Singularities in Linear Wave Propagation

Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann Subseries: Nankai Institute of Mathematics, Tianjin, RR. China vol. 2 Adviser: S.S. Chern

1241 Lars G&rding

Singularities in Linear Wave Propagation I

III

Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo

Author

Lars G&rding Department of Mathematics, University of Lund Box 118, 2 2 1 0 0 Lund, S w e d e n

Mathematics Subject Classification (1980): 35-xx; 3 5 L x x ISBN 3 - 5 4 0 - 1 8 0 0 1 - X Springer-Verlag Berlin Heidelberg N e w Yo'rk ISBN 0 - 3 8 7 - 1 8 0 0 1 - X Springer-Verlag N e w York Berlin Heidelberg

Library of Congress Cataloging-in-Publication Data. G&rding, Lars, 1919- Singularities in linear wave propagation. (Lecture notes in mathematics; 1241) Includes index. 1. Differential equations, Hyperbolic. 2. Wave motion. Theory of. 3. Singularities (Mathematics) I. Title. II, Series: Lecture notes in mathematics (Springer-Verlag); 1241. OA3.L28 no. 1241 510 s 8?-16397 [QA3??] [515.3'53] ISBN 0~387-18001-X (U.S.) This work is subject to copyright. AIt rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1987 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210

TABLE OF CONTENTS Historical introduction Chapter I H y p e r b o l i c o p e r a t o r s w i t h constant c o e f f i c i e n t s 1.1 A l g e b r a i c h y p e r b o l i c i t y 1.2 D i s t r i b u t i o n s associated w i t h the inverses of a homogeneous h y p e r b o l i c polynomial 1.3 I n t r i n s i c h y p e r b o l i c i t y 1.4 Fundamental s o l u t i o n s of homogeneous h y p e r b o l i c o p e r a t o r s . Propagation cones. L o c a l i z a t i o n . General c o n i c a l r e f r a c t i o n 1.5 The H e r g l o t z - P e t r o v s k y f o r m u l a Chapter 2 Wave f r o n t sets and o s c i l l a t o r y i n t e g r a l s 2.1 Wave f r o n t sets 2.2 The r e g u l a r i t y f u n c t i o n 2.3 O s c i l l a t o r y i n t e g r a l s 2.4 F o u r i e r i n t e g r a l o p e r a t o r s 2.5 A p p l i c a t i o n s Chapter 3 P s e u d o d i f f e r e n t i a l o p e r a t o r s 3.1 The c a l c u l u s of p s e u d o d i f f e r e n t i a l o p e r a t o r s 3.2 L= e s t i m a t e s . R e g u l a r i t y p r o p e r t i e s of s o l u t i o n s of p s e u d o d i f f e r e n t i a l e q u a t i o n s 3.3 L a x ' s c o n s t r u c t i o n f o r Cauchy's problem and a f i r s t order d i f f e r e n t i a l o p e r a t o r Chapter 4 The Hamilton-Jacobi e q u a t i o n and s y m p l e c t i c geometry 4.1 Hamilton systems 4.2 Symplectic spaces and La9rangian planes 4.3 Lagrangian submanifolds of the cotangent bundle of a m a n i f o l d 4.4 Hamilton f l o w s on the cotangent bundle. Very r e g u l a r phase f u n c t i o n s Chapter 5 A g l o b a l p a r a m e t r i x f o r the fundamental s o l u t i o n of a f i r s t order h y p e r b o l i c p s e u d o d i f f e r e n t i a l operator 5 . 1 C a u c h y ' s problem f o r a f i r s t order h y p e r b o l i c p s e u d o d i f f e r e n t i a l operator 5.2 Cauchy's problem on the product of a l i n e and a m a n i f o l d 5.3 A g l o b a l p a r a m e t r i x Chapter 6 Changes of v a r i a b l e s and d u a l i t y f o r general oscillatory integrals 6 . 1 H ~ r m a n d e r ' s equivalence theorem f o r o s c i l l a t o r y i n t e g r a l s w i t h a r e g u l a r phase f u n c t i o n 6.2 Reduction of the number of v a r i a b l e s a.3 D u a l i t y and r e d u c t i o n of the number of v a r i a b l e s Chapter ? Sharp and d i f f u s e f r o n t s of ~aired oscillatory integrals 7 . 1 A f a m i l y of d i s t r i b u t i o n s ?.2 P o l a r c o o r d i n a t e s in o s c i l l a t o r y i n t e g r a l s 7.3 Almost a n a l y t i c e x t e n s i o n s 7.4 S i n g u l a r i t i e s of paired o s c i l l a t o r y i n t e g r a l s w i t h Hessians of corank 2 7.5 The general case. Petrovsky chains and c y c l e s . The Petrovsky c o n d i t i o n References Index

I i0 I0

15 17

20 25 34 34 37 39 44 45 51 53 62 70 73 73 75 78 81

85 85 88 89 95 96 78 I01 104 I05 I08 111 112 121 124 125

SINGULARITIES

IN LINEAR WAVE PROPAGATION

o

Lars Gardin9

Historical

introduction

HuY9hens~s t h e o r y o f Its

first

light

in

The t h e o r y o f

wave f r o n t

wave p r o p a g a t i o n s t a r t e d

s e t s as e n v e l o p e s o f

s u c c e s s was t h e p r o p e r e x p l a n a t i o n o f refracting

media.

boundary problems for

Its

e q u a t i o n s . The d e v e l o p m e n t which search f o r

lead to

the theory of

partial

this

e l e m e n t a r y waves.

the p r o p a g a t i o n of

modern s u c c e s s o r i s

h y p e r b o l i c s y s t e m s of

differential

theory

is

a story

chapter is

the d i s c o v e r y in

the eighteenth century of

p a r a d o x . The wave e q u a t i o n u ~ - u ~ = O

in one t i m e

and one space

d i m e n s i o n e x p r e s s e s t h e movement o f

the deviation

u from r e s t

an i d e a l i z e d

g arbitrary

is

directions.

But

string

fixed

string.

Its

t h e sum o f

at

it

general solution

two t r a v e l l i n g

series with

f(x-t)+g(x+t)

waves w i t h

a

its

end p o i n t s

as an i n f i n i t e

solution

the theory of

with

of

this

f

and

opposite

sum o f

a

sine functions.

functions

smooth t e r m s c o u l d e x p r e s s a r b i t r a r y

a

position

was a l s o p o s s i b l e t o e x p r e s s t h e movements o f

This r a i s e d the q u e s t i o n about the n a t u r e of

efficient

of

proper mathematical t o o l s .

The f i r s t

for

with

and how a

functions.

The f i r s t

p r o b l e m came two hundred y e a r s l a t e r

with

distributions.

The n i n e t e e n t h c e n t u r y made i m p o r t a n t d i s c o v e r i e s a b o u t wave propagation. Gemetrical optics Hamilton.

It

rays rather fronts the

is

a theory of

t h a n waves o f

or c a u s t i c s but

intensity

of

light

normalized to

normals of

light.

It

wave f r o n t s ,

I,

our n o t a t i o n

Other e f f o r t s and w i t h

by

o t h e r words o f

idea about t h e i r

outside the fronts. in

in

9ave a v e r y good i d e a o f

not a v e r y c l e a r

a r o u n d t h e wave e q u a t i o n , velocity

was d e v e l o p e d t o g r e a t p e r f e c t i o n

wave

intensity centered

the propagation

or

u t t - d u = O, where d i s L a p l a c e ' s o p e r a t o r in n space v a r i a b l e s x = ( x 1 , . . . , x n ) .

The

physically

the

interesting

nineteenth century it u=f(t-lxl)/Ixl

case i s o f

course n=3.

was observed t h a t

are s o l u t i o n s f o r

u(t,x)

=(4~) - I

the s p h e r i c a l waves

arbitrary

d i s c o v e r e d the r e m a r k a b l e f o r m u l a ,

i s p r e c i s e l y the envelope of

the s u p p o r t o f

f.

and Poisson

J f(y)~(t-lx-yl)dy/Ix-yl,

w i t h g e o m e t r i c a l o p t i c s was p e r f e c t ,

in

functions f

in modern n o t a t i o n ,

which s o l v e s Cauchy's problem u ~ - d u = O ,

time t

In the b e g i n n i n g o f

u=O, u t = f ( x )

the support of

for

this

Its

fit

the s o l u t i o n at

spheres w i t h r a d i u s t

For almost a c e n t u r y ,

t=O.

and c e n t e r s

cemented t h e

g e o m e t r i c a l o p t i c s c o n t a i n s a l m o s t the whole s t o r y o f

idea t h a t

wave

propagation. In modern

lan9uage we can i n t e r p r e t

P o i s s o n ' s f o r m u l a by s a y i n g t h a t

the d i s t r i b u t i o n (I)

E(t,x)

= H(t)

~(t-lxl)/4~txl,

H(t)=l

when t>O and 0 o t h e r w i s e ,

which s o l v e s t h e wave e q u a t i o n E ~ - d E = ~ ( t ) ~ ( x ) and v a n i s h e s when tI and b e h a v e s l i k e

&c'~-=''='(d(t,x))

cone o u t s i d e t h e o r i 9 i n . coefficients

by Hadamard

Tedone's results in

Problem and H y p e r b o l i c L i n e a r P a r t i a l p u b l i s h e d in

1923 w i t h

were e x t e n d e d

a famous b o o k , The Cauchy

Differential

a French e d i t i o n

in

1932.

Equations, He r e p l a c e d t h e wave

o p e r a t o r by an o p e r a t o r P=~ g l k ( X ) U i ~ + l o w e r terms where 9 = ( 9 1 k )

is

a s y m m e t r i c nXn m a t r i x w i t h

p l u s and t h e r e s t derivatives of

with

minus,

and t h e

indices of

respect to the variables

Lorentz si9nature, u indicate

(x~,...,x.).

If

d(x,y}

c o r r e s p o n d i n g d i s t a n c e from x t o equation of half

H of

solution

t h e c o r r e s p o n d i n g cone, E(x,y)

properties.

It

with

pole at

y,

the

denotes the square of

a given point

a two-sheeted conoid with

second o r d e r The l i g h t

t h e wave e q u a t i o n a r e r e p l a c e d by t h e e x t r e m a l s o f

m e t r i c c o r r e s p o n d i n 9 t o 9-

its

y,

y.

rays

indefinite the

d(x,y)=O is

vertex at

one

the

For a g i v e n

Hadamard c o n s t r u c t e d a f u n d a m e n t a l

PE(x,y)=&(x-y),

with

the following

v a n i s h e s o u t s i d e H and b e h a v e s i n s i d e H as i n

constant coefficient

case e x c e p t t h a t

when n i s even ( i . e .

odd i n

the

the v a n i s h i n g i n s i d e the conoid

the p r e v i o u s n o t a t i o n )

is

r e p l a c e d by a

smooth b e h a v i o r up t o the boundary. Hadamard guessed the shape of the fundamental s o l u t i o n in the form of an a s y m p t o t i c s e r i e s . To v e r i f y t h a t the c o n s t r u c t i o n y i e l d e d a fundamental s o l u t i o n ~ Green's formula had t o be used. This led t o d i f f i c u l t i e s

w i t h the s i n 9 u l a r i t i e s on the

cone were avoided by a l i m i t i n 9 procedure c a l l e d the method of finite

part.

A few years l a t e r ,

to replace i t

the

Marcel Riesz (1937 and 1949} managed

by a more p a l a t a b l e a n a l y t i c c o n t i n u a t i o n w i t h r e s p e c t

t o a parameter. It

seemed hopeless t o extend Hadamard's method t o higher o r d e r

h y p e r b o l i c e q u a t i o n s . Even an e x i s t e n c e p r o o f f o r with

it

Cauchy's problem and

the e x i s t e n c e of fundamental s o l u t i o n s presented problems.

Petrovsky mana9ed a v e r y c o m p l i c a t e d e x i s t e n c e p r o o f in the t h i r t i e s , but an e a s i e r one using f u n c t i o n a l a n a l y s i s was found in the f i f t i e s by G~rding. S t i l l ,

an a n a l y s i s of

the s i n 9 u l a r i t i e s of the fundamental

s o l u t i o n s remained. The b r e a k - t h r o u g h came in 1957 w i t h a paper by Lax. He found out how t o make the F o u r i e r method work f o r general o s c i l l a t o r y

v a r i a b l e c o e f f i c i e n t s by usin9

i n t e g r a l s , i g n o r i n 9 low f r e q u e n c i e s and keepin9

the hi9h ones which are r e s p o n s i b l e f o r was not new in

itself.

It

the s i n 9 u l a r i t i e s .

The method

had been used by p h y s i c i s t s under the name

of the g e o m e t r i c a l o p t i c s a p p r o x i m a t i o n and, s e m i - c l a s s i c a l a p p r o x i m a t i o n . But i t s

use f o r

in quantum p h y s i c s , the h y p e r b o l i c systems and

o p e r a t o r s of high order was a n o v e l t y . The c o n s t r u c t i o n s of Hadamard and Lax shared one d e f e c t not present in the e x i s t e n c e proo÷s: both were r e s t r i c t e d t o a neighborhood of the p o l e of

the fundamental

solution. L a x ' s paper was one of

the f i r s t

t h a t aroused the i n t e r e s t of

mathematicians in the a n a l y s i s of s i n g u l a r i t i e s o÷ o s c i l l a t o r y i n t e g r a l s . The outcome has been a v a s t t h e o r y of

prime importance

whose i n g r e d i e n t s are p s e u d o d i f f e r e n t i a l o p e r a t o r s , the n o t i o n of wave f r o n t s e t or s i n g u l a r i t y spectrum f o r

an a r b i t r a r y d i s t r i b u t i o n or

h y p e r f u n c t i o n and a t h e o r y of

p r o p a g a t i o n of s i n g u l a r i t i e s .

i s m i c r o l o c a l a n a l y s i s and i t

has a host of

Its

name

a p p l i c a t i o n s . In one of

them, H~rmander and Duistermaat succeded in makin9 the c o n s t r u c t i o n of Hadamard and Lax 91obal. The t h e o r y of m i c r o l o c a l a n a l y s i s i n c l u d i n 9 many recent r e s u l t s are to be found

in Hormander~s books The A n a l y i s

of L i n e a r P a r t i a l D i f f e r e n t i a l Operators I - I V The aim of

t h i s s e r i e s of

( S p r i n g e r 1983-85).

l e c t u r e s i s t o present the use of

m i c r o l o c a l t h e o r y in the a n a l y s i s i f

singularities

in

l i n e a r wave

p r o p a g a t i o n , in the m a j o r i t y of cases represented by the fundamental s o l u t i o n s of

linear hyperbolic p a r t i a l d i f f e r e n t i a l

and

p s e u d o d i f f e r e n t i a l o p e r a t o r s . Chapter I d e a l s w i t h forward fundamental s o l u t i o n s of h y p e r b o l i c d i f f e r e n t i a l coefficients. It

operators wit

presents the t h e o r y of

constant

lacunas in a 9eneral form and

has one a p p l i c a t i o n t o a 9eneral form of c o n i c a l r e f r a c t i o n . Chapter 2 about o s c i l l a t i n 9

i n t e g r a l s and wave f r o n t s e t s ,

Chapter 2 about

p s e u d o d i f f e r e n t i a l o p e r a t o r s and Chapter 4 about s y m p l e c t i c 9eometry present known m a t e r i a l necessary f o r

the sequel d e a l i n 9 w i t h the

s i n g u l a r i t i e s of fundamental s o l u t i o n s of s t r o n g l y h y p e r b o l i c o p e r a t o r s and o s c i l l a t i n 9

i n t e g r a l s in 9 e n e r a l . In Chapter 5 t h e r e i s

a new simple c o n s t r u c t i o n of a 91obal p a r a m e t r i x of s o l u t i o n of a f i r s t

the fundamental

o r d e r p s e u d o d i f f e r e n t i a l o p e r a t o r . This

c o n s t r u c t i o n i s b a s i c since parametrices of s t r o n g l y h y p e r b o l i c differential

o p e r a t o r s are sums of such p a r a m e t r i c e s p a i r e d in a

c e r t a i n w a y . The f i n a l

chapters 6 and 7 9 i r e a d e t a i l e d a n a l y s i s of

the s i n g u l a r i t i e s of such p a i r e d o s c i l l a t o r y

integrals.

These l e c t u r e s were d e l i v e r e d in A p r i l and May of

1986 at the

Mahematics I n s t i t u t e of Nankai U n i v e r s i t y , T i a n j i n . The author wants t o take t h i s o p p o r t u n i t y t o thank the I n s t i t u t e f o r and h i s audience f o r

its

patience.

its

hospitality

References t o t h e i n t r o d u c t i o n in h i s t o r i c a l

order

Huygens C. Abhandlun9 ~ber das L i c h t . Ostwalds K l a s s i k e r 20(1913) Fresnel A.J. Ouevres completes (1866-70). I m p r l m e r l e I m p e r l a l e P a r i s . Lame G. Lemons sur l a t h ~ o r l e mathematlque de I e l a s t l c l t e des corps s o l i d e s . Deu×ieme e d i t i o n (1866), P a r i s , G a u t h i e r - V i l l a r s . K o v a l e v s k y 8.V. Uber d i e Brechun9 des L i c h t e s in c r i s t a l l i s c h e n M i t t e l n . Acta Math. 6 (1985)249-304 Tedone O. S u l l ' i n t e g r a z i o n e d e l l ' e q u a z i o n e . . . Ann. d i Mat. S e t . 3 v o l . 1 (1889) V o l t e r r a V. Sur l e s v i b r a t i o n s lumineuses darts l e m i l i e u x b i r ~ f r i n g e n t s . Acta Math. 16(1892)154-21 Z e i l o n N. Sur l e s e q u a t l o n s aux d e r l v e e s p a r t i e l l e s a q u a t r e dimensions e t l e probleme o p t i q u e des m i l i e u x b i r e f r l n g e n t s I , I I . Acta Reg. Soc. So. U p s a l i e n s i s Ser. IV v o l . 5 No 3 ( 1 9 1 9 ) ~ i - 5 7 and No 4(1921),I-130 / / Hadamard J. Le probleme de Cauchy e t l e s e q u a t i o n s aux d e r i v e e s p a r t i e l l e s h y p e r b o l i q u e s . Hermann e t Cie, P a r i s 1932 . ( O r i g i n a l l y l e c t u r e s a t Yale U n i v . 1922) H e r g l o t z G. Uber d i e I n t e 9 r a t i o n l i n e a r e r p a r t i e l l e r D i f f e r e n t i a l 9 1 e c h u n g e n m i t k o n s t a n t e n K o e 4 f i z i e n t e n . Ber. Sachs. Akad. Wissa 78 (1926).~ 93-126, 287-318, 80 (1928}69-114 P e t r o v s k y I.G. Uber das C a u c h y s c h e P r o b e l m fur Systeme yon partiellen D i f f e r e n t a l g l e i c h u n g e n . Mat. Sb. 2(44) (1937)815-870 On t h e d i f f u s i o n of waves and lacunas f o r h y p e r b o l i c e q u a t i o n s . Mat. Sb. 17(59) (1945)145-215 Riesz M. L ' i n t e 9 r a l e de R i e m a n n - L i o u v i l l e e t l e probleme de Cauchy. Acta Math 81 (1949) 1-223. 8 ~ r d i n 9 L. S o l u t i o n d i r e c t e du probleme de Cauchy pour le s e q u a t i o n s h y p e r b o l i q u e s . C o l l . I n t . CNRS Nancy 1956, 71-90 Lax P. A s y m p t o t i c s o l u t i o n s of o s c i l l a t o r y i n i t i a l v a l u e problems. Duke Math. J. 24 (1957) 627-646 Ludwi9 G. C o n i c a l r e f r a c t i o n in C r y s t a l O p t i c s and Hydroma9netics. Comm. Pure Appl Math ×IV (1961)113-124 B u i s t e r m a a t J . J . and H~rmander L. F o u r i e r i n t e 9 r a l o p e r a t o r s I I . Acta Math. 128 (1972) 183-269 A t i y a h M . F . , B o t t R., G~rdin9 L. Lacunas f o r h y p e r b o l i c d i f f e r e n t i a l operators with constant c o e f f i c i e n t s I , I I . Acta Math. 124(1970)~109-189 and 131(1973)145-206 •

t

.

P

,

.

y

/

-

7

.

.

CHAPTER i

HYPERBOLIC OPERATORS WITH CONSTANT COEFFICIENTS

Introduction

The main o b j e c t of

this

chapter is

t o e x p r e s s the

fundamental s o l u t i o n s of

homogeneous h y p e r b o l i c d i f f e r ~ n t i a l

as i n t e g r a l s o f

forms o v e r c e r t a i n c y c l e s .

rational

Petrovsky condition for algebraic hyperbolicity.

l a c u n a s . The f i r s t

hyperbolicity,

This y i e l d s

the

s t e p i s a s e c t i o n on

In a second s e c t i o n i n v e r s e s o f

p o l y n o m i a l s a r e s t u d i e d . The t h i r d

operators

section deals with

hyperbolic

intrinsic

t h e f o u r t h w i t h fundamental s o l u t i o n s and i n t h e f i f t h

a f o r m u l a by Gelfand i s used t o d e r i v e t h e d e s i r e d r e s u l t s .

1.1 A l g e b r a i c h y p e r b o l i c i t y

Let f ( x ) = f ( x ~ . . . , x . )

be a n a l y t i c

for

small

x and l e t

a~O be f i x e d

i n Rn . Definition

The f u n c t i o n f

i s s a i d t o be m i c r o h y p e r b o l i c w i t h r e s p e c t

to a if Im t for

all

>O=> ÷ ( x + t a ) ~ 0

sufficiently

L e t us d e v e l o p f

small

at

t and a l l

sufficiently

x=O i n s e r i e s o f

terms o f

small

real

x.

increasing

homogeneity, f(x)

The f i r s t f

and w i l l

=

fo(x)+f~(x)+...+f~(x)+

n o n - v a n i s h i n g term,

....

say fm,

i s c a l l e d the p r i n c i p a l

p a r t of

be denoted by Pr f .

Examples. When m=O, f(O) ~43 and r e s p e c t t o any a.

f

is

When m=l and ÷ i s

trivially real,

÷ is

microhyperbolic with locally

hyperbolic with

11

r e s p e c t t o any a w i t h

Lemma

Put h ( t , s ) =

Pr f ( a ) ¢ 0 .

f(ta+sx)

is microhyperbolic ~ith (I.I.i)

h(t,s)

where H ( t , s )

is

analytic

small

for

small

=H(t,sl at

the ori9in,

H(O,O)¢ 0 and

s and v a n i s h when s=O.

t h e numbers ck a r e r e a l (1.1.3)

and s .

~ (tmcks)

part of

r e s p e c t b o t h a and - a ,

r e s p e c t t o a.

Pr f

p r o p e r t y when f

is microhyperbolic with

real

for

real

hyperbolic with

and h ( t , s ) the

least

a r e t h e same. k for

(1.1.4) for

small

s and t .

with

Im t

>0,

in

= go(s)

Without

s.

exponent i s e a s i l y de9ree of H(O,O) of

Pr h ( s , t )

= Pr h ( l , O )

+ t91(s)

of

generality,

power s e r i e s ,

seen t o c o n t r a d i c t i s m and e q u a l

requiring

does n o t v a n i s h so t h a t

that (1)

real.

o f m,

parts of let

f

m be

we may a l s o assume t h a t (1)

to

holds, real

power s e r i e s . with

t h e dk b e i n g

and t

is

In f a c t ,

the

a fractional

a s s u m p t i o n . Hence t h e

that

of

Pr f .

In p a r t i c u l a r ,

a statement independent

does n o t v a n i s h .

We can t h e n 9o t o

Pr f ( x ) ¢ O and a r e t h e n s u r e t h a t (2)

small

this

does n o t v a n i s h ,

and

that

t~gk(s)+...

the Puiseux s e r i e s

Pr f ( x )

(3)

said

r e s p e c t t o a.

Then t h e p r i n c i p a l

+...

are actually

= Pr f ( a )

from

is

the expansion

loss of

term o f

the assumption t h a t

again without

is

it

t h e numbers dk a r e

But s i n c e h ( t , s ) ¢ O when s i s

these series

e x i s t e n c e o÷ a f i r s t

follows

Disregarding the definition

m>O. Then, by t h e p r o p e r t i e s Puiseux s e r i e s

f(a)

Pr f ( x ) 4 ~ .

which gk(O)¢O i n h(t,s)

It

r e s p e c t t o a,

x and s and hence f ( x ) / P r

P r o o f . Choose an x such t h a t

is

= Pr f ( t a + s x ) .

is microhyperbolic with

locally

t h e dk a r e

h(t,s)

hyperbolic with

is

f

If

t o be l o c a l l y

When f

if

+ h i g h e r terms,

and t h e p r i n c i p a l

H(O,O)

has t h i s

Then,

~ (t+ d k ( S x , a ) )

d k ( S X , a ) = Ok(S)

When f

complex t

r e s p e c t t o a,

analytic

(1.1.2)

Note.

with

follow,

t h e f o r m u l a (3)

9m(O)

being a

(4)

12

consequence o f

t h e s e two.

This finishes

the p r o o f .

L e t us n o t e t h a t (1.1.5) for

all

Pr f ( x ) x.

In

Definition

H(O,O)~ c k .

t h e s e q u e l we s h a l l

Let C ( f ~ a ) ,

component o f

=

called

assume t h a t

Pr f ( a ) = l .

the h y p e r b o l i c i t y

t h e complement o f

the real

cone o f

f,

be t h e

h y p e r s u r f a c e Pr f ( x ) = O which

c o n t a i n s a.

Accordin9 to

(5)

this

C k ( a , x ) > O on C ( a , f ) .

Theorem

C(f,a)

÷ is uniformly precisely, (1.1.6)

is

means t h a t In f a c t ,

b i n K,

is

when x=a, a l l

hyperbolic with

a positive

IsI,Ixl

P r o o f . L e t us w r i t e

in C ( f , a )

an open c o n v e x cone.

locally

there

x is

O => f ( x + s b ) # O .

the formula

(2)

with

x r e p l a c e d by x+sb and w i t h

t a and sb i n t e r c h a n g e d , ÷(ta+sb+x)= H(t,s,x) Here H ( t , s , x } Further, t>O,

~ (s + d k ( b , x + t a ) ) .

does n o t v a n i s h f o r

since the

none o f

left

sufficiently

small

arguments.

s i d e does n o t v a n i s h when s i s

t h e numbers dk c r o s s e s t h e r e a l

axis.

real

and Im

Hence, s i n c e

d k ( a , s b ) = ck s + s m a l l e r where t h e ck a r e p o s i t i v e ,

we have

Im t>O => Im d k ( b , x + t a ) when x and s a r e s m a l l follows.

Pr f ( t a + s b ) It

follows

that

> O.

enough. Hence t h e second p a r t

To p r o v e t h e f i r s t

part,

of

t h e theorem

note t h a t

= Pr f ( a )

B ( t

+sck(a,b)).

C = C ( a , f ) c o n t a i n s t a + s b when b i s

This completes the p r o o f .

it,

in

C and t , s

>0.

13

Translates For s m a l l

real

y,

let

f~(x)

=

be t h e t r a n s l a t e o f

f

f(x+y) by y .

Our l a s t

theorem has t h e f o l l o w i n 9

corollary

Theorem .

If

f

i s m i c r o h y p e r b o l i c w i t h r e s p e c t t o a,

so i s f y .

The

function

y-> C ( f y , a ) is

i n n e r c o n t i n u o u s i n the sense t h a t

if

y t e n d s t o z,

s i d e above c o n t a i n s any compact s u b s e t o f sufficiently

Proof.

It

C(f,a)

then t h e r i 9 h t

when z i s

c l o s e t o y.

suffices

t o prove t h e theorem when z=O i n which case i t

f o l l o w s from the p r e v i o u s one.

Homogeneous h y p e r b o l i c p o l y n o m i a l s . When f ( x )

has a p r i n c i p a l

p a r t Pr f

0 => r - ~ f ( r x )

-> Pr f ( x ) .

r-> It

f o l l o w s from t h i s

then P ( x ) = Pr f ( x )

that,

if

f

of

o r d e r m, then

is microhyperbolic with

r e s p e c t t o a,

i s a p o l y n o m i a l , homo9eneous o f o r d e r m w i t h t h e

property that (1.1.7)

Im s>O, x r e a l

=> P ( s a + x ) f O .

Such p o l y n o m i a l s a r e s a i d t o be h y p e r b o l i c w i t h r e s p e c t t o a. of

those w i l l

homo9eneous,

The s e t

be denoted by H y p ( a , m ) . Note t h a t s i n c e P i s (7)

holds with

Im s>O r e p l a c e d by Im s¢O so t h a t P i s

also h y p e r b o l i c with respect to -a.

It

is obvious that

if

two

homogeneous p o l y n o m i a l s P,Q a r e h y p e r b o l i c w i t h r e s p e c t t o a, has t h e same p r o p e r t y and C(PQ,a)=C(P,a )A C(Q,a).

then PQ

14

Examples. Let m be the degree o f

a real

m=O, P i s h y p e r b o l i c i f

if

and o n l y

r e s e p e c t t o any a~O. When m=l, w i t h P(a)#O and C(P~a)

is

homogeneous p o l y n o m i a l P.

P~O and then

it

is hyperbolic with

P i s h y p e r b o l i c w i t h r e s p e c t t o any a

the h a l f - s p a c e P(x)~O c o n t a i n i n g a.

q u a d r a t i c forms P which a r e h y p e r b o l i c and not p r o d u c t s o f f a c t o r s a r e t h o s e such t h a t r e s t minus o r z e r o ,

i.e.

If

The o n l y

linear

± P has a s i g n a t u r e w i t h one p l u s and the

i n normal form,

P ( x ) = x ~ = - x = = - . . . - x k Z,

lO and C(P,a)

the p a r t of

t h e d o u b l e cone P(x)>O c o n t a i n i n g a.

C o l l e c t i n g some of our e a r l i e r Theorem

results,

A homogenous p o l y n o m i a l P,

also hyperbolic with

h y p e r b o l i c w i t h r e s p e c t t o a,

locally

i s the c o r r e s p o n d i n g h y p e r b o l i c i t y

Cy(P,a)

is

a t y and t h e p r i n c i p a l s - k P ( y + s x ) , s->O,

be c a l l e d the

p a r t of

where k i s

localization

of

the

the f u n c t i o n

if y->

linear

local

hyperbolicity

cone of P

×-> P ( x + y ) , d e f i n e d by Py(x) = l i m

the o r d e r of

y as a z e r o of P, w i l l

be

P a t y.

Examples. When k=O, Py=P(y) is

cone,

is

inner continuous.

The cone Cy=Cy(P,a) w i l l

Py(x)

For any y,

h y p e r b o l i c w i t h r e s p e c t t o a and,

Cy(P,a)

c a l l e d the

we have

r e s p e c t t o any b in C ( P , a ) .

f u n c t i o n x->P(x+y) i s

is

i s a c o n s t a n t and C~ i s Rn\O.

and Cy i s a h a l f - s p a c e . When k=2,

Py(x)

When m=1,

has degree 2

and Cy i s a cone o r a h a l f - s p a c e .

Lineality The l i n e a l i t y y for

L(P)

of

a p o l y n o m i a l P i s d e f i n e d t o be the s e t o f

which P ( x + t y ) = P ( x ) f o r

a linear

space and i f

its

all

x and a l l

dimension i s k,

numbers t .

The l i n e a l i t y

P i s a p o l y n o m i a l on t h e

all is

15

quotient R"IL(P),

a l i n e a r space o f

Very s i m p l y one can say t h a t P i s number of v a r i a b l e s .

dimension n-k where k= dim L ( P ) .

a p o l y n o m i a l i n n-k but no l e s s

A p o l y n o m i a l whose l i n e a l i t y

vanishes is said to

be c o m p l e t e . When P i s homogeneous and h y p e r b o l i c , C ( P , a ) + L ( P ) = C(P,a) trivially.

Hence a h y p e r b o l i c i t y

cone i s open i f

and o n l y i f

c o r r e s p o n d i n g p o l y n o m i a l i s c o m p l e t e . When P has a z e r o of y,

P ( x + y ) = P~(x)+ h i g h e r terms i n x ,

the

order k at

Py not z e r o as a p o l y n o m i a l i n x ,

we have P~(x+y)= l i m s - k P ( y + s ( x + y ) ) = P y ( x ) . Hence y i s

in

cone a t y i s

the

lineality

invariant

of

P~ and hence the

under t r a n s l a t i o n s

in

local

hyperbolicity

the y d i r e c t i o n .

1.2 D i s t r i b u t i o n s associated with the inverses of a homogeneous hyperbolic polynomial

It

is well

Q(y-N)

known t h a t

there for

if

f(z)

some N, then f(x+iO)

of

(H~rmander

limit,

Here B i s

-i

limit for

y ~ 0 any ~ > 0 . An easy p r o o f the

64), -

If(x+ib)g(x)dx

=

Jj f ( x + i ( y + c ) ) 9 = ( x , y i d x d y

the s t r i p

s u p p o r t and 9 ( x , Y )

O,

the o r i g i n

1983 I ,

is analytic

66),

when when z

Rn and y t o an open cone w i t h

and b e l o n g i n g t o a b a l l I f ( x + i y ) l~const

Then the

(H~rmander

z = x + i y in Cn,

b e l o n g s t o an open s u b s e t of at

kO. Suppose a l s o t h a t

l y t -N.

limit f(x+iO)

=

lim

f(x+iy),

y->O,

e x i s t s as a d i s t r i b u t i o n .

Proof.

Change v a r i a b l e s

a p p l y the f o r m u l a

(I)

that

so

the y ~ - a x i s p o i n t s i n t o the cone,

to the f i r s t

variable with

the o t h e r s as

parameters and i n t e g r a t e w i t h r e s p e c t t o x = , . . . , x . .

Usin9 t h e of

lemma we can now i n t r o d u c e two i n v e r s e s o u t s i d e the o r i g i n

a homogeneous p o l y n o m i a l P ( x ) o f

degree m, h y p e r b o l i c w i t h r e s p e c t

t o a v e c t o r a~43 in Rn.

Theorem . L e t x - > c { x ) be a smooth f u n c t i o n from R"\O, degree I

and chosen so t h a t c(x)

for

all

homo9eneous o f

x.

b e l o n g s t o C~(P,a)

Then t h e l i m i t s l l P ( x + ¢ i O ) = l i m P ( x + i s c ( x ) as O<s->O, ¢=I o r - I ,

exist

independently of

o u t s i d e the o r i g i n

Proof.

the c h o i c e of

c(x)

and d e f i n e d i s t r i b u t i o n s

which a r e homogeneous of

Since 1 / P ( x + i y ) s a t i s f i e s

degree -m.

the h y p o t h e s i s of

t h e p r e c e d i n g lemma

17

locally, unity.

the existence of

t h e two i n v e r s e s ~ o l l o w s by a p a r t i t i o n

The h o m o g e n e i t y f o l l o w s

Note t h a t ,

since P is

When P ( D ) ,

to radii

D=~/ibx the

differential

P

P is

said

fundamental s o l u t i o n vertex at

R~.

is

P(~)

said

is

be i t s

a constant characteristic

t o be a f u n d a m e n t a l s o l u t i o n

such an E i s

~(x).

E with

s u p p o r t in

hyperbolic

if

it

has a

a p r o p e r c l o s e d cone K w i t h

its

supports for

t(x)

which

The i n t u i t i v e

x=O in an e l a s t i c

is

positive

meanin9 o f

i.e.

K will

Without

the directions

be c a l l e d

restriction

We w r i t e

C is

definition

in

finite

velocity

be c a l l e d

is

that

them

in

in

all

the hyperplane t ( x ) = O .

a

a time

a shock a t

space In view of

P.

we may assume K t o be c o n v e x .

r e q u i r e d t o be i n v a r i a n t that

this

on K\O w i l l

t h e p r o p a g a t i o n cone o f

i n t r o d u c e an open cone C d u a l

functions.

s u p p o r t s in

medium whose movements a r e g o v e r n e d by P as a

law s h o u l d p r o p a g a t e w i t h

directions,

convolutions with

unique.

function

function.

physical

=

the o r i g i n .

A linear

clear

let

t o be i n t r i n s i c a l l y

N o t e . By t h e t h e o r e m o f

to

E(x)

P(D)E(x)

Definition

this,

in

if

(1.3,1)

cone,

imaginary gradient,

operator,

polynomial. A distribution of

from the o r i g i n

to

hyperbolicity

with

coefficient

P and c ( x ) .

homogeneous, P has u n i q u e r e s t r i c t i o n s

manifolds transversal

1.3 I n t r i n s i c

from the homogeneities of

of

It

is convenient

t o K and d e f i n e d as t h e s e t o f

time

the form

under r e a l

open and c o n v e x .

It

linear will

transformations.

be c a l l e d

It

is

the hyperbolicity

18

cone.

Theorem

P is

intrinsically

h y p e r b o l i c w i t h cone K i f

an o n l y i f ,

x

9 i v e n a compact subset B of C, t h e r e

is a continuous real

function

s(9)=s(-9)

d e f i n e d in C and homogeneous o÷ degree 1 such t h a t

(1.3.2)

9 i n C, s(9)>1 = > I P ( ~ - i 5 ) - ~ l ~ c o n s t ( l + l ~ - i 9 ) I ==n.~.

When t h i s

condition is satisfied,

E is given explicitly

as an i n v e r s e

Fourier-Laplace transform (1.3.3)

E(x)

where 3=~+ig, side is Note. (3)

=

(2~) -~

g is

in

(I)

P(5)-~d5

Under t h e s e c o n d i t i o n s ,

g and t h e s u p p o r t o f

i s weakened t o

(4)

below and the c h a i n o f

and (1)

Proof.

is

Let

a distribution

the r i g h t

E i s c o n t a i n e d i n K. integration

i s m o d i f i e d a c c o r d i n 9 1 y , the theorem h o l d s when P(5)

F o u r i e r - L a p l a c e t r a n s f o r m of

h(x)

x.S

and s ( g ) > l .

-C

independent of

If

J exp

fix)

is

in

the

w i t h compact s u p p o r t

r e p l a c e d by the c o n v o l u t i o n f ~ g ( x ) = ~ ( x ) .

g be i n C and denote t h e t i m e f u n c t i o n

be a smooth f u n c t i o n which

is

1 when t ( x ) < l

g . x by t ( x ) . and 0 f o r

Let

t(x)>2.

We

have ~(x)

=

P(D)h(x)E(x)

+ P(D)

(1-h(x))E(x)

where g ( x ) = h ( x ) E ( x ) and the second f u n c t i o n above, say k ( x ) ~

have

compact s u p p o r t s . Let G(5)

=

I

exp

-ix.5

9(x)

be the F o u r i e r - L a p l a c e t r a n s f o r m of 1 =

The lower bound o f

dx

9 and K t h a t of

k.

We have

P(5)G(5)+K(S).

g . x on t h e s u p p o r t o f

c o n t i n u o u s f u n c t i o n u(9)

of

9 which i s

k for

g in C i s a p o s i t i v e ,

homo9eneous o f

degree 1.

For

i n -C we have IK(S)I

~ A(I+ISI)

for

some p o s i t i v e

numbers A and N.

I/2

and hence (3)

h o l d s when

N exp

It

-u(9)

follows that

IK(S)I

i s a t most

9

19

lu(~)l>const Io9(2+I~I). Here t h e

lo9arithm is

a l 9 e b r a i c theorem, 364)f

not necessary which f o l l o w s from a 9 e n e r a l

t h e S e i d e n b e r g - T a r s k i lemma (Hormander 1983 I I

With a s u i t a b l e c h o i c e o f

s,

this

theorem. To p r o v e t h e second p a r t , distribution

sense and l e t

f(×)

let

p r o v e s the f i r s t

E be d e f i n e d by (3)

Since F i s of

fast

Cauchy's theorem, s(9)>I,

does not

=(2~) -~

When

inte9ral

Then

i s a b s o l u t e l y c o n v e r g e n t . By

integral

to P(-D)f

5=~+i9 w i t h

9 fixed

in -C,

as Ion9 as 9 s a t i s ÷ i e s t h e proves t h a t

x i s o u t s i d e K, we can f i n d

x.9>O. With t h e s u p p o r t o÷ f

in the

F(~)P(5)-~dS.

the c h a i n o~ i n t e g r a t i o n ,

c o n d i t i o n s s t a t e d . Cangin9 f solution.

j

d e c r e a s e , the

i n f l e u n c e the

the

be a smooth f u n c t i o n w i t h compact

s u p p o r t and the F o u r i e r - L a p l a c e t r a n s f o r m F ( 5 ) . l~(-x)E(x)d×

p a r t of

c l o s e to x,

E i s a fundamental

a permitted ~ for

we a r e then

which

in a s i t u a t i o n

when t h e second i n t e g r a n d above tends t o z e r o u n i f o r m l y when 9 i s r e p l a c e d by t9 and t support of

tends t o plus

infinity.

E i s c o n t a i n e d i n K and f i n i s h e s

This proves t h a t this

the

sketchy proof.

Non-homogeneous h y p e r b o l i c p o l y n o m i a l s The c o n d i t i o n for

(2)

can be s i m p l i f i e d

some t i m e f u n c t i o n

has de9ree m and i t s

e.x,

considerably. It

that,

the f o l l o w i n g c o n d i t i o n h o l d s where P(~)

principal

part

i s denoted by pm,

(1.3.4)

Pm(e)~O and P ( ~ + t e ) # 0 when Im t

When t h i s

condition is satisfied,

respect to

suffices

< some t o .

we say t h a t P i s

e. Since Q=Pm i s t h e p r i n c i p a l

p a r t of

hyperbolic with P, we have

s->~ => s - ~ P ( s ( ~ + t e ) ) - > Q ( ~ + t e ) and i t

follows that

hyperbolic wit applies.

(4)

Q w i t h to=O. Hence Q i s a l s o

r e s p e c t t o e and the a l g e b r a i c t h e o r y o f

In p a r t i c u l a r ,

hyperbolicity

holds f o r

Q is

cone C(~,e)

section 1

h y p e r b o l i c w i t h r e s p e c t t o any 9 i n the

o f 6.

It

can be shown t h a t a l s o P has t h i s

p r o p e r t y b u t s i n c e homo9eneous h y p e r b o l i c p o l y n o m i a l s a r e our main

20

interest,

we s h a l l

h e r e l e a v e t h e non-homogeneous c a s e .

1 . 4 Fundamental s o l u t i o n s

of

homogeneous h y p e r b o l i c o p e r a t o r s .

P r o p a g a t i o n cones. L o c a l i z a t i o n s .

When P ( { )

is

to

C=C(P,e) be i t s

B,

let

consisting

a homogeneous p o l y n o m i a l which

of

all

x for

hyperbolicity

which x.C~O,

o b v i o u s l y c l o s e d and c o n v e x . P.

General c o n i c a l

The r e a s o n i s

that,

It

according to

cone.

i.e.

will

is

refraction.

hyperbolic with

respect

The d u a l cone K=K(P,e)

x.{~O for

be c a l l e d

all

{

i n C,

is

t h e p r o p a g a t i o n cone o f

the preceding s e c t i o n ,

P has t h e

fundamental s o l u t i o n E(x) with

= (2~)--

s u p p o r t i n K.

In

To g i v e an i d e a o f is

the degree of

P,

J exp i x . S

particular,

When m=l,

P(e)$O, C i s

P is

and

P is

When m=2, P i s

hyperbolic with

hyperbolic.

h e r e a r e some examples where m polynomial.

e may be a n y t h i n g , C = R " and

respect to

e if

and o n l y

with

a q u a d r a t i c f o r m and i t

respect to

some e i f

one p l u s and t h e r e s t

if

generated is

easy t o

and o n l y P o r minus o r

-P

zero.

In

K is

the

c o o r d i n a t e s we t h e n have

6=(1,0,...,0).

closed

$0,

hyperbolic with

has L o r e n t z s i g n a t u r e , suitable

intrinsically

supposed t o be a r e a l

a constant

S=~-ie,

t h e h a l f - s p a c e P ( e ) P ( { ) > O and K t h e h a l f - l i n e

by P ( e ) g r a d P ( e ) . see t h a t

P is

p r o p a g a t i o n cones,

Examples. When m=O, P i s K={O}.

d{/P(S),

cone

Here

C is

the

open

cone

x,>O,

P(¢)>O

and

x=~O, C-=Xz =

-Xz=-...

--Xk =

~0

i n t e r s e c t e d by t h e h y p e r p l a n e s x , ÷ = = O , . . . , x . = O .

Our l a s t

example i l l u s t r a t e s

hyperbolic polynomial P is

the fact

that

orthogonal to

t h e p r o p a g a t i o n cone K o f

the

lineality

L of

P.

In

a

21

fact,

the hyperbolicity

x . L = - x . L ~ O when x i s of

two p o l y n o m i a l s

the propagation

c o n e C has t h e

in

is

K.

Since

the

cone o f

the

that

hyperbolicity

intersection

the

property

product

of

is

their

C+L=C so t h a t

cone o f

the

hyperbolicity

the union of

their

product cones,

propagation

cones.

Localization.

The wave f r o n t

L e t Q be t h e

localization

surface.

of

a hyperbolic

polynomial

P at

a point

9 so

that (1.4.1)

P(tg+S)

where k i s

the order

= t~-kQ(S)+

of

O(tk-~),

9 as a z e r o o f

fundamental solution

E(x)

of

fact,

real

and c o n s i d e r

E(x)

=

let

t

(2hi) n

be l a r g e exp-ix.t9

=Jexp i S . x w h e r e S= ~ - i e .

P.

We s h a l l

see t h a t

P has a c o r r e s p o n d i n 9

J exp i ( S - t g ) d ~ / P ( S )

the

localization.

In

=

d~/P(tg+s)

Multiplyin9

by a smooth f u n c t i o n

h(x)

and

integratin9

we 9 e t (1.4.2)

(2~i)~

i E(x)h(x)exp-itx.9

dx = j

H(-5)d~IP(tg+S),

where H(S) is

= J exp - i x S

the Fourier-Laplace

IP(tg+S)l

~ IP(iel

G(-S)

is

limit

t->-

(1.4.3)

fast

h(x)dx

transform

f 0 and,

decreasin9

in

of

since ~.

h.

9 is

Hence,

Since P is

hyperbolic,

smooth w i t h usin9

(I),

compact s u p p o r t ,

we can p a s s t o

the

and g e t

t ~-k

lexp-ix.t9

E(x)h(x)dx

->

i F(x)h(x)dx

where F(x) is

(2~} -~

the fundamental solution

K(g) 9.

=

dual

to

the

for

9@0 i s

of

hyperbolicity

The p r o p a 9 a t i o n

K(9)

i exp ix.3

cone K ( 9 )

called

the

Q with

d~IQ(S) support

cone C(g) is

said

wave f r o n t

of

in

the

localization

t o be l o c a l . surface

a propagation

Q of

The u n i o n o f

W=W(P,e)

of

cone

all

P (with

P at

22

respect to e).

Its

p r o p e r t i e s are 9iven in

Lemma . The wave f r o n t p r o p a g a t i o n cone of

Proof.

It

suffices

hyperbolicity

surface

t o c o n s i d e r c o m p l e t e p o l y n o m i a l s P. L e t C(5) the l o c a l i z a t i o n

c o n t a i n s the h y p e r b o l i c i t y the

of C(9),

K(~)

Note.

P ( O ) f O, K(9)

If

P, K(9)

lies

the

is outer continuous, (conically)

local

the o r i g i n

i.e.

as 9 tends t o

c l o s e to K(S).

spanned by a m u l t i p l e of

lineality

the p r o o f .

9 i s a s i m p l e z e r o of 9rad P. Hence most of

p r o p a g a t i o n cones are j u s t

shown t h a t W i s c o n t a i n e d i n t h e dual of

half-lines.

It

can be

t h e h y p e r s u r f a c e P(~)=O,

t h e h y p e r s u r f a c e p a r a m e t r i z e d by 9 t a d P(~)

S,

This proves

This f i n i s h e s

and i f

is

are c o n t a i n e d in

K and s i n c e ~ i s c o n t a i n e d in t h e

is just

be the

5. Since e v e r y C(9)

cones K(5)

i n a h y p e r p l a n e when ~ 0 .

is a half-line

the n o n - t r i v i a l

P at

hyperbolicity

come a r b i t r a r i l y

t h a t W i s a closed p a r t of

of

cone C o f P and t h e f u n c t i o n 9->C(9)

local

K and t h e f u n c t i o n 9->K(9) t h e cones K(9)

i s a closed s e m i a l g e b r a i c p a r t of

codimension 1.

cone o f

inner continuous,

the f o l l o w i n 9

w i t h P(~)=O which

i.e.

includes

t h e h y p e r p l a n e s x.~=O when ~ i s a n o n - s i m p l e p o i n t o f P ( ( ) = O .

In

Figure I

, where n=3,

some r e a l

t h e c o r r e s p o n d i n 9 wave f r o n t

h y p e r s u r f a c e s P=O are p a i r e d w i t h

surfaces,

L e t us now r e t u r n t o the f o r m u l a

(3).

both in p r o j e c t i v e

If

y is

a point

clothing.

in K o u t s i d e W

and h i s a smooth f u n c t i o n w i t h s u p p o r t y b u t o u t s i d e W, a l l s i d e s of

(3)

transform of

vanish.

E ( x ) h ( x ) tends t o z e r o f o r

In the n e x t s e c t i o n , the e f f e c t E(x)

Hence t h e f o r m u l a

that

this

shows t h a t

t h e F o u r i e r t r a n s f o r m has f a s t

i s smooth ( a c t u a l l y r e a l

analytic)

the F o u r i e r

l a r g e v a l u e s of

v e r y weak c o n c l u s i o n w i l l

the r i g h t

the argument.

be s t r e n g t h e n e d t o

d e c r e a s e and hence t h a t

in K o u t s i d e W. I f

we

23

'~.J ne ? = o

///I

~

~ I

I / I

~ . ~

Z

~''--~

~

Ip

.,~ ~v

0

~.

0

IX

~

/

C~ Fig.

I.

Hyperbolicity cones and propagation cones when n=3. Some

local hyp~rbolicity cones and the corresponding local propagation cones are also indicated.

24

interpret

in

(3)

the n e g a t i v e sense,

support close to support of solutions

the support of

F,

E c o n t a i n s the union of belonging to

the

taking

the f u n c t i o n

we can c o n c l u d e t h a t the supports of

Iocalizations

of

P at

local

this

union

is

t h e wave f r o n t

surface.

p r o p a g a t i o n cones a r e h a l f - l i n e s

and

E x c e p t i o n s may o c c u r when a l o c a l i z a t i o n solution

with

a lacuna in

its

the s i n g u l a r

the fundamental

points

t h e s e s u p p o r t s a r e t h e same as t h e c o r r e s p o n d i n g l o c a l cones,

h with

9~0.

When a l l

propagation

T h i s happens when t h e

i n many o t h e r c a s e s .

of

P has a f u n d a m e n t a l

p r o p a g a t i o n cone.

Conical refraction. Consider the following (1.4.4)

t - > ~ => t ~ - k l

formula,

proved in

E(x-y)h(y)exp -iy.t9

This f o r m u l a i s connected w i t h a 9 e n e r a l way.

t h e same way as

When v

is

dy ->

IF(x-y)h(y)dy.

t h e phenomenon o f

a smooth f u n c t i o n

with

(3),

double refraction compact s u p p o r t ,

in

the

function u(x)

=

i E(x-y)v(y)dy

has s u p p o r t i n K+supp v and s o l v e s t h e e q u a t i o n P ( D ) u ( x ) = v ( x ) . let

one o f

the v a r i a b l e s

source S of

vibration

and u ( x )

where p r o p a g a t i o n o f

waves i s

monochromatic s o u r c e of space and t i m e ,

the

then the limit

suitable by t h e

of

integral

localization

t

as t h e r e s u l t i n g

=

h(x)exp ix.t5

of

the

is

with

side of

vibration

P to

~.

o r t h o g o n a l t o an o p t i c a l

with

This

refraction.

entering a double refracting

(4).

a medium

To i m i t a t e

a small

The r e s u l t i n g

and t

is

vibration

The f o r m u l a s a y s t h a t multiplied

source h of precisely

transversally rise

a

region in

by a

a medium g o v e r n e d the s i t u a t i o n

A r a y of monochromatic l i g h t

crystal

axis 9ires

is

of

where 9 i s f i x e d

s o u r c e h exp i x . t ~

a vibration

Q of

encountered in c o n i c a l

left

the o r i g i n .

we

can be c o n s i d e r e d as a

g o v e r n e d by t h e o p e r a t o r P.

support at

a vibration

power o f

v(x)

high f r e q u e n c y l o c a t e d at

we p u t v ( x )

l a r g e and h has s m a l l is

x represent time,

If

to a side

t o a monochromatic s o u r c e

25

i n s i d e the c r y s t a l

which t h e n ~ p r o p a g a t e s a c c o r d i n g t o r u l e s o f wave

p r o p a 9 a t i o n in the c r y s t a l .

Now s i n c e l i g h t

o n l y the h i g h f r e q u e n c y l i m i t the c r y s t a l ,

the

seen in c r y s t a l s K(g)

has a v e r y high f r e q u e n c y ,

becomes c l e a r l y

visible.

In the case o f

l o c a l i z e d o p e r a t o r Q i s a wave o p e r a t o r and what i s is

the p r o j e c t i o n o n t o space of

in space and t i m e .

Since the s i n g u l a r i t i e s

its of

p r o p a g a t i o n cone

the c o r r e s p o n d i n g

fundamental s o l u t i o n a r e s i t u a t e d on t h e boundary of K(g) i s much b r i g h t e r

than t h e i n s i d e .

t h e p r o p a g a t i o n cone h i t s

Note. Ludwi9

(1961)

the boundary

In the a c t u a l e x p e r i m e n t , l i g h t

from

a s c r e e n where one can see a luminous r i n g .

deduced c o n i c a l r e f r a c t i o n

from M a x w e l l ' s

e q u a t i o n s . The e x p e r i m e n t d e s c r i b e d above can never be r e a l i z e d precisely since

it

is

i m p o s s i b l e t o have p u r e l y monochromatic l i g h t .

When t h e e x p e r i m e n t i s performed v e r y c a r e f u l l y d i s s o l v e s i n t o two w i t h a s m a l l dark r i n g Wolf

1975 f o r

the

luminous r i n g

i n between (see Born and

r e f e r e n c e s and d i s c u s s i o n ) . A g e n e r a l t h e o r e t i c a l

explanation for

this

phenomenon was g i v e n by Uhlmann

(1982).

1.5 The H e r 9 1 o t z - P e t r o v s k y f o r m u l a

In t h i s

s e c t i o n we s h a l l

deduce f o r m u l a s f o r

fundamental s o l u t i o n s o f

homogeneous h y p e r b o l i c c o m p l e t e p o l y n o m i a l s i n t h e p r o p a g a t i o n cone b u t o u t s i d e the wave f r o n t cycles.

s u r f a c e as i n t e g r a l s o f

T h i s c o u l d be done s t a r t i n g

Fourier-Laplace transforms,

from t h e i r

but a f o r m u l a of

O-function in

terms of

departure. It

uses the f o l l o w i n g f a m i l y of

(1.5.1)

where s i s

H(s,z)

real

=

rational

forms o v e r

e x p r e s s i o n s as i n v e r s e

Gelfand e x p r e s s i n g the

p l a n e waves i s a more c o n v e n i e n t p o i n t of

J e-r=r-=-~dr

=

F(-p-t)=F(1-t)/(-p-t)

in one v a r i a b l e ,

F ( - s ) z =,

O. The F - f a c t o r

when s i s an i n t e 9 e r p = O , l , 2 , . . . .

functions

We have

. . . . (-t)

i s meromorphic w i t h p o l e s

26

so t h a t = -(-1)=

H(p+t,z)

zp~l(p+t)

.... ( 1 + t ) t

and hence H(p+t,z)

(1.5.2)

=

-(-z)"/p!t

+H(p,z)+O(t)

where (1.5.3)

H(p,z)

Since H ( s , z )

i s u n d e f i n e d w h e n s=p,

a definition. (1.5.4)

z +F' ( 1 ) - 1 - 1 / 2 - . . . - i / p ) p ! .

=-(-z)"(iog

we a r e f r e e

t o use

(2)

and

(3)

as

Note t h a t

t>O => H ( p , t z )

= tPH(p,z)

+ ( - z ) p Io9 t / p ! ,

so t h a t (1.5.5)

We s h a l l cut

dH(s,z)/dz

= -H(s-l,z),

dH(p,z)/dz

= -H(p-l,z)

use H ( s , z )

is

as an a n a l y t i c

along the n e g a t i v e real

below t h e r e a l real,

~(~)

axis.

function Its

is

of

z in

which we s h a l l

d e f i n e d as t h e d i s t r i b u t i o n

lemma we s h a l l

=d~...d~.,

0k(1)

with

a l s o use.

When t

H(s,it+O).

need some d i f f e r e n t i a l = 0({)

t h e complex p l a n e

b o u n d a r y v a l u e s f r o m above and

axis are distributions

H(s, i t )

For t h e n e x t

-(-z)~-~/(p-1)!.

forms,

d~k t a k e n away,

and t h e K r o n e c k e r f o r m ~({)

with

=

~

(--1)k--l~k({),

k=l,2,...,n,

the p r o p e r t y t h a t

(1.5.6)

df(~)w(~)

=

(~,~fl~

When f

is

homo9eneous o f

when f

is

quasihomogeneous o f f(t~)

de9ree -n,

a p o l y n o m i a l of

degree

then

df(~)g(~)w({)

Lemma

side vanishes. Also, t h e sense t h a t

+ p({)

d e 9 r e e m-n,

and 9 ( 4 )

is

homogeneous o f

= p(~)~(~).

( G e l f a n d ) . Let h(~)

homogeneous o f

the ri9ht

d e g r e e m-n~O i n

= t~-"f({)

where p i s -m,

+n)~(~).

d e g r e e 1.

be any smooth r e a l

Then,

in

positive

the d i s t r i b u t i o n

function

sense,

27

(1.5.2) with

~(x)

=

integration

Note.

(2Ki) - "

] H(-n,-ix.{)

over h(~)=l.

This f o r m u l a appears in G e l f a n d - S h i l o v

Proof.

We s h a l l

see t h a t

~(x) The r i g h t

= (2K) - "

side

is

the

j exp i x . ~ limit

o b t a i n e d by a d d i n 9 - ¢ h ( ~ ) integral

(1958).

Gelfand's formula results

p o l a r c o o r d i n a t e s and i n t e g r a t i n g

the

w(~)

out radially

zero of

the e x p o n e n t i a l .

we r e p l a c e ~ by r~ w i t h

in

~(~).

as 0O tends t o O. But s i n c e the

We s h a l l

side is

x.9 E ( x )

Proof.

respect to

result

x i n one

where t h e P e t r o v s k y

role.

l a c u n a t h e o r e m . When x i s

is

a polynomial of

is

homologous t o

d e g r e e m-n t h e r e

i n such a if

the Petrovsky

cycle (1.5.16) Note.

~(x)

Components where o f

homogeneous o f be c a l l e d

d e g r e e m-n)

z e r o i n X*

K\W where E ( x ) are called

the Petrovsky c r i t e r i o n .

It

o u t s i d e P*.

is a polynomial

(necessarily

l a c u n a s . The c r i t e r i o n

(2)

will

comes v e r y c l o s e t o b e i n g

necessary. The aim o f coefficient

this

chapter- to

introduce hyperbolicity

o p e r a t o r s and t o s t u d y t h e i r

as making P e t r o v s k y ' s We round o f f

with

for

fundamental solutions

l a c u n a theorem u n d e r s t a n d a b l e - i s

an a p p l i c a t i o n

to

constant so f a r

now a c h i e v e d .

the fundamental s o l u t i o n

E(x)

of

t h e wave o p e r a t o r

P ( D ) = D ~ = - D = = - . . . - D n ~, hyperbolic

with

respect

to 8 = ( I , 0 , . . . , 0 ) .

The p r o p a g a t i o n c o n e

is

d e f i n e d by

x1~O,

x~-xz=-...-x~

the wave s u r f a c e

= ~0,

is its boundary.

When

x=(l,O,..,O),

Re X* n P*

is

33

empty.

Hence, by t h e P e t r o v s k y c r i t e r i o n ,

propagation

cone

is

a

lacuna

when

n

the

is e v e n

interior

and,

by

o÷ t h e

its

converse,

not

a l a c u n a when n i s odd. N o t e . The P e t r o v s k y c r i t e r i o n f o r m u l a t e d as f o l l o w s . its

has a l o c a l

L e t L be a component o f

b o u n d a r y . L e t Y be t h e s e t o f

local E(x)

hyperbolicity

meet Y*.

We s h a l l

extension at is

homologous i n X*%P*

The use o f

in

on

which t h e c o r r e s p o n d i n 9 h y p e r p l a n e x.~=O.

this

form,

Then

L provided

the c r i t e r i o n

smooth c o e f f i c i e n t s .

chapter is Atiyah-Bott

Gelfand's decomposition of

p l a n e waves (Ge-S l , p . H~rmander ( 1 9 8 3 ) .

this

K\W and y a p o i n t

t o a c y c l e which does n o t

hyperbolic operators with

The main r e f e r e n c e f o r

(I,1971).

~ for

which can be

y a c r o s s the boundary of

prove in Chapter 7 t h a t

s u r v i v e s a passage t o Note.

points

cones do n o t meet t h e r e a l

has an a n a l y t i c

the Petrovsky cycle

variant

-G~rdin9

the b-function

118) makes o u r p r e s e n t a t i o n c l o s e t o

that

into of

CHAPTER 2

WAVE FRONT SETS AND OSCILLATORY INTEGRALS

The F o u r i e r t r a n s f o r m i s differential

t h e supreme t o o l

equations with

only for Fourier

equations with

si9ht

transform retains

1957) o f

variables

much o f

a parametrix for

construction

is

microlocal

inte9rals.

analysis.

2 . 1 Wave f r o n t

L e t us s t a r t set

But

this

is

true

with

is

Lax's construction of

a first

coefficients.

(Lax order

This

an e s t a b l i s h e d b r a n c h o f

mathematics

In

its

this

chapter,

t h e wave f r o n t

some of

basic

s e t s and t h e o s c i l l a t o r y

Lax's construction

a convenient definition.

is

a non-empty i n t e r i o r ,

s a i d t o be f a s t =

e v e r y N>O. S i m i l a r l y ,

When f

the f i r s t

applied to a

sets

f(X)

means f a s t

power. One o f

hi9h o r d e r .

i n R~ w i t h

defined there

its

variable

The c h a p t e r ends w i t h

sin91e equation of

set R(f)

coefficients.

the fundamental s o l u t i o n

now p a r t o f

concepts are introduced,

for

especially

seems t o be u s e l e s s f o r

became a p p a r e n t was i n

s t r o n 9 1 y h y p e r b o l i c system w i t h

conical

it

partial

phenomena i n v o l v i n 9 low f r e q u e n c i e s . F o r h i 9 h f r e q u e n c i e s t h e

i n s t a n c e s when t h i s

called

the study of

constant coefficients,

h y p e r b o l i c e q u a t i o n s . At f i r s t differential

in

C is

IX1->

with

open and c o n i c a l ,

J f(x)

f

e×p-ix.~

dx

decrease

subset.

s e t where i t s

transform =

fast

compact s u p p o r t , d e f i n e

t o be t h e maximal open c o n i c a l

f^(~)

a smooth f u n c t i o n

~,

decrease in e v e r y c l o s e d c o n i c a l a distribution

a closed

decreasin9 if

O(;xI--N),

if

When C i s

its

re9ularity

Fourier

35 is

fast

d e c r e a s i n 9 . When R ( f )

complement S ( f )

of

is

R~\O,

the re9ularity

f

is

set will

a smooth f u n c t i o n . be c a l l e d

The

the singularity

set. Lemma

Multiplication

by a smooth f u n c t i o n

does n o t d e c r e a s e t h e r e g u l a r i t y

(2.1.1)

R(hf)

Proof.

The F o u r i e r

compact s u p p o r t

set,

~ R(T).

transform of

(hf)^(~)=

h with

h$,

Jh^(~-g)f^(g)d 9

i s m a j o r i z e d by CN J ( l + i { - g l )

for

all

nei9hborhood C of

function

of

C and w r i t e

S i n c e 9 ( f ^)

prove t h a t

is

f^

f = f ~ + f = where f ~

inte9ral

transform of

infinity

above w i t h

f

fast

hf2

decr~easin9 i n

a narrow

9 be t h e c h a r a c t e r i s t i c has t h e F o u r i e r

d e c r e a s i n g , so i s is

fast

this,

transform

( h f , ) ^ and i t

remains to

d e c r e a s i n 9 when i t s

i n a some c l o s e d c o n i c a l

4= c o n t a i n e d i n C. When v e r i f y i n g

the

is

s o m e { o and l e t

fast

the F o u r i e r

argument { t e n d s t o of

I f ^ ( g ) Idg

N and some CN. Suppose t h a t

conical

9(f^).

-N

neighborhood D

we may keep g o u t o f

r e p l a c e d by f = .

But

then

C in

4-g n e v e r v a n i s h e s

and we have

I~IZI9 => l~-g12 Cl~l,

for

some c>O d e p e n d i n 9 on C and D~C. T h i s

estimate of

where in

leads to

the f o l l o w i n 9

( h f = ) ^,

const

i

( 1 ÷ I ~ i ) -N)

const

I

(I+igl)-N(I+IOI)Mdg

Igl~l~l

holds in

t h e second one.

proves the

(l+Igl)Mdg

the f i r s t

Here M i s

+

integral

fixed

and t h e o p p o s i t e i n e q u a l i t y

and N i s

arbitrily

lemma.

Since t h e F o u r i e r

transform of

Dkf(x), D=~/i~x, is

I~IS191 => l~-g12 clgl

a derivative

k=(k~,...,k,)

~ w f ^ ( ~ ) , we have i n a d d i t i o n

to

(1)~

large.

This

36

(2.1.2)

R(hDkf)

IT u i s

any d i s t r i b u t i o n

I R(f). and h l , h z

a r e smooth f u n c t i o n s

s u p p o r t s and supp h~ 33 supp hz a r e compact, R(h~u) Lettin9 point

the

of

sets of all

is

called

rays,

positive

set of

the s i n g u l a r i t y

i.e.

x in

of

it.

this

u at

its =

way,

x.

Both a r e o f

course

content of

r e p r e s e n t e d by r a y s ,

S~(u)

is

the

which a r e n e c e s s a r y t o

x.

singularity

s e t WF(u) o f

a distribution

u

sets,

U Sx(~). the

t h e wave f r o n t

The complement S~(u)

an e l e m e n t ~ t h e y c o n t a i n

The i n t u i t i v e

(H~rmander) The wave f r o n t

WF(u) In

set of

do n o t c o n t a i n 0 and w i t h

multiples

the product of

Note.

h(x)~O tend t o the

the approximatin9 family.

high f r e q u e n c i e s ,

Definition

h with

= l i m R(hu)

synthesize u close to

is

smooth f u n c t i o n s

independently of

R~(u)

lemma,

lemma shows t h a t R~(u)

exists

by t h e

compact

~ R(h=u).

the supports of

x,

then,

with

singularity

set

Sx(u)

appears

as

the

fiber

over

set.

For r e f e r e n c e we now s t a t e . Lemma

E v e r y wave f r o n t

Differentiation the the f i b e r s

all

is

a closed conical

and m u l t i p l i c a t i o n of

t h e wave f r o n t

Sx(hDku) for

set

by smooth f u n c t i o n s set of

The

projection

o n t o R~ i s

its

singular

of

the

wave

front

front

do n o t u,

increase

one has

set

of

a distribution

u

support.

Examples. S i n c e t h e F o u r i e r t r a n s f o r m o f one and i t s

a distribution

RnX R-\O.

c S~(u)

x.

identically

subset of

support is

s e t a r e empty e x c e p t a t

we have t h e d e c o m p o s i t i o n

the O - f u n c t i o n ~(x)

the o r i g i n ,

the fibers

x=O where t h e f i b e r

is

R~\O.

of

is its

wave

When n = l ,

37

2~i~(x) Since the F o u r i e r

= ( x - i O ) -~

transforms of

t h e n e g a t i v e and p o s i t i v e their It

wave f r o n t

follows

wave f r o n t

set of

points

consistent with

(1.4.3)

of

o p e r a t o r P(D) (x,~)

4.

In

the fact

such t h a t

over zero of

the preceding chapter t h a t E(x)

of

the

a homogeneous

constructed there is contained x belongs to H({),

particular, that

the fibers

v a n i s h on

and n e g a t i v e a x i s .

the fundamental s o l u t i o n

p r o p a g a t i o n cone a t

inclusion

axes r e s p e c t i v e l y ,

from the f o r m u l a

the set of

t h e two terms on t h e r i 9 h t

sets are the positive

hyperbolic differential in

+ (x+iO) -~.

the f i b e r

the

over 0 is

local

Rn\O,

P ( D ) E ( x ) = ~ ( x ) . I n most c a s e s t h e

i s an e q u a l i t y .

Convolutions The wave f r o n t

set of

f~g(x) of

This

(x,~)

i s easy t o

i-space, then, that

if(x-y)9(Y)dy

one o f in

which has compact s u p p o r t c o n s i s t s o f

t h e wave f r o n t

prove if

Fourier transform of

in

=

two d i s t r i b u t i o n s

(x+y,~) with

the convolution

h is

obviously,

we s t a r t

f

and

(y,~)

in

that

of

if

the

from the o b s e r v a t i o n t h a t

the c h a r a c t e r i s t i c t h e wave f r o n t

function

set of

f~h

of

g-

a cone i n

has a l l

its

fibers

cone.

2 . 2 The r e g u l a r i t y

A more d e t a i l e d than t h a t regularity

function

i n f o r m a t i o n on t h e s i n g u l a r i t i e s

o b t a i n a b l e f r o m t h e wave f r o n t function.

In o r d e r t o d e f i n e

C o n s i d e r t h e S o b o l e v s p a c e s Hp, finite

set of

all

J

lf"(~)12(l+l~t)=~

it,

p any r e a l

S o b o l e v norm s q u a r e

Itfflp m =

set

d~.

is

of

a distribution

g i v e n by i t s

we need some p r e p a r a t i o n . number, o f

functions

with

38

We s h a l l

see t h a t c o n v o l u t i o n by a smooth f u n c t i o n h w i t h compact

support i s a continuous self-map of 9(g)=(l+Igl)Pf^(g),

HR. In f a c t ,

if

then

J(l+I{l)mPd~ISh(~-g)f(g)dglZ~ where M i s t h e maximum o f

MJlg(g) I~dg

the f u n c t i o n

l(l+l~l)=P(l+lgl)-ZP[h(~-g)l Estimating lh(~-g)l

=p

by c o n s t ( l + l ~ - g l ) -N w i t h

d~. l a r g e N and u s i n g t h e

inequality ( 1 + 1 ~ 1 ) mm ~ ( l + l ~ - g l ) Z P ( l + l g l ) when p>0 and t h e same i n e q u a l i t y w i t h that M is finite. flhfll. Next,

let

Ilfll~.c Lemma

If

relatively

~ and g permuted when p

~hfn~,= finite

when h i s smooth w i t h compact s u p p o r t .

Proof.

Let b and c be the c h a r a c t e r i s t i c

f u n c t i o n s of B and C. By the

p r e c e d i n g theorem, lh^(~-g)(l-c(g)lf^(g)dg is fast

d e c r e a s i n g in B and, by t h e p r e l i m i n a r i e s nhbfnp ~ c o n s t

The r e g u l a r i t y

lifll..=.

function

The lemmas above show t h a t f o r r~(x,~)

every d i s t r i b u t i o n

which e x p r e s s the f o l l o w i n 9

p r o p e r t y of u:

u t h e r e a r e numbers when h i s smooth

w i t h compact s u p p o r t c l o s e t o x and h ( x ) ~ 0 and C i s a s m a l l open cone around ~, then l ~ l - ( h u ) ^ b e l o n g s t o L z in C

39

where m comes a r b i t r a r i l y to

close to

x and t h e cone C t e n d s t o

l e a s t u p p e r bound o f

4.

r=(x,~)

when the s u p p o r t of

Of c o u r s e ,

r~(x,~)

is

t h e numbers m. To e x p r e s s t h i s

in

h tends

t a k e n t o be t h e a natural

way,

we can say t h a t u is The f u n c t i o n function

of

r~(x,()

u.

is

i n H~ a t

It

is

that

r~

that

t h e wave f r o n t

(x,(),

is called

the r e g u l a r i t y

obvious that

identically

minus

set of

r = r~(x,().

r~

is

infinity

u is

(or singularity)

homogeneous o f

precisely

the closure of

degree 0 in

when u i s

~,

smooth and

t h e s e t where r~

is

finite.

2.3

Oscillatory

One o f

integrals.

t h e main o b j e c t s o f

singularities

of

distributions

t h e s i m p l e s t case f o r m a l (2.3.1) with

F(x)

a(x,e)

I

a(x,e)

exp

(2.3.2)

s(x,e)

(2.3.3)

sx(x,e)~K~

(2.3.4) The l a s t

is

s(x,e)

real

when

degree of

a,

and a l l

phase f u n c t i o n properties

listed

S i n c e we a r e o n l y

the

inte9rals,

in

and t h e a m p l i t u d e f u n c t i o n

di~ferentiable

J)

supposed t o

for

i n XXRN\ 0 and

crucial

properties

de9ree 1 in

e

in

in

small

a fixed

uniformly

in

the sequel,

the product of

a phase f u n c t i o n

interested

e->~.

hold f o r

locally

a b o v e . Note t h a t

a vanishes for

also the region of

e#O,

~ and 5,

is

is

and homogeneous o f

and a m p l i t u d e o c c u r

d e g r e e s m and m'

assume t h a t

is

the study of

de

have t h e f o l l o w i n g

D-~Dm~a(x,8) = O ( l e l ' - ' ~ inequality

is(x~e)

R~ and e i n RN which

a r e assumed t o be i n f i n i t e l y and t o

is

d e f i n e d by o s c i l l a t o r y

The phase f u n c t i o n

××RN r e s p e c t i v e l y

of

=

analysis

integrals

x i n an open s e t X o f

integration.

microlocal

of

ms c a l l e d x.

the

When t h e t e r m s

they refer

to

the

two phase f u n c t i o n s

d e 9 r e e m+m'

the s i n g u l a r i t i e s e. Note t h a t

of

F,

we s h a l l

(3) means t h a t

s.(x,e)

40

is equivalent to We s h a l l

lel

locally

Ju(x)f(x)dx

where f

and × a r e i n f i n i t e l y

= lim

i n a n e i g h b o r h o o d of

In f a c t ,

in x .

see t h a t F i s a d i s t r i b u t i o n

(2.3.5)

×=I

uniformly

d e f i n e d by the f o r m u l a

JJa(x,8)f(x)×(~/t)

exp

differentiable

is(x,8)

dxde,

t->m,

w i t h compact s u p p o r t s and

the o r i g i n .

by the c o n d i t i o n s (2)

and

(3),

the d i f f e r e n t i a l

operator

L~ d e f i n e d by L~ = I s ~ ( x , 8 ) l - z ~ . ~ / i ~ x e x i s t s and r e p r o d u c e s t h e e x p o n e n t i a l exp i s ( x , m ) i n t e 9 r a l of

(5)

does not change i f

M~ of L~ t o the r e s t of and M~ a r e O ( l e ~ - ~ ) ,

this

l a r g e n e g a t i v e degree. the c h o i c e of (3) If

integral

immediately v e r i f i e d of

(5).

Hence t h e

Conical

the o s c i l l a t o r y

a product of

of

that

this

Note t h a t ,

(I)

oscillatory

rules.

formally,

is

integrals exist

and

functions.

f u n c t i o n s c(x,8)

which are

I in

in X and a c l o s e d c o n i c a l s e t C in RN and

small c o n i c a l neighborhood of to a p a r t i t i o n

(l-c(x,~>)a(x,e)

decreasin9 f o r

all

x,

of

Y×~. E v e r y such

t h e a m p l i t u d e a,

+c(x,e)a(x,~),

L e t F=G+H be t h e c o r r e s p o n d i n 9 o s c i l l a t o r y

But t h i s

we 9et an

The same h o l d s f o r

differentiable

amplitude f u n c t i o n 9 i r e s r i s e a(x,8)=

so f a r ,

o p e r a t i o n commutes w i t h the passage t o

there are a m p l i t u d e

v a n i s h i n an a r b i t r a r i l y

differentiable.

r e g a r d l e s s of

support.

a compact s e t

amplitude is fast

exists

F i n ×.

integral

Hence d e r i v a t i v e s of

singular

that

(5)

arbitrarily

w i t h a a n o t h e r a m p l i t u d e f u n c t i o n and i t

by i n f i n i t e l y

support,

It is clear

limit

of L~

~=0.

a r e o b t a i n e d by t h e usual f o r m a l multiplication

the a d j o i n t

i n t e g r a n d . Since the c o e f f i c i e n t s

× and d e f i n e s a d i s t r i b u t i o n

we d i f f e r e n t i a t e

limit

we a p p l y any power o f

i n t r o d u c e s a new a m p l i t u d e of

has been used o n l y f o r

oscillatory

the

the

and hence the

integrals.

G is

If

t h e second

infinitely

happens a l s o when t h e 9 r a d i e n t s~(x~e) does

41

not v a n i s h on the s u p p o r t S o f

ca.

from below and t h e d i f f e r e n t i a l Le =

In f a c t ,

then

infinitely

Ise(x,e) l-=~e.~eli

G and i t s

coefficients

differentiable

for

the a d j o i n t

of

f o l l o w s from

union o f

adjoint

are

degree S0 in a

(5)

that applying

the o s c i l l a t o r y

integral

8

d e c r e a s i n g a m p l i t u d e and hence

above in g e n e r a l terms,

the a m p l i t u d e f u n c t i o n a ( x , e )

d e f i n e the c o n i c a l

as the complement of

the

s e t s Y×C where Y i s open in × and C i s open and c o n i c a l

RN and a ( x , e )

i s of

fast

decrease u n i f o r m l y f o r

Y and 8 i n c l o s e d c o n i c a l p a r t s o f conical,

its

differentiable.

To f o r m u l a t e the r e s u l t s support of

reproduces the

L~ t o the a m p l i t u d e ca d i m i n i s h e s i t s

e q u a l s a n o t h e r one w i t h an a r b i t r a r i l y infinitely

It

and t h o s e o f

degree by any i n t e g e r . We c o n c l u d e t h a t

is

S.

e~43 and homogeneous o f

c o n i c a l n e i g h b o r h o o d o f Y×C. Hence i t l a r 9 e powers o f

norm i s bounded

operator

i s d e f i n e d i n an open c o n i c a l n e i g h b o r h o o d of e x p o n e n t i a l of

its

in

x i n compact p a r t s o f

C. L e t S be t h e s e t ,

c l o s e d and

where se(x,e)=O.

It f o l l o w s

from

which e q u a l s I with

the

for

above

that

if c(×~e)

a(x,e)

n e i g h b o r h o o d , then t h e o s c i l l a t o r y inte9ral

differentiable when × i s

in

function.

integral

In p a r t i c u l a r ,

the s i n g u l a r support of

It

in

front

is n o w

(5)

set

easy

of

an

We can w r i t e

oscillatory

to m a j o r i z e

by f ( x ) e x p - i x . ~ ,

the

function

i n t e r s e c t i o n of S

and v a n i s h e s o u t s i d e a n o t h e r F differs

F,

wave

from the

by an i n f i n i t e l y

se(x,e)=0 for

some n o n - z e r o e

r e g a r d l e s s of

the nature of

this

o b s e r v a t i o n as

s i n g supp F ~ { x , s e ( × , e ) = 0 , x , e

wave

the

with amplitude c(x,e)a(x,e)

the a m p l i t u d e f u n c t i o n .

The

amplitude

l a r g e e in a n e i 9 b o r h o o d o f

the c o n i c a l s u p p o r t of

oscillatory

is an

i n con supp s ( x , 8 ) }

inte9ral. front

we g e t an o s c i l l a t i n 9

set

of

F.

integral

Replacing

f(x)

with amplitude

42

f(x)a(x,O)

and phase f u n c t i o n

zero c o n s i d e r i t s

, lel>[).

the r a y s 9 e n e r a t e d by the two terms

positive

numbers)

are separate f o r

in c l o s e d c o n i c a l s e t s ,

it

u n i f o r m l y e q u i v a l e n t to

I01+I~I.

adjoint

of

Keeping 6 and ~ away from

9radient

s~(x,S)-~ If

s(x,e)-x.~.

(under m u l i p l i c a t i o n

x in some compact s e t and 8 and

is clear that

the norm of

the two c o n i c a l s e t s .

decreasing for

exp

ix.~

e and ~ i n

the

follows that dx

x in a neighborhood of

suppf.

Combining

w i t h what we know about t h e s i n g u l a r s u p p o r t of F p r o v e s

Theorem. the set of

The wave f r o n t pairs

(x,~)

= s~(x,O) Note.

It

in

~ in some c o n i c a l c l o s e d s e t C p r o v i d e d

s x ( x , 0 ) - ~ does not v a n i s h f o r this

the

t h e c o r r e s p o n d i n g L~ t o the a m p l i t u d e f u n c t i o n we g e t a new

IF(x)f(x)

is fast

the 9 r a d i e n t i s

Hence, a p p l y i n g l a r g e powers o f

a m p l i t u d e f u n c t i o n which d e c r e a s e s a r b i t r a r i l y product of

by

The f i r s t

s~(x,O)

s e t of

for

and

the d i s t r i b u t i o n

(I)

i s c o n t a i n e d in

which

se(x,e)=O.

condition

says

precisely

that

the

rays generated

by

and ~ are the same.

Note. The theorem h o l d s f o r individual

one i s

s e t of F f u r t h e r

every amplitude function a(x,e).

taken i n t o a c c o u n t , we can r e s t r i c t by r e s t r i c t i n g

(x,e)

to

When an

t h e wave f r o n t

the c o n i c a l s u p p o r t o f

a(x,0).

Examples. The wave f r o n t

s e t of

lexp i x . ~ with

d~

the o s c i l l a t o r y and

Jexp i ( x - y ) . S

~ and S in R" are~ r e s p e c t i v e l y ,

(O,R"

\0)

in R " \ O ) .

The wave f r o n t

s e t of

the o s c i l l a t o r y

inte9ral

integrals dS and

(x=y,~=S,g=-S, S

43

J exp i ( x . ~ - l ~ l )

d{,

dim x=n,

points

(x,~)

i s c o n t a i n e d in

t h e s e t of

particular,

s i n g u l a r support i s the u n i t

It

will

its

t u r n out

later

i n t e g r a l s are those of u(x)

=

for

which x = ~ / l ~ l .

In

ball.

t h a t v e r y 9 e n e r a l examples o f o s c i l l a t o r y

t h e form

j a(x,~)

exp

i(x.~-H(~))

d~

where a v a n i s h e s when ~ i s o u t s i d e som open c o n i c a l s e t C i n R~. The wave f r o n t for

set of

u i s c o n t a i n e d in

the s e t o f

pairs

(x,{)

with

~ in C

which x=H'(~).

I n 9 e o m e t r i c language, t h i s

means t h a t

the h y p e r s u r f a c e H(5)=1 a t

the p o i n t x .

of

the wave f r o n t

is well

In o t h e r words,

the p r o j e c t i o n

s e t on x - s p a c e i s a h y p e r s u r f a c e dual t o t h e

hypersurfaces H(5)= It

t h e h y p e r p l a n e x . ~ = l touches

const.

known t h a t

the d u a l s of v e r y r e g u l a r h y p e r s u r f a c e s can

have v e r y c o m p l i c a t e d s i n g u l a r i t i e s .

Suppose f o r

instance that

~

does

not v a n i s h on C\O and put

where

F=F(t)=F(t2,...,t.)

i s any smooth f u n c t i o n .

That x=H' ( { )

then

take

=t

means t h a t x= = F = ( t ) , . . .

xi=F(t)-t=F=(t)-..., where

Fj

equals

~F(t)/~xj.

To

see

what

this

means,

n=2,

tz

and F(t)

We

=

l+at+bt=+ctm+

....

9et x1=l-btZ-2ct~+..., x2

= a+2bt+3ct~+

When b$O, t h e dual

....

is approximately a parabola for

c~O, we 9et a cusp a t

the o r i g i n

the w e l l - k n o w n r e s u l t

is

that

in

small

t.

When b=O,

the x ~ , x = - p l a n e . T h i s i l l u s t r a t e s

the dual h y p e r s u r f a c e i s smooth u n l e s s

44

the Hessian H ' ' ( { )

is degenerate, i . e .

e i g e n v a l u e z e r o than t h e o b v i o u s one, since H'({)

has more e i g e n v e c t o r s w i t h namely {

the

(note t h a t H ' ' ( { ) . { = O

i s homogeneous o4 degree 0 ) .

2.4 Fourier

integral

operators.

Associated with o s c i l l a t o r y

i n t e g r a l s a r e the F o u r i e r

integral

operators (2.4.1)

Fu(x)

= J a ( x , e ) u ^ ( e ) exp i s ( x , 8 )

i n t r o d u c e d by Hb'rmander. distribution

u(tl

Here u ^ ( e )

is the F o u r i e r t r a n s f o r m of

in RN w i t h compact s u p p o r t , s ( x , e )

f u n c t i o n and a ( x , e )

a

i s a phase

an a m p l i t u d e . The p r o d u c t a ( x , e ) u ^ ( e ) f a i l s

an a m p l i t u d e o n l y i n t h a t operating with

de

(2.3.4}

the d i f f e r e n t i a l

t h a t Fu(x)

is a distribution

infinitely

di4ferentiable

definition

(2.3.5)

holds only f o r

~=0.

t o be

Hence,

o p e r a t o r c o r r e s p o n d i n g t o s~ p r o v e s

for

e v e r y u.

When v ( t )

and f ( x )

are

f u n c t i o n s w i t h compact s u p p o r t s , t h e

shows t h a t we can w r i t e

(2.4.2)

iF(uv)(x)÷(x)

exp -ix,~

dx

as (2.4.3) and,

if

JJf(x)a(x,e)(uv)^(B) u is

infinitely

s u p p o r t of v , (2.4.4)

differentiable

i(s(×,e)-x.~)

dxde~

in a neighborhood of

dxJla(x,e)(uv)^(t)exp

f o r m u l a shows t h a t

(2.4.3)

is fast

i(s(x,O)-e.t)

decreasing f o r

c l o s e d c o n i c a l s e t when the r a y s g e n e r a t e d by s ~ ( x , e ) separate for

x in

the c l o s u r e of

o f course d i s r e g a r d a l l fast

in

oscillatory

all

t,x

in

f.

integral

the usual way, an a m p l i t u d e o f

provided the g r a d i e n t s e ( x , e ) - t for

the s u p p o r t o f

dtdo ~ i n some

and ~ are

In a l l

this

we may

e in open c o n i c a l s e t s where a ( x , B ) ( u v ) ^ ( e )

d e c r e a s i n g . The i n t e r i o r

acquires,

the

as

Jf(x)exp-ix.~

The f i r s t

exp

the product of

of

is

(2.4.4)

any n e g a t i v e degree

i s d i f ÷ e r e n t from z e r o f o r

the c l o s u r e s of

all

the s u p p o r t s o f

e and v and ÷.

45

Hence, fast

in

that

case,

d e c r e a s i n g in

it

infinitely

4. Combining a l l

a b o u t t h e wave f r o n t

Theorem

is

sets of

The wave f r o n t

differentiable this

Fourier

set of

so t h a t

(4)

is

p r o v e s an i m p o r t a n t theorem

integral

a Fourier

operators.

integral

operator

S a ( × , e ) u ^ ( e ) exp i s ( × , e ) d 8 consists of

pairs

(x,~)

~ = s ~ ( x , e ) and

for

which

(se(s,6),e)

is

i n WF(u).

2.5 A p p l i c a t i o n s

The wave f r o n t

sets of

distributions

To e v e r y smooth b i ] e c t i o n bi]ection

u->v of

distributions.

a Fourier

=

the F o u r i e r

Fourier

Theorem

a corresponding

u(f(x))

=

(x,~)

(2~)-~lu^(g)exp

transform of

integral

A pair

distribution

is

as

operator v(x)

sets of

R~ t h e r e

When u has compact s u p p o r t , we can e x p r e s s v ( x )

integral

where u ^ i s

of

distributions, v(x)

of

x->f(×)

on m a n i f o l d s .

if(x),9

u.

d9,

The theorem on t h e wave f r o n t

o p e r a t o r s has t h e f o l l o w i n g

belongs to

v(x)=u(f(x))

if

t h e wave f r o n t

and o n l y

if

application.

set of

a

the pair

(f(x),~f' (x)-1~) belongs to Note.

The m o r a l o f

distribution X.

In f a c t ,

Proof. of

t h e wave f r o n t this

set

of

result

is

on a m a n i f o l d X i s if

(y,5)

is

(x,()

with

the p a i r

4= t f , ( × ) g

that

part of

By t h e p r e c e d i n g t h e o r e m ,

pairs

u. t h e wave f r o n t

a

t h e c o t a n g e n t b u n d l e T*(X)

a b o v e , we have

t h e wave f r o n t

and

set of

(f(×),g)

in

of

~.d×=~.dy.

set of

v(x)

consists

t h e wave f r o n t

set of

46

u.

Chan9in9 the r o l e s o f

u , v and chan9in9 f

the i n c l u s i o n i s a b i j e c t i o n .

to

its

inverse proves t h a t

T h i s p r o v e s the theorem when u ha

compact s u p p o r t and hence i n 9 e n e r a l .

Parametrices of The f i r s t

fundamental s o l u t i o n s

time t h a t F o u r i e r

inte9ral

was in P e t e r L a x ' s c o n s t r u c t i o n o f s o l u t i o n s of

o p e r a t o r s appeared e x p l i c i t l y

p a r a m e t r i c e s of

stron91y h y p e r b o l i c f i r s t

fundamental

o r d e r systems. The c o n s t r u c t i o n

extends to stron91y h y p e r b o l i c d i f f e r e n t i a l

o p e r a t o r s w i t h smooth

coefficients

P(x,D)=I Here

x=(xo,...,x~)

IJl£m.

a a ( x ) D ~,

stands

for

n+l

real

variables,

i m a 9 i n a r y 9 r a d i e n t . The m u l t i i n d e × J = ( J o , . . . , J n ) components ZO and Da, IJl

its

principal

a product of

(2.5.1)

integral of

the c h a r a c t e r i s t i c

= I a ~ ~,

order

polynomial

IJl=m,

a non-zero f u n c t i o n po(x)

and m f a c t o r s

~o - q k ( x , ~ )

where t h e qk a r e independent o~ ~o and r e a l In t h e sequel we s h a l l

i n s t a n c e in Hormander s o l u t i o n E(x)

the

part,

p~ ( x , { ) =

and not z e r o .

is

= Jo+...+Jn.

p(x,~) is

has n+l

defined accordin91y is a d e r i v a t i v e

That P i s s t r o n 9 1 y h y p e r b o l i c means t h a t of

D=~/i~x

1985 ( I V ,

and s e p a r a t e f o r

~ real

put p o = l . As e x p l a i n e d f o r

394-395),

P has a u n i q u e fundamental

w i t h p o l e i n x=O which v a n i s h e s f o r

Xo < O. The Cauchy

problem w i t h d a t a on the h y p e r p l a n e xo=O, (2.5.2)

PF(x)=O,

DokF=O, Dom-~F=i&

( x ~ ) . . . & ( x M ) when xo=O, kKm-l,

a l s o has a unique s o l u t i o n .

By d i r e c t

c o m p u t a t i o n one f i n d s

E(x)=H(xo)F(x) where H i s

the H e a v i s i d e f u n c t i o n . P=Do=-D~:-...

-D, m

When

that

47

is

t h e wave o p e r a t o r

the function

F can be computed e x p l i c i t l y .

In

fact, F(x)

=

(2~) -n

J ((exp ixot+ix.

-

exp - i x o t + i x .

)/2t

dO

where go=O and

t=(g Lax's

~:+...+

i d e a was t o

O~=)~,m

imitate

this

f o r m u l a by p u t t i n g ,

in

the general

case,

(2.5.3) with

F(x)

=

~ J ak(x,g)exp

isk(x,g)

do,

a m p l i t u d e s ak and phases sk d e t e r m i n e d so t h a t

sense o f

oscillatory

formulated

If

42.5.4)

integrals.

the

phase f u n c t i o n s

are formal

the

approach i s

Sk a r e chosen so t h a t

On w h e n

= lak~ ( x , g )

whose t e r m s a r e smooth f u n c t i o n s homogeneous o f

xo=O,

degree j

in

,

j=l-m,-m,-l-m,...

when

O such t h a t

the formal

(exp - i s k ( x , O ) ) P ( x , D ) a k ( × , O ) e x p

vanish of

infinite

(2.5.6)

order

and t h e f o r m a l

(exp - - i s k ( x , g ) ) D o k a k 4 x , g )

vanishes

when

Since

k<m-I

the

ak are

amplitudes

in the next

chapter

one

neighborhood

the

numbers

tj

and

of

equals formal

in the that

technichal if

×4g)

the o r i g i n

increase

isk(x,g) k=m-1.

strictly sense.

I outside

sufficiently

fast,

homogeneous

However,

is a s m o o t h

and

sums

sum

exp

of

~ 0 and

isk(x,g))

i(2n) -~ when sums

,

g =(g~,...,g~)

(2.5.5)

not

in

sums

ak(x,g)

Note.

this

holds

pk(x,skx)=O,

sk(x,g)=x~o~+...+x. there

The s u c c e s s o f

(i)

in

Theorem

are

go=O,

function another

then

terms,

it will

be

which

is 0

they

proved in

neighborhood

and

the s u m s

~ × ( O / t j ) a k j (x,g) converge t o t h e s e bk,

proper amplitudes bk(x,O).

we g e t

a distribution

G(x)

If

we r e p l a c e t h e ak

which s o l v e s

the

in

problem

(2) 41)

by

48

modulo smooth f u n c t i o n s . parametrix of Note.

Restrictin9

it

to

the fundamental s o l u t i o n

S i n c e t h e Pk a r e homogeneous o f

same p r o p e r t y .

parametrix is

E(x). degree I

in

The H a m i l t o n - J a c o b i d i f f e r e n t i a l

solvable only for

x close to

only

local.

we 9 e t a t r u e

Xo>O,

the origin.

~,

t h e sk have t h e

e q u a t i o n s (4)

T h i s means t h a t

are

our

That a 9 1 o b a l p a r a m e t r i x e x i s t s

will

be

p r o v e d i n C h a p t e r 5.

P r o o f . The p r o o f o f

t h e theorem i s

a straightforward

verification

based on t h e f o l l o w i n 9 Lemma

When s ( x , 9 )

is

a phase f u n c t i o n ,

there are differential

operators

Qj(x,9,D) homo9eneous o f (2.5.7) for

de9ree j

exp - i s ( x , 9 )

, k=O,...,m, in

P(x,D)a(x)exp is(x,g)

e v e r y smooth f u n c t i o n

Qm

O such t h a t

a(x).

In

= IQj(x,g,D)a(x)

particular,

p(x,s~),

=

Qm-1 = pCJ~ ( x , s ~ ) B j

+P~-~ ( x , s x )

where p'J'(x,O) and P ~ - 1 ( x , 9 )

is

polynomial P(x,9)

This

lem~a i s

reader.

Its

t h e sum o f

of

verification

the c h a r a c t e r i s t i c

into

(5)

which

show t h a t

differential

(x,9)

the differential

equations called

is

left

to

the

the vanishin9 of

the formal sum, ordered a c c o r d i n 9 to

Lk(x,5,D)akj

their

degrees in

transport

all 5,

equations,

= bkj (x,9) o p e r a t o r Qm-~ w i t h

s i d e depends o n l y on t h e ak~ w i t h every akj

d e g r e e m-1 o f

P(x,D).

p r o v e d by a d i r e c t

linear

where Lk i s

terms o f

formulas inserted

t h e terms o f amounts t o

= ~p(x,O)/~gj

l<j

f r e e when xo=O. These i n i t i a l

s=sk and t h e r i g h t

already constructed.

This

leaves

v a l u e s on t h e o t h e r hand can

49

be chosen so t h a t principal

t h e s u m s (5)

terms Ck=akj w i t h

get

j=l-m

qkJck=O,(j<m-l), which,

the desired properties. we g e t

qk~-~Ck

=

equations

i(2~)-~,

s i n c e t h e qk a r e s e p a r a t e , have u n i q u e s o l u t i o n s

homo9eneous o f

d e g r e e l-m

in

9.

to

Ck which a r e

The o t h e r t e r m s akS a r e put e q u a l

z e r o when xo=O. These a r e t h e e s s e n t i a l s left

the f o l l o w i n g

For o u r

of

the proof.

to

The d e t a i l s

are

the r e a d e r .

The p a i r i n g The f a c t of

that

we a r e d e a l i n g w i t h

s p e c i a l c i r c u m s t a n c e s . The p o l y n o m i a l p ( x , 9 )

d e g r e e m and a l s o t h e p r o d u c t o f to

polynomials P(x,3)

-9

we g e t p a i r i n g

k->k'

the f a c t o r s

such t h a t

is

(i).

c r e a t e s a number

homogeneous o f Hence, c h a n g i n 9

Pk'(×,-9)

= -pk(x,9),

in

9 other

words

(2.5.8) If

qk.(x,-9)

we o r d e r t h e qk so t h a t

reflection

in

According to

is

(4),

(2.5.9)

paired to

the pairing sk.(x,9)

S i n c e t h e e q u a t i o n s (5) change k - > k '

and

(2.5.10)

9-> -9

is

which

for

some k.

By t h e p a i r i n g ,

t h e wave f r o n t set of

set

Hence t h e r e

itself

m i s even o r odd.

if

and o n l y

if

is

no

implies that

(6) it

=

(I0)

by o u r g e n e r a l r u l e s

for

front

and

is simply

t h e sequence l , . . . , m .

are

is

invariant

not d i f f i c u l t

under the s i m u l t a n e o u s to

see t h a t

(-l)~akj (x,-9)

the expansion of

The e q u a t i o n s (9) F,

,

the p a i r i n g

= --Sk(X,-9).

and

ak.j(x,9)

where ak =lak~

set of

q~>...>qm,

the midpoint of

and one qk which

= -qk(x,g).

ak i n

terms o f

homo9eneity j .

have c o n s e q u e n c e s f o r contained in

b o t h sk and s k .

the set

t h e wave f r o n t o~ p o i n t s

contribute

to

o v e r a g i v e n × • Hence t h e p r o j e c t i o n

F on x - s p a c e a p p e a r s as [ m / 2 ]

(x,~)

the fiber

of

on t h e wave

connected sheets.

In

50

particular, of

for

t h e wave e q u a t i o n t h e r e

o r d e r 3 and 4 t h e r e a r e two o f

consequences f o r solution

at

its

the fact

that

the nature of

constant coefficients coincides with principle.

its

variables

N o t e . The m a t e r i a l

of

Hormmnder

has s i n c e

La×'s

paper

is L a x

1957.

is

the

the effects

operators will

and

for

light

instance, for

cone and hence

s u p p o r t , a phenomenon c a l l e d

pseudodifferential

1971

have

the fundamental

They a r e r e s p o n s i b l e ,

I n C h a p t e r s 6 and 7,

this

The e q u a t i o n s (8)

equations

t h e homogeneous wave e q u a t i o n w i t h

in f o u r

singular

one s h e e t and f o r

the b e h a v i o r of

singularities.

the support of

them.

is

of

Huygens ~

a 9eneral pairin9

be i n v e s t i 9 a t e d .

c h a p t e r appeared f o r

become

standard

the first

microlocal

time

in

analysis.

for

Chapter 3

PSEUDODIFFERENTIAL OPERATORS

Introduction:

dif÷erential

The c a l c u l u s o f of

differential

o p e r a t o r s and t h e i r

pseudodiifferential

operators with

symbols .

operators is

smooth,

i.e,

an e x t e n s i o n o f

infinitely

that

differentiable

coefficients a(x,D) i n some open s u b s e t X o f D=(D~,...,Dn).

introduce their

~ i n R~ and x . ~

a(x,D)->a(x,~)

of

is

a l s o use t h e n o t a t i o n

and p r o d u c t s o f

~!

such o p e r a t o r s i t

and

=~!...~n!. is

convenient to

p o l y n o m i a l s o r symbols

= la~(x)~ =

= exp - i x . ~

= x~+...+x~n.

a linear

are multiindices

bijection.

It

a ( x , D ) exp i x . { ,

is clear

that

t h e map

A few moments o f

reflection

form

Su(x)E(x)d×

has

also

write

show

the a d j o i n t

a with

respect

characteristic

Usin9

to the s e s q u i l i n e a r

Since

the m u l t i n o m i a l

a(x,D)exp

a product

the

polynomial

theorem,

(exp

of

Here ~ = ( ~ 1 , . . . , ~ n )

characteristic

a(x,~)

that

Rn.

L a t e r we s h a l l

To h a n d l e a d j o i n t s

with

= I a ~ ( x ) D~ , D = ~ l i ~ x ,

ix.~

we can

it as

iD~.D~)a(x,~). = exp

a(×,D)b(x,D)

ix.~ of

a(x,D+~),

the c h a r a c t e r i s t i c

two d i f f e r e n t i a l

operators

polynomial

is

where

A n o t h e r way o f

writin9

the c h a r a c t e r i s t i c

exp iD~.D~ a ( y , g ) b ( x , ~ )

polynomial of for

y=x,

9=~.

a product is

52

Finally, of

Rn,

if

y=f(x)

i s a smooth b i j e c t i o n

the d i f f e r e n t i a l

characteristic

t r a n s p o r t e d t o Y has the

= exp - i f ( x ) . o

a ( x , D ) exp i f ( x ) . o .

see i n t h e n e x t s e c t i o n t h a t a l l

formulas f o r

X t o a n o t h e r open subset Y

polynomial b(y,9)

We s h a l l

o p e r a t o r a(×,D)

of

chan9es o f

t h e s e f o r m u l a s and a l s o t h e

variables survive essentially

for

pseudodifferential operators a(x,D)u(x)

=(2~}-~S

where u i s a d i s t r i b u t i o n transform,

a(x,~)

a ( x , D ) and the

integral

front

s e t s of

is oscillatory.

side is actually

d~,

oscillatory

When a ( x , D )

its

Fourier

the o p e r a t o r

is a differential

a ( x , D ) u ( x ) by t h e F o u r i e r

the n o t a t i o n i s c o n s i s t e n t .

9 r a d i e n t of

o p e r a t o r s do not

ix.~

i n X w i t h compact s u p p o r t , u ^ i s

i n v e r s i o n theorem so t h a t ~

exp

i s an a m p l i t u d e c a l l e d the symbol of

o p e r a t o r , the r i g h t

Since the

a(x,~)u^(~)

x.~

i s x,

inte9rals

the r u l e s f o r shows t h a t

i n c r e a s e wave f r o n t

sets,

computin9 the wave

pseudodifferential

the wave f r o n t

s e t of

a ( x , D ) u i s c o n t a i n e d i n t h a t of u. If

we make t h e F o u r i e r t r a n s f o r m of u e x p l i c i t

the p e u d o d i f f e r e n t i a l o p e r a t o r a ( x , D ) , a(x,D)u(x) where

the k e r n e l

A(x,y)

=

i n the d e f i n i t i o n

of

we can w r i t e

$ A(x,y)u(y)dy

is a d i s t r i b u t i o n

defined

by

the o s c i l l a t o r y

inte9ral A(x,y) which When and It

=

is s m o o t h

(2~)-nla(x,~) in e v e r y

the a m p l i t u d e

every

N,

locally

is i m p o r t a n t

operators

to be

open

uniformly in m i n d

presented

below

o p e r a t o r s w i t h smooth k e r n e l s . sin9ularities.

set

has d e 9 r e e

to k e e p

exp

-~,

i(x-y).~

where i.e.

in x, that

d~,

x is not a(x,{)

equal

= O(i~l -w)

the k e r n e l

In t h i s

for

is a s m o o t h

the c a l c u l u s

is a c a l c u l u s

to y. lar9e

function.

of

pseudodifferential

modulo

pseudodifferential

sense i t

i s a c a l c u l u s of

53

3.1

The c a l c u l u s o f

pseudodi÷Terential operators

In o r d e r t o e x t e n d t h e f o r m u l a s above f r o m d i f f e r e n t i a l pseudodifferential We l e t

o p e r a t o r s , we need some t e c h n i c a l

S~ be t h e space o f D(~,~)a(x,~)

locally

uniformly

a l s o be used f o r y.

for

amplitudes of

information.

d e 9 r e e a t most m,

i.e.

= O(i~I~-'~'),

x in compact s u b s e t s o f

a m p l i t u d e s where x i s

×.

This notation will

r e p l a c e d by two v a r i a b l e s

x and

T o p o l o g i z e d by t h e c o r r e s p o n d i n 9 seminorms, S~ becomes a F r e c h e t

space. F i r s t ,

we s h a l l

a m p l i t u d e s aj

whose d e 9 r e e s mj

When a i s

when a i s

(3.1.1) for all shall

Proof.

(a-a~-,,.-aj)

(I)

X.

Let

bj

×(~)

Moreover, for

In view of the f o l l o w i n 9 and

its a s y m p t o t i c

there

is

on X,

we can r e s t r i c t

lemma,

we

expansion.

an a m p l i t u d e a o f

d e g r e e m~

of

unity

be smooth and 0 f o r

I~I2.

Put

= ×(~/t~)aj(x,~)

I~I+161 a ' t a k i n g the a d j o i n t

p.

them. Here

polyhomogeneous

i s r e m a r k a b l y s t a b l e under o p e r a t i o n s such as

or c h a n g i n g v a r i a b l e s .

Adjoints Theorem

The a d j o i n t

of a p s e u d o d i f f e r e n t i a l o p e r a t o r s w i t h r e s p e c t t o

a sesquilinear duality

(u,v)

= Su(x)~(x)d× ~ is a p s e u d o d i f f e r e n t i a b l e

o p e r a t o r w i t h t h e symbol

exp

Proof.

Writing

iD~.D~(x,~).

(u,a(x,D)v)

(2~)-"

explicitly

fu(y)$(y,~)~(x)

exp

we g e t

i(x-y).~

d~dxdy

when u an v a r e smooth w i t h compact s u p p o r t s . Hence t h e a d j o i n t a(x,D)

of

i s a p s e u d o d i f f e r e n t i a l o p e r a t o r w i t h the kernel (2~) - "

so t h a t explicit

f~(y,~)

the d e s i r e d r e s u l t form of

exp i ( x - y ) . ~

d~

f o l l o w s form the p r e c e d i n g lemma. The

t h e symbol of

the a d j o i n t

shows a t once t h a t

a->a'

commutes w i t h t a k i n 9 the a d j o i n t .

Products When computing the p r o d u c t o f symbols a ( x , ~ )

and b ( x , ~ ) ,

two p s u d o d i f f e r e n t i a l

operators with

we have t o suppose t h a t b ( × , ~ )

vanishes f o r

56

x outside

Theorem

some

If

compact

subset

of

X.

b ( x , D ) conserves compact s u p p o r t s ,

the p r o d u c t o f

p s e u d o d i f f e r e n t i a l o p e r a t o r s w i t h symbols a ( x , ~ ) pseudodifferential operator with

and b ( x , ~ )

two

is a

the symbol

exp iDx. D9 a ( y , o ) b ( x , ~ ) l y = x , g = ~ . Note. The e x p a n s i o n o f

this

symbol has t h e terms

( i B ~ ) ~ a k ( y , 9 ) D ~ b k ( x , ~ ) / ~ ! l y= ,g=~. R e p l a c i n g a , b by a ' , b ' which

is at

Proof.

The

multiplies

t h e same t i m e i t s

function

b(x,D)u(x)

this

term by - I

t o t h e power j + k + l ~ l

homogeneity. Hence ( a b ) ' = a ' b ' .

has

the

Fourier

transform

(bu)^(O) = (2~)-~S b ( y , g ) u ^ ( ~ ) e x p i y ( ~ - g ) dyd~, and hence a ( x , D ) b ( x , D ) has the f o r m a l (3.1.2)

( 2 ~ ) - " S a ( x , o ) e x p i x . (o-~)

Since the

last

integral

symbol i n x and { and t h i s

is

fast

symbol

do f b ( y , { ) e x p

decreasing in

proves t h a t

i y . (~-g)

~-O,

this

dy. is

really

a

the product i s a

p s e u d o d i f f e r e n t i a l o p e r a t o r . The c o m p u t a t i o n s which f o l l o w reduce t h e e x p r e s s i o n above t o normal form. (2~)-"

With

~-9=5,

x - y = z , we can w r i t e

fa(x,~-5)d5 Sb(x-z,~)exp ix.5

With a ' ' ~ ( x , ~ ) = ( i D ~ ) ~ a ( x , ~ ) ,

let

it

dz.

us d e v e l o p a ( x , ~ - 5 )

in a T a y l o r

series I + where the f i r s t

a~ I

(x,~) (-5)~I~!

+

Ia c~ ( x , ~ - t $ ) (-5)~tN-~dt/~!N!.

sum runs o v e r J~l o f

(Y,O)

of N(O)

c o n t a i n i n g p.

introduce re9ular canonical coordinates

a t q and r e g u l a r c a n o n i c a l c o o r d i n a t e s ( x , ~ ) i s a v e r y r e g u l a r phase f u n c t i o n f o r

L(O)

at

p such t h a t ,

i n N(O),

if

the f u n c t i o n

f(s,x,~)=9(y,o) where (Y,O)

and

(x,~)

are connected by the H a m i l t o n f l o w ,

r e 9 u l a r phase f u n c t i o n f o r

Proof.

Assume t h a t

L(s)

a t p.

the c h o i c e o f

c o o r d i n a t e s has been made so t h a t

v a r i a b l e s ~ can be chosen as p a r a m e t e r s on L ( s ) . is

invertible

and a t

is a very

The map ( y , o ) - > ( x , ~ )

p , q we have

dx=xydy+x~ds, d~=~dx+~gdo. We should l i k e

t o have the

the

i m p l i c a t i o n dx,do=O dy,do=O, i . e .

xy

84

should be n o n - s i n g u l a r . This can be achieved by a change of coordinates

y~'=y~+¢z,

y='=y=

etc

where ~ i s small and

z=(yim+...+y~Z)/2 In fact,

then d g = d o ' + ~ d y ~ so that

the end of s e c t i o n 4.2,

xy changes

t h e r e are a r b i t r a r i l y

to x~+cx~. small

As remarked

at

e f o r which x~+ex~

i s n o n - s i n 9 u l a r . Hence the d e s i r e d i m p l i c a t i o n can be achieved by a change of v a r i a b l e s and when i t

i s s a t i s f i e d , we can d e f i n e a f u n c t i o n

h(x,O) by

h(x,o)=9(y,o). Then dh=h~dx+h~dO=gydy+g~dg. On L(O),

g~ vanishes and gy=9 so t h a t the r i g h t s i d e equals ody=~.dx.

Hence h~d9=O f o r

all

do which means t h a t h~=O. Hence

(x,h~), parametrices L(s) the b i j e c t i o n L(s)

h~=O

c l o s e t o p. Since ( x , o ) - > ( x , ~ ) i s a b i j e c t i o n

(via

( y , o ) - > ( x , ~ ) ) , we have ÷ ( s , x , ~ ) = h ( x , 0 ) c l o s e t o p. On

we have h~=O which means t h a t df=h~dx. Hence

(x,~=f~), parametrices L ( s ) ,

i.e.

f~=O ÷ i s a v e r y r e g u l a r phase f u n c t i o n ÷or L ( s ) .

The proof i s f i n i s h e d . References All

the m a t e r i a l of

t h i s c h a p t e r except the n o t i o n of v e r y r e g u l a r

phase f u n c t i o n i s standard (see H6~mander 1985, I l l differential E l i e Caftan.

Ch. XXI).

Invariant

forms were i n t r o d u c e d by Poincare and e x t e n s i v e l y used by

CHAPTER 5

A GLOBAL PARAMETRIX FOR THE FUNDAMENTAL SOLUTION OF A FIRST ORDER HYPERBOLIC PSEUDODIFFERENTIAL OPERATOR.

Introduction Fourier to

The m o t i v e s f o r

integral

equations with In

this

( M a s l o v 1965)

coefficients

c h a p t e r we s h a l l

illustrates

a first

most o f

5.1Cauchy's

and t h e c o n s t r u c t i o n

the fundamental s o l u t i o n s variable

parametrix for

L a g r a n g i a n m a n i f o l d s and

o p e r a t o r s have been t h e s e m i c l a s s i c a l

quantum p h y s i c s

parametrices of

the study of

of

approximations of

91obal

hyperbolic differential

( D u i s t e r m a a t and H~rmander 1 9 7 2 ) .

9ive a simple construction

of

such a

order hyperbolic pseudodifferential

the d i f f i c u l t i e s

problem

for

a first

and t h e t e c h n i q u e s o f

order

hyperbolic

operator.

It

the f i e l d .

pseudodifferential

operator

A first

order pseudodifferential

(x~,...,xn)

is said

to be

o p e r a t o r in

hyperbolic

with

the v a r i a b l e s

respect

to t if

t

and ×=

it has

the

degree

1 and

the

polyhomogeneous.

Such

form P = Dt+Q(t,x,D) where

the

complete

principal symbol

operators hyperbolic

Q(t,x,~)

appear

differential

the c o r r e s p o n d i n g

of

how

to h a n d l e Cauchy

(5.1.1) and c o n s t r u c t

Pu=O

q(t,x,~)

of Q is real

is s u p p o s e d

in L a x ' s

of

following

symbol

to be

construction

operators.

first

order

of

They

a parametrices

are

essential

differential

pseudodifferential

of

operators

strongly

generalizations

operators. we shall

for

As an e x a m p l e

solve

the

problem when

t>O,

a parametrix

u=w for

when

t=O

the c o r r e s p o n d i n g

fundamental

solutlon.

86

To s i m p l i f y

we s h a l l

symbol Q ( t , x , ~ ) t.

with

u such t h a t

if

I

(5.1.2) all

is

the

special

number. in H ' ,

u(t)

interval

H u ( T ) H . < exp cT( s and T where c i s

in

t h e sense t h a t

the

e x c e e d s some c o n t i n u o u s f u n c t i o n

denotes u(t,x)

v a l u e s i n H" and w i s

addition,

for

u(t)

L e t s be any r e a l

weak s o l u t i o n

Q is

v a n i s h e s w h e n I×I

In what f o l l o w s ,

Theorem

assume t h a t

as a d i s t r i b u t i o n

If

v(t)

is

is

on R~.

inte9rable

Cauchy's problem ( I ) continuous with

of

as a f u n c t i o n

has a u n i q u e

v a l u e s in H ' .

In

f r o m 0 t o T,

#wll.

+ Sz Uvll. d r )

a locally

bounded f u n c t i o n

of

these

variables. Note.

The theorem a l s o a p p l i e s t o

distribution

which

is

a solution

(I) for

in

t h e r e 9 i o n tO and t u ( O ) , P u ( t ) with

u continuously

the direct

differentiable

and w i t h

values

in

H=÷~

is

dense i n

sum H= m L •

where t h e are

last

term r e f e r s

integrable

in

to

functions

some i n t e r v a l

of

I=(O,T).

t

with

The d u a l

values of

in

this

H= w h i c h

direct

sum

is H-= where L

refers

to

bounded.

Hence i t

(4.1.3) vanishes for

• L~ functions

suffices

to

(u(O),w) all

v a n i s h n e a r 0.

u above,

from

I

to

H-" which a r e e s s e n t i a l l y

show t h a t

+

if

fz(Pu(t),v(t)

w+v b e l o n g s t o

v(t)

is

sum and

dt

t h e n w and v v a n i s h .

T h i s means t h a t

this

First,

a weak s o l u t i o n

we s h a l l of

let

the equation

(D~ + Q ~ ) v ( t ) = O . Here Q*v i s

inte9rable

with

values

in

H- ~ - ~

and

it

follows

u

that

the

88

derivative

of

v i s an i n t e g r a b l e ÷ u n c t i o n w i t h v a l u e s in H- = - I .

then we can i n t e 9 r a t e by p a r t s i n 0 = Since

u(T)

H ".I

of

is a r b i t r a r y

H -'-I,

applies

to

we

the

lift

Hence the

our

limit

(u(T),v(T))

+

v and

the

that

statement

in

also

the

this

v(T)

=0.

I and

that

density (i),

and

interval

restriction

9ettin9

Sz(u(t),P~v(t)).

in H"

it f o l l o w s

(3)

But

space But

shows

u(O)=O,

is d e n s e

then

that

v=O.

it f o l l o w s

is p r o v e d

and

with

the

in t h e

energy Then,

from that,

(2) by

dual

inequality finally,

that

if

w=O.

a passa9e

to

theorem.

5 . 2 C a u c h y ' s problem on t h e p r o d u c t o f

a line

and a m a n i f o l d

L e t X be a n - d i m e n s i o n a l o r i e n t a b l e m a n i f o l d and P ( t , x ~ D ) a p s e u d o d i f f e r e n t i a l o p e r a t o r of principal

degree I ,

polyhomogeneous and w i t h

symbol q ( t ~ × , ~ ) ~ d e f i n e d on the p r o d u c t Y = X X R. C o n s i d e r

the f o l l o w i n 9 Cauchy problem f o r

Pu=v when t>O,

(5.2.1)

P=D~+Q(t,x,D) on Y,

u:w when t=O.

A c c o r d i n 9 t o Hormander's p r o p a g a t i o n of s i n 9 u l a r i t i e s ÷ f o n t set of

a s o l u t i o n u i s c o n t a i n e d in the s e t of

theorem,

the wave

nul

bicharacteristics (5.2.2)

x~ = q ~ ( t , x , ~ ) ,

~

i s s u i n 9 from the wave f r o n t

=

-q.(t,x,~),

s e t s of

o c c u r s when the nul

and w. The Cauchy problem ( I )

v

cannot be s o l v e d i n 9enera! even f o r difficulties

T=-q(t,x,~),

differential

o p e r a t o r s . One o f

bicharacteristics

approach the

boundary of X. T h i s can be a v o i d e d by t h e assumption t h a t a l l maps x ( a ) , ~ ( a ) sets for

all

-> x ( b ) , ~ ( b )

a and b.

send compact s u b s e t s o f

A difficulty

for

is

T*(×)

Hamilton

i n t o compact

pseudodifferential operators is

the e x i s t e n c e of Pu when X i s not compact. However, (I)

the

if

taken modulo smooth f u n c t i o n s we can a s s e r t t h a t

Cauchy's problem there is a

8g

unique s o l u t o n f o r proof,

which w i l l

s m a l l s t e p s in

all

distribution

not be 9 i v e n i n d e t a i l ,

time.

the supports of

t i m e s and a l l

In t h i s

d a t a v and w. The

uses p a r t i t i o n s

way the problem i s

of

u n i t y and

reduced t o cases when

v and w a r e c o n t a i n e d i n c o o r d i n a t e n e i g h b o r h o o d s and

t h e p r e v i o u s theorem can be used. In o r d e r t o s o l v e

(2)

modulo smooth f u n c t i o n s ,

know fundamental s o l u t i o n s E ( t , s , x , y ) P(t,x,D~,D~)E(t,x,s,y) where 6 ( x , y )

is a distribution

is s u f f i c i e n t

to

the p r o p e r t y t h a t

= 6(t-s)6(x,y),

E=O when t < s .

such t h a t

S f(x)6(x,y)w(y)

with

of P w i t h

it

= f(y).

~ some f i x e d smooth p o s i t i v e

n-form on X.

In p a r t i c u l a r ,

the

distribution

u(t,x)= solves

the Cauchy

(5.2.3)

t,x

f E(t,O,x,y)w(y)~(y}

problem

-> F ( t , x , y )

above

with

v=O.

The d i s t r i b u t i o n

= E(t,O,x,y).

s o l v e s the same problem w i t h w = 6 ( x , y ) . The wave f r o n t

set of

F is

c o n t a i n e d i n t h e H a m i l t o n o u t f l o w from t h e L a g r a n g i a n m a n i f o l d L ( O ) = ( ( O , y l , R ~ \ O ) . The image o f To g e t the c o m p l e t e wave f r o n t both t

and x ,

the p a i r

For s m a l l t , section 3.3).

L(O) set of

(t,T=-q(t,x,~))

at time t

will

be denoted by L ( t ) .

F c o n s i d e r e d as a f u n c t i o n of s h o u l d be j o i n e d t o L ( t ) .

L a x ' s c o n s t r u c t i o n p r o v i d e s a p a r a m e t r i x of

In the next s e c t i o n we s h a l l

F (see

construct a global

parametrix.

5 . 3 A 91obal p a r a m e t r i x

In the f o l l o w i n g

theorem L a x ' s c o n s t r u c t i o n i s 9 e n e r a l i z e d t o a 91obal

c o n s t r u c t i o n on t h e p r o d u c t of

Theorem

Consider, f o r

t=to,

an o r i e n t e d m a n i f o l d X and a r e a l

c a n o n i c a l c o o r d i n a t e s (y, ol

in a

line.

90

neighborhood M of a p o i n t p=(yo,Oo) on L ( t o )

and assume t h a t 9(Y,9)

d e f i n e d in M i s a v e r y r e g u l a r phase f u n c t i o n f o r

L(to)

t h a t the v a r i a b l e s 0 can be taken as parameters on L ( t o )

there,

i.e.

in M and t h a t

( y , g y = 0 ) , 97=0, parametrizes L(to)

in M. Let

( t o , y , 9 ) ->

(t,x,{)

be the Hamilton f l o w

and l e t B be the b i c h a r a c t e r i s t i c i s s u i n g from p t o q = ( t , x o , ~ o ) . Then t h e r e i s a number s>O and a c o n i c a l neighborhood N of that,

if

p such

O < t - t o <s,

A) t h e r e are c o n i c a l neighborhoods N~ and N= of (Xo,0o) and (xo,~o} r e s p e c t i v e l y such t h a t

( x , o ) - > ( y , o ) i s a smooth map from N~ whose image

c o n t a i n s N and ( x , o ) - > ( x , ~ )

i s a smooth map from N= whose image

c o n t a i n s N~. B) t h e r e i s a neighborhood of N of p such t h a t the e q u a t i o n s 9(Y,O)

= h(t,x,o)

= f(t,x,~)

d e f i n e f u n c t i o n s in N~ and N2 r e s p e c t i v e l y , h s a t i s f i e s the Hamilton-Jacobi equation h~+q(t,x,h~)=O, h ( t , x , s ~ ) = g ( y , o ) and f

when t = t o .

i s a v e r y r e g u l a r phase f u n c t i o n f o r L ( t )

Corollary

If

a(y,9)

in N=.

i s a polyhomogeneous a m p l i t u d e f u n c t i o n s w i t h

c o n i c a l l y compact support in M and u(x) is a c o r r e s p o n d i n g

=

S a(y,g)

oscillatory

exp

ig(y,o)do

integral,

there

is a p o l y h o m o g e n e o u s

amplitude f u n c t i o n a ( t , x , ~ ) w i t h c o n i c a l l y compact support in N2 such that v(t,x) solves

the

Cauchy

Note.

From

this

to=O,

9=y.0,

=

S a(t,x,~)

if(t,x,~)

d~

problem

Pv

= smooth,

we

can

there

exp

v-u=

recover

is a n u m b e r

a

smooth local

s such

when

t=to.

arametrix. that

N

In fact,

is j u s t

the

if w e

product

take of

a

91

neighborhood of

0 in y - s p a c e an a l l

9et a p a r a m e t r i x f o r

Proof.

The p o i n t A)

from A)

that

Theorem of f

small

t

of

R"\O.

p r e c i s e l y as a t t h e end o f c h a p t e r 2.

i s o b v i o u s s i n c e x=y,

~=g

t h e f u n c t i o n s h and ÷ a r e w e l l

Section 4.1,

We can t a k e a=l and we

when t = t o .

It

follows

d e f i n e d . According to the

h s o l v e s the H a m i l t o n - J a c o b i e q u a t i o n and t h a t

i s a v e r y r e g u l a r phase f u n c t i o n f o l l o w s from the Theorem o f

section

4.3. Proof of of

the c o r o l l a r y .

Section 3.3,

A c c o r d i n g t o B) of

the theorem and the theorem

t h e Cauchy problem Pu= smooth, u - v smooth when t = t o .

has a s o l u t i o n

u(t,x) with b(t,x,o)

=

exp ih(t,x,g)

S b(t,x,g)

polyhomogeneous. A change o f

b(t,x,g)

do

variables g=g(t,x,~)

produces the d e s i r e d r e s u l t .

In the p r o o f above, o n l y p a r t of Using i t s

Theorem (0,0o)

full

S e c t i o n 3 . 3 was used.

power we can p r o v e

Given go~O and s > O , t h e r e i s a c o n i c a l neighborhood N o f

such t h a t f o r

conically

the Theorem o f

any polyhomogeneous a m p l i t u d e a ( y , g )

with

compact s u p p o r t i n N, t h e e q u a t i o n Pu(t,x)

= smooth, u - v smooth f o r

t=O.

where v=

$ a(y,5)

exp

iy.g

do

has a u n i q u e s o l u t i o n d e f i n e d when OO.

a l a r g e r s.

By

92

Hormander~s p r o p a g a t i o n o f u tends to

singularities

the b i c h a r a c t e r i s t i c

t h e r a y g e n e r a t e d by 0o. L e t for in

which u e x i s t s this

theorem o f such t h a t

r

be t h e

q be t h e p o i n t

c h a p t e r 4,

l e a s t upper bound o f of

image o f

on B c o r r e s p o n d i n g t o

L(r)

q.

r.

shrunk

By t h e

t h e r e are then c a n o n i c a l c o o r d i n a t e s x,~

By t h e p r e c e d i n g t h e o r e m ,

there

is

at

last q

and a phase f u n c t i o n

which

is very regular

a neighboorhood M of

and a number b>O such t h a t

any

amplitude having conically

compact s u p p o r t i n M can be c o n t i n u e d t o

solution

w~ o f

Carrying this small,

situation

permits a similar

This contradicts

it

global

w with

oscillatory

to

to

t-c

inte9rals

t o beyond r

the distribution E(t,x)

a finite

the following

iii)

f r o m t h e sum o f

with

c very

of

r

when c i s s u f f i c i e n t l y

and p r o v e s t h e t h e o r e m .

F(t,x) of

with conically

d e f i n e d by

P with

its

(5.2.3)

pole at

y,

and has a

number o f

functions U{t,x)

properties

t u b e T where i t

U(t,x)= locally (z,Si

t h e U by a smooth f u n c t i o n

U vanishes outside the projection

bicharacteristic

in

t.

Here t h e t , z = z ( x )

are canonical.

on t , x - s p a c e o f

some

has t h e f o r m

S a(t,x,5)

a

d e f i n e d when r < t < r + b .

PU i s a smooth f u n c t i o n F differs

q

w i t h a n e i g h b o r h o o o d M= which

the definition

For any s>O t h e r e i s

for

phase 9 and

parametrix.

d e f i n e d when t < s w i t h

ii)

of

the fundamental s o l u t i o n

Theorem

i)

situation

M= f r o m r - c

We can now p r o v e t h a t

t=r

back by t h e H a m i l t o n f l o w

continuation

compact s u p p o r t s i n

with

integral

Pw~=smooth, w ' - w smooth f o r

produces a s i m i l a r

small.

oscillatory

to

numbers t

B on X × R when N i s

y . o under t h e H a m i l t o n f l o w ,

set of

(O,Oo) when N s h r i n k s

t h e ~ c o o r d i n a t e s a r e p a r a m e t e r s on L ( r )

9(x,~), at

B i s s u i n g from

near the p r o j e c t i o n

way and l e t

t h e o r e m , t h e wave f r o n t

exp if(t,x,$)

d5

are coordinates in

The phase f u n t i o n

f(t,z,$)

the projection,

is very regular

for

93

L(t)

and a ( t , z , 5 )

i s a polyhomogeneous a m p l i t u d e w i t h c o n i c a l l y

support in T f o r

Proof. at

compact

t=const.

By t h e p r e c e d i n g theorem, we can c o v e r t h e wave f r o n t

t h e p o i n t t=O,x=O by a f i n i t e

that to every N there

s e t of F

number o f c o n i c a l n e i g h b o r h o o d s N such

i s a f u n c t i o n U w i t h the p r o p e r t i e s i )

and i i )

and such t h a t U(O,x) = S a ( x , y , 9 ) where a has c o n i c a l l y degree 0 such t h a t sum o f

compact s u p p o r t i n N.

their

sum i s

I for

all

If

dO

we choose t h e s e a o f

O and a l l

x close t o y,

the

t h e U s o l v e s the same Cauchy problem as F modulo smooth

functions. Note.

exp i ( x - y ) . 9

Hence i i )

f o l l o w s and t h i s

p r o v e s t h e theorem.

The theorem c o u l d be e x p r e s s e d o t h e r w i s e , namely t h a t

any p o i n t p in t h e wave f r o n t t h e r e i s an o s c i l l a t o r y wave f r o n t

s e t of

Parametrix f o r We s h a l l

integral

F(t,x)

U(t,x)

of

(with

t

the form above such t h a t

now prove t h a t

the t a k i n g of

U(t,x)

p.

d u a l s commutes w i t h t h e

a g l o b a l p a r a m e t r i x . T o g e t h e r w i t h P=D¢+Q(t,x,D)

the amplitudes a ( t , x , 5 )

Lemma L e t F'

the

pseudodifferential operator

c o n s i d e r the d u a l o p e r a t o r P ' = D ¢ + Q ' ( t , x , D ) and l e t duals of

to

as a parameter)

F-U does not c o n t a i n a c o n i c a l neighborhood o f

a dual

c o n s t r u c t i o n of

s e t of

so t h a t

of

a'(t,x,;)

be t h e

t h e p r e v i o u s theorem.

be t h e fundamental s o l u t i o n o f P'

w i t h p o l e a t y and l e t

be as i n t h e theorem. Then F ~ i s r e p r e s e n t e d by t h e

distributions U'(t,x)

= $ a'(t,x,~)exp-if(t,x,5)d~

i n t h e same way as F i s r e p r e s e n t e d by t h e c o r r e s p o n d i n g d i s t r i b u t i o n s

U(t,x). Proof.

A c c o r d i n g t o Chapter 3,

(PU)'=P'U'. Further,

commutes w i t h changes o f v a r i a b l e s and,

as i t

the map U->U'

i s easy t o v e r i f y ,

also

94

w i t h c o n t i n u a t i o n in the t and w i t h of

this

the homogeneous map ( x , o ) - > ( x , ~ )

it

under A) of

the f i r s t

s e c t i o n . Since these are the o n l y o p e r a t i o n s i n v o l v e d i n

c o n s t r u c t i o n of t=,

v a r i a b l e by t h e H a m i l t o n - J a c o b i e q u a t i o n ,

is

a p a r a m e t r i x of F and s i n c e the

proved i n 9 e n e r a l .

theorem the

lemma i s o b v i o u s f o r

CHAPTER 6

CHANGES OF VARIABLES AND DUALITY FOR GENERAL OSCILLATORY INTEGRALS

Introduction

Consider a general o s c i l l a t o r y u(x)

= $ a(x,e)

is(x,e)

exp

integral

de

w i t h x i n R" and s i n RN and s a r e g u l a r phase f u n c t i o n . that

t h e number N o f

(1971).

It

may happen

i n t e g r a t i o n v a r i a b l e s can be reduced l o c a l l y

leading to another o s c i l l a t o r y the o r i g i n a l

It

integral

one. The t h e o r y of

with

the same wave f r o n t

s e t as

such changes i s due t o H~rmander

occupies the the f i r s t

two s e c t i o n s o f

be a p p l i e d t o fundamental s o l u t i o n s

in

this

c h a p t e r and w i l l

the n e x t c h a p t e r .

When a(x,~}

= I

ak(x,~)

i s a polyhomogeneous a m p l i t u d e in R~ x Rn, a'(x,~) and l e t

t h e dual o f

let

its

dual be

= ~ (-l)kak(x,~) the o s c i l l a t o r y

u(x)

= S a(x,~)

integral

exp i s ( x , ~ )

d~

be u'(x) The dual P '

of

= S a'(x,~)

exp - i s ( x , ~ )

d{.

a polyhomogeneous p s e u d o d i f f e r e n t i a l o p e r a t o r P i s

d e f i n e d as the o p e r a t o r b e l o n g i n g t o the dual of connection w i t h the v a r i o u s p r o p e r t i e s of shown i n Chapter 3 we have v e r i f i e d (P*)'=(P')*,

(PQ)'=P'Q',

In t h e second p a r t of number o f oscillatory

this

symbol.

In

pseudodifferential operators

that

(Pu)'=P'u'.

c h a p t e r we s h a l l

integration variables affects integrals

its

d e f i n e d in

see how r e d u c t i o n o f

the d u a l i t y

t h e same way.

of general

the

96

6.1

H~rmander's e q u i v a l e n c e theorem f o r

oscillatory

integrals with

r e g u l a r phase f u n c t i o n s

C o n s i d e r v a r i a b l e s x and e in R~ and RN r e s p e c t i v e l y and phase functions f(x,8)

homo9eneous o f

s e t N which i s c o n i c a l s a i d t o be r e g u l a r i f

de9ree 1 i n

e and d e f i n e d i n some open

i n t h e second v a r i a b l e . the N d i f f e r e n t i a l s

Such a phase f u n c t i o n

df~ a r e l i n e a r l y

independent

when fe = O. Close t o such a p o i n t and when f~=O,

the p a i r

(x,fx)

p a r a m e t r i c e s a homogeneous L a g r a n g i a n m a n i f o l d L.

In f a c t ,

the

dimension of

the mani÷old i s

f00de =0 so t h a t de =0.

n and when dx=dfx=O, then f~mde =0 and

Further,

on the m a n i f o l d we have e.fe=f=O so

t h a t O=df=f~dx and chan9in9 e t o t e , is

is

t>O,

chan9es f~

to t f ~

so t h a t L

homo9eneous. It

f o l l o w s form these c o n s i d e r a t i o n s t h a t wave f r o n t

oscillatory

integrals

(6.1.I)

u = S a(x,e)

are c o n t a i n e d in We s h a l l and 9 ( x , e )

exp i f ( x , 8 )

d8

homo9eneous L a g r a n 9 i a n m a n i f o l d s .

now c o n s i d e r a s i t u a t i o n

where two phase f u n c t i o n s f ( x , e )

d e ÷ i n e the same La9ran9ian a t a p o i n t p in the sense t h a t

(x,fx=9~) at

the same p o i n t x , e f o r

equivalent equations. It and the n o t i o n o f

s i g n a t u r e of

second one t h e number o f

Deformation lemma

(6.1.2)

a real

symmetric m a t r i x ,

one i s t h e number o f

n e g a t i v e ei9envalues of

c l o s e t o p where b ( x , e )

9(x,8)

b(l,x,e)=b(x,8)

and 9

a p a i r of

p o s i t i v e and t h e the m a t r i x .

+ 9o.b(x,e)9m/2

i s a symmetric m a t r i x .

and 9 have the same s i g n a t u r e a t p,

function b(t)=b(t,x,8)

i.e.

f

Under the above c o n d i t i o n s ,

f(x,8)=

f

which f e =0 and Be =0 a r e

i n v o l v e s the Hessians f0e and 9me of

i n t e g e r s o f which t h e f i r s t

and 9e~ of

s e t s of

When the Hessians f e e there is a continuous

whose v a l u e s a r e symmetric m a t r i c e s such t h a t

and b ( O , x , ~ ) = O and t h e s i g n a t u r e s o f

t h e Hessians o f

97

the correspondin9 f ( t , x , ~ )

are constant.

Proof.

some b i s

That

follows

(2)

from

so t h a t ,

holds with

(2)

that

÷m~

= 9~e

+ 9eeb9e9

clear

f~o

so t h a t ,

It

also follows

9m~ and t h e r i g h t

from

(2)

side of

(6.1.3)

I+b900 i s

and 9e =0 t h a t

dfe are linearly

independent,

invertible.

The d e s i r e d s t a t e m e n t t h e n amounts t o

the following:

9iven a symmetric

t h e n two o t h e r s y m m e t r i c m a t r i c e s B~ and B= can be

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

each o t h e r r e s p e c t i n 9 t h e c o n d i t i o n

det(I+AB)~O i f

B+BAB has t h e same s i 9 n a t u r e f o r

When d e t A i s sin9ular

at

and o n l y not

zero,

if

this

is

evident for

then

B=B~,Bz.

I+AB and A÷BAB a r e

t h e same t i m e and t h e map B->A+ABA i s

s y m m e t r i c m a t r i c e s B. that

the

= 9~e(l+gee)

since the differentials

m a t r i x A,

It

= 9ee(l+bgee),

by h y p o t h e s i s , t h e s i 9 n a t u r e s o f

f o r m u l a a r e t h e same.

s i n c e d ( f - g ) = O when 9e=O.

bijective

Hence t h e s t a t e m e n t amounts t o

for

t h e known f a c t

two n o n - s i n g u l a r s y m m e t r i c m a t r i c e s can be deformed i n t o

other throu9h non-singular matrices if

and o n l y

if

their

each

si9natures are

t h e same. When d e t A v a n i s h e s ,

let

P be t h e p r o j e c t i o n

on t h e r a n 9 e o f

Then B can be r e p l a c e d by PBP w i t h o u t c h a n 9 i n 9 any o f c o n d i t i o n s so t h a t proof also

if

we o b s e r v e t h a t

in a c o n i c a l

Next,

we s h a l l

oscillatory

Theorem conical

we a r e back if

in

case.

a deformation t->b(t)

nei9hborhood of prove part of

integrals

the f i r s t

This finishes works at

p,

it

the works

H~rmander's e q u i v a l e n c e theorem f o r

(1971, Theorem 3 . 2 . 1 ) .

L e t ÷ and 9 be two r e 9 u l a r

nei9hborhood of

and 9 a t

t h e two

p.

a point

phase f u n c t i o n s

d e f i n e d in a

p = ( x o , e o ) and assume t h a t f~=O and 9e=O

d e f i n e t h e s~me L a 9 r a n 9 i a n m a n i f o l d L c l o s e t o o~ f

A.

p have t h e same s i 9 n a t u r e s .

p and t h a t

Then t h e r e

is

the Hessians

a bijection

98

e->h(x,8) defined close to

Proof.

We have t h e f o r m u l a (2)

(6.1.4)

and,

f(x,e)=g(x,h(x,e)).

by T a y l o r ' s

formula,

9(x,e')=g(x,e)+(e'-e).9e+(e'-e).Gee(x,e',8)(8'-8),

where Ge0 i s

a symmetric m a t r i x of

t o d e t e r m i n e e'

so t h a t 8'

with

p such t h a t

=

8

homo9eneity - I

f(x,e)=g(x,8')

in

e,e'.

We s h a l l

try

by p u t t i n g

÷W(X, ~}ge

some s y m m e t r i c w t o be d e t e r m i n e d . Then,

combining

(2)

and

(4)

we

get z.wz

+ wz.G(x,e,e+wz)wz

where z=ge. T h i s h o l d s f o r w+

which is

is

small

f=f(t)

a complicated equation for and t h e r e i s

w. However,

a unique s o l u t i o n

After

this,

we can

Hence a c l a s s i c a l

6.2 Reduction of

Suppose t h a t

let

f

is

neighborhood N of

variables

t h e number o f

a re9ular a point

u(x) defined

oscillatory

t

is

of

play the part of

phase f u n c t i o n

9 and p r o c e e d

with

some f ( s )

with

the proof.

variables

d e f i n e d in

p = ( x o , e o ) and suppose t h a t conically

the form

9 r e p l a c e d by any f ( t )

connecting f(t)

integration

b=b(t)

and a c o r r e s p o n d i n g

e variable

with

small,

a conical a(x,8)

compact s u p p o r t i n

is

a

N. Then t h e

integral

(6.2.1) is w e l l

the

if

c o v e r i n g argument f i n i s h e s

polyhomogeneous a m p l i t u d e w i t h oscillatory

f(t)

w=w(t)

We c o u l d a l s o have s t a r t e d

and o b t a i n e d a change o f

but

integral

it

we can i n

this

= I

exp i f ( x , e )

a(x,e)

is s o m e t i m e s

with

p l u s a smooth f u n c t i o n . p,

if

wG(x,e,8+wge)w=b/2

another step.

at

z

c o n n e c t e d t o 9 by a change o f

desired.

s>t.

all

= z.bzl2

possible

a s m a l l e r number o f

We s h a l l

see t h a t

way e l i m i n a t e

if

de to

express

u

integration

locally

as

an

variables

t h e h e s s i a n f e e has r a n k r

r variables.

When A i s

a symmetric

99

matrix,

we d e f i n e

difference

its

sign,

that

t o be

between t h e t h e number o f

negative eigenvalues of

Theorem

sgn A,

Suppose t h a t

the partial

Then t h e r e i s

( - I ) d where d i s

positive

the

and t h e number o f

A.

there is

a division

H e s s i a n Q= f 0 . . e . ,

a function

h(x,~'>,

of

f

of

at

variables

e',e''

such

p has r a n k r = dim

homogeneous o f

degree I

in

e''

8',

such

that 9(x,e')=f(x,e',h(x,8')) is a regular

phase f u n c t i o n

L a g r a n g i a n L t h e r e as f (6.2.2)

v(x)

differs

c(Q)

=

p which d e f i n e s t h e same

and an a m p l i t u d e b ( × , e ' )

= c(Q)

from u c l o s e to

(6.2.3)

close to

$ b(x,e')

exp

i9(x,8')

p by a smooth f u n c t i o n .

(2~) ~/=

Idet

gl

-1,=

such t h a t

it,.

de'

Here mr=

When a has t h e e x p a n s i o n (6.2.4)

aj

, j=k,k-l,...,

b has an e x p a n s i o n (6.2.5)

b~1=÷j,

where t h e

indices

indicate

P r o o f . S i n c e Q has r a n k r , function

h(x,8')

8''+h(x,m')

of

degree of

x and e'

(4).

homogeneity.

the e q u a t i o n s re=8 determine e ' ' c l o s e t o p.

we may assume t h a t

homogeneities in

j=k,k-l,...

Hence, c h a n g i n g e ' '

as a to

h = O , an o p e r a t i o n which p r e s e r v e s t h e

L e t P be any q u a d r a t i c f o r m

in

the v a r i a b l e s

e''

which has t h e same s i g n a t u r e as Q. Then t h e two phase f u n c t i o n s f(x,8)

and f ' ( x , 8 ' , 8 ' ' ) =

f(x,e',O)+

d e f i n e t h e same L a g r a n g i a n m a n i f o l d a t there.

p and have t h e same s i 9 n a t u r e

Hence, by t h e p r e v i o u s t h e o r e m , t h e y a r e c o n n e c t e d by a

transformation of is conically a'(x,e)

P(e~')/2te'l

the e variables

c l o s e enough t o

such t h a t

u(x)

differs

p,

close to

there is from

p.

Hence i f

the support of

a polyhomogeneous a m p l i t u d e

a

100

(6.2.6) by

u' (x)

a smooth

function.

=

In

f a' order

x,O',O'') to

exp

compute

i÷'(x,e',O'')

the

right

side

dO of

this

f o r m u l a we need a d i v e r s i o n .

Diversion to Lemma

t h e method o f

Let y,9

f o r m and f ( y ) following

be r e a l

variables

a smooth f u n c t i o n

S f{y)exp

itQ(y)/2

according to

(3)

c~ = c=(Q)= v a n i s h e s u n l e s s I~I

Proof.

The F o u r i e r

which

with

is verified

t h e case r = l . (2~) - r

compact s u p p o r t .

Then we have t h e

l a r 9 e t>O,

dy =

C(Q)

~ t ~-'~'tm

c=D=f(O)

and where for

0 =0

even.

transform of

e×p i Q ( y ) / 2

is

exp - i Q - 1 ( O ) ,

by a d i a g o n a l i z a t i o n

Hence t h e t rt=

Q a nondegenerate q u a d r a t i c

D~exp(-iQ-~(9)/2)/~!

is

C(Q)

phase

i n Rr ,

asymptotic expansion for

(6.2.7) w i t h C(Q)

stationary

left

Sf^(9)

side of

of (7)

exp(-iQ-~(0)/2t)

Expanding t h e e x p o n e n t i a l 9 i r e s

A,

reducing the f o r m u l a to

i s C(Q)

times

do.

a series of

terms

c=9=t-~ =~ / z f r o m which t h e d e s i r e d r e s u l t

Return t o

the proof of

By t h e d i v e r s i o n (6.2.8)

C(Q)

t h e theorem

(and i f

d e t P = d e t Q),

~le'l ¢~ .... ~t=

where 5=0 a f t e r

follows.

c~(Q)D

the o p e r a t i o n s .

the right

side of

Hence, c o l l e c t i n g

br~=÷j=~ c~(Q)D~=a'~(x,e',~) for

T h i s proves t h e theorem.

equals

~a,j(x,e,,~)

terms w i t h

h o m o 9 e n e i t y , we g e t (6.2.9)

(6)

j=

q-3[~[/2.

t h e same

101

&.3 D u a l i t y

We s h a l l

and r e d u c t i o n o÷ t h e number o f

now c o n s i d e r d u a l i t y

genera! settin9 reduction of to

oscillatory

t h a n we have done so f a r .

t h e number o f

introduce classes of

half-integral

of

inte9ration

indices

with N v a r i a b l e s r e d u c t i o n of (6.3.1)

it turns out

I((÷,a,x)

chan9es s i 9 n , to

a(x,e)

with

of

j=0,1,2,... terms of

I ,

t h e more

our r e s u l t s is

integral

when m i s

on t h e

then c o n v e n i e n t or

any r e a l

with e x p a n s i o n s of

number,

the form

in e. For o s c i l l a t o r y

to be natural

integration

=(2~) -N/2

it

in

p=O,l,...

indicate h o m o g e n e i t y

the number

where q = j - N / 2 ,

by - I

variables,

h o m o 9 e n e i t i e s . More g e n e r a l l y ,

a~-p(x,e), the

inte9rals

In view of

amplitude functions

let S ~ be the c l a s s of a m p l i t u d e s

where

variables

integrals

to n o r m a l i z e both

the

v a r i a b l e s by p u t t i n g

de,

~ e ( E ( q - m ) ) a m - ~ - N ( X , e ) exp i f ( x , e ) c= 1 o r

-I

and e ( j ) = i J .

Note t h a t

as

¢

c o r r e s p o n d i n g h o m o g e n e i t y m-q-N g e t m u l t i p l i e d

t h e power q-m.

Hence t h e d u a l i t y

differs

f r o m t h e one we have

c o n s i d e r e d b e f o r e o n l y by a n o r m a l i z a t i o n and t h e f a c t

that

m may be a

half-integer. If

b(x,e')

is

t h e a m p l i t u d e computed i n

t h e t h e o r e m above and i f

we

put a'(x,e')

= I d e t QI - I t =

the o s c i l l a t i n g

(6.3.2)

inte9ral

I((f',a',x)

b(x,B'),

(1)

and i t s

these notations,

particular, shall

number o f matrix, •p o s i t i v e

d u a l can be w r i t t e n

(I)

and

expiCf'(x,e') (2)

with

t h e d i f ÷ e r e n c e between ( I )

be i n t e r e s t e d variables

let

r=r(A)

, N'=dim

e',

as

=

(2~)--N'/2S ~ e ( ¢ ( q ' - m ) ) a ' m - ~ . - N , With

f'(x,e')=9(x,e')

in

the r e s u l t

also to

(1)

of

with

be t h e r a n k o f

¢=I g i v e amd (2)

(6.2.1) is

and

(6.2.2).

a smooth f u n c t i o n .

a p p l y i n g our r e d u c t i o n of ¢= - I .

A and r + ( A )

and n e g a t i v e e i g e n v a l u e s o f

de'.

A and d(A)

their

We

the the

When A i s a s y m m e t r i c and t - ( A )

In

t h e number o f

difference.

102

Lemma

The d i f f e r e n c e

I-(f,a,x)is

(-1)~I-(f',a~,x)

smooth w h e n t = r _ ( f e e ) - r m ( f ' e . e . ) ,

where t h e H e s s i a n s a r e t a k e n a t

( x o , e o ) and t h e c o r r e s p o n d i n 9 p o i n t

Proof.

In

o u r new n o r m a l i z a t i o n s ,

e(c(j'-N'/2-m)) I

e(c(j-N/2-m))

for

a'm-j.-N./=

c~(ce)le'l

~=I and j ' = j +

for

e'.

(6.2.9) =

r e a d s as

e(¢d(Q)/2))

(~-,°,,tm

Ds"

am-N/=-j(x~8's~)l~=0

3 1 ~ I / 2 making t h e h o m o 9 e n e i t i e s e q u a l s i n c e

r ( Q ) = N - N ' . T h i s f o r m u l a can a l s o be w r i t t e n e(¢j')

bm-j.-N./=

=e(c(d(e)+r(e))/2) with

I

by - I

to

by - I

to

is

view of

t h e way t h a t

lemma. T h i s f i n i s h e s

Paired oscillatory I((f,a,x)

is

t h e number o f

and

the factor

I~I

si9n

in

t h e power r - ( Q ) ,

is in

front

to

which

Q was o b t a i n e d ,

is

t h e same as t h e number t

the of

in of

the

integrals

h o m o 9 e n e i t y . By o u r in

is

the p r o o f .

s i m u l t a n e u s change o f

variables

c=(Q)

t h e change o f

si9n of

side

-I

be d u a l

of

left

t h e sum on t h e r i g h t

(-I) 'g''z

mod 2,

the

T h i s change i s

oscillatory

t o be p a i r e d .

operation,

since c=(-Q)=

congruent to j'

J¢(f,a,x) are said

si9n,

and t h e t e r m s o f

d e t e r m i n e d by t h e change o f

t h e sum on t h e r i g h t .

D~ = a m - j _ N / z l ~ = 0 ,

. When ¢ changes i t s

t h e power j + l ~ I / 2

formula is

Let

e(~j)le'l(~-'"~tzc.(¢e)

t h e power j

even. Since j + l ~ I / 2

as

=

sum o v e r j ' = j + 3 1 ~ I / 2

multiplied

times

It

inte9rals.

= I÷(÷,a,x) is

+ ¢I-(f,a,x)

obvious that

variables

of

The sums

the p a i r i n g

integration

which p r e s e r v e s

lemma a b o v e , a s i m u l t a n e o u s r e d u c t i o n o f

t h e two t e r m s p r o d u c e s a change o÷ the

s u r v i v e s any

lemma. Hence t h e p a i r i n 9

u n a f ÷ e c t e d by changes o f

variables

¢ to

t h e number

(-I)~¢

wh'ere t

a p p e a r s as a v e r y s t a b l e and a f f e c t e d

o n l y by a

103

s i g n when the number of

i n t e g r a t i o n v a r i a b l e s are chan9ed.

As an example we s h a l l first

c o n s i d e r fundamental s o l u t i o n s of

o r d e r h y p e r b o l i c p s e u d o d i f f e r e n t i a l o p e r a t o r s P and P ' .

U(t,x) is

the l o c a l

= S a(x,~)

expression for

modulo smooth f u n c t i o n s , chapter that E'

= I

(-l)kak

was proved a t

d~

t h e end o f

the preceding

t h e fundamental s o l u t i o n

is

f a'(x,~)exp-if(t,x,~)

d~.

when a=~ ak w i t h ak homogeneous of

,k=O,-l,...

degree k and a n a l o g o u s l y f o r ¢=I,

If

a fundamental s o l u t i o n E of P = B t + Q ( t , x , D )

it

U'(t,x)= Here a'

expif(t,x,~)

the c o r r e s p o n d i n g e x p r e s s i o n f o r

of P ' = D ~ + Q ' t , x , D )

with

two dual

P and P'

If

U i s denoted by I C ( f , a , x )

the sum U+U' c o r r e s p o n d s in our new n o r m a l i z a t i o n t o J((f,a,×)

where ¢ = ( - I ) - . changes o f

Using the

t h e number of v a r i a b l e s .

c o n s t r u c t i o n of

a parametrix for

hyperbolic differential be e x p r e s s e d in k=l,2,...,m

lemma above we can t r a c e

the fundamental s o l u t i o n o f

the new n o r m a l i z a t i o n . I f

again,

e is -I raised

space v a r i a b l e s .

differential

in L a x ' s a stron91y can a l s o

fk.~k for in

the

t a k e s t h e form

( J((fl,al,x)+...+J¢

b e h a v i o r of

Chapter 3)

are the phase f u n c t i o n s and a m p l i t u d e s t h a t occur

parametrix, it

where,

b e h a v i o u r under

The p a i r i n g s which occur

o p e r a t o r s (see t h e end o f

terms of

its

to the

This v a l u e of

the s i n g u l a r i t i e s

(fm,am,x))/2 power

n which

~ accounts f o r of

here

is the

number

the v e r y d i f f e r e n t

fundamental s o l u t i o n of

hyperbolic

o p e r a t o r s in even and odd d i m e n s i o n .

The coming c h a p t e r paired oscillatory

i s d e v o t e d t o the a n a l y s i s o f

of

integrals.

Note. The m a t e r i a l o f a p p l i c a t i o n t o dual

singularities

this

chapter is

taken from H~rmander

and p a i r e d i n t e g r a l s

c h a p t e r g i v e s an expanded v e r s i o n o f

that

1971, i t s

from G~rdin9 1977. The n e x t paper.

of

CHAPTER 7

SHARP AND DIFFUSE FRONTS OF PAIRED OSCILLATORY INTEGRALS

Introduction

In Chapter 5 we have c o n s t r u c t e d a 91obal p a r a m e t r i x f o r

the fundamental s o l u t i o n of o p e r a t o r and w i t h differential

it

a hyperbolic first

a 9obal p a r a m e t r i x f o r

operator. Locally,

a strongly hyperbolic

these p a r a m e t r i c e s a r e o s c i l l a t i n 9

i n t e 9 r a l s w i t h v e r y r e 9 u l a r phase f u n c t i o n s . p o s s i b l e to analyze e x p l i c i t l y As s a i d

in

the h i s t o r i c a l

ago. The f i r s t

order pseudodi÷ferential

Hence i t

s h o u l d be

how t h e y behave near any s i n 9 u l a r p o i n t .

introduction,

such q u e s t i o n s were r a i s e d

o b s e r v a t i o n came w i t h P o i s s o n ' s f o r m u l a f o r

fundamental s o l u t i o n o f cone which means t h a t

the wave e q u a t i o n .

the s i n 9 u l a r i t y

cone by the b e h a v i o r of

Its

support

can not be f e l t

the fundamental s o l u t i o n

o c c u r s i n a weaker form f o r

Here the

li9ht

t h e fundamental s o l u t i o n o f

s l n 9 u l a r s u p p o r t of solution

cone i s

no l o n g e r s t r a i g h t

the fundamental s o l u t i o n .

has smooth e x t e n s i o n s o v e r t h e l i g h t

+ time)

c a l l e d a sharp f r o n t ,

cone from e i t h e r

To c a t a l o 9 u e a l l phase f u n c t i o n s to 9 i r e

reoccurs f o r

criteria

oscillatory

small

singularities

inte9rals

a l o c a l v e r s i o n of

sharp f r o n t s . (this

side,

by

inside. This

an even number o f

(space

variables is

local,

these

time o n l y .

of o s c i l l a t o r y

integrals with regular

i s p r o b a b l y i m p o s s i b l e . The aim of for

i s o n l y the

But the fundamental

s i n c e Hadamard's c o n s t r u c t i o n i s o n l y

a s s e r t i o n s have been proved f o r

t h e wave c o n s t r u c t e d by

v a r i a b l e s and does not occur when the number of

odd. F u r t h e r ,

li9ht

T h i s phenomenon

and i t

z e r o from t h e o u t s i d e and by somethin9 e l s e from t h e phenomenon,

light

o u t s i d e the

e q u a t i o n in f o u r v a r i a b l e s w i t h v a r i a b l e c o e f f i c i e n t s , Hadamard.

the forward

i s the

there.

Ion9

this

chapter is only

They seem t o be r e s t r i c t e d

to paired

may even be p r o v e d ) . The main c r i t e r i o n

the Petrovsky t o p o l o 9 i c a l c r i t e r i o n

which e x t e n d s t o fundamental s o l u t i o n s of

for

lacunas

hyperbolic differential

is

105

operators with integrals. 1)

it

2.1

variable

Precisely

applies after

A family

So*

where e i s

I

following

and t o

a radial

integration

integrations

i (= e x p ( i x r

or

-1~

x is

large r.

paired oscillatory

the constant c o e f f i c i e n t

real

integral.

meet d i s t r i b u t i o n s

integrals

of

in

the form

f(r)dr

and f ( r )

To t h i s

(see Chapter

the o s c i l l a t o r y

t o come we s h a l l

r ---I

standard integral

in

case

i n one v a r i a b l e

x d e f i n e d by o s c i l l a t i n g

(7.1.1)

constant for

as i n

o$ d i s t r i b u t i o n s

In our r a d i a l one v a r i a b l e

coefficients

end,

vanishes for

it

met w i t h

is

small

r

and i s

convenient to c o n s i d e r the

in S e c t i o n 1.5

in c o n n e c t i o n w i t h

the Herglotz-Petrovsky formula, (7.1.2)

H(s,z)

where s i s

real

=So ~ e x p - r z r - . - I 0.

meromorphic f u n c t i o n function s=p,

of

z in

we d e f i n e H ( p , z )

all

Lemma equals I

z=

poles at

s=p=O,l,.,

H(s,z)

extends to

and t o an a n a l y t i c

along the negative axis.

by t h e f o r m u l a ( 1 . 5 . 3 ) .

a

For

L e t us n o t e t h a t

dH(s,z)/dz =-H(s-l,z) values of

s.

When f ( r )

is

for

(7.1.4) is

s with

= r(-s)

As e x p l a i n e d t h e r e ,

t h e complex p l a n e c u t

(7.1.3) for

of

dr

large r, H(s,z)

an e n t i r e

a smooth f u n c t i o n

-

compact s u p p o r t

r

and

So ~ e x p - r z f ( r ) r - ' - I d r function

of

z.

integral

of

(4)

i n O0 on

i n t e g r a t i o n produces t h e d i f f e r e n t i a l

Since we are o n l y i n t e r e s t e d in s i n g u l a r i t i e s of

c(O)=l.

the s i n g u l a r f a c t o r

under t h e s i g n o f

when q i n c r e a s e s by an i n t e g e r , we s h a l l

r~-q-~dr.

and t h e s i n g u l a r i t i e s

i n t e g r a t i o n d e c r e a s e one s t e p use

I H ( q , f (x, 8 ) + i ¢ 0 ) ) a - m - N ( × , 8 )

as an a s y m p t o t i c sum f o r the s i n 9 u l a r i t i e s (7.2.1)

of

singularities

in the sequel.

the p a i r e d o s c i l l a t o r y

3c(f,a,x)

=

l÷(÷,a,x)

+

T h i s means t h a t

integral

~I-(f,a,x)

a r e r e p r e s e n t e d by t h e a s y m p t o t i c sum (7.2.2) where

$ ~ HC(q,f(x,e) q=j-N/2,

a-Q-N(X,8)W(e)

j=p,p+i, ....

To the sum above one can a p p l y r e d u c t i o n o f

the number of

integration

v a r i a b l e s a c c o r d i n 9 t o the f o l l o w i n 9

R e d u c t i o n theorem

Suppose t h a t

in

the sum above the Hessian o f

f

at a

109

point

(Xo,eo) has rank r and r -

n e g a t i v e i g e n v a l u e s . Then i t

a z e r o e x p a n s i o n from an e x p a n s i o n of

S ~ H(' (q',f' ( X , 8 ' ) b - q . - N . where

N'=dim -I

8'=N-r,

factor

of

Proof.

This r e s u l t

q'=j'-N'/2,

differs

by

the form

( X , 8 ' ) W ( 8 ')

j'=p,p+l,

....

and

¢,~'

differ

by a

r a i s e d t o t h e power r _ .

differs

from the r e d u c t i o n theorem of

s e c t i o n 6.3

o n l y in the n o t a t i o n s . Example

G l o b a l p a r a m e t r i x of

t h e fundamental s o l u t i o n of

a strongly

hyperbolic operator

We can now s t a t e t h e p r o p e r t i e s o f L a x ' s l o c a l solution F(t,x) stands f o r

xo,

of

the Cauchy problem o f

x for

p a r a m e t r i x of

section 3.3.

the o t h e r n v a r i a b l e s and ~ f o r

the

For s i m p l i c i t y ,

t

the dual

variables.

Theorem

There a r e a s y m p t o t i c e x p a n s i o n s f o r

3k(t,x)= with

5 I ak,1-~-j (t,x,~)

¢=(-1) ~ and

sin9ularities Proof.

of

j=0,1,2,..,

whose

small

t,

H¢(j+m-l-n.fk(t,x,~)) sum

divided

w(~)

by 2 r e p r e s e n t s

the

F.

By L a x ' s c o n s t r u c t i o n ,

t h e p a r a m e t r i x i s a sum f o r

k=l,...,m

of

terms Fk = S ~ bkj(t,x, where j = l - m , - m , . . . . .

(7.2.3) If

) expifk(t,x,~)d~

Under t h e p a i r i n g

bk.j

=

,

k->k'

we have

(-1)Jbkj.

we r e w r i t e Fk as

$ I (bkje(-j-n) the s i n g u l a r i t y

e(j+n)

expansion of

S ~ (bkje(-j-n))

expif~(t,x,~)

d~

,

Fk reads

H(j+n,f(x,~+i0))

w(~)

Changing k t o k '

we have t o change e ( j + n )

view o f

means t h a t we g e t the same e x p r e s s i o n as b e f o r e w i t h a

(3)

this

to e(-j-n)

and +iO t o - i O .

In

110

factor

(-ll"

and +iO changed t o -iO.

change the summation index from j

This proves the theorem i f

we

t o 1-m-j.

P a i r i n g and the g l o b a l p a r a m e t r i x S t a r t i n g w i t h the form 9iven above, the c o n s t r u c t i o n of a 91obal p a r a m e t r i x f o r our s t r o n g l y h y p e r b o l i c o p e r a t o r proceeds as f o r

a first

order h y p e r b o l i c p s e u d o d i f f e r e n t i a l o p e r a t o r by p a r t i t i o n s of u n i t y in ~-space, c a n o n i c a l changes of v a r i a b l e s and e x t e n s i o n t o

l a r g e r and

larger t

along the b i c h a r a c t e r i s t i c s by the H a m i l t o n - J a c o b i e q u a t i o n .

In f a c t ,

t h i s c o n s t r u c t i o n a p p l i e s when the p a r a m e t r i x i s w r i t t e n as a

sum of o s c i l l a t o r y

i n t e g r a l s and we have seen t h a t

r e w r i t i n g in terms of

paired o s c i l l a t o r y

it

commutes w i t h the

i n t e g r a l s . Hence i t

applies

a l s o t o a p a r a m e t r i x in the form given above. The advantage of t h i s form i s t h a t , terms,

knowing the Hessian of the phase f u n c t i o n of any of

we know how t o a p p l y r e d u c t i o n of v a r i a b l e s t i l l

its

we get a phase

f u n c t i o n s whose Hessian vanishes a t the p o i n t we are i n t e r e s t e d i n . Below,

t h e r e are some simple examples of

t h i s a n a l y s i s . To have more

complicated examples, we need t o review the t h e o r y of almost a n a l y t i c e x t e n s i o n s in the next s e c t i o n .

Examples Let us c o n s i d e r an o s c i l l a t o r y

integral

(7.2.1)

c o o r d i n a t e s and suppose t h a t the Hessian of rank N-I

in a c o n i c a l neighborhood P of a

w r i t t e n in p o l a r

f(x,8)

has corank i ,

i.e.

p o i n t p=(xo,eo) and t h a t the

supports of the amplitudes are c o n i c a l l y compact subsets of P. Then, by the r e d u c t i o n theorem, the s i n g u l a r i t i e s of

the i n t e g r a l have an

expansion of the form

I H¢(j-I/2,f' (x,8'))b-~-~,=(x,e'l w i t h no i n t e g r a t i o n . A p p l y i n 9 t h i s strongly hyperbolic d i f f e r e n t i a l distributions

, c(8')=I,

j=p,p+1,...

t o the fundamental s o l u t i o n of a

o p e r a t o r , we 9et expansions where the

111

H¢(j+m-l-n/2,f'(t,x,e')), are m u l t i p l i e d

j=0,I,2,...

by smooth f u n c t i o n s .

p l u s t h e number o f o u t e r sheet of

The s i g n

n e g a t i v e e i g e n v a l u e s of

t h e s i n g u l a r s u p p o r t and f o r

n e g a t i v e e i g e n v a l u e s and the f r o n t s and d - s when n i s odd where the known t o be s h a r p . of s i n g u l a r i t i e s e i g e n v a l u e s of

t o t h e power n - I

t h e Hessian i n v o l v e d . At t h e small t ,

t h e r e a r e no

have the t y p e s - s when n i s even

last

p l a c e i n d i c a t e s the o u t e r f r o n t ,

I n s i d e t h e o u t e r s h e e t and f o r

depends e s s e n t i a l l y

l a r g e t~

on t h e number o f

the nature

negative

t h e Hessians.

7 . 3 Almost a n a l y t i c

Let f ( x )

¢ is -I

extensions

be a smooth f u n c t i o n and c o n s i d e r

(7.3.1)

f(x+iy)

= I

f~k'(x)(iy)k/k!

as an a s y m p t o t i c e x p a n s i o n i n y .

As such i t

satisfies

the equation

( d / d ~ ) f ( × + i y ) = O , d/d~= d / d x + i d / d y , for

a p p l y i n g d / d x we g e t the s e r i e s f'(x)

+iyf''(x)

+...

and a p p l y i n 9 i d / d y we g e t t h e s e r i e s -f'(x)

-iyf''(x)

By a s i m p l e e x t e n s i o n of Whitney),

+ ....

a result

the T a y l o r s e r i e s

(I)

by B o r e l

for

(or by a theorem by

a smooth f u n c t i o n can be p r o v i d e d

w i t h f a c t o r s depending on x w i t h the p r o p e r t y o f c o n v e r g e n t and h a v i n g the same d e r i v a t i v e s a t real

a x i s as the f u n c t i o n f .

c a l l e d almost a n a l y t i c of

infinite

the

f

for

which

(d/dZ)f(x+iy)

vanishes

o r d e r when y=O..

variables x~,...,xn, a function f(x+iy) and

t h e same p o i n t s o t

Hence t h e r e a r e smooth f u n c t i o n s f ( x + i y ) ,

e x t e n s i o n s of

In the g e n e r a l case, f o r

f(x+iy)

r e n d e r i n g the s e r i e s

smooth r e a l

functions f(x)

we d e f i n e an almost a n a l y t i c for

which f ( x - i y )

is

of

several

e x t e n s i o n of

f

t h e complex c o n j u g a t e o f

t o be

112

dld~kf(x+iy) vanish of

infinite

order for

theorem e x t e n d s t o f u n c t i o n s e x t e n s i o n s always e x i s t . AE(f),

it

is

clear

for of

y=O and k = 1 , . . . , n .

Since B o r e l ' s

several variables,

Denotin9 almost a n a l y t i c

almost a n a l y t i c extensions of

f

by

that

A E ( f + 9 )= A E ( f ) + A E ( 9 ) , A E ( f g ) = A E ( f ) A E ( 9) and t h a t clear

AE(1/f)

that.

arbitrarily

= I/AE(f)

9iven f, small

the support of

exists

where f

does n o t v a n i s h .

we can choose 9 = A E ( f )

neighborhood of

so t h a t

R" and 9 ( x + i y )

It

is

9 v a n i s h e s in

vanishes for

also an

x outside

f.

7.4 S i n g u l a r i t i e s of paired o s c i l l a t o r y i n t e g r a l s with Hessians of corank 2

C o n s i d e r a term o f integral

the expansion ( 7 . 2 . 2 )

of

a paired oscillatory

(7.2.1),

H¢(q,f(x,o))a(x,e)w(e),

I

(7.4.1)

q=j-Nl2.

L e t p = ( x o , e o ) be a p o i n t

where t h e H e s s i a n o f

has i t s

support close to

p,

rewrite

the

p o l a r form let

(7.4.1)

integral into

dim e=1,

represent this

w(e)=de and a ( x , e ) By a s s u m p t i o n , f

one w i t h

dim e=2, o r

and w i t h

another

new s i t u a t i o n

a regular

phase f u n c t i o n

d e f i n e d as t h e s e t o f

c o d i m e n s i o n 1. simple zeros of

In our case, f(x,e),

L e t Y be a component o f reached.

f(x)

some o f

e for is

so t h a t

we t a k e t h e

fe

a finite

f~

is

X be i t s

We

not zero. projection

does n o t v a n i s h so

which f ( x , e ) = O ,

just

we can

line.

and l e t

on x - s p a c e . Then × c o n t a i n s Xo and o u t s i d e X, f(x),

a(x,8)

e and a n o t h e r q.

has compact s u p p o r t on t h e r e a l is

if

If

and choose c o o r d i n a t e s so t h a t

L e t L be t h e L a g r a n g i a n m a n i f o l d 9 i v e n by f

that

has c o r a n k 2.

t h e r e d u c t i o n theorem s a y s t h a t

to a similar

account,

f

is

a manifold of

collection

of

points,

which come t o g e t h e r as x a p p r o a c h e s x o .

t h e complement o f

× f r o m which xo can be

113

To see what happens when x a p p r o a c h e s xo i n Y, analytic

e x t e n s i o n s in

letters.

L e t M be a n a r r o w s t r i p

plane.

e of

our f u n c t i o n s

f,

we s h a l l

use a l m o s t

d e n o t e d by t h e same

around t h e r e a l

axis R in

t h e complex

Then t h e d i f f e r e n t i a l

(7.4.2) and i t s

F(x,e)

=

H(q,f(x,e))a(x,8)de

differential

dF(x,e) are well

=

(H(q,f(x,e)(~a/~B)(x,e)

defined locally

the f u n c t i o n s

M as l o n g as Im f ( x , e )

involved are analytic,

g e n e r a l case i t which w i l l

in

+H'(q,f(x,e))(~f/~)(x,e))d0d~

vanishes of

be a s u b s t i t u t e

the d i f f e r e n t i a l

infinite for

the zero set of

f

i n M, Re f ( x )

its

imaginary part.

The p o i n t s o f

not zero. is

zero.

When In

the

o r d e r when Im 8=0, a p r o p e r t y

analyticity.

for

is

for

In

its

Im f ( x )

the sequel, f(x)

real

stands

p a r t and Im f ( x )

then occur

for

in c o n j u g a t e

pairs. Our program i s (7.4.3)

to

rewrite

the f u n c t i o n s

$ H(q,f(x~e)+i~O)a(x,elde

as i n t e g r a l s

in

t h e complex p l a n e modulo smooth f u n c t i o n s .

shall

use S t o k e ' s f o r m u l a a p p l i e d t o dF and F and c e r t a i n

their

b o u n d a r i e s . The v a r i a b l e

for

a real

t

in

the d e f i n i t i o n

that

For t h i s

r e 9 i o n s and

follows

doubles

e.

De÷inition

(7.4.4)

F o r O<s v ( x , t , s )

be smooth c u r v e s w i t h

Im v v e r y s m a l l

chosen t o have t h e f o l l o w i n g

properties

(6.4.5)

(i)

t=O => v=t

(ii)

t=O,

(iii)

s > 8 => f ( x , t ) ~ O .

That such f u n c t i o n s

f(x,t)=O

exist

is

=>

Im ÷~ v.

clear.

p o i n t s wher,e f

n e i g h b o r h o o d s Im v s m a l l

with

close to

these points,

>0,

We o n l y have t o choose v=8

outside neighborhoods of

v a n i s h e s and i n

the property that

(ii)

we

these

holds.

In f a c t ,

114

f(x,v) Lemma

= f(x,

Let

c(x)

Re be

v)

+

the

t,s

i Im

-> v ( x , t , s ) , its

and i t s

boundary at

a half-integer,

c(x)

=

O((Im

f(x,e),

s=l

v)Z).

o r i e n t e d by dt>O.

$=c~F(x,e)

boundary b ( x )

b e l o w , where t h e z e r o s e t o f is

+

O<sO and b ( × ) (7.4.6)

v ft(x,Re

-

$=cx~dF(x,8).

a r e shown i n

e real

is

the fisure

d e n o t e d by f ( x ) .

depends on t h e c h o i c e o f

Then

When q

the square r o o t

at

one

point.

F i 9 u r e 1 Crosses denote f ( x )

P r o o f . By S t o k e s " f o r m u l a , the

limit

zero.

of

it

suffices

to prove that

F inte9rated over the curve t->e=v(x,t,s)

In o t h e r words,

if

f(x,v(x,t,s)=f{x,t)

we have t o v e r i f y

to

the

left

since u(x,t,s)

+ u(x,t,s),

side

of

(6)

and w ( x , t , s ) in

the

as

v(x,t,s)

addition,

last

x a c r o s s Xo. Hence t h e

s tends

tend t o

Since d F ( x , e ) v a n i s h e s of

of

side is

as s t e n d s t o

= t+

w(x,t,s),

that

t e n d s t o z e r o and,

is obvious that

left

we put

$ H(q,f(x,t)+u(s,t,x))a(x,t+w(x,t,s) tends

the

to

d(t+w(x,t,s) zero.

zero with

all

Im u ( x , t , s ) > O .

infinite

term on t h e

But

this

is c l e a r

derivatives

The p r o o f

is finished.

o r d e r when Im e t e n d s t o left

of

lemma p r o v e s t h a t

(6) the

when s

zero it

i s a smooth f u n c t i o n left

s i d e and t h e f i r s t

115

term on the r i 9 h t

of

(6)

We a r e now r e a d y t o (1).

In doing t h i s

half-integral

Theorem.

As x t e n d s

to Xo

S H((q,f(x,8))

has a s h a r p f r o n t

ii)

i n v e s t i g a t e the s i n 9 u l a r i t i e s

of

Xo. the

between i n t e g r a l

integrals and

q.

q

i)

by a smooth f u n c t i o n a t

we have t o d i s t i n g u i s h

Integral

(7.4.7)

differ

in Y,

the

inte9ral

a(x,e)de

a t Xo i f

c=l and no complex z e r o s o f ¢=-I no p o i n t s

of

f(x,e)

Re f ( x )

appear

come t o g e t h e r o r meet

incoming p a r t s of

Im f ( x ) . Note. The r e a l

p o i n t s of

f(x)

may come t o 9 e t h e r under

complex p o i n t s may come t o g e t h e r under i i ) converge t o t h e r e a l a,

p o i n t s of

the c o n d i t i o n s f o r

f(Xo).

sharp f r o n t s

proved by s i m p l e c a l c u l a t i o n s f o r

i)

and the

but t h e y are not a l l o w e d t o

For g e n e r a l a m p l i t u d e f u n c t i o n s

are a l s o n e c e s s a r y . T h i s can be analytic

data.

The same remark

a p p l i e s t o the n e x t theorem. Note. Since t h e theorem i s all

terms of

(7.2.2)

the same f o r

all

integral

reformulated conditions,

f o ! l o w i n 9 one h o l d a l s o f o r See t h e end o f

Proof. the

to the

of F(x,e) b(x)

Hence

the

a p p l i e s to

paired oscillatory

q.

this

theorem and t h e

i n t e g r a l s w i t h dim e > I .

s e c t i o n 7.5.

Accordin9

inte9ral

it

when N i s even. The c o r r e s p o n d i n9 remark a p p l i e s

t o the n e x t theorem which concerns h a l f - i n t e g r a l Note. With s l i g h t l y

q,

theorem

lemma,

(7) d i f f e r s

by a s m o o t h

the f i 9 u r e s

below.

over + ~(x).

follows

from

function

from

116

oO~

q inte9er, detached

c=1=> b ( x ) + ~ ( x )

from Re

q integer,

Half-integral

t o 2 Re M

f(x)

¢=-I=>

around Re f ( x )

i s homolo9ous i n M \ f ( x )

b(x)-~(x)

with alternating

is h o m o l o g o u s

in M\f(x)

to

loops

orientations

q

We have t o m o d i f y our former machinery by i n t r o d u c i n g a t w o - s h e e t e d c o v e r M~ ' z of M r a m i f i e d around Re f ( x ) .

Its

p o i n t s are p a i r s

(8, e ~)

where 8 'm = f ( x , 8 ) .

Definition and

let c(x)

c+(x)

and

When

f~(x)

and

its c o n j u g a t e

c-(x)

half-turn the

Let

of

M

of

5' at o n e

H(q,f)

has

zeros

interval

of

if we

of

f÷(x),

~(x)

by

way

The

signs

f(x,e) H will

and be

on

of

be

carries

in t h i s

point.

project

part

the

real

the s a m e lifting

line w h e r e

chains

cx

one

belong

to d i f f e r e n t

sheet

is u n a m b i g u o u s means

two c y c l e s

if we c h o o s e

O

in the

and

b-(x)

Since

sheets

and

other,

of M Itm.

from

the

the f u n c t i o n

and b-ix)

on b+(x)

next

Let

every

to the

apart

that

~f(x,8)>O

as b e f o r e .

to M Ir~.

from

construction

that

the c y c l e s

the

(e,e')

the c o n s t r u c t i o n

alternating

the real

the

obtained

b(x)

obtained

is c o n n e c t e d ,

choice

Hence,

be c h a i n s

the c y c l e

two c h a i n s

denote

over

interval their

between

of

an f÷(x).

conjugate

117

q half-integer,

(=1.

homologous in M \ f ( x )

to

The c y c l e b ~ ( x ) + ~ ÷ ( x ) p r o j e c t e d down t o M i s

l o o p s around f ÷ ( x )

with alternatin9

orientations

q half-integer ,

(=-i.

homologous i n M \ f i x )

to

The c y c l e b - ( x ) + ~ - ( x ) l o o p s around f - ( x )

p r o j e c t e d down t o M i s

with alternatin9

orientations.

The p i c t u r e s above p r o v i d e a p r o o f of

Theorem

Wen q i s a h a l f - i n t e g e r ,

the f o l l o w i n 9 r e s u l t .

the d i s t r i b u t i o n s

£ H¢(q,f(x,8))a(x,6)d8

have sharp f r o n t s

at

xo i f

collapses involve just

new z e r o s appear o n l y o u t s i d e f ( ( x )

one component o f

and a l l

f~(×).

A p p l i c a t i o n s t o cusps and s w a l l o w ' s t a i l Cusps appear when the phase f u n c t i o n has a t r i p l e case, we can assume t h a t

z e r o . To c o v e r t h i s

t h e r e are smooth v a r i a b l e s s = s ( x , e ) and y = y ( x )

such t h a t

f(×,8)

= S=/3

yls

+

c l o s e t o a p o i n t where f = f ~ = f ~ e f=O,

f.=O has a cusp d i v i d i n 9

+ Ym

=0 c o r r e s p o n d i n 9 t o s=y=O.

the y p l a n e in

The c u r v e

two r e g i o n s where

118

s->f(x,m)

has one o r t h r e e r e a l

p o s i t i o n s of the

intervals

zmros. The f i g u r e

below

shows t h e

t h e z e r o s i n t h e v a r i o u s p a r t s and, when r e l e v a n t , a l s o o~ f ~ ( x )

paired oscillatory

and f - ( x ) .

A c c o r d i n g t o the r u l e s above, the

integral

f HC(q,f(x,e))a(x,e)de has sharp and d i f f u s e

fronts

a c c o r d i n g t o F i g u r e 4 below.

÷

F I Q

Sharp and d i f f u s e f r o n t m a t cusps. The o r i B i n o f t h e c o o r d i n a t e s y~,Y2 i s a t the v e r t e x , the y~-a×is i s v e r t i c a l . There a r e f o u r cases where q i s an i n t e g e r o r a h a l f - i n t e g e r and e=~l. The d o t s i n d i c a t e t h e p o s i t i o n s o f t h e z e r o s in the complex p l a n e o f the p o l y n o m i a l f , e i t h e r a l l t h r e e r e a l o r e l s e one r e a l and a c o n j u g a t e p a i r .

119

N e x t , assume t h a t of

o r d e r 4.

such t h a t ,

f(x,8)

has an i s o l a t e d s i n g u l a r i t y

as b e f o r e b u t one

T h e n t h e r e are new smooth v a r i a b l e s s = s ( x , 8 ) after

e v e n t u a l l y changing f

f(x,e)= The s u r f a c e f = f . = O

is

to - f ,

s4/4 + y l s m / 2 +y=s+y~. then a s w a l l o w ' s t a i l

whose s e c t i o n s w i t h

p l a n e y , = c o n s t a r e s k e t c h e d i n the n e x t f i g u r e p o s i t i o n s of

and y = y ( x )

the z e r o s of

the

t o g e t h e r w i t h the

f.

i 0

•

The arrows a r e a t t h e o r i g i n o f t h e c o o r d i n a t e s Y2,y= w i t h the y z - a x i s vertical. The l e f t f i g u r e r e f e r s t o y~O.

120

A p p l y i n 9 our c r i t e r i a , b e f o r e . To i t s distribution tail

first

the f r o n t

below t o be i n t e r p r e t e d as

column one m i g h t add t h a t

i s s h a r p from

appears a t

s o l u t i o n of

we 9 e t t h e f i g u r e

i n s i d e the t a i l

t h e o u t e r wave f r o n t

of

the correspondin9

a t y~=8.

I n case a s w a l l o w ' s

t h e f o r w a r d fundamental

a second o r d e r s t r o n g l y h y p e r b o l i c d i f f e r e n t i a l

o u t s i d e o$ the t a i l

complement o f four apply,

must have an s on t h e s i d e l a t i n 9

the

t h e s u p p o r t . T h i s means t h a t o n l y t h e columns two and

the f i r s t

one f o r

odd and t h e o t h e r one f o r

even d i m e n s i o n .

I

integer,+

operator,

integer,-

Sharp and d i f f u s e

fronts

half-integer,+ half-inte9er,at a s w a l l o w ' s t a i l

121

7 . 5 The 9 e n e r a l c a s e , P e t r o v s k y c h a i n s and c y c l e s ,

the Petrovsky

condition

The r e s u l t s dim 8.

of

the precedin9 s e c t i o n extend immediately to a r b i t r a r y

Consider a paired oscillatory

(7.5.1)

inte9ral

p o l a r form

I H(q,f(x,e))a(x,e)w(8)

and l e t

p = ( x o , e o ) be a p o i n t

conical

s u p p o r t c l o s e t o some d i r e c t i o n ,

where f e

inhomo9eneous c o o r d i n a t e s e w i t h the function which,

in

we m i g h t

as w e l l

dim e=N-I and w ( e ) = d e . The s u p p o r t o f

Xo,

i s c o n t a i n e d in

open s e t Q. L e t X be t h e p r o j e c t i o n manifold associated with f,

let

integral

o u t s i d e ×,

codimension I,

on x - s p a c e o f

investigate

of

its

complement f r o m

the behavior of

f(x,~)

s e p a r a t i n 9 p a r t s where f ( x , 8 ) > 0

bounded

the La9rangian

when x a p p r o a c h e s xo i n

the zero set f ( x )

RN - i ,

some f i x e d

Y be a component o f

which Xo can be r e a c h e d . We s h a l l paired oscillatory

a has

introduce

e - > a ( x , e ) t h e n a p p e a r s as a bounded open s e t o f

when we keep x c l o s e t o

When x i s

v a n i s h e s . Assumin9 t h a t

the

Y. is a manifold of

and v(x,t,s)

and v a l u e s i n M. T h i s a l l o w s us t o

(7.4.6),

with

i n t r o d u c e the

i.e.

term on t h e r i g h t Y at

is

the chain still

well

= $bcx~

F(x,8)

t,s -> v(t,x,s) illustrated

a smooth f u n c t i o n

of

Hence, by a theorem o f

Whitney's it

- I=c~

and b(x)

by F i 9 u r e

dF(x,e) its b o u n d a r y

I. The second

x up t o and i n c l u d i n 9

xo s i n c e d F ( x , e ) v a n i s h e s o f

f

be t h e z e r o s e t o f

= H(q,f(x,e)de

c(x)

t->v(x,t,l),

boundary of

f(x)

we can c o n s t r u c t f u n c t i o n s

$ H(q,f(x,8)+iO))a(x,e)d8

holds with

at s=l,

CN-~ and l e t

e x t e n s i o n s in

(N-1)-form

F(x,e) in M\Q.

denote almost a n a l y t i c

infinite

order at

the Q.

has a smooth e x t e n s i o n a c r o s s X a t

122

XOI

We can a l s o d e f i n e t h e c h a i n s c ( q , x , ¢ ) precisely

as b e f o r e c o r r e s p o n d i n g t o

half-integral call

q and t h e s i g n o f

¢.

the f o u r

In

this

boundaries b(q,x,~)

cases i n t e g r a l

orientations

in

oscillatory

integral

i)

e=l and

Im f ( x )

ii)

e=-l,

still

q is

if

of

f o r m u l a t e d as f o l l o w s .

as x t e n d s t o

Re f ( x )

collapses,

The

Xo i n Y i f

no two components o f

opposite orientations

fc(x)

Re

meet and no component o f

half-integral

carrying

q are:

cycles of

opposite orientations

does n o t meet r e ( x ) .

t h e c a s e s a b o v e , s h a r p n e s s has t h e same s o u r c e ,

The P e t r o v s k y c o n d i t i o n for

There i s

x i n Y and s u f f i c i e n t l y

b(x)=b(q,x,¢)

is

a ( N - 2 ) - c y c l e B in M \ f ( × o ) close to

a c h a i n whose b o u n d a r y i s =

When ÷ and a a r e a n a l y t i c ,

xo,

S=c~,

B -

such

the Petrovsky cycles

F(x,e)

b(q,x,c), -

Sbc~

the cases above, the s i t u a t i o n

F(x,e)

components o f

f(x)

implies

depends on x .

is very simple.

c h o o s e s B as t w i c e M moved away f r o m M i n t o enclosing all

we have

dF=O and t h e P e t r o v s k y c o n d i t i o n

s h a r p n e s s . O t h e r w i s e , one has t o know how C ( x )

(N-l)-cycle

namely

a r e homologous t o B i n M \ f ( x ) .

Scc~dF(x,e)

In

the description

meets Re f ( x ) .

meet and Im f ( x )

C(x)

used by

does n o t meet Re M

cycles of

no two components o f

that,

true

has a s h a r p f r o n t

The c o r r e s p o n d i n g c o n d i t i o n s f o r

In a l l

r e l e v a n t but

we s h a l l

2 and 3 becomes more c o m p l i c a t e d . The

integral

no component o f

carrying

Im ÷ ( x )

is

the figures

s h a r p n e s s theorem f o r

÷(x)

I

or

general situations

them P e t r o v s k y c h a i n s and c y c l e s s i n c e t h e y were f i r s t

P e t r o v s k y . The f i g u r e s

If

and t h e i r

CN-~,

Under i )

under i i )

one

as a

which come t o g e t h e r when x

123

tends to

Xo and a r e e n c l o s e d by s u b c y c l e s o f

orientation.

In

t h e case o f

t h e same way w i t h of

C(x)

function to for

is of

half-integral

respect to fc(x).

certainly

such t h a t

In a l l

It

B is

cases,

with

a cycle

may be p o s s i b l e t o

Xo.

The f o r m a l

prove that

t h e same chosen i n

t h e dependence on ×

the corresponding integral

x across a neighborhood of

the r e a d e r .

q,

b(q,x,-l)

is

a smooth

proofs are

the c o n d i t i o n s

left above

s h a r p n e s s a r e n e c e s s a r y i n many c a s e s .

Note.

This chapter is

a somewhat expanded v e r s i o n o f

G i r d i n g 1977.

References

A t i y a h M.F., B o t t R., Gardln9 L. Lacunas f o r h y p e r b o l i c d i f f e r e n t i a l operators with constant c o e f f i c i e n t s I , I I . Acta Math. 125 (1970) 109-189 and 131 (1973) 145-206. Born M. and Wolf E. P r i n c i p l e s o f o p t i c s . 1975. Duistermaat J.J. Math. 128 (1972) D

and H6rmander L. 183-269.

Fifth.

ed. Pergamon Press

F o u r i e r I n t e g r a l Operators I I .

•

Gardln9 L. Sharp f r o n t s of p a i r e d o s c i l l a t o r y (1927) s u p p l . C o r r e c t i o n i b i d . 13 (1977) 821.

integrals.

G e l f a n d I.M. and S h i l o v G.E. G e n e r a l i z e d f u n c t i o n s . 1958) E n g l i s h t r a n s l a t i o n Academic Press 1964.

H~rmander L. (1983)

The A n a l y s i s of L i n e a r P a r t i a l

-"-

Differential

-"-

-"Linear Differential (19Y0),121-133.

Vol

P u b l . RIMS 12,

I.

(Moscow

Operators I , I I

III,IV O p e r a t o r s . Actes Congr.

F o u r i e r I n t e 9 r a l Operators I .

Acta Math.

Acta

Int.

(1985) Math. Nice

127 (1971) 79-183.

Ludwi9 D. C o n i c a l r e f r a c t i o n in C r y s t a l O p t i c s and Hydromagnetics. Comm. Pure and A p p l . Math. XIV (1961) 113-124. Maslov V.P. Theory of U n i v . Moscow 1965.

p e r t u r b a t i o n s and a s y m p t o t i c methods. Mosc. Gos.

Uhlmann G.A. L i 9 h t i n t e n s i t y d i s t r i b u t i o n Pure App. Math. ××XV (1982) 69-80.

in c o n i c a l r e f r a c t i o n .

Comm.

Index almost a n a l y t i c extension I I I conical r e f r a c t i o n 3~24 crystal optics 3 double r e f r a c t i o n 4 Fourier i n t e g r a l operators 44 f r o n t , sharp or d i f f u s e i07,115,117,118,120,122 fundamental s o l u t i o n 6 Hadamard 6 Herglotz-Petrovsky formula 25,31 homo9eneous hyperbolic 13 H~rmander 16,33,36~55,6S,96 h y p e r b o l i c i t y cone 14 l h t r i n s i c h y p e r b o l i c i t y 17 microhyperbolic I0 Kovalevskaya 4 Lagrangian planes 77 Lax 7,46,70 l o c a l i z a t i o n 21 o s c i l l a t o r y i n t e g r a l s 39 -equivalence of 96 - d u a l i t y of I01 - p a i r e d 102 parametrices 46 - g l o b a l 89,110 Petrovsky c r i t e r i o n , c o n d i t i o n 32,104 polyhomogeneous operators 54 propagation cone 20 propagation of s i n g u l a r i t i e s 6B p s e u d o d i f f e r e n t i a l operators 52 -on manifolds 62 -Cauchy's problem for 85,88 symplectic geometry 75 wave equation 2 wave f r o n t set 34 wave f r o n t surface 21 very r e g u l a r phase f u n c t i o n 83 Volterra 4 Zeilon 4