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 t
light
from a p o i n t s o u r c e .
a l s o be d e s c r i b e d as t h e f o r w a r d fundamental s o l u t i o n of e q u a t i o n in
of
double r e f r a c t i o n
of
light
problem o f
the wave
t o the c r y s t a l .
g e o m e t r i c a l o p t i c s was t h e problem
observed and a n a l y z e d a l r e a d y by Huygens. A r a y
entering certain
whose d i r e c t i o n s
k i n d s of
vary with
crystals
the d i r e c t i o n
of
is refracted into the
A c c o r d i n g t o Huygens's t h e o r y of
light
in the c r y s t a l
refraction,
has two v e l o c i t i e s ,
D i r e c t i o n s where the two v e l o c i t i e s axes. They appear as d o u b l e p o i n t s of o b t a i n e d by t a k i n g v e l o c i t y For c r y s t a l s
this
both d i r e c t i o n the r e f r a c t i o n .
a r e t h e same are c a l l e d o p t i c a l the v e l o c i t y
s u r f a c e which i s
as d i s t a n c e t o an o r i g i n
w i t h one o p t i c a l
two r a y s
i n c i d e n t ray r e l a t i v e
dependent, which can be measured by the s t r e n g t h o f
ray.
can
t h r e e space v a r i a b l e s .
The most i n t e r e s t i n g
means t h a t
It
along a v a r i a b l e
a x i s , Huygens found the v e l o c i t y
s u r f a c e t o be a s p h e r e and an e l l i p s o i d Crystals with
two o p t i c a l
French p h y s i c i s t
with
f o u r d o u b l e p o i n t s on t h e o p t i c a l Associated with
the v e l o c i t y
wave f r o n t s
point source of
li9ht
at
in
the envelope of
analytical
form.
By a f r e a k o f
amon9 o t h e r s ,
form a c i r c l e
that
light.
identical
to
the velocity
has t h e same
the v e l o c i t y
surface throu9h a
inlet
t h e wave s u r f a c e to a double p o i n t .
light
refraction,
He
whose d i r e c t i o n
a x i s o u g h t t o be b r o k e n i n t o
d i s c o v e r y in
The f i r s t
was v e r i f i e d
a cone o f
rays.
by e x p e r i m e n t a
with
time
with
late
1820's.
t h e aim o f
The f o l l o w i n 9
understandin9 the
amon9 o t h e r t h i n 9 s
in
the e q u a t i o n s of
second o r d e r d i f f e r e n t i a l
and t h r e e space v a r i a b l e s .
initial
when t h e m a g n e t i c f i e l d v a l u e problem f o r
is
equations
These e q u a t i o n s a r e
what one 9 e t s f r o m M a x w e l l ' s e q u a t i o n s f o r
a dielectricum
To s o l v e t h e
the
a t t e m p t s were based on a n a l o g y w i t h
t h e o r y and r e s u l t e d
four variables,
in
F r e s n e l 9uessed i t s
The c o m p u t a t i o n s were c a r r i e d
an o u t s i d e r a y o f
Lame, a 3x3 h y p e r b o l i c system o f
field
is
c o v e r i n 9 the
decades saw e x t e n s i v e a c t i v i t y
identical
it
Huy9ens, t h e wave
later.
H a m i l t o n made h i s
in
surface.
on t h e o u t e r s h e e t o f
T h i s phenomenon, t h e c o n i c a l
elasticity
Accordin9 to
surface.
disc
c o i n c i d e s w i t h an o p t i c a l
nature of
t h e wave s u r f a c e
e m a n a t i n 9 f r o m an i n s t a n t a n e o u s
nature,
the tangent planes to
p r e d i c t e d from t h i s
time
is
H a m i l t o n . He added an i m p o r t a n t complement~
which bounds a c i r c u l a r
short
t=l
the velocity
g e n e r a l f o r m as t h e v e l o c i t y
double point
axes.
surface there
time
The s u r f a c e s a r e
t h e t h r e e c o n s t a n t s i n v e r t e d and hence i t
observin9 that
them.
two s h e e t s which come t o g e t h e r a t
the c r y s t a l .
surface is
o u t by,
surfaces for
t h e n a t u r e o÷ t h e c r y s t a l .
symmetric around the o r i g i n
surface with
velocity
the the
t o be a l g e b r a i c o÷ d e g r e e 4 d e p e n d i n 9 on t h r e e
constants varyin9 with
of
it.
a x e s remained a m y s t e r y u n t i l
Fresnel found e x p l i c i t
They t u r n e d o u t
consistin9
tangent to
the electric
elimininated.
Lame~s system was a 9 t e a t
c h a l l e n g e taken up by Sonya K o v a l e v s k a y a . She had a model t o 9o by, W e i e r s t r a s s ' s s o l u t i o n of
the Cauchy problem f o r
a s s o c i a t e d w i t h the product of speeds o f
li9ht.
an analogous system
two wave o p e r a t o r s w i t h d i f f e r e n t
Led by g e o m e t r i c a l o p t i c s ,
she assumed t h a t
li9ht
from a p o i n t source ought t o p r o p a g a t e between t h e two s h e e t s o f wave s u r f a c e l e a v i n g no t r a c e b e h i n d . The l a t t e r b u t she d i d not r e a l i z e sheet and i t s fact
that
that
convex h u l l .
the wave f r o n t
there
is
light
Her f o r m a l
that
t h e f o r m u l a s b u t d i d not a r r i v e
(I)
for
H(t =
source i n a medium w i t h
li9ht
the analogue of
the
front
on t h e c i r c l e
In
190a,
V o l t e r r a p o i n t e d out t h a t
Ixl=t.
to treat
listeners,
from an i n s t a n t a n e o u s p o i n t
two space v a r i a b l e s . there
notice.
Cauchy's p r o b l e m .
-JxlZ)-l/2/2~,
does not v a n i s h when i × J < t ,
his
He c o r r e c t e d
two space v a r i a b l e s i s
which d e s c r i b e s p r o p a 9 a t i o n o f
One o f
identically
in g e o m e t r i c a l o p t i c s as a c o m p l e t e c l u e t o wave
H(t)
not s u f f i c i e n t
by V o l t e r r a .
at a s o l u t i o n of
p r o p a g a t i o n was shaken when he found t h a t distribution
she deduced i s
the C a u c h y - K o v a l e v s k y a theorem. Her
m i s t a k e was p o i n t e d out a few y e a r s l a t e r
his faith
a l s o between t h e o u t e r
s u r f a c e can be p a r a m e t r i z e d by e l l i p t i c
a clear contradiction with
Earlier,
assumption i s c o r r e c t
c a l c u l a t i o n s where she used the
f u n c t i o n s led her a s t r a y . The s o l u t i o n zero,
the
Since t h i s
distribution
i s an a f t e r g l o w behind t h e wave
lectures that
he gave i n Stockholm in
the a n a l y t i c a l
Lame's e q u a t i o n s f o r
tools
tried
so f a r
were
the double r e f r a c t i o n .
a young mathematics s t u d e n t N i l s
Zeilon,
took
His admired t e a c h e r I v a r Fredholm had c o n s t r u c t e d fundamental
s o l u t i o n s of
elliptic
abelian integrals.
differential
o p e r a t o r s in t h r e e v a r i a b l e s using
Z e i l o n c o n t i n u e d h i s work f o r
other types of
e q u a t i o n s but u s i n g a n o t h e r p o i n t of d e p a r t u r e , namely t h e remark t h a t if
P(~)
(2)
is,
i s a p o l y n o m i a l in n v a r i a b l e s , E(x)=(2~)
at
least formally,
-~
i
exp
ix.~
the
integral
d~/P(~)
a fundamental s o l u t i o n f o r
the o p e r a t o r P(D)
where D = ~ l i ~ x .
In f a c t ,
P ( D ) E ( x ) = ( 2 R ) - ~ J exp i x . ~ The problem i s
t o make sense of
O, a t
and a t
infinity
d~ = ~ ( x ) .
the
inte9ral
(2)
which may d i v e r 9 e a t
the z e r o s of P. A p a r t from t h i s ,
the f o r m a l
machinery works a l s o when P(~)
i s a square m a t e i x whose e l e m e n t s are
p o l y n o m i a l s , in p a r t i c u l a r
t h e Lame system. Z e i l o n ' s mathod o f
avoidin9 sin9ularities
for
was t o move t h e c h a i n R~ o f
C~. T h i s can be done i n v a r i o u s ways. Z e i l o n ' s ri9ht,
b u t h i s ar9uments are shady,
a p p l i e s t o h i s ma9num opus, problem o f of
the fundamental s o l u t i o n
wave s u r f a c e and i t s refraction,
intuition
read by a c r i t i c a l
two Ion9 a r t i c l e s
conical refraction.
inte9ration
into
l e d him eye.
This also
around 1920 on the
But h i s r e s u l t s
are r i s h t .
The s u p p o r t
i n c l u d e s the space t h e o u t e r s h e e t o f
convex h u l l .
This f a c t
the
has t o do w i t h c o n i c a l
b u t t h e p r e c i s e c o n n e c t i o n was not c l a r i f i e d
until
1961
w i t h a paper by Ludwi9. Z e i l o n ' s work d i d not 9et much a t t e n t i o n .
Some y e a r s l a t e r
c o n s t r u c t e d f o r w a r d fundamental s o l u t i o n s o f o p e r a t o r s wih c o n s t a n t c o e f f i c i e n t s them,
the v e l o c i t y
Her91otz
hyperbolic differential
i n any number o f
variables.
For
s u r f a c e has m s h e e t s c o r r e s p o n d i n 9 t o m d i f f e r e n t
propa9ation velocities.
He a p p l i e d t h e F o u r i e r t r a n s f o r m t o t h e space
v a r i a b l e s and a r r i v e d a t v e r y s i m p l e f o r m u l a s c o v e r i n 9 a l s o t h e Lame system. He showed t h a t
the wave s u r f a c e
system o f c r i s s - c r o s s i n 9 s u r f a c e s o f
in
the 9 e n e r a l case i s a
v a r y i n 9 d i m e n s i o n s near which t h e
fundamental s o l u t i o n may have a v e r y c o m p l i c a t e d b e h a v i o r . O u t s i d e t h e wave s u r f a c e , is
zero, but
i s does f o r
i.e. it
outside its
front,
t h e fundamental s o l u t i o n
may a l s o v a n i s h i n r e g i o n s i n s i d e t h e f a s t e s t
p r o p a g a t i o n of
l a c u n a s , a t t r a c t e d the the
li9ht
in f r e e space. These r e 9 i o n s ,
interest
fundamental paper about them in existence if
fastest
of
the
P e t r o v s k y who p u b l i s h e d a
the f o r t i e s
where he t i e d
lacunas t o t o p o l o g i c a l p r o p e r t i e s of
t h e complex v e l o c i t y
front
the
p l a n e s e c t i o n s of
s u r f a c e . H i s work was e x t e n d e d t o t h e 9 e n e r a l
as
6 o
case o f
degenerate velocity
the e a r l y
in
En o f
space v a r i a b l e s
the form of
terms o f
Bott
and G a r d i n 9 i n
seventies.
Fundamental s o l u t i o n s number o f
s u r f a c e s by A t i y a h ,
solutions
t h e wave e q u a t i o n w i t h
an a r b i t r a r y
were c o n s t r u c t e d by Tedone a l r e a d y i n of
t h e c o r r e s p o n d i n g Cauchy p r o b l e m s .
1889 In
the function
d ( t , x ) = t = - I x l =, their
main p r o p e r t i e s
the forward light
can be d e s c r i b e d as f o l l o w s ,
cone where t~O,
d(t,x)~O.
They v a n i s h o u t s i d e
I n s i d e the
light
cone E.
behaves l i k e const
when n i s e v e n .
d(t,x)
It
const on t h e
light
to variable
c~-~'~=
v a n i s h e s t h e r e when n i s odd >I 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
If
precisely
when a l l
ck=
t h e numbers ck a r e I .
K is
a compact p a r t
r e s p e c t any b i n K.
of
More
number A such t h a t
Im s>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,
l
Such a 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 w i t h P(a)>O 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
g(x)
i s a smooth f u n c t i o n w i t h compact
a smooth e x t e n s i o n o f
by t h e f o r m u l a d9=g=dz+g~dZ. The p r o o f of f= =0,
the upper h a l f - p l a n e and
b~O, i n G r e e n ' s f o r m u l a in
1983 I ,
i f(x+i(b+c))g(x,c)dx =
in
o r d e r < N+l+c f o r
depends on a passage t o the
(1.2.1)
the
= lim f ( x + i y )
e x i s t s as a d i s t r i b u t i o n
f o l l o w i n g form
is analytic
9 for
Oiy~c and 9s i s d e f i n e d
the f o r m u l a i s easy. Since
we have df(x+i(y+c))g(x,y)dz
Note t h a t
the d i f f e r e n t i a l
9(x,y)=g(x+iy).
= 2i
f(x+i(y+c)g~(x,y)dxdy.
v a n i s h e s when g i s a n a l y t i c
I n the g e n e r a l case one d e f i n e s g ( x , y )
f o l l o w i n g f o r m u l a whose r i g h t
side
and by t h e
i s 9 ( x + i y ) when 9 i s a p o l y n o m i a l
16
of
degree a t most N, 9(x,Y)
A passage t o the
limit
= I 9ck~(x)
(iy)k/k!,
in G r e e n ' s f o r m u l a then g i v e s a c o n v e r g e n t
i n t e 9 r a l on the r i g h t
and t h i s
p r o v e s the s t a t e m e n t .
L e t us now pass t o s e v e r a l v a r i a b l e s
Lemma Suppose t h a t
f(z>,
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
k
its
vertex
~yISc>O. 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 =,
<0 and Re 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 0<¢ t e n d s t o in
by i n t r o d u c i n g
~ restricted
the
inte9ral
In
the r e s u l t i n g
to
h(~)=l.
The r e s u l t
integral (2~) -~
i
w(~)
J exp i ( x . ~
+i¢)r
= (2R)-"S H(-n,-i(x.~+i~) This proves the
r~-~dr =
w(~).
lemma.
I n our n e x t theorem we s h a l l Ho(p,z)=
use t h e n o t a t i o n
-(-z)P/p!
when p~O i s an i n t e g e r and z e r o o t h e r w i s e . Then t h e f o r m u l a written
for
H(p+t,z)
all
(5) can be
as
(1.5.8)
= H(s,z)
+ Ho(s,z)/t
+
O(t)
z.
Now l e t of
is
P(~)
be h y p e r b o l i c w i t h
d e g r e e m. L e t C ( P , e )
be i t s
respect to
hyperbolicity
e in
Rn\O and homogeneus
cone and K ( P , e )
its
p r o p a g a t i o n cone.
Theorem A s u p p o r t in (1.5.9)
When P b e l o n 9 s t o H y p ( 6 , m ) , t h e p r o p a g a t i o n cone K ( P , 8 )
E(x) -J
with
= (2~) - n
its
fundamental s o l u t i o n
is
J w(~)(H(m-n,ix.~)/P(~-iOe)
w(~)Ho(m-n, i x . ~ ) ( l o g P / m P ) ( ~ - i O B ) )
integration
over h(~)=l.
-
with
28
Note. The second term on the r i g h t
i s a polynomial of
hence v a n i s h e s when m-n
degree m-n and
s e c t i o n 1.2,
the r i g h t
s i d e does not change when e i s r e p l a c e d by any element o f C ( P , e ) . f o r m u l a can be d i f f e r e n t i a t e d the r i g h t
under t h e s i g n o f
inte9ration.
Hence~
s i d e i s a fundamental s o l u t i o n by G e l f a n d ' s f o r m u l a . suffices
Proof.
the
i n t e 9 r a n d i s o n l y quasihomo9eneous when m-n~O. To
let
s be c l o s e t o
avoid t h i s (I.5.10) with
case,
Eo(x)=
(2~)-nl
independent o f such t h a t
h.
is
and c o n s i d e r the d i s t r i b u t i o n
it
Since the
integrand is strictly
i s c l o s e d and hence t h e r i g h t
When x i s o u t s i d e K ( P , B ) ,
there
=
lim
J ~(5)H(ms-n,
inte9ral
is
independent of
can r e w r i t e
it
infinity
follows that
it
To a r r i v e
at
with
S = {/t
=
I
the
-ig
(9)
w(~)(H(p,
see t h a t E(x)
homo9eneous of
t h r o u g h o u t . Hence,
degree O, we
lettin9
t
tend t o
side vanishes.
in case m - n = p ~ , ix.{)+Ho(p,
consider
ix.~)m/t+0(t))lP({-iOe)
~÷~.
as t->O.
In
i n t e g r a n d of
i s homo9eneous o f
must v a n i s h .
Ho(p, i x . ~ ) / P ( { - i O e )
de9ree 0 so t h a t
i s r e p l a c e d by { - i t e
the
integral
does not chan9e o f
with t
large.
But t h e n ,
Insertin9 this
result
i n t o t h e p r e v i o u s f o r m u l a and
t->O p r o v e s the d e s i r e d f o r m u l a
fundamental s o l u t i o n f o l l o w s from s i g n of
Since i t
integrand is closed,
i s o b t a i n e d by t a k i n 9 t h e l i m i t
Jw(~)
lettin9
t.
the r i g h t
the f o r m u l a
(2~)~E~÷~(x)
fact,
i s an 9 i n C(P,B)
ix.5)/P(~)-"
where 5 = ~ - i t 9 and t>O tends t o O. But s i n c e the
We s h a l l
side is
x.9
(2~)"E.(x)
the
supported in K ( P , e ) .
~(~)H(ms - n , i x { ) ( P ( ~ - i O e ) - °
i n t e g r a t i o n over h ( ~ ) = l .
homogeneous o f degree 0,
I
it
To
prove the theorem i t
Note t h a t
t o show t h a t
The
integration.
(8)
(9).
by a p r e v i o u s argument i t
That the r i g h t
and d i f f e r e n t i a t i o n s
side is a under the
29
We s h a l l
now see t h a t
E.(x)
and hence a l s o E ( x )
p r o p a g a t i o n cone o u t s i d e t h e wave f r o n t
is analytic
in
the
s u r f a c e W(P,B)
provided P is a
c o m p l e t e p o l y n o m i a l so t h a t
K has a non-empty i n t e r i o r .
The p r o o f u s e s
Theorem A and t h e f o l l o w i n g
lemma.
Lemma When x i s
in
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 t h e wave f r o n t
surface,
there are continuous functions
degree I
such t h a t
(i)
c(~).x
(ii)
O
If
we want,
(i)
all
=> t o ( { )
4-> c ( { ) = c ( - { ) ,
4, in
-C({)
O t c ( ~ ) . x =
cones and t h e f a c t
continuity
Definition
that
the r e a l
the p r o p e r t i e s
for
all
Let
It
(i)
of
the
local
hyperbolicity
p l a n e s x . g =O meet e v e r y such cone. and
(ii)
hold f o r
one x,
t h e y h o l d by
x in a n e i g h b o r h o o d .
~ ( x ) be t h e map {->
from
O.
O.
P r o o f . F o l l o w s from the o u t e r c o n t i n u i t y
if
and P ( { + i t c ( { ) ) ¢
can a l s o be r e p l a c e d by
(iii)
Note t h a t
homogeneous o f
{+it({),
h({)=l.
is clear
that
t o move t h e
a(x)
is
integration
Theorem B . When x i s
a c y c l e homologous t o h ( ~ ) = 1 . We s h a l l in
t h e f o r m u l a (9)
of
theorem A i n t o
i n K b u t o u t s i d e W and w i t h
use i t
C".
integration
over
~ ( x ) we have E.(x) E(x)
where @ i s
= = i
i
w(S)H(ms-n,ix.S)IP(S)', w(S)H(m-n, i x . S ) / P ( S )
a polynomial of
+Q(x),
d e g r e e m-n.
In
particular:
E.(x)
and E ( x )
30
are a n a l y t i c
Proof.
i n x i n K\W.
The i n t e g r a n d of
(I0)
leads i n t o the a n a l y t i c i t y p a r t of
locally
follows since
indepedent of
the c o m p l i c a t i o n t h a t But then
its
x.
The p r o o f
the
last
the
i n t e 9 r a n d . Hence the f i r s t theorem o f
s e c t i o n 1.2.
~(x) w i t h the p r o p e r t i e s
The p r o o f o f
is
(i)
t h e second p a r t
i n t e g r a n d of
differential
e x p l a i n s the presence of E(x).
domain o f
the theorem f o l l o w s by the
E= i s a n a l y t i c
m-riO.
i s c l o s e d and r e p l a c i n 9 { by ~ ÷ i t c ( { )
(9)
0(~)
and
(ii)
is similar
fact
of
with
t i m e s a p o l y n o m i a l which
the p o l y n o m i a l Q in
the f o r m u l a above f o r
is finished
a rational
t h a t E(x)
is
i s no l o n g e r c l o s e d when
When m-n
That
f u n c t i o n over a c e r t a i n c y c l e ~ ( x ) .
vanishes o u t s i d e K(P,8),
a l s o when m-nlO.
We s h a l l
When q i s an i n t e 9 e r ,
first
let
this
situation
as the
Usin9 the
can be a c h i e v e d
work w i t h Theorem A.
us put
M ( q , t ) = H(q, i t ) - ( - l ) Q H ( q , - i t ) . so t h a t (1.5.11)
qlO
(1.5.12) Usin9
(1.5.13)
In o u r At
the
space
qM(q,t)=
Theorem
Theorem
C
E(x)
inte9rals
=
next
of of
f(m-n,5)
theorem,
n-1 the
-
E(x)-(-l)qE(-x)
(t÷iO)q) we 9 e t
x.8>O, (2~)-"
time
t)/q!,
iq(-q-l)!((t-iO)q
A to c o m p u t e
When
same C*
=>M(q,t)=-~i(it)q(s9n
J w(~)M(q,x,{)IP({-i08).
we
we s h a l l
shall be
dimensions
9o
able
and
out
into
to o p e r a t e
to e x p r e s s
the
C ~ with
in c o m p l e x fundamental
(n-l)-form
= ( 2 ~ ) -~
(ix.S} m - n
w(S)IP(3),
the
S=~+i9,
formula
(13).
projective solution
as
31
over c e r t a i n c y c l e s . 5 belongs to
-C(~)
hyperbolicity orient
it
The form
is defined,
and 5 i s s u f f i c i e n t l y
cone C=C(P,0).
Let
h o l o m o r p h i c and c l o s e d when s m a l l when o u t s i d e t h e 91obal
~ ( x ) * be the
image of
~(x)
in C* and
so t h a t
(1.5.14)
~(~)
x.~
>0.
L e t X* and P~ be the complex p r o j e c t i v e
h y p e r s u r f a c e s x.5=O and P(~)=O
respectively. To see what the new c y c l e i s , maps ~(x) Hence,
into
( - 1 ) ~-~
~(x)
i n complex space,
Petrovsky cycle,
note t h a t
t h e a n t i p o d a l map ~->-~
where the bar denotes complex c o n j u g a t i o n .
t h e boundary ~(x)
of
~(x)*,
c a l l e d the
appears as
(I.5.15)
t w i c e Re X* detached from Re P* when n i s odd
(1.5.15)
t u b e s around Re P* ¥ Re ×~ when n i s even.
We can now l e t
the
i n t e 9 r a t i o n of
Theorem C 9o o u t
Theorem D. 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
i n t o the complex.
Let P be a c o m p l e t e
32
polynomial hyperbolic with with
s u p p o r t in
in K but
=-J~if(q,x.5)/q!,
q E ( x )
= J f(p,x.5),
As t h e Theorem B,
integration
now s t a r t i n g
s p a c e , and t h e t o p o l o g y o f
Theorem E.
a crucial
Petrovsky's
component, E(×)
Then,
if
q=m-n,
and x i s
over
~(×)*.
o v e r a tube around ~ ( x ) .
f r o m t h e o r e m C.
rational
f o r m s on complex p r o j e c t i v e
P*nX* does n o t chan9e f o r
K\W we have t h e f o l l o w i n g
c y c l e appears in
fundamental s o l u t i o n
s u r f a c e W, integration
Since we a r e d e a l i n g w i t h
component o f
8 and E i t s
t h e p r o p a g a t i o n cone K ( P , e ) .
o u t s i d e t h e wave f r o n t
q lO => 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<0 p r o v e s
Hence ~ C~MfUp
.
C be a c o n i c a l s u b s e t o f =
:p
(lolf(~)
I=
G-space and c o n s i d e r i n t e g r a l s
(l+l~[)=Pd~
B and C a r e open c o n i c a l s u b s e t s and B i s c o n i c a l l y compact i n C, then
|fnp.c finite =>
~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~I
and I
x t o a compact for
I~I>2.
Put
= ×(~/t~)aj(x,~)
I~I+161<3+i. This t h e bj
in x.
as a b o v e ,
and choose t h e numbers t j
sum o f
infinity.
holds.
By a p a r t i t i o n
subset of
t o minus
of
mj÷i
=
identify an a m p l i t u d e
Given a ~ , . . ,
such t h a t
decrease s t r i c t l y
a~+az + . . .
~a~+a~+...
locally u n i f o r m l y
sometimes
Lemma
on a s y m p t o t i c s e r i e s
an a m p l i t u d e and
degree ],
need a r e s u l t
an a m p l i t u d e , we say t h a t a
for
operators to
is all
is
locally
tendin9 to
clearly finite
so f a s t
that
p o s s i b l e and can be done so t h a t
the
and hence d e f i n e s a smooth f u n c t i o n
N,
D(~,~) ( a - b i - . . . - b N ÷ i ) when I ~ I + I P I < N + 2 .
infinity
Since a l l
=
O(J~j~-'
bk--ak a r e i n
;'
)~
NS-~ f o r
m=mw
k=l,2,..,
and b j ÷ i
a.
54
and the f o l l o w i n g Next, we s h a l l
terms are of
Bu(x)
If
a(x,~)
(I)
follows.
c o n s i 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 s d e f i n e d by
amplitudes b(x,y,~)
Lemma
degree m j ÷ ~ ,
depending on x and y ,
= ( 2 T t ) - ~ S b ( x , y , ~ ) u ( y ) exp i ( x - y ) . ~
dyd~.
i s an a m p l i t u d e w i t h the a s y m p t o t i c e x p a n s i o n
exp iDyD~ b ( x , y , { ) I x = y then a ( x , D ) - B has a smooth k e r n e l .
Proof.
By T a y l o r ' s f o r m u l a we have b(x,y,~)= I((iDy)=b(x,x,~))(y-x)~/=! + Z
(iB~)=c~(x,y,~)
w i t h summations o v e r a r e a m p l i t u d e s of for
+
(x-y)~/~!
I~I
I~I=N r e s p e c t i v e l y .
the same degree as b.
Inserting this
Bu and p e r f o r m i n g i n t e g r a t i o n s by p a r t s w i t h
turns
i(x-y)
Here the c ~ ( x , y , { ) i n t o the f o r m u l a
respect to
i n t o D~, 9 i r e s an a m p l i t u d e
~((iDy)=D~b(x,y,~))u(y)dyd~/~! The a m p l i t u d e s o f
the
last
+~
((iD)~D~=c~(x,y,~)u(y).
term have t h e degrees m-N which means t h a t
the c o r r e s p o n d i n g k e r n e l s a r e c o n t i n u o u s l y d i f f e r e n t i a b l e times.
The k e r n e l of
the kernel of kernel of
~, which
a(×,~)
the f i r s t
term,
by a k e r n e l o f
on t h e o t h e r
N-m-n+l
hand, d i f f e r s
from
the same smoothness, Hence t h e
a ( × , D ) - B i s smooth and t h i s
p r o v e s the
lemma.
Polyhomo9eneous o p e r a t o r s . A pseudodifferential operator a(x,B)
and i t s
de9ree m are s a i d t o be polyhomogeneous i f , o p e r a t o r s , the a m p l i t u d e i s
amplitude a(x,~)
as f o r
the a s y m p t o t i c sum f o r
differential j=O,I,..,
a m p l i t u d e s a ~ _ j ( x , ~ ) which are homogeneous of degree m-j v a l u e s of k=l,2, .....
~. Such a m p l i t u d e s are of
of
for
of lar9e
course unique modulo -k f o r
A polyhomogeneous o p e r a t o r a(×,D)
w i t h symbol ~ I
ak(x,~)
55
has a k i n d o f
dual a ' ( x , D )
a'(x,~)
When a ( x , ~ )
w i t h symbol
~ ~ (-l)kak(x,-~).
i s a p o l y n o m i a l , we have a ' ( x , D ) = a ( x , D ) .
said to s a t i s f y
the
'transmission condition'
Such o p e r a t o r s are
(H~rmander
1985 I I I
110) which g u a r a n t e e s t h a t boundary problems make sense f o r we s h a l l
use a n o t h e r p r o p e r t y , namely t h a t f o r
o p e r a t o r s t h e map a - > 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
I n s e r t i n g the f i r s t
sum i n t o
(2)
F o u r i e r t r a n s f o r m g i v e s a sum o f
i~l=n.
and u s i n g t h e p r o p e r t i e s o f
the
terms
a('~(x,~)(iD~)~b(x,~)/~!, I ~ l < N , which a r e t h e f i r s t
ones of our proposed a s y m p t o t i c sum. The second
term above when i n s e r t e d i n t o
(2)
gives rise
t o terms
constStN-~dt S a ( ' ~ ( x , ~ - t ~ ) d ~ $ ( i D x ) ~ b ( x - z , { ) e x p i x . ~
dz.
as
57
Here a ~=~ has t h e m a j o r a n t const
(I
+l~-t~l) m-''' ,
where m=m. i s the degree o f
a,
and the
const((l+l~l)-P(l+l~l) where m=mb i s the degree o f
b.
in
l a r g e and
in the
integration
the m a j o r a n t
the terms of
Note. P r i n c i p a l It
has the m a j o r a n t
m,
and t l ~ l < l ~ I / 2
c o n s t ( ( l + l ~ l ) ~-N, for
integral
Taking p s u f f i c i e n t l y
s e p a r a t i n g the cases t l ~ l > I ~ I / 2 results
last
m=m,+m=,
the second sum of
(2).
symbols of a d j o i n t s ,
i s easy t o see t h a t
This f i n i s h e s
the p r o o f .
p r o d u c t s and commutators.
the c l a s s o f
polyhomogeneous o p e r a t o r s i s
i n v a r i a n t under a d j o i n t s and p r o d u c t s . An a m p l i t u d e a ( x , ~ )
for
which
t h e r e e x i s t s a n o n - z e r o homo9eneous a m p l i t u d e a ~ ( x , ~ ) of degree m such t h a t a - a~ has degree <m i s s a i d t o have t h e p r i n c i p a l o b v i o u s l y u n i q u e l y d e t e r m i n e d and i t of
is
the f i r s t
a when a i s polyhomogeneous. When P ( x , ~ )
principal
part p(x,~),
P ( x , D ) and p ( x , ~ )
its
p(x,D)
term of
principal
symbol.
p(x,D) with
symbol p ( x , ~ )
adjoint
an o p e r a t o r whose p r i n c i p a l
It
is
and i
then p ( x , ~ ) q ( x , ~ )
is
the expansion
p a r t of
f o l l o w s from our theorems
the p r i n c i p a l
symbol i s
the p r i n c i p a l
p a r t of
~(×,~).
and Q a r e 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 p r i n c i p a l and q ( x , ~ ) ,
is
i s an a m p l i t u d e w i t h
i s s a i d t o be the p r i n c i p a l
above t h a t of
p a r t am. I t
the
Further,
if
P
symbols p ( x , ~ )
symbol of
the p r o d u c t PQ
t i m e s the Poisson commutator o r p a r e n t h e s i s
{p,q}=
p~(x,~).q~(x,~)
-p~(x,~).q~(x,~),
where an i n d e x d e n o t e s t h e c o r r e s p o n d i n g g r a d i e n t , symbol o f
i s the p r i n c i p a l
the commutator PQ-gP.
Note. P r o p e r l y s u p p o r t e d 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 . Since t h e s i n g u l a r s u p p o r t o f
the k e r n e l K ( x , y ) of
p s e u d o d i f f e r e n t i a l operator a(x,D)
a
i n an open s e t X i s c o n t a i n e d i n t h e
58
dia9onal X)CK, m u l t i p l y i n g the kernel by any smooth f u n c t i o n f ( x ~ y ) which equals I
in a neighborhood of the d i a g o n a l amounts t o s u b t r a c t i n g
a smooth k e r n e l . Hence, w i t h o u t changing i t s its
e f f e c t on s i n g u l a r i t i e s or
symbol we can r e p l a c e any 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 a p r o p e r l y
supported one,
i.e.
an o p e r a t o r whose kernel has i t s
t o the d i a g o n a l t h a t the s e t s of S in compact s e t s .
It
support S so c l o s e
p o i n t s ( x , y ) w i t h x or y constant meet
i s c l e a r t h a t the product of
two p r o p e r l y
supported o p e r a t o r s i s again p r o p e r l y supported and t h a t the symbols of the products of such o p e r a t o r s can be computed a c c o r d i n g t o the theorem above.
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 a p p l i e d to o s c i l l a t o r y
integrals
Sometimes one has to apply 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 t o an o s c i l l a t o r y i n t e g r a l . Under s u i t a b l e r e s t r i c t i o n s , another o s c i l l a t o r y
the r e s u l t i s
i n t e g r a l w i t h the same phase f u n c t i o n and another
a m p l i t u d e . The p r e c i s e r e s u l t
i s as f o l l o w s .
Let s ( x , ~ ) be a r e g u l a r phase f u n c t i o n f o r
x and ~ in Rn,
let a(×,~l
be a polyhomo9eneous phase f u n c t i o n and l e t
u(x)
=
f a(x,~)
exp
be the c o r r e s p o n d i n g o s c i l l a t o r y
Theorem
Let P(x,D)
polyhomogeneous
be a first
symbbol
P(x,~).
is(x~)
d~
integral.
order
pseudodifferential
If a(x,~)
vanishes
for
operator
large enough
then, modulo smooth f u n c t i o n s , P ( x , D ) u ( x ) i s an o s c i l l a t o r y
S b(x,~)
exp
is(x,~)
with x,
integral
d~
where (3.1.3)
b(x,~)
~ ~ PC~(x,sx(x,~)(D~+r~(x,y,~)=a(x
~))ly=x,
with
r(x,y,~)
= s(x,~i-s(y,~)
-s~(x,~).(x-y).
Note. Since ry=O when x=y, the c o e f f i c i e n t s of most [ ~ / 2 ] + I
in
~. Hence the formula f o r
(Dy + i r y ) ~ have o r d e r at
b 9 i r e s a polyhomogeneous
59
expansion.
Note.
If
the oscillatory
chan9in9 a to a'
and s t o
is
homogeneous o f
change s t o j+l~I+l+k.
To g e t P ' u ' ,
in
The F o u r i e r
u^(o)
r.
In
we change s t o
that
To compute
t h e term above by - i
t h e term above by - I
comes f r o m t h e f a c t
i s o b t a i n e d from u ( x )
we have ( P u ) ' = P ' u ~.
de9ree i
- s and m u l t i p l y
and m u l t i p l y
Proof.
-s,
u'(x)
fact,
by
consider a
(3),
term in the e x p a n s i o n
where f
integral
to
-s,
=S a(x,~)
exp
change 0 i n
u(x)
is
-s,
Pk(x,9)
t o -0
+k+l where t h e
r changes t o
the o s c i l l a t o r y
i(s(x,~)-ix.o)
we have t o
t h e power
t h e power j + i ~ l
as s changes t o
transform of
to
(Pu)'
1
-r.
integral
dxd~.
Since Pu(y)=(2~)-" Pu(y)
equals the oscillatory
(2~)-" The r u l e s that
$ P(y,0)u^(9)
contribute
to
(3.1.4)
the
P(y,9)
in
some s m o o t h
supposed t o inserted
s e t s of
PC=' ( y , ~ , O )
the
i~I
integral
integrals
c l o s e t o where x=y and
o f Pu.
(g-s~(×,~) of
dxdod~.
oscillatory
shows
O=sx(x,~)
T h i s makes t h e f o l l o w i n g a natural
=IP~'~ (y,s~(x,~)) (g-s~(x,~))=/~!
run o v e r
into
set
+is(x,~)
t e r m s o÷ g - s x ( x , ~ )
P~(y~o,s~(×,~) for
i(y-x).9
integral
t h e wave f r o n t
P(Y,9)
expansion of
exp
c o m p u t i n g wave f r o n t
o n l y the p a r t s of
do,
integral
S P(y,o)a(x,~)
for
exp i y . o
starting
point,
+
=
degree
1-1~1
t h e second o v e r above and w i t h
in
~.
The f i r s t
I~t=N.
The f i r s t
sum i s sum
t h e e x p o n e n t i a l r e a r r a n g e d as
follows,
exp(i(y-x).o calls of
for
integration
terms w i t h
I~I
+is(y,~) by p a r t s
+ is~(x,~).(x-y) in
x with
+ir(y,x,~))
the r e s u l t
t h a t we g e t a sum
60
(2~)-"
S E(x,x,~,O)
ir(y,x,~)/~!dxdod~
P~a~(y,s~(x,~)(Dx)~a(x,~)exp
where E = exp
This
can be
i(y-x).o
rewritten
(2~}-~S
exp
+is(x,~)
as
is(x,~)
d~
$ F(x,y,~)
where F i s t h e g e n e r a l term o f with respect to shown t h a t
exp
i(y-x).o
the expansion (3).
d0
Hence an i n t e 9 r a t i o n
9 g i v e s the d e s i r e d expansion except t h a t
t h e second term o f
lengthy exercise is
Changes of
- ir(x,x,~).
left
variables.
(4)
it
has t o be
behaves p r o p e r l y . T h i s somewhat
t o the r e a 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 s on m a n i f o l d s .
Let f : × - > Y be a smooth d i f f e o m o r p h i s m form an open s e t × i n R~ t o a n o t h e r open s e t Y. We s h a l l
see t h a t
f
induces a l i n e a r b i j e c t i o n
between c o r r e s p o n d i n g p s e u d o d i f f e r e n t i a ! o p e r a t o r s .
Theorem
The map f
induces a map
a(x,D) where the symbol of
a,(y,g)=
-> a ÷ ( y , D x ) , y = f ( x ) ,
the o p e r a t o r a÷ i s g i v e n by the f o r m u l a
exp
iDy..D~
b(y,y',o)ly'=y,
with b(y,y',9) = a(g(y),~9' (y')-~o)Idet Here
9(Y)
is the
inverse
of
f(x)
and
F(y,y')
g' (y')
F(y,y') I.
is an nXn m a t r i x
defined
by (9(Y)-9(Y')).F(y,y')~ for
y and y '
symbol o f
p b e i n g the p r i n c i p a l
of
In the expansion of
powers o f
y-y'
symbol o f
a,
the
t h e o p e r a t o r a÷ i s
P(X,~f'(x)9), Note.
(y-y').o.
c l o s e t o each o t h e r .
Note. The theorem shows t h a t , principal
=
x=9(y).
bw(y,y',o)
have the form
i n powers o f y - y '
the c o e f f i c i e n t s
61
C(y,g) where A i s m of
= A(y,9)9 =
homogeneous in
such a f u n c t i o n
P and
degree - k - i ~ l .
a sum o f
8 are
g-derivatives
(-I) k and c h a n g i n g
is a l s o
what
time multiplying dual
9 to -O c h a n g e s
a polyhomogeneous
The
one by
and c h a n g i n g
Proof.
of
order
terms
of o r d e r s
(_i),-k .... B(y This
An g - d e r i v a t i v e
(PA(y,o))Qg=
B(y,g)= where
is
o of
gets
variables
a(x,D)
K(x,x')=
g to -g
homogeneity
are
has
$ a(x,~)
A by
-g).
m-k
commutative
pseudodifferential
operator
Multiplying
B(y, g) to
by c h a n g i n g
-I to the
n and m-n.
the exp
in B w h i l e
at
of B. H e n c e ,
operations
the s a m e
taking
when
the
applied
to
operator.
kernel i(x-×').~
d~.
Hence t h e t r a n s p o r t e d o p e r a t o r has t h e k e r n e l
K(g(y),g(y'))
Jdet g' (Y') l
which e q u a l s $ a(g(y),~)
Let
G(y,y')
be a m a t r i x
exp i ( g ( y ) - 9 ( y ' ) . ~
defined
((9(Y)-9(Y'))-~ so that and y'
G(y,y)=~g'(y) close
to e a c h
well
defined.
has
the k e r n e l
Putting
for
y'=y.
other
and
That
this
gives
with
adjoints.
the d e s i r e d This finishes
Id~.
by = We
(Y-Y').G(Y,Y')~
need
therefore
G ( y , y ' ) ~=g and
I b(y,y',g)exp
Idet 9'(y')
the
consider inverse
substituting
i(y-y').9
kernel
only
the
kernel
F(y,y')
shows
that
of
for G is
a$(y,Dy)
do.
is c l e a r
the p r o o f .
.
from
the s e c t i o n
deali~g
y
62
Manifolds
It
i s c l e a r from our l a s t theorem t h a t
it
makes sense t o t a l k 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 on a smooth m a n i f o l d M and t h a t the p r i n c i p a l symbols of such o p e r a t o r s are f u n c t i o n s on i t s
cotan9ent
bundle T*M. In f a c t ,
becomes
given a p r i n c i p a l symbol p ( x , ~ ) ,
q ( y , 0 ) = p ( x , ~ ) in terms of c o o r d i n a t e s ( y , 0 ) the p r i n c i p a l symbol i { p , q } of
for
which
the commutator of
it
~.dx= 0.dy. Also
two o p e r a t o r s w i t h
p r i n c i p a l symbols p and q i s a f u n c t i o n on the cotangent bundle.
3.2 L = e s t i m a t e s . R e 9 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 ÷ e r e n t i a l equations
The main r e s u l t about the s i z e 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 i s t h a t those of o r d e r 0 map square i n t e g r a b l e f u n c t i o n s w i t h compact supports i n t o l o c a l l y square i n t e g r a b l e f u n c t i o n s . This w i l l
be proved below as
p a r t of a more general r e s u l t . For t h i s we need two lemmas.
Lemma
If
N>max(n,p+n)~ then, w i t h S
for
any r e a l
(l+Is-tl)-N(l+Itl)=dt
=
i n t e g r a t i o n over R ~, Q((I÷Isl)=)
p.
Proof. F i r s t ,
let
p > 0 . I n t e 9 r a t i n 9 over
Itl
majorant s t r a i g h t away. I n t e g r a t i n 9 over I t l > I s l / 2 , and i t
we 9et the d e s i r e d we have I s - t l > I s l / 2
s u ÷ f i c e s t o e s t i m a t e the i n t e 9 r a l of (l+Itl)P-Ndt
over I t l > I s l / 2 .
Since N-p>n,
f u n c t i o n of s.
When p<0 and I t l < I s l l 2 ,
( l + I s - t l ) -N
the i n t e g r a l converges t o a bounded
< const(l+Is-tl)N-~(l+Isl)
and an i n t e g r a t i o n w i t h r e s p e c t t o t
p.
9 i r e s the d e s i r e d r e s u l t . When
63
Itl>lsl/2
we can p r o c e e d as b e f o r e . -
Schur's
lemma. I f
K(s,t)
$1K(s,t)Idt
is
S A,
Our n e x t t o o l
integrable
is
and
flK(s,t)Ids
£ B,
then
~ AB 5 1 f ( s )
f(lSK(s,t)f(t)dtl=ds Proof.
The l e f t
side of
S K(s,t)K(s,t')l and t h e r e s u l t
t h e c o n c l u s i o n i s m a j o r i z e d by (If(t)12+lf(t')lZ)dsdtdt'/2
follows.
L e t H = be t h e space o f with
integration Uun.
is
finite.
:
=
distributions
is
any r e a l
with
a sufficiently
with
compact s u p p o r t .
and f
number. When s i n c r e a s e s , t h e
all
P(x,D)
is
i s smooth w i t h
Proof.
s and a l l
a scale of
s p a c e s H= where t h o s e
l a r g e n e g a t i v e s c o n t a i n any g i v e n d i s t r i b u t i o n
a pseudodifferential
compact s u p p o r t ,
llfP(x,D)un. for
which t h e norm s q u a r e
flu(~)iZ(l+l~I)Z=d~
Here s
If
u for
o v e r R",
c o r r e s p o n d i n g space d e c r e a s e s . We g e t
Theorem
t = ds.
~ const
operator of
o r d e r m i n R~
then
Ilu~÷.
distributions
u.
We have v(x)
= f(x)P(x,D)u(x)
Taking the F o u r i e r
v^(o) where K i s
of
the
left
side gives
decrease in
its
first
variab]e
Hence
(l+lOI)=v^(g) where, f o r
transform of
ix.~ ÷ ( x ) P ( x , ~ ) u ( ~ ) d ~ .
= $ K(~-O,~)u^(~)d~,
fast
t h e second one.
=(2n)-"Sexp
any N>O,
= S G(9,~)(l+l~l)~÷=u^(~)d~
and o f
degree m in
64
G(g,~)
= O((I+lol)'(I+I~-91)-N(I+I~I)-°).
By t h e f i r s t
of
our
lemmas a b o v e , t h i s
k e r n e l meets t h e r e q u i r e m e n t s o f
t h e second one. Hence t h e d e s i r e d r e s u l t
The wave f r o n t
sets of
solutions
We have seen t h a t u when P i s
9ire
the wellknown f a c t distribution
(x,~)
points
if
P is
R~×R~ a r e s a i d
in Y.
(x,~/1~I)
conically of
is
It
with
is
Pu i s
said
a partial
equations
c o n t a i n e d in
o p e r a t o r and u i s
and a l s o 9 i r e that
set of
statement a quantitative
and Pu=O t h e n u i s
Subsets Y o f and
this
function
pseudodifferential
t h e wave f r o n t
a pseudodifferential
Below we s h a l l regularity
of
follows.
form
in
terms o f
it
differential
the
containin9
operator,
u a
a smooth f u n c t i o n .
t o be c o n i c a l
if
t o be c o n i c a l l y
(x,t~) compact
(x,~)
in Y i s
compact.
c l o s e when one o f
them i s
c o n t a i n e d in
is
in
Y when t>1
if
the set
of
Two p o i n t s a r e s a i d t o be a conical
neighborhood
the o t h e r . Given a c o n i c a l
pseudodifferential T=T(U)
where f
and A(~)
l a r g e ~,
their
neighborhood U of
a pair
consider
f(x)A(D),
a r e smooth and A i s
homogeneous o f
product vanishes outside a conically
and f ( x ) = A ( { ) = l
when ( x , { )
is
conically
close to
such o p e r a t o r s we can d e f i n e t h e r e g u l a r i t y as t h e
(y,9),
operators =
l e a s t u p p e r bound as U t e n d s
(Y,9)
of
de9ree z e r o f o r compact s u b s e t U
(Y,9).
function
In
r~ o f
numbers s f o r
terms o f u at
(y,9)
which Tu
belongs to H'.
of
Similarly,
if
P at
as t h e 9 r e a t e s t
(y,9)
P(x,~)
is
an a m p l i t u d e , we d e f i n e t h e d e 9 r e e m P ( y , 9 ) l o w e r bound as U t e n d s t o
(y,o)
TP.
Theorem
If
distribution,
P
is
then
of
a distribution.
converse of
an e l l i p t i c
that
a pseudodiffferential
operator
and
u
is a
of
de9
65
for
all
x and ~. There i s e q u a l i t y w i t h m a t
principal
p a r t of
the
last
place i f
degree m which does not v a n i s h c o n i c a l l y
P has a
close to
(x,~). Note.
If
P is elliptic
in
the sense t h a t
it
which never v a n i s h e s , the r e q u i r e m e n t f o r everywhere and we get the c l a s s i c a l
Proof.
Let T=T(U)
and S=T(V)
that
symbol
is satisfied
a distribution
is a
Pu.
be as above,
t h a t S ( x , ~ ) = l on the s u p p o r t of
equality
result
smooth f u n c t i o n o u t s i d e the s u p p o r t of
has a p r i n c i p a l
T(x,~).
let
V be f i x e d and U so s m a l l
By t h e p r e c e d i n g theorem,
IITPSuII. ~ c o n s t I I S u I I . ~ , where m i s
t h e degree of
symbols o f
products,
TP. F u r t h e r ,
t h a t of
TP(I-S)
by t h e r u l e f o r is fast
d e c r e s i n 9 . Hence t h e
i n e q u a l i t y above h o l d s f o r
TPu i f
we add a c o n s t a n t t o
By l e t t i n g
(y,0),
it
s+m i s
U and V tend t o
l e s s than b u t a r b i t r a r i l y
arbitrarily
follows that
Lemma . Suppose t h a t
an a m p l i t u d e a ( x , ~ )
p a r t of
(y,9).
I/a~ close to
Then t h e r e (y,9)
when
the
need the f o l l o w i n g r e s u l t .
has a p r i n c i p a l
p a r t am of
degree m which does not v a n i s h i n some c o n i c a l neighborhood U o f point
side.
w i t h m l a r g e r but
c l o s e t o m p ( y , o ) . T h i s proves the f i r s t we s h a l l
the r i g h t
RulJ. i s f i n i t e
close to r~(y,O)
theorem. To prove the second p a r t ,
c~mputin9 the
i s an a m p l i t u d e b ( x , ~ )
with principal
a
part
such t h a t
a(x,B)b(x,B)-I has an a m p l i t u d e
Proof.
of fast
Let us denote
by the r u l e s f o r
decrease
close
by a(x,~)ob(x,~)
to
(y,g).
the symbol
computing the symbols of
the o t h e r of
given
the p r o d u c t 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 . Keeping ( x , ~ ) after
of a(x,D)b(x,D)
in U we s h a l l
compute one
the terms in an a s y m p t o t i c s e r i e s f o r b ( x , ~ )
with
66
terms b - ~ - j
of
degree - m - j ,
b-m(x,~)
+b-m-1(x,~)+...
Letting Bj be the sum of the first j+I terms of the series,
assume
that a(x,~) where c j ~
Bj(x,~)
has degree - j - l .
=I +cj÷~(x,~) For j=O,
this
i s a c h i e v e d by p u t t i n g
b~(x,~)=I/a~(x,~). am b e i n g t h e p r i n c i p a l
p a r t of
a.
In the s t e p from j
to j+l,
b-~-~-~
should s o l v e the e q u a t i o n
a(x, ~)o(Bj ( x , ~ ) + b - m - j - 1 ( x , ~ ) ) = l+c~÷~(x,~)b-j-~-~(x,~)+R where R has the degree - 3 - 2 .
b-~-j-~(x,~)= achieves
Putting
-cj÷~(x,~)/a~(x,~)
the step of the induction.
asymptotic
expansion
above and
which e q u a l s 1 in a c o n i c a l
t h e second p a r t o f
let f(x,~)
(Y,O)
c o n t a i n e d in U. The
meets the r e q u i r e m e n t s of
(y,g)
where the p r i n c i p a l
P has f i x e d degree m and does not v a n i s h .
lemma, t h e r e
the
in W. By the p r o o f of
part of
Hence,
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 Q such t h a t
symbol PoQ(x,~) e q u a l s I
lemma.
the theorem. By a s s u m p t i o n , t h e r e i s a
f i x e d c o n i c a l neighborhood W o f symbol o f
be an a m p l i t u d e with the
be an a m p l i t u d e of degree 0
neighborhood of
amplitude b(x,~)=f(x,~)B(x,~)
Proof of
Let B(x,~)
the f i r s t
the
by t h e that
the
p a r t of
the
theorem, #TQSuH. ~ c o n s t ~ u l l . - p , when t h e s u p p o r t s o f
T and S a r e s u f f i c i e n t l y
close to
(y,g)
any number >m. Here we can r e p l a c e u by Pu. By t h e r u l e s f o r t h e symbols o f
products,
the symbol of
degree - ~ on the s u p p o r t of
T(x,~).
w i t h compact s u p p o r t . Hence t h e left
and p i s computing
the commutator of P and S has
Hence T [ P , S ] u i s a smooth f u n c t i o n
i n e q u a l i t y above h o l d s w i t h TSu on t h e
and a c o n s t a n t added on t h e r i g h t
so t h a t ,
finally,
67
r~(y~o) ~ r ~ ( y ~ o ) and t h i s
finishes
+m
the p r o o f o f
the theorem.
Pseudodifferential operators with type.
Propa9ation of
singularities
When a d i s t r i b u t i o n when u i s is
x~.
where f
symbols o f
see t h a t
e q u a t i o n D~u=Oj i . e . its
re9ularity
t o measure the r e g u l a r i t y
exp i x . ~
9 ( x ) = f ( x + y ) chan9es the
inte9ral
f u n c t i o n over
dx
inte9ral
f exp i x . ~
×~ , a s h i f t
Putting
to
9(x)u(x+y)dx
where 9 i s smooth w i t h s u p p o r t c l o s e t o 0 and 9 ( 0 ) ~ 0 . independent of
function
integrals
i s smooth w i t h s u p p o r t c l o s e t o y and f ( y ) ~ O .
exp i y . ~
principal
theorem.
x~, we s h a l l
In f a c t ,
a p o i n t y~ one l o o k s a t S f(x)u(x)
principal
u s o l v e s the d i f f e r e n t i a l
independent of
independent o f
real
of
y i n the x~ d i r e c t i o n
above by an o s c i l l a t i n g
factor with
Hence, just
when u i s
chan9es t h e
no i n f l u e n c e on s i z e .
This
p r o v e s our s t a t e m e n t . The s i m p l e f a c t s o f situations. principal of
symbol p ( x , ~ )
vanishes ( f o r
t y p e in
which
is
9ires rise
order d i f f e r e n t i a l xt
non-linear partial
=p~(x,~),
its
degree m and
~ gradient p~(x~)
This p r i n c i p a l
symbol,
in
never the example
t o a n o n - d e g e n e r a t e H a m i l t o n i a n system o f
in
~
=-p~(x,~)
the c o t a n g e n t b u n d l e o f
o÷ p and P (and c h a r a c t e r i s t i c s differential
from t h e e q u a t i o n s t h a t
When p = ~
of
Rn~ c a l l e d the ÷ i r m t o r d e r
equations p(x,vx)= const).
the f u n c t i o n p ( x , ~ )
connected b i c h a r a c t e r i s t i c . bicharacteristics.
and homo9eneous of
order m with
equations,
curves t - > ( x ( t ) , ~ ( t ) )
bicharacteristics
real
t h e sense t h a t
n o n - v a n i s h i n g ~).
above equal t o ~
for
above e x t e n d t o much more g e n e r a l
L e t P ( x , D ) be 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 o f
principal
first
the s i t u a t i o n
Those f o r
It
follows
i s c o n s t a n t on e v e r y
which p v a n i s h e s are c a l l e d nul
the b i c h a r a c t e r i s t i c s
are l i n e s p a r a l l e l
68
to
t h e x~ a x i s w i t h
zero.
In
L e t P be a p s e u d o d i f f e r e n t i a l
symbol p o f Pu,
principal
and t h a t
Note.
of
u is
If
a point
set
of
of
the regularity
t h e theorem i s
t h e main r e s u l t
of
Since n e i t h e r
if
we m u l t i p l y
of
p
is
this
a real
principal
u i s c o n s t a n t a l o n g any n u l
B,
of
Pu i s
at
least s at
then the r e g u l a r i t y
B
function
of
function
(3.2.1) if
k i n d and
P by an e l l i p t i c
u is
infinite.
This
I n an o b v i u o s sense t h e
it
is
singularities
last result.
It
due t o L a r s H~rmander ( 1 9 7 0 ) .
t h e theorem nor operator,
its
c o n c l u s i o n chan9e
we may assume t h a t
along bicharacteristics
bicharacteristic
the proof
Fm, G2 o f (y,O)
of
the degree
I.
B be a nul
idea of
Pu and on t h e n o n - n u l
a propagation of
the data of
L e t T be t h e t r a n s p o r t
Then,
operator with
function
from the p r e c e d i n g theorem.
statement of
to
of
the r e g u l a r i t y
O u t s i d e t h e wave f r o n t
Proof.
is
l e a s t smm-I e v e r y w h e r e on B.
follows
let
function
s+m-1 a t
bicharacteristics,
is
~:
t y p e and d e g r e e m. Then, o u t s i d e t h e wave f r o n t
the regularity
bicharacteristic.
u is at
bicharacteristics,
t h e s e q u e l we o n l y c o n s i d e r c o n n e c t e d b i c h a r a c t e r i s t i c s .
Theorem
set of
~ c o n s t a n t . On t h e nul
is
with
to construct
i[Q,R]
(Ty,T0),
end p o i n t s
(y,~)
pseudodifferential
t h e same d e 9 r e e such t h a t and 8 c l o s e t o
from t=t~
Q is
to
and
t=t2
(Ty,To).
The
o p e r a t o r s Q and
s u p p o r t e d c l o s e t o B,
and such t h a t ,
and
F close
approximately,
= F~ -Gz.
P,F,G a r e s e l f a d j o i n t
and Pu=O, we have, s t i l l
approximately,
0 =i((PQu,u)-(RQu,u))=i([P,Q]u,u) = JIFull= -JIGuIK=. If
F can be chosen
more o r
c o n c l u s i o n must be t h a t nul
bicharacteristics.
less arbitrarily
the r e g u l a r i t y We s h a l l
and t h e same f o r
function
of
G,
the
u is constant along
show i n what f o l l o w s
that
it
suffices
69
to satisfy
(I)
on t h e symbol
level.
L e t R be a p s e u d o d i f f e r e n t i a l its
s u p p o r t c l o s e t o B. -(Rv,Ru)
If
operator of
d e 9 r e e s whose symbol has
Pu=v we 9 e t
=-(RPu,Ru) = ( [ P , R ] u , R u )
where each term has a sense and t h e e q u a l i t y sufficiently
l a r 9 e n e g a t i v e . We s h a l l
Poisson p a r e n t h e s i s { p , r } differential
o p e r a t o r on r .
are selfadjoint. is the
ima9inary part
At
To do t h i s ,
this
point
it
convenient to
is
the t l
end o f
an a m p l i t u d e 9 ( x , ~ )
non-ne9ative amplitude q of joinin9 {p,q}-cq
B,
a chanBe o f
If
state
and p r o v e a lemma a b o u t t h e
the
B and l e t
left
side.
de9ree s w i t h c be a r e a l
number. Then
=fm _9=
b e i n 9 n o n n e g a t i v e and smooth, has a s q u a r e r o o t
r
t h e same s u p p o r t .
a(x,{)
{p~q)
is
differentiation
i s a smooth f u n c t i o n
unknown f u n c t i o n s
which case t h e r e e x i s t s 9 be f
support
d e 9 r e e s s u p p o r t e d i n TS and a
q->q exp - c a ,
a solution
t r a n s p o r t e d by T.
u with
of
with
f->f
e x p - c a / 2 chan9es t h e e q u a t i o n above t o a s i m i l a r
let
its
d e 9 r e e 2s s u p p o r t e d on t h e
the f o r m u l a above,
bicharacteristic.
of
B
T and ST such t h a t
an a m p l i t u d e w i t h
In
for
homo9eneous o f
there exist
Proof.
symbol p and t h e d e g r e e o f
~ (Rv~Rv)
R i n h e r e n t in
close to
is
P=A+iB where b o t h A and B
we 9 e t
f(x,~)
q,
R, as a
t h e above and u s i n 9 S c h w a r z ' s i n e q u a l i t y
S conically
N o t e . Observe t h a t
symbol o f
Takin9
equation for
bicharacteristics
is
s o m e c o n s t a n t b.
is
Lemma. Suppose t h a t
t r u e when s
to determine R usin9 the
let
J ( B R u , R u ) l ~ b(Ru,Ru)
of
is
the p r i n c i p a l
Im ( R * [ P ~ R ] u , u ) - ( l + b ) (Ru~Ru)
differential
which
is
Then A has t h e p r i n c i p a l
a t most 0 so t h a t
(3.2.2)
, where r
try
+(PRu,Ru),
q alon9 a
{p,a)=l
exp - c a / 2 ,
equation with
close to 9->9 c=O i n
the required properties
This proves the
lemma.
if
we
70
End of
the p r o o f of
t h e e q u a t i o n (2)
the theorem.
If
R has a r e a l
where R= has degree s - I / 2 .
r(x,~),
const((Rv,Rv)+(R=u,R°u)),
In v i w o f
the
lemma, t h i s
9ives
(Fu,Fu)-(Gu,Gu)~ const((Rv,Rv)+R=u,R=u))+const
(3.2.2)
where F = f ( x , D ) ,
to
part
amounts t o
([p(x,D),r2(x,D)]-cr2(x,D)u,u)~
first
prinicpal
G=9(x,D)
and where t h e
l a s t c o n s t depends on s.
t h a t v i s smooth c l o s e t o B. Then,
(y,o)
we g e t ,
noting that
regularity
letting
the s u p p o r t of
Suppose f
tend
f u n c t i o n s are c o n t i n u o u s from
below, r~(y,9)
Z min
(r~(Ty,Tg),I/2
V a r y i n g B o v e r p a r t s of f u n c t i o n of at
itself,
this
u i s c o n s t a n t a l o n g B.
If
+ min r~ o v e r B). proves t h a t
the r e g u l a r i t y
the r e g u l a r i t y
f u n c t i o n of Pu i s
l e a s t s a t B and t h a t of u i s equal t o s a t a p o i n t of B,
formula that
(2)
a p p l i e d t o a p a r t of B near the p o i n t
the r e g u l a r i t y
finishes
f u n c t i o n of
u is
at
in
the
q u e s t i o n shows
least s at a l l
of
B. T h i s
the p r o o f .
3 . 3 Lax's c o n s t r u c t i o n f o r Cauchy's problem and a hyperbolic f i r s t order d i f f e r e n t i a l
operator
L a x ' s c o n s t r u c t i o n extends to h y p e r b o l i c f i r s t
order p s e u d o d i f f e r e n t i a l
operators P = Dt
+ Q(t,x~D.)
and g e n e r a l Cauchy problems. Here Q i s polyhomogeneous o f ~(t,x~)
with real
=
principal
~ ~k(t,x~)
(k=l~O~.-.)~
part
q(t,x,~) the
~
degree I ,
= Q1(t,x,~).
This means
that
principal
part
T+q(t,x,~)
with first
order hyperbolic differential
of P
operators.
is real,
in a n a l o g y
71
Theorem
Let
v(x)
= S b(x,9)
be an o s c i l l a t o r y r.
If
exp i
integral
t h e phase f u n c t i o n s~
+ q(t,x,~)
there exists
with
=
a polyhomogeneous a m p l i t u d e a o f
s(t,×,o) =0,
degree
solves the Hamilton-~acobi equation,
s(O,x,o)
an o s c i l l a t o r y
u(t,x) with
s(x,o)
= s(x,g),
integral
$ a(t,x,o)
exp
is(t,x,o)dg
a polyhomogeneous a m p l i t u d e a o f
d e g r e e r which s o l v e s t h e Cauchy
problem Pu= smooth, u - v = smooth when t=O for
small
Note.
t
With
and x . s(x,o)=x.o,
r=O
v(t,x)=H(t)u(t,x)
with
distribution
is s m o o t h
pole
at
Proof.
Pv-6
and
H(t)=O
a(x,9) when
and
we
=
tI(2~)
~,
v(x)=6(x)
t
and
I otherwise,
get
a fundamental
and the
solution
of
P with
0.
Let
us
expand
a(t,x,o)
a = ~ ak, By t h e r u l e s oscillatory
for
in a s e r i e s
(k=r,r-l,...
with
terms,
).
applying a pseudodifferential
i n t e 9 r a l , Pu i s
in h o m o 9 e n e o u s
an o s c i l l a t o r y
o p e r a t o r t o an
integral
with
phase f u n c t i o n
s and a m p l i t u d e sta+sa~ + I
Qc=~(t,x,s~(t,x,~)(Dx+rx(t,x,y,
)~a(t,x
~)/~!ly=x
where r(t,x,y,~) The
term
of
= s(t,x,~)-sy(t,x,~)
degree (s~
r+l
-sx(t,x,~).(x-y).
is
+q(t,s,s~))a~
and v a n i s h e s by a s s u m p t i o n . The term o f L = ~t + a ~ ( t , x , s ~ ) . i D ~ is
a linear
differential
unique solution b~(x,o).
The t e r m s o f
is
La~ where
+ Qo
operator of
ao when ao i s
degree r
fixed
degree I . for
t=O.
The e q u a t i o n La~=O has a
We g i v e
it
the value
degree k i n v o l v e the p a r t s a o , . . . , a k
of
a and has
72
the
form Lak
where
H does
ak w i t h
+ M not
k>O.
sum w i t h
The r e s u l t
ak.
is
Putting
adjusted at
d e g r e e 0 which f u l f i l s
finished.
ak
=bk
i s a sequence o f
t h e terms s u i t a b l e
amplitude a of The p r o o f
involve
for
all
k and
terms a o , . . ,
~=0 i s
t=O
fixes
such t h a t
all
their
a polyhomogeneous
the r e q u i r e m e n t s of
the theorem.
CHAPTER 4
THE HAMILTON-JACOBI EQUATION AND SYMPLECTIC GEOMETRY
We have a l r e a d y met the H a m i l t o n - J a c o b i equation in La×'s c o n s t r u c t i o n of a l o c 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 h y p e r b o l i c
e q u a t i o n . The c o n s t r u c t i o n of g l o b a l p a r a m e t r i c e s r e q u i r e s a f a i r amount of d i f f e r e n t i a l
geometry connected w i t h t h i s e q u a t i o n , in
p a r t i c u l a r the s i m p l e s t f a c t s about Lagrangian m a n i f o l d s . This m a t e r i a l will
be reviewed below.
Hamilton systems
A Hamilton system of o r d i n a r y e q u a t i o n s i s one of (4.1.1)
the form
x~=Hp(x,p,t), p~=-H~(t,x,p)
where x = ( x ~ , . . . , x , )
and p = ( p ~ , . . . , p . ) are f u n c t i o n s of
g i v e n smooth r e a l f u n c t i o n of
t,x,p.
t and H i s a
The i n d i c e s i n d i c a t e the
corresponding g r a d i e n t s . By the l o c a l t h e o r y of such systems,
the
Hamilton f l o w T : ( s , y ~ q ) - > ( t , x ~ p ) ( x , p = y , q when t=s) i s a l o c a l smooth b i j e c t i o n f o r
f i x e d s and t .
The t h e o r y of Hamilton
systems depend on the f o l l o w i n g
Theorem
The d i f f e r e n t i a l w=p.dx-Hdt,
is
invariant
When
under
of
the d i f f e r e n t i a l
(p.dx=pldx1+...+p.dx.)
the H a m i l t o n
H is h o m o g e n e o u s
P r o o f . For s i m p l i c i t y ,
form
of d e g r e e
flow,
i.e.
I in p, w
dw(t,×(t),p(t)=dw(s,y,q). itself
is
invariant.
l e t s=O. Using an obvious n o t a t i o n we have
d×=×ydy+x,dq+x~dt, dp=pydy+pqdq+p~dt, dH=H~dx+Hmdp+H~dt. Hence,
if
dR and dp denote dx and dp w i t h dr=O, we get
74
dw=dp~dx-dH^dt
= d~d%
+p~dt^d.Z+d~^x~dt-
-H~dZAdt-H,dp^dt. Here e v e r y t h i n g i n v o l v i n g dt disappears due t o the e q u a t i o n s ( I ) . f o l l o w s t h a t dw i s a l i n e a r combination of
the d i f f e r e n t i a l s
dyAdq and dq^dq w i t h c o e f f i c i e n t s depending, maybe, on t . ddw=O, a l l dw i s
the t - d e r i v a t i v e s of
independent of
t.
It
dy^dy,
But s i n c e
the c o e f f i c i e n t s of dw vanish and hence
When H i s homogeneous of degree I ,
p. Hm=H so
that Hxdx+Hpdp=dH=Hpdp+pdHp. Then H~dx=pdHp and hence,
with
d i f f e r e n t i a t i o n s along the f l o w , d(p.x)/dt=ptdx+pdx~=-Hxdx+pdHp =O. This f i n i s h e s the p r o o f . Our theorem can be used t o f i n d more or
less e x p l i c i t
s o l u t i o n s of
H a m i l t o n - J a c o b i ' s equation
(4.I.2)
f~+H(t,x,f~)=O,
w i t h a given 9 ( x ) . t=O,
In f a c t ,
f(O,x)=9(x)
w i s closed on every m a n i f o l d Y of
q=g'(Y) w i t h y in some open bounded p a r t of R.,
the Hamilton o u t f l o w T(Y) hyperplanes t=const
in m a n i f o l d s Y ( t )
c o o r d i n a t e s . In terms of that w=df(t,x) for
of Y. When t
equation (2).
where we can choose x=Y(t) as
these c o o r d i n a t e s , w=pdx, x = x ( t ) , p=p(t)
f~(t,x)=-H(t,x,p), It
and hence a l s o on
i s small enough, T(Y) meets the
some smooth f u n c t i o n f ( t , x )
were x = x ( t ) , p = p ( t ) .
the form
so
with
f~(t,x)=p,
follows that f
s o l v e s the H a m i l t o n - J a c i b i
By general t h e o r y , the s o l u t i o n i s unique. Along the f l o w
we have ( d / d t ) f ( t , x ) = f ~ + f ~ d x = -H+pH, so t h a t (4.1.3) Hence, its
f(t,x(t)=g(x(0))+ if
intial
H(t,x,p)
5°~(p. H p - H ) ( s , x ( s ) , p ( s ) ) d s .
i s homogeneous of degree 1 in P, f
v a l u e 9(x) by p r o p a g a t i o n along the f l o w .
i s o b t a i n e d from The formula ceases
t o hold when the f l o w from Y no longer has a s u r j e c t i v e p r o j e c t i o n on
75
x - s p a c e . We f o r m u l a t e t h e r e s u l t s
in
a theorem.
L e t A be a bounded p a r t
of
R",
Theorem
A and Y t h e m a n i f o l d t=O,
q=g'(Y),
w=p.dx-H(t,p,x)dt
is
t h e number a>O i s
so s m a l l
9 a smooth r e a l
y i n A.
when I t I < a ,
satisfying
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 (2.
(3)
i s constant along the f l o w
de9ree I
in
In t h i s solution later
do t h i s
the dif÷erential
if
of It
f r o m Y.
H(t,x,p)
form When
a r e smooth
a functions is
form
s(t,x)
given explicitly is
by
homogeneous o f
p.
theorem, of
that
w is
t h e maps y - > x ( t , Y )
bijections
and i t
Then t h e d i f f e r e n t i a l
c l o s e d on t h e H a m i l t o n o u f l o w T(Y) that
function
the d i f f e r e n t i a l
form i s
the Hamilton-Jacobi equation is also the solution
we s h a l l
s
is
the g l o b a l a local
a g l o b a l one i n
object while
one,
We s h a l l
a certain
need some g e o m e t r i c p r e r e q u i s i t e s
given
in
the
se
s e n s e . To the next
section.
4 . 2 S y m p l e c t i c s p a c e s and L a g r a n g i a n p l a n e s
A linear
space S o f
d i m e n s i o n 2n i s s a i d t o be s y m p l e c t i c w h e n e q u i p p e d
by a n o n - d e g e n e r a t e skewsymmetric f o r m W(X,Y)
in the s e n s e
Theorem
e~,...,e,,
that
The space S has b a s e s , c a l l e d
~,,..., E, such
P r o o f . Since the form i s
linearly
is
non-degenerate
W(X,S)=O=>X=O.
W(×,Y)=I.
It
canonical,
with
elements
that
W(ej , e k ) = O ~ W ( Cj Ck)=O,
such t h a t
which
W(ej , Ck)=bjk.
n o n - d e g e n e r a t e , e v e r y X~O has some companion Y
follows
from t h i s
formula that
i n d e p e n d e n t . Hence t h e r e a r e e l e m e n t s e~ and
t h e f o r m u l a s above when j , k = l .
S i n c e t h e subspace o f
X and Y a r e ¢~ which s a t i s f y S for
which w
76 v a n i s h e s on t h e span o f
el
and
d i m e n s i o n 2 n - 2 on which w i s repeated till A linear
we 9 e t a n u l
bijection
~
is
a g a i n a s y m p l e c t i c space o f
a non-degenerate, the construction
can be
space. This completes the p r o o f .
U:S->S i s s a i d
t o be s y m p l e c t i c i f
it
p r e s e r v e s W,
i.e.
W(X,Y)=W(UX,UY) for
all
X and Y.
canonical basis relative
For t h i s into
it
is
n e c e s s a r y and s u f f i c i e n t
a canonical basis.
In
terms o f
that
U maps a
the matrix of
U
to a canonical basis,
Ue~=~a~kej + ~ b j k c k , U~=Icj~ek this
+ Idjk~,
means t h a t AC'=O,B'=O,AD'-BC'=I
for
the c o r r e s p o n d i n 9 m a t r i c e s .
transferred
to
its
The s y m p l e c t i c s t r u c t u r e
d u a l S* c o n s i s t i n 9
and G a r e two such f o r m s ,
of
linear
of
S can be
f o r m s F on S.
we d e f i n e F~G as a b i l i n e a r
When F
skewsymmetric
f o r m on SXS d e f i n e d by
(F^G) ( X , Y ) = F ( X ) G ( Y ) - F ( Y ) G ( X ) for
all
X and Y i n S.
(4.2.1)
With
this
n o t a t i o n we have
W=f~^9~+...+fn^gn,
where
fl,..o,f~,gl,...,9~ is
a basis of
S* b i o r t h o g o n a l t o
such a way t h a t i s mapped t o
it
f~ maps a l l
1 etc.
In
fact,
×=
lalel
+
Y=
~clel
÷ ~diEi ,
W(×,Y)
the b a s i s elements to if
Ibi¢i,
simply expresses the f a c t = laid~
-Ib~c~
Puttin9
UF(X)=F(UX),
a canonical basis e ~ , . . . , ¢ n of
that
S in
zero except that
e~
77
every linear that
U is
right
map o f
S i n d u c e s one o f
symplectic
s i d e of
if
and o n l y o f
S*.
The f o r m u l a
U, a c t i n g on S*,
(I)
above shows
preserves the
the formula.
Lagrangian planes L i n e a r subspaces o f
a s y m p l e c t i c space where W v a n i s h e s i d e n t i c a l l y
a r e s a i d t o be i s o t r o p i c . such a subspace o f W(f,M)=O w i t h
f
d i m e n s i o n p>n and M i s
p>n,
a solution
f
Isotropic
implies that
subspaces of
S is
W(f,S)=O,
exists
if
L is
then
p variables i.e.f=O.
which i s
But
a
Hencep~n. maximal d i m e n s i o n a r e s a i d t o be L a g r a n g i a n .
Examples. Any p l a n e spanned by one h a l f basis of
e q u a t i o n s in
t h e n 2n-p
contradiction.
In f a c t ,
a complement i n S,
i n L amounts t o 2 n - p l i n e a r
and t h e e x i s t e n c e o f if
They must have d i m e n s i o n ~n.
Lagrangian. It
L a 9 r a n g i a n p l a n e has t h i s
is
of
t h e members o f
not d i f f i c u l t
f o r m when r e f e r r e d
t o show t h a t to
a suitable
a canonical every canonical
basis.
Parametrization of
Lagrangian planes
L e t C" be complex n - s p a c e w i t h metric Q of
basis e~,...,e,
the form
Q(Z,W)=
Izk~
where z = z ~ e ~ + . . , and t h e same f o r
Lemma
Write
z=x+i~,
W(z,w) is a bilinear
is
and c o n s i d e r a u n i t a r y
w=y+i9.
w.
Then
= Im @(z,w)
non-degenerate s y m p l e c t i c form f o r
which
a canonical basis.
P r o o f . The supposed b a s i s i s
linearly
independent over the r e a l s
and,
78
s i n c e W ( z , w ) = ~ . y - o . x , they form a c a n o n i c a l b a s i s .
Lemma
The image o f
R" C C" u n d e r a u n i t a r y
map i s
and e v e r y L a g r a n g i a n p l a n e can be o b t a i n e d i n P r o o f . That U i s
unitary
z and w a r e r e a l , is
k.
the right
a Lagran9ian plane.
the real
span o f
means t h a t side
real
Conversely, let
so t h a t
Re Q ( f ~ , f k ) = 6 ~ k .
e~,..
by a u n i t a r y
Note.
S i n c e UR~=Rn i f
such t h a t
Lagran9ian planes is
When z = U w
if
U is
homeo~orphic t o
Im UO,
y=y',
+t
+tlm
contradiction. of
by a change o f O=
a certain
In
fact,
with
is
important for
Im U ) y ~,
~ =
(Im U
+t
U)=O
all
t,
also
for
Hence Re U -
so
tlm U i s
us t h a t
Re U can
Re
U)y' for
+ Re
t=-i,
invertible
U
0'
i.e.
det
U=O,
e x c e p t when t
is
a a zero
polynomial.
the cotangent bundle of
L e t X be a smooth m a n i f o l d o f
functions
are obtained from
then
4.3 La9rangian submanifolds of
a point p of
can be
9'+ty
is s y m p l e c t i c .
U
and
~ = Im Uy + Re UO.
variables
det(Re
t h e f~
j
is
the quotient U(n)/O(n).
be made i n v e r t i b l e
U
metric,
all
It
o r t h o g o n a l , t h e space o f
It
x=(Re
Hence URN
lemma.
Here Re U need n o t be i n v e r t i b l e .
If
W(fj,fk>=O for
If
have
we
x= Re Uy -
which
z and w.
W(Uz,Uw)=O.
But t h e n f ~ , . . ,
map. T h i s p r o v e s t h e and o n l y
all
L be a L a g r a n g i a n p l a n e .
S i n c e Re Q ( z , w ) = x . y ÷ ~ . O d e f i n e s a d e f i n i t e
made t o s a t i s f y
way.
Q(Uz,Uw)=Q(z,w) f o r
is
vectors f1,...~f,
this
a Lagrangian plane
X is
d i m e n s i o n n.
spanned by t h e d i f f e r e n t i a l s
from X r e s t r i c t e d
p c o r r e s p o n d i n9 t o 0,
to
p.
If
a manifold
The c o t a n 9 e n t space o v e r df
x=(xl,...,x.)
of
smooth r e a l
are coordinates at
t h e c o t a n g e n t space o v e r p i s
the
linear
p
79
span of
the c o o r d i n a t e
restricted plane. p.
It
to p. Here
The dual
~=(~,...,~.)
of the c o t a n g e n t
consists of
thou9ht of
differentials
linear
t h e g e o m e t r i c meaning o f can a l s o be t h o u g h t o f
plane
÷orms i n
as t h e span o f
are c o o r d i n a t e s is c a l l e d
in the c o t a n g e n t
the t a n g e n t
plane over
t h e c o o r d i n a t e s ~ and can a l s o be
the differentiations
~/~xk a t
the tangent plane at
as t h e space o f
p.
linear
x=O.
This
is
The c o t a n g e n t p l a n e
f o r m s on t h e t a n g e n t
plane. Combining a m a n i f o l d × and a l l manifold,
its
c o t a n g e n t p l a n e s we g e t a new
the c o t a n 9 e n t bundle T*(×)
of
(or over)
t h e base ×.
it
has c o o r d i n a t e s x and ~ and a p p e a r s as t h e p r o d u c t o f
of
R~ and a l i n e a r
the differential
space.
It
gets its
structure
f o r m w be i n v a r i a n t ,
i.e.
Locally,
an open s u b s e t
by t h e r e q u i r e m e n t t h a t
that
~.dx=g.dy when t h e c o o r d i n a t e s x,~ and Y,O o v e r l a p . canonical 1-form.
Since
it
dw=d~^dx = I is
also
invariant
smooth f .
When f=O,
In a c e r t a i n submanifolds of
t o w as t h e
differential
t h e c o t a n g e n t b u n d l e has a
everyone of
its
cotangent planes or
a s y m p l e c t i c space. t h e c o t a n g e n t b u n d e l has s u b m a n i f o l d s , f o r
sections,
submanifold of
its
refer
d~kAdxk
and t h a t
L i k e any m a n i f o l d , instance its
invariant,
which means t h a t
symplectic structure tan9ent planes is
is
We s h a l l
sunmanifolds given
this
de÷inition
is global
the cotangent bundle c a l l e d sense o p p o s i t e t o
locally
its
by ~ = f ( x ) f o r
and d e f i n e s X as a zero section.
the s e c t i o n s are the Lagrangian
the cotangent bundle.
They a r e s u b m a n i f o l d s o f
d i m e n s i o n n on which
t h e c a n o n i c a l f o r m ~.dx and hence a l s o
differential
It
vanish.
Lagrangian planes in
follows
that
their
its
tan9ent planes are
t h e t a n g e n t p l a n e s o~ t h e c o t a n 9 e n t b u n d l e .
Lagrangian manifold L is
said
some
t o be homogeneous o f
(x,t~)
is
in
A L when
80
(x,~)
is
and t>O.
Examples. Any s u b m a n i f o l d o f
the cotangent bundle l o c a l l y
of
the form
x=H~(~) where H i s manifold.
homogeneous o f In
fact,
homogeneous o f
its
degree I
is
dimension is
a homogeneous L a g r a n g i a n
n and ~ . d x = ~ H ~ d x = O s i n c e H~ i s
d e g r e e O. The word l o c a l
s t a y s c l o s e t o some x= and ~ c o n i c a l l y that
~ is
n o t z e r o and r e s t r i c t e d
refer
to
Note.
The z e r o s e c t i o n o f
this
situation
has t o be i n t e r p r e t e d c l o s e t o some ~ = ~ 0 i n
close to
the cotangent bundle is
homogeneous L a g r a n g i a n m a n i f o l d .
In
x
t h e sense
t o an open cone a r o u n d ~=.
as x,~ b e i n g c o n i c a l l y
so t h a t
We s h a l l
x=,~=.
obviously a
t h e s e q u e l we s h a l l
be i n t e r e s t e d
in Lagran9ian m a n i f o l d s o u t s i d e the zero s e c t i o n . Before proving that
Lemma
our
last
example i s
g e n e r a l we need a
L e t L be a homogeneous L a 9 r a n g i a n s u b m a n i f o l d of
suppose t h a t restricted
the d i f f e r e n t i a l s
d{
are l i n e a r l y
t o a tangent plane T(L)
of
L at
function
Proof.
H homogeneous o f
p.
gradient of ~.H~=O,
i.e.
Inserting
a function H3 i s
p with
a unique
degree i .
this H(~)
into
hj
such t h a t
d~^dx=O shows t h a t
and t h e c o n d i t i o n
homogeneous o f
that
degree I.
He s h a l l
o u r example i s
xj=hj
(h~,...,hn)
is
on the
~.dx=O shows t h a t
d e g r e e z e r o an t h e n H i s
chosen homogeneous o f
now see t h a t
p with coordinates
close to
By h y p o t h e s i s , t h e r e a r e smooth f u n c t i o n s
L close to
and
i n d e p e n d e n t when
a point
x = , { = . Then L has t h e f o r m x = H ~ ( { ) c o n i c a l l y
T*(X)
unique if
general o u t s i d e the zero s e c t i o n
o$ t h e c o t a n g e n t b u n d l e .
Lemma
L e t L be a homogeneous L a g r a n g i a n m a n i f o l d L o f
the cotangent
81
bundle T~(X).
Under any p o i n t
p of
c o o r d i n a t e s x on X and a f u n c t i o n c o r r e s p o n d i n 9 c o o r d i n a t e s ~ in x=H~(~) c o n i c a l l y
Proof. that at
close to
By an a f f i n e
p is
p is
linear
restrictions
of
degree 1 in
o v e r p such t h a t
L is
the
g i v e n by
c o o r d i n a t e s on X u n d e r p we can a c h i e v e
9=(I,0,...,0).
of
homogeneous o f
the f i b e r
d i m e n s i o n 2n,
to T(L)
H(~),
there are
p.
change o f
g i v e n by y=O,
L o u t s i d e the zero s e c t i o n
Since the t a n g e n t p l a n e of
we can f i n d
a k such t h a t
T*(×)
the
the d i f f e r e n t i a l s
dgz,..-,dok,dyk÷l,...,dyn are linearly
i n d e p e n d e n t . Now make a change o f
xz=yi+z,
x2=y2
variables,
etc
where z=(yk+i2+...+y~2)/2. The
rule
9j=~j Hence,
It
o.dy=~.dx
whenj l k , at
then
Oj:~j+~zyj
the
new
~ coordinates
when j > k .
p we have
d~j=doj
when j Ek,
follows
that
appeal to
gives
d~j=doj-dyj
d~,...,d~
are
when j > k . linearly
t h e p r e v i o u s lemma f i n i s h e s
4.4 Hamilton
flows
on
the c o t a n g e n t
i n d e p e n d e n t on T ( L ) .
Hence an
the p r o o f .
bundle.
Very
regular
phase
functions
Let H ( t , x , ~ )
be a smooth r e a l
on t h e p r o d u c t o f of
d i m e n s i o n n.
a real
then
is
degree I
and t h e c o t a n 9 e n t b u n d l e o f
in
~,
a manifold X
invariant,
i.e.
~t=-H~(t,x,~) the form of
coordinate transformations this
homo9eneous o f
The c o r r e s p o n d i n 9 H a m i l t o n s y s t e m ,
xt=H~ ( t , x , ~ ) , is
line
function,
immediate.
t h e e q u a t i o n does n o t change under
l e a v i n 9 ~.dx
invariant.
The v e r i f i c a t i o n
Hence t h e c o r r e s p o n d i n 9 H a m i l t o n f l o w
o÷
82
(x,~i(s)->(x,~)(t) i s 9 1 o b a l l y d e f i n e d on T * ( X ) .
Since H i s
t h e f l o w commutes w i t h maps ~->const
homogeneous o f
Finally,
let
remains a t
form
~.dx i s
us n o t e t h a t an o r b i t
that
point.
all
By a p r e v i o u s r e s u l t ,
from a p o i n t where ~ v a n i s h e s
t h e c o t a n 9 e n t b u n d l e remains
time. ~.dx i s
i n v a r i a n t under the H a m i l t o n f l o w ,
maps a L a g r a n 9 i a n m a n i f o l d L g i v e n f o r
La9rangian m a n i f o l d s L ( t ) .
We s h a l l
t=O i n t o a f a m i l y of
look i n t o
the d e t a i l s
Very r e g u l a r phase f u n c t i o n s a l o n g a b i c h a r a c t e r i s t i c In the c o n s t r u c t i o n of
global parametrices for
p s e u d o d i f f e e n t i a l e q u a t i o n s in definition
the
Hence t h e H a m i l t o n f l o w from a L a 9 r a n 9 i a n
Since the c a n o n i c a l form it
~,
i n v a r i a n t under the H a m i l t o n f l o w .
m a n i f o l d o u t s i d e the z e r o s e c t i o n o f outside for
in
~ and hence p r e s e r v e s t h e
homogeneity s t r u c t u r e of c o t a n 9 e n t p l a n e s . canonical differential
degree I
the n e x t c h a p t e r ,
of
this
map.
tube
hyperbolic the f o l l o w i n 9
is crucial.
Definition
Canonical c o o r d i n a t e s (x,~)
s a i d t o be r e g u l a r f o r
near a p o i n t p in T*(X)
are
a homogeneous L a g r a n g i a n m a n i f o l d L t h r o u g h p i f
p has a neighborhood where t h e v a r i a b l e s
~ can be taken as p a r a m e t e r s
on L. Arbitrary locally of
for
L by a change of v a r i a b l e s .
section 4.2.
Definition at
canonical coordinates (x,~)
p if
p can be made r e 9 u l a r
T h i s f o l l o w s from the Theorem
We can now d e f i n e v e r y r e g u l a r phase f u n c t i o n s .
A phase f u n c t i o n 9 ( x , ~ )
i s s a i d t o be v e r y r e g u l a r f o r
t h e e q u a t i o n s 9~=0 d e f i n e x as a ~ u n c t i o n of
such a way t h a t 9~=O=>dv=~.dx to
at
and
(x,~=9~),
~ c l o s e t o p in
9~=0 p a r a m e t r i c e s L c l o s e
p.
E×ampleT A phase f u n c t i o n ×.~-H(~)
L
w i t h H homogeneous o f
degree I
is
83
very
regular
every
for
the k~omoeneous
homogeneous
Lagrangian
Lagrangian
manifold
manifold
has a very
x=H'(~).
re9ular
Hence
phase
function
locally. Note. The n o t i o n of changes of In f a c t , ~/~'
v e r y r e g u l a r phase f u n c t i o n
is
i n v a r i a n t under
t h e x v a r i a b l e s and c o n c o m i t a n t changes of
if
×'
a r e the new v a r i a b l e s ,
x=x(x')
and ~ ' . d x ' = ~ . d x and
i s not d e g e n e r a t e , then 93 and 9 ' ~ . = g ~ / ~ '
t i m e and when t h e y v a n i s h , g i v e s some p l e n t y o f
vanish at
then d g = ~ . d x = ~ ' . d x ' = d 9' .
examples o f
the ~ v a r i a b l e s .
t h e same
This o b s e r v a t i o n
v e r y r e g u l a r phase f u n c t i o n s .
Very r e g u l a r phase f u n c t i o n s under the H a m i l t o n f l o w L e t B be a nul b i c h a r a c t e r i s t i c flow starting
for
from a p o i n t q f o r
canonical coordinates for
L(O)
H,
i.e.
the t r a c e of
t=O t o a p o i n t p f o r
t=s.
the H a m i l t o n Let
bicharacteristics
p and t h e o u t f l o w o f
(y,9) 9(Y,O)
It
around B. A L a g r a n g i a n m a n i f o l d L(O)
is possible to
q to a
N(O) i s a t u b e o f
c o n t a i n i n g q i s mapped t o a L a g r a n 9 i a n m a n i f o l d L ( s )
Theorem
be
around q=(Y=,O=) and x , ~ c o o r d i n a t e s
around p = ( x = , ~ = ) . The f l o w maps a c o n i c a l neighborhood N(O) of c o n i c a l n e i g h b o r h o o d N(s> 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 t
t>O and t
all
t.
Proof.
We n o t e f i r s t
that
Qu(t,x)
since
= ( 2 n ) - n 5 exp i x . ~
Q o p e r a t e s f r o m tempered d i s t r i b u t i o n s compact s u p p o r t s . More p r e c i s e l y , pseudodifferential more 9 e n e r a l l y ,
operators,
functions u(t)
inte9rable or essentially functions to
with
v a l u e s in
in
u~(~)d~,
x to distributions
by t h e c o n t i n u i t y
with
properties
Q maps H" c o n t i n u o u s l y i n t o H" - I which a r e d i f f e r e n t i a b l e ,
bounded w i t h H= - I .
Q(x,t,~)
v a l u e s i n H= t o
S i n c e H-=
is
the dual o f
of and,
continuous, t h e same Rind o f H= w i t h
respect
the duality (u,v)
the adjoint
Q* o f
first
differentiable
in
the proof
is
the ener9y i n e q u a l i t y
under the assumption t h a t with
v a l u e s i n H=÷~
A(D) so t h a t
S u(x)~(x)dx,
Q has t h e same p r o p e r t i e s .
The main i n g r e d i e n t prove i t
=
u(t)
Put
= l+(D~m+...+D~m) ~ =
is
(2).
continuously
We
87
UuH== = where t h e
right
side
is
(Amu,A=u)
the usual
L= norm s q u a r e o v e r R - .
With
u as
above we t h e n have (d/dt)A=u(t)
=-iA=Qu(t)
+iA=v(t),
v=Pu.
Hence (dldt)
(u(t),u(t)m
+ 2 Im (u,v)~.
=-i(Bu(t),u(t))
where B = Am~Q - Q ~ A m~.
S i n c e Q and Q* have t h e same p r i n c i p a l this
the order of
B is
2s and
proves that (d/dt)Su(t)n,
with
part,
c bounded f o r
a passage t o inte9rable proof,
the
t
and s bounded.
limit
function
~ c #u(t)~m
it
with
also follows values
the energy i n e q u a l i t y
To p r o v e e x i s t e n c e ,
Hence
in
+ Mv(t)U (I)
in
this
when t h e d e r i v a t i v e
H~.
We n o t e t h a t ,
holds also for
we s h a l l
follows
see t h a t
the
the
of
case. u is
by v i r t u e
adjoint
image o f
of
of
By an
its
P.
t h e map
u -> 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 O
the nul
bicharacteristic
the
of P i s s u i n g from
(O, g o ) .
Proof.
By t h e p r e v i o u s theorem~ the d e s i r e d r e s u l t
And i t
is clear that
if
N is
shrunk,
holds f o r
we can hope f o r
some s>O.
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 and Re z >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 O
it
suffices
o t h e r w i s e . Assume f i r s t difference
small
the difference
analytic
P r o o f . Since the
which v a n i s h e s f o r
that
above e q u a l s t h e
$olexp-rz r-=-Ids
s is
is
an e n t i r e to
function
take f(r)=l
n o t an i n t e 9 e r
integral
of
z when f
when r > l
p=O,l,2, ....
has
and z e r o Then t h e
106
continued a n a l y t i c a l l y expansion of
in f a c t
case. N e x t ,
T h i s can be done by an
t h e e x p o n e n t i a l . The a n a l y t i c a l I
and is
in s from Re s <0.
(-z)klk!(k-s)
entire
let
analytic
c o n t i n u a t i o n equals
, k=O,l,2,...
i n z.
s=p be an i n t e g e r ~0.
Hence t h e lemma i s proved i n It
suffices
this
t o prove t h a t
S~* e x p - r z r - p - ~ d r + ( - z ) P l o 9 z / p ! has an a n a l y t i c v a r i a b l e s in
e x t e n s i o n t o z=O. When z>O i s
the
integral
in
this
real,
a change o f
f o r m u l a shows t h a t
it
equals
zp S=* e x p - r r - P - ~ d r and the e q u a l i t y last
integral
h o l d s by a n a l y t i c a l
differs
zpf=~
exp-r
z¢O. Now our
r-P-ldr
and d o i n g the I
all
from
by a c o n s t a n t t i m e s z " . integral
continuation for
Expanding the e x p o n e n t i a l in t h i s
last
i n t e g r a t i o n s produces the c o n v e r g e n t sum , k=O,l,...
(-l)k(l-zk-~)/(k-p)k!
,
where t h e term w i t h k=p has t o be i n t e r p r e t e d as ( - l ) P ( l - l o 9 whose s i n g u l a r i t y
is
p r e c i s e l y t h a t of H ( p , z ) .
We can now compute the
Theorem
The
i Cm where
for
f
So*
integral
exp~ixr
is a s m o o t h
large r,
integral
(I), r -°-~
analytic
We may assume t h a t
integral
( i e ) ' = i c-.
close
to
the
ori9in
and
constant
equals
modulo an e n t i r e
the
noting that
lemma.
dr
vanishing
H(s,x+i¢O)
Proof.
This proves the
i.e. f(r)
function
(I)
z)/p!.
f(~).
f u n c t i o n of f(~)
x.
e q u a l s 1. When Re icx
i.e.
Im ~x>O,
converges and, by the p r e v i o u s lemma e q u a l s
(i~) m H ( s , - i c x ) = ( i ¢ ) - ( - i ~ x ) m F ( - s ) = H ( s , x )
modulo an e n t i r e this
f u n c t i o n of
equals H(s,x)
analytic
so t h a t
continuation.
x.
When s i s
not an i n t e g e r p = O , l , 2 , . . . ,
t h e theorem i s proved i n t h a t case by
When s=p t h e e x p r e s s i o n above e q u a l s H ( p , x )
107
modulo a p o l y n o m i a l . T h i s p r o v e s the theorem.
The d i s t r i b u t i o n s It
is
H(s,×+i(O)
and t h e i r
sums and d i f f e r e n c e s
i m p o r t a n t t o have a c l e a r view o f
distributions
the b e h a v i o r of
the
H ( s , x + i ¢ O ) . They a r e boundary v a l u e s of f u n c t i o n s o f a
complex v a r i a b l e z which a r e a n a l y t i c the n e g a t i v e r e a l
axis.
in t h e complex p l a n e c u t a l o n 9
Their s i n g u l a r support
is
the o r i g i n
and t h e y
a r e equal when x > O . Since (iz)"
=
when Im z
real
r-'-~dr,
$o ® e x p - i z r
o v e r z e r o i n t h e wave f r o n t
a x i s times
¢. Since
H(O,z)
s e t of
H(s,x+i~O)
is
= l o g 1 / z = Io9 I / I z l
we have H(O,x+i¢O) = Io9 I / t x I - e i ~ h ( - x )
where h i s H e a v i s i d e ' s f u n c t i o n , the p o s i t i v e
axis.
Hence,
i.e.
i n view o f
H ( - 1 , x + i ¢ O ) = Pv I / x For
inte9ral
s,
the d i s t r i b u t i o n s
d e r i v a t i v e s and i n t e g r a l s o f H(1/2,x+i~O) = - 2 ~ I , ( and f o r
of
this
f u n c t i o n of
(3), + i~¢h(-x).
H(s,x)
are j u s t
the s u c c e s s i v e
H ( O , x + i ¢ O ) . For s = I / 2 , x~,=h(x)
general h a l f - i n t e g r a l
differentiation
the c h a r a c t e r i s t i c
+ i ¢ ( - x ) ~,~ h ( - x ) )
s t h e y a r e o b t a i n e d by i n t e g r a t i o n
and
distribution.
In c o n n e c t i o n w i t h p a i r e d o s c i l l a t o r y
integrals
we use t h e
following definition.
Definition
The d i s t r i b u t i o n s
H¢(s,x),
c=l o r
-I,
are d e f i n e d as
H(s,x+iO)+~H(s,x-iO). When ¢=-1,
these d i s t r i b u t i o n s
s h a r p and d i f f u s e , ( i . e , the o r i g i n
~=I
v a n i s h on t h e p o s i t i v e
axis.
They a r e
n o n - s h a r p ) , from n e g a t i v e and p o s i t i v e s i d e o f
a c c o r d i n g t o the f o l l o w i n g
table
s inte9er
s half-integer
d-d
s-d
i
108
~=-I
S-S
d-s
7.2 Polar coordinates in paired o s c i l l a t o r y
C o n s i d e r dual o s c i l l a t o r y IC(f,a,x) where
=
e(q)=iq
introduce
c=c(8),
(2~}-N/=
and
coordinates
homogeneous of
given at
S ~ expif(x,e)
, q=j-N/2
polar
integrals
degree 1.
t h e end o f
c h a p t e r 6,
e(eq)a-q-N(x,e)de
j=p,p+1,.., with
integrals
for
respect
some
to
Then,
integer
a positive
p.
We
smooth
shall function
r e p l a c i n g 8 by r8 where c ( 8 ) = I ,
t h e p r e c e d i n g theorem shows t h a t each term in
t h e sum above has the
form S H ( q , f ( x , 8 ) + i ¢ 0 ) a - ~ - N ( X , e ) ~(e) if
we d i s r e g a r d the powers o f
~(8)
In f a c t ,
=
the r a d i a l
+ a smooth f u n c t i o n
~ i n v o l v e d . Here
81d82...dSN
- 8~d81...
>0 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
x,t
let
-> 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<s
~ H(q,f(x,e)+iO)a(x,e)de
N o t e . The c h a i n c ( x )
v)
chain
o r i e n t e d by dtds>O 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
<0 on b÷(x) b÷((x)
down t o M, we s e t t h e f o l l o w i n g f i g u r e s
b÷(x) H>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 <0.
On f ( x ) ,
the
9 r a d i e n t f e does n o t v a n i s h and d e f i n e s n o r m a l s t o f ( x ) . Now l e t
f(x,e)
and a ( x , e )
and a t o a s t r i p f(x,e)
M around Q in
i n M. As b e f o r e ,
the properties differential
(7.4.5)
(7.5.2) still
The f o r m u l a
e of
x,t->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
