Analytic hypoellipticity for sums of squares and the Treves conjecture, II

Abstract

We are concerned with the problem of real analytic regularity of the solutions of sums of squares with real analytic coefficients. The Treves conjecture defines a stratification and states that an operator of this type is analytic hypoelliptic if and only if all the strata in the stratification are symplectic manifolds.Albano, Bove, and Mughetti (2016) produced an example where the operator has a single symplectic stratum, according to the conjecture, but is not analytic hypoelliptic.If the characteristic manifold has codimension 2 and if it consists of a single symplectic stratum, defined again according to the conjecture, it has been shown that the operator is analytic hypoelliptic.We show here that the above assertion is true only if the stratum is single, by producing an example with two symplectic strata which is not analytic hypoelliptic.

On the Gevrey regularity for sums of squares of vector fields, study of some models

“…More precisely in [2] and [3] the authors produced and studied the first models which are not consistent with the Treves conjecture, [13]. However, contrary to the cases of [2] and [3], the operators studied here have no exceptional strata because the symbols do not depend on the tangent variables of the "inner most" stratum. Our results can be stated as follows:…”

“…In 1999 Treves formulated a conjecture which related the analytic hypoellipticity with geometrical properties of the characteristic variety of P , see [12] and [13]. In recent papers Albano, Bove and Mughetti, [2], and Bove and Mughetti, [3], showed that the sufficient part of the Treves' conjecture does not hold neither locally nor microlocally. More precisely in [2] and [3] the authors produced and studied the first models which are not consistent with the Treves conjecture, [13].…”

Analytic Hypoellipticity for Sums of Squares in the Presence of Symplectic Non Treves Strata

“…In [1,3] it was shown that the Treves conjecture does not hold by producing an example contradicting the conjecture. The example is the operator in (1.3).…”

(Semi-)global analytic hypoellipticity for a class of “sums of squares” which fail to be locally analytic hypoelliptic

“…They showed that even if P ABM has a single symplectic stratum, in meaning of the Poisson-Treves stratification, it is Gevrey hypoelliptic of order s = r(q − 1)[q − 1 + (r − 1)(p − 1)] −1 and not better. In [2] the authors investigated the following operator (1.2) P BM (x, D) = D 2 1 +x 2(r+ℓ−1) 1…”