Remarks on Dunkl Translations of Non-radial Kernels

Abstract

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