Optimal Domain of Generalized Volterra Operators

• The paper provides a precise characterization of the optimal domain for generalized Volterra operators, establishing continuity criteria in various analytic spaces.
• It utilizes explicit norm estimates and sharp inclusion results in spaces such as Korenblum growth, weighted sup-norm, and Hardy spaces.
• The study highlights functional and geometric properties of the optimal domain, offering new perspectives on analytic extension and operator theory.

The optimal domain space for a generalized Volterra operator is, given a Banach space $X$ of analytic functions and a linear operator $T: X \to X$ (typically a Volterra-type integration operator), the largest Banach space $[T, X]$ sitting between $X$ and the ambient Fréchet space $H(\mathbb{D})$ such that $T$ extends continuously and linearly as a map from $[T, X]$ into $X$. This "optimal domain" formalizes the maximal analytic extension principle for $T$ within the analytic category, revealing new function spaces depending both on the action of $T$ and the geometry of $X$ (Albanese et al., 6 Dec 2025).

1. General Setting and Definitions

Let $\mathbb{D} = \{z \in \mathbb{C}: |z| < 1\}$. For an analytic weight $\mu \in H(\mathbb{D})$, the generalized Volterra operator is defined as

$$(T_{\mu}f)(z) = \int_0^z f(\zeta) \mu(\zeta) \, d\zeta, \qquad f \in H(\mathbb{D}).$$

Given a Banach space $X \subset H(\mathbb{D})$ and a continuous linear operator $T$ with $T(X) \subset X$, the optimal domain of $T$ is

$$[T, X] = \{f \in H(\mathbb{D}) : Tf \in X\}, \qquad \|f\|_{[T, X]} := \|Tf\|_X.$$

Injectivity of $T$ on $H(\mathbb{D})$ ensures $\|\cdot\|_{[T,X]}$ is a norm, and closed-graph arguments show that $[T, X]$ is the largest (maximal) Banach space to which $T$ extends continuously as a map into $X$ (Albanese et al., 2 Feb 2025, Albanese et al., 6 Dec 2025).

2. Optimal Domains in Growth, Weighted, and Hardy Spaces

The structure of the optimal domain $[T, X]$ is governed by the interplay between $X$ and the analytic weight $\mu$ (or symbol $g$, when $T_g f = \int_0^z f(\zeta) g'(\zeta) d\zeta$).

• Korenblum Growth Spaces: For $$X = A^{-\gamma} = \{ f \in H(\mathbb{D}) : \sup_{z \in \mathbb{D}} |f(z)|(1-|z|)^\gamma < \infty \}$$ and $\gamma > 0$, the main result is:
  • $T_\mu$ maps $A^{-\gamma}$ into itself continuously if and only if $\mu \in \mathcal{B}$ (the Bloch space: $$\sup_{z \in \mathbb{D}} (1 - |z|)|\mu'(z)| < \infty$$).
  • In that case, $$[T_{\mu}, A^{-\gamma}] = \{f \in H(\mathbb{D}) : f \mu \in A^{-(\gamma+1)}\}$$, with $$\|f\|_{[T_{\mu}, A^{-\gamma}]} \simeq \|f \mu\|_{A^{-(\gamma+1)}}$$. The identification is via explicit two-sided norm estimates; any larger domain violates continuity (Albanese et al., 2 Feb 2025).
• Weighted Sup-norm Spaces: If $X = H^\infty_v$ for log-convex weights $v$, and $w(r) = (1 - r)v(r)$, then for $T_g$ with $g', 1/g' \in H^\infty$,

$[T_g, H^\infty_v] = H^\infty_w,$

with equivalent norms. This construction covers both general weights and the Korenblum setting ($v(r) = (1 - r)^\gamma$) (Albanese et al., 6 Dec 2025).

• Hardy Spaces $H^p$: For $1 \leq p < \infty$, the Volterra operator $V_g$ is bounded if and only if $g \in \mathrm{BMOA}$ (analytic functions of bounded mean oscillation). The optimal domain $[V_g, H^p] = [T_g, H^p]$ always forms a Banach space, and for compact $V_g$, strict inclusion $H^p \subsetneq [V_g, H^p]$ occurs (Albanese et al., 6 Dec 2025).

3. Sharpness, Examples, and Structural Results

Sharpness phenomena are a central feature:

• For classical and weighted Banach spaces (e.g., $H_{v_\alpha}^\infty$), the critical integrability/growth index for weighted norms is optimal: given a non-constant symbol $g$, there is a unique $\alpha_0$ such that $T_g$ is bounded on $H_{v_\alpha}^\infty$ if and only if $\alpha > \alpha_0$, and not for any $\alpha' < \alpha_0$ (Eklund et al., 2018).
• In Korenblum spaces, since $A^{-\gamma} \subsetneq A^{-(\gamma+1)}$, the optimal domain is genuinely larger than the original for nontrivial weights. When $g', 1/g' \in H^\infty$, $[T_g, A^{-\gamma}] = A^{-(\gamma+1)}$, with strict inclusion and equivalent norm (Albanese et al., 6 Dec 2025).

Table: Optimal Domain Characterizations in Analytic Banach Spaces

Operator/Space	Continuity Criterion	Optimal Domain
$T_\mu$, $A^{-\gamma}$	$\mu \in \mathcal{B}$	$\{f: f\mu \in A^{-(\gamma+1)}\}$
$T_g$, $H^\infty_v$	$g', 1/g' \in H^\infty$	$H^\infty_w$, $w(r)=(1-r)v(r)$
$T_g$, $H^p$	$g \in \mathrm{BMOA}$	$[T_g, H^p]$, explicit for $p=2$
$T_g$, $A^p_\mu$ (Dirichlet)	$g \in \mathrm{Bloch}_\mu$	$A^p_\mu$

A salient feature is that, under mild regularity/hypotheses (log-convexity of the weight, boundedness of $g'$), the optimal domain space is expressible as a classical weighted-type analytic Banach space associated to a new "shifted" or "improved" weight (Albanese et al., 6 Dec 2025).

4. Functional and Geometric Properties

The optimal domain spaces $[T, X]$ exhibit a rich geometric structure:

• Banach Space Structure: $[T, X]$ is a Banach space if $T$ is injective and the original point-evaluation functionals are continuous (Albanese et al., 2 Feb 2025).
• Multiplier Algebra: For analytic settings, the multiplier algebra of $[T_g, H^p]$ (or the corresponding meromorphic domain) is precisely $H^\infty$ (Bellavita et al., 22 Nov 2024).
• Duality and Reflexivity: For reflexive target $X$ and injective $T$, $[T, X]$ can be reflexive (e.g., in Hardy spaces with invertible multiplication operators), but in classical function space settings, lack of reflexivity can occur (e.g., Volterra in function spaces on $[0,1]$ contains a complemented copy of $L^1$) (Kiwerski et al., 2019).

Meromorphic extensions are possible: when considering $T_g$ with $g$ whose derivative vanishes inside $\mathbb{D}$, the meromorphic optimal domain $(T_g,H^p)$ describes the largest meromorphic class with $fg' \in \mathrm{Hol}(\mathbb{D})$ and $T_g(f) \in H^p$, and this can be strictly larger than the holomorphic optimal domain (Bellavita et al., 22 Nov 2024).

5. Domains for Generalizations: Dirichlet Series and Bergman Spaces

For classical and Dirichlet-series Hardy and Bergman spaces:

• In Hardy spaces $\mathcal{H}^p$ of Dirichlet series, the operator $T_g$ is bounded if and only if an explicit Carleson measure condition on $g'$ holds, admitting a sharp description for the associated "space of symbols" (Brevig et al., 2016).
• In weighted Bergman spaces $A^p_\omega$ with doubling, especially regular, weights, the optimal domain is maximal precisely when the symbol criterion reduces to membership in the Bloch space: for regular $\omega$, $T_g$ is bounded iff $g \in \mathcal{B}$ (Du et al., 2021).

6. Applications, Examples, and Extensions

• Volterra-type Operators in Function Spaces: For Banach function spaces $X$ on $[0,1]$, the Volterra optimal domain is precisely $D(V) = \{f: Vf \in X \}$ and can be identified with Cesàro-type spaces. This domain contains isomorphic copies of $L^1$ and fails both reflexivity and the Radon–Nikodym property, and its Köthe dual contains isomorphic $L^\infty$ (Kiwerski et al., 2019).
• Generalized Cesàro: Applying the same techniques, one obtains a full description for optimal domains of the Cesàro and generalized Cesàro operators in analytic and weighted spaces (Albanese et al., 6 Dec 2025).

7. Concluding Remarks and Open Problems

The optimal domain construction exposes new analytic Banach spaces, often larger and more functionally versatile than the original space $X$, and admits precise structural, density, and multiplier characterizations depending on the operator and the underlying function space. In Korenblum, weighted, and Hardy settings the optimal domain can often be described explicitly via weighted space identifications or explicit norm estimates, but for some settings (e.g., $[T_g,H^p]$ with $p\ne 2$), a fully intrinsic characterization remains open (Bellavita et al., 12 Apr 2024, Albanese et al., 6 Dec 2025).

Optimal domain spaces thus serve as a natural analytic envelope for the action of Volterra-type operators, with deep connections to Carleson measures, function-theoretic operator theory, and Banach space geometry (Albanese et al., 6 Dec 2025, Albanese et al., 2 Feb 2025, Bellavita et al., 12 Apr 2024, Eklund et al., 2018).