Van der Corput lemmas for Mittag-Leffler functions. II. $\alpha$-directions

Abstract

The paper is devoted to study analogues of the van der Corput lemmas involving Mittag-Leffler functions. The generalisation is that we replace the exponential function with the Mittag-Leffler-type function, to study oscillatory integrals appearing in the analysis of time-fractional partial differential equations. More specifically, we study integral of the form $I_{\alpha ,\beta }(\lambda )={\int }_{\mathbb{R}}{E}_{\alpha ,\beta }\left(i^{\alpha }\lambda \varphi (x)\right)\psi (x)dx,$ for the range $0<\alpha \le 2,\beta >0$. This extends the variety of estimates obtained in the first part, where integrals with functions $E_{\alpha ,\beta }\left(i\lambda \varphi (x)\right)$ have been studied. Several generalisations of the van der Corput lemmas are proved. As an application of the above results, the generalised Riemann-Lebesgue lemma, the Cauchy problem for the time-fractional Klein-Gordon and time-fractional Schrödinger equations are considered.

19 pages