Decomposition and characterization of VMO via vanishing Carleson measures

TL;DR

This paper characterizes the space VMO using vanishing Carleson measures, providing new decompositions and connections to boundary value problems, enhancing understanding of function spaces in harmonic analysis.

Contribution

It introduces two equivalent characterizations of VMO via vanishing Carleson measures and offers a new perspective on its decomposition.

Findings

VMO functions can be decomposed into boundary and integral parts.

Characterization of VMO through boundary values of smooth functions.

Recovery of the VMO = VLO - VLO decomposition.

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.