Absolutely continuous mappings on doubling metric measure spaces

TL;DR

This paper explores the relationship between absolutely continuous mappings and Sobolev mappings on doubling metric measure spaces, establishing conditions under which these classes coincide or imply each other, especially involving differentiability and pseudomonotonicity.

Contribution

It characterizes when $Q$-absolutely continuous mappings are Sobolev and vice versa, introducing new links between absolute continuity, differentiability, and pseudomonotonicity in metric measure spaces.

Findings

Locally $Q$-absolutely continuous mappings are in $N^{1,Q}_{ m{loc}}(X;V)$ and differentiable almost everywhere.

Continuous Sobolev mappings are not necessarily $Q$-absolutely continuous unless pseudomonotone.

Pseudomonotone mappings with relaxed quasiconformality are $Q$-absolutely continuous.

Abstract

Following Mal\'y's definition of absolutely continuous functions of several variables, we consider $Q$-absolutely continuous mappings $f\colon X\to V$ 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 $p\ge Q$ is investigated. In particular, a locally $Q$-absolutely continuous mapping on an Ahlfors $Q$-regular space is a continuous mapping in $N^{1,Q}_{\rm{loc}}(X;V)$, as well as differentiable almost everywhere in terms of Cheeger derivatives provided $V$ satisfies the Radon-Nikodym property. Conversely, though a continuous Sobolev mapping $f\in N^{1,Q}_{\rm{loc}}(X;V)$ 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.