On improved fractional Sobolev-Poincar, inequalities

We study a certain improved fractional Sobolev-Poincar, inequality on domains, which can be considered as a fractional counterpart of the classical Sobolev-Poincar, 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-Poincar, inequality, in particular, we show that a domain of finite measure, satisfying this inequality and a 'separation property', is a John domain.