  We prove that $L^2$ weak solutions to hypoelliptic equations with bounded
measurable coefficients are H\"older continuous. The proof relies on classical
techniques developed by De Giorgi and Moser together with the averaging lemma
and regularity transfers developed in kinetic theory. The latter tool is used
repeatedly: first in the proof of the local gain of integrability of
sub-solutions; second in proving that the gradient with respect to the velocity
variable is $L^{2+\epsilon}_{\mathrm{loc}}$; third, in the proof of an
"hypoelliptic isoperimetric De Giorgi lemma." To get such a lemma, we develop a
new method which combines the classical isoperimetric inequality on the
diffusive variable with the structure of the integral curves of the first-order
part of the operator. It also uses that the gradient of solutions w.r.t. $v$ is
$L^{2+\epsilon}_{\mathrm{loc}}$.
