Weighted norm inequalities for oscillatory integrals with finite type phases on the line

We obtain two-weighted L2 norm inequalities for oscillatory integral operators of convolution type on the line whose phases are of finite type. The conditions imposed on the weights involve geometrically-defined maximal functions, and the inequalities are best-possible in the sense that they imply the full Lp(R)→Lq(R) mapping properties of the oscillatory integrals. Our results build on work of Carbery, Soria, Vargas and the first author.