We show that p-harmonic functions in the plane satisfy a nonlinear asymptotic mean value property for p > 1. This extends previous results of Manfredi and Lindqvist for certain range of p's. div(|∇u| p−2 ∇u) = 0

When 1 < p < ∞ and p = 2 the p-harmonic measure on the boundary of the half plane R 2

Let (X, d, µ) be a proper metric measure space and let Ω ⊂ X be a bounded domain. For each x ∈ Ω, we choose a radius 0 < ̺(x) ≤ dist(x, ∂Ω) and let Bx be the closed ball centered at x with radius ̺(x). If α ∈ R, consider the following operator in C(Ω),Under appropriate assumptions on α, X, µ and the radius function ̺ we show that solutions u ∈ C(Ω) of the functional equation Tαu = u satisfy a local Hölder or Lipschitz condition in Ω. The motivation comes from the so called p-harmonious functions in euclidean domains.

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