Optimal Domain Spaces in Operator Theory

• Optimal domain spaces are defined as the maximal Banach or rearrangement-invariant spaces where an operator is bounded into a target space.
• They are key in analyzing operators such as Volterra, Cesàro, and various embeddings in Sobolev or Orlicz spaces, providing sharp norm equivalences.
• Their applications extend to spectral shape optimization and geometric analysis, offering a bridge between operator theory and practical embedding designs.

Optimal domain spaces constitute a foundational concept across function and operator theory, variational calculus, and the study of Sobolev embeddings. They single out the largest possible domain (often a Banach or rearrangement-invariant space) on which a given operator, typically sublinear or linear, is bounded into a fixed target space. This notion arises systematically in the analysis of operators on analytic function spaces (such as generalized Volterra or Cesàro operators), in spectral shape optimization problems, and in the optimal design of Sobolev or Orlicz embedding theorems for function spaces over geometric domains.

1. Abstract Definition and Theoretical Framework

Given a Banach space $X$ of functions (analytic, measurable, etc.) and a continuous linear operator $T \colon X \to X$, the optimal domain of $T$ (relative to $X$) is defined as

$[T, X] := \{ f \in \mathcal{B} : T f \in X \}$

where $\mathcal{B}$ is a larger ambient space (e.g., all analytic or all measurable functions on a fixed domain). The space is equipped with the norm

$\| f \|_{[T, X]} := \| T f \|_X.$

If $T$ is injective, this is a norm, and under mild completeness or closed range conditions, $[T, X]$ is a Banach space containing $X$ continuously. This domain is maximal: if $Z$ is any Banach space with $X \subset Z \subset \mathcal{B}$ and $T: Z \to X$ is bounded, then $Z \subset [T, X]$ as a normed space (Albanese et al., 2 Feb 2025, Albanese et al., 6 Dec 2025).

In rearrangement-invariant or weighted settings, the optimal domain is characterized by operator-specific invariants, boundedness criteria, or duality properties, with sharp minimality/maximality results.

2. Analytic Function Spaces and Operators

2.1 Volterra and Cesàro Operators

For $\mathbb{D}$ the unit disk, the generalized Volterra operator $V_g$ acts as

$(V_g f)(z) = \int_0^z f(\zeta) g'(\zeta)\, d\zeta$

and the classical Cesàro operator can be viewed as $$C f(z) = f(0) + \int_0^z f(\zeta)/(1-\zeta) \, d\zeta$$ (Albanese et al., 2 Feb 2025, Albanese et al., 6 Dec 2025).

Korenblum Growth Spaces

Consider $$A^{-\gamma} = \{ f \in H(\mathbb{D}) : \sup_{z \in \mathbb{D}} (1-|z|)^\gamma |f(z)| < \infty \}$$. For nonconstant $g$ in the Bloch space $\mathcal{B}$, the optimal domain satisfies: $$[T_g, A^{-\gamma}] = \{ f \in H(\mathbb{D}) : f g' \in A^{-(\gamma+1)} \}$$ and

$$\| f \|_{[V_g, A^{-\gamma}]} \approx \| f g' \|_{A^{-(\gamma+1)}}.$$

Hence, functions whose product with $g'$ grows as in $A^{-(\gamma+1)}$ form the optimal domain. In the Cesàro case, the optimal domain is

$$[C, A^{-\gamma}] = \{ f \in H(\mathbb{D}) : f(z)/(1-z) \in A^{-(\gamma+1)} \}$$

and is strictly larger than $A^{-\gamma}$ (Albanese et al., 2 Feb 2025, Albanese et al., 6 Dec 2025).

Hardy Spaces and More General Banach Analytic Spaces

For $H^p$, the optimal domain of the Volterra operator is

$$[T_g, H^p] = \{ f \in H(\mathbb{D}) : T_g(f) \in H^p \}$$

where $T_g$ is bounded if and only if $g \in \mathrm{BMOA}$. Substantially, the optimal domain is always strictly larger than $H^p$, and its multiplier algebra is exactly $H^\infty$ (Bellavita et al., 2024). For weighted sup-norm spaces or growth classes, the identification

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

with $w(r) = (1-r)v(r)$, is obtained whenever $g', 1/g' \in H^\infty$ (Albanese et al., 6 Dec 2025).

3. Rearrangement-Invariant Spaces, Sobolev and Orlicz Embeddings

3.1 Sobolev Embeddings and Isoperimetric Profiles

Let $\Omega \subset \mathbb{R}^n$ be a domain (possibly with weights), and $V^mX(\Omega)$ the Sobolev space based on a rearrangement-invariant (r.i.) space $X$. For an embedding

$\| u \|_Y \lesssim \| \nabla^m u \|_X$

the optimal domain (for fixed target $Y$) is explicitly characterized via an associated "Hardy-type" operator

$H_I f(t) = \int_t^1 \frac{f(s)}{I(s)} s \, ds$

where $I$ is the isoperimetric profile of the domain (Kubíček, 2024, Drážný, 2024, Musil, 2019). The optimal domain is then given by the norm

$$\|f\|_{X_Y} \approx \left\| \int_t^1 \frac{f^*(s)}{I(s)} s ds \right\|_Y$$

and for John domains or power-law isoperimetric classes, sharp exponents for Lebesgue, Lorentz, or Orlicz targets are recovered. A rearrangement-invariant domain $X$ is optimal for target $Y$ if and only if the supremum operator

$$S_I f(t) = \frac{1}{I(t)} \sup_{0 < s \le t} I(s) f^*(s)$$

is bounded on $X'$, the associate space (Kubíček, 2024).

3.2 Orlicz–Sobolev Theory

Given two Young functions $A, B$, the embedding

$W^{m, A}_0(\Omega) \hookrightarrow L^B(\Omega)$

admits an optimal Orlicz domain if and only if the local upper Boyd index $I_{B_n} < n/m$, where $B_n$ is constructed via an explicit formula (see the data), and in that case the optimal domain is $W^{m, B_n}_0(\Omega)$ (Cianchi et al., 2017, Musil, 2019). At the critical case, e.g., the Brezis–Wainger endpoint for John domains, no largest Orlicz domain exists, reflecting failure of maximality in borderline embedding situations (Beránek, 2024).

3.3 Weighted and Product Spaces

For Sobolev embeddings on cones with $\alpha$-homogeneous weights, the optimal domain is given by a specific functional construction

$D_{\mathrm{opt}} = U_m X,$

with

$$\|f\|_{U_m X} \simeq \| t^{m/(n+\alpha)} f^{**}(t) \|_{X'}.$$

In the Lorentz–Karamata setting, this recovers all classical and critical weighted Sobolev domains (Drážný, 2024).

4. Abstract Cesàro Spaces: Operator Theory in Ideal Spaces

For Banach-ideal spaces $X \subset L^0(I)$ (with $I = [0,\infty)$ or $[0,1]$), the abstract Cesàro space is

$$C X := \{ f \in L^0(I) : C|f| \in X \}, \quad C f(x) = \frac{1}{x} \int_0^x f(t) dt.$$

$C X$ is then the largest Banach ideal space for which $C : C X \to X$ is bounded (Leśnik et al., 2014). On $[0,1]$, the optimal range and domain must be described in terms of the weight $v(x) = 1-x$: $C X \to X(1/v)(v)$

where $X(1/v)(v)$ is the two-weighted space. The optimal domain is strictly between weighted and unweighted spaces in general: $X(v) \subsetneq C X \subsetneq X(1/v)(v).$ An improved Hardy inequality underpins these identification results and provides new minimality and maximality criteria for such operator domains.

5. Optimal Domain Spaces in Shape and Spectral Optimization

In spectral optimization (Dirichlet energy, first eigenvalue, etc.), the optimal domain refers to a quasi-open set (in capacity-theoretic terms) minimizing a variational or spectral cost among all admissible domains with bounded measure: $$\min_{\Omega \subset X,\, m(\Omega) \leq M} E_f(\Omega) \quad \text{or} \quad \lambda_k(\Omega)$$ with $E_f$ or $\lambda_k$ defined on Sobolev or similarly constructed spaces (Buttazzo et al., 2013, Buttazzo et al., 2018). Under mild regularity, existence of an optimizer in the quasi-open class is guaranteed; with additional uniform growth assumptions on the data, optimal domains can be shown to be open sets or have finite perimeter.

In metric graphs, the optimal domain is a compact graph or network minimizing the Dirichlet or torsional energy under length constraints, with explicit minimizers constructed under geometric constraints.

6. Operator-Theoretic and Functional Analysis Consequences

Optimal domain spaces serve as a bridge between the operator-theoretic extension, maximal inequalities, and functional analysis. Key results include:

• The domain characterizations for various integral, Toeplitz, or transfer operators,
• Explicit norm equivalence or isometric extension of the operator to the optimal domain,
• Duality and interpolation results (as in spaces arising for parabolic PDEs or Schwarz preconditioners), where Kolmogorov n-width and singular value decomposition produce local spaces optimal for prescribed approximation order (Schleuß et al., 2020, Heinlein et al., 2022).

7. Examples, Counterexamples, and Open Problems

Concrete identifications include:

• For $g(z) = z^n$, the optimal domain in Hardy spaces is a weighted $\ell^p$ sequence space.
• For Cesàro and Volterra operators on Korenblum spaces, the optimal domain is another Korenblum-type class of order increased by one (Albanese et al., 2 Feb 2025, Albanese et al., 6 Dec 2025).
• In critical Orlicz-Sobolev or limiting Sobolev embeddings on irregular domains, non-existence of an optimal domain is established via Boyd index or fundamental function analysis, and families of near-optimal scales must be used (Beránek, 2024).

Open directions include the explicit determination of optimal domains for more general or less regular operators, the extension of these theories to vector-valued or non-commutative settings, and the analysis of density or spectral properties (such as when polynomials are dense in an optimal analytic domain).

---

In summary, the theory of optimal domain spaces provides a systematic, operator-centric method for extending the boundedness of operators and embeddings to their maximal functional context—yielding new spaces, sharper norm structures, and a deep interplay with the geometry of underlying domains and analytic or measure-theoretic structure (Albanese et al., 2 Feb 2025, Albanese et al., 6 Dec 2025, Bellavita et al., 2024, Albanese et al., 6 Dec 2025, Musil, 2019, Kubíček, 2024, Drážný, 2024, Cianchi et al., 2017, Leśnik et al., 2014, Buttazzo et al., 2013, Buttazzo et al., 2018, Beránek, 2024).