  We prove a uniform C^alpha estimate for collapsing Calabi-Yau metrics on the
total space of a proper holomorphic submersion over the unit ball in C^m. The
usual methods of Calabi, Evans-Krylov, and Caffarelli do not apply to this
setting because the background geometry degenerates. We instead rely on blowup
arguments and on linear and nonlinear Liouville theorems on cylinders. In
particular, as an intermediate step, we use such arguments to prove sharp new
Schauder estimates for the Laplacian on cylinders. If the fibers of the
submersion are pairwise biholomorphic, our method yields a uniform C^infinity
estimate. We then apply these local results to the case of collapsing
Calabi-Yau metrics on compact Calabi-Yau manifolds. In this global setting, the
C^0 estimate required as a hypothesis in our new local C^alpha and C^infinity
estimates is known to hold thanks to earlier work of the second-named author.
