Reverse Carleson inequalities for weighted Fock spaces with $A_{\mathrm{\infty }}$-type weights

We characterize dominating sets for weighted Fock spaces $F_{\alpha ,w}^p$ induced by Muckenhoupt $A_{\mathrm{\infty }}$ weights, which leads to a sufficient condition for reverse Carleson measures for the spaces $F_{\alpha ,w}^p$ with restricted $A_{\mathrm{\infty }}$ weights. Applications to invertibility of Toeplitz operators, closed range property of Volterra operators, and atomic decomposition of $F_{\alpha ,w}^p$ are given.