Questions tagged [besov-space]

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

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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}=\{...
4
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0answers
353 views

Fractional Sobolev Space Trace Inequality

Let $\phi\in\mathcal{S}(\mathbb{R}^{n})$ be a Schwartz function, such that ${\phi}\equiv 1$ on the unit ball $|\xi|\leq 1$ and $\text{supp}({\phi})\subset B_{2}(0)$. Set $\phi_{0}=\phi$ and $\phi_{j}=\...
4
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1answer
367 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
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0answers
23 views

Are Besov embeddings strict?

Let $B^{\alpha}_p:=B^{\alpha}_{p,\infty}$ be the Besov space of regularity $\alpha<0$ and integrability $p\ge1$. Recall that a distribution $f$ from the dual Schwarz space is in $B^{\alpha}_p$ if ...
3
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0answers
65 views

Fractional Sobolev space on Union of Sets

I have a question on fractional Sobolev Hilbert space $H^s$ with fractional $s$: I have a bounded Lipschitz domain $\Omega\subseteq\mathbb{R}^d$ that is the union of two Lipschitz domains $\Omega_1,\...
3
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0answers
337 views

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 ...
3
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0answers
337 views

Besov–Zygmund spaces and the Inverse Function Theorem, is the Inverse Zygmund?

Preliminary Definitions Let $\Omega \subset \mathbb{R}^n$ be open. We define the Zygmund spaces $C^r_{*}(\Omega)$ with $r>0$, $r \in \mathbb{R}$ in the following way: (all the functions are ...
2
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0answers
47 views

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 $p≥ 2$ $$ \...
2
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0answers
21 views

Equivalent Besov seminorm - change of integral limits

Let $f \in L^p(\mathbb{R}), p\ge1.$ $\omega_k(f;t)_p,$ the $k^{th}$ order modulus of smoothness of $f,$ is defined by $$\omega_k(f;t)_p = \sup_{0<|h| \le t} \|\Delta_h^kf\|_p,$$ where $$\Delta_h^...
2
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0answers
32 views

Examples of Besov functions of power-logarithmic type $|x|^{\alpha} |\log |x||^{\beta}$

I'm stuck on the following exercise 17.9 from Leoni's text A first course in Sobolev Spaces (second edition). This is from the chapter on Besov spaces, but this is really just a integral inequality ...
2
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0answers
78 views

Embedding property of homogeneous periodic Besov spaces

Let's consider homogeneous periodic Besov spaces $\dot{B}^s_{p,r}(\mathbb{S}^1)$ defined on the circle. The definition is similar to that defined on the real line, except that we use Fourier expansion ...
2
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0answers
73 views

How to show the lifting property of an operator described in its ellipticity

Consider Soblev spaces as special cases of Besov spaces characterized by wavelets, $H^s = B^s_{2,2}$. I want to prove the following statement. For a positive definite self-adjoint operator $A$, if ...
2
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0answers
65 views

Dense subset of Nikol'skii spaces?

I'm looking for a dense subset of the Nikol'skii space $N^{s,p}(\mathbb{R}^N)=B_{p,\infty}^s(\mathbb{R}^N)$, $s\in(0,1)$, $p\in(1,\infty)$, the definition of which is recalled below: $N^{s,p}(\mathbb{...
1
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0answers
24 views

Can $L^p$ space be imbedded in some Besov space $B_{q,2}^\sigma$?

Let \begin{equation} \lVert f \rVert_{B_{q,2}^\sigma} = \lVert P_0 f \rVert_p + ( \sum_{j> 0} 2^{2 \sigma j} \lVert P_j f \rVert_p^2 )^{\frac{1}{2}}. \end{equation} For $\sigma \ge0$ and $p \ge 2$, ...
1
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0answers
51 views

A function space satisfying $ |f(x) -f(y)| \le ||x|^\theta -|y|^\theta| $

Consider a class of function on the real line such that $$ |f(x) -f(y)| \le ||x|^\theta -|y|^\theta | $$ for a $\theta >0$. Does this class of function space have a name When $ |f(x) -f(y)| \le ...
1
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1answer
73 views

On the Besov space

I want to prove this: $$ C^{-1}\lambda^{s-\frac dp}\lVert u\rVert\smash{\dot B}_{p,r}^s \,\le \, \lVert u(\lambda\cdot)\rVert_{\smash{\dot B}_{p,r}^s} \,\le\, C\lambda^{s-\frac dp}\lVert u\rVert\...
0
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1answer
140 views

$L^p$ norm of Littlewood-Paley block of a smooth function

I'm working on a thesis concerning the Littlewood-Paley decomposition of distributions and the use of paraproducts. I'm referring to the book of Bahouri, Chemin, Danchin and during a proof (page 88 ...
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44 views

Search for a reference on characterising Lebesgue spaces in terms of (inhomogeneous) Besov spaces.

I am trying to find a proof that $||f||_{L^{p}}\sim{}||f||_{B^0_{p,2}}$ where $||f||_{B^s_{p,r}}=||(2^{js}||f_j||_{L^p})_{j\in\mathrm{Z}_{\geq{}-1}}||_{\ell^r}$ is the norm characterising the ...