Bochner-Sobolev Spaces

• Bochner-Sobolev spaces are function spaces that extend classical Sobolev spaces to Banach-valued settings, incorporating weak differentiability and integrability.
• They underpin the analysis of vector-valued PDEs, geometric problems on singular spaces, and time-space mixed regularity in evolution equations.
• Their robust functional-analytic properties, including compactness and density, enable precise error estimates in numerical approximations and neural network methods.

A Bochner-Sobolev space is a function space that generalizes classical Sobolev spaces to the setting of Banach space-valued functions. The extension of weak differentiability and integrability concepts to vector-valued and Banach-valued functions, and its connections to the geometry of the target space, yields two competing, but closely related, frameworks: the classical Bochner-Sobolev (or weak-derivative) space and the Sobolev–Reshetnyak space. Bochner-Sobolev spaces are central to modern analysis on vector-valued PDEs, the functional calculus of evolution equations, numerical approximation, and geometric analysis on singular spaces.

1. Definitions and Basic Structure

Given an open set $\Omega\subset\mathbb{R}^N$, $1\le p<\infty$, and a Banach space $V$ with norm $\|\cdot\|_V$, the Bochner-Sobolev space $W^{1,p}(\Omega;V)$ consists of all strongly measurable functions $u:\Omega\to V$ such that $u\in L^p(\Omega;V)$ and every weak partial derivative $\partial_{x_i}u$ exists and lies in $L^p(\Omega;V)$. Specifically,

$$\|u\|_{W^{1,p}(\Omega;V)} = \|u\|_{L^p(\Omega;V)} + \|\nabla u\|_{L^p(\Omega;V)},$$

with $\nabla u = (\partial_{x_1}u,\dots,\partial_{x_N}u)$ defined in the distributional sense. The vector-valued weak derivatives are characterized via duality: $$\int_\Omega \langle u(x),\,\partial_{x_i}\varphi(x)\rangle_{V,V^*} \,dx = - \int_\Omega \langle \partial_{x_i}u(x),\,\varphi(x)\rangle_{V,V^*}\,dx$$ for test functions $\varphi\in C_c^\infty(\Omega)$ and all $i$.

For time-dependent or mixed regularity settings, if $I\subset\mathbb{R}$ is a time interval and $X$ a Banach space, the space $W^{m,p}(I;X)$ comprises all $u$ in $L^p(I;X)$ whose weak $k$th derivatives belong to $L^p(I;X)$ for $k=1,\dots,m$ (Abdeljawad et al., 2020). The norm is

$$\|u\|_{W^{m,p}(I;X)}^p = \sum_{k=0}^m \|u^{(k)}\|_{L^p(I;X)}^p.$$

For non-integer $s>0$, fractional smoothness is included through an appropriate semi-norm involving difference quotients of the $m$-th derivative.

2. Comparison to Sobolev–Reshetnyak Spaces and the Radon–Nikodým Property

The Sobolev–Reshetnyak space $R^{1,p}(\Omega;V)$ provides an alternative by imposing requirements only on scalar-valued compositions $v^*\circ u$, where $v^*\in V^*$. A function $u\in L^p(\Omega;V)$ lies in $R^{1,p}(\Omega;V)$ if for every $v^*\in V^*$ with $\|v^*\|_{V^*}\le 1$, $v^*\circ u\in W^{1,p}(\Omega)$, and there is a Borel function $g\in L^p(\Omega)$ such that $|D(v^*\circ u)(x)|\leq g(x)$ for almost every $x$ (Caamaño et al., 2020).

The main structural result is: $W^{1,p}(\Omega;V) = R^{1,p}(\Omega;V) \text{ (with equivalent norms) if and only if %%%%33%%%% has the Radon–Nikodým property (RNP)}.$ Here, the RNP is a geometric property of Banach spaces: $V$ has RNP if every Lipschitz $f:[a,b]\to V$ is differentiable almost everywhere. Reflexive Banach spaces have the RNP, guaranteeing equality of the two spaces. For spaces such as $V = \ell^\infty$ or $V = C([0,1])$, which lack the RNP, $W^{1,p}$ is strictly contained in $R^{1,p}$ (Caamaño et al., 2020).

3. Functional-Analytic Properties

Bochner-Sobolev spaces inherit several fundamental properties from the scalar-valued case, with necessary adjustments for the Banach-valued setting:

• Banach Structure: $W^{s,p}(I;X)$ is a Banach space for all $1\le p\le\infty$ and $s\ge0$ (Abdeljawad et al., 2020).
• Density: $C^\infty(I;X)$ is dense in $W^{s,p}(I;X)$ when $p<\infty$.
• Reflexivity: If $1<p<\infty$ and $X$ is reflexive, then $W^{s,p}(I;X)$ is reflexive.
• Sobolev Embeddings: For $s_1>s_2\ge0$, $W^{s_1,p}(I;X)\hookrightarrow W^{s_2,p}(I;X)$ continuously. If $s>1/p$, $W^{s,p}(I;X)$ embeds continuously into $C^{s-1/p}(\bar{I};X)$.
• Compactness: Aubin–Lions type compactness results hold for suitable triples of Banach spaces $X\hookrightarrow\hookrightarrow Y\hookrightarrow Z$ and exponents $1<p,q<\infty$.

A plausible implication is that these properties are essential for the well-posedness and regularity theory of PDEs in vector-valued settings.

4. Bochner–Sobolev Spaces in Geometric and Singular Settings

On manifolds and in geometric analysis, Bochner-Sobolev spaces are defined for sections of hermitian vector bundles. For a (possibly singular) Riemannian manifold $(M,g)$ and a complex vector bundle $E\to M$ equipped with metric connection $\nabla$, the spaces $W^{k,2}(M,E)$ consist of all $L^2$ sections with up to $k$ covariant derivatives in $L^2$. Specifically,

$$W^{k,2}(M,E) = \{ s\in L^2(M,E) : \nabla^j s\in L^2(M, T^*M^{\otimes j}\otimes E),\, 0\leq j\leq k \},$$

with norm

$\|s\|_{W^{k,2}(M,E)}^2 = \sum_{j=0}^k \int_M |\nabla^j s|^2_{h,g}\, d\vol_g$

(Bei, 2015).

In the context of irreducible complex projective varieties with the induced Kähler metric, $$W^{1,2}_0(\mathrm{reg}(V),E)=W^{1,2}(\mathrm{reg}(V),E)$$ (the closure with respect to compactly supported smooth sections coincides with the full space) and the Sobolev embedding is compact. The associated Bochner Laplacian admits a Friedrichs extension with discrete spectrum, and its heat semigroup is trace-class.

5. Bochner-Sobolev Spaces with Mixed (Anisotropic) Regularity

Bochner-Sobolev spaces are crucial in problems where the regularity and integrability in different variables (e.g., time and space in evolution equations) are distinct. Let $I\subset\mathbb{R}$ (time) and $\Omega\subset\mathbb{R}^d$ (space), and consider functions $u: I \to W^{n,p}(\Omega)$. The mixed-norm Bochner–Sobolev space is defined by

$$W_{m,p}^{n,q}(I,\Omega) := \left\{ u \in L^p\big(I; W^{n,q}(\Omega)\big) : \partial_t^k u \in L^p\big(I; W^{n,q}(\Omega)\big),\, 0\leq k \leq m \right\}$$

with norm

$$\|u\|_{W_{m,p}^{n,q}(I,\Omega)} = \left( \sum_{k=0}^m \|\partial_t^k u\|_{L^p(I; W^{n,q}(\Omega))}^p \right)^{1/p}.$$

Anisotropic Bochner–Sobolev spaces capture properties when regularity is significantly different in each variable, which typically arises for time-dependent PDEs and their approximation theory (Abdeljawad et al., 2023, Abdeljawad et al., 2020).

6. Applications in PDE Theory and Approximation

Bochner–Sobolev spaces are the natural environment for solutions of vector-valued and time-dependent PDEs, supporting the formulation of weak derivatives and mixed regularity. In contemporary approximation theory, these spaces serve as the setting for measuring the error of neural network approximations for solutions of PDEs. Shallow neural networks (SNNs) and deep networks have been shown to achieve quantifiable and sometimes dimension-independent approximation rates in these norms (Abdeljawad et al., 2023, Abdeljawad et al., 2020). For instance, for target functions in suitable weighted Fourier–Lebesgue spaces, SNNs achieve rates of

$$\inf_{u_N\in\Sigma^{N}} \|u-u_N\|_{W_{n_1,p_1}^{n_2,p_2}(I,\Omega)} \leq C N^{-1/2} T^{1/p_1} 2^{d/p_2} \|u\|_{\mathscr FL^{q_1,q_2}(\tilde{\omega})}$$

and deep networks with Rectified Cubic Unit activation match the best-possible algebraic rates in Bochner-Sobolev norms (Abdeljawad et al., 2023, Abdeljawad et al., 2020).

7. Summary Table: Key Types of Bochner-Sobolev Spaces

Space Type	Definition (brief)	Key Property / Condition
$W^{1,p}(\Omega;V)$	Banach-valued weakly differentiable functions	Closed Banach space
$R^{1,p}(\Omega;V)$	Scalar-projection-based weak differentiability	May be larger than $W^{1,p}$
Equivalence $W^{1,p}=R^{1,p}$	Holds if and only if target $V$ has RNP	RNP: a.e. differentiability of Lipschitz $V$-valued maps
$W^{m,p}(I;X)$	Time-wise Bochner-Sobolev for Banach space $X$	Fractional smoothness via difference quotients
$W_{m,p}^{n,q}(I,\Omega)$	Mixed space–time Bochner–Sobolev (anisotropic)	Crucial for PDEs, neural network approximation

• "Sobolev spaces of vector-valued functions" (Caamaño et al., 2020)
• "Space-Time Approximation with Shallow Neural Networks in Fourier Lebesgue spaces" (Abdeljawad et al., 2023)
• "Sobolev spaces and Bochner Laplacian on complex projective varieties and stratified pseudomanifolds" (Bei, 2015)
• "Approximations with deep neural networks in Sobolev time-space" (Abdeljawad et al., 2020)