This talk is devoted to studying the weighted annealed Calderon-Zygmund estimates, there are three approaches to develop this theory. The first one was based upon Shen’s real methods (kernel-free arguments), which was applied to stochastic homogenization by M. Duerinckx and F. Otto; The second one was rooted upon approximating arguments coupled with real and complex interpolations, developed by M. Josien and F. Otto; and the last one was appealing to two-scale expansions of fundamental solutions for large-scales together with spherical harmonic decomposition techniques for small-scales (which could be replaced with a PDE argument given by F. Otto), recently developed by Q. Xu, inspired from M. Avellaneda and F. Lin’s work on singular integrals in a periodic setting.

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