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

Michael Lacey, Xiaochun Li

Research output: Contribution to journalArticlepeer-review


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 (Equation) where e 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 e 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.

Original languageEnglish (US)
Pages (from-to)1-80
Number of pages80
JournalMemoirs of the American Mathematical Society
Issue number965
StatePublished - May 2010
Externally publishedYes


  • Carleson theorem
  • Fourier series
  • Hilbert transform
  • Kakeya set
  • Maximal function
  • Phase plane
  • Vector field

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics


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