Decomposition and characterization of VMO via vanishing Carleson measures

Abstract

We establish two equivalent characterizations of $\mathrm{VMO}$ in terms of vanishing Carleson measures. First, we show that any $\mathrm{VMO}$ function admits a decomposition into a continuous boundary term and an integral operator associated with a vanishing Carleson measure. Second, motivated by Varopoulos's work on the $\bar{\partial}$-equation, we characterize $\mathrm{VMO}$ via the boundary values of smooth functions whose gradients induce vanishing Carleson measures. As a consequence, we recover the known representation \[ \mathrm{VMO}=\mathrm{VLO}-\mathrm{VLO}, \] thereby providing a new perspective on this decomposition.