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 $A^p_α(\B_n)$ and $A^q_β(\B_n)$. 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 $q/p={\hat{I}}^2/\hat{I}\pm$, we obtain a deficit contraction near the extremizer $f\equiv 1$: if $f=1+\ddot{I}•$ with $\ddot{I}•$ small and its weighted level sets are nearly spherical (after recentering), then the $A_{\hat{I}}^2$-deficit is controlled by the $A_{\hat{I}}^p\pm$-deficit, and the same deficit quantitatively controls the deviation of the level sets from spheres.