Skip to main content
Log in

Singular integrals in the rational Dunkl setting

  • Published:
Revista Matemática Complutense Aims and scope Submit manuscript

Abstract

On \(\mathbb {R}^N\) equipped with a normalized root system R and a multiplicity function \(k\ge 0\) let us consider a (not necessarily radial) kernel \(K({\mathbf {x}})\) satisfying \(|\partial ^\beta K({\mathbf {x}})|\lesssim \Vert {\mathbf {x}}\Vert ^{-{\mathbf {N}}-|\beta |}\) for \(|\beta |\le s\), where \({\mathbf {N}}\) is the homogeneous dimension of the system \(({\mathbb {R}}^N,R,k)\). We additionally assume that

$$\begin{aligned} \sup _{0<a<b<\infty }\Big |\int _{a<\Vert {\mathbf {x}}\Vert<b} K({\mathbf {x}})\, dw({\mathbf {x}})\Big |<\infty , \end{aligned}$$

where dw is the associated measure. We prove that if s large enough then a singular integral Dunkl convolution operator associated with the kernel \(K({\mathbf {x}})\) is bounded on \(L^p(dw)\) for \(1<p<\infty \) and of weak-type (1,1). Furthermore, we study a maximal function related to the Dunkl convolutions with truncation of K.

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

References

  1. Anker, J.-P., Dziubański, J., Hejna, A.: Harmonic functions, conjugate harmonic functions and the Hardy space \(H^1\) in the rational Dunkl setting. J. Fourier Anal. Appl. 25, 2356–2418 (2019)

    Article  MathSciNet  Google Scholar 

  2. Amri, B., Hammi, A.: Dunkl-Schrödinger operators, Complex Anal. Oper. Theory (2018)

  3. Amri, B., Sifi, M.: Singular integral operators in the Dunkl setting. J. Lie Theory 22, 723–739 (2012)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  5. de Jeu, M., Rösler, M.: Asymptotic analysis for the Dunkl kernel. J. Approx. Theory 119(1), 110–126 (2002)

    Article  MathSciNet  Google Scholar 

  6. Dunkl, C.F.: Reflection groups and orthogonal polynomials on the sphere. Math. Z. 197(1), 33–60 (1988)

    Article  MathSciNet  Google Scholar 

  7. Dunkl, C.F.: Differential-difference operators associated to reflection groups. Trans. Amer. Math. 311(1), 167–183 (1989)

    Article  MathSciNet  Google Scholar 

  8. Dunkl, C.F.: Hankel transforms associated to finite reflection groups, in: Proc. of the special session on hypergeometric functions on domains of positivity, Jack polynomials and applications, Proceedings, Tampa 1991. Contemp. Math. 138, 123–138 (1989)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  10. Duoandikoetxea, J.: Fourier Analysis, Graduate Studies in Mathematics, 29. American Mathematical Society, Providence, RI, (2001)

  11. Dziubański, J., Hejna, A.: Remark on atomic decompositions for Hardy space \(H^1\) in the rational Dunkl setting. Studia Math. 251, 89–110 (2020)

    Article  MathSciNet  Google Scholar 

  12. Dziubański, J., Hejna, A.: Hörmander’s multiplier theorem for the Dunkl transform. J. Funct. Anal. 277, 2133–2159 (2019)

    Article  MathSciNet  Google Scholar 

  13. Gallardo, L., Rejeb, C.: A new mean value property for harmonic functions relative to the Dunkl-Laplacian operator and applications. Trans. Amer. Math. Soc. 368(5), 3727–3753 (2015)

    Article  MathSciNet  Google Scholar 

  14. Grafakos, L.: Modern Fourier Analysis, 3rd edition, Graduate Texts in Mathematics, 250. Springer, New York (2014)

    Google Scholar 

  15. Rösler, M.: Generalized Hermite polynomials and the heat equation for Dunkl operators. Comm. Math. Phys. 192, 519–542 (1998)

    Article  MathSciNet  Google Scholar 

  16. Rösler, M.: Positivity of Dunkl’s intertwining operator. Duke Math. J. 98(3), 445–463 (1999)

    Article  MathSciNet  Google Scholar 

  17. Rösler, M.: A positive radial product formula for the Dunkl kernel. Trans. Amer. Math. Soc. 355(6), 2413–2438 (2003)

    Article  MathSciNet  Google Scholar 

  18. Rösler, M.: Dunkl operators (theory and applications). In: Koelink, E., Van Assche, W. (eds.) Orthogonal polynomials and special functions (Leuven, 2002), 93–135. Lect. Notes Math. 1817, Springer-Verlag (2003)

  19. Rösler, M., Voit, M.: Dunkl theory, convolution algebras, and related Markov processes, in Harmonic and stochastic analysis of Dunkl processes, P. Graczyk, M. Rösler, M. Yor (eds.), 1–112, Travaux en cours 71, Hermann, Paris, (2008)

  20. Stein, E.M.: Singular integral and differentiability properties of functions, Princeton Math. Series 30, Princeton University Press, New Jersy (1970)

    Google Scholar 

  21. Stein, E.M.: Harmonic analysis (real variable methods, orthogonality and oscillatory integrals), Princeton Math. Series 43, Princeton University Press, (1993)

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

    Article  MathSciNet  Google Scholar 

  23. Thangavelu, S., Xu, Y.: Riesz transforms and Riesz potentials for the Dunkl transform. J. Comp. Appl. Math. 199, 181–195 (2007)

    Article  MathSciNet  Google Scholar 

  24. Triméche, K.: Paley-Wiener theorems for the Dunkl transform and Dunkl translation operators. Integral Transforms Spec. Funct. 13(1), 17–38 (2002)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors want to thank the reviewers for a careful reading of the manuscript and for helpful comments and suggestions which improved the presentation of the paper.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Jacek Dziubański.

Additional information

Publisher's Note

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

Research supported by the National Science Centre, Poland (Narodowe Centrum Nauki), Grant 2017/25/B/ST1/00599.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dziubański, J., Hejna, A. Singular integrals in the rational Dunkl setting. Rev Mat Complut 35, 711–737 (2022). https://doi.org/10.1007/s13163-021-00402-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13163-021-00402-1

Keywords

Mathematics Subject Classification

Navigation