Questions tagged [besov-spaces]
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Why does the integrand $\frac{f(x)-f(y)}{|x-y|^\alpha}$ often appear when studying regularity?
Certain spaces (such as Besov, Holder, or Sobolev spaces) often measure regularity by function increments or difference quotients, and their norms contain integrands similar to
$$\frac{f(x)-f(y)}{|x-y|...
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Holder-Besov space and time continuity
Let $\mathbb{T}^d$ be the $d$-dimensional torus, $\mathscr{S}:=C^\infty(\mathbb{T}^d)$ the Schwartz space, $\mathscr{S}'$ the space of tempered distributions.
We consider a dyadic partition of unity $(...
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Does the complex interpolation space $(L^1(\mathbb{R}),W^{2,1}(\mathbb{R}))_{\frac{1}{2}}$ continuously embed into $L^\infty(\mathbb{R})$?
The complex interpolation space between $(L^p(\mathbb{R}),W^{2,p}(\mathbb{R}))_\theta$ with interpolation parameter $\theta=\frac{1}{2}$ is known to be $W^{1,p}(\mathbb{R})$ for $1<p<\infty$. As ...
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Is there any example of linear operator which is bounded on all Besov spaces but not on Triebel-Lizorkin spaces
Is there any linear operator $T:S'(\mathbb R^n)\to S'(\mathbb R^n)$ such that $T:B_{pq}^s(\mathbb R^n)\to B_{pq}^s(\mathbb R^n)$ for all $0<p,q\le\infty$ and $s\in\mathbb R$, but there exist a $F_{...
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Equivalent Littlewood-Paley-type decompositions
The theory of Besov and Triebel-Lizorkin spaces usually proceeds by taking a dyadic decomposition of unity, i.e. some non-negative functions $\psi_0,\psi \in C_c^\infty(\mathbb{R})$ such that
\begin{...
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How to get $\int_{\mathbb R^d} |\partial_i\partial_j(1-\Delta)^{-\frac{\delta}{2}}p_t(\cdot-y)(x)| \, \mathrm d x \lesssim t^{\frac{\delta}{2}-1}$?
We consider the Gaussian heat kernel $p_t$ on $\mathbb R^d$, i.e.,
$$
p_t (x) := (4\pi t)^{-\frac{d}{2}} e^{-\frac{|x|^2}{4t}},
\quad t>0, x \in \mathbb R^d,
$$
and define the operator $P_t$ by
$$
...
- 1,179
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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 ...
- 3,113
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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, ...
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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://...
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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. ...
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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
$$\Vert v\Vert_{\ell^p_\omega} := \left(\sum_{k=1}^\infty \omega_k^p |v|_k^p\right)...
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Besov space norms
We need to recall some Besov space norms to formulate the question.
Let $d \in \mathbb N$, $0<s<2, 1 \le p,q \le \infty$. Then the Besov space $B^s_{p,q}(\mathbb R^d)$ is given by the norm
$$ \...
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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 ...
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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
$$K(f,t,X_0,X_1)=\inf\{\|f_0\|_{X_0}+t\|f_1\|_{X_1}:...
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Interpolation of Sobolev/Besov spaces in the limiting case q = ∞
I'm interested in the interpolation space ($1\le p_0,p_1\le\infty$, $0<\theta<1$)
$$
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 ...
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