Korenblum Growth Spaces

• Korenblum growth spaces are analytic function spaces defined by weighted norms that control pointwise growth near the boundary of the unit disk and other domains.
• They bridge classical spaces such as Hardy, Bergman, and Bloch, and support rigorous operator theory, cyclicity, and spectral analysis.
• Weighted variants extend their applicability to upper half-plane and tube domains, offering explicit sequence-space models and insights into PDEs.

Korenblum growth spaces are analytic function spaces encoding precise pointwise growth rates near the domain boundary, central to interpolation, operator theory, and invariant subspace analysis in several complex variables and function theory. In the standard unit disk $\mathbb{D} = \{z \in \mathbb{C} : |z| < 1\}$, the prototypical Korenblum space $A^{-\gamma}$ consists of analytic functions whose magnitude is controlled by a power of the boundary distance, fitting naturally between classical spaces such as Hardy, Bergman, and Bloch spaces. The theory extends via weighted variants and generalizations to upper half-plane domains, incorporates premeasure and entropy constructs for cyclicity and spectral classification, and yields explicit models for operator domains ranging from Volterra and Cesàro operators to composition and Hankel-type maps.

1. Fundamental Definitions and Topological Structure

For $\gamma > 0$, the Banach Korenblum space is

$$A^{-\gamma} = \left\{ f \in H(\mathbb{D}) : \|f\|_{(-\gamma)} := \sup_{z \in \mathbb{D}} (1-|z|)^\gamma |f(z)| < \infty \right\}.$$

Its closed subspace $A^{-\gamma}_0$ comprises functions vanishing at the boundary in the weighted norm: $$A^{-\gamma}_0 = \left\{ f \in A^{-\gamma} : \lim_{|z| \to 1^-} (1-|z|)^\gamma |f(z)| = 0 \right\}.$$ More generally, Fréchet and (LB)-type inductive limits over chains of weighted Banach spaces yield projective intersections $A_+^{-\gamma} = \bigcap_{\mu > \gamma} A^{-\mu}$ and inductive unions $A_-^{-\gamma} = \bigcup_{\mu < \gamma} A^{-\mu}$ (Bonet et al., 2017), possessing topologies generated by seminorms $p_\mu(f) := \|f\|_\mu$ and admitting Schauder monomial bases.

The weighted spaces extend naturally to the upper half-plane $\mathbb{H} = \{z \in \mathbb{C} : \Im z > 0\}$, with

$$K_{\omega,t}(\mathbb{H}) = \left\{ f \in \mathcal{H}(\mathbb{H}) : \sup_{z \in \mathbb{H}} (\Im z)^t \omega(z) |f(z)| < \infty \right\}$$

for suitable weight functions $\omega$ (Sehba, 10 Aug 2025), and multi-indexed variants on tube domains and products.

2. Growth Conditions, Norms, and Sequence-Space Representations

A function $f \in A^{-\gamma}$ exhibits boundary behavior $|f(z)| = O((1-|z|)^{-\gamma})$ as $|z| \to 1^-$; monomials $f(z) = z^n$ yield coefficients $|a_n| = O(n^{\gamma - 1})$ by reproducing kernel estimates (Albanese et al., 2 Feb 2025).

Weighted sup-norm and radial maximal function descriptions include

$$\|f\|_{A^{-\gamma}} = \sup_{0 \leq r < 1} (1-r)^\gamma \sup_{|z|=r} |f(z)|.$$

Sequence-space models identify $A_+^{-\gamma}$ as Köthe echelon spaces $\lambda^\infty(A_\gamma)$ and $A_-^{-\gamma}$ as co-echelon spaces $k_\infty(V_\gamma)$ via explicit weight formulas $S_\mu(j) = \sup_{0<r<1} r^j (1-r)^\mu$ (Bonet et al., 2017).

Logarithmic and more general weights permit

$$H^\infty_{\alpha, \log} = \left\{ f \in H(\mathbb{D}) : \sup_{z \in \mathbb{D}} (1-|z|)^\alpha \log \left( \frac{e}{1-|z|} \right) |f(z)| < \infty \right\}$$

(Hu et al., 22 Oct 2024), or ad hoc majorants $w(t)$ for Besov-type variants (Limani, 2023).

3. Operator Theory: Volterra, Cesàro, Composition, and Hilbert Matrix Operators

The action of generalized Volterra operators $V_g f(z) = \int_0^z f(\xi) g'(\xi) d\xi$ on $A^{-\gamma}$ spaces is characterized by optimal domain lifting: $$[V_g, A^{-\gamma}] = \{ f \in H(\mathbb{D}) : f g' \in A^{-(\gamma+1)} \},$$ with norm equivalence $$\|f\|_{[V_g, A^{-\gamma}]} \asymp \|f g'\|_{(-(\gamma+1))}$$ (Albanese et al., 2 Feb 2025, Albanese et al., 6 Dec 2025). Cesàro-type operators $C_\mu$ defined via moment sequences $\{\mu_n\}$ act continuously on $A_y^-, A_y^{0+}$ iff $\mu$ is $s$–Carleson for all $0Meneu et al., 17 Jan 2024). No nontrivial compactness is possible in these spaces.

On Korenblum spaces over tube domains, compactness and boundedness of composition operator differences is governed by explicit pseudo-hyperbolic formulas: $$\sup_{z}\left[ \prod_{k=1}^n \left( \frac{\Im z_k}{\Im \varphi_k(z)} \right)^{\gamma_k} + \prod_{k=1}^n \left( \frac{\Im z_k}{\Im \psi_k(z)} \right)^{\gamma_k} \right] \rho(z) < \infty,$$ with vanishing conditions for compactness (Liang et al., 5 Nov 2024).

The Hilbert matrix operator $\mathcal{H}$ achieves sharp norm estimates between weighted and Korenblum/Bloch-type spaces, e.g.,

$$\| \mathcal{H} \|_{H^\infty_{\alpha,\log} \to H^\infty_\alpha} = \sup_{0\le r<1} \int_0^1 \ldots dt,$$

precisely formulated via integral representations and extremal functions such as $f_\alpha(z) = (1-z)^{-\alpha} \log(2/(1-z))$ (Hu et al., 22 Oct 2024).

4. Cyclicity, Invariant Subspaces, and Spectral Theory

Cyclicity in Korenblum-type growth spaces hinges on measure-theoretic premeasures, entropy conditions, and outer-inner factorization. For spaces $\mathcal{A}_\Lambda^{-\infty}$ determined by a majorant $\Lambda$, Hanine’s cyclicity theorem asserts that

$$f_\mu(z) = \exp \left\{ \int_0^{2\pi} \frac{e^{i\theta} + z}{e^{i\theta} - z} d\mu(\theta) \right\}$$

is cyclic if and only if its $\Lambda$–singular part vanishes (Hanine, 2012).

Classification of shift-invariant subspaces in $G_w$ (growth spaces with majorant $w$) depends on whether the singular measure’s support is a finite $w$–entropy set; Beurling-type theorem generalizations apply (Limani, 2023). For Nevanlinna-class functions, all zero-free elements are cyclic provided the associated negative singular measure does not charge such sets (Bergqvist et al., 31 Mar 2025).

Spectral theory for Cesàro-type and related operators on $A^{-}$ and projective Korenblum spaces yields explicit point spectrum $\sigma_{\rm pt}(C_\mu, A^-)=\{\mu_n\}$ and full spectrum $\sigma(C_\mu, A^-)$ depending on growth rates of $\{\mu_n\}$ (Meneu et al., 17 Jan 2024).

5. Radial Growth, Regularity, and PDE Extensions

Korenblum’s radial growth estimate for harmonic functions, $|u(re^{i\theta})| = O(\log|\log(1-r)|)$, generalizes to solutions of elliptic equations on higher-dimensional balls $B^n$: $$M_v(u, r)^2 \leq |u(0)|^2 + v(v-1)\| |\nabla u|^2 + u \Delta u \|_{L_{v,w}^B} w(1) \ldots$$ for majorant $w$ and parameter $v$ (Chen et al., 2016). This connects radial boundary behavior to Dirichlet-type energy integrals and regularity results for analytic, harmonic, and elliptic PDEs.

6. Basis Properties and Sequence-Space Realizations

Monomial Schauder bases $(z^n)$ exist for Fréchet ($A_+^{-\gamma}$) and (LB)-type ($A_-^{-\gamma}$) spaces for all $\gamma > 0$, contrasting sharply with the lack of such bases in single weighted Banach spaces (Bonet et al., 2017). Functions are uniquely representable by Taylor series in explicit Köthe echelon and co-echelon sequence spaces, according to the maximal growth permitted by the Korenblum norm.

7. Connections and Classical Cases

Korenblum growth spaces interpolate between Hardy, Bergman, and Bloch spaces. Polynomial weights yield classical $A^{-\gamma}$, logarithmic weights yield finer scales, and model spaces $K_\Theta$ with inner function $\Theta$ exhibit interrelated boundary regularity depending on entropy of carrier sets (Limani, 2023). The Korenblum scale is strictly increasing in $\gamma$; as $\gamma \to \infty$, only polynomials remain.

Weighted variants on the upper half-plane and higher dimensions extend the growth-space paradigm; operator theory and entropy methods generalize cyclicity and spectral results from classical analytic function spaces.

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In summary, Korenblum growth spaces are robust analytic frameworks for controlling pointwise boundary growth, admitting precise normed and sequence-space structures, powerful operator-theoretic characterizations, and deep connections to cyclicity, spectral theory, and partial differential equations. The fine interplay between growth indices, weights, operator domains, and measure-theoretic entropy distinguishes this family from classical settings and provides structural insight for modern complex analysis and operator theory.