Absolutely minimal extensions of functions on metric spaces

Extensions of a real-valued function from the boundary $\partial X_0$ of an open subset $X_0$ of a metric space ${(X,d)}$ to $X_0$ are discussed. For the broad class of initial data coming under discussion (linearly bounded functions) locally Lipschitz extensions to $X_0$ that preserve localized moduli of continuity are constructed. In the set of these extensions an absolutely minimal extension is selected, which was considered before by Aronsson for Lipschitz initial functions in the case $X_0\subset\mathbb R^n$. An absolutely minimal extension can be regarded as an $\infty$-harmonic function, that is, a limit of $p$-harmonic functions as $p\to+\infty$. The proof of the existence of absolutely minimal extensions in a metric space with intrinsic metric is carried out by the Perron method. To this end, $\infty$-subharmonic, $\infty$-superharmonic, and $\infty$-harmonic functions on a metric space are defined and their properties are established.