On fractional Poincar\'e inequalities
  We show that fractional (p,p)-Poincar\'e inequalities and even fractional Sobolev-Poincar\'e inequalities hold for bounded John domains, and especially for bounded Lipschitz domains. We also prove sharp fractional (1,p)-Poincar\'e inequalities for s-John domains. 
Ritva Hurri-Syrj\"anen and Antti V. V\"ah\"akangas
math.FA