On the maximal directional Hilbert transform in three dimensions

TL;DR

This paper determines the exact growth rate of the $L^p$ norms of the maximal Hilbert transform in three dimensions along finite lacunary directions, providing the first sharp estimate for such operators in dimensions higher than two.

Contribution

It establishes the first sharp bounds for maximal directional singular integrals in three dimensions, answering a longstanding open question.

Findings

Sharp growth rate of $L^p$ norms established

Representation of the transform via two-dimensional angular multipliers

Utilization of weighted norm inequalities and extrapolation

Abstract

We establish the sharp growth rate, in terms of cardinality, of the $L^p$ norms of the maximal Hilbert transform $H_\Omega$ along finite subsets of a finite order lacunary set of directions $\Omega \subset \mathbb R^3$, answering a question of Parcet and Rogers in dimension $n=3$. Our result is the first sharp estimate for maximal directional singular integrals in dimensions greater than 2. The proof relies on a representation of the maximal directional Hilbert transform in terms of a model maximal operator associated to compositions of two-dimensional angular multipliers, as well as on the usage of weighted norm inequalities, and their extrapolation, in the directional setting.