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A note on commutators of singular integrals with ${\rm BMO}$ and ${\rm VMO}$ functions in the Dunkl setting (2302.00790v1)

Published 1 Feb 2023 in math.FA

Abstract: On $\mathbb RN$ equipped with a root system $R$, multiplicity function $k \geq 0$, and the associated measure $dw(\mathbf{x})=\prod_{\alpha \in R}|\langle \mathbf{x},\alpha\rangle|{k(\alpha)}\,d\mathbf{x}$, 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 $\mathbf{T}f=f*K$ associated with ${K}$. Assuming that $b$ belongs to the ${\rm BMO}$ space on the space of homogeneous type $X=(\mathbb{R}N,|\cdot|,dw)$, 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 $Lp(dw)$ for all $1<p<\infty$. Moreover, $[b,\mathbf T]$ is compact on $Lp(dw)$, provided $b\in {\rm VMO} (X)$. The paper extents results of Han, Lee, Li and Wick.

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