On the nonlinear Brascamp-Lieb inequality

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Colleges, School and Institutes

Abstract

We prove a nonlinear variant of the general Brascamp-Lieb inequality. Instances of this inequality are quite prevalent in analysis, and we illustrate this with substantial applications in harmonic analysis and partial differential equations. Our proof consists of running an efficient, or "tight", induction on scales argument, which uses the existence of gaussian near-extremisers 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-minimisers of weighted sums of exponential functions.

Bibliographic note

39 pages. This article subsumes the results of arXiv:1801.05214 (https://arxiv.org/abs/1801.05214)

Details

Original languageEnglish
JournalDuke Mathematical Journal
Publication statusAccepted/In press - 19 Mar 2020

Keywords

  • math.CA, 42B37, 44A12, 52A40