Log Subharmonic Weighted Bergman Spaces

• Logarithmically subharmonic weighted Bergman spaces are function spaces of holomorphic functions with integrability against logarithmic or log-subharmonic weights, effectively merging classical Bergman theory with potential and subharmonic function analysis.
• They are characterized by subtle weight behaviors near boundaries and at infinity, which drive precise kernel estimates, duality relations, and operator-theoretic results in harmonic and geometric analysis.
• Methodologies include the use of isoperimetric inequalities, Stein-type phenomena, and refined asymptotic analyses to delineate the borderline between polynomial and exponential growth regimes.

Logarithmically subharmonic weighted Bergman spaces are function spaces of holomorphic or real-analytic functions possessing square-integrability or $L^p$-integrability against weights defined by logarithmic or log-subharmonic functions. The study of these spaces merges classical Bergman theory, potential theory, subharmonic function theory, and fine asymptotics of reproducing kernels under slow, logarithmic-type weights. The motivation arises from harmonic analysis, operator theory, and geometric function theory, particularly in characterizing function-theoretic and operator-theoretic phenomena at the borderline of polynomial and exponential growth regimes.

1. Logarithmic Weights and Subharmonicity

Logarithmic weights are defined in terms of the positive-part logarithm and encode subtle growth both near the boundary and at infinity. On the upper half-plane, the prototypical weight is

$$\omega(z) = 1 + \ln_{+}\left(\frac{1}{\Im z}\right) + \ln_{+}(|z|), \qquad z=x+iy \in \mathbb{C}_{+},$$

where $\ln_{+}(t) = \max\{0, \ln t\}$. For a parameter $k \in \mathbb{R}$, the powers $w_k(z) = \omega(z)^k$ are relevant weights.

Subharmonicity is central: each of $1$, $\ln_{+}(1/\Im z)$, and $\ln_{+}(|z|)$ are subharmonic on $\mathbb{C}_+$, so $\omega(z)$ inherits subharmonicity. For $k \geq 1$, the function $t \mapsto t^k$ is convex and increasing, making $w_k$ subharmonic; for $0 < k < 1$, convexity fails, implying the need for alternative arguments in this range (Bonami et al., 23 Jun 2025).

2. Logarithmically Subharmonic Weighted Bergman Spaces: Definitions

For $1 \leq p < \infty$ and $k \in \mathbb{R}$, the weighted $L^p$-space on the upper half-plane is

$$L^p_{\,\omega^k}(\mathbb{C}_+) = \left\{\, f : \|f\|_{L^p_{\omega^k}}^p = \int_{\mathbb{C}_+} |f(z)|^p\, \omega(z)^k\, dV(z) < \infty \, \right\}.$$

The logarithmically subharmonic weighted Bergman space is

$$A^p(\omega^k) = L^p_{\omega^k}(\mathbb{C}_+) \cap \{\text{holomorphic functions on } \mathbb{C}_+\}.$$

For classical cases $p=2$, $k = \alpha > -1$, these specialize to standard weighted Bergman spaces with $y^\alpha$ weights, but $\omega(z)^k$ grows only logarithmically as $y \to 0$ or $|z| \to \infty$ (Bonami et al., 23 Jun 2025).

On the unit sphere or ball, log-subharmonic weights appear via the real-analytic framework of $\mathcal{B}_{\alpha,p}$, the space of analytic functions $f$ for which $|f|$ is $\Delta_S$-log-subharmonic (Laplacian on the sphere), and

$$\|f\|_{\alpha,p}^p = \frac{1}{c(\alpha)} \int_{\mathbb{R}^n} |f(x)|^p\, W_n(x)^\alpha\, dm_S(x) < \infty,$$

where $W_n$ is a radial weight solving $\Delta_S \log W_n = -1$ and decays like $\exp\{-C|x|^2\}$, ensuring control at infinity (Jaguzović et al., 17 Dec 2025).

3. Kernel Estimates, Duality, and Operator Theory

For weighted Bergman spaces $AL^2_\phi(\mathbb{D})$ on the unit disk with $C^2$–subharmonic weights $\phi$ (including logarithmic and exponential type), sharp two-sided estimates for the Bergman kernel $K_\phi(z, w)$ are available: $$|K_\phi(z,w)| e^{-\phi(z)-\phi(w)} \leq C (T(z) T(w))^{-1} \exp\{-\sigma d_\phi(z,w)\},$$ where $T(z) = (\Delta\phi(z))^{-1/2}$ and $d_\phi$ is the metric associated to $ds^2 = T(z)^{-2}|dz|^2$ (Asserda et al., 2017). The diagonal estimate

$K_\phi(z,z) \leq C T(z)^{-2} e^{2\phi(z)}$

demonstrates sharp control in the logarithmic and exponential-weight regimes.

In the upper half-plane context, spaces $A^1_{\omega^{-k}}$ are in duality (under a canonical pairing) with logarithmic Bloch-type spaces $\mathcal{B}_{\omega^k}$, consisting of holomorphic functions $g$ such that

$$\|g\|_{\mathcal{B}_{\omega^k}} = |g(i)| + \sup_{z \in \mathbb{C}_+} (\Im z) \omega(z)^k |g'(z)| < \infty.$$

This duality isomorphism parallels classical $A^1$–Bloch space duality but is sensitive to the subtlety of logarithmic weight growth and the corresponding operator kernels (Bonami et al., 23 Jun 2025).

For $k<1$, pointwise products $fg$ of $f \in A^1_{\omega^l}$ and $g \in \mathcal{B}_{\omega^k}$ fall into $A^1_{\omega^{l+k-1}}$. The pointwise multipliers of these Bloch-type spaces are likewise governed by explicit logarithmic growth and derivative conditions, with thresholds at $k=1$ distinguishing the algebraic structure.

Hankel operators $h_b(f) = P(b\bar{f})$, where $P$ is the Bergman projection, extend boundedly on $A^1_b$ precisely when $b$ lies in the Bloch-type space, with kernel estimates and boundedness conditions again controlled by the logarithmic weight structure (Bonami et al., 23 Jun 2025).

4. Extremal Properties, Isoperimetry, and Concentration

On the unit sphere, $\mathcal{B}_{\alpha,p}$ spaces of log-subharmonic functions admit sharp geometric concentration properties:

• For $f \in \mathcal{B}_{\alpha,p}$, the measure of super-level sets $\rho(t) = m_S\{ |f|^p W_n^\alpha > t \}$ is absolutely continuous, satisfying the differential inequality

$\alpha \Theta(\rho(t))\, \rho'(t) + 1/t \leq 0,$

where $\Theta(\rho)$ encodes the geometric profile of spheres via the isoperimetric inequality (Jaguzović et al., 17 Dec 2025).

• The extremal (maximizing) functions for convex functionals of the form $\int G(|f|^pW_n^\alpha)\,dm_S$ under normalization are the constant functions, reminiscent of Wehrl-type entropy bounds.

Quantitative concentration results demonstrate: if $f$ nearly achieves the maximal possible localization on balls, then $|f|$ must be close (in norm) to a sharply peaked extremizer, reflecting stability of extremality under log-subharmonic constraints (Jaguzović et al., 17 Dec 2025).

5. Stein-Type Phenomena and Logarithmic Sharpness

A fundamental insight is that for logarithmic weight regimes, the convergence or divergence of Bergman kernel integrals, projections, and operator norms is governed by critical exponents arising from integrals of the form

$$\int_2^R \frac{(\ln t)^k}{t}\,dt \sim \begin{cases} (\ln R)^{1+k}, & k > -1, \ \ln\ln R, & k = -1, \ \text{const}, & k < -1, \end{cases}$$

and similarly at the boundary, $\int_0^1 \frac{dy}{y[\ln(1/y)]^m}$.

For $A^p(\omega^k)$ spaces:

• For $k > -1$, integrability of Bergman projections requires higher logarithmic moments.
• For $k = -1$, two-step logarithmic integrability is needed.
• For $k < -1$, no extra moment condition is necessary; the weight itself ensures convergence (Bonami et al., 23 Jun 2025).

This aligns with the classical Stein phenomenon, capturing the precise borderline between integrable and non-integrable phenomena for projections and kernels under logarithmic weights.

6. Connections, Generalizations, and Examples

Classical weighted Bergman spaces with $y^\alpha$ or $(1 - |z|^2)^\alpha$ weights appear as limiting cases. For subharmonic or log-subharmonic weights with more rapid (e.g., exponential) growth, kernel estimates transition from logarithmic to exponential-type decay. Operators, multipliers, and duality phenomena deform continuously across this spectrum.

On the unit sphere, when $n=2$, the $\mathcal{B}_{\alpha,p}$ framework recovers holomorphic polynomial Bergman spaces with explicit weight $W_2(z) = (1 + |z|^2)^{-1}$, and degree cutoff $< \frac{2\alpha+2}{p}$. For $n>2$, the inclusion of general real-analytic log-subharmonic functions significantly expands the analytic and geometric landscape (Jaguzović et al., 17 Dec 2025).

Prototypical examples include:

• $\phi(z) = -A \log(1 - |z|^2)$, yielding $T(z) = (1 - |z|^2)/\sqrt{A}$ and reproducing classical kernel formulas (Asserda et al., 2017).
• Weights combining logarithmic and negative power terms, e.g., $$\phi(z) = -A \log(1 - |z|^2) + B(1 - |z|^2)^{-\alpha}$$, broadening the class to complex exponential-type behaviors.
• On the upper half-plane, the pure logarithmic weight $\omega$ exemplifies the slowest nontrivial growth ensuring subharmonicity and fine operator-theoretic transitions (Bonami et al., 23 Jun 2025).

A plausible implication is that logarithmically subharmonic weighted Bergman spaces represent the critical threshold for function-theoretic and operator-theoretic behavior between polynomial and exponential regimes, making them central objects in complex analysis, harmonic analysis, and geometric analysis.