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Classics in Mathematics Lars Hormander
The Analysis of Linear Partial Differential Operators I
Springer
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Lars Hormander
The Analysis of Linear Partial Differential Operators I Distribution Theory and Fourier Analysis
Reprint of the 2nd Edition 1990
Springer
Lars Hormander Department of Mathematics University of Lund Box 118
SE-22100 Lund, Sweden
e-mail: [email protected]
Originally published as Vol. 256 in the series: Grundlehren dermathematischen Wissenschaften in 1983 and 1990, and thereafter as a Springer Study Edition in 1990 Library of Congress Cataloging.in·Publication Data HOrmander, Lars. The analysis of linear partial differential operato", I L. HOrmander. p. cm. - (Classics in mathematics, ISSN 1431-0821) Originally published: 2nd ed. Berlin; Now Yark : Springer-Verlag, c1990- . Includes bibliographical ref"""""", and index... Contents: 1. Distribution theoIy and Fourier analysis ISBN-13: 978-3-540-00662-6
1. Differential equations. Partial 2. Partial differential operators. l Title. II. Series. QA377.H5782003 515',7242-dc21
2003050516
Mathematics Subject Classification (2000): 46F, 46E, 26A, 42A, 35A, 35J, 35L
ISSN 1431-0821 ISBN-13: 978-3-540-00662-6 DOl: 10.1007/978-3-642-61497-2
e-ISBN-13: 978-3-642-61497-2
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Springer-Verlag BerlinHeidelberg 2003
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41/3142ck-543210
Preface to the Second Edition
The main change in this edition is the inclusion of exercises with answers and hints. This is meant to emphasize that this volume has been written as a general course in modern analysis on a graduate student level and not only as the beginning of a specialized course in partial differential equations. In particular, it could also serve as an introduction to harmonic analysis. Exercises are given primarily to the sections of general interest; there are none to the last two chapters. Most of the exercises are just routine problems meant to give some familiarity with standard use of the tools introduced in the text. Others are extensions of the theory presented there. As a rule rather complete though brief solutions are then given in the answers and hints. To a large extent the exercises have been taken over from courses or examinations given by Anders Melin or myself at the University of Lund. I am grateful to Anders Melin for letting me use the problems originating from him and for numerous valuable comments on this collection. As in the revised printing of Volume II, a number of minor flaws have also been corrected in this edition. Many of these have been called to my attention by the Russian translators of the first edition, and I wish to thank them for our excellent collaboration. Lund, October 1989
Lars Hormander
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 development has made it difficult to bring the book up to date. However, 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 differential equations up to that time, and it has provided an ideal framework for later developments. "Linear partial differential operators" began with a brief summary of distribution theory for this was still unfamiliar to many analysts 20 years ago. The presentation then proceeded directly to the most general results available on partial differential operators. Thus the reader was expected to have some prior familiarity with the classical theory although it was not appealed to explicitly. Today it may no longer be necessary to include basic distribution theory but it does not seem reasonable to assume a classical background in the theory of partial differential equations since modern 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 expanded 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 knowledge of the classical theory to serve as an introduction to the system-
VIII
Preface
atic study in the later volumes. Volume I should also be a useful introduction to harmonic analysis. A chapter on hyperfunctions at the end is intended to give an introduction in the spirit of Schwartz distributions 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 respectively. 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 references. 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 considerable difficulties in establishing priorities correctly, and these problems tend to be emotionally charged. All this makes it tempting to omit references altogether. However, rather than doing so I have chosen to give at the end of each chapter a number of references indicating recent 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 JI1lrgen Kokholm took very active part in the seminar and has also read all the proofs. In doing so he has found a number of mistakes and suggested many improvements. His help has been invaluable 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 . . . . . . 1.1. A review of Differential Calculus 1.2. Existence of Test Functions 1.3. Convolution . . . . . . . . . 1.4. Cutoff Functions and Partitions of Unity Notes . . . . . . . . . . . . . . . . . Chapter II. Definition and Basic Properties of Distributions Summary . . . . . 2.1. Basic Definitions 2.2. Localization . . 2.3. Distributions with Notes . . . . . . .
. . . . . . . . . . . . . . . . . . Compact Support . . . . . . . . .
Chapter III. Differentiation and Multiplication by Functions Summary . . . . . . . . . . 3.1. Definition and Examples 3.2. Homogeneous Distributions 3.3. Some Fundamental Solutions 3.4. Evaluation of Some Integrals Notes . . . . . . . Chapter IV. Convolution Summary . . . . . 4.1. Convolution with a Smooth Function 4.2. Convolution of Distributions . . . 4.3. The Theorem of Supports 4.4. The Role of Fundamental Solutions
5 5 14 16 25 31 33 33 33 41 44 52 54 54 54 68 79 84 86 87 87 88 100 105 109
X
Contents
4.5. Basic I! Estimates for Convolutions Notes . . . . . . . . . . . . . . .
116 124
Chapter V. Distributions in Product Spaces
126
Summary . . . . . . . 5.1. Tensor Products 5.2. The Kernel Theorem Notes . . . . . . . . .
126 126 128 132
Chapter VI. Composition with Smooth Maps
133
Summary . . . . . . . . . . 6.1. Definitions . . . . . . . . 6.2. Some Fundamental Solutions 6.3. Distributions on a Manifold 6.4. The Tangent and Cotangent Bundles Notes . . . . . . . . . . . . . . .
133 133 137 142 146 156
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 If' 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)' I! and HOlder Estimates 240 Notes . . . . . . . . . . . . . 248 Chapter VIII. Spectral Analysis of Singularities Summary . . . . . . . . . . . . . . . 8.1. The Wave Front Set . . . . . . . . 8.2. A Review of Operations with Distributions 8.3. The Wave Front Set of Solutions of Partial Differential Equations. . . . . . . . . . . . . . 8.4. The Wave Front Set with Respect to CL . . . . • 8.5. Rules of Computation for WFL • . . • . • . • • 8.6. WFL for Solutions of Partial Differential Equations 8.7. Microhyperbolicity Notes . . . . . . . . . . . . . . . . . . . . . .
251 251 252 261 271 280 296 305 317 322
Contents
Chapter IX. Hyperfunctions Summary . . . . . . . 9.1. Analytic Functionals 9.2. General Hyperfunctions 9.3. The Analytic Wave Front Set of a Hyperfunction 9.4. The Analytic Cauchy Problem . . . . . . . . 9.5. Hyperfunction Solutions of Partial Differential Equations 9.6. The Analytic Wave Front Set and the Support Notes . . . . . . . . . .
XI
325 325 326 335 338 346 353 358 368
Exercises
371
Answers and Hints to All the Exercises.
394
Bibliography.
419
Index . . . .
437
Index of Notation.
439
Introduction
In differential calculus one encounters immediately the unpleasant fact that not every function is 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,
~E1R.
-00
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)
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)
v(x, 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)
82uj8x 2 + 8 2uj8y2 =F,
(6)
82vj8x 2 -8 2vj8y2 =F
where F is a continuous function vanishing outside a bounded set. If F is continuously differentiable a solution of (6) is given by
(7)
V(x,y)=
H
-F(~,rJ)d~drJj2.
~-y+lx-~IO for allj unless Mj* = 0 for every j > O. Moreover, (1.3.21) is a decreasing sequence. (We define aj = + 00 if M; =0.) Theorem 1.3.8 (Denjoy-Carleman). The following conditions are equivalent: (i) CM is quasi-analytic. 00
(ii)
L I/L o
j
= 00.
L (Mj)- Iii = 00. 00
(iii)
o
00
(iv)
L a = 00. j
1
First we consider the rather uninteresting case where for every j, that is, limMWi~L for some sequence kr-+ 00. Letting I run through this sequence in (1.3.20) and taking k = 0 there we obtain IiMo?;,Mj*=Mo/a l ... aj?;,Mo/(a l ... ai_l)a:- j - l , i~j, Proof
Lj~L< 00
and this implies ai-I ~L. Thus the conditions (ii), (iii), (iv) are fulfilled. If u satisfies (1.3.18) and uU)(xo)=O for every j, then Taylor's formula gives for every xE[a, b] Ju(x)J ~Kki+ 1 JX-xoJkiMk/k j ! -+ 0
when j
-+00.
Thus (i) is also valid. Assume now that L j -+ 00 when j -+ 00, that is, that M:/ k -+ 00 as k -+ 00. Then the points (k, log M k ) will lie above lines with arbitrarily high slope so Mj is positive and aj -+ O. Thus J = U; aj+ I < aj} is an infinite set where the graph of log M: as a function of k has a corner. This implies that Mj = Mj when jEi; analytically this follows since for k (iv). On the other hand, if (iv) is valid we can apply (1.3.17) with m equal to an element of J. The set J in (1.3.16), defined in Lemma 1.3.6, is then the set of elements ~m in J, so (1.3.16) follows from (1.3.18) since M j =Mj=l/a l ... aj when jEJ. Hence (iv) => (i). The remaining proof that (iii) => (iv) follows from
Lemma 1.3.9 (Carleman's inequality). If aj>O then 00
(1.3.22)
00
~)ala2'" an)l/n~eLan' 1
1
Proof With cj to be chosen later we have by the inequality between geometric and arithmetic means n (a l .. , an)l/n =(c l ... cn)-l/n(c l a l '" Cnan) l/n ~(Cl ... cn)-l/nn- l L cmam • 1
00
L(al .. ·an)l/n~
L
00
cmam/n(n+l)=Lcmam/m~Leam 1
l~m~n
1
which proves (1.3.22). When Mn=n! then Taylor's formula gives for JECM([a,b]) that 00
J(x) = L JU)(y)(x - y)j/j! o
if x,YE[a,b] and Ix-yl1 =4>1/1 1 ,4>2=4>1/12(1-1/11)' ···,4>k=4>l/Ik(l-l/Il)···(l-l/Ik-l) have the required properties since k
"'i4>j-4>= -4> 1
n(1-1/1)=0 k
1
because either 4> or some 1 -1/1j is zero at any point.
28
L Test Functions
Combining Theorems 1.4.4 and 1.4.1 we obtain Theorem 1.4.5. Let Xl' ... ' X k be open sets in 1Rnand K a compact subset of k
k
UXj.
Then one can find (x)dx= v(x)cf>(sx)dx --> cf>(0) v(x)dx,
if cf>EC~(1R"). (Compare Theorem 1.3.2.) Example 2.1.11. Let wEC~(1Rn), assume that Jxaw(x)dx=O when lod icf>(O),
o
as t -->00,
Example 2.1.14. Let Ut(x)=tl/keitX", xE1R, where k is an integer >1. To determine the limit we first examine x
F(x)= Jeiykdy. o
When x>O we shift the integration to a segment of the line argz=n/2k and a circular are, on which Imzk~clzlk-l Imz where c>O.1t follows that F(x)-->e"i/2k
Hence
Je-ykdy
00
o
00
F(x)-->_e"i/2k Jrykdy o
as x --> 00.
as x-->-oo ifkiseven
-e-"i/ 2k Je- yk dy o
as x-->-oo ifkisodd.
2.2. Localization
41
Now we have if cPECo(1R.)
J
JF(x tl/k)cP'(x)dx o J cP'(x)dx-F(oo) J cP'(x)dx=(F(oo)-F(-oo))cP(O).
ut(cP) = t l /kF'(xtl/k)cP(x)dx = -
00
--+
-F(-oo)
0
-00
Here
J e-ykdy
00
F( 00)- F( - 00)=2 cos n/2k
J rykdy
00
=2e"i/2k
o
o
if k is odd if k is even,
which is not O. (In terms of the r function defined in Section 3.2 the integral is equal to r«l + k)/k).) Note the contrast with the case k = 1.
2.2. Localization If Y eX e1R.n and UE.@'(X), we can restrict u to a distribution u y in Y by setting Uy(cP) = u(cP), cPECo(Y).
Our next purpose is to prove the less trivial fact that a distribution is determined by the restrictions to the sets in an open covering:
Theorem 2.2.1. If UE.@'(X) and every point in X has a neighborhood to which the restriction of U is 0, then U = O. Proof If cPECo(X) we can for every XESUPPcP find an open neighborhood Y e X such that the restriction of U to Y is O. By the BorelLebesgue lemma we can choose a finite number of such open sets lj e X which cover supp cP. But according to Theorem 1.4.4 we can then write cP=I.cPj where cPjECo(lj). Thus u(cPj)=O which proves that u(cP) = I. u(cPj) =0.
Theorem 2.2.1 makes it natural to extend Definition 1.2.2 as follows:
Definition 2.2.2. If UE.@'(X) then the support of u, denoted supp u, is the set of points in X having no open neighborhood to which the restriction of u is O.
42
II. Definition and Basic Properties of Distributions
Thus X ....... supp U is the open set of points having a neighborhood in which U vanishes, so U vanishes in X . . . . supp U by Theorem 2.2.1, and X ....... supp U contains every open set where U vanishes. Thus we have (2.2.1)
U(~X} such that (i) u(c/»=u(c/» if fjJEC't;'(X), (ii) u(c/» =0 if c/>ECOO(X) and Fllsuppc/>=0. The domain of it is of course largest when F = supp u, but we need the uniqueness statement also for other sets F. Proof a) Uniqueness. Let fjJECOO(X) and let FllsuppfjJ=K be a com-
pact subset of X. By Theorem 1.4.1 we can find t/lEC't;'(X) so that t/I =1 in a neighborhood of K. Then we have fjJ=fjJO+fjJI where fjJo
44
II. Definition and Basic Properties of Distributions
=t/I¢EC'~(X)
(ii) we obtain
and cPI =(1- t/I)cP, so that F nsupp cPI =0. Using (i) and u(cP) = u(cPo)+ u(cP I) = u(cPo),
which proves the uniqueness of U. b) Existence. We have seen in a) that every cP E COO(X) with F nsuppcP compact can be written cP =cPo +cPI with cPoEC~(X) and F n supp cP 1 = 0. If cP = ¢~ + cP'l is another such decomposition, then X =cPo-¢~EC~(X) and FnsuPPx=FnsuPP(cPl-cP'l)=0 so it follows from (2.2.1) that O=u(X)=u(cPo)-u(cP~). Setting u(cP)=u(cPo) therefore gives a unique definition of a linear form u which obviously has the required properties. From now on we write u(¢) instead of u(cP) and thus consider u(cP) as defined for all UEg&'(X) and all cPECOO(X) satisfying (2.2.3). In view of the symmetry of (2.2.3) we shall sometimes write (u, cP) instead of u(¢).
2.3. Distributions with Compact Support If UE~'(X) has compact support we have seen that u(cP) can be defined for all cPECOO(X). When t/lEC~(X) and t/I= 1 in a neighborhood of supp u, we have u(¢)=u(t/I¢)+u«1-t/l)cP)=u(t/lcP),
cPECOO(X).
Hence it follows from (2.1.2) that (2.3.1)
lu(¢)1 ~ C
L
lal;iik
sup laacPl,
cPECOO(X),
K
where K is the support of t/I and C, k are constants. Conversely, suppose that we have a linear form v on COO(X) such that for some constants C and k and some compact set LeX (2.3.2)
Iv(¢)I~C
L
lal;iik
sup laa¢l,
cPECOO(X).
L
Then the restriction of v to C~(X) is a distribution u with support contained in L. Since it follows from (2.3.2) that v(cP)=O if LnsuppcP =0, we obtain from Theorem 2.2.5 that v(cP) = u(cP) for every ¢ECOO(X). Hence we have proved
Theorem 2.3.1. The set of distributions in X with compact support is identical with the dual space of COO(X) with the topology defined by the semi-norms
2.3. Distributions with Compact Support
¢
I
-+
45
sup lo"¢I,
l"l~k
K
where K ranges over all compact subsets of X and k over all integers ~O.
Schwartz used the notation @,,(X) for the space Coo(X) equipped with this topology. Accordingly the space of distributions with compact support in X is denoted by @"'(X). From the proof of Theorem 2.3.1 it follows that @"'(X) can be identified with the set of distributions in @"'(1R") with supports contained in X. We may therefore use the notation @"'(A) also when A is an arbitrary subset of JR" to denote the set of distributions in @"'(JR") with supports contained in A. We write @"'k(A)=@"'(A)(l~'k(JR"). The smallest k which can be used in (2.3.1) is of course the order of the distribution u. For K one can take any neighborhood of supp u but usually not the support itself. Example 2.3.2. Let K be a compact set in JR" which is not the union of finitely many compact connected sets. Then one can find uE@"'(K) of order 1 so that (2.3.1) is not valid for any C and k. In fact, the
hypothesis means that we can find a sequence of disjoint non-empty compact subsets K j of K such that K'-.(K1u ... uKj) is compact. Choose xjEK j , let Xo be a limit point of {xJ and set u(¢)=
I
mi¢(x)-¢(xo))
where mj is a positive sequence such that Imjlxj-xol=1,
Imj=oo.
Such a sequence exists since liminflxj-xol=O. Then lu(¢)1 ~sup WI so u is a distribution. On the other hand, if (2.3.1) is valid and we choose ¢ E COO equal to 1 in a neighborhood of K 1 u ... u K j and 0 near K '-. (K 1 u ... u K j ), hence at x o, then we obtain
I
mi~C
i~j
which is a contradiction when j
-+ 00.
Although (2.3.1) is not in general valid with K =supp u we can prove that the left-hand side must vanish then if the right-hand side does:
46
II. Definition and Basic Properties of Distributions
Theorem 2.3.3. If UEtf' is of order ~k and (2.3.3)
8a4>(x) =0
for
lal~k
if 4>ECk,
and XESUPPU,
then it follows that u( 4» = o. Recall that u(4)) was defined for all 4>EC~ in Theorem 2.1.6, and as in Theorem 2.2.5 we have a unique extension to all 4>ECk, for which u(4))=O when suppunsupp4>=0. The estimate (2.3.1) is valid for all 4> E Ck if K is a neighborhood of supp u.
Proof of Theorem 2.3.3. By Theorem 1.4.1 and the remarks after it we can choose X.ECO' so that X.=1 in a neighborhood of suppu, X.=O outside M.={y; Ix-yl~e for some XESUppU} and 18a.x.I~Ce-la.l,
lod~k.
Since suppu and supp(1-X')4> do not intersect, we have
u(4)) = u(4) X.) + u(4)(1- X.» = u(4) X.) so using (2.3.1) we obtain
I ~ C" I
lu(4))I~C
supI8a.(4)x.)I~C'
lal~k la.l~k
I
supI8a.4>118 Px.1
la.l+IPI~k
Bla.H sup 1811.4>1. M,
To show that the right-hand side tends to 0 with
B
(2.3.4)
if lad ~ k.
BlaH SUp 1811.4>1 ..... 0 M,
when
B .....
0
we must prove that
By the definition of M. we can for every YEM. choose XESUPP u so that Ix - yl ~ B. This gives (2.3.4) for IIXI = k since 8a 4> is uniformly continuous and vanishes on Supp u. If lal < k and YEM. we obtain by Taylor's formula 1 I811. 4> (y)1 ~ k I I sup l(d/dt)k-la.l (811. 4» (x + t(y - x»1 ( - a)! 0 of order k, which proves (2.3.4). An important consequence of Theorem 2.3.3 is the following
Theorem 2.3.4. If u is a distribution of order k with support equal to {y}, then u has the form
2.3. Distributions with Compact Support
(2.3.5)
u( according to Theorem 1.3.2 with ¢ of so small support that ulj> is defined in a neighborhood of K. In view of (2.3.15) we have IUIj>(x) - u",(y)1 ~ Ix - yl in a neighborhood of K when Ix - yl is small, hence lai ulj>l ~ 1 on K. With l/I = ulj> we obtain from (2.3.14) IUIj>(xo)-ulj>(Yo)1 ~ Ixo - Yol C(d(yo) +n). Letting supp ¢
-+
{OJ we conclude that d(Yo)~lxo-yol C(d(Yo)+n).
When Ixo - Yol ~ 1/2 C it follows that d(Yo)~2 n Clxo - Yol. For any &>0 the set K,={x; Ix-ylz. By the localization Theorem 2.2.4 it follows that there is a unique distribution jUE.@'(X) with restriction (fu}y to Y for every Y. 3) If jECOO(X), UE.@'(X) and (3.1.3)'
supp u n supp j
~X
then the definitions at the end of Section 2.2 mean that E C't'(X) ,
which is true since 8k(fc{»=(8kf)C{>+f8kc{>. Example 3.1.2. The function H(x)=1 for x>O, H(x)=O for
x~O,
on 1R
is called the Heaviside function. The derivative is by definition 00
H'(c{» = -H(c{>') = -
Sc{>'(x)dx=c{>(O).
o
One defines the Dirac measure ba at aE1R" by
that is, as the unit mass at a. With this notation H' = boo The derivatives of the Dirac measure are (8 a b.)(c{» =( _1)la l 8ac{>(a) ,
c{>E COO,
so Theorem 2.3.4 means that linear combinations of ba and its derivatives are the only distributions with support at a. Theorem 3.1.3. If u is a function in the open set Xc 1R which is in C1(X'- {x o}) for some XoEX, and if the function v which is equal to u' for X=FXo is integrable in a neighborhood of x o, then the limits
u(xo±O)= lim u(x) exist and
x-xo±o
u' = v +(u(xo +O)-u(xo -0)) bxo . Proof If xo(x,y)/oz(z-,)-ldxdy
Y>~/2
+ (2ni)-1 Se/>(x, '1/2) (x -
e - i'1/2)-1 f(x + i'1/2)dx.
The hypothesis implies a uniform bound for f(. 0 -1. If we integrate by parts k times
70
III. Differentiation and Multiplication by Functions
and use Taylor's formula to express ¢Ul(e) in terms of derivatives of ¢ at 0, we obtain an identity of the form 00
(3.2.6)
J xa+k¢(kl(x)/«a + 1) ... (a+ k»dx
Ha .• (¢) =( _1)k k-1
+ L: o
o
A j ¢U)(0)ea+1+j +o(1),
e-+O.
There can be no other decomposition of the form
Ha,.(¢)=Bo+ L: Bj e- Aj +o(1), where the sum is finite and
ReAj~O,
e -+ 0,
A/tO. In fact, we have
Lemma 3.2.1. If Co, ... , Ck and A1' ... , Ak are different complex numbers with Re Aj~O and Aj=l=O, then k
Co + L: Cje- Aj -+ 0,
e -+ 0,
1
implies that C o = ... = Ck=O. Proof. Assume first that all Aj are purely imaginary. Replace e by erl and let e -+ through a sequence such that e- Aj has a limit Yj for every j. Then IYjl = 1 and
°
C o + L: CjYje-Ajl=O
for all real t, hence for all complex t. When t -+ 00 on the imaginary axis one term dominates so this is not possible unless all Cj = 0. If max Re Aj = (J > in the general case, we have
°
L:
Cje(q-Aj) -+ 0,
e -+ 0,
ReAj=q
so all the coefficients here must vanish which completes the proof.
°
By Lemma 3.2.1 the terms in the expansion (3.2.6) with Re a + 1 are therefore uniquely determined, so it is legitimate to discard the singular terms and define the finite part of the integral
+j ~
00
Jxa¢(x)dx to be
o
(_1)k
JxQ+k¢(k)(x)/«a+1) ... (a+k»dx.
00
o
But this agrees with our previous definition of O,
x~
=Ixl a
if x -1. This is the reflection of x~ with respect to the origin, = '
cfJ(x)dx/x.
The last integral, where a symmetric neighborhood of the singularity tending to 0 has been removed, is called a principal value. Thus (3.2.14)
J
=lim
•~o Ixl>'
cfJ(x)dx/x=PVJcfJ(x)dx/x,
cfJEq .
The problems we have encountered in the discussion of x':. when a is a negative integer were caused by the factor a in (3.2.2). By a change of normalizations they can be made to disappear. First note that (3.2.2)' assumes a particularly simple form if cfJ' = - cfJ, that is, cfJ(x) =e- x • This is not a function of compact support but it decreases so fast at + 00 that the proof of (3.2.2)' is valid for it. Set 00
(3.2.15)
r(a)=
Jxa-1e-xdx,
o
Rea>O,
which in our old notation is Ia_ 1 (e-"). Then (3.2.2)' means that (3.2.16)
r(a+l)=ar(a)
if Rea>O.
Using (3.2.16) we can extend r(a) analytically to a meromorphic function in -1
is analytic when Re a > -1. Since (3.2.2)' gives, when combined with (3.2.16), (3.2.2)'"
x':. (cfJ') = -X':.-I(cfJ)
74
III. Differentiation and Multiplication by Functions
it is now clear that X':. can be continued analytically to all aE(rw)drdw.
0
This suggests that for arbitrary ¢ E Cg"(lR") we should introduce (3.2.21)
76
III. Differentiation and Multiplication by Functions
It follows from (3.2.7) that RA> is a homogeneous function of degree -n-a. By Theorem 2.1.3 Ra is a continuous map from C~(K) to coo(IRn" 0) for every compact set K c]Rn. Choose a fixed function IjJEC~(IRn"o) such that 00
JljJ(tx) dt/t = 1,
(3.2.22)
x =1=0.
o
It suffices to take ljJ(x) as a function of one x. Then IjJRacf>EC~(JR.n"o) and
Ra(IjJRacf>)(x) =
Ixl so that (3.2.22) is valid for
J ta+n-lljJ(tx)(R.cf»(tx) dt
00
o
00
J
=(R.cf>)(x) ljJ(tx) dt/t = (Racf»(x). o
Hence it follows from (3.2.20) that u(IjJRacf» is always independent of the choice of IjJ and that u(IjJR a4»=u(4» if cf>EC~(JR.n"o). Thus (3.2.23)
t)(x) = (trx» =t"-I+k+ 1 «r:;:k-l,4>(rX»-IOgt :~ cf>(rX)/k!lr=o) = rk+n(R_n_kcf>(X) -log tu in ~'(Rn) as suppcfJ-> {O}.
Proof To clarify the computations we note that u(l/I) = u * t/f(O) if I/IEC,(;(R n) and t/f(x) = 1/1 ( -x). This gives u",(I/1) = u'" * t/f(O) = u* cfJ * t/f(O) = u(
O
is a convex function of x and an increasing function of e when e > 0 since d dy (v(x+ y)+v(x- y»=v'(x+ y)-v'(x- y)~O if y~O. b) In general we choose ¢ as in Theorem 4.1.4 and form the regularizations u'" = u * ¢, vtI> = V*¢. Assume u' ~ 0, v" ~ O. Then u¢ = u' * ¢ ~ 0, v~ = v" * ¢ ~ 0 so uti> * l/I: (x) is an increasing function of x and a decreasing function of 6 while vtl>*l/I=(x) is a convex function of x which increases with 6. Letting supp -+ {O} we conclude that u* l/I: and v*l/I~ have the same properties, so when 610 we obtain u*l/I:ruo,
where
Uo
is increasing,
v*l/I~lvo
Vo
satisfies (4.1.6), and
. Thus u4,~O and v~~O which gives u'~O and v"~O when supp -+ {O}. Theorem 4.1.7. If VE.@'(X) where X is open in 1R.n and
(4.1.7)
L LYjYkal\v~O
if
for all YE1R. n,
then v is defined by a continuous function satisfying (4.1.6) on every line segment in X and conversely. One calls v a convex function.
92
IV. Convolution
Proof We may assume that X is convex. If VEC OO then (4.1.7) means
precisely that d2 dt 2 v(x+ty)~O
when X+tYEX,
so the statement follows from Theorem 4.1.6. If O~t/!EC'f/ is an even function with Jt/! dx = 1, then
J
v*t/!.(x) = v(x-ey)t/!(y)dy
is also a convex function and it increases with e. If v is just known to be in .@'(X) we can now argue exactly as in the proof of Theorem 4.1.6. v satisfies (4.1.7) so v*t/!. is a convex function which increases with e, hence v*t/!. is convex and increases with e. The decreasing limit Vo as e!O defines v and satisfies (4.1.6) so Vo is finite everywhere and upper semicontinuous. This implies continuity since for sufficiently small Iyl v(x+hy)-v(x)~h(v(x)-v(x- y))~
- Ch,
OO,
100
IV. Convolution
where I" is a homogeneous polynomial of degree 0 which identically O. Put F.(z) = f(rz)/rk.
IS
not
Then F. converges locally uniformly to I" as r --+ 0, so log IF.I --+ log II"I in fi)' in view of Theorem 4.1.9. Hence the same is true for the Laplaceans and we conclude as in part c) of the proof of Theorem 4.1.13 that when r --+ 0 e(log IF.I, 1,0) --+ e(log Ifkl, 1,0) = k where the last equality is a consequence of Proposition 4.1.14. Now e(log IF.I, 1,0) = e(log If I, r, 0) which follows immediately if log If I is replaced by a smooth approximation. This proves (4.1.15). If f is a polynomial we can let r --+ 00 instead and obtain as limit of F. the homogeneous part of highest degree, which proves the last statement.
4.2. Convolution of Distributions To define the convolution of two distributions we shall use the properties of the convolution CO'(IRn)3c/>
--+
u*cPEcoo(IR n)
defined in Section 4.1 when UEfi)'(IR n). It is obvious in view of (4.1.2) that u* c/>j --+ 0 in coo(IRn) if cPj --+ 0 in CO'(IR n). (See Theorems 2.3.1 and 2.1.4 for the definition of convergence.) If hEIRn we define the translation operator 'h by ('hcP)(x)=cP(x-h) (which is convolution by (\) and obtain U*('hc/» = 'h(U* cPl· Thus u* commutes with translations. Conversely, we have Theorem 4.2.1. If U is a linear map from CO'(IRn) to C"(IRn) which is continuous in the sense that U cPj --+ 0 in C(IRn) when cPj --+ 0 in CO'(IR n), and if U commutes with all translations, then there exists a unique distribution U such that U cP=u*c/>, c/>ECO'(IR n).
Proof If such a distribution exists we must have u(;P) = U cP(O). (We recall the notation ;P(x) = c/>( -x).) Now the linear form C0'3cP
--+
(U ;P)(O)
4.2. Convolution of Distributions
101
is by hypothesis a distribution u. From the fact that (U 1/»(O)=(u*I/>)(O) we obtain by replacing I/> by 'hI/> and using the commutativity with 'h that (U 1/>)( - h) =('h U 1/»(0) =(U 'hl/>)(O) = (U*'hl/>)(O) = (u* 1/>)( - h)
which proves that UI/>=u*l/>,
I/>EC~(lRn).
The proof is complete.
If uE~"(lRn) it follows from (4.1.1) that I/> --+ u*1/> is a continuous map from C~(lRn) to C~(lRn), that is, sequences converging to 0 are mapped to other such sequences. The convolution u * I/> is also defined for arbitrary I/>Ec oo (lR n) then and gives a continuous map from c oo (lRn) to Coo (lRn). There is a unique way to define the convolution of two distributions U 1 and u 2 , one of which has compact support, so that the associativity (u 1 *U 2 )*I/>=U 1 *(u z *1/»
remains valid for
I/>EC~(lRn).
In fact, the mapping
C~(lRn)31/> --+
U1
*(uz*l/»
is linear, translation invariant and continuous because it is the composition of two such mappings. Hence there is a unique uE~'(lRn) such that (4.2.1) Definition 4.2.2. The convolution u 1 *u 2 of two distributions
U 1 and u 2 one of which has compact support is defined to be the unique distribution u such that (4.2.1) is valid.
By Theorem 4.1.2 the definition is consistent with our original one when u 2 E C~, and a simple modification of Theorem 4.1.2 shows that it is also consistent with our earlier definition when u 1 EC'(lRn) and u 2 Ec oo (lRn). Somewhat more generally we have U t E~'k(lRn), u 2 ECt(lRn) (or U t EC'k(lRn), u 2 EC k (lRn» then u t *u 2 is the continuous function x --+ u t (u 2 (x-
Theorem 4.2.3. If
.».
Proof If this function is denoted by u then the proof of Theorem 4.1.2 shows without change that when t/JEC~, U t E~'k(lRn), u 2 ECt(lRn) then
u*t/J=u t *(uz*t/J)·
This proves the first part of the statement and the other follows in the same way.
102
IV. Convolution
By its definition the convolution is associative, that is, u 1*(U 2*U 3)=(U 1*U 2)*U 3
if all the distributions uj except at most one have compact support.
Theorem 4.2.4. The convolution is commutative, that is,
if one of the distributions u 1 , u 2 (4.2.2)
has compact support. We have
sUPP(U 1 *U 2 )c:suPpu 1 +suppu 2 .
Proof To prove that two distributions V 1 and V 2 are equal it suffices to show that v 1*(¢*I/I)=v 2 *(qJ*I/I) when ¢, I/IEC~.
For then we have (Vl *¢)*I/I=(v 2 *¢)*1/1 by Theorem 4.1.2, hence V 1*¢ =v 2 *¢ and so V 1=v 2 • Now we have (u 1*u 2 )*(¢*I{!)=u 1*(u 2 *(¢*I{!))=u 1*((u 2 *¢)*I{!) =U 1*(I{!*(u 2 *¢))=(u 1*I/I)*(u z*¢)
where in addition to Theorem 4.1.2 we have used the commutativity of convolution of functions. In the same way we obtain (u 2 *u 1)*(¢* 1/1) = (u 2 *u1)*(I{!* ¢) =(u 2 * ¢)*(u 1* I{!) =(u 1*u 2 )*(¢*I{!)
which proves that U1*u 2 =U Z *U 1. To prove the last statement we choose ¢ as in Theorem 4.1.4 and note that since (u 1*u z )*¢ =U 1 *(u z *¢) we have supp ((u 1 *u 2 )*¢)c:supp U 1 +supp U z +supp ¢ by (4.1.1). When supp ¢ theorem is proved.
-+
{OJ it follows that (4.2.2) is valid. The
If U z has compact support it follows from (4.2.2) that U 1 * u2 is determined in a neighborhood of x if U 1 is known only in a neighborhood of {x} -supp u z . Hence the convolution U1*u z is defined in
{x; x - YEX for all yESUpp u z }
if u 1 E.@'(X).
Theorem 4.2.5. If
U1
compact support, then
(4.2.3)
and
Uz
are distributions in IR", one of which has
sing sUPP (u 1*uz)c:sing supp u 1 + sing supp u2.
4.2. Convolution of Distributions
103
Proof Assume u 2 Etf', choose t/lEC,(? equal to 1 near singsuppu 2, and U2 =V 2 +W2 where v 2 =t/lu 2 and w2 =(1-t/l)u 2EC,(? Then u 1*W 2EC oo and U1*v 2 is a Coo function in
set
{x; {x} -supp v 2 C Csing supp ud· This means that sing supp u 1*u 2 = sing supp u 1* v2 c sing supp u 1+ supp V 2 and since supp v2 c supp t/I can be taken as close to sing supp u 2 as we wish, we obtain (4.2.3). Differentiation can be interpreted as a convolution, for we have (4.2.4)
aau=(aabo)*u,
uE.@'(IRn).
In fact, using (4.1.2) we obtain for ljJEC,(?(IR n) (oau) * ljJ = u*(aaljJ) = u*(b o * (aaljJ)) = u*(aabo)*ljJ
which proves (4.2.4). Note in particular that convolution with 15 0 is the identity operator. If u1 and u2 are two distributions, one of which has compact support, it follows that (4.2.5) In fact, if the differentiations are rewritten as convolutions with aab o this follows from the associativity and commutativity of the convolution. More generally, if P=~ L... aa aa , where aaE
Ixi ~ C and
Iyl ~ C.
Then the restriction of (cPl Ul)*(cP2U2) to X is independent of the choice of cPjEC;;'(lR") provided that cPj= 1 in a neighborhood of {x; Ixi ~ q. In fact, suppose that we change cPl by adding a function l/J 1 vanishing near this ball. By (4.2.2) supp (l/J 1 U1 )*(cP2 u 2) csupp (l/J I Ul ) +supp (cP2 u 2) which contains no point in X by (4.2.7), so ((cPl +l/JI)U l )*(cP2 U2)=(cPl Ul)*(cP2 U2)
in X.
We take this as definition of U1 *U 2 in X and note that Theorem 2.2.4 shows that these local definitions together define u l *u 2 in !0'(lR"). More generally, u 1 *u z is defined in X if X is the largest open set such that (4.2.6) is proper if lR" is replaced by X; and (4.2.2) remains valid. As a~ example we observe that if uj E!0'(lR), j= 1, 2, and suppujclR+, then Ul *U 2 is defined and suPP(U l *U 2 )clR+. More generally, let rclRn be a closed convex cone which is proper in the sense that it does not contain any straight line. Then {(x, Y)Er x r, Ix + yl ~ q
is bounded, for if (x j , Y)Er x rand Ix j + Yjl ~ c, IXjl-+ 00, we can pass to a subsequence such that x)lxjl-+ XEr, hence Y)lxjl-+ -XEr. Then
4.3. The Theorem of Supports
105
the straight line 1Rx lies in r which is a contradiction. For every such cone the convolution of distributions in {uE.@'(1R"), supp ucr}
is therefore always defined and makes this set an algebra.
4.3. The Theorem of Supports In this section we shall prove an inclusion opposite to (4.2.2) for the support of a convolution. This is not possible without restrictions, for if we take u 1 = 1 and u 2 with Ju 2 dx = 0 then u 1 *u 2 = O. We shall therefore have to assume that both U 1 and U 2 have compact supports. Even so, if U 1 is the characteristic function of a bounded open set X, and supp u 2 cB for some ball B, then supp u 1 *u 2 c(oX)+B, which does not contain X + supp u2 if B is small enough. This forces us to take convex hulls of the supports, and we digress to discuss this concept before returning to the main topic. Definition 4.3.1. If E is a compact set in 1R" then chE is the closed convex hull of E, that is, the intersection of all closed convex sets containing E.
Equivalently, chE is the closure of the convex set of centers of gravity By the Hahn-Banach theorem, if y¢:chE then one can separate y from chE by a hyperplane a. If we balance the terms by choosing R so that Rnlp= Ilullp/liulloo then W la ' = Ilull:/a' Ilull~pla' and (4.5.8) follows. Next we need a fundamental covering lemma of Calderon and Zygmund: Lemma 4.5.5. Let uELl(]R.n) and let s be a number >0. Then we can write
(4.5.9) where all terms are in Ll, 00
(4.5.10)
Ilvlll +I Ilwklll ~31Iulll' 1
(4.5.11)
almost everywhere,
Iv(x)I~2ns
and for certain disjoint cubes I k
(4.5.12) 00
(4.5.13)
sI
1
m(lk)~
Ilu11 1 •
If u has compact support, the supports of v and all a fixed compact set.
Wk
are contained in
Proof Divide the whole space ]R.n into a mesh of cubes of volume > s- 1 Slui dx. The mean value of lui over every cube is thus < s. Divide each cube into 2n equal cubes, and let 111' 1 12 , 113 , ... be those (open) cubes so obtained over which the mean value of lui is ~s. We have (4.5.14)
sm(llk)~
S luldxa, and when s = lu(x)1 this is proportional to lu(x)l(q-a)p/q = lu(x)la p/p'. Altogether we have therefore Ilka*ull:~ C 3
(J lu(xW+P/P' dx)a= C (J lu(x)IPdx)a= C 3
3,
which completes the proof of (4.5.5)'. As an application we shall now prove the Sobolev embedding theorems, for which Theorem 4.5.3 was in fact originally intended. First we give a local form. Theorem 4.5.8. Let UE.@'(X) where X is an open set in IRn, and assume that 0jUEn;oc(X), j=l, ... ,n, where l{X} xIRN ->IRN is .a linear isomorphism. In the case of T(X) we have defined isomorphisms n- 1 XIC->XIC xIRn->x lC xIRn with these properties. One calls T(X) the tangent bundle of X. Let V be any vector bundle over X and choose an open covering {XJiEI of X such that for each i there is a Coo map I/Ii of n- 1 (X i ) onto Xi x IR N with the properties listed above. Then gij = I/Ii I/Ij 1 0
can be regarded as a Coo map from X;rlXj to the group GL(N,IR) of invertible N x N matrices with real entries, and we have (6.4.1)
gijgji=identity in XinX j; gijgjkgki=identity in XinXjnX k.
A system of such N x N matrices gij with Coo coefficients is called a system of transition matrices. One can recover the bundle V from them by forming the set V' of all (i, x, t)EI x X x IR N such that XEX i, and defining (i, x, t) to be equivalent to (i', x', t') if X=X' and t' = gi,;t. It follows from (6.4.1) that this is an equivalence relation, and it is easily proved that the quotient of V' by this equivalence relation is a vector bundle. It is isomorphic to V if gij were obtained from local trivializations of the vector bundle V as explained above. We shall sometimes find it convenient to look at a vector bundle in this way which is directly suitable for calculations. A vector bundle is thus a family of vector spaces Vx , XEX, varying smoothly with x. If Y c X then a section u of V over Y is a map Y3Y-> U(Y)E v" that is, U is a map from Y to V with nou=identity. Since V is a Coo manifold the set Ck(y, V) of C k sections is well defined for k=O,l, ... or k=oo. If we have a covering X=UX i as above with local trivializations I/Ii of l'Ix, then Ui=l/IiouECk(YnXi,IRN) and (6.4.2)
ui = gijuj
in Y n Xi n Xj'
Conversely, any system UiE Ck(y n Xi' IRN) satisfying (6.4.2) defines a section of the vector bundle. We can therefore also define say the space of distribution sections !i&'(Y, V) (if Y is open) as the space of all
148
VI. Composition with Smooth Maps
systems UiE.@'(Y n Xi' JR N) satisfying (6.4.2). We could also allow the distributions u i to be complex valued which strictly speaking means that we complexify the bundle V. (The definition of complex vector bundles is obtained by substituting u(x)dx.
In Section 7.1 we extend the definition to all uE[I", the space of temperate distributions, which is the smallest subspace of !!}' containing L1 which is invariant under differentiation and multiplication by polynomials. That this is possible is not surprising since the Fourier transformation exchanges differentiation and multiplication by coordinates. (See also the introduction.) It is technically preferable though to define [1" as the dual of the space [I' of rapidly decreasing test functions. After proving the Fourier inversion formula and basic rules of computation, we study in Section 7.1 the Fourier transforms of L2 functions, distributions of compact support, homogeneous distributions and densities on submanifolds. As an application fundamental solutions of elliptic equations are discussed. Section 7.2 is devoted to Poisson's summation formula and Fourier series expansions. We return to the Fourier-Laplace transform of distributions with compact support in Section 7.3. After proving the Paley-Wiener-Schwartz theorem we give applications such as the existence of fundamental solutions for arbitrary differential operators with constant coefficients, Asgeirsson's mean value theorem and Kirchoffs formulas for solutions of the wave equation. The Fourier-Laplace transform of distributions which do not necessarily have compact support is studied in Section 7.4. In particular we compute the Fourier-Laplace transform of the advanced fundamental solution of the wave equation. The Fourier transformation gives a convenient method for approximating COO functions by analytic functions. This is used in Section 7.5 to prove the Malgrange preparation theorem after we have recalled the classical analytical counterpart of Weierstrass. Section 7.6 is devoted to the Fourier transform of Gaussian functions and the convolution operators which they define. This prepares
7.1. The Fourier Transformation in .'l' and in .'l"
159
for a rather detailed discussion in Section 7.7 of the method of stationary phase, which is a fundamental tool in the study of pseudodifferential and Fourier integral operators in Chapters XVIII and XXV. The Malgrange preparation theorem plays an essential role in many of the proofs. As an application of the simplest form of the method of stationary phase we introduce in Section 7.8 the notion of oscillatory integral. This gives a precise meaning to equations such as b(~)=(21t)-n
f ei(x.odx
and will simplify notation later on. In Section 7.9 finally we continue the proof of IJ' estimates for convolution operators started in Section 4.5. Applications are given concerning the regularity of solutions of elliptic differential equations with constant coefficients. Although the results are very important in the study of non-linear elliptic differential equations they will not be essential in this book so the reader can skip Section 7.9 without any loss of continuity.
7.1. The Fourier Transformation in [/ and in [/' The purpose of Fourier analysis in lR" is to decompose arbitrary functions into usually continuous sums of characters. By a character one means an eigenfunction for the translations, that is, a function f such that for every YElRn f(x + y)= f(x)c(y),
x ElRn ,
for some c(y). If f(O)=O we conclude that f vanishes identically. Excluding this uninteresting case we can normalize f so that f(O) = 1. Then x=O gives f(y)=c(y), hence (7.1.1)
f(x+y)=f(x)f(y),
f(O) = 1.
Assuming that f is continuous we obtain if gEC~ and ffgdy=l f(x) = f f(x+ y)g(y)dYECoo
(Theorem 1.3.1). Differentiation of (7.1.1) with respect to y gives when y=O
oJ=aJ,
and since f(0) = 1 it follows that (7.1.2)
f(x)=exp (x)t/J(~)e-idxd~. To prove (7.1.8) we set x=(2n)-ntP and obtain using the Fourier inversion formula
xm =(2n)-" JtP(x)ei( -x)(2n)"t/J( -x) in view of (7.1.9) and the Fourier inversion formula. The proof is complete. Definition 7.1.7. A continuous linear form u on [/' is called a temperate distribution. The set of all temperate distributions is denoted by [/".
The restriction of a temperate distribution to C~(lR") is obviously a distribution in ~'(IR"). We can in fact identify [/" with a subspace of ~'(lRn) since the following lemma shows that a distribution UE[/" which vanishes on C~(lR") must also vanish on !I'. Lemma 7.1.8.
C~
is dense in !I'.
Proof Let 4>E[/' and take t/JEC~ such that t/J(x)=l when 4>.(x) = 4>(x) t/J(ex). Then it is clear that 4>.E C~, and since 4>.(x)-4>(x)=4>(x)(t/J(ex)-l)=O
Ixl~1.
Put
if Ixl.-+4> in [/' when e-+O. It is obvious that Iff' c [/". Other examples of elements in [/" are measures dll such that for some m
W+ Ixl)-mldll(X)1 < 00.
164
VII. The Fourier Transformation
In particular, this implies that lJ'(1R n) c [/' for every p. It is also clear that [/' is closed under differentiation and under multiplication by polynomials or functions in Yo Definition 7.1.9. If UE[/', the Fourier transform U is defined by (7.1.11) It follows from Lemma 7.1.3 that UE[/', and since the proof of (7.1.7) is valid for all cP, t/JELl the preceding definition agrees with (7.1.3) if fELl.
Fourier's inversion formula as proved in Theorem 7.1.5 states that ¢=(2ntcp
if cPE[/,
cp(x) = cP( -x).
If U is in [/' we obtain
Here it is of course the composition of u and x ~ - x. Thus. we have Theorem 7.1.10. The Fourier transformation is an isomorphism of [/' (with the weak topology), and Fourier's inversion formula li=(2ntu is valid for every UE[/'.
In particular, fELl and JEV then the inversion formula (7.1.4) is valid for almost every x. Theorem 7.1.11. If uEL2(1Rn) then the Fourier transform U is also in L2(1Rn) and Parsevars formula (7.1.8) is valid for all cP, t/JEL2.
Proof Choose a sequence UjEC~ such that Uj~U in L2 norm. Then
Iluj - uk lli2 = (2n)n IIUj -Uk IIi2 ~ 0 by (7.1.8) which is already proved in Yo In view of the Riesz-Fischer theorem it follows that there is a function UEL2 with Uj~ U in L2, and U =u by the continuity of the Fourier transformation in [/'. Now both sides of (7.1.8) are continuous functions of cP and t/J in the L2 norms, so (7.1.8) is valid for arbitrary L2 functions. If UElJ' and 1 ;£p;£2, we can write U as the sum of a function in L2 and one in Ll, so the Fourier transform is in L7oc' A better result follows from the Riesz-Thorin convexity theorem:
Theorem 7.1.12. If T is a linear map from lJ" n lJ'2 to H' n H2 such that
7.1. The Fourier Transformation in ;? and in ;?'
165
(7.1.12) and
if
l/P=t/Pi +(1-t)/P2' l/q=t/qi +(1-t)/q2' for some tE(O, 1), then
(7.1.12)'
IITfllq~MtiMi-t
Ilfll p,
fEIfI(1If2.
Proof We may assume P < 00 for otherwise Pi = P2 = 00 and (7.1.12)' follows then from Holder's inequality. The method of proof is similar to that of Theorem 4.5.1. First we write (7.1.12) in the form I 2. Already Lemma 7.1.3 indicates that the Fourier transformation exchanges local smoothness properties and growth properties at 00. Another case of this is the following Theorem 7.1.14. The Fourier transform of a distribution UEtS"(1R.n) is the function
(7.1.14) The right-hand side is also defined for every complex vector ~Eccn and is an entire analytic function of ~, called the Fourier-Laplace transform ofu.
166
VII. The Fourier Transformation
Proof. If (t.» )/2, 4>E C~(IRn).
The formula extends by continuity to all 4>EY'. To compute (11,4» =(u, ¢) we first calculate (Ck-l, ¢(tx» in terms of 4> when x = (1, 0, ... , 0). Then ¢(t, 0, ... , 0) is the Fourier transform of S4>(~l,~')d~' where ~'=(~2""'~n)' The Fourier transform of r k- 1 IS 2ni- 1 - k(Jk where (7.1.23)
(Jir)=2-1(sgn7:)7:k/k!, (Jk(7:)=O(-k-l),
k=0,1, ... ;
k=-1,-2, ...
by example 7.1.17. Hence (rk-l, ¢(tx» = 2n i- 1 =2ni- 1 -
J
«(Jk(7:), 4>(7:, ~'» d~' k«(Jk(x, 0), 4>(~».
k
Note that (J k is homogeneous of degree k so the last expression is homogeneous in x of degree k. The orthogonal invariance shows that it is equal to (l-k-l, ¢(tx» for every x=l=O, so we obtain for all 4>EY (7.1.24)
or formally (7.1.24)'
In Sections 8.2 and 12.6 we shall give a precise meaning to such formulas. Note that when k(~/R)d~/Rk =(2n)"(u* EC o(IRn+l), Then
(7.1.31)
(2n)-n
IS
1J.-;(~W4>(~/R, t/R)d~ dt/R
t~O
---+
SS Iv(xW4>(tF'(x), t)dtdx. t~O
Proof With s= t/R as new variable the left-hand side can be written
If we replace t by R, F by sF and (~) by 4>(~, s) in Theorem 7.1.29 it follows that the inner integral converges boundedly to (2n)" SIv(xW4>(sF'(x), s) dx.
This implies (7.1.31) since the integration with respect to s is taken over a finite interval.
7.2. Poisson's Summation Formula and Periodic Distributions
177
7.2. Poisson's Summation Formula and Periodic Distributions In Section 7.1 we determined the Fourier transform of the Lebesgue measure on any linear subspace of ]R.n. Our first purpose here is to determine the Fourier transform of the sum of the Dirac measures at the points in a discrete subgroup of ]R.n.
Theorem 7.2.1. If U a is the sum of Dirac measures ua=
I
bag,
OoFaE]R.,
gEZ»
Proof. Since bag * U a= U a if g E ?In, we have (ei(ag")-l)ua=O,
gE?l".
All the factors do not vanish except at points in (2 n/a)?l". At the origin for example a must just be a multiple of the Dirac measure, for (sinaxp)ua=O,j=l, ... ,n, so this follows from Theorem 3.1.16 if we take sinax/2 as new variables. Now a is invariant under translations in (2n/a)?l" because e21ti(.,g)/aua=ua' This implies that ua is a measure with the same mass at every point in (2 n/a) ?In, thus
u
u
Explicitly this means that (7.2.1) or if we replace 1> by a translation of 1>
I
¢(ag) ei(ag,x) = CaI 1>(2ng/a + x),
1> E.9'.
Now integrate both sides for O (2n g/a),
4> E /7,
and is called Poisson's summation formula. Note that as a ..... O or 00 we obtain as special cases the Fourier transform of 1 and c5 o. Let us now consider a distribution u which is periodic with period 1 in each variable, that is, u(x-g)=u(x),
gE71n •
Such a distribution is automatically temperate. For let 4> be a function in CO' (1R.n) with (7.2.2)
L4>(x-g)=l
(see Theorem 1.4.6). If 1/1 E CO' then (x) L .(i(x + g) = (2nt L I/I(2n g) e- h i (X,g) 4>(x). Hence where
cg=e- hi (.,g».
Note that if u is a continuous function then
Ju 4> e- 21ti (.,g) dx = Jue- 21ti(.,g) dx I
where l={x; O~xjl.
7.3. The Fourier-Laplace Transformation in 8'
189
If k?;Rm the exponential can be estimated by (1 +l(l)k in suppdJl.
Hence the integral (7.3.14) converges and
Iu(4)) I ~ C R
L
sup IDa¢l,
4>ECO'({X; Ixl ~R}).
lal~N+k
Thus (7.3.14) defines a distribution which is of order ~N +Rm+ 1 when Ixl ECO'( UEY" and has Fourier transform U(. + i'1) for every 1]EI: Proof. Only the last statement remains to be proved. Let u~ be the inverse Fourier transform of U(. + i1]). Then ou~/oYJv is the inverse Fourier transform of o U(~ + i1])/o1]v = io U(~ + i1])/o~V'
so oU~/01]v=xvu~ and e-u~=u is independent of 1]. The proof is complete.
7.4. More General Fourier-Laplace Transforms
193
Now assume that u has support in a convex closed set K which is no longer assumed to be compact. We can still define the supporting function by HK(~)=SUp '7 + 8EI;.
if H K(8) < 00,
and {8;HK(8)k
IIDaull p +
H(Hk)
is equivalent to
L (HiDa u(x)-Da u(y)llx-yl-n- 2s dxdy)t.
lal=k
Having described the H(s) spaces with s~O we observe that !/ is dense in L;, hence in H(s)' for every s. If UE!/' it follows that
is finite if and only if uEH(s) and then it is equal to lIull(s)' Thus H(s) is the dual of H(_s) which gives a description also when s n(l/q -1/2).
if
and only
if
I? of H(S) to IJ'spaces.
q=2 and s~O or 1~q 2. When 1 ~ q < 2 we obtain by Holder's inequality
SIvlqd~ = S(lvI2)q/2 d~ ~(S Iv1 2 (1 + 1~12)Sd~)q/2(J (1 + 1~12)-qs/(2-q)d~)1-q/2 ~
C(SlvI2(1 + 1~12)Sd~~/2
if 2qs/(2-q»n, that is, s>n(1/q-1/2). If s=n(1/q-1/2) we can take v=(1+1~12)-n/2q(log(2+1~I))-a and obtain VEH if and only if qa>l, and vEL~ if and only if 2a > 1. When we take aE(1/2, l/q) we find that L~ is not contained in H. Theorem 7.9.3. The Fourier transform of H(s) is contained in H if 1~qn(1/q-l/2). The Fourier transform of IJ' is contained in H(_S) if 2n(1/2-1/p). Proof The first statement follows immediately from Lemma 7.9.2. To prove the second one let UEIJ' and note that when 4>Eff'
I(u, 4»1 =I(u, 4»1 ~ Ilulbll4>ll v ' ~ C114>II(s) by the first part of the proof, so uEH(_s)' If we combine the second part of Theorem 7.9.3 with (7.9.7) we conclude that for p>2 the Fourier transform of 11' is in ~/j if j> n(1/2 -l/p). (Recall that by Theorem 7.6.6 this would be false if j < n(1/2-1/p).) We have actually proved a great deal more for there is a considerable margin in the inclusion (7.9.7).
7.9. H(,), IJ and HOlder Spaces
243
A particularly important special case of Theorem 7.9.3 is the following Bernstein theorem:
Corollary 7.9.4. The Fourier transform of H(S) is contained in L1 if s>nI2, and H(S) is then contained in the space of continuous functions on 1R." tending to 0 at 00. Corollary 7.9.4 is of course a slightly stronger version of Lemma 7.6.3. In estimates such as (7.6.10) we could therefore have used H(S) norms for any s > nl2 instead of the smallest integer s > nl2 as we actually did. Our discussion so far shows that one cannot express the IJ norm of u very well in terms of the Fourier transform u. To prove continuity of maps in IJ spaces one can therefore seldom use Fourier transforms except in L2. However, we shall now prove some rather precise estimates supplementing those in Section 4.5 by combining the methods used there with the Fourier transformation in L2. Theorem 7.9.5. Let kE9"(1R.") and assume that kELloe' (7.9.8)
L
J
lal ~s RI2 < I~I < 2R
IRlaIDak(~Wd~/R"~CO,
where s is an integer > n12. Then it follows that for 1 < p < 00
(7.9.9) In addition
(7.9.10) Proof Choose a function ljJEC~({~; I~I ~ 1. Then we have for ~ =1= 0
(7.9.11)
1=
1~1~2})
which is equal to 1 when
L (ljJ(2-j~)-ljJ(21-j~)), 00
-00
which we shall use to decompose
k. If we set
kR(~) =(ljJ(~) -ljJ(2~)) k(R~)
it follows from (7.9.8) that (with another C) (7.9.12)
L JIDakR(~Wd~~C.
lal ~s
Hence sup IkRI ~ C' by Lemma 7.6.3, so Ik(R ~)I ~ C' when I~I = 1, which means that (7.9.13)
244
VII. The Fourier Transformation
when ~ =FO. Since we have assumed that kEL~oc it follows that kEL'XJ. Parseval's formula now gives (7.9.9) when p = 2 with C 2 = C'. Moreover, Corollary 7.9.4 shows that kR is the Fourier transform of a function kREV with IlkRIILl ~ C" and kRECoo. More precisely we have Slk R(xW(1 + IxI2)Sdx~ C3
so Cauchy-Schwarz' inequality gives (7.9.14)
S
Ixl>t
IkR(X)ldx~
C 4 t"/2-s.
Bounds of the same form are valid for ~ikR' hence for DiR' so we have (7.9.15) which implies (7.9.16) We are now ready to prove the analogue of (4.5.16), (7.9.17)
ifwECg'(I)andSwdx=O.
Slhwldx~CSlwldx P*
Here I is a cube and 1* the "doubled cube" as in Lemma 4.5.6. We may assume in the proof that the center is at 0 and that the norm in lRn is the maximum norm so that I is defined by Ixl -r/2} ~4I1hvlli2~ CIIvlli2~ C'-rllvIl LI.
If 0 =
UIt then -rm(O)~2"lIullL' by (4.5.13), and (7.9.17) gives
m{x; x¢O'L Ihwix)1 >-r/2}rr/2 ~
S Llhw) dx ~ C JLlwjl dx ~ 3 quilL"
CO
Since Ik*u(x)I~-r unless Ik*v(x)I>-r/2 or XEO or x¢O and Llhwix)1 >-r/2, we have proved the weak type estimate (7.9.10). It suffices to prove (7.9.9) when UEC,/:,. If (7.9.9) is known for some p then it follows for the conjugate exponent p', l/p + l/p' = 1. In fact Ihu*v(O)1 =lhv*u(O)1 ~ IIhvllLPliull v ' ~ CIIvllvllullv'
when u, VEC,/:" This implies k*UElf' and that (7.9.9) is valid with p replaced by p'. Thus we may assume 1 -r/2} +m{x; Ih l!..(x)I>-r/2} ~ C(-r- 2 I1u,lIi2 +c 111 U,IIL')'
by (7.9.9) with p=2 and by (7.9.10). Hence 00
IIhullf.,=p J -rP-Im{x; Ihu(x)1 >-r} d-r o ~C(
Sf
lu(xW-rP - 3 dxd-r+
lu(x)1 C.
Since VE!/' it follows that for some N, C', C" we have when ~Er, I~I> C,R~el~1
lu(~)I~C'
I
IHPI;;;N
supl'1
(0, -x)EWF(u)
if x =1=0,
(8.1.19)
~ESUppU
¢>
(0, ~)EWF(u)
if
~
~
=1=0.
°
Proof Assume first that u is homogeneous in JR.n. To prove (8.1.17) it is sufficient to show that if Xo =1= 0, ~ 0 =1= then
(8.1.17)'
u
u
for is also homogeneous and (8.1.17)' applied to gives the reversed implication since ~=(2n)nu. Choose XEC~(JR.n) equal to 1 in a neighborhood of ~o and I/tEC~(JR.n) equal to 1 in a neighborhood of Xo so small that (8.1.20)
(supp I/t x supp x)n WF(u) =0.
We have to estimate the Fourier transform of v = XU in a conic neighborhood of -xo' Let l/t(x)=l when Ix-xol2r in supp(l-I/I(y)), hence t~tly-xl/r and lyl~ly-xl+ IXol + r. Since XEY this completes the proof of (8.1.17)'. In general it follows from (7.1.19) and (7.1.18) that we can write u =w+wo+Q(D)Wl' u=w+wo+Q(~)Wl where w is homogeneous, suppwoC:{O}, w1 (x)=lxl-"/c" when x*o, Wl(~)= -logl~l, and Q is a polynomial. Since u - wand 12 - ware in Coo except at the origin, we obtain (8.1.17) in general. To prove (8.1.18) we first observe that since a=(2n)"u it follows from Lemma 8.1.7 with v=u that x¢suppu => (0, -x)¢WF(u). Assume now that (0, -xo)¢WF(u). Choose XECg' equal to 1 at so that the Fourier transform of Xu is rapidly decreasing in a conic neighborhood r of -x o' Adding to u a term with support at does not affect (8.1.18) so we may assume that u is homogeneous of degree a in JR" unless a = -n-k and (3.2.24) is valid for an integer k~O. Hence the Fourier transform of Xu at tx is
°
°
x*u(tx)=(u, X(. +tx) =t"(u, ¢t(' +x) +logt
L ca(aaX)(tx)/a! lal~k
°
where ¢t(x)=t"X(tx) and the sum should be omitted unless k= -n-a is an integer ~O. When XEr the left-hand side tends rapidly to as t -+ 00, and so does the sum. Thus (u,¢t(' +x)=u*¢t(x)-+O
in
r as t-+oo.
The convolution converges to (2n)"u in Y'(lR"). Hence U=O in r so Xo ¢supp u and (8.1.18) is proved. If u and therefore 12 is homogeneous in JR" then (8.1.19) follows if (8.1.18) is applied to U. If u is not homogeneous then u(t.)_ta u is a
260
VIII. Spectral Analysis of Singularities
distribution =1= 0 supported by 0 for some t > O. Hence (O,~) is in WF(u) for every ~ =1= 0, and ~ ESUPP a since a= U + V where U is of the form (7.1.19) with Uo and Q homogeneous, Q$O and V is a polynomial. The proof is complete. Our final example concerns the distributions defined by oscillatory integrals in Section 7.8. Theorem 8.1.9. For the distribution
A= Jei c/>(.,8)a(., e)de
defined in Theorem 7.8.2 we have (8.1.21)
W F(A) c {(x, ¢~(x, e)); (x, e)EF and ¢~(x, e) =O}.
Before the proof we observe that ¢~(x, e) = 0 implies ¢(x, e) = 0 since ¢ is homogeneous of degree 1 with respect to e. By hypothesis Im¢~O so it follows that Im¢~(x, e)=o. Thus ¢~(x, e) is real in (8.1.21). Proof Let t/lE C'~(X). Then the definition of A means that f,4.(~)=
JJei (c/>(x,8)-(X,O)t/I(x)a(x, e)dxde
as an oscillatory integral. We want to show that this is rapidly decreasing in any closed cone VelR n which does not intersect e); (x, e)EF, xESUppt/l, ¢~(x, e)=O}.
{¢~(x,
Then we have for some c > 0 e)1 + lell¢~(x, e)1 ~ c(I~1 + leI) if (x, e)EF, XESUPP t/I, ~EV.
(8.1.22)
I~ -¢~(x,
To prove (8.1.22) we first observe that ¢~(x, e) and lei ¢~(x, e) are continuous in F with the value 0 when e=o. By the homogeneity it suffices to prove (8.1.22) when I~ I+ lei = 1. By the compactness we only have to show then that the left-hand side is never 0 when (x, e)EF, xESUppt/l, ~EV. If e=o we have 1~-¢~(x,e)l=l, and when e=l=O, ¢~(x, e) = 0 we have ~ =1= ¢~(x, e) since ~ EV, which proves (8.1.22). Expressing the oscillatory integral by means of the partition of unity in e used in the proof of Theorem 7.8.2 we have
f
f,4(~)= JJei (c/>(x,8)- (x, mt/l(x)Xv(e)a(x, e)dxde. o
Each term is in fI'. With R=2 v (8.1.23)
RN
l
the terms with v=l=O can be written
JJe (Rc/>(x,8)- (x, ~»t/I(x) Xl (e)a(x, R e)dxde. i
8.2. A Review of Operations with Distributions
261
If cJ>(X, O)=(ReP(x, O)-~1 + 1cJ>~1 ~ c(R 101 + I¢I)/(R + I¢I) ~ c
in the support of l/I (x) XI (O)a(x, RO). With y=max(1-p,b)x- YERn.
Theorem 8.2.4 gives WF(K)c{(x,y, (, -(); (x-y, ()EWF(k)).
For any constant c we have k= fc* K where fc(x)=(x+c, c), thus 1.,'(x)«(, 1'/)=(. Hence Theorem 8.2.4 also gives WF(k)c {(x,
0; (x+c, c, (,
-()EWF(K))
so there is in fact equality, (8.2.15)
WF(K) = {(x, y, (, - e); (x - y, ()E WF(k)}.
Since the two frequency components vanish simultaneously it follows that convolution with k maps C~ into Coo and has a continuous extension to a map tff' -> !!i/. Furthermore, we have (8.2.16) WF(hu)c{(x+y,();(X,()EWF(k) and (y,e)EWF(u)},
uEtff'.
This improves Theorem 4.2.5 a great deal. (A direct proof of (8.2.16) is easily obtained from Theorem 4.2.5 and the obvious fact that E(hu) is contained in E(k)rlE(u) when k, uEtff'.)
8.3. The Wave Front Set of Solutions of Partial Differential Equations
271
8.3. The Wave Front Set of Solutions of Partial Differential Equations A differential operator with COO coefficients of order m in an open set X e lRn is of the form (8.3.1)
I
P=P(x, D)=
a~(x)D~.
1~I~m
The principal part (or symbol) Pm is defined by (8.3.2)
Pm(x, ~)=
I
a~(x) ~~.
1~I=m
Note that the definition differs from that in Section 6.4 by a factor i"'. Corresponding to (6.4.6)' we have (8.3.2)' If X is a Coo manifold then a differential operator of order m on X is by definition an operator which has the form (8.3.1) in local coordinate systems. From (8.3.2)' it follows that the principal symbol is invariantly defined in the cotangent bundle. We shall now prove a weak converse of (8.1.11).
Tbeorem 8.3.1. If P is a differential operator of order m with Coo coefficients on a manifold X, then (8.3.3)
WF(u) e Char Pu WF(Pu),
UE~/(X),
where the characteristic set Char P is defined by
(8.3.4)
Char P= {(x,
~)ET*(X)'- 0, Pm(x,~) =o}.
Corollary 8.3.2. If P is elliptic, that is, Hence
WF(u) = WF(Pu),
Pm(x,~)=1= 0
in T*(X) '- 0, then
UE~/(X).
singsuppu=singsuppPu,
UE~/(X).
Proof of Theorem 8.3.1. We have stated the result for a manifold but it is purely local so we may assume that X elRn in the proof. If Pm(X O ' ~ 0) =1= 0 we can choose a neighborhood U e X of Xo and an open cone V3~0 such that
(8.3.5) for some C. Later on another condition will be imposed on U and V. Choose a fixed n + 1, for if E ± has the properties stated in the theorem for ,1kP(D), then ,1kE± has these properties for P(D). (,1 is the Laplace operator.) Replacing P by - P interchanges E + and E _ so it suffices to construct E _. Guided by the second order case as indicated above we let r be the chain
(8.3.12) where C 3 and t are given by Lemma 8.3.6. Noting that by (8.3.10) ImPm~ C 1(1 +IRe (It+ 1 on r we set (8.3.13)
E_(x)=(2n)-n
Jei (x,OjP(()d(1 A ... Ad(n'
r
In terms of the parameters d(l
~ l' ... , ~ n
on
xERn.
r we have explicitly
A ... Ad(n=Jd~l A ... Ad~n'
J =D(~ 1 + itV 1(~), ... , ~n + itvn(~))fD(~ l'
... , ~n) --+
1 at
00,
8.3. The Wave Front Set of Solutions of Partial Differential Equations
277
so the integral (8.3.13) is locally absolutely and uniformly convergent. When ¢EC't' we have with l/t=P( -D)¢ (P(D)E _, ¢) = (E_,
l/t) =(2n)-n JJ l/t(x)ei(x. O /P(()dxd(l A
... A
d(n.
T
Here we integrate first with respect to x and use that t/I(-()=P(()$(-Q. Now F(() d( 1 A .•. A d(n is a closed differential form for every analytic function F, since dF is a linear combination of d( l ' ... , d(n. Thus $( - ()d( 1 A ... A d(n is a closed differential form which decreases rapidly at infinity, so Stokes' formula gives (P(D)E _, ¢) =(2n)-n
S $( -
()d(l
A ..• A
d(n
T
=(2n)-n
Here
S $( -
~)d~ -(2n)-n S $( - ()d( 1 A
.•. A
d(".
To
ro is the chain 1R.n3~~~+itvo(~),
1~I~C3'
where Vo is a smooth extension of v from I~I = C3 to I~I ~ C 3, which makes ruro homotopic to 1R.". Thus P(D)E_ =b+w_ where w_(x)= -(2n)-n
S ei(x' O d(l
A ..•
Ad(n
To
is an entire analytic function since ro is compact. Hence Theorem 8.3.1 shows that Pm@=O if (x, ~)EWF(E_) and x*O. To complete the proof we must show that (xo, ~o)¢WF(E_) if xo¢1R._P~(~o). This condition means precisely that we can find VE1R." with IVI = 1 and (8.3.14) Choosing a conic neighborhood W of ~ 0 such that for some c > 0 (P~(~), V»cl~lm-l,
we obtain from (8.3.10) when
~EW,
~EW
1m P(~ + itv(~) + is V) ~ C1 (1 +1~lt-l +cl~lm-ls_
Cz(s+ l)s(l~1 +s)m-Z
~Cl(I+I~l)m-l
if 0 < s< eI~ I and (8.3.15)
I~ I is large enough. Replacing V by e V we have 1m P(~ + itv(~) + is V)~ C1 (1 + 1~lt-l, ~EW,
O~s~I~I,
I~I ~ C~.
Choose XEC (1R." . . . . 0) homogeneous of degree 0 with support in W so that 0 ~ X ~ 1, and X= 1 in a conic neighborhood Wo of ~ o. If OO
278
VIII. Spectral Analysis of Singularities
(x, V) >0, which is true in a neighborhood of x o, we obtain using Stokes' formula (8.3.13)'
E_(x)=(2n)-n
J
ei(x,P/P«()d(lA ... Ad(n
r'urQ
where r' is the chain JRn3~
and
r~
-> ~ + itv(~) + iI~1 X(~) V,
I~I ~ C'3'
is the union of the part of r where C 3 < I~I < C'3 with the chain {(~, s); I~I
= C'3' O ~ + itv(~) + isX(~)V
with suitable orientations. The contribution to (8.3.13)' when (Er~ or Re (EWo is an analytic function of x when (x, V) >0. If M is a measurable conic set contained in a closed proper convex cone G, then the wave front set of the function
x->
J
ei(x,o/P«()d(lA ... Ad(n'
(x,V»O,
{Er', Re{EM
is contained in {(x,~); (x,V»O, ~EG}. This follows from Theorem 8.1.6. In fact, replacing x by z = x + i Y we obtain a bounded analytic function when Ixl is bounded, (x, V) > 0, and y is in the interior of the dual cone of G, for Re i(z, 0= -(x, im O-(y, ReO~ -t(x, v(~»< Clxl. We can cover CWo with a finite number of such cones G which do not contain ~o, so it follows that (xo, ~o)¢ WF(E _). The proof is complete. Repetition of the proof of Theorem 8.3.3 gives now
Theorem 8.3.3/. Let P(D) be of real principal type. If uEEd/(X), P(D)u = f and (x, ~)EWF(u),- WF(f), then Pm(~)=O and I x {O eWF(u)
if I eX is a line segment containing x with direction I x {~} does not meet WF(f).
l!:m
such that
Finally we shall give a general version of Example 8.3.4.
Theorem 8.3.8. Let P(D) be of real principal type, O=!=~EJRn and =0. Then one can find uEcm (1R.n) such that P(D)UEcoo(JRn) and (8.3.16)
WF(u)={(tP~(~),s~);
Pm(~)
tEJR, s>O}.
Proof Set L=JRP~(~) and let fF be the set of all UEcm(JRn) with pUEc oo (1R.n), uECOO(CL) and WF(u)eJRn x (1R.+ ~). The theorem states
8.3. The Wave Front Set of Solutions of Partial Differential Equations
279
that there is an element uEff which is not in Coo, for uEff implies WF(u)cIRP~
x IR+ ~
and by Theorem 8.3.3' UEC OO if the inclusion is strict. Now ff is a Frechet space with the seminorms (i) supID"ul, 1Q(I~m, K a compact subset ofIR", K
(ii) sup ID"ul, Q( arbitrary, K a compact subset of CL, K
(iii) sup ID" P(D)ul, Q( arbitrary, K a compact subset of IR", ...-....
K
(iv) supl'1I N I¢u('1)I, N=l, 2, ... , ¢EC(f(IR"). erN
Here
rN is a sequence of conic neighborhoods of ~ in IR" shrinking to
We need only use a countable number of compact sets K and functions ¢ since the semi-norms (iv) can be estimated by the corresponding ones with ¢ replaced by a function t/I which is 1 in supp ¢. (See the proof of Lemma 8.1.1.) The proof of completeness is an exercise for the reader. If ff c cm+ 1 then the closed graph theorem shows that the inclusion ff "-+ cm + 1 is continuous. Thus one can find N, ¢EC(f(IR"), Kl ~IR" and K2~C L so that
IR+~.
(8.3.17)
I
l"l=m+ 1
+
ID"u(O)1 ~ C {
I
I
1,,1 ~m
sup ID"ul + K,
I
1,,1 ~N
sup ID"ul K2
sup ID" P(D)ul + sup (1 + 1'1lt Iqm('1)I, erN
I"I~N K,
uEff.
To show that (8.3.17) is not valid we need to construct approximate solutions of the equation Pu =0 concentrated close to L, thus away from K 2 • To make the last term small the Fourier transform of u should be concentrated close to the direction ~. It is therefore natural to set for t > 0 Then
P(D)ut(x) = eit(x,~) P(D + t~)Vt(X) =('"-1 eit(x,O
(t p~j)(~)DjVt +
Pm - 1(~)Vt + ... )
where terms indicated by dots contain a negative power of t, and p;!) =8j Pm • A formal solution Vt =V O +t- 1 V1+ ... may be found by solving the first order equation (8.3.18)
"
Lvo = I P;!)(~)Djvo + Pm - 1(~)Vo =0 1
280
VIII. Spectral Analysis of Singularities
and then successively equations (8.3.19)
where Jj is determined by Vo, •.. , Vj-I. The support of Vo is a cylinder with the axis in the direction ~(e); we can choose Vo with vo(O) = 1 and support close to L by prescribing such values on a plane 1: orthogonal to p~(e). If the other functions Vj are determined by the boundary condition vj=O on 1:, it is clear that suppvjcsuPPVo for j'*'O. For v/ =
"L.
j (x(t), ~ (t» for p through (XO, ~O) remains in Ne(F). If p~(XO, eO) =1= there is a function tPECOO(X) such that for some e>O
°
tP (x (t» =0,
and tP(x) < x¢=r.
e
dtP(x(t» = (t)
if It I< e,
°when x is in a neighborhood of r = {x(t), Itl < e} in F but
Proof We may assume that X e]Rn since the result is local. Choose fECoo with f(xO)=O, df(xO)=eo, and f(x)O, the derivative at t=O of f(y(t)) must be -:i:,0, that is, (v(x), 0-:i:,0. To prove that (ii) => (i) we may assume that X =1R.n and begin by proving an elementary lemma: Lemma 8.5.12. Let F be a closed set in 1R.n and set
f(x)=min Ix-zl 2 ZEF
where
I I is the Euclidean norm. Then we have f(x+ y)= f(x)+f' (x, y)+o(iyi), f'(x, y)=min {(2y, x-z); zEF, Ix-zI 2 = f(x)}.
Proof We may assume in the proof that x = O. Set q.(y)=inf { - 2(y, z); zEF, Izl-:i:,(f(O))t + B}.
8.6. WFL for Solutions of Partial Differential Equations
q. is homogeneous of degree 1, and q./, qo as therefore uniform on the unit sphere, so
The limit is
c.--+O as 8--+0.
qo(y)~q.(y)~qo(y)-c.lyl,
Now
dO.
305
ly-zl2 =lzl2 -2(y, z) +lyl2
which gives immediately f(y)~f(O)+qo(Y) + lyl2.
On the other hand, when Iyl ~8j2 the minimum in the definition of fey) is assumed for some z with Izl~f(0)t+8j2, so f(y)~f(0)+q.(Y)+lyI2,
Iyl ~8j2.
The lemma is proved. Proof of Theorem 8.5.11. With the notation in (i) and Lemma 8.5.12 we have if t< T
lim (f(x(s» - f(x(t»)j(s - t) = f'(x(t), v(x(t»).
s-t+O
Since the result to be proved is local we may assume that for all x and y Iv(x)-v(y)1 ~ Clx- YI. When ZEF and Ix(t)-zI2 = f(x(t» we have 2(v(x(t», x(t) -z) = 2(v(z), x(t) - z) - 2( v(z) - v (x (t», x(t) - z).
The last term is ~2Cf(x(t». From the proof of Proposition 8.5.8 we recall that if f(x(t» > 0 then (z, x(t) - Z)ENe(F) for every z such that Ix(t)-zI2=f(x(t». Hence the first term on the right is ~O by condition (ii) so the right-hand derivative of f(x(t» is ~ 2 C f(x(t». Thus the right-hand derivative of f(x(t» e- 2et is ~ 0 so this is a decreasing function by a simple modification of the proof of Theorem 1.1.1. (Note that f is continuous.) If f(x(O»=O we obtain f(x(t»=O, O dx = JU(X) eN (X, ~)e-i(x,O dx + Jf(x)e-i(x,~> wN(x, WPm(x,~) dx.
Here f =P(x, D)u. To estimate the right-hand side of (8.6.7) we first prove a simple lemma. Lemma 8.6.2. There is a constant C' such that,
if j = j 1 + ... +jk and j
+!PI~2N,
(8.6.8)
!DJlRj, ... RjkX2N!~C'N+INj+IJlI!~!-j,
~EV.
Proof By the homogeneity it suffices to prove the lemma when !~! = 1. All coefficients occurring in some R j when !~! = 1, ~E V, have a fixed bound in a fixed complex neighborhood of K, so the lemma is a consequence of the following one. Lemma 8.6.3. Let K be a compact set in 1R.n and K' a neighborhood of K in (2M)P (8.6.15)
u(x)=
J
i1"+oo ei(x,sN+t(s)Oe-(sli)Pds.
i't-oo
Here we define (s/i)P so that it is real and positive when s is on the positive imaginary axis, and we choose a fixed branch of slip in the upper half plane. The integral is convergent and independent of 't, for when x is in a fixed bounded set we have in view of (8.6.14) (8.6.16)
Re(i(x, sN + t(s) 0 -(s/i)P) ~ -'t(x, N) + Clxll~llsll-I/P -lslP cos (np/2) ~ -'t(x, N) -clsl P
if 0 < c < cos (n p/2) and lsi is large. This estimate also shows that, when x is in a compact set, the integral·,(8.6.15) is uniformly convergent even after an arbitrary number of differentiations with respect to x. Hence UECoo and using (8.6.12) we conclude that P(D)u=O. From (8.6.16) we also obtain lu(x)1 ~e-t(x,N>
J e-clulP du.
00
-00
Hence it follows when 't -+ + 00 that u(x) =0 if (x, N) >0. (Compare the proof of Theorem 7.3.1.) When (x, N) < 0 we can replace the integration contour in (8.6.15) by the negatively oriented boundary of the set {s; lsi < (2M)P or Ims (ii) Assume that n is a characteristic hyperplane which does not intersect Xl' Let H be the half space bounded by n which does not intersect Xl' By Theorem 8.6.7 we can find a solution u of the equation P(D)u=O with suppu=H, so (i) shows that HnX 2 =0, hence nnX2=0.
(ii) => (i). Let Y2 be a point in X 2. Choose a point Y1EXl and denote by I the line segment between Y1 and Y2' We can find an open convex set X ~ Xl such that every characteristic plane intersecting I also meets X. In fact, if xoEI and ~oE1R.", Pm(~O) =0, I~ol = 1, we can choose an open ball 17~Xl which meets the plane =O and consequently meets every characteristic plane with normal close to ~o passing through a point near xo' By the Borel-Lebesgue lemma a set X with the required properties can therefore be constructed by taking the convex hull of a finite number of open balls 17 ~ Xl' Let 1"; be the interior of the convex hull of X and Yt = Yl + t(Y2 -Yl)' O~t~1. For small t we have Yt EX 1, hence 1";cX1 and u=O in 1";. Let T be the supremum of all tE[O, 1] such that u=O in 1";. Then u =0 in YT and h¢X1 • If n is a supporting plane of YT then n is noncharacteristic if hEn since n intersects I but not X, and if h¢n then n n ¥reX ~ Xl' Hence it follows from Proposition 8.5.8 and Theorem 8.6.5 that aYT n supp u = 0. Hence T = 1, and u = 0 in a neighborhood of the arbitrarily chosen point Y2 EX 2' This proves condition (i). Corollary 8.6.9. If the support of a solution UE.@'(1R.") of the equation P(D)u = 0 is contained in a half space with non-characteristic boundary, then u=O.
8.6. WFL for Solutions of Partial Differential Equations
313
Proof Every characteristic plane intersects the half space.
Corollary 8.6.10. Let Nl and N2 be real vectors such that
(8.6.17)
Pm(rlNl+r2N2)=l=0
when rl>O and r2~0.
Set Xa),az = {x; O. Set
Now choose a cutoff function XEC~ which is 1 near the segment [y,x] so that l_rlsuppdxcV. We can choose X so that X(x) =",«x,'1»), XEV, for some "'ECce and some '1 with C} for some C such that iP (i. = 1 in Vc "JR.n and (8.6.20) for some Nand C 1 • We shall denote by Char Jl the complement of r in JR.n ...... {OJ. Theorem 8.6.15. If JlEY"(1R.n) and uEt9"(1R.n), then
(8.6.21) Proof We shall use the interpretation of W FA in Theorem 8.4.11. With the notation in that theorem we must show that u*K(z) is analytic at xo-i¢o if ¢o¢;CharJl, l¢ol=1 and (xo,¢o)¢;WFA(f), f=Jl*u. Choose V and iP as above so that (8.6.20) is valid and iP (i. = 1 in Vc "JR.n. Let W' and W" be closed conic neighborhoods of ¢o in JR.n ...... {OJ such that W" is contained in the interior of W' and W' c V. Choose XE COO with O~X~ 1 equal to 1 in a neighborhood of W~'c and supp XC W;c so that X is homogeneous of degree 0 when I¢I > 3 C. Then the Fourier transform of u * K(. + iy), Iyl < 1, can be decomposed as follows
If we introduce the inverse Fourier transforms
K 1 (z) =(2n)-n J(1- x(¢))ei /I@d¢, K 2 (z)=(2n)-n JX(¢)iP(¢)ei 0 there is an analytic continuation of K 2 to
{z; 11m zl < I-b +b(l + IRe ZI2)t, 11m z + ~ol O}. (Cf. Lemma 8.4.12.) If we recall that (xo'~O)¢WFA(fl) it follows by the analogue of (8.2.16) for WFA that xo¢singsuPPA 11 * Kz(zo) JE/zo - x) X(x) dxl ~ Co JU/nt/2 e-3jlx-xoI2/4Ix_xol dx ~ C 1 j-t. Now
1- JEj(zo - x)x(x)dx= J(1- X(x»E/zo - x)dx
is exponentially decreasing as j-HXJ since Re is a harmonic function in JRn+ 1 and X E C~(JRn+ 1) is equal to 1 in a neighborhood of K x {OJ, then
(9.1.3)
Proof. U(X) is defined when XrtKx{O} for y-+P(X-(y, 0)) is then analytic in a neighborhood of K. The continuity of u implies that U is continuous and that we may compute the derivatives of U by differentiating on P in (9.1.2). Since P is harmonic outside 0 it follows that U is harmonic. To prove (9.1.3) we note that if YEJRn+l and X=1 near Y then
SP(X -
Y) LI (XcJ>)(X)dX = (X)
= (Y). By the uniqueness of analytic continuation this remains true for all Y in a complex neighborhood of K x {OJ in =O,On+lcJ>=¢ when zn+l=O. n+ 1
1
For every R> 1 there is a constant CR such that
(9.1.5)
1cJ>(Z,Zn+l)I~CRlzn+ll
sup
I_-zi satisfies (9.1.4) then u(t, x)=cJ>(z+iZn+ lX, Zn+ It);
tEJR, xEJRn;
satisfies the wave equation 02 ujot 2 = LI xu, and u=O,
oujot=Zn+ 1 ¢(z+izn+ lX) when t=O.
Hence it follows from Theorem 6.2.4 that cJ>(z, Zn+ 1) =Zn+ 1 is any harmonic function in 1R.n + 1. We have
where H is harmonic in 1R.n + 1, and H vanishes identically U~O at 00.
if and only if
Proof. The right-hand side of (9.1.3) is independent of the choice of X, for it is equal to 0 if XEC~(1R.n+l'-(Kx{O})). For any (bO we may therefore choose X so that every point in supp X has distance < (j to K x {O}, and we can always take X even as a function of xn+ l' Then (9.1.3) is automatically true if cI> is even as a function of xn+ l' When ¢ is a polynomial in (Cn we now define
(9.1.6)
u(¢)= -
JU A(XcI»dX
where cI> is given by Lemma 9.1.4. Taking R=4j3 in the lemma we obtain IA (xcI»1 ~ C~ sup I¢I, K76
for if Ix-YI2+X;+1 and for all exponential solutions 4> of A in JR"+1 we have 0= JH 1 A (X 4» dX = J(A H l)4>dX. Hence it follows from Theorem 7.3.2 and Lemma 7.3.7 that AH 1 =Af for some fE C~. This means that H 1 - f is a harmonic function which is equal to H outside a compact set, and therefore outside K x {O}. Thus H has been extended to a function which is harmonic in JR"+1. Since H = 0 is equivalent to H -+0 at 00 by the maximum principle, and since U1-+0 at 00, the last statement in the proposition follows. We obtain (9.1.3) for every harmonic function in JR"+ 1 since this is true with U replaced by U1 and since
JH A(X4»dX =0. of
We are now ready to prove some important facts on the elements A'(JR")=
U A'(K).
KelRn
Theorem 9.1.6. IfuEA'(JRn) then there is a smallest compact set KcJR" such that uEA'(K); it is called the support of u. Proof. Let K be the intersection of all compact sets K' cJR" such that uEA'(K'). By Proposition 9.1.3 a harmonic function in JRn+l . . . . (K' X {O}) is defined by (9.1.2). It is uniquely determined by its restriction to the complement of the plane x"+ 1 =0. The functions obtained for different choices of K' must therefore agree in their common domain of definition and give together a harmonic function in JRn+ 1 ....... (K x {O}). Hence uEA'(K) by Proposition 9.1.5.
Next we shall prove a completeness theorem for analytic functionalso In order to prepare for the construction of boundary values of analytic functions in Section 9.3 we shall then consider some analytic functionals which are carried by compact sets close to JR" but not contained in JRn. This requires another look at Propositions 9.1.3 and 9.1.5. If uEA'(KJ where K, is defined in Proposition 9.1.2, then (9.1.2)
332
IX. Hyperfunctions
defines a harmonic function U in the complement of K.={XEIRn+l; Ix-YI2+X;+1~e2 for some YEK}. This will follow if we just show that (9.1.7)
if X¢K. and zEK•.
Re(x-z, x-Z)+X;+l>O
The left-hand side is equal to Ix-RezI 2 +x;+1-IImzI 2 and for some Y E K we have IRez- yl +2lImzl ~e. Since X ¢ K. it follows that e2 = 1 near K1 '-.(K1 nK 2). Then U1 =4>U-v,
U2 =(1-4»U+v
have the required properties ifvEc oo (IRn+1'-.(K 1nK 2 )) and Llv=Ll(4)U).
Here we define 4>U=O near K2'-.(K1nK2) and (l-4»U=O near K1 '-.(K1nK2)' Since Ll(4)U) vanishes near (K1 UK2)'-.(K1nK2) it is a COO function outside K1 nK 2 • The existence of v is therefore a consequence of Theorem 4.4.6. (Note that this is based on another application of Runge's approximation theorem.) Finally we note that if U EIf' (JR n) then U defines an element in A'(JR n) with the same support. In fact, the harmonic function U(x,x n + 1) defined by (9.1.2) has the~' limit ±u/2 as Xn+ 1--+ ±O, so continuation of U as a harmonic function is only possible outside supp U x to}. Thus we have an injection preserving supports If' (JR n) "-+ A' (JR n).
The operations defined for distributions in Chapters III to VII carryover easily to A'(JRn). We shall just recall them briefly and leave all details for the reader. a) If uEA'(JR n) then ojuEA'(JRn) can be defined by (OjU)(4» = -u(Oj4»
when 4> is an entire function, for sup IOj4>1 can be estimated by the (fJ
supremum of 14>1 over a neighborhood of w. b) If uEA'(K), KeJRn, and f is analytic in a neighborhood w of K, then we define the product fu by (ju)(4» = u(j 4»
when 4> is analytic in w. Here it is of course important that Proposition 9.1.2 allowed us to extend U to all functions analytic in w. c) If U EA'(JRn) and v E A'(JRm) then U Q9 v E A'(JRn+m) is defined by (UQ9v)(4» =ux (v y (4>(x, y))) = Vy (u x (4) (x, y)))
when 4> is a polynomial in (Cn+m. The second equality is obvious then and the first is a definition; it is clear that we obtain an analytic functional supported by supp U x supp v. d) If KEA'(JRnxJRm) and v is analytic in a neighborhood of the projection of supp K in JRm, then % v E A' (JR n) is defined by (% v)(4)) =K(4) Q9 v)
for every entire function 4> in (Cn.
9.2. General Hyperfunctions
335
e) If U E A' (K), where K is a compact set in 1Rn, and if f is a real analytic bijection of an open set wc1Rn on a neighborhood of K with inverse h, then the pullback f* U E A' (f* K) is defined by (f* u)(4)) = u«4> h) Idet h'!) 0
when 4> is analytic in w. f) If u, V E A' (1R n) then c), d), e) allow us to define U * v by letting the pullback of U ® v by the map (x, y)--+(y, x - y) act on the function 1. g) If uEA'(K), K compact in 1Rn, then the Fourier-Laplace transform am=U(exp -i(.,
0)
is an entire analytic function such that for every e > 0
laml ~ C. exp(HK(Im ')+eIW,
(E(Cn.
This follows at once from the definition of A'(K). Conversely, every entire function satisfying these estimates is the Fourier-Laplace transform of a unique element in A'(K). The uniqueness follows from the fact that U(P) = P( - D) 12(0)
for every polynomial P. The existence proof will be given as Theorem 15.1.5, and the result will not be used in the meantime.
9.2. General Hyperfunctions We want to define hyperfunctions in 1Rn in such a way that they are locally equivalent to analytic functionals with compact support in 1Rn. This will be done in two steps. Definition 9.2.1. If Xc 1Rn is open and bounded we define the space of hyperfunctions B(X) in X by
(9.2.1)
B(X) =A'(X)jA'(oX).
The reader might object here that this does not give the desired result in the case of distributions, for tC'(X)/tC'(OX)L-+!0'F(X).
However, the definition will be justified in a moment when we prove that the analogue of the Localization Theorem 2.2.4 is valid.
336
IX. Hyperfunctions
If u, vEA'(X) and u-vEA'(aX), then X nsuppu=X nsuppv since supp u e supp V u ax and supp v e supp U u ax. Thus it is legitimate to define the support of the class u· of u in B(X) by
suppu'=X nsuppu. If YeX and X, Yare open and bounded, we can for every uEA'(X) find vEA'(Y) so that Ynsupp(u-v)=0. This follows from
Theorem 9.1.8 applied to u and the compact sets Y and (X'- Y). The class v· of v in B(Y) is uniquely determined by the class u· of u in B(X) and is called the restriction of u· to Y. Note that the restriction of u· to Y is 0 if and only if Ynsuppu=0. As in the case of distributions the support of a hyperfunction is therefore the smallest closed set such that the restriction of the hyperfunction to the complement is equal to O. The definition of sing supp u is also extended to hyperfunctions with no change. We can now state and prove the localization theorem.
Theorem 9.2.2. Let Xj be open sets in JR." with bounded union X. If ujEB(X) and for all i,j we have ui=u j in XinX j (that is, the restrictions are equal) then there is a unique u E B(X) such that the restriction of u to Xj is equal to U j for every j. Proof. The uniqueness is clear for if v has the same property as u then the support of u - v is empty so u - v = O. To prove the existence we begin with the case of just two open sets X 1 and X 2' Choose ~j E A'(X j) defining u j for j = 1, 2. The support of Vi - V 2 is contained III
(Xl uX 2)'-(X 1 nX 2)e(CX 1 nX 2)U(X 1 n
CX 2)'
so Theorem 9.1.8 gives a decomposition Vi - V 2 = Vi - V2 ,
Vi EA'(CX i nX 2)'
Now V= Vi - Vi
= V 2-
V2 EA'(X 1 n
CX 2)'
V2 EA'(X 1 uX 2 )
defines an element in B(X 1 U X 2) which restricts to u j in Xj for j =1,2. Next we assume that we have countably many sets Xj' j = 1,2, .... Repeated use of the special case just proved gives a sequence Vj in B(X 1 u ... u X j) with restriction ui to Xi for i ~j. Let l-jEA'(X 1 u ... uX j ) be in the class of Vj' Since supp(l-j- Y,;)eX '- (X 1 U ... U X) whe.? k > j, it follows from Theo~m 9.1.7 that there is an element VEA'(X) such that supp(V-l-j)eX,-(Xiu ... uX j) for every j. Hence the class u of V has the desired restrictions.
9.2. General Hyperfunctions
337
If we have more than countably many Xj we just choose countably many of them with the same union and· then a corresponding u. The uniqueness established at the beginning of the proof shows that the restriction of U to Xj is then uj for every j. It follows from Theorem 2.2.4 and the remarks at the end of Section 9.1 that we have an injection ~'(X)--+B(X). Let us also note here that the elements with compact support in B(X) can be identified with the elements in A' (lR") having support in X. In fact, let U E A' (X) and assume that the class u· has compact support K eX. Then supp U e K U 0 X so Theorem 9.1.8 gives a decomposition
u 1 EA'(K), u2 EA'(oX)
u=u 1 +u 2 ,
which is unique since K and ax are disjoint. This means that u· =u~ for a unique u 1 EA'(K). It is easy to extend the operations on A'(lR") discussed at the end of Section 9.1 to operations on B(X). First it is clear that if X and Y are bounded open sets in lRn and f is a real analytic diffeomorphism of a neighborhood of Y on a neighborhood of X, then we obtain a bijection from the bijections
1*:
f*: A'(X)--+A'(y)
B(X)--+B(Y)
and
1*:
A'(aX)--+A'(ay).
The easy proof that (fg)* =g*f*
is left for the reader. We can now define B(X) for any real analytic manifold X. First we choose an atlas !#" of analytic diffeomorphisms of coordinate patches X ,/t:::::, X on open sets XK ~ lR" such that K has an analytic extension to a neighborhood of the closures. Then KK'-l: K'(XKnXK,)--+K(XKnX K ,),
K, K'E!#",
has an analytic extension to a neighborhood of the closures, so (K K'-l)*: B(K(X K n X K')) --+ B(K'(X K n X K'))
is defined. We can therefore define a hyperfunction U E B(X) as a collection of hyperfunctions UK E B(X K) for every K E!#" such that (6.3.3) is valid. The easy but tedious proof that B(X) is independent of the choice of atlas and that it agrees with our previous definition when X ~ lR" is left for the conscientious reader. The notion of support and restriction carryover immediately to the general case. A final justification of Definition 9.2.1 is given by
338
IX. Hyperfunctions
Theorem 9.2.3. If X is a real analytic manifold and Y an open subset then every uEB(Y) is the restriction to Y of a hyperfunction vEB(X) with support in Y. Proof Let Ie X" -> X" be a coordinate system E ff on X. Then u"EB(K(YnX.,)) is the class of an element UEA'(K(YnX,J) which also defines a hyperfunction VEB(X,,) since K(YnX")cX,,. The restriction of V to K(YnX,,) is equal to u". The desired extension of u to YuX" is now obtained immediately if to an atlas for Y with coordinate patches ~ Y we add the coordinate system K with the hyperfunction V. Continuing in this way we can successively extend u to all of X. (If X is not countable one should use Zorn's lemma but we are
not interested in such generality.) The extension of Theorem 9.2.2 to a real analytic manifold X with open subsets Xj is immediate. So is the definition of the product fu of a hyperfunction u EB(X) by a function f which is real analytic in a neighborhood of supp u, as well as the definition of the tensor product.
9.3. The Analytic Wave Front Set of a Hyperfunction Definitions 8.1.2 and 8.4.3 of WF and WFL make no sense for hyperfunctions but it is possible to use the equivalent characterization in Theorem 8.4.11. For the sake of brevity we shall only discuss WFA . With K still denoting the analytic function in {z; 11m Zl2 < 1 + IRe Z12} constructed in Lemma 8.4.10 we first prove an analogue of a part of Theorem 8.4.11.
Proposition 9.3.1. If uEA'(JR") then U(z)=K * u(z)=utK(z -t) is an analytic function in Z={z; IImzI 2 0, ... , fJk >0, fJk+ 1 = ... = fJn =0, then 1 1 k U(Z) =ZP J'" U(t1 Zl' ... , tkZk, Zk+ 1"'" Zn) (1- t/i-1/(fJj-1)! dt o
0
TI 1
is the unique solution of the boundary problem. It is obvious that (9.4.3) follows. A slightly weaker existence theorem is valid for small perturbations of (9.4.1). We take R = 1 for the sake of simplicity. Theorem 9.4.2. Let fJ be a fixed multi-index, IfJ I= m, and let a"', and f be bounded analytic functions in Q 1 with A =(2n etsup Lla"'l < 1. a,
I I~ m, Q(
9.4. The Analytic Cauchy Problem
Then the equation
(9.4.1)'
L
DJl u =
347
aaDau+f
lal~m
has a unique solution satisfying (9.4.2) in Qt' and
sup lui ~(1- A)-l sup Ifl/P!.
(9.4.3)'
The proof requires two elementary lemmas.
Lemma 9.4.3. If v is an analytic function of one complex variable , when 1'1 < 1, such that Iv'ml~C(1-1'1)-a,
1'1 1 the statement follows from the fact that r
(1-r)a J(1-t)-adt=((1-r)-(1-r)a)/(a-1) o
takes its maximum when a(1- r)a-l = 1, for the maximum value is (1- r)/a < l/a. Letting a-+ 1 we obtain the statement when a = 1.
Lemma 9.4.4. If v is analytic when
1'1 < 1 and
Ivml~C(l-I'I)-a,
1'1O, and this completes the proof that analytic continuation is possible from to (O,zn) ifOO,
for every compact set K e CC" and every continuous function cP on K such that (9.6.5)
cP(z) > (11m Zl2 -d(Re z, CX)2)/2,
(9.6.6)
cP(z»d(Imz,Q)2/2
if RezeX,
zeK, zeK.
If u = 0 in X then (9.6.6) can be omitted. Proof We may assume that X is bounded since (9.6.5) remains valid if X is replaced by the intersection with a sufficiently large ball. The
9.6. The Analytic Wave Front Set and the Support
359
estimate (9.6.4) follows from (9.6.2) and (9.6.5) if Re Z is in a neighborhood of Cx, so we may assume in the proof that ReK cx. Write u=u l +u o where suppu l cX and suppuocCX. By (9.6.5) and (9.6.2) Re-(z-y)2/2O.
Proof Take any covering of JRn" {OJ with small convex proper cones as in the proof of Theorem 9.6.3. Then u=u o + ~>j where uo=O in a neighborhood X of and uj=bGjij in X with ij analytic in X +iQ j • Here Qj is a neighborhood of if IjnWo=0, and Qj generates the open convex cone Gj with dual lj otherwise. We have U6 = L U j6 where U 06 vanishes in XI1> and uj6 restricted to Xlf> is the boundary value of ij(1)z),j=t=O, which is analytic in Xjf>+iQ/f>. For small 1> the estimate (9.6.12) follows from Proposition 9.6.1 if
°
°
4>(z) >d(Im z,Q/1»2j2,
zEK, j=t=O, Ijn Wo=t=0.
Since d(Imz,Q/1»'\.d(Imz, G) when 1>--+0, this is true for small f> if (9.6.13) Now we have a kind of Pythagorean theorem d(~,Gj)2+d(~, _lj)2=1~12.
Since Gj n ( -lj) = {OJ and the relation b~tween these cones is sy~ metric it suffices to prove this when ~¢Gj' If ~* is the point in Gj closest to ~ then ~O, l1EGj' hence ~*-~Elj since Gj is a cone, and =O~, OE-lj. Thus ~-~* is the point in -lj closest to ~ and (see Fig. 5). d(~,
Gy +d(~,
_lj)2=1~ _~*12+1~*12 = 1~12.
Fig. 5
364
IX. Hyperfunctions
Hence (9.6.13) means that (9.6.13)'
j(W)DU)
o
has a limit in f!)' (R) as H
:3
W --+
O.
Exercise 2.13. Put Xe(x) = x(x/e)/e where X E CO'(R) and JXdx = 1. Determine a constant C and a distribution U such that the distribution Ke = Xe(xy) + ClogeDo(x,y) --+ U when e --+ +0.
374
Exercises
Exercise 2.14. Define ua,e(x) = 8/(82 + (x - aJe)2), x E R, where a E R and 8 > O. Determine lime->o Ua,e and lim.-.o Ua,eUb,e when they exist in .@'(R). Exercise 2.15. Determine the limit in !?&'(R2 \ 0) as t
-+ +00
of
Does the limit exist in !?'&'(R2 )? Exercise 2.16. Set ftAx) = e-itX(x + i8)-1, x following limits in .@'(R): a)
E
R, and determine the
b)
Section 3.1 Exercise 3.1.1. Let f E !?&' (1) where I is an open interval on R. Show that there is a solution u E !?&'(I) of the differential equation u' = f, and that the difference between two such primitive distributions is a constant. Exercise 3.1.2. Let u E !?&' (I), where I is an open interval c R. Show that if U has order k > 0, then u' has order k + 1. Exercise 3.1.3. Let u E !?&' (I) where I is a finite open interval c R. Show that if u is the restriction to I of a distribution of order k in a neighborhood of Y, then lu(cp) I :s; C
I
sup IcpUlj,
cp E C(f(I),
j5.k
and that conversely this estimate implies that there is a measure dJ.l on R with support in Y such that u is the restriction to I of its kth derivative. Exercise 3.1.4. Show that if f is a measurable function in (-1,1) and
where m is a positive integer, then there is a distribution FE tB',m([-I, 1]) with restriction f to (-1,1).
Section 3.1
Exercise 3.1.5. Does there exist a distribution u E to the function x ..... el / x exp (ie l / x ), x > O?
~'(R)
375
which restricts
Exercise 3.1.6. For which a E R does there exist a distribution U E ~'k(R) with the restriction x ..... e l / x exp (itt/X) to R+ ? Exercise 3.1.7. Prove that the limit
(vp(l/x), q>} = lim .-+0
1
Ixl>.
q> E CO'(R),
q>(X) dx/x,
exists. What is the order of the distribution vp(l/x)? Exercise 3.1.8. J is an odd locally integrable function on R such that xJ(x) ~ 1 as x ~ 00. Prove that gt(x) = t 2 f'(tx) has a limit in ~'(R) and determine it. Exercise 3.1.9. Determine real numbers at.
a2
such that the integral
exists when q> E CO'(R) and u = d2(vp(1/x»/dx2. Exercise 3.1.10. Define log(x + iO) = log Ixl + niH(-x), x E R, calculate the derivative, and compare it with vp(l/x). Exercise 3.1.H. Show that the function
has a limit in
~'(R)
when
Il ~
0 and give it in a simple form.
Exercise 3.1.12. Compute the nth derivative of x ..... C"(R+). Exercise 3.1.13. C"(R).
JE
J (Ixl) when J
E
Compute the nth derivative of x ..... IxlJ (x) when
Exercise 3.1.14. Let 1 be an open interval c: R, and let a E I. a) Show that for every J E ~'(/) there is a solution u E ~'(I) to the equation (x - a)u = J, and that two solutions of this division problem differ by a multiple of ~a. b) Give a solution when J = ~Jfl. Exercise 3.1.15. Show that if 1 is an open interval on Rand F E CaV) has no zero of infinite order, then the equation Fu = g has a solution u E ~'(/) for every g E ~/(I). Describe the solutions when g = O.
376
Exercises
Exercise 3.1.16. Show that if F E C lxI}, and calculate v = 0 U, W = 0 v, where 0 = 8t - A is the wave operator.
Section 4.1 Exercise 4.1.1. Show that if Ue = sgn t X~ (t2 - 1. Calculate Jl * E where E(z) = (2n)~110g Izl.
f (x) = Ixl~5 I,J,k=l ajk XjXk, x E R3 \ 0, where (ajk) is a constant symmetric matrix. a) What is the condition for the existence
Exercise 4.4.2. Set
382
Exercises
of the limit F(cp) = lim £->0
1
Ixl>£
f(x)cp(x) dx
for arbitrary cp E CO(R3)? b) Calculate AF when this condition is fulfilled and prove that F = E * AF where E(x) = -1/(4nlxl). Exercise 4.4.3. V is a function on R3 such that V (x) ~ 0 as x ~ 00, and V is harmonic outside a compact set. Show that -4nlxlV(x) ~ (AV, 1), x ~ 00. Is the hypothesis Vex) ~ 0 essential? Exercise 4.4.4. Let u E ~' (R) and assume that for some integer k 2 0 we have u * f * f E COO(R) for every f E C§(R). Show that u E COO(R). Exercise 4.4.5. Let u be a distribution in Rn with compact support, and assume that fk = (a l ... an)k u is a continuous function for k = 1,2,3, .... Show that u E C;)(Rn). Exercise 4.4.6. Show that a differential equation P(d/dx)u = f, where E 6"'(R) and P is a polynomial, has a solution u E 6"'(R) if and only if (f, cp) = 0 for every solution cp of the adjoint differential equation P(-d/dx)cp = O.
f
Exercise 4.4.7. Let X be the characteristic function of (-1, 1). Determine a number a E (0,1) and a function u E CJ(R) such that X- ba - b_ a
Show that u 2 0, compute I =
il
1
I
= d4 u/dx 4 .
S u dx and show that
f dx - f(a) - f (-a) 1:::; I max If(4)1,
f E C 4 ([-1, 1]).
Exercise 4.4.8. Let f E 6'" (Rn) and let (1. = ((1.1, ... , (1.n) be a multi-index. Show that there exists some u E 6"'(Rn ) with aau = f if and only if (f, x P) = 0 for all multi-indices not satisfying the condition f3 2 (1.. Exercise 4.4.9. Show that if f E 6'" (R2) then the equation Au = f has a solution u E 6'" (R2) if and only if (f, cp) = 0 when cp(x, y) = (x ± iy)n, n = 0, 1,2, ...
Section 5.1 Exercise 5.1.1. Show that if u E ~'k, V E ~'l, then u ® v E ~'k+l, and that u ® v E ~'N implies u E ~'N, v E ~'N unless u or v equals O.
Section 5.2
383
Exercise 5.1.2. Construct for given positive integers k and 1 two distributions U E .@lk(R) and v E .@II(R) such that U ® v is not of order k+l-l. Exercise 5.1.3. Construct for a given positive integer N two distributions UI on R which are not of order N - 1 such that UO ® UI is of order
Uo,
N.
Section 5.2 Exercise 5.2.1. Let f be a continuous function from R to R. Which operator has the distribution kernel oH(y - f(x»/oy? Exercise 5.2.2. What is the kernel of the operator fcp(x)
= cp(x) +
1:
a(x,y)cp'(y)dy,
cp
E
CO'(R),
where a E C(RZ)? Exercise 5.2.3. K is a measurable function in X I open subset of Rnj, such that
j j
X Z where Xj is an
X
IK(X,Y)1 dX::5: A, for almost all y
E Xz;
Xl
IK(X,Y)1 dy ::5: B, for almost all x E XI.
X2
Prove that for the corresponding operator f
Section 6.1 Exercise 6.1.1. Calculate ba(cosx) when -1 < a < 1. Exercise 6.1.2. Calculate
U
= b~(f) in RZ \ 0 when f(x) = XIXZ.
Exercise 6.1.3. Determine the limit of CPe(x z - yZ)CPe(Y - 1) in .@/(Rz) as e ---+ +0, where cp E CO', f cp(x) dx = 1, and CPe(t) = qJ(t/e)/e.
384
Exercises
Exercise 6.1.4. Let f,g be real valued functions in COO(X), X open in Rn , such that df and dg are linearly independent when f = g = O. Determine U = (j(f,g). Exercise 6.1.5. Let f,g E coo(Rn) be real valued, df =1= 0 when f = 0 and dg =1= 0 when g = o. Show that if q> E CO'(Rn), q> E CO'(Rn) and U = (q>(j(f» * (q>(j(g», then sing suppu is contained in {x + y;x E rl(O) n suppq>,y E g-I(O) n suppq> and df(x),dg(y) are linearly dependent}. Give an integral formula for u valid in the complement of this set. Exercise 6.1.6. Set ut(x) = (f(x) + ie)-l where f E coo(R) is real valued. Determine the condition on f required for the existence of the limits u± = limt-+±o U t , and calculate u+ - u_ then. Exercise 6.1.7. Show that if s > 0 and k is a positive integer, then the function x ~ (.x2 k - s2k + ie)-l has a limit fs E ~'(R) as e -. +0. Show that one can find UQ, .•• , Uk E ~' (R) such that
I
k-l
fs -
o
i
HI - 2k Uj -. Uk,
S -.
0,
and determine support and order for these distributions.
Section 6.2 Exercise 6.2.1. In RxRn, with variables denoted (t, x), let 0 = iJ2 /at 2 -l1x be the wave operator. Calculate the fundamental solution Ek of Ok+l with support in the forward light cone {(t,x);t ~ Ixl} for every integer k~O.
Exercise 6.2.2. Find the forward fundamental solution Fa of 0 every a E C, with notation as in the preceding exercise.
- a
for
Exercise 6.2.3. Find the forward fundamental solution F of the operator o + 2boa t + 2 'L7 bjaj + c for arbitrary complex bo, . .. , bn, c, with notation as in the preceding exercises.
Section 7.1 Exercise 7.1.1. For which even positive integers m and n is f (x) = exp(xn + i exp(xm» in 9" (R)?
Section 7.1
385
Exercise 7.1.2. Let M be an unbounded subset of Rn. Show that for every integer m there is a distribution u E 9'" (Rn) with supp u c: M such that the order of uin the unit ball is > m. Exercise 7.1.3. Show that if u is a measurable function on Rn and m is a positive integer, then u E 9'" and uE !!)'m if Jlu(x)1 2(1 + IxI2)-m dx < 00. Exercise 7.1.4. Prove that if K is a compact subset of Rn and I~j - ~kl ~ 1, j 1= k, then
I
00
1«p(~j)12 ~ CK
1
J
IqJ(x)1 2 dx,
with CK independent of the sequence
~j
ERn,
qJ E CO'(K),
~j.
Exercise 7.1.5. Show that if ~j E R I~j - ~kl ~ 1, j 1= k, and m is a non-negative integer, then Lajei(X,~j) converges in 9'" to a sum of order ~ m if aj E C and L lajl2(1 + l~jI2)-m < 00. n,
Exercise 7.1.6. Let u E 9'" (Rn). When does there exist a function f E 9" such that u = u * f? Exercise 7.1.7. Show that if u E LP(Rn) and I~I ~ A when then IIu'liv ~ CAliuliv where C only depends on n.
~ E
supp U,
Exercise 7.1.8. Show that if u E Loo(Rn) and ~ E supp U, then one can find a sequence qJj E 9" such that lu * qJjl ~ 1 and u * qJj(x) ~ ei(x,l;) uniformly on every compact set. Exercise 7.1.9. When does a differential equation P(D)u = 0 with constant coefficients have a solution 1= 0 in a)!!)' b) 9'" c) Iff' d) COO e) 9".
Exercise 7.1.10. Let f
E Ll(Rn)
and f * f = f. Find f·
Exercise 7.1.ll. Show that the equation u - u * f = f for a given E 9"(Rn) has a solution u E 9" if and only if! 1= 1.
f
Exercise 7.1.12. Show that if u, v E 9"'(R) have supports on the positive half axis, then u * v E 9"'(R). Exercise 7.1.13. Let ua(x) = 1/llogxla when 0 < x < ~, u(x) = 0 when x < 0 or x > 1, and u E COO when x > 0; here a > O. Determine the limit of u(~)~(log I~ I)a as ~ ~ 00. Exercise 7.1.14. What is the Fourier transform of the space Z (a, k) in Exercise 3.2.2? Exercise 7.US. Set!Fu = (2n)-n/2u when u E 9"(Rn). Prove that a) !F4 = I, the identity, and that every u E 9"(Rn) has a unique
386
Exercises
decomposition
b) Show that the differential operators Lvu = XvU + ovu, V = 1, ... , n, are surjective on 9'(Rn), determine the kernels and show that § Lv Uk = jk+l Lv Uk for the terms in the decomposition. Exercise 7.1.16. Show that if K is a continuous function in Rn then the following conditions are equivalent:
(i) The convolution operator
0 and b > O. Prove that the Fourier-Laplace transform U(O
=
J
e-i(X+iy,Ou(x+iy)dx
is independent of y and that
Show that u = 0 if b > a.
Section 7.6 Exercise 7.6.1. Find the Fourier transform of R3
:3
x 1--+ exp i(xi+x~-x~)
Exercise 7.6.2. Let f E CO'(R'). Prov~ that for every t > 0 there is a function ft E !/' such that ft(~) = f(~)exp(itl~12), and prove that 1ft (x) I :::;; Ct- n/ 2 for x E Rn and t > 1. Use this to decide for which p E [1,00] that the Fourier transform of LP consists of measures. Exercise 7.6.3. Find the Fourier transform of the distribution u = t5(X2 - xi} in R2. Exercise 7.6.4. Find a fundamental solution E E !/"(R3) of the differential operator Exercise 7.6.5. Find a real number a and u E 2&'(R) such that the sequence fn(x) = n a sin (nx 2 ), n = 1,2, ... has the limit u =1= 0 in 2&'(R) as n --+ 00.
Section 7.6
393
Exercise 7.6.6. Find a real number a and u E f!fi' (R2) such that the sequence un{x) = na sin (nxlX2), n = 1,2, ... , has the limit u =F 0 in f!fi' (R2) as n ~ 00. Exercise 7.6.7. For which positive real numbers a and which p E [1,00] is the Fourier transform of fa (X) = (I + x2)-a/2e ix2 in LP? Exercise 7.6.8. Let p be a polynomial in x E R of degree m > 1 and real coefficients. Prove that the Fourier transform F of eip(x) is an entire analytic function, and determine a homogeneous differential equation of order m - 1 with linear coefficients which it satisfies. Exercise 7.6.9. Let A be a symmetric n x n matrix with Re A positive semi-definite and II 1m A II :s:: 1. Prove that if n < Jl < n + 1 and u is Holder continuous of order Jl in Rn , then e-(AD,D)u is continuous and with C independent of A and u sup le-(AD,D)ul :s:: C(Jl- n)-lluI Jl ; lul Jl
=
I
io:i,,;n
sup loO:ul +
I
sup loO:u{x) - oO:u(y)llx - yln- Jl .
io:i=n xioy
Exercise 7.6.10. Prove that if in addition to the assumptions in the preceding lemma we know that A{D)ju is Holder continuous of order Jl for 0 :s:: j :s:: N, then le-(AD,D)U(X) -
I
j O.
0 and extend g to R.
1.2. Use the preceding exercise. 1.3. Review the solution of the preceding exercises. 1.4. Iterate the result in the preceding exercise. 1.5. Introduce
Xl
± X2
as new coordinates and use Exercise 1.3.
1.6. Choose for example bk = _2k and solve the equations (iii) with n ::; N, k ::; N first, which gives (N)
ak
=
II
k\2k+I)7t. (Hint: By the periodicity it suffices to study u,. when It I < 2n. Taylor expansion of 1 - cos t at reduces the first question to a study of (IX - a)t2,. at 0. The dominating contributions to v occur when cos t = -1, so look at the Taylor expansion there, which leads to Exercise 2.5 b).)
2.10. a
°
2.11. Show that every qJ qJ(X)
with CPj
E
E
CO'(R n) can be written in the form
= qJI (x) + ... + qJn(x) + XI ... xnqJ(x)
CO'(R n) even in
Xj
and cP
E
CO'(Rn).
ao(w) = nw-3/4/J2, al(w) = 0, a2(w) = nw- I / 4 /v'8, where n/2. Hint: It suffices to study Jf w(x)cp(x) dx when cP is an even test function. Write cp(x) = cp(o) + X2qJ"(0)/2 + qJ(x) and note that Icp(x)/(x4 +w)1 : : ; Icp (x)/x 4 I which is integrable. Use residue calculus
2.12.
Iarg wi
j. 3.1.16. Hint: Use test functions cp«x - a)/e) if a is a zero of infinite order.
Section 3.1
399
3.1.17. (_I)i h~k-i) k !/(k - j)! if j ::;; k and 0 otherwise.
3.1.1S. L~=o(-I)i (;)fU)(O)h~k-j). 3.1.19. a) x+ + C; d) H(x) + C; e)
b) !x! + C; log Ixl + C.
c)
(eX -1)H(x) + C;
3.1.20. a) U = -ho + C, + C2 H b) U = c, vp(l/x) + C 2 h c) u = Cf(x) where f(x) = exp{l/x) when x < 0 and f(x) = 0 when x~O.
u = h~ + C,x::;:' + C2x=' + C3hO u = ho - h~ + Ch, f) u = -ho + C,h, + C2h_, g) u=LcA h) u = H(x) -2(x-l)H(x-l) + C'x+ C2 i) u = H(x -1) + H(x + 1) + C,H(x) + C2 j) u = H(x) exp«(x+ 1)-2-1)/2). (As in c) we have u = 0 in (-00, -1) and (-1,0), and Exercise 3.1.17 shows that there is no contribution with
d)
e)
support at -1.) k) u = Cf(x) where f(x) = exp (1/3x 3 ), x < 0 and f(x) = 0 when x ~ 0 (compare with c». 1) u = LcA,/2+j1t + '[,dj H(n/2 -Ix - U+ l)nl}. 3.1.21. u(rp) = '[,j 1. (Motivate the distribution convergence near the unit circle carefully!) Au is the arc length measure on the unit circle.
3.1.28. This is minus the arc length measure on the unit circle. 3.1.29. The order is 0 in cases a) and c); it is 1 in case b). (Use polar coordinates in case c) to show that the distribution is -1 times the arc length measure on the unit circle.) 3.1.30. We must have a > b. If f(x) = x a sin (x-b) then (ou/oy, qJ) = - S;' qJ(X,j(x» dx, qJ E CO', so ou/oy is always a measure. When x > 0 then ou/ox = -f'(x)ou/oy has infinite measure near the origin unless a > b. When this condition is fulfilled verify that ou/ ox = -f'(x)ou/oy + b(x)H(-y). 3.1.31. Either f vanishes identically or else all zeros of f have finite order. The sufficiency is close to the one dimensional case (Exercise 3.1.15). To prove necessity test with functions of the form qJ(x)qJ«y-a)/Il) where a is an endpoint of an interval where f > 0, and estimate the order of a as a zero of f by means of the order of u as a distribution! 3.1.32. qJ(t) = C.jt - t where C is an arbitrary constant. - Direct calculation shows that we have a solution outside the curve x = qJ(t). Taking the jumps into account we obtain qJ(t) = -t or the differential equation qJ'(t) + (1 - qJ(t)/t)/2 = o.
3.1.33. The limit is 2ni«x _1)-3 H(-x) + b~ (x) + !b, (x». Hint: Take the Taylor expansion at z = 1.
Section 3.2
401
Section 3.2 3.2.1. Verify (i),(ii),(iii) by direct computation. Then (v),(vi) follow if Re a > -1. The first part of (iv) is clear. Z (0) consists of functions constant on each half axis. Now u E Z(-l) means that xu = Co and u = Covp(l/x)+ ClbO which proves that dimZ(-l) = 2. The other statements follow now from the first part of (iv). (See also Exercises 2.1 and 3.1.2.) 3.2.2. Argue as in the preceding exercise. To prove (iv) note that since xtZ(O,k) c Z(O,k - 1) and xtZ(O,l) = {O}, the dimensions of the spaces show that the inclusion is an equality, which implies the statement on tZ(O,k). Since txZ(-l,k) c Z(-1,k-1) we conclude from (v) for lower k that the dimension of Z(-l,k) is at most 2k. We have xZ(-l, k)
::J
xtZ(O, k) = Z(O, k - 1).
The inclusion is strict, for if w,,(x) = (log Ix!)" E Z (0, k + 1) then xw~ = kWk-l is in Z(0,k)\Z(0,k-1) although w~EZ(-l,k) since we have (xt + l)ktwk = t(xt)kWk = O. Hence dimZ(-l,k) = 2k and the other statements follow. 3.2.3. The dimension is 2m. Hint: Write Lajr j = amII(r -Av)k" and use the preceding exercise. 3.2.4. Hint: Elaborate the solution of Exercise 2.1. 3.2.5. The order of u is 1, the order of v is 2, and the degree of homogeneity is -2 for both u and v. That the order is at most 2 is obvious. That the order of v is not 1 follows using test functions of the form 0 and some z E suppv; if x E suppu, that is, y E supp u + supp v, it follows that (u * v, qJ) > O. 4.3.2. No, we have for example c50* 1 = O.
4.3.3. Hint: A convex set K is contained in an affine hyperplane with normal ~ if and only if R 3 t ~ H (t~) is linear, where H is the supporting function of K. Now apply the theorem of supports. 4.3.4. supp 1 is the square when P (0) =1= 0; when P (0) = 0 it is the boundary with the interior of the sides parallel to the Xj axis removed if P(iJ) is divisible by OJ, and it is empty when P = O. Generalize to an arbitrary polygon!
Section 4.4 4.4.1. I(z)
= z + v'z2 -1
continuous and
=1=
0 outside [-1,1] so Jl. = 0 there; log I/(z)1 is
(iJ/ox-,iiJ/oy) log III
=1'/1 = 1/Jz 2_1
Section 4.4
405
also in the sense of distribution theory. The distribution boundary values at x±iO are =Fi/Vl-x2 , hence Jl = iu(x)t5(y) where u(x) = -2i/Vl-x2 if Ixl < 1, u(x) = 0 if Ixl > 1, that is, Jl( sUPcesuppf "I and conclude that u(z) = 0 when Izl > R.
Section 5.1 5.1.1. Immediate consequence of the definitions. 5.1.2. Try U = I.f'v-22kvei2'x and a similar definition of v; test with cp(x, y)e- i(x+ y )2'. 5.1.3. Take
Uj
=
If' A~v~}ei}.2HjX
where Av = 2v!.
The idea is that
I
A~_l ~ AJ for large v which makes one amplitude factor dominate in any term in the product. (To prove that Uj is not of order N - 1 note I I 2 .eiJ. 2,+jx were a measure dll then J2n e-iJ. 2Hj x dll = 2nA 2 . that if "L A~~ r 0 r ~~ which is absurd.)
Section 5.2 5.2.1. The composition cp 1-+ cp
0
f.
5.2.2. o(H(y - x) - a(x,y))/oy. 5.2.3. This is obvious by Fubini's theorem if p = 1 or p = 00. For 1 < p < 00 write IK(x,y)cp(Y)1 = IK(x,y)11-1/p(IK(x,yW/Plcp(y)1) and use
HOlder's inequality.
Section 6.1 6.1.1. cos x = a when x = 2kn ± arccos a with integer k, and then we have sin x = ±,J1 - a2 • The answer is therefore 00
(1 -
a2 )-!
Z)c5 2kn+arccosa + c52kn-arccosa). -00
Section 6.1
6.1.2. U = b'(xt} ® (1/x2Ix2i) + (l/xIlxIl) ® b'(X2). v = bO(XlX2) first and note that OIV = X2U, 02V = XlU.
6.1.3. The limit is coordinates.)
!(b(1,I)
+ b(-l,l).
(Take x 2
-
407
Hint: Calculate
y2 and y - 1 as local
6.1.4. U = du / vlf'1 21g'1 2- (f', g')2 where du is the Euclidean surface measure on L = {x;f(x) = g(x) = O}. Hint: Assume coordinates labelled so that Yl = f (x), Y2 = g(x), Yj = xb j > 2, is a local coordinate system at a chosen point xO E L. Then
(U,t:p) =
f
t:p(x(O,O,y'))IDx/Dyldy',
y' = (Y3,···,Yn);
t:p
E
Cg";
ifsupp t:p is close to xo. Here IDy /Dxl = lof /OXlOg/OX2-of/ox2og/0Xll. At a point where the tangent plane is dXl = dX2 = 0, this is equal to vlf'1 2 1g'1 2 - (f',g,)2 since of /OXj = og/OXj = 0, j > 2, so we have the asserted density at such a point. Orthogonal invariance proves that it is true everywhere. 6.1.5. If Lx = {y; f (y) = g(x - y) = O}, then u(x) =
1
t:p(y)t:p(x - y) du/\/lf'(Y)1 2Ig'(X - Y)12 - (f'(y),g'(x - y»2,
Ex
where du is the surface measure of Lx. (Use the preceding example and compare with Exercise 4.2.8.) 6.1.6. The condition is that f(x) = 0 implies f'(x) =f. O. (Consider 1m (u B , t:p) with t:p ~ 0.) The limits are then rv± where v± = (t ± iO)-I, so U+ - u_ = -2niLf(x)=O bx/If'(x)l·
6.1.7. The first assertion follows since t ~ (t + iB)-1 --+ (t + iO)-1 in fi}' as B --+ +0 and since x ~ x 2k - s2k does not have 0 as a critical value when s > O. Examination of (fs, t:p) for even test functions t:p shows after Taylor expansion that the stated formula holds with Uj equal to a constant times ba2j ), of order 2j and support {O}, when j < k, whereas Uk is a constant times (d/dx)2k-1 vp (l/x), thus of order 2k and support equal to R.
Section 6.2 6.2.1. Ek = !n(l-n)/24-k X:+(I-n)/2(A)/kl for t > 0, where A = t 2 _lxI 2 ; Ek is extended as a homogeneous distribution of degree 2k + 1 - n. - Note
408
Answers and Hints
that Ek is a constant times the characteristic function of the forward light cone if n is odd and k = (n - 1)/2. (Cf. Exercise 3.3.13.) 6.2.2. Fa = ~ akEk with the notation in the preceding exercise. The sum converges, for the terms with k + (1 - n)/2 > 0 are continuous and
I
00
(A)
=
o
4- kAk /(k!r(k + 1 + (1 - n)/2))
converges to an entire analytic function. We have Fa = 0 when t < Ixi and Fa = 4n(1-n)/2(aA)A(1-n)/2 when t > lxi, and the singularities at the light cone are described by Lk«n-l)/2 akEk. (Cf. Exercise 3.3.12.) 6.2.3. F = e-huHL; bjxj F-a, a = L~ bJ preceding exercise.
b6 + c,
with the notation in the
Section 7.1 7.1.1. The condition is m ;;::: n. Hint: Compare with exercise 3.1.5. Note that a function may be in g' although the absolute value is very large. 7.1.2. Hint: Choose a sequence Xj E M with IXjl > j and set u = Prove that u E g' and derive a contradiction if u is of order m in the unit ball by looking at 11P12 * u where IP E g and I~I < 4 if ~ E supp QJ.
L IXjlm+lbxj"
7.1.3. Hint: Write u(x) = LIIXI:sm XIXUIX(X) with 7.1.4. Hint: Choose X E CO'(R implies (2n)nQJ = X* QJ,
n)
Urx E L2.
equal to 1 in K and use that IP = XIP
7.1.5. Hint: Reduce to m = 0 and apply the preceding exercise. 7.1.6. supp u must be compact, for the equation is equivalent to (1-j)u = 0, and 1 - i= 0 outside a compact set. Conversely, we can always take for f the inverse Fourier transform of a function in CO' equal to 1 on supp uif this is a compact set.
i
7.1.7. Use the answer to the preceding exercise. 7.1.8. Hint: Reduce to the case ~ = 0 by passing to ue-i(x,~). Choose X E g so that X has support in the unit ball and X(O) i= O. Then
Section 7.1
409
U,s = X,s '" U =1= 0, if X,s (x) = X(!5x)· Now take CPj(x) = CjXl/j(X + Xj) where ICjl = l/sup IU1/jl and Xj is chosen so that IU"'cpj(O)-11 < 1/j. Conclude using the preceding exercise that Iu'" CPj(x) - 11 < (1 + Clxl}/j. (The result is due to Beuding.)
7.1.9. a) and d) P not a constant and e) Only when P
7.1.10.
f
= O.
= O.
=1=
0
b) When P has a real zero
Since / is continuous and ~ 0 at == O.
/(1- /) = 0 implies /
00,
c)
the equation
7.1.11. U = //(1 - j) is in [/ precisely when the denominator never vanishes. - The statement is also valid with [/ replaced by L 1, but the sufficiency of the condition is a much harder theorem of Wiener then.
7.1.12. Hint: Use that (u '" v, cp) = (u ® v, ) where (e, '1) = cp(e
+ '1),
cP E CO'. Replace by X for a suitable X E CCX:> which is 1 in the first quadrant and vanishes when lei + I'll > 2(1 + Ie + '11}. Generalize to higher dimensions!
7.1.13. The limit is -i. Hint: We may assume that u is real valued; then u(-e)
= u(e) so we may take e > O. Using (3.1.13) we obtain
use dominated convergence to get the result. Note that u is continuous but u ~ L 1 if a :::;; 1.
7.1.14. Z(-I-a,k). Use this to simplify the answer to Exercise 3.2.2!
7.1.15. That fF4 = I is a consequence of the Fourier inversion formula. Let Pk, k = 0, 1,2,3, be the interpolation polynomials defined by Pk(-.) = (-.4 _ 1)/(4i3k (r - ik» and show that the decomposition holds precisely when Uk = Pk(fF)u. Show that fF Lv = iLvfF and solve the differential equation Lvu = f E [/ explicitly. The kernel of Lv is the set of functions u E [/ such that u(x)~~/2 is independent of xv.
L tjb xj by functions cP E CO' to prove that (i) ==> (ii), and approximate integrals by Riemann sums to prove the converse. Note that (ii) with N = 2 implies that K(x) = K(-x) and that IK(x)1 :::;; K(O), so K E [/'. (iii) ==> (i) since
7.1.16. Hint: Approximate
(K '" cP, cp)
= (K, cP '" fp) = (fl,I«$1 2),
fp(x)
= cp(-x).
410
Answers and Hints
(i),(ii) ==> (iii), for (K, Icp12) 20, cP E [I' implies (K,X) 20 for all X E Co with X 2 0 (approximate X by the square of (X + ee- 1xI2 )! E [1'). Hence K is a positive measure. If CPb(X) = cp(xN)fJ-n , cP E f cpdx = 1, then
Co'
so K has finite mass and equality holds. - K is called positive definite and the result is Bochner's theorem. 7.1.17. Hint: Choose CPb as in the preceding answer. Show that (i) implies that Kb = K * CPb * (Pb = fib where J1-b is a positive measure with total mass Kb(O) = O(fJ-N ) if K is of order N in a neighborhood of O. Show that J1-bl$(e')1 2 = J1-el$(fJW and conclude that J1-b = 1$(fJ')1 2 J1where J1- is a measure with fl~l
