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

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.