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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On finite Hilbert transforms


Author: Kevin F. Clancey
Journal: Trans. Amer. Math. Soc. 212 (1975), 347-354
MSC: Primary 47G05; Secondary 44A15
MathSciNet review: 0377598
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Abstract: Let E be a bounded measurable subset of the real line. The finite Hilbert transform is the operator $ {H_E}$ defined on one of the spaces $ {L^p}(E)(1 < p < \infty )$ by

$\displaystyle {H_E}f(x) = {(\pi i)^{ - 1}}\int_E {f(t){{(t - x)}^{ - 1}}\;dt;} $

here, the singular integral is interpreted as a Cauchy principal value. The main result establishes that for $ {H_E}$ to be Fredholm on $ {L^p}(E)$, when $ p \ne 2$, it is necessary and sufficient that E be equal almost everywhere to a finite union of intervals. The sufficiency of this condition was established in 1960 by H. Widom. In the case where E is not a finite union of intervals and $ p < 2$ it is shown that the operator $ {H_E}$ has an infinite dimensional null space. The method of proof is constructive.

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DOI: http://dx.doi.org/10.1090/S0002-9947-1975-0377598-5
PII: S 0002-9947(1975)0377598-5
Article copyright: © Copyright 1975 American Mathematical Society