Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Sobolev embeddings for kinetic Fokker-Planck equations

Andrea Pascucci, Antonello Pesce

Abstract

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

Sobolev embeddings for kinetic Fokker-Planck equations

Abstract

We introduce intrinsic Sobolev-Slobodeckij spaces for a class of ultra-parabolic Kolmogorov type operators satisfying the weak Hörmander condition. We prove continuous embeddings into Lorentz and intrinsic Hölder spaces. We also prove approximation and interpolation inequalities by means of an intrinsic Taylor expansion, extending analogous results for Hölder 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.
Paper Structure (18 sections, 25 theorems, 203 equations)

This paper contains 18 sections, 25 theorems, 203 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Assumptions
  4. Intrinsic Hölder and Sobolev spaces
  5. Dilation and translation groups
  6. Alternative Sobolev norms and a first interpolation result
  7. Intrinsic weak derivatives in $W^{n,p}_B$
  8. Interpolation inequality and equivalence of the norms $\|\cdot\|_{W^{n,p}_B}$ and $\vvvert\cdot \vvvert_{W^{n,p}_B}$
  9. Taylor expansion in $W^{n,p}_B$
  10. Approximation and interpolation
  11. Approximation in $W^{n,p}_B$
  12. Interpolation on the degree of smoothness
  13. Proof of Theorem \ref{['Embedd_1']}
  14. Higher orders embeddings
  15. Fundamental solution of Fokker-Planck equations
  16. ...and 3 more sections

Key Result

Theorem 1.1

Theorems & Definitions (37)

  • Theorem 1.1: $W^{1,p}_B$ embeddings
  • Definition 2.3
  • Definition 2.4: Intrinsic Hölder spaces
  • Definition 2.5
  • Definition 2.6: Intrinsic Sobolev spaces
  • Definition 2.7
  • Remark 2.8
  • Lemma 2.9: MR1751429, Proposition $5.1$
  • Remark 2.10
  • Remark 2.11
  • ...and 27 more