Harmonic Measure in Simply Connected Domains Revisited

Let Ω be a bounded simply connected domain in the complex plane, C. Let N be a neighborhood of ∂Ω, let p be fixed, 1 < p < ∞, and let û be a positive weak solution to the p Laplace equation in Ω ∩N. Assume that û has zero boundary values on ∂Ω in the Sobolev sense and extend û to N \ Ω by putting û ≡ 0 on N \ Ω. Then there exists a positive finite Borel measure μ̂ on C with support contained in ∂Ω and such that ∫ |∇û|p−2 〈∇û,∇φ〉 dA = − ∫ φdμ̂ whenever φ ∈ C∞ 0 (N). In this paper we continue our studies in [BL05], [L06], [LNP11], [LNV] by establishing endpoint type results for the Hausdorff dimension of this measure in simply connected domains. Our results are similar to the well known result of Makarov [M85] concerning harmonic measure in simply connected domains. 2000 Mathematics Subject Classification. Primary 35J25, 35J70.