On Semigroups Generated by Sums of Even Powers of Dunkl Operators

On the Euclidean space $${\mathbb {R}}^N$$ equipped with a normalized root system R, a multiplicity function $$k\ge 0$$ , and the associated measure $$dw({\mathbf {x}})=\prod _{\alpha \in R} |\langle {\mathbf {x}},\alpha \rangle |^{k(\alpha )}d{\mathbf {x}}$$ we consider the differential-difference operator $$\begin{aligned} L=(-1)^{\ell +1} \sum _{j=1}^m T_{\zeta _j}^{2\ell }, \end{aligned}$$ where $$\zeta _1,\ldots ,\zeta _m$$ are nonzero vectors in $${\mathbb {R}}^N$$ , which span $${\mathbb {R}}^N$$ , and $$T_{\zeta _j}$$ are the Dunkl operators. The operator L is essentially self-adjoint on $$L^2(dw)$$ and generates a semigroup $$\{S_t\}_{t \ge 0}$$ of linear self-adjoint contractions, which has the form $$S_tf({\mathbf {x}})=f*q_t({\mathbf {x}})$$ , $$q_t({\mathbf {x}})=t^{-{\mathbf {N}}/(2\ell )}q({\mathbf {x}}/t^{1/(2\ell )})$$ , where $$q({\mathbf {x}})$$ is the Dunkl transform of the function $$ \exp (-\sum _{j=1}^m \langle \zeta _j,\xi \rangle ^{2\ell })$$ and $$*$$ stands for the Dunkl convolution. We prove that $$q({\mathbf {x}})$$ satisfies the following exponential decay: $$\begin{aligned} |q({\mathbf {x}})| \lesssim \exp (-c \Vert {\mathbf {x}}\Vert ^{2\ell /(2\ell -1)}) \end{aligned}$$ for a certain constant $$c>0$$ . Moreover, if $$q({\mathbf {x}},{\mathbf {y}})=\tau _{{\mathbf {x}}}q(-{\mathbf {y}})$$ , then $$\begin{aligned} |q({\mathbf {x}},{\mathbf {y}})|\lesssim w(B({\mathbf {x}},1))^{-1} \exp (-c d({\mathbf {x}},{\mathbf {y}})^{2\ell /(2\ell -1)}), \end{aligned}$$ where $$d({\mathbf {x}},{\mathbf {y}})=\min _{\sigma \in G}\Vert {\mathbf {x}}- \sigma ({\mathbf {y}})\Vert $$ , G is the reflection group for R, and $$\tau _{{\mathbf {x}}}$$ denotes the Dunkl translation.