On unboundedness of maximal operators for directional Hilbert transforms

TL;DR

This paper proves that the maximal operator associated with directional Hilbert transforms is unbounded on L^2 for any infinite set of directions in the plane, highlighting limitations in controlling such operators.

Contribution

It establishes the unboundedness of the maximal operator for all infinite sets of directions in two-dimensional space, a significant extension of previous results.

Findings

Maximal operator is unbounded on L^2 for any infinite set of directions.

Unboundedness holds for all infinite directional sets in rac12;^2.

Results demonstrate fundamental limitations in directional harmonic analysis.

Abstract

We show that for any infinite set of unit vectors $U$ in $\ZR^2$ the maximal operator defined by $$ H_Uf(x)=\sup_{u\in U}\bigg|\pv\int_{-\infty}^\infty \frac{f(x-tu)}{t}dt\bigg|,\quad x\in \ZR^2, $$ is not bounded in $L^2(\ZR^2)$.