Carathéodory-type extension theorem with respect to prime end boundaries

Abstract

We prove a Carathéodory-type extension of BQS homeomorphisms between two domains in proper, locally path-connected metric spaces as homeomorphisms between their prime end closures. We also give a Carathéodory-type extension of geometric quasiconformal mappings between two such domains provided the two domains are both Ahlfors $Q$-regular and support a $Q$-Poincaré inequality when equipped with their respective Mazurkiewicz metrics. We also provide examples to demonstrate the strengths and weaknesses of prime end closures in this context.

43 pages