Microlocal maximal hypoellipticity from the geometric viewpoint: I

TL;DR

This paper develops a new bi-graded pseudo-differential calculus tailored to sub-Riemannian structures, enabling the proof of microlocal maximal hypoellipticity and resolving a conjecture by Helffer and Nourrigat.

Contribution

It introduces a comprehensive geometric pseudo-differential calculus that unifies classical and sub-Riemannian analysis, proving a key hypoellipticity conjecture.

Findings

Established a new calculus combining classical and sub-Riemannian pseudo-differential operators.

Proved that invertibility of the principal symbol implies microlocal maximal hypoellipticity.

Resolved a long-standing conjecture of Helffer and Nourrigat regarding hypoellipticity.

Abstract

Given some vector fields on a smooth manifold satisfying H\"ormander's condition, we define a bi-graded pseudo-differential calculus which contains the classical pseudo-differential calculus and a pseudo-differential calculus adapted to the sub-Riemannian structure induced by the vector fields. Our approach is based on geometric constructions (resolution of singularities) together with methods from operators algebras. We develop this calculus in full generality, including Sobolev spaces, the wavefront set, and the principal symbol, etc. In particular, using this calculus, we prove that invertibility of the principal symbol implies microlocal maximal hypoellipticity. This allows us to resolve affirmatively the microlocal version of a conjecture of Helffer and Nourrigat.