Czechoslovak Mathematical Journal, first online, pp. 1-12
A simple proof of Fefferman-Stein type characterization of ${\rm CMO}(\mathbb{R}^n)$ space
Qingdong Guo, Zeqiang Linli, Kang Hu
Received August 7, 2024. Published online February 6, 2025.
Abstract: We give a simple proof of Fefferman-Stein type characterization of the space ${\rm CMO}(\mathbb{R}^n)$, that is, $f\in{\rm CMO} (\mathbb{R}^n)$ if and only if $f=\phi+\sum_{j=1}^nR_j\varphi_j,$ where $\phi,\varphi_j\in{C_0(\mathbb{R}^n)}$ and $R_j$, $j=1,2,\ldots,n$, are the Riesz transforms. Notice that this result was established by G. Bourdaud (2002), but his proof depends on the Fefferman-Stein type decomposition of the space ${\rm VMO}(\mathbb{R}^n)$ obtained by D. Sarason (1975). We will provide a direct method to prove this conclusion.
Affiliations: Qingdong Guo, School of Mathematical Sciences, Xiamen University, Xiamen 361005, P. R. China, e-mail: qingdongmath@stu.xmu.edu.cn; Zeqiang Linli, School of Mathematical and Statistics, Guangdong University of Foreign Studies, 21 Luntou Road, Guangzhou 510004, P. R. China, e-mail: linlizeqiang@gdufs.edu.cn; Kang Hu (corresponding author), School of Information Engineering, Wuhan Business University, No. 816 Dongfeng Avenue, Wuhan Economic and Technological Development Zone, Wuhan 430056, P. R. China, e-mail: huk2016423@163.com
