A pseudodifferential calculus for maximally hypoelliptic operators and the Helffer-Nourrigat conjecture

TL;DR

This paper develops a pseudodifferential calculus for maximally hypoelliptic operators, extending classical elliptic regularity results to a broader class of differential operators satisfying Hörmander's condition, and proves a conjecture by Helffer and Nourrigat.

Contribution

It introduces a new principal symbol for differential operators based on vector fields and their commutators, proving invertibility characterizes maximal hypoellipticity, thus generalizing Hörmander's sum of squares theorem.

Findings

Established a symbol invertibility criterion for maximal hypoellipticity.

Extended Hörmander's sum of squares theorem to higher order polynomials.

Provided a positive resolution to the Helffer-Nourrigat conjecture.

Abstract

We extend the classical regularity theorem of elliptic operators to maximally hypoelliptic differential operators. More precisely, given vector fields $X_1,\ldots,X_m$ on a smooth manifold which satisfy H\"ormander's bracket generating condition, we define a principal symbol for \textit{any} linear differential operator. Our symbol takes into account the vector fields $X_i$ and their commutators. We show that for an arbitrary differential operator, its principal symbol is invertible if and only if the operator is maximally hypoelliptic. This answers affirmatively a conjecture due to Helffer and Nourrigat. Our result is proven in a more general setting, where we allow each one of the vector fields $X_1,\ldots,X_m$ to have an arbitrary weight. In particular, our theorem generalizes H\"ormander's sum of squares theorem to higher order polynomials.