Skip to main content
Log in

Almost everywhere convergence of the Bochner–Riesz means for the Dunkl transforms of weighted \(L^{p}\)-functions

  • Original Paper
  • Published:
Annals of Functional Analysis Aims and scope Submit manuscript

Abstract

For the Dunkl transforms associated with the weight functions \(h_{\kappa }^2(x)=\prod _{j=1}^d |x_j|^{2{\kappa }_j}\), \({\kappa }_1,\ldots , {\kappa }_d\ge 0\) on \({{\mathbb {R}}}^d\), it is proved that if \(p\ge 2\) and \({\delta }>{\delta }_{\kappa }(p):=\max \{(2l_{\kappa }+1) |\frac{1}{2}-\frac{1}{p}|-\frac{1}{2},0\}\), the Bochner–Riesz means \(B_{R}^{\delta }(h_{\kappa }^2; f)\) converges to f at almost everywhere for all \(f\in L^p({{\mathbb {R}}}^d; h_{\kappa }^2dx)\). This extends a well known result of Carbery et al. for the classical Fourier transforms (J Lond Math Soc 38:513–524, 1988).

This is a preview of subscription content, log in via an institution to check access.

Access this article

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. The result of [15] holds for more general weights invariant under a finite reflection group as well.

References

  1. Carbery, A., De Francia, J.L.R., Vega, L.: Almost everywhere summability of Fourier integrals. J. Lond. Math. Soc. 38(2), 513–524 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  2. Christ, M.: On almost everywhere convergence of Bochner–Riesz means in Higher dimensions. Proc. Am. Math. Soc. 95(1), 16–20 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  3. Coifman, R., Weiss, G.: Analyse harmonique non-commutative sur certains espaces homogénes. (French) Étude de certaines intgrales singulières. Lecture Notes in Mathematics, vol. 242. Springer, Berlin (1971)

    Book  Google Scholar 

  4. Dai, F., Wang, H.P.: A transference theorem for the Dunkl transform and its applications. J. Funct. Anal. 258(12), 4052–4074 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Dai, F., Xu, Y.: Analysis on \(h\)-harmonics and Dunkl transforms. In: Tikhonov, S. (ed.) Advanced Courses in Mathematics CRM Barcelona. Birkh user/Springer, Basel (2015)

    Google Scholar 

  6. Dai, F., Ye, W.: Local restriction theorem and maximal Bochner–Riesz operators for the Dunkl transforms. Trans. Am. Math. Soc. 371, 641–679 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  7. de Jeu, M.F.E.: The Dunkl transform. Invent. Math. 113, 147–162 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  8. Dunkl, C.F.: Differential-difference operators associated to reflection groups. Trans. Am. Math. Soc. 311, 167–183 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dunkl, C.F.: Integral kernels with reflection group invariance. Can. J. Math. 43(6), 1213–1227 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dunkl, C.F., Xu, Y.: Orthogonal Polynomials of Several Variables, 2nd edn. Cambridge University Press, Cambridge (2014)

    Book  MATH  Google Scholar 

  11. Hytönen, T., Pèrez, C., Rela, E.: Sharp reverse Hölder property for \(A_\infty \) weights on spaces of homogeneous type. J. Funct. Anal. 263(12), 3883–3899 (2012)

    MathSciNet  MATH  Google Scholar 

  12. Rösler, M.: Positivity of Dunkl intertwining operator. Duke Math. J. 98, 445–463 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  13. Rösler, M.: A positive radial product formula for the Dunkl kernel. Trans. Am. Math. Soc. 355, 2413–2438 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  14. Rösler, M.: Dunkl Operators: Theory and Applications. Orthogonal Polynomials and Special Functions (Leuven, 2002). Lecture Notes in Math, vol. 1817, pp. 93–135. Springer, Berlin (2003)

    Google Scholar 

  15. Thangavelu, S., Xu, Y.: Convolution operator and maximal function for the Dunkl transform. J. Anal. Math. 97, 25–55 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  16. Thangavelu, S., Yuan, X., Riesz, : Transform and Riesz potentials for Dunkl transform. J. Comput. Appl. Math. 199(1), 181–195 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Tomas, P.: Restriction theorems for the Fourier transform. In: Harmonic Analysis in Euclidean space, Proceedings of Symposium Pure Math., vol. 35, Part I, pp. 111–114. Amer. Math. Soc., Providence (1979)

  18. Xu, Y.: Orthogonal polynomials for a family of product weight functions on the spheres. Can. J. Math. 49, 175–192 (1997)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This paper is completed under the guidance of my Ph.D. supervisor Dr. Feng Dai. I wish to express my deepest gratitude to him. And this work is supported by the National Natural Science Foundation of China (Project no. 11701082).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Wenrui Ye.

Additional information

Communicated by Feng Dai.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ye, W. Almost everywhere convergence of the Bochner–Riesz means for the Dunkl transforms of weighted \(L^{p}\)-functions. Ann. Funct. Anal. 11, 981–1006 (2020). https://doi.org/10.1007/s43034-020-00065-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s43034-020-00065-x

Keywords

Mathematics Subject Classification

Navigation