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A purely metric characterization of supercritical Sobolev spaces

  • Ryan Alvarado
  • Efstathios Konstantinos Chrontsios Garitsis
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Abstract

In this paper we provide a purely metric and derivative-free characterization of continuous mappings in the local supercritical Sobolev space. More specifically, we show that a continuous mapping f:ℝn→ℝm lies in the local Sobolev space W1,ploc(ℝn:ℝm) if and only if f is a $(p,1-n/p)$-compactly Hölder mapping, where p>n. In fact, we prove the following more general result: whenever X is a complete Q-Ahlfors regular metric space supporting a p-Poincaré inequality with p>Q, and V is any Banach space, the compactly Hölder class CHp,1−Q/p(X: V) consists precisely of the continuous representatives of the local Hajłasz and Newtonian Sobolev spaces M1,ploc (X : V) and N1,ploc(X: V). We also establish quantitative comparisons between the corresponding seminorms.

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Cite

BibTeXRIS
BibTeX
@misc{alvarado2026purely,
  title = {{A purely metric characterization of supercritical Sobolev spaces}},
  author = {Ryan Alvarado and Efstathios Konstantinos Chrontsios Garitsis},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.24520},
  eprint = {2609.24520},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.24520},
}
RIS
TY  - UNPB
TI  - A purely metric characterization of supercritical Sobolev spaces
AU  - Ryan Alvarado
AU  - Efstathios Konstantinos Chrontsios Garitsis
PY  - 2026
DA  - 2026/09/21
PB  - arXiv
DO  - 10.48550/arxiv.2609.24520
UR  - https://arxiv.org/abs/2609.24520
AB  - In this paper we provide a purely metric and derivative-free characterization of continuous mappings in the local supercritical Sobolev space. More specifically, we show that a continuous mapping f:ℝ^n→ℝ^m lies in the local Sobolev space W^1,p_loc(ℝ^n:ℝ^m) if and only if f is a $(p,1-n/p)$-compactly Hölder mapping, where p>n. In fact, we prove the following more general result: whenever X is a complete Q-Ahlfors regular metric space supporting a p-Poincaré inequality with p>Q, and V is any Banach space, the compactly Hölder class CH^p,1−Q/p(X: V) consists precisely of the continuous representatives of the local Hajłasz and Newtonian Sobolev spaces M^1,p_loc (X : V) and N^1,p_loc(X: V). We also establish quantitative comparisons between the corresponding seminorms.
ER  -