Stability of Wehrl-type Functionals and Concentration Estimates on Bergman Spaces of Log-Subharmonic Functions on the Unit Sphere
Vladan Jaguzović, Petar Melentijević
TL;DR
The paper extends Wehrl-type concentration theory to Bergman spaces on the unit sphere by introducing a radial Bergman weight $\mathcal{W}_n$ from the equation $\Delta_S \log \mathcal{W}_n = -1$ and defining $\mathcal{B}_{\alpha,p}$. It proves a monotonicity principle for super-level sets of $|f|^p\mathcal{W}_n^{\alpha}$, deriving a Faber–Krahn-type bound that concentrates mass in a geodesic ball and showing that the constant function maximizes convex functionals under a unit-norm constraint. The authors establish stability results: near-extremizers are quantitatively close to the radial profile, with explicit rates depending on dimension, and they extend these stability findings to general convex Wehrl-type functionals. Technically, the work blends spherical isoperimetry, rearrangement inequalities, and the subharmonic structure with a detailed construction of the Bergman weight on $\mathbb{S}^n$, providing a higher-dimensional analogue to known two-dimensional results and enabling precise concentration estimates on Bergman spaces of log-subharmonic functions on the sphere.
Abstract
In this paper, we consider weighted Bergman spaces $\mathcal{B}_{α,p}$ of log-subharmonic functions on the unit sphere. Using the isoperimetric inequality for the spherical metric we prove certain monotonicity property for super-level sets of $|f(x)|^p\mathcal{W}_n^α(x),$ where $f\in \mathcal{B}_{α,p}$ and $\mathcal{W}_n^α(x)$ is the Bergman weight. As a consequence, we solve a maximization problem for certain Wehrl-type (convex) functionals and concentration estimates. Moreover, we show the stability of these estimates, proving that near-extremizing values are achieved for near-extremizing functions.