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memo_has_moved_text();On a conjecture of E. M. Stein on the Hilbert transform on vector fields

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)
DOI: http://dx.doi.org/10.1090/S0065-9266-10-00572-7
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

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Chapters

• 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

Abstract

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.