Skip to main content
SpringerLink
Log in

Hardy Spaces Associated to Critical Functions and Applications to T1 Theorems

  • Published:
Journal of Fourier Analysis and Applications Aims and scope Submit manuscript

Abstract

Let X be a metric space equipped with a measure satisfying the doubling and reverse doubling conditions. In this paper, we develop the theory of new localized Hardy spaces \(H^p_\rho (X)\) for \(\frac{n}{n+1}<p\le 1\) associated to critical functions \(\rho \) defined on X where n is the doubling order. Our results include the atomic decomposition characterization and the maximal function characterization associated to an approximation of the identity. We then study the T1 criteria for singular integrals to be bounded on our Hardy spaces.

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. Auscher, P., Duong, X.T., McIntosh, A.: Boundedness of Banach space valued singular integral operators and Hardy spaces, unpublished preprint (2005)

  2. Betancor, J.J., Crescimbeni, R., Farina, J.C., Stinga, P.R., Torrea, J.L.: A \(T1\) criterion for Hermite-Calderón-Zygmund operators on the \(\text{ BMO }_H({\mathbb{R}}^n)\) space and applications. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 12(1), 157–187 (2013)

    MathSciNet  MATH  Google Scholar 

  3. Bongioanni, B., Harboure, E., Salinas, O.: Weighted inequalities for negative powers of Schrödinger operators. J. Math. Anal. Appl. 348(1), 12–27 (2008)

    Article  MathSciNet  Google Scholar 

  4. Bongioanni, B., Harboure, E., Salinas, O.: Weighted inequalities for commutators of Schrödinger-Riesz transforms. J. Math. Anal. Appl. 392(1), 6–22 (2012)

    Article  MathSciNet  Google Scholar 

  5. Bui, T.A.: Weighted estimates for commutators of some singular integrals related to Schrödinger operator. Bull. Sci. Math. 138, 270–292 (2014)

    Article  MathSciNet  Google Scholar 

  6. Bui, T.A., Cao, J., Ky, L.D., Yang, D., Yang, S.: Musielak-Orlicz-Hardy spaces associated with operators satisfying reinforced off-diagonal estimates. Anal. Geom. Metr. Spaces 1, 69–129 (2013)

    Article  MathSciNet  Google Scholar 

  7. Bui, T.A., Duong, X.T., Ly, F.K.: Maximal function characterizations for new local Hardy type spaces on spaces of homogeneous type. Trans. Am. Math. Soc. 370(10), 7229–7292 (2018)

    Article  MathSciNet  Google Scholar 

  8. Bui, T.A., Li, J., Ly, F.K.: \(T1\) criteria for generalised Calderón-Zygmund type operators on Hardy and BMO spaces associated to Schrödinger operators and applications. Ann. Sc. Norm. Super. Pisa Cl. Sci. 5(18), 203–239 (2018)

    MATH  Google Scholar 

  9. Campanato, S.: Proprieta di holderianita di alcune classi di funzioni. Ann. Scuola Norm. Sup. Pisa 17, 175–188 (1963)

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  11. Coulhon, T., Duong, X.T.: Maximal regularity and kernel bounds: observations on a theorem by Hieber and Prüss. Adv. Differ. Eq. 5(1–3), 343–368 (2000)

    MATH  Google Scholar 

  12. Coulhon, T., Kerkyacharian, G., Petrushev, P.: Heat kernel generated frames in the setting of Dirichlet spaces. J. Fourier Anal. Appl. 18(5), 995–1066 (2012)

    Article  MathSciNet  Google Scholar 

  13. David, G., Journé, J.-L.: A boundedness criterion for generalized Calderón-Zygmund operators. Ann. Math. (2) 120(2), 371–397 (1984)

    Article  MathSciNet  Google Scholar 

  14. Dinh, T.D., Ha, D.H., Luong, D.K.: On weak\(^*\)-convergence in the localized Hardy spaces \(H^1_\rho (X)\) and its application. Taiwan. J. Math. 20(4), 897–907 (2016)

    Article  Google Scholar 

  15. Duong, X.T., Yan, L.X.: Duality of Hardy and BMO spaces associated with operators with heat kernel bounds. J. Am. Math. Soc. 18, 943–973 (2005)

    Article  MathSciNet  Google Scholar 

  16. Dziubański, J., Zienkiewicz, J.: \(H^p\) spaces for Schrödinger operators. In: Fourier Analysis and Related Topics (Bedlewo, 2000). Banach Center Publication, vol. 56, pp. 45–53. Polish Academic Science, Warsaw (2002)

  17. Dziubański, J.: Atomic decomposition of Hardy spaces associated with certain Laguerre expansions. J. Fourier Anal. Appl. 15(2), 129–152 (2009)

    Article  MathSciNet  Google Scholar 

  18. Dziubański, J., Zienkiewicz, J.: Hardy space \(H^1\) associated to Schrödinger operator with potential satisfying reverse Hölder inequality. Rev. Mat. Iberoam. 15(2), 279–296 (1999)

    Article  Google Scholar 

  19. Fefferman, C.: The uncertainty principle. Bull. Am. Math. Soc. 9, 129–206 (1983)

    Article  MathSciNet  Google Scholar 

  20. Grafakos, L.: Classical and Modern Fourier Analysis. Pearson Education Inc, Upper Saddle River (2004)

    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 51, 2253–2284 (2008)

    Article  MathSciNet  Google Scholar 

  22. Guo, Z., Li, P., Peng, L.: \(L^p\) boundedness of commutators of Riesz transforms associated to Schrödinger operator. J. Math. Anal. Appl. 341(1), 421–432 (2008)

    Article  MathSciNet  Google Scholar 

  23. 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). https://doi.org/10.1155/2008/893409

    Article  MATH  Google Scholar 

  24. He, Z., Han, Y., Li, J., Liu, L., Yang, D., Yuan, W.: A complete real-variable theory of Hardy spaces on spaces of homogeneous type. J. Fourier Anal. Appl. 25, 2197–2267 (2019)

    Article  MathSciNet  Google Scholar 

  25. He, Z., Liu, L., Yang, D., Yuan, W.: New Calderón reproducing formulae with exponential decay on spaces of homogeneous type. Sci. China Math. 62, 283–350 (2019)

    Article  MathSciNet  Google Scholar 

  26. Hofmann, S., Lu, G., Mitrea, D., Mitrea, M., Yan, L.: Hardy spaces associated with non-negative self adjoint operators satisfying Davies–Gaffney estimates. Mem. Am. Math. Soc. 214, no. 1007 (2011)

  27. Ky, L.D.: On weak\(*\)-convergence in \(H^1_L({\mathbb{R}}^d)\). Potential Anal. 39(4), 355–368 (2013)

    Article  MathSciNet  Google Scholar 

  28. Lebedev, N.N.: Special Functions and Their Applications. Dover, New York (1972)

    MATH  Google Scholar 

  29. Lin, C., Liu, H.: \(BMO_L({\mathbb{H}}^n)\) spaces and Carleson measures for Schrödinger operators. Adv. Math. 228, 1631–688 (2011)

    Article  MathSciNet  Google Scholar 

  30. Lin, H., Yang, D.: Pointwise multipliers for localized Morrey-Campanato spaces on RD-spaces. Acta Math. Sci. 34B, 1677–1694 (2014)

    Article  MathSciNet  Google Scholar 

  31. Ly, F.K.: Second order Riesz transforms associated to the Schrödinger operator for \(p\le 1\). J. Math. Anal. Appl. 410(1), 391–402 (2014)

    Article  MathSciNet  Google Scholar 

  32. Ly, F.K.: Classes of weights and second order Riesz transforms associated to Schrödinger operators. J. Math. Soc. Japan 68(2), 489–533 (2016)

    Article  MathSciNet  Google Scholar 

  33. Ma, T., Stinga, P.R., Torrea, J., Zhang, C.: Regularity estimates in Hölder spaces for Schrödinger operators via a \(T1\) theorem. Ann. Mat. Pura. Appl. (4) 193(2), 561–589 (2014)

    Article  MathSciNet  Google Scholar 

  34. Nowak, A., Stempak, K.: Riesz transforms for multi-dimensional Laguerre function expansions. Adv. Math. 215(2), 642–678 (2007)

    Article  MathSciNet  Google Scholar 

  35. Shen, Z.: \(L^p\) estimates for Schrödinger operators with certain potentials. Ann. Inst. Fourier (Grenoble) 45, 513–546 (1995)

    Article  MathSciNet  Google Scholar 

  36. Shen, Z.: Estimates in \(L^p\) for magnetic Schrödinger operators. Indiana Univ. Math. J. 45(3), 817–841 (1996)

    Article  MathSciNet  Google Scholar 

  37. Stein, E.M.: Harmonic analysis: real variable methods, orthogonality and oscillatory integrals. Princeton University Press, Princeton (1993)

    MATH  Google Scholar 

  38. Sugano, S.: Estimates for the Operators \(V^\alpha (-\Delta +V)^{-\beta }\) and \(V^\alpha \nabla (-\Delta +V)^{-\beta }\) with Certain Non-negative Potentials \(V\). Tokyo J. Math. 21(2), 441–452 (1998)

    Article  MathSciNet  Google Scholar 

  39. Tang, L.: Weighted local Hardy spaces and their applications. Illinois J. Math. 56(2), 453–495 (2012)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  42. Yang, D., Zhou, Y.: Localized Hardy spaces \(H^1\) related to admissible functions on RD-spaces and applications to Schrödinger operators. Trans. Am. Math. Soc. 363, 1197–1239 (2011)

    Article  Google Scholar 

  43. Yang, D., Yang, D., Zhou, Y.: Localized Morrey-Campanato spaces on metric measure spaces and applications to Schrödinger operators. Nagoya Math. J. 198, 77–119 (2010)

    Article  MathSciNet  Google Scholar 

  44. Yang, D., Yang, D., Zhou, Y.: Localized BMO and BLO spaces on RD-spaces and applications to Schrödinger operators. Commun. Pure Appl. Anal. 9(3), 779–812 (2010)

    Article  MathSciNet  Google Scholar 

  45. Zhong, J.: “Harmonic analysis for some Schrödinger type operators”, Ph. D. Thesis, Princeton University (1993)

Download references

Acknowledgements

Xuan Thinh Duong was supported by the Australian Research Council through the Grant DP190100970. Luong Dang Ky is supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant Number 101.02-2019.02 and Vingroup Innovation Foundation (VINIF) annual research support program in project code VINIF.2019.DA09. The authors would like to thank the referees for their useful comments and suggestions which helped to improve the paper.

Author information

Authors and Affiliations

  1. Department of Mathematics, Macquarie University, NSW, 2109, Australia

    The Anh Bui & Xuan Thinh Duong

  2. Department of Mathematics, Quy Nhon University, 170 An Duong Vuong, Quy Nhon, Binh Dinh, Viet Nam

    Luong Dang Ky

Authors
  1. The Anh Bui
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xuan Thinh Duong
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Luong Dang Ky
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to The Anh Bui.

Additional information

Communicated by Dachun Yang.

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bui, T.A., Duong, X.T. & Ky, L.D. Hardy Spaces Associated to Critical Functions and Applications to T1 Theorems. J Fourier Anal Appl 26, 27 (2020). https://doi.org/10.1007/s00041-020-09731-z

Download citation

  • Received:

  • Revised:

  • Published:

  • DOI: https://doi.org/10.1007/s00041-020-09731-z

Keywords

Mathematics Subject Classification

Search

Navigation