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

Let $0<\alpha,\beta,t<\infty$ and $\mu$ be a positive Borel measure on $\mathbb{C}^n$. We consider the Berezin-type operator $S^{t,\alpha,\beta}_{\mu}$ defined by $$S^{t,\alpha,\beta}_{\mu}f(z):=\left(\int_{\mathbb{C}^n}e^{-\frac{\beta}{2}|z-u|^2}|f(u)|^te^{-\frac{\alpha t}{2}|u|^2}d\mu(u)\right)^{1/t},\quad z\in\mathbb{C}^n.$$ We completely characterize the boundedness and compactness of $S^{t,\alpha,\beta}_{\mu}$ from the weighted Fock space $F^p_{\alpha,w}$ into the Lebesgue space $L^q(wdv)$ for all possible indices, where $w$ is a weight on $\mathbb{C}^n$ that satisfies an $A_{\infty}$-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 $A_{\infty}$-type weights.