A New Proof of Okaji’s Theorem for a Class of Sum of Squares Operators
Annales de l'Institut Fourier, Volume 59 (2009) no. 2, p. 595-619

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.

Soit $P$ un opérateur différentiel analytique, de la forme “somme de carrés”, avec la condition d’Hörmander réalisée. Soit $q$ un point caractéristique de $P$. On suppose que $q$ est un point d’un “symplectic Poisson stratum” de codimension deux (au sens de Treves). D’après le théorème d’Okaji, $P$ est hypoelliptique analytique en $q$. Autrement dit, la conjecture de Treves est vraie en codimension deux. On donne dans ce travail une preuve élémentaire de ce fait.

DOI : https://doi.org/10.5802/aif.2442
Classification:  35H10,  35H20,  35A17,  35A20,  35A27
Keywords: Analytic hypoelliptic, sum of squares
@article{AIF_2009__59_2_595_0,
author = {Cordaro, Paulo D. and Hanges, Nicholas},
title = {A New Proof of Okaji's Theorem  for a Class of Sum of Squares Operators},
journal = {Annales de l'Institut Fourier},
publisher = {Association des Annales de l'institut Fourier},
volume = {59},
number = {2},
year = {2009},
pages = {595-619},
doi = {10.5802/aif.2442},
mrnumber = {2521430},
zbl = {1178.35138},
language = {en},
url = {http://www.numdam.org/item/AIF_2009__59_2_595_0}
}

Cordaro, Paulo D.; Hanges, Nicholas. A New Proof of Okaji’s Theorem  for a Class of Sum of Squares Operators. Annales de l'Institut Fourier, Volume 59 (2009) no. 2, pp. 595-619. doi : 10.5802/aif.2442. http://www.numdam.org/item/AIF_2009__59_2_595_0/

[1] Bove, A.; Derridj, M.; Tartakoff, D. Analytic hypoellipticity in the presence of non–symplectic characteristic points, J. Funct. Anal., Tome 234 (2006) no. 2, pp. 464-472 | Article | MR 2216906 | Zbl 1098.35057

[2] Cordaro, P. D.; Hanges, N.; Soc., Amer. Math. Impact of lower order terms on a model PDE in two variables, Geometric analysis of PDE and several complex variables, Contemp. Math., Providence, RI, Tome 368 (2005), pp. 157-176 | MR 2126468 | Zbl 1071.35035

[3] Cordaro, P. D.; Hanges, N.; Birkhäuser Symplectic strata and analytic hypoellipticity, Progresss in Nonlinear Differential Equations and Their Applications (Phase Space Analysis of Partial Differential Equations) Tome 69 (2006), pp. 83-94 | MR 2263208 | Zbl 1213.35189 | Zbl pre05184369

[4] Cordaro, P. D.; Himonas, A. A. Global analytic regularity for sums of squares of vector fields, Trans. Amer. Math. Soc., Tome 350 (1998) no. 12, pp. 4993-5001 | Article | MR 1433115 | Zbl 0914.35087

[5] Hanges, N. Analytic regularity for an operator with Treves curves, J. Funct. Anal., Tome 210 (2004) no. 2, pp. 295-320 | Article | MR 2053489 | Zbl 1053.35046

[6] Hanges, N.; Himonas, A. A. Non–analytic hypoellipticity in the presence of symplecticity, Proceedings of the AMS, Tome 126 (1998) no. 2, pp. 405-409 | Article | MR 1422872 | Zbl 0906.35027

[7] Hörmander, L. Hypoelliptic second order differential equations, Acta Math., Tome 119 (1967), pp. 147-171 | Article | MR 222474 | Zbl 0156.10701

[8] Hörmander, L. The analysis of linear partial differential operators I, Springer–Verlag (1983) | Zbl 0521.35001

[9] Menikoff, A. Some examples of hypoelliptic partial differential equations, Math. Ann., Tome 221 (1976), pp. 167-181 | Article | MR 481452 | Zbl 0323.35019

[10] Métivier, G. Analytic hypoellipticity for operators with multiple characteristics, Commun. Partial Differ. Equations, Tome 6 (1981) no. 1, pp. 1-90 | MR 597752 | Zbl 0455.35040

[11] Okaji, T. Analytic hypoellipticity for operators with symplectic characteristics, J. Math. Kyoto Univ., Tome 25 (1985) no. 3, pp. 489-514 | MR 807494 | Zbl 0593.35027

[12] Oleinik, O. A. On the analyticity of solutions of partial differential equations and systems, Astérisque, Tome 2/3 (1973), pp. 272-285 | MR 399640 | Zbl 0291.35013

[13] Sibuya, Y. Global Theory of a Second Order Linear Ordinary Differential Equation with a Polynomial Coefficient, North–Holland (1975) | MR 486867 | Zbl 0322.34006

[14] Sjöstrand, J. Singularités analytiques microlocales, Astérisque, Tome 95 (1982), pp. 1-166 | MR 699623 | Zbl 0524.35007

[15] Tartakoff, D. On the local real analyticity of solutions to ${\square }_{b}$ and the $\overline{\partial }-$Neumann problem, Acta Math., Tome 145 (1980), pp. 117-204 | Article | MR 590289 | Zbl 0456.35019

[16] Treves, F. Analytic hypoellipticity of a class of pseudodifferential operators with double characteristics and applications to the $\overline{\partial }$–Neumann problem, Commun. Partial Differ. Equations, Tome 3 (1978), pp. 475-642 | Article | MR 492802 | Zbl 0384.35055

[17] Treves, F. Symplectic geometry and analytic hypo-ellipticity, Proceedings of Symposia in Pure Mathematics, Tome 65 (1999), pp. 201-219 | MR 1662756 | Zbl 0938.35038

[18] Treves, F.; Birkhäuser On the analyticity of solutions of sums of squares of vector fields, Progress in Nonlinear Differential Equations and Their Applications (Phase space analysis of partial differential equations) Tome 69 (2006), pp. 315-329 | MR 2263217 | Zbl pre05184379