ANALYTIC HYPOELLIPTICITY for SUMS of SQUARES in the PRESENCE of SYMPLECTIC NON TREVES STRATA

In Albano, Bove and Mughetti [J. Funct. Anal. 274(10) (2018), 2725-2753]; Bove and Mughetti [Anal. PDE 10(7) (2017), 1613-1635] it was shown that Treves conjecture for the real analytic hypoellipticity of sums of squares operators does not hold. Models were proposed where the critical points causing a non-analytic regularity might be interpreted as strata. We stress that up to now there is no notion of stratum which could replace the original Treves stratum. In the proposed models such 'strata' were non-symplectic analytic submanifolds of the characteristic variety. In this note we modify one of those models in such a way that the critical points are a symplectic submanifold of the characteristic variety while still not being a Treves stratum. We show that the operator is analytic hypoelliptic.