Uniform $C^k$ bounds from a Laplacian bound.

Question

In a paper I am reading, I am given smooth solutions to a family of Monge-Ampere equations on a smooth, compact subset of a variety. The paper claims that uniform $C^k$ bounds on the family of solutions follow from a uniform Laplacian bound and "standard linear elliptic theory after linearizing the Monge-Ampere equation." I am somewhat familiar with bootstrapping to obtain $C^{k,\alpha}$ bounds but not $C^k$ bounds. If someone could point me in the right direction that would be much appreciated. Thanks!