Tangent measures and absolute continuity of harmonic measure

• Jonas Azzam ^[1] ; Mihalis Mourgoglou ^[1]
  1. [1] Universitat Autònoma de Barcelona

     Universitat Autònoma de Barcelona

     Barcelona, España

     
• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 34, Nº 1, 2018, págs. 305-330
• Idioma: inglés
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• Resumen

  • We show that for uniform domains Ω⊆Rd+1 whose boundaries satisfy a certain nondegeneracy condition that harmonic measure cannot be mutually absolutely continuous with respect to α-dimensional Hausdorff measure unless α≤d. We employ a lemma that shows that, at almost every non-degenerate point, we may find a tangent measure of harmonic measure whose support is the boundary of yet another uniform domain whose harmonic measure resembles the tangent measure.