Calderon-type commutators and chamber lifting in the Dunkl setting

Abstract

Let $G$ be a finite reflection group, and let $T_j$, $\Delta_\kappa$, and $d\omega$ denote its Dunkl operators, Laplacian, and measure, respectively. Write $\mathcal R_j=-T_j(-\Delta_\kappa)^{-1/2}$ for the $j$th Dunkl Riesz transform. We study the Calder\'on-type commutator $[M_b,T_i\mathcal R_j]$ on the full space $L^p(\R^N,d\omega)$, without assuming $G$-invariance of the functions. For $b\in\Lipd$, a full-space Dunkl $T1$ argument for $[M_b,(-\Delta_\kappa)^{1/2}]$, combined with the exact factorization $$ [M_b,T_i\mathcal R_j] =-M_{\partial_i b}\mathcal R_j-\mathcal R_iM_{\partial_jb} +\mathcal R_i[M_b,(-\Delta_\kappa)^{1/2}]\mathcal R_j, $$ implies boundedness on $L^p(d\omega)$ for every $1<p<\infty$. We also prove that the prescribed heat truncations are uniformly bounded on $L^2(d\omega)$ and converge jointly to this operator. To control these truncations, we lift all reflected values of an arbitrary function to separate coordinates on a fixed Weyl chamber. This chamber lifting retains all non-$G$-invariant information and places every possible orbit singularity on the ordinary chamber diagonal of a finite matrix operator. Wall-layer estimates and scalar $T1$ bounds for the integrated entries are uniform in both heat endpoints. A dense-core argument then proves joint, path-independent weak-operator convergence to the factorized commutator. The lifted limit is associated, in the separated-support sense, with a finite matrix of scalar Calder\'on--Zygmund kernels on the chamber.