Hilbert-type operator induced by radial weight

Abstract

We consider the Hilbert-type operator defined by $$H_{\ddot{I}}‰(f)(z)={\int }_0^{1}f(t)\left(\frac{1}{z}{\int }_0^z{B}^{\ddot{I}}{‰}_t(u)du\right)\ddot{I}‰(t)dt,$$ where $\{B^{\ddot{I}}{‰}_{\hat{I}}\P {\}}_{\hat{I}\P \in \mathbb{D}}$ are the reproducing kernels of the Bergman space $A_{\ddot{I}}^2‰$ induced by a radial weight $\ddot{I}‰$ in the unit disc $\mathbb{D}$. We prove that $H_{\ddot{I}}‰$ is bounded from $H^{\infty }$ to the Bloch space if and only if $\ddot{I}‰$ belongs to the class $\hat{\mathcal{D}}$, which consists of radial weights $\ddot{I}‰$ satisfying the doubling condition ${\sup }_{0\le r<1}\frac{{\int }_r^1\ddot{I}‰(s)ds}{{\int }_{\frac{1+r}{2}}^1\ddot{I}‰(s)ds}<\infty$. Further, we describe the weights $\ddot{I}‰\in \hat{\mathcal{D}}$ such that $H_{\ddot{I}}‰$ is bounded on the Hardy space $H^1$, and we show that for any $\ddot{I}‰\in \hat{\mathcal{D}}$ and $p\in (1,\infty )$, $H_{\ddot{I}}‰:L_{[0,1)}^p\to H^p$ is bounded if and only if the Muckenhoupt type condition \begin{equation*} \sup\limits_{0<r<1}\left(1+\int_0^r \frac{1}{\widehatω(t)^p} dt\right)^{\frac{1}{p}} \left(\int_r^1 ω(t)^{p'}\,dt\right)^{\frac{1}{p'}} <\infty, \end{equation*} holds. Moreover, we address the analogous question about the action of $H_{\ddot{I}}‰$ on weighted Bergman spaces $A_{\hat{I}}^p½$.

Accepted manuscript (postprint)