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Let be a bounded simply connected domain in the complex plane, . Let be a neighborhood of , let be fixed, and let be a positive weak solution to the Laplace equation in Assume that has zero boundary values on in the Sobolev sense and extend to by putting on Then there exists a positive finite Borel measure on with support contained in and such thatwhenever If and if is the Green function for with pole at then the measure coincides with harmonic measure...
When and the -harmonic measure on the boundary of the half plane is not additive on null sets. In fact, there are finitely many sets , ,..., in , of -harmonic measure zero, such that .
In this paper we survey some recent results in connection with the so called Painlevé's problem and the semiadditivity of analytic capacity γ. In particular, we give the detailed proof of the semiadditivity of the capacity γ+, and we show almost completely all the arguments for the proof of the comparability between γ and γ+.
We prove that generalized Cantor sets of class α, α ≠ 2 have the extension property iff α < 2. Thus belonging of a compact set K to some finite class α cannot be a characterization for the existence of an extension operator. The result has some interconnection with potential theory.
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