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Journal of Function Spaces and Applications
Volume 9, Issue 1, Pages 1-16
http://dx.doi.org/10.1155/2011/637083

Weighted holomorphic Besov spaces on the polydisk

1Fac. for Inf. and Appl. Math., University of Yerevan, Alek Manukian 1, Yerevan 25, Armenia
2Inst. for Math., University of Paderborn, Warburger Str. 100, D-33098 Paderborn, Germany

Received 1 March 2008

Academic Editor: Jurgen Appell

Copyright © 2011 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 1p<,ω=(ω1,,ωn),ωjS(1jn) and fH(Un). The function f is said to be an element of the holomorphic Besov space Bp(ω) if fBp(ω)p=Un|Df(z)|pj=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.