On a conjecture of E. M. Stein on the Hilbert transform on vector fields

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

$$\operatorname H_{v, \epsilon }f(x) :=$$    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. M. 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 $v$, namely 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 vector fields. The relationship between our results and other known sufficient conditions is explored.