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Riesz transforms and commutators in the Dunkl setting

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Abstract

In this paper we characterise the pointwise size and regularity estimates for the Dunkl Riesz transform kernel involving both the Euclidean metric and the Dunkl metric, where these two metrics are not equivalent. We further establish a suitable version of the pointwise kernel lower bound of the Dunkl Riesz transform via the Euclidean metric only. Then we show that the lower bound of commutator of the Dunkl Riesz transform is with respect to the BMO space associated with the Euclidean metric, and that the upper bound is respect to the BMO space associated with the Dunkl metric. Moreover, the compactness and the two types of VMO are also addressed.

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Acknowledgements

The authors would like to thank the referee for carefully reading and checking the paper and for helpful suggestions, which made the paper more accurate and readable. Ji Li would like to thank Jorge Betancor for helpful discussions.

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Authors and Affiliations

  1. Department of Mathematics, Auburn University, Auburn, AL, 36849-5310, USA

    Yongsheng Han

  2. Department of Mathematics, National Central University, Chung-Li, 320, Taiwan, Republic of China

    Ming-Yi Lee

  3. School of Mathematical and Physical Sciences, Macquarie University, Sydney, NSW, 2109, Australia

    Ji Li

  4. Department of Mathematics, Washington University - St. Louis, St. Louis, MO, 63130-4899, USA

    Brett D. Wick

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  1. Yongsheng Han
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Correspondence to Ming-Yi Lee.

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The second author is supported by NSTC 112-2115-M-008-001-MY2. The third author is supported by the Australian Research Council under Grant No. ARC DP220100285 and NNSF 12171221. B.D. Wick’s research partially supported in part by NSF Grant NSF-DMS-1800057 as well as ARC DP190100970.

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Han, Y., Lee, MY., Li, J. et al. Riesz transforms and commutators in the Dunkl setting. Anal.Math.Phys. 14, 46 (2024). https://doi.org/10.1007/s13324-024-00911-4

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