A note on the weak regularity theory for degenerate Kolmogorov equations

TL;DR

This paper establishes a weak Harnack inequality and Hölder continuity for weak solutions to degenerate Kolmogorov equations with measurable coefficients, introducing a new functional space and extending classical inequalities to ultraparabolic equations.

Contribution

It introduces a novel functional space for weak solutions and extends the weak Harnack inequality to ultraparabolic equations, advancing regularity theory for degenerate PDEs.

Findings

Proved a weak Harnack inequality for non-negative solutions.

Established Hölder continuity for weak solutions.

Extended the Ink-Spots Theorem to ultraparabolic equations.

Abstract

The aim of this work is to prove a Harnack inequality and the H\"older continuity for weak solutions to the Kolmogorov equation $\mathscr{L} u = f$ 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 $\mathscr{L}u = f$, that allows us to prove a weak Poincar\'e 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.