The Hilbert Transform of a Measure
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...
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| Sprache: | eng |
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| Zusammenfassung: | 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 $\mu$. |
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| DOI: | 10.48550/arxiv.0811.0791 |