On the Gevrey hypo-ellipticity of sums of squares of vector fields
Annales de l'Institut Fourier, Volume 54 (2004) no. 5, pp. 1443-1475.

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 σL to 2D or 4D symplectic manifolds associated to each bicharateristic curve in a nonsymplectic stratum.

On étudie un opérateur différentiel du second ordre du type -L= X 1 2 ++X r 2 , où les X i sont des champs vectoriels réels et analytiques. On décrit, en termes analytiques et géométriques simples, la stratification de Poisson de la variété caractéristique de L et on rappelle la conjecture selon laquelle une condition nécessaire et suffisante pour l’hypo-ellipticité analytique de L serait que chaque strate de Poisson soit symplectique. Les auteurs formulent une conjecture nouvelle sur l’hypo-ellipticité Gevrey de L selon laquelle cette propriété dépendrait de la restriction du symbole principal σL à certaines sous-variétés bi- ou quadri-dimensionnelles contenant une courbe bicaratéristique d’une strate non symplectique.

DOI: 10.5802/aif.2055
Classification: 35H05, 35A20
Bove, Antonio 1; Treves, François 

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)
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Bove, Antonio; Treves, François. On the Gevrey hypo-ellipticity of sums of squares of vector fields. Annales de l'Institut Fourier, Volume 54 (2004) no. 5, pp. 1443-1475. doi : 10.5802/aif.2055. http://archive.numdam.org/articles/10.5802/aif.2055/

[A-B, 2004] P. Albano & A. Bove, Analytic stratifications and the cutlocus of a class of distance functions, Preprint

[B-G, 1972] M. S. Baouendi & Ch. Goulaouic, Non analytic-hypoellipticity for some degenerate operators, Bull. A. M. S 78 (1972) p. 483-486 | MR | Zbl

[Be-Bo-Ta,] E. Bernardi, A. Bove & D.S. Tartakoff, On a conjecture of Treves: analytic hypoellipticity and Poisson strata, Indiana Univ. Math. J 47 (1998) p. 401-417 | MR | Zbl

[Bo-Ta, 199] A. Bove & D.S. Tartakoff, Optimal non-isotropic Gevrey exponent for sums of squares of vector fields, Comm. in P. D. E 22 (1997) p. 1263-1282 | MR | Zbl

[Bo-Ta, 200] A. Bove & D.S. Tartakoff, On a class of sums of squares with a given Poisson-Treves stratification, J. Geom. Analysis 13 (2003) p. 391-420 | MR | Zbl

[C, 1992] M. Christ, A class of hypoelliptic PDE admitting nonanalytic solutions, Contemporary Math. A. M. S 137 (1992) p. 155-167 | MR | Zbl

[C, 1994] M. Christ, A necessary condition for analytic hypoellipticity, Math. Research Letters 1 (1994) p. 241-248 | MR | Zbl

[C, 2003] M. Christ, Hypoellipticity: geometrization and speculation, Progress in Math. A. M. S., Birkhäuser, 1997, p. 91-109 | Zbl

[Ch, 2001] S. Chanillo, Kirillov theory, Treves strata, Schrödinger equations and analytic hypoellipticity of sums of squares, e-print, http://arxiv.org/pdf/math.AP/0107106, August 2001

[Ch-H-L, 20] S. Chanillo, B. Helffer & A. Laptev, Nonlinear eigenvalues and analytic hypoellipticity, J. of Functional Analysis, to appear, 2003 | Zbl

[Cost, 2003] O. Costin & R. Costin, Failure of Analytic Hypo-ellipticity in a Class of Differential Operators, Annali Sc. Normale Sup. Pisa Cl. Sci 5 (2003) no.2 p. 21-45 | Zbl

[D-Z, 1973] M. Derridj & C. Zuily, Régularité analytique et Gevrey d'opérateurs elliptiques dégénérés, J. Math. Pures et Appl 52 (1973) p. 65-80 | MR | Zbl

[G-S, 1985] A. Grigis & J. Sjöstrand, Front d'onde analytique et sommes de carrés de champs de vecteurs, Duke Math. J 52 (1985) p. 35-51 | MR | Zbl

[Gr, 1971] V. V. Grušin, On a class of elliptic pseudodifferential operators degenerate on a submanifold, Math. USSR Sbornik 13 (1971) p. 155-185 | Zbl

[Hanges, 20] N. Hanges, Analytic regularity for an operator with Treves curves, to appear | MR | Zbl

[H-H, 1991] N. Hanges & A.A. Himonas, Singular solutions for sums of squares of vector fields, Comm. in PDE 16 (1991) p. 1503-1511 | MR | Zbl

[H-H, 1995] N. Hanges & A.A. Himonas, Non-analytic hypoellipticity in the presence of symplecticity, Proceed. A.M.S 126 (1998) p. 1549-1557 | MR | Zbl

[H-H, 1996] N. Hanges & A.A. Himonas, Singular solutions for a class of Grusin type operators, Proceed. A. M. S 124 (1996) p. 1549-1557 | MR | Zbl

[Hlf, 1979] B. Helffer, Hypoellipticité analytique sur des groupes nilpotents de rang 2, p. I.1-I.13 | Numdam | Zbl

[Hlf, 1982] B. Helffer, Conditions nécessaires d'hypoanalyticité pour des opérateurs invariants à gauche homogènes sur un groupe nilpotent gradué, J. Diff. Equations 44 (1982) p. 460-481 | MR | Zbl

[H, 1967] L. Hörmander, Hypoelliptic second order differential equations, Acta Math 119 (1967) p. 147-171 | MR | Zbl

[Hosh, 1995] T. Hoshiro, Failure of analytic hypoellipticity for some operators of X 2 +Y 2 type, J. Math. Kyoto Univ. 35 (1995) p. 569-581 | MR | Zbl

[K, 1973] J.J. Kohn, Pseudo-differential operators and hypoellipticity, Proceed. Symposia in Pure Math. XXIII (1973) p. 61-70 | MR | Zbl

[Ma, 1998] T. Matsuzawa, Optimal Gevrey esponents for some degenerate elliptic operators, J. Korean Math. Soc 35 (1998) p. 981-997 | MR | Zbl

[M, 1980,1] G. Métivier, Analytic hypoellipticity for operators with multiple characteristics, Comm. in PDE 6 (1980) p. 1-90 | MR | Zbl

[M, 1980,2] G. Métivier, Une classe d'opérateurs non hypoelliptiques analytiques, Indiana Univ. Math. J 29 (1980) p. 169-186 | MR | Zbl

[M, 1981] G. Métivier, Non-hypoellipticité analytique pour D x 2 +(x 2 +y 2 )D y 2 , C. R. Acad. Sci. Paris 292 (1981) p. 401-404 | MR | Zbl

[N, 1966] T. Nagano, Linear differential systems with singularities and applications to transitive Lie algebras, J. Math. Soc. Japan 18 (1966) p. 398-404 | MR | Zbl

[O, 1973] O. Oleinik, On the analyticity of solutions of partial differential equations and systems, Astérisque 2,3 (1973) p. 272-285 | Numdam | MR | Zbl

[O-R, 1973] O. A. Oleinik & E.V. Radkevic, Second order equations with nonnegative characteristic form, AMS and Plenum Press, 1973 | MR

[R-S, 1977] L.P. Rothschild & E.M. Stein, Hypoelliptic differential operators and nilpotent groups, Acta Math 137 (1977) p. 247-320 | MR | Zbl

[S, 1974] J. Sjöstrand, Parametrices for pseudodifferential operators with multiple characteristics, Arkiv för Mat 12 (1974) p. 85-130 | MR | Zbl

[S, 1982] J. Sjöstrand, Analytic wavefront sets and operators with multiple characteristics, Hokkaido Math. J 12 (1983) p. 392-433 | MR | Zbl

[Su, 1990] H.J. Sussmann, Real-analytic desingularization and subanalytic sets: an elementary approach, Trans. A. M. S 317 (1990) p. 417-461 | MR | Zbl

[Ta, 1980] D.S Tartakoff, On the local real analyticity of solutions to U and the ¯-Neumann problem, Acta Math. 145 (1980) p. 117-204 | MR | Zbl

[Tr, 1984] J.-M. Trépreau, Sur l'hypoellipticité analytique microlocale des opérateurs de type principal, Comm. in PDE 9 (1984) p. 1119-1146 | MR | Zbl

[T, 1971] F. Treves, Analytic hypo-elliptic PDEs of principal type, Comm. Pure Applied Math. XXIV (1971) p. 537-570 | MR | Zbl

[T, 1978] F. Treves, Analytic hypo-ellipticity of a class of pseudodifferential operators with double characteristics and applications to the ¯-Neumann problem, Comm. in PDE 3 (1978) p. 476-642 | MR | Zbl

[T, 1999] F. Treves, Symplectic geometry and analytic hypo-ellipticity, Proceed. Sympos. Pure Math. 65, A.M.S., 1999, p. 201-219 | Zbl

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