Sobolev estimates for fractional parabolic equations with space-time non-local operators

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.