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

TL;DR

This paper investigates $L^p$--$L^q$ boundedness of Shimorin-type integral operators on the unit disk, identifying the critical boundary in the $(1/p,1/q)$-plane and establishing conditions for boundedness and endpoint estimates.

Contribution

It introduces a new quantity $c_ u$ to determine the critical boundary for boundedness of $T_ u$ and provides necessary and sufficient conditions on this boundary, including endpoint estimates.

Findings

Determined the critical boundary in the $(1/p,1/q)$-plane for boundedness.

Established necessary and sufficient conditions for $T_ u$ on the critical line.

Provided weak-type and BMO-type estimates at endpoints.

Abstract

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 \[ T_\nu f(z) = \int_{\mathbb{D}} \frac{1}{1 - z\overline{\lambda}} \left( \int_0^1 \frac{d\nu(r)}{1 - r z \overline{\lambda}} \right) f(\lambda) \, dA(\lambda), \] 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 $(1/p,1/q)$-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 $(1/p,1/q)$-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}