Regularity of Degenerate Hessian Equation

We show a second order a priori estimate for solutions to the complex $k$-Hessian equation on a compact Kähler 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. An example is provided to show that the exponent is sharp.