Besov Spaces: Compact Embedding In Hölder Spaces Explained

Aug 26, 2025 by Lucas 59 views

Reference Request: Compact Embedding of Besov Spaces into Hölder Spaces

Let's dive into a fascinating topic in functional analysis: the compact embedding of Besov spaces into Hölder spaces. This concept pops up in various areas, particularly when dealing with the smoothness and regularity of functions. Today, we're going to break down what this means, why it's important, and where you can find the references to back it all up. Guys, trust me, it’s more interesting than it sounds!

Understanding Besov and Hölder Spaces

Before we get into the nitty-gritty, let’s define our terms. Besov spaces and Hölder spaces are types of function spaces that characterize the smoothness of functions. They quantify how many derivatives a function has, and how well-behaved those derivatives are. Think of them as measuring how "smooth" a function is. To truly understand the compact embedding, we need a solid grasp of what these spaces entail.

Besov Spaces

Besov spaces, denoted as ${B_{p,q}^s(\mathbb{R}^n)}$ , are a family of function spaces that generalize Hölder spaces and Sobolev spaces. They are defined using decomposition techniques, often involving wavelet decompositions or Littlewood-Paley decompositions. The parameters ${s}$ , ${p}$ , and ${q}$ play crucial roles:

${s}$ represents the smoothness index. It tells you how many derivatives (possibly fractional) the functions in the space possess.
${p}$ is related to the ${L^p}$ integrability of the function.
${q}$ is a finer index that controls the integrability of the differences in the function's derivatives.

In simple terms, ${B_{p,q}^s(\mathbb{R}^n)}$ contains functions that have ${s}$ derivatives, with their ${p}$ -th power being integrable, and the differences of these derivatives satisfying a certain integrability condition governed by ${q}$ . Besov spaces are incredibly versatile, allowing us to describe a wide range of smoothness properties. They are particularly useful when dealing with functions that have different degrees of smoothness in different directions or at different scales.

Hölder Spaces

Hölder spaces, denoted as ${C^{k,\alpha}(\Omega)}$ , are another class of function spaces that characterize the smoothness of functions. Here, ${k}$ is a non-negative integer representing the number of continuous derivatives, and ${\alpha}$ is a real number between 0 and 1, representing the Hölder exponent. A function ${f}$ belongs to ${C^{k,\alpha}(\Omega)}$ if its ${k}$ -th derivative is Hölder continuous with exponent ${\alpha}$ . Mathematically, this means there exists a constant ${C}$ such that for all ${x, y \in \Omega}$ :

${ |D^k f(x) - D^k f(y)| \leq C |x - y|^{\alpha} }$

where ${D^k f}$ represents the ${k}$ -th derivative of ${f}$ . When ${\alpha = 0}$ , the space ${C^{k,0}(\Omega)}$ is simply the space of functions with ${k}$ continuous and bounded derivatives. When ${\alpha > 0}$ , it implies a certain level of uniformity in the continuity of the ${k}$ -th derivative. Hölder spaces are intuitive and widely used because they provide a direct measure of how well a function can be approximated by polynomials. They are particularly useful in the study of partial differential equations and regularity theory.

Compact Embedding: What Does It Mean?

Now that we have a handle on Besov and Hölder spaces, let’s talk about embedding. An embedding is essentially an inclusion between two function spaces. We say that a space ${X}$ is embedded in a space ${Y}$ , denoted as ${X \hookrightarrow Y}$ , if every function in ${X}$ is also in ${Y}$ , and the inclusion map is continuous. In other words, if a function is in ${X}$ , it's automatically in ${Y}$ , and small changes in ${X}$ lead to small changes in ${Y}$ .

A compact embedding is a stronger condition. We say that ${X}$ is compactly embedded in ${Y}$ , denoted as ${X \hookrightarrow\!\hookrightarrow Y}$ , if every bounded sequence in ${X}$ has a subsequence that converges in ${Y}$ . Compact embeddings are extremely useful because they allow us to approximate functions in ${X}$ by functions in ${Y}$ in a controlled way. This is particularly important in applications like numerical analysis and approximation theory.

Why Compact Embeddings Matter

Compact embeddings are crucial in various areas of analysis and applications:

PDEs: They ensure the existence of solutions to partial differential equations by providing a way to obtain convergent subsequences of approximate solutions.
Approximation Theory: They guarantee that functions in a certain space can be well-approximated by functions in another space, which is essential for numerical methods.
Functional Analysis: They provide valuable insights into the structure and properties of function spaces.

The Specific Embedding: Besov into Hölder

The specific statement we're interested in is that Besov spaces are compactly embedded in Hölder spaces under certain conditions. This can be stated as:

${ B_{p,q}^s(\mathbb{R}^n) \hookrightarrow\!\hookrightarrow C^{k,\alpha}(\mathbb{R}^n) }$

provided that ${s > k + \alpha}$ . This condition is crucial because it ensures that the Besov space has enough smoothness to be included in the Hölder space.

Lemma 3.3 in Wasserstein GANs

In the paper "Wasserstein GANs are Minimax Optimal Distribution Estimators," Lemma 3.3 states this embedding result. The authors likely use this result to establish certain properties of the function spaces involved in their analysis of GANs. Specifically, it allows them to control the regularity of the discriminator function, which is essential for proving convergence and optimality results. The compact embedding ensures that even if the discriminator function is complex and high-dimensional, it can be approximated by smoother functions in a controlled manner.

The precise statement in the paper likely involves specific conditions on the parameters ${s}$ , ${p}$ , ${q}$ , ${k}$ , and ${\alpha}$ . For instance, it might require ${s > k + \alpha}$ and certain relationships between ${p}$ , ${q}$ , and ${n}$ to ensure the embedding holds.

Finding the Right References

Now, where can you find solid references to back up this claim? Here are some suggestions:

Original Paper: Start by carefully examining the cited references in the "Wasserstein GANs" paper. The authors likely cite a standard textbook or research article that proves the embedding result.
Textbooks on Function Spaces: Look into classic textbooks on function spaces, such as:
- "Theory of Function Spaces" by Hans Triebel: This book is a comprehensive resource on Besov spaces, Hölder spaces, and other related function spaces. It contains detailed proofs of embedding theorems and other important results.
- "Function Spaces and Partial Differential Equations" by David Adams and John Fournier: This book provides a thorough treatment of function spaces with a focus on applications to PDEs. It includes detailed discussions of embeddings and compactness results.
- "Interpolation Theory, Function Spaces, Differential Operators" by Hans Triebel: Another excellent book by Triebel, offering deep insights into interpolation theory and its applications to function spaces.
Research Articles: Search for research articles that specifically address the embedding of Besov spaces into Hölder spaces. You can use keywords like "Besov embedding," "Hölder embedding," and "compact embedding" to narrow down your search.
Online Resources: Check online resources like MathOverflow and StackExchange. Often, experts in the field will provide useful references and explanations of key concepts.

Conclusion

The compact embedding of Besov spaces into Hölder spaces is a powerful result with significant implications in various areas of analysis. By understanding the definitions of these spaces and the concept of compact embedding, you can gain a deeper appreciation for the smoothness and regularity of functions. So, next time you encounter this result in a paper, you'll know exactly what it means and where to find the references to support it. Keep exploring, guys, and happy reading!