i would to like know what fact in the pde theory is used in the following article , pg 15
"Since $Dw$ is Hölder continuous, we can see this equation as a linear ellip- tic equation with $C^{\alpha}$ coefficients. The standard Calderon-Zygmund theory provides local $C^{2,\alpha}$ regularity on $w$ (bootstrapping the argument gives even C∞ regularity on w)."
https://arxiv.org/abs/1603.06391
Thnaks
The result the paper references is commonly referred to as Schauder estimates, which are a collection of results which (roughly) assert that if $u$ is a solution to a linear elliptic PDE of the form $Lu=f$ whose coefficients in $C^{\alpha},$ then $f \in C^{\alpha}$ implies that $u \in C^{2,\alpha}.$
You can find the details in most texts on elliptic PDE, such as Chapter 6 of Gilbarg and Trudinger.