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).
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Notes
The result of [15] holds for more general weights invariant under a finite reflection group as well.
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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).
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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
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DOI: https://doi.org/10.1007/s43034-020-00065-x