Bergman-Toeplitz operators on fat Hartogs triangles

TL;DR

This paper investigates the boundedness and mapping properties of Bergman-Toeplitz operators on fat Hartogs triangles, providing new insights into their behavior in complex analysis and operator theory.

Contribution

It establishes $L^{p}$ mapping properties of Bergman-Toeplitz operators on fat Hartogs triangles, a class of complex domains, extending previous understanding in several complex variables.

Findings

Derived $L^{p}$ boundedness conditions for the operators

Characterized the operators' behavior depending on parameter $eta$

Extended known results to a new class of complex domains

Abstract

In this paper, we obtain some $L^{p}$ mapping properties of the Bergman-Toeplitz operator \[ f\longrightarrow T_{K^{-\alpha}}\left(f\right):=\intop_{\Omega}K_{\Omega}\left(\cdot,w\right)K^{-\alpha}\left(w,w\right)f\left(w\right)dV(w) \] on fat Hartogs triangles $\Omega_{k}:=\left\{ \left(z_{1},z_{2}\right)\in\mathbb{C}^{2}:\left|z_{1}\right|^{k}<\left|z_{2}\right|<1\right\} $, where $\alpha\in\mathbb{R}$ and $k\in \mathbb Z^+$.