  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 $A(x)$ is a $\mathbb{Z}^d$-periodic, coercive, and H\"older
continuous matrix, and $n$ is a large integer. We prove here pointwise
estimates on $G_n(x,y)$, $\nabla_x G_n(x,y)$, $\nabla_y G_n(x,y)$ and $\nabla_x
\nabla_y G_n(x,y)$ in dimensions $d \geq 2$. Moreover, we derive an explicit
decomposition of this Green function, which is of independent interest. These
results also apply for systems.
