An approach to metric space-valued Sobolev maps via weak* derivatives

Creutz, Paul; Evseev, Nikita

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An approach to metric space-valued Sobolev maps via weak* derivatives

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Creutz, Paul
Max Planck Institute for Mathematics, Max Planck Society;

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https://doi.org/10.48550/arXiv.2106.15449
(Preprint)

https://doi.org/10.1515/agms-2023-0107
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2106.15449.pdf
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Creutz-Evseev_An approach to metric space-valued Sobolev maps via weak derivatives_2024.pdf
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Citation

Creutz, P., & Evseev, N. (2024). An approach to metric space-valued Sobolev maps via weak* derivatives. Analysis and Geometry in Metric Spaces, 12(1): 20230107.

Cite as: https://hdl.handle.net/21.11116/0000-000F-76CD-6

Abstract

We give a characterization of metric space-valued Sobolev maps in terms of weak* derivatives. More precisely, we show that Sobolev maps with values in dual-to-separable Banach spaces can be defined in terms of classical weak derivatives in a weak* sense. Since every separable metric space X embeds isometrically into ℓ∞, we conclude that Sobolev maps with values in X can be characterized by postcomposition with such embedding and the mentioned weak gradients. A slight variation on our definition was proposed previously by Hajłasz and Tyson. However, we show that their definition does not work in the sense that for technical reasons the arising Sobolev space is essentially empty.