Modulus of continuity of solutions to complex Hessian equations

Abstract

We give a sharp estimate of the modulus of continuity of the solution to the Dirichlet problem for the complex Hessian equation of order $m$ ($1\le m\le n$) with a continuous right hand side and a continuous boundary data in a bounded strongly $m$-pseudoconvex domain $\Om \Subset \C^n$. Moreover when the right hand side is in $L^p(\Om) $, for some $p>n/m$ and the boundary value function is $C^{1,1}$ we prove that the solution is Hölder continuous.

We prove the Hölder continuity of the solution when the right hand side is in $L^p$ for $p>1$ without any condition on the growth near the boundary