Hilbert-type operator induced by radial weight on Hardy spaces

Abstract

We consider the Hilbert-type operator defined by $$H_{\omega }(f)(z)={\int }_0^{1}f(t)\left(\frac{1}{z}{\int }_0^z{B}_t^{\omega }(u)du\right)\omega (t)dt,$$ where $\{B_{\zeta }^{\omega }{\}}_{\zeta \in \mathbb{D}}$ are the reproducing kernels of the Bergman space $A_{\omega }^2$ 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{\widehatω(r)}{\widehatω\left( \frac{1+r}{2}\right)}<\infty, \tag† \end{equation} and \begin{equation*} \sup\limits_{0<r<1}\left(\int_0^r \frac{1}{\widehatω(t)^p} dt\right)^{\frac{1}{p}} \left(\int_r^1 \left(\frac{\widehatω(t)}{1-t}\right)^{p'}\,dt\right)^{\frac{1}{p'}} <\infty, \end{equation*} where $\hat{\omega }(r)={\int }_r^1\omega (s)ds$. We also prove that $H_{\omega }:H^1\to H^1$ is bounded if and only if \eqref{abs1} holds and $$\underset{r\in [0,1)}{\sup }\frac{\hat{\omega }(r)}{1-r}\left({\int }_0^r\frac{ds}{\hat{\omega }(s)}\right)<\infty .$$ 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.