# A New Proof of Okaji’s Theorem for a Class of Sum of Squares Operators

• [1] Universidade de São Paulo São Paulo, SP (Brazil)
• [2] Lehman College Department of Mathematics/CUNY Bronx, New York 10468 (USA)
• Volume: 59, Issue: 2, page 595-619
• ISSN: 0373-0956

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## Abstract

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Let $P$ be a linear partial differential operator with analytic coefficients. We assume that $P$ is of the form “sum of squares”, satisfying Hörmander’s bracket condition. Let $q$ be a characteristic point for $P$. We assume that $q$ lies on a symplectic Poisson stratum of codimension two. General results of Okaji show that $P$ is analytic hypoelliptic at $q$. Hence Okaji has established the validity of Treves’ conjecture in the codimension two case. Our goal here is to give a simple, self-contained proof of this fact.

## How to cite

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Cordaro, Paulo D., and Hanges, Nicholas. "A New Proof of Okaji’s Theorem for a Class of Sum of Squares Operators." Annales de l’institut Fourier 59.2 (2009): 595-619. <http://eudml.org/doc/10406>.

@article{Cordaro2009,
abstract = {Let $P$ be a linear partial differential operator with analytic coefficients. We assume that $P$ is of the form “sum of squares”, satisfying Hörmander’s bracket condition. Let $q$ be a characteristic point for $P$. We assume that $q$ lies on a symplectic Poisson stratum of codimension two. General results of Okaji show that $P$ is analytic hypoelliptic at $q$. Hence Okaji has established the validity of Treves’ conjecture in the codimension two case. Our goal here is to give a simple, self-contained proof of this fact.},
affiliation = {Universidade de São Paulo São Paulo, SP (Brazil); Lehman College Department of Mathematics/CUNY Bronx, New York 10468 (USA)},
author = {Cordaro, Paulo D., Hanges, Nicholas},
journal = {Annales de l’institut Fourier},
keywords = {Analytic hypoelliptic; sum of squares; analytic hypoelliptic; symplectic Poisson strata},
language = {eng},
number = {2},
pages = {595-619},
publisher = {Association des Annales de l’institut Fourier},
title = {A New Proof of Okaji’s Theorem for a Class of Sum of Squares Operators},
url = {http://eudml.org/doc/10406},
volume = {59},
year = {2009},
}

TY - JOUR
AU - Cordaro, Paulo D.
AU - Hanges, Nicholas
TI - A New Proof of Okaji’s Theorem for a Class of Sum of Squares Operators
JO - Annales de l’institut Fourier
PY - 2009
PB - Association des Annales de l’institut Fourier
VL - 59
IS - 2
SP - 595
EP - 619
AB - Let $P$ be a linear partial differential operator with analytic coefficients. We assume that $P$ is of the form “sum of squares”, satisfying Hörmander’s bracket condition. Let $q$ be a characteristic point for $P$. We assume that $q$ lies on a symplectic Poisson stratum of codimension two. General results of Okaji show that $P$ is analytic hypoelliptic at $q$. Hence Okaji has established the validity of Treves’ conjecture in the codimension two case. Our goal here is to give a simple, self-contained proof of this fact.
LA - eng
KW - Analytic hypoelliptic; sum of squares; analytic hypoelliptic; symplectic Poisson strata
UR - http://eudml.org/doc/10406
ER -

## References

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14. J. Sjöstrand, Singularités analytiques microlocales, Astérisque 95 (1982), 1-166 Zbl0524.35007MR699623
15. D. Tartakoff, On the local real analyticity of solutions to ${\square }_{b}$ and the $\overline{\partial }-$Neumann problem, Acta Math. 145 (1980), 117-204 Zbl0456.35019MR590289
16. F. Treves, Analytic hypoellipticity of a class of pseudodifferential operators with double characteristics and applications to the $\overline{\partial }$–Neumann problem, Commun. Partial Differ. Equations 3 (1978), 475-642 Zbl0384.35055MR492802
17. F. Treves, Symplectic geometry and analytic hypo-ellipticity, Proceedings of Symposia in Pure Mathematics 65 (1999), 201-219 Zbl0938.35038MR1662756
18. F. Treves, On the analyticity of solutions of sums of squares of vector fields, Progress in Nonlinear Differential Equations and Their Applications 69 (2006), 315-329, Birkhäuser Zbl1214.35012MR2263217

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