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<p>We prove that a plurisubharmonic function on a domain in the complex Euclidean space is a locally VMO (vanishing mean oscillation) function if and only if its Lelong number at each point vanishes. We also give a global version of this result when the boundary of the domain satisfies the <i>interior sphere condition</i>. An example emphasizes the importance of this condition. These equivalences contribute to a better understanding of the behavior of singular plurisubharmonic functions. We end the paper by discussing the link between the residual Monge–Ampère mass and VMO functions, by providing examples.</p>

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Equivalence between VMO functions and plurisubharmonic functions with zero Lelong numbers

  • 期刊论文

  • 发表日期：

    2026年1月27日

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  • Séverine Biard,
  • Jujie Wu

摘要

We prove that a plurisubharmonic function on a domain in the complex Euclidean space is a locally VMO (vanishing mean oscillation) function if and only if its Lelong number at each point vanishes. We also give a global version of this result when the boundary of the domain satisfies the interior sphere condition. An example emphasizes the importance of this condition. These equivalences contribute to a better understanding of the behavior of singular plurisubharmonic functions. We end the paper by discussing the link between the residual Monge–Ampère mass and VMO functions, by providing examples.