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.

Major revision. Author list updated. After a substantial revision of the paper, some sections and results related to the higner dimentional case have been removed in order to maintain the coherence of the presentation. Consequently, the author list has been adjusted accordingly, with the agreement of all authors