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

Let $\alpha \in (0,2)$ and let $\beta >0$. Fix $-\pi <\phi \le \pi$ such that $\mathrm{\mid }\phi \mathrm{\mid }>\alpha \pi \mathrm{/}2$. 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} \varphi} |x|^σ), \end{equation*} for $\sigma >(n-1)\mathrm{/}2$, where $E_{\alpha ,\beta }$ is the two-parameter Mittag-Leffler function. As an application, we obtain some values of the Lebesgue exponent $p=p(\sigma )$, $\sigma >(n-1)\mathrm{/}2$, 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 $\sigma \le n\mathrm{/}2$.