Complex vector fields and hypoelliptic partial differential operators

TL;DR

This paper establishes subelliptic estimates for complex vector fields, extending classical conditions, and demonstrates their implications for hypoellipticity of differential operators, including applications to specific CR manifolds.

Contribution

It generalizes pseudoconcavity and H"ormander's bracket condition to complex vector fields, proving subelliptic estimates and hypoellipticity results.

Findings

Proved subelliptic estimates under generalized conditions.

Established hypoellipticity for certain differential operators.

Characterized a class of CR manifolds with subelliptic estimates.

Abstract

We prove a subelliptic estimate for systems of complex vector fields under some assumptions that generalize the essential pseudoconcavity for $CR$ manifolds and H\"ormander's bracket condition for real vector fields. Applications are given to prove the hypoellipticity of first order systems and second order partial differential operators. Finally we describe a class of compact homogeneous CR manifolds for which the distribution of $(0,1)$ vector fields satisfies a subelliptic estimate. v2: minor revision, to appear in Ann. Inst. Fourier