  In this paper, we find some error estimates for periodic homogenization of
p-Laplace type equations under the same structure assumption on homogenized
equations. The main idea is that by adjusting the size of the difference
quotient of the correctors to make the convergence rate visible. In order to
reach our goal, the corresponding flux corrector with some properties are
developed. Meanwhile, the shift-arguments is in fact applied down to
$\varepsilon$ scale, which leads to a new weighted type inequality for
smoothing operator with the weight satisfying Harnack's inequality in small
scales. As a result, it is possible to develop some large-scale estimates. We
finally mention that our approach brought in a systematic error (this
phenomenon will disappear in linear and non-degenerated cases), which was
fortunately a quantity $o(\varepsilon)$ here.
