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Absolutely continuous mappings on doubling metric measure spaces

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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.

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Acknowledgements

We are grateful to the referee for the careful reading of the paper and for the comments and suggestions which helped to improve the exposition of the paper. The work of the second author was supported by JSPS Grant-in-Aid for Early-Career Scientists (No. 22K13947).

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Authors and Affiliations

  1. Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, People’s Republic of China

    Panu Lahti

  2. Analysis on Metric Spaces Unit, Okinawa Institute of Science and Technology, 1919-1 Tancha, Onna-son, Kunigami-gun, Okinawa, 904-0495, Japan

    Xiaodan Zhou

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  1. Panu Lahti
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Correspondence to Panu Lahti.

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Lahti, P., Zhou, X. Absolutely continuous mappings on doubling metric measure spaces. manuscripta math. 173, 1–21 (2024). https://doi.org/10.1007/s00229-023-01460-z

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