Abstract maximal hypoellipticity and applications

TL;DR

This paper establishes a general criterion for maximal hypoellipticity in an abstract setting, linking it to principal symbol invertibility, and applies it to resolve several longstanding conjectures in analysis.

Contribution

The paper introduces an abstract maximal hypoellipticity theorem connecting operator regularity to principal symbol invertibility, unifying and extending previous results.

Findings

Proves an abstract maximal hypoellipticity theorem.

Shows equivalence between hypoellipticity and principal symbol invertibility.

Resolves the Helffer-Nourrigat conjecture and related problems.

Abstract

We prove an abstract theorem of maximal hypoellipticy showing that in an abstract calculus under some natural assumptions, an operator is maximally hypoelliptic if and only if its principal symbol is left invertible. We then show that our theorem implies various known results in the literature like regularity theorem for elliptic operators, Helffer and Nourrigat's resolution of the Rockland conjecture, Rodino's theorem on regularity of operators on products of manifolds, and our resolution of the Helffer-Nourrigat conjecture. Other examples like our resolution of the microlocal Helffer-Nourrigat conjecture will be given in a sequel to this paper. Our arguments are based on the theory of $C^*$-algebras of Type I.