$\mathrm{C}^2$ estimates for general $p$-Hessian equations on closed Riemannian manifolds

We study the $\mathrm{C}^2$ estimates for $p$-Hessian equations with general left-hand and right-hand terms on closed Riemannian manifolds of dimension $n$. To overcome the constraints of closed manifolds, we advance a new kind of "subsolution", called pseudo-solution, which generalizes "$\mathcal{C}$-subsolution" to some extent and is well-defined for fully general $p$-Hessian equations. Based on pseudo-solutions, we prove the $\mathrm{C}^1$ estimates for general $p$-Hessian equations, and the corresponding second-order estimates when $p\in\{2, n-1, n\}$, under sharp conditions -- we don't impose curvature restrictions, convexity conditions or "MTW condition" on our main results. Some other conclusions related to a priori estimates and different kinds of "subsolutions" are also given, including estimates for "semi-convex" solutions and when there exists a pseudo-solution.