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Extension properties of planar subsets supporting a weak (1,1)-Poincaré inequality

  • Miguel García-Bravo
  • Tapio Rajala
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Abstract

We study Sobolev and BV-extension properties of planar subsets. In particular, we prove that fat Sierpiński carpets that support a weak (1,1)-Poincaré inequality are W1,1-extension sets, and we provide an explicit linear extension operator for them. We also show that for planar domains satisfying a weak (1,1)-Poincaré inequality the W1,1-extension property and the BV-extension property are equivalent. Finally, as a consequence of the previous result, we show that if a simply-connected planar domain is Ahlfors regular and supports a weak (1,1)-Poincaré inequality, then it is a (linear) W1,1-extension domain.

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Cite

BibTeXRIS
BibTeX
@misc{garciabravo2026extension,
  title = {{Extension properties of planar subsets supporting a weak $(1,1)$-Poincaré inequality}},
  author = {Miguel García-Bravo and Tapio Rajala},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.07790},
  eprint = {2609.07790},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.07790},
}
RIS
TY  - UNPB
TI  - Extension properties of planar subsets supporting a weak (1,1)-Poincaré inequality
AU  - Miguel García-Bravo
AU  - Tapio Rajala
PY  - 2026
DA  - 2026/09/07
PB  - arXiv
DO  - 10.48550/arxiv.2609.07790
UR  - https://arxiv.org/abs/2609.07790
AB  - We study Sobolev and BV-extension properties of planar subsets. In particular, we prove that fat Sierpiński carpets that support a weak (1,1)-Poincaré inequality are W^1,1-extension sets, and we provide an explicit linear extension operator for them. We also show that for planar domains satisfying a weak (1,1)-Poincaré inequality the W^1,1-extension property and the BV-extension property are equivalent. Finally, as a consequence of the previous result, we show that if a simply-connected planar domain is Ahlfors regular and supports a weak (1,1)-Poincaré inequality, then it is a (linear) W^1,1-extension domain.
ER  -