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

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)=α2supBxu+infBxu+1−αμ(Bx)∫Bxudμ.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.