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On the p-norm of the truncated Hilbert transform

  • W. McLean (a1) and D. Elliott (a1)
Abstract

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.

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References
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[1]Bergh J. and Löfström J., Interpolation Spaces: an Introduction (Springer, Berlin, 1976).
[2]Cohn D. L., Measure Theory (Birkhäuser, Boston, 1980).
[3]Gohberg I. Ts. and Krupnik N. Ya., ‘Norm of the Hilbert transformation in the Lp space’, Functional Analysis and its Applications 2 (1968), 180181.
[4]Krupnik N. Ya., Banach Algebras with Symbols and Singular Integral Operators (Birkhäuser Verlag, Basel, 1987).
[5]O'Neil R. and Weiss G., ‘The Hilbert transform and rearrangement of functions’, Studia Math. 23 (1963), 189198.
[6]Pichorides S. K., ‘On the best values of the constants arising in the theorems of M. Riesz, Zygmund and Kolmogorov’, Studia Math. 44 (1972), 165179.
[7]Riesz M., ‘Sur les functions conguées’, Math. Z. 27 (1927), 218244.
[8]Stein E. M., Singular Integrals and Differentiability Properties of Functions (Princeton University Press, Princeton, New Jersey, 1970).
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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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