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

Abstract

Let $α\in (0,2)$ and let $β>0$. Fix $-π<φ\leq π$ such that $|φ|>απ/2$. We obtain asymptotic upper bounds on the Fourier transform of the radially symmetric tempered distribution \begin{equation*} \mathbb{R}^n\ni x\mapsto E_{α,β}(e^{\dot{\imath} φ} |x|^σ), \end{equation*} for $σ>(n-1)/2$, where $E_{α,β}$ is the two-parameter Mittag-Leffler function. As an application, we obtain some values of the Lebesgue exponent $p=p(σ)$, $σ>(n-1)/2$, 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 $E_{α,β}$ and the Hausdorff-Young inequality, when $σ\leq n/2$.