On the Gevrey hypo-ellipticity of sums of squares of vector fields

Antonio Bove[1]; François Treves

  • [1] Università di Bologna, Dipartimento di Matematica, Piazza di porta S. Donato 5, 40127 Bologna (Italy), Rutgers University, Department of Mathematics, 110 Frelinghuysen RD, Piscataway, N.J. 08854-8019 (USA)

Annales de l’institut Fourier (2004)

  • Volume: 54, Issue: 5, page 1443-1475
  • ISSN: 0373-0956

Abstract

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The article studies a second-order linear differential operator of the type - L = X 1 2 + ⋯ + X r 2 , i. e., a sum of squares of real, and real-analytic, vector fields X i . The conjectured necessary and sufficient condition for analytic hypo- ellipticity, based on the Poisson stratification of the characteristic variety, is recalled in simple analytic and geometric terms. It is conjectured that the microlocal Gevrey hypo-ellipticity of L depends on the restrictions of the principal symbol σ L to 2 D or 4 D symplectic manifolds associated to each bicharateristic curve in a nonsymplectic stratum.

How to cite

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Bove, Antonio, and Treves, François. "On the Gevrey hypo-ellipticity of sums of squares of vector fields." Annales de l’institut Fourier 54.5 (2004): 1443-1475. <http://eudml.org/doc/116148>.

@article{Bove2004,
abstract = {The article studies a second-order linear differential operator of the type $ -L=$$X_\{1\}^\{2\}+\cdots +X_\{r\}^\{2\}$, i. e., a sum of squares of real, and real-analytic, vector fields $X_\{i\}$. The conjectured necessary and sufficient condition for analytic hypo- ellipticity, based on the Poisson stratification of the characteristic variety, is recalled in simple analytic and geometric terms. It is conjectured that the microlocal Gevrey hypo-ellipticity of $L$ depends on the restrictions of the principal symbol $ \sigma \left( L\right) $ to $2D$ or $4D$ symplectic manifolds associated to each bicharateristic curve in a nonsymplectic stratum.},
affiliation = {Università di Bologna, Dipartimento di Matematica, Piazza di porta S. Donato 5, 40127 Bologna (Italy), Rutgers University, Department of Mathematics, 110 Frelinghuysen RD, Piscataway, N.J. 08854-8019 (USA)},
author = {Bove, Antonio, Treves, François},
journal = {Annales de l’institut Fourier},
keywords = {stratification; symplectic; sums of squares of vector fields; analytic and Gevrey hypo-ellipticity; sum of squares; bicharacteristic curves},
language = {eng},
number = {5},
pages = {1443-1475},
publisher = {Association des Annales de l'Institut Fourier},
title = {On the Gevrey hypo-ellipticity of sums of squares of vector fields},
url = {http://eudml.org/doc/116148},
volume = {54},
year = {2004},
}

TY - JOUR
AU - Bove, Antonio
AU - Treves, François
TI - On the Gevrey hypo-ellipticity of sums of squares of vector fields
JO - Annales de l’institut Fourier
PY - 2004
PB - Association des Annales de l'Institut Fourier
VL - 54
IS - 5
SP - 1443
EP - 1475
AB - The article studies a second-order linear differential operator of the type $ -L=$$X_{1}^{2}+\cdots +X_{r}^{2}$, i. e., a sum of squares of real, and real-analytic, vector fields $X_{i}$. The conjectured necessary and sufficient condition for analytic hypo- ellipticity, based on the Poisson stratification of the characteristic variety, is recalled in simple analytic and geometric terms. It is conjectured that the microlocal Gevrey hypo-ellipticity of $L$ depends on the restrictions of the principal symbol $ \sigma \left( L\right) $ to $2D$ or $4D$ symplectic manifolds associated to each bicharateristic curve in a nonsymplectic stratum.
LA - eng
KW - stratification; symplectic; sums of squares of vector fields; analytic and Gevrey hypo-ellipticity; sum of squares; bicharacteristic curves
UR - http://eudml.org/doc/116148
ER -

References

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