Advertisement

A Gevrey Differential Complex on the Torus

  • 14 Accesses

Abstract

To a system of n closed one-forms on the torus \({\mathbb {T}}^{m+n}\), we associate a differential complex and compute the induced cohomology groups in the s-Gevrey category, provided that a related matrix of periods satisfies a Diophantine condition. Also, we present a complete characterization for the s-global hypoellipticity at the level of q-forms for \(q\ge 0\).

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

Access options

Buy single article

Instant unlimited access to the full article PDF.

US$ 39.95

Price includes VAT for USA

References

  1. 1.

    Bergamasco, A., Cordaro, P., Malagutti, P.: Globally hypoelliptic systems of vector fields. J. Funct. Anal. 114(2), 267–285 (1993)

  2. 2.

    Bergamasco, A., Dattori da Silva, P., Gonzalez, R.: Global solvability and global hypoellipticity in Gevrey classes for vector fields on the torus. J. Differ. Equ. 264(5), 3500–3526 (2018)

  3. 3.

    Constantine, G., Savits, T.: A multivariate Faà di Bruno formula with applications. Trans. Am. Math. Soc. 348(2), 503–520 (1996)

  4. 4.

    Dattori da Silva, P., Meziani, A.: Cohomology relative to a system of closed form on the torus. Math. Nachr. 289(17–18), 2147–2158 (2016)

  5. 5.

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

  6. 6.

    Meziani, A.: Hypoellipticity of nonsingular closed 1-forms on compact manifolds. Commun. Partial Differ. Equ. 27(7–8), 1255–1269 (2002)

  7. 7.

    Rodino, L.: Linear Partial Differential Operators in Gevrey Spaces. World Scientific Publishing Co. Pte. Ltd., Singapore (1993)

Download references

Author information

Correspondence to A. Meziani.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

P. L. Dattori da Silva was supported in part by CNPq (grant 309496/2018-7) and FAPESP (grants 2018/15046-0 and 2018/14316-3).

Communicated by Fabio Nicola.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

da Silva, P.L.D., Meziani, A. A Gevrey Differential Complex on the Torus. J Fourier Anal Appl 26, 8 (2020) doi:10.1007/s00041-019-09713-w

Download citation

Keywords

  • Differential complex
  • Solvability
  • Hypoellipticity
  • Fourier series
  • Diophantine condition

Mathematics Subject Classification

  • Primary 58J10
  • Secondary 35F35