On the nonlinear Brascamp–Lieb inequality

Abstract

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.