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

Abstract

Let $0<α<2$, $β>0$ and $α/2<|s|\leq 1$. In a previous work, we obtained all possible values of the Lebesgue exponent $p=p(γ)$ for which the Fourier transform of $ E_{α,β}(e^{\dot{\imath}πs} |\cdot|^γ )$ is an $L^{p}(\mathbb{R}^d)$ function, when $γ>(d-1)/2$. We recover the more interesting lower regularity case $0<γ\leq (d-1)/2$, 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 $α\in (0,1)$, $β=α,1$, 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 $β,γ>0$ and $s\in (-1,1]\setminus [-α/2,α/2]$. This enabled us to prove various key estimates for a general class of nonlocal space-time problems.