Improved resolvent approximations in homogenization of second order operators with periodic coefficients

For elliptic divergent self-adjoint second-order operators with $\varepsilon$-periodic measurable coefficients acting on the whole space $\mathbb{R}^d$, resolvent approximations in the operator norm $\|\!\,\boldsymbol\cdot\,\!\|_{H^1\to H^1}$ with remainder of order $\varepsilon^2$ as $\varepsilon\to 0$ are found by the method of two-scale expansions with the use of smoothing.