Riesz Potential in Generalized Hölder Spaces

Riesz Potential in Generalized Hölder Spaces

Boris Grigorievich Vakulov (Southern Federal University, Russia), Galina Sergeevna Kostetskaya (Southern Federal University, Russia) and Yuri Evgenievich Drobotov (Southern Federal University, Russia & Azov Sea Research Fisheries Institute, Russia & SPE Vibrobit LLC, Russia)
DOI: 10.4018/978-1-5225-3767-0.ch013

Abstract

The chapter provides an overview of the advanced researches on the multidimensional Riesz potential operator in the generalized Hölder spaces. While being of interest within mathematical modeling in economics, theoretical physics, and other areas of knowledge, the Riesz potential plays a significant role for analysis on fractal sets, and this aspect is briefly outlined. The generalized Hölder spaces provide convenient terminology for formalizing the smoothness concept, which is described here. There are constant and variable order potential type operators considered, including a two-pole spherical one. As a sphere is, in some sense, a convenient set for analysis, there are two results, proved in detail: the conditions for the spherical fractional integral of variable order to be bounded in the generalized Hölder spaces, whose local continuity modulus has a dominant, which may vary from point to point, and the ones for the constant-order two-pole spherical potential type operator.
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Introduction

The chapter called “Potentials and Their Basic Properties” of a monograph by N.S. Landkof (1973) begins with basic definition of a potential as “some integral operator, acting on a space of signed measures” (p. 43). Such an operator is defined by a kernel which is usually dependent only on the distance. It is emphasized, that M. Riesz kernels include the ones of the classical theory (Newtonian and logarithmic kernels) and Green kernels associated with a region as special and limiting cases. This is what makes the Riesz potential to be of a great interest for mathematical physics, but while fractal models are being considered, the new reasons for the operator to be important are being discovered.

Fractal sets significantly widen the scope of functional analysis and its applications as they generalize the classical vision and approaches, which are based on the concept of a smooth manifold. According to this, fractional calculus appears to provide a convenient theory of integration over fractal sets and non-integer-dimensional spaces, as well as to define key methods for solving specific problems. Let the definitions of functions and integrals on fractals be recalled here following (Tarasov, 2010) in order to outline briefly the motivation for fractional calculus operators to be considered while studying fractal models in any field of science.

Let be a non-empty subset of the -dimensional Euclidian space , and let be a countable family of disjoint subsets of diameter at most , covering : , for all , where

,
denotes a metric. A characteristic function of a subset is defined as follows:

A function, defined on , is called a simple one, if

, and .

Let the simple function be integrable on , i.e. , where

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