On the nonlinear Brascamp-Lieb inequality

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.

29 pages. This article subsumes the results of arXiv:1801.05214. To appear in the Duke Mathematical Journal

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• An Algebraic Brascamp-Lieb Inequality
• A new approach to the Fourier extension problem for the paraboloid
• Equality in Liakopoulos's generalized dual Loomis-Whitney inequality via Barthe's Reverse Brascamp-Lieb inequality