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

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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, εf(x):= p.v.∫₋ε εf(x-yv(x)) dyy where ε 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 ε 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.