(Semi-)Global Analytic Hypoellipticity for a class of "sums of squares" which fail to be locally analytic hypoelliptic

TL;DR

This paper proves global and semi-global analytic hypoellipticity on the torus for certain sums of squares operators that satisfy H"ormander's condition but are not locally or microlocally analytic hypoelliptic, expanding understanding of hypoellipticity in these cases.

Contribution

It establishes the global and semi-global analytic hypoellipticity for specific sums of squares operators that do not exhibit local or microlocal analytic hypoellipticity, addressing cases related to the Treves conjecture.

Findings

Proves global analytic hypoellipticity on the torus for these operators.

Shows semi-global hypoellipticity under certain conditions.

Identifies classes of sums of squares operators that fail local and microlocal analytic hypoellipticity.

Abstract

The global and semi-global analytic hypoellipticity on the torus is proved for two classes of sums of squares operators, introduced in "Analytic Hypoellipticity for Sums of Squares and the Treves Conjecture" by P. Albano and A. Bove and M. Mughetti, and in "Analytic Hypoellipticity for Sums of Squares and the Treves Conjecture. II" by A. Bove and M. Mughetti, satisfying the H\"ormander condition and which fail to be neither locally nor microlocally analytic hypoelliptic.