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Weighted composition operators from Hardy spaces into logarithmic Bloch spaces. (English) Zbl 1250.47027
Summary: The logarithmic Bloch space ${ℬ}_{log}$ is the Banach space of analytic functions on the open unit disk $𝔻$ whose elements $f$ satisfy the condition $||f||={sup}_{z\in 𝔻}{\left(1-|z|}^{2}{\right)log\left(2/\left(1-|z|}^{2}\right)\right)|{f}^{\text{'}}\left(z\right)|<\infty$. In this work, we characterize the bounded and the compact weighted composition operators from the Hardy space ${H}^{p}$ (with $1\le p\le \infty$) into the logarithmic Bloch space. We also provide boundedness and compactness criteria for weighted composition operators mapping ${H}^{p}$ into the little logarithmic Bloch space defined as the subspace of ${ℬ}_{log}$ consisting of the functions $f$ such that ${lim}_{|t|\to 1}{\left(1-|z|}^{2}{\right)log\left(2/\left(1-|z|}^{2}\right)\right)|{f}^{\text{'}}\left(z\right)|=0$.
MSC:
 47B33 Composition operators 30H10 Hardy spaces 30H30 Bloch spaces 46E15 Banach spaces of continuous, differentiable or analytic functions