Hilbert transform along measurable vector fields constant on Lipschitz curves: $L^p$ boundedness

TL;DR

This paper proves the boundedness of a directional Hilbert transform in the plane for certain measurable vector fields that are constant on Lipschitz curves, expanding understanding of harmonic analysis in geometric contexts.

Contribution

It establishes $L^p$ boundedness of the Hilbert transform along measurable vector fields constant on Lipschitz curves, a novel result in harmonic analysis.

Findings

Proves $L^p$ boundedness for $p > 3/2$.

Extends the class of vector fields for which the Hilbert transform is bounded.

Provides new techniques for analyzing directional singular integrals.

Abstract

We prove the $L^p (p > 3/2)$ boundedness of the directional Hilbert transform in the plane relative to measurable vector fields which are constant on suitable Lipschitz curves.