Dimension‐free square function estimates for Dunkl operators

Huaiqian Li and Mingfeng Zhao

Mathematische Nachrichten, 2023, vol. 296, issue 3, 1225-1243

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 Rd$\mathbb {R}^d$ is introduced by employing the full “gradient” induced by the corresponding carré du champ operator and then the Lp$L^p$ boundedness is studied for all p∈(1,∞)$p\in (1,\infty )$. For p∈(1,2]$p\in (1,2]$, we successfully adapt Stein's heat flows approach to overcome the difficulty caused by the difference part of the Dunkl operator and establish the Lp$L^p$ boundedness, while for p∈[2,∞)$p\in [2,\infty )$, we restrict to a particular case when the corresponding Weyl group is isomorphic to Z2d$\mathbb {Z}_2^d$ and apply a probabilistic method to prove the Lp$L^p$ 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.

Date: 2023
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