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.
Categories: math.FA