ScalingStacks

Theorem 4.8. Fix $k\geq 2$ and $0<\gamma<1$. On the unit ball, let $v$ be a given smooth solution to the complex Monge-Ampère equation $(dd^{c}v)^{n}=1$. Then there are constants $0<\epsilon\ll 1$ and $C$ depending on $n,k,\gamma,\left\lVert v\right\rVert_{C^{k,\gamma}}$, such that if

$$(dd^{c}(u+v))^{n}=1+f,\quad\left\lVert f\right\rVert_{C^{k-2,\gamma}}<\epsilon,$$

and $\left\lVert u\right\rVert_{C^{0}}<\epsilon$, then $\left\lVert u\right\rVert_{C^{k,\gamma}(B_{1/2})}\leq C\epsilon$.