Dimension free weak-type endpoint estimates for the vectors of the Dunkl--Riesz transform
Abstract
Let R⊂ℝd be a reduced root system, G the associated finite reflection group, and k≥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∈ L1(νk)∩ L2(νk), and λ>0, we obtain a decomposition f=μ+(−Δk)s/2u, 0≤μ≤λ, with μ=λ on Ω={u>0} and λνk(Ω)≤|f|L1(νk). As an application, we prove that the vector Dunkl--Riesz transform ℛk=∇k(−Δk)−1/2 is of weak type (1,1) with constant at most (Mk+2), where Mk=#{α∈ 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.
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BibTeX
@misc{mukherjee2026dimension,
title = {{Dimension free weak-type endpoint estimates for the vectors of the Dunkl--Riesz transform}},
author = {Suman Mukherjee},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08492},
eprint = {2609.08492},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08492},
}
RIS
TY - UNPB
TI - Dimension free weak-type endpoint estimates for the vectors of the Dunkl--Riesz transform
AU - Suman Mukherjee
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.08492
UR - https://arxiv.org/abs/2609.08492
AB - Let R⊂ℝ^d be a reduced root system, G the associated finite reflection group, and k≥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∈ L^1(ν_k)∩ L^2(ν_k), and λ>0, we obtain a decomposition f=μ+(−Δ_k)^s/2u, 0≤μ≤λ, with μ=λ on Ω={u>0} and λν_k(Ω)≤|f|_L^1(ν_k). As an application, we prove that the vector Dunkl--Riesz transform ℛ_k=∇_k(−Δ_k)^−1/2 is of weak type (1,1) with constant at most (M_k+2), where M_k=#{α∈ 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.
ER -