Weighted theory of Toeplitz operators on the Fock spaces

We study the weighted compactness and boundedness of Toeplitz operators on the Fock spaces. Fix $\alpha>0$. Let $T_{\varphi}$ be the Toeplitz operator on the Fock space $F^2_{\alpha}$ over $\mathbb{C}^n$ with symbol $\varphi\in L^{\infty}$. For $1<p<\infty$ and any finite sum $T$ of finite products of Toeplitz operators $T_{\varphi}$'s, we show that $T$ is compact on the weighted Fock space $F^p_{\alpha,w}$ 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 $1\leq p<\infty$, we characterize the $r$-doubling weights $w$ such that $T_{\varphi}$ is bounded on the weighted spaces $L^p_{\alpha,w}$ via a $\varphi$-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].