Abstract
We study a certain improved fractional Sobolev–Poincaré inequality on domains, which can be considered as a fractional counterpart of the classical Sobolev–Poincaré inequality. We prove the equivalence of the corresponding weak and strong type inequalities; this leads to a simple proof of a strong type inequality on John domains. We also give necessary conditions for the validity of an improved fractional Sobolev–Poincaré inequality, in particular, we show that a domain of finite measure, satisfying this inequality and a ‘separation property’, is a John domain.
Funding Statement
L.I. and A.V.V. were supported by the Finnish Academy of Science and Letters, Vilho, Yrjö and Kalle Väisälä Foundation. B.D. was supported in part by NCN grant 2012/07/B/ST1/03356. The authors would like to thank the referee for a careful reading of the manuscript and for the comments.
Citation
Bartłomiej Dyda. Lizaveta Ihnatsyeva. Antti V. Vähäkangas. "On improved fractional Sobolev–Poincaré inequalities." Ark. Mat. 54 (2) 437 - 454, October 2016. https://doi.org/10.1007/s11512-015-0227-x
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