Tagged Questions

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Differentiation in Besov-Zigmund spaces

This is my second question in a short time on Besov spaces. I apologize. I am having a rough time with them and I really need to understand this spaces quickly. The besov spaces ...
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Which functions lies in $H^{loc}_{s}\setminus H_{s}$?

We put $H^{s}=$The Sobolev spaces, and $H^{loc}_{s}=$The localized Sobolev spaces. We note that, $H_{s}\subset H^{loc}_{s};$ also this. Bit roughly speaking, I am interested in knowing that how big ...
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Is $C(\mathbb R) \subset H_{s}^{loc}$ (localized Sobolev space)?

We put, Sobolev space $$H_{s}=H_{s}(\mathbb R)=\{f\in L^{2}(\mathbb R):[\int_{\mathbb R} |\hat{f}(\xi)|^{2}(1+|\xi|^{2})^{s}d\xi]^{1/2}<\infty \}.$$ If $U$ is an open set in $\mathbb R,$ the ...
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About a property of weighted Sobolev spaces

If we define the weighted Sobolev norm as $$\|f\|_{H^{s,k}(\mathbb R^n)} := \|(1+|x|^2)^{k/2} (\sqrt{1-\Delta})^s f \|_{L^2(\mathbb R^n)}$$ where $f(x):\mathbb R^n \to \mathbb R$ and $\Delta$ is ...
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Bounding Sobolev Norms *Below*

Firstly, apologies if this is a duplicate question - I've looked, but can't find this question on SE (or elsewhere online); if it is, please let me know and I'll remove it. I am trying to bound the ...
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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 ...
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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 ...
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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 $\mathcal{S}(\mathbb{R}^d)$ ...
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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: $$\| W(t ) *g\|_{H^s} \leq c \sqrt2 (1 + |t|)\| g\|_{H^{s-1}}$$ ...
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$u \in H^n$ 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 ...
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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 $\alpha$-HÃ¶lder-continuous functions) and vice-versa. My question is for which exponents $s, \alpha$ can ...
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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 $$\|f\|_{H^{-k}(\mathbb{R}^n)}=\|(I-\triangle)^{-k}f\|_{H^{k}(\mathbb{R}^n)}.$$
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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 ...
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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 $f\in W^{1,n}(B(0,1))$?
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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 ...
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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, ...
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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 ...
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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 $u \in L^2(\mathbb{R}^n)$ ...
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: (1 + |\xi|)^{2t} \leq \varepsilon ...
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 ...