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}$$beaboundeddomain\mathrm{.}Foreach$x\in \Omega$, we choose a radius $0<\varrho (x)\le \mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(x,\mathrm{\partial }\mathrm{\Omega })$ and let $B_x$ be the closed ball centered at $x$ with radius $\varrho (x)$. If $\alpha \in $\mathbb{R}$$,considerthefollowingoperatorin$C( \overline{\Omega} )$,$$ $\mathcal{T}${\alpha}u(x)=\frac{\alpha}{2}\left(\sup{B_x } u+\inf_{B_x } u\right)+(1-\alpha),\frac{1}{\mu(B_x)}\int_{B_x}\hspace{-0.1cm} u\ d\mu. $$Underappropriateassumptionson$\alpha$,$$\mathbb{X}$$,$\mu$ and the radius function $\varrho$ we show that solutions $u\in C(\bar{\mathrm{\Omega }})$ of the functional equation ${\mathcal{T}}_{\alpha }u=u$ satisfy a local H"{o}lder or Lipschitz condition in $\mathrm{\Omega }$. The motivation comes from the so called $p$-harmonious functions in euclidean domains.