Trace Inequalities, isocapacitary inequalities and regularity of the complex Hessian equations

Abstract: In this paper, we study the relations between trace inequalities(Sobolev and Moser-Trudinger type), isocapacitary inequalities and the regularity of the complex Hessian and Monge-Amp`ere equations with respect to a general positive Borel measure. We obtain a quantitative characterization for these relations through properties of the capacity minimizing functions.