Calderon-type commutators and chamber lifting in the Dunkl setting

Abstract

We study Calderón-type commutators $[M_b,T_i\mathcal R_j]$ in the rational Dunkl setting with a finite reflection group $G$. If $b$ belongs to the orbit Lipschitz class $\operatorname{Lip}_d$, then for every $1<p<\infty$ we prove $$\|[M_b,T_i\mathcal R_j]f\|_{L^p(\mathbb{R}^N,dω)}\le C_p\|b\|_{\operatorname{Lip}_d}\|f\|_{L^p(\mathbb{R}^N,dω)}.$$ No $G$-invariance is imposed on the input function $f$. The key is a chamber lifting: fix a closed Weyl chamber $\mathcal C$ and set $Uf(x)=(f(σ_1x),\dots,f(σ_{|G|}x))$ for $x\in\mathcal C$. This identifies $L^p(\mathbb{R}^N,dω)$ with $L^p(\mathcal C,dω;\ell_{|G|}^p)$. Under this lifting, the orbit singularity becomes the ordinary diagonal on $\mathcal C$ and the commutator becomes a finite matrix singular integral on $\mathcal C$. We construct it via heat-scale regularizations, prove component $T1$ testing for chamber indicators, and then apply scalar Calderón--Zygmund theory to obtain the $L^p$ bounds.