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Weighted composition operators from Hardy spaces into logarithmic Bloch spaces. (English) Zbl 1250.47027
Summary: The logarithmic Bloch space log is the Banach space of analytic functions on the open unit disk 𝔻 whose elements f satisfy the condition ||f||=sup z𝔻 (1-|z| 2 )log(2/(1-|z| 2 ))|f ' (z)|<. In this work, we characterize the bounded and the compact weighted composition operators from the Hardy space H p (with 1p) 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 log consisting of the functions f such that lim |t|1 (1-|z| 2 )log(2/(1-|z| 2 ))|f ' (z)|=0.
47B33Composition operators
30H10Hardy spaces
30H30Bloch spaces
46E15Banach spaces of continuous, differentiable or analytic functions