Weighted composition operators from Hardy spaces into logarithmic Bloch spaces (Q442597)

Summary: The logarithmic Bloch space \(\mathcal B_{\log}\) is the Banach space of analytic functions on the open unit disk \(\mathbb D\) whose elements \(f\) satisfy the condition \(||f|| = \sup _{z \in \mathbb D}(1 - |z|^2)\log (2/(1 - |z|^2))|f'(z)| < \infty\). In this work, we characterize the bounded and the compact weighted composition operators from the Hardy space \(H^p\) (with \(1 \leq p \leq \infty\)) 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 \(\mathcal B_{\log}\) consisting of the functions \(f\) such that \(\lim_{|t| \rightarrow 1}(1 - |z|^2)\log (2/(1 - |z|^2))|f'(z)| = 0\).