Optimal Gevrey Regularity for Certain Sums of Squares in Two Variables

TL;DR

This paper establishes the optimal Gevrey regularity for a class of sums of squares operators in two variables, demonstrating that the known hypoelliptic regularity exponent cannot be improved.

Contribution

It proves the optimality of the Gevrey regularity exponent for certain sums of squares operators, extending previous methods and analyzing the characteristic manifold.

Findings

The operator is Gevrey s0 hypoelliptic with s0^{-1} = 1 - a^{-1}(q - 1)/q.

Solutions can be more regular than G^{s0} but not better than Gevrey s0.

The characteristic manifold and its relation to Treves conjecture are described.

Abstract

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.