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Existence and uniqueness of ∞ -harmonic functions under assumption of ∞ -Poincaré inequality

Given a complete metric measure space whose measure is doubling and supports an ∞ -Poincaré inequality, and a bounded domain (cid:2) in such a space together with a Lipschitz function f : ∂(cid:2) → R , we show the existence and uniqueness of an ∞ -harmonic extension of f to (cid:2) . To do so, we show that there is a metric that is bi-Lipschitz equivalent to the original metric, such that with respect to this new metric the metric space satisﬁes an ∞ -weak Fubini property and that a function which is ∞ -harmonic in the original metric must also be ∞ -harmonic with respect to the new metric. We also show that if the metric on the metric space satisﬁes an ∞ -weak Fubini property, then the notion of ∞ -harmonic functions coincide with the notion of AMLEs proposed by Aronsson. The notion of ∞ -harmonicity is in general distinct from the notion of strongly absolutely minimizing Lipschitz extensions found in Crandall et al. (Calc Var Partial Differ Equ 13: 123–139, 2001), Juutinen (Ann Acad Sci Fenn Math 27(1):57–67, 2002), Juutinen and Shanmugalingam (Math Nachr 279(9–10):1083– 1098, 2006), but coincides when the metric space supports a p -Poincaré inequality for some ﬁnite p ≥ 1.