Holomorphic functions on the mixed norm spaces on the polydisc
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We generalize several integral inequalities for analytic functions on the open unit polydisc Un = {z ∈ Cn ||z|| < 1, j = 1,...,n}. It is shown that if a holomorphic function on Un belongs to the mixed norm space [수식], where wj(ㆍ), 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 wj (zj) = (1 - |zj|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.
Abstract
We generalize several integral inequalities for analytic functions on the open unit polydisc Un = {z ∈ Cn ||z|| < 1, j = 1,...,n}. It is shown that if a holomorphic function on Un belongs to the mixed norm space [수식], where wj(ㆍ), 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 wj (zj) = (1 - |zj|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.
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