Modulus of continuity of solutions to complex Hessian equations (Q2790322)

It is known that the Dirichlet problem for the complex Hessian equation in a strongly \(m\)-pseudoconvex domain in \(\mathbb C ^n\), with \(C^{1,1}\) boundary data and the density in \(L^p\), \(p>n/m\), has a continuous solution. The bound for \(p\) is optimal. In a recent paper, \textit{N. C. Nguyen} [Potential Anal. 41, No. 3, 887--902 (2014; Zbl 1302.32034)] proved Hölder continuity of those solutions with an explicit exponent, but assuming a controlled growth of the right hand side near the boundary. In the present paper the author is able to remove this extra assumption. For \(p>2m/n\) he obtains also a better Hölder exponent. On the way, a sharp result on the modulus of continuity of the solution, when the right hand side is continuous, is shown. The proof uses a rather delicate barrier construction.