TY - JOUR
T1 - On singularity of harmonic measure in space
AU - Wu, Jang Mei
PY - 1986/2
Y1 - 1986/2
N2 - 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.
AB - 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.
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U2 - 10.2140/pjm.1986.121.485
DO - 10.2140/pjm.1986.121.485
M3 - Article
AN - SCOPUS:84972574188
SN - 0030-8730
VL - 121
SP - 485
EP - 496
JO - Pacific Journal of Mathematics
JF - Pacific Journal of Mathematics
IS - 2
ER -