  In this paper, we present an $L_q(L_p)$-regularity theory for parabolic
equations of the form: $$ \partial_t
u(t,x)=\mathcal{L}^{\vec{a},\vec{b}}(t)u(t,x)+f(t,x),\quad u(0,x)=0. $$ 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 Calder\'on-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.
