AI 中文总结
本文在邓克利-薛定谔分析中，建立两类利普希茨空间的等价关系，解耦空间光滑分量与对称结构，给出热核导数上界，解决非G-不变势带来的几何难点。
AI 中文摘要
本文研究适配于邓克利-薛定谔算子$\boldsymbol{\textit{L}} = -\boldsymbol{\textit{Δ}}_k + V$的利普希茨空间，其中$\boldsymbol{\textit{Δ}}_k$为邓克利拉普拉斯算子，$V \boldsymbol{\textit{≥}} 0$且属于逆赫尔德类$\boldsymbol{\textit{RH}}^q(dw)$，$q \boldsymbol{\textit{>}} \boldsymbol{\textit{max}}(1, \boldsymbol{\textit{N}}/2)$。对于$\boldsymbol{0 < \beta < 2}$，我们建立了基于热半群的利普希茨空间$\boldsymbol{\textit{Λ}}_{\boldsymbol{\textit{L}},k}^{\beta/2}(\boldsymbol{\textit{R}}^N)$与通过临界半径函数$\boldsymbol{\textit{m}}(\boldsymbol{\textit{x}})$定义的逐点加权齐格蒙类$\boldsymbol{\textit{Λ}}_{\boldsymbol{\textit{L}},k}^{\beta}(\boldsymbol{\textit{R}}^N)$之间的完全等价关系。值得注意的是，该刻画表明空间光滑分量与基础反射群对称性及根系完全解耦。作为中间结果，我们建立了与未受扰邓克利拉普拉斯算子$\boldsymbol{\textit{Δ}}_k$相关的非齐次加权利普希茨空间的新等价关系。该语境中的一个核心难点——源于需将欧氏距离与轨道度量$\boldsymbol{\textit{d}}(\boldsymbol{\textit{x}},\boldsymbol{\textit{y}})$解耦——通过详细的几何分析得到解决。最后，我们给出邓克利-薛定谔热核时间导数的逐点高斯上界，这对邓克利结构上的调和分析未来研究具有独立意义。关键的是，我们未假设势函数$V$在外尔群作用下是$\boldsymbol{\textit{G}}$-不变的，这引入的重大几何难点通过轨道度量的局部分析得以解决。
英文摘要
In this paper, we study Lipschitz spaces adapted to Dunkl--Schrödinger operators $\mathcal{L} = -Δ_k + V$, where $Δ_k$ is the Dunkl Laplacian and $V \ge 0$ belongs to the reverse Hölder class $\mathrm{RH}^q(dw)$ with $q > \max(1, \mathbf{N}/2)$. For $0 < β< 2$, we establish the full equivalence between the heat semigroup-based Lipschitz spaces $\widetildeΛ_{\mathcal{L},k}^{β/2}(\mathbb{R}^N)$ and the pointwise weighted Zygmund classes $Λ_{\mathcal{L},k}^β(\mathbb{R}^N)$ defined via the critical radius function $m(\mathbf{x})$. Remarkably, this characterization shows that the spatial smoothness component decouples completely from the underlying reflection group symmetries and root systems. As an intermediate result, we establish new equivalences for inhomogeneous weighted Lipschitz spaces associated with the unperturbed Dunkl Laplacian $Δ_k$. A central difficulty in this context --- arising from the necessity to disentangle the Euclidean distance from the orbit metric $d(\mathbf{x},\mathbf{y})$ --- is resolved through a detailed geometric analysis. Finally, we provide pointwise Gaussian upper bounds for the time derivatives of the Dunkl--Schrödinger heat kernel, which are of independent interest for future work in harmonic analysis on Dunkl structures. Crucially, we do not assume the potential $V$ to be $G$-invariant under the action of the Weyl group, which introduces major geometric difficulties resolved here via a local analysis of the orbit metric.
