Dini-Campanato spaces and applications to nonlinear elliptic equations

We generalize a result due to Campanato [C] and use this to obtain regularity results for classical solutions of fully nonlinear elliptic equations. We demonstrate this technique in two settings. First, in the simplest setting of Poisson's equation $Delta u=f$ in $B$, where $f$ is Dini continuous in $B$, we obtain known estimates on the modulus of continuity of second derivatives $D^2u$ in a way that does not depend on either differentiating the equation or appealing to integral representations of solutions. Second, we use this result in the concave, fully nonlinear setting $F(D^2u,x)=f(x)$ to obtain estimates on the modulus of continuity of $D^2u$ when the $L^n$ averages of $f$ satisfy the Dini condition