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In this paper, we are interested in the reiterated homogenization of linearelliptic equations of the form $-\frac{\partial}{\partial x_{i}} \left(a_{i j}\left(\frac{x}{\varepsilon}, \frac{x}{\varepsilon^{2}}\right) \frac{\partialu_\varepsilon}{\partial x_{j}}\right)=f$ in $\Omega$ with Dirichlet boundaryconditions. We obtain error estimates $O(\varepsilon)$ for a bounded $C^{1,1}$domain for this equation as well as the interior Lipschitz estimates at (very)large scale. Compared to the general homogenization problems, the difficulty inthe reiterated homogenization is that we need to handle different scales of$x$. To overcome this difficulty, we firstly introduce the Fourier transform inthe homogenization theory to separate these different scales. We also note thatthis method may be adapted to the following reiterated homogenization problem:$-\frac{\partial}{\partial x_{i}} \left(a_{i j}\left(\frac{x}{\varepsilon},\cdots, \frac{x}{\varepsilon^{N}}\right)\frac{\partial u_\varepsilon}{\partial x_{j}} \right) = f$ in $\Omega$ withDirichlet boundary conditions. Moreover, our results may be extended to therelated Neumann boundary problems without any real difficulty.

Quantitative Estimates on Reiterated Homogenization of Linear Elliptic operators Using Fourier Transform Methods