The H\"older continuous subsolution theorem for complex Hessian equations

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