Bergman-Toeplitz operators on weakly pseudoconvex domains

TL;DR

This paper investigates the boundedness of Bergman-Toeplitz operators with specific symbols on certain pseudoconvex domains, extending previous results from strongly pseudoconvex to finite type domains.

Contribution

It generalizes known boundedness criteria for Bergman-Toeplitz operators from strongly pseudoconvex to finite type pseudoconvex domains.

Findings

Boundedness of $T_{\psi}$ characterized by $\alpha \ge \frac{1}{p} - \frac{1}{q}$

Extension of results to finite type pseudoconvex domains

Conditions for $L^p$ to $L^q$ mapping of Bergman-Toeplitz operators

Abstract

We prove that for certain classes of pseudoconvex domains of finite type, the Bergman-Toeplitz operator $T_{\psi}$ with symbol $\psi=K^{-\alpha}$ maps from $L^{p}$ to $L^{q}$ continuously with $1< p\le q<\infty$ if and only if $\alpha\ge\frac{1}{p}-\frac{1}{q}$, where $K$ is the Bergman kernel on diagonal. This work generalises the results on strongly pseudoconvex domains by \v{C}u\v{c}kovi\'{c} and McNeal, and Abeta, Raissy and Saracco.