A class of Berezin-type operators on weighted Fock spaces with A∞-type weights

Let 0<α,β,t<∞ and μ be a positive Borel measure on Cⁿ. We consider the Berezin-type operator St,α,βμ defined by St,α,βμf(z):=(∫Cⁿe-β/2|z-u|²|f(u)|ᵗe-α t/2|u|²dμ(u))1/t, zⁿ. We completely characterize the boundedness and compactness of St,α,βμ from the weighted Fock space Fᵖα,w into the Lebesgue space Lq(wdv) for all possible indices, where w is a weight on Cⁿ that satisfies an A∞-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∞-type weights.