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

$$f(z) = \sum_{m=0}^{\infty} \sum_{|\alpha| = m} a_\alpha z^\alpha + \sum_{m=1}^{\infty} \sum_{|\alpha| = m} b_\alpha^* \bar{z}^\alpha,$$

where $a_\alpha, b_\alpha \in X$, and the series converges uniformly on compacta. The Banach space $\mathcal{PH}(\Omega, X)$ consists of all such bounded pluriharmonic maps, with

$$\|f\|_{\Omega, X} := \sup_{z\in\Omega} \|f(z)\|_X < \infty.$$

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

Given $Z = (\mathbb{C}^n, \|\cdot\|)$ as above, $B_Z$ its unit ball, and $f \in \mathcal{PH}(B_Z, X)$, write the homogeneous expansion $f(z) = \sum_{m=0}^{\infty} P_m(z)$, with $P_m$ being the $m$-homogeneous component. For each $m\geq 1$ and $z\in B_Z$:

• Operator-norm bounds:

$$\left\|\sum_{|\alpha|=m} (a_\alpha + b_\alpha) z^\alpha \right\|_X \leq 4 \|f\|_{B_Z, X} \|I_H - a_0\|_X^{-1},$$

$$\left\|\sum_{|\alpha|=m} (a_\alpha - b_\alpha) z^\alpha \right\|_X \leq 4 \|f\|_{B_Z, X} \|I_H - a_0\|_X^{-1}.$$

• When $Z = \ell_q^n$ (so $B_Z = \{z : \sum |z_j|^q < 1\}$ for $1\leq q\leq \infty$), for each multi-index $\alpha$ with $|\alpha| = m$:

$$\|a_\alpha + b_\alpha\|_X \leq \frac{4}{\pi} \rho_\alpha \sup_{z\in B_{\ell_q^n}} \left\| \sum_{|\beta|=m} (a_\beta + b_\beta) z^\beta \right\|_X,$$

$$\|a_\alpha - b_\alpha\|_X \leq \frac{4}{\pi} \rho_\alpha \sup_{z\in B_{\ell_q^n}} \left\| \sum_{|\beta|=m} (a_\beta - b_\beta) z^\beta \right\|_X,$$

where $$\rho_\alpha := \left(\frac{m^m}{\alpha_1^{\alpha_1} \cdots \alpha_n^{\alpha_n}}\right)^{1/q}$$.

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 $z \in B_Z$, consider the map $g(\omega) = f(\omega z)$ for $\omega\in\mathbb{D}$. The expansion for $g$ 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 $\ell_q^n$. For $Z = \ell_q^n$, duality with linear functionals $\varphi \in X^*, \|\varphi\|\leq 1$ reduces the operator case to the scalar case. The explicit sharp constant $4/\pi$ and the $\rho_\alpha$ distortion factor are inherited from the optimal pluriharmonic Pick inequality for scalar maps on $B_{\ell_q^n}$.

4. Universal Constants and Asymptotic Behavior

The constant $4$ appearing in the one-variable bound is sharp for the scalar harmonic case; the $4/\pi$ factor is similarly sharp and inherited from extremal scalar cases. The distortion factor $\rho_\alpha$ encodes "monomial distortion" depending on $q$ and the structure of the multi-index in $\ell_q^n$ balls, with $\rho_\alpha \equiv 1$ when $q = \infty$ (the classical cube case).

These coefficient bounds serve as critical input for obtaining explicit asymptotic estimates for Bohr radii. For the unit ball of $\ell_q^n$ and as $n\to\infty$:

• $$R_\lambda(\ell_q^n, 1) \asymp \left(\frac{\log n}{n}\right)^{1 - 1/\min\{q, 2\}}$$ in the finite-dimensional case.
• $$R_\lambda(\ell_q^n, p, X) \gtrsim n^{(1/\mathrm{Cot}(X) - 1/p)}$$ for infinite-dimensional $X$ with cotype $\mathrm{Cot}(X)$.

5. Connection to Classical and Recent Results

When $X = \mathbb{C}$ and $m=1$, 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 $r\Omega$ for $f\in \mathcal{PH}(\Omega,X)$ with $\|f\|\leq 1$:

$$\sum_{m,|\alpha|=m} \|a_\alpha\| r^m + \sum_{m,|\alpha|=m} \|b_\alpha\| r^m \leq \sum_{m=0}^\infty \left(\sum_{|\alpha|=m} \frac{4}{\pi} \rho_\alpha\right) r^m.$$

Estimating $$\sum_{|\alpha|=m} \rho_\alpha = \binom{n+m-1}{m}^{1/q}$$ provides lower bounds for the operator-valued Bohr radius, as in Theorem 1.2 of (Halder, 9 Dec 2025). Analogous bounds extend to higher $p$ and nontrivial operator coefficients $U$, 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).