Analytic Hypoellipticity for Sums of Squares and the Treves Conjecture

TL;DR

This paper investigates the real analytic regularity of sums of squares operators, providing a counterexample to Treves conjecture's sufficiency by demonstrating a model operator with a single symplectic stratum that is Gevrey hypoelliptic but not analytic.

Contribution

The authors construct a specific model operator with one symplectic stratum that is Gevrey hypoelliptic but not analytic, challenging the sufficiency part of Treves conjecture.

Findings

Counterexample to Treves conjecture's sufficiency

Model operator with a single symplectic stratum

Operator is Gevrey hypoelliptic but not analytic

Abstract

We are concerned with the problem of real analytic regularity of the solutions of sums of squares with real analytic coefficients. Treves conjecture states that an operator of this type is analytic hypoelliptic if and only if all the strata in the Poisson-Treves stratification are symplectic. We produce a model operator, $ P_{1} $, having a single symplectic stratum and prove that it is Gevrey $ s_{0} $ hypoelliptic and not better. See Theorem \ref{th:1} for a definition of $ s_{0} $. We also show that this phenomenon has a microlocal character. We point out explicitly that this is a counterexample to the sufficient part of Treves conjecture and not to the necessary part, which is still an open problem.