Sobolev embeddings for kinetic Fokker-Planck equations

TL;DR

This paper develops Sobolev embedding theorems for kinetic Fokker-Planck equations using intrinsic Sobolev spaces, extending classical results to ultra-parabolic operators under the weak H"ormander condition.

Contribution

It introduces intrinsic Sobolev-Slobodeckij spaces for kinetic equations and proves new embedding, approximation, and interpolation results, extending classical analysis to this setting.

Findings

Continuous embeddings into Lorentz and H"older spaces.

Extension of Taylor expansion and interpolation inequalities.

Method adaptation for higher-order embeddings using kernel estimates.

Abstract

We introduce intrinsic Sobolev-Slobodeckij spaces for a class of ultra-parabolic Kolmogorov type operators satisfying the weak H\"ormander condition. We prove continuous embeddings into Lorentz and intrinsic H\"older spaces. We also prove approximation and interpolation inequalities by means of an intrinsic Taylor expansion, extending analogous results for H\"older 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.