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arXiv · 2609.07790

Extension properties of planar subsets supporting a weak $(1,1)$-Poincar\'e 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 Sierpi\'nski carpets that support a weak $(1,1)$-Poincar\'e 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 $(1,1)$-Poincar\'e 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 $(1,1)$-Poincar\'e inequality, then it is a (linear) $W^{1,1}$-extension domain.

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Miguel García-Bravo, Tapio Rajala. 2026-09-07. Extension properties of planar subsets supporting a weak $(1,1)$-Poincar\'e inequality. https://arxiv.org/abs/2609.07790

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