Toeplitz operators on weighted Fock spaces with $A_{\mathrm{\infty }}$-type weights

By establishing some reproducing kernel estimates, we characterize the bounded, compact and Schatten $p$-class Toeplitz operators with positive measure symbols on the weighted Fock space $F_{\alpha ,w}^2$ for $p\ge 1$, where $w$ is a weight on the complex plane satisfying an $A_{\mathrm{\infty }}$-type condition. Applications to Volterra operators and weighted composition operators are given.