COMPOSITION OPERATORS ON THE PRIVALOV SPACES OF THE UNIT BALL OF ℂn

Title & Authors
COMPOSITION OPERATORS ON THE PRIVALOV SPACES OF THE UNIT BALL OF ℂn
UEKI SEI-ICHIRO;

Abstract
Let B and S be the unit ball and the unit sphere in $\small{\mathbb{C}^n}$, respectively. Let $\small{{\sigma}}$ be the normalized Lebesgue measure on S. Define the Privalov spaces $N^P(B)\;(1\;<\;p\;<\;{\infty})$ by N^P(B)\;=\;\{\;f\;{\in}\;H(B) : \sup_{0 be a holomorphic self-map of B. Let $\small{{\mu}}$ denote the pull-back measure $\small{{\sigma}o({\varphi}^{\ast})^{-1}}$. In this paper, we prove that the composition operator $\small{C_{\varphi}}$ is metrically bounded on $\small{N^P}$(B) if and only if $\small{{\mu}(S(\zeta,\delta)){\le}C{\delta}^n}$ for some constant C and $\small{C_{\varphi}}$ is metrically compact on $\small{N^P(B)}$ if and only if $\small{{\mu}(S(\zeta,\delta))=o({\delta}^n)}$ as $\small{{\delta}\;{\downarrow}\;0}$ uniformly in $\small{{\zeta}\;\in\;S}$. Our results are an analogous results for Mac Cluer's Carleson-measure criterion for the boundedness or compactness of $\small{C_{\varphi}}$ on the Hardy spaces $\small{H^P(B)}$.
Keywords
Hardy spaces;Privalov spaces;composition operators;unit ball of $\small{\mathbb{C}^n}$;
Language
English
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