  The paper extends the results obtained by C. Kenig, F. Lin and Z. Shen in
\cite{SZW2} to more general elliptic homogenization problems in two
perspectives: lower order terms in the operator and no smoothness on the
coefficients. We do not repeat their arguments. Instead we find the new
weighted-type estimates for the smoothing operator at scale $\varepsilon$, and
combining some techniques developed by Z. Shen in \cite{SZW12} leads to our
main results. In addition, we also obtain sharp $O(\varepsilon)$ convergence
rates in $L^{p}$ with $p=2d/(d-1)$, which were originally established by Z.
Shen for elasticity systems in \cite{SZW12}. Also, this work may be regarded as
the extension of \cite{TS,TS2} developed by T. Suslina concerned with the
bounded Lipschitz domain.
