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 Hv,ǫ f(x) := p.v. ∫ ǫ −ǫ f(x− yv(x)) dy y where ǫ 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 ǫ for which the transform above is bounded on L. 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, for some 1 < p < 2. We show that the maximal function is bounded from L to weak L for all Lipschitz vector fields. The relationship between our results and other known sufficient conditions is explored.