  This article is concerned with the Schauder estimate for linear kinetic
Fokker-Planck equations with H\"older continuous coefficients. This equation
has an hypoelliptic structure. As an application of this Schauder estimate, we
prove the global well-posedness of a toy nonlinear model in kinetic theory.
This nonlinear model consists in a non-linear kinetic Fokker-Planck equation
whose steady states are Maxwellian and whose diffusion in the velocity variable
is proportional to the mass of the solution.
