# On a Conjecture of EM Stein on the Hilbert Transform on Vector Fields

## TL;DR

This paper investigates conditions under which a Hilbert transform along vector fields is bounded, providing a new sufficient condition related to a maximal function, and explores its implications for Stein's conjecture.

## Contribution

It introduces a new maximal function condition that guarantees boundedness of the Hilbert transform along vector fields, advancing understanding of Stein's conjecture.

## Key findings

- Maximal function boundedness from L^2 to weak L^2 for Lipschitz vector fields.
- Sufficient condition for Hilbert transform boundedness based on maximal function.
- Relationship between new condition and existing criteria explored.

## Abstract

Let $ v$ be a smooth vector field on the plane, that is a map from the plane to the unit circle. We study sufficient conditions for the boundedness of the Hilbert transform \operatorname H_{v, \epsilon}f(x) := \text{p.v.}\int_{-\epsilon}^ \epsilon f(x-yv(x)) \frac{dy}y where $ \epsilon $ is a suitably chosen parameter, determined by the smoothness properties of the vector field. It is a conjecture, due to E.\thinspace M.\thinspace Stein, that if $ v$ is Lipschitz, there is a positive $ \epsilon $ for which the transform above is bounded on $ L ^{2}$. Our principal result gives a sufficient condition in terms of the boundedness of a maximal function associated to $ v$. This sufficient condition is that this new maximal function be bounded on some $ L ^{p}$, for some $ 1<p<2$. We show that the maximal function is bounded from $ L ^{2}$ to weak $ L ^{2}$ for all Lipschitz maximal function. The relationship between our results and other known sufficient conditions is explored.