$C^{1,α}$ and $W^{2,1+\varepsilon}$ regularity for the $2$-Hessian equation

In this paper, for every $\varepsilon>0$, we establish interior and global a priori $C^{1,α}$ estimates for some $α\in(0,1)$ from $W^{2,1+\varepsilon}$ bounds for bounded smooth $2$-admissible solutions of the $2$-Hessian equation, with the interior estimate requiring only an upper bound on the density. Under positive upper and lower density bounds, we prove global $W^{2,1+\varepsilon}$ estimates for some $\varepsilon>0$ on bounded strictly mean convex $C^{2,β}$ domains with $C^{2,β}$ boundary values. For densities bounded above and below by positive constants, the interior $C^{1,α}$ estimate follows by combining the first result with an interior $W^{2,1+\varepsilon}$ estimate obtained by a slight modification of an existing argument. Combining the first result with the global $W^{2,1+\varepsilon}$ estimate yields the global $C^{1,α}$ estimate. Known examples show that the $C^{2,β}$ regularity assumptions on the boundary and boundary values are optimal in the Hölder scale for both the global $W^{2,1+\varepsilon}$ estimate and the global $C^{1,α}$ estimate.