Weighted Composition Operators from Hardy Spaces into Logarithmic Bloch Spaces

Abstract

The logarithmic Bloch space Blog⁡ 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 Hp (with 1≤p≤∞) into the logarithmic Bloch space. We also provide boundedness and compactness criteria for the weighted composition operator mapping Hp into the little logarithmic Bloch space defined as the subspace of Blog⁡ consisting of the functions f such that lim⁡|z|→1(1-|z|2)log⁡ (2/(1-|z|2))|f'(z)|=0.