If U U U Is In H S ( T ) H^s(\mathbb{T}) H S ( T ) , Then Which (fractional) Sobolev Space Is U 2 U^2 U 2 In?

Sobolev Spaces and Fractional Sobolev Spaces: Understanding the Relationship Between $H^s(\mathbb{T})$ and $u^2$

Introduction

The study of Sobolev spaces and fractional Sobolev spaces has been a crucial area of research in the field of functional analysis and partial differential equations. These spaces play a vital role in understanding the regularity of functions and their properties. In this article, we will explore the relationship between the Sobolev space $H^s(\mathbb{T})$ and the fractional Sobolev space $H^r(\mathbb{T})$ when $u^2$ is involved. We will delve into the properties of these spaces, the conditions under which $u^2$ belongs to $H^r(\mathbb{T})$, and the implications of these findings.

Sobolev Spaces and Fractional Sobolev Spaces

Sobolev spaces are a class of function spaces that are defined based on the regularity of functions. They are used to study the properties of functions and their derivatives. The Sobolev space $H^s(\mathbb{T})$ is a Hilbert space that consists of all functions $u$ that are square-integrable and have a certain number of derivatives. The space is defined as:

$$H^s(\mathbb{T}) = \left\{ u \in L^2(\mathbb{T}) \, \Big| \, \int_{\mathbb{T}} (1 + |\xi|^2)^s |\hat{u}(\xi)|^2 d\xi < \infty \right\}$$

where $\hat{u}(\xi)$ is the Fourier transform of $u$.

Fractional Sobolev spaces are a generalization of Sobolev spaces. They are used to study functions that have a certain degree of regularity, but not necessarily a certain number of derivatives. The fractional Sobolev space $H^r(\mathbb{T})$ is defined as:

$$H^r(\mathbb{T}) = \left\{ u \in L^2(\mathbb{T}) \, \Big| \, \int_{\mathbb{T}} (1 + |\xi|^2)^r |\hat{u}(\xi)|^2 d\xi < \infty \right\}$$

where $r$ is a real number.

The Relationship Between $H^s(\mathbb{T})$ and $u^2$

We are interested in determining the fractional Sobolev space to which $u^2$ belongs when $u$ is in $H^s(\mathbb{T})$. To do this, we need to analyze the properties of $u^2$ and its relationship with $H^r(\mathbb{T})$.

Let $u \in H^s(\mathbb{T})$. Then, we can write:

$$u^2 = u \cdot u$$

Using the Fourier transform, we can write:

$$\hat{u^2}(\xi) = \hat{u}(\xi) \cdot \hat{u}(\xi)$$

Now, we need to determine the properties of $\hat{u^2}(\xi)$ and its relationship with $H^r(\mathbb{T})$.

The Condition $s > 1/4$

We are interested in determining the condition under which $u^2$ belongs to $H^r(\mathbb{T})$ for some $r > 0$ depending only on $s$. To do this, we need to analyze the properties of $\hat{u^2}(\xi)$ and its relationship with $H^r(\mathbb{T})$.

Let $s > 1/4$. Then, we can write:

$$\int_{\mathbb{T}} (1 + |\xi|^2)^r |\hat{u^2}(\xi)|^2 d\xi = \int_{\mathbb{T}} (1 + |\xi|^2)^r |\hat{u}(\xi)|^2 |\hat{u}(\xi)|^2 d\xi$$

Using the Cauchy-Schwarz inequality, we can write:

$$\int_{\mathbb{T}} (1 + |\xi|^2)^r |\hat{u}(\xi)|^2 |\hat{u}(\xi)|^2 d\xi \leq \left( \int_{\mathbb{T}} (1 + |\xi|^2)^{2s} |\hat{u}(\xi)|^2 d\xi \right)^{1/2} \left( \int_{\mathbb{T}} (1 + |\xi|^2)^r |\hat{u}(\xi)|^2 d\xi \right)^{1/2}$$

Now, we need to determine the properties of the two integrals on the right-hand side of the inequality.

The First Integral

The first integral on the right-hand side of the inequality is:

$$\int_{\mathbb{T}} (1 + |\xi|^2)^{2s} |\hat{u}(\xi)|^2 d\xi$$

Since $u \in H^s(\mathbb{T})$, we know that:

$$\int_{\mathbb{T}} (1 + |\xi|^2)^s |\hat{u}(\xi)|^2 d\xi < \infty$$

Using the fact that $s > 1/4$, we can write:

$$(1 + |\xi|^2)^{2s} \leq (1 + |\xi|^2)^s (1 + |\xi|^2)^s$$

Using the Cauchy-Schwarz inequality, we can write:

$$\int_{\mathbb{T}} (1 + |\xi|^2)^{2s} |\hat{u}(\xi)|^2 d\xi \leq \left( \int_{\mathbb{T}} (1 + |\xi|^2)^s |\hat{u}(\xi)|^2 d\xi \right)^2$$

Since $u \in H^s(\mathbb{T})$, we know that:

$$\int_{\mathbb{T}} (1 + |\xi|^2)^s |\hat{u}(\xi)|^2 d\xi < \infty$$

Therefore, we can conclude that:

$$\int_{\mathbb{T}} (1 + |\xi|^2)^{2s} |\hat{u}(\xi)|^2 d\xi < \infty$$

The Second Integral

The second integral on the right-hand side of the inequality is:

$$\int_{\mathbb{T}} (1 + |\xi|^2)^r |\hat{u}(\xi)|^2 d\xi$$

Since $u \in H^s(\mathbb{T})$, we know that:

$$\int_{\mathbb{T}} (1 + |\xi|^2)^s |\hat{u}(\xi)|^2 d\xi < \infty$$

Using the fact that $s > 1/4$, we can write:

$$(1 + |\xi|^2)^r \leq (1 + |\xi|^2)^s$$

Using the Cauchy-Schwarz inequality, we can write:

$$\int_{\mathbb{T}} (1 + |\xi|^2)^r |\hat{u}(\xi)|^2 d\xi \leq \int_{\mathbb{T}} (1 + |\xi|^2)^s |\hat{u}(\xi)|^2 d\xi$$

Since $u \in H^s(\mathbb{T})$, we know that:

$$\int_{\mathbb{T}} (1 + |\xi|^2)^s |\hat{u}(\xi)|^2 d\xi < \infty$$

Therefore, we can conclude that:

$$\int_{\mathbb{T}} (1 + |\xi|^2)^r |\hat{u}(\xi)|^2 d\xi < \infty$$

Conclusion

We have shown that if $u \in H^s(\mathbb{T})$ and $s > 1/4$, then $u^2 \in H^r(\mathbb{T})$ for some $r > 0$ depending only on $s$. This result has important implications for the study of Sobolev spaces and fractional Sobolev spaces.

References

• Adams, R. A., & Fournier, J. J. F. (2003). Sobolev spaces. Academic Press.
• Triebel, H. (1992). Theory of function spaces. Birkhäuser.
• Stein, E. M. (1970). Singular integrals and differentiability properties of functions. Princeton University Press.
  Q&A: Understanding Sobolev Spaces and Fractional Sobolev Spaces

Introduction

In our previous article, we explored the relationship between the Sobolev space $H^s(\mathbb{T})$ and the fractional Sobolev space $H^r(\mathbb{T})$ when $u^2$ is involved. We showed that if $u \in H^s(\mathbb{T})$ and $s > 1/4$, then $u^2 \in H^r(\mathbb{T})$ for some $r > 0$ depending only on $s$. In this article, we will answer some frequently asked questions about Sobolev spaces and fractional Sobolev spaces.

Q: What is the definition of a Sobolev space?

A: A Sobolev space is a class of function spaces that are defined based on the regularity of functions. The Sobolev space $H^s(\mathbb{T})$ is a Hilbert space that consists of all functions $u$ that are square-integrable and have a certain number of derivatives.

Q: What is the difference between a Sobolev space and a fractional Sobolev space?

A: A Sobolev space is a space of functions that have a certain number of derivatives, while a fractional Sobolev space is a space of functions that have a certain degree of regularity, but not necessarily a certain number of derivatives.

Q: What is the relationship between the Sobolev space $H^s(\mathbb{T})$ and the fractional Sobolev space $H^r(\mathbb{T})$?

A: We have shown that if $u \in H^s(\mathbb{T})$ and $s > 1/4$, then $u^2 \in H^r(\mathbb{T})$ for some $r > 0$ depending only on $s$.

Q: What is the significance of the condition $s > 1/4$?

A: The condition $s > 1/4$ is necessary for the result to hold. If $s \leq 1/4$, then the result does not hold.

Q: Can you provide an example of a function that belongs to the Sobolev space $H^s(\mathbb{T})$?

A: Yes, the function $u(x) = x^s$ belongs to the Sobolev space $H^s(\mathbb{T})$.

Q: Can you provide an example of a function that belongs to the fractional Sobolev space $H^r(\mathbb{T})$?

A: Yes, the function $u(x) = x^r$ belongs to the fractional Sobolev space $H^r(\mathbb{T})$.

Q: What are some applications of Sobolev spaces and fractional Sobolev spaces?

A: Sobolev spaces and fractional Sobolev spaces have many applications in mathematics and physics, including the study of partial differential equations, the theory of functions of several variables, and the study of harmonic analysis.

Q: Can you recommend some resources for learning more about Sobolev spaces and fractional Sobolev spaces?

A: Yes, some recommended resources include the books "Sobolev Spaces" by R. A. Adams and J. J. F. Fournier, "Theory of Function Spaces" by H. Triebel, and "Singular Integrals and Differentiability Properties of Functions" by E. M. Stein.

Conclusion

We hope that this Q&A article has provided a helpful introduction to Sobolev spaces and fractional Sobolev spaces. If you have any further questions, please don't hesitate to ask.

References

• Adams, R. A., & Fournier, J. J. F. (2003). Sobolev spaces. Academic Press.
• Triebel, H. (1992). Theory of function spaces. Birkhäuser.
• Stein, E. M. (1970). Singular integrals and differentiability properties of functions. Princeton University Press.