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 Poincare inequality. These measures include ones arising from power weights d(x, ∂Ω) α and need not be doubling. 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 Ω 1 × Ω 2 of John domains for A p (ℝ n x ℝ m ) weights and power weights of the type w(x, y) = dist(x, G 1 ) α dist(y, G 2 ) β , where G 1 C ∂Ω 1 and G 2 C ∂Ω 2 . For certain cases, we obtain sharp conditions.