$L^p$--$L^q$ estimates for Shimorin-type integral operators

Let $\nu$ be a positive measure on $[0,1]$. A Shimorin-type operator $T_{\nu }$ is an integral operator on the unit disk given by $$T_{\nu }f(z)={\int }_{\mathbb{D}}\frac{1}{1-z\bar{\lambda }}\left({\int }_0^1\frac{d\nu (r)}{1-rz\bar{\lambda }}\right)f(\lambda )dA(\lambda ),$$ which originates from Shimorin’s work on Bergman-type kernel representations for logarithmically subharmonic weighted Bergman spaces. In this paper, we study $L^p$–$L^q$ estimates for $T_{\nu }$. Unlike classical Bergman-type operators, the critical line on the $(1\mathrm{/}p,1\mathrm{/}q)$-plane that separates the boundedness and unboundedness regions of $T_{\nu }$ is not immediately evident. Moreover, even along this line, new phenomena arise. In the present work, by introducing a quantity $c_{\nu }$, \begin{itemize} \item we first determine the critical boundary in the $(1\mathrm{/}p,1\mathrm{/}q)$-plane for bounded $T_{\nu }$; \item furthermore, on this critical line, we establish necessary and sufficient conditions for $T_{\nu }$ which have standard Bergman-type $L^p$–$L^q$ estimates, meaning that it is bounded in the interior of the region and admits weak-type and BMO-type estimates at endpoints. \end{itemize}