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On the p-norm of the truncated Hilbert transform

Published online by Cambridge University Press:  17 April 2009

W. McLean  and
D. Elliott
W. McLean
Affiliation:
Department of Mathematics, University of Tasmania, Box 252C, G.P.O., Hobart, Tasmania 7001, Australia
D. Elliott
Affiliation:
Department of Mathematics, University of Tasmania, Box 252C, G.P.O., Hobart, Tasmania 7001, Australia
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Abstract

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The p-norm of the Hilbert transform is the same as the p-norm of its truncation to any Lebesgue measurable set with strictly positive measure. This fact follows from two symmetry properties, the joint presence of which is essentially unique to the Hilbert transform. Our result applies, in particular, to the finite Hilbert transform taken over (−1, 1), and to the one-sided Hilbert transform taken over (0, ∞). A related weaker property holds for integral operators with Hardy kernels.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 38 , Issue 3 , December 1988 , pp. 413 - 420
Copyright
Copyright © Australian Mathematical Society 1988

References

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