On the sharp Gevrey regularity for a generalization of the Métivier operator

Abstract

The sharp Gevrey hypoellipticity is provided for the following generalization of the Métivier operator, "Non-hypoellipticité analytique pour $D_x^2+\left(x^2+y^2\right)D_y^2$" by G. Métivier, \begin{align*} D_{x}^{2}+\left(x^{2n+1}D_{y}\right)^{2}+\left(x^{n}y^{m}D_{y}\right)^{2}, \end{align*} in $\Omega$ open neighborhood of the origin in ${\mathbb{R}}^2$, where $n$ and $m$ are positive integers.