Year: 2026
Type: preprint
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 $W^{1,1}$-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.
Source: arXiv (Cornell University)
Authors: García-Bravo, Miguel, Rajala, Tapio
Institutions: Universidad Complutense de Madrid, University of Jyväskylä
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G1747273737, G8920660035
OpenAlex ID: W7212009608