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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A remark on singular Calderón-Zygmund theory


Authors: Michael Christ and Elias M. Stein
Journal: Proc. Amer. Math. Soc. 99 (1987), 71-75
MSC: Primary 42B25
MathSciNet review: 866432
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Abstract: It is shown that in $ {{\mathbf{R}}^n}$ the operator

$\displaystyle Hf\left( x \right) = pv\int_{ - \infty }^{ + \infty } {f\left( {{x_1} - t, \ldots {x_n} - {t^n}} \right){t^{ - 1}}dt} $

maps $ L\left( {\log L} \right)$ to weak $ {L^1}$ locally. A slight variant of the Calderón-Zygmund procedure provides a new approach to the previously known $ {L^p}$ boundedness of $ H,1 < p < \infty $. Relatively sharp bounds are obtained as $ p \to {1^ + }$, and extrapolation produces the result for $ L\left( {\log L} \right)$.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1987-0866432-6
PII: S 0002-9939(1987)0866432-6
Article copyright: © Copyright 1987 American Mathematical Society