Doubling Argument of the Hessian Estimate for the Hessian Quotient Equations
In this paper, we establish a doubling argument to obtain Hessian estimates for convex solutions to the Hessian quotient equation $\frac{\sigma_n}{\sigma_k}(D^2u) = f(x,u,Du)$ for $k=n-1$ and $k=n-2$ under the condition that $1/f$ is concave in the $Du$ variable. In particular, our approach is pointwise and does not make use of the Legendre transform or integral-based local maximum principles. We provide a counterexample demonstrating that interior estimates can fail if no structural assumption is imposed on $f$ in the $Du$ variable. Finally, we extend our doubling argument to general Hessian quotient equations $\frac{\sigma_l}{\sigma_k}(D^2u) = f(x,u,Du)$ for $k \in \{l-1, l-2\}$, under a similar structural condition imposed on $f$ in the $Du$ variable, alongside an additional structural concavity assumption on the operator introduced by Lu-Tsai 2026.