  Stein conjectured that the Hilbert transform in the direction of a vector
field is bounded on, say, $L^2$ whenever $v$ is Lipschitz. We establish a wide
range of $L^p$ estimates for this operator when $v$ is a measurable,
non-vanishing, one-variable vector field in $\bbr ^2$. Aside from an $L^2$
estimate following from a simple trick with Carleson's theorem, these estimates
were unknown previously. This paper is closely related to a recent paper of the
first author (\cite{B2}).
