The Hilbert Transform of a Measure

TL;DR

This paper establishes a condition involving the Hilbert transform of a measure that ensures the measure's singular part is zero on a specific homogeneous set, advancing understanding of measure decomposition.

Contribution

It introduces a new criterion linking the decay of the Hilbert transform's level sets to the absence of singular measure components on homogeneous sets.

Findings

If the measure's Hilbert transform's level sets decay sufficiently fast, then the measure has no singular part on the set.

The result applies to homogeneous subsets of the real line, extending classical measure decomposition results.

Provides a new analytical condition for the absolute continuity of measures on certain sets.

Abstract

Let $\fre$ be a homogeneous subset of $\bbR$ in the sense of Carleson. Let $\mu$ be a finite positive measure on $\bbR$ and $H_\mu(x)$ its Hilbert transform. We prove that if $\lim_{t\to\infty} t \abs{\fre\cap\{x\mid\abs{H_\mu(x)}>t\}}=0$, then $\mu_s(\fre)=0$, where $\mu_\s$ is the singular part of $\mu$.