Tohoku Mathematical Journal
2022
December
SECOND SERIES VOL. 74, NO. 4

Tohoku Math. J.
74 (2022), 569-596

Title KOLMOGOROV OPERATOR WITH THE VECTOR FIELD IN NASH CLASS

Author Damir Kinzebulatov and Yuliy A. Semënov

(Received April 5, 2021, revised August 24, 2021)
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 $L_{\loc}^{2+\varepsilon}$, $\varepsilon>0$. We establish Hölder continuity of the bounded soutions, sharp two-sided Gaussian bound on the heat kernel, Harnack inequality.

Mathematics Subject Classification. Primary 35K08; Secondary 47D07, 60J35.
Key words and phrases. Heat kernel bounds, De Giorgi-Nash theory, Harnack inequality, strong solutions, singular drift, Feller semigroups.

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