Some advances in analytic hypoellipticity for sums of squares operators

In this talk we discuss the problem of the real analytic regularity for the solutions of sums of squares of vector fields. While the problem of the C^\infty hypoellipticity has been settled from the very beginning by Hörmander, the problem of the analytic hypoellipticity is still open and seems much more involved. Treves conjecture states that a “sum of squares”-type operator is analytic hypoelliptic if and only if all the Poisson strata of its characteristic set are symplectic. We show that this conjecture, as stated, does not hold. However, we briefly discuss some model examples which would suggest that the analytic regularity still depends on a suitable stratification of the characteristic variety of the operator.