Arithmetic Bohr radius and Local Banach space theory

TL;DR

This paper introduces the arithmetic Bohr radius for operator-valued pluriharmonic functions on Reinhardt domains, analyzing its asymptotic behavior using local Banach space theory, and extends classical results to broader sequence spaces.

Contribution

It develops a new concept of arithmetic Bohr radius for operator-valued functions and extends Bohr's theorem to a wide class of Banach sequence spaces.

Findings

Asymptotic estimates for the arithmetic Bohr radius in various domains.

Extension of Bohr's theorem to mixed Minkowski, Lorentz, and Orlicz spaces.

Unified framework for scalar and operator-valued pluriharmonic functions.

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.