Hausdorff–Berezin operators on weighted spaces

Abstract

The research presented in this paper is a continuation of the recent studies of new classes of integral operators on the unit disc D, which are called Hausdorff– Berezin operators. We consider general operators of Hausdorff–Berezin type constructed with an arbitrary positive Radon measure on the unit disc within the framework of weighted spaces with the so-called Möbius weights. This is a fairly wide class of weights in the unit disc which includes classical radial weights with singularities or zeroes at the boundary. Sufficient conditions for the boundedness of the Hausdorff–Berezin operators in these spaces are obtained. In the case of nonnegative kernels, a boundedness criterion is obtained in Lp(D). Approximation properties of operators of Hausdorff–Berezin type are also studied. We also discuss some applications.