  This paper is concerned with the quantitative homogenization of $2m$-order
elliptic systems with bounded measurable, rapidly oscillating periodic
coefficients. We establish the sharp $O(\varepsilon)$ convergence rate in
$W^{m-1, p_0}$ with $p_0=\frac{2d}{d-1}$ in a bounded Lipschitz domain in
$\mathbb{R}^d$ as well as the uniform large-scale interior $C^{m-1, 1}$
estimate. With additional smoothness assumptions, the uniform interior $C^{m-1,
1}$, $W^{m,p}$ and $C^{m-1, \alpha}$ estimates are also obtained. As
applications of the regularity estimates, we establish asymptotic expansions
for fundamental solutions.
