Vol. 6, No. 7, 2013

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$L^p$ estimates for the Hilbert transforms along a one-variable vector field

Michael Bateman and Christoph Thiele

Vol. 6 (2013), No. 7, 1577–1600
Abstract

Stein conjectured that the Hilbert transform in the direction of a vector field v is bounded on, say, L2 whenever v is Lipschitz. We establish a wide range of Lp estimates for this operator when v is a measurable, nonvanishing, one-variable vector field in R2. Aside from an L2 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
Mathematical Subject Classification 2010
Primary: 42B20, 42B25
Milestones
Received: 18 October 2011
Revised: 2 April 2013
Accepted: 21 May 2013
Published: 27 December 2013
Authors
Michael Bateman
Department of Pure Mathematics and Mathematical Statistics
Centre for Mathematical Sciences
University of Cambridge
Wilberforce Road
Cambridge, CB3 0WB
United Kingdom
Christoph Thiele
Mathematisches Institut
Universität Bonn
Endenicher Allee 60
D-53115 Bonn
Germany