Hankel and Berezin Type Operators on Weighted Besov Spaces of Holomorphic Functions on Polydisks

Abstract

Let S be the space of functions of regular variation and let ω = (ω₁,..., ω_n), ω_j ∈ S. The weighted Besov space of holomorphic functions on polydisks, denoted by B_p(ω) (0 < p < +∞), is defined to be the class of all holomorphic functions f defined on the polydisk Uⁿ such that \(||f||_{{B_{P(\omega )}}}^P = \int_{{U^n}} {|Df(z){|^p}\prod\limits_{j = 1}^n {{\omega _j}{{(1 - |{z_j}{|^2})}^{P - 2}}dm{a_{2n}}(z) < \infty } } \), where dm_2n(z) is the 2ndimensional Lebesgue measure on Uⁿ and D stands for a special fractional derivative of f.We prove some theorems concerning boundedness of the generalized little Hankel and Berezin type operators on the spaces B_p(ω) and L_p(ω) (the weighted L_p-space).