  Let $ v$ be a smooth vector field on the plane, that is a map from the plane
to the unit circle. We study sufficient conditions for the boundedness of the
Hilbert transform
  \operatorname H_{v, \epsilon}f(x) := \text{p.v.}\int_{-\epsilon}^ \epsilon
f(x-yv(x)) \frac{dy}y where $ \epsilon $ is a suitably chosen parameter,
determined by the smoothness properties of the vector field. It is a
conjecture, due to E.\thinspace M.\thinspace Stein, that if $ v$ is Lipschitz,
there is a positive $ \epsilon $ for which the transform above is bounded on $
L ^{2}$. Our principal result gives a sufficient condition in terms of the
boundedness of a maximal function associated to $ v$. This sufficient condition
is that this new maximal function be bounded on some $ L ^{p}$, for some $
1<p<2$. We show that the maximal function is bounded from $ L ^{2}$ to weak $ L
^{2}$ for all Lipschitz maximal function. The relationship between our results
and other known sufficient conditions is explored.
