Nikolai Karapetiants Stefan Samko

Equations with Involutive Operators

Springer Science+Business Media, LLC

Nikolai Karapetiants Department of Mathematics Rostov State University 344711 Rostov-na-Donu Russia

Library of Congress Cataloging-in-Publication Data

Karapetiants, N.K. (Nikolai Karapetovich)

Stefan Samko Faculdade de Ciencias e Tecnologia Universidade do Algarve 8000 Faro Portugal

Equations with involutive operators 1 Nikolai Karapetiants, Stefan Sarnko. p. cm.

Includes bibliographical references and index. ISBN 978-1-4612-6651-8 ISBN 978-1-46l2-0183-0 (eBook) DOI 10.1007/978-1-46l2-0183-0 1. Fredholm operators. 2. Integral equations. 1 Sarnko, S. G. (Stefan Grigor'evich) 11

Title.

QA329.2.K36 2001 515'.7246-dc21 2001035385

CIP

AMS Subject Classifications: 47A53, 45E05, 47835, 47838, 47010, 47N20

Printed on acid-free paper. ©200 1 Springer Science+Business Media New York Originally published by Birkhäuser Boston in 2001 Softcover reprint ofthe hardcover Ist edition 2001

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Reformatted from authors' files in "0TEX2e by TEXniques, Inc., Cambridge, MA Cover design by Typesrnith Graphics, Westwood, MA

987 6 5 4 3 2 1

To the memory of our teachers

Contents

Introduction

Notation

1 On Fredholmness of Singular Type Operators 1 Fredholm operators . . . . . . . . . . . . . . . .

2

3

1.1 Basics on Fredholm operators ..... . 1.2 Fredholmness of operators with projectors 1.3 Fredholmness of matrix operators ..... Singular integral operators with piecewise continuous coefficients in the space Lp (r) . . . . . . . . . . . . . . . . . . . . 2.1 Singular integral operators in the space Lp(f) . 2.2 The notion of p-index ............ . 2.3 Fredholmness criterion and the index formula 2.4 Approximation of non-Fredholm operators

by Fredholm ones. . . . . . . . . . . . . . . . 2.5 On compactness of some "composite" singular operators On Fredholmness of convolution type operators .... 3.1 Fourier transforms and the Wiener algebra .. 3.2 3.3 3.4

General preliminaries on convolution operators Theorems on Fredholmness . . . . . . . . . . . On compactness of some convolution type operators

xv

xxi

1 1 1 5 9

14 14 16 21

24 25 26 26 30 34 36

viii

3.5 On Fredholmness of convolution type operators with "variable" coefficients .....

3.6 Discrete convolution type equations 3.7 On Hankel operators ...

4 Bibliographic notes to Chapter 1 . . . . . .

Contents

41 43 45 48

2 On Fredholmness of Other Singular-type Operators 51 5 On operators with homogeneous kernels; the one-dimensional case 51

5.1 Connection with convolution operators; Lp-boundedness 52 5.2 On Fredholmness of the operators ),,1 - K . . . . . . 53 5.3 The Carleman equation and notion of the standard

a-lemniscate ...................... 65 6 Operators with homogeneous kernels; the multi-dimensional case 68

6.1 Lp-boundedness.................. 69 6.2 On spherical harmonics . . . . . . . . . . . . . . 72 6.3 Formal reduction to a system of one-dimensional

equations with homogeneous kernels . . . . . . . 75 6.4 Justification of the reduction and the main result

for rotation invariant homogeneous kernels 77 6.5 Some cases of non-rotation invariant kernels and other

types of homogeneity. . . . . . . . . . . . . . . . 82 7 Convolution-type operators with discontinuous symbols 84

7.1 Setting of the problem . . . . . . . . . . . . . . . 85 7.2 Lizorkin space ................... 86 7.3 Fractional integration of Liouville and Bessel type 87 7.4 The case of a continuous symbol . . . . . . 90 7.5 The case of a discontinuous coefficient G(x) 93 7.6 Fourier transform of a certain function. . . 94 7.7 On a special group of operators. . . . . . . 95 7.8 The model operator with a discontinuous symbol 98 7.9 The case G(x) == 1 . . . . . . . . . . . . . . 100 7.10 The general case of a discontinuous symbol 101

8 Bibliographic notes to Chapter 2 . . . . . . . 109

3 Functional and Singular Integral Equations with Carleman Shifts in the Case of Continuous Coefficients 111 9 Carleman and generalized Carleman shifts 111

9.1 On some properties of shifts. . . . . . . . . 111 9.2 On a(t)-factorization of functions. . . . . . 114 9.3 The winding number of functions, invariant

and anti-invariant with respect to shift. . . . . . . . .. 118 9.4 On a(t)-factorization in the case of the shift a(t) = teiw

on the unit circle . . . . . . . . . . . 119 9.5 Logarithmic means and factorization. . . . . . . . . .. 124

Contents ix

9.6 Factorization with the shift x + h on the real line . . .. 127 10 A functional equation with shift. . . . . . . . . . . . . . . . .. 128

10.1 Connection between a functional equation with shift and an algebraic system ............... 129

10.2 Functional equations in the degenerate case . . 131 10.3 Functional equations with the shift a(t) = teiw 134 10.4 Some generalizations . . . . . . . . . . . . . . . 139

11 Singular integral equations with Carleman shift on a closed curve; the case of continuous coefficients. . . . . . . . . . . . . 142 11.1 Accompanying and associated operators . . . . . 142 11.2 Fredholmness theorem; the case of preservation of the

orientation ................ . 144 11.3 Fredholmness theorem; the case of change

~t~orie~ation ............. . 145 11.4 On regularizers of singular operators with shift . . . .. 146

12 Singular integral equations with Carleman shift on an open curve; the case of continuous coefficients. . . . . . . . . . . . . . . .. 147 12.1 The passage to a "composite" singular integral operator 147 12.2 Representation of a "composite" singular integral

operator as a composition of usual singular operators. 149 12.3 Fredholmness theorem for the operator (12.1) 150

13 Bibliographic notes to Chapter 3 . . . . . . . . . . . . . . 151

4 Two-term Equations (A + QB)rp = f with an Involutive Operator Q; an Abstract Approach and Applications 153 14 Fredholmness of an abstract equation with an involutive

operator; non-matrix approach . . . . 154 14.1 Prompting ideas . . . . . . . . 154 14.2 A system of axioms. Examples 155 14.3 Fredholmness theorem . . . . . 157 14.4 Invertibility theorem . . . . . . 161

15 Application to singular integral equations with complex conjugate unknowns . . . . . . . . . . . . . . . . . . . . 162 15.1 Anti-quasicommutation of the operators Q and S . 162 15.2 The case of a closed curve . . . . . . . . . . . . . . 163 15.3 The case of an open curve. . . . . . . . . . . . . . 164

16 Applications to integral equations on the real line with reflection or InVerSIOn . . . . . . . . . . . . . . . . . . . . . . 165 16.1 Convolution type equations with reflection. . 165 16.2 The case of weighted spaces . . . . . . . . . . 167 16.3 Singular convolution operators with reflection 172 16.4 Convolution type operators with a discontinuous symbol

and reflection . . . . . . . . . . . . . . . . . . . . . . 173 16.5 Equations with homogeneous kernels and the shift ~ .. 176

x Contents

16.6 Discrete analogue of convolution equations with reflection 178 17 Application to singular integral equations with Carleman shift

on an open curve; the case of discontinuous coefficients . . . 179 17.1 The normed operator ................. 180 17.2 Reduction of the normed operator to a "composite"

singular operator . . . . . . . . . . . . . . . . . . . . 180 17.3 Representation of a "composite" singular operator as a

composition of usual singular operators and the theorem on Fredholmness of the "composite" operator . . . 182

17.4 Theorem on Fredholmness of the normed operator . .. 186 17.5 Theorem on Fredholmness in the general case. . . . .. 186

18 Singular integral equations with a fractional linear shift in the space Lp with a special weight .................. 188 18.1 The choice of the weighted space and construction of the

involutive operator . . . . . . . . . . . . . . . . . . . 189 18.2 The case of shift preserving the orientation (D < 0) 1912 18.3 The case of shift changing the orientation (D > 0). 192

19 Fredholmness of abstract equations with a generalized involutive operator (non-matrix approach) . . . . . . . . . . . . . . . . 193 19.1 System of axioms and the theorem on Fredholmness 193 19.2 On deficiency numbers of the operator K = A + QB 197 19.3 Application to discrete Wiener-Hopf equations

with oscillating coefficients .............. 199 19.4 Application to paired discrete convolution type equations

with oscillating coefficients ..... 202 19.5 The case of irrational oscillation .. 203

20 Abstract equations with algebraic operators 206 20.1 On a partition of unity. . . . . . . . 207 20.2 Algebraic operators. Equations with such operators in

the case of constant coefficients . . . . . . . . . . . . .. 208 20.3 The nilpotent case . . . . . . . . . . . . . . . . . . . .. 212 20.4 Regularization of equations with quasi-algebraic opera-

tors in the case of operator coefficients . . . . . . . . . . . . . . . . . 213

20.5 The characteristic part of equations with algebraic operators . . . . . . . . . . . . . . . . . . . . . . . 215

20.6 Application to a functional equation with the Fourier transform . . . . . . . . . 218

21 Bibliographic notes to Chapter 4 . . . . . . . . . . . . . . .. 219

5 Equations with Several Generalized Involutive Operators. Matrix Abstract Approach and Applications 223 22 Fredholmness of abstract equations with generalized involutive

operators (matrix approach) . . . . . . . . . . . . . . . . . . .. 223

Contents xi

22.1 The case of one generalized involutive operator . . .. 224 22.2 A scheme of investigation of equations

with two generalized involutive operators (reduction) . 231 22.3 Connections between matrix and non-matrix approaches 239

23 Singular integral equations with a finite group of shifts in the case of continuous coefficients . . . . . . . . . . . . . . . . 244 23.1 The equation with one generalized Carleman shift 245 23.2 Classification of a finite group of shifts on closed

or open curves . . . . . . . . . . . . . . . . . . . . 246 23.3 Singular integral equations on a closed curve with two

shifts. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 248 24 Singular integral equations with a finite group of shifts (the case

of piecewise continuous coefficients) . . . . . . . . . . . . . . ., 253 24.1 The case of a closed curve ................ , 252 24.2 The case of one shift, changing the orientation on a closed

curve .................... 256 24.3 The case of an open curve . . . . . . . . . 258 24.4 Singular integral equations with shift and

complex conjugation . . . . . . . . . . . . . . . . . . .. 259 25 Convolution type equations with shifts and complex conjugation 261

25.1 Discrete convolution operators with almost stabilizing coefficients ....................... 261

25.2 Fredholmness of discrete convolution operators with oscillation and reflection . . . . . . . . . . . . . . . . 265

25.3 Convolution type equations with reflection and complex conjugation . . . . . . . . . . . . . . . . . . . . . . . .. 267

25.4 Integral equations with homogeneous kernels involving terms with inversion and complex conjugation. 270

26 Bibliographic notes to Chapter 5 . . . . . . . . . . . . . . . .. 273

6 Application of the Abstract Approach to Singular Equation on the Real Line with Fractional Linear Shift 275 27 Singular integral operators perturbed by integral operators with

homogeneous kernels . . . . . . . . . . . . . . . . . . . . . .. 277 27.1 Some necessary conditions. . . . . . . . . . . . . . .. 278 27.2 Reduction to a system of paired convolution equations 279 27.3 Systems of singular integral equations perturbed

by integrals with homogeneous kernels . . . . 284 28 Singular integral operators with a fractional linear Carleman

shift in the weighted space L;(Rl) . . . . . . . . . . . . . . .. 287 28.1 Setting of the problem and introduction of the involutive

operator Qv . . . . . . . . . . . . . . . . . . . . . . . .. 288 28.2 Some connections between the involutive operator Qv

and the singular integral operators S, Sa. and Sa. . . .. 289

xii Contents

28.3 Reduction of singular integral equations with a fractional linear shift to a system of perturbed singular equations without shift . . . . . . . . . . . . . . . . . . . . . 294

28.4 The case of preservation of the orientation (D < 0) 297 28.5 The case of change of the orientation (D > 0) . 304 28.6 The investigation of equation (B) . . . . . . . . . . 305 28.7 The case of a nonfractionallinear shift on Rl. . . . 309

29 Equations including operators with homogeneous kernels, the singular integral operators and the inversion shift . . . . . . .. 311

30 Potential type operators on the reall ine with a fractional linear Carleman shift . . . . . . . . . . . . . . . . . . . . . . . . . .. 316 30.1 On normal solvability of potential type operators with

discontinuous characteristics .......... 317 30.2 The case of linear shift (the case of reflection) . . . .. 319 30.3 The case of fractional linear shift . . . . . . . . . . .. 321

31 Generalized Carleman fractional linear shifts on the real line . 326 31.1 Iterations of fractional linear shifts and its properties. 326 31.2 Some additional properties of fractional linear shifts. 330 31.3 On a finite group of fractional linear transformations. 331

32 Singular integral equations with a generalized Carleman fractional linear shift . . . . . . . 333

33 Bibliographic Notes to Chapter 6 . . . . . . . . . . . 337

7 Application to Hankel Type and Multidimensional Integral Equations 339

339 340

34 Convolution integral equations of Hankel type . . . . 34.1 Formulation of the main result . . . . . . . . 34.2 The scheme of the proof and some basic ideas . 343 34.3 Proof of Theorem 34.1. The case p = 2, 'Y = 0 . 347 34.4 Proof of Theorem 34.1. The general case. . . . 349 34.5 The case of the equation corresponding to the diffraction

problem . . . . . . . . . . . . . . . . . . . . . . . . . 359 35 Some multidimensional singular type equations with shifts. 361

35.1 Some properties of linear involutive transformations in Rn .......................... 361

35.2 Wiener-Hopf operators with reflection in sectors on the plane ............................ 366

35.3 Convolution operators with Carleman linear transform. 370 35.4 Equations with homogeneous kernels and the inversion

shift in Rn ......................... 379 35.5 Functional equations with shifts in different variables.. 385 35.6 On a functional equation on the torus and factorization

with shift . . . . . . . . . 388 36 Bibliographic Notes to Chapter 7 . . . . . . . . . . . . . . . .. 393

Contents

Bibliography

Index

List of Symbols

Xlll

395

419

425

Introduction

This book is based on the original Russian edition (see Karapetiants and Samko [127]), but it should not be considered either as a mere translation or an enlarge­ment. The authors have completely rearranged the presentation and rewritten the content to reflect results obtained since publication of the original.

The book is devoted to some general approaches to investigating Fredholm­ness (= Noetherity) of various equations (singular integral equations, continual or discrete convolution type equations, equations with homogeneous kernels and others, one-dimensional and multidimensional), unified by the same idea. Namely, they involve some involutive operator Q, that is, an operator Q, such that Q2 = lor, more generally, Qn = I. Such equations can be written in the form

K<p: = (A + QB)<p = f (1)

or, more generally,

(2)

Aj being linear bounded operators in a Banach space X under consideration. The typical situation is such that the operators Aj form an algebra or, perhaps a lineal .c, for the elements for which the question of Fredholmness is clear.

Interest in equations of this kind appeared long ago, first in the theory of singular integral equations since the singular operator 8 along a closed curve has the property 8 2 = I. However, the most important impulse for development of the general theory of equations with involutive operators came from the theory of singular integral equations with a Carleman or generalized Carleman shift,

XVI Introduction

developed by G .S. Litvinchuk; see the books [170]' [155]. Of course, one may consider not only singular integral equations, but also differential or pseudodif­ferential equations with involutive shifts. Our interests in this book are mainly connected with singular type equations.

It was natural to have a theory that would allow us to construct, for a given operator K, another operator MEL not "containing" the operator Q, for which the criteria of Fredholmness is known, and such that there exists a simple connection between K and M that provides their simultaneous Fredholmness. This idea, in particular, is realized in different settings in different parts of the book. We present two abstract schemes of investigation of equations of types (1) and (2). One of them is given in Sections 14 and 19 and provides for our presentation of some simple axioms (Axioms 1-4) without the use of matrix operators. This scheme covers equations of the form (1), with n = 2 in Section 14 and an arbitrary n in Section 19. In the case n = 2 it proves possible to give some additional results on the necessity of Fredholmness conditions. Another scheme presented in Section 22 is based on fewer axioms, but requires the use of matrix operators. In applications, the first approach leads as usual to more effective Fredholmness criteria and the index formulae.

The abstract matrix approach has as a prototype the well-known idea of pas­sage from singular integral equations with Carleman or generalized Carleman shift to a system of such equations without shift, which was widely used in the theory of singular equations with shifts; see Litvinchuk [170]. In the form of the corresponding exact matrix identity, this idea was given in Gohberg and Krupnik [83] in the case n = 2. In this book, the generalization of this abstract identity to the case of an arbitrary n is the basis for a general approach to investigation of equations with iterations of a generalized involutive operator of order n.

Choosing different realizations of the operators Aj and Q, we get various concrete equations:

a) singular integral equations with Carleman or generalized Carleman shifts on a closed or open curve;

b) singular integral equations or convolution type operators, containing, together with the unknown function, its complex conjugate as well;

c) discrete convolutions with oscillating coefficients;

d) singular integral equations in convolutions with reflection;

e) equations with kernels homogeneous of degree -1 with a similar shift;

f) multidimensional convolution type equations with linear shifts;

g) multidimensional equations with homogeneous kernels and inversion, and others.

Introduction XVll

Therefore, the statements of their Ftedholmness are obtained as an immediate corollary to the general theorems of Sections 14, 19, and 22, relating to the operators in equations (1) and (2). The only thing that needs to be done to derive this corollary is to verify the validity of the axioms in every concrete case. Applications to equations (a)-(e) are considered in Sections 15-18. We remark that while dealing, for example, with equations (a) and (b), general theorems for equations (1) and (2) are realized directly in the case of continuous coefficients. In the more difficult case of piecewise coefficients, the equations should be modified before applying the abstract theory of Sections 14 or 19. Multidimensional equations (f) and (g) are treated in Section 35.

We consider the equations

m n

Kip: == L L pj-lQk-l Ajkip = f, (3) j=lk=l

which are more general than (2), where pm = Qn = I, and it is assumed that the operators P and Q are connected by the condition that there exists a number 'Y E C 1 and an integer 1/ such that

(4)

Condition (4) guarantees that the algebra generated by the operators P and Q has finite dimension. Some general theorems on the Ftedholmness of equation (3) under the assumption (4) are also obtained; see Section 22.

It goes without saying that this more general result provides wider possi­bilities for applications, in particular, to convolution type operators with both reflection and complex conjugation, to singular integral equations on closed or open curves with a finite group of shifts in the case of piecewise coefficients, to discrete convolution equations with both oscillation in the coefficients and reflection under the sign of the unknown sequence. Such applications are dealt with in Sections 23-25, Chapter 5.

A special place in the book is occupied by Chapter 6, devoted to singular integral equations with shift on Rl, which is an example of an unbounded curve. In this case the results on Fredholmness are also obtained by means of the abstract approach presented in Sections 14 and 22. However, this requires some modification of the way to introduce a bounded shift operator, and there arises the preliminary problem of considering singular integral operators perturbed by integral operators with a kernel homogeneous of degree -1. The final results depend heavily on the way the involutive shift operator is introduced and on different ways leading to different kinds of assumptions about the coefficients of the equation. In Section 30 we give an application to potential type operators with shift.

We also single out a special presentation of Hankel-type integral equations that generalize an equation in diffraction theory; see Section 34. The final results here are given in terms of the so-called standard lemniscates.

xviii Introduction

The above topics - the abstract approaches presented in Sections 14, 19, and 22, and their application to Fredholmness of various concrete equations -constitute the content of Chapters 4-7. Chapter 1 contains some preliminaries. Most of them are given with proofs for the reader's convenience. In Chapter 2 we present the Fredholm theory of equations with kernels homogeneous of degree -n in n-dimensional space and one-dimensional convolution type equations with discontinuous symbols. These theories are needed to consider more general equations of such type with involutions. In Chapter 3 we consider the simplest object, which is the singular integral equation with Carleman shift in the case of continuous coefficients. Although this object is well known and the results for it follow from the general theorem of Section 14, we examine it in Sections 11 and 12 independently of the abstract scheme in order to show by this example the genesis of the ideas which led us to the abstract approach of Sections 14 and 19.

The abstract scheme and its applications, as presented in this book, are based mainly on investigations reported in a series of the authors' papers. The authors, in studying some classes of singular integral equations with shift and discrete Wiener-Hopf operators with oscillating coefficients, arrived at the con­struction of an abstract theory for the Fredholm (= N oether) properties of the equations (A+QB)cp = j, Qn = I, within the framework of an abstract Banach space setting in the general noncommutative case. Part of the applications is connected with singular integral equations with generalized Carleman shifts. Considering these equations as a realization of the abstract equations (1)-(3), we show that for them it is possible to get both the known and new results by a unique method that is suitable for many other types of equations.

Going back to earlier investigations of abstract equations with an operator satisfying the condition Q2 = I, we note that the first one was undertaken by Z. Khalilov within the framework of normed rings; see his book [141]. The theory developed there was a direct treatment of the theory of singular inte­gral equations with continuous coefficients within the framework of an abstract normed ring. And it was algebraic in the sense that it was based just on the idea of regularization and did not include any method to calculate the index.

A significant step in the abstract theory was made by Cherskii [32], who constructed the abstract theory of the characteristic singular integral equation. He was the first to give the formula for the index in terms of the so-called factorization index of the operator coefficient corresponding to the abstract Riemann boundary problem.

Przeworska-Rolewicz [213] investigated equations with an involutive operator or algebraic (P( Q) = 0, P being a polynomial) or almost algebraic operator (P(Q) is compact). This leads to an algebraic approach that allowed us to construct regularizers and, in some cases, to obtain solutions of the equations. A number of papers by Przeworska-Rolewicz were reflected in the book of Przeworska-Rolewicz [213].

Introduction xix

The investigations carried out in the above mentioned research were, in fact, based on a model of singular integral operators. For this reason, the theories developed there did not cover other types of equations with an involutive oper­ator, e.g., singular integral equations with a Carleman shift and even equations with complex conjugate unknowns. In abstract terms this means that in equa­tions of the form (A + QB)<p = f with Q2 = I, the operators A and B were assumed to commute with the involutive operator Q up to compact terms. In the case of singular integral equations with a Carleman shift, this immediately requires invariance of the coefficients of the equations with respect to shift.

Every chapter is completed by bibliographic notes in which one can find the history of the question and references to some related investigations not cov­ered in the book. In particular, more detailed information on investigations of abstract equations with involutive operators is presented in Section 21. De­tailed references on equations with shifts may be found in Litvinchuk [170] and Kravchenko and Litvinchuk [155]; see also the survey paper Karlovich, Kravchenko and Litvinchuk [135] on investigations of singular equations with shift.

We also mention the recent books Antonevich [4], Antonevich and Lebedev [7], Antonevich, Belousov and Lebedev [5]-[6]' close to the topic of this book, which treat different kinds of functional differential equations within an L 2-

setting, based on the C* approach. The authors are grateful to one another for the many years of happy collab­

oration, although at intervals and in general not always coinciding in research interests. This period had its origin at the beginning of the 1970s at Rostov State University. It began soon after the Fredholm theory of singular equa­tions with generalized Carleman shift was constructed by Litvinchuk, then a professor at Rostov State University. The authors wish to emphasize that the theory developed by Litvinchuk and constant communication with him strongly influenced their interest in equations with involutive operators.

The authors owe the suggestion and encouragement to undertake this project to Francisco Sepulveda Teixeira, Instituto Superior Tecnico of Lisbon, who was in fact the initiator of the idea of writing this book for the English reading audience.

The preparation of this work was supported by Fundagao para a Ciencia e a Tecnologia, Ministry of Science and Technology of Portugal, under the grant PRAXIS XXI/BCC/1977/98 during the first author's visit to Centro de Matematica Aplicada of Instituto Superior Tecnico in Lisbon. The authors are thankful to Centro de Matematica Aplicada for providing facilities for the work on the book and in particular for the possibility of preparing the manuscript in l5\'IEX2e.

The authors will always remember the general friendly scientific atmosphere at the Department of Mathematics and Mechanics of Rostov State University where the first Russian version of the book was written and from which both

xx Introduction

authors originate. They also highly appreciate the long leave from Rostov Uni­versity given to the first author for writing the book.

Thanks go also to numerous friends for support. Discussions of various ques­tions with Yu.I. Karlovich, V.S. Rabinovich, and N.L. Vasilevski were helpful. It is the authors' pleasure to acknowledge their thanks to all of them. We also express our thanks to G.Yu. Vinogradova who read a significant part of the first version of the book which helped to eliminate many misprints.

Our last but not least gratitude is to our families who not only were very patient but constantly gave us moral support and shared our enthusiasm during the long period of preparation of the book.

Nikolai Karapetiants Rostov State University Russia

Stefan Samko University of Algarve Portugal

January 28, 2000

Notation

In the context of the book the following notation is adopted:

N = {l, 2, 3, 0 0 o}, Z+ = {a, 1,2,3,000}, Z = {a, ±1, ±2, ±3, 0 0 o};

lp = {{CPn}':'=: Ilcpllt = 2::=-=ICPnIP < oo}; (r, m) denotes the maximal common divider of the positive integers rand m;

o is the empty set;

C is the complex plane;

Rn is the Euclidean space, x = (Xl, 0 0 0 , xn) ERn;

R'+. = {X = (Xl,ooo,Xn): Xn > a}, el = (1,0,000,0), en = (0,000,0,1), 1 = (1,1,000 ,1);

X 0 Y = XlYl + 000 + XnYn, Ixi = (x 0 X)!, x' = I~I; < x >= VI + Ix12; j = (jl, 000 ,jn) is a multi-index with Ijl = jl + 000 + jn; {O,l}n is the set of multi-indices j = (jl, 0 0 0 , jn) with jk taking values ° or 1

only;

Zm = Z X Z x 0 0 0 x Z;

101 is the Lebesgue measure of the set 0 ERn;

B(xo,r) = Bn(xo,r) = {x E Rn: Ix - xol < r};

sn-l = {x E Rn: Ixl = I}, Isn-ll = if;), ISol = 1;

SO(n) is the group of rotations in Rn;

Xn(x) is the characteristic function of the set 0, equal to 1 for x E 0 and ° for x 1. 0, X(x) = XB(O,l)(X);

xxii Notation

Pncp(x) := xn(x)cp(x); iln is the compactification of Rn by the unique infinite point;

D - (~ ....!L) Dj _ alll . - aXl"'" 8xn' - 8xil ... ax~n'

Cff' = Cff'(Rn) is the space of infinitely differentiable functions with a compact support;

S = S(Rn) is the Schwartzian test function space;

Bcm(Rn) is the space of functions in Rn differentiable up to the order m and bounded in Rn with all their derivatives;

Lp(n; p) = {J(x) : In If(x)IP p(x)dx < oo}, 1 '5:. p < 00, p(x) ;:: ° being a weight function;

Lp(n) = Lp(n; 1), Loo(n) = {f(x) : esssuPnlf(x)1 < oo};

IlfIILp(n) = Un If(x)IPdx)l/P, 1'5:. p < 00, IlfIIL=(n) = esssuPnlf(x)l;

IlfIILp(n;p) = Ilfpl/PIILp(n), Ilfllp = IlfIILp(Rn); L;,(n) is the space of vector functions 'I' = ('1'1, ... ,'I'm) with components in

Lp(n) and natural norm;

PC(r) is the space of piecewise continuous functions on r with a finite number of discontinuity points;

In writing of the type f (.), by . we denote the variable with respect to which the norm is calculated, integration is carried out, etc;

(Thf)(X) = f(x - h) is the translation operator;

(a) r(a+1). h b' . 1 £Ii' k = k!r(a+l-k) IS t e momla coe clent;

[a] is the integer part of a, [a] '5:. a < [a] + 1;

a _ { t a , t > ° . a _ {O, t > ° . t+ - 0, t < ° ,t_ - Itl a , t < ° ' (to~td denotes an oriented arc of a curve starting at the point to and ending at t 1 ;

o denotes the end of proof.

Equations with Involutive Operators