Operator Schwarz-Pick Lemma for Pluriharmonic Maps

• The paper establishes explicit operator-norm bounds on the homogeneous coefficients of pluriharmonic maps using a coefficient-type Schwarz–Pick lemma.
• It employs a reduction to one-variable harmonic maps and duality arguments over ℓq^n domains to derive sharp estimates with constants like 4 and 4/π.
• These results extend classical scalar inequalities to operator-valued settings, providing key insights into the Bohr phenomenon in several complex variables.

A coefficient-type Schwarz–Pick lemma for operator-valued pluriharmonic maps provides explicit operator-norm bounds on the coefficients in the multivariable, operator-valued power series expansion of pluriharmonic maps defined on complete Reinhardt domains. These results generalize classical Schwarz–Pick-type inequalities to the setting of bounded pluriharmonic maps with values in the algebra of bounded linear operators on a Hilbert space, illuminating connections to the Bohr phenomenon and quantitative coefficient estimates in several complex variables (Halder, 9 Dec 2025).

1. Preliminaries and Framework

Let $\Omega \subset \mathbb{C}^n$ be a complete Reinhardt domain, i.e., a domain invariant under coordinate-wise rotations and dilations such that $(\theta_1 z_1, \ldots, \theta_n z_n) \in \Omega$ for every $z \in \Omega$ and $|\theta_j|\leq 1$. The open unit ball $B_Z$ of any finite-dimensional Banach space $Z = (\mathbb{C}^n, \|\cdot\|)$ whose canonical basis is 1-unconditional is a complete Reinhardt domain.

Let $\mathcal{H}$ be a complex Hilbert space and $X = \mathcal{B}(\mathcal{H})$ the algebra of bounded linear operators on $\mathcal{H}$, equipped with the operator norm. A bounded map $f : \Omega \to X$ is pluriharmonic if it admits an expansion

$(\theta_1 z_1, \ldots, \theta_n z_n) \in \Omega$0

where $(\theta_1 z_1, \ldots, \theta_n z_n) \in \Omega$1, and the series converges uniformly on compacta. The Banach space $(\theta_1 z_1, \ldots, \theta_n z_n) \in \Omega$2 consists of all such bounded pluriharmonic maps, with

$(\theta_1 z_1, \ldots, \theta_n z_n) \in \Omega$3

2. Statement of the Coefficient-Type Schwarz–Pick Lemma

Given $(\theta_1 z_1, \ldots, \theta_n z_n) \in \Omega$4 as above, $(\theta_1 z_1, \ldots, \theta_n z_n) \in \Omega$5 its unit ball, and $(\theta_1 z_1, \ldots, \theta_n z_n) \in \Omega$6, write the homogeneous expansion $(\theta_1 z_1, \ldots, \theta_n z_n) \in \Omega$7, with $(\theta_1 z_1, \ldots, \theta_n z_n) \in \Omega$8 being the $(\theta_1 z_1, \ldots, \theta_n z_n) \in \Omega$9-homogeneous component. For each $z \in \Omega$0 and $z \in \Omega$1:

• Operator-norm bounds:

$z \in \Omega$2

$z \in \Omega$3

• When $z \in \Omega$4 (so $z \in \Omega$5 for $z \in \Omega$6), for each multi-index $z \in \Omega$7 with $z \in \Omega$8:

$z \in \Omega$9

$|\theta_j|\leq 1$0

where $|\theta_j|\leq 1$1.

3. Methodology and Proof Outline

The proof employs a reduction to one-variable harmonic operator-valued maps and a scalarization argument:

• Step 1: One-variable reduction. Fix $|\theta_j|\leq 1$2, consider the map $|\theta_j|\leq 1$3 for $|\theta_j|\leq 1$4. The expansion for $|\theta_j|\leq 1$5 enables application of an operator-valued Schwarz–Pick lemma for harmonic maps on the unit disk, yielding the initial coefficient bounds with constant 4.
• Step 2: Scalarization for $|\theta_j|\leq 1$6. For $|\theta_j|\leq 1$7, duality with linear functionals $|\theta_j|\leq 1$8 reduces the operator case to the scalar case. The explicit sharp constant $|\theta_j|\leq 1$9 and the $B_Z$0 distortion factor are inherited from the optimal pluriharmonic Pick inequality for scalar maps on $B_Z$1.

4. Universal Constants and Asymptotic Behavior

The constant $B_Z$2 appearing in the one-variable bound is sharp for the scalar harmonic case; the $B_Z$3 factor is similarly sharp and inherited from extremal scalar cases. The distortion factor $B_Z$4 encodes "monomial distortion" depending on $B_Z$5 and the structure of the multi-index in $B_Z$6 balls, with $B_Z$7 when $B_Z$8 (the classical cube case).

These coefficient bounds serve as critical input for obtaining explicit asymptotic estimates for Bohr radii. For the unit ball of $B_Z$9 and as $Z = (\mathbb{C}^n, \|\cdot\|)$0:

• $Z = (\mathbb{C}^n, \|\cdot\|)$1 in the finite-dimensional case.
• $Z = (\mathbb{C}^n, \|\cdot\|)$2 for infinite-dimensional $Z = (\mathbb{C}^n, \|\cdot\|)$3 with cotype $Z = (\mathbb{C}^n, \|\cdot\|)$4.

5. Connection to Classical and Recent Results

When $Z = (\mathbb{C}^n, \|\cdot\|)$5 and $Z = (\mathbb{C}^n, \|\cdot\|)$6, the lemma recovers the classical Bohr–Carathéodory bounds for scalar harmonic maps. The scalar versions of these inequalities for pluriharmonic mappings were established by Hamada–Pellegrino (J. Funct. Anal., 2022). For vector-valued holomorphic functions (no conjugate terms), related coefficient bounds underlie the Bohr phenomenon results of Defant–Maestre–Schwarting (Adv. Math., 2012). The present operator-valued generalization, and the methods based on local Banach space invariants, extend and unify these frameworks in the pluriharmonic and multivariable domains (Halder, 9 Dec 2025).

6. Application: Explicit Bounds for Bohr Radii

The coefficient-type Schwarz–Pick lemma enables explicit estimation of operator-norm sums over homogeneous components on dilations $Z = (\mathbb{C}^n, \|\cdot\|)$7 for $Z = (\mathbb{C}^n, \|\cdot\|)$8 with $Z = (\mathbb{C}^n, \|\cdot\|)$9:

$\mathcal{H}$0

Estimating $\mathcal{H}$1 provides lower bounds for the operator-valued Bohr radius, as in Theorem 1.2 of (Halder, 9 Dec 2025). Analogous bounds extend to higher $\mathcal{H}$2 and nontrivial operator coefficients $\mathcal{H}$3, yielding a broad class of Bohr-phenomenon estimates for operator-valued, pluriharmonic, and holomorphic maps in several variables.

7. Significance and Structural Role

The coefficient-type Schwarz–Pick lemma for operator-valued pluriharmonic maps is a key structural result, providing the analytic machinery to link Banach space-theoretic invariants with precise coefficient and radius bounds in operator-valued function theory on multidimensional domains. This connection is instrumental in advancing quantitative theories of the Bohr phenomenon and in extending classic scalar and vector-valued results to the full operator and pluriharmonic regime (Halder, 9 Dec 2025).