  In this paper, we study the estimates of resolvents $
R(\lambda,\mathcal{L}_{\varepsilon})=(\mathcal{L}_{\varepsilon}-\lambda I)^{-1}
$, where $$
\mathcal{L}_{\varepsilon}=-\operatorname{div}(A(x/\varepsilon)\nabla) $$ is a
family of second elliptic operators with symmetric, periodic and oscillating
coefficients defined on a bounded domain $ \Omega $ with $ \varepsilon>0 $. For
$ 1<p<\infty $, we will establish uniform $ L^p\to L^p $, $ L^p\to W_0^{1,p} $,
$ W^{-1,p}\to L^p $ and $ W^{-1,p}\to W_0^{1,p} $ estimates by using the real
variable method. Meanwhile, we use Green functions for operators $
\mathcal{L}_{\varepsilon}-\lambda I $ to study the asymptotic behavior of $
R(\lambda,\mathcal{L}_{\varepsilon}) $ and obtain convergence estimates in $
L^p\to L^p $, $ L^p\to W_0^{1,p} $ norm.

Analysis of PDEs