Inequalities that carry some kind of transversality condition are ubiquitous in multilinear harmonic analysis, among which the Brascamp-Lieb inequalities are fundamental examples. In this talk, we shall discuss some of the history of the topic, focusing in particular on nonlinear Brascamp-Lieb inequalities, and how it relates to other areas in harmonic analysis.

One of the main properties of functions of Bounded Mean Oscillation is the self-improvement of their integrability, a fact that is encoded within the John-Nirenberg theorem. In this talk, we will present an estimate for the Hardy-Littlewood maximal function that generalizes the JN theorem. This estimate can be applied to obtaining sharp weighted estimates between the Hardy-Littlewood and the sharp maximal functions. We will also discuss minimal conditions for defining BMO. These conditions describe weak local oscillations that are measured with concave functions via Luxemburg-type expressions.
This talk is based on joint work with Carlos Pérez (UPV/EHU and BCAM) and Ezequiel Rela (Universidad de Buenos Aires).

In this talk, we consider the periodic generalized Korteweg-de Vries equations (gKdV). In particular, we study gKdV with the Gibbs measure initial data. The main difficulty lies in constructing local-in-time dynamics in the support of the measure. Since gKdV is analytically ill-posed in the $L^2$-based Sobolev support, we instead prove deterministic local well-posedness in some Fourier-Lebesgue spaces containing the support of the Gibbs measure. New key ingredients are bilinear and trilinear Strichartz estimates adapted to the Fourier-Lebesgue setting. Once we construct local-in-time dynamics, we apply Bourgain's invariant measure argument to prove almost sure global well-posedness of the defocusing gKdV and invariance of the Gibbs measure.
Our result completes the program initiated by Bourgain (1994) on the invariance of the Gibbs measures for periodic gKdV equations. This talk is based on joint work with Nobu Kishimoto (RIMS, University of Kyoto).