  This article is concerned with uniform $C^{1,\alpha}$ and $C^{1,1}$ estimates
in periodic homogenization of fully nonlinear elliptic equations. The analysis
is based on the compactness method, which involves linearization of the
operator at each approximation step. Due to the nonlinearity of the equations,
the linearized operators involve the Hessian of correctors, which appear in the
previous step. The involvement of the Hessian of the correctors deteriorates
the regularity of the linearized operator, and sometimes even changes its
oscillating pattern. These issues are resolved with new approximation
techniques, which yield a precise decomposition of the regular part and the
irregular part of the homogenization process, along with a uniform control of
the Hessian of the correctors in an intermediate level. The approximation
techniques are even new in the context of linear equations. Our argument can be
applied not only to concave operators, but also to certain class of non-concave
operators.
