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

Jacek Dziubański and Agnieszka Hejna

Mathematische Nachrichten, 2024, vol. 297, issue 2, 629-643

Abstract: On RN$\mathbb {R}^N$ equipped with a root system R, multiplicity function k≥0$k \ge 0$, and the associated measure dw(x)=∏α∈R|⟨x,α⟩|k(α)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,∥·∥,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

Date: 2024
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