A note on commutators of singular integrals with $\mathrm{B}\mathrm{M}\mathrm{O}$ and $\mathrm{V}\mathrm{M}\mathrm{O}$ functions in the Dunkl setting

On ${\mathbb{R}}^N$ equipped with a root system $R$, multiplicity function $k\ge 0$, and the associated measure $dw(\mathbf{x})={\prod }_{\alpha \in R}\mathrm{\mid }⟨\mathbf{x},\alpha ⟩{\mathrm{\mid }}^{k(\alpha )}d\mathbf{x}$, we consider a (non-radial) kernel $K(\mathbf{x})$ which has properties similar to those from the classical theory of singular integrals and the Dunkl convolution operator $\mathbf{T}f=f\ast K$ associated with $K$. Assuming that $b$ belongs to the $\mathrm{B}\mathrm{M}\mathrm{O}$ space on the space of homogeneous type $X=({\mathbb{R}}^N,\mathrm{\parallel }\cdot \mathrm{\parallel },dw)$, we prove that the commutator $[b,\mathbf{T}]f(\mathbf{x})=b(\mathbf{x})\mathbf{T}f(\mathbf{x})-\mathbf{T}(bf)(\mathbf{x})$ is a bounded operator on $L^p(dw)$ for all $1<p<\mathrm{\infty }$. Moreover, $[b,\mathbf{T}]$ is compact on $L^p(dw)$, provided $b\in \mathrm{V}\mathrm{M}\mathrm{O}(X)$. The paper extents results of Han, Lee, Li and Wick.