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Let $ v$ be a smooth vector field on the plane, that is a map from the planeto the unit circle. We study sufficient conditions for the boundedness of theHilbert transform  \operatorname H_{v, \epsilon}f(x) := \text{p.v.}\int_{-\epsilon}^ \epsilonf(x-yv(x)) \frac{dy}y where $ \epsilon $ is a suitably chosen parameter,determined by the smoothness properties of the vector field. It is aconjecture, 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 theboundedness of a maximal function associated to $ v$. This sufficient conditionis 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 resultsand other known sufficient conditions is explored.

On a Conjecture of EM Stein on the Hilbert Transform on Vector Fields