Weak Compactness in W^k,∞

Abstract

We characterize weak compactness in the Sobolev space $W^{k,\infty }\left(\mathrm{\Omega }\right)$. For non-reflexive spaces like $W^{k,\infty }$, criteria beyond boundedness are required. By exploiting the von Neumann algebra structure of $L^{\infty }$ via Gelfand duality, we establish a unified theory. Our main result is a necessary and sufficient condition: a subset is relatively weakly compact if and only if it is bounded and its weak derivatives up to order k have uniformly small oscillation on a finite measurable partition of $\mathrm{\Omega }$. This provides a tool for analyzing nonlinear problems in these spaces.

1. Introduction

In functional analysis and its applications, compactness in function spaces is a cornerstone for studying the existence of solutions to nonlinear problems, analyzing approximation schemes, and understanding the geometric structure of spaces. For reflexive Banach spaces (such as $L^p$ spaces and $W^{k,p}$ spaces for $1<p<\infty$), the Eberlein–Šmulian theorem [1] [p. 141] establishes that relative weak compactness is equivalent to boundedness. However, the characterization of weak compactness in non-reflexive spaces, such as $L^1$ and $L^{\infty }$, is profoundly more intricate, typically requiring conditions that go far beyond mere boundedness.

In the theory of $L^1$ spaces, the classical Dunford–Pettis theorem [2] [Theorem 3.2.1, p. 141] provides an elegant characterization: a subset is relatively weakly compact if and only if it is uniformly bounded, integrable, and tight. This theorem beautifully links weak compactness to the “uniform behavior” of the set of functions and stands as a paradigm in the field.

In contrast, the study of weak compactness in $L^{\infty }$ spaces presents unique challenges. On the one hand, $L^{\infty }$ is the dual space of $L^1$, yet its own predual ($L^1$) is separable, leading to specific behaviors of its weak topology. On the other hand, $L^{\infty }$ possesses a rich algebraic structure; it is a commutative von Neumann algebra. This key fact allows us to employ Gelfand duality: any $L^{\infty }(X;\mathbb{C})$ is isometrically *-isomorphic to the algebra of continuous functions $C\left(\mathrm{\Delta }\right)$ on a compact Hausdorff space $\mathrm{\Delta }$ (its Gelfand spectrum). Crucially, when $L^{\infty }$ is considered a von Neumann algebra, its Gelfand spectrum $\mathrm{\Delta }$ possesses a stronger topological property—it is a hyper-Stonean space (in short, it is stronger than Stonean; it is an extremally disconnected compact Hausdorff space). This profound connection implies that many problems in $L^{\infty }$ can be transformed into corresponding problems for $C\left(\mathrm{\Delta }\right)$ on this special compact space $\mathrm{\Delta }$.

Regarding the characterization of weak compactness in $L^{\infty }$, a distinct line of inquiry exists in the literature. Schlüchtermann [3] (1994) provided a complete characterization of relatively weakly compact sets in $L^{\infty }(\mu ,X)$ through a parametrization of operators approach. Subsequently, Khurana [4] (2012) offered simplified proofs for the criteria governing weakly compact subsets and weak Cauchy sequences in $L^{\infty }\left(E\right)$ by employing the theory of liftings. The present work explores a different path. Rather than focusing on the characterization within $L^{\infty }$ itself, we leverage the Gelfand duality—which allows the translation of $L^{\infty }$ problems into the setting of $C\left(\mathrm{\Delta }\right)$—to investigate weak compactness in a broader context.

Within this framework, Grothendieck’s pioneering work [5] established the fundamental criterion for weak compactness in $C_b\left(E\right)$ for a general topological space E—the double limit criterion (Theorem 1). This criterion is in the same spirit as the Dunford–Pettis theorem but applies to a broader class of function spaces. When we specialize this theorem to $C\left(\mathrm{\Delta }\right)$, where $\mathrm{\Delta }$ is hyper-Stonean (i.e., the spectrum of an $L^{\infty }$ space), we can deduce a more concrete and verifiable necessary and sufficient condition for weak compactness, presented in this paper as Theorem 8. It requires that the functions have uniformly small oscillation over a finite clopen partition of the spectrum. This uniform oscillation refinement condition can be seen as the analog of “uniform integrability” in the world of $L^{\infty }$ and is the central concept of this paper.

The primary aim of this paper is to apply the profound theories from functional analysis and operator algebras, as discussed above, to the study of weak compactness in the Sobolev spaces $W^{k,\infty }\left(\mathrm{\Omega }\right)$. The $W^{k,\infty }$ spaces, essential for describing bounded differentiable functions, play a critical role in the calculus of variations, the theory of nonlinear PDEs (such as hyperbolic conservation laws), and geometric analysis. However, due to their non-reflexivity, bounded sequences in $W^{k,\infty }$ do not necessarily possess weakly convergent subsequences. Therefore, establishing a practical criterion for weak compactness is of fundamental importance.

The criterion we establish in Theorem 11, characterized by the uniform oscillation refinement condition, thus provides a verifiable substitute for the missing reflexivity. To illustrate its potential utility in nonlinear analysis, consider the problem of proving the existence of solutions to a first-order nonlinear hyperbolic conservation law, where natural a priori estimates yield sequences bounded in $W^{1,\infty }$ but not in a reflexive space. In such a setting, our theorem offers a direct path: one can now attempt to verify that the derivatives of the approximate solutions satisfy the uniform oscillation condition on some finite partition of the domain. If this condition holds, it guarantees the weak compactness of the sequence in $W^{1,\infty }$, enabling the extraction of a weakly convergent subsequence—a critical step in many existence proofs via compactness methods. This example underscores how our abstract characterization translates into a concrete analytical tool.

Our central strategy hinges on constructing an isometric embedding $\iota :W^{k,\infty }\left(\mathrm{\Omega }\right)\hookrightarrow L^{\infty }\left({\mathrm{\Omega }}^{\left(k\right)}\right)$ (see Definition 10), where ${\mathrm{\Omega }}^{\left(k\right)}$ is a measure space obtained by “unfolding” $\mathrm{\Omega }$ with respect to its partial derivatives up to order k (Definition 9). This construction cleverly transforms the problem of weak compactness in $W^{k,\infty }\left(\mathrm{\Omega }\right)$ into the problem of weak compactness of its image in $L^{\infty }\left({\mathrm{\Omega }}^{\left(k\right)}\right)$. We then lift the problem to the $C\left(\mathrm{\Delta }\right)$ space via the Gelfand transform and apply the weak compactness criterion (Theorem 3) valid on the hyper-Stonean spectrum $\mathrm{\Delta }$. The final step is to translate the result back to the $W^{k,\infty }$ setting, yielding a criterion expressed in terms of the oscillation of the functions themselves and their weak derivatives of all orders up to k.

The main results of this paper are as follows:

• Theorem 3: Establishes a characterization of relative weak compactness in $C(\mathrm{\Delta };\mathbb{C})$ for a Stonean space $\mathrm{\Delta }$ via the uniform oscillation refinement.

This result on Stonean spaces provides the foundation for our analysis in $L^{\infty }$ spaces. By recognizing that the Gelfand spectrum of a commutative von Neumann algebra is hyper-Stonean (hence Stonean), we can immediately obtain the following:

• Theorem 8: Presents a complete characterization of relative weak compactness in $L^{\infty }(X;\mathbb{C})$ as a direct consequence of Theorem 3.

The bridge between $L^{\infty }$ and Sobolev spaces is constructed through an isometric embedding that encodes derivative information. This leads to our central result in the Sobolev setting:

• Theorem 11: The core result of this paper. It characterizes the necessary and sufficient conditions for a subset of $W^{k,\infty }\left(\mathrm{\Omega }\right)$, where $\mathrm{\Omega }\subset {\mathbb{R}}^d$, to be relatively weakly compact. The conditions are that the subset is uniformly bounded and that all its weak derivatives of order up to k have uniformly small essential oscillation on some finite measurable partition of $\mathrm{\Omega }$.

For domains with additional regularity, we can strengthen this characterization by requiring control only on the highest-order derivatives, which significantly simplifies practical verification:

• Theorem 12: For domains $\mathrm{\Omega }$ with better regularity (satisfying the strong local Lipschitz condition), we obtain a refined result. It shows that to ensure weak compactness in $W^{k,\infty }\left(\mathrm{\Omega }\right)$, it suffices to verify that only the highest-order derivatives satisfy the uniform oscillation refinement condition. The corresponding property for lower-order derivatives can be automatically derived via embedding theorems and the compactness property of Sobolev spaces. This significantly simplifies the application of Theorem 11 in practice.

The structure of this paper is as follows:

• In Section 2, we state necessary preliminaries and the main results. We also provide systematic proofs of the theorems, progressing from the abstract theory of $C\left(\mathrm{\Delta }\right)$ spaces, gradually delving into the specific structures of $L^{\infty }$ and $W^{k,\infty }$ spaces, thereby clearly demonstrating how tools from operator algebra resolve classical problems in Sobolev spaces.
• In Section 3, we present a brief conclusion and discuss potential directions for future work.

2. Preliminaries and Main Results

This section assembles the definitions required for the proofs and the main results.

Theorem 1

(Grothendieck compactness principle: weak compactness criterion in $C_b(E;\mathbb{C})$). [5] [Theorem 6, p. 182] Let E be an arbitrary topological space. A uniformly bounded subset $\mathcal{F}$ of the bounded continuous function space $C_b(E;\mathbb{C})$ is weakly relatively compact if and only if the following condition is satisfied.

• Double limit criterion: There do not exist any sequence $\left\{x_m\right\}$ extracted from E and any sequence $\left\{f_n\right\}$ extracted from $\mathcal{F}$ such that the iterated limits

  $$\underset{n\to \infty }{lim}\underset{m\to \infty }{lim}{f}_n\left(x_m\right)\mathit{and}\underset{m\to \infty }{lim}\underset{n\to \infty }{lim}{f}_n\left(x_m\right)$$
  both exist and are different.

Definition 2

(Extremally disconnected space and Stonean space). [6] [Definition 1.6, p. 104] A topological space is called extremally disconnected if the closure of every open set is open. A compact, extremally disconnected Hausdorff space is also called a Stonean space.

A surprising fact is that Stonean spaces are zero-dimensional [7] [p. 360 and Theorem 6.2.25, p. 368] (note that compact Hausdorff spaces are regular), which means that a Stonean space has a base consisting of clopen sets.

Theorem 3

(weak compactness criterion in $C(\mathrm{\Delta };\mathbb{C})$ on a Stonean $\mathrm{\Delta }$). Let $\mathcal{F}$ be a uniformly bounded subset of the continuous function space $C(\mathrm{\Delta };\mathbb{C})$, where Δ is a Stonean space (recall Definition 2). Then, $\mathcal{F}$ is relatively weakly compact if and only if $\mathcal{F}$ satisfies the following condition.

• Uniform oscillation refinement: For every $\epsilon \in {\mathbb{R}}_{>0}$, there exists a finite clopen partition $\{U_1,\dots ,U_m\}$ of Δ such that for every $f\in \mathcal{F}$ and every $i\in \{1,\dots ,m\}$,

  $$\underset{x,y\in U_i}{sup}|f\left(x\right)-f\left(y\right)|<\epsilon .$$

Proof. 

The proof proceeds in two complementary directions. The converse direction utilizes Grothendieck’s double limit criterion, while the forward direction employs a contradiction argument that leverages the topological structure of Stonean spaces.

• (⇐): Suppose $\mathcal{F}$ satisfies the uniform oscillation refinement condition. We aim to show that $\mathcal{F}$ is relatively weakly compact. According to Grothendieck’s double limit criterion, it suffices to prove that for any sequence ${\left\{f_n\right\}}_{n\in \mathbb{N}}\subset \mathcal{F}$ and any sequence ${\left\{x_m\right\}}_{m\in \mathbb{N}}\subset \mathrm{\Delta }$, if the iterated limits

  $$\underset{n\to \infty }{lim}\underset{m\to \infty }{lim}{f}_n\left(x_m\right),\underset{m\to \infty }{lim}\underset{n\to \infty }{lim}{f}_n\left(x_m\right)$$

  both exist, then they are equal.

Fix an arbitrary $\epsilon \in {\mathbb{R}}_{>0}$. By the uniform oscillation refinement condition, choose a finite clopen partition $\{U_1,\dots ,U_m\}$ of $\mathrm{\Delta }$ such that for every $f\in \mathcal{F}$ and every $i\in \{1,\dots ,m\}$,

$$\underset{x,y\in U_i}{sup}|f\left(x\right)-f\left(y\right)|<\epsilon .$$

Since the partition is finite, at least one block, say $U_{i_0}$, contains infinitely many terms of the sequence $\left\{x_m\right\}$. Extract a subsequence $\left\{x_{m_k}\right\}$ entirely contained in $U_{i_0}$. Because the existence and values of the iterated limits are unchanged by passing to a subsequence (provided both limits exist for the original sequences), we may assume without loss of generality that the entire sequence $\left\{x_m\right\}$ lies in $U_{i_0}$. (Otherwise, replace $\left\{x_m\right\}$ with the subsequence and relabel.)

For each $f\in \mathcal{F}$, we have

$$\underset{x,y\in U_{i_0}}{sup}|f\left(x\right)-f\left(y\right)|<\epsilon .$$

In particular, for each fixed $n\in \mathbb{N}$,

$$|f_n\left(x_m\right)-f_n\left(x_{m^{\prime }}\right)|<\epsilon \mathrm{for}\mathrm{all}m,m^{\prime }\in \mathbb{N}.$$

(1)

Thus, for each n, the sequence ${\left\{f_n\left(x_m\right)\right\}}_m$ is Cauchy. Hence, the limit

$$a_n:=\underset{m\to \infty }{lim}{f}_n\left(x_m\right)$$

exists. Moreover, letting $m^{\prime }\to \infty$ in (1) yields the following for every $m\in \mathbb{N}$:

$$|f_n\left(x_m\right)-a_n|\le \epsilon .$$

(2)

Now, fix an arbitrary point $x\in U_{i_0}$. By the uniform oscillation refinement condition, $|f_n\left(x_m\right)-f_n\left(x\right)|<\epsilon$ for all $m\in \mathbb{N}$. Letting $m\to \infty$ yields

$$|a_n-f_n\left(x\right)|\le \epsilon .$$

(3)

By assumption, the limit ${lim}_{n\to \infty }{a}_n={lim}_{n\to \infty }{lim}_{m\to \infty }{f}_n\left(x_m\right)$ exists, so $\left\{a_n\right\}$ is a Cauchy sequence. Choose N such that for all $n,n^{\prime }\ge N$,

$$|a_n-a_{n^{\prime }}|<\epsilon .$$

Using (3) for the fixed $x\in U_{i_0}$, we obtain for any $n,n^{\prime }\ge N$,

$$|f_n\left(x\right)-f_{n^{\prime }}\left(x\right)|\le |f_n\left(x\right)-a_n|+|a_n-a_{n^{\prime }}|+|a_{n^{\prime }}-f_{n^{\prime }}\left(x\right)|\le 3\epsilon .$$

Hence, ${\left\{f_n\left(x\right)\right\}}_n$ is a Cauchy sequence. Denote the limit of ${\left\{f_n\left(x\right)\right\}}_n$ by $b\left(x\right)$. Taking $n\to \infty$ in (3) gives

$$|b\left(x\right)-\underset{n\to \infty }{lim}\underset{m\to \infty }{lim}{f}_n\left(x_m\right)|\le \epsilon .$$

(4)

For each fixed m, consider the limit $c_m:={lim}_{n\to \infty }{f}_n\left(x_m\right)$ (which exists by hypothesis). Since $x_m\in U_{i_0}$, the uniform oscillation refinement condition implies $|f_n\left(x_m\right)-f_n\left(x\right)|<\epsilon$ for all n. Letting $n\to \infty$, we obtain

$$|c_m-b\left(x\right)|\le \epsilon .$$

(5)

Consequently, the sequence $\left\{c_m\right\}$ is bounded, and all its terms lie within $\epsilon$ of $b\left(x\right)$. The limit ${lim}_{m\to \infty }{c}_m={lim}_{m\to \infty }{lim}_{n\to \infty }{f}_n\left(x_m\right)$ is assumed to exist. Letting $m\to \infty$ in (5) yields

$$|\underset{m\to \infty }{lim}\underset{n\to \infty }{lim}{f}_n\left(x_m\right)-b\left(x\right)|\le \epsilon .$$

(6)

Combining (4) and (6), we have

$$|\underset{n\to \infty }{lim}\underset{m\to \infty }{lim}{f}_n\left(x_m\right)-\underset{m\to \infty }{lim}\underset{n\to \infty }{lim}{f}_n\left(x_m\right)|\le 2\epsilon .$$

Since $\epsilon \in {\mathbb{R}}_{>0}$ was arbitrary, letting $\epsilon \to 0$, we conclude that

$$\underset{n\to \infty }{lim}\underset{m\to \infty }{lim}{f}_n\left(x_m\right)=\underset{m\to \infty }{lim}\underset{n\to \infty }{lim}{f}_n\left(x_m\right).$$

Then, by Grothendieck’s double limit criterion, $\mathcal{F}$ is relatively weakly compact in $C(\mathrm{\Delta };\mathbb{C})$.

Having established the converse direction, we now prove the forward direction by constructing a counterexample under the assumption that the uniform oscillation refinement condition fails.

• (⇒): We proceed by contradiction. Suppose the uniform oscillation refinement condition fails. Then, there exists some positive number ${\epsilon }_0$ such that for every finite clopen partition $\mathcal{P}$ of $\mathrm{\Delta }$, one can find a function $f_{\mathcal{P}}\in \mathcal{F}$, a set $U_{\mathcal{P}}\in \mathcal{P}$, and two points $x_{\mathcal{P}}$, $y_{\mathcal{P}}\in U_{\mathcal{P}}$ satisfying

  $$|f_{\mathcal{P}}\left(x_{\mathcal{P}}\right)-f_{\mathcal{P}}\left(y_{\mathcal{P}}\right)|\ge {\epsilon }_0.$$

  (7)

Let $\mathfrak{P}$ be the collection of all finite clopen partitions of $\mathrm{\Delta }$. Define a partial order on $\mathfrak{P}$ by ${\mathcal{P}}_1⪯{\mathcal{P}}_2$ if and only if ${\mathcal{P}}_2$ refines ${\mathcal{P}}_1$ (i.e., every member of ${\mathcal{P}}_2$ is contained in some member of ${\mathcal{P}}_1$). Then, $\mathfrak{P}$ becomes a directed set (since any two partitions have a common refinement; for instance, the partition formed by all nonempty intersections of their members).

For each $\mathcal{P}\in \mathfrak{P}$, choose $f_{\mathcal{P}}$, $x_{\mathcal{P}}$, $y_{\mathcal{P}}$ as above. This yields three nets:

$${\left\{f_{\mathcal{P}}\right\}}_{\mathcal{P}\in \mathfrak{P}}\subset \mathcal{F},{\left\{x_{\mathcal{P}}\right\}}_{\mathcal{P}\in \mathfrak{P}}\subset \mathrm{\Delta },{\left\{y_{\mathcal{P}}\right\}}_{\mathcal{P}\in \mathfrak{P}}\subset \mathrm{\Delta }.$$

Because $\mathcal{F}$ is relatively weakly compact, the net $\left\{f_{\mathcal{P}}\right\}$ has a weakly convergent subnet. Denote this subnet by ${\left\{f_{{\mathcal{P}}_{\alpha }}\right\}}_{\alpha \in A}$, where A is a directed set, and suppose it converges weakly to some function $f\in C(\mathrm{\Delta };\mathbb{C})$ (i.e., for every $\phi \in C(\mathrm{\Delta };\mathbb{C})\ast$, $\phi \left(f_{{\mathcal{P}}_{\alpha }}\right)\to \phi \left(f\right)$.

Since $\mathrm{\Delta }$ is compact, the net $\left\{x_{{\mathcal{P}}_{\alpha }}\right\}$ has a convergent subnet. To simplify notation, we may assume (by passing to a subnet) that $\left\{x_{{\mathcal{P}}_{\alpha }}\right\}$ itself converges to a point $x\in \mathrm{\Delta }$. Similarly, after possibly taking a further subnet, we may assume $\left\{y_{{\mathcal{P}}_{\alpha }}\right\}$ converges to a point $y\in \mathrm{\Delta }$.

Take any $g\in C(\mathrm{\Delta };\mathbb{C})$. We show that $g\left(x\right)=g\left(y\right)$.

Given an arbitrary $\eta \in {\mathbb{R}}_{>0}$, since f is uniformly continuous on the compact space $\mathrm{\Delta }$ (here, uniform continuity means that there exists a finite open cover of $\mathrm{\Delta }$ such that the oscillation of f on each open set is less than $\eta$), we can choose a finite open cover $\{V_1,\dots ,V_r\}$ with ${sup}_{x,y\in V_j}|f\left(x\right)-f\left(y\right)|<\eta$ for each j.

Since $\mathrm{\Delta }$ is zero-dimensional by Definition 2, each open set contains a clopen set, and finite Boolean combinations (i.e., ∪, ∩ and ${(·)}^c$) of clopen sets remain clopen. Hence, we can construct a finite clopen partition $\mathcal{Q}=\{W_1,\dots ,W_s\}$ that refines $\{V_1,\dots ,V_r\}$, i.e., each $W_i$ is contained in some $V_j$. Consequently, for each i,

$$\underset{u,v\in W_i}{sup}|g\left(u\right)-g\left(v\right)|<\eta .$$

(8)

Since the net $\left\{{\mathcal{P}}_{\alpha }\right\}$ is directed by refinement, there exists an index ${\alpha }_0\in A$ such that for all $\alpha \ge {\alpha }_0$, ${\mathcal{P}}_{\alpha }$ refines $\mathcal{Q}$. For such $\alpha$, the points $x_{{\mathcal{P}}_{\alpha }}$ and $y_{{\mathcal{P}}_{\alpha }}$ belong to the same member $U_{{\mathcal{P}}_{\alpha }}$ of ${\mathcal{P}}_{\alpha }$, and because ${\mathcal{P}}_{\alpha }$ refines $\mathcal{Q}$, this $U_{{\mathcal{P}}_{\alpha }}$ is contained in some $W_i$. Hence, by (8),

$$|g\left(x_{{\mathcal{P}}_{\alpha }}\right)-g\left(y_{{\mathcal{P}}_{\alpha }}\right)|<\eta (\forall \alpha \ge {\alpha }_0).$$

Letting $\alpha$ increase along the directed set, continuity of g gives $g\left(x_{{\mathcal{P}}_{\alpha }}\right)\to g\left(x\right)$ and $g\left(y_{{\mathcal{P}}_{\alpha }}\right)\to g\left(y\right)$. Thus,

$$\left|g\right(x)-g(y\left)\right|\le \eta .$$

Since the above inequality holds for any $\eta \in {\mathbb{R}}_{>0}$ and every continuous function g, we must have $g\left(x\right)=g\left(y\right)$ for all $g\in C(\mathrm{\Delta };\mathbb{C})$. Because the continuous functions on a compact Hausdorff space separate points (by Urysohn’s lemma), this forces $x=y$.

Note that weak convergence implies pointwise convergence (because for each $z\in \mathrm{\Delta }$, the evaluation map $f\mapsto f\left(z\right)$ is a continuous linear functional). Therefore,

$$f_{{\mathcal{P}}_{\alpha }}\left(x_{{\mathcal{P}}_{\alpha }}\right)\to f\left(x\right),f_{{\mathcal{P}}_{\alpha }}\left(y_{{\mathcal{P}}_{\alpha }}\right)\to f\left(y\right)=f\left(x\right).$$

But condition (7) gives $|f_{{\mathcal{P}}_{\alpha }}\left(x_{{\mathcal{P}}_{\alpha }}\right)-f_{{\mathcal{P}}_{\alpha }}\left(y_{{\mathcal{P}}_{\alpha }}\right)|\ge {\epsilon }_0$ for every $\alpha$. Passing to the limit yields $0\ge {\epsilon }_0$, a contradiction. Hence, our initial assumption must be false, and the theorem is proved. □

The following algebraic structures will provide the framework for understanding $L^{\infty }$ as a function space. We begin with the general concept of a Banach algebra.

Definition 4

(Banach algebra). [6] [Definition 1.1, p. 2] Let $\mathcal{B}$ be an algebra over the complex field $\mathbb{C}$. If $\mathcal{B}$ is also a Banach space (a complete normed linear space) whose norm satisfies

$$\parallel ab\parallel \le \parallel a\parallel \parallel b\parallel$$

for all $a,b\in \mathcal{B}$, then $\mathcal{B}$ is called a Banach algebra.

If there exists an element $e\in \mathcal{B}$ such that $ea=ae=a$ for all $a\in \mathcal{B}$ and $\parallel e\parallel =1$, then $\mathcal{B}$ is called a unital Banach algebra.

Definition 5

(C*-algebra). [8] [Definition 4.26, p. 83] Let $\mathcal{C}$ be a unital Banach algebra equipped with a map $\ast :\mathcal{C}\to \mathcal{C}$ (called an involution) satisfying that for all $a,b\in \mathcal{C}$ and $\lambda ,\mu \in \mathbb{C}$, the following four conditions are satisfied.

• ${\left(a^{*}\right)}^{*}=a$.
• ${\left(ab\right)}^{*}=b^{*}{a}^{*}$.
• ${(\lambda a+\mu b)}^{*}=\overline{\lambda }{a}^{*}+\overline{\mu }{b}^{*}$.
• C*-identity: $\parallel a^{*}a\parallel ={\parallel a\parallel }^2$.
  Then, $\mathcal{C}$ is called a C*-algebra.

A key feature of C*-algebras is the Gelfand–Naimark theorem [8] [Theorem 4.29, p. 84]: every commutative unital C*-algebra is isometrically ∗-isomorphic to the algebra $C\left(\mathrm{\Delta }\right)$ of continuous functions on a compact Hausdorff space, which is called the Gelfand spectrum of $\mathcal{C}$ denoted by $\mathrm{\Delta }$. This isomorphism is called the Gelfand transform, which represents abstract algebraic elements as concrete functions (see Definition 7).

A particularly important subclass of C*-algebras, characterized by stronger topological closure properties, is that of von Neumann algebras. Their additional structure leads to a richer theory for spaces like $L^{\infty }$.

Definition 6

(von Neumann algebra). [6] [first paragraph of Preface and Definition 3.2, p. 72] A von Neumann algebra is a C*-algebra with stronger topological closure properties. Let H be a Hilbert space and $B\left(H\right)$ the algebra of all bounded operators on H. A von Neumann algebra $\mathcal{M}$ is a *-subalgebra (a subalgebra closed under the involution ∗) of $B\left(H\right)$ satisfying the following two conditions.

• $\mathcal{M}$ is closed in the weak operator topology [8] [Definition 4.44, p. 91].
• $\mathcal{M}$ contains the identity operator I.

An equivalent definition  (von Neumann’s double commutant theorem [6] [Theorem 3.9, p. 74]): $\mathcal{M}$ is a von Neumann algebra if and only if $\mathcal{M}={\mathcal{M}}^{\prime \prime }$, where ${\mathcal{M}}^{\prime }$ is the commutant of $\mathcal{M}$.

Example 1.

$L^{\infty }(X;\mathbb{C})$ is a commutative von Neumann algebra (regarded as multiplication operators on $L^2(X;\mathbb{C}))$. In fact, every commutative von Neumann algebra $\mathcal{M}$ is isometrically *-isomorphic (i.e., there exists an isometry which preserves both multiplication and involution) to some space of essentially bounded measurable functions $L^{\infty }(X;\mathbb{C})$ defined on a locally compact space X with a positive Radon measure [6] [Theorem 1.18, p. 109].

Definition 7

(Gelfand spectrum and Gelfand transform). [8] [Definition 2.21, p. 36 and Definition 2.24, p. 37] Let $\mathcal{C}$ be a commutative unital C*-algebra. Its Gelfand spectrum (or maximal two-sided ideal space [8] [Proposition 2.33, p. 40]), denoted ${\mathrm{\Delta }}_{\mathcal{C}}$, is defined as the set of all nonzero multiplicative linear functionals $\phi :\mathcal{C}\to \mathbb{C}$, i.e., nonzero continuous linear functionals satisfying

$$\phi \left(ab\right)=\phi \left(a\right)\phi \left(b\right),\phi \left(1\right)=1.$$

Equipping ${\mathrm{\Delta }}_{\mathcal{C}}$ with the weak* topology induced by $\mathcal{C}$ (the weakest topology making each map $\phi \mapsto \phi \left(a\right)$ continuous) makes it a compact Hausdorff space.

For each element $a\in \mathcal{C}$, its Gelfand transform Γ is defined as the continuous function $\widehat{a}:{\mathrm{\Delta }}_{\mathcal{C}}\to \mathbb{C}$, where $\widehat{a}\left(\phi \right)=\phi \left(a\right)$. The mapping

$$\mathrm{\Gamma }:\mathcal{C}\to C\left({\mathrm{\Delta }}_{\mathcal{C}}\right),a\mapsto \widehat{a}$$

is an isometric *-isomorphism (the Gelfand–Naimark theorem).

Thus, every commutative unital C*-algebra can be concretely realized as the algebra of all continuous functions on some compact Hausdorff space.

• The von Neumann case:   When the commutative C*-algebra $\mathcal{C}$ is furthermore a von Neumann algebra, its spectrum ${\mathrm{\Delta }}_{\mathcal{C}}$ possesses a stronger topological property and becomes a hyper-Stonean space [6] [Definition 1.14, p. 107] which is even stronger than the Stonean space.

In fact, a commutative C*-algebra is a von Neumann algebra if and only if its Gelfand spectrum is a hyper-Stonean space [6] [Theorem 1.18, p. 109].

In a commutative von Neumann algebra, projections (elements satisfying $p^2=p=p^{*}$) correspond to characteristic functions, and the extremal disconnectedness of the spectrum precisely reflects the completeness of this lattice: the supremum (closure of the union) of any family of clopen sets remains clopen.

Equipped with the Gelfand transform, we can now return to the weak compactness problem in $L^{\infty }$. The following theorem is a direct translation of Theorem 3 to the $L^{\infty }$ setting via this isomorphism.

Theorem 8

(weak compactness criterion in $L^{\infty }(X;\mathbb{C})$). Let $(X,\mu )$ be a σ-finite measure space and $\mathcal{F}\subset L^{\infty }(X;\mathbb{C})$ a uniformly bounded subset. Then, $\mathcal{F}$ is relatively compact in the weak topology $\sigma (L^{\infty },{\left(L^{\infty }\right)}^{*})$ if and only if the following condition is satisfied.

• Uniform oscillation refinement: For every $\epsilon \in {\mathbb{R}}_{>0}$, there exist finitely many pairwise disjoint measurable sets $E_1$, …, $E_m$ with $\mu \left(X\setminus {\bigcup }_{i=1}^m{E}_i\right)=0$ such that for every $f\in \mathcal{F}$ and every $i\in \{1,\dots ,m\}$,

  $${osc}_{E_i}\left(f\right)<\epsilon ,$$
  where

  $${osc}_E\left(f\right):=\underset{x,y\in E}{ess\; sup}|f\left(x\right)-f\left(y\right)|.$$

Proof. 

The proof strategy is to establish an equivalence loop: we start with $L^{\infty }$, transfer to $C\left(\mathrm{\Delta }\right)$ via the Gelfand transform $\mathrm{\Gamma }$ (recall Definition 7), apply Theorem 3 there, and then transfer the result back to $L^{\infty }$ via the correspondence between clopen and measurable sets.

• (⇒): Suppose that $\mathcal{F}$ is relatively weakly compact in $L^{\infty }(X;\mathbb{C})$. By the Gelfand transform $\mathrm{\Gamma }$, $\mathrm{\Gamma }\mathcal{F}:=\{\widehat{f}\subset C(\mathrm{\Delta };\mathbb{C})\mid f\in \mathcal{F}\}$ is relatively weakly compact in $C(\mathrm{\Delta };\mathbb{C})$, where $\mathrm{\Delta }$ denotes the Gelfand spectrum of $L^{\infty }(X;\mathbb{C})$. Applying Theorem 3, $\mathrm{\Gamma }\mathcal{F}$ satisfies the uniform oscillation refinement condition in $C(\mathrm{\Delta };\mathbb{C})$, that is, for every $\epsilon \in {\mathbb{R}}_{>0}$, there exists a finite clopen partition $\{U_1,\dots ,U_m\}$ of $\mathrm{\Delta }$ such that for every $\widehat{f}\in \mathrm{\Gamma }\mathcal{F}$ and every $i\in \{1,\dots ,m\}$,

  $$\underset{\phi ,\psi \in U_i}{sup}|\widehat{f}\left(\phi \right)-\widehat{f}\left(\psi \right)|<\epsilon .$$

  (9)

  By the Gelfand transform for the commutative von Neumann algebra $L^{\infty }(X;\mathbb{C})$ (recall Definition 7), the clopen set $U_i$ corresponds uniquely (up to null sets) to a measurable set $E_i\subset X$ such that

  $$\mathrm{\Gamma }\left({\chi }_{E_i}\right)={\chi }_{U_i}.$$

  (10)

  The properties of clopen partition $\left\{U_i\right\}$ then translate to the desired properties of the measurable partition $\left\{E_i\right\}$. Here, we verify that $\left\{E_i\right\}$ is a measurable partition of X:

• Disjointness: For $i\ne j$, since $U_i\cap U_j=⌀$, we have ${\chi }_{U_i}{\chi }_{U_j}=0$. Because $\mathrm{\Gamma }$ is an algebra homomorphism, $\mathrm{\Gamma }\left({\chi }_{E_i}{\chi }_{E_j}\right)=\mathrm{\Gamma }\left({\chi }_{E_i}\right)\mathrm{\Gamma }\left({\chi }_{E_j}\right)={\chi }_{U_i}{\chi }_{U_j}=0$. Since $\mathrm{\Gamma }$ is injective, it follows that ${\chi }_{E_i}{\chi }_{E_j}=0$ almost everywhere, i.e., $\mu (E_i\cap E_j)=0$.
• Coverage: Since ${\bigcup }_i{U}_i=\mathrm{\Delta }$, we have ${\sum }_i{\chi }_{U_i}=1_{\mathrm{\Delta }}$ (the constant function 1). Moreover, $\mathrm{\Gamma }\left(1_X\right)=1_{\mathrm{\Delta }}$, and

  $$\mathrm{\Gamma }(\sum _{i=1}^m{\chi }_{E_i})=\sum _{i=1}^m\mathrm{\Gamma }\left({\chi }_{E_i}\right)=\sum _{i=1}^m{\chi }_{U_i}=1_{\mathrm{\Delta }}=\mathrm{\Gamma }\left(1_X\right).$$
  By injectivity of $\mathrm{\Gamma }$, we obtain ${\sum }_{i=1}^m{\chi }_{E_i}=1_X$ almost everywhere, meaning

  $$\mu (X\setminus (\bigcup _{i=1}^m{E}_i))=0.$$

Then, for any $f\in \mathcal{F}$, by functional calculus [8] [4.31, p. 85] and (9),

$${osc}_{E_i}\left(f\right)=\underset{x,y\in E_i}{ess\; sup}|f\left(x\right)-f\left(y\right)|=\underset{\phi ,\psi \in U_i}{sup}|\widehat{f}\left(\phi \right)-\widehat{f}\left(\psi \right)|<\epsilon .$$

Thus, $\mathcal{F}$ satisfies the uniform oscillation refinement condition in $L^{\infty }(X;\mathbb{C})$.

The converse direction completes the equivalence by showing that the uniform oscillation condition in $L^{\infty }$ implies weak compactness through the same Gelfand correspondence.

• (⇐): Assume $\mathcal{F}$ satisfies the uniform oscillation refinement in $L^{\infty }(X;\mathbb{C})$. We want to construct the clopen partition of $\mathrm{\Delta }$. For each $E_i$, let $U_i$ be the corresponding clopen set, i.e., the unique clopen set satisfying $\mathrm{\Gamma }\left({\chi }_{E_i}\right)={\chi }_{U_i}$. By (10) and the previous argument, $\left\{U_i\right\}$ forms a clopen partition of $\mathrm{\Delta }$. Then, for any $\widehat{f}\in \mathrm{\Gamma }\mathcal{F}$, by functional calculus and the assumption,

  $$\underset{\phi ,\psi \in U_i}{sup}|\widehat{f}\left(\phi \right)-\widehat{f}\left(\psi \right)|={osc}_{E_i}\left(f\right)<\epsilon .$$

  Hence, $\mathrm{\Gamma }\mathcal{F}$ is relatively weakly compact in $C(\mathrm{\Delta };\mathbb{C})$ by Theorem 3 and $\mathcal{F}$ is relatively weakly compact in $L^{\infty }(X;\mathbb{C})$ through ${\mathrm{\Gamma }}^{-1}$. □

Remark 1.

The above theorems also hold for the real-valued function space because the real space can be isometrically embedded into the complex space by identifying each real function with its complex extension (i.e., $f\mapsto f+0·i$), and both the weak topology (by applying Theorem 3.10, p. 61 of [9]) and the uniform oscillation refinement condition are preserved under this embedding.

Example 2

(a non-weakly compact sequence in $L^{\infty }$). Let $\left\{f_n\right\}$ be a sequence in $L^{\infty }(0,1)$ and each term has the following form:

$$f_n\left(x\right)=\left\{\begin{array}{cc}0,\hfill & x\in (0,{\displaystyle \frac{n-1}{n}});\hfill \\ 1,\hfill & x\in [{\displaystyle \frac{n-1}{n}},1).\hfill \end{array}\right.$$

This sequence $\left\{f_n\right\}$ does not converge strongly (in norm) to 0 because ${\parallel f_n-0\parallel }_{L^{\infty }(0,1)}=1$ for all n. And using the Mazur lemma [10] [Lemma 4.3.1, p. 116], one can show that $\left\{f_n\right\}$ does not converge weakly to 0 since no finite convex combination of its terms converges strongly to 0.

More strongly, the sequence $\left\{f_n\right\}$ is not relatively weakly compact in $L^{\infty }(0, 1)$ because it fails the uniform oscillation refinement condition required in Theorem 8. For any given finite measurable partition of $(0, 1)$, when n is sufficiently large, the "jump" of $f_n$ from 0 to 1 will necessarily occur within (or across) at least one block of that partition. On that block, the essential oscillation of $f_n$ is 1. Therefore, it is impossible to find a single, fixed finite partition that makes the oscillation of all $f_n$ uniformly small.

Next, we extend Theorem 8 to Sobolev spaces $W^{k,\infty }\left(\mathrm{\Omega }\right)$ and first introduce a notation as follows.

Notation 1

(multi-index). [11] [p. 2] Given integers $d\ge 1$ and $k\ge 0$, let

$$\alpha :=({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_d)\in {\mathbb{N}}^d$$

be a multi-index such that

$$\left|\alpha \right|:=\sum _{i=1}^d{\alpha }_i\le k.$$

For any $f\in W^{k,\infty }\left(\mathrm{\Omega }\right)$ where Ω is an open subset of ${\mathbb{R}}^d$, we write

$$D^{\alpha }f={\displaystyle \frac{{\partial }^{{\alpha }_1}}{\partial x_1^{{\alpha }_1}}}{\displaystyle \frac{{\partial }^{{\alpha }_2}}{\partial x_2^{{\alpha }_2}}}\dots {\displaystyle \frac{{\partial }^{{\alpha }_d}}{\partial x_d^{{\alpha }_d}}}f$$

to denote the weak generalized partial derivative of f with respect to α.

The following two definitions are key to connecting $L^{\infty }$ with the Sobolev space $W^{k,\infty }$.

Definition 9

(the disjoint union ${\mathrm{\Omega }}^{\left(k\right)}$). [11] [3.5, p. 61] Let Ω be a non-empty open set in ${\mathbb{R}}^d$. For each multi-index α, let ${\mathrm{\Omega }}_{\alpha }$ be a different copy of Ω lying in ${\mathbb{R}}_{\alpha }^d$ which is a different copy of ${\mathbb{R}}^d$ with respect to α. Let $N:={\sum }_{\left|\alpha \right|\le k}1$ be the number of these multi-indices. Thus these N non-empty open sets are mutually disjoint. More mathematically, an isomorphism between Ω and ${\mathrm{\Omega }}_{\alpha }$ refers to an isometry $I_{\alpha }:\mathrm{\Omega }\to {\mathrm{\Omega }}_{\alpha }$ that also preserves the measure:

$${\lambda }_{\alpha }\left(\{I_{\alpha }\left(x\right)\mid x\in \omega \}\right)=:{\lambda }_{\alpha }\circ I_{\alpha }\left(\omega \right)=\lambda \left(\omega \right),\mathit{for}\mathit{all}\omega \subset \mathrm{\Omega }\mathit{with}\omega \mathit{measurable}.$$

Here, ${\lambda }_{\alpha }$ denotes the Lebesgue measure in ${\mathbb{R}}_{\alpha }^d$.

The union of these N sets is denoted as

$${\mathrm{\Omega }}^{\left(k\right)}:=\underset{\left|\alpha \right|\le k}{⨆}\mathrm{\Omega }=\bigcup _{\left|\alpha \right|\le k}{\mathrm{\Omega }}_{\alpha }.$$

The space ${\mathrm{\Omega }}^{\left(k\right)}$ naturally inherits the following structures from the Euclidean spaces:

• Measure: A subset $\omega \subseteq {\mathrm{\Omega }}^{\left(k\right)}$ is measurable if $\omega \cap {\mathrm{\Omega }}_{\alpha }$ is Lebesgue measurable in ${\mathbb{R}}_{\alpha }^d$ for all α. The measure of such ω in ${\mathrm{\Omega }}^{\left(k\right)}$ is defined as the sum of the Lebesgue measures of $\omega \cap {\mathrm{\Omega }}_{\alpha }$ taken over α, i.e.,

  $$\mu \left(\omega \right):=\sum _{\left|\alpha \right|\le k}{\lambda }_{\alpha }(\omega \cap {\mathrm{\Omega }}_{\alpha }).$$

  Note that ${\mathrm{\Omega }}^{\left(k\right)}$ equipped with μ is a σ-finite measure space.
• Topology: The space ${\mathrm{\Omega }}^{\left(k\right)}$ is equipped with the disjoint union topology, i.e., a set ω is open in ${\mathrm{\Omega }}^{\left(k\right)}$ if and only if $\omega \cap {\mathrm{\Omega }}_{\alpha }$ is open in ${\mathbb{R}}_{\alpha }^d$ for every α.

Now, we can define the isometry operator $\iota$ from the Sobolev space $W^{k,\infty }\left(\mathrm{\Omega }\right)$ to the Lebesgue space $L^{\infty }\left({\mathrm{\Omega }}^{\left(k\right)}\right)$. In this way, we transform the discussion on the compactness of a subset of $W^{k,\infty }\left(\mathrm{\Omega }\right)$ into the discussion on the compactness of its image in space $L^{\infty }\left({\mathrm{\Omega }}^{\left(k\right)}\right)$.

Definition 10

(the isometric operator $\iota$). [11] [3.5, p. 61] We construct an operator ι from $W^{k,\infty }\left(\mathrm{\Omega }\right)$ to $L^{\infty }\left({\mathrm{\Omega }}^{\left(k\right)}\right)$ as follows:

$$\begin{array}{cc}\hfill \iota :W^{k,\infty }\left(\mathrm{\Omega }\right)& \to L^{\infty }\left({\mathrm{\Omega }}^{\left(k\right)}\right)\hfill \\ \hfill u& \mapsto \iota u,\hfill \end{array}$$

where $\iota u:{\mathrm{\Omega }}^{\left(k\right)}\to \mathbb{R}$ is given by

$$\iota u\left(y\right)=\left(D^{\alpha }u\right)\circ I_{\alpha }^{-1}\left(y\right),\mathrm{if}y\in {\mathrm{\Omega }}_{\alpha }.$$

That is, the restriction of function $\iota u$ on each component of ${\mathrm{\Omega }}_{\alpha }$ in ${\mathrm{\Omega }}^{\left(k\right)}$ is given by the function $D^{\alpha }u$.

Lemma 1.

The operator ι is a well-defined isometry. Let W be the range of ι. W is a closed linear subspace of $L^{\infty }\left({\mathrm{\Omega }}^{\left(k\right)}\right)$. Furthermore, $W^{k,\infty }\left(\mathrm{\Omega }\right)$ and W are homeomorphic both in their norm and weak topologies. Here the norm (and weak) topology of W is considered the subspace topology induced from the norm (and weak) topology of $L^{\infty }\left({\mathrm{\Omega }}^{\left(k\right)}\right)$.

Proof. 

We first check that $\iota$ is well-defined. Notice that for any $y\in {\mathrm{\Omega }}^{\left(k\right)}$, there must be only one multi-index $\alpha$ such that $y\in {\mathrm{\Omega }}_{\alpha }$. Furthermore, for each $\alpha$, the weak generalized partial derivative $D^{\alpha }u$ is unique up to sets of measure zero in $L^{\infty }\left(\mathrm{\Omega }\right)$. Thus, $\iota$ is well-defined.

Now, we will check that $\iota$ is an isometric embedding. Indeed, we have the following equalities:

$${\parallel \iota u\parallel }_{L^{\infty }\left({\mathrm{\Omega }}^{\left(k\right)}\right)}=\underset{\left|\alpha \right|\le k}{max}{\parallel \iota u{|}_{{\mathrm{\Omega }}_{\alpha }}\parallel }_{L^{\infty }\left({\mathrm{\Omega }}_{\alpha }\right)}=\underset{\left|\alpha \right|\le k}{max}\underset{x\in \mathrm{\Omega }}{ess\; sup}\left|D^{\alpha }u\left(x\right)\right|={\parallel u\parallel }_{W^{k,\infty }\left(\mathrm{\Omega }\right)}.$$

Since $\iota$ is an isometric embedding and $W^{k,\infty }\left(\mathrm{\Omega }\right)$ and $L^{\infty }\left({\mathrm{\Omega }}^{\left(k\right)}\right)$ are both complete spaces, W is closed in $L^{\infty }\left({\mathrm{\Omega }}^{\left(k\right)}\right)$ and W is homeomorphic with $W^{k,\infty }\left(\mathrm{\Omega }\right)$ in their norm topologies. Finally, it follows from [9] [Theorem 3.10, p. 61] that $W^{k,\infty }\left(\mathrm{\Omega }\right)$ and W are homeomorphic in their respective weak topologies. □

Theorem 11

(weak compactness criterion in $W^{k,\infty }\left(\mathrm{\Omega }\right)$). Let $\mathcal{F}$ be a uniformly bounded subset of $W^{k,\infty }\left(\mathrm{\Omega }\right)$, where Ω is a non-empty open set in ${\mathbb{R}}^d$. Then, $\mathcal{F}$ is relatively weakly compact if and only if $\mathcal{F}$ satisfies the following condition:

• Uniform oscillation refinement: for every $\epsilon \in {\mathbb{R}}_{>0}$, there exist finitely many pairwise disjoint measurable sets $E_1$, …, $E_m$ with $\mu \left(\mathrm{\Omega }\setminus {\bigcup }_{i=1}^m{E}_i\right)=0$ such that for every $f\in \mathcal{F}$ and each multi-index α and each $i\in \{1,\dots ,m\}$,

  $${osc}_{E_i}\left(D^{\alpha }f\right)<\epsilon .$$

  (11)

Proof. 

The proof strategy synthesizes our previous results. The isometric embedding $\iota$ allows us to reduce the problem from $W^{k,\infty }$ to $L^{\infty }$, where we can apply Theorem 8. The idea is to simply put the following two pieces together.

• By Lemma 1, $\mathcal{F}$ is relatively weakly compact in $W^{k,\infty }\left(\mathrm{\Omega }\right)$ if and only if $\iota \mathcal{F}$ is relatively weakly compact in $L^{\infty }\left({\mathrm{\Omega }}^{\left(k\right)}\right)$.
• By Theorem 8, a relatively weakly compact subset of $L^{\infty }\left({\mathrm{\Omega }}^{\left(k\right)}\right)$ coincides with a uniformly bounded set satisfying the uniform oscillation refinement condition.

We now make this strategy precise by establishing the equivalence between the oscillation conditions in $W^{k,\infty }$ and its image under $\iota$ in $L^{\infty }$.

• (⇒): Let $\mathcal{F}$ be a uniformly bounded subset of $W^{k,\infty }\left(\mathrm{\Omega }\right)$ satisfying (11). On the one hand, since $\iota$ is an isometric embedding, $\iota \mathcal{F}$ is uniformly bounded. On the other hand, the partition ${\{I_{\alpha }\left(E_1\right),\dots ,I_{\alpha }\left(E_m\right)\}}_{\left|\alpha \right|\le k}$ implies that $\iota \mathcal{F}$ satisfies the uniform oscillation refinement condition in $L^{\infty }\left({\mathrm{\Omega }}^{\left(k\right)}\right)$.

The converse direction requires careful construction of appropriate partitions that respect the multi-index structure of the Sobolev space.

• (⇐): Let $\iota \mathcal{F}$ be a uniformly bounded subset of $L^{\infty }\left({\mathrm{\Omega }}^{\left(k\right)}\right)$ satisfying the uniform oscillation refinement condition. Recalling Definitions 9 and 10, we list the condition in our special case below.

• Uniform oscillation refinement: for every $\epsilon \in {\mathbb{R}}_{>0}$, there exist finitely many pairwise disjoint measurable sets $E_1$, …, $E_m$ with $\mu \left({\mathrm{\Omega }}^{\left(k\right)}\setminus {\bigcup }_{i=1}^m{E}_i\right)=0$ such that for every $\iota f\in \iota \mathcal{F}$ and every $i\in \{1,\dots ,m\}$,

  $${osc}_{E_i}\left(\iota f\right)<\epsilon .$$

For each $\alpha$, denote a partition of $\mathrm{\Omega }$ as

$${\mathcal{P}}_{\alpha }:=I_{\alpha }^{-1}{\{E_i\cap {\mathrm{\Omega }}_{\alpha }\}}_{i\in \{1,\dots ,m\}}.$$

Each ${\mathcal{P}}_{\alpha }$ contains finitely many blocks. Then, choose a partition of $\mathcal{F}$ as

$$\mathcal{P}:=\{\bigcap _{\left|\alpha \right|\le k}{A}_{\alpha }\mid A_{\alpha }\in \mathcal{P}\}.$$

According to the form of $\mathcal{P}$, this $\mathcal{P}$ refines each ${\mathcal{P}}_{\alpha }$ and contains fewer than $m^N$ blocks, where $N:={\sum }_{\left|\alpha \right|\le k}1$ denotes the total number of these multi-indices $\alpha$. Let P be an arbitrary block of $\mathcal{P}$. For each $\alpha$, we obtain

$${osc}_P\left(D^{\alpha }f\right)=\underset{x_{\alpha },y_{\alpha }\in I_{\alpha }\left(P\right)}{ess\; sup}|\iota f\left(x_{\alpha }\right)-\iota f\left(y_{\alpha }\right)|<\epsilon ,$$

which implies (11) and completes the proof. □

Moreover, if $\mathrm{\Omega }$ is a bounded open subset of ${\mathbb{R}}^d$ and satisfies the strong local Lipschitz condition [11] [4.9, p. 83], by applying the Stein extension theorem [11] [Theorem 5.24, p. 154] to Theorem 4 on page 279 of [12], the Sobolev space $W^{1,\infty }\left(\mathrm{\Omega }\right)$ is isometrically isomorphic to the Lipschitz function space $C^{0,1}\left(\mathrm{\Omega }\right)$. Based on this, we have the following theorem.

Theorem 12.

Let Ω be a non-empty bounded open subset of ${\mathbb{R}}^d$ satisfying the strong local Lipschitz condition and let $\mathcal{F}$ be a subset of $W^{k,\infty }\left(\mathrm{\Omega }\right)$. Then, $\mathcal{F}$ is relatively weakly compact in $W^{k,\infty }\left(\mathrm{\Omega }\right)$ if and only if the following three conditions are satisfied.

• $\mathcal{F}$ is uniformly bounded in $L^{\infty }\left(\mathrm{\Omega }\right)$.
• The kth weak generalized partial derivatives of each $f\in \mathcal{F}$ denoted by $D^{\alpha }f$ (with $\left|\alpha \right|=k$) are uniformly bounded in $L^1\left(\mathrm{\Omega }\right)$.
• Uniform oscillation refinement*: the kth derivatives satisfy the uniform oscillation refinement in Theorem 8, that is, for every $\epsilon \in {\mathbb{R}}_{>0}$, there exists finitely many pairwise disjoint measurable sets $E_1$, …, $E_m$ with $\mu \left(\mathrm{\Omega }\setminus {\bigcup }_{i=1}^m{E}_i\right)=0$ such that for every $f\in \mathcal{F}$ and each $i\in \{1,\dots ,m\}$, its kth derivatives $D^{\alpha }f$ satisfies

  $${osc}_{E_i}\left(D^{\alpha }f\right)<\epsilon .$$

  (12)

Proof. 

This theorem represents a refinement of Theorem 11 under additional regularity assumptions. The key observation is that for sufficiently regular domains, control of the highest-order derivatives automatically implies control of lower-order derivatives through compact embedding theorems.

The forward direction is obvious by the Hölder’s inequality and Theorem 11.

The converse direction requires the most technical work, as we need to establish uniform boundedness in $W^{k,\infty }$ from the weaker $L^1$ bound on highest-order derivatives and then leverage the compact embedding properties of Sobolev spaces.

Therefore, it suffices to prove the converse direction. Let $\mathcal{F}$ be a subset of $W^{k,\infty }\left(\mathrm{\Omega }\right)$ that is uniformly bounded in $L^{\infty }\left(\mathrm{\Omega }\right)$, with the kth derivatives of each $f\in \mathcal{F}$ that are uniformly bounded in $L^1\left(\mathrm{\Omega }\right)$ and satisfy (12). Our purpose is to show that $\mathcal{F}$ is relatively weakly compact in $W^{k,\infty }\left(\mathrm{\Omega }\right)$.

When $k=1$, this theorem is just a special case of Theorem 8. Hence, we only prove the case of $k\ge 2$.

In fact, the 2nd and 3rd conditions imply the uniform boundedness of the kth derivatives in $L^{\infty }\left(\mathrm{\Omega }\right)$. Fix $\epsilon =1$. By assumption, there exists a partition of $\mathrm{\Omega }$ containing a finite number (say $m_1$) of blocks $E_1$, …, $E_{m_1}$. For each block $E_i$ and f, the inequality (12) holds (without loss of generality, each $E_i$ has a positive measure). Let $\lambda$ be the Lebesgue measure in ${\mathbb{R}}^d$ and denote $M:={sup}_{f\in \mathcal{F}}{\parallel D^{\alpha }f\parallel }_{L^1\left(\mathrm{\Omega }\right)}<+\infty$. We have

$${\parallel D^{\alpha }f\parallel }_{L^1\left(\mathrm{\Omega }\right)}\le \sum _{i=1}^{m_1}\lambda \left(E_i\right)\underset{x_i\in E_i}{ess\; sup}\left|D^{\alpha }f\left(x_i\right)\right|\le M+\lambda \left(\mathrm{\Omega }\right)\mathrm{for}\mathrm{all}f\in \mathcal{F}.$$

It follows that

$$\underset{f\in \mathcal{F}}{sup}{\parallel D^{\alpha }f\parallel }_{L^{\infty }\left(\mathrm{\Omega }\right)}\le {\displaystyle \frac{M+\lambda \left(\mathrm{\Omega }\right)}{{min}_{i\in \{1,\dots ,m_1\}}\lambda \left(E_i\right)}}<+\infty .$$

(13)

Combining the 1st condition with (13) and applying the interpolation inequality [11] [Theorem 5.2, p. 135], we obtain that $\mathcal{F}$ is uniformly bounded in $W^{k,\infty }\left(\mathrm{\Omega }\right)$. By the Arzelà-Ascoli theorem [9] [Theorem 4.25, p. 111], we have

$$W^{1,\infty }\left(\mathrm{\Omega }\right)\cong C^{0,1}\left(\mathrm{\Omega }\right)\hookrightarrow \hookrightarrow C_b\left(\mathrm{\Omega }\right)\hookrightarrow L^{\infty }\left(\mathrm{\Omega }\right).$$

For an arbitrary bounded set $\mathcal{G}$ in $W^{k,\infty }\left(\mathrm{\Omega }\right)$, the weak derivatives of functions in $\mathcal{G}$ up to order $k-1$ are uniformly bounded in $W^{1,\infty }\left(\mathrm{\Omega }\right)$. Hence, using the above compact embedding and by Definition 10, we have that

$${\iota \mathcal{G}|}_{{\mathrm{\Omega }}^{(k-1)}}:=\{\iota g{|}_{{\mathrm{\Omega }}^{(k-1)}}\mid g\in \mathcal{G}\}$$

is relatively compact in $L^{\infty }\left({\mathrm{\Omega }}^{(k-1)}\right)$. Applying Lemma 1, $\mathcal{G}$ is relatively compact in $W^{k-1,\infty }\left(\mathrm{\Omega }\right)$, that is, we have the compact embedding

$$W^{k,\infty }\left(\mathrm{\Omega }\right)\hookrightarrow \hookrightarrow W^{k-1,\infty }\left(\mathrm{\Omega }\right),\mathrm{for}k\ge 2.$$

Since $\mathcal{F}$ is uniformly bounded in $W^{k,\infty }\left(\mathrm{\Omega }\right)$, as we have just proved, $\mathcal{F}$ is relatively compact (and, of course, relatively weakly compact) in $W^{k-1,\infty }\left(\mathrm{\Omega }\right)$ by definition. Applying Theorem 11, it means that there exists a measurable partition of $\mathrm{\Omega }$ containing a finite number of blocks such that for each block P in this partition and for any multi-index $\alpha$ with $\left|\alpha \right|\le k-1$, the following holds:

$${osc}_P\left(D^{\alpha }f\right)<\epsilon .$$

(14)

The proof concludes by iterating the compact embedding argument to establish the required oscillation control for all derivative orders. Combining (14) with (12) and applying Theorem 11 again, we get the relative weak compactness of $\mathcal{F}$ in $W^{k,\infty }\left(\mathrm{\Omega }\right)$, which completes the proof. □

Example 3

(a non-weakly compact sequence in $W^{1,\infty }(0,1)$). Let $\left\{g_n\right\}$ be a sequence in $W^{1,\infty }(0,1)$ and each term has the following form:

$$g_n\left(x\right)=\left\{\begin{array}{cc}0,\hfill & x\in (0,{\displaystyle \frac{n-1}{n}});\hfill \\ x-{\displaystyle \frac{n-1}{n}},\hfill & x\in [{\displaystyle \frac{n-1}{n}},1).\hfill \end{array}\right.$$

Recalling Example 2, we immediately obtain that $\left\{g_n\right\}$ fails the uniform oscillation refinement* condition since the weak generalized derivative of each term is exactly $f_n$. Hence, $\left\{g_n\right\}$ is not relatively weakly compact by Theorem 12.

3. Conclusions and Future Work

In this paper, we have established a characterization of weak compactness in the non-reflexive Sobolev spaces $W^{k,\infty }\left(\mathrm{\Omega }\right)$. Our main result, Theorem 11, provides a necessary and sufficient condition: a subset is relatively weakly compact if and only if it is bounded and all its weak derivatives up to order k have uniformly small essential oscillation on a finite measurable partition of $\mathrm{\Omega }$. The key to this result lies in bridging functional analysis with operator algebra theory. By constructing an isometric embedding $\iota :W^{k,\infty }\left(\mathrm{\Omega }\right)\hookrightarrow L^{\infty }\left({\mathrm{\Omega }}^{\left(k\right)}\right)$ and leveraging the von Neumann algebra structure of $L^{\infty }$ via Gelfand duality, we transformed the problem into one concerning $C\left(\mathrm{\Delta }\right)$ on a hyper-Stonean space, where Grothendieck’s abstract criterion could be concretely applied. For domains with sufficient regularity, Theorem 12 offers a significant simplification, showing that control over the highest-order derivatives alone is sufficient.

The criterion developed here, expressed through the uniform oscillation refinement condition, serves as an analog of the Dunford–Pettis theorem for $L^1$ spaces, now tailored for the $L^{\infty }$–Sobolev setting. It provides a tool for analyzing convergence in nonlinear problems where standard compact embeddings fail due to the lack of reflexivity.

Several promising directions for future research emerge from this work:

• Applications to nonlinear PDEs and calculus of variations: The immediate application lies in proving existence theorems for problems formulated in $W^{k,\infty }$. For instance, in the study of contact Hamilton–Jacobi equations, viscosity solutions typically possess Lipschitz continuity and thus naturally belong to some $W^{1,\infty }$ space (see Wu & Zhang [13], 2026). The criterion for weak compactness in $W^{k,\infty }$ spaces established in this work may therefore serve as a relevant functional-analytic tool for analyzing the convergence of solution sets in such equations.
• Extension to more general function Spaces: It is natural to investigate whether a similar strategy can be applied to characterize weak compactness in other non-reflexive function spaces closely related to $L^{\infty }$.
• The non-commutative setting: Given the central role of commutative von Neumann algebras, a generalization would be to explore weak compactness in the non-commutative analogs of $W^{k,\infty }$, such as Sobolev spaces associated with non-commutative $L^{\infty }$–algebras [14] [Chapter 6, p. 391]. However, in this situation, the underlying Gelfand spectrum is no longer available.

The in-depth analysis of mathematical structures is an important aspect for understanding the related theories. This is also reflected in the exploration of compactness conditions for various spaces in other fields, such as the study of the weak* extensibility of subspaces in Banach spaces ([15], 2022) and the investigation of the compactness properties concerning topological orders and neutrosophic sets ([16], 2025).