Pogorelov interior estimates for sum-of-Hessians equations

Abstract

In this paper, we develop a new approach to sum-of-Hessians equations involving multiple $k$-Hessian operators of distinct orders. By exploiting the concavity of sums of Hessian operators, we derive Pogorelov estimates for the corresponding equations under the dynamic semi-convexity condition. Moreover, when the highest order is $n-1$ or $n$, we establish such estimates for admissible solutions without relying on this condition, and they are therefore optimal. As an application, when the right-hand side is identically $1$, we prove that every entire admissible solution in ${\mathbb{R}}^n$ with quadratic growth is necessarily a quadratic polynomial.

32 pages, comments welcome