Weighted holomorphic Besov spaces on the polydisk

Abstract

This work is an introduction of weighted Besov spaces of holomorphic functions on the polydisk. Let Un be the unit polydisk in Cn and S be the space of functions of regular variation. Let 1≤p<∞,ω=(ω1,…,ωn),ωj∈S(1≤j≤n) and f∈H(Un). The function f is said to be an element of the holomorphic Besov space Bp(ω) if ‖f‖Bp(ω)p=∫Un|Df(z)|p∏j=1nωj(1-|zj|)/(1-|zj|2)2-pdm2n(z)<+∞, where dm2n(z) is the 2n-dimensional Lebesgue measure on Un and D stands for a special fractional derivative of f defined in the paper. For example, if n=1 then Df is the derivative of the function zf(z).We describe the holomorphic Besov space in terms of Lp(ω) space. Moreover projection theorems and theorems of the existence of a right inverse are proved.