Second-order estimates for degenerate complex $k$-Hessian and Christoffel-Minkowski equations

It is known that the complex $k$-Hessian equation admits almost $C^{1,1}$ regularity (i.e., $\sup \mathrm{\Delta }u<\mathrm{\infty }$) and the Christoffel-Minkowski equation admits $C^{1,1}$ regularity under the sharp degenerate condition $f^{1\mathrm{/}(k-1)}\in C^{1,1}$ for a nonnegative right-hand side $f$. Assuming instead the alternative sharp degenerate condition $f^{3\mathrm{/}(2k-2)}\in C^{2,1}$, we prove almost $C^{1,1}$ regularity for the complex $k$-Hessian equation when $k\ge 5$ and $C^{1,1}$ regularity for the Christoffel-Minkowski equation. The argument deeply exploits various concavity properties of the operators under the stronger regularity assumption on $f$.