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Questions tagged [besov-space]

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

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The scaling effect on the Besov norm

The Besov spaces can be defined using the dyadic decomposition. For this purpose, let $\varphi \in C^\infty_c(\mathbb R^n)$ be such $ {\rm supp}(\varphi) \subseteq \left\{x; \, \frac 12< |x| &...
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For a bounded domain $\Omega \subset \mathbb{R}^3$, does the Besov space $B^{1/2,6/5}_{4/3}(\Omega)$ belong to $L^2$?

According to the formula, we have the real interpolation \begin{equation} (L^{6/5}(\Omega), W^{2,6/5}\Omega))_{1/4,4/3} = B^{1/2,6/5}_{4/3}(\Omega). \end{equation} for any bounded domain $\Omega \...
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Definition of the Besov space

The Besov spaces can be defined using the dyadic decomposition. For this purpose, let $\varphi \in C^\infty_c(\mathbb R^n)$ be such $ {\rm supp}(\varphi) \subseteq \left\{x; \, \frac 12< |x| &...
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Sobolev-type embeddings for Bessel potential spaces

I am reading the paper A Zvonkin's transformation for stochastic differential equations with singular drift and applications where the authors introduce the following function spaces: We fix $\theta \...
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Is there a constant $c>0$ such that $\| (1-\Delta)^{-\alpha} P^\kappa_t f\|_{C^{n,\beta}_b}\le c t^{-(\frac{n}{2} - \alpha)^+} \| f \|_{C^{\beta}_b}$?

For $n \in \mathbb N$ and $\alpha \in (0, 1)$, let $C^{n, \alpha}_b (\mathbb R^d)$ be the space of $n$-times continuously differentiable real-valued functions $f$ on $\mathbb R^d$ that admit the ...
Analyst's user avatar
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Smoothness of the remainder in the Littlewood-Paley theory

Assume $f\in L^1(\mathbb{R}^d)$ and $\eta\in C_c^\infty(\mathbb{R}^d)$. What can I know about the smoothness of the non-homogeneous remainders $\mathbf{R}(f,\eta):=\sum_{|j-k|\le 2}\Delta_j f\Delta_k \...
Alexander Khan's user avatar
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Real Interpolation Constant for Besov Spaces

In the book Fourier Analysis and Nonlinear PDE, B-C-D establish the following Proposition 2.22. A constant $C$ exists which satisfies the following properties. If $s_1$ and $s_2$ are real numbers ...
newbie's user avatar
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Is the Sobolev space $\dot{W}^{s,1}$ continuously embedded in the Besov space $\dot{B}^{s,1}_1$?

Do we have the continuous embedding $\|u\|_{\dot{B}^{s,1}_1}\leq C\|u\|_{\dot{W}^{s,1}}$? Here $$\|u\|_{\dot{B}^{s,1}_1} = \sum_{k=1}^{\infty}2^{ks}\|P_ku\|_{L^1}\\ \|u\|_{\dot{W}^{s,1}} = \||\nabla|^...
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Seeking Assistance on Embeddings of Non-Homogeneous Besov Spaces and a Relation with Hölder Continuous Functions

Hello esteemed members, I am currently grappling with some properties of non-homogeneous Besov spaces and am seeking guidance or references to aid my understanding. Embedding of Besov Spaces: I aim to ...
Sara Testori's user avatar
2 votes
1 answer
129 views

Bounding an integral by a Besov norm

Let $f: \mathbb{R}^3 \rightarrow \mathbb{R}^3$, $\eta^\epsilon$ a mollifier, and $B_3^{\alpha, \infty}$ the Besov space. Define $$I_\epsilon(x) = \int \eta^\epsilon(y)\big(f(x-y)-f(x)\big)^2 dy$$ ...
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1 answer
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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 ...
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Is any homogeneous Sobolev space $\dot{W}^{s,p}$ a "homogeneous Banach space", as the the name suggests?

A homoegeneous Sobolev space $\dot{W}^{s,p}(\mathbb{R}^n)$ is defined to be the completion of the Schwartz space $\mathcal{S}(\mathbb{R}^n)$ under the norm \begin{equation} \lVert (-\Delta)^{s/2} f \...
Keith's user avatar
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Injecting $L_1$ into a Besov space

Can someone help me to find a counter-example that proove that $L_1$ is not injected in the homogeneous Besov space $B^{0,2}_1$ ?
Lin's user avatar
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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}}=\...
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Is $\mathbb{R}^{d_1}\times (0,\infty)^{d_2}$ a $(\varepsilon, \delta)$-locally uniform domain?

Is $\mathbb{R}^{d_1}\times (0,\infty)^{d_2}$ a $(\varepsilon, \delta)$-locally uniform domain, for integer $d_1 \geq 0, d_2 \geq 0$? I am trying to get a grasp on these domains and am having a hard ...
Athere's user avatar
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Is $\mathbb{R}^n_+$ an $(\varepsilon, \delta)$-domain?

Is it true that $\mathbb{R}^n_+=\mathbb{R}^{n-1}\times (0,\infty)$ is trivially $(\varepsilon, \delta)$-locally uniform, for $\varepsilon=1$ and any $\delta$ and the arc chosen as the segment of a ...
Athere's user avatar
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Showing a function belongs to a Besov space.

In this paper, just before Theorem 3 it is stated that the kernels given in equations 4a-f are in $\mathbb{B}_{pq}^s$ for all $p,q\geq 1$ and some $s>0$. This is stated without proof/reference and ...
Dylan Zammit's user avatar
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Does there exist a function in $B_{\infty,1}^{1/2}\setminus H^{1/2}$?

Does there exist a function in $B_{\infty,1}^{1/2}([0,1])\setminus H^{1/2}([0,1])$? If so can it be constructed explicitly? A reference will be much appreciated.
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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 ...
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Proof of Besov space embeddings: $B_{p, \infty}^{s+\varepsilon}$ is continuously embedded in $B_{p, 1}^s$

Let's consider the Besov spaces $B_{p,q}^s$, $p,q \in [1,\infty]$, $s\in \mathbb{R}$ and $\varepsilon >0$. I keep coming across that $B_{p, \infty}^{s+\varepsilon}$ is continuously embedded in $B_{...
Nils's user avatar
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2 votes
1 answer
277 views

Embedding from homogeneous Besov spaces to homogeneous Sobolev spaces.

I am looking for references about embeddings from the homogeneous besov space $\dot{B}^s_{p,r}(\mathbb{R}^3)$ (or $\dot{B}^s_{p,r}(\mathbb{R}^d)$ for any $d>0$) into sobolev space $\dot{W}^{\alpha,\...
Velobos's user avatar
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Paley-Littlewood blocks - $| u^ j|_{B^{-1}_2} \leq C 2^{-j\sigma} | u^j |_{B^{\sigma-1}_2} $ if $\hat{u^j} \subset 2^j B$

As in the title, I'd like to know if it's true that: $$| u^j|_{B^{-1}_{2,\infty}} \leq C 2^{-j\sigma} | u^j |_{B^{\sigma-1}_{2,\infty}} $$ if $\hat{u^j} \subset 2^j B$ ($B$ being a ball, $| \cdot |_{B^...
defenestrator's user avatar
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1 answer
303 views

Does absolute value of a function belong to the same non-positive Besov/Sobolev space

Let $f$ be a measurable function belonging to a Besov space of non-positive regularity $B^{\gamma}_{p,\infty}$, where $\gamma\le 0$, $p\in[1,\infty]$. Is it true that $|f|$ belongs to the same space ...
Oleg's user avatar
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Reference for interpolation theory

In the book "Interpolation theory, function spaces, differential operators" by Hans Triebel, I tried to understand the result Theorem~1(a) of section~2.4.2. In particular, I am interested in ...
kaushik's user avatar
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Why $t^{\frac{N}{2}\left(\frac{2}{N}-\frac{1}{q}\right)}N(\cdot) \in BC_w([0,\infty);L^q)$?

In the article of "Kozono et. all (2016) pg. 11" with title: Existence and uniqueness theorem on mild solution to the Keller- Segel system coupled with the Navier Stokes fluid, the author ...
Jarbas Dantas Silva's user avatar
6 votes
1 answer
264 views

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, $0 \leq \hat{\theta} \leq 1$ ...
Desura's user avatar
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3 votes
1 answer
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Besov spaces: Regularization by convolution

Consider either a Besov space $B_{p,q}^s$ or a Triebel-Lizorkin space $F_{p,q}^s$ with parameters $p,q \in [1,\infty)$ and $s \in \mathbb{R}$. It is known that the Schwartz space $\mathcal{S}(\mathbb{...
saz's user avatar
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0 answers
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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 ...
Oleg's user avatar
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1 vote
2 answers
80 views

Is there a nice relationship between $\dot{B}^{-1}_{p,1}$ and $\dot{B}^{0}_{p,1}$?

In particular, I'm hoping there is an embedding/inequality relating elements of the two. For background, I have a particular function, $f \in W^{\infty,p}(\mathbb{R}^2)$, for all $p \in [1,\infty]$, ...
gbnhgbnhg's user avatar
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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 $p \ge 2$, ...
Tao's user avatar
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3 votes
1 answer
224 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 $p≥ 2$ $$ \...
LL 3.14's user avatar
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3 votes
1 answer
535 views

Reference on periodic Besov spaces

I am looking for a reference for the construction and the study of the main properties of Besov spaces on the torus (i.e. $\mathrm{B}_{p,q}^s(\mathbb{T}^d))$. Indeed, I know the classical $\mathrm{B}_{...
SELM's user avatar
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1 answer
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How are these two Banach spaces related ? (weighted $L2$ type space involving a logarithm and Besov type space)

First the standard $L2$ space : $$L^2(\mathbb{R}) = \Big \{ f : \| f \|_2 = \left( \int_{\mathbb{R}} | f(x) |^2 dx \right)^{1/2} < \infty \Big \}.$$ Let $s \geqslant 1/2$. Define a weighted $L2$ ...
Marc_Adrien's user avatar
1 vote
1 answer
108 views

Basic question about Bony decompositions - summation indices

I'm trying to understand some sort of inequality in a larger calculation. I believe my only issue is in counting correctly, so I've also tagged combinatorics. Suppose I have a function $f$. Let $\...
Calvin Khor's user avatar
2 votes
0 answers
107 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^...
Birendra Singh's user avatar
1 vote
1 answer
228 views

Does this inclusion for Besov function spaces hold and how to prove it?

Consider the Besov spaces $B_{p,q} ^s$ consisting of those functions $f$ such that $$f \in W^{n,p} (\mathbb{R}), \quad \int_{0} ^{\infty} \Big | \frac{\omega_p ^2(f^{(n)}, t)}{t^{a}} \Big |^q \frac{...
Marc_Adrien's user avatar
3 votes
0 answers
163 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 ...
ktoi's user avatar
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3 votes
1 answer
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Does $C_c(X)$ separate points in $X$ when $X$ is a Banach space?

Suppose that $X$ is a separable, infinite dimensional Banach space. We say that a set of functions $\{f_\alpha\}_{\alpha \in A}$ separates points in $X$ if for every $x,y \in X$, there is an $\alpha$...
Rhys Steele's user avatar
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6 votes
1 answer
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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 ...
Oleg's user avatar
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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 ...
user92646's user avatar
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2 votes
0 answers
127 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 ...
Cohen Lu's user avatar
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1 vote
1 answer
287 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 ...
Riccardo Ceccon's user avatar
6 votes
1 answer
850 views

$\mathcal{C}^{\alpha}$ 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_{...
Riccardo Ceccon's user avatar
3 votes
0 answers
106 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,\...
user500801's user avatar
6 votes
1 answer
1k 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 ...
πr8's user avatar
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1 vote
1 answer
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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\...
Motaka's user avatar
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0 answers
62 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 ...
Mathmo's user avatar
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3 votes
1 answer
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Different definitions of Besov norm/space

I'm following two "different" approaches to the Besov Spaces, but I don't get if the two definitions given are equivalent. Victor I. Burenkov - Sobolev Spaces On Domains. Given $f:\mathbb{R}^n \to \...
gangrene's user avatar
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2 votes
0 answers
80 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 ...
newbie's user avatar
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2 votes
0 answers
72 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{...
SchlaggosZero's user avatar