Kolmogorov operator with the vector field in Nash class

TL;DR

This paper proves regularity and heat kernel estimates for divergence-form parabolic equations with vector fields in Nash class, extending results to less regular vector fields beyond traditional integrability conditions.

Contribution

It establishes Hölder continuity, Gaussian bounds, and Harnack inequality for solutions with vector fields in the Nash class, including those not in standard $L^p$ spaces.

Findings

Hölder continuity of solutions

Sharp Gaussian bounds on heat kernel

Harnack inequality for solutions

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.