Hilbert-type operator induced by radial weight on Hardy spaces

TL;DR

This paper characterizes the boundedness of a Hilbert-type operator induced by radial weights on Hardy spaces, providing necessary and sufficient conditions for various p-values and function spaces.

Contribution

It introduces new criteria for the boundedness of the operator on Hardy spaces and related spaces, extending understanding of weighted integral operators in complex analysis.

Findings

Bounded on H^p for 1<p<∞ under condition (∗).

Bounded on H^1 under a specific weight condition.

No compactness of H_ω on H^p for 1≤p<∞.

Abstract

We consider the Hilbert-type operator defined by $$ H_{\omega}(f)(z)=\int_0^1 f(t)\left(\frac{1}{z}\int_0^z B^{\omega}_t(u)\,du\right)\,\omega(t)dt,$$ where $\{B^{\omega}_\zeta\}_{\zeta\in\mathbb{D}}$ are the reproducing kernels of the Bergman space $A^2_\omega$ induced by a radial weight $\omega$ in the unit disc $\mathbb{D}$. We prove that $H_{\omega}$ is bounded on the Hardy space $H^p$, $1<p<\infty$, if and only if \begin{equation} \label{abs1} \sup_{0\le r<1} \frac{\widehat{\omega}(r)}{\widehat{\omega}\left( \frac{1+r}{2}\right)}<\infty, \tag{\dag} \end{equation} and \begin{equation*} \sup\limits_{0<r<1}\left(\int_0^r \frac{1}{\widehat{\omega}(t)^p} dt\right)^{\frac{1}{p}} \left(\int_r^1 \left(\frac{\widehat{\omega}(t)}{1-t}\right)^{p'}\,dt\right)^{\frac{1}{p'}} <\infty, \end{equation*} where $\widehat{\omega}(r)=\int_r^1 \omega(s)\,ds$. We also prove that $H_\omega: H^1\to H^1$ is bounded if and only if \eqref{abs1} holds and $$ \sup\limits_{r \in [0,1)} \frac{\widehat{\omega}(r)}{1-r} \left(\int_0^r \frac{ds}{\widehat{\omega}(s)}\right)<\infty.$$ As for the case $p=\infty$, $H_\omega$ is bounded from $H^\infty$ to $BMOA$, or to the Bloch space, if and only if \eqref{abs1} holds. In addition, we prove that there does not exist radial weights $\omega$ such that $H_{\omega}: H^p \to H^p $, $1\le p<\infty$, is compact and we consider the action of $H_{\omega}$ on some spaces of analytic functions closely related to Hardy spaces.