论文标题
关于非radial内核的dunkl翻译的评论
Remarks on Dunkl translations of non-radial kernels
论文作者
论文摘要
在$ \ mathbb r^n $配备了根系$ r $和多重功能$ k> 0 $上，我们研究了普遍的（dunkl）翻译$τ_ {\ mathbf x} g（ - \ \ m athbf y）$，不一定是radial kernels $ g $。在$ g $上的某些规律性假设下，我们以$τ_ {\ mathbf x} g（ - \ mathbf y）$得出界限，这意味着Euclidean距离$ \ | \ | \ Mathbf x- \ Mathbf x- \ Mathbf y \ | $ | \ Mathbf x-σ（\ Mathbf y）\ | $，其中$ g $是与$ r $相关的反射组。此外，我们证明$τ$没有保留积极性，也就是说，有一个非负SCHWARTZ类函数$φ$，因此$τ_{\ Mathbf x}φ（ - \ Mathbf y）<0 $ for某些点$ \ Mathbf x，\ Mathbf x，\ Mathbf x，\ MathBf x，\ Mathbf y \ in \ Mathb in \ MathB r^n $。
On $\mathbb R^N$ equipped with a root system $R$ and a multiplicity function $k>0$, we study the generalized (Dunkl) translations $τ_{\mathbf x}g(-\mathbf y)$ of not necessarily radial kernels $g$. Under certain regularity assumptions on $g$, we derive bounds for $τ_{\mathbf x}g(-\mathbf y)$ by means the Euclidean distance $\|\mathbf x-\mathbf y\|$ and the distance $d(\mathbf x,\mathbf y)=\min_{σ\in G} \| \mathbf x-σ(\mathbf y)\|$, where $G$ is the reflection group associated with $R$. Moreover, we prove that $τ$ does not preserve positivity, that is, there is a non-negative Schwartz class function $φ$, such that $τ_{\mathbf x}φ(-\mathbf y)<0$ for some points $\mathbf x,\mathbf y\in\mathbb R^N$.
