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Questions tagged [besov-space]

For questions on Besov spaces, which are complete quasinormal spaces.

A Besov space named after Oleg Vladimirovich Besov, is a complete quasinormal space which is a Banach space when $1 \leq p, q \leq \infty$. These spaces, as well as the similarly defined Triebel–Lizorkin spaces, serve to generalize more elementary function spaces such as Sobolev spaces and are effective at measuring regularity properties of functions.

68 questions
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Besov spaces---concrete description of spatial inhomogeneity

Some very pedestrian questions about Besov spaces. Just to fix notation: 1.Let $f \in \mathcal{S}'$, the space of tempered distributions. 2.$\Psi, \{ \Phi_n \}_{n \geq 0} \subset \mathcal{S}$ such that their Fourier transforms $\hat{\Psi}, \{…
Michael
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Rellich–Kondrachov theorem for traces

Let $W^{1,p}(\Omega)$ be the Sobolev space of weakly differentiable functions whose weak derivatives are $p$-integrable, where $\Omega \subset \mathbb R^n$ is a domain with Lipschitz boundary. Let furthermore $\gamma$ be the trace map. I am looking…
anonymous
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What is the motivation for Besov spaces?

I am trying to understand the definition of Besov spaces. With such a complicated definition I wonder what is the motivation behind them and why are they so often used in PDE? What advantage do they give over Sobolev spaces? Are there any nice…
CBBAM
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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 $\Omega_{rh}=\{x\in\Omega:[x+rh]\subset\Omega\}$, and…
timur
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Completeness of Besov spaces

Problem: Let us recall that: $$\dot{B}^{-\sigma}_{\infty,\infty}=\{u \in S'(\mathbb{R}^d): \|u\|_{\dot{B}^{-\sigma}_{\infty,\infty}} < \infty\}$$ where $$\|u\|_{\dot{B}^{-\sigma}_{\infty,\infty}}=\sup\limits_{A>0}\{A^{d-\sigma}\| \theta(A\cdot)*…
Filippo Giovagnini
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Interpretation of Besov Space parameters

I've been reading about Besov spaces (my reference thus far has been "Mathematical foundations of infinite-dimensional statistical models" (Nickl & Gine), and I've been struggling a bit with the interpretation of the parameters given when describing…
πr8
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Differentiation in Besov–Zygmund 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 $B^s_{\infty,\infty}(\mathbb{R^n})$ for $s \in \mathbb{R}$ are defined…
juan rojo
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Reference request: Lebesgue/Sobolev spaces on the boundary

I am interested in the Boundary Lebesgue/Sobolev/Besov Spaces $L^p(\partial\Omega;\mathcal{H}^{N-1}), \ W^{k,p}(\partial\Omega),\ B^{s,p}(\partial\Omega)$ where $\Omega\subseteq\mathbb{R}^N$ is a bounded Lipschitz domain. I found in the book of…
Bogdan
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Homogeneous Besov space $\dot B^{-\sigma}(\mathbb R^d)$ is a Banach space

Let $\sigma > 0$ and $\theta \in S(\mathbb R^d)$ (Schwartz function) be fixed, where $\theta$ is such that its Fourier transform $\hat{\theta}$ is compactly supported, $0 \leq \hat{\theta} \leq 1$ and $\hat{\theta} = 1$ in a neighborhood of $0$. Let…
Desura
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How to prove that a delta function belongs to the Besov space $B^{-1}$?

$\def\R{\mathbb{R}}$ $\DeclareMathOperator{\supp}{supp}$ I am trying to understand the definition of the Besov space and to prove that the delta function in $\R^1$. However I got stuck. First, let us recall that a sequence of smooth functions…
Oleg
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$\mathcal{C}^{\alpha}$ Besov spaces: Definition

I'm reading an article for my future thesis (I'm a third-year undergraduate) where the authors define the generalized Holder Spaces as a special class of Besov Spaces. Define $\chi,\tilde{\chi}\in C_{c}^{\infty}(\mathbb{R}^d)$ such that…
Riccardo Ceccon
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Periodic vs. real distributions?

I need some help understanding the relationship between distributions $\mathcal{D}^\prime(\mathbb{T})$, where $\mathbb{T}=\mathbb{R}/\mathbb{Z}$, and the subset of distributions that are invariant under shifts in $\mathbb{Z}$. Is there a one-to-one…
Dasi
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Dense subspace of Zygmund space or Hölder space?

Do we know any function spaces dense in Zygmund space $C_*^s$(a special case of Besov space, i.e. $C_*^s = B^s_{\infty,\infty}$) or Hölder space$C^{k,r}$, with underlying field $\mathbb{R}^d$? Will $C^\infty$ do the job?
newbie
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Besov or Triebel-Lizorkin spaces versus Lorentz spaces

At the $0$ order of derivatives of Sobolev spaces, we find Besov spaces $\dot{B}^0_{p,q}$, Triebel Lizorkin spaces $\dot{F}^0_{p,q}$ and Lorentz spaces $L^{p,q}$, with in particular if $p≥ 2$ $$ \begin{align*} \dot{B}^0_{p,1} ⊂ \dot{B}^0_{p,2} ⊂ L^p…
LL 3.14
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