Hilbert transforms along Lipschitz direction fields: A lacunary model

TL;DR

This paper establishes bounds for the truncated directional Hilbert transform in two dimensions under combined Lipschitz and lacunarity conditions, advancing understanding of boundedness in harmonic analysis.

Contribution

It proves boundedness results for the directional Hilbert transform with combined Lipschitz and lacunarity assumptions, addressing a key open question.

Findings

Boundedness of the transform for all 1<p<∞

Lacunarity alone is insufficient for p=2

Lipschitz assumption contributes to boundedness

Abstract

We prove bounds for the truncated directional Hilbert transform in $L^p(\mathbb{R}^2)$ for any $1<p<\infty$ under a combination of a Lipschitz assumption and a lacunarity assumption. It is known that a lacunarity assumption alone is not sufficient to yield boundedness for $p=2$, and it is a major question in the field whether a Lipschitz assumption alone suffices, at least for some $p$.