2
votes
1answer
34 views

What would be to show $(1+|\xi|^2)^{s/2}\hat{u}\in \mathcal{S}^{'}(\mathbb R^n)$ is a function?

If $s\in\mathbb R$ define the sobolev space $H^s(\mathbb R^n)$ as $$H^s(\mathbb R^n):=\{u\in \mathcal{S}^{'}(\mathbb R^n): (1+|\xi|^2)^{s/2}\hat{u}\in L^2(\mathbb R^n)\}.$$ I have a doubt concearning ...
2
votes
1answer
85 views

Smooth function becomes analytic

Let $f$ be a smooth function ,defined on unit interval $[0,1]$.Moreover $\Vert f^{(k)}\Vert_2\leq \alpha,\:\forall k\in\mathbb{N}_o$. Can we conclude that $f$ is analytic. More generally when ...
3
votes
0answers
70 views

Closure of Schwartz space in homogeneous Besov space

Let $\dot{B}^s_{\infty,\infty}(\mathbb{R}^d)$ denote the homogeneous Besov space of order $s$ with second and third index $\infty$, i. e. the homogeneous Zygmund space. Let $\mathcal{S}(\mathbb{R}^d)$ ...
3
votes
0answers
53 views

$f, f|f| \in L^{1} (\mathbb R) \cap C_{0} (\mathbb R) \implies |f| \in H_{1} (\mathbb R)$?

Suppose $f \ \text{and} \ f|f|\in L^{1}(\mathbb R).$ Then, clearly, $|f|\in L^{2}(\mathbb R)$ and therefore by Plancheral theorem, we get, $\widehat{|f|} \in L^{2}(\mathbb R).$ Also, assume, $f, ...
2
votes
1answer
46 views

Sobolev spaces and integrability of Fourier transforms

I have a Lemma from a text that states that if $g\in W^{1,2}(\mathbb{R})$ ($W^{k,p}$ a Sobolev space) and the weak derivative $Dg\in L^2(\mathbb{R})$ then the Fourier transform $\mathcal{F}g\in ...
1
vote
0answers
36 views

Estimative in Hs spaces

Consider $W(t)$ defined by $\widehat{W(t)} = (2 \pi)^{\frac{-n}{2}} \cos (t|\xi|)$. Consider $g \in H^{s-1} (R^{n-1})$. Show that: $$ \| W(t ) *g\|_{H^s} \leq c \sqrt2 (1 + |t|)\| g\|_{H^{s-1}}$$ ...
0
votes
0answers
28 views

$u \in H^n$ then also $|u|^2 \in H^n$ for complex valued function in Sobolev space

Suppose that a complex valued function $u$ is in the Sobolev space $H^n(R^n) = W^{n,2}$. Is it necessarily true that $|u|^2 \in H^n$ ? The result is true for real - valued functions since we know ...
1
vote
1answer
46 views

Relation between Schwartz space and Sobolev space $H_{1}$

The Schwartz space, $S(\mathbb R): = \{f\in C^{\infty}(\mathbb R): \sup_{x\in \mathbb R} |x^{\alpha} D^{\beta}f(x)|< \infty , \forall \alpha, \beta \in \mathbb N \cup \{0\} \}$ and $S'(\mathbb R) ...
0
votes
0answers
21 views

estimate for a function in the H^s space

Consiser $f \in H^s(R^n)$ $(s> n/2)$. Show that : $$ |f(x) - f(y)| \leq \gamma_s(|x-y|) || f||_s , \ \ \forall x,y \in R^n .$$ Where $\gamma_s(|x-y|) = 2 (2 ...
1
vote
1answer
191 views

Compact, continuous embeddings of $H^s := W^{s,2} \leftrightarrow C^{(\alpha)}$

The sobolev-space $H^s([-\pi,\pi])$ can be embedded into $C^{(\alpha)}([-\pi,\pi])$ (space of $\alpha$-Hölder-continuous functions) and vice-versa. My question is for which exponents $s, \alpha$ can ...
1
vote
1answer
32 views

What is the definition of $H^{-k}(\mathbb{R}^n)$ and its norm?

What is the definition of $H^{-k}(\mathbb{R}^n)$ and its norm? How can I understand the fact $$\|f\|_{H^{-k}(\mathbb{R}^n)}=\|(I-\triangle)^{-k}f\|_{H^{k}(\mathbb{R}^n)}.$$
2
votes
2answers
60 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
55 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
146 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
90 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
182 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 ...
9
votes
2answers
660 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
180 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
55 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
206 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
194 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
91 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
126 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
279 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
98 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
87 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 ...
6
votes
1answer
686 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 ...