Abstract maximal hypoellipticity and applications

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.