Weighted holomorphic Besov spaces on the polydisk

Abstract

This work is an introduction of weighted Besov spaces of holomorphic functions on the polydisk. Let U n be the unit polydisk in C n and S be the space of functions of regular variation. Let 1 ≤ p < ∞ , ω = ( ω 1 , …, ω n ), ω j ∈ S (1 ≤ j ≤ n ) and f ∈ H ( U n ) . The function f is said to be an element of the holomorphic Besov space B p ( ω ) if , where d m 2 n ( z ) is the 2 n ‐dimensional Lebesgue measure on U n and D stands for a special fractional derivative of f defined in the paper. For example, if n = 1 then D f is the derivative of the function z f ( z ). We describe the holomorphic Besov space in terms of L p ( ω ) space. Moreover projection theorems and theorems of the existence of a right inverse are proved.