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

Yuerang Li, Zipeng Wang, Kenan Zhang

TL;DR

This work characterizes the $L^p$–$L^q$ boundedness of Shimorin-type integral operators $T_\nu$ on the unit disk via a new critical index $c_\nu$ that captures how the measure $\nu$ concentrates near 1. A two-fold approach is developed: (i) kernel-norm estimates and a coefficient-multiplier analysis yield sharp interior $L^p$–$L^q$ bounds and a precise boundary line $\frac{1}{q}=\frac{1}{p}+\frac{1}{c_\nu}-1$, with endpoint weak-type/BMO behavior, and (ii) the operator is shown to be a $2/c_\nu$-Bergman–CZ operator under Carleson-type or hyperbolic integrability conditions, enabling standard Bergman-type estimates. The results establish a complete Bergman-space–style picture for Shimorin-type operators, including sufficiency/necessity, weak-type endpoints, and Bloch/BMO dual endpoints, and connect these estimates to multiplier growth $m_n$ via $m_n\sim (n+1)^{-s_0}$ with $s_0=2/c_\nu'$. Overall, the paper provides a sharp, measure-dependent L^p-L^q theory that generalizes classical Forelli–Rudin-type operators to Shimorin kernels.

Abstract

Let $ν$ be a positive measure on $[0,1]$. A Shimorin-type operator $T_ν$ is an integral operator on the unit disk given by \[ T_νf(z) = \int_{\mathbb{D}} \frac{1}{1 - z\overlineλ} \left( \int_0^1 \frac{dν(r)}{1 - r z \overlineλ} \right) f(λ) \, dA(λ), \] 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 $T_ν$. Unlike classical Bergman-type operators, the critical line on the $(1/p,1/q)$-plane that separates the boundedness and unboundedness regions of $T_ν$ is not immediately evident. Moreover, even along this line, new phenomena arise. In the present work, by introducing a quantity $c_ν$, \begin{itemize} \item we first determine the critical boundary in the $(1/p,1/q)$-plane for bounded $T_ν$; \item furthermore, on this critical line, we establish necessary and sufficient conditions for $T_ν$ 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}

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

TL;DR

This work characterizes the – boundedness of Shimorin-type integral operators on the unit disk via a new critical index that captures how the measure concentrates near 1. A two-fold approach is developed: (i) kernel-norm estimates and a coefficient-multiplier analysis yield sharp interior – bounds and a precise boundary line , with endpoint weak-type/BMO behavior, and (ii) the operator is shown to be a -Bergman–CZ operator under Carleson-type or hyperbolic integrability conditions, enabling standard Bergman-type estimates. The results establish a complete Bergman-space–style picture for Shimorin-type operators, including sufficiency/necessity, weak-type endpoints, and Bloch/BMO dual endpoints, and connect these estimates to multiplier growth via with . Overall, the paper provides a sharp, measure-dependent L^p-L^q theory that generalizes classical Forelli–Rudin-type operators to Shimorin kernels.

Abstract

Let be a positive measure on . A Shimorin-type operator is an integral operator on the unit disk given by which originates from Shimorin's work on Bergman-type kernel representations for logarithmically subharmonic weighted Bergman spaces. In this paper, we study -- estimates for . Unlike classical Bergman-type operators, the critical line on the -plane that separates the boundedness and unboundedness regions of is not immediately evident. Moreover, even along this line, new phenomena arise. In the present work, by introducing a quantity , \begin{itemize} \item we first determine the critical boundary in the -plane for bounded ; \item furthermore, on this critical line, we establish necessary and sufficient conditions for which have standard Bergman-type -- estimates, meaning that it is bounded in the interior of the region and admits weak-type and BMO-type estimates at endpoints. \end{itemize}
Paper Structure (11 sections, 23 theorems, 336 equations, 3 figures)

This paper contains 11 sections, 23 theorems, 336 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Two-sided $L^p$-norm estimates for Shimorin-type kernels
  3. The multiplier coefficient related to $T_\nu$
  4. The $\alpha$-Bergman-CZOs on the unit disk
  5. The proof of Theorem \ref{['pq-necessary']}
  6. Proof of sufficiency
  7. Proof of necessity
  8. The proof of Theorem \ref{['thm:new2']}
  9. Proof of Theorem \ref{['weak bd on the line']}
  10. The proof of Theorem \ref{['weak bd']}
  11. The proof of Theorem \ref{['dual of weak 1 alpha']}

Key Result

Theorem 1.1

Let $\nu$ be a finite positive Borel measure on $[0,1]$ and $(p,q)\in [1,\infty]^2\setminus \mathcal{C}$. Then $T_{\nu} : L^p(\mathbb{D}) \to L^q(\mathbb{D})$ is bounded if and only if $(p,q)$ satisfies one of the following conditions:

Figures (3)

  • Figure 1: $K_\alpha\ (0 < \alpha \leq 2)$
  • Figure 2: $K_\alpha\ (2 < \alpha < 3)$
  • Figure 3: Critical boundary lines

Theorems & Definitions (51)

  • Conjecture 1
  • Conjecture 2
  • Theorem 1.1
  • Remark 1.1
  • Definition 1.1
  • Theorem 1.2
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Proposition 2.1
  • ...and 41 more