The complex Sobolev Space and H\"older continuous solutions to Monge-Amp\`ere equations

TL;DR

This paper establishes a characterization of H"older continuous solutions to complex Monge-Amp ext`ere equations on compact K"ahler manifolds and domains in ^n, linking solution regularity to measure regularity in a Sobolev space.

Contribution

It provides a necessary and sufficient condition for H"older continuity of solutions based on the measure's H"older continuity as a functional on a complex Sobolev space.

Findings

H"older continuous solutions exist iff the measure is H"older continuous as a functional.

Results extend to Monge-Amp ext`ere equations on domains in ^n.

Characterizes the regularity of solutions via measure properties.

Abstract

Let $X$ be a compact K\"ahler manifold of dimension $n$ and $\omega$ a K\"ahler form on $X$. We consider the complex Monge-Amp\`ere equation $(dd^c u+\omega)^n=\mu$, where $\mu$ is a given positive measure on $X$ of suitable mass and $u$ is an $\omega$-plurisubharmonic function. We show that the equation admits a H\"older continuous solution {\it if and only if} the measure $\mu$, seen as a functional on a complex Sobolev space $W^*(X)$, is H\"older continuous. A similar result is also obtained for the complex Monge-Amp\`ere equations on domains of $\mathbb{C}^n$.