A note on commutators of singular integrals with BMO and VMO functions in the Dunkl setting

On RN$\mathbb {R}<^>N$ equipped with a root system R, multiplicity function k & GE;0$k \ge 0$, and the associated measure dw(x)=∏& alpha;& ISIN;R|& LeftAngleBracket;x,& alpha;& RightAngleBracket;|k(& alpha;)dx$dw(\mathbf {x})=\prod _{\alpha \in R}|\langle \mathbf {x},\alpha \rangle |<^>{k(\alpha )}\,d\mathbf {x}$, we consider a (nonradial) kernel K(x)${K}(\mathbf {x})$, which has properties similar to those from the classical theory of singular integrals and the Dunkl convolution operator Tf=f*K$\mathbf {T}f=f*K$ associated with K. Assuming that b belongs to the BMO space on the space of homogeneous type X=(RN,& PAR;& BULL;& PAR;,dw)$X=(\mathbb {R}<^>N,\Vert \cdot \Vert ,dw)$, we prove that the commutator [b,T]f(x)=b(x)Tf(x)-T(bf)(x)$[b,\mathbf {T}]f(\mathbf {x})=b(\mathbf {x})\mathbf {T}f(\mathbf {x})-\mathbf {T}(bf)(\mathbf {x})$ is a bounded operator on Lp(dw)$L<^>p(dw)$ for all 1<p<& INFIN;$1<p<\infty$. Moreover, [b,T]$[b,\mathbf {T}]$ is compact on Lp(dw)$L<^>p(dw)$, provided b & ISIN;VMO(X)$b\in {\rm VMO} (X)$. The paper extends results of Han, Lee, Li, and Wick.