On the Gevrey hypo-ellipticity of sums of squares of vector fields.
Treves' conjecture for the analytic hypoellipticity of ``sum of squares'': The sum of squares \(-L= X^2_1+ X^2_2+\cdots+ X^2_r\) is analytic hypoelliptic if and only if every Poisson stratum of the set \(\text{Char\,} L\) of common zeros of the symbols \(F_j= i\text{\,symb\,}(X_j)\), \(j= 1,\dots,r\), in phase space \(M= \Omega\times(\mathbb{R}^n\setminus\{0\})\) is symplectic. A key tool to prove this conjecture (it is still an open problem) is the Poisson stratification of an analytic variety. Section 2 of this paper explains this stratification by the following steps: \(\bullet\) Analytic stratification of an analytic set; \(\bullet\) symplectic stratification of an analytic submanifold; \(\bullet\) Poisson stratification; \(\bullet\) Poisson stratification associated to vector fields. In Section 3 an approach using ideas of stratification is described which leads to results for Gevrey hypoellipticity of \(-L\). If a stratum of the analytic stratification is not symplectic it has a foliation by bicharacteristic curves. Microlocal neighbourhoods are described in terms of higher-order microlocalization. It is conjectured that microlocal Gevrey hypoellipticity of \(-L\) depends on the restriction of \(\sigma(L)\) to \(2- d\) or \(4- d\) symplectic manifolds associated to each bicharacteristic curve.
scientific article; zbMATH DE number 2162430
scientific article