Decomposition and characterization of VMO via vanishing Carleson measures

We establish two equivalent characterizations of functions in V M O \mathrm {VMO} (vanishing mean oscillation) in terms of vanishing Carleson measures. First, we show that any V M O \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 V M O \mathrm {VMO} via the boundary values of smooth functions whose gradients induce vanishing Carleson measures. As a consequence, we recover the known representation \[ V M O = V L O − V L O , \mathrm {VMO}=\mathrm {VLO}-\mathrm {VLO}, \] thereby providing a new perspective on this decomposition.