the mellin transformation and fuchsian type partial differential equations

Mathematics and Its Applications ( East European Series)
Managing Editor:
M. HAZEWINKEL
Editorial Board:
A. BIAL YNICKI-BIRULA, Institute of Mathematics, Warsaw University, Poland H. KURKE, Humboldt University, Berlin, Germany J. KURZWEIL, Mathematics Institute, Academy of Sciences, Prague, Czechoslovakia L. LEINDLER, Bolyai Institute, Szeged, Hungary L. LOV Asz, Bolyai Institute, Szeged, Hungary D. S. MITRINOVIC, University of Belgrade, Yugoslavia S. ROLEWICZ, Polish Academy of Sciences, Warsaw, Poland BL. H. SENDOV, Bulgarian Academy of Sciences, Sofia, Bulgaria I. T. TODOROV, Bulgarian Academy of Sciences, Sofia, Bulgaria H. TRIEBEL, University of lena, Germany
Volume 56
by
and
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
Library of Congress Cataloging-in-Publication Data
Szmydt, Zofla. The Mellin transformat ton and Fuchsian type partial dtfferential
equations I by Zofia Sz.ydt and Bogdan Ztemian. p. cm. -- (Mathematics and its appltcatl0ns (Kluwer Academtc
Publishers). East European Series ; v. 56) Includes tndex. ISBN 978-94-010-5069-2 ISBN 978-94-011-2424-9 (eBook) DOI 10.1007/978-94-011-2424-9 1. Dtfferential equations, Partial. 2. Melltn transform.
1. Ztemtan, Bogdan. II. Tttle. III. Title: Fuchsian type partial dtfferenttal equations. IV. Sertes. OA377.S969 1992 515' .353--dc20 92-4675
ISBN 978-94-010-5069-2
AlI Rights Reserved @ 1992 Springer Science+Business Media Dordrecht Origina1ly published by Kluwer Academic Publishers in 1992 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.
To the memory of our great friend and adviser Professor Andrzej Plis
SERIES EDITOR'S PREFACE
'Et moi, .. Of si j'avail su comment en revenir. je n'y semis point alll!.'
Jules Verne
The series is divergent; therefore we may be able to do something with iL
O. Heaviside
One selVice mathematics has rendered the human race. It has put common sense back when: it belongs, on the topmon shelf next to the dusty canister labelled 'discarded nonsense'.
Eric T. Bell
Mathematics is a tool for thought A highly necessary tool in a world where both feedback and nonlineari ties abound, Similarly. all kinds of parts of mathematics serve as tools for other parts and for other sci ences,
Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One ser vice topology has rendered mathematical physics .. , '; 'One service logic has rendered computer science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series.
This series, Mathematics and Its Applications, started in 1977. Now that over one hundred volumes have appeared it seems opportune to reexamine its scope. At the time I wrote
"Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the 'tree' of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sci ences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma. coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as 'experi mental mathematics', 'CFD', 'completely integrable systems', 'chaos, synergetics and large scale order', which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics. "
By and large, all this still applies today. It is still true that at first sight mathematics seems rather frag mented and that to find, see, and exploit the deeper underlying interrelations more effort is needed and so are books that can help mathematicians and scientists do so. Accordingly MIA will continue to try to make such books available.
If anything, the description I gave in 1977 is now an understatement. To the examples of interaction areas one should add string theory where Riemann surfaces, algebraic geometry, modular functions, knots, quantum field theory, Kac-Moody algebras, monstrous moonshine (and more) all come together. And to the examples of things which can be usefully applied let me add the topic 'finite geometry'; a combination of words which sounds like it might not even exist, let alone be applicable. And yet it is being applied: to statistics via designs, to radar/sonar detection arrays (via finite projective planes), and to bus connections of VLSI chips (via difference sets). There seems to be no part of (so-called pure) mathematics that is not in immediate danger of being applied. And, accordingly, the applied mathematician needs to be aware of much more. Besides analysis and numerics, the traditional workhorses, he may need all kinds of combina torics, algebra, probability, and so on.
In addition, the applied scientist needs to cope increasingly with the nonlinear world and the extra
viii
mathematical sophistication that this requires. For that is where the rewards are. Linear models are honest and a bit sad and depressing: proportional efforts and results. It is in the nonlinear world that infinitesimal inputs may result in macroscopic outputs (or vice versa). To appreciate what I am hinting at: if electronics were linear we would have no fun with transistors and computers; we would have no TV; in fact you would not be reading these lines.
There is also no safety in ignoring such outlandish things as nonstandard analysis, superspace and anticommuting integration, p-adic and ultrametric space. All three have applications in both electrical engineering and physics. Once, complex numbers were equally outlandish, but they frequently proved the shortest path between 'real' results. Similarly, the first two topics named have already provided a number of 'wormhole' paths. There is no telling where all this is leading - fortunately.
Thus the original scope of the series, which for various (sound) reasons now comprises five subseries: white (Japan), yellow (China), red (USSR), blue (Eastern Europe), and green (everything else), still applies. It has been enlarged a bit to include books treating of the tools from one subdiscipline which are used in others. Thus the series still aims at books dealing with:
a central concept which plays an important role in several different mathematical and/or scientific specialization areas; new applications of the results and ideas from one area of scientific endeavour into another; influences which the results, problems and concepts of one field of enquiry have, and have had, on the development of another.
The present volume in the series is a book about two things, maybe two and a half. The two are: the theory of the Mellin transform, a very useful integral transform that, till now, has had no systematic treatment (in more than one dimension), and Fuchsian type singular differential equations, the subject of Chapter III which includes the authors' own important results. The half is an appendix on Ecalle's resurgent functions, a most significant topic in my view, which can do with a few extra clear expositions here and there.
This is deep and up-to-date mathematics at the cutting edge of research, but, thanks to the authors, still accessible to all those with a standard background. That, as one of my teachers once remarked, is a sign of good research mathematics; within a few years of when they were obtained, the results should be explain able to graduate students. All this gives me something like 3/1.2 reasons to welcome this volume in this series, and I do so with pleasure.
The shortest path between two truths in the real
domain passes through the complex domain.
J. Hadamard
nous fait pressentir la solution.
H. Poincare
the only books I have in my library are books
that other folk have lent me.
Anatole France
The function of an expert is not to be more right
than other people, but to be wrong for more
sophisticated reasons.
David Butler
Michiel Hazewinkel
Chapter I. Introduction
§1. Terminology and notation ........................................... . §2. Elementary facts on complex topological vector spaces ............... .
1. Multinormed complex vector spaces and their duals ............. . 2. Inductive and projective limits .................................. . 3. Subspaces. The Hahn-Banach theorem ......................... .
Exercise
§3. A review of basic facts in the theory of distributions ................. . 1. Spaces DK and (DK)' .......................................... . 2. Spaces D(A) and D'(A) .......................................... .
3. Spaces S and S' ................................................ . 4. Spaces E and E' ............................................... . 5. Substitution in distributions. Homogeneous distributions ........ . 6. Classical order of a distribution and extendibility theorems for
distributions ................................................... . 7. Convolution of distributions 8. Tensor product of distributions
Exercises
§4. The Fourier and the Fourier-Mellin transformations ................. . 1. The Fourier transformation in S' ............................... . 2. The Fourier-Mellin transformation in the space of Mellin
distributions with support in R+. ............................... . Exercises ........................................................... .
§5. The spaces of Mellin distributions with support in a polyinterval 1. Spaces Ma «0, t]) and M~ «0, t]) ............................... . 2. Spaces M(w) «0, t]) and M(w) «0, t]) ............................ .
Exercises
§6. Operations of multiplication and differentiation in the space of Mellin distributions
1 5 5 7 8 9
9
x CONTENTS
1. Multiplication and differentiation in M a , M(w) and their duals 2. Mellin multipliers .............................................. .
Exercises
52
§7. The Mellin transformation in the space of Mellin distributions ....... . 54 1. The Mellin transformation in the space of Mellin distributions
and its relations with the Fourier-Laplace transformation ....... . 2. Examples of Mellin transforms of some functions ................ .
3. Mellin transforms of certain cut-off functions .................... .
3.1. One-dimensional smooth cut-off functions .................. . 3.2. n-Dimensional smooth cut-off functions with a parameter
Exercises
55 60
67 71 73
76 1. Characterizations of Mellin distributions .......................... 76 2. Substitution in a Mellin distribution 3. Mellin order of a Mellin distribution
Exercises
82
§ 10. Mellin transforms of cut-off functions (continued) . . . . . . . . . . . . . . . . . . . 100
1. Conical cut-off functions . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . 100 2. The K -inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 3. The "tangent cones" EK and related cut-off functions .......... 106 4. Further investigation of the Mellin transform of a conical cut-off
function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Exercises 114
1. Subspaces M(~) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
2. Subspaces SPr(s, s') of Mellin distributions ..................... 119
3. Spaces M(nj e) and Zd(nj e) of distributions with continuous radial asymptotics . . . . . . . . . . . .. . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . 121
Exercises 124
§12. The modified Cauchy transformation 125 1. Modified Cauchy and Hilbert transformations in dimension 1 125
2. The case with parameters ...................................... 128 Exercises 137
CONTENTS Xl
§13. Fuchsian type ordinary differential operators . . . . . . . . . . . . . . . . . . . . . . . . 139 1. Asymptotic expansions . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . 139 2. The equation P( x ddx)u = f and definition of ordinary Fuchsian
type differential operators . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . 144 3. Case of smooth coefficients ..................................... 146 4. Case of analytic coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 5. Special functions as generalized analytic functions . . . . . . . . . . . . . . 162
Exercises 174
§14. Elliptic Fuchsian type partial differential equations in spaces M(:) 175 1. Existence and regularity of solutions on tangent cones 5 K 176 2. Case of a proper cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
Exercise 187
§15. Fuchsian type partial differential equations in spaces with continuous radial asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
1. The radial characteristic set Chaq P . . . . . . . . . . . . . . . . . . . . . . . . . . 190 2. Regularity of solutions in spaces M(n; e) and Zd(n; e) ......... 196
Appendix. Generalized smooth functions and theory of resurgent functions of Jean Ecalle ................................................ .
1. Introduction 2. 3.
Generalized Taylor expansions ................................ . Algebra of resurgent functions of Jean Ecalle .................. .
4. Applications .................................................. .
205
PREFACE
The purpose of this book is to provide a systematic introduction which paves the way to the results of Chapter IlIon Fuchsian type singular differential equations. This chapter consists of 3 sections. The first of these, Section 13, is devoted to ordinary Fuchsian equations. It contains known results as well as their extensions recently obtained by the authors. Elliptic Fuchsian partial differential equations are treated in Sections 14 and 15. Section 14 presents the authors' results ([Sz Ziel]' [Sz-Zie2), [Sz-Zie3)) concerning the local existence and regularity of solutions in suitable weighted spaces. However, it is only Section 15 that presents a complete geometric regularity result for such solutions, obtained in the recent papers [Zie4], [Zie5], [Zie6) of the second author.
The main tool used in Chapter III is the Mellin transformation, and in Sec tion 15 also the generalized Cauchy and Hilbert transformations originating from [Zie3) and presented in Section 12.
Ordinary Fuchsian differential operators, their generalizations and "parameter versions" have been investigated by many authors in different contexts (see [EI], [Rem-S], [T), [B-G), [Le-P], [Ml), [W], [Mel-Men) to quote but a few). It should be underlined that the operators which we study in dimensions higher than 2 are not of the Baouendi-Goulaouic type [B-G). They arise naturally, for example, as Laplace-Beltrami operators on manifolds with corners, and fall within the scope of the global theories developed in [Mel). Systems of such operators (holonornic systems) were considered in [K-K), see also [Schl).
Due to numerous applications of the Mellin transformation (see e.g. [I), [Le P], [La-M], [Mel-Men), [Mel], [Z], [C-L], [P-B-M], [Sn], [Wi], [OJ) and the lack of text-book treatment of the Mellin transformation in several dimensions, we give its systematic presentation in Chapter II. In contrast to the classical texts on the Mellin and Laplace transformations (see for instance [Sm], [Wi), [C-L), [La-M), [Z)) we are interested in the local properties of the Mellin transforms, i.e. those properties of the Mellin transform of functions f which are preserved under multiplication of f by cut-off functions (of various types). We treat only certain aspects. Other important topics, including the relation with the theory of resurgent functions of J. Ecalle, are outlined in the Appendix.
The reading of the book requires no knowledge which exceeds the first 3 years of university studies of mathematics. Among the theorems which may not be included in standard university courses and whose proofs are not given in the text are the "division theorem" and the theorem on the Fourier transform of a convolution. We have confined ourselves to stating such theorems and indicating the pertinent references. Basic theorems and definitions of the theory of distributions and of
PREFACE xiv
the Fourier transfonnation used in the book are collected in Chapter I with proofs partially transformed into exercises with hints. The proofs omitted can be found in most textbooks on distributions (e.g. [Szl]).
Sections 12 and 15 are more difficult due to the application of more advanced tools of complex analysis.
The book originated from a seminar conducted by the authors in the years 1983-1990 at Warsaw University. The authors wish to thank Dr. G. Lysik for his active participation in the seminar and valuable help in the preparation of the book.
The text was typeset in '!EX at the Institute of Mathematics of the Polish Academy of Sciences in Warsaw.
Chapter I
§1. TERMINOLOGY AND NOTATION
We employ the usual notation of set theory. The union, the intersection and the difference of sets A and B are denoted by AU B, An B, A \ B, respectively. 0 is the empty set. We write a E A if a is an element of A, otherwise we write a rt A. The notation A C B (or B :> A) means that A is a subset of B.
The set of all elements of a set A satisfying condition R( .) is denoted by {a E A: R(a)}, or more concisely {a: R(a)} or {R(a)} if the set A is fixed and no confusion is likely to arise.
The product of sets A and B is the set A X B consisting of all ordered pairs (a, b) with a E A, bE B.
The notation
f: A~B or A :3 a 1-+ f(a) E B
means that f is a function of the set A into the set B. A functional is a number valued function. The supremum of a real-valued function on a set A is denoted by supu or supu, or supu(a). Similarly the infimum of u on A is written as infu or
A aEA infu, or inf u(a). Analogous notation is employed for maxu and minu on A. A aEA
If f: A ~ B is any mapping and if A is a subset of A, then the restriction of f from A to A, denoted by f11' is the mapping: A:3 a 1-+ f(a) E B.
If f: A ~ B, g: B ~ C, then the superposition of f and g, i.e. the mapping A:3 a 1-+ g(f(a)) E C, is denoted by go f.
Suppose that T is a function defined in the product of two sets A, B and has values in a set C:
A X B :3 ( a, b) 1-+ T( a, b) E C.
1
2 I INTRODUCTION
Then for any fixed values b E B, a E A there are well defined mappings
A:3 a 1--+ T(a, b) E C, B:3 b 1--+ T(a,b) E c.
We denote these mappings by the symbols T( " b), T(a, . ), respectively. R denotes the set of real numbers. We denote by R U {oo} the right-sided
compactification of R. The n-dimensional Euclidean space - denoted by Rn - is the product of n copies of the set R. We also consider (R U {oo})n, i.e., the set of all points x = (Xl,"" x n ), where xl,"" Xn are real numbers or 00. By ei we denote the vector
(1) with e1 = 0 for j =/: i, e~ = 1.
The norm IIxll of a vector X = (Xl,'" ,Xn ) E Rn is given by
We also denote (x) = l+lxII+" ·+Ixnl. If X = (Xl"'" Xn) E Rn, Y = (YI,"" Yn) E Rn , we write
xy = XIYI + ... + XnYn
and call this number the scalar product of X and y. The smallest closed set containing a given set A C Rn is called the closure of A
and is denoted by if. The boundary of A is the set vA = An Rn \ A. The biggest open set contained in A is called the interior of A and denoted by Int A.
We call a set Q C Rn connected, if Q admits no decomposition of the form Q = A U B with An B = 0 = A n B.
Any set of the form
{( X I , ... , X n ): ai < Xi < bi for i = 1, ... , n},
where al, ... , an, bl , ... , bn are given real numbers or ±oo with ai < bi for i 1, ... , n is called an open polyinterval denoted by (a, b). Any set of the form
{(X}, ... ,xn): -00 < ai ~ Xi ~ bi < +00 for i = 1, ... ,n}
is called a closed polyinterval denoted by [a, bJ. In an analogous way, we define polyintervals (a, bJ and [a, b).
N denotes the set of positive integers (natural numbers), No is the set of non negative integers, and N~ is the set of all multi-indices a = (aI, ... ,an), ai E No for i = 1, ... ,n. We write:
§l. TERMINOLOGY AND NOTATION 3
Let a, (3 E N~, a = (al, ... ,an ), ((31, ... ,(3n). The Newton symbol
( n) n! k - k!(n - k)!
for n, kENo, n ~ k
extends to multi-indices as follows:
(;) = (;:) ..... (;:) for a,(3EN~, aj~(3j for j=l, ... ,n.
By Z we denote the set of integers, e denotes the set of complex numbers z = a + i b, a, b E R. We write z = a - ib (the conjugate of z), a = Rez (the real part),
b = Imz (the imaginary part), Izl = va2 + b2 (the modulus of z). If z E en we write Rez = (Rezt, ... ,Rezn) and similarly Imz = (ImZl, ... ,Imzn). We also define
which agrees with the convention for x E Rn. For convenience, we recall the basic notation and facts of the theory of the
Lebesgue integral. By £leA) we denote the space of integrable functions on a measurable set A C Rn and by L2(A) the space of square integrable functions. The respective norms are denoted by 11·11£1 and 11·11£2' The symbol LOO(A) stands for the space of essentially bounded measurable functions on A.
The Lebesgue dominated convergence theorem. Let A be a measurable set in Rn, let {fv} be a sequence of integrable complex-valued functions defined a.e. on A and convergent a.e. to a function f. Suppose that there exists an integrable function 9 ~ 0 which is a common majorant for all fv's: Ifvl ~ 9 (II = 1,2, .. .). Then f is integrable and we have
[ f dx = lim [ f v dx. JA v-+oo JA
The Fubini-Tonelli theorem. Let A C Rm , BeRn, m, n E N, be measurable sets and let f: A x B --+ C be a measurable function. Suppose that either (i) f is integrable, or (ii) f is positive-valued. Then the following three quantities exist and are equal:
(In case (ii) their common value may be infinite).
4 I INTRODUCTION
Throughout the book we use the following vector notation: if a, bERn, a =
(al, ... ,an), b = (bl, ... ,bn) then a < b (a:::; b, resp.) denotes aj < bj (aj :::; bj , resp.) for j = l, ... ,n. We denote R+. = {x ERn: 0 < x}, R_ = {x E R: x < O}, 1= (O,t] = {x E R+.: x:::; t} where t E R+.. B(x,r) = {y ERn: Ily-xll < r}, r > O.We write shortly B(r) instead of B(O,r) and put B = B(l), B+ = B n R+. sn-l = {x ERn: IIxll = I} , S+-l = sn-l n R+..
If r E R we write r = (rl, ... ,rn) E Rn where rl = r2 = ... = rn = r. For instance 1 = (1, ... ,1) E Rn and e = (e, ... , e) E Rn for the constant e.
For x E R+. and Z = (Zl' ... ' zn) E en we write
Thus for y = (Yl, ... , Yn) E Rn, e-Y = e-Y1 ••••• e-Yn where R :3 r I-t er E R+ is the exponential function. We also denote e-Y = (e- Y1 , ••• , e-Yn ) for Y E Rn and similarly, if x E R+., In x = (In Xl, ... , In xn) for the logarithmic function In r inverse to er . In particular for x E R+. and a EN;: (lnxy' = (lnxdOl ••••• (lnxn)On. Vector notation is also used for differentiations. Namely we write
tx = (8~1 ' ... , 8~n) , and if v E N; then
For points a E Rn we write a = (aI, a') where al E R, a' E Rn-l, similarly for (E en, (= ((1,('), (1 E e, (' E en-I, we also consider sets Ween of the form W = WI X W' where WI c e, w' c en-I. For a set Ween and a vector a E Rn
we write W + a = {z E en: Z - a E W}. Let U be an open subset of Rn. Let f be a real-valued or complex-valued
function defined in U and let m be an integer, m ~ O. We say that f is of class cm in U (in short, f is a Cm(U)-function) iff all the derivatives (txt f, lal :::; m, exist and are continuous functions in Uj f is of class COO(U) (smooth function) iff it is of class Cm(U) for all m E No. By Cgo(U) we denote the set of all functions in COO(U) vanishing outside a compact set.
Among all COO-functions on U we distinguish the important class A(U) of analytic functions: a (real- or complex-valued) function f is analytic in U iff for every point x E U there exists r > 0, so that f can be expanded into a power series:
f(x) = L ao(x - xY' for IIx - xii < r. oEN~
§2. ELEMENTARY FACTS ON COMPLEX TOPOLOGICAL VECTOR SPACES 5
Similarly, if il c en we denote by V(il) the set of holomorphic functions on il i.e. the functions F such that for every point i E il, F can be expanded into a power series
F(z) = L aa(z - i)a aEN~
convergent for liz - ill < r for some r > 0 (depending on i).
§2. ELEMENTARY FACTS ON COMPLEX TOPOLOGICAL VECTOR SPACES
In this section, we recall some basic notions and theorems, referring the reader to [SzI] for the proofs of the theorems quoted.
L Multinormed Complex Vector Spaces and Their Duals
Let P be a complex vector space. A functional q on P is called a seminorm provided the following two conditions hold:
1) q( ~ + Tf) ~ q( 0 + q( Tf ) for ~, Tf E P,
2) q(A~)=IAlq(e) for eEP,AEC.
A seminorm q is a norm if and only if q( 0 = 0 for a e E P implies e = O.
We equip P with a convergence topology defined by a sequence of seminorms {qk}k:,o' Since P is a linear space it is sufficient to define the convergence to zero.
Definition 1. Let P be a complex vector space and {qk}k:,O a sequence of semi norms on P. We say that a sequence {e8}~1 of elements of P converges to zero if
(1) (k = 0,1, ... ).
The space P with the convergence defined above is denoted by p{qk} or by P for short. It is called a multinormed space.
It is easy to note that a necessary and sufficient condition in order that every convergent sequence had a unique limit, is that for every ~ E P, e =f 0 there exist kENo such that qkCe) =f O. In particular, this is the case if at least one of the seminorms qk is a norm. In the following, we consider only the sequences of semi norms possessing the above property.
6 I INTRODUCTION
It can be proved (see [H] §§ 3 and 4) that P is a Hausdorff topological space.
The sequences of seminonns {qk} considered in the sequel will be increasing in the sense that
for ~ E P (k = 0,1,2, ... ).
This can always be achieved by replacing an arbitrary sequence of seminorms {qk} by the sequence {r d:
k
without affecting the topology of P.
The theorem below characterizes linear continuous functionals on P with values in C.
Theorem 1. A linear functional Ion P{qk} is continuous if and only if there exists a constant C < 00 and kENo such that
for ~ E P.
We recall the Banach-Steinhaus theorem in the framework of multinormed spaces.
Theorem 2. Suppose P is a complete multinormed space. Let {IT }TET, where T is a set of indices, be a family of continuous linear functionals on P. We assume that for every ~ E P the set {fT(~): T E T} is bounded in C. Then there exists a constant C < 00 and kENo such that
for ~ E P (T E T) .
Corollary 1. Let P be as in Theorem 2. Let {l1/} ~1 be a sequence of continuous
linear functionals on P. Suppose that for every ~ E P the limit 1(0 ~f lim 11/(0 1/-+00
is finite. Then there exists a C < 00 and kENo such that
for ~ E P (v = 1,2, ... ).
By pI we denote the space of continuous linear functionals on P with the topology of pointwise convergence. It is called the dual space of P. Observe that it follows from Corollary 1 and Theorem 1 that the space pI is complete.
§2. ELEMENTARY FACTS ON COMPLEX TOPOLOGICAL VECTOR SPACES 7
In applications we shall need the following theorem on the separate continuity of 2-linear functionals
Theorem 3 (cf. [Ho2j or [Th]). Let PI, P2 be two complete linear topological spaces with the topologies given by non-decreasing sequences of seminorms q~ (i = 1,2; k = 0,1, .. .). Let I be a separately continuous 2-linear functional on PI X P2
i.e. such that for every fixed 6 , 1(6,') E P~ and for every fixed 6, Ie-. 6) E Pf· Then I is continuous as a mapping from PI X P2 into C, i.e. for some constants C < 00 and kENo
2. Inductive and Projective Limits
Let {Pr }rET be a family of multinormed vector spaces. By the inductive limit P = lim P r of the spaces P r we understand the vector
;:e1 space P = U Pr with the convergence topology defined as follows:
rET
A sequence es E P (8 = 1,2, ... ) is convergent to zero if there exists a TO E T such that es E Pro (8 = 1,2, ... ) and for every Pr such that all es E Pr we have lim es = 0 in P r .
s .... co
By the projective limit R = lim Pr of the spaces Pr we understand the vector f-- rET
space R = n Pr with the following convergence topology: rET
A sequence es E R (8 = 1,2, ... ) tends to zero if es ---t 0 as 8 ---t 00 in every Pr, T E T.
Note that I E pI (= the dual space of P with the pointwise topology) if and only if it is linear on P and its restriction to any Pr is in (Pr ), .
From Corollary 1 we get:
Corollary 2. Suppose P is an inductive limit of complete multinormed spaces P r .
Let Iv E pI and let the limit J(e) = lim Iv(O exist for every e E P and be finite. v .... co
Then I E P'.
8 I INTRODUCTION
In Section 3 we present examples of complex multinormed vector spaces and their inductive limits. Some of them are well-known from the theory of distributions.
3. Subspaces. The Hahn-Banach Theorem
We consider pairs Q, P of topological vector spaces (not necessarily multinormed) with topologies given by convergence of sequences to zero. We write Q c P if for the underlying sets we have Q C P and the convergence to zero in Q implies the convergence to zero in P. If f E pI we denote by Zf = flQ the restriction of f to Q.
Proposition 1. Suppose Q C P and the set Q is dense in P. Then Z(PI) C Q' and the restriction mapping Z: pI -+ QI is a homeomorphism onto the image. We identify pI with Z(PI) under this bijection and write pI C Q'.
Remark 1. Clearly if Q c P and f E pI then flQ E Q'. However to conclude that pI C Q' it is essential to assume the density of Q in P.
Theorem 4 (The Hahn-Banach extension theorem; [Rul]). Let Q C P be topolo gical vector spaces. Let q be a seminorm on P and f E Q' be such that
for e E Q.
Then there exists a linear functional j on P which extends f, i.e.
for e E Q
and satisfies the inequality
for e E P.
Remark 2. The extension j is in general, not unique. However if Q is dense in P the uniqueness holds. If, in addition, P is multinormed then the proof of the Hahn-Banach theorem becomes very simple.
Corollary 3 ([Tr]). It follows from Theorem 4 that if Q c P are multinormed vector spaces and eo E P is not in the closure of Q in P then there exists a functional f E P' such that f(eo) = I and fee) = 0 for e E Q.
Corollary 4. H Q is a topological subspace of a multinormed vector space P then P' is a subspace of Q' (see Proposition 1) if and only if Q is dense in P. Equivalently, Q is not dense in P if and only if there exists an f E P' , f =1= 0 such that flQ = O.
§3. BASIC FACTS IN THE THEORY OF DISTRIBUTIONS 9
Exercise
1. Let A be a subset of a vector space P with topology defined by a sequence {qdk::l of seminonns. Prove that a point e belongs to the closure of A if and only if, for every c > 0 and kENo there exists a point eEk E A such that qk( e~k - e) < c.
§3. A REVIEW OF BASIC FACTS IN THE THEORY OF DISTRIBUTIONS
1. Spaces DK and (DK)'
We denote by C'K the set of smooth functions on Rn with supports in a compact set J{. Observe that the formula
for tpEC'K (k=0,1, ... )
defines an increasing sequence of norms on C'K. The set C'K equipped with the topology defined by this sequence of (semi )norms is denoted by D K. The space D K
is complete (see e.g. [Sz1] or Proposition 5.1 for a similar proof). Let u E (D K )' . The value of u on a function tp E C'K is denoted by u [tp]. It
follows from Theorem 2.1 that a linear functional u on C'K belongs to (DK)' if and only if one of the following equivalent conditions holds:
Condition WI: If tpll E C'K (v = 1,2, ... ) and lim IlItplllllk = 0 (k = 0,1, ... ) 11--+00
then lim U[tpll] = O. 11--+00
Condition W 2: There exist constants C < 00 and p E No such that
for tp E C'K.
Corollary 2.1 asserts that (D K )' equipped with the pointwise convergence topology is complete.
2. Spaces D(A) and D'(A)
Let A C Q c Rn be such that Q is open in Rn and A is relatively closed in Q. We denote by C(o')(A) (COO(A), resp.) the space of restrictions to A of functions in Cg<'(Q) (COO(Q), resp.). We equip C(o')(A) with the inductive limit topology as follows:
10 I INTRODUCTION
where K ranges over all compact subsets of Q and DKIA denotes the space of restrictions to A of elements of D K with the topology induced from D K .
The dual space D'(A) is called the space of distributions on A (see Exer cises 1, 2, 3).
The convergence in the space D' will always be understood as a pointwise convergence. Hence by Corollary 2.2 the limit of a sequence of distributions is itself a distribution.
Note that if A is open then we take A = Q and D'(A) is the "usual" space of distributions on an open set (see Exercise 6).
In applications we take A = (0, tj C R~ , or A = [0, tj C Rn with t > 0, tERn. To verify that a linear functional u on D(A) is a distribution it is enough to
check (in view of the inductive limit topology and Condition WI) that:
(i) for every sequence {If>i}~l of functions in Co(Q) with supports in a
compact set K C Q and such that .lim (: t If> i = ° uniformly on Q for ,-"00 x
a E N~, we have
or (in view of Condition W2 ) that:
(ii) for every compact K C Q
lu[c,oIAlI::; C(K) L sup \ (txf c,o(x) \ JaJ$k(K) xEK
for some constants C(K) < 00, k(K) E No.
for c,o E CK
If the set A is sufficiently regular then the inequality in (ii) can be improved so that the supremum is taken over the set A (see Theorem 1 below). We prove this fact in the case of a polyinterval (0, tj (See Theorem 8.3 and Exercise 8.8).
Definition 1. A distribution u is said to be of finite order on A iff there exists a kENo such that k(K) ~ k for every K C Q (see (ii) above). The smallest k satisfying this inequality is called the order of the distribution u on A.
As it is well known distributions in D'(Q) enjoy the localization property: u E D'(Q) is zero on Q if u is zero in a neighbourhood of every point in Q. This property allows us to define the support of u E D'(Q) (denoted by suppu) as the largest relatively closed subset of Q such that u is zero on its complement in Q.
Denote by D~(Q) the space of distributions on Q with support in A. We have the following characterization of D~ (Q):
Proposition 1. The spaces D~(D) and D'(A) are isomorphic (denoted D~(D) ~ D'(A)) under a canonical linear isomorphism L: D~(D) -+ D'(A), given by the assignment
D~(D) :3 u I-t U E D'(A),
where u[epIA] = u[ep] for ep E Co(D) with u E D'.4(D).
Proof: We start by showing that the definition of u is correct. To this end we have to verify that if ep E Co(D) and ep = 0 on A then u[ep] = o. We take a E Co(Rn ),
a = 1 on suppep. Then ep = aep, u[ep] = au[ep]. Thus by Exercise 8 applied to au we have u[ep] = o.
It is clear that u E D'(A) and that the assignment u I-t u is an isomorphism.
In view of this isomorphism we often identify u and u.
Theorem 1. Let u E DK(Rn ) where J{ is a connected compact set in Rn such that any two points x, y E J{ can be joined by a rectifiable curve in J{ of length ::; Clx - YI, C < 00, then there exists a constant C < 00 and kENo such that
lu[1f1] I ::; C 2: sup I(txr 1f1(x) I lal~kxEK
The proof of this theorem, based on Whitney extension theorem, is given in [Ho 2].
As examples of distribution we note:
(a) the Dirac delta o(y) at a point yEA defined as
o(y)[ep] = ep(y) for ep E C~)(A),
(b) locally integrable functions are imbedded into distributions by means of the identity
f[ep] = L f(x)ep(x) dx
if f E Lloc(A).
(c) We introduce a functional Pf ~ by the formula
Pf -[ep] = lim + ep x dx 1 J- e 1+00 ()
x e ..... O+ -00 e X for ep E D(R).
Let ep E Co(R), ep(x) = 0 for Ixl ~ R. Write cp(x) = ep(O) + xep(x) then
I Pf .![ep]1 = I (R ep(x) dxl ::; 2R sup lep'(x)l, x LR Ixl<R
which shows that Pf ~ is a distribution on R of order 1. Clearly for x -=1= 0 it coincides with the function ;.
12 I INTRODUCTION
The spaces of distributions are closed under differentiation which is defined as (see Exercise 4)
(1) for u E D'(A), 'P E D(A)
and under multiplication by Coo (A)-functions a defined as
aU['PJ = u[a'PJ.
We shall prove now a theorem which characterizes distributions supported by a single point {y}.
Theorem 2. u E D{y} (Rn) if and only if it is a tinite linear combination of the Dirac distribution Sty) and its derivatives.
Proof: Let u E D'(Rn ), suppu = {y} and let k ~ order of u. By the Taylor formula for 'IjJ E Coo there exists a function a E Coo such that
'IjJ(x) = L ~! ((txr 'IjJ) (y)(x - yt + a(x), lal9
(txr a(y) = 0 for 10:1 ~ k.
Thus by Theorem 1 (or equivalently by Exercise 8) ural = 0 and by the linearity of u we get
[ (._y)a] where aa = u o:! .
Since 'IjJ E Coo was arbitrary the above formula denotes precisely that
_" lal (a)a u - ~ (-1) aa ax Sty)·
lal9
The second part of the theorem is obvious.
Certain distributions can be obtained as boundary values of holomorphic func tions. Having in mind applications to Sections 13 and 15 we state the following theorem which also generalizes the classical theorem of Painleve:
Theorem 3. Let Q be an open subset of R and V an open subset in C such that V n R = Q. Let F E O(V \ Q) be such that for every compact set K c Q there exist constants N = N(K) E No, C = C(K) < 00 such that
(2) IF(a + i,8)1 ~ C 'I,BI-N for a E K, 0 < 1,81 ~ 1], 1] small enough.
Then there exist distributions F(. ±iO) E D'(Q) (called the boundary values of F) such that
F(·±iO)[<p] = lim [F(a±ie)<p(a)da, t:-+O+ in <p E COO(Q).
Moreover if F( . + iO) = F( . - iO) then F extends holomorphically to V.
We denote the difference F( . + iO) - F(. - iO) by b(F) and call the jump of F across R.
Proof: We may assume that Q is an open interval. Let K be a fixed closed subinterval of Q. Choose 0 < 1] < 1 so that the set
A = {z = a + i,8 E C: a E K, 0 <,8 ~ 1]} c V+ = V n {Imz > O}
and (2) holds for a + i,8 E A. Let z be a fixed point in A with 1m z = 1]. Define
J+F(z) = 1 F(B) dB, "Y.
where 'Yz is the curve in V+ joining z to z = a + i,8 and consisting of two segments [z,a + i1]] and [a + i1],a + i,8]. Clearly fzJ+F(z) = F(z) for z E V+ and from (2) and the fact that F is bounded on K + i1] we get for z E A
for z E A.
Hence there exists a constant C1(K) such that for a + i,8 E A
Iterating the above operation N + 1 times we find J!;!+lF E O(V+) such that
ddz~+:l J!;!+! F = F on V+ and J!;!+l F(a + ie) converges uniformly, as e ~ 0+, to a continuous function F.f+1(a) on K. Thus J!;!+l F extends to a continuous function on A = {z E C: a E K, 0 ~,8 :::; 1]}.
14 I INTRODUCTION
Let cp E CQ"(Q), suppcp C 1<. Integrating by parts we get from the Lebesgue theorem
lim f F(a+ic:)cp(a)da= lim (_1)N+1 f J.f+1F(a+iC:)(dd )N+1cp(a)da .<-+0+ in .<-+0+ in a
= (_1)N+1 In F.f+1(a) (ia)N+1 cp(a) da.
Thus the limit F( a + iO) exists and is a distribution. By symmetry one defines J_F and J~+1 F on V_ = V n {Imz < O} and one proves the existence of the limit F(a - iO). Now if F(a - iO) = F(a + iO) it follows (see Exercise 10.1 in [Szl]) that
F.f+1(a) - F~+1(a) = w(a)
where w is a polynomial of degree at most N. Hence by the classical Painleve theorem [Ko] we infer that the function
{ J.f+1 F(z) for z E V+
F N+1(z) = F.f+1(a) for z = a
J~+1 F(z) + w(z) for z E V_
is holomorphic on V. Hence (lz)N+1 FN+1 is the desired extension of FE O(V\Q) to V.
Example 1. With Pf ~ being the distribution of example (c) we have
lim _±1. = ~i7rb'o + Pf .! .<-+0+ X ze: x
and hence
I.e. 1 (-1) -.b - = b'o. 27rz Z
Proof: Let cp E CQ"(R), cp(x) = 0 for Ixl > R. Then
li J cp( x) d l' jR x - ie: () d m -- x = 1m cp x x .<-0+ X + ie: e-+O+ -R x 2 + t:2
j R x - it: = cp(O) lim 2 2 dx+
e-+O+ -R x + t:
j R X - ic: + lim 2 2(cp(x)-cp(0))dx e-+O+ -R X + e:
= -2icp(0) lim arctg R + jR cp(x) - cp(O) dx e-+O+ t: -R X
= -i7rcp(O) + Pf '![cp]. x
The proof for _1_. is analogous. x-Ie
3. Spaces Sand S'
By S = S(Rn) we denote the set of functions u E coo(Rn) such that
(3) (k = 0,1,2, ... )
is finite. A function u E S is called a rapidly decreasing function. It is clear that
Cgo eSc Coo
and the set S is indeed bigger then Cgo since it contains e.g. the function Rn :3 x ~ e- lIxll2 which is not in Co.
It is easy to note that qk is an increasing sequence of (semi) norms on S, and S endowed with the topology defined by the sequence of norms qk (denoted by the same letter S) is complete (see e.g. [Szl] Exercise 18.1). According to the general scheme we define the dual S' = S'(Rn) of S(Rn). It follows that if u = lim U v in
v-+oo
S' then there exist constants C < 00 and kENo such that for all v
lu[ull, luv[ull ~ C L sup (1 + IIxll2)k I (txf u(x)1 for u E S. lal:5k
Further, the convergence in D implies the convergence in S which we write as DeS. Moreover, Cgo is dense in S. These facts imply, in view of Corollary 2.4, that S' c D' i.e. S' is a subspace of the space of distributions. We call it the space of tempered distributions. As in the case of (D K ) " Corollary 2.1 implies that S' equipped with the pointwise topology is complete.
The following theorem characterizes tempered distributions.
Theorem 4. A distribution u E D'(Rn) is tempered if and only ifit is a derivative (:x r, for some a E N~, of a Co -function slowly increasing at infinity i.e. of a
continuous function f majorized by the function x ~ C (1 + Ilxll2) k for some kEN:
The "if" part of Theorem 4 is clear: any CO-function slowly increasing at in finity is a tempered distribution and so is any derivative of a tempered distribution. The "only if" part can be derived from the Hahn-Banach theorem (see for example [Tr] or Exercises 10-12).
16 I INTRODUCTION
We end this subsection with the famous "division theorem" which emphasizes the importance of the spaces S':
Theorem 5 ([L1l. [L2], [HoI]). Let u E S'(Rn) and P be a polynomial in Rn. Then there exists a distribution f E S'(Rn) such that p. f = u on Rn.
Some simple cases of the division problem are indicated in Exercises 14-16.
4. Spaces E and E'
By E(Rn) we denote the set of functions (J' E coo(Rn) and define
(k = 0,1,2, ... ).
It is easy to note that {qk} is an increasing sequence of seminorms on E. Hence the space E endowed with the topology defined by this sequence of seminorms (denoted by the same letter E) is complete (see e.g [Szl], Theorem 7.2). Further ego is dense in E and the convergence in D implies the convergence in E hence in view of Corollary 2.4 it follows that E' c D'. More precisely it can be shown that E' is the subspace of D' consisting of all distributions with bounded supports (cf. e.g. [Szl], Theorem 7.1). We call it the space of distributions with bounded support. Clearly E' C S' as topological spaces.
5. Substitution in Distributions. Homogeneous Distributions
Let A C il C Rn be such that il is open in Rn and A is relatively closed in il. Let x = f(y) be a one-to-one mapping of a ill C Rn onto il, of class Coo with a non-vanishing Jacobian Jf = det *. One easily checks that Al = f-I(A) is relatively closed in ill and that for every u E D'(A) the formal definition
(4)
defines u 0 f E D'(At}. Moreover the mapping D'(A) 3 u 1--+ U 0 f E D'(At} is an isomorphism of D'(A) onto D'(AI ).
In the case where u is a continuous function on A the distribution u 0 f is a continuous function Al 3 Y 1--+ U (f(y)).
Definition 2. A distribution u E D'(Rn ) is called homogeneous of order oX if and only if for every homothety Rn 3 y 1--+ f(y) = ry (r > 0) we have
f - A u 0 - r u.
It can be proved that a distribution u E D' (Rn) is homogeneous of order A if and only if u satisfies the Euler equation (see e.g. [Szl] Exercise 9.7)
Example (Homogeneous distributions on R cf. [G-S]). For A E C ReA> -1 we define x~ as the regular distribution (=function)
x~[cp] = 100 xAcp(x) dx for cp E Cge'(R).
Subject to a suitable regularization at zero, this definition extends to all A ft -N:
100 ( n-l (i)(O) ) x~[cp] = 0 x A cp(x) - t; t xi dx for cp E Cge'(R)
if n is such that -n - 1 < Re A < -no By means of the substitution x 1--+ -x
we define the distributions x~. Observe that for A ft N x~ E D~+ (R), x~ E
D'---- (R) are homogeneous of order A. The distributions x~ do not exhaust the list R_
of homogeneous distributions on R (see [G-S]). For example t5~n) is homogeneous of order -n - 1 and the distribution Pf ~ of example (c) is homogeneous of order -1.
6. Classical Order of a Distribution and Extendibility Theorems for Distributions
Let Q be an open set in Rn. By Definition 1 (cf. also Exercise 6) u E D' (Q) is of finite order::; k on Q, kENo if and only if for every compact K C Q there exists a constant C(K) such that
(5) for cp E C'K, where
IIlcplllk = L sUPn I (Ix r cp(x)\. lol:::;k zER
(6)
Observe that the distribution defined by a locally integrable function is of order zero and that every tempered distribution is of finite order. It is also clear that:
(7) 00 .
for cp E Cge' ((0,1))
defines u E D' ((0,1)). It can be proved (cf. Exercises 17 and 18) that u is of infinite order.
18 I INTRODUCTION
Proposition 2. Let ilo C il C Rn be open sets and lio a compact subset of il. If u E D'(il) then there exist constants C < +00, kENo such that
(8) for c.p E CgoUZo).
Thus the restriction of u to ilo is a distribution of order ~ k.
Proof: Since u E D'(il), by Exercise 6 there exist constants C k = k (li 0) satisfying (8) as asserted.
C(lio) and
Proposition 3. Let u be a distribution on a bounded open set ilo c Rn and let ill :::) lio. Then the following conditions are equivalent:
(i) u is extendible to ill,
(ii) there exist constants C < +00, kENo satisfying (8),
(iii) the condition Co(ilo) 3 c.p" -t 0 in D(Rn) (equivalently c.p" E Co(ilo) (v = 1,2, ... ), 1I1c.p"llIk ~ 0 (k = 0,1,2, ... )) implies u[c.p"J -t O. ,,-+=
Proof: The implication (i) =? (ii) is just Proposition 2. For the proof of (ii) =? (i) let u satisfy (8). Since 1II·llIk is a norm in Co ( ill) the functional u can be extended by the Hahn-Banach theorem to a linear functional on Co ( ill) continuous in this norm. The extended functional is a distribution on ill' The implication (ii) =? (iii) is clear. The proof of (iii) =? (ii) follows from Theorem 2.1.
Proposition 3 implies immediately
Corollary 1. A distribution u admitting an extension from a bounded open set ilo to a set ill :::) lio is also extendible to the whole of Rn.
By Proposition 2 every distribution on a bounded open set which is of infinite order cannot be extended to Rn hence Exercise 18 implies
Corollary 2. There exists a distribution on a bounded open set which cannot be extended to Rn.
The next proposition concerns the extendibility to Rn of a distribution defined on any open set il C Rn.
Its proof requires the following partition of unity:
Lemma 1 ([Schw]). There exists a locally finite covering of Rn by open bounded sets il" and functions a" E Co(il,,), v E A such that I: a" == 1 in Rn. (Note that
"EA for every x E Rn only finitely many a,,(x) are different from zero).
Proposition 4 ([L2], Proposition 1). Let u E D'(n). In order tbat u be extendible to Rn it is necessary and sufficient tbat u restricted to an arbitrary open set be extendible to Rn.
Proof: The condition is clearly necessary so it is enough to prove its sufficiency. By Lemma 1 there exists a locally finite covering of Rn by bounded open sets nv and a partition of unity O:'v E ego ( nv). Let Uv be an extension to Rn of the restriction of u to nv n n (we take Uv = 0 if nv n n = 0). Since O:'vUv = O:'vU in n we have L: O:'vUv = L: O:'vU = u in n. On the other hand for every test function <p E D(Rn) v v
the sum L: O:'vuv[<p] is finite which easily gives that L: O:'vU E D'(Rn). v v
We end this subsection with the theorem which states that distributions can be regarded as the derivatives of continuous functions.
Theorem 6. Let u be a distribution on a bounded open set n c Rn extendible to Rn. Tben tbere exist a multi-index 0:' and a continuous function G (wbose support may be cbosen to be contained in any prescribed neigbbourbood of ?i) such tbat
u = (txtG on n.
The proof ofthis theorem (see e.g. [Szl]) is sketched here in Exercises 20 and 21.
7. Convolution of Distributions
We begin with the definition of sequences {T/v} of ego(Rn )-functions convergent to one in Rn (written as T/v -+ 1 in Rn) in the following sense (see [V2]):
1° for every compact set J( C Rn there exists N E No such that T/v(x) = 1 for x E J( if v 2:: N,
20 for every 0:' E N~ there exists an Ma < 00 such that
(v = 1,2, ... ).
(9) (v=1,2, ... ),
where T/v -+ 1 in R2n. It is easy to see that Wv E D' (Rn) (v = 1,2, ... ) and hence if {wv } is convergent its limit also belongs to D' (Rn).
20 I INTRODUCTION
Definition 3. We say that a distribution w E D'(Rn ) is the convolution of distri butions u, v E D'(Rn) (denoted w = u*v) if it is the limit in D'(Rn) of distributions Wv (v = 1,2, ... ) defined by (9).
Remark 1. The existence of u * v implies the existence of v * u and u * v = v * u.
Example 2. If u,v E Ll(Rn) we get
(u * v)[cp] = J u(x)v(y)cp(x + y) d(x, y) = J cp(x) ( J v(y)u(x - y) dY) dx
i.e. (u * v)( x) = J v(y )u( x - y) dy which agrees with the clasical definition of the convolution.
Example 3. If u E E'(R n ), v E D'(Rn ) then
(u * v)[cp] = u[v[cp(x + y)]]
Example 4. Assume one of the following conditions:
Then the convolution u * 0' is a COO-function defined by
(0' * u)(e) = u[O'(e - x)]
The following simple properties of convolution will be used later on (here and later on we write 8 for 8(0»): u * 8 = 8 * u = u for u E D'(R n ) and the Dirac delta distribution; if u, v E D'(Rn ) and u * v exists, then for every k = 1, ... , n
for every distribution u E D' (Rn) and every differential operator
with constant coefficients we have
8. Tensor Product of Distributions
Let cp E S(Rn), 't/J E S(Rm). We denote by cp 129 't/J the function
Rn X Rm 3 (x, y) ~ cp(x)· 't/J(y).
Now, given u E S'(Rn), v E S'(Rm) we denote by w = u 0 v a unique element of S'(Rn x Rm) such that
w[cp 0 tP] = u[cp] . v[tP] for cp E S(Rn), tP E S(Rm)
and call it the tensor product of u and v.
Exercises
1. Give an example showing that the space (D K )' defined in Subsection 1 is different from the space D'(K) defined in Subsection 2 (K -a compact set).
2. Let A c il c Rn , A, il open and denote by K a compact set. Show that
if and only if A = il.
3. Let A C il C Rn , il open. Show that
4. Let A be relatively closed in an open set il C Rn. For u E D' (A), a E N~ let (txtu be the derivative defined by (1). (If A = il formula (1) gives the "usual" distributional derivative of the "usual" distribution u E D' (il)).
Prove that (txt L = L(txt where L is the isomorphism given by Proposi tion 1.
5. Let X be a linear functional on £1 «0, t)) defined by
X[cp] = It cp(x)dx for cp E Ll «0, t)).
Prove that
(i) XED' «0, t]), D' «0, t]) I"V D(O,t)(R+), * = -t5(tb
(ii) XED' ([0, t]), D' ([0, t]) I"V D{o,t)(R), * = 15(0) - b(t) (X is a distribution on R defined by the function equal to 1 on [0, t] and to zero outside),
(iii) X E (D[o,t))', .!!x - ° dx - .
6. Let il be an open set in Rn. that
Deduce from the definitions given in this section
(a) CP/l --+ ° in D(il) iff there exists a compact set K C il such that CP/l E C'K and for any a E N~ .lim (aa r cp /I = ° uniformly in il,
)-+00 X
22 I INTRODUCTION
(b) u E D'(Q) iff u is linear on CO"(Q) and for every compact K c Q there exists a constant C(K) < 00 and a non-negative integer k(K) such that for any
<p E CII
7. Let Y be the Heaviside function on R: i.e. Y(x) = 1 for x > 0, Y(x) = 0 for x < o. Show that ~~ = 8(0).
8. Suppose that u E DK(Rn), K compact, is of order:::; k and that a E ck(Rn), (:xta(x) = 0 for lal :::; k and x E K. Show that ural = 0 (see e.g. [Sz1] Theo rem 7.4). HINT. Let 1jJ E CO", J 1jJ(x)dx = 1, 1jJ(x) 2: 0 for x ERn, supp1jJ C B(O, 1). Denote
Kr = U B(x, r) and define xEK
for x E Rn , 0 < c < co.
Prove that X E CO", Xe = 1 on K e, SUPPXe C K3e and that for every f3 E N~:
as c -+ o.
Next show that for every function <p E C k : u[<p] = U [Xe<P] and that there exists a constant C such that
for 0 < c :::; co .
Now it suffices to prove that for every lal:::; k lim sup Ictxt (Xe(x)a(x)) I = o. e-O xEKac
For this aim apply the Leibniz rule to Xea and the Taylor formula for (:x fa (ITI :::; k) to deduce that
(txra(x) = o(lx - ylk-hl ) for 1,1:::; k.
9. Let u E D'(Rn ) be a distribution of order:::; k. Show that ural = 0 for every function a E C~(Rn) such that (:xta(x) = 0 for lal:::; k and x E supp u.
10. Let u E S'(R n ). Show that there exist kENo and functions hOt E Loo such that
HINT. If u E S'(Rn) then there exist kENo and G < 00 such that
IU[<p)l ::; G L sup 1(1 + IIxll2)k (tx r <p(x)1 l"l~k
Denote <Pk(X) = (1 + IIxll2)k <p(x) for x ERn. Observe that there exists a constant Gl = Gl(a) such that
and hence with some constant G2 we have
Apply the Hahn-Banach theorem and the Riesz theorem. (See the proof of Propo sition 8.4).
11. Let kENo, (3 E N; and let 9 be a continuous function. Show that there exist polynomials P"'t such that
12. Let U E S/{Rn). Show that there exist a GO(Rn)-function f slowly increasing at infinity and a E N; such that U = (:X)" f. HINT. 1) Show that there exist kENo and functions gp, 1(31 ::; k + 2n continuous, slowly increasing at infinity such that
(10)
To this aim apply Exercise 10 and consider the functions
g,,{x) = Lx1 •• ·Lxn
h,,(Yl,"" Yn) dYl ... dYn for lal::; k + n.
2) Using Exercise 11 deduces from (10) that there exist continuous functions f"'t slowly increasing at infinity such that
24 I INTRODUCTION
3) Denoting Jjf'Y = Jox; f'Y (Xl, ... , Xj-l, t, Xj+b ... , Xn) dt (j = 1, ... , n) put f = L: J;n-'Yl ... J::'-'Yn f'Y and prove that u = (tx r f where m = (m, ... , m).
hl:::;m
13. Show that the transformation -aa : S' - S' is surjective. Xk
14. Prove that if the polynomial Q has zeros in Rn then the equation Qv = 0 has non-zero distributional solutions.
15. Let 9 E S'(Rl). Show that there exist infinitely many distributions v E S'(Rl) satisfying the equation xv = g. Any two of them differ by a multiple of the Dirac distribution o. 16. Show that the general solution of the equation (x~ + ... + x;)u = 0 is the distribution u = L: aOl(txto where aOi (10'1 ~ 1, a E N~) are any constants and
10119 8 E D'(Rn) is the Dirac distribution.
17. Show that for every x E R, r > 0, pEN there exists a function f E COO(R) such that
f(q)(x) = {ro for q = p for q < p,
where f(q) = (lx)q f·
and If(q)(x)1 < 1 for Ix - xl < 1, q < p,
HINT. Let 9 E Cgo (( x-I, x+ 1)) be a non-negative function such that J g( x) dx < 1, g(x) = r. Define f by f(p)(x) = g(x), f(q)(x) = 0 for q < p.
00
18. Let u[cp] = L: cp(j)(~) for cp E Cgo ((0, 1)). Show that u E D' ((0, 1)) and that j=2 J
U has infinite order. OUTLINE OF THE PROOF. Suppose conversely that the distribution u is of order
~ m (m E No). Take x = m~l' 0 < 'f/ < m~l - m~2' K = [x - 'f/, x + 'f/l and let C be a constant such that lu[cpll ~ ClIlCPlllm for cp E C'K. Choose a function X E Cgo (( -1,1)), X = 1 for Ixi ~ t. Let fr be a function from Exercise 17 corresponding to p = m + 1 and x = m~l. Take '¢r(x) = fr(X)x(x~l) for x E R. Observe that '¢r E Cft, u['¢r] = r and that IU['¢rll ~ CIII'¢rlllm ~ CM/1]n where M is a constant depending only on the function X, and independent of r. Taking r > CM/'f/n we get a contradiction.
19. Give an example of a distribution of finite order on an open proper subset of R, which is not extendible.
20. Let A be a bounded open cube. Let Hk(A), k = 1, ... , n, (H'Y(A)" E N~ resp.) be the image of Cgo(A) under the differentiation a~k ((:J'Y resp.). Define operations: h: Hk(A) 3 '¢ 1--+ (] = Jk('¢) E Cgo(A) where k = 1, ... ,n, (](x) =
J~:' 'Ij;(X1,' .. ,Xk-1, t, Xk+1, . .. ,xn) dt for x E En, and J'Y = JJn ... J't: H'Y(A) -+
CoCA) where J2 = identity, J1 = h, Jf = hJf-1 for p = 2,3, ... Show that
cp = J'Y'Ij; E C~(A), for 'Ij; E H'Y(A)
and that for any m E No, 7 = m + 1 E N~ and any 'Ij; E H"r(A) there exists a constant C < 00 such that IIlcplllm ::; C 11¢11£1 where cp = J"r¢.
21. Prove Theorem 6. HINT. Observe first that there exist a bounded open cube A :J li and constants C > 0, mEN such that lu[cpll ::; ClllCPlllm for cp E CoCA). Adopt the notation of Exercise 20 and consider the linear functional w[¢] = u[J"r¢] for ¢ E H"r(A). Show that there exists a constant C* > 0 such that Iw[¢]1 ::; C* 1I¢11£1 for ¢ E H"r(A) C L1(A). Apply the Hahn-Banach theorem and by the familiar theorem on
the conjugate space to L1 get u = (_1)(m+1)n(I.,)"r g on A where 9 is a bounded measurable function on A. Let De be any neighbourhood of li, ¢ E Co( De), ¢ = 1 for x E li. Take a continuous function F such that Ix F = g. Show that G = (-1 )(m+1)n F¢ is the required function corresponding to the multi-index
a=7+l. 22. Let u,v E D'(Rn ), A = suppu, B = suppv. Suppose that for every compact set H c Rn the set
(11) Ii = {(x,y): x E A, y E B, x + y E H}
is bounded. Show that the convolution u * v exists and that the condition (11) is satisfied
if one of the sets A, B is bounded (cf. Example 3).
Chapter II
§4. THE FOURIER AND THE FOURIER-MELLIN TRANSFORMATIONS
1. The Fourier Transformation in S'
The Fourier transform Fa of a function a E S(Rn) is defined by
The transformation F is an isomorphism of S onto S with the inverse
Transformations F and F- 1 preserve the L2-norm of functions a E S, i.e. for every a E S the following equalities, known as the Parseval equalities hold:
By means of the relations
Fu[aJ ~f u[FaJ
F- 1u[aJ ~ u[F- 1 aJ for u E S', a E S
the Fourier transformation and the inverse Fourier transformation extend to iso morphism of S' onto S'. Below we recall basic theorems and spaces occurring in the theory of the Fourier transformation.
27
28 II MELLIN DISTRIBUTIONS AND MELLIN TRANSFORMATION
Definition 1. By OM(Rn) we denote the vector space ofthe functions mE coo(Rn) such that for every f3 E N; there exist constants C f3 < 00 and S f3 > 0 fulfilling the estimation
The elements of OM are called smooth functions slowly increasing at infinity.
OM is the class of multipliers in S(Rn) and consequently in S'(Rn ). More precisely m E OM if and only if the mapping S :3 .,p f-t m . .,p E S is continuous.
Denote by 06 the image of OM under F. Clearly 06 c S'. N ow we recall the following exchange formulas for the Fourier transformation
(cf. [Tr] or [Schw]).
Theorem L Ifv E S', w E OM, U E 06 then
F(u * v) = (27r)~ Fu· Fv.
The same equalities hold with F replaced by F- 1 . If v E S', a E S then .,p = v * a E
OM and .,p(e) = v [ace - x)] for e E Rn. Consequently v· a E 06.
The scale of the spaces OM C OM introduced below will play an important role in Section Ii.
Definition 2. Let s E R. We denote by OM the class of functions m E COO(Rn) such that for every 6 EN; there exists a constant C6 = C6(m) < 00 such that
OM is equipped with a natural topology of uniform convergence given by the small est constants C 6.
Theorem 2. (see e.g. [SzI]). The Fourier transform Fu of a distribution u E E'(R n ) is a function of class OM given by
Similarly F- 1u E OM and
§4. FOURIER AND FOURIER-MELLIN TRANSFORMATIONS 29
We end this subsection with an S'-version of the Schwartz kernel theorem. We start with definitions:
Denote by S' (Rn, S'(Rm)) n, mEN the space of S'(Rm)-valued tempered distributions on Rn i.e. T E S' (Rn, S'(Rm)) iffor every rp E S(Rn), Trp E S'(Rm) and the assignment
is linear and continuous. Then it follows from Theorem 2.3 that there exist constants C < 00 and kENo such that
(1) for rp E S(Rn), 'Ij; E S(Rm),
where qk is the seminorm given by (3.3) in Rn and Rm.
Theorem 3. The spaces S' (Rn, S'(Rm)) and S' (Rn+m) are isomorphic in a canon ical way. The isomorphism is given by the assignment
where
(2)
and rp ® 'Ij; denotes the function Rn x Rm 3 (x, y) ~ rp(x) . 'Ij;(y).
Proof: Let T E S' (Rn, S'(Rm)). We shall find a distribution T E S' (Rn+m) such that (2) holds. Observe that (by Exercise 1 and Hahn-Banach theorem) (1) holds for those functions rp and 'Ij; for which the norm qk is finite. Thus by Exercise 2 we have for some constant 0 < C < 00
Let cf! E S(Rn+m) and denote t.P(x, y) = (1 + IlxI12)k+1(1 + lIyI12)k+1cf!(x, y) for
(x, y) E Rn+m (with k from (3)). T is defined explicitly as follows:
(4)
where g is given by (3) and hence the integral in (4) makes sense and f is a tempered distribution. If!p = r.p 0.,p with r.p E S(Rn), .,p E S(Rm) then by (4) we get
f[r.p 0.,pj = (27r)-~ r (F((l + IIxI12)kr.p ))(Ox JRn X {lm gee, 1])F((l + IIYI12)k.,p(Y))(1]) d1]} de,
and hence by Exercises 2-4 we get
f[r.p 0.,pj = (27r)-~ In (F(l + IIxIl2)kr.p))(O(TxC1 +e;I:1I2)k) )r.,pjde
= (Tr.p)[.,pj.
2. The Fourier-Mellin Transformation in the Space of Mellin Distributions with Support in R+
Below we present basic properties of the Fourier transformation in logarithmic variables, which we call the Fourier-Mellin transformation.
Denote by p,: Rn ~ R+ the diffeomorphism (see Exercise 5)
() -y def ( -Yl -Y ) p,y=e = e , ... ,e n.
We define the space of Mellin distributions on R+ for every a E Rn as the dual of the space
with a natural topology in 9)1<> induced from S(Rn). Note that 9)1~(R+) is a subspace of D'(R+) (cf. Exercise 6). Hence and from the definition of substitution in a distribution (cf. Section 3.5) follows
Remark 1. u E 9)1~(R+) if and only if e<>Y(u 0 p,) E S'(Rn).
Definition 3. Let a E Rn. Define
M tr . S ~ on Q- ,;,/.1 I. co (M~) -1: 9)1<> ~ S,
+
Proposition 1. Tbe transformations M~ and (M~)-1 are continuous and mutu ally inverse.
The proof follows from the properties of the Fourier transformation on S in view of the relations
(5) M!:~ = (27rt/2x-Ot-l(J:-1~) 0 11-- 1
(M!:)-1 a = (27r)-n/2 F((x Ot+1 a ) 0 11-)
for ~ E S,
for a E 9)1Ot.
Definition 4. Let U E 9)1~. The MOt Mellin transform of U is the distribution MOtU E S' defined by the duality
or equivalently (see (5))
(MOt)-1T[a] = T[(M!:)-1a]
for ~ E S,
Proposition 2. MOt: 9)1~ ~ S' is an isomorpbism witb inverse (M Ot )-1.
The proof is obvious in view of Proposition 1.
Proposition 3. If U E 9)1~ tben x j a~. U E 9)1~ and )
MOt (Xj a~. u) = (aj + i,Bj)MOtu (j = 1, ... , n). J
Proof: In the proof of the first part we apply Remark 1 and the identity
(7) for u E D'(R~) and v E N~.
The second part follows by Definition 4 and 3.
Similarly to the case of the Fourier transformation we have the following ex
cbange formula for the MOtMellin transformation
Proposition 4. Let ~ E 9)1-1. Ifu E 9)1~(R~) tben
(27rt MOt(~u) = Mo1/J * MOtU,
wbere tbe rigbt band side is tbe convolution of a function Mo1/J E S witb a tempered
distribution MOtU (i.e. can be written as MOtu[Mo1/J(,B - ,)]).
Proof: Since 'I/J E roLl it follows from the definition of roLl that 'I/J 0 p E S(Rn) and by Remark 1 and (6) we get Mo'I/J E s. As it was noted the assumption u E 9J1~ means that ee>y ( u 0 p) E S'. Hence
ee>U(('l/Ju) 0 p) = 'I/J 0 p' ee>y(u 0 p) E S'
since 'l/Jop E S. Therefore 'l/Ju E 9J1~. From the definition ofthe transformation Me>
and by the exchange formula for the Fourier transformation we get
which ends the proof.
Exercises
1. Let k,p E No. Denote by Sk = Sk(Rn) the space of functions 0' E CP(Rn) such that for every c > 0 there exists an R > 0 fulfilling the inequality
for IIxll > R, lad:::; p. Define a norm qk in Sk putting
qk(O') = L sup 1(1 + IIx112)k (txt O'(x)1 for 0' E Sk' !e>!::;p
Show the following inclusions between the topological vector spaces D, S, Sr, s'/ and their duals
DeS c sy c Sk
(SD' c (S'f)' c S' c D' if p, k, q, 1 E No, p:::; q, k:::; 1.
Observe that u E S' iff there exists a kENo such that u E (Sk)" where Sk = st. (See [Szl], Subsection 18.2.)
2. Show that for every e E Rn and kEN the function Rn 3 x ~ eiXe (1 + IIxI12)-k belongs to the space Sk-l (see Exercise 1).
3. Let T E S'(Rn, S'(Rm). Show that there exists kENo such that the assignment
is linear and continuous.
4. Let Sk = Sf, (Sk)' = (Sf), (see Exercise 1). Let A be a function defined on Rn+m satisfying the conditions:
(i) A(ry, .) E sk(Rm) for ry ERn,
(ii) for any a E N;:', lal ::::; k, the function
n m (1 + IIYIl2)k (a)OI R x R 3 (ry, y) ~ (1 + 117711)1 011 ay A(ry, y)
is continuous and bounded in Rn+m,
(iii) For any c > 0, compact set KeRn and a E N;:' there exists 0 < ~ = ~(c, K, a) such that
y~':fm (1 + IIYIl2)k I (ty r A(ry, y) - (ty r A( 17, y) I < c
for ry, 17 E K, Iry - 171 < ~.
HINT. Assume first that a E CO"(Rn). Consider Riemann sums for the integral J(y) = JRn A(ry,y)a(ry)dry and show that they converge to J in Sk (see Theo rem 18.11 in [Szl)). In the case of a E S(Rn) choose a sequence of functions Xi E CO"(Rn) such that a . Xi --t a in S(Rn) and pass to the limit in the already proved formula for aXi E CO"(Rn).
5. Prove that the transformation {t: Rn --t R+ defined in Section 2 is a diffeomor phism with an inverse {t-I: R+ --t Rn, {t-I(x) = -lnx = (-lnxI, ... ,-lnxn) for x = (XI, .. . , x n ) E R+. Also prove that the operations of composition with {t and {t-I are continuous
Ott: D(R+) --t D(Rn),
Ott: D'(R+) --t D'(Rn),
Ott-I: D(Rn) --t D(R+),
Ott-I: D'(Rn) --t D'(R+).
where for u E D'(R~) (cf. Section 3.5)
and for v E D'(Rn )
(see also Exercise 5.9).
6. Prove that Cn(W:.) is dense in 9Jta for any 0: ERn.
7. Give an example of a function a E 9Jta(R~) with unbounded support.
8. Let U E 9Jt~(R~), 1jJ E 9Jt-a- 1 (R+.). Show that (27r)n Ma(1jJu) = Ma1jJ * Mou, Ma1jJ E S(Rn), Mou E S'(Rn).
§5. THE SPACES OF MELLIN DISTRIBUTIONS WITH SUPPORT IN A POLYINTERVAL
A natural setting for the local theory of the Mellin transformation are the spaces Mew) (( 0, t]) of Mellin distributions, defined below, whose definition is modelled upon that introduced by Zemanian [Z].
1. Spaces Ma((O, i]) and M~((O, i])
Let a, bERn. Recall that we write a < b (a ::; b resp.) to denote that aj < bj
(aj ::; bj resp.) for j = 1, ... , n. Take an arbitrary 0 < tERn. We denote by I the polyinterval
I = (O,t] = {x ERn: 0 < x::; t}.
Recall (d. 3.2) that COCCI) denotes the space of restrictions to I of smooth functions on COC(R+).
Definition 1. For a E Rn we introduce the space
Ma = Ma(I) = {<p E COCCI): ~~}lxa+aH(txr <p(x)1 < 00, 0: E N~}
equipped with the convergence topology defined by a sequence of seminorms
{ea,a}aEN~ :
§5. MELLIN DISTRIBUTIONS WITH SUPPORT IN A POLYINTERVAL 35
As noted in Section 2 {ea,,,,} can be replaced by an increasing sequence of (semi )norms
qk( c.p) = L ea,,,,( c.p), 1"'19
without changing the topology in Ma. Since the linear spaces spanned by the operators xP (Ix t and (x Ix t coincide
(see Exercise 1) it follows that Definition 1 may be replaced by the equivalent
Definition 1'. For a E Rn
Ma = {c.p E coo(1): ~~}lxa+1(x tx)'" c.p(x) I < 00 for a E N~}
with the topology defined by the sequence of seminorms(l)
ea,,,,(c.p) = ~~}lxa+1 (x tx)'" c.p(x) \ , a E N~.
Note that the topology in Ma(1) is defined in "inner" terms i.e. in terms of supremum on I, in contrast to the "outer" topology in terms of extensions, where the supremum is taken on larger sets (c.f. the definition of D(A) in Subsection 3.2). To prove that those topologies coincide we shall construct a linear continuous ex tension mapping
£: Ma«O, t]) --+ Ma«O, i]) for any i> t.
We begin with a lemma which is a parameter version of the Seeley extension theorem from a half-line to the real line (see [Mel] and Exercise 14).
Let a ERn, x ERn, t E R~ and write a = (al,a'), x = (XI,X'), t = (t1,t') with a' = (a2, ... , an), x' = (X2, ... , xn), t' = (t2, ... , tn). Take 5 E R+ and denote
800([0,5)) = {c.p E C OO «0,5)):
~ } dxi c.p extends continuously to [0,5) (j = 0, 1,2, ... ) ,(2)
800 ([0,5); Mal«O,t'])) = {c.p E Coo «0,5) X (O,t']):
sup l(x't'+1 (x' 88,)",' (!'l8 )"'1 c.p(XI' x')\ < 00 (O,e)x(O,t'] X UXI
for a = (al,a') E N~, c.p(. ,x') E 800([0,5) for every x' E (O,t']}.
Analogously we define Coo« -5,5); Mal«O, t'])) and C0,~]«0, 00); Mal «0, t'])).
(1) See Exercise 2 for other examples of equivalent seminorms on Ma. (2)In an analogous way we define 6 00 ((0, eo]). The "tilde notation" is used exclusively in Lemma 1. It will result therefrom (see Exercise 4(ii» that 6 00 ([0, eo)] = COO([O, eo».
Lemma 1. There exists a linear extension mapping
such that for every a E N~ there exists a constant COil such that
(1)
for every 'P E COO ([0, c); Mal ((0, t'])).
Proof: Let X E Cgo(R) be 1 in a neighbourhood of zero and X( Xl) = ° for IX11 ~ ~. Define
00
-00
for'P E C ([O,c); Mal((O, t'])), where {ad is a sequence of real numbers. Observe that for each Xl > ° only finitely many terms on the right hand side of (2) are non
-1 zero and (£ 'P)(x) = ° for Xl ~ ~. Clearly
and this map is linear. We shall choose the sequence {ad to satisfy
(3) for pEN
Assuming this for a moment we find by differentiating (2)
for pEN.
Thus from the properties of X and from (3) we get the estimation
(6) sup I(X't'+l (x' aa I)"" (aa )"',(£'\?)(Xl' X')I (O,e)x(O,t'j x Xl
with some constant C"'l < 00, and on the other hand we find
where
Now by (4)
x!~+ (Xl a~/)"" ((a~J "''(l\?))(XI,XI) = (-1)"'1 (X' a~/)"" ((a~J"'l ~ )(O'X')
for every ex E N~. Put
for ex E N~. The desired extension is obtained by taking
1 I {~(Xl'XI) for O<Xl<e:, O<XI:::;t' (£ ~)(Xl'X) = -1
(£ ~)(-Xl'X') for -e: < Xl :::; 0, ° < X':::; t'
-00
and (1) follows from (6) and the assumption that ~ E C ([O,e:)jMa,((O,t' ])). Finally to find the sequence {a,} satisfying (3) and ( 4) we note that the function
h(z) = cos (7rez 21)) is entire and h(p) = (-l)P for pEN. We take a, (l = 0,1, ... ) as the coefficients of the power series expansion of the function C 3 w f-+ COS7r(w;-l) i.e. h(z) = 00
2: al(3 Z )'.
~l
Remark 1. (£\?)(XI,X') = 0 for Xl ~ -~ since (£ cp)(x) = 0 for Xl ~ ~.
Theorem 1. Let a ERn, 0 < t < i E Ri.. Then to every 0 < C < i - t, c < t there exists a linear extension mapping
continuous in the respective topologies and such that for every cp E Ma((O, t]) (£,.cp)(x) = 0 iftj + Cj < Xj ~ ij for some 1 ~ j ~ n.
Proof: Let cp E Ma((O, t]), choose 0 < C < t; C < i - t and observe that the function
~oo
belongs to C ([O,cd; Mal((O, t'])). Thus by Lemma 1 and Remark 1
which yields the correctness of the following definition:
It is clear that £lr:p is an extension of r:p to Ma((O, (tl + Cl, t')]) and in fact to Ma((O, (iI, t')]) since
for Xl < - T' The continuity of £El follows from the continuity of £1.
If n ~ 2 we iterate the above procedure starting with £El cp defined above instead of cp and (iI, t') instead of t.
Proposition 1. The space Ma(I) is complete.
Proof: Let {'Pi}~l be a Cauchy sequence in Ma(I). Take i> t, 0 < c < i - t, c < t and a continuous extension map £e from Theorem 1. Let 'Pi = £,,'Pi (j = 1,2, ... ). By the continuity of £e {'Pi}~l is also a Cauchy sequence in Ma«O,i]). This means that for every a E N~ the sequence {xa+O'+l (tJO' 'P i};:l satisfies
the Cauchy condition for the uniform convergence on J = (0, il. Thus there exist functions hO' (a E N~) continuous on J vanishing near the boundary J \ I and such that
(7)
This in view of (7) implies that
(8) almost uniformly on I.
Denote 'P(x) = ho/xa+I, x E Int I and 'P = 'Plr Then'P E COO(I) and from (8)
(txt'P = hO'/xa+O'+1 for x E IntI, a E N; and consequently
sup lxa+O'+l(2...)O''Pi -hO'I- 0, xEI ax ) ..... 00
which proves that 'Pi -t 'P in Ma(I).
Example 1. Let fz be a function depending on a complex parameter Z E en defined by
13 x t-+ fz(x) = x-z- 1 .
Let a E Rn. The function fz is in Ma (1) if and only if Re z ::; a. Indeed for a fixed a E N; there exists a polynomial WO' of degree lal such that
ea,O'(fz) = IwO'(z)1 sup xa- Re z, xEI
and ea,O'(fz) is finite if and only if Re z ::; a.
Recall that for x E R+ lnx denotes the function lnx = (lnx1,'" ,lnxn ) and consequently for a E N~, (ln x)O' = (In xt}O'1 ••••• (In xn)O'n •
Example 2. Let for Z E en gz be the function I:3 x I---? gz(x) = (Inx)Px- z- 1 for some j3 E N~. The function gz is in Ma(I) if and only if Rezj < aj if j3j > 0 and Re Zj ~ aj if j3j = O. This is seen immediately from the following two facts:
(i) For arbitrary ct, j3 E N~ there exist polynomials w Ol !3'Y such that
(txf ((lnxfx- Z - 1 ) = x-z- 0I - 1 L(1nx)'YwOl!3'Y(z),
'Y5.!3
(ii) xj'-Rezj Ilnxjl!3j = O(Xj) as Xj -+ 0 if and only if
and Rezj < aj if j3j > O.
Other examples of functions belonging to Ma are given in Exercise 3.
Returning to the spaces Ma we note the following topological inclusions:
valid if a, bERn a ~ b, where D(I) was defined in Section 3.2 for A = I. Next, it follows from Theorem 2.1 that U E M~(I) -the dual space of Ma(I),
if and only if u is linear and for some m E No and e < 00
(9)
We then say that u is of Mellin order less or equal to m. For a fixed m E No the space of the elements of M~ (I) whose Mellin order is not greater than m can be defined as a dual of the space M;:'(I) of complex functions rp E em(I) for which the norm L: f!a,OI is finite. The dual space of M;:'(I) is denoted by M~m(I). Note
10015. m
that it follows from (9) that
M~(I) = U M~m(I). mENo
Proposition 2. The set e~)(I) is not dense in Ma(I) and consequently M~(I) is not a subspace of the space of distributions D'(I) for any a ERn.
Proof: On account of Corollary 2.4 it is enough to construct a non-zero functional u E M~(I) whose restriction to e~)(I) is zero.
Consider a subspace Ba of Ma defined as
Ba = {rp E Ma: lim xaHr.p(x) exists and is finite}. x ..... o
We define on Ba a linear functional u; u[cp] = lim xa+1cp(x). The estimation x-+o
Ixa+lcp(x)1 ~ !la,O(CP) for x E I implies that lu[cpll ~ !la,O(CP). Now from the Hahn Banach Theorem 2.4 it follows that u extends to an u E M~(I) such that lu[cp] I ~ !la,o(CP) for cp E Ma(I). Clearly u is non-zero since u[fa] = 1 where fa(x) = x-a- 1 for x E I. However for cp E C(~)(I) we have u[cp] = u[cp] = lim xa+1cp(x) = 0, which
x-+O
is what we wanted.
In Exercise 4 (cf. also Exercises 5 and 6) the reader will find a direct proof that C~)(I) is not dense in Ma(I).
In Subsection 2 we introduce a subspace M(a) C Ma such that C~)(I) is dense in M(a).
2. Spaces M(w) «0, t]) and M(w) «0, t])
As in Subsection 1 we take I = (0, t] for some t E R~. Let w E (R U {oo})n. We define the function space M(w)(I) as the inductive limit
M(w)(I) = ~ Ma(I). a<w
It follows from the definition of the inductive topology that a sequence {<f'v}, CPv E
M(w) (v = 1,2, ... ) converges to zero in M(w) if there exists a < w such that CPv E Ma (v = 1,2, ... ) and for every such a the sequence {CPv} is convergent to zero in M a •
The following topological inclusion is clear:
(10)
Proposition 3. The set C~)(I) is dense in M(w)(I).
Proof: Take an arbitrary cp E M(w). Thus there exist a < b < w such that cp E Ma(I) C Mb(I). To prove the proposition it is enough to construct a sequence {<f'v}~l of C~)(I) functions such that CPv -t cp in Mb(l). We introduce auxiliary functions fi E C=(R+), 0 ~ fi(S) :::; 1, S E R+ (i = 1, ... , n)
{ 0 for 0 < S < ~ti
fi(S) = 1 1 for 2ti < S < 00,
(l)One does not have convergence in Ma since Ct'l:) (I) in not dense there.
and we define
for x E I (v=I,2, ... ).
Let A" = {x E I: 0 < Xj < 21"tj for some j, 1 ~ j ~ n}. We check that
(11) tP" E coo(I), 0 ~ tP,,(x) ~ 1, tP" = 1 on I\A" and sup Ixb-al ----+ o. xEA. "--+00
We shall prove that the sequence
for x E I
has the desired properties. Fix arbitrarily c¥ E N;. Let c be a constant such that
(12) for k~C¥i' 8>0 (i=I, ... ,n).
It follows from the Leibniz formula and (11) that
(13) (!b,c.(<P" - <p) = ~~~lxb+a+1(tx)''(<P(x)(tP,,(x) -1))1
~ ~~~Ixb+a+l(txr <p(x)· (tP,,(x) -1)1+
+ L (;) ~~~lxa+~+l(txt <p(x)(txr-~(tPv(x) _1)xb-a+a-~1 ~5:.a ~#a
~ (!a,a(<P) sup Ixb-a(tP,,(x) -1)1+ xEA.
It is seen from (11) that the first summand above tends to zero as v -+ 00 (which proves that (!b,O( <Pv - <p) -+ 0). If c¥ i= 0 then in each of the remaining summands there is a j, 1 ~ j ~ n with Pj < C¥j. The assumptions on J; imply that
(14)
which in view of the relations (11) and (12) leads to
1 and Xk > -tk,
Hence there exists a constant c* < 00 such that
(15)
Assuming that /3j < (Xj we have by (14) and (15) the estimates
suplxb-a+cx-i3(2..-)CX-p(1/Jv(x) -1)1 = xU ax
:s c* sup xEI
Xj'5,!vtj
43
which imply that the left hand side tends to zero as v -7 00, which proves the proposi tion.
By Mew) = Mew) (I), w E (R U {oo})n we denote as usual the dual space of M(w) endowed with the pointwise convergence topology. By (10), Proposition 3 and Proposition 2.1 we derive that M(w)(I) is a subspace of D'(I). Therefore the elements of M(w) are called Mellin distributions. The totality of Mellin distributions is denoted by M'(I):
M'(I) = u wE(RU{oo})n
Definition 2. We say that u E D'(I) (u E D1(R+.))) is a Mellin distribution if there exists an w E (R U {oo})n such that u extends to M(w) as a continuous linear functional on M(w).
Remark 1. Observe that a functional u E D'(!) may extend to different spaces M(w), however for a given w if an extension exists then it is unique due to the density of C~)(I) in M(w). Moreover the extensions are compatible in the sense
that for any extensions u W , U W of u to M(w) and M(w) respectively
Definition 2 may be reformulated equivalently as
Definition 2'. We say that u E D' (I) (u E D1(R+.)) is a Mellin distribution ifthere exists an a E Rn such that u extends (clearly, non-uniquely) to a linear continuous functional on Ma.
Indeed if U E M(w) then for any a < w, U E M~ since M(w) C M~. On the other hand given a U E M~ it follows from the topological inclusion, M(w) C Ma if
w < a, that U E M(w)' We shall yet introduce a space M[a](I) (considered in detail in Subsection 6.2)
M[a](I) = ~Mb(I) b>a
equipped with the projective limit topology: 'Pv -+ 0 in M[a] if and only if 'Pv -+ 0 in every space Mb for b > a.
Let o:,a,b ERn, W E (R U {oo})n, a < b < w. The topological inclusions for the spaces Ma given in Subsection 1 can now be completed (see Exercise 7 for the proof) as follows:
(16) D(!) C M(a)(I) C Ma(I) C M[a](I) C Mb(I) c M(w)(I),
M(w) (I) C D'(!).
Also note that it follows from (16) and the definitions of the spaces M[a], M(a) that
(17) M(w) = lim M(c), ~
b>a
As it was shown in Subsection 1 M~ is not a subspace of D'(I). Nevertheless, due to the topological inclusion D C Ma, the restriction of a functional U E M~ to D(I) belongs to D'(!). The same is also valid for a pair of spaces M(a)' M~. Also note that 91ta ct 91tb (see Exercise 10).
We shall distinguish now another important dense subset of M(w)(!) different from C(,;:)(I). We start with the notation:
For a E Rn we denote Xa = span {X- b- 1 h<a the space of finite linear combina tions of elements of the set {x-b- 1 h::;a with c~mplex coefficients, X(w) = U Xa.
From Example 1 Xa C Ma and X(w) C M(w)' We have a<w
Proposition 4. X(w) is dense in M(w)'
The proof is immediate from the following
Lemma 2. Let a, bERn, b < a. Then Mb is contained in the closure of Xa in Ma.
Proof: Let cp E M b• It follows from Exercise 2.1 that it is enough to show that for every c > 0 and m E No there exists a 1/1 E Xa such that
(18) em( cp -1/1) < c,
where em = :E ea,a with ea,a given in Definition 1. Since cp E M(w) if w > b lal~m
and the space C~)(I) is dense in M(w) for any w, again by Exercise 2.1, there exists
"p E C~) (I) such that em (cp - "p) < ~. To prove (18) we thus need find a 1/1 E X a such that em (1/1 - "p) < ~. Since"p
is in C~) (I), the function I :3 x 1---+ xa+1"p( x) is smooth and from the Weierstrass
theorem for every 8 > 0 one can find a polynomial W6 with the property
~~}I (tx) p (x a+1"p(x) - W6(X)) I < 8 for j3 $ Q.
We shall prove that subject to an appropriate choice of 8, the function 1/I(x) = X-a-1W6(X) has the desired property. Indeed 1/1 E Xa and by Exercise 1 there exist constants ca,a,p such that
em(1/I-"p) $ L L Ca,a,pSUpl (tx)p(xa+1(1/I- "p))jtP $ c8 lal~m p~a xEI
for C = :E :E Ca,a,pt p. This ends the proof. lal~mp~a
In the same way that locally integrable functions were identified with distri butions we can identify functions in Ma(!) with distributions in Ml- a- 1). Indeed given an f E Ma we define
f[1/I] = 1 f(x)1/I(x) dx for 1/1 E Me with C < -a - 1.
Note that the above formal definition makes sense and defines f E M~. Due to the arbitrariness of C < -a -1 we have f E M(-a_l) C D'(I). Thus
(19) Ma C M( -a-I)'
Analogously
M[al C Ml -a-I)'
We shall complete inclusions (16) with those between the spaces M(w), Mw, M(w) and the spa

the mellin transformation and fuchsian type partial differential equations

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