Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Global hypoellipticity, global solvability and normal form for a class of real vector fields on a torus and application


Author: G. Petronilho
Journal: Trans. Amer. Math. Soc. 363 (2011), 6337-6349
MSC (2010): Primary 35A01, 35F05, 35H10
DOI: https://doi.org/10.1090/S0002-9947-2011-05359-6
Published electronically: July 20, 2011
MathSciNet review: 2833557
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The main purpose of this paper is to present a class of real vector fields defined on a torus for which the concepts of global hypoellipticity and global smooth solvability are equivalent. Furthermore, such a vector field is globally hypoelliptic if and only if its adjoint is globally hypoelliptic, and therefore we can reduce it to its normal form. As an application, we study global $ C^\infty$ solvability for certain classes of sub-Laplacians.


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

  • [AZ] A. A. Albanese and L. Zanghirati, Global hypoellipticity and global solvability in Gevrey classes on the $ n$-dimensional torus, J. Differential Equations 199, (2004), 256-268. MR 2047910 (2005a:35049)
  • [A] K. Amano, The global hypoellipticity of a class of degenerate elliptic-parabolic operators, Proc. Japan Acad. 60, Ser. 4, (1984), 312-314. MR 778515 (87f:35054)
  • [BCP] A. P. Bergamasco, P. D. Cordaro and G. Petronilho, Global solvability for a class of complex vector fields on the two-torus, Comm. in PDE 29, no. 5 and 6, (2004), 785-819. MR 2059148 (2005k:35034)
  • [Br] A. D. Brjuno, Analytic form of differential equations, Trans. Moscow Math. Soc. 25, (1971), 131-288 and 26, (1972), 199-239. MR 0377192 (51:13365)
  • [CC] W. Chen and M.Y. Chi, Hypoelliptic vector fields and almost periodic motions on the torus $ T^n$, Comm. in PDE 25(1-2), (2000), 337-354. MR 1737551 (2000m:35042)
  • [FO] D. Fujiwara and H. Omori, An example of a globally hypoelliptic operator, Hokkaido Math. J. 12, (1983), 293-297. MR 719969 (86a:35038)
  • [GPY1] T. Gramchev, P. Popivanov and M. Yoshino, Global solvability and hypoellipticity on the torus for a class of differential operators with variable coefficients, Proc. Japan Acad., 68, (1992), 53-57. MR 1167986 (93g:35025)
  • [GPY2] T. Gramchev, P. Popivanov and M. Yoshino, Global properties in spaces of generalized functions on the torus for second order differential operators with variable coefficients, Rend. Sem. Mat. Pol. Torino 51, no.2, (1993), 145-172. MR 1289385 (95k:35047)
  • [GW1] S. J. Greenfield and N. R. Wallach, Global hypoellipticity and Liouville numbers, Proc. Amer. Math. Soc., 31 (1972), 112-114. MR 0296508 (45:5568)
  • [GW2] S. J. Greenfield and N. R. Wallach, Globally hypoelliptic vector field, Topology 12, (1973), 247-253. MR 0320502 (47:9039)
  • [He] M. R. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Publ. I.H.E.S. 49, (1979), 5-233. MR 538680 (81h:58039)
  • [Hi] A. A. Himonas, On degenerate elliptic operators of infinite type, Math. Z. 220, (1995), 449-460. MR 1362255 (96j:35089)
  • [HP1] A. A. Himonas and G. Petronilho, Global hypoellipticity and simultaneous approximability, J. Funct. Anal., 170, (2000), 356-365. MR 1740656 (2000m:35043)
  • [HP2] A. A. Himonas and G. Petronilho, On $ C^\infty$ and Gevrey regularity of sublaplacians, Trans. Amer. Math. Soc. 358, (2006), no. 11, 4809-4820. MR 2231873 (2007b:35057)
  • [HPS] A. A. Himonas, G. Petronilho and L. A. C. dos Santos, Analytic, Gevrey and $ C^\infty$ global regularity for a class of subLaplacinas, pre-printed.
  • [Hou] J. Hounie, Globally hypoelliptic vector field on compact surfaces, Comm. in PDE 7(4), (1982), 343-370. MR 652813 (83k:35025)
  • [OK] H. Omori and T. Kobayashi, Global hypoellipticity of subelliptic operators on closed manifolds, Hokkaido Math. J., 28, (1999), 613-633. MR 1723457 (2000k:35054)
  • [P1] G. Petronilho, Global solvability and simultaneously approximable vectors, J. Differential Equations, 184, (2002), 48-61. MR 1929145 (2004g:35047)
  • [P2] G. Petronilho, On Gevrey solvability and regularity, Math. Nachr., 282, no. 3, (2009), 470-481. MR 2503164 (2010d:35007)
  • [Y] J.C. Yoccoz, Conjugaison différentiable des difféomorphismes du cercle dont le nombre de rotation vérifie une condition Diophantienne, Ann. Sci. École Norm. Sup. IV Sér. 17, (1984), 333-359. MR 777374 (86j:58086)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 35A01, 35F05, 35H10

Retrieve articles in all journals with MSC (2010): 35A01, 35F05, 35H10


Additional Information

G. Petronilho
Affiliation: Departamento de Matemática, Universidade Federal de São Carlos São Carlos, SP 13565-905, Brazil
Email: gersonpetro@gmail.com

DOI: https://doi.org/10.1090/S0002-9947-2011-05359-6
Keywords: Global hypoellipticity, global solvability, normal form.
Received by editor(s): October 20, 2009
Published electronically: July 20, 2011
Additional Notes: The author was partially supported by CNPq and Fapesp.
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society