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$H^2-$Corona problem on $δ-$regular domains

Bo-Yong Chen, Xu Xing

Abstract

We prove an $H^2-$Corona theorem with estimate $C(δ)=Cδ^{-1-q}|\log δ|$ for $δ\ll 1$ on delta-regular domains, where $q=\min\{n,m-1\}$ and $m$ is the number of generators. This class of domains includes smooth bounded domains with defining functions that are plurisubharmonic on boundaries and pseudoconvex domains of D'Angelo finite type.

$H^2-$Corona problem on $δ-$regular domains

Abstract

We prove an Corona theorem with estimate for on delta-regular domains, where and is the number of generators. This class of domains includes smooth bounded domains with defining functions that are plurisubharmonic on boundaries and pseudoconvex domains of D'Angelo finite type.
Paper Structure (8 sections, 10 theorems, 132 equations)

This paper contains 8 sections, 10 theorems, 132 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Carleson-type inequalities
  4. Hardy space
  5. A sufficient condition for delta-regularity
  6. The case of two generators
  7. The general case
  8. A heuristic remark on the three-weights technique

Key Result

Theorem 1.1

If $\Omega\subset \mathbb C^n$ admits a plurisubharmonic (psh) defining function on $\Omega$, then the $H^2-$corona problem is solvable in the positive sense with estimate $C(\delta)=C\delta^{-1-q}|\log\delta|$ for $\delta\ll 1$, where $q=\min\{n,m-1\}$ and $C$ is independent of $m$.

Theorems & Definitions (18)

  • Theorem 1.1: cf. Andersson, Andersson2
  • Definition 1.1: cf. ChenFu
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2: compare AnderssonBook
  • proof
  • Proposition 2.3
  • proof
  • Lemma 4.1: cf. Ohsawa, p. 89
  • ...and 8 more