Volterra-type operators mapping weighted Dirichlet space into $H^\infty$

TL;DR

This paper characterizes when certain integral operators, defined by functions with non-negative Maclaurin coefficients, are bounded or compact from weighted Dirichlet spaces to the space of bounded analytic functions, based on properties of the weights.

Contribution

It provides a complete description of the boundedness and compactness of Volterra-type operators from weighted Dirichlet spaces to $H^ $^ ext{infty}$, especially for weights satisfying upper doubling conditions.

Findings

Characterization of boundedness and compactness of $T_g$ for non-negative Maclaurin coefficient functions.

Identification of weight conditions ensuring $T_g$ is bounded or compact only if $g$ is constant.

Conditions on weights $ ext{omega}$ that guarantee the operator's properties.

Abstract

The problem of describing the analytic functions $g$ on the unit disc such that the integral operator $T_g(f)(z)=\int_0^zf(\zeta)g'(\zeta)\,d\zeta$ is bounded (or compact) from a Banach space (or complete metric space) $X$ of analytic functions to the Hardy space $H^\infty$ is a tough problem and remains unsettled in many cases. For analytic functions $g$ with non-negative Maclaurin coefficients, we describe the boundedness and compactness of $T_g$ acting from a weighted Dirichlet space $D^p_\omega$, induced by an upper doubling weight $\omega$, to $H^\infty$. We also characterize, in terms of neat conditions on $\omega$, the upper doubling weights for which $T_g: D^p_\omega\to H^\infty$ is bounded (or compact) only if $g$ is constant.