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Arithmetic Bohr radius and Local Banach space theory

Himadri Halder

Abstract

This article introduces the notion of arithmetic Bohr radius for operator valued pluriharmonic functions on complete Reinhardt domains in $\mathbb{C}^n$. Using tools from local Banach space theory, we determine its asymptotic behavior in both finite and infinite dimensions. Asymptotic estimates for this constant are derived for both convex and non-convex complete Reinhardt domains. The framework developed in this article extends the classical Minkowski-space setting to a much broader class of sequence spaces, such as mixed Minkowski, Lorentz, and Orlicz spaces. Our results also apply to a wide class of Banach sequence spaces, including symmetric and convex Banach spaces. This generality allows for a unified and systematic investigation of Bohr's theorem for both holomorphic and pluriharmonic functions. As an application of our results, we obtain several consequences extending known results in the scalar valued setting and in the existing literature.

Arithmetic Bohr radius and Local Banach space theory

Abstract

This article introduces the notion of arithmetic Bohr radius for operator valued pluriharmonic functions on complete Reinhardt domains in . Using tools from local Banach space theory, we determine its asymptotic behavior in both finite and infinite dimensions. Asymptotic estimates for this constant are derived for both convex and non-convex complete Reinhardt domains. The framework developed in this article extends the classical Minkowski-space setting to a much broader class of sequence spaces, such as mixed Minkowski, Lorentz, and Orlicz spaces. Our results also apply to a wide class of Banach sequence spaces, including symmetric and convex Banach spaces. This generality allows for a unified and systematic investigation of Bohr's theorem for both holomorphic and pluriharmonic functions. As an application of our results, we obtain several consequences extending known results in the scalar valued setting and in the existing literature.
Paper Structure (5 sections, 28 theorems, 139 equations)

This paper contains 5 sections, 28 theorems, 139 equations.

Table of Contents

  1. Introduction and the main results
  2. Application
  3. Preliminaries
  4. Auxiliary Lemmas
  5. Proof of the main results

Key Result

Theorem 1.1

Let $Z=(\mathbb{C}^n, ||.||)$ be a Banach space such that $\chi(\{e_k\}^n_{k=1})=1$. Then Moreover, if $\left\lVert U\right\rVert<\lambda$, then $AP_\lambda(B_{Z},p,U) \geq D.\frac{\left\lVert Id:Z \rightarrow \ell^n_1\right\rVert}{n} \,. \frac{1}{\sup_{z \in B_{Z}}\left\lVert z\right\rVert_{p}}$, where

Theorems & Definitions (34)

  • Theorem 1.1
  • Corollary 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Remark 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.4
  • Theorem 1.5
  • ...and 24 more