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

David Kalaj, Jian-Feng Zhu

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

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 and . 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 , we obtain a deficit contraction near the extremizer : if with small and its weighted level sets are nearly spherical (after recentering), then the -deficit is controlled by the -deficit, and the same deficit quantitatively controls the deviation of the level sets from spheres.
Paper Structure (27 sections, 14 theorems, 227 equations)

This paper contains 27 sections, 14 theorems, 227 equations.

Table of Contents

  1. Introduction
  2. Introduction to the local theory
  3. Preliminaries
  4. Hardy and Bergman preliminaries on the disk
  5. Hardy and weighted Bergman spaces on the unit ball
  6. Hardy spaces
  7. Invariant (Bergman) measure and normalized weighted Bergman spaces
  8. The Bergman ball: metric, measure, and perimeter of level sets
  9. A coarea-type perimeter formula
  10. Fuglede-type stability in the Bergman ball
  11. Main results
  12. A local contraction theorem near the extremizer $f\equiv1$
  13. Local centered interval hypothesis.
  14. Proof of the local dominance and contraction estimates
  15. Endpoint local contraction and the Hardy bridge
  16. ...and 12 more sections

Key Result

Theorem 2.1

Let $G:[0,\infty)\to\mathbb{R}$ be convex. Then the maximum of over all $f\in A^{p}_{\alpha}$ with $\|f\|_{A^{p}_{\alpha}}=1$ is attained at constant functions (equivalently, at normalized reproducing kernels via automorphisms).

Theorems & Definitions (32)

  • Conjecture 1.1
  • Theorem 2.1: KulikovGAFA2022 Extremizers in the disk Bergman space
  • Corollary 2.2: KulikovGAFA2022 Disk contraction chain
  • Remark 2.3
  • Theorem 2.4: KalajArxiv Perimeter of a level set in the Bergman ball
  • Remark 2.5
  • Theorem 2.6: KalajArxiv Fuglede-type theorem in the Bergman ball
  • Remark 2.7
  • Remark 3.1
  • Theorem 3.2: Local dominance near $f\equiv 1$
  • ...and 22 more