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Harmonic measures supported on curves. (English) Zbl 0677.30017
The following nice result is proved: Let $$\Omega_ 1$$, $$\Omega_ 2$$ be the two complementary domains of a Jordan curve $$\Gamma$$ in the complex plane and let $$\omega_ 1$$, $$\omega_ 2$$ be the harmonic measures on $$\Gamma$$ for the two regions $$\Omega_ 1$$, $$\Omega_ 2$$ with respect to two given points $$z_ 1\in \Omega_ 1$$ and $$z_ 2\in \Omega_ 2$$. Then (1) and (2) below are equivalent: (1) $$\omega_ 1\perp \omega_ 2$$; (2) $$T_ 1\cap T_ 2$$ has zero length (1-dimensional Hausdorff measure) where $$T_ j$$ is the set of inner tangent points with respect to $$\Omega_ j$$; $$j=1,2$$.
Reviewer: B.Øksendal

##### MSC:
 30C85 Capacity and harmonic measure in the complex plane 31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions
##### Keywords:
Jordan domain; inner tangent points; harmonic measures
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