Jonathan Bennett (Birmingham)

The nonlinear Brascamp-Lieb inequality and applications

The Brascamp--Lieb inequality is a broad generalisation of many well-known multilinear inequalities in analysis, including the multilinear H\"older, Loomis--Whitney and sharp Young convolution inequalities. There is by now a rich theory surrounding this inequality, along with diverse applications in convex geometry, partial differential equations, number theory and beyond. Of particular importance is Lieb's Theorem (1990), which states that the best constant in this inequality is exhausted by centred gaussian functions. In this talk we present a recent "nonlinear" variant of the Brascamp--Lieb inequality, and describe some of its applications in harmonic analysis and PDE. A key ingredient in our proof is a certain effective version of Lieb's theorem, providing information about the shapes of gaussian near-extremisers for the classical Brascamp--Lieb inequality. This is joint work with Stefan Buschenhenke, Neal Bez, Michael Cowling and Taryn Flock.