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introduction to the theory of pseudodifferential operators studies in advanced mathematics

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
of references are listed. Every reasonable effort has been made to give reliable data
and information, but the author and the publisher cannot assume responsibility for the
validity of all materials or for the consequences of their use.
All rights reserved. This book, or any parts thereof, may not be reproduced in any form
without written consent from the publisher.

This book was formatted with L TEX by Archetype Publishing Inc., P.O. Box 6567,
Champaign, IL 61821.
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
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!ty) satisfies vk(x, y, 1) = u(x + y), vk(x, y,0) _ EI I(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)!

if0a

C

b

otherwise. Then for a, O E Z+, one defines the binomial coefficients as the
products

It is easy to check that

ifQ
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
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)!


! (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 ntI < 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,
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 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- (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 I(u - v, v)I = I(u - v, V - We)I <- COeIIPIN+2
It follows that (u, gyp) = (v,
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.


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