On semigroups generated by sums of even powers of Dunkl operators

Abstract

On the Euclidean space ${\mathbb{R}}^N$ equipped with a normalized root system $R$, a multiplicity function $k\ge 0$, and the associated measure $dw(\mathbf{x})={\prod }_{\alpha \in R}|⟨\mathbf{x},\alpha ⟩{|}^{k(\alpha )}d\mathbf{x}$ 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 $\{S_t{\}}_{t\ge 0}$ of linear self-adjoint contractions, which has the form $S_{t}f(\mathbf{x})=f\ast q_t(\mathbf{x})$, $q_t(\mathbf{x})=t^{-\mathbf{N}/(2\ell )}q(\mathbf{x}/t^{1/(2\ell )})$, where $q(\mathbf{x})$ is the Dunkl transform of the function $\exp (-{\sum }_{j=1}^m⟨{\zeta }_j,\xi {⟩}^{2\ell })$. We prove that $q(\mathbf{x})$ satisfies the following exponential decay: $$|q(\mathbf{x})|\lesssim \exp (-c\Vert \mathbf{x}{\Vert }^{2\ell /(2\ell -1)})$$ for a certain constant $c>0$. Moreover, if $q(\mathbf{x},\mathbf{y})={\tau }_{\mathbf{x}}q(-\mathbf{y})$, then $|q(\mathbf{x},\mathbf{y})|\lesssim w(B(\mathbf{x},1){)}^{-1}\exp (-cd(\mathbf{x},\mathbf{y}{)}^{2\ell /(2\ell -1)})$, 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.

26 pages