On the asymptotic behaviour of the Fourier transform of the Mittag-Leffler function

Abstract

Let $\alpha \in (0,2)$ and let $\beta>0$. Fix $-\pi<\varphi\leq \pi$ such that $|\varphi|>\alpha \pi/2$. We obtain asymptotic upper bounds on the Fourier transform of the radially symmetric tempered distribution \begin{equation*} \mathbb{R}^n\ni x\mapsto E_{\alpha,\beta}(e^{\dot{\imath} \varphi} |x|^{\sigma}), \end{equation*} for $\sigma>(n-1)/2$, where $E_{\alpha,\beta}$ is the two-parameter Mittag-Leffler function. As an application, we obtain some values of the Lebesgue exponent $p=p(\sigma)$, $\sigma>(n-1)/2$, for which the Fourier transform is in $L^{p}(\mathbb{R}^{n})$. Such values cannot be obtained via the well-known $L^{p}(\mathbb{R}^{n})$ properties of $E_{\alpha,\beta}$ and the Hausdorff-Young inequality, when $\sigma\leq n/2$.