The Hölder continuous subsolution theorem for complex Hessian equations

Abstract

Let $\Omega \Subset \mathbb C^n$ be a bounded strongly $m$-pseudoconvex domain ($1\leq m\leq n$) and $\mu$ a positive Borel measure with finite mass on $\Omega$. Then we solve the Holder continuous subsolution problem for the complex Hessian equation $(dd^c u)^m \wedge \beta^{n - m} = \mu$ on $\Omega$. Namely, we show that this equation admits a unique Holder continuous solution on $\Omega$ with a given Holder continuous boundary values if it admits a Holder continuous subsolution on $\Omega$. The main step in solving the problem is to establish a new capacity estimate showing that the $m$-Hessian measure of a Holder continuous $m$-subharmonic function on $\Omega$ with zero boundary values is dominated by the $m$-Hessian capacity with respect to $\Omega$ with an (explicit) exponent $\tau > 1$.