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) := \text{ 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.\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 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.
Categories: math.CA