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On an Inequality for the Hilbert Transform
Author(s) -
Andersen Kenneth F.
Publication year - 1977
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-16.2.290
Subject(s) - mathematics , pure mathematics , hilbert transform , inequality , norm (philosophy) , lorentz transformation , lebesgue integration , mathematical analysis , law , physics , statistics , spectral density , classical mechanics , political science
If f ˜ denotes the Hilbert transform of f , weighted rearrangement invariant norm inequalities for convex power combinations |f| r | f ˜ | 1−r , 0 ⩽ r ⩽ 1, are derived. The class of norms under consideration includes the Lorentz and Orlicz norms as well as the usual Lebesgue L p norms, so that our results generalize, in particular, a recent inequality of S. K. Pichorides. In the course of the proof, a mild extension of the Tricomi relation for the Hilbert transform is given. The corresponding results for the periodic and discrete analogues of the Hilbert transformation are indicated.