%0 Journal Article
%T $L^p$ estimates for the Hilbert transforms along a one-variable vector field
%A Michael Bateman
%A Christoph Thiele
%J Mathematics
%D 2011
%I arXiv
%X Stein conjectured that the Hilbert transform in the direction of a vector field 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, non-vanishing, one-variable vector field in $\bbr ^2$. 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 (\cite{B2}).
%U http://arxiv.org/abs/1109.6396v1