HOLOMORPHIC FUNCTIONS ON THE MIXED NORM SPACES ON THE POLYDISC

Title & Authors
HOLOMORPHIC FUNCTIONS ON THE MIXED NORM SPACES ON THE POLYDISC
Stevic, Stevo;

Abstract
We generalize several integral inequalities for analytic functions on the open unit polydisc $\small{U^n={\{}z{\in}C^n||zj|}$<$\small{1,\;j=1,...,n{\}}}$. It is shown that if a holomorphic function on $\small{U^n}$ belongs to the mixed norm space $\small{A_{\vec{\omega}}^{p,q}(U^n)}$, where $\small{{\omega}_j(\cdot)}$,j=1,...,n, are admissible weights, then all weighted derivations of order $\small{|k|}$ (with positive orders of derivations) belong to a related mixed norm space. The converse of the result is proved when, p, q $\small{{\in}\;[1,\;{\infty})}$ and when the order is equal to one. The equivalence of these conditions is given for all p, q $\small{{\in}\;(0,\;{\infty})}$ if $\small{{\omega}_j(z_j)=(1-|z_j|^2)^{{\alpha}j},\;{\alpha}_j}$>$\small{-1}$, 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.
Keywords
holomorphic function;mixed norm space;polydisc;weighted derivations;admissible weight;
Language
English
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