Stability of Wehrl-type Functionals and Concentration Estimates on Bergman Spaces of Log-Subharmonic Functions on the Unit Sphere

TL;DR

This paper investigates stability and concentration properties of Wehrl-type functionals within Bergman spaces of log-subharmonic functions on the unit sphere, utilizing isoperimetric inequalities to establish monotonicity and extremal stability.

Contribution

It introduces new stability results for Wehrl-type functionals and concentration estimates in Bergman spaces of log-subharmonic functions, based on spherical isoperimetric inequalities.

Findings

Proved monotonicity of super-level sets of weighted functions.

Solved a maximization problem for Wehrl-type functionals.

Established stability of concentration estimates near extremizers.

Abstract

In this paper, we consider weighted Bergman spaces $\mathcal{B}_{\alpha,p}$ 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 $|f(x)|^p\mathcal{W}_n^{\alpha}(x),$ where $f\in \mathcal{B}_{\alpha,p}$ and $\mathcal{W}_n^{\alpha}(x)$ 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.