The Continuous Subsolution Problem for Complex Hessian Equations

Abstract

Let $\Omega \subset {\mathbb{C}}^n$ be a bounded strictly $m$-pseudoconvex domain ($1\le m\le n$) and $\mu$ a positive Borel measure on $\Omega$. We study the Dirichlet problem for the complex Hessian equation $(d{d}^{c}u{)}^m\wedge {\beta }^{n-m}=\mu$ 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 $\phi$ in $\Omega$ with zero boundary with respect to the $m$-Hessian capacity in terms of the modulus of continuity of $\phi$. Another important ingredient is a new weak stability estimate for the $m$-Hessian measure of a continuous $m$-subharmonic function in $\Omega$.

This is the final version accepted for publication in IUMJ. arXiv admin note: text overlap with arXiv:2004.06952

References in corpus (4)

• The smoothing of $m$-subharmonic functions
• The Hölder continuous subsolution theorem for complex Hessian equations
• A remark on the continuous subsolution problem for the complex Monge-Ampère equation
• Remarks on the modulus of continuity of subharmonic functions