An Algebraic Brascamp–Lieb Inequality

The Brascamp–Lieb inequalities are a very general class of classical multilinear inequalities, well-known examples of which being Hölder’s inequality, Young’s convolution inequality, and the Loomis–Whitney inequality. Conventionally, a Brascamp–Lieb inequality is defined as a multilinear Lebesgue bo...

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Veröffentlicht in:The Journal of geometric analysis Jg. 31; H. 10; S. 10136 - 10163
1. Verfasser: Duncan, Jennifer
Format: Journal Article
Sprache:Englisch
Veröffentlicht: New York Springer US 01.10.2021
Springer Nature B.V
Schlagworte:
Abstract Harmonic Analysis
Algebra
Convex and Discrete Geometry
Convolution
Differential Geometry
Dynamical Systems and Ergodic Theory
Fourier Analysis
Geometry
Global Analysis and Analysis on Manifolds
Group theory
Inequalities
Inequality
Mathematical analysis
Mathematics
Mathematics and Statistics
Polynomials
42B99
Multilinear harmonic analysis
Kakeya inequalities
Affine-invariance
Brascamp–Lieb inequalities
58C99
ISSN:1050-6926, 1559-002X
Online-Zugang:Volltext
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Zusammenfassung:The Brascamp–Lieb inequalities are a very general class of classical multilinear inequalities, well-known examples of which being Hölder’s inequality, Young’s convolution inequality, and the Loomis–Whitney inequality. Conventionally, a Brascamp–Lieb inequality is defined as a multilinear Lebesgue bound on the product of the pullbacks of a collection of functions f j ∈ L q j ( R n j ) , for j = 1 , … , m , under some corresponding linear maps B j . This regime is now fairly well understood (Bennett et al. in Geom Funct Anal 17(5):1343–1415, 2008), and moving forward there has been interest in nonlinear generalisations, where B j is now taken to belong to some suitable class of nonlinear maps. While there has been great recent progress on the question of local nonlinear Brascamp–Lieb inequalities (Bennett et al. in Duke Math J 169(17):3291–3338, 2020), there has been relatively little regarding global results; this paper represents some progress along this line of enquiry. We prove a global nonlinear Brascamp–Lieb inequality for ‘quasialgebraic’ maps, a class that encompasses polynomial and rational maps, as a consequence of the multilinear Kakeya-type inequalities of Zhang and Zorin-Kranich. We incorporate a natural affine-invariant weight that both compensates for local degeneracies and yields a constant with minimal dependence on the underlying maps. We then show that this inequality generalises Young’s convolution inequality on algebraic groups with suboptimal constant.
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ISSN:1050-6926
1559-002X
DOI:10.1007/s12220-021-00638-9