Weighted theory of Toeplitz operators on the Fock spaces

Abstract

We study the weighted compactness and boundedness of Toeplitz operators on the Fock spaces. Fix $\hat{I}\pm >0$. Let $T_{\ddot{I}}†$ be the Toeplitz operator on the Fock space $F_{\hat{I}}^2\pm$ over ${\mathbb{C}}^n$ with symbol $\ddot{I}†\in L^{\infty }$. For $1<p<\infty$ and any finite sum $T$ of finite products of Toeplitz operators $T_{\ddot{I}}†$'s, we show that $T$ is compact on the weighted Fock space $F_{\hat{I}\pm ,w}^p$ if and only if its Berezin transform vanishes at infinity, where $w$ is a restricted $A_p$-weight on ${\mathbb{C}}^n$. Concerning boundedness, for $1\le p<\infty$, we characterize the $r$-doubling weights $w$ such that $T_{\ddot{I}}†$ is bounded on the weighted spaces $L_{\hat{I}\pm ,w}^p$ via a $\ddot{I}†$-adapted $A_p$-type condition. Our method also establishes a two weight inequality for the Fock projections in the case of $r$-doubling weights. Moreover, we characterize the corresponding weighted compactness of Bergman--Toeplitz operators, which answers a question raised by Stockdale and Wagner [Math. Z. 305 (2023), no. 1, Paper No. 10].