colin and Guy M´etivier

Remark 2.2 The system can be reduced to first order in t, introducing (∂x e, ∂t n, ∂x n) as unknowns, but it is not first order in x, because of the Schr¨ odinger part of the system. However, there is a good analogy with the analysis of a weakly hyperbolic system. Indeed, the analysis of the symbol (2.3) shows that for ζ = −|ξ|2 − |ξ|, there is a double eigenvalue with a 2 × 2 Jordan block. The existence of non-real eigenvalues (2.8), simply means that the natural analogue of the Levi condition for a first order system is not satisfied. Pursuing the analogy, the exponential growth (4.4) indicates that the Cauchy problem should be well posed in Gevrey classes Gs for s ≤ 2.

3 Scheme of the proof It is certainly sufficient to prove the theorem with functions of x = (x1 , x2 ) independent of x2 . To simplify notation, we assume from now on that x is one real variable. Consider spatially periodic solutions of (1.3), with period 2π in x and 2πZ in z. Moreover, we look for solutions n and E of the form n = n(kx − mz, t), E = E + e(kx − mz, t),

(3.1)

with new functions n(θ, t) and e(θ, t) being 2π periodic in θ. For the functions to be 2π periodic in x and 2π/Z periodic in z, it is sufficient that k ∈ N,

mZ ∈ N.

(3.2) √ To be close to the unstable frequencies, we require that |m − k − k 2 |  k and therefore we choose m ∈ N/Z such that (k 2 + k) − 1/Z < m ≤ (k 2 + k). The new equations read i(∂t − m∂θ )e + k 2 ∂θ2 e − En = ne, (∂t2 − k 2 ∂θ2 )n − k 2 ∂θ2 (E e + E e) = k 2 ∂θ2 |e|2 ,

(3.3)

(3.4)

written in short Lk (∂t , ∂θ )U = Nk (U ),

(3.5)

where U = (e, n), Lk is the linear operator defined in the left-hand side of (3.4), and Nk (u) the quadratic term in the right-hand side. The first step concerns the homogeneous equation Lk U = 0,

(3.6)

which is studied using Fourier series expansions in θ. The choice (3.3) together with the spectral analysis of Section 2 and the choice (3.3) imply that for k large, the harmonic 1 is unstable:

Instabilities in Zakharov equations for laser propagation in a plasma

69

Proposition 3.1 There is k0 such that for k ≥ k0 , there are solutions U a = (ea , na ) of (3.6) such that a e = eˆa1 (t)eiθ + eˆa−1 (t)e−iθ , (3.7) na = sinh(tσ) cos(t Re λ + θ), with

eˆa±1 (t) = (ea±1,+ etγ + ea±1,− e−tγ )eitλ ,

(3.8)

where the parameters λ, σ, e±1,± depend on k, λ and σ being real positive and satisfying, as k → +∞, ea+1,+ ∼ −iE/4σ, λ ∼ k,

ea+1,− ∼ −iE/4σ,

σ ∼ |E|

ea−1,± = O(k −2 ),

k / 2.

(3.9) (3.10)

The proof is given in Section 4. Next, we consider δU a as a first approximation of the solution of (3.5) to be constructed, with δ a small parameter to be chosen. More precisely look for solutions of (3.5) as U = δ(U a + u),

u = (e, n),

(3.11)

with the same initial data as δU a . Because the nonlinearity is exactly quadratic, the equation for u reads Lk (∂t , ∂θ )u = δNk (U a + u) ,

e|t=0 = n|t=0 = ∂t n|t=0 = 0.

(3.12)

This equation is solved by Picard’s iteration; therefore the main step is to solve the linear equation Lk U = F,

e|t=0 = n|t=0 = ∂t n|t=0 = 0

(3.13)

in Banach spaces which are also well adapted to the nonlinearity. The choice of these spaces, more precisely of their norm, is technical and dictated by the computations detailed in the next sections. We just give here their definition. For a periodic function v of θ, we denote by vˆp its Fourier coefficients so that
vˆp eipθ . (3.14) v= p∈Z

The first Fourier coefficient eˆ1 plays a special role and we use the notation e(t, θ) = eˆ1 (t)eiθ + e
(t, θ).

(3.15)

For s ≥ 1 and T > 0, we denote by E1 (T ) the space of u = (e, n) with n real valued, such that e ∈ C 0 ([0, T ]; H s+2 ) ∩ C 1 ([0, T ]; H s ),

n ∈ C 1 ([0, T ]; H s ),

(3.16)

70

Thierry Colin and Guy M´etivier

equipped with the norm 1

1

e1 (t)| + k − 2 |∂t eˆ1 (t)| uE1(T ) = supt∈[0,T ] e−σt { k 2 |ˆ 3

1

+k 4 e
(t)H s+2 + k − 2 ∂t e
(t)H s

(3.17)

+n(t)H s + k −1 ∂t n(t)H s }, where σ is defined in Proposition 3.1. The norm depends on k ≥ 1 and s, but, to lighten the text, we do not mention this dependence explicitly in the notation. We denote by E2 (T ) the same space (3.16), equipped with the norm uE2 (T ) = supt∈[0,T ] e−2σt { k|ˆ e1 (t)| + |∂t eˆ1 (t)| 1

+ke
(t)H s+2 + k − 4 ∂t e
(t)H s 1 2

+k n(t)H s + k

− 12

(3.18)

∂t n(t)H s }.

There are two differences between (3.17) and (3.18): first the weight e−σt is replaced by e−2σt and second, all the powers of k in the coefficients are increased, at least by a factor 14 . In particular, 1

uE1 (T ) ≤ k − 4 eσT uE2 (T ) .

(3.19)

For the right-hand sides, we denote by F2 (T ) the space of F = (f, g) with g real valued such that f ∈ C 1 ([0, T ]; H s ),

g ∈ C 0 ([0, T ]; H s ) with gˆ0 = 0,

(3.20)

equipped with the norm 1

1

F E1 (T ) = supt∈[0,T ] e−2σt { k 2 f (t)H s + k − 2 ∂t f (t)H s 3

+k − 4 g(t)H s }.

(3.21)

The next three results justify the choices of these norms. We assume that the parameter s ≥ 1 is fixed. The first estimate is an immediate consequence of Proposition 3.1, (3.9), and (3.10). Lemma 3.1 There is a constant K a such that for all k ≥ k0 and all T ≤ 1, the approximate solution U a of Proposition 3.1 satisfies U a E1 (T ) ≤ K a .

(3.22)

The next two propositions are proved in Section 6. Proposition 3.2 There is C1 > 0, such that for all k ≥ k0 , all T ≤ 1 and all F ∈ F2 (T ), the Cauchy problem (3.13) has a unique solution U ∈ E2 (T ) and U E2(T ) ≤ C1 F F2 (T ) .

(3.23)

Instabilities in Zakharov equations for laser propagation in a plasma

71

The nonlinearity Nk (U ) occurring in (3.5) is quadratic. Denote by Nk (U, V ) the associated bilinear form such that Nk (U ) = Nk (U, U ). Proposition 3.3 There is C2 > 0, such that for all k ≥ k0 , all T ≤ 1 and all U and V in E1 (T ), there holds Nk (U, V ) ∈ F2 (T ) and Nk (U, V )F2 (T ) ≤ C2 U E1 (T ) V E1 (T ) .

(3.24)

These estimates easily imply the following: Corollary 3.1 There are c0 > 0, C and k0 , such that for all k ≥ k0 and all δ ∈]0, 1], the problem (3.12) has a unique solution u = (e, n) in the unit ball of E1 (T ), provided that 1 (3.25) δk − 4 eσT ≤ c0 . Moreover, the solution satisfies 1

n(t)H s ≤ Ck − 4 eσt .

(3.26)

Proof. Denote by L−1 k F the solution of (3.13), and consider the mapping a u → T u := δL−1 k Nk (u + u),

which, by the lemma and propositions above, is well defined from E1 (T ) to E1 (T ). Moreover, 1

T uE1 (T ) ≤ C1 C2 δk − 4 eσT (K a + uE(T ))2 . Thus it maps the unit ball to of E1 (T ) to itself, if (3.25) holds with c0 small enough. Similarly, decreasing c0 if necessary, one shows that this mapping is contractive on the unit ball, implying the existence and uniqueness of the solution of u = T u in the unit ball. The equation u = T u and the estimates also imply that 1

1

n(t)H s ≤ k − 2 e2σt uE2(T ) ≤ C1 C2 δk − 2 e2σt (K a + 1)2 1

≤ C1 C2 c0 k − 4 eσt (K a + 1)2 finishing the proof of the corollary. We end this section by proving that the main Theorem 1.1 is a consequence of this analysis. Proof (Proof of Theorem 1.1). We fix an integer s. With δ = k −(2s+2) ,

(3.27)

Corollary 3.1 provides us with solutions of (3.5), Uk = U + δ(U a + uk ), with 1 uk in the unit ball of E1 (Tk ), and Tk = σ1 ln(k 2s+2+ 4 /c0 ) satisfies δk −1/4 eσTk = c0 .

(3.28)

72

Thierry Colin and Guy M´etivier 1

Since σ is of order k 2 by (5.7), Tk tends to 0 as k tends to infinity, at the rate ln k Tk ≈ √ . k

(3.29)

Going back to the (z, x) variables, according to the change of variables ˜k = U + u˜k , of the original Zakharov (3.1), we obtain solutions, denoted by U system (1.3). Set u ˜k = (˜ ek , n ˜ k ); these functions are deduced from δ(U a + uk ) by the change of variables (3.1). Since m ≤ k 2 + k, we can evaluate the H s norm (in the variables (z, x)) of the Cauchy data ˜ k|t=0 , ∂t n ˜ k|t=0 H s (T) ≤ C δ k 2s+1 U a + uk E1(T ) ˜ ek|t=0 , n ≤ C δ k 2s+1 (K a + 1). Note that there is no Jacobian factor because the L2 norms are taken for (z, x) ∈ T in the left-hand side and for θ ∈ R/2πZ in the right-hand side so that   measT 2π v(kx − mz)dzdx = v(θ)dθ. (3.30) 2π 0 T Therefore, with our choice of δ, the left-hand side tends to zero as k tends to infinity. ˜ k at time Tk . Using (3.30) and (3.7) Finally we compute the L2 norm of n we see that ˜ nk (Tk )L2 (T) ≥ c1 δ sinh(Tk σ) − δnk (Tk )L2 , with c1 > 0 independent of k. Therefore, (3.26) (3.28) imply that 1 1 c1 δeσTk − Cδk − 4 eσTk − O(δe−σTk ) 2 1 1 ≥ c1 c0 k 4 − Cc0 − o(1). 2

˜ nk (Tk )L2 (T) ≥

Therefore this L2 norm tends to +∞ and the proof of the theorem is complete.

4 The linear instability We study the linear equation for U = (e, n) and F = (f, g), Lk U = F, using Fourier series expansions in θ:
e(θ, t) = eˆp (t)eipθ ,

n(θ, t) =

(4.1)


n ˆ p (t)eipθ .

(4.2)

Instabilities in Zakharov equations for laser propagation in a plasma

73

Since n and g are real, n ˆ −p = n ˆp, and (4.1) reduces to



˜ k (∂t , 0)U0 := L and for p ≥ 1

gˆ−p = gˆp ,

∂t eˆ0 − E0 n ˆ0 ∂t2 n ˆ0

(4.3) 

 = F0 :=

fˆ0



gˆ0 ,

⎧ (i∂t + mp − k 2 p2 )ˆ ep − E n ˆ p = fˆp , ⎪ ⎪ ⎨ ep + E n ˆ p = f˜p , (i∂t + mp + k 2 p2 )˜ ⎪ ⎪ ⎩ 2 np + k 2 p2 (E eˆp + E˜ ep ) = gˆp , (∂t + k 2 p2 )ˆ

(4.4)

(4.5)

with e˜p = e−p ,

f˜p = −f−p ,

(4.6)

are the Fourier coefficients of e and −f respectively. For p > 0, we denote by k (∂t , p) the linear operator in the left-hand side of (4.5). L In the remaining part of this section we concentrate on the case p = 1 and prove Proposition 3.1. We reduce (4.5) for p = 1 to a first order system by ˆ 1 . The equation reads introducing v1 = −ik −1 ∂t n i∂t V1 + AV1 = F1 , with V1 = (ˆ e1 , e˜1 , n ˆ 1 , v1 ), ⎛

and

⎜ ⎜ A=⎜ ⎜ ⎝

m − k2 0 0 kE

(4.7)

F1 = (fˆ1 , f˜1 , 0, k −1 gˆ1 ) −E 0

0

(4.8)

⎞

⎟ m + k2 E 0 ⎟ ⎟. 0 0 k⎟ ⎠ kE k 0

(4.9)

Lemma 4.1 If E = 0 and k is large enough, A has four distinct eigenvalues; two, called λ1 and λ2 are real and the other two, λ3 and λ4 , are non-real and complex conjugated. There holds λ1 ∼ 2k 2 ,

λ2 ∼ −k,

Re λ3 ∼ k,

σ := Im λ3 ∼ |E| k/2.

(4.10)

Proof. This follows from the analysis of the determinant equation in Section 2. The eigenvalue equation is P = (λ2 − k 2 )((λ − m)2 − k 4 ) − 2|E|2 k 4 = 0. 2




Following (3.3), we write m = k + k + m , and the equation reads (λ2 − k 2 )(λ − k + m
)(λ − 2k 2 − k + m
) = 2|E|2 k 4 .

(4.11)

74

Thierry Colin and Guy M´etivier

Because m
= O(1), the lemma easily follows by perturbation analysis of the roots of (λ2 − k 2 )(λ − k + m
)(λ − 2k 2 − k + m
) = 0. Next, to evaluate eitA , we need to analyze the eigenprojectors of A. Denote by rj [resp. lj ] right [resp. left] eigenvectors of A associated to the eigenvalue λj . Then 4
(lj · Φ) itA rj . e Φ= eitλj (4.12) (l j · rj ) j=1 A detailed inspection of the eigenvector equations implies that ⎡

O(k −4 )

⎤

⎢ 1 ⎥ ⎥ ⎢ r1 = ⎢ ⎥, −2 ⎣ O(k ) ⎦

  l1 = O(k −4 ), 1, O(k −2 ), O(k −3 ) ,

(4.13)

  l2 ∼ O(1), O(k −1 ), 1, −1 ,

(4.14)

O(k −1 ) ⎡

O(k −1 )

⎤

⎢ O(k −2 ) ⎥ ⎢ ⎥ r2 ∼ ⎢ ⎥, ⎣ 1 ⎦ −1

where, for vectors a and b, a ∼ b means that all the components satisfy ak ∼ bk . Moreover, ⎡

iE/σ

⎤

⎢ O(k −2 ) ⎥ ⎥ ⎢ r3 ∼ ⎢ ⎥, ⎣ 1 ⎦

  l3 ∼ kE/iσ, O(k −1 ), 1, 1 ,

(4.15)

1 r4 = r 3 ,

l4 = l 3 ,

(4.16)

√ where σ 2 = k|E|2√/2 ≈ k. Note that r3,4 = O(1) and r3 − r4 = O(|E|/ k) and l3,4 = O(|E| k) while r3,4 · l3,4 ∼ 4. This reflects that for E = 0, the corresponding matrix has a Jordan block. Proof (Proof of Proposition 3.1). With notation as above, ⎛

eˆa1

⎞

⎜ e˜a ⎟ 1 ⎜ 1⎟ V1a = ⎜ a ⎟ := (eitλ4 r4 − eitλ3 r3 ) 4 ⎝n ˆ1 ⎠ v1a

(4.17)

Instabilities in Zakharov equations for laser propagation in a plasma

75

is a solution of (4.7) with F1 = 0. It corresponds to a solution (ˆ ea1 , e˜a1 , na1 ) of a = 0 and therefore to a solution ˜ 1U L 1

ea = eˆa1 eiθ + e˜a1 e−iθ ,

na = n ˆ a1 eiθ + na1 e−iθ

(4.18)

a

of L1 U = 0. Choosing, as we may, r3 and r4 such that the third component is exactly equal to 1, we obtain that na (t, θ) = sinh(tσ) cos(t Re λ3 + θ) and the estimate (3.9) follows from the estimates of the eigenvectors above. Moreover, (3.10) follows from Lemma 4.1. Next we turn to the analysis of (4.7). The solution with vanishing initial data is 4  t
(lj · F1 (s)) rj ds. ei(t−s)λj (4.19) V1 (t) = (lj · rj ) j=1 0 Introduce Φj = lj · F1 . With f denoting (fˆ1 , f˜1 ) and g = gˆ1 it holds that Φ1 = ∗f + ∗k −4 g, Φ2 = ∗f + ∗k −1 g, √ Φ3,4 = ∗ kf + ∗k −1 g

(4.20)

where ∗ denotes constant coefficients that are uniformly bounded in k. Let  t eiλj (t−s) Φj (s)ds. (4.21) Ψj (t) = 0

e1 , e˜1 , n ˆ 1 , v1 ) The properties of the rj ’s and (4.19) imply that the components (ˆ of V1 satisfy: eˆ1 = ∗k −4 Ψ1 + ∗k −1 Ψ2 + ∗k −1/2 Ψ3,4 , e˜1 = ∗Ψ1 + ∗k −2 Ψ2 + k −2 Ψ3,4 , n ˆ 1 = ∗k −2 Ψ1 + ∗Ψ2 + ∗Ψ3,4 ,

(4.22)

v1 = ∗k −1 Ψ1 + ∗Ψ2 + ∗Ψ3,4 . We use the following elementary estimates: Lemma 4.2 Let



t

eiλ(t−s) φj (s)ds.

ψ(t) =

(4.23)

0

Then it holds that |ψ(t)|

≤

't 0

e− Im λ(t−s) |φ(s)|ds,

|∂t ψ(t)| ≤ |λj | |ψ(t)| + |φ(t)|, 't |∂t ψ(t)| ≤ e− Im λt |φ(0)| + 0 e− Im λ(t−s) |∂t φ(s)|ds, |λ| |ψ(t)| ≤ |∂t ψ(t)| + |φ(t)|.

(4.24)

76

Thierry Colin and Guy M´etivier

To simplify notation, we write A
B to mean that there is a constant C independent of k such that A ≤ CB. We use the first and second estimate of Lemma 4.2 to bound the contributions of g to the integrals in (4.19), and we use the third and fourth estimate, when necessary, to bound the contributions of f . Therefore 't |Ψ1 (t)|
0 |f (s), k −4 g(s)|ds, 't (4.25) |∂t Ψ1 (t)|
|f (0)| + |k −4 g(t)| + 0 |∂t f (s), k −2 g(s)|ds, ' t k 2 |Ψ1 (t)|
|f (0)| + |f (t)| + 0 |∂t f (s), k −2 g(s)|ds, 't

|f (s), k −1 g(s)|ds, 't (4.26) |∂t Ψ2 (t)|
|f (0)| + |k −1 g(t)| + 0 |∂t f (s), g(s)|ds, 't k|Ψ2 (t)|
|f (0)| + |f (t)| + 0 |∂t f (s), g(s)|ds, √ 't
0 e(t−s)σ | kf (s), k −1 g(s)|ds, |Ψ3,4 (t)| √ √ 't |∂t Ψ3,4 (t)|
etσ | kf (0)| + |k −1 g(t)| + 0 e(t−s)σ | k∂t f (s), g(s)|ds, (4.27) √ √ √ 't k|Ψ3,4 (t)|
etσ | kf (0)| + | kf (t)| + 0 e(t−s)σ | k∂t f (s), g(s)|ds. |Ψ2 (t)|




0

Adding up the various estimates, we obtain: ˆ 1 ) of (4.5) with vanishing Proposition 4.1 For p = 1, the solution (ˆ e1 , e˜1 , n initial data satisfies: 't 3 |ˆ e1 (t)|
0 eσ(t−s) |f1 (s), k − 2 gˆ1 (s)|ds, 3

|∂t eˆ1 (t)|
eσt |f1 (0)| + |k − 2 gˆ1 (t)| 't 1 + 0 eσ(t−s) |∂t f1 (s), k − 2 gˆ1 (s)|ds, e1 (t)| + |∂t e˜1 (t)|
eσt |f1 (0)| + |f1 (t)| + |k −3 gˆ1 (t)| k 2 |˜ 't + 0 eσ(t−s) |∂t f1 (s), k −1 gˆ1 (s)|ds, 1

(4.28)

(4.29)

1

k|ˆ n1 (t)| + |∂t n ˆ 1 (t)|
eσt k 2 |f1 (0)| + |k 2 f1 (t)| + |k −1 gˆ1 (t)| 't 1 + 0 eσ(t−s) |f1 (s), k 2 ∂t f1 (s), gˆ1 (s)|ds,

(4.30)

where f1 = (fˆ1 , f˜1 ). Corollary 4.1 There are k0 and C such that for all k ≥ k0 , K, T > 0, and all f1 = (fˆ1 , f˜1 ), g1 satisfying for t ∈ [0, T ], 1

1

3

k 2 |f1 (t)| + k − 2 |∂t f1 (t)| + k − 4 |ˆ g1 (t)| ≤ Ke2σt ,

Instabilities in Zakharov equations for laser propagation in a plasma

77

then the solution of (4.5) for p = 1 with vanishing initial data satisfies k|ˆ e1 (t)| + |∂t eˆ1 (t)| ≤ CKe2σt , 1

k|˜ e−1 (t)| + k − 4 |∂t e˜−1 (t)| ≤ CKe2σt , 1

1

n1 (t)| + k − 2 |∂t n ˆ 1 (t)| ≤ CKe2σt . k 2 |ˆ Proof. a) From Proposition 4.1 we deduce that √  t σ(t−t
) 2σt

k|ˆ e1 (t)| ≤ CK1 k e e dt ≤ CKe2σt ,

(4.31)

0

√ where we have used that σ ≈ k. Similarly,    t√ −1/2 σt −3/4 2σt σ(t−t
) 2σt

e +k e + ke e dt ≤ CKe2σt . |∂t eˆ1 (t)| ≤ CK1 k 0

(4.32) This implies the first estimate. b) Similarly, (4.29) implies that    t√

k 2 |ˆ e−1 (t)| + |∂t eˆ−1 (t)| ≤ CK1 eσt + e2σt + keσ(t−t ) e2σt dt
0

≤ CKe

2σt

(4.33)

.

c) The estimate (4.30) implies that    t

k|ˆ n1 (t)| + |∂t n ˆ 1 (t)| ≤ CK1 eσt + e2σt + keσ(t−t ) e2σt dt
√ ≤ CK ke2σt ,

0

(4.34)

and the lemma is proved.

5 The linear equation We continue the analysis of the linear equation (4.1). As seen in (4.5), when expanded in Fourier series, this equation couples the coefficients of indices p and −p. The case of indices +1 and −1 is studied in the previous section. Using the notation (5.1) v = vˆ1 eiθ + vˆ−1 e−iθ + v

, we consider the equation (4.1) for functions with vanishing Fourier coefficients of indices ±1: (5.2) L

k U

= F

, which reduces to the analysis of equations (4.5) for Fourier p = 1.

78

Thierry Colin and Guy M´etivier

k (∂t , p) is The symbol of L ⎛ 0 −τ + mp − k 2 p2 ⎜ k (iτ , p) = ⎝ L 0 −τ + mp + k 2 p2 2 2

k p E0

2 2

k p E0

⎞

−E0

⎟ ⎠,

E0 k p −τ 2 2

(5.3)

2

which is of course equal to the symbol (2.3) with ξ = kp, ζ = −mp, up to a change of sign in the first line. k (∂t , p) as a perturbaAssume first that p > 1. In this case, we consider L tion of ⎞ ⎛ 0 0 i∂t + mp − k 2 p2 ⎟ ⎜ 0 i∂t + mp + k 2 p2 0 Mk (∂t , p) := ⎝ (5.4) ⎠. 0

0

∂t2 + k 2 p2

For the wave operator, we use the classical estimates: Lemma 5.1 There is C > 0, such that for all k ≥ 1 and p ≥ 1, the solution n of (5.5) ∂t2 n + k 2 p2 n = g, n(0) = ∂t n(0) = 0, satisfies kp|n(t)| + |∂t n(t)| ≤ CgL1 ([0,t]) .

(5.6)

For the Schr¨ odinger equations, we use the following estimates. Lemma 5.2 There are C > 0 and k0 ≥ 1, such that for all k ≥ k0 and p ≥ 2, the solutions of (i∂t + mp ± k 2 p2 )e = f, e(0), (5.7) satisfy k 2 p2 |e(t)| + |∂t e(t)| ≤ C(f L1 ([0,t]) + ∂t f L1 ([0,t]) + |f (0)|).

(5.8)

Proof. Standard energy estimates imply that |e(t)| ≤ C(|e(0)| + f L1 ([0,t]) ).

(5.9)

Differentiating the equation in time, we obtain |∂t e(t)| ≤ C(|∂t e(0)| + ∂t f L1 ([0,t]) ).

(5.10)

The initial condition in (5.7) implies that ∂t e(0) = −if (0). Therefore, |(k 2 p2 ± mp)e(t)| + |∂t e(t)| ≤ C(f L1 ([0,t]) + ∂t f L1 ([0,t]) + |f (0)| + |f (t)|). (5.11) Recall that m is linked to k through (3.3). Thus mp ≤ k 2 p+kp and k 2 p2 −mp ≥ k 2 (p2 − p) − kp ≥ ck 2 p2 for all p ≥ 2 if k is large enough.

Instabilities in Zakharov equations for laser propagation in a plasma

79

Proposition 5.1 Consider the equation (4.5) with initial data eˆp (0) = e˜p (0) = n ˆ p (0) = ∂t n ˆ p (0) = 0.

(5.12)

Then, for p ≥ 2, k ≥ k0 , there holds for t ∈ [0, 1]: ep (t), e˜p (t)| + |∂t eˆp (t), ∂t e˜p (t)| + kp|ˆ np (t)| + |∂t n ˆ p (t)| k 2 p2 |ˆ ≤ C(fˆp , f˜p L1 ([0,t]) + ∂t fˆp , ∂t f˜p L1 ([0,t])

(5.13)

+|fˆp (0), f˜p (0)| + |fˆp (t), f˜p (t)| + ˆ gp L1 ([0,t]) ). Proof. The lemmas above imply that the left-hand side is estimated by the right-hand side plus C(|ˆ np (t)| + ˆ np , ∂t n ˆ p , k 2 p2 eˆp (t), k 2 p2 e˜p L1 ([0,t]) ).

(5.14)

The first term is absorbed in the left-hand side by kp|ˆ np (t)| for k large enough. With Gronwall’s lemma, this implies (5.13) for t ∈ [0, 1], with a larger constant C. When p = 0, the following lemma holds. Lemma 5.3 When gˆ0 = 0, the solution of (4.4) with vanishing initial data is  n ˆ 0 = 0,

eˆ0 (t) =

t

fˆ0 (t
)dt
.

(5.15)

0

With the estimates (5.13), one deduces the following result. Corollary 5.1 There are k0 and C such that for all k ≥ k0 , K, T > 0, and all (f

, g

) with gˆ0 = 0, satisfying for t ∈ [0, T ], 1

1

k 2 f

(t)H s + k − 2 ∂t f

(t)H s ≤ Ke2σt , g

(t)H s ≤ Kk 3/4 e2σt , the solution of (5.2) with vanishing initial data satisfies 1

ke

(t)H s+2 + k − 4 ∂t e

(t)H s ≤ CKe2σt , 1

1

k 2 n

(t)H s + k − 2 ∂t n

(t)H s ≤ CKe2σt . Proof. By Lemma 5.3, it holds that    t√

k|ˆ e0 (t)| + |∂t eˆ0 (t)| ≤ CK1 e2σt + keσ(t−t ) e2σt dt
≤ CKe2σt . (5.16) 0

80

Thierry Colin and Guy M´etivier

Next, Proposition 5.1 implies that e

satisfies k 2 ∂θ2 e

(t)H s + ∂t e

(t)H s    t 2σt 3/4 σ(t−t
) 2σt

≤ CK (1 + e ) + k e e dt ≤ k 1/4 CKe2σt .

(5.17)

0

Together with (5.16) this implies the first estimate. Moreover, Proposition 5.1 implies that n

satisfies kn

(t)H s + ∂t n

(t)H s    t

≤ CK (1 + e2σt ) + k 3/4 eσ(t−t ) e2σt dt
≤ k 1/4 CKe2σt .

(5.18)

0

Since n ˆ 0 = 0, this implies the second estimate.

6 End of proofs First, we note that Proposition 3.2 is an immediate consequence of Lemma 4.1 and Corollary 5.1. It remains to prove Proposition 3.3. With U = (e, n) and U ∗ = (e∗ , n∗ ), it holds that (6.1) Nk (U, U ∗ ) = (f, g), with f = ne∗ + n∗ e, g=

k 2 ∂θ2 {Re(ee∗ )}.

(6.2) (6.3)

Proposition 3.3 follows from the next estimates. Lemma 6.1 There is a constant C, independent of k, such that √

1 kf (t)H s + √ ∂t f (t)H s ≤ Ce2σt U E1 (T ) U ∗ E1 (T ) , k g(t)H s ≤ Ck 3/4 e2σt U E1 (T ) U ∗ E1 (T ) .

(6.4) (6.5)

Moreover, the mean value gˆ0 of g vanishes. Proof. The first estimate follows directly from the definitions and the inequality abH s ≤ CaH s bH s . (6.6) Next, we note that for e = eˆ1 eiθ + e
and e∗ = eˆ∗1 eiθ + e∗
∂θ2 (ee∗ ) = ∂θ2 (e
e∗
) + eˆ1 ∂θ2 (e∗
e−iθ ) + eˆ∗1 ∂θ2 (e
eiθ ).

(6.7)

Instabilities in Zakharov equations for laser propagation in a plasma

81

Hence, in H s norms, it holds that ∂θ2 (ee∗ )H s
∂θ2 e
H s (e∗
 + ∂θ e∗
2 ) + ∂θ2 e∗
H s (e
 + ∂θ e
2 ) e∗1 |(∂θ2 e
 + e
), +|ˆ e1 |(∂θ2 e∗
 + e∗
) + |ˆ (6.8) and (6.5) follows. In addition, the θ-mean value gˆ0 vanishes since g is a θ-derivative.

References [CC1]

[CC2] [GTV] [LPS] [OT] [RDR]

[R]

[S]

[ZMR]

M. Colin and T. Colin, On a quasilinear Zakharov system describing laser-plasma interactions, Differential Integral Equations 17(2004), 297– 330. M. Colin and T. Colin, A numerical model for the Raman Amplification for laser-plasma interaction, J. Comput. Appl. Math. 193(2006), 535–562. J. Ginibre, Y. Tsutsumi and G. Velo, On the Cauchy problem for the Zakharov system, J. Funct. Anal. 151(1997), 384–436. P. Linares, G. Ponce and J.-C. Saut, On a degenerate Zakharov system, Bull. Braz. Math. Soc. (N.S.) 36(2005), 1–23. T. Ozawa and Y. Tsutsumi, Existence and smoothing effect of solutions for the Zakharov equations, Publ. Res. Inst. Math. Sci. 28(1992), 329–361. D. A. Russel, D. F. Dubois and H. A. Rose, Nonlinear saturation of simulated Raman scattering in laser hot spots, Physics of Plasmas 6(1999), 1294–1317. G. Riazuelo, Etude th´eorique et num´erique de l’influence du lissage optique sur la filamentation des faisceaux lasers dans les plasmas souscritiques de fusion inertielle, Th`ese, Universit´e Paris XI, 2001. C. Sulem and P.-L. Sulem, The nonlinear Schr¨ odinger Equation. Self-Focusing and Wave Collapse, Applied Mathematical Sciences 139, Springer-Verlag, New York, 1999. V. E. Zakharov, S. L. Musher and A. M. Rubenchik, Hamiltonian approach to the description of nonlinear plasma phenomena, Phys. Reports 129(1985), 285–366.

Symplectic strata and analytic hypoellipticity Paulo D. Cordaro1 and Nicholas Hanges2 1 2

Departamento de Matem´ atica Aplicada, Instituto de Matem´ atica e Estat´ıstica, Universidade de S˜ ao Paulo, SP, Brazil Lehman College, CUNY, New York, USA

Summary. We review various classical results on analytic hypoellipticity for operators with double characteristics. Several examples will be discussed to motivate Treves’ conjecture. Finally we announce regularity results obtained recently.

2000 Mathematics Subject Classification: 35H10, 35H20, 35A17, 35A20, 35A27. Key words: Analytic regularity, degenerate elliptic equations, Poisson stratification.

1 Introduction Let M be a real analytic manifold and let X0 , . . . , Xν be real analytic, real-valued vector fields on M. We study an operator P of the form “sum of squares”. That is P has the form P = X0 2 + · · · + Xν 2 . Definition 1.1 We say that P is analytic hypoelliptic (in the strong sense) on M if for every open O ⊂ M we have the following: P u analytic on O implies that u is analytic on O. Here u is a distribution on O. We always assume that the Xj satisfy a “finite type” condition. That is, at each point of M, the Lie algebra generated by the Xj (under the commutation bracket) has dimension equal to dimM. Under these conditions a classical result of H¨ ormander [24] guarantees the hypoellipticity of P . However analytic hypoellipticity will not hold unless further assumptions are made.

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Paulo D. Cordaro and Nicholas Hanges

2 The symplectic case If we assume that Σ, the characteristic set of P , is a symplectic manifold and that the principal symbol of P vanishes precisely to second order on Σ, then P is analytic hypoelliptic. This follows from results of Treves [37] and Tartakoff [36]. Using the concatenation method, Treves gave precise conditions on the subprincipal symbol to guarantee regularity for operators such as P . This is related to work with Boutet de Monvel [3], [4] and work with Gilioli [16]. Even lower order terms can influence regularity. This was studied by Kwon [26], who was also influenced by the work of Stein [35]. Further work in this direction around this time was done by M´etivier [28], [30] and Sj¨ ostrand [34]. This generalized to the analytic category classical results of Boutet–Grigis–Helffer [2]. These ideas go back to Grusin’s work [18].

3 The example of Baouendi–Goulaouic When Σ is not symplectic, analytic hypoellipticity may fail. We have the example due to Baouendi–Goulaouic [1]. Consider the operator on R3 given by (3.1) B = ∂t 2 + t2 ∂x 2 + ∂y 2 . The characteristic set Σ for B is defined by the equations1 t = τ = η = 0. Σ is not symplectic and B is not analytic hypoelliptic on any open set that intersects t = 0. To show that analytic hypoellipticity fails in this case, one may proceed in the following way. Let u be defined by  ∞ √ 2 u(x, y, t) = eiρ x A(ρt)e λρy e−ρ dρ. (3.2) 0

Then u is a smooth solution to Bu = 0 (near y = 0), provided that A is an eigenfunction of the Hermite operator, with corresponding eigenvalue λ. We have  ∞ ρ2j e−ρ dρ = A(0)ij (2j)!

∂x j u(0, 0, 0) = A(0)ij

(3.3)

0

and hence u is not analytic. Formula 3.2 slightly generalizes one that can be found in H¨ ormander’s book [23]. We note here that the integrand in (3.2) appears in the work of Oleinik [32]. 1

Here and in the sequel we shall denote the dual coordinates of x, y, t by ξ, η, τ .

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This method has been generalized by many people including: Chanillo [6], [8], Christ [9], [10], [11], Costin–Costin [15], Hanges–Himonas [20], [21], [22], Hoshiro [25], M´etivier [28], [30], Pham The Lai–Robert [33]. Of particular interest is the work on nonlinear eigenvalue problems by Chanillo, Helffer and Laptev [7]. Returning to the Baouendi–Goulaouic example B = ∂t 2 + t2 ∂x 2 + ∂y 2 ,

(3.4)

we see that T Σ, the tangent bundle of Σ, is spanned by the vector fields ∂x , ∂y , ∂ξ . Also T Σ ⊥ , the orthogonal bundle with respect to the symplectic form, is spanned by ∂t , ∂y , ∂τ . Note that

∂y ∈ T Σ ∩ T Σ ⊥ .

Hence Σ is not symplectic. Indeed if we let γ be a particular y line, then we have Tp γ = Tp Σ ∩ Tp Σ ⊥ for each p ∈ γ. These y lines are examples of what are now called bicharacteristic curves.

4 Treves’ original conjecture Let (0, 1) ⊂ R denote the open unit interval. We have the following: Definition 4.1 Let Σ ⊂ T ∗ M be an analytic submanifold and let γ : (0, 1) → Σ be a non-constant analytic curve. We call γ a bicharacteristic curve for Σ if dγ (t) ∈ (Tγ(t) Σ)⊥ (4.1) dt for all t ∈ (0, 1). In [37], Treves conjectured that when the characteristic set Σ is a manifold, and contains such curves, then the associated operator is not analytic hypoelliptic. Later, in [38], Treves extended his conjecture. We will discuss this later. Certainly the Baouendi–Goulaouic example is consistent with the conjecture. More evidence is supplied by the example of M´etivier [29]: M = ∂x 2 + x2 ∂y 2 + (y ∂y )2 .

(4.2)

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Paulo D. Cordaro and Nicholas Hanges

M is not analytic hypoelliptic on any open set containing the origin. (M is elliptic away from the origin.) Note that the characteristic set Σ is given by Σ = {ξ = x = y = 0}, and hence Σ itself is a bicharacteristic curve. On the other hand, operators with nonsymplectic characteristic set may still be analytic hypoelliptic. Consider the operator    2 4 3 2 2 x + 4xy ∂t . P = ∂x + ∂y + 3 This is the principal part of the Kohn Laplacian for the domain {(z, w) ∈ C2 : Im w > |z|4 }. In this case, the set where Σ fails to be symplectic is itself a symplectic submanifold. There are no bicharacteristics. P is analytic hypoelliptic by work of Sj¨ ostrand [34] (cf. also [17]). Compare this with the following example of Oleinik [32]. Let p, r be integers ≥ 1 and consider P = ∂t 2 + t2p ∂x 2 + t2r ∂y 2 . P is analytic hypoelliptic if and only if p = r. Note that the characteristic set is given by τ = 0 = t. This is always a symplectic manifold. Hence we see that analytic hypoellipticity can fail in the symplectic case, when the order of vanishing of the principal symbol is not uniform.

5 The Poisson stratification of Σ Let U ⊂ Rm be open and let X0 , . . . , Xν be real analytic vector fields on U . Let P have the form P = X0 2 + · · · + Xν 2 . We assume P satisfies H¨ormander’s condition. Let fj , j = 0, . . . , ν denote the symbols of the Xj . The characteristic set of P is defined as Σ = {p ∈ T ∗ (U ) \ 0 : fj (p) = 0, j = 0, . . . , ν}. By a Poisson stratum of Σ we shall mean a subset Σ
⊂ Σ satisfying the following properties: 1. Σ
is a connected, embedded analytic submanifold of T ∗ (U ); ⊥ 2. p →  dim{Tp Σ
∩ Tp Σ
} is constant on Σ
; 3. There exists an integer n such that fI vanishes on Σ
for all |I| < n, but for each p ∈ Σ
, there exists I with |I| = n such that fI (p) = 0. Note that if I = (i1 , . . . , iq ), then fI = {fi1 , . . . , {fiq−1 , fiq } . . .}. 4. Σ
is maximal with respect to properties (1), (2) and (3).

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87

It is a Theorem of Treves [39] that Σ can be decomposed as a locally finite union Σ = ∪Σj of pairwise disjoint Poisson strata Σj . The family {Σj } is called the Poisson stratification of Σ.

6 Examples We study the stratification for    2 4 3 x + 4xy 2 ∂t . P = ∂x 2 + ∂y + 3 We see that the characteristic set is stratified in the following way:     4 3 2 2 2 x + 4xy τ , x + y = 0 , Σ1 = ξ = 0 = η + 3 Σ2j = {x = ξ = 0 = η = y, (−1)j τ > 0}, for j = 1, 2. All strata are symplectic and the operator is analytic hypoelliptic. Now consider the Oleinik operator. P = ∂t 2 + t2p ∂x 2 + t2r ∂y 2 , with p, r both ≥ 1. In all cases, Σ = {t = 0 = τ }. If p = r, Σ is the only stratum, which is symplectic. P is analytic hypoelliptic. However when p < r we have, for j = 1, 2, Σ1j = {t = 0 = τ , (−1)j ξ > 0}, Σ2j = {t = τ = ξ = 0, (−1)j η > 0}. The Σ1j are symplectic, while the Σ2j are not. The operator is not analytic hypoelliptic.

7 Treves’ conjecture Let U ⊂ Rm be open and let X0 , . . . , Xν be real analytic vector fields on U . Let P have the form P = X0 2 + · · · + Xν 2 .

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Paulo D. Cordaro and Nicholas Hanges

We assume P satisfies H¨ormander’s condition. Let Σ denote the characteristic set of P and let Σj be one of the strata of the Poisson stratification of Σ. By definition T Σj ∩T Σj ⊥ is an involutive (in the sense of Frobenius) subbundle of T Σj and consequently it defines a foliation on Σj . The leaves of such foliations will be called bicharacteristic leaves. Treves’ original conjecture implied the following statement: (S) The operator P is analytic hypoelliptic on U if every Poisson stratum of Σ is symplectic. Such a statement is consistent with all known results. However, its converse is not true in the global sense or in the sense of germs. See Cordaro–Himonas [14] and Hanges [19]. Also, the contribution of Bove, Derridj, Tartakoff [5] is a very interesting generalization of [19]. Indeed, these papers have motivated Treves to give a more precise conjecture, see [39]. (C) For P to be analytic hypoelliptic on U it is necessary and sufficient that every bicharacteristic leaf is vertical2 and relatively compact in T ∗ U . Of course the validity of (C) implies (S).

8 Symplectic strata of codimension 2 We have the following recent result of Cordaro and Hanges [13], which establishes Treves’ conjecture (S) in the codimension 2 case.3 Theorem 8.1 Let U ⊂ Rm be open and let X0 , . . . , Xν be real analytic vector fields on U which satisfy H¨ ormander’s condition. Let P have the form P = X0 2 + · · · + Xν 2 , with Σ the characteristic set of P . Let q ∈ Σ. We assume that near q, Σ is a symplectic Poisson stratum of codimension 2. Then P is analytic hypoelliptic at q. / W FA (P u), it follows that This means that whenever u ∈ D
(U ), with q ∈ q∈ / W FA (u).

2

A subset S of T ∗ U is vertical if it is either empty or else its image under the canonical projection T ∗ U → U reduces to a single point. 3 Very recently we became aware of the article [32]. Theorem 8.1 follows from the main result of that paper. The methods of proof are completely different. We believe that the techniques sketched here can be applied in more general situations.

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89

9 Sketch of the proof Let (x, t) ∈ Rm × R be coordinates near (0, 0). We may assume that P has the form ν
P = ∂t2 + Xj2 , j=1

where, for j = 1, . . . , ν we have Xj = bj (x, t)∂t +

m


aj,k (x, t)∂ k .

k=1

We write Xj = bj (x, t)∂t + fj (x, t, ∂x ). We see that we have Σ = {τ = 0 = fj , j = 1, . . . , ν}. We work near the point p0 = (0, 0; ξ 0 , 0). Near p0 , Σ is symplectic of codimension 2. Hence we may write Σ = {τ = 0 = t − λ(x, ξ)} near p0 . Here λ is real analytic and positively homogeneous of degree 0, defined in a conic neighborhood of (0, ξ 0 ). But Σ is also a Poisson stratum. Hence there exists an integer k ≥ 1 such that all brackets formed from τ , fj of order less than k vanish on Σ. Also, at each point of Σ, some bracket of order k does not vanish. Hence, it follows that we can write, in a conic neighborhood of (0, 0; ξ 0 , 0), fj (x, t, ξ) = (t − λ(x, ξ))k Ej (x, ξ) + O((t − λ(x, ξ))k+1 |ξ|) where Ej is positively homogeneous of degree 1 and E(x, ξ) =

m


Ej (x, ξ)2

j=1

is an elliptic symbol of degree 2 defined in a conic neighborhood of (0, ξ 0 ). After a real analytic change of coordinates preserving (0, 0), we may assume that P has the form P = ∂t2 +

ν


ajl (x, t)Xj Xl + b(x, t)∂t + Y.

j,l=1

Here Xl =

m
j=1

blj (x, t)∂j ,

l = 1, . . . ν

(9.1)

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Paulo D. Cordaro and Nicholas Hanges

and Y =

m


bj (x, t)∂j .

j=1

We also assume that there exists C > 0 such that ν


ajl (x, t)ξ j ξ l ≥ C|ξ|2

(9.2)

j,l=1

for all ξ ∈ Rν and all (x, t) near (0, 0). Furthermore, we have Xl (x, t, ξ) = i

m


blj (x, t)ξ j

j=1

= i(t − λ(x, ξ))k El (x, ξ) + O((t − λ(x, ξ))k+1 |ξ|),

(9.3)

in a conic neighborhood of p0 . Furthermore, we assume that there exists an l, 1 ≤ l ≤ ν such that (9.4) El = 0 in a conic neighborhood of (0; ξ 0 ). Note that we also have λ(0, ξ 0 ) = 0. We also have Y (x, t, ξ) = O((t − λ(x, ξ))k−1 |ξ|). Now we introduce the FBI transform. If v is a smooth function with compact support, defined near (0, 0), we define  2 I[v](z, t, ξ) = e−iy·ξ−|ξ|(z−y) /2 v(y, t)dy. (9.5) Note that I[v] is entire on Cm (as (Rm \ 0) × R. We assume that we are given a support near (0, 0) such that P u is that there exists an ε > 0 such that

a function of z), for each fixed (ξ, t) ∈ smooth function u with small, compact analytic near (0, 0). It follows from this if |z| < ε and |t| < ε, then

|I[P u](z, t, ξ)| ≤

1 −ε|ξ| e , ε

(9.6)

for all ξ ∈ Rm . We have the following formulas for v ∈ C0∞ (Rm × R): I[∂j v] = (∂j + iξ j )I[v], 

and I[yj v] =

 1 ∂j + zj I[v]. |ξ|

(9.7)

(9.8)

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91

It follows that if a(x, t) is real analytic near the support of v we have I[av](z, t, ξ) = a(z, t)I[v](z, t, ξ) + O(1/|ξ|). Using these formulas, we see that there exists a partial differential operator Q = Q(z, t, ξ, Dz , Dt ) such that I[P u](z, t, ξ) = QI[u](z, t, ξ) + R1

(9.9)

for (z, t) near (0, 0) and ξ ∈ Rm . We see that Q can be written as Q(z, t, ξ, Dz , Dt ) ν


= ∂t2 +

ajl (z, t)Xj (z, t, ξ)Xl (z, t, ξ) + Y (z, t, ξ)

j,l=1

+

ν


ajl (z, t)Xj (z, t, ξ)Xl +

j,l=1

+

ν


ν


ajl (z, t)Xj Xl (z, t, ξ)

j,l=1

ajl (z, t)Xj Xl + b(z, t)∂t + Y.

j,l=1

Using our earlier estimates we see that Q(z, t, ξ, Dz , Dt ) = ∂t2 − |ξ|2 (t − λ(z, ξ))2k E(z, ξ) − i|ξ|ϕ(z, ξ)(t − λ(z, ξ))k−1 +

ν


ajl (z, t)Xj Xl + b(z, t)∂t + Y + O((t − λ(z, ξ))k |ξ|∂x )

j,l=1

+ O((t − λ(z, ξ))2k+1 |ξ|2 ) + O((t − λ(z, ξ))k |ξ|). We make the change of variable 1

s = t|ξ| k+1 , and denote by Q# (z, s, ξ, Dz , Ds ) the transformed operator. We have the following −2

|ξ| k+1 Q# (z, s, ξ, Dz , Ds ) 1

1

= ∂s2 − (s − λ(z, ξ)|ξ| k+1 )2k E(z, ξ) − iϕ(z, ξ)(s − λ(z, ξ)|ξ| k+1 )k−1 −1

1

−1

+ O(|ξ| k+1 ) + O((s − λ(z, ξ)|ξ| k+1 )k |ξ| k+1 ∂x ) 1

−1

1

−1

+ O((s − λ(z, ξ)|ξ| k+1 )2k+1 |ξ| k+1 ) + O((s − λ(z, ξ)|ξ| k+1 )k |ξ| k+1 ).

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We study the rescaled operator ∂s2 − s2k − iE(x, ξ)

−1 2

ϕ(x, ξ)sk−1 .

(9.10)

The Green’s function is constructed from two solutions to the homogeneous equation, f # , g # . For example, f # (s; x, ξ) has the behavior (1/2)s−(k−iϕ(x,ξ)E(x,ξ)

−1 2

)/2

e

|s|k+1 k+1

,

as s → +∞, and (−1/2)(−s)−(k−iϕ(x,ξ)E(x,ξ)

−1 2

)/2

e

−|s|k+1 k+1

,

as s → −∞. There are analogous formulas for g # . Let G# denote the Green’s function constructed from f # , g # . Following arguments of Menikoff [27] (cf. also [12]), we have C > 0 such that  +∞ |s
|2k G# (s, s
; x, ξ)ds
≤ C, (9.11) −∞

for all s ∈ R and (x, ξ) in a conic neighborhood of (0, ξ 0 ). This estimate is important in absorbing the remainders.

References 1. M. S. Baouendi and C. Goulaouic, Nonanalytic hypoellipticity for some degenerate elliptic operators, Bull. Amer. Math. Soc. 78(1972), 483–486. 2. L. Boutet de Monvel, A. Grigis and B. Helffer, Parametrixes d’op´ erateurs pseudo-diff´ erentiels ´ a caract´eristiques multiples, Journ´ees ´equations aux D´eriv´ees Partielles de Rennes (1975), 93–121. Ast´erisque, No. 34–35, Soc. Math. France, Paris, 1976. 3. L. Boutet de Monvel and F. Treves, On a class of pseudodifferential operators with double characteristics, Invent. Math. 24(1974), 1–34. 4. L. Boutet de Monvel and F. Treves, On a class of systems of pseudodifferential equations with double characteristics, Comm. Pure Appl. Math. 27(1974), 59–89. 5. A. Bove, M. Derridj and D. Tartakoff, Analytic hypoellipticity in the presence of non-symplectic characteristic points, preprint. 6. S. Chanillo, Analytic hypoellipticity and spectral problems for Schr¨ odinger’s equation, Geometric analysis of PDE and several complex variables, 101–120, Contemp. Math. 368(2005), Amer. Math. Soc., Providence, RI. 7. S. Chanillo, B. Helffer and A. Laptev, Nonlinear eigenvalues and analytic hypoellipticity, J. Funct. Anal. 209(2004), 425–443. 8. S. Chanillo, Kirillov theory, Treves strata, Schr¨ odinger equations and analytic hypoellipticity of sums of squares, preprint. 9. M. Christ, A class of hypoelliptic PDE admitting non-analytic solutions, Contemp. Math. 137(1992), 155–167, Amer. Math. Soc., Providence, RI.

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10. M. Christ, A necessary condition for analytic hypoellipticity, Math. Res. Lett. 1(1994), 241–248. o kernels, 11. M. Christ, Remarks on the breakdown of analyticity for ∂¯b and Szeg˝ in “Harmonic Analysis”, ICM–90 Satellite Conference Proceedings, SpringerVerlag, 1991, 61–78. 12. P. D. Cordaro and N. Hanges, Impact of lower order terms on a model PDE in two variables, Geometric analysis of PDE and several complex variables, 157– 176, Contemp. Math. 368(2005), Amer. Math. Soc., Providence, RI. 13. P. D. Cordaro and N. Hanges, Analytic hypoellipticity for operators with symplectic strata of codimension two, in preparation. 14. P. D. Cordaro and A. A. Himonas, Global analytic hypoellipticity of a class of degenerate elliptic operators on the torus, Math. Res. Lett. 1(1994), 501–510. 15. O. Costin and R. Costin, Failure of analytic hypoellipticity in a class of differential operators, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 2(2003), 21–45. 16. A. Gilioli and F. Treves, An example in the solvability theory of linear PDE’s, Amer. J. Math. 96(1974), 367–385. 17. A. Grigis and J. Sjostrand, Front d’onde analytique et sommes de carres de champs de vecteurs, Duke Math. J. 52(1985), 35–51. 18. V. V. Grusin, A certain class of hypoelliptic operators, Mat. Sb. 83(1970), 456-473. English trasl. Math. USSR-Sb. 12(1970), 458–476. 19. N. Hanges, Analytic regularity for an operator with Treves curves, J. Funct. Anal. 210(2004), 295–320. 20. N. Hanges and A. A. Himonas, Singular solutions for sums of squares of vector fields, Comm. Partial Differential Equations 16(1991), 1503–1511. 21. N. Hanges and A. A. Himonas, Analytic hypoellipticity for generalized Baouendi–Goulaouic operators, J. Funct. Anal. 125(1994), 309–325. 22. N. Hanges and A. A. Himonas, Non-analytic hypoellipticity in the presence of symplecticity, Proc. Amer. Math. Soc. 126(1998), 405–409. ¨ rmander, The analysis of linear partial differential operators I, Springer– 23. L. Ho Verlag, Berlin,1983. ¨ rmander, Hypoelliptic second order differential equations, Acta Math. 24. L. Ho 119(1967), 147–171. 25. T. Hoshiro, Failure of analytic hypoellipticity for some operators of X 2 + Y 2 type, J. Math. Kyoto Univ. 35(1995), 569–581. 26. K. H. Kwon, Concatenations applied to analytic hypoellipticity of operators with double characteristics, Trans. Amer. Mat. Soc. 283(1984), 753–763. 27. A. Menikoff, Some examples of hypoelliptic partial differential equations, Math. Ann. 221(1976), 167–181. 28. G. M´ etivier, Une classe d’op´erateurs non-hypoelliptiques analytiques, Indiana Univ. Math. J. 29(1980), 823–860. 29. G. M´ etivier, Non-hypoellipticit´e analytique pour Dx2 +(x2 +y 2 )Dy2 , C. R. Acad. Sci. Paris 292(1981), 401–404. 30. G. M´ etivier, Non-hypoellipticit´e analytique pour des op´erateurs a ` caract´eristiques doubles, S´eminaire Goulaouic-Meyer-Schwartz, 1981–82, No. 12. 31. T. Okaji, Analytic hypoellipticity for operators with symplectic characteristics, J. Math. Kyoto Univ. 25(1985), 489–514. 32. O. A. Oleinik, On the analyticity of solutions of partial differential equations and systems, Ast´erisque 2/3(1973), 272–285. 33. Pham The Lai and D. Robert, Sur un probl`eme aux valeurs propres nonlin´eaire, Israel J. Math. 36(1980), 169–186.

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¨ strand, Analytic wavefront sets and operators with multiple characteris34. J. Sjo tics, Hokkaido Math. J. 12(1983), 392–433. 35. E. M. Stein, An example on the Heisenberg group related to the Lewy operator, Invent. Math. 69(1982), 209–216. ¯ 36. D. Tartakoff, On the local real analyticity of solutions to b and the ∂Neumann problem, Acta Math. 145(1980), 117–204. 37. F. Treves, Analytic hypoellipticity of a class of pseudodifferential operators ¯ with double characteristics and applications to the ∂-Neumann problem, Comm. Partial Differential Equations 3(1978), 475–642. 38. F. Treves, Symplectic geometry and analytic hypo-ellipticity, Differential equations: La Pietra 1996 (Florence), 201–219, Proc. Sympos. Pure Math., 65, Amer. Math. Soc., Providence, RI, 1999. 39. F. Treves, On the analyticity of solutions of sums of squares of vector fields, this volume.

On the backward uniqueness property for a class of parabolic operators Daniele Del Santo and Martino Prizzi Dipartimento di Matematica e Informatica, Universit` a di Trieste, Trieste, Italia

Summary. We give sharp regularity conditions, ensuring the backward uniqueness property to a class of parabolic operators.

2000 Mathematics Subject Classification: 35K25, 35K35, 35A05. Key words: Parabolic operator, backward uniqueness, modulus of continuity, Osgood condition.

1 Introduction, statements and remarks In this note we illustrate some new results concerning the backward uniqueness property for a class of parabolic operators, whose coefficients are non-Lipschitz continuous in time. Namely, we consider parabolic operators of the form
(−1)|α| ∂xα (ραβ (t, x)∂xβ ); (1.1) P := ∂t + 0≤|α|,|β|≤m

here m ∈ N, (t, x) ∈ [0, T ] × Rn , and α and β are n-multiindices with weights |α| and |β| ≤ m. We assume that ραβ = ρβα for all α’s and β’s (formal selfadjointness), that ραβ is real when |α| = |β| = m, and that there exists  c0 > 0 such that |α|=|β|=m ραβ (t, x)ξ α ξ β ≥ c0 |ξ|2m for all ξ ∈ Rn (strong ellipticity). Given a functional space H, we say that P enjoys the backward uniqueness property in H iff, whenever u ∈ H satisfies P u ≡ 0 (in the sense of distributions) in [0, T ] × Rn , and u(T, ·) ≡ 0 in Rn , then u ≡ 0 in [0, T ] × Rn . Our aim is to find conditions on the coefficients ραβ ’s, ensuring that P enjoys the backward uniqueness property in some given functional space H. As a preliminary observation, we notice that in [13] Tychonoff constructed a function u ∈ C ∞ (R × Rn ) satisfying

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Daniele Del Santo and Martino Prizzi

∂t u − ∆u ≡ 0 u(0, ·) ≡ 0

in R × Rn in Rn ,

(1.2)

but u ≡ 0 in any open subset of R × Rn . It follows that, whether P enjoys the backward uniqueness property in H or not, depends first of all on the choice of H. We are interested here in the case H = H1m , where H1m := H 1 ([0, T ], L2 (Rn )) ∩ L2 ([0, T ], H 2m(Rn )).

(1.3)

The reason for this choice is essentially due to its historical background, but other choices are possible as well. In [7] Lions and Malgrange proved that P enjoys the backward uniqueness property in H1m , provided the ραβ ’s are sufficiently smooth with respect to x and Lipschitz continuous with respect to t. They work in an abstract Hilbert space setting and their proof is based on a Carleman type estimate. The required smoothness of the ραβ ’s with respect to x is related to the regularity theory for elliptic equations and is needed to let P fall in the abstract Hilbert space setting. The required Lipschitz continuity with respect to t seems to be more intrinsically connected with the backward uniqueness property. In fact, in the same paper Lions and Malgrange raised the question, whether Lipschitz continuity could be replaced by, say, simple continuity. As a first step in this direction, in [3] Bardos and Tartar proved that P enjoys the backward uniqueness property in H1m , provided the ραβ ’s are absolutely continuous with respect to t. Their proof exploits a sort of logarithmic convexity property satisfied by the norm u of any nontrivial solution of P u = 0. Later, in [6] Ghidaglia, by using the same technique, extended the results of [3], so as to cover also some classes of nonlinear parabolic equations. We stress that, in all the above mentioned results, it is required that the ραβ ’s be differentiable with respect to t, at least in a weak sense. The reason is that, at a certain point, one needs to perform some integration by parts. Although this latter seems to be just a technical obstruction, the possibility of replacing Lipschitz continuity by simple continuity was finally ruled out by Miller in [9]. He exhibited an example of an operator P which does not enjoy the backward uniqueness property in H1m . The operator is of second order in space and its coefficients are of class C ∞ with respect to x and H¨ older continuous of exponent 1/6 with respect to t. Recently, in [8] Mandache improved the result of Miller constructing a similar nonuniqueness example in which the older continuous of every coefficients are of class C ∞ with respect to x and H¨ exponent less than 1 with respect to t. More precisely, in the result of Mandache the regularity with respect to t is expressed in terms of a modulus of continuity. Our goal is to find a sharp condition on the modulus of continuity of the ραβ ’s, ensuring that P enjoys the backward uniqueness property in H1m . Let I ⊂ R be a closed bounded interval, let B be a Banach space and let f : I → B be a continuous function. The modulus of continuity of f is the function µ(f, ·) : [0, 1] → R defined by

Backward uniqueness for parabolic operators

µ(f, τ ) :=

sup t,s∈I 0≤|t−s|≤τ

f (t) − f (s)B .

97

(1.4)

Notice that µ(f, ·) is nondecreasing and µ(f, 0) = 0. Since f is uniformly continuous on I, it follows that µ(f, τ ) → 0 as τ → 0. If f is nonconstant, then µ(f, τ ) > 0 for τ > 0 and µ(·) = µ(f, ·) satisfies sup t,s∈I 0<|t−s|≤1

f (t) − f (s)B < +∞. µ(|t − s|)

(1.5)

Moreover, µ(f, ·) is minimal with respect to this latter property, in the sense that, whenever a nondecreasing function µ : [0, 1] → R satisfies (1.5), then Cµ(τ ) ≥ µ(f, τ ) for some positive constant C. It is easy to check that µ(f, ·) is subadditive, that is µ(f, τ 1 + τ 2 ) ≤ µ(f, τ 1 ) + µ(f, τ 2 ) whenever 0 ≤ τ 1 ≤ τ 2 ≤ τ 1 + τ 2 ≤ 1. Then, by a result of Efimov ([5], Lemma 4), there exists a concave (hence continuous) nondecreasing function µ : [0, 1] → R such that µ(τ ) ≤ µ(f, τ ) ≤ 2µ(τ ), τ ∈ [0, 1]. If f is nonconstant, µ can be chosen to be strictly increasing. Therefore it is natural to make the following Definition 1.1 Let I ⊂ R be a closed bounded interval, let B be a Banach space and let µ : [0, 1] → [0, 1] be a concave strictly increasing function, with µ(0) = 0. We say that a function f : I → B is µ-continuous (and we write f ∈ C µ (I, B)) iff (1.5) is satisfied. Whenever f : I → B is a continuous function, then certainly f ∈ C µ (I, B) for some concave strictly increasing function µ, with µ(0) = 0 . Moreover, if f is nonconstant, µ can be chosen in such a way that µ(τ ) ≤ µ(f, τ ) ≤ 2µ(τ ), τ ∈ [0, 1]. If µ(τ ) = τ , then C µ (I, B) = Lip(I, B); if µ(τ ) = τ α , 0 < α < 1, then C µ (I, B) = C α (I, B). Definition 1.2 Let ω : [0, 1] → R be a nondecreasing function, with ω(0) = 0. We say that ω satisfies the Osgood condition iff  1 1 ds = +∞. (1.6) 0 ω(s) Condition (1.6) was introduced by Osgood in [10], while proving uniqueness for ordinary differential equations with non-Lipschitz continuous nonlinearities. If ω(τ ) = τ , then ω satisfies the Osgood condition. If ω(τ ) = τ α , 0 < α < 1, then ω does not satisfy the Osgood condition. If ω(τ ) = τ | log τ |, then ω satisfies the Osgood condition. The result of Mandache states that if ω is a concave strictly increasing function with ω(0) = 0 and ω does not satisfy the Osgood condition, then there exists a parabolic operator of type (1.1) with m = 1 such that the coefficients are C ∞ with respect to x and C ω with respect to t and the backward uniqueness property does not hold.

98

Daniele Del Santo and Martino Prizzi

Let us come to our result. We shall consider here only operators whose coefficients ραβ ’s are independent of the spatial variable x. If m = 1, the general case can be recovered by a microlocal approximation procedure similar to the one exploited in [2], provided the ραβ ’s are sufficiently smooth in x (see [4] for details). We do not know whether it is possible to extend the result of [4] to the case m > 1. If the coefficients are independent of x, the operator P takes the simpler form
i |α| ρα (t)∂xα , (1.7) P = ∂t + 0≤|α|≤2m

where ρα ∈ R for all α. Let A be the set of all n-multiindices whose weight is smaller than or equal to 2m, let  be the cardinality of A and let R : [0, T ] → R , R(t) := (ρα (t))α∈A , be a continuous mapping. Setting  α |α|=k ρα (t)ξ k , (t, ξ) ∈ [0, T ] × (Rn \ {0}), k = 0, . . . , 2m, ρk (t, ξ) := (−1) |ξ|k (1.8) we assume that there exists Λ > 0 such that, for (t, ξ) ∈ [0, T ] × (Rn \ {0}), |ρk (t, ξ)| ≤ Λ,

k = 0, . . . , 2m − 1,

and 1/Λ ≤ ρ2m (t, ξ) ≤ Λ.

Consider the following backward-parabolic inequality:    
  |α| α  ∂t u − ˜ i ρα (t)∂x u ≤ Cu Hm .    0≤|α|≤2m 2

(1.9)

(1.10)

L

The main result of the paper is the following. Theorem 1.1 Let the modulus of continuity µ(R, ·) of R(·) := (ρα (·))α∈A satisfy the Osgood condition. If u ∈ H1m satisfies (1.10) and u(0, ·) ≡ 0 in Rn , then u ≡ 0 in [0, T ] × Rn . Notice that, if R(·) is constant, then R ∈ C µ ([0, T ], R) with µ(τ ) = τ . If R(·) is nonconstant, then we can find a concave strictly increasing function µ : [0, 1] → R such that µ(τ ) ≤ µ(R, τ ) ≤ 2µ(τ ), τ ∈ [0, 1]. It follows that in both cases R ∈ C µ ([0, T ], R ) for some µ which satisfies the Osgood condition. This observation is crucial for the proof of Theorem 1.1. Remark 1.1 Theorem 1.1 allows us to treat also operators with x-dependent coefficients up to the order m. Indeed, all terms up to the order m are absorbed by the right-hand side of the inequality (1.10). It is very likely that Theorem 1.1 is sharp. Indeed not only the example of Mandache confirms it in the case of m = 1, but, by modifying a well-known elliptic counterexample of Pli´s [11], we can prove the following.

Backward uniqueness for parabolic operators

99

Theorem 1.2 Let µ : [0, 1] → [0, 1] be a concave strictly increasing function with µ(0) = 0. If µ does not satisfy the Osgood condition, then for all m ∈ N there exist l ∈ C µ ([0, 1], R), with 1/2 ≤ l(t) ≤ 3/2 for all t ∈ [0, 1], b1 , b2 , c ∈ Cb∞ ([0, 1] × R2 , R), and u ∈ Cb∞ ([0, 1] × R2 , R), with u(1, ·) ≡ 0 in R2 but u ≡ 0 in [0, 1] × R2 , such that ∂t u + (−1)m (∂x2m u + l(t)∂x2m u) + b1 (t, x)∂x1 u + b2 (t, x)∂x2 u 1 2 in [0, 1] × R2 .

+ c(t, x)u = 0

(1.11)

Remark 1.2 If m = 1, we can take any function ψ ∈ C ∞ (R2 ) such that ψ(x) = e−|x| for |x| ≥ 1, and defining v(t, x) := ψ(x)u(t, x) we obtain a counterexample to the backward uniqueness property in H11 . However if m > 1, by the same procedure we get only a non-selfadjoint counterexample, with x-dependent coefficients up to the order 2m − 1. In the next sections we give sketches of the proofs of Theorems 1.1 and 1.2.

2 Proof of Theorem 1.1 Theorem 1.1 is a consequence of the following. Proposition 2.1 Let µ : [0, 1] → [0, 1] be a concave strictly increasing function with µ(0) = 0. Let T > 0 and let R(·) ∈ C µ ([0, T ], R ), R(·) := (ρα (·))α∈A , be a function satisfying (1.8)–(1.9). There exist C > 0, γ 0 > 0 and a strictly increasing C 2 -function Φ : [0, +∞[→ [0, +∞[ such that, for all γ ≥ γ 0 and for all u ∈ C0∞ (R × Rn ) with supp u ⊂ [0, T /2] × Rn , the following Carleman estimate holds:  2    T /2
  2 |α| α  e γ Φ(γ(T −t))  u − i ρ (t)∂ u dt ∂ α x   t 0   2 0≤|α|≤2m L

 ≥ Cγ 1/2

T /2

2

e γ Φ(γ(T −t)) u2H m dt.

(2.1)

0

Let us briefly sketch how to prove Theorem 1.1 from the Carleman estimate (2.1). First, we notice that if µ(R, ·) satisfies the Osgood condition, then R ∈ C µ ([0, T ], R ) for some concave strictly increasing function µ which satisfies the Osgood condition. Second, by a density argument we have that (2.1) holds for any u ∈ H1m such that u(0, ·) ≡ 0 and u(t, ·) ≡ 0 for t ∈ [T /2, T ]. Now if u ∈ H1m satisfies (1.10) and u(0, ·) ≡ 0, we take ϑ ∈ C ∞ (R), ϑ ≡ 0 on [T /2, +∞], ϑ ≡ 1 on [0, T /3] and we apply (2.1) to the function ϑu. We obtain

100

Daniele Del Santo and Martino Prizzi



T /2

e

2 γ Φ(γ(T −t))

0

 2  
  |α| α ∂t (ϑu) − i ρα (t)∂x (ϑu)   dt   2 0≤|α|≤2m L



T /2

≥ Cγ 1/2

2

e γ Φ(γ(T −t)) ϑuH m dt.

(2.2)

0

Since ϑ ≡ 1 for t ∈ [0, T /3], (1.10) and (2.2) imply 

T /2

e

2 γ Φ(γ(T −t))

T /3

2   
  |α| α ∂t (ϑu) − i ρα (t)∂x (ϑu)  dt   2  0≤|α|≤2m L



T /3

˜ ≥ (Cγ 1/2 − C)

2

e γ Φ(γ(T −t)) uH m dt.

(2.3)

0

Since Φ is increasing, for all sufficiently large γ we have 2    T /2  
  C 1/2 T /3 |α| α  ∂t (ϑu) − γ i ρ (t)∂ (ϑu) dt ≥ u2H m dt. α x   2 T /3  0  2 0≤|α|≤2m L (2.4) Letting γ → ∞, we get u ≡ 0 in [0, T /3] × Rn . Finally, a standard connection argument implies that u ≡ 0 in [0, T ] × Rn . Let us come to the proof of Proposition 2.1. Let Φ : [0, +∞[→ [0, +∞[ 1 be of class C 2 and increasing. Setting v(t, x) := e γ Φ(γ(T −t)) u(t, x) and denoting by vˆ(t, ξ) the Fourier transform of v(t, x) with respect to x, (2.1) becomes 2  2m   T /2   
  ρk (t, ξ)|ξ|k − Φ
(γ(T − t)) vˆ(t, ξ) dξ dt ∂t vˆ(t, ξ) −   n 0 R k=0



T /2

≥ Cγ 1/2

 Rn

0

(|ξ|2m |ˆ v (t, ξ)|2 + |ˆ v (t, ξ)|2 ) dξ dt.

(2.5)

Denoting by Ξ the left member of (2.5), direct computation and integration by parts give 

T /2

 |∂t vˆ(t, ξ)|2 dξ dt

Ξ= Rn

0

 + 0

T /2

 2 2m 
   k
ρk (t, ξ)|ξ| − Φ (γ(T − t)) |ˆ v (t, ξ)|2 dξ dt    n R



k=0

Backward uniqueness for parabolic operators



T /2 

γΦ

(γ(T − t))|ˆ v (t, ξ)|2 dξ dt

+ Rn

0



101

T /2



− 2Re

Rn

0

∂t vˆ(t, ξ)

 2m


 k

ρk (t, ξ)|ξ|

vˆ(t, ξ) dξ dt.

k=0

If the ρk (·, ξ)’s are Lipschitz continuous (that is: if R(·) is Lipschitz continuous), one could just take Φ(τ ) := τ 2 , integrate by parts the double product and get the desired estimate (see [7] for details). If the ρk (·, ξ)’s are not Lipschitz continuous, we exploit a standard approximation pro∞ cedure. ' We extend ρk (·, ξ) on the whole R, we take φ ∈ C0 (R) such that R φ(s) ds = 1, φ ≥ 0 and supp φ ⊂ [−1/2, 1/2], and then we define    t−s 1 ρk,ε (t, ξ) := ρk (s, ξ) φ (2.6) ds, (t, ξ) ∈ R × (Rn \ {0}). ε ε R It follows that ρk,ε (·, ξ) ∈ C ∞ for every ξ ∈ Rn \ {0}. Moreover |ρk,ε (t, ξ) − ρk (t, ξ)| ≤ Kµ(ε),

(t, ξ) ∈ R × (Rn \ {0})

(2.7)

and

µ(ε) , (t, ξ) ∈ R × (Rn \ {0}) (2.8) ε (here “
” indicates derivation with respect to t). Now let ε1 , . . . , ε2m be approximation parameters to be chosen later. Then, adding and subtracting ρk,εk and integrating by parts with respect to t, we get:  2m   T /2 
k − 2Re ∂t vˆ(t, ξ) ρk (t, ξ)|ξ| |ξ|2 vˆ(t, ξ) dξ dt |ρ
k,ε (t, ξ)| ≤ K

Rn

0



T /2



≥−

Rn

0



T /2

k=0

|∂t vˆ(t, ξ)|2 dξ dt

 2m 
µ(εk ) k |ξ| |ˆ v (t, ξ)|2 dξ dt ε n k R



−K 0

k=0



T /2

− K2 0

 Rn

 2m


 µ(εk )2 |ξ|2k

|ˆ v (t, ξ)|2 dξ dt.

(2.9)

k=0

Now the first key idea is to let the approximation parameters εk depend on ξ (cf [1]). First, we observe that, by (1.9), there exist N0 ≥ 1 and Λ0 > 0 such that, for all |ξ| ≥ N0 , 1 2m
|ξ| ≤ ρk (t, ξ)|ξ|k ≤ Λ0 |ξ|2m . Λ0 2m

k=0

(2.10)

102

Daniele Del Santo and Martino Prizzi



Then we take εk :=

|ξ|−k

if |ξ| ≥ N0 ,

N0−k

if |ξ| ≤ N0 .

(2.11)

With this choice, noticing also that s2 µ(1/s) is increasing on [1, +∞], we ˜ such that: obtain that there exists a positive constant K 2  2m   T /2   
  k
ρk (t, ξ)|ξ| − Φ (γ(T − t)) vˆ(t, ξ) dξ dt ∂t vˆ(t, ξ) −   n 0 R k=0

≥ (i) + (ii) + (iii) + (iv), where 



T /2

(i) = γ Rn

0



Φ

(γ(T − t))|ˆ v (t, ξ)|2 dξ dt,

 2m 2 
   ρk (t, ξ)|ξ|k − Φ
(γ(T − t)) |ˆ v (t, ξ)|2 dξ dt,    n R



T /2

(ii) = 0

k=0



T /2



˜ (iii) = −K

Rn

0



T /2

˜ (iv) = −K 0

|ˆ v (t, ξ)|2 dξ dt,

 {|ξ|≥N0 }

µ(1/|ξ|2m )|ξ|4m |ˆ v (t, ξ)|2 dξ dt.

Now we observe that: • • • • •

the summand (i) behaves well, provided Φ

(τ ) ≥ 1 for large τ ; under the same condition the summand (iii) is absorbed by (i); when |ξ|2m ≥ 2Λ0 Φ
(γ(T − t)), then the integrand in (ii) behaves like |ξ|4m , which is enough to compensate the integrand in (iv); when N02m ≤ |ξ|2m ≤ (1/2Λ0 )Φ
(γ(T − t)), then again the integrand in (ii) behaves like |ξ|4m , which is enough to compensate the integrand in (iv); the difficult case is when |ξ|2m ∼ Φ
(γ(T − t)).

At this point the second key idea is to modulate the weight Φ on the function µ (cf Tarama [12]). Roughly speaking, we ask that, when |ξ|2m ∼ Φ
(γ(T − t)), then the integrand in (iv) must be compensated by the integrand in (i). More precisely, we ask that Φ

(γ(T − t)) ∼ µ(1/|ξ|2m )|ξ|4m . In other words, the Carleman estimates (2.1) will follow, provided Φ satisfies the ordinary differential equation Φ

= µ(1/Φ
)(Φ
)2 .

(2.12)

Backward uniqueness for parabolic operators

103

All we have to do then is to find a solution of (2.12) and to check that: • • •

Φ is defined on [0 + ∞[, i.e., it does not blow up in finite time; Φ is positive and increasing; Φ

(τ ) ≥ 1 for all sufficiently large τ .

Equation (2.12) can be easily solved by separation of variables. The explicit solution of the Cauchy problem with initial values Φ(0) = 0 and Φ
(0) = 1 is given by:  1 1 ds, t ≥ 1, η(t) := 1/t µ(s)  τ Φ(τ ) := η −1 (r)dr, τ ≥ 0. 0

The Osgood condition precisely guarantees that Φ is defined on [0, +∞[. The other properties that we require for Φ follow by easy computation. With this choice of Φ we finally get the desired Carleman estimate (2.1). The details are left to the reader.

3 Proof of Theorem 1.2 The proof of Theorem 1.2 is very similar to that one of Theorem 3 in [4]. Also in this case we will follow closely the construction of the example in [11]. Let A, B, C, J be four C ∞ functions defined in R with 0 ≤ A(s), B(s), C(s) ≤ 1, −2 ≤ J(s) ≤ 2 for all s ∈ R and A(s) = 1 for s ≤

1 , 5

B(s) = 0 for s ≤ 0 or s ≥ 1, 1 , 4 1 1 J(s) = −2 for s ≤ or s ≥ , 6 2 C(s) = 0 for s ≤

A(s) = 0

for s ≥

B(s) = 1

for

C(s) = 1 J(s) = 2

1 , 4

1 ≤s≤ 6 1 for s ≥ , 3 1 for ≤ s ≤ 5

1 , 2

1 . 3

Let (an )n , (zn )n be two real sequences such that −1 < an < an+1 1 < zn < zn+1

for all n ≥ 1, lim an = 0,

(3.1)

for all n ≥ 1, lim zn = +∞;

(3.2)

n

n

n and let us define rn = an+1 − an , q1 = 0, qn = k=2 zk rk−1 for all n ≥ 2, and pn = (zn+1 − zn )rn . We suppose moreover that

104

Daniele Del Santo and Martino Prizzi

pn > 1 for all n ≥ 1. (3.3)  t−1−an   t−1−an   t−1−an  , Bn (t) = B , Cn (t) = C and We set An (t) = A rn rn rn  t−1−a  n Jn (t) = J . We define rn vn (t, x1 ) = exp(−qn − zn (t − 1 − an )) cos

√ zn x1 ,

2m

wn (t, x2 ) = exp(−qn − zn (t − 1 − an ) + Jn (t)pn ) cos

√

2m

zn x2 ,

and u(t, x1 , x2 ) ⎧ v1 (t, x1 ) ⎪ ⎪ ⎪ ⎪ ⎨An (t)vn (t, x1 ) + Bn (t)wn (t, x2 ) = ⎪ +Cn (t)vn+1 (t, x1 ) ⎪ ⎪ ⎪ ⎩ 0

for 0 ≤ t ≤ 1 + a1 , for 1 + an ≤ t ≤ 1 + an+1 , for t = 1.

If for all α, β γ > 0, α pβn rn−γ = 0, lim exp(−qn + 2pn )zn+1 n

(3.4)

then u is a Cb∞ ([0, 1] × R2 , R) function. We define 1 for t ≤ 1 + a1 or t = 1, l(t) =
−1 1 + Jn (t)pn zn for 1 + an ≤ t ≤ 1 + an+1 . The condition sup {pn rn−1 zn−1 } ≤ n

1 2J
L∞

(3.5)

guarantees that the operator L  = ∂t + (−1)m (∂x2m − l(t)∂x2m ) is parabolic. 1 2 µ Moreover l is a C function under the condition   pn rn−1 zn−1 sup < +∞. (3.6) µ(rn ) n Finally we define

L
u 2 +(∂ 2 ∂x1 u, x2 u)

b1 = − u2 +(∂x

1 u)

b2 = − u2 +(∂x

1 u)

c = − u2 +(∂x

1 u)

L
u 2 +(∂ 2 ∂x2 u, x2 u) L
u 2 +(∂ 2 u. x2 u)

As in [11], or similarly in [4], the coefficients b1 , b2 , c will be in Cb∞ if for all α, β, γ > 0, α pβn rn−γ = 0. (3.7) lim exp(−pn )zn+1 n

Backward uniqueness for parabolic operators

105

We choose an = −

+∞


1

j=n

)2 µ

(j + k0



1 j+k0

,

zn = (n + k0 )3

(3.8)

with k0 sufficiently large. To conclude the proof it will be sufficient to verify in the same way as in [4] that, with the choice (3.8), the conditions (3.1), . . . , (3.7) hold. We leave it to the reader.

References 1. F. Colombini, E. De Giorgi and S. Spagnolo, Sur les ´equations hyperboliques avec des coefficients qui ne d´ependent que du temps, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 6(1979), 511–559. 2. F. Colombini and N. Lerner, Hyperbolic operators having non-Lipschitz coefficients, Duke Math. J. 77(1995), 657–698. 3. C. Bardos and L. Tartar, Sur l’unicit´e r´etrograde des ´ equations paraboliques et quelques questions voisines, Arch. Rational Mech. Anal. 50(1973), 10–25. 4. D. Del Santo and M. Prizzi, Backward uniqueness for parabolic operators whose coefficients are non-Lipschitz continuous in time, J. Math. Pures Appl. 84(2005), 471–491. 5. A.V. Efimov, Linear methods of approximation of continuous periodic functions, (Russian) Mat. Sb. (N. S.) 54 (96)(1961), 51–90. 6. J.-M. Ghidaglia, Some backward uniqueness results, Nonlinear Anal. 10(1986), 777–790. 7. J.-L. Lions and B. Malgrange, Sur l’unicit´e r´etrograde dans les probl`emes mixtes paraboliques, Math. Scand. 8(1960), 277–286. 8. N. Mandache, On a counterexample concerning unique continuation for elliptic equations in divergence form, Math. Phys. Anal. Geom. 1(1998), 273–292. 9. K. Miller, Nonunique continuation for uniformly parabolic and elliptic equations in selfadjoint divergence form with H¨ older continuous coefficients, Arch. Rational Mech. Anal. 54(1974), 105–117. 10. W.F. Osgood, Beweis der Existenz einer L¨ osung der Differentialgleichung dy/dx = f (x, y) ohne Hinzunahme der Cauchy-Lipschitz’schen Bedingung, Monatsh. Math. 9(1898), 331–345. 11. A. Pli´s, On non-uniqueness in Cauchy problem for an elliptic second order differential equation, Bull. Acad. Polon. Sci. S´er. Sci. Math. Astronom. Phys. 11(1963), 95–100. 12. S. Tarama, Local uniqueness in the Cauchy problem for second order elliptic equations with non-Lipschitzian coefficients, Publ. Res. Inst. Math. Sci. 33(1997), 167–188. 13. A. Tychonoff, Th´eor`eme d’unicit´e pour l’´equation de la chaleur, Rec. Math. Moscou 42(1935), 199–215.

Inverse problems for hyperbolic equations Gregory Eskin Department of Mathematics, University of California, Los Angeles, USA

Summary. We present a new approach to the unique determination of the coefficients of the second order hyperbolic equations modulo diffeomorphisms and gauge transformations, assuming that the time-dependent Dirichlet-to-Neumann operator is given on a part of the boundary. We consider also the case of multi-connected domains with obstacles. The interest in this case is spurred by the Aharonov–Bohm effect.

2000 Mathematics Subject Classification: 35L20, 35J10. Key words: Inverse problems, hyperbolic equations, broken rays.

1 Formulation of the problem and the main theorem Let Ω be a smooth bounded domain in Rn , n ≥ 2. Consider in the cylinder Ω × (0, T0 ) the following hyperbolic equation:  2 ∂ def Lu = −i + A0 (x, t) u(x, t) (1.1) ∂t   n
1 ∂ − + Aj (x, t) g(x)g jk (x) −i ∂xj g(x) j,k=1   ∂ + Ak (x, t) u − V (x, t)u = 0, × −i ∂xk where Aj (x, t), 0 ≤ j ≤ n, V (x, t) are C ∞ (Ω × [0, T0 ]) functions, analytic in t, g jk (x)−1 is the metric tensor in Ω, g(x) = det g jk −1 . We consider the initial-boundary value problem for (1.1) in Ω × (0, T0 ): u(x, 0) = ut (x, 0) = 0, u(x, t)|∂Ω×(0,T0 ) = f (x, t).

x ∈ Ω,

(1.2) (1.3)

108

Gregory Eskin

The following operator is called the Dirichlet-to-Neumann (D-to-N) operator: − 12    
n n 
∂u def  Λf = g jk (x) + iAj (x, t)u ν k g pr (x)ν p ν r ,   ∂xj p,r=1 j,k=1

∂Ω×(0,T0 )

(1.4) where u(x, t) is the solution of the initial-boundary value problem (1.1), (1.2), (1.3), ν = (ν 1 , . . . , ν n ) is the unit exterior normal vector at x ∈ ∂Ω with respect to the Euclidian metric. If F (x) = 0 is the equation of ∂Ω in some neighborhood of a point x0 ∈ ∂Ω, then Λf has the following form in this neighborhood:   n
∂u jk g (x) + iAj (x, t)u Fxj (x) (1.5) Λf = ∂xj j,k=1

 ·

n
p,r=1

− 12    g (x)Fxp Fxr   pr

.

F (x)=0,0
Let Γ0 be an open subset of ∂Ω. We shall consider smooth f (x, t) such that supp f ⊂ Γ0 × (0, T0 ]. The inverse problem consists of recovering the coefficients of (1.1) knowing the restriction of Λf to Γ0 × (0, T0 ) for all smooth f with supports in Γ0 × (0, T0 ]. There is a built-in nonuniqueness of this inverse problem: def a) Let y = ϕ(x) be a diffeomorphism of Ω onto Ω0 = ϕ(Ω) such that Γ0 ⊂ ∂Ω0 and ϕ = I on Γ0 . ˆ u = 0 be the equation (1.1) in y-coordinates and let Λˆ be the new Let Lˆ D-to-N operator. It follows from (1.5) that Λˆ = Λ on Γ0 × (0, T ). Therefore ˆ |Γ ×(0,T ) = Λf |Γ ×(0,T ) for all f, supp f ⊂ Γ0 × (0, T0 ], i.e., the D-to-N Λf 0 0 0 0 operator on Γ0 × (0, T0 ) cannot distinguish between Lu = 0 in Ω × (0, T0 ) and ˆ u = 0 in Ω0 × (0, T0 ). Lˆ b) Let G0 (Ω × [0, T0 ]) be a group of C ∞ (Ω × [0, T0 ]) complex-valued functions c(x, t) such that c(x, t) = 0 in Ω × [0, T0 ], c(x, t) = 1 on Γ0 × [0, T0 ]. We say that potentials A(x, t) = (A0 (x, t), A1 (x, t), . . . , An (x, t)) and A
(x, t) = (A
0 (x, t), A
1 (x, t), . . . , A
n (x, t)) are gauge equivalent if there exists c(x, t) ∈ G0 (Ω × [0, T0 ]) such that ∂c (1.6) A
0 (x, t) = A0 (x, t) − ic−1 (x, t) , ∂t ∂c A
j (x, t) = Aj (x, t) − ic−1 (x, t) , 1 ≤ j ≤ n. ∂xj Note that if Lu = 0 and u
= c(x, t)u, then L
u
= 0 where L
is an operator of the form (1.1) with Aj (x, t), 0 ≤ j ≤ n, replaced by A
j (x, t), 0 ≤ j ≤ n. We shall write for brevity L
= c ◦ L.

Inverse problems for hyperbolic equations

109

It is easy to show that if Λ
is the D-to-N operator for L
, then Λ
= Λ on Γ0 × (0, T0 ), i.e., all potentials A(x, t) in the same gauge equivalence class correspond to the same D-to-N operator on Γ0 × (0, T0 ). Note that if we consider real-valued potentials only, then the gauge group G0 should be reduced to c(x, t) such that |c(x, t)| = 1. If Ω is simply-connected, then any c(x, t) ∈ G0 has a form c(x, t) = eiϕ(x,t) where ϕ(x, t) ∈ C ∞ (Ω × [0, T ]). Also if coefficients of L are independent of t, it is natural that the group G0 consists of c(x) independent of t. Then A
0 (x) = A0 (x) (see (1.6)). Denote T∗ = max d(x, Γ0 ), x∈Ω

where d(x, Γ0 ) is the distance in Ω with respect to the metric g jk (x)−1 from x ∈ Ω to Γ0 . We shall assume L and Γ0 satisfy the BLR-condition (see [BLR92]) for t = T∗∗ . This means roughly speaking that any nullbicharacteristic of L in (Ω × [0, T∗∗]) × (Rn+1 \ {0}) intersects (Γ0 × [0, T∗∗]) × (Rn+1 \{0}). It was proven in [BLR92] that the BLR-condition implies that the (bounded) map of f ∈ H01 (Γ0 × (0, T0 )) to (u(x, T∗∗ ), ut (x, T∗∗ )) ∈ H 1 (Ω) × L2 (Ω) is onto. Here H01 (Γ0 × (0, T∗∗ ) is the subspace of H 1 (∂Ω × (0, T∗∗ )) such that f |t=0 = 0 and supp f ⊂ Γ0 × (0, T∗∗ ], u(x, t) is the solution of (1.1), (1.2), (1.3). The following theorem was proven in [E06]: Theorem 1.1 Let L and L0 be two operators of the form (1.1) in domains Ω and Ω0 , respectively, with coefficients A(x, t), V (x, t) and A0 (x, t), V0 (x, t) analytic in t and real-valued. Suppose Γ0 ⊂ ∂Ω ∩ ∂Ω0 and suppose that L and Γ0 satisfy the BLR-condition when t = T∗∗ . Suppose that D-to-N operators Λ and Λ0 , corresponding to L and L0 , respectively, are equal on Γ0 × (0, T0 ) for all smooth f with supports on Γ0 × (0, T0 ]. Let T0 > 2T∗ + T∗∗ . Then there exists a diffeomorphism ϕ of Ω onto Ω0 , ϕ = I on Γ0 , and there exists a gauge transformation c0 (x, t) ∈ G0 (Ω × [0, T0 ]) such that c0 ◦ ϕ−1 ◦ L0 = L on Ω × (0, T0 ). Denote by L∗ the formally adjoint operator to L. Note that L∗ has the form (1.1) with Aj (x, t), 0 ≤ j ≤ n, V (x, t) replaced by Aj (x, t), 0 ≤ j ≤ n, V (x, t). To prove Theorem 1.1 we need to know also the D-to-N operator Λ∗ corresponding to L∗ . If L∗ = L, then obviously Λ∗ = Λ. In the case when A0 = 0 and potentials Aj (x), 1 ≤ j ≤ n, V (x) are independent of t, one can show that Λ determines Λ∗ on Γ0 × (0, T0 ) (c.f. [KL00] and Section 2 below) even when Aj (x), 1 ≤ j ≤ n, V (x) are complex-valued. Therefore Theorem 1.1 holds in this case and gives a new proof of the corresponding result in [KL00], [KL2 97]. When A0 = 0, Aj (x), 1 ≤ j ≤ n, V (x) are realvalued and independent of t, i.e., in the selfadjoint case, the BLR-condition is not needed. In this case Theorem 1.1 is true with T∗∗ = 0. This result was

110

Gregory Eskin

first obtained by the BC (Boundary Control) method (see [B97]) (see also [B1 02], [B2 97], [KKL01]. [K93], [KK98], [KL1 02]). In [E1 06] we gave a new proof for the time-independent selfadjoint case. The proof in [E06] is based on the new approach in [E1 06]. The inverse problem for wave equations with time-dependent potentials in the case when Γ0 = ∂Ω was considered in [St89], [RS91] (see also [I98]). A crucial step of the proof of Theorem 1.1 uses the unique continuation theorem by Tataru [T95]. This theorem requires that Aj (x, t), 0 ≤ j ≤ n, V (x, t) depend analytically on t. The proof of Theorem 1.1 consists of two steps: the local step and the global step. In the local step we recover the coefficients of L (up to a diffeomorphism and a gauge transformation) in the domain Γδ × [0, T0 ] where Γ is an open connected subset of Γ0 and Γδ in a small neighborhood of Γ in Ω. The main novelty of the proof here is the study of the restrictions of the solutions of Lu = 0 to the characteristic surfaces, instead of the restrictions to the hyperplanes t = constant as in the BC-method. The main part of the global step is the following lemma that reduced the inverse problem in the domain to the inverse problem in a smaller domain (c.f. [KKL1 04]): Lemma 1.1 Let L(p) , p = 1, 2 be two operators of the form (1.1) in domains Ωp , p = 1, 2, respectively, satisfying the initial-boundary conditions (1.2), (1.3). We assume that Γ0 ⊂ ∂Ω1 ∩∂Ω2 , supp f ⊂ Γ0 ×(0, T0 ] and Λ1 = Λ2 on Γ0 ×(0, T0 ) where Λp are the D-to-N operators corresponding to L(p) , p = 1, 2. Let B ⊂ Ω1 ∩ Ω2 be such that the domains Ωp \ B are smooth, L(1) = L(2) def

in B and S1 = ∂B ∩ ∂Ωp ⊂ Γ0 , p = 1, 2. Let δ = maxx∈B d(x, Γ0 ) where d(x, Γ0 ) is the distance in B from x ∈ B to Γ0 . Denote by Λˆp the D-to-N operators corresponding to L(p) in domains (Ωp \ B) × (δ, T0 − δ), p = 1, 2. Let S2 = ∂B \ S1 and let Γ1 = (Γ0 \ S1 )∪S2 . Then Λˆ1 = Λˆ2 on Γ1 × (δ, T0 − δ).

2 Hyperbolic systems with Yang–Mills potentials and domains with obstacles Consider in Ω × (0, T0 ) a system of the form (c.f. [E2 05])  2 ∂ −i Im + A0 (x, t) u(x, t) ∂t   n
∂ 1 −i − Im + Aj (x, t) ∂xj g(x) def

Lu =

g(x)g jk (x)

j,k=1

  ∂ × −i Im + Ak (x, t) u − V (x, t)u = 0, ∂xk

(2.1)

Inverse problems for hyperbolic equations

111

where u(x, t), Aj (x, t), 0 ≤ j ≤ n, V (x, t) are m× m matrices, Im is the identity m×m matrix. Assume that the initial-boundary conditions (1.2), (1.3) are satisfied. Let Γ0 ⊂ ∂Ω and let G0 (Ω×[0, T ]) be the gauge group of nonsingular C ∞ m×m matrices C(x, t) in Ω ×[0, T0 ] analytic in t and such that C(x, t) = Im on Γ0 ×[0, T0 ]. Matrices A(x, t) = (A0 (x, t), . . . , An (x, t)), V (x, t) are called Yang–Mills potentials. We say that (A(x, t), V (x, t)) and (A
(x, t), V
(x, t)) are gauge equivalent if there exists C(x, t) ∈ G0 (Ω0 × [0, T0 ]) such that A
0 (x, t) = C −1 (x, t)A0 (x, t)C(x, t) − iC −1 (x, t) A
j (x, t) = C −1 Aj (x, t)C − iC −1

∂C(x, t) , ∂t

∂C , 1 ≤ j ≤ n, ∂xj

V
(x, t) = C −1 V (x, t)C.

(2.2)

When we consider selfadjoint operators of the form (2.1), i.e., when matrices Aj (x, t), 0 ≤ j ≤ n, V (x, t) are selfadjoint, the group G0 (Ω × [0, T0 ]) consists of unitary matrices C(x, t). A generalization of the proof of Theorem 1.1 leads to the following result (c.f. [E2 05]): Theorem 2.1 Theorem 1.1 holds for equations of the form (2.1) with Yang– Mills potentials. Consider now the system of the form (2.1) when the Yang–Mills potentials are independent of t but not necessarily selfadjoint matrices:  2 ∂ Lu = −i Im + A0 (x) u(x, t) ∂t   n
1 ∂ − Im + Aj (x) −i ∂xj g(x) def

g(x)g jk (x)

j,k=1

  ∂ × −i Im + Ak (x) u(x, t) − V (x)u(x, t) = 0. ∂xk

(2.3)

We also assume that T0 = +∞, i.e., (2.3) and the boundary condition (1.3) hold for t ∈ (0, +∞). Let L∗ be formally adjoint to L, i.e., when Aj (x), 0 ≤ j ≤ n, V (x) are replaced by the adjoint matrices A∗j (x), 0 ≤ j ≤ n, V ∗ (x). Consider the initial-boundary value problem adjoint to (2.3), (1.2), (1.3) on some interval (0, T ): L∗ v = 0 v|t=T =

on Ω × (0, T ),

∂v  = 0,  ∂t t=T

v|∂Ω×(0,T ) = g,

(2.4) (2.5)

112

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where supp g ⊂ Γ0 × [0, T ). Let Λ∗ be the D-to-N operator corresponding to (2.4), (2.5). We have 0 = (Lu, v) − (u, L∗ v) = (Λf, g) − (f, Λ∗ g) for any smooth f and g, supp f ⊂ Γ0 ×(0, T ], supp g ⊂ Γ0 ×[0, T0). Therefore Λ∗ is an adjoint operator to Λ and we can determine Λ∗ on Γ0 × [0, T ) if we know Λ on Γ0 × (0, T ). Change, in (2.4), (2.5), t to T − t. Then we get an initial-boundary value problem L∗1 w = 0

on Ω × (0, T ),

w(x, 0) = wt (x, 0) = 0,

(2.6)

w|∂Ω×(0,T ) = g1 (x, t),

(2.7)

where w(x, t) = v(x, T − t), g1 (x, t) = g(x, T − t), 0 < t < T, L∗1 is obtained from L∗ by changing A∗0 (x) to −A∗0 (x). It is clear that the D-to-N operator Λ1∗ on Γ0 × (0, T ) corresponding to (2.6), (2.7) is determined by Λ∗ . Consider also the initial-boundary value problem L∗ u = 0

on Ω × (0, T ),

u(x, 0) = ut (x, 0) = 0,

(2.8)

u|∂Ω×(0,T ) = f (x, t).

(2.9)

Denote by Λ∗ the D-to-N operator corresponding to (2.8), (2.9). Here T > 0 is arbitrary, i.e., (2.6), (2.7) and (2.8), (2.9) hold on (0, +∞). We assume that f (x, t) and g1 (x, t) belong to C0∞ (Γ × (0, +∞)). Performing the Fourier– Laplace transform in t in (2.6), (2.7) and in (2.8), (2.9) when T = +∞ we get: L∗ (k)˜ u(x, k) = 0,

x ∈ Ω,

u ˜(x, k)|∂Ω = f˜(x, k),

(2.10) (2.11)

and L∗ (−k)w(x, ˜ k) = 0,

x ∈ Ω,

w(x, ˜ k)|∂Ω = g˜1 (x, k),

(2.12) (2.13)

where u ˜(x, k), w(x, ˜ k) are analytic in k for k < −C0 for some C0 > 0, L∗ (k) ∂ is obtained from L∗ by replacing −i ∂t by k. Let Λ∗ (k) be the D-to-N operator on Γ corresponding to the boundary value problem (2.10), (2.11), depending on parameter k. Note that Λ∗ (k) is the Fourier–Laplace transform in t of the D-to-N operator Λ∗ corresponding to (2.8), (2.9) on (0, +∞). Since Ω is a bounded domain, Λ∗ (k) has an analytic continuation from k < −C0 to C \ Z where Z is a discrete set. Note that the Fourier–Laplace transform of Λ1∗ is Λ∗ (−k). Since Λ∗ (k) is analytic in C \ Z, Λ∗ (−k) determines Λ∗ (k). Therefore when T0 = +∞ we get that the D-to-N operator Λ on Γ0 ×(0, +∞) determines the D-to-N operator Λ∗ on Γ0 × (0, +∞). Therefore the proof of Theorem 2.1 applies and we have the following result (c.f. [KL 00], [KL2 97]):

Inverse problems for hyperbolic equations

113

Theorem 2.2 Let Lp be two operators of the form (2.3) in domains Ωp × (0, +∞), p = 1, 2. Suppose Γ0 ⊂ ∂Ω1 ∩ ∂Ω2 and Λ1 = Λ2 on Γ0 × (0, +∞) where Λp are the D-to-N operators corresponding to Lp , p = 1, 2. Suppose that L1 and Γ0 satisfy the BLR-condition for some t = T∗∗ . Then there exists a diffeomorphism y = ϕ(x) of Ω1 onto Ω2 and a gauge transformation c0 (x) ∈ G0 (Ω1 ) such that c0 ◦ ϕ−1 ◦ L2 = L1 , x ∈ Ω. We do not assume here that Lp , p = 1, 2, are formally selfadjoint. Note that domains Ω can be multi-connected and Γ0 ⊂ ∂Ω can be not connected. An important example of inverse problems with the boundary data prescribed on a part of the boundary are the inverse problems in domains with obstacles. In this case Ω = Ω0 \ (∪rj=1 Ωj ), where Ω1 , . . . , Ωr are nonintersecting domains inside Ω0 , called obstacles, Γ0 = ∂Ω0 and the zero Dirichlet boundary conditions are prescribed on ∂Ωj , 1 ≤ j ≤ r, i.e., we have Lu = 0 on

Ω × (0, T0 ),

u(x, 0) = ut (x, 0) = 0 u|∂Ω0 ×(0,T0 ) = f (x, t),

(2.14)

on Ω,

(2.15)

u|∂Ωj ×(0,T0 ) = 0, 1 ≤ j ≤ r.

Unfortunately, the BLR-condition is not satisfied for domains with more than one smooth obstacle. Therefore we shall assume that L is a formally selfadjoint operator of the form (2.3), i.e., when Aj (x), 0 ≤ j ≤ n, V (x) are selfadjoint matrices, and initial-boundary conditions (2.15) are satisfied. In this case Theorem 2.1 holds for any T0 > 2T∗ and for any number of obstacles. Finally consider the following particular case: T0 = +∞, g jk (x) = δ jk , A0 (x) = 0, Aj (x), 1 ≤ j ≤ n, V (x) are selfadjoint. Making the Fourier– Laplace transform in (2.14) we get the Schr¨ odinger equation with Yang–Mills potentials in Ω: 2 n 
∂ Im + Aj (x) w(x) + V (x)w(x) − k 2 w(x) = 0, (2.16) −i ∂xj j=1 where we omitted the dependence of w on k in (2.16). When m = 1 we have the Schr¨ odinger equation with electromagnetic potentials. The boundary conditions for (2.16) have the form w|∂Ω0 = h(x),

w|∂Ωj = 0,

1 ≤ j ≤ r.

The D-to-N operator for (2.16), (2.17) has the form    ∂w + i(A · ν)w  , Λ(k)h = ∂ν ∂Ω0

(2.17)

(2.18)

where ν is the exterior unit normal vector to ∂Ω0 . Knowing the hyperbolic D-to-N operator for (2.14), (2.15) for the arbitrary T0 > 0 we can find Λ(k)

114

Gregory Eskin

for all k ∈ C \ Z, and vice versa. Since Λ(k) is analytic, knowing Λ(k) on any interval (k0 − ε, k0 + ε) of analyticity determines Λ(k) for all k ∈ C \ Z. Therefore Theorem 2.1 implies that Λ(k) given on (k0 − ε, k0 + ε), k0 > 0, ε > 0, determines the location of all obstacles Ωj , 1 ≤ j ≤ r, since the metric is fixed, and determines potentials Aj (x), 1 ≤ j ≤ n, V (x) in Ω up to a gauge transformation C(x) ∈ G0 (Ω), i.e., C(x) = Im on ∂Ω0 , C(x) is a unitary matrix in Ω. The interest in considering multi-connected domains with obstacles was spurred by the Aharonov–Bohm effect. It was shown by Aharonov and Bohm [AB59] that the presence of distinct gauge equivalence classes of potentials can be detected in an experiment and this phenomenon is called the Aharonov–Bohm effect. As it was shown above the D-to-N Λ(k) on ∂Ω0 given for all k ∈ (k0 − ε, k0 + ε) allows us to detect the gauge equivalent class of Yang–Mills (or electromagnetic) potentials.

3 A geometric optics approach Consider the Schr¨ odinger equation with electromagnetic potentials in the domain Ω = Ω0 \ (∪rj=1 Ωj ) with obstacles, i.e., consider (2.16) when m = 1, with boundary conditions (2.17). Assume that the D-to-N operator Λ(k) on ∂Ω0 (see (2.18) ) is given for all k ∈ C \ Z. Another approach to the inverse problem for (2.16), (2.17) is based on geometric optics constructions and the reduction to the integral geometry (tomography) problems. We say that γ = γ 1 ∪ γ 2 ∪ · · · ∪ γ N is a broken ray with legs γ 1 .γ 2 , . . . , γ N if γ k , 1 ≤ k ≤ N , are geodesics, γ starts at point x0 ∈ ∂Ω0 , γ has N − 1 nontangential points of reflection at the obstacles and γ ends at a point xN ∈ ∂Ω0 . One can construct geometric optics solutions supported in a small neighborhood of γ (c.f. [E3 04], [E2 05]). Consider two Schr¨ odinger equations with electromagnetic potentials A(p) (x), V (p) (x), p = 1, 2, with the Euclidian metric g jk = δ jk in a plane domain with convex obstacles. Let Λp (k) be the corresponding D-to-N operators, p = 1, 2. Using the geometric optics solutions one can prove that if the D-to-N operators are equal on ∂Ω0 , then       (1) (2) exp i A (x) · dx = exp i A (x) · dx , (3.1) γ



 V (1) (x)ds = γ

γ

V (2) (x)ds

(3.2)

γ

for any broken ray (c.f. [E3 04], [E2 05]). The geometric optics construction and equalities (3.1), (3.2) hold in any dimension n ≥ 2 and for any broken ray even when the broken rays are passing through generic caustics. Having (3.1),

Inverse problems for hyperbolic equations

115

(3.2), we reduce the inverse problem for the Schr¨odinger equation to the inverse problem of the integral geometry of broken rays, i.e., the recovery of potentials from integrals over broken rays. This is a difficult problem. Some results in this direction were obtained in [E3 04] for n = 2 under the geometric restriction that there are no trapped rays. This condition is not satisfied when there exist more than one smooth obstacle. However, there are piecewise smooth convex obstacles that satisfy these conditions. In this case it was shown in [E3 04] that if (3.1), (3.2) hold for all broken rays in Ω0 , then V (1) = V (2) and A(1) and A(2) are gauge equivalent. Despite this approach being much more restrictive than the hyperbolic equations approach, it has an advantage that it allows us to prove the stability results in some cases. It also does not require the BLR-condition in the non-selfadjoint case. Consider the following example: Let Ω1 ⊂ Ω0 be the only convex obstacle in Ω0 and let ' f (x) be a smooth function in Ω0 \ Ω1 . It is well known (c.f. [He80]) that if γ f (x)ds = 0 for all lines γ not intersecting Ω1 , then f (x) = 0. This problem is severely ill posed. ' If one uses the broken rays, i.e., if one computes γ f ds for all broken rays γ, then the inverse problem is well posed and there is a stability estimate. More precisely, let γ x,θ be the broken ray starting on ∂Ω0 and ending at x ∈ Ω0 \ Ω1 . Here ' θ is the direction of the ray at the endpoint x. We assume that w(x, θ) = γ f ds is known when x ∈ ∂Ω0 , ∀θ ∈ S 1 . The following x,θ stability estimate holds (c.f. [E3 04] and [M77] in the case of no obstacles):      l0      ∂w(x(s), θ)   ∂w 2 2 +   |f (x)| dx ≤ C   ∂θ  dsdθ,  ∂s Ω0 \Ω1 0 S1 where x = x(s) is the equation of ∂Ω0 , l0 is the arclength of ∂Ω0 .

References [AB59] [BLR92]

[B97] [B2 97]

[B1 02]

Y. Aharonov and D. Bohm, Significance of electromagnetic potentials in the quantum theory, Phys. Rev. (2) 115(1959), 485–491. C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary, SIAM J. Control Optim. 30(1992), 1024–1065. M. Belishev, Boundary control in reconstruction of manifolds and metrics (the BC method), Inverse Problems 13(1997), R1–R45. M. Belishev, On the uniqueness of the reconstruction of lower-order terms of the wave equation from dynamic boundary data, (Russian), Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 249(1997), 55–76. Translation in J. Math. Sci. 101 (2000), 3408–3421. M. Belishev, How to see waves under the Earth surface (the BC-method for geophysicists), Ill-Posed and Inverse Problems, 67–84, VSP, Zeist, The Nederlands, 2002.

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G. Eskin, Inverse boundary value problems in domains with several obstacles, Inverse Problems 20(2004), 1497–1516. [E2 05] G. Eskin, Inverse problems for Schr¨ odinger equations with Yang–Mills potentials in domains with obstacles and the Aharonov–Bohm effect, Institute of Physics Conference Series 12, 23–32, ArXiv:math. AP/0505554 (2005). [E06] G. Eskin, Inverse hyperbolic problems with time-dependent coefficients, ArXiv:math.AP/0508161 (2006). [E1 06] G. Eskin, A new approach to hyperbolic inverse problems, ArXiv: math. AP/0505452 (2006) Inverse Problems 22(2006), 815–831. [He80] S. Helgason, The Radon transform, Progress in Mathematics, v. 5, Birkh¨ auser, Boston, MA, 1980. [I98] V. Isakov, Inverse problems for partial differential equations, Applied Mathematical Sciences, 127, Springer-Verlag, New York, 1998. [KK98] A. Katchalov and Y. Kurylev, Multidimensional inverse problems with incomplete boundary spectral data, Comm. Partial Differential Equations 23(1998), 55–95. [KKL01] A. Katchalov, Y. Kurylev and M. Lassas, Inverse boundary spectral problems, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 123, Boca Raton, FL, 2001. [KKL1 04] A. Katchalov, Y. Kurylev and M. Lassas, Energy measurements and equivalence of boundary data for inverse problems on noncompact manifolds, Geometric methods in inverse problems and PDE control, 183–213, IMA Vol. Math. Appl., 137, Springer-Verlag, New York, 2004. [K93] Y. Kurylev, Multi-dimensional inverse boundary problems by BCmethod: groups of transformations and uniqueness results, Math. Comput. Modelling 18(1993), 33–45. [KL2 97] Y. Kurylev and M. Lassas, The multidimensional Gel’fand inverse problem for non-selfadjoint operators, Inverse Problems 13(1997), 1495– 1501. [KL00] Y. Kurylev and M. Lassas, Hyperbolic inverse problems with data on a part of the boundary, AMS/1P Stud. Adv. Math. 16 Amer. Math. Soc., Providence, RI, 2000, 259–272. [KL1 02] Y. Kurylev and M. Lassas, Hyperbolic inverse boundary-value problem and time-continuation of the non-stationary Dirichlet-to-Neumann map, Proc. Roy. Soc. Edinburgh Sect. A 132 (2002), 931–949. [M77] R. Mukhometov, The reconstruction problem of two-dimensional Riemannian metric and integral geometry, Sov. Math. Dokl. 18(1977), 27–31. ¨ strand, An inverse problem of the wave equation, [RS91] A. Ramm and J. Sjo Math. Z. 206(1991), 119–130. [St89] P. Stefanov, Uniqueness of multidimensional inverse scattering problem with time-dependent potentials, Math. Z. 201(1989), 541–549. [T95] D. Tataru, Unique continuation for solutions to PDE, Comm. Partial Differential Equations 20(1995), 855–884.

On the optimality of some observability inequalities for plate systems with potentials∗ Xiaoyu Fu1 , Xu Zhang2 and Enrique Zuazua3 1 2

3

School of Mathematics, Sichuan University, Chengdu, China Yangtze Center of Mathematics, Sichuan University, Chengdu, China and Departamento de Matem´ aticas, Facultad de Ciencias, Universidad Aut´ onoma de Madrid, Madrid, Spain Departamento de Matem´ aticas, Facultad de Ciencias, Universidad Aut´ onoma de Madrid, Madrid, Spain

Summary. In this paper, we derive sharp observability inequalities for plate equations with lower order terms. More precisely, for any T > 0 and suitable observation domains (satisfying the geometric conditions that the multiplier method imposes), we prove an estimate with an explicit observability constant for plate systems with an arbitrary finite number of components and in any space dimension with lower order bounded potentials. These inequalities are relevant for control theoretical purposes and also in the context of inverse problems. We also prove the optimality of this estimate for plate systems with bounded potentials in even space dimensions n ≥ 2. This is done by extending a construction due to Meshkov to the bi-Laplacian equation, to build a suitable complex-valued bounded potential q = q(x), with a non-trivial solution u of ∆2 u = qu in lR2 , with the decay property |u(x)| + |∇u(x)| + |∇∆u(x)| ≤ exp(−|x|4/3 ) for all x ∈ lR2 .

2000 Mathematics Subject Classification: Primary: 93B07; Secondary: 93B05, 35B37. Key words: Plate system, Meshkov’s construction, Carleman inequalities, observability constant, optimality, potential.

∗

The work is supported by Grant MTM2005-00714 of the Spanish MEC, the DOMINO Project CIT-370200-2005-10 in the PROFIT program of the MEC (Spain), the SIMUMAT project of the CAM (Spain), the FANEDD of China (Project No: 200119), the EU TMR Project “Smart Systems”, the NCET of China under grant NCET-04-0882, and the NSF of China under grants 10371084 and 10525105.

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Xiaoyu Fu, Xu Zhang and Enrique Zuazua

1 Introduction 1.1 Formulation of the problem Let n ≥ 1 and N ≥ 1 be two integers. Let T > 0 be given, Ω be a bounded domain in lRn with C 4 boundary Γ , and ω be a nonempty open subset of Ω. Put Q = (0, T )×Ω and Σ = (0, T )×Γ . For simplicity, we will use the notation ∂y yj = ∂xj , where xj is the jth coordinate of a generic point x = (x1 , . . . , xn ) in lRn . Throughout this paper, we will use C = C(Ω, ω) and C ∗ = C ∗ (T, Ω, ω) to denote generic positive constants which may vary from line to line. Set Y = {y ∈ H 3 (Ω)| y|Γ = ∆y|Γ = 0}. We consider the following lRN -valued plate system with a potential a ∈ L∞ (0, T ; W 1,p (Ω; lRN ×N )) for some p ∈ [n, ∞]: ⎧ 2 in Q, ⎪ ⎨ ytt + ∆ y + ay = 0 y = ∆y = 0 on Σ, (1.1) ⎪ ⎩ 0 1 in Ω, y(0) = y , yt (0) = y where y = (y1 , . . . , yN ) , and the initial datum (y 0 , y 1 ) is supposed to belong to Y N × (H01 (Ω))N , the state space of the system (1.1). It is easy to show that the system (1.1) admits a unique weak solution y ∈ C([0, T ]; Y N ) ∩ C 1 ([0, T ]; (H01 (Ω))N ). In what follows, we shall denote by | · |, | · |p ,  · p and ||| · |||p the (usual) norms on lRN , W 1,∞ (0, T ; Lp(Ω; lRN ×N )), L∞ (0, T ; Lp (Ω; lRN ×N )) and L∞ (0, T ; W 1,p (Ω; lRN ×N )), respectively. We shall study the observability constant P (T, a) of the system (1.1), defined as the smallest (possibly infinite) constant such that the following observability estimate for system (1.1) holds: ∆y 0 2(H 1 (Ω))N + y 1 2(H 1 (Ω))N 0



T

0



≤ P (T, a)

(|∇y|2 + |∇∆y|2 )dtdx, 0

∀ (y 0 , y 1 ) ∈ Y N × (H01 (Ω))N .

ω

(1.2) This inequality, the observability inequality, allows estimating the total energy of solutions in terms of the energy localized in the observation subdomain ω. It is relevant for control problems. In particular, in the linear setting, it is equivalent to the so-called exact controllability problem, i.e., that of driving solutions to rest by means of control forces localized in ω×(0, T ) (see [8]). This type of inequality, with explicit estimates on the observability constant, is also relevant for the control of semilinear problems (see [11]). Similar inequalities are also useful for solving a variety of inverse problems.

Observability inequality for plate systems with potentials

119

The main purpose of this paper is to analyze the dependence of P (T, a) on the potential a. The main tools to derive this kind of explicit observability estimates are the so-called Carleman inequalities. Here we have chosen to work in the space Y N × (H01 (Ω))N in which Carleman inequalities can be applied more naturally. But some other choices of the state space are possible. For example, one may consider similar problems in state spaces of the form (H01 (Ω))N × (H −1 (Ω))N or (H 2 (Ω) ∩ H01 (Ω))N × (L2 (Ω))N where the plate system is also well posed. But the corresponding analysis, in turn, is technically more involved and will be treated elsewhere ([6]). 1.2 Preliminaries on the heat and wave equations Similar problems have been considered for the heat and wave equations in [3]. Consider the following heat and wave equations (or systems) with potentials: •

•

The heat equation/system with potential: ⎧ ⎪ ⎨ zt − ∆z + az = 0, z = 0, ⎪ ⎩ z(0) = z 0 ,

in Q, on Σ,

(1.3)

in Ω.

The wave equation/system with potential: ⎧ ⎪ ⎨ wtt − ∆w + aw = 0, w = 0, ⎪ ⎩ w(0) = w0 , wt (0) = w1 ,

in Q, on Σ,

(1.4)

in Ω.

In [3] the observability inequalities for these equations were analyzed in the state spaces (L2 (Ω))N and (L2 (Ω))N × (H −1 (Ω))N , respectively. More precisely, by definition, the heat and wave observability constants H(T, a) and W (T, a) are the smallest (possibly infinite) constants such that the following observability estimates hold: •

The heat equation/system with potential:  T 2 |z|2 dtdx, z(T )(L2(Ω))N ≤ H(T, a) 0

•

∀ z 0 ∈ (L2 (Ω))N

(1.5)

ω

for systems of the form (1.3). The wave equation/system with potential: w0 2(L2 (Ω))N + w1 2(H −1 (Ω))N 

T



≤ W (T, a)

|w|2 dtdx, 0

∀ (w0 , w1 ) ∈ (L2 (Ω))N × (H −1 (Ω))N

ω

(1.6) for systems of the form (1.4).

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For systems (1.3), due to the infinite speed of propagation and its parabolic nature, it is shown that for any T > 0 and ω, the observability constant H(T, a) is bounded by (see [3, Theorem 2.1])    1 1 3/2−n/p H(T, a) ≤ exp C 1 + + T ap + ap ; (1.7) T while for system (1.4), due to the finite speed of propagation and its hyperbolic nature, it is shown that for any fixed triple (T, ω, Ω) satisfying suitable geometric assumptions (say, the classical ones arising when applying multiplier methods ([8])), the observability constant W (T, a) is bounded by (see [3, Theorem 2.2 ii)]) 1

W (T, a) ≤ exp[C ∗ (1 + ap3/2−n/p )].

(1.8)

In particular the observability constant in (1.7) includes three different terms, i.e., (1.7) can be rewritten as H(T, a) = H1 (T, a)H2 (T, a)H3 (T, a), where    1 H1 (T, a) = exp C 1 + , T 1 3/2−n/p

H3 (T, a) = exp(Cap

H2 (T, a) = exp(CT ap), (1.9)

).

As explained in [3, 12], the role that each of these constants plays in the observability inequality is of a different nature: H1 (T, a), which blows up exponentially as T ↓ 0, is the observability constant for the special case that a ≡ 0; H2 (T, a) is the constant which arises naturally when applying Gronwall’s inequality to establish the energy estimate for solutions of system (1.3); while H3 (T, a) is the one arising when using global Carleman estimates (see [7], [4]) to derive the observability inequality (1.5) by absorbing the undesired lower order terms. In a similar spirit, by [3, Theorem 2.2 ii)], it is easy to see that one can decompose the right-hand side of (1.8) into two different terms, i.e., (1.8) can be rewritten as W (T, a) ≤ W1 (T, a)W2 (T, a), where ∗

W1 (T, a) = eC ,

1

W2 (T, a) = exp(C ∗ ap3/2−n/p ).

(1.10)

Here, W1 (T, a) is the observability constant in (1.6) for the special case that a ≡ 0 but it is finite only, for suitable subsets ω and for T large enough (unless ω = Ω); and W2 (T, a) is the one arising when using a global Carleman estimate to derive the observability inequality (1.6) by absorbing the undesired

Observability inequality for plate systems with potentials

121

lower order terms. We emphasize again that for this purpose one needs some geometric assumptions on the triple (T, Ω, ω). In this case the sharp observability constant does not involve the term related with the Gronwall estimate for evolution of energy in time, since it may be bounded above in terms of W1 (T, a) and W2 (T, a). Based on the construction by Meshkov ([10]), it is shown in [3, Theorems 1.1 and 1.2] that both H3 (T, a) and W2 (T, a) are optimal for systems (N ≥ 2) with bounded potentials (i.e., p = ∞), in even dimensions n ≥ 2 for certain ranges of the observability time T . In [3] an extension of Meshkov’s construction is also given for odd dimensions, showing that the above estimates are almost sharp in that case. 1.3 The sharp observability constant for plate systems Plate systems can be viewed as intermediate ones between the heat and the wave systems. Indeed, on the one hand, system (1.1) is time-reversible, which is, typically, a hyperbolic property; on the other hand, similar to the heat system, the solutions of system (1.1) propagate with infinite velocity. As we shall see, under suitable geometric conditions on the pair (Ω, ω), P (T, a) is finite with the following decomposition: P (T, a) = P1 (T, a)P2 (T, a)P3 (T, a).

(1.11)

Here P1 (T, a) is the observability constant in (1.2) for the special case that a ≡ 0; P2 (T, a) is the constant which arises when applying Gronwall’s inequality to establish an energy estimate for solutions of system (1.1); while P3 (T, a) is the one arising when using global Carleman estimates to derive the observability inequality (1.2) by absorbing the undesired lower order terms. More precisely, the first main purpose of this paper is to show that    1 1 P1 (T, a) = exp C 1 + , P2 (T, a) = exp(CT |||a|||p2−n/2p ), T (1.12) 1

P3 (T, a) = exp(Cap3−5n/2p ). Note that the power of |||a|||p in P2 (T, a) is always less than 1 for p ∈ [n, ∞]. This may be achieved as a consequence of a modified energy estimate, because the plate system is second order in time. But this term may not be absorbed by P3 (T, a) when p > 2n. In this sense the estimate we get is closer to that on the heat equation H(T, a) since the observability constant contains three different terms. Another important analogy of this estimate with the heat equation is that the observability inequality holds for all T > 0. Note however that, for plate systems, the subdomain ω needs to fulfill suitable geometric conditions for the observability to hold.

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Xiaoyu Fu, Xu Zhang and Enrique Zuazua

There are two other important differences between (1.12) and (1.9)–(1.10), i.e., 1) The power of ap in P3 (T, a) is 1/(3 − 5n/2p), but the ones in H3 (T, a) and W2 (T, a) are 1/(3/2 − n/p). This is due to the different scaling of the various terms arising in the Carleman inequality for plate systems. The main reason for that is that the plate system is fourth order in space, while the heat and wave systems are of order 2. 2) There is only one norm ap in (1.9)–(1.10) but one needs to use two different norms, ap and |||a|||p , in (1.12). The reason is as follows. For the well-posedness of the heat and wave systems (1.3) and (1.4) in (L2 (Ω))N and (H01 (Ω))N × (L2 (Ω))N respectively, it is sufficient to assume the potential a to belong to L∞ (0, T ; Lp (Ω; lRN ×N )). But, for system (1.1), one needs a to be more regular, i.e., to be in L∞ (0, T ; W 1,p(Ω; lRN ×N )) for establishing its well-posedness in Y N × (H01 (Ω))N , as we shall see below. This extra regularity assumption on the potential can be replaced by a ∈ W 1,∞ (0, T ; Lp(Ω; lRN ×N )). But one always needs to assume that one of the derivatives of the potential a (either in space or time) belong to L∞ (0, T ; Lp (Ω; lRN ×N )). Of course, in (1.11), one may replace the right-hand side of P3 (T, a) in 1

(1.12) by exp(C|||a|||p3−5n/2p ), through which one ends up with only one norm |||a|||p in (1.12). But, the observability estimate that one obtains that way fails to be optimal. Actually, the second main purpose of this paper is to show the optimality of P3 (T, a). The rest of this paper is organized as follows. In Section 2 we recall some preliminary results concerning energy and boundary trace estimates for plate systems, and weighted pointwise estimates for the Schr¨ odinger equation. In Section 3 we state the sharp observability estimate for the plate system. In Section 4 we recall the construction by Meshkov [10], indicating its consequences for the bi-harmonic operator. In Section 5 we prove the optimality of the observability estimate. We conclude in Section 6 indicating some closely related issues and open problems. We refer to [6] for the details of the proofs of the results in this paper and other results in this context.

2 Preliminaries In this section, we recall some preliminary results. 2.1 Energy estimates for plate systems Denote the energy of the system (1.1) by E(t) =

1 [|yt (t, ·)|2(H 1 (Ω))N + |∆y(t, ·)|2(H 1 (Ω))N ], 0 0 2

(2.1)

Observability inequality for plate systems with potentials

123

for the solutions y of the system (1.1). Consider also a modified energy of (1.1): E(t) =

4 1 [|yt (t, ·)|2(H 1 (Ω))N + |∆y(t, ·)|2(H 1 (Ω))N + |||a|||p4−n/p |y(t, ·)|2(H 1 (Ω))N ]. 0 0 0 2

It is clear that both energies are equivalent 4

E(t) ≤ E(t) ≤ C(1 + |||a|||p4−n/p )E(t).

(2.2)

The following estimates hold for the modified energy: Lemma 2.1 i) Let a ∈ L∞ (0, T ; W 1,p(Ω; lRN ×N )) for some p ∈ [n, ∞]. Then 2

E(t) ≤ Ce

CT |||a|||p4−n/p

E(s),

∀ t, s ∈ [0, T ].

(2.3)

ii) Let a ∈ W 1,∞ (0, T ; Lp (Ω; lRN ×N )) for some p ∈ [n, ∞]. Then 2 4−n/p

E(t) ≤ CeCT |a|p

E(s),

∀ t, s ∈ [0, T ].

(2.4)

Clearly, 2/(4 − n/p) < 1 for any p ∈ [n, ∞]. Therefore, the modified estimate in (2.3) is finer than the usual energy estimate (which gives CT |||a|||p E(t) ≤ Ce E(s) for all t, s ∈ [0, T ]). However, the optimality of the estimates above is still to be investigated. The problem of the well- and illposedness of wave equations with low regularity coefficients in the principal part has been intensively investigated (see, for instance, [2] and [1]). Obtaining examples of equations with constant coefficients in the principal part, and low regularity zero order potentials for which the energy estimates of the above form are shown to be optimal, seems to be open both in the context of wave and plate models. Unlike in the state spaces of the form (H01 (Ω))N ×(H −1 (Ω))N or (H 2 (Ω)∩ 1 H0 (Ω))N × (L2 (Ω))N , to derive energy estimates in Y N × (H01 (Ω))N further regularity assumptions on the potential a are needed. This is due to the fact that, for deriving energy estimates in this space, 'one needs to multiply the equation by ∆yt . In this way we get the term Ω ay∆yt that can not be estimated directly using the terms entering in the energy since the latter only involves the norm of yt in H 1 (Ω) and not in H 2 (Ω). Thus, we have to integrate by parts:   ay∆yt dx = − Ω

∇(ay) · ∇yt dx. Ω

Once this is done the integral can be estimated in terms of the energy but at the price of using an Lp estimate on ∇a. A similar argument can be done, after integration in time, to get energy estimates under an Lp assumption on the time derivative at of the potential.

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Xiaoyu Fu, Xu Zhang and Enrique Zuazua

2.2 Boundary trace estimates for plate systems For proving the optimality of the observability estimates which we shall derive in Section 3, one needs to solve non-homogeneous boundary-value problems for plate systems. This, by transposition, requires a fine analysis of the boundary traces of solutions of the homogeneous system. In particular, in the class of solutions under consideration, one needs sharp estimates on the traces of the normal derivatives ∂∆y/∂ν, ∂yt /∂ν and ∂y/∂ν (Here and henceforth, ν ≡ ν(x) denotes the unit outward normal vector of Ω at x ∈ Γ ), which are the complementary boundary conditions for our problem. This is done typically using multiplier techniques as in [8]. The estimates obtained in this way are often referred to as “hidden regularity results”. The following holds: Lemma 2.2 Assume that Γ0 is an open subset of Γ , and ω is an open subset of Ω, intersection of Ω with a neighborhood of Γ0 . Given T > 0, 0 ≤ s1 < s0 < s
0 < s
1 ≤ T and a ∈ L∞ (0, T ; Lp(Ω; lRN ×N )) for some p ∈ [n, ∞]. Then 

s
0

s0

 |∇yt |2 dtdx ≤ Ω

and 

s
0

CT 2 (1 + T 2 )(1 + a2p ) (s0 − s1 )2 (s
1 − s
0 )2



s
1

s1

 |∇∆y|2 dtdx, Ω

        ∂y 2  ∂yt 2  ∂∆y 2   +  +   ∂ν   ∂ν   ∂ν  dtdx Γ0



s0

≤

CT 4 (1 + T 2 )(1 + a3p ) (s0 − s1 )3 (s
1 − s
0 )3

 (|∇∆y|2 + |∇y|2 )dtdx. ω

The first conclusion in the above lemma follows from the usual energy method; while the second one can be proved, as we mentioned above, by using multiplier techniques similar to those in [8, Chapter IV]. More precisely, to show the second conclusion, by multiplying the equation (or system) by (t − s1 )(s
1 − t)η · ∇∆y, where η is a smooth extension of the normal vector field to the interior of Ω, we obtain an estimate for ∂∆y/∂ν and ∂yt /∂ν. In order to get the estimate for ∂y/∂ν we observe that y can be viewed as a solution of a Schr¨ odinger equation of the form iyt + ∆y = z, satisfying Dirichlet boundary conditions. Using the multiplier η · ∇y as above (see [9]), one gets an estimate on ∂y/∂ν in L2 (Σ) in terms of the L2 (Q)-norm of z and the L2 (0, T ; H 1(Ω))-norm of y. Obviously, both quantities are bounded above in terms of the energy. 2.3 Pointwise weighted estimates for the Schr¨ odinger operator In this section, we present some pointwise weighted estimates for the Schr¨ odinger equation that will play a key role when deriving the sharp observability estimate for the plate system. In fact, the estimates for the plate system

Observability inequality for plate systems with potentials

125

will be obtained applying, in an iterative manner, these pointwise estimates for the Schr¨ odinger equation. The underlying fact is the possibility of decomposing the plate operator ∂t2 + ∆2 as two conjugate Schr¨ odinger operators: ∂t2 + ∆2 = (i∂t + ∆)(−i∂t + ∆). First, we show a pointwise weighted estimate for the Schr¨ odinger operator “i∂t + ∆”. For this, for any λ > 0, x0 ∈ lRn and c ∈ lR, set   2  λ T (t, x) = |x − x0 |2 − c t − . (2.5) 2 2 By taking a = 0, b = 1 and (ajk )n×n = I the identity matrix, and θ = e with  given by (2.5) in [5, Theorem 1.1], and using H¨ older’s inequality, one gets the following result. Lemma 2.3 Let z ∈ C 2 (lR1+n ; C), l θ = e and v = θz. Then 4λ|∇v|2 + B|v|2 ≤ θ2 |izt + ∆z|2 + Mt +

n


Vj ,

(2.6)

j=1

where ⎧ n
⎪ 2 ⎪ M =  |v| + i j (v j v − vj v), ⎪ t ⎪ ⎪ ⎪ j=1 ⎪ ⎪ ⎪ n ⎪ ⎪
⎨V = {−ij (v t v − vvt ) − it (vj v − v j v) + ∆(vj v + v j v) j k=1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ +2j (vj v k + v j vk − |∇v|2 ) + (2j |∇|2 − ∆j )|v|2 }j , ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 3 B = 4λ |x − x0 |2 − λc.

(2.7)

Noting the obvious fact that |izt + ∆z| = | − iz t + ∆z|, Lemma 2.3 also gives a pointwise estimate for the conjugate Schr¨odinger operator “−i∂t + ∆”. Note also that Lemma 2.3 simplifies a similar pointwise estimate in [11].

3 The sharp observability estimate In this section we state the sharp observability estimate for system (1.1). For this purpose, for any fixed x0 ∈ lRn and δ > 0, we introduce the following set: ⎧ 1 ⎨ ω = Oδ (Γ0 ) Ω, (3.1) ⎩ Γ0 = {x ∈ Γ | (x − x0 ) · ν(x) > 0}, where Oδ (Γ0 ) = {x ∈ lRn | |x − x
| < δ for some x
∈ Γ0 }. One of the main results in this paper is the following observability inequality with explicit dependence of the observability constant on the potential a for system (1.1):

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Xiaoyu Fu, Xu Zhang and Enrique Zuazua

Theorem 3.1 Let ω be given by (3.1) for some x0 ∈ lRn , δ > 0, and p ∈ [n, ∞]. Then there is a constant C > 0 such that for any T > 0 and any a ∈ L∞ (0, T ; W 1,p (Ω; lRN ×N )), the weak solution y of system (1.1) satisfies estimate (1.2) with the observability constant P (T, a) > 0 verifying    1 1 1 2−n/2p 3−5n/2p P (T, a) ≤ exp C 1 + + T |||a|||p + ap . (3.2) T If the potential a ≡ a(x) ∈ Lp (Ω; lRN ×N ) is assumed to be time-independent, then (3.2) can be improved to    1 1 1 P (T, a) ≤ exp C 1 + + T ap2−n/2p + ap3−5n/2p . (3.3) T Inequality (3.2) provides the estimates (1.11) and (1.12) we discussed in the introduction. Note that we have used two norms on the potential a in (3.2). But, as we mentioned above, the use of the norm in L∞ (0, T ; W 1,p (Ω; lRN ×N )) is only due to the need for performing energy estimates. In the special case of time-independent potentials, we use only one norm (see (3.3)), because, as we mentioned above, the additional regularity assumption that a belongs to L∞ (0, T ; W 1,p (Ω; lRN ×N )) can be replaced by the fact that it belongs to W 1,∞ (0, T ; Lp (Ω; lRN ×N )), a fact that automatically holds when a = a(x) belongs to Lp (Ω; lRN ×N ). As in [11], to prove Theorem 3.1, one needs to decompose the plate equation into two Schr¨ odinger systems and to apply the pointwise estimate for the later one in cascade. The main point in the proof of Theorem 3.1 is as follows: For simplicity, we assume that x0 ∈ lRn \ Ω. Hence,



R1 = max |x − x0 | > R0 = min |x − x0 | > 0. x∈Ω

x∈Ω

First, set z = −iyt + ∆y, and note that −ay = ytt + ∆2 y = izt + ∆z. We see that y and z solve ⎧ ⎧ −iyt + ∆y = z in Q, ⎪ in Q, ⎪ ⎪ ⎨ ⎨ izt + ∆z = −ay y=0 on Σ, z=0 on Σ, ⎪ ⎪ ⎪ ⎩ 0 ⎩ y(0) = y 0 in Ω, z(0) = −iy1 + ∆y in Ω. Next, choose the constant c in  defined by (2.5) of subsection 2.3 such that R12 − cT 2 /4 < 0. This gives the desired weight function θ = e . One may find T1 and T1
satisfying 0 < T1 < T1
< T , independent of λ, so that

 2 (t, x) < 0, ∀ (t, x) ∈ [0, T1 ] [T1
, T ] × Ω. (3.4)

Observability inequality for plate systems with potentials

127

We now apply Lemma 2.3 to z. Integrating (2.6) in (T1 , T1
) × Ω, noting (3.4), recalling the definition of E(t) in (2.1), one may deduce that 

T1




 θ |∇z| + λ 2

λ T1

3

2

Ω

T1




T1

θ2 |z|2 dtdx Ω

≤C

θ(izt +

∆z)2L2 (Q)

+ λ[E(T1 ) +

E(T1
)]

 + λe

Cλ

 2  ∂z    dtdx   Γ0 ∂ν

T1


T1

≤ C θay2L2 (Q) + λ[E(T1 ) + E(T1
)]  + λeCλ



T1


T1

Γ0



     ∂yt 2  ∂∆y 2      ∂ν  +  ∂ν  dtdx ,

(3.5)

with Γ0 being given in (3.1). Obviously, for the above to be true we need to take λ > 0 large enough so that the constant B in Lemma 2.3 is positive. Similarly, applying Lemma 2.3 to y, we deduce that 

T1




 θ |∇y| + λ 2

λ T1

2

Ω

3

T1


T1



 θ2 |y|2 dtdx Ω

≤ C θ(−iyt + ∆y)2L2 (Q) + λ[E(T1 ) + E(T1
)] 

T1


+ λeCλ

T1

 2  ∂y    dtdx   Γ0 ∂ν

(3.6)



θz2L2 (Q)

=C

+ λ[E(T1 ) +

E(T1
)]

 + λe

T1


Cλ

 2  ∂y    dtdx .   Γ0 ∂ν



T1

Combining (3.5) and (3.6), we arrive at    2 4 2 6 2 2 2 λ θ (|∇yt | + |∇∆y| )dtdx + λ θ |∇y| dtdx + λ θ2 |y|2 dtdx Q

Q

Q

≤ C θay2L2 (Q) + λ [E(T1 ) + E(T1
)] + λ 4



6



T1

E(t)dt + 0

 + λe

Cλ

T1


T1

 

      ∂y 2  ∂yt 2  ∂∆y 2   +     ∂ν   ∂ν  +  ∂ν  dtdx . Γ0



T

E(t)dt T1




(3.7)

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At this level it is useful to observe that, since, without loss of generality, one may assume y to be a real function, an estimate on z as in (3.5) yields an estimate on both yt and ∆y simultaneously. We have now to get rid of the terms on the right-hand side of (3.7): •

We consider first the term θay2L2 (Q) . By the proof of [3, Theorem 2.2], for any ! > 0, we have θay2L2 (Q) ≤ !λθy2L2 (0,T ; H 1 (Ω)) + !−n/(p−n) a2p/(n−p) λ−n/(p−n) θy2L2 (Q) . p 0 (3.8)

•

By taking ! small enough the first term !λθy2L2 (0,T ; H 1 (Ω)) can be ab0 sorbed by the left-hand side of (3.7). Then, for this choice of ! and taking 2p/(n−p) −n/(p−n) λ sufficiently large, the term !−n/(p−n) ap λ θy2L2 (Q) can be absorbed similarly. Concerning the boundary integrals we proceed as follows. Noting the definition of ω in (3.1), using Lemma 2.2, one has 2  2   T1
  2     ∂y      +  ∂yt  +  ∂∆y  dtdx  ∂ν   ∂ν   ∂ν  T1 Γ0 (3.9)  T (|∇y|2 + |∇∆y|2 )dtdx. ≤ C(1 + a3p ) 0

•

ω

The energy terms in the right-hand side can be absorbed by the terms on the left side. This can be done, once more, by taking λ large enough and exploiting the exponential growth of the weight function θ (near t = T /2) on λ. Note however that in this argument one has to use carefully the energy estimates, which grow exponentially with a suitable power of the norm of the potential.

4 Extension of Meshkov’s construction to the bi-Laplacian equation In this section, we construct a very special time-independent complex-valued solution u of the following bi-Laplacian equation: ∆2 u = qu,

in lRn ,

(4.1)

which decays at infinity sufficiently fast, for some bounded complex-valued potential q = 0. For x in lRn , we shall write r = |x|. The main result of this section is stated as follows:

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Theorem 4.1 Let n ≥ 2 be even and c∗ > 0. Then there exist two non-trivial complex-valued functions: 1 u ∈ C ∞ (lRn ; C), l q ∈ C ∞ (lRn ; C) l L∞ (lRn ; C) l such that (4.1) is satisfied, and for some constant C: |u(x)| + |∇u(x)| + |∇∆u(x)| ≤ Ce−c∗ r

4/3

,

∀ x ∈ lRn .

(4.2)

The general case of any even dimension can easily be derived by separation of variables. The construction needed for proving Theorem 4.1 is in fact the same as in Meshkov’s paper and [3]. There, a complex-valued smooth function decaying as in (4.2) is built such that |∆u| ≤ C|u| for some finite C > 0. The proof of the present theorem is based on the observation that, in fact, |∆2 u| ≤ C
|u| as well for some other finite C
> 0. The construction above is sharp. More precisely, one can not build nontrivial solutions u of equations (scalar or systems) of the form ∆2 u = q(x)u with q = q(x) bounded for which u decays at infinity faster than exp(−|x|4/3 ). This can be proved as in [10, Theorem 1]. The proof there uses the following Carleman inequality: For some constant τ 0 > 0, and C > 0,   τ3 |v|2 r1−n exp (2τ r4/3 )dx ≤ C |∆v|2 r1−n exp(2τ r4/3 )dx, lRn

lRn

∀ τ ≥ τ 0, Applying this inequality twice we get   |v|2 r1−n exp (2τ r4/3 )dx ≤ C τ6 lRn

lRn

v ∈ C0∞ ({r > 1}).

|∆2 v|2 r1−n exp(2τ r4/3 )dx,

∀ τ ≥ τ 0,

v ∈ C0∞ ({r > 1}).

Starting from this inequality the argument in the proof of [10, Theorem 1] shows the optimality of the decay rate (4.2) for the bi-harmonic operator, too. It is worth noticing that, despite the different order of the bi-harmonic equation considered here, which is of order 4, the sharp superexponential decay is the same as that for the Laplacian.

5 Optimality of the observability constant for plate systems 1

This section is devoted to showing that when p = ∞, the term ap3−5n/2p 1/3 (i.e., ap ) in the estimate (3.3) is sharp in what concerns the dependence on the potential a in even space dimensions n ≥ 2 for systems with at least two equations. More precisely, the following holds:

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Theorem 5.1 Assume that n ≥ 2 is even and that N ≥ 2. Let ω be any given open non-empty subset of Ω such that Ω \ ω = ∅. Then, there exist two constants c > 0 and µ > 0, a family of time-independent potentials {aR }R>0 ⊂ L∞ (Ω; lRN ×N ) satisfying aR ∞ → ∞,

as R → ∞

0 1 , yR )}R>0 ∈ Y N × (H01 (Ω))N such that the and a family of initial data {(yR corresponding weak solutions {yR }R>0 of (1.1) satisfy

0 2 1 2 ∆yR (H 1 (Ω))N + yR (H 1 (Ω))N 0 0 lim inf = ∞, (5.1) ' ' R→∞ T ∈Iµ exp(ca 1/3 ) T 2 + |∇∆y|2 )dtdx (|∇y| R ∞ 0 ω

−1/6

where Iµ = (0, µaR ∞

].

The main idea in order to prove Theorem 5.1 is the same as that in [3]. Based on the construction of u and q in Theorem 4.1, by suitable scaling and localization arguments, one can find a family of rescaled potentials aR (x) = R4 q(Rx) with an L∞ -norm of the order of R4 and a family of solutions uR (x) = u(Rx) of the corresponding bi-harmonic problem, with a decay of the order of |uR (x)| ≤ Cexp(−R4/3 |x|4/3 ). Without loss of generality we may assume that both, the observation subdomain ω and the exterior boundary Γ , are included in the region |x| ≥ 1. This yields a sequence of solutions of the elliptic systems ∆2 uR = aR uR in which the ratio between total energy and the energy concentrated in ω and the norm of the boundary traces is of the order of exp(−R4/3 ). Taking into account that 1/3 aR ∞ ∼ R4 , this ratio turns to be of the order of exp(−aR ∞ ). These solutions of the above-mentioned elliptic system can be regarded also as solutions of the plate system for suitable initial data. However, they do not fully satisfy the requirements in our optimality theorem because they do not fulfill homogeneous boundary conditions. This can be compensated by subtracting the solution taking their boundary data and zero initial ones. These solutions turn out to be exponentially small in the energy space Y N × (H01 (Ω))N during −1/6 a time interval of the order of T ≤ µaR ∞ . This can be shown to hold because of the estimates in Lemma 2.2 and standard energy and transposition arguments. Note that Theorem 5.1 looks more like the situation described in [3, Theorem 1.1] for the heat system rather than that in [3, Theorem 1.2] for the wave equation. Indeed, as for the optimality for the heat observability constant H(T, a), one has to take T to be small enough to compensate the time evolution of the energy and make sure that the concentration of the solution of the evolution plate system suffices for guaranteeing that (5.1) holds. This is however not necessary for the wave system. It is an unsolved problem

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whether one can prove Theorem 5.1 for T > 0 fixed and replacing (5.1) by the following 0 2 1 2 ∆yR (H 1 (Ω))N + yR (H 1 (Ω))N 0 0 = ∞. lim ' ' R→∞ exp(ca 1/3 ) T 2 + |∇∆y|2 )dtdx (|∇y| R ∞ 0 ω

(5.2)

6 Further remarks and open problems In this paper we have indicated some open problems and closely related issues that remain to be clarified. We summarize here some of them: •

•

•

•

According to the construction in [3] one can adapt Meshkov’s solutions to odd space dimensions and get the quasi-optimality of the observability estimates for heat and wave systems. One could expect the same to hold for the plate systems as well. The optimality result in this paper does not apply, either for scalar equations or for 1−d problems. The same happens for heat and wave equations. This is a completely open subject. Note however that the potentials we use depend only on x. Very likely other constructions could be made using time-dependent potentials, but this remains to be explored. It is very likely that our results, both with respect obtaining explicit observability estimates and their optimality, can be extended to plate systems with other boundary conditions, like, for instance, those corresponding to clamped plates: y = ∂y/∂ν = 0. A systematic analysis of this issue remains to be done. Similar problems arise for other plate systems, including those containing the rotational inertia term: ytt − γ∆ytt + ∆2 y + ay = 0.

• •

As we mentioned above, all these questions can be analyzed in other energy spaces. Similar questions arise for the Schr¨ odinger equation also. The analysis in this paper can be adapted in a straightforward way to the observability of that model in H01 (Ω). But the issue is more subtle and remains to be investigated in the L2 (Ω) context.

References 1. F. Colombini and N. Lerner, Hyperbolic operators with non-Lipschitz coefficients, Duke Math. J. 77(1995), 657–698. 2. F. Colombini and S. Spagnolo, Some examples of hyperbolic equations with´ out local solvability, Ann. Sci. Ecole Norm. Sup. (4) 22(1989), 109–125.

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3. D. Duyckaerts, X. Zhang and E. Zuazua, On the optimality of the observability inequality for parabolic and hyperbolic systems with potentials, Ann. Inst. H. Poincar´e Anal. Non Lin´eaire, to appear. ´ ndez-Cara and E. Zuazua, The cost of approximate controllability 4. E. Ferna for heat equations: The linear case, Adv. Differential Equations 5(2000), 465– 514. 5. X. Fu, A weighted identity for partial differential operators of second order and its applications, C. R. Math. Acad. Sci. Paris 342(2006), 579–584. 6. X. Fu, X. Zhang and E. Zuazua, Analysis on the optimality of the observability inequalities for Schr¨ odinger and plate systems with potentials, in preparation. 7. A. V. Fursikov and O. Yu. Imanuvilov, Controllability of evolution equations, Lecture Notes Series 34, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996. 8. J.-L. Lions, Contrˆ olabilit´e exacte, perturbations et stabilisation de syst`emes distribu´ es, Tome 1, Recherches en Math´ematiques Appliqu´ees 8. Masson, Paris, 1988. 9. E. Machtyngier, Exact controllability for the Schr¨ odinger equation, SIAM J. Control Optim. 32(1994), 24–34. 10. V. Z. Meshkov, On the possible rate of decrease at infinity of the solutions of second-order partial differential equations, (Russian), Mat. Sb. 182(1991), 364–383; translation in Math. USSR-Sb. 72(1992), 343–361. 11. X. Zhang, Exact controllability of the semilinear plate equations, Asymptot. Anal. 27(2001), 95–125. 12. E. Zuazua, Controllability and observability of partial differential equations: some results and open problems, in “Handbook of Differential Equations: Evolutionary Differential Equations”, vol. 3, C. Dafermos and E. Feireisl, eds., Elsevier Science, to appear.

Some geometric evolution equations arising as geodesic equations on groups of diffeomorphisms including the Hamiltonian approach Peter W. Michor∗ Fakult¨ at f¨ ur Mathematik, Universit¨ at Wien, and Erwin Schr¨ odinger International Institute of Mathematical Physics, Wien, Austria

2000 Mathematics Subject Classification: 58B20, 58D05, 58D15, 58F07, 58E12, 35Q53. Key words: Diffeomorphism group, connection, Jacobi field, symplectic structure, Burgers’ equation, KdV equation.

Introduction This is the extended version of a lecture course given at the University of Vienna in the spring term 2005. Many thanks to the audience of this course for many keen questions. The main aim of this course was to understand the papers [12] and [13]. The purpose of this review article is to give a complete account of existence and uniqueness of the solutions of the members of higher order of the hierarchies of Burgers’ equation and the Korteweg–de Vries equation, including their derivation and all the necessary background. We do this both on the circle and on the real line in the setting of rapidly decreasing functions. These are all geodesic equations of infinite-dimensional regular Lie groups, namely the diffeomorphism group of the line or the circle and the corresponding Virasoro group. Let us describe the content: Appendix A is a short description of convenient calculus in infinite dimensions (beyond Banach spaces) where everything is based on smooth curves: A mapping is C ∞ if it maps smooth curves ∗ Supported by “Fonds zur F¨ orderung der wissenschaftlichen Forschung”, Projekt P 17108. Work partly done at the Program for Evolutionary Dynamics, Harvard University.

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to smooth curves. It is a theorem that smooth curves in a space of smooth functions are just smooth functions of one variable more; this is the basic assumption of variational calculus. Appendix B gives a short account of infinitedimensional regular Lie groups. Here regularity means that a smooth curve in the Lie algebra can be integrated to a smooth curve in the group whose right (or left) logarithmic derivative equals the given curve. No infinite-dimensional Lie group is known which is not regular. Section 1, as a motivating example, computes the geodesics and the curvature of the most naive Riemannian metric on the space of embeddings of the real line to itself and shows that this can be converted into Burgers’ equation. Section 2 treats Hamiltonian mechanics on infinite-dimensional weak symplectic manifolds. Here ‘weak’ means that the symplectic 2-form is injective as a mapping from the tangent bundle to the cotangent bundle. Section 3 computes geodesics and curvatures of right invariant Riemannian metrics on regular Lie groups as done by Arnold [3]. Section 4 redoes this in the symplectic approach and computes the associated momentum mappings and conserved quantities. Section 5 shows that the geodesic distance vanishes on any full diffeomorphic group for the right invariant metric coming from the L2 -metric on the Lie algebra of vector fields for a given Riemannian metric on a manifold. In particular, Burgers’ equation is the geodesic equation of such a metric. Section 6 treats the group of diffeomorphisms of the real line which decrease rapidly to the identity as a regular Lie group. This will be important for Burgers’ equation as a geodesic equation on this group, and also for the KdV equation. Here we also give a short presentation of Sobolev spaces on the real line and of the scale of HC n -spaces for which we were able to give simple proofs of the results we will need later. Section 7 treats geodesic equations on the diffeomorphism groups of the real line or S 1 which leads to Burgers’ hierarchy. We solve these equations starting at certain higher order, following [13]. Section 8 does this for the Virasoro groups on the real line or S 1 . For solutions of higher order equations we follow [12]. Note that in this paper we concentrate on the smooth (= C ∞ ) aspect. We also do not treat complete integrability for the Burgers and KdV equations, although we prepared almost all of the necessary background.

1 A general setting and a motivating example 1.1 The principal bundle of embeddings Let M and N be smooth finite-dimensional manifolds, connected and second countable without boundary, such that dim M ≤ dim N . Then the space Emb(M, N ) of all embeddings (immersions which are homeomorphisms on their images) from M into N is an open submanifold of C ∞ (M, N ) which is stable under the right action of the diffeomorphism group of M . Here C ∞ (M, N ) is a smooth manifold modeled on spaces of sections with compact support Γc (f ∗ T N ). In particular the tangent space at f is canonically

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isomorphic to the space of vector fields along f with compact support in M . If f and g differ on a non-compact set, then they belong to different connected components of C ∞ (M, N ). See [31] and [37]. Then Emb(M, N ) is the total space of a smooth principal fiber bundle with structure group the diffeomorphism group of M ; the base is called B(M, N ), it is a Hausdorff smooth manifold modeled on nuclear (LF)-spaces. It can be thought of as the “nonlinear Grassmannian” or “differentiable Chow variety” of all submanifolds of N which are of type M . This result is based on an idea implicitly contained in [51]; it was fully proved in [7] for compact M and for general M in [36]. See also [37], section 13 and [31]. If we take a Hilbert space H instead of N , then B(M, H) is the classifying space for Diff(M ) if M is compact, and the classifying bundle Emb(M, H) carries also a universal connection. This is shown in [38]. 1.2 If (N, g) is a Riemannian manifold, then on the manifold Emb(M, N ) there is a naturally induced weak Riemannian metric given, for s1 , s2 ∈ Γc (f ∗ T N ) and ϕ ∈ Emb(M, N ), by  Gϕ (s1 , s2 ) = g(s1 , s2 ) vol(ϕ∗ g), ϕ ∈ Emb(M, N ), M

where vol(g) denotes the volume form on N induced by the Riemannian metric g and vol(ϕ∗ g) the volume form on M induced by the pull-back metric ϕ∗ g. The covariant derivative and curvature of the Levi–Civita connection induced by G were investigated in [6] if N = Rdim M+1 (endowed with the standard inner product) and in [25] for the general case. In [40] it was shown that the geodesic distance (topological metric) on the base manifold B(M, N ) = Emb(M, N )/ Diff(M ) induced by this Riemannian metric vanishes. This weak Riemannian metric is invariant under the action of the diffeomorphism group Diff(M ) by composition from the right and hence it induces a Riemannian metric on the base manifold B(M, N ). 1.3 Example Let us consider the special case M = N = R, that is, the space Emb(R, R) of all embeddings of the real line into itself, which contains the diffeomorphism group Diff(R) as an open subset. The case M = N = S 1 is treated in a similar fashion and the results of this paper are also valid in this situation, where Emb(S 1 , S 1 ) = Diff(S 1 ). For our purposes, we may restrict attention to the space of orientation-preserving embeddings, denoted by Emb+ (R, R). The weak Riemannian metric has thus the expression  Gf (h, k) = h(x)k(x)|f
(x)| dx, f ∈ Emb(R, R), h, k ∈ Cc∞ (R, R). R

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We shall compute the geodesic equation for this metric by variational calculus. The energy of a curve f of embeddings is E(f ) =

1 2



b

Gf (ft , ft )dt = a

1 2

 b a

R

ft2 fx dxdt.

If we assume that f (x, t, s) is a smooth function and that the variations are with fixed endpoints, then the derivative with respect to s of the energy is   1 b f 2 fx dxdt 2 a R t   1 b = (2ft fts fx + ft2 fxs )dxdt 2 a R   1 b (2ftt fs fx + 2ft fs ftx + 2ft ftx fs )dxdt =− 2 a R   b  ft ftx =− ftt + 2 fs fx dxdt, fx a R

∂s|0 E(f ( , s)) = ∂s|0

so that the geodesic equation with its initial data is: ft ftx , f ( , 0) ∈ Emb+ (R, R), fx =: Γf (ft , ft ),

ftt = −2

ft ( , 0) ∈ Cc∞ (R, R)

(1.3.1)

where the Christoffel symbol Γ : Emb(R, R) × Cc∞ (R, R) × Cc∞ (R, R) → Cc∞ (R, R) is given by symmetrisation: Γf (h, k) := −

hkx + hx k (hk)x =− . fx fx

(1.3.2)

For vector fields X, Y on Emb(R, R) the covariant derivative is given by the expression ∇Emb X Y = dY (X) − Γ (X, Y ). The Riemannian curvature R(X, Y )Z = (∇X ∇Y − ∇Y ∇X − ∇[X,Y ] )Z is then determined in terms of the Christoffel form by R(X, Y )Z = (∇X ∇Y − ∇Y ∇X − ∇[X,Y ] )Z = ∇X (dZ(Y ) − Γ (Y, Z)) − ∇Y (dZ(X) − Γ (X, Z)) − dZ([X, Y ]) + Γ ([X, Y ], Z) = d2 Z(X, Y ) + dZ(dY (X)) − Γ (X, dZ(Y )) − dΓ (X)(Y, Z) − Γ (dY (X), Z) − Γ (Y, dZ(X)) + Γ (X, Γ (Y, Z)) − d2 Z(Y, X) − dZ(dX(Y )) + Γ (Y, dZ(X))

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+ dΓ (Y )(X, Z) + Γ (dX(Y ), Z) + Γ (X, dZ(Y )) − Γ (Y, Γ (X, Z)) − dZ(dY (X) − dX(Y )) + Γ (dY (X) − dX(Y ), Z) = −dΓ (X)(Y, Z) + Γ (X, Γ (Y, Z)) + dΓ (Y )(X, Z) − Γ (Y, Γ (X, Z) so that Rf (h, k) = −dΓ (f )(h)(k, ) + dΓ (f )(k)(h, ) + Γf (h, Γf (k, )) − Γf (k, Γf (h, ))  



(k)x (h)x h k f f kx (h)x hx (k)x x x x x + + − (1.3.3) =− fx2 fx2 fx fx =

1 (fxx hx k − fxx hkx  + fx hkxx  − fx hxx k + 2fx hkx x − 2fx hx kx ). fx3

Now let us consider the trivialisation of T Emb(R, R) by right translation (this is most useful for Diff(R)). The derivative of the inversion Inv : g → g −1 is given by h ◦ g −1 Tg (Inv)h = −T (g −1 ) ◦ h ◦ g −1 = − gx ◦ g −1 for g ∈ Emb(R, R), h ∈ Cc∞ (R, R). Defining u := ft ◦ f −1 ,

or, in more detail,

u(t, x) = ft (t, f (t,

)−1 (x)),

we have ux = (ft ◦ f −1 )x = (ftx ◦ f −1 )

1 ftx = ◦ f −1 , −1 fx ◦ f fx

ut = (ft ◦ f −1 )t = ftt ◦ f −1 + (ftx ◦ f −1 )(f −1 )t = ftt ◦ f −1 − (ftx ◦ f −1 )

1 (ft f −1 ) fx f −1

which, by (1.3.1) and the first equation becomes     ftx ft ftx ft ut = ftt ◦ f −1 − ◦ f −1 = −3 ◦ f −1 = −3uxu. fx fx The geodesic equation on Emb(R, R) in right trivialization, that is, in Eulerian formulation, is hence (1.3.4) ut = −3uxu , which is just Burgers’ equation. Finally let us solve Burgers’ equation and also describe its universal completion, see [10], [1], or [26].

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In R2 with coordinates (x, y) consider the vector field Y (x, y) = (3y, 0) = 3y∂x with differential equation x˙ = 3y, y˙ = 0. It has the complete flow FlYt (x, y) = (x + 3ty, y). Let now t → u(t, x) be a curve of functions on R. We ask when the graph of u can be reparametrized in such a way that it becomes a solution curve of the push-forward vector field Y∗ : f → Y ◦ f on the space of embeddings Emb(R, R2 ). Thus consider a time dependent reparametrization z → x(t, z), i.e., x ∈ C ∞ (R2 , R). The curve t → (x(t, z), u(t, x(z, t))) in R2 is an integral curve of Y if and only if       x xt 3u ◦ x = = ∂t ut ◦ x + (ux ◦ x) · xt u◦x 0 xt = 3u ◦ x, ⇐⇒ 0 = (ut + 3uux) ◦ x. This implies that the graph of u(t, ·), namely the curve t → (x → (x, u(t, x))), may be parameterized as a solution curve of the vector field Y∗ on the space of embeddings Emb(R, R2 ) starting at x → (x, u(0, x)) if and only if u is a solution of the partial differential equation ut + 3uux = 0. The parameterization z → x(z, t) is then given by xt (z, t) = 3u(x(t, z)) with x(0, z) = z ∈ R.

The characteristic flow of the inviscid Burgers’ equation tilts the plane.

This has a simple physical meaning. Consider freely flying particles in R, and trace a trajectory x(t) of one of the particles. Denote the velocity of a particle at the position x at the moment t by u(t, x), or rather, by 3u(t, x) := x(t). ˙ Due to the absence of interaction, the Newton equation of any particle is x ¨(t) = 0. Let us illustrate this: The flow of the vector field Y = 3u∂x is tilting the plane to the right with constant speed. The illustration shows how a graph of

Geometric evolution equations

139

an honest function is moved through a shock (when the derivatives become infinite) towards the graph of a multivalued function; each piece of it is still a local solution.

2 Weak symplectic manifolds 2.1 Review For a finite-dimensional symplectic manifold (M, ω) we have the following exact sequence of Lie algebras: ω

0 → H 0 (M ) → C ∞ (M, R) −grad → X(M, ω) −→ H 1 (M ) → 0. Here H ∗ (M ) is the real De Rham cohomology of M , the space C ∞ (M, R) is equipped with the Poisson bracket { , }, X(M, ω) consists of all vector fields ξ with Lξ ω = 0 (the locally Hamiltonian vector fields), which is a Lie algebra for the Lie bracket. Furthermore, gradω f is the Hamiltonian vector field for f ∈ C ∞ (M, R) given by i(gradω f )ω = df and γ(ξ) = [iξ ω]. The spaces H 0 (M ) and H 1 (M ) are equipped with the zero bracket. Consider a symplectic right action r : M × G → M of a connected Lie group G on M ; we use the notation r(x, g) = rg (x) = rx (g) = x.g. By ζ X (x) = Te (rx )X we get a mapping ζ : g → X(M, ω) which sends each element X of the Lie algebra g of G to the fundamental vector field X. This is a Lie algebra homomorphism (for right actions!). H 0 (M )

i

ω

grad / C ∞ (M, R) / X(M, ω) dII w; II ww II w ww j III I www ζ g

γ

/ H 1 (M )

A linear lift j : g → C ∞ (M, R) of ζ with gradω ◦j = ζ exists if and only if γ ◦ ζ = 0 in H 1 (M ). This lift j may be changed to a Lie algebra homomorphism if and only if the 2-cocycle j¯ : g × g → H 0 (M ), given by (i ◦ j¯)(X, Y ) = {j(X), j(Y )} − j([X, Y ]), vanishes in the Lie algebra cohomology H 2 (g, H 0 (M )), for if j¯ = δα then j − i ◦ α is a Lie algebra homomorphism. If j : g → C ∞ (M, R) is a Lie algebra homomorphism, we may associate the moment mapping µ : M → g
= L(g, R) to it, which is given by µ(x)(X) = χ(X)(x) for x ∈ M and X ∈ g. It is G-equivariant for a suitably chosen (in general affine) action of G on g
. 2.2 We now want to carry over to infinite-dimensional manifolds the procedure of subsection (2.1). First we need the appropriate notions in infinite dimensions. So let M be a manifold, which in general is infinite dimensional.

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A 2-form ω ∈ Ω 2 (M ) is called a weak symplectic structure on M if it is closed (dω = 0) and if its associated vector bundle homomorphism ωˇ: T M → T ∗ M is injective. A 2-form ω ∈ Ω 2 (M ) is called a strong symplectic structure on M if it is closed (dω = 0) and if its associated vector bundle homomorphism ωˇ: T M → T ∗ M is invertible with smooth inverse. In this case, the vector bundle T M has reflexive fibers Tx M . Let i : Tx M → (Tx M )

be the canonical mapping onto the bidual. Skew symmetry of ω is equivalent to the fact that the transposed (ωˇ)t = (ωˇ)∗ ◦ i : Tx M → (Tx M )
satisfies (ωˇ)t = −ωˇ. Thus, i = −((ωˇ)−1 )∗ ◦ ωˇ is an isomorphism. 2.3 Every cotangent bundle T ∗ M , viewed as a manifold, carries a canonical weak symplectic structure ωM ∈ Ω 2 (T ∗ M ), which is defined as follows. Let π ∗M : T ∗ M → M be the projection. Then the Liouville form θM ∈ Ω 1 (T ∗ M ) is given by θ M (X) = π T ∗ M (X), T (π ∗M )(X) for X ∈ T (T ∗ M ), where ,

denotes the duality pairing T ∗ M ×M T M → R. Then the symplectic structure on T ∗ M is given by ω M = −dθM , which of course in a local chart looks like ω E ((v, v
), (w, w
)) = w
, v E − v
, w E . The associated mapping ωˇ : T(0,0) (E × E
) = E × E
→ E
× E

is given by (v, v
) → (−v
, iE (v)), where iE : E → E

is the embedding into the bidual. So the canonical symplectic structure on T ∗ M is strong if and only if all model spaces of the manifold M are reflexive. 2.4 Let M be a weak symplectic manifold. The first thing to note is that the Hamiltonian mapping gradω : C ∞ (M, R) → X(M, ω) does not make sense in general, since ωˇ: T M → T ∗ M is not invertible. Namely, gradω f = (ωˇ)−1 ◦ df is defined only for those f ∈ C ∞ (M, R) with df (x) in the image of ωˇ for all x ∈ M . A similar difficulty arises for the definition of the Poisson bracket on C ∞ (M, R). Definition For a weak symplectic manifold (M, ω) let Txω M denote the real linear subspace Txω M = ωˇx (Tx M ) ⊂ Tx∗ M = L(Tx M, R), and let us call it the smooth cotangent space with respect to the symplectic structure ω of M at x, in view of the embedding of test functions into distributions. These vector spaces fit together to form a subbundle of T ∗ M which is isomorphic to the tangent bundle T M via ωˇ : T M → T ω M ⊆ T ∗ M . It is in general not a splitting subbundle. 2.5 Definition For a weak symplectic vector space (E, ω) let Cω∞ (E, R) ⊂ C ∞ (E, R)

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denote the linear subspace consisting of all smooth functions f : E → R such that each iterated derivative dk f (x) ∈ Lksym (E; R) has the property that dk f (x)(

, y 2 , . . . , yk ) ∈ E ω

is actually in the smooth dual E ω ⊂ E
for all x, y2 , . . . , yk ∈ E, and that the mapping k 3

E→E

(x, y2 , . . . , yk ) → (ωˇ)−1 (df (x)(

, y2 , . . . , yk ))

is smooth. By the symmetry of higher derivatives, this is then true for all entries of dk f (x), for all x. 2.6 Lemma For f ∈ C ∞ (E, R) the following assertions are equivalent: (2.6.1) df : E → E
factors to a smooth mapping E → E ω . (2.6.2) f has a smooth ω-gradient gradω f ∈ X(E) = C ∞ (E, E) which satisfies df (x)y = ω(gradω f (x), y). (2.6.3) f ∈ Cω∞ (E, R). Proof. Clearly, (2.6.3) ⇒ (2.6.2) ⇔ (2.6.1). We have to show that (2.6.2) ⇒ (2.6.3). Suppose that f : E → R is smooth and df (x)y = ω(gradω f (x), y). Then dk f (x)(y1 , . . . , yk ) = dk f (x)(y2 , . . . , yk , y1 ) = (dk−1 (df ))(x)(y2 , . . . , yk )(y1 ) = ω(dk−1 (gradω f )(x)(y2 , . . . , yk ), y1 ).

2

2.7 For a weak symplectic manifold (M, ω) let Cω∞ (M, R) ⊂ C ∞ (M, R) denote the linear subspace consisting of all smooth functions f : M → R such that the differential df : M → T ∗ M factors to a smooth mapping M → T ω M . In view of lemma (2.6) these are exactly those smooth functions on M which admit a smooth ω-gradient gradω f ∈ X(M ). Also the condition (2.6.1) translates to a local differential condition describing the functions in Cω∞ (M, R).

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2.8 Theorem The Hamiltonian mapping gradω : Cω∞ (M, R) → X(M, ω), which is given by igradω f ω = df

or

gradω f := (ωˇ)−1 ◦ df,

is well defined. Also the Poisson bracket {

,

} : Cω∞ (M, R) × Cω∞ (M, R) → Cω∞ (M, R),

{f, g} := igradω f igradω g ω = ω(gradω g, gradω f ) = dg(gradω f ) = (gradω f )(g) is well defined and gives a Lie algebra structure to the space Cω∞ (M, R), which also fulfills {f, gh} = {f, g}h + g{f, h}. We have the following long exact sequence of Lie algebras and Lie algebra homomorphisms: ω

0 → H 0 (M ) → Cω∞ (M, R) −grad → X(M, ω) −γ→ Hω1 (M ) → 0, where H 0 (M ) is the space of locally constant functions, and Hω1 (M ) =

{ϕ ∈ C ∞ (M ← T ω M ) : dϕ = 0} {df : f ∈ Cω∞ (M, R)}

is the first symplectic cohomology space of (M, ω), a linear subspace of the De Rham cohomology space H 1 (M ). Proof. It is clear from lemma (2.6), that the Hamiltonian mapping gradω is well defined and has values in X(M, ω), since by [31], 34.18.6 we have Lgradω f ω = igradω f dω + digradω f ω = ddf = 0. By [31], 34.18.7, the space X(M, ω) is a Lie subalgebra of X(M ). The Poisson bracket is well defined as a mapping { , } : Cω∞ (M, R) × Cω∞ (M, R) → C ∞ (M, R); it only remains to check that it has values in the subspace Cω∞ (M, R). This is a local question, so we may assume that M is an open subset of a convenient vector space equipped with a (nonconstant) weak symplectic structure. So let f , g ∈ Cω∞ (M, R), then {f, g}(x) = dg(x)(gradω f (x)), and we have d({f, g})(x)y = d(dg(

)y)(x). gradω f (x) + dg(x)(d(gradω f )(x)y)

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= d(ω(gradω g( ), y)(x). gradω f (x) + ω(gradω g(x), d(gradω f )(x)y) = ω(d(gradω g)(x)(gradω f (x)) − d(gradω f )(x)(gradω g(x)), y), since gradω f ∈ X(M, ω) and for any X ∈ X(M, ω) the condition LX ω = 0 implies ω(dX(x)y1 , y2 ) = −ω(y1 , dX(x)y2 ). So (2.6.2) is satisfied, and thus {f, g} ∈ Cω∞ (M, R). If X ∈ X(M, ω) then diX ω = LX ω = 0, so [iX ω] ∈ H 1 (M ) is well defined, and by iX ω = ωˇ oX we even have γ(X) := [iX ω] ∈ Hω1 (M ), so γ is well defined. Now we show that the sequence is exact. Obviously, it is exact at H 0 (M ) and at Cω∞ (M, R), since the kernel of gradω consists of the locally constant functions. If γ(X) = 0, then ωˇ◦ X = iX ω = df for f ∈ Cω∞ (M, R), and clearly X = gradω f . Now let us suppose that ϕ ∈ Γ (T ω M ) ⊂ Ω 1 (M ) with dϕ = 0. Then X := (ωˇ)−1 ◦ ϕ ∈ X(M ) is well defined and LX ω = diX ω = dϕ = 0, so X ∈ X(M, ω) and γ(X) = [ϕ]. Moreover, Hω1 (M ) is a linear subspace of H 1 (M ) since, for ϕ ∈ Γ (T ω M ) ⊂ 1 Ω (M ) with ϕ = df for f ∈ C ∞ (M, R), the vector field X := (ωˇ)−1 ◦ϕ ∈ X(M ) is well defined, and since ωˇ oX = ϕ = df by (2.6.1) we have f ∈ Cω∞ (M, R) with X = gradω f . The mapping gradω maps the Poisson bracket into the Lie bracket, since by [31], 34.18 we have igradω {f,g} ω = d{f, g} = dLgradω f g = Lgradω f dg = Lgradω f igradω g ω − igradω g Lgradω f ω = [Lgradω f , igradω g ]ω = i[gradω f,gradω g] ω. Let us now check the properties of the Poisson bracket. By definition, it is skew symmetric, and we have {{f, g}, h} = Lgradω {f,g} h = L[gradω f,gradω g] h = [Lgradω f , Lgradω g ]h = Lgradω f Lgradω g h − Lgradω g Lgradω f h = {f, {g, h}} − {g, {f, h}}, {f, gh} = Lgradω f (gh) = (Lgradω f g)h + gLgradω f h = {f, g}h + g{f, h}. Finally, it remains to show that all mappings in the sequence are Lie algebra homomorphisms, where we put the zero bracket on both cohomology spaces. For locally constant functions we have {c1 , c2 } = Lgradω c1 c2 = 0. We have already checked that gradω is a Lie algebra homomorphism. For X, Y ∈ X(M, ω) i[X,Y ] ω = [LX , iY ]ω = LX iY ω + 0 = diX iY ω + iX LY ω = diX iY ω is exact.

2

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2.9 Weakly symplectic group actions Let us suppose that an infinite-dimensional regular Lie group G with Lie algebra g acts from the right on a weak symplectic manifold (M, ω) by r : M × G → M in a way which respects ω, so that each transformation rg is a symplectomorphism. This is called a symplectic group action. We shall use the notation r(x, g) = rg (x) = rx (g). Let us list some immediate consequences: (2.9.1) The space Cω∞ (M )G of G-invariant smooth functions with ωgradients is a Lie subalgebra for the Poisson bracket, since for each g ∈ G and f, h ∈ C ∞ (M )G we have (rg )∗ {f, h} = {(rg )∗ f, (rg )∗ h} = {f, h}. (2.9.2) For x ∈ M the pull-back of ω to the orbit x.G is a 2-form, invariant under the action of G on the orbit. In the finite-dimensional case the orbit is an initial submanifold. In our case this has to be checked directly in each example. In any case we have something like a tangent bundle Tx (x.G) = T (rx )g. If i : x.G → M is the embedding of the orbit then rg ◦ i = i ◦ rg , so that i∗ ω = i∗ (rg )∗ ω = (rg )∗ i∗ ω holds for each g ∈ G and thus i∗ ω is invariant. (2.9.3) The fundamental vector field mapping ζ : g → X(M, ω), given by ζ X (x) = Te (rx )X for X ∈ g and x ∈ M , is a homomorphism of Lie algebras, where g is the Lie algebra of G (for a left action we get an anti-homomorphism of Lie algebras). Moreover, ζ takes values in X(M, ω). Let us consider again the exact sequence of Lie algebra homomorphisms from (2.8): 0

/ H 0 (M )

α

/ C ∞ (M ) ω f

gradω

j

/ X(M, ω) O

γ

/ H 1 (M ) ω

/0

ζ

g One can lift ζ to a linear mapping j : g → C ∞ (M ) if and only if γ ◦ ζ = 0. In this case the action of G is called a Hamiltonian group action, and the linear mapping j : g → C ∞ (M ) is called a generalized Hamiltonian function for the group action. It is unique up to addition of a mapping α◦τ for τ : g → H 0 (M ). (2.9.4) If Hω1 (M ) = 0, then any symplectic action on (M, ω) is a Hamiltonian action. But if γ ◦ ζ = 0, we can replace g by its Lie subalgebra ker(γ ◦ ζ) ⊂ g and consider the corresponding Lie subgroup G which then admits a Hamiltonian action. (2.9.5) If the Lie algebra g is equal to its commutator subalgebra [g, g], the linear span of all [X, Y ] for X, Y ∈ g (true for all full diffeomorphism groups), then any infinitesimal symplectic action ζ : g → X(M,  ω) is a Hamiltonian action, since then any Z ∈ g can be written as Z = i [Xi , Yi ] so that ζ Z =  [ζ Xi , ζ Yi ] ∈ im(gradω ) since γ : X(M, ω) → H 1 (M ) is a homomorphism into the zero Lie bracket. (2.9.6) If j : g → (Cω∞ (M ), { , }) happens to not be a homomorphism of Lie algebras, then c(X, Y ) = {j(X), j(Y )} − j([X, Y ]) lies in H 0 (M ), and indeed c : g × g → H 0 (M ) is a cocycle for the Lie algebra cohomology: c([X, Y ], Z) + c([Y, Z], X) + c([Z, X], Y ) = 0. If c is a coboundary, i.e.,

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c(X, Y ) = −b([X, Y ]), then j + α ◦ b is a Lie algebra homomorphism. If the cocycle c is nontrivial we can use the central extension H 0 (M ) ×c g with bracket [(a, X), (b, Y )] = (c(X, Y ), [X, Y ]) in the diagram / H 0 (M )

0

α

/ C ∞ (M ) ω O

gradω

/ X(M, ω) O

j¯

γ

/ H 1 (M ) ω

/0

ζ

H 1 (M ) ×c g

pr2

/g

where j¯(a, X) = j(X) + α(a). Then j¯ is a homomorphism of Lie algebras. 2.10 Momentum mapping For an infinitesimal symplectic action, i.e., a homomorphism ζ : g → X(M, ω) of Lie algebras, we can find a linear lift j : g → Cω∞ (M ) if and only if there exists a mapping J ∈ Cω∞ (M, g∗ ) := {f ∈ C ∞ (M, g∗ ) : f (

), X ∈ Cω∞ (M ) for all X ∈ g}

such that gradω ( J, X ) = ζ X

for all X ∈ g.

The mapping J ∈ Cω∞ (M, g∗ ) is called the momentum mapping for the infinitesimal action ζ : g → X(M, ω). Let us note again the relations between the generalized Hamiltonian j and the momentum mapping J: J : M → g∗ ,

j : g → Cω∞ (M ),

J, X = j(X) ∈ Cω∞ (M ),

ζ : g → X(M, ω),

gradω (j(X)) = ζ(X),

X ∈ g,

(2.10.1)

iζ(X) ω = dj(X) = d J, X , where , : g∗ × g → R is the duality pairing. 2.11 Basic properties of the momentum mapping Let r : M × G → M be a Hamiltonian right action of an infinite-dimensional regular Lie group G on a weak symplectic manifold M , let j : g → Cω∞ (M ) be a generalized Hamiltonian and let J ∈ Cω∞ (M, g∗ ) be the associated momentum mapping. (2.11.1) For x ∈ M , the transposed mapping of the linear mapping dJ(x) : Tx M → g∗ is dJ(x) : g → Tx∗ M,

dJ(x) = ω ˇ x ◦ ζ,

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Peter W. Michor

since for ξ ∈ Tx M and X ∈ g we have ω x (ζ X (x)), ξ . dJ(ξ), X = iξ dJ, X = iξ d J, X = iξ iζ X ω = ˇ (2.11.2) The closure of the image dJ(Tx M ) of dJ(x) : Tx M → g∗ is the annihilator g◦x of the isotropy Lie algebra gx := {X ∈ g : ζ X (x) = 0} in g∗ , since the annihilator of the image is the kernel of the transposed mapping, im(dJ(x))◦ = ker(dJ(x) ) = ker(ˇ ω x ◦ ζ) = ker(evx ◦ζ) = gx . (2.11.3) The kernel of dJ(x) is the symplectic orthogonal (T (rx )g)⊥,ω = (Tx (x.G))⊥,ω ⊆ Tx M, since for the annihilator of the kernel we have ω x ◦ ζ) ker(dJ(x))◦ = im(dJ(x) ) = im(ˇ = {ˇ ωx (ζ X (x)) : X ∈ g} = ω ˇ x (Tx (x.G)). (2.11.4) If G is connected, x ∈ M is a fixed point for the G-action if and only if x is a critical point of J, i.e., dJ(x) = 0. (2.11.5) (Emmy Noether’s theorem) Let h ∈ Cω∞ (M ) be a Hamiltonian function which is invariant under the Hamiltonian G action. Then dJ(gradω (h)) = 0. Thus the momentum mapping J : M → g∗ is constant on each trajectory (if it exists) of the Hamiltonian vector field gradω (h). Namely, dJ(gradω (h)), X = d J, X (gradω (h)) = dj(X)(gradω (h)) = {h, j(X)} = −dh(gradω j(X)) = dh(ζ X ) = 0. E. Noether’s theorem admits the following generalization. 2.12 Theorem Let G1 and G2 be two regular Lie groups which act by Hamiltonian actions r1 and r2 on the weakly symplectic manifold (M, ω), with momentum mappings J1 and J2 , respectively. We assume that J2 is G1 -invariant, i.e., J2 is constant along all G1 -orbits, and that G2 is connected. Then J1 is constant on the G2 -orbits and the two actions commute. Proof. Let ζ i : gi → X(M, ω) be the two infinitesimal actions. Then for X1 ∈ g1 and X2 ∈ g2 we have Lζ 2X J1 , X1 = iζ 2X d J1 , X1 = iζ 2X iζ 1X ω = { J2 , X2 , J1 , X1 } 2

2

2

1

= −{ J1 , X1 , J2 , X2 } = −iζ 1X d J2 , X2 = −Lζ 1X J2 , X2 = 0 1

1

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since J2 is constant along each G1 -orbit. Since G2 is assumed to be connected, J1 is also constant along each G2 -orbit. We also saw that each Poisson bracket { J2 , X2 , J1 , X1 } vanishes; by gradω Ji , Xi = ζ iXi we conclude that [ζ 1X1 , ζ 2X2 ] = 0 for all Xi ∈ gi , which implies the result if also G1 is connected. In the general case we can argue as follows: ω −1 d J2 , X2 ) (r1g1 )∗ ζ 2X2 = (r1g1 )∗ gradω J2 , X2 = (r1g1 )∗ (ˇ = (((r1g1 )∗ ω)ˇ)−1 d (r1g1 )∗ J2 , X2 = (ˇ ω −1 d J2 , X2

= gradω J2 , X2 = ζ 2X2 . exp(tX2 )

Thus r1g1 commutes with each r2 connected.

and thus with each r2g2 , since G2 is 2

3 Right invariant weak Riemannian metrics on Lie groups 3.1 Notation on Lie groups Let G be a Lie group which may be infinite dimensional, but then is supposed to be regular, with Lie algebra g. See appendix (B) for more information. Let µ : G × G → G be the multiplication, let µx be left translation and µy be right translation, given by µx (y) = µy (x) = xy = µ(x, y). Let L, R : g → X(G) be the left and right invariant vector field mappings, given by LX (g) = Te (µg ).X and RX = Te (µg ).X, respectively. They are related by LX (g) = RAd(g)X (g). Their flows are given by exp(tX) X (g), FlL t (g) = g. exp(tX) = µ

X FlR t (g) = exp(tX).g = µexp(tX) (g).

We also need the right Maurer–Cartan form κ = κr ∈ Ω 1 (G, g), given −1 by κx (ξ) := Tx (µx ) · ξ. It satisfies the right Maurer–Cartan equation dκ − 12 [κ, κ]∧ = 0, where [ , ]∧ denotes the wedge product of g-valued forms on G induced by the Lie bracket. Note that 12 [κ, κ]∧ (ξ, η) = [κ(ξ), κ(η)]. The (exterior) derivative of the function Ad : G → GL(g) can be expressed by d Ad = Ad .(ad ◦κl ) = (ad ◦κr ). Ad, since we have d Ad(T µg .X) =

d dt |0

Ad(g. exp(tX)) = Ad(g). ad(κl (T µg .X)).

3.2 Geodesics of a right invariant metric on a Lie group Let γ = , : g × g → R be a positive definite bounded (weak) inner product. Then −1 −1 γ x (ξ, η) = T (µx ) · ξ, T (µx ) · η) = κ(ξ), κ(η)

(3.2.1)

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is a right invariant (weak) Riemannian metric on G, and any (weak) right invariant bounded Riemannian metric is of this form, for suitable , . Let g : [a, b] → G be a smooth curve. The velocity field of g, viewed in the right trivializations, coincides with the right logarithmic derivative δ r (g) = T (µg

−1

) · ∂t g = κ(∂t g) = (g ∗ κ)(∂t ), where ∂t =

∂ . ∂t

The energy of the curve g(t) is given by 1 E(g) = 2

 a

b

1 Gg (g , g )dt = 2







b

(g ∗ κ)(∂t ), (g ∗ κ)(∂t ) dt.

a

For a variation g(s, t) with fixed endpoints we have then, using the right Maurer–Cartan equation and integration by parts,  1 b ∂s E(g) = 2 ∂s (g ∗ κ)(∂t ), (g ∗ κ)(∂t ) dt 2 a  b = ∂t (g ∗ κ)(∂s ) − d(g ∗ κ)(∂t , ∂s ), (g ∗ κ)(∂t ) dt a

 =

b

(− (g ∗ κ)(∂s ), ∂t (g ∗ κ)(∂t )

a

− [(g ∗ κ)(∂t ), (g ∗ κ)(∂s )], (g ∗ κ)(∂t ) ) dt  b (g ∗ κ)(∂s ), ∂t (g ∗ κ)(∂t ) + ad((g ∗ κ)(∂t )) ((g ∗ κ)(∂t )) dt =− a

where ad((g ∗ κ)(∂t )) : g → g is the adjoint of ad((g ∗ κ)(∂t )) with respect to the inner product , . In infinite dimensions one also has to check the existence of this adjoint. In terms of the right logarithmic derivative u : [a, b] → g −1 of g : [a, b] → G, given by u(t) := g ∗ κ(∂t ) = Tg(t) (µg(t) ) · g
(t), the geodesic equation has the expression: ut = − ad(u) u

.

(3.2.2)

This is, of course, just the Euler–Poincar´e equation for right invariant systems using the Lagrangian given by the kinetic energy (see [34], section 13). 3.3 The covariant derivative Our next aim is to derive the Riemannian curvature and for that we develop the basis-free version of Cartan’s method of moving frames in this setting, which also works in infinite dimensions. The right trivialization, or framing, (π G , κ) : T G → G × g induces the isomorphism R : C ∞ (G, g) → X(G), given

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by R(X)(x) := RX (x) := Te (µx ) · X(x), for X ∈ C ∞ (G, g) and x ∈ G. Here X(G) := Γ (T G) denote the Lie algebra of all vector fields. For the Lie bracket and the Riemannian metric we have [RX , RY ] = R(−[X, Y ]g + dY · RX − dX · RY ),

(3.3.1)

R−1 [RX , RY ] = −[X, Y ]g + RX (Y ) − RY (X), γ x (RX (x), RY (x)) = γ(X(x), Y (x)) , x ∈ G. In the sequel we shall compute in C ∞ (G, g) instead of X(G). In particular, we shall use the convention ∇X Y := R−1 (∇RX RY )

for X, Y ∈ C ∞ (G, g).

to express the Levi–Civita covariant derivative. Lemma Assume that for all ξ ∈ g the adjoint ad(ξ) with respect to the inner product , exists and that ξ → ad(ξ) is bounded. Then the Levi– Civita covariant derivative of the metric (3.2.1) exists and is given for any X, Y ∈ C ∞ (G, g) in terms of the isomorphism R by ∇X Y = dY.RX +

1 1 1 ad(X) Y + ad(Y ) X − ad(X)Y. 2 2 2

(3.3.2)

Proof. Easy computations show that this formula satisfies the axioms of a covariant derivative, that relative to it the Riemannian metric is covariantly constant, since RX γ(Y, Z) = γ(dY.RX , Z) + γ(Y, dZ.RX ) = γ(∇X Y, Z) + γ(Y, ∇X Z), and that it is torsion free, since 2

∇X Y − ∇Y X + [X, Y ]g − dY.RX + dX.RY = 0.

For ξ ∈ g define α(ξ) : g → g by α(ξ)η := ad(η) ξ. With this notation, the previous lemma states that for all X ∈ C ∞ (G, g) the covariant derivative of the Levi–Civita connection has the expression ∇X = RX +

1 1 1 ad(X) + α(X) − ad(X). 2 2 2

(3.3.3)

3.4 The curvature First note that we have the following relations: [RX , ad(Y )] = ad(RX (Y )),

[RX , α(Y )] = α(RX (Y )),

(3.4.1)

[RX , ad(Y ) ] = ad(RX (Y )) , [ad(X) , ad(Y ) ] = − ad([X, Y ]g ) .

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The Riemannian curvature is then computed by R(X, Y ) = [∇X , ∇Y ] − ∇−[X,Y ]g +RX (Y )−RY (X)  1 1 1 = RX + ad(X) + α(X) − ad(X), RY 2 2 2  1 1 1

+ ad(Y ) + α(Y ) − ad(Y ) 2 2 2 − R−[X,Y ]g +RX (Y )−RY (X) −

1 ad(−[X, Y ]g + RX (Y ) − RY (X)) 2

1 − α(−[X, Y ]g + RX (Y ) − RY (X)) 2 +

1 ad(−[X, Y ]g + RX (Y ) − RY (X)) 2

1 = − [ad(X) + ad(X), ad(Y ) + ad(Y )] 4

(3.4.2)

1 1 + [ad(X) − ad(X), α(Y )] + [α(X), ad(Y ) − ad(Y )] 4 4 1 1 + [α(X), α(Y )] + α([X, Y ]g ). 4 2 If we plug in all definitions and use the Jacobi identity four times we get the following expression: γ(4R(X, Y )Z, U ) = +2γ([X, Y ], [Z, U ]) − γ([Y, Z], [X, U ]) + γ([X, Z], [Y, U ]) − γ(Z, [U, [X, Y ]]) + γ(U, [Z, [X, Y ]]) − γ(Y, [X, [U, Z]]) − γ(X, [Y, [Z, U ]]) + γ(ad(X) Z, ad(Y ) U ) + γ(ad(X) Z, ad(U ) Y ) + γ(ad(Z) X, ad(Y ) U )

(3.4.3)

− γ(ad(U ) X, ad(Y ) Z) − γ(ad(Y ) Z, ad(X) U ) − γ(ad(Z) Y, ad(X) U ) − γ(ad(U ) X, ad(Z) Y ) + γ(ad(U ) Y, ad(Z) X). This yields the following expression which is useful for computing the sectional curvature:

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4γ(R(X, Y )X, Y ) = 3γ(ad(X)Y, ad(X)Y ) − 2γ(ad(Y ) X, ad(X)Y ) − 2γ(ad(X) Y, ad(Y )X) + 4γ(ad(X) X, ad(Y ) Y ) (3.4.4) − γ(ad(X) Y + ad(Y ) X, ad(X) Y + ad(Y ) X). 3.5 Jacobi fields, I We compute first the Jacobi equation directly via variations of geodesics. So let g : R2 → G be smooth, t → g(t, s) a geodesic for each s. Let again u = κ(∂t g) = (g ∗ κ)(∂t ) be the velocity field along the geodesic in right trivialization which satisfies the geodesic equation ut = − ad(u) u. Then y := κ(∂s g) = (g ∗ κ)(∂s ) is the Jacobi field corresponding to this variation, written in the right trivialization. From the right Maurer–Cartan equation we then have yt = ∂t (g ∗ κ)(∂s ) = d(g ∗ κ)(∂t , ∂s ) + ∂s (g ∗ κ)(∂t ) + 0 = [(g ∗ κ)(∂t ), (g ∗ κ)(∂s )]g + us = [u, y] + us . Using the geodesic equation, the definition of α, and the fourth relation in (3.4.1), this identity implies ust = uts = ∂s ut = −∂s (ad(u) u) = − ad(us ) u − ad(u) us = − ad(yt + [y, u]) u − ad(u) (yt + [y, u]) = −α(u)yt − ad([y, u]) u − ad(u) yt − ad(u) ([y, u]) = − ad(u) yt − α(u)yt + [ad(y) , ad(u) ]u − ad(u) ad(y)u . Finally we get the Jacobi equation as ytt = [ut , y] + [u, yt ] + ust = ad(y) ad(u) u + ad(u)yt − ad(u) yt − α(u)yt + [ad(y) , ad(u) ]u − ad(u) ad(y)u , ytt = [ad(y) + ad(y), ad(u) ]u − ad(u) yt − α(u)yt + ad(u)yt .

(3.5.1)

3.6 Jacobi fields, II Let y be a Jacobi field along a geodesic g with right trivialized velocity field u. Then y should satisfy the analogue of the finite dimensional Jacobi equation ∇∂t ∇∂t y + R(y, u)u = 0.

152

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We want to show that this leads to the same equation as (3.5.1). First note that from (3.3.2) we have ∇∂t y = yt +

1 1 1 ad(u) y + α(u)y − ad(u)y 2 2 2

so that, using ut = − ad(u) u, we get:   1 1 1 ∇∂t ∇∂t y = ∇∂t yt + ad(u) y + α(u)y − ad(u)y 2 2 2 = ytt +

1 1 1 ad(ut ) y + ad(u) yt α(ut )y 2 2 2

1 1 1 + α(u)yt − ad(ut )y − ad(u)yt 2 2 2   1 1 1 1

yt + ad(u) y + α(u)y − ad(u)y + ad(u) 2 2 2 2   1 1 1 1

+ α(u) yt + ad(u) y + α(u)y − ad(u)y 2 2 2 2   1 1 1 1 − ad(u) yt + ad(u) y + α(u)y − ad(u)y 2 2 2 2 = ytt + ad(u) yt + α(u)yt − ad(u)yt 1 1 1 − α(y) ad(u) u − ad(y) ad(u) u − ad(y) ad(u) u 2 2 2   1 1 1 1

α(y)u + ad(y) u + ad(y)u + ad(u) 2 2 2 2   1 1 1 1

+ α(u) α(y)u + ad(y) u + ad(y)u 2 2 2 2   1 1 1 1 − ad(u) α(y)u + ad(y) u + ad(y)u . 2 2 2 2 In the second line of the last expression we use 1 1 1 − α(y) ad(u) u = − α(y) ad(u) u − α(y)α(u)u 2 4 4 and similar forms for the other two terms to get: ∇∂t ∇∂t y = ytt + ad(u) yt + α(u)yt − ad(u)yt 1 1 1 + [ad(u) , α(y)]u + [ad(u) , ad(y) ]u + [ad(u) , ad(y)]u 4 4 4

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1 1 1 + [α(u), α(y)]u + [α(u), ad(y) ]u + [α(u), ad(y)]u 4 4 4 1 1 − [ad(u), α(y)]u − [ad(u), ad(y) + ad(y)]u, 4 4 where in the last line we also used ad(u)u = 0. We now compute the curvature term using (3.4.2): 1 R(y, u)u = − [ad(y) + ad(y), ad(u) + ad(u)]u 4 1 1 + [ad(y) − ad(y), α(u)]u + [α(y), ad(u) − ad(u)]u 4 4 1 1 + [α(y), α(u)] + α([y, u])u 4 2 1 1 = − [ad(y) + ad(y), ad(u) ]u − [ad(y) + ad(y), ad(u)]u 4 4 1 1 1 + [ad(y) , α(u)]u − [ad(y), α(u)]u + [α(y), ad(u) − ad(u)]u 4 4 4 1 1 + [α(y), α(u)]u + ad(u) ad(y)u . 4 2 Summing up we get ∇∂t ∇∂t y + R(y, u)u = ytt + ad(u) yt + α(u)yt − ad(u)yt 1 − [ad(y) + ad(y), ad(u) ]u 2 1 1 + [α(u), ad(y)]u + ad(u) ad(y)u . 2 2 Finally we need the following computation using (3.4.1): 1 1 1 [α(u), ad(y)]u = α(u)[y, u] − ad(y)α(u)u 2 2 2 =

1 1 ad([y, u]) u − ad(y) ad(u) u 2 2

1 1 = − [ad(y) , ad(u) ]u − ad(y) ad(u) u . 2 2 Inserting we get the desired result: ∇∂t ∇∂t y + R(y, u)u = ytt + ad(u) yt + α(u)yt − ad(u)yt − [ad(y) + ad(y), ad(u) ]u.

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3.7 The weak symplectic structure on the space of Jacobi fields Let us assume now that the geodesic equation in g, ut = − ad(u) u, admits a unique solution for some time interval, depending smoothly on the choice of the initial value u(0). Furthermore we assume that G is a regular Lie group (B.9) so that each smooth curve u in g is the right logarithmic derivative of a smooth curve g in G which depends smoothly on u, so that u = (g ∗ κ)(∂t ). Furthermore we have to assume that the Jacobi equation along u admits a unique solution for some time, depending smoothly on the initial values y(0) and yt (0). These are non-trivial assumptions: in (A.4) there are examples of ordinary linear differential equations ‘with constant coefficients’ which violate existence or uniqueness. These assumptions have to be checked in the special situations. Then the space Ju of all Jacobi fields along the geodesic g described by u is isomorphic to the space g × g of all initial data. There is the well-known symplectic structure on the space Ju of all Jacobi fields along a fixed geodesic with velocity field u, see e.g., [28], II, p.70. It is given by the following expression which is constant in time t: ω(y, z) : = y, ∇∂t z − ∇∂t y, z

4 5 1 1 1 = y, zt + ad(u) z + α(u)z − ad(u)z 2 2 2 4 5 1 1 1

− yt + ad(u) y + α(u)y − ad(u)y, z 2 2 2 = y, zt − yt , z + [u, y], z − y, [u, z] − [y, z], u

5 4 5 4 1 1 = y, zt − ad(u)z + α(u)z − yt − ad(u)y + α(u)y, z . 2 2 It is worth while to check directly from the Jacobi field equation (3.5.1) that ω(y, z) is indeed constant in t. Clearly ω is a weak symplectic structure on the relevant vector space Ju ∼ = g × g, i.e., ω gives an injective (but in general not surjective) linear mapping Ju → Ju∗ . This is seen most easily by writing ω(y, z) = y, zt − Γg (u, z) |t=0 − yt − Γg (u, y), z |t=0 which is induced from the standard symplectic structure on g×g∗ by applying first the automorphism (a, b) → (a, b − Γg (u, a)) to g × g and then by injecting the second factor g into its dual g∗ . For regular (infinite-dimensional) Lie groups, variations of geodesics exist, but there is no general theorem stating that they are uniquely determined by y(0) and yt (0). For concrete regular Lie groups, this needs to be shown directly.

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4 The Hamiltonian approach 4.1 The symplectic form on T ∗ G and G × g∗ For an (infinite-dimensional regular) Lie group G with Lie algebra g, elements in the cotangent bundle π : (T ∗ G, ω G ) → G are said to be in material or Lagrangian representation. The cotangent bundle T ∗ G has two trivializations, the left one (π G , κl ) : T ∗ G → G × g∗ , Tg∗ G  αg → (g, Te (µg )∗ αg = Tg∗ (µg−1 )αg ), also called the body coordinate chart, and the right one, (π G , κr ) : T ∗ G → G × g∗ , T ∗ G  αg → (g, Te (µg )∗ αg = Tg∗ (µg )αg ), Tg (µg

−1

(4.1.1)

)∗ α ← (g, α) ∈ G × g∗ ,

also called the space or Eulerian coordinate chart. We will use only this from now on. The canonical 1-form in the Eulerian chart is given by (where , : g∗ × g → R is the duality pairing): θ G×g∗ (ξ g , α, β) := (((π, κr )−1 )∗ θG )(g,α) (ξ g , α, β) = θG (T(g,α) (π, κr )−1 (ξ g , α, β)) = π T ∗ G (T(g,α) (π, κr )−1 (ξ g , α, β)), T (π)(T(g,α) (π, κr )−1 (ξ g , α, β))

= (π, κr )−1 (π G , π g∗ )(ξ g , α, β), T (π ◦ (π, κr )−1 )(ξ g , α, β))

= (π, κr )−1 (g, α), T (pr1 )(ξ g , α, β)) = Tg (µg = α, Tg (µg

−1

)ξ g = α, κr (ξ g ) .

−1

)∗ α, ξ g

(4.1.2)

Now it is easy to take the exterior derivative: For Xi ∈ G, thus RXi ∈ X(G) right invariant vector fields, and g∗  β i ∈ X(g∗ ) constant vector fields, we have θ G×g∗ (RXi (g), (α, β i )) = α, Xi , θ G×g∗ (RXi , β i ) = Idg∗ , Xi = , Xi , ω G×g∗ ((RX1 , β 1 ), (RX2 , β 2 )) = −dθG×g∗ ((RX1 , β 1 ), (RX2 , β 2 ))

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= −(RX1 , β 1 )(θ G×g∗ (RX2 , β 2 )) + (RX2 , β 2 )(θ G×g∗ (RX1 , β 1 )) + (θ G×g∗ ([(RX1 , β 1 ), RX2 , β 2 )]) = −(RX1 , β 1 )( , X2 ) + (RX2 , β 2 )(

, X1 )

+ (θ G×g∗ (−R[X1 ,X2 ] , 0g∗ ) = − β 1 , X2 + β 2 , X1 − , [X1 , X2 ] , (ω G×g∗ )(g,α) ((T (µg ).X1 , β 1 ), (T (µg )X2 , β 2 )) = β 2 , X1 − β 1 , X2 − α, [X1 , X2 ] .

(4.1.3)

4.2 The symplectic form on T G and G × g and the momentum mapping We consider an (infinite-dimensional regular) Lie group G with Lie algebra g and a bounded weak inner product γ : g × g → R with the property that the transpose of the adjoint action of G on g, γ(Ad(g) X, Y ) = γ(X, Ad(g)X), exists. It is then unique and a right action of G on g. By differentiating it follows that then also the transpose of the adjoint operation of g exists: γ(ad(X) Y, Z) = ∂t |0 γ(Ad(exp(tX)) Y, Z) = γ(Y, ad(X)Z) exists. We extend γ to a right invariant Riemannian metric, again called γ on G and consider γ : T G → T ∗ G. Then we pull-back the canonical symplectic structure ω G to G × g in the right or Eulerian trivialization: γ : G × g → G × g∗ , (g, X) → (g, γ(X)) × (γ ∗ ω)(g,X) ((T (µg ).X1 , X, Y1 ), (T (µg )X2 , X, Y2 )) = ω(g,γ(X)) ((T (µg ).X1 , γ(X), γ(Y1 )), (T (µg )X2 , γ(X), γ(Y2 ))) = γ(Y2 ), X1 − γ(Y1 ), X2 − γ(X), [X1 , X2 ]

= γ(Y2 , X1 ) − γ(Y1 , X2 ) − γ(X, [X1 , X2 ]).

(4.2.1)

Since γ is a weak inner product, γ ∗ ω is again a weak symplectic structure on TG ∼ = G × g. We compute the Hamiltonian vector field mapping (symplectic gradient) for functions f ∈ Cγ∞∗ ω (G × g) admitting such gradients: (γ ∗ ω)(g,X) (gradγ

∗

ω

(f )(g, X), (T (µg )X2 , X, Y2 ))

= df (T (µg )X2 ; X, Y2 )

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= d1 f (g, X)(T (µg )X2 ) + d2 f (g, X)(Y2 ) = γ(κr (gradγ1 (f )(g, X)), X2 ) + γ(gradγ2 (f )(g, X), Y2 ) = γ(X1 , Y2 ) + γ(−Y1 − ad(X1 ) X, X2 ) Thus the Hamiltonian vector field of f ∈ gradγ

∗

ω

Cγ∞∗ ω (G

by ((4.2.1)).

× g) = Cγ∞ (G × g) is

(f )(g, X)

(4.2.2)

= (T (µg ) gradγ2 (f )(g, X), X,− ad(gradγ2 (f )(g, X)) X −κr (gradγ1 (f )(g, X))). In particular, the Hamiltonian vector field of the function (g, X) → γ(X, X) = X2γ on T G is given by   ∗ 1  2γ (g, X) = (T (µg )X; X, − ad(X) X). gradγ ω (4.2.3) 2 We can now compute again the flow equation of the Hamiltonian vector field   ∗ gradγ ω 12  2γ : For gt (t) ∈ T G we have −1

(π G , κr )(gt (t)) = (g(t), u(t)) = (g(t), T (µg(t) )gt (t)) and γ∗ ω

∂t (g, u) = grad



1  2

 2γ

(g, u) = (T (µg )u, u, − ad(u) u)

(4.2.4)

which reproduces the geodesic equation from (3.2). 4.3 The momentum mapping Under the assumptions of (4.2), consider the right action of G on G and its prolongation to a right action of G on T G in the Eulerian chart. The corresponding fundamental vector fields are then given by: T (µg ) : T G → T G, (π, κr )T (µg )T (µh )X = (π, κr )T (µhg )X = (h.g, X),

(h, X) → (hg, X)

(h, Y ) = ∂t |0 (h. exp(tX), Y ) = (T (µh )X, 0Y ) ∈ T G × T g. ζ G×g X (4.3.1) Consider now the diagram from (2.1) in the case of the weak symplectic manifold (M = G × g, γ ∗ ω): γ∗ ω

H0

grad / Cγ∞∗ ω (G × g, R) / X(G × g, γ ∗ ω) 9 eLLL ss LLL ss s LLL ss j LL ssss ζ g

/ Hγ1∗ ω

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From the formulas derived above we see that for j(X)(h, Y ) := γ(Ad(h)X, Y ) we have: γ(gradγ2 (j(X))(h, Y ), Z) = d2 (j(X))(h, Y )(Z) = γ(Ad(h)X, Z), gradγ2 (j(X))(h, Y ) = Ad(h)X, γ(gradγ1 (j(X))(h, Y ), T (µh )Z) = d(j(X))(T (µh )Z, Y, 0) = γ(d Ad(T (µh )Z)(X), Y ) = γ(((ad ◦κr ) Ad)(T (µh )Z)(X), Y ) = γ(ad(Z) Ad(h)X, Y ) = −γ([Ad(h)X, Z], Y ) = −γ(Z, ad(Ad(h)X) Y ), κr (gradγ1 (j(X))(h, Y )) = − ad(Ad(h)X) Y. Thus the momentum mapping is J : G × g → g∗ ,

J ∈ Cγ∞∗ ω (G × g, g∗ )

= {f ∈ C ∞ (G × g, g∗ ) : f (

), X ∈ Cγ∞∗ ω (G × g) ∀X ∈ g},

J(h, Y ), X = j(X)(h, Y ) = γ(Ad(h)X, Y ) = γ(Ad(h) Y, X) = γ(Ad(h) Y ), X , J(h, Y ) = γ(Ad(h) Y ) ∈ g∗, J¯ := γ −1 ◦ J : G × g → g, ¯ Y ) = Ad(h) Y ∈ g. J(h,

(4.3.2)

(4.3.3) Note that the momentum mapping J : G × g → g∗ is equivariant for the right G-action and the coadjoint action, and that J¯ : G × g → g is equivariant for the right action Ad( ) on g: J(hg, Y ), X = γ(Ad(hg) Y ), X = γ(Ad(g) Ad(h) Y, X) = γ(Ad(h) Y, Ad(g)X) = γ(Ad(h) Y ), Ad(g)X

= Ad(g)∗ γ(Ad(h) Y ), X = Ad(g)∗ J(h, Y ), X , ¯ ¯ Y ). J(hg, Y ) = Ad(hg) Y = Ad(g) J(h, ¯ (4.3.4) For x ∈ G × g, the transposed mapping of dJ(x) : Tx (G × g) → g is ¯ : g → T ∗ (G × g), dJ(x) x

dJ¯(x) = (γ ∗ ω)x ◦ ζ,

since for ξ ∈ Tx (G × g) and X ∈ g we have ¯ X)(ξ) = dj(X)(ξ) = (γ ∗ ω)(ζ ), ξ . γ(dJ¯(ξ), X) = dγ(J, X

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¯ (4.3.5) For x ∈ G × g, the closure dJ¯(Tx (G × g)) of the image of dJ(x) : ⊥,γ Tx (G × g) → g is the γ-orthogonal space gx of the isotropy Lie algebra gx := {X ∈ g : ζ X (x) = 0} in g, since the annihilator of the image is the kernel of the transposed mapping, im(dJ(x))◦ = ker(dJ(x) ) = ker((γ ∗ ω)x ◦ ζ) = ker(evx ◦ζ) = gx . Attention: the orthogonal space with respect to a weak inner product need not be a complement. (4.3.6) For (h, Y ) ∈ G×g, the G-orbit (h, Y ).G = G×{Y } is a submanifold of G × g. The kernel of dJ¯(h, Y ) is the symplectic orthogonal space (T(h,Y ) (G × {Y }))⊥,γ

∗

ω

⊂ T (µh )g × g

since for the annihilator of the kernel we have ker(dJ¯(h, Y ))◦ ¯ Y ) ) = im((γ ∗ ω(h,Y ) ◦ ζ), = im(dJ(h,

by ((4.3.4)),

= {(γ ∗ ω)(h,Y ) (ζ X (x)) : X ∈ g} = (γ ∗ ω)(h,Y ) (T(h,Y ) (G × {Y })), = ((T(h,Y ) (G × {Y }))⊥,γ

∗

ω ◦

) .

The last equality holds by the bipolar theorem for the usual duality pairing. (4.3.7) Thus, for (h, Y ) ∈ G × g, ¯ Y )) T (µh )X1 , Y1 ) ∈ ker(dJ(h, ⇐⇒ (γ ∗ ω)(h,Y ) ((T (µh )X1 , Y1 ), (T (µh )Z, 0)) = 0for all Z ∈ g ⇐⇒ 0 = 0 − γ(Y1 , Z) − γ(Y, [X1 , Z]) = −γ(Y1 + ad(X1 ) Y, Z) ∀ Z ∈ g ⇐⇒ Y1 = − ad(X1 ) Y. (4.3.8) (Emmy Noether’s theorem) Let h ∈ Cω∞ (G × g) be a Hamiltonian γ∗ ω ¯ (h)) = function which is invariant under the right G-action. Then dJ(grad γ∗ω ∗ 0 ∈ g and also dJ(grad (h)) = 0 ∈ γ(g) ⊆ g . Thus the momentum mappings J¯ : G× g → g and J : G× g → γ(g) ⊂ g∗ are constant on each trajectory ∗ (if it exists) of the Hamiltonian vector field gradγ ω (h). Namely, consider the ¯ X) = J, X = j(X). function γ(J, γ(dJ¯(gradγ

∗

ω

(h)), X) = gradγ

∗

ω

¯ X)) (h)(γ(J,

= {h, γ(J¯, X)} = −{j(X), h} = −ζ X (h) = 0, dJ(gradγ

∗

ω

(h)), X = gradγ

∗

ω

(h)( J, X )

= {h, j(X)} = −{j(X), h} = −ζ X (h) = 0.

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4.4 The geodesic equation via conserved momentum We consider a smooth curve t → g(t) in G and (π G , κr )gt (t) = (g(t), u(t)) = −1 (g(t), T (µg(t) )gt (t)) as in (4.2.4). Applying J¯ : G × g → g to it we get ¯ u) = Ad(g) u. We claim that the curves t → g(t) in G for which J(g, ¯ J(g(t), u(t)) is constant in t are exactly the geodesics in (G, γ). Namely, by (3.1) we have 0 = ∂t Ad(g(t)) u(t) = ((ad ◦κr )(∂t g(t)). Ad(g(t))) u(t) + Ad(g(t)) ∂t u(t) = Ad(g(t)) (ad(u(t)) u(t) + ut (t)) ⇐⇒

ut = − ad(u) u.

4.5 Symplectic reduction to transposed adjoint orbits Under the assumptions of (4.2) we have the following: ¯ (4.5.1) For X ∈ J(G× g) the inverse image J¯−1 (X) ⊂ G× g is a manifold. Namely, it is the graph of a smooth mapping: J¯−1 (X) = {(h, Y ) ∈ G × g : Ad(h) Y = X} ∼

= {(h, Ad(h−1 ) X) : h ∈ G} ←=− G.

2

(4.5.2) At any point of J¯−1 (X), the kernel of the pull-back of the symplectic form γ ∗ ω on G × g from (4.2.1) equals the tangent space to the orbit of the isotropy group GX := {g ∈ G : Ad(g) X = X} through that point. For (h, Y = Ad(h−1 ) X) ∈ J¯−1 (X) the GX -orbit is h.GX × {Y } and its tangent space at (h, Y ) is T (µh )gX × 0 where gX = {Z ∈ g : ad(Z) X = 0}. The tangent space at (h, Y ) of J¯−1 (X) is T(h,Ad(h−1 ) X) J¯−1 (X) = {∂t |0 (exp(tZ).h, Ad((exp(tZ).h)−1 ) X) : Z ∈ g} = {(T (µh )Z,− ad(Z) Ad(h−1 ) X) : Z ∈ g} ⊂ Th G× g. For Z1 , Z2 ∈ g consider the tangent vectors (T (µh ) Ad(h)Z1 , Y, − ad(Z1 )X) and (T (µh )Z, Y, − ad(Z) Ad(h−1 ) X) in T(h,Y ) J¯−1 (X). From (4.2.1), we get (γ ∗ ω)(h,Y ) ((T (µh ) Ad(h)Z1 ,−ad(Z1 ) X), (T (µh )Z2 ,−ad(Z2 ) Ad(h−1 ) X)) = γ(− ad(Z2 ) Ad(h−1 ) X, Ad(h)Z1 ) − γ(− ad(Z1 ) X, Z2 ) − γ(Y, [Ad(h)Z1 , Z2 ]) = −γ(Ad(h−1 ) X, ad(Z2 ) Ad(h)Z1 ) + γ(ad(Z1 ) X, Z2 ) − γ(Ad(h−1 ) X, [Ad(h)Z1 , Z2 ]) = γ(ad(Z1 ) X, Z2 ) = 0

∀Z2 ∈ g ⇐⇒ Z1 ∈ gX .

2

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(4.5.3) The reduced symplectic manifold J¯−1 (X)/GX with symplectic form induced by γ ∗ ω|J¯−1 (X) is symplectomorphic to the adjoint orbit Ad(G) X ⊂ g with symplectic form the pull-back via γ : g → g∗ of the Kostant–Kirillov– Souriou form ω α (ad(Y1 )∗ α, ad(Y2 )∗ α) = α, [Y1 , Y2 ]

which is given by ωZ (ad(Y1 ) Z, ad(Y2 ) Z) = ωγ(Z) (γ ad(Y1 ) Z, γ ad(Y2 ) Z) = ωγ(Z) (ad(Y1 )∗ γZ, ad(Y2 )∗ γZ) = γ(Z), [Y1 , Y2 ]

= γ(Z, [Y1 , Y2 ]), since for Y, Z, U ∈ g we get γ ad(Y ) Z, U = γ(ad(Y ) Z, U ) = γ(Z, ad(Y )U ) = γ(Z), ad(Y )U = ad(Y )∗ γ(Z), U . ∼ The quotient space is J¯−1 (X)/GX = {(h.GX , Ad(h−1 ) X) : h ∈ G} = Ad(G) X ∼ = G/GX . The 2-form γ ∗ ω|J¯−1 (X) induces a symplectic form on the quotient by (4.5.2) and it remains to check that it agrees with the pullback of the Kirillov–Kostant–Souriou symplectic form. But this is obvious from the last computation in (4.5.2) (for the special case h = e if the reader insists). 2 (4.5.4) Reconsider the geodesic equation on the reduced space J¯−1 (X)/GX ∼ = Ad(G) X. The energy function is E(Ad(g) X) = 12  Ad(g) X2γ . For Z = Ad(g) X ∈ Ad(G) X the tangent space is given by TZ (Ad(G) X) = {ad(Y ) Z : Y ∈ g}. We look for the Hamiltonian vector field of E in the form gradω E(Z) = ad(HE (Z)) Z, for a vector field HE . The differential of the energy function is dE(Z)(ad(Y ) Z) = γ(Z, ad(Y ) Z) = γ([Y, Z], Z) which equals ω Z (gradω E(Z), ad(Y ) Z) = ω Z (ad(HE (Z)) Z, ad(Y ) Z) = γ(Z, [HE (Z), Y ]) from which we conclude that HE (Z) = −Z will do (which is defined up to an annihilator of Z). Thus gradω E(Z) = − ad(Z) Z, which again leads us back to the geodesic equation ut = − ad(u) u.

5 Vanishing H 0 -geodesic distance on groups of diffeomorphisms This section is based on [40]. 5.1 The H 0 -metric on groups of diffeomorphisms Let (N, g) be a smooth connected Riemannian manifold, and let Diff c (N ) be the group of all diffeomorphisms with compact support on N , and let Diff 0 (N )

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be the subgroup of those which are diffeotopic in Diff c (N ) to the identity; this is the connected component of the identity in Diff c (N ), which is a regular Lie group in the sense of [31], section 38. This is proved in [31], section 42. The Lie algebra is Xc (N ), the space of all smooth vector fields with compact support on N , with the negative of the usual bracket of vector fields as Lie bracket. Moreover, Diff 0 (N ) is a simple group (has no nontrivial normal subgroups), see [18], [50], [35]. The right invariant H 0 -metric on Diff 0 (N ) is then given as follows, where h, k : N → T N are vector fields with compact support along ϕ and where X = h ◦ ϕ−1 , Y = k ◦ ψ −1 ∈ Xc (N ):   0 ∗ γ ϕ (h, k) = g(h, k) vol(ϕ g) = g(X ◦ ϕ, Y ◦ ϕ)ϕ∗ vol(g) N

N



g(X, Y ) vol(g).

=

(5.1.1)

N

5.2 Theorem Geodesic distance on Diff 0 (N ) with respect to the H 0 -metric vanishes. Proof. Let [0, 1]  t → ϕ(t, ) be a smooth curve in Diff 0 (N ) between ϕ0 and ϕ1 . Consider the curve u = ϕt ◦ ϕ−1 in Xc (N ), the right logarithmic derivative. Then for the length and the energy we have:  6 1

u2g vol(g) dt,

Lγ 0 (ϕ) = 0



1

 u2g vol(g) dt,

E (ϕ) = γ0

0

(5.2.1)

N

(5.2.2)

N

Lγ 0 (ϕ)2 ≤ Eγ 0 (ϕ).

(5.2.3)

(5.2.4) Let us denote by Diff 0 (N )E=0 the set of all diffeomorphisms ϕ ∈ Diff 0 (N ) with the following property: For each ε > 0 there exists a smooth curve from the identity to ϕ in Diff 0 (N ) with energy ≤ ε. (5.2.5) We claim that Diff 0 (N )E=0 coincides with the set of all diffeomorphisms which can be reached from the identity by a smooth curve of arbitrarily short γ 0 -length. This follows by (5.2.3). (5.2.6) We claim that Diff 0 (N )E=0 is a normal subgroup of Diff 0 (N ). Let ϕ1 ∈ Diff 0 (N )E=0 and ψ ∈ Diff 0 (N ). For any smooth curve t → ϕ(t, ) from the identity to ϕ1 with energy Eγ 0 (ϕ) < ε we have Eγ 0 (ψ −1 ◦ ϕ ◦ ψ)  1 T ψ−1 ◦ ϕt ◦ ψ2g vol((ψ −1 ◦ ϕ ◦ ψ)∗ g) = 0

N

Geometric evolution equations

≤ sup Tx ψ −1 2 · x∈N



1

0

≤ sup Tx ψ −1 2 · sup x∈N

x∈N

≤ sup Tx ψ −1 2 · sup x∈N

x∈N

163

ϕt ◦ ψ2g (ϕ ◦ ψ)∗ vol((ψ −1 )∗ g)

N

vol((ψ −1 )∗ g) · vol(g)

 0

1



ϕt ◦ ψ2g (ϕ ◦ ψ)∗ vol(g)

N

−1 ∗

vol((ψ ) g) · Eγ 0 (ϕ). vol(g)

Since ψ is a diffeomorphism with compact support, the two suprema are bounded. Thus ψ −1 ◦ ϕ1 ◦ ψ ∈ Diff 0 (N )E=0 . (5.2.7) We claim that Diff 0 (N )E=0 is a nontrivial subgroup. In view of the simplicity of Diff 0 (N ) mentioned in (5.1) this concludes the proof. It remains to find a nontrivial diffeomorphism in Diff 0 (N )E=0 . The idea is to use compression waves. The basic case is this: take any nondecreasing smooth function f : R → R such that f (x) ≡ 0 if x  0 and f (x) ≡ 1 if x  0. Define ϕ(t, x) = x + f (t − λx) where λ < 1/ max(f
). Note that ϕx (t, x) = 1 − λf
(t − λx) > 0, hence each map ϕ(t, ) is a diffeomorphism of R and we have a path in the group of diffeomorphisms of R. These maps are not the identity outside a compact set however. In fact, ϕ(x) = x + 1 if x  0 and ϕ(x) = x if x  0. As t → −∞, the map ϕ(t, ) approaches the identity uniformly on compact subsets, while as t → +∞, the map approaches translation by 1. This path is a moving compression wave which pushes all points forward by a distance 1 as it passes. We calculate its energy between two times t0 and t1 :  t1   t1  ϕt (t, ϕ(t, )−1 (x))2 dx dt = ϕt (t, y)2 ϕy (t, y)dy dt Ett01 (ϕ) = t0



R

t1 

= t0

R

t0

f
(z)2 · (1 − λf
(z))

max f
≤ · (t1 − t0 ) · λ 2

R

dz dt λ

 supp(f
)

(1 − λf
(z))dz.

If we let λ = 1 − ε and consider the specific f given by the convolution f (z) = max(0, min(1, z))  Gε (z), where Gε is a smoothing kernel supported on [−ε, +ε], then the integral is bounded by 3ε, hence Ett01 (ϕ) ≤ (t1 − t0 )

3ε . 1−ε

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Peter W. Michor

We next need to adapt this path so that it has compact support. To do this we have to start and stop the compression wave, which we do by giving it variable length. Let fε (z, a) = max(0, min(a, z))  (Gε (z)Gε (a)). The starting wave can be defined by ϕε (t, x) = x + fε (t − λx, g(x)),

λ < 1,

g increasing.

Note that the path of an individual particle x hits the wave at t = λx − ε and leaves it at t = λx + g(x) + ε, having moved forward to x + g(x). Calculate the derivatives: (fε )z = I0≤z≤a  (Gε (z)Gε (a)) ∈ [0, 1], (fε )a = I0≤a≤z  (Gε (z)Gε (a)) ∈ [0, 1], (ϕε )t = (fε )z (t − λx, g(x)), (ϕε )x = 1 − λ(fε )z (t − λx, g(x)) + (fε )a (t − λx, g(x)) · g
(x) > 0. This gives us: Ett01 (ϕ) =



t1

 R

t0

 ≤

t1

 R

t0



(ϕε )2t (ϕε )x dx dt

t1

(fε )2z (t − λx, g(x)) · (1 − λ(fε )z (t − λx, g(x)))dx dt



+ t0

R

(fε )2z (t − λx, g(x)) · (fε )a (t − λx, g(x))g
(x)dx dt.

The first integral can be bounded as in the original discussion. The second integral is also small because the support of the z-derivative is −ε ≤ t − λx ≤ g(x) + ε, while the support of the a-derivative is −ε ≤ g(x) ≤ t − λx + ε, so together |g(x) − (t − λx)| ≤ ε. Now define x1 and x2 by g(x1 ) + λx1 = t + ε and g(x0 ) + λx0 = t − ε. Then the inner integral is bounded by  g
(x)dx = g(x1 ) − g(x0 ) ≤ 2ε, |g(x)+λx−t|≤ε

and the whole second term is bounded by 2ε(t1 − t0 ). Thus the length is O(ε). The end of the wave can be handled by playing the beginning backwards. If the distance that a point x moves when the wave passes it is to be g(x), so that the final diffeomorphism is x → x + g(x), then let b = max(g) and use the above definition of ϕ while g
> 0. The modification when g
< 0 (but g
> −1 in order for x → x + g(x) to have positive derivative) is given by: ϕε (t, x) = x + fε (t − λx − (1 − λ)(b − g(x)), g(x)).

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165

Particle trajectories under φ, λ = 0.6 3.5

3

2.5

Space x

2

1.5

1

0.5

0

−0.5 −0.5

0

0.5

1

1.5

2

2.5

3

Time t

Consider the figure showing the trajectories ϕε (t, x) for sample values of x. It remains to show that Diff 0 (N )E=0 is a nontrivial subgroup for an arbitrary Riemannian manifold. We choose a piece of a unit speed geodesic containing no conjugate points in N and Fermi coordinates along this geodesic; so we can assume that we are in an open set in Rm which is a tube around a piece of the u1 -axis. Now we use a small bump function in the slice orthogonal to the u1 -axis and multiply it with the construction from above for the coordinate u1 . Then it follows that we get a nontrivial diffeomorphism in Diff 0 (N )E=0 again. 2 Remark Theorem (5.2) can be proved directly without the help of the simplicity of Diff 0 (N ). For N = R one can use the method of (5.2.7) in the parameter space of a curve, and for general N one can use a Morse function on N to produce a special coordinate for applying the same method. 5.3 Geodesics and sectional curvature for γ 0 on Diff (N ) According to (3.2), (3.4), or (4.4), for a right invariant weak Riemannian metric G on a (possibly infinite-dimensional) Lie group, the geodesic equation and the curvature are given in terms of the transposed operator (with respect to G, if it exists) of the Lie bracket by the following formulas: ut = − ad(u)∗ u,

u = ϕt ◦ ϕ−1 ,

G(ad(X)∗ Y, Z) := G(Y, ad(X)Z), 4G(R(X, Y )X, Y ) = 3G(ad(X)Y, ad(X)Y ) − 2G(ad(Y )∗ X, ad(X)Y )

166

Peter W. Michor

− 2G(ad(X)∗ Y, ad(Y )X) + 4G(ad(X)∗ X, ad(Y )∗ Y ) − G(ad(X)∗ Y + ad(Y )∗ X, ad(X)∗ Y + ad(Y )∗ X). In our case, for Diff 0 (N ), we have ad(X)Y = −[X, Y ] (the bracket on the Lie algebra Xc (N ) of vector fields with compact support is the negative of the usual one), and:  γ 0 (X, Y ) = g(X, Y ) vol(g), N

γ 0 (ad(Y )∗ X, Z) = γ 0 (X, −[Y, Z]) =

 g(X, −LY Z) vol(g) N



g(LY X + (g −1 LY g)X + divg (Y )X, Z) vol(g),

= N

ad(Y )∗ = LY + g −1 LY (g) + divg (Y ) IdT N = LY + β(Y ), where the tensor field β(Y ) = g −1 LY (g) + divg (Y ) Id : T N → T N is selfadjoint with respect to g. Thus the geodesic equation is ut = −(g −1 Lu (g))(u) − divg (u)u = −β(u)u,

u = ϕt ◦ ϕ−1 .

The main part of the sectional curvature is given by: 4G(R(X, Y )X, Y )  = (3[X, Y ]2g +2g((LY + β(Y ))X, [X, Y ]) +2g((LX + β(X))Y, [Y, X]) N

+ 4g(β(X)X, β(Y )Y ) − β(X)Y + β(Y )X2g ) vol(g)  (−β(X)Y − β(Y )X + [X, Y ]2g − 4g([β(X), β(Y )]X, Y )) vol(g). = N

So sectional curvature consists of a part which is visibly nonnegative, and another part which is difficult to decompose further. 5.4 Example: n-dimensional analog of Burgers’ equation For (N, g) = (Rn , can) or ((S 1 )n , can) we have:
((∂i X k )Y i − X i (∂i Y k )), (ad(X)Y )k = i

(ad(X)∗ Z)k =


i

((∂k X i )Z i + (∂i X i )Z k + X i (∂i Z k )),

Geometric evolution equations

167

so that the geodesic equation is given by
((∂k ui )ui + (∂i ui )uk + ui (∂i uk )), ∂t uk = −(ad(u) u)k = − i

the n-dimensional analog of Burgers’ equation. 5.5 Stronger metrics on Diff 0 (N ) A very small strengthening of the weak Riemannian H 0 -metric on Diff 0 (N ) makes it into a true metric. We define the stronger right invariant semiRiemannian metric by the formula  A (g(X, Y ) + A divg (X). divg (Y )) vol(g). Gϕ (X ◦ ϕ, Y ◦ ϕ) = N

Then the following holds: Theorem For any distinct diffeomorphisms ϕ0 , ϕ1 , the infimum of the lengths of all paths from ϕ0 to ϕ1 with respect to GA is positive. Proof. We may suppose that ϕ0 = IdN . If ϕ1 = IdN , there are two functions ρ and f on N with compact support such that   ρ(y)f (ϕ1 (y)) vol(g)(y) = ρ(y)f (y) vol(g)(y). N

N

Now consider any path ϕ(t, y) between ϕ0 = IdN to ϕ1 with left logarithmic derivative u = T (ϕ)−1 ◦ ϕt and a path in Xc (N ). Then we have: 

 ρ(f ◦ ϕ1 ) vol(g) −

N





1





1



N 1

) vol(g)dt

N



ρ(df.ϕt ) vol(g) dt = 0

 ρ∂tf (ϕ(t,

0

N

=

1

ρf vol(g) =

ρ(df.T ϕ.u) vol(g)dt 0

N

 (df.T ϕ.(ϕu)) vol(g)dt.

= 0

N

Locally, on orientable pieces of N , we have: div((f ◦ ϕ)ρu) vol(g) = L(f ◦ϕ)ρu vol(g) = (i(f ◦ϕ)ρu d + di(f ◦ϕ)ρu ) vol(g) = d((f ◦ ϕ)iρu vol(g)) = d(f ◦ ϕ) ∧ iρu vol(g) + ρ div(u) vol(g),

168

Peter W. Michor

= d(f ◦ ϕ)(ρu) vol(g)+(f ◦ ϕ) div(ρu) vol(g),

since

d(f ◦ ϕ) ∧ iρu vol(g) = −iρu (d(f ◦ ϕ) ∧ vol(g))+(iρu d(f ◦ ϕ)) vol(g)). Thus on N we have:  0= div((f ◦ ϕ)ρu) vol(g) N



 d(f ◦ ϕ)(ρu) vol(g) +

= N

(f ◦ ϕ) div(ρu) vol(g) N

and hence         0 ≤  ρ(f ◦ ϕ1 ) vol(g) − ρf vol(g) =  N

  = 

0

N

1 N

1 0

 N

  d(f ◦ ϕ)(ϕu)) vol(g)dt

  −(f ◦ ϕ) div(ρu) vol(g)dt 

1

6 Cρ u2 + Cρ
| div(u)|2 vol(g) dt

≤ sup |f | · 0

N

for constants Cρ , Cρ
depending only on ρ. Clearly the right-hand side gives a lower bound for the length of any path from ϕ0 to ϕ1 . 2 5.6 Geodesics and sectional curvature for GA on Diff (R) We consider the groups Diff c (R) or Diff(S 1 ) with Lie algebras Xc (R) or X(S 1 ) whose Lie brackets are ad(X)Y = −[X, Y ] = X
Y − XY
. The GA -metric equals the H 1 -metric on Xc (R), and we have:   GA (X, Y ) = (XY + AX
Y
)dx = X(1 − A∂x2 )Y dx, R

R

 GA (ad(X)∗ Y, Z) = (Y X
Z − Y XZ
+ AY
(X
Z − XZ
)
)dx R

 = Z(1−∂x2 )(1−∂x2)−1 (2Y X
+Y
X −2AY

X
−AY


X)dx, R

ad(X)∗ Y = (1 − ∂x2 )−1 (2Y X
+ Y
X − 2AY

X
− AY


X), ad(X)∗ = (1 − ∂x2 )−1 (2X
+ X∂x )(1 − A∂x2 ), so that the geodesic equation in Eulerian representation u = (∂t f ) ◦ f −1 ∈ Xc (R) or X(S 1 ) is

Geometric evolution equations

169

∂t u = − ad(u)∗ u = −(1 − ∂x2 )−1 (3uu
− 2Au

u
− Au


u), or ut − utxx = Auxxx .u + 2Auxx .ux − 3ux .u, which for A = 1 is the dispersionless version of the Camassa–Holm equation, see (7.3.4). Note that here geodesic distance is a well-defined metric describing the topology.

6 The regular Lie group of rapidly decreasing diffeomorphisms 6.1 Lemma For smooth functions of one variable we have: (f ◦ g)(p) (x) = p!


f (m) (g(x)) m!

m≥0

=




i=1 α∈Nm >0 α1 +···+αm =p




f (m) (g(x))

 

m≥0

m 3 g (αi ) (x)




N

>0 λ=(λn )∈N≥0 λ =m n n n λn n=p

αi !

 λn p! 3 g (n) (x) . λ! n>0 n!

Let f ∈ C ∞ (Rk ) and let g = (g1 , . . . , gk ) ∈ C ∞ (Rn , Rk ). Then for a multiindex γ ∈ Nn the partial derivative ∂ γ (f ◦ g)(x) of the composition is given by the following formula, where we use multiindex notation heavily. ∂ γ (f ◦ g)(x) =




(∂ f )(g(x))

β∈Nk

=

 

k×(Nn \0)

λ=(λiα )∈N α λiα =β i iα λiα α=γ






λ




β

n \0)

λ=(λiα )∈Nk×(N iα λiα α=γ

  γ! 3 1 λ! α! n

i

iα

α∈N α>0

λ

  γ! 3 1 λ! α! n

i

iα

(∂



α

λα

f )(g(x))

3

(∂ α gi (x))λiα

i,α>0

3

(∂ α gi (x))λiα

i,α>0

α∈N α>0

The one-dimensional version is due to Fa`a di Bruno [19], the only beatified mathematician. Proof. We compose the Taylor expansions of ∞ f (g(x) + h) : jg(x) f (h) =


f (m) (g(x)) hm , m!

m≥0

170

Peter W. Michor

g(x + t) : jx∞ g(t) = g(x) +


g (n) (x) tn , n!

n≥1

⎛ ⎞m
f (m) (g(x))
g (n) (x) ⎝ f (g(x + t)) : jx∞ (f ◦ g)(t) = tn ⎠ m! n!

=

m≥0

n≥1


f (m) (g(x))




m!

m≥0

α1 ,...,αm

Or we use the multinomial expansion ⎞m ⎛ q

⎝ aj ⎠ = j=1

λ1 ,...,λq ∈N≥0 λ1 +···+λq =m

 m 3 g (αi ) (x) tα1 +···+αm . α ! i >0 i=1

m! aλ1 . . . aλq q λ1 ! . . . λq ! 1

to get jx∞ (f


f (m) (g(x)) ◦ g)(t) = m! m≥0






N

>0 λ=(λn )∈N≥0 λ =m n n

m! λ!



3  g (n) (x) λn

n>0

n!

 t



n

λn n

where λ! = λ1 ! λ2 ! . . . ; most of the λi are 0. The multidimensional formula just uses more indices. 2 6.2 The space S(R) of all rapidly decreasing smooth functions f for which x → (1 + |x|2 )k ∂xn f (x) is bounded for all k ∈ N and all n ∈ N≥0 , with the locally convex topology described by these conditions, is a nuclear Fr´echet space. The dual space S
(R) is the space of tempered distributions. S(R) is a commutative algebra under pointwise multiplication and convo' lution (u ∗ v)(x) = u(x − y)v(y)dy. The Fourier transform   1 F (u)(ξ) = u ˆ(ξ) = e−ixξ u(x)dx, F −1 (a)(x) = eixξ a(ξ)dξ 2π is an isomorphism of S(R) and also of L2 (R) and has the following further properties: ∂7 ˆ(ξ), x u(ξ) = −iξ · u
u(x − a)(ξ) = eiaξ uˆ(ξ),

x7 · u(ξ) = −i∂ξ u ˆ(ξ),
eiax u(x)(ξ) = eiaξ u ˆ(ξ),

Geometric evolution equations

  ξ 1
u ˆ u(ax)(ξ) = , |a| a u7 ·v = u ˆ ∗ vˆ,

171


u(−x)(ξ) =u ˆ(−ξ),

u ∗v =u ˆ · vˆ.

In particular, for any polynomial P with constant coefficients we have u(ξ). F (P (−i∂x )u)(ξ) = P (ξ)ˆ S(R) satisfies the uniform V-boundedness principle for every point separating set V of bounded linear functionals by [31], 5.24, since it is a Fr´echet space; in particular for the set of all point evaluations {evx : S(R) → R, x ∈ R}. Thus a linear mapping  : E → S(R) is bounded (smooth) if and only if evx ◦f is bounded for each x ∈ R. 6.3 Lemma The space C ∞ (R, S(R)) of smooth curves in S(R) consists of all functions f ∈ C ∞ (R2 , R) satisfying the following property: • For all n, m ∈ N≥0 and each t ∈ R the expression (1 + |x|2 )k ∂tn ∂xm f (t, x) is uniformly bounded in x, locally in t. Proof. We use (A.3) for the set {evx : x ∈ R} of point evaluations in S
(R). 2 Note that S(R) is reflexive. Here ck (t) = ∂tk f (t, ). 6.4 Diffeomorphisms which decrease rapidly to the identity Any orientation preserving diffeomorphism R → R can be written as Id +f for f a smooth function with f
(x) > −1 for all x ∈ R. Let us denote by Diff S (R)0 the space of all diffeomorphisms Id +f : R → R (so f
(x) > −1 for all x ∈ R) for f ∈ S(R). Theorem Diff S (R)0 is a regular Lie group. Proof. Let us first check that Diff S (R)0 is closed under multiplication. We have ((Id +f ) ◦ (Id +g))(x) = x + g(x) + f (x + g(x)), (6.4.1) and x → f (x + g(x)) is in S(R) by the Fa` a di Bruno formula (6.1) and the following estimate:     1 1 f (m) (x + g(x)) = O = O (6.4.2) (1 + |x + g(x)|2 )k (1 + |x|2 )k which holds since g(x) → 0 for |x| → ∞ and thus 1 + |x|2 1 + |x + g(x)|2

is globally bounded.

172

Peter W. Michor

Let us check next that multiplication is smooth. Suppose that the curves t → Id +f (t, ), Id +g(t, ) are in C ∞ (R, Diff S (R)0 ), which means that the functions f, g ∈ C ∞ (R2 , R) satisfy the conditions of lemma (6.2). Then (1 + |x|2 )k ∂tn ∂xm f (t, x + g(t, x)) is bounded in x ∈ R, locally in t, by the 2-dimensional Fa´a di Bruno formula (6.1) and the more elaborate version of estimate (6.4.2)     1 1 (n,m) f )(t, x + g(t, x)) = O (∂ =O (1 + |x + g(t, x)|2 )k (1 + |x|2 )k (6.4.3) which follows from (6.3) for f and g. Thus the multiplication respects smooth curves and is smooth. To check that the inverse (Id +g)−1 is again an element in Diff S (R)0 for g ∈ S(R), we write (Id +g)−1 = Id +f and we have to check that f ∈ S(R). (Id +f ) ◦ (Id +g) = Id =⇒ x + g(x) + f (x + g(x)) = x =⇒ x → f (x + g(x)) = −g(x) is in S(R).

(6.4.4)

Now consider ∂x (f (x + g(x))) = f
(x + g(x))(1 + g
(x)), ∂x2 (f (x + g(x))) = f

(x + g(x))(1 + g
(x))2 + f
(x + g(x))g

(x), ∂x3 (f (x + g(x))) = f (3) (x + g(x))(1 + g
(x))3

(6.4.5)

+ 3f

(x + g(x))(1 + g
(x))g

(x) + f
(x + g(x))g (3) (x), ∂xm (f (x + g(x))) = f (m) (x + g(x))(1 + g
(x))m +

m−1


f (m−k) (x + g(x))amk (x),

k=1

where ank ∈ S(R) for n ≥ k ≥ 1. We have 1 + g
(x) ≥ ε > 0, thus 1 1+g
(x) is bounded and its derivative is in S(R). Hence we can conclude that, (1 + |x|2 )k f (n) (x + g(x)) is bounded for each k. Since (1 + |x + g(x)|2 )k = O(1 + |x|2 ) we conclude that (1 + |x + g(x)|2 )k f (n) (x + g(x)) is bounded for all k and n. Inserting y = x + g(x) it follows that f ∈ S(R). Thus inversion maps Diff S (R) into itself. Let us check that inversion is also smooth. So we assume that g(t, x) is a smooth curve in S(R), satisfies (6.3), and we have to check that then f does the same. Retracing our considerations we see from (6.4.4) that

Geometric evolution equations

173

f (t, x + g(t, x)) = −g(t, x) satisfies (6.3) as a function of t, x, and we claim that f then does the same. Applying ∂tn to the equations in (6.4.5) we get ∂tn ∂xm (f (t, x + g(t, x))) = (∂ (n,m) f )(t, x + g(t, x))(1 + ∂x g(t, x))m
+ (∂ (k1 ,k2 ) f )(t, x + g(t, x))ak1 ,k2 (t, x), k1 ≤n k2 ≤m+n

  1 uniformly in x and locally in t. Again 1 + where ak1 ,k2 (t, x) = O (1+|x| 2 )k ∂x g(t, x) ≥ ε > 0, locally in t and uniformly in x, thus the function 1+∂x1g(t,x) is bounded with any derivative in S(R) with respect to x. Thus we can conclude f satisfies (6.3). So the inversion is smooth and Diff S (R) is a Lie group. We claim that Diff S (R) is also a regular Lie group. So let t → X(t, ) be a smooth curve in the Lie algebra S(R)∂x , i.e., X satisfies (6.3). The evolution of this time dependent vector field is the function given by the ODE Evol(X)(t, x) = x + f (t, x),

∂t (x + f (t, x)) = ft (t, x) = X(t, x + f (t, x)),

(6.4.6)

f (0, x) = 0. We have to show that f satisfies (6.3). For 0 ≤ t ≤ C we consider  |f (t, x)| ≤



t

|ft (s, x)|ds = 0

t

|X(s, x + f (s, x))| ds.

(6.4.7)

0

Since X(t, x) is uniformly bounded in x, locally in t, the same  is true for f (t, x) by (6.4.7). But then we may insert X(s, x + f (s, x)) = O (1+|x+f1(s,x)|2 )k =     1 1 O (1+|x| into (6.4.7) and can conclude that f (t, x) = O (1+|x| globally 2 )k 2 )k in x, locally in t, for each k. For ∂tn ∂xm f (t, x) we differentiate equation (6.4.6) and arrive at a system of ODEs with functions in S(R) which we can estimate in the same way. 2 6.5 Sobolev spaces and HC n -spaces The differential operator Ak = Pk (−i∂x ) =

k
i=0

(−1)i ∂x2i ,

P (ξ) =

k


ξ 2i ,

i=0

will play an important role later on. We consider the Sobolev spaces, namely the Hilbert spaces H n (R) = {f ∈ S
(R) : f, f
, f (2) , . . . , f (n) ∈ L2 (R)}.

174

Peter W. Michor

In terms of the Fourier transform fˆ we have, by the properties listed in (6.2): f ∈ H n ⇐⇒ (1 + |ξ|)n fˆ(ξ) ∈ L2 ⇐⇒ (1 + |ξ|2 )n/2 fˆ(ξ)) ∈ L2 ⇐⇒ (1 + |ξ|)n−2k Pk (ξ)fˆ(ξ) ∈ L2 ⇐⇒ Ak (f ) ∈ H n−2k . We shall use the norm f H n := fˆ(ξ)(1 + |ξ|)n L2 on H n (R). Moreover, for 0 < α ≤ 1 we consider the Banach space

|f (x) − f (y)| 0,α <∞ Cb (R) = f ∈ C 0 (R) : sup |f (x)| + sup |x − y|α x∈R x=y∈R of bounded H¨ older continuous functions on R, and the Banach spaces Cbn,α (R) = {f ∈ C n (R) : f, f
, . . . , f (n−1) bounded, and f (n) ∈ Cb0,α (R)}. Finally we shall consider the space HC n (R) = H n (R) ∩ Cbn (R),

f HC c = f H n + f Cbn .

6.6 Lemma Consider the differential operator Ak =

k

i 2i i=0 (−1) ∂x .

(6.6.1) Ak : S(R) → S(R) is a linear isomorphism of the Fr´echet space of rapidly decreasing smooth functions. (6.6.2) Ak : H n+2k (S 1 ) → H n (S 1 ) is a linear isomorphism of Hilbert spaces for each n ∈ Z, where H n (S 1 ) = {f ∈ L2 (S 1 ) : An (f ) ∈ L2 (S 1 )}. Note that H n (S 1 ) ⊆ C k (S 1 ) if n > k + 1/2 (Sobolev inequality). (6.6.3) Ak : C ∞ (S 1 ) → C ∞ (S 1 ) is a linear isomorphism. (6.6.4) Ak : HC n+2k (R) → HC n (R) is a linear isomorphism of Banach spaces for each n ≥ 0. Proof. Without loss of generality we may consider complex-valued functions. (6.6.1) Let F : C ∞ (S 1 ) → s(Z) be the Fourier transform which is an isomorphism on the space of rapidly decreasing sequences. Since F (fxx )(n) = −(2πn)2 F (f )(n) we have F ◦Ak ◦F −1 : (cn ) → ((1+(2πn)2 +· · ·+(2πn)2k ) cn ) which is a linear bibounded isomorphism. (6.6.2) This is obvious from the definition. (6.6.3) can be proved similarly to (6.6.1), using that the Fourier series expansion is an isomorphism between C ∞ (S 1 ) and the space ∫ of rapidly decreasing sequences. (6.6.4) follows from (6.6.2). 2

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6.7 Sobolev inequality We have bounded linear embeddings (0 < α ≤ 1): 1 H n (R) ⊂ Cbk (R) if n > k + , 2 H n (R) ⊂ Cbk,α (R) if n > k +

1 + α. 2

Proof. Since ∂xk : H n (R) → H n−k (R) is bounded we may assume that k = 0. So let n > 12 . Then we use the Cauchy–Schwartz inequality:       1 ixξ   u(ξ)| dξ = |ˆ u(ξ)|(1 + |ξ|)n 2π|u(x)| =  e u ˆ(ξ) dξ  ≤ |ˆ dξ (1 + |ξ|)n  |ˆ(ξ)|2 (1 + |ξ|)2n dξ

≤ where

 C=

 12 

1 dξ (1 + |ξ|)2n

1 dξ (1 + |ξ|)2n

 12

 12 = CuH n

<∞

depends only on n > 12 . For the second assertion we use x > y and  1 eixξ − eiyξ = (x − y) iξei(y+t(x−y))ξ dt, 0

|e

ixξ

−e

iyξ

to obtain     u(x) − u(y)  ≤ 2π  (x − y)α  

≤2

| ≤ |x − y|.|ξ|

  ixξ  e − eiyξ α ixξ iyξ 1−α   |ˆ u(ξ)| dξ  x − y  .|e − e |

|ˆ u(ξ)|(1 + |ξ|)n

|ξ|α dξ (1 + |ξ|)n

 ≤2

|ˆ u(ξ)|2 (1 + |ξ|)2n dξ

 12 

where C1 depends only on n − α > 12 .

|ξ|2α dξ (1 + |ξ|)2n

 12

= C1 uH n 2

6.8 Banach algebra property If n > 12 , then pointwise multiplication S(R) × S(R) → S(R) extends to a bounded bilinear mapping H n (R) × H n (R) → H n (R). For n ≥ 0 multiplication HC n (R) × HC n (R) → HC n (R) is bounded bilinear. See [17] for the most general version of this property on open Riemannian manifolds with bounded geometry.

176

Peter W. Michor

Proof. For f, g ∈ H n (R) we have to show that for 0 ≤ k ≤ n we have k  
k (l) (k−l) (k) (f.g) = ∈ L2 (R) f .g l l=0

with norm bounded by a constant times f H n .gH n . If l < n, then f (l) ∈ Cb0 (R) by the Sobolev inequality and g (k−l) ∈ H l ⊂ L2 , so the product is in L2 with the required bound on the norm. If l = 0 we exchange f and g. In the case of HC n , the L2 -norm of each product in the sum is bounded by the sup-norm of the first factor times the L2 -norm of the second one. And the sup-norm is clearly submultiplicative. 2 6.9 Differentiability of composition If n ≥ 0, then composition S(R) × S(R) → S(R) extends to a weakly, C k -mapping HC n+k (R) × (IdR +HC n (R)) → HC n (R). A mapping f : E → F is weakly C 1 for Banach spaces E, F if df : E ×E → F exists and is continuous. We call it strongly C 1 if df : E → L(E, F ) is continuous for the operator norm on the image space. Similarly for C k . Since I could not find a convincing proof of this result for the spaces H n under the assumption n > 12 , I decided to use the spaces HC n (R). This also improves on the degree n which we need. Proof. We consider the Taylor expansion f (x + g(x)) =

k
1 (p) f (x).g(x)p p! p=0



1

+ 0

(1 − t)k−1 (k) (f (x + tg(x)) − f (k) (x)) dt .g(x)k . (k − 1)!

For fixed f this is weakly C k in g by invoking the Banach algebra property and by estimating the integral in the remainder term. We have to show that the integrand is continuous at (f (k) , g = 0) as a mapping H n × H n → H n . The integral from 0 to 1 does not disturb this so we disregard it. By (6.1) we have ∂xp (f (k) (x + g(x)) − f (k) (x)) = p!

p
f (k+m) (x + g(x)) m=0

m!


α1 ,...,αm >0 α1 +···+αm =p

∂ αm (x + g(x)) ∂xα1 (x + g(x)) ... x . α1 ! αm !

The most dangerous term is the one for p = n. As soon as a derivative of g of order ≥ 2 is present, this is easily estimated. The most difficult term is f (k+n) (x + g(x)) − f (k+n) (x)

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177

which should go to 0 in L2 ∩ Cb0 for fixed f and for g → 0 in HC n . f (k) is continuous and in L2 . Off some big compact interval it has small H n -norm and small sup-norm (the latter by the lemma of Riemann–Lebesgue). On this compact interval f (k) is uniformly continuous and if we choose gC n small enough, f (k) (x + tg(x)) − f (k) (x) is uniformly small there, thus small in the sup-norm, and also small in L2 (which involves the length of the compact interval—but we can still choose g smaller). 2 The last result cannot be improved to strongly C k since we have: 6.10 Attention Composition HC n (R) × (IdR +HC n (R)) → HC n (R) is only continuous and not Lipschitz in the first variable. Proof. To see this, consider (f, t) → f ( −t.g) for a given bump function g which equals 1 on a large interval. For each t > 0 we consider a bump function f with support in (− 2t , 2t ) with f L2 = 1. Then we have √ f −f ( −t)L2 = √2 by Pythagoras, and consequently f −f ( −t.g)HC n ≥ f − f ( −t)L2 = 2. 2 6.11 The topological group Diff (R) For n ≥ 1 we consider f : R → R of the form f (x) = x + g(x) for g ∈ HC n . Then f is a C n -diffeomorphism iff g
(x) > −1 for all x. The inverse is also of the form f −1 (y) = y + h(y) for h ∈ HC n (R) iff g
(x) ≥ −1 + ε for a constant ε. Indeed, h(y) = −g(f −1 (y)). Let us call DiffHCn (R) the group of all these diffeomorphsms. Lemma Inversion DiffHCn+k (R) → DiffHCn (R) is weakly C k . Proof. As we saw above, DiffHCn+k (R) is stable under inversion. (f, g) → f ◦g is a weak C k submersion by (6.9). So we can use the implicit function theorem 2 for the equation f ◦ f −1 = Id. 6.12 Proposition For n ≥ 1 and a ∈ HC n (R), the mapping HC n (R) × DiffHCn (R) → HC n−1 (R) given by (f, g) → (a∂x (f ◦ g −1 )) ◦ g is continuous and Lipschitz in f. For n > k + 12 and for each linear differential operator D of order k, the mapping HC n (R) × DiffHCn (R) → HC n−k (R) given by (f, g) → (D(f ◦ g −1 )) ◦ g is continuous and Lipschitz in f . Here Diff(R) = {IdR +h : h
Cb0 > −1}.

178

Peter W. Michor

Proof. We have (a∂x (f ◦ g −1 )) ◦ g =

 a.(fx ◦ g −1 )

1 gx ◦ g −1

 ◦ g = (a ◦ g).fx .

1 gx 2

which is Lipschitz by the results above. 6.13 Proposition

k For the operator Ak = i=0 (−1)i ∂x2i and for n ≥ 2k, the mapping (f, g) → −1 (A−1 )) ◦ g is Lipschitz HC n (R) × DiffHCn (R) → HC n+2k (R). k (f ◦ g Proof. The inverse of Ak is given by the pseudo differential operator  1 f )(x) = ei(x−y)ξ f (y) dξ dy. (A−1 k 2 1 + ξ + +ξ 2n R2 Thus the mapping is given by  −1 (A−1 (f ◦ g ))(g(x)) = ei(g(x)−y)ξ f (g −1 (y)) k R2



ei(g(x)−g(z))ξ f (z)

= R2

1 1+ξ + 2

+ξ 2n

dξ dy

g
(z) dξ dz 1 + ξ + +ξ 2n 2

which is a genuine Fourier integral operator. By the foregoing results this is visibly locally Lipschitz. 2

7 The diffeomorphism group of S 1 or R, and Burgers’ hierarchy 7.1 Burgers’ equation and its curvature We consider the Lie groups Diff S (R) and Diff(S 1 ) with Lie algebras XS (R) and X(S 1 ) where the Lie bracket [X, Y ] = X
Y − XY
is the ' negative of the usual one. For the L2 -inner product γ(X, Y ) = X, Y 0 = X(x)Y (x) dx integration by parts gives  [X, Y ], Z 0 = (X
Y Z − XY
Z)dx R

 =

R

(2X
Y Z + XY Z
)dx = Y, ad(X) Z ,

which in turn gives rise to ad(X) Z = 2X
Z + XZ
,







α(X)Z = ad(Z) X = 2Z X + ZX ,

(7.1.1) (7.1.2)

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179

(ad(X) + ad(X))Z = 3X
Z,

(7.1.3)

(ad(X) − ad(X))Z = X
Z + 2XZ
= α(X)Z.

(7.1.4)

Equation (7.1.4) states that − 12 α(X) is the skew-symmetrization of ad(X) with respect to the inner product , 0 . From the theory of symmetric spaces one then expects that − 12 α is a Lie algebra homomorphism and indeed one can check that   1 1 1 − α([X, Y ]) = − α(X), − α(Y ) 2 2 2 holds for any vector fields X, Y . From (7.1.1) we get the geodesic equation, whose second part is Burgers’ equation [10]: gt (t, x) = u(t, g(t, x)), (7.1.5) ut = − ad(u) u = −3uxu. Using the above relations and the general curvature formula (3.4.2), we get R(X, Y )Z = −X

Y Z + XY

Z − 2X
Y Z
+ 2XY
Z
= −2[X, Y ]Z
− [X, Y ]
Z = −α([X, Y ])Z.

(7.1.6)

Sectional curvature is nonnegative and unbounded: −G0a (R(X, Y )X, Y ) = α([X, Y ])(X), Y = ad(X) ([X, Y ]), Y

= [X, Y ], [X, Y ] = [X, Y ]2 , k(X ∧ Y ) = − =

G0a (R(X, Y )X, Y ) X2Y 2 − G0a (X, Y )2 [X, Y ]2 ≥ 0. 2 − X, Y 2

(7.1.7)

X2 Y

Let us check invariance of the momentum mapping J¯ from (4.3):  ¯ X), Y ) = γ(Ad(g) X, Y ) = γ(X, Ad(g)Y ) = X((g
Y ) ◦ g −1 )dx γ(J(g,  =

X(g
◦ g −1 )(Y ◦ g −1 )dx = sign(g
)



(X ◦ g)(g
)2 Y dx

= sign(g
)γ((g
)2 (X ◦ g), Y ) ¯ X) = sign(gx ).(gx )2 (X ◦ g). J(g, Along a geodesic t → g(t,

(7.1.8)

), according to (7.1.5) and (4.3), the momentum

¯ u = gt ◦ g −1 ) = g 2 gt J(g, x This is what we found in (1.3) by chance.

is constant.

(7.1.9)

180

Peter W. Michor

7.2 Jacobi fields for Burgers’ equation A Jacobi field y along a geodesic g with velocity field u is a solution of the partial differential equation (3.5.1), which in our case becomes: ytt = [ad(y) + ad(y), ad(u) ]u − ad(u) yt − α(u)yt + ad(u)yt

(7.2.1)

= −3u2yxx − 4uytx − 2ux yt , ut = −3uxu. If the geodesic equation has smooth solutions locally in time, it is to be expected that the space of all Jacobi fields exists and is isomorphic to the space of all initial data (y(0), yt (0)) ∈ C ∞ (S 1 , R)2 or Cc∞ (R, R)2 , respectively. The weak symplectic structure on it is given by (3.7): 5 4 5 4 1 1 ω(y, z) = y, zt − ux z + 2uzx − yt − ux y + 2uyx , z 2 2  = (yzt − yt z + 2u(yzx − yx z)) dx. (7.2.2) S 1 or R

7.3 The Sobolev H k -metric on Diff (S 1 ) and Diff (R) On the Lie algebras Xc (R) and X(S 1 ) with Lie bracket [X, Y ] = X
Y − XY
we consider the H k -inner product γ(X, Y ) = X, Y k =

k 


 (∂xi X)(∂xi Y

) dx =

Ak (X)(Y ) dx

i=0

 =

XAk (Y ) dx,

where

Ak =

k


(−1)i ∂x2i (7.3.1)

i=0

is a linear isomorphism Xc (R) → Xc (R) or X(S 1 ) → X(S 1 ) whose inverse is a pseudodifferential operator. Ak is also a bounded linear isomorphism between the Sobolev spaces H l+2k (S 1 ) → H l (S 1 ), see lemma (6.5). On the real line we have to consider functions with fixed support in some compact set [−K, K] ⊂ R. Integration by parts gives   [X, Y ], Z k = (X
Y − XY
)Ak (Z)dx = (2X
Y Ak (Z) + XY Ak (Z
))dx R

 =

R

R





,k Y Ak A−1 , Z k , k (2X Ak (Z) + XAk (Z ))dx = Y, ad(X)

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181

which in turn gives rise to

ad(X) ,k Z = A−1 k (2X Ak (Z) + XAk (Z )),

αk (X)Z = ad(Z) ,k (X) = A−1 k (2Z Ak (X) + ZAk (X )).

Thus the geodesic equation is ⎧ g (t, x) = u(t, g(t, x)), ⎪ ⎪ ⎨ t ut = − ad(u) ,k u = −A−1 k (2ux Ak (u) + uAk (ux )) ⎪ ⎪ k k ⎩ −1 = −Ak (2ux i=0 (−1)i ∂x2i u + u i=0 (−1)i ∂x2i+1 u).

(7.3.2)

(7.3.3)

For k = 0 the second part is Burgers’ equation, and for k = 1 it becomes ut − utxx = −3uux + 2ux uxx + uuxxx   1 =0 ⇐⇒ ut + uux + (1 − ∂x2 )−1 u2 + u2x 2 x

(7.3.4)

which is the dispersion-free version of the Camassa–Holm equation, see [11], [44], [29]. We met it already in (5.6), and will meet the full equation in (8.7). Let us check the invariant momentum mapping from (4.3.2): γ(J¯(g, X), Y ) = Ad(g) X, Y k = X, Ad(g)Y k  = Ak (X)(g
◦ g −1 )(Y ◦ g −1 )dx


= sign(g )



(Ak (X) ◦ g)(g
)2 Y dx


2 = sign(g
) A−1 k ((g ) (Ak (X) ◦ g)), Y k ,

¯ X) = sign(gx ).A−1 ((gx )2 (Ak (X) ◦ g)). J(g, k Along a geodesic t → g(t,

(7.3.5)

), by (7.3.3) and (4.3), the expressions

2 sign(gx )J¯(g, u = gt ◦ g −1 ) = A−1 k ((gx ) (Ak (u) ◦ g))

(7.3.6)

and thus also (gx )2 (Ak (u) ◦ g) are constant in t. 7.4 Theorem Let k ≥ 1. There exists an HC 2k+1 -open neighborhood V of (Id, 0) in Diff(S 1 ) × X(S 1 ) such that for each (g0 , u0 ) ∈ V there exists a unique C 3 geodesic g ∈ C 3 ((−2, 2), Diff(S 1 )) for the right invariant H k Riemann metric, starting at g(0) = g0 in the direction gt (0) = u0 ◦ g0 ∈ Tg0 Diff(S 1 ). Moreover, the solution depends C 1 on the initial data (g0 , u0 ) ∈ V .

182

Peter W. Michor

The same result holds if we replace Diff(S 1 ) by Diff S(R) and X(S 1 ) by XS (R) = S(R)∂x . This result is stated in [13], and also this proof follows essentially [13]. But there is a mistake in [13], p. 795, where the authors assume that composition and inversion on H n (S 1 ) are smooth. This is wrong. One needs to use (6.12) and (6.13). The mistake was corrected in [12], for the more general case of the Virasoro group. In the following proof, Diff, X, DiffHCn , HC n should stand for either Diff(S 1 ), X(S 1 ), DiffHCn (S 1 ), HC n (S 1 ) or for Diff S (R), XS (R), DiffHCn (R), HC n (R), respectively. Proof. For u ∈ HC n , n ≥ 2k + 1, we have Ak (uux ) =

k


(−1)i ∂x2i (uux ) =

i=0

= uAk (ux ) +

k


(−1)i

i=0 k


2i
2i j 2i−j+1 u) j (∂x u)(∂x j=0

2i
2i j 2i−j+1 (−1) u) j (∂x u)(∂x i

i=0

j=1

=: u Ak (ux ) + Bk (u), where Bk : HC n → HC n−2k is a bounded quadratic operator. Recall that we have to solve ut = − ad(u) ,k u = −A−1 k (2ux Ak (u) + uAk (ux )) = −A−1 k (2ux Ak (u) + Ak (uux ) − Bk (u)) = −uux − A−1 k (2ux Ak (u) − Bk (u)) =: −uux + A−1 k Ck (u), where Ck : HC n → HC n−2k is a bounded quadratic operator, and where u = gt ◦ g −1 ∈ X. Note that Ck (u) = −2ux Ak (u) + Bk (u) = −2ux Ak (u) +

k
i=0

(−1)i

2i
2i j 2i−j+1 u). j (∂x u)(∂x j=1

We put ⎧ ⎪ ⎨gt =: v = u ◦ g, vt = ut ◦ g + (ux ◦ g)gt = ut ◦ g + (uux ) ◦ g = A−1 k Ck (u) ◦ g ⎪ ⎩ −1 −1 where = Ak Ck (v ◦ g ) ◦ g =: pr2 (Dk ◦ Ek )(g, v), Ek (g, v) = (g, Ck (v ◦ g −1 ) ◦ g),

(7.4.1)

−1 Dk (g, v) = (g, A−1 ) ◦ g). k (v ◦ g

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183

Now consider the topological group and Banach manifold DiffHCn described in (6.11). (7.4.2) Claim. The mapping Dk : DiffHCn ×HC n−2k → DiffHCn ×HC n is strongly C 1 . First we check that all directional derivatives exist and are in the right spaces. For w ∈ HC n we have ∂s |0 (u ◦ (g + sw)) = (ux ◦ g)w, ∂s |0 (g + sw)−1 = −

w ◦ g −1 , gx ◦ g −1

∂s |0 pr2 Dk (g + sw, v) −1 −1 = ∂s |0 A−1 ) ◦ (g + sw) + ∂s |0 (A−1 )) ◦ g k (v ◦ g k (v ◦ (g + sw) −1

−1 −1 w◦g )) ◦ g) w − (A−1 ) gx ◦g−1 )) ◦ g = ((∂x A−1 k (v ◦ g k ((vx ◦ g −1 −1 )x .(w ◦ g −1 )) ◦ g − (A−1 )x (w ◦ g −1 ))) ◦ g. = (A−1 k (v ◦ g k ((v ◦ g

Therefore, Ak ((∂s |0 pr2 Dk (g + sw, v)) ◦ g −1 ) −1 = Ak (A−1 )x .(w ◦ g −1 )) − (v ◦ g −1 )x (w ◦ g −1 ) k (v ◦ g

= (v ◦ g −1 )x .(w ◦ g −1 ) +

k 2i−1

  2i j+1 −1 −1 ).∂x2k−j (w ◦ g −1 ) j ∂x Ak (v ◦ g i=0 j=0

− (v ◦ g

−1

)x (w ◦ g

−1

) ∈ HC n−2k .

By (6.12) and (6.13) this is locally Lipschitz jointly in v, g, w. Moreover we have ∂s |0 pr2 Dk (g + sw, v) ∈ HC n , and Dk is linear in v. Thus Dk is strongly C1. (7.4.3) Claim. The mapping Ek : DiffHCn ×HC n → DiffHCn ×HC n−2k is strongly C 1 . This can be proved similarly, again using (6.12) and (6.13). By the two claims equation (7.4.1) can be viewed as the flow equation of a C 1 -vector field on the Hilbert manifold DiffHCn ×HC n . Here an existence and uniqueness theorem holds. Since v = 0 is a stationary point, there exists an open neighborhood Wn of (Id, 0) in DiffHCn ×HC n such that for each initial point (g0 , v0 ) ∈ Wn equation (7.4.1) has a unique solution Flnt (g0 , v0 ) = (g(t), v(t)) defined and C 2 in t ∈ (−2, 2). Note that v(t) = gt (t), thus g(t) is even C 3 in t. Moreover, the solution depends C 1 on the initial data. We start with the neighborhood W2k+1 ⊂ DiffHC2k+1 ×HC 2k+1 ⊃ DiffHCn ×HC n

for n ≥ 2k + 1

and consider the neighborhood Vn := W2k+1 ∩ DiffHCn ×HC n of (Id, 0)

184

Peter W. Michor

(7.4.4) Claim. For any initial point (g0 , v0 ) ∈ Vn the unique solution Flnt (g0 , v0 ) = (g(t), v(t)) exists, is C 2 in t ∈ (−2, 2), and depends C 1 on the initial point in Vn . We use induction on n ≥ 2k + 1. For n = 2k + 1 the claim holds since (g0 , v0 ) = (˜ g (t), v˜(t)) V2k+1 = W2k+1 . Let (g0 , v0 ) ∈ V2k+2 and let Fl2k+2 t be maximally defined for t ∈ (t1 , t2 )  0. Suppose for contradiction that t2 < 2. Since (g0 , v0 ) ∈ V2k+2 ⊂ V2k+1 , the curve Fl2k+2 (g0 , v0 ) = (˜ g (t), v˜(t)) t solves (7.4.1) also in DiffHC2k+1 ×HC 2k+1 , thus Fl2k+2 (g , v ) = (˜ g (t), v˜(t)) = 0 0 t (g , v ) for t ∈ (t , t )∩(−2, 2). By (7.3.6), the expression (g(t), v(t)) := Fl2k+1 0 0 1 2 t ˜ = J(g, ˜ v, t) = gx (t)2 Ak (u(t))◦g(t) = gx (t)2 Ak (v(t)◦g(t)−1 )◦g(t) (7.4.5) J(t) is constant in t ∈ (−2, 2). Actually, since we used C ∞ -theory for deriving this, one should check it again by differentiating. Since u = gt ◦ g −1 we get the following (the exact formulas can be computed with the help of Fa` a di Bruno’s formula (6.1)). ux = (gtx ◦ g −1 )(g −1 )x =  ∂x2 u = ∂x (g −1 ) =



◦ g −1 ,

1 ◦ g −1 , gx

∂x2 (g −1 ) ◦ g = − ∂x2k (g −1 ) ◦ g = − (∂x2k u) ◦ g =

∂x2 gt ∂2g − gtx x3 2 gx gx

gtx ◦ g −1 , gx

∂x2 g , gx3 ∂x2k g + lower order terms in g, gx2k+1

∂x2k gt ∂x2k g − g + lower order terms in g, gt = v. tx gx2k gx2k+1

Thus ˜ = gx ∂ 2k gt − gtx ∂ 2k g + lower order terms in g, gt = v. (−1)k gx2k−1 J(t) x x Hence for each t ∈ (−2, 2): ˜ + Pk (g, v)), where gx ∂x2k gt − gtx ∂x2k g = (−1)k gx2 (gx2k−3 J(t) Pk (g, v) =

Qk (g, ∂x g, . . . , ∂x2k−1 g, v, ∂x v, . . . , ∂x2k−1 v) gx2

˜ = J(0) ˜ for a polynomial Qk . Since J(t) we obtain that  2k  ∂x g(t) = (−1)k (gx2k−3 (t)J˜(0) + Pk (g(t), v(t))) for all t ∈ (−2, 2). gx (t) t

Geometric evolution equations

This implies ∂ 2k g(0) ∂x2k g(t) = x + (−1)k gx (t) gx (0)



185

t

˜ + Pk (g(s), v(s))) ds. (gx2k−3 (s)J(0) 0

For t ∈ (t1 , t2 ) we have ∂x2k g˜(t) =

∂x2k g0 gx (t) ∂x g0

(7.4.6) 

t

(gx2k−3 (s)J˜(0) + Pk (g(s), v(s))) ds.

+ (−1)k gx (t) 0

˜ ˜ 0 , v0 , 0) ∈ HC 2 by (7.4.5). Since (g0 , v0 ) ∈ V2k+2 we have J(0) = J(g 2k Since k ≥ 1, by (7.4.6) we see that ∂x g˜(t) ∈ HC 2 . Moreover, since t2 < 2, limt→t2 − ∂x2k g˜(t) exists in HC 2 , so limt→t2 − g˜(t) exists in HC 2k+2 . As this limit equals g(t2 ), we conclude that g(t2 ) ∈ DiffHC2k+2 . Now v˜ = g˜t ; so we may differentiate both sides of (7.4.6) in t and obtain similarly, that limt→t2 − v˜(t) exists in HC 2k+2 and equals v(t2 ). But then we can prolong the flow line (˜ g , v˜) in DiffHC2k+2 ×HC 2k+2 beyond t2 , so (t1 , t2 ) is not maximal. By the same method we can iterate the induction. 2

8 The Virasoro–Bott group and the Korteweg–de Vries hierarchy 8.1 The Virasoro–Bott group Let Diff denote any of the groups DiffHC+ (S 1 ), Diff(R)0 (diffeomorphisms with compact support), or Diff S (R) of section (6). For ϕ ∈ Diff let ϕ
: S 1 or R → R+ be the mapping given by Tx ϕ · ∂x = ϕ
(x)∂x . Then c : Diff × Diff → R c(ϕ, ψ) :=

1 2

 S1

log(ϕ ◦ ψ)
d log ψ
=

1 2

 S1

log(ϕ
◦ ψ)d log ψ


satisfies c(ϕ, ϕ−1 ) = 0, c(Id, ψ) = 0, c(ϕ, Id) = 0, and is a smooth group cocycle, i.e., c(ϕ2 , ϕ3 ) − c(ϕ1 ◦ ϕ2 , ϕ3 ) + c(ϕ1 , ϕ2 ◦ ϕ3 ) − c(ϕ1 , ϕ2 ) = 0, called the Bott cocycle. Proof. Let us check first:   log(ϕ ◦ ψ)
d log ψ
= log((ϕ
◦ ψ)ψ
)d log ψ
 =

log(ϕ
◦ ψ)d log ψ
+



log(ψ
)d log ψ
,

186

Peter W. Michor



 1 log(ψ )d log ψ = d log(ψ
)2 = 0, 2  2c(Id, ψ) = log(1)d log ψ
= 0,





 2c(ϕ, Id) = 2c(ϕ−1 , ϕ) =



log(ϕ
)d log(1) = 0, log((ϕ−1 ◦ ϕ)
)d log ϕ
=



log(1)d log ϕ
= 0,

c(ϕ, ϕ−1 ) = 0. For the cocycle condition we add the following terms:  2c(ϕ2 , ϕ3 ) = log(ϕ
2 ◦ ϕ3 )d log ϕ
3 

log((ϕ1 ◦ ϕ2 )
◦ ϕ3 )d log ϕ
3

− 2c(ϕ1 ◦ ϕ2 , ϕ3 ) = −  =−  =−  2c(ϕ1 , ϕ2 ◦ ϕ3 ) =  =  =

log(ϕ
2 ◦ ϕ3 )d log ϕ
3 ,

log(ϕ
1 ◦ ϕ2 ◦ ϕ3 )d log(ϕ
2 ◦ ϕ3 ) log(ϕ
1 ◦ ϕ2 ◦ ϕ3 )d log ϕ
3

log(ϕ
1 ◦ ϕ2 )d log ϕ
2 + 

−2c(ϕ1 , ϕ2 ) = −



log(ϕ
1 ◦ ϕ2 ◦ ϕ3 )d log((ϕ
2 ◦ ϕ3 )ϕ
3 )

 

log(ϕ
1 ◦ ϕ2 ◦ ϕ3 )d log ϕ
3 −

log(ϕ
1 ◦ ϕ2 ◦ ϕ3 )d log(ϕ2 ◦ ϕ3 )


+ =

log((ϕ
1 ◦ ϕ2 ◦ ϕ3 )(ϕ
2 ◦ ϕ3 ))d log ϕ
3



log(ϕ
1 ◦ ϕ2 )d log ϕ
2 .

log(ϕ
1 ◦ ϕ2 ◦ ϕ3 )d log ϕ
3 2

The corresponding central extension group S 1 ×c DiffHC+ (S 1 ), called the periodic Virasoro–Bott group, is a trivial S 1 -bundle S 1 × DiffHC+ (S 1 ) that becomes a regular Lie group relative to the operations       −1  −1  ϕ ϕ ψ ϕ ϕ◦ψ = , = 2πic(ϕ,ψ) α α β αβ e α−1

Geometric evolution equations

187

for ϕ, ψ ∈ DiffHC+ (S 1 ) and α, β ∈ S 1 . Likewise we have the central extension group with compact supports R ×c Diff(R)0 with group operations       −1  −1  ϕ ϕ ψ ϕ◦ψ ϕ = = , α β α + β + c(ϕ, ψ) −α α for ϕ, ψ ∈ DiffHC+ (R) and α, β ∈ R. Finally there is the central extension of the rapidly decreasing Virasoro–Bott group R ×c Diff + S (R) which is given by the same formulas. 8.2 The Virasoro Lie algebra Let us compute the Lie algebra of the two versions of the Virasoro–Bott group. Consider R ×c Diff, where again Diff denotes any one of the groups DiffHC+ (S 1 ), Diff(R)0 , or Diff S (R). So let ϕ, ψ : R → Diff with ϕ(0) = ψ(0) = Id and ϕt (0) = X, ψ t (0) = Y ∈ Xc (R), X(S 1 ), or S(R)∂x . For completeness’ sake we also consider α, β : R → R with α(0) = 0, β(0) = 0. Then we compute:    ϕ(t) Y Ad α(t) β
(0)     ϕ(t) ψ(s) ϕ(t)−1 = ∂s |0 β(s) α(t) −α(t)   ϕ(t) ◦ ψ(s) ◦ ϕ(t)−1 = ∂s |0 0 α(t) + β(s) + c(ϕ(t), ψ(s)) − α(t) + c(ϕ(t) ◦ ψ(s), ϕ(t)−1 )   ϕ(t)∗ Y = Ad(ϕ(t))Y = , (8.2.1) β t (0) + ∂s |0 c(ϕ(t), ψ(s)) + ∂s |0 c(ϕ(t) ◦ ψ(s), ϕ(t)−1 )     X Y , αt (0) β t (0)   (FlX t )∗ Y = Ad(ϕ(t))Y = ∂t |0 β t (0) + ∂s |0 c(ϕ(t), ψ(s)) + ∂s |0 c(ϕ(t) ◦ ψ(s), ϕ(t)−1 )   −[X, Y ] = . (8.2.2) ∂t |0 ∂s |0 c(ϕ(t), ψ(s)) + ∂t |0 ∂s |0 c(ϕ(t) ◦ ψ(s), ϕ(t)−1 ) Now we differentiate the Bott cocycle, where sometimes f
= ∂x f :  2∂s |0 c(ϕ(t), ψ(s)) = ∂s |0 log(ϕ(t)
◦ ψ(s)) d log(ψ(s)
)  =

(ϕ(t)

◦ ψ(0))Y d log(ψ(0)
) + 8 9: ; ϕ(t)
◦ ψ(0) =1



log(ϕ(t)
) dY


188

Peter W. Michor

 =

log(ϕ(t)
)Y

dx, 

2∂t |0 ∂s |0 c(ϕ(t), ψ(s)) = ∂t |0

log(ϕ(t)
)Y

dx =



X
Y

dx = ϕ(0)




X
Y

dx.

For the second term we first check: (ϕ−1 )x =

1 , ϕx ◦ ϕ−1

ϕxx ◦ ϕ−1 , (ϕx ◦ ϕ−1 )3

1 dx = dy, ϕx ◦ ϕ−1

ϕ−1 (x) = y, d log((ϕ−1 )x ) = −

(ϕ−1 )xx = −

ϕ

◦ ϕ−1 ϕ

dx = −
dy
−1 2 (ϕ ◦ ϕ ) ϕ

and continue to compute −1

2∂s |0 c(ϕ(t) ◦ ψ(s), ϕ(t)

 ) = ∂s |0

log((ϕ(t) ◦ ψ(s))x ◦ ϕ(t)−1 ) d log(ϕ(t)−1 x )



(ϕ(t)

◦ ϕ(t)−1 )(Y ◦ ϕ(t)−1 )+(ϕ(t)
◦ ϕ(t)−1 )(Y
◦ ϕ(t)−1 ) d log(ϕ(t)−1 x ) (ϕ(t)
◦ ϕ(t)−1 )(ψ(0)
◦ ϕ(t)−1 )  (ϕ(t)

)2 Y + ϕ(t)
ϕ(t)

Y
=− dy, (ϕ(t)
)2  (ϕ(t)

)2 Y + ϕ(t)
ϕ(t)

Y
−1 dy 2∂t |0 ∂s |0 c(ϕ(t) ◦ ψ(s), ϕ(t) ) = −∂t |0 (ϕ(t)
)2  0 + 0 + ϕ(0)
X

Y
− 0 dy =− (ϕ(0)
= 1)4  


= − X Y dy = X
Y

dx. =

Finally we get from (8.2.2):       
 X Y −[X, Y ] X Y − XY
, = = a b ω(X, Y ) ω(X, Y )

(8.2.3)

where  ω(X, Y ) = ω(X)Y =

X
dY
=



X
Y

dx =

1 2



 det

X
Y
X

Y



 dx,

Geometric evolution equations

189

is the Gelfand–Fuchs Lie algebra cocycle ω : g × g → R, which is a bounded skew-symmetric bilinear mapping satisfying the cocycle condition ω([X, Y ], Z) + ω([Y, Z], X) + ω([Z, X], Y ) = 0. It is a generator of the 1-dimensional bounded Chevalley cohomology H 2 (g, R) for any of the Lie algebras g = X(S 1 ), Xc (R), or S(R)∂x . The Lie algebra of the Virasoro–Bott Lie group is thus the central extension R ×ω g of g induced by this cocycle. We have H 2 (Xc (M ), R) = 0 for each finite-dimensional manifold of dimension ≥ 2 (see [21]), which blocks the way to find a higher-dimensional analog of the Korteweg–de Vries equation in a way similar to that sketched below. For further use we also note the expression for the adjoint action on the Virasoro–Bott groups which we computed along the way. For the integral in the central term in (8.2.1) we have:    (ϕ

)2 Y + ϕ
ϕ

Y
1


dx log(ϕ )Y − 2 (ϕ
)2 

2    ϕ ϕ


1 −2
Y − Y dx = 2 ϕ ϕ
 2   


 ϕ 1 ϕ

= − Y dx = S(ϕ)Y dx, ϕ
2 ϕ
where a new character appears on stage, the Schwartzian derivative: 

ϕ

ϕ





−

1 2



ϕ

ϕ


2

ϕ


3 − ϕ
2



ϕ

ϕ


2

1 = log(ϕ
)

− (log(ϕ
)
)2 2 (8.2.4) which measures the deviation of ϕ from being a M¨ obius transformation:   ax + b ab for S(ϕ) = 0 ⇐⇒ ϕ(x) = ∈ SL(2, R). cd cx + d S(ϕ) =

=

ϕ

ϕ
satisfies the differential d x+ c which means log(ϕ
)
(x) =

Indeed, S(ϕ) = 0 if and only if g = log(ϕ
)
=

−2 equation g
= g 2 /2, so that 2gdg 2 = dx or g = ' −2 −2dx , or again log(ϕ
(x)) = x+d/c = −2 log(x + d/c) − 2 log(c) = g(x) = x+d/c  1  1 ax+b
. log (cx+d)2 . Therefore, ϕ (x) = (cx+d)2 = ∂x cx+d For completeness’ sake, let us note here the Schwartzian derivative of a composition and an inverse (which follow since the adjoint action (8.2.5) below is an action):

S(ϕ ◦ ψ) = (S(ϕ) ◦ ψ)(ψ
)2 + S(ψ),

S(ϕ−1 ) = −

S(ϕ) ◦ ϕ−1 . (ϕ
)2

190

Peter W. Michor

So finally, the adjoint action is given by      ϕ Y Ad(ϕ)Y = ϕ∗ Y = T ϕ ◦ Y ◦ ϕ−1 ' Ad = . α b b + S(ϕ)Y dx

(8.2.5)

8.3 H 0 -Geodesics on the Virasoro–Bott groups We shall use the L2 -inner product on R×ω g, where g = X(S 1 ), Xc (R), S(R)∂x : 4   5  X Y := XY dx + ab. (8.3.1) , a b 0 Integrating by parts we get 4     5 4
  5 X Y X Y − XY
Z Z ad = , , a b c ω(X, Y ) c 0 0  = (X
Y Z − XY
Z + cX
Y

) dx  =

(2X
Z + XZ
+ cX


)Y dx

<     = Y X Z = , ad , b a c

where

0

ad

    
 X 2X Z + XZ
+ cX


Z = . a c 0

Using matrix notation we get therefore (where ∂ := ∂x )   
 X X − X∂ 0 ad = , ω(X) 0 a ad

   
X 2X + X∂ X


= , 0 0 a       
 X X X + 2X∂ + a∂ 3 0 α = ad = , 0 0 a a

      X X 3X
X


+ ad = ad , ω(X) 0 a a    
  X X X + 2X∂ X


. − ad = ad −ω(X) 0 a a

Geometric evolution equations

191

Formula (3.2.2) gives the H 0 geodesic equation on the Virasoro–Bott group:         ut u u −3uxu − auxxx = − ad = a a at 0      ϕ(t)−1  ϕ(s) u(t) . = ∂s α(s) −α(t) s=t a(t)    ϕ(s) ◦ ϕ(t)−1  = ∂s −1 α(s) − α(t) + c(ϕ(s), ϕ(t) ) s=t   ϕt ◦ ϕ−1 ' ϕtx ϕxx = αt − 2ϕ2 dx

where

(8.3.2)

x

since we have −1

2∂s c(ϕ(s), ϕ(t)



log(ϕ(s)
◦ ϕ(t)−1 ) d log((ϕ(t)−1 )
)|s=t

)|s=t = ∂s

  ϕt (t)
◦ ϕ(t)−1 ϕ(t)

◦ ϕ(t)−1 − dx ϕ(t)
◦ ϕ(t)−1 (ϕ(t)
◦ ϕ(t)−1 )2  


ϕtx ϕxx ϕt ϕ dy = − dx. =−
2 (ϕ ) ϕ2x 

=

by (8.2)

Thus a is a constant in time and the geodesic equation is hence the Korteweg– de Vries equation (8.3.3) ut + 3uxu + auxxx = 0 with its natural companions  ϕt = u ◦ ϕ,

αt = a +

ϕtx ϕxx dx. 2ϕ2x

It is the periodic equation, if we work on S 1 . The derivation above is direct and does not use the Euler–Poincar´e equations; for a derivation of the Korteweg–de Vries equation from this point of view see [34], section 13.8. Let us compute the invariant momentum mapping from (4.3.2). First we need the transpose of the adjoint action (8.2.5): <      = 4     5

Y ϕ Z Y Z ϕ = , Ad , Ad b α c b c α 0 0

=

4   5 Y ϕ Z ' ∗ , b c + S(ϕ)Z dx 0

192

Peter W. Michor



Y ((ϕ ◦ ϕ

=  =

Ad




−1

)(Z ◦ ϕ

−1

 ) dx + bc+

bS(ϕ)Z dx

((Y ◦ ϕ)(ϕ
)2 + bS(ϕ))Z dx + bc,

      ϕ (Y ◦ ϕ)(ϕ
)2 + bS(ϕ) Y = . b α b

Thus the invariant momentum mapping (4.3.2) turns out as J¯

         (Y ◦ ϕ)(ϕ
)2 + bS(ϕ) Y ϕ Y ϕ , = Ad . b α b α b

(8.3.4)

 ) , according to (8.3.3) and (4.3), the Along a geodesic t → g(t, ) = ϕ(t, α(t) momentum         ϕ u = ϕt ◦ ϕ−1 ϕt ϕ2x + aS(ϕ) (u ◦ ϕ)ϕ2x + aS(ϕ) ¯ J , = = α a a a (8.3.5) is constant in t. 8.4 The curvature The computation of the curvature at the identity element has been done independently by [41] and Misiolek [42]. Here we proceed with a completely general computation that takes advantage of the formalism introduced so far.   X  Inserting the matrices of differential and integral operators ad X , α a , a X  and ad a etc., given above, into formula (3.4.2) and recalling that the matrix Z  is applied to vectors of the form c , where c is a constant, we see that

    X1 X2 is the following 2 × 2-matrix whose entries are differential 4R a1 , a2 and integral operators: ⎛

4(X1 X2

− X1

X2 ) + 2(a1 X2 − a2 X1 ) ⎜ +(8(X X
− X
X ) + 10(a X


− a X


))∂ 1 2 1 2 2 1 ⎜ 1 2 ⎜ ⎜ +18(a1 X2

− a2 X1

)∂ 2 ⎜ ⎜ +(12(a1 X2
− a2 X1
) + 2ω(X1 , X2 ))∂ 3 ⎜ ⎜ −X


ω(X ) + X


ω(X ) 2 1 ⎜ 1 2 ⎜ ⎜ ⎜ ⎝ ω(X2 )(4X1
+ 2X1 ∂ + a1 ∂ 3 ) −ω(X1 )(4X2
+ 2X2 ∂ + a2 ∂ 3 ) (4)

(4)

⎞ 2(X1


X2
− X1
X2


) ⎟ ⎟ ⎟ (4) (4) +2(X1 X2 − X1 X2 )⎟ ⎟ (6) (6) +(a1 X2 − a2 X1 ) ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ 0

Geometric evolution equations

Therefore, 4R ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

193

 X2  X3  X1 a1 , a2 a3 has the following expression:



4(X1 X2

− X1

X2 )X3 + 2(a1 X2 − a2 X1 )X3 (4)

(4)

⎞

⎟ ⎟ ⎟ ⎟










⎟ + 18(a1 X2 − a2 X1 )X3 + 12(a1 X2 − a2 X1 )X3 ⎟ ⎟    ⎟ ⎟ + 2X3


X1
X2

dx − X1


X2
X3

dx + X2


X1
X3

dx ⎟ ⎟ ⎟ (4) (4) (6) (6) ⎟







+ 2a3 (X1 X2 −X1 X2 ) + 2a3 (X1 X2 −X1 X2 ) + a3 (a1 X2 −a2 X1 )⎟ ⎟ ⎟  ⎟ ⎟








X3 (a1 X2 − a2 X1 )dx ⎟ ⎟ ⎟  ⎠












+ 2X3 (X1 X2 − X1 X2 − 2X1 X2 + 2X1 X2 )dx + (8(X1 X2
− X1
X2 ) + 10(a1 X2


− a2 X1


))X3


which coincides with formula (2.3) in Misiolek [42]. This in turn leads to the following expression for the sectional curvature: >

       ? X2 X1 X2 1 4R X a1 , a2 a1 , a2 0  = (4(X1 X2

− X1

X2 )X1 X2 + 8(X1 X2
− X1
X2 )X1
X2 + 2(a1 X2 − a2 X1 )X1 X2 + 10(a1 X2


− a2 X1


)X1
X2 (4)

(4)

+ 18(a1 X2

− a2 X1

)X1

X2 + 12(a1 X2
− a2 X1
)X1


X2 + 2ω(X1 , X2 )X1


X2 − X1


ω(X2 , X1 )X2 + X2


ω(X1 , X1 )X2 + 2(X1


X2
− X1
X2


)a1 X2 (4)

(4)

+ 2(X1 X2 − X1 X2 )a1 X2 (6)

(6)

+ (a1 X2 − a2 X1 )a1 X2 + (4X1
X1 X2


+ 2X1 X1
X2


+ a1 X1


X2


− 4X2
X1 X1


− 2X2 X1
X1


− a2 X1


X1


)a2 ) dx  (4) = (−4[X1 , X2 ]2 + 4(a1 X2 − a2 X1 )(X1 X2 − X1
X2


+ X1


X2
− X1 X2 ) (4)

− (X2


)2 a21 + 2X1


X2


a1 a2 − (X1


)2 a22 ) dx + 3ω(X1 , X2 )2 .

194

Peter W. Michor

This formula shows that the sign of the sectional curvature is not constant. Indeed, choosing h1 (x) = sin x, h2 (x) = cos x we get −π(8 + a21 + a22 − 3π) which can be positive or negative by choosing the constants a1 , a2 judiciously. 8.5 Jacobi fields     A Jacobi field y = yb along a geodesic with velocity field ua is a solution of the partial differential equation (3.5.1) which in our case looks as follows.             u y u ytt y + ad , ad = ad b a b a btt           u yt u yt u yt −α + ad a a a bt bt bt       u 3yx yxxx 2ux + u∂x uxxx = , ω(y) 0 0 0 a     yt −2ux − 4u∂x − a∂x3 −uxxx + , ω(u) 0 bt − ad

which leads to ytt = −u(4ytx + 3uyxx + ayxxxx ) − ux (2yt + 2ayxxx)

(8.5.1)

− uxxx (bt + ω(y, u) − 3ayx ) − aytxxx , btt = ω(u, yt ) + ω(y, 3ux u) + ω(y, auxxx).

(8.5.2)

Equation (8.5.2) is equivalent to  btt = (−ytxxx u + yxxx(3ux u + auxxx))dx.

(8.5.2
)

Next, let us show that the integral term in equation (8.5.1) is constant:  bt + ω(y, u) = bt + yxxx u dx =: B1 . (8.5.3) Indeed its t-derivative along the geodesic for u (that is, u satisfies the Korteweg–de Vries equation) coincides with (8.5.2
):   btt + (ytxxx u + yxxxut ) dx = btt + (ytxxx u + yxxx(−3ux u − auxxx)) dx = 0. Thus b(t) can be explicitly solved from (8.5.3) as  t yxxxu dx dt. b(t) = B0 + B1 t − a

(8.5.4)

Geometric evolution equations

195

The first component of the Jacobi equation on the Virasoro–Bott group is a genuine partial differential equation. Thus the Jacobi equations are given by the following system: ytt = −u(4ytx + 3uyxx + ayxxxx) − ux (2yt + 2ayxxx) − uxxx (B1 − 3ayx) − aytxxx ,

(8.5.5)

ut = −3ux u − auxxx, a = constant, where u(t, x), y(t, x) are either smooth functions in (t, x) ∈ I ×S 1 or in (t, x) ∈ I × R, where I is an interval or R, and where in the latter case u, y, yt have compact support with respect to x. Choosing u = c ∈ R, a constant, these equations coincide with (3.1) in Misiolek [42] where it is shown by direct inspection that there are solutions of this equation which vanish at nonzero values of t, thereby concluding that there are conjugate points along geodesics emanating from the identity element of the Virasoro–Bott group on S 1 . 8.6 The weak symplectic structure on the space of Jacobi fields on the Virasoro Lie algebra Since the Korteweg–de Vries equation has local solutions depending smoothly on the initial conditions (and global solutions if a = 0), we expect that the space of all Jacobi fields exists and is isomorphic to the space of all initial data (R ×ω X(S 1 )) × (R ×ω X(S 1 )). The weak symplectic structure is given in section (3.7):     4   5 4   5 4     5 y z y zt yt u y z z ω , = , − + , , , b c b a c b c ct b t 0 0 0 4     5 4     5 y y u z z u − − , , , , b b a c b a 0 0  = (yzt − yt z + 2u(yzx − yx z)) dx + b(ct + ω(z, u)) − c(bt + ω(y, u)) − aω(y, z)  = (yzt − yt z + 2u(yzx − yx z)) dx  + bC1 − cB1 − a

(8.6.1)

y
z

dx,

where the constant C1 relates to c as B1 does to b, see (8.5.3) and (8.5.4).

196

Peter W. Michor

8.7 The geodesics of the H k -metric on the Virasoro group We shall use the H k -inner product on R×ω g, where g is any of the Lie algebras X(S 1 ) or XS (R) = S(R)∂x . The Lie algebra Xc (R) does not work here any  more since Ak = kj=0 (−1)j ∂x2j is no longer a linear isomorphism here. 4   5  X Y : = (XY + X
Y
+ · · · + X (k) Y (k) ) dx + ab (8.7.1) , a b k   where = Ak (X)Y dx + ab = XAk (Y ) dx + ab, v

Ak =

k


(−1)i ∂x2i as in (7.3.1).

i=0

Integrating by parts we get 4
4     5   5 X Y − XY
X Y Z Z = ad , , a b c ω(X, Y ) c k k  = (X
Y Ak (Z) − XY
Ak (Z) + cX
Y

) dx  =  =

(2X
Y Ak (Z) + XY Ak (Z
) + cX


) dx




Y Ak A−1 k (2X Ak (Z) + XAk (Z ) + cX ) dx

<     = X Y Z , ad = , a b c

where

0

     −1  X Ak (2X
Ak (Z) + XAk (Z
) + cX


) Z ad = . (8.7.2) a c 0 Using matrix notation we get therefore (where ∂ := ∂x )   
 X X − X∂ 0 ad = , ω(X) 0 a ad

   −1 


X Ak .(2X
.Ak + XAk .∂x ) A−1 k (X ) , = 0 0 a        −1  X X Ak .(Ak (X
) + 2Ak (X)∂x + a∂ 3 ) 0 α = ad = . 0 0 a a

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197

Formula (3.2.2) gives the geodesic equation on the Virasoro–Bott group:        −1  ut u u −Ak (2ux Ak (u) + uAk (ux ) + auxxx) = − ad = , where a a 0 at (8.7.3)     u(t) ϕt ◦ ϕ−1 ' ϕtx ϕxx = αt − a(t) 2ϕ2 dx x

as in (8.3.2) Thus a is a constant in time and the geodesic equation contains the equation from the Korteweg–de Vries hierarchy: Ak (ut ) = −2uxAk (u) − uAk (ux ) − auxxx.

(8.7.4)

For k = 0 this gives the Korteweg–de Vries equation. For k = 1 we get the equation ut − utxx = −3uux + 2ux uxx + uuxxx − auxxx , the Camassa–Holm equation, [11], [43], [44]. See (7.3.4) for the dispersion-free version. Let us compute the invariant momentum mapping from (4.3.2). First we need the transpose of the adjoint action (8.2.5): <      = 4     5

ϕ Y ϕ Z Z Y Ad = , Ad , α c b b α c k k

=

4   5 Y ϕ Z ' ∗ , b c + S(ϕ)Z dx k 

=  = 

bS(ϕ)Z dx

Ak (Y )((ϕ
Z) ◦ ϕ−1 ) dx + bc +
2

(Ak (Y ) ◦ ϕ)(ϕ ) Z dy + bc +

=  =  =

 Ak (Y )(ϕ∗ Z) dx + bc +

 bS(ϕ)Z dx

 bS(ϕ)Z dx

((Ak (Y ) ◦ ϕ)(ϕ
)2 + bS(ϕ))Z dx + bc
2 Ak A−1 k ((Ak (Y ) ◦ ϕ)(ϕ ) + bS(ϕ))Z dx + bc

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Peter W. Michor

=

4 −1   5 Ak ((Ak (Y ) ◦ ϕ)(ϕ
)2 + bS(ϕ)) Z , , b c

     −1  Ak ((Ak (Y ) ◦ ϕ)(ϕ
)2 + bS(ϕ)) Y ϕ = Ad . b α b Thus the invariant momentum mapping (4.3.2) turns out as          −1  Ak ((Ak (Y ) ◦ ϕ)(ϕ
)2 + bS(ϕ)) Y ϕ Y ϕ = , = Ad . b α b α b (8.7.5)  ) , according to (8.7.4) and (4.3), the Along a geodesic t → g(t, ) = ϕ(t, α(t) momentum       −1  ϕ u = ϕt ◦ ϕ−1 Ak Ak (u) ◦ ϕ)ϕ2x + aS(ϕ) ¯ J , = α a a   −1  Ak (Ak (ϕt ◦ ϕ−1 ) ◦ ϕ)ϕ2x + aS(ϕ) = a (8.7.6) J¯

is constant in t, and thus also ˜ ϕ) := (Ak (ϕt ◦ ϕ−1 ) ◦ ϕ)ϕ2 + aS(ϕ) J(a, x

(8.7.7)

is constant in t. 8.8 Theorem [12] Let k ≥ 2. There exists an HC 2k+1 -open neighborhood V of (Id, 0) in the space (S 1 ×c Diff(S 1 )) × (R ×ω X(S 1 )) such that for each (g0 , α, u0 , a) ∈ V there exists a unique C 3 geodesic g ∈ C 3 ((−2, 2), S 1 ×c Diff(S 1 )) for the right invariant H k Riemann metric, starting at g(0) = g0 in the direction gt (0) = u0 ◦ g0 ∈ Tg0 Diff(S 1 ). Moreover, the solution depends C 1 on the initial data (g0 , u0 ) ∈ V . The same result holds if we replace S 1 ×c Diff(S 1 ) by R ×c Diff S (R) and X(S 1 ) by S(R)∂x = XS (R). In the following proof Diff, X, DiffHCn , HC n will mean either Diff(S 1 ), X(S 1 ), DiffHCn (S 1 ), HC n (S 1 ), or Diff S (R), XS (R), DiffHCn (R), HC n (R), respectively. Proof. For u ∈ HC n , n ≥ 2k + 1, we have, as in the proof of (7.4), Ak (uux ) = uAk (ux ) +

k
i=0

(−1)i

2i
2i j 2i−j+1 u) =: u Ak (ux ) + Bk (u), j (∂x u)(∂x j=1

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199

where Bk : HC n → HC n−2k is a bounded quadratic operator. Recall from (8.7.4) that we have to solve (where a is a real constant) ut = −A−1 k (2ux Ak (u) + uAk (ux ) + auxxx ) = −A−1 k (2ux Ak (u) + Ak (uux ) − Bk (u) + auxxx ) = −uux − A−1 k (2ux Ak (u) − Bk (u) + auxxx ) =: −uux + A−1 k Ck (u, a), where u = gt ◦ g −1 ∈ X, and where Ck : HC n → HC n−2k is a bounded polynomial operator, given by Ck (a, u) = −2uxAk (u) + Bk (u) − auxxx = −2uxAk (u) +

k


(−1)i

i=0

2i
2i j 2i−j+1 u) − auxxx . j (∂x u)(∂x j=1

Note that here we need 2k ≥ 3. In [43] this result was obtained for k ≥ 3/2. We put ⎧ g =: v = u ◦ g, ⎪ ⎪ ⎨ t (8.8.1) vt = ut ◦ g + (ux ◦ g)gt = ut ◦ g + (uux ) ◦ g = A−1 k Ck (a, u) ◦ g ⎪ ⎪ ⎩ −1 −1 = Ak Ck (a, v ◦ g ) ◦ g =: pr2 (Dk ◦ Ek )(g, v), where Ek (a, g, v) = (g, Ck (a, v ◦ g −1 ) ◦ g),

−1 Dk (g, v) = (g, A−1 ) ◦ g). k (v ◦ g

Now consider the topological group and Banach manifold DiffHCn . Claim The mapping Dk : DiffHCn ×HC n−2k → DiffHCn ×HC n is strongly C1. Let us assume that we have C 1 -curves s → g(s) ∈ DiffHCn and s → v(s) ∈ HC n−2k . Then we have: −1 ∂s pr2 Dk (a, g(s), v(s)) = ∂s A−1 )◦g k (v ◦ g    gs ◦ g −1 −1 −1 −1 = A−1 (v ◦ g ) ◦ g + A ◦ g ) − (v ◦g s x k k gx ◦ g −1 −1 )x ◦ g)gs , + (A−1 k (v ◦ g

Ak ((∂s pr2 Dk (a, g(s), v(s))) ◦ g −1 ) −1 )x (gs ◦ g −1 )) = vs ◦ g −1 − (v ◦ g −1 )x (gs ◦ g −1 ) + Ak (A−1 k (v ◦ g

= vs ◦ g −1 − (v ◦ g −1 )x (gs ◦ g −1 ) + (v ◦ g −1 )x (gs ◦ g −1 )

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Peter W. Michor

+

k 2i−1

2i −1 ))∂x2i−j (gs ◦ g −1 ) ∈ HC n−2k , (∂xj+1 (A−1 k (v ◦ g j i=0 j=0

−1 )◦g ∂s pr2 Dk (a, g(s), v(s)) = A−1 k (vs ◦ g k 2i−1

2i −1 j+1 −1 + ))∂x2i−j (gs ◦ g −1 )) ◦ g, A−1 k ((∂x (Ak (v ◦ g j i=0 j=0

and by (6.12) and (6.13) we can conclude that this is continuous in a, g, gs , v, vs jointly and Lipschitz in gs and vs . Thus Dk is strongly C 1 . Claim The mapping Ek : DiffHCn ×HC n → DiffHCn ×HC n−2k is strongly C1. This can be proved in a similar way as the last claim. By the two claims equation (8.8.1) can be viewed as the flow equation of a C 1 -vector field on the Hilbert manifold DiffHCn ×HC n . Here an existence and uniqueness theorem holds. Since v = 0 is a stationary point, there exists an open neighborhood Wn of (Id, 0) in DiffHCn ×HC n such that for each initial point (g0 , v0 ) ∈ Wn , equation (8.8.1) has a unique solution Flnt (g0 , v0 ) = (g(t), v(t)), defined and C 2 in t ∈ (−2, 2). Note that v(t) = gt (t), thus g(t) is even C 3 in t. Moreover, the solution depends C 1 on the initial data. We start with the neighborhood W2k+1 ⊂ DiffHC2k+1 ×HC 2k+1 ⊃ DiffHCn ×HC n

for n ≥ 2k + 1

and consider the neighborhood Vn := W2k+1 ∩ DiffHCn ×HC n of (Id, 0). Claim For any initial point (g0 , v0 ) ∈ Vn the solution Flnt (g0 , v0 ) = (g(t), v(t)) exists, is unique, is C 2 in t ∈ (−2, 2), and depends C 1 on the initial point in Vn . We use induction on n ≥ 2k + 1. For n = 2k + 1 the claim holds since (g0 , v0 ) = (˜ g (t), v˜(t)) V2k+1 = W2k+1 . Let (g0 , v0 ) ∈ V2k+2 and let Fl2k+2 t be maximally defined for t ∈ (t1 , t2 )  0. Suppose for contradiction that (g0 , v0 ) = (˜ g (t), v˜(t)) t2 < 2. Since (g0 , v0 ) ∈ V2k+2 ⊂ V2k+1 the curve Fl2k+2 t solves (8.8.1) also in DiffHC2k+1 ×HC 2k+1 , thus Fl2k+2 (g0 , v0 ) = (˜ g (t), v˜(t)) = t (g0 , v0 ) for t ∈ (t1 , t2 ) ∩ (−2, 2). By (7.3.6), the (g(t), v(t)) := Fl2k+1 t expression ˜ = J(g, ˜ v, t) = gx (t)2 Ak (u(t)) ◦ g(t) = gx (t)t Ak (v(t) ◦ g(t)) ◦ g(t) (8.8.2) J(t) is constant in t ∈ (−2, 2). Actually, since we used C ∞ -theory for deriving this, one should check it again by differentiating. Since u = gt ◦ g −1 we get the following (the exact formulas can be computed with the help of Fa` a di Bruno’s formula (6.1)):

Geometric evolution equations

ux = (gtx ◦ g −1 )(g −1 )x =  ∂x2 U = ∂x (g −1 ) =



gtx ◦ g −1 , gx

◦ g −1 ,

1 ◦ g −1 , gx

∂x2 (g −1 ) ◦ g = − ∂x2k (g −1 ) ◦ g = − (∂x2k u) ◦ g =

∂x2 gt ∂2g − gtx x3 2 gx gx

201

∂x2 g , gx3 ∂x2k g + lower order terms in g, gx2k+1

∂x2k g ∂x2k gt − gtx 2k+1 + lower order terms in g, gt = v. 2k gx gx

Thus ˜ = gx ∂x2k gt − gtx ∂x2k g + lower order terms in g, gt = v. (−1)k gx2k−1 J(t) Hence for each t ∈ (−2, 2): ˜ + Pk (g, v)), where gx ∂x2k gt − gtx ∂x2k g = (−1)k gx2 (gx2k−3 J(t) Pk (g, v) =

Qk (g, ∂x g, . . . , ∂x2k−1 g, v, ∂x v, . . . , ∂x2k−1 v) gx2

˜ = J(0) ˜ for a polynomial Qk . Since J(t) we obtain that  2k  ∂x g(t) = (−1)k (gx2k−3 (t)J˜(0) + Pk (g(t), v(t))) for all t ∈ (−2, 2). gx (t) t This implies ∂x2k g(t) ∂ 2k g(0) = x + (−1)k gx (t) gx (0)



t

˜ + Pk (g(s), v(s))) ds. (gx2k−3 (s)J(0) 0

For t ∈ (t1 , t2 ) we have ∂x2k g˜(t) =

∂x2k g0 gx (t) ∂x g0

(8.8.3) 

t

˜ + Pk (g(s), v(s))) ds. (gx2k−3 (s)J(0)

k

+ (−1) gx (t) 0

˜ = J(g ˜ 0 , v0 , 0) ∈ HC 2 by (2). Since k ≥ 1, Since (g0 , v0 ) ∈ V2k+2 we have J(0) 2k 2 by (8.8.3) we see that ∂x g˜(t) ∈ HC . Moreover, since t2 < 2, limt→t2 − ∂x2k g˜(t) exists in HC 2 , so limt→t2 − g˜(t) exists in HC 2k+2 . As this limit equals g(t2 ),

202

Peter W. Michor

we conclude that g(t2 ) ∈ DiffHC2k+2 . Now v˜ = g˜t ; so we may differentiate both sides of (8.8.3) in t and obtain similarly that limt→t2 − v˜(t) exists in HC 2k+2 and equals v(t2 ). But then we can prolong the flow line (˜ g , v˜) in DiffHC2k+2 ×HC 2k+2 beyond t2 , so (t1 , t2 ) was not maximal. By the same method we can iterate the induction. 2

Appendix A Smooth calculus beyond Banach spaces The traditional differential calculus works well for finite-dimensional vector spaces and for Banach spaces. For more general locally convex spaces we sketch here the convenient approach explained in [20] and [31]. The main difficulty is that the composition of linear mappings stops being jointly continuous at the level of Banach spaces, for any compatible topology. We use the notation of [31] and this is the main reference for the whole appendix. We list results in the order in which one can prove them, without proofs for which we refer to [31]. This should explain how to use these results. Later we also explain the fundamentals about regular infinite-dimensional Lie groups. A.1 Convenient vector spaces Let E be a locally convex vector space. A curve c : R → E is called smooth or C ∞ if all derivatives exist and are continuous — this is a concept without problems. Let C ∞ (R, E) be the space of smooth functions. It can be shown that C ∞ (R, E) does not depend on the locally convex topology of E, but only on its associated bornology (system of bounded sets). E is said to be a convenient vector space if one of the following equivalent conditions is satisfied (called c∞ -completeness): '1 1. For any c ∈ C ∞ (R, E) the (Riemann) integral 0 c(t)dt exists in E. 2. A curve c : R → E is smooth if and only if λ ◦ c is smooth for all λ ∈ E
, where E
is the dual consisting of all continuous linear functionals on E. 3. Any Mackey–Cauchy–sequence (i.e., tnm (xn −xm ) → 0 for some tnm → ∞ in R) converges in E. This is visibly a weak completeness requirement. The final topology with respect to all smooth curves is called the c∞ -topology on E, which then is denoted by c∞ E. For Fr´echet spaces it coincides with the given locally convex topology, but on the space D of test functions with compact support on R it is strictly finer. A.2 Smooth mappings Let E and F be locally convex vector spaces, and let U ⊂ E be c∞ -open. A mapping f : U → F is called smooth or C ∞ , if f ◦ c ∈ C ∞ (R, F ) for all c ∈ C ∞ (R, U ). The main properties of smooth calculus are the following.

Geometric evolution equations

203

1. For mappings on Fr´echet spaces this notion of smoothness coincides with all other reasonable definitions. Even on R2 this is nontrivial. 2. Multilinear mappings are smooth if and only if they are bounded. 3. If f : E ⊇ U → F is smooth, then the derivative df : U ×E → F is smooth, and also df : U → L(E, F ) is smooth where L(E, F ) denotes the space of all bounded linear mappings with the topology of uniform convergence on bounded subsets. 4. The chain rule holds. 5. The space C ∞ (U, F ) is again a convenient vector space where the structure is given by the obvious injection 3 3 C ∞ (R, F ) → C ∞ (R, R). C ∞ (U, F ) → c∈C ∞ (R,U )

c∈C ∞ (R,U ),λ∈F


6. The exponential law holds: C ∞ (U, C ∞ (V, G)) ∼ = C ∞ (U × V, G) is a linear diffeomeorphism of convenient vector spaces. Note that this is the main assumption of variational calculus. 7. A linear mapping f : E → C ∞ (V, G) is smooth (bounded) if and only if E −f→ C ∞ (V, G) −evv→ G is smooth for each v ∈ V . This is called the smooth uniform boundedness theorem and it is quite applicable. A.3 Theorem [20], 4.1.19 Let c : R → E be a curve in a convenient vector space E. Let V ⊂ E
be a subset of bounded linear functionals such that the bornology of E has a basis of σ(E, V)-closed sets. Then the following are equivalent: 1. c is smooth. 2. There exist locally bounded curves ck : R → E such that  ◦ c is smooth R → R with ( ◦ c)(k) =  ◦ ck . If E is reflexive, then for any point separating subset V ⊂ E
the bornology of E has a basis of σ(E, V)-closed subsets, by [20], 4.1.23. A.4 Counterexamples in infinite dimensions against common beliefs on ordinary differential equations Let E := s be the Fr´echet space of rapidly decreasing sequences; note that by the theory of Fourier series we have s = C ∞ (S 1 , R). Consider the continuous linear operator T : E → E given by T (x0 , x1 , x2 , . . . ) := (0, 12 x1 , 22 x2 , 32 x3 , . . . ). The ordinary linear differential equation x
(t) = T (x(t)) with constant coefficients has no solution in s for certain initial values. By recursion one sees that the general solution should be given by

204

Peter W. Michor

xn (t) =

n
 n! 2 i!

xi (0)

i=0

tn−i . (n − i)!

If the initial value is a finite sequence, say xn (0) = 0 for n > N and xN (0) = 0, then xn (t) =

2 N 
n! i=0

i!

xi (0)

tn−i (n − i)!


(n!)2 tn−N = (n − N )! i=0 N

 2 1 (n − N )! N −i t xi (0) , i! (n − i)!

   2 N −1 2
1 1 (n!)2 (n − N )! N −i n−N |xn (t)| ≥ |t| |t| |xN (0)| − |xi (0)| (n − N )! N! i! (n − i)! i=0 (n!)2 ≥ |t|n−N (n − N )!



 |xN (0)|

1 N!

2 −

N −1 
i=0

1 i!

2

 N −i

|xi (0)||t|

where the first factor does not lie in the space s of rapidly decreasing sequences and where the second factor is larger than ε > 0 for t small enough. So at least for a dense set of initial values this differential equation has no local solution. This shows also, that the theorem of Frobenius is wrong, in the following sense: The vector field x → T (x) generates a 1-dimensional subbundle E of the tangent bundle on the open subset s \ {0}. It is involutive since it is 1-dimensional. But through points representing finite sequences there exist no local integral submanifolds (M with T M = E|M ). Namely, if c were a smooth nonconstant curve with c
(t) = f (t).T (c(t)) for some smooth function f , then x(t) := c(h(t)) would satisfy x
(t) = T (x(t)), where h is a solution of h
(t) = 1/f (h(t)). As our next example consider E := RN and the continuous linear operator T : E → E given by T (x0 , x1 , . . . ) := (x1 , x2 , . . . ). The corresponding differential equation has solutions for every initial value x(0), since the coordinates must satisfy the reclusive relations xk+1 (t) = x
k (t) and hence any smooth (k) function x0 : R → R gives rise to a solution x(t) := (x0 (t))k with initial (k) value x(0) = (x0 (0))k . So by Borel’s theorem there exist solutions to this equation for any initial value and the difference of any two functions with the same initial value is an arbitrary infinite flat function. Thus the solutions are far from being unique. Note that RN is a topological direct summand in C ∞ (R, R) via the projection f → (f (n))n , and hence the same situation occurs in C ∞ (R, R). Let now E := C ∞ (R, R) and consider the continuous linear operator T : E → E given by T (x) := x
. Let x : R → C ∞ (R, R) be a solution of the ∂ equation x
(t) = T (x(t)). In terms of x ˆ : R2 → R this says ∂t xˆ(t, s) =

Geometric evolution equations

205

Hence r → x ˆ(t − r, s + r) has a vanishing derivative everywhere and so this function is constant, and in particular x(t)(s) = x ˆ(t, s) = xˆ(0, s + t) = x(0)(s + t). Thus we have a smooth solution x uniquely determined by the initial value x(0) ∈ C ∞ (R, R) which even describes a flow for the vector field T in the sense of (A.6) below. In general this solution is however not real-analytic, since for any x(0) ∈ C ∞ (R, R), which is not real-analytic in a neighborhood of a point s the composite evs ◦x = x(s+ ) is not real-analytic around 0. ∂ ˆ(t, s). ∂s x

A.5 Manifolds and vector fields In the sequel we shall use smooth manifolds M modeled on c∞ -open subsets of convenient vector spaces. Since we shall need it we also include some results on vector fields and their flows. Consider vector fields Xi ∈ C ∞ (T M ) and Yi ∈ Γ (T N ) for i = 1, 2, and a smooth mapping f : M → N . If Xi and Yi are f -related for i = 1, 2, i.e., T f ◦ Xi = Yi ◦ f , then also [X1 , X2 ] and [Y1 , Y2 ] are f -related. In particular if f : M → N is a local diffeomorphism (so (Tx f )−1 makes sense for each x ∈ M ), then for Y ∈ Γ (T N ) a vector field f ∗ Y ∈ Γ (T M ) is defined by (f ∗ Y )(x) = (Tx f )−1 .Y (f (x)). The linear mapping f ∗ : Γ (T N ) → Γ (T M ) is then a Lie algebra homomorphism. A.6 The flow of a vector field Let X ∈ Γ (T M ) be a vector field. A local flow FlX for X is a smooth mapping FlX : M × R ⊃ U → M defined on a c∞ -open neighborhood U of M × 0 such that: X d 1. dt FlX t (x) = X(Flt (x)). X 2. Fl0 (x) = x for all x ∈ M . 3. U ∩ ({x} × R) is a connected open interval. X X 4. FlX t+s = Flt ◦ Fls holds in the following sense. If the right-hand side exists, then also the left-hand side exists and we have equality. Moreover: If FlX s exists, then the existence of both sides is equivalent and they are equal.

Let X ∈ Γ (T M ) be a vector field which admits a local flow FlX t . Then for each integral curve c of X we have c(t) = FlX t (c(0)), thus there exists a unique X maximal flow. Furthermore, X is FlX t -related to itself, i.e., T (Flt ) ◦ X = X X ◦ Flt . Let X ∈ Γ (T M ) and Y ∈ Γ (T N ) be f -related vector fields for a smooth mapping f : M → N which have local flows FlX and FlY . Then we have Y f ◦ FlX t = Flt ◦f , whenever both sides are defined. ∗ Moreover, if f is a diffeomorphism we have Flft Y = f −1 ◦ FlYt ◦f in the following sense: If one side exists, then also the other side exists, and they are equal.

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Peter W. Michor

For f = IdM this implies that if there exists a flow, then there exists a unique maximal flow FlX t . A.7 The Lie derivative There are situations where we do not know that the flow of X exists but where we will be able to produce the following assumption: Suppose that ϕ : R × M ⊃ U → M is a smooth mapping such that (t, x) → (t, ϕ(t, x) = ϕt (x)) is a diffeomorphism U → V , where U and V are open neighborhoods of {0} × M in R × M , and such that ϕ0 = IdM and ∂t ϕt = X ∈ Γ (T M ). Then again ∂t |0 (ϕt )∗ f = ∂t |0 f ◦ ϕt = df ◦ X = X(f ). In this situation we have for Y ∈ Γ (T M ), and for a k-form ω ∈ Ω k (M ): ∂t |0 (ϕt )∗ Y = [X, Y ], ∂t |0 (ϕt )∗ ω = LX ω.

Appendix B Regular infinite-dimensional Lie groups B.1 Lie groups A Lie group G is a smooth manifold modeled on c∞ -open subsets of a convenient vector space, and a group such that the multiplication µ : G × G → G and the inversion ν : G → G are smooth. We shall use the following notation: µ : G × G → G, multiplication, µ(x, y) = x.y. µa : G → G, left translation, µa (x) = a.x. µa : G → G, right translation, µa (x) = x.a. ν : G → G, inversion, ν(x) = x−1 . e ∈ G, the unit element. The tangent mapping T(a,b) µ : Ta G × Tb G → Tab G is given by T(a,b) µ.(Xa , Yb ) = Ta (µb ).Xa + Tb (µa ).Yb and Ta ν : Ta G → Ta−1 G is given by −1

−1

Ta ν = −Te (µa ).Ta (µa−1 ) = −Te (µa−1 ).Ta (µa ).

B.2 Invariant vector fields and Lie algebras Let G be a (real) Lie group. A vector field ξ on G is called left invariant, if µ∗a ξ = ξ for all a ∈ G, where µ∗a ξ = T (µa−1 ) ◦ ξ ◦ µa . Since we have µ∗a [ξ, η] = [µ∗a ξ, µ∗a η], the space XL (G) of all left invariant vector fields on G is closed under the Lie bracket, so it is a sub-Lie algebra of X(G). Any left invariant

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207

vector field ξ is uniquely determined by ξ(e) ∈ Te G, since ξ(a) = Te (µa ).ξ(e). Thus the Lie algebra XL (G) of left invariant vector fields is linearly isomorphic to Te G, and on Te G the Lie bracket on XL (G) induces a Lie algebra structure, whose bracket is again denoted by [ , ]. This Lie algebra will be denoted as usual by g, sometimes by Lie(G). We will also give a name to the isomorphism with the space of left invariant vector fields: L : g → XL (G), X → LX , where LX (a) = Te µa .X. Thus [X, Y ] = [LX , LY ](e). Similarly a vector field η on G is called right invariant, if (µa )∗ η = η for all a ∈ G. If ξ is left invariant, then ν ∗ ξ is right invariant. The right invariant vector fields form a sub-Lie algebra XR (G) of X(G), which is again linearly isomorphic to Te G and induces the negative of the Lie algebra structure on Te G. We will denote by R : g = Te G → XR (G) the isomorphism discussed, which is given by RX (a) = Te (µa ).X. If LX is a left invariant vector field and RY is a right invariant vector field, then [LX , RY ] = 0. So if the flows of LX and RY exist, they commute. Let ϕ : G → H be a smooth homomorphism of Lie groups. Then ϕ
:= Te ϕ : g = Te G → h = Te H is a Lie algebra homomorphism. B.3 One parameter subgroups Let G be a Lie group with Lie algebra g. A one parameter subgroup of G is a Lie group homomorphism α : (R, +) → G, i.e., a smooth curve α in G with α(s + t) = α(s).α(t), and hence α(0) = e. Note that a smooth mapping β : (−ε, ε) → G satisfying β(t)β(s) = β(t+s) for |t|, |s|, |t + s| < ε is the restriction of a one-parameter subgroup. Namely, choose 0 < t0 < ε/2. Any t ∈ R can be uniquely written as t = N.t0 + t
for 0 ≤ t
< t0 and N ∈ Z. Put α(t) = β(t0 )N β(t
). The required properties are easy to check. Let α : R → G be a smooth curve with α(0) = e. Let X ∈ g. Then the following assertions are equivalent. 1. α is a one-parameter subgroup with X = ∂t α(t). 2. α(t) is an integral curve of the left invariant vector field LX , and also an integral curve of the right invariant vector field RX . X 3. FlLX (t, x) := x.α(t) (or FlL = µα(t) ) is the (unique by (A.6)) global flow t of LX in the sense of (A.6). X = µα(t) ) is the (unique) global flow of RX . 4. FlRX (t, x) := α(t).x (or FlR t Moreover, each of these properties determines α uniquely. B.4 Exponential mapping Let G be a Lie group with Lie algebra g. We say that G admits an exponential mapping if there exists a smooth mapping exp : g → G such that t → exp(tX) is the (unique by (B.3)) 1-parameter subgroup with tangent vector X at 0. Then we have by (B.3):

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Peter W. Michor

1. FlLX (t, x) = x. exp(tX). 2. FlRX (t, x) = exp(tX).x. 3. exp(0) = e and T0 exp = Id : T0 g = g → Te G = g since T0 exp .X = ∂t |0 exp(0 + t.X) = ∂t |0 FlLX (t, e) = X. 4. Let ϕ : G → H be a smooth homomorphism between Lie groups admitting exponential mappings. Then the diagram g

ϕ


expG

 G

/h expH

ϕ

 /H

commutes, since t → ϕ(expG (tX)) is a one-parameter subgroup of H and ∂t |0 ϕ(expG tX) = ϕ
(X), so ϕ(expG tX) = expH (tϕ
(X)). We shall strengthen this notion in (B.9) below and call it a ‘regular Fr´echet Lie group’. If G admits an exponential mapping, it follows from (B.4).(3) that exp is a diffeomorphism from a neighborhood of 0 in g onto a neighborhood of e in G, if a suitable inverse function theorem is applicable. This is true for example for smooth Banach Lie groups, also for gauge groups, but it is wrong for diffeomorphism groups. If E is a Banach space, then in the Banach Lie group GL(E) of all bounded linear automorphisms of Ethe exponential mapping is given by the von ∞ Neumann series exp(X) = i=0 i!1 X i . If G is connected with exponential mapping and U ⊂ g is open with 0 ∈ U , then one may ask whether the group generated by exp(U ) equals G. Note that this is a normal subgroup. So if G is simple, the answer is yes. This is true for connected components of diffeomorphism groups and many of their important subgroups. B.5 The adjoint representation Let G be a Lie group with Lie algebra g. For a ∈ G we define conja : G → G by conja (x) = axa−1 . It is called the conjugation or the inner automorphism by a ∈ G. This defines a smooth action of G on itself by automorphisms. The adjoint representation Ad : G → GL(g) ⊂ L(g, g) is given by Ad(a) = (conja )
= Te (conja ) : g → g for a ∈ G. By (B.2) Ad(a) is a Lie algebra −1 homomorphism. By (B.1) we have Ad(a) = Te (conja ) = Ta (µa ).Te (µa ) = −1 Ta−1 (µa ).Te (µa ). Finally we define the (lower case) adjoint representation of the Lie algebra g, ad : g → gl(g) := L(g, g), by ad := Ad
= Te Ad. We shall also use the right Maurer–Cartan form κr ∈ Ω 1 (G, g), given −1 by κrg = Tg (µg ) : Tg G → g; similarly the left Maurer–Cartan form κl ∈ Ω 1 (G, g) is given by κlg = Tg (µg−1 ) : Tg G → g.

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1. LX (a) = RAd(a)X (a) for X ∈ g and a ∈ G. 2. ad(X)Y = [X, Y ] for X, Y ∈ g. 3. d Ad = (ad ◦κr ). Ad = Ad .(ad ◦κl ) : T G → L(g, g). B.6 Right actions Let r : M × G → M be a right action, so rˇ : G → Diff(M ) is a group anti-homomorphism. We will use the following notation: ra : M → M and rx : G → M , given by rx (a) = ra (x) = r(x, a) = x.a. For any X ∈ g we define the fundamental vector field ζ X = ζ M X ∈ X(M ) by ζ X (x) = Te (rx ).X = T(x,e) r.(0x , X). In this situation the following assertions hold: 1. ζ : g → X(M ) is a Lie algebra homomorphism. 2. Tx (ra ).ζ X (x) = ζ Ad(a−1 )X (x.a). 3. 0M × LX ∈ X(M × G) is r-related to ζ X ∈ X(M ). B.7 The right and left logarithmic derivatives Let M be a manifold and let f : M → G be a smooth mapping into a Lie group G with Lie algebra g. We define the mapping δ r f : T M → g by the formula −1

δ r f (ξ x ) := Tf (x) (µf (x) ).Tx f.ξ x for ξ x ∈ Tx M. Then δ r f is a g-valued 1-form on M , δ r f ∈ Ω 1 (M ; g). We call δ r f the right logarithmic derivative of f , since for f : R → (R+ , ·) we have δ r f (x).1 = f
(x)
f (x) = (log ◦f ) (x). Similarly the left logarithmic derivative δ l f ∈ Ω 1 (M, g) of a smooth mapping f : M → G is given by δ l f.ξ x = Tf (x) (µf (x)−1 ).Tx f.ξ x . Let f, g : M → G be smooth. Then the Leibniz rule holds: δ r (f.g)(x) = δ r f (x) + Ad(f (x)).δ r g(x). Moreover, the differential form δ r f ∈ Ω 1 (M ; g) satisfies the ‘left Maurer– Cartan equation’ (left because it stems from the left action of G on itself ) dδ r f (ξ, η) − [δ r f (ξ), δ r f (η)]g = 0, or

1 dδ r f − [δ r f, δ r f ]g∧ = 0, 2

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Peter W. Michor

where ξ, η ∈ Tx M , and where for ϕ ∈ Ω p (M ; g), ψ ∈ Ω q (M ; g) one puts [ϕ, ψ]g∧ (ξ 1 , . . . , ξ p+q ) :=

1
sign(σ)[ϕ(ξ σ1 , . . . ), ψ(ξ σ(p+1) , . . . )]g . p!q! σ

For the left logarithmic derivative the corresponding Leibniz rule is uglier, and it satisfies the ‘right Maurer–Cartan equation’: δ l (f g)(x) = δ l g(x) + Ad(g(x)−1 )δ l f (x), 1 dδ l f + [δ l f, δ l f ]g∧ = 0. 2 For ‘regular Lie groups’ a converse to this statement holds, see [30], 40.2. The proof of this result in infinite dimensions uses principal bundle geometry for the trivial principal bundle pr1 : M × G → M with right principal action. Then the submanifolds {(x, f (x).g) : x ∈ M } for g ∈ G form a foliation of M × G whose tangent distribution is complementary to the vertical bundle M × T G ⊆ T (M × G) and is invariant under the principal right G-action. So it is the horizontal distribution of a principal connection on M × G → G. Thus this principal connection has vanishing curvature which translates into the result for the right logarithmic derivative. (1) Let G be a Lie group with Lie algebra g. For a closed interval I ⊂ R and for X ∈ C ∞ (I, g) we consider the ordinary differential equation

g(t0 ) = e, ∂t g(t) = Te (µg(t) )X(t) = RX(t) (g(t)),

or κr (∂t g(t)) = X(t),

for local smooth curves g in G, where t0 ∈ I. (2) Local solution curves g of the differential equation (1) are unique. (3) If for fixed X the differential equation (1) has a local solution near each t0 ∈ I, then it has also a global solution g ∈ C ∞ (I, G). (4) If for all X ∈ C ∞ (I, g) the differential equation (1) has a local solution near one fixed t0 ∈ I, then it has also a global solution g ∈ C ∞ (I, G) for each X. Moreover, if the local solutions near t0 depend smoothly on the vector fields X, then so does the global solution. (5) The curve t → g(t)−1 is the unique local smooth curve h in G which satisfies h(t0 ) = e, ∂t h(t) = Te (µh(t) )(−X(t)) = L−X(t) (h(t)),

or κl (∂t h(t)) = −X(t).

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B.8 Regular Lie groups If for each X ∈ C ∞ (R, g) there exists g ∈ C ∞ (R, G) satisfying ⎧ g(0) = e, ⎪ ⎪ ⎨ ∂t g(t) = Te (µg(t) )X(t) = RX(t) (g(t)), ⎪ ⎪ ⎩ or κr (∂t g(t)) = δ r g(∂t ) = X(t),

(B.8.1)

then we write evolrG (X) = evolG (X) := g(1), EvolrG (X)(t) := evolG (s → tX(ts)) = g(t), and call it the right evolution of the curve X in G. By Lemma (B.8) the solution of the differential equation (1) is unique, and for global existence it is sufficient that it has a local solution. Then EvolrG : C ∞ (R, g) → {g ∈ C ∞ (R, G) : g(0) = e} is bijective with inverse the right logarithmic derivative δ r . The Lie group G is called a regular Lie group if evolr : C ∞ (R, g) → G exists and is smooth. We also write evollG (X) = evolG (X) := h(1), EvollG (X)(t) := evollG (s → tX(ts)) = h(t), if h is the (unique) solution of ⎧ ⎪ ⎪h(0) = e, ⎨ ∂t h(t) = Te (µh(t) )(X(t)) = LX(t) (h(t)), ⎪ ⎪ ⎩ or κl (∂t h(t)) = δ l h(∂t ) = X(t).

(B.8.2)

Clearly evoll : C ∞ (R, g) → G exists and is also smooth if evolr does, since we have evoll (X) = evolr (−X)−1 by lemma (B.8). Let us collect some easily seen properties of the evolution mappings. If f ∈ C ∞ (R, R), then we have Evolr (X)(f (t)) = Evolr (f
.(X ◦ f ))(t). Evolr (X)(f (0)), Evoll (X)(f (t)) = Evoll (X)(f (0)). Evoll (f
.(X ◦ f ))(t).

212

Peter W. Michor

If ϕ : G → H is a smooth homomorphism between regular Lie groups, then the diagram C ∞ (R, g)

ϕ
∗

evolG

 G

/ C ∞ (R, h) evolH

ϕ

 /H

commutes, since ∂t ϕ(g(t)) = T ϕ.T (µg(t) ).X(t) = T (µϕ(g(t)) ).ϕ
.X(t). Note that each regular Lie group admits an exponential mapping, namely the restriction of evolr to the constant curves R → g. A Lie group is regular if and only if its universal covering group is regular. Up to now the following statement holds: All known Lie groups are regular. Any Banach Lie group is regular, since we may consider the time dependent right invariant vector field RX(t) on G and its integral curve g(t) starting at e, which exists and depends smoothly on (a further parameter in) X. In particular finite-dimensional Lie groups are regular. For diffeomorphism groups the evolution operator is just integration of time dependent vector fields with compact support. B.9 Extensions of Lie groups Let H and K be Lie groups. A Lie group G is called a smooth extension of H with kernel K if we have a short exact sequence of groups {e} → K −i→ G −p→ H → {e},

(B.9.1)

such that i and p are smooth and one of the following two equivalent conditions is satisfied: 2 p admits a local smooth section s near e (equivalently near any point), and i is initial (i.e., any f into K is smooth if and only if i ◦ f is smooth). 1. i admits a local smooth retraction r near e (equivalently near any point), and p is final (i.e., f from H is smooth if and only if f ◦ p is smooth). Of course by s(p(x))i(r(x)) = x the two conditions are equivalent, and then G is locally diffeomorphic to K × H via (r, p) with local inverse (i ◦ pr1 ).(s ◦ pr2 ). Not every smooth exact sequence of Lie groups admits local sections as required in (2). Let for example K be a closed linear subspace in a convenient vector space G which is not a direct summand, and let H be G/K. Then the tangent mapping at 0 of a local smooth splitting would make K a direct summand. Let {e} → K −i→ G −p→ H → {e} be a smooth extension of Lie groups. Then G is regular if and only if both K and H are regular.

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B.10 Subgroups of regular Lie groups Let G and K be Lie groups, let G be regular and let i : K → G be a smooth homomorphism which is initial (see (B.10)) with Te i = i
: k → g injective. We suspect that K is then regular, but we know a proof for this only under the following assumption. There is an open neighborhood U ⊂ G of e and a smooth mapping p : U → E into a convenient vector space E such that p−1 (0) = K ∩ U and p is constant on left cosets Kg ∩ U .

References 1. V. I. Arnold, Geometrical methods in the theory of ordinary differential equations, Springer-Verlag, New York, 1983. Fundamental Principles of Mathematical Sciences, 250. 2. V.I. Arnold, An a priori estimate in the theory of hydrodynamic stability, Izvestia Vyssh. Uchebn. Zaved. Matematicka 54(1966), 3–5. (Russian). 3. V.I. Arnold, Sur la g´eometrie diff´ erentielle des groupes de Lie de dimension infinie et ses applications a ` l’hydrodynamique des fluides parfaits, Ann. Inst. Fourier (Grenoble) 16(1966), 319–361. 4. V.I. Arnold, Mathematical methods of classical mechanics, Springer-Verlag, Berlin, Heidelberg, New York, 1978. Graduate Texts in Math. 60. 5. V. Arnold and B. Khesin, Topological methods in fluid dynamics, Applied Math. Sciences. Springer-Verlag, Berlin, New York, 1998. 6. E. Binz, Two natural metrics and their covariant derivatives on a manifold of embeddings, Monatsh. Math. 89(1980), 275–288. 7. E. Binz and H.R. Fischer, The manifold of embeddings of a closed manifold, Proc. Differential geometric methods in theoretical physics, Clausthal 1978. Springer-Verlag, Berlin, New York, Lecture Notes in Physics 139. 1981. 8. R. Bott, On the characteristic classes of groups of diffeomorphisms, Enseign. Math. 23(1977), 209–220. 9. Jean-Luc Brylinski, Loop spaces, characteristic classes, and geometric quantization, Birkh¨ auser, Boston, Basel, Berlin, 1993. Progress in Math. 107. 10. J. Burgers, A mathematical model illustrating the theory of turbulence, Adv. Appl. Mech., 171–199. Academic Press, New York, 1948. 11. R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett. 71(1993), 1661–1664. 12. A. Constantin, T. Kappeler, B. Kolev, and P. Topalov, On geodesic exponential maps of the Virasoro group, Preprint 13-2004, University of Z¨ urich. 13. A. Constantin and B. Kolev, Geodesic flow on the diffeomorphism group of the circle, Comm. Math. Helv. 78(2003), 787–804. 14. K. Deimling, Ordinary differential equations in Banach spaces. Lecture Notes in Mathematics, Vol. 596. Springer-Verlag, Berlin, New York, 1977. 15. J. Eichhorn, The manifold structure of maps between open manifolds, Ann. Global Anal. Geom. 11(1993), 253–300. 16. J. Eichhorn, Diffeomorphism groups on noncompact manifolds, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 234(1996), Differ. Geom. Gruppy Li i Mekh., 15-1, 41–64, 262; translation in J. Math. Sci. (New York) 94(1999), 1162–1176.

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17. J. Eichhorn and J. Fricke, The module structure theorem for Sobolev spaces on open manifolds, Math. Nachr. 194(1998), 35–47. 18. D.B.A. Epstein, The simplicity of certain groups of homeomorphisms, Compositio Math. 22(1970), 165–173. ` di Bruno, Note sur une nouvelle formule du calcul diff´erentiel, Quart. 19. C.F. Faa J. Math. 1(1855), 359–360. ¨ licher and A. Kriegl, Linear spaces and differentiation theory, Pure 20. A. Fro and Applied Mathematics, J. Wiley, Chichester, 1988. 21. D. Fuks, Cohomology of infinite dimensional Lie algebras. Nauka, Moscow, 1984 (Russian). Transl. English, Contemporary Soviet Mathematics. Consultants Bureau (Plenum Press), New York, 1986. 22. I.M. Gelfand and I.Y. Dorfman, Hamiltonian operators and algebraic structures associated with them,(Russian) Funktsional. Anal. i Prilozhen. 13(1979), 13–30. 23. I. M. Gelfand and D. B. Fuks, Cohomologies of the Lie algebra of vector fields on the circle,(Russian) Funktsional. Anal. i Prilozhen. 2(1968), 92–93. ¨ rmander, The Analysis of linear partial differential operators I, II, III, 24. L. Ho IV. Springer-Verlag, Berlin, Heidelberg, New York, 1983, 1983, 1985, 1985. Grundlehren 256, 257, 274, 275. 25. G. Kainz, A note on the manifold of immersions and its Riemannian curvature, Monatsh. Math. 98(1984), 211–217. 26. B. Khesin and P.W. Michor, The flow completion of Burgers’ equation, De Gruyter, Berlin, 2004, 17–26. IRMA Lectures in Mathematics and Theoretical Physics 5. 27. A.A. Kirillov, The orbits of the group of diffeomorphisms of the circle, and local Lie superalgebras, (Russian) Funktsional. Anal. i Prilozhen. 15(1981), 75– 76. 28. S. Kobayashi and K. Nomizu, Foundations of Differential Geometry. Vol. I, Vol. II, J. Wiley-Interscience, 1963, 1969. 29. S. Kouranbaeva, The Camassa–Holm equation on as a geodesic flow on the diffeomorphism group, J. Math. Phys. 40(1999), 857–868. 30. A. Kriegl and P. W. Michor, A convenient setting for real analytic mappings, Acta Math. 165(1990), 105–159. 31. A. Kriegl and P. W. Michor, The Convenient Setting of Global Analysis. AMS, Providence, RI, 1997. Mathematical Surveys and Monographs 53. 32. A. Kriegl and P. W. Michor, Regular infinite dimensional Lie groups, J. Lie Theory 7(1997), 61–99. 33. A. Lasota and J. A. Yorke, The generic property of existence of solutions of differential equations in Banach space, J. Differential Equations 13(1973), 1–12. 34. J. Marsden and T. Ratiu, Introduction to mechanics and symmetry. SpringerVerlag, New York, Berlin, Heidelberg, 1994. 35. J. N. Mather, Commutators of diffeomorphisms I, II, Comment. Math. Helv., 49 (1974), 512–528; 50 (1975), 33–40. 36. P. W. Michor, Manifolds of smooth maps III: The principal bundle of embeddings of a non-compact smooth manifold, Cah. Topol. G´eom. Diff´er. Cat´eg., 21(1980), 325–337. 37. P. W. Michor, Manifolds of differentiable mappings, Shiva Mathematics Series, 3, Shiva Publishing Ltd., Nantwich, 1980.

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lek, A shallow water equation as a geodesic flow on the Bott–Virasoro group, J. Geom. Phys. 24(1998), 203–208. 45. J. P. Ortega and T. Ratiu, Momentum maps and Hamiltonian reduction. Birkh¨ auser, Boston, 2004. Progress in Mathematics 222. 46. V. Y. Ovsienko and B. A. Khesin, The super Korteweg–de Vries equations as an Euler equation, (Russian) Funktsional. Anal. i Prilozhen. 21(1987), 81–82. 47. A. Pressley and G. Segal, Loop groups. Oxford University Press, 1986. Oxford Mathematical Monographs. 48. T. Ratiu, The motion of the free n-dimensional rigid body, Indiana Univ. Math. J. 29(1980), 609–629. 49. G. Segal, The geometry of the KdV equation, Topological methods in quantum field theory (Trieste, 1990). Internat. J. Modern Phys. A 6(1991), 2859–2869. 50. W. Thurston, Foliations and groups of diffeomorphisms, Bull. Amer. Math. Soc. 80(1974), 304–307. 51. A. Weinstein, Symplectic manifolds and their Lagrangian manifolds, Adv. Math. 6(1971), 329–345.

Non-effectively hyperbolic operators and bicharacteristics Tatsuo Nishitani Department of Mathematics, Osaka University, Osaka, Japan

Summary. We shall present here a survey concerning the (microlocal) well-posedness of the Cauchy problem for a second order hyperbolic operator P around a non-effectively hyperbolic double characteristic ρ and its close relations to the behavior of the bicharacteristics of p, the principal symbol of P , near ρ. Assuming that p vanishes of second order on the smooth doubly characteristic manifold Σ and that the rank of the canonical symplectic two form is constant on Σ, the microlocal Cauchy problem is C ∞ well posed if and only if p admits an elementary decomposition, and moreover p admits an elementary decomposition if and only if there is no bicharacteristic issuing from a simple characteristic point which has a limit point in Σ. Thus the behavior of bicharacteristics is a real object which controls the (microlocal) C ∞ well-posedness of the Cauchy problem, while the spectral properties of the Hamilton map Hp , defined through Hesse matrix of p, cannot determine the behavior of bicharacteristics completely in general.

2000 Mathematics Subject Classification: Primary, 35L15; Secondary, 35L30 Key words: Non-effectively hyperbolic, bicharacteristics, elementary decomposition, Hamilton map, well-posedness, Cauchy problem

1 Introduction Let P be a second order hyperbolic differential operator with respect to the x0 direction with principal symbol p(x, ξ). Let ρ be a double characteristic of p, that is p(ρ) = 0, dp(ρ) = 0. Since ρ is a singular point of the Hamilton vector field Hp of p, then, to investigate the behavior of bicharacteristics, we are naturally led to consider the linearization of Hp at ρ which is called the Hamilton map Fp of p at ρ defined as ([IP74], [H77]) 1 Q(X, Y ) = σ(X, Fp Y ), 2

X, Y ∈ Tρ (T ∗ Ω)

218

Tatsuo Nishitani

 where Q is the polar form of the Hessian of p at ρ and σ = dξ j ∧ dxj is the canonical symplectic two form on T ∗ Ω. It is well known that all eigenvalues of Fp are on the imaginary axis, possibly except for a pair of non-zero real eigenvalues ([IP74], [H77]). If there exists a pair of non-zero real eigenvalues, then p is called effectively hyperbolic at ρ and otherwise not effectively hyperbolic at ρ. When p is effectively hyperbolic at every double characteristic, then the Cauchy problem for P is C ∞ well posed for any lower order term, that is P is strongly hyperbolic ([N84bis], [Iw84], [Iv78bis]). If p is not effectively hyperbolic at ρ, then according to the spectral property of Fp we divide the cases into two: KerFp2 (ρ) ∩ ImFp2 (ρ) = {0}

(1.1)

KerFp2 (ρ) ∩ ImFp2 (ρ) = {0}.

(1.2)

It is very natural to expect that the behavior of bicharacteristics plays an important role for the well-posedness of the (microlocal) Cauchy problem. In the first case (1.1) bicharacteristics are stable with respect to the doubly characteristic manifold, that is a bicharacteristic issuing from a simple characteristic does not touch the doubly characteristic manifold. In this case, assuming the constancy of the rank of the symplectic form σ on the doubly characteristic manifold, the Cauchy problem is C ∞ well posed (of course assuming suitable conditions on lower order terms, called the Levi condition and the Ivrii–Petkov–H¨ ormander condition [Iv78], [H77], [N90]). On the other hand, in the latter case (1.2), the spectral property of Fp is not sufficient to determine completely the behavior of bicharacteristics. In some cases there exists a bicharacteristic issuing from a simple characteristic point which touches the double characteristic manifold whereas in other cases there is no such bicharacteristic ([N83]). This is a survey concerning the behavior of bicharacteristic around a double characteristic verifying (1.2) and its close relations to the well-posedness of the (microlocal) Cauchy problem.

2 Non-effectively hyperbolic symbols, elementary decomposition and a priori estimates 2.1 A lemma Let P (x, D) be a second order differential operator with the principal symbol p(x, ξ): p(x, ξ) = −ξ 20 + q(x, ξ
), q(x, ξ
) ≥ 0 where x = (x0 , x1 , . . . , xn ) = (x0 , x
), ξ = (ξ 0 , ξ 1 , . . . , ξ n ) = (ξ 0 , ξ
). Let ρ ˆ = (0, ˆξ) be a double characteristic of p(x, ξ), that is p(0, ˆξ) = 0, dp(0, ˆξ) = 0. We assume that p(x, ξ) is not effectively hyperbolic at ρ ˆ.

(2.1)

Non-effectively hyperbolic operators and bicharacteristics

219

We look for a system of local homogeneous canonical coordinates around ˆρ where p(x, ξ) takes a special form. Let us denote x(p) = (xp , . . . , xn ), ξ (p) = (ξ p , . . . , ξ n ), x = x(0) , ξ = ξ (0) , 0 ≤ p ≤ n. We also denote by Fp (x, ξ) the Hamilton map of p at (x, ξ). We divide the cases ρ) ∩ ImFp2 (ˆ ρ) = {0} KerFp2 (ˆ

(2.2)

ρ) ∩ ImFp2 (ˆ ρ) = {0}. KerFp2 (ˆ

(2.3)

Lemma 2.1 ([N86]) Let us assume (2.1) and (2.2) (resp. (2.3)). Then there are homogeneous local canonical coordinates such that ρ ˆ = (0, 0, en ) with which q(x, ξ
) takes the form p


qi (x, ξ
)(xi−1 − xi )2 +

i=1

p


ri (x, ξ
)ξ 2i + g(xp , x(p+1) , ξ (p+1) )rp+1 (x, ξ
)

i=1

with {ξ p , {ξ p , g}}(0, en) = 0,

p


ri (0, en )−1 < 1, 0 ≤ p ≤ n − 1

i=1

(resp. {ξ p , {ξ p , g}}(0, en) = 0,

p


ri (0, en )−1 = 1, 1 ≤ p ≤ n − 1)

i=1

where qi , ri are positive, homogeneous of degree 2, 0 respectively, g is nonnegative, vanishes at ˆ ρ, homogeneous of degree 2 and en = (0, . . . , 0, 1). 2.2 Elementary decomposition In this subsection we consider p = −ξ 20 + a1 (x, ξ
)ξ 0 + a2 (x, ξ
) which is hyperbolic with respect to dx0 , that is a1 (x, ξ
)2 + 4a2 (x, ξ
) ≥ 0. Definition 2.1 ([Iv78]) We say that p(x, ξ) admits an elementary decomposition near ρ if there exist λ, µ, Q symbols in (x, ξ
) defined in a conic neighborhood of ρ depending smoothly on x0 , homogeneous of degree 1, 1, 2 respectively, Q(x, ξ
) ≥ 0 such that p(x, ξ) = −Λ(x, ξ)M (x, ξ) + Q(x, ξ
) Λ(x, ξ) = ξ 0 − λ(x, ξ
), M (x, ξ) = ξ 0 − µ(x, ξ
) |{Λ(x, ξ), Q(x, ξ
)}| ≤ CQ(x, ξ
) |{Λ(x, ξ), M (x, ξ)}| ≤ C with some constant C.

Q(x, ξ)

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Lemma 2.2 ([Iv79]) Assume that p admits an elementary decomposition. Then there is no bicharacteristic which has a limit point in Σ. Proof. Note that Σ = {(x, ξ) | Λ(x, ξ) = M (x, ξ) = Q(x, ξ
) = 0} because ∂ξ0 p = Λ(x, ξ) + M (x, ξ) = 0 and p(x, ξ) = 0 implies Λ(x, ξ)2 + Q(x, ξ
) = 0. Let γ(s) be a bicharacteristic of p such that γ(0) ∈ Σ and lims→∞ γ(s) = ρ ˆ ∈ Σ. Since p(γ(s)) = 0 and γ(s) ∈ Σ, one has M (γ(s)) + Λ(γ(s)) = 0 and hence we may assume that Λ(γ(s)) + M (γ(s)) > 0. Thus we see M (γ(s)) ≥ 0, Λ(γ(s)) ≥ 0. Since x˙ 0 (s) = −(Λ(γ(s)) + M (γ(s))) < 0 we can take x0 as a parameter: s = s(x0 ) with s(¯ x0 ) = 0, limx0 →ˆx0 s(x0 ) = ∞. We now have ds d d ds Λ(γ(s)) Λ(γ(x0 )) = = {p, Λ}(γ(s)) . dx0 ds dx0 dx0 √ √ √ Since |{p, Λ}| ≤ C(Q + Λ Q) = CΛ(M + ΛM ), (M + ΛM )/(Λ + M ) ≤ 2 one has    d     dx0 Λ(γ(x0 )) ≤ CΛ(γ(x0 )). This proves that e−C(¯x0 −x0 ) Λ(γ(¯ x0 )) ≤ Λ(γ(x0 )) ≤ eC(¯x0 −x0 ) Λ(γ(¯ x0 )).

(2.4)

If Λ(γ(¯ x0 )) = 0, then from (2.4) we have Λ(γ(ˆ x0 )) = 0 and hence ρ ˆ ∈ Σ. If Λ(γ(¯ x0 )) = 0, then M (γ(¯ x0 )) > 0 because γ(¯ x0 ) ∈ Σ. It follows from (2.4) that Λ(γ(x0 )) = 0 and Q(γ(x0 )) = 0. This shows that    d     dx0 M (γ(x0 )) ≤ C[M (γ(x0 )) Q(γ(x0 )) + Q(γ(x0 ))] = 0 x0 )) > 0 which proves that ρ ˆ ∈ Σ. and hence we conclude M (γ(ˆ x0 )) = M (γ(¯ 2 2.3 A priori estimates Assume that p(x, ξ) admits an elementary decomposition p = −ΛM + Q and consider P = −(D0 − µ(x, D
))(D0 − λ(x, D
)) + Q(x, D
) where λ(x, D
), µ(x, D
), Q(x, D
) are Weyl quantization of real λ(x, ξ
) ∈ S( ξ
, g), µ(x, ξ
) ∈ S( ξ
, g) and Q(x, ξ
) ∈ S( ξ
2 , g). Here g = |dx
|2 + ξ
−2 |dξ
|2 . Let us put Pθ (x, D) = P (x, D0 − iθ, D
) so that P (eθx0 u) = eθx0 Pθ u. We also put Mθ = M − iθ, Λθ = Λ − iθ. We denote by (u, v) the inner product in L2 (Rn ). Note that

Non-effectively hyperbolic operators and bicharacteristics

221

2Im (Qu, Λθ u) = 2θRe (Qu, u) +

d Re (Qu, u) + Im ([Λ, Q]u, u) dx0

and −2Im (Mθ Λθ u, Λθ u) = 2θΛθ u2 +

d Λθ u2 dx0

because Q∗ = Q and λ∗ = λ, µ∗ = µ where u denotes the L2 (Rn ) norm of u. This gives Lemma 2.3 We have, with E(u; x0 ) = Λθ u2 + Re (Qu, u), 2Im (Pθ u, Λθ u) =

d E(u; x0 ) + 2θE(u; x0 ) + Im ([Λ, Q]u, u). dx0

From this we have θ−1 Pθ u2 ≥

d E(u; x0 ) dx0 + θ{Λθ u2 + 2Re (Qu, u)} + Im ([Λ, Q]u, u).

(2.5)

On the other hand, from −2Im (Λθ u, u) = 2θu2 +

d u2 dx0

one gets Λθ u2 ≥ θ2 u2 + θ

d u2. dx0

(2.6)

Replacing Λθ u2 /2 in (2.5) by the estimate (2.6) we have   d θ2 θ −1 Pθ u2 ≥ E(u; x0 ) + u2 dx0 2 + θ{Λθ u2 /2 + θ2 u2/2 + 2Re (Qu, u)} + Im ([Λ, Q]u, u). Here we note that Im ([Λ, Q]u, u) ≥ −Re ({Λ, Q}w (x, D
)u, u) − Cu2 because a#b − b#a = −i{a, b} + S(1, g) if a ∈ S( ξ
, g) and b ∈ S( ξ
2 , g). Then from the Fefferman–Phong inequality it follows that 2θRe (Qu, u) + Im ([Λ, Q]u, u) ≥ −Cu2 for θ ≥ θ0 because 2θQ−{Λ, Q} ≥ 0 by the assumption. The Fefferman–Phong inequality again shows 2θRe (Qu, u) + θ2 u2 /2 ≥ cθ2 u2

(2.7)

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Tatsuo Nishitani

with some c > 0 for θ ≥ θ1 . Thus we obtain   d θ2 θ−1 Pθ u2 ≥ E(u; x0 ) + u2 ≥ cθ(Λθ u2 + θ2 u2 ). dx0 2 Integrating this inequality and remarking (2.7) we get Proposition 2.1 Assume that p(x, ξ) admits an elementary decomposition. Then we have  t  t 2 −1 2 2 θ (Λθ u(s, ·) + θ u(s, ·) )ds ≤ Cθ Pθ u(s, ·)2 ds. −∞

−∞

Here we have used Lemma 2.4 Let u ∈ H(1,) (Rn+1 ). Then we have limt→±∞ u(t, ·)2 = 0 for any  ∈ R.

3 Conditions for elementary decomposition Recall that we are working with p(x, ξ) = −ξ 20 + q(x, ξ
),

q(x, ξ
) ≥ 0.

We introduce the following hypotheses: the doubly characteristic set Σ = {(x, ξ) | p(x, ξ) = dp(x, ξ) = 0} is a smooth manifold near ρˆ such that dimTρ Σ = dimKerFp (ρ),

ρ ∈ Σ,

(3.1)

that is, the codimension of Σ is equal to the rank of the Hessian of p at every point on Σ and rank σ|Σ = constant. (3.2) 3.1 Case ImFp2 ∩ KerFp2 = {0} In this subsection we study the case KerFp2 (ρ) ∩ ImFp2 (ρ) = {0}, ∀ρ ∈ Σ. From Lemma 2.1 one can write q as p
i=1

qi (x, ξ
)(xi−1 − xi )2 +

p
i=1

ri (x, ξ
)ξ 2i + g(x(p) , ξ (p+1) )rp+1 (x, ξ
)

(3.3)

Non-effectively hyperbolic operators and bicharacteristics

223

with {ξ p , {ξ p , g}}(0, en) = 0,

p


ri (0, en )−1 < 1, 0 ≤ p ≤ n − 1.

i=1

From the Morse lemma there are ni (x(p) , ξ (p+1) ) such that g(x(p) , ξ (p+1) ) =

h


ni (x(p) , ξ (p+1) )2

i=1 (p+1)

where dni (0, en

) are linearly independent. Note that one has ∂ ni (0, e(p+1) ) = 0, i = 1, . . . , h n ∂xp

(3.4)

and Σ = {ξ i = 0, 0 ≤ i ≤ p, xi−1 − xi = 0, 1 ≤ i ≤ p, ni = 0, 1 ≤ i ≤ h}. ⎧ ⎪ ⎨ φi = ξ i , 0 ≤ i ≤ p, φp+i = xi−1 − xi , 1 ≤ i ≤ p, ⎪ ⎩ φ2p+i = ni , 1 ≤ i ≤ h.

Let us set

We first recall Lemma 3.1 The condition (3.2) is equivalent to rank ({φi , φj })(ρ) = constant,

ρ ∈ Σ.

Proof. Note that the σ orthogonal space of Tρ Σ is given as (Tρ Σ)σ = Hφ0 (ρ), . . . , Hφr (ρ) ,

r = 2p + h + 1

and σ(Hφi (ρ), Hφj (ρ)) = {φi , φj }(ρ). From this it is enough to show that rank σ|(Tρ Σ)σ = constant. Let us consider the map L : Tρ Σ  v →

s


σ(v, fj (ρ))fj (ρ) ∈ Tρ Σ

j=1

where Tρ Σ = f1 (ρ), . . . , fs (ρ) . The assumption implies that the rank of the matrix (σ(fi (ρ), fj (ρ))) is constant and hence   dim KerL = dim Tρ Σ ∩ (Tρ Σ)σ = constant.

224

Tatsuo Nishitani

This proves the desired assertion because the kernel of ˜ : (Tρ Σ)σ  v → L

r


σ(v, Hφj (ρ))Hφj (ρ) ∈ (Tρ Σ)σ

j=0

2

is just KerL. Lemma 3.2 We have ⎛

0

⎜ ⎜ ⎜ −∂xp n1 rank ⎜ ⎜ .. ⎜ . ⎝ −∂xp nh

···

∂xp n1 ..

∂xp nh

. {ni , nj }

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

= rank(m0 , m1 , . . . , mh ) = const on Σ where mi denotes the (i + 1)th column of the matrix. (p+1)

) that rank({ni , nj }(ˆ ρ)) = 2k. Since Let us assume with ρ ˆ = (0, en ∂xp ni (ˆ ρ) = 0, then from Lemma 3.2 it follows that ⎞ ⎛ 0 ∂xp n1 · · · ∂xp nh ⎟ ⎜ . ⎟ ⎜ ⎟ ⎜ −∂xp n1 . . ⎟ (ρ) = 2k, ρ ∈ Σ. ⎜ rank ⎜ ⎟ .. ⎟ ⎜ . {ni , nj } ⎠ ⎝ −∂xp nh Then one can choose J ⊂ {1, 2, . . . , h} so that |J| = 2k and 2k columns mi (ρ), i ∈ J are linearly independent for ρ ∈ Σ. Then there are smooth αi (ρ), ρ ∈ Σ, i ∈ J such that ⎞ ⎛ 0 ⎟ ⎜ ⎜ ∂xp n1 ⎟
⎟ ⎜ αi (ρ)mi (ρ). (3.5) ⎜ . ⎟= ⎜ .. ⎟ ⎠ i∈J ⎝ ∂xp nh Note that αi (ˆ ρ) = 0 because ∂xp ni (ˆ ρ) = 0. Introduce Λ = ξ0 + ξ1 + · · · + ξp +




αj (ρ)nj (ρ) = ξ 0 + λ

j∈J

and write p = −(ξ 0 + λ)(ξ 0 − λ) + q − λ2 = −ΛM + Q.

Non-effectively hyperbolic operators and bicharacteristics

225

Lemma 3.3 We have p


ξ 2i +

i=1

p


qi (xi−1 − xi )2 +

i=1

h


n2i ≤ CQ,

i=1

|{Λ, Q}| ≤ CQ,

|Λ − M | ≤ C

Q

with some C for ρ ∈ Σ near ˆ ρ. Proof. We check the first assertion. Recall that Q = q − λ2 . Note that  2

λ =

p


2 ξi

+2

i=1

p


ξi




α i ni +




2 αi ni

i=1

2


2


≤ (1 + !) ξ i + (1 + !−1 ) α i ni ≤ (1 + !)




ri (ρ)−1




ri (ρ)ξ 2i + (1 + !−1 )




2 α i ni

.

 From the assumption we have ri (ˆ ρ)−1 < 1 and αi (ˆ ρ) = 0. Then taking ! > 0 small it is clear that we have λ2 ≤ cq with some c < 1 in a small neighborhood of ρˆ and hence the first assertion. The third assertion is clear. Note that (3.5) implies that  i∈J αi ∂xp ni = 0, (3.6)  ∂xp nj + i∈J αi {ni , nj } = 0, 1 ≤ j ≤ h on Σ. Consider {Λ, Q} = {Λ, q − λ2 }. Note that {Λ, λ2 } = {ξ 0 + λ, λ2 } = {ξ 0 , λ2 } = 0 because αi (x, ξ) = αi (x(p) , ξ (p+1) ). Note

p p


2 2 2 ni ri ξ i + qi (xi−1 − xi ) + rp+1 {Λ, Q} = ξ 0 + λ, i=1

≡ ≡

ξ 0 + λ, @


≡ −2rp

p


i=1

ri ξ 2i

+ rp+1




n2i

i=1

A @

A α i ni , n2i αi ni , rp ξ 2p + rp+1 ξ p +


αi ∂xp ni ξ p + 2rp+1





∂xp ni + αk {nk , ni } ni

on Σ modulo terms vanishing on Σ of order 2. Thanks to (3.6) the right-hand side is estimated by a constant times Q. 2

226

Tatsuo Nishitani

3.2 Case ImFp2 ∩ KerFp2 = {0} Here we study the case ImFp2 ∩ KerFp2 = {0}.

(3.7)

From the hypothesis (3.1) one can write p(x, ξ) = −ξ 20 +

r


φj (x, ξ
)2

j=1

where dφj are linearly independent at ρ and Σ is given by Σ = {φj (x, ξ) = 0, j = 0, . . . , r}. Assume (3.7), then by Theorem 21.5.3 in [H85] the Hessian Q of p at ρ takes the form, in suitable symplectic coordinates: k k+

√ µj (x2j + ξ 2j ) + ξ 2j . Q = (−ξ 20 + 2ξ 0 ξ 1 + x21 )/ 2 + j=2

(3.8)

j=k+1

Since dim(KerFp (ρ) ∩ ImFp3 (ρ)) = 1, which is easily verified examining (3.8), then one can choose a smooth z1 (ρ) so that z1 (ρ) = KerFp (ρ) ∩ ImFp3 (ρ), ρ ∈ Σ. Note Lemma 3.4 There is a smooth z2 (ρ) such that KerFp2 (ρ) ∩ ImFp2 (ρ) = z1 (ρ), z2 (ρ) , Fp z2 (ρ) = 0, ρ ∈ Σ. Proof. Since dim(KerFp2 (ρ) ∩ Im Fp2 (ρ)) = 2, ρ ∈ Σ there are smooth h1 (ρ), h2 (ρ) such that KerFp2 (ρ) ∩ Im Fp2 (ρ) = h1 (ρ), h2 (ρ) . Noting that z1 (ρ) ∈ KerFp (ρ) ∩ Im Fp3 (ρ) ⊂ KerFp2 (ρ) ∩ Im Fp2 (ρ), there exist smooth α(ρ), β(ρ) such that z1 (ρ) = α(ρ)h1 (ρ)+ β(ρ)h2 (ρ). It suffices to take, for instance, z2 (ρ) = β(ρ)h1 (ρ) − α(ρ)h2 (ρ). Since Fp (ρ)h1 (ρ), Fp (ρ)h2 (ρ) = KerFp (ρ) ∩ Im Fp3 (ρ) then we conclude Fp z2 (ρ) = 0. 2 Lemma 3.5 There exists a smooth S(ρ) vanishing on Σ such that HS (ρ) = z2 (ρ),

ρ ∈ Σ.

Non-effectively hyperbolic operators and bicharacteristics

227

Proof. Recall that Im Fp (ρ) = Hφ0 (ρ), Hφ1 (ρ), . . . , Hφr (ρ) with φ0 = ξ 0 . Since dim Im Fp2 (ρ) = constant which follows easily by examining (3.8), one can choose smooth g1 (ρ),. . . ,gs (ρ) such that g1 (ρ), . . . , gs (ρ) = Im Fp2 (ρ) ⊂ Im Fp (ρ).  From this one can write gi = rk=0αik (ρ)Hφk (ρ) with smooth αik (ρ). Since r z2 (ρ) ∈ Im Fp2 (ρ) we have z2 (ρ) = k=0 αk (ρ)Hφk (ρ) with smooth αk (ρ). It is enough to take r
S(ρ) = α ˜ k (ρ)φk (ρ) k=0

where α ˜ k (ρ) is a smooth extension of αk outside Σ.

2

Let S(x, ξ) be a smooth function vanishing on Σ such that HS (ρ) ∈ KerFp2 (ρ) ∩ Im Fp2 (ρ), ρ ∈ Σ.

(3.9)

We are concerned with the following condition: Condition ([N86], [BBP93]) For any smooth S(ρ) vanishing on Σ verifying (3.9) we have HS3 p(ρ) = 0, ρ ∈ Σ. (3.10) The next result is proved in a little bit restrictive form in [N86] and in full generality in [BBP93]. Theorem 3.1 ([N86], [BBP93]) Let S be a smooth function verifying (3.9) and Fp HS = 0. If HS3 p(ρ) = 0, ρ ∈ Σ, then p admits an elementary decomposition. Here we give a naive proof of the above result for a model case which will be useful for studies on the Gevrey well-posedness of the Cauchy problem without the condition (3.10). We start with p=

−ξ 20

+

p


qi (x, ξ
)(xi−1 − xi )2

i=1

+

p


ri (x, ξ
)ξ 2i + rp+1 (x, ξ
)

i=1

h


ni (x(p) , ξ (p+1) )2

i=1

where we assume that (3.4) is verified at every point in Σ; ∂ ni (ρ) = 0, ∂xp

ρ ∈ Σ.

Making a linear change of coordinates: y0 = x0 , yi = xi−1 − xi , i = 1, . . . , p, yi = xi , i = p + 1, . . . , n

(3.11)

228

Tatsuo Nishitani

one can write p in the form p(x, ξ) = −(ξ 0 + ξ 1 )2 +

p


qj (x, ξ
)x2j

j=1

+

p−1


rj (x, ξ
)(ξ j − ξ j+1 )2 + rp (x, ξ
)ξ 2p

j=1


+ rp+1 (x, ξ )

h


 n2j

x0 −

p


 (p+1)

xs , x

,ξ

(p+1)

.

s=1

j=1

We now explicitly write down Im Fp3 (ρ) ∩ KerFp (ρ) and Im Fp2 (ρ) ∩ KerFp2 (ρ) for ρ ∈ Σ. Lemma 3.6 We have Im Fp3 (ρ) ∩ KerFp (ρ) = Hξ 0 . We turn to Im F 2 ∩ KerF 2 . We now write p−1


rj (x, ξ
)(ξ j − ξ j+1 )2 + rp (x, ξ
)ξ 2p − ξ 21 = A(x, ξ
)ξ (p) , ξ (p)

j=1

with ξ (p) = (ξ 1 , . . . , ξ p ). Lemma 3.7 Let 0 = v ∈ Hx1 , . . . , Hxp be such that v ∈ KerA(ρ). Then Im Fp2 (ρ) ∩ KerFp2 (ρ) = Hξ0 , v

and Fp (ρ)v is proportional to Hξ0 . We take a more precise look at KerA(ρ). Let us set  rj−1 (x, ξ
) ps=j−1 rs (x, ξ
)−1
p aj (x, ξ ) =
−1 s=j rs (x, ξ ) p

and


cj (x, ξ ) = p

s=j

rs (x, ξ
)−1

s=j−1 rs (x, ξ


−1 , )

2≤j≤p

so that aj (x, ξ
) = rj−1 (x, ξ
)/cj (x, ξ
). We now have Lemma 3.8 We have A(x, ξ
)ξ (p) , ξ (p) =

p
j=2

aj mj (x, ξ
)2 + R(x, ξ
)ξ 21

Non-effectively hyperbolic operators and bicharacteristics

229

where mj (x, ξ
) = ξ j − cj (x, ξ
)ξ j−1 and r1 (x, ξ
)2 =0 a2 (x, ξ
)

R(x, ξ
) = r1 (x, ξ
) − 1 −

on Σ.

In particular KerA(ρ) = (1, c2 (ρ), (c2 c3 )(ρ), . . . , (c2 · · · cp )(ρ))

is given by {(x, ξ) | dmj (ρ)(x, ξ) = 0, j = 1, . . . , p} for ρ ∈ Σ. As observed above we can write p(x, ξ) = −ξ 20 − 2ξ 0 ξ 1 +

p


qj (x, ξ
)x2j +

p


j=1

aj (x, ξ
)mj (x, ξ
)2

j=2

+ R(x, ξ
)ξ 21 + rp+1 (x, ξ
)

h


 n2j

x0 −

p


 xs , x(p+1) , ξ (p+1)

s=1

j=1


where m1 (x, ξ ) = ξ 1 and R = 0 on Σ hence   p p p h



(p+1) (p+1) β j mj + 2 γ j xj + 2 δ j nj x0 − xs , x ,ξ . R=2 j=1

j=1

s=1

j=1

Let us consider S(x, ξ) vanishing on Σ verifying (3.9) and Fp HS = 0. Recall that Σ = {ξ j = 0, 0 ≤ j ≤ p, xj = 0, 1 ≤ j ≤ p, nj = 0, 1 ≤ j ≤ h} and hence one can write S=

p
j=1

aj xj +

p


bj ξ j +

j=0

h


dj nj

j=1

p where nj = nj (x0 − s=1 xs , x(p+1) , ξ (p+1) ). From Lemma 3.7 and (3.9) it is clear that bj (ρ) = 0, 1 ≤ j ≤ p, dj (ρ) = 0, 1 ≤ j ≤ h and (a1 (ρ), . . . , ap (ρ)) is proportional to (1, c1 (ρ), . . . , (c2 · · · cp )(ρ)) for ρ ∈ Σ by Lemma 3.8. Then we have ∂dj ∂bj (ˆ ρ) = 0, 1 ≤ j ≤ p, (ˆ ρ) = 0, 1 ≤ j ≤ h ∂x0 ∂x0 by (3.11). Hence one can write HS = L0 + L1 , L0 =

p


a j Hx j + b 0 Hξ 0

j=1

where L1 is a vector field withcoefficients vanishing at ρ ˆ. Moreover the coefp ficient of ∂/∂ξ 0 has the form j=1 ej xj + e0 where e0 vanishes of order 2 at ρ ˆ.

230

Tatsuo Nishitani

Lemma 3.9 We have HS3 p(ˆ ρ) = −12β 1 (ˆ ρ)a1 (ˆ ρ)3 . Proof. Note that L0 xj , 1 ≤ j ≤ p, L0 mj , 2 ≤ j ≤ p, L0 nj , 1 ≤ j ≤ h vanish at ρ ˆ. Let us denote by Fj functions vanishing at ρˆ of order j and by Gj a linear combination of xj , 1 ≤ j ≤ p, mj , 2 ≤ j and nj , 1 ≤ j ≤ h with coefficients 3 2 vanishing at ρ ˆ of order j − 1. We first  remark that HS (ξ 0 + 2ξ 0 ξ 1 )|ρ=ˆρ = 0 2 because HS (ξ 0 + 2ξ 0 ξ 1 ) = aξ 0 − 2( ej xj + e0 )ξ 1 with some a. It is easy to see that  p  p h


HS qj x2j + aj m2j + rp+1 n2j = G2 1

2

and HS2 G2 = F2 +G1 . On the other so that HS2 (Rξ 21 ) = 12β 1 a21 ξ 1 + F2 .

1

hand we see that HS (Rξ 21 ) = F3 −6a1 β 1 ξ 21 Thus we have HS3 p(ˆ ρ) = −12β 1 (ˆ ρ)a1 (ˆ ρ)3 . 2

Recall that p−1


rj (ξ j − ξ j+1 )2 + rp ξ 2p − ξ 21 =

j=1

p


aj mj (x, ξ
)2 + R(x, ξ
)ξ 21

j=2

where mj = ξ j − cj ξ j−1 . Since β 1 = 0 on Σ we can write R = a
1 ξ 21 + 2

p


a
j mj +

j=2

p


γ
j xj +

j=1

h


δ
j nj .

j=1

Replacing mj by mj + (a
j /aj )ξ 21 we see that p = −ξ 0 (ξ 0 + 2ξ 1 ) +

p


qj x2j

j=1

+

p


aj m2j + rp+1

j=2

where R =

p 1

γ
j xj +

h


nj (x, ξ (p+1) )2 + Rξ 21 + cξ 41

j=1

h 1

δ
j nj . Let us set Φ=

p


κj xj ξ 1

j=2

where κj will be determined later. We still rewrite as −ξ 0 (ξ 0 + 2ξ 1 ) = −(ξ 0 + Φ + Kξ 31 )(ξ 0 + 2ξ 1 − Φ − Kξ 31 ) + 2(Φ + Kξ 31 )ξ 1 − (Φ + Kξ 31 )2

Non-effectively hyperbolic operators and bicharacteristics

231

and hence p = −ΛM + Q with Q=

p


p


qj x2j +

j=1

+

Rξ 21

aj m2j + rp+1

j=2

h


n2j + 2Kξ 41

j=1

+ 2Φξ 1 − (Φ +

Kξ 31 )2

+ cξ 41

where Λ = ξ 0 + Φ + Kξ 31 , M = ξ 0 + 2ξ 1 − Φ − Kξ 31 . Let us denote g 2 = p 2 p 2 h 2 4 2 mj + 1 xj + 1 nj + ξ 1 . Note that {ξ j , n } = O(g), 0 ≤ j ≤ p, {xj , n } = 0, 0 ≤ j ≤ p, {Φ, n } = O(g), {Φ, ξ 1 } = O(g)|ξ 1 |. Noting that ξ k ≡ fk
ξ 1 mod O(g) with some fk
we see that {ξ 0 , Q} ≡

p


f j mj ξ 1

j=2

mod O(g 2 ). A similar argument shows that {Φ, Q} ≡ 2ξ 1

p


κj (cj+1 mj+1 − mj ), mp+1 = 0

j=2

mod O(g 2 ). We now choose κj so that {ξ 0 + Φ, Q} ≡ 0 mod O(g 2 ). Thus we conclude that {Λ, Q} ≡ 0 mod O(g 2 ). On the other hand it is easy to check that, taking K large and choosing a neighborhood of ρˆ small, we have CQ ≥

p
j=1

x2j +

p
j=2

m2j +

h


n2j + ξ 41 .

j=1

Thus we have√|{Λ, Q}| ≤ CQ with some C > 0. It is also easy to check that |{Λ, M }| ≤ C Q.

4 Behavior of bicharacteristics and elementary decomposition We first recall Theorem 4.1 ([N83]) Assume that Σ is a smooth manifold of codimension 3 and (3.7). Then p admits an elementary decomposition if and only if there is no bicharacteristic of p issuing from a simple characteristic having a limit point in Σ.

232

Tatsuo Nishitani

This result can be extended in full generality; Theorem 4.2 ([N04]) Assume (3.1), (3.2) and (3.7). Then the following assertions are equivalent: (i) p admits an elementary decomposition, (ii) there is no bicharacteristic of p issuing from a simple characteristic having a limit point in Σ. Thanks to Lemma 2.2 it suffices to prove Theorem 4.3 ([N04]) There exists a null bicharacteristic having a limit point in the doubly characteristic set if the assumption in Theorem 3.1 fails. Since the proof of Theorem 4.1, based on an argument available for a 2dimensional autonomous system, works only for the case of codimension 3, then to prove Theorem 4.3 we argue in a completely different way. 4.1 Hamilton system Proposition 4.1 ([N84]) Assume (3.7). For any small conic neighborhood V of (0, en ) there exist ρ ˆ ∈ V , 1 ≤ p ≤ n − 1 and a symplectic local chart {U, (x, ξ)} around ρ ˆ, such that p(x, ξ) =

ξ 20

−

p


qi (x, ξ (1) )(xi−1 − xi )2

i=1

−

p


ri (x, ξ
)ξ 2i − rp+1 (x, ξ
)

i=1

where

and

h


ni (x(p) , ξ (p+1) )2

(4.1)

i=1

∂ ni (x(p) , ξ (p+1) ) = 0, ∂xp p


ri (x, ξ
)−1 = 1

on

on Σ ∩ U

Σ ∩ U.

i=1

We study the Hamilton system with the Hamiltonian p in (4.1). Let us assume (p+1 (p+1) nj (ˆ x(p+1) , ˆξ ) = 0, 1 ≤ j ≤ h so that (x0 , 0, . . . , 0, x ˆ(p+1) , 0, . . . , 0, ˆξ )∈ Σ. In what follows, since the homogeneity in ξ is irrelevant in the study of (p+1) bicharacteristics, replacing (x(p+1) , ξ (p+1) ) by (ˆ x(p+1) + x(p+1) , ˆξ + ξ (p+1) ) we are led to study the Hamilton system with Hamiltonian p where nj (0, 0) = 0, 1 ≤ j ≤ h.

Non-effectively hyperbolic operators and bicharacteristics

233

Lemma 4.1 We may assume that nα (t, x(p+1) , ξ (p+1) ) = xp+α + uα + O(n2 ), 1 ≤ α ≤ k, nk+α (t, x(p+1) , ξ (p+1) ) = ξ p+α + uk+α + O(n2 ), 1 ≤ α ≤ k +  where 2k +  = h, n2 = |x(p+1) |2 + |ξ (p+1) |2 and uβ , 1 ≤ β ≤ h has the form uβ =

k


cβj (t)xp+j +

k+


j=1

dβj (t)ξ p+j

j=1

with cβj (0) = dβj (0) = 0. Proof. From ∂nα (t, x(p+1) , ξ (p+1) )/∂t = 0 on Σ it follows that nα (t, 0, 0) = 0 for any small t. Hence one can write nα (t, x(p+1) , ξ (p+1) ) n


=

cαj (t)xj + dαj (t)ξ j + O(n2 )

j=p+1 n


=

n


cαj (0)xj + dαj (0)ξ j +

j=p+1

c˜αj (t)xj + d˜αj (t)ξ j + O(n2 ).

j=p+1

Making a linear symplectic change of coordinates (x(p+1) , ξ (p+1) ) we may assume that n


cαj (0)xj + dαj (0)ξ j = xp+α + O(n2 ),

1 ≤ α ≤ k,

j=p+1 n


ck+α,j (0)xj + dk+α,j (0)ξ j = ξ p+α + O(n2 ),

1 ≤ α ≤ k + .

j=p+1

Thus we have nα (t, x(p+1) , ξ (p+1) ) = xp+α +

n


c˜αj (t)xj + d˜αj (t)ξ j + O(n2 ),

j=p+1

nk+α (t, x(p+1) , ξ (p+1) ) = ξ p+α +

n


c˜k+α,j (t)xj + d˜k+α,j (t)ξ j + O(n2 ).

j=p+1

Here we recall again that ∂nα (t, x(p+1) , ξ (p+1) )/∂t = 0 on Σ and then one can write
∂nα = eα (t, x(p+1) , ξ (p+1) )nα (0, x(p+1) , ξ (p+1) ). ∂t

234

Tatsuo Nishitani

This proves that ∂˜ cαj (t) ∂ d˜αj (t) = 0, p + k + 1 ≤ j, = 0, p + k +  ≤ j. ∂t ∂t Since c˜αj (0) = 0, d˜αj (0) = 0 we have the desired assertion.

2

We now have Lemma 4.2 One can write p = −ξ 20 − 2ξ 0 ξ 1 +

p


qj 2j +

p


j=1

rj m2j +

j=2

h


bj n2j − β ∗ ξ 31 + Φ(x, ξ
)

j=1

where qj , rj , bj , β ∗ ∈ R and ⎧
2 ⎪ ⎨ mj = ξ j − cj ξ j−1 − gj (x, ξ ), j = xj − dj ξ 1 nj = xp+j + uj − ej ξ 21 , 1 ≤ j ≤ k ⎪ ⎩ nk+j = ξ p+j + uk+j − ek+j ξ 21 , 1 ≤ j ≤ k +  with cj , dj , ej ∈ R. Here gj (x, ξ
) = O(ρ2 ), gj (x, 0) = 0 with ρ = |(x, ξ
)| and uα =

k


 cαj

x0 −

p


 xs

xp+j +

s=1

j=1

k+


 dαj

x0 −

p


 xs

ξ p+j .

s=1

j=1

Moreover Φ(x, ξ
) =

p


αj0 (x, ξ
)m2j + αj1 (x, ξ
)mj +

j=2

+ β j1 (x, ξ
)j +

p


β j0 (x, ξ
)2j

j=1 h


γ j0 (x, ξ
)n2j + γ j1 (x, ξ
)nj + ∆(x, ξ
)

j=1

where αj0 = O(ρ), β j0 = O(ρ), γ j0 = O(ρ) αj1 = O(ρ3 )O(|ξ|), β j1 = O(ρ)O(|ξ|2 ), γ j1 = O(n2 ) + O(ρ)O(|ξ|2 ) ∆ = O(ρ4 )O(|ξ|2 ) + O(ρ)O(ξ 31 ) + O(ρ)O(n2 )O(|ξ|2 ) +O(n4 ) + O(ρ2 )O(|ξ|4 ) with m2 =

p j=2

mj (x, ξ
)2 , 2 =

p

j=1 j (x, ξ


2

) .

Non-effectively hyperbolic operators and bicharacteristics

Our Hamilton system is: ⎧ x˙ 0 = −2ξ 0 − 2ξ 1 ⎪ ⎪ ⎪ p ⎪ ⎪ x˙ 1 = −2ξ 0 − 4 k=1 qk dk ξ 1 k − 2c2 r2 m2 − 3β ∗ ξ 21 ⎪ ⎪ ⎪ ⎪ p h ⎪ ⎪ ∂Φ k ⎪ −2 k=2 rk mk ∂g ⎪ k=1 bk ek ξ 1 nk + ∂ξ 1 ∂ξ 1 − 4 ⎪ ⎪ ⎪ ⎪ ⎪ x˙ j = 2rj mj − 2rj+1 cj+1 mj+1 ⎪ ⎪ ⎪ ⎪ p ⎪ ∂Φ k ⎪ −2 k=2 rk mk ∂g 2≤j≤p ⎪ ∂ξ j + ∂ξ j , ⎪ ⎪ ⎪ ⎪˙ p h ⎪ ∂gk ∂Φ α ⎪ ξ 0 = 2 k=2 rk mk ∂x − 2 α=1 nα ∂u ⎪ ∂x0 − ∂x0 ⎪ 0 ⎪ ⎪ p ⎪ ∂gk ⎪ ⎪ = −2qj xj + 2qj dj ξ 21 + 2 k=2 rk mk ∂x ξ˙ ⎪ j ⎨ j h ∂uα ∂Φ +2 n − , 1 ≤ j ≤ p α=1 α ∂xj ∂xj ⎪ ⎪ ⎪ p ⎪ ⎪ ⎪ x˙ p+j = 2bk+j (ξ p+j + uk+j − ek+j ξ 21 ) − 2 i=2 ri mi ∂ξ∂gi ⎪ ⎪ p+j ⎪ ⎪ h ⎪ ⎪ ∂uα ∂Φ ⎪ +2 α=1 nα ∂ξ + ∂ξ , 1 ≤ j ≤ k +  ⎪ ⎪ p+j p+j ⎪ ⎪  ⎪ 2 i ⎪ ˙ ⎪ ξ p+j = −2bj (xp+j + uj − ej ξ 1 ) + 2 pi=2 ri mi ∂x∂gp+j ⎪ ⎪ ⎪ ⎪  ⎪ ∂uα ⎪ ⎪ −2 hα=1 nα ∂x − ∂x∂Φ , 1≤j≤k ⎪ p+j p+j ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ x˙ p+j = −2 pi=2 ri mi ∂ξ∂gi + ∂ξ∂Φ , k +  + 1 ≤ j ⎪ ⎪ p+j p+j ⎪ ⎪ ⎪ p ⎩ ξ˙ ∂gi ∂Φ k + 1 ≤ j. p+j = 2 i=2 ri mi ∂xp+j − ∂xp+j ,

235

(4.2)

Suppose that mj are also unknowns and (x(s), ξ(s), m(s)) verifies (4.2). From (4.2) one can write 2rj mj − 2rj+1 cj+1 mj+1 = x˙ j + rj (x, ξ, m),

2≤j≤p

where we have set mp+1 = 0 and rj = 2

p


rk mk

k=2

∂gk ∂Φ − . ∂ξ j ∂ξ j

Similarly from (4.2) we can write xj = − with

1 ˙ ξ + sj (x, ξ), 2qj j

2≤j≤p

  p
∂gk 1 ∂Φ 2 sj = rk mk − qj dj ξ 1 + . qj ∂xj ∂xj k=2

(4.3)

236

Tatsuo Nishitani

Let us set

d gj (x, ξ
) = hj (x, ξ
) ds

j and put kj (x, ξ
) = =2 a h . Taking mj = ξ j − cj ξ j−1 − gj into account, we introduce the following equations for mj : 2rj mj (s) − 2rj+1 cj+1 mj+1 (s) j 1
1 d a m ¨  (s) − a1 ¨ ξ 1 (s) + rj + sj (x(s), ξ(s), m(s)) 2qj 2qj ds =2   d 1 − kj (x(s), ξ(s)), 2 ≤ j ≤ p. 2qj ds

=−

(4.4)

If (x, ξ, m) verifies (4.2) and (4.4) with (x, ξ) = O(s−1 ) mj = O(s−1 ),

(4.5)

then (x(s), ξ(s)) is a solution to the Hamilton system (4.2). Now our question is reduced to looking for (x, ξ, m) verifying (4.2) and (4.4) with (4.5). 4.2 Reduction of Hamilton system We further simplify the equations (4.2) and (4.4). We make the change of independent variable s: 1 s= t and put ⎧ x0 (s) = tX0 (t), ξ 0 (s) = t4 Ξ0 (t), m(s) = t4 M (t) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ xj (s) = t3 Xj (t), ξ j (s) = t2 Ξj (t), 1 ≤ j ≤ p ⎪ ⎪ ⎨ xj (s) = t3 Xj (t), ξ j (s) = t3 Ξj (t), p + 1 ≤ j ≤ p + k ⎪ ⎪ ⎪ ⎪ xj (s) = t3 Xj (t), ξ j (s) = t4 Ξj (t), p + k + 1 ≤ j ≤ p + k +  ⎪ ⎪ ⎪ ⎪ ⎩ xj (s) = t4 Xj (t), ξ j (s) = t4 Ξj (t), p + k +  + 1 ≤ j and denote V = (X, Ξ), V(p) = (X0 , . . . , Xp , Ξ0 , . . . , Ξp ) and for f (x, ξ, m) we put f  (t, V, M ) = f (tX0 , t3 X
, t3 Xp+1 , . . . , t3 Xp+k+ , t4 Xp+k++1 , . . . , t4 Xn , t4 Ξ0 , t2 Ξ
, t3 Ξp+1 , . . . , t3 Ξp+k , t4 Ξp+k+1 , . . . , t4 Ξn , t4 M ) where X
= (X1 , . . . , Xp ), Ξ
= (Ξ1 , . . . , Ξp ).

Non-effectively hyperbolic operators and bicharacteristics

Lemma 4.3 We have ⎧

 ⎪ ∂Φ ⎪ = O(t4 ), ⎪ ∂xj ⎪ ⎪ ⎪ ⎪

 ⎪ ⎪ ∂Φ ⎪ ⎪ = O(t5 ), ⎪ ⎪ ∂ξj ⎪ ⎨  ∂Φ = O(t5 ), ∂ξ j ⎪ ⎪ ⎪

 ⎪ ⎪ ⎪ ∂Φ 6 ⎪ ⎪ ⎪ ∂ξj = O(t ), ⎪ ⎪ ⎪ ⎪ ∂Φ  ⎪ ⎩ = O(t6 ) ∂x0

237



∂Φ 1 ≤ j ≤ p + k, ∂x = O(t6 ), p + k + 1 ≤ j j

 ∂Φ 1 ≤ j ≤ p, ∂ξ = O(t4 ), p + 1 ≤ j ≤ p + k j

p+k+1≤j ≤p+k+ p+k++1≤j

where by O(ts ) we denote a term which is of the form ts R(t, V, M ) with a smooth function R(t, V, M ). Let us set

d . dt Then since tD(t G) = t+1 (DG + G), d/ds = −tD thanks to Lemma 4.3 the equation (4.2) is transformed to ⎧ DX0 = −X0 + 2Ξ1 + t2 φ0 (t, V ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ DX1 = −3X1 + 2Ξ0 + 2r2 c2 M2 + 3β ∗ Ξ12 + tφ1 (t, V, M ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ DXj = −3Xj − 2rj Mj + 2rj+1 cj+1 Mj+1 + tφj (t, V, M ), 2 ≤ j ≤ p ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ DΞ0 = −4Ξ0 + tψ 0 (t, V, M ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ DΞj = −2Ξj + 2qj Xj + tψ j (t, V, M ), 1 ≤ j ≤ p ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ tDXp+j = −3tXp+j − 2bk+j Ξp+j + tφp+j (t, V, M ), 1 ≤ j ≤ k (4.6) ⎪ tDΞp+j = −3tΞp+j + 2bj Xp+j + tψ p+j (t, V, M ), 1 ≤ j ≤ k ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ DXp+j = −3Xp+j − 2bk+j Ξp+j + 2bk+j ek+j Ξ1 ⎪ ⎪ ⎪ k ⎪ ⎪ ⎪ + α=1 cp+j,α X0 Xp+α + dp+j,α X0 Ξp+α ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ +tφp+j (t, V, M ), k + 1 ≤ j ≤ k +  ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ DXp+j = −4Xp+j + tφp+j (t, V, M ), k +  + 1 ≤ j ⎪ ⎪ ⎪ ⎪ ⎩ DΞp+j = −4Ξp+j + tψ p+j (t, V, M ), k + 1 ≤ j. D=t

We turn to (4.4). It is also easy to see that  2 d −4 (t2 Ξ1 ) = 6Ξ1 + 5DΞ1 + D2 Ξ1 = LΞ1 t ds

238

Tatsuo Nishitani

and t−4



d ds

2

(t4 M ) = t−4 (tD)2 (t4 M ) = (tD + 4t)2 M.

Thus the equation (4.4) turns to 2rj Mj − 2rj+1 cj+1 Mj+1 = −

j 1
1 (c · · · cj )(tD + 4t)2 M 2qj c =2

−

1 (c2 · · · cj )LΞ1 + R1 (t, V(p) , tDV(p) ) (4.7) 2qj

+ tR2 (t, V, M, tDV, tDM ),

2 ≤ j ≤ p,

where R1 (t, V(p) , 0) = 0.

(4.8)

We further rewrite (4.7) removing the D2 Ξ1 term in LΞ1 . Using (4.6) we get 2rj Mj − 2rj+1 cj+1 Mj+1 +

2q1 r2 2 (c2 c3 · · · cj )M2 qj

j 1
1 =− (c · · · cj )(tD + 4t)2 M 2qj c

(4.9)

=2

+ R1 (t, V(p) , tDV(p) ) + tR2 (t, V, M, tDV, tDM ). For later use we give here another less precise expression of (4.7). From (4.8) ˜ 1 (t, V, DV ) and hence we can rewrite one can write R1 (t, V(p) , tDV(p) ) = tR (4.7) in the form 2rj Mj − 2rj+1 cj+1 Mj+1 = −

1 (c2 · · · cj )LΞ1 2qj

+ tθj (t, V, M, DV, DM ).

(4.10)

If we have a solution (X, Ξ, M ) of (4.6) and (4.10) which is bounded as t ↓ 0 then (x, ξ, m), defined by (4.2), satisfies (4.2) and (4.4) with (4.5) and hence (x, ξ) is a solution to the original Hamilton system. 4.3 Formal solutions We first look for a formal solution to (4.6) and (4.10). Let us define the class of formal series in t and log 1/t in which we look for formal solutions: Definition 4.1 For k ∈ N we set ⎫ ⎧ ⎬ ⎨
E k = tk ti (log t)j Vij | Vij ∈ CN . ⎭ ⎩ 0≤j≤i

Non-effectively hyperbolic operators and bicharacteristics

239

We further rewrite the equation (4.10). Note that one has Mj = κj [6β ∗ κ−1 Ξ12 + 4κ−1 Ξ0 ] + tfj (t, V, M, DV, DM ),

2 ≤ j ≤ p (4.11)

where fj are affine linear in (M, DM ). Thus we conclude that our problem is reduced to finding a solution (X, Ξ, M ) verifying (4.6) and (4.11). Lemma 4.4 Assume that (X, Ξ, M ) ∈ E0 satisfies (4.6) and (4.11) formally and Ξ1 (0) = 0. Then X(0), Ξ(0) and M (0) are uniquely determined. We now show that there exists a formal solution (X, Ξ, M ) ∈ E0 verifying (4.6) and (4.11). If such a solution exists, then (X(0), Ξ(0), M (0)) is uniquely determined by Lemma 4.4. Taking this fact into account let us put ¯ Ξ, ¯ M ¯ ) = (X(0), Ξ(0), M (0)) (X, ¯ + X, Ξ ¯ + Ξ, M ¯ + M ) for (X, Ξ, M ) in (4.11) to get the and substitute (X equation for (X, Ξ, M ). Theorem 4.4 There exists a formal solution (X, Ξ, M ) ∈ E0 verifying Ξ1 (0) = 0 and (4.6), (4.11).  Let E  = { 1≤i,0≤j≤i Vij ti (log 1/t)j } and start with Lemma 4.5 For any V = (X, Ξ) ∈ E  there is a unique M ∈ E  such that ¯ + X, Ξ ¯ + Ξ, M ¯ + M ) satisfies (4.11) where M has the form (X Mj = 12κj Ξ1 + 4κj κ−1 Ξ0 + tFj + Cj , with a constant vector Fj and
(j) p Cpq t (log 1/t)q , Cj =

2≤j≤p

(j) (j) Cpq = Cpq (Vµν | µ ≤ p − 1).

2≤p,0≤q≤p−1

¯ + X, Ξ ¯ + Ξ, M ¯ + M (X, Ξ)) for (X, Ξ, M ) in (4.11). Here Substitute (X M (X, Ξ) is given by Lemma 4.5. Let ⎧ I t V = (X0 , . . . , Xp , Ξ0 , . . . , Ξp ) ⎪ ⎪ ⎨ V II = t (Xp+1 , . . . , Xp+k , Ξp+1 , . . . , Ξp+k ) ⎪ ⎪ ⎩ III t = (Xp+k+1 , . . . , Xn , Ξp+k+1 , . . . , Ξn ). V Then (4.6) becomes ⎧ I I ⎪ ⎪ DV = AI V + FI t + GI (t, V ) ⎨ 0 = AII V II + FII t + GII (t, V ) ⎪ ⎪ ⎩ DV III = AIII V III + KΞ1 + BIII V II + FIII t + GIII (t, V )

(4.12)

240

Tatsuo Nishitani

where G∗ (t, V ) =




G∗ij ti (log 1/t)j , G∗ij = G∗ij (Vpq | p ≤ i − 1)

2≤i,0≤j≤i

and F∗ , K are constant vectors. Let us write HDV = AV + tF + G(t, V ) where

⎡

EOO

⎤

⎢ ⎥ H = ⎣O O O⎦,

⎡

AI

O

⎢ A = ⎣ O AII

OOE

 BIII

Lemma 4.6 We have spec(AI ) = {−6, −4, −3, −2, −1, 1},

(4.13) O

⎤

⎥ O ⎦. AIII

spec(AII ) ⊂ iR \ {0},

spec(AIII ) = {−3, −4}. Proof of Theorem 4.4. Note that (4.13) implies that H(iVij − (j + 1)Vij+1 ) = AVij + δ i1 δ j0 F + Gij where Gij = 0 for i = 0, 1. Then we have (H − A)V11 = 0 (H − A)V10 = V11 + F. Choose V11 ∈ Ker(H − A) so that F + V11 ∈ Im (H − A). Then we can take V10 = 0 so that (H − A)V10 = F + V11 since Ker(H − A) = {0} by Lemma 4.6. We turn to the case i ≥ 2: (iH − A)Vij = (j + 1)HVij+1 + Gij .

(4.14)

With j = i, (4.14) becomes (iH − A)Vii = Gii (Vpq | p ≤ i − 1). Since iH − A is non-singular for i ≥ 2, by Lemma 4.6 one has Vii = (iH − A)−1 Gii (Vpq | p ≤ i − 1). Recurrently one can solve Vij by Vij = (iH − A)−1 [(j + 1)HVij+1 + Gij (Vpq | p ≤ i − 1)] for j = i − 1, i − 2, . . . , 0. This proves the assertion.

2

Non-effectively hyperbolic operators and bicharacteristics

241

4.4 A coupled model system of ODEs In this subsection we study the next system of ordinary differential equations   2 d t dt − iΛ u = −tK1 u + L1 (t)v + Q1 (t, u, v) + tR1 (t, u, v) + tF1 (4.15) d t dt v = −K2 v + Lu + L2 (t)v + Q2 (t, u, v) + tR2 (t, u, v) + tF2 where Qj (t, u, v) and Rj (t, u, v) are C 1 functions defined in a neighborhood of (0, 0, 0) ∈ R × CN1 × CN2 such that ˜j (|u| + |v|) |Qj (t, u, v)| ≤ Bj (|u|2 + |v|2 ), |Rj (t, u, v)| ≤ B for (t, u, v) ∈ {|t| ≤ T1 } × {|u| ≤ C1 T1 } × {|v| ≤ C1 T1 } and L2 (t) ∈ C 1 ((0, T ]), L1 (t) ∈ C 1 ((0, T ]) are N2 × N2 and N1 × N2 matrix-valued functions respectively which verifies Lj (t)C((0,T ]) ,

tL
j (t)C((0,T ]) ≤ B

and L is a constant N2 × N1 matrix. We assume that Λ is a constant nonsingular real diagonal matrix Λ = diag(λ1 , . . . , λN1 ),

λj ∈ R \ {0}

and Ki are real diagonal matrices: Ki = diag(mi1 , . . . , miNi ), i = 1, 2. We also assume that |Ki | ≤ 2m,

m=

min

i=1,2,j=1,...,Ni

mij .

(4.16)

Our aim in this subsection is to prove: Theorem 4.5 If m > 0 is sufficiently large, then (4.15) has a solution (u, v) such that u(0) = 0, v(0) = 0. Let m > 0. We define  −K1  t  t  −K2 t t i i 1 1 H[f ] = h(s)ds e− t Λ+ s Λ f (s)ds, G[h] = 2 s s s s 0 0 for f ∈ C([0, T ]) with f (t) = O(t) as t ↓ 0 and for h ∈ C([0, T ]) so that   d d t2 − iΛ H[f ] = −K1 tH[f ] + f, t G[h] = −K2 G[h] + h. (4.17) dt dt We start with Lemma 4.7 Let f (t) ∈ C 1 ((0, T ]) be such that f (t) = O(t) and tf
(t) = O(1) as t ↓ 0 and let h ∈ C((0, T ]). Assume m > 0. Then we have H[f ](t) = −(iΛ)−1 f (t) + K1 (iΛ)−1 H[tf ](t) + (iΛ)−1 H[t2 f
](t), |H[f ](t)| ≤ m−1 s−1 f C((0,t]), |G[h](t)| ≤ m−1 hC((0,t]).

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Using (4.17) we rewrite (4.15) as an integral equation: u = H[L1 (t)v + Q1 (t, u, v) + tR1 (t, u, v) + tF1 ] v = G[Lu + L2 (t)v + Q2 (t, u, v) + tR2 (t, u, v) + tF2 ]. Let u0 (t) = 0, v0 (t) = 0 and define un (t), vn (t) successively by un+1 (t) = H[L1 (t)vn + Q1 (t, un , vn ) + tR1 (t, un , vn ) + tF1 ] vn+1 (t) = G[Lun + L2 (t)vn + Q2 (t, un , vn ) + tR2 (t, un , vn ) + tF2 ]. Lemma 4.8 For large m we have |vn − vn−1 | ≤ m−1 A
(un−1 − un−2 C([0,t]) + vn−1 − vn−2 C([0,t]) )
| ≤ 2A
(un−1 − un−2 C([0,t]) + vn−1 − vn−2 C([0,t]) ). t|vn
− vn−1

Proof of Theorem 4.5. We show that un , vn converges to some u, v in C([0, T ]). Set Wn (t) = un − un−1 C([0,t]) + vn − vn−1 C([0,t]). We now take m large so that we have Wn+1 (t) ≤ δ{Wn (t) + Wn−1 (t)},

0≤t≤T n−2 with 0 < δ < 1/2. It is easy to check that Wn (t) ≤ k=1 (2δ)k (W2 +W1 ). This proves that {un }, {vn } converges in C([0, T ]) to some u(t), v(t) ∈ C([0, T ]). 2 4.5 Proof of theorem 4.3 To prove the existence of a bicharacteristic which falls into the doubly characteristic set, we show that we can apply Theorem 4.5 to conclude this. One can express the equation (4.9) as BM = −A(tD + 4t)2 M + R1 (t, V(p) , tDV(p) ) + tR2 (t, V, M, tDV, tDM ) and hence (tD + 4t)2 M = −A−1 BM + R1 (t, V(p) , tDV(p) ) + tR2 (t, V, M, tDV, tDM ). (4.18) In this subsection Rj stands for smooth functions which may differ on each line. Lemma 4.9 Every eigenvalue of A−1 B is positive and A−1 B is diagonalizable. Let us set N = (tD + 4t)M

(4.19)

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243

and denote u = t (N, M ), va = t (V I , tDV I , V III , tDV III ), vb = t (V II , tDV II ) and v = (va , vb ). Then one can rewrite (4.18) and (4.19) as   O −A−1 B u + R1 (t, va ) + tR2 (t, u, v). (tD + 4t)u = I O Lemma 4.10 The matrix



O −A−1 B I

(4.20)



O

is diagonalizable and all its eigenvalues are pure imaginary. By this lemma there is a non-singular T such that ⎡ ⎤ λ1   ⎢ ⎥ O −A−1 B .. ⎥ = iΛ1 T = i⎢ T −1 . ⎣ ⎦ I O λ2(p−1) where λi ∈ R \ {0}. Denoting T −1 u by u, again the equation (4.20) becomes (tD + 4t)u = iΛ1 u + Φ1 (t, va ) + tΦ2 (t, u, v). We turn to the equation (4.6) which can be written as DV I = AI V I + A˜I M + QI (V I ) + tΨI (t, V, M ) DV III = AIII V III + QIII (V I , V II ) + tΨIII (t, V, M )

(4.21)

and tDV II = −3tV II + AII V II + tΨII (t, V, M ) where AJ are constant matrices and QJ are quadratic forms. Since every eigenvalue of AII is non-zero pure imaginary and AII is diagonalizable, there is a non-singular constant matrix S such that S −1 AII S = iΛ2 where Λ2 is a non-singular real diagonal matrix. Denoting S −1 V II by V II and (V II , tDV II ) by vb again we get tDV II = −3tV II + iΛ2 V II + tΨ˜II (t, V, M ).

(4.22)

Applying tD to (4.22) we get tD(tDV II ) = −3t(tDV II ) + iΛ2 (tDV II ) + tΨb (t, u, v).

(4.23)

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Combining (4.22) and (4.23) we get tDvb = −3tvb + iΛ2 vb + tΨb
(t, u, v). We now multiply (4.21) by t and then apply D to get ⎧ ˜ I (t, V I , tDV I ) + Ψ˜I (t, u, v) D(tDV I ) = AI (tDV I ) + A˜I N + Q ⎪ ⎪ ⎨ ˜ III (t, V I , V II , tDV I , tDV II ) D(tDV III ) = AIII (tDV III ) + Q ⎪ ⎪ ⎩ +Ψ˜III (t, u, v).

(4.24)

Combining (4.21) and (4.24) one gets ˜ + Q(t, va ) + tΨa (t, u, v). Dva = Ava + Au We now denote (u, vb ) by u and va by v to get tDu = −tKu + iΛu + Φ1 (t, v) + tΦ2 (t, u, v) Dv = A1 u + A2 v + Q(t, u, v) + tΨ (t, u, v) where Ai are constant matrices and   Λ1 O Λ= , O Λ2

 K=

4I O O 3I

(4.25)

 .

By Theorem 4.4 there exists a non-trivial formal solution

uij ti (log 1/t)j , v = vij ti (log 1/t)j . u= 0≤j≤i

0≤j≤i

This shows that for any m ∈ N there is N = N (m) such that

uN = uij ti (log 1/t)j , vN = vij ti (log 1/t)j 0≤j≤i≤N

0≤j≤i≤N

verifies (4.25) modulo O(tm+1 ). Substituting (uN + tm u, vN + tm v) into (4.25) and dividing the resulting equation by tm one has (tD − iΛ)u = −t(mI + K)u + L1 (t)v + tR1 (t, u, v) + tF1 (4.26) Dv = −mv + Lu + L2 (t)v + tR2 (t, u, v) + tF2 where L is a constant matrix. Since it is clear that (4.16) is verified for large m we can now apply Theorem 4.5 to conclude that there exist u, v verifying (4.26).

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5 Remarks In this issue, Bernardi and Bove [BB05] showed that the Cauchy problem is not C ∞ well posed for a model operator verifying (3.7): P = −D02 + 2x1 D0 Dn + D12 + bx31 Dn2

(5.1)

with b = 0 so that the condition (3.10) is not satisfied. We note that P is an operator with polynomial coefficients and the subprincipal symbol Psub of P vanishes identically. Moreover after making a change of variables: y0 = x0 , y1 = x1 , yn = xn + x0 x1 one can express P in divergence-free form: P = −D02 + Dn (x21 (1 + bx1 ))Dn + (D1 + x0 Dn )2 . This is in sharp contrast with the fact that in the two-dimensional case, the Cauchy problem is always C ∞ well posed for any second order hyperbolic operators in divergence-free form and real analytic coefficients ([N80]). Now it is very natural to study the (microlocal) Cauchy problem around ρ verifying (3.7) but not (3.10) in the Gevrey classes and determine the optimal Gevrey class in which the Cauchy problem is well posed. In [BB03], it is proved that the Cauchy problem for the model operator (5.1) is well posed in the Gevrey class of order 5.

References [BBP93] E. Bernardi, A. Bove and C. Parenti, Geometric results for a class of hyperbolic operators with double characteristics, II, J. Func. Anal., 116(1993), 62–82. [BB03] E. Bernardi and A. Bove, A remark on the Cauchy problem for a model hyperbolic operator, in “Hyperbolic Differential Operators and Related Problems”, V. Ancona and J. Vaillant eds., Marcel Dekker, 2003. [BB05] E. Bernardi and A. Bove, On the Cauchy problem for some hyperbolic operators with double characteristics, This volume. ¨ rmander, The Analysis of Linear Partial Differential Operators, [H85] L. Ho III, Springer, Berlin-Heidelberg-New York-Tokyo, 1985. ¨ rmander, The Cauchy problem for differential equations with dou[H77] L. Ho ble characteristics, J. Anal. Math., 32(1977), 118–196. [Iv78] V. Ja. Ivrii, The well posedness of the Cauchy problem for non-strictly hyperbolic operators III, Trans. Moscow Math. Soc., 34(1978), 149–168. [Iv78bis] V. Ja. Ivrii, Sufficient conditions for regular and completely regular hyperbolicity, Trans. Moscow Math. Soc., 33(1978), 1–65. [Iv79] V. Ja. Ivrii, Wave fronts of solutions of certain hyperbolic pseudodifferential equations, Trans. Moscow Math. Soc., 39(1979), 87–119. [IP74] V. Ja. Ivrii and V. M. Petkov, Necessary conditions for the Cauchy problem for non-strictly hyperbolic equations to be well posed, Russian Math. Surveys, 29(1974), 1–70.

246 [Iw84]

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N. Iwasaki, The Cauchy problem for effectively hyperbolic equations (standard type), Publ. Res. Inst. Math. Sci., 20(1984), 551–592. [N80] T. Nishitani, The Cauchy problem for weakly hyperbolic equations of second order, Comm. Partial Differential Equations, 5(1980), 1273–1296. [N83] T. Nishitani, Note on some non-effectively hyperbolic operators, Sci. Rep. College Gen. Ed. Osaka Univ. 32(1983), 9–17. [N84bis] T. Nishitani, Local energy integrals for effectively hyperbolic operators, I, II, J. Math. Kyoto Univ. 24(1984), 623–658, 659–666. [N84] T. Nishitani, A note on reduced forms of effectively hyperbolic operators and energy integrals, Osaka J. Math. 21(1984), 843–850. [N86] T. Nishitani, Microlocal energy estimates for hyperbolic operators with double characteristics, in “Hyperbolic Equations and Related Topics”, S. Mizohata ed., Kinokuniya, Tokyo, 1986. [N90] T. Nishitani, Note on Ivrii–Petkov–H¨ ormander condition of hyperbolicity, Sci. Rep. College Gen. Ed. Osaka Univ. 39(1990), 7–9. [N04] T. Nishitani, Non-effectively hyperbolic operators, Hamilton map and bicharacteristics, J. Math. Kyoto Univ. 44(2004), 55–98.

On the Fefferman–Phong inequality for systems of PDEs Alberto Parmeggiani Dipartimento di Matematica, Universit` a di Bologna, Bologna, Italia

Summary. We extend the Fefferman–Phong inequality to certain N × N systems of PDEs, and hence generalize Sung’s result in [S86], that was obtained for systems of ODEs. Our proof uses a Fefferman–Phong Calder´ on–Zygmund decomposition of the phase-space and induction on the size N of the system.

2000 Mathematics Subject Classification: Primary 35S05; Secondary 35B45, 35A30. Key words: Lower bounds, Systems of PDEs, Fefferman–Phong inequality, Calder´ on–Zygmund decomposition.

1 Introduction Let a ∈ S 2 be a second-order symbol in Rn , that is a ∈ C ∞ (Rn × Rn ) and for any given α, β ∈ Zn+ (Z+ = {0, 1, 2, . . .}) there exists Cαβ > 0 such that |∂xα ∂ξβ a(x, ξ)| ≤ Cαβ (1 + |ξ|)2−|β| , ∀(x, ξ) ∈ Rn × Rn . In [FP78], C. Fefferman and D.H. Phong proved their famous inequality which is stated as follows. Theorem 1.1 Let a ∈ S 2 be a (scalar) symbol with a ≥ −c for some constant c ≥ 0. Then there exists a constant C > 0 such that (aw (x, D)u, u) ≥ −Cu20 , ∀u ∈ S(Rn ).

(1.1)

Here (·, ·) and ·0 denote the scalar product and norm in L2 (Rn ), respectively, and aw denotes the Weyl quantization    x+y w −n ix−y,ξ , ξ u(y)dydξ, u ∈ S(Rn ). a e a (x, D)u(x) = (2π) 2

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Alberto Parmeggiani

Note that in Theorem 1.1 no regularity assumption is made on the zero-set of the symbol. It should also be mentioned that in the case of classical pseudodifferential operators and under some assumptions on the geometry of the zeroset of the principal symbol (the characteristic set), L. H¨ ormander proved in [H77] that (1.1) holds even when the symbol tends to −∞ in some (characteristic) directions, provided the famous “trace-+” condition of A. Melin holds. There has been a great deal of work, in the scalar case, to extend the Fefferman–Phong inequality (1.1) in various directions: the reader is addressed to the papers of L. H¨ ormander [H79] (see also H¨ormander’s book [H83–85]), F. Herau [He01], D. Tataru [T02], N. Lerner and Y. Morimoto [LM05], C. Parenti and A. Parmeggiani [PP06], and M. Mughetti, Parenti and Parmeggiani [MPP06]. In the case of systems, the inequality is in general false as shown by R. Brummelhuis in [B92]. Brummelhuis’ counterexample was then generalized to a geometrically characterized class of systems by Parmeggiani in [Pa04] (see also [Pa02]). These counterexamples are based on the example, due to H¨ ormander [H79], of a nonnegative Hermitian matrix whose Weylquantization cannot be nonnegative. It is also worth mentioning that for these counterexamples the sharp G˚ arding inequality cannot be improved, not even to the Melin inequality (work on the Melin inequality for systems has been done by Brummelhuis in [B01], by Brummelhuis and J. Nourrigat in [BN01], and by Parenti and Parmeggiani in [PP02]; see also [Pa02] and references therein). Sufficient conditions for the Fefferman–Phong inequality for certain systems of classical pseudodifferential operators with double characteristics were given by H¨ ormander in [H77], and by Parenti and Parmeggiani in [PP02] (see also Parmeggiani [Pa04] for a perturbative result in the case of symbols in S 2 ). However, Brummelhuis’ counterexample and all the counterexamples given in [Pa04] to inequality (1.1) for systems require at least two variables, i.e., n ≥ 2. This is not by chance. In fact, L.-Y. Sung proved in [S86] that if p = p(x, ξ) is a Hermitian N × N system of ordinary differential equations (that is, x and ξ are one dimensional) which is nonnegative (in the sense of Hermitian matrices) then inequality (1.1) holds for pw (x, D). His proof is based on the use of Fourier series to reduce the problem to an estimate from below of an infinite-size matrix. The aim of this paper is to give a different proof of Sung’s result by actually generalizing the Fefferman–Phong inequality to systems of partial differential equations in Rn of the kind p(x, ξ) = A(x)e(ξ) +

n


B (x)ξ  + C(x) = p(x, ξ)∗ ≥ −cI, ∀(x, ξ) ∈ Rn × Rn ,

=1

where e ∈ S 2 is a positive homogeneous quadratic form in ξ ∈ Rn , and the N × N matrices A, B , C are smooth and bounded on Rn along with their

On the Fefferman–Phong inequality for systems of PDEs

249

derivatives of all orders. Our proof is in the spirit of the Fefferman–Phong Calder´ on–Zygmund decomposition of the phase-space Rn × Rn introduced in [FP83], which makes it possible to use an induction on the size N of the system. We will use here the well-known machinery of the Weyl–H¨ormander calculus (see H¨ormander’s book [H83–85], Vol. III; or H¨ ormander’s paper [H79]), whose basic facts will be recalled in the next section. In Section 3 we shall prove that inequality (1.1) holds for the class of systems of PDEs we consider here.

2 Background on the Weyl–H¨ ormander calculus We recall in this section a few basic facts about admissible metrics and weightfunctions (see [H83–85], Sections 18.4 and 18.5). We shall denote by σ =  n n n j=1 dξ j ∧ dxj the canonical symplectic 2-form in R × R . Definition 2.1 An admissible metric in Rn × Rn is a function Rn × Rn  (x, ξ) −→ gx,ξ where gx,ξ is a positive-definite quadratic form on Rn × Rn such that: •

Slowness: There exists C0 > 0 (the constant of slowness) such that for any given (x, ξ), (y, η) ∈ Rn × Rn one has gx,ξ (y − x, η − ξ) ≤ C0−1 =⇒ C0−1 gy,η ≤ gx,ξ ≤ C0 gy,η ;

•

Uncertainty: For any given (x, ξ) ∈ Rn × Rn one has σ gx,ξ ≤ gx,ξ , σ is the dual metric defined by where gx,ξ σ gx,ξ (y, η) =

•

σ((y, η), (z, ζ))2 ; gx,ξ (z, ζ) (z,ζ)=(0,0) sup

Temperateness: There exists C1 > 0 and N0 ∈ Z+ such that for all (x, ξ), (y, η) ∈ Rn × Rn one has σ gx,ξ ≤ C1 gy,η (1 + gx,ξ (x − y, ξ − η))N0 .

The Planck function associated with g is by definition h(x, ξ)2 =

sup (z,ζ)=(0,0)

gx,ξ (z, ζ) σ (z, ζ) . gx,ξ

Remark that by the uncertainty property one always has h ≤ 1.

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Alberto Parmeggiani

Definition 2.2 Given an admissible metric g, a g-admissible weight is a positive function m on Rn × Rn for which there exist constants c, C, C
> 0 and N1 ∈ Z+ such that for all (x, ξ), (y, η) ∈ Rn × Rn , gx,ξ (x − y, ξ − η) ≤ c =⇒ C −1 ≤ and

m(x, ξ) ≤ C, m(y, η)

m(x, ξ) σ ≤ C
(1 + gx,ξ (y − x, η − ξ))N1 . m(y, η)

Remark 2.1 In particular, given an admissible metric g, by possibly shrinking the slowness constant C0−1 of g, one always has that the Planck function h associated with g is a g-admissible weight. Definition 2.3 Let g be an admissible metric and m be a g-admissible weight. Let a ∈ C ∞ (Rn × Rn ). Denote by a(k) (x, ξ; v1 , . . . , vk ) the kth differential of a at (x, ξ) in the directions v1 , . . . , vk of R2n . Define |a|gk (x, ξ) :=

sup 0=v1 ,...,vk ∈R2n

|a(k) (x, ξ; v1 , . . . , vk )| . Bk 1/2 j=1 gx,ξ (vj )

We say that a ∈ S(m, g) if for any given integer k ∈ Z+ the following seminorms are finite: |a|g (x, ξ) < +∞. (2.1) sup ≤k, (x,ξ)∈Rn ×Rn m(x, ξ) Given µ ∈ R, we shall say that a ∈ S µ (g) if a ∈ S(h−µ , g). Remark 2.2 It is important to observe that if g1 , g2 are admissible metrics with g1 ≤ Cg2 , for some constant C > 0, then a ∈ S(1, g1 ) =⇒ a ∈ S(1, g2 ). This remark will be useful when considering cut-off functions related to different admissible metrics. As regards the composition, one has the following result (see [H83–85]). Theorem 2.1 Given a ∈ S(m1 , g), b ∈ S(m2 , g), then aw (x, D) ◦ bw (x, D) = (a$b)w (x, D), where for any given N ∈ Z+ , (a$b)(x, ξ) =

j  N
 1 i σ(Dx , Dξ ; Dy , Dη ) a(x, ξ)b(y, η)(x,ξ)=(y,η)+rN +1 (x, ξ), j! 2 j=0 (2.2)

with rN +1 ∈ S(hN +1 m1 m2 , g).

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251

Associated with an admissible metric g one has a partition of unity as follows (see [H83–85]). Lemma 2.1 Let g be an admissible metric, and let r2 < C0−1 . Then there exists a sequence of centers {(xν , ξ ν )}ν∈Z+ , a covering of Rn ×Rn made of g-balls g = {(x, ξ); gxν ,ξν (x−xν , ξ−ξ ν ) < r2 } centered at (xν , ξ ν ) and radius r, and Bν,r g a sequence of functions {ϕν } uniformly in S(1, g), with supp ϕν ⊂ Bν,r , such  2 2 2 that ν∈Z+ ϕν = 1. Moreover, for any given r∗ such that r ≤ r∗ < C0−1 , g there exists an absolute integer N∗ such that no more than N∗ balls Bν,r can ∗ intersect at each time (i.e., one has an a priori finite number of overlappings g ). of the dilates by r∗ /r of the Bν,r In the case of matrix-valued symbols, Definitions 2.2 and 2.3, and the composition formula (2.2) hold (keeping track, where necessary, of the order of the terms). Upon denoting by MN the set of N × N complex matrices, we shall write S(m, g; MN ), and S µ (g; MN ), for the matrix-valued analogue of the symbol spaces S(m, g) and S µ (g) considered above. (Analogous notation will be used for the spaces S(m, g; CN ) etc.) In the sequel, given A, B > 0, we write A
B when A ≤ CB for some absolute constant C > 0, and A ∼ B when A
B and B
A.

3 A proof by induction on the size of the system We now prove the following theorem, that in the case n = 1 recaptures Sung’s result in [S86]. Theorem 3.1 Let N ≥ 1 be an integer. For (x, ξ) ∈ Rn × Rn consider the N × N system p(x, ξ) = A(x)e(ξ) + B(x, ξ) + C(x), 2 quadratic form in ξ, B(x, ξ) = where n e ∈ S is a positive homogeneous ∞ n B (x)ξ , and where A, B , C ∈ C (R ; MN ) (1 ≤  ≤ n) are bounded    =1 along with their derivatives of all orders, that is for any given α ∈ Zn+ ,

∂xα AL∞ (Rn ;MN ) +

n


∂xα B L∞ (Rn ;MN ) + ∂xα CL∞ (Rn ;MN ) < +∞.

=1

Suppose that, for some constant c ≥ 0, p(x, ξ) = p(x, ξ)∗ ≥ −cI, ∀(x, ξ) ∈ Rn × Rn .

(3.1)

Then the Fefferman–Phong inequality holds: there exists a constant C > 0 such that (pw (x, D)u, u) ≥ −Cu20 , ∀u ∈ S(Rn ; CN ). (3.2)

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Proof. Our proof is based on a Calder´ on–Zygmund decomposition of Rn ×Rn , similar to that of Fefferman and Phong in [FP83], that makes it possible to use an induction on the size N of the system. In the first place, we reduce matters to the case e(ξ) = |ξ|2 . Take in fact a linear isomorphism T : Rn −→ Rn (a rotation and dilation) such that e(t T −1 ξ) = |ξ|2 . Consider next the linear symplectic transformation χ : Rn × Rn  (x, ξ) −→ (T x, t T −1 ξ) ∈ Rn × Rn , and the metaplectic operator Uχ : u(x) −→ |det T |−1/2 u(T −1 x) associated with χ. We then have, by considering U := Uχ ⊗ I, a metaplectic operator on L2 (Rn ; CN ), that U ∗ pw (x, D)U = (p ◦ χ)w (x, D), where now, for new matrices A, B , C with the same properties as in the hypotheses, (p ◦ χ)(x, ξ) = A(x)|ξ|2 + B(x, ξ) + C(x), B(x, ξ) =

n


B (x)ξ  .

=1

We may hence suppose p(x, ξ) = A(x)|ξ|2 + B(x, ξ) + C(x) = p(x, ξ)∗ ≥ −cI, ∀(x, ξ) ∈ Rn × Rn . (3.3) In the second place, we establish a few consequences of hypothesis (3.1) (in the form (3.3)) that we may suppose, as we shall do from now on without loss of generality, to hold for c = 0. We shall always denote by ·, · the Hermitian product in CN and by | · | the relative norm, regardless of the dimension N , since there will never be risk of confusion. Lemma 3.1 One has A(x) = A(x)∗ ≥ 0, C(x) = C(x)∗ ≥ 0, ∀x ∈ Rn ; B (x) = B (x)∗ , ∀x ∈ Rn ,  = 1, . . . , n;

(3.4) (3.5)

B (x)v, v 2 ≤ 4 C(x)v, v A(x)v, v , ∀x ∈ Rn , ∀v ∈ CN ,  = 1, . . . , n. (3.6) Proof. Since p(x, 0)∗ = p(x, 0) ≥ 0 one immediately has that C(x) = C(x)∗ ≥ 0. To establish the property for A, we note that A(x) =

p(x, ξ) p(x, ξ)∗ = lim = A(x)∗ , |ξ|2 |ξ|→+∞ |ξ|2 |ξ|→+∞ lim

whence the conclusion follows, being |ξ|−2 p(x, ξ) ≥ 0 for all ξ = 0. As regards (3.5), by the hypothesis p = p∗ and (3.4) we have, for all x ∈ Rn and  = 1, . . . , n, B (x)v, w = (p(x, e ) − A(x) − C(x))v, w = v, B (x)w , ∀v, w ∈ CN ,

On the Fefferman–Phong inequality for systems of PDEs

253

where {e1 , . . . , en } is the canonical basis of Rn . As regards (3.6), take any fixed v ∈ CN and x ∈ Rn and consider, for  = 1, . . . , n, the polynomials () (τ ) := p(x, τ e )v, v ≥ 0, ∀τ ∈ R. Px,v When A(x)v, v = 0, then it must be that B (x)v, v = 0 and (3.6) holds. () When A(x)v, v = 0, then the discriminant of Px,v must be non-positive, that is B (x)v, v 2 ≤ 4 C(x)v, v A(x)v, v , and (3.6) holds again. This concludes the proof of the lemma.

' &

Corollary 3.1 Upon writing A(x) = (ajk (x))1≤j,k≤N and likewise B (x) = N (b,jk (x))1≤j,k≤N , and denoting by t(x) = j=1 ajj (x) the trace of A(x), we have |ajk (x)|
t(x), ∀j, k = 1, . . . , N, ∀x ∈ Rn ; |b,jk (x)|2
t(x), ∀j, k = 1, . . . , N, ∀x ∈ Rn , 1 ≤  ≤ n,

(3.7) (3.8)

where the constant in (3.7) depends only on N , and that in (3.8) depends only on CL∞ (Rn ;MN ) . Proof. The proof of (3.7) follows immediately from the fact that, since A = A∗ ≥ 0, one has |ajk (x)|2 ≤ 2ajj (x)akk (x). That of (3.8) follows immediately from (3.6) by considering special choices of v ∈ CN . In fact, considering v = ej , with {e1 , . . . , eN } the canonical basis of CN , and using (3.6) yields |b,jj (x)|2
t(x). Considering v = ej + ek , and using (3.6) and (3.7) gives (b,jj (x) + 2Re b,jk (x) + b,kk (x))2
t(x), so that 2|Re b,jk (x)| ≤ |b,jj (x)+2Re b,jk (x)+b,kk (x)|+|b,jj (x)+b,kk (x)|
t(x)1/2 . Similarly, considering v = ej − iek and once again using (3.6) and (3.7) gives ' & |Im b,jk (x)|
t(x)1/2 , and hence (3.8). The foregoing corollary hence says that whenever the trace of A can be localized to a certain scale, then also all the entries of the matrices A and B can be localized to the same scale. (This is crucial when further microlocalizing the second-order symbol A(x)|ξ|2 and the first-order symbol B(x, ξ). The microlocalization of the zeroth-order symbol C(x) never gives problems.) The next lemma will be used to ensure that in the induction step the resulting (N − 1) × (N − 1) system is still non-negative. Lemma 3.2 Let p(ξ) = A|ξ|2 + B(ξ) + C be an N × N matrix, B(ξ) = n ∗ n =1 B ξ  , with p(ξ) = p(ξ) ≥ 0, for all ξ ∈ R . Write

254

Alberto Parmeggiani

 A=

   a11 a∗1 p11 (ξ) p1 (ξ)∗ , , p(ξ) = a1 A
p1 (ξ) p
(ξ)

where a11 ∈ R, a1 ∈ CN −1 is a column vector (N −1)×1, a∗1 = t a1 ∈ (CN −1 )∗ is a row vector 1 × (N − 1) (i.e., a linear complex form in CN −1 ) and A
is an (N − 1) × (N − 1) matrix (and likewise for p(ξ) etc.). Suppose a11 > 0. Then p
(ξ) ≥ 0 and p
0 (ξ) ≥ 0, ∀ξ ∈ Rn , where p
0 (ξ) := p
(ξ) −

a∗1 ⊗ p1 (ξ) + p1 (ξ)∗ ⊗ a1 p11 (ξ) + 2 a∗1 ⊗ a1 . a11 a11

(3.9)

Here a∗1 ⊗ a1 (etc.) is the (N − 1) × (N − 1) matrix a1 t a1 , invariantly defined as (a∗1 ⊗ a1 )v
= v
, a1 a1 , v
∈ CN −1 .   0 Proof. The first point follows immediately by considering v = v
, v
∈ CN −1 . As regards the second point, define   1 −a∗1 /a11 E= : CN = C ⊕ CN −1 −→ C ⊕ CN −1 . 0 IN −1

(3.10)

Of course, E is an isomorphism with inverse  ∗  1 a1 /a11 E −1 = . 0 IN −1 Since E∗ = one computes ⎡ E ∗ p(ξ)E = ⎣



 1 0 , −a1 /a11 IN −1

p1 (ξ) −

p11 (ξ) a11 a1

 so that considering v = E

0 w


4 0 ≤ p(ξ)v, v =

p
(ξ) −

p11 (ξ) ∗ a11 a1 ∗ a∗ (ξ) ∗ 1 ⊗p1 (ξ)+p1 (ξ) ⊗a1 + p11 a1 a11 a211

⎤

p1 (ξ)∗ −

p11 (ξ)

⊗ a1

⎦,

 , w
∈ CN −1 , gives E ∗ p(ξ)E



  5 0 0 , w
w


= p
0 (ξ)w
, w
, ∀w
∈ CN −1 , ∀ξ ∈ Rn , which concludes the proof of the lemma.

' &

On the Fefferman–Phong inequality for systems of PDEs

255

We now make a first microlocalization that fixes the size of ξ. By hypothesis, letting |dξ|2 Gx,ξ = |dx|2 + 1 + |ξ|2 (so that the Planck function associated with G is hG (x, ξ) = (1+|ξ|2 )−1/2 ), we have p ∈ S 2 (G; MN ). Let {BνG }ν∈Z+ be a covering of Rn ×Rn and {ϕν }ν∈Z+ be a partition of unity associated with G as in Lemma 2.1. Take χν ∈ C0∞ (BνG ) which are also uniformly in S 0 (G), with χν ϕν = ϕν , 0 ≤ χν ≤ 1 (such functions χν can be constructed by using the translates and dilates of a fixed cut-off function χ). Then (the sum being locally finite)

ϕν χν pϕν =: ϕν pν ϕν . p(x, ξ) = ν∈Z+

ν∈Z+

Since {ϕν }ν∈Z+ is a symbol in S 0 (G) with values in 2 (Z+ ), it follows from the Cotlar–Stein lemma that the map 2 n 2 N Φ : u −→ {ϕw ν (x, D)u}ν∈Z+ ∈ L (R ;  (Z+ ; C ))

is bounded, that is




ϕw ν (x, D)u0
u0 .

ν∈Z+

Since the ϕν are scalar functions, we obtain from Theorem 2.1, as it is well known,
∗ w w w ϕw (3.11) pw (x, D) = ν (x, D) pν (x, D)ϕν (x, D) + r (x, D), ν∈Z+

where r ∈ S 0 (G; MN ) has any desired number of seminorms bounded. Hence
w w 2 (pw (pw (x, D)u, u) = ν (x, D)ϕν (x, D)u, ϕν (x, D)u) + O(u0 ), ν∈Z+

where O(·) has the usual meaning, with uniform estimates. We have therefore reduced matters to proving inequality (3.2) for each pν , with a constant independent of ν. By putting Mν := (1 + |ξ ν |2 )1/2 , where (xν , ξ ν ) is the center of BνG , we may therefore drop the subscript ν and suppose that we are working with the admissible metric Gx,ξ = |dx|2 +

|dξ|2 , M2

(3.12)

depending on the parameter M ≥ 1, where 0 ≤ p = p∗ and p ∈ S 2 (G; MN ), that is for any given α, β ∈ Zn+ one has |∂xα ∂ξβ p(x, ξ)|
M 2−|β| , ∀(x, ξ) ∈ Rn × Rn .

(3.13)

256

Alberto Parmeggiani

We hence want to prove that there exists an absolute constant C > 0 such that (pw (x, D)u, u) ≥ −Cu20 , ∀u ∈ S(Rn ; CN ), (3.14) where p(x, ξ) = p(x, ξ)∗ = χ(x, ξ)(A(x)|ξ|2 + B(x, ξ) + C(x)) ≥ 0,  with, recall, B(x, ξ) = n=1 B (x)ξ  , and

(3.15)

⎧ χ ∈ C0∞ (R2n ), 0 ≤ χ ≤ 1, ⎪ ⎪ ⎨ (x, ξ) ∈ supp χ =⇒ |x|
1, |ξ|
M, ⎪ ⎪ ⎩ χ ∈ S 0 (G), that is for all α, β ∈ Zn+ one has |∂xα ∂ξβ χ(x, ξ)|
M −|β| . (3.16) Without loss of generality we may suppose that N


ajk C 2 (Rn ) ≤ 1.

(3.17)

j,k=1

Note that t(x) ∼ max ajj (x), ∀x ∈ Rn . 1≤j≤N

(3.18)

The next and crucial step in the proof is to make a Calder´ on–Zygmund decomposition in x as in [FP83], to localize x to cubes on which at least one of the ajj is “elliptic”, however stopping the localization procedure whenever the diameter of the cube becomes “too small” with respect to M −1 . More precisely, one repeatedly (dyadically) cuts Rn into cubes Qν of varying diameters δ ν and centers xν , stopping at Qν whenever either max max ajj (x) ≥ 20C∗ δ 2ν ,

1≤j≤N x∈Q∗ ν

or

δ ν ≤ c∗ M −1 ,

where Q∗ν is the dilate of Qν by a suitable constant, C∗  1 and 0 < c∗  1. The first condition grants the “ellipticity” of some ajj on Qν and the fact that by Corollary 3.1 all the ajk and b,jk can be localized to Qν (the cjk give no problem), whereas the second condition says that we are in an “uncertainty” (i.e., volume ∼ 1) block in phase-space, which contributes to the lower bound (3.2) by an L2 error (all these errors may then be re-summed through a Cotlar– Stein argument). To rephrase this within the framework of admissible metrics, and hence have a further microlocalized Weyl-calculus, we have to prove the following elementary lemma. This will allow us to construct a Fefferman– Phong metric g (admissible), that will take care of the Calder´ on–Zygmund microlocalization.

On the Fefferman–Phong inequality for systems of PDEs

257

Lemma 3.3 Let e be the Euclidean metric in Rn , and let B2 be the (open) Euclidean ball of radius 2 centered at the origin. Let f ≥ 0 belong to C ∞ (B2 ), and suppose that |f |e2 (x) ≤ 1 for all x ∈ B2 . Suppose f (0) = 1. Then there exists r0 > 0, independent of f , such that 1 < f (x) < 2 and |f |e1 (x) < 2, ∀x ∈ Br0 . 2 Proof. Let f1 (x) be the first-order Taylor polynomial of f at 0. Then 0 ≤ f (x) ≤ 1 + f1 (x) +

|x|2 , |x| < 2, 2

and also 0 ≤ f (−x) ≤ 1 − f1 (x) +

|x|2 , |x| < 2. 2

Hence it follows that |f1 (x)| ≤ 1 +

|x|2 , |x| ≤ 2. 2

When |x| = 1 we obtain |f1 (x)| ≤ 3/2 which yields |f |e1 (0) ≤ |f |e1 (x) < 2 for all x ∈ Br1 provided r1 is chosen sufficiently with 0 < r0 ≤ r1 to be picked, again from Taylor’s formula |x| ≤ r0 1 r2 r2 ≤ 1 − 2r0 − 0 ≤ f (x) ≤ 1 + 2r0 + 0 ≤ 2, 2 2 2 provided r0 is sufficiently small. This concludes the proof of the

3/2, so that small. Now, we have for

lemma.

' &

Corollary 3.2 With the same notation as in Lemma 3.3, f =  N suppose N ∞ e j=1 fj , with fj ≥ 0, fj ∈ C (B2 ), j = 1, . . . , N, and j=1 |fj |2 (x) ≤ 1 for all x ∈ B2 . If f (0) = 1, there exists j0 ∈ {1, . . . , N } and r0 > 0 independent of fj for all j, such that 1 < fj0 (x) < 2, ∀x ∈ Br0 , 2N fj (x) < 2, and |fj |e1 (x) < 2, ∀x ∈ Br0 , ∀j = 1, . . . , N. The proof of the corollary follows immediately from Lemma 3.3, once one notes that f (0) = 1 yields fj0 (0) > 1/N for some j0 ∈ {1, . . . , N }, and by possibly shrinking r0 . Next, to carry out the Calder´ on–Zygmund decomposition of Fefferman and Phong, set   1 , t(x)1/2 , H(x)−1 := max (3.19) M and put

 f (z) = H(x) t x + 2

z H(x)

 .

(3.20)

258

Alberto Parmeggiani

By Lemma 3.3, and Corollary 3.2 (and (3.17)), considering the function H(x) amounts to localizing x to cubes Qν on which H(x)−1 ∼ diam(Qν ) for x ∈ Qν , and either ajj (x)  diam(Qν )2 on Qν , for some j, or M diam(Qν )
1. Lemma 3.4 Define the Fefferman–Phong metric |dξ|2 . M2

(3.21)

|dξ|2 , H(x)2

(3.22)

gx,ξ = H(x)2 |dx|2 + Then g is an admissible metric. Proof. Since σ = M 2 |dx|2 + gx,ξ

we have h(x, ξ)2 =

H(x)2 . M2

(3.23)

⎧ ⎪1, if t(x) ≤ M −2 , H(x) ⎨ = h(x, ξ) = 1 ⎪ M , if t(x) ≥ M −2 , ⎩ t(x)1/2 M σ . so that gx,ξ ≤ gx,ξ We now prove that Hence

|x − x
|2 H(x)2 < r2 =⇒

2 1 ≤ , H(x) H(x
)

where r is the r0 given by Lemma 3.3. In fact, let x
= x + Then •

(3.24) z H(x) ,

|z| < r.

when f (0) = 1 we have, by the definition of f, H(x)−1 = t(x)1/2 , and 1 2 H(x)2 1 < f (z) = H(x)2 t(x
) ≤ ≤ ; , i.e., 2 H(x
)2 H(x) H(x
)

•

when f (0) ≤ 1, that is H(x)−1 = 1/M, we apply Lemma 3.3 to f (z) + 1 − f (0) so as to obtain 1 < f (z) + 1 − f (0), ∀z ∈ Br , 2 i.e., being f (0) ≥ 0, 1 1 1 1 ≤ + f (0) < f (z) + 1 ⇐⇒ ≤ + H(x)2 t(x) < H(x)2 t(x
) + 1, 2 2 2 2 that is, on dividing by H(x)2 , 1 1 1 2 = ≤ t(x
) + 2 ≤ , 2H(x)2 2M 2 M H(x
)2 which again implies H(x)−1 ≤ 2H(x
)−1 .

On the Fefferman–Phong inequality for systems of PDEs

259

Using (3.24) we now prove that g is slowly varying. In fact, on supposing gx,ξ (x
− x, ξ
− ξ) < r2 , we then have in particular that H(x)2 |x − x
|2 < r2 , whence   |ζ|2 |ζ|2 gx
,ξ
(z, ζ) = H(x
)|z|2 + 2 ≤ 4 H(x)2 |z|2 + 2 = 4gx,ξ (z, ζ), M M and this suffices (in view of [H83–85], Definition 18.4.1) to prove that g is slowly varying. At last we prove that g is temperate. We must find a universal constant C > 0 and integer N ≥ 0 such that σ (x
− x, ξ
− ξ))N , gx,ξ ≤ Cgx
,ξ
(1 + gx,ξ

that is   N |ζ|2 |ζ|2 |ξ − ξ
|2
2 2 2
2 . 1 + M |x − x | + H(x) |z| + 2 ≤ C H(x ) |z| + 2 M M H(x)2 (3.25) When H(x
)2 |x − x
|2 < r2 , by (3.24) we have H(x) ≤ 2H(x
), so that (3.25) trivially holds with C = 4 and any chosen N ∈ Z+ . Hence we may suppose that (3.26) H(x
)2 |x − x
|2 ≥ r2 . 2

2

The inequality (3.25) is of course equivalent to H(x)2 2 |ζ|2 |z| + 4 ≤ C M2 M



H(x
)2 2 |ζ|2 |z| + 4 M2 M

 N |ξ − ξ
|2 , 1 + M 2 |x − x
|2 + H(x)2 (3.27)

whence it follows that • •

when z = 0, (3.27) holds with any chosen C ≥ 1 and N ∈ Z+ ; when ζ = 0, being H(x)/M ≤ 1 (by definition), we get that H(x
)2 H(x
)2 |x − x
|2 H(x
)2 H(x)2 ≤1 ≤ 1+ ≤ (by (3.26)) ≤ + 2 2 2 2 M M r r M 2 r2   1 H(x
)2 H(x
)2 |ξ − ξ
|2
2
2 ≤ 2 + H(x ) |x − x | + r M2 H(x)2 M 2 =

H(x
)2 M 2 r2

  |ξ − ξ
|2 1 + M 2 |x − x
|2 + , H(x)2

whence the temperateness of g follows, with (say) C = 410 + r−2 and N = 1. This concludes the proof of the lemma. & ' Remark 3.1 By Remark 2.2 we have that S(1, G) ⊂ S(1, g), since G ≤ g.

260

Alberto Parmeggiani

We next take, as before by virtue of Lemma 2.1, a covering of Rn ×Rn made of g-balls {Bν }ν∈Z+ of radius r0 /102 , where r0 is given by Corollary 3.2, and  a partition of unity ϕν ∈ C0∞ (Bν ) uniformly in S 0 (g), with ν∈Z+ ϕ2ν = 1. Also take χν ∈ C0∞ (Bν ) uniformly in S 0 (g), with 0 ≤ χν ≤ 1 and χν ϕν = ϕν . It is now very important to note that by Corollary 3.1, Lemma 3.3, Corollary 3.2 and Remark 3.1, we have (with bjk (x, ξ) the jk entry of B(x, ξ)) χν χajk |ξ|2 ∈ S 2 (g), χν χbjk (x, ξ) ∈ S 1 (g), χν χcjk ∈ S 0 (g), ∀j, k = 1, . . . , N, uniformly in ν ∈ Z+ . Then (recall that p(x, ξ) = χ(x, ξ)(A(x)|ξ|2 + B(x, ξ) + C(x)))
w w w pw (x, D) = ϕw ν (x, D)(χν p) (x, D)ϕν (x, D) + R (x, D), ν∈Z+

where R ∈ S 0 (g; MN ), with uniform bounds on the seminorms of any desired order. Hence (Rw u, u) = O(u20 ). Now, in each ball Bν either ajj (x) is “big” for some j ∈ {1, . . . , N } with all the coefficients in x in the entries of χν p bounded in terms of (powers of) ajj , or all the entries of χν p are uniformly bounded by an absolute constant (along with any fixed number of seminorms). In the latter case, an application of the Cotlar–Stein lemma gives a lower bound by −Cu20 , for some absolute constant C > 0. We may therefore assume, without loss of generality, that the summation is restricted only to those balls Bν , on which some ajj (x) is big (i.e., ajj (x)  H(xν )−2 ), and it is no restriction to suppose that the entry a11 (x) is big (in fact, each time it suffices to use a permutation matrix, which has constant entries and brings the jjth entry of A to the 11 position). Now, following Lemma 3.2, set   1 −a1 (x)∗ /a11 (x) (3.28) , (x, ξ) ∈ Bν . Eν = Eν (x) = 0 IN −1 Then αν := Eν−1 ϕν ∈ S 0 (g; MN ), pν := Eν∗ χν pEν ∈ S 2 (g; MN ), with uniform estimates. It is important to notice that, by the Cotlar–Stein lemma,
2 2 n N αw (3.29) ν u0
u0 , ∀u ∈ S(R ; C ). ν∈Z+

Hence (the sum being locally finite)
p= α∗ν pν αν . ν∈Z+

On the Fefferman–Phong inequality for systems of PDEs

We now have

261

w ∗ w (α∗ν )w pw ν αν = (αν $pν $αν ) ,

where i (3.30) α∗ν $pν $αν = α∗ν pν αν − (α∗ν {pν , αν } + {α∗ν , pν αν }) + rν , 2  with ν rνw bounded on L2 (Rn ; CN ). The second term on the right-hand side of (3.30) is Hermitian and may be rewritten, on gathering lower order terms into a new rν , as i ∗ (2) β ν := − (α∗ν {p(2) ν , αν } + {αν , pν αν }), 2 where

∗ 2 p(2) ν (x, ξ) = χν (x, ξ)χ(x, ξ)Eν (x) A(x)Eν (x)|ξ| .

The main (first-order) contribution in β ν is therefore given by   n (2) (2) −1 i
∂(Eν∗ )−1 ∂pν ∗ −1 ∂pν ∂Eν −1 − (Eν ) ϕ2ν − E 2 ∂ξ  ∂x ∂x ∂ξ  ν =1

   −1 ∗ −1 ∂E ) ∂(E ν ν −1 + (Eν∗ )−1 p(2) − p(2) ϕν ∂ξ ϕν . ν ν Eν ∂x ∂x We hence obtain that α∗ν $pν $αν = α∗ν pν αν − β 1,ν − β 2,ν + rν , where β 1,ν

 ν

rνw is bounded on L2 (R; CN ), and

   ∗  n 
∂Eν−1 ∂Eν−1 ∗ := i ϕ − ϕ $q,ν $αν , αν $q,ν $ ∂x ν ∂x ν =1

β 2,ν

   ∗  n  i
∂Eν−1 ∂Eν−1 ∗ := ξ
∂ξ ϕ − ξ
∂ξ ϕ $q
,ν $αν , αν $q
,ν $ 2
∂x   ν ∂x   ν , =1

where q,ν := χν χξ  Eν∗ AEν = χν χξ 



a11 0 A
−

0 a∗ 1 ⊗a1 a11

 ∈ S 1 (g; MN ),  = 1, . . . , n,

(3.31) uniformly in ν. Since ξ 
∂ξ  ϕν ∈ S 0 (g) uniformly in ν (in fact, |ξ 
∂ξ  ϕν |
M M −1 ), 1 ≤ , 
≤ n, we have β j,ν ∈ S 1 (g; MN ), j = 1, 2, uniformly in ν and with any desired number of seminorms universally bounded. It is also important to note that

262

Alberto Parmeggiani

  ∂Eν−1 0 ∂x (a1 (x)∗ /a11 (x)) = , (x, ξ) ∈ Bν ,  = 1, . . . , n, 0 0 ∂x and that, as a consequence,   ∂E −1 ∂Eν−1 0 ∂x (a1 (x)∗ /a11 (x)) , (x, ξ) ∈ Bν . Eν±1 ν = = 0 0 ∂x ∂x

(3.32)

(3.33)

Hence (pw (x, D)u, u) =


  w w w w 2 (pw ν αν u, αν u) + (β 1,ν u, u) + (β 2,ν u, u) + O(u0 ). ν∈Z+

One now writes (using Lemma 3.2)   p11 (x, ξ) p1 (x, ξ)∗ , (x, ξ) ∈ Bν , pν (x, ξ) = χν (x, ξ)χ(x, ξ) p1 (x, ξ) p
(x, ξ)  where (recall that we write A =

p11 (x, ξ) = a11 (x)|ξ|2 +

n


a11 a∗1 a1 A

(3.34)

 etc.)

b,11 (x)ξ  + c11 (x),

(3.35)

=1

p1 (x, ξ) =

   n 
b,11 (x) c11 (x) a1 (x) ξ  + c1 (x) − a1 (x) , b,1 (x) − a11 (x) a11 (x) =1

(3.36) and   a1 (x)∗ ⊗ a1 (x)
p (x, ξ) = A (x) − |ξ|2 a11 (x)  n
a1 (x)∗ ⊗ b,1 (x) + b,1 (x)∗ ⊗ a1 (x) + B
(x) − a11 (x)


=1

 b,11 (x) ∗ + a1 (x) ⊗ a1 (x) ξ  a11 (x)2   c11 (x) a1 (x)∗ ⊗ c1 (x) + c1 (x)∗ ⊗ a1 (x)
∗ + + C (x) − a1 (x) ⊗ a1 (x) . a11 (x) a11 (x)2

On the Fefferman–Phong inequality for systems of PDEs

263

Remark that, uniformly in ν,

and, by Lemma 3.2, Write αw νu =

 (αw u)  ν

1


(αw ν u)

0 ≤ χν χp11 ∈ S 2 (g),

(3.37)

χν χp1 ∈ S 1 (g; CN −1 ),

(3.38)

0 ≤ χν χp
∈ S 2 (g; MN −1 ).

(3.39)

∈ L2 (Rn ; C ⊕ CN −1 ) (and likewise for u). By (3.31),

(3.32) and (3.33), one has   w n 
∂Eν−1 w ϕν u, αw u q,ν ν ∂x =1

  ∗ w  n 
a1 w
w = u , (αν u)1 (χν χa11 ξ  ) ϕν ∂x a11 =1

 ∗ w  n 
a1 1/2 1/2 u
, (χν χa11 ξ  )w (αw u) a11 ϕν ∂x + (˜ rν u, u), 1 ν a11

=

=1

where

 ν

r˜ν is bounded in L2 (Rn ; CN ), and similarly that

  w n 
∂Eν−1 w w u, αν u q
,ν ξ 
∂ξ ϕν ∂x


, =1

=

 n
=1

1/2 a11 ξ 

n


 ∂ξ
ϕν ∂x



=1

 w a∗1 1/2
w w rν
u, u), u , (χν χa11 ξ  ) (αν u)1 +(˜ a11

 where ν r˜ν
is bounded in L2 (Rn ; CN ). By the same token, using the expression obtained from (3.36) for p∗1 in (3.34), w
2Re((χν χp1 )w (αw ν u)1 , (αν u) )

= 2Re

n



((χν χa11 ξ  )w (αw ˜w rν

u, u), ν u)1 , γ ,ν u ) + (˜ 1/2

=1

with γ˜ ,ν ∈ S 0 (g; (CN −1 )∗ ), 1 ≤  ≤ n, uniformly in ν, and where the op 

erators ν γ˜w ˜ν are bounded in L2 (R; (CN −1 )∗ ) and ,ν (1 ≤  ≤ n) and νr 2 N L (R; C ), respectively.

264

Alberto Parmeggiani

Hence, by the Cotlar–Stein lemma, we may write (for new γ w ,ν ) w

(p u, u) =




 w
w w
w
((χν χp11 )w (αw ν u)1 , (αν u)1 ) + ((χν χp ) (αν u) , (αν u) )

ν∈Z+

+ Re

n


 1/2 w
((χν χa11 ξ  )w (αw ν u)1 , γ ,ν u )

+ O(u20 ),

=1



2 N −1 ∗ where ν γ w ) ), for all  = 1, . . . , n. ,ν is bounded in L (R; (C Now, by the Cauchy–Schwarz inequality we have   n  
 1/2 w
  ((χν χa11 ξ  )w (αw ν u)1 , γ ,ν u )   =1

w (1) (2)

≤ ε((χ2ν χ2 a11 |ξ|2 )w (αw ν u)1 , (αν u)1 ) + ε(rν u1 , u1 ) + Cε (rν u , u ),

 (1) (2) where ε > 0 will be picked later, and ν diag(rν , rν ) is bounded in 2 N −1 ). L (R; C ⊕ C We finally obtain, for 0 < ε ≤ 1 to be picked, w
w w
w
((χν χp11 )w (αw ν u)1 , (αν u)1 ) + ((χν χp ) (αν u) , (αν u) )

+ Re

n


w
((χν χa11 ξ  )w (αw ν u)1 , γ ,ν u ) 1/2

=1 w ≥ ((χν χp11 − εχ2ν χ2 a11 |ξ|2 )w (αw ν u)1 , (αν u)1 )
w
(1) (2)

+ ((χν χp
)w (αw ν u) , (αν u) ) − ε(rν u1 , u1 ) − Cε (rν u , u ).

By choosing ε sufficiently small, uniformly in ν and M (ε = 1/2 suffices), we get, recalling (3.35), that for some universal constant c > 0, χν (x, ξ)χ(x, ξ)p11 (x, ξ)−εχν (x, ξ)2 χ(x, ξ)2 a11 (x)|ξ|2 ≥ −c, ∀(x, ξ) ∈ Rn ×Rn , for all ν ∈ Z+ , and may therefore apply the scalar Fefferman–Phong inequality (1.1) to obtain, with C > 0 a universal constant (for the symbols have estimates uniform in ν), w w 2 ((χν χp11 − εχ2ν χ2 a11 |ξ|2 )w (αw ν u)1 , (αν u)1 ) ≥ −C(αν u)1 0 ,

and hence, by (3.29), that
w
2 ((χν χp11 − εχ2ν χ2 a11 |ξ|2 )w (αw ν u)1 , (αν u)1 ) ≥ −C u0 . ν∈Z+

On the Fefferman–Phong inequality for systems of PDEs

265

At this point we may use induction on the size N of the system, for we now have that 0 ≤ χν χp
∈ S 2 (g; MN −1 ), uniformly in ν ∈ Z+ , the initial step of the induction being taken care of once more by the scalar Fefferman–Phong inequality. This concludes the proof of the theorem. ' & Remark 3.2 Note that the same proof yields also a semi-classical analogue of Theorem 3.1 for the h-Weyl–H¨ormander quantization pw (x, hD) (see, for instance, [DS99]) of p(x, ξ).

References [B92]

R. Brummelhuis, Sur les in´egalit´es de G˚ arding pour les syst` emes d’op´ erateurs pseudo-diff´ erentiels, C. R. Acad. Sci. Paris, S´erie I 315 (1992), 149–152. [B01] R. Brummelhuis, On Melin’s inequality for systems, Comm. Partial Differential Equations 26(2001), 1559–1606. [BN01] R. Brummelhuis and J. Nourrigat, A necessary and sufficient condition for Melin’s inequality for a class of systems, J. Anal. Math. 85(2001), 195–211. ¨ strand, Spectral asymptotics and the semi[DS99] M. Dimassi and J. Sjo classical limit, London Mathematical Society Lecture Note Series, 268. Cambridge University Press, Cambridge, 1999. [FP78] C. L. Fefferman and D. H. Phong, On positivity of pseudo-differential operators, Proc. Natl. Acad. Sci. USA 75(1978), 4673–4674. [FP83] C. L. Fefferman and D. H. Phong, Subelliptic eigenvalue problems, Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I (Chicago, Ill., 1981), 590–606, Wadsworth Math. Ser., Wadsworth, Belmont, CA, 1983. [He01] H. Herau, Melin-H¨ ormander inequality in a Wiener type pseudo-differential algebra, Ark. Mat. 39(2001), 311–338. ¨ rmander, The Cauchy problem for differential equations with dou[H77] L. Ho ble characteristics, J. Anal. Math. 32(1977), 118–196. ¨ rmander, The Weyl Calculus of Pseudodifferential Operators, [H79] L. Ho Comm. Pure Appl. Math. 32(1979), 359–443. ¨ rmander, The Analysis of Linear Partial Differential Opera[H83–85] L. Ho tors, Vol. I–IV, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1983/85. [LM05] L. Lerner and Y. Morimoto, On the Fefferman–Phong inequality and a Wiener-type algebra of pseudodifferential operators, preprint, 2005. [MPP06] M. Mughetti, C. Parenti and A. Parmeggiani, Lower bound estimates without transversal ellipticity, preprint, 2006. [PP02] C. Parenti and A. Parmeggiani, Lower bounds for systems with double characteristics, J. Anal. Math. 86(2002), 49–91. ormander inequal[PP06] C. Parenti and A. Parmeggiani, A remark on the H¨ ity, Comm. Partial Differential Equations 31(2006), 1071–1084.

266 [Pa02]

[Pa04]

[S86] [T02]

Alberto Parmeggiani A. Parmeggiani, On lower bounds of pseudodifferential systems. Hyperbolic problems and related topics, 269–293, Grad. Ser. Anal., Int. Press, Somerville, MA, 2003. A. Parmeggiani, A class of counterexamples to the Fefferman–Phong inequality for systems, Comm. Partial Differential Equations 29(2004), 1281–1303. L.-Y. Sung, Semi-boundedness of Systems of Differential Operators, J. Differential Equations 65(1986), 427–434. D. Tataru, On the Fefferman–Phong inequality and related problems, Comm. Partial Differential Equations 27(2002), 2101–2138.

Local energy decay and Strichartz estimates for the wave equation with time-periodic perturbations Vesselin Petkov D´epartement de Math´ematiques Appliqu´ees, Universit´e Bordeaux 1, Talence, France Summary. We examine the memorphic continuation of the cut-off resolvent Rχ (z) = χ(U (T, 0) − z)−1 χ, χ(x) ∈ C0∞ (Rn ), where U (t, s) is the propagator related to the wave equation with non-trapping time-periodic perturbations (potential V (t, x) or a periodically moving obstacle) and T > 0 is the period. Assuming that Rχ (z) has no poles z with |z| ≥ 1, we establish a local energy decay and we obtain global Strichartz estimates. We discuss the case of trapping moving obstacles and we present some results and conjectures concerning the behavior of Rχ (z) for |z| > 1.

2000 Mathematics Subject Classification: 35P25, 35L05, 47A40. Key words: Monodromy operator, local energy decay, Strichartz estimates.

1 Introduction In this paper we present a survey of some recent results concerning two problems for the wave equation with time-periodic perturbations. The first one is the Cauchy problem with time-periodic potential 2 ∂t u − ∆u + V (t, x)u = F (t, x), (t, x) ∈ R × Rn , (1.1) u(τ , x) = f0 (x), ut (τ , x) = f1 (x), x ∈ Rn , where the potential V (t, x) ∈ C ∞ (Rn+1 ), n ≥ 2, satisfies the conditions: (H1 ) There exists R0 > 0 such that V (t, x) = 0 for |x| ≥ R0 , ∀t ∈ R, (H2 ) V (t + T, x) = V (t, x), ∀(t, x) ∈ Rn+1 with T > 0. Consider the homogeneous Sobolev spaces H˙ γ (Rn ) = Λ−γ L2 (Rn ), where √ Λ = −∆ and −∆ is the Laplacian in Rn and set H˙ γ (Rn ) = H˙ γ (Rn ) ⊕ H˙ γ−1 (Rn ). The solution of (1.1) with F = 0 is given by the propagator

268

Vesselin Petkov

U (t, τ ) : H˙ γ (Rn )  (f0 , f1 ) −→ U (t, τ )(f0 , f1 ) = (u(t, x), ut (t, x)) ∈ H˙ γ (Rn ). Let U0 (t) = eitG0 be the unitary group in H˙ γ (Rn ) related to the Cauchy problem (1.1) with V = 0, F = 0, τ = 0 and let U (T ) = U (T, 0). Let χ, ψ 1 be functions in C0∞ (Rn ) such that χ(x) = ψ 1 (x) = 1 for |x| ≤ R0 + T. We suppose also that (1 − ψ 1 )U (0, s)Q(s) = 0, 0 ≤ s ≤ T, where

(1.2)

 Q(s) =

 0 0 . V (s, x) 0

Consider the cut-off resolvent Rχ (θ) = χ(U (T ) − e−iθ I)−1 ψ 1 : H˙ 1 (Rn ) → H˙ 1 (Rn ), where Im θ ≥ A > 0, −π < Re θ ≤ π and ψ 1 is fixed. We show that Rχ (θ) admits a meromorphic extension in C for n ≥ 3, n odd, and to


C = {θ ∈ C : θ = 2πk − iµ, µ ≥ 0, k ∈ Z} for n ≥ 2, n even. The poles of Rχ (θ) play an essential role in the problems of local energy decay, global Strichartz estimates, trace formulae and blow-up of the local energy (see [7], [1], [2], [15], [21]). The second problem we deal with is the Dirichlet problem for the wave equation outside a time-periodic moving obstacle. Let Q ⊂ Rn+1 , n ≥ 3, be an open domain with C ∞ smooth boundary ∂Q. Set Ω(t) = {x ∈ Rn : (t, x) ∈ Q}, ∅ ≡ K(t) = {x ∈ Rn : (t, x) ∈ / Q} ⊂ {x : |x| ≤ R0 }. We suppose that the obstacle is periodically moving K(t + T ) = K(t), ∀t ∈ R, T > 0 and for each (t, x) ∈ ∂Q the exterior unit normal (ν t , ν x ) to ∂Q at (t, x) satisfies |ν t | < |ν x |. We study the problem ⎧ (∂ 2 − ∆x )u = 0 in Q, ⎪ ⎪ ⎨ t (1.3) u = 0 on ∂Q, ⎪ ⎪ ⎩ u(τ , x) = f0 (x), ut (τ , x) = f1 (x). The solution is given by a propagator U (t, τ ) : H(τ ) −→ H(t), where H(t) is the energy space related to Ω(t) (see [7], [14] for a precise definition). As

Time-periodic perturbations

269

above we introduce the monodromy operator U (T ) = U (T, 0) and the cut-off resolvent Rχ (θ) = χ(U (T ) − e−iθ I)−1 χ with χ = 1 on {x : |x| ≤ R0 + T }. We examine the problem of the meromorphic continuation of the cutoff resolvents Rχ (θ) for time-periodic potentials and non-trapping moving obstacles. In contrast to stationary perturbations, the absence of trapping rays is not sufficient to guarantee a uniform local energy decay. To obtain the last property, we must exclude the existence of poles of Rχ (θ) with Im θ ≥ 0 and for this purpose we introduce the condition (R) in Section 2. In Section 3 we show that the local energy decay of solutions with initial data having compact support leads to an L2 -integrability of the local energy of solutions with data in the energy space. This is the crucial point in the proof of global Strichartz estimates for time-periodic non-trapping perturbations. The investigation of trapping moving obstacles is more complicated and many problems are still open. In some recent works (see [3], [4]) it is proved that for stationary trapping obstacles the cut-off resolvent χ(U (t) − z)−1 χ has a singularity as z → z0 , |z| > 1, for every z0 ∈ S and almost all t ∈ R+ (see Theorem 5.1). Thus we do not have a meromorphic extension across the unit circle S as in the case of non-trapping perturbations. Moreover, it is not known for trapping moving obstacles whether χ(U (T ) − z)−1 χ has a meromorphic continuation from {z ∈ C : |z| ≥ A  1} to {z ∈ C : eT ≤ |z| ≤ A}, ! > 0. We conjecture that for obstacles having at least one δ-trapping bicharacteristic the cut-off resolvent χ(U (T ) − z)−1 χ is not meromorphic in {z ∈ C : eT ≤ |z|}, 0 < ! < δ (see Section 5 for the notation).

2 Resonances for time-periodic potentials In this section we study the problem (1.1) and U (t, s) denotes the corresponding propagator. Let ψ ∈ C0∞ (Rn ) be a fixed cut-off such that ψ(x) = 1 for |x| ≤ R0 + T. By a finite speed of propagation argument we get (1 − ψ)U (T, s)Q(s) = 0, Q(s)U0 (s)(1 − ψ) = 0, 0 ≤ s ≤ T. For A > 0 large enough and Im θ ≥ A, and if the resolvents (U0 (T )−e (U (T ) − e−iθ I)−1 exist, we have the equality U (T ) − zI 



I)−1 ,



T

−1

U (T, s)Q(s)U0 (s)dsψ(U0 (T ) − zI)

= I−ψ

(2.1) −iθ

(U0 (T ) − zI), z = e−iθ

0

and (U0 (T ) − zI)−1

 −1

= (U (T ) − zI)



T

 −1

U (T, s)Q(s)U0 (s)dsψ(U0 (T ) − zI)

I −ψ 0

.

270

Vesselin Petkov

Assume that ψ 1 ∈ C0∞ (Rn ) satisfies (1.2) and let ψ 1 (x) = 1 on supp ψ. We take an arbitrary cut-off function χ ∈ C0∞ (Rn ) so that χ = 1 on supp ψ and multiply the above equality by χ and ψ 1 to get χ(U0 (T ) − zI)−1 ψ 1 = χ(U (T ) − zI)−1 ψ 1   T

× I −ψ



U (T, s)Q(s)U0 (s)dsψ(U0 (T ) − zI)−1 ψ 1 .

0

Introduce the operator 

T

K(z) = ψ

U (T, s)Q(s)U0 (s)dsψ(U0 (T ) − zI)−1 ψ 1 .

0

For n ≥ 3, n odd, the operator ψ(U0 (T ) − e−iθ I)−1 ψ 1 admits an analytic continuation with respect to θ in C and this follows immediately from the Huygens principle and the expansion ψ(U0 (T ) − e−iθ I)−1 ψ 1 = −




N (ψ,ψ1 )

ψU0 (kT )ψ 1 ei(k+1)θ

k=0

which holds for Im θ ≥ A > 0. On the other hand, the operator K(z) is compact in H˙ 1 (Rn ) and an application of the analytic Fredholm theorem leads to a meromorphic continuation of Rχ (θ) in C. For n even a similar argument leads to a meromorphic continuation of Rχ (θ) in


C = {z ∈ C : z = 2πk − iµ, µ ≥ 0, k ∈ Z},


but the analysis of the analytic extension of ψ(U0 (T ) − e−iθ I)−1 ψ 1 in C is more complicated (see [20], [21], [15]). Thus we have the following Proposition 2.1 The cut-off resolvent Rχ (θ) admits a meromorphic contin
uation in C for n odd and in C for n even. The time-periodic potentials are non-trapping perturbations. Nevertheless, some exponentially growing modes could exist. To establish a local energy decay, we introduce the following condition. (R) The operator Rχ (θ) admits a holomorphic extension from {θ ∈ C : Im θ ≥ A > 0} to {θ ∈ C : Im θ ≥ 0}, for n ≥ 3, odd, and to {θ ∈ C : Im θ ≥ 0, θ = 2πk, k ∈ Z} for n ≥ 2, even . Moreover, for n even we have lim

λ→0, λ>0

Rχ (iλ)H˙ 1 →H˙ 1 < ∞.

This condition is independent of the choice of χ, ψ 1 . Let ϕ ∈ C0∞ (Rn ), f ∈ H˙ 1 , f = 0 for |x| ≤ R. We denote the norm in H˙ 1 (Rn ) by . and we use the same notation for the norm of bounded operators in H˙ 1 (Rn ).

Time-periodic perturbations

271

Theorem 2.1 ([15]) Assume that the condition (R) is fulfilled. Then for 0 ≤ s ≤ t, t − s ≥ t0 > 1 we have ϕU (t, s)f  ≤ C(n, ϕ, R)p(t − s)f , where

p(t) =

e−δt , δ > 0, n ≥ 3, odd , t−1 (ln t)−2 , n ≥ 2, even.

The local energy decay has been established for n odd, by Bachelot and Petkov [1] assuming that the Lax–Phillips operator Z b (T ) = P+b U (T )P−b , b > R0 + T has no eigenvalues z ∈ C, |z| ≥ 1, (see Section 4 for the definition of the projectors P±b ) and by Vainberg [21] for n ≥ 2 assuming a similar condition for an operator R(θ) having a complicated form. The novelty of our approach is the role of the cut-off resolvent Rχ (θ). It is worth remarking that the resolvent of the monodromy operator plays an essential role in the analysis of timeperiodic perturbations of the Schr¨ odinger operator (see for example, [8]). On the other hand, the link between the poles of Rχ (θ) and the spectrum of Z b (T ) has been established in [2]. Sketch of the proof. We have the representation  t U (t, s)Q(s)U0 (s)f ds, U (t, 0)f = U0 (t)f − 0

and we will deal with 

t

I(ϕ, f ) = −∞

ϕU (t, s)Q(s)U0 (s)f ds

extending U0 (s)f by 0 for s < 0. Introduce the Fourier–Block–Gelfand transform ∞
g(θ, s) = F (U0 (s)f )(θ, s) = U0 (kT + s)eikθ f k=−∞

which is well defined for Im θ ≥ α > 0. Applying the inverse transform of F , we are going to examine   t 1 J(t) = ϕU (t, s)Q(s) g(θ, s)dθds, 2π −∞ dα where dα = [iα − π, iα + π] and α > 0 will be chosen large enough in the following. Choose an integer m ∈ Z so that t
= t − mT ∈ [0, T [. Then J(t) has the form

272

Vesselin Petkov

1 2π



t
















ϕU (t , s )Q(s )U0 (s ) 0

e−imθ g(θ, 0)dθds


dα

 ∞  1
−kT


+ ϕU (t , s )Q(s ) e−imθ g(θ, s
)dθds
2π −kT −T dα k=0

= I1 (t) + I2 (t). We write I2 (t) as 

 dα

T

ϕU (t
+ T, 0)χ(e−iθ I − U (T ))−1 ψ 1

0

× U (0, ξ)Q(ξ)U0 (ξ)e−imθ ψg(θ, 0)dξdθ, where χ = 1 on supp ψ and ϕU (t
+ T, 0)(1 − χ) = 0. Assume n ≥ 3, n odd. Then the condition (R) implies that Rχ (θ) has no poles θ with Im θ ≥ 0 and we can choose δ > 0 so that Rχ (θ) has no poles θ with Im θ ≥ −δT, −π < Re θ ≤ π. Let d−δT = [−iδT − π, −iδT + π]. Recall that t = mT + t
, so that e−mδT ≤ Ce−δt with C > 0 independent of m and t. On the other hand, ψg(θ, 0) = e−iθ ψ(e−iθ − U0 (T ))−1 f, Im θ > 0 and we conclude that ψg(θ, 0) admits an analytic continuation in C. We shift the contour of the integration from dα to d−δT (see the figure) and we obtain I2 (t) ≤ C1 e−δt f , t ≥ 0.

iα − π

−iδT − π

iα + π

−iδT + π

By the same argument we get an estimate for I1 (t) and we conclude that |ϕU (t, s)f  ≤ C(n, ϕ, f )e−δ(t−s) f , t − s ≥ 1.

Time-periodic perturbations

273

For n even we apply a similar argument by shifting the contour of integration to a curve γ going around 0 (see [15]). For the analysis of the integral in a neighborhood of 0 we use the hypothesis on the behavior of Rχ (θ) and a result of Vainberg [20], to obtain Ik (t) ≤ C2 t−1 (ln t)−2 f , t ≥ t0 > 1, k = 1, 2. We refer to [15] for more details.

3 Strichartz estimates We say that the real numbers 1 ≤ p˜, q˜ ≤ 2 ≤ p, q ≤ +∞, 0 ≤ γ ≤ 1 are admissible for the free wave equation if the following estimate holds: For data (f0 , f1 ) ∈ H˙ γ (Rn ), F ∈ Lpt˜(R; Lqx˜ (Rn )) and u(t, x) a solution of (1.1) with τ = 0, V = 0 we have uLpt(R; Lqx (Rn )) + u(t, x)H˙ xγ + ∂t u(t, x)H˙ xγ−1 ≤ C(f0 H˙ γ + f1 H˙ γ−1 + F Lp˜(R; Lqx˜ (Rn )) )

(3.1)

t

with a constant C = C(n, p, q, p˜, q˜, γ) > 0 independent of t ∈ R. We refer to Lindblad–Sogge [11] and Keel–Tao [12] and to the references given there for global Strichartz estimates for the free wave equation and to [18] for some results for perturbations depending only on t. Notice that if q, q˜
< 2(n−1) ˜, q˜, γ are admissible if the following n−3 , then p, q, p conditions hold: n 1 n 1 n + = −γ = + p q 2 p˜ q˜   n−1 1 1 ≤ − p 2 2

− 2, 1 q

 ,

1 ≤ p˜




n−1 2



1 1 − 2 q˜


 .

Theorem 3.1 ([15]) Let the condition (R) be fulfilled and let 1 ≤ p˜, q˜ ≤ 2 ≤ p, q ≤ +∞, 0 ≤ γ ≤ min{1, (n − 1)/2}, p > 2 be admissible for the free wave equation. Moreover, if n is even assume that p˜ < 2. Then for data (f0 , f1 ) ∈ H˙ γ (Rn ), F ∈ Lpt˜(R; Lqx˜ (Rn )) and u(t, x) a solution of (1.1) with τ = 0 we have the estimate uLpt(R; Lqx (Rn )) + u(t, x)H˙ xγ + ∂t u(t, x)H˙ xγ−1 ≤ C(f0 H˙ γ + f1 H˙ γ−1 + F Lp˜(R; Lqx˜ (Rn )) ) t

with a constant C = C(n, p, q, p˜, q˜, γ) > 0 independent of t ∈ R.

(3.2)

274

Vesselin Petkov

Sketch of the proof. The proof is based on the following propositions and to the approach in [5]. Proposition 3.1 ([15]) Assume that (R) is fulfilled and 0 ≤ γ ≤ min{1, (n − 1)/2}. Let (f0 , f1 ) ∈ H˙ γ (Rn ) and let F ∈ L2t (R; H˙ xγ (Rn )) be supported in {x : |x| ≤ R}. Then for every fixed ϕ ∈ C0∞ (Rn ) the solution u(t, x) of (1.1) with τ = 0 satisfies the estimate  ∞ (ϕu(t, x), ϕ∂t u(t, x))2H˙ γ (Rn ) dt −∞

≤ C(n, ϕ, R)(f0 H˙ γ (Rn ) + f1 H˙ γ−1 (Rn ) + F L2(R;H˙ xγ (Rn )) )2 . t

Proposition 3.2 ([19], [15]) Let (p, q, p˜, q˜, γ), f0 , f1 , F be as in Theorem 3.1. Let u0 (t, x) be the solution of (1.1) with τ = 0, V = 0. Then for every ϕ ∈ C0∞ (Rn ) we have  ∞ (ϕu0 (t, x), ϕ∂t u0 (t, x))2H˙ γ (Rn ) dt −∞

≤ C(n, ϕ)(f0 H˙ γ + f1 H˙ γ−1 + F Lp˜(R;Lqx˜ (Rn )) )2 . t

For n odd and 1 ≤ p˜ ≤ 2, Proposition 3.2 has been established in [19]. To obtain the L2 -integrability of the local energy in Proposition 3.1, we use the local energy decay given by Theorem 3.1 and for this purpose we need the condition (R). To prove the estimate (3.2), we write the solution of (1.1) as a sum u = u0 + v, where u0 is the solution of the free problem 2 (∂t − ∆)u0 = F, u0 |t=0 = f0 , ∂t u0 |t=0 = f1 , while v is the solution of the problem with the potential 2 (∂t − ∆ + V )v = −V u0 , v|t=0 = ∂t v|t=0 = 0. Applying Proposition 3.2 for V u0 , we obtain the estimate V u0 L2 (R; H˙ xγ (Rn )) ≤ C0 (f0 H˙ γ + f1 H˙ γ−1 + F Lp˜(R; Lqx˜ (Rn )) ). t

t

(3.3)

In fact, choosing a function β ∈ C0∞ (Rn ) such that β = 1 on suppx V (t, x), we have V (t, x)u0 H˙ xγ (Rn ) ≤ Cγ,V βu0 H˙ xγ (Rn ) . The estimate of u0 Lpt (R; Lqx (Rn )) follows from (3.1). Next we have  v(t, x) = − 0

t

sin((t − s)Λ) (V u0 + V v)(s, x)ds. Λ

Time-periodic perturbations

275

The function V u0 satisfies the estimate (3.3) and by Proposition 3.1 applied to the equation (∂t2 − ∆ + V )v = −V u0 we deduce V u0 + V vL2 (R; H˙ xγ (Rn )) ≤ C1 (f0 H˙ γ + f1 H˙ γ−1 + F Lp˜(R; Lqx˜ (Rn )) ). (3.4) t

t

We wish to show that  t    sin((t − s)Λ)  (V u0 + V v)(s, x)ds   p + q n Λ 0 L (R ; Lx (R )) t

≤ C2 V u0 + V vL2 (R+ ; H˙ xγ (Rn )) .

(3.5)

t

Following the argument of [19], we conclude that the operator T : H˙ −γ (Rn )  g → βe±itΛ g ∈ L2t (R+ ; H˙ x−γ (Rn )) is bounded. The adjoint operator (T ∗ G)(x) =



∞

e∓isΛ βG(s, x)ds

0

is bounded as an operator from L2t (R+ ; H˙ xγ (Rn )) to H˙ xγ (Rn ) and this yields   ∞   ±isΛ  e βh(s, x)(s, x)ds (3.6)  ˙ γ n ≤ C2 hL2t (R+ ; H˙ xγ (Rn )) .  0

H (R )

Consider the integral operators 

t

J : L2t (R+ ; H˙ xγ (Rn ))  h(t, x) −→

K(s, t)h(s, x)ds ∈ Lpt (R+ ; Lqx (Rn )), 0

where K(s, t) = Λ−1 sin((t − s)Λ)β. To apply the Christ–Kiselev lemma [6], it is sufficient to have an estimate for   ∞   sin((t − s)Λ)  βh(s, x)ds  p + q n .  Λ 0 L (R ;Lx (R )) t

By (3.1) and (3.6), we get     ±itΛ −1 ∞ ±isΛ  e Λ e βh(s, x)ds     ≤ C3  

0

0

q + n Lp t (R ; Lx (R ))

 ∞  ±isΛ e βh(s, x)ds  ˙ γ−1 H

(Rn )

≤ C2 C3 hL2 (R+ ; H˙ xγ (Rn )) . t

We take h = V u0 + V v and we use the addition formula for sin((t − s)Λ) to conclude that

276

Vesselin Petkov

   

∞

0

  sin((t − s)Λ) (V u0 + V v)ds  p + q n Λ L (R ;Lx (R )) t

≤ C4 V u0 + V vL2 (R+ ; H˙ xγ (Rn )) .

(3.7)

t

By hypothesis p > 2, and hence an application of the Christ–Kiselev lemma [6] yields immediately (3.5). Consequently, (3.4) implies an estimate for vLpt (R+ ;Lqx (Rn )) and, similarly, we deal with the norm vLpt(R− ;Lqx (Rn )) . To estimate the quantity v(t0 , x)H˙ γ (Rn ) uniformly with respect to t0 , notice x that      ±itΛ −1 t0 ±isΛ  e Λ e (V u + V v)(s, x)ds 0  ˙γ n  0

  ≤ C5  

0

t0

H (R )

  ±isΛ e (V u0 + V v)(s, x)ds  ˙ γ−1 H

(Rn )

with a constant C5 > 0 independent of t0 . As above, we can estimate the right-hand side by V u0 + V vL2 (R; H˙ xγ (Rn )) uniformly with respect to t0 and t apply (3.4). A similar argument works for ∂t v(t0 , x)H˙ γ−1 (Rn ) and the proof x of Theorem 3.1 is complete.

4 Non-trapping moving obstacles Throughout this and the following sections we assume that n is odd. To make a precise definition of non-trapping obstacles we must consider the generalized bicharacteristics of the wave operator  = ∂t2 − ∆x determined as the trajectories of the generalized Hamiltonian flow Fσ in Q related to the symn bol i=1 ξ 2i − τ 2 of  (see [13] for a precise definition). In general, Fσ is not smooth and in some cases there may exist two different integral curves issued from the same point in the phase space. To avoid this situation, we assume that for every (t, x, τ , ξ) ∈ T ∗ (Q) \ {0} the flow Fσ is uniquely determined. To deal with a continuous flow, following [13] we consider the compressed cotangent bundle T˜ ∗ (Q) which for (t, x) ∈ ∂Q can be identified with ∗ Tt,x (Q)/Nt,x (∂Q), ∗ Nt,x (∂Q) being the fiber of the cotangent spaces Tt,x (Q) vanishing on Tt,x (∂Q). ∗ ∗ ˙ ˜ Thus given ρ = (t, x, τ , ξ) ∈ T (Q) \ {0} = T (Q), there exists a unique generalized (compressed) bicharacteristic γ(σ) = (t(σ), x(σ), τ (σ), ξ(σ)) ∈ T˙ ∗ (Q) such that γ(0) = ρ and we define Fσ (ρ) = γ(σ) for all σ ∈ R (see [13]). We obtain a flow Fσ : T˙ ∗ (Q) −→ T˙ ∗ (Q) which is also called generalized geodesic flow on T˙ ∗ (Q). The projections of the compressed generalized bicharacteristics on Q are called generalized geodesics.

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277

Definition 4.1 The obstacle Q is called non-trapping if for each R > R0 there exists TR > 0 such that there are no generalized geodesics of  with length TR lying entirely in Q ∩ {(t, x) : |x| ≤ R}. Let P±b be the orthogonal projections on the orthogonal complements of the Lax–Phillips spaces b = {f ∈ H˙ 1 : U0 (t)f = 0, |x| < ±t + b, ±t > 0}, D±

where U0 (t) is the unitary group introduced in Section 1. Set Z b (T ) = P+b U (T, 0)P−b . Following the general results on propagation of singularities (see [13]), it is not difficult to show that if Q is non-trapping, given a function ϕ ∈ C0∞ (Rnx ) with supp ϕ ⊂ {x : |x| ≤ a}, a ≥ R0 , the operator ϕU (t, 0)P−a : H(0) −→ H(t) for t > 4a + T4a is compact (see [7], [14]). In fact, set M (t, s) = U (t, s) − U0 (t − s) and let Φ ∈ C0∞ (Rn ) be a cut-off such that Φ = 1 for |x| ≤ 3a, Φ = 0 for |x| ≥ 4a. Then for t > 4a + T4a we have ϕU (t, 0)P−a = ϕM (t, t − 2a)ΦU (t − 2a, 2a)ΦM (2a, 0)P−a and the operator on the right-hand side is compact. Next we take a = R0 and by a similar argument choosing kT > 4a + T4a , we deduce that the operator (Z a (T ))k is compact. This implies that the spectrum of the operator Z a (T ) is discrete with finite multiplicity. For b ≥ a we can use the same argument and show that (Z b (T ))m(b) is compact for some integer m(b) ∈ N depending on b. Consequently, the spectrum of Z b (T ) is also discrete and with finite multiplicity. According to [7], the eigenvalues of Z b (T ) and their multiplicities are independent of b. Next, given a cut-off χ ∈ C0∞ (Rn ) such that χ = 1 for |x| ≤ R0 , supp χ ⊂ {x : |x| ≤ b}, b > a, we deduce P±b χ = χ = χP±b . It is clear that for |z| ≥ A  1 we have χ(Z b (T ) − z)−1 χ = χ(U (T ) − z)−1 χ. The left-hand side admits a meromorphic continuation for |z| ≤ A and the same is true for the cut-off resolvent χ(U (T ) − z)−1 χ, hence the poles of χ(U (T )−z)−1χ are between the poles of (Z b (T )−z)−1 which are independent of b. To prove that the poles of χ(U (T ) − z)−1 χ coincide with those of b (Z (T ) − z)−1 , we apply with some modification an argument used in [3] for stationary obstacles. Choose a function ψ ∈ C0∞ (Rn ) so that ψ = 1 for |x| ≤ R0 + 1, ψ = 0 for |x| ≥ R0 + 2 and consider the operator Lψ (g, h) = (0, ∇x ψ, ∇x g + (∆ψ)g). In particular, we define Lψ (U (t, s)f ) and Lψ (U0 (t)f ) and will write simply Lψ U (t, s) and Lψ U0 (t). It is easy to see that we have

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Vesselin Petkov



t

(1 − ψ)U (t, 0) = U0 (t)(1 − ψ) +

U0 (s)Lψ U (t, s)ds,

(4.1)

U (t, s)Lψ U0 (s)ds.

(4.2)

0

 U (t, 0)(1 − ψ) = (1 − ψ)U0 (t) +

t

0

An application of these equalities yields  t U (t, 0) = U (t, 0)ψ + (1 − ψ)U0 (t) + ψU (t, s)Lψ U0 (s)ds 

0 t

U0 (t − s)(1 − ψ)Lψ U0 (s)ds

+ 0

 t

t−s

U0 (τ )Lψ U (t − s, τ )Lψ U0 (s)dτ ds

+ 0

0



= ψU (t, 0)ψ + U0 (t)ψ(1 − ψ) + (1 − ψ)U0 (t) +  +

t

U0 (t − s)(1 − ψ)Lψ U0 (s)ds

U0 (s)Lψ U (t, s)ψds + 0

 t

ψU (t, s)Lψ U0 (s)ds 0



t

t

0 t−s

U0 (τ )Lψ U (t − s, τ )Lψ U0 (s)dτ ds.

+ 0

0

Let g ∈ C0∞ (BR0 +3 ) be a cut-off function equal to 1 on BR0 +2 . We choose the projectors P±b so that P±b ψ = ψ = ψP±b , P±b g = g = gP±b . Next we fix b > 0 and the projectors P±b with these properties and will write P± , Z(T ) instead of P±b , Z b (T ). Note that gLψ = Lψ = Lψ g and let T0 > 0 be chosen so that P+ U0 (t)P− = 0 for t ≥ T0 . For A large enough and z ∈ C, |z| ≥ A, we have (Z(T ) − z)−1 = −

∞


z −j−1 P+ U (jT, 0)P− .

j=0

Now we apply the above representation of U (jT, 0) for P+ U (jT, 0)P− , j ∈ N, and write (Z(T ) − z)−1 = ψ(U (T ) − z)−1 ψ
z −j−1 P+ U0 (jT )ψ(1 − ψ)P− − jt≤T0

−




jT ≤T0

z −j−1 P+ (1 − ψ)U0 (jT )P−

Time-periodic perturbations



T0

+

279

P+ U0 (s)Lψ (U (T ) − z)−1 ΦU (0, s)ψP− ds

0



T0

+

P+ ψ(U (T ) − z)−1 ΦU (0, s)Lψ U0 (s)P− ds

0

−




T0



+ 0

z −j−1 P+ U0 (jT − s)(1 − ψ)Lψ U0 (s)P− ds

0

jT ≤T1



min(jT,T0 )

T0

P+ U0 (τ )Lψ U (−s, 0)Φ(U (T ) − z)−1 Φ

0

× U (0, τ )Lψ U0 (s)P− dτ ds + G(z) with an operator G(z) holomorphic for z = 0. Here Φ is a cut-off function with compact support determined by the finite speed of propagation so that (1 − Φ)U0 (t)g = 0 and (1 − Φ)U (t, τ )g = gU (t, τ )(1 − Φ) = 0 for |t| ≤ 2T0 , 0 ≤ τ ≤ T0 . The terms given by finite sums are holomorphic operators with respect to z = 0. Choose a function Ψ ∈ C0∞ (|x| ≤ c + 1) equal to 1 for |x| ≤ c and fix c > b large enough. Thus we conclude that if Ψ (U (T ) − z)−1 Ψ is analytic in a neighborhood of z0 , 0 < |z0 | < A, the same is true for (Z(T ) − z)−1 , hence Ψ (U (T ) − z)−1 Ψ and (Z(T ) − z)−1 have the same poles. The analysis of the multiplicities of the corresponding poles is more difficult and we refer to [2] for the results in this direction. To study the local energy decay for non-trapping obstacles, we can follow the approach in [7] (see also Chapter 6 in [14]). In fact, assume that Ψ (U (T ) − z)−1 Ψ has no poles z ∈ C, |z| ≥ 1, for a cut-off function Ψ given above. Then choosing b > R0 large enough, we get σ(Z b (T )) ∩ {z ∈ C : |z| ≥ 1} = ∅, where σ(L) denotes the spectrum of the operator L. The same property of σ(Z a (T )) holds for all a ≥ R0 and we deduce Z a (t, s) ≤ Ca e−δa (t−s) , t ≥ s

(4.3)

with Ca > 0, δ a > 0 independent of t and s. Thus given a function f ∈ H(s) with supp f ∈ {|x| ≤ R} and ϕ ∈ C0∞ (Rn ), ϕ = 1 for |x| ≤ R0 , we conclude that ϕU (t, s)f H(t) ≤ C(ϕ, R)e−γ(t−s) f H(s) , t ≥ s with γ > 0 independent of t and s. For this purpose we choose suitably b and apply (4.3) with a = b.

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Finally, to establish the L2 -integrability of the local energy, we exploit (4.2) and using the notation in (4.2), we write  t U (t, 0)f = U (t, 0)ψf + (1 − ψ)U0 (t)f + U (t, s)Lψ U0 (s)ds. The estimate of

'∞ 0

0

ϕU (t, 0)ψf 2H(t) dt is trivial, while for 

∞

0

 t 2    ϕU (t, s)Lψ U0 (s)ds   0

dt

H(t)

we apply Young’s inequality. Thus we obtain  ∞ ϕU (t, 0)f 2H(t) dt ≤ C(ϕ)f 2H(0) . 0

Under the condition that we have no poles z ∈ C with |z| ≥ 1 of the cutoff resolvent, we can obtain Strichartz estimates modifying the arguments of Section 3.

5 Trapping moving obstacles First let us consider a stationary obstacle K(t) = K, ∀t ∈ R and set Ω = Rn \ K. Let U (t) = eitG be the unitary group related to the Dirichlet problem (1.2) in R × Ω and let H = HD (Ω) ⊕ L2 (Ω) be the energy space (see [10]). Let χ ∈ C0∞ (Rn ) be a cut-off function equal to 1 on K and let Rχ (λ) = χ(−∆D − λ2 )−1 χ be the cut-off resolvent of the Dirichlet Laplacian ∆D in Ω which is bounded in L2 (Ω) for Im λ > 0. For non-trapping obstacles K we have the estimate (see for instance, [20]) λRχ (λ)L2 (Ω)→L2 (Ω) ≤ C, ∀λ ∈ R.

(5.1)

On the other hand, the existence of at least one trapped ray leads to the following Proposition 5.1 ([4]) If the generalized compressed Hamiltonian flow Fσ in R × Ω is continuous and if we have at least one (generalized) trapping ray in Ω, then (5.2) sup λRχ (λ)L2 (Ω)→L2 (Ω) = +∞. λ∈R

Proof. Our hypotheses imply the existence of a sequence of ordinary reflecting rays γ n with sojourn times Tγ n → ∞ (see for instance, [13]) and we may apply the result of Ralston [17] which states that we do not have a uniform decay of local energy. On the other hand, according to the results in [22], the uniform decay of the local energy is equivalent to (5.1) and we deduce that the estimate (5.1) fails. Consequently, we get (5.2).

Time-periodic perturbations

281

The existence of one trapping ray γ leads to several results (see [3], [4]) which hold without having any knowledge of the geometry of K outside a small neighborhood of γ. In particular, we are interested in the analytic properties of the cut-off resolvent of the monodromy operator U (T ) introduced in Section 1. Since a stationary obstacle K is periodic with period every t > 0, it is natural to study the analytic properties of the cut-off resolvent Ψ (U (t) − z)−1 Ψ with Ψ ∈ C0∞ (|x| ≤ c + 1), Ψ = 1 for |x| ≤ c, where c > R0 is large and fixed. For trapping obstacles we cannot obtain a meromorphic continuation across the unit circle S1 and we have the following Theorem 5.1 ([3]) Assume the obstacle K stationary and the condition (5.2) fulfilled. Then for almost all t ∈ R+ and all z0 ∈ S1 we have lim

z→z0 , |z|>1

Ψ (U (t) − z)−1 Ψ H→H = +∞.

The proof is based on the following idea. Taking b ≥ c + 1, we have P±b Ψ = Ψ = Ψ P±b , where P±b have been introduced in the previous section. Consider the Lax–Phillips semigroup Z b (t) = P+b U (t)P−b . We fix b with the above property and for simplicity of notation we write Z(t) instead of Z b (t). Let B be the generator of Z(t), that is Z(t) = etB . Therefore, it is easy to see that the condition (5.2) implies sup (iB − λ)−1 H→H = +∞.

λ∈R

By applying a result of I. Herbst [9], we deduce that for almost all t ∈ R+ we have the inclusion (5.3) S1 ⊂ σ(Z(t)). Next we obtain a representation of (Z(t) − z)−1 , |z| > 1, as a sum of terms involving the cut-off resolvent Ψ

∞


z −j−1 U (jt)Ψ = −Ψ (U (t) − z)−1 Ψ

j=0

as we have done this for the operator Z(T ) and the propagator U (T, 0) in the previous section. Consequently, if the norm of Ψ (U (t) − z)−1 Ψ has a limit as z → z0 ∈ S1 , |z| > 1, we obtain a contradiction to (5.3). Passing to trapping moving obstacles, introduce the normal speed of ∂Q by ν t (z) ν x (z) v(z) = . |ν x (z)| |ν x (z)| Given a point z = (t, x) ∈ ∂Q, and a bicharacteristic γ = (t(σ), x(σ), τ (σ), ξ(σ)) ∈ T ∗ (Q)

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Vesselin Petkov

reflecting at z, denote the incident direction of γ by direction by

−ξ r τr

with |ξ i | = 2

τ 2i ,

|ξ r | = 2

τ 2r .

−ξ i τi

and the reflecting

Then τ r = µ(z)τ i and

(1 − 2|v(z)| cos ϕ + |v(z)|2 ) > 0, (1 − |v(z)|2 )−1

µ(z) =

i where 0 ≤ ϕ ≤ π is the angle between −ξ τ i and v(z). We say that a bicharacter∗ ˙ istic (ray) γ issued from (s, y, τ , η) ∈ T (Q) with infinite number of reflection points zj ∈ ∂Q, j ∈ N, at times tj → ∞ is δ-trapping if 3 µ(zj ) ≥ Ceδt , t ∈ [0, ∞], δ > 0. (5.4)

0≤tj ≤t

It turns out that for stationary obstacles we have always µ(z) = 1 and the existence of δ-trapping rays is possible only for trapping moving obstacles. Next we consider an example examined by Popov and Rangelov. Example 5.1 (see [16]) Let K(t) = O1 ∪ O2 (t), O1 ∩ O2 (t) = ∅, O2 (t + T ) = O2 (t), ∀t ∈ R. Suppose that for all t the obstacles O1 and O2 (t) are strictly convex and set d(t) = dist (O1 , O2 (t)), d1 = min d(t), d2 = max d(t). Assume that the obstacle K(t) and its exterior normal satisfy the hypothesis in Section 1 and the conditions: (i) d1 < T /2 < d2 , (ii) there exists y1 ∈ ∂O1 and y2 (t) ∈ ∂O2 (t) so that d(t) = |y1 − y2 (t)|, ∀t ∈ R, (iii) the normal speed v(t, y2 (t)) of O2 (t) vanishes only if d(t) = di , i = 1, 2. We have |d
(t)| < 1 and by our assumptions there exists s0 > 0 so that d(s0 ) = T /2, d
(s0 ) < 0. We choose s < s0 and set y = y2 (s0 ) + (s − s0 )ω, ω = y2 (t)−y1 |y2 (t)−y1 | . The bicharacteristic γ(σ) = (t(σ), x(σ), τ (σ), ξ(σ)) issued from (s, y, 1 − ω) has an infinite number of reflections at zk = (tk , xk ), k ∈ N, with tk = s0 + (k − 1)T /2, x2k−1 = y2 (s0 ), x2k = y1 and µ(z2k ) = 1, µ(z2k+1 ) =

1 + |d
(s0 )| > 1. 1 − |d
(s0 )|

Moreover, γ(σ) is δ-trapping with δ=

1 (ln(1 + |d
(s0 )|) − ln(1 − |d
(s0 )|)) > 0. T

Time-periodic perturbations

283

The following general result of Popov and Rangelov leading to solutions with exponentially growing local energy can be considered as a generalization of that of Ralston [17] for stationary obstacles. Theorem 5.2 ([16]) Assume that there exists a δ-trapping bicharacteristic γ(σ) issued from (s, y, τ , η) ∈ T˙ ∗ (Q). Then for every neighborhood W of y in Ω(s) and every 0 < ! < δ there exists f = (f0 , f1 ) ∈ H(s) with supp f ⊂ W so that for R ≥ R0 + T we have U (t + s, s)f HΩ(t+s)∩{|x|≤R} ≥ C(!, s, f )et , t ∈ [s, ∞[,

(5.5)

.HΩ(t+s)∩{|x|≤R} being the energy norm over Ω(t + s) ∩ {|x| ≤ R}. In particular, the above result shows that if we have a δ-trapping bicharacteristic γ(σ), then the spectral radius of Z b (T ) = P+b U (T, 0)P−b for b > R0 + T is greater than or equal to eδT . Following the argument of the previous section, we may compare the analytic singularities of (Z b (T ) − z)−1 and those of the cut-off resolvent Ψ (U (T ) − z)−1 Ψ, where Ψ ∈ C0∞ (|x| ≤ c + 1) and c > R0 is large enough and fixed. Theorem 5.3 Under the hypothesis of Theorem 5.2, for every 0 < ! < δ the cut-off resolvent of the monodromy operator Ψ (U (T ) − z)−1 Ψ does not have an analytic continuation from {z ∈ C : |z| ≥ A  1} to {z ∈ C : eT ≤ |z| ≤ A}. The analysis of the spectrum of Z(T ) = Z b (T ) for |z| > 1 is an open problem. We conjecture that the existence of a δ-trapping bicharacteristic implies that (Z(T ) − z)−1 does not have a meromorphic continuation in {z ∈ C : eT ≤ |z| ≤ A}, 0 < ! < δ. More precisely, we expect that the continuous spectrum of the operator Z(T ) is not empty. In this direction it is interesting to note that for two strictly convex disjoint stationary obstacles Ki , i = 1, 2, for almost all t ∈ R+ we have the inclusion (5.3). In fact, a much stronger result holds. Theorem 5.4 ([4]) Let K = K1 ∪ K2 , where Ki , i = 1, 2, are strictly convex and disjoint and let Ω = Rn \ K. Consider the semigroup Z b (t) = P+b U (t)P−b , b > R0 , where U (t) is the unitary group related to the Dirichlet problem (1.3) in R × Ω. Then, for almost all t ∈ R+ , we have {z ∈ C : |z| ≤ 1} = σ(Z b (t)).

(5.6)

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References 1. A. Bachelot and V. Petkov, Existence des op´ erateurs d’ondes pour les syst`emes hyperboliques avec un potentiel p´eriodique en temps, Ann. Inst. H. Poincar´e, Phys. Th´eor. 47(1987), 383–428. 2. J.-F. Bony and V. Petkov, Resonances for non-trapping time-periodic perturbations, J. Phys. A 37(2004), 9439–9449. 3. J.-F. Bony and V. Petkov, Resolvent estimates and local energy decay of hyperbolic equations, Around Hyperbolic Systems, Conference in memory of Stefano Benvenuti, Ferrara 2005, to appear in Annali Universita di Ferrara, Sec. VII – Sci. Math. (2006), Springer. 4. J.-F. Bony and V. Petkov, Estimates for the cut-off resolvent of the Laplacian for trapping obstacles, Expos´e S´eminaire EDP, 2005–2006, Centre de ´ Math´ematiques, Ecole Polytechnique. 5. N. Burq, Global Strichartz estimates for non-trapping geometries: about an article by H. F. Smith and C. D. Sogge, Comm. Partial Differential Equations 28(2003), 1675–1683. 6. M. Christ and A. Kiselev, Maximal functions associated to filtrations, J. Funct. Anal. 179(2001), 409–425. 7. J. Cooper and W. Strauss, Scattering of waves by periodically moving bodies, J. Funct. Anal. 47(1982), 180–229. 8. A. Galtbayar, A. Jensen and K. Yajima, Local time-decay of solutions to Schr¨ odinger equations with time-periodic potentials, J. Statist. Phys. 116(2004), 231–281. 9. I. Herbst, Contraction semigroups and the spectrum of A1 ⊗ I + I ⊗ A2 , J. Operator Theory 7(1982), 61–78. 10. P. D. Lax and R. S. Phillips, Scattering Theory, 2nd Edition, Academic Press, New York, 1989. 11. H. Lindblad and C. D. Sogge, On existence and scattering with minimal regularity for semilinear wave equation, J. Funct. Anal. 130(1995), 357–426. 12. M. Keel and T. Tao, Endpoint Strichartz Estimates, Amer. J. Math. 120(1998), 955–980. ¨ strand, Singularities of boundary value problems, 13. R. Melrose and J. Sjo Comm. Pure Appl. Math. I, 31(1978), 593–617, II, 35(1982), 129–168. 14. V. Petkov, Scattering Theory for Hyperbolic Operators, North Holland, Amsterdam, 1989. 15. V. Petkov, Global Strichartz estimates for the wave equation with time-periodic potentials, J. Funct. Anal. 235(2006), 357–376. 16. G. Popov and Tz. Rangelov, Exponential growth of the local energy for moving obstacles, Osaka J. Math. 26(1989), 881–895. 17. J. Ralston, Solutions of the wave equation with localized energy, Comm. Pure Appl. Math. 22(1969), 807–823. 18. M. Reissig and K. Yagdjian, Lp − Lq estimates for the solutions of strictly hyperbolic equations of second order with increasing in time coefficients, Math. Nachr. 214(2000), 71–104. 19. H. F. Smith and C. Sogge, Global Strichartz estimates for non-trapping perturbations of the Laplacian, Comm. Partial Differential Equations 25(2000), 2171–2183. 20. B. R. Vainberg, Asymptotic methods in equations of mathematical physics, Gordon and Breach, New York, 1989.

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21. B. R. Vainberg, On the local energy of solutions of exterior mixed problems that are periodic with respect to t, (Russian), Trudy Moskov. Mat. Obshch. 54(1992), 213–242, 279; translation in Trans. Moscow Math. Soc. 1993, 191–216. 22. G. Vodev, On the uniform decay of local energy, Serdica Math. J. 25(1999), 191–206.

An elementary proof of Fedi˘ı’s theorem and extensions David S. Tartakoff Department of Mathematics, University of Illinois at Chicago

Summary. We present an elementary, L2 , proof of Fedi˘ı’s theorem on arbitrary (e.g., infinite order) degeneracy and extensions. In particular, the proof allows and shows C ∞ , Gevrey, and real analytic hypoellipticity, and allows the coefficents to depend on the remaining variable as well.

2000 Mathematics Subject Classification: 35H10, 35B45, 35B65, 35H20. Key words: Hypoellipticity, infinite order degeneracy, sum of squares.

1 Introduction In 1971, V.S. Fedi˘ı [Fed71] proved local hypoellipticity for the operator Dx2 + a2 (x)Dt2 where a(x) ≥ 0, and a(x) = 0 for x = 0. Related and more recent results include those of Kusuoko and Strook [KuStr85], Morimoto [Mori87], Christ [Christ95] and Bell and Mohammed [BellMo95]. Here, thanks in part to helpful conversations with A. Bove, we will give a flexible and utterly elementary proof of Fedi˘ı’s result which proves hypoellipticity in the smooth, Gevrey, and real analytic categories rapidly, when appropriate. Theorem 1.1 Let a(x) have the above properties and b(t) be a smooth (resp. real analytic) non-zero function of t near t0 . Then the operator P = Dx2 + a2 (x)b2 (t)Dt2 = X 2 + Y 2 is hypoelliptic at (0, t0 ) in the C ∞ , Gevrey, and real analytic categories, assuming, of course, that the coefficients belong to that class.

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David S. Tartakoff

2 Proof of the theorem We make a few preliminary observations. First, for x = 0, the operator is elliptic, where the results are known. Thus our localization will be assumed to be in a neighborhood of x = 0 and the associated localizing function(s) may be taken to depend on t alone, since using a product of a cut-off in x as well would only clutter up the notation, and whenever such a function received a derivative, we would be thrown into the elliptic region. Second, we will estimate derivatives of a solution u in L2 norm, using the Sobolev embedding theorem. Third, using the pseudodifferential calculus and microlocalizing in the standard ways, we shall demonstrate only that derivatives in the variable t grow as desired. The restrictions of this microlocalization are that if a(x) belongs to a given differentiability class, then we will be able to prove hypoellipticity in that class (in x,) but, as we will see below, the regularity in t will be limited only by that of the coefficient b(t). Fourth, taking all inner products in L2 , and using the identity 1 = Dx x we have, for smooth v supported near x = 0, v2L2 = |((Dx x)v, v)| ≤ |(xDx v, v)| + |(Dx v, xv)| ≤

1 1 v2L2 + CDx v2L2 ≤ v2L2 + CDx v2L2 + CabDt v2L2 2 2

≤

3 v2L2 + C  |(P v, v)| 4

so that we have the following a priori inequality (in L2 norms) for v of small x-support: v2 + Dx v2 + abDt v2 = v2 + Xv2 + Y v2
|(P v, v)|. It is important to note that the estimate is not subelliptic in the usual sense (which would require v2ε on the left), and of course this corresponds to the fact that for general a(x), which may degenerate to infinite order at x = 0, H¨ormander’s bracket condition may be violated. We will concentrate on the analytic hypoellipticity of P, assuming the solution is already smooth; showing that a distribution solution is smooth can be accomplished by introducing a cut-off function and a mollifier and observing that any brackets with P are rapidly handled by using a weighted Schwarz inequality and maximality of the estimate. We shall see more of this below as we handle a solution u known to be smooth. To explore high derivatives, we start with powers of Dt , localized by a function ϕ(t) (see above). We have, in L2 norms and inner product, since ϕx = 0 near the point in question,

Fedi˘ı’s theorem and extensions

289

ϕDtr u2 + Dx ϕDtr u2 + abDt ϕDtr u2

(∗ϕDtr ) :

≤ |(P ϕDtr u, ϕDtr u)| which can be estimated ≤ |(ϕDtr P u, ϕDtr u)| + |([P, ϕDtr ]u, ϕDtr u)| ≤ |(ϕDtr P u, ϕDtr u)| + |([Y 2 , ϕDtr ]u, ϕDtr u)| ≤ Cε ϕDtr P u2 + εϕDtr u + 2|([Y, ϕDtr ]u, Y ∗ ϕDtr u)| + |([Y, [Y, ϕDtr ]]u, ϕDtr u)|. Now Y ∗ ϕDtr u2 may be added to the left side of the inequality for |x| small, since Y ∗ = −Y − ab and ab will be small for |x| small, and [Y, ϕDtr ] = abϕt Dtr − ϕa[Dtr , b]Dt = abϕt Dtr − rϕab Dtr + · · · , [Y, [Y, ϕDtr ]] = [abDt , abϕt Dtr − rϕab Dtr + · · · ] = ababϕtt Dtr − rab abϕt Dtr − rabab ϕDtr − r2 ab ab ϕDtr + · · · . Now since b = 0, b or b can be estimated by b. And modulo terms with one
fewer Dt and one additional derivative on ϕ or b, we may move one ab( ) Dt to the right-hand side in the inner product and estimate it by a Y. That is, including Y ∗ ϕDtr u2 in (∗ϕDtr ), |([Y, ϕDtr ]u, Y ∗ ϕDtr u)|
|(abDt ϕ Dtr−1 u, ϕDtr u)| + r|(ab Dt ϕDtr−1 u, ϕDtr u)| + · · ·


1 (∗ϕDtr ) + Cε (∗ϕt Dr−1 ) + r2 (∗ϕDr−1 ) + · · · t t 2

and |([Y, [Y, ϕDtr ]]u, ϕDtr u)|
|(ababϕtt Dtr u, ϕDtr u)| + r|(ab abϕt Dtr u, ϕDtr u)| + r|(abab ϕDtr u, ϕDtr u)| + r2 |(ab ab ϕDtr u, ϕDtr u)| + · · ·


1 (∗ϕDtr ) + Cε (∗ϕtt Dr−2 ) + Cε r2 (∗ϕt Dr−2 ) + Cε r4 (∗ϕDr−2 ) + · · · t t t 2

or, in all, (∗ϕDtr )
(∗ϕt Dr−1 ) + (∗ϕtt Dr−2 ) + r2 (∗ϕt Dr−2 ) + r4 (∗ϕDr−2 ) + · · · t

t

t

t

where under · · · we include terms where we must move one Dt across a ϕ, thus increasing the number of derivatives on ϕ by one but decreasing r by one.

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David S. Tartakoff

All of this may be iterated until we have C r terms each with r reduced to zero and at most r derivatives on the localizing function ϕ(t). The result is hypoellipticity in (x, t) in the appropriate spaces. Remark 2.1 We have not emphasized the C ∞ hypoellipticity of P. In the case of b(t) ≡ 1, as in the paper of Kohn [Koh05], one may introduce a pseudodifferential cut-off in the variable τ dual to t which is equal to one for |τ | ≤ N and then smoothly to zero by the time |τ | ≥ 2N, and, since the resulting function is smooth in t, apply the a priori estimates and derivatives, then let N → ∞ to see that the corresponding norms are finite. When the coefficient b(t) is not constant, one must introduce a mollifier in the variable t, treat the brackets of functions with the mollifier as in the classical works of Friedrichs, H¨ ormander and others, and then let the mollifier approach the identity. Note that it is important here that b(t) is never zero. Remark 2.2 When one works in the real analytic category, the localizing function ϕ(t) must be taken to belong to the Ehrenpreis class: ϕ(t) is the convolution of N identical bump functions with derivative proportional to N with the characteristic function of an intermediate set. Such a function will depend on N but have the property that, with C independent of N, ϕ = ϕN ≡ 1 on I0 , ϕ ∈ C0∞ (I2 ), and |Dk ϕ| ≤ C k+1 N k ,

k ≤ N.

This is enough to prove analyticity (when the coefficients are analytic).

References [BellMo95] D. Bell and S. Mohammed An extension of H¨ ormander’s theorem for infinitely degenerate differential operators, Duke Math. J. 78(1995), 453–475. [Christ95] M. Christ Hypoellipticity in the infinitely degenerate regime, Complex analysis and geometry, de Gruyter, Berlin, New York. [Fed71] V. S. Fedi˘i, On a criterion for hypoellipticity, Math. USSR Sb. 14(1971), 14–45. [Koh05] J. J. Kohn, Hypoellipticity and loss of derivatives, Ann. Math. 162 (2005), 943–986. [KuStr85] S. Kusuoka and D. Strook, Applications of the Malliavin calculus II, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 32(1985), 1–76. [Mori87] Y. Morimoto, Hypoellipticity for infinitely degenerate elliptic operators, Osaka J. Math. 24(1987), 13–35.

Outgoing parametrices and global Strichartz estimates for Schr¨ odinger equations with variable coefficients Daniel Tataru∗ Department of Mathematics, University of California, Berkeley, USA Summary. In these notes we discuss recent results concerning the long time evolution for variable coefficient time dependent Schr¨ odinger evolutions in Rn . Precisely, we use phase space methods to construct global in time outgoing parametrices and to prove Strichartz type estimates. This is done in the context of C 2 metrics which satisfy a weak asymptotic flatness condition at infinity.

2000 Mathematics Subject Classification: 81Q05, 35A17, 35S10. Key words: Schr¨ odinger equations, outgoing parametrices, Strichartz estimates, phase space transforms.

1 Introduction Consider first solutions to the homogeneous Schr¨ odinger equation in R × Rn (i∂t − ∆)u = 0

u(0) = u0 .

Their energy is preserved, u(t)L2 = u(0)L2 . At the same time there is uniform decay for spatially localized initial data, u(t)L∞
t− 2 u(0)L1 . n

(1.1)

This can be viewed as a consequence of uniform bounds for the fundamental solution, x2

K(t, x) = cn t− 2 ei 4t . n

∗

The author was partially supported by NSF grants DMS0354539 and DMS 0301122 and also by MSRI for Fall 2005

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Daniel Tataru

From (1.1) one can also obtain time averaged decay estimates for merely L2 initial data. These are called Strichartz estimates, and have the form uLp(Lq )
∇u0 L2 .

(1.2)

This holds for all pairs (p, q) satisfying the relations 2 ≤ p ≤ ∞, 2 ≤ q ≤ ∞ and n 2 n + ≤ (1.3) p q 2 with the exception of the endpoint (2, ∞) for n = 2. In the sequel such pairs are called Strichartz pairs. A consequence of (1.2) is an estimate for solutions to the inhomogeneous problem (i∂t − ∆)u = f u(0) = 0 ut (0) = 0, namely uLp(Lq )
f L1 (L2 ) .

(1.4)

The simplest case of (1.4) is the well-known energy estimate ∇uL∞ (L2 ) ≤ f L1 (L2 ) .

(1.5)

However, there is a larger family of estimates for solutions to the inhomogeneous wave equation where we also vary the norms in the right-hand side, uLp(0,T ;Lq )
f Lp
1 (Lq1
) .

(1.6)

This holds for all Strichartz pairs (p, q), (p1 , q1 ). For more information  2nwe refer the reader to the expository article [6]. The endpoint (p, q) = 2, n−2 was obtained later in [8] (n ≥ 3). In this article we are interested in the variable coefficient case of these estimates, where we replace −∆ by a second order elliptic operator of the form A(t, x, D) = Di aij (t, x)Dj . Thus we consider evolutions of the form Pu = f

u(0) = u0

(1.7)

where P = Dt + A(t, x, D). This is a considerably more delicate problem, which has several new features tied to the nontrivial behavior of its Hamilton flow. The first of these is that dispersive estimates such as (1.1) do not hold in general, even if we restrict ourselves to coefficients aij which are sufficiently small smooth compactly supported perturbations of the identity. This is because even a small perturbation of the identity suffices in order to refocus

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a group of Hamilton flow rays originating at the same point. This produces some caustics-like concentration for the fundamental solution. A second feature is related to the long time behavior of the bicharacteristics. In the flat case, all bicharacteristics are straight so they escape to infinity both forward and backward in time. However, in the variable coefficient case it is possible to have trapped rays, which are confined to a bounded spatial region. These correspond to singularities which are largely confined to a bounded region, and destroy not only the dispersive estimates (1.1), but also the Strichartz estimates in (1.2). On the positive side, the existence of trapped rays is a more stable phenomena; in particular, it cannot happen for small perturbations of the identity. The first work in this direction [13] considers the case of a C 2 compactly supported perturbation of the identity, subject to a nontrapping condition. Then Strichartz estimates are proved locally in time. An essential part of the argument is to take advantage of the local smoothing estimates for variable coefficient Schr¨ odinger equations. These allow one to stably split the estimates in two, one part which is localized to a compact set and another which lives on a flat background. In the simplest form (see [3]) they are stated as 1

1

 x − 2 + D 2 uL2 ([0,1]×Rn )
u(0)L2 . Hence they give a gain of 1/2 derivative within a compact spatial region. Heuristically, this is a reflection of the fact that waves with high frequency λ move at high speed O(λ) and thus spend a short time O(λ−1 ) within a bounded spatial region. Square averaging in time, one then obtains the half 1 derivative gain λ− 2 . The results in [13] are based on a phase space analysis of the spatially localized part of the Schr¨ odinger waves, following earlier work of Smith [12] and the author [14], [15] on the similar problem for the wave equation. In the meantime this type of local analysis has been recast in a semiclassical language in [1], which further considered various properties of Schr¨ odinger evolutions on compact manifolds. Simplified presentations of localized wave packet type parametrix constructions are now available in [9], [17]. These apply to evolutions of the form (Dt + aw (t, x, D))u = 0,

u(0) = u0

0 on the unit time scale, for symbols a which satisfy a partial S00 type condition

|∂xα ∂xβ a(t, x, ξ)| ≤ cαβ ,

|α| + |β| ≥ 2.

These parametrices are often useful in rescaled forms. However due to their finite time horizon they cannot be directly applied to obtain optimal results for metrics which are not compactly supported perturbations of the identity. More recently, two versions of parametrix constructions have been obtained for metrics which are asymptotically flat; both imply local in time Strichartz estimates.

294

Daniel Tataru

Robbiano and Zuily [10] consider smooth asymptotically flat metrics in Rn of the short range type and which satisfy a nontrapping assumption. Their approach uses a parametrix which is a Fourier integral operator with complex phase and relies considerably on Sj¨ ostrand’s theory of the FBI transform. Hassell–Tao–Wunsch [7] instead have a more direct parametrix construction emulating the model of the constant coefficient fundamental solution. A sharper version of the localized energy estimates is then used to control the errors. Their setup is of smooth asymptotically conic manifolds with short range scattering metrics, extended shortly afterward to long range scattering metrics. In the present article we consider global in time parametrices and Strichartz estimates for metrics in Rn which are merely of class C 2 and which are asymptotically flat only in a very weak sense. Due to the global nature of the result it is convenient to consider scale invariant assumptions on the coefficients. Such a scale invariant assumption is |a − In | + |a−1 − In | + |x||∂x a(x, t)| + |x|2 (|∂x2 a(x, t)| + |∂t a(x, t)|) ≤ C. If C is small this prevents trapping, but some heuristic computations seem to indicate that the sharp pointwise decay of outgoing waves may fail because of repeated caustics formation along geodesics. Hence it is conceivable that one might be able to construct solutions which are localized along certain geodesics for a long time. Thus we are led to introduce a slightly stronger assumption, namely
sup |x|2 (|∂x2 a(t, x)| + |∂t a(x, t)|) + |x||∂x a(t, x)| + |a(t, x) − In | ≤ ! (1.8) j∈Z

Aj

where Aj is the dyadic region Aj = R × {2j ≤ |x| ≤ 2j+1 }. If ! is small enough, then this precludes the existence of trapped rays, while for arbitrary ! it restricts the trapped rays to finitely many dyadic regions. Because of the reduced coefficient regularity for small x, it seems virtually impossible to control the Hamilton flow and to construct parametrices along incoming rays, i.e., which approach the origin. However, the situation improves considerably in the case of outgoing rays. Thus the main part of the article is devoted to an outgoing parametrix construction. This suffices in order to capture the full behavior of the Schr¨ odinger equation due to the nontrapping assumption, which guarantees that each ray can be split into two parts, one of which is outgoing forward in time while the other is outgoing backward in time. Our parametrix construction is based on the use of a time dependent FBI transform. However we do not use Sj¨ ostrand’s theory [11]. Instead, we take advantage of the simpler approach introduced by the author in [14], [15], [16],

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[17]; the latter is recommended to the reader as a good starting point. In this analysis the FBI transform is used to turn the equation into a degenerate parabolic evolution in the phase space. Bounds for this evolution are then obtained using the maximum principle. For more information about phase space transforms we refer to [5] and [2]. One of the main starting points in the phase space analysis of PDEs is Fefferman’s article [4]. Even though our parametrix is very precise, there are still errors which need to be controlled and this is done using localized energy estimates, otherwise known as local smoothing estimates. We prove such estimates in the case when the parameter ! in (1.8) is sufficiently small. If ! is large, then nontrapping may fail, and thus the localized energy estimates may fail. With a nontrapping assumption it is likely that the localized energy estimates hold locally in time, but it is not clear what happens globally in time. To avoid being distracted from the main purpose of this paper we have decided to brush aside this problem and simply use the localized energy estimates as an assumption for large !. Scaling plays an essential role in our analysis. Modulo rescaling and Littlewood–Paley theory, all our analysis is reduced to waves which have fixed frequency of size O(1). Such waves have a propagation speed of size O(1), therefore our study of outgoing waves can be largely localized to cones of the form {|x| ≈ |t|}. Certainly the exact flow cannot have a precise localization of this type due to the uncertainty principle. To compensate for this, we introduce an artificial damping term which produces rapid decay of waves that do not have the above localization. This allows us to restrict our attention to the above cone modulo rapidly decreasing errors. Before we state our main results we need to introduce the function spaces for the localized energy estimates. We consider a dyadic partition of unity in frequency, ∞
Sk (D) 1= k=−∞

and for each k ∈ Z we measure functions of frequency 2k using the norm 1

uXk = 2k uL2(A<−k ) + 2 2 sup (|x| + 2−k )− 2 uL2 (Aj ) k

j≥−k

where A<j = R × {|x| ≤ 2j }. To measure the regularity of solutions to the Schr¨ odinger equation we use the global localized energy space X defined by the norm u2X

=

∞
k=−∞

Sk u2Xk .

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Daniel Tataru

If n ≥ 3, then one can think of this as a space of distributions, and the following Hardy type inequality holds: |x|−1 uL2
uX .

(1.9)

On the other hand if n = 1, 2 one has a BMO type structure, i.e., X is a space of distributions modulo time dependent constants. For the inhomogeneous term in the equation, on the other hand, we use the dual space X
with norm f 2X
=

∞


Sk f 2X
. k

k=−∞

Following the discussion above, if n ≥ 3 then X
is dense in S
(Rn ) and uX

xuL2 . If n = 1, 2 then functions in X
must satisfy the cancellation condition  f (x)dx = 0 Rn

Definition 1.1 We say that the operator P satisfies the localized energy estimates if for each initial data u0 ∈ L2 and each inhomogeneous term f ∈ L1 L2 ∩ X
there exists a unique solution u to (1.7) which satisfies the bound (1.10) uL∞ L2 ∩X
u0 L2 + f L1 L2 +X
. The localized energy estimates hold under the assumption that the coefficients aij are a small perturbation of the identity: Theorem 1.1 Assume that the coefficients aij satisfy (1.8) with an ! which is sufficiently small. Then the operator P satisfies the localized energy estimates globally in time. This leads to our main scale invariant Strichartz estimate: Theorem 1.2 Assume that the coefficients aij satisfy (1.8) with an ! which is sufficiently small. Let (p1 , q1 ) and (p2 , q2 ) be two Strichartz pairs. Then the solution u to (1.7) satisfies uLp1 (Lq1 )∩X
u0 L2 + f Lp
2 (Lq2
)+X
.

(1.11)

If ! is large, then any localized energy estimates require an additional nontrapping condition. Even then the nontrapping can at most guarantee local in time bounds. However, we can still prove a conditional result:

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Theorem 1.3 a) Let −∞ ≤ T − < T + ≤ ∞. Assume that the coefficients aij satisfy (1.8) in (T − , T + ). Then for every Strichartz pair (p, q) we have uLpLq
uX + P uX
.

(1.12)

In addition there is a parametrix K for P which satisfies Kf Lp1 Lq1 ∩X + (P K − I)f X

f Lp
2 (Lq2
)

(1.13)

for any two Strichartz pairs (p1 , q1 ) and (p2 , q2 ). b) Assume that, in addition, the operator P satisfies the localized energy estimates in [T − , T + ]. Then the solution u to (1.7) satisfies the full Strichartz estimates in (1.11). Finally, suppose we add first and zero order terms to P , P = Dt + Di aij Dj + bi Di + c. Consistent with (1.8) we introduce the following condition on the coefficients b and c:
sup |x|2 |∂x b(t, x)| + |x||b(t, x)| ≤ ! (1.14) j∈Z Aj

sup |x|2 |c(t, x)| ≤ !.

(1.15)

R×Rn

Then we have Remark 1.1 If n ≥ 3 and b, c satisfy (1.14), (1.15) then the following estimate holds: (bi Di + c)uX

!uX . (1.16) Consequently the results in Theorems 1.1, 1.2, 1.3 remain valid when such lower order terms are added to P . If n = 1, 2 then (1.16) cannot hold for any nonzero b, c due to the BMO type structure of X. One can still add lower order terms to the equation but these must have some decay in time if one is to be able to take advantage of the dispersive estimates.

2 Outline of the proofs 2.1 Notation We consider a smooth spatial Littlewood–Paley decomposition 1=

∞
j=−∞

χj (x)

supp χj ⊂ {2j−1 < |x| < 2j+1 }.

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Daniel Tataru

We also set χ<j =




χk .

k<j

We consider a frequency Littlewood–Paley decomposition 1=

∞


Sj (D)

j=−∞

where supp sj ⊂ {2j−1 < |ξ| < 2j+1 }. We also use the related notation Sk , etc. We say that a function f is localized at frequency 2k if fˆ is supported in j−1 {2 < |ξ| < 2j+1 }. An operator K is localized at frequency 2k if for any f localized at frequency 2k its image Kf is frequency localized in {2j−10 < |ξ| < 2j+10 }. 2.2 The paradifferential calculus The first step in the proof is to mollify the coefficients in order to reduce our results to a frequency localized context. Given a frequency scale 2k we define the regularized coefficients
S
which are roughly localized at frequencies 1

|ξ|  2k (1 + 2k |x|)− 2 . Correspondingly, we define the mollified operators A(k) = Di aij (k) Dj which are used on functions of frequency 2k , as well as a global mollified operator ∞
A˜ = A(k) Sk . k=−∞

The operator A(k) maps functions of frequency 2k to functions of frequency 2 . The operator A˜ maps functions of frequency 2k to functions of frequency 2k for all integers k. The following result shows that we can freely replace A by A˜ in Theorems 1.1, 1.3(a). It also shows that at frequency 2k the operators A˜ and A(k) are interchangeable. k

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Proposition 2.1 Assume that the coefficients aij satisfy (1.8). Then the following estimates hold: ˜ X

!uX (A − A)u

(2.2.1)

(A˜ − A(k) )Sl uXk

!Sl uXk ,

|l − k| ≤ 2

[A(k) , Sk ]uXk

!uXk .

(2.2.2) (2.2.3)

2.3 Localized energy estimates Here we outline the proof of Theorem 1.1. Using a Littlewood–Paley decomposition the problem reduces to a frequency localized context with A replaced by A(k) and u localized at frequency 2k . By rescaling we can take k = 0. Let (αm )m∈N be a positive slowly varying sequence with αm = 1. Correspondingly, we define the space X0,α with norm
1 u2X0,α = αj (|x| + 1)− 2 u2L2 (Aj ) j>0

and the dual space = u2X0,α





1

2 2 α−1 j (|x| + 1) uL2 (Aj ) .

j>0

Then the frequency localized version of Theorem 1.1 follows by optimizing with respect to α in the following: Proposition 2.2 Assume that ! is sufficiently small. Then the bound
uL∞L2 ∩X0,α
u(0)L2 + (Dt + A(0) )uL1 L2 +X0,α

(2.3.1)

holds for all functions u ∈ L∞ L2 ∩ X0,α localized at frequency 1 uniformly with respect to all slowly varying sequences (αm ) as above. The proposition is proved using a positive commutator method. Let Q be an L2 bounded selfadjoint operator in Rn . Then the operator C = i[A(0) , Q] is also selfadjoint and we have 2Im A(0) u, Qu = Cu, u . For the Schr¨ odinger equation we obtain d u, Qu = −2 (Dt + A(0) )u, Qu + Cu, u . dt

300

Daniel Tataru

When Q = I this gives the energy estimate d u2L2 = −2 (Dt + A(0) )u, u . dt If δ is a small parameter, then the modified energy E(u) = u2L2 − δ u, Qu

is positive definite and satisfies d E(u) = 2 (Dt + A(0) )u, (1 − δQ)u − δ Cu, u . dt Integrating in time we obtain u2L∞L2 + δ Cu, u



u(0)2L2 + (Dt + A(0) )uL1 L2 +X0,α (1 − δQ)uL∞ L2 ∩X0,α .

Then the conclusion of the proposition follows from the Cauchy–Schwartz inequality provided Q is chosen so that for all functions u localized at frequency 1 we have Cu, u  u2X0,α QuX0,α
uX0,α .

(2.3.2) (2.3.3)

To achieve this it suffices to let Q be the differential operator Q(x, D) = δ(Dxφ(δ|x|) + φ(δ|x|)xD) where δ is a small parameter and φ is an even function with the following properties: (i) φ(s) ≈ (1 + s)−1 for s > 0. (ii) φ(s) + sφ
(s) > αj for 2j < s < 2j+1 . (iii) φ(|x|) is localized at frequency at most O(1). 2.4 Outgoing parametrices and Strichartz estimates Here we reduce the proof of Theorem 1.2 to the construction of a suitable parametrix for Dt + A(0) . Our main result concerning parametrices is Proposition 2.3 Assume that ! is sufficiently small. Then there is a parametrix K0 for Dt + A(0) which is localized at frequency 1 and has the following properties: (i) L2 bound: K0 (t, s)L2 →L2
1.

(2.4.1)

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301

(ii)Error estimate: (1 + |x|)N (Dt + A(0) )K(t, s)L2 →L2
(1 + |t − s|)−N .

(2.4.2)

0 type pseudodiffe(iii) Jump condition: K0 (s + 0, s) and K0 (s − 0, s) are S1,0 rential operators satisfying

(K0 (s + 0, s) − K0 (s − 0, s))S0 = S0 . (iv) Outgoing parametrix: 1{|x|<2−10 |t−s|} K0 (t, s)L2 →L2
(1 + |t − s|)−N .

(2.4.3)

(v) Pointwise decay: K0 (t, s)L1 →L∞
(1 + |t − s|)− 2 . n

(2.4.4)

We postpone the proof of this result and show that it implies Theorems 1.2, 1.3. As an intermediate step we obtain the following localized Strichartz estimates for the parametrix: Proposition 2.4 The parametrix K0 given by Proposition 2.3 has the following properties: (i) (regularity) For any Strichartz pairs (p1 , q1 ) respectively (p2 , q2 ) with q1 ≤ q2 we have K0 f Lp1 Lq1 ∩X0
f Lp
2 Lq2
. (2.4.5) (ii) (error estimate) For any Strichartz pair (p, q) we have [(Dt + A(0) )K0 − 1]f X0

f Lp
Lq
.

(2.4.6)

This is obtained in a standard manner by interpolation, the Hardy– Littlewood–Sobolev inequality and the T T ∗ argument. The latter requires a flow reversibility bound for the parametrix, K0∗ (s1 , t)K0 (t, s2 ) − K0∗ (s1 + 0, s1 )K0 (s1 , s2 )L2 →L2
(1 + |s1 − s2 |)−N where t > s1 > s2 . The X0 bound is obtained from the L∞ L2 estimate via Holder’s inequality; this is where the fact that the parametrix is outgoing is used in an essential manner. Proposition 2.4 applies only for small !. However, a similar result holds even if ! is not small: Proposition 2.5 Assume that the coefficients aij satisfy (1.8) in a time interval [T − , T + ]. Then there is a parametrix K0 for A(0) localized at frequency 1 and which satisfies

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Daniel Tataru

(i) (regularity) For any Strichartz pairs (p1 , q1 ) respectively (p2 , q2 ) with q1 ≤ q2 we have K0 f Lp1 Lq1 ∩X0
f Lp
2 Lq2
. (2.4.7) (ii) (error estimate) For any Strichartz pair (p, q) we have [(Dt + A(0) )K0 − 1]f X0

f Lp
Lq
.

(2.4.8)

This is proved by dividing Rn into finitely many shells on which the small ! result in Theorem 2.4 can be applied. The spatially localized parametrices are then assembled together using a spatial partition of unity. Details are omitted. Rescaling the result in either Proposition 2.4 or Proposition 2.5 we obtain similar parametrices Kj at any dyadic frequency 2j . Then the global parametrix K is defined by ∞


K=

K j Sj .

K=−∞

Using Proposition 2.3 and Littlewood–Paley theory one shows that K has the following properties: (i) (regularity) For any Strichartz pairs (p1 , q1 ) respectively (p2 , q2 ) with q1 ≤ q2 we have Kf Lp1 Lq1 ∩X
f Lp
2 Lq2
. (2.4.9) (ii) (error estimate) For any Strichartz pair (p, q) we have ((Dt + A)K − I)f X

f Lp
Lq
.

(2.4.10)

Using these properties and duality we can establish an L2 → Lp Lq bound. Lemma 2.1 If there is a parametrix K for Dt + A as above and (p, q) is a Strichartz pair, then uLpLq
uL∞ L2 ∩X + (Dt + A)uX
.

(2.4.11)

Finally we prove (1.11). Without any restriction in generality we assume that q1 ≤ q2 ; the opposite case follows by duality. If (Dt + A)u = f + g,







f ∈ Lp2 Lq2 , g ∈ X


then we write u = Kf + v. We use (2.4.9) to bound Kf in Lp1 Lq1 ∩ X. It remains to bound v, which solves (Dt + A)v = (1 − (Dt + A)K)f + g.

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303

In the case of Theorem 1.2 we use successively (2.4.11) and Theorem 1.1 to obtain vLp1 Lq1
vL∞ L2 ∩X + (Dt + A)vX

v(0)L2 + (Dt + A)vX

u(0)L2 + Kf L∞L2 + (1 − (Dt + A)K)f X
+ gX

u(0)L2 + f Lp
2 Lq2
+ gX
. Then (1.11) follows. In the case of Theorem 1.3 the argument is similar, but instead of using Theorem 1.1 we assume that the localized energy estimates hold. 2.5 Pseudodifferential operators and phase space transforms In preparation for the outgoing parametrix construction we introduce the required microlocal analysis setup. 0 The simplest class of pseudodifferential operators which we use is S00 and 0(k) some variations of it S00 defined by 0,(k)

a ∈ S00

≡ |∂xα ∂ξβ a(x, ξ)| ≤ cαβ

|α| + |β| ≥ k.

These correspond to the euclidean metric in R2n , g = dx2 + dξ 2 . The indices 0 and 00 are not useful here so we simply drop them, and use instead the shorter notation S (k) . In the sequel k will take only the values 0, 1 and 2. In our analysis we have to work on a varying time dependent scale. Thus (k) for t > 0 we introduce the classes St defined by (k)

a ∈ St

≡ |∂xα ∂ξβ a(x, ξ)| ≤ cαβ t

|β|−|α| 2

|α| + |β| ≥ k.

These are obtained from S (k) by rescaling, so all the results we need are (k) quickly transferred from S (k) to St . They correspond to the metric gt = t−1 dx2 + tdξ 2 . The distance with respect to this metric is denoted by dt . Finally we also use time dependent pseudodifferential operators in R × T ∗ Rn , with the corresponding symbol classes
|α|−|β| (k) a ∈ l1 S ⇐⇒ 2j(1+ 2 ) ∂xα ∂ξβ a(t, x, ξ)L∞ ({t≈2j }) ≤ cαβ , (2.5.1) j

where |α| + |β| ≥ k.

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We note that these symbol classes are invariant with respect to the parabolic scaling a(t, x, ξ) → λ2 a(λ2 t, λx, λ−1 t). (2)

A special role in our analysis is played by the class l1 S . In this context we introduce a notion of smallness. Precisely, for a small parameter ! > 0 we (2) (2) define l1 S  to consist of those l1 S symbols for which


2j(1+

|α|−|β| ) 2

∂xα ∂ξβ a(t, x, ξ)L∞ ({t≈2j }) ≤ !,

|α| + |β| = 2.

(2.5.2)

j

To study the microlocal regularity of solutions to Schr¨ odinger type equations we use phase space transforms. Corresponding to the unit scale we have the Bargmann transform  (x−y)2 T u(x, ξ) = cn e− 2 eiξ(x−y) u(y)dy. The value T u(x, ξ) roughly measures how much of the function u is concentrated near position x and frequency ξ on the unit scale. This is an isometry from L2 (Rn ) into L2 (R2n ), which implies the inversion formula T ∗ T = I. However T is not onto; its range consists of those L2 functions which satisfy a Cauchy–Riemann type equation, i∂ξ T = (∂x − iξ)T. The corresponding transform on the gt scale is obtained by rescaling, and is sometimes called the FBI transform:  (x−y)2 n T 1t u(x, ξ) = cn t− 4 e− 2t eiξ(x−y) u(u)dy. The Cauchy–Riemann type equation has now the form i ∂ξ T 1t = (∂x − iξ)T 1t . t

(2.5.3)

The main idea in our approach to long time dynamics for Schr¨odinger type evolutions is to use a time dependent phase space transform to turn the equation into an evolution equation in the phase space. This requires results on conjugating pseudodifferential operators with respect to phase space transforms. Such results were first proved in [14], [15], [16]. However, for what is needed here we refer the reader to the expository paper [17]. For convenience the results below are stated including the parameter t. However, by rescaling they all reduce to the case when t = 1.

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305 (k)

Given a pseudodifferential operator in the Weyl calculus aw ∈ OP St define its phase space image

we

A˜ = T 1t aw T 1∗ . t

The kernel of A˜ is called the phase space kernel of aw . We begin our discussion with the case k = 0. Proposition 2.6 a) Let A : S(Rn ) → S ∗ (Rn ). Then A ∈ OP St phase space kernel K is rapidly decreasing away from the diagonal,

(0)

|K(x1 , ξ 1 ; x2 , ξ 2 )| ≤ cN (1 + dt ((x1 , ξ 1 ), (x2 , ξ 2 )))−N .

iff its (2.5.4)

(0)

b) Let a ∈ St be a symbol supported in a set D. Then its phase space kernel K satisfies the stronger bound |K(x1 , ξ 1 ; x2 , ξ 2 )| ≤ cN (1 + dt ((x1 , ξ 1 ), (x2 , ξ 2 )) + dt ((x1 , ξ 1 ), D)−N . (2.5.5) Part (a) is proved in [17]. Part (b) is an easy variation on the same theme which is left for the reader. As a consequence of part (a) one obtains that (0) OP St operators are L2 bounded, which is the Calderon–Vaillancourt theorem. (1) For OP St the L2 boundedness is lost. However the next result asserts (0) that modulo OP St such operators can be replaced with the multiplication by their symbol in the phase space. (1)

Proposition 2.7 a) Let a ∈ St . Then we have the conjugation result T 1t aw = (a + E)T 1t

(2.5.6)

where the kernel Ke of E satisfies (2.5.4). b) Assume in addition that a is supported in a set D. Then its phase space kernel K satisfies |K(x1 , ξ 1 ; x2 , ξ 2 )| ≤ cN (|a(x1 , ξ 1 )| + (1 + dt ((x1 , ξ 1 ), D)−N )(1 + dt ((x1 , ξ 1 ), (x2 , ξ 2 )))−N . (2.5.7) If D = R2n , then part (b) follows from part (a) which is proved in [17]. Otherwise it is again a fairly straightforward variation on the same theme. A direct consequence of part (a) is the sharp Garding inequality, (1)

Corollary 2.1 Let a ∈ St

be a real nonnegative symbol. Then

aw u, u ≥ −Cu2L2 . Finally in the case k = 2 we have (see [17]):

(2.5.8)

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Daniel Tataru (2)

Proposition 2.8 a) Let a ∈ St . Then we have the conjugation result T 1t aw = (a + i(aξ (∂x − iξ) − ax ∂ξ ) + E)T 1t

(2.5.9)

where the kernel Ke of E satisfies (2.5.4). Last but not least we consider an evolution equation which is a good model for short time Schr¨ odinger dynamics, (Dt + aw (t, x, D))u = 0,

u(0) = u0

(2.5.10)

where a is a real symbol in S (2) , uniformly in t ∈ [0, 1]. For the next results we refer the reader to [17] and also [9] to some extent. We begin with the corresponding Hamilton flow, x˙ = aξ (t, x, ξ) ξ˙ = −ax (t, x, ξ). We denote the time evolution maps by χ(t, s). These are characterized by Proposition 2.9 Assume that a is a real symbol in S (2) , uniformly in t ∈ [0, 1]. Then χ(t, s) are bi-Lipschitz symplectic maps. Now we turn our attention to the evolution (2.5.10). Proposition 2.10 Assume that a is a real symbol in S (2) , uniformly in t ∈ [0, 1]. Then (2.5.10) is L2 well posed forward and backward in time. We denote by S(t, s) the corresponding evolution operators. These are characterized using the Bargmann transform as follows: Proposition 2.11 Assume that a is a real symbol in S (2) , uniformly in t ∈ [0, 1]. Then the phase space kernels K(t, s) of S(t, s) satisfy |K(t, x, ξ, s, y, η)| ≤ cN (1 + |(x, ξ) − χ(t, s)(y, η)|)−N . In the terminology of [17] we say that S(t, s) is an S (0) type FIO associated to the canonical transformation χ(t, s). We also have a corresponding Egorov theorem. Given a pdo q w (0) at the initial time we define its conjugates along the flow by q w (t) = S(t, 0)q w (0)S(0, t). Then Proposition 2.12 Assume that a is a real symbol in S (2) , uniformly in t ∈ [0, 1]. a) Let q(0) ∈ S (0) . Then q(t) ∈ S (0) uniformly in t. b) Let q(0) ∈ S (1) . Then q(t) ∈ S (1) uniformly in t, and q(t, x, ξ) − q(0) ◦ χ(0, t) ∈ S (0) .

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307

We also need an improvement of this result in a special case. Proposition 2.13 Assume that a(t, x, ξ) = ξ 2 . Let q(0) ∈ S (2) . Then q(t) ∈ S (2) uniformly in t and q(t, x, ξ) − q(0) ◦ χ(0, t) ∈ S (0) . 2.6 A long time phase space parametrix In this section we obtain global in time phase space bounds for a class of equations governed by pseudodifferential operators aw (t, x, D) satisfying a smallness condition, (2) !  1. a ∈ l1 S  , This class does not include the operator A(0) which we are interested in. However, it does include the operator −∆ − A(0) in the phase space region {|ξ| ≈ 1, |x| ≈ |t|, t ≥ 1} This will allow us in the next section to make the transition to A(0) via a conjugation with respect to the flat Schr¨ odinger flow. In our analysis we add a damping term to the L2 conservative equation. Its role will ultimately be to kill all the waves which stray away from the above phase space region. Thus we consider the L2 well-posed forward evolution equation (Dt + aw (t, x, D) − ibw (t, x, D) + cw (t, x, D))u = 0, (2)

t>0

(2.6.1)

(1)

are real symbols with b ≥ 0 while where a ∈ l1 S  , respectively b ∈ l1 S (0) is a complex symbol. We think of aw as the operator driving the c ∈ l1 S evolution while bw is a damping term and cw is a negligible error. We are interested in obtaining much more precise bounds on the phase space localization of the solutions. The phase space image of the evolution S(t, s) is the family of evolution operators ˜ s) = T 1 S(t, s)T 1∗ S(t, t s

whose kernels we want to study. These are described in terms of two geometric quantities: (i) The Hamilton flow of Dt + aw . This is described by the ODEs x˙ = aξ (t, x, ξ) ξ˙ = −ax (t, x, ξ). We denote the trajectories of the Hamilton flow by t → (xt , ξ t ) and the flow map by χ(t, s). The regularity of the flow is computed using the linearized equations:

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Daniel Tataru (2)

Proposition 2.14 If a ∈ l1 S  and t > s, then the Hamilton flow has the Lipschitz regularity       In + !O st !O 1s ∂(xt , ξ t ) = (2.6.2) ∂(xs , ξ s ) !O(t) In + !O(1) respectively, ∂(xs , ξ s ) = ∂(xt , ξ t )





1

1 + !O(1)

!O

!O(t)

In + !O

s

t

.

(2.6.3)

s

We note that if ! is small and s < t, then for fixed xt the map ξ s → ξ t is a diffeomorphism. Then it is more convenient to parametrize the graph of χ(t, s) using the variables (xt , ξ s ). This choice of independent variables yields the better relation     In + !O(1) !O 1s ∂(xt , ξ s ) = . (2.6.4) ∂(xs , ξ t ) !O(t) In + !O(1) (ii) The exponential decay along the flow determined by the damping. Along each trajectory (xt , ξ t ) we define the weight function  ψ(t, xt , ξ t ) =

t

b(s, xs , ξ s )ds. 1

Heuristically we expect e−ψ(t,xt ,ξt ) to describe the behavior of the energy along the flow. The lower limit of integration is set arbitrarily to 1. In our analysis we only care about the differences ψ(xt , ξ t ) − ψ(xs , ξ s ). Their Lipschitz dependence on the (xs , ξ t ) variables is described in the following (2)

Proposition 2.15 If a ∈ l1 S 

with ! small, b ∈ l1 S

(1)

and t > s, then

1 1 ∂(ψ(xt , ξ t ) − ψ(xs , ξ s )) = (O(s− 2 ), O(t 2 )). ∂(xs , ξ t )

(2.6.5)

Now we can state the main result, namely a sharp pointwise bound on the ˜ s). kernel of the phase space operator S(t, (2)

(1)

be real symbols with b ≥ 0 and Theorem 2.1 Let a ∈ l1 S  , b ∈ l1 S 1 (0) ˜ s) satisfies the c ∈ l S . Then for s < t the kernel K of the operator S(t, bound

Schr¨ odinger equations with variable coefficients

309

|K(t, x, ξ t , s, xs , ξ)|  −N n n (x − xt )2 + s(ξ − ξ s )2
t− 4 s 4 1 + (ψ(xs , ξ s ) − ψ(xt , ξ t ))2 + . t (2.6.6) If u is the forward solution to (2.6.1) with initial data 1

u(s, y) = cn s− 4 e−

(y−xs )2 2s

eiξ(y−xs ) ,

then the kernel K is given by K(t, x, ξ t , s, xs , ξ) = (T 1t u(t))(x, ξ t ). At time t = s a direct computation gives an initial data for K, K(s, x, ξ s , s, xs , ξ) = cn e−

(x−xs )2 4s

e−

(ξ−ξ s )2 4s

1

ei 2 (x−xs )(ξ+ξs ) .

From (2.6.1) we have 0 = T 1t (∂t + iaw (t, x, D) + bw (t, x, ξ) + icw (t, x, D))u. To obtain an equation for K we need to conjugate the above pseudodifferential operators with respect to the phase space transform T 1t . For the time derivative a direct computation yields   1 n ∂t T 1t = − − 2 ∂ξ2 T 1t . 4t 2t Using the Cauchy–Riemann type equation (2.2) this can be rewritten in the form   n 1 + (∂x − iξ)2 T 1t . ∂t T 1t = 4t 2 For the pseudodifferential operators aw , bw and cw we use the conjugation results in Propositions 2.6, 2.7, 2.8. Adding the pieces together we can write an equation for the phase space function K(t) = T 1t u(t):  ∂t + ia + b(t, x, ξ) − ax ∂ξ + aξ (∂x − iξ) −

 n 1 − (∂x − iξ)2 + E K(t, x, ξ) = 0 4t 2 (0)

where E is a negligible error term with l1 S type kernel bounds. From this, one deduces that |K| is a subsolution for a degenerate parabolic equation, n 1 + aξ ∂x − ax ∂ξ − ∂x2 . (2.6.7) 4t 2 Here we can assume that E has a positive kernel. Then the bound (2.6.6) is obtained from the maximum principle by constructing an appropriate supersolution for L − E. L|K| ≤ E|K|,

L = ∂t + b −

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Daniel Tataru

2.7 A perturbation of the Schr¨ odinger equation Here we consider the evolution equation w (Dt − ∆ + aw 0 (t, x, D) − ib0 (t, x, D))u = 0 (2)

(1)

where a0 ∈ l1 S  , b0 ∈ l1 S are real symbols with b ≥ 0. This will serve as the model for our outgoing parametrix. We denote by S0 (t, s) the L2 evolution generated by the above equation, and by S˜0 (t, s) its phase space image S˜0 (t, s) = T 1t S0 (t, s)T 1∗ . s

We claim that the kernel of S˜0 (t, S) satisfies the same bounds as in Theorem 2.1. This can be done directly, but for our purposes it is easier to reduce it to the case considered in the previous section. odinger flow Precisely, we conjugate S0 (t, s) with respect to the flat Schr¨ and set 2 2 S(t, s) = e−itD S0 (t, s)eisD . In the phase space this corresponds to a conjugation with respect to the spacetime symplectic map µ(t, τ , x, ξ) = (t, τ − ξ 2 , x + 2tξ, ξ). Using rescaled versions of Proposition 2.12, 2.13, the evolution S(t, s) is governed by the operator aw − ibw + cw where a(t, x, ξ) = a0 (t, x + 2tξ, ξ) ∈ l1 S b(t, x, ξ) = b0 (t, x + 2tξ, ξ) ∈ l1 S

(2)

(1)

,

.

c ∈ l1 S

(0)

w The Hamilton flow χ0 for Dt +aw 0 is the conjugate of the flow χ for Dt +a with respect to the canonical transformation µ. Hence from Proposition 2.14 we obtain (2)

Proposition 2.16 If a0 ∈ l1 S  with ! sufficiently small and t > s, then the Hamilton flow χ0 (t, s) has the Lipschitz regularity     !O 1s In + !O(1) ∂(xt , ξ s ) = . (2.7.1) ∂(xs , ξ t ) 2tIn + !O(t) In + !O(1) Similarly, the integral ψ 0 of b0 along the χ0 flow is the µ conjugate of the integral ψ of b along the χ flow. Hence we also trivially obtain the analog of Proposition 2.15, namely (2)

Proposition 2.17 If a0 ∈ l1 S  then for t > s we have

with ! sufficiently small and b0 ∈ l1 S

1 1 ∂(ψ 0 (xt , ξ t ) − ψ 0 (xs , ξ s )) = (O(s− 2 ), O(t 2 )). ∂(xs , ξ t )

(1)

,

(2.7.2)

Schr¨ odinger equations with variable coefficients

311

Now we can apply Theorem 2.1 to obtain bounds for the phase space kernel S(t, s), and then return to S0 (t, s) using Proposition 2.11 rescaled for the flat Schr¨ odinger flow: (2)

(1)

Theorem 2.2 Let a0 ∈ l1 S  , b0 ∈ l1 S be real symbols with b0 ≥ 0 with ! sufficiently small. Then for s < t the kernel K0 of the operator S˜0 (t, s) satisfies the bound |K0 (t, x, ξ t , s, xs , ξ)|  −N n (x − xt )2 −n 2 2 4 4 + s(ξ − ξ s )
t s . 1 + (ψ 0 (xs , ξ s ) − ψ 0 (xt , ξ t )) + t (2.7.3) 2.8 The parametrix construction Here we outline the proof of Proposition 2.3. We begin with a dyadic partition of the initial data with respect to the distance from the origin. At frequency 1 we consider a phase space decomposition of the initial data u0 =

∞



w (p± j ) u0

± j=0

where the symbols p± j have the support properties −2
|ξ|
22 , 2j−1 < |x| < 2j+1 , ±xξ ≥ −2−5 |x|}. supp p± j ⊂ {2

The signs ± correspond to waves which are outgoing forward, respectively backward in time. Fix the “+” sign. We want to approximately solve the forward problem w with the initial data (p+ j ) u0 . After a time translation we can assume that j the initial time is s = 2 . Then we expect our approximate solution to be localized in the region |x| ≈ t. Within this region the symbol of A(0) has the (2)

right behavior a(0) ∈ l1 S . We modify it outside this region so that it keeps the same regularity. To insure that the output outside the above region is negligible we add a damping term b0 to the equation and solve instead (Dt + A(0) − ibw )u = 0,

w u(2j ) = (p+ j ) u0 .

(2.8.1)

We construct b0 so that its symbol has the following properties: (b1) At the initial time we have b(2j , x, ξ) = 0 in {2−3 < |ξ| < 23 , 2j−2 < |x| < 2j+2 , xξ > −2−4 |x|}.

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Daniel Tataru

(b2) At any time t ≥ 2j we have 3

b(t, x, ξ) = t− 4

outside {2−4 < |ξ| < 24 , 2−6 t < |x| < 26 t, xξ > −2−4 |x|}.

3

(b3) t 4 b0 is nonincreasing along the Hamilton flow for Dt + Dx2 + aw , and 3

0 < t 4 b0 (xt , ξ t ) < 1 =⇒ b0 (x2t , ξ 2t ) = 0. Here (b1) says that b is supported away from the initial data, and (b2) guarantees that all waves which stray outside the desired localization region are damped. The role of (b3) is more subtle; it insures that the expression bw 0 u is both small and rapidly decaying in time, j −N bw u0 L2 0 uL2
(2 + |t|)

so it can be included in the error term. The power 34 is somewhat arbitrary, anything between 12 and 1 works. Finally, for the evolution (2.8.1) we can directly use the phase space kernel bounds in (2.6.6) to prove the estimates in Proposition 2.3.

References ´rard and N. Tzvetkov, Strichartz inequalities and the non1. N. Burq, P. Ge linear Schr¨ odinger equation on compact manifolds, Amer. J. Math. 126(2004), 569–605. 2. Jean-Marc Delort, F.B.I. transformation. Second microlocalization and semilinear caustics, Springer-Verlag, Berlin, 1992. 3. Shin-ichi Doi, Smoothing effects for Schr¨ odinger evolution equation and global behavior of geodesic flow, Math. Ann. 318(2000), 355–389. 4. Charles L. Fefferman, The uncertainty principle, Bull. Amer. Math. Soc. (N.S.) 9(1983), 129–206. 5. Gerald B. Folland, Harmonic analysis in phase space, Princeton University Press, Princeton, NJ, 1989. 6. J. Ginibre and G. Velo, Smoothing properties and retarded estimates for some dispersive evolution equations, Comm. Math. Phys. 144(1992), 163–188. 7. Andrew Hassell, Terence Tao and Jared Wunsch, A Strichartz inequality for the Schr¨ odinger equation on nontrapping asymptotically conic manifolds, Comm. Partial Differential Equations 30(2005), 157–205. 8. Markus Keel and Terence Tao, Endpoint Strichartz estimates, Amer. J. Math. 120(1998), 955–980. 9. Herbert Koch and Daniel Tataru, Dispersive estimates for principally normal pseudodifferential operators, Comm. Pure Appl. Math. 58(2005), 217–284. 10. L. Robbiano and C. Zuily, Strichartz estimates for the Schr¨ odinger equation with variable coefficients, Preprint. ¨ strand, Singularit´es analytiques microlocales, in Ast´erisque 95, 11. Johannes Sjo 1–166, Soc. Math. France, Paris, 1982.

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12. Hart F. Smith, A parametrix construction for wave equations with C 1,1 coefficients, Ann. Inst. Fourier (Grenoble)48 (1998), 797–835. 13. Gigliola Staffilani and Daniel Tataru, Strichartz estimates for a Schr¨ odinger operator with nonsmooth coefficients, Comm. Partial Differential Equations 27(2002), 1337–1372. 14. Daniel Tataru, Strichartz estimates for operators with nonsmooth coefficients and the nonlinear wave equation, Amer. J. Math. 122(2000), 349–376. 15. Daniel Tataru, Strichartz estimates for second order hyperbolic operators with nonsmooth coefficients. II, Amer. J. Math. 123(2001), 385–423. 16. Daniel Tataru, On the Fefferman–Phong inequality and related problems, Comm. Partial Differential Equations 27(2002), 2101–2138. 17. Daniel Tataru, Phase space transforms and microlocal analysis, Phase space analysis of partial differential equations. Vol. II, 505–524, Pubbl. Cent. Ric. Mat. Ennio De Giorgi, Scuola Norm. Sup., Pisa, 2004.

On the analyticity of solutions of sums of squares of vector fields Fran¸cois Treves Mathematics Department, Rutgers University, New Brunswick, USA

Summary. The note describes, in simple analytic and geometric terms, the global Poisson stratification of the characteristic variety Char L of a second-order linear differential operator −L = X12 + · · · + Xr2 , i.e., a sum-of-squares of real-analytic, real vector fields Xi on an analytic manifold Ω. It is conjectured that the leaves in the bicharacteristic foliation of each Poisson stratum of Char L propagate the analytic singularities of the solutions of the equation Lu = f ∈ C ω . Closely related conjectures of necessary and sufficient conditions for local, germ and global analytic hypoellipticity, respectively, are stated. It is an open question whether the new conjecture regarding local analytic hypoellipticity is equivalent to that put forward by the author in earlier articles.

2000 Mathematics Subject Classification: Primary: 35H05; secondary: 35A20. Key words: Stratification, symplectic, sums of squares of vector fields, analytic, hypoellipticity. 0.1 Introduction Let Xj (j = 1, . . . , r) be real vector fields of class C ω (i.e., real-analytic, always abbreviated to analytic) in a C ω manifold Ω. The purpose of this note is to state formally each of the conditions on these vector fields, conjectured by the author to be necessary and sufficient for the second-order differential operator L = −X12 − · · · − Xr2 to be analytic hypoelliptic (henceforth abbreviated to ahe) in the strict sense; germ ahe; globally ahe. We recall the definitions. For any open set U ⊂ Ω let D
(U ) denote the space of distributions in U .

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Fran¸cois Treves

Definition 0.1 The differential operator L is said to be 1. globally ahe in Ω if for all u ∈ D
(Ω) , Lu ∈ C ω (Ω) =⇒ u ∈ C ω (Ω); 2. germ ahe at a compact set K ⊂ Ω if to each open set U ⊃ K there is an open set V ⊃ K such that for all u ∈ D
(U ) , Lu ∈ C ω (U ) =⇒ u ∈ C ω (U ∩ V ); 3. ahe in Ω if it is germ ahe at every compact subset of Ω; 4. ahe at a point x◦ ∈ Ω if there is an open neighborhood of x◦ in which L is ahe. Each of the above definitions has its microlocal counterpart, whose statement is self-evident. Naturally, we say that L is germ ahe at a point x◦ ∈ Ω if L is germ ahe at K = {x◦ }. The Cauchy–Kovalewski theorem implies that germ analytic hypoellipticity of L at x◦ ∈ Ω is equivalent to the following property: •

to each open neighborhood U of x◦ there is an open neighborhood V of x◦ such that ∀u ∈ D
(U ) , Lu = 0 =⇒ u ∈ C ω (U ∩ V ).

Analytic hypoellipticity at a point is an open property: it is automatically valid at all nearby points. Not so with germ analytic hypoellipticity. The recent paper [Hanges, 2004] gives the example of an operator L which is germ ahe at a point without being ahe at that point. Hanges’ example has provided much of the motivation for this note. Analytic hypoellipticity at every point implies global analytic hypoellipticity. Proposition 0.1, related to the extension of Hanges’ example in [Bo–De–Ta, 2005], is self-evident. Proposition 0.1 If there is a basis of open neighborhoods Uk (k = 1, 2, . . .) of a compact subset K ⊂ Ω such that L is globally ahe in Uk for every k then L is germ ahe at K. Examples of sum-of-squares operators that are globally ahe but not ahe were first given in [C–Him, 1994] (see also [Ta, 1996], [C–Him, 1998]). In order to state our conjectures we need the Poisson stratification of the characteristic set of L, Char L, introduced in [Tr, 1999] and somewhat simplified at the local level in [Bo–Tr, 2004]. In Section 1 we give a streamlined description that works well in the global set-up. The “philosophy” of the conjectures stated in Subsection 2.3 conforms to the common view that bicharacteristics in the characteristic set propagate the analytic wave-front set (cf. e.g., [G–S, 1985]). The difference here is that the bicharacteristics must be those of some Poisson stratum of Char L (Conjecture 2.5). Analytic hypoellipticity will hold if (and only if) the loci of possible singularities in the data, be it in the base Ω or at infinite frequencies, cannot be reached by such bicharacteristics. I should say, however, that the conjecture

On the analyticity of solutions of sums of squares of vector fields

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put forward here, about analytic hypoellipticity stricto sensu, is an apparent, and perhaps real, weakening of that stated in [Tr, 1999] and [Bo–Tr, 2004]. The present article deals with sums of squares of analytic vector fields acting on distributions. There is no visible obstacle to extend the conjectures to sums of squares of arbitrary first-order analytic pseudodifferential operators acting on hyperfunctions. Of course, proving them is another matter. I wish to thank Antonio Bove and Paulo Cordaro for their suggestions and help.

1 Global Poisson stratification 1.1 Step 1: Global analytic stratification Initially we shall be dealing with subsets of an arbitrary C ω manifold M countable at infinity (and soon assumed to be symplectic) with dim M = N ≥ 2. By T M (resp., T ∗ M) we mean the tangent (resp., cotangent) bundle of M. Let V be an analytic subvariety of M, meaning that each point x◦ ∈ V has an open neighborhood N (x◦ ) in M such that V∩N (x◦ ) = {x ∈ N (x◦ ); fj (x) = 0, j = 1, . . . , ν} with fj ∈ C ω (N (x◦ )) real-valued (and ν allowed to vary with x◦ ). We recall rapidly the analytic stratification of V (see, e.g., [L, 1965], [Hardt, 1975], [Su, 1990], [Sim, 2003]). Call R(V) the regular part of V: x◦ ∈ R(V) means that x◦ has an open neighborhood N (x◦ ) in M such that V∩N (x◦ ) is a C ω submanifold of M; R(V) is an open and dense subset of V. In general the complement V(1) = V\R(V) is not an analytic subvariety of M, only a closed semi-analytic subset of M: each point x◦ ∈ V(1) has an open neighborhood N (x◦ ) in M such that V(1) ∩N (x◦ ) is a finite union of sets {x ∈ N (x◦ ); gj (x) = 0, hk (x) > 0, j = 1, . . . , ν
, k = 1, . . . , ν

}, with gj , hk ∈ C ω (N (x◦ )) real-valued (possibly with gj ≡ 0 or hk ≡ 1). The regular part R(V(1) ) of V(1) is well defined, in the same manner as for V; it is an open and dense subset of V(1) . The complement V(2) = V(1) \R(V(1) ) is a closed semi-analytic subset of M ([L, 1965], pp. 150–153). Repeating the procedure indefinitely leads to the decomposition into disjoint analytic submanifolds, ∞ 2 V= R(V(k) ). (1.1) k=0

Observe that R(V ) ⊂ ∂R(V ) for every k ∈ Z+ (∂S is the boundary of the subset S of M, i.e., the complement of S in its closure S). By decomposing each R(V(k) ) into its connected components we end up with the decomposition (k+1)

(k)

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V=

∞ 2

Λα

(1.2)

α=0

into connected, pairwise disjoint, analytic submanifolds Λα of M. It is not difficult to show that the partition (1.2) is locally finite (see, e.g., Finiteness Theorem, p. 9, [Sim, 2003]). The hypothesis that M is countable at infinity ensures then that there are “only” countably many Λα . A basic property of the partition (1.2) is that Λα ∩ Λβ = ∅ =⇒ Λα ⊂ ∂Λβ . Definition 1.1 The partition (1.2) will be called the analytic stratification of V and each submanifold Λα will be referred to as an analytic stratum of V. 1.2 Step 2: Global symplectic stratification From now on we assume M to be a symplectic manifold of class C ω ; per force dim M is even: N = 2n. We denote by % the fundamental symplectic form on M; % is an analytic section of Λ2 T ∗ M, closed and nondegenerate. If x ∈ M we denote by %x the nondegenerate skew-symmetric bilinear form induced by % on the tangent space Tx M. If f is a real-valued analytic function in some open subset U of M, we denote by Hf the Hamiltonian vector field of f , defined by the property that, for any vector field v in U, %(Hf , v) = − df, v . Here df is the differential of f and , is the duality bracket between tangent and cotangent vectors. If also g ∈ C ω (U), we denote the Poisson bracket of f and g by {f, g} = %(Hf , Hg ) = Hf g = −Hg f . We denote by %|Λα the restriction of % to a submanifold Λα in the partition (1.2) and by (%|Λα )x the (possibly degenerate) bilinear form on Tx Λα defined by the two-form %|Λα (x ∈ Λα ). An arbitrary point x◦ ∈ Λα has an open neighborhood N (x◦ ) in M such that Λα ∩N (x◦ ) = {x ∈ N (x◦ ); ϕi (x) = 0, i = 1, . . . , κ = codim Λα }

(1.3)

with ϕi ∈ C ω (N (x◦ )) and dϕ1 ∧ · · · ∧ dϕκ nowhere zero in N (x◦ ). For every x ∈ Λα ∩N (x◦ ), rank(%Λα )x + codim Λα = rank({ϕi , ϕj }(x))1≤i,j≤κ + dim Λα . (Both ranks are even numbers.) We refer to rank(%|Λα )x as the symplectic rank of the submanifold Λα at the point x. Denote by Λα,0 the open and dense subset of Λα consisting of the points x at which the symplectic rank of Λα is maximum, say equal to µ ≥ 0. Each connected component of Λα,0 is a submanifold of M of class C ω whose symplectic rank is everywhere equal to µ. The subset Λα \Λα,0 is an analytic subvariety of Λα . Indeed, if N (x◦ ) is the neighborhood of (1.3), then N (x◦ ) ∩ (Λα \Λα,0 ) can be defined in Λα ∩N (x◦ ) as the set of zeros of all the ν × ν minors of the matrix ({ϕi , ϕj })1≤i,j≤κ where ν = µ + codim Λα − dim Λα . It ensues that Λα \Λα,0 admits an analytic stratification of type (1.2) in Λα . The dimension of each analytic stratum of

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Λα \Λα,0 is strictly less than dim Λα . This means that we can repeat with each one of them the construction started with Λα ; and that it will suffice to repeatCthis same construction a finite number of times, to obtain a partition ∞ Λα = j=1 Λα,j in which each Λα,j is a connected C ω submanifold of M whose symplectic rank is constant. We end up with a new locally finite partition V=

∞ 2 ∞ 2

Λα,j .

(1.4)

α=1 j=1

Definition 1.2 The decomposition (1.4) will be called the symplectic stratification of the analytic set V. 1.3 Step 3: Global Poisson stratification So far we have dealt with an arbitrary analytic subvariety V. From now on −1

we take V = F (0), the set of zeros of an analytic map F = (F1 , . . . , Fr ) : M −→ Rr . For each multiindex I = (i1 , . . . , iν ) with 1 ≤ i1 , . . . , iν ≤ r , ν ≥ 2, we shall use the notation FI = {Fi1 , . . . , Fiν } = {Fi1 , . . . {Fiν−1 , Fiν } . . .}, where {·, ·} is the Poisson bracket (ν = |I|, the length of the multiindex I). When |I| = 1, i.e., when I = {i} for some i, 1 ≤ i ≤ r, we write Fi rather than FI . Definition 1.3 We say that the functions F1 , . . . , Fr ∈ C ω (M) satisfy the finite type condition if for every x ∈ M there is a multiindex I, |I| ≥ 1, such that FI (x) = 0. We can define the following monotone decreasing sequence of analytic subvarieties of M: for each ν ≥ 1, (ν) = V∩{x ∈ M; ∀I, |I| ≤ ν, FI (x) = 0}. V

(1.5)

(ν) = ∅. (1) . The finite type condition states that D∞ V In particular V = V ν=1 Note that there is a subsequence of integers 1 = ν 1 < ν 2 < · · · such that (ν p ) ; (ν p+1 ) = V 1. V (ν
) = V (ν) for every ν
, ν p ≤ ν
< ν p+1 . 2. if ν p < ν p+1 , then V Now consider, for any given integer p ≥ 1, the symplectic stratification of (ν p ) (Definition 1.2): the analytic variety V (ν p ) = V

∞ 2 ∞ 2 α=1 j=1

(ν )

Λα,jp .

(1.6)

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(ν ) (ν ) (ν p ) \V (ν p+1 ) is either In each stratum Λα,jp the set Λ α,jp of points x ∈ V (ν )

empty or it is an open and dense subset of Λα,jp (as the latter is a connected (ν ) (ν p ) C ω submanifold). If Λ α,jp = ∅ we denote by Λα,j,γ its connected components. We obtain the decomposition ∞ 2

(ν p ) =V (ν p+1 ) ∪ V

(ν )

p Λα,j,γ .

α,j,γ=1

Letting p range over the set of positive integers yields a decomposition into pairwise connected and disjoint C ω submanifolds: V=

∞ 2

∞ 2

(ν )

p Λα,j,γ .

(1.7)

p=1 α,j,γ=1

The partition (1.7) is locally finite, as a consequence of the local finiteness of the partition (1.6). Definition 1.4 The decomposition (1.7) will be called the Poisson stratifica(ν p ) tion of V defined by the functions F1 , . . . , Fr and each submanifold Λα,j,γ will be called a Poisson stratum of V defined by these functions. If Σ is a Poisson stratum of V, then 1. Σ is a connected, embedded analytic submanifold of M contained in V; 2. dim((T Σ) ∩ (T Σ) ⊥ ) is constant throughout Σ and dim Σ − dim((T Σ) ∩ (T Σ)

⊥

)

is an even integer ; 3. at each point of Σ all Poisson brackets FI of length ν < ν p+1 (for some p ≥ 1) vanish but at least one of length ν p+1 does not; 4. Σ is maximal for Properties 1,2,3 conjoined. The vector bundle T Σ ∩ (T Σ) ⊥ satisfies the Frobenius condition: the commutation bracket of two smooth sections is also a section. As a consequence T Σ ∩ (T Σ) ⊥ defines a foliation on Σ in which all the leaves have the same dimension. We refer to the leaves of this foliation as the bicharacteristic leaves and to any analytic curve contained in a bicharacteristic leaf as a bicharacteristic curve. The bicharacteristic leaves are immersed, not necessarily embedded, submanifolds of Σ (see Example 2.2 below). Remark 1.1 It follows immediately from the elementary properties of the Poisson bracket that the Poisson stratification of V defined by the functions F1 , . . . , Fr is invariant under substitutions Fj! =

r


akj Fk , j = 1, . . . , r,

k=1

with

akj

∈ C (M) and ω

det(akj )1≤j,k≤r

= 0 at every point of M.

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2 The analyticity conjectures 2.1 Poisson stratification associated to vector fields In this section we consider r real vector fields X1 , . . . , Xr of class C ω in an analytic manifold Ω and the associated “sum of squares” operator −L = X12 + · · · + Xr2 . We assume that the C ω manifold Ω is without a boundary and countable at infinity (possibly compact); it is convenient to assume also that Ω is connected. The symplectic manifold M of the preceding section will be the cotangent bundle of Ω with the zero section deleted, T ∗ Ω\0; thus n = dim Ω. We denote by π the base projection M = T ∗ Ω\0 → Ω. The fundamental ntwo-form % on n M is exact: % = i=1 dξ i ∧ dxi = d(ξ · dx) where ξ · dx = i=1 ξ i dxi in any local coordinates system x1 , . . . , xn , with ξ 1 , . . . , ξ n the dual coordinates on the fibres of T ∗ Ω. We recall that a subset of phase-space T ∗ Ω is said to be conic if it is invariant under the dilations (x, ξ) → (x, λξ), λ > 0. The variety V will be the set of common zeros of the symbols σ(Xj ) of the vector fields Xj ; in other words, V = Char L, the characteristic variety of the operator L. Of course Char L is conic. In accordance to established custom the symbol σ(X) of a real vector field X is obtained by substituting √ ∂ −1ξ j for the partial derivative ∂x and therefore σ(X) is purely imaginary. j 2 2 We have equated X1 + · · · + Xr to −L to ensure that the principal symbol of L is nonnegative: σ(L) = |σ(X1 )|2 + · · · + |σ(Xr )|2 . √ We apply the concepts of Subsection 1.3 with the choice of Fj = −1σ(Xj ), j = 1, . . . , r. This choice will define once and for all the meaning of the Poisson strata of Char L. We can repeat the constructions in Subsections 1.1, 1.2, 1.3, making use only of functions F (x, ξ) that are homogeneous with respect to ξ i.e., F (x, λξ) = λm F (x, ξ) for some integer m and all λ ∈ R. We see immediately that every Poisson stratum of Char L is conic. A theorem of Nagano (see [N, 1966]) states that the base Ω is foliated by immersed analytic submanifolds whose tangent space at any point is equal to the “freezing” at that point of the Lie algebra g(X1 , . . . , Xr ) generated by the vector fields X1 , . . . , Xr for the commutation bracket [X, Y ] = XY − Y X. One says that the vector fields X1 , . . . , Xr and the differential operator L = −(X12 +· · ·+Xr2 ) satisfy the finite type condition if dim g(X1 , . . . , Xr ) = n at √ every point of Ω. This is equivalent to saying that the set of functions −1σ(Xj ), j = 1, . . . , r, satisfy the finite type condition in Definition 1.3. It is also equivalent to saying that there is only one Nagano leaf, Ω itself (since Ω is connected). Another name often used for the finite type condition is H¨ ormander’s condition, in reference to the classical theorem of [H, 1967]: the differential operator L is C ∞ hypoelliptic if the Lie algebra g(X1 , . . . , Xr ), “frozen” at an arbitrary point x ∈ Ω, is equal to the tangent space Tx Ω. On the other hand, to say that L does not satisfy the finite type condition at some point x◦ ∈ Ω is to say that the Nagano leaf F through x◦ is a

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proper submanifold of Ω and thus dim F < n. The conormal bundle N ∗ F \0 of F with its zero section excised is a Lagrangian submanifold of T ∗ Ω\0 contained in Char L. Some nonempty, relatively open subset of N ∗ F \0 must be contained in a Poisson stratum of Char L. Since this stratum has dimension < 2n it cannot be symplectic. As shown in [D, 1971], in the analytic category H¨ormander’s condition is also necessary for L to be C ∞ hypoelliptic and ahe. 2.2 Types of bicharacteristic foliations We shall use the following terminology: an analytic submanifold S of T ∗ Ω will be said to be vertical at a point (x, ξ) if T(x,ξ) S ⊂ T(x,ξ) (Tx∗ Ω); it will be said to be vertical if it is vertical at every one of its points; when S is connected this means that π(S) = {x} for some x ∈ Ω. The set of points (x, ξ) ∈ S at which S is vertical is an analytic subset S
of S. Indeed, S
is the set of points at which rank(π|S ) = 0. If S
= S, then S is vertical. To say that a bicharacteristic leaf Λ of Σ is vertical at a point (x, ξ) means that (2.1) T(x,ξ) Σ ∩ T(x,ξ) Σ ⊥ ⊂ T(x,ξ) (Tx Ω). In the Baouendi–Goulaouic example (the first example of a sum-of-squares operator L which is C ∞ but not C ω hypoelliptic, [B–G, 1972]) Char L = {(x, ξ) ∈ T ∗ R3 ; x1 = ξ 1 = ξ 2 = 0, ξ 3 = 0} consists of two Poisson strata, Σ ± defined by x1 = ξ 1 = ξ 2 = 0, ξ 3 ≷ 0; the bicharacteristics are not vertical: they are the “horizontal” lines R  x2 −→ (0, x2 , x◦3 , 0, 0, ξ ◦3 ) with (x◦3 , ξ ◦3 ) ∈ R2 fixed, ξ ◦3 ≷ 0. In general Char L may have symplectic and nonsymplectic Poisson strata, the latter either with vertical or nonvertical bicharacteristic leaves. Example 2.1 Suppose σ(L) is the sum of the squares of the functions ξ 1 , x1 ξ 2 , x1 ξ 3 , x3 ξ 2 .

(2.2)

The analytic subvariety Char L = {(x, ξ) ∈ T ∗ R3 ; x1 = ξ 1 = x3 ξ 2 = 0, ξ 22 + ξ 23 > 0} = { (x, ξ) ∈ T ∗ R3 ; x1 = ξ 1 = ξ 2 = 0, x3 ξ 3 = 0} ∪ { (x, ξ) ∈ T ∗ R3 ; x1 = x3 = ξ 1 = 0, ξ 2 ξ 3 = 0} ∪ { (x, ξ) ∈ T ∗ R3 ; x1 = x3 = ξ 1 = ξ 3 = 0, ξ 2 = 0} consists of ten analytic strata, each a Poisson stratum: two symplectic ones, Σ0± defined by x1 = x3 = ξ 1 = ξ 3 = 0, ξ 2 ≷ 0; two quartets of nonsymplectic (i) strata: the quadrants Σ1 defined by x1 = ξ 1 = ξ 2 = 0, x3 ≷ 0, ξ 3 ≷ (i) 0, and Σ2 those defined by x1 = x3 = ξ 1 = 0, ξ 2 ≷ 0, ξ 3 ≷ 0 (i =

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323

(i)

1, 2, 3, 4). The bicharacteristic leaves in Σ1 are the same “horizontal” lines (i) as in the Baouendi–Goulaouic example. The bicharacteristic leaves in Σ2 ◦ ◦ are the “vertical” lines R  t −→ (0, x◦2 , 0, 0, ξ 2 , t), with (x◦2 , ξ 2 ) ∈ R2 fixed, ξ ◦2 = 0. In the remainder of this subsection Σ will be a nonsymplectic Poisson stratum (below simply referred to as “the stratum Σ”) of Char L with, as usual, L = −(X12 + · · · + Xr2 ). Proposition 2.1 The set Σ
of points (x, ξ) ∈ Σ at which the bicharacteristic leaf of Σ through (x, ξ) is vertical at (x, ξ) is an analytic subvariety of Σ. Proof. Let Σ be defined in some open subset U of T ∗ Ω\0 by analytic real equations ϕi (x, ξ) = 0, i = 1, . . . , κ, with dϕ1 ∧ · · · ∧ dϕκ = 0 at every point of U; thus dim Σ = 2n − κ. The Hamiltonian fields Hϕi (i = 1, . . . , κ) are linearly independent and span the symplectic orthogonal T Σ ⊥ of T Σ. We can choose U and the ϕi so that Hϕi , i = 1, . . . , d ≤ κ, span T Σ ∩ T Σ ⊥ over Σ ∩ U. Let Λ be a characteristic leaf of Σ that intersects U; dim Λ = d and T Λ = T Σ ∩ T Σ ⊥ over Λ. Thus Λ will be tangent to Tx∗ Ω at a point (x, ξ) ∈ Λ∩U if and only if the tangent vectors Hϕi (i = 1, . . . , d) are “vertical” at that point. This is equivalent to saying that dξ ϕi (x, ξ) = 0, i = 1, . . . , d. Corollary 2.1 Either Σ is entirely foliated by vertical leaves or π(Σ\Σ
) is open and dense in π(Σ). Proof. If Σ
= Σ, every bicharacteristic leaf of Σ must be vertical. If Σ
= Σ, then Σ\Σ
is open and dense in Σ. In Example 2.1 the strata Σ2± are entirely foliated by vertical leaves. An extreme example of vertical foliation is a foliation by bicharacteristic rays. By a ray in T ∗ Ω\0 we mean a set of points (x◦ , λξ ◦ ) with λ > 0 arbitrary and (x◦ , ξ ◦ ) fixed, ξ ◦ = 0. In the classical example of G. M´etivier ([M, 1981]) the characteristic manifold Char L = {(x, ξ) ∈ T ∗ R2 ; x1 = x2 = ξ 1 = 0, ξ 2 ≷ 0} consists of two opposite rays. In passing, note that a ray γ ⊂ Σ is a bicharacteristic of the stratum Σ if and only if there is a real-valued function ϕ, defined and analytic in an open set U ⊂ T ∗ Ω\0 containing γ, such that ϕ ≡ 0 on U ∩ Σ and dϕ = ξ · dx along γ. In a sense, at the opposite of radial bicharacteristics stand compact ones. N. Hanges (see [Hanges, 2004]) has given an example (in R3 ) in which every bicharacteristic leaf of Char L, the single Poisson stratum, is compact but no bicharacteristic curve is vertical at any point except those contained in T0∗ R3 . Hanges’ example can be slightly elaborated to produce strata in which the bicharacteristic curves are dense geodesics of tori (and thus provide examples of immersed but not embedded bicharacteristics):

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Example 2.2 Let σ(L) be the sum of the squares of the functions ξ 1 , x1 ξ j (j = 2, 3, 4, 5), F = x3 ξ 2 − x2 ξ 3 − α(x5 ξ 4 − x4 ξ 5 ) with α ∈ R; Σ = Char L is the 7D analytic submanifold of T ∗ R5 defined by the equations x1 = ξ 1 = 0, x3 ξ 2 − x2 ξ 3 = α(x5 ξ 4 − x4 ξ 5 ), 5 5 under the condition j=2 ξ 2j = 0. We have j=2 |{ξ 1 , x1 ξ j }| > 0 everywhere in Σ and T Σ ∩ (T Σ) ⊥ is a line-bundle, spanned by HF . There is a single Poisson stratum, Σ itself, and the bicharacteristics of Σ are the integral curves of HF . Using polar coordinates in the complement of the origin, respectively (r, θ) in the (x2 , x3 )-plane, (r
, θ
) in the (x4 , x5 )-plane, (ρ, ω) in the (ξ 2 , ξ 3 )plane and (ρ
, ω
) in the (ξ 4 , ξ 5 )-plane, we can write HF =

∂ ∂ ∂ ∂ −α
+ −α
∂θ ∂ω ∂ω ∂θ

in the region rr
ρρ
> 0. If r = r
= 0, ρρ
> 0, i.e., in the product of deleted planes defined by x = 0, ξ 1 = 0, ξ 22 + ξ 23 = 0, ξ 24 + ξ 25 = 0, we have HF =

∂ ∂ − α
. ∂ω ∂ω

The bicharacteristics are geodesics of the two-torus T(ρ, ρ
) = {(0, ξ) ∈ R5 × R5 ; ξ 1 = 0, ξ 22 + ξ 23 = ρ2 , ξ 24 + ξ 25 = ρ
2 }. If α is irrational those geodesics are dense in T(ρ, ρ
). Similar configurations occur also when at least one number in each pair (r, ρ) and (r
, ρ
) is nonzero. But if rρ > 0 and r
+ ρ
= 0 then ℘◦ = (0, r cos θ◦ , r sin θ ◦ , 0, 0, 0, ρ cos ω◦ , ρ sin ω ◦ , 0, 0) belongs to Σ if and only if θ◦ − ω ◦ = 0 or π. The bicharacteristic through ℘◦ consists entirely of points ℘ = (0, r cos θ, r sin θ, 0, 0, 0, ρ cos ω, ρ sin ω, 0, 0) such that θ −ω = θ ◦ −ω ◦ (= 0 or π). If α ∈ Q all bicharacteristics are compact. 2.3 Conjectures and questions As before −L = X12 + · · · + Xr2 . Conjecture 2.1 For L to be globally ahe in Ω it is necessary and sufficient that every Poisson stratum Σ of Char L have the following property: (•) The closure in T ∗ Ω of every bicharacteristic leaf of Σ is compact.

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Needless to say Conjecture 2.1 agrees with the examples known to the author (in [C–Him, 1998] and [Ta, 1996], for instance). As further “circumstantial evidence” for Conjecture 2.1 we mention the following (personal communication from P. D. Cordaro, based on an idea of J. M. Tr´epreau). There are sum-of-squares operators L on analytic manifolds Ω whose characteristic set is equal to the union of two rays (x◦ , λξ ◦ ), x◦ ∈ Ω, 0 = ξ ◦ ∈ Tx◦ Ω, λ ≷ 0, and which are not ahe: e.g., the M´etivier operator (see [M, 1981]). Since L is elliptic in Ω\{x◦ } there must exist an open neighborhood U of x◦ and a distribution u in U such that Lu = 0 in U , u ∈ C ω (U \{x◦ }) but u ∈ C ω (U ), i.e., the analytic singular support of u, singsupp a u, is exactly equal to {x◦ }. We are in a position to apply the following general result. Theorem 2.1 Let P (x, D) be a linear partial differential operator with C ω coefficients defined in a paracompact manifold Ω of class C ω . If there exist an open subset U of Ω, a point x◦ ∈ U and a distribution u in U such that P (x, D)u ∈ C ω (U ) and singsuppa u = {x◦ }, then P (x, D) is not globally ahe in Ω. Proof. Let V be an open neighborhood of x◦ whose closure V is a compact subset of U . Since u ∈ C ω (U \V ) and since the cohomology of Ω with values in the sheaf of germs of real-analytic functions vanishes (cf. [Gr, 1958]) we can write u = v −w in U \V with v ∈ C ω (U ) and w ∈ C ω (Ω\V ). Define u1 ∈ D
(U ) by the equations u1 = u − v in U , u1 = −w in Ω\V . It is clear that P (x, D)u1 ∈ C ω (Ω) and singsupp a u1 = {x◦ }. The next statement is self-evident (cf. Proposition 0.1). Recall that π : T ∗ Ω\0 −→ Ω is the base projection. Proposition 2.2 Suppose that Conjecture 2.1 is true. Suppose, moreover, that there is a basis of open neighborhoods Uk (k = 1, 2, . . .) of the compact set K ⊂ Ω such that every Poisson stratum Σ of Char L has the following property, for every k: −1 −1 (•)k The closure in π (Uk ) of every bicharacteristic leaf of Σ ∩ π (Uk ) is compact. Then L is germ ahe at K. Our second conjecture concerns germ analytic hypoellipticity and states a converse of sorts to Proposition 2.2. Conjecture 2.2 For L to be germ ahe at a compact set K ⊂ Ω (Definition 0.1), it is necessary and sufficient that to each open set U ⊃ K there be an open set V ⊂ U , V ⊃ K, such that every bicharacteristic leaf of every Poisson stratum −1 −1 Σ of Char L intersecting π (V ) have compact closure contained in π (U ).

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Proposition 2.3 Suppose that Conjectures 2.1 and 2.2 are both true. If there is a bicharacteristic curve γ in some Poisson stratum of Char L whose base projection π(γ) is not a single point, then L is not ahe in Ω. Proof. If π(γ) is not a single point, then it contains some open arc of curve c. Let Bδ (x◦ ) be an open ball centered at x◦ ∈ c such that c ∩ Bδ (x◦ ) is not compact. In this case, whatever the open neighborhood U ⊂ Bδ (x◦ ) of x◦ , −1

γ ∩ π (U ) cannot be compact. It follows that L cannot be germ ahe at x◦ . Thirdly we look at “true” analytic hypoellipticity. We recall the conjecture first formulated in [Tr, 1999] (see also [Bo–Tr, 2004]): For L to be analytic hypoelliptic in Ω, it is necessary and sufficient that every Poisson stratum of Char L be symplectic. The Hanges example and the above considerations (in particular Theorem 2.1 and Proposition 2.3) suggest that the conjecture of [Tr, 1999] be modified as follows: Conjecture 2.3 For L to be ahe in Ω, it is necessary and sufficient that every bicharacteristic leaf of every Poisson stratum of Char L be vertical and have compact closure in T ∗ Ω. As the next example shows, submanifolds of T ∗ Ω\0 entirely foliated by vertical and compact bicharacteristic leaves do exist. Note however that the submanifold S in Example 2.3 is not realized as the Poisson stratum of Char L for some choice of analytic vector fields Xj . Example 2.3 In T ∗ R3 \0 consider the submanifold S defined by the equations x1 = x2 = 0, ξ 21 +ξ 22 = ξ 23 . The base projection π(S) is the x3 -axis and the rank of π|S is everywhere equal to 1. The vector bundle T S ∩(T S) ⊥ is spanned by the single vector field ξ 1 ∂ξ∂ −ξ 2 ∂ξ∂ (we cannot have ξ 1 = ξ 2 = 0). The bichar2 1 acteristic curves are the vertical circles R  θ −→ (0, 0, x◦3 , r cos θ, r sin θ, ±r), with x◦3 ∈ R, r > 0 fixed. If it could be proved that a Poisson stratum Σ cannot have all its bicharacteristic leaves both vertical and relatively compact unless Σ is symplectic, then Conjecture 2.3 would turn out to be equivalent to the “old” conjecture of [Tr, 1999]. On the other hand, finding examples of characteristic varieties whose Poisson strata are either symplectic or else are entirely foliated by vertical and relatively compact bicharacteristic leaves would show that the two conjectures are not equivalent. It might also provide examples of sums of square operators whose analytic hypoellipticity, or lack of it, would compel us to discard one (at least) of the two conjectures. Remark 2.1 Whereas (Remark 1.1) the conditions on the Poisson stratification of Char L in each one of Conjectures 2.1, 2.2 or 2.3 are all invariant under nonsingular C ω substitutions

On the analyticity of solutions of sums of squares of vector fields

Xj! =

r


327

akj Xk , j = 1, . . . , r

k=1

[akj ∈ C ω (Ω), det(akj (x))1≤j,k≤r = 0 for all x ∈ Ω], such invariance has not been proved, to our knowledge, of Definitions 0.1. Remark 2.2 The validity of Conjecture 2.3 would imply the following result: if Char L is a symplectic analytic submanifold consisting of a single Poisson stratum then L is ahe. In particular, if Char L is a symplectic analytic submanifold and if at each point of Char L at least one bracket {σ(Xi ), σ(Xj )} does not vanish (1 ≤ i < j ≤ r) then L is ahe. Note that this claim is much stronger than the results in [Ta, 1980] and [Tr, 1978]. As evidence for its validity we recall the theorem of Oleinik: the differential operator L = ∂x2 + x2p ∂y2 + x2q ∂z2 in R3 (p, q ∈ Z+ ) is ahe if and only if p = q. Note that Char L is the symplectic submanifold of T ∗ R3 \0 defined by x = ξ = 0; it consists of a single Poisson stratum if and only if p = q. We return to our generic sum-of-squares operator L. It might be worthwhile to state one microlocal version of Conjecture 2.3: Conjecture 2.4 Let Γ be an open and conic subset of T ∗ Ω\0 and let u ∈ D
(Ω) be such that the analytic wave-front set Wa (Lu) of Lu does not intersect Γ . Then Γ ∩ Wa (u) is contained in the intersection of Γ with the union of the nonsymplectic Poisson strata of Char L. For the definition of the analytic wave-front set of a distribution we refer the reader to texts on microlocal analysis (e.g., [G–S, 1985] or Ch. 5, [Tr, 1980]). The intersection of Γ with the union of the nonsymplectic Poisson strata of Char L is a closed (semi-analytic) subset of Γ . We conclude this note with a conjecture about the propagation of analytic singularities, closely related to Conjecture 2.4: Conjecture 2.5 Let Γ be an open and conic subset of T ∗ Ω\0 and let u ∈ D
(Ω) be such that Γ ∩ Wa (Lu) = ∅. Let Λ be any bicharacteristic leaf of a Poisson stratum of Char L whose intersection with Γ is connected. If Γ ∩ Λ ∩ Wa (u) = ∅, then Γ ∩ Λ ⊂ Γ ∩ Wa (u). Example 2.4 Going back to the Oleinik operator L = ∂x2 + x2p ∂y2 + x2q ∂z2 in R3 for 1 ≤ p < q, it is worth mentioning the following consequence of the main theorem in the article [C–Hanges, 2006] in this volume: if u ∈ D
(Ω) satisfies the hypothesis of Conjecture 2.4, then Γ ∩ Wa (u) ⊂ Σ1+ ∪ Σ1− = {((x, y, z), (y, η, ζ)) ∈ R3 × R3 ; x = ξ = η = 0, ζ = 0};

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Σ1± are the two nonsymplectic strata of Char L. Indeed, Σ0+ ∪ Σ0− = {((x, y, z), (y, η, ζ)) ∈ R3 × R3 ; x = ξ = 0, η = 0} is a symplectic submanifold of Char L of codimension 2 on which all the Poisson brackets of length ≤ p of the defining symbols ξ, xp η, xq ζ vanish, whereas one of length p + 1 does not vanish at any point. Moreover, according to Conjecture 2.5 the microlocal singularities of any distribution u in R3 such that Lu ∈ C ω (R3 ) must be propagated by the straight-lines t −→ ((0, y ◦ + t, z ◦ ), (0, 0, ζ ◦ )).

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