Abstract maximal hypoellipticity and applications
Omar Mohsen
TL;DR
The article develops a unified, abstract framework for maximal hypoellipticity using $C^*$-algebras of Type I. It proves that maximal hypoellipticity in an abstract calculus with a principal symbol is equivalent to the left-invertibility of the symbol, under three natural assumptions, enabling a broad range of applications. This approach recovers classical results on elliptic regularity, Rockland's criterion, and Rodino's theorems, and sets the stage for microlocal and further conjectural advances. The method hinges on embedding the differential operator into a Type I $C^*$-calculus and leveraging representation theory to connect symbol invertibility with operator regularity. The framework promises a versatile route to new hypoellipticity results and microlocal analyses beyond traditional orbit-method techniques.
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.