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Concerned with elliptic operator with stationary random coefficients ofintegrable correlations and bounded Lipschitz domains, arising from stochastichomogenization theory, this paper is mainly devoted to studyingCalder\'on-Zygmund estimates. As an application, we obtain the homogenizationerror in the sense of oscillation and fluctuation, and these results areoptimal up to a quantity $O(\ln(1/\varepsilon))$, which is coming from thequantified sublinearity of correctors in dimension two and less smoothness ofthe boundary of the domain.  The main scheme relies on (weighted) annealed Calder\'on-Zygmund estimates,which was recently developed by Josien and Otto \cite{Josien-Otto22} via arobust non-perturbative argument, independent of the quenchedCalder\'on-Zygmund estimate that originally developed by Armstrong and Daniel\cite{Armstrong-Daniel16} and Gloria, Neukamm and Otto\cite{Gloria-Neukamm-Otto20}.  Enlightened by Duerinckx and Otto's job \cite{Duerinckx-Otto20}, the maininnovation of the present work is to apply Shen's real arguments (of weightedversion) to study elliptic systems with Dirichlet or Neumann boundaryconditions. We start from a qualitative description of homogenization theoremin a local way (without using boundary correctors), and find a new form of\emph{minimal radius}, which proved to be a suitable key to open quantitativehomogenization on the boundary value problems, if adoptingGloria-Neukamm-Otto's strategy that originally inspired by Naddaf and Spencer'spioneering work.

Calderon-Zygmund estimates for stochastic elliptic systems on bounded Lipschitz domains