Calderon-type commutators and chamber lifting in the Dunkl setting

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 ${\mathrm{Lip}}_d$, then for every $1<p<\mathrm{\infty }$ we prove $$\mathrm{\parallel }[M_b,T_i{\mathcal{R}}_j]f{\mathrm{\parallel }}_{L^p({\mathbb{R}}^N,d\omega )}\le C_p\mathrm{\parallel }b{\mathrm{\parallel }}_{{\mathrm{Lip}}_d}\mathrm{\parallel }f{\mathrm{\parallel }}_{L^p({\mathbb{R}}^N,d\omega )}\mathrm{.}$$ 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 $Uf(x)=(f({\sigma }_{1}x),\dots ,f({\sigma }_{\mathrm{\mid }G\mathrm{\mid }}x))$ for $x\in \mathcal{C}$. This identifies $L^p({\mathbb{R}}^N,d\omega )$ with $L^p(\mathcal{C},d\omega ;{\mathrm{\ell }}_{\mathrm{\mid }G\mathrm{\mid }}^p)$. 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.