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Fractional integrals and potentials. (English) Zbl 0864.26004
Pitman Monographs and Surveys in Pure and Applied Mathematics. 82. Harlow: Addison Wesley Longman Ltd. xiv, 409 p. (1996).
In this research monograph the author gives a detailed and rigorous account of the analytical theory of fractional integrals, Riesz potentials and related topics in one and several dimensions, also in balls and on spheres, and corresponding integral equations of first kind. Integral equations of second kind and fractional differential equations are outside the scope of the book which consists of eight chapters.
In Chapter 1 the basic analytic tools are described (e.g., Fourier analysis, distributions and spaces of test functions, spaces of Lizorkin type). The second, rather voluminous chapter, is devoted to one-dimensional fractional integrals whose theory is fundamental for that of fractional integrals and potentials in several dimensions. The treatment comprises fractional integrals with variable upper or lower limit as well as fractional integrals with fixed limits (Riesz potentials, Feller potentials), generally with order of fractional integration as a complex number with positive real part (in some instances also with zero real part). Fourier and Mellin transforms are investigated, and the actions of the fractional integral operators on various function spaces are considered. Some further topics analyzed are relations between fractional and singular integrals, problems of restriction and extension, local properties of fractional integrals of generalized functions, in particular of finite Borel measures, composition properties (semigroup structures), to name only a few. Great attention is paid to the Marchaud method of rewriting fractional integrals by aid of generalized differences that leads to a natural way of analytic continuation to “integration” orders with negative real parts, thus to a method of inverting fractional integral operators via hypersingular integrals (fractional differentiation). The idea of truncating such hypersingular integrals near the singularity helps to find methods of regularized inversion of fractional integral operators in important function spaces and in spaces of finite Borel measures (in bounded and unbounded intervals), and the author uses the opportunity to develop the principles of locally controllable regularization. In Chapter 3 the author develops his theory of one-dimensional fractional integro-differentiation via continuous wavelet transforms, starting from the Calderón reproducing formula, and exhibits corresponding representations of fractional derivatives and integrals. In Chapter 4 begins the treatment of problems in more than one dimension, first of Riesz potentials in a multidimensional real space. Inversion formulas by means of Poisson and Gauss-Weierstrass integrals and wavelet type representations of the Riesz fractional derivative are derived. In Chapter 5 oscillatory potentials (Helmholtz and Bessel potentials) and methods to invert them are investigated. Again, continuation methods play an essential role. In Chapter 6 potentials on a half-space (one-sided Riesz and one-sided oscillatory potentials) are discussed in detail. The final Chapters 7 and 8 are devoted to Riesz potentials in a ball and to fractional integrals on a sphere, respectively.
Although even more books on fractional integration and differentiation and their applications appear on the market, the author has convincingly succeeded in writing a monograph clearly distinct from all existing ones. In an impressive manner he has condensed, reworked and coherently presented material published in many sources and also interesting nice results of his own research some of which appearing in this book for the first time. Thanks to the author’s clear notational distinction between integration and differentiation operators his work is well readable (the lack of this distinction is a big obstacle to easy and smooth study of the works of several other authors). Naturally, in the topics treated by Rubin there are intersections with the big monograph (the “bible”) by S. G. Samko, A. A. Kilbas and O. I. Marichev [Fractional integrals and derivatives: theory and applications (Russian) (1987; Zbl 0617.26004; Engl. translation 1993; Zbl 0818.26003)], but the author’s method, style and viewpoint are his own, and by this and by the amount of additional material and of new results his book is a valuable complementation to the above mentioned monograph to which he also had contributed.
Rubin’s book closes with a bibliography of 193 titles, but regrettably does not contain an index.

26A33 Fractional derivatives and integrals
26-02 Research exposition (monographs, survey articles) pertaining to real functions
45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions
31B15 Potentials and capacities, extremal length and related notions in higher dimensions
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42C15 General harmonic expansions, frames
65R30 Numerical methods for ill-posed problems for integral equations
45L05 Theoretical approximation of solutions to integral equations