An Algebraic Brascamp–Lieb Inequality

Jennifer Duncan

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Abstract

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 fj∈Lqj(Rnj), for j=1,…,m, under some corresponding linear maps Bj. 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 Bj 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.
Original languageEnglish
Pages (from-to)10136-10163
JournalJournal of Geometric Analysis
Volume31
Issue number10
Early online date29 Mar 2021
DOIs
Publication statusE-pub ahead of print - 29 Mar 2021

Keywords

  • Affine-invariance
  • Brascamp–Lieb inequalities
  • Kakeya inequalities
  • Multilinear harmonic analysis

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