Holomorphic functions on the mixed norm spaces on the polydisc

Abstract

We generalize several integral inequalities for analytic functions on the open unit polydisc Un = {z ∈ ℂn ||zj| < 1, j = 1,...,n}. It is shown that if a holomorphic function on U n belongs to the mixed norm space Aω→p,q(Un), where ωj(·) j=1,...,n, are admissible weights, then all weighted derivations of order |k| (with positive orders of derivations) belong to a related mixed norm space. The converse of the result is proved when, p, q ∈ [1, ∞) and when the order is equal to one. The equivalence of these conditions is given for all p, q ∈ (0, ∞) if ωj(zj) = (1 - |z j|2)αj, αj > -1, j = 1,... ,n (the classical weights.) The main results here improve our results in Z. Anal. Anwendungen 23 (3) (2004), no. 3, 577-587 and Z. Anal. Anwendungen 23 (2004), no. 4, 775-782.