The mutual singularity of harmonic measure and Hausdorff measure of\n codimension smaller than one

2019/01/23 by Xavier Tolsa, Tolsa, Xavier
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analytic and geometric function theory #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.1901.07783

Abstract

Let \Ω\⊂ mathbb Rⁿ⁺¹ be open and let E\⊂ \∂\Ω\nwith 0<H^s(E)<\∞, for some s\∈(n,n+1), satisfy a local capacity\ndensity condition. In this paper it is shown that the harmonic measure cannot\nbe mutually absolutely continuous with H^s on E. This answers a question of\nAzzam and Mourgoglou, who had proved the same result under the additional\nassumption that \Ω is a uniform domain.\n

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