Absolutely continuous mappings on doubling metric measure spaces

2021/09/28 by Panu Lahti, Lahti, Panu, Xiaodan Zhou +1
Mathematics · #26B30 #30L10 #46E36 #Analytic and geometric function theory #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2109.13615

openalex publication_date 2021/09/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Following Malý's definition of absolutely continuous functions of several variables, we consider Q-absolutely continuous mappings f\colon 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 is investigated. In particular, a locally Q-absolutely continuous mapping on an Ahlfors Q-regular space is a continuous mapping in N^1,Q_\rmloc(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^1,Q_\rmloc(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.

Citations

• Quasiconformal maps in metric spaces with controlled geometry
• The Poincaré inequality is an open ended condition

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