Contraction properties for holomorphic functions via isoperimetric stability on the Bergman ball

Abstract

We prove a local contraction property for holomorphic functions that are nearly constant, relating weighted Bergman spaces $A^p_\alpha(\B_n)$ and $A^q_\beta(\B_n)$. Our approach converts geometric information on weighted superlevel sets into analytic deficit inequalities and rests crucially on a quantitative stability result (of Fuglede type) for the isoperimetric inequality in the Bergman ball. As an application, along the contractive line $q/p=\beta/\alpha$, we obtain a deficit contraction near the extremizer $f\equiv 1$: if $f=1+\phi$ with $\phi$ small and its weighted level sets are nearly spherical (after recentering), then the $A^q_\beta$-deficit is controlled by the $A^p_\alpha$-deficit, and the same deficit quantitatively controls the deviation of the level sets from spheres.