I am assuming that the reference quoted by the OP is the book by L. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations (American Mathematical Society, 1995), henceforth referred to as Caffarelli-Cabré. As such, the Cordes-Nirenberg estimate as spelled above is actually cited in the Introduction (estimate (a) right after formula (0.3) in page 2) and does not assume the lhs to be in divergence form - the aforementioned partial differential equation (0.3) actually has the form $$a_{ij}(x)\partial_i\partial_j u=f(x)$$ (we assume above the Einstein summation convention over repetated indices) - and $u$ is assumed to be a classical (hence twice continuously differentiable) solution of the above equation. This does not change in the fully nonlinear case considered in Chapter 8 (the lhs then becomes a function of $x$ and the Hessian of $u$ in $x$). I will also take the opportunity to fill in the missing details regarding hypotheses and notation. In what follows, $B_r$ is the open Euclidean ball of radius $r>0$ centered at some fixed $x_0\in\mathbb{R}^n$ and the above partial differential equation is assumed to be uniformly elliptic in the given domain $D\subset\mathbb{R}^n$, that is, there are constants $0<m<M$ such that $$m\|\xi\|^2\leq a_{ij}(x)\xi_i\xi_j\leq M\|\xi\|^2$$ for all $\xi\in\mathbb{R}^n$, $x\in D$. The precise statement in Caffarelli-Cabré then reads:
(Cordes-Nirenberg) Let $0<\alpha<1$ and $u\in C^2(B_1)$ be a solution of the uniformly elliptic partial differential equation in $D=B_1$ $$a_{ij}(x)\partial_i\partial_j u=f(x)$$ with $f\in L^\infty(B_1)$ and $\|a_{ij}-\delta_{ij}\|_{L^\infty(B_1)}\leq\delta$ for some $\delta>0$, where $\delta_{ij}$ is the $n$-dimensional Kronecker delta symbol. Then $u\in C^{1,\alpha}(\overline{B_{1/2}})$ and there is a constant $C>0$ such that $$\|u\|_{C^{1,\alpha}(\overline{B_{1/2}})}\leq C(\|u\|_{L^\infty(B_1)}+\|f\|_{L^\infty(B_1)})\ .$$
The latter inequality is the so-called Cordes-Nirenberg estimate (which, by the way, is missing in the OP as well). As stated, Caffarelli-Cabré indeed does not provide references for this result, but the following paper by Caffarelli et alli (which quotes Caffarelli-Cabré) does:
- L. Caffarelli, L. Silvestre, Regularity Results for Nonlocal Equations by Approximation. Arch. Rat. Mech. Anal. 200 (2011) 59-88, arXiv:0902.4030 [math].
More precisely, there the result is provided with the following original references:
- H. O. Cordes, Über die erste Randwertaufgabe bei quasilinearen Differentialgleichungen zweiter Ordnung in mehr als zwei Variablen, Math. Ann. 131 (1956) 278-312 (see Satz 8, pp. 303-306).
- L. Nirenberg, On a generalization of quasi-conformal mappings and its application to elliptic partial differential equations. In: Contributions to the Theory of Partial Differential Equations, Annals of Mathematics Studies no. 33 (Princeton University Press, 1954), pp. 95-100 (see Theorem 2, pp. 97-99).
Nirenberg requires $a_{ij}(x)$ to be continuous in $\overline{B_1}$ and Cordes allows for lower-order terms in the equation, whose coefficients should then also belong to $L^\infty(B_1)$ (one can extract from Cordes's result a suitable set of hypotheses on $a_{ij}(x)$ when the equation is in divergence form as in the OP). A two-dimensional version of this result had been previously proven in
- Morrey, C. B., On the Solutions of Quasilinear Elliptic Partial Differential Equations, Trans. Amer. Math. Soc. 43 (1938) 126-166.
- L. Nirenberg, On Nonlinear Elliptic Partial Differential Equations and Hölder Continuity, Comm. Pure Appl. Math. 6 (1953) 103-156.
I do not know who first provided the extension of the Cordes-Nirenberg estimates to $H^1$ weak solutions (if the equation is in divergence form, of course), see however this Math.SE question.
