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 $\mu$.Comment: 18 page

Poltoratski, Alexei

Simon, Barry

Zinchenko, Maxim

English

arXiv

Let e be a homogeneous subset of R in the sense of Carleson. Let μ be a finite positive measure on R and H_μ(x) its Hilbert transform. We prove that if lim_(t→∞)t|e∩{x||H_μ(x)|>t}| = 0, then μ_s(e) = 0, where μ_s is the singular part of μ

Caltech Authors - Main

The Hilbert transform of a measure

https://authors.library.caltech.edu/21947/3/0811.0791

The Hilbert Transform of a Measure

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Caltech Authors - Main