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In this note we consider the homogenization problem for a matrix locally periodic elliptic operator on R d  of the form A  ε  = −divA(x, x/ε)∇. The function A is assumed to be Hölder continuous with exponent s ∈ [0, 1] in the “slow” variable and bounded in the “fast” variable. We construct approximations for (A  ε  − μ)−1, including one with a corrector, and for (−Δ) s/2(A  ε  − μ)−1 in the operator norm on L 2(R d ) n . For s ≠ 0, we also give estimates of the rates of approximation.

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