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. The authors study sufficient conditions for the boundedness of the Hilbert transform $mathrm{H}_{v, epsilon }f(x) := ext{p.v.}int_{-epsilon}^{epsilon} f(x-yv(x));frac{dy}y$ where $epsilon$ is a suitably chosen parameter, determined by the smoothness properties of the vector field.