*Preprint*

**Inserted:** 20 jul 2016

**Last Updated:** 27 feb 2018

**Year:** 2016

**Abstract:**

Given a complete metric measure space whose measure is doubling and supports an $\infty$-Poincaré inequality, and a bounded domain $\Omega$ in such a space together with a Lipschitz function $f:\partial\Omega\to\mathbb{R}$ we show the existence and uniqueness of an $\infty$-harmonic extension of $f$ to $\Omega$. We also show that in the event that the metric on the metric space has an $\infty$-weak Fubini property, the notion of $\infty$-harmonic functions coincide with the notion of AMLEs proposed by Aronsson. As an auxiliary tool we show that given that the measure on the metric space is doubling and supports an $\infty$-Poincar\'e inequality, one can construct a metric bi-Lipschitz equivalent to the original one, with respect to which the metric space has an $\infty$-weak Fubini property. The notion of $\infty$-harmonicity is in general distinct from the notion of strongly absolutely minimizing Lipschitz extensions found in Crandall-Evans-Gariepy,Juutinen,Juutinen-Shanmugalingam, but coincides when the metric space supports a $p$-Poincaré inequality for some finite $p\ge1$.

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