Hausdorff dimension and σσ finiteness of pp harmonic measures in space when p≥np≥n

In this paper we study a measure, μˆ, associated with a positive pp harmonic function uˆ defined in an open set O⊂RnO⊂Rn and vanishing on a portion ΓΓ of ∂O∂O. If p>np>n we show μˆ is concentrated on a set of σσ finite Hn−1Hn−1 measure while if p=np=n the same conclusion holds provided ΓΓ is uniformly fat in the sense of nn 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.