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

This paper is concerned with the elliptic equation $-\text{div} (A_\varepsilon \nabla u_\varepsilon) = \text{div} f$ in a bounded $C^1$ domain, where $A_\varepsilon$ takes a form of $A_\varepsilon(x) = A(x/\varepsilon_1, x/\varepsilon_2,\cdots, x/\varepsilon_n)$, with $A(y_1,y_2,\cdots,y_n)$ being 1-periodic in each $y_i$. We prove the uniform Calder\'{o}n-Zygmund estimate, namely, the uniform $L^p$ boundedness of the linear map $f\mapsto \nabla u_\varepsilon$ for any $p\in (1,\infty)$ with a constant independent of small parameters $(\varepsilon_1,\varepsilon_2,\cdots, \varepsilon_n) \in (0,1]^n$. 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.