  We prove weighted estimates for singular integral operators which operate on
function spaces on a half-line. The class of admissible weights includes
Muckenhoupt weights and weights satisfying Sawyer's one-sided conditions. The
kernels of the operators satisfy relaxed Dini conditions. We apply the weighted
estimates to extrapolation of maximal $L^p$ regularity of first order, second
order and fractional order Cauchy problems into weighted rearrangement
invariant Banach function spaces. In particular, we provide extensions, as well
as a unification of recent results due to Auscher and Axelsson, and Chill and
Fiorenza.
