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On the Gevrey hypo-ellipticity of sums of squares of vector fields. (English) Zbl 1073.35067
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.

35H10 Hypoelliptic equations
35A20 Analyticity in context of PDEs
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