Memoirs of the American Mathematical Society 2010; 72 pp; softcover Volume: 205 ISBN10: 0821845403 ISBN13: 9780821845400 List Price: US$68 Individual Members: US$40.80 Institutional Members: US$54.40 Order Code: MEMO/205/965
 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 \[\mathrm{H}_{v, \epsilon }f(x) := \text{p.v.}\int_{\epsilon}^{\epsilon} f(xyv(x))\;\frac{dy}y\] 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 \(L^2\) estimate
 Almost orthogonality between annuli
