  As a significant application of the duality theory between real Hardy spaces
$H^1(\mathbb{R}^n)$ and $\mathrm{BMO}(\mathbb{R}^n)$, Fefferman and Stein
developed a representation theorem for $\mathrm{BMO}(\mathbb{R}^n)$ by
utilizing the Riesz transforms $(n\geq 1)$. L. Carleson provided a constructive
proof for the case $n=1$. In this article, we propose a representation theorem
for $\mathrm{VMO}(\mathbb{S})$ using Carleson's construction and demonstrate a
representation theorem for $\mathrm{VMO}(\mathbb{R}^n)$ through an iterative
process. Additionally, we provide a brand-new characterization of
$\mathrm{VMO}$ as an application of our results.

Complex Variables