Гладкостные свойства оператора типа потенциала Рисса с логарифмической характеристикой

Abstract

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.