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Convolution operator and maximal function for the Dunkl transform

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Abstract

For a family of weight functionsh K invariant under a finite reflection group onR d, analysis related to the Dunkl transform is carried out for the weightedL p spaces. Making use of the generalized translation operator and the weighted convolution, we study the summability of the inverse Dunkl transform, including as examples the Poisson integrals and the Bochner-Riesz means. We also define a maximal function and use it to prove the almost everywhere convergence.

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Author notes

  1. Sundaram Thangavelu

    Present address: Department of Mathematics, Indian Institute of Science, 560012, Bangalore, India

Authors and Affiliations

  1. Stat-Math Division, Indian Statistical Institute, 8th Mile, Mysore Road, 560 059, Bangalore, India

    Sundaram Thangavelu

  2. Department of Mathematics, University of Oregon, 97403-1222, Eugene, OR, USA

    Yuan Xu

Authors
  1. Sundaram Thangavelu
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  2. Yuan Xu
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Additional information

ST wishes to thank YX for the warm hospitality during his stay in Eugene. The work of YX was supported in part by the National Science Foundation under Grant DMS-0201669.

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Thangavelu, S., Xu, Y. Convolution operator and maximal function for the Dunkl transform. J. Anal. Math. 97, 25–55 (2005). https://doi.org/10.1007/BF02807401

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  • DOI: https://doi.org/10.1007/BF02807401

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