Fefferman–Stein type decomposition of CMO spaces in the Dunkl setting and an application

Abstract

In this paper, we establish the Fefferman–Stein type decomposition of the $\mathrm{CMO}$ space in the Dunkl setting. That is $f\in \mathrm{CMO}(R^d,d\omega )$ if and only if $f=f_0+\sum _{j=1}^d{\tilde{R}}_j{f}_j,$where $f_0,f_1,\dots ,f_d\in C_0\left(R^d\right)$ and ${\tilde{R}}_j$, $j=0,1,\dots ,d$, represent the Dunkl–Riesz transforms. Our main tool is to characterize $\mathrm{CMO}(R^d,d\omega )$ via two approximations, which are new even for the classical space $\mathrm{CMO}\left(R^d\right)$. As a direct application of our characterization of $\mathrm{CMO}(R^d,d\omega )$, we prove the duality of $\mathrm{CMO}(R^d,d\omega )$ with $H^1(R^d,d\omega )$.