Remote access

How to Order

For AMS eBook frontlist subscriptions or backfile collection purchases:

   1a. To purchase any ebook backfile or to subscibe to the current year of Contemporary Mathematics, please download this required license agreement,

   1b. To subscribe to the current year of Memoirs of the AMS, please download this required license agreement.

   2. Complete and sign the license agreement.

   3. Email, fax, or send via postal mail to:

Customer Services
American Mathematical Society
201 Charles Street Providence, RI 02904-2294  USA
Phone: 1-800-321-4AMS (4267)
Fax: 1-401-455-4046

Visit the AMS Bookstore for individual volume purchases.

Browse the current eBook Collections price list

Powered by MathJax

On a conjecture of E. M. Stein on the Hilbert transform on vector fields

About this Title

Michael Lacey, School of Mathematics, Georgia Institute of Technology, Atlanta Georgia 30332 and Xiaochun Li, Department of Mathematics, University of Illinois, Urbana, Illinois 61801

Publication: Memoirs of the American Mathematical Society
Publication Year: 2010; Volume 205, Number 965
ISBNs: 978-0-8218-4540-0 (print); 978-1-4704-0579-3 (online)
Published electronically: January 7, 2010
MathSciNet review: 2654385
Keywords:Hilbert transform, Carleson Theorem, Fourier series, Kakeya set, vector field, Maximal Function, phase plane
MSC: Primary 42A50; Secondary 42B25

View full volume PDF

View other years and numbers:

Table of Contents


  • Preface
  • Chapter 1. Overview of principal results
  • Chapter 2. Besicovitch set and Carleson’s Theorem
  • Chapter 3. The Lipschitz Kakeya maximal function
  • Chapter 4. The $L^2$ estimate
  • Chapter 5. Almost orthogonality between annuli


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

$\displaystyle \operatorname H_{v, \epsilon }f(x) :=$   p.v.$\displaystyle \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. M. 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 $v$, namely 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 vector fields. The relationship between our results and other known sufficient conditions is explored.

References [Enhancements On Off] (What's this?)