Weighted Composition Operators from Hardy Spaces into Logarithmic Bloch Spaces. (16th May 2012) - Colonna, Flavia; Li, Songxiao
Abstract : 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 ∈ 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 space H p (with 1 ≤ p ≤ ∞ ) into the logarithmic Bloch space. We also provide boundedness and compactness criteria for the weighted composition operator mapping H p into the little logarithmic Bloch space defined as the subspace of B log ⁡ consisting of the functions f such that lim ⁡ | z | → 1 ( 1 - | z | 2 ) log ⁡ (2 / ( 1 - | z | 2 )) | f ' ( z ) | = 0 .