A regularity theory for parabolic equations with anisotropic non-local operators in $L_q(L_p)$ spaces

Abstract

In this paper, we present an $L_q(L_p)$-regularity theory for parabolic equations of the form: $${\partial }_{t}u(t,x)={\mathcal{L}}^{\vec{a},\vec{b}}(t)u(t,x)+f(t,x),u(0,x)=0.$$ Here, ${\mathcal{L}}^{\vec{a},\vec{b}}(t)$ represents anisotropic non-local operators encompassing the singular anisotropic fractional Laplacian with measurable coefficients: $${\mathcal{L}}^{\vec{a},\vec{0}}(t)u(x)=\sum _{i=1}^d{\int }_{\mathbb{R}}\left(u(x^1,\dots ,x^{i-1},x^i+y^i,x^{i+1},\dots ,x^d)-u(x)\right)\frac{a_i(t,y^i)}{|y^i{|}^{1+{\alpha }_i}}\mathrm{d}{y}^i.$$ To address the anisotropy of the operator, we employ a probabilistic representation of the solution and Calderón-Zygmund theory. As applications of our results, we demonstrate the solvability of elliptic equations with anisotropic non-local operators and parabolic equations with isotropic non-local operators.