Advertisement

The Journal of Geometric Analysis

, Volume 29, Issue 2, pp 1571–1582 | Cite as

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

  • Philippe JamingEmail author
  • Felipe Negreira
Article
  • 100 Downloads

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).

Keywords

Besov spaces Sampling theory Plancherel–Polya inequality Spaces of homogeneous type 

Mathematics Subject Classification

Primary 94A20 Secondary 30H25 43A85 

Notes

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).

References

  1. 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)Google Scholar
  2. 2.
    Coifman, R., Weiss, G.: Analyse Harmonique Non-commutative sur Certains Espaces Homogénes. Lecture Notes in Mathematics, vol. 242. Springer, Berlin (1971)CrossRefzbMATHGoogle Scholar
  3. 3.
    Deng, D., Han, Y.: Harmonic Analysis on Spaces of Homogeneous Type. Lecture Notes in Mathematics, vol. 1966. Springer, Berlin (2009)Google Scholar
  4. 4.
    Führ, H., Gröchenig, K.: Sampling theorems on locally compact groups from oscillation estimates. Math. Z. 255, 177–194 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Gogatishvili, A., Koskela, P., Shanmugalingam, N.: Interpolation properties of Besov spaces defined on metric spaces. Math. Nachr. 283, 215–231 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Geller, D., Pesenson, I.: Band-limited localized Parseval frames and Besov spaces on compact homogeneous manifolds. J. Geom. Anal. 21, 334–371 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Hytönen, T., Kairema, A.: Systems of dyadic cubes in a doubling metric space. Colloq. Math. 126, 1–33 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 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)Google Scholar
  9. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 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)zbMATHGoogle Scholar
  11. 11.
    Jaming, Ph, Malinnikova, E.: An uncertainty principle and sampling inequalities in Besov spaces. J. Fourier Anal. Appl. 22, 768–786 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 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)Google Scholar
  13. 13.
    Meyer, Y.: Ondelettes et Opérateurs. Hermann, Paris (1990)zbMATHGoogle Scholar
  14. 14.
    Mhaskar, H., Narcowich, F., Ward, J.: Spherical Marcinkiewicz–Zygmund inequalities and positive quadrature. J. Math. Comput. 70, 1113–1130 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Macias, R., Segovia, C.: Lipschitz functions on spaces of homogeneous type. Adv. Math. 33, 257–270 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Müller, D., Yang, D.: A difference characterization of Besov and Triebel–Lizorkin spaces on RD-spaces. Forum Math. 21, 259–298 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Narcowich, F., Petrushev, P., Ward, J.: Decomposition of Besov and Triebel–Lizorkin spaces on the sphere. J. Funct. Anal. 238, 530–64 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Pesenson, I.: Sampling of Paley–Wiener functions on stratified groups. J. Fourier Anal. Appl. 4, 271–281 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Pesenson, I.: Poincaré-type inequalities and reconstruction of Paley–Wiener functions on manifolds. J. Geom. Anal. 4, 101–121 (2004)CrossRefzbMATHGoogle Scholar
  20. 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)Google Scholar
  21. 21.
    Triebel, H.: Theory of Function Spaces III. Monographs in Mathematics, vol. 100. Birkhäuser, Basel (2006)zbMATHGoogle Scholar
  22. 22.
    Triebel, H.: A new approach to function spaces on quasi-metric spaces. Rev. Mat. Complut. 18, 7–48 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Unser, M.: Sampling-50 years after Shannon. Proc. IEEE 88, 569–587 (2000)CrossRefzbMATHGoogle Scholar
  24. 24.
    Zayed, A.I.: Advances in Shannon’s Sampling Theory. CRC Press, Boca Raton (1993)zbMATHGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2018

Authors and Affiliations

  1. 1.Univ. Bordeaux, IMB, CNRS, UMR 5251TalenceFrance

Personalised recommendations