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 $ extrm{H}_{v, psilon }f(x): = ext{p.v.}int_{- psilon} DEGREES{ psilon} f(x-yv(x)); rac{dy}y$ where $ psilon$ is a suitably chosen parameter, determined by the smoothness properties of the vector field. Table of Contents: Overview of principal results; Besicovitch set and Carleson's theorem; The Lipschitz Kakeya maximal function; The $L DEGREES2$ estimate; Almost orthogonality between annuli.