Gevrey hypoellipticity for sums of squares of vector fields in $ \R^2 $ with quasi-homogeneous polynomial vanishing

Abstract

Analytic and Gevrey hypo-ellipticity are studied for operators of the form P (x, y, D x , D y) = D 2 x + N j=1 (p j (x, y)D y) 2 , in R 2. We assume that the vector fields D x and p j (x, y)D y satisfy Hörmander's condition, that is, that they as well as their Poisson brackets generate a two-dimensional vector space. It is also assumed that the polynomials p j are quasi-homogeneous of degree m j , that is, that p j (λx, λ θ y) = λ m j p j (x, y), for every positive number λ. We prove that if the associated Poisson-Trèves stratification is not symplectic, then P is Gevrey s hypo-elliptic for an s which can be explicitly computed. On the other hand, if the stratification is symplectic, then P is analytic hypo-elliptic.

Analytic regularity for solutions to sums of squares: an assessment

“…∂ k x b j (0, 0) = 0 when 2 ≤ j ≤ N and k < m. It is also evident that X 1 = D x is the only field that we can meaningfully use to form brackets of vector fields, i.e. we have to consider only brackets of the form ad(X 1 ) k X j , since any other vector field has a vanishing coefficient in front (see also [14].) Set…”