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$L^p$ estimates for the
Hilbert transforms along a one-variable vector field
Michael Bateman and Christoph Thiele |
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Vol. 6 (2013), No. 7, 1577–1600
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Abstract |
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Stein conjectured that the Hilbert transform in the direction of a vector field is bounded on, say, whenever is Lipschitz. We establish a wide range of estimates for this operator when is a measurable, nonvanishing, one-variable vector field in . Aside from an estimate following from a simple trick with Carleson’s theorem, these estimates were unknown previously. This paper is closely related to a recent paper of the first author (Rev. Mat. Iberoam. 29:3 (2013), 1021–1069). |
Keywords
singular integrals, differentiation theory, Carleson's
theorem, maximal operators, Stein's conjecture, Zygmund's
conjecture
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Mathematical Subject Classification 2010
Primary: 42B20, 42B25
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Milestones
Received: 18 October 2011
Revised: 2 April 2013
Accepted: 21 May 2013
Published: 27 December 2013
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Authors |
| Michael Bateman | |
| Department of Pure Mathematics and
Mathematical Statistics Centre for Mathematical Sciences University of Cambridge Wilberforce Road Cambridge, CB3 0WB United Kingdom |
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| Christoph Thiele | |
| Mathematisches Institut Universität Bonn Endenicher Allee 60 D-53115 Bonn Germany |
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