Questions tagged [besov-spaces]

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Is there any example of linear operator which is bounded on all Besov spaces but not on Triebel-Lizorkin spaces

Is there any linear operator $T:S'(\mathbb R^n)\to S'(\mathbb R^n)$ such that $T:B_{pq}^s(\mathbb R^n)\to B_{pq}^s(\mathbb R^n)$ for all $0<p,q\le\infty$ and $s\in\mathbb R$, but there exist a $F_{...
Liding Yao's user avatar
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1 answer
164 views

Equivalent Littlewood-Paley-type decompositions

The theory of Besov and Triebel-Lizorkin spaces usually proceeds by taking a dyadic decomposition of unity, i.e. some non-negative functions $\psi_0,\psi \in C_c^\infty(\mathbb{R})$ such that \begin{...
vmist's user avatar
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4 votes
1 answer
743 views

How to get $\int_{\mathbb R^d} |\partial_i\partial_j(1-\Delta)^{-\frac{\delta}{2}}p_t(\cdot-y)(x)| \, \mathrm d x \lesssim t^{\frac{\delta}{2}-1}$?

We consider the Gaussian heat kernel $p_t$ on $\mathbb R^d$, i.e., $$ p_t (x) := (4\pi t)^{-\frac{d}{2}} e^{-\frac{|x|^2}{4t}}, \quad t>0, x \in \mathbb R^d, $$ and define the operator $P_t$ by $$ ...
Akira's user avatar
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Validity of Hölder inequality for the homogeneous Besov spaces $\dot{B}^0_{1,2}(\mathbb{R}^n)$ and $\dot{B}^0_{2,2}(\mathbb{R}^n)=L^2(\mathbb{R}^n)$

I am looking at Corollary 1. in p.244-245 of the book "Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations" (1996) by Thomas Runst Winfried ...
Isaac's user avatar
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1 vote
1 answer
184 views

What is the exact description of the homogeneous Besov space $\smash{\dot{B}}^0_{1,1}(\mathbb{R})$?

The Besov space is defined briefly in Wikipedia and I looked for a number of references to find some information on the homogeneous Besov space $\smash{\dot{B}}^0_{1,1}(\mathbb{R})$. However, ...
Isaac's user avatar
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2 votes
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Why do we work on homogeneous Besov/Triebel-Lizorkin spaces?

This question is mainly for understanding the history behind homogeneous spaces. There is extensive literature on Besov and Triebel-Lizorkin spaces. For instance, see the standard textbook: https://...
fast_and_fourier's user avatar
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83 views

Extension for fractional Sobolev spaces, s>0

In their paper, Fractional Sobolev extension and imbedding, the author describes all extension domains for $s \in (0,1)$ -- meaning spaces functions in which are not required to have weak derivatives. ...
Athere's user avatar
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Can a weighted $\ell^p$ norm be bounded by an unweighted $\ell^q$ norm?

For any sequence $\omega\in[1, \infty)^{\mathbb{N}}$, define the weighted $\ell^p_\omega$-norm of the sequence $v$ by $$\Vert v\Vert_{\ell^p_\omega} := \left(\sum_{k=1}^\infty \omega_k^p |v|_k^p\right)...
Philipp Trunschke's user avatar
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148 views

Besov space norms

We need to recall some Besov space norms to formulate the question. Let $d \in \mathbb N$, $0<s<2, 1 \le p,q \le \infty$. Then the Besov space $B^s_{p,q}(\mathbb R^d)$ is given by the norm $$ \...
Paul Pfeiffer's user avatar
6 votes
1 answer
895 views

Does the embedding $W^{2,1}(\mathbb R^2) \to L^\infty(\mathbb R^2)$ factor through some space that is "slightly better" than $W^{1,2}(\mathbb R^2)$?

Using the fundamental theorem of calculus, we can show that the Sobolev space $W^{2,1}(\mathbb R^2)$ embeds into $L^\infty(\mathbb R^2)$. If we attempt to prove this by applying Sobolev embedding ...
Alan C's user avatar
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Extreme case of K-interpolation

Suppose $X_0$ and $X_1$ are Banach spaces living in a larger Banach space $X$. The $K$-functional is defined for each $f\in X_0+X_1$ and $t>0$ as $$K(f,t,X_0,X_1)=\inf\{\|f_0\|_{X_0}+t\|f_1\|_{X_1}:...
pipenauss's user avatar
  • 297
2 votes
0 answers
94 views

Interpolation of Sobolev/Besov spaces in the limiting case q = ∞

I'm interested in the interpolation space ($1\le p_0,p_1\le\infty$, $0<\theta<1$) $$ X=(L_{p_0}(0;1),W^1_{p_1}(0,1))_{\theta,q}\quad\text{with}\quad q=\infty\ \ \text{and}\ \ p_0\ne p_1 . $$ It ...
Lev's user avatar
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218 views

Sobolev (Triebel-Lizorkin) norm estimate for $F \circ u - F \circ v$

Let $F \in C^1(\mathbb R^d;\mathbb R)$ be such that $F(0) = 0$ and $$|F'(\tau v + (1 - \tau)w)| \leq \mu(\tau)(G(v) + G(w))$$ for some $\mu \in L^1([0,1])$ and some non-negative $G \in C^0(\mathbb R^d;...
Desura's user avatar
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3 votes
1 answer
198 views

Smooth cut-off in homogeneous Besov space

Given a Littlewood-Paley decomposition $$1 = \chi(\xi) + \sum_{j \geq 0}\varphi(2^{-j} \xi), \quad \xi \in \mathbb R^n$$ where $\chi$ is smooth, supported on a ball, and $\varphi$ is smooth, supported ...
Desura's user avatar
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Imbed Sobolev spaces of fractional order into Holder spaces?

This result exist (https://encyclopediaofmath.org/wiki/Imbedding_theorems ) for regular (i.e. not fractional) Sobolev spaces; looks like it's provable for fractional spaces through results for Besov ...
Athere's user avatar
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1 answer
306 views

Is the Besov space $B_{\infty,1}^0(\mathbb{R}^d)$ a multiplication algebra?

Let $s\in\mathbb{R}$ and $1\leq p,q\leq\infty$. Consider the Besov scale of spaces $B_{p,q}^s(\mathbb{R}^d)$ defined by the norm $$\|f\|_{B_{p,q}^s} := (\sum_{j=0}^\infty \|P_{j} f\|_{L^p}^q)^{1/q},$$ ...
Matt Rosenzweig's user avatar
1 vote
1 answer
128 views

Show a inequality in homogeneous Besov space

How to prove $$ \lVert uv\rVert_{\dot{B}^{\frac{N}{p}-1}_{p,1}}\leqslant C \lVert u\rVert_{\dot{B}^{\frac{N}{p}}_{p,1}} \lVert v\rVert_{\dot{B}^{\frac{N}{p}-1}_{p,1}}$$ when $N\geqslant2 $and$1\...
Bob's user avatar
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2 votes
0 answers
111 views

Hilbert transform on a Besov space

Consider the usual Hilbert transform of periodic functions $$H(f) = \frac{1}{2\pi}P.V.\int_{-\pi}^{\pi}\cot(\frac{x-y}{2})f(y)dy.$$ We know $H$ does not map $L^\infty$ continuously to $L^\infty$. Now ...
Jacob Lu's user avatar
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5 votes
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136 views

Characterizing Besov spaces in terms of p-variation

For $s>1/p$ the Besov space $B_{p,q}^s([0,1])$ can be characterized in terms of the $p$-variation: Let $p,q \in (1,\infty)$ and $s \in (0,1)$, $s>1/p$. A function $f:[0,1] \to \mathbb{R}$ is in ...
saz's user avatar
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1 vote
1 answer
88 views

Inequality regarding a probability measure

First of all, I am sorry for the ''not clear title' for this question but I cannot find a better way to describe this seemingly very simple and standard inequality, So.. I am reading a paper 'Two-...
Lev Bahn's user avatar
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5 votes
1 answer
163 views

Critical Smoothness on Besov Spaces $B^s_{p}$: how does it evolved with $p$?

We denote by $B_{p}^s(\mathbb{T}) := B_{p,p}^s(\mathbb{T})$ the Besov space over the circle $\mathbb{T}$ with parameters $p=q \in (0, \infty]$ and smoothness $s \in \mathbb{R}$. For $p>0$ fixed and ...
Goulifet's user avatar
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1 vote
1 answer
129 views

Box counting dimension and Besov spaces on $\mathbb R^2$

I found a lemma in this paper of Constantin and Wu, stated with no proof: Lemma 3.2. Let $b=\chi_{D}$ be the characteristic function of a bounded domain $D\subset\mathbb R^2$ whose boundary has box-...
Calvin Khor's user avatar
2 votes
0 answers
181 views

Besov or Triebel-Lizorkin spaces versus Lorentz spaces

I first asked this question on math.stackexchange here but it seems it is more a research level question ... At the $0$ order of derivatives of Sobolev spaces and for a fixed integrability order $p$, ...
LL 3.14's user avatar
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3 votes
0 answers
167 views

Space contained in the Interpolation of $L^\infty$ and the Wiener Algebra $\mathcal{F}(L^1)$

Let $\ell^p$ be the space of sequences with power $p$ summable to $\ell^\infty$, $L^p = L^p(\mathbb{R^d})$ be the Lebesgue spaces and $\mathcal{F}$ be the Fourier $d$-dimensional Fourier transform. ...
LL 3.14's user avatar
  • 230
2 votes
2 answers
215 views

Why is this estimate about Besov norms true

For reference, I am reading the paper "Uniqueness of Finite Energy Solutions for Maxwell-Dirac and Maxwell-Klein-Gordon Equations" by Masmoudi and Nakanishi. Let $A_0$ be a scalar function satisfying ...
Jakob Möller's user avatar
3 votes
1 answer
319 views

Characterization of Besov space with Lp-modulus of continuity

When reading the characterization of Besov space with $L_p$-modulus of continuity in the 7th chapter “Fractional Order Space” of Sobolev space written by Adams(Page 243), I encounter some small ...
JohnLee's user avatar
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4 votes
0 answers
95 views

Fractional Hajłasz-Besov-like similar to the Korevaar-Schoen-Sobolev spaces?

Suppose that $(X,\mu,d)$ and $(Y,\nu,\rho)$ are doubling metric measure spaces. Fix $\alpha>0$ and define the space, analogously to this paper, as the collection of all measurable functions $f:X\...
ABIM's user avatar
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1 vote
0 answers
99 views

Choosing the weight in a particular definition of Besov spaces

Following Giovanni Leoni's excellent book (or the Wikipedia article) one possible way to define the Besov spaces $B^{s,p,\theta}(\mathbb R ^d)$, with $s\in(0,1)$ the fractional "order of derivative" ...
leo monsaingeon's user avatar
3 votes
1 answer
449 views

Injection of Besov spaces in $L^p$

I believe that for $p\ge 2$, we have the continuous injection (for $p=2$, it is an equality), $$ B^0_{p,2}(\mathbb R^n)\subset L^p(\mathbb R^n), $$ where $B^0_{p,2}(\mathbb R^n)$ is the Besov space. ...
Bazin's user avatar
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4 votes
0 answers
84 views

Sharp asymptotic behavior of the metric entropy for the unit ball in Besov space

For $s>0$ and $1 \leq p,q \leq \infty$ let $B^s_{p,q}$ be the Besov space defined on $[0,1]^d$, and assume $ s > d( \frac 1 p - \frac 1 2)_+$, such that $B^s_{p,q}$ is compactly embedded in $L^2(...
andreas klotz's user avatar
5 votes
1 answer
327 views

Estimate of the difference quotients in terms of an $L^{1,\infty}$ function

Let $f \colon \mathbb R^d \to \mathbb R$ be a measurable function. Consider the following property: (P) there exist a negligible set $N \subset \mathbb R^d$ and function $T_f \in L^p(\mathbb R^d)$ ...
Romeo's user avatar
  • 960
1 vote
1 answer
245 views

Relation between a norm and norm of Besov spaces

Let $(H, \|\cdot\|)$ be a Hilbert space, $A \colon D(A)\subset H \longrightarrow H$ generates an analytic semigroup $T(t)$ on $H$. We define the following Banach space with the respect norm $$F=\{x\in ...
S. Maths's user avatar
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2 votes
0 answers
334 views

Hölder-Zygmund spaces of negative order

In the equation (1.1.17) in Proposition 1.1.6 (ii) in Alazard and Delort's Sobolev Estimates for Two Dimensional Water Waves, there appears a norm named $C^{-1}$, but in Chapter 6 (Appendix) of the ...
Fan Zheng's user avatar
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2 votes
0 answers
63 views

Generalized Besov spaces with different integrability and smoothness in space and time?

Consider the family of Besov spaces $B_{p,q}^{s}(\mathbb{R})$ with $0<p,q \leq \infty$ and $s \in \mathbb{R}$. Is there a natural way to define spaces of generalized functions $f(t,x) \in \mathcal{...
Goulifet's user avatar
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3 votes
1 answer
398 views

An inequality from Bessel potential space to Besov space

I'm not sure this question is suitable for MathOverflow. Currently, I'm reading a paper "Inhomogeneous Dirichlet Problem in Lipschitz domain" by Jerison and Kenig. I have a question on some ...
Will Kwon's user avatar
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1 vote
0 answers
54 views

Continuous mapping on Besov spaces?

Consider $f\in B_{p,q}^s(\Omega)$, $\Omega$ compact in $\mathbb{R}^d$, with $p,q\geq 1$ and $s>d/p$ (so the elements in the space are regular enough to be continuous functions), such that $\|f\|_{...
newbie's user avatar
  • 319
7 votes
2 answers
509 views

"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 ...
Hannes's user avatar
  • 2,195
2 votes
0 answers
120 views

Imbedding Theorems between Besov Spaces and space of continuos functions on the unit circle

I'll try to be brief. Let us consider the Besov Space $B^{1/p}_{p, p}(\mathbb{T})$, where $1\leq p<\infty $ and $\mathbb{T}$ is the unit circle in the complex plane. I would like to know for which ...
John's user avatar
  • 141
4 votes
1 answer
195 views

Besov regularity of càdlàg functions?

Let $D(\mathbb{R})$ be the space of functions from $\mathbb{R}$ to $\mathbb{R}$ that are right continuous with left limits (also referred to as càdlàg functions). $D(\mathbb{R})$ is often called the ...
Goulifet's user avatar
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6 votes
0 answers
322 views

Embeddings between weighted Besov spaces

Consider the Besov spaces $B_{p,q}^s(\mathbb{R}^d)$ for parameters $0<p,q\leq \infty$ and $s\in \mathbb{R}$. The weighted Besov space $B_{p,q}^s(\mathbb{R}^d;\mu)$ is defined for $\mu \in \mathbb{R}...
Goulifet's user avatar
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12 votes
0 answers
446 views

Are Sobolev trace spaces equal from both sides of the boundary?

Let $\Omega\subset\mathbb R^n$ be a bounded open set and $\Omega'$ the complement of its closure. Assume $\partial\Omega=\partial\Omega'$. Are the quotient spaces $W^{1,p}(\Omega)/W^{1,p}_0(\Omega)$ ...
Joonas Ilmavirta's user avatar
3 votes
1 answer
3k views

Besov and Triebel-Lizorkin spaces

Let me start with a couple of notational reminders. For $\xi\in \mathbb R^n$, $$ 1=\varphi_{0}(\xi)+\sum_{\nu \ge 1}\varphi_{\nu}(\xi),\quad \varphi_{0}\in C^\infty_c(\mathbb R^{n}),\quad \varphi_{\nu}...
Bazin's user avatar
  • 15.2k
6 votes
0 answers
157 views

Real interpolation space between the Wiener algebra and $L^2$

The Wiener algebra $W_n$ is the image by the Fourier transform of $L^1(\mathbb R^n)$. What is the (complex) interpolation space between $W_n$ and $L^2(\mathbb R^n)$? It is probably not true that for $\...
Bazin's user avatar
  • 15.2k
5 votes
3 answers
305 views

Reference request : Besov spaces on ubounded domains

As I am relatively new to these matters, I would like to know if you could provide me a reference for Besov spaces on unbounded domains, because when I checked the first tome of Triebel's Theory of ...
Paul-Benjamin's user avatar