A note on the weak regularity theory for degenerate Kolmogorov equations

The aim of this work is to prove a Harnack inequality and the Hölder continuity for weak solutions to the Kolmogorov equation L u = f with measurable coefficients, integrable lower order terms and nonzero source term. We introduce a function space W, suitable for the study of weak solutions to 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. Krŭzkov (1963). Then this functional inequality is combined with a classical covering argument (Ink-Spots Theorem) that we extend for the first time to the case of ultraparabolic equations.

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