Asymptotic analysis of oscillatory integrals with the Mittag-Leffler function as an oscillatory kernel

Abstract

Suppose $\phi \in C^{\infty }{\left( {\mathbb {R}}\rightarrow [0,\infty )\right) }$ and $\psi \in C^{\infty }_{c}{\left( {\mathbb {R}}\rightarrow {\mathbb {C}}\right) }$. Let $E_{\alpha }$, $0<\alpha \le 1$, denote the one-parameter Mittag-Leffler function. We show that the integral ${{\mathcal {I}}}_{\alpha }(\lambda ):=\int _{{\mathbb {R}}} {E}_{\alpha }{\left( \left( {\dot{\imath }} \lambda \phi (x)\right) ^{\alpha }\right) } \psi (x)dx$ is an oscillatory integral perturbation of the oscillatory integral $\frac{1}{\alpha }\int _{{\mathbb {R}}} e^{{\dot{\imath }} \lambda \phi (x)} \psi (x)dx$. We obtain an asymptotic expansion for ${{\mathcal {I}}}_{\alpha }(\lambda )$ as $\lambda \rightarrow \infty$ both when $\phi$ is non-stationary and when it has a non-degenerate stationary point in the support of $\psi$. Interestingly, the asymptotic behaviour of ${{\mathcal {I}}}_{\alpha }$ in the latter case depends on whether $\alpha <1/2$, $\alpha =1/2$, or $\alpha >1/2$. The term of main contribution to the expansion is given explicitly in each case. Then we generalize this expansion to the case when there is a single point $x_{0}$ in the support of $\psi$ such that $\phi ^{(j)}(x_{0})=0$, $j=0,1,\dots , k-1$, while $\phi ^{(k)}\ne 0$. We also obtain Van der Corput estimates for the oscillatory integral $\int _{a}^{b} {E}_{\alpha }{\left( \left( {\dot{\imath }} \lambda \phi (x)\right) ^{\alpha }\right) } \psi (x)dx$, where $-\infty<a<b<\infty$. The obtained estimates are sharp with respect to the order of decay.

Appendix A: Proof of Lemma 1

Proof

Observe that $D_{\phi }^{m}{e^{-a\lambda \phi (x) r}} =\left( -a\lambda r\right) ^{m}{e^{-a\lambda \phi (x) r}}$. Therefore

$$\begin{aligned} \int _{{\mathbb {R}}} {e^{-a\lambda \phi (x) r}}\psi (x)dx= & {} \frac{1}{\left( -a\lambda r\right) ^{m}} \int _{{\mathbb {R}}}\psi (x) D_{\phi }^{m}{e^{-a\lambda \phi (x) r}}dx\nonumber \\= & {} \frac{1}{\left( -a\lambda r\right) ^{m}} \int _{{\mathbb {R}}} {e^{-a\lambda \phi (x) r}} \left( ^tD_{\phi }\right) ^{m}\psi (x)dx. \end{aligned}$$

This gives the identity (4.1). The estimate (4.2) follows then by the integrability of $r\mapsto r^{b-m}$ on $[c,\infty )$ for $m>1+b$. $\square$