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

Abstract: Let $R\subset\mathbb{R}<sup>d$ 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 $0&lt;s\&lt;2$, nonnegative $f\in L^1(ν_k)\cap L^2(ν_k)$, and $$λ\&gt;0$$, we obtain a decomposition [ f=μ+(-Δk)^{s/2}u, \qquad 0\leμ\leλ, ] with $μ=λ$ on $Ω={u&gt;0}$ and [ λν_k(Ω)\le|f|{L^1(ν_k)}. ] As an application, we prove that the vector Dunkl--Riesz transform $\mathcal R_k=\nabla_k(-Δ<em>k)<sup>{-1/2}$ 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.