On singularity of harmonic measure in space

Jang Mei Wu

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We construct a topological ball D in R3, and a set E on ∂ D lying on a 2-diniensional hyperpiane so that E has Hausdorff dimension one and has positive harmonic measure with respect to D. This shows that a theorem of Øksendal on harmonic measure in R2 is not true in R3. Suppose D is a bounded domain in Rm, m ≥ 2, Rm\D satisfies the corkscrew condition at each point on ∂ D; and E is a set on ∂ D lying also on a BMO1 surface, which is more general than a hyperpiane, then we can prove that if E has m-1 dimensional Hausdorff measure zero then it must have harmonic measure zero with respect to D.

Original languageEnglish (US)
Pages (from-to)485-496
Number of pages12
JournalPacific Journal of Mathematics
Issue number2
StatePublished - Feb 1986

ASJC Scopus subject areas

  • General Mathematics

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