An $L_q(L_p)$-theory for time-fractional diffusion equations with nonlocal operators generated by L\'evy 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 $\partial^{\alpha}_{t}u = \mathcal{L}u +f$. Here, $\partial^{\alpha}_{t}$ is the Caputo fractional derivative of order $\alpha\in(0,1)$ 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 L\'evy 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 $$ C_{1}\left(\frac{R}{r}\right)^{\delta_{1}} \leq \frac{\ell(R)}{\ell(r)} \leq C_{2} \left( \frac{R}{r} \right)^{\delta_{2}} \quad \text{for}\;\; \,1\leq r\leq R<\infty, $$ where $0\leq \delta_{1}\leq \delta_{2}<2$. Hence, $\ell$ can be slowly varying at infinity. Our result covers $\mathcal{L}$ whose Fourier multiplier $\Psi(\xi)$ satisfies $\Psi(\xi)\asymp -\log{(1+|\xi|^{\beta})}$ for $\beta \in (0,2]$ and $\Psi(\xi) \asymp-(\log(1+|\xi|^{\beta/4}))^{2}$ for $\beta\in(0,2)$ by taking $\ell(r) \asymp 1$ and $\ell(r) \asymp \log{(1+r^{\beta})}$ for $r\geq1$ respectively. In this article, we use the Calder\'on-Zygmund approach and function space theory for operators having slowly varying symbols.