  We extend the De Giorgi--Nash--Moser theory to a class of kinetic
Fokker-Planck equations and deduce new results on the Landau-Coulomb equation.
More precisely, we first study the H{\"o}lder regularity and establish a
Harnack inequality for solutions to a general linear equation of Fokker-Planck
type whose coefficients are merely measurable and essentially bounded, i.e.
assuming no regularity on the coefficients in order to later derive results for
non-linear problems. This general equation has the formal structure of the
hypoelliptic equations "of type II" , sometimes also called ultraparabolic
equations of Kolmogorov type, but with rough coefficients: it combines a
first-order skew-symmetric operator with a second-order elliptic operator
involving derivatives along only part of the coordinates and with rough
coefficients. These general results are then applied to the non-negative
essentially bounded weak solutions of the Landau equation with inverse-power
law $\gamma$ $\in$ [--d, 1] whose mass, energy and entropy density are bounded
and mass is bounded away from 0, and we deduce the H{\"o}lder regularity of
these solutions.
