We study the operator $$ H_vf(x):=\text{p.v.}\int_{-1}^1f(x-yv(x))\;\frac{dy}y$$ defined for smooth functions on the plane and measurable vector fields $v$ from the plane into the unit circle. We prove that if $v$ has $1+\ze$ derivatives, then $H_v$ extends to a bounded map from $L^2(\ZR^2)$ into itself. What is noteworthy is that this result holds in the absence of some additional geometric condition imposed upon $v$, and that the smoothness condition is nearly optimal. Whereas $H_v$ is a Radon transform, for which there is an extensive theory, see e.g.~\cite{MR2000j:42023}, our methods of proof are necessarily those associated to Carleson's theorem on Fourier series \cite{car}, and the proof given by Lacey and Thiele \cite{laceythiele}. A previous paper of the authors \cite{laceyli}, has shown how to adapt these ideas to $H_v$; herein these ideas are combined with a crucial maximal function estimate that is particular to the smooth vector field in question.
31 pages