## 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 (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 language | English (US) |
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Pages (from-to) | 1-80 |

Number of pages | 80 |

Journal | Memoirs of the American Mathematical Society |

Volume | 205 |

Issue number | 965 |

DOIs | |

State | Published - May 2010 |

Externally published | Yes |

## Keywords

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

## ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics