Hausdorff operators on some classical spaces of analytic functions

Abstract

In this note, we start on the study of the sufficient conditions for the boundedness of Hausdorff operators $$ \begin{align*}(\mathcal{H}_{K,\mu}f)(z):=\int_{\mathbb{D}}K(w)f(\sigma_w(z))d\mu(w)\end{align*} $$on three important function spaces (i.e., derivative Hardy spaces, weighted Dirichlet spaces, and Bloch type spaces), which is a continuation of the previous works of Mirotin et al. Here, $\mu $ is a positive Radon measure, K is a $\mu $-measurable function on the open unit disk $\mathbb {D}$, and $\sigma _w(z)$ is the classical Möbius transform of $\mathbb {D}$.