Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-23T02:22:34.900Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

[1]Bergh, J. and Löfström, J., Interpolation Spaces: an Introduction (Springer, Berlin, 1976).Google Scholar
[2]Cohn, D. L., Measure Theory (Birkhäuser, Boston, 1980).Google Scholar
[3]Gohberg, I. Ts. and Krupnik, N. Ya., ‘Norm of the Hilbert transformation in the Lp space’, Functional Analysis and its Applications 2 (1968), 180–181.Google Scholar
[4]Krupnik, N. Ya., Banach Algebras with Symbols and Singular Integral Operators (Birkhäuser Verlag, Basel, 1987).Google Scholar
[5]O'Neil, R. and Weiss, G., ‘The Hilbert transform and rearrangement of functions’, Studia Math. 23 (1963), 189–198.Google Scholar
[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), 165–179.Google Scholar
[7]Riesz, M., ‘Sur les functions conguées’, Math. Z. 27 (1927), 218–244.Google Scholar
[8]Stein, E. M., Singular Integrals and Differentiability Properties of Functions (Princeton University Press, Princeton, New Jersey, 1970).Google Scholar