  We prove two H\"older regularity results for solutions of generated Jacobian
equations. First, that under the A3 condition and the assumption of nonnegative
$L^p$ valued data solutions are $C^{1,\alpha}$ for an $\alpha$ that is sharp.
Then, under the additional assumption of positive Dini continuous data, we
prove a $C^{2}$ estimate. Thus the equation is uniformly elliptic and when the
data is H\"older continuous solutions are in $C^{2,\alpha}$.
