Analytic hypoellipticity in the presence of nonsymplectic characteristic points

Recently, N. Hanges proved that the operator P = ∂2 t + t2 x +∂2 θ(x) in R3 is analytic hypoelliptic in the sense of germs at the origin and yet fails to be analytic hypoelliptic ‘in the strong sense’ in any neighborhood of the origin (there is no neighborhood U of the origin such that for every open subset V of U and distribution u in U, Pu analytic in V implies that u is analytic in V ). Here ∂θ(x) = x1∂/∂x2 − x2∂/∂x1. We give a short L2 proof of this result which generalizes easily and suggestively to other operators with nonsymplectic characteristic varieties.  2005 Elsevier Inc. All rights reserved.