An interior second derivative bound for solutions of Hessian equations

In previous work we showed that weak solutions in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$W^{2,p}(\Omega)$\end{document} of the k-Hessian equation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$F_k[u]=g(x)$\end{document} have locally bounded second derivatives if g is positive and sufficiently smooth and p > kn/2. Here we improve this result to p > k(n-1)/2, which is known to be sharp in the Monge-Ampère case k=n > 2.