The book presents a comprehensive exposition of methods and results on hypersingular integrals (HSI). A typical HSI for the whole Euclidean space has the form \[ (D^\alpha_\Omega f)(x)=\int_{R^n}\frac{(\Delta_y^\ell f)(x)} {|y|^{n+\alpha}}\Omega(y) dy, \quad\operatorname{Re} \alpha >0, \tag{1} \] where \((\Delta_y^\ell f)(x)\) is the finite difference of the function \(f(x)\). The difference may be a usual or generalized one. The function \(\Omega(y)\) is referred to as the “characteristic” of the HSI (1). The book also deals with various applications of HSI to the inversion of potential-type operators and description of some function spaces of fractional smoothness. In the simplest case \(\Omega(y)\equiv \text{const.}\) the operator \(D^\alpha_\Omega \equiv D^\alpha\) is known as the operator of Riesz fractional differentiation. It realises the positive fractional powers \((-\Delta)^{\frac{\alpha}{2}}\) of the Laplace operator. The Riesz fractional differentiation was introduced by E. Stein for \(0<\alpha<2\) to describe the spaces of Bessel potentials. A generalization of Stein’s result to the case of arbitrary \(\alpha>0\) (and more generally, to the anisotropic case) is due to P. I. Lizorkin. HSI (1) with a homogeneous characteristic were treated by W. Trebels, M. Fisher, and R. L. Wheeden within the frameworks of Bessel potentials. The investigation of HSI were continued by S. G. Samko within the framework of the spaces of Riesz potentials. They were presented in his book under the same title, which appeared in Russian in 1984 (Zbl 0577.42016). The reviewed book, although partially based on that edition, is in fact, written anew. It reflects the modern state of the results and methods related to HSI and their applications. It contains both the results of the author and the members of the research group created by him at the Rostov State University.
The main topics of the book are the following: 1) inversion and regularization of multidimensional potential type operators; 2) construction of explicit expressions for the positive fractional powers of some second order differential operators in \(R^n\); 3) investigation of the influence of the characteristic \(\Omega(y)\) of HSI (1) on the convergence of the HSI in the \(L_p\)-norm and almost everywhere; 4) applications of the method of approximative inverse operators for inverting potentials.
Part I of the book (Chapters 1-6) deals with the theory of HSI itself (normalizing constants, the Fourier transform, generalized differences, representation of homogeneous differential operators by HSI with homogeneous characteristics, convergence of HSI with characteristics in various classes etc.). Part II (Chapters 7-11) is devoted to some applications of HSI. Chapter 7 deals with the characterization of the spaces \(L_{p,r}^\alpha (R^n)\), \(\alpha>0\), \(1\leq p,r <\infty\), containing the spaces both of Bessel and Riesz potentials. They consist of functions \(f(x)\in L_r\) for which \(D^\alpha f \in L_p\). As is proved, \[ L_{p,r}^\alpha =L_r \cap I^\alpha (L_p), \qquad 1< p < \infty, \quad 1<r<\infty, \tag{2} \] where \(I^\alpha (L_p)\) is the space of Riesz potentials. The spaces of the type (2) are widely used nowadays. Domains of positive powers of degenerate differential operators, in particular, of some classical operators of mathematical physics, have such a structure with \(I^\alpha\) denoting the corresponding potential. Chapter 7 is specially devoted to spherical convolutions, in particular to spherical potential operators and spherical hypersingular integrals. In Chapter 8, the central one in the book, the author presents his results on the inversion of generalized Riesz potentials of the form \[ (K^\alpha_\theta\varphi)(x)= \int_{R^n}\frac{\theta(x-t)}{|x-t|^{n-\alpha}}\varphi(t) dt, \qquad 0< \alpha < n, \tag{3} \] with a homogeneous characteristic \(\theta (t)=\theta(t/|t|)\) in the elliptic case by means of HSI (1) with the so-called associated characteristic \(\Omega (t)\), which is effectively constructed by the symbol of the operator (3). These results gave rise to further applications of HSI as an effective tool for inverting potential-type operators. In case of radial characteristics \(\theta (t)=\theta(|t|)\) the author exploits his general method of the inversion of potentials via HSI, which is based on the solution of a certain functional equation which has some quasi-polynomial as its symbol. Of special interest is Chapter 9, in which the author shows that positive fractional powers of many classical differential operators in partial derivatives (heat, wave, Schrödinger operators and others) may be effectively realized in terms of some hypersingular integrals. Chapter 11 presents the so-called method of approximative inverse operators and its application for inverting potentials. This may be considered as an alternative approach to hypersingular integrals, when the hypersingular construction, whose convergence is guaranteed by using finite differences, is replaced by a limit of a sequence of operators with “nice” kernels. The author demonstrates the effectiveness of this approach by the examples of the Riesz and Bessel potentials and others. This method is especially of interest when applied to potentials in the non-elliptic case, when the symbol of the potential vanishes (on a set of zero measure in \(R^n\)).
The book is written in a very clear manner: the author first speaks about the main results and ideas, and only after that he pays attention to details.
The book combines features of a valuable scientific monograph and an excellent source both for experts in the field and beginners. The large useful bibliography includes 318 items. The book is rich of original ideas, which can give an impulse to further investigations of potentials, HSI, and function spaces related to such operators.
Contents. Part 1. Chapter 1. Some basics from the theory of special functions and operator theory. Chapter 2. The Riesz potential operator and Lizorkin type invariant spaces \(\Phi_V\). Chapter 3. Hypersingular integrals with constant characteristics. Chapter 4. Potentials and hypersingular integrals with homogeneous characteristics. Chapter 5. Hypersingular integrals with non-homogeneous characteristics. Chapter 6. Hypersingular integrals on the unit sphere. Part 2. Chapter 7. Characterization of some function spaces in terms of hypersingular integrals. Chapter 8. Solution of multidimensional integral equations of the first kind with a potential type kernel. Chapter 9. Hypersingular operators as positive fractional powers of some operators of mathematical physics. Chapter 10. Regularization of multidimensional integral equations of the first kind with a potential type kernel. Chapter 11. Some modification of hypersingular integrals and their applications. References. Author index. Subject index. Index of symbols.