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 $\alpha/2<|s|\leq 1$. In a previous work, we obtained all possible values of the Lebesgue exponent $p=p(\gamma)$ for which the Fourier transform of $E_{\alpha,\beta}(e^{\dot{\imath}\pi s} |\cdot|^{\gamma} )$ is an $L^{p}(\mathbb{R}^d)$ function, when $\gamma>(d-1)/2$. We recover the more interesting lower regularity case $0<\gamma\leq (d-1)/2$, using tools from the Littlewood-Paley theory. This question arises in the analysis of certain space-time fractional diffusion and Schr\"{o}dinger problems and has been solved for the particular cases $\alpha\in (0,1)$, $\beta=\alpha,1$, and $s=-1/2,1$ 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 $s\in (-1,1]\setminus [-\alpha/2,\alpha/2]$. This enabled us to prove various key estimates for a general class of nonlocal space-time problems.