Abstract
We show that every uniform domain of \({{{\mathbb {R}}}^n}\) with \(n\ge 2\) is a Morrey–Sobolev \({\mathscr {W}}^{1,\,p}\)-extension domain for all \(p\in [1,\,n)\), and moreover, that this result is essentially the best possible for each \(p\in [1,\,n)\) in the sense that, given a simply connected planar domain or a domain of \({{{\mathbb {R}}}^n}\) with \(n\ge 3\) that is quasiconformal equivalent to a uniform domain, if it is a \({\mathscr {W}}^{1,\,p} \)-extension domain, then it must be uniform.
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Acknowledgments
Pekka Koskela and Yi Ru-Ya Zhang were supported by the Academy of Finland Grant 120972; Yuan Zhou (corresponding author) was supported by National Natural Science Foundation of China ( No. 11522102).
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Dedicated to Professor Vladimir G. Maz’ya on his 75th birthday.
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Koskela, P., Zhang, Y.RY. & Zhou, Y. Morrey–Sobolev Extension Domains. J Geom Anal 27, 1413–1434 (2017). https://doi.org/10.1007/s12220-016-9724-9
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DOI: https://doi.org/10.1007/s12220-016-9724-9