Non-divergence Parabolic Equations of Second Order with Critical Drift in Morrey Spaces

We consider uniformly parabolic equations and inequalities of second order in the non-divergence form with drift $$-u_t+Lu=-u_t+\sum _{ij}{a}_{ij}{D}_{ij}u+\sum b_i{D}_{i}u=0(\ge 0,\le 0)$$ in some domain $\mathrm{\Omega }\subset {\mathbb{R}}^{n+1}$. We prove a variant of Aleksandrov-Bakelman-Pucci-Krylov-Tso estimate with $L^p$ norm of the inhomogeneous term for some number $p<n+1$. Based on it, we derive the growth theorems and the interior Harnack inequality. In this paper, we will only assume the drift $b$ is in certain Morrey spaces defined below which are critical under the parabolic scaling but not necessarily to be bounded. This is a continuation of the work in \cite{GC}.