The Riesz Potential Type Operator with a Power-Logarithmic Kernel in the Generalized Hölder Spaces on a Sphere

Abstract

Marcel Riesz kernels include the ones of the classical theory as special and limiting cases. Therefore, and due to their link with the fractional powers of the Laplace operator, Riesz potential type operators are of significant interest. Along with hypersingular integrals they form a valuable toolkit for considering various problems of mathematical physics. Riesz fractional integro-differentiation is studied in connection with fractal media, that can be, for instance, a porous material or a polymer. When researching integral equations on their solvability and the sustainability of the solutions, the issue on a relationship between the integrability of a pre-image and the integral operator’s smoothness is essential. For a spherical convolution operator, such a study can be based on two approaches: the Fourier—Laplace multipliers theory, and Zygmund type estimates, which are used to describe the behavior of the continuity modulus. In this chapter, both are applied to consider the Riesz potential type operator with a power-logarithmic kernel on a sphere.