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

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 $\mathrm{\mid }f(x){\mathrm{\mid }}^p{\mathcal{W}}_n^{\alpha }(x),$ 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.