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

Abstract

Let $0<\hat{I}\pm ,{\hat{I}}^2,t<\infty$ and $\hat{I}¼$ be a positive Borel measure on ${\mathbb{C}}^n$. We consider the Berezin-type operator $S_{\hat{I}}^{t,\hat{I}\pm ,{\hat{I}}^2}¼$ defined by $$S_{\hat{I}}^{t,\hat{I}\pm ,{\hat{I}}^2}¼f(z):={\left({\int }_{{\mathbb{C}}^n}{e}^{-\frac{\hat{I}}{²}2|z-u{|}^2}|f(u){|}^t{e}^{-\frac{\hat{I}\pm t}{2}|u{|}^2}d\hat{I}¼(u)\right)}^{1/t},z\in {\mathbb{C}}^n.$$ We completely characterize the boundedness and compactness of $S_{\hat{I}}^{t,\hat{I}\pm ,{\hat{I}}^2}¼$ from the weighted Fock space $F_{\hat{I}\pm ,w}^p$ into the Lebesgue space $L^q(wdv)$ for all possible indices, where $w$ is a weight on ${\mathbb{C}}^n$ that satisfies an $A_{\infty }$-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.

23 pages