The Hilbert Transform of a Measure

Abstract

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

18 pages