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.

Boundedness conditions for spherical and spatial variable-order Riesz potential-type operators with integrable density in variable-exponent Hölder spaces are proved.

Доказаны условия ограниченности сферического и пространственного операторов типа потенциала Рисса переменного порядка с суммируемой плотностью в пространствах переменной гельдеровости. Библиография: 20 названий.