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In the present work, we established almost-sharp error estimates for linearelasticity 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 boundedsmooth domain. It followed from weighted Hardy-Sobolev's inequalities and asuboptimal error estimate for the square function of the first-orderapproximating corrector (which was earliest investigated by C. Kenig, F. Lin,Z. Shen \cite{KLS} under additional regularity assumption on coefficients). Thenew 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 homogenizationtheory). The second effort was $L^2$-error estimates$O\big(\varepsilon^{\frac{5}{6}}\ln^{\frac{2}{3}}(1/\varepsilon)\big)$ for aLipschitz domain, followed from a new duality scheme coupled with interpolationinequalities. Also, we developed a new weighted extension theorem forperforated domains, and a real method imposed by Z. Shen \cite{S3} played afundamental role in the whole project.

suboptimal error estimates for homogenization of linear elasticity systems on perforated domains