Non-radial Dunkl multipliers at the L2-Sobolev threshold
Abstract
We prove a Hörmander multiplier theorem for the Dunkl transform associated with an arbitrary finite reflection group. Uniform Hσ(ℝN) bounds for the normalised dyadic pieces of a measurable symbol, with σ>N/2 and N the homogeneous dimension, imply Lp(dω) boundedness for 1<p<∞ and weak type (1,1). No radiality or reflection-group invariance is assumed, and the dyadic pieces may meet the reflecting hyperplanes. The main new estimate controls the Dunkl kernel Eκ(iξ,x) and its first Euclidean derivatives in x, uniformly near arbitrary intersections of reflecting hyperplanes. Here ξ lies in a fixed compact annulus. Writing wκ for the Dunkl weight and Ωκ for its inhomogeneous counterpart, it gives the bound CΩκ(x)−1/2wκ(ξ)−1/2. The proof decomposes each root contribution into Hermitian two-dimensional blocks, applies oscillatory changes of variables near the rootwise transition points, and uses a commutator identity to cancel the non-integrable phase derivative. Chamber lifting also yields endpoint bounds for the pullbacks of the componentwise Coifman--Weiss atomic H1 space and the corresponding BMO space on a fixed chamber.
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BibTeX
@misc{chang2026nonradial,
title = {{Non-radial Dunkl multipliers at the $L^2$-Sobolev threshold}},
author = {Der-Chen Chang and Ji Li and Chaojie Wen and Liangchuan Wu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14298},
eprint = {2609.14298},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14298},
}
RIS
TY - UNPB
TI - Non-radial Dunkl multipliers at the L^2-Sobolev threshold
AU - Der-Chen Chang
AU - Ji Li
AU - Chaojie Wen
AU - Liangchuan Wu
PY - 2026
DA - 2026/09/13
PB - arXiv
DO - 10.48550/arxiv.2609.14298
UR - https://arxiv.org/abs/2609.14298
AB - We prove a Hörmander multiplier theorem for the Dunkl transform associated with an arbitrary finite reflection group. Uniform H^σ(ℝ^N) bounds for the normalised dyadic pieces of a measurable symbol, with σ>N/2 and N the homogeneous dimension, imply L^p(dω) boundedness for 1<p<∞ and weak type (1,1). No radiality or reflection-group invariance is assumed, and the dyadic pieces may meet the reflecting hyperplanes. The main new estimate controls the Dunkl kernel E_κ(iξ,x) and its first Euclidean derivatives in x, uniformly near arbitrary intersections of reflecting hyperplanes. Here ξ lies in a fixed compact annulus. Writing w_κ for the Dunkl weight and Ω_κ for its inhomogeneous counterpart, it gives the bound CΩ_κ(x)^−1/2w_κ(ξ)^−1/2. The proof decomposes each root contribution into Hermitian two-dimensional blocks, applies oscillatory changes of variables near the rootwise transition points, and uses a commutator identity to cancel the non-integrable phase derivative. Chamber lifting also yields endpoint bounds for the pullbacks of the componentwise Coifman--Weiss atomic H^1 space and the corresponding BMO space on a fixed chamber.
ER -