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6 Preliminaries.- 6.1 The operator of singular integration.- 6.2
The space Lp(?, ?).- 6.3 Singular integral operators.- 6.4 The
spaces $$L_{p}^{ + }(\Gamma, \rho ), L_{p}^{ - }(\Gamma, \rho ) and
\mathop{{L_{p}^{ - }}}\limits^{^\circ } (\Gamma, \rho )$$.-
6.5 Factorization.- 6.6 One-sided invertibility of singular
integral operators.- 6.7 Fredholm operators.- 6.8 The local
principle for singular integral operators.- 6.9 The interpolation
theorem.- 7 General theorems.- 7.1 Change of the curve.- 7.2 The
quotient norm of singular integral operators.- 7.3 The principle of
separation of singularities.- 7.4 A necessary condition.- 7.5
Theorems on kernel and cokernel of singular integral operators.-
7.6 Two theorems on connections between singular integral
operators.- 7.7 Index cancellation and approximative inversion of
singular integral operators.- 7.8 Exercises.- Comments and
references.- 8 The generalized factorization of bounded measurable
functions and its applications.- 8.1 Sketch of the problem.- 8.2
Functions admitting a generalized factorization with respect to a
curve in Lp(?, ?).- 8.3 Factorization in the spaces Lp(?, ?).- 8.4
Application of the factorization to the inversion of singular
integral operators.- 8.5 Exercises.- Comments and references.- 9
Singular integral operators with piecewise continuous coefficients
and their applications.- 9.1 Non-singular functions and their
index.- 9.2 Criteria for the generalized factorizability of power
functions.- 9.3 The inversion of singular integral operators on a
closed curve.- 9.4 Composed curves.- 9.5 Singular integral
operators with continuous coefficients on a composed curve.- 9.6
The case of the real axis.- 9.7 Another method of inversion.- 9.8
Singular integral operators with regel functions coefficients.- 9.9
Estimates for the norms of the operators P?, Q? and S?.- 9.10
Singular operators on spaces H?o(?, ?).- 9.11 Singular operators on
symmetric spaces.- 9.12 Fredholm conditions in the case of
arbitrary weights.- 9.13 Technical lemmas.- 9.14 Toeplitz and
paired operators with piecewise continuous coefficients on the
spaces lp and ?p.- 9.15 Some applications.- 9.16 Exercises.-
Comments and references.- 10 Singular integral operators on
non-simple curves.- 10.1 Technical lemmas.- 10.2 A preliminary
theorem.- 10.3 The main theorem.- 10.4 Exercises.- Comments and
references.- 11 Singular integral operators with coefficients
having discontinuities of almost periodic type.- 11.1 Almost
periodic functions and their factorization.- 11.2 Lemmas on
functions with discontinuities of almost periodic type.- 11.3 The
main theorem.- 11.4 Operators with continuous coefficients - the
degenerate case.- 11.5 Exercises.- Comments and references.- 12
Singular integral operators with bounded measurable coefficients.-
12.1 Singular operators with measurable coefficients in the space
L2(?).- 12.2 Necessary conditions in the space L2(?).- 12.3
Lemmas.- 12.4 Singular operators with coefficients in ?p(?).
Sufficient conditions.- 12.5 The Helson-Szegoe theorem and its
generalization.- 12.6 On the necessity of the condition a ? Sp.-
12.7 Extension of the class of coefficients.- 12.8 Exercises.-
Comments and references.- 13 Exact constants in theorems on the
boundedness of singular operators.- 13.1 Norm and quotient norm of
the operator of singular integration.- 13.2 A second proof of
Theorem 4.1 of Chapter 12.- 13.3 Norm and quotient norm of the
operator S? on weighted spaces.- 13.4 Conditions for Fredholmness
in spaces Lp(?, ?).- 13.5 Norms and quotient norm of the operator
aI + bS?.- 13.6 Exercises.- Comments and references.- References.
This book is an introduction to the theory of linear
one-dimensional singular integral equations. It is essentually a
graduate textbook. Singular integral equations have attracted more
and more attention, because, on one hand, this class of equations
appears in many applications and, on the other, it is one of a few
classes of equations which can be solved in explicit form. In this
book material of the monograph [2] of the authors on
one-dimensional singular integral operators is widely used. This
monograph appeared in 1973 in Russian and later in German
translation [3]. In the final text version the authors included
many addenda and changes which have in essence changed character,
structure and contents of the book and have, in our opinion, made
it more suitable for a wider range of readers. Only the case of
singular integral operators with continuous coefficients on a
closed contour is considered herein. The case of discontinuous
coefficients and more general contours will be considered in the
second volume. We are grateful to the editor Professor G. Heinig of
the volume and to the translators Dr. B. Luderer and Dr. S. Roch,
and to G. Lillack, who did the typing of the manuscript, for the
work they have done on this volume.
About fifty years aga S. G. Mikhlin, in solving the regularization
problem for two-dimensional singular integral operators [56],
assigned to each such operator a func tion which he called a
symbol, and showed that regularization is possible if the infimum
of the modulus of the symbol is positive. Later, the notion of a
symbol was extended to multidimensional singular integral operators
(of arbitrary dimension) [57, 58, 21, 22]. Subsequently, the
synthesis of singular integral, and differential operators [2, 8,
9]led to the theory of pseudodifferential operators [17, 35] (see
also [35(1)-35(17)]*), which are naturally characterized by their
symbols. An important role in the construction of symbols for many
classes of operators was played by Gelfand's theory of maximal
ideals of Banach algebras [201. Using this the ory, criteria were
obtained for Fredholmness of one-dimensional singular integral
operators with continuous coefficients [34 (42)], Wiener-Hopf
operators [37], and multidimensional singular integral operators
[38 (2)]. The investigation of systems of equations involving such
operators has led to the notion of matrix symbol [59, 12 (14), 39,
41]. This notion plays an essential role not only for systems, but
also for singular integral operators with piecewise-continuous
(scalar) coefficients [44 (4)]. At the same time, attempts to
introduce a (scalar or matrix) symbol for other algebras have
failed.
This book is an introduction to the theory of linear
one-dimensional singular integral equations. It is essentually a
graduate textbook. Singular integral equations have attracted more
and more attention, because, on one hand, this class of equations
appears in many applications and, on the other, it is one of a few
classes of equations which can be solved in explicit form. In this
book material of the monograph [2] of the authors on
one-dimensional singular integral operators is widely used. This
monograph appeared in 1973 in Russian and later in German
translation [3]. In the final text version the authors included
many addenda and changes which have in essence changed character,
structure and contents of the book and have, in our opinion, made
it more suitable for a wider range of readers. Only the case of
singular integral operators with continuous coefficients on a
closed contour is considered herein. The case of discontinuous
coefficients and more general contours will be considered in the
second volume. We are grateful to the editor Professor G. Heinig of
the volume and to the translators Dr. B. Luderer and Dr. S. Roch,
and to G. Lillack, who did the typing of the manuscript, for the
work they have done on this volume.
This monograph is the second volume of a graduate text book on the
modern theory of linear one-dimensional singular integral
equations. Both volumes may be regarded as unique graduate text
books. Singular integral equations attract more and more attention
since this class of equations appears in numerous applications, and
also because they form one of the few classes of equations which
can be solved explicitly. The present book is to a great extent
based upon material contained in the second part of the authors'
monograph 6] which appeared in 1973 in Russian, and in 1979 in
German translation. The present text includes a large number of
additions and complementary material, essentially changing the
character, structure and contents of the book, and making it
accessible to a wider audience. Our main subject in the first
volume was the case of closed curves and continuous coeffi cients.
Here, in the second volume, we turn to general curves and
discontinuous coefficients. We are deeply grateful to the editor
Professor G. Heinig, to the translator Dr. S. Roeh, and to the
typist Mr. G. Lillack, for their patient work. The authors
Ramat-Aviv, Ramat-Gan, May 26, 1991 11 Introduction This book is
the second volume of an introduction to the theory of linear
one-dimensional singular integral operators. The main topics of
both parts of the book are the invertibility and Fredholmness of
these operators. Special attention is paid to inversion methods."
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