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On Semigroups Generated by Sums of Even Powers of Dunkl Operators
Integral Equations and Operator Theory ( IF 0.8 ) Pub Date : 2021-05-31 , DOI: 10.1007/s00020-021-02646-4
Jacek Dziubański , Agnieszka Hejna

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.



中文翻译:

由 Dunkl 算子的偶次幂和产生的半群

在欧几里德空间\({\mathbb {R}}^N\)配备归一化根系统R、多重函数\(k\ge 0\)和相关测度\(dw({\mathbf {x }})=\prod _{\alpha \in R} |\langle {\mathbf {x}},\alpha \rangle |^{k(\alpha )}d{\mathbf {x}}\)我们考虑微分-差分算子

$$\begin{aligned} L=(-1)^{\ell +1} \sum _{j=1}^m T_{\zeta _j}^{2\ell }, \end{aligned}$$

其中\(\zeta _1,\ldots ,\zeta _m\)\({\mathbb {R}}^N \)中的非零向量,其跨度\({\mathbb {R}}^N\),并且\(T_{\zeta _j}\)是 Dunkl 算子。算子L本质上是\(L^2(dw)\)上的自伴随,并生成线性自伴随收缩的半群\(\{S_t\}_{t \ge 0}\),其形式为\(S_tf({\mathbf {x}})=f*q_t({\mathbf {x}})\) , \(q_t({\mathbf {x}})=t^{-{\mathbf {N }}/(2\ell )}q({\mathbf {x}}/t^{1/(2\ell )})\),其中\(q({\mathbf {x}})\)是函数的 Dunkl 变换\( \exp (-\sum _{j=1}^m \langle \zeta _j,\xi \rangle ^{2\ell })\)\(*\)代表 Dunkl 卷积。我们证明\(q({\mathbf {x}})\)满足以下指数衰减:

$$\begin{对齐} |q({\mathbf {x}})| \lesssim \exp (-c \Vert {\mathbf {x}}\Vert ^{2\ell /(2\ell -1)}) \end{aligned}$$

对于某个常数\(c>0\)。此外,如果\(q({\mathbf {x}},{\mathbf {y}})=\tau _{{\mathbf {x}}}q(-{\mathbf {y}})\),然后

$$\begin{aligned} |q({\mathbf {x}},{\mathbf {y}})|\lesssim w(B({\mathbf {x}},1))^{-1} \ exp (-cd({\mathbf {x}},{\mathbf {y}})^{2\ell /(2\ell -1)}), \end{aligned}$$

其中\(d({\mathbf {x}},{\mathbf {y}})=\min _{\sigma \in G}\Vert {\mathbf {x}}- \sigma ({\mathbf {y}} }})\Vert \)GR的反射群,\(\tau _{{\mathbf {x}}}\)表示 Dunkl 平移。

更新日期:2021-05-31
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