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In this paper, we present an $L_q(L_p)$-regularity theory for parabolicequations of the form: $$ \partial_tu(t,x)=\mathcal{L}^{\vec{a},\vec{b}}(t)u(t,x)+f(t,x),\quad u(0,x)=0. $$ Here,$\mathcal{L}^{\vec{a},\vec{b}}(t)$ represents anisotropic non-local operatorsencompassing the singular anisotropic fractional Laplacian with measurablecoefficients: $$ -\sum_{i=1}^{\ell}a_{i}(t)(-\Delta_{x_i})^{\alpha_i/2}. $$ Toaddress the anisotropy of the operator, we employ a probabilisticrepresentation of the solution and Calder\'on-Zygmund theory. As applicationsof our results, we demonstrate the solvability of elliptic equations withanisotropic non-local operators and parabolic equations with isotropicnon-local operators.

A regularity theory for parabolic equations with anisotropic non-local operators in $L_{q}(L_{p})$ spaces