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25 April 2024

» arxiv » 0811.0791

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The Hilbert Transform of a Measure
Alexei Poltoratski ; Barry Simon ; Maxim Zinchenko ;
Date:	5 Nov 2008
Abstract:	Let $frak{e}$ be a homogeneous subset of $mathbb{R}$ in the sense of Carleson. Let $mu$ be a finite positive measure on $mathbb{R}$ and $H_mu(x)$ its Hilbert transform. We prove that if $lim_{t oinfty} t\|frak{e}cap{xmid \|H_mu(x)\|>t}}=0$, then $mu_s(frak{e})=0$, where $mu_s$ is the singular part of $mu$.
Source:	arXiv, 0811.0791
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