Questions tagged [besov-space]

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

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Dual of homogeneous Besov space

I am trying to prove a trace theorem from the homogeneous Besov space $\overset{\cdot}{B}_{2,1}^{1/2}(\mathbb{R}^n)$ to $L^2(K)$, where $K$ is a compact euclidean hypersurface. The homogeneous Besov ...
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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 ...
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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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Show that homogeneous distribution of degree $-1$, bounded in $S^2$ satisfies that $\left| S_{0} v(x) \right| \leq \frac{C}{1+ |x|} $

Let $\varphi(x)$ be a rotation-invariant function such that $\varphi \in \mathcal{S}(\mathbb{R}^3)$ (Schwartz space) and \begin{equation} 0 \leq \hat{\varphi}(\xi) \leq 1 , \quad \quad \begin{cases} ...
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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_{...
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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,\...
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  • 1,996
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29 views

Products in Besov spaces

Consider the Besov spaces $B^s_{p,q}(\mathbb R^n)$ (I am thinking about the Fourier analytic definition given in, e.g., “Theory of function spaces” by Hans Triebel). I know that $$B^0_{3,1}(\mathbb R^...
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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^...
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118 views

Does absolute value of a function belong to the same negative Besov/Sobolev space

Let $f$ be a measurable function belonging to a negative Besov space $B^{\gamma}_{p,q}$, where $\gamma<0$. Is it true that $|f|$ belongs to the same space? Note that if $\gamma=0$, then the answer ...
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Riemann Lebesgue equivalent of Fourier Transform for Hölder Continuity

I am currently working with a paper using the following norm: \begin{equation} \|F\|_{B^{s/2}}=\int_{-\infty}^{\infty}|\hat{F}(\tau)|(1+|\tau|)^{s/2}\,\mathrm{d}\tau\\ \end{equation} for all $C_c^\...
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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 ...
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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 ...
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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, $0 \leq \hat{\theta} \leq 1$ ...
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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{...
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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 ...
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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 $p \in [1,\infty]$, ...
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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 $p \ge 2$, ...
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  • 301
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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 $p≥ 2$ $$ \...
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3 votes
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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}_{...
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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 $L2$ ...
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1 answer
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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 $\...
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2 votes
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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^...
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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{...
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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 ...
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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 $\alpha$...
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5 votes
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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 ...
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  • 668
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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 ...
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2 votes
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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 ...
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  • 243
1 vote
1 answer
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$L^p$ 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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4 votes
1 answer
498 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_{...
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3 votes
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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,\...
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4 votes
1 answer
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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 ...
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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\...
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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 ...
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  • 895
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 \...
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2 votes
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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 ...
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  • 3,261
2 votes
0 answers
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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{...
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4 votes
1 answer
1k views

Relation between Besov and Sobolev spaces (Littlewood-Paley-theory)

For the Sobolev-norm there holds $\Vert f\Vert_{W^{s,p}(\mathbb{R}^n)}\sim_{n,p,s}\left\Vert\left(\sum_{k\in\mathbb{N}_0}\left\vert (1+2^k)^sP_kf(x)\right\vert^2\right)^\frac{1}{2}\right\Vert_{L^p}$ ...
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4 votes
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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}=\...
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2 votes
1 answer
139 views

Simple Inequality for Proving Equivalent Besov Seminorms

For $f\in L^{p}(\mathbb{R}^{n})$, $1\leq p<\infty$, and $h\in\mathbb{R}^{n}$, define the quantity $$I_{p}(h):=\left(\int_{\mathbb{R}^{n}}\left|f(x+h)-f(x)\right|^{p}dx\right)^{1/p}$$ and define ...
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2 votes
1 answer
357 views

Reference request on Besov spaces and bounded variation functions.

Would someone have any good references which address Besov spaces, and functions of bounded variations. I don't know anything about theses spaces so I am looking for something that would introduce me ...
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4 votes
1 answer
444 views

question about property of $L^p$ Lipschitz space

$f\in L^p$ is said to satisfy $L^p$ Lipschitz condition of order $\alpha$ if there exists $C>0$ such that $\displaystyle|h|^{-\alpha}\Big(\int_{\mathbb{R}^d}|f(x-h)-f(x)|^p \,dx\Big)^\frac{1}{p}\...
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4 votes
1 answer
412 views

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}(\...
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3 votes
0 answers
430 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 ...
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  • 269
12 votes
1 answer
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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 ...
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  • 2,908
4 votes
1 answer
458 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 $\mathcal{S}(\mathbb{R}^d)$ ...
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  • 1,119
7 votes
0 answers
461 views

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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0 votes
1 answer
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Functions on Besov spaces and Hölder conditions

Consider $f_1(x)=-1/\log(|x|)$ and $f_2(x)=1/\sqrt{-\log(|x|)}$ around $x=0$. It can be shown, just by contradiction, that the previous functions do not verify any (positive) Hölder condition: $$ |...
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5 votes
1 answer
969 views

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 $...
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