  We obtain $L_p$ estimates for fractional parabolic equations with space-time
non-local operators $$ \partial_t^\alpha u - Lu + \lambda u= f \quad
\mathrm{in} \quad (0,T) \times \mathbb{R}^d,$$ where $\partial_t^\alpha u$ is
the Caputo fractional derivative of order $\alpha \in (0,1]$, $\lambda \ge 0$,
$T\in (0,\infty)$, and $$Lu(t,x) := \int_{ \mathbb{R}^d} \bigg( u(t,x+y)-u(t,x)
- y\cdot \nabla_xu(t,x)\chi^{(\sigma)}(y)\bigg)K(t,x,y)\,dy $$ is an
integro-differential operator in the spatial variables. Here we do not impose
any regularity assumption on the kernel $K$ with respect to $t$ and $y$. We
also derive a weighted mixed-norm estimate for the equations with operators
that are local in time, i.e., $\alpha = 1$, which extend the previous results
by using a quite different method.
