Let us consider two notions of concentration for homogeneous polynomials in $d$ complex variables on the unit sphere: a local notion measuring the fraction of the $L^2$-norm supported on a measurable subset and a global notion given by the generalized Wehrl entropy. Lieb and Solovej proved that the extremizers in both cases are monomials up to a unitary rotation. Their result generalizes the one by Lieb in 1978 on the Wehrl entropy conjecture for coherent states in representations of the Heisenberg group to symmetric representations of the groups $\operatorname{SU}(d)$.

In this talk, we will focus on the stability of the previous inequalities. Namely, if the concentration is close to the optimal one, we will quantify how close the polynomial is to the extremizers. This is obtained in full generality in the case $d=2$, while in the case of higher dimensions restrictions on the size of the subset or on the degree of the polynomials arise. We will finally recover analogous stability results in the Bargmann–Fock space.

This is a joint work with Joaquim Ortega-Cerdà (UB-CRM).

The Zakharov-Kuznetsov equation (ZK) is a model for the propagation of waves in the context of plasma physics and can be viewed as a two-dimensional analogue of the celebrated Korteweg-de Vries equation (KdV). In this talk, we study the Cauchy problem associated with the $k$-generalized Zakharov-Kuznetsov equation (gZK) posed on $\mathbb{R} \times \mathbb{T}$, where $k \geq 2$ is an integer. We establish several new Strichartz-type estimates in the framework of Jean Bourgain's $X_{s,b}$ spaces, with the main contributions being an almost optimal linear $L^4$-estimate and a family of bilinear refinements of this bound. As a direct application, we prove multilinear $X_{s,b}$-estimates that lead to improved local well-posedness thresholds for gZK via a fixed-point iteration.