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

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).