Improved homogenization estimates for high order elliptic systems

In the whole space $R^d$ ($d\ge 2$), we study homogenization of a divergence-form matrix elliptic operator $L_{\epsilon }$ of an arbitrary even order larger than 2 with measurable $\epsilon$-periodic coefficients, where $\epsilon$ is a small parameter. We constuct an approximation for the resolvent of $L_{\epsilon }$ with the remainder term of order ${\epsilon }^2$ in the operator $L^2$-norm. We impose no regularity conditions on the operator beyond ellipticity and boundedness of coefficients. We use two scale expansions with correctors regularized by the Steklov smoothing.