论文标题
Mittag-Leffler功能的Van der Corput引理。 ii。 $α$ - 导向
Van der Corput lemmas for Mittag-Leffler functions. II. $α$-directions
论文作者
论文摘要
该论文致力于研究涉及Mittag-Leffler功能的Van der Corput引理的类似物。概括是我们用Mittag-Leffler型函数替换指数函数，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，我们研究了$ i_ {α，β}（λ）= \ int_ \ mathbb {r} e_ {α，β} \ left（i^αλ（x）\ right）ψ（x）ψ（x）dx，$ 0 <α\ leq 2，$ 0 $ 0 $ 0 $ 0.0 $ 0.0 $ 0.0 $ 0. 0 $ 0。这扩展了第一部分获得的各种估计值，其中已经研究了具有函数$ e_ {α，β} \ left（iλct（x）\ right）$的积分。证明了范德尔的几种概括。作为上述结果的应用，考虑了时间分数的klein-gordon和时间折叠的schrödinger方程，这是广义的riemann-lebesgue引理。
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 $I_{α,β}(λ)=\int_\mathbb{R}E_{α,β}\left(i^αλϕ(x)\right)ψ(x)dx,$ for the range $0<α\leq 2,\,β>0$. This extends the variety of estimates obtained in the first part, where integrals with functions $E_{α,β}\left(i λϕ(x)\right)$ 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.
