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 $I_{\alpha ,\beta }(\lambda )={\int }_{\mathbb{R}}{E}_{\alpha ,\beta }(i^{\alpha }\lambda \varphi (x))\psi (x)dx$, for the range $0<\alpha \le 2,\beta >0$. This extends the variety of estimates obtained in the first part, where integrals with functions $E_{\alpha ,\beta }(i\lambda \varphi (x))$ 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.

该论文致力于研究涉及 Mittag-Leffler 函数的 van der Corput 引理的类似物。概括地说，我们用 Mittag-Leffler 型函数代替指数函数，以研究出现在时间分数偏微分方程分析中的振荡积分。更具体地说，我们研究形式的积分${\mathrm{一世}}_{\alpha ,\beta }(\lambda )={\int }_{\mathbb{电阻}}{乙}_{\alpha ,\beta }({\mathrm{一世}}^{\alpha }\lambda \phi (X))\psi (X)dX$，对于范围 $0<\alpha \le 2,\beta >0$. 这扩展了第一部分中获得的各种估计，其中函数积分${乙}_{\alpha ,\beta }(\mathrm{一世}\lambda \phi (X))$已被研究。证明了 van der Corput 引理的几个推广。作为上述结果的应用，考虑了广义黎曼-勒贝格引理、时间分数阶 Klein-Gordon 和时间分数阶薛定谔方程的柯西问题。