  This paper is concerned with periodic homogenization of second-order elliptic
systems in divergence form with oscillating Dirichlet data or Neumann data of
first order. We prove that the homogenized boundary data belong to $W^{1, p}$
for any $1<p<\infty$. In particular, this implies that the boundary layer tails
are H\"older continuous of order $\alpha$ for any $\alpha \in (0,1)$.
