Abstract
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 limt→∞ t|e∩{x ‖H µ(x)| > t}| = 0, then µ s (e) = 0, where µs is the singular part of µ.
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Supported in part by NSF grant DMS-0800300.
Supported in part by NSF grant DMS-0652919.
Supported in part by NSF grant DMS-0965411.
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Poltoratski, A., Simon, B. & Zinchenko, M. The Hilbert transform of a measure. JAMA 111, 247–265 (2010). https://doi.org/10.1007/s11854-010-0017-0
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DOI: https://doi.org/10.1007/s11854-010-0017-0