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.
Categories: math.AP