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

TL;DR

This paper proves new $L^p$ bounds for Hilbert transforms along certain vector fields in the plane, extending previous results and confirming conjectures for a broad class of measurable, non-vanishing fields.

Contribution

It establishes a wide range of $L^p$ estimates for Hilbert transforms along measurable, non-vanishing, one-variable vector fields, advancing understanding beyond prior known cases.

Findings

Established $L^p$ bounds for the operator for various $p$

Extended the range of vector fields for which bounds are known

Connected results to recent developments in harmonic analysis

Abstract

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}).