An $L_q(L_p)$-theory for time-fractional diffusion equations with nonlocal operators generated by Lévy processes with low intensity of small jumps

Abstract

We investigate an $L_{q}(L_{p})$-regularity ($1<p,q<\infty$) theory for space-time nonlocal equations of the type $\partial^α_{t}u = \mathcal{L}u +f$. Here, $\partial^α_{t}$ is the Caputo fractional derivative of order $α\in(0,1)$ and $\mathcal{L}$ is an integro-differential operator $$ \mathcal{L}u(x) = \int_{\mathbb{R}^{d}} \left( u(x)-u(x+y) -\nabla u (x) \cdot y \mathbf{1}_{|y|\leq 1} \right) j_{d}(|y|)dy $$ which is the infinitesimal generator of an isotropic unimodal Lévy process. We assume that the jump kernel $j_{d}(r)$ is comparable to $r^{-d} \ell(r^{-1})$, where $\ell$ is a continuous function satisfying $$ C_{1}\left(\frac{R}{r}\right)^{δ_{1}} \leq \frac{\ell(R)}{\ell(r)} \leq C_{2} \left( \frac{R}{r} \right)^{δ_{2}} \quad \text{for}\;\; \,1\leq r\leq R<\infty, $$ where $0\leq δ_{1}\leq δ_{2}<2$. Hence, $\ell$ can be slowly varying at infinity. Our result covers $\mathcal{L}$ whose Fourier multiplier $Ψ(ξ)$ satisfies $Ψ(ξ)\asymp -\log{(1+|ξ|^β)}$ for $β\in (0,2]$ and $Ψ(ξ) \asymp-(\log(1+|ξ|^{β/4}))^{2}$ for $β\in(0,2)$ by taking $\ell(r) \asymp 1$ and $\ell(r) \asymp \log{(1+r^β)}$ for $r\geq1$ respectively. In this article, we use the Calderón-Zygmund approach and function space theory for operators having slowly varying symbols.