H\"older continuous solutions to complex Hessian equations

We prove the H\"older continuity of the solution to complex Hessian equation with the right hand side in $L^p$, $p>\frac{n}{m}$, $1< m< n$, in a $m$-strongly pseudoconvex domain in $\mathbb{C}^n$ under some additional conditions on the density near the boundary and on the boundary data.Comment: 19 pages. Added Theorem 3.7: when the boundary is Holder continuous, there exists a Holder continuous $m$-sh extension to the domai

Subsolution theorem for the complex Hessian equation

We prove the subsolution theorem for the complex Hessian equations in a smoothly bounded strongly $m$-pseudoconvex domain, $1 < m < n$, in \bC^n.Comment: 18 pages. Corrected typos. To appear in Universitatis Iagellonicae Acta Mathematic

Stability and regularity of solutions of the Monge-Amp\`ere equation on Hermitian manifolds

We prove stability of solutions of the complex Monge-Amp\`ere equation on compact Hermitian manifolds, when the right hand side varies in a bounded set in $L^p, p>1$ and it is bounded away from zero. Such solutions are shown to be H\"older continuous. As an application we extend a recent result of Sz\'ekelyhidi and Tosatti on K\"ahler-Einstein equation from K\"ahler to Hermitian manifolds.Comment: v4 final version incorporated the referee repor