The Continuous Subsolution Problem for Complex Hessian Equations

TL;DR

This paper investigates conditions under which continuous solutions exist for the complex Hessian equation on certain domains, providing new capacity estimates and stability results that extend the understanding of the equation's solvability.

Contribution

It introduces a sufficient condition based on the measure's modulus of diffusion for the existence of continuous solutions, and establishes new capacity and stability estimates for the complex Hessian equation.

Findings

Existence of continuous solutions under a modulus of diffusion condition.

A new capacity estimate relating the measure's diffusion to the solution's modulus of continuity.

A weak stability estimate for the $m$-Hessian measure of continuous $m$-subharmonic functions.

Abstract

Let $\Omega \subset \mathbb C^n$ be a bounded strictly $m$-pseudoconvex domain ($1\leq m\leq n$) and $\mu$ a positive Borel measure on $\Omega$. We study the Dirichlet problem for the complex Hessian equation $(dd^c u)^m \wedge \beta^{n - m} = \mu$ on $\Omega$. First we give a sufficient condition on the "modulus of diffusion" of the measure $\mu$ with respect to the $m$-Hessian capacity which guarantees the existence of a continuous solution to the associated Dirichlet problem with a continuous boundary datum. As an application, we prove that if the equation has a continuous $m$-subharmonic subsolution whose modulus of continuity satisfies a Dini type condition, then the equation has a continuous solution with an arbitrary continuous boundary datum. Moreover when the measure has a finite mass on $\Omega$, we give a precise quantitative estimate on the modulus of continuity of the solution. One of the main steps in our proof is to establish a new capacity estimate providing a precise estimate of the modulus of diffusion of the $m$-Hessian measure of a continuous $m$-subharmonic function $\varphi$ in $\Omega$ with zero boundary with respect to the $m$-Hessian capacity in terms of the modulus of continuity of $\varphi$. Another important ingredient is a new weak stability estimate for the $m$-Hessian measure of a continuous $m$-subharmonic function in $\Omega$.