Vector Valued Pluriharmonic Functions

• Vector valued pluriharmonic functions are mappings from complete Reinhardt domains into Banach spaces defined via holomorphic components and their adjoints.
• They are analyzed through detailed coefficient expansions and local Banach invariants to establish sharp, quantitative Bohr radius estimates.
• The framework extends classical Bohr radius theorems to operator-valued cases, linking complex analysis with modern Banach space theory.

Vector valued pluriharmonic functions constitute a central object of study in several complex variables, operator theory, and modern Banach space theory. These functions generalize the classical notion of pluriharmonicity studied in scalar-valued function theory, extending it to the setting where the codomain is a Banach space. Recent advances focus on the quantitative aspects of their coefficient expansions, with particular emphasis on Bohr-type phenomena on multidimensional Reinhardt domains, operator (or Banach space) targets, and the interplay with local Banach space invariants (Halder, 18 Jan 2026, Halder, 9 Dec 2025).

1. Definitions and Structure of Vector Valued Pluriharmonic Functions

Given a simply-connected complete Reinhardt domain Ω in $\mathbb{C}^n$, let $X$ denote a complex Banach space—often taken as the algebra $\mathcal{B}(\mathcal{H})$ of bounded operators on a Hilbert space $\mathcal{H}$. A continuous function $f: \Omega \to X$ is pluriharmonic if there exist $X$-valued holomorphic functions $h(z)$ and $g(z)$ such that

$f(z) = h(z) + g(z)^*$

where $*$ represents the adjoint in $\mathcal{B}(\mathcal{H})$. Explicitly, the expansions take the form

$$h(z) = \sum_{m=0}^\infty \sum_{|\alpha|=m} a_\alpha z^\alpha, \qquad g(z) = \sum_{m=1}^\infty \sum_{|\alpha|=m} b_\alpha z^\alpha,$$

with $a_\alpha, b_\alpha \in X$ and $z^\alpha = z_1^{\alpha_1}\cdots z_n^{\alpha_n}$. This covers both operator-valued and scalar-valued pluriharmonic functions, the latter by taking $X = \mathbb{C}$ (Halder, 18 Jan 2026, Halder, 9 Dec 2025).

2. Complete Reinhardt Domains, Banach Space Structures, and Local Invariants

A domain $\Omega \subset \mathbb{C}^n$ is Reinhardt if it is closed under separate rotations in each variable, and complete Reinhardt if it is further closed under coordinatewise domination. Convex, bounded, complete Reinhardt domains can be identified with unit balls $B_Z$ of Banach spaces $Z = (\mathbb{C}^n, \|\cdot\|)$ with 1-unconditional bases. Structural properties of these domains and the underlying Banach spaces—such as unconditional basis constants, projection and summing norms, and cotype—influence the behavior of vector valued pluriharmonic functions defined therein. For instance, the unconditionally constant (equal to 1) implies optimal symmetry and facilitates coefficient norm estimates in Bohr-type inequalities (Halder, 9 Dec 2025).

3. The Second Bohr Radius: Definition and Positivity

The second powered Bohr radius for vector valued pluriharmonic functions, denoted $L_\lambda(\Omega, p, U)$ for $1 \leq p < \infty$ and a bounded operator $U: X \to Y$ with $\|U\| \leq \lambda$, is the largest $r \in [0,1]$ for which the following inequality holds for all $f \in PH(\Omega, X)$ with $\|f\|_{\Omega, X} < \infty$:

$$\sum_{m=0}^\infty \sum_{|\alpha|=m} \sup_{z \in r\Omega} \left( \|U(a_\alpha)\|_Y^p + \|U(b_\alpha)\|_Y^p \right) |z^\alpha|^p \leq \lambda^p \|f\|_{\Omega, X}^p.$$

For $\lambda=1$ and $U = \mathrm{Id}$ the notation simplifies to $L(\Omega,p)$.

A key result is strict positivity: For $X = \mathcal{B}(\mathcal{H})$ and $U:X \to Y$ nonzero with $\|U\| < \lambda$, there exists $D>0$ (explicitly computable in terms of $\lambda$, $p$, and $\|U\|$) such that for any $n$,

$$L_\lambda(\Omega, p, U) \geq D \cdot n^{-1/p \cdot 1/(S(\Omega, \mathbb{D}^n) \cdot S(\mathbb{D}^n, \Omega))},$$

where $$S(\Omega_1, \Omega_2) = \inf\{s > 0: \Omega_1 \subset s \Omega_2\}$$. Thus, the Bohr radius is always strictly positive under the stated operator norm restriction (Halder, 18 Jan 2026, Halder, 9 Dec 2025).

4. Asymptotic Behavior and Banach Space Parameters

Refined asymptotics for $L_\lambda(\Omega, p, U)$ are controlled by local invariants of the Banach space $Z$ corresponding to the domain, such as

• $s = \| \mathrm{Id}: Z \to \ell_q^n \|$
• $t = \| \mathrm{Id} : \ell_q^n \to Z \|$
• $I_2 = \| \mathrm{Id} : \ell_2^n \to Z \|$
• $I_1 = \| \mathrm{Id} : Z \to \ell_1^n \|$

For $X = \mathcal{B}(\mathcal{H})$ finite-dimensional and $\lambda > 1$:

• For $p = 1$:

$$L_\lambda(B_Z, 1, X) \gtrsim E_1(X) \frac{1}{s t n^{1/q}} \max \left\{ \frac{1}{\sqrt{n} I_2}, \frac{1}{e I_1} \right\} \left( \frac{\lambda^p - 1}{2 \lambda^p - 1} \right)^{1/p}$$

• For $p \geq 2$:

$$L_\lambda(B_Z, p, X) \gtrsim E_3 \frac{1}{s} I_2^{-2/p} \left( \frac{\lambda^p - 1}{2 \lambda^p - 1} \right)^{1/p}$$

• For $1 < p < 2$: analogous formulas involve power interpolation between $I_2$ and $I_1$ (Halder, 18 Jan 2026).

For classical sequence spaces $Z = \ell_q^n$, this yields

$$L_\lambda(B_{\ell_q^n}, p, X) \gtrsim c \cdot n^{-(1/p + 1/q)} \left( \frac{\lambda^p - 1}{2 \lambda^p - 1} \right)^{1/p}$$

with upper bounds of order $n^{-1/p}$ up to logarithmic corrections (Halder, 9 Dec 2025).

In non-convex domains, or domains modeled by other Banach sequence spaces (e.g., Lorentz or Orlicz spaces), the same asymptotics apply with explicit adjustments via the corresponding embedding constants and basis parameters.

5. Homogeneous Bohr Radii and Extension to Holomorphic Functions

The $m$-homogeneous versions $L^m_\lambda(\Omega, p, U)$ restrict to $m$-homogeneous pluriharmonic polynomials, and the general Bohr radius satisfies

$$L_\lambda(\Omega, p, U) = \inf_m L^m_\lambda(\Omega, p, U)$$

within explicit $\lambda$-dependent factors. There exists a direct analogue for $X$-valued holomorphic functions $f(z) = \sum a_\alpha z^\alpha$, with the holomorphic second Bohr radius $B_\lambda(\Omega, p, U)$ exhibiting the same positivity and finite-dimensional asymptotics as above (Halder, 18 Jan 2026).

6. Key Analytical Tools and Mechanisms

The analysis of vector valued pluriharmonic and holomorphic Bohr radii exploits a suite of local Banach space methods:

• Coefficient-type Schwarz–Pick lemmas for pluriharmonic and holomorphic coefficients, giving direct $L^p$ bounds in terms of local Banach space invariants.
• Reduction to the $m$-homogeneous case to control monomial growth and optimize over degrees.
• Two-sided domain comparison principles, transferring estimates between general Reinhardt domains and standard reference bodies (e.g., $\mathbb{D}^n$, $B_{\ell_q^n}$) via the $S(\cdot, \cdot)$ constants.
• Use of unconditional basis constants, absolutely summing norms, and Stirling/combinatorial estimates to handle sums over multi-indices (Halder, 9 Dec 2025).

These mechanisms yield sharp quantitative theorems and unify scalar/vector, holomorphic/pluriharmonic Bohr phenomena across diverse geometric settings.

7. Applications and Extensions

The generalized framework encompasses classical results such as the multidimensional Bohr radius of Boas–Khavinson and Aizenberg for scalar holomorphic functions, as well as extensions to operator-valued cases and all finite-dimensional subspaces of symmetric Banach sequence spaces (including mixed Minkowski, Lorentz, and Orlicz spaces). In infinite-dimensional settings, the asymptotics involve the cotype of the target space and the parameters of the sequence space involved, while domain embedding and comparison theorems enable the transfer of Bohr radius asymptotics between various Reinhardt-type domains.

In summary, the study of vector valued pluriharmonic functions and their Bohr phenomena is governed by an overview of analytic function theory and local Banach space geometry, resulting in robust general positivity, explicit asymptotics, and the unification of results across the holomorphic and pluriharmonic settings (Halder, 18 Jan 2026, Halder, 9 Dec 2025).