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PreprintCC BY 4.0arXiv

An Exact Spherical-Solid Mean Formula and Vanishing Criteria for Lelong Numbers

  • Fusheng Deng
  • Yizhe Yang
  • Zhanpeng You
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Abstract

We establish an exact additive formula expressing the Lelong number of a plurisubharmonic function in terms of the limiting difference between its spherical and solid means. Building on this identity, we derive several equivalent criteria for the vanishing of the Lelong number, formulated in terms of fixed-scale increments and the asymptotic agreement of radial maxima, spherical and solid means, and radial regularizations. We also give a pointwise L1 mean-oscillation formulation of the VMO characterization for plurisubharmonic functions with zero Lelong number, together with a streamlined proof of the corresponding global result of Biard and Wu. Finally, we establish a nonlinear characterization in terms of the relative extremal depth and the Bedford-Taylor capacity of deep sublevel sets.

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Cite

BibTeXRIS
BibTeX
@misc{deng2026exact,
  title = {{An Exact Spherical-Solid Mean Formula and Vanishing Criteria for Lelong Numbers}},
  author = {Fusheng Deng and Yizhe Yang and Zhanpeng You},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.30999},
  eprint = {2609.30999},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.30999},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - An Exact Spherical-Solid Mean Formula and Vanishing Criteria for Lelong Numbers
AU  - Fusheng Deng
AU  - Yizhe Yang
AU  - Zhanpeng You
PY  - 2026
DA  - 2026/09/25
PB  - arXiv
DO  - 10.48550/arxiv.2609.30999
UR  - https://arxiv.org/abs/2609.30999
AB  - We establish an exact additive formula expressing the Lelong number of a plurisubharmonic function in terms of the limiting difference between its spherical and solid means. Building on this identity, we derive several equivalent criteria for the vanishing of the Lelong number, formulated in terms of fixed-scale increments and the asymptotic agreement of radial maxima, spherical and solid means, and radial regularizations. We also give a pointwise L^1 mean-oscillation formulation of the VMO characterization for plurisubharmonic functions with zero Lelong number, together with a streamlined proof of the corresponding global result of Biard and Wu. Finally, we establish a nonlinear characterization in terms of the relative extremal depth and the Bedford-Taylor capacity of deep sublevel sets.
ER  -