Skip to main content
SpringerLink
Log in

Global hypoellipticity for first-order operators on closed smooth manifolds

  • Published:
Journal d'Analyse Mathématique Aims and scope

Abstract

The main goal of this paper is to address global hypoellipticity issues for the class of first-order pseudo-differential operators L = Dt + C(t, x,Dx), where (t, x) ∈ T × M, T is the one-dimensional torus, M is a closed manifold, and C(t, x,Dx) is a first-order pseudo-differential operator on M, smoothly depending on the periodic variable t. In the case of separation of variables, when C(t, x,Dx) = a(t)p(x,Dx) + ib(t)q(x,Dx), we give necessary and sufficient conditions for the global hypoellipticity of L. In particular, we show that the famous (P) condition of Nirenberg-Treves is neither necessary nor sufficient to guarantee the global hypoellipticity of L.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. V. Beresnevich, D. Dickinson, and S. Velani, Diophantine approximation on planar curves and the distribution of rational points, Ann. of Math. (2) 166 (2007), 367–426.

    Article  MathSciNet  MATH  Google Scholar 

  2. A. P. Bergamasco, Remarks about global analytic hypoellipticity, Trans. Amer. Math. Soc. 351, (1999), 4113–4126.

    Article  MathSciNet  MATH  Google Scholar 

  3. A. P. Bergamasco, P. D. Cordaro, and G. Petronilho, Global solvability for a class of complex vector fields on the two-torus, Comm. Partial Differential Equations 29 (2004), 785–819.

    Article  MathSciNet  MATH  Google Scholar 

  4. A. P. Bergamasco, P. D. da Silva, R. B. Gonzalez, and A. Kirilov, Global solvability and global hypoellipticity for a class of complex vector fields on the 3-torus, J. Pseudo-Differ Oper. Appl. 6 (2015), 341–360.

    Article  MathSciNet  MATH  Google Scholar 

  5. A. P. Bergamasco, A. Kirilov, S. L. Zani, and W. V. L. Nunes, Global solutions to involutive systems, Proc. Amer. Math. Soc. 143 (2015), 4851–4862.

    Article  MathSciNet  MATH  Google Scholar 

  6. F. Cardoso and J. Hounie, Global solvability of an abstract complex, Proc. Amer. Math. Soc. 65 (1977), 117–124.

    Article  MathSciNet  MATH  Google Scholar 

  7. W. Chen and M. Chi, Hypoelliptic vector fields and almost periodic motions on the torus T n Comm. Partial Differential Equations 25, (2000), 337–354.

    Article  MathSciNet  MATH  Google Scholar 

  8. J. Delgado and M. Ruzhansky, Fourier multipliers, symbols and nuclearity on compact manifolds, J. Anal. Math. 135 (2018),757–800.

    Article  MathSciNet  Google Scholar 

  9. J. Delgado and M. Ruzhansky, Kernel and symbol criteria for Schatten classes and r-nuclearity on compact manifolds, C. R. Acad. Sci. Paris Ser. I 352 (2014), 779–784.

    Article  MathSciNet  MATH  Google Scholar 

  10. J. Delgado and M. Ruzhansky, Schatten classes on compact manifolds: Kernel conditions, J. Funct. Anal. 267 (2014), 772–798.

    Article  MathSciNet  MATH  Google Scholar 

  11. D. Dickinson, T. Gramchev, and M. Yoshino, First order pseudodifferential operators on the torus: normal forms, diophantine phenomena and global hypoellipticity Ann, Univ. Ferrara Sez. VII (N.S.) 41 (1997), 51–64.

    MATH  Google Scholar 

  12. G. Forni, On the Greenfield-Wallach and Katok conjectures in dimension three, in Geometric and Probabilistic Structures in Dynamics, Contemp. Math. 469 (2008), 197–213.

    Article  MATH  Google Scholar 

  13. A. Gorodnik and A. Nevo, Quantitative ergodic theorems and their number-theoretic applications, Bull. Amer. Math. Soc. 52 (2015), 65–113.

    Article  MathSciNet  MATH  Google Scholar 

  14. T. Gramchev, S. Pilipovic, and L. Rodino, Eigenfunction expansions in Rn Proc. Amer. Math. Soc. 139, (2011), 4361–4368.

    Article  MATH  Google Scholar 

  15. S. Greenfield and N. R. Wallach, Global hypoellipticity and Liouville numbers, Proc. Amer.Math. Soc. 31 (1972), 112–114.

    Article  MathSciNet  MATH  Google Scholar 

  16. S. Greenfield and N. R. Wallach, Globally hypoelliptic vector fields, Topology 12 (1973), 247–253.

    Article  MathSciNet  MATH  Google Scholar 

  17. S. Greenfield and N. R. Wallach, Remarks on global hypoellipticity, Trans. Amer. Math. Soc. 183 (1973), 153–164.

    Article  MathSciNet  MATH  Google Scholar 

  18. J. Hounie, Globally hypoelliptic and globally solvable first-order evolution equations, Trans. Amer. Math. Soc. 252 (1979), 233–248.

    MathSciNet  MATH  Google Scholar 

  19. J. Hounie, Globally hypoelliptic vector fields on compact surfaces, Comm. Partial Differential Equations 7 (1982), 343–370.

    Article  MathSciNet  MATH  Google Scholar 

  20. A. Katok, Combinatorial Constructions in Ergodic Theory and Dynamics, Amer. Math. Soc., Providence, RI, 2003.

    Book  MATH  Google Scholar 

  21. A. Katok and E. A. Robinson, Cocycles, cohomology and combinatorial constructions in ergodic theory, in Smooth Ergodic Theory and its Applications, Amer. Math. Soc., Providence, RI, 2001, pp. 107–173.

    MATH  Google Scholar 

  22. A. Kocsard, Cohomologically rigid vector fields: the Katok conjecture in dimension 3, Ann. Inst. H. Poincare Analyse Nonlinéaire 26 (2009), 1165–1182.

    Article  MathSciNet  MATH  Google Scholar 

  23. L. Nirenberg and F. Treves, On local solvability of linear partial differential equations, I. Necessary conditions, Comm. Pure and Appl. Math. 23, (1970), 1–38.

    Article  MathSciNet  MATH  Google Scholar 

  24. L. Nirenberg and F. Treves, Remarks on the solvability of linear equations of evolution, Symposia Math. 7 (1971), 325–338.

    MathSciNet  MATH  Google Scholar 

  25. G. Petronilho, Global hypoellipticity, global solvability and normal form for a class of real vector fields on a torus and application, Trans. Amer. Math. Soc. 363, (2011), 6337–6349.

    Article  MathSciNet  MATH  Google Scholar 

  26. M. Ruzhansky and V. Turunen, Pseudo-Differential Operators and Symmetries, Pseudo-Differential Operators Theory and Applications, vol. 2, Birkhäuser, Basel, 2010.

    Book  MATH  Google Scholar 

  27. M. Ruzhansky and V. Turunen, Quantization of pseudo-differential operators on the torus, J. Fourier Anal. Appl. 16 (2010), 943–982.

    Article  MathSciNet  MATH  Google Scholar 

  28. M. Ruzhansky, V. Turunen, and J. Wirth, Hörmander class of pseudo-differential operators on compact Lie groups and global hypoellipticity, J. Fourier Anal. Appl. 20, (2014), 476–499.

    Article  MathSciNet  MATH  Google Scholar 

  29. R. T. Seeley, Integro-differential operators on vector bundles, Trans. Amer. Math. Soc. 117 (1965), 167–204.

    Article  MathSciNet  MATH  Google Scholar 

  30. R. T. Seeley, Eigenfunction expansions of analytic functions, Proc. Amer. Math. Soc. 21 (1969), 734–738.

    Article  MathSciNet  MATH  Google Scholar 

  31. M. A. Shubin, Pseudodifferential Operators and Spectral Theory, vol. 2. Springer-Verlag, Berlin, 2001.

    Book  MATH  Google Scholar 

  32. F. Treves, Hamiltonian fields, bicharacteristic strips in relation with existence and regularity of solutions of linear partial differential equations, Actes du Congrés International des Mathématiciens,Gauthier-Villars, Paris, 1971, pp. 803–811.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Matemática, Universidade Federal do Paraná, Caixa Postal 19081, Curitiba, PR, 81531-990, Brazil

    Fernando de Ávila Silva, Todor Gramchev & Alexandre Kirilov

Authors
  1. Fernando de Ávila Silva
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Todor Gramchev
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Alexandre Kirilov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexandre Kirilov.

Additional information

Dedicated to the memory of Todor Gramchev

Supported by CAPES Foundation, Ministry of Education of Brazil.

Todor Gramchev, friend, teacher, and coauthor, passed away on October 18th, 2015, shortly after completion of this article.

Partially supported by IMI–UFPR and CAPES Foundation.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

de Ávila Silva, F., Gramchev, T. & Kirilov, A. Global hypoellipticity for first-order operators on closed smooth manifolds. JAMA 135, 527–573 (2018). https://doi.org/10.1007/s11854-018-0039-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11854-018-0039-6

Search

Navigation