Families of Monge-Ampere measures with Holder continuous potentials

TL;DR

This paper proves that solutions to complex Monge-Ampere equations on compact Kahler manifolds are Holder continuous when the measures involved have Holder continuous superpotentials with uniform exponents and constants.

Contribution

It establishes the Holder continuity of solutions for a family of Monge-Ampere equations with measures having uniformly Holder continuous superpotentials.

Findings

Solutions are Holder continuous under specified conditions.

Uniform Holder continuity of measures implies Holder continuity of solutions.

Extends regularity results for Monge-Ampere equations on Kahler manifolds.

Abstract

Let X be a compact Kahler manifold. Let F be a family of probability measures on X whose superpotentials are Holder continuous with uniform Holder exponent and Holder constant. We prove that the corresponding family of solutions of the complex Monge-Ampere equations whose right-hand sides are measures in F is Holder continuous.