SOBOLEV INTERPOLATION INEQUALITIES ON GENERALIZED JOHN DOMAINS

We obtain weighted Sobolev interpolation inequalities on generalized John domains that include John domains (bounded or unbounded) for delta-doubling measures satisfying a weighted Poincare inequality. These measures include ones arising from power weights d(x, partial derivative Omega)(alpha) 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 Omega(1) x Omega(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))(alpha) dist(y, G(2))(beta), where G(1) subset of partial derivative Omega(1) and G(2) subset of partial derivative Omega(2). For certain cases, we obtain sharp conditions.