Table of contents for this issue | Previous article | Next article
Antonio Bove; François Treves
On the Gevrey hypo-ellipticity of sums of squares of vector fields
(Hypo-ellipticité Gevrey de sommes de carrés de champs vectoriels)
Annales de l'institut Fourier, 54 no. 5 (2004), p. 1443-1475, doi: 10.5802/aif.2055
Article PDF | Reviews MR 2127854 | Zbl 1073.35067
Class. Math.: 35H05, 35A20
Keywords: stratification, symplectic, sums of squares of vector fields, analytic and Gevrey hypo-ellipticity
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.
[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 Article | MR 296507 | Zbl 0276.35023
[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 1647900 | Zbl 0937.35023
[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 1466316 | Zbl 0921.35043
[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 1984848 | Zbl 1036.35059
[C, 1992] M. Christ, “A class of hypoelliptic PDE admitting nonanalytic solutions”, Contemporary Math. A. M. S 137 (1992), p. 155-167 MR 1190978 | Zbl 0804.35022
[C, 1994] M. Christ, “A necessary condition for analytic hypoellipticity”, Math. Research Letters 1 (1994), p. 241-248 MR 1266762 | Zbl 0841.35026
[C, 2003] M. Christ, Hypoellipticity: geometrization and speculation, Progress in Math. A. M. S., Birkhäuser, 1997, p. 91-109 Zbl 0965.47033
[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 arXiv
[Ch-H-L, 20] S. Chanillo, B. Helffer & A. Laptev, “Nonlinear eigenvalues and analytic hypoellipticity”, J. of Functional Analysis, to appear, 2003 arXiv | Zbl 1053.35045
[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 05019602
[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 390474 | Zbl 0263.35020
[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 Article | MR 791290 | Zbl 0581.35009
[Gr, 1971] V. V. Gru#x0161;in, “On a class of elliptic pseudodifferential operators degenerate on a submanifold”, Math. USSR Sbornik 13 (1971), p. 155-185 Zbl 0238.47038
[Hanges, 20] N. Hanges, “Analytic regularity for an operator with Treves curves”, to appear MR 2053489 | Zbl 1053.35046
[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 1132794 | Zbl 0745.35011
[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 1422872 | Zbl 0906.35027
[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 1307525 | Zbl 0858.35025
[Hlf, 1979] B. Helffer, Hypoellipticité analytique sur des groupes nilpotents de rang 2, p. I.1-I.13 Cedram | Zbl 0471.35022
[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 661164 | Zbl 0458.35019
[H, 1967] L. Hörmander, “Hypoelliptic second order differential equations”, Acta Math 119 (1967), p. 147-171 MR 222474 | Zbl 0156.10701
[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 Article | MR 1365248 | Zbl 0846.35034
[K, 1973] J.J. Kohn, “Pseudo-differential operators and hypoellipticity”, Proceed. Symposia in Pure Math. XXIII (1973), p. 61-70 MR 338592 | Zbl 0262.35007
[Ma, 1998] T. Matsuzawa, “Optimal Gevrey esponents for some degenerate elliptic operators”, J. Korean Math. Soc 35 (1998), p. 981-997 MR 1666486 | Zbl 0924.35031
[M, 1980,1] G. Métivier, “Analytic hypoellipticity for operators with multiple characteristics”, Comm. in PDE 6 (1980), p. 1-90 MR 597752 | Zbl 0455.35040
[M, 1980,2] G. Métivier, “Une classe d'opérateurs non hypoelliptiques analytiques”, Indiana Univ. Math. J 29 (1980), p. 169-186 MR 589650 | Zbl 0455.35041
[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 609762 | Zbl 0481.35033
[N, 1966] T. Nagano, “Linear differential systems with singularities and applications to transitive Lie algebras”, J. Math. Soc. Japan 18 (1966), p. 398-404 Article | MR 199865 | Zbl 0147.23502
[O, 1973] O. Oleinik, “On the analyticity of solutions of partial differential equations and systems”, Astérisque 2,3 (1973), p. 272-285 MR 399640 | Zbl 0291.35013
[O-R, 1973] O. A. Oleinik & E.V. Radkevic, Second order equations with nonnegative characteristic form, AMS and Plenum Press, 1973 MR 457908
[R-S, 1977] L.P. Rothschild & E.M. Stein, “Hypoelliptic differential operators and nilpotent groups”, Acta Math 137 (1977), p. 247-320 MR 436223 | Zbl 0346.35030
[S, 1974] J. Sjöstrand, “Parametrices for pseudodifferential operators with multiple characteristics”, Arkiv för Mat 12 (1974), p. 85-130 MR 352749 | Zbl 0317.35076
[S, 1982] J. Sjöstrand, “Analytic wavefront sets and operators with multiple characteristics”, Hokkaido Math. J 12 (1983), p. 392-433 MR 725588 | Zbl 0531.35022
[Su, 1990] H.J. Sussmann, “Real-analytic desingularization and subanalytic sets: an elementary approach”, Trans. A. M. S 317 (1990), p. 417-461 MR 943608 | Zbl 0696.32005
[Ta, 1980] D.S Tartakoff, “On the local real analyticity of solutions to $\scriptstyle\square _U$ and the $\overline \partial $-Neumann problem”, Acta Math. 145 (1980), p. 117-204 MR 590289 | Zbl 0456.35019
[Tr, 1984] J.-M. Trépreau, “Sur l'hypoellipticité analytique microlocale des opérateurs de type principal”, Comm. in PDE 9 (1984), p. 1119-1146 MR 759240 | Zbl 0566.35027
[T, 1971] F. Treves, “Analytic hypo-elliptic PDEs of principal type”, Comm. Pure Applied Math. XXIV (1971), p. 537-570 MR 296509 | Zbl 0222.35014
[T, 1978] F. Treves, “Analytic hypo-ellipticity of a class of pseudodifferential operators with double characteristics and applications to the $\overline\partial $-Neumann problem”, Comm. in PDE 3 (1978), p. 476-642 MR 492802 | Zbl 0384.35055
[T, 1999] F. Treves, Symplectic geometry and analytic hypo-ellipticity, Proceed. Sympos. Pure Math., A.M.S., 1999, p. 201-219 Zbl 0938.35038
Antonio Bove; François Treves
On the Gevrey hypo-ellipticity of sums of squares of vector fields
(Hypo-ellipticité Gevrey de sommes de carrés de champs vectoriels)
Annales de l'institut Fourier, 54 no. 5 (2004), p. 1443-1475, doi: 10.5802/aif.2055
Article PDF | Reviews MR 2127854 | Zbl 1073.35067
Class. Math.: 35H05, 35A20
Keywords: stratification, symplectic, sums of squares of vector fields, analytic and Gevrey hypo-ellipticity
Résumé - Abstract
Bibliography
[B-G, 1972] M. S. Baouendi & Ch. Goulaouic, “Non analytic-hypoellipticity for some degenerate operators”, Bull. A. M. S 78 (1972), p. 483-486 Article | MR 296507 | Zbl 0276.35023
[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 1647900 | Zbl 0937.35023
[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 1466316 | Zbl 0921.35043
[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 1984848 | Zbl 1036.35059
[C, 1992] M. Christ, “A class of hypoelliptic PDE admitting nonanalytic solutions”, Contemporary Math. A. M. S 137 (1992), p. 155-167 MR 1190978 | Zbl 0804.35022
[C, 1994] M. Christ, “A necessary condition for analytic hypoellipticity”, Math. Research Letters 1 (1994), p. 241-248 MR 1266762 | Zbl 0841.35026
[C, 2003] M. Christ, Hypoellipticity: geometrization and speculation, Progress in Math. A. M. S., Birkhäuser, 1997, p. 91-109 Zbl 0965.47033
[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 arXiv
[Ch-H-L, 20] S. Chanillo, B. Helffer & A. Laptev, “Nonlinear eigenvalues and analytic hypoellipticity”, J. of Functional Analysis, to appear, 2003 arXiv | Zbl 1053.35045
[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 05019602
[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 390474 | Zbl 0263.35020
[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 Article | MR 791290 | Zbl 0581.35009
[Gr, 1971] V. V. Gru#x0161;in, “On a class of elliptic pseudodifferential operators degenerate on a submanifold”, Math. USSR Sbornik 13 (1971), p. 155-185 Zbl 0238.47038
[Hanges, 20] N. Hanges, “Analytic regularity for an operator with Treves curves”, to appear MR 2053489 | Zbl 1053.35046
[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 1132794 | Zbl 0745.35011
[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 1422872 | Zbl 0906.35027
[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 1307525 | Zbl 0858.35025
[Hlf, 1979] B. Helffer, Hypoellipticité analytique sur des groupes nilpotents de rang 2, p. I.1-I.13 Cedram | Zbl 0471.35022
[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 661164 | Zbl 0458.35019
[H, 1967] L. Hörmander, “Hypoelliptic second order differential equations”, Acta Math 119 (1967), p. 147-171 MR 222474 | Zbl 0156.10701
[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 Article | MR 1365248 | Zbl 0846.35034
[K, 1973] J.J. Kohn, “Pseudo-differential operators and hypoellipticity”, Proceed. Symposia in Pure Math. XXIII (1973), p. 61-70 MR 338592 | Zbl 0262.35007
[Ma, 1998] T. Matsuzawa, “Optimal Gevrey esponents for some degenerate elliptic operators”, J. Korean Math. Soc 35 (1998), p. 981-997 MR 1666486 | Zbl 0924.35031
[M, 1980,1] G. Métivier, “Analytic hypoellipticity for operators with multiple characteristics”, Comm. in PDE 6 (1980), p. 1-90 MR 597752 | Zbl 0455.35040
[M, 1980,2] G. Métivier, “Une classe d'opérateurs non hypoelliptiques analytiques”, Indiana Univ. Math. J 29 (1980), p. 169-186 MR 589650 | Zbl 0455.35041
[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 609762 | Zbl 0481.35033
[N, 1966] T. Nagano, “Linear differential systems with singularities and applications to transitive Lie algebras”, J. Math. Soc. Japan 18 (1966), p. 398-404 Article | MR 199865 | Zbl 0147.23502
[O, 1973] O. Oleinik, “On the analyticity of solutions of partial differential equations and systems”, Astérisque 2,3 (1973), p. 272-285 MR 399640 | Zbl 0291.35013
[O-R, 1973] O. A. Oleinik & E.V. Radkevic, Second order equations with nonnegative characteristic form, AMS and Plenum Press, 1973 MR 457908
[R-S, 1977] L.P. Rothschild & E.M. Stein, “Hypoelliptic differential operators and nilpotent groups”, Acta Math 137 (1977), p. 247-320 MR 436223 | Zbl 0346.35030
[S, 1974] J. Sjöstrand, “Parametrices for pseudodifferential operators with multiple characteristics”, Arkiv för Mat 12 (1974), p. 85-130 MR 352749 | Zbl 0317.35076
[S, 1982] J. Sjöstrand, “Analytic wavefront sets and operators with multiple characteristics”, Hokkaido Math. J 12 (1983), p. 392-433 MR 725588 | Zbl 0531.35022
[Su, 1990] H.J. Sussmann, “Real-analytic desingularization and subanalytic sets: an elementary approach”, Trans. A. M. S 317 (1990), p. 417-461 MR 943608 | Zbl 0696.32005
[Ta, 1980] D.S Tartakoff, “On the local real analyticity of solutions to $\scriptstyle\square _U$ and the $\overline \partial $-Neumann problem”, Acta Math. 145 (1980), p. 117-204 MR 590289 | Zbl 0456.35019
[Tr, 1984] J.-M. Trépreau, “Sur l'hypoellipticité analytique microlocale des opérateurs de type principal”, Comm. in PDE 9 (1984), p. 1119-1146 MR 759240 | Zbl 0566.35027
[T, 1971] F. Treves, “Analytic hypo-elliptic PDEs of principal type”, Comm. Pure Applied Math. XXIV (1971), p. 537-570 MR 296509 | Zbl 0222.35014
[T, 1978] F. Treves, “Analytic hypo-ellipticity of a class of pseudodifferential operators with double characteristics and applications to the $\overline\partial $-Neumann problem”, Comm. in PDE 3 (1978), p. 476-642 MR 492802 | Zbl 0384.35055
[T, 1999] F. Treves, Symplectic geometry and analytic hypo-ellipticity, Proceed. Sympos. Pure Math., A.M.S., 1999, p. 201-219 Zbl 0938.35038
© Annales de L'Institut Fourier - ISSN (électronique) : 1777-5310