Singular integrals in the rational Dunkl setting

On R-N equipped with a normalized root system R and amultiplicity function k >= 0 let us consider a (not necessarily radial) kernel K(x) satisfying vertical bar partial derivative K-beta(x)vertical bar less than or similar to parallel to x parallel to(-N-vertical bar beta vertical bar) for vertical bar beta vertical bar <= s, where N is the homogeneous dimension of the system (R-N, R, k). We additionally assume that sup(0<a<b<infinity) vertical bar integral(a<parallel to x parallel to<b) K(x)dw(x)vertical bar < infinity, 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(x) is bounded on L-p(dw) for 1 < p < infinity and of weak-type (1,1). Furthermore, we study a maximal function related to the Dunkl convolutions with truncation of K.