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

Abstract

Let $0<\alpha <2$, $\beta >0$ and $\alpha /2<|s|\le 1$. In a previous work, we obtained all possible values of the Lebesgue exponent $p=p(\gamma )$ for which the Fourier transform of $E_{\alpha ,\beta }(e^{\dot{ı}\pi s}|\cdot {|}^{\gamma })$ is an $L^p({\mathbb{R}}^d)$ function, when $\gamma >(d-1)/2$. We recover the more interesting lower regularity case $0<\gamma \le (d-1)/2$, 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 $s=-1/2,1$ 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.