On the sharp Gevrey regularity for a generalization of the M\'etivier operator

The sharp Gevrey hypoellipticity is provided for the following generalization of the M'etivier operator, “Non-hypoellipticit'e analytique pour $D_x^2+(x^2+y^2)D_y^2$” by G. M'etivier, \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 $\mathrm{\Omega }$ open neighborhood of the origin in ${\mathbb{R}}^2$, where $n$ and $m$ are positive integers.