Tagged Questions
2
votes
2answers
27 views
Definition of Sobolev Space
I have a definition that says that the space of functions that satisfy$$\|u\|_{H^m}^2=\sum_{k\in\mathbb{Z}}(1+|k|^2)^m|\hat{u}_k|^2<\infty$$is called Sobolev Space and when $m=1$, this is ...
2
votes
1answer
43 views
Proving that a certain function is in $W^{1,n}(B(0,1))$
Fix $0<\alpha<1-\frac{1}{n}$ and let $f\colon\mathbb{R}^n \rightarrow \mathbb{R}$ be the function $f(x)=(\log(\frac{1}{|x|}))^{\alpha}$.
How can I prove that $f\in W^{1,n}(B(0,1))$?
5
votes
2answers
134 views
Is $\{g(n)\}_{n\in\mathbb{Z}}\in\ell_2$ if $g$ is a Sobolev function on the real line?
If we are given a function $g\in W_2^k(\mathbb{R})$ (even consider $k=1$ for simplicity), then is it true or not that $\{g(n)\}_{n\in\mathbb{Z}}\in\ell_2$? That is, do we have ...
0
votes
1answer
74 views
Fourier transform in $L^2$
I have a function $\phi\in L^2(\mathbb{R}^3)$ and I know that its Fourier transform satisfies the following equation:
$$(p^2-A)\hat{\phi}(p)=Q\frac{A+\lambda}{p^2+\lambda}$$
where $Q$ is a constant, ...
2
votes
2answers
96 views
Poincaré's inequality in Fourier space
If $f\in C^{\infty}_c(\mathbb{R})$ is supported in the interval $[-R, R]$, then by means of the fundamental theorem of calculus one can show that
$$ \lVert f\rVert_{L^2(\mathbb{R})}\le ...
7
votes
2answers
221 views
Sobolev space is an algebra
How do you prove that the Sobolev space $H^s(\mathbb{R}^n)$ is an algebra if $s>\frac{n}{2}$, i.e. if $u,v$ are in $H^s(\mathbb{R}^n)$, then so is $uv$? Actually I think we should also have ...
2
votes
1answer
80 views
The Sobolev norm for vector-valued functions
For a compactly supported function $f: \mathbb{R}^n \to \mathbb{C}$, the Sobolev norm is defined by
$$\|f\|_s^2 = \int |\hat{f}(y)|^2(1+|y|^2)^sdy.$$
Here $\hat{f}$ is the Fourier transform of $f$, ...
2
votes
0answers
38 views
Inequality involving Bessel potential.
I'm not able to prove the following inequality: Fix $s>0$
$$\|fg\|_{H^s}\lesssim \|fJ^sg\|_{L^2}+\|gJ^sf\|_{L^2},$$
where $\widehat{J^sf}(\xi)=(1+|\xi|^2)^{s/2}\hat{f}(\xi)$ (Bessel potential).
...
2
votes
1answer
142 views
Sobolev inequality
If $f\in H^2(\mathbb R^2)$, I want to show that
$||f||_{L^\infty}\le c||f||_{H^2}$
$||f||_{L^\infty}\le c||f||_{H^1} [1+\ln(1+||f||_{H^2})]$
For 1, I use $||f||_{L^\infty}\le \sup_{x\in \mathbb ...
3
votes
1answer
142 views
Sobolev space exercise
I need to show $|f|_{L^\infty}\leq c|f|_{H^2} = c(\int_{\mathbb R^n} (1+|\xi|^2)^2|\hat f(\xi)|^2 d\xi )^{1/2}$, assume $f\in H^2(\mathbb R^2)$
I think I can trasnfer $f\ = \int \hat f(\xi)e^{2\pi i ...
1
vote
1answer
69 views
Fourier transform of function defined on subset of $\mathbb{R}^n$
If I have a function $f:\Omega \to \mathbb{R}$ in $H^k(\Omega)$ where $\Omega \subset \mathbb{R}^n$ is compact, then what is known about the Fourier transform $\hat{f}$? What space does it lie in? I ...
2
votes
0answers
99 views
Sobolev-type inequality.
Let $0<\alpha<n$, $1<p<q<\infty$ and $\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{n}$. Then:
$$\left\|\int_{\mathbb{R}^n} \frac{f(y)dy}{|x-y|^{n-\alpha}} \right\|_{L^q(\mathbb{R}^n)} \leq ...
0
votes
2answers
162 views
Slobodeckij and Bessel definitions of fractional Sobolev spaces on Lipschitz domains
Let $\Omega \subset \mathbb R^n$ be a Lipschitz domain, let $s \in \mathbb R$. Let $W^s(\Omega)$ denote the Sobolev-Slobodeckij space on $\Omega$, and let $H^s(\Omega)$ denote the Bessel-potential ...
3
votes
0answers
87 views
Truncation in singular integrals
After some thinking, I have a terrible headache caused by the following problem. Imagine we have a function $u \colon \mathbb{R}^n \to \mathbb{R}$ such that $u \in L^2(\mathbb{R}^n)$ ...
3
votes
2answers
66 views
A basic estimate for Sobolev spaces
Here is a statement that I came upon whilst studying Sobolev spaces, which I cannot quite fill in the gaps:
If $s>t>u$ then we can estimate:
\begin{equation}
(1 + |\xi|)^{2t} \leq \varepsilon ...
5
votes
1answer
507 views
Schwartz Space is a subspace of Sobolev Space, but how can I show that?
How can I see that $S(\mathbb{R}) \subset H^s(\mathbb{R})$, where the former is Schwartz and the latter is Sobolev space ? This should be obvious according to my notes but unfortunately I can't make ...
