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Estimates of the Second Bohr radius for vector-valued Holomorphic functions

Vibhuti Arora, Vinayak M

TL;DR

The paper defines and analyzes the lambda-second Bohr radius for vector-valued holomorphic functions on bounded complete Reinhardt domains, extending the classical Bohr phenomenon to multivariable, Banach-space-valued settings. It establishes that the radius can vanish in the polydisk for target spaces like $\overline{B_p^m}$ with $m\ge2$, while providing universal and logarithmic lower bounds, as well as finite-dimensional upper bounds and asymptotic decay results tied to the geometry of the underlying Banach spaces. The results connect the radius to symmetric Banach space structure, summing operators, and Schauder bases, and show that certain product radii involving dual spaces tend to zero, highlighting intricate interactions between domain geometry and target-space geometry. Overall, the work extends second Bohr radius theory to vector-valued holomorphic functions on Reinhardt domains and offers tools for comparing radii across domains and dimensions.

Abstract

This paper introduces the second Bohr radius for vector-valued holomorphic functions defined on arbitrary complete Reinhardt domains. We aim to establish the lower and upper bounds of the second Bohr radius in both finite and infinite-dimensional settings. Additionally, we provide specific estimates that connect the Second Bohr radius to a symmetric Banach space. We also explore the relationships between our findings and certain existing results.

Estimates of the Second Bohr radius for vector-valued Holomorphic functions

TL;DR

The paper defines and analyzes the lambda-second Bohr radius for vector-valued holomorphic functions on bounded complete Reinhardt domains, extending the classical Bohr phenomenon to multivariable, Banach-space-valued settings. It establishes that the radius can vanish in the polydisk for target spaces like with , while providing universal and logarithmic lower bounds, as well as finite-dimensional upper bounds and asymptotic decay results tied to the geometry of the underlying Banach spaces. The results connect the radius to symmetric Banach space structure, summing operators, and Schauder bases, and show that certain product radii involving dual spaces tend to zero, highlighting intricate interactions between domain geometry and target-space geometry. Overall, the work extends second Bohr radius theory to vector-valued holomorphic functions on Reinhardt domains and offers tools for comparing radii across domains and dimensions.

Abstract

This paper introduces the second Bohr radius for vector-valued holomorphic functions defined on arbitrary complete Reinhardt domains. We aim to establish the lower and upper bounds of the second Bohr radius in both finite and infinite-dimensional settings. Additionally, we provide specific estimates that connect the Second Bohr radius to a symmetric Banach space. We also explore the relationships between our findings and certain existing results.

Paper Structure

This paper contains 10 sections, 10 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. Definitions and Main Results
  3. Preliminary Results
  4. Proof of the Main Results
  5. Proof of Theorem \ref{['3Tm1']}:
  6. Proof of Theorem \ref{['3T1']}
  7. Proof of Theorem \ref{['LOGTHM']}.
  8. Proof of Theorem \ref{['THM2.12']}
  9. Proof of Theorem \ref{['T3.4.3']}
  10. Proof of Theorem \ref{['THM2.15']}

Key Result

Proposition 2.5

The first and second $\lambda$-Bohr radii of the unit polydisk in $\mathbb{C}^n$ are the same. That is, $K(\mathbb{D}^n, T, \lambda) = B(\mathbb{D}^n, T, \lambda).$ In particular, $B(\mathbb{D},\mathbb{C},1)=1/3$.

Theorems & Definitions (20)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Proposition 2.5
  • Theorem 2.6
  • Remark 2.7
  • Theorem 2.8
  • Corollary 2.9
  • proof
  • ...and 10 more