A Littlewood-Paley approach to the Mittag-Leffler function in the frequency space and applications to nonlocal problems

Let $0<\alpha <2$, $\beta >0$ and $\alpha \mathrm{/}2<\mathrm{\mid }s\mathrm{\mid }\le 1$. In a previous work, we obtained all possible values of the Lebesgue exponent $p=p(\gamma )$ for which the Fourier transform of $E_{\alpha ,\beta }(e^{\dot{\mathrm{ı}}\pi s}\mathrm{\mid }\cdot {\mathrm{\mid }}^{\gamma })$ is an $L^p({\mathbb{R}}^d)$ function, when $\gamma >(d-1)\mathrm{/}2$. We recover the more interesting lower regularity case $0<\gamma \le (d-1)\mathrm{/}2$, using tools from the Littlewood-Paley theory. This question arises in the analysis of certain space-time fractional diffusion and Schrödinger problems and has been solved for the particular cases $\alpha \in (0,1)$, $\beta =\alpha ,1$, and $s=-1\mathrm{/}2,1$ via asymptotic analysis of Fox $H$-functions. The Littlewood-Paley theory provides a simpler proof that allows considering all values of $\beta ,\gamma >0$ and $s\in (-1,1]\setminus [-\alpha \mathrm{/}2,\alpha \mathrm{/}2]$. This enabled us to prove various key estimates for a general class of nonlocal space-time problems.