(2=)-.fe Au =/~ C'~(n") FOURIER INTEGRAL OPERATORS. I

FOURIER INTEGRAL OPERATORS. I

BY

LARS H O R M A N D E R University of Lund, Sweden

Preface

Pseudo-differential operators have been developed as a tool for the s t u d y of elliptic differential equations. Suitably extended versions are also applicable to hypoelliptic equations, b u t their value is rather limited in genuinely non-elliptic problems. I n this paper we shall therefore discuss some more general classes of operators which are a d a p t e d to such applications. F o r these operators we shall develop a calculus which is almost as smooth as t h a t of pseudo-differential operators. I t also seems t h a t one gains some more insight into the theory of pseudo-differential operators b y considering t h e m f r o m the point of view of the wider classes of operators to be discussed here so we shall take the oppor- t u n i t y to include a short exposition.

Pseudo-differential operators as well as our Fourier integral operators are intended to m a k e it possible to handle differential operators with variable coefficients roughly as one would handle differential operators with constant coefficients using the Fourier trans- formation. For example, the inhomogeneous Laplace equation

is for n > 2 solved b y

Au =/~ C'~(n")

J

where

= f e -'<x'`v(x) dx

is the Fourier transform o f / . To be able to solve a r b i t r a r y elliptic equations with variable coefficients one is led to consider more general operators of the form

A/(x) =

(2=)-.fe

~ a(x, ~) f(~) d~, (0.1)

8 0 LARS ItORMANDER

where a behaves as a sum of homogeneous functions when 4-~ ~ . These are the (classical) pseudo-differential operators. On the other hand, suppose t h a t we want to solve the Cauchy problem

Au-~u/~t ~

= 0;

u = O, ~u/~t

=/EC~C(R ~) when t = 0.

Then the solution is given by

u(x~)=(2~)-`fe`~x~+~)(2i~4~)-~(4)d4-(27t)-~ fe.(~x.~-~)(2i~4])~1~(4)d4.

(0.2) Each of the terms on the right-hand side is similar to (0.1) except for the fact t h a t the func- tion (x, 4) in the exponent has been replaced by (x, 4 ) i t [4l. This is a homogeneous function of 4 with critical points as a function of 4 where

x= •

thus [ x l 2 = t ~ which is the light cone. The function (x, 4), on the other hand, has no critical point except when x = 0. These observations reflect the fact t h a t the fundamental solution of the wave equation is singular on the light cone whereas the fundamental solution of the Laplacean is singular only at the origin.

As a generalization of (0.1), (0.2) it is natural to consider operators of the form

Al(x ) = f e ~s(x' ~)a(x, 4) I(4) d4.

(0.3) L a x [21] showed t h a t for any strictly hyperbolic equation the solution of the Cauchy problem is for small values of the time variable a sum of operators of this form where S is obtained b y solving the characteristic equation with initial data

x-~(x, 4).

Related global results were proved by Ludwig [22]. A more systematic study of operators of the form (0.3) was made b y Maslov [23] under the hypothesis t h a t det S ~ : 0, and his results have subsequently been extended and applied b y E~kin [9], Egorov [7, 8] in connection with studies of non-elliptic pscudo-dfferential Operators.

Introduction of the definition of the Fourier transform in (0.3) gives formally

A/(x)= f f e'c(x'~.~) a(x, y, 4) /(y) dy d4,

(0.4) where r y, 4) =

S(x, 4) - (Y, 4),

and a is independent of y of course. Quite general operators of the form (0.4) were discussed b y the author [14] and the term Fourier integral operator was introduced for them. The purpose was a study of the asymptotic properties of the eigenfunctions of elliptic operators, which is actually a problem involving a related hyper- bolic operator. A more systematic development with applications to differential operators of principal type with real principal part was given in mimeographed lecture notes from the

F O U R I E R II~T]~GRAL O P E R A T O R S . I 81 Nordic Summer School of Mathematics 1969 (see also [15, section 5]). Originally this p a p e r was intended as a finished version of those notes b u t in fact it has been completely revised and v e r y much extended in order to take into account the v e r y i m p o r t a n t observation of Egorov [7] t h a t if A is an operator of the form (0.3) and P, Q are pseudo-differential opera- tors with P A = A Q , t h e n the principal symbols of P and Q are related b y the canonical transformation corresponding to the generating function S. Now it turns out t h a t with a n y operator of the form (0.4) where r satisfies a certain regularity condition one can also associate a canonical transformation and prove t h a t the class of operators of the form (0.4) is determined b y the canonical transformation alone. I t is t h e n possible to develop a fairly complete calculus of such operators where the result of Egorov is imbedded in a nat- ural way. As a result one can for example give a reinterpretation of the result of L a x [21]

mentioned above which is valid globally in the time variable. The results indicated in [15]

concerning operators of principal t y p e with real principal p a r t can also be made global under suitable convexity assumptions weaker t h a n those discussed in [17, Chapter V I I I ] . These applications are left for the second p a r t of the paper which is being written in colla- boration with J. J. Duistcrmaat. However, we wish to call attention to the papers of Egorov [8] and Nirenberg-Trbves [25] which use operators of the form (0.3) in a v e r y essential w a y in studies concerning existence and regularity theorems for general operators of principal type.

The work of Egorov is actually an application of ideas from Maslov [23] who stated at the International Congress in Nice t h a t his book actually contains the ideas attributed here to Egorov [7] and Arnold [1] as well as a more general and precise operator calculus t h a n ours. Since the book is highly inaccessible and does not appear to be quite rigorous we can only pass this information on to the reader, adding a reference to the explanations of Maslov's work given b y Buslaev [5]. I n this context we should also mention t h a t the

"Maslov index" which plays an essential role in Chapters I I I and I V was already con- sidered quite explicitly b y J. Keller [18]. I t expresses the classical observation in geo- metrical optics t h a t a phase shift of ~/2 takes place at a caustic. The purpose of the present paper is not to extend the more or less formal methods used in geometrical optics b u t to extract from t h e m a precise operator theory which can be applied to the theory of partial differential operators. I n fact, we only use the simplest expansions which occur in geo- metrical optics, and a wealth of other ideas remain to be investigated.

The plan of the paper is as follows. Chapter I presents generalities concerning Fourier integral operators. Actually this is mainly a more systematic version of the introductory chapter of [14]. I n Chapter I I we review the calculus of pseudo-differential operators from this more general point of view and give some applications. The kernels of pseudo-differen-

6 - 712906 A c t a mathematica 127. Imprim@ lo 1 J u i n 1971

82 LARS ~RMA~DER

tial operators are certain distributions in a product X x X with singularities only on the diagonal. With a n y manifold X and submanifold Y there is similarly associated in a natural w a y a class of distributions with singularities only on Y which is discussed at the end of Chapter I I . Actually, these distributions are connected with the normal bundle of Y in the sense t h a t they have symbols living on t h a t bundle. I n Chapter I I I we study more general classes of distributions which are associated with any conic Lagrangean submani- fold of

T*(X).

If X is replaced b y a product X x Y one can interpret these as classes of operators from functions on Y to functions on X. I n particular there is such a class of operators associated with a n y canonical diffeomorphism of

T*(Y)~O

on

T*(X)~O.

Composition of such operators corresponds to composition of the canonical transformations.

Pseudo-differential operators are obtained when X = Y and the canonical transformation is the identity. This general operator calculus contains the result of Egorov [7] referred to above and also leads immediately to estimates for the norm of the operators. I t is developed in Chapter IV.

A summary of the results of this paper has been given in [16] which can also be read as an introduction giving additional background material.

Finally I would like to t h a n k J. J. Duistermaat for m a n y discussions concerning symplectic geometry which have improved the exposition.

I. Oscillatory integrals

1.0.

Introduction

I n this chapter we shall give precise definitions of integrals of the form (0.4) and discuss some of their most elementary properties. Concerning the amplitude a in (0.4) we shall usually make essentially the same hypotheses as in earlier studies of pseudo-differential operators (see [13]). The basic facts are collected in section 1.1. I n section 1.2 we can then give a precise definition of the corresponding integrals of the form (0.4) by means of essenti- ally the same methods as in [14]. However, the hypotheses of sections 1.1 and 1.2 are some- what too special for some purposes. I n section 1.3 we shall therefore relax the conditions on the amplitude a in (0.4). Basic facts concerning operators of the form (0.4) are then given in section 1.4. Under suitable additional assumptions concerning the phase function

we shall give much more precise results in Chapters I I and IV.

For standard notation not explained in the text we refer to H6rmander [17].

1.1.

Symbols

The theory of distributions gives a meaning to the Fourier transform

F O U R I E R I N T E G R A L O P E R A T O R S . I 83

w h e n ](x)= O(]x I a) for some m as x ~ ~ . T h e d e f i n i t i o n of (0.4) is s o m e w h a t m o r e d e h c a t e so we h a v e t o i m p o s e s u i t a b l e g r o w t h c o n d i t i o n s of t h i s t y p e o n all d e r i v a t i v e s of t h e func- t i o n a(x, y, 0). T h e r e is no r e a s o n for us a t t h i s t i m e t o consider t h e v a r i a b l e s x a n d y s e p a r a t e l y , so in t h e following d e f i n i t i o n we consider c o m p l e x v a l u e d f u n c t i o n s a d e f i n e d in X • R ~ w h e r e X is a n o p e n s u b s e t of R n (we allow n to b e 0).

Definition 1.1.1. L e t m, ~, ~ b e r e a l n u m b e r s w i t h 0 < ~ < 1 , 0~<5~<1. T h e n we d e n o t e b y S~.~ ( X • R N) t h e set of all a E C ~ ( X • R y) s u c h t h a t for e v e r y c o m p a c t set K ~ X a n d all m u l t i o r d e r s a, fl t h e e s t i m a t e

ID/Do~a(x,O)l

< ~ , , , ~ ( 1 +

101)

^{x~ K, 0E R N, (1.1.1)}

is v a l i d for s o m e c o n s t a n t C~.~.K. T h e e l e m e n t s of S~.~ are called s y m b o l s of o r d e r m a n d t y p e ~, 5. I f ~ + ( ~ = 1 we also use t h e n o t a t i o n S~ n i n s t e a d of Se~t, a n d w h e n ~ = 1 , ( ~ = 0 we s o m e t i m e s w r i t e o n l y S m a n d t a l k a b o u t s y m b o l s of o r d e r m. I f (l.1.1) is o n l y v a l i d for large

IOl,

we s a y t h a t a E S~'~ for l a r g e

101.

F i n a l l y we set

T/~ m

B y t h e conic s u p p o r t of a, d e n o t e d cone s u p p a we d e n o t e t h e closure in X • t t ~ of {(x, tO);

(x, O) E s u p p a, t ~> 0}.

S o m e w h a t i n c o r r e c t l y we shall also s a y t h a t a set M c X x R ~ is conic if (x, O)EM i m p l i e s (x, tO)EM w h e n t > 0 . T h e conic s u p p o r t of a is t h u s t h e s m a l l e s t closed conic s u b s e t of X x R N such t h a t a v a n i s h e s in t h e c o m p l e m e n t .

Example 1.1.2. I f a E C ~ a n d a is a h o m o g e n e o u s f u n c t i o n of d e g r e e m w i t h r e s p e c t t o 0 for large ]0I, t h e n a is a s y m b o l of o r d e r m ( a n d t y p e 1, 0).

Example 1.1.3. I f a is s e m i - h o m o g e n e o u s in t h e sense t h a t a(x, 01t ~, .... ONt m Jr) = tma(x, 01 .... , ON)

for s o m e m j > 0 a n d m E R , a n d if aEC ~~ for 0 # 0 , t h e n a is for large 0 a s y m b o l of de- gree m a x j m/mj a n d t y p e mini. k mj/m,, O.

Example 1.1.4. I f Z E 5P(Rn), t h e S c h w a r t z space, t h e n a(x, O) = Z(x[OI ~) is in S ~ ~(R n • R N) for large 101.

84 L A R S H O R M A N D E R

E x a m p l e 1.1.5. I f 0 < t < 1, the function (x, O) ~ e x p ic(x) [0] ~-t, where c is a real v a l u e d function, is in sg.~ for large I Of if a n d only if ~ < t , ($~>l-t, or c is c o n s t a n t a n d ~ < t , 5 arbitrary, or c = 0 a n d b o t h Q a n d (~ arbitrary.

P R 0 P 0 S I T I 0 N 1.1.6. S~. o(X • l~ N) is a Fre:chet space with the topology defined by taking as seminorms the best constants C~.~,K which can be used in (1.1.1). This space increases when (5 and m increase and Q decreases. I f aCS~,~ it follows that

a(~) -- (iDo) ~ (iD~)~ a E S~.~ QI ~ I +till (fl) ^-- and i / b E S~'~ it follows that ab E ~+~"

The proof is obvious. N o t e t h a t to prove (1.1.1) for ab one needs to k n o w only t h a t (1.1.1) holds for a a n d for b when the differentiations involved are of order ~<[~ +fl]. This is i m p o r t a n t in some proofs b y induction.

I t follows i m m e d i a t e l y f r o m Definition 1.1.1 t h a t S~,~ is i n v a r i a n t for diffeomorphisms in t h e x variable, so the definition makes sense also if X is a manifold. I n order to be able to consider more general fiber spaces t h a n X • R N over X we shall need some fact con- cerning the action on symbols of more general maps. I n doing so it is useful to w o r k locally so we first e x t e n d Definition 1.1.1 somewhat. T h u s let F be an open conic set c X • R N.

I f K is a c o m p a c t subset of P we set K ~ tO); (x, O)EK, t~>l}. A function in C~(F) is n o w said to be in S~,~(F) if (1.1.1) is valid w h e n (x, 0 ) E K c for a n y choice of the c o m p a c t set K. This agrees with Definition 1.1.1 when F = X • ~. N o t e t h a t if 1 ~ does n o t m e e t X • t h e n there are no restrictions on t h e g r o w t h of the derivatives of a(x, O) w h e n 0-~0.

L e t F 1 c X • M, F 2 c Y • N be open conic sets disjoint f r o m X • a n d Y • a n d let y~: Fx-~F2 be a C ~ m a p which is positively h o m o g e n e o u s of degree one, t h a t is, com- m u t e s with multiplication b y positive scalars in the fibers of X • R M a n d Y x R N.

P R O P O S I T I O N 1.1.7. Under the preceding hypotheses we have aoyJES~,~(F1) /or all a ES~.~(F~) provided that either

(i) Q + ~ = I ; or

(ii) ~ + 8 ~> 1 and ~o is fiber preserving, that is the projection o/~v(x, O) on Y depends only on x; or

(iii) ~) and ~ are unrestricted but y) is the direct product o / a m a p Y ~ X and a homo- geneous m a p from a cone in R M to a cone in R N.

Proof. A p a r t of t h e last s t a t e m e n t was a l r e a d y pointed o u t above. I n t h e general proof we shall use t h e n o t a t i o n x, ~ for the variables in X • R M a n d y, ~ for t h e variables in

FOURIER INTEGRAL OPERATORS. I 85 Y • R N. T h e n y~(x, ~) = (y(x, ~), ~(x, ~)) where y a n d ~ are h o m o g e n e o u s of degree 0 a n d 1 w i t h respect to ~. I f K is a c o m p a c t subset of 1"~ we h~ve for some positive c o n s t a n t s

C11~1 ~ l~(x, ~)] ~ 2 1 ~ ] (1.1.2)

w h e n (x, ~) 6 K for b y hypothesis ~(x, ~) ~=0 then. Since (1.1.2) is h o m o g e n e o u s it is also valid w h e n (x, ~ ) 6 K ~. W r i t i n g b = a o F , b~=(3a/3~)o~, b~= (3a/3y~)o~f, we h a v e

~ b / ~ = Z b~y,~/~ + Y. b ~ v k / ~ . J

(1.1.3)

H e r e ~yk/3xj, 3~k/Oxj, ~yk/~j, ~ k / ~ j are h o m o g e n e o u s of degree 0, 1, - 1, 0 with respect to ~.

T h e e s t i m a t e s (1.1.1) for (x, ~ ) 6 K c n o w follow i m m e d i a t e l y w h e n l a + ~ l ~<1 if we use (1.1.2). (Note t h a t in case (ii) we h a v e ~yk/~j =0 a n d in case (iii) also a~]k/3xj = 0 . ) Assuming t h a t (1.1.1) is p r o v e d w h e n ] ~ + f l [ ~<k for a n y s y m b o l a, we conclude t h a t (1.1.1) is also valid w h e n l~ +ill =/c + 1 if we use t h e r e m a r k concerning t h e multiplieative properties of s y m b o l s m a d e i m m e d i a t e l y after P r o p o s i t i o n 1.1.6.

Remark. I f one t a k e s for a one of t h e e x a m p l e s 1.1.4 a n d 1.1.5 it is e a s y t o see t h a t t h e h y p o t h e s e s on Q a n d (~ in t h e p r o p o s i t i o n c a n n o t be i m p r o v e d ( a p a r t f r o m a case where 3~k/3xj = 0 a n d Q + ~ ~< 1 which we h a v e o m i t t e d as of no interest). T h e h y p o t h e s i s on homo- geneity could be s o m e w h a t relaxed to %0 6S~.0 a n d a n a p p r o p r i a t e s u b s t i t u t e for (1.1.2).

W e can also compose s y m b o l s t o t h e left with suitable functions:

P I ~ O P O S l T I O ~ 1.1.8. Let a I . . . a k be real valued/unctions in S~ • N) and let / be a C OO/unction in a neighborhood in R k o/all limit points o/(al(x, 0) .... , ak(x, 0)) when 0-+ oo while x may vary in X . Then it/ollows that (x, O)--+/(al(x , O) ... ak(x, 0)) is in S~.~(X • ~) /or large O.

Proo/. Choose C so large t h a t / is C ~ in a n e i g h b o r h o o d of t h e closure of {(al(x, O) ...

ak(x, 0)), x 6 X , ]0] > C ) . F o r 10] > C it is t h e n clear t h a t /(al, ..., ak) is bounded. Since k

~/(a I . . . ak)/~(x , ~) = ~ (a//~aj) (~aJ~(x, ~) ) 1

a n d ~//~aj are b o u n d e d functions, it is clear t h a t ](a 1 ... a~) satisfies t h e e s t i m a t e s (1.1.1) w h e n I ~ + ~ ] ~<1. As in t h e proof of P r o p o s i t i o n 1.1.7 it follows b y induction t h a t t h e y are valid w h e n I ~ + E l ~<?" for 1" = 1, 2 ... for we can use t h e m u l t i p l i c a t i v e properties of s y m b o l s in P r o p o s i t i o n 1.1.6 a n d t h e r e m a r k following t h a t s t a t e m e n t .

86 LARS HORMANDER

P r o p o s i t i o n 1.1.7 m a k e s i t possible t o define S~.~ (V) w h e n ~ § (~ >/1 if V is a cone bundle o v e r a m a n i f o l d X in t h e sense t h a t t h e following t h r e e c o n d i t i o n s are fulfilled:

(i) V, X a r e manifolds(1) a n d we a r e g i v e n a Coo p r o j e c t i o n p : V ~ X w i t h s u r j e c t i v e differential.

(ii) T h e r e is g i v e n a Coo a c t i o n of R+ on V w h i c h p r e s e r v e s t h e fibers.

(iii) E v e r y p o i n t in V h a s a n e i g h b o r h o o d N O i n v a r i a n t u n d e r t h e a c t i o n of R+ such t h a t t h e r e is a fiber p r e s e r v i n g d i f f e o m o r p h i s m u: No-> F c o m m u t i n g w i t h t h e g r o u p actions, w h e r e F is a n o p e n set in R n • ( R N ~ 0 ) i n v a r i a n t u n d e r t h e g r o u p a c t i o n (t, (x, 0))-~

(x, tO); t E R + , x E R n, 0 E l t N, a n d w i t h t h e p r o j e c t i o n (x, O)-+x. H e r e n = d i m X .

I f N1 a n d N~ are t w o such n e i g h b o r h o o d s w i t h d i f f e o m o r p h i s m s ~i a n d ~2, t h e n u = ~ l o ~ 1 is a d i f f e o m o r p h i s m F~l->F12 w h e r e F~I a n d F12 a r e o p e n conic s u b s e t of F z a n d F1 r e s p e c t i v e l y , a n d ~ satisfies t h e h y p o t h e s e s i n p a r t (ii) of P r o p o s i t i o n 1.1.7. Composi- t i o n w i t h x t h e r e f o r e m a p s 2~.~(F12 ) t o S~.t(F~I ) c o n t i n u o u s l y if # +(~ >/1. F o r such ~ a n d we define S~.~(V) as t h e set of f u n c t i o n s a on V for w h i c h a o u -1 is in S~.~(F) ( a n d v a n i s h e s n e a r R ~ • 0) if ~ is a local t r i v i a l i z a t i o n w i t h t h e p r o p e r t i e s l i s t e d in (iii). B y t h e p r e c e d i n g r e m a r k s i t suffices t o m a k e t h i s h y p o t h e s i s for a set of such n e i g h b o r h o o d s iVj w h i c h covers V.

L e t V~ be a cone b u n d l e o v e r X j , ] = 1, 2, a n d let v 2 be a fiber p r e s e r v i n g C ~ m a p VI-~ V2 c o m m u t i n g w i t h t h e a c t i o n of R+. I f a C S~.o(V~), p + 8 / > 1, i t follows f r o m p a r t (ii) of P r o p o s i t i o n 1.1.7 t h a t a o ~ E S~.s(V1). I f ~ +(~ = 1, p a r t (i) of t h e s a m e r e s u l t shows t h a t t h e s a m e conclusion is v a l i d e v e n if y~ is n o t f i b e r p r e s e r v i n g .

L e t V b e a cone b u n d l e o v e r X a n d l e t X b e a f i b e r s p a c e o v e r a n o t h e r m a n i f o l d Y so t h a t w e h a v e a Coo m a p P r : X - + Y w i t h s u r j e c t i v e d i f f e r e n t i a l . T h e n V is also a cone b u n d l e o v e r Y if we r e p l a c e t h e p r o j e c t i o n p : V - ~ X b y PrP: V--'t Y. To p r o v e t h i s i t suffices t o n o t e t h a t if U is a n o p e n set in t t n, F a n o p e n cone in R N ~ 0 , a n d p t h e p r o j e c t i o n .

V x F ~ (x, O)--->x ' = (x 1 .. . . . xv) EI~ v,

Where r ~< n, t h e n U • F is a cone b u n d l e o v e r R ~. T h i s follows f r o m t h e f a c t t h a t we h a v e t h e local h o m o g e n e o u s d i f f e o m o r p h i s m

(x, O)-+ (x', x"[OI,

0 ) c R " • ~§

w h e r e x" = (x,+l . . . . , Xn). W e d e n o t e t h i s cone b u n d l e b y Vr. T h e m a p V ~ V r ( b u t n o t t h e m a p V r -> V) is t h e n fiber p r e s e r v i n g , so if Q § ~> 1 we h a v e t h e inclusions

S~, l_q (V) c S~..~(Vy)c=S~,,~(V).

(1) By a manifold we shall always mean a paraeompact G ~ manifold.

F O U R I E R I I ~ T E G R A L O P E R A T O R S . I 87 W h e n Y is a p o i n t t h e left i n c l u s i o n b e c o m e s a n e q u a l i t y . N o t e t h a t i n t h a t case Vr is V w i t h t h e p r o j e c t i o n p f o r g o t t e n e n t i r e l y . A cone b u n d l e o v e r a p o i n t will also be called a conic mani/old in t h i s p a p e r .

W i t h V still d e n o t i n g a cone b u n d l e o v e r a m a n i f o l d X we a s s u m e t h a t we h a v e a c o m p l e x v e c t o r b u n d l e W o v e r V in w h i c h t h e r e is g i v e n a n e q u i v a r i a n t C ~ a c t i o n of It+.

(cf. A t i y a h [2].) T h u s t h e p r o j e c t i o n on V of t h e v e c t o r b u n d l e m a p t: W ~ W , t E R + , is a s s u m e d t o b e t h e a l r e a d y g i v e n m a p t: V ~ V. W e c a n n o w i n t r o d u c e in a n a t u r a l w a y a class Sq~(V, W) of sections of W o v e r V. F o r in a conic n e i g h b o r h o o d of a n y p o i n t v0E V we c a n f i n d a basis s I . . . ~N of C ~~ sections of W w h i c h are i n v a r i a n t u n d e r It+. I n fact, i t suffices t o choose t h e m on a m a n i f o l d t r a n s v e r s a l t o t h e o r b i t of It§ a t v 0 a n d e x t e n d t h e m b y h o m o g e n e i t y . T w o such bases differ b y m u l t i p l i c a t i o n w i t h a N • IV m a t r i x of f u n c t i o n s h o m o g e n e o u s of d e g r e e 0 in a conic n e i g h b o r h o o d of v 0. A s e c t i o n s of W o v e r V is n o w s a i d t o b e in S ~ ( V , W) if for t h e local r e p r e s e n t a t i o n s = Z a~sj w i t h such b a s e s we h a v e a j E S~.~ i n a conic n e i g h b o r h o o d of v 0. I t is clear t h a t t h e d e f i n i t i o n does n o t d e p e n d o n t h e choice of bases.

All r e m a r k s m a d e p r e v i o u s l y c o n c e r n i n g t h e b e h a v i o r of SQm~ u n d e r m a p p i n g s c a r r y o v e r w i t h o b v i o u s m o d i f i c a t i o n s t o S~.~(V, W). W e l e a v e t h e s t a t e m e n t s for t h e r e a d e r in o r d e r n o t t o b u r d e n t h e e x p o s i t i o n f u r t h e r . I n t h e following r e s u l t s we o n l y consider s y m - b o l s i n X • It~ for s i m p l i c i t y in t h e s t a t e m e n t s b u t i t s h o u l d b e clear t h a t t h e y c a r r y o v e r e a s i l y t o s y m b o l s in t h e spaces S~.o(V, W).

N o w we recall a n e l e m e n t a r y b u t i m p o r t a n t c o m p l e t e n e s s p r o p e r t y of t h e s p a c e of s y m b o l s , p r o v e d for e x a m p l e i n [13, T h e o r e m 2.7].

P R O P O S I T I O N 1.1.9. Let aj~ S~.~(X • Itr~), ~ = 0 , 1, 2 . . . . and assume that mj--> - c~ as

~ oo. Set m'~ =maxj~>kmj. T h e n one can lind aE S~.:~(X • R N) such t h a t / o r every k

a - a j e • Its). (1.1.4)

J < k

T h e / u n c t i o n a is uniquely determined modulo S - ~ ( X • R N) and has the same property relative to a n y rearrangement o/ the series F~ aj. We write a,,~ ~ a r

T h e c o n d i t i o n (1.1.4) i n v o l v e s b o u n d s on all d e r i v a t i v e s of t h e f u n c t i o n on t h e left.

I n o r d e r t o s i m p l i f y a v e r i f i c a t i o n of (1.1.4) i t is t h e r e f o r e useful t o h a v e t h e following r e s u l t , w h i c h is T h e o r e m 2.9 in [13].

P R O P O S I T I O N 1.1.10. Let aj E S~,o ( X • Itn), j = 0, 1, 2 . . . . and assume that m r + - oo

88 LARS It6RMANDEI~

when ] ~ c~ . Let a 6 C~r ( X • R N) and assume that /or all multiorders ~, fl and compact sets K c X we h a v e / o r some C and # depending on ~, fl and K

la(~,(~,o)l<C(l +]Ol)., x6K.

(~)

I / t h e r e exist numbers/~k + - ~ such that/or arbitrary K and k ]a(x,O)--~aj(x,O)[<~CK.k(I +[O[) € x 6 K ,

t < k

it ]ollows that a 6 S~ n ~ (X • R N) where m = supj ms, and that a ,,, ~ aj.

Finally we shall make some remarks on the topology of the Fr4chet space S~,~ (X • RN).

Recall t h a t a set M c S~,~ is bounded if (1.1.1) is valid with C~,~.K independent of a when a 6M. On a bounded set in S~.~ the topology of pointwise convergence, the topology of C ~ (X • R N) and the toplogy of S~,~ (X • RN), m ' > m, all coincide. This is an immediate consequence of Ascoli's theorem.

P ~ O P O S I T I O ~ 1.1.11. Let a 6 S ~ . ~ ( X x R N) and let g 6 5Z(R N) be equal to 1 at O. I ] a~ (x, O) = )~(eO) a(x, 0), i t / o U o w s that a~ 6 S~.7 ( X • RN) and that a~ ~ a in S~'~ ( X • R N) when

~-->0 i / m ' > m .

Proo/. I t suffices to note t h a t the functions (x, O)->X(80 ) form a bounded set in S~

when 0 ~<8 ~< 1 (see Example 1.1.4), for the continuity of multiplication of symbols then shows t h a t the functions a~ form a bounded set too.

I n particular, we can take • with compact support. Then we obtain

CO~OLZAI~Y 1.1.12. Let L be a linear m a p ]rom ]unctions in C~176 > R N) v a n i s h i n g / o r large [0] to a Frdchet space F such that, /or every m 6 R, the m a p L is c o n t i n u o u s / o r the topology induced by S~.o(X • RN). Then there is a unique extension o / L to S~,a(X • R N) which is continuous on S~,~ ( X • R N) /or every m.

1.2. Oscillatory integrals

We shall now discuss the definition of integrals of the form

I (au) : f fe' "O aIx, o) uIx) x o, u (x),

^(1.2.1)

where a 6 S ~ ( X • From now on we assume t h a t Q > 0 and t h a t ~ < 1 . For the sake of simplicity it will be assumed t h a t ~ is real valued and positively homogeneous of degree 1

F O U R I E R I N T E G R A L O P E R A T O R S . I 89 with respect to 0, and t h a t r E C ~ for 0 # 0 . However, this hypothesis could easily be relaxed (see also [14] for a somewhat weaker hypothesis).

The integral (1.2.1) is absolutely convergent for every a E S ~ (X • R N) provided t h a t m + N < 0. I n particular, it is well defined if a(x, 0) = 0 for large [ 0 ]. We wish to extend the definition of (1.2.1) to a r b i t r a r y a E S~.o using Corollary 1.1.12. This is not always possible - - f o r example it cannot be done if r vanishes in an open s e t - - b u t we shall prove t h a t the definition of (1.2.1) is always possible if r has no critical point with 0 # 0 . The proof depends on partial integrations in (1.2.1). I n order to avoid having to split (1.2.1) into a sum of t e r m s where integration b y parts with respect to a fixed variable will do, it is convenient to use the following

L ] ~ x 1.2.1. I f r has no critical point (x, 0) with 0 # 0 , then one can find a first order differential operator

L = Z aj~/~Oj + Z bj~/~xj + c

with ar E S ~ • R N) and be, c E S - I ( X >( R N) 8uch that tLe~r = e ~r if tL is the ad]oint of L.

Proof. B y hypothesis the sum

I o I ~ y~ (~r ~ + 2 (ar ~

is homogeneous of degree 2 with respect to 0 and # 0 for 0 # 0 . L e t ~ be the reciprocal of this sum which is then homogeneous of degree - 2 and Coo for 0 # 0 . With zEC~C(R N) chosen so t h a t Z = 1 near 0, we set

where a~ = - i ( 1 -Z)~0[0128r ES 0, b; = - i ( 1 -X)~f~r -1.

The coefficients are chosen so t h a t M e ir = e ~, so L = t M has the required properties since

! ! ! ! 1

a j = - - a j , b j = - b r c = z - ~ a r The l e m m a is proved.

I f a vanishes for large b y tLdr This gives

10], we can integrate b y parts in (1.2.1) after replacing e ~r

I~(au)=ffe~*(x'~

or after iteration

Ir162176

k = 0 , 1 , 2 . . . (1.2.2) Now L is a continuous m a p of S~m~ into S~m~ t if t = m i n ( Q , 1 - ~ ) . Hence L k m a p s Sem.~

90 L A R S H O R M A N D E R

c o n t i n u o u s l y into

S~.~ kt.

I f

m - k t < - N ,

t h e integral (1.2.2) is t h u s defined a n d con- tinuous on all of

S~,~(X •

RN). I n view of Corollary 1.1.12 we h a v e therefore p r o v e d

P R 0 P 0 S I T I 0 N 1.2.2. I[ ~ has no critical points and

Q > 0, ~ < 1,

then the de/inition o/the integral

(1.2.1)

can be extended in one and only one way to all

a E S~.~ (X x R N) a n d u E Cff (X)

so that I~(au) is a continuous /unction o/ aES~.~ /or every fixed m. The linear /orm A:

u~Ir is a distribution o/ order <~ k i/aES~,~ and m - k ~ < - N , m - k ( 1 - ( ~ ) < - h

r.

F o r t h e e x t e n d e d f o r m I r we h a v e t h e r e p r e s e n t a t i o n (1.2.2) if k is sufficiently large.

According to Proposition 1.1.11 we also h a v e

I~ (au)

= lim~_~0

f f e~r ~ a(x' O) z(eO) u(x) dxdO (1.2.3)

if Z ESP a n d Z(0) = 1. W e shall keep the n o t a t i o n (1.2.1) for t h e continuous extension of t h e f o r m I r which we have just defined a n d refer to t h e generalized integral as a n

oscillatory integral.

I f r a n d a are continuous functions of a p a r a m e t e r t with values in

C~(X

• ( R N ~ { 0 ) ) ) a n d S~.~ (X • R N) respectively, t h e n an inspection of t h e proof of L e m m a 1.2.1 a n d P r o p o - sition 1.2.2 shows t h a t

Ir

is a continuous function of t. N o t e t h a t if a is a continuous f u n c t i o n of t with values in

C~(X

• R N) whose range is a b o u n d e d subset of S~.o (X • RN), t h e n a is a continuous f u n c t i o n of t with values in S~.~ (X • R N) w h e n m ' > m. These r e m a r k s allow us to pass to the limit in the oscillatory integral (1.2.2) if there is continuous dependence on a parameter. I n particular we can differentiate with respect to p a r a m e t e r s u n d e r t h e integral sign.

N o w let r be a C ~176 f u n c t i o n in X x Y • ( R N ~ ( 0 ) ) where X a n d Y are o p e n subsets of some E u c l i d e a n spaces, a n d assume t h a t r has no critical point even w h e n considered as a function in X • (RN~{0}) depending on t h e p a r a m e t e r yE Y. If

aES~,~(X

• Y • RN),

~ > 0 , ~ < 1 , a n d

uEC~176 • Y),

we can t h e n prove a F u b i n i t h e o r e m

f f f eir176 u(x,y)dxdydO= f d y ( f f e~(x.~.~ dxdO).

(1.2.4)

Indeed, this follows if we i n t r o d u c e a f a c t o r Z(e0) as in (1.2.3) in b o t h sides a n d t h e n let e ~ 0 .

After these r e m a r k s we r e t u r n to t h e oscillatory integral (1.2.1). L e t X~ be t h e o p e n set of all

x E X

such t h a t t h e f u n c t i o n

O-~r O)

has no critical point 0 # 0 . I f

uEC~176162

we can regard x as a p a r a m e t e r a n d rewrite (1.2.1) in t h e f o r m

F O U R I E R I N T E G R A L O P E R A T O R S . I 91

I r =

f A(.)u(x)dx, ueC

where A

(x) = f

~ O) dO, x E Xr (1.2.5)

B y our preceding remarks on oscillatory integrals, A is a continuous function of x EXr and since we can differentiate under the sign of (oscillatory) integration as often as we like, we conclude t h a t A E COO(Xr I f we recall t h a t the singular support (written sing supp) of a distribution is the completment of the largest open set where it is a C ~ function, we have proved

PROPOSITION 1.2.3. tZor the distribution A : u ~ I r de]ined by (1.2.1) we have sing supp A c { x e X ; r 0) = 0 / o r some 0 ~=0}. (1.2.6) The formula (1.2.5) also makes sense for all x E X provided t h a t r has no critical point as a function of 0 in cone supp a, for this is clearly all ^that is required in the proof of Propo- sition 1.2.3. Thus we have the following simple result which shows t h a t the singularities of the distribution A are uniquely determined b y the behavior of the symbol a in a conical neighborhood of the set of points where r is critical with respect to the 0 variables.

PROPOSITION 1.2.4. I] aESo.$(X • N) and a vanishes in some conic neighborhood o/the set

C = {(x, 0); xE X, 0 E R ~ { 0 } , r 0) --0}, (1.2.7) then the distribution u ~ I r de/ined by (1.2.1) is a C ~ /unction.

If one looks more carefully into the proof of Proposition 1.2.2 one finds easily t h a t the conclusion of Proposition 1.2.4 remains valid if we assume only t h a t for some C and e with e < m i n (@, 1/2) we have a(x, 0 ) = 0 when [r 0)[ [0[~<C. The proof is left to t h e reader, b u t we shall prove a stronger result under some hypotheses on r which guarantee t h a t C is a smooth manifold.

L e t F be an open conic set in X • R N and let r be a positively homogeneous function of degree 1 with respect to 0 which is in C ~ and has no critical point in F ~ ( X • {0~).

Such a function will be called a phase/unction from now on. I t is clear t h a t the definition of (1.2.1) given above is still valid for such a function r provided t h a t we require t h a t cone supp a c F (J (X • (0}).

We shall say t h a t r is non-degenerate if at a n y point in the set C defined b y (1.2.7) the differentials d(~r ] = 1, ..., N, are linearly independent. This implies of course t h a t

92 LARS H O R M A N D E R

C is a manifold of dimension dim X. F o r such phase functions we can i m p r o v e P r o p o s i t i o n 1.2.4.

P R O P O S I T I O N 1.2.5. Let r be a non-degenerate phase /unction in F c X • N and let a E S~.~ (X • RN), cone supp a C F U (X • 0). We assume that ~ > (i and that either r is a linear /unction o/ 0 or that p + ( ~ = l . T h e n the distribution u--->Ir defined by (1.2.1) is a C ~ /unction i/ a vanishes o / i n f i n i t e order on

c={(x,o)er; r =0}.

I / a just vanishes on C we can find b e S~. ~ - 0 ( X x It N) with cone supp b ~ F U ( X • O) such that Ir (au) = Ir (bu), u e C~ (X).

F o r t h e proof we need a lemma.

L E M ~ A 1.2.6. Let (~1 . . .

Ck

be real valued C ~ in F ~ ( X • {0)) which are homo- geneous o/degree 0, and assume that the di//erentials de j, ] = 1, ..., k are linearly independent in

C = {(x, 0 ) E F , 0 # 0 , Cj(x, 0) = 0 , ] = 1, ..., k).

Let a E S~.~ (X • R N) where we assume that ~ + (~ = 1 unless r .... , Ck a r e / u n c t i o n s o / x only.

I / a = 0 in a neighborhood o/ X • {0} and a vanishes (o/infinite order) on C, cone s u p p a c F U (X • {0}), one can find a j e S ~ + ~ ( X • RN), ] = 1, ..., Ic, with cone supp a j c F U (X • (0}) such that (aj vanishes o/ infinite order on C and)

k

a = ~ aj Cj. (1.2.8)

1

P r o o / t h a t the lemma implies the proposition. I f we a p p l y the l e m m a with r 1 6 2 s, which are functions i n d e p e n d e n t of 0 precisely w h e n r is a linear function of 0, an integra- t i o n b y parts gives

Ir (au) = eiCCx'~ ~ iOaj(x,O)/OOju(x) dxdO.

1

H e r e t h e new a m p l i t u d e f u n c t i o n is of order m + (~ - 0 a n d in case a vanishes of infinite order on C it will also vanish of infinite order there. I f t h e a r g u m e n t is r e p e a t e d k times we find t h a t a can be replaced b y a s y m b o l of order m + k ( ( ~ - O ) - > - 0% k-> oo, so t h e dis- t r i b u t i o n u-+Ir is a Coo function.

P r o o / o / t h e lemma. I t is sufficient to find a local solution of (1.2.8) a n d t h e n a p p l y a p a r t i t i o n of u n i t y on t h e sphere bundle in F c X • R N, extended to a s y s t e m of homo-

F O U R I E R I N T E G R A L O P E R A T O R S . I 93 geneous functions of degree 0, to o b t a i n a global solution. I f (x0,00)eF,

0o#0,

t h e existence of a solution in a conic n e i g h b o r h o o d is obvious unless (x0, 00)EC as we shall n o w assume. T h e functions r ..., r are t h e n i n d e p e n d e n t functions on t h e u n i t sphere bundle at

(Xo,

0o/]00[ ) so we can find additional h o m o g e n e o u s functions 6k+1, ..-, r of degree 0, / = d i m

X + N - 1 ,

all vanishing at (x0, 00) so t h a t Cx, ..., r is a local coordinate s y s t e m at (x0, 00/100 I) on the sphere bundle. B u t t h e n t h e m a p

(x, 0)+ (r

^{. . . ,}

101) •

is a h o m o g e n e o u s diffeomorphism of a conic n e i g h b o r h o o d of (x0, 00) on U • It+ where U is an open ball in R z with center at 0. B y p a r t (i) of Proposition 1.1.7 symbols of t y p e ~, (~

are preserved b y such m a p s if ~ + ~ = 1 so w h e n this is assumed we h a v e reduced t h e proof to t h e case w h e n Cx .... , Ck are equal to the first coordinates x 1 .... , xk in X a n d X is a ball with center at 0. I f r .... , Ck are i n d e p e n d e n t of 0 t h e same result is achieved b y a substitu- t i o n which only affects t h e x variables a n d t h u s preserves symbols of t y p e Q, (~ for a r b i t r a r y Q, 6. The l e m m a n o w follows f r o m T a y l o r ' s f o r m u l a which gives, since a vanishes w h e n

Z 1 = . . . = X k = 0

~ f 0

a(x, O) = ~ xj a(j)(txl .... , txk, xk+l . . . O) dr,

where a<j)=

~a/Oxj E oo.c~m+'~

^. T h e proof is complete.

S u m m i n g up, w h e n r is non-degenerate t h e singularities of t h e distribution

u ~ Ir

only d e p e n d on t h e T a y l o r expansion of a on t h e set C defined b y (1.2.7), p r o v i d e d t h a t suitable assumptions are m a d e concerning Q a n d d. I n Chapters I I a n d I I I we shall s t u d y the consequences of this more closely a n d also discuss how essential t h e choice of t h e phase f u n c t i o n ~b really is.

1.3. Singular symbols and oscillatory integrals

T h e definition of t h e oscillatory integral (1.2.1) given in t h e preceding section did n o t fully use t h e hypothesis t h a t a is a s y m b o l in t h e sense of section 1.1. Indeed, we only used the fact t h a t for some first order differential o p e r a t o r L with t h e properties s t a t e d in L e m m a 1.2.1 we can conclude t h a t

Lk(au)

is an integrable f u n c t i o n for sufficiently large values of k.

This we shall exploit in w h a t follows.

I n some constructions of f u n d a m e n t a l solutions one needs to be able to define inte- grals of t h e f o r m

f f e~r176 a(x, O)/q(x, O) dx dO ,

(1.3.1) where q is h o m o g e n e o u s with respect t o 0 of degree

m,

say, a n d m a y h a v e simple real zeros.

94 LARS HORMANDER

These form an obvious difficulty in defining the integral. To bypass this singularity we would like to integrate over a suitable cycle in the complex domain instead of over R n.

Assume for simplicity t h a t r q and a are analytic with respect to 0 in a neighborhood of the real domain and t h a t there is a vector ~ such t h a t <grad 6

q(x,

0), U) # 0 when

q(x, O) =0

and 0 e R N ~ 0 ,

x e X .

(It will in fact be necessary to let the direction of ~ v a r y and in ease the d a t a a, q, r are not analytic m a k e suitable " a l m o s t analytic" continuations of them.

These questions will be discussed elsewhere, b u t here we only wish to m o t i v a t e w h a t follows.) Our hypotheses imply t h a t

Iq( , o+i )l = 101ml q(x, 0/101 +i /101)l

^i>

101 m-1

for large ]0], so if we replace 0 b y

O+i~

in (1.3.1) we shall for large 0 no longer have a n y infinities in the integrand. We shall now examine to what extent the function

(x, O)-+a(x, O+iu)/q(x,

O+iu) has the properties of a symbol. We have for e x a m p l e

~q-l(x, 0 + i~)/~xj = - q-2~q/~xr

a n d we can only be certain t h a t this can be bounded b y ]0] m-2(m-1)=

IO] 2-m.

Pursuing this a r g u m e n t one will find t h a t

1/q(x, 0

+iu)ES~7~ which does not suffice for application of Proposition 1.2.2. However, we can say more about the action on q-1 of some operators

L = 2 aj(x, O) ~/~Oj + 2 bj(x, O) ~/~xj + c

(1.3.2) with

ajES ~

and

bj, c 6 S -1.

Indeed, if

Z aj~q/~Oj+Zbj~q/~xj=O

when

q=O

we obtain [L~

<~C]q] [01 -~

if L ~ is the principal p a r t of L, and this leads easily to a proof t h a t

L(1/q(x, 0 +i~))6S~.'~.

More generally, application of k operators of this t y p e will always give an element in r *90,1

We have made the preceding discussion rather brief for it has to be reexamined after a precise definition of " a l m o s t analytic" continuation has been introduced. How- ever, the preceding arguments should suffice to m o t i v a t e the interest of the following developments.

L e t Lf be the set of all first order differential operators of the form (1.3.2) with

a j 6 S ~ 2 1 5 N)

and

bj, c 6 S - I ( X •

This is a module over the ring S~ I f

a E S ~ ( X • N)

and L6.Lf we have

L a 6 S ' ~ - e ( X • N)

in view of Proposition 1.1.6. B y iteration it follows t h a t

L 1 ... LkafiS~-~e(X •

R N) if L 1 .. . . . L ~ 6 ~ . Conversely, if we just k n o w t h a t ~5~ ...

L~aE S~-~e(X •

R N) for all L1 . . . L~6~f, taking these operators to be differentiations with respect to xr and 0, variables we conclude t h a t

a r

• RN). This connects our earlier definitions with the following one.

De/inition 1.3.1.

I f F is a subset of .Lf we denote b y

FS'~(X •

R N) the set of all

a 6 S ~ ( X

x R ~) such t h a t for arbitrary L1 . . .

L~6F,

k = 1, 2 . . . . we have

F O U R I E R I N T E G R A L O P E R A T O R S . I 95

L 1 ... Lea6 S ~ - ~ ~ • R~). (1.3.3)

I t is clear b y the definition t h a t La 6 ~S~ -~ if L 6 F a n d a 6 ~S~. If we set . ~ = S m ~ we also h a v e .~m ~S~' c ~S~ n + ~'. F o r let L E .~q~m, a 6 ~S~ n'. T h e n we have a 6 S~" so La 6 S~ + m,.

F u r t h e r m o r e , if L~ E F,

k

L ~ . . . L ~ L a = L L ~ . . . L e a + ~ L~...L~-I[L~,L]L~+~...Lea.

1=1

We shall prove b y i n d u c t i o n over /c t h a t this is in S~ +'n'-kr This is clear for t h e first t e r m since L 1 . . . L e a E S~ '-~. Since Lj+I...LkaEFS~'-(e-J)Q a n d the c o m m u t a t o r [Lj, L]E.~e m-l, the inductive h y p o t h e s i s shows t h a t t h e terms in t h e s u m are in S m-l+~'-(k-s)~-(j-1)Q c S~ n+m'-k~, which proves t h e assertion. I f b 6 S m it follows similarly t h a t ba E FS~'+ m, for w i t h t h e same n o t a t i o n we have

k

L1... Leba = bL1... Lea + ~ L I . . . Lj_ 1 [Lj, b]Lj+l... L~a, (1.3.4)

1

a n d [Lj, b] 6 S m-l, so t h e same proof b y i n d u c t i o n can be applied. I t follows in p a r t i c u l a r t h a t if F ' is t h e S o module g e n e r a t e d b y F a n d S -1, t h e n F'Sg ~ F S g . W e can therefore always assume w i t h o u t restriction t h a t F is a m o d u l e containing S -1. If L j 6 F it follows t h e n t h a t t h e principal p a r t L ~ 6 F , hence [Lj, b] = LOb 6 vSg -~ if b E FSg. A n i n d u c t i o n proof based again on (1.3.4) therefore gives t h a t ba E Fsg+m" if b E Fs~n a n d a 6 FS~ n'. S u m m i n g up, we have p r o v e d

P R O P O S I T I O ~ 1.3.2. I / F' is the S o submodule o/ C generated by F and S -1 (consid- ered as a set o/ di//erential operators o~ order 0), then FSy = F'S~. I / a 6 FS~, b E FS~', we have abE FS~ +'n', and i / L E .if we have La6 ~S~.

N o t e t h a t t h e proposition shows t h a t

LI...

L e a 6 S m-le if L 1 . . . L k 6 C a n d ?" of these operators belong t o F . This could also h a v e been t a k e n as a definition a n d saved m u c h of the proof of P r o p o s i t i o n 1.3.2.

I f r is a real v a l u e d function 6S1(X • R g) (possibly only for large ]0[) a n d if for some L 6 F we h a v e tLe~r = e ~r we can define the oscillatory integral (1.2.1) for all a E~S~(X • R~).

I t is a distribution of order ~< k if m - k 9 < - N . I n d e e d t h e proof of Proposition 1.2.2 does n o t require a n y change.

Example 1.3.3. L e t Z be a closed conic C ~ submanifold of X • ( R ~ { 0 } ) a n d let F be the set of all L E ~ whose principal p a r t defines a v e c t o r field t a n g e n t i a l to F, at every p o i n t

96 LARS H()RMANDER

in F=. (Equivalently, if L 0 is t h e principal p a r t of an operator in F , t h e n L~ vanishes on E if a does. T h u s we are considering a generalization of the situation discussed at t h e beginning of t h e section.) T h e n there exists an o p e r a t o r L E F with t h e required properties if a n d only if neither r nor the restriction o / ~ to Z has a critical point. Indeed, as in t h e proof of L e m m a 1.2.1 we h a v e L E F if a n d only if M = t L E F , a n d if we write M = Z aj~/~Oj + Z bj~/~xj + c t h e problem is t o choose t h e v e c t o r field (a, b) tangential to Z on $;

a n d such t h a t

i(Z aj~r + E bj~r + c = 1. (1.3.5)

Since aj E S O a n d b~, c E S -1 this requires t h a t r has no critical point a n d t h a t t h e restriction t o E has no critical point. Conversely, w h e n this is t r u e we can find a solution of (1.3.5) in a conic n e i g h b o r h o o d of a n y point, such t h a t c = 0 a n d a j, bj are homogeneous of degree 0 a n d - 1 respectively, b y just choosing a v e c t o r field tangential to E which does n o t annihilate r at t h e point in question. Application of a p a r t i t i o n of unity, introducing t h e solution e = 1, a ~ = b j = 0 near 0, t h e n gives the assertion.

A special case is o b t a i n e d w h e n r 0 ) = ( x , 0} a n d for a splitting of t h e 0 variables in 0' - (01 .... , 0 N _ k ) , 0 " = ( 0 N _ k + 1 . . . ON) we h a v e F. = {(x, 0', 0"); (0',0") =~0 a n d 0' = 0 or 0" =0}.

This occurs in t h e s t u d y of t h e multiplicative properties of the index of elliptic pseudo- differential operators (cf. PalMs [26], pp. 206-209).

F m

W e shall use t h e n o t a t i o n zS~ instead of S e w h e n F is defined b y E as in E x a m p l e 1.3.3. This should cause no ambiguity.

W h e n a EXS~, the singularities of t h e distribution defined b y (1.2.1) m a y be caused either b y points with r 0 or b y points in X with r ~= 0. W e shall investigate t h e latter contributions. I n doing so we assume t h a t t h e manifold X is transversal t o t h e fibers x = c o n s t a n t so t h a t the sets Ex = {0; (x, 0)E E} are manifolds of t h e same codimension as E at e v e r y point. L e t (x0, 0o)EX , r 0 o ) 4 0 , a n d let /c be t h e codimension of E there.

Choose a labelling of t h e 0 coordinates so t h a t with 0' ~ (01 ... ON-k) a n d 0" = (0N-k+1 .... ,0N) t h e plane dx =dO' = 0 is transversal to E. T h e n we h a v e 0~ 4 0 , for E~, being a cone, t h e v e c t o r (0, 00) would otherwise lie in the t a n g e n t plane of Ex~ at 00. I n a n e i g h b o r h o o d of (%, 0o) t h e manifold E is therefore of t h e f o r m O"=y~(x, 0') where yJ is h o m o g e n e o u s of degree 1 with respect to 0' a n d defined in a conical n e i g h b o r h o o d of 0~. I n t h e integral

(.A, u} =

f f O,a(x O) u(x)

^{dx dO}

we assume t h a t t h e s u p p o r t of a belongs to such a small conic n e i g h b o r h o o d of (x0,

00)

t h a t we can i n t r o d u c e 0"-yJ(x, 0') as a new variable instead of 0" there. This t r a n s f o r m s X t o t h e manifold E0: 0" = 0 . The new a m p l i t u d e al(x , O) =a(x, 0', 0" +~v(x, 0')) will belong t o

F O U R I E R I N T E G R A L O P E R A T O R S . I 97 z ' S ~ a n d vanish w h e n [ 0 " ] > 10'l , say. L e t F ~ be t h e corresponding subspace of S~ de- fined in E x a m p l e 1.3.3. The operators in F ~ are t h e n those which w h e n 0" = 0 do n o t involve differentiation with respect to t h e O" v a r i a b l e s . - - T h e new phase f u n c t i o n r 0) = r 0', 0" +V(x, 0')) will be defined in a conic n e i g h b o r h o o d of cone supp al, we h a v e r 0~, O) # 0 a n d

I f r 0, 0~, 0 ) # 0 a n d t h e s u p p o r t of a is so small t h a t this inequality remains valid in cone supp al, we can prove t h a t A EC ~ b y using t h e proof of Prop. 1.2~3. I n fact, t h e derivatives of a 1 with respect to the 0' variables are as well b e h a v e d as if al were in

. . . . 0o, O) # 0 we

S~. If, on t h e other hand,

r o,

0o, O) = 0 we h a v e instead

r o, '

a n d m a y assume t h a t this is t r u e in a n e i g h b o r h o o d of cone supp a r N o w we can write

(A, u) = f f e*r176176 0') u(x) dxdO',

where

b(x, 0') = f a l (x, 0',0") dO"

Here we m a y assume t h a t a 1 = 0 w h e n [ 0' ] < 1, for modification of a 1 on a c o m p a c t set only changes A b y a C ~ term. W e wish to prove t h a t b e S em+k. Since ]0' [ ~< ]0] < 2 [ 0 ' [ in cone supp a 1 a n d

]al(x ,

0)] ~< C(1 + [0]) m, it is clear t h a t

I b(x,

0')] ~< C(1 + ]0']) ~+k. To estimate t h e derivatives we note t h a t

~(r162 Y, aj,~(x,O) O,Or

N

]Ljr

N - k + 1

where LjC F ~ A similar result w i t h o u t t h e f a c t o r [0' I is v a h d for t h e derivative with respect to 0 s when

] <~N-k.

N o w we o b t a i n after an integration b y p a r t s

~b(x, O')/~xr f e~(r162176162176176 [ (tLj + [O'l-~/~xj)a~(x,O)dO"

a n d a similar f o r m u l a for

3b(x, O')~Oj

w h e n ~ ~ < N - k . H e r e t L j + ]0' [-l~/~x s belongs t o F ~ (after suitable modification w h e n [ 0 ' l < 1). Application of

~/~Oj, ] = 1, ..., N - k ,

or

~/~xj

t o b is t h u s equivalent to o p e r a t i o n on % b y a n operator in F ~ followed b y multiplication b y 10'l in the case of

~/~xj.

This gives i m m e d i a t e l y t h a t b C S e m+k . S u m m i n g up, we h a v e p r o v e d

T ~ E O R E M 1.3.4.

Let aC~S~ vanish in a conic neighborhood o] the set where

r

assume that in a neighborhood of

cone supp

a the restriction o/r to E has no critical point and that the mani]old E can be expressed in the /orm 0"

=yJ(x, 0')

where 0'= (01, ..., ON-k) and

0 " = (0N-k+x ...

0~,). Then the distribution A defined by

7 -- 7 1 2 9 0 6 A c t a mathematica 127. I m p r i m 6 le 2 J u i n 1971

98 LARS H()RMA:NDER

(A, u) = J Je i*(x' ~ O) u(x) dx dO, u E C~ r (X)

can also be de/ined by means o/the phase/unction r 0', ~f(x, 0')) and an amplitude bESe'~+k.

F r o m t h e p r e c e d i n g r e s u l t one s h o u l d n o t c o n c l u d e t h a t t h e r e is no n e e d t o consider s i n g u l a r s y m b o l s . T h e n e w f e a t u r e s are c a u s e d b y p o i n t s in t h e set Z w h e r e r = 0. W e l e a v e for t h e r e a d e r to c o n s t r u c t a n e x a m p l e of t h i s for e x a m p l e b y m e a n s of t h e special case of E x a m p l e 1.3.3 m e n t i o n e d a t t h e e n d of it. W e shall also e n c o u n t e r n a t u r a l e x a m p l e s in p a r t I I .

As in s e c t i o n 1.2 i t is e a s y t o e x t e n d t h e p r e c e d i n g discussion to o p e r a t o r s d e p e n d i n g o n p a r a m e t e r s . I n d e e d , l e t r be a c o n t i n u o u s f u n c t i o n of a p a r a m e t e r t C T w i t h v a l u e s in

Coo(X

x ( R N ~ ( 0 } ) ) a n d l e t Z t b e a conic s u b m a n i f o l d of Z x ( R N ~ ( 0 } ) w h i c h also d e p e n d s c o n t i n u o u s l y on t. This m e a n s t h a t l o c a l l y in T • X x ( R N ~ ( 0 } ) we can define Z~ b y e q u a - t i o n s

q~(x, O) . . . q~(x,

0 ) = 0 where qJ is a c o n t i n u o u s f u n c t i o n of t whose v a l u e s a r e C OO f u n c t i o n s of (x, 0) w i t h t h e d i f f e r e n t i a l s of ql . . . . , qk l i n e a r l y i n d e p e n d e n t . W e a s s u m e t h a t n e i t h e r Zt n o r t h e r e s t r i c t i o n of 4~ t o Zt has a n y critical p o i n t . L e t F be t h e set of all c o n t i n u o u s m a p s L: T-+oL~ a such t h a t t h e c o r r e s p o n d i n g v e c t o r field is t a n g e n t i a l t o Zt for e v e r y t. I t is e a s y t o see t h a t one can choose L E F so t h a t L t e x p i4~ = e x p i f t for e v e r y t.

D e f i n i n g z S ~ in t h e o b v i o u s w a y using t h e o p e r a t o r s in F , we c o n c l u d e t h a t (x,O) u(x) dO, u

~ ^{dx C}

is a c o n t i n u o u s f u n c t i o n of t. A g a i n t h i s allows p a s s a g e t o t h e l i m i t u n d e r t h e i n t e g r a l sign, d i f f e r e n t i a t i o n w i t h r e s p e c t t o p a r a m e t e r s a n d so on.

1.4. Definition of Fourier integral operators

L e t X , Y be o p e n sets in R "~ a n d R ~r a n d let r be a r e a l v a l u e d f u n c t i o n of

(x, y, O) E

X • Y • R N w h i c h is p o s i t i v e l y h o m o g e n e o u s of d e g r e e 1 w i t h r e s p e c t to 0 a n d i n f i n i t e l y d i f f e r e n t i a b l e for 0:4: 0. W i t h a s y m b o l a E S~.~ (X • Y • RN), ~ > 0, 5 < 1, we wish to consider t h e o p e r a t o r d e f i n e d b y t h e i n t e g r a l

Au(x) = I r e ~r ~ y, O) u(y) dydO, u E C~ (Y), x E X,

(1.4.1) o r a w e a k f o r m of (1.3.1)

(Au, v)=fffe~(x.~.~ ueC~(r),veC~(X).

^(1.4.2)

F O U r I E R I N T E G r A L O ~ E R A T O R S . I 99 To give a meaning to these integrals we apply the results on oscillatory integrals (depending on parameters in the case (1.4.1)) proved in section 1.2. The conclusions are as follows:

T~EO~EM 1.4.1. (i) I f ~ has no critical point as a/unction o] (x, y, O) with 0:~0 then the oscillatory integral (1.4.2) exists and is a continuous bilinear form/or the C~ topo- logies on u, v if

m - k ~ < - i V , m - k ( 1 - 5 ) < -1V. (1.4.3) When (1.4.3) is valid we thus obtain a continuous linear map A ]rom C~( Y) to ~'k(X) which has a distribution kernel KA C~'k(X x Y) given by the oscillatory integral

KA(u)=fffe' ( ' '~ uCC (Xx y).

^(1.4.4)

(if) I / / o r each fixed x the function r has no critical point (y, 0) with 0 40, then (1.4.1) is defined as an oscillatory integral. When (1.4.3) is valid we obtain a continuous map A: C~( Y) ---> C(X). B y differentiation under the integral sign it follows that A is also a continuous map from C~(Y) to CJ(X) i]

m + N + ] < k ~ , m + N + j </c(1-5). (1.4.5) ( i i i ) / / / o r each fixed y the/unction r has no critical point (x, O) with 0 4 0 , then the ad]oint of A has the properties listed in (if) so A is a continuous map o] g'J(Y) into ~'k(X) when (1.4.5) is ]ul/iUed. I n particular, A defines a continuous map ]tom N'(Y) to ~ ' ( X ) .

(iv) Let Re be the open set o / a l l (x, y) E X • Y such that r y, O) has no critical point 0 =~0 as a function of O. Then the oscillatory integral

K~(x,y)= fe~(x'"~ ^(x,y)ER~,

^(1.4.6)

defines a/unction in C~176162 which is equal to the distribution (1.4.4) in Re. I f Re = X • Y, it [ollows that A is an integral operator with a Coo kernel, so A is a continuous map of ~' ( Y) into C~176

The proof is an immediate consequence of Proposition 1.2.2 and the remarks fol- lowing it.

Example 1.4.2. Pseudo-differential operators correspond to the function r y, 0) ( x - y , O } ( n x = n r - N ). Then (i), (if), (iii) are fulfilled and Re is the complement of the diagonal if we take X - Y. We shall study this case extensively in Chapter II.

100 LARS HORMANDER

Example 1.4.3. I n the introduction we saw t h a t the s t u d y of the Cauchy problem for the wave equation leads to the function

r t; y, o) = ( x - y , o ~ + t l o I.

Here n x - l = n r = N and the variable in X is denoted b y (x, t). Then (i), (ii), (iii) are fulfilled and ~Rr consists of all (x, t; y) with ] x - y ] ~ = t 2. This means t h a t (x, t) lies on the light cone with vertex at (y, 0).

Definition 1.4.4. A real valued function r of (x, y, 0) E X • Y • R N which is a C ~ func- tion for 0 =~ 0 and positively homogeneous of degree 1 with respect to 0 will be called an operator phase function if for each fixed x (or y) it has no critical point (y, 0) (or (x, 0)) with 0 4 0 .

When r is an operator phase function the hypotheses of parts (i), (ii), (iii) of Theorem 1.4.1 are thus fulfilled. L e t Cr denote the complement of Re in X • Y, t h a t is, the projection on X • Y of the conic set

= {(x, y, 0) EX • Y • (RN\{0}), r y, 0) = 0}. (1.4.7) F r o m (iv) in Theorem 1.4.1 it follows then t h a t

sing supp Au c Cr supp u, uE#'(Y) (1.4.8)

where the right-hand side is defined b y considering 0r as a relation between points in Y and in X, thus

Cr K = {x; (x, y)E Cr for some y E K}.

I n fact, if K = s u p p u and K ' is a c o m p a c t subset of X which does not intersect CcK, we have K' x K c R r so we can find neighborhoods s ~ K such t h a t ~ ' • 1 6 3 1 6 2 Hence AuECm(~ ') which proves (1.4.8). Using (fi) in Theorem 1.4.1 we can i m p r o v e (1.4.8) further..For if s is a n y neighborhood of sing supp u, we can m a k e a decomposition u = v + w where supp v c s and wEC% Since AwEC ~ we obtain

sing supp Au = sing supp A v ~ Cr supp v, so we have proved

THEOREM 1.4.5. I/ u E S ' ( Y ) , then

sing supp Au c Cr sing supp u. (1.4.9)

Example 1.4.6. For pseudo-differential operators (see E x a m p l e 1.4.2) this means t h a t sing supp Au c sing supp u,

which is usually called the pseudo-local property.