Weighted Bergman Spaces Overview

• Weighted Bergman spaces are Hilbert or Banach spaces of holomorphic functions defined on complex domains using weighted Lp norms, offering a rich analytic structure.
• They exhibit reproducing kernels, support sharp interpolation and Carleson measure characterizations, and allow norm equivalences through derivative formulations.
• Applications span operator theory, invariant subspace analysis, and complex geometry, driving advanced research and practical extension theorems.

Weighted Bergman spaces are Hilbert or Banach spaces of holomorphic functions defined on domains in complex Euclidean spaces, equipped with symmetry-invariant $L^p$ norms against a positive weight. They provide a central setting for function theory, operator theory, and complex geometry, unifying classical analytic frameworks and a broad family of analytic function spaces through the selection of weights and domains.

1. Definition, Fundamental Properties, and Reproducing Kernels

Given a domain $\Omega\subseteq\mathbb{C}^n$ and a non-negative, locally integrable weight function $w:\Omega\to[0,\infty)$, the weighted Bergman space $A^p_w(\Omega)$ is the space of holomorphic functions $f$ such that

$$\|f\|_{A^p_w}^p = \int_{\Omega}|f(z)|^p w(z) dV(z) < \infty,$$

where $dV$ is Lebesgue measure, and $p\in (0,+\infty)$. The prototypical case is the unit disk $\mathbb{D}$ or the unit ball $\mathbb{B}_n$, with radial weights $w(z) = (1-|z|^2)^\alpha$, $\alpha > -1$; in this case, $A^p_\alpha$ denotes the classical weighted Bergman space.

These spaces are Banach for $p\geq1$, and for $p=2$, Hilbert. The point-evaluation functional is always bounded, so $A^2_w(\Omega)$ admits a unique reproducing kernel $K_w(z,\zeta)$, satisfying

$$f(z) = \langle f, K_w(\cdot,z) \rangle_{A^2_w}, \quad \forall f\in A^2_w.$$

Explicit kernel formulas are available for a range of weights and canonical domains, including the disk, the ball, tube domains, and certain cones (Deng et al., 2020).

For the weighted Bergman space $A^2_\alpha(\mathbb{D})$: $$K_\alpha(z,w) = \frac{\Gamma(\alpha+2)}{\pi\, \Gamma(\alpha+1)}(1 - z\bar{w})^{-2-\alpha}.$$

2. Weighted Bergman Spaces: Interpolation, Carleson Measures, Factorization

Weighted Bergman spaces admit sharp characterizations of interpolating and sampling sequences, zero sets, and Carleson measures in terms of geometric box/counting criteria involving the weight. For standard weights on the disk, a separated sequence $\{z_j\}$ is interpolating for $A^p_\alpha$ if and only if its upper uniform (Beurling) density $D^+(\{z_j\}) < (\alpha + 1)/p$ (Luecking, 2014).

Given an admissible interpolation scheme, the necessary and sufficient condition is that the density $S_\varphi^+(\{z_j\})<\alpha$, where $\varphi$ captures the logarithmic or more general weight.

Factorization holds under weak invariance conditions on the weight: if $f \in A^p_\omega$ and $1/p = 1/p_1 + 1/p_2$, there exist $f_1 \in A^{p_1}_\omega$, $f_2 \in A^{p_2}_\omega$ with $f = f_1 f_2$ and norm control (Peláez et al., 2012).

3. Derivative Norms, Littlewood-Paley Inequalities, and Maximal Functions

Weighted Bergman norms admit equivalent formulations in terms of derivatives: for $k\in\mathbb{N}$ and under broad (Bekollé–Bonami type) assumptions on the weight,

$$\|f\|_{A^p_\omega}^p \approx \int_D |f^{(k)}(z)|^p (1-|z|)^{k p} \omega(z)\, dA(z) + \sum_{j=0}^{k-1} |f^{(j)}(0)|^p,$$

with comparability constants uniform in $f$ (Rättyä, 2021).

Characterizations of Carleson measures, embedding results, and estimates for maximal functions are similarly available in geometric terms, e.g., via box or Carleson-square tests involving the weight.

4. Generalizations: Nonstandard Weights, Domains, and Spaces

Beyond algebraic powers, weights of exponential-type, rapidly increasing or logarithmic type, and arbitrary subharmonic or strictly plurisubharmonic $\varphi$ have been extensively studied, leading to spaces that interpolate between Hardy and Bergman regimes (Park, 2017, Peláez et al., 2012, Bonami et al., 23 Jun 2025).

Examples include:

• Exponential-type weights: $\omega(z) = e^{-\varphi(z)}$ with $\varphi$ smooth subharmonic, leading to "large" weighted Bergman spaces characterized by fast vanishing.
• Logarithmic weights: e.g., on the upper-half plane, $$\omega(z) = (1 + \ln_+(1/\operatorname{Im}z) + \ln_+|z|)^k$$, which lead to fine variants of Stein-type theorems for projection and duality (Bonami et al., 23 Jun 2025).
• Subharmonic weights $w(z) = e^{-p\varphi(z)} (1 - |z|^2)^{\alpha p - 1}$, for $C^2$ smooth $\varphi$ (Luecking, 2014).

Generalizations to several complex variables include polydiscs, balls, symmetric domains, and domains with finite group symmetry, where transformation rules for reproducing kernels under proper holomorphic mappings are available (Ghosh, 2021, Misra et al., 2011).

Weighted Bergman spaces on quotient domains (e.g., symmetrized polydiscs, Rudin's domains) can be described via unitary isomorphisms with relative invariants, and explicit kernel transformation formulas (Ghosh, 2021, Misra et al., 2011).

5. Invariant Subspaces, Shift Operators, and Functional Models

Invariant subspace theory for weighted Bergman spaces reveals greater structural variety than in the Hardy case. For $A^2_{n-2}$ (index $n$), the forward shift $S_n$ and adjoint $S_n^*$ generate several types of invariant subspaces:

• Partially-isometric inner multipliers generalize classical Beurling–Lax theory.
• Wandering-subspace techniques, time-varying linear systems, and realization through Stein identities and transfer functions supply a functional-model approach (Ball et al., 2012).

Weighted forward/backward shift operators and their hypercyclicity/mixing properties have explicit descriptions in terms of moment integrals involving the weight and the product of the weight-sequence; the mixing/hypercyclic dichotomy is encoded in the spectral radius relative to these moment integrals (Das et al., 20 Mar 2025).

6. Operator Theory: Hankel, Toeplitz, and Composition Operators

The study of bounded, compact, and Schatten class operators—including (weighted) composition, Hankel, and Toeplitz operators—on weighted Bergman spaces is highly developed.

• Hankel forms and operators: The boundedness of Hankel forms $H_\mu$ is equivalent to the membership of the Poisson transform $P_\omega(\mu)$ in appropriate spaces—$A^r_\omega$, $BMOA_\omega$, or a Bloch-type space, depending on $r = (1/p + 1/q)^{-1}$ (Eskandari et al., 26 Sep 2024).
• Volterra-type operators: $T_g(f)(z)=\int_{0}^{z}f(\zeta)\,g'(\zeta)\,d\zeta$ is bounded if and only if $g$ lies in a weight-dependent function space, often not conformally invariant. Schatten class membership is characterized via analytic Besov spaces (Peláez et al., 2012).
• Weighted composition operators: Boundedness and compactness on large weighted Bergman spaces correspond to weighted Carleson measure conditions, or equivalently supremum conditions on the weight-pullbacks. The Berezin transform provides Hilbert-space ($p=2$) compactness criteria (Park, 2017).

7. Poly-Bergman, Sub-Bergman, and Generalized Spaces

Weighted poly-Bergman spaces, consisting of $m$-analytic functions, admit explicit disc-polynomial orthonormal bases and hypergeometric reproducing kernels, interpolating between analytic ($m=1$) and full polyanalytic regimes (Harti et al., 2020).

Sub-Bergman Hilbert spaces are analogues of the de Branges–Rovnyak spaces, reproducing kernel spaces contractively contained in the weighted Bergman space, characterized by modifications of the kernel. Contractive containment and norm equivalence with Hardy-type spaces occur precisely for finite Blaschke symbols (Chu, 2018).

Poletsky–Stessin weighted Bergman spaces on hyperconvex domains (generalizing classical weighted Bergman and Dirichlet spaces) are constructed in terms of Monge–Ampère measures and exhaustions. Main features include parameter shift equivalence, relations to Dirichlet-type spaces, and delicate cyclicity criteria for polynomials in several variables (Ziarati, 4 Jul 2025).

8. Applications in Complex Geometry, Extension Theorems, and Open Directions

Weighted Bergman spaces furnish precise analytic tools for extension and interpolation problems in several complex variables, for instance via explicit $L^2$-jet extension with optimal constants from compact Riemann surfaces to 1-convex domains with Levi-flat boundary (Adachi, 2017).

Geometric characterizations of Bergman space membership for conformal maps, via harmonic measure and Euclidean area, provide sharp analogues of classical Hardy-theoretic statements (Karafyllia et al., 2021).

Ongoing research targets the full description of Carleson measures, zero sets, capacities, and operator spectra for spaces with general, rapidly growing, or nonstandard weights, as well as the full classification of cyclic vectors and domains where the structure of Bergman-type functional spaces is especially rich or subtle (Ziarati, 4 Jul 2025, Peláez et al., 2012).

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References:

• Weighted poly-Bergman spaces, orthonormal disc polynomial bases, and reproducing kernels (Harti et al., 2020)
• General theory of weighted Bergman spaces, reproducing kernels, and fundamental formulas (Deng et al., 2020)
• Interpolation, zero sets, and Carleson measure characterizations in weighted settings (Luecking, 2014, Rättyä, 2021, Peláez et al., 2012)
• Operator theory: boundedness of Hankel forms, Volterra/Toeplitz/Schatten class, composition (Eskandari et al., 26 Sep 2024, Park, 2017, Peláez et al., 2012)
• Generalizations to higher dimensions, pseudoreflection groups, and quotient domains (Ghosh, 2021, Misra et al., 2011)
• Cyclicity, Dirichlet-type and Poletsky–Stessin spaces (Ziarati, 4 Jul 2025)
• Shift-invariant subspaces and functional model correspondence (Ball et al., 2012)
• Levi-flat domains, extension theorems (Adachi, 2017)
• Geometric embeddings and conformal mappings (Karafyllia et al., 2021)
• Large exponential-type and rapidly increasing weights (Park, 2017, Peláez et al., 2012, Bonami et al., 23 Jun 2025)