  This paper is concerned with the regularity theory of a transmission problem
arising in composite materials. We give a new self-contained proof for the
$C^{k,\alpha}$ estimates on both sides of the interface under the minimal
assumptions on the interface and data. Moreover, we prove the uniform Lipschitz
estimate across a $C^{1,\alpha}$ interface when the coefficients on both sides
of the interface are periodic with independent structures and oscillating at
different microscopic scales.
