Uniform Calder\'{o}n-Zygmund estimates in multiscale elliptic homogenization

This paper is concerned with the elliptic equation -div (A_ε ∇ u_ε) = div f in a bounded C¹ domain, where A_ε takes a form of A_ε(x) = A(x/ε₁, x/ε₂,⋯, x/εₙ), with A(y₁,y₂,⋯,yₙ) being 1-periodic in each yᵢ. We prove the uniform Calder\'{o}n-Zygmund estimate, namely, the uniform Lᵖ boundedness of the linear map f↦ ∇ u_ε for any p∈ (1,∞) with a constant independent of small parameters (ε₁,ε₂,⋯, εₙ) ∈ (0,1]ⁿ. Our result includes the uniform Calder\'{o}n-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.