Absolutely continuous mappings on doubling metric measure spaces

Abstract

We consider Q-absolutely continuous mappings \(f:X\rightarrow V\) between a doubling metric measure space X and a Banach space V. The relation between these mappings and Sobolev mappings \(f\in N^{1,p}(X;V)\) for \(p\ge Q\ge 1\) is investigated. In particular, a locally Q-absolutely continuous mapping on an Ahlfors Q-regular space is a continuous mapping in \(N^{1,Q}_\textrm{loc}\,(X;V)\), as well as differentiable almost everywhere in terms of Cheeger derivatives provided V satisfies the Radon-Nikodym property. Conversely, though a continuous Sobolev mapping \(f\in N^{1,Q}_\textrm{loc}\,(X;V)\) 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.