Hilbert transform along measurable vector fields constant on Lipschitz curves: $L^2$ boundedness

TL;DR

This paper proves the $L^2$ boundedness of the directional Hilbert transform in the plane for measurable vector fields that are constant on Lipschitz curves, advancing understanding of singular integral operators in harmonic analysis.

Contribution

It establishes the $L^2$ boundedness of the Hilbert transform along measurable vector fields constant on Lipschitz curves, a novel result in the study of directional singular integrals.

Findings

Proves $L^2$ boundedness of the transform

Extends previous results to measurable vector fields

Provides new techniques for analyzing directional operators

Abstract

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