Skip to main content
SpringerLink
Log in

New Calderón reproducing formulae with exponential decay on spaces of homogeneous type

  • Articles
  • Published:
Science China Mathematics Aims and scope Submit manuscript

Abstract

Assume that (X, d, μ) is a space of homogeneous type in the sense of Coifman and Weiss (1971, 1977). In this article, motivated by the breakthrough work of Auscher and Hytönen (2013) on orthonormal bases of regular wavelets on spaces of homogeneous type, we introduce a new kind of approximations of the identity with exponential decay (for short, exp-ATI). Via such an exp-ATI, motivated by another creative idea of Han et al. (2018) to merge the aforementioned orthonormal bases of regular wavelets into the frame of the existed distributional theory on spaces of homogeneous type, we establish the homogeneous continuous/discrete Calderón reproducing formulae on (X, d, μ), as well as their inhomogeneous counterparts. The novelty of this article exists in that d is only assumed to be a quasi-metric and the underlying measure μ a doubling measure, not necessary to satisfy the reverse doubling condition. It is well known that Calderón reproducing formulae are the cornerstone to develop analysis and, especially, harmonic analysis on spaces of homogeneous type.

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

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Auscher P, Hytönen T. Orthonormal bases of regular wavelets in spaces of homogeneous type. Appl Comput Harmon Anal, 2013, 34: 266–296

    Article  MathSciNet  MATH  Google Scholar 

  2. Bownik M. Atomic and molecular decompositions of anisotropic Besov spaces. Math Z, 2005, 250: 539–571

    Article  MathSciNet  MATH  Google Scholar 

  3. Bownik M, Ho K-P. Atomic and molecular decompositions of anisotropic Triebel-Lizorkin spaces. Trans Amer Math Soc, 2006, 358: 1469–1510

    Article  MathSciNet  MATH  Google Scholar 

  4. Bownik M, Li B, Yang D, et al. Weighted anisotropic Hardy spaces and their applications in boundedness of sublinear operators. Indiana Univ Math J, 2008, 57: 3065–3100

    Article  MathSciNet  MATH  Google Scholar 

  5. Bui T A, Duong X T. Hardy spaces associated to the discrete Laplacians on graphs and boundedness of singular integrals. Trans Amer Math Soc, 2014, 366: 3451–3485

    Article  MathSciNet  MATH  Google Scholar 

  6. Bui T A, Duong X T, Ly F K, Maximal function characterizations for new local Hardy type spaces on spaces of homogeneous type. Trans Amer Math Soc, 2018, 370: 7229–7292

    Google Scholar 

  7. Calderón A-P. Intermediate spaces and interpolation, the complex method. Studia Math, 1964, 24: 113–190

    Article  MathSciNet  MATH  Google Scholar 

  8. Calderón A-P. An atomic decomposition of distributions in parabolic Hp spaces. Adv Math, 1977, 25: 216–225

    Article  MATH  Google Scholar 

  9. Calderón A-P, Torchinsky A. Parabolic maximal functions associated with a distribution. Adv Math, 1975, 16: 1–64

    Article  MathSciNet  MATH  Google Scholar 

  10. Coifman R R, Weiss G. Analyse Harmonique Non-commutative sur Certains Espaces Homogenes (in French). Étude de certaines intégrales singulieres. Lecture Notes in Mathematics, vol. 242. Berlin-New York: Springer-Verlag, 1971

    Book  MATH  Google Scholar 

  11. Coifman R R, Weiss G. Extensions of Hardy spaces and their use in analysis. Bull Amer Math Soc (NS), 1977, 83: 569–645

    Article  MathSciNet  MATH  Google Scholar 

  12. David G, Journé J-L, Semmes S. Opérateurs de Calderón-Zygmund, fonctions para-accrétives et interpolation. Rev Mat Iberoamericana, 1985, 1: 1–56

    Article  MathSciNet  MATH  Google Scholar 

  13. Deng D, Han Y. Harmonic Analysis on Spaces of Homogeneous Type. With a preface by Yves Meyer. Lecture Notes in Mathematics, vol. 1966. Berlin: Springer-Verlag, 2009

    MATH  Google Scholar 

  14. Duong X Y, Yan L. Hardy spaces of spaces of homogeneous type. Proc Amer Math Soc, 2003, 131: 3181–3189

    Article  MathSciNet  MATH  Google Scholar 

  15. Fan X, He J, Li B, et al. Real-variable characterizations of anisotropic product Musielak-Orlicz Hardy spaces. Sci China Math, 2017, 60: 2093–2154

    Article  MathSciNet  MATH  Google Scholar 

  16. Frazier M, Jawerth B. Decomposition of Besov spaces. Indiana Univ Math J, 1985, 34: 777–799

    Article  MathSciNet  MATH  Google Scholar 

  17. Frazier M, Jawerth B. A discrete transform and decompositions of distribution spaces. J Funct Anal, 1990, 93: 34–170

    Article  MathSciNet  MATH  Google Scholar 

  18. Frazier M, Jawerth B, Weiss G. Littlewood-Paley Theory and the Study of Function Spaces. CBMS Regional Conference Series in Mathematics, vol. 79. Providence: Amer Math Soc, 1991

    Book  MATH  Google Scholar 

  19. Fu X, Chang D-C, Yang D. Recent progress in bilinear decompositions. Appl Anal Optim, 2017, 1: 153–210

    MathSciNet  Google Scholar 

  20. Grafakos L, Liu L, Maldonado D, et al. Multilinear analysis on metric spaces. Dissertationes Math (Rozprawy Mat), 2014, 497: 1–121

    Article  MathSciNet  MATH  Google Scholar 

  21. Grafakos L, Liu L, Yang D. Maximal function characterizations of Hardy spaces on RD-spaces and their applications. Sci China Ser A, 2008, 51: 2253–2284

    Article  MathSciNet  MATH  Google Scholar 

  22. Grafakos L, Liu L, Yang D. Vector-valued singular integrals and maximal functions on spaces of homogeneous type. Math Scand, 2009, 104: 296–310

    Article  MathSciNet  MATH  Google Scholar 

  23. Han Y. Calderón-type reproducing formula and the Tb theorem. Rev Mat Iberoamericana, 1994, 10: 51–91

    Article  MathSciNet  MATH  Google Scholar 

  24. Han Y. Inhomogeneous Calderón reproducing formula on spaces of homogeneous type. J Geom Anal, 1997, 7: 259–284

    Article  MathSciNet  MATH  Google Scholar 

  25. Han Y, Han Y, Li J. Criterion of the boundedness of singular integrals on spaces of homogeneous type. J Funct Anal, 2016, 271: 3423–3464

    Article  MathSciNet  MATH  Google Scholar 

  26. Han Y, Han Y, Li J. Geometry and Hardy spaces on spaces of homogeneous type in the sense of Coifman and Weiss. Sci China Math, 2017, 60: 2199–2218

    Article  MathSciNet  MATH  Google Scholar 

  27. Han Y, Li J, Ward L D. Hardy space theory on spaces of homogeneous type via orthonormal wavelet bases. Appl Comput Harmon Anal, 2018, 45: 120–169

    Article  MathSciNet  MATH  Google Scholar 

  28. Han Y, Lu S, Yang D. Inhomogeneous discrete Calderón reproducing formulae for spaces of homogeneous type. J Fourier Anal Appl, 2001, 7: 571–600

    Article  MathSciNet  MATH  Google Scholar 

  29. Han Y, Müller D, Yang D. Littlewood-Paley characterizations for Hardy spaces on spaces of homogeneous type. Math Nachr, 2006, 279: 1505–1537

    Article  MathSciNet  MATH  Google Scholar 

  30. Han Y, Müller D, Yang D. A theory of Besov and Triebel-Lizorkin spaces on metric measure spaces modeled on Carnot-Carathéodory spaces. Abstr Appl Anal, 2008, Article ID 893409, 250pp

    Book  MATH  Google Scholar 

  31. Han Y S, Sawyer E T. Littlewood-Paley Theory on Spaces of Homogeneous Type and the Classical Function Spaces. Memoirs of the American Mathematical Society, vol. 110. Providence: Amer Math Soc, 1994

    Book  MATH  Google Scholar 

  32. He Z, Han Y, Li J, et al. A complete real-variable theory of Hardy spaces on spaces of homogeneous type. ArXiv: 1803.10394, 2018

    Book  Google Scholar 

  33. Heideman N J H. Duality and fractional integration in Lipschitz spaces. Studia Math, 1974, 50: 65–85

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  35. Koskela P, Yang D, Zhou Y. A characterization of Haj lasz-Sobolev and Triebel-Lizorkin spaces via grand Littlewood-Paley functions. J Funct Anal, 2010, 258: 2637–2661

    Article  MathSciNet  MATH  Google Scholar 

  36. Koskela P, Yang D, Zhou Y. Pointwise characterizations of Besov and Triebel-Lizorkin spaces and quasiconformal mappings. Adv Math, 2011, 226: 3579–3621

    Article  MathSciNet  MATH  Google Scholar 

  37. Liang Y, Huang J, Yang D. New real-variable characterizations of Musielak-Orlicz Hardy spaces. J Math Anal Appl, 2012, 395: 413–428

    Article  MathSciNet  MATH  Google Scholar 

  38. Liu J, Weisz F, Yang D, et al. Variable anisotropic Hardy spaces and their applications. Taiwanese J Math, 2017, doi: 10.11650/tjm/171101

    Google Scholar 

  39. Liu L, Chang D-C, Fu X, et al. Endpoint boundedness of commutators on spaces of homogeneous type. Appl Anal, 2017, 96: 2408–2433

    Article  MathSciNet  MATH  Google Scholar 

  40. Liu L, Chang D-C, Fu X, et al. Endpoint estimates of linear commutators on Hardy spaces over spaces of homogeneous type. Math Methods Appl Sci, 2018, 41: 5951–5984

    Article  MathSciNet  MATH  Google Scholar 

  41. Liu L, Xiao J. A trace law for the Hardy-Morrey-Sobolev space. J Funct Anal, 2018, 274: 80–120

    Article  MathSciNet  MATH  Google Scholar 

  42. Nakai E, Sawano Y. Hardy spaces with variable exponents and generalized Campanato spaces. J Funct Anal, 2012, 262: 3665–3748

    Article  MathSciNet  MATH  Google Scholar 

  43. Nakamura S, Noi T, Sawano Y. Generalized Morrey spaces and trace operator. Sci China Math, 2016, 59: 281–336

    Article  MathSciNet  MATH  Google Scholar 

  44. Peetre J. New Thoughts on Besov Spaces. Duke University Mathematics Series, vol. 1. Durham: Mathematics Department, Duke University, 1976

    MATH  Google Scholar 

  45. Rochberg R. Eigenvalue estimates for Calderón-Toeplitz operators. In: Function Spaces. Lecture Notes in Pure and Applied Mathematics, vol. 136. New York: Dekker, 1992, 345–356

    MATH  Google Scholar 

  46. Sawano Y. An observation of the subspaces of S′. In: Generalized Functions and Fourier Analysis. Operator Theory: Advances and Applications, vol. 260. Cham: Birkhäuser/Springer, 2017, 185–192

    MATH  Google Scholar 

  47. Stein E M. Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton Mathematical Series, vol. 43, Princeton: Princeton University Press, 1993

    MATH  Google Scholar 

  48. Uchiyama A. Characterization of Hp(Rn) in terms of generalized Littlewood-Paley g-functions. Studia Math, 1985, 81: 135–158

    Google Scholar 

  49. Yang D, Liang Y, Ky L D. Real-Variable Theory of Musielak-Orlicz Hardy Spaces. Lecture Notes in Mathematics, vol. 2182. Cham: Springer, 2017

    Book  MATH  Google Scholar 

  50. Yang D, Yuan W. A new class of function spaces connecting Triebel-Lizorkin spaces and Q spaces. J Funct Anal, 2008, 255: 2760–2809

    Article  MathSciNet  MATH  Google Scholar 

  51. Yang D, Yuan W. New Besov-type spaces and Triebel-Lizorkin-type spaces including Q spaces. Math Z, 2010, 265: 451–480

    Article  MathSciNet  MATH  Google Scholar 

  52. Yang D, Zhou Y. Boundedness of sublinear operators in Hardy spaces on RD-spaces via atoms. J Math Anal Appl, 2008, 339: 622–635

    Article  MathSciNet  MATH  Google Scholar 

  53. Yang D, Zhou Y. Radial maximal function characterizations of Hardy spaces on RD-spaces and their applications. Math Ann, 2010, 346: 307–333

    Article  MathSciNet  MATH  Google Scholar 

  54. Yang D, Zhou Y. New properties of Besov and Triebel-Lizorkin spaces on RD-spaces. Manuscripta Math, 2011, 134: 59–90

    Article  MathSciNet  MATH  Google Scholar 

  55. Yuan W, Sickel W, Yang D. Morrey and Campanato Meet Besov, Lizorkin and Triebel. Lecture Notes in Mathematics, vol. 2005. Berlin: Springer-Verlag, 2010

    Book  MATH  Google Scholar 

  56. Zhuo C, Sawano Y, Yang D. Hardy spaces with variable exponents on RD-spaces and applications. Dissertationes Math (Rozprawy Mat), 2016, 520: 1–74

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 11771446, 11571039, 11726621, 11761131002 and 11871100).

Author information

Authors and Affiliations

  1. Laboratory of Mathematics and Complex Systems (Ministry of Education of China), School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China

    Ziyi He, Dachun Yang & Wen Yuan

  2. School of Mathematics, Renmin University of China, Beijing, 100872, China

    Liguang Liu

Authors
  1. Ziyi He
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Liguang Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Dachun Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Wen Yuan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dachun Yang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

He, Z., Liu, L., Yang, D. et al. New Calderón reproducing formulae with exponential decay on spaces of homogeneous type. Sci. China Math. 62, 283–350 (2019). https://doi.org/10.1007/s11425-018-9346-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-018-9346-4

Keywords

MSC(2010)

Search

Navigation