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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.

##### MSC:
 35H10 Hypoelliptic equations 35A20 Analyticity in context of PDEs
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