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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 generalized Kolmogorov inequality for the Hilbert transform


Author: Mark A. Pinsky
Journal: Proc. Amer. Math. Soc. 130 (2002), 753-758
MSC (2000): Primary 42A50; Secondary 44A15
Published electronically: August 28, 2001
MathSciNet review: 1866030
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Abstract:

If $f\in L^1(\mathbf{R}^1;(1+\vert x\vert)^{-1}dx)$ we can define the Hilbert transform $Hf$ almost everywhere (Lebesgue) and obtain an estimate for $\mu\{x:\,\vert Hf(x)\vert\ge \alpha\}$ where $\mu$ is a suitable finite measure. The classical Kolmogorov inequality for the Lebesgue measure of $\{x:\,\vert Hf(x)\vert\ge\alpha\}$ is obtained by a scaling argument.


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Additional Information

Mark A. Pinsky
Affiliation: Department of Mathematics, Northwestern University, Evanston, Illinois 60208-2730
Email: pinsky@math.nwu.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-01-06122-6
PII: S 0002-9939(01)06122-6
Received by editor(s): March 14, 2000
Received by editor(s) in revised form: September 11, 2000
Published electronically: August 28, 2001
Communicated by: Christopher D. Sogge
Article copyright: © Copyright 2001 American Mathematical Society