Stevo Stević

Copyright © 2005 Stevo Stević. 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

Let f be a measurable function defined on the unit polydisc Un in Cn and let ωj(zj), j=1,…,n, be admissible weights on the unit disk U, with distortion functions ψj(zj), ℒω→,Np,q(Un)={f|‖f‖ℒω→,Np,q<∞}, where ‖f‖ℒω→,Np,qq=∫[0,1)nMpq(f,r)Πj=1nωj(rj)drj, and 𝒜ω→,Np,q(Un)=ℒω→,Np,q(Un)∩H(Un). We prove the following result: if p,q∈[1,∞) and for all j=1,…,n, ψj(zj)(∂f/∂zj)(z)∈ℒω→,Np,q, then f∈𝒜ω→,Np,q and there is a positive constant C=C(p,q,ωj,n) such that ‖f‖𝒜ω→,Np,q≤C(|f(0)|+∑j=1n‖ψj(∂f/∂zj)‖ℒω→,Np,q).