Integral representations in general weighted Bergman spaces

Abstract

We introduce fractional -integro-differentiation for functions holomorphic in the upper half-plane. It gives us a tool to construct Cauchy-Bergman type kernels associated with the weights Some estimates of the kernels enable us to obtain reproducing integral formulas for Bergman spaces with general weights which may decrease to zero with arbitrary rate near the origin. Accordingly, such Bergman functions have arbitrary growth near the real axis.