Dimension?free square function estimates for Dunkl operators

Dunkl operators may be regarded as differential-difference parameterized by finite reflection groups and multiplicity functions. In this paper, the Littlewood--Paley square function for heat flows in $\mathbb{R}^d$ is introduced employing full "gradient" induced corresponding carr\'{e} du champ operator then $L^p$ boundedness studied all $p\in(1,\infty)$. For $p\in(1,2]$, we successfully adapt Stein's approach to overcome difficulty caused difference part of establish boundedness, while $p\in[2,\infty)$, restrict a particular case when Weyl group isomorphic $\mathbb{Z}_2^d$ apply probabilistic method prove boundedness. latter case, curvature-dimension inequality sense Bakry--Emery, which independent interest, plays crucial role. The results are dimension-free.