Arkiv för Matematik
- Ark. Mat.
- Volume 54, Number 2 (2016), 437-454.
On improved fractional Sobolev–Poincaré inequalities
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.
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.
Ark. Mat. Volume 54, Number 2 (2016), 437-454.
Received: 24 May 2014
Revised: 21 September 2015
First available in Project Euclid: 30 January 2017
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Dyda, Bartłomiej; Ihnatsyeva, Lizaveta; Vähäkangas, Antti V. On improved fractional Sobolev–Poincaré inequalities. Ark. Mat. 54 (2016), no. 2, 437--454. doi:10.1007/s11512-015-0227-x. https://projecteuclid.org/euclid.afm/1485802743