Decomposition and pointwise estimates of periodic Green functions of some elliptic equations with periodic oscillatory coefficients

This article is about the ${\mathbb{Z}}^d$-periodic Green function $G_n(x,y)$ of the multiscale elliptic operator $Lu=-{\rm div}\left( A(n\cdot) \cdot \nabla u \right)$,where$A(x)$isa$$\mathbb{Z}$^d$-periodic, coercive, and H"older continuous matrix, and $n$ is a large integer. We prove here pointwise estimates on $G_n(x,y)$, ${\mathrm{\nabla }}_x{G}_n(x,y)$, ${\mathrm{\nabla }}_y{G}_n(x,y)$ and $\nabla_x \nabla_y G_n(x,y)$indimensions$d \geq 2$. Moreover, we derive an explicit decomposition of this Green function, which is of independent interest. These results also apply for systems.