Universal moduli of continuity for solutions to fully nonlinear elliptic equations

This paper provides universal, optimal moduli of continuity for viscosity solutions to fully nonlinear elliptic equations $F(X, D^2u) = f(X)$, based on weakest integrability properties of $f$ in different scenarios. The primary result established in this work is a sharp Log-Lipschitz estimate on $u$ based on the $L^n$ norm of $f$, which corresponds to optimal regularity bounds for the critical threshold case. Optimal $C^{1,\alpha}$ regularity estimates are delivered when $f\in L^{n+\epsilon}$. The limiting upper borderline case, $f\in L^\infty$, also has transcendental importance to elliptic regularity theory and its applications. In this paper we show, under convexity assumption on $F$, that $u \in C^{1,\mathrm{Log-Lip}}$, provided $f$ has bounded mean oscillation. Once more, such an estimate is optimal. For the lower borderline integrability condition allowed by the theory, we establish interior \textit{a priori} estimates on the $C^{0,\frac{n-2\epsilon}{n-\epsilon}}$ norm of $u$ based on the $L^{n-\epsilon}$ norm of $f$, where $\epsilon$ is the Escauriaza universal constant. The exponent $\frac{n-2\epsilon}{n-\epsilon}$ is optimal. When the source function $f$ lies in $L^q$, $n > q > n-\epsilon$, we also obtain the exact, improved sharp H\"older exponent of continuity.