Operator estimates for homogenization of higher-order elliptic operators with periodic coefficients

In $L_2(\mathbb{R}^d;\mathbb{C}^n)$, we study a selfadjoint strongly elliptic differential operator $\mathcal{A}_\varepsilon$ of order $2p$ with periodic coefficients depending on $\mathbf{x}/\varepsilon$. We obtain the following approximation for the resolvent $( {\mathcal A}_\varepsilon+I)^{-1}$ in the operator norm on $L_2(\mathbb{R}^d;\mathbb{C}^n)$:
$$( {\mathcal A}_\varepsilon+I)^{-1} = ( {\mathcal A}^0+I)^{-1} + \sum_{j=1}^{2p-1} \varepsilon^{j} {\mathcal K}_{j,\varepsilon} + O(\varepsilon^{2p}).$$
Here ${\mathcal A}^0$ is the effective operator with constant coefficients and ${\mathcal K}_{j,\varepsilon}$, $j=1,\dots,2p-1$, are suitable correctors.