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.

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.

Equations

$$H_{v,ϵ}f(x):=\text{p.v.}{\int }_{-ϵ}^{ϵ}f(x-yv(x))\frac{dy}{y}$$

Full text

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where $\epsilon$ is a suitably chosen parameter, determined by the smoothness properties of the vector field. It is a conjecture, due to E. M. 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 $v$, namely 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 vector fields. The relationship between our results and other known sufficient conditions is explored.

Preface

This memoir is devoted to a question in planar Harmonic Analysis, a subject which is a circle of problems all related to the Besicovitch set. This anomalous set has zero Lebesgue measure, yet contains a line segment of unit length in each direction of the plane. It is a known, since the 1970’s, that such sets must necessarily have full Hausdorff dimension. The existence of these sets, and the full Hausdorff dimension, are intimately related to other, independently interesting issues [stein]. An important tool to study these questions is the so-called Kakeya Maximal Function, in which one computes the maximal average of a function over rectangles of a fixed eccentricity and arbitrary orientation.

Most famously, Charles Fefferman showed [MR45:5661] that the Besicovitch set is the obstacle to the boundedness of the disc multiplier in the plane. But as well, this set is intimately related to finer questions of Bochner-Riesz summability of Fourier series in higher dimensions and space-time regularity of solutions of the wave equation.

This memoir concerns one of the finer questions which center around the Besicovitch set in the plane. (There are not so many of these questions, but our purpose here is not to catalog them!) It concerns a certain degenerate Radon transform. Given a vector field $v$ on $\mathbb{R}^{2}$, one considers a Hilbert transform computed in the one dimensional line segment determined by $v$, namely the Hilbert transform of a function on the plane computed on the line segment $\{x+tv(x)\mid\lvert t\rvert\leq 1\}$.

The Besicovitch set itself says that choice of $v$ cannot be just measurable, for you can choose the vector field to always point into the set. Finer constructions show that one cannot take it to be Hölder continuous of any index strictly less than one. Is the sharp condition of Hölder continuity of index one enough? This is the question of E. M. Stein, motivated by an earlier question of A. Zygmund, who asked the same for the question of differentiation of integrals.

The answer is not known under any condition of just smoothness of the vector field. Indeed, as is known, and we explain, a positive answer would necessarily imply Carleson’s famous theorem on the convergence of Fourier series, [car]. This memoir is concerned with reversing this implication: Given the striking recent successes related to Carleson’s Theorem, what can one say about Stein’s Conjecture? In this direction, we introduce a new object into the study, a Lipschitz Kakeya Maximal Function, which is a variant of the more familiar Kakeya Maximal Function, which links the vector field $v$ to the ‘Besicovitch sets’ associated to the vector field. One averages a function over rectangles of arbitrary orientation and—in contrast to the classical setting—arbitrary eccentricity. But, the rectangle must suitably localize the directions in which the vector field points. This Maximal Function admits a favorable estimate on $L^{2}$, and this is one of the main results of the Memoir.

On Stein’s Conjecture, we prove a conditional result: If the Lipschitz Kakeya Maximal Function associated with $v$ maps is an estimate a little better than our $L^{2}$ estimate, then the associated Hilbert transform is indeed bounded. Thus, the main question left open concerns the behavior of these novel Maximal Functions.

While the main result is conditional, it does contain many of the prior results on the subject, and greatly narrows the possible avenues of a resolution of this conjecture.

The principal results and conjectures are stated in the Chapter 1; following that we collect some of the background material for this subject, and prove some of the folklore results known about the subject. The remainder of the Memoir is taken up with the proofs of the Theorems stated in the Chapter 1.

Acknowledgment**.**

The efforts of a strikingly generous referee has resulted in corrections of arguments, and improvements in presentation throughout this manuscript. We are indebted to that person.

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Michael T. Lacey and Xiaochun Li