Classes of singular integral operators along variable lines

We prove estimates for classes of singular integral operators along variable lines in the plane, for which the usual assumption of nondegenerate rotational curvature may not be satisfied. The main L^p estimates are proved by interpolating L² bounds with suitable bounds in Hardy spaces on product domains. The L² bounds are derived by almost-orthogonality arguments. In an appendix we derive an estimate for the Hilbert transform along the radial vector field and prove an interpolation lemma related to restricted weak type inequalities.