A generalized Kolmogorov inequality for the Hilbert transform

Mark A. Pinsky

doi:10.1090/S0002-9939-01-06122-6

A generalized Kolmogorov inequality for the Hilbert transform

Mark A. Pinsky^*

^*Corresponding author for this work

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

If f ∈ L¹(R¹; (1 + |x|)^-1dx) we can define the Hilbert transform H f almost everywhere (Lebesgue) and obtain an estimate for μ{x : |H f(x)| ≥ α} where μ is a suitable finite measure. The classical Kolmogorov inequality for the Lebesgue measure of {x: |H f(x) | ≥ α} is obtained by a scaling argument.

Original language	English (US)
Pages (from-to)	753-758
Number of pages	6
Journal	Proceedings of the American Mathematical Society
Volume	130
Issue number	3
DOIs	https://doi.org/10.1090/S0002-9939-01-06122-6
State	Published - 2002

ASJC Scopus subject areas

Mathematics(all)
Applied Mathematics

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10.1090/S0002-9939-01-06122-6

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AB - If f ∈ L1(R1; (1 + |x|)-1dx) we can define the Hilbert transform H f almost everywhere (Lebesgue) and obtain an estimate for μ{x : |H f(x)| ≥ α} where μ is a suitable finite measure. The classical Kolmogorov inequality for the Lebesgue measure of {x: |H f(x) | ≥ α} is obtained by a scaling argument.

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A generalized Kolmogorov inequality for the Hilbert transform

Abstract

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