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 $dw(\mathbf x)=\prod_{\alpha\in R} |\langle \mathbf x,\alpha\rangle|^{k(\alpha)}d\mathbf x$weconsiderthedifferential-differenceoperator$$L=(-1)^{\ell+1} \sum_{j=1}^m T_{\zeta_j}^{2\ell},where $\zeta_1,...,\zeta_m$ are nonzero vectors in $\mathbb R^N$, which span $\mathbb R^N$, and $T_{\zeta_j}$ are the Dunkl operators. The operator $L$ is essentially self-adjoint on $L^2(dw)$ and generates a semigroup $\{S_t\}_{t \geq 0}$ of linear self-adjoint contractions, which has the form $S_tf(\mathbf x)=f*q_t(\mathbf{x})$, $q_t(\mathbf x)=t^{-\mathbf N/ (2\ell)}q(\mathbf x/ t^{1/ (2\ell)})$, where $q(\mathbf x)$ is the Dunkl transform of the function $ \exp(-\sum_{j=1}^m \langle \zeta_j,\xi\rangle^{2\ell})$. We prove that $q(\mathbf x)$ satisfies the following exponential decay: |q(\mathbf x)| \lesssim \exp(-c | \mathbf x|^{2\ell/ (2\ell-1)})$$foracertainconstant$c>0$\mathrm{.}Moreover,if$q(\mathbf x,\mathbf y)=\tau_{\mathbf x}q(-\mathbf y)$,then$|q(\mathbf x,\mathbf y)|\lesssim w(B(\mathbf x,1))^{-1} \exp(-c d(\mathbf x,\mathbf y)^{2\ell / (2\ell-1)})$,where$d(\mathbf x,\mathbf y)=\min_{\sigma\in G}| \mathbf x- \sigma(\mathbf y)| $,$G$isthereflectiongroupfor$R$,and$\tau_{\mathbf x}$ denotes the Dunkl translation.