Pluripolarity of sets with small Hausdorff measure

Denis A. Labutin

doi:10.1007/s002291020163

Pluripolarity of sets with small Hausdorff measure

Denis A. Labutin^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

4 Citations (Scopus)

Abstract

We show that any set E ⊂ Cⁿ, n ≥ 2, with finite Hausdorff measure Λ_{(log 1/r)-n} (E) < + ∞ is pluripolar. The result is sharp with respect to the measuring function. The new idea in the proof is to combine a construction from potential theory, related to the real variational integral ∫_Ω|∇u|^m, Ω ⊂ R^m, with properties of the pluricomplex relative extremal function for the Bedford-Taylor capacity.

Original language	English
Pages (from-to)	163-167
Number of pages	5
Journal	Manuscripta Mathematica
Volume	102
Issue number	2
DOIs	https://doi.org/10.1007/s002291020163
Publication status	Published - Jun 2000

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abstract = "We show that any set E ⊂ Cn, n ≥ 2, with finite Hausdorff measure Λ(log 1/r)-n (E) < + ∞ is pluripolar. The result is sharp with respect to the measuring function. The new idea in the proof is to combine a construction from potential theory, related to the real variational integral ∫Ω|∇u|m, Ω ⊂ Rm, with properties of the pluricomplex relative extremal function for the Bedford-Taylor capacity.",

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AB - We show that any set E ⊂ Cn, n ≥ 2, with finite Hausdorff measure Λ(log 1/r)-n (E) < + ∞ is pluripolar. The result is sharp with respect to the measuring function. The new idea in the proof is to combine a construction from potential theory, related to the real variational integral ∫Ω|∇u|m, Ω ⊂ Rm, with properties of the pluricomplex relative extremal function for the Bedford-Taylor capacity.

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JO - Manuscripta Mathematica

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Pluripolarity of sets with small Hausdorff measure

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