Skip to main content
SpringerLink
Log in

A Plancherel–Polya Inequality in Besov Spaces on Spaces of Homogeneous Type

  • Published:
The Journal of Geometric Analysis Aims and scope Submit manuscript

Abstract

In this work, we establish a Plancherel–Polya inequality for functions in Besov spaces on spaces of homogeneous type as defined in Han and Yang (Diss Math 403:1–102, 2002) in the spirit of their recent counterpart for \({\mathbb {R}}^d\) established by Jaming and Malinnikova (J Fourier Anal Appl 22:768–786, 2016. The main tool is the wavelet decomposition presented by Deng and Han (Harmonic Analysis on Spaces of Homogeneous Type, Springer, New York, 2009).

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

Notes

  1. An orthonormal wavelet system on spaces of homogeneous type has been constructed in [1]. The construction is much more involved than the wavelet frames considered here and does not bring any significant improvement in our results.

References

  1. Auscher, P., Hytönen, T.: Orthonormal bases of regular wavelets in spaces of homogeneous type. Appl. Comput. Harmon. Anal. 34, 266–296. Addendum in Same Journal 39(2015), 568–569 (2013)

  2. Coifman, R., Weiss, G.: Analyse Harmonique Non-commutative sur Certains Espaces Homogénes. Lecture Notes in Mathematics, vol. 242. Springer, Berlin (1971)

    Book  MATH  Google Scholar 

  3. Deng, D., Han, Y.: Harmonic Analysis on Spaces of Homogeneous Type. Lecture Notes in Mathematics, vol. 1966. Springer, Berlin (2009)

  4. Führ, H., Gröchenig, K.: Sampling theorems on locally compact groups from oscillation estimates. Math. Z. 255, 177–194 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Gogatishvili, A., Koskela, P., Shanmugalingam, N.: Interpolation properties of Besov spaces defined on metric spaces. Math. Nachr. 283, 215–231 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Geller, D., Pesenson, I.: Band-limited localized Parseval frames and Besov spaces on compact homogeneous manifolds. J. Geom. Anal. 21, 334–371 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Hytönen, T., Kairema, A.: Systems of dyadic cubes in a doubling metric space. Colloq. Math. 126, 1–33 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  8. Han, Y., Müller, D., Yang, D.: A Theory of Besov and Triebel-Lizorkin Spaces on Metric Measure Spaces Modeled on Carnot-Carathéodory Spaces. Abstract and Applied Analysis (2008)

  9. Han, Y., Xu, D.: New characterizations of Besov and Triebel–Lizorkin spaces over spaces of homogeneous type. J. Math. Anal. Appl. 325, 305–318 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Han, Y., Yang, D.: New characterizations and applications of inhomogeneous Besov and Triebel–Lizorkin space on spaces of homogeneous type and fractals. Diss. Math. 403, 1–102 (2002)

    MATH  Google Scholar 

  11. Jaming, Ph, Malinnikova, E.: An uncertainty principle and sampling inequalities in Besov spaces. J. Fourier Anal. Appl. 22, 768–786 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  12. Jaming, Ph., Negreira, F.: A sampling theorem for functions in Besov spaces on the sphere. In: 2017 International Conference on Sampling Theory and Applications, Tallinn (Estonia) (2017)

  13. Meyer, Y.: Ondelettes et Opérateurs. Hermann, Paris (1990)

    MATH  Google Scholar 

  14. Mhaskar, H., Narcowich, F., Ward, J.: Spherical Marcinkiewicz–Zygmund inequalities and positive quadrature. J. Math. Comput. 70, 1113–1130 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  15. Macias, R., Segovia, C.: Lipschitz functions on spaces of homogeneous type. Adv. Math. 33, 257–270 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  16. Müller, D., Yang, D.: A difference characterization of Besov and Triebel–Lizorkin spaces on RD-spaces. Forum Math. 21, 259–298 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Narcowich, F., Petrushev, P., Ward, J.: Decomposition of Besov and Triebel–Lizorkin spaces on the sphere. J. Funct. Anal. 238, 530–64 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  18. Pesenson, I.: Sampling of Paley–Wiener functions on stratified groups. J. Fourier Anal. Appl. 4, 271–281 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  19. Pesenson, I.: Poincaré-type inequalities and reconstruction of Paley–Wiener functions on manifolds. J. Geom. Anal. 4, 101–121 (2004)

    Article  MATH  Google Scholar 

  20. Pesenson, I.: Parseval space-frequency localized frames on sub-Riemannian compact homogeneous manifolds. In: Pesenson I., Le Gia Q., Mayeli A., Mhaskar H., Zhou D.X. (eds.) Frames and Other Bases in Abstract and Function Spaces. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham (2017)

  21. Triebel, H.: Theory of Function Spaces III. Monographs in Mathematics, vol. 100. Birkhäuser, Basel (2006)

    MATH  Google Scholar 

  22. Triebel, H.: A new approach to function spaces on quasi-metric spaces. Rev. Mat. Complut. 18, 7–48 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  23. Unser, M.: Sampling-50 years after Shannon. Proc. IEEE 88, 569–587 (2000)

    Article  MATH  Google Scholar 

  24. Zayed, A.I.: Advances in Shannon’s Sampling Theory. CRC Press, Boca Raton (1993)

    MATH  Google Scholar 

Download references

Acknowledgements

The authors wish to thank the anonymous referees for a careful reading of the manuscript and constructive remarks. The first author kindly acknowledges the financial support from the French ANR program, ANR-12-BS01-0001 (Aventures), the Austrian-French AMA-DEUS project 35598VB-ChargeDisq, and the French-Tunisian CMCU/UTIQUE project 32701UB Popart. This study has been carried out with the financial support from the French State, managed by the French National Research Agency (ANR) in the framework of the Investments for the Future Program IdEx Bordeaux—CPU (ANR-10-IDEX-03-02). The second author is supported by the doctoral grant POS-CFRA-2015-1-125008 of the Agencia Nacional de Innovación e Investigación (Uruguay) and the Campus France (France).

Author information

Authors and Affiliations

  1. Univ. Bordeaux, IMB, CNRS, UMR 5251, 33400, Talence, France

    Philippe Jaming & Felipe Negreira

Authors
  1. Philippe Jaming
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Felipe Negreira
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Philippe Jaming.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jaming, P., Negreira, F. A Plancherel–Polya Inequality in Besov Spaces on Spaces of Homogeneous Type. J Geom Anal 29, 1571–1582 (2019). https://doi.org/10.1007/s12220-018-0052-0

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12220-018-0052-0

Keywords

Mathematics Subject Classification

Search

Navigation