Interior Hessian estimates for k-convex solutions of Hessian quotient equations with 2k>n

Abstract

We establish interior Hessian estimates for $k$ convex solutions of $\frac{{\sigma }_k(D^{2}u)}{{\sigma }_l(D^{2}u)}=1$, where $1\le l<k<n$, $k-l\in \{1,2\}$, and $2k>n$.The proof combines a concavity inequality with a pointwise doubling argument.

15 pages