Interior $C^2$ estimates for a class of sum Hessian equation

TL;DR

This paper establishes interior second derivative estimates for a class of sum Hessian equations, including Pogorelov type estimates, advancing the understanding of regularity in these nonlinear PDEs.

Contribution

The paper provides new interior $C^2$ estimates and Pogorelov type estimates for sum Hessian equations, including a weaker version when $k=n$, filling gaps in regularity theory.

Findings

Established interior $C^2$ estimates for sum Hessian equations.

Derived Pogorelov type estimates for $0<k<n$.

Obtained weaker Pogorelov estimates for the case $k=n$.

Abstract

In this paper, we mainly study the interior $C^2$ estimates for a class of sum Hessian equations. We establish the interior estimates and the Pogorelov type estimates for $0<k<n$. If $k=n$, we derive a weaker Pogorelov type estimates.