On homogenization for non-self-adjoint locally periodic elliptic operators

In this note we consider the homogenization problem for a matrix locally periodic elliptic operator on $R^d$ of the form $\mathcal A^\varepsilon=-\operatorname{div}A(x,x/\varepsilon)\nabla$. The function $A$ is assumed to be Hölder continuous with exponent $s\in[0,1]$ in the “slow” variable and bounded in the “fast” variable. We construct approximations for $(A^\varepsilon-\mu)^{-1}$, including one with a corrector, and for $(-\Delta)^{s/2}(A^\varepsilon-\mu)^{-1}$ in the operator norm on $L_2(R^d)^n$. For $s\ne0$, we also give estimates of the rates of approximation.