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 Calderón-type commutator $[M_b,T_i{\mathcal{R}}_j]$ on the full space $L^p({\mathbb{R}}^N,d\omega )$, 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 $$[M_b,T_i{\mathcal{R}}_j]=-M_{{\partial }_{i}b}{\mathcal{R}}_j-{\mathcal{R}}_i{M}_{{\partial }_{j}b}+{\mathcal{R}}_i[M_b,(-{\Delta }_{\kappa }{)}^{1/2}]{\mathcal{R}}_j,$$ 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 Calderón--Zygmund kernels on the chamber.

This is an update of V2, typos fixed