Year: 2026
Type: preprint
Abstract: Let $R$ be a normalized root system in $\mathbb{R}^d$ with reflection group $G$, let $k$ be a $G$-invariant multiplicity function, and let $\mathcal{F}_k$ be the associated Dunkl transform. We write $k_α=k(α)>0$ and $\mathrm{d}μ_k(x)=w(x)\,\mathrm{d}x$, where $w(x)=\prod_{α\in R_+}|\langle x,α\rangle|^{2k_α}$. We study observability, Hölder-type interpolation, and spectral inequalities for the Dunkl heat equation on $\mathbb{R}^d$. 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 $\overline{B(0,N)}$. 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.
Source: arXiv (Cornell University)
Authors: Zhao, Xingyu, Wei, Longben, Duan, Zhiwen
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214330100