A priori Hölder and Lipschitz regularity for generalized p-harmonious functions in metric measure spaces

Abstract

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 < ϱ(x) ≤ dist(x, ∂Ω) and let Bx be the closed ball centered at x with radius ϱ(x). If α ∈ R, consider the following operator in C(Ω), Tαu(x) = α 2 (sup Bx u + inf Bx u) + 1 – α μ(Bx) ∫ Bx u dμ. Under appropriate assumptions on α, X, μ and the radius function ϱ 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.