Fourier Integrals in Classical Analysis (Cambridge Tracts in Mathematics)

is a Lagrangian submanifold ofT* X \ 0.

e·

e;

Proof: Since w = dx = }:j=1 dx; satisfies dA.J = u it suffices to show that the restriction of the canonical one form, w, to AI(> vanishes identically. But the pullback of w to E4> under the map K. in (0.5.9) is


= d
Notes This vanishes identically on E~, since, by Euler's homogeneity relations, 4> = (f//o, 8}, 4> must vanish identically onE~. Since K is a diffeomorphism, I it follows that w vanishes identically on A~. We shall call J~(x) a Lagrangian distribution if A~ is Lagrangian. We remark in passing that it is not necessary for 4> to be non-degenerate in order for J~(x) to be Lagrangian. For instance, if we set R.n+l 3 8 = (8',8n+l) and define 4J(x,8) = (x,O'} in some cone avoiding the 8n+l axis then A~ is Lagrangian, while 4> of course cannot be a non-degenerate phase function since it is independent of 8n+l·

Remark. Notice that A~ is homogeneous. We shall also encounter Lagrangian submanifolds which are C 00 sections ofT* X. A C 00 section is a submanifold of the form {(x, ,P(x)} with ,P E C 00 • Since the pullback of w under the diffeomorphism x -+ (x, ,P(x)) is ,P · dx it follows from Poincare's lemma that the section is Lagrangian if and only if locally ,P(x) = Vcp(x) for some C 00 function cp. For ,P · dx is closed if and only if ,P has this form. When we study non-homogeneous oscillatory integrals of the form T_xu(x) = [

}Rn

ei.X~(x,y)a(x,y)u(.y)dy,

the (non-homogeneous) Lagrangian associated to cp will play a role which is similar to those associated to homogeneous phase functions.

Notes As we pointed out before, most of the material in this chdpter is standard so we refer the reader to Stein [2), Stein and Weiss [1), and Hormander [7) for historical comments. On the other hand, the argument showing how then-dimensional Hardy-Littlewood-Sobolev inequality follows from the one-dimensional version is taken from Strichartz [1), where it was used to prove a sharp embedding theorem for the wave operator in an.

39

Chapter 1 Stationary Phase

In this chapter we continue to present background material that will be needed later. We start out in Section 1 by going over basic results from the theory of stationary phase, including the stationary phase formula. This will play an important role in everything to follow, especially in the composition theorems for pseudo-differential and Fourier integral operators and in the proof of estimates for oscillatory integral operators. In the next section we use the stationary phase formula to compute the Fourier transform of measures carried on smooth hypersurfaces with don-vanishing Gaussian curvature. The first application of this result is a classical theorem of Hardy and Littlewood, and Hlawka on the distribution of lattice points in IRn.

1.1. Stationary Phase Estimates Stationary phase is of central importance in classical analysis since integrals of the form l(A)

= { ei.\~(Y)a(y)dy, }Rn

A> 0,

(1.1.1)

arise naturally in many problems. The purpose of this section is to study such integrals when ~ E C 00 is real and a E C(f. If~ is not constant, the factor ei.\~(y) becomes highly oscillatory as A-+ oo, and, on account of this, one expects l(A) to decay as A becomes large. Stationary phase is a tool that allows one to make rather precise estimates for these integrals when the phase function, ~, satisfies natural

41

1.1. Stationary Phase Estimates conditions involving its derivatives. The simplest result is Lemma 0.4. 7 which says that (1.1.2) On the other hand, if \74> = 0 somewhere but the determinant of the n x n matrix (8 2 4>j8y;8Yk) never vanishes, then l(A) = O(A-n/2).

The one-dimensional case For simplicity, let us first see that this is the case when n = 1. Specifically, we shall consider one-dimensional oscillatory integrals involving phase functions with non-degenerate critical points. For later use, it will be convenient to work with amplitudes that involve the parameter A. The natural condition to place on such functions a( A, y) is that they always vanish when y does not belong to a fixed compact set, and that

l(~r(:Ara(A,y)l ~ca-y(1+A)-o,

(1.1.3)

for all a, "Y· Under these conditions we have following.

Theorem 1.1.1: Suppose that 4.>(0), 4>1(0) suppa(A, · )\{0}. Set

I( A) = Then i/4>11 (0)

"# 0

0, and 4>'(y)

"#

0 on

fooo ei.M>(y)a(A, y) dy.

and o: = 0, 1, 2, ... , I (:Ar I(A)I

Example.

=

~ ca(1 + A)-1/2-a.

(1.1.4)

The Bessel functions

Jk(A) = (211")-1

Jo{21r i.>.sin8eik8 d9,

k=0,1, ... ,

are a model case. Notice that Theorem 1.1.1 implies that IJk(A)I ~ CkA- 112 .

To prove (1.1.4) notice that, by the product rule, (:Ar l(A) =

L

·~~!

i+k=a J

roo ei.>.~(Y)(i4>(y))i. (:At a( A, y) dy.

Jo

But the hypotheses on 4> and Taylor's theorem imply that there is a nonzero C 00 function 11 such that

4>(y)

= Y2"1(Y)

42

1. Stationary Phase on the support of a, and, hence,

-,a.

~a = Y2a 1

Thus, by (1.1.3}, one sees that (1.1.4} is implied by the following: Lemma 1.1.2 (Vander Corput): Let cp be as in Theorem 1.1.1. Then for k = 0, 1, 2, ... (1.1.5) Proof: Let Ik denote the integral in (1.1.5). To estimate it we fix a C 00 function p(y) which equals 1 when y < 1 but equals 0 for y > 2. Then for6>0

Ik =

fooo ei>.~yka(>.., y) p(y/6) dy + fooo ei>.~ yka(>.., y) ( 1- p(y/6}) dy

=I +II.

The first integral is easy to handle. Just by taking absolute values we get {26

IllS. c lo

ykdy = C'6I+k.

To estimate the second one, we shall need to integrate by parts. As in the proof of Lemma 0.4.7, let

L*(x,D) = :y

i>..~'·

Then for N = 0, 1, 2, ...

IIII

=I/ ei>.~ 5:.

· (L*(x,D})N (yka(>..,y}(1- p(y/6))) dyl

i>ol (

I

L*(x, D)) N (yka(>.., y}(1- p(y/6))) dy. (1.1.6)

However,

Icp'~y) Is.

c;.

and so the product rule for differentiation implies that the last integrand in (1.1.6} is majorized by

>._-N max.(yk-2N,yk-N6-N)

43

1.1. Stationary Phase Estimates for any N. However, if the support of a is contained in [-c, c], where c < oo, this means that, for N ;;::: k + 2, we can dominate II II by

).-N

foe (yk-2N + yk-N rN) dy :5 c>.-N 61+k-2N.

Putting together our estimates, we conclude that

Ilk! ::; c (61+k + >. -N 61+k-2N). However, the right side is smallest when the two summands agree, that is, 6 = >. -l/2 , which gives

I

as desired.

For many problems it is useful to have a variable coefficient version of Theorem 1.1.1. Suppose that 4>(x, y) is a C 00 real phase function such that 4>~(0, 0) = 0,

but 4>~(0, 0)

-1 0.

Then, by the implicit function theorem, there must be a smooth solution y( x) to the equation 4>~(x, y(x)) = 0,

(1.1. 7)

with y(O) = 0, when x is small enough. For such phase functions, we shall study oscillatory integrals

I(x, >.)

=

i:

ei.X., x, y) dy,

(1.1.8)

where a now is a C 00 function having small enough y-support so that y(x) is the only solution to (1.1.7). Then if, in addition, a satisfies

l( l:J)o(l:J){3l(O)f32 o>.

ox

oy

I

a(>., x, y) :5 C0

f3(1 + >..)

_ 0

,

(1.1.9}

we have the following result. Corollary 1.1.3: Let a and 4> be as above. Then for a, {3 = 0, 1, 2, ...

when x is small.

44

1. Stationary Phase Proof: First, set

i(x,y) = ~(x,y)- ~(x,y(x)), and note that this phase function equals zero when y = y(x). Then, if {3 = 0, the quantity we need to estimate is

'J

•" L.J' j!0:.k!

(

a ) '·(ei>.4>(x - y) ) a>. I

•

(

a ) k a( \ a>.

/\l

X,

y) dy.

J+k=a

However, by Theorem 1.1.1 and (1.1.9), >.k times such a summand is 0(>.-1/2-i), which implies (1.1.10) for this special case. To handle the cases involving non-zero {3, note that i has a zero of order two in the y variable when y = y(x). On account of this, one sees that the above arguments show that the inequalites for arbitrary {3 follow from (1.1.5). I To finish the section we remark that the proof of the Van der Corput lemma can be adapted to show the following variant of (1.1.5): If ~(j)(O) = 0 for 0 ~ j ~ m- 1 but ~(m)(o) -1 0 then

Iloco ei~4>(y) a(y) dyl ~ c>.-1/m

(1.1.11)

provided that a has small enough support.

We leave the proof as an exercise for the interested reader. Stationary phase in higher dimensions We shall now consider n-dimensional oscillatory integrals

I(>.)

= f ei~4>(y) a(>., y) dy, >.-> 0,

JR"

(1.1.12)

which involve C 00 functions ~ and a, with ~ being real-valued and a having compact support. Our main task will be to extend Theorem 1.1.1. We will be working with phase functions~ having non-degenerate critical points. Recall that YO is said to be a non-degenerate critical point if V~(Yo)

= 0,

but (1.1.13)

Notice that non-degenerate critical points must be isolated since, by

1.1. Stationary Phase Estimates

45

Taylor's theorem, if we let H be the Hessian matrix in (1.1.13}, then near a non-degenerate critical point Yo,

~(y) = !{H(y- Yo), (y- Yo))+ O(ly- Yol 3 ), and hence

V~(y) = H(y- Yo)+ O(IY- Yol 2 ). Finally, we shall say that ~ is a non-degenerate phase function if all of its critical points are non-degenerate. As before, we shall work with amplitudes a(>., y) whose y-support is contained in a fixed compact set, and now we shall also require that

for all a and 'Y. Our main result then is the following.

Theorem 1.1.4: Suppose that a is as above, " degenerate critical point of~. Set I(>.) = Then, ifV~(y)

~(0}

= 0, and 0 is a non-

f ei.\~(Y)a(>., y) dy.

}Rn

"I 0 on supp a(>.,· )\{0},

I(:.>.r I(.>.)l ~ Ca(1 + >.)-ni2-a.

(1.1.14}

We shall prove (1.1.14} by first establishing the result in the model case where the phase function equals a non-degenerate quadratic form

2 - · · · - Yn2) · Q( Y ) -- 2"1 ( YI2 + · · · + Yj2 - Yj+I

(1.1.15}

Q is obviously non-degenerate since the Hessian of Q is a diagonal matrix where the diagonal entries equal 1 or -1.

Proposition 1.1.5: If Q and a are as above (:>.) 0

J

ei.\Q(y)

a(>., y) dy = 0(>. -n/2-lol).

(1.1.16}

By repeating the arguments in the previous section, however, it is easy to see that (1.1.16} would be a consequence of the following result.

48

1. Stationary Phase

a

where is a function having the same properties as a and Q is a quadratic form as in (1.1.15). (Actually, we have cheated since this change of variables only works in a neighborhood of 0; however, since v~ ¥- 0 when y -::f 0, (1.1.2) implies that we can always assume that the support of a is sufficiently small.) Next we will want to state a variable coefficient version of Theorem 1.1.4. Suppose that ~(x, y) is a real C 00 function satisfying

V 11 ~(0, 0) = 0, det

(a2 ~jayiayk) -:f:.O

when x,y = 0.

Then, as above, the implicit function theorem implies that there is a smooth solution to the equation V 11 ~(x,y(x)) =

(1.1.18)

0

when x is small. Since (1.1.18) holds, y(x) is called a stationary point of~.

For such functions we will consider oscillatory integrals I(x, A) =

f

}Rn

ei.M/(x,y)a(A, x, y) dy,

• #

where we assume that a has small enough y-support so that y(x) is the only solution to (1.1.18) and, in addition, satisfies

for all a, "Yj·

Corollary 1.1.8: Suppose that a a= 0, 1, 2, ... and multi-index "Y

I

(:A) o

and~

are as above. Then for every

(:X) "Y ( e-i>.~(x,y(x)) I(x, A)) I:S Co-y(1 + A)-n/2-o.

(1.1.19)

The proof of this result is the same as the one-dimensional version, Corollary 1.1.3. Let us conclude this section by showing that the estimates we have obtained for oscillatory integrals are sharp. To do so we notice that Lemma 0.1.7 implies the formula

I:

ei>.x2/2dx = (A/27ri)-l/2.

1.2. Fourier T'ransfonn of Surface-carried Measures

49

Suppose that

Q(x) = !(x~ + · · · + x~- x~+l- · · ·- x~); then eiQ is even. Taking this into account, one can see that the above formula together with (1.1.2) and Proposition 1.1.5 implies that, for >. > 0,

r ei>.Q(x) 71(x) dx

JR.,

= (.>./211")-n/277(0) e!!JsgnQ"

+ 0(>. -n/2-1 ),

where Q" denotes the Hessian of Q. But, by using the Morse lemma, we find that this gives that { ei>.4>(x>11(x)dx = (.>./27r)-nl 2 ei>.4>(o)77(0)Idet ell"(O)I- 112 e!!fsgn4>"

'R"

+ 0(>. -n/2-1 ),

(1.1.20)

if ell has a non-degenerate critical point at 0 and 71 has small support. Notice that, in this case, sgn ell" is constant on supp 71, so the first term on the right side is well defined. Similar considerations show that the estimates on the derivatives of oscillatory integrals are also sharp.

1.2. Fourier Transform of Surface-carried Measures Suppose that S is a smooth hypersurface in R.n and let du be the induced Lebesgue measure on S. Then if f3 E C«f(:IRn) we can form the compactly supported measures d11(x)

= f3(x) du(x).

The main goal of this section is to study the Fourier transform of this measure, that is, (1.2.1) when the Gaussian curvature of S never vanishes. The chief result will be that, under this assumption, the decay of the Fourier transform in all directions is 0(1{1-(n- 1)1 2). The key role that curvature plays is evident from the observation that if S were a hyperplane then would have no decay in the normal direction (cf. Theorem 0.4.4). Locally, we can always choose coordinates so that S is the graph of a smooth function h(x'). That is,

d/l

S = {(h(x'), x') : x'

= (x2, ... , Xn) E R.n- 1}.

1. Stationary Phase

46

Lemma 1.1.6: If, for a given multi-index a, we set y 0 = yr 1 then

If

i.\Q(y) a( A, y) y0 dyl

~ C0 (1 + A)-(n+lo.IJ/2.

Furthermore, if lo:l is odd, the integral is 0((1

• • •

y~",

(1.1.17)

+ A)-(n+lo.l+l)/2).

Proof: We shall use an induction argument. In Lemma 1.1.2 we saw that (1.1.17) holds when n = 1. Therefore, let us now assume that the result is true when the dimension equals n - 1. For the induction, we first write the integral in (1.1.17) as

Ln-1 ei.\Q(y') {/_: ei.\yU2 a( A, Yt. y')yr1 dy1} (y')o.' dy', where we have set Q(y') = Q(y) - y~ /2 (which of course is also a nondegenerate function). By Lemma 1.1.2, however, we can control the inner integral. In fact, (1.1.5) implies that the inner integral equals a function a( A, y') which has compact support in they' variable and satisfies

I(:A)j (a~' r' a(A, y')l ~ Cj-y'(1 + A)-(I+o.i)/2-j.

But, by the induction hypothesis, this means that

IA(I+o.i)/2 Ln-1 ei.\Q(y') a( A, y')(y')o.' dy'l which implies (1.1.17). To prove the assertion when the estimate that if 0:1 is odd,

i:

lo:l

~ C(1 + A)-((n-1)+1o.'IJ/2,

is odd one argues as above but uses

ei.\yU2a(A,Yt.Y')yo.1 dy1

= 0(A-1/2-(o.1+1)/2).

We leave the proof of this as an exercise for the reader.

I

The next thing we shall do is to show that Theorem 1.1.4 can be deduced from Proposition 1.1.5. To do so we will make use of the following result which is the classical Morse lemma.

Lemma 1.1.7: Suppose that ~ has a non-degenerate critical point at the origin and that ~(0) = 0. Then near y = 0 there is a smooth change of variables, y -+ y, such that, in the new coordinates,

A"..(;;'\ Y1 --

'¥

1(:-:2 :-:2 Y1 + · .. + Yj -

~

:-:2 -2) · Yj+l - .. ·- Yn

1.1. Stationary Phase Estimates

47

Proof: After making an initial (linear) change of variables, we can always assume that 1

0

1 -1 -1

0

when y = 0. Then, since aiPjay 1 = 0 and a 2 iPf8yr =F 0 at the origin, it follows from the implicit function theorem that there must be a smooth function fh = ih (y 1 ) solving the equation

a iP (th, YI ) = -a

o.

YI

However, if we make the change of variables Y -+ (YI - fh, Y1 ),

we can always assume that fh = 0, that is,

a (O,y) = I

-iP a YI

0.

But, by Taylor's theorem this means that we are now in the situation where

iP(y) = iP(O, y 1 )

+ c(y) yi/2,

where c is a smooth function which is positive near the origin. Finally, if we then let

1h (y)

=

Vc(Y} Yb

it follows that

iP(y)

= !Yi + i(y

1 ),

which by induction establishes the lemma.

I

The Morse lemma and (1.1.16) imply Theorem 1.1.4, since we can rewrite the oscillatory integral there,

I( A) = as

J

ei.\4>(y)a(A, y) dy,

1. Stationary Phase

50

If S is parameterized in this way

du(x)

=

Jt + 1Vhl

2 dx'

and the curvature is (1.2.2)

Also,

The unit normals to S at v E sn- 1 satisfying

(h(x'), x')

are the two antipodal points

(1.2.3) Let

~

be the function inside the gradient, that is, ~(v, x') = (v, (h(x'), x')}.

(1.2.4)

Then, if Vh = 0, v = (1, 0, ... , 0) would be one of the normals, and, furthermore, in this case we would have the identity

(a2 ~foxjoxk) = (o2 h/oxjoxk)· By a rotation argument and (1.2.2) we can conclude that if S has nonvanishing curvature det

(a2 ~(v,x')foxjoxk) ¥:-0

(1.2.5)

when vis normal to Sat (h(x'), x'). From this and the implicit function theorem we see that the Gauss map which sends a point x E S to its normal v(x) E sn- 1 is a local diffeomorphism when K :f:. 0 on S. Having made these remarks, we can now see that the following result follows from Corollary 1.1.8 and Theorem 1.1.4. Theorem 1.2.1: As above letS be a smooth hypersurface in IR.n with non-vanishing Gaussian curvature and dp. a Cif measure on S. Then (1.2.6)

Moreover, suppose that

r

C IR.n\0 is the cone consisting of all~ which

51

1.2. Fourier Transform of Surface-carried Measures are normal to some point x E S belonging to a fixed relatively compact neighborhood N of supp dJ.L. Then,

(~r ~(e)= 0((1 + lei)-N)

vN,

if

e¢ r, (1.2.7)

~(e)= 2.>-i(x;,{)a;(e),

if

eE f,

where the (finite) sum is taken over all points Xj E normal and

N having

e as a (1.2.8)

Remark. Note that (1.1.20) implies that the estimate (1.2.6) is sharp. That is, if dJ.L :f:. 0, there must be a nonempty open cone r c IR.n such that, for large one has the bounds from below

e.

for some c > 0, provided that dJ.L has sqtficiently small support so that the sum in (1.2.7) involves only one term. Corollary 1.2.2: Let x(x) denote the characteristic function of the unit ball in IR.n. Then

lx(e)l $

c(I + lw-(n+I)/ 2 .

•

Proof: Let {3(r) E C 00 (1R.) equal one when r Then, if we let

>

~ and zero when r <

XI(x) = f3(1xl)x(x) and

Xo(x) = x(x)- XI(x) it follows that xo E C
lxo(e)l $

(1.2.9)

cN(l + lw-N

for any N. Consequently, we are only left with showing that

l·

52

1. Stationary Phase If we use polar coordinates,

;b(e) =

f

1

X=

rw, r

> 0, w

Esn- 1 '

we get that

.B(Ixl) e-i(x,e) dx

JlxiE[4,1] =

f1

}1/4

,B(r) (

f

Jsn-1

e-ir(w,e) du(w)) rn- 1 dr.

However, ( 1.2. 7) implies that the inner integral equals

e-irlela1(re) + eirlel a2(re), where

(~r a;(e) = o((l + lw-(n-1)/2-lol). But this, along with (1.1.11), gives that

11;4 .B(r) e-irlel a1(re)rn-1drl ~ C(l + lel)-(n-1)/2. (1 + lel)-1 = C(l + lw-(n+I)/2,

and since the other term satisfies the same estimate, we are done.

I

We are now ready to give the first application of the stationary phase method which will be a classical result of Hardy and Littlewood, and Hlawka concerning the distribution of lattice points in an. Theorem 1.2.3: Let N(>.) denote the number of lattice points j E zn satisfying Iii ~ >.. Then, if B is the unit ball in an'

N(>.) =Vol (B)· >.n +

o(>.n- 2+2/(n+l)).

(1.2.10)

Proof: The strategy will be to estimate N(>.) indirectly by comparing it to a slightly smoother function of >.. With this in mind, let us fix a non-negative C 00 bump function .B which is supported in B(!) and satisfies

f

,B(y) dy = 1.

(1.2.11)

Here B(>.) denotes the ball {x: lxl ~>.}in an. Next, if X>. denotes the characteristic function ofB(>.), then for a number e to be specified later, we let x.x(e,x) = (e-n.B(. /e)* x.x)(x) =

f

e-n,a((x- y)/e) x.x(y)dy

53

1.2. Fourier Transform of Surface-carried Measures and N(e,>.) =

L

ix(e,x).

(1.2.12)

To compare N and N note that the support properties of {3 imply that when lxl~[>.-e,>.+e],

n(x)=x.x(e,x) and, since N(>.) =

LjEZ"

n(j), this implies

N(e,)..- e)$ N(>.) $ N(e,>. +e).

(1.2.13)

To estimate N we need to use the following form of the Poisson summation formula for an' which follows from (0.1.17) via a change of scale:

2: /(j) = 2: /(27rj). jEZ"

jEZ"

Since the Fourier transform of x.x(e,x) equals :h(~) · ,B(e~) the formula and (1.2.12) give that N(e, >.) =

L

h

(21rj),8(21rej)

jEZ"

=

2: x(27r >.j),8(27rej),

)..n

(1.2.14)

jEZ"

where x is the characteristic function of the unit ball. But, if we recall that

/(0) =

J

fdx,

we see that (1.2.14) implies that N(e, >.)=Vol (B).

)..n

+

)..n

2: x(27r>.j),8(27rej). jEZ"

#0 However, by Corollary 1.2.2, we have the estimate

and since {3 E Cifl it follows that

1.8(~)1 :5 C(1 +

IW-N

for any N.

(1.2.15)

1. Stationary Phase

54

This means that we can majorize the second summand in (1.2.15) by >.n times

L

(1 + l>.jl)-(n+l)/2 (1 + leji}-N

jEZ..

#0

~ {

(1 + IAel)-(n+l)/2 (1 + leei}-N cJ.e

11{12:1

={

lt~l{l~e-l

+ {

J1e12:e-l

::=; C [>. -(n+l)/2 E-(n-1)/2 + (>./e)-(n+l)/2E-n]

= 2c>. -(n+l)/2 e-(n-1)/2. Putting everything together, we get

N(e,>.) = Vol(B). >.n +0(>.(n-1)/2E-(n-1)/2). But, if we consider this result for >. ± E and note that (>. ± e)n = >.n O(e>.n- 1), then we see that (1.2.13) implies the estimate

+

N(>.) =Vol (B). >.n + o(e>.n-1 + >,(n-1)/2e-(n-1)/2). The remainder term is minimized when e>.n-1 = >,(n-1)/2g-(n-1)/2, that is,

E

= >.- :~~. And for this choice of E one has (1.2.10).

Remarks.

I

The reader may check that one can never have

N(>.) =Vol (B)>.n + O(>.n- 2). Also, notice that, if one considers the standa;d Laplacian on the n-torus ~ = 8 1 X .•. X 8 1 , that is, "£'1=1(8/88;) 2 = d, then the eigenvalues of -d are all integers that are the sum of the squares of n integers. Thus, (1.2.10) implies that the number of eigenvalues of -d that are ::=; >. 2 equals Vol (B)>.n +O(>.n-2+ 2 /n). Later, we shall see that a weaker result holds for all Riemannian manifolds, and the proof of this result will be similar in spirit to the proof of Theorem 1.2.3.

Notes The material presented here is standard and we have followed the expositions of Beals, Fefferman, and Grossman [1], Stein [4], and Hormander [7]. Most of the material from Section 2 was first proved by Hlawka [1], including the theorem on the distribution of lattice points which extended earlier results of Hardy and Littlewood.

Chapter 2 Non-homogeneous Oscillatory Integral Operators

Recall that in the last chapter we studied non-homogeneous oscillatory integrals of the form I(>.)=

f

JR..

ei.\cf>(y)a(y) dy.

If

II(>.)I-:::::, ).-n/2

as>.-+

+oo,

whenever a(O) =F 0. In this chapter we shall study some natural generalizations of these results, where the integrals now will take their values in LP spaces. Specifically, we shall consider operators of the form T.xf(x) = [

JR..

ei.\cf>(x,y) a(x, y) f(y) dy,

>.

> 0,

where now a is a smooth cutoff function and

det (OXjOYk) =F 0, then we shall find that

56

2. Non-homogeneous Oscillatory Integral Operators This result obviously has the same flavor as the estimates for I(A), and, in fact, one can see that, for every A, there are functions for which IIT.x/ll2/llfll2 ~ (1 + A)-n/2 if a is non-trivial. There are many natural situations where the non-degeneracy condition is not met. For instance, if
2.1. Non-degenerate Oscillatory Integral Operators The main result of this section is the following.

Theorem 2.1.1: Suppose that
on the support of a(x, y) E Clf(IR.n

X

(2.1.1)

"# 0

R.n). Then for A> 0,

{ ei.\1/>(x,y)a(x, y)f(y) dyll L2(Rn) ~ CA -n/2llfiiL2(Rn)· II }Rn

(2.1.2)

If we let T,x be the operator in (2.1.2), then

IIT.xfiiL<"' ~ CllfiiLl· Therefore, by applying the M. Riesz interpolation theorem, we get the following result.

Corollary 2.1.2: If 1 ~ p

~

2 then

IIT.x/IILP'(Rn) ~ CA-n/p'IIJIILP(R")•

(2.1.3)

if p 1 denotes the conjugate exponent.

Remark. Notice that the phase function

Ill

ei(x,y) a(x/.f>.., yjv'>..)

f(y) dyt

~ Cll/llp,

we see that (2.1.3) implies the Hausdorff-Young inequality:

llfllp• ~ Cllfllp·

57

2.1. Non-degenerate Oscillatory Integral Operators Before proving the theorem it is illustrative to restate the non-degeneracy condition in an equivalent form. Let ct/> = {(x,t/>~(x,y),y,-tf>~(x,y))}

c

T*IRn

X

T*IRn

be the canonical relation associated to the non-homogeneous phase function tf>. Then by the remark at the end of Section 0.5, Ct/> is Lagrangian with respect to the symplectic form~ A dx- dTJ A dy. Moreover, the condition (2.1.1) is equivalent to the condition that the two projections ct/>

./

'\.

(x,t/>~(x,y)) E T*IRn

(y, -t/>~(x, y)) E T*IRn

be local diffeomorphisms (i.e., have surjective differentials). In results to follow we shall encounter variations on this geometric condition. It of course means that

IVx[tf>(x,y)- t/>(x,z)]l ~ IY- zl,

IY- zl small.

(2.1.1')

This is what we shall~ in the proof of Theorem 2.1.1 and it of course just follows from the fact that

8 2 tf>(x,y)) 2 Vx[tf>(x,y)-t/>(x,z)] = ( ax;aYk (y-z)+O(Iy-zl ). Proof of Theorem 2.1.1: B_v using a smooth partition of unity we can decompose a(x,y) into a finite number of pieces each of which has the property that (2.1.1 1) holds on its support. So there is no loss of generality in assuming that

IVx[tf>(x,y)- t/>(x,z)] I~ ely- zl on supp a,

(2.1.4)

for some c > 0. To use this we notice that

IIT.x/11~ = where

K,x(y,z) =

1

Jf

K,x(y,z)f(y)f(z)dydz,

ei.X(tf>(x,y)-t/>(x,z)]

a(x,y)a(x,z)dx.

R"

However, (2.1.4) and Lemma 0.4.7 imply that

IK_x(y,z)l $ CN(1 + -XIy- zi)-N 'VN.

(2.1.5)

58

2. Non-homogeneous Oscillatory Integral Operators Consequently, by Young's inequality, the operator with kernel K>. sends £ 2 into itself with norm O(A-n). This along with (2.1.5) yields

IIT>./11~ $CA-n 11/11~,

I

as desired.

2.2. Oscillatory Integral Operators Related to the Restricted Theorem AB we pointed out in the introduction, there are very important situations where the non-degeneracy hypothesis of Theorem 2.1.1 is not fulfilled. For instance, if tjJ(x, y) = lx - Yl then the mixed Hessian of tjJ cannot have full rank n since for fixed x the image of y -+ ¢>~ ( x, y) is sn- 1 (and hence this map can not be a submersion). In fact, one can check that for this phase function the differentials of the projections from Ct/> to T*IRn have corank 1 everywhere. Nonetheless, since the image of y-+ t/J~(x,y) has non-vanishing Gaussian curvature, we shall see that the oscillatory integrals T>. associated to this phase function map .LP(IRn)-+ Lq(IRn) with norm O(A-nfq) for certain p and q. Moreover, we shall actually prove a stronger result which says that certain oscillatory integral operators sending functions of n - 1 variables to functions of n variables satisfy the same type of estimates. AB before, these oscillatory integrals will be of the form T>.f(z) =

J

ei>.tf>(z,y) a(z, y) f(y) dy,

(2.2.1)

except, now, a E COO(IRn x IRn- 1) and tjJ is real and C 00 in a neighborhood of supp a. Thus, in what follows, z shall always denote a vector in IRn and y one in IRn- 1 . The canonical relation associated to tjJ is now a subset of T*IRn x T*IRn- 1 . The hypotheses in the oscillatory theorem will be based on the properties of the projections of Ct/> into T*IRn- 1 and the fibers of T*IRn. First, the non-degeneracy hypothesis is that rankdliT•Rn-1

=2(n- 1},

(2.2.2)

if JIT•Rn-1 : C4> -+ T*IRn- 1 ia the natural projection. Thus, (2.2.2) says that the differentials of this projection must have full rank everywhere,

2. 2. Opemtors Related to the Restricted Theorem

59

or, to put it another way, the mixed Hessian always has maximal rank, that is, 1)2q,

rank (ay;l)zk)

= n- 1.

(2.2.2')

Assumption (2.2.2) of course is an analogue of the hypothesis in Theorem 2.1.1 and it is enough to guarantee that T.\: V(IR.n- 1 )-+ Lq(IR.n) with norm O(,x-(n- 1)/q) if q ~ 2 and p ~ q'. To get better results where the norm is O(.x-nfq) a curvature hypothesis is needed. To state it, we first notice that, since Ct/> = {(z,
Bz0 = IIT;0 Rn (Ctl>) = {} (2.2.3) is a C 00 (immersed) hypersurfa.ce in r;OJR.n. We can identify T.:OIR.n with IR.n and the curvature hypothesis is just that SZo C T;0 JR.n

has everywhere non-vanishing Gaussian curvature.

(2.2.4) .Since (0.4.9 says that changes of coordinates induce changes of coordinates in the cotangent bundle which are linear in the fibers, one concludes that, like (2.2.2), (2.2.4) is an invariant condition. We just pointed out that (2.2.2) is a condition involving second derivatives of the phase function; momentarily we shall see that (2.2.4) is equivalent to a condition involving third derivatives of the phase function. If the two conditions (2.2.2) and (2.2.4) are met, then we say that the phase function satisfies the Carleson-Sjolin condition. The main result of this section concerns oscillatory integral operators with such phase functions. 1)

Theorem 2.2.1: Let T.\ be as in (2.2.1) and suppose that the nondegeneracy hypothesis (2.2.2) and the curvature hypothesis (2.2.4) both hold. Then

(2.2.5) if q = ~:!:lP' and

(1) 1 ~ p

~ 2 for n ~ 3;

(2) 1 ~ p

< 4 for n

= 2.

Before turning to the proof, let us go over a couple of important consequences which will help illustrate the hypotheses. The first one is the restriction theorem for the Fourier transform.

60

2. Non-homogeneous Oscillatory Integral Operators Cor()llary 2.2.2: Suppose that S c an, n ~ 2, is a coo hypersurface with everywhere non-vanishing Gaussian curvature. Then if da is Lebesgue measure on S and if dp. = {3da with {3 E Cff, it follows that

Js lf(~)lr dp.(~) {

(

A

)1/r $ Cs llfiiL•(R")•

(2.2.6)

l

provided that r = ~+ s' and (1) 1$ s $ 2 ~:J> forn (2) 1 $ s < ~ for n = 2.

~ 3;

Notice that the exponents in the Corollary are conjugate to those in Theorem 2.2.1. Keeping this in mind, let us see how (2.2.6) follows from (2.2.5). We first notice that there is no loss of generality in assuming that

s= for some C

00

(1Rn- 1 )

{(y,h(y))}

function h. If we then let

<jJ(z,y) = (z, (y,h(y))} it follows that

(aZ'Izk)

T,xg(z) =

Ln-1

eiA{z,(y,h(y)))

a(z, y) g(y) dy

satisfies (2.2.5). By the change of variables z-+ zf >., this means that if p and q are as in (2.2.5)

{ ei{z,(y,h(y))) a(zf >., y) g(y) dyll Lq(R") $ II }Rn-1

CpiiYIILP(R"-1)

for every>.. Thus, if a(z,y) = f3o(z){3(y) with f3o,{3 E Cff, we conclude that (2.2.6') Since this is the dual version of (2.2.6), the Corollary follows. Later on it will be useful to have oscillatory integral theorems involving phase functions like 1/J(z, w) = lz - wl. Specifically, we shall use the following result which involves a natural analogue of the Carleson-Sjolin condition, which we shall call the n x n Carleson-Sjolin condition.

2.2. Operators Related to the Restricted Theorem

61

Corollary 2.2.3: Suppose t/J E C 00 (1R.n x IR.n) is real and that a(z, w) E C~(IR.n x IR.n). Then if the differentials of the projections of C.,p into T*IR.n have constant rank 2n-1 and if the C 00 (immersed} hypersurfaces Sz0 = {'1/Jio(zo,w) : a(zo,w) "I 0} C T.:a!R.n and Sw0 = {t/J:00 (z,wo) : a(z, wo) "I 0} C T~0 1R.n satisfy (2.2.4), it follows that

111

ei>..,P(z,w)a(z,w)f(w)dwll

R"

~CpA-n/qllfiiLP(R")

£q(R")

(2.2.7)

provided that p and q are as in Theorem 2.2.1. Notice that the n x n Carleson-Sjolin condition is symmetric in the two variables. The proof of (2.2.7) is easy. For the hypotheses imply that we can always choose local coordinates w = (y, t) E IR.n- 1 x JR. such that for fixed t, tjJ(z, w) = t{J(z; y, t) satisfies the hypotheses of Theorem 2.2.1. Thus, since the left side of (2.2.7) is majorized by

/111

ei>..,P(z;y,t) a(z; y, t) f(y, t) dyll

R"-1

dt, Lq(R")

the desired result follows from Theorem 2.2.1 and Holder's inequality since we may assume that f has fixed compact support. Before turning to the proof of Theorem 2.2.1, let us reformulate the curvature condition (2.2.4). Since the image of y-+ t/Ji(zo, y) is a smooth hypersurfa.ce it follows that, if y = Yo is fixed, there must be unique antipodal points ±v(yo, zo) E sn- 1 such that 'Vy( t/Ji(zo, y),

v(yo, zo)) = 0 at y =YO·

Specifically, ±v(zo,YO) are just the unit normals at t/Ji(zo,yo) E Szo· As we saw in Section 1.2, the Gaussian curvature is nonzero there if and only if

a2

det ( ay;aYk (t/JAzo,y), v(zo,YO)) ,

)

"I 0 at y = y0 .

(2.2.41)

In proving Part (2) of Theorem 2.2.1, it will be useful to note that, when n = 2, conditions (2.2.21) and (2.2.41 ) can be combined into the equivalent condition that

f,Y,z

2 )

"I O.

(2.2.8)

'l"yyz2

Proof of Theorem 2.2.1, Part (1): Since (2.2.5) clearly holds for

2. Non-homogeneous Oscillatory Integral Opemtors

62

p = 1, by interpolation, it suffices to prove the inequality for the other endpoint, that is, IIT~/11

n{n-1)

L

2 ( " \1 ) n-

(R")

$C)..-

11/IIP(R"- 1 }•

2 {"+ 1 )

(2.2.9)

By duality, an equivalent problem is to show that the adjoint operator 2{n+1) T~ maps L n+3 (R.n) into L 2(R.n-l) with the same bounds. However, if we use the observation of Tomas,

IIT~gll~ = {

}Rn-1

T~ gT~gdy = { T~T~ggdz JR"

$ IITAT~gll

L

21ny> n-

(R")

11911 L 21"•U> , " (R")

we conclude that (2.2.9) holds if and only if (2.2.10) An advantage of this reduction is that, unlike (2.2.9), both sides involve functions of the same number of variables. In proving this inequality we may assume that z E R.n splits into variables z = (x, t) E R.n-l x R. such that

det(ax~~k (x,t;y)) #0

(2.2.11)

onsuppa.

With this in mind, we define the frozen operators

rt f(x)

=

L. -ei~tP(x.t;y)a(x. 1

t; y) f(y) dy.

Then

TAT~g(x, t) = /_: r((rl~)* [g( ·, t')] (x) dt'.

(2.2.12)

Using the argument in our observation that the n-dimensional HardyLittlewood-Sobolev inequality follows from the one-dimensional version, we see that (2.2.10) would follow from Proposition 0.3.6 and the estimate

_ 2(n+l)

p-

n+

.

(2.2.10')

2.2. Operators Related to the Restricted Theorem

63

In fa.Ct, this inequality, (2.2.12), and Minkowski's integral inequality give

IIT~Ttg( • I t)IILP1 {Rn-1) n(n-1)

$C>..- n+ 1

100 e 1) _ 1t-t'l-1+;-;r l!g(·,t')IILP(R"-1)dt'.

00

Raising this inequality to the p' power, integrating with respect to t, and applying the one-dimensional ij:ardy-Littlewood-Sobolev inequality leads to

IIT~T~giiLP'(Rn)

(100 -oo 1100 _ 00 1t- t'l-1+ e ;-;r llg( ·, t')IILP(R"-1) dt' lp'dt)~ 1/p' n(n-1) (100 ) = C')..- n(n-1) $ C')..- n+ 1 llg( t)II~"(Rn-1) df n+ 1 IIYIILP(Rn)•

$ C>..- !!..1.!!..=!1 n+ 1

1 )

1/p

-OO

•!

as desired. So we are left with proving (2.2.10'). We shall do this by interpolating between £ 2 -+ L 2 and £ 1 -+ L estimates. The first is easy since (2.2.11) and the non-degenerate oscillatory integral theorem give ~ n-1 1 " 2 !ITt IIIP(R"-1) ::; c>..-11/IIP(R"-1)·

00

Since the adjoint operator satisfies the same bounds, we conclude that (2.2.13) So far we have only used the non-degeneracy hypothesis. To prove the L estimate we must now use the curvature hypothesis. To do this, we first notice that the kernel of T((Tt )* is

£ 1 -+

00

K~t' (x, x')

= {

}Rn-1

ei~(tl>(x,t;y)-1/>(x',t';y)] a(x, t; y) a(x', t'; y) dy.

We may assume that a has small support and we claim that

!K~t'(x,x')!

$ C(>..!(x,t)- (x',t')l)-"; 1

(2.2.14)

To prove this, we use Taylor's formula to write

t/J(x,t;y)- t/J(x',t';y) = (Vx,tt/J(x,t;y), ((x,t)- (x',t')))

-+ O(l(x,t)- (x',t')l 2). Thus, if (x, t)- (x', t') belongs to a small conic neighborhood of the unit vectors ±v(x, t) in (2.2.4'), we can apply the stationary phase formula (1.1.19) to get (2.2.14) for this case, assuming, as we may, that (x', t')

2. Non-homogeneous Oscillatory Integral Operators

64

is close to (x, t). On the other hand, if (x, t) - (x', t') is outside of this conic neighborhood, it follows from the definition of v(x, t) that

IVy[(x, t;y)- (x', t';y)] I~ cl(x, t)- (x',t')l,

for some c > 0,

provided again that (x', t') is close to (x, t). So in this case, Lemma 0.4.7 implies that a stronger estimate holds, where in the right side of (2.2.14) we may replace (n - 1) /2 by any power. The estimate (2.2.14) of course implies that n-1 n-1 II Tt (Tt') *I 11 Loo(Rn-1) ~ c>..-2 llfii£1(R"-1)· It-t ~-~

~

1

2

(2.2.15)

n-1 1 n-1 d . Smce 2(n+1) = 2" n+l an

). - "~";11) =

(>.. -(n-1)) =~: (>..- n;1) n~1'

It - t'l- 1+( ;-;,)

=

(It - t'l- "; 1 ) "~ 1 1

the missing inequality, (2.2.10'), follows from interpolating between (2.2.13) and (2.2.15). I

Proof of Theorem 2.2.1, Part (2): Here n = 2 and we must show that forq=3p1 and 1 ~p<4

111

00

-oo

ei~,P(z,y) a(z, y) f(y) dyll

£q(R2 )

~ Cp >.. - 2 /q 11/IILP(R)·

(2.2.16)

To take advantage of the fact that q > 4, we write

(T~f(z) ) 2

=IIei~[tfJ(z,y)+tfJ(z,y')]

a(z, y)a(z, y') f(y)f(y') dydy'. (2.2.17)

We would like to use the non-degenerate oscillatory integral theorem to estimate the Lqf 2 (JR 2 ) norm of this quantity, However, the mixed Hessian of the phase function ~(z; y, y')

= ¢(z, y) + 4>(z, y')

has determinant

~ "'"1 y l '+'z2y

1-"'

11 ( 11 11 ( r/J~ ( z, y1 ) - '+'Z1Y' A.11 ( z, y1 )"' "'"1 Y' - '+'Z1Y z, y )"' '+'Z2Y '+'Z2Y z, y ) I (2 •2• 18) '+'z2y' 1

and so the assumptions are not verified as the determinant vanishes on the diagonal y = y'. On the other hand, the Carleson-Sjolin assumption (2.2.8) implies that

8 iP Idet (aza(y, y')) I~ ely- y'l 2

(2.2.19)

2.9. Riesz Means in llln

65

for some c > 0 if !y-y'l is small. In fact, at the diagonal, they' derivative of (2.2.18) equals the determinant in (2.2.8). There is no loss of generality in assuming that (2.2.19) holds whenever the above integrand is nonzero. To exploit (2.2.19) as well as the fact that the integrand in (2.2.17) is symmetric in (y, y'), it is convenient to make the change of variables

u= (y-y',y+y'). Then since

!dujdyl

= 2, it follows that

a2~ )I ~ c!utl· I (8z8u det

(2.2.19')

Also, since ~(z;u) is an even function of the diagonal variable u1, it must be a C 00 function of u~. So we make the change of variables

v = (~u~,u2)· Then since

!dv/dul =!uti, it follows that ldet

(:::JI ~c.

(2.2.19")

We can now apply Corollary 2.1.2. If r' = q/2 and if r is conjugate to it follows that

r',

IIT~JIIiq(R2)

=

II(T~/) 2 11Lr'(R2)

=

11//

f(y)f(y') !dv/d(y,y')l-l dviL

ei'P(z,v)

~ c>.-2/r' (//lt(y)f(y') ldv/d(y,y')l-llr dvy/r ~ c>.-4/q

(If

!f(yW lf(y'W IY- y'l-l+a dydy') l/r'

with a: = 2 - r. Since a:=

[p/r] -1- [(p/r)'rl,

(2.2.16) follows from applying the Hardy-Littlewood-Sobolev inequality.

I 2.3. Riesz Means in llln Let q({) be homogeneous of degree one, C 00 ' and nonnegative in an\ 0, with n ~ 2. For 6 ~ 0, we define the Riesz means of a given function by

S~f(x) =

(21r)-n

f

}Rn

ei(x,e) (1-

q({/>.)) 6 j({) d{. +

(2.3.1)

2. Non-homogeneous Oscillatory Integral Operators

66

tt = t

for t > 0 and zero otherwise. Iff E S (and thus j E S) it follows from Fourier's inversion theorem that S~f(x) -+ f(x) for every x as >. -+ +oo. In this section we shall consider the convergence of the Riesz means of LP functions. To apply the oscillatory integral theorems of the previous section, we shall assume that the "cospheres" associated to q, Here

6

E

= {~: q(~) = 1},

(2.3.2)

have non-vanishing Gaussian curvature. Note that, since q is smooth and V q ¥- 0, E is a C 00 hypersurfa.ce. The assumption regarding the curvature of E is equivalent to (

a2q )

rank a~ja~k

(2.3.3)

= n- 1. The most important case is when q(~) = 1~1· For a given 1 ~ p

~

oo we define the critical index for LP(JR.n):

6(p)

= max{nl~- ~1- ~.o}.

Note that 6(p) > 0 <===> p a necessary condition for

S~f

-+

f

(2.3.4)

fl. [2n/(n + 1), 2nf(n- 1)). It is known that , in V

when

f

E V, p # 2,'

is that 6 > 6(p). When 6(p) = 0 this is a theorem of C. Fefferman. The other cases follow from the fact that the kernel of S~ is in LP only when 6 > 6(p) for 1 ~ p ~ 2nf(n + 1). (This can be seen from Lemma 2.3.3 below.)

Sf.

Theorem 2.3.1: Let 8 6 = Then if the cospheres E associated to q(~) have non-vanishing Gaussian curvature, and (1) n ~ 3 and p E [1, 2 ~:l)J U [ 2 ~~l), oo), or (2) n = 2 and 1 ~ p ~ oo,

it follows that

1186 IIILP(Rn) ~ Cp,6llfiiLP(Rn)

when 6 > 6(p).

(2.3.5)

Corollary 2.3.2: If p is as in Theorem 2.3.1 then

S~f

-+

f

in V(JR.n),

when f E LP(IR.n) and 6 > 6(p). The proof of the corollary is easy. First (2.3.5) implies that the means S~ are uniformly bounded on LP when 6 > 6(p). Next, given f E LP and

2.9. Riesz Means in Rn

67

e > 0 there is a g E S such that II/-gllv < e, and hence IIS~f- S~gllv = O(e). Since both g and g are inS, Fourier's inversion theorem and the uniform boundedness of S~ : V' --+ V' imply that S~g --+ g in V'. This implies the assertion, since, by Minkowski's inequality, one sees that IIS~f- fllv = 0(3e) when A is large. The proof of Theorem 2.3.1 requires knowledge of the convolution kernel of the operator S 6 :

This kernel will be a sum of two terms, each involving an oscillatory factor and an amplitude. To describe the phases, note that our assumptions imply that given X E an \ 0, there are exactly two points 6(x),{2(x) E E such that the differential of the map

vanishes when e = e;(x). In fact, e;(x) are just the two points in E with normal x. We now define (2.3.6) Since e;(x) is homogeneous of degree zero and smooth in an\ 0, it follows that '1/J; is also smooth and it is of course homogeneous of degree one. Note that in the model case where q = 1e1, we would have '1/J; = ±lxl. Lemma 2.3.3: The kernel of S 6 can be written as al(x)ei1/Jl(x)

6

K (x) =

(1 + lxl)

~ 6 2

+

+

a2(x)ei1/12(x) ~ 6' (1 + lxl) 2 +

(2.3.7)

where the a; are bounded below near infinity and satisfy

I(:xrx a;(x)l :::; Calxl-lal

'Va.

(2.3.8)

The proof of the lemma is a straightforward application of the stationary phase method. First, let TJ E C~(an) be a function which is constant on dilates of E and equals one near E but vanishes near the origin. Then the difference of K 6 and

is in S, since the Fourier !_ransform of the difference is in C~. Therefore we need only show that K 6 can be written as in (2.3.7). ·

2. Non-homogeneous Oscillatory Integral Operotors

68

If we recall that Theorem 1.2.1 says that the inverse Fourier transform of surface measure on the cosphere satisfies

-

(27r)

n

{ "( ) b1(x)ei1/ll(x) JE e' x,w du(w) = (1 + lxl)(n-1)/2

~(x)ei1/12(x)

+ (1 + lxl)(n-1)/2'

for functions bj satisfying (2.3.8), then it follows that two terms. The first is

I

K6 is the sum of

b1(px)eiP'Pl(x) 1 6 d (1 + lpxl)(n-1)/271(P)( - p)+ p = ei1/ll(x)

I

i

bt((p + 1)x) ij(p + 1) eiP'Pl(x) dp, (1 + (p + 1)lxl) (n- 1)/ 2 +

where ij is 71 times a smooth function coming from the polar coordinates. Since lt/11(x)l ~ clxl for some c > 0, and since the Fourier transform of is homogeneous of degree -6 - 1 and smooth away from the origin, it follows that the last integral is of the form a1(x)/(1 + lxl)(n+l)/ 2+6 with a1 satisfying (2.3.8). Since the other term has the same form, we are done. To apply Lemma 2.3.3 we shall use a scaling argument and the following.

Pt

Lemma 2.3.4: Ifa(x,y) is in C[f and suppa(x,y) [~,2]} then

C

{(x,y): !x-y! E

f ei>.,P;(x-y)a(x,y)f(y)dyll L•(Rn) ~CqA-n/qllfiiLP(Rn)• II }Rn

(2.3.9)

if q = ~:!;}p' and (1) 1 ~ p ~ 2 for n ~ 3; (2) 1 ~ p < 4 for n = 2.

Furthermore, for a given tPj the constants depend only on the size of finitely many derivatives of a. To verify this result we need to check that tPj(x, y) = tPj(x-y) satisfies the n x n Carleson-Sjolin condition in Corollary 2.2.3. This is easy since, by construction,

Vxt/lj(x,y) = ej(X- y). This implies that for every xo, Sx0 = {VxtPj(xo,y)} = E, and, hence, the curvature condition holds. Since the Gauss map sn- 1 --+ E is a local diffeomorphism, it follows that, for fixed y, the differential of x--+ ~ .( ,._.,) t= E has constant rank n-1. So the non-degeneracy condition is

2.3. Riesz Means in R.n

69

also satisfied. In view of these facts, Lemma 2.3.4 follows from Corollary 2.2.3. Notice that p < q in (2.3.9). So if we use Holder's inequality and the fact that a vanishes for y outside of a compact set, we get

{ ei>..P;(x-y) a(x, y) f(y) dyll Lq(Rn) ~ Cq A-n/q llfiiLQ(Rn)· II }Rn To apply this, we fix {3 E Ccf(IR.) supported in {8: satisfying }:~00 {3(2-l s) = 1, 8 > 0. We then define

Kf(x) = {3(2-1lxi)K6 (x),

8

(2.3.9')

E [~,2]} and

l = 1, 2, 3, ... ,

and Kg(x) = K 6 (x)- L:i Kf(x). We claim that if q is as in (2.3.91), then we have the following estimate for convolution with these dyadic kernels:

(2.3.10) The estimate for l = 0 is easy since Kg is bounded and supported in the ball of radius 2. Since this implies that Kg is in £ 1 , Young's inequality implies that convolution with Kg is bounded. To prove the estimate for l > 0 we note that the Lq -+ Lq operator norm of convolution with Kf equals the noJin of convolution with the dilated kernels

Kf(x) = AnKf(Ax),

A= 21•

But this function equals n-1 " a 1 (Ax)ei>..p 1 (x) n-1 " .,.1, ( ) A-2 -v {3(! I} = A-2 -v (">. ) '"Y'1 X x (A -1 + lxl)(n+l)/2+6 a1 , x e

plus a similar term involving tP2· Since (2.3.8) 1(/xt'a1(A,x}j ~Co. for every a, (2.3.91) implies that

implies that

IIKf * /llq ~ Cq An; 1 -~- 6 11/llq•

i

But ~ = 6(q) and so (2.3.10) follows. Finally, by summing a geometric series we conclude that (2.3.10) implies that for q as above

IIS6 /llq ~ Cq,oll/llq, 6 > 6(q). This proves (2.3.5) for q ~ 2(n + 1)/(n- 1),n ~ 3, and q > 4 for n = 2. The remaining cases follow from duality and interpolating with the trivial inequality IIS6 /112 ~ 11/112· If we drop the assumption that E has non-vanishing Gaussian curvature, it becomes much harder to analyze the I.l' mapping properties

70

2. Non-homogeneous Oscillatory Integral Operators of S 6 . This is because the stationary phase methods used before break down, and, except in special cases, it is impossible to compute K 6 . Nonetheless, it is not hard to prove the following. Theorem 2.3.5: Assume only that q(e) is homogeneous of degree one and both C 00 and nonnegative in an\ 0. Then, if S 6 is the Riesz mean associated to q, it follows that (2.3.11)

Proof: The operator we are trying to estimate is

s6 f(x) = (2tr)-n }Rn f ei(x,e) (1- q(e))t i<e) ~· If {3 is as above and we now let

sf f(x) = (2tr)-n

1

ei(x.e) /3(2l(1-

q(e)))(1- q(e)) 6 i<e)~. l

= 1, 2, ...•

00

sgf(x)=S6 J(x)- LSff(x), 1 #

I

then sg : £ 1 - £ 1 since its kernel is in we could show that, for any e > 0,

II Sf /1!1

s.

Hence we would be done if

~ Ce 2-(o- n;l-e]lll/1!1·

(2.3.11 1)

To prove this we now let Kf be the kernel of this operator. Then, for every N,

Kf(x) = (2tr)-nlxi-2N

I

ei(x,e) (

-~)N {f3(2l(1-q(e)))(1-q(e))6} ~·

Since ( -~)N {/3(2l(1 - q(e)))(1- q(e)) 6} that

= 0(22Nl T

6l),

it follows

IKf(x)l ~eN 2- 6l!xf2li- 2N, and so, for large N and fixed e > 0,

{

Jlxl>2<1+•)1

!Kf(x)! dx

~ CN,e 2-Nl. .

The last inequality says that Kf(x) is essentially supported in the ball of radius 2(1+e)l centered at the origin. In fact, if .Kf is Kf times the characteristic function of this ball, then convolution with the difference has norm 0(2-lN). So if sf I = .Kf * /, then it suffices to show that

iLl. Kakeya Maximal Functions, Maximal Riesz Means in JR2 71 this operator satisfies the bounds in (2.3.11 1). But if we write f = E /;, where the/; are supported in non-overlapping cubes Q; of side-length 2(l+e)l I it follows that I; is supported in Qj' the cube with the same center but 5 times the side-length. The cubes Qj have finite overlap and so

sf

IISf /1!1 ~ CL IISf !;llu(Qj)· j

Sf

sf

Since the difference between and has norm 0(2-lN), this implies that the desired result must follow from

nsf 111Ll(B(2(1+<)1)) ~ c T[c5- n;l-e)lll/llu(Rn)•

(2.3.11 11 )

But the Schwarz inequality implies that

nsf IIIL1(B(2(l+<)l)) ~ 2~(l+e)lnsf IIIL2(Rn)· We can estimate the right side since

nsf Ill~= (21r)-n /lf3(2l(1- q(e))) (1- q(e)) 6l2 1i(e)l2 df. ~ c2- 2 l 6 uiu~.

r

j {{:l-q({)E[2-1-1 ,2-1+1)}

elf.

~ C2- 2lc5 2-lii/11~-

Combining the last two inequalities gives (2.3.11 11 ).

I

2.4. Kakeya Maximal Functions and Maximal Riesz Means in R.2 We shall show here that, in two dimensions, the maximal Riesz means operators

S~f(x) =sup IS~f(x)l

.>.>0 enjoy the same mapping properties as S 6 when p > 2.

Theorem 2.4.1: If p > 2, 6 > 6(p), and E = {e : q(e) = 1} has non-vanishing Gaussian curuature

IIS~/IILP(R2) ~ Cp,ollfiiLP(R2)· Remark. If one repeats the arguments used in the proof of Corollary 2.3.2, one sees that

S~f(x)

--+

f(x)

almost everywhere as >.--+ oo,

iff E .LP(JR2), p ~ 2, and 6 > 6(p).

72

2. Non-homogeneous Oscillatory Integml Opemtors The proof of the theorem will be based on interpolating with estimates that arise in the special case where p = 4: (2.4.1) The proof of this inequality is based on first dominating s~ I by the Hardy-Littlewood maximal function and certain square functions involving the half-wave operators associated to q({): (2.4.2) Estimating the square functions is the most difficult part of the proof and it is done in two steps. First, there is an orthogonality argument which uses elementary wave front analysis and involves breaking the operators eitQ into simpler ones whose mapping properties are much easier to analyze. After this step, we dominate the pieces which arise from this decomposition by means of a "Kakeya maximal operator." This auxiliary maximal operator involves averages over thin tubular neighborhoods of rays on light cones associated to Q. At the end of this section we shall see that our arguments can b& modified to show that the following related maximal operator # supl [ f(x- ty) oo(y)l t>O

}E

is bounded on .LP(JR2 ) for all p > 2. In subsequent chapters we shall return to variations on this result and we shall make use of techniques used in its proof. Turning to the proof, let us first see how, using _the Hardy-Littlewood maximal function and the Littiewood-Paley estimate, we can deduce (2.4.1) from the seemingly weaker inequality where the supremum is only taken over >. E [1, 2). To do this we let 17 E Clf(IR2 ) be as in the proof of Lemma 2.3.3. Then if we set

it follows that the absolute value of Rff(x) = Sff(x) -T!f(x) is pointwise dominated by the Hardy-Littlewood maximal function independently of >. since

2.4. Kakeya Maximal FUnctions, Maximal Riesz Means in JR2 73 Therefore, if we use the Hardy-Littlewood maximal theorem, we conclude that (2.4.1) would be a consequence of (2.4.1') Notice that, since 17 vanishes near the origin, there must be an integer

ko such that (2.4.3) To exploit this, we use a Littlewood-Paley decomposition off: 00

f

=

L J;,

where i;(~) = ,6(2-ijel)/(e),

-oo

with ,6 E

C~(IR)

satisfying 00

supp,6c[!,2J,

L.6(2-j~)=1,

s>O.

-oo

It then must follow that

L

'I'!. I= rt. (

!; )

if >. E [2k, 2k+l].

li-kl:5ko+2

Using this we get 00

IT! f(x)l 4 S:

L k=-oo

sup

I'I'!.f(x)l 4

AE[2•,2•+1)

Based on this, we claim that (2.4.1') follows from the localized esti-

mate sup I'Tff(x)lll S: C6ll/ll4, II AE(2•,2Hl) 4

k E Z.

(2.4.4)

This claim is not difficult to verify. In fact, since, by Theorem 0.2.10,

74

2. Non-homogeneous Oscillatory Integral Opemtors

II(L: lfkl 2 ) 112 114 :::; Cll/114, we see that (2.4.4) and Minkowski's inequality yield

l

supl7'ff(x)l 4 dx::; .bO

f: I f: I lfkl

sup

k=-oo

lrf(

>.E[2•,2"+1]

:::; C

L !;)(x)l li-kl:5ko+2

I

L: 00

dx

4 dx

k=-oo

:::; c (

4

lfk1 2 )

14

2•

dx:::;

c'

I

ltl 4 dx.

k=-oo

By dilation invariance, (2.4.4) must follow from the special case where k = 0. Furthermore, if we again use the fact that the difference between Tf.f(x) and S1J(x) is pointwise dominated by the Hardy-Littlewood maximal function and recall (2.4.3), we deduce that this in turn would be a consequence of

II

sup IS!f(x)lll :::; AE[1,2]

4

C.sll/114.

if suppj c {e: lei E [2-ko- 2 ,2ko+ 2)}. (2.4.111 )

This completes the first reduction. We now turn to the main step in the proof which involves the use of the half-wave operators eitQ defined in (2.4.2). To use them we need to know about the Fourier transform of the distribution T~ = lri 6X(-oo,O] E S'(R.). By results in Section 0.1, the Fourier transform must be homogeneous of degree -6 - 1 and C 00 away from the origin. This is all that will be used in the proof; however, we record that the Fourier transform is actually the distribution

where

Using this we first notice that

2.4,. Kakeya Maximal Functions, Maximal Riesz Means in .R2 75 Based on this we make the decomposition 00

sit

= si,o I + 2: si,k /, k=1

where, if f3 is as above, then for k = 1, 2, ...

si,k I = (21r)-1c6 A-6

I:

e-iAt /3(2-kltl)

(t + io)-6-1 eitQ I dt. (2.4.5)

These are Fourier multiplier operators, of course, and the multiplier,

mi k• behaves like 2- 6kp(2k(A- q(,))), with p a fixed Ccr'(IR) function. In ~articular, mi k becomes an increasingly singular function around A· E, but, on the' other hand, its L 00 norm is decreasing like 2-k6 as k

-+

+oo. Taking this into account, we naturally expect that for any

E>O sup ISikf(x)lll $ Ce2-( 6 -e)k 11/114, II AE[1,2) ' 4

for

f

as in (2.4.111 ). (2.4.6)

By summing a geometric series this of course implies (2.4.1") .• The inequality is trivial when k = 0. In fact, the Fourier' transform of the distribution [1 - 'Ek:: 1 /3(2-kt)] (t + i0)- 6- 1 must be smooth since the latter is in E'. Consequently, 0 must be a C 00 multiplier.

mi,

Therefore, since we are assuming that / has fixed compact support, the special case where k = 0 in (2.4.6) follows from the Hardy-Littlewood • maximal theorem. To estimate the terms involving k > 0 we shall use the following elementary result.

Lemma 2.4.2: Suppose that F is C 1 (JR). Then if p > 1 and 1/p +

1/r/ = 1

s~p IF(A)jP $

IF(O)jP +

P(

I

IF(A)IP

dA) 1/p' ·

(I

IF' (A)jP

dA) 1/P

To prove the lemma one just writes

IF_(A)jP = IF(O)jP

+loA ~IF(s)IP ds = IF(O)IP +p loA IFip-1·F' ds

and then uses HOlder's inequality to estimate the last term. To apply the lemma in the special case where p =. 2, we now fix p E Ccr'(IR) satisfying suppp C [~,4] and p = 1 on [1,2]. Then, using

76

2. Non-homogeneous Oscillatory Integral Operators (2.4.5) and Holder's inequality, we conclude that the left side of (2.4.6) is majorized by

I (/1 j x I (/I j -Jx

p(A)e-i.Xt.B(2-kltl)(t + i0)-6-1eitQ f dt 12

dA) 1/2 L

(p(A)e-i.Xt).B(2-kltl)(t + i0)-6-1eitQ f dt 12

dA) 1/2 L

Plancherel's theorem implies that this expression is controlled by

r I 2 dt ) 11211 I (lltie[2•-1,2k+1J e' !I 1t12+26 r I 2 dt )11211 I( Jlt1E[2•-1,2k+l) x e' !I 1t126 "tQ

4

.tQ

4

Thus (2.4.6) would be a consequence of the estimate

I( r

I "tQ 12

JltiE[2•,2k+l) e' f

dt ) 112 ltil+e

~ Cell/114,

~

4

suppj C {': 1!1 E [1,2]},

E

> 0.

By taking complex conjugates one sees that we need only estimate the expression involving t integration over [2k, 2k+l]. Moreover, by a change of scale argument, one sees that this in turn is equivalent to proving the following result. ' Proposition 2.4.3: For which

E

> 0 and r > 1 there is a constant Ce for

(2.4.7) Remark.

In Chapter 6 we shall see that the operators

1·

are bounded on L 4 (R.2 ) if and only if u ~ On the other hand, (2.4.7) implies that for any u > 0 this expression is in £ 4 (£2 ([1, 2])).

2.4. Kakeya Maximal Functions, Maximal Riesz Means in lll2 77 Let us introduce some notation that will be used in the proof of the proposition. First, if p and {3 are as above and if we set :Frf(x, t)

= p(t) f ei{x,e) eitq({) /3(1el/r) /(e) d{, JR2

then it is clear that (2.4.7) would be a consequence of

II(/ I:Frf(x, t)1 2 dtf 12 IIL'(R2 )~ ee Te llfiiL'(R2)·

(2.4.7')

The first step in the proof of this inequality is to decompose the square function inside the L 4 norm for fixed x. To do this we write

where

:F~f(x, t) = p(t) f

JR2

ei{x,e)eitq(e>a(r- 112 q(e)-

j) /3(1el/r) /(e) d{

for some function a: E e
suppo: C [-1, 1]

and

L

a(s- j)

=1.

j=-oo

Note that there are O(r 112) nonzero terms :F~f. Moreover, if .. -, denotes the partial Fourier transform with respect to the t variable and if we set itf(x, t)

= (21r)- 1

1 .eit71 (:F~fr<x, 77El~

Jt =

TJ) dTJ,

[r112j - 2r 112, r 112 j

+ 2r112],

(2.4.8)

then we have the following. Lemma 2.4.4: Given any N, there is a uniform constant eN such that if we set f(x, t) = :F~ f(x, t) - ~~ f(x, t), then

Rt

II(/ lll?f(x, t)1 2 dt) 112 IIL'(R2) ~eN T-NIIfiiL'(R2)· Proof: The Fourier transform of :F~f(x,t) is (21T) 2/(e) times m?<e. TJ) = /3(1el/r)

L

eit(q({)-'7) p(t) o:(r- 1' 2 q(e)

_ j) dt.

Note that o:(r- 112q(e)- j) = o if q(e) ¢ [r 1/2j- rl/2,rl/2j + rl/2]. So, for TJ outside of lf, the double of this interval, m? and all of its

2. Non-homogeneous Oscillatory Integral Opemtors

78

derivatives are 0 ( (T

Rt. (211")-1

+ I(e' 1J)I)- N) . But this implies that

r

.

lce,,.,)ER2x(n)c

ei((x-y,e)+t'l)

the kernel of

m?(e, TJ) d{d1J,

is O(T-N (1 + ltl)- 1 (1 + lx- yi)-N). Using this one deduces the lemma.

I By applying the lemma twice and using Plancherel's theorem one gets

II (IIFT/(x, t)l2 dt) 1/2114

~ II(II~FJ/(x,t)l 2 dt) 112 11 4+ C'NT-N 11/114 J

~ 411 (I~ IFJJ(x, t)1 2dt) 112 11 4+ C'N T-N 11/114 J

~ 411 (I~ IFJJ(x, t)1 2dt) 112 11 4+ C'N T-N 11/114· J

On account of this and the fact that FJJ(x,t) vanishes fort fl. !!,4], we can apply Holder's inequality to conclude that (2.4.7') would follow from the p~ely L 4 estimate

II(~ IFJ/12) 1/211L'(R2xR) ~ Ce Te llfiiL'(R2)·

(2.4.711 )

J

So far we have used a decomposition of the Fourier integral operator F with respect to the radial frequency variables. Now we must make a decomposition involving the angular variables. To make this decomposition, we shall need to define homogeneous partitions of unity of JR2 \0 that depend on the scale T. Specifically, we choose C 00 functions x~,v = 1, ... ,N(T) ~ T 112 , satisfying LvX~ = 1 and having the following properties. First, the X~ are to be homogeneous of degree zero and satisfy the uniform estimates

ID"~x~(e)l ~

c"f Thl/2

for all 'Y when lei = 1. Furthermore, if unit vectors e~ are chosen so that x~(e~) =F 0, the x~ are to have the natural support properties associated to these estimates, that is,

x~<e> =

o

if

lei= 1

and

It- e~l ~ cT- 112.

2.4. Kakeya Maximal Functions, Maximal Riesz Means in JR2 79 Using these functions we put (2.4.9)

and define the decomposed operators

It then follows that L:~.~.r;'•j I= :Ftl. Uwe set

we record, for later use, that Qv,j is comparable to a square of size -r 112 x -r 112 and that the sets {Q~.~,;};,v have finite overlap. Moreover, a key observation is that the Fourier transform of (x, t) -+ F,:',j l(x, t) is concentrated on

This is consistent with the results of Section 0.5 since the wave front set of (x, t) -+ eitQ 1 must be contained in {(x, t, e. 77) : 11 = q(e)}. To make our statement about the Fourier transform more precise, let litE C«f(IR) equal one near the origin. Then the difference, E;.•j l(x, t), between F,:',j l(x, t) and

J:•j l(x, t) =

(211")-31

. ei[{x,{)+t7J)

a

({,7J)ER 2 X

satisfies (2.4.10}

One proves this using the argument which was used to estimate the other error term. First of all, we notice that the Fourier transform of F,:"·i l(x, t) is (211") 2 /(e) times

80

2. Non-homogeneous Oscillatory Integral Operators and hence the kernel of Jt;.•i must be

But if the integrand is nonzero, it follows that lq(e) -171 ::?: c(r + I11De, for some c > 0. Since this implies that for such (e, 17), m~J and all of its derivatives are O((r + l(e,f1)1}-N), the estimate (2.4.10) follows as before. In view of (2.4.10), the desired inequality would follow from showing that (2.4.11)

The key observation behind this inequality is that, if 'Ill has small enough support and if we let

then the ~ourier transform of (x, t) -+ ~,j f(x, t) vanishes outside of A~,j. To exploit this we shall use the following geometric lemma.

Lemma 2.4.5: lfe > 0 is as in the definition of A~,j' there is a uniform constant C so that, if j and j' are fixed,

•

Here A~,; + A~, ,j' denotes the algebraic sum of the two sets, and if we identify R.2 and c in the usual way, arg e~ is the argument of the unit vector e~. We postpone the proof of this result. It is based on the fact that our assumption that the cospheres associated to q(e) have non-vanishing Gaussian curvature implies that the cones {(e, q(e))} c R.3 \0 have one non-vanishing principal curvature. This allows the overlap to be finite rather than just O(rl/2). To apply this result, by symmetry, we may clearly restrict the v summation in (2.4.11) to indices as in Lemma 2.4.5. It then follows that the fourth power of the resulting expression can be estimated in the

2.4. Kakeya Maximal Functions, Maximal Riesz Means in IR2 81 following way using Plancherel's theorem and the overlap lemma:

=I L:!L:i:·j I i:',j' ,,

2

dxdt

j,j' v,v'

~I L:IL:(xA:.;+A:,.)(e, 11)(.r:J !)" * (i:',j' /)"1 2 ded11 j,j' v,v'

~ere IL:L:!i:·iti:',i't! 2 dxdt j,j' v,v'

Consequently, if we use (2.4.10) again, we conclude that we would be done if we could show that

II(~ IF:•j /1 2) 112 I L4(R ~ Clogr llfii£4(R2)· 3)

(2.4.12)

V,]

This completes the orthogonality arguments. The proof of (2.4.12) will be based on estimating the L2(R.3 ) -+ L 2(R.2) norm of a "Kakeya maximal function" which involves averages over rays on light cones associated to q(e). To use the Kakeya maximal estimates to follow, we shall need to make use of a variant of Littlewood-Paley theory which involves a decomposition of the Fourier transform into functions supported in a lattice of cubes. To describe what is needed, we first recall that the e-support of the symbols o:~,j of the operators :F;•i are sets Q 11,j all of which are comparable to cubes of side-length r 112 . In particular, each intersects at most a fixed number of cubes in a r 112 lattice of R.2. With this in mind, we let a be as above and set

fm(e)

= o:(r-l/26- m1)o:(r-l/ 2e2- m2)f(e),

m

= (mt,m2) E 'l'}.

Thus EmEZ2 /m =f. In addition, if for a given (v,j), we let Iv,j c Z 2 be those m for which :F;•i fm is not identically zero, it follows from the above discussion that Card (I11,j) ~ C,

(2.4.13)

82

2. Non-homogeneous Oscillatory Integral Operators with C being a uniform constant. Also since the sets Q11,j have finite overlap there is an absolute constant C such that Card {(v,j): mE I 11J} :5 C 'Vm E Z2.

(2.4.14)

We now turn to (2.4.12). Let

f ei(x-y,{) eitq({) o:~,j (t, e) cl{ JR2

K!;J (x, t; y) =

be the kernel of :F;•i. In a moment we shall see that

f IK~J(x,t;y)ldy :5 C JR2

(2.4.15)

uniformly. Thus, the Schwarz inequality and (2.4.13) give

1.1"!/-,j f(x,t)1 2 :5 C

/1 2:: f 2::

fm(Y)I 2 1K!;,j(x,t;y)ldy

mEI~.;

:5 C'

l/m(Y)I 21K!;..i(x, t;y)l dy.

mEI~.;

If we now use (2.4.14) we see that, for a given g(x, t),

1/ L; 1.1"!/-,j J2::

f(x, t)l 2 g(x, t) dxdtl

II,J

:5 C

m

l/m(Y)I 2 su~{ "•3

/rJ{IK~·j

(x, t; y)llg(x, t)l dx dt} dy.

Since the square of the left side of (2.4.12) is dominated by the supremum over allllgll2 = 1 of the last quantity, by applying the Schwarz inequality, we see that we would be done if we could prove

(L, •:JI J/ IK;J (

y

x, t; )I g(x, t) dxdt

I' dy)

112

:5 C llogrl 312 llgii£2(R3)•

(2.4.16)

as well as the following.

Lemma 2.4.6: For 2 :5 p :5 oo,

II ( 2:: l/mi 2 ) 112 IILP(R2) :5 C II/IILP(R2)· m

Proof of Lemma 2.4.6: When p = 2 the inequality holds because of Plancherel's theorem. If we apply a vector-valued version of the M. Riesz interpolation theorem (which follows from the same proof) we conclude

!LI. Kakeya Maximal Functions, Maximal Riesz Means in JR2 83 that it suffices to prove the inequality for p = oo. By dilation invariance we may take T = 1. Then, if Q = [-1r, 1r] 2 and if a is the inverse Fourier transform of a,

(L

lfm(0)1 2) 112

mEZ2

:5

L (L nEZ2 mEZ2

11

f(y- 21rn)a(21rn1- yl)a(21rn 2 - y2)e-i(m,y) dyi 2

Q

Y

12

But the terms in the absolute values of the last expression are the Fourier coefficients of the function there. So Parseval's formula says that the last expression is

211"

L (lif(y- 21rn) a(21rn1- Y1) a(21rn2- Y2)1 2 dy) 112 nEZ2

Q

:::; Cllflloo .

L (lia(21rn1 - Y1) a(21rn2 - Y2) 12 dy) 112 nEZ2

Since

aE S

Q

the last sum converges and this finishes the proof.

I

To prove (2.4.15) and (2.4.16) let us introduce some notation. Given a direction (cos 8, sin 8) E S 1 we let 'Y8 C R.3 be the ray defined by

'Y8 = { (x, t):

X+ tq{(cos8,sin8) =

0}.

(2.4.17)

Clearly these light rays depend smoothly on 8 and our assumption that rank ( a[;l~c) = 1 implies that 'Y8 -=f. 'Y8' if 8 -=f. 81 • We also remark that the union of all the rays is the dual cone to {(e,q(e))} c R.3 \0. The estimate about the kernels which we require is the following lemma.

Lemma 2.4. 7: If e~ are the unit vectors occurring in the definition of F;J and if (cos 811 , sin 811 ) = (~, then given N there is an absolute constant C N for which

IK~,j (x, t; y)l :::; eN r( 1 + r112 dist ((x- y, t), 'Yo..)) -N.

(2.4.18)

Note that (2.4.18) implies that, for fixed y, the kernels are essentially supported in a tubular neighborhood of width r- 1/ 2 around 'Y8~ + (y, 0). With this in mind one gets (2.4.15).

84

2. Non-homogeneous Oscillatory Integral Opemtors Proof of Lemma 2.4.7: To prove (2.4.18}, we recall that o:~J has e-support contained in a sector r~.~ of IR2 \0 of angle O(r- 112 ). Thus, since q( is homogeneous of degree zero,

This implies that for some c > 0

provided that dist ( (x - y, t}, 'Yo.,) is larger than a fixed constant times r- 1/2. By applying Lemma 0.4.7 we get (2.4.18}, since l(&t'o:~'jl $

C0 r-lal/ 2 and l{supp~o:~'j}l $Cr.

I

Using (2.4.18}, one sees that (2.4.16} follows from the following Kakeya maximal theorem.

Proposition 2.4.8: Let 'YO be defined by (2.4.17}. If, for a given 0 < 6 < ~, we let

'Ro = { (x, t) E IR2 x [0, 1] : dist ( (x, t), 'YO) < 6}, it follows that

Proof of Proposition 2.4.8: First we fix a E Cox>(R.2 ) satisfying 0 $ a and alsop E Cox>(JR) vanishing outside a small neighborhood of(}= 0. We set

where X[o, 1] denotes the characteristic function of [0, 1]. Then it suffices to prove that Aog(y)

(211"}-211 ei((y-x,~)+t(q((cosO,sinO),~)] ao(8; t, e)g(x, t) dedxdt = (211")-211 ei((y.~)+t(q((cosO,sinO),~)] ao(8; t, e)9(e, t) dedt

=

R3

R2

R R2

2.4. Kakeya Maximal Functions, Maximal Riesz Means in IR2 85 satisfies

IA9g(y)lll L2(R2) $ C llog6I 3/ 2 11YII£2(R3)· ll sup 9

(2.4.19)

Here "-" denotes the partial Fourier transform with respect to x. Just as before, to apply Lemma 2.4.2, we need to make a couple of reductions. First, if we define dyadic operators

A9"g(y) = (21!")-2

II

ei[(y,~)+t(q((cos9,sin9),~)] X

where

.B(Iel/r) ao(8; t, erq·(e, t) dedt,

.B is as above, then it suffices to prove that IA9g(y)lll ::; Clogr IIYII£2(R3)• ll sup 9 £2(R2 )

T

> 2.

(2.4.19'}

E A~"+~. where Cis a fixed constant l
This is because A9 =

&(qe(cos8,sin9},e} =

o <==>

(cos8,sin8}

=±fer,

l~(qe(cos8,sin8},e}l ~ clel for some c > 0 if (cos8,sin9)

=±fer. (2.4.20}

For instance, to see the first half, we notice that at, say 8 = 0, 8/88(~(cos8,sin8),e} = 6q'{2 ~2 , and q'{2 ~2 does not vanish at (1,0) since q{~ has rank 1, by assumption, and Q'l;e,. (1, 0} = 0 unless j = k = 2, by the homogeneity of q. This argument also gives the second half. Keeping (2.4.20} in mind, we define fork= 1, 2, ...

A~·k g(y)

= (21r}-2 X

II

ei[(y,e)+t(q((cos9,sin9),e)Jp(lel/r}

,8{2-kTl/2 1((-sin 9, COS 8),

fer }I) ac5(8; t, e) g(e, t) dedt.

Thus the symbol of this operator vanishes unless the angle between (cos9,sin9} and the line through and the origin is~ 2kr- 112 . So A;•k vanishes when 2k is larger than a fixed multiple of r 112 • Consequently, if

e

1

86

2. Non-homogeneous Oscillatory Integral Operators

A;•

A Ek>O A;•k, then (2.4.19

0 = we define 0of the uniform estimates

1)

would be a consequence

(2.4.1911 ) If we apply Lemma 2.4.2 we see that this would follow if we could show that

(II!A;•kg(y)l2d8dyf/2 ~ Cr-l/42-k/2119IIL2(R3)• (ll1ioAtg(y)! 2d8dyf 12 ~ Cr114 2k/ 2 IIYIIL2(R3)·

(2.4.21)

In fact, the lemma and the Schwarz inequality imply that the square of the left side of (2.4.19") is controlled by the product of the left sides of (2.4.21). Notice that (2.4.20) implies that on the support of the symbols io(q((cos8,sin8),e} = O(r- 112 2k). With this in mind one sees that

lo A;•k

A;•k,

behaves like r 1l 2 2k so we shall only prove the first inequality. To prove.,it, we first assume k > 0 and notice that the square of the left side equals

I {fo1fol

2

HT,k(t, t'; e) I.B(Iel/r)a(6e)l g(e, t) g(e, t') dtdt'} de, (2.4.22)

where

HT·k(t, t'; e) =

I

ei(t-t')(q((cos8,sin8),{) 1,8{2-krl/21((- sin 8, cos 8),

fer }l)p(8) 12 d8.

We claim that

After applying Schwarz's inequality to estimate the expression inside the braces in (2.4.22), we see that this gives the desired result. To verify the claim, let us assume for simplicity that { is on the first coordinate axis since, in view of the support properties of p, this is the worst case. Then (2.4.20) implies that [q(1 (cos8,sin8)- q( 1 (1,0))6 is a non-vanishing function times r8 2 . Therefore, if we make a change of

!LI. Kakeya Maximal Functions, Maximal Riesz Means in JR2 87 variables, the estimate follows from the fact that if near 0 then

I

ei(t-t')T8 2 p( 2 -kr1/2(J) d(J = T-1/2 2k

I

PE Cij"(R.) vanishes

ei(t-t')2 2 k82 p((J) d(J

= O(r-1/2 2k(1 + 22klt- t'I)-N),

where the constants depend only on the support properties of {3 and the size of finitely many of its derivatives (depending on N). This finishes the proof for k > 0. The proof for k = 0 is simpler: It 0 vanishes just follows from the fact that for fixed the symbol of I for (J outside of an interval of width~ r- 1/ 2 .

A;·

e

To finish the proof of the L 4 maximal theorem for Riesz means we still have to prove the overlap lemma. Proof of Lemma 2.4.5: It is convenient to prove a scaled version of the lemma. To do this, we define sectors r v = { E R.2 \ 0 : arg E [r- 112v,r- 112(v+1))}. In what follows we only consider 0 ~ v < ~r 1 1 2 . Next we let

e

uv,;

=

{(e, 17) E R.2 for some

e

x R.: dist ( (e, 17), (e, q(e))) ~ r- 1

eE r v with q(e) E [1 + jr- 112 , 1 + (j + 1)r- 112J}.

Thus, Uv,j is basically a r- 1 x r- 1/ 2 x r- 1/ 2 rectangle lying on the cone q( Its shortest side points in the normal direction to this cone. The scaled version of the lemma is that there is a uniform constant C such that

{(e' e>)}.

E xu,.;+U,,,j, (e, 17) ~c.

(2.4.23)

v,v'

First, though, we shall prove the dyadic version:

E

xu,.;+u,, .;' (e, 17) ~ c.

(2.4.23')

lv-v'IR::2k To deduce the latter, we first notice that for lv-v'l ~ 2k, Uv,;+Uv',j' is comparable to a rectangle of size 2kr- 1 x r- 1/ 2 x r- 112 . This is because the curvature of E implies that for e~ E r v, e~' E r v' the angle between the normals to the cone at (e~,q(e~)) and (e~',q(e~')) is~ 2kr- 112 if lv - v' I ~ 2k. This means that (2.4.24)

88

2. Non-homogeneous Oscillatory Integral Operators Next, we notice that we are summing in (2.4.23') over ~ 2kr 112 pairs (v, v 1). To see how these two facts yield (2.4.23') we let

n = {(e. TJ)

=

:L:

xu,,;+U,'.;' (e. TJ)

lv-v' 1::::::2• 1 > -2" max

lv-v' 1::::::2•

Using the curvature of the cosphere as above shows that Vol (n) ~ r-1/2. (2kr-112)2 = 22kr-3/2. This is because (non-empty) cross sections with vertical planes have height ~ r- 112 and length~ 22kr- 1, the latter coming from the curvature of E and the fact that the summation involves pairs of sectors of angle~ 2kr- 112. Finally, since the sum is equally distributed on n, the estimate for the volume of n together with (2.4.24) implies (2.4.231) since, as we pointed out, the sum involves~ 2kr112 pairs. To conclude, one notices again by using the curvature of E that the sets Ulv-v' 1:::::2•Uv,j +Uv' ,j' are disjoint for different indices k and k' with lk- k'llarger than a fixed constant. Thus (2.4.231) implies (2.4.23). I End of proof of Theorem 2.4.1: Repeating the arguments at the beginning of the proof of the L 4 estimate shows that the lJl estimate would be a consequence of

II

sup .>.E(1,2]

IS~,kf(x)lll :5 Ce,6 2-(6-6(p)-e)k 11/llp, P

for k = 1, 2, ... , f as in (2.4.1 11 ).

(2.4.61)

However, by using the definition of t5(p) we see that, by interpolating with the L 4 estimate, this would follow if we could prove the inequality for the special cases of p = 2 and p = oo. The inequality for p = 2 trivially holds by repeating the arguments used for L 4 since a stronger version of (2.4. 7) holds:

II (121eitQ /12 dtf'2112 :511/112. due to the fact that eitQ maps £ 2 to itself with norm 1. To prove the £ 00 estimate, one notices that the proof of Lemma 2.3.3 shows that for>. E (1, 2J the kernels of the operators S~ k are dominated by ( n-1 Ce2- 6-,--e)k(1 + lxl)-n-e, n = 2, I

which of course implies (2.4.6') for p = oo.

I

2..4. Kakeya Maximal Functions, Maximal Riesz Means in JR2 89 To conclude this section let us see how the arguments which were used to prove the maximal theorems involving singular multipliers can also be used to prove maximal theorems for operators with singular kernels. Specifically, as above let E = { x E R.2 : q(x) = 1}. Then we have the following "circular maximal theorem."

Theorem 2.4.9: If we assume that q is as above and if we assume that E has non-vanishing curvature, then for p > 2

(2.4.25) Here dn denotes Lebesgue measure on E.

Remark. This result is sharp in the sense that (2.4.25) can never hold for p $ 2. To see this there is no loss of generality in assuming that (-1,0) E E and that the normal there is (-1,0). If we take fv to be the characteristic function of the ball of radius times lx21-l/Pilog lx2!1-I, it follows that fv E LP for p > 1. But the nonzero curvature of E implies that for p ~ 2

!

h

fv(x- ty) dn

=+oo

for x =(xi. t),

lx1l

< 1.

On the other hand, a limiting argument shows that for p > 2 the maximal function in (2.4.25) extends to a bounded operator on LP(R.2) even though functions in this space are only defined almost everywhere. To prove the maximal theorem, we notice that Theorem 1.2.1 implies that the Fourier transform of dn is the sum of two terms each of which is of the form a(,)eiq(e)

(1 + lel)l/2, where ±q(') has the same properties as q and

Thus if we abuse notation a bit and replace

q by q, and then set

Qf(x t) = (21T)-2/ei(x,e)eitq(e) a(t') , (1 + ltel)l/2

j(')~

,

2. Non-homogeneous Oscillatory Integml Opemtors

90

then (2.4.25) would follow from

~~~~giQ/(x,t)IIILP(R2 ) ~ Cpii/IILP(R2)•

P > 2.

To prove this we define the dyadic operators

Then since the supremum over t of the absolute value of the difference beoo

tween g f(x, t) and }: Q2" f(x, t) is dominated by the Hardy-Littlewood k=l

maximal function of f, it suffices to prove that for

T

>1

As before, since these are dyadic operators, one can use Littlewood-Paley theory to see that the inequality holds if and only if II sup IQT f(x, t)lll tE(1,2) P

~ Cp T-e(p) 11/llp.

2

< p < oo.

(2.4.26)

As with Riesz means, the proof is based on establishing the inequality for the special case of p = 4 and then interpolating with easy £ 2 and £ 00 estimates. The main estimate then turns out to be II sup IQd(x, t)ill tE(1,2) 4

~ Ce T-l/B+e 11/114 Ve > 0.

(2.4.26')

If we let p E C3"((l, 2)) then we can apply Lemma 2.4.2 in the case where p = 4 to see that II supt lp(t)Qd(x, t)1111 is dominated by

a(te) /A(t) ricll3 II/ ei(x,~) eitq(~) f3(1cl/r) ._ (1 + t!ei)l/2 ._ ....._ L4 (R2 x(I,2)) x II/ ei(x.~) eitq(~) f3(1el/r) a(te) d ( p(t)a(te) ) } II {. lq(e)p(t) (1+tlel)l/2 + dt (1+tlel)l/2 f(e)cte L'(R2x(l,2J>. A

x

Thus, since q(e) ~ron supp.B(Iel/r), if we now let

Fd(x, t) = p(t)

f ei(x,~) eitq(~).B(Iel/r)

a(t,e) J(e) de,

!L4. Kakeya Maximal Functions, Maximal Riesz Means in R.2 91 where we 88Sume 1(8t)i(~ta(t,e)l::; C;0 (1 + lel)-lol, then it suffices to prove the estimate (2.4.27) This of course should be similar to (2.4.7'). In fact, if n is the function occurring in the proof of this inequality and if we now set

:F~f(x, t) = p(t)

j

ei(x,{) eitq({) .B(Iel/r) n(r_ 1, 2q(e) -

i) a(t, e> /(e) cte,

then the difference between this operator and F~/(x, t) defined in (2.4.8) has L 4 (R.2)-+ L 4 (R.3 ) norm O(r-N) for any N. The important thing about F~f(x, t) is that its partial Fourier transform with respect tot vanishes for TJ fl. I~= [r 112j-2rl/2,r 1/2j+2rl/2j. Note that, as j and j' vary over the ~ r 112 indices for which :F~ =F 0, XIt+It' (TJ) ::; Cr 112. Thus, Plancherel's theorem and Schwarz's in-

k

J,3

equality yield

i:

~~ ~~f(x, t) 14 dt = 3

i:

14

~~f(x, t)F{ f(x, t) 12 dt

3.3

= (27r)- 1/ _ :

14x

~

1t+It'

(TJ}(F~ff*(J-t' !)1 2 dt

3·3

::; Cr 112 / _ : 41 (F~ff*(ij' !)1 2 dt 3.3

Therefore, since the difference between :F~ and ~~ has rapidly decreasing norm, we conclude that

II:FT IIIL4 (R3)

::; Cr 118 ll (

L I:F~/1 2 ) 112 IIL'(R3) +eN T-N IIIIIL (R 4

2 )·

j

Finally, since the proof of (2.4. 7") applies to the slightly different operators in this context we can estimate the right side and get (2.4.26'). To finish the proof of (2.4.26) we see, by interpolating with the £ 4 estimate, that this inequality would follow from

II tE(l,2) sup lgd(x,t)lll :=;CIIfllv. P

p=2orob.

92

2. Non-homogeneous Oscillatory Integral Operators The inequality for p = 2 follows from applying Lemma 2.4.2 and using the £ 2 boundedness of eitQ. The inequality for p = oo follows from the fact that the kernels off-+ Qd( ·, t) are uniformly in L 1 (JR 2 ).

Notes Theorem 2.1.1 is due to Hormander [6]. Theorem 2.2.1 is due to Carleson and Sjolin [1] and Hormander [1] in the two-dimensional case and to Stein [4] in the higher-dimensional case. We have given a slightly different proof of Stein's oscillatory integral theorem which uses an argument in Journe, Soffer, and Sogge [1]. See also Oberlin [1]. Recently, Bourgain [3] has shown that this higher-dimensional oscillatory integral theorem cannot be improved in the sense that there are phase functions satisfying the Carleson-Sjolin condition for which (2.2.5) does not hold for p > 2. On the other hand, it is not known whether the range of exponents in Corollary 2.2.3 is sharp. The two-dimensional restriction theorem is due to Fefferman and Stein (Fefferman [1]) and Zygmund [2]. The £ 2 restriction theorem is due to Stein and Tomas (Tomas [1]). Bourgain [2] has recently improved Corollary 2.2.2 slightly in higher dimensions. T):vl maximal theorem for Riesz means in two dimensions is due to Ca.rbery [1], although the proof given here is slightly different, as it is based on the alternate proof of the circular maximal theorem of Bourgain [1] given in Mockenhaupt, Seeger, and Sogge [1].

Chapter 3 Pseudo-differential Operators

The rest of this course will mainly be concerned with "variable coefficient Fourier analysis" -that is, finding natural variable coefficient versions of the restriction theorem, and so forth. One of our ultimate goals will be to extend these results to the setting of eigenfunction expansions given by the spectral decomposition of a self-adjoint pseudo-differential operator. To state the results, however, and to develop the necessary tools for their study, we need to go over some of the main elements in the theory of pseudo-differential operators. These will be given in Section 1 and our presentation will be a bit sketchy but essentially self-contained. For a more thorough treatment, we refer the reader to the books of Hormander [7], Taylor [2], and Treves [1]. In Section 2 we present the equivalence of phase function theorem for pseudo-differential operators. This will play an important role in the parametrix construction for the (variable coefficient) half-wave operator. Finally, in Section 3, we present background needed for the study of Fourier analysis on manifolds, such as basic facts about the spectral function. We also present a theorem of Seeley on powers of elliptic differential operators which allows one to reduce questions about the Fourier analysis of higher order elliptic operators to questions about first order operators.

3.1. Some Basics We start out by defining pseudo-differential operators on an. We say that a function P(x,e) in C 00 (an X an) is a symbol of orderm 'or more

3. Pseudo-differential Operators

94

succinctly P(x,e) E

sm, if, for all multi-indices a, {3, (3.1.1)

To a given symbol we associate the operator

P(x, D)u(x) = (21r)-n

= (21T)-n

ff f

ei(x-y,{) P(x, e) u(y) d(.dy

ei{x,{) P(x, e) u(e) d(..

By the second formula it is clear that P(x, D)u is well-defined and C 00 0 has symbol 1 • • • e~n. when u E S; also notice that D 0 = ( = We shall say that an operator P : COO -+ C 00 is a pseudo-differential operator of order m if it equals P(x, D), for some P(x, e) E sm. Finally, an operator R which is in s-oo = nmsm is called a smoothing operator since all derivatives of its kernel are 0((1 + lx- yi)-N) for any N, and, hence, R: S' -+ C 00 (1Rn). It is not hard to see that Puis well-defined when u is a distribution. Note that the distribution kernel of P(x, D) is the oscillatory integral

t Jx )

(21r)-n

f

ei(x-y,{) P(x, e) d(. = lim (21r)-n e--+0

eo er

f

ei{x-y,{) p(ee)P(x, e) d(.,

(3.1.2) where p E COO equals one near the origin. By the results in Section 0.5, this definition does not depend on the particular choice of p. Moreover, away from the diagonal, { (x, y) : x = y}, the kernel of P is C 00 , and all of its derivatives are O(lx-yi-N) when lx-yl is larger than a fixed positive constant. Taking this into account one can see that pseudo-differential operators are pseudo-local: If u is C 00 in an open set 0, then so is Pu. Thus, they do not increase the "singular support" of u. However, unlike differential operators, they are usually not local-that is, it is not usually true that supp Pu C supp u. A chief result is that pseudo-differential operators are closed under composition. Theorem 3.1.1: Suppose that P(x, e) E sm and Q(x, e) E SP.. Then P(x, D) o Q(x, D) is a pseudo-differential operator having a symbol P o Q E sm+p. given by

(3.1.3)

3.1. Some Basics

95

By this we mean that p

0

Q-

~~De p

L

.

(!r'Q

E

sm+J1-N

'VN.

(3.1.3')

loi
I(x,y)

= (21r)- 2n = (21r)-n

JJJ ei({x-z,'7)+{z-y,e)Jp(x,f1)Q(z,~)dfldZ~

j ei{x-y,e)(PoQ)(x,~)~

if

(PoQ)(x,~) = (21T)-n

(3.1.4)

JJ ei({x-z,'l)+{z-x,e)lp(x,f1)Q(z,~)dfldZ

= (>./21T)n

JJ

ei.\((x-z),(11-Z)) P(x, >.17)Q(z, ~) d17dz, (3.1.5)

where we have set >. = 1~1 and ~ = ~/ >.. Note that the phase function appearing in the last oscillatory integral,

~ = ((x- z), (17- ~)), satisfies Th~,

(3.1.6) V 71 ,z~ = (x- z,~ -17). by Lemma 0.4.7, if p(s) E C~(IR) equals 1 near 0 and vanishes

when lsi > ~' we see that, modulo a function which is smooth and rapidly decreasing in ~, P o Q equals (>./21T)n

JJei.X{(x-z),(11-Z)) p(lx- zl)p(l -1171)P(x, >.17)Q(z, ~) d17dz.

We can estimate this using the method of stationary phase. First, notice that (3.1.6) implies that the unique stationary point of ~ is (17,z) = (~,x). The Hessian of (17,z)-+ ~is the 2n x 2n matrix ( 0

-1

-1) 0

(I= n x n identity matrix),

which has determinant 1. Thus, since ~ vanishes at the stationary point and since the integration is over JR.2n, we see that formula (1.1.20) implies that, modulo a symbol in sm+~£- 1 , (Po Q)(x,~) equals (>./21T)- 2n/ 2 • (>./21T)n P(x, >.~) Q(x, ~) = P(x, ~) Q(x, ~).

Thus. we have nroved (3.1.3') for N =

1

9. Pseudo-differential Operators

96

The proof of the formula for the general case is' similar, except that one needs to use Taylor's formula: P(x, TJ) Q(z, ~)

=

[~, (:~ rk P(x, ~)(TJ- ~) 0]

L O~lai,I.BI
x

[;,(:x).BQ(x,~)(z-x).B] +RN(x,z,TJ,~).

Taking into account Proposition 1.1.5, one can argue as above to see that (27r)-2n / / ei((x-z),(TJ-{))RNd1Jdz E

sm+~-N,

since RN vanishes of order 2N - 1 at the stationary point. Finally, by using the fact that 1 (27r)-n /r { ei{(x-z),(TJ-{))(11- ~)a(z- x).Bd1]dz = { a!jilal, J 0,

p, -1 {J,

a=

a

one sees that the contribution from the other term has the desired form. We were able to see that PoQ was a pseudo-differential operator with symbol satisfying (3.1.3') directly from the Vander Corput lemma. On the other hand, one often would like to know whether a "formal series" L:~o Pj (x, ~) can be used to build a true symbol of a pseudo-differential operator. Lemma 3.1.2: Suppose that Pj(x,~) E sm;, where mo ~ m1 ~ ··· and mi -+ -oo. Then there exists a symbol P(x, ~) E 8fno such that P "' L:~o Pi, that is, N-1

P(x,~)- L

Pj(x,~) E

smN

VN.

(3.1.7)

j=O

Furthermore, if P "' L:~o Pj and Q "' L:~o Pj, it follows that P(x, D) - Q(x, D) is smoothing.

The proof is simple. Let x(~) = 1- p(~), where pis as in (3.1.2). Then we claim that if we choose ei -+ 0 rapidly, then 00

P(x,~) = LX(ej~)Pj(x,~)

(3.1.8)

j=O 1 This is a consequence of the fact that the kth derivative of the delta function satisfies (211')- 1 J~oo eit((i{)k d{ = 6~k)(t).

3.1. Some Basics

97

works. Specifically, notice that for any Ej the symbol fj(x,~) x(e;~)P;(x,~) is compactly supported in~ and hence insfor any M. Thus, if the Ej satisfy

I(~)

Q

(:x) 13 x(e;~)P;(x,~)l :5 2-j(l + IWm;+I-Iol, when

lal, 1.81 :5 j,

one can check that the series in (3.1.8) converges in the C 00 topology and satisfies (3.1.7). The last part of the lemma follows from the fact that (3.1.7) implies that P(x, D)P;(x, D) E smN. Hence P(x, D)Q(x, D) E smN for all N, which implies that the difference is in s-oo since mN -+ -oo. We now shall see that operators with compound symbols, that is,

Ef=r/

Pu(x) = (21r)-n

Jf ei{x-y,{)P(x,y,~)u(y)d{dy,

(3.1.9)

with (3.1.10) are actually pseudo-differential operators. This follows from the fact thl!t

P(x,y,~)=

L

~~(~tP(x,x,~)·(y-x) 0 +RN(x,y,~),

ioi~N

where RN vanishes of order N + 1 when x = y. Using the last fact, one sees from stationary phase that the kernel associated to RN,

(21T)-n

f

ei{x-y,{) RN(x, y, ~) ~'

is C 0 , a = N- (n + m), near the diagonal, and C 00 and rapidly decreasing away from this set. Thus, if one notes that

j (x- y)aei<x-y,{) (:Yr P(x,x,~) tte (-l)lol j ei{x-y,{)De(:y) 0 P(x,x,~)d{, =

and lets

P(x,~) ~ L ~~ D{(~)a P(x,x,~),

(3.1.11)

Q

one concludes that the operator Pin (3.1.9) agrees with P(x, D) modulo a smoothing operator.

98

9. Pseudo-differential Operators Note that the adjoint of P(x, D) is the operator (27r)-n

If

ei(x-y,{)

P(y,~)u(y)~dy.

Thus, as a special case of the above we have the following. Theorem 3.1.3: If Pis a pseudo-differential operator of order m then so is its adjoint p•. Moreover, the symbol of P*(x, D) is given by the formal series (3.1.12)

We now point out that (3.1.12) implies the following fact which will be useful later. Corollary 3.1.4: If P is a self-adjoint pseudo-differential operator of order m, then the symbol of P satisfies P(x,~)- ReP(x,~) E

sm- 1.

Our next result concerns parametrices, that is, approximate inverses, of operators with symbols satisfying 1

C1(1 + l~l)m ~ IP(x,~)l ~ C2(1 + lel)m,

I

0 < C; < oo,

1~1 ~ C,

(3.1.13)

where C is some constant. Theorem 3.1.5: Suppose that P E sm satisfies (3.1.13). Then P(x, D) has a parametrix E in s-m. That is, modulo smoothing operators,

P(x, D) o E(x, D) = E(x, D) o P(x, D) = I, where I is the identity operator.

To constructE, we first notice that (3.1.13) implies that P(x,~) =F 0 when ~ does not belong to a fixed compact set. Therefore, if x E C 00 equals 0 in a neighborhood of this set but 1 near oo, it follows that E1(x,~) = x(~)/P(x,~) E s-m. But then (3.1.31) implies that

P(x, D) o E1(x, D)= I+ r1, where r1 is a pseudo-differential operator of order -1. Next, if r1(x,~) is its symbol and we put E2(x,D) = -x(~)rl(x,~)/P(x,~), then

P(x, D) o [E1(x, D)+ E2(x, D)] =I+ r2, where r2 is of order -2. Thus, if for j = 2, 3, ... , E; is chosen inductively

9.1. Some Basics

99

so that p 0 E; = -r;-t modulo s-i' where r;-1 is the error from the (j - 1)st stage, we can conclude that

satisfies Po E = I modulo smoothing operators. The s~e arguments of course_ imply that P(x, D) has a left parametrix E E s-m, meaning that E o P = I up to smoothing operators. But then since

E= E0

(P 0 E) =

(E 0 P) 0

E

=E

modulo

s-oo'

we see that we can actually take E =E. This completes the proof. We conclude the section by discussing the £P boundedness of pseudodifferential operators. Theorem 3.1.6: Suppose that P is a pseudo-differential operator of

order zero. Then 1
< oo.

(3.1.14)

We claim that we can assume that P(x, {)is compactly supported in x. This is the case since the kernel K(x, y) of P(x, D) is rapidly decreasing away from the diagonal. Consequently,

f

IKidx,

./1x-yj?_l

f

IKidy

J1x-yj?_l

~ c,

and so, ifwe let Po be the operator withkernelKo = K, when lx-yl and 0 otherwise, it follows that

II(P- Po)/llp ~ C 11/llp·

~

1,

(3.1.15)

Next, if {Q;} denotes the collection of cubes in R.n of unit side length whose vertices have integer coordinates, and if we let f;(x) = f(x) when x E Q; and 0 otherwise, then it follows that

since the supports of the functions Pof; have finite overlap. Thus (3.1.14) would follow if we could prove that

IIPo/;llv ~ C 11/;llv uniformly. However, since Pof;(x) = 0 when x ¢ Qj, where Qj is the cube with the same center but four times the side-length, we see that

3. Pseudo-differential Operators

100

(3.1.15) implies that the last inequality would be a consequence of

IIPJ;IILP(Qj) $ Cll/jiiLP, which establishes the claim. To summarize, we have just seen that it suffices to prove that 1
< oo,

(3.1.14')

whenever P(x, e) E so vanishes for X outside the unit ball. Let us first see that this inequality holds in the one-dimensional case. We first write

Thus,

IIP(x, D)/(x)llp

$11/ ei(x,e) P(O, e)f(e) cieiiP + /11/ ei(x,() P'(t,e)f(e) ciellp dt.

But both P(O, e) and P'(t, e) satisfy the hypothesis of Theorem 0.2.6, since P is a symbol of order zero. Therefore since we are assuming that P( t, e) has compact support in t, the desired result follows from this multiplier theorem. The argument for higher dimensions is similar. Arguing as above shows that

IIP(x, D)/IILP(Rn)

$ II/ ei(x,e) P(O, e) !(e) ciellp + /_:11/ ei(x,() 0: 1P(y1,o,e) f(e)cieiiP dy1 + ... + f 11/ei(x,e)~ ···~P(y,e) f(e)ciell }Rn OYl Oyn

dy. P

And since the multipliers inside the lJl norms satisfy the conditions of the multiplier theorem, the right side is $ Cll/llp. giving us (3.1.141).

3.2. Equivalence of Phase Functions In this section we shall study linear operators of the form

(P'I'u)(x)

= (27r)-n

II

ei'l'(x,y,e) P(x, y, e)u(y) dedy,

(3.2.1)

3.2. Equivalence of.Phase Functions

101

where P(x,y,e) E sm (i.e., it satisfies (3.1.10)) and is assumed to vanish when lx- Yl is larger than some fixed positive constant. 2 Since we shall want to compare such operators to pseudo-differential operators it is natural to assume that rp is real and rp

IV'e'PI

e s1 , ~

cix- Yl

on supp P,

for some c > 0.

(3.2.2)

Clearly rp = (x- y, e} satisfies (3.2.2), and we have seen that in this case Prp is a pseudo-differential operator of order m. Our main result is that this is true for other phase functions as well. Theorem 3.2.1: Suppose that rp is as above and that

cp(x, y, 0 = (x - y, e} + O(lx- Yl 2 1el).

(3.2.3)

Then, if P(x, y, e) E sm, Prp is a pseudo-differential operator of order m, and, moreover, if we set P(x,e) = P(x, x,e), it follows that Prp-P(x, D) is a pseudo-differential operator of order (m- 1). Conversely, given a pseudo-differential operator P there is an operator Prp such that P- Prp is a smoothing operator. "

By (3.2.3) we of course mean that

for every n. Before turning to the proof, let us first see that this result implies that compactly supported pseudo-differential operators are invariant under changes of coordinates. Specifically, suppose that n and nit are open subsets of R.n and that K. : n -+ nit is a diffeomorphism; then one can define a map sending functions in n to functions in nit by setting X

En.

Our next result says that there is also a push-forward map for pseudodifferential operators. Corollary 3.2.2: Let tt, n, and nit be as above. Then if P(x, e) E sm vanishes for X not belonging to a compact subset of n, there is a 2 We have placed this condition only to allow our assumptions on rp to be of a local nature in what follows.

3. Pseudo-differential Operators

102

pseudo-differential opemtor Prt(x, D) which is compactly supported in Ott such that, modulo smoothing opemtors, y

Furthermore, if ~'(x)

= (a~;ax)

= ~(x).

(3.2.4)

is the Jacobian matrix,

Plt(~(x),
E

sm- 1 I

(3.2.5)

and, thus Prt(~(x), e)- P(x, t~'(x){) is of order m- 1. To prove this we first put P(x,y,~) = p(x-y)P(x,~), where p E C(f equals one near the origin. Then, modulo a smoothing error, P(x, D)u equals

JJei{x-y,{) P(x, y, e) u(y) d{dy = (21r)-n JJ ei{x-y,{)P(x,y,{)urt(~(y))d{dy.

(21r)-n

However, if we let x = ~(x), then, by changing variables in the integrations, we see that this integral equals (21r)-n times

jj ei{rt- (x}-rt- (z},'rt'(x) 1

1

11}

x { P(~- 1 (x), ~- 1 (z), t~'(x)f1)1 8ij; 1 ll~l }ult(z) d17dz. When z = fore, since

x the expression inside the braces equals P(x, t~'(x)f1). There( 8~~)) (~- 1 (x)- ~- 1 (z)) = x- z + O(lx- zl 2 ),

Theorem 3.2.1 implies the result.

Proof of Theorem 3.2.1: Let rt~o(x, y, e) = rt~(x, y, e) and rt11 (x, y, e) (x- y, {). Then for 0 :::; t :::; 1 set rt't(X, y, e) = (1 - t)rt~o + trt~1 and

(Ptu)(x) = (21r)-n

JJei
where P(x, y, e) E sm is as in (3.2.1). Since rt~o (3.2.3), we can assume that IVert~tl ~

=

= ft1 satisfies (3.2.2) and

clx- Yi on supp P(x, y, e).

(3.2.6)

We may have to decrease the support of P near the diagonal; however, since (3.2.6) holds for t = 0, Po - P'P would then be smoothing.

3.2. Equivalence of Phase Functions Next, notice that d ( dt

)j Ptu = (211")-n !rJr[i(cp1 -

..

cp0 )]3e'ep• P(x,y,e) u(y) ~dy.

The symbol here,

(i( IP1

-

cpo)]i P(x, y, e),

is in sm+i, but recall that we are assuming that IP1 -cpo = O(lx-yl 2 1el). Thus, by the arguments of the previous section, when t = 1, the operator with this compound symbol is actually a pseudo-differential operator of order m - j, and for all t the kernel of Pt becomes arbitrarily smooth, as j --+ oo, since (3.2.6) holds. To use this, set Qi

(-1)i ( d )j I = -.,dt Pt . J. t=1

Then, by Taylor's formula, k-1

Po=

1

L Q; + (-1)k /k! lor tk- 1(£)k Pt dt. 0

Thus, if we let P(x, D) be a pseudo-differential operator defined by the formal series EO" Q;, it follows that P- Po has a smoothing kernel. Furthermore, if we let Q(x, D) have symbol P(x,x, e), then Po-Q(x, D) is a pseudo-differential operator of order m-1. This proves the first part of the theorem. To prove the converse assertion one simply repeats the argument reversing the roles of cpo and fP1· Then, since it is clear that (3.2.2) implies that Lemma 3.1.2 can be extended to include operators of the form (3.2.1), one sees that any pseudo-differential operator can be written as an operator Pep, modulo a smoothing error. I Since the composition of pseudo-differential operators is a pseudodifferential operator, it follows that, if Pep is as above and if Q(x, D) is a pseudo-differential operator of order J.L, then Q(x, D) o Pep is an operator of the form (3.2.1) which, modulo an operator of order m+ J.L -1, equals

(21r)-n!! eiep(x,y,~) Q(x, e) P(x, y, e) u(y) ~dy. In the next chapter we shall need to know more about the lower order terms. One could evidently find this by a closer examination of the proof of Theorem 3.1.1; however, it is easier to do things directly. We shall assume that the symbol Q(x, e) vanishes when X fl. K, K compact, since this is all we shall need. The first step in computing a

103

104

3. Pseudo-differential Operators composition formula for Q o P'P is to notice that when

::1 0

(3.2.7)

on K,

then Vander Corput's lemma can be used to compute ..X when

f

(3.2.8)

large,

is smooth. To see this we first introduce the notation

(3.2.9)



(21T)-n

II

ei((x-y,e)-.X((x)-(y))) Q(x,e) f(y) tJedy.

(3.2.10)

Since we are assuming (3.2.7), the only significant contributions in this integral occur when lei ~ ..X. Therefore, we may assume that Q(x, e) vanishes when lei fl. [c1..X, c2..X] for certain c; > 0, because if we multiplied Q by the appropriate cutoff function, the difference between the ana1ogue of (3.2.10) and the above integral would be O(..x-N) for all N. Next observe that we can make a change of variables to rewrite (3.2.10) as

(21T)-n

II

ei((x-y,e)-.X2(x,y)) Q(x, e + ..XV<jJ(x)) f(y) tJedy.

We use Taylor's formula to write

Q(x,e + ..XV<JJ(x)) =

I:

~~ (~r· Q(x,..XV
0

+ RN,.x(x,e),

loi
II

ei((x-y,e)+.x2(x,y))RN,.xf tJedy =

,.xn

II

ei.x((x-y,e)+2(x,y))RN,.x(x, ..xe) f(y) tJedy

is O(..X~£-N/2 ), and moreover O(,.X~£-(N+I)/ 2 ) if N is odd. For each N, the constants involved depend only on the size of finitely many derivatives of

9.2. Equivalence of Phase FUnctions

105

Thus, since

we see from the above that

e-i~Q(x, D) [ei~4>

1]

= "" ..!..(~) 0 Q(x,~V(x))Dy0 {ei~4>2 (x,y)f(y)}l y=x L....J a! at;. lai
+ ~~-&-N/2 RN(x, ~), where RN(x, ~) is bounded for all N and ~ 1 / 2 times the function is bounded if N is odd. Now, let cp and P(x,y,f;,) be as in (3.2.1). Then if we take .A = lt;.l, f = t;./lt;.l, and set

f(x)

= ~-mP(x,y,~f),

(x)

= ~- 1 cp(x,y,~f),

we can conclude that

e-i'PQ(x, D) [Peirp]

=ILl

~! ( ; r

Q(x, v xiP) Dr; { P(z, y,t;,)ei~4>2(x,z)} L=x

a
+ RN(x,y,t;,),

(3.2.11)

where RN E sm+~-&-N/ 2 when N is even and sm+~-&-(N+l)/ 2 when N is odd. We shall want to apply this formula for N = 3. To do this, notice that since 2(x, z) vanishes of order 2 when z = x, it follows that for lal = 2, Dr;{P(z,y,t;,)ei~2}z=x = P(x,y,t;,){Dr;ei~4>2 }z=x• modulo a symbol of order m. Also, since the first term on the right side of (3.2.9) is linear, it follows from our choice of 4> that {Dr;ei~4>2 }z=x = D~(icp(x,y,t;,)) when Ia I = 2. Consequently, if we let

Q1(x,y,t;,) =

I:

~~ (:erQ(x, Vxcp)D~icp E s~-'- 1 ,

lal=2 then (3.2.11) gives the following result.

(3.2.12)

3. Pseudo-differential Operators

106

Theorem 3.2.3: Assume that P(x, y, e) E sm and that cp(x, y, e) is as above. Then if Q(x, D) is a pseudo-differential operator of order p. and Q1 is defined by (3.2.12), e-icpQ(x, D) [Peicp] =

L lo!<2

~! ( ;

r

Q(x, v xiP) D~ P+QIP+R(x, y, e), (3.2.13)

where R(x,y,e) E sm+p.- 2.

(3.2.14)

By writing out the integrals, one sees that Theorem 3.2.3 can be used to compute Q(x, D) o Pep modulo sm+p.-2.

3.3. Self-adjoint Elliptic Pseudo-differential Operators on Compact Manifolds For later use, we now define pseudo-differential operators on a C 00 compact manifold M. A continuous map P: C 00 (M)---+ C 00 (M) is called a pseudo-differential operator of order m if it satisfies the following conditions: Whenever Ov is a local coordinate patch and 1/Jo and 1/JI are in C8" (Ov) and 1/J1 = 1 in a neighborhood of supp 1/Jo, the operator Pvu(y) = 1/Jo(x). P(1/Jl . u 0 ll:v) (x),

y = Kv(x),

u E C 00 (flv ),

is a pseudo-differential operator of order m in R.n. The operator Pv is of

nv.

course supported in the compact set ll:v(supp'I/Jo) c From this definition we see that, in local coordinates, P is (modulo operators with C 00 kernels) of the form P(x,D) where P(x,e) E sm. In what follows we shall always assume that m > 0, and, in addition, it will be convenient to assume that Pis classical which means that, in every local coordinate system, P(x,e) "'LPm-;(x,{),

where Pm-; is homogeneous of degree m - j. When this happens, we shall write P E Wci(M). If we use the coordinates (x,e) E T*M\0 ---+ (~~:v(x),ev) E R.n x R.n \ 0, then we can define the principal symbol of P by setting

It follows from the change of variables formula for the cotangent bundle, (0.4.91), and (3.2.5) that the principal symbol is a well-defined C 00

107

3.3. Self-adjoint Elliptic Pseudo-differential Operators function on T* M \0. P will be called elliptic if p(x, e) does not vanish on this set. Next, P E "111~ is said to be self-adjoint with respect to a positive C 00 densityl dx if

(Pu, v} = (u, Pv}

V u,v E C 00 (M),

where the inner product is defined as

(/,g)=

JM fgdx.

Locally, we can always choose coordinates on M so that dx is identified with Lebesgue measure on R.n and, consequently, we can conclude from Corollary 3.1.4 that if P E Wci is elliptic and self-adjoint , then its principal symbol must be real. Moreover, if the dimension of M is greater or equal to two, as we shall assume throughout, the principal symbol must be identically positive or negative on T* M \ 0. Recall that in R.n, the Sobolev spaces L~ (R.n) are defined by the norms

There is a natural way of defining Sobolev spaces on M using this. If { t/Jv} is a partition of unity subordinate to a finite covering U f!v = M, then we shall put v

where fv(x) = (t/Jv/)(~~:;; 1 (x)). Clearly, different partitions of unity give comparable norms. For the moment let us assume for the sake of simplicity that P is a first order classical self-adjoint elliptic pseudo-differential operator. Then we can assume that its principal symbol p(x,e) is positive, and let Q be an elliptic pseudo-differential operator with principal symbol (p(x, e)) 112 . Then Theorems 3.1.5 and 3.1.6 imply that

llulli~ 12 (M) ~ C1 11Qulli2(M) + C2llulli2(M)· On the other hand, Theorems 3.1.1 and 3.1.3 imply that P- Q*Q is a pseudo-differential operator of order zero and since Theorem 3.1.6 3 dx is a C 00 positive density if it is a distribution that always projects to positive measures on compact subsets of coordinate patches.

coo

108

3. Pseudo-differential Operators implies that such operators are bounded on £ 2 , we see that the Schwarz inequality implies

IJ

Puudx- j Q*Quudxl S: Cllull~·

However, since

IIQull~ =

j Q*Quudx,

the last two inequalities give

llulll2

S:

1/2

C1 jPuudx + C' !lull~·

Therefore, if c is large enough, it follows that

llulli2

1/2

S:

C1 j(P + c)uudx.

(3.3.1)

We shall assume that c = 0 since replacing P by P + c will not affect any of the results in what follows. From this inequality we can conclude that P is invertible. By the Rellich embedding theorem, the inverse must be a compact operator on L 2 (M) since M is compact. Consequently, the spectral theorem implies that ~ 00

P= L>.;E;, j=l 00

I=

(3.3.2)

LE;, j=l

where E; : £ 2 -+ L 2 are the projection operators that project onto the one-dimensional eigenspace E; with eigenvalue >.;. By (3.3.1) the eigenvalues are all positive, and, since +oo is the only limit point, we may assume that they are ordered so that

>.1 $ >.2 S: · · · . Since the eigenspaces are mutually orthogonal (3.3.2) gives 00

ll/lli2(M) =I: IIE;/IIi2(M)• j=l

Let {ej ( x)} be the orthonormal basis associated to the spectral decomposition. Then of course

E;f(x) = e;(x)

JM f(y) ej(Y) dy.

9.3. Self-adjoint Elliptic Pseudo-differential Operators

109

We claim that e;(x) E C 00 (M). To see this, we notice that one can use Theorems 3.1.5 and 3.1.6 to prove that for every k = 1, 2, ... there is an inequality (3.3.3)

This inequality and the Sobolev embedding theorem imply that e;(x) is C 00 since it belongs to every Sobolev space L~. It also shows that the spectrum of any C 00 ( M) function is rapidly decreasing, that is,

>.f

I/

VN

u(y) e;(Y) dyl-+ 0

if u E C 00 (M).

(3.3.4)

In the next chapter, we shall investigate the distribution of the eigenvalues of P, or, more specifically, the behavior of the function

N(>.) = #{j : >.; :::; >.}. Let us see now that this function is tempered. To do this we let

L

S>.l(x) =

E;l(x)

>.; $>. be the projection operator onto U>.;$>.£;. Then (3.3.3} implies'

IIS>.IIIL: :::; c>.klllll£2· By the Sobolev embedding theorem we see from this that if IE U>.;$>.£; then

lllllvx• :::; CIIIIIL[n/2+11 <- C>.n/ 2+111111£2, 2

where [ ·] denotes the greatest-integer function. However, if we define the spectral function

L

S>.(x,y) =

e;(x)e;(y),

>.; $>. then since this is the kernel of S >.,

I/

S>,(x,y)l(y)dyl = IS>.I(x)l:::; c>.ni 2+111111L2·

Since this inequality holds for all that

I

E L2 , by duality, we can conclude

( / IS>.(x,y)l2dyf/2:::; c>.n/2+1, which by the above gives the pointwise estimate x,yEM.

9. Pseudo-differential Operators

110 Finally, since

N(>.) =

JM S>,(x,x)dx,

it follows that this function is tempered. Later on we shall improve these estimates and see that N(>.) = c>.n + O(>.n- 1 ) when P satisfies some natural assumptions. So far we have been discussing first order operators; however, the following result reduces the study of properties of the spectrum etc. of elliptic self-adjoint operators of arbitrary order to the first order case.

Theorem 3.3.1: Let P E 'W~(M) be self-adjoint and positive with m > 0. Then the operator P 11m defined by the spectral theorem is in w~ 1 . Its principal symbol is (p(x, e)) 11m, if p(x, e) is the principal symbol of P. Let us sketch the proof. In local coordinates we choose

where x E C 00 vanishes near the origin but equals one near infinity. If we let Q1 = (Q1 + Qi)/2 then Q1 is self-adjoint, and it is also classical by Theorem 3.1.3. In addition, Theorem 3.1.1 implies that (QI)m- P is in wcl'- 1 . By the arguments of Section 3.1 we can recursively choose self-adjoint classical pseudo-differential operators Q; of order 2 - j such that P- (Q1 + · · · + QN)m is in wcl-N for every N. Thus, if we let Q E W~ be a representative of the formal series E Q;, we conclude that P- Qm is smoothing. Since each Q; is self-adjoint, Q equals its adjoint up to a smoothing operator, and so, after possibly adding such an operator, we can assume that the Q .constructed is self-adjoint. Since it is elliptic and first order (3.3.1) implies that it has at most finitely many non-positive eigenvalues, and, therefore, after possibly modifying it on a finite-dimensional space, we can assume that Q is positive as well. To summarize, we have seen that there is a positive first order self-adjoint elliptic Q E W~ such that

P-Qm=R, where R is smoothing. We claim that Q- P 11m is smoothing as well. To see this let 'Y C C be the contour shown:

3.3. Self-adjoint Elliptic Pseudo-differential Operators

Then, by Cauchy's formula and (3.3.2),

p-1/"'1 = ~ ~

211"Z

1

z-1/m (z- P)-1 dz,

'Y

and

=~ 2n

1

z- 1/m(z- P+ R)- 1 dz.

'Y

Therefore,

Q-1- p-1/m =

-1·1 ;11"~ i

211"Z

=

z-1/m [(z- p + R)-1- (z- P)-1] dz

'Y

z- 1/m [(z- P + R)- 1R(z- P)- 1] dz.

However, since R is smoothing one can see that the operator inside the brackets is smoothing and that the integral converges and defines a smoothing operator R1. But this implies the desired result, since Q _ p1/m = -QR1 P 11m is smoothing.

111

112

3. Pseudo-differential Operators

Notes The theory of pseudo-differential operators goes back to Hormander [2) and Kohn and Nirenberg [1) and it has as its roots earlier work of Calderon and Zygmund [1) (see also Calderon [1) and Seeley[1)). The equivalence of phase function theorem presented in Section 3.2 is taken from Hormander [4) and the important Theorem 3.3.1 is from Seeley [2).

Chapter 4 The Half-wave Operator and Functions of Pseudo-differential Operators

Let M be a compact manifold, and suppose that Pis in \11~1 (M) with positive principal symbol p(x, e). Then if P is self-adjoint, as before, let N(>.) denote the number of eigenvalues of P which are ~ >.. The main result of this chapter is the sharp Weyl formula,

where c denotes the "volume" of M, that is, c = (211")-n /r { cl{dx. l{(x,e)ET· M :p(x,e):'5:1}

The proof of the sharp Weyl formula begins with the observation that N(>.) =

JM S>.(x,x)dx,

where S>.(x, y) is the kernel of the summation operator S>.f = L>.,_ ·<>. E;f (see Section 3.3 for the notation). This operator is an example of a function of P since if we let m(r) = m>.(r) be the characteristic function of the interval ( -oo, >.], then 00

S>.f = m(P)f =

L m(>.;)E;f· j=l

4. The Half-wave Opemtor

114

If we let m(t) be the Fourier transform of m, then Fourier's inversion formula gives

S,f ~

(2~)- 1

= (211")-1

1: L:

ffl(t)[t,•"''E;t] dl m(t)eitP I dt.

The operator eitP is the solution operator to the Cauchy problem for the half-wave operator (i8/fJt + P), and since we shall see that we can compute the kernel of this operator very precisely when It! is small, it is natural to also study

S>.l = (211")- 1

L:

p(t) m(t) eitP I dt

when p is a bump function with small support. If p equals 1 near the origin then a Tauberian argument involving sharp £ 00 estimates for £ 2-normalized eigenfunctions will show that the difference between the kernels of S>. and S>. is O(.~n- 1 ). Finally, we shall be able to compute the kernel of the "local operator" S>. and see that S>.(x,x) = c(x)An + O(An- 1 ), -

#

for the appropriate constant c(x). This along with the estimate for the remainder yields the sharp Weyl formula. Variations on this argument will be used throughout much of the monograph, and at the end of the chapter we shall see "that if m( A) E S~-', then m(P) is a pseudo-differential operator of order p. on M whose principal symbol is m(p(x,e)).

4.1. The Half-wave Operator In this section we shall construct a parametrix Q(t) for the Cauchy problem: (i8/fJt

+ P) u(x, t) =

0,

u(x,O)

= l(x).

( 4.1.1)

Since 00

u(x, t) =

L eit>.; E;l, j=1

we shall denote the operator sending 1 to u(x,t) by eitP. The main result here is that eitP can be represented by a Fourier integral. More

4.1. The Half-wave Opemtor

115

specifically, in local coordinates, we shall find that, for small times t, modulo a smoothing operator, eitP equals Q(t)l = (211")-n

JJei~(x,y,e)eitp(y,e) q(t, x, y, e) l(y) ~dy,

(4.1.2)

where rp is the type of phase function studied in Section 3.2 and q E S 0 , that is,

Using stationary phase, we shall observe that the singularities of the kernel of eitP are very close to the diagonal when t is near 0, and it is precisely this fact that will allow us to reduce global problems, such as proving the sharp Weyl formula or obtaining sharp estimates for the size of eigenfunctions, to localized versions which lend themselves directly to the techniques developed before. Let us now turn to the details. We shall first work locally constructing the parametrix for functions with small support, and then, at the end, using a partition of unity, glue together the pieces. In local coordinates on a patch n c M, the self-adjoint elliptic operator P E \11~1 is (modulo an integral operator with C 00 kernel) of the form P(x, D) where 00

P(x, e)

"' L

PI-j(X, e),

j=O

with the P; being homogeneous of degree j and P1 = p(x, e) the principal symbol. To be able to apply the results of the last section, we assume that the density dx on M agrees with Lebesgue measure in the local coordinates. This can always be achieved after possibly contracting n. Let us fix a relatively compact open subset w of n and try to construct an operator of the form (4.1.2) so that Q(t)l is a parametrix for eitP I whenever I E C3"(w) and t is small. In order to apply Theorems 3.2.1 and 3.2.3, we shall want rp to be in 8 1 and also satisfy (4.1.3)

while q E SO must have small enough support around the diagonal {(x,y): x = y} so that

IVecpl ;:::: cix - Yi

on supp q.

( 4.1.4)

4. The Half-wave Operator

116

The first step in achieving this for small time t is to construct rp. If Q(t) is to be an approximate solution to (4.1.2), then (iaj&t + P)Q(t) must be smoothing, and this will be the case if is in

s-oo,

(4.1.5)

where we have set

4'(t, x, y, e> = cp(x, y, e> + tp(y, e). But, by Theorem 3.2.3, the quantity in (4.1.5) equals [p(y,e)- p(x, Vxrp)] · q +lower order terms. Thus, it is natural to require that rp solves the eikonal equation

p(x, Vxcp) = p(y,e),

lx - Yl

small.

(4.1.6)

To see that a solution verifying (4.1.3) always exists, we shall need a fundamental result from the theory of Hamilton-Jacobi equations, whose proof will be postponed until the end of this section. Lemma 4.1.1: Letp be a realC00 function in a neighborhood of(O,T/) E R.n X R.n such that p(O, 71) = 0,

a

~ p(O, 71)

"# 0,

and suppose that 1/J is a real-valued C 00 function in R.n-l satisfying

a axJ_'1/J(O) = '1i'

j = 1, ... ,n -1.

Then there is a neighborhood of the origin and a unique real-valued solution ¢ E C 00 of the equation

p(x, Vx¢) = 0 satisfying

¢(x1 , 0) = '1/J(x'),

Vx¢(0)

= '1·

Here x' = (xt, ... , Xn-d·

Since the principal symbol is real and homogeneous of degree one, we need only apply the lemma to see that (4.1.6) can be solved when lei = 1; for then, if we extend rp to be homogeneous of degree one, (4.1.6) will be satisfied for all The resulting phase function need not be smooth ate= 0; however, this is irrelevant since the contribution in

e.

4.1. The Half-wave Operotor

117

the integral (4.1.2) coming from small' is smooth. With this in mind, if we fix the parameters y E w and ', then Lemma 4.1.1 implies that there is a unique function rp = rp(x, y, ')solving the nonlinear equation (4.1.6) that satisfies the boundary conditions

rp(x,y,,)

=0

when (x- y,,)

=0

and Vxrp

='

when x

= y, (4.1.3')

when xis close toy. Since (4.1.3') clearly implies (4.1.3), rp has the right properties. Having chosen rp, we need to impose a condition on the symbol q so that when t = 0, Q(t)- I is smoothing if I is the identity operator. To do this we recall that Theorem 3.2.1 implies that there is a symbol

vanishing outside a small enough neighborhood of the diagonal so that rp is defined there and

(211")-n

II

ei
~dy- /,

f

E Ccf(w),

(4.1.7)

is an integral operator with..C"00 kernel. On account of this, we require that

q(t, x, y, ')

= I(x, y, ')

when t

= 0.

(4.1.8)

For later use, we note that Theorem 3.2.1 implies also that •

q(O,x,x,,) -1 E S

-1

.

(4.1.9)

We now return to (4.1.5). Note that since rp solves the eikonal equation, Theorem 3.2.3 implies that if b E S~ then

L

e-i4>(iit+P(x, D)) [ei4>b] -(i&b+ p< 0 >(x, v zrp)D~b+ab) lol=1

E

s~- 1 '

(4.1.10)

if we set

and define a E

a(x, y, ')

sO by

=

L lol=2

Ap<0 >(x, V zrp)D~irp + P(x, V xlfJ)- p(y, ,). a.

( 4.1.11)

4·

118

The Half-wave Operator

We shall use (4.1.10) to solve the transport equation (4.1.5) by successive approximations. First, we pick qo(t,x,y,e) so that for small timet and large 1e1

i&qo +

L p(o}(x, Vxcp)D~qo + aqo = 0, lol=l

qo(O, x, y, e) = I(x, y, e). Note that

8t- L

P(o)(x, Vxcp)(fxt

lol=l is a real vector field. Therefore, since p(o)(z, V xiP) and a are bounded, there is a solution qo E SO to this linear differential equation that vanishes outside a small neighborhood of the diagonal, when ltl <e. If we then let b = qo in (4.1.10), it follows that if Ro is determined by

e-i~(iit + P(x,D)) [ei~qo] + Ro(t,x,y,e)

= 0,

then Ro E s- 1. Since P(x, D) is not local, the remainder Ro need not vanish outside a neighborhood of the diagonal; however, this will not be a major obstacle since Pis pseudo-local. In fact, let x(x, y) be a C[f bump function that equals one in a small enough neighborhood of the diagonal in w x w so that cp is defined and satisfies (4.1.4) near its support. We then argue as follows to determine the formal series for q. Suppose that we have chosen symbols Qj and associated remainders R; for 0 :5 j < k. One then requires that Qk solves the boundary value problem

i&qk +

L

p(o)(x, Vxcp)D~qk

+ aqk

= xRk-1>

lol=l qk(o,x,y,e)

(4.1.12)

= o;

as above, since xRk-l is supported near the diagonal, this ordinary differential equation can be solved for t E [ -e, e] with Qk supported near the diagonal. Then, by (4.1.10), if the symbol Qk is in Sll-, and if we let the symbol Rk be determined by

e-i~(iit+P(x,D))[ei~qk]- (i&qk+

L

p(o)(x,Vxcp)D~qk+aqk)

lol=l

+ Rk(t, x, y, e) = o,

4.1. The Half-wave Operator

119

then Rk E S~J.- 1 . To determine the order of these symbols, note that when k = 1, one must have q1 E s- 1 in (4.1.12) 88 Ro E s- 1, but then we have seen that we must have R1 E s- 2 ; proceeding by induction, we find that qk E s-k and Rk E s-k- 1 for all k. Next, if we add these equations, we find that k-1

e-ici>(i£+P(x,D))[eici>(qo+···+qk)]

=(x-1)~R;-Rk (4.1.13) 0

and qo(O, x, y, e)+···+ qk(O, x, y, e)

= I(x, y, e).

(4.1.14)

With this in mind, let q ,.._Eo qk (see Lemma 3.1.2). We then claim that (4.1.13) implies that

(i£ + P)Q(t)l =

J

Ko(t,x,y,e) l(y) dy,

1 E C(f(w) (4.1.15)

=Kol,

where the kernel Ko is a smooth function of x E M, y E w, and t E [-e, e] for some small e > 0. To see this we notice that the kernel of Q( t) is (4.1.16)

By Theorem 0.5.1, the kernel is C 00 at (t,x,y) if

for all large 1e1. But, since cp satisfies (4.1.3), we can conclude that given any neighborhood N of the diagonal, Q(t, x, y) is C 00 when t is small and (x,y) fl. N. Thus, if e is small enough, (4.1.15) follows from (4.1.13) and the fact that Pis pseudo-local. The next thing we need is that Q(O)I =I+ K1l,

IE C(f(w),

(4.1.17)

where K 1 is an integral operator with C 00 kernel. This of course follows from (4.1.7) and (4.1.14). To see that (4.1.15) and (4.1.17) imply that Q(t) is a local parametrix for eitP, we recall that if u E C 00 (M) then .A_fiiE;ull2 -+ 0 88 j -+ +oo for all N. Consequently, eitP : C 00 (M) -+ C 00 (1Rx M), since inequality (3.3.3) implies that (djdt)ieitP u E L~ Vj, k

4. The Half-wave Operator

120

ifu E C 00 • On the other hand, one checks by differentiating that (4.1.15) and (4.1.17) imply that Q(t)f

= eitP(f +Kif)+ lot eisP Ko(t- s)f ds,

IE c.r(w),

which means that Q(t)-eitP=eitPKt+ foteisPKo(t-s)ds. Thus, R(t) = eitP - Q(t) is an integral operator with a kernel that is C 00 on [-e,e] x M x w. This proves that we can construct a parametrix for (4.1.1) whenever f has small support. To construct a global parametrix, fix a (finite) partition of unity {'1/J;} subordinate to a covering of M by coordinate patches. Then if the '1/J; have small enough support, we can construct operators Qj ( t) as above that provide a parametrix for functions whose support is contained in that of '1/J;, when t is in some small interval. If we then let Q(t)f = "£Q;(t)(t/J;/), clearly Q(O)- I and (i8/8t + P)Q(t) are smoothing. Putting everything together, we see that we have proved the following. ~

Theorem 4.1.2: Let M be a compact C 00 manifold and let P E \11~1 (M) be elliptic and self-adjoint with respect to a positive C 00 density dx. Then there is an e > 0 such that when ltl < e, eitP = Q(t)

+ R(t),

where the remainder has kernel R(t, x, y) E 0 00 ([-e, e] xMxM) and the kernel Q(t, x, y) is supported in a small neighborhood of the diagonal in M x M. Furthermore, suppose that local coordinates are chosen in a patch 0 c M so that dx agrees with Lebesgue measure in the corresponding open subset c R.n; then, if w c n is relatively compact, Q( t, x, y) takes the form (4.1.16) when (t,x,y) E (-e,e] x M x w.

n

To finish matters we must prove Lemma 4.1.1. We start out by making some definitions that will be used in the proof. Definition: The Hamilton vector field associated to p(x, {) is (4.1.18)

One calls the integral curves of Hp the bicharacteristics of p. These

121

4.1. The Half-wave Operator are solutions

(x(t),~(t))

E T*IRn \0 of the Hamilton equations

dx; _ 8p(x,~) dt ac. '

~j _ _ 8p(x,~)

dt -

~J

ax . ' J

._ 1

J -

(4.1.19)

' ..• 'n.

The phase flow associated to the Hamilton vector field is the one-parameter group of diffeomorphisms ~t : T*IRn \0 -+ T*IRn \ 0 defined by ~t(Y, 71)

= (x, ~)

if

(x(O), ~(0))

= (y, 71)

(x(t), ~(t))

and

= (x, ~).

Definition: Let u = ~ 1\ dx be the standard symplectic form on T* X \ 0. Then we say that a (local) diffeomorphism xis a canonical transformation if the pullback of u with respect to x satisfies x• u = u. The main tool that will be used in the proof of Lemma 4.1.1 is the following. Proposition 4.1.3:

( 1) p( x, ~) is constant on its bicharacteristics. (2) The phase flow associated to p(x, ~) is a canonical transformation ofT*IRn\0. Proof of Proposition 4.1.3: The first part is easy. The derivative of t-+ p(x(t),~(t)) is

n 8p dx; 8x· dt

L

j=l

J

~ 8p ~j + L.J dt' j=l

8~·J

and this vanishes identically by (4.1.19). To prove the second part we note that ~t+s

Thus, since

~o

= ~to ~s·

is the identity, it suffices to show that d ..Y.* dt'J!'tU=O

whent=O.

(4.1.20)

= 0, ~t(x,~) = (x + t 8p~~), ~- t 8p~~)) + O(t2 ).

To see this we use the fact that near t

Thus

~iu = L(~j- td ::.) 1\ ( dx; + td ::.) + O(t2 ) ~J

J

= 0' + t(L~i 1\ d : . ~J

d ::. J

1\

dx;) + O(t

2)

4. The Half-wave Operator

122

But d2p = 0 and so we get (4.1.20).

I

Proof of Lemma 4.1.1: Recall from Section 0.5 that a (local) C 00 section of T*an is Lagrangian if and only if it is locally of the form (x, Vt/>(x)) for some 4> E C 00 • Thus, we must show that there is a Lagrangian section S over a neighborhood of 0 E an which has the following property. First, S is to lie in the zero set of p, {(x, 17) : p(x, 17) = 0}. Second, to fulfill the boundary conditions, we shall need (0, 17) E S and (x', Vx•'l/l(x')) C S' if we set S' = {(x',fl') : (x1,0,f11,fln) E S for some fln}. The transversality condition ~p(O, 17) =F 0 and the implicit function Ufln theorem imply that, when lx'l is small, there is a unique solution en(x') of

with en(O) = fln· Let

=

Then ulso 0, since the symplectic form de' A dx1 vanishes on {(x', V x•'l/l(x') )}. Also, since the transversality condition implies that Hp is not a tangent vector to So, it follows that if we let S be the union of all bicharacteristics starting at So, then S is a C 00 section over a neighborhood of the origin in an. Since Proposition 4.1.3 implies that uls 0, and p = 0 on S, it follows that S has the desired properties. Specifically, for x near 0, S must be of the form {(x, Vxtf>(x))} with 4> satisfying

=

;:,(x',o) = Vx•'l/l(x'),

Vt/>(0) = 11·

The first condition means that tf>(x',O) = ,P(x') + C for some constant C, and hence 4> - C solves the Hamilton-Jacobi equations with the preI scribed boundary conditions. Further remarks. It follows from the proof of Lemma 4.1.1 that the wave front set of the Schwartz kernel off--+ u(x, t) = eitP f is WF(eitP) = {(x,t,y,e,r,fl): ~t(Y, -17) = (x,e),r = p(x,e)}.

Here, as above,

~t

is the phase flow associated to the Hamilton vector

123

4.1. The Half-wave Opemtor field of p(x,{). So the canonical relation of the wave group is 1

C = {(x,t,{,r,y,71): 4»t(Y,11)

= (x,{),r = p(x,{)} C T*(M x IR) \0 x T* M\ 0.

(4.1.21)

This is of course Lagrangian with respect to the symplectic form O'MxRO'M·

A notable special case occurs when P = J -!1 9 + c2, with !1 9 being a Laplace-Beltrami operator on M. In this case ~t : T* M \ 0 -+ T* M \ 0 is geodesic flow in the cotangent bundle. Furthermore, if, in local coordinates, !1 9 = L,gik(x)-/x;ix;; +lower order terms, and if

L,gik(Y)71j1lk = 1 then

-y(t)

= {X : (x, {) = ~t(Y, 71)}

is the geodesic traveling at unit speed which starts at y in the direction 9 E TyM satisfying gik(y)9 = {. See Sternberg [1, Chpt. IV] for details concerning the Legendre transformation which relates geodesic flow in the cotangent bundle to the usual formulation involving the EulerLagrange equation. As a final remark, let us point out how the parg.metri.x construction can be modified to construct parametrices for mt;h order strictly hyperbolic differential operators. Let P(x, D)= DF + L,j!:l vr;-j P;(x, Dx) with P; being a polynomial of degree j in Dx depending smoothly on x E R.n. Then, if Pm(x,{, r) is the principal symbol, Pis said to be strictly hyperbolic if for every fixed (x, {) withe =f. 0 the mth order homogeneous polynomial T -+ Pm (X, { 1 T) has m distinct real roots. If this condition is satisfied the Cauchy problem Pu=O, {

alult=O =

j = 0, ...

/;,

,m -1,

is well posed. Moreover, the methods used to construct the parametri.x for the half-wave operator apply to this problem as well. For simplicity, let us assume that /; = 0 for j = 1, ... , m - 1 and set f = fo. Since Pis strictly hyperbolic we can factor its principal symbol, m

Pm(x,{,r) =

IJ (r- A;(x,{)), j=l

1 Strictly speaking, since the parametrix construction is only valid for small times, we can only make this conclusion for small t. However, the group property, ei(t+•)P =

eitP ei•P, and the composition formula for Fourier integral operators fn Chapter 6 show that the canonical relation has this form for all times.

124

4. The Half-wave Operator where the>..; are real and coo in T*IR.n \ o. If we assume that supp I c n, with n being a fixed relatively compact open set, and if cp;(x,t,e) are chosen to satisfy the eikonal equations

&If'; = >.;(x, V xiPj ), {

IP;It=O = {x,e} in f! X (-E,Ej, it turns OUt that, for

u(x, t) = f(21r)-n

J

ltl < e

1

eilf';(x,t,{)

a;(x, t, e) J(e) ~

j=1

plus a smoothing error. The zero order symbols a; are constructed as before from transport equations involving the roots >..; and the phase functions IPj.

4.2. The Sharp Weyl Formula AP. before let P E w~1 (M) be elliptic and self-adjoint with respect to a positive C 00 density dx. Then if we assume that the principal symbol p(x, e) is positive on T* M \0 and let

c=(21r)-njrf

~dx,

(4.2.1)

J{(x,{)ET• M:p(x,{)9}

our main result is the following.

Theorem 4.2.1: If N(>.) denotes the number of eigenvalues of P which are ::; >.. (and counted with respect to multiplicity), then, as>..-+

+oo,

N(>.) = c>.n + O(>.n-1).

By recalling Theorem 3.3.1, we see that Theorem 4.2.1 implies the following result.

Corollary 4.2.2: Suppose that Pm is an mth order self-adjoint elliptic differential operator whose principal symbol is positive. Then N(>.)

= c>.n/m + O(>..(n-1)/m),

if c is defined as in (4.2.1) with p = Pm being the principal symbol of Pm.

Recall that Theorem 1.2.3 shows that the estimates for the remainder can be improved in the special case where - F.2 is the standard Laplacian

4.2. The Sharp Weyl Formula

125

on the n-torus; however, in many cases the estimate is sharp. For instance, consider then-sphere sn and let P:z = -ll.s denote the standard spherical Laplacian which arises from restricting the Laplace operator in JR.n+l to sn. The distinct eigenvalues of -ll.s are v(v + n- 1), and they repeat with multiplicity

_ (n + v)! _ (n + v- 2)! _ n- 1( ( / ))/( _ )I n!v! n!(v _ 2)! - v 2+0 1 v n 1 ..

dv Thus, if e

> 0 is small and>.= v(v + n- 1) is large, N(>. +e)- N(>.- e)= dv ~ vn-1 ~ >,(n-1)/2.

Since this holds for all small e, it is clear that in this case the estimates for the remainder term in the Weyl formula cannot be improved. We now turn to the proof of Theorem 4.2.1. Recall that N(>.) is the trace of the partial summation operator S>.f = L>.;~>. Ej/. that is, N(>.) =

JM S>., (x, x) dx.

Thus, the formula would follow if we could show that, if dx agrees with Lebesgue measure in a local coordinate system around x E M, then the kernels satisfy d{ + O(>.n-1)

S>.,(x, x) = (21r)-n {

(4.2.2)

J{(:p(:z:,()9}

= c(x)>.n

+ O(>.n-1),

with

c(x) = (211")-n

r

J{(:p(x,()~1}

d{.

To try to prove this, we recall that if X( -oo,>.] denotes the characteristic function of the interval ( -oo, >.], then S>., = (211")- 1

I:

(X(-oo,>.J)"(t)eit? dt.

The Fourier transform of the characteristic function of ( -oo, 0] is the distribution

i(t+i0)- 1 =i lim - 1-. e--+0+ t + Ze

=1r6o(t)+i~, t

(4.2.3)

and so (4.2.4)

4·

126

The Half-wave Operator

Note that (t + i0)- 1 is only singular at t = 0; therefore, since Theorem 4.1.2 allows us to compute the kernel of eitP very precisely when ltl < c, it might be reasonable to compareS>. with

S>. = (211")- 1

I:

p(t) i(t + i0)- 1 e-i>.t eitP dt,

(4.2.5)

if p is the Fourier transform of some p E S and satisfies

ltl < c/2,

p(t) = 1,

ltl >c.

and p(t) = 0,

If we let X( -oo,>.] = X( -oo,>.] * p, note that the "approximate summation operators" are given by 00

(4.2.51)

S>.f = LX(-oo,>.J(A;)E;f. 1

Let us now use Theorem 4.1.2 to compute S>.(x,x). We assume that coordinates are chosen so that, around x EM, dx agrees with Lebesgue measure. If this is the case then (4.2.5) implies that

S>.(x,x) = (21r)-n- 1

+ (211")- 1

jj p(t)i(t+i0)- 1 q(t,x,x,e)eit(p(x,e)->.)~dt

I:

p(t)i(t + i0)- 1 R(t, x, x)e-i>.t dt, (4.2.6)

where q and R are as in Theorem 4.1.2. However, since R is C 00 , Corollary 0.1.15 implies that the last term in (4.2.6) is 0(1). To prove that the first term on the right side of (4.2.6) is equal to c(x)An +O(An- 1 ), note that Taylor's formula implies that

q(t, x, x, e)= q(O, x, x, e)+ tr(t, x, x, e) where r E SO. However, it only contributes O(An- 1 ) to the error since (4.2.3) gives

JJp(t)(t + io)-

1 tr(t, x, e) eit(p(x,e)->.)

=

jj p(t) r(t, x,e)

=

J

eit(p(x,e)->.)

r(A- p(x,e),x,e) ~.

dt~

dt~

127

4.2. The Sharp Weyl Formula where, with an abuse of notation, f( ·, x, ~)denotes the Fourier transform of p( · )r( · , x, ~). But the last integral is clearly dominated by

(4.2.7) (The last estimate is easy to prove after recalling that ~ -+ p(x, ~) is positive and homogeneous of degree one in IR.n \ 0.) By putting together these estimates, we see that we have shown that

S>.(x,x)

= (21r)-n- 1 X

If

p(t)i(t + i0)- 1

q(O, X 1 x, ~) eit(p(z,()->.) d{dt + O(.~n- 1 ).

(4.2.8) However, (4.1.9) implies that the main term here is

(211")-n-1 =

II

p(t)i(t + i0)-1 eit(p(z,()->.) d{dt

(211")-n

j

(x(-oo,O]

= c(x)>..n + (211")-n

*P)(p(x,~)- >..)cl{

J

(x(-oo,O]

* (p -1)v)(p(x,~)- >..)cl{. (4.2.9)

To estimate the last term, note that (p-1)v(s) = -Jst/J(s), where ~(t) = (1- p(t))fit. Since t/J is a bounded function which is rapidly decreasing at infinity, and since, by (4.2.3), (x(-oo,O]

* (p-1)v)(s) = -t/J(s),

one can use (4.2.7) to see that the last term in (4.2.9) is O(>..n- 1). Finally, since, by (4.1.9), q(O,x,x,~)-1 = 0(1~1- 1 ), it is clear that this argument also gives

and this means that we have proved the desired estimate

Therefore, we would be done if we could show that

IS>.(x,x)- S>.(x,x)l::::; c>..n- 1,

xEM.

(4.2.10)

4. The Half-wave Operator

128

To prove this, we note that (4.2.4) and (4.2.5) imply that the Fourier transform of the function A-+ S.>.(x,x)- S.>,(x,x)

vanishes when It! Tauberian lemma.

< e/2. To exploit this we shall use the following

Lemma 4.2.3: Let g(A) be a piecewise continuous tempered function of JR. Assume that for A > 0

lg(A + s)- g(A)I::; C(1 + A)a,

O<s::;l.

(4.2.11)

Then, if g(t) = 0, when iti ::; 1, one must have (4.2.12)

Remark. Clearly, the assumption that g(t) vanish for small t is needed. For, if g(A) = amAm + · · · + ao is a polynomial of degree m, then (4.2.11) holds for a= m- 1, while clearly (4.2.12) cannot hold if am =F 0. On the other hand, in this case g is, in some sense, concentrated at the origin, since sing supp g = {0}. # • Proof of Lemma 4.2.3: Let {>.+I

G(A)

= l.>.

g(s) ds. I

Then G is absolutely continuous and, except on possibly a set of measure zero, G' exists and satisfies

IG'(A)I

= lg(A + 1)- g(A)I ::; C(1 + A)a.

It is also clear that the Fourier transform of G vanishes in [-1,1]. Next since (4.2.11) and the triangle inequality imply that

it follows that (4.2.12) would follow from the estimate

!,

To prove this last inequality let TJ E S satisfy TJ(t) = 0, when It! < and TJ(t) = 1, for It! > 1, and let t/J be defined by ~(t) = (it)- 1TJ(t). Then, since t/J is bounded and rapidly decreasing at infinity, it is easy to

129

4.2. The Sharp Weyl Formula check that G' * 1/J =G. Consequently, the estimate for G' gives

IG(A)I = I(G' * 1/J)(A)I

J

~ C(1 + A)a ~

I1/J(s)l(1 + lsl)a ds

C(1 + A}a,

in view of the rapid decrease of 1/J.

I

If we apply the lemma to g(A) = S.>,(x,x)- S.>.(x,x}, then, since we have already observed that g vanishes near 0, we need only prove the following two estimates:

IS.>.+s(x,x}- S.>,(x,x)l ~ C(1 + A)n- 1 ,

0

< s ~ 1, (4.2.13)

IS.>.+ 8 (X, x) - S.>,(x, x)l

~

C(1 +At- 1 ,

O<s~l.

(4.2.14} However, we claim that these inequalities are a corollary of the following estimates for the L 00 norm of eigenfunctions. Lemma 4.2.4: Let X>. be the spectml projection opemtor X>.!=

L

(4.2.15}

E;f.

.>.;E(.>.,.>.+I]

Then, for A ~ 0,

llnfiiL""(M) ~ C(1 + A)(n- 1)/2 11/IIP(M)·

(4.2.16}

To see why (4.2.16} implies (4.2.13} and (4.2.14), note that (4.2.16} holds if and only if the kernel of the spectral projection operator satisfies sup

f ln(x,y)l 2 dy~C(1+A)n- 1 •

(4.2.17}

xeMJM

However, since

n(x,y} =

L

e;(x} e;(y),

.>.;E(.>.,.>.+1]

where {e;(x)} is an orthonormal basis associated to the spectral decomposition, we see that

L

jln(x,y)l 2 dy=

le;(x}l 2 •

.>.;E(.>.,.>.+I]

Consequently, since

S.>.+ 8 (x,x}- S.>.(x,x) =

L .>.;E(.>.,.>.+s]

le;(x}l 2 ,

130

4- The Half-wave Operator it is obvious that (4.2.17) implies (4.2.13). On the other hand, to see that (4.2.14) follows as well, note that B>.+s- B>. = (211")- 1

J

p(t) i(t + iO)-l ( e-ist-

1) eit(P->.) dt.

Thus, since (e-ist- 1) vanishes at t = 0, one can use (4.2.3) and argue as above to see that, for every N, IS>.+s(x,x)S>.(x,x)l:::;

eN ""' L..,(1 + lA- Ajl}- N le;(x)l 2 . j

But since the estimates for X>. imply that this term is 0( An- 1 ) as well, we need only to prove (4.2.16) to finish the proof of the Weyl formula. Proof of Lemma 4.2.4: The first step in trying to apply the above arguments is to notice that it suffices to prove the dual version of (4.2.16):

lln/II£2(M) :::; C(1 + A)(n- 1)/ 2 11/llu(M)·

(4.2.18)

Next, to exploit Theorem 4.1.2, we notice that it suffices to prove the analogous inequality for certain "approximate spectral projection operators." Namely, for a fixed XES, satisfying x(O) > 0, X~ 0, and x(t) = 0 unless ltl :::; e, we define X>.!= LX(Aj -A)E;f·

(4.2.151)

j

(Such a function always exists, since if the function p in (4.2.5) is real, then x(A) = (p(A/4)) 2 has the right properties.) It is useful to have the £ 2 norm on the left side, since orthogonality and the fact that x(O) "I 0 imply that (4.2.18) would be a consequence of the analogous estimates for the approximate operators:

llx>./112 :::; C(1 + A)(n- 1)/211/111·

(4.2.18')

Moreover, the above arguments also apply here since x( t) = 0 when ltl ~ e and X>. = (211")-1

L:

x(t) e-it>. eitP dt.

Next, notice that if X>.(x, y) is the kernel for x>.. then X>.(x, y) =

L x(Aj -A) e;(x) e;(y). j

(4.2.19)

.pJ. Smooth Functions of Pseudo-differential Operators

131

Hence, it is easy to see from the Schwarz inequality that the L1 (M)-+ L2 (M) operator norm of X.\ satisfies

llx.\II~Ll(M}P(M)) $:~~I lx.\(x, y)l 2 dx = sup 2)x(A; -

(4.2.20)

A)) 2 1e;(Y)I 2

yEM j

$llxllv"'(R} · sup X.\(y,y). yEM

In the last inequality we have used the fact that

x ;::: 0. Finally, if we let

c(t, x, e) = X(t)q(t, x, x, e), where q E S0 is the symbol in the parametrix for eitP, then (4.2.19) and Theorem 4.1.2 give lx.\(x, x)l

$1Ln $eN

c(A- p(x, e), x, e) del+

I

11.:

x(t)e-it.\ R(t,x, x) dtl

(1 + lA- p(x, em-N de+ 0(1)

$ C(1 + A)n- 1

I

(by (4.2.7)).

I

Combining this with (4.2.20) completes the proof.

Remark. Lemma 4.2.4 is a generalization of the (£ 1 , £ 2 ) restriction theorem for the Fourier transform in Rn. In fact, duality and a scaling argument show that the latter is equivalent to the uniform estimates

•

f ei(x,() f(e)dell $ C(1 + A)(n- 1}/2 11/IIP(Rn)· 11 (211")-n ji(IE(.\,.\+1) L""(Rn} The next chapter will be devoted to proving a generalization of the full restriction theorem under the assumption that the principal symbol p(x, e) satisfies certain natural curvature conditions. These estimates will allow us to extend the Tauberian argument used above to handle other situations, such as proving estimates for Riesz means on M.

4.3. Smooth Functions of Pseudo-differential Operators Let m E C 00 (R) belong to the symbol class SP., that is, assume that (4.3.1)

4·

132

The Half-wave Opemtor

Then, using the spectral decomposition of P, we can define an operator m(P) that sends C 00 (M) to COO(M) by 00

L m(>.;) E;f,

m(P)f =

(4.3.2)

j=l

if Pis as above. Using the ideas in the proof of the sharp Weyl formula, we shall see that m(P) is actually a pseudo-differential operator.

Theorem 4.3.1: Let P E \11~1 (M) be elliptic and self-adjoint with respect to a positive C 00 density dx. Then, if m E Sf.l., m(P) is a pseudodifferential operator of order p., and the principal symbol of m(P) is

m(p(x,,)). As before, one can see that the same result holds for pseudo-differential operators of arbitrary positive order as well. Also, since Theorem 3.1.6 says that zero order pseudo-differential operators are bounded on l.Jl, we also have the following.

Corollary 4.3.2: Let P be as above. Then if m E llm(P)fiiLP(M) ~ Cp llfiiLP(M)•

SO,

1 < P < oo.

(4.3.3)

Proof of Theorem 4.3.1: We assume that local coordinates are chosen as in Theorem 4.1.2, and, as in the proof of Theorem 4.2.1, we fix p E S(IR) satisfying p(t) = 1, ltl ~ e/2, and p(t) = 0, ltl >E. Then, using the half-wave operator, we make the decomposition

/_:fJ(t) m(t)

m(P) = (27r)- 1 = m(P)

eitP dt

L:(l-

+ (27r)- 1

fJ(t)) m(t) eitP dt

+ r(P),

where, if 1/J is defined by

,P =

1-

m(>.) = (m * p)(>.)

p, and

r(>.) = (m * 1/J)(>.).

Since m satisfies (4.3.1), m(t) is C 00 away from t = 0, and rapidly decreasing at oo; thus r(>.) E S. Since the kernel of r(P) equals 00

r(P)(x, y) =

L r(>.;) e;(x) e;(y), 1

one can, therefore, use the crude estimates from Section 3.3 for the size of the derivatives of the eigenfunctions e;(x) together with the Weyl formula to see that r(P)(x,y) is C 00 •

Notes

133

On the other hand, Theorem 4.1.2 implies that, in our local coordinate system, the kernel of m(P) equals

(21r)-n- 1

11

p(t) m(t)q(t, x, y, {) eitp(y.{) eicp(x.y,{) dedt

+ (211")- 1

I

p(t) m(t) R(t, x, y) dt.

Since R is C 00 and since p(t)m(t) E e'(JR), the second term must be in C 00 (M x M). Thus, we would be done if we could show that

(21r)-n- 1

11

p(t) m(t) q(t, x. y, {) eitp(y.{) eicp(x,y.{) dedt

(4.3.4)

is the kernel of a pseudo-differential operator with principal symbol m(p(x,{)). To see this we note that, as before, we can write

q(t, x, y, {) = q(O, x, y, {) + tr(t, x, y, {),

so.

where r E Since m is singular at the origin, we would, therefore, expect that the main term in (4.3.4) would be

(211")-n- 1

II

p(t) m(t) q(O, x, y, Q,eitp(y,{) eicp(x,y,{) dedt

= (211")-n

I

m(p(y, {)) q(O, x, y, {) eicp(x,y,{) cte.

Since, q(O,x,x, {)-1 E s- 1 , Theorem 3.2.1 implies that this is the kernel of a pseudo-differential operator having principal symbol m(p( X,{)), and since m-m E S, we conclude that this kernel has the desired form. Thus, we would be done if we could show that

(21r)-n- 1

11

p(t) m(t) tr(t, x, y, {) eitp eicp dedt

is the kernel of a pseudo-differential operator of order :5 p. - 1. But this too follows from Theorem 3.2.1 after checking that

(211")- 1

L:

p(t) m(t) tr(t, x, y, {) eitp(y,{) dt E s~'- 1 •

I

Notes The parametrix construction for the half-wave operator is taken from Hormander [4]. This was a modification of the related construction of Lax [1] which used generating functions. We have used Lax's construction at the end of Section 4.1 to construct parametrices for strictly

134

4·

The Half-wave Opemtor

hyperbolic differential operators. As the reader can tell, Lax's approach is somewhat more elementary, but, since the phase functions in the Lax construction need not be linear in t, it is harder to use in the study of eigenvalues and eigenfunctions. The proof of the sharp Weyl formula is from Hormander (4], except that here the Tauberian argument uses the bounds for the spectral projection operators in Lemma 4.2.4 which are due to Sogge (2] and Christ and Sogge (1]. The Tauberian arguments which were used in the proof go back to Avakumovic [1] and Levitan [1], where they were used to prove the sharp Weyl formula for second order elliptic self-adjoint differential operators. The material from Section 4.3 is due to Strichartz (2] and Taylor (1].

Chapter 5

LP Estimates of Eigenfunctions

In Chapter 2 we saw that if E C R.n is a compact C 00 hypersurfa.ce with non-vanishing curvature then

1< <2(n+l)· _p_ n+ If, in addition, say, E = {e: q(e) = 1}, where q is homogeneous of degree one, C 00 , and positive in R.n \ 0, this inequality is equivalent to

({

he:q(E)E(l,l+e)}

i/(e)l 2 ~) 112 ~ Ce 1/ 2 11/IILP(Rn)•

If we define projection operators

X>.f(x) = (27r)-n {

ei(x,e) /(e)~,

he:q(E)E[.>.,>.+l)}

then taking e = 1/ >. and applying a scaling argument shows that the last inequality is equivalent to the uniform estimate

IIX>./IIL2(Rn) ~ C(1 + A)c5(p) 11/IILP(Rn)>

1 ~ P ~ 2 ~-tl) l

with 6(p) = nl~- ~~-~Notice that, for this range of exponents, 6(p) agrees with the critical index for lliesz summation (see Section 2.3). The operators X>. of course are the translation-invariant analogues of the spectral projection operators which were introduced in the proof of the sharp Weyl formula. The main goal of this ch~pter is to show

136

5. LP Estimates of Eigenfunctions that these operators satisfy the same bounds as their Euclidean versions under the assumption that the cospheres associated to the principal symbols Ex= {e: p(x,e) = 1} c T;M\0 have everywhere non-vanishing curvature. Since 6(1) = (n- 1)/2 this is the natural extension of the estimate (4.2.18). After establishing this "discrete L 2 restriction theorem" we shall give a few applications. First, we shall show that the special case having to do with spherical harmonics can be used to give a simple proof of a sharp unique continuation theorem for the Laplacian in an. Then we shall see how the Tauberian arguments that were used in the proof of the sharp Weyl formula can be adapted to help prove sharp multiplier theorems for functions of pseud{)odifferential operators. Specifically, we shall see that estimates for Riesz means and the Hormander multiplier theorem carry over to this setting.

5.1. The Discrete L 2 Restriction Theorem Let M be a C 00 compact manifold of dimension n ~ 2. We assume that P = P(x, D) E w~1 (M) is self-adjoint, with principal symbol p(x, e) positive on T* M \ 0. Then if n are the spectral projection operaJ;o~s defined in (4.2.15), we have the following result.

Theorem 5.1.1: Assume that, for each x E M, the cospheres {e : p( X' e) = 1} c T; M \ 0 have everywhere non-vanishing curvature. Then if 6(p) = nl~- ~~- ~ and A> 0 1< < 2(n+l) _p_ n+ '

(5.1.1)

llnfii£2(M) ~ C(1 + A)(n-l)( 2-p)/4p llfiiLP(M)•

2(n+l) < < 2. --n+3 _p-

(5.1.2) Furthermore, these estimates are sharp. If we use Theorem 3.3.1, then we see that the dual versions of these inequalities yield the following estimates for the "size" of eigenfunctions on compact Riemannian manifolds.

Corollary 5.1.2: Let t1 9 be the Laplace-Beltrami operator on a compact C 00 Riemannian manifold (M,g). Then, if {A;} are the eigenvalues of

5.1. The Discrete L 2 Restriction Theorem -6.9 , and if one defines projection operators R>.f = Ly:X;e[>.,>.H] E;f, one has the sharp estimates IIR>.fllLv(M) $ C(1 + A)c5(q) llfiiL2(M)• IIR>.fllLv(M) $ C(1 +A) (n - 1}(2 -q')/4q llfiiL2(M)• 1

2< <2(n+l)· - qn-

In particular, if {e;} is an orthonormal basis corresponding to the spectral decomposition of -6., then 2 ~~l) $ q $ oo,

lle;IILv(M) $ C(1 + .X;)o(q)/2 , lle;IILv(M) $ C(1 + .X;)(n- 1)( 2 -q')/Bq',

2$ q$

2 ~~p.

Proof of Theorem 5.1.1: We shall first prove the positive results and then show that the estimates cannot be improved. Since we have already seen that (5.1.1) holds when p = 1 and since X>. : L 2 - L 2 with norm one, the M. lliesz interpolation theorem implies that we need only prove the special case where p = 2(n + 1)/(n + 3); that is, it suffices to show that when p = 2 ~-tl>. (5.1.3) To prove this we shall use the idea from the proof of Lemma 4.2.4 of proving an equivalent version of this inequality which involves operators whose kernels can be computed very explicitly. The operators X>. used in the proof of Lemma 4.2.4 have kernels which are badly behaved in the diagonal {x = y}. So, in the present context, it is convenient to modify their definitions slightly. To do this we let E > 0 be as in Theorem 4.1.2. Thus, in suitable local coordinate systems, eitP has a parametrix of the form (4.1.16) as long as It! <E. Then for 0 <Eo< E to be specified later we fix 0 -=f. X E S(IR) satisfying x(t) := 0 if t ~ (eo/2,eo), and define approximate projection operators x>.f

= x(P- .X)f = L:x(.X;- .X)E;f. j

Then, as before, orthogonality arguments show that the uniform bounds (5.1.3) hold if and only if llx>.fllL2(M> $ c(1 + .x)o(p) llfiiLP(M>·

p = 2 ~ti>,

.x > o. (5.1.3')

137

5. LP Estimates of Eigenfunctions

138

Let Q(t) and R(t) be as in Theorem 4.1.2. Then

X>.f =

..!... I 2~

Q(t)fe-it>. x(t) dt +..!...I R(t)fe-it>. x(t) dt. 2~

Since R( t) has a C kernel, the last term has a kernel which is 0( >.- N) for any N and hence it satisfies much better bounds than those in (5.1.3'). Therefore, if we work in local coordinates so that dx agrees with Lebesgue measure, it suffices to show that 00

T>.f(x)

= (2~)-n-1 I f f ei[. x x(t) q(t, x, y, e) f(y) ctedtdy

satisfies the bounds in (5.1.31) if cp and q are as in (4.1.16). Notice that since c- 1 1x- Yi :5 IVecp(x,y,e)l :5 Clxconstant C, it follows that on the support of the integrand

IVe[cp(x,y,e)+tp(y,e)]l #0 if

Yi

for some

lx-yl f/. [C0 1eo,Coeo]

for some constant Co. Therefore, by Theorem 0.5.1, if we let a(t, x, y, e) = 71(x, y)x(t)q(t, x, y, e), where '7 E C 00 equals 1 when lx - Yl E [C01eo, Coco] and 0 when lx - Yl f/. [(2Co)- 1eo, 2Coeo], it follows that the difference between T>.f and

T>.f(x) =

(2~)-n- 1 I I f ei[. a(t, x, y, e) f(y) dedtdy

has a kernel with norm 0(>.-N) for any N. On account of these reductions, we would be done if we could show that (5.1.3")

The proof of this follows immediately from the estimates for non-homogeneous oscillatory integrals and the following result. Lemma 5.1.3: Let a(t, x, y, e) E S0 be as above and set

K>.(x,y) =

JJ

ei[
e-i>.ta(t,x,y,e)dedt,

Then, if eo is chosen small enough we can write n-1 K>.(x, y) = >. ----r a>.(X, y) ei>.,P(x,y),

>. > 1.

(5.1.4)

where t/J and a>. have the following properties. First, the phase function t/J is real and C 00 and satisfies the n x n Carleson-Sjolin condition on suppK>.. In addition, a>. E C 00 with uniform bounds

l~,ya>.(x,y)l :5 Co.

139

5.1. The Discrete L 2 Restriction Theorem If we apply Corollary 2.2.3 we see from the lemma that the oscillatory integral operator with kernel K>. is bounded from £ 2 norm

-+

L

~

with

Since the n x n Carleson-Sjolin condition is symmetric, the adjoint ofT>. must also satisfy this condition. So, by duality, we see that the operators n-1 2(n+1) T>. are bounded from L 7i+3 to L 2 with norm O(.X 2(n+l) ). But 6(p) = 2 ~+11 ) when p = 2 ~tl> and hence the proof of Theorem 5.1.1 would

I

be complete once Lemma 5.1.3 has been established.

Proof of Lemma 5.1.3: The proof will have two steps. First we shall show that K>. is of the form (5.1.4), and then we shall show that the phase function satisfies the n x n Carleson-Sjolin condition. The first thing we notice is that, if p(y, e) fl. (.X/2, 2-X],

I

eit(p(y,{)->.la(t, x, y, e) dt = O((lp(y, e) I+ .x)-N)

"

for any N. But p(y, e) ~ 1e1. So if {3 E C8"(1Rn \ O)"equals one when 1e1 E (c- 1, C], with C sufficiently large, the difference between K>. and

II

ei[rp(x.y,{)+tp(y.{)) e-i>.t f3(e/ .X) a(t, x, y, e) d{dt

(5.1.5)

is C 00 , with all derivatives being O(.X-N) for any N. On account of this, it suffices to show that we can write (5.1.5) as in (5.1.4). But if we set

\ll(x, y; t, e) = rp(x, y, e)+ t(p(y, e)

- 1)

and

then, after making a change of variables, we can rewrite (5.1.5) as (5.1.51) Notice that a>. is compactly supported and that all of its derivatives are bounded independently of .X. So we can use stationary phase to evaluate this integral if the Hessian with respect to the n + 1 variables (e, t) is non-degenerate on suppa>._. By symmetry it suffices to show that this is

1

140

5. LP Estimates of Eigenfunctions the case when e lies on the en axis. If we write

e = (e2 •... 'en) then

0

0

Ifen To see that this matrix is non-degenerate we first notice that, by homogeneity and our assumption that e lie on the en axis, we have p{Jy, e) = p(y, e)/en· So the non-degeneracy of the Hessian is equivalent to (5.1.6)

I

,

But our curvature hypothesis implies that o2 pfoe' 2 is non-singular and (4.1.3) implies that o2cpfoe' 2 consists of elements which are O(lx-yl 2 ) = O(e~) on suppa~. So, if eo in the statement of the lemma is sufficiently small, (5.1.6) holds and consequently det o2 'ifl fo(e, t) 2 =F 0 as desired. Furthermore, since cp satisfies (4.1.3) one can modify the argument before the proof of Lemma 2.3.3 to see that, if eo is sufficiently small, then, on suppa~, (e, t) -+ 'if! has a unique stationary point for every fixed (x, y). Hence Corollary 1.1.8 implies that (5.1.51) can be written as in (5.1.4). What remains to be checked is that the phase function satisfies the n x n Carleson-Sjolin condition. That is, we must show that, for every fixed x,

Sx = {Vx,P(x,y): a~(x,y) =F 0} c

T;x

(5.1.7)

is a C 00 hypersurface with everywhere non-vanishing Gaussian curvature, and also that

a2,p

rank oxf)y

=n- 1

on suppa~.

(5.1.8)

Since the adjoint ofT~ has a similar form, the proof of (5.1.7) also gives that, for each y, Sy

= {Vy,P(x,y): a~(x,y) =F 0} c T;x

has everywhere non-vanishing Gaussian curvature. This, together with (5.1.7) and (5.1.8), implies that the n x n Carleson-Sjolin condition is satisfied. To prove (5.1.7) and (5.1.8), let us first compute ,P. We note that V~,t'ifl = ( IPe(x,y,e)

+ tpe(y,e),p(y,e) -1).

5.1. The Discrete L 2 Restriction Theorem

141

Thus, if (t(x, y), e(x, y)) is the solution to the equations

{

cp~(x, y, e)+ tp~(y, e) = 0, p(y,e) = 1,

(5.1.9)

it follows from Corollary 1.1.8 that our phase function must be given by

1/J(x,y) = w(x,y;t(x,y),e(x,y)) = cp(x,y,e(x,y)).

(5.1.10)

To check that the curvature condition is satisfied, notice that

Vxt/J(x,y)

= cp~(x,y,e(x,y)) + ae~~y) cp~(x,y,e(x,y)).

However, the first half of (5.1.9) implies that

ae(x,y) ax Pt.'( y,e (x,y )) ax cp£.'( x,y,e (x,y )) = -t (x,y )ae(x,y)

a {p(y,e(x,y))}. = -t(x,y) ax Since the second half of (5.1.9) implies that the last expression vanishes identically we conclude that

Vxt/J(x,y) = cp~(x,y,e(x,y)). But this along with the eikonal equation (4.1.6) implies that

p(x, Vxt/J(x,y)) = p(y,e(x,y)). So if we use the second half of (5.1.9) again, we conclude that the hypersurfaces in (5.1.7) are just the cospheres Ex= {e: p(x,e) = 1}, which have non-vanishing curvature by assumption. To verify (5.1.8) we fix y and let p(e) = p(y, e). Then if ~(x- y) is determined by the analogue of (5.1.9) and (5.1.10) where cp is replaced by the Euclidean phase function (x- y, e) and p(y, e) by p(e) we have already seen (in the proof of Lemma 2.3.4) that rank a 2 ~(x-y)jaxay::::: n -1. But 1/J(x, y) = ~(x- y) + O(lx- y! 2 ) as x- y which implies that a2 .,pjaxay = a2 ~(x- y)jaxay + 0(1). From this and the fact that ~is homogeneous of degree one, we conclude that rank a2 .,p1axay ~ n - 1 on suppa~ if eo is small enough. However, since we have just seen that y- Vxt/J(x,y) is contained in the hypersurface Ex, the rank can be at most n- 1 and so we get (5.1.8). I

Remark. The proof of Lemma 5.1.3 shows that the operators x~ inherit their structure from the wave group eitP. In fact, if C c T*(M x IR) \0 x T* M \0 is the canonical relation associated to the wave

5. LP Estimates of Eigenfunctions

142

group, which is given in (4.1.21), it follows that the canonical relation associated to t/J satisfies

C,p

c {(x,e,y,TJ): (x,t,e,p(x,e),y,TJ) e c, for some 0 < t < e and (x,e) such that p(x,e) = 1}. (5.1.11)

To see this, notice that, for x close toy, there is a unique positive small timet such that (x,t,y) is in the singular support of the kernel of eitP. But, if we call this time t(x, y), it follows from Euler's homogeneity relations, cp = {cp(,e},p = {p(,e}, along with (5.1.9) and (5.1.10) that .,P(x, y) = -t(x, y). We should also point out that this implies that, when P = ~, t/J is just minus the Riemannian distance between x and y.

Proof of sharpness: We first prove that for 1 ~ p

~

2

lim sup sup A-n(~-~)+~ llx>./112 > c >.--++oo /ELP(M) II/IlP - 1

(5.1.12)

for some c > 0. This of course implies that the estimate (5.1.1) must be sharp. To prove (5.1.12) we fix {3 E S(R.) satisfying /3(1) "I 0 and /3(r) = 0 if T ¢ [-e,e], where e is as in Theorem 4.1.2. We then fix xo EM and define

f>.(x)

= Lf3(Aj/A)e;(xo)e;(x). j

Notice that this is the kernel of {3(P/A) evaluated at (xo,x). So, if we argue as in Section 4.3, we conclude that, given N, there must be an absolute constant CN such that 1/>.(x)l ~ CNAn(1 +A dist (x,xo))-N. Consequently, 11/>.llp ~ CbAn-nfp. On the other hand, since /3(1) "I 0, we conclude that there must be a co > 0 such that, if A is large enough,

lln/>.11~ =

L

I/3(Aj/A)I 2 Ie;(xo)l 2 lle;(x)ll~

>.; E [>.,).+ 1)

~CO

L

>.;E[>.,).+l)

le;(xo)l 2·

143

5.1. The Discrete L 2 Restriction Theorem However,

~

h

L

lej(x)l 2 dx

~ N(>.. + 1) -

N(>..).

M >.;E(>.,.Hl]

(5.1.121)

Combining this with the lower bounds for the norm of f>. yields sup lln/ll2

II/IlP

/ELP

for some c

~c>..-n+n/p{N(>..+ 1 )-N(>..)}l/2

> 0. However, since N(>..)

~

>..n,

limsup>..-(n-l){N(>.. + 1)- N(>..)} >.-++oo

> 0,

and hence (5.1.12) follows from (5.1.121). The proof that the other estimate in Theorem 5.1.1 is sharp is more difficult and requires a special choice of local coordinates in M. Before specifying this, let us first notice that it suffices to show that, for large enough>.., sup llx>./ll2 /ELP

> c>..(n-t)(1p-i>

II/IlP -

'

1 :5 P :5 2,

(5.1.13)

if X>. are the approximate spectral projection operators occurring in the proof of Theorem 5.1.1. To prove this lower bound we shall use the last remark. That is we shall use the fact that -'1/J(x, y) = t(x, y), where, if xis close but not equal toy, t(x,y) is the unique small positive number such that JIM(It(x,y)(Y,TJ) = x for some TJ E T; M \ 0. Here 41t : T* M \ 0 --+ T* M \ 0 is the flow out for time t along the Hamilton vector field associated to p(x, e), and II M : T* M \ 0 --+ M is the natural projection operator. Keeping this in mind, there is a natural local coordinate system vanishing at given Yo E M which is adapted to '1/J(x, y). This is just given by ~(z)=x

e

if1IM4llxi(YO,e)=z

withe=x/lxl-

(5.1.14)

Here denotes the coordinates in T;0 M \ 0 which are given by an initial choice of local coordinates around Yo (see Section 0.4). Note that, if P = FfS:;, the local coordinates would just be geodesic normal coordinates around Yo and so one should think of (5.1.14) as the na.tural "polar coordinates" associated to p(x, e). They are of course well defined for z

5. LP Estimates of Eigenfunctions

144

near Yo and C 00 (away from possibly z =Yo) since, for a given initial choice of local coordinates vanishing at YO,

and since, by assumption, Pee has maximal rank n- 1. The reason that these coordinates are useful for proving (5.1.13) is that, in these coordinates, '1/J(x, 0) = Ixi- Moreover, using the semigroup property of ~t, one sees that

'1/J(x,y)

=-(XI-

YI) and - Vxt/l(x,y)

= Vyt/l(x,y) = (1,0),

if x = (xi,O),y = (yt.O), and 0 <

YI <XI·

(5.1.15)

From this and Taylor's theorem we deduce that if x' = (x2, ... , xn) and 0 < YI < XI then

'1/J(x, y) = -(xi - YI) + O(lx'- y'l 2 ).

(5.1.15')

Since '1/J is smooth away from the diagonal, the constants in the error term remain, bounded if XI - YI is larger than a fixed constant. We can now deduce (5.1.13). We first fix a nontrivial o: E ccr{IR.+) and set

where 0 < ei «:eo, with eo being the number occurring in the definition If we then define of

x.x.

we shall show that, if ei is small enough, we can obtain favorable lower

bounds for llx.xhll2· Notice that, for large A, I>. is supported in a small neighborhood of the positive XI axis, which, by construction, is the projection of a bicharacteristic for p(x, ,). To establish the lower bound, let Co be the constant occurring in the proof of (5.1.3) and set

n.x = {x: eo/2Co :5 XI :5 2Coeo,

lx'l :5 ei_x-I/ 2 }.

Then, if ei is sufficiently small, given any N there is a constant CN such that, if a.x(x,y) is as in Lemma 5.1.3, then

5.1. The Discrete L 2 Restriction Theorem

145

But the left side can be rewritten as An-1

In>.

II

ei~((,P(x,y)-y1)-(,P(x,y)-y1)) x

o:~(y) o:~(jj) a~(x, y) a~(x, jj) dydydx.

Notice that, by (5.1.151), the term in the exponential is O(e~) on the support of the integrand. Using this and the fact that the a~ belong to a bounded subset of C 00 shows that, for small enough e1, the last expression is bounded from below by a fixed positive constant times

e~nAn-l(A-(n- 1 )1 2 ) 2

{

lrh

la~(x,O)I 2 dx-Ce~n+liO~I,

where Cis a fixed constant. However, since q(O,x,x,e) = 1 + O(lel- 1), it follows from the proof of Lemma 5.1.3 that, if eo is small and A large, then there must be positive constants such that {

lrh

la~(x,O)I 2 dx ~ ciO~I ~ c' A-(n- 1)/2.

Thus, if A is large and if e1 is small, we reach the desired concltision that #

llx~hll2 ~ CQA-(n- 1)/ 4 ,

some

co> 0.

Finally, since llhiiP::; CA-(n-1)/2p, we conclude that (5.1.13) must hold.

I

Application: Unique continuation for the Laplacian A special case of Corollary 5.1.2 concerns spherical harmonics. If M = 1, n ~ 3, and A 9 = As is the usual Laplacian on sn- 1 the eigenvalues of the conformal Laplacian, -As + [(n - 2)/2] 2, are {(k + (n- 2)/2) 2}, k = 0, 1, 2, .... The eigenspace corresponding to the kth eigenvalue is called the space of spherical harmonics of degree k and it has dimension ~ kn- 2 for large k. If we let Hk be the projection onto this eigenspace, then Corollary 5.1.2, duality, and the fact that Hk = Hk o Hk give the bounds

sn-

IIHk/IILP'(Sn-1) :S C(1 + k) 1- 2/n 11/IILP(Sn-1)• Notice that for this value of p we have 1/p-1/p'

P=

n~2·

(5.1.16)

= 2/n, so the exponents

5. LP Estimates of Eigenfunctions

146

in (5.1.16) correspond to the dual exponents in the classical Sobolev inequality for the Laplacian in an: lluiiLP'(R") ~ C 116-uiiLP(R")•

P = n~2•

u E Ccf(an).

We claim that (5.1.16) along with the Hardy-Littlewood-Sobolev inequality yield a weighted version of this. Namely, if p and p1 are as above there is a uniform constant C for which lllxi-TuiiLP'(R") ~ Clllxi-T 6-uiiLP(R")• ifuEC«f(an\0) and dist(r,Z+~)=~.

(5.1.17)

The condition on T is of course related to the spectrum of the conformal Laplacian on sn-l. Before proving (5.1.17) let us see how it yields the following unique continuation theorem. Theorem 5.1.4: Let n ~ 3 and let X be a connected open subset of an containing the origin. Suppose that D 0 u E 0 c{X) for ial ~ 2 where p = 2nf(n + 2). Assume further that

Lf

I

{5.1.18)

for some potential V E L':!c2 {X). Then if u vanishes of infinite order at the origin in the LP mean, that is, if for all N {

Jlxi
iujPdx=O(RN),

+2•

p= n2

(5.1.19)

it follows that u vanishes identically in X. Remark. u =

The condition on V cannot be relaxed. For, if one takes

e-llog lxW+•, then u vanishes of infinite order at the origin and

16-uful = V ~ I log lxii2E · lxl- 2 is in L[oc for every r

< nf2.

Proof of Theorem 5.1.4: It suffices to show that u vanishes in a small ball centered at 0. To see this we first recall that our hypotheses imply that we actually have

{

Jlxi
ID

0

ujP dx = O(RN) VN if 0

~ lal ~ 1

(5.1.19')

{see, e.g., Hormander [7, Vol III, Theorem 17.1.3]). To use (5.1.17) let TJ E Ccf{an) equal1 near the origin and put w(x) = TJ(x)u(x). Then if

147

5.1. The Discrete L 2 Restriction Theorem we fix 0 $ p $ 1 E C 00 (1R.n) satisfying p = 0 for

lxl

< ~ and p = 1 for

lxl > 1, it follows that the functions w;(x)

= p(jx)w(x),

= 1, 2, ... ,

j

satisfy (5.1.17). If C is the constant in this differential inequality, we choose R small enough so that 17(x) = 1 when lxl
However, Liebnitz's rule and (5.1.19') imply that as j --. oo

lllxl-r 6-w; IILP(BR) = lllxl-r p(jx)tl.u IILP(BR) + o(1). Furthermore, since 1/p- 1/p' = 2/n, Holder's inequality, (5.1.20), and our assumption (5.1.18) imply that

Clllxl-r p(jx)tl.u IILP(BR)

$ ~lllxl-ru IILP'(BR)·

So by letting j --. oo we conclude that

If we letT--. +oo in this inequality, we conclude that u

=0 in BR.

I

Proof of (5.1.17): We first recall that, in polar coordinates,

-6. = (~ ~)2 tOr

_n-r 1 ~ _ 6-s. ar

So if we make the change of variables t = logr,

X=

etw,

wE

sn- 1,

minus the Euclidean Laplacian becomes

Also, in these exponential coordinates,

Since p

= 2n/(n + 2),

it follows that nfp'

= (n- 2)/2.

Keeping this in

5. LP Estimates of Eigenfunctions

148

mind, one sees that (5.1.17) is equivalent to the statement that, forT as in (5.1.17), lle-(T-(n- 2)/ 2Jtvii£P 1 (dtdw) ~ Q lle-(T-(n- 2)/ 2]tQVII£P(dtdw)'

v E Clf(IR. x with

1a Q = ( i at

sn- 1 ),

(5.1.17')

)2 - (n- 2) ata As.

But if we let QT = e-[T-(n-2)/2Jtqe!T-(n-2)/2]t 1

this is equivalent to llwiiLP'(dtdw) ~ C IIQTwiiLP(dtdw)·

(5.1.17")

The symbol of QT is

(17- i(k + ~- r))(1J + i(k + ~ + r)). So if we define TT by

Td(t,w) =

2~

i:i: t.{(

1J - i( k +

~ - T)) (1J + i(k + ~ + r))} - 1

x Hkf(s,w)ei(t-s)TJ d7Jds, it suffices to show that (5.1.21) We suppose from now on that T > 0 since the argument for T < 0 is similar. Following the proof that the one-dimensional Hardy-LittlewoodSobolev inequality implies the higher-dimensional versions, we first fix t and estimate the v' norm over sn- 1 :

where

mT(t, s; k)

=~ 211"

1

00

-oo

ei(t-s)TJ{ (17- i(k

+ ~- r)) (17 + i(k + ~ + r))} - 1 d7J.

149

5.2. Estimates for Riesz Means Since we are assuming that lk + ~ N,

- rl

~ ~, it follows that, for any

lmT(t, s; k)l :$ CN(1 + k)- 1 {1 + lk + ~- rl· It- sl) -N. Hence, if we use the estimates for spherical harmonics (5.1.16), we get

:$ C

i: t.(l X

+ k)- 21n(1 + lk + ~- rl·lt- si)-N

llf(s, . )IILP(dw) ds.

But our assumption on

T

implies that if N

>2

00

~)1 + k)- 2 /n{l + lk + ~- rl·lt- si)-N = O{lt- sl-1+ 2/n). k=O

Hence we get IITd(t, · )IILP'(dw) :$ C

i:

It- sl-1+ 2/n llf(s, · )IILP(dw) ds,

which leads to (5.1.21) after an application of the Hardy-LittlewoodSobolev inequality (i.e., Proposition 0.3.6) as 1/p- 1/p' = 2/n. 1

5.2. Estimates for Riesz Means In this section we shall study the Riesz means of index to P(x,D):

S~f(x) =

L {1- >.;/>.) 6E;f.

/j ~

0 associated (5.2.1)

.>.;$.>.

As in IR.n, these operators can never be uniformly bounded in V(M), p 'f:. 2, when /j :$ 6(p) if 6(p) is the critical index 6(p)

= max{nl~- ~~- ~,0}.

(5.2.2)

On the other hand, we shall prove the following positive result which extends Euclidean estimates in Section 2.3.

Theorem 5.2.1: Let P(x, D) E w~1 (M) be positive and self-adjoint. Then there is a uniform constant C5 such that for all>.> 0 (5.2.3) IfwealsoassumethatthecospheresassociatedtoP,'Ex = {e :p(x,e) = 1}

5. LP Estimates of Eigenfunctions

150

c

T; M \ 0 have non-vanishing Gaussian curvature, then for p E [1, 2(n + 1)/(n + 3)] U [2(n + 1)/(n- 1), oo] and A> 0

IIS~fiiLP(M) ~ cp,6llfiiLP(M)•

6 > 6(p).

(5.2.4)

As a consequence of this result, we get that, for p < oo as in the theorem, S~f--+ f in the l.Jl topology if 6 > 6(p). Notice that, for the exponents in both (5.2.3) and (5.2.4), 6{p) = ~- ~ ~· Hence, by Lemma 4.2.4, Theorem 5.1.1, and duality, both inequalities are a consequence of the following.

nl

1-

Proposition 5.2.2: Suppose that P(x, D) E \11~ 1 is positive and selfadjoint. Suppose also that for a given 1 ~ p < 2 there is a uniform constant C such that

llnfii£2(M) ~ C(1 + A)n(1/p-1/2)-1/211fiiLP(M)•

A~ 0.

(5.2.5)

Then (5.2.4) holds.

Proof of Proposition 5.2.2: Since the Fourier transform ofT~ is c6(t + iO) - 6- 1 , we can write

S~f =

(21r)-1c6A-6

I

eitP fe-it>.(t

+ i0)-6-1 dt.

To use the parametrix for eitP we let e be as in Theorem 4.1.2 and fix p E C~(IR) which equals 1 for ltl < e/2 and 0 for ltl >e. We then set 6

-6

...Fi

S>.f = S>.f + .«).f,

where

S~f =

(21r)-1c6A-6

I

eitP fe-it>.p(t)(t

We would be done if we could show that, for 6

+ i0)-6-1 dt.

> 6(p),

-6

IIS>.fiiP ~ C6llfiiP•

(5.2.6)

~ C6llfllp·

(5.2.7)

II.RifiiP

We first estimate the remainder term and in fact show that it satisfies a much stronger estimate:

IIR~fll2 ~ C(1 + A)[n(1/p-1/2)-1/2J-611fllp·

{5.2.7')

This is not difficult. We note that, for 6 ~ 0, the Fourier transform of ( 1 - p( t)) (t + iO) - 6- 1 is bounded and rapidly decreasing at infinity. This means that = A- 6 r6(A- Aj)Ej/ j

Rif

L

151

5.2. Estimates for Riesz Means for some function r6 satisfying lr6{.X)I :5 CN{l + 1-XI)-N for all N. Hence, for large N, 00

II.Ri/11~ :5 cN-X- 26 L:(t +I-X- ki)-NIIxkfll~ k=1 00

:5 c'_x-26 L(1 +I-X- ki)-N (1 + k)2(n(1/p-1/2)-1/2]11/ll~ k=1

:5 c" (1 +.X) -2[6-n(1/p-1/2)+1/2]11/ll~. as desired. To estimate the main term we decompose S~ as in the proof of Theorem 2.4.1. To this end, we fix /1 E C~(IR \ 0) satisfying E~-oo .B(2ks) = 1, s '# 0. We then write -6 -6 ~ -6 S>.f = s>.,of + L....J s>.,kf, k~1

where for k = 1, 2, ...

stkf = {21r)- 1c6.x- 6

I

eitP fe-it>./1(-XTkt) p(t)(t + i0)- 6- 1 dt.

=

Note that S~ kf 0 if 2k is larger than a fixed multiple of .X. We claim that (5.2.5) and the finite propagation speed of the singularities of ( eitP)(x, y) can be used to prove that, for any e > 0, (5.2.8) By summing a geometric series this leads to (5.2.6). Actually the estimate fork= 0 just follows from Theorem 4.3.1. This is because where mi 0 (r) is the convolution of {1- r/.X)t with .x- 1,.,(r/.X) if 1J is

'

00

the inverse Fourier transform of {1-

E

/1{2-kr)) E C~(IR.). Hence, for

k=1

anyN,

I(:Tr

mi,o(r)l :5 Ca,N_x-lal {1 +r/A)-N.

Thus, if p > 1 the estimate is a special case of (4.3.3). Since pseudodifferential operators of order -1 are bounded on £ 1, the case of p = 1 follows via Theorem 4.3.1 and the easy fact that the pseudo-differential operators with symbol mi_ 0 (p(x,~)) are uniformly bounded on £ 1.

5. LP Estimates of Eigenfunctions

152

To handle the terms with k

s~lkf =

(2?r)- 1c6A - 6

I

+ (2?r)- 1c6A - 6

~

1 in (5.2.8) we write

Q(t)fe-it>.{3(A2-kt) p(t)(t + i0)_ 6_ 1 dt

I

R(t)fe-it>.{3(ATkt) p(t)(t + i0)- 6- 1 dt

1

where Q(t) is the parametrix for eitP. Since the kernel of R(t) is C 00 one sees that the kernel of the second operator is 0(2-kN) for any N. Hence it suffices to show that the integral operator with kernel

Htk(x~ y) =

(2?r)-n-1c6A -6

II

ei(rp(x~y~e)+t(p(y~e)->.)]

x q(tl x, y, e> f3(ATkt> p(t> (t + io)- 6 - 1 dedt

satisfies the bounds in (5.2.8). A main step in the proof of this is to show that, if E > 0 is fixed, then for every N

r

Jlx-yl>>.-12k(l+•) #

IH~ k(x,y)l dy, I

"rJlx-yl>><-12k(l+•) IH~ Ik(x, Y)l dx ~ CeiNTkN.

(5.2.9)

If we could prove this inequality then, by repeating the arguments in the proof of Theorem 2.3.5, we would find that (5.2.8) would follow from showing that one always has I

ns~lk/IILP(B(xol>.-12•)) ~ CTk(6-n( 1/p- 1/ 2)+1/ 2lllfiiLP(M)• (5.2.8') if B(xo, r) denotes the ball ofradius r around xo with respect to a fixed smooth metric. But, if we use HOlder's inequ~ity, we can dominate the left side by {2k /At( 1/p- 1/ 2) IIS~Ikfii£2(M)·

Next, we observe that S~ kf = eN 2-k6 (1

-6

mi k(A -

+ l2kTI AI) -N f~r any N.

P)/ where

lienee (5.2.5) gives

2

IIS>.Ik/IIL2(M) 00

~ CT2k6

L(1 + 2k A-11A _ ii)-N (1 + j)2(n(1/p-1/2)-1/2)11/ll~ j=O

~

C' T2k6 A2(n(1/p-1/2)-1/2) (A/ 2k) 11/11~.

5.3. More General Multiplier Theorems

153

So, if we combine this with the last estimate we conclude that the left side of (5.2.8') is always majorized by 2-k6 (2k / >..)n(1/p-1/2) >._n(1/p-1/2)-1/2(>../2k)1/211/llp

= Tk[6-n(1/p-1/2)+1/2) 11/llp, as desired.

To finish matters and prove (5.2.9) we let atk( ·, x, y, ')denote the inverse Fourier transform oft---+ c0 >.. - 6q(t, x, y, 0.8(>..2-kt)p(t)(t+i0)- 6- 1. Then one sees that

ID~DJai,k(r,x,y,,)l :$ Ca-yNTk 6 (2kf>..)a(1+12krf>..I)-N (1+1,1)-hl. (5.2.10) But

Htk(x,y)

= (211")-n

I

ei
>..,x,y,,)~,

and so (5.2.10) (and the fact that IVepl ~ cix- Yl on the support of the symbol) gives the bounds

IH~ k(x,y)l :$ CNTk>..n i>..Tk(x- Y)I-N, I

I

which of course yield (5.2.9).

Remark. One could also prove (5.2.4) in a more constructive way by combining arguments from the proofs of Theorems 5.1.1 and 2.3.1. In fact, the stationary phase arguments that were used in the proof of Lemma 5.1.3 show that the kernel of S~ is 0(>..-N) outside of a fixed neighborhood of the diagonal, while near the diagonal it is of the form

s~ (X, y) where

t/J;

= >.. (n-1)/2-6 ( ei.X,Pl(x,y) ai,1 (x, y) + e-i.X1/J2(y,x)Qt2(X, y))

I

is as in Lemma 5.1.3 and a 6 ·( x, y )I< ID x,ya_x, _ Ca (d"IS t (x, y ))-(n+l)/2-6-lal . 3

Since the phase functions satisfy the n x n Carleson-Sjolin condition one can therefore break up the kernel dyadically near the diagonal and argue as in the proof of Theorem 2.3.1 to obtain (5.2.4).

5.3. More General Multiplier Theorems Suppose that mE £CX>(JR). Fix .BE C~( (1/2, 2)) satisfying L:~oo ,8(2ir) = 1, T > 0, and suppose also that sup>..- 1 -X>O

1-oo l>..aD~(.B(r/>..)m(r))l 00

2 dr < oo,

0 :$a:$ s,

(5.3.1)

5. LP Estimates of Eigenfunctions

154

where sis an integer> n/2. Then if p(e) is homogeneous of degree one, positive, and C 00 in Rn \0 it follows that m(p(e)) satisfies the hypotheses in the Hormander multiplier theorem, Theorem 0.2.6. Therefore (m(p(D))f)(x) = (211")-n

f ei(x,~)m(p(e)) i(e) de

}Rn

is a bounded operator on LP(Rn) for all1 < p < oo. The purpose of this section is to extend this result to the setting of compact manifolds of dimension n ~ 2.

Theorem 5.3.1: Let m E L00 (R) satisfy (5.3.1). Then if P(x, D) E \11~1 (M) is positive and self-adjoint it follows that

llm(P)fiiLP(M) :::; Cp llfiiLP(M)•

1 < P < oo.

(5.3.2)

Proof: Since the complex conjugate of m satisfies the same hypotheses we need only prove (5.3.2) for exponents 1 < p :::; 2. This will allow us to exploit orthogonality and also reduce (5.3.2) to showing that m(P) is weak-type (1,1): JL{x: lm(P)f(x)l > o:} :::; Co:- 1 11flh·

(5.3.2')

Here JL(E) denotes the dx measure of E C M. Since m(P) is bounded on £ 2 , (5.3.21) implies (5.3.2) by the Marcinkiewicz interpolation theorem. The proof of the weak-type estimate will involve a splitting of m(P) into two pieces: a main piece to which the Euclidean arguments apply, plus a remainder which can be shown to satisfy much better bounds than are needed using the estimates for the spectral projection operators. Specifically, if p E CQ"(R) is as in the proof of Theorem 5.2.1, we write

m(P)

= m(P) + r(P),

where

m(P) = (m * p) (P) =

_.!:.._I eitP p(t) m(t) dt. 211"

To estimate the remainder we define for>.= 2i ,j = 1, 2, ... , m~(r) = {3(rf>.)m(r). Then we put

r~(P) = 2~ and notice that ro(P)

I

eitP(1- p(t)) m~(t) dt,

= r(P) -

E r 2 (P) t

is a bounded and rapidly

k~1

decreasing function of P. Hence ro(P) is bounded from £ 1 to any £P space. Therefore, we would have

llr(P)fll2 :::; C llfll1,

(5.3.3)

155

5.3. More General Multiplier Theorems which is much stronger than the analogue of (5.3.21), if we could show that (5.3.3')

To prove this we use the L 1 operators to get

-+

L2 bounds for the spectral projection

00

llr.x(P)/11~ ::;

L

llr.x(P)Xkfll~

k=O 00

::; C

L TE(k,k+l) sup lr.x(r)l(1 + k)n-

1

11/11~.

k=O

Hence (5.3.3') would be a consequence of 00

L: k=O

sup

lr.x(r)l2(1 + k)n-1::; c.xn-2s.

(5.3.4)

TE(k,k+l)

We claim that this follows from our assumption (5.3.1). The first thing to notice, though, is that since m,x(r) = 0 forT fl_ (A/2,2Aj both m,x(r) and r,x(r) are 0((1 + lrl + 1-XI)-N) for any N if T fl. [.X/4,4-X]. Hence .£S:3.4) would follow from

L:

sup

lr.x(r)l2::; c.x1-2s.

(5.3.41)

kE(.\/4,4.\) TE(k,k+l) But if we use the fundamental theorem of calculus and the CauchySchwarz inequality, we find that we can dominate this by I

jlr.x(r)l 2 dr+ jlr~(r)l 2 dr=

2~ jlm.x(t)(1-p(t))1 2 dt +

2~ j 1tm.x(t)1 21(1- p(t))l 2 dt.

Recall that p = 1 for ltl < e/2, so a change of variables shows that this is majorized by

_!_,x -1-2s 211"

j ltsm.x (tf .X)I2 dt

= _x-1-2s jlv:(.xm,x(.Xr)) 12 d; = _x1-2s. {.x-t jl.xsv:(f3(rf.X)m(r))l2dr}. By (5.3.1) the expression inside the braces is bounded independently of A, giving us (5.3.4').

5. LP Estimates of Eigenfunctions

156

Since we have established (5.3.3), it suffices to show that m(P) m(P)- r(P) is weak-type (1,1}:

JL{x: lm(P)I(x)l >a}~ Ca- 1 111111·

=

(5.3.5)

But if we argue as before, this would follow from showing that the integral operator with kernel

K(x, y)

= (211")-n- 1 = (211")-n

If ei[rp(x,y,~)+tp(y.~)]

1eirp(x,y.~)

p(t) m(t) q(t, x, y, ') ~dt

m(p(y, ,), x, y, ,) ~

(5.3.6)

is weak-type (1,1}. Here we have abused notation a bit by letting m(r,x,y,,) denote the inverse Fourier transform oft ---+ p(t)m(r) X q(t, x, y, ,), that is,

m(r,x,y,O = ([p( · )q( · ,x,y,OJv * m)(r).

(5.3.7)

Notice that, if we define m_x(r,x,y,,) in a similar manner, then this function is C 00 and, moreover, (5.3.1) and the fact that q E SO imply

L

sup

l!Di'De(m_x(p(y,A,),x,y,A'})I 2 ~
(5.3.8)

.X O$lal$s Rn

If we let

K_x(x, y) =

Jeirp(x/.X,y/.X,.X~)m_x(p(yjA,

A,), xjA, yjA, A')~.

we claim that this gives the bounds

{

!K.x(x,y)! dx

J{x:lx-yi>R}

J

IK.x(x, Yo)- K_x(x, yi)I dx

~ C(1 + R)n/ 2 -

8,

~ CIYO-=- Y11·

(5.3.9)

Before proving these, though, let us see why they imply that

Tl(x) =

J

K(x,y)l(y)dy

is weak-type (1,1). We let I= g + Ek:, 1 bk be the Calder6n-Zygmund decomposition of I at level a given in Lemma 0.2.7. Let Qk ::> suppbk be the cube associated to bk. Then, since T is bounded on L 2 , if one repeats the arguments at the end of the proof of Theorem 0.2.6, one concludes that the weak-type boundedness ofT would be a consequence of the estimates

(5.3.51)

5.3. More General Multiplier Theorems

157

if Qic denotes the double of Qk. To prove this, we may assume that Qk is centered at the origin and we let R be its side-length. Then using the first estimate in (5.3.9) we get

1~Q;.II ,xnK,x(.Xx,.Xy)bk(y)dyldx :$

llbkllt sup

r

Y l{x: jx-yj >R}

I.Xn K.x(Ax, .Xy)i dx :$ C(.XR)n/ 2-sllbklll·

Using the cancellation property

I

_xn K,x(.Xx, .Xy)bk(Y) dy

=I

J bk dy = 0, we have _xn [K.x(.Xx, .Xy)- K,x(.Xx, O)]bk(Y) dy.

This and the second inequality in (5.3.9) lead to

1~Q;.II _xn K,x(.Xx, .Xy)bk(Y) dyl dx :$1 yEQk

:$

1

x~Qi.

_xniK.x(.Xx, .Xy)- K.x(.Xx, O)libk(Y)i dxdy

C(.XR)IIbkllt·

But K(x,y) = L:~ 1 2niK2;(2ix,2iy)+Ko(x,y), where Ko is bounded and vanishes when lx- Yi is larger than a fixed constant. Therefore, if we combine the last two estimates we conclude that

1

.ITbk(x)i dx :$ Cllbkllt.

x~Q.

:$

(

L

(2j R)n/ 2-s

2;R~l

+

L 2j n)

2;R
C'llbkllt.

which establishes (5.3.5'). To finish matters and prove (5.3.9), we notice that both inequalities would follow via Schwarz's inequality and

liD~ K,x(x, y)l 2 (1 + lx- yi) 28 dx :$ C,

0 :$

lo:l :$ 1.

(5.3.9')

We shall only prove the case o: = 0 since the others follow from the same argument, after observing that

Drm,x(r,x,y,e)

=I

eitritp(t)m,x(t)q(t,x,y,e)dt = fft,x(r,x,y,e),

with m,x having the same properties as Let us first show that

I

m,x.

IK.x(x,y)i 2 dx :$C.

(5.3.10)

5. LP Estimates of Eigenfunctions

158

By possibly making e smaller we may assume that on the support of q (5.3.11) for some positive constant c. To use this we write

J

IK.x(x,y)i 2 dx =

>.n x

JJJ

J

= >.n IK.x(>.x,y)i 2 dx

ei.X{
m.x (p(y/ >.,>.e), x, yf >.,>.e) m.x (p(y/ >., >.,.,), x, yf >., >.,.,) dx ded,.,.

However, if we use (5.3.11) we see that, for any N, this is dominated by

2: JJJ >.n(1 + >.ie- ,.,1) -NIDJ m.x (p(y/ >.,>.e), x, yf >.,>.e) 1

1

O:Sh;I:SN

x

IDJ m.x (p(y/ >., >.,.,), x, yf >.,>.,.,)I ded17dx. 2

Therefore, if we apply the Schwarz inequality and (5.3.8), we get (5.3.10), since the integrand in the last expression is compactly supported in x. To finish the proof we must show that when Ia: I= s

jl(x- y)aK,x(x,y)l 2 dx

~ ~-

But IVePI ~ clx- Yi on suppq. Therefore, the above arguments show that the integral is dominated by

L JJJ >.n(1 + >.ie- '71) -NIDil Dt (m,x(p(yf>., ~),x, yf>., >.e)) I O:Sia;I:Ss O:Sh;I:SN

x 1Di2 D~ 2 (m.x (p(y/ >., >.,.,), x, yf >., >.17)) lded17dx.

As before, this along with (5.3.8) and an application of the Schwarz I inequality implies the missing inequality.

Notes Theorem 5.1.1 is due to Sogge [2], Christ and Sogge [1], and Seeger and Sogge [2]. The argument showing that the estimates in this theorem are sharp is a variable coefficient version of an argument of Knapp (see Tomas [1]) which applied to the £ 2 restriction theorem for the Fourier transform in an' and the Riemannian version of this argument was given in Sogge [4]. See also Stanton and Weinstein [1]. Davies [2] has shown that the estimates in Corollary 5.1.2 need not hold if the metric is not

Notes assumed to be C 00 • Even sharp L 00 estimates are not known when the for 0 < a < 2. However, using heat kernel metric is assumed to be techniques (see Davies [1), [2)), one can show that, for L 00 uniformly elliptic metrics, IIXAII(£2,£"") = O(.~n/ 2 ). A reasonable conjecture might metrics, with 0 .II(£2,£oo) = O(A(n-a)/2 ). be that for Using the Hadamard parametrix and the proof of Lemma 4.2.4, one can show that the bounds are 0(A(n-l)(2 ) for C 1•1 metrics. It would also be interesting to find the appropriate extension of Theorem 5.1.1 to compact manifolds with boundary. In two dimensions, Grieser [1) has shown that if P = ~and if the boundary is geodesically concave (i.e., diffractive), then the estimates in Theorem 5.1.1 hold; on the other hand, (5.1.1) can only hold for a smaller range of exponents for manifolds with convex (i.e., gliding) boundary. The strong unique continuation theorem for the Laplacian is due to Jerison and Kenig [1), and independently to Sawyer [1) in three dimensions. The simplified proof we have used, though, is from Jerison [1). For related arguments which show how the L 2 restriction theorem for the Fourier transform in Rn and related oscillatory integral theorems can be used to prove uniqueness theorems and embedding theorems see Hormander [8), Kenig, Ruiz, and Sogge [1), Sogge [5), and Wolff [1], [2). The estimates for Riesz means are due to Sogge [3) and Christ and Sogge [1). The best prior results were due to Hormander [3) and Berard [1). The extension of the Hormander multiplier theorem to the setting of compact manifolds was proved in Seeger and Sogge [1).

ca

ca

159

Chapter 6 Fourier Integral Operators

We start out with a rapid and somewhat sketchy introduction to Fourier integral operators, emphasizing the role of stationary phase and only presenting material that will be needed later. In Section 2 we give the standard proof of the L 2 boundedness of Fourier integral operators whose canonical relations are locally a canonical graph and we state and prove a special case of the composition theorem in which one of the operators is assumed to be of this form. The same proof of course shows that this theorem holds under the weaker assumption that C1 x C2 intersects {(x, y, TJ, y, TJ, z, () : (x, E T* X\ 0, (y, TJ) E T*Y \0, (z, () E T* Z \ 0} transversally, although it is a little harder to check here that the phase function arising in the proof of the composition theorem is non-degenerate. The next thing we do is to prove the pointwise and LP regularity theorems for Fourier integral operators and show that these are sharp if the operators are conormal with largest possible singular supports. Although this theorem came first, its proof uses the decomposition used in the proof of the maximal theorems for Riesz means and the circular maximal theorem given in Section 2.4. In the last section we apply the estimates for Fourier integral operators to give a proof of Stein's spherical maximal theorem and its variable coefficient generalizations involving the assumption of rotational curvature. In anticipation of the last chapter, we point out how this assumption is inadequate for variable coefficient maximal theorems in the plane.

e,

e)

6.1. Lagrangian Distributions

161

6.1. Lagrangian Distributions In Section 0.5 we studied certain types of homogeneous oscillatory integrals whose wave front sets turned out to be Lagrangian submanifolds of the cotangent bundle. In this section we return to the study of such distributions, this time taking a somewhat more global point of view. We start out with a few definitions. First of all, the Besov space 00Hu(R.n) is defined to be the space of all u E S'(R.n) for which u E L~c (R.n) and, moreover,

llullooH

"

(Rn) = ( {

l1e19

lft(,)l 2 ~) 112 f

+sup( j?:_O

J2i$Jei$2Hl

j2uiu(,)j 2 ~) 112 < oo.

(6.1.1)

If X is a C 00 manifold of dimension n we can extend this definition by using local coordinates. We define 00H~c(X) to be all u E V'(X) for which (t/Ju) 01\';-l is always in 00Hu(R.n) whenever n c X is a coordinate patch with coordinates ~t and t/J E Clf(O). Next, if A C T* X\ 0 is a C 00 closed conic (immersed) Lagrangian submanifold, we define the space of all I;..airangian distributions of order m which are associated to A, 1m( X, A), as follows. We say that u E Im(X,A) if N

IT P;u

E 00H~~-n;4 (X),

(6.1.2)

j=l

whenever P; E '111~ 1 (X) are properly supported pseudo-differential opertors1 whose principal symbols P;(x,,) vanish on A. The reason for the strange convention concerning the order will become apparent if one considers pseudo-differential operators. Specifically, if a(x, ') E sm(R.n) then it follows from Theorem 0.5.1 that the Schwartz kernel of a(x, D),

u(x,y) = {21r)-n

Jei(x-y,e)a(x,,)~,

satisfies WF(u) C A= {(x,x,,,-,)}. Moreover, Theorem 6.1.4 below 1 P is said to be properly supported if, given a compact set K C X, there is always a compact set K' C X such that supp u C K ~ supp Pu C K' and u 0 inK' ~ Pu. = 0 inK. The reader can check that any pseudo-differential operator can be written as the sum of a properly supported pseudo-differential operator plus a smoothing error. Notice also that if X is compact then every pseudo-differential operator is properly supported.

=

6. Fourier Integral Operators

162

shows that (6.1.2) is exactly the right normalization so that the order of the Lagrangian distribution u is the same as the order of the associated pseudo-differential operator, that is, u E rn(R.n x IRn, A). Another remark is that ifu E /m(X, A) then WF(u) CA. To see this, we pick (x 0 , 'o) fl. A and let r be a small conic neighborhood of (xo, 'o) satisfying f n A = 0. Then, if P; E w~1 ,j = 1, ... , N, have principal symbols supported inside r, (6.1.2) must hold. From this one deduces WF(u) = 0 which gives us the claim. To prepare for the main result of this section, the equivalence of phase function theorem for Lagrangian distributions, we need a few preliminary results. The first one concerns the Fourier transform of u E /m(R.n, A) when A takes a special form. We saw in Section 0.5 that if H(') E C 00 (1Rn \ 0) is real and homogeneous of degree one, then A= {(H'(,), ')} is a conic Lagrangian submanifold of T*IRn \0. In this setting we have:

nr

Proposition 6.1.1: lfu E I~mp (IRn, A) with A of the form {(H'(,), ')} then for 1,1 ~ 1, u(') = e-iH(e)v(') with v E sm-n/4(1Rn). Proof: Let p E Clf(IRn) equal !me near the origin and let h = pHo, with Ho = (1- p)H. Then Hrr- hE S and so it suffices to show that

v(') = eih(e) u(') E sm-n/4. Set h; = oh/8,;. Then, by construction, h;(D) is properly supported since this operator is convolution with the inverse Fourier transform of h;(') which is compactly supported. Hence, since the principal symbol of Dk(x; - h;(D)) vanishes on A, we get from (6.1.2) that

v.BIJ(x;- h;(D))a;u E 00H_m-n/4 if la:l

= I.BI-

This means that for R > 1

{

1R/2::;iei::;2R

l,.a

IJ {-D.h ·('))a;u(,)l2 cl{ :S CaR2(m+n/4), 3 3 la:l =

I.BI,

or, equivalently,

{

1R/2::;iei9R

1'12lai1Dav(,)l2 cl{ :S CaR2(m+n/4).

By rescaling, we see that vn(') = v(RI,.)/ Rm-n/4 satisfy the uniform estimates

6.1. Lagmngian Distributions By the Sobolev embedding theorem, this implies that ID0 vR(e)l when lei= 1, or, equivalently, ID0 vl ~ Ca(1 + len-m+n/ 4 -lal.

163 ~

Ca

1

We need one other result for the proof of the equivalence of phase function theorem. Recall that in Section 0.5 we saw that every Lagrangian section ofT*R.n is locally the gradient of a C 00 function. A similar result holds for homogeneous Lagrangian submanifolds. Proposition 6.1.2: Let 'YO = (xo, eo) E A C T* X\ 0, with A being a C 00 conic Lagrangian manifold. Then local coordinates vanishing at xo can be chosen such that

e

( 1) A 3 (X' e) -+ is a local diffeomorphism, (2) and there is a unique real homogeneous HE C 00 such that, near (xo, eo),

A= {(H'(e), en. Let us first assume that (1) holds and then see that this implies (2). This is easy, for, near 'Yo, A = {(if>(e), en for some l/> which is homogeneous of degree zero and C 00 near eo. We saw in the proof of Proposition o.5.4 that the canonical one form w = E e;dx; must vanish identically on A. This means that if 4>; denotes the jth coordinate,

L:e; dlf>;(e)

= o.

Or, if we set H(e) = Ee;l/>;(e), then

dH(e) = L:4>;(e)de;. that is, l/>;(e) = aH(e)Jae;. giving us (2). To prove that local coordinates can be chosen so that ( 1) holds we need an elementary result from the theory of symplectic vector spaces whose proof will be given in an appendix. Lemma 6.1.3: lfVo and V1 are two Lagrangian subspaces ofT*R.n one can always find a third Lagrangian subspace V which is transverse to both Vo and V1. Recall that two C 00 submanifolds Y, Z of a C 00 manifold X are said to intersect transversally at xo E Y n Z if Tx 0 X = Tx 0 Y + Tx 0 Z. Proof of (1): We first choose local coordinates y so that 'YO = (0, cl} with E'l = (1, 0, ... , 0). The tangent plane to A at 'Yo, Vo, must be a Lagrangian plane. If it is transverse to the plane W = {(y, cl)} = {(y, dyl)}, we can take y as our coordinates since the transversali ty of Vo and W is equivalent to (1). If not, we use Lemma 6.1.3 to pick a Lagrangian plane V which is transverse to both Vo and V1 = {(0, en.

164

6. Fourier Integral Operators The transversality of V and V1 means that V is a section passing through 'Yo and hence V = {(y,d(yl + Q(y)))} for some real quadratic form Q. If we now take x1 = Yl + Q(y),x; = Y;,j = 2, ... ,n, as our new coordinates, it follows that, in these coordinates, the tangent plane at 'Yo, Vo, and V = {(x,dxi)} are transverse, giving us (1). I We now come to the main result. Recall that the homogeneous phase function ¢(x, 9) is said to be non-degenerate if d¢ '# 0 and when ¢ = 0 the N differentials d(o¢fo9;) are linearly independent. We saw before that this implies that :Ecf> = {(x, 9) : ¢ (x, 9) = 0} is a C 00 submanifold of X x (aN\ 0) and that A= {(x, ¢~(x, 9)) : (x, 9) E :Ecf>} is Lagrangian.

0

0

Theorem 6.1.4: Let¢ be a non-degenerate phase function in an open conic neighborhood of (xo, lio) E an X (aN\ 0). Then, if a E S~£(an X aN), p. = m+ n/4- N /2, is supported in a sufficiently small conic neighborhood r of(xo,9o) itfollows that

u(x) = (211")-(n+ 2N)/4 {

JRN

eicf>(x,(J) a(x, 9) d9

(6.1.3)

is in Im(an,A) with A as above. If we also assume that coordinates are chosen so that A= {(H'(,), ,)}, then

eiH(e)u(')- (21r)nf 4a(x, 9)ldet ¢"1-l/2e¥ sgncf>" E sm-n/ 4 - l (6.1.4) for 1,1 > 1 near'o = ¢~(xo,9o), where (x,9) is the solution of¢ (x,9) = 0, ¢~(x, 9) = ,, and

0

"'" = (

"+'

rP~x ¢~(} ) "'" "+'(Jx

"'"(}(} "+'

.

Conversely, every u E Im(an, A) with W F( u) contained in a small neighborhood of (xo, 'o) can be written as (6.1.3) modulo C 00 • One should notice that this result contains the equivalence of phase function theorem for pseudo-differential operators, Theorem 3.2.1, since the phase function rp(x, y, ') there and the Euclidean phase function (x -y,,} both parameterize the trivial Lagrangian {(x,x,,, _,)}.Similarly, if u E I~mp (an x an, A), with A the trivial Lagrangian, it follows that u(x, y) is the kernel of a pseudo-diffE'ft~tial operator of order m. So this clarifies the remark made earlier that the order of the distribution kernel of a pseudo-differential operator agrees with the order of the pseudo-differential operator. Using (6.1.4) we can define ellipticity. We say that u E F(X, A) is elliptic if, when coordinates are chosen so that A = {( H' ('), ')}, the absolute value of either of the terms on the left side of (6.1.4) is bounded

165

6.1. Lagrangian Distributions from below by l'lm-n/ 4 for large '· Since det t/J11 is homogeneous of degree -(N- n) this just means that a(x,8) is bounded from below by l8lm+n/ 4-N/2 when 181 is large and (x, 8) E :Eel>.

Proof of Theorem 6.1.4: We may assume that a(x, 8) vanishes when x is outside of a compact set and that coordinates have been chosen so that A= {(H'(,), ,)}. We shall then use stationary phase to evaluate eiH(e>u(') = (21r)-(n+2N)/4

JJ i[cf>(x,B)-(x,e)+H(e)J a(x, 8) d8dx. (6.1.5)

The first thing we must check is that the Hessian (with respect to the variables of integration) of the phase function is non-degenerate, that is, det t/J" '# 0. But this follows from the fact that the maps

Eel>

3

(x, 8) - (x, t/J~(x, 8))

E

A and A 3 (x, ') - '

are both diffeomorphisms. Since t~J9 = 0 on Ecf>, this means that the map r 3 (x, 8) - (t/J~. t~J{J) has surjective differential on :Eel> and hence in r, if this set is small enough. But this is just the statement that det t/J" '# 0. Since the stationary points, depending on the parameter ', are nondegenerate, we can assu~e, ..after possibly contracting r, that, for every 'near there is a unique stationary point. Since we may also assume that t/J~ '# 0 in r it follows that the difference between (6.1.5) and

,o,

(21r)-(n+2N)/4

JJei[cf>(x,B)-(x,e)+H(e)lp(8/lel) a(x, 8) d8dx

(6.1.5')

is rapidly decreasing if .B E CQ"(IRn \ 0) equals one for 181 E re-I C] with C sufficiently large-in particular, large enough so that .8(8/IW = 1 at a stationary point. Next, if we set >. = lei and w = '/1,1, we can rewrite (6.1.51) as I

(21r) -(n+2N)/4 >. N

JJei.X[cf>(x,B)-(x,w)+H(w)] .8(8) a(x, >.8) d8dx.

Notice that the integration is over a fixed compact set and that, at a stationary point, we have t/Jo = O,x = H'(w), and hence, by Euler's homogeneity relations,

tfJ(x,8) = (tfJ{J(x,8),8} = 0 and

(x,w} = (H1(w),w} = H(w).

Thus one reaches the conclusion that the phase function always vanishes at the stationary point. Therefore, the stationary phase formula (1.1.20) tells us that the difference between (6.1.51) and

(21r)n1 4 >.(N-n)/2 a(x, >.O)Idet t/J11 (x, 8) ~-l/ 2 e¥ sgncf>"

(6.1.6)

6. Fourier Integral Operators

166

is in sm-n/4- 1I since (m + n/4- N/2) + (N- n)/2 dettf>" is homogeneous of degree -(N- n) and so

= m- n/4.

But

A(N-n)/21det 4>" (x, 8)1-1/2 = ldet 4>" (x, .X8)1-1/2.

Hence, (6.1.6) is the second term in (6.1.4), which gives us the first half of the theorem. To prove the converse, we use Propositions 6.1.1 and 6.1.2 to see that it suffices to consider u E Im(!Rn I A) having the property that v = ueiH E sm-n/4 is supported in a small conic neighborhood of eo. We then let ~(x, 8) = 8tj>f8x and put

ao(x,8)

= (21r)-n/ 4 vo~(x,8)ldettf>"l 1 / 2 e-7 8 gn¢" E sm+(n-2N)/4.

If we then define uo by the analogue of (6.1.3) where a(x, 8) is replaced by ao(x, 8), it follows that u-uo E Jm- 1. Continuing, we construct Uj from

aj(x,8) E sm+(n- 2N)/ 4-i so that u-(uo+·· ·+uj) E lm-i- 1(1Rn,A). If we then pick a"" Eai, it then follows that (6.1.4) holds mod C 00 •

I Let us end this discussion by making a few miscellaneous remarks concerning the many ways that one can write Lagrangian distributions. First, if u is given by an oscillatory integral as in (6.1.3), one can add as many 8 variables as one wishes and get a similar expression involving the new variables. In fact, if Q(8NH• ... , 8N') is a non-degenerate quadratic form in N' - N variables, one checks that the phase function ;[,(8) = 4>(8) +Q(8N+I•····8N,)/I81, defined in the region I(8NH·····8N')I < 181, is also non-degenerate and parameterizes A. So by_ Theorem 6.1.4 we can express u by an oscillatory integral involving 4> and a symbol a(x,B) E sm+n/4-N'/2. A more interesting construction involves the reduction of theta variables. As a preliminary, we need an observation about the rank of the Hessian of the phase function with respect to the theta variables. Specifically, let /IA : A 3 (x, e) ---+ X and /IE~ : Eq, 3 (x, 8) ---+ X denote projection onto the base variable, and let K. : Eq, 3 (x, 8) ---+ (x, t/>~(x, 8)) E A. Then, if K.(xo, 8o) = (xo, eo), we can compare the rank of t/>~o(xo, 8o) with the rank of d/IA(xo,eo). In fact, since K. is locally a diffeomorphism and Eq, is n-dimensional-both because we are assuming that 4> is non-degenerate-we have the formula dim

Kerd/IE~ =

n- rankd/IA·

(6.1.7)

But if a tangent vector is in Kerd/IE~ it must necessarily be of the form v = Ei vj8/88j. And it must also satisfy Ei vj8 2¢jaoiaok = 0,

6.1. Lagrangian Distributions

167

= 1, ... , N, since this is a necessary and sufficient condition for v to be a tangent vector to E4>. Since the dimension of such tangent vectors at (xo, 8o) is N- rank 4>~9 (xo, Oo), (6.1.7) implies the following:

k

Proposition 6.1.5: If (xo, 'o) E A we have

4> is non-degenerate and (xo, t/>~(xo, 8o))

N- rankt/>~9 (xo,8o) ~ n- rankdllA(xo,,o).

=

(6.1.8)

Let us now see that this result implies that we can reduce the number of theta variables ifrankdllA is large. Let r = rankt/>~9 (xo,8o). We may assume that tf>~, 9 ,(xo,8o) is invertible where 81 = (8t, ... ,8N-r) and 811 = (8N-r+b ... ,ON)· By the implicit function theorem, near (xo,,o), there is a unique solution 811 = g(x, 81) to the equation tf>~,(x, 81, 811 ) = 0. Clearly g must be smooth and homogeneous of degree one in 8'. We then lt:t 8 = (81, 8''- g(x, 81)) and define~ by ~(x, 0) = tf>(x, 0). Since'#:: = 0 if and only if 011 = 0 we conclude that ~"- = 0 and 411: x,9"

011 =

9',9"

9" = 0 when

0. Therefore, if we set

1/J(x,O')

= tf>(x,81,g(x,81)) = ~(x,O',o),

we get

·'·" 'l'x9' = ;." xB' IB"=O

I

;." B"=O" and ·'·" '1'9'9' = "'8•8• These conditions imply that 1/J is non-degenerate. Furthermore, the first condition, along with Euler's homogeneity relations and the fact that ~11- = 0 when 011 = 0, gives that 1/J~ = 4>~ when 011 = 0. Consequently, x9" 1/J also parameterizes the Lagrangian A near (xo, 'o). Therefore, if u(x) is given by (6.1.3), with a(x, 0) having small enough support, there must be a symbol bE sm+n/4 -(N-r)/2 such that, modulo C 00 , 'I'

u(x) = ( 21r)-[n+2(N-r)]/4 { ei'I/J(x,9') b(x, 8') d8'. }RN-r

(6.1.9)

Notice that the order of the symbol here has increased by r/2 from the order of the symbol a(x, 8) in (6.1.3) to compensate for the fact that, in (6.1.9), the integration involves r fewer variables. From this we see that if rankdllA = r and if u E I~mp(X, A) then we can write u as a finite sum of oscillatory integrals of the form (6.1.9) modulo C 00 • If dllA has constant rank then we say that u E Im(x, A) is conormal since A is the conormal bundle of Y = llA(A), which must be an r-dimensional smooth submanifold of X by the constant rank theorem.

6. Fourier Integral Operators

168

A concrete example which illustrates the remarks about the reduction of theta variables concerns the Fourier integral operators eitP when the cospheres associated to the principal symbol of P, {e: p(x,e) = 1} c T; X, have nonvanishing Gaussian curvature. Under this assumption, we saw in the proof of Lemma 5.1.3 that the phase functions ~t(x,y,e) = rp(x,y,e) + tp(y,e) satisfy rank82 ~t!aeiaek n. -1 for small nonzero times t. This means that, modulo a smoothing error, for such times one can write

=

(eitP f)(x) = {

JM

1oo ei'I/J,(x,y,9) at(x, y, 8) f(y) d8dy -oo

for some at (x, y, 8) E s(n-I )/2 . The symbol and phase function depend smoothly on the time parameter when t ranges over compact subintervals of [-co,c]\0; however, not as t-+ 0 since eitPit=O is the identity operator and hence can not be expressed by oscillatory integrals involving one theta variable. In the special case where P = ~ one can use (4.1.21) to see that, fort> 0, t/Jt(x,y,8) must be a nonvanishing function of (x, t, y) times 8(ltl- dist (x, y)), where dist (x, y) denotes the distance with respect to the Riemannian metric g. The solution to the Cauchy problem for 82 jat 2 -l:l. 9 of-course can be written in a similar form, involving two oscillatory integrals both having this same phase function.

6.2. Regularity Properties Here we shall study the mapping properties•of a special class of Fourier integral operators. If X and Y are C 00 manifolds, then we shall say that an integral operator :F with kernel :F(x,y) E Im(x x Y,A) is a Fourier integral operator of orderm if A c {(x!y,e, 17) E T*(X x Y) \0: i' 0, 1J i' 0}. We shall usually write things, though, in terms of the associated canonical relation,

e

C = {(x,

e, y, 17) : (x, y, e, -17) E A} C (T* X\ 0) x (T*Y \ 0),

(6.2.1)

and from now on use the notation :FE Im(x, Y;C). The reason that it is more natural to express things in terms of the canonical relation, rather than the Lagrangian associated to the distribution kernel, will become more apparent in the composition formula below and the formulation of various hypotheses concerning the operators. Notice that, since A is Lagrangian with respect to the symplectic from ox +oy, the minus sign in (6.2.1) implies that C must be Lagrangian with respect to the symplectic form ox - oy = E df.j 1\ dxj - E d1Jk 1\ dyk in (T* X\ 0) X (T*Y \ o).

6.2. Regularity Properties

169

In this 5{ ction we shall study the mapping properties of Fourier integral operato 'S whose canonical relation is locally the graph of a canonical transformation-or locally a canonical graph for short. By this we mean that if "YO = (xo, Yo, 110) E C then there must by a symplectomorphism x defined near (yo, 110) so that, near ')'o, C is of the form

eo,

{(x,e,y,7J): (x,e)

= x(y,7J)}.

(6.2.2)

Notice that this forces dim X = dim Y. In addition, this condition is equivalent to the condition that either of the natural projections C -+ T* X\ 0 or C -+ T*Y \0 (and hence both) are local diffeomorphisms. Clearly, if C is locally a canonical graph then the projections are local diffeomorphisms. To see the converse we notice that if, say, C -+ T*Y \ 0 is a local diffeomorphism, then, near -yo, we can use (y, 71) as coordinates for C and hence the canonical relation must locally be of the form (6.2.2). The fact that X must then be canonical is a consequence of the fact that ox - oy vanishes identically on C which forces oy = x• (u x). Let us also, for future use, express this condition in terms of phase functions which locally parameterize C. If ¢(x, y, 8) is a non-degenerate phase function parameterizing A, then, by (6.2.1), C = {(x, ¢~(x, y, 8), y, -¢~(x, y, 8)) : (x, y, 8) E E}·

We claim that C being locally a canonical graph is equivalent to the condition that

¢" det ( ¢~:

¢" ) ¢~: "f:. 0 on E.

To see this, we notice that this is the Jacobian of (y, 8)

(6.2.3) -+

(¢~, ¢~). So

if (6.2.3) holds, then we can solve the equations r/J~(x, y, 8) = 0,

e= ¢~(x, y, 8)

with respect to (y, 8) and hence use (x, e) as local coordinates on E. Thus, (6.2.3) is equivalent to the condition that the projection C -+ T* X \ 0 is a local diffeomorphism, which establishes the claim. Having characterized the hypothesis, let us turn to the main result.

Theorem 6.2.1: Let X andY ben-dimensional C 00 manifolds and let FE JTR(X, Y;C), with C being locally the graph of a canonical transformation. Then

(1) F: L~omp(Y) -+ L~0c(X) if m ~ 0, (2) F: L~mp(Y)-+ Lfoc(X) if 1 < p < oo and m ~ -(n -1)1~- ~I, (3) F: LiPcomp(Y, o:) -+ LiPloc(X, o:) if m ~ -(n- 1)/2.

170

6. Fourier Integral Operators Furthermore, none of these result.s can be improved if :F is elliptic and if corank dllx x y = 1 somewhere, where llx x y : C -+ X x Y is the natural projection operator.

Notice that the non-degeneracy hypothesis that C be locally a canonical graph is the homogeneous version of the non-degeneracy hypothesis in the non-degenerate oscillatory integral theorem, Theorem 2.1.1. In both cases, the non-degeneracy hypothesis is equivalent to the condition that the projection from the canonical relation toT* X be non-singular. Also, by the discussion at the end of the previous section, the last condition in the theorem means that sing supp :F( x, y) contains a C 00 hypersurface. We shall first prove the most important part, the £ 2 estimate of Eskin and Hormander. 1b do this we shall need the following composition theorem. Theorem 6.2.2: Let :F E J~mp(X, Y; Cl) and g E IComp(Y, Z; C2). Then, if C1 is locally the graph of a canonical transformation, :Fog E pn+~£(X, Z;C) where

C = C1 oC2

= {(x,e,z,(): (x,e,Y,'1) E C1 and (y, 71, z, () E C2

for some (y, 71) E T*Y \ 0}.

Also, if both :F and g are elliptic, then so is :F o g.

By taking adjoints (see below), one sees of course that the same result holds if we instead assume that C2 is locally a canonical graph.

Remark. In Chapter 4 we saw that, for small t, the canonical relation of the operator eitP is

c = {(x,t,e,r,y,'1): (x,e) = 4't(Y,'1), r = p(x,e)}, where cllt is the canonical transformation defined by flowing along the Hamilton vector field associated to p(x, e) for time t. However, Theorem 6.2.2 shows that if this holds for small time then it must hold for all times since ei(t+s)P = eitP o eisP has canonical relation

c

0

Cs

= {(x, t, e, T, y, 11) : (x, e) = o { (y, 71, z, () :

4't(Y, '1),

T

= p(x, e)}

(y, 71) = 4'8 (z, ()}

= {(x, t, e, T, y, 71) : (x, e)

= 4't+s(Y, 71),

T

= p(x, e)}.

Hence if C has the stated form for t in [-e, c] then it must also have this form fort in [-e,e] + [-e,e] = [-3e,3e], and by iterating this one concludes that C is as above for all time.

171

6.2. Regularity Properties If we assume the composition theorem for the moment, it is easy to prove part (1) of Theorem 6.2.1. After perhaps breaking up the operator we may assume that :FE ~omp(X, Y;C) with Cas in (6.2.2). But then :F* E ~mp(Y,X;C•) where

c• = {(y,7J,x,e): (x,e,y,7J) E C} = {(y,1J,x,e):

(x,e)

= x(y,7J)}.

c•

Hence, C o must be the trivial relation {(y, 1J, y, 17)} and therefore TheoreiJ15 6.2.2 and 6.1.4 imply that':F* :F must be a pseudo-differential operator of order 0. So the L 2 boundedness of pseudo-differential operators of order 0 gives

j 1Ful 2dx = j r Fu u dy $ 11.r•Full2llull2 $ Cllull~, proving ( 1). Notice that Theorem 6.2.2 implies that PF E Jm+jj(X, Y;C) if Pis a pseudo-differential operator of order JL on X. Using this and Theorem 6.2.1 one obtains the following regularity theorem.

Corollary 6.2.3: Let .r E (1) F : L~mp(Y)

-+

I

~(X, Y; C)

be as in Theorem 6.2.1. Then

Lfoc m-o p (X), if 1 < p < t

00

and ap =

(n- 1)11/P- 1/21, (2) F: LiPcomp(Y, a)-+ LiPloc(X, a- a 00 ), with aoo = (n- 1)/2. As before, all of these results are sharp if .r is elliptic and corank diixxY = 1 somewhere. Notice that the Corollary says that, compared to the L 2 estimates, • one in general loses (n- 1)11/p- 1/21 derivatives in LP and (n- 1)/2 derivatives in the pointwise sense.

Proof of Theorem 6.2.2: We may assume that Ct is parameterized by a non-degenerate phase function t/J(x, y, 9) which is defined in a conic region of (X x Y) x (IRN1 \ 0) and that C2 is parameterized by a non-degenerate phase function cp(y, z, u) defined in a conic region of (Y x Z) x (IRN2 \ 0). It then follows that Ct o C2

= {(x, t/J~, z, -cp~) : t~J{J(x, y, 9) = 0, cp~(y, z, u) = 0, t/J~(x, y, 9) = -cp~(y, z, u)}.

Equivalently, if we set

e = ((191 2 + lul 2)112y,9,u), and define the homogeneous phase function ~(x,

z, 8) = tjJ(x, y, 9)

+ cp(y, z, u),

6. Fourier Integral Operators

172 we have

Ct o C2

= {(x, ell~, z, -ell~) : (x, z, 9) E E4> }.

Notice that ell is defined in a conic region of (X x Z) x (IRN \ 0), where N = N1 + N2 + n, with n = dim Y = dim X. We claim that ell is nondegenerate. This will show that Ct o C2 is a smooth canonical relation which is Lagrangian with respect to ux- uz. To prove the claim we must show that the differentials d(oellfo9j), j = 1, ... , N, are linearly independent on E4>. One can rephrase this as the requirement that the N X ( N + n +dim Z) matrix IJ2 ell I aea( e, X' z) have full rank N here. This in turn just means that

(

,1,11 + """' 'l'yy ryy ,1,11 'I'(Jy (f'l/1

Tt7y

,1,11 'l'y(J ,1,11 'I'(J(J 0

rya

"""'

0 (f'l/1

Tt7t7

,1,11 'l'yx

ryz )

(f'l/1

,1,11

0

0

(f'l/1

'I'(Jz

(6.2.4)

Tt7Z

has full rank when 4J9 = 0, cp~ = 0, and l/J~ + cp~ = 0. But we have already seen that Ct being locally a canonical graph is equivalent to the condition that the (Nt + n) x (Nt + n) submatrix I

be non-singular when degenerate ~orces

4J9

= 0. In addition, the fact that cp is non-

( (f'l/1

(f'l/1 (f'l/1 ) rayraaraz

to have full rank N2. By combining these two facts, and noting the form of the matrix, one sees that (6.2.4) must have rank Nt + n + N2 = N, giving us the claim. It is now a simple matter to finish the proof. Using a partition of unity, we may assume that a1(x,y,8) E sm-Nd 2+n/ 2 and a2(y,z,u) E SP-N2/2+(n+nz)/4 are supported in small compactly based cones in (IRn x R.n) x (IRN1 \0) and (R.n x JRnz) x (IRN2 \0), respectively. Here nz = dimZ. We must show that if :F(x,y) = [ JRN1

ei
6.2. Regularity Properties then

J =IfJ

(.r o Q)(x, z) =

173

F(x, y)Q(y, z) dy ei((x,y,B)+<,e(y,z,u))

a1(x, y, 9) a2(y, z,u) d9dudy

defines an element of Im(X,Z;C). If the wave front sets of y-+ F(x,y) and y -+ Q(y, z) are always disjoint, the inner product defines a Cif kernel and the result trivially holds. If not-that is, if C "I 0-and if we assume in addition, as we may, that the a; have small conic supports it is clear from Theorem 0.4.5 that the inner product is well defined. To show that {.r o Q)(x, z) has the desired form we first notice that, if {3 E Cif(IR \ 0) equals one for s E [c- 1, C] with C sufficiently large, and if we set

a(x, z; y, 9, u)

= /3(191/lul) a1 (x, y, 9) a2(y, z, u),

then the difference between

III

(.r o Q)(x, z)

ei((x,y,9)+<,e(y,z,u))

and

a(x, z; y, 9, u) dyd9du

(6.2.5)

is C 00 • To prove this one just notices that, on the support of the integrand, if 4>~ + rp~ = 0 then 191 and lui must be comparable. Hence, if C above is large enough IV11 (t/>+rp)l must be bounded below by a multiple of 191 +lui on the support of a- a1a2 which leads to the claim. To finish, we change variables and write (6.2.5) as

f

JRN

eici>(x,z;e)

a(x, z; 9)(191 2 + lul 2) -n/ 2 d9.

(6.2.5')

Note that since a1 and a2 are symbols of order m - NI/2 + n/2 and J.L- N2/2+ (n+nz)/4, respectively, and since their product is therefore a symbol of order m + J.L- (NI + N2)/2 + 3n/4 + nz/4 in a conic region

where 191 and lui are comparable, we conclude that the symbol in (6.2.5') is of order m + J.L- N/2 + (n +nz)/4.

Hence (6.2.51) defines an element of Jm+P.(X, Z;C), by Theorem 6.1.4.

I We now tum to the proof of parts (2) and (3) of Theorem 6.2.1. It will be convenient to represent the Fourier integrals there in a manner which is close to that of pseudo-differential operators. To do this we shall use the following result.

6. Fourier Integral Operators

174

Proposition 6.2.4: Let F E J~mp(Rn, Rn; C) where C is a canonical graph. Then, modulo C 00 , one can write the kernel of F as the sum of finitely many terms, each of which in appropriate local coordinate systems is of the form (21T)-n

r

}Rn

ei(cp(x,,.,)-(y,fJ)]

b(x, 17) d17.

(6.2.6)

Each term is of course in Jffl(Rn, Rn; C), and the symbols are compactly supported in X and are in sm. Moreover, on supp b, (6.2.7)

Proof: The first step is to see that local coordinates can be chosen so that C is given by a generating function. Using Lemma 6.1.3 and the fact that Ax 0 = {(y,17): (xo,e,y,17) E Cforsomee} c T"'Y\0 must be Lagrangian, we can adapt the proof of Proposition 6.1.2 to see that local y COOrdinates can be chosen SO that C 3 (X, y, 17) -+ (X, 17) is a diffeomorphism, if, as we may, we assume that C is small enough. This means that and y are functions of (X' 17) on c. To use this, we notice that since the canonical one form vanishes on C we have

e,

e

o= (e, dx}- (17, dy} = (e, dx} -

+ (y, d17}. Therefore, if we set S(x,17) = (y,17}, we must have 8Sf8x = e, 88/877 = d( (y, 17})

y, i.e.

C = {(x, 8Sf8x, 88/817, 17)}. By Theorem 6.1.4, this means that, if we set rp(x, 17) = S(x, -17), and, if WF(F(x,y)) is sufficiently small, we can write, modulo C 00 ,

F(x, y)

= (21r )-n

{ ei(cp(x,,.,)-(y,,.,)] a(x, y, 17) d17

}Rn

{6.2.8)

for some symbol a E sm. But Theorem 6.1.4 also implies that, if we set ao(x,17) = a(x,rp~(x,17),17) E sm, then (21T)-n

J

ei(cp(x,fJ)-(y,fJ)]

Continuing, we can choose (21T)-n

J

a;

E

(a(x,y,17)- ao(x,17)) d17 E pn-l.

sm-j so that

ei(cp(x,fJ)-(y,fJ)] (a(x, y, 17)

-

L

a;(x, 17)) d17 E Im-N

j
If we then choose b"' }:a;, the difference between (6.2.6) and (6.2.8) is C 00 • Finally, (6.2. 7) must hold since 4J(x, y, 17) = rp(x, 17) - (y, 17} must satisfy (6.2.3). I

175

6.2. Regularity Properties End of proof of Theorem 6.2.1 We shall first prove (2) and (3) and then conclude by discussing the sharpness of the estimates. In proving (2), by duality, it suffices to show that the estimates hold for 1 < p < 2. This is because, as we have already seen in the proof of (1), F* E 1m (Y, X; C*), where C* just comes from interchanging T* X \ 0 and T*Y \0 in C. So, c• is locally a canonical graph if C is, and, therefore, if (2) holds for exponents 1 < p < 2; it must also hold for 2 < p < oo. In contrast to the case of p > 2, when proving estimates involving V' for p < 2, it turns out to be more convenient to use the adjoint of the microlocal representation (6.2.6) of the kernels. However, by the above discussion, if we apply Proposition 6.2.4 to F* and then take adjoints, we see that it suffices to show that if b(y, ') E sm vanishes for y outside of a compact set of an and if r.p(y, e) satisfies (6.2. 7), then

II/I ei((x,~)-<,o(y,~))

b(y, ')f(y) dedyiiLP(Rn)

S: CllfiiLP(Rn),

m S: -(n- 1)11/p- 1/21. (6.2.9)

We may assume that b vanishes for 1e1 < 10. Then if (3 E satisfies E~oo (3(2-k s) = 1, s > 0, we set

C~((1/2, 2))

where

If we call :F the operator in (6.2.9) then we of course have

(6.2.10) To prove the Lipschitz estimates (3), we shall use the following:

Proposition 6.2.5: Let b E the uniform bounds

sm

be as above. Then for

>. > 1 we have (6.2.11)

Furthermore, the constant C remains bounded if b has fixed support and belongs to a bounded subset of sm.

176

6. Fourier Integral Operators Before proving (6.2.11), let us see how it implies the Lipschitz estimates. In fact, if we define the Littlewood-Paley operators P:>.. by

(P:>..Jt<e)

= P
and recall the definition of Lip (a:) from Section 0.2, we conclude that part (3) follows by duality from the uniform estimates IIP:>..~FP:>..2/IIu(R") ~ C,\~,\2° llfiiLl(R")•

if m = -(n -1)/2. (6.2.12)

~ lcp~(y,e)l ~ c- 1 1el,

But using the fact that, on suppb, clel one can argue as in the proof of Lemma 2.4.4 to see that there is a uniform constant Co such that IIP:>.. 1 FP:>.. 2 11(Ll,Ll) ~ CNX!N_x2N for any N unless

C0 1 ~ ,\If A2 ~ Co. The operators P:>.. are uniformly bounded on £ 1 and so we now see that (6.2.12) must be a consequence of IIP:>..F/III ~

c 11111~.

m

= -(n -1)/2.

(6.2.12')

But if we recall the form ofF, we see that

P:>..F = P:>.. (

L

.r2.)'

1/49/2•5:_4 and, therefore, (6.2.11) implies (6.2.12') and hence part (3) of the Theorem. Before moving on, let us notice that (6.2.11) can be used to prove everything else except for the endpoint estimate in (6.2.9). In fact, the L 2 estimate for Fourier integral operators shows that the dyadic operators F:>.. are bounded on £ 2 with norm 0(,\m). So, by applying the M. Riesz interpolation theorem, we conclude that (6.2.11) yields the dyadic estimates IIF:>..fiiLP(R") ~ c,xm+(n-I)II/p-l/2 111fiiLP(R")• Consequently, for m strictly smaller than -(n- 1)11/p- 1/21, we get (6.2.9) by summing a geometric series. To prove the endpoint estimate, we shall need to know more precise information about the kernels of the operators F:>.. than is given in Proposition 6.2.5. In addition, we shall need to use some basic facts from Hardy space theory. Using the Hardy space H 1 , we shall be able to prove the sharp estimate (6.2.9) by interpolating between the £ 2 estimate and an estimate involving £ 1 . This of course will be reminiscent of the proof of the multiplier theorem in Section 0.2. First, we define the Hardy space H 1 (Rn). It consists of all £ 1 functions which have the additional property that all of their Riesz

177

6.2. Regularity Properties transforms, R;J, also belong to £ 1 . Here R; is defined by

(R;f)"(e)

= /(e)e;/lel,

j

= 1, ... , n.

So the norm is given by n

II/IIH = llfiiLl(RR) + L IIR;fiiLl(RR)• 1

j=1

Notice that, since (R;f)"(e) must be continuous if it belongs to £ 1 , it follows that /(0) must vanish-in other words, iff E H 1 then f must have mean value zero. There is a decomposition off E H 1 which is similar to the CalderonZygmund decomposition off E £ 1. However, in the present sett~ng, the decomposition involves elements which possess the salient features of the good and bad functions-namely, they are bounded, localized, and have mean value zero. Specifically, given f E H 1, one can write

!= Lo:QaQ,

(6.2.13)

Q

where the sum ranges over all the dyadic cubes coming from a fixed lattice in an'

Llo:QI ~ II/IIH

1,

(6.2.14)

Q

and the individual atoms satisfy suppaQ C Q,

J

aQdX = 0,

llaQIIv"' ::; IQI- 1 ·

(6.2.15) (6.2.16)

This is called the atomic decomposition of H 1 and a proof can be found, for instance, in Coifman and Weiss [1]. Using (6.2.13)-(6.2.16) and more detailed information about the kernels of the dyadic operators :F>., we shall prove the following substitute for an £ 1 -+ £ 1 estimate. Proposition 6.2.6: If m = -(n- 1)/2 and if :F is as above (6.2.17)

Furthermore, the constants remain bounded if a has fixed support and belongs to a bounded subset of s-(n- 1)/2.

Momentarily, we shall see that this implies (6.2.9) if we apply the following interpolation lemma which, for later use, we state in more generality than is needed now.

6. Fourier Integral Operators

178

Lemma 6.2.7: Suppose that when 0 :5 Re(z) :5 1, Tz is a family of linear operators on an which has the property that, whenever 1 and 9 are fixed simple functions which vanish outside of a set of finite measure, the map

is bounded and analytic in the strip 0 < Re( z) < 1 and continuous in the closure. Then: {1} If, for Re(z) = j = 0, 1, IITz/llq; :5 C;ll/llp;, with 1 :5 P;, q; :5 oo, and if, for 0 < t < 1 we define Pt and qt by 1/Pt = (1- t)/Po + tjp1 and 1/qt = (1- t)fqo + tjq1, it follows that IITtfllq, :5

cJ-t cf 11/llp,.

(6.2.18)

{2} If, instead, we have IITzfiiLl :5 Coii/IIHl when Re(z) = 0, then (6.2.18) holds with Pt and qt satisfying 1/qt = (1- t) + tjq1 and 1/Pt = (1- t) + tfpt. if 11Tz/llq1:5 C1ll/llp1 for Re(z) = 1.

Part (1) of the interpolation theorem is due to Stein and follows from modifying the argument that was used to prove the M. Riesz interpolation theore~, Theorem 0.1.13. In fact, if one changes the definition of F(z) in the proof to be F(z) = JTzfz9z dx, and then repeats the arguments, one gets (6.2.18). The complex interpolation theorem for H 1 is due to Fefferman and Stein [1]. The proof uses an argument which reduces (2) to (1) via the duality of H 1 and BMO and the characterization of V in terms of the sharp function of Fefferman and Stein. To see how the interpolation lemma implies (6.2.9), we set ap = (n- 1}11/p- 1/21 and then put

Fz f(x) = e
JJ ei((x,~)-.p(y,~))

b(y,e)lelap-(n-1)(1-z)/2 f(y)

~dy.

Notice that when Re(z) = t the symbols involved belong to a bounded subset of s-(n- 1H1 -t)/2 . So we have

IIFz /11£1 :5 CllfiiHl, Re(z) = 0, IIFz/11£2 :5 Cll/11£2, Re(z) = 1, which gives the sharp estimate (6.2.9) as P = F, t = 2(1 - 1/p). We now turn to the proof of Proposition 6.2.6. In the course of the proof we shall prove an estimate which implies Proposition 6.2.5.

179

6.2. Regularity Properties To prove the H 1 estimate (6.2.17), we see by (6.2.13) and (6.2.14) that it suffices to check that for an individual atom (6.2.17')

It is easy to see that this inequality holds when Q is a cube of sidelength ~ 1. First, since the symbol b(y, €) vanishes for y outside of a fixed compact set, one checks that the kernel ofF satisfies IF(x, y)l ::; CNixi-N for any N when lxl is larger than some fixed constant depending on band rp. Hence, to prove (6.2.171), it suffices to check that the inequality holds when the £ 1 norm is taken over a fixed ball B. But if we use the £ 2 boundedness ofF along with (6.2.16) we see that IIFaQii£1(B)::; IBin/ 2 11FaQII£2(R")::; C llaQII£2(Q)::; C IQI- 1/ 2 . Since the right side is bounded when Q is large, we may assume from now on that Q has side-length R = 2-j ::; 1. We shall now construct a small exceptional set NQ on which we can obtain favorable estimates for the £ 1 norm of FaQ using the Schwarz inequality and the £ 2 boundedness of Fourier integral operators. To do this, we first choose for>.= 2k unit vectors €r, v = 1, ... ,N(>.), such that 1er - €r' I ~ C()A -I/2 , v =F v', for some positive constant C() and such that balls ofradius >. -I/2 centered at €r cover sn-I. We observe that

N(>.) ~ ).(n-1)/2.

(6.2.19)

For a fixed y E Q, we let

IX,v = {x: l((x- rp~(y,€r)),€r)l::; >.- 1 and

l1If.v(x- rpe(Y,€r))l ::;·>.-I/2 },

(6.2.20)

f

where li v denotes the projection onto the subspace orthogonal to €r. Thus IX,v' is a rectangle centered at rp~(y, er) with n- 1 sides of length >.-I / 2 and one of length >.-I. The thinnest side points in the direction er. We now define N(>.)

Nf=

U 1X.v•

v=l

and, if the side-length of Q is R, (6.2.21)

6. Fourier Integml Opemtors

180

Since II~.... ,v I= A-(n+l)/ 2 it follows immediately that INQI :5 CR(n+l)/2 R-(n-1)/2 =CR.

(6.2.22)

Note that, if~ E sn- 1 and I~- ~~I :5 A- 1/ 2 ' then ct'e(y, ~) belongs to a fixed dilate of I~ v (depending on cp). Hence the singular support of x-+ :F(x,y) is basi~ally contained in NQ, since the latter is a subset of E 11

= {x: x = ft'e(y,~)

for some (y,~) E suppb}.

In fact, if the rank of the differential of the projection c -+ an X an is maximal, NQ is basically a tubular neighborhood of width R around E 11 • If the rank is not maximal, the exceptional set may be larger than a tubular neighborhood of this size. Let us first estimate :FaQ on the exceptional set. Since :F is of order -(n -1)/2 it follows that :F(I- d)(n- 1)/4 is bounded on £ 2 and hence II:FaQIILl(No)

:5 CR112 II:FaQII2 :5 CR112 11(I- d)-(n- 1)1 4aQII2·

But, if we let Pn = 2n/(2n- 1) so that n(1/Pn- 1/2) = (n- 1)/2, the Hardy-Littlewood-Sobolev inequality gives II(I- d)-(n-1)/4aQII2

:5 c llaQIIvn :5 R-n+n/Pn = cR-1/2.

Combining these two inequalities leads to the favorable estimate II:FaQIILl(No) :5 C.

Now we shall prove the inequality off of the exceptional set:

(6.2.23) If :F~ are the kernels of the dyadic pieces of :F we claim that this would be a consequence of the following estimates valid for y, y' E Q:

f

}R"\No

I:F~(x,y)l dx :5 C(ru)- 1

f I:F~(x,y) -:F~(x,y')ldx :5 CRA

JR"

if A> R- 1 ,

(6.2.24) if A :5

R- 1 .

Clearly these two estimates yield (6.2.23). In fact, the first one gives II:F2.aQIILl(R" \No) :5 C(R2k)- 1 if R2k

> 1.

While, if we use the mean value zero property (6.2.15), we get that if y' E Q

6.2. Regularity Properties

181

and hence the second estimate implies IIF2"aQII£1(R" \.NQ) :5 CR2k

if R2k

< 1.

Recalling (6.2.10), one sees that (6.2.23) follows by combining these two estimates and summing a geometric series. To prove (6.2.24) it is necessary to introduce homogeneous partitions of unity of IR" \ 0 that depend on the scale >.. Specifically, we choose coo functions x~. v = 1, ... IN(>.), 'satisfying Ev X~ = 1 and having the following additional properties. First, the x~ are to be homogeneous of degree zero and satisfy the uniform estimates

ID'Yx~(e)l :5 C-y>.hl/2 V>. when lei= 1.

(6.2.25)

Second, x~(en f. 0 and the x~ are to have the natural support properties associated to (6.2.25), that is,

x~(e) = 0 if lei= 1 and le- e~l ~ c>.- 112 • These are just the n-dimensional versions of the partitions of unity used to prove the maximal theorems in Section 2.4. We now define the new kernels Ff(x, y), 1 :5 v :5 N(>.), by

Ff(x,y) = jei[(x,e)-
where

b~(y,e) = x~(e)b>.(Y,e). (6.2.26)

The idea behind this decomposition is that the x~ have the largest possible angular support so that -+ cp behaves like a linear function in,the appropriate scale on suppb~. By Euler's homogeneity relations, the closest linear approximation of supp X~ 3 e -+ cp(y, e) should be (cp((y, en. e}, and, in fact, if we let

e

r~(y,e) = cp(y,e)- (cp((y,en.e}

denote the difference, we claim that the following estimates are valid in suppb~ if N ~ 1:

l{(ve.e~})Nrny.e)l::; cN>.- 1 Iei 1-N, ID~r~(y, e) I :5 eN min{ >.- 1/ 2 , lei 1-N},

(6.2.27) lo:l = N. (6.2.28)

Thus, by considering the case N = 1, one sees that this remainder term is much better behaved in the ''radial direction" e~ I as Opposed to the "angular directions" rr_t v(IR"). To prove (6.2.27), it s~ftices, by homogeneity, to show that the same estimate holds when e E sn- 1 n suppx~- But when e~ = e, Euler's

6. Fourier Integml Opemtors

182

formula gives rr(y, ~r) = 0, or, equivalently, Verr(y, ~) = 0. But Verr is homogeneous of degree zero and hence ((V~,~r}) Verr(y,~n = 0, which leads to the desired estimate ( (Ve, ~r}) rny, ~) = O(.X - 1) since the first two terms in the Taylor expansion around ~r must vanish and dist(~~~n ~ c.x- 1/ 2 for~ E sn- 1 nsuppx~· To prove (6.2.28), we notice that the estimate trivially holds for N > 1 as 1~1 1 -N ~ c.x- 1/ 2 for ~ E suppb~. So (6.2.28) would follow from the estimate Verr(y,~) = cp{(y,~)- cp{(y,~r) = O(.x- 112). Since this term is homogeneous of degree zero, we need only show that the estimate holds for ~ E sn- 1 n supp X~, which is apparent. We shall use (6.2.27)-(6.2.28) and an integration by parts argument to estimate the kernels .ry(x, y). After performing a rotation, we may assume that ~r =

(1,o, ... ,o),

ntv(~) = (0, 6, ... 'en) = (0, ~'). From the integration by parts, we split ~ = (6, ~') and define the selfadjoint operator

L~ =(I- .X2 82 f8d)(I- .X(Ve'• Vd)· Our assumptions on the symbol together with (6.2.25) and (6.2.26) imply that (6.2.29)

Furthermore, (6.2.27) and (6.2.28) imply that the same estimate holds if we replace b~ by b~ (y, ~) = eirA (y,{) b~ (y, ~).

So we integrate by parts and obtain

:Ff(x,y) = H'fv,>.(x,y)

j ei((x-tp((y,eA)).e)(L~)Nb~(y,~)~,

where the weight factor is

.

H'fv >.(x,y) = ( 1 + l.x(x- ft'e(y,

~r))11 2 ) -N ( 1 + l.x 112 (x- ft'e(y, er))'l 2) -N.

Since the symbol of the last integrand is supported inside a region of volume O(.X(n+l)/2) and since it satisfies the estimates in (6.2.29), we conclude that (6.2.30)

6.2. Regularity Properties

183

Consequently, we get, for fixed y,

f IFr<x,y)ldx::; CA-(n-1)/2 JR.,

(6.2.31)

uniformly in v, which implies Proposition 6.2.5 in view of (6.2.19). To prove the first estimate in (6.2.24), we notice that, given N' and yEQ,

f IAHN .\ (x, y)l dx ::; eN, A-(n-1)/2(RA)-N' JR.. \.No '

I

A> R-1. (6.2.32)

This is because if A is large compared to R- 1 and if N is larger than N', then HKr,.x(x,y) is O((RA)-N') times HN-N',.\(x,y) in the compliment of NQ· Clearly (6.2.32) leads to the first estimate by summing over v. The second estimate in (6.2.24) follows immediately from

f IVy.rnx,y)l dx::; CA. A-(n-1)/2. JR.. But this follows from the proof of (6.2.31) as D~F!(x,y) behaves like A.rf(x,y) when lo:l = 1. This finishes the proof of the positive results in Theorem 6.2.1. #

Sharpness of results Let us now see that the estimates in Theorem 6.2.1 cannot be improved for elliptic .r E Jffl(X, Y;C) if dllxxY has full rank somewhere, where IIxxY : C-+ X x Y is the natural projection operator. Note that if it does have full rank somewhere then the singular support of the kernel is as large as it can be in the sense that it must contain a piece of a C 00 hypersurface. One sees from the bounds for INQ I in the proof how this enters into the V' estimates. Returning to the issue at hand, we may assume that C is the graph of a canonical transformation and that xo belongs to the projection onto X. It then follows that

Ax0 = {(y, 71) : (xo, ~. y, 71) E C some ~} C T*Y \0

=

must be Lagrangian. Moreover, if corank dllx x y 1, then Ax0 is the conormal bundle of a smooth hypersurface Sx 0 C Y. By choosing appropriate local coordinates we may always assume that X = Y = JR.", Xo = 0, Sxo c sn- 1 I and Axu = H77/l711. 77H· If we then let

f~~.(y) =

(21r)-n

f

JR..

ei((y,e)+lell ( 1 + 1~12)-p./2

tte,

184

6. Fourier Integral Operators it follows from the composition theorem for Fourier integrals-taking Z to be the empty set-that :Ff,.,.(x) is an elliptic Lagrangian distribution of order m- J.L + n/4 whose wave front set is contained in the conormal bundle of the origin-that is, {(0, e)}. In other words, in some nonempty open cone based at the origin, there must be a c > 0 such that as x --+ 0, I:F/,.,.(x)l > clxlp.-m-n. Thus, we have

:Ff,.,.

fl. Lfoc

if J.L- m- n ~ -nfp.

(6.2.33)

On the other hand, using stationary phase (cf. Lemma 2.3.3) one sees that for y near sn- 1 1/,.,.(y)l ~ (dist(y,sn- 1 ))"'-(n+l)/2 ,

J.L

< (n+ 1)/2,

meaning that

!,.,. E I?

{:=:?

J.L > (n + 1)/2- 1/p.

Substituting this into (6.2.33) shows that :F cannot be bounded on £P for p 2::: 2 if (m + n + 1/p- (n + 1)/2) > nfp, that is,

-(n -1)(1/2- 1/p) < m. This shows tpat the estimates in Theorem 6.2.1 are sharp for p 2::: 2. By duality, the same holds for p < 2 and one can adapt the above arguments to see that the Lipschitz estimates are also sharp.

Remark. If X = Y is a compact C 00 manifold of dimension n and if P E '111~ 1 (X) is elliptic then, for a given t, eitP E f>(X,X;Ct) where Ct is the canonical rel~ion described in the remark after Theorem 6.2.2. Since Ct is a canonical graph we conclude that Otp =

(n- 1)11/p- 1/21, (6.2.34)

lleitP !IILip(o-(n-1)/2)

~ c IIIII Lip (o)•

where C remains bounded for t in any compact time interval. We claim that, for all but a discrete set of times t, these estimates cannot be improved. In view of the above discussion, this amounts to showing that the differential of Ct --+ X x Y must have full rank somewhere unless t belongs to a discrete exceptional set. To see that this is the case, we note that the condition T = p(x, e) in the full canonical relation of eitP means that if (x, e) - cP(y, t, e) is a (local) phase function for the operator, then ¢~ = p(¢e(y, t, e), e). Let us suppose that dllxxY does not have full rank for a given time to. We

6. 2. Regularity Properties

185

shall then see how these facts imply that for all t near to the rank of Ct -+ X x Y has to be maximal somewhere. To show this, we first note that, given xo, we can always choose ~o E Ex0 = {~ E T;0 X\O: p(xo,~) = 1} so that this cosphere has all n - 1 principal curvatures positive at ~0· If coordinates are chosen so that ~o = (0, ... , 0, 1), this is equivalent to the statement that (82 pf8~ja~k) 1 <. k~(yo, to,~o)) E Ct0 , let n + r denote the rank of the differential of the projection at this point. In view of Proposition 6.1.5, after perhaps performing a rotation in the first n - 1 variables, we may assume that cl>e'f.' (yo, to, ~o) is invertible if~ = (e' ~"), = (6' ... '~r ), and cl>'/.;f.• (yo, to, ~o) = 0 if either j or k is > r. (In the case where r = 0 one would modify this in the obvious way taking ~" = 0 To make use of this, we notice that at (YO, t, ~o) we must have

e

, _ (cl>'/.'f.' + O(t- to) tPf.f. O(t- to)

O(t- to) ) (t- to)p'{"f." + O((t- t 0 ) 2 )

·

( 6·2·35 )

But this matrix must have the largest possible rank, n -1, fort close to

t0 , because the (n- r) x (n- r) Hessian of the homogeneous function ~11 -+

p(xo, 0, ~") must have largest rank n- r - 1, due to the fact that {~": (0, ~") E Ex0 } has non-vanishing Gaussian curvature. One reaches this conclusion about the rank of cl>'/.f. after noting that det (

~: ~:) ~ tm-r,

t small,

if At= A+O(t), with A being a non-singular rxr matrix, Bt, Ct = O(t) and if Dt = tD + O(t2 ) with D being an (m- r) x (m- r) non-singular matrix. Since Proposition 6.1.5 says that the rank of the differential of the projection of Ct at 'YO = (ci>~(YO, t, ~o), ~o, YQ, cl>~(yo, t, ~o)) E Ct is n plus the rank of cl>'/.f.(YO, t, ~o), we conclude that this differential must have maximal rank 2n- 1 at -ro, which finishes the proof. Similar remarks apply to the regularity properties of solutions to strictly hyperbolic differential equations. Specifically, let L(x, t, Dx,t) = D~ + Ej!: 1 Pj(x, t, Dx)D;n-j be a strictly hyperbolic differential operator of order m. Then if {/j} j!:(/ are the data for the Cauchy problem

186

6. Fourier Integral Operators we have, for instance, that the solution satisfies m-1

lluiiLP(X) :5 C

L

11/;IIL:p_/X).

j=O

ll.i=1 (

Moreover, if T - Aj (x, t, e)) is a factorization of the principal symbol of L and if, for every t, at least one of the roots >.; is a nonzero function of T* X \ Q-that is, elliptic-this result cannot be improved.

6.3. Spherical Maximal Theorems: Take 1 In this section we shall present some maximal theorems which are related to Stein's spherical maximal theorem. The latter is the higherdimensional version of the circular maximal theorem proved at the end of Section 2.4. It says that, if n ~ 3 and p > nf(n- 1), f(x- ty)dcT(y)'P dx)lfp :5 Cpii/IILP(Rn)• ( { supl { }Rn t>O lsn-1

f E S. (6.3.1)

Notice that the spherical means operators which are involved are Fourier integral operators of order -(n-1)/2. This is because their distribution kernels are t-"6o(1-lx- Ylft), where 6o is the Dirac delta distribution. Consequently, we can write the spherical mean operator corresponding to the dilation t as

(211")-1 {

}Rn

1

00

t-neiO(l-lx-ylft] f(y) d(}dy.

-oo

Since there is just one theta variable, the order must be 1/2- 2n/4 = -(n-1)/2 as claimed. Notice also that the phase function satisfies (6.2.3) and hence the spherical means operators are a smooth family of Fourier integral operators of order -(n-1)/2, each of which is locally a canonical graph. In this section we shall always deal with Fourier integral operators Ft E I~mp(X,Y;Ct),t E I, where X andY are C 00 manifolds of a common dimension n and I C R. is an interval. We say that {Ft} is a smooth family of operators in I~mp if there are finitely many nondegenerate phase functions tPtJ(x, y, 9) and symbols a;,t(x, y, 9) so that, fortE I, we can write Ftf(x)

='~}y}RNi "" { { ei¢•.;(x,y,O)at,;(x,y,9)f(y)d9dy. 1

We require that both the ¢t,j and the atJ be smooth functions of t

6.3. Spherical Maximal Theorems: Take 1

187

with values in S 1 (r;) and sm-N;/2+n/2(r;), respectively, where r; is an open conic subset of (X x Y) x (JR.N; \ 0). In addition, we require that the symbols be supported in r; and vanish for (x, y) not belonging to a fixed compact subset of X x Y. Usually, we shall deal with smooth bounded families in 1:mp> by which we mean that, in addition to the above, we have the following uniform bounds for (t,x,y,9) E I X r;:

ID~,y,tD~t,j(X, y, 9)1 ~ Ca-y(1 + 191) 1-lal, ID~,y,tD~at,j(X, y, 9)1 ~ Ca-y(1 + l91)m-(N;-n)/2-lal.

(6.3.2)

We can now state the main result of this section. Later on we shall see that it can be used to recover Stein's theorem (6.3.1). Theorem 6.3.1: Let :Ft E /:mp(X, Y; Ct), t E [1, 2], be a smooth family of Fourier integral operators which belongs to a bounded subset of 1:mp· Assume also that, for every t E [1, 2], Ct is locally a canonical graph. Then if ap = (n- 1)11/p- 1/21, we have for p > 1

(J

sup I:Ftf(x)IP dx) 1/P

~ CllfiiLP,

m

< -ap- 1/p.

(6.3.3)

tE(1,2J

In particular, if n ~ 3 and p > nf(n- 1), the maximal inequality holds if m = -(n- 1)/2.

The proof involves a straightforward application of Lemma 2.4.2. We fix {3 E Clf((1/2,2)) satisfying E/3(2-ks) = 1, s > 0, and define for k = 1,2, ... :Fk,tf(x)

=~If eitP•.;(x,y,fJ) nt,;(x, y, 9) /3(19l/2k) f(y) d9dy. 3

Then, if we set :Fo,t = :Ft- E~ 1 :Fk,t• it follows that :Fo,t has a bounded compactly supported kernel and hence the maximal operator associated to it is trivially bounded on all £P spaces. On account of this, (6.3.3) would follow from showing that fork= 1, 2, ... ( / sup I:Fk,tf(x)IPdx) 11P ~ C2k(m+ap+l/p) llfiiLP·

(6.3.31)

tE(1,2J

But if we apply Lemma 2.4.2, we can dominate the pth power of the left side by

lxf I:Fk 'd(x)IP dx

188

6. Fourier Integral Operators Theorem 6.2.1 implies that the first term satisfies the desired bounds and also that

( [ IFk,tf(x)IPdx f/p' :S C{2k(m+op) = C(2k(m+op)

11/llpt/p' 11/llp)p-1.

Since (6.3.2) implies that 2-kdt_.rk,t is a bounded family in I:mp> we can also estimate the last factor:

If we integrate in t and combine these two estimates, we get (6.3.31). The last part of the theorem just follows from the fact that, for n ~ 3, O:p + 1/p is smaller than (n -1)/2 precisely when p > nf(n -1). Notice that when n = 2 there are no exponents for which this is true. In fact, for p < 2, a:p + 1/p is > 1/2, while for p ~ 2, a:p + 1/p 1/2. This explains why maximal theorems like the circular maximal theorem are harder to prove than their higher-dimensional counterpart. Momentarily, we shall see that the last statement in Theorem 6.3.1 cannot hold in this level of generality when n = 2. An additional condition is needed which takes into account the t dependence of the canonical relations Ct. To make this more precise, let us state a corollary of Theorem 6.3.1. This deals with averaging over hypersurfaces given by

=

Sx,t

= {y E IR.n: 4't(x,y) = 0}.

Here, the defining function is assumed to be a smooth function of (t, x, y) E [1, 2] X IR.n X IR.n. We also assume that both V' x4't and V' y4't never vanish, and, moreover, that the Monge-Ampere matrix associated to 4't is non-singular: det (

~ afxi~) 1 o.

(6.3.4)

If this is the case, we say that the family of surfaces satisfies the rotational curvature condition of Phong and Stein, and we have the following:

Corollary 6.3.2: Let Sx,t be as above and let dux,t denote Lebesgue measure on the hypersurface. Fix71(x, y) E CSO(JR.n xlR.n) and (for f E S) set

Atf(x)

=

1

S.,,,

71(x, y)f(y) dux,t(y).

189

6.3. Spherical Maximal Theorems: Take 1 Then, if n

~

3, it follows that

II tE(1,2) sup IAtf(x)lllv•(R") ~ Cp 11/IILP(R")•

if P > nf(n- 1).

(6.3.5)

To see that this follows from the last part of Theorem 6.3.1, we write Atf(x) = (211")- 1

Ji: ei9~•(x,y)170(x,

y) f(y) d9dy,

where 710 is a C 00 function times TJ· This is a bounded family in ~~~p- 1 >1 2 . By (6.2.3), each operator is locally a canonical graph since the Monge-Ampere condition is satisfied, and hence (6.3.5) is just a special case of Theorem 6.3.1. There are two extreme cases which should be pointed out. One is when Sx,t = { x} + t · S, where S is a fixed hypersurface. In this case the operators are (essentially) translation-invariant and (6.3.4) is satisfied if and only if the hypersurface S has nonvanishing Gaussian curvature. The other extreme case is when the rotational curvature hypothesis is fulfilled because of the way the surfaces change with x and y and not because of the presence of Gaussian curvature. For instance, if ~t(x,

y) = (x, y} - t,

(6.3.6)

the operator At involves averaging over the hyperplane Sx,t = {y : (x, y} = t}. IfTJ(x, y) vanishes near the origin, this family still satisfies the curvature hypothesis (6.3.4) despite the fact that the Gaussian curvature and all the principal curvatures vanish identically on Sx,t·

Remark. The last example shows how the above results cannot extend in this level of generality to two dimensions. For if ~t is given by (6.3.6) and 0 $ TJ E Cif is nontrivial then (6.3.5) can never hold for a finite p if n = 2. This is because, given E > 0, there is a measurable subset n~ = [-1, 1] X (-1, 1] satisfying IO~I < E but having the property that a unit line segment can be continously moved in n~ until its orientation is reversed. Thus, n~ contains a unit line segment in every direction. If one takes f to be the characteristic function of an appropriate translate and dilate of this Kakeya set n~, the left side is ~ 1 while the right side is < e 1/P, providing a counterexample to the possibility of the twodimensional theorem. Let us conclude this section by showing how Theorem 6.3.1 can be used to prove maximal theorems involving smooth hypersurfaces which shrink to a point as t --+ 0. Specifically, we assume that Bx.t c R.n is a

I

190

6. Fourier Integral Operators family of C 00 hypersurfaces which depend smoothly on the parameters (x,t) E R.n x [0, 1]. We then put Sx,t

=X+ tSx,t·

Our assumptions are that: (i) Given any subinterval [a, 1] with a > 0 the surfaces Sx,t satisfy the rotational curvature hypothesis in Corollary 6.3.2, and (ii) the "initial hypersurface" Bx,o always has everywhere nonvanishing Gaussian curvature. Corollary 6.3.3: Let TJ E cr(IRn x R.n) and suppose that Sx,t satisfies the two conditions described above. Then if n ~ 3 and if dnx,t now denotes Lebesgue measure on Bx,t we have

r_

sup I II O
:5 Cpii/IILP(Rn)• p > nf(n -1). (6.3.7) Remark. Taking Bx,t

=

sn-l one sees that this implies the seemingly weaker version of Stein's spherical maximal theorem where the dilations 0 < t < oo are replaced by 0 < t < 1. However, the full result follows from the estimate involving 0 < t < 1 by a scaling and a limiting argument. Corollary 6.3.3 also implies a maximal theorem for averages over geodesic spheres. Specifically, if (X, g) is a compact Riemannian manifold then, fort> 0 smaller than the injectivity radius T, we let Atf(x) denote the average off over the geodesic sphere of radius t around x. It then follows that supO nf(n- 1). One deduces t~is from the corollary by_noting that in local coordinates Sx,t = x + tSx,t where the surfaces Sx,t tend to the ellipsoid Bx,O = {y : 'EY;k(x)YjYk = 1}. FUrthermore, the rotational curvature hypothesis is satisfied for all intervals of the form [a, b] with a > 0 and b < T since the canonical relation associated to At is Ct = {(x,e,y,TJ): (x,e) = Xt(Y,TJ)}, where Xt: T*X\0--+ T*X\0 is the canonical transformation obtained by flowing for time t along the I

Hamilton vector field associated to V'Egik(x)f.;f.k, if Egik(x)f.;f.k is the cometric. To prove Corollary 6.3.3 we notice that Theorem 1.2.1 implies that, for small t, we can write the operators in (6.3. 7) as the sum of two terms, each of which is of the form :Ftf(x)

={

}Rn

eicp(x,t,e) {1 + 1tf.12) -(n-1)/4 a(x, t, tf.) ](f.) df..

6.9. Spherical Maximal Theorems: Take 1 The symbol satisfies ID~ItD{a(x, t, e)l ::::; Ca-y(1 phase function must be of the form

+ len-lal,

191 while the

where 1/J(x, e) is one of the two phase functions occurring in the Fourier transform of surface measure on Bx 0· Since we can estimate the maximal operator corresponding to dilations t E [2-io, 1], it suffices to show that if :Ft is as above then 1

II

I:Ftf(x)IIILP(R") $

sup . 0
Cp 11/IILP(R")•

P > nf(n- 1).

(6.3.7')

Different arguments are needed to handle the cases p ~ 2 and nf(n - 1) < p < 2. To handle the latter case we shall need to interpolate with stronger estimates for exponents p ~ 2. To this end we define the analytic family of operators

:Ft f(x)

= ez2

I ei'l?(x~t~e)

(1 + 1te12)z/2-(n-l)/4a(x,

t, te> J(e> cte.

The first step is to show that we have

II

sup . 0
I:Ft J(x>lllp::::; cp~Re(z) ll!llp. if p

~

2 and

- (n- 1)/2 + Re(z) < -o:p- 1fp.

(6.3.8)

Since .1'P = :Ft, this gives (6.3.7') for p ~ 2, and we shall handle the remaining exponents by interpolating with this stronger estimate. To prove (6.3.8), we define :F: t• for k ~ 1, by I

:F: tf(x) I

= ez2 f ei'~'(x,t,e) .B(Itel/2k)(1 + 1te12)z/2-(n-l)/4 a(x, t, te) f(e) cte. }Rn

where .B is as above. Then :Fff t = :Ff - Ek>I :F: t is dominated by the Hardy-Littlewood maximal Gperator and so-it suffices to show that for k = 1,2, ...

II

sup 0
I:F: tf(x)lll 1

$ C2k(Re(z)-(n-l)/2+ap+l/p) 11/llp·

(6.3.9)

p

We claim that this follows from the dyadic maximal estimates valid for j ~jo:

II

sup 2-j-l
I:F: tf(x)lll I

p

$ C2k(Re(z)-(n-l)/2+ap+l/P) 11/llp·

(6.3.9')

192

6. Fourier Integral Operators To prove these, we let ft be defined by ft(~) = .B(I~I/2 1 )](~). Then

L

FZ,tf(x) =

FZ,t(fk+j+m)(x),

if t E [Ti-l,Ti].

lml90

Also, after rescaling, we see that Theorem 6.3.1 implies that there is a uniform constant C such that

So, we get

J

sup

O
IFZ tf(x)IP dx '

~ c 2 kp(Re(z)-(n-1)/2+ap+l/p)

Jf: 1/;IP

dx.

-oo

liCE

But if p ~ 2 we can dominate the last integral by l/;1 2 ) 1 1 2 11~, and since Littlewood-Paley theory shows that this is bounded by II/II~, we get (6.3.9') and hence (6.3.8). To prove the remaining maximal estimates for exponents p < 2 we shall need to apply Stein's analytic interpolation theorem (Lemma 6.2. 7, (1)). Since the interpolation theorem deals with linear operators, we must first linearize the maximal function. More precisely, we notice that (6.3.7') holds if and only if there is an absolute constant Cp such that, whenever t(x) :an-+ (0, 2-io] is measurable,

IIFt(x)fiiP ~ Cp II/IlP•

P

> nf(n- 1).

(6.3. 7")

However, by (6.3.8), Ft(x) : £ 2 -+ L2 with a constant independent of the dilations if Re(z) - (n- 1)/2 + 1/2 < 0. Also, it is not hard to check that, for Re(z) < -1, IFt(x/(x)l is dominated by the HardyLittlewood maximal function (cf. Lemma 2.3.3). Hence, for any p > 1, Ft(x) is bounded on LP with a fixed constant Cp,Re(z) if Re(z) < -1. Since Ft(x) = ~x) an application of the analytic interpolation lemma yields (6.3.711 ). This completes the proof of Corollary 6.3.3.

Notes

Notes For historical comments about the development of the theory of Fourier integral operators we refer the reader to Hormander [5]. We have consciously presented here only the bare minimum of material for use in the next chapter and Hormander's paper is also an excellent source for the interested noninitiated reader who wishes to go further. See also Treves [1] and Hormander [7] whose expositions we have also followed in the first section. The L 2 regularity theorem for Fourier integral operators goes back to Eskin [1] and Hormander [5]. Eskin proved a local version, while Hormander proved the global version we have stated. The LP and pointwise regularity theorem is more recent and is due to Seeger, Sogge, and Stein [1], although many partial results were known, including those of Beals [1], Littman [1], Miyachi [1], and Peral [1]. See also Sugimoto [1]. The proof of the regularity theorem used ideas from the study of Riesz means in Fefferman [3], Cordoba [1], and Christ and Sogge [1]. In some ways the basic idea behind the decomposition of the operators in the proof is also similar to that used in the analysis of the solution to the wave equation using plane waves (see John [1]). The spherical maximal,.theorem is due to Stein [3] and the variable coefficient versions of the spherical maximal theorem are due to Sogge and Stein [3]. See also Oberlin and Stein [1], where the mapping properties of the operator corresponding to cf>t(x, y) = t - (x, y} were studied. For background about the Kakeya set and discussions about applications of such maximal theorems we refer the reader to Falconer [1]. The role of rotational curvature in Fourier ahalysis was introduced by Phong and Stein [1] in their study of singular Radon transforms.

193

Chapter 7 Local Smoothing of Fourier Integral Operators

In this chapter we shall prove estimates for certain Fourier integral operators which send functions of n variables to functions of n + 1 variables. We shall deal with a special class which contains the solution operators for the Cauchy problem associated to variable coefficient wave equations. The estimates we obtain are better than those which follow trivially from the sharp regularity estimates for Fourier integral operators in Theorem 6.2.1. We call these estimates local smoothing estimates. In Section 3 we shall see that these local smoothing estimates can be used to improve many of the estimates for maximal operators in Section 6.3. In particular, we shall be able to prove the natural variable coefficient version of the circular maximal theorem proved in Section 2.4, which includes estimates for averages over geodesic circles. The argument will involve an adaptation of the arguments in Section 2.4, and, among other things, we shall require a variable coefficient Kakeya maximal theorem which contains estimates for a maximal operator involving averages over thin "geodesic rectangles." The reason for the terminology is because the phrase "local smoothing" was first used for certain types of estimates for dispersive equations which go back to Kato, Sjolin, and Vega. In the case of the solution to the Schrooinger equation, w(x,t) = (eit~ J)(x), the local smoothing estimate says that if f E L2(1Rn), then w(x, t) E L~; 2 ,loc(!Rn X IR). Note that this is a big improvement over the fixed-time estimate llw(' 1 t)11£2(Rn) = II/II£2(Rn)·

7.1. Local Smoothing in Two Dimensions On the other hand, if one considers the seemingly related hyperbolic operator I -+ u(x,t) = eit~ I, there can be no local smoothing in L 2 • This is because-in sharp contrast to the solution operator for Schrooinger's equation-the hyperbolic operator is of a local nature, by which we mean that the kernel K(x, t; y) and all of its derivatives are O(lx- Yi-N) for any N if lx- Yi > 2t. Because of this, and the fact that here we also have llu( · ,t)IIL2(Rn) IIIIIP(Rn)• one concludes that u(x, t) can, in general, only be in L?oc(IR.n x JR.) if I is in L 2 (JR.n). It is not also hard to see that if O:p = (n -1)11/p -1/21, then, for 1 < p < 2, one can in general only say that u(x, t) E £P-ap, 1oc (IR.n x JR.) if I E LP(IR.n), which is just a trivial consequence of Theorem 6.2.1. For 2 < p < oo, the situation is much different. For every p there is an e = e(p, n) such that u(x, t) E L~-ap,loc(IR.n x JR.) if I E LP(IR.n). Since we saw at the end of Section 6.2 that, in general, u( · , t) only belongs to L~ap,loc(JR.n) if I E LP(IR.n), we conclude that, when measured in

=

LP, p > 2, eit~ I has much better properties as a distribution in (x, t), rather than in x alone. In two dimensions this local smoothing estimate just follows from inequality (2.4.27); however, we shall see that this estimate holds for variable coefficient Laplacians in all dimensions. The class of Fourier integral operators we shall deal with satisfy the so-called cinematic curvature condition. This is just the natural homogeneous version of the Carleson-Sjolin condition for non-homogeneous oscillatory integral operators which was stated in Section 2.2.

7.1. Local Smoothing in Two Dimensions and Variable Coefficient Kakeya Maximal Theorems Before we can state the local smoothing estimates we need to go over the hypotheses. From now on Y and Z are to be C 00 paracompact manifolds of dimension nand n + 1, respectively. As usual, we assume that n ~ 2. We shall consider a class of Fourier integral operators 1 JP.-l/ 4 (Z, Y;C), which is determined by the properties of the canonical relation C. Notice that our assumptions imply that C c T* Z \0 xT*Y \0 is a conic submanifold of dimension 2n + 1. 1 To be consistent with the convention regarding the orders of Fourier integral operators given in Chapter 6, we prefer to state things in terms of order p. - 1/4 so that when the operators are written in terms of oscillatory integrals with n theta variables, such as when one uses a generating function, the symbols will have order p.. The possible confusion arises from the fact that Z and Y have different dimensions.

195

196

7. Local Smoothing of Fourier Integral Operators To guarantee nontrivial local regularity properties of operators

r

E

p~- 1 14 (Z, Y;C), we shall impose conditions on C which are based on the

properties of the following three projections:

c /

l

(7.1.1)

z

T*Y\0

The condition has two parts: first, a natural non-degeneracy condition which involves the first two projections, and, second, a condition involving the principal curvatures of the images of the projection of C onto the fibers of T* Z \ 0. To describe the first condition, let nT·Y and llz denote the first two projections in (7.1.1). We require that they both be submersions, that is,

rankdllT·Y rankdllz

=2n,

(7.1.2)

= n + 1.

(7.1.3)

Together, these make up the non-degenerjl-Cy requirement. As a side remark, let us point out that if Y and Jl were of the same dimension, then, as we saw in the last chapter, (7.1.2) would imply that C is locally a canonical graph. Also, in this case the differential in (7.1.3) would automatically be surjective; however, since we are assuming that the dimensions are different, (7.1.2) does not imply that llz is a submersion. In order to describe the curvature condition, l~t zo E llzC and let llT· z be the projection of C onto the fiber T.zo* Z \ 0. Then, clearly, •o

r .zo

(7.1.4)

= llT·•o z(C)

is always a conic subset of T;0 Z \ 0. In fact, r Zo is a smooth immersed hypersurface in T;0 Z \0. In order to see this, note that the first assumption, (7.1.2), implies that the differential of the projection of C onto the whole space T* Z \0 must have constant rank 2n+ 1. (See formulas (7.1.7) and (7.1.21) below.) Furthermore, since the differential of C -+ T* Z \0 splits into the differential in the Z direction and in the fiber direction, we see that in view of (7.1.3)

rank dllT·•o z

=n

and therefore, by the constant rank theorem, dimensional hypersurface.

(7.1.5)

r zo

is a smooth conic n-

197

7.1. Local Smoothing in Two Dimensions Now in addition to the non-degeneracy assumption we shall impose the following Cone condition: For every ( E do not vanish.

r zo, n -

1 principal curvatures (7.1.6)

Since r zo is conic this is the maximum number of curvatures which can be nonzero. Clearly (7.1.6) does not depend on the choice of local coordinates in Z since changes of variables in Z induce changes of variables in the cotangent bundle which are linear in the fibers. If (7.1.2), (7.1.3), and (7.1.6) are all met then we say that C satisfies the cinematic curvature condition. This condition is of course related to the Carleson-Sjolin condition since the non-degeneracy condition is just the homogeneous analogue of (2.2.2) and the cone condition is just the homogeneous replacement for the curvature condition (2.2.4). Let us now see how our assumptions can be reformulated in two useful ways, if we use local coordinates. First, we note that we can use the proof of Proposition 6.2.4 to see that (7.1.2) and (7.1.3) imply that, near a given (zo, (o, yo, 170) E C, local coordinates can be chosen so that C is given by a generating function. Specifically, there is a phase function rt~(z, 71) such that Cis parameterized by rt~(z, 71)- (y, 71}, that is, C can be written (locally) as { (z, rt~~(z, 71), rt~~(z, 17), 71) : 71 E IRn \0 in a conic neighborhood of 170}. (7.1.7) In this case, condition (7.1.2) becomes rank rt~z" 71 -= n,

(7.1.2')

which of course means that if we fix zo, then, as before,

r Zo

= { ft'~ ( zo, 71) : 71 E IRn \ 0 in a conic neighborhood of 170} C T;0 Z \ 0

must be a smooth conic submanifold of dimension n. So if

and (} E sn is normal to directions for which

r Zo

at

(I

it follows that

±(}

are the unique

(7.1.8)

198

7. Local Smoothing of Fourier Integml Opemtors The condition that n - 1 principal curvatures be nonzero at ( then is just that

a2

'

rank ( 817;a1Jk) (rpz(zo, 17), 8} = n- 1,

if 7J, 8 are as in (7.1.8). (7.1.6')

It is also convenient to give a formulation which is in the spirit of the wave equation. This involves a splitting of the z variables into "timelike" and "spacelike" directions. If, as above, we work locally, then (7.1.2') guarantees that we can choose coordinates z = (x, t) E IRn x lR vanishing at a given point zo such that, first of all, " = r ank 'Px'1

(7.1.9)

n,

and second rp~ ~ 0,

if 1J :f: 0.

In other words, ro must be of the form ro = {(rp~(0,7J),q(rp~(0,7J))}, for some q satisfying q(e) ~ 0 if e :f: 0. This is because, if (e, r) are the variables dual to (x, t), then ro does not intersect the T axis. The cone condition, (7.1.6), just translates here to the condition that rankqee = n- 1. Since r x,t must have the same form for small (x, t), we see that local coordinates can always be chosen so that C is of the form C = {(x, t,

e,

T,

y, 1]~.: (x, ~)

= Xt(Y,_1J), T = q(x, t, e) :f: 0},

(7.1.10)

where, by (7.1.9),

Xt

is a canonical transformation

(7.1.2")

and " = - n- 1. rank qee

(7.1.6")

We can now state the main result of this chapter. Theorem 7.1.1: Suppose that :F E [P.- 114 (Z, Y;C) where, as before, C satisfies the non-degeneracy condition (7.1.2), (7.1.3), and the cone condition (7.1.6). Then :F: L~omp(Y) -+ Lf0 c(Z) if JL $ -(n- 1}(1/21/p) + e and e < e(p), with

e(p) = {

,Jp.

4 $p

< oo, (7.1.11)

H! - ~),

2 < p $ 4.

Remark. This result is not sharp-at least in higher dimensions. However, since the result has the most interesting consequences in two

199

7.1. Local Smoothing in Two Dimensions dimensions we just state it this way for the sake of simplicity. A natural conjecture would be that for p ~ 2n/(n- 1) one should be able to take e(p) = 1/p in the theorem. If this result were true, one could use it to prove sharp estimates for Riesz means in R.n by estimating the operators in (2.4.5) using Minkowski's integral inequality and a scaling argument. Also, one can use the counterexample which was used to prove the sharpness of Theorem 6.2.1 to see that for 2 < p < 2n/(n-1) there cannot be local smoothing of all orders < 1/p, and in fact the best possible result would just follow from interpolating with the trivial £ 2 estimate if the conjecture for 2n/(n- 1) were true. Different arguments are needed to handle two dimensions versus higher dimensions. So we shall prove the two-dimensional case in this section and then turn to the case of higher dimensions in Section 2. Before turning to the proofs, though, let us state a corollary. If M is either a C 00 compact manifold of dimension n or R.n we consider the Cauchy problem

((8/&t) 2 - ag)u(x, t) = 0, {

(7.1.12}

ult=o= /, (aj&t)ult=O =g. Here a9 is a Laplace-Beltrami operator which is assumed to be the usual Laplacian in the Euclidean case. By the results at the end of Section 6.2, it follows that iff E L~(M) and g E £~_ 1 (M) then u( ·, t) E ~-orp (M) if O:p = (n-1)11/p-1/21. Furthermore, this result is sharp for all nonzero times in the Euclidean and all but a discrete set of times in the compact case. Since the canonical relation for the solution to the wave equation has the form (7.1.10) with q = J'"£gik(x)e;ek, the solution operator satisfies the cinematic curvature condition. Therefore, Theorem 7.1.1 gives local smoothing for the wave equation. Corollary 7.1.2: Let u be the solution to the Cauchy problem (7.1.12). Then, if I CR. is a compact interval and if e < e(p),

Similarly, eitP f belongs to L~-orp+c,loc(M x R.) iff E L~(M), provided that P E \II ~1 (M) is self-adjoint and elliptic and the cospher:es Ex = {e : p(x, e) = ±1} c T; M \ 0 all have non-vanishing Gaussian curvature.

200

7. Local Smoothing of Fourier Integral Operators Orthogonality arguments in two dimensions We now turn to the first step in the proof of the local smoothing estimates for n = 2. Since local coordinates can be chosen so that C is of the form (7.1.7), we can use the proof of Proposition 6.1.4 to conclude that we may assume that F is of the form

F l(x, t) = [

ei
JR2

a(x, t, 17) j(17) d17,

where the phase function cp satisfies (7.1.21) and (7.1.61). Since we are now dealing with the two-dimensional case, a is assumed to be a symbol of order zero which vanishes for x E R.2 and t E R. outside fixed compact sets and 17 outside a fixed neighborhood of the 172 axis. With this normalization, we shall show that, fore> 0, (7.1.13)

In proving this, we may assume that I is supported in a fixed compact set since the kernel K(x, t; y) ofF is O(IYI-N) for sufficiently large y. Since 0:4 = 1/4, (7.1.13) implies the two-dimensional local smoothing for L 4 • The other estimates follow by analytic interpolation using IIFIIIL2(R3) ~ CIIIIIL2(R2) IIFIIILP(R3) ~ CIIIIIL~p(R2)· Both of these estimates follow from Theorem 6.2.1 since, for fixed times -+ Fl( · ,t) is a Fourier integral operator of order zero which is locally a canonical ~aph. Following the proof of (2.4.27), we fix {3 E Ccf((1/2, 2)) and set a~(x,t,17) = f30171/..X)a(x,t,17)· If we define

t, I

F~l(x, t) =

J

ei
a~(x, t, 17) j(17) d17,

then, by summing a geometric series, we see that (7.1.13) must follow from the estimates (7.1.14) To prove (7.1.14) we need to use the angular decomposition that was used in the proof of Theorem 6.2.1. So, if we identify R.2 and C in the obvious way, then for v = 0, 1, ... , [log ,.xl/2] we let 11K = e21fivf~ 112 • Then, we choose functions x~ E C 00 (R. 2 \ 0) which are homogeneous of degree zero, satisfy Ev X~ (17) = 1, and have the property that x~ (17~) :f:0 but xK (17) = 0 if 1171 = 1 and 111-11KI ~ C.\ -l/2 , and, lastly, IOOx~ (17)1 ~ Ca..XIal/2 , for every a: if 1171 = 1.

201

7.1. Local Smoothing in Two Dimensions As before, using the operators by setting

:Fff(x, t)

x~

=I

we make an angular decomposition of the

eirp(x,t,TJ)

x~(17) a>.(x, t, 17) f(f1) df1.

The square of the left side of (7.1.14) is dominated by

Lll lv1-v21:::::2k L :f'rl 10.

2tiiL2(R3) ·

k

Since we are assuming that a has small conic support the summation only involves indices k < log ..X 1/ 2 . We shall need to make a further decomposition based on k. To this end, we fix p E C8"((-1, 1}) satisfying p(u) = 1 for lui < 1/4 and LjEzp(u- j) 1. We then set

=

:F'{;{f(x, t) =

I

eirp(x,t,TJ)

a~·,{(x, t, 17) j(17) d17,

with (7.1.15) Note that this decomposition involves localizing T = rp~ to intervals of height ~ ..\2-k. These are larger than the intervals of height ..X 112 which were used in Section 2.4 to prove the constant coefficient local smoothing estimates. Since rp~ is not assumed to be constant in x and t, this different localization is needed for the integration by parts arguments which are to follow. Let ""'" denote the partial Fourier transform in the t variable. Then if I{ k is the interval of length 2-k ,xi+e and center j2-k ..X, the proof of Lem~a 2.4.4 shows that

:F'{•{ (f)(x, t) = 2_ f '

.

21T }TEl'.>.,k

where R~1 has an £ 4

-+

eiTt

(:F~·{ (!))"' (x, r) dr + R~·i(f)(x, t), '

'

L 4 norm which is O(..x-N). Thus, since :Ff,kf =

E;:::::2 k :F~1 (/), one can use Plancherel's theorem in the t variable, as in (2.4.28), to reach the desired conclusion II:F>..flli4(R3)

1 ~ c..xe'\:" L....J 2k 211("' L....J I ""' L....J k

it J2 lvt -v21:::::2k

1 2 :f":t.i1U):f":2,;2U)I2) ' 11 >.,k >.,k L2(R3)

202

7. Local Smoothing of Fourier Integml Opemtors On account of this, the desired inequality, (7.1.14), would follow if we could show that there is a fixed constant C, independent of ..X and k, for which

(7.1.16) This is the main inequality. Notice that, in going from (7.1.14) to (7.1.16), we have lost a factor of 2k/ 4 in the L 4 bounds. To prove (7.1.16), we shall need to exploit the cone condition. Recall that this condition says that, if 0 is a small conic neighborhood of supp71 a, then rx,t

= {(rp~(x,t,71),rp~(x,t,71)): 71 E 0}

is a C 00 conic hypersurface in R.3 \ 0 which has the property that at every point (e, r) E r x,t one principal curvature is nonzero. To use this, let us define subsets (7.1.17) which depend on the scale ..X and also on k. By the non-degeneracy hypothesis (7.1.9), r~:{ is contained in a sector around ( (rp~(x, t, 11r), rp~(x, t, TJr)) which has angle~ ..X - 1/ 2 . Here 11r are the unit vectors occurring in the defiThe sets r~:{ have height ~ ..\Tk. So r~:{ is basically a ..X1 / 2 X nition of

xr.

..XTk rectangle in the tangent plane tor~~ at (rp~(x, t, 11r), rp~(x, t, 11r)).

On account of this, it is a geometrical fact that, if we consider algebraic sums of the sets in (7.1.17), then if lv- v'l = dist ((vt, V2), (z1, v~)) is larger than a fixed constant, dist (rVt,jl x,t

+ rV2,j2 rV~,jl + rV~,h) x,t • x,t x,t

> c..\j(nVl .,>.

-

+ .,v2) _ "I).

(TJV~ ),.

+ TJV~), ),. • (7.1.18)

where if, as we are assuming, a has small conic support, the constant c > 0 depends only on rp. To see this, let us fix "heights" Tt, T2 ~ ..X so that the cross-sections f~~t1 = {( T) E r~i : T = Tf} are nonempty. Here we are assuming, as we may, that 0 < rp~. It is not hard to check that the sets r~·{ have been chosen to have the largest height so that { ( T) E ' r~i 1 + r~~ih : T = Tl + 1'2} is contained in a tubular neighborhood of

e,

e,

7.1. Local Smoothing in Two Dimensions

203

width 0(2k) around f~i 1 + f~~th. Using the curvature of r z 1t one can prove that if lv- v'l is larger than a fixed constant dist (fvtJt + fv21i2 fv~Jt + fv~J2) ~ ~~(TJv1 + TJv2) _ (TJv~ + TJv~)~

X1t

X1t ' x 1t

x 1t

~

>.

>.

>.

which, by the previous observations, leads to (7.1.18), since

~I(TJ~1

>.

'

+TJ~2 )­

(TJ~~ + TJ~~)i ~ 2klv- v'l. Next, we claim that (7.1.18) leads to the bounds

1/

:F':l .it (!):F':21j2 (!)F.'; ljl (!).r!'2J2 (!) dxdtl >.lk

>.lk

>.lk

>.lk

(7.1.19)

One sees this by first noticing that the left side equals the absolute value of

J

eici> a~ttt (x, t, TJ)a~2t2 (x, t,{) I

I

with ~ = rt~(x, t, TJ)

+ rt~(x, t, {) -

rt~(x, t, TJ') - rt~(x, t, {').

To estimate this term we need the following lower bounds for the gradient of the phase function on the support of the integrand:

1Vx 1 t~l ~ c~I(TJ~1 + T/~2 )- (TJ~~ + TJ~~)I ~ ~(~ -l/21v- v'l· max ITJ~; - TJ~~ 1),

lv- v'l ~CO (7.1.20)

IVx~l ~ cd(TJ + {)- (TJ1 +{')I- c2~(max ITJ~; - TJ~~ 1) 2. (7.1.21)

Here c; are fixed positive constants. The first inequality in the first lower bound is equivalent to (7.1.18), while the second inequality there follows from the fact that the unit vectors {TJn are separated by angle ~ ~-l/ 2 . To prove (7.1.21) one notices that, by homogeneity, rt~~(x, t, TJ) = rt~~11 (x, t, TJ) · TJ· Hence we can write Vx~ = rt~~71 (x,t,TJ~1 )((TJ+{)- (TJ1 +e'))

+ (rt~~11 (x, t, TJ)- rt~~11 (x, t, TJ~1 ))TJ + ·· ·- (rt~~(x,t,{')- rt~~11 (x,t,TJ~1 }){'.

204

7. Local Smoothing of Fourier Integral Operators The first term has absolute value ~ ell (17 + e) - (77' + e') I because det
l~,t~l :S Ca(l(77+e)- (77' +e')l +~(maxl77~; -77~~1) 2 ).

(7.1.22)

Let us first see that this holds when o: = 0. To do so, we use Euler's identity to write ~ = (
+ ((
l~,t a~',i(x, t, 77)1 :S Ca 2klol.

(7.1.23)

By putting (7.1.20)-(7.1.23) together, one gets the claim (7.1.19) via a partial integration. In fact, we can dominate the left side of (7.1.19) by CN~-NII/11~. which leads to (7.1.19) since we are assuming that f has fixed compact support. Returning to (7.1.16), one sees after an application of (7.1.19) and the Schwarz inequality that the left side is majorized by

~ell (L 10:{U)I 2 ) 112 IIi4(R3) +

ll/lli4(R2)·

v,j

This means that we would be done if we could show that II(L 10:{U)I 2 ) 112 II£4(R3):::;

c~c(2-k~l/ 2 ) 114 II/IIL4(R2)·

v,j

This is the main reduction. To be able to apply the variable coefficient Kakeya maximal estimates which are to follow, we need to make one more reduction, which this time is much easier since it relies only on the fact that
7.1. Local Smoothing in Two Dimensions

205

square function involving operators whose kernels, for fixed y and v, are concentrated in small tubular neighborhoods of curves in JR2 x JR. To this end, if pis as in (7.1.15), we define Fourier multiplier operators, pm, acting on functions of three variables by setting

(Pmg)"(e, r)

= p(>.-lf2r- m)g(e, r).

Using the definition (7.1.15) and the proof of Lemma 2.4.4 one sees that, if j is fixed, there must be an integer m(j) such that pm ~1 has I

LP(JR2 )-+ LP(JR3 ) norm 0(>.-N) if lm- m(j)l ~ >. 112 Tk>.c. To prove this claim, one merely uses the fact that the symbol in (7.1.15) vanishes if cp~ does not belong to an interval of length 2-k >. which depends only on j. Using this claim, one repeats the arguments which lead to (7.1.16) to verify that

(L IF~:iU>I 2 ) 112 IIL'(R3) ~ c>.c(Tk>.l/2) 114 11( 2: IPm~:iU)I 2 ) 112 IIL'(R3) + c IIIIIL (R II

v,j

4

2 )·

v,j,m

Combining this with the last inequality means that we would be done if we could prove that

II(

2: IPm~:iU)I 2 ) 112 IIL'(R3) :5 c>.c IIIIIL (R 4

(7.1.24)

2 )·

v,j,m

1·

To prove this, we recall that suppp C ( -1, 1) and p(u) = 1 for lui < So, if we define a~1m = a~·,{;(x, t, TJ)p((>.- 1 12 cp~(x, t, 17)-m) /8) and then set

then

_ pm:F':,j,mll ll pm:P:.i >.,k >.,k LP-+LP = 0(>.-N). Consequently, (7.1.24) must follow from

II(

2: 1Pm~:i·m(J)I 2 ) 112 IIL'(R3) ~ c>.c IIIIIL (R 4

(7.1.24')

2 )·

v,j,m

To prove this, a couple of observations are in order. We first notice that if m is fixed then there are 0(1) indices j for which ~·i·m =F 0. I

Also, the symbols, a~·~m(x, t, TJ), of these operators vanish if cp~ [>.1/2m- 8>.1/2, >.112m'+ 8>. 112]. In addition, if we let Q(x, t, v, m) = {TJ: x~(TJ)p((>. -l/ 2 cp~(x, t, 17)- m)/8) =F

0},

fl.

206

7. Local Smoothing of Fourier Integral Operators then this set is comparable to a cube of side-length >.. 1/ 2 • In particular, each one intersects at most Co cubes in a given >.. 1/ 2 lattice of cubes in R2 . Since It'~ is smooth, Co can be chosen to be independent of the relatively compact set K = {(x, t) : a(x, t, 71) # 0}. With this in mind, we set

Thus I = E I w In addition, if, for a given (x, t, v, m), we let I(x,t,v,m) c Z 2 denote those JL for which .r~:t·ml~£(x,t) # 0, it follows from the properties of the Q(x, t, v, m) that Card(I(x,t,v,m)) ~C1,

(7.1.25)

where C1 is an absolute constant. Similarly, for fixed v and (x, t) E K there must be a uniform constant c2 such that (7.1.26) Finally, we let .1(v) d..enote those JL for which .B(I711/>..)x~(71) x p(>.-l/211l_JLI)p(>.- 1 1~-JL2) does not vanish identically and note that JL belongs to only finitely many of the sets .1(v) since supp,B(I711/>..)x~(71) is contained in the set {71: 1711 ~ >.., and 171~ -71/17111 ~ c>..- 112 }. We now turn to (7.1.241). Let K~·fm(x, t; y) be the kernel of I

pm.rrfm. We shall see that we .have the uniform bounds (7.1.27) Thus, since, for fixed m, .rr:t·m vanishes for all but finitely many j, the Schwarz inequality, (7.1.25), and (7.1.26) yield LIPm.rr:fml(x,t)1 2 mj

I

~ c:E I m,j

I~£(Y)I 2 1K~:fm(x,t;y)ldy

2:

~£EI(x,t,v,m)

~£E.7(v)

~ c'f2:

2:

II~£(Y)I 2 1K~:fm(x, t; y)l dy

m,j ~£EI(x,t,v,m) ~£E.7(v)

~ C"

I

L ~£E.7(v)

II~£(Y)I 2 su~ IK~;{•m(x, t; y)l dy. m,J

7.1. Local Smoothing in Two Dimensions

207

From this we see that for a given g(x, t)

1/ ~

1Pm-0i'mf(x,t)l 2 g(x,t)dxdtl

v,J,m

5, C

JL

lf14 (y)l 2 sup{ v

14

f

sup 1Kr1·m(x, t; y)llg(x, t)l dxdt} dy.

JR3 j,m

'

(7.1.28)

We have already seen in Lemma 2.4.6 that IICE 14 lf14 12)112114 5, Cllfll4· Therefore, since the left side of (7.1.24') is dominated by the supremum over all llgll2 = 1 of the left side of (7.1.28), we would be done if we could prove the maximal inequality

(f

supl

JR2 v

f

supiKr·fm(x,t;y)lg(x,t)dxdtl 2 dy) 112

}Ra j,m

'

5, CllogAI 312 IIgiiL2(R3)·

(7.1.29)

To prove the missing inequalities (7.1.27) and (7.1.29) we need to introduce some notation. If N C JR.3 x IR.2 \ 0 is a small conic neighborhood of suppa~, we define the smooth curves 'Yy,17

= {(x, t) : cp~(x, t, 17) = y, (x, t, 17) EN}.

(7.1.30)

Then one estimate we need is the following. Lemma 7.1.3: If 11r are the unit vectors occurring in the definition of

:Ff, then, given any N, (7.1.31)

Proof: Since the kernel of pm is 6o(x-y)Al/2p(Alf2(s-t))eim~- 112 (t-s), it suffices to show that the kernel .Kr·fm(x, t; y) of .rr·fm satisfies the estimates in (7.1.31). But ' '

.Kr·fm(x, t; y) = f '

~2

ei(.p(x,t,TJ)-(y,TJ)) a~·tm(x, t, 17) d11. '

After using a finite partition of unity, we may assume that the (x, t)support of the symbol is small. Since we are assuming that cp~ f; 0 and hence cp~17 f; 0, it follows from the implicit function theorem that 'Yy,17A is of the form (x(t), t) near K = SUPPx,ta if this set is small enough. Also, we are assuming that det cp~17 f; 0, so if K is small enough there must be a c > 0 such that

208

7. Local Smoothing of Fourier Integral Operators But V 11 rp is homogeneous of degree zero and therefore

IV11 rp(x,t,f1)- V11 rp(x,t,11nl ~ c>.- 112 ,

11 E suppx~·

Combining these two bounds shows that there is a d

IV11 [rp(x, t, 17) -

> 0 such that

I

(y, 17}] ~ c' dist ((x, t)!Yy, 11;:),

provided that dist ((x,t),'Yy, 11;:) is larger than a fixed multiple of >.- 1/ 2. Since O::a~1m(x, t, 17) = 0(>.-lol/2) and since, for fixed (x, t), this symbol vanishes for 11 outside a set of measure 0(>.), the desired estimate I for K~·fm follows from integration by parts. ' It is clear that (7.1.31) implies the uniform L 1 estimates (7.1.27) for the kernels. Using (7.1.31) again, we shall see that the maximal estimates (7.1.29) follow from the Kakeya maximal estimates in the next subsection. Variable coefficient Kakeya maximal functions For later use in proving the higher-dimensional local smoothing estimates, and since the result is of independent interest, we shall state a maximal theorem which is more general than the one needed to prove (7.1.29). We now assume that Z and Y are as in Theorem 7.1.1, with the dimension of Y being n ~ 2 and dim Z = n + 1. To state the hypotheses in an invariant way, let C satisfy the non-degeneracy conditions (7.1.2) and (7.1.3). It then follows that

'Yy, 11

= {z E Z: (z,(,y,17) E C,

some(},

is a C 00 immersed curve in Z which depends smoothly on the parameters (y, 17) E llT·Y(C). Let us fix a smooth metric on Z and define the "6-tube"

Rt,

11 =

{z: dist (z,-yy, 11 ) < 6}.

The variable coefficient maximal theorem then is the following. Theorem 7.1.4: Let C satisfy the non-degeneracy conditions (7.1.2) and (7.1.3) as well as the cone condition (7.1.6). Then, i/0 < 6 < ~ and a E C~(Y x Z) ( {

sup

}y TJEIIT;Y(C)

Iy, 1(~6 0

y,T]

) {

6

lR,,., 5:

a(y, z) g(z) dzl2 dy) 1/2 n-2

3

cr-2 llog6l2llgiiL2(Z)·

(7.1.32)

7.1. Local Smoothing in Two Dimensions

209

Since the canonical relation associated to the operator in the proof of the two-dimensional version of Theorem 7.1.1 is given by (7.1.7) (with z = (x,t)), it is clear from (7.1.31) that (7.1.32) implies (7.1.29) by taking 6 = >.- 112 ,2>.- 112 , .... Before turning to the proof, let us state one more consequence. If (Y, g) is a compact Riemannian manifold, then for a given y E Y and 8 E TyY, let 'Yy,e(t) be the geodesic starting at yin the direction 8 which is parameterized by arclength. Then, if we fix 0 < T < oo and let

Rt,

11 =

{(x,t): dist (x,'Yy,B(t))

< 6, 0 ~ t ~ T},

one can use the Legendre transform, mapping TY -+ T*Y, (see, e.g., Sternberg [1]) to see that a special case of Theorem 7.1.4 is: sup 8 Vol

(~

y,B

)1

lg(z)idxdt

£2(Y)

R!. 9

~C6_n; 2 llog6l~llgiiL2(YxR)·

Proof of Theorem 7 .1.4: For the sake of clarity, let us first present the arguments in the case where n = 2 and then explain the modifications that are need to handle the higher-dimensional case. We write z = (x, t) in what fOllows. We may work locally and assume that 'Yy, 11 is of the form (7.1.30) where of course we are now assuming that N C JR3 x JR.2 \ 0. We may assume that (0,0,0; 1,0) E N. Also, by (7.1.2') and (7.1.61 ), we may assume for the sake of convenience that coordinates have been chosen so that 2

and

a8 2 1Pz3, =F 0 '12

at z = 0, 1J = (1, 0).

(7.1.33)

To proceed, we fix a E C~(JR.2 ) satisfying 0. ~ 0. Then for o:(z, 8) E C~ supported in a small neighborhood of (0, 0) we put

o:0 (z, 8; TJ) = o:(z, 8) a(617) and define Aog(y, 8) =

11 ei((<,e~(z,cos8,sin8),1J)-(y,7J)]

(7.1.34)

o:o(z, 8; TJ) g(z) dTJdZ.

R3 R2

(7.1.35)

This operator dominates the averaging operator in (7.1.32) if the bump function there has small support and if 6 is replaced by a multiple of 6

210

7. Local Smoothing of Fourier Integral Operators (depending on a). So it suffices to show that

(L s~piA.sg(y, 2

8) 12 dy) 112 $ Cllog 6l 312 llgii£2(R3)·

(7.1.36)

To prove this we shall want to use the following special case of Lemma 2.4.4: sup IF(8)1 2 $ 9ER

2(1 IFI 2ds) 112

(I

IF'I 2

ds) 112

ifF E C 1 (R) and F(O) = 0.

{7.1.37)

Before applying this, though, we need to break up the operators A.s. To do this, as before, we let {3 E Clf(R \ 0) satisfy Ek ,8(2-ks) = 1,s "I 0. We then define dyadic operators

A~g(y, 8)

=IIei((<,e~(z,cos9,sin9),7J)-(y,7J)].B(I171/

>.) o:.s(z, 8; 17) g(z) d17dZ.

Then, using Plancherel's theorem one sees that (7.1.36) would follow from Schwarz's inequality and

IA~g(y,8)1ll £2(R2) ll sup (J

~ Clog>.llgii£2(R3)• >. > 2.

{7.1.36')

We need to make one more reduction. If we set ~(z;17,8) = (cp~(z,cos8,sin8),17),

(7.1.38)

it is based on the observation that (7.1.8), (7.1.33), and homogeneity imply that, at z = 0,

~~3 9

=0 ~~~(}(} ¥ 0

when 8 = 0 {::::::::::} (1, 0)

= ±77/1171,

when 8 = 0 and (1,0) = ±77/1171·

Hence, (7.1.33) also implies that, for ±77/1171 = (1,0) and 8 E suppoo:.s, ldet (

oz:~:. 8 )) I~ 1171181.

More generally, if we change variables (17, 8)-+ (17, 8) = (17, 8- arcsin{1J2/I77D), where arcsin is the inverse with values in [-1r /2, 11/2], we get ldet (

az:;: e)) I~

1171181,

(7.1.39)

provided that 17 is outside a conic neighborhood of the 1J2 axis such that C 00 in these coordinates. On the other hand, if {8, 17) E suppo, 71 o:.s is

~is

7.1. Local Smoothing in Two Dimensions

211

in a sufficiently small neighborhood of the TJ2 axis, ldet 8 2 ~j8z8(17, 9)1 ~ 1171· Another important fact, which follows from the homogeneity of cp, is that

a

ae ~ =

(7.1.40)

0 when e = 0.

Based on (7.1.39) and (7.1.40) we set for k = 1, 2, ...

A~·kg(y, 9) =

JJei((cp~(z,cos9,s.in9),1J)-(y,1J)] x {3(2-k A112e) /3(1171/ A) a 6 (z, 9; 17) g(z) d17dz, (7.1.41)

and A~· 0 = A~ - Ek>1 A~·k. Since e is bounded, there are O(log A) terms and hence (7.1.36') would follow from the uniform estimates

lls~p IA~·kg(y, 9)ljjL2(R2) ~ C II911£2(R3)•

k

= 0, 1, 2,... .

(7.1.3611 )

We can now invoke (7.1.37). By applying it and Schwarz's inequality we see that (7.1.3611 ) would be a consequence of the estimates

(//1 (!)j A~·kg(y,9)12

dOdy) 1/2

~ C(A-1/42-k/2)1-2j II9II£2(R3)• j

= 0, 1. (7.1.42)

However, (7.1.40) implies that, on the supports of the symbols (8/89)(cp~(z,cos9,sin9),17) = 0(A 1122k) and hence (8/89)A~,k behaves like A1/22k A~·k, so we shall only prove the estimate for j = 0. It turns out to be easier to prove the estimate for the adjoint operator because this allows us to use the Fourier transform. Specifically, if " - " denotes the partial Fourier transform with respect toy, then the desired estimate is equivalent to

II/Jei(cp~(z,cos9,sin9),1J) /3(

2-k A1/2e)

x f3(1171/A)ao(z,9;17)i(17,9)d17d911

< CA- 1/ 4 2-k/2 11/11£2·

L 2 (dz) -

However, if we make the change of variables 17-+ A17, 8 would be a consequence of the following.

-+

e, this in turn

I

Lemma 7.1.5: Let bk(z; 17, e) satisfy l~bkl ~ C0 Va, as well as bk = 0 ifeitherlzl > 1,1171 fl. [1,2], ore fl. [2k- 1A- 1/2,.2kA- 1/2] when k = 1,2, ... ,[21l'logA 112] and e fl. [O,A- 112] when k = 0. Then, if we

212

7. Local Smoothing of Fourier Integral Operators also assume that bk vanishes in a neighborhood of the 112 axis (so that

(>(z;71,8) is C 00 on suppbk}, it follows that T>.,k f(z) =

f

JR3

ei>.4>(z;7J,e) bk(z; 71, 8) f(71, 8) d71d8

satisfies

(7.1.43) On the other hand, if r( z; 71, 8) = 0 for 1711 fl. [1, 2] and 71 outside a small neighborhood of the 112 axis, or for 8 outside a small neighborhood of the origin, then

n>< f(z)

= {

JR3

ei>.4>(z;7J,B) r(z; 71. 8) f(11. 8) d77d8,

satisfies IIR>. fiiL2(R3) $ CA -a/ 2 11fiiL2(R3)·

Proof: Let us first prove the estimates for the operators T>.,k. We use a modification of the argument used in the proof of the Carleson-Sjolin theorem (Theorem 2.2.1, (2)). We first note that, after perhaps using a smooth partition of unity, we may assume that bk has small support. Then the square of tl& left side of (7.1.43) equals #

f f

JR3 JR3

H><,k( 71 ,8;77'.8')f( 71 ,8)f(71',8')d77d8d77'd8',

(7.1.44)

with

To estimate this kernel, we note that (7.1.40) implies that, for 8 ~ 0, 4> is a C 1 function of 8 2 . But (7.1.39) implies that, in the coordinates where 8 ~ 0 is replaced by 8 2 , the Hessian of 4> is non-singular. This and the fact that (> is C 1 in (71. 8 2 ) implies that" there must be a c > 0 such that

IVz [4>(z; 71. 8)- 4>(z; 77'. 8')] I ~ cl (77- 77 ,82 1

(8') 2 ) I

on the support of the symbol if bk has small enough support. Consequently, since both 4> and the bk are uniformly smooth in the z variables, a partial integration gives the bounds IH>.,kl $ CN(1 + Al71- 77'1)-N (1

+ Al82 -

(8') 2 1)-N,

for any N. But if k =F 0, 182 - (8') 2 1~ A- 112 2kl8- 8'1 on the support of the kernel and hence the L 1 norm with respect to either of the pairs of variables is O(A- 2 2-kA- 112 ). One reaches the same conclusion for

213

7.1. Local Smoothing in Two Dimensions k = 0 if one recalls that in this case the kernel vanishes for 181 > >. - 1/ 2. Using these estimates, one concludes that (7.1.44) must be :5 c>.- 2 2-k>.- 1 1 2 11/11~. which finishes the proof of {7.1.43). If werecall that our assumptions imply that deta2 ~(z;TJ,8)/8z8(TJ,8) "I 0 on supp r, modifications of this argument give the estimate for R>.. I The proof of the higher-dimensional maximal estimates follows the same lines. First, if w(8) are local coordinates near w(O) = (1, 0, ... , 0) E sn- 1, then one sets

A6 g(y, 8) =

f ei((rp~(z,w(B)),11)-(y,17)] o:0 (z, 8; TJ) g(z) dTJdz,

f

}Rn+l }Rn

where now o:0 (z, 8; TJ) = o:(z, 8)a(6TJ) with o: E ego supported near the origin and a E Cif(!Rn) satisfying a~ 0. The maximal inequality would then follow from IAog(y, 8)111 :5 c6-(n-2)/2llog 6I3/2119IIL2(Rn+l)· II sup 8 L2(Rn) This time we shall want to use the following higher-dimensional version of (7.1.37}: sup IF(8)1 2 BERn-1

:5 Cn-1

L lal+l/3l$n-1

(I

188 Fl 2

d(J) 112 (I~~ Fl 2 d(J) 112 , (7.1.371 )

if F(O) = 0 when 8; = 0 for some 1 :5 j :5 n - 1. To apply it, as before, we must break up the operators. First, if A~ are the dyadic operators, it suffices to show that the associated maximal operators send L2(JRn+l) -+ L2(1Rn) with norm O(>.(n- 2)/ 2 log >.). To see this, we note as before that (7.1.2') and (7.1.6') imply that ~(z;TJ,8) = (
satisfies rank82 ~/8z8(TJ,8) = n+ 1 unless w(8) and TJ are colinear. So if r is the compliment of a conic neighborhood of (1, 0, ... , 0) = w(O) which contains {w(8): 8 E suppeo:}, and if we set

R~g(y, 0) = then

II ei~(z;1},B)

(IIlao R~g(y,

.B(ITJI/>.)xr(TJ) o:0 (z, 8; TJ) g(z) dTJdz,

8) 12 d8dy) 1/2 :5 c>.lal-1/2119112·

This follows from the argument used to estimate the remainder terms in Lemma 7.1.5. By (7.1.37') the maximal operator associated to R~ has

214

7. Local Smoothing of Fourier Integml Opemtors £ 2 bound O(.~(n- 2 )1 2 ). So, if we let and it suffices to show that

Rg,

.Ag

be the difference between

lls~p IAgg(y, 9)1112 $ C>. (n-2)/2

log >.llgll2·

Ag

(7.1.45)

If we let f!(9, TJ) =min± dist (w(9), ±TJ/ITJI), then, by the above discussion, we may assume that f!(9, TJ) is small on the support of the symbol, ii0 , of .Ag, and, hence, 18(Jf!l $ Calfll 1-lal there. With this in mind, we then set for k = 1, 2, ... -.>.k

A 6 • g(y,9)

=II

ei'P(z;1J,9)

{3(2-k >.1/2n(9, TJ)) fJ(ITJI/ >..) iio(z, 9; TJ) g(z) d1Jdz,

-.>.o -.>. -.>. k and A6 • = A6 - Lk> 1 A6 • . Then arguing as before shows that (7.1.45) follows from the estimates

(lllfYo A~·kg(y,9)i2

d9dy) 1/2:::;; C(>..-1/4 2-k/2)1-2laiiiYII2·

-.>. •k ~ (>. 1 /2 2k ) 1a 1A -.>. •k , the arguments for o = 0 give the Since 8(j A 6 6 bounds for general o. To prove this, just like before, one ~timates the adjoint operator in order to use the Fourier transform. Since we are assuming that cp satisfies (7.1.2') and (7.1.61), the arguments that were used to handle the two-dimensional case can easily be modified to show that this operator satisfies the desired estimates. I

7.2. Local Smoothing in Higher Dimensions In this section we shall prove the local smoothing estimates in Theorem 7.1.1 corresponding to n ~ 3. The orthogonality arguments are much simpler here because we can use a variable coefficient version of Strichartz's £ 2 restriction theorem for the light cone in JRn+I. Applying this £ 2 -+ Lq local smoothing theorem, we use a variation of the arguments in Section 5.2 that showed how, in the favorable range of exponents for the lJl -+ £ 2 spectral projection theorem, one could deduce sharp lJl -+ lJl estimates for Riesz means from lJl -+ £ 2 estimates for the spectral projection operators. To prove the higher-dimensional lJl -+ lJl local smoothing theorem, in addition to the the Kakeya maximal estimates just proved, we shall need the following sharp £ 2 -+ Lq local smoothing estimates whose straightforward proof will be given at the end of this section.

7. 2. Local Smoothing in Higher Dimensions

21

Theorem 7.2.1: Suppose that :FE JP.-l/ 4(Z, Y;C) and that C satisfies the non-degeneracy assumptions (7.1.2)- (7.1.3) and the cone condition (7.1.6). Then :F: L~omp(Y)-+ Li!,c(Z) if2(n+ 1)/(n-1) $ q < oo and JL ~ -n(1/2- 1/q) + 1/q. Remark. Using the Sobolev embedding theorem and the £ 2 boundedness of Fourier integral operators it is not hard to see that if one just assumes (7.1.2) and (7.1.3) then :F : L~omp(Y) -+ Li!,c(Z) if JL $ -n(1/2 - 1/q) and 2 ~ q < oo. Thus, Theorem 7.2.1 says that, under cinematic curvature, there is local smoothing of order 1/q for q as in the theorem. Using the counterexample that was used to prove the sharpness of Theorem 6.2.1, one can see that the above £ 2 -+ Lq local smoothing theorem is sharp. In the model case, where C is the canonical relation for the solution to the wave equation in IR.n, it is equivalent to Strichartz's restriction theorem

The reason for this is that, by duality and Plancherel's theorem, the last inequality is equivalent to

f ei((x,{)+tlell j(e)__!!{_ I II }Rn leil/2 L•(Rn+i) $ CII/IIL2(Rn)•

q = P1 = 2 <;:~~)

•

However, 6ince -n(1/2- 1/q) -1/q = -1/2 for this value of q, one can use the model case of Theorem 7.2.1, together with Littlewood-Paley theory (cf. the proof of Theorem 7.2.1 below) and scaling arguments, to deduce the last inequality.

Orthogonality arguments in higher dimensions We let f3 E Clf((~, 2)) be as before. Then, if a(z, 71) is a symbol of order zero which vanishes for z outside of a compact set in JRn+l, we define dyadic operators

:F>,/(z) =

f

}Rn

ei
a>,(~. 71) j(71) d71,

a>,(z, 71) =

/3(1'71/ A)a(z, 71).

If we assume that cp satisfies (7.1.21) and (7.1.6'), then, by summing a geometric series, it suffices to show that for e > 0

(7.2.1)

216

7. Local Smoothing of Fourier Integral Operators To prove this we need to make an angular decomposition of these operators. So we let {xn. v = 1, ... , N(A) ~ A(n- 1)/ 2 , be the homogeneous partition of unity that was used in the proof of Theorem 6.2.1. We then put

:Frf(z)

=I

eicp(z,TJ) x~(17)a>.(z,17)/(17)d17.

We now make one further decomposition so that the resulting operators will have symbols that have rrsupports that are comparable to A112 cubes. To this end, if p E C~(( -1, 1}) are the functions which were used in the two-dimensional orthogonality arguments, we set

F;.•j /(z)

=I eicp(z,TJ)a~,j(z,

17) /(17) d17,

with

a~,j (z, 17) =X~ (17)P(A - 1/ 2 1171 - j) a>.(z, 17).

A couple of important observations are in order. First, if nv,j ~>.

= suppTJ a>.v,j( z, 17) ,

(7.. 2 2)

then the Q~,j are all comparable to cubes of side-length A1/ 2 which are contained in the annulus {17 : 1171 ~ A}. This, along with the fact that the symbols satisfy the natural estimates associated to this support property,

!a]~a~•j(z, 11 )!::; Ca,"f(1 + 117 1)-lal/2 ~ A-lal/2,

(7.2.3)

makes the decomposed operators much easier to handle. The square function that will be used in the proof of (7 .2.1) will involve operators which are related to the F;..,j operators. In fact, the main step in the proof of the orthogonality argument is to establish the following result which, for reasons of exposition, is stated in more generality than what is needed here. Proposition 7.2.2: Fix c > 0. Then, given any N, there are finite M(N) and CN such that whenever Q C JRn+l is a cube of side-length A-1/2-e and 2(n + 1)/(n- 1) $ q < oo II:F>.IIIL•(Q) $ CNIQI-! A-~+n(~-!>-!

L lli~M(N)

II(L IF;..;~/1 2 ) 112 IIL•(Q) v,j

+ CNA-NII/IIL•(R") · (7.2.4)

21

7.2. Local Smoothing in Higher Dimensions Here

-0;~f(z)

=I

ei
a~1(z, 17) j(17) d17,

with symbols satisfying v,j( ) 0 a.>.,i z,17 =

;I

'J

d Qv,j 17 'F .>. ,

l~a~',{(z, 17)1 :5 CaA -lal/ 2 'Vo..

(7.2.5)

If the phase function cp is fixed, the constants in (7.2.4) and (7.2.5) depend on only finitely many of those in (7.2.3). The first operator -0;~ will just be an oscillatory factor times -0';, while, for i ~ 1, the operators in (7.2.4) will involve derivatives of the symbol and the phase function. '1\uning to the proof, it is clear that (7.2.4) would be a consequence of the following uniform upperbounds valid for z E Q:

IIF.>.fiiL•(Q) :5

eN( A-~ An(!-~)-~ L (L 1-0;~f(z)l 2 ) 112 +A-NIIfiiL•(R")). lii~M(N) v,j

1

(7.2.4') We may assume that 0 E Q and we shall prove the estimate for z = 0. To proceed, given v, j for which Q~·j is nonempty, we choose 17~,j E Q~·j and set

c~·j (z)

=I

ei[
Note that

I

~~ { ei[
(7.2.6)

So, if we let

it follows that

c~.i(z) =

L

zic~·.{(o) + RM(z),

lii~M(N)

where, if M is large enough and if z E Q (and hence lzl = O(A- 1/ 2-c)),

IRM(z)l :5 CNA-NIIfllq·

I

218

7. Local Smoothing of Fourier Integral Operators Notice that :F>.f(z) = Ev,j eicp(z,f1~';)c~'j (z). Therefore, if we set

~;~f(z) =

A-lll/2 ~I ei[cp(z,f1)-cp(z,f1~';)] a~,j (z, 17) /(77) d7J,

then since lziiAill/ 2 :5 1, the left side of (7.2.4') is dominated by CA-NIIfllq plus

L

IlL eicp(z,f1~';) ~~f(O)IILq

lli~M v,j

Since 117 - 17~,; I :5 C A112 for 17

· (Q)

E

supp17 a~,j it is clear that the symbol

of~;~ satisfies (7.2.5). Therefore, by the last majorization, we would be done if we had the following discrete version of the £ 2 -+ Lq local smoothing theorem.

Lemma 7 .2.3: Let Q and 17~,; be as above. Then if Q is contained in a small relatively compact neighborhood of supp17 a>. (so that cp is well defined}

ll~eicp(z,f1~·;>cv,;IILq(Q) :5 CA-~ An-~(~ 1~,;12f/2. V 1J

V 1J

1

Proof: As before we""assume 0 E Q. If we then normalize the phase function by setting

t/>(z, 17) = cp(z, 17) - cp(O, 17), then the desired estimate is equivalent to the following: s 1 1 1 " ei~(z,f1~';)~,jll < CA-~ An(2"-q>-q lcv,j12)1/2. (7.2.7) II ~ ~~.

(L

V 1J

V 1J

To see that this is a corollary of the £ 2 -+ Lq local smoothing theorem, we let Q~,j be the cube of side-length A1/ 2 in IR.n that is centered at 17~,;. If A is sufficiently large and if we let

bv,j(z,7J) = { { . ei[~(z,f1)-~(Z,f1~';)] d7J}-1. X v,;(7J), >. JQ~·' Q,. it follows that, for z E Q,

lb~,j (z, 17)1 :5 CIQ~,j 1- 1 = CA -n/2 • To see this, note that t/>(z, 17) - t/>(z, 17~,;) = 0 when 17 = 17~,; and V 17 ¢(z,7J) = O(A- 112-e) for z in Q. Hence, lt/>(z,7J)- t/>(z,7J~,j)l < ! for large A, giving the estimate. Similar considerations show that

~~b~'i(z,7J)I :5 CaA-n/ 2 Alal/ 2,

z E Q.

(7.2.8)

21

7.2. Local Smoothing in Higher Dimensions These estimates are relevant since the quantity inside the Lq norm in (7.2.7) equals

J

eicf>(z,TJ)

2; b~,j (z, 17) ~,j d17. V,]

So, if we write

.

b~'1 (z, 17)

a b~J.(u1, 0, ... , 0, 17) du1 = b~·1.(0, 17) + loz' -a 0

+ .. ·+

U1

rl ... rn+l _!!_.,,_a_b~·j(u,17)du,

lo

lo

aul

aun+l

then it follows that the left side of (7.2.7) is dominated by

+

I ... I

lukl~.>.-l-•

II/ eicf>(z,TJ) ""_!!__ ~ au1

... _a_bv,j (u, 17)cv,j d7711 8un+l .>.

Lq(Q)

du.

v,J

Since b~,j vanishes unless 1171 ~A, Theorem 7.2.1 and Plancherel's theorem imply that this is majorized by

+An(i-~)-~

/···/

lukl9-l-•

ll"'_!!_ ... aun+l ~ au1

_a_bv,j(u,17)cvJII .>.

£2(dTJ)

du.

v,J

However, using (7.2.8) and the fact that the sets Q~·j have finite overlap shows that this in turn is

~cAn<~-~)-~ A-~ (L ~~,ji21Q~·jl )1/2 vj

1 1

1

= C'A -~ An(:1-q)-q (L lcv,jl2 )1/2, v,j as desired.

I

End of proof of Theorem 7.1.1: Returning to (7.2.1), let us fix a lattice of cubes in Rn+l of side-length A- 1/ 2-c. We theli consider the

7. Local Smoothing of Fourier Integml Opemtors

220

special case of (7.2.4) corresponding to q = 4. If we raise this inequality to the fourth power and then sum over the cubes which intersect suppza~, we conclude that, for large enough M, II.1"~/IIL4 (Rn+l)

+ c - n n-1 """' ~ CA !!±l -g- 4A 4 A""4 ~ lti:5M = CA(n- 1)/S+c/ 4

II ("""' ....v j 2) 1/211 £4(Rn+l) + CII/II£4(Rn) ~ ~J'"~:~dl v,j

L II(L ~~~/1 2 ) 112 11£4(Rn+1) + CII/IIL4(Rn)• lli:5M

v,j

Next, if p is as above, then we define

/~('1) = p(A- 1/ 2'71- JLI) · • • p(A- 1/ 2'7n- JLn)i('1)· As before, L:I'Ezn '" = /, and, if for a given v,j we let r;..·j those JL for which suppp(A- 1/ 271- JL) n Q~,j =F 0, we have

c

zn be

CardT;..'j ~ C. Conversely, we also have, for a given fixed JL, Card { (v, j) : JL E T;..,j} ~ C. The final ingredient in the proof is that if

K~~(z, y) =

J

ei[.p(z,7J)-{y,7J))

a~·.{(z, 71) d71

denotes the kernel of ~1. then, since the symbols satisfy (7.2.5), the proof of Lemma 7.1.3 giv'es

IK~:~(z, y)l ~eN An/ 2 (1 + A112 dist (z, 'Yy,1j~.; )) -N,

(7.2.9)

where 'Yy,11 is given by (7.1.30). In particular, J IK~·~(z, y)l dy ~ C uniformly. ' If we put these observations together, we see that, if g(z) ~ 0, Ln+l

~ l~;~f(z)l 2 g(z) dz V,]

But II(L: l/1'1 2) 112 114 ~ Cll/114- So, if we take the supremum over all g as above satisfying 119112 = 1, we conclude that (7.2.1) would now follow

7.2. Local Smoothing in Higher Dimensions from

I

(}Rn f su~ f IKr·~(z, y)l g(z) dzl 2 dy V,J }Rn+l '

22

r

12

$ C).(n-2)/4pog..\l3/2llgii£2(Rn+l)•

Since this follows from Theorem 7.1.4, by taking 6 = >. -1/2, 2>. -1/2, ... , we are done. I Proof of Theorem 7.2.1: Let a(x, t, 17) be a symbol of order J.Lq = -n(1/2- 1/q) + 1/q which vanishes for (x, t) outside of a compact set K C Rn+l or 1171 $ 10. Then we must show that

II/ ei'l'(x,t,'l)a(x, t, 17) f(17) d1711 Lq(Rn+l) $ Cllfii£2(Rn),

2 ~~11 ) $ q

< oo,

(7.2.10)

provided that the canonical relation C associated to this Fourier integral operator satisfies (7.1.10), (7.1.211 ), and (7.1.611 ). The first step is to notice that it suffices to make dyadic estimates. Specifically, if {3 E Cgc'((l/2, 2)) is the Littlewood-Paley bump function u8ed before, and if we set a>. (x, t, 17) = /3(1171/ >.)a(x, t, 17), then we claim that (7.2.10) follows from the uniform estimates

II/ ei'l'(x,t,'l)a>.(X, t, 17)f(17) d1711 Lq(Rn+l) $ CII/II£2(Rn)•

2 ~~l) $ q $

00.

(7.2.10')

To see this, we let L; be the Littlewood-Paley operators in Rn+l defined by (L;g)"(() = /3(l(lf2i)g((). Then, since It'~ "I 0 for 17 "I 0, one sees that there must be an absolute constant Co such that, if :F>. is the dyadic operator in (7.2.101 ), then IIL;:F>.fiiLq(Rn+l) $eN 2-Nj >.-NII/II£2(Rn)

'V N,

if 2; I>.¢ [C01, Co].

(7.2.11)

To apply this, we notice that, if :F is the operator in (7.2.10), then, by Littlewood-Paley theory and the fact that q > 2, we have 00

II:F/llq $ Cqll (~= j=O

00

IL;:F/1 2) 112 llq $ Cq (~= IIL;:F/II~) 112 . j=O

Note that :F = L::F2k· So, if/; is defined by }j(17) = /3(117lf2i)f(17), we

222

7. Local Smoothing of Fourier Integral Operators can use (7.2.11) to see that, if (7.2.101) were true, then the last term would be majorized by

(~= ll/i11~) 112

00

+L

j

2-Nj

11/112 :::; Cll/112·

j=O

Since we have verified the claim we are left with proving (7.2.10'). We shall actually prove the dual version (7.2.10")

However, since

I IF~g(y)

12

dy =

I F>.F~g(x,

t) g(x, t) dxdt

:::; IIF>.J19IILP (R"+l) IIYIILP(Rn+l), 1

this in turn would follow from

IIF>..1"~giiLP 1 (Rn+l) :5 CllgiiLP(R"+l)

l

1

:5 P :5 2 ~:l)

•

(7.2.12)

Recall that .1">. is of order JLq-1/4 with JLq = -nl1/q-1/21+1/q. Thus, since the canonical relation of .r is given by (7.1.10), the composition theorem implies that F>.J-1 is a Fourier integral operator of order 2JLq1/2 with canonical relation

C 0 C* = {(x, t, ~~ T, y, S, 1J, a) : (x, ~) = Xt o x_;- 1 (y, 1J), T = q(x, t, ~),a = q(y, s, 17) }.

(7.2.13)

So, if we assume, as we may, that a(x, t, 17) is supported in a small conic set, it follows that the kernel .1">..1"). is of the form {

ei[¢(x,t,s,7J)-{y,7J)]

b>,(X, t, S, 1J) d7J,

'R"

modulo C 00 , where b>. E S2~-'Q vanishes unless 1171 ~ >. and t - s is small. Consequently, if we let O:>.(x,t,s,7J) = >.- 21J.qb>.(x,t,s,7J) E SO, then we see that (7.2.12) would be a consequence of the following estimates. I Lemma 7.2.4: Let l/J be as above and suppose that o: E SO vanishes unless and t - s is small. Then, if we define the dyadic operators g>.g(x, t)

=

III

ei[¢(x,t,s,7J)-{y,7J)]p(I7JI/ >.) o:(x, t,

s, 17) g(y, s) d7Jdyds,

7.2. Local Smoothing in Higher Dimensions

22

it follows that IIQ>.giiLP'(R"+l):::;

C>.

1 1 2 2n( .. - - ) - ~ p' p' llgiiLP(R"+l)•

1:::; p:::;

2(n+ll) --fttr.

(7.2.14)

Furthermore, the constants remain bounded if o as above belongs to a bounded subset of SO. Proof: Since the symbol of 9>. vanishes unless 1171 be O(.~n). Hence

~

>., the kernel must

ll9>.giiL""(R"+l) $ C).n llgiiLl(R"+l)·

So, if we apply theM. Riesz interpolation theorem, we find that (7.2.14) would follow from the other endpoint estimate:

ll9>.giiLP (Rn+l) $ C>.llgiiLP(R"+l)• 1

P = 2 ~:31 ) •

(7.2.141 )

To prove this we need to use (7.2.13) to read off the properties of the phase function tf>. First, since Xt o x;- 1 = Identity when t = s, it follows that (x, y, 17) -+ tf>(x, t, t, 17)- (y, 17} must parameterize the trivial Lagrangian. In other words,

t/>(x, t, s, 17) = (x, 17}

if t = s.

Also, since T = q(x, t, e) in (7.2.13), it follows that, when t q(x, t, 17). So we conclude further that

t/>(x, t, s, 17)

= s,

= (x, 17} + (t- s)q(x, s, 17) + (t- s) 2 r(x, t, y, s, 17).

4>~

=

(7.2.15)

Using this we can estimate 9>.. More precisely, we claim that if we set

Tt,sf(x)

=II

ei[4>(x,t,s,7J)-(y,7J)].B(I771/>.)o(x, t, s, 17)f(y) d17dy,

and if o is as above, then !ttl

n-1

I!Tt,sfiiL""(R") :5 C>. ---r It- si--r llfiiLl(R")·

(7.2.16)

This estimate is relevant because we also have 11Tt,sfiiL2(R")

:5 C llfiiP(R")•

since the zero order Fourier integral operators Tt,s have canonical relations which are canonical graphs. By interpolating between these two estimates we get

224

7. Local Smoothing of Fourier Integral Operators And since 1 - (1/p- 1/p') = (n - 1)/(n + 1) for this value of p, we get (7.2.141 ) by applying the Hardy-Littlewood-Sobolev inequality-or, more precisely, Proposition 0.3.6. The proof of (7.2.16) just uses stationary phase. Assuming that t-s > 0, we can make a change of variables and write the kernel of Tt,s as (t _ s)-n

·~

! e'[{ t-s ,,.,)+q(x,s,,.,)+(t-s)r(x,t,y,s,,.,)) x

.8(1771/A(t- s)) a(x, t, s, 11/(t- s)) d17.

(7.2.17)

=

To estimate the integral we first recall that, by assumption, rank q:;,., n - 1. So, if N is a small conic neighborhood of supp71 o and if t is close to s, the sets Sx,t,y,s = { 17 EN: q(x, s, 17)

+ (t- s )r(x, t, y, s, 17) = ±1}

have non-vanishing Gaussian curvature. In addition, since q =F 0 for 17 =F 0, it follows that q~ =F 0 there as well. Consequently, for t close to s x-y

V,.,[( t _ S ,7J) + q(x, S,1J) + (t- s)r(x,t,y,s,7J)] =j; 0, unless

lx- Yl ~ It- sl.

If we put these two facts together and use the polar coordinates associated to Sx,t,y,s, we conclude that Theorem 1.2.1 implies that (7.2.17) must be

Since this of course implies (7.2.16), we are done.

I

7.3. Spherical Maximal Theorems Revisited Using the local smoothing estimates in Theorem 7.1.1 we can improve many of the maximal theorems in Section 6.3. We shall deal here with smooth families of Fourier integrals :Ft E rn(X, X; Ct), t E I, having the property that if we set :Ff(x, t) = :Ftf(x), then :F E pn-l I4 (X x I, X; C), where the full canonical relation C satisfies the non-degeneracy conditions and the cone condition described in Section 7.1. Our main result then is the following. Theorem 7.3.1: Let X be a smooth n-dimensional manifold and suppose that :Ft E Im(X, X;Ct), t E [1, 2J, is a smooth family of Fourier integral operators which belongs to a bounded subset of I~mp· Suppose

7.3. Spherical Maximal Theorems Revisited

22

further that the full canonical relation C C T*(X x [1, 2]) \0 x T* X\ 0 associated to this family satisfies the non-degeneracy conditions (7.1.2)(7.1.3) and the cone condition (7.1.6). Then, if e(p) is as in Theorem 7.1.1 and 2 < p < oo, II sup IFtf(x)lllv•(X) S: Cm,p llfiiLP(X)• tE[l,2]

+ e(p).

if m < -(n -1)(!- ~)- ~

(7.3.1)

If we define .rk,t as in the proof of Theorem 6.3.1, then Theorem 7.1.1 yields

(1 /)(ft)i.rk,tf(x)lp dxdtr'p 2

S: Ce 2 kj 2k[m+(n-l)(l/2-l/p)-e]llfllp,

E

< e(p).

By substituting this into the proof of Theorem 6.3.1, we get (7.3.1). Remark. A reasonable conjecture would be that for p ~ 2n/(n-1) the mapping properties of the maximal operators associated to the family of operators should be essentially the same as the mapping properties of the individual operators. By this we mean that for p ~ 2n/(n- 1) (7.3.1) should actually hold for all m < -(n- 1)(1/2 - 1/p). This of course would follow from showing that there is local smoothing of all orders< 1/p for this range of exponents. Using Theorem 7.3.1 we can give an important extension of Corollary 6.3.2 which allows us to handle the case of n = 2 under the assumption of cinematic curvature. Specifically, if we consider averaging operators

Atf(x) =

1s.,,.

f(y) TJ(x, y) dax,t(Y),

TJ E C(f,

associated to C 00 curves Sx,t in the plane which vary smoothly with the parameters, then we have the following result. Corollary 7.3.2: Let C C T*(R.2 x [1, 2]) \0 x T*R. 2 \0 be the conormal bundle of the C 00 hypersurface

S

= {(x,t,y): y E Sx,t} C

(R.2

X

[1,2])

X

R.2 .

Then if C satisfies the non-degeneracy condition (7.1.2) and the cone condition (7.1.6)

II tE[1,2] sup IAtf(x)IIILP(R2) S: Cp llfiiLP(R2)•

P

> 2.

(7.3.2)

226

7. Local Smoothing of Fourier Integml Opemtors Note that, as we saw in Section 6.3, At is a Fourier integral operator of order-!. Also, when n = 2, -(n-1)(!-1/p)-1/p =-!.So (7.3.2) follows from Theorem 7.3.1 since, if we set Ff(x, t) = Atf(x), then F is a conormal operator whose canonical relation is the conormal bundle of S. It is not hard to adapt the counterexample that was used to show that the circular maximal operator can never be bounded on .LP(R.2 ) when p $ 2 and see that the same applies to (7.3.2). Also, as we pointed out in Section 6.3, the Kakeya set precludes the possibility of Corollary 7.3.2 holding without the assumption of cinematic curvature. More precisely, we saw that (7.3.2) cannot hold for any finite exponents if Sx,t = {y : (x, y} = t}. ButS= {(x, t, y): (x, y} = t} is a subspace of R.5 and hence the cone condition cannot hold for the rotating lines operators, since, if C is the conormal bundle of S, then the images of the projection of C onto the fibers of T*(R.2 x R.) are just subspaces, meaning that the cones in (7.1.6) are just linear subspaces which of course cannot have any non-vanishing principal curvatures. We can also estimate maximal theorems corresponding to curves in R.2 which shrink to a point. Specifically, we now let Sx,t

=X

+ t Sx,t,

where Sx,t are C 00 curves depending smoothly on (x, t) E R.2 x [0, 1J. If we define new averaging operators by setting Atf(x) =

~-

f(x- ty) dux,t(y),

S,.,,

where dux,t denotes Lebesgue measure on Sx,t, then we have the following result. Corollary 7.3.3: Let C C T*(R.2 x (0, 1J) \0 x T*R.2 \0 be the conormal bundle of S = {(x,t,y): Y E Sx,t,t

> 0}

C (R. 2 X

(0, 1J) X R.2 .

Assume thatC satisfies the non-degeneracy condition (7.1.2) and the cone condition (7.1.6). Then, if the initial curves Sx,O have non-vanishing curvature,

II sup 1Atf(x)IIILP(R2) O
$ Cp

llfiiLP(R2)•

P > 2.

We single out an important special case which is the Riemannian version of the circular maximal theorem for R.2:

22

Notes Corollary 7.3.4: Let (M, g) be a two-dimensional compact Riemannian manifold with injectivity radius T. Then, if for 0 < t < T, Atf(x) denotes the mean value off over the geodesic circle of radius t around x,

II sup IAtf(x)IIILP(M) S: Cp,T0 11/IILP(M)• O
P > 2, 0
< T.

To see this follows from Corollary 7.3.3 we first note that, in local coordinates,

Sx,O = {y: Lgjk(X)YjYk = 1}. The other condition is satisfied, since, in the Riemannian case, the full canonical relation associated to the geodesic spherical means is the same as the relation associated to the Cauchy problem for the wave equation, that is, C=

{(x,t,~,r,y,TJ): (x,~) =

Xt(Y,TJ),

T

=

±VLgik(x)~j~k},

where Xt : T* M \ 0 - T* M \ 0 is the canonical transformation given by flowing for time t along the Hamilton vector field associated to

/•£gik(x)~j~k· Notes Local smoothing estimates for dispersive operators were first obtained by Kato (1], Sjolin (1], and Vega (1]. Constantin and Saut (1] later came up with an independent proof of the local smoothing estimates for Schrooinger's equation. For a typical application of local smoothing estimates see Journe, Soffer, and Sogge (1]. The role of cinematic curvature in problems in Fourier analysis was discussed in Sogge (6]. Here the first lJl - lJl local smoothing estimates for hyperbolic operators were obtained and the maximal theorems for variable coefficient curves were proved. Later Mockenhaupt, Seeger, and Sogge (1], [2] extended these results to all dimensions and simplified the proofs. The conormality assumption in the maximal theorem for Fourier integral operators was also removed. The proofs given here are slight modifications of those in the latter paper. Kapitanskii (1] independently obtained the £ 2 - Lq estimates in Theorem 7.2.1 for operators that arise in the solution of hyperbolic equations; for applications, see Kapitanskii (2].

Appendix: Lagrangian Subspaces of T* lRn

Here we shall prove Lemma 6.1.3. That is, we shall show that, i/Vo and V1 are two Lagmngian subspaces of T*R.n, then there must be a third Lagrangian subspace V which is transverse to both. To do this we need to introduce some terminology. We shall call 2n linearly independent unit vectors {e;}j=l• {e;}j=l a symplectic basis if, for all j, k = 1, ... n,

Here u is the symplectic form on T*R.n. The standard symplectic basis of course is when ej and Ej are the unit vectors along the Xj and ej axes, respectively. The main step in the proof of Lemma 6.1.3 is to show that a partial symplectic basis can always be extended to a full symplectic basis. Proposition A.l: Suppose that {e;}j,!, 1 and {e;}j! 1 are linearly independent unit vectors satisfying (A.1). Then one can choose additional unit vectors so that e1, ... , en, EI, ... , En becomes a full symplectic basis. Proof: Let us first assume that n1 < n2. We then claim that we can choose a unit vector en 1 +I so that {e;}~ 1 +1, {e;}~ 2 satisfy (A.1) and are linearly independent. To do this we notice that, since u is nondegenerate, we can always choose a unit vector e so that

u(e,e;) = 0, 1 $ j $ n1

and u(Ek,e) = 6k,n 1 +b 1 $ k $ n2.

Furthermore, if

then taking the u product with cn 1 +I shows that x = 0. Hence, setting en 1 +I = e gives us the claim.

Since the same argument applies to the case where n2 < n1 we may assume that n1 = n2. If then n1 = n we are done, otherwise we can argue as above to see that we can choose a unit vector e so that u(e,e;) = u(e,e-k) = 0, 1$ j,k::; n1. From this, we conclude as before that {e;} ~ 1 +I, { c;} ~ 1 must be a partial symplectic basis if we now set

Appendix: Lagrangian Subspaces of T*.Rn en1 +1 = e. Since we can next add a unit vector en 1 +1 to our partial symplectic basis, we complete the proof by induction. I Proof of Lemma 6.1.3: It suffices to show that we can choose a symplectic basis e1, ... , en, ct. ... , en of T*IRn so that Vo is spanned by ct. ... , en and V1 is spanned by ek+lo ... , en, ct. ... , ek· For we can then take V to be the subspace

v=

{Lxiei

+ I:~jej

=

with

x" = ~".

e = o,

x' = (xi. ... , xk), x" = (xk+Io ... , xn)}.

To verify this assertion we first choose a basis et, ... ek for the subspace Von V1 and extend it to a full basis et, ... , en of Vo. If we restrict u to Vo x Vt, we see that it gives a duality between Vo/(Vo n VI) and VI/ (Vo n VI). This is because both Vo and V1 are Lagrangian and hence V;l. = V;, j = 0, 1. On account of this we can choose ek+l, ... , en E V1 so that u(ei, ej) = 6i,j if k + 1 ~ i, j ~ n. If we then extend the partial symplectic basis {ej}J=l• {ej}J=k+l to a full symplectic basis, using I Proposition A.l, we are done.

229

Bibliography

Arnold, V.I. [1] Mathematical methods of classical mechanics, Springer-Verlag, Berlin, 1978. Avakumovic, V. G. [1] Uber die Eigenfunktionen auf geschlossenen Riemannschen Mannigfaltigkeiten, Math. Z. 65 (1956), 327-44. Beals, M. [1] £P boundedness of Fourier integra~ Mem. Amer. Math. Soc. 264 (1982). Beals, M., Fefferman, C., and Grossman, R. [1] Strictly pseudoconvex domains in en I Amer. Math. Soc. 8 (1983), 125-322. Berard, P. [1] Riesz means on Riemannian manifolds, Proc. Symp. Pure Math. XXXVI, Amer. Math. Soc., Providence, Rl, 1980, pp. 1-12. Besse, A. [1] Manifolds all of whose geodesics are closed, Springer-Verlag, Berlin, 1978. Bonami, A., and Clerc J. L. [1] Sommes de 'Cesaro et multiplicateures des developpements en harmonics sphirfques, Trans. Amer. Math. Soc. 183 (1973), 223--63. Bourgain, J. [1] Averages in the plane over convex curves and maximal operators, J. Analyse Math. 47 (1986), 69--85. [2] Besicovitch type maximal operators and applications to Fourier analysis, Geometric and Funct. Anal. 1 (1991), 147-87. [3] £P estimates for oseillatory integrals in several variables, Geometric and FUnct. Anal. 1 (1991), 321-74. Calderon A. P. [1] Uniqueness in the Cauchy problem for partial differential equations, Amer. J. Math. 80 (1958), 16-36. Calderon, A. P., and Zygmund, A. [1] On the existence of certain singular integral operators, Acta Math. 88 (1952), 85-139. Carbery, A. [1] The boundedness of the maximal Bochner-Riesz operator on L4 (lt2 ), Duke Math. J. 50 (1983), 409--16. Carleson, L., and Sjolin, P. [1] Oscillatory integrals and a multiplier problem for the disk, Studia Math. 44 (1972), 287-99. Christ, F. M. [1] On the almost everywhere converyence of Bochner-Riesz means in higher dimensions, Proc. Amer. Math. Soc. 95 (1985), 16-20. Christ, F. M., and Sogge, C. D. [1] The weak type L 1 converyence of eigenfunction expansions for pseudodifferential operators, Invent. Math. 94 (1988), 421-53.

Bibliogmphy Coifman, R., and Weiss, G. [1] Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), 569--645. Colin de Verdiere, Y., and Frisch, M. [1] Regularite Lipschitzienne et solutions de l'eqv.ation des ondes sur une variete Riemannienne compacte, Ann. scient. Ecole Norm. Sup. 9 (1976), 539--65. Constantin, P., and Saut J. [1] Local smoothing properties of dispersive equations, J. Amer. Math. Soc. 1 (1988), 413-46. Cordoba, A. [1] A note on Bochner-Riesz operators, Duke Math. J. 46 (1979), 505-11. [2] Geometric Fourier analysis, Ann. Inst. Fourier 32 (1982), 215-26. Davies, E. B. [1] Heat kernels and spectral theory, Cambridge Univ. Press, Cambridge, 1989. [2] Spectral properties of compact manifolds and changes of metric, Amer. J. Math. 21 (1990), 15-39. Duistermaat, J. J. [1] Fourier integral operators, Courant Institute of Mathematical Sciences, New York, 1973. Duistermaat, J. J., and Guillemin, V. W. [1] The spectrum of positive elliptic operators and periodic bicharacteristics, Invent. Math. 29 (1975), 39-79. Egorov, Y. P. [1] The canonical transformations of pseudo-differential operators (Russian), Uspehi Mat. Nauk 24,5 (149) (1969), 235-6. Eskin, G. I. [1] Degenerate elliptic pseudo-differential operators of principal type (Russian), Mat. Sbornik 82 (124} (1970), 585-628; English translation, Math. USSR Sbornik 11 (1970), 539-82. Falconer, K. J. [1] The geometry of fmctal sets, Cambridge Univ. Press, Cambridge, 1985. Fefferman, C. [1] Inequalities for strongly singular convolution operators, Acta Math. 124 (1970), 9-36. [2] The multiplier problem for the ball, Annals of Math. 94 (1971), 330-6. [3] A note on spherical summation multipliers, Israel J. Math. 15 (1973), 44-52. Fefferman, C., and Stein, E. M. [1] HP spaces of several variables, Acta Math. 129 (1972), 137-93. Garcia-Cuerva, J., and Rubio de Francia, J. L. [1] Weighted norm inequalities and related topics, North-Holland, Amsterdam 1985. Gelfand, I. M. and Shilov, G. E. [1] Generalized functions. Volume 1: Properties and operations, Academic Press, New York, 1964. Greenleaf, A. [1] Principal curvature and harmonic analysis, Indiana Math. J. 30 (1982), 519-37.

231

232

Bibliography Greenleaf, A., and Uhlmann, G. (1] Estimates for singular Radon transforms and pseudo-differential operators with singular symbols, J. Funct. Anal. 89 (1990), 202-32. Grieser, D. (1] LP bounds for eigenfunctions and spectral projections of the Laplacian near a concave boundary. Thesis, UCLA, 1992. Guillemin, V., and Sternberg, S. (1] Geometric asymptotics, Amer. Math. Soc. Surveys, Providence, RI, 1977. Hadamard, J. (1] Lectures on Cauchy's problem in linear partial differential equations, Yale Univ. Press, New Haven, CT 1923. Hardy, G. H., and Littlewood J. E. (1] Some properties of fractional integrals, I, Math. Z. 27 (1928), 565606. Hlawka, E. (1] Uber Integrale auf konvexen Kiirpern I, Monatsh. Math. 54 {1950), 1-36. Hormander, L. (1] Estimates for translation invariant operators in LP spaces, Acta Math. 104 (1960), 93-140. (2] Pseudo-differential operators, Comm. Pure and Appl. Math. 18 (1965), 501-17. (3] On the Riesz means of spectral functions and eigenfunction expansions for elliptic differential operators, Some recent advances in the basic sciences, Yeshiva Univ., New York, 1966. (4] The spectral function of an elliptic operator, Acta Math. 121 (1968), 193-218. (5] Fourier integral operators I, Acta Math. 127 (1971), 79-183. (6] Oscillatory integrals and multipliers on FLP, Ark. Mat. 11 (1971), 1-11. (7] The analysis of linear partial differential operators Vols. I-IV, SpringerVerlag, Berlin, 1983, 1985. (8] Uniqueness theorems for second order elliptic differential equations, Comm. Partial Differential Equations 8 (1983), 21-64. Jerison, D. [1] Carleman inequalities for the Dirac and Laplace operators and unique continuation, Adv. Math. 63 (1986), 118-34. Jerison, D., and Kenig, C. E. (1] Unique continuation and absence of positive eigenvalues for Schriidinger operators, Annals of Math. 121 (1985), 463-94. John, F. (1] Plane waves and spherical means applied to differential equations, Interscience, New York, 1955. Journe, J.-L., Soffer, A., and Sogge, C. D. (1] Decay estimates for Schriidinger operators, Comm. Pure Appl. Math. 44 (1991), 573-604. Kaneko, M., and Sunouchi, G. [1] On the Littlewood-Paley and Marcinkiewicz functions in higher dimensions, Tohoku Math. J. 37 (1985), 343-{)5.

Bibliography Kapitanskii, L. V. (1] Norm estimates in Besov and Lizorkin-1Hebel spaces for the solutions of second-oder linear hyperbolic equations, J. Soviet Math. 56 (1991), 2347-2389. (2] The Cauchy problem for semilinear wave equations (to appear). Kato, T. (1] On the Cauchy problem for the (generalized} Kortweg-de Vries equation, Studies in Applied Math., vol. 8, Academic Press, 1983, pp. 93128. Kenig, C. E., Ruiz, A., and Sogge, C. D. (1] Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators, Duke Math. J. 55 {1987), 329-48. Kenig, C. E., Stanton, R., and Tomas, P. [1] Divergence of eigenfunction expansions, J. FUnct. Anal. 46 (1982), 28-44. Kohn, J. J., and Nirenberg, L. (1] On the algebra of pseudo-differential operators, Comm. Pure and Appl. Math. 18 (1965), 269-305. Lax, P. D. (1] On Cauchy's problem for hyperbolic equations and the differentiability of solutions of elliptic equations, Comm. Pure Appl. Math. 8 (1955), 615-33. (2] Asymptotic solutions of oscillatory intitial value problems, Duke Math. J. 24 (1957), 627-46. Levitan, B. M. (1] On the asymptotic behavior of the spectral function of a self-adjoint differential equation of second order, lsv. Akad. Nauk SSSR Ser. Mat. 16 (1952), 325-52. Littman, W. [1J LP --> Lq estimates for singular integral operators, Proc. Symp. Pure and Appl. Math. Amer. Math. Soc. 23 {1973), 479-81. Marcinkiewicz, J. (1] Sur l'interpolation d'operations, C. R. Acad. Sci. 208 (1939), 1272-3. (2] Surles muliplicateurs des series de Fourier, Studia Math. 8 (1939), 78-91. Maslov, V. P. (1] Thiorie des perturbations et methodes asymptotiques, French translation, Dunod, Paris, 1972. Mihlin, S. G. (1] Multidimensional Singular Integrals and Integral Equations, lnternat. Series of Monographs in Pure and Applied Math., Vol 83, Pergamon Press, Elmsford, NY, 1965. Miyachi, A. (1] On some estimates for the wave equation in LP and HP, J. Fac. Sci. Tokyo 27 {1980), 331-54. Mockenhaupt, G., Seeger, A., and Sogge, C. D. (1] Wave front sets, local smoothing and Bourgain's circular maximal theorem, Annals Math. 136 (1992), 207-18. [2] Local smoothing of Fourier integral operators and Carleson-Sjolin estimates, J. Amer. Math. Soc. 6 (1993), to appear. 0

233

234

Bibliography Oberlin, D. [1] Convolution estimates for some distributions with singularities on the light cone, Duke Math. J. 59 (1989), 747-58. Oberlin, D., and Stein, E. M. [1] Mapping properties of the Radon transform, Indiana Math. J. 31 {1982), 641-50. Pan, Y., and Sogge, C. D. [1] Oscillatory integrnls associated to folding canonical relations, Colloquium Mathematicum 60 (1990), 413-19. Peral, J. [1] £P estimates for the wave equation, J. Funct. Anal. 36 (1980), 114--45. Phong, D. H., and Stein, E. M. [1] Hilbert integrals, singular integrnls and Radon transforms I, Acta Math. 157 (1986), 99-157. Riesz, M. [1] Surles maxima des formes bilineaires et sur les fonctionnelles lineaires, Acta Math. 49 (1926), 465-97. [2] L 'integrale de Reimann-Liouville et le probleme de Cauchy, Acta Math. 81 (1949), 1-223. Sato, M. [1] Hyperfunctions and partial differential equations, Proc. Int. Conf. on Funct. Anal. and Rei. Topics, Tokyo Univ. Press (1969), 91--4. Sawyer, E. [1] Unique continuation for Schrodinger operators in dimensions three or less, Ann. Inst. Fourier (Grenoble) 33 (1984), 189-200. Seeger, A., and Sogge, C. D. [1] On the boundedness of functions of (pseudo)-differential operators on compact manifolds, Duke Math. J. 59 (1989). 709-36. [2] Bounds for eigenfunctions of differential operators, Indiana Math. J. 38 (1989), 669-82. Seeger, A., Sogge, C. D., and Stein, E. M. [1] Regularity properties of Fourier integral operators, Annals of Math. 134 (1991), 231-51. Seeley, R. T. [1] Singular integrals and boundary value problems, Amer. J. Math. 88 (1966), 781-809. [2] Complex powers of an elliptic operator, Proc. Symp. Pure Math. 10 (1968), 288-307. Sjolin, P. [1] Regularity of solutions to the Schrodinger equation, Duke Math. J. 55 (1987), 699-715. Sobolev, S. L. [1] Sur un theorem d'analyse fonctionnelle (Russian; French summary), Mat. Sb. 46 (1938). 471-97. Sogge, C. D. [1] Oscillatory integrals and spherical harmonics, Duke Math. J. 53 (1986), 43-65. [2] Concerning the LP norm of spectral clusters for second order elliptic operators on compact manifolds, J. Funct. Anal. 77 (1988), 123-34. [3] On the convergence of Riesz means on compact manifolds, Annals of Math. 126 (1987), 439--47.

Bibliography (4J Remarks on L 2 restriction theorems for Riemannian manifolds, An~ysis at Urbana 1, Cambridge Univ. Press, Cambridge, 1989, pp. 416-22. (5J Oscillatory integrals and unique continuation for second order elliptic differential equations, J. Amer. Math. Soc. 2 (1989), 489-515. (6J Propagation of singularities and maximal functions in the plane, Invent. Math. 104 (1991), 349-76.

Sogge, C. D., and Stein, E. M. [1J Averages over hypersurfaces in It'\ Invent. Math. 82 (1985), 543-56. (2J Averages over hypersurfaces II, Invent. Math. 86 (1986), 233--42. (3J Averages of functions over hypersurfaces: Smoothness of generalized Radon transforms, J. Analyse Math. 54 (1990), 165-88. Stanton, R., and Weinstein, A. (1J On the L 4 norm of spherical harmonics, Math. Proc. Camb. Phil. Soc. 89 (1981), 343-58.

Stein, E. M. (1J Interpolation of linear operators, Trans. Amer. Math. Soc. 83 (1956), 482-92. [2J Singular integrals and differentiablity properties of functions, Princeton Univ. Press, Princeton, NJ, 1970. (3J Maximal functions: spherical means, Proc. Nat. Acad. Sci. 73 (1976), 2174-5. (4J Oscillatory integrals in Fourier analysis, Beijing Lectures in Harmonic Analysis, Princeton Univ. Press, Princeton, NJ, 1986, pp. 307-56.

Stein, E. M., and Wainger, S. (1J Problems in harmonic analysis related to curvature, Bull. Amer. Math. Soc. 84 (1978), 1239-95. Stein, E. M., and Weiss, G. (1J Introduction to Fourier analysis on Euclidean spaces, Princeton Univ. Press Princeton, NJ, 1971. Sternberg, S. (1J Lectures on Differential Geometry, Chelsea, New York, 1964, 1983. Strichartz, R. (1J A priori estimates for the wave equation and some applications, J. Funct. Analysis 5 (1970), 218-35. (2J A functional calculus for elliptic pseudo-differential operators, Amer. J. Math. 94 (1972), 711-22. (3J Restriction of Fourier transform to quadratic surfaces, Duke Math. J. 44 (1977), 705-14. Sugimoto, M. (1J On some LP-estimates for hyperbolic operators, Arkiv Mat. 30 (1992), 149-63.

Taylor, M. [1J Fourier integral operators and harmonic analysis on compact manifolds, Proc. Symp. Pure Math. 35 (1979), 115-36. (2J Pseudodifferential operators, Princeton Univ. Press, Princeton, NJ, 1981.

Thorin, 0. (1J An extension of a convexity theorem due to M. Riesz, Kungl. Fys. Sallsk. Lund. Forh. 8 (1939).

235

236

Bibliography Tomas, P. (1] Restriction theorems for the Fourier transform, Proc. Symp. Pure Math. 35 (1979), 111-14. Treves, F. (1] Introduction to pseudodifferential and Fourier integral operators. Volume 1: Pseudodifferential operators. Volume 2: Fourier integral operators, Plenum Press, New York and London, 1980. Vega, L. (1] Schriidinger equations: pointwise converyence to the initial data, Proc Amer. Math. Soc. 102 (1988), 874-8. Wiener, N. (1] The eryodic theorem, Duke Math. J. 5 (1939), 1-18. Wolff, T. (1] Unique continuation for l6ul $ VIVul and related problems, Revista Math. Iber. 6 (1990), 155--200. [2] A property of measures in Rn and an application to unique continuation, Geometric and Funct. Anal. 2 (1992), 225--84. Zygmund, A. [1] On a theorem of Marcinkiewicz concerning interpolation of operators, J. Math. Pures Appl. 35 (1956), 223--48. (2] On Fourier coefficients and transforms of two variables, Studia Math 50 (1974), 189--201. (3] 7Hgonometric Series, Cambridge Univ. Press, Cambridge, 1979.

Index

amplitude, 36 bicharacteristics, 120 Calderon-Zygmund decomposition, 15 canonical one form, 38 canonical relation, 57, 168 canonical transformation, 121 Carleson-Sjolin condition, 59 n x n condition, 60 cinematic curvature, 197 circular maximal theorem, 89, 226 conormal distribution, 167 cone condition, 197 cosphere, 66, 136 cotangent bundle, 35 critical index, 66 eikonal equation, 116 equivalence of phase function theorem, 101, 164 Euler's homogeneity relations, 39 formal series, 96 Fourier integral operator, 168 Fourier transform, 1 fractional integrals, 23 half-wave operator, 74, 114 Hamilton vector field, 120 O.ow, 121 Hardy-Littlewood maximal theorem, 10 Hardy-Littlewood-Sobolev inequality, 23 Hardy space, 176 Hausdorff-Young inequality, 5 Hessian matrix, 45, 55 Holder continuity, 19 Kakeya maximal theorem 84, 208 Lagrangian distribution, 39, 161 Lagrangian submanifold, 39 Littlewood-Paley operator, 20 local smoothing, 194, 214 Marcinkiewicz interpolation theorem, 12 Morse lemma, 46

oscillatory integral operators, 55 non-degenerate, 56 oscillatory integrals, 36, 40 . parametrix, 98, 115 phase function, 36 non-degenerate, 38, 45, 56 Plancherel's theorem, 5 Poisson summation formula, 8 principal symbol, 106 pseud