Heat kernel bounds for parabolic equations with singular (form-bounded) vector fields

Abstract

We consider Kolmogorov operator $-\nabla \cdot a\cdot \nabla +b\cdot \nabla$ with measurable uniformly elliptic matrix $a$ and prove Gaussian lower and upper bounds on its heat kernel under minimal assumptions on the vector field $b$ and its divergence $\mathrm{div}b$. More precisely, we prove: (1) Gaussian lower bound, provided that $\mathrm{div}b\ge 0$, and $b$ is in the class of form-bounded vector fields (containing e.g.\,the class $L^d$, the weak $L^d$ class, as well as some vector fields that are not even in $L_{\mathrm{loc}}^{2+\epsilon }$, $\epsilon >0$); in these assumptions, the Gaussian upper bound is in general invalid; (2) Gaussian upper bound, provided that $b$ is form-bounded, and the positive part of $\mathrm{div}b$ is in the Kato class; in these assumptions, the Gaussian lower bound is in general invalid; (3) Gaussian upper and lower bounds, provided that $b$ is form-bounded, $\mathrm{div}b$ is in the Kato class; (4) A priori Gaussian upper and lower bounds, provided that $b$ is in a large class containing the class of form-bounded vector fields, $\mathrm{div}b$ is in the Kato class.

Added a priori counterparts of Theorems 2 and 3 (upper Gaussian bound and two-sided Gaussian bounds) for an even larger class of vector fields