Hypoellipticity: Geometrization And Speculation

. To any finite collection of smooth real vector fields X j in R n we associate a metric in the phase space T R n . The relation between the asymptotic behavior of this metric and hypoellipticity of P X 2 j , in the smooth, real analytic, and Gevrey categories, is explored. To Professor P. Lelong, on the occasion of his 85th birthday, in admiration. 1. Introduction Let fX j g be a collection of real vector fields with C 1 or C ! coefficients, defined in a neighborhood of a point x 0 2 R d . Consider a second order differential operator L = P j X 2 j . Under what conditions is L hypoelliptic, in C 1 or C ! ? As is well known, certain of these sums of squares operators provide a good model for aspects of the ¯ @-- Neumann problem in C 2 , and a grossly oversimplified but still illuminating window into the higher dimensional case. Indeed, application of the method of boundary reduction to the ¯ @--Neumann problem leads to a pseudodifferential equation on the..