A class of Berezin-type operators on weighted Fock spaces with $A_{\mathrm{\infty }}$-type weights

Let $0<\alpha ,\beta ,t<\mathrm{\infty }$ and $\mu$ be a positive Borel measure on ${\mathbb{C}}^n$. We consider the Berezin-type operator $S_{\mu }^{t,\alpha ,\beta }$ defined by $$S_{\mu }^{t,\alpha ,\beta }f(z):={({\int }_{{\mathbb{C}}^n}{e}^{-\frac{\beta }{2}\mathrm{\mid }z-u{\mathrm{\mid }}^2}\mathrm{\mid }f(u){\mathrm{\mid }}^t{e}^{-\frac{\alpha t}{2}\mathrm{\mid }u{\mathrm{\mid }}^2}d\mu (u))}^{1\mathrm{/}t},z\in {\mathbb{C}}^n\mathrm{.}$$ We completely characterize the boundedness and compactness of $S_{\mu }^{t,\alpha ,\beta }$ from the weighted Fock space $F_{\alpha ,w}^p$ into the Lebesgue space $L^q(wdv)$ for all possible indices, where $w$ is a weight on ${\mathbb{C}}^n$ that satisfies an $A_{\mathrm{\infty }}$-type condition. This solves an open problem raised by Zhou, Zhao and Tang [Banach J. Math. Anal. 18 (2024), Paper No. 20]. As an application, we obtain the description of the boundedness and compactness of Toeplitz-type operators acting between weighted Fock spaces induced by $A_{\mathrm{\infty }}$-type weights.