Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Global analytic and Gevrey hypoellipticity of sublaplacians under Diophantine conditions

Author: A. Alexandrou Himonas
Journal: Proc. Amer. Math. Soc. 129 (2001), 2061-2067
MSC (1991): Primary 35H05
Published electronically: December 4, 2000
MathSciNet review: 1825918
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information


In this paper we consider the problem of global Gevrey and analytic regularity for a class of partial differential operators on a torus in the form of a sum of squares of vector fields, which may not satisfy the bracket condition. We show that these operators are globally Gevrey or analytic hypoelliptic on the torus if and only if the coefficients satisfy certain Diophantine approximation properties.

References [Enhancements On Off] (What's this?)

  • [BG] M.S. Baouendi and C. Goulaouic, Nonanalytic-hypoellipticity for some degenerate elliptic operators, Bull. AMS 78, (1972), 483-486. MR 45:5567
  • [BT1] B. E. Bove and D. Tartakoff, Optimal non-isotropic Gevrey exponents for sums of squares of vector fields, Comm. Partial Differential Equations 22, no. 7-8, (1997), 1263-1282. MR 98f:35026
  • [BT2] B. E. Bove and D. Tartakoff, On a conjecture of Treves: analytic hypoellipticity and Poisson strata, Indiana Univ. Math. J. 47, no. 2, (1998), 401-417. MR 2000d:35022
  • [C1] M. Christ, Certain sums of squares of vector fields fail to be analytic hypoelliptic, Comm. in PDE 16, No. 10, (1991), 1695-1707. MR 92k:35056
  • [C2] M. Christ, Global analytic hypoellipticity in the presence of symmetry, Math. Res. Lett. 1, (1994), 559-563. MR 95j:35047
  • [C3] M. Christ, Intermediate optimal Gevrey exponents occur, Comm. Partial Differential Equations 22 , no. 3-4, (1997), 359-379. MR 98c:35028
  • [CH] P. D. Cordaro and A. A. Himonas, Global analytic hypoellipticity for a class of degenerate elliptic operators on the torus, Math. Res. Lett. 1, (1994), 501-510. MR 95j:35048
  • [GW] S. J. Greenfield and N. R. Wallach, Global hypoellipticity and Liouville numbers, Proceedings of the AMS 31, (1972), 112-114. MR 45:5568
  • [HH1] N. Hanges and A.A. Himonas, Singular solutions for sums of squares of vector fields, Comm. in PDE 16, No. 8 & 9, (1991), 1503-1511. MR 92i:35031
  • [HH2] N. Hanges and A.A. Himonas, Analytic hypoellipticity for generalized Baouendi-Goulaouic operators, J. Funct. Anal. 125, No. 1, (1994), 309-325. MR 95j:35049
  • [HH3] N. Hanges and A.A. Himonas, Non-analytic hypoellipticity in the presence of symplecticity, Proceedings of AMS 126, No. 2, (1998), 405-409. MR 98d:35031
  • [H] B. Helffer, Conditions nécessaires d'hypoanalyticité pour des opérateurs invariants à gauche homogènes sur un groupe nilpotent gradué, J.D. Equations 44, (1982), 460-481. MR 84c:35026
  • [He] R. Herman, Sur le groupe des diffeomorphismes du tore, Ann. Inst. Fourier, Grenoble, vol 23, (1973), 75-86. MR 52:11988
  • [Ho] L. Hörmander, Hypoelliptic second order differential equations, Acta Math. 119, (1967), 147-171. MR 36:5526
  • [HP] A. A. Himonas and G. Petronilho, Global hypoellipticity and simultaneous approximability, J. Funct. Anal. 170, (2000), 356-365. CMP 2000:08
  • [K] J.J. Kohn, Pseudo-differential operators and hypoellipticity, Proceedings of Symposia in Pure Mathematics XXIII, (1973), 61-70. MR 49:3356
  • [M1] G. Metivier, Analytic hypoellipticity for operators with multiple characteristics, Comm. Partial Differential Equations 1 (1981), 1-90. MR 82g:35030
  • [M2] G. Metivier, Une class d'operateurs non hypoelliptiques analytiques, Indiana Univ. Math. J. 29 (1980), 823-860. MR 82a:35029
  • [OR] O.A. Oleinik and E. V. Radkevic, Second order equations with nonnegative characteristic form, AMS and Plenum Press, (1973). MR 56:16112
  • [PR] Pham The Lai and D. Robert, Sur un problème aux valeurs propres non linéaire, Israel J. of Math. 36, (1980), 169-186. MR 80b:35132
  • [R] L. Rodino, Linear partial differential operators in Gevrey spaces, World Scientific, (1993). MR 95c:35001
  • [RS] L.P. Rothschild and E.M. Stein, Hypoelliptic differential operators and nilpotent groups, Acta Math 137, (1977), 247-320. MR 55:9171
  • [S] J. Sjöstrand, Analytic wavefront sets and operators with multiple characteristics, Hokkaido Mathematical Journal 12, (1983), 392-433. MR 85e:35022
  • [T1] D. S. Tartakoff, Local analytic hypoellipticity for $\square_b$ on non-degenerate Cauchy-Riemann manifolds, Proc. Nat. Acad. Sci. U.S.A., 75:7, (1978), 3027-3028. MR 80g:58045
  • [T2] D. S. Tartakoff, On the local real analyticity of solutions to $\square_b$ and the $\bar{\partial}$-Neumann problem, Acta Math. 145, (1980), 117-204. MR 81k:35033
  • [T3] D. S. Tartakoff, Global (and local) analyticity for second order orerators constructed from rigid vector fields on products of tori, Trans. AMS 348, No 7, (1996), 2577-2583. MR 96i:35018
  • [Tr1] F. Treves, Analytic hypo-ellipticity of a class of pseudodifferential operators with double characteristics and applications to the $\bar\partial$-Neuman problem, Comm. Partial Differential Equations 3 (1978), 475-642. MR 58:11867
  • [Tr2] F. Treves, Symplectic geometry and analytic hypo-ellipticity, La Pietra 1996 (Florence), Proc. Sympos. Pure Math., 65, Amer. Math. Soc., Providence, RI, (1999), 201-219. MR 2000b:35031
  • [Y] J. C. Yoccoz, Recent development in dynamics, Proc. of the International Congress of Mathematicians, Zurich, Switzerland, August 1994, Birkhauser, (1995), 246-265. MR 98e:58146

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 35H05

Retrieve articles in all journals with MSC (1991): 35H05

Additional Information

A. Alexandrou Himonas
Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556

Keywords: Global, analytic and Gevrey hypoellipticity, Diophantine conditions, torus, Fourier transform, bracket condition
Received by editor(s): November 15, 1999
Published electronically: December 4, 2000
Additional Notes: The author was supported in part by NSF Grant DMS-9970857.
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society