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On the log-hyperconvexity index of pseudoconvex H$\ddot{o}$lder domains in $\mathbb{C}^n$

Tianlong Yu

TL;DR

This paper proves that every bounded pseudoconvex domain in $\mathbb{C}^n$ with Hölder boundary has a positive global log-hyperconvexity index $\alpha_l(\Omega)>0$. The authors construct a continuous plurisubharmonic exhaustion function $w$ with improved boundary growth, showing $-\frac{M_1}{(-\log d(z))^{\tau}} \frac{1}{\log(-\log d(z))} \le w(z) \le -\frac{M_2}{(-\log d(z))^{\tau}} \frac{1}{\log(-\log d(z))}$, where $d(z)$ is the boundary distance and $\tau>0$. The proof combines a detailed local analysis near the Hölder boundary with a two-stage Richberg patching: first locally on boundary charts to obtain psh functions with controlled $d_t$-dependent growth, then globally to glue these into a single exhaustion on the whole domain. This extends localized results on Hölder boundaries to a global construction, providing a stronger tool for the study of plurisubharmonic functions on Hölder pseudoconvex domains.

Abstract

In this note we prove that every bounded pseudoconvex domain in $\mathbb{C}^n$ with H$\ddot{o}$lder boundary has positive log-hyperconvexity index.

On the log-hyperconvexity index of pseudoconvex H$\ddot{o}$lder domains in $\mathbb{C}^n$

TL;DR

This paper proves that every bounded pseudoconvex domain in with Hölder boundary has a positive global log-hyperconvexity index . The authors construct a continuous plurisubharmonic exhaustion function with improved boundary growth, showing , where is the boundary distance and . The proof combines a detailed local analysis near the Hölder boundary with a two-stage Richberg patching: first locally on boundary charts to obtain psh functions with controlled -dependent growth, then globally to glue these into a single exhaustion on the whole domain. This extends localized results on Hölder boundaries to a global construction, providing a stronger tool for the study of plurisubharmonic functions on Hölder pseudoconvex domains.

Abstract

In this note we prove that every bounded pseudoconvex domain in with Hlder boundary has positive log-hyperconvexity index.
Paper Structure (2 sections, 1 theorem, 43 equations)

This paper contains 2 sections, 1 theorem, 43 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem 1

Key Result

theorem 1

Let $\Omega$ be a bounded pseudoconvex domain in $\mathbb{C}^n$ with Hölder boundary. Then for some $\tau>0$, there exists a continuous psh function $w(z)$ on $\Omega$ and $M_1, M_2>0$ such that

Theorems & Definitions (2)

  • definition 1
  • theorem 1