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PreprintarXiv licencearXiv

Another characterization of Ẇ1,p(ℝd)

  • Rupert L. Frank
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Abstract

We improve an earlier characterization of the Sobolev space Ẇ1,p(ℝd), 1<p<∞, in terms of the oscillations of a function on balls of varying centers and radii. In contrast to earlier work, finiteness of the supremum is replaced by the formally weaker assumption of finiteness of the liminf, thereby answering a question by Brezis.

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Cite

BibTeXRIS
BibTeX
@misc{frank2026another,
  title = {{Another characterization of $\dot W^1,p(\mathbb R^d)$}},
  author = {Rupert L. Frank},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.03426},
  eprint = {2610.03426},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.03426},
}
RIS
TY  - UNPB
TI  - Another characterization of Ẇ^1,p(ℝ^d)
AU  - Rupert L. Frank
PY  - 2026
DA  - 2026/10/02
PB  - arXiv
DO  - 10.48550/arxiv.2610.03426
UR  - https://arxiv.org/abs/2610.03426
AB  - We improve an earlier characterization of the Sobolev space Ẇ^1,p(ℝ^d), 1<p<∞, in terms of the oscillations of a function on balls of varying centers and radii. In contrast to earlier work, finiteness of the supremum is replaced by the formally weaker assumption of finiteness of the liminf, thereby answering a question by Brezis.
ER  -