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.