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On a Conjecture of E. M. Stein on the Hilbert Transform on Vector Fields
Michael Lacey, Georgia Institute of Technology, Atlanta, GA, and Xiaochun Li, University of Illinois, Urbana, IL

Memoirs of the American Mathematical Society
2010; 72 pp; softcover
Volume: 205
ISBN-10: 0-8218-4540-3
ISBN-13: 978-0-8218-4540-0
List Price: US$68
Individual Members: US$40.80
Institutional Members: US$54.40
Order Code: MEMO/205/965
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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 \[\mathrm{H}_{v, \epsilon }f(x) := \text{p.v.}\int_{-\epsilon}^{\epsilon} f(x-yv(x))\;\frac{dy}y\] 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 \(L^2\) estimate
  • Almost orthogonality between annuli
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