  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.
