Hausdorff dimension and σ finiteness of p harmonic measures in space when p ≥ n

In this paper we study a measure, it, associated with a positive p harmonic function (u) over cap defined in an open set O subset of R-n and vanishing on a portion Gamma of partial derivative O. If p > n we show it is concentrated on a set of sigma finite Hn-1 measure while if p = n the same conclusion holds provided Gamma is uniformly fat in the sense of n capacity. Our work nearly answers in the affirmative a conjecture in Lewis (2015) and also appears to be the natural extension of Jones and Wolff (1988), Wolff (1993), to higher dimensions. (C) 2015 Elsevier Ltd. All rights reserved.