The Sobolev embedding theorem is a fundamental result in function space theory. It concerns the inclusion of Sobolev spaces into spaces of continuous or HÃ¶lder continuous functions. This theorem has significant implications in functional analysis, partial differential equations, and variational problems. It allows us to understand the regularity properties of solution functions.

The Sobolev embedding theorem establishes a strong connection between the smoothness and integrability of functions. It states that under certain conditions, functions with sufficiently high derivatives also possess desirable integrability characteristics. This theorem plays a crucial role in mathematics and physics, shedding light on the interplay between smoothness and integrability with rigor.

The theorem asserts that if a function belongs to a Sobolev space, which includes functions with a prescribed number of weak derivatives, it also belongs to a specific Lebesgue space. It provides conditions for embedding the Sobolev space into a higher integrability Lebesgue space. This embedding property is important as it bridges the gap between function regularity and integrability behavior.

For example, in three-dimensional space, the Sobolev embedding theorem implies that a function with enough weak derivatives will also possess certain integral properties. This allows us to derive meaningful results about its behavior, convergence, continuity, and decay rates based on its smoothness.

The theorem’s formulation requires careful consideration of the conditions under which the embedding holds. These conditions involve assumptions about spatial dimension, derivative order, and integrability requirements. By establishing these conditions, the theorem ensures that the embedded function inherits the desired integrability properties, facilitating rigorous analysis of various mathematical problems.

In summary, the Sobolev embedding theorem is a foundational result in functional analysis, connecting the smoothness and integrability of functions. Its significance resonates in mathematics and physics, offering valuable insights into function behavior and enabling the development of advanced mathematical techniques and tools.

# Definition of Sobolev Space

The Sobolev space \(W^{k,p}(\Omega)\), where \(\Omega \subset \mathbb{R}^n\) is a bounded domain, \(k \in \mathbb{N}\), and \(p \in [1, \infty]\) is defined as the set of all functions \(u \in L^p(\Omega)\) such that all weak derivatives of \(u\) up to order \(k\) are also in \(L^p(\Omega)\). In other words, \(u \in W^{k,p}(\Omega)\) if

\[u \in L^p(\Omega), \hspace{2mm} D^{\alpha}u \in L^p(\Omega) \quad \text{for all } |\alpha| \leq k,\]

where \(\alpha = (\alpha_1, \alpha_2, \ldots, \alpha_n)\) is a multi-index representing partial derivatives, and \(D^{\alpha}u\) denotes the weak derivative of \(u\) of order \(|\alpha|\).

# Definition of Continuous and HÃ¶lder continuous functions

Here are the definitions of continuous and HÃ¶lder continuous functions:

- Continuous function: A function \(f: X \to Y\) is continuous if for every point \(x_0\) in the domain \(X\) and every small positive number \(\epsilon\), there exists a small positive number \(\delta\) such that for all points \(x\) in the domain satisfying \(|x – x_0| < \delta\), it follows that \(|f(x) – f(x_0)| < \epsilon\), where \(|\cdot|\) denotes the distance or norm between points in the respective spaces \(X\) and \(Y\).
- HÃ¶lder continuous function: A function \(f: X \to Y\) is HÃ¶lder continuous with exponent \(\alpha\), \(0 < \alpha \leq 1\), if there exists a positive constant \(C\) such that for all points \(x, y\) in the domain \(X\), it holds that \(|f(x) – f(y)| \leq C |x – y|^\alpha\), where \(|\cdot|\) denotes the distance or norm between points in the respective spaces \(X\) and \(Y\).

These definitions capture the notions of continuity and HÃ¶lder continuity, which describe the behavior and smoothness of functions in mathematical analysis.

The main motivation behind the Sobolev embedding theorem is to establish a link between the smoothness of functions and their integrability. In many applications, it is important to determine the regularity of solutions to partial differential equations, which can be described in terms of derivatives or weak derivatives. The Sobolev spaces provide a natural framework for measuring the smoothness or regularity of functions based on their weak derivatives.

The embedding theorem addresses the question of whether functions with certain degrees of smoothness also possess certain integrability properties. It establishes conditions under which functions in a Sobolev space can be continuously or HÃ¶lder continuously extended to larger function spaces. This is particularly relevant because continuous or HÃ¶lder continuous functions have desirable properties for various applications, such as being suitable for interpolation or being well-defined pointwise.

By establishing the embedding theorem, one can relate the regularity properties of solutions to PDEs to their integrability, allowing for a deeper understanding of the behavior of solutions. This result has significant implications in the analysis of PDEs, harmonic analysis, and approximation theory. It provides a bridge between different function spaces, allowing for the transfer of results and techniques from one space to another.

In summary, the motivation behind the Sobolev embedding theorem lies in understanding the relationship between the smoothness and integrability of functions. By establishing this connection, the theorem provides insights into the regularity properties of solutions to PDEs and variational problems and facilitates the application of tools and techniques from different function spaces.

# Definition of Regularity and Integrability

In the context of the Sobolev embedding theorem, regularity and integrability refer to two different aspects of the behavior of functions:

- Regularity: Regularity refers to the smoothness or differentiability properties of functions. In the study of partial differential equations (PDEs) and variational problems, it is often important to understand how “well-behaved” or smooth the solutions are. Regularity can be quantified by considering the existence and behavior of derivatives or weak derivatives of functions. Functions with higher regularity possess more derivatives or higher order weak derivatives, indicating a higher degree of smoothness.
- Integrability: Integrability refers to the behavior of functions with respect to their integrals. Integrability properties describe how functions behave when integrated over a certain domain or measure. In the context of the Sobolev embedding theorem, integrability is typically measured by considering the square integrability or Lebesgue integrability of functions. Functions with higher integrability have finite integral values over certain domains or measures.

The Sobolev embedding theorem establishes a connection between these two aspects. It provides conditions under which functions with a certain level of regularity, typically measured in terms of the Sobolev norm, also possess a certain level of integrability. This result indicates that functions with a certain smoothness or regularity have well-defined integral properties.

For example, the classical Sobolev embedding theorem states that if a function belongs to a Sobolev space with a certain order of weak derivatives, then it can be continuously embedded into a space of continuous functions or HÃ¶lder continuous functions. This means that functions with a certain degree of smoothness (regularity) also possess certain integrability properties, such as being continuous or having bounded oscillation.

In summary, regularity refers to the smoothness or differentiability properties of functions, while integrability refers to the behavior of functions with respect to their integrals. The Sobolev embedding theorem establishes a relationship between these two aspects by providing conditions under which functions with a certain regularity also possess certain integrability properties.

# Sobolev Embedding Theorems Simplified

Imagine you have a big box of different shapes like squares, circles, and triangles. Some shapes are smooth and nicely rounded, while others have sharp edges or corners. Now, let’s say we want to put these shapes inside different containers. Some containers can hold all the shapes, while others can only hold certain types.

The Sobolev embedding theorem is like a rule that tells us which containers can hold different shapes. It helps us understand how smooth or bumpy the shapes are and whether they can fit into certain containers.

In mathematics, instead of shapes, we have functions. Functions are like mathematical shapes that describe how things change or behave. The Sobolev embedding theorem helps us understand how smooth or bumpy these functions are and which “containers” they can fit into.

One way we measure smoothness is by looking at how many times we can differentiate a function. Just like we can calculate the area of a shape, we can calculate the derivative of a function. The more times we can differentiate a function, the smoother it is.

The Sobolev embedding theorem tells us that if a function is very smooth, meaning we can differentiate it many times, then it can fit into different types of “containers” called function spaces. These function spaces are like special boxes that hold functions with specific smoothness properties.

By using the Sobolev embedding theorem, mathematicians can understand how smooth a function is and whether it belongs to certain function spaces. This helps us solve problems involving functions, like equations or calculations, and it gives us a way to describe how smooth or bumpy a function is in a mathematical way.

So, just like the shapes can fit into different containers depending on how smooth or bumpy they are, the Sobolev embedding theorem tells us which function spaces can hold different functions depending on their smoothness.

# Difference between embedding and containing

Embedding states that $U$ is embedded in $V$ if the identical map is bounded from $U$ to $V$, i.e., $|x|_{V} \leq C|x|_{U}$.

Containing asserts that $U \subset V$ if $x \in V$ for all $x \in U$.

It seems to me that embedding implies containing between normed spaces?

Embedding and containing are very similar^{1}. However, embedding is a bit more “general” in that containing is embedding with the identity map. An embedding of $A$ into $B$ is using an injective homomorphism $\phi:A\to B$. In this sense, $A$ is almost like a subset of $B$, but it lives in a different world.

For example, $\mathbb Z$ is contained in $\mathbb R$, since the map $f:\mathbb Z\to\mathbb R$ defined by $f(n)=n$ is the identity.

On the other hand, $\mathbb R$ can be embedded into $\mathbb R^2$ by the natural map $g:\mathbb R\to \mathbb R^2$ defined by $g(x)=(x,0)$. The big difference here is that $\mathbb R$ is not a subset of $\mathbb R^2$, since they do not have the same kinds of elements. However, the image of $g$ (which is $\mathbb R\times 0$) *is* contained in $\mathbb R^2$.

Said another way, if $A$ is contained in $B$, we can map all of the elements of $A$ to themselves in $B$, which is clearly an injective mapping. If $A$ can be embedded into $B$, then there exists an injective homomorphism $\phi:A\to B$. We can call the image of this map $\phi[A]$, and all of the elements of $\phi[A]$ could be directly mapped to themselves in $B$.

# Rewording what we just said

When we say that showing something can be embedded means you can basically treat it like a subset, we are referring to the idea that if one mathematical structure (such as a set, space, or algebraic system) can be embedded into another, then the properties and operations defined on the smaller structure can be interpreted within the larger one.

## Embedding and Subset in Mathematical Context

**Embedding Definition**

- An embedding is a function that maps one mathematical structure into another in a way that preserves the structure. This means the function is injective (one-to-one) and maintains the operations and relations of the embedded structure within the larger structure.

**Subset Interpretation**

- When a structure $A$ is embedded into a structure $B$, the image of $A$ under the embedding function can be viewed as a subset of $B$. This allows us to work with $A$ as if it were a part of $B$, utilizing the same operations and relations defined in $B$.

### Examples

**Vector Spaces**

- Consider vector spaces $V$ and $W$ where $V$ is a subspace of $W$. The embedding of $V$ into $W$ is simply the inclusion map. Once $V$ is embedded in $W$, every vector in $V$ is treated as a vector in $W$, and operations like addition and scalar multiplication are inherited from $W$.

**Topological Spaces**

- For topological spaces, an embedding is a continuous injective map that is also a homeomorphism onto its image. If a space $X$ is embedded into a space $Y$, then $X$ can be treated as a subspace of $Y$ with the topology induced by $Y$.

**Groups**

- In group theory, if a group $G$ can be embedded into a group $H$, the embedding is a homomorphism that is injective. This means the group $G$ operates within $H$ while preserving its group structure, allowing us to treat $G$ as a subgroup of $H$.

## Conclusion

Thus, when we say that an embedding allows us to treat a structure like a subset, we mean that the embedded structure retains its identity and operations within the larger structure. This concept is crucial across various fields of mathematics, as it enables the seamless integration and interaction of different structures.