Regularity of degenerate Hessian equations

We show a second order a priori estimate for solutions to the complex k-Hessian equation on a compact Kahler manifold provided the \((k-1)\)st root of the right hand side is \(\mathcal C^{1,1}\). This improves an estimate of Hou–Ma–Wu (Math Res Lett 17:547–561, 2010). An example is provided to show that the exponent is sharp.