Year: 2026
Type: preprint
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 $00$, we obtain a decomposition \[ f=μ+(-Δ_k)^{s/2}u, \qquad 0\leμ\leλ, \] with $μ=λ$ on $Ω=\{u>0\}$ and \[ λν_k(Ω)\le\|f\|_{L^1(ν_k)}. \] As an application, we prove that the vector Dunkl--Riesz transform $\mathcal R_k=\nabla_k(-Δ_k)^{-1/2}$ 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.
Source: arXiv (Cornell University)
Author: Mukherjee, Suman
Institution: Indian Institute of Technology Bombay
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7212000651