Oscillatory integrals for Mittag–Leffler functions

Variations of the van der Corput lemmas that involve Mittag-Leffler functions are studied in this paper. The extension involves replacing the exponential function with a Mittag–Leffler-type function. It allows one to analyze oscillatory integrals encountered in the study of time-fractional partial differential equations. Several generalizations of both the first and second van der Corput lemmas are established. Optimal estimates for decay orders in specific cases of Mittag–Leffler functions are also derived.