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 \(dw({\mathbf {x}})=\prod _{\alpha \in R} |\langle {\mathbf {x}},\alpha \rangle |^{k(\alpha )}d{\mathbf {x}}\) 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.