The book is devoted to a systematic consideration of the problem of solvability of singular integral equations and boundary value problems for analytic functions with a shift. Let \(\Gamma\) be a simple (closed) contour and \(W\varphi (t)=\varphi[\alpha(t)]\) be the shift operator with diffeomorphism \(\alpha(t)\) of \(\Gamma\) onto itself which preserves (or changes) the orientation on \(\Gamma\). Usually it is supposed that \(\alpha^\prime(t)\in H_\mu (\Gamma)\) and \(\alpha^\prime(t)\neq 0, t\in \Gamma\) and, as a rule in this book, \(\alpha (t)\) is a Carleman shift that is \(\alpha[\alpha(t)]\equiv t\) for all \(t\in\Gamma\). Let further \(S\varphi= \int_\Gamma [\pi i(\tau-t)]^{-1}\varphi (\tau)d\tau\) denote the Cauchy singular integration operator and let \(P_\pm=\frac 12(I\pm S)\) be projectors connecting with Cauchy operator \(S\) so that \(P_{\pm}^2=I, P_++P_-=I\) where \(I\) is the identity operator. The main spaces considered are \(L_p(\Gamma), 1<p<\infty\) and \(H_\mu (\Gamma), 0<\mu\leq 1\).
The basic object which is studied in the book is the simplest polynomial operator with shift \[ K=(aI+bW)P_++(cI+dW)P_- .\tag{1} \] The Fredholm theory of the singular integral equations with shift was given in the case of Carleman shift in the book of G. S. Litvinchuk [Boundary value problems and singular integral equations with shift (Russian), Nauka, Moscow (1977; Zbl 0462.30029)] and in the case of non-Carleman shift in the book of V. G. Kravchenko and G. S. Litvinchuk [Introduction to the theory of singular integral operators with shift (1994; Zbl 0811.47049)]. Some aspects relevant to the theory developed in these books may be found in the books of A. B. Antonevich [Linear functional equations. Operator approach. Operator theory: Advances and Applications (1996; Zbl 0841.47001)] and of N. K. Karapetiants and S. Samko [Equations with involutive operators (2001; Zbl 0990.47011)], etc.
The main difference of the investigations in the book under review from the ones mentioned above is the systematic consideration of the more difficult problem of solvability instead only of the Fredholm problem. Of course, the author needs the Fredholm theory and some of its aspects are given in the preliminaries in chapter 1 as well as other necessary results. The solvability problem of some degenerated cases (so-called binomial operators) of (1) (for example, for operators of Carleman type \(K=aP_++bWP_+, \;K=cP_-+dWP_-\)) was considered in many papers or surveys and was reflected in the mentioned books by G. S. Litvinchuk and V. G. Kravchenko and G. S. Litvinchuk (see also the references in the book under review).
Here the author studies general operators of the form (1). The main aim is the following: a) construction of a solvability theory for binomial operators with shift (chapters 2,3,8); b) construction of a solvability theory for polynomial operators with Carleman shift (chapters 4-7) under some additional restrictions on the shift and coefficients.
The main problems considered in chapters 2,3,8 are the Haseman and related boundary value problems, the Carleman or Carleman type boundary value problem and Noether and solvability problems with a Carleman shift and complex conjugation for functions analytic in a multiply connected domain. The chapters 4-7 are devoted in principal to the solvability theory of generalized Riemann, Hilbert and Carleman boundary value problems and general characteristic singular integral equation with a Carleman fractional linear shift. Finally, chapter 9 is devoted to the analogous problems in the case of non-Carleman shift.
All these chapters present the history of the considered problem, references and a survey of closely related results. Some unsolved problems are also discussed.
The book is an important union of new results with known classical results. It can be very interesting and useful for various specialists in different domains of pure and applied mathematic and may be recommended to scientific investigators and postgraduate students.
Contents: Introduction. 1. Preliminaries. 2. Binomial boundary value problems with shift for a piecewise analytic function and for a pair of functions analytic in the same domain. 3. Carleman boundary value problems and boundary value problems of Carleman type. 4. Solvability theory of the generalized Riemann boundary value problem. 5. Solvability theory of singular integral equations with a Carleman shift and complex conjugated values in the degenerated and stable cases. 6. Solvability theory of general characteristic singular integral equations with a Carleman fractional linear shift on the unit circle. 7. Generalized Hilbert and Carleman boundary value problems for functions analytic in a simply connected domain. 8. Boundary value problems with a Carleman shift and complex conjugation for functions analytic in a multiply connected domain. 9. On solvability theory for singular integral equations with a non-Carleman shift. References. Subject index.