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 Lu=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 Lu=f , that allows us to prove a weak Poincaré inequality. Our analysis is based on a weak Harnack inequality, a weak Poincaré inequality combined with a $L^2-L^{\infty}$ estimate and a classical covering argument (Ink-Spots Theorem).