Optimal Schauder estimates for kinetic Kolmogorov equations with time measurable coefficients

We prove global Schauder estimates for kinetic Kolmogorov equations with coefficients that are Hölder continuous in the spatial variables but only measurable in time. Compared to other available results in the literature, our estimates are optimal in the sense that the inherent Hölder spaces are the strongest possible under the given assumptions: in particular, under a parabolic Hörmander condition, we introduce Hölder norms defined in terms of the intrinsic geometry that the operator induces on the space-time variables. The technique is based on the existence and the regularity estimates of the fundamental solution of the equation. These results are essential for studying backward Kolmogorov equations associated with kinetic-type diffusions, e.g. stochastic Langevin equation.