Weighted holomorphic Besov spaces on the polydisk

摘要

This work is an introduction of weighted Besov spaces of holomorphic functions on the polydisk. LetUnbe the unit polydisk inCnandSbe the space of functions of regular variation. Let1≤p<∞,ω=(ω1,…,ωn),ωj∈S(1≤j≤n)andf∈H(Un).The functionfis said to be an element of the holomorphic Besov spaceBp(ω)if‖f‖Bp(ω)p=∫Un|Df(z)|p∏j=1nωj(1-|zj|)/(1-|zj|2)2-pdm2n(z)<+∞, wheredm2n(z)is the2n-dimensional Lebesgue measure onUnandDstands for a special fractional derivative offdefined in the paper. For example, ifn=1thenDfis the derivative of the functionzf(z).We describe the holomorphic Besov space in terms ofLp(ω)space. Moreover projection theorems and theorems of the existence of a right inverse are proved.