On a Conjecture of E. M. Stein on the Hilbert Transform on Vector Fields

by Michael Lacey, Xiaochun Li

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 $\textrm{H}_{v, \epsilon }f(x): = \text{p.v.}\int_{-\epsilon} DEGREES{\epsilon} f(x-yv(x))\;\frac{dy}y$ where $\epsilon$ 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.

72 pages, Paperback

First published April 30, 2010