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

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, ε}f(x) := \text{p.v.}\int_{-ε}^ εf(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.

92 pages, 20+ figures. Final version of the paper. To appear in Memoirs AMS

References in corpus (1)

• $L^p$ estimates for the bilinear Hilbert transform for $1/2<p\le 2/3$: A counterexample and generalizations to non-smooth symbols

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• Square functions for bi-Lipschitz maps and directional operators
• Directional square functions
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• Singular integrals along lacunary directions in ${\mathbb{R}}^n$