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

TL;DR

This paper establishes a local contraction property for nearly constant holomorphic functions in weighted Bergman spaces, linking geometric stability of level sets to analytic deficit inequalities via isoperimetric stability in the Bergman ball.

Contribution

It introduces a novel approach connecting geometric isoperimetric stability with analytic deficits for holomorphic functions in Bergman spaces, providing quantitative contraction results.

Findings

Near-constant holomorphic functions exhibit contraction in weighted Bergman spaces.

Weighted level sets close to spheres imply controlled deficits and deviations.

Quantitative stability results relate geometric and analytic properties of functions.

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.