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Weighted composition operators from Bergman-Privalov-type spaces to weighted-type spaces on the unit ball. (English) Zbl 1250.47030

Necessary and sufficient conditions for boundedness and for compactness of weighted composition operators $\mu {C}_{\varphi }$ from the weighted Bergman-Privalov-type space $A{N}_{p,\alpha }$ to a weighted type space or a little weighted-type space on the unit ball in the $n$-dimensional complex vector space ${ℂ}^{n}$ are given. The spaces and the composition operator mentioned above are defined in the following way. Let $B$ be the unit ball in ${ℂ}^{n}$, $H\left(B\right)$ be the space of all holomorphic functions on $B$, $dV\left(z\right)$ be the normalized Lebesgue measure on $B$ and $d{V}_{\alpha }\left(z\right)={c}_{\alpha }{\left(1-|z|}^{2}{\right)}^{\alpha }dV\left(z\right)$, where $\alpha >-1$ and ${c}_{\alpha }$ is a normalization constant, that is, ${V}_{\alpha }\left(B\right)=1$. For each $p\ge 1$ and $\alpha >-1$, the weighted Bergman-Privalov-type space $A{N}_{p,\alpha }$ is defined in the following way:

$A{N}_{p,\alpha }=\left\{f\in H\left(B\right):{\int }_{B}{ln}^{p}\left(1+|f\left(z\right)|\right)\phantom{\rule{0.166667em}{0ex}}d{V}_{\alpha }\left(z\right)<\infty \right\}·$

The weighted-type space ${H}_{\mu }^{\infty }$ is defined as the space that consists of all $f\in H\left(B\right)$ such that $\underset{z\in B}{sup}\mu \left(z\right)|f\left(z\right)|<\infty$, where $\mu \left(z\right)$ is a positive continuous function (a weight) on $B$. The little weighted-type space ${H}_{\mu ,o}^{\infty }$ consists of all $f\in H\left(B\right)$ such that

$\underset{|z|\to 1}{lim}\mu \left(z\right)|f\left(z\right)|=0·$

For $f\in H\left(B\right)$, the corresponding weighted composition operator is defined by

$\left(\mu {C}_{\varphi }\right)\left(f\right)\left(z\right)=\mu \left(z\right)f\left(\varphi \left(z\right)\right),\phantom{\rule{0.166667em}{0ex}}z\in B,$

where $\varphi$ is a holomorphic self-map of $B$ and $\mu \in H\left(B\right)$ is fixed.

##### MSC:
 47B33 Composition operators 32A37 Spaces of holomorphic functions (several variables) 46E15 Banach spaces of continuous, differentiable or analytic functions 47B38 Operators on function spaces (general)