Weighted holomorphic Besov spaces on the polydisk

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 ∥f∥ bp(ω) p = ∫ un |df(z)| pπ j=1 n ωj(1-|z j|)/(1-|z j| 2) 2-pdm 2n(z)<+∞,where dm 2n(z) is the 2n-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 df is the derivative of the function zf(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. copyright © 2011 hindawi publishing corporation.