A priori H\"older and Lipschitz regularity for generalized $p$-harmonious functions in metric measure spaces

Let $(\mathbb{X} , d, \mu )$ be a proper metric measure space and let $\Omega \subset \mathbb{X}$ be a bounded domain. For each $x\in \Omega$, we choose a radius $0< \varrho (x) \leq \mathrm{dist}(x, \partial \Omega )$ and let $B_x$ be the closed ball centered at $x$ with radius $\varrho (x)$. If $\alpha \in \mathbb{R}$, consider the following operator in $C( \overline{\Omega} )$, $$\mathcal{T}_{\alpha}u(x)=\frac{\alpha}{2}\left(\sup_{B_x } u+\inf_{B_x } u\right)+(1-\alpha)\,\frac{1}{\mu(B_x)}\int_{B_x}\hspace{-0.1cm} u\ d\mu.$$ Under appropriate assumptions on $\alpha$, $\mathbb{X}$, $\mu$ and the radius function $\varrho$ we show that solutions $u\in C( \overline{\Omega} )$ of the functional equation $\mathcal{T}_{\alpha}u = u$ satisfy a local H\"{o}lder or Lipschitz condition in $\Omega$. The motivation comes from the so called $p$-harmonious functions in euclidean domains.