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).