Homogenization of elliptic systems with periodic coefficients: Weighted $L^p$ and $L^\infty$ estimates for asymptotic remainders

The difference between the fundamental matrix for a second order selfadjoint elliptic system with sufficiently smooth periodic coefficients and the fundamental matrix for the corresponding homogenized system in $\mathbb R^n$ is shown to decay as $O(1+|x|^{1-n}$) at infinity, $n\ge 2$. As a consequence, weighted $L^p$ and $L^\infty$ estimates are obtained for the difference $u^\varepsilon-u^0$ of the solutions of a system with rapidly oscillating periodic coefficients and the homogenized system in $\mathbb R^n$ with right-hand side belonging to an appropriate weighted $L^p$-class in $\mathbb R^n$.