Abstract
The logarithmic Bloch spaceBlogis the Banach space of analytic functions on the open unit disk&#x1D53B;whose elementsfsatisfy the condition∥f∥=supz∈&#x1D53B;(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 spaceHp(with1≤p≤∞) into the logarithmic Bloch space. We also provide boundedness and compactness criteria for the weighted composition operator mappingHpinto the little logarithmic Bloch space defined as the subspace ofBlogconsisting of the functionsfsuch thatlim|z|→1(1-|z|2)log (2/(1-|z|2))|f'(z)|=0.
Highlights
Let X and Y be Banach spaces of analytic functions on a domain Ω in C, ψ an analytic function on Ω, and let φ be an analytic function mapping Ω into itself
We 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
Let H D be the set of analytic functions on D {z ∈ C : |z| < 1}
Summary
The logarithmic Bloch space Blog is the Banach space of analytic functions on the open unit disk. 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 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
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