Remarks on hypoelliptic equations

TL;DR

This paper discusses the current understanding and gaps in hypoelliptic equations, focusing on systems, regularity levels, and $L^1$-hypoellipticity, providing examples and counterexamples.

Contribution

It offers new examples and counterexamples highlighting unresolved issues in hypoelliptic equations, especially for systems and lower regularity levels.

Findings

Current understanding of scalar hypoellipticity is comprehensive.

Gaps exist in the theory for systems and $L^1$-hypoellipticity.

The paper presents specific examples and counterexamples related to these gaps.

Abstract

Smooth hypoellipticity for scalar equations is quite well understood presently. On the other hand, much remains to be done for systems and/or at different levels of regularity and in particular for $L^1$-hypoellipticity. In this article we provide some examples and counter-examples.