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.
"On improved fractional Sobolev–Poincaré inequalities." Ark. Mat. 54 (2) 437 - 454, October 2016. https://doi.org/10.1007/s11512-015-0227-x