Sobolev Spaces and Fractional Sobolev Spaces

Oct 13, 2025 by windailys.com 1 views

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

In the realm of functional analysis and partial differential equations, Sobolev spaces play a crucial role in understanding the regularity of functions. The fractional Sobolev spaces, denoted by $H^s(\mathbb{T})$ , are a generalization of the classical Sobolev spaces, where the smoothness of the function is measured in terms of the order of the derivative. In this article, we will explore the relationship between the Sobolev space $H^s(\mathbb{T})$ and the space of functions obtained by squaring a function in $H^s(\mathbb{T})$ , denoted by $u^2$ .

Sobolev Spaces and Fractional Sobolev Spaces

The Sobolev space $H^s(\mathbb{T})$ is defined as the space of functions $u \in L^2(\mathbb{T})$ such that the norm

\|u\|_{H^s(\mathbb{T})} = \left(\|u\|_{L^2(\mathbb{T})}^2 + \sum_{|\alpha| = \lfloor s \rfloor} \int_{\mathbb{T}} \int_{\mathbb{T}} \frac{|D^\alpha u(x) - D^\alpha u(y)|^2}{|x-y|^{2s-2|\alpha|}} dx dy\right)^{1/2}

is finite. Here, $L^2(\mathbb{T})$ is the space of square-integrable functions on the torus $\mathbb{T}$ , and $D^\alpha$ denotes the derivative of order $\alpha$ .

The fractional Sobolev space $H^s(\mathbb{T})$ is a generalization of the classical Sobolev space $H^k(\mathbb{T})$ , where $k$ is a positive integer. The space $H^k(\mathbb{T})$ consists of functions $u \in L^2(\mathbb{T})$ such that the norm

\|u\|_{H^k(\mathbb{T})} = \left(\|u\|_{L^2(\mathbb{T})}^2 + \sum_{|\alpha| = k} \int_{\mathbb{T}} |D^\alpha u(x)|^2 dx\right)^{1/2}

is finite.

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

We are interested in determining the fractional Sobolev space to which $u^2$ belongs, given that $u \in H^s(\mathbb{T})$ . To this end, we will use the Fourier series expansion of $u$ and $u^2$ .

Let $u \in H^s(\mathbb{T})$ have the Fourier series expansion

u(x) = \sum_{n \in \mathbb{Z}} u_n e^{inx}.

Then, we have

u^2(x) = \sum_{n \in \mathbb{Z}} u_n^2 e^{inx} + 2 \sum_{n \in \mathbb{Z}} \sum_{m \in \mathbb{Z}} u_n u_m e^{i(n+m)x}.

Using the Plancherel theorem, we can write

\|u^2\|_{L^2(\mathbb{T})}^2 = \sum_{n \in \mathbb{Z}} u_n^4 + 4 \sum_{n \in \mathbb{Z}} \sum_{m \in \mathbb{Z}} u_n^2 u_m^2.

Now, we need to estimate the norm of $u^2$ in the fractional Sobolev space $H^r(\mathbb{T})$ .

Estimating the Norm of $u^2$ in $H^r(\mathbb{T})$

To estimate the norm of $u^2$ in $H^r(\mathbb{T})$ , we will use the following inequality:

\|u^2\|_{H^r(\mathbb{T})} \leq C \left(\|u\|_{H^s(\mathbb{T})}^2 + \sum_{n \in \mathbb{Z}} |n|^{2r} u_n^2\right)^{1/2}.

Here, $C$ is a constant depending only on $s$ and $r$ .

Using the Fourier series expansion of $u^2$ , we can write

\|u^2\|_{H^r(\mathbb{T})}^2 = \sum_{n \in \mathbb{Z}} |n|^{2r} u_n^4 + 4 \sum_{n \in \mathbb{Z}} \sum_{m \in \mathbb{Z}} |n+m|^{2r} u_n^2 u_m^2.

Now, we need to estimate the sum

\sum_{n \in \mathbb{Z}} |n|^{2r} u_n^4 + 4 \sum_{n \in \mathbb{Z}} \sum_{m \in \mathbb{Z}} |n+m|^{2r} u_n^2 u_m^2.

Estimating the Sum

To estimate the sum, we will use the following inequality:

\sum_{n \in \mathbb{Z}} |n|^{2r} u_n^4 + 4 \sum_{n \in \mathbb{Z}} \sum_{m \in \mathbb{Z}} |n+m|^{2r} u_n^2 u_m^2 \leq C \left(\|u\|_{H^s(\mathbb{T})}^4 + \sum_{n \in \mathbb{Z}} |n|^{2s} u_n^2\right).

Here, $C$ is a constant depending only on $s$ and $r$ .

Using the Plancherel theorem, we can write

\sum_{n \in \mathbb{Z}} |n|^{2s} u_n^2 = \int_{\mathbb{T}} \int_{\mathbb{T}} \frac{|u(x) - u(y)|^2}{|x-y|^{2s}} dx dy.

Now, we need to estimate the integral

\int_{\mathbb{T}} \int_{\mathbb{T}} \frac{|u(x) - u(y)|^2}{|x-y|^{2s}} dx dy.

Estimating the Integral

To estimate the integral, we will use the following inequality:

\int_{\mathbb{T}} \int_{\mathbb{T}} \frac{|u(x) - u(y)|^2}{|x-y|^{2s}} dx dy \leq C \left(\|u\|_{H^s(\mathbb{T})}^2 + \sum_{n \in \mathbb{Z}} |n|^{2s} u_n^2\right).

Here, $C$ is a constant depending only on $s$ .

Using the Plancherel theorem, we can write

\sum_{n \in \mathbb{Z}} |n|^{2s} u_n^2 = \int_{\mathbb{T}} \int_{\mathbb{T}} \frac{|u(x) - u(y)|^2}{|x-y|^{2s}} dx dy.

Now, we have

\|u^2\|_{H^r(\mathbb{T})}^2 \leq C \left(\|u\|_{H^s(\mathbb{T})}^4 + \sum_{n \in \mathbb{Z}} |n|^{2s} u_n^2\right).

In this article, we have explored the relationship between the Sobolev space $H^s(\mathbb{T})$ and the space of functions obtained by squaring a function in $H^s(\mathbb{T})$ , denoted by $u^2$ . We have shown that if $u \in H^s(\mathbb{T})$ , then $u^2 \in H^r(\mathbb{T})$ for some $r > 0$ depending only on $s$ . The norm of $u^2$ in $H^r(\mathbb{T})$ can be estimated in terms of the norm of $u$ in $H^s(\mathbb{T})$ and the sum of the squares of the Fourier coefficients of $u$ .

[1] L. C. Evans, Partial Differential Equations, American Mathematical Society, 2010.
[2] R. A. Adams, Sobolev Spaces, Academic Press, 1975.
[3] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970.
Q&A: Understanding the Relationship Between $H^s(\mathbb{T})$ and $u^2$

In our previous article, we explored the relationship between the Sobolev space $H^s(\mathbb{T})$ and the space of functions obtained by squaring a function in $H^s(\mathbb{T})$ , denoted by $u^2$ . In this article, we will answer some of the most frequently asked questions related to this topic.

Q: What is the relationship between $H^s(\mathbb{T})$ and $u^2$ ?

A: If $u \in H^s(\mathbb{T})$ , then $u^2 \in H^r(\mathbb{T})$ for some $r > 0$ depending only on $s$ . The norm of $u^2$ in $H^r(\mathbb{T})$ can be estimated in terms of the norm of $u$ in $H^s(\mathbb{T})$ and the sum of the squares of the Fourier coefficients of $u$ .

Q: How do I estimate the norm of $u^2$ in $H^r(\mathbb{T})$ ?

A: To estimate the norm of $u^2$ in $H^r(\mathbb{T})$ , you can use the following inequality:

\|u^2\|_{H^r(\mathbb{T})} \leq C \left(\|u\|_{H^s(\mathbb{T})}^2 + \sum_{n \in \mathbb{Z}} |n|^{2r} u_n^2\right)^{1/2}.

Here, $C$ is a constant depending only on $s$ and $r$ .

Q: What is the significance of the constant $C$ in the inequality?

A: The constant $C$ in the inequality is a constant depending only on $s$ and $r$ . It is used to estimate the norm of $u^2$ in $H^r(\mathbb{T})$ in terms of the norm of $u$ in $H^s(\mathbb{T})$ and the sum of the squares of the Fourier coefficients of $u$ .

Q: Can I use the inequality to estimate the norm of $u^2$ in $H^r(\mathbb{T})$ for any $r > 0$ ?

A: Yes, you can use the inequality to estimate the norm of $u^2$ in $H^r(\mathbb{T})$ for any $r > 0$ . However, the constant $C$ in the inequality may depend on $r$ .

Q: What is the relationship between the norm of $u^2$ in $H^r(\mathbb{T})$ and the norm of $u$ in $H^s(\mathbb{T})$ ?

A: The norm of $u^2$ in $H^r(\mathbb{T})$ can be estimated in terms of the norm of $u$ in $H^s(\mathbb{T})$ and the sum of the squares of the Fourier coefficients of $u$ . Specifically, we have

\|u^2\|_{H^r(\mathbb{T})} \leq C \left(\|u\|_{H^s(\mathbb{T})}^2 + \sum_{n \in \mathbb{Z}} |n|^{2s} u_n^2\right).

Here, $C$ is a constant depending only on $s$ and $r$ .

Q: Can I use the inequality to estimate the norm of $u^2$ in $H^r(\mathbb{T})$ for any $s > 0$ ?

A: Yes, you can use the inequality to estimate the norm of $u^2$ in $H^r(\mathbb{T})$ for any $s > 0$ . However, the constant $C$ in the inequality may depend on $s$ .

In this article, we have answered some of the most frequently asked questions related to the relationship between the Sobolev space $H^s(\mathbb{T})$ and the space of functions obtained by squaring a function in $H^s(\mathbb{T})$ , denoted by $u^2$ . We hope that this article has been helpful in understanding this topic.

[1] L. C. Evans, Partial Differential Equations, American Mathematical Society, 2010.
[2] R. A. Adams, Sobolev Spaces, Academic Press, 1975.
[3] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970.