Skip to main content
SpringerLink
Log in

Bochner–Riesz Means on Hardy Spaces Associated with Ball Quasi-Banach Function Spaces

  • Published:
Mediterranean Journal of Mathematics Aims and scope Submit manuscript

Abstract

Let X be a ball quasi-Banach function space on \(\mathbb {R}^{n}\). In this article, we obtain the boundedness of the Bochner–Riesz means \(B_{1/\varepsilon }^{\delta }\) and the maximal Bochner–Riesz means \(B_{*}^{\delta }\) on Hardy spaces associated with X. Furthermore, by weakening the assumption that X has an absolutely continuous quasi-norm, the main results in this article can be applied to many concrete spaces such as Morrey spaces. This shows that the results obtained in this article have a wide range of generality.

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

Data Availability

Not applicable.

References

  1. Bennett, C., Sharpley, R.: Interpolation of Operators, Pure and Applied Mathematics, vol. 129. Academic Press, Boston (1988)

    MATH  Google Scholar 

  2. Bochner, S.: Summation of multiple Fourier series by spherical means. Trans. Am. Math. Soc. 40(2), 175–207 (1936)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chang, D.-C., Wang, S., Yang, D., Zhang, Y.: Littlewood–Paley characterizations of Hardy-type spaces associated with ball quasi-Banach function spaces. Complex Anal. Oper. Theory 14, 1–33 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  4. Coifman, R., Rochberg, R., Weiss, G.: Fractorization theorems for Hardy spaces in several variables. Ann. Math. 103, 611–635 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cruz-Uribe, D.V., Wang, L.-A.D.: Variable Hardy spaces. Indiana Univ. Math. J. 63, 447–493 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  6. Fefferman, C., Stein, E.M.: Some maximal inequalities. Am. J. Math. 93, 107–115 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  7. Fefferman, C., Stein, E.M.: \(H^{p}\) spaces of several variables. Acta Math. 129, 137–193 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  8. Ho, K.-P.: Sublinear operators on weighted Hardy spaces with variable exponents. Forum Math. 31, 607–617 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ho, K.-P.: Calderón–Zygmund operators, Bochner–Riesz means and Parametric Marcinkiewicz integrals on Hardy-Morrey spaces with variable exponents. Kyoto J. Math. 63(2), 335–351 (2023)

    Article  MathSciNet  MATH  Google Scholar 

  10. Huang, L., Chang, D.-C., Yang, D.: Fourier transform of Hardy spaces associated with ball quasi-Banach function spaces. Appl. Anal. 101, 3825–3840 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  11. Jia, H., Wang, H.: Decomposition of Hardy–Morrey spaces. J. Math. Anal. Appl. 354, 99–110 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Lee, M.-Y.: Weighted norm inequalities of Bochner–Riesz means. J. Math. Anal. Appl. 324, 1274–1281 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  13. Lu, S., Yang, D.: The local versions of \(H^{p}\left(\mathbb{R} ^{n}\right)\) spaces at the origin. Studia Math. 116, 103–131 (1995)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  15. Nakai, E., Sawano, Y.: Orlicz–Hardy spaces and their duals. Sci. China Math. 57, 903–962 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  16. Sato, S.: Weak type estimates for some maximal operators on the weighted Hardy spaces. Ark. Mat. 33, 377–384 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  17. Sawano, Y., Ho, K.-P., Yang, D., Yang, S.: Hardy spaces for ball quasi-Banach function spaces. Dissertationes Math. 525, 1–102 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  18. Shi, X.L., Sun, Q.: Weighted norm inequalities for Bochner–Riesz operators and singular integral operators. Proc. Am. Math. Soc. 116, 665–673 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  19. Sjölin, P.: Convolution with oscillating kernels in \(H^{p}\) spaces. J. Lond. Math. Soc. 23, 442–454 (1981)

    MathSciNet  MATH  Google Scholar 

  20. Stein, E.M., Taibleson, M.H., Weiss, G.: Weak type estimates for maximal operators on certain \(H^{p}\) classes. Rend. Circ. Mat. Palermo 2(Suppl. 1), 81–97 (1981)

    MATH  Google Scholar 

  21. Stein, E.M., Weiss, G.: On the theory of harmonic functions of several variables, I: the theory of \(H^{p}\) spaces. Acta Math. 103, 25–62 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  22. Sun, J., Yang, D., Yuan, W.: Weak Hardy spaces associated with ball quasi-Banach function spaces on spaces of homogeneous type: Decompositions, real interpolation, and Calderón–Zygmund operators. J. Geom. Anal. 32, 1–85 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  23. Tan, J.: Atomic decompositions of localized Hardy spaces with variable exponents and applications. J. Geom. Anal. 29(1), 799–827 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  24. Tan, J.: Some Hardy and Carleson measure spaces estimates for Bochner–Riesz means. Math. Inequal. Appl. 23(3), 1027–1039 (2020)

    MathSciNet  MATH  Google Scholar 

  25. Tan, J.: Weighted Hardy and Carleson measure spaces estimates for fractional integrations. Publ. Math. Debr. 98(3–4), 313–330 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  26. Tan, J.: Weighted variable Hardy spaces associated with para-accretive functions and boundedness of Calderón–Zygmund operators. J. Geom. Anal. 33(2), 1–32 (2023)

  27. Wang, F., Yang, D., Yang, S.: Applications of Hardy spaces associated with ball quasi-Banach function spaces. Results Math. 75(26), 1–58 (2020)

  28. Yan, X., He, Z., Yang, D., Yuan, W.: Hardy spaces associated with ball quasi-Banach function spaces on spaces of homogeneous type: Littlewood–Paley characterizations with applications to boundedness of Calderón–Zygmund operators. Acta Math. Sin. (Engl. Ser.) 38, 1133–1184 (2022)

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  30. Yang, D.-C., Zhou, Y.: A boundedness criterion via atoms for linear operators in Hardy spaces. Constr. Approx. 29(2), 207–218 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  31. Zhang, Y., Yang, D., Yuan, W., Wang, S.: Weak Hardy-type spaces associated with ball quasi-Banach function spaces I: decompositions with applications to boundedness of Calderón–Zygmund operators. Sci. China Math. 64, 2007–2064 (2021)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Science, Nanjing University of Posts and Telecommunications, Nanjing, 210023, People’s Republic of China

    Jian Tan & Linjing Zhang

Authors
  1. Jian Tan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Linjing Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

The authors contributed equally to the paper. They read the whole paper and approved it.

Corresponding author

Correspondence to Jian Tan.

Ethics declarations

Conflict of Interest

The authors declare no competing interests.

Additional information

Publisher's Note

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

This project is supported by the National Natural Science Foundation of China (Grant no. 11901309), Natural Science Foundation of Jiangsu Province of China (Grant no. BK20180734) and Natural Science Foundation of Nanjing University of Posts and Telecommunications (Grant no. NY222168).

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tan, J., Zhang, L. Bochner–Riesz Means on Hardy Spaces Associated with Ball Quasi-Banach Function Spaces. Mediterr. J. Math. 20, 240 (2023). https://doi.org/10.1007/s00009-023-02449-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00009-023-02449-4

Keywords

Mathematics Subject Classification

Search

Navigation