The article studies a second-order linear differential operator of the type , i. e., a sum of squares of real, and real-analytic, vector fields . 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 depends on the restrictions of the principal symbol to or symplectic manifolds associated to each bicharateristic curve in a nonsymplectic stratum.
On étudie un opérateur différentiel du second ordre du type , où les sont des champs vectoriels réels et analytiques. On décrit, en termes analytiques et géométriques simples, la stratification de Poisson de la variété caractéristique de et on rappelle la conjecture selon laquelle une condition nécessaire et suffisante pour l’hypo-ellipticité analytique de serait que chaque strate de Poisson soit symplectique. Les auteurs formulent une conjecture nouvelle sur l’hypo-ellipticité Gevrey de selon laquelle cette propriété dépendrait de la restriction du symbole principal à certaines sous-variétés bi- ou quadri-dimensionnelles contenant une courbe bicaratéristique d’une strate non symplectique.
@article{AIF_2004__54_5_1443_0, author = {Bove, Antonio and Treves, Fran\c{c}ois}, title = {On the {Gevrey} hypo-ellipticity of sums of squares of vector fields}, journal = {Annales de l'Institut Fourier}, pages = {1443--1475}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {54}, number = {5}, year = {2004}, doi = {10.5802/aif.2055}, mrnumber = {2127854}, zbl = {1073.35067}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/aif.2055/} }
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 SP - 1443 EP - 1475 VL - 54 IS - 5 PB - Association des Annales de l’institut Fourier UR - http://archive.numdam.org/articles/10.5802/aif.2055/ DO - 10.5802/aif.2055 LA - en ID - AIF_2004__54_5_1443_0 ER -
%0 Journal Article %A Bove, Antonio %A Treves, François %T On the Gevrey hypo-ellipticity of sums of squares of vector fields %J Annales de l'Institut Fourier %D 2004 %P 1443-1475 %V 54 %N 5 %I Association des Annales de l’institut Fourier %U http://archive.numdam.org/articles/10.5802/aif.2055/ %R 10.5802/aif.2055 %G en %F AIF_2004__54_5_1443_0
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] Analytic stratifications and the cutlocus of a class of distance functions, Preprint
& ,[B-G, 1972] Non analytic-hypoellipticity for some degenerate operators, Bull. A. M. S 78 (1972) p. 483-486 | MR | Zbl
& ,[Be-Bo-Ta,] On a conjecture of Treves: analytic hypoellipticity and Poisson strata, Indiana Univ. Math. J 47 (1998) p. 401-417 | MR | Zbl
, & ,[Bo-Ta, 199] 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] 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] A class of hypoelliptic PDE admitting nonanalytic solutions, Contemporary Math. A. M. S 137 (1992) p. 155-167 | MR | Zbl
,[C, 1994] A necessary condition for analytic hypoellipticity, Math. Research Letters 1 (1994) p. 241-248 | MR | Zbl
,[C, 2003] Hypoellipticity: geometrization and speculation, Progress in Math. A. M. S., Birkhäuser, 1997, p. 91-109 | Zbl
,[Ch, 2001] 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] Nonlinear eigenvalues and analytic hypoellipticity, J. of Functional Analysis, to appear, 2003 | Zbl
, & ,[Cost, 2003] 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] Régularité analytique et Gevrey d'opérateurs elliptiques dégénérés, J. Math. Pures et Appl 52 (1973) p. 65-80 | MR | Zbl
& ,[G-S, 1985] Front d'onde analytique et sommes de carrés de champs de vecteurs, Duke Math. J 52 (1985) p. 35-51 | MR | Zbl
& ,[Gr, 1971] On a class of elliptic pseudodifferential operators degenerate on a submanifold, Math. USSR Sbornik 13 (1971) p. 155-185 | Zbl
,[Hanges, 20] Analytic regularity for an operator with Treves curves, to appear | MR | Zbl
,[H-H, 1991] Singular solutions for sums of squares of vector fields, Comm. in PDE 16 (1991) p. 1503-1511 | MR | Zbl
& ,[H-H, 1995] Non-analytic hypoellipticity in the presence of symplecticity, Proceed. A.M.S 126 (1998) p. 1549-1557 | MR | Zbl
& ,[H-H, 1996] Singular solutions for a class of Grusin type operators, Proceed. A. M. S 124 (1996) p. 1549-1557 | MR | Zbl
& ,[Hlf, 1979] Hypoellipticité analytique sur des groupes nilpotents de rang 2, p. I.1-I.13 | Numdam | Zbl
,[Hlf, 1982] 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] Hypoelliptic second order differential equations, Acta Math 119 (1967) p. 147-171 | MR | Zbl
,[Hosh, 1995] Failure of analytic hypoellipticity for some operators of type, J. Math. Kyoto Univ. 35 (1995) p. 569-581 | MR | Zbl
,[K, 1973] Pseudo-differential operators and hypoellipticity, Proceed. Symposia in Pure Math. XXIII (1973) p. 61-70 | MR | Zbl
,[Ma, 1998] Optimal Gevrey esponents for some degenerate elliptic operators, J. Korean Math. Soc 35 (1998) p. 981-997 | MR | Zbl
,[M, 1980,1] Analytic hypoellipticity for operators with multiple characteristics, Comm. in PDE 6 (1980) p. 1-90 | MR | Zbl
,[M, 1980,2] Une classe d'opérateurs non hypoelliptiques analytiques, Indiana Univ. Math. J 29 (1980) p. 169-186 | MR | Zbl
,[M, 1981] Non-hypoellipticité analytique pour , C. R. Acad. Sci. Paris 292 (1981) p. 401-404 | MR | Zbl
,[N, 1966] Linear differential systems with singularities and applications to transitive Lie algebras, J. Math. Soc. Japan 18 (1966) p. 398-404 | MR | Zbl
,[O, 1973] On the analyticity of solutions of partial differential equations and systems, Astérisque 2,3 (1973) p. 272-285 | Numdam | MR | Zbl
,[O-R, 1973] Second order equations with nonnegative characteristic form, AMS and Plenum Press, 1973 | MR
& ,[R-S, 1977] Hypoelliptic differential operators and nilpotent groups, Acta Math 137 (1977) p. 247-320 | MR | Zbl
& ,[S, 1974] Parametrices for pseudodifferential operators with multiple characteristics, Arkiv för Mat 12 (1974) p. 85-130 | MR | Zbl
,[S, 1982] Analytic wavefront sets and operators with multiple characteristics, Hokkaido Math. J 12 (1983) p. 392-433 | MR | Zbl
,[Su, 1990] Real-analytic desingularization and subanalytic sets: an elementary approach, Trans. A. M. S 317 (1990) p. 417-461 | MR | Zbl
,[Ta, 1980] On the local real analyticity of solutions to and the -Neumann problem, Acta Math. 145 (1980) p. 117-204 | MR | Zbl
,[Tr, 1984] Sur l'hypoellipticité analytique microlocale des opérateurs de type principal, Comm. in PDE 9 (1984) p. 1119-1146 | MR | Zbl
,[T, 1971] Analytic hypo-elliptic PDEs of principal type, Comm. Pure Applied Math. XXIV (1971) p. 537-570 | MR | Zbl
,[T, 1978] 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] Symplectic geometry and analytic hypo-ellipticity, Proceed. Sympos. Pure Math. 65, A.M.S., 1999, p. 201-219 | Zbl
,Cited by Sources: