Absolutely continuous mappings on doubling metric measure spaces

We consider Q -absolutely continuous mappings f : X → V between a doubling metric measure space X and a Banach space V . The relation between these mappings and Sobolev mappings f ∈ N 1 , p ( X ; V ) for p ≥ Q ≥ 1 is investigated. In particular, a locally Q -absolutely continuous mapping on an Ahlfors Q -regular space is a continuous mapping in N loc 1 , Q ( X ; V ) , as well as differentiable almost everywhere in terms of Cheeger derivatives provided V satisfies the Radon-Nikodym property. Conversely, though a continuous Sobolev mapping f ∈ N loc 1 , Q ( X ; V ) is generally not locally Q -absolutely continuous, this implication holds if f is further assumed to be pseudomonotone. It follows that pseudomonotone mappings satisfying a relaxed quasiconformality condition are also Q -absolutely continuous.