Sobolev embeddings for kinetic Fokker-Planck equations

We introduce intrinsic Sobolev-Slobodeckij spaces for a class of ultra-parabolic Kolmogorov type operators satisfying the weak Hormander condition. We prove continuous embeddings into Lorentz and intrinsic Holder spaces. We also prove approximation and interpolation inequalities by means of an intrinsic Taylor expansion, extending analogous results for Holder spaces. The embedding at first order is proved by adapting a method by Luc Tartar which only exploits scaling properties of the intrinsic quasi-norm, while for higher orders we use uniform kernel estimates.