On the nonlinear Brascamp-Lieb inequality

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We prove a nonlinear variant of the general Brascamp–Lieb inequality. Our proof consists of running an efficient, or “tight,” induction-on-scales argument, which uses the existence of Gaussian near-extremizers to the underlying linear Brascamp–Lieb inequality (Lieb’s theorem) in a fundamental way. A key ingredient is an effective version of Lieb’s theorem, which we establish via a careful analysis of near-minimizers of weighted sums of exponential functions. Instances of this inequality are quite prevalent in mathematics, and we illustrate this with some applications in harmonic analysis.

Bibliographic note

39 pages. This article subsumes the results of arXiv:1801.05214 (https://arxiv.org/abs/1801.05214). Not yet published as of 02/11/2020.


Original languageEnglish
Pages (from-to)3291-3338
Number of pages48
JournalDuke Mathematical Journal
Issue number17
Early online date16 Oct 2020
Publication statusPublished - 15 Nov 2020


  • math.CA, 42B37, 44A12, 52A40