Distributional estimates for the bilinear Hilbert transform

Dmitriy Bilyk; Loukas Grafakos

doi:10.1007/BF02922131

Distributional estimates for the bilinear Hilbert transform

Dmitriy Bilyk, Loukas Grafakos

Mathematics

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Abstract

We obtain size estimates for the distribution function of the bilinear Hilbert transform acting on a pair of characteristic functions of sets of finite measure, that yield exponential decay at infinity and blowup near zero to the power -2/3 (modulo some logarithmic factors). These results yield all known L ^p bounds for the bilinear Hilbert transform and provide new restricted weak type endpoint estimates on L ^p1 × L ^p2 when either 1/p ₁ + 1/p ₂ = 3/2 or one of p ₁, p ₂ is equal to 1. As a consequence of this work we also obtain that the square root of the bilinear Hilbert transform of two characteristic functions is exponentially integrable over any compact set.

Original language	English (US)
Pages (from-to)	563-584
Number of pages	22
Journal	Journal of Geometric Analysis
Volume	16
Issue number	4
DOIs	https://doi.org/10.1007/BF02922131
State	Published - 2006

Bibliographical note

Funding Information:
Math Subject Classifications. 46B70, 42B99. Key Words and Phrases. Multilinear operators, distributional estimates. Acknowledgements andNotes. Both authors have been supported by the National Science Foundation under grant DMS 0400387 and by the University of Missouri Research Council.

Keywords

Multilinear operators
distributional estimates

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abstract = "We obtain size estimates for the distribution function of the bilinear Hilbert transform acting on a pair of characteristic functions of sets of finite measure, that yield exponential decay at infinity and blowup near zero to the power -2/3 (modulo some logarithmic factors). These results yield all known L p bounds for the bilinear Hilbert transform and provide new restricted weak type endpoint estimates on L p1 × L p2 when either 1/p 1 + 1/p 2 = 3/2 or one of p 1, p 2 is equal to 1. As a consequence of this work we also obtain that the square root of the bilinear Hilbert transform of two characteristic functions is exponentially integrable over any compact set.",

keywords = "Multilinear operators, distributional estimates",

author = "Dmitriy Bilyk and Loukas Grafakos",

note = "Funding Information: Math Subject Classifications. 46B70, 42B99. Key Words and Phrases. Multilinear operators, distributional estimates. Acknowledgements andNotes. Both authors have been supported by the National Science Foundation under grant DMS 0400387 and by the University of Missouri Research Council.",

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doi = "10.1007/BF02922131",

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AU - Bilyk, Dmitriy

AU - Grafakos, Loukas

N1 - Funding Information: Math Subject Classifications. 46B70, 42B99. Key Words and Phrases. Multilinear operators, distributional estimates. Acknowledgements andNotes. Both authors have been supported by the National Science Foundation under grant DMS 0400387 and by the University of Missouri Research Council.

PY - 2006

Y1 - 2006

N2 - We obtain size estimates for the distribution function of the bilinear Hilbert transform acting on a pair of characteristic functions of sets of finite measure, that yield exponential decay at infinity and blowup near zero to the power -2/3 (modulo some logarithmic factors). These results yield all known L p bounds for the bilinear Hilbert transform and provide new restricted weak type endpoint estimates on L p1 × L p2 when either 1/p 1 + 1/p 2 = 3/2 or one of p 1, p 2 is equal to 1. As a consequence of this work we also obtain that the square root of the bilinear Hilbert transform of two characteristic functions is exponentially integrable over any compact set.

AB - We obtain size estimates for the distribution function of the bilinear Hilbert transform acting on a pair of characteristic functions of sets of finite measure, that yield exponential decay at infinity and blowup near zero to the power -2/3 (modulo some logarithmic factors). These results yield all known L p bounds for the bilinear Hilbert transform and provide new restricted weak type endpoint estimates on L p1 × L p2 when either 1/p 1 + 1/p 2 = 3/2 or one of p 1, p 2 is equal to 1. As a consequence of this work we also obtain that the square root of the bilinear Hilbert transform of two characteristic functions is exponentially integrable over any compact set.

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KW - distributional estimates

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JO - Journal of Geometric Analysis

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Distributional estimates for the bilinear Hilbert transform

Abstract

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