Uniform Calder\'on-Zygmund estimates in multiscale elliptic homogenization

This paper is concerned with the elliptic equation -div (A_ u_) = div f in a bounded C¹ domain, where A_ takes a form of A_ (x) = A (x/₁, x/₂, , x/ₙ), with A (y₁, y₂, , yₙ) being 1-periodic in each yᵢ. We prove the uniform Calder\'on-Zygmund estimate, namely, the uniform Lᵖ boundedness of the linear map f u_ for any p (1, ) with a constant independent of small parameters (₁, ₂, , ₙ) (0, 1]ⁿ. Our result includes the uniform Calder\'on-Zygmund estimate in quasiperiodic elliptic homogenization (even without the Diophantine condition), which was previously unknown. The proof novelly combines the Dirichlet's theorem on the simultaneous Diophantine approximation from number theory, a technique of reperiodization, reiterated periodic homogenization and a large-scale real-variable argument. Using the idea of reperiodization, we also obtain some large-scale or mesoscopic-scale Lipschitz estimates.