Fractional Sobolev-Poincare Inequalities in Irregular Domains

This paper is devoted to the study of fractional (q,p)-Sobolev-Poincare inequalities in irregular domains. In particular, the author establishes (essentially) sharp fractional (q, p)-Sobolev-Poincare inequalities in 8-John domains and in domains satisfying the quasihyperbolic boundary conditions. When the order of the fractional derivative tends to 1, our results tend to the results for the usual derivatives. Furthermore, the author verifies that those domains which support the fractional (q,p)-Sobolev-Poincare inequalities together with a separation property are s-diam John domains for certain s, depending only on the associated data. An inaccurate statement in [Buckley, S. and Koskela, P., Sobolev-Poincare implies John, Math. Res. Lett., 2(5), 1995, 577-593] is also pointed out.