A note on the weak regularity theory for degenerate Kolmogorov equations

Abstract

The aim of this work is to prove a Harnack inequality and the Hölder continuity for weak solutions to the Kolmogorov equation $\mathcal{L}u=f$ with measurable coefficients, integrable lower order terms and nonzero source term. We introduce a functional space $\mathcal{W}$, suitable for the study of weak solutions to $\mathcal{L}u=f$, that allows us to prove a weak Poincaré inequality. More precisely, our goal is to prove a weak Harnack inequality for non-negative super-solutions by considering their Log-transform and following S. N. Kruzkov (1963). Then this functional inequality is combined with a classical covering argument (Ink-Spots Theorem) that we extend for the fist time to the case of ultraparabolic equations.

36 pages