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 $u^2$

The study of Sobolev spaces and fractional Sobolev spaces has been a crucial area of research in functional analysis and partial differential equations. These spaces provide a powerful tool for analyzing the regularity of functions and their derivatives. 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.

Sobolev spaces are a class of function spaces that are defined in terms of the regularity of functions and their derivatives. The Sobolev space $H^s(\mathbb{T})$ is a Hilbert space that consists of all functions $u \in L^2(\mathbb{T})$ such that the distributional derivative $u'$ belongs to $L^2(\mathbb{T})$. The norm of $u$ in $H^s(\mathbb{T})$ is defined as $\|u\|_{H^s(\mathbb{T})} = \left(\|u\|_{L^2(\mathbb{T})}^2 + \|u'\|_{L^2(\mathbb{T})}^2\right)^{1/2}$.

Fractional Sobolev spaces are a generalization of Sobolev spaces that allow for the analysis of functions with non-integer regularity. The fractional Sobolev space $H^r(\mathbb{T})$ is a Hilbert space that consists of all functions $u \in L^2(\mathbb{T})$ such that the distributional derivative $u'$ belongs to $H^{r-1}(\mathbb{T})$. The norm of $u$ in $H^r(\mathbb{T})$ is defined as $\|u\|_{H^r(\mathbb{T})} = \left(\|u\|_{L^2(\mathbb{T})}^2 + \|u'\|_{H^{r-1}(\mathbb{T})}^2\right)^{1/2}$.

Now, let's consider 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 are given that $u \in H^s(\mathbb{T})$, and we want to determine whether $u^2 \in H^r(\mathbb{T})$ for some $r > 0$ depending only on $s$.

The Case $s > 1/4$

For $s > 1/4$, we can use the following inequality to bound the norm of $u^2$ in $H^r(\mathbb{T})$:

$$\|u^2\|_{H^r(\mathbb{T})} \leq C\|u\|_{H^s(\mathbb{T})}^2$$

where $C$ is a constant depending only on $s$. This inequality follows from the fact that $u^2$ can be written as a convolution of $u$ with itself, and the convolution operator is bounded from $H^s(\mathbb{T})$ to $H^r(\mathbb{T})$ for $s > 1/4$.

The Case $s \leq 1/4$

For $s \leq 1/4$, we can use the following inequality to bound the norm of $u^2$ in $H^r(\mathbb{T})$:

$$\|u^2\|_{H^r(\mathbb{T})} \leq C\|u\|_{H^s(\mathbb{T})}^2\log(1/\|u\|_{H^s(\mathbb{T})})$$

where $C$ is a constant depending only on $s$. This inequality follows from the fact that $u^2$ can be written as a convolution of $u$ with itself, and the convolution operator is bounded from $H^s(\mathbb{T})$ to $H^r(\mathbb{T})$ for $s \leq 1/4$.

In conclusion, 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 relationship between $H^s(\mathbb{T})$ and $u^2$ depends on the value of $s$, and we have provided explicit bounds on the norm of $u^2$ in $H^r(\mathbb{T})$ for both cases $s > 1/4$ and $s \leq 1/4$.

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

There are several directions for future research on the relationship between $H^s(\mathbb{T})$ and $u^2$. One possible direction is to investigate the relationship between $H^s(\mathbb{T})$ and $u^2$ for more general domains than the torus $\mathbb{T}$. Another possible direction is to study the relationship between $H^s(\mathbb{T})$ and $u^2$ for more general types of functions than the ones considered in this article.
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 fractional Sobolev space $H^r(\mathbb{T})$ when $u^2$ is involved. We provided explicit bounds on the norm of $u^2$ in $H^r(\mathbb{T})$ for both cases $s > 1/4$ and $s \leq 1/4$. In this article, we will answer some of the most frequently asked questions about the relationship between $H^s(\mathbb{T})$ and $u^2$.

Q: What is the significance of the value $s = 1/4$ in the relationship between $H^s(\mathbb{T})$ and $u^2$?

A: The value $s = 1/4$ is significant because it marks the boundary between the cases $s > 1/4$ and $s \leq 1/4$. For $s > 1/4$, the norm of $u^2$ in $H^r(\mathbb{T})$ can be bounded by a constant times the norm of $u$ in $H^s(\mathbb{T})$. For $s \leq 1/4$, the norm of $u^2$ in $H^r(\mathbb{T})$ can be bounded by a constant times the norm of $u$ in $H^s(\mathbb{T})$ times a logarithmic factor.

Q: Can you provide more details about the logarithmic factor in the bound for $s \leq 1/4$?

A: The logarithmic factor in the bound for $s \leq 1/4$ is given by $\log(1/\|u\|_{H^s(\mathbb{T})})$. This factor arises because the convolution operator is not bounded from $H^s(\mathbb{T})$ to $H^r(\mathbb{T})$ for $s \leq 1/4$, and the logarithmic factor is needed to compensate for this lack of boundedness.

Q: How does the relationship between $H^s(\mathbb{T})$ and $u^2$ depend on the dimension of the domain?

A: The relationship between $H^s(\mathbb{T})$ and $u^2$ depends on the dimension of the domain in a subtle way. For higher-dimensional domains, the norm of $u^2$ in $H^r(\mathbb{T})$ can be bounded by a constant times the norm of $u$ in $H^s(\mathbb{T})$ times a factor that depends on the dimension of the domain. For lower-dimensional domains, the norm of $u^2$ in $H^r(\mathbb{T})$ can be bounded by a constant times the norm of $u$ in $H^s(\mathbb{T})$ times a factor that depends on the dimension of the domain.

Q: Can you provide more details about the factor that depends on the dimension of the domain?

A: The factor that depends on the dimension of the domain is given by a power of the dimension of the domain. Specifically, for a $d$-dimensional domain, the factor is given by $(1 + \|u\|_{H^s(\mathbb{T})}^2)^{d/2}$. This factor arises because the convolution operator is not bounded from $H^s(\mathbb{T})$ to $H^r(\mathbb{T})$ for lower-dimensional domains, and the power of the dimension of the domain is needed to compensate for this lack of boundedness.

Q: How does the relationship between $H^s(\mathbb{T})$ and $u^2$ depend on the type of function?

A: The relationship between $H^s(\mathbb{T})$ and $u^2$ depends on the type of function in a subtle way. For smooth functions, the norm of $u^2$ in $H^r(\mathbb{T})$ can be bounded by a constant times the norm of $u$ in $H^s(\mathbb{T})$. For non-smooth functions, the norm of $u^2$ in $H^r(\mathbb{T})$ can be bounded by a constant times the norm of $u$ in $H^s(\mathbb{T})$ times a factor that depends on the type of function.

Q: Can you provide more details about the factor that depends on the type of function?

A: The factor that depends on the type of function is given by a power of the norm of $u$ in $H^s(\mathbb{T})$. Specifically, for a non-smooth function, the factor is given by $(1 + \|u\|_{H^s(\mathbb{T})}^2)^{1/2}$. This factor arises because the convolution operator is not bounded from $H^s(\mathbb{T})$ to $H^r(\mathbb{T})$ for non-smooth functions, and the power of the norm of $u$ in $H^s(\mathbb{T})$ is needed to compensate for this lack of boundedness.

In conclusion, we have answered some of the most frequently asked questions about the relationship between $H^s(\mathbb{T})$ and $u^2$. We hope that this article has provided a useful overview of the relationship between these two spaces, and has helped to clarify some of the subtleties involved.