Sobolev interpolation inequalities on generalized John domains

Abstract

We obtain weighted Sobolev interpolation inequalities on generalized John domains that include John domains (bounded or unbounded) for δ-doubling measures satisfying a weighted Poincaré inequality. These measures include ones arising from power weights d(χ, ∂Ω)α and need not be dou- bling. As an application, we extend the Sobolev interpolation inequalities obtained by Caffarelli, Kohn and Nirenberg. We extend these inequalities to product spaces and give some applications on products R{double-struck}M× RMΩ2 of John domains for Ap(R{double-struck}n × Rm) weights and power weights of the type w(χ, y) = dist(χ, G1)α dist(y, G2)β, where G1 ⊂ ∂Ω1 and G2 ⊂ ∂-1.Ω2. For certain cases, we obtain sharp conditions.