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

TL;DR

This paper explores the connections between trace inequalities, isocapacitary inequalities, and the regularity of complex Hessian and Monge-Ampère equations, providing a quantitative framework involving capacity minimizing functions.

Contribution

It offers a new quantitative characterization of the relationships between inequalities and regularity in complex Hessian equations using capacity minimizing functions.

Findings

Established links between trace and isocapacitary inequalities

Characterized regularity conditions for complex Hessian equations

Developed a framework involving capacity minimizing functions

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.