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A New insight into the Ambrosio--Reshetnyak approach to Sobolev maps

  • Alexander Tyulenev
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Abstract

Given p ∈ (1,∞), let X=(X,ρ,μ) be a metric measure space such that the measure μ is uniformly locally doubling and X supports a weak local (1,p)-Poincaré inequality. Let (Y,d,underliney) be a complete pointed metric space. We prove that the equivalence class of a Borel map $ u:\operatorname{X} \to \operatorname{Y}$ (modulo coincidence μ-a.e.) belongs to the Ambrosio--Reshetnyak--Sobolev class Wp1(X,Y) if and only if h ∘ u belongs to the Sobolev space Wp1(X) for every $1$-Lipschitz function h:Y → ℝ.

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Cite

BibTeXRIS
BibTeX
@misc{tyulenev2026insight,
  title = {{A New insight into the Ambrosio--Reshetnyak approach to Sobolev maps}},
  author = {Alexander Tyulenev},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.04384},
  eprint = {2610.04384},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.04384},
}
RIS
TY  - UNPB
TI  - A New insight into the Ambrosio--Reshetnyak approach to Sobolev maps
AU  - Alexander Tyulenev
PY  - 2026
DA  - 2026/10/12
PB  - arXiv
DO  - 10.48550/arxiv.2610.04384
UR  - https://arxiv.org/abs/2610.04384
AB  - Given p ∈ (1,∞), let X=(X,ρ,μ) be a metric measure space such that the measure μ is uniformly locally doubling and X supports a weak local (1,p)-Poincaré inequality. Let (Y,d,underliney) be a complete pointed metric space. We prove that the equivalence class of a Borel map $ u:\operatorname{X} \to \operatorname{Y}$ (modulo coincidence μ-a.e.) belongs to the Ambrosio--Reshetnyak--Sobolev class W_p^1(X,Y) if and only if h ∘ u belongs to the Sobolev space W_p^1(X) for every $1$-Lipschitz function h:Y → ℝ.
ER  -