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

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 $I_{\alpha,\beta}(\lambda)=\int_\mathbb{R}E_{\alpha,\beta}\left(i^\alpha\lambda \phi(x)\right)\psi(x)dx,$ for the range $0<\alpha\leq 2,\,\beta>0$. This extends the variety of estimates obtained in the first part, where integrals with functions $E_{\alpha,\beta}\left(i \lambda \phi(x)\right)$ 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 Schr\"{o}dinger equations are considered.