发表机构
School of Mathematics and Statistics, Huazhong University of Science and Technology; School of Mathematical Sciences, Guizhou Normal University（华中科技大学数学与统计学院; 贵州师范大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对Dunkl热方程建立了可观测性、Hölder型插值及谱不等式的等价性，并给出了Bernstein不等式与谱常数为$e^{C(1+N)}$的Logvinenko--Sereda定理。
AI 中文摘要
设$R$为$\mathbb{R}^d$中具有反射群$G$的归一化根系，$k$为$G$-不变重数函数，$\mathcal{F}_k$为相应的Dunkl变换。记$k_\alpha=k(\alpha)>0$，$\mathrm{d}\mu_k(x)=w(x)\\\\,\mathrm{d}x$，其中$w(x)=\prod_{\alpha\in R_+}|\langle x,\alpha\rangle|^{2k_\alpha}$。我们研究$\mathbb{R}^d$上Dunkl热方程的可观测性、Hölder型插值不等式和谱不等式。我们建立了普通导数的Bernstein不等式，以及一个谱常数为$e^{C(1+N)}$的Logvinenko--Sereda定理，适用于Dunkl变换支在$\overline{B(0,N)}$中的函数。我们将可观测集刻画为关于$\mu_k$的厚可测集，并证明了可观测性、Hölder型插值不等式和谱不等式的等价性。
英文摘要
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.
