Hessian Estimates for the Sigma-2 Equation with Variable Right-Hand Side Terms in Dimension 4

We derive a priori interior Hessian estimates and regularity for the sigma-2 Hessian equation ${\sigma }_2(D^{2}u)=f(x,u,Du)$ with positive $C^{1,1}$ right hand side in dimension 4. In higher dimensions, the same result holds under an additional dynamic semi-convexity condition on solutions. This generalizes Qiu’s and Shankar-Yuan’s results.