Remarks on Dunkl translations of non-radial kernels

On ${\mathbb{R}}^N$ equipped with a root system $R$ and a multiplicity function $k>0$, we study the generalized (Dunkl) translations ${\tau }_{\mathbf{x}}g(-\mathbf{y})$ of not necessarily radial kernels $g$. Under certain regularity assumptions on $g$, we derive bounds for ${\tau }_{\mathbf{x}}g(-\mathbf{y})$ by means the Euclidean distance $\mathrm{\parallel }\mathbf{x}-\mathbf{y}\mathrm{\parallel }$ and the distance $d(\mathbf{x},\mathbf{y})={\min }_{\sigma \in G}\mathrm{\parallel }\mathbf{x}-\sigma (\mathbf{y})\mathrm{\parallel }$, where $G$ is the reflection group associated with $R$. Moreover, we prove that $\tau$ does not preserve positivity, that is, there is a non-negative Schwartz class function $\phi$, such that ${\tau }_{\mathbf{x}}\phi (-\mathbf{y})<0$ for some points $\mathbf{x},\mathbf{y}\in {\mathbb{R}}^N$.