Hp spaces in the Dunkl setting meet Coifman--Weiss--type atoms
Abstract
Let Δk be the Dunkl Laplacian associated with an arbitrary root system and a nonnegative multiplicity function. For every 0<p≤1, a Coifman-Weiss type atomic characterization of the Hardy space HpDunkl is established. More precisely, it is proved that the space HpDunkl, which is originally defined by a relevant square function, coincides with the space generated by (CW,p,2)-atoms, that is, atoms supported on Euclidean balls and satisfying cancellation conditions against all polynomials of degree $ \leq s_p$, where sp=⌊ N(1/p−1)⌋ and N is the homogeneous dimension of the underlying Dunkl measure. The corresponding quasi-norms are equivalent. We also show that the same space is obtained when the L2 size condition in the definition of atoms is replaced by the L∞ size condition. The strategy of the proof is to use an operator-type atomic decomposition associated with the Dunkl Laplacian, and then prove that each such operator atom can be written a linear combination of (CW,p,2)-atoms.
Related
- A self-dual Dunkl wavelet frame and Besov--Triebel--Lizorkin spaces
- Riesz transform characterization of Hardy spaces associated with Dunkl--Schrödinger operators
- A Remark on Hörmander Multipliers on Dunkl Hardy Spaces
- On Lipschitz Spaces, Semigroup Transference, and Heat Kernel Derivative Estimates in Dunkl--Schrödinger Analysis
- Order convergence in ordered vector spaces: compactness, atoms, pervasiveness, and ice cream spaces
BibTeX
@misc{dziubanski2026spaces,
title = {{$H^p$ spaces in the Dunkl setting meet Coifman--Weiss--type atoms}},
author = {Jacek Dziubański and Agnieszka Hejna-Łyżwa},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.17070},
eprint = {2609.17070},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.17070},
}
RIS
TY - UNPB
TI - H^p spaces in the Dunkl setting meet Coifman--Weiss--type atoms
AU - Jacek Dziubański
AU - Agnieszka Hejna-Łyżwa
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.17070
UR - https://arxiv.org/abs/2609.17070
AB - Let Δ_k be the Dunkl Laplacian associated with an arbitrary root system and a nonnegative multiplicity function. For every 0<p≤1, a Coifman-Weiss type atomic characterization of the Hardy space H^p_Dunkl is established. More precisely, it is proved that the space H^p_Dunkl, which is originally defined by a relevant square function, coincides with the space generated by (CW,p,2)-atoms, that is, atoms supported on Euclidean balls and satisfying cancellation conditions against all polynomials of degree $ \leq s_p$, where s_p=⌊ N(1/p−1)⌋ and N is the homogeneous dimension of the underlying Dunkl measure. The corresponding quasi-norms are equivalent. We also show that the same space is obtained when the L^2 size condition in the definition of atoms is replaced by the L^∞ size condition. The strategy of the proof is to use an operator-type atomic decomposition associated with the Dunkl Laplacian, and then prove that each such operator atom can be written a linear combination of (CW,p,2)-atoms.
ER -