article

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

Authors: David Tartakoff and Antonio Bove

Issue: Volume 64 (2015), Issue 2, 613-633

Abstract:

$Analytic and Gevrey hypo-ellipticity are studied for operators of the form \[P(x,y,\DD_x,\DD_y)=\DD_x^2+\sum_{j=1}^N(p_j(x,y)\DD_y)^2,\] in $\mathbb{R}^2$. We assume that the vector fields $\DD_x$ and $p_j(x,y)\DD_y$ satisfy H\"or\-man\-der'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 $p_j(\lambda x,\lambda^{\theta}y)=\lambda^{m_j}p_j(x,y)$, for every positive number $\lambda$. We prove that if the associated Poisson-Tr\`eves 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.$