  This paper is devoted to the study of uniform $W^{1,\frac{np}{n-p}}$- and
$W^{2,p}$-estimates for viscosity solutions to fully nonlinear, uniformly
elliptic, periodic homogenization problems, up to boundaries, subject to
Dirichlet boundary conditions. We characterize the size of "effective" Hessian
and gradient of viscosity solutions to homogenization problems, and prove its
uniform integrability without any regularity assumption on the governing
functionals. Our estimates are new even for the standard problems. Our analysis
applies to a large class of non-convex functionals.
