On \(\mathbb {R}^N\) equipped with a normalized root system R and a multiplicity function \(k\ge 0\) let us consider a (not necessarily radial) kernel \(K({\mathbf {x}})\) satisfying \(|\partial ^\beta K({\mathbf {x}})|\lesssim \Vert {\mathbf {x}}\Vert ^{-{\mathbf {N}}-|\beta |}\) for \(|\beta |\le s\), where \({\mathbf {N}}\) is the homogeneous dimension of the system \(({\mathbb {R}}^N,R,k)\). We additionally assume that

where dw is the associated measure. We prove that if s large enough then a singular integral Dunkl convolution operator associated with the kernel \(K({\mathbf {x}})\) is bounded on \(L^p(dw)\) for \(1<p<\infty \) and of weak-type (1,1). Furthermore, we study a maximal function related to the Dunkl convolutions with truncation of K.

在\(\mathbb {R}^N\)配备归一化根系统R和多重函数\(k\ge 0\)让我们考虑一个（不一定是径向的）核\(K({\mathbf {x }})\)满足\(|\partial ^\beta K({\mathbf {x}})|\lesssim \Vert {\mathbf {x}}\Vert ^{-{\mathbf {N}}-| \beta |}\)为\(|\beta |\le s\)，其中\({\mathbf {N}}\)是系统的齐次维度\(({\mathbb {R}}^N ,R,k)\)。我们另外假设

其中dw是相关度量。我们证明，如果s足够大，那么与内核\(K({\mathbf {x}})\)相关的奇异积分 Dunkl 卷积算子在\(L^p(dw)\)上有界，对于\(1< p<\infty \)和弱类型 (1,1)。此外，我们研究了与截断K的 Dunkl 卷积相关的最大函数。