I
Elementary
Introduction to the
Theory of
Pseudodifferential
Operators
STUDIES IN.tDtiANCED MATHEMATICS
Studies in Advanced Mathematics
Elementary Introduction to the Theory of
Pseudodifferential Operators
Studies in Advanced Mathematics
Series Editor
Steven G. Krantz
Washington University in St. Louis
Editorial Board
R. Michael Beals
Gerald B. Folland
Rutgers University
University of Washington
Dennis de Turck
William Helton
University of Pennsylvania
University of California at San Diego
Ronald DeVore
Norberto Salinas
University of South Carolina
University of Kansas
L. Craig Evans
Michael E. Taylor
University of California at Berkeley
University of North Carolina
Volumes in the Series
Real Analysis and Foundations, Steven G. Krantz
CR Manifolds and the Tangential Cauchy-Riemann Complex, Albert Boggess
Elementary Introduction to the Theory of Pseudodifferential Operators,
Xavier Saint Raymond
Fast Fourier Transforms, James S. Walker
Measure Theory and Fine Properties of Functions, L. Craig Evans and
Ronald Gariepy
XAVIER SAINT RAYMOND
Universite de Paris-Sud, Departemettt de Mathematiques
Elementary Introduction to the Theory
of Pseudodifferential Operators
CRC PRESS
Boca Raton Ann Arbor Boston London
Library of Congress Cataloging-in-Publication Data
Saint Raymond, Xavier.
Elementary introduction to the theory of pseudodifferential
operators / Xavier Saint Raymond.
cm.
p.
Includes bibliographical references (p. ) and indexes.
ISBN 0-8493-7158-9
1. Pseudodifferential operators.
I. Title.
QA329.7.S25 1991
515'.7242-1c20
91-25184
CIP
This book represents information obtained from authentic and highly regarded sources.
Reprinted material is quoted with permission, and sources are indicated. A wide variety
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validity of all materials or for the consequences of their use.
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Direct all inquiries to CRC Press, Inc., 2000 Corporate Blvd., N.W., Boca Raton, Florida,
33431.
© 1991 by CRC Press, Inc.
International Standard Book Number 0-8493-7158-9
Printed in the United States of America 1 2 3 4 5 6 7 8 9 0
Contents
Preface
vii
1
Fourier Transformation and Sobolev Spaces
I
1.1
1.2
Introduction
Functions in IR"
Fourier transformation and distributions in R"
9
1.3
Sobolev spaces
17
Exercises
23
Notes on Chapter I
27
Pseudodifferential Symbols
28
Introduction to Chapters 2 and 3
Definition and approximation of symbols
Oscillatory integrals
Operations on symbols
Exercises
28
Pseudodifferential Operators
Action in S and S'
47
Action in Sobolev spaces
Invariance under a change of variables
Exercises
Notes on Chapters 2 and 3
52
2
2.1
2.2
2.3
3
3.1
3.2
3.3
2
29
32
37
43
47
58
61
67
V
Vi
4
4.1
4.2
4.3
Applications
69
Introduction
Local solvability of linear differential operators
Wave front sets of solutions of partial
differential equations
The Cauchy problem for the wave equation
Exercises
Notes on Chapter 4
69
70
Bibliography
97
76
83
89
94
Index of Notation
103
Index
107
Preface
These notes correspond to about one-third of a one-year graduate course entitled
"Introduction to Linear Partial Differential Equations," taught at Purdue University during Fall 1989 and Spring 1990. It is an attempt to present in a very
elementary setting the main properties of basic pseudodifferential operators.
It is the author's conviction that the development of this theory has reached
such a state that the basic results can be considered as a complete whole and
should be mastered by all mathematicians, especially those involved in analysis.
Unfortunately, the beginning student is immediately faced with a technical difficulty that forms the heart of the theory, namely the extensive use of oscillatory
integrals, that is, non-absolutely convergent integrals over ll2". Indeed, all the
texts written on these pseudodifferential operators assume explicitly, and even
more often implicitly, a good familiarity with such integrals, the theory of which
is based on the rather difficult results known as stationary phase formulas, and
the authors perform changes of variables, integrations by parts, or interversions
of the f exactly as if the integrals were absolutely convergent while the allowed
rules are probably not quite clear for the uninitiated reader.
The main originality of these notes, maybe the only one, is to restrict the
use of such oscillatory integrals to the case of real quadratic phases for which
the theory is both simple and pleasant. Of course, this restriction prevents a
full proof of the fundamental result of invariance of pseudodifferential operators
under a change of variables. Many other important aspects of the theory are not
even mentioned in this course: properties of distribution kernels of the operators;
precise description of their local action (properly supported operators); definition
of wider classes of symbols and operators such as in Coifman and Meyer [6),
Hbrmander [8), or more recently Bony and Lerner [4]. But the goal of the
following pages will be reached if this simple setting and the few applications
given in the last chapter convince the reader of the fundamental importance of
the topic and give sufficient motivations for reading more complete texts.
The exposition begins with a chapter devoted to the Fourier transformation and
Sobolev spaces in R", which both play a central role in the theory. A sufficient
knowledge in classic integration theory (properties of Lebesgue measure and
related LP spaces in R) is assumed, and Chapter 1 will provide all the additional
vii
Preface
viii
background needed to take up the next chapters. For the more advanced reader
who has encountered these topics before, a quick reading is recommended to get
adjusted to the notation used throughout the book. Chapters 2 and 3, respectively
devoted to basic symbols and basic operators, form the theory itself. Chapter 4
provides applications to local solvability of linear partial differential equations
and to the study of singularities of solutions of such equations .
To avoid any ambiguity, it is emphasized that nothing is original in the topics
presented here: the text has been based mainly on Hormander [8, Section 18.11
and to some extent on Alinhac and G6rard [ 1, Chap. I I (in particular, the origin
of the use of oscillatory integrals as given in Chapter 2 and the origin of several
exercises can be found in this latter reference). Thus, the specific features of this
text lie only in the exposition: it is self-contained with very light prerequisites
and all the complements that were not strictly necessary to reach the main results
have been avoided, so that it should be considered merely as a first introduction
to the topic.
It is my pleasure to thank the Department of Mathematics of Purdue University
for the opportunity I had to teach this course. I also wish to thank Mrs. Judy
Mitchell, who with great competence and patience typed the manuscript of this
course.
- X. Saint Raymond
West Lafayette, March 1991
1
Fourier Transformation and Sobolev Spaces
Introduction
The main purpose of Chapter 1 is to fix the notation used throughout this course;
most of the notation is classic, but some is probably unusual, e.g., the notation
P for the space of C' functions with polynomial growths at infinity. This is
why a quick reading is recommended, even if the student is already aware of
the topics presented here.
The central notion is that of Fourier transformation: for each function a defined on R' (with a controlled growth at infinity), one can define its Fourier
transform u., also defined on III, with the following properties: (i) differentiations on u correspond to multiplication by polynomials on u (which is a simpler
operation, particularly with respect to the inversion of such an operation); (ii) one
can recover u from u essentially by achieving the same transformation a second
time; (iii) the Fourier transform of an L2 function is an L2 function. Thus, in
order to study the properties of this transformation, it is more convenient to work
in spaces that are closed under operations of differentiation and multiplication
by polynomials, and this leads to the introduction of the Schwartz space S and
of the larger space of temperate distributions S', which contains L2.
Since there is this correspondence between differentiation of u and multi-
plication of u by a polynomial, there is also a correspondence between the
smoothness of u and the growth of u at infinity (and by symmetry between the
growth of u at infinity and the smoothness of u). This fact is used to define
the so-called Sobolev spaces, which are much more convenient than the classic
classes Ck of k-times continuously differentiable functions, especially when one
deals with L2 estimates.
I
Fourier Transformation and Sobolev Spaces
2
1.1
Functions in !R"
Throughout this course, we are going to study properties of complex-valued
functions of n independent real variables and their various derivatives. Therefore, we need to develop convenient notation for these variables, functions, and
derivatives.
The variables will be denoted by x1, ... , x", or in short by x. A function u
of these variables can thus be considered as defined on (a domain of) R", and
we will write u(x) and x E W. For any multiindex a = (a 1, ... , a") E Z"+,
+ an and its factorial as the
we define its length as the sum j al = al +
E Z+ if one has
product a! = (a1!)...(a"!). Moreover, we will write a
aj <3, forallj=I,...,n.
These multiindices a are used to write polynomials: for x E 1t" and a E Z.
one defines x' as the product x' = xi' ...xn'. Similarly, if a; denotes the
operation of taking the partial derivative with respect to x3 (i.e., a; = 9/ax;),
one will write
t ip, = V 1
1
... aQn
"
-
alai
axal
I
... axon
"
Recall that a function u(x) is said to be of class Ck if it has continuous derivatives up to order k, and these derivatives do not depend on the order used to
achieve the differentiation (i.e., 8;ak = aka; when acting on C2 functions).
These derivatives are therefore denoted by an u without possible confusion.
To show how this notation can be conveniently used, we prove the following
classic result known as Taylor's formula.
THEOREM 1.1
TAYLOR'S FORMULA
Let u be a Ck function defined on R; then for any x and y E R" one has
u(x + y)
Ial<k
a.
aau(x) +
k
IaI=k
-
a I Jp
I
(1 _ t)k- Iaau(x + ty) dt.
I}={aEZ'+;IaI= 1131-1 and
PROOF Forall0EZ+, let us write
a < Q}; thus one has
a
d
dt
a!aau(x+ty)
(Qk_1
I$t=k
cEtl3-I}
I$I=k
7=1
a.
ypapu(x + ty)
yQ81 u(x + ty)
A
ky O'v(x+ty).
IAI=k
QI
Functions in R'
3
Then, the C' function in t: vk(x, y, t) _ E Q!<k((l - t)!QI y°/a!)8°u(x +
ty) satisfies vk(x, y, 1) = u(x + y), vk(x, y,0) _ EI I<k(y"/a!)8°u(x) and
(8vk/8t)(x, y, t) = Ej.j=k k(y°/a!)(1 - t)k-'& u(x + ty) so that the result
simply follows from the fundamental theorem of calculus
Vk (X, y, 1) = vk (x, y, 0) +
f0 '
J (x, y, t) dt.
We will also need the so-called binomial coefficients, defined as follows: for
a,b E Z+,
a
a!
b
b ((a-b)!
if0<b<a,
a
C
b
otherwise. Then for a, O E Z+, one defines the binomial coefficients as the
products
It is easy to check that
ifQ<aand
otherwise. The fundamental relation
a
b- I)
+
a
( b)
_
+ (a - b + 1) (a!)
b(al)
b!(a-b+1)!
b!(a-b+1)!
is the key to the following result.
THEOREM 1.2 BINOMIAL AND LEIBNIZ'S FORMULAS
Let a E Z+ ; then
(i)
(Binomial formula). For any x and y E R1,
(x + y)" _
t
a
a f xPy°-a.
Q
a+
b
4
Fourier Transformation and Sobolev Spaces
(ii) (Leibniz's formula). For any C1°1 functions u and v,
e (UV) = E (13) (O
u)(e-Av)
PROOF By induction on a. The two properties are obvious for a = 0. Then,
assuming they hold for a, we prove they hold for a + b,, where b; is the
multiindex of length 1 with jth component 1, as follows:
(xp+6;y°-A+xQy°-A+6;
(x3 +yi)(x+y)° _
Q
(13)
((y
())+-
bi) +
(a+63) xly y°+6, -7
=
=
7
y
and similarly
19je(uv) = a;
(p)
(a
(a b ) - (y a bi ) + (y )
according to the fundamental relation given above.
It is a classic remark that when one takes x = y = (1,1, ... ,1) in formula (i),
one gets
Although all the notions we introduce below are actually invariant under the
most general linear transformations on the variables x, it is convenient to use the
euclidean structure of R" when the system of coordinates is fixed, in particular
to measure the growth of functions at infinity. Thus, for any vectors x and
F E R", we will consider the scalar product (x, 0 = X1 l;1 +
+ x"E and
Functions in R"
S
the euclidean noun IxI = (x, x)'/2. We will denote by Br the closed ball with
radius r, i.e., Br = {x E Rn; lxi < r}.
To measure growths at infinity, the norm lx1 is convenient because it satisfies
the triangular inequality, while its square Ix12 is convenient because it is smooth
(even at 0). Therefore, we will use powers of (1 + IxI) as well as powers of
(1 + IxI2); the following elementary lemma is then useful.
LEMMA 1.3
The integrals over Rn
+
IxI)-8 dx
and
/ (t + IxI2)-°/2 dx
are convergent if and only if s > n; in particular, we have the precise estimate
f(1+IxI2)-ndx<7rn.
PROOF One can write for s > 0 and x E Rn
(I +
Ixl)-8
= (1 + 2Ixl + 1x12)-8/2 < (I + 1x12)-8/2
+x2)-s/2n...(I +x2,)-s/2n
< (1 +x2i)-s/2n(1
and the result follows from the classic one-dimensional case. In particular, for
s = 2n this gives the precise estimate since f c(1 +x2)-l dxt = 2r. We will
not use the "only if" part of this lemma, which can be proved by writing the
integrals in polar coordinates.
I
The notation being thus fixed, we now introduce the Schwartz space S of
C°° functions that are rapidly decreasing at infinity. More precisely, the C°°
function cp belongs to S if the functions xa89cp(x) are bounded on Rn for all
pairs a, 3 of multiindices. If we denote by lcplo the supremum over R' of a
bounded continuous function cp, the implicit topology of S is that defined by
the norms'
I'P1k =
sup
lx0B1cplo
= sup{Ix°8Qcp(x)I; x E Rn and Ia + i31 <_ k}
to+aI <k
where k E Z.. Obviously, S is closed under the operations of differentiation
and multiplication by polynomials. As a matter of fact, it is even closed under
multiplication by C°° functions with polynomial growths at infinity: if a continIx12)N
uous function 1 satisfies an estimate ltli(x)1 < C(1 +
for some constants
C and N, we will write r/i E P°, and if rp is a C°° function such that Oat' E P°
'Equipped with these norms, S is then a Frbchet space. Throughout the text, we will use
some words from functional analysis (such as Frdchet, Banach, or Hilbert spaces) but we will not
use any result from this theory. The purpose is just to give additional information to the more
advanced student. Similarly, we will use the words continuous and continuity (e.g.. in the statement
of Lemma 1.4) only as synonyms of an inequality between norms.
Fourier Transformation and Sobolev Spaces
6
for all a E Z we will write -0 E P (space of CO0 functions with polynomial
growths at infinity). The following continuity properties will be important.
LEMMA 1.4
One has
(i)
(ii)
and cp E S, & E S
(Continuity of differentiation). For any a E
with
k Ik+I°I for all k E Z.
(Continuity of multiplication by a V) E P). For any V, E P there exist
two sequences Ck and Nk such that cp E S * z E S with I?GcpIk <
for all k E Z+; in particular. if O(x) = x° one has Ix°wIk <
2k(a!)1V1k+I°I
PROOF Property (i) is clear, property (ii) follows from Leibniz's formula (cf.
Theorem 1.2), which can be written here
and thus implies the estimate if we take Nk then Ck such that
I8' (x)I :5 2k(n -Fkl)Nk (I +
IxI2)Nk
for I'yl :5k.
Similarly, one gets the more precise estimate for Ix°Vlk by substituting 031(x°) _
(a!/(a - y)!)x°-'r in Leibniz's formula, then using the estimate
any)!
<O \
! (a
the proof of which is left to the reader.
< 21QI(&)
I
The student will benefit by determining himself which of the following funcE Rn, cos IxI. Wee are
tions are in S or P : e-'Ixl for a E C, e'(x,t) for a
now going to define an important subspace of S, the space of C°` functions
with compact support, also called test functions.
Recall that the support of a continuous function p can be defined as follows:
x V supp cp if and only if cp = 0 in a neighborhood of x. The support of 0 is
thus always a closed set, and it is compact if and only if it is bounded. Then
the space Co of test functions is defined as the space of C" functions with
compact support. If cp E Co and Sl is an open set containing supp gyp, we will
write more precisely p E Co (fi) (thus, Co = Co (R")). It is clear that test
functions are automatically in S, but it is less obvious that Co : 0. The classic
example of a test function is that of cp(x) = f(Ixl2 - 1) where f(t) = ell' if
t < 0, f (t) = 0 if t > 0. Indeed, this function is in CO° since this is true for
f (classic exercise), and its support is B, = {x E Rn; IxI < 11. Moreover, if
Functions in R"
7
we divide this function by the constant f cp(x) dx > 0, we get a new function
p satisfying
cpECo ,
J'P(x)dx=1 ,
c p>0,
suppVCB1,
and a function with these properties will be called a unit test function.
These unit test functions can be used to construct partitions of unity, as in
the following result which will be used to reduce proofs of global properties to
local proofs.
LEMMA 1.5 PARTITIONS OF UNITY
Let K C R be a compact set contained in a union of open sets 52;. Then there
exist a finite number of functions cp,j E Co (52,)(I < j < k) such that cps > 0,
Ej=1
7 < 1 and X:j=1 pj = I in a neighborhood of K.
(i) First assume that K C 521. Then for e > 0, let us denote by KE the
set of points at distance < e from K, and set 7E(x) = E'"cp(x/E) where cp is a
unit test function as above. (Thus, cp, satisfies the same properties as cp but the
last one to be replaced with supp TE C Be.) We choose e = one-fourth of the
distance from K to the complement of 521, then we set
PROOF
*(x) =
'K2,
'E( x - y) dy
which is a CO° function (take derivatives under f) satisfying ?U = I on KE and
supp C K3E C 521 as required.
(ii) In the general case, the compact K is actually contained in a finite union
521 U ... U 52k of open sets 52,, and K = UkI K; for some K3 C 1l that are
compact. For each j < k, let j E Co (52 j, satisfying iP. = I near K. as in
part (i) of this proof, then let
'P1 = ,L1,
'P2 ='02(1 -0l), ..., 'Pk =V)k(I -'01)...(I -'bk-I)
These functions solve our problem because they satisfy cpj E Co A)- cpj > 0,
and
k
F, cpj = 1 - (I -'+Gl)(1 - 02)...(I -'0k)
I
j=1
We finally end this section by pointing out that the main motivation for the
introduction of the Schwartz space S lies in the fact that when dealing with
integrals of such functions, all the difficult operations of integration theory (integration by parts, differentiation under f or interversion of f) will be obviously
valid thanks to the good decreasing of Schwartz functions at infinity. We re-
mind that for I < p < oo, the Lebesgue space LP is defined as the space of
8
Fourier Transformation and Sobolev Spaces
measurable functions2 u on R!' satisfying NormLP(u) < oc where
NormLP (u) =
(Jlu(xvdx)
I /p
if p < 00,
NormL-(u) = inf{U E 1R lu(x)I < U almost everywhere}.
For p = 2 and p = oo we will use the simpler notation
llullo = NormL2(u)
and
lulo = NorrLs(u)
(note that lulo corresponds to the previous definition when u is continuous).
These spaces are Banach spaces; whenever u and v are two measurable functions
such that uv E L', we will use the notation
(u,v) = fu(x)i(x)dx.
This product is linear in u and semi-linear in v (i.e. one has the relation (u,
v +µw) = (u, v) + p(u, w)), and since llullo = (u, u) for u E L2, (u, v) is a
scalar product that defines a Hilbert space structure on L2.
The following statement gives the properties of the Schwartz space S that
can be obtained directly from integration theory.
THEOREM 1.6
One has s C nt<p<ooL' with NormLP(cp) < (2ir)"I02" for any p E S and
I < p < oo. Moreover,
(i)
For any 1 < p < oo, u E LP, and ep E S. one has ueo E L' and
l(u,co)l S (22r)"NormLP(u)IWI2n
(ii)
For any measurable u such that up E L' for all ep E S,
(u, <p) = 0 for all W E S
u = 0 a.e.
() If cp H U(cp) is a semilinear form on S satisfying JU(p)I < Cllpllo, then
there exists a unique u E L2 such that U(cp) = (u, cp) for ep E S, and one
has IuIIo <C.
2lndeed, we will always consider two functions as equal if they are equal almost everywhere:
actually, it would be more correct here to speak of classes of equivalent measurable functions, where
u and v are equivalent if and only if u = v almost everywhere.
Fourier transformation and distributions in R"
9
For p = oo and p E S, it is clear that NormL- (<p) = Io
any 1 < p < oo and cp E S, one can write
PROOF
oc. For
n
kP(x)1 n <_
sup O(x)11(1 + xi'
xER"
J=1
fl (1 + X.11) -I
?=1
n
r.
<(2"IV12n)p JJ(1 + x
j=1
Thus by integration (cf. Lemma 1.3) we get NormLP (gyp) < irn/p2n IWI2n. Prop-
erty (i) then follows by using Holder's inequality I(u,cp)I < NormLP(u)
NormLQ (gyp) where q = p/(p - 1) is the conjugate exponent of p.
If a measurable u satisfies (u, gyp) = 0 for all cp E S, we will prove that one
has (u, XE) = 0 for any bounded measurable set E with characteristic function
XE, for this classically implies that u = 0 almost everywhere. First, if E is an
open set, the sets Kj C E of points at distance larger than or equal to 1 I j from
the complement of E are compact, and the sequence W. E Co (E) of functions
satisfying cpj = 1 on Kj as in Lemma 1.5 satisfies also
V. = XE pointwise, so that (u, XE) = limt , (u, V j) = 0 by dominated convergence. Now, a
general bounded measurable E is, up to a negligible error, the limit of a sequence
of open sets with characteristic functions Xj. Therefore, XE = limy-"" Xj al-
most everywhere, then (u, XE) = limj.. (u, X j) = 0 again by dominated
convergence. Thus, we get (ii).
Finally, given a semi-linear form U as in (iii), the existence of a u E L2 such
that U(cp) = (u, cp) and Ilullo < C follows from Riesz's representation theorem,
while the uniqueness comes from property (ii). I
1.2
Fourier transformation and distributions in R"
For any u E L', the Fourier transform u of u, defined by the formula
u(0 = Jc_i()u(x)dx,
is a bounded continuous function since obviously
NormL, (u) for all i; E
R", while the continuity follows from dominated convergence. The purpose of
this section is to extend this transformation to a large class of objects (containing,
in particular, L2 functions) called temperate distributions and to establish its
basic properties. Let us begin with an example.
Fourier Transformation and Sobokv Spaces
10
Example 1.7
The Fourier transform of cp(x) = e-IS12/2
(21r)f12e-IEI2/2.
0
PROOF By definition, one can write
e-'(s,f)e-1x12/z dx
(J e-ixifie-xi/Z dx
...
/
/J
1
1 e-ix,.f
dx
1\J
R
so that it is sufficient to prove the result when the dimension is n = 1. Thus,
assuming n = 1, we have for any fixed E R
e-f2/2 r
00
f
e-(x+to2/2 dx.
Now, the integrand is a holomorphic function of x + k E C and we can use
Cauchy's integral formula with the path
a
-A
A
0
to get
e-(x+if)2/2 dx -
1 AA
jA
a-x2/2
dx <
A
which yields Ell e-(x+`f)2/2 dx = f . e-x2/2 dx by taking the limit for A -a
oo. Thus the result comes from the identity f !:. e-x2/2dx = (2Tr)h/2, which
can be proved by computing its square with polar coordinates as follows:
f
oo
z
e-x212 dx)
0o
/
= f e-(x2+y2)/2 dx dy
JR2
=
/2x
Jo
oe
d9 /
2
e-r /2r dr = 2ir.
JJJo
As a first step, we establish properties of Fourier transformation in the
Schwartz space S. To simplify the formulas, we introduce the operators Dj =
Fourier transformation and distributions in R"
-i8;, their powers D° =
11
and the notation u(x) = u(-x). Then,
we can state the following theorem.
THEOREM 1.8
For any 0 E S, one has cp E S with IWIk < (8ir)n(k + 1)!IWI2n+k (continuity
of Fourier transformation). Moreover, the Fourier transform cp of a cp E S
satisfies
(i)
(ii)
(iii)
(iv)
For any a E Z+, Dx cp(s)
and
For any u E L', (u,
(Inversion formula). P = (2ir)"cp or in other words, cp(x) = (21r)-n
f
dC.
(Parseval's formula). For any '0 E S,
(21r) n
NormLl(cp) <
Since W ELI, is bounded and continuous with
(27r)nIWI2n (cf. Theorem 1.6). Since the integrand
is in S, we can
differentiate under f or integrate by parts, and these operations give for a E Zn+
PROOF
DF
JD(e())W(x)dx =
L5
fe2D(x)dx = f
fe_i()(_x)W(x)cix
= (-x
p(x)(-Dx)'(e-i(x,f))dx
which are formulas (i). In addition, these formulas prove that aO
is the
Fourier transform of some function in S; therefore, it is bounded and continuous
with the estimate
ICav3;3Io = IOxOcoIo <
(cf. Lemma 1.4), and this gives E S with I Ik S (8ir)' (k+
If u E L' and cP E S, then u(x)cp(4) E L' (R2n) and by Fubini's theorem and
the change of variables y = -x one gets
(fl, W) = f
=
(fe'>u(x)dx) P(0 4
Ju(_)(Je1(YJw()de) dy = (u, 0).
For the inversion formula, the proof is more difficult because e`(x-v,E>cp(y)
L' (R2') as a function of y and C. To overcome this difficulty, we introduce a
factor
e-l0/2 to obtain absolute convergence, and by Fubini's theorem
Fourier Transformation and Sobokv Spaces
12
_ (/e we get
and the change of variables y = x + ez,
d=J
'(e
J
=
(EE)
f
(y)e`(=-y.e) dy
V (()V(x + ez)e-1(z.() dz d(
ez) dz.
=J
Now we can take the limit for c - 0 by dominated convergence to get
'+,(0)
f
dd = v(x)
J
t(z) dz = o(40(0)
and this is the result according to the formula given in Example 1.7.
Finally, Parseval's formula easily follows from (ii) and (iii) since these properties imply
P, V,) = (,P,?) =
(2n)"(4p,V)).
Property (iii) in Theorem 1.8 will allow us to extend the Fourier transformation as announced. Indeed, on one hand we remarked in Theorem 1.6(i)
and (ii) that each LP space can be considered as a subspace of the space of
semi-linear forms on S. On the other hand, the relations (u, cp) = (u, gyp) (which
follows from the change of variables y = -x when u is any function) and
(u, cp) = (u, c) (which holds for any integrable u according to Theorem 1.8(ii))
still make sense for general semi-linear forms cp '- (u, gyp) on S, even not defined by a function u, and can be taken as definitions of new semi-linear forms
u and u. Actually, to get a good theory where we can also take limits, we
must restrict ourselves to continuous semi-linear forms, and this leads to the
following definition: we say that u is a temperate distribution, and we write
u E S', if u is a semi-linear form cp f-, (u, cp) on S (not necessarily defined by
a function u, even if we keep the same notation) with two constants C E R and
N E Z+ such that
I(uMI < CkOI N
for cp E S.
It follows from Theorem 1.6(i) and (ii) that every Lebesgue space LP is a
subspace of S. Thus, the extension of the Fourier transformation to S' will
give a meaning to u for all u in any LP (but this it will merely be a continuous
semi-linear form, not always a function). The first properties of the Fourier
transformation in S' can be stated as follows.
THEOREM 1.9
Let U E S'; then the formulas
(u, w) = (u, 0)
and
(u, p)
for cpES
Fourier transformation and distributions in R"
13
define distributions u and u E S. Moreover, one has u = (2a)"u (inversion
formula), and u E L2 implies u E L2 with Parseval's formula:
(u, v) = (2n)"(u,v)
for u,v E L2.
(The similar formula (u, cp) = (27r)" (u, gyp) for u E S' and cp E S follows
directly from the inversion formula.)
PROOF To prove that u and u E S', we just have to check that cp and cp
depend continuously on cp E S. This is obvious for cp, and for cp this follows
from Theorem 1.8.
The inversion formula also comes from results of Theorem 1.8, since for all
cp E S, the function
E S satisfies r' = v and
(u,
(u, ) = (u, i) = (27r)" (u, +b) = (27r)" (u, +Il) = (27r)"(u, cp)
According to Theorem 1.8(iv) we have for u E L2
I (u,'p)I = I (u, v)I <- IIullollcvllo =
(2ir)n'2IIulloIIcpIIo
so that the semi-linear form U(W) = (u, cp) satisfies the assumptions in Theorem 1.6(iii) with the constant C = (27r)"'2IIullo Therefore the distribution u is
equal to a square integrable function with IIuIIo < (27r)"/2IIulIo; thus if we use
the inversion formula we get
(27r )"/2IIuIIO = (2r)"/2IIuIlo = (27r) ""IIuIIo 5 IIuIIo <- (2ir)""2IIulI0
Finally, Parseval's
with the equality all along, which gives IIuIIo =
formula then follows since any pair u, v of square integrable functions satisfies
the elementary identity
(u, v) = I (Ilu + vllo - Ilu - vllo + i11 U + ivllo - illu - ivllo)
I
As a matter of fact, we can also extend to S' several other simple operations
on functions.
First, the operation of differentiation: indeed, as soon as a function u is
smooth enough to define Dau without ambiguity, we can integrate by parts in
(Dau, cp), and this gives
(Dau, cp) = (u, Da p)
for cp E Co
since the integrated terms vanish. If, moreover, the functions u and Dau define
continuous semi-linear forms on S, the semi-linear form cp '--, (u, Dace) is also
in S' (cf. Lemma 1.4(i)) and agrees with cp H (Dau, cp) on Co and then even
on S thanks to the following result.
14
Fourier Transformation and Sobokv Spaces
LEMMA 1.10
If u and v E S' satisfy (u, gyp) = (v, cp) for all c E C01, then u = v (i.e.,
(u,cp)=(v,cp) forV ES).
PROOF Choose a * E Co such that io = I on B1, then for 0 < e < 1 set
zji,(x) = '(ex), and also for cp E S set c', = -0fcp E Co. One can estimate the
norms of cp - ct
by writing
x°e(sv - 0E) = (I - 00x°O
+E(a)O
7#0
For -y # 0, 6P7 P, = 0(e) so that the sum is bounded by
As for the
first term, one has 11 - AEI < e21xI2 since 0 < I - ipE < I and lexl > I on
supp (1 - iE ). Thus we get the estimate
IMP - S' Ik 5 ne2lwIk+2 +
fork E Z.
Now, u - v E S' satisfies an estimate I(u - v, p)I < CI API N for some constants
C and N, and since (u - v, wE) = 0 for <p, E Co, one has
I(u - v, v)I = I(u - v, V - We)I <- COeIIPIN+2
It follows that (u, gyp) = (v, <p) for all W E S by taking the limit for e -i 0.
U
Thus, if u E S', it follows from Lemma 1.4(i) that the formula
(D* u, gyp) = (u, D°cp)
for p E S
defines a distribution D°u E S' for any a E Z+, and from the discussion
given above it is clear that this operation extends the usual differentiation of
functions. The student will remark that differentiation is always possible in
the space of distributions, and this is an important improvement of the classic
theory of functions: we can now always differentiate a function, even when it
is not "classically" differentiable (but in that case, of course, the result will not
be a function, but merely a continuous semi-linear form). Also notice that we
always have D2 Dk = Dk Dj, since this is true for C°° functions.
These wonderful properties, however, are not compatible with a good multiplication theory: indeed, it has been proved that it is impossible to define in
general the product of two distributions with the usual properties of products of
functions (e.g., see Exercise 4.5(a)). Here, we start from the formula we can
write in the case of two functions and u,
(tu, 0) = f V)(x)u(x)cc(x) dx = (u, V),
so that we see such a formula will define a distribution Ou E S' for any u E S'
only if ,p E S for all cp E S. Thus, the operation of multiplication will
Fourier
and distributions in R"
15
be restricted to the following two situations: (i) when & and u are functions,
,ou is defined in the usual way; (ii) when E P and u E S', the formula
(mu, cp) = (u, cp) for cp E S defines a distribution iiu E S' according to
Lemma 1.4(ii), and this agrees with the usual definition when u is also a function.
We complete this list of elementary operations on distributions by giving the
following two: if u is a function and p E S then
(u, gyp) =
f
u(x)cp(x) dx =
f u(x)V(x) dx =
and if ru denotes the function ryu(x) = u(x + y),
(ryu, cp) =
f u(x + y)(x) dx = J u(z)c3(z - y) dz = (u,T_yW)
/
so that we can define distributions u and ryu by these formulas for a general
uES'. Relations such as fi=u=u,u=u, rru=rytZuandsoforth are
completely obvious; however, the student is strongly encouraged to prove the
following collection of less obvious, but still easy, useful formulas.
PROPOSITION 1.11
Let u ES', 4.' EP,aEZ ,andy,77ER';then
)
D°(V)u) _
(D3V,)(D"-'3u),
C
ryu=&('>u,
PROOF
u=u=u
e'(x.n)u=T-77)U
Left to the reader as an exercise.
Actually, considering only semi-linear forms on the Schwartz space S, which
contains functions with noncompact supports, is equivalent to a certain control
of the "growth" at infinity of temperate distributions. To have a good theory of
Fourier transformation, we need such a control, since a very wild growth of u at
infinity would correspond to a very singular local behavior of u, and too singular
an object cannot even be a distribution. However, if one gives up the Fourier
transformation to keep only the operations of differentiation and multiplication
by smooth functions, one can consider much wider classes of distributions, and
even distributions defined only locally.
Indeed, if ! is any open set in R", one can define a "distribution in f2" as
follows: u E D'(il) (the space of distributions in 1) if u is a semilinear form
on CI (Q) continuous in the sense that for each compact set K C 9, there exist
Fourier Transformation and Sobolev Spaces
16
two constants CK and NK such that
for cp E Co (S2) and supp, C K.
I(u,cp)J < CKIcpINK
(We will write just D' for D'(R" ).) The same formula (4'u, (p) = (u, W) as
above allows us to define the product 4'u E D'(S2) of any u E V(Q) and
1P E Coo (Q).
It is clear from Lemma 1.10 that S' can be considered as a subspace of V.
Moreover, given a distribution u in S2 C R, we can define its restriction u,
to a smaller open set w C 1 simply by restricting the semi-linear form u to
Co (w). One then says that u and v E D'(tl) satisfy u = v in w C 9 if one has
ul,, = vow,. These considerations give meaning to the notion of local behavior of
a distribution; the following result shows that the local behavior of a distribution
determines it completely.
PROPOSITION 1.12
Let 0 be an open set in R"; if it and v are two distributions in SZ such that
every point x E 0 has a neighborhood where u = v, then u = v in Q.
For any cp E Co (S2), K = supp p is covered by open sets Q., C
SI where u = v by assumption. Then, using the partition of unity E VJ of
PROOF
Lemma 1.5, one can write
(u,VEWJ) =
J
J
_
(v, w2 ') = (v,'P
since cpjW has its support in S2, where u = v.
VJ) _ (v, 4%)
I
This property gives clear meaning to the notions of support and singular
support of a distribution, which we now introduce. Indeed, if u E V(Q) and
x E Q, we say that x V supp u if x has a neighborhood where u = 0 (i.e.,
the same definition as in the case of a function u) and we say that x V sing
supp u if x has a neighborhood where u is a smooth function (i.e., if there exist
an w C St with x E w and a 4 E C°° (w) such that (u, cp) = (4', cp) for all
cp E C0 (w)). It is clear that supp u and sing supp u are closed subsets of 0,
and
supp (4u) C (supp 4/i) fl (supp u)
and
sing supp (4'u) C sing supp it
if 4' E C°°(1) and u E D'(Il). The following characterizations are also useful,
but we leave their easy proofs to the student as exercises: x i supp u (resp.
x 0 sing supp u) if and only if x has a neighborhood w such that cpu = 0 (resp.
cpu E Co) for every cp E Co (w); if F is a closed subset of Q, supp u C F if
and only if (u, cp) = 0 for every V E Co' (11) with supp cp fl F = 0.
17
Sobokv spaces
Of course, these classes D'(1) of distributions contain distributions that are
not "temperate" for two reasons: because they are not defined on the whole
of R", nor even when 1 = lR", because there is no control of their "growth"
at infinity. Locally, however, these distributions are "temperate": by that we
mean that distributions with compact supports can be extended to the whole of
IR" as temperate distributions. (Actually, any distribution u can be written as a
locally finite sum of distributions with compact supports if one multiplies u by
a partition of unity slightly more general than that from Lemma 1.5.)
Indeed, if u E D'(Sl) has a compact support K and if we choose a 0 E
Co (Sl) such that 1' = 1 near K (cf. Lemma 1.5), one has (u, gyp) = (u, iV) for
all V E Co (fl) since (u, (1- ')gyp) = 0, and thus one can extend the semi-linear
form u to S (and even to C°° (IY" )) by setting (u, gyp) = (u, ?Pp) for cp E S.
Moreover, this extended form satisfies
I(u,w)I <CKI'01PINA <CVCKICINK
if K = supp b without any constraint on supp cp, so that u E S'. (This extension
consists essentially in setting u = 0 outside Q.) The subspace of D'(1) formed
by distributions with compact supports is denoted by £'(f) (just £' for £'(R" ));
if wCfl,one has £'(w)C£'(Q)CS'.
Finally, we end this section by stating the classic Paley-Wiener-Schwartz
theorem, which shows that distributions (resp. smooth functions) with compact
support can be recognized on their Fourier transforms. Since we will use this
result only in the very last application (in Section 4.3), we provide the proof in
Exercises 1.7 and 1.8 rather than here.
THEOREM 1.13 PALEY-WIENER-SCHWARTZ THEOREM
Let U(() be a function defined on R'. Then, it is the Fourier transform of
a distribution (resp. a C°° function) with support contained in BA = {x E
R"; Ixl 5 A} if and only if U can be extended as an entire function U(() on
C" satisfying an estimate
IU(()I 5 Co +
I(I2)NeAIlm(I
for some constants C and N (resp. satisfying estimates
IU(()I < CN(1 +
I(I2)-NeAI1mfl
for all N E Z+ and some sequence (CN)NEZ+)
13 Sobolev spaces
It is a remarkable property of Fourier transformation that for a temperate distribution u, u E L is equivalent to u E L2. Since, moreover, differentiations on u
bution
Fourier Transformation and Sobokv Spaces
18
correspond to multiplication by polynomials on u, it is clear that the smoothness
of u can be measured by the growth of u at infinity. The definition of Sobolev
spaces is based on these properties, and this way of measuring smoothness will
be convenient especially when we will deal with L2 estimates.
From now on, the Greek letter A will denote the function A(e) = (1 + 112)1/2
defined on R", and more generally we will write A" (t) = (1 + 1e12) 8/2 for s E R
and C E R. For any s E JR, we say that u E H8 (the Sobolev space of exponent
s) if u E S' and A'u E L2. In other words, u E H' if u is a function satisfying
IIuI12 = (27r)"
f
(I + 1C12)81u(C)I2 d < 00
(the factor (27r)-" is introduced here to keep the convenient relation 11ullo =
NormL: (u), cf. Parseval's formula in Theorem 1.9) or more shortly satisfying
11u11, = (21r)-n/211Aeu11o < oo. Since H' C Ht if s > t, we will also use
the notation H-OO = U8H' aqd H°° = fl,H'. The inclusions S C HO° C
H-°° C S' are immediate.
The first properties of these spaces that we prove show that they measure
essentially the same smoothness as the classic classes Ck up to a fixed shift of
exponents. Indeed, they clearly imply that u E Ck = u E HIM (which means
Vu E Hk for all So E C01), and, for example, that u E H"+k = u E Ck.
PROPOSITION 1.14
For all s E R one has
uEH'+' q u,Diu,..., andD"uEH8
with the equality 11u118+i = 1ju118 + >, 11 Diu118.
Z+ U {oo},
(i)
(ii)
Moreover, for any k E
uEHkt* D"uEL2forall lal <k.
D' u are bounded continuous funcs > (n/2) + k and u E H'
tions for la1 < k (with 1D'ulo < C,.k1Mu)i,: for example, one has
lulo S 2-"'211ul1" for u E H").
PROOF
Since A2 (l;) = 1 + (1;12 = 1 + E, ,2, one has for any function u
IAe+l,u12
= A21aeu12 =
1A'u12 +
lA'Ciu12 = 1ABu12 + E 1A'D,j"u12
from which the equivalence and the equality of norms follow. Properties (i)
and (ii) can then be proved by induction. Indeed, (i) is true for k = 0 in view
of Theorem 1.9. As for (ii) for k = 0, s > n/2 implies \-" E L2 thanks to
Lemma 1.3, so that u E H' implies u = (A_')(a'u) E L' as a product of two
L2 functions, therefore u E co fl L°°. Finally, the estimate lulo < C8,o11u118
19
Sobolev spaces
follows from the Cauchy-Schwarz inequality: in the case s = n, for example,
Iulo <- (21r)-nNorMLI(u) <
(27r)-nlllla-n1Io(21r)-n,2ll,\null0
since III-nllo < irn/2 as proved in Lemma 1.3.
<-
2-n/2llulin
1
Actually, property (ii) can be improved as follows: one has u E
H' = u E CE (the Holder space of exponent e, i.e., the space of functions u
such that Iu(x) - u(y)I < CIx - yl' - it is assumed implicitly that e < 1) as
long as s - e > n/2 (cf. Exercise 1.9); one can also prove that for any s > n/2
1
functions in H' tend to 0 at infinity (cf. Exercise 1.10).
REMARK
Through Riesz's representation theorem, the H' distributions can also be
characterized as the continuous semi-linear forms on H'. This property can
be stated more precisely as follows.
PROPOSITION 1.1 S
If u E H' and V E S, then
I (u, W)I <
Conversely, if u E S'
satisfies I (u, W) I < CII VII -, for some constant C and a!1 W E S, then u E H'
with IIull, < C.
PROOF We get the first conclusion from Parseval's formula and the CauchySchwarz inequality:
I(u, O)I = (2 r)-nl(uov)I = (21r)-nl(A'u, A-',v)I < Ilull,llpll-,.
Conversely, if one has the estimate R u, co)l <
for all cp E S, then
(Au, p)I = I(u, ))l = I(u, A'V)I
C(27r)-n/2lla-ea'wllo = C(2a) n/211 wvllo.
Thus .1'ui is a semilinear form on S satisfying the assumption in Theorem 1.6(iii),
so that .\'u E L2 with IPki llo < C(21r)n/2, i.e., u E H' with llull, < C.
I
We already mentioned the inclusions S C H°° C H-°° C S'; they are all
strict. It is not true that S = H°° (take u(x) = (1 + Ix12)-n, which satisfies
u E H°'D but u S), nor is it true that H-OD = S'. For the latter, the reason is
that the control of the "growth" of u at infinity is not sufficient, or on the Fourier
side the smoothness of u is not sufficient, i.e., u is not a function. Indeed, any
temperate distribution can be thought as an H' distribution for some s E R if
we force the control at infinity, as in the following lemma.
Fourier Transformation and Sobokv Spaces
20
LEMMA 1.16
For any u E S' there exists an N E Z. such that ? u E H-N if 1' E S or
V)(x)
= (1 + IzI2)-N
Since U E S', one has an estimate I (u, w)I < Clwlk for some C and
k, and all o E S; one can then write
PROOF
C2kl,lk
101:5k
tO
Io
by using Leibniz's formula (cf. Theorem 1.2) for expressing derivatives of
Furthermore, 11ilk is a finite constant if '0 E S, and also if O(x) = (1 +
lxl2)-N with N > k/2. On the other hand,
I1,PIIn.+I81 in view
of Proposition 1.14(ii), so that the estimate becomes l('u,V)l <
Hence, thanks to Proposition 1.15, we get z'u E H-N with N = n + k.
I
When using H' distributions in the study of linear partial differential equations with variable coefficients, we will have to consider products of such distributions with the coefficients of the equation. To measure the smoothness of
these products, we first compute their Fourier transforms.
LEMMA 1.17
One has
uv(C) = (27r)-" f u(C - rl)v(rl) dr1= (27r)-"
f u(()v(C - () d(
for any u and v E L2, as well as for any u E H°° and v E HMoreover,
Leibniz's formula
D°(uv) _
I
c' ) (D3u)(D°-0v)
holds for any u and v E H'°I (and also for u E H°° and v E H-°°, but this
already follows from Proposition 1.11 since H°° C 1'; cf. Proposition 1.14(11)).
PROOF
uv E L', we have the formula
Since u and v E L2
uv(C) =
J
e
i(x.4)u(x)v(x) dx = (u{, v)
where u{(x) = e-`(x'0u(x) Using Parseval's formula from Theorem 1.9 and
various formulas from Proposition 1.11, we get
uv(C) _ (2ir)-"(u£, v) = (27r)-"( r
, v)
= (27r)-" f u(C - r!)v(r!) drl
21
Sobokv spaces
as required. If U E H°° and ' E S, they are both in L2 so that we can write as
above
(27r)-' f UW
uVG(n) = (2ir) " f u(77
- 76(0 A.
(21r)-n (-C) (so
Thus, for u E H°°, v E H'°°, and cp E S, setting
that V = 1') and writing
(27r)"(uv, z/)) _ (27r)' (v, iiV,)
(uv, gyp) = (uv,
(27r)-" f v(rl)u( - WOW A drl
=
J
(_n f u( - n)v(r1) dill
dC
The other equality with the integral in
again give the same formula for
((E R") comes from the change of variables n+(' = t, while Leibniz's formula
follows from multiplication of v(t) by
(cf. Theorem 1.2(i)).
I
To get estimates on integrals as in Lemma 1.17, we will need the following
elementary result, known as Peetre's inequality.
LEMMA 1.18 PEETRE'S INEQUALITY
For any aERandall{,77 ER", one has
A8(e)
2181,\I81(C - 7n)A8(n).
PROOF The triangular inequality for the euclidean norm in R" gives
(I + ICI) <- (1 + IC -101 + 1771) <- (1 + IC - nl)(I + Inl)
so that
a2(C) <- (I + ltl)2 < (I + IC - n1)2(l +
I771)2.
On the other hand, (I + I1)I)2 < (I + 1171 )1 + (1 - I771)2 = 2\2 (71). and estimating
(1 + IC - 77I)2 in the same way, we thus get
A2(S)
22A2(C - 71) \2(77)
We obtain the conclusion of Lemma 1.18 for s > 0 simply by raising this
inequality to the power s/2. As for s < 0, we just have to exchange f and n to
22
Fourier Transformation and Sobolev Spaces
see that
which can be rewritten
I
From these lemmas we get the following continuity properties.
COROLLARY 1.19
Let
E H°O ands E R; then u E H8
2181-(n/2) ilVlli°i+nlluli°
Vu E H8 with II'uII° <
Moreover, if a(x,D) _ E,Q,<maa(x)Da is a lin-
ear partial differential operator of order m with coefficients aQ E HO*, a(x, D)
maps continuously H8 into H8-1 for any s E llt
PROOF From Lemma 1.18 we get )28(x) < 4iela2Iel( -,)A2e(i1); thus, using
the Cauchy-Schwarz inequality in the integral of Lemma 1.17, we can write
2
la°(t)
(x)12 = I(2ir)-" f a°(f);5(e
<
IIolI1°i+n(2ir)-"
- i7)u(71) drj
f A-21°i-2'
(2ir)-n
i)A2a(ii)lu(i )12 dn.
Integrating this estimate with respect to (the integrations in the right-hand side
are achieved first with respect to using f a-2n(C-q) dt < 2r" (cf. Lemma 1.3)
and then with respect to 71) gives the estimate of
From Proposition 1.14, it is clear that each D° maps continuously H8 into
H°-1Q1 for any a E Z and s E R. Therefore, the result for a(x, D) follows
from the first part of this corollary.
I
Finally, we will close this introductory chapter by giving our first examples of
pseudodifferential operators, which here provide merely another way of writing
the H8 norm. Indeed, if we denote by )8(D) the operator from S' into S'
D)u = au for u E S', we will have Ilull8 = (2n)-ni211A°ullo =
(2ir)-"'2IIA8(D)ullo
= Ilae(D)ullo, thanks to Parseval's formula. The identity
`+t(D)u
As(D)At(D)u =
is immediate, and the following properties are not
defined by
more difficult.
PROPOSITION 120
ForanysER, coES,anduES',
A'(D)V(x) = (2-yr)-n f e'(x,l) A"
dC
23
Exercises
and
()"(D)u,V) = (u, A'(D)W)
Moreover, for any t E R, is E H' if and only if At(D)u E H°-t, with 11ullr =
IPPt(D)uIIs-t.
PROOF The formula for )t'(D)co follows from the inversion formula since
0ES
A-4(3 E S, while the formula for A'(D)u comes from
()R(D)u,V)
_
_
(2x')-"(a'u,(G)
(27r)-"(u,
A',P) = (u,)1'(D)4p)
Finally, the identity A8-tat(D)u = a'-'A'u = A'ti shows the equivalence
between u E H' and At(D)u E H'-t, and also the equality of norms.
I
Exercises
1.1 A Banach algebra of holomorphic functions. The goal of this exercise is to prove
properties of multiindices a E Z. and to use them to construct a Banach algebra of
holomorphic functions (i.e., a Banach space with a continuous product). Such an
algebra is useful when studying nonlinear problems with holomorphic functions.
(a) Use an induction on the dimension n to show that for any z = (z1,... , z") E
C",aEZn, andNEZ+,
101=N.
(
)_(' I)
and
z
(Z, + ....r z") N
QI
N!
IBI=N
(b) Using some symmetry, show that for any N E Z+.
Nv
16
1
1
L(j+I)2(N_j.+..l)'
j --0
- (tV+1)2
(c) One considers the space of formal power series u(z)
u(z°/a!)
(i.e., the space of sequences uo of complex numbers indexed by Z. and
equipped with the formal product
z°
z
z
wo,
a!
wa
a+,=o
a
Q
C'
uov«-0
and one defines
NormB(u) = sup
IuaI(IciI + l)2
l al!
24
Fourier Transformation and Sobokv Spaces
Compute the norm of u(z) = (zO//3!) (i.e., uo = 0 if a 36 /3, up = 1).
Show that NormB (u) is a norm on B = {u; Norm3 (u) < oo} and that B
is a Banach space for this norm. Show that for any u and v E B, uv E B
with NormB (uv) < 16Nonn8 (u)NormB (v).
(d) Show that u E B implies that E. u,,(z°/a!) is absolutely convergent
in 1z, I +
1. Conversely, if F. u0(z' /a!) is convergent in
+
max{lz,l, j < n} < I + f for some f > 0, show that u E B.
More generally, show that if F(z, Z) =
F,,3 (z" /a!) (Z1113!) is
convergent in max{lz,1, j < n; JZkl, k < N} < 1 + f for some f > 0 and
if u = (u1, ... , UN) E BN with NormB (uk) < 1/16 for all k < N. then
the function f (z) = F(z, u(z)) satisfies f E B.
1.2 A distribution u E S' is said to be "real valued" if ii = u. Show that u is real
valued if and only if (u, cp) E R for all real-valued V E S. Show that iz is real
valued if and only if u = u.
1.3 Let b be the semilinear form S 3 '-. (6, p) = (P(0). Show that 6 E S' and that
supp 6 = {0}. Compute b and determine all the s E R such that 6 E H'.
1.4 For a < b E R and c E C, one defines on R(n = 1) the function
f (x) = e`= if x E [a, b];
f (x) = 0 if x
[a, b].
If, moreover, Re c > 0, one also defines on R
g(x) = e-'Izl
and
h(x) =
e-"'/2.
Show that these functions are in L' and compute their Fourier transforms (for h,
depends holomorphically on c).
first study the case c E R. then prove that
1.5 Forc E C, Rec > 0and E R(n = 1). one sets G(4) =
Show that G E L' and compute C (without using the results of Exercise 1.4) by
the following method.
For x < 0, compute C(x) by using Cauchy's integral formula with the path
for large R.
For x = 0, compute d(O) by a direct integration if c is real, then remark that
G(0) depends holomorphically on c.
For x > 0, use the same kind of device as for x < 0.
Finally, compare your results with those of Exercise 1.4.
1.6
(a) For c E C\{0} and Rec > 0, the functions defined on R(n = 1)h(x) _
e-`Z2/2 are all bounded by I (they are uniformly in L°°). By taking the
limit for Re c - 0* in the formula (h, rp) = (h, gyp), use the results of
Exercise 1.4 to find the expression of h also when Re c = 0 (but c # 0).
25
Exercises
(b) If A is a real symmetric nonsingular n x n matrix, the function H(x) =
e`(A=,=)/2 defined on R" is obviously bounded. Show that its Fourier transform is given by the formula
k(f) = (2a)"J2IdetAI-1/2eIf (spA)e-:(A
where sgn A is the signature of A, that is, the number of positive eigenvalues minus the number of negative eigenvalues. Finally, determine all the
s E R for which H E H.
1.7 The Paley-Wiener theorem for smooth functions. In questions (b) and (c), the
number A > 0 is fixed.
(a) Show by direct calculation that the Fourier transform of e-1=12/2 can be
extended to C" as an entire function, but that it does not satisfy any estimate
of the types given in Theorem 1.13.
(b) Let u E C'° such that supp u C BA. Set U(() = f
dx and
show that U is an entire function on C' extending the Fourier transform of
u and satisfying estimates IU(()I < CN(1+I(I2)-NeAltm(l for all N E Z+
and some sequence CN.
(c) Let U be an entire function on C" satisfying estimates iU(()I < CN
(I +
I(I2)-NeAllmll for all N E Z+ and some sequence CN. Set u(x) _
A.
Show that u(() = U(() and u E H°°. Show that for any e > 0,
(2r)-" f
u(x) = (21r)-" f e'(=
i(x/e)) d{, then prove that u(x) = 0
for Ixi > A and finally give the conclusion.
1.8 The Paley-Wiener-Schwartz theorem for distributions. In this exercise, use the
results of Exercise 1.7. In particular, if tG E Co , denote by r1'(() the entire
extension of '. In questions (b) and (c), the number A > 0 is fixed.
(a) Let ', E S, u E S', and So be a unit test function.
Show that for any e > 0, gyp(-e()t/, (resp. c'(e()u) is the Fourier transform of a function r' , E S (resp. of a distribution u, E S'), that lim,.o
10 - rb-, Ik = 0 for all k E Z+, and that lim, .o(u,, iP) = (u, 0).
Show that supp V), Csuppr(i+B,={x+yER";xEsupp,pandyE
B,}, then that supp u, C supp u + B,.
(b) Let u E S' such that supp u C BA.
Using Lemma 1.16, show that the distribution u, defined in (a) satisfies
u, E H'C and supp u, C BA+,. Then show that U(() = lim,-ou,(()
is an entire function and that its restriction to R" is the Fourier transform
of u.
Let 0 be a C°C function of one variable t satisfying fi(t) = 0 for t > I
and 0(t) = I for t < 1/2, and set *((x) = 0(I(I(IxI A))e'(=.<) E Co .
Show that for e < 1/2I(I one has u,(() = (u,,,y(), then U(() = (u,r(b().
Finally, show that IU(()I < C(1 +
for some constants C
and N.
(c) Let U be an entire function on Cl satisfying an estimate IU(()I < C
(I +
Show that there exists a u E S' such that it = UIR" . Show that the
I(I2)NeAUm(I
distribution u, defined in (a) satisfies u, E C°° with supp u, C BA+E, then
that supp u C BA, and give the conclusion.
1.9 For 0 < e < 1, one defines the HSlder space C' as the space of functions u
Fourier Transformation and Sobolev Spaces
26
defined on R" such that lu(x) - u(y)l < Clx - yl' for some constant C and any
xandyER".
Show that for any 0 < e < I and x, y,£ E R",le`(',() - e'(11,0 I < 2'
lx - y1`a`({). Show that if it E H' with s > (n/2) + e, then it E C.
1.10 Let s > n/2, u E H', and cp E C°° such that ,p(0) = I and supp cp C B1. For
any U E Lz, one sets U,(C) = (21r) -" f U({ - en)0(77) dj7.
Show that (fi), is the Fourier transform of a bounded continuous function u` (x)
and that supsx,,18 1U(x)1 < lu - 00.
Show that for any , rt E R" one has
A'(n)l <- s2'-' If - nja'-' (,7)A'-' ( -'1).
Show that for any U E LZ one has )-'U E fL2 and that one can write
UU(f) a kernel K,(t,'i) satisfying for some constant Co
f IKf(f,71)ldC < Coe
and
JlK('i)ldii
Coe.
Use Schur's lemma (i.e., Lemma 3.7) to prove that for any U E Lz, lim,.,o
IIUe - A'0`17)(110 = 0.
Using the L2 function U = a'u, prove that
u(x) = 0.
1.11 The algebra H' for s > n/2. The following properties of Sobolev spaces lead to
their use in the study of nonlinear problems as well.
(a) Let r, s, t E R+ be such that r < s, r < t, and r < s + t - (n/2).
Show that if l -'ii <_ InI, one has A2r(4) < 5rA2t('i)Azr-zt( - ii).
Similarly, prove \2"(e) < 5ra2'(e - p)A2r-2a(r!) for l - r71 ? Inl.
Let it E H' and V E Ht. Write the integral of Lemma 1.17 as a sum of
two integrals on the domains 1 - rll S Inl and l - 7I ? I'l respectively,
then show that uv E Hr With Iluvjlr < CIIuIIa1IvIIt where the constant C
depends only on r, s, and t.
Let it and v E H' for some s > n/2: show that uv E H' with Iluvlla <
Callullallulla-
(b) More generally, let F(x, X) be a C' function defined on R" x RN, u =
(it,,. .. , UN) a function defined on R" and valued in RN satisfying u, E
; E H' for all cp E Coo),
Hi° for all j < N and some s > n/2 (i.e.,
and set Fu (x) = F(x, u(x)). Choose a unit test function cp, and for any
locally integrable v set v, (x) = f v(x - ey)cp(y)dy fore > 0. Prove that
va E C°°.
If v is continuous, show that v = lime_o v, uniformly on every compact
set of R". If v is square integrable, compute the Fourier transform of v,
(see Lemma 1.17) and show that limf_o 11va - v110 = 0.
Assuming s > I and using similar arguments, show the validity of the
generalized chain rule
(F", Dkib) = ((DkF)u
+>(D,,u,) (F)
+G)
for all (,EC0 .
Prove by induction that for any t E Z+ with t < s, F E Ct implies
F°EHH.
Notes on Chapter 1
27
Notes on Chapter 1
During the eighteenth century, trigonometric series were introduced in the problems of interpolation (Euler), astronomy (Clairaut), and sound (Lagrange). By
the end of the century, they played a central role in the famous controversy over
the vibrating string problem, which would lead eventually to the revision of
the bases of analysis initiated by Cauchy [28]. The integral transformation also
is introduced in Fourier's me moire [35], considered a fundamental contribution
to the theory of trigonometric series despite its lack of rigor. (Actually, the
same results were obtained concurrently by Cauchy and Poisson.) The Fourier
transformation was then extended, thanks especially to the Lebesgue integration
theory, but it is the introduction of distributions by Schwartz [61] that simplified
and unified the theory.
The best account on the origins of distribution theory is to be found in the
introduction of Schwartz [61]: this theory has roots in the symbolic calculus of
engineers initiated by Heaviside [39] and continued by the physicist Dirac [29],
in the turbulent solutions of Leray [51], in the derivatives of Sobolev [65], in
the finite parts of Hadamard [38], in the Fourier transformation as extended by
Bochner [18], etc. Expository texts on distribution theory are Schwartz [61],
Treves [67, Part II], and Gelfand and Silov [37]. Extensions of these ideas can
be found in Beurling [17] and in Sato's theory of hyperfunctions [60] (see also
Hormander [8, Chap. 9]).
Finally, Sobolev spaces were first introduced for positive integral exponents
by Sobolev [64,65]; they now play an increasingly important role in the theory
of partial differential equations. The student will find a systematic study of
these (and related) spaces in Adams [14].
2
Pseudodifferential Symbols
Introduction to Chapters 2 and 3
Elementary properties of Fourier transformation allow us to write for (p E S
then
D°co(x) =
(27r)-n
by the Fourier inversion formula. For a linear partial differential operator
a(x, D)
a°(x)D°, these formulas thus lead to the following expression:
a(x, D)V(x) =
(27r)_n
J e'(x,Oa(x,
for cp E S
where the "symbol" a(x, ) of the operator a(x, D) is simply the polynomial
et
a(x, S)
= L{a{<m a.(x)S°.
On the other hand, we can remark that the operators a'(D) introduced in
Section 1.3 can actually be defined by a similar formula where the symbol
a(x,e) is replaced with the symbol
(1 + ICf2)'/2, which is no longer
a polynomial (see Proposition 1.20). Moreover, the operator A-2(D) is a twosided inverse of A2(D) = 1+Ej Dj' = 1-A where A = Fj OJ2 is the euclidean
Laplacian operator. This simple remark shows that the operator A-2(D) can be
used to study the equation (1 - O)u = f. Indeed, for any f E S', this equation
has at least the solution u = A-2(D) f (property of solvability), and if f E S,
then any solution u E S' is actually in S since u = A-2(D)(1 - A)u =
A-2(D) f E S (property of hypoellipticity, i.e., the solutions are smooth as soon
as the right side is).
The purpose of the theory of pseudodifferential operators is to extend this kind
of proof to more general linear partial differential equations than (I - O)u = f.
28
Definition and approximation of symbols
29
The key idea is to replace all the computations on the operators with algebraic
calculations on their symbols. If we consider sufficiently large classes of symbols (which will contain functions that are not polynomials, corresponding to
operators that are not differential, therefore called pseudodifferential operators),
we will be able to find inverses of these operators, at least for the best of them,
called elliptic operators. However, one can notice that the problem with variable
coefficients is much harder than with constant coefficients, since a(x, )c (f) is
no longer the Fourier transform of a(x, D)p, and it turns out that the operator
is not an exact inverse of the
we would get by using the symbol
operator a(x, D). Thus, in general, we will construct only approximate inverses
of elliptic operators, but this will be sufficient for the study of solvability and
hypoellipticity of these operators.
To construct a good theory of pseudodifferential operators, one must restrict
the class of allowed symbols, and this is the main topic treated in Chapter 2.
Here we present the basic classes S', also known as S o. One of the main further developments of the theory has been to extend the fundamental properties
of these basic pseudodifferential operators to larger classes of symbols, which
can be adapted to the study of various problems in partial differential equations,
but this is definitely beyond the scope of this course. (We simply refer to Coifman and Meyer [6], Hormander 18, Chaps. 18.5-18.6], and Bony and Lerner [4]
for such extensions.) After a section devoted to simple oscillatory integrals, we
close Chapter 2 by defining two fundamental operations on symbols, the use of
which will be essential in the next chapter.
The only motivations for the results presented in this chapter lie in the description given in Chapter 3 of the essential features of the theory. Chapter 3
will thus provide the definition of the operators and the proof of their continuity
in Sobolev spaces, a description of the symbolic calculus (i.e., the correspondence between operations or estimates on symbols and operations or estimates
on operators), and a sketch of the invariance property under a change of variables (which allows us to define the corresponding operators on a manifold).
This set of results can be considered as the most basic properties of pseudodifferential operators, and we will see in Chapter 4 that it is already sufficient to
be conveniently used in the solution of difficult problems of partial differential
equations.
2.1
Definition and approximation of symbols
As in Section 1.3, we will keep the notation a" for the function )a(£) _
(1+ICI2)8/2 where CElR and sE R.
Let m E R and a(x, C) be a C°° complex-valued function defined on 1[2" x RT'
Then we say that a is a symbol of order m, and we write a E S', if the functions
30
Pseudodiferential Symbols
A1131-`&Oa
are bounded on R" x Rn for all multiindices a, >3 E Z. If one
prefers to write estimates, this means that there exist constants Cap such that
C
(1 +
for
(X, C) E Rn x R",
a,0 E Z.
Since St C S"' when f < m, we will also use Sc °= U,"S" and S
=
f1,nS'. The student will prove that if a E S', b E St, and a,,3 E Zn+,
then OOa E S-- 1,31 and ab E S. As a consequence of the inclusions
St C S'" for £ < m, we can consider computations in S" modulo Sf with a
lower f. Thus, besides an exact calculus (without rests), we will also develop an
approximate calculus (modulo terms of lower order than principal terms). When
looking for progressively more precise approximations, this point of view will
lead to the development of an asymptotic calculus (modulo S-°°).
Our first example is that of symbols of differential operators. If a(x,
a polynomial with coefficients aQ E H'°, then a E S'n (cf.
Proposition 1.14(ii)). If the coefficients are "only" of class C°° and we want to
study local properties, we can reduce the problem to the previous case where
the symbol is in a class S' merely by multiplying the coefficients by a cut-off
function locally equal to I since the modified operator thus has its coefficients in
Co C H°°. A second example is that of the functions a', which clearly satisfy
A' E S'". Our third example will be used in Section 4.1 to study the local
solvability of differential operators: if a(x, %) is a function with compact support
in x, (positively) homogeneous of degree m in l; and C°° outside = 0 (if a is
also C°° at = 0, it is a polynomial), then there is a symbol b E S"' (uniquely
determined modulo S-OC)t such that b(x, t;) = a(x, ) for ICI > 1. Indeed, to
transform a into an actual symbol satisfying the definition, it is sufficient to take
b(x, i;) = (1 near
where V E Co (R") with supp V C B, and V = I
= 0, and it is clear that if b(x, f) = a(x,.) = c(x, ) for ICI > I, the
difference b - c is in Co (R" x R") C S-°O. In this situation, we will usually
use the same letter a to denote the modified symbol b = (I - yo)a, since this
will not bring too much confusion.
Classes S'n can be characterized by the following equivalence: a E S' --'"a
E S°, so that it would be sufficient, from a theoretical point of view, to
study only the class S° of zero-order symbols. However, from a practical point
of view, it is better to have at hand all the orders m E R. We already remarked
that S° is closed under multiplication (it is an algebra); one can even prove the
following result.
'We say that a symbol possessing certain properties is uniquely determined modulo S-°° if,
given two symbols with these properties, their difference is always in S.
Definition and approximation of symbols
31
LEMMA 2.1
If a E S° and F E C' (C), then F(a) E S°.
Let us write a = b + is where b and c are real valued. Since a E S°,
we have b and c E S° C C° fl L°°, and therefore the function F(a) = F(b, c)
PROOF
satisfies 1((9'rF)(b,c)j < C., for all ry E Z2+. The estimates on rO[F(a)]
can then be proved through an easy induction, which is left to the reader as an
exercise. r
In setting up an asymptotic calculus as announced above, we will use the
following lemma as a substitute for the summation of a series.
LEMMA 2.2
Let aj E S'"-' for j E Z+; then there exists a symbol a E S' (unique modulo
S-'°) such that for any k E Z+
a - E a, E
Sm-k.
I<k
Moreover, a can be chosen so that supp a C U j supp a3. We will write a - E. aj .
Since the series E a. has no reason to be convergent, we will define
a as a convergent series E bj where the bj's will be approximations of the aj's.
Thus let cp E Co (R") such that cp = I in B, and cp = 0 outside B2; then for
a sequence of real numbers 0 < e j < 1 tending to 0, set
PROOF
b,(x,C) = (1 -
Since bj = a, for C outside a compact set in R" we have bj - a, E S-°° so
that b, E S'"-J. However, we will need some more precise estimates before
taking the sum of the series.
For JCJ < 2/f1, one has Ac, < v" whence
C'
IO&'b,l <
A--,-101
for some constants CQ0, and we have the same kind of estimates for JCJ > 2/e,
since b, = a. there. Moreover, since 1 < E., ICI in supp (1 - gyp) D supp b., we
can refine this estimate as follows:
It Obit < 3AI8 ebjI -< e3CJ
00 A"-j-01.
Thus, if we choose Ej < min{l/CQO; Ia + 31 < j} we will have
IA101-moitb3 I <
Al-j
for 1a +,Q1 :5j.
0, the sum a(x, ) = Ej>° b,(x, ) is a finite sum near any fixed 1;o,
and therefore this formula defines a function a E C°°. If k E Z+ and a,,3 E Z"+
Since Ej
Pseudodiferential Symbols
32
are fixed, we set N = max(Ia + 01, k + 1) and write
a-Eaj=J(bj-a,)+ E b3+1: bj.
j<k
j<k
k<,,<N
j?N
The sums E,<k and Fk<,<N are in S'-k since these are finite sums of term
in Sm-k (one even has bj - aj E S- O°). As for the sum Ej >,v, we have
AI$I-(m-k)000$
r I'\l,9l-,n+kO
b
31ibj
j> N
j>N
<
2
<
j>k+l
'--
1
because in this sum one has Ja +,31 < j and A > f on supp b3. We thus get
a - E3<k aj E S,"-k, and for k = 0 this gives a E S". The relation on the
supports comes from the explicit construction we gave here.
I
As a particular example of this construction, we take functions a. with com-
pact support in x, (positively) homogeneous of degree m. - j in and Cx
outside
= 0. As shown above, we can associate symbols b., E S'"-.' to
such functions, and the construction of Lemma 2.2 will give us a new sym-
bol a E S', for which we will still write a - >j a3 (a is clearly uniquely
determined by the functions aj modulo S-O°). Such symbols are called polyhomogeneous (some authors say "classic"). These polyhomogeneous symbols
form a subclass of S°° containing the differential symbols (polynomials in ),
the symbols Am(1;) =
+ 1)-/2 (indeed, the second factor can be
written as a power series in Ifs-2 convergent for
1) and closed under
differentiation and multiplication (and also, as we will see, under adjunction,
composition, and inversion).
2.2
Oscillatory integrals
In this section, we just want to sketch a theory for some integrals of functions that
are not absolutely integrable. These integrals will be used in the next section to
define operations on the symbols (symbols of the adjoint and of the compound).
Thus, we do not try to give here any idea of what a general theory could be, but
only to establish the fundamental properties of these integrals in the particular
case we will use most. The functions of which we want to define the integrals
are products of an oscillatory term and an amplitude with a controlled growth
at infinity, exactly as in the classic integral f 00 (sin x/x) dx. More precisely,
33
Oscillatory integrals
the oscillatory term will be a trigonometric function of a quadratic form, while
the amplitudes will be taken in the following spaces: for m >_ 0, a E A"` (the
space of amplitudes of order m) if a is a Cx complex-valued function defined
on R" such that the functions (1 + IxI2)-+"/2Oaa(x) are bounded on R' for all
a E Z. It is clear that polynomials are such amplitudes, and that derivatives
or products of amplitudes are amplitudes. On the space A', we will use the
norms IIIaIIIk = maxlaI<k I(1 + IxI2)-m/2O"aIo where m is implicitly fixed.
THEOREM 2.3 DEFINITION OF OSCILLATORY INTEGRALS
Let q be a nondegenerate real quadratic form on R, a E A' and p E S such
that cp(0) = 1. Then the limit
lim
J
e`Q(x)a(x)cp(ex) dx
exists, is independent of 'p (as long as 'p(0) = 1), and is equal to f e'q(z)a(x) dx
when a E V. When a L' we continue to denote this limit by f e'Q(x)a(x) dx,
and one has an estimate
if
e=q(x)a(x)dx 1 G Cq,mIIIaIIIm+n+I
where the constant Cq,,,, depends only on the quadratic form q and the order m.
PROOF When a E L' the result simply comes from dominated convergence.
In the general case, let us choose a function 0 E Co (R" ), such that b = 1 in
B, and V) = 0 outside B2, and set I. = f e`9(x)a(x)V)(2-)x) dx. We are going
to show that limi-,,,, I, exists and is equal to limf,o f e'q(')a(x)'p(ex) dx, and
this will prove both the existence of the limit for any 'p E S and its independence
with respect to V. Furthermore, since for any fixed c > 0,
J
e`Q(=)a(x)V(ex) dx = lim
J
e'q(x)a(x)'(ex)0(2-3x) dx
by dominated convergence, we will just set
li(e) = f e+q(z)a(x)(I
-
A(Ex))V,(2-3x)dx
and show that lim3.o,, Ii exists (resp. that limi_m IJ(e) = 0(e)).
To get these results, we use the change of variables y = 2-3x:
li - ii_, =
f
e'Q(x)a(x)(,(2-ix)
- V,(2'-Jx)) dx
= r ei2''9(y)a(21 y)(,0(y) - 0(2y))23' dy
(resp. li(e)-I,_,(e) =1 et22jq(y)a(2iy)(I-'P(e23y))(il'(y)-rl'(2y))2'"dy
.
34
Pseudodiferential Symbols
The function X(y) = i(y) - '(2y) satisfies X E Co and supp X C {y; 1/2 <
IyI < 2}. Moreover, for y E supp x, I(&a)(23y)I S
IIIaiIIIQi(1+22jlY12)m/2 <
IIaI I Ilal2m('+2) (resp. I(d°b()(2'y)I <- EC2(*"+i )2 where bE (x) =
and the constant C depends on everything but one and j, for I1 - cp(e2'y)I <
I E2iyI sup I p'I < eC23 ). Thus, one gets the following estimates, which imply
the theorem:
IIj - h-11 <_
-
(resp. II,(E)
q,m2-'IIIaIII,1+,.+1
- I,-i(E)I <- (C2-J)
by using the next lemma with µ = 2J and N = m + n + 1 (resp. N = m+n+2).
1
LEMMA 2.4
Let q be a nondegenerate real quadratic form on Rn and X E Co with x = 0
near 0; then for all N E Z+,
l
eill 24(11)b(py)X(y) dyl
CNA-N
sup
yEsuPPX,IaI<N
I (0'b)(uy)I
where the constants CN do not depend on µ > I nor on b E C.
PROOF One can perform a linear change of variables so that q(y) = Iy' I2 - ly" I2
with y = (y',y"). Then the operator L = (1/2Iy12)((y',c') - (y",8")) is well
defined on supp X (with CO° coefficients) and satisfies Lq = 1. Integrations
by parts will involve the transpose of L : tL = (8", y"/2Iy12) - (8', y'/2IyI2),
which is also a first-order differential operator with C°° coefficients, and N
such integrations by parts thus give
f etvZ((y)b(µy)X(y)dy =
(aµ2)-N f (LNetµ2q(y))b(µy)X(y)dy
_ (iµ 2)-N
J
dy
_ (ii, 2)-N
where cµ,N is a linear combination with C°° coefficients of terms of the form
µ1 a1((8ab)(py))(81 X(y)) for Ia +,31 < N. The result follows since supp X is
compact.
I
The main result of this section is that oscillatory integrals behave essentially
as absolutely convergent integrals.
35
Oscillatory integrals
THEOREM 2S
The integrals defined in Theorem 2.3 satisfy the following properties:
(i)
Change of variaables. if A is an invertible real matrix.
J eiq(AV)a(Ay)I det Al dy = J eiq(x)a(x) dx.
(ii) Integrationn by parts: if a E A", b E A', and a E r+,
(iii)
f
ei9(x)a(x)8'b(x) dx = f b(x)(-a)"(e`q(x)a(x)) dx.
Differentiation under f : if a E A' (1R" x RP), then f eiq(x)a(x, y) dx E
Am(1RP)/ and
a°
(iv)
J
for all a E Z.
eiq(x) a(x, y) dx = Je'Oa(xY)dx
Interversion of the f : if a E At(1R" x RP) as in (iii) and if r is a
nondegenerate real quadratic form on RP,
f
eir(y)
(f elq(x)a(x, y) dx) dy =
fe*()+)a(x,y)dxdy.
PROOF The proof of this theorem consists essentially in checking that all the
integrals written in the statement are oscillatory integrals, then in taking the limit
0 as in the definition from Theorem 2.3. Thus, property (i) follows
for f
dx,
from the change of variables x = Ay in the integral f
since '(y) = p(Ay) E S and satisfies &(O) = cp(0) = 1 and since b(y) =
I det AIa(Ay) is an amplitude of order m.. Similarly, integrations by parts in the
right-hand side of (ii) where we add a factor cp(Ex) give a factor
&'(so(Ex)b(x))
),Ell" (013 )(Ex)e-'b(x),
and for 0 # 0, the EIAl gives 0, when taking the limit c -. 0, while for = 0
we get the left-hand side of (ii). (The details are left to the reader.)
For (iii) and (iv), the proof is not as easy. Coming back to the proof of
Theorem 2.3, one considers the integrals
Ij (y) = Je"a(xY)IJ(2'x)dx
which satisfy a&I3(y) = f eiq(x)a&a(x, y)i,t(2-ix) dx thanks to the absolute
convergence given by the factor ''(2-l x). Since I a_0a& a(pz, y) I < CQ3
(1 + Ipz12 + Iy(2)m/2 _< CCp5m/2µ'"(l + Iy12)m/2 for Izi < 2, we get from
Lemma 2.4 estimates
Ia& I2(y) - 8&Il_1(y)I <_ CC2-1(l +
IYI2)m/2
Pseudodiferential Symbols
36
which imply uniform convergence on every compact set for the sequence 80 Ij (y).
It follows that the limit I(y) of the sequence Ij (y) is in Am(RP) and satisfies
0I(y) = limj-. c' I) (y), which is formula (iii). Moreover, this estimate also
shows that
Ir(I(y) - Ij(y))I : C°2-1 (1 +
Iy12)m/2
so that the functions b3(y) = 0(2-3y)(I(y) - Ij(y)) satisfy bj E Am(RP) with
lllbjlllm+p+i <_ Co2'j. Then one can write
fei'(y) (feiQa(x,y)dx) dy = Jim00 fe1t1)I(Y)1,(2_iy)dY
and
f eir(v)I(y)V)(2-Jy)dy= f eir(y)Ij(y)VG(2-2y)dy+
Je
ib(y
thus get property (iv) since
jx J
lim
eir(y) Ij
(y)'(2-' y) dy =
r ei(e(x)+r(y))a(x, y) dx dy
f
and
if
eir(y)b,
(y) dy < Cr,rIllb)Illm+p+l < Cr,mCo2-'
I
Example 2.6
(i) If a E A' (RI), then
(27r)-n
J e-i(y,n)a(y) dy dr) = (2ir)-"
J
e-`(y.n)a(7l) dy d77 = a(0).
(ii) If a and 0 E Z+,
y° p dydri _
(27r)-n
J
a! Q!
=
0 if a 0
)3;
a
if a
PROOF The quadratic form on R2n E) (y, r)) F--i (y, 7l) is obviously nondegenerate since the identity (y, q) = (1/4)(Iy+qI2- Iy-7i12) shows its n positive and
n negative eigenvalues. On the other hand, the polynomial y°rla is in Al°+131 so
that all the integrals written in the statement are oscillatory integrals, as defined
in Theorem 2.3.
(i) The first equality follows from exchanging y and 77. Then take a p E S
such that p(0) = 1. By definition,
r e-'(y.n)a(71) dy drl = li m
r
J
dy drr.
37
Operations on symbols
Through the change of variables ey = z, e(= rl then integration in z we get
f
f
dz d(
For f < 1,
d(.
I(I2)m12lwlo, which
IIIatlIo(1 +
is inte-
grable, so that by dominated convergence we get
(21r)-"
f e-;(b,») a(q) dy d77 = (2ir)-" f P(()a(0) d( = V(O)a(O) = a(0).
(ii) For a and 0 E Z+, yae-'(b,'7) = (-D,,)ae-'(Y,')), from which
e-i(b,n) yQ
(2.r)-"
a dyd77 =
(27r)-"
e-;ly,n)
Dn (
1
1 dydi7.
The function
a() =
satisfies a(O) = 0 if Q
follows from (i). I
2.3
(iii)
(Z)k
a
()7s-Q
a and a(0) = (-i)1°I /a! if 3 = a, so that the result
Operations on symbols
We can now define the symbols a* and a#b, which will be the symbols of the
adjoint and the compound.
THEOREM 2.7
Let a E S' and b E St; then the oscillatory integrals
- y,( - 77) dy d77
a* (x, () = (27r) -"
a#b(x, () = (27r)-" r e-'(",v)a(x, - tl)b(x - y, C) dy dij
define symbols a* E S' and a#b E S'"+t with the following asymptotic expansions:
Ot'Daa
aa
and
± &t aDz b.
a#b
a
PROOF The quadratic form (y, 77) is nondegenerate (see the proof of Example 2.6). To see that the function bx, (y, 77) = a(x - y, - r7) is an amplitude,
Pseudodiferenrial Symbols
38
we simply remark that Peetre's inequality (Lemma 1.18) gives the estimates
18" 8» a(x - y, - 0 1 S
C°3\, (C - n)
< CR,321"'lAm(S)(l + Iy12 +
I71I2)Im1/2
for all a,$ E Z', from which bx, E AI-I(lY2") With Illbx,{IIIImj+2,'+I
CoV (l;). Thanks to the estimate given in Theorem 2.3, it follows that a-"`a'
S"'-1131, we get
and 8x 3 a E
is bounded, and since 82 8 (a`)
the boundedness of A I0I -'8x 8. a' for any a, 0 E Z. by the same argument, so that a' E S'. The proof of a#b E S`+P is similar, since the
function cx,{ (y, rl) = a(x,e - rl)b(x - y,{) satisfies c,.,f E AI'nl (R2') With
and since
IIICX,
8°(a#b) _
I (a`ja)#(a°-fib)
for all aEZ+.
To get the asymptotic expansions, we use Taylor's formula (cf. Theorem 1.1):
(a) ()F
Ox - Y'!; - n) _
a(x, ) + rk (x, , y, rl)
I°+131 <2k
with
rk (x, C y, rl) =
1 °+RI =2k
2k(-y)°
(-rl)a
a!
,0!
r°Q(x, , y, rl )
and
1I
r.0 (x, , y,'7) =
Jo
(1 -
t)2k-18y 8{ a(x
- ty, - trl) dt.
The terms for ka + of < 2k give after integration the terms of the expansion
in view of Example 2.6(ii). For rk, which is clearly in AIm1+2k in (y, 77), we
integrate by parts as in Example 2.6(ii):
J
e
a!
Q!
r°,5(x, , y, n) dy drt
p
1
(e-`(b,")) dy dq
y, rl) Dn
J(a fe_'")(-D,')((_Dt)°-r°y(x,,y,))dydn
&
Or
7
(a)1-Y I
/\
0)
/e-;(b,+r)(_)A-7(-Dn)°-7r°p(x,,y,rl)dydrl-
39
Operations on symbols
A second integration by parts then gives
(-i)171y!
7
y
\a/
a!/3!
Y
V e-
Recalling the definition of ra,q, one has
S, y, rl)
f(l 1
=
- ty, - tq) dt.
Since y < a and -y < 0, one also has Iry(< k and Ia + Q - -yl > k, then
+v-_Yaf +f1-'a E Sm-k. Thus, the calculations given above can be rephrased
f e`(y.n)rk(x,C y,rl)dyd?1 =
f
where now the amplitude sk satisfies sk E AI"'-kl with IIISkIIIIm-kl+2n+1 <
CkA' -k(t ). We thus get the boundedness of
Ak_m(S) f e-1(y.^'rk(x,C y, z) dydrl,
and using the same argument as above, we get
f
f, y, rl) dydrl E Sm-k
just by remarking that i3Ot rk is the rest of index 2k in Taylor's expansion
Sm-101. Finally, the
of c,'8 a(x - y, - r)), for which one has
E
asymptotic expansion of a#b can be obtained along the same lines, but its
technical verification is left to the student.
I
REMARK AND EXAMPLE 2.8
If a is a differential symbol (that is, a polynomial
in t: with coefficients in HO0), we can follow the previous proof with k =
m + 1, which leads to the terms
which are identically 0
since J a + 3 - y I _> k > m. Thus the asymptotic formula for a' is exact
and contains only the terms with Ial < m. For the same reason, we also have
a#b = E{al<,n(I/a!)O I) b for any b E St.
It would be interesting to give the same remark if a is a polynomial in x with
coefficients depending on l;, but such functions are symbols only if the degree of
the polynomial is 0, that is, if a depends only on . Yet, the argument still works
and gives a" = a and b#a = ba for any b E St. In particular, this is the case
for the symbols A' which thus satisfy (A'")' = A"' and At#Am = .1t+` (and
more generally, b#Am = ba'n for any b E St). As for higher order polynomials
in x, which are not symbols, we will see (in Example 3.5(ii)) a formula quite
similar to the finite development of a#b we could expect. I
Pseudodifferential Symbols
40
PROPOSITION 2.9
The operations defined in Theorem 2.7 satisfy the properties
(i) (a*)* = a.
(ii) a#1 = 1#a = a.
(iii)
a#(b#c) = (a#b)#c.
(iv)
(a#b)' = b'#a'.
PROOF One has
(a*)+(x,() _
(2Tr)-2n
(27r)
f
(fe-;(Y'h1)a(x_z_Y_(_77)dYd1)) dzd(
-2n f e=((b,n)-((z,())a(x
- z - y, - ( - 77) dy dij dz d(.
Using, then, the change of variables Y = -y, H = 77 + (, Z = z + y, Z = (,
for which (y, 77) - (z, () _ -(Y, H) - (Z, Z) and dy d&J dz d(= dY dH dZ dZ,
we get
(a*)* (x, O = (27r)-2nJ
e-i((YH)+(z.Z))a(x
- Z, ( - H) dY dH dZ dZ
(27r)-2n re-i(z.z) (fe_"1a(x_Z,
_
= (27r)-n J e-'(z,z)a(x
H) dY dH) dZ dZ
- Z, )dZdZ =
where the last two equalities come from Example 2.6(i). Formulas a#1 = 1#a =
a simply follow from Remark 2.8 above.
To prove (iii), one writes
a#(b#c) (x, () _
(27r)-zn
f e-(1,I) a(x, ( - rf)
(Jeb(x _ (2r)-2nJ
y,
- ()c(x - y - z, () dz d() dy di
e''((v,n)+(z,S>)a(x,
- r!)b(x - y,( - ()
c(x - y - z, () dy dri dz d(,
Operations on symbols
41
then
(27r)
-2n f e-+(Z.Z)
2 - H)b(x - Y,
CJ
e-i((YH)+(Z,Z))a(x,
_ (27r) -2n
f
.2) dY dH) c(x - Z, ) dZ dZ
- 2 - H)
b(x - Y, - Z)c(x - Z, ) dY dH dZ dZ,
and these two quantities are equal through the change of variables y = Y,
77=H+Z,z=Z-Y,c=Z.
Finally, for the last formula we have
b* #a*
(27r) -3n J e-+(t'T)
e-=(z,S)b(x
\J
- z,
rr - () dz do/
(Je_"a(x - t - y,
= (27r)
-3n Je-
- 7]) dy d7)) dt d7-
t - y, - 7l)
c)dyd7/dzdOdtdr
= (27r) -3n r e-'(-(Y,H)+(Z,Z)+(x,=))a(x
b(x-Z-Y,e-Z)dYdHdZdZdXdr
J
_ (27r)-2n
J
- Z, - 2 - H)
e-'(Z,Z)
(fetc'.h1)a(x - Z, - Z - H)b(x - Z - Y, - Z) dY dH)
dZ dZ
after a change of variables (Y = z - t - y, H = 77--r-C, Z = t + y, 2 = 7-+(,
X = z - t, E = 77 - r) then integration in (X, E-) (cf. Example 2.6). This ends
the proof of Proposition 2.9, since
(27r) -n
J
e-'(YH)a(x - Z, - 2 - H)b(x - Z - Y, - Z) dY dH.
I
42
Pseudod{ferential Symbols
To close Chapter 2, we examine the problem of inverting a symbol for the
operation #, that is, to solve - at least approximately - the equation a#b = 1
(or b#a = 1) for a given a E S'". Since b E St implies a#b E S-+' and
1 E So, it seems natural to seek in S-' the inverse b. The result can then be
stated as follows.
THEOREM 2.10 ELLIPTIC SYMBOLS
If a E S', the following four statements are equivalent:
There exists a b E S-' such that a#b - 1 E S.
(ii) There exists a b E S-' such that b#a - I E S.
(i)
(iii)
(iv)
There exists a b o E S
such that abo - 1 E S-1.
There exists an e > 0 such that Ia(x, C)I > eA"'(l;) for ICI >
/e.
Moreover, when these conditions are fulfilled, a is said to be elliptic, and there
exists an ae E S-' such that
b solves (i)
PROOF
a
b solves (ii)
a
b - a0 E S-°°.
If a E S"' and b E S-'n, then a#b = b#a = ab modulo S-1 in
view of the asymptotic expansion given in Theorem 2.7. Thus each of the
statements (i) and (ii) implies (iii); then (iii) implies the existence of an e > 0
such that
11 < 1/2 for ICI > 1/e, so that we have for such :
1/2, then
since bo E S-'n
Ambo
bounded, and this is (iv).
Conversely, if (iv) is satisfied, then c = A-'"a E So and satisfies Ic(x, ) I > e
for Il;') > 1/e. If F(z) is a C°° function defined on C and equal to 1/z for
Izi > e, F(c) E So thanks to Lemma 2.1, and then bo = A-mF(c) E S-'
satisfies abo(x,l;) = 1 for ICI > 1/e, which implies (iii). Now, if (iii) is
satisfied, then a#bo = I - rl and bo#a = 1 - sl with rl and sl E S-1. Let
us set rj = rl#rj_l E S-j, sj = S,_l#sl E S-J, b, = bo#rj E S-"'-I,
cj =sj#b0ES-'"-j, and finally bNF_12!ob, ES-' and c,,bo+>,2,lcj E
S-' by using the construction given in Lemma 2.2. One then has for any fixed
k E Z the following equalities modulo S-k:
a#b=a#l:bj=(1-rl)#(1+ > rj I =1-rk=1
j<k
c#a= (b0+
0<j<k
I
E Cj #a= I+ E s, #(1-s1)=I-sk=1
0<j<k
0<j<k
since rk and Sk E S-k. We thus get a#b - 1 E S-O° and c#a - 1 E S-O°, for
k is arbitrary.
43
Exercises
Finally, if b solves (i) and c satisfies c#a - 1 E S-°°, one has modulo S-1
b = (c#a)#b = c#(a#b) = c
which implies the uniqueness of the inverse modulo S- O°. Conversely, if b-a",
a#a# - 1, and a"#a - I are in S', this is also true for
aft-1 = (a#aM-1)+(a#(b-a#))
and
b#a-l = (aa#a-1)+((b-ae)#a)
since this is true for the four terms appearing in the right-hand sides.
I
The symbols A"' are all elliptic since they satisfy A"#A-' = 1. One
can remark that they even have exact inverses. For a differential symbol
a(x,f) = Ejai<ma"(x)e (and more generally for a polyhomogeneous symbol), the principal symbol is defined as the homogeneous part of degree m, that
is, the function p(x, t;) =
a,, (x)t;". By using the characterization (iv),
it is easy to show that such a symbol is elliptic if and only if Ip(x,
e>0
forxEW' and ICI= 1.
Exercises
2.1
Let e > 0, f? _ {S E C"; Jim (1 < elRe(I} and a(() a holomorphic function
on f1 satisfying an estimate Ia(()I < C(1 + ICI2)'"/Z for large (in 1. By using
Cauchy's integral formula, show that (1 - :p(t;))a(i;') E S°' for some function
p EC. with cp=l near l;=0.
2.2
The goal of this exercise is to show that a C°` function a(x,t;) is a symbol of
order m as soon as the functions A-'&.',a and ak-'".9E1a are bounded for all
j < n and k E Z+. (Notice that there is no assumption on the mixed derivatives.)
Let cp and
be two C°° functions of one real variable t satisfying
V(t)=l fort<1,
yp(t)=0fort> f,
ra(t) = I fort E (f/2, 11,
(t) = 0 for t (1/2, 03).
Then, for a E C°°(RZ") satisfying the assumptions given above, x E R", p > 1,
and(y,i)ER" xR",set
a..,. (y, 17) = v(IyI)'(InI)i °'a(x+y,p71)
if µ>2,
if p < 2.
= v(IyI)v(IiiI)p ma(x + y, p i)
Show that for (y, rj) E supp a,,,µ one has (1 /2)A(pi1) < p < 2A(µ7j). Show
that there exists a sequence Ck independent of x E R" and u > I such that
Iay,ax.pl° < Ck and IO,k,,ax.µIo < ck for all j < n and k E 4+. Show that there
exists a sequence Ck independent of x E R" and p > I such that ax,, E H°°(R" x
ax.,.(y, )7)
ar
R") with Ilax,,,Ilk < Ck for all k E 4. Show that
a
R", the function a(x) =
f) _
that
E
satisfies a E A°. If
A is a real symmetric nonsingular n x n matrix, show that the oscillatory integral
Pseudodiferential Symbols
2.4
e'(^=._)l=
UW = f e;('t= =)/2a(x) dx is in P° and satisfies (U, gyp) = f
p(x) dx
for all V E S. Then compute U(.) by using Exercise 1.6(b).
More general oscillatory integrals. Let tP be a real-valued C°° function defined
on 9t" \ {0} such that Op never vanishes on It" \ {O}.
(a)
Show that there exists a first-order linear partial differential operator L with
C"° coefficients in R" \ {0} such that Lv = 1.
Let k E JR and X E Co be such that X = 0 near 0. Show that
J
elakw(v)b(py)X(y) dyl
CNI.tN('-k)
sup
I (Ob)(µy)I
LEsuppx.jnjGN
for all N E Z+ where the constants CN do not depend on u j> 1 nor on
b E C°°.
(b)
Let k and i E lR be such that k + i > 1, and assume that p is also
homogeneous of degree k. Let a be any amplitude taken from the space
AL = {a E C°`; At* -m(x)O"a(x) is bounded on 1R" for all a E Z. }.
Show that for any t' E S such that t;,(0) = 1. the limit
J
lim I e"0(=)a(x)y)(ex) dx
o
exists, is independent of io (as long as t!)(0) = 1) and is equal to
J e"°t )a(x) dx
when a E L) (this limit will also be denoted by f e"')=)a(x) dx when
(c)
Give an estimate of this integral, as in Theorem 2.3.
If 96 0, show that the function W4 (x)
(x, ) satisfies the assumptions
of question (b).
If a E A' for some m and i > 0, show that the function
f e" °f (=)a(x) dx satisfies A E P.
Show that if t' E S satisfies ib = 0 near
2.5
= 0. then (A,ty)
conclusion?
Let a E S'" and b E St. Rewrite the asymptotic expansions of Theorem 2.7 to get
simple expressions of a' and a#b modulo S` and
Sm+r-2, respectively. Then,
relate the symbol a#b - b#a to the Poisson bracket of a and b, which is defined
as (a, b} = (efa, 8=b) - (8=a, O (b).
2.6
Let a and b be two elliptic symbols. Prove that a' and a#b are elliptic and express
their inverses (a')' and (a#b)' in terms of the inverses a' and V.
2.7
Conversely, assume that a and b E S°° are such that a#b is elliptic. Show that
a and b are elliptic. What conclusion could you give when you assume that a#b#c
is elliptic?
Let a E S' satisfying
eX"'({) for all
I/e and some e > 0,
and let k E Z.+ \ {0}. Show how to construct a symbol b E Sm/k such that
a = b#b#...#b (k terms) modulo S.
2.8
This problem is made up of two exercises: in questions (a) and (b), one proves
that if a symbol a of unknown order satisfies a single estimate Ial < Cat, it is
automatically in St+ = n>IS,; in questions (c) and (d), one uses this property
in the study of nilpotent and idempotent symbols.
45
Exercises
(a)
Let k and q be two positive integers. Using Holder inequality, show that
IIaIIk <- IIaIIo-(1/°)IIaIIk/° for all a E Hoc.
Let K be a compact set. Show that there exist constants C depending
on K, k, and q but not on a such that IIaIIo 5 CIabo and
1/q
Sup Ia°alo
IIaIIk < CIa11-(I/9) (Ial!5kq
(b)
for all a E C' satisfying supp a C K.
Let f < m < p and assume that a E S' is such that A-ta is also bounded.
For x E R" and µ > 1, define the same functions a..,, as in Exercise 2.2.
Show that there exist constants CO independent of x E R" and p > I
and loO a=,,,Io < C°Qµp-".
such that Ia:,,Io 5 Cooµt
Use question (a) to show that there exists a sequence Ck independent of
x E R" and µ > I such that IIa,,,,Ilk 5 Ck for all k E Z+.
As in Exercise 2.2, conclude that a E S'".
(c) A symbol a E S°° is said to be nilpotent if there exists a k E Z+ such that
a#a# ... #a = 0 (k terms). Show that if this relation holds for an a E S',
then a E S"'-('/2k), and conclude that nilpotent symbols belong to S-°`.
(d) A symbol a E S°` is said to be idempotent if a#a = a.
Show that a is idempotent if and only if I - a is idempotent.
Show that if a is idempotent, then a E S' for some m > 0 implies that
a E S2m/3 U S"'-(1/3), and a E Sm for some m < 0 implies that a E S2m.
Show that an idempotent symbol a satisfies a E Sol = fl,">OSm and that
a2 - a = b for some b E S-213. Then, show that there exists an e > 0 such
1/4,Re(1 +4b(x,e)) > 0 (so that
that for ICI > 1/e one has
you can take the usual definition of (1 + 4b)' /2), 11 - (1 + 4b(x, ) )' /2I <
Finally, conclude that
4Ib(x,t)i and I1+(1+4b(x,C))'/2I >
a E S-'13 or 1- a E S-'/3 (one assumes n> 1).
2.9
Prove that an idempotent symbol a satisfies a E S-°° or 1- a E S-O°.
This long exercise will be continued in Exercise 3.6.
(a)
Quasi-elliptic operators. For any p E Z+ such that µ, > 0 for all j < n,
one sets Ia : µI =
(a,/µ1)+...+(a"lµ") and Ii;(N)I = (Ej 2"' )'l2. The
differential operator a(x, D) is said to be quasi-elliptic at xo if there exists
a p as above such that a(x, D) = >I°:,AI<l a°(x)D° and its quasi-principal
symbol p(z,C) =
vanishes at xo only for = 0.
Show that elliptic operators are quasi-elliptic and that the heat operator
Ej=2 (which is not elliptic) is also quasi-elliptic.
81 -
Let a(x, D) be quasi-elliptic at as and p(x,l:) = FI° Fl_i
be
its quasi-principal symbol. Show that there exists a constant C and a
neighborhood Il of xo such that
CIC(,.)I
for x E S2.
Then, prove that the functions (Ama)'' and a -' AP1 I gs ea are bounded
hand ICI > 1/e} for some mE R,e>0,andp>0,and
on
all a,l3EZ+.
(b)
The S,7,0 calculus. Let p > 0 and m E R, then define the class S,7,,
of symbols as the set of smooth functions a(x, l;) such that the functions
APIQI--O."
a are bounded on R" x R" for all a and,3 E Z.
Pseudodifer+ential Symbols
46
" for j E Z+ , one can construct an a E
Show that if a, E oS"
that a - E2<k a, E Spo °k (cf. Lemma 2.2).
SP o such
Show that if a E SP0 and b E SP, 0, one can define symbols a' E S'"110
and a#b E So o by the same formulas and with the same asymptotic
expansions as in Theorem 2.7.
(c)
Inversion of quasi-elliptic symbols. Let p > 0 be fixed, then let a E So
be such that the functions (,\'a)-' and a-' )PlB1OOa are bounded on
ICI > 1/e for some m E R and c > 0, and all a, 8 E Z. (cf. question (a)).
Show that there exists a E Co (R") such that the function bo(x, =
co(x,l;) _ (1 -,(l;))/a(x,satisfies bo = co E SP0 and .1P10+61(0,0 a)
(&'Obo) bounded for all a, )3, y, b E Z.
Show that the symbols b, and c, defined by induction as
(a)(Dzbk)
bj = -bo
Inl+k=j
k<j
and
(Mt ck)(Dx a)
c' = -co
1a1+k=
<j
satisfy b, and c, E Sp o P2 and
and
AP(J+Ip+61)(ooc,)(8
a)
b E Z+ .
bounded for all a,
Show how to construct symbols b and c E S" no such that a#b - I and
c#a - 1 are in S_0°1
3
Pseudodifferential Operators
3.1
Action in S and S'
For a E SO° and V E S, the integral defining a(x, D)V(x) in the introduction
to Chapters 2 and 3 (see also the statement of Theorem 3.1 below) is absolutely
convergent, and we are going to show that it even defines a function in S.
THEOREM 3.1
If a E SO` and cp E S, the formula
a(x, D),p(x) = (27r)
n
r el(=.f)a(x,
)(P(C) d
defines a function a(x, D)So E S. and there exist constants N E Z+ and Ck
for k E Z+ depending on a such that Ia(x, D)wlk < CkIwik+N (continuity
property).
PROOF
Since St C S'n for £ < m, one can assume that a E S2,n for some
m E Z+. Then wp E S implies cp E S and one can write
l a(x, D)w(x)I <_
(27r)-n r
which shows (cf. Lemma 1.3) that a(x, D)cp is bounded with (a(x, D)cplo <
then la(x, D)cwlo < Colcpl N with N = 2m + 4n in view of Theorem 1.8. Moreover, one gets
O) (a(x,
a(x, D)(a,w)(x) + (0 ,a)(x, D)Ax)
by differentiating under f, and also
xi(a(x, D)V(x)) = a(x, D)(xicp)(x) + i(Ot,a)(x, D)cp(x)
47
Pseudodiferential Operators
48
by integrating by parts. Thus, x"Oa(a(x, D)cp(x)) can be written as a linear
combination of terms (bid{a)(x, D)(xa-6OQ-7cp)(x) so that a(x, D)cp E S
I
with Ia(x,D)plk !5
The next step consists in proving that the operations of adjunction and composition of operators a(x, D) correspond to the operations introduced in Section 2.3.
THEOREM 3.2
For any aand bES°°and 'pand 1P ESone has
(a*(x, D)p,
(i)
(ii)
t) = ( a(x, D)V,)
,
(a#b(x, D)cp, bb) = (a(x, D)b(x, D)cp, ).
PROOF The quantity Io = (a' (x, D)cp, ) is equal to the oscillatory integral
Io = (21r) -2n
=
f
y,
(f
/ e,((=,e>-(=-=.e-<>)a(z,
(2T)-2n
- rl) dy d7l ) 0(e)0(x) dx dl;
dx Adz d(.
Similarly, Io = (a#b(x, D)cp, 0) is equal to the oscillatory integral
1o
=
(27r)-2n
e=(t=
o)b(z, )c3() {x) dx d dz do.
On the other hand, the quantities 1' = (<p, a(x, D)O) = (27r)-n(b, a(x, D)V))
and I" = (a(x, D)b(x, D)cp,,O) are given by the formulas
I' =
(22r)-2n
f Ab)
(Jei> (Je(z() (fe'(x)dx) doJ dz) d
I" = (27r)-2n
f
(Je"a(x,) (fe>
(fei'
We thus have to prove IS = I' and 100 = it.
b(z,
4) dz) do) f(x) dx.
Action in S and S'
49
Here we will provide the proof only for * since it is a good exercise for the
student to write the details for #. First, we write that Io is the limit for e
0
of the integral
IE _ (227r)-2n
t
J
where x E S can be chosen so that x = I in B1. Then 1' - I. = IE + IE + IE
with
IE =
(21r)-n
fe
(l
- X(e)X(Ez))a(z, D)O(z) d. dz,
x(E())0(()d4 dzd(,
1, = (27r) -2n
IE = (21r) -2n
ei((=,()+(_,()-(z,())g()a(z,
J
()x(e )X(Ez)x(E()
(1 - x(ex))(x) Adz d(dx.
The integral IE tends to 0 with a by dominated convergence. The integrals IE
and IE also tend to 0 with e, thanks to the following result.
I
LEMMA 3.3
Let a(x, y) E A'n(Rn x RP), cp a real-valued function, and x, -0, and w E S
with x = 1 in B1. Then
dx dy = 0.
y)w(Ex)(1 -
lim I
E---o
PROOF The change of variables z = ex gives
f
ei'P(=/(,Y)a(z/E,
y)w(z)(1 - X(Ey)) (y)e-n dz dy,
then a(z/e, y) is estimated by
IIIaIIIo(1 + Iz/EI2 + Iy12)m/2 <
IIIaIIIoe-m(1 +
IZI2)m/2(1
+
IyI2)m/2.
For y E supp (1 - x(ey)), one has IyI > 1/c, and this gives
11 + IyI2
1
1 +p
m-n-p
)
<
IyI2)-
Pseudodifferential Operators
SO
for Y E supp (1 - X(ey)), so that finally
Y)a(z/E,y)w(z)(1 - X(Ey))'+G(y)C' I
C EplllailIoC,0(1 + jz12)m12lw(z)I(1 + IyI2)-
which gives the result after integration (cf. Lemma 1.3).
1
Theorems 3.1 and 3.2 allow us to extend the operator a(x, D) : S - S as an
operator from S' into S' as follows.
DEFINITION 3.4 Given an a E S°°, we call pseudodifferential operator of
symbol a the operator a(x, D) : S' - S' defined by
(a(x, D)u, cp) = (u, a*(x, D)p)
for u E S', ,p E S.
If a E S", a(x, D) is said to have order m. The set of pseudodifferential
operators of order m will be denoted by W'. The set of all pseudodifferential
operators is q" = U,,,%P m and the elements of-°° = fl,,,4m are called
smoothing operators (because of the result in Corollary 3.8).
for is that of A' (D), which
was defined in Section 1.3 by the formula a"'(D)u = At u. Indeed, ProposiA simple example of a pseudodifferential
tion 1.20 shows that this operator is the same as the pseudodifferential operator
of symbol A'° as defined in Definition 3.4; of course, Am(D) has order m.
Another example is that of differential operators with coefficients a,, E H°O:
the student will benefit by showing that in this case the definition of a(x, D)u
through Definition 3.4 coincides with the usual definition (through differentiations and multiplication by coefficients as defined in S').
Even if these pseudodifferential operators are very similar to the differential
operators, one must take care that they do not share all their properties. In
particular, this is the case for the local property (control of supports): indeed,
in general, pseudodifferential operators do not satisfy
supp (a(x, D)u) C supp u
for all u E S'.
(One can even prove that the only operators a(x, D) E W°° possessing this
property are the differential operators; cf. Exercise 3.1.) However, it is clear
that if p E C°° satisfies p = I in a neighborhood of supp u, then one has
a(x, D)u = a(x, D) (cpu) = a#W(x, D)u,
and it follows from the asymptotic formula for the operation # (Theorem 2.7)
and from Lemma 2.2 that a#cp = b+r with supp b(x, D)u C supp xb C supp cp
and r E S-O°. Thus one will keep some control on the supports provided that
the computations are achieved modulo smoothing operators.
Si
Action in S and S'
Instead of the local property, we will see in Corollary 3.8 that pseudodifferential operators satisfy the so-called pseudolocal property (control of singular
supports), i.e., they do not increase singular supports:
sing supp (a(x, D)u) c sing supp u
for all u E S.
Actually, we will eventually show (in Section 4.2) a more precise result, namely
the microlocal property (control of wave front sets), which implies the pseudolocal property.
Before beginning the next section, let us show that the rather implicit Definition 3.4 allows us to carry out explicit computations: formula (i) below will
show that one can recover the symbol of a pseudodifferential operator from the
operator itself (thus the correspondence between S°° and 'P°° is one to one),
while formula (ii) extends Remark 2.8 to the case of polynomials in x. Indeed,
if
E(aI<kxaba(t;)( S°°) for some
E S°°, one could define
an operator b(x, D1 : S' - S' by the formula
b(x, D)u = E xa(ba(D)u)
for u E S'
Ial<k
since multiplication by polynomials is well defined on S'. Formula (ii) below
then means that for such an operator and any a E SOD, a(x, D)b(x, D) = c(x, D)
where c is formally given by the expansion of a#b obtained in Theorem 2.7 with
only a finite number of nonzero terms.
Example 3.5
(i) Let a E S°°. Then, for any fixed
E R, one has
a(x,D)e'(x,f) =
a(x,S)ei(x,f>.
(ii) For any polynomial p(x), a E S°°, and u E S', one has
Dzp(x)(O a)(x,D)u.
a(x, D)(p(x)u)
PROOF
(
a
(i) We first compute a* (x, D)cp for cp E S as follows.
i, a* (x, D)cp)
=
a* (x, D)cp) = (a(x, D)'i', cp)
for ' E S,
and since the Fourier transform of r/' is (2rr)nm , this is also equal to
r (ir -n J et(x,')a(x,
e)(2ir)'
de) ;P(x) dx
=
f(e)
(fea(x)(x)dx)
dC
52
Pseudodiferential Operators
f e'(x,{>a(x,
by Fubini's theorem, so that a (x,
orem 1.6(ii)). Then one has for every W E S
(a(x,D)e'(=,f>,v) =
dx (cf. The-
(x, D)(x) dx
Je'a(x)(x)dx
= a* (x,
which proves (i).
(ii) Using the linearity of the operator a(x, D) it is sufficient to prove the
formula for p(x) = xR. First, for p E S,
xpa'(x,
(27r)-n
f
(21r)-"fe(_D(a*(x,())d
( l ((-DF)"a')(x,
= (27r)-" J e'(2,E>
\
a
J
a! (Dt a)"(x,
since
(aaJ xR
/
=
CEO xR-
Then we get
(a(x, D)(x'u), cc) = (u,?Ra*(x, D)c) _
a! (u, (DE a)"(x, D)((ol x' )c))
_ Ea a! ((a°xI)(DF a)(x, D)u,'p)
which gives the result, since the factor (-i)I°I can be transferred from the D{
to the 8.
3.2
1
Action in Sobolev spaces
For the definition and elementary properties of Sobolev spaces, we refer to
Section 1.3. In particular, we showed in Corollary 1.19 that differential operators
of order m with coefficients in HOD map continuously H' into H'-' for any
s E R. This property can be extended to any pseudodifferential operator of any
order m E R
Action in Sobolev spaces
53
THEOREM 3.6
Let a E S"`; then for every s E R there exists a constant C, such that
a(x, D)u E H'-m for all u E H', with IIa(x, D)ulla-m < Ca Ilufl8.
PROOF Assume first that we have proved that for any b E So there exists a
constant Cb such that
for all r' 1E S.
IIb(x, D) 'IIo <- CbIIV IIo
Then, if a E S' and s E R, we can set b = A-'#a'#A'-' E So, and using
Proposition 1.20, one gets for W E S
Ila* (x,
IlA-`(D)a'(x,
IIb(x,
< CbIlAm-'(D)WIlo = CbllCppll
o
-s
so that for all u E H' and still p E S
I(a(x,D)u,')I = I(u,a'(x,D)W)I <- Ilullslla*(x,D)VII
<CbllullallVllm-8
Thanks to Proposition 1.15, this implies that a(x, D)u E H'-m with
IIa(x, D)ulla-m S Cbllulls as required.
Thus, all we have to prove is the estimate IIb(x, D)0110 S CbIIV'IIo, t' E S,
for any b E So. This will be done in three steps: first when b E S-n-', then
when b E S' for some m < 0, and finally in the general case b E So.
(i) Operators of order -n - 1. Let b E S-"-' and t' 1E S. One has
(21r)-n J e"x,l) b(x, f)tG(e) df
b(x, D)tP(x) =
= (21r)-" J e=(x-v,N(x, )V)(y) dy 4
=
f
K(x, y)',(y) dy
where
K(x, y) = (27r) -' f
e'(x-u,f)b(x,C) dC
since all these integrals are absolutely convergent thanks to the condition b E
S-"-' (cf. Lemma 1.3). This kernel K(x, y) satisfies for any a E ?+
I(x - y)QK(x,y)I = (27r)-" J(x -
(2a)-" t ei(x-v,f)
y)°e`(x-v,E)b(x,
0A
b(x,t)dal <Ca
Pseudodiferential Operators
54
after integration by parts since A"+1+I'IOb is bounded. Therefore, one has
(I + Ix - yi2)" I K(x, y)I < C/ir" for some constant C, so that
f IK(x,y)Idx<C
and
The result then follows from the next lemma, known as Schur's lemma.
LEMMA 3.7 SCHUR'S LEMMA
Let K(x, y) be a measurable function defined on R" x lR" and satisfying
f
I K(x, y)I dx < C
and
f
for some uniform bound C. Then for any IP E L2, one has K(x, y)yP(y) E
L' (dy) for almost every x, and the function p(x) = f K(x, y)zb(y) dy is square
integrable over R' with
C110110
PROOF The function K(x, y)Vi(y)K(x, z) ,b(z) is integrable over R" x R' x R1
since 1,0(y)V(z)I < 1(It'(y)I2 + 1,0(Z) 12) and
Jk(y)I2 (JIK(xY)I (JIK(xz)Idz) dx)
2
00.
dy <
Thus, by Fubini's theorem, K(x, y)z'(y) E L'(dy) for almost every x and
f Ip(x)I2 dx =
1(1
K(x, y)V(y)
dy)
U
K(x, z)''(z)
dz) dx
< C111,0112
0*
(ii) Operators of negative order. Since U,<OSm = UkEZS-i/2k, it is sufficient to prove the property IIb(x, D)V)Ilo S CbIITI'IIo for the symbols b E
S-1/2k,
for all k E Z, and this is done by induction on k. Indeed, this is true for some
negative k, say for k = -n, according to step (i). Then, if b E
S-'/2k+,,
b*#b E S-;12k, and one can write
IIb(x, D),O112 = (b(x, D),O, b(x, D),O) = (b*#b(x, D)O,,O)
< IIb*#b(x,D)'IIo11'IIo 5
by using the recurrence hypothesis, so that IIb(x, D),0IIo <- CbIIiPIIo with Cb =
1/2
(iii) Operators of order zero. Now if b E So, b is bounded and IbIo - Ib12 E S°
is nonnegative. We can choose a function F E C°°(C) such that F(z) _
-
(1 + z)1/2 for z E R+, and it follows from Lemma 2.1 that a = (1 + Ib1o
IbI2)'/2 = F'(Ib1o - Ib12) E So. One has a* = a and a*#a = Ia12 modulo S'1,
then
a*#a + b*#b = 1+ Iblo + c
for some c E S-'.
55
Action in Sobokv spaces
Thus one can write
Iib(x, D)iIIo < IIa(x, D),PIIo + IIb(x, D)'IIo = ((a'#a + b'#b)(x, D)?P, 0)
((I +
(c(x, D)1, VG) <- (1 + Iblo + CC)Ik1'IIo
since c E S-' satisfies IIc(x, D)-+'I10 <- CeII'IIo, thanks to step (ii).
I
As a consequence of Theorem 3.6, we can show that operators of order
-oo are smoothing (this explains the terminology used in Definition 3.4), and
that pseudodifferential operators do not increase singular supports (the so-called
pseudolocal property). Since elliptic operators (i.e., operators with elliptic symbols) are invertible modulo smoothing operators, they do not decrease either
singular supports. This latter property is called hypoellipticity.
COROLLARY 3.8
If a E S-°°, then a(x, D) maps E' into S and S' into P; similarly, if a E S°°,
then a(x, D) maps S into S and P into P. Finally, any pseudodifferential
operator a(x, D) E 41°° has the pseudolocal property
sing supp (a(x, D)u) C sing supp u
for all u E S',
and one even has sing supp (a(x, D)u) = sing supp u if a is elliptic.
PROOF For a E S-°° and u E E' one has u = i4u for some & E Co (provided
that * = 1 near supp u) from which u E H-N for some N E Z+ thanks to
Lemma 1.16. From Theorem 3.6 we then get a(x, D)u E H" which contains
only bounded continuous functions (cf. Proposition 1.14(ii)). Using the same
argument as in the proof of Theorem 3.1, we can now write x'BA(a(x, D)u) as
a linear combination of terms (8T8{a)(x, D)(x°-b81-yu) which are bounded
continuous functions for the same reason, and thus we get a(x, D)u E S.
If now u E S', one can write
D' (a(x, D)u) = ba (x, D)u
where ba = r°#a E S-O°.
Thanks to Lemma 1.16, there exists an N E Z+ such that v = (1 + IxI2)-Nu E
H-N. Thus,
ba(x, D)u = ba(x, D) ((I + IxI2)`vv)
(DO (I + IXI2)N)(e, b,,,) (x, D)v
(cf. Example 3.5(ii)), and one still has 8{ b(k E S-°° for all a and,3. It follows
that (&ba)(x, D)v is a bounded continuous function, since it is in H" thanks
to Theorem 3.6, from which Da(a(x, D)u) = b,, (x, D)u E P°.
Then, if a E S', Theorem 3.1 shows that a(x, D) maps S into S. If
u E P, we first remark that \2k(D)u E P° for any k E Z+, since a2k(D)
is then a differential operator with constant coefficients. Therefore there exists
56
Pseudodifferential Operators
an N E Z+ depending on k such that v = ( 1 + I x t 2) - N A2k (D) u E L2 (cf.
Lemma 1.3). Thus, for any fixed a E Z+, we take k such that 2k > m + Ica I + n,
which implies that ba = fa#a#A -2k ES-". We then write
D"(a(x, D)u) = ba(x, D) A 2k(D)u = b,, (x, D)((1 + Ix12)NV)
(DO(1 + Ix12)N)(O ba)(x, D) v.
Here, we have 4ba E S-' and v E L2 = H°, so that (Obn)(x, D)v E H",
thanks to Theorem 3.6, and we can conclude as above.
For the pseudolocal property, let a E S°°, U E S', and set St = R" \
sing supp u. Then 'Ou E Co for all ' E Co (1l), and for any
can find a V) E Co (S2) with = I near supp ;p and write
E Ca (SZ) one
cpa(x, D)u = cpa(x, D)('u) + cpa(x, D)((1 - iP)u).
The first term is in S since 7Pu E Co C S, and the second term can be
rewritten b(x, D)u with b = cpa#(1 - P) E S-'°, thanks to the asymptotic
expansion for the operation # (Theorem 2.7) since the supports of cpa and 1 - iP
do not meet. We thus get cpa(x, D)u E C'° for all <p E Co (1), from which
a(x, D)u E C'°(Q) or sing supp (a(x, D)u) C sing supp it.
Finally, if a is elliptic, there exist some b E S'° and r E S
such that
b#a - 1 = r, then it = b#a(x, D)u - r(x, D)u with sing supp (b#a(x, D)u) C
sing supp (a(x, D)u) and r(x, D)u E P as proved above, and this shows that
sing supp u C sing supp (a(x, D)u) also.
I
In Theorem 3.2 we established a link between operations on operators and
operations on symbols. The next statement will give a link between estimates
on symbols and estimates on operators. In the simplest version of this result
(known as Girding's inequality), the symbol a E S2," is assumed to satisfy the
elliptic type estimate Rea(x, ) >
for all ICS > 1/E and some E > 0 (cf.
Theorem 2.10(iv)). Since on ICI < I /f a is bounded (from above) and A2ni' 1 is
bounded from below, this assumption clearly implies Re a + Coa'-'"-1 _> EVrit
everywhere for some large constant Co, and conversely if this latter estimate
holds, Re a can be proved to satisfy an elliptic type estimate with a smaller
f > 0. In our statement we will therefore use this more convenient form of the
assumption.
THEOREM 3.9 GARDING'S INEQUALITY
Let a E S2m and assume that for some Co and c > 0 one has Re
E)I2m (assumption satisfied in particular when Re a(x, C) >
I /E). Then for any N > 0 there exists a constant CN such that
2Re (a(x, D)V, gyp) > EIIV1122 - CNIIc I12 _N
a+CoA2m- i >
for 1 I >
for all cp E S.
Action in Sobolev spaces
57
Let us set b =
\-m#a#a-m E So. Since b = A-2ma modulo S-1, the
assumption on a implies that Re b + (Co + CI )A- I > for some constant C1, so
that b itself satisfies the assumption in the theorem with m = 0. If we assume
momentarily that the result is proved when m = 0, we can write for ,p E S
PROOF
2Re (a(x, D),p, ,p) = 2Re (b(x, D)Am (D),,, A' (D),p)
?EIIA"`(D)WIIO-CNIIA'(D)wII? N=EIIwIIm-CNIIwII,,,-N
(cf. Proposition 1.20). Therefore, it is sufficient to prove the theorem in the case
m = 0.
Thus assume a E So with Re a + CoA-' > E. We can choose a function
F E C°°(C) such that F(z) =
z)1/2 for z E lll, and since 2(Rea +
CoA-' - ) E So is nonnegative, it follows from Lemma 2.1 that b = (2Re a +
F(2(Rea + CoA-' - E)) E So. We can write modulo
S-1: b*#b = 2Rea - (3/2) = a + a* - (3/2)E, which implies
2CoA-1
a+ a*
for some cE S-1.
Then for w E S,
2Re (a(x, D),p,,p) = (a(x, D),p,,p) + (,p, a(x, D),p) = ((a + a')(x, D)w, w)
= (b"#b(x, D)w, V) +
(23
'Y' 1P
+ (c(x, D)w, V)
> Ilb(x, D)wll0 + 2EIlwll0 - II c(x, D)wlII/211wI1-1/2
>_ Ellwll0 + (211w110 - C112114' 1/2)
for some constant CI/2 since c E S'. Thus the result will finally come from
the estimate
CI/2IIwIL1/2 !5 211wII0 + CNI1wII2-N
with CN =
(2C)2N
which can be proved as follows: when CI/2A-V) > /2, one has
then
C1/2A-V) =
so that
CI/2A-1 <
+G,NA-2hr
from which one gets the estimate after multiplication by I,p12 and integration.
I
58
Pseudofifferential Operators
To end this section, we simply point out that the result of Theorem 3.9 is still
true with N = 1/2, when a is replaced with 0 in the assumption. The proof of
this stronger version, known as the sharp Girding's inequality, does not require
more theory than what we have here, but it is too long and technical to be
given in this elementary course. Thus, we simply refer the interested reader to
Hdrmander [8, Theorem 18.1.14]. Even sharper estimates are due to Melin and
to Fefferman and Phong (see references in the Notes to Chapters 2 and 3).
3.3
Invariance under a change of variables
The first obstacle one meets when one wants to prove the invariance of this
theory under a change of variables is the lack of invariance of the spaces S,
S. and H8 where everything was done up to now. However, these spaces are
clearly invariant under a linear change of variables.
In this framework, let us introduce the following notation: if X(x) = Ax + b
is an invertible linear change of variables in R',' and if V E 8, one defines
the transform V. E S of ep under the change of variables X by the formula
px(y) = 'p o X-'(y) = cp(A-'(y - b)). When performing the change of
variables y = Ax + b in the integral, one finds that for all V and /i E S.
(W,,, ,O) =
where IX'I = IdetX'I = IdetAI (also denoted by JAI)
and z/ix-. (x) = V, o X(x) = ry(Ax + b). Therefore, if u E S', we define a
distribution ux E S' by the same formula (ux, cp) = (u, JX'Jcpx-. ).
Example 3.5(i) allows us to guess what the symbol of the transformed operator
will be. Indeed, one must have ax(y,1J) = e-'(y,'')ax(y, D)e'(y"n), and this
leads to the following computations:
ax (y,rl) _
(e-'(Az+b,n)a(x,D)ei(Ax+b,q))
Ix=A '(y-b)
(e-t(a,A7)a(x,
D)e'(x,`Av7)llx=A-'(y-b)
=
a(A-'(y - b), Arl)
And, indeed, one can state the following proposition.
PROPOSITION 3.10
Let X(x) = Ax + b be an invertible linear change of variables in R. Then
a E S'° if and only if ax E S"' where ax is defined as
ax (y, r!) = a(A-' (y - b), Arl) = a(X-' (y), tX'rl)
'This only means that the matrix A is invertible. Likewise, we say that X : n
0x is an
invertible C' change of variables to mean that x is one to one, indefinitely differentiable, and that
its Jacobian matrix X' is invertible at every point of Q.
Invariance under a change of variables
59
Moreover, one has for u E S'
(a(x, D)u)x = ax(y, D)ux
PROOF We have eA(77) < A(%,7) < A(71)/e for some e > 0 depending only
on the matrix A. Since the derivatives of ax are equal, up to multiplicative
constants, to the corresponding derivatives of a at the point (A-' (y - b). Ari).
we thus get the equivalence a E S'
ax E S"'.
For cp E S, one can write
- '(y,>>)cp(A-'(y - b)) dy
cPx(77) =
= e-i(b.77) f e-i(x,An)W(x)IAI
dx = e-"(b,n)43(Arl)I AI
through the change of variables y = Ax+ b. Similarly, the change C=%? gives
ax(y, D)cox(y) = (27r)-" f e'(y,n)a(A-'(y - b),
= (27r)-" f
ei(A-'(y-b),t)a(A-'(y
Arl)e-'(b.,,)
(All)IAI dr7
- b),d
= (a(x, D)cp)x(y)
Taking the scalar product with a V) E S, we get the identity a* (x, D)(z'x-,) _
((ax)*(y, D)1i)x-c, which gives
((a(x, D)u)x+'f') _ (a(x, D)u, IAI x-c) _ (u, I AI a* (x, D)('x-c ))
_ (u,IAI((ax)*(y,D)'+b)x-c) _ (ux,(ax)*(y,D)'+b)
_ (ax(y, D)ux,'c')
The case of a nonlinear change of variables is much more intricate. Indeed,
since such a transformation is usually only locally defined, we first have to define
the local action of a pseudodifferential operator if Q is an open set of R" one
says that a E Sa(S2) if a is defined on fl x R" and satisfies cpa E S' for any
cp E Co (fl). Then, for a E Sa(Sl), cp E Co (SZ), and U E S', the distribution
cpa(x, D)u E S' satisfies supp (cpa(x, D)u) C supp cp, and therefore one can
define an operator a(x, D) : S' - D'(cl) by the formulas
(a(x, D)u, cp) =
D)u, 1) = (u, (<pa)If=O)
for cp E Co (Sl)
where the last equality comes from Example 3.5(i). When restricting this operator to e'(Sl) C S', one gets an operator a(x, D) : V(fl) D'(S2) for which a
good theory of invariance under a change of variables can be expected.
Indeed, if X : fl --+ S2x is an invertible C°° change of variables, one defines
cpx E Co (lx) by the formula cpx(y) = cp o X-'(y) for any cp E Co (fl),
Pseudodiferentia/ Operators
60
then, as above, uX E D'(1 ) by the formula (uX, cp) = (u,
for any
u E D'(1), where IX'I = I det X'I is now a C°° function. Again, in view of
Example 3.5(i), a natural guess for ax would be
a,, (X(x), rl) = e-i(X(=).n)a(x,
but this formula does not mean anything since e'(x(2),7) is defined only on SZ
and does not have a compact support. Thus, to get a good definition of ax, we
will have to modify this expression by introducing cut-off functions.
A theory of oscillatory integrals, quite similar to the results given in Section 2.2 but more general and known as the stationary phase method (or formula), is the key of the following result, the proof of which is out of reach in
an elementary introduction such as ours (we refer the reader to Hbrmander [8,
Chap. 7 and Theorem 18.1.17]).
THEOREM 3.11
Q. be an invertible C°° change of variables where Sl and Q. are
two open sets of R". Then a E S. (Q) if and only if (a#cp)X E Sa(QX) for all
V E Co (Q), where we set
Let X : Sl
(a#W)X(X(x), n) =
e-i(X(2),7)a(x,
D)(o(x)e`(X(Z).n)).
Moreover, for any u E E'(1) and cp E Co (St) such that cp = 1 on supp u.
(a(x, D)u)X = (a#,)X(y, D)ux
Finally, one can even write an asymptotic expansion for (a#cp)X which shows in
where
particular that (a#cp)X(y,q) = (a#cp)(X-'(y),'X'rl) modulo S
tX' is taken at the point X-' (y).
Even if the proof of this theorem is difficult in its full generality, there is
at least one situation where it can be written through elementary computations:
when a(x, D) is a differential operator, i.e., when a(x, ) is a polynomial in .
The student is invited to provide, in this case, the details of the proof (which
only makes use of the chain rule).
Such a result on the effect of a change of variables naturally leads to a theory
of pseudodifferential operators on a manifold M of dimension n. Since we
cannot provide here the basic definitions of analysis on manifolds, we simply
remind the student of some of its features: if p and ' are two C' functions
8v'i of
defined near a point m on the manifold M, then the equality
their gradients at m for a choice of local coordinates y implies the same equality
O2V = a2' for any choice of local coordinates x. The chain rule even gives the
m o r e precise result O (cp o X) (xo) = tX'(xo)(8yW)(yo) (here xo and yo = X(xo)
are the coordinates of m in the two systems), which also shows that the vector
space structure of gradients at m of C' functions is preserved after a change of
variables. This intrinsic vector space of gradients at m of C' functions, T,, M,
Exercises
61
is called the cotangent space of M at m (it is naturally the dual space of the
tangent space
at m), and the set of all (x, ) with x E M and E Ti M
is called the cotangent bundle over M and is denoted by T'M.
In the construction of a theory of pseudodifferential operators on a manifold
M, the symbols will be functions defined on the cotangent bundle T' M, but
the asymptotic expansion for (a#W)x (see Theorem 3.11) suggests that a symbol
a will be intrinsically related to the mth order operator a(x, D) only modulo
Sr'. In particular, in the case of polyhomogeneous operators, the principal
symbol is a true function defined on T*M, but the following terms are not
invariant under a change of variables. This phenomenon is very easy to check
on simple examples of differential operators, and we recommend these computations as a useful exercise (take for example a(x, D) = D1 - D2 and y = X(x)
defined by y, = x,, 112 = X2 + (xI/2)).
To go a little bit further in this theory, we should add that the next problem is
then to define the compound of two pseudodifferential operators with symbols
a and b E Sa(S1). Indeed, since a(x, D) and b(x, D) are operators from V(Q)
into D'(S1), one cannot in general define a(x, D)(b(x, D)u) for u E P(Q).
Thus, the operation # will be restricted to a subclass of Sa(1l), namely, the
subclass of operators that map V(Q) into E'(1l). The use of adjoints will then
lead to a smaller subclass of operators, the so-called properly supported operators, which can be extended as operators from V(Q) into D'(l), and the
necessity of dealing only with properly supported operators is balanced by the
fact that any operator in Ik'10C(f1) is equal to a properly supported operator modulo 'i
Therefore, the counterpart of the use of adjoints and compounds
will just be to achieve the calculations modulo smoothing operators.
Actually, as far as we are interested only in local properties of solutions of
partial differential equations, it is probably simpler to substitute a practical use
of cut-off functions in place of these theoretical constructions, and that is the
point of view we will take when presenting applications in Chapter 4.
Exercises
3.1
Remember that a(x, D) is said to have the local property if supp (a(x, D)u) C
supp u for all u E S'. The goal of this exercise is to prove that differential
operators are the only pseudodifferential operators possessing this property.
(a) Let a E S-" such that a(x, D) has the local property.
Let t G E S and xp E III" ; for a fixed function 1p E Co such that V = 1
in B1, set fork E Z+ tPk(x) = (1-cp(k(x-xo)))iP(x). Using Proposition
1.14(ii) and Theorem 3.6, show that la(x, D)(r& - v'k)Io < C111y -'kIIo for
a constant C independent of k E Z+. Then prove that a(x, D)r/ik(xo) = 0
for all k and finally that a(x, D)i(xo) = 0.
Show that a(x, D)ri = 0 for all tp E S.
(b) Let a E S' for some m < k E Z+ such that a(x, D) has the local property.
Pseudodifer+ential Operators
62
Show by induction that for any a, p E Z+, (c7z 3 a)(x, D) has the local
property. Similarly, show that a' (x, D) also has the local property.
For E S, write a Taylor formula up to order k+n, then use Example 3.5
and question (a) to prove that
a(x, D)'lb(x)
r
a(x, 0)D°,r(x)
fal<k+n
3.2
Show that the symbol a is a polynomial in (or in other words, that
a(x, D) is a differential operator).
The elliptic estimate. Throughout the exercise, a E Sm will be assumed to be
elliptic. One wants to prove that there exist constants C, and CN,8 such that
u E H-N and a(x, D)u E H' imply it E H'+m with the estimate
flulls+m < C.11a(x, D)ulla + CN.,II uII -N.
Questions (a) and (b) provide two independent proofs, while in question (c) it is
proved that a(x, D)u E H' no longer implies that it E H'+' if one does not
assume u E H-0O (but of course one still has it E HL m).
(a) In this question one proves only the estimate when it E S since the general
case follows from this one by means of standard techniques of approximations (cf. Exercise 3.8(a) for example). Show that the symbol a'#a2'#a
satisfies the assumptions in Theorem 3.9, and show that this implies the
elliptic estimate for all it E S.
(b)
By using the inverse a" of a, show that it E H-N and a(x, D)u E H'
imply u E H'+m with the elliptic estimate.
In this question, it is any temperate distribution. Show that a(x, D)u E H'
implies it E H'+m and that the elliptic estimate holds with CN,3 = 0 if a
has an exact inverse b (i.e., b#a = 1). However, show that the Laplacian
operator 0 = Fn _I has an elliptic symbol of order 2, that the function
I is not in H-°°, and that A has no exact inverse.
Another proof of continuity in Sobolev spaces. The major part of the proof of
Theorem 3.6 was devoted to establish an inequality IIb(x,D)VGIIo 5 CbI) 'IIo.
1' E S, for any b E S°. The present exercise provides a different proof of this
fundamental result.
E S.
(a) One sets XE(x) =
f -,(1
(c)
3.3
+then, for
J
Show that if E L2(lRIn) with II'I'IIo = 2-"/'(27r )n110110.
''(x, ) =
XE(x - y)'tl'(y) dy.
Lemma 1.17 to write f
(Hint: Use
as an integral involving Xo and Vii.)
41(x, )) E
Using the same method, show that for a E {0, I in, M.
L2(R2n) with a norm equal to 2-(n+i°1)/2(27r)nI[
IIo.
(r1n ,(1 +OE,)) 4Y(x,e) =
Show that
Similarly, for V E S, show that the function
(2ir) " / x-:( - 17)0(77) d+
satisfies 7 E L2(1R2") with 11 0110 = 7rnJ2IIWIIo and
C
(1
- a=i) I
a
/
(x, e) =
W(x).
63
Exercises
(b)
0,6b is bounded for all a, 0 E {0,1
Let b E C2' (R211 ) be such that
(any symbol of order 0 satisfies such an assumption, but also symbols taken
in much wider classes, as the So,o studied in Exercises 2.9 and 3.6). One
sets for gyp, V) E S
(b(x,D)tb,,p) = (2ir)'" J
Use results from (a) and integrations by parts to show that (b(x, D)ip, cp)
is also equal to
(2tr)-" ` (-l)I'
` /f J
Ca
rt,/3,yE{0,11^
Conclude that b(x, D)1 E L2(R,) with an estimate
Ilb(x, D)tlll0 <- CbIJV'IIo
where the constant Cb will be computed explicitly.
3.4
An extension of Schur's lemma. Assume as in Lemma 3.7 that K(x, y) is a
measurable function defined on R" x R" and satisfying
I K(x, y)l dx < C and
J
E Lp(R") (where I < p < oc)
for some uniform bound C, and define for
1P(x) = J K(x, y)t4'(y) dy
whenever K(x,y)tp(y) E L'(dy), and
oo otherwise. Show that
I(x)I C (JIKixv)iIYw'dY)
l/p
and conclude that cp E LP with NormLp (gyp) < C NormLP (w). (Remark in addition
that for p = I you need only the assumption f I K(x, Y) I dx < C, while only
f I K(x, Y) I dy < C is needed when p = oo.)
REMARK
This exercise thus shows that the operators a(x, D) are bounded on
the Lebesgue spaces LP when a E S-"-'. One can even prove that this is still
3.5
true when a E S° and p E (1,001(cf. Coifman and Meyer (6, Theorem 9, p. 38]),
but the proof is much longer.
L2 continuity for certain symbols decreasing at infinity in x. Assume that b E
C"+' (R2i) satisfies f li b(x, )I dx is bounded for all Ial < n + I (remark that
no smoothness is assumed with respect to t), and let ' E S.
Show that b(x, {) c({) E L' (dC) for almost all x E R" , and that
f ei(z.f)b(x,
b(x, D)tIc(x) =
)t 4
(21r)-n
defines a function b(x, D)V) E L' (R").
Show that
b(x
(tl) =
f
K(r1,
dC
will be written explicitly.
where the kernel
Show that .\n+1
-17)K(q,.) is bounded and that
((
J IK(t1,C)Id71<C and{< C
Pseudod{,ferential Operators
64
for some uniform bound C, and conclude that b(x, D)* E L2 With IIb(x, D)iO IIo <
C,II'IIO.
3.6
Continuation of Exercise 2.9.
(a) The S,7,0 calculus. For any symbol a E S,7,, (classes of symbols defined in
(b)
Exercise 2.9), one defines an operator a(x, D) E if' o by the formula given
in Theorem 3.1. Extend the results of Chapter 3 up to Theorem 3.6 and
Corollary 3.8 to these classes of operators.
Hypoellipticityy. Let a and c E Sp o be such that c#a - I E SP°. For any
'p E C.1, choose a ' E Co such that ?P = I on supp gyp. Show that there
exist b E SPo and r E SS,°° such that 'p = (b#r/i#a) + r, then prove that a
has the property of hypoellipticity:
,
a(x,D)uES'nC°°
uES'nC°°.
.
More precisely, prove sing supp (a(x, D)u) = sing supp u for any u E S'.
(c)
,
Local solvability. When a has a right inverse (i.e., a#b - I E S,-.' for
some b E SP o). then the equation a(x, D)u = f is locally solvable. This
last result will be proved in Exercise 4.1.
3.7
Converses of Theorems 3.6 and 3.9.
(a)
"Microlocalization" at (y, q). Let a E S' for some rn E R and 'p(x) _
(2n)-"/4
then simply writing ) for .S(ri), set p,.,,(x) =
V((x e-I=I2i4,
Compute
then show that
and
a(x. D)'p,,.»(x) =
e'(:.n)(ba.,,(z,
D)P(z))i:_(Z-v)a'n
where br,,, E S"' is given by b,,,,, (z, () = a(y + A-1J2z, )? +
Coming back to the notation \(77) instead of A, show that for any s E 1[P,
a°(17 + /2(i)!)
(treat the situations 12171 ? A'/2(1))I(I
and 12771 < a'/2(;7)I(I separately), then prove that b,.,,(z,() = a(y,ri) +
cv,,,(z, () where c,,.,, satisfies an estimate
\(Z))
ICN,7(Z, ()I
uniformly for y E R', and finally conclude that
(a(x, D),pv,,,,'py.,,) = (a(y, n) + r(y,
17))11
vb.n I10
where the function r satisfies an estimate Ir(y,ri)I < CA--(1i2)(77) uniformly for y E R".
(b) Applications. Let a E S2' and assume that for some f > 0 and b > 0,
2Re (a(x, D)'p.'p) > CIIV112 - CIpIIm_6 for V E S
(cf. Theorem 3.9). Show that b = (A-'"#a#A-"') + CA-26 satisfies
2Re (b(x, D)'p,'p) > EII'v1l2
for 'v E S,
and use results of part (a) to show that there exists a constant Co such that
CoA'm-' > ,\2m /3.
Re a +
Let a E S'" and assume that for some s and e E R,
IIa(x, D)'p1l° < CIkvII°+e
(cf. Theorem 3.6). Show that b =
IIb(x, D)'pIIo <- CII'pjIo
for p E S
satisfies
for cp E S,
65
Exercises
3.8
+Am-t-(1/2)). Then
and use results of part (a) to show that IA-CaI < C'(1
use results of Exercise 2.8(b) to conclude that A -'a is bounded and that
a E St+ = nm>tSm
Friedrichs's lemma and subellipticity.
(a) Friedrichs's lemma. If V is a unit test function, p, will denote the symbol
cp(el;)(E S-°°), and the corresponding operator c',(D) is called a
Friedrichs mollifier.
then show that u E
Show that for it E
H' if and only if II,p,(D)uII, is bounded uniformly fore E (0, 1).
Using the proof of Example 3.5, show that for a E S' and ) E S,
(a*#cp,
- 0,#a')(x, D)'+b(x) =
(2ir)-n
r e'(=-v.() (a
(x, ) - a(y, ))
dy d ,
and show that this quantity can be rewritten
(b#O )(x, D)t1(x) +
(b)
fK(xY)l&(Y)dy
where b E S° and K, (x, y) _
(27r)-n f a+(x-v.() (a(x, 0 -a(y, fOK) A.
Then use a Taylor formula and integration by parts to show that this kernel K, satisfies the assumptions of Schur's lemma (Lemma 3.7) with a
constant C independent of e, and finally prove that if it E L2,
cp,#a)(x, D)uII o is bounded uniformly fore E (0,1).
Use Theorem 1.13 to show that supp u C BR implies that ;!,(D)u E C"
with Supp (,p,(D)u) C BR+,.
Subellipticity and hypoellipticity. A differential operator
a.(x)Do
a(x, D) _
aI<m
with complex-valued coefficients ao E H°° is said to be subelliptic at x°
if there exists a neighborhood 0 of xo, a b > 0, and a constant CO such
that
IIt1IIm-1+6 C Co(IIa(x, D)'Ilo + ft1IIm-1) for t' E Co (Q).
Here, a(x, D) is assumed to be subelliptic at xo.
Show that if w has a compact closure in fl, there exist constants (C,),ER
such that for any s E R
for zV E C0 00(w).
IIYIla+m-1+6 !5 C,(IIa(x, D) 'II, +
Use the functions 7P, = p,(D)(Xu), where 0,(D) is a Friedrichs mollifier
as in part (a) and X is a cut-off function, to show that if it E H'+'4-' and
a(x, D)u E H' in some ball centered at xo, then it E
H'+--1+6 in
any
smaller ball.2
Show that if it E S' satisfies a(x, D)u E C° near xo, then u E C°° near
xo (property of hypoellipticity).
(c)
A subelliptic operator. For k E Z4., let a2k(x,C) = 1 + ix2 kl;2. For x1
(xik+' - y2k+1)/(2k + 1), then for t2 E R,
and y E R define B(xi, y) =
eB(x1,v)(2
if 52(x1 - y) < 0
K2(x1,y) _ 0'(z
if {2(x l - y) ? 0,
2We say that u E H' in a ball B if there exists a v E H' such that it = v in B.
66
Pseudod(ferential Operators
and finally for 0 E S = S(W), set
(y,l;z) = J e-"={=,L(y,z)dz,
/
?0(xi,t2) = J
and
K,O(xi,x2) = (27)-i / eix2fz,I0(xl,6)42.
J
Show that for any ?1' E S, Kii is aCOO function and -0 = Ka2k(x, D) V,.
Show that there exists a constant C such that
f IKe2(xj,y)I dx1 <
and
where 6 = 11(k + 1). Similarly, show that
fxlK(xiY)Ida
i
<_ CI&I
and
/
J
dy 5 Cj6l-
Show that for any r/' E S,
J
(1 +
then show that
d < C J (lazk(x, D)v(x)I z + IVG(x)I2) dx,
L'(x1,£z) =
f iI+G( )Izd{<C
f
and
(lazk(x,D)V,(x)Iz+IV,(x)J2)dx,
and finally conclude that there is a subelliptic estimate for azk(x, D).
Use the function u(x) = ((x,k/2k) + ixz)31z to show that the operator
azk_I(x, D). where
CI +ix2k-16, is not hypoelliptic.
Notes on Chapters 2 and 3
67
Notes on Chapters 2 and 3
The notion of the pseudodifferential operator has old roots: as early as 1927,
Weyl [69] suggested associating to any symbol a(x, ) an operator a(x, D)
defined by a variant of the formula we used in the text (see Theorem 3.1).
One could also quote Hadamard [38] for his finite parts of (singular) integrals
and his elementary solutions of elliptic or hyperbolic second-order equations.
However, the direct origin of the theory lies in the study, taken up by Mikhlin
[58], Calderdn and Zygmund [27], Seeley [62], and others, of singular integrals
occurring in elliptic problems. The consideration of an algebra containing both
singular integral operators and partial differential operators then led Kohn and
Nirenberg [48] (see also Hormander [41 ]) to the definition of pseudodifferential
operators and the proof of their basic properties. These were essentially the
operators with polyhomogeneous symbols as defined in Chapters 2 and 3.
Eventually, the theory of pseudodifferential operators was extended in a lot
of various ways. We now describe some of these extensions (with references),
but we apologize for the incompleteness of the picture; the applications will be
discussed in the Notes on Chapter 4.
First, it was natural to adapt pseudodifferential operators to the analytic framework. This was done by Boutet de Monvel and Kree [21 ] and Boutet de Monvel
[20], while an algebraic approach was developed by Sato et al. [60]. The theory
presented in Treves [13, Chap. 5], equivalent to that of [211, makes use of the
"analytic cut-off functions" of Andersson and Hormander (as in [46]). Another
approach, avoiding the use of such cut-off functions but based on the methods
of stationary (complex) phase, was initiated by Bros and Iagolnitzer [22] and
developed by Sjostrand [63]. A link between these divergent points of view
was established by Bony [19). The student must be informed that none of the
references quoted here is elementary or easy to read.
Another generalization led to the so-called Fourier integral operators. If the
definition of O(C) is carried into the formula of Theorem 3.1, we can formally
write
a(x, D)v(x) =
(21r)-"
f
dy A.
Fourier integral operators are essentially given by the same formula where th
exponent (x - y, ) is replaced with a more general phase function 4 (x, y, ).
The use of such a formula to represent solutions of hyperbolic Cauchy problems
appears explicitly in Lax [49], but it has a long tradition that goes back at least
to Poisson [59] . These operators were also used in Hormander [41] to prove
the invariance of pseudodifferential operators under a change of variables, then
by Egorov [32] to prove the same invariance under the more general canonical
transformations. A systematic presentation of the theory, including the introduction of very general oscillatory integrals, is given in Hormander [45] and
Duistermaat and Hdrmander [31]. The interested student will find a nice expo-
68
Notes on Chapters 2 and 3
sition in the lecture notes of Duistermaat [30] (see also Treves [68, Chaps. 6,8],
Taylor [11, Chap. 8], and Kumano-Go [9, Chap. 10]). Further developments of
the theory, allowing phase functions with nonnegative imaginary part, can be
found in Melin and Sjostrand [56,57].
In [43], H6rmander enlarged the algebra of pseudodifferential operators to
include the fundamental solutions of hypoelliptic operators of constant strength.
This was obtained by allowing more general symbols, called symbols of type
p, 6, and the proof of continuity of these new operators was completed by
Calderdn and Vaillancourt [25,26]. An elementary discussion of the LP and
Lipschitz theory of pseudodifferential operators is given in Folland [34], and a
systematic investigation of minimal conditions implying continuity is taken up
in Coifman and Meyer [6], where nonsmooth symbols are considered (see also
Stein [66] and Bony [3]). In another direction, the study of locally solvable and
subelliptic operators led Beals and Fefferman [2], Beals [16], and H6rmander
[47] to very general classes of (smooth) symbols which can be adapted to the
operator being studied. One by-product of these generalizations is an improvement by Fefferman and Phong [33] (see also H6rmander [8, Theorem 18.6.8])
of the Girding inequalities (i.e., inequalities of the type considered in Theorem 3.9) previously obtained by H6rmander [42], Lax and Nirenberg [50], and
Melin [55].
4
Applications
Introduction
There are so many applications of pseudodifferential operators that it seems
impossible to describe them completely. Books have been written on this
subject (cf. [8,9,11,13]) but none claims to give a general description of all
possible applications. Moreover, this introductory course was not intended to
develop applications but only to describe the very basic elements of the theory itself. However, it would have been very artificial to drop the applications
completely, since the main motivations of a theory lie in its ability to solve
problems.
Thus we now present a few applications without pretending that they well
represent the power of the theory. Our goal here is just to give examples of
how to use pseudodifferential operators to solve problems of partial differential
equations, and we hope that the very few results described will convince the
reader of the great convenience of the whole theory. In this chapter we will
change the style of exposition slightly to avoid overly long developments: we
will give fewer details, more freely use results from functional analysis, and
sometimes give only references instead of proofs.
The first section is devoted to the study of local solvability of linear differential
equations. The elementary result we present in Section 4.1 was essentially
obtained before (sic!) Nirenberg and Treves [10] began to use pseudodifferential
operators to take up this problem. However, it is thanks to this tool that we are
able to give here a short proof of it. In Section 4.2 we describe the bases of the
study of microlocal singularities of solutions of partial differential equations.
This topic could be considered as a part of the theory itself rather than as
an application, and it is certainly one of the main motivations of this whole
construction. Finally, in the last section we illustrate these latter results in the
study of the Cauchy problem for the wave equation.
69
Applications
70
4.1
Local solvability of linear differential operators
Consider a linear differential operator a(x, D) =
a,(x)D° with complex-valued C°° coefficients aa. Remember that its principal symbol p(x, ) =
a complex-valued COD function on T`R". We want to
study the following property: the operator a(x, D) is said to be locally solvable
at xo if there exists a neighborhood Sl of xo such that a(x, D)u = f has a
solution u E D'(cl) for any f E Co (0).
It can be proved that this property is equivalent to some a priori estimate'
for the operator a' (x, D). The next problem then is to translate this a priori
estimate into a checkable geometric condition on the symbol a. To state such a
condition, we introduce the Poisson bracket {p, q} of two complex-valued C1
functions on T`R" defined as
{p, q}(x, ) = (O p(x, ), 05q(x, )) - (axp(x, ), (9fq(x, c)).
This quantity appears naturally as the principal symbol (up to a factor i) o
the commutator [a(x, D), b(x, D)] = (a#b - b#a)(x, D) when p and q are the
principal symbols of a(x, D) and b(x, D) (cf. the asymptotic formula in Theorem 2.7). In [7, Chap.6], Harmander proved that the principal symbol p of
a locally solvable operator a(x, D) must satisfy {p, p} = 0 on p = 0, but the
proof is too long and technical to be rewritten here. In this section, we will just
prove a converse of this result, the statement of which requires the following
definitions.
The operator a(x, D) is said to be of principal type at xo if the c-gradient of
its principal symbol at xo vanishes only for = 0. It is said to be principally
normal at xo if there exists a function q E C°°(T'R" \ 0) homogeneous of
degree m - 1 in C such that the principal symbol p satisfies
{p, p} (x, ) = 2iRe (q(x, i;)p(x, )) for i; E R" \ 0 and x close to xo.
REMARK
In order to prove that a(x, D) is of principal type at xo, it is sufficient
to check that Ot p(xo, C) 96 0 when p(xo, l;) = 0 in view of Euler's identity
p(x, i;) = (I /m) (8fp(x, i; ), C). In order to prove that a(x, D) is principally
normal at xo, it is also sufficient to check that property near the zeroes of p.
p} = 2iRe (qp) near (xo, to) if p(xo, to) -A 0
Indeed, one can always write
because it suffices to take q = {p, p}/2ip as long as p 0 0. Thus, if one
can also write {p, p} = 2iRe (qp) near any zero of p, the compact set K =
{(x, £) E R" x R"; x = xo and 1i; I = 1 } can be covered by open sets S2, where
one has {p, p} = 2iRe (qjp). Then, using the partition of unity constructed
'The Latin words a priori mean "before"; an a priori estimate for an operator A is an inequality
between a norm of A,p and a norm of V proved for a whole class of functions ,p (typically for all
w E CO '(0) for some 0) before assuming anything on ,p; it is only eventually that it will be used
for special <p's.
Local solvability of linear differential operators
71
in Lemma 1.5, the function qo = > ojq, satisfies {p, p} = 2iRe (q0p) in a
neighborhood of K, which finally gives {p, p} = 2iRe (qp) for f E ' \ 0 and
x close to xo by homogeneity if we set q(x, ) = I I m_ I qo(x, /ICI). Principally
normal operators obviously satisfy Hormander's condition {p, p} = 0 on p = 0.
The converse will be discussed at the end of this section (see Corollary 4.4 and
its comments).
I
The main result of this section is the following.
THEOREM 4.1
Let a(x, D) be an mth order principally normal operator of principal type at xo.
Then there exists a neighborhood S2 of xo such that the equation a(x, D)u = f
(in S2) has a solution u E L2 (Q) for any f E H'-'.
Example
Operators with real or constant coefficients in the principal part satisfy {p, p}
0, and therefore they are principally normal (take q = 0). Elliptic operators
have nonvanishing principal symbols (except at = 0), so that - thanks to the
previous remark - they satisfy the assumptions of Theorem 4.1. The student
will find other operators satisfying these assumptions by solving the following
questions. If a is of principal type, is a' also of principal type? If a is principally
normal, is a* also principally normal? Is a#b of principal type when a and b are
of principal type? What about the converse? Assuming that a#b is of principal
type, that a and b are principally normal, and that the order of c is smaller than
the order of a#b, does a#b + c satisfy the assumptions of Theorem 4.1? 0
In the proof of Theorem 4.1, our first step will be to show that the principal
type and the normality correspond to some a priori estimates, but before stating
and proving this result, we give a lemma that provides estimates for functions
with small support. In the following, we denote by S26 = {x E R"; Ixi < b}
the open ball of radius b > 0.
LEMMA 4.2
For allb>0andmEZ
,
for all p E Co (SZ6).
II'PIIm <_
Moreover, if Q and R are differential operators of orders m and 2m respectively,
there exists a constant C such that for all E Co (S26)
and
I(ixj p,RV)I S C6IIPIIm
Applications
72
PROOF The first inequality is proved by induction. Indeed, it is sufficient to
prove it for m = 0 in view of Proposition 1.14. and since
one
can write
Ilsall0 = (tG, gyp) = (D1 (ixl (p), (p) - (ixI (D1 gyp), gyp)
= (ix1 p, Di p) + (Dip, ixl'p) S 211ixivIIoIlDiwIIo <- 2blIwIloflw
.
For the second inequality, let us write Q(ixjy2) = [Q , ixj]yp+ix3 (QV), which
gives
5 II[Q,ixJ]'IIo + llix2(Q')Ilo <
CbII'PII,n
since [Q, ixj] has order m - 1, and we get the result from the first inequality.
Finally, for the third inequality, we write R as a sum of differential monomials
of order 2m, i.e., R = Ek QkQk for some mth order operators Qk and Qk,
and the result comes from the second inequality since
I(ix.,p, Rw)l = E(Qk(tx.JV),Qk' )
k
1
k
PROPOSITION 4.3
Let a(x, D) be a linear differential operator of order m. Then,
(i) If a(x, D) is of principal type at 0, there exist a bo > 0 and a Co such
that for all 6 < bo and E Co (S26 ),
n-I < Cob(Ila(x,
Ila*(x,
D) is principally normal at 0, there exist a 6 > 0 and a C such
that for all p E Co (f2b),
Ila(x,D)Vblo <- C(Ila'(x,D)<pllo+
(iii)
If a(x, D) is both principally normal and of principal type at 0, there
exists a 6 > 0 such that for all p E C o (11 ),
D)VIIo
PROOF
(i) Let A=a(x,D), Qj=[A,ixj]=(8f,a)(x,D), B=Ej_I Q,Qj=
late pI2 modulo S2ni-3, and since A
b(x, D). Thanks to Theorem 2.7, b =
EJ=1
is of principal type we have by homogeneity
I at,p(x, )I2 > 2E1e12m-2 for
some e > 0 and all x in some S22N. Therefore, the symbol b + EA2m-2#(1 - V))
satisfies the assumption of Theorem 3.9 (GArding's inequality) if 0 E Co (126o)
and 0 < 1. Moreover, if 0 = 1 in !
and 6 < bo, (1 - V,)Ip = 0 for all
Local solvability of linear differential operators
73
W E Co (f26) then (b(x, D) + EA2m-2(D)(1 - 0))y7 = Bye. It follows that we
have
n
2Re (By7, p) ? EIIsvIIm-i - C'IIVIIm-2
for some constant C.
On the other hand, for each operator Qj one can write
= (A(ix.iV) - ix3(Ap),Q3p)
(ix3sc, A`Q, ) - (ixj (Afw), Qif')
(ixi'p, [A',Qi] 'p) + (Q(ix.,'p),
For V E Co (06), this can be estimated by using Lemma 4.2:
IIQ2wII <
-i + Ci,26Ik IIm IIA',pIIo +
CC6(IIApII0 1 + IIA'wllo + IpIIm_1),
and since the same Lemma 4.2 also implies IIVII -2 <
C
E E IIQ,"'IIo + f
j=1
Cob(IIApII0 +
this gives
II,pII;,
)
as required.
(ii) Let us modify the function q near = 0 so that q is now C'° everywhere
while the relation {p, p} = 2iRe (qp) holds only for ICI > I and x in some f226.
Then for '0 E Co (ft26) we set
b=zbaESt,
c= q+i{a,ii} E S'-1
and
r = b'#b - b#b* - b#c' - c#b' E S2m.
Actually, the symbol r is in S2+,-2 because by using the asymptotic formulas
of Theorem 2.7, we can write modulo S2,n-2: b' = b (where we use
the notation a(x,E) = F_i 8,x,8{,a; similarly, we will use the notation ax = 8.'a
and at Ota), then
r = (b - ib(1,4))b - i(b{,bx) - b(b -
+ i(bt,bx) - be - cb
_ -i{b, b} - 2Re (eb) = -i({b, b} - 2iRe (eb))
_ -iil12({a, a) - 2iRe (qa)) = -ii2({p, p} - 2iRe (qp))
and this is identically zero for ICI > 1.
74
Applications
Therefore, if we take such that 0 = 1 in S26 and we write A, B, Q, and
R for a(x, D), b(x, D), c(x, D), and r(x, D), then Bp = Acp and B'cp = A'cp
for all cp E Co (Q6) since A has the local property, and we can write
IIAwllo== (B'Bp,') = (Rye, p) + (BB'V,1v) + (BQ'So,,v) + (QB'1v, o)
(Rp, gyp) + IIA',pIlo + 2Re
A'4p)
<_ IIRwlI1-mllcvllm-1 + 2IIA'1pl1o + !IQ'1vllo
< 21IA'Vllo +
as required, since R E W2",-2 and Q' E IV--1.
(iii) Finally, when both assumptions hold, we get from (i) and (ii) that for
small 6 > 0 and for p E C0,A),
IIVII n-1 < C1b(lla'(x, D)wllo +
Ilwll;,,-1)
for a new constant C1, and if we now choose 6 < 1 /2CI, one has
IlVllm.-1 = 2llcvllm-1
-
11o111
M-1
< 2C16(Ila'(x, D)cpllo + lllvllm-1)
- IIcIIm-1 < Ila'(x, D)wllo
PROOF OF THEOREM 4.1
The property of local solvability now follows from
the a priori estimate thanks to a classic argument from functional analysis.
Indeed, using a translation, we can assume that x0 = 0, and we take b > 0 as in
Proposition 4.3(iii). Then, the operator a*(x, D) is injective on Co (S16), and
we can consider its inverse (A')-1 which is well defined on the space
E = {zL' E Co (Slb); 3cp E Co (16) with iP = a'(x, D)y'}.
For f E H1-" we define on E the semi-linear form U(ii) = (f, (A')-1 ')
which satisfies
IU(0)I =1(f, p)l <_
11f111-m1l'vllm-I <_ 11f111-rIla'(x, D)cp110 =11f111-mll'Ilo
where we used the estimate (iii) of Proposition 4.3 for V = (A')-1 0. Thus, U
is continuous for the L2 norm. We can then extend this form as a continuous
form on L2(1)6) (by the Hahn-Banach theorem, or in a more elementary way
by the following Hilbertian arguments: by continuity to the closure F of E,
then by setting U(O) = U(rrF*) where lrF denotes the Hilbertian projection on
the closed subspace F), and by using Riesz's representation theorem we get the
existence of a u E L2(06) such that (u, V') = U('r') for 10 E E, i.e.,
(u, a* (x,
(f, w)
for all 1v E Co (Slb ),
which means a(x, D)u = f in S26 by definition.
Local solvability of linear differential operators
75
To close this section, we finally compare Hdrmander's result quoted at the
beginning of this section with the result of Theorem 4.1.
The first situation we consider is when the sets Rep = 0 and Imp = 0
intersect transversally. Technically, this leads to the following transversality
assumption: p(xo, ) = 0 and # 0 = Re OO p(xo, l;) and lm OE p(xo, e) are
linearly independent. Then we have:
COROLLARY 4.4
Let a(x, D) be a linear differential operator with principal symbol p satisfying
the following statement: the real and imaginary parts of the -gradient of p
are linearly independent at (xo, ) for all solutions 0 0 of p(xo, ) = 0. Then
a(x, D) of principal type at xo, and the following three conditions are equivalent:
(i)
(ii)
(iii)
a(x, D) is principally normal at xo.
a(x, D) is locally solvable at xo.
a(x, D) satisfies Hdrmander's condition { p, p} = 0 on p = 0 in a neighborhood of xo.
PROOF Under the transversality assumption, a(x, D) is obviously of principal
type at x0 in view of the remark following the definitions given above.
The implications (i) = (ii)
(iii) follow respectively from Theorem 4.1 and
(1)
from Hdrmander [7, Theorem 6.1.1], so that we only need to prove (iii)
under the transversality assumption. Again, thanks to the remarks follow-
ing the definitions given above, it is sufficient to prove that one can write
{p, p} = 2iRe (qp) near the zeroes of p; at such points, the transversality assumption shows that Rep and Imp can be taken as local coordinates in R2" and
a Taylor formula thus gives
1
1
2i{p,p} =
2i{p",p}IP=o+giRep+g2Imp
for some coefficients ql and q2 E C°°(R2n). Finally, Hdrmander's condition
(iii) allows to rewrite this relation {p, p} = 2iRe (qp) by setting q = q, + iq2.
The next step would be to weaken the transversality assumption in that corollary, and a natural substitute is to require that the c-gradient of p never vanishes
at zeroes of p, in other words, to assume that a(x, D) is an operator of principal
type. In this situation, Hdrmander's condition is no longer equivalent to the normality of a(x, D). As a matter of fact, it can even be proved that Hdrmander's
condition is not sufficient and the normality is not necessary for the local solvability. However, a more precise characterization of local solvability is known
for these operators of principal type (the so-called condition (P)), but the proof
is much more difficult and requires the introduction of much wider classes of
pseudodifferential operators. Indeed, this result, mainly due to Nirenberg and
Applications
76
Treves [10] and Beals and Fefferman [2], was one of the main motivations in
later developments of the theory of pseudodifferential operators.
When any transversality assumption is removed, the question becomes even
more difficult because the lower order terms then play an important role, and
there is no complete answer to this question.
4.2
Wave front sets of solutions of partial differential equations
In Corollary 3.8 we established the pseudolocal property: pseudodifferential
operators do not increase singular supports. In particular, operators of order
-oc erase singular supports while elliptic operators preserve them.
In this section, we are going to present the notion of wave front set of a distribution, a refinement of the notion of singular support that is well adapted to the
study of singularities of solutions of partial differential equations. As explained
at the end of the next section, this point of view will lead to much more precise
results than when one considers only singular supports. The convenience of this
notion lies in the following basic three properties: (i) One can compute exactly
the singular support from the wave front set (Theorem 4.6). (ii) The action of a
pseudodifferential operator does not increase wave front sets (microlocal property) and actually preserve them where there is some ellipticity (Theorem 4.7).
(iii) There is a propagation property for wave front sets of solutions of partial
differential equations (Theorem 4.8).
Before defining the wave front set of a distribution u E S', recall a characterization of its singular support: xo 19 sing supp u if and only if there exists a
neighborhood S2 of xo such that Vu E C°° for all cp E Co (S1). The definition
of the wave front set will be very similar to this characterization, but it will first
require us to "microlocalize" the notion of a pseudodifferentia] operator, i.e., to
consider localizations in the cotangent space.
A subset r of T* R' \ 0 is said to be conic if
E F and p > 0
imply (x, p) E F. If r is a conic open set in T' R" \ 0, a E S'" is
a compact set K C F such that
supp a C {(x, p ); (x, 1;) E K and p > 0). (Using the asymptotic expansion
for the operation # and Lemma 2.2, one gets that if a E
and b E St,
then a#b and b # a can be written as sums of a term in ScornP (F) and a term in
S-OD.) a E S°° is said to satisfy a E Sa(F) if ab E S'" for all b E S°°,,,p(r)
(this also implies that ab, a#b, and b#a are in S°'+t for all b E S ...P(17)). The
symbol a E S"' is said to be elliptic at (xo, Co) E T*R" \0, or (xo, o) is said to
be noncharacteristic for a, if there exist a b E S-'" and a conic neighborhood I'
of (xo, o) such that ab - I E S;.' (t). The student will check that the proof of
Theorem 2.10 can then be adapted to construct a b E S-' such that a#b- 1 and
b#a -1 are in Sj;°°(F), maybe for a smaller F. Finally, the set of characteristic
said to satisfy a E S
Wave front sets of solutions of partial differential equations
77
points for a will be denoted by Char a, and from its definition it is thus a conic
closed subset of T'R1 \ 0.
Example
If a(x, D)
p(x,.) =
a,, (x)D' is a differential operator with principal symbol
a,,(x)t , the characteristic set of a is simply Char a =
{ (x, l;) E T* R' \ 0; p(x, t;) = 0}. Indeed, if p(xo, to) # 0, one can define
EJa1_,,,
b(x, t;) = 1 /p(x, f) in a conic neighborhood r of (xo, CO) since p is homogenous,
and define b E S-"` anyhow out of r, and it is then clear that ab - I E Si-.' (I').
Conversely, if p(xo, to) = 0 and b E S-'°, a(xo, µl o)b(xo, 14o) = O(µ-1) and
this shows that ab - 1 ¢ Sj;' (F) for any conic neighborhood r of (xo, co). 0
The wave front set of a distribution is then defined as follows.
DEFINITION 4S Let u E S'. One says that the point (xo, to) E T'R" \ 0
is not in the wave front set of u, or (xo, to) 0 WFu, if there exists a conic
neighborhood r of (xo, to) such that a(x, D)u E C°° for all a E S P(r).
From its definition, WFu is thus a conic closed subset of T*R \ 0.
The wave front set is related to the singular support through the following
result.
THEOREM 4.6 PROJECTION THEOREM
Let u E S'; then one has
sing supp u = {x E lR
;
there exists a C # 0 with (x, f) E WFu}.
PROOF Let xo ¢ sing supp u, V E Co such that Vu E C°` and cp = 1 in a
neighborhood f of xo, and r = St x (Rn \ 0) which is a conic neighborhood of
(xo,l;) for every t 0. Then if a E S(r) one writes
a(x, D)u = a(x, D%pu) + a(x, D)((l - V)u).
Since Vu E Co, one has a(x, D)(Vu) E S. Moreover, a(x, D)((1 - cp)u) _
b(x, D)u where b = a#(1 - c') E S-OC thanks to the asymptotic expansion of
the operation # since the supports of a and (1 - cp) do not meet. It follows that
a(x, D)((l - V)u) E P (cf. Corollary 3.8) and a(x, D)u E C°°.
Conversely assume that (xo,1;) it WFu for all t E R'a \ 0. For every such
£, there exists a conic neighborhood r(t) of (xo,t;) as in Definition 4.5. The
compact set { (xo,1; ); It I = 1 } is covered by these neighborhoods and one can
find a finite number of them F1,..., rk and some functions cps E Co (F3) such
that
Vi (x,.) = 1 in KE = {(x, t;); Ix - xol + IIt;I - 1I < e} for some
e > 0 (cf. Lemma 1.5). Choosing also a function -0 E Co(k') satisfying 'o = 1
near t = 0, one sets aj(x,l) = (1 -''(t;))cp?(x,t;/I1;I) E S.' P(FD). Therefore
Applications
78
F,, aj
1 for Ix - xoI < e and E R1. We then take
S2={x;Ix-xoI <e}; for any WECo (Q),cp=v7P+J:,ypajand
one has vp
<pu = cp ,(D)u + p
aj (x, D)u.
One has i4i(D)u E P since z' E S-'° (cf. Corollary 3.8), and aj (x, D)u E C°°
by assumption, so that cpu E COD.
The next result links singularities of a(x, D)u and singularities of u.
THEOREM 4.7 MICROLOCAL PROPERTY
LetuES'andaES°°; then
WF(a(x, D)u) C WFu C WF(a(x, D)u) U Char a.
PROOF Assume that (xo, o) WF(a(x, D)u) U Char a. Then, there exist a
conic neighborhood r of (xo, eo) and a b E S°O such that
c#a(x, D)u E C°°
for all c E S mp(r) and b#a - 1 E S
(F).
Then for any d E S°°,,,P(17). one has d#(1 - b#a) E S-O° and one can write
d#b = c + r with c E S P(r) and r E S-0°, from which one gets
d(x, D)u = d#(1 - b#a)(x, D)u + c#a(x, D)u + r#a(x, D)u.
The first and third terms are in P since the operators are in %V°°, while the
central term is also in C°° by assumption, so that d(x, D)u E C°°.
Assume now that (xo, fo) it WFu, which means b(x, D)u E C°° for some
conic neighborhood r of (xo, to) and all b E S a,,,p(I ). Then, for all c E
c#a = b+r with b E S
and r E S-O°, from which c#a(x, D)u =
b(x, D)u + r(x, D)u where the two terms are smooth since r E S-°° and
b(x, D)u E C°° by assumption.
I
Let a(x, D) =
aQ(x)D° be a linear differential operator with realvalued coefficients aQ E C°° in the principal part (i.e., for jai = m), and u E S'
a solution of a(x, D)u = f where f E C°°. Thanks to Theorem 4.7 one has
WFu C Char a = { (x, ); p(x, C) = 0} where p(x, C) =
a,, (x)Ca is
the principal symbol of a. Now, we are going to prove a more precise result,
namely that WFu is actually a union of curves, called "bicharacteristic curves,"
contained in Char a: this property is known as the "propagation of singularities."
Let p E C°° (T' R) be a real-valued function; its Hamiltonian vector field
Hp = (8Ep, 8x) - (,9.p, .Qt) is clearly tangent to the level surfaces of p since
one has Hpp = (0g, 8xp) - (8xp, O p) = 0. Its integral curves, that is, the
Wave front sets of solutions of partial differential equations
79
solutions (x(t),z;(t)) of the equations
dx/dt = Ot;p(x,l;)
dl;/dt =
(which exist locally thanks to the Cauchy-Lipschitz theorem) are therefore contained in the level surfaces of p. In particular, those curves that start at a point
(x(0), e(0)) where p vanishes satisfy p(x(t), fi(t)) = 0. and they are then called
bicharacteristic curves of p. If p and q E C°° (T'1R') are real valued and if
p(xo, o) = 0 while q(xo, o)
0, then one has Hpq = qHp on p = 0 near
(xo, i;o) so that bicharacteristic curves of p or of pq are geometrically the same
near (xo, CO), although parameterized in a different way. One can then state the
propagation result.
THEOREM 4.8 PROPAGATION THEOREM
Let a(x, D) =
a,(x)D° be a linear differential operator with a real-
valued principal symbol p(x,t) = c m
If u E S' satisfies an
equation a(x, D)u = f with f E C°O, then its wave front set is a union of
I
bicharacteristic curves of p.
Before giving the (rather long) proof of this theorem, we will prove a technical
lemma that forms the basis of the study of the Cauchy problem for hyperbolic
operators with variable coefficients. In return, this latter problem will then be
the typical situation where the result of Theorem 4.8 can be used, as we will
see in Section 4.3. If I is a compact interval of liP, we will use the notation
C°(I; Hk) for the space of functions w(t), which are continuous functions of t
valued in the Sobolev space Hk. Similarly, we will use C' (1; H k ) and C°(I; S)
in the same way.
LEMMA 4.9
Let b E S' be a symbol satisfying b - b" E S°, s > 0, and I = [-s, s]; then for
any k E Z one has
sup II e``tw(t)II k <- II e"8w(s)II k + 2
J
1Ie"'(Ot - ib(x, D))w(t)II k dt,
and
IIe-"W(t)IIk
SUP
< Ile1'w(-s)II k + 2 if
IIe-,t(Ot
- ib*(x, D))w(t)IIk dt
for a constant p depending on k and for all w E CO (1; Hk+') n C' (1; Hk).
Moreover, for any g E C°(1;S) and X E S, the Cauchy problem
(Ot - ib(x, D))w(t) = g(t)
w(s)=X
has only one solution w E UkC°(I; Hk), and this solution actually satisfies
w E nkC°(I; Hk).
80
Applications
PROOF We will give the proof in three steps.
(i) Energy estimates. Since b - b` E So, one has for any w E C°(1; H' )
12Re (e"tib(x, D)w(t), e"tw(t))I = I (i(b - b')(x, D)e"t w(t), a"t w(t))I
< CIIe"tw(t)Ilo
Thus one can write for w E Co(I; H') fl C' (I; H°)
Me (e"tatw(t), e"tw(t))
dt Ile"tw(t)IIo =
> (2tt - C)Ile"tw(t)II2 + Me (e"" (at - ib(x, D))w(t), e"tw(t))
> -2lle"t(Ot - ib(x, D))w(t)IIoIIe"tw(t)IIo
if one has chosen p > C/2. Using the notation W(t) = IIet'tw(t)1io, M =
suptEI W(t), and Int = f, IIe"t(Ot - ib(x, D))w(t) 11o dt, we can integrate this
estimate over the interval It, s) then take the supremum for t E I, and this gives
M2 < W(s)2 + 2MInt, so that (M - Int)2 < W(s)2 + Int2 then M - Int <
W (s) + Int, which is our first estimate with k = 0.
For k 0 one has
((fit - i(ak#b#X-k)(x, D))Ak(D)w(t) = \k(D)(8t - ib(x, D))w(t),
and since bk = Ak#b#X-k E S' still satisfies bk - bk E So, one can use the
previous result (now p depends on k), which gives
sup IIe"tAk(D)w(t)IIo <- IIe"-'Xk(D)w(s)Ilo
tEl
+2
1IIe"t'\k(D)(8t
- ib(x.D))w(t)Ilodt,
and this is exactly our first energy estimate.
Finally, the second estimate in terms of w(-s) and b' can be obtained in the
same way by changing tin -t since b' also satisfies b* - (b*)* E So.
(ii) Uniqueness. If the Cauchy problem had two solutions in UkC0(1; Hk) for
the same data g and X, then their difference w(t) would satisfy w E C°(1; Hk+' )
for some k E Z, Otw(t) = ib(x, D)w(t), and w(s) = 0. The equation for c9tw(t)
shows that w E C' (I; Hk), then the energy estimate implies that w(t) = 0 for
t E 1, and this is the uniqueness property.
With the second energy estimate, one would prove in the same way that the
Cauchy problem
f (8t - ib'(x, D))ip(t) ='p(t)
0(-s) = 0
has at most one solution ' E UkCo(I; Hk)
Wave front sets of solutions of partial 4fifferential equations
81
(iii) Existence and smoothness of the solution. Actually, we will show that
for any g E C°(I; S), X E S, and k E Z, the Cauchy problem with data g
and X has a solution wk E C°(1; Hk-' ). However, thanks to the uniqueness
property we got in step (ii), all these solutions wk are equal to a unique solution
to E lkC°(I; Hk).
For any cp taken in the space
E = {(8t-ib'(x,D)) (t);IP E Co (RxWz) with supp TP C {(t,x);t > -s}},
the Cauchy problem
(t - ib' (x, D))z'(t) = cp(t)
,0(-s) = 0
has a solution E C°(I; S), namely the restriction to I of the Vi given in the
definition of E. This solution is unique according to step (ii) and it satisfies the
energy estimate SUP El II (t)II-k < Ck f j
dt for any k E Z. Then,
for g E C°(I; S) and X E S, one defines a semilinear form W on E by
W(A _ (X, -005)) -
J
(9(t), 0(t)) dt
where zji is the unique solution of the previous Cauchy problem. One has
IWMI <-
(iiIk + J
)
119(t) Ilk dtSUP II
tE/
tII-k <Ck
J
IIv(tII-k dt,
so that W is continuous for the norm of the space L' (I; H-k), and thanks
to the Hahn-Banach theorem, there exists an element w E L°° (I; Hk)
(L' (I; H-k))' such that
(X, '(s)) - j(9(t),ib(t)) dt =
for all
r
Jr
(w(t)(t - ib(x,
dt
ECC (RxR')with supp+P C{(t,x);t>-s}.
When we restrict ourselves to functions 7P E Co with supp zb C H =
{(t, x); Iti < s}, the term (X, -i(s)) vanishes and the previous equation then
means that (t-ib(x, D))w(t) = g(t) in Q. It follows that 8tw E L°°(I; Hk-')
whence w E C' (I; Hk-' ), and similarly w E C' (I; Hk-2). Moreover, for any
cF E Co (l[l;") one can construct a 10 E Co (R x R") with supp C {(t,x);
t > -s} and V'(s) = cp, so that the integration of f, (w(t), O (t)) dt by parts
gives (w(s), co) = (X, cp), and this proves that the function w E C°(I; Hk-') is
a solution to the Cauchy problem with data g and X. I
REMARK Of course, we could have solved the Cauchy problem with data on
t = -s as well since all the assumptions are preserved when reversing the
time. To use this result in the study of the Cauchy problem for a hyperbolic
82
Applications
operator with variable coefficients, we would have to consider more generally
symbols b depending also on t, but this would not affect the proof very much
(cf. Hdrmander [8, Chap. 23], where we took the proof of Lemma 4.9). 1
Since WFu is a closed set and a bicharacteristic
curve ry is a connected set, the only fact to prove is that WFu fl y is an open
subset of y. Thus, given a
E WFu C Char a, we want to prove that
the bicharacteristic curve (x(t),.(t)) starting at this point is locally contained
in WFu. We will proceed by contradiction, assuming that this is not true.
does not modify WFu
Since the action of an operator elliptic at
near this point (cf. Theorem 4.7), we will deal with the distribution v =
),'- t (D) (pu) rather than with u itself where the function z!, E Co satisfies 0 = l near x0. The advantage is twofold: on one hand, we have iu E Et
from which we get r(x, D)v E S for all r E S-°` (cf. Corollary 3.8). On the
other hand, if cp E Co is chosen such that p = 1 near xo and V) = 1 near
PROOF OF THEOREM 4.8
supp gyp, the distribution v will satisfy an equation b(x, D) v = pf E Co where
b = coaA` is a first-order polyhomogeneous symbol (the three factors are
polyhomogeneous) whose principal symbol q(x, t:) = cp(x)p(x, t )JC'Jt -' is real
valued and has the same bicharacteristic curves as p.
WFv = WFu, and this will
Our contradiction will then be that (xo, Co)
follow from the construction of a symbol co E S° elliptic at (xo, eo) and such
that co(x, D)v E C. To prove this latter property, we will construct a symbol
c(t, x, ) satisfying c(0, x, ) = co(x, C) and such that w(t) = c(t, x, D)v is a
solution of a Cauchy problem as in Lemma 4.9. Since this w E C°(I; H-N)
for some N E Z, it will then follow from Lemma 4.9 that w E lkC°(I; Hk),
so that co(x, D)v = w(0) E Cc".
The symbol c(t, x, C), a C' function of t valued in So, is going to be constructed so that
r(t) = 8tc(t) - i(b#c(t) - c(t)#b)
is a continuous function of t valued in S'°°. To achieve this, we just have to
choose a co homogeneous in of degree zero, and to look for a polyhomogeneous symbol c(t, x, ) - F_, c, (t, x, ) where the c) are homogeneous in a of
degree -j. Following this point of view and ordering the terms in the asymptotic expansions of b#c(t) and c(t)#b according to their degrees of homogeneity,
we then have to solve the sequence of problems
{
8tco-Hgco=0
co (0,x,0 = co(x,0,
and for
j >0
9c, -Hqc. =rj
c.7 (O,x,0 = 0
where the vector field Of - Hq is real. These problems can be solved in a
same neighborhood of (xo, o) thanks to the Cauchy-Lipschitz theorem, and
their solutions cj(t) are there homogeneous in l;' of degree -j and have their
support contained in the support of co(t). If we have chosen co E Soomp(F) for
The Cauchy problem for the wave equation
83
some small conic neighborhood r of (xo, co), it thus follows that for small t,
the c3(t) can be extended by homogeneity as symbols c,(t) E S (F), and
the symbol c(t, x, ) is then constructed as in Lemma 2.2. Since we assume
that the bicharacteristic curve is not locally contained in WFu, there is a point
(x(s),C(s)) ¢ WFu for an s in the domain of definition of c, and one has
supp c(s) C supp co(s) C e'H9 supp co
where e'H, supp co denotes the image of supp co under the flow of the Hamiltonian vector field HQ. Therefore, if the (conic) support of co has been chosen
sufficiently small around (xo, to), the support of c(s) will be small around
(x(s), C(s)) and we will have w(s) = c(s, x, D)v = X E C'° (and actually
X E Co since the support of c(s) is compact in x).
Thanks to Lemma 1.16, the distribution ipu is in some Sobolev space so that
v = a'-1(D)(i)u) E H-N for some N E Z+ and then w(t) = c(t, x, D)v E
C°(I; H-N) since c E C°(I; S°). Moreover, this distribution w(t) satisfies
Otw(t) - ib(x, D)w(t) = g(t) where
g(t) = r(t, x, D)v - ic(t, x, D)b(x, D)v
by construction of c(t), and these two terms are in C°(I; S) because r E
C°(I;S-OQ) and v = ,\'"-I(D)(,u) with r1'u E £' (cf. Corollary 3.8), and
b(x, D)v = V f E C000. It follows that w is a CO(I; H-N) solution of a Cauchy
problem as in Lemma 4.9 (indeed, b E S' satisfies b - b' E S° since it is
polyhomogeneous with a real principal symbol), therefore w E lkC°(I; Hk),
then ca(x, D)v = w(O) E HOC C C°° as claimed. I
43 The Cauchy problem for the wave equation
This section actually has little to do with pseudodifferential operators. Following
an idea of Treves [121, the Cauchy problem for the wave equation is treated
through very simple computations, which allow us to prove the main results. We
have added this section because the last theorem provides a pleasant illustration
of the general results given in the previous section.
Consider in the space R"+' = {(t, x); t E R, X E R" } the wave operator,
i.e., the operator u '-4 Ou - Du where 0 = E' 102 with 0, = 0/Ox,
is the Laplacian operator in R". The present treatment will be completely
dissymmetric in x and t: we will consider solutions u such that for each fixed t,
u(t) E S' (space of temperate distributions in x only).
The spaces Ck(R, H') and Ck(R; S) are the standard spaces of Ck functions
of t valued in the Hilbert space H' (Sobolev space in x only) or in the Fr6chet
space S (Schwartz space in x only). We will write u E C°(R;S') if for each
fixed
E S the expression (u(t), gyp) is a continuous function of t, and actually,
Applications
84
according to general results from functional analysis, these separate continuities
imply the continuity of u : R x SE) (t, cp) '-. (u (t), V). As a consequence of
this remark, if u E C°(R, S'), the formula
(U(t), cp) =
J0
t(u(s), cp) ds
for p E S
defines a U E Co (R; S') also. Then, for k > 0 we will write u E Ck (R; S' )
if for each fixed cp E S, (u(t), cp) is a differentiable function of t such that the
formula
Ptu(t), 0 = a (U (t), V)
defines a distribution Btu E Ck_ I (R; S'). It is clear that the previous U(t) _
fo u(s) ds satisfies U E Ck+' (R; S') if u E Ck (R; S').
Most of the proofs of this section will be based on the following, very elementary lemma on trigonometric functions.
LEMMA 4.10
The functions
C(t,()
and
S(t,() =
s
ICI
are smooth on R"+' and for any k E Z+ and fixed t, one has 8i C(t) E P
and Ot S(t) E P as functions of ( E Rn. Moreover, they satisfy 8tC(t) =
(I - A2)S(t) and 8tS(t) = C(t), they can both be extended to the whole of Cn
as entire functions with the estimates
I C(t, OI 5 eltl hmCI
and
I S(t, ()I <- Itleltl IIm(I,
and finally they also satisfy (for ( E Rn )
IC(t)I < I
and
XIS(t)l < (1
+t2)1/2.
Ek>o(-t2IeI2)k/(2k)! so that C
Indeed, we can write C(t, f) =
is obviously smooth and can be extended as an entire function by setting
PROOF
C(t,() _ Ek>O(-t2(2)/C/(2k)! Where (2 = (j + +(n. Moreover, C(t,() _
(e'tz + e-'tz)72 if z E
has been chosen such that z2 = (2, and since
n
n
2(Im z)2 = Iz2I - Re (z2) 5 E(I(; I - Re ((()) = 2 E(Im(()2 = 2IIm(12,
j=1
j=1
one gets the estimate IC(t,()I < ellmt=I < eltlllm(l. The Paley-Wiener theorem (Theorem 1.13) then shows that C(t, () is the Fourier transform of some
distribution c(t) with support contained in the ball of radius Iti, and since
D C(t, () is the Fourier transform of (-x)ac(t) which also has its support
in the same ball, D C(t, f) also has a polynomial growth at infinity, i.e.,
85
The Cauchy problem for the wave equation
C(t) E P. One can similarly prove the same facts for S, or simply remark
that S(t) = fo C(s) ds.
(1 - \2(C))S(t'C) and
The relations
C(t,1;) are immediate and they also imply that Ot C(t) E P and 8i S(t) E P
simply by induction. Finally, the estimate AIS(tt )I < (1 +t2)'/2 comes from
a2(C)IS(t,C)I2 =
sin tici
+ sine tIeI < t2 (sup
sER
2
nl
s
J
+ 1 = 1 + t2.
1
The first result we give here is an existence and uniqueness theorem for the
Cauchy problem. It also shows that the smoother the data, the smoother the
solution.
THEOREM 4.11
Let uo and ul E S' and f E Ck(R;S'). Then the Cauchy Problem
(8c
- A)u(t) = f (t),
u(0) = uo,
8iu(0) = ui
has a unique solution u E C2(R; S'). Moreover, this solution satisfies
u(t) = C(t)vo + S(t)ir + J S(t - s) f (s) ds,
c
0
u E Ck+2 (R; S') and the following:
(i)
(ii)
If uo and ui E S and f E Ck(R,S), then u E Ck+2(RS).
If uo E Hs+', ul E Hs, and f E C°(R; Hs), then u E C°(R, H3+') fl
C' (R; Hs) f1 C2(R; Ha-1).
PROOF Through Fourier transformation in x only, the operator -0 =
- Ejol 82 is transformed into the operator of multiplication by the function
Fn q2 =
)12(1;) - 1. Thus, this Cauchy problem is equivalent to the
j_l
following:
(8r + (J12
- 1))u(t) = f (t),
u(0) = uo,
8tu(0) = ul,
where the only variable is t since there is no derivative with respect to the j's
(they are merely parameters).
In view of Lemma 4.10, it is now obvious that the given formula for u(t)
defines a temperate distribution that solves this problem. For the uniqueness, we
have to prove that u(t) must be given by this formula. Indeed, if u E C2 (K S,)
is a solution of the problem, we can use the distribution U(s, t) = C(t-s)u(s)+
S(t - s)8tu(s) to write
ft(t) = U(t, t) = U(0, t) +
= C(t)Lo + S(t)ii +
0sU(s, t) ds
J0
J0
t
S(t - s) f (s) ds
Applications
86
since 83U(s, t) = S(t - 8)(O u(s) + (A2 - 1)u(s)) = S(t - s) f (s).
To prove u E Ck+2 (fly; S') we simply compute
Otu(t) = (I - A2)S(t)uo + C(t)ui + f tQt - s) f (s) ds,
0
Ot u(t) = (1
- A2)u(t) + j (t)
and, because f E 0(1R; S'), this gives the result.
Finally, the claim (i) is obvious in view of the previous formulas, and for (ii)
we remark, using estimates from Lemma 4.10, that the function
Ae+1
A8+, IC(t)'uol + as+, IS(t)u, I + \8+1 I f t
I u(t)i
S(t - r)f (r) dr
o
t
AS(t - r).A8 f (r) dr
< a8+, Iuol + (1 + t2)1i2a81u, I +
J0
is square integrable in (for the integral, use the Cauchy-Schwarz inequality),
so that u E C°(K- H8+1). The proof of u E CI (X- H8)f1C2(l H8-,) is similar
when using the previous formulas for Btu and 8i u. I
When f = 0, the same computations lead to the property of conservation of
energy.
COROLLARY 4.12
If uo E H8+1, u1 E H8, and f = 0, the energy
E.(t) = IIOiu(t)Ils +
It
IID,u(t)II
=1
of the solution u E C°(X- H8+1) n C' (R; H8) obtained in Theorem 4.11(ii) is
independent oft and therefore is equal to E8 = 11u, I I2 + F- , I I D, uo 112
PROOF
Using the expression of u given in Theorem 4.11,
n
ID-,U(t)12 (1- A2)S(t)uo + C(t)u112
l0tu(t)I2 +
9=1
+ (A2 - 1)IC(t)uo + S(t)u112
(A2
= (C(t)2 +
= Iu1 I2 + (A2 -
- 1)S(t)2)(1u112 + (A2 - 1)Iuo12)
1,&o 12
since C(t)2 + (A2 - 1)S(t)2 = cost tI l + sin2
1. The conservation of
energy follows by multiplying by \2a and integrating in . I
The Cauchy problem for the wave equation
87
The last result will describe the phenomenon of finite propagation speed. It
shows that a modification of the data cannot immediately affect the values of
the solution far from the domain where the data are modified. For the sake of
simplicity, we continue to assume f = 0.
THEOREM 4.13
Let uo and ul E S', f = 0 and u E C°°(IRY;S') the solution of the Cauchy
problem as in Theorem 4.11. Then
(i)
If supp uo U supp ul does not intersect the closed ball l y E 1Rn ; l y - x I <
ItI}, then (t,x) ¢ supp u.
(ii) If sing supp uo U sing supp ul does not intersect the sphere {y E RI;
(y - xI = ItI}, then (t, x) ¢ sing supp u.
PROOF We will assume x = 0 and t > 0 since the wave operator is invariant
under translations in x and reversion of the time.
(i) Let us denote by Bi the closed ball of radius t and by Qt the open ball
of the same radius; if supp up U supp ul does not intersect the closed ball BI,
we actually have uo = ul = 0 in some 11T for a T > t. Then we will prove
that for 0 < s < T, u(s) vanishes in QT-,, and this will obviously imply that
(t, 0) 0 supp U.
Thus let 0 < s < T be fixed and let E Co (S2T_s). From Theorem 4.11
we know that u(s) = C(s)uo + S(s)ul, and we can write
(u(s),,p) =
(2ir)-n(uo,C(s)o)+(2ir)-n(ul,S(s)'4
Actually one has supp
C BT_e_, for some positive c, and thanks to the
Paley-Wiener theorem (Theorem 1.13), 0 can be extended as an entire function
I(I2)-Ne(T- c)1Im(I for all N E Z+
satisfying estimates lo(C)l < CN(1 +
and some sequence of constants CN. Using the results of Lemma 4.10, we see
that the functions 4o = C(s)y and 4Dl = S(s)y' can also be extended as entire
functions satisfying estimates
CN(1 +
I(I2)-Ne(T-E)IhnCt
for all N E Z and a new sequence C. Again using the Paley-Wiener theorem,
it follows that there exist two functions Wo and cpl E Co with supp cpy C BT_,,
cpo = C(s)o and c = S(s)cp, so that we can write
(u(s), p) = (27r)-'(u0, 0) + (27r)-n(ul, 01) = (uo, po) + (ul, WI),
and this is zero since uo and ul vanish in S1T. We thus get u(s) = 0 in SIT-.,
as claimed.
(ii) We will first prove that u is smooth in a neighborhood of {(s, y) E
R x 1Rn; s = 0 and jyj = t}. Indeed, if lyI = t, one has y ¢ sing supp uo U
sing supp ul by assumption, and one can find two functions vo and vi E S such
that no = vo and ul = vl in some neighborhood of y. The Cauchy problem
88
Applications
with data vo and v, has a solution v E C'°(R;S) (cf. Theorem 4.11(1)), and
thanks to the part (i) already proved, (0, y) ¢ supp (u - v) since u - v is the
solution of the Cauchy problem with data uo - vo and ul - v, the supports of
which do not intersect {y}. Therefore u is smooth near (0, y).
To prove that (t, 0) ¢ sing supp u, we now use the results of the previous
section. If u were singular at (t, 0), there would be a direction (r, f) E R" +I \ 0
such that (t, 0; r, ) E W Fu and since (o - O)u = 0, this point (t, 0; r, )
should lie in the characteristic set of the wave operator by Theorems 4.6 and 4.7.
This characteristic set is described by the equation 1t 12 - rZ = 0 so that we get
r = ±jjj # 0. According to the propagation theorem (Theorem 4.8), the
whole bicharacteristic curve starting at this point should then lie in WFu. This
bicharacteristic curve (s(r), y(r); a (r), q(r)) is the solution of
ds
dr
= -2a(r)
dy
dr
= 2q(r),
da
dq
dr = dr = 0,
(3(0), y(o); o(0), 77(0)) = (t, 0; r, t)
i.e., (s(r), y(r); a(r), q(r)) = (t-27-r, 4r; r, ). For r = t/2r we get s(r) = 0
while y(r) = tC/r satisfies l y(r)l = tle/rl = t, so that this point, which is in
sing supp u because of Theorems 4.7 and 4.8, lies on {(s, y) E JR x ]It"; s = 0
and jyl = t} where we proved that u is smooth. This contradiction shows that
(t,0) ¢ sing supp u.
I
A consequence of Theorem 4.13(ii) is that one has the following estimate for
the singular support of the solution u:
sing supp it C
U
W, X). Ix - yi = (tl },
yEE.UE.
where E, = sing supp uj. In other words, the singular support of u is contained
in the union of all light cones with vertices located at the singularities of the
data. However, the reader will easily understand that the results of Section 4.2
give much more than this estimate.
Indeed, while this inclusion is the most precise one that can be proved when
one knows only the singular supports of the data, it is not true that it is an
equality. Actually, the singular support of u is much smaller than this union
of cones, at least in most of the cases. On the contrary, if we take the point
of view of wave front sets, we can give an exact description of the location of
singularities of it (and therefore of its singular support). Indeed, denoting the
(polyhomogeneous) pseudodifferential operator of symbol ICI by IDJ, it can be
proved that (WFu) fl {t = 0} is equal to
{(0,x;(x,C) E WF(ul +ilDIuo)}
U {(0,x;
(x,C) E WF(ui - ijDjuo)}
89
Exercises
and then the exact description of WFu follows by using the propagation theorem
(Theorem 4.8); cf. Exercise 4.7.
We hope that this simple example will convince the reader of the great usefulness of the microlocal point of view in the study of singularities of solutions of
partial differential equations. To close this course, we simply point out that very
similar results have been proved for the nonlinear hyperbolic Cauchy problem,
and the essential tool in the proof was a well-adapted variant of the theory of
pseudodifferential operators. For such results, we refer the reader to Bony [31
and Chemin [5].
Exercises
4.1 Local solvability of (approximately) invertible operators. Assume that a and b E
S°° satisfy a#b-1 E S-°° (or more generally a and b E So with aft-1 E So.°`.
as in Exercises 2.9 and 3.6).
Let H6 = {x E R"; txI < b} as in Lemma 4.2; show that for any E > 0 there
is a 6 > 0 such that
11,P11_1 < ik1'lIo
for all WE Co (t1 ).
(Hint: Show by contradiction that otherwise you could find a square integrable
function with support equal to {0}.)
Show that you can find a 6 > 0 and a function E Co (f26) with o = I near
0 and such that the operator c(x, D) = p(x) (I - a#b(x, D)) satisfies
IIc(x, D)iPIj-, 5 1 M-
for all 1P E Ca (S26).
Let s E IR and f E H"; explain why the formulas
W'o = c(x, D) f,
ip,+i = c(x, D)>yr,
and ?G., _
TV
3>o
define functions ?, = Co (f26) (for j = oo, first prove that ?Pa E H-1, then
observe that tj°° = o + c(x, D)?Pa). Finally, show that the formula
u = b(x,D)(f +tV)
defines an H"-' distribution solution of a(x, D)u = f in a neighborhood of 0
(here, m denotes the order of b).
4.2 Let b : 1(P -. R be a C°° function, and let a(x, ) = t;i + ib(xi )l;2. We want to
study the local solvability of a(x, D) at the origin of R.
(a) Determine conditions on b equivalent to the following properties:
- a(x, D) is of principal type.
- a(x, D) is principally normal.
- a(x, D) satisfies Hdrmander's condition p = 0 = {p, p} = 0, where
p is the principal symbol.
Then describe for what functions b the operator a(x, D) is locally solvable
(resp. is not locally solvable) at the origin thanks to the results given in
Applications
90
Section 4.1, but exclude the case where b would have a zero of infinite
order at x, = 0.
(b) Assume that b does not change sign for lx, I < E < 1, and set 9 =
(-E, E) X (-E, E).
Take B(x3) = fo' tlb(t)I dt and prove that the operator c(x, D) defined
by c(x, D)t,(x) = eB(x
D)(e-B(nl)?i)(x) satisfies the a priori estimate
II
for all
'IIo
E CO '(Q).
(Hint: Integrate by parts the scalar product Im (c(x, D)tli, (x, ± D2)tP).)
Show that there exists a constant C such that
IkcIIo <_ Clla'(x, D),pll,
for all W E CO '(9).
(Hint: Use 0 = eB(n1)gyp.) Conclude that the equation a(x, D)u = fin f'?
has a solution u E H'3 for all f E H°. The converse of this result will be
proved in the next exercise.
4.3 Necessary conditions for local solvability. As was said in Section 4.1, the property
of local solvability for an operator a(x, D) is equivalent to an a priori estimate for
the operator a' (x, D). In this exercise, it is proved that the solvability property
implies such an estimate, and then this result is used to complete the study of
the operator D, + ib(x, )D2 at the origin of R2. (Hormander's theorem [7, Theorem 6.1.11, i.e., (ii) r (iii) in Corollary 4.4, can also be proved by using this
estimate.)
(a) Let a(x, D) = LloI<m a°(x)D° be a differential operator with complex-
valued coefficients a° E H°"(R").
Let K be a compact set of R", denote by Co (K) the space of C"0 functions with support contained in K, and consider on CI (K) the topology
defined by the Sobolev norms II 11. for s E 7+. Show that COI(K) is a
Fr+ chet space.
With f E C, 00(K) as above, assume that a(x, D)u = f has a solution
u E D'(Sl) where Il is a neighborhood of K. Show that there exists an
r E Z+ (depending on f) such that
I(w, f )l <_ rll a*(x,
for all p E CO '(K).
Show that 52,. = if E Co (K); I (gyp, f )I > rll a' (x, D)Vll r for some
p E CI(K) I is open in CO(K).
Assume that a(x, D)u = f has a solution u E 1Y (Q) for all f E Co (K)
where K C S1 are two neighborhoods of the origin (K is compact). Use
Baire's theorem to prove that there exist fo r= CI (K), c > 0, and r and
s E Z+ such that
Ilf - foil, < 2E =* f V flr.
Finally, prove that if a(x, D) is locally solvable at the origin, there exist
a compact neighborhood K of the origin, a constant C. and two integers
r, s E 7L+ such that
for all cp,fECo (K).
1(p,f)l<CIIfII,IIa'(x,D)wIIr.
(Hint: For any fixed f 34 0. write f = (Ilf Its/E)(g - fo) with g = fo +
(Ef/IIfIIa))
91
Exercises
(b) An example. Assume that a(x, D) = D, + ib(x, )D2 is locally solvable at
the origin of R2 (here, b is a smooth, real-valued function as in Exercise 4.2),
and let K, C, r, and s be as in part (a). Assume in addition that b changes
sign in every neighborhood of 0.
Show that K contains a rectangle R = [x_,x+J x [-E,EJ with a xo E
(x_, x+) such that B(xl) = f b(t) dt does not change sign and satisfies
1131 < 1/2 on [x_,x+], and does not vanish at x_ or x+. Then set
O(x) = B(xi) - ix2 - (B(x,) - ix2)2
if B > 0 on [x_, x+],
O(x) = ix2 - B(xi) - (ix2 - B(xi ))2
if B < 0 on [x-,x+],
and show that in both cases a* (x, D)ct = 0 and Re 0(x) > x2+(1/2) B(xj)
in R.
Choose (carefully) two functions Wo E Co (R) and f E Co (with <po = 1
near (xo,0) and f(0,0) = 1). Then use Vµ = e'µmcpo and f,(x) =
f (U(xi - xo), µ2x2) to show that the a priori estimate of part (a) cannot
hold for all p and f E Co (K). Conclusion?
4.4 In part (a) of this exercise, we prove that if u E S' and (xo, o) E T' R" \ 0, then
WFu if and only if there exists a V E Co satisfying cp = 1 near xo
and a conic neighborhood r of o such that the functions ,\kWpu are bounded in r
for all k E Z+. This characterization is then used in part (b) to determine some
wave front sets.
(a) Characterization of the wave front set. Let u E S' and (xo, to) E T'1R" \0.
Use Remark 2.8 to show that if the symbol a E S°° does not depend
on x,then aEPand a(D)v=avforall vES'.
Assume that for some wp E Co such that cp = I near xo, the functions
Akcpu are bounded in some conic neighborhood r of to for all k E Z.
Prove that there exists an a E S°° independent of x such that a#cp is elliptic
at (xo,lo) and
E S, and conclude that (xo,lo) ¢ WFu.
Conversely assume that (xo,1;o) V WFu. Show that one can construct
a W E Co with W = I near xo and a symbol a E S°° independent of x
with a(C) = 1 for large in some conic neighborhood r of c o such that
a#cp(x, D)u E S, and conclude that the functions Apu are bounded in r
for all k E Z+.
(b) Applications. Using the characterization proved in (a), determine WF6
where b E S' is defined by (b, r[)) = (0) for ' E S. In the following
questions, the dimension is n = 1. For tji E S = S(R), one sets
r
(pv ! , ) = lim
,-o+ J xl?
X
r
`x + i0 '
Irm
i_.0+
(x
E-.0+
lim
+G(x)
dx,
X
f !(x)
dx
f
dx.
x + if
X_%(
Show that these formulas define temperate distributions satisfying
1
pv
1
1
x+i0+irb=
x- i0
-i7rb
Applications
92
and
1
=x-i0
=
x+ i0 _
1
_
1
1.
xpv x x
x
Show that if V is a unit test function, then P(x) - (f 0(t) dt)V(x) is the
derivative of a function tli, E S for all ' E S. Then, prove that if u E S'
satisfies 02u = 0, then u is a constant.
Show that DEpv(l/x) = -2irb, and observing that pv(I /x) = -pv(l/x),
conclude that pv(l /x) is the L'° function with value iri on < 0 and value
-ai on > 0. Compute also the Fourier transforms of 1 /(x + i0) and
1/(x - i0).
Prove that if cp E Co satisfies p = 1 near x = 0, then ((cp(x) - 1)/x) E
H°O, and use the characterization proved in (a) to determine the wave front
sets of the distributions pv(l /x), 1/(x + i0), and 1/(x - i0).
4.5 Continuation of Exercise 4.4: Products and restrictions of distributions. In this
exercise, we want to discuss the possibility of defining the product of two distributions in R", or the restriction of such a distribution to a submanifold RP
of R". The properties of these notions we expect are that if b E P. then
(" ,u)v = O(uv) = -i(vu) = (a/iv)u and ('tl'u)IRP = ('+LIRP )(uIRP ).
(a) Consider in R(n = 1) the distributions u = pv(1/x) and v = b, defined in
Exercise 4.4(b), and ?P(x) = x, and show that you cannot define both (tpu)v
and (z'v)u, nor both uI:=o and (t'u)j=..o with the expected properties.
To overcome this difficulty, one can assume some smoothness on u:
indeed, uv is a L' function if u and v are L2 functions. Similarly, let
(x, y) denote the elements of R" where in E RP and y E R"-', assume
that u E H8(R") for some s > (n - p)/2 and show that the formula for
Wp E S(RP)
(uIRP, o) = (21r) " J u(C,,7)W-(f) d6 dp
defines a distribution UIRP
E H''(("-P)/2)(RP) satisfying (r(iu),RP =
(tPIp,)(u1R,,) for all V, E HOO.
In parts (b) and (c), we will investigate another way of defining products
and restrictions where some geometric properties of the wave front sets are
required instead of smoothness.
(b) Products of distributions. Let u and v E S' be such that WFun WFi = 0.
Relate WFv and WFv. Prove that for any in E R" there exists a
function cp= E Co with V. = I in a neighborhood !l of in and a v ucpzii
bounded for all k E Z.
Show that for any -0 E Co (fly), cp u(C)c
that the formula
(uv,'+1') =
(21r)_2"
E L'(R2") and
r)) dt drl
defines a distribution uv E D'(12 ).
Explain how to define a distribution uv E D'(R"), show that this definition has the expected properties, and determine the wave front set of the
product uv.
What products can you write with the distributions pv(1/x), I/(x+i0),
and I/ (x - i0) (cf. Exercise 4.4(b))?
Exercises
93
(c) Restrictions of distributions. Let (x, y) denote the elements of R", where
xE RP and yER"-P, and set N'RP={(x,y;g,n)ET'R";y={=0}.
that WFunN'RP=0.
Prove that for any x E RP there exists a function gyp= E Co (R") with
V.r = 1 in a neighborhood of (x, 0) and A';p u bounded in some conic
neighborhood of
q); = 0} for all k E Z+. In the following, Q..
denotes a neighborhood of x in RP where V_,Iv=o = 1.
Show that for any ip E Co (12=), pur E L'(R"), and show that the
formula
dt; drr
(uIRP, i,) = (21r)-° J
defines a distribution uIRP E D'(1l ).
Explain how to define a distribution uIRP E D'(RP), show that this
definition has the expected properties, and determine the wave front set of
the restriction.
4.6 Extend the theorems of Section 4.3 to the Klein-Gordon equation (8 - 0+µ)
u(t) = f (t), where p is any fixed complex number. More precisely:
Write the solution u(t) given by the method of variation of parameters and show
that the coefficients of uo and 1i1 can be extended as entire functions U,,(t,()
satisfying estimates
I or Itl).
IUN(t,()l
Prove the existence and uniqueness theorem, and a regularity theorem with
Sobolev spaces.
Assume f = 0 and it E R and prove the conservation of energy for a modified
definition of the energy.
Come back to any µ E C but keep the assumption f = 0, and prove theorems
of propagation of the support and singular support.
4.7 Propagation of the wave front set in the wave equation. Let uo and u, E S', and
let u be the solution of the Cauchy problem
(' - o)u(t) = 0,
u(0) = uo,
Otu(0) = u,
(cf. Theorem 4.11). Denote by IDI the polyhomogeneous pseudodifferential operator of symbol
and set
v = (8, +iIDl)u
and
w = (8, - iIDI)u.
Show that ((9t -zIDI)v, (Be+il DI )w, and (B,+iI DI )v+(Ot -iI Dl )w-2(8+0)u
are smooth.
Show that WFu = WFv U WFw.
Show that
l,VFvl,=o = {(0,x;
(x,t;) E WF(u, +iIDluo)}
and
WFwie=D = {(0,z;
(z,.) E WF(u, - iIDluo)}
(cf. Exercise 4.5(c)), then give an exact description of WFu involving only
WF(ui + iIDluo) and WF(u, - ilDluo).
94
Notes on Chapter 4
Notes on Chapter 4
Singular integral operators were first introduced in the study of elliptic problems (see e.g. Calder6n and Zygmund [27]), and Calder6n [231 used them later
to prove his celebrated uniqueness theorem. However, these operators did not
play an essential role in these questions, and it is mainly the proof of the index
formula by Atiyah and Singer [15] that convinced analysts of the importance
of this tool. These first results were soon followed by many other theorems
proved with the use of pseudodifferential operators theory. Among them, let
us quote the boundary problems treated by Calder6n [24], some hypoellipticity
and subellipticity results obtained by H6rmander [43,44], the results on microlocal singularities of solutions of partial differential equations as found in
Hi rmander [45] and Duistermaat and Hormander 1311, for example. We now
discuss more precisely the topics treated in Chapter 4.
During the nineteenth and half of the twentieth centuries, the problem of existence of solutions of partial differential equations was reduced to find how
many additional conditions had to be given in order to insure uniqueness of the
solution. This led essentially to the development of the study of the hyperbolic
Cauchy problem and of the elliptic Dirichlet problem. Things changed drastically in 1957 when Lewy [53] discovered a simple first-order equation (even
with analytic coefficients) admitting no solution for most C°° right-hand sides.
The first general results on local solvability are due to Hdrmander [40], whom
we followed in Section 4.1, and a characterization in the case of differential operators of principal type was obtained in Nirenberg and Treves [10) and Beals
and Fefferman [2]. (The corresponding result for pseudodifferential operators of
principal type is still an open problem.) For operators that are not of principal
type, some results have been obtained in the framework of nilpotent Lie groups
(see Levy-Bruhl [52] and the references therein).
It is difficult to decide who first proved that elliptic equations with smooth
coefficients and right sides admit only smooth solutions, since this was proved in
greater and greater generality by many authors. One main step was the famous
Weyl's lemma [70], while the first explicit statement of the result for general
linear elliptic equations is probably that of Friedrichs [36]. On the other hand,
the result on propagation of "discontinuities" in the hyperbolic Cauchy problem
seems to be due to Hadamard [38]. The microlocal versions of these results
were first given by Sato in the framework of hyperfunctions (see [60]), then in
Hbnnander [45, Section 2.51 and Duistermaat and Hbrmander [31, Section 6.1]
in the form we give in Section 4.2. The use of Fourier integral operators in
the latter reference is replaced in our text with Lemma 4.9, which is very close
to a theory of these operators. For the corresponding results in the analytic
framework, we refer to Sato et al. [60], Hormander [46], or Sjostrand [63]. In
the case of nonlinear equations, similar results were also obtained by Bony [3].
It is from the original memoire of Poisson [59], who treated the Cauchy
Notes on Chapter 4
95
problem for the wave equation, that the hyperbolic theory was developed. The
results of Section 4.3 were essentially already given in Hadamard [38] for general second-order hyperbolic equations, but the microlocal point of view we use
here simplifies the proof. For similar results in the nonlinear hyperbolic Cauchy
problem, we refer to Majda [54] (existence of a local solution) and to Bony [3]
and Chemin [5] (propagation of singularities).
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Index of Notation
Multiindices and geometry of R"
a! = (al!)...(a"!),
For a = (al,..., an) E Z+, Ial =
0<a/j <a, for all j,
if /3<a,
otherwise.
xn)1/Z,x"=xT...x"-,BR
{xElR';jxl<R}, and
Forj=l,...,n,a,=a/ax,,D2=-i03=(t/i)(a/ax,),a"=a;'...
ann = 01"1/8x1' ...axR^, D° = (-i)IOIa
Spaces of functions defined on W1
S: Schwartz space, i.e., space of COO functions cP such that all the norms
E R and Ia+/31 < k}
kPIk =
for k E Z+ are finite.
Co : space of test functions, i.e., of C°° functions with compact support. When
E Co and supp cP C S1, we also write cP E Co (S2).
P: space of C°G functions with polynomial growths at infinity: we write
/' E P° if /, is continuous and (1 + IxI2)-NV,(x) is bounded for some
NEZ+; we write V) EPif a°iI'EP°for all aEZ+.
LP: Lebesgue space, i.e., space of measurable functions u such that the norm
NormLP (u) =
(f1ux1'dx
103
Index of Notation
104
if I <p<oc,or
Norm- (u) = julo = inf{U E R; Iu(x)I < U almost everywhere}
is finite. We also use the notation
(u, Ia) =
J
u(x)v(,r) dx
if uv E L',
if u E L2.
IIuII1i = (u. IC) = J Iu(x)I2dx
Spaces of distributions defined on R"
S': space of temperate distributions (topological dual of S), i.e., space of
semi-linear forms on S 3 ,; - (u. ) E C such that
I (a,
)I<C1k_'I.v
for all
ES
and some CEll andNEZ+.
D': space of distributions (topological dual of C°`); we also use the notation
D'(S2) for the dual of Co (( ).
E': space of distributions with compact support (topological dual of C"); we
also use the notation E'(Q) for the dual of C"(SZ).
H': Sobolev space of exponent s E P, i.e., subspace of S' formed by distributions u such that ASir E L2 where AS(S) = (1 +
They are Hilbert
spaces when equipped with the norms
IIuII' = (2ir)-"IIA'hIIi1= (27r)"
1(1 +
We also use H-" = U5H5 and H" = f1SH'.
Other spaces on IR"
S"': space of symbols of order in, i.e., of C' functions a(x, ) defined on
IR" x R" such that the functions All3l-' c7Qa are bounded for all a.,3 E
Z" , where A113I-' is the function (I +Il;I2)(1,SI-"')/2. We also use S-" _
S' =
and S ( ) _ {a E C" (St x W'); yea E S'" for
all ,^ E Co (i2)}. For S,".. F) and Sa(F), see page 76.
A"': space of amplitudes of order in, i.e., of C" functions a(x) defined on 11P"
such that all the norms
IIIaIIIk = sup{I(1 +
IxI2)-m"2d'ra(x)I;x
E R" and Ial < k}
(in III 111k in is implicitly fixed) for k E 7L+ are finite.
Index of Notation
105
space of pseudodifferential operators of order m as defined on page 50.
We also use %F-OC = n,,41, (smoothing operators), *" = U,,, %' and
W
(1l) (resp. 'comp("),'Y"'(I')) for operators with symbols in Sla(Sl)
(resp. Scomp(r), Sia(r))Miscellaneous
A unit test function is a nonnegative C" function
supp V C tai
satisfying
J2(x)dx=l.
and
Operations on distributions: for u E S' and ' E S
(u, V) _ (u,,) (symmetry)
Ax) = 0(-x)
p(x) = f
(Fourier transformation)
d
(D' u, y') = (u, D" p) (differentiation)
(Vu, gyp) = (u,
gyp) (multiplication by a && E P)
(u, 0 = (u, ) (conjugation)
°a*
(Tyu,,P) = (u, r_ yp) (translation by a y E R")
supp u = support of u, see page 16
sing supp u = singular support of u, see page 16
WFu = wave front set of u, see page 77
r_yyJ(x) = y:(x - y)
Operations on symbols: for a and b E Sc
(x, ) =
(27r)_ f e-i(y,77)a(x
- y,
71) dy di (adjunction)
r!)b(x - y, ) dy d? (composition)
a#b(x, = (27r) -n f
Char a = characteristic set of a, see page 76-77
Index
adjoins, adjunction, 37, 48
amplitudes, 33
a priori estimates, 70-71
symbols, operators, 42-43, 55, 62, 71,
76
energy
conservation of, 86
estimates, 80
bicharacteristic curve, 79
binomial coefficients, formula, 3
Cauchy problem, 79, 85
characteristic set, 76-77
classic symbols, 32
composition, compound, 37, 48
conic, 76
cotangent bundle, 60-61
differential operator, symbol, 22, 30, 32,
39, 43, 50, 60, 61-62
differentiation of distributions, 13-14
distributions, 12-23
in an open set, 15-17
products of, 14-15, 26, 92
real-valued, 24
restrictions of, 92-93
temperate, 12
with compact support, 16-17
elliptic
estimate, 62
finite propagation speed, 87
Fourier integral operators, 67
Fourier transform(ation), 9-13
Friedrichs's lemma, 65
GArding's inequality, 56
Hamiltonian vector field, 78
Holder spaces, 19, 25-26
holomorphic functions. 23-24
hypoellipticity, 28, 55, 64, 65
inversion formula, 11, 13
Klein-Gordon equation, 93
Lebesgue spaces, 7-8
Leibniz's formula, 4
local property, 50, 61-62
107
108
Index
local solvability, locally solvable, 28, 64,
70, 89-91
symbol, 29-30
pseudolocal property, 51, 55
microlocalization, 64, 76
microlocal property, 51, 76, 78
multiindex, 2
quasi-elliptic, quasiprincipal symbols,
45-46
restrictions of distributions, 92-93
noncharacteristic, 76
S,11.1,, calculus, 45-46, 64, 89
operators, 47-61
Schur's lemma, 54, 63
differential, 22, 30, 50, 60, 61-62
elliptic, 55, 62, 71
Schwartz space, 5
Fourier integral, 67
of principal type, 70-71
principally normal, 70-71
properly supported, 61
pseudodifferential, 50
smoothing, 50, 55
wave, 83
singular support. 16
smoothing operator, 50, 55
Sobolev spaces, 18
subellipticity, 65-66
support, 6, 16
symbols, 28-43
differential, 30, 32, 39, 43
oscillatory integrals, 32-37, 44
Paley-Wiener-Schwartz theorem, 17, 25
Parseval's formula. 11, 13
partitions of unity, 7
Peetre's inequality, 21
Poisson brackets, 44, 70
polyhomogenous symbols. 32
polynomial growths at infinity, 5-6
principally normal, 70-71
principal symbol, 43, 70
principal type, 70-71
products of distributions, 14-15, 26. 92
projection theorem, 77
propagation of singularities, 78-79
properly supported, 61
pseudodifferential
operator, 50
semi-linear form, 8
elliptic, 42-43, 55, 62, 76
idempotent, nilpotent, 45
of the adjoint, compound operator. 37,
48
polyhomogenous (or classic), 32
principal, 43
(pseudodifferential), 29-30
quasi-elliptic, quasi-principal, 45-46
Taylor's formula, 2
test functions, 6
unit test function, 7
wave equation, operator, 83
wave front set, 77, 91
7158
ISBN 0-8493-7158-9