The multidimensional Riesz potential type operators are of interest within mathematical modelling in economics, mathematical physics, and other, both theoretical and applied, disciplines as they play a significant role for analysis on fractal sets. Approaches of operator theory are relevant to researching various equations, which are widespread in financial analysis. In this chapter, integral equations with potential type operators are considered for functions from generalized Hölder spaces, which provide content terminology for formalizing the concept of smoothness, briefly described in the presented chapter. Results on potentials defined on the unit sphere are described for convenience of the analysis. An inverse operator for the Riesz potential with a logarithmic kernel is carried out, and the isomorphisms between generalized Hölder spaces are proven.
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.
Marcel Riesz kernels generalize the ones of classical theory, and convolution with them implements the negative frac-tional powers of the Laplace operator. Along with hypersingular integrals, the Riesz potential type operators arise in new areas of analysis and its applications, for example, in various problems of mathematical physics. In the study of such problems, a significant role is played by the conditions of solvability of the corresponding multidimensional integral equations in certain function spaces, often non-classical but postulating the required analytical properties of solutions. In the presented paper, the Hardy-Littlewood type theorem is proved, considering the boundedness con-ditions for the potential with a power-logarithmic kernel and a density, integrable by Lebesgue with a specific power weight. It is shown that for the higher orders of potential, the image of a function from this class belongs to the weighted generalized Hölder space with a power-logarithmic characteristic. The action of this operator in the gener-alized Hölder spaces with a weight from the scale of power functions is also considered, including the isomorphisms of these spaces in special cases. Stereographic projection and theorems proved for the spherical Riesz potential type operators are applied. Consequently, the solvability conditions of a Poisson-type equation with a negative fractional power of the Laplace operator are obtained.
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