On regularity of averaged Green's functions in homogenization of elliptic PDE

This paper is concerned with establishing estimates on averaged Green’s functions for a uniformly elliptic divergence form partial difference equation with random coefficients on the $d$ dimensional integer lattice ${\mathbb{Z}}^d$. It has previously been shown that the averaged Green’s function is point-wise well approximated by the homogenized Green’s function at large length scales. Corresponding results also hold for (fractional) derivatives of the averaged Green’s function up to but not including the second derivative. Second derivative results have been established when the elliptic equation generates a positive semi-group, as in the diagonal case. Here the second derivative result is shown more generally by using an insight of Bourgain.