论文标题
dunkl运算符的无维平方功能估算
Dimension-Free Square Function Estimates for Dunkl Operators
论文作者
论文摘要
DUNKL操作员可以被视为通过有限反射组和多重功能参数化的差分差异操作员。在本文中，$ \ mathbb {r}^d $中的dunkl热量流的Littlewood-paley Square功能是通过使用相应的CarréDuChamp Operator诱导的完整“渐变”来引入的，然后研究了$ l^p $界面的所有$ l^p $界面，以（1，\ iffty）$ in（1，\ infty）$。对于$ p \，在（1,2] $中，我们成功地适应了Stein的热流方法来克服由Dunkl操作员的差异部分引起的难度，并建立了$ l^p $界限，而对于[2，\ infty）$中的$ p \ in [2，\ infty）$，我们限制了当相应的weyl oil osomorphic and $ \ mathbb的usim of $ \ bb的特定情况时证明$ l^p $有限。在后一种情况下，从巴克里（Bakry）的意义上讲，dunkl操作员的曲率维度不平等 - 可能具有独立利益的emery起着至关重要的作用。结果不含尺寸。
Dunkl operators may be regarded as differential-difference operators parameterized by finite reflection groups and multiplicity functions. In this paper, the Littlewood--Paley square function for Dunkl heat flows in $\mathbb{R}^d$ is introduced by employing the full "gradient" induced by the corresponding carré du champ operator and then the $L^p$ boundedness is studied for all $p\in(1,\infty)$. For $p\in(1,2]$, we successfully adapt Stein's heat flows approach to overcome the difficulty caused by the difference part of the Dunkl operator and establish the $L^p$ boundedness, while for $p\in[2,\infty)$, we restrict to a particular case when the corresponding Weyl group is isomorphic to $\mathbb{Z}_2^d$ and apply a probabilistic method to prove the $L^p$ boundedness. In the latter case, the curvature-dimension inequality for Dunkl operators in the sense of Bakry--Emery, which may be of independent interest, plays a crucial role. The results are dimension-free.
