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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
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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$64
Individual Members: US$38.40
Institutional Members: US$51.20
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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