Improved approximations of resolvents in homogenization of fourth-order operators with periodic coefficients

In the whole space $R^d$, $d\ge 2$, we study homogenization of a divergence form elliptic fourth-order operator $A_{\epsilon }$ with measurable $\epsilon$-periodic coefficients, where $\epsilon$ is a small parameter. For the resolvent $(A_{\epsilon }+1{)}^{-1}$, acting as an operator from $L^2$ to $H^2$, we find an approximation with remainder term of order $O({\epsilon }^2)$ as $\epsilon$ tends to $0$. Relying on this result, we construct the resolvent approximation with remainder of order $O({\epsilon }^3)$ in the operator $L^2$-norm. We employ two-scale expansions that involve smoothing.