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 $k\ge 0$ 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 $0<s<2$, nonnegative $f\in L^1({\nu }_k)\cap L^2({\nu }_k)$, and $\lambda >0$, we obtain a decomposition \[ f=μ+(-Δ_k)^{s/2}u, \qquad 0\leμ\leλ, \] with $\mu =\lambda$ on $\Omega =\{u>0\}$ and \[ λν_k(Ω)\le\|f\|_{L^1(ν_k)}. \] 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 \[ M_k=\#\{α\in R_+:k(α)>0\}. \] 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--Schrödinger Riesz transform.

18 pages