  We obtain two-weighted $L^2$ 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 $L^p(\mathbb{R})\rightarrow L^q(\mathbb{R})$ mapping properties of the
oscillatory integrals. Our results build on work of Carbery, Soria, Vargas and
the first author.
