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 $k\geq 0$, 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 $$L=(-1)^{\ell+1} \sum_{j=1}^m T_{\zeta_j}^{2\ell},$$ 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: $$ |q(\mathbf x)| \lesssim \exp(-c \| \mathbf x\|^{2\ell / (2\ell-1)})$$ for certain constant $c>0$. 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)\| $ and $\tau_{\mathbf x}$ denotes the Dunkl translation.