Kolmogorov operator with the vector field in Nash class

Abstract

We consider divergence-form parabolic equation with measurable uniformly elliptic matrix and the vector field in a large class containing, in particular, the vector fields in $L^p$, $p>d$, as well as some vector fields that are not even in $L_{\rm loc}^{2+\varepsilon}$, $\varepsilon>0$. We establish H\"{o}lder continuity of the bounded soutions, sharp two-sided Gaussian bound on the heat kernel, Harnack inequality.