A compactness result for the Sobolev embedding via potential theory

TL;DR

This paper presents a novel proof of Sobolev and Morrey embedding theorems using potential theory and fundamental solutions, extending to degenerate equations and establishing compactness of embeddings.

Contribution

It introduces a potential theory-based approach to Sobolev embeddings, including degenerate cases, and proves their compactness, with applications to kinetic equations.

Findings

Classical Sobolev and Morrey theorems recovered via potential theory.

Established compactness of Sobolev embeddings in new settings.

Applied results to a kinetic equation.

Abstract

In this note we give a proof of the Sobolev and Morrey embedding theorems based on the representation of functions in terms of the fundamental solution of suitable partial differential operators. We also prove the compactness of the Sobolev embedding. We first describe this method in the classical setting, where the fundamental solution of the Laplace equation is used, to recover the classical Sobolev and Morrey theorems. We next consider degenerate Kolmogorov equations. In this case, the fundamental solution is invariant with respect to a non-Euclidean translation group and the usual convolution is replaced by an operation that is defined in accordance with this geometry. We recover some known embedding results and we prove the compactness of the Sobolev embedding. We finally apply our regularity results to a kinetic equation.