PreprintCC BY 4.0arXiv
Observability Inequalities and the Logvinenko--Sereda Theorem for the Dunkl Transform
Abstract
Let R be a normalized root system in ℝd with reflection group G, let k be a G-invariant multiplicity function, and let ℱk be the associated Dunkl transform. We write kα=k(α)>0 and dμk(x)=w(x) dx, where w(x)=∏α∈ R+|⟨ x,α⟩|2kα. We study observability, Hölder-type interpolation, and spectral inequalities for the Dunkl heat equation on ℝd. We establish a Bernstein inequality for ordinary derivatives and a Logvinenko--Sereda theorem with a spectral constant of the form eC(1+N) for functions whose Dunkl transforms are supported in B(0,N)̅. We characterize observable sets as the measurable sets that are thick with respect to μk, and prove the equivalence of the observability, Hölder-type interpolation, and spectral inequalities.
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BibTeX
@misc{zhao2026observability,
title = {{Observability Inequalities and the Logvinenko--Sereda Theorem for the Dunkl Transform}},
author = {Xingyu Zhao and Longben Wei and Zhiwen Duan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28993},
eprint = {2609.28993},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28993},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Observability Inequalities and the Logvinenko--Sereda Theorem for the Dunkl Transform
AU - Xingyu Zhao
AU - Longben Wei
AU - Zhiwen Duan
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.28993
UR - https://arxiv.org/abs/2609.28993
AB - Let R be a normalized root system in ℝ^d with reflection group G, let k be a G-invariant multiplicity function, and let ℱ_k be the associated Dunkl transform. We write k_α=k(α)>0 and dμ_k(x)=w(x) dx, where w(x)=∏_α∈ R_+|⟨ x,α⟩|^2k_α. We study observability, Hölder-type interpolation, and spectral inequalities for the Dunkl heat equation on ℝ^d. We establish a Bernstein inequality for ordinary derivatives and a Logvinenko--Sereda theorem with a spectral constant of the form e^C(1+N) for functions whose Dunkl transforms are supported in B(0,N)̅. We characterize observable sets as the measurable sets that are thick with respect to μ_k, and prove the equivalence of the observability, Hölder-type interpolation, and spectral inequalities.
ER -