Some problems in Holder and Besov classes

In recent decades, the study of integral operators with Bergman kernels in spaces of smooth functions in complex and functional analysis has not lost its relevance. The article deals with the above-named operators in analytic spaces of the functions extended smoothly to the boundary of the domain, which boundary values belong to the Holder and Besov classes. We have described the behavior of such operators in a circle and a half-plane. It is established that an integral operator with Bergman kernels projects Holder classes in the case of a circle, and Besov classes in the case of a half-plane, onto the corresponding classes of analytic functions, that is, Bergman integral operator leaves the indicated classes invariant.