A note on the weak regularity theory for degenerate Kolmogorov equations

The aim of this work is to prove a Harnack inequality and the Holder 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 functional space W, suitable for the study of weak solutions to L u = f, that allows us to prove a weak Poincare inequality. Our analysis is based on a weak Harnack inequality, a weak Poincare inequality combined with a L2 - L infinity estimate and a classical covering argument (Ink-Spots (c) 2022 Elsevier Inc. All rights reserved.

Automated summary

The aim of this work is to prove a Harnack inequality and Holder continuity for weak solutions to the Kolmogorov equation with measurable coefficients, integrable lower order terms, and nonzero source term. A functional space W is introduced for the study of weak solutions, which enables the proof of a weak Poincare inequality. The analysis is based on a weak Harnack inequality, a weak Poincare inequality combined with an L2 - L infinity estimate, and a classical covering argument.