On the second Bohr radius for vector valued pluriharmonic functions

TL;DR

This work introduces and analyzes the second Bohr radius for vector-valued pluriharmonic functions on complete Reinhardt domains, focusing on the operator-valued setting with a bounded linear map $U:\mathcal{B}(\mathcal{H})\to Y$ and $\lVert U\rVert\le\lambda$. It establishes that the second Bohr radius constants $L_{\lambda}(\Omega, p, U)$ are strictly positive under the natural condition $\lambda>1$ and $\lVert U\rVert<\lambda$, and derives explicit finite-dimensional asymptotics using local Banach-space invariants. The paper develops a coefficient-based, Bogoliubov-style analysis for homogeneous pluriharmonic polynomials, obtaining sharp lower bounds in all $p$-regimes and matching upper bounds that yield asymptotic rates in $n$ (domain dimension) and in terms of domain-embeddings $S(\cdot,\cdot)$ and unconditional-basis constants. The framework extends to Banach sequence spaces and to the holomorphic vector-valued setting, yielding consequences that generalize and unify known scalar results and connections to first Bohr radii, providing a robust toolkit for second Bohr phenomena in several complex variables and Banach-space-valued function theory.

Abstract

In this paper, we introduce the notion of the second Bohr radius for vector valued pluriharmonic functions on complete Reinhardt domains in $\mathbb{C}^n$. This investigation is motivated by the work of Lev Aizenberg [Proc. Amer. Math. Soc. 128 (2000), 1147-1155], where the corresponding problem was studied for complex valued holomorphic functions. We show that the second Bohr radius constant for pluriharmonic functions is strictly positive under suitable condition. In addition, we obtain its asymptotic behavior in the finite-dimensional settings using invariants from local Banach space theory. Asymptotic estimates for this constant are obtained on both convex and non-convex complete Reinhardt domains. Our results also apply to a broad class of Banach sequence spaces, including symmetric and convex Banach spaces. The framework developed here also includes the second Bohr radius problem for vector valued holomorphic functions. As an application of our results, we derive several consequences that extend known results in the scalar valued setting as well as existing results in the literature.

Theorems & Definitions (19)

• Lemma 2.1
• Theorem 2.1
• proof : Proof
• Corollary 2.1
• Corollary 2.2
• Remark 2.1
• Example 2.1
• Lemma 2.2
• Theorem 2.2
• Corollary 2.3