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Tagged with function-spaces sobolev-spaces
9 questions
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Fractional Sobolev embedding
Let $s\in (0,1)$ and $1<p<\infty$. Let $H^{s,p}(\mathbb{R}^n)=H^{s,p}$ the Bessel potential space, defined as the image of $L^p(\mathbb{R^n})$ by the Bessel potential. It is known that these ...
8
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Understanding spaces of negative regularity
I apologize if this question is too basic for this site, but I posted it on mathSE and did not get any responses (link can be found here) so I'm crossposting it here.
Let $C^k(\mathbb{R}^n$) be the ...
- 721
2
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Dual of homogeneous Triebel-Lizorkin
Let $ p, q \in (1,\infty)$ and consider the homogeneous Triebel- Lizorkin space $\dot{F}^{s}_{p,q}$ to be the space of all tempered distributions (modulo polynomials) with
$$
[f]^{p}_{\dot{F}^{s}_{p,q}...
- 31
1
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Is the product of $u \in W^{\sigma,1}(\Omega)$ and $v \in C^{0,\sigma}(\Omega)$ again in $W^{\sigma,1}(\Omega)$?
The following startles me. Let $\Omega \subseteq \mathbb R^n$ and write $W^{\sigma,1}(\Omega)$ for the fractional Sobolev space with norm
$$|u|_{W^{\sigma,1}(\Omega)} := \iint \frac{|u(x) - u(y)|}{|x-...
- 5,327
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Is $C^\infty(\Omega) \cap W^{s_1,p_1}(\Omega)$ dense in $W^{s_2,p_2}(\Omega)$ if $W^{s_1,p_1}(\Omega) \subset W^{s_2,p_2}(\Omega)$?
Background: The proof of Theorem 6.4 in http://mate.dm.uba.ar/~jrossi/Fractional-1-lapla-07_02_2015.pdf, I want to use the density that $C^\infty(\Omega) \cap W^{r_0,q_0}(\Omega) \cap L^2(\Omega)$ is ...
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Analogue of Lipschitz continuity of $W^{1,\infty}$ for Hölder continuity and Sobolev-Slobodeckij spaces
A function $u : U \rightarrow \mathbb R$ is an element of the Hölder space $C^{\alpha}(U)$ if
$\sup\limits_{x \in U} |u(x)| < \infty$
$\sup\limits_{x,y \in U} \dfrac{|u(x) - u(y)|}{|x-y|^\alpha} &...
- 5,327
7
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2
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536
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"Reversion" of class $J(\theta)$ interpolation property for Besov spaces
In (function space) interpolation theory, a Banach space $E$ is of class $J(\theta)$ (for $0 < \theta < 1$) if $$X \cap Y \subseteq E \subseteq X+Y,$$ where $(X,Y)$ are Banach spaces and form an ...
- 2,670
3
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Wavelet characterization of Sobolev spaces
We know that there exist wavelets generating orthonormal bases in Sobolev spaces $W^{p,s}(\mathbb R^n)$, where $p$ is the index of integral and $s$ is the index of smoothness. Consider the orthonormal ...
- 231
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Moduli of smoothness, Besov spaces, and Sobolev spaces
For $1\leq p\leq\infty$, the $r$-th order $L^p$-modulus of smoothness is
\begin{equation}
\omega_r(u,t,\Omega)_p=\sup_{|h|\leq t}\|\Delta_h^ru\|_{L^p(\Omega_{rh})}
\end{equation}
where $\Omega_{rh}=\{...
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