Weighted integrals of holomorphic functions in the unit polydisc

Abstract

Let f be a measurable function defined on the unit polydisc U n in C n and let ω j (z j ), j = 1,...,n, be admissible weights on the unit disk U, with distortion functions ψ j (z j ), ℒ ω→,Np,q (U n ) = {f | || f || ℒω→,N p,q where || f || ℒω→,N p,qq = ∫ [01)n M pq (f, r) ∏ j=1n ω j (r j )dr j , and script A sign ω→,Np,q (U n ) = ℒ ω→,Np,q (U n ) ∩ H(U n ). We prove the following result: if p, q ∈ [1, ∞) and for all j = 1,...,n, ψ j (z j )(∂f/∂z j ) (z) ∈ ℒ ω→,Np,q , then f ∈ script A sign ω→,Np,q and there is a positive constant C = C(p,q,ω j ,n) such that || f || script A signω→,Np,q ≤ C(| f(0) | + ∑ j=1n || ψ j (∂f/∂z j ) || ℒω→,N p,q ).