Table Of ContentGrundlehren der
mathematischen Wissenschaften 256
A Series of Comprehensive Studies in Mathematics
Editors
M. Artin S.S. Chern J.L. Doob A. Grothendieck
E. Heinz F. Hirzebruch L. H6rmander
S.MacLane WMagnus C.C.Moore IK.Moser
M.Nagata WSchmidt D.S.Scott ITits
RL. van der Waerden
Managing Editors
M. Berger R Eckmann S.R.S. Varadhan
Lars Hormander
The Analysis of Linear
Partial
Differential Operators I
Distribution Theory and Fourier Analysis
Springer-Verlag
Berlin Heidelberg NewY ork Tokyo
1983
J. David Logan
Department of Mathematics and Statistics
University of Nebraska at Lincoln
Lincoln, NE 68588-0323
USA
Editorial Board
S. Axler F.W. Gehring K.A. Ribet
Mathematics Department Mathematics Department Department of
San Francisco State East Hall Mathematics
University University of Michigan University of California
San Francisco, CA 94132 Ann Arbor, MI 48109 at Berkeley
USA USA Berkeley, CA 94720-3840
USA
Mathematics Subject Classification (1991): 35-01
Library of Congress Cataloging-in-Publication Data
Logan, J. David (John David)
Applied partial differential equations / J. David Logan.
p. cm.-(Undergraduate texts in mathematics)
Includes bibliographical references and index.
ISBN·I3: 978·3·642·96752·8 .·ISBN·I3: 978·3·642·96750·4
DOl: 10.1007/978·3·642·96750-4
1. Differential equations, Partial. I. Title. II. Series.
QA377.L578 1998
515'.353-dc21 97-48861
Printed on acid-free paper.
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1998 Springer-Verlag New York, Inc.
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ISBN-I3: 978·3·642·96752·8
Preface
In 1963 my book entitled "Linear partial differential operators" was
published in the Grundlehren series. Some parts of it have aged well
but others have been made obsolete for quite some time by techniques
using pseudo-differential and Fourier integral operators. The rapid de
velopment has made it difficult to bring the book up to date. Howev
er, the new methods seem to have matured enough now to make an
attempt worth while.
The progress in the theory of linear partial differential equations
during the past 30 years owes much to the theory of distributions
created by Laurent Schwartz at the end of the 1940's. It summed up a
great deal of the experience accumulated in the study of partial differ
ential equations up to that time, and it has provided an ideal frame
work for later developments. "Linear partial differential operators" be
gan with a brief summary of distribution theory for this was still un
familiar to many analysts 20 years ago. The presentation then pro
ceeded directly to the most general results available on partial differ
ential operators. Thus the reader was expected to have some prior fa
miliarity with the classical theory although it was not appealed to ex
plicitly. Today it may no longer be necessary to include basic distribu
tion theory but it does not seem reasonable to assume a classical
background in the theory of partial differential equations since mod
ern treatments are rare. Now the techniques developed in the study
of singularities of solutions of differential equations make it possible
to regard a fair amount of this material as consequences of extensions
of distribution theory. Rather than omitting distribution theory I have
therefore decided to make the first volume of this book a greatly ex
panded treatment of it. The title has been modified so that it indicates
the general analytical contents of this volume. Special emphasis is put
on Fourier analysis, particularly results related to the stationary phase
method and Fourier analysis of singularities. The theory is illustrated
throughout with examples taken from the theory of partial differential
equations. These scattered examples should give a sufficient know
ledge of the classical theory to serve as an introduction to the system-
VI Preface
atic study in the later volumes. Volume I should also be a useful in
troduction to harmonic analysis. A chapter on hyperfunctions at the
end is intended to give an introduction in the spirit of Schwartz distri
butions to this subject and to the analytic theory of partial differential
equations. The great progress in this area due primarily to the school
of Sato is beyond the scope of this book, however.
The second and the third volumes will be devoted to the theory of
differential equations with constant and with variable coefficients re
spectively. Their prefaces will describe their contents in greater detail.
Volume II will appear almost simultaneously with Volume I, and Volume
III will hopefully be published not much more than two years later.
In a work of this kind it is not easy to provide adequate refer
ences. Many ideas and methods have evolved slowly for centuries, and
it is a task for a historian of mathematics to uncover the development
completely. Also the more recent history provides of course consider
able difficulties in establishing priorities correctly, and these problems
tend to be emotionally charged. All this makes it tempting to omit ref
erences altogether. However, rather than doing so I have chosen to
give at the end of each chapter a number of references indicating re
cent sources for the material presented or closely related topics. Some
references to the earlier literature are also given. I hope this will be
helpful to the reader interested in examining the background of the
results presented, and I also hope to be informed when my references
are found quite inadequate so that they can be improved in a later
edition.
Many colleagues and students have helped to improve this book,
and I should like to thank them all. The discussion of the analytic
wave front sets owes much to remarks by Louis Boutet de Monvel, Pierre
Schapira and Johannes Sjostrand. A large part of the manuscript was
read and commented on by Anders Melin and Ragnar Sigurdsson in
Lund, and Professor Wang Rou-hwai of Jilin University has read a
large part of the proofs. The detailed and constructive criticism given
by the participants in a seminar on the book conducted by Gerd
Grubb at the University of Copenhagen has been a very great help.
Niels J0rgen Kokholm took very active part in the seminar and has
also read all the proofs. In doing so he has found a number of mis
takes and suggested many improvements. His help has been invalu
able to me.
Finally, I wish to express my gratitude to the Springer Verlag for
encouraging me over a period of years to undertake this project and
for first rate and patient technical help in its execution.
Lund, January 1983 Lars Hormander
Contents
Introduction . . . . . .
Chapter I. Test Functions 5
Summary ...... 5
1.1. A review of Differential Calculus 5
1.2. Existence of Test Functions 14
1.3. Convolution . . . . . . . . . 16
1.4. Cutoff Functions and Partitions of Unity 25
Notes . . . . . . . . . . . . . . . . . 31
Chapter II. Definition and Basic Properties of Distributions 33
Summary ..... 33
2.1. Basic Definitions 33
2.2. Localization 41
2.3. Distributions with Compact Support 44
Notes . . . . . . . . . . . . . . . . 52
Chapter III. Differentiation and Multiplication by Functions 54
Summary .......... 54
3.1. Definition and Examples 54
3.2. Homogeneous Distributions 68
3.3. Some Fundamental Solutions 79
3.4. Evaluation of Some Integrals 84
Notes . . . . . . . 86
Chapter IV. Convolution 87
Summary ..... 87
4.1. Convolution with a Smooth Function 88
4.2. Convolution of Distributions . . . 100
4.3. The Theorem of Supports 105
4.4. The Role of Fundamental Solutions 109
VIII Contents
4.5. Basic IJ' Estimates for Convolutions 116
Notes . . . . . . . . . . . . . . . 124
Chapter V. Distributions in Product Spaces 126
Summary ....... 126
5.1. Tensor Products 126
5.2. The Kernel Theorem 128
Notes . . . . . . . . . 132
Chapter VI. Composition with Smooth Maps 133
Summary .......... 133
6.1. Definitions . . . . . . . . 133
6.2. Some Fundamental Solutions 137
6.3. Distributions on a Manifold 142
6.4. The Tangent and Cotangent Bundles 146
Notes . . . . . . . . . . . . . . . 156
Chapter VII. The Fourier Transformation 158
Summary ............. 158
7.1. The Fourier Transformation in [/ and in [/' 159
7.2. Poisson's Summation Formula and Periodic Distributions 177
7.3. The Fourier-Laplace Transformation in cff' 181
7.4. More General Fourier-Laplace Transforms 191
7.5. The Malgrange Preparation Theorem . . 195
7.6. Fourier Transforms of Gaussian Functions 205
7.7. The Method of Stationary Phase 215
7.8. Oscillatory Integrals 236
7.9. H(S)' IJ' and HOlder Estimates 240
Notes . . . . . . . . . . . . . 248
Chapter VIII. Spectral Analysis of Singularities 251
Summary ............... 251
8.1. The Wave Front Set ........ 252
8.2. A Review of Operations with Distributions 261
8.3. The Wave Front Set of Solutions of Partial Differential
Equations. . . . . . . . . . . . . . 271
8.4. The Wave Front Set with Respect to CL • • . . . 280
8.5. Rules of Computation for WFL • • • • • • • • • 296
8.6. WF for Solutions of Partial Differential Equations 305
L
8.7. Microhyperbolicity 317
Notes . . . . . . . . . . . . . . . . . . . . . . 322
Contents IX
Chapter IX. Hyperfunctions 325
Summary ....... 325
9.1. Analytic Functionals 326
9.2. General Hyperfunctions 335
9.3. The Analytic Wave Front Set of a Hyperfunction 338
9.4. The Analytic Cauchy Problem . . . . . . . . 346
9.5. Hyperfunction Solutions of Partial Differential Equations 353
9.6. The Analytic Wave Front Set and the Support 358
Notes . . 368
Bibliography 371
Index . . . 388
Index of Notation 390
Introduction
In differential calculus one encounters immediately the unpleasant fact
that every function is not differentiable. The purpose of distribution
theory is to remedy this flaw; indeed, the space of distributions is
essentially the smallest extension of the space of continuous functions
where differentiation is always well defined. Perhaps it is therefore self
evident that it is desirable to make such an extension, but let us
anyway discuss some examples of how awkward it is not to be
allowed to differentiate.
Our first example is the Fourier transformation which will be
studied in Chapter VII. If v is an integrable function on the real line
then the Fourier transform Fv is the continuous function defined by
00
(Fv)(¢)= S riX~v(x)dx, ¢EIR.
- co
It has the important property that
(1) F(Dv)=MFv, F(Mv) = -DFv
whenever both sides are defined; here Dv(x)= -idv/dx and Mv(x)
=xv(x). In the first formula the multiplication operator M is always
well defined so the same ought to be true for D. Incidentally the
second formula (1) then suggests that one should also define F for
functions of polynomial increase.
Next we shall examine some examples from the theory of partial
differential equations which also show the need for a more general
definition of derivatives. Classical solutions of the Laplace equation
(2)
or the wave equation (in two variables)
(3) 32v/3x2-cPy/3y2=O
are twice continuously differentiable functions satisfying the equations
everywhere. It is easily shown that uniform limits of classical solutions
2 Introduction
of the Laplace equation are classical solutions. On the other hand, the
classical solutions of the wave equation are all functions of the form
(4) vex, y)= f(x+ y)+ g(x-y)
with twice continuously differentiable f and g, and they have as
uniform limits all functions of the form (4) with f and g continuous.
All such functions ought therefore to be recognized as solutions of (3)
so the definition of a classical solution is too restrictive.
Let us now consider the corresponding inhomogeneous equations
(5) c2u/8x2 + 82u/8y2 =F,
(6) 82v/8x2 -82v/8y2 =F
where F is a continuous function vanishing outside a bounded set. If
F is continuously differentiable a solution of (6) is given by
H
(7) v(x,y)= -F(~,I1)d~dl1/2.
~-y+lx-~I<O
However, (7) defines a continuously differentiable function v even if F
is just continuous. Clearly we must accept v as a solution of (6) even if
second order derivatives do not exist. Similarly (5) has the classical
solution
(8) u(x, y) =(4n)-1 HF (~, l1)log((x - ~)2 + (y -11)2) d~ dl1
provided that F is continuously differentiable. Again (8) defines a
continuously differentiable function u even if F is just continuous, and
we should be able to accept u as a solution of (5) also in that case.
The difficulties which are illustrated in their simplest form by the
preceding examples were eliminated already by the concept of weak
solution which preceded distribution theory. The idea is to rewrite the
equation considered in a form where the unknown function u is no
longer differentiated. Consider as an example the equation (6). If u is a
classical solution it follows that
(6 )'
for every continuous function ¢ vanishing outside a compact, that is,
closed and bounded, set. Conversely, if (6)' is fulfilled for all such ¢
which are say twice continuously differentiable then (6) is fulfilled. In
fact, if (6) were not satisfied at a point (xo, Yo) we could take ¢ non
negative and 0 outside a small neighborhood of (xo, Yo) and conclude
that (6)' is not fulfilled either. For such "test functions" ¢ we can
integrate by parts twice in the left-hand side of (6)' which gives the
equivalent formula
