Dimension-free square function estimates for Dunkl operators |
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Authors: | Huaiqian Li Mingfeng Zhao |
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Institution: | Center for Applied Mathematics, Tianjin University, Tianjin, P. R. China |
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Abstract: | Dunkl operators may be regarded as differential-difference operators parameterized by finite reflection groups and multiplicity functions. In this paper, the Littlewood–Paley square function for Dunkl heat flows in is introduced by employing the full “gradient” induced by the corresponding carré du champ operator and then the boundedness is studied for all . For , we successfully adapt Stein's heat flows approach to overcome the difficulty caused by the difference part of the Dunkl operator and establish the boundedness, while for , we restrict to a particular case when the corresponding Weyl group is isomorphic to and apply a probabilistic method to prove the boundedness. In the latter case, the curvature-dimension inequality for Dunkl operators in the sense of Bakry–Emery, which may be of independent interest, plays a crucial role. The results are dimension-free. |
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Keywords: | curvature-dimension condition Dunkl heat flow Dunkl operator Dunkl process Littlewood–Paley square function |
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