Skip to main content
SpringerLink
Log in

Weighted restriction type estimates for Grushin operators and application to spectral multipliers and Bochner–Riesz summability

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

We prove weighted restriction type estimates for Grushin operators. These estimates are then used to prove sharp spectral multiplier theorems as well as Bochner–Riesz summability results with sharp exponent.

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. Chen, P., Ouhabaz, E.M., Sikora, A., Yan, L.X.: Restriction estimates, sharp spectral multipliers and endpoint estimates for Bochner–Riesz means. J. Anal. Math.arXiv:1202.4052

  2. Chen, P., Sikora, A.: Sharp spectral multipliers for a new class of Grushin type operators. J. Fourier Anal. Appl. 19, 1274–1293 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. Duong, X.T., Ouhabaz, E.M., Sikora, A.: Plancherel-type estimates and sharp spectral multipliers. J. Funct. Anal. 196, 443–485 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  4. Gadziński, P.: On a semigroup of measures with irregular densities. Colloq. Math. 83(1), 85–99 (2000)

    MathSciNet  MATH  Google Scholar 

  5. Hebisch, W.: Multiplier theorem on generalized Heisenberg groups. Colloq. Math. 65(2), 231–239 (1993)

    MathSciNet  MATH  Google Scholar 

  6. Kenig, C.E., Stanton, R.J., Tomas, P.A.: Divergence of eigenfunction expansions. J. Funct. Anal. 46(1), 28–44 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  7. Koch, H., Tataru, D.: \(L^p\) eigenfunction bounds for the Hermite operator. Duke Math. J. 128(2), 369–392 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. Liu, H., Song, M.: The restriction theorem for the Grushin operators (2014). arXiv:1402.5298

  9. Martini, A., Müller, D.: A sharp multiplier theorem for Grushin operators in arbitrary dimensions. Rev. Mat. Iberoam. 30(4), 1265–1280 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  10. Martini, A., Sikora, A.: Weighted Plancherel estimates and sharp spectral multipliers for the Grushin operators. Math. Res. Lett. 19(5), 1075–1088 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. Müller, D., Stein, E.M.: On spectral multipliers for Heisenberg and related groups. J. Math. Pures Appl. (9) 73(4), 413–440 (1994)

    MathSciNet  MATH  Google Scholar 

  12. Robinson, D.W., Sikora, A.: Analysis of degenerate elliptic operators of Grushin type. Math. Z. 260(3), 475–508 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Sikora, A., Yan, L.X., Yao, X.H.: Sharp spectral multipliers for operators satisfying generalized Gaussian estimates. J. Funct. Anal. 266(1), 368–409 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  14. Stein, E.M.: Harmonic analysis: Real variable methods, orthogonality and oscillatory integrals. With the assistance of Timothy S. Murphy. Princeton Mathematical Series, Monographs in Harmonic Analysis, III, vol. 43. Princeton University Press, Princeton (1993)

  15. Tao, T.: Some recent progress on the restriction conjecture. In: Fourier Analysis and Convexity, Appl. Numer. Harmon. Anal., pp 217–243, Birkhäuser Boston, Boston (2004)

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Sun Yat-sen (Zhongshan) University, Guangzhou, 510275, People’s Republic of China

    Peng Chen

  2. Institut de Mathématiques de Bordeaux, UMR CNRS 5251, Univ. Bordeaux, 351, Cours de la Libération, 33405, Talence, France

    Peng Chen & El Maati Ouhabaz

Authors
  1. Peng Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. El Maati Ouhabaz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to El Maati Ouhabaz.

Additional information

The research of both authors was partially supported by the ANR Project HAB, ANR-12-BS01-0013-02. Peng Chen was also partially supported by the NNSF of China, Grant No. 11501583.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, P., Ouhabaz, E.M. Weighted restriction type estimates for Grushin operators and application to spectral multipliers and Bochner–Riesz summability. Math. Z. 282, 663–678 (2016). https://doi.org/10.1007/s00209-015-1558-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-015-1558-9

Keywords

Search

Navigation