Stability of the generalized Wehrl entropy and the local concentration of homogeneous polynomials

Abstract

We study two notions of concentration for homogeneous polynomials of degree $N$ in $d+1$ 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. In both cases, the extremizers are known to be reproducing kernels, that is, monomials up to a unitary rotation, by results of Lieb--Solovej. We establish stability results for both inequalities in higher dimensions. For the local concentration, we show that for sets of sufficiently small measure, the almost-maximizers are quantitatively close to reproducing kernels, both in the polynomial and in the domain, extending previous resuls in one dimension. For the generalized Wehrl entropy, we prove that for any non-linear convex function $\Phi$ and all sufficiently large degree $N$, the reproducing kernels are the unique minimizers up to stability, complementing recent results by Nicola-Riccardi-Tilli, which require a non-linearity condition near $1$ that exclude key examples such as the concentration functional. As a consequence, by passing to the large-$N$ limit, we recover stability results for both problems in the Bargmann-Fock space.