On the Gevrey hypo-ellipticity of sums of squares of vector fields.
type: Item
instance of: scholarly article
title: On the Gevrey hypo-ellipticity of sums of squares of vector fields.
zbMATH Open document ID: 1073.35067
author: Antonio Bove
author: François Treves
publication date: 2005-04-29T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/116148
review text: 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.
reviewed by: Michael Reissig
Mathematics Subject Classification ID: 35H10
Mathematics Subject Classification ID: 35A20
zbMATH DE Number: 2162430
zbMATH Keywords: sum of squares
zbMATH Keywords: stratification
zbMATH Keywords: symplectic
zbMATH Keywords: bicharacteristic curves
MaRDI profile type: Publication
OpenAlex ID: W2527487528
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DOI: 10.5802/AIF.2055
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published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 2162430