Optimal Gevrey Regularity for Certain Sums of Squares in Two Variables

For $q$, $a$ integers such that $a \geq 1$, $1 < q$, $(x, y) \in U$, $U$ a neighborhood of the origin in $\mathbb{R}^{2}$, we consider the operator $$D_{x}^{2} + x^{2(q-1)} D_{y}^{2} + y^{2a} D_{y}^{2} .$$ Slightly modifying the method of proof of \cite{monom} we can see that it is Gevrey $s_{0}$ hypoelliptic, where $s_{0}^{-1} = 1 - a^{-1} (q - 1) q^{-1}$. Here we show that this value is optimal, i.e. that there are solutions to $P u = f$ with $f$ more regular than $G^{s_{0}}$ that are not better than Gevrey $s_{0}$. The above operator reduces to the M\'etivier operator (\cite{metivier81}) when $a = 1$, $q = 2$. We give a description of the characteristic manifold of the operator and of its relation with the Treves conjecture on the real analytic regularity for sums of squares.