Analytic Hypoellipticity for a Class of Sums of Squares of Vector Fields with Non-Symplectic Characteristic Variety

The recent example of Hanges: $P = \partial_t^2 + t^2\Delta_x + \partial^2_{\theta(x)}$in$R^3$ is analytic hypoelliptic in the sense of germs but not in the strong sense in any neighborhood of the origin. And its characteristic variety is non-symplectic. We give a purely $L^2,$ and hence quite flexible, proof of this result and generalizations, and link it to, and contrast it with, the celebrated Baouendi-Goulaouic operator. We point out that the results are consistent with the conjecture of Treves.