Bergman-Toeplitz operators between weighted $L^p$-spaces on weakly pseudoconvex domains

TL;DR

This paper investigates the boundedness, compactness, and Schatten class membership of Bergman-Toeplitz operators on weighted $L^p$-spaces over pseudoconvex domains, extending recent results and providing new estimates and characterizations.

Contribution

It provides necessary and sufficient conditions for boundedness, estimates for essential norm, and Schatten class criteria for Bergman-Toeplitz operators on a broad class of pseudoconvex domains.

Findings

Established criteria for boundedness of $T_$

Derived estimates for the essential norm and compactness

Characterized Schatten class membership on $L^2()$

Abstract

In this paper we study the Bergman-Toeplitz operator $T_{\psi}$ induced by $\psi(w) = K_{\Omega}^{-\alpha}(w,w)d_{\Omega}^{\beta}(w)$ with $\alpha, \beta \geq 0$ acting from a weighted $L^p$-space $L_a^p(\Omega)$ to another one $L_a^q(\Omega)$ on a large class of pseudoconvex domains of finite type. In the case $1 < p \leq q < \infty$, the following results are established: \\ - Necessary and sufficient conditions for boundedness, which generalize the recent results obtained by Khanh, Liu and Thuc.\\ - Upper and lower estimates for essential norm, in particular, a criterion for compactness.\\ - A characterization of Schatten class membership of this operator on Hilbert space $L^2(\Omega)$.