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MAXIMAL SEMIGROUP SYMMETRY AND DISCRETE RIESZ TRANSFORMS

Part of: Harmonic analysis in several variables Lie groups

Published online by Cambridge University Press:  14 December 2015

TOSHIYUKI KOBAYASHI ,
ANDREAS NILSSON  and
FUMIHIRO SATO
TOSHIYUKI KOBAYASHI*
Affiliation:
Kavli IPMU, Japan Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba Meguro-ku, Tokyo 153-8914, Japan email toshi@ms.u-tokyo.ac.jp
ANDREAS NILSSON
Affiliation:
SAAB AB, Bröderna Ugglas gata, 58254 Linköping, Sweden email andreas.nilsson@saabgroup.com
FUMIHIRO SATO
Affiliation:
Department of Mathematics, Rikkyo University, 3-34-1 Nishi-Ikebukuro Toshima-ku, Tokyo 171-8501, Japan email sato@rikkyo.ac.jp
*
✉toshi@ms.u-tokyo.ac.jp
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Abstract

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We raise a question of whether the Riesz transform on $\mathbb{T}^{n}$ or $\mathbb{Z}^{n}$ is characterized by the ‘maximal semigroup symmetry’ that the transform satisfies. We prove that this is the case if and only if the dimension is one, two or a multiple of four. This generalizes a theorem of Edwards and Gaudry for the Hilbert transform on $\mathbb{T}$ and $\mathbb{Z}$ in the one-dimensional case, and extends a theorem of Stein for the Riesz transform on $\mathbb{R}^{n}$. Unlike the $\mathbb{R}^{n}$ case, we show that there exist infinitely many linearly independent multiplier operators that enjoy the same maximal semigroup symmetry as the Riesz transforms on $\mathbb{T}^{n}$ and $\mathbb{Z}^{n}$ if the dimension $n$ is greater than or equal to three and is not a multiple of four.

Keywords

multipliertranslation invariant operatorRiesz transformHilbert transformFourier seriessemigroup representationHasse–Minkowski principle

MSC classification

Primary: 42B15: Multipliers
Secondary: 22E46: Semisimple Lie groups and their representations 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 100 , Issue 2 , April 2016 , pp. 216 - 240
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

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