Contraction property on complex hyperbolic ball

TL;DR

This work develops an isoperimetric and rearrangement framework on the complex hyperbolic ball $(\\mathbb B_n,g)$ to study contraction properties of holomorphic function spaces. By proving two isoperimetric inequalities via total-curvature comparisons and establishing a hyperbolic Polya–Szegő theory, the authors derive monotonicity of superlevel-set measures and weak-type estimates for Hardy spaces. They then show that, for convex increasing $G$, the extremal problem $\\max \\int G(|f|^p(1-|z|^2)^{\alpha}) dv_g$ under $\\|f\\|_{A_\alpha^p}=1$ is achieved at the constant function, yielding a contraction of embeddings between Hardy and weighted Bergman spaces in higher dimensions. The results generalize Kulikov’s two-dimensional contraction results to $n\ge1$, with potential implications for holomorphic function estimates in complex hyperbolic geometry.

Abstract

We prove an isoperimetric inequalitie on the complex hyperbolic ball with Assumption \ref{assumption}}. As an application, we prove a contraction property for the holomorphic functions in Hardy and weighted Bergman spaces on the complex hyperbolic ball with this assumption. The results can be seen as partial generalization of Kulikov's result on the complex hyperbolic plane.

Theorems & Definitions (26)

• Definition 1.1
• Definition 1.2
• Theorem 1.3
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Corollary 1.8
• Theorem 2.1
• Lemma 2.2
• Theorem 2.3