#### 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, ${L}^{2}$ whenever $v$ is Lipschitz. We establish a wide range of ${L}^{p}$ estimates for this operator when $v$ is a measurable, nonvanishing, one-variable vector field in ${R}^{2}$. Aside from an ${L}^{2}$ 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