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 (Equation) where e 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 e 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.