  In this article, we present the existence, uniqueness and regularity of
solutions to parabolic equations with non-local operator $$ \partial_{t}u(t,x)
= \mathcal{L}u(t,x) + f(t,x), \quad t>0 $$ in $L_{q}(L_{p})$ space. Our spatial
operator $\mathcal{L}$ is an integro-differential operator of the form $$
\int_{\mathbb{R}^{d}} \left( u(x+y)-u(x) -\nabla u(x) \cdot y
\mathrm{1}_{|y|\leq 1} \right) j_{d}(|y|)dy. $$ We investigate conditions on
$j_{d}(r)$ which yield $L_{q}(L_{p})$-regularity of solution. Our assumptions
on $j_d$ are general so that $j_d(r)$ may be comparable to $r^{-d}\ell(r^{-1})$
for a function $\ell$ which is slowly varying at infinity. For example, we can
take $\ell(r)=\log{(1+r^{\alpha})}$ or $\ell(r) = \min{\{r^{\alpha},1\}}$
($\alpha\in(0,2)$). Indeed, our result covers the operators $\mathcal{L}$,
whose Fourier multiplier $\psi(\xi)$ does not have any scaling condition for
$|\xi|\geq 1$. Furthermore, we give some examples of $\mathcal{L}$, which
cannot be covered by previous results where smoothness or scaling conditions on
$\psi$ are considered.

Analysis of PDEs;Probability