Uniform estimates for Mittag–Leffler functions with smooth phase

In this paper we consider the problem on uniform estimates for Mittag–Leffler functions with the smooth phase functions having singularities $D_{\infty} $, $D_{4}^{\pm}$ and $A_{r}$. The generalisation is that we replace the exponential function with the Mittag–Leffler-type function, to study oscillatory type integrals.