Decomposition and characterization of VMO via vanishing Carleson measures

We establish two equivalent characterizations of $\mathrm{V}\mathrm{M}\mathrm{O}$ in terms of vanishing Carleson measures. First, we show that any $\mathrm{V}\mathrm{M}\mathrm{O}$ 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 $\bar{\mathrm{\partial }}$-equation, we characterize $\mathrm{V}\mathrm{M}\mathrm{O}$ via the boundary values of smooth functions whose gradients induce vanishing Carleson measures. As a consequence, we recover the known representation $$\mathrm{V}\mathrm{M}\mathrm{O}=\mathrm{V}\mathrm{L}\mathrm{O}-\mathrm{V}\mathrm{L}\mathrm{O},$$ thereby providing a new perspective on this decomposition.