Optimal Gevrey Regularity for Certain Sums of Squares in Two Variables

Abstract

For $q$, $a$ integers such that $a\ge 1$, $1<q$, $(x,y)\in U$, $U$ 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 $Pu=f$ with $f$ more regular than $G^{s_0}$ that are not better than Gevrey $s_0$. The above operator reduces to the Métivier 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.