  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.

Classical Analysis and ODEs