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arXiv · 2309.06155

Decomposition and characterization of VMO via vanishing Carleson measures

Fei Tao·Yaosong Yang
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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.

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Fei Tao, Yaosong Yang. 2023-09-12. Decomposition and characterization of VMO via vanishing Carleson measures. https://arxiv.org/abs/2309.06155

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