On improved fractional Sobolev-Poincar\'e inequalities

We prove a certain improved fractional Sobolev-Poincar\'e inequality on John domains; the proof is based on the equivalence of the corresponding weak and strong type inequalities. We also give necessary conditions for the validity of an improved fractional Sobolev-Poincar\'e inequality, in particular, we show that a domain having a finite measure and satisfying this inequality, and a `separation property', is a John domain.