  In the present work, we established almost-sharp error estimates for linear
elasticity systems in periodically perforated domains. The first result was
$L^{\frac{2d}{d-1-\tau}}$-error estimates
$O\big(\varepsilon^{1-\frac{\tau}{2}}\big)$ with $0<\tau<1$ for a bounded
smooth domain. It followed from weighted Hardy-Sobolev's inequalities and a
suboptimal error estimate for the square function of the first-order
approximating corrector (which was earliest investigated by C. Kenig, F. Lin,
Z. Shen \cite{KLS} under additional regularity assumption on coefficients). The
new approach relied on the weighted quenched Calder\'on-Zygmund estimate
(initially appeared in A. Gloria, S. Neukamm, F. Otto's work
\cite{Gloria_Neukamm_Otto_2015} for a quantitative stochastic homogenization
theory). The second effort was $L^2$-error estimates
$O\big(\varepsilon^{\frac{5}{6}}\ln^{\frac{2}{3}}(1/\varepsilon)\big)$ for a
Lipschitz domain, followed from a new duality scheme coupled with interpolation
inequalities. Also, we developed a new weighted extension theorem for
perforated domains, and a real method imposed by Z. Shen \cite{S3} played a
fundamental role in the whole project.
