Infinity thick quasiconvexity and applications

We investigate geometric properties of a metric measure space where every function in the Newton--Sobolev space $N^{1,\infty}(Z)$ has a Lipschitz representative. We prove that when the metric space is locally complete and the reference measure is infinitesimally doubling, the above property is equivalent to the space being very $\infty$-thick quasiconvex up to a scale. That is, up to some scale, every pair of points can be joined by a family of quasiconvex curves that is not negligible for the $\infty$-modulus. As a first application, we prove a local-to-global improvement for the weak $(1,\infty)$-Poincar\'e inequality for locally complete quasiconvex metric spaces that have a doubling reference measure. As a second application, we apply our results to the existence and uniqueness of $\infty$-harmonic extensions with Lipschitz boundary data for precompact domains in a large class of metric measure spaces. As a final application, we illustrate that in the context of Sobolev extension sets, very $\infty$-thick quasiconvexity up to a scale plays an analogous role as local uniform quasiconvexity does in the Euclidean space. Our assumptions are adapted to the analysis of Sobolev extension sets and thus avoid stronger assumptions such as the doubling property of the measure. Examples satisfying our assumptions naturally occur as simplicial complexes, GCBA spaces, and metric quotients of Euclidean spaces.