Weighted Composition Operators from Hardy Spaces into Logarithmic Bloch Spaces. (16th May 2012) - Colonna, Flavia; Li, Songxiao
Abstract : The logarithmic Bloch spaceℬ log is the Banach space of analytic functions on the open unit disk𝔻 whose elementsf satisfy the condition∥ f ∥ = sup z ∈ 𝔻 ( 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 spaceH p (with1 ≤ p ≤ ∞ ) into the logarithmic Bloch space. We also provide boundedness and compactness criteria for the weighted composition operator mappingH p into the little logarithmic Bloch space defined as the subspace ofℬ log consisting of the functionsf such thatlim | z | → 1 ( 1 - | z | 2 ) log (2 / ( 1 - | z | 2 )) | f ′ ( z ) | = 0 .