Differential Properties of Generalized Potentialsof the Type Bessel and Riesz Type

In this paper we study differential properties of convolutions of functions with kernels thatgeneralize the classical Bessel-Macdonald kernels ... The theory ofclassical Bessel potentials is an important section of the general theory of spaces of differentiablefunctions of fractional smoothness and its applications in the theory of partial differentialequations. The properties of the classical Bessel-Macdonald kernels are studied in detail in thebooks of Bennett and Sharpley, S. M. Nikolskii, I. M. Stein, V. G. Mazya. The local behavior ofthe Bessel-Macdonald kernels in the neighborhood of the origin is characterized by the presenceof a power-type singularity ||-. At infinity, they tend to zero at an exponential rate. Therecent work of M. L. Goldman, A. V. Malysheva, and D. Haroske was devoted to the investigationof the differential properties of generalized Bessel-Riesz potentials.In this paper we study the differential properties of potentials that generalize the classicalBessel-Riesz potentials. Potential kernels can have nonpower singularities in the neighborhoodof the origin. Their behavior at infinity is related only to the integrability condition, so thatkernels with a compact support are included. In this connection, the spaces of generalized Besselpotentials generated by them belong to the so-called spaces of generalized smoothness. The casewith the satisfied criterion for embedding potentials in the space of continuous bounded functionsis considered. In this case, the differential properties of the potentials are expressed in termsof the behavior of their module of continuity in the uniform metric. Criteria for embedding ofpotentials in Calderon spaces are established and explicit descriptions of the module of continuityof potentials and optimal spaces for such embeddings are obtained in the case when the basespace for potentials is the Lorentz weight space. These results specify the general constructionsestablished in previous works.