Dimension free weak-type endpoint estimates for the vectors of the Dunkl--Riesz transform

Abstract

Let $R\subset\mathbb{R}^d$ be a reduced root system, $G$ the associated finite reflection group, and $k\ge0$ a $G$-invariant multiplicity function. We develop a Dunkl analogue of the obstacle/partial-balayage method of Ouyang, Spector, and Stockdale (https://arxiv.org/abs/2608.18068) for the Euclidean fractional Laplacian. For $0 0$, we obtain a decomposition \[ f=\mu+(-\Delta_k)^{s/2}u, \qquad 0\le\mu\le\lambda, \] with $\mu=\lambda$ on $\Omega=\{u>0\}$ and \[ \lambda\nu_k(\Omega)\le\|f\|_{L^1(\nu_k)}. \] As an application, we prove that the vector Dunkl--Riesz transform $\mathcal R_k=\nabla_k(-\Delta_k)^{-1/2}$ is of weak type $(1,1)$ with constant at most $(M_k+2)$, where \[ M_k=\#\{\alpha\in R_+:k(\alpha)>0\}. \] For $G$-invariant functions, the reflection terms vanish and the same argument gives the universal constant $2$. We further establish a dimension-free weak-type $(1,1)$ estimate for the Dunkl--Schr\"odinger Riesz transform.