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Fourier duality in the Brascamp–Lieb inequality

Part of: Integral transforms, operational calculus Harmonic analysis in several variables General convexity

Published online by Cambridge University Press:  27 September 2021

JONATHAN BENNETT  and
EUNHEE JEONG
JONATHAN BENNETT
Affiliation:
School of Mathematics, University of Birmingham, Edgbaston, Birmingham, B15 2TT, United Kingdom e-mail: [email protected]
EUNHEE JEONG
Affiliation:
Department of Mathematics Education and Institute of Pure and Applied Mathematics, Jeonbuk National University, Jeonbuk 54896, Republic of Korea. e-mail: [email protected]
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Abstract

It was observed recently in work of Bez, Buschenhenke, Cowling, Flock and the first author, that the euclidean Brascamp–Lieb inequality satisfies a natural and useful Fourier duality property. The purpose of this paper is to establish an appropriate discrete analogue of this. Our main result identifies the Brascamp–Lieb constants on (finitely-generated) discrete abelian groups with Brascamp–Lieb constants on their (Pontryagin) duals. As will become apparent, the natural setting for this duality principle is that of locally compact abelian groups, and this raises basic questions about Brascamp–Lieb constants formulated in this generality.

MSC classification

Primary: 42B37: Harmonic analysis and PDE 44A12: Radon transform 52A40: Inequalities and extremum problems
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 173 , Issue 2 , September 2022 , pp. 387 - 409
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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References

Ball, K.. Volumes of sections of cubes and related problems. in: Geometric Aspects of Functional Analysis (Lindenstrauss, J., Milman, V. D., eds). Springer Lecture Notes in Math. 1376 (1989), 251–260.10.1007/BFb0090058CrossRefGoogle Scholar
Beckner, W.. Inequalities in Fourier analysis. Ann. of Math. 102 (1975), 159–182.10.2307/1970980CrossRefGoogle Scholar
Bennett, J. and Bez, N.. Some nonlinear Brascamp–Lieb inequalities and applications to harmonic analysis. J. Funct. Anal. 259 (2010), 2520–2556.10.1016/j.jfa.2010.07.015CrossRefGoogle Scholar
J. Bennett and N.Bez. Higher order transversality in harmonic analysis. To appear in RIMS Kôkyûroku Bessatsu.Google Scholar
Bennett, J., Bez, N., Buschenhenke, S., Cowling, M. G. and Flock, T. C.. On the nonlinear Brascamp–Lieb inequality. Duke Math. J. 169, no. 17 (2020), 3291–3338.10.1215/00127094-2020-0027CrossRefGoogle Scholar
Bennett, J., Carbery, A., Christ, M. and Tao, T.. The Brascamp–Lieb inequalities: finiteness, structure and extremals. Geom. Funct. Anal. 17 (2007), 1343–1415.10.1007/s00039-007-0619-6CrossRefGoogle Scholar
Bennett, J., Carbery, A., Christ, M. and Tao, T.. Finite bounds for Hölder–Brascamp–Lieb multilinear inequalities. Math. Res. Lett. 17 (2010), 647–666.10.4310/MRL.2010.v17.n4.a6CrossRefGoogle Scholar
Bez, N., Lee, S., Nakamura, S. and Sawano, Y.. Sharpness of the Brascamp–Lieb inequality in Lorentz spaces. Electron. Res. Announc. Math. Sci. 24 (2017), 53–63.Google Scholar
Brascamp, H. J. and Lieb, E. H.. Best constants in Young’s inequality, its converse, and its generalization to more than three functions. Adv. Math. 20 (1976), 151–173.10.1016/0001-8708(76)90184-5CrossRefGoogle Scholar
Bramati, R.. Brascamp–Lieb inequalities on compact homogeneous spaces. Anal. Geom. Metr. Spaces 7 (2019), no. 1, 130–157.10.1515/agms-2019-0007CrossRefGoogle Scholar
Carlen, E. A., Lieb, E. H. and Loss, M.. A sharp analog of Young’s inequality on $S^N$ and related entropy inequalities. J. Geom. Anal. 14 (2004), 487–520.10.1007/BF02922101CrossRefGoogle Scholar
Christ, M.. The optimal constants in Hölder–Brascamp–Lieb inequalities for discrete Abelian groups. arXiv:1307.8442.Google Scholar
Christ, M., Demmel, J., Knight, N., Scanlon, T. and Yelick, K.. Communication lower bounds and optimal algorithms for programs that reference arrays — Part 1. arXiv:1308.0068.Google Scholar
Christ, M., Demmel, J., Knight, N., Scanlon, T. and Yelick, K.. On Hölder–Brascamp–Lieb inequalities for torsion-free discrete abelian groups. arXiv:1510.04190.Google Scholar
Deitmar, A. and Echterhoff, S.. Principles of Harmonic Analysis (Universitext. Springer, New York, 2009).Google Scholar
Durcik, P. and Thiele, C.. Singular Brascamp–Lieb: a survey. arXiv:1904.08844.Google Scholar
Eisner, T. and Tao, T.. Large values of the Gowers–Host–Kra seminorms. J. Anal. Math. 117 (2012), 133–186.10.1007/s11854-012-0018-2CrossRefGoogle Scholar
Gowers, W. T.. A new proof of Szemerédi’s theorem. Geom. Funct. Anal. 11 (2001), no. 3, 465–588.10.1007/s00039-001-0332-9CrossRefGoogle Scholar
Hewitt, E. and Ross, K. A.. Abstract Harmonic Analysis I (Springer–Verlag, Berlin–Heidelberg–New York, 1963).Google Scholar
Lee, J. M.. Introduction to Smooth Manifolds (Springer–Verlag, New York, 2003).10.1007/978-0-387-21752-9CrossRefGoogle Scholar
Lieb, E. H.. Gaussian kernels have only Gaussian maximizers. Invent. Math. 102 (1990), 179–208.10.1007/BF01233426CrossRefGoogle Scholar
Martina Neuman, A.. Functions of nearly maximal Gowers–Host–Kra norms on euclidean spaces. J. Geom. Anal. 30 (2020), no. 1, 1042–1099.10.1007/s12220-018-00125-8CrossRefGoogle Scholar
Perry., P. Global well-posedness and long-time asymptotics for the defocussing Davey-Stewartson II equation in $H^{1,1}({\mathbb{C}})$ , with an appendix by M. Christ, J. Spectral Theory 6 (2016), 429–481.10.4171/JST/129CrossRefGoogle Scholar
Roth., K. On certain sets of integers. J. Lond. Math. Soc. 28 (1953),104–109.10.1112/jlms/s1-28.1.104CrossRefGoogle Scholar
Ceccherini–Silberstein, T., Scarabotti, F. and Tolli., F. Harmonic Analysis on Finite Groups (Cambridge University Press, Cambridge, 2008).10.1017/CBO9780511619823CrossRefGoogle Scholar
Tao., T. Higher order Fourier analysis (American Mathematical Society, Providence, RI, 2012).10.1090/gsm/142CrossRefGoogle Scholar