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PreprintarXiv licencearXiv

On Two Questions by Brezis et al Concerning the Critical Difference Quotient Characterization of First-Order Sobolev Spaces

  • Yiqun Chen
  • Dachun Yang
  • Wen Yuan
  • Yangyang Zhang
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Abstract

Let N∈ℕ and γ∈[−1,0). In [Anal. PDE 17 (2024)], Brezis, Seeger, Van~Schaftingen, and Yung asked how, in the exceptional range γ∈[−1,0), BV̇(γ) and Ẇ1,1(γ) on ℝN are related to other function spaces, especially to Hardy--Sobolev spaces, and whether these spaces are normable. In this article, we prove that the homogeneous Hardy--Sobolev space is strictly embedded, respectively, into BV̇(γ) and Ẇ1,1(γ), and neither Ẇ1,1(γ)/ℝ nor BV̇(γ)/ℝ is normable, which answers the above two questions.

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Cite

BibTeXRIS
BibTeX
@misc{chen2026questions,
  title = {{On Two Questions by Brezis et al Concerning the Critical Difference Quotient Characterization of First-Order Sobolev Spaces}},
  author = {Yiqun Chen and Dachun Yang and Wen Yuan and Yangyang Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.19029},
  eprint = {2609.19029},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.19029},
}
RIS
TY  - UNPB
TI  - On Two Questions by Brezis et al Concerning the Critical Difference Quotient Characterization of First-Order Sobolev Spaces
AU  - Yiqun Chen
AU  - Dachun Yang
AU  - Wen Yuan
AU  - Yangyang Zhang
PY  - 2026
DA  - 2026/09/18
PB  - arXiv
DO  - 10.48550/arxiv.2609.19029
UR  - https://arxiv.org/abs/2609.19029
AB  - Let N∈ℕ and γ∈[−1,0). In [Anal. PDE 17 (2024)], Brezis, Seeger, Van~Schaftingen, and Yung asked how, in the exceptional range γ∈[−1,0), BV̇(γ) and Ẇ^1,1(γ) on ℝ^N are related to other function spaces, especially to Hardy--Sobolev spaces, and whether these spaces are normable. In this article, we prove that the homogeneous Hardy--Sobolev space is strictly embedded, respectively, into BV̇(γ) and Ẇ^1,1(γ), and neither Ẇ^1,1(γ)/ℝ nor BV̇(γ)/ℝ is normable, which answers the above two questions.
ER  -