The Hölder continuous subsolution theorem for complex Hessian equations

Abstract

Let $\Omega ⋐{\mathbb{C}}^n$ be a bounded strongly $m$-pseudoconvex domain ($1\le m\le n$) and $\mu$ a positive Borel measure with finite mass on $\Omega$. Then we solve the Hölder continuous subsolution problem for the complex Hessian equation $(d{d}^{c}u{)}^m\wedge {\beta }^{n-m}=\mu$ on $\Omega$. Namely, we show that this equation admits a unique Hölder continuous solution on $\Omega$ with a given Hölder continuous boundary values if it admits a Hölder 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 Hölder 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 $\tau >1$.

This is a corrected version of a published paper where a new correct versions of Theorem B and Lemma 4.2 was provided