Description of some weighted spaces of holomorphic functions in terms of fractional derivatives

Abstract

Assuming that Un is the unit polydisk in the n-dimensional complex plane , H(U n ) is the set of holomorphic functions in U n , S is the space of functions of regular variation and ω =(ω1, …,ω n ), ω j ∈S, by Lp (ω) we denote the class of all measurable functions defined in Un and such that where dm 2n (z) is the 2n-dimensional Lebesgue measure in U n . In this article, for any p∈(0,∞) we prove that if and only if , where D β stands for the fractional derivative of f of some multi-index order β.