The Hölder continuous subsolution theorem for complex Hessian equations

Abstract

Let $Ω\Subset \mathbb C^n$ be a bounded strongly $m$-pseudoconvex domain ($1\leq m\leq n$) and $μ$ a positive Borel measure with finite mass on $Ω$. Then we solve the Hölder continuous subsolution problem for the complex Hessian equation $(dd^c u)^m \wedge β^{n - m} = μ$ on $Ω$. Namely, we show that this equation admits a unique Hölder continuous solution on $Ω$ with a given Hölder continuous boundary values if it admits a Hölder continuous subsolution on $Ω$. 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 $Ω$ with zero boundary values is dominated by the $m$-Hessian capacity with respect to $Ω$ with an (explicit) exponent $τ> 1$.