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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: J.Bennett@bham.ac.uk
EUNHEE JEONG
Affiliation:
Department of Mathematics Education and Institute of Pure and Applied Mathematics, Jeonbuk National University, Jeonbuk 54896, Republic of Korea. e-mail: eunhee@jbnu.ac.kr
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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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