The maximal singular integral: Estimates in terms of the singular integral

Joan Verdera

doi:10.1007/978-88-470-2853-1_17

The maximal singular integral: Estimates in terms of the singular integral

Joan Verdera

Department of Mathematics

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

Abstract

© Springer-Verlag Italia 2013. This paper considers estimates of the maximal singular integral T ⋆ f in terms of the singular integral Tf only. The most basic instance of the estimates we look for is the L 2 (ℝ n ) inequality ║T ⋆ f ║ 2 ≤ C║Tf║ 2 . We present the complete characterization, recently obtained by Mateu, Orobitg, Pérez and the author, of the smooth homogeneous convolution Calderón–Zygmund operators for which such inequality holds. We focus attention on special cases of the general statement to convey the main ideas of the proofs in a transparent way, as free as possible of the technical complications inherent to the general case. Particular attention is devoted to higher Riesz transforms.

Original language	English
Pages (from-to)	425-447
Journal	Springer INdAM Series
Volume	3
DOIs	https://doi.org/10.1007/978-88-470-2853-1_17
Publication status	Published - 1 Jan 2017

Keywords

Calderón–Zygmund operators
Fourier multipliers
Maximal singular integrals

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Verdera, J. (2017). The maximal singular integral: Estimates in terms of the singular integral. Springer INdAM Series, 3, 425-447. https://doi.org/10.1007/978-88-470-2853-1_17

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AB - © Springer-Verlag Italia 2013. This paper considers estimates of the maximal singular integral T ⋆ f in terms of the singular integral Tf only. The most basic instance of the estimates we look for is the L 2 (ℝ n ) inequality ║T ⋆ f ║ 2 ≤ C║Tf║ 2 . We present the complete characterization, recently obtained by Mateu, Orobitg, Pérez and the author, of the smooth homogeneous convolution Calderón–Zygmund operators for which such inequality holds. We focus attention on special cases of the general statement to convey the main ideas of the proofs in a transparent way, as free as possible of the technical complications inherent to the general case. Particular attention is devoted to higher Riesz transforms.

KW - Calderón–Zygmund operators

KW - Fourier multipliers

KW - Maximal singular integrals

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EP - 447

JO - Springer INdAM Series

JF - Springer INdAM Series

SN - 2281-518X

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