  The absolutely continuous spectrum of one-dimensional Schr\"odinger operators
is proved to be stable under perturbation by potentials satisfying mild decay
conditions. In particular, the absolutely continuous spectrum of free and
periodic Schr\"odinger operators is preserved under all perturbations $V(x)$
satisfying $|V(x)|\leq C(1+x)^{-\alpha}$, $\alpha >\frac{1}{2}.$ This result is
optimal in the power scale. More general classes of perturbing potentials which
are not necessarily power decaying are also treated. A general criterion for
stability of the absolutely continuous spectrum of one-dimensional
Schr\"odinger operators is established. In all cases analyzed, the main term of
the asymptotic behavior of the generalized eigenfunctions is shown to have WKB
form for almost all energies. The proofs rely on new maximal function and norm
estimates and almost everywhere convergence results for certain multilinear
integral operators.
