# Questions tagged [besov-space]

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

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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 ...
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### 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 ...
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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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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 $\mathcal{S}(\mathbb{... • 120k 3 votes 0 answers 67 views ### Are Besov embeddings strict? Let$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$if ... • 740 1 vote 2 answers 80 views ### Is there a nice relationship between$\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]$, ... • 457 1 vote 0 answers 108 views ### Can$L^p$space be imbedded in some Besov space$B_{q,2}^\sigma$? Let $$\lVert f \rVert_{B_{q,2}^\sigma} = \lVert P_0 f \rVert_p + ( \sum_{j> 0} 2^{2 \sigma j} \lVert P_j f \rVert_p^2 )^{\frac{1}{2}}.$$ For$\sigma \ge0$and$p \ge 2$, ... • 353 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$$$\... • 12.7k 3 votes 1 answer 535 views ### Reference on periodic Besov spaces I am looking for a reference for the construction and the study of the main properties of Besov spaces on the torus (i.e. \mathrm{B}_{p,q}^s(\mathbb{T}^d)). Indeed, I know the classical \mathrm{B}_{... • 440 1 vote 1 answer 133 views ### How are these two Banach spaces related ? (weighted L2 type space involving a logarithm and Besov type space) First the standard L2 space :$$L^2(\mathbb{R}) = \Big \{ f : \| f \|_2 = \left( \int_{\mathbb{R}} | f(x) |^2 dx \right)^{1/2} < \infty \Big \}.$$Let s \geqslant 1/2. Define a weighted L2 ... • 417 1 vote 1 answer 108 views ### Basic question about Bony decompositions - summation indices I'm trying to understand some sort of inequality in a larger calculation. I believe my only issue is in counting correctly, so I've also tagged combinatorics. Suppose I have a function f. Let \... • 35k 2 votes 0 answers 107 views ### Equivalent Besov seminorm - change of integral limits Let f \in L^p(\mathbb{R}), p\ge1. \omega_k(f;t)_p, the k^{th} order modulus of smoothness of f, is defined by$$\omega_k(f;t)_p = \sup_{0<|h| \le t} \|\Delta_h^kf\|_p,$$where$$\Delta_h^... 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 $$f \in W^{n,p} (\mathbb{R}), \quad \int_{0} ^{\infty} \Big | \frac{\omega_p ^2(f^{(n)}, t)}{t^{a}} \Big |^q \frac{... • 417 3 votes 0 answers 163 views ### Examples of Besov functions of power-logarithmic type |x|^{\alpha} |\log |x||^{\beta} I'm stuck on the following exercise 17.9 from Leoni's text A first course in Sobolev Spaces (second edition). This is from the chapter on Besov spaces, but this is really just a integral inequality ... • 7,392 3 votes 1 answer 88 views ### Does C_c(X) separate points in X when X is a Banach space? Suppose that X is a separable, infinite dimensional Banach space. We say that a set of functions \{f_\alpha\}_{\alpha \in A} separates points in X if for every x,y \in X, there is an \alpha... • 19.7k 6 votes 1 answer 493 views ### How to prove that a delta function belongs to the Besov space B^{-1}? \def\R{\mathbb{R}} \DeclareMathOperator{\supp}{supp} I am trying to understand the definition of the Besov space and to prove that the delta function in \R^1. However I got stuck. First, let ... • 740 1 vote 0 answers 58 views ### A function space satisfying |f(x) -f(y)| \le ||x|^\theta -|y|^\theta| Consider a class of function on the real line such that$$ |f(x) -f(y)| \le ||x|^\theta -|y|^\theta | $$for a \theta >0. Does this class of function space have a name When |f(x) -f(y)| \le ... • 1,348 2 votes 0 answers 127 views ### Embedding property of homogeneous periodic Besov spaces Let's consider homogeneous periodic Besov spaces \dot{B}^s_{p,r}(\mathbb{S}^1) defined on the circle. The definition is similar to that defined on the real line, except that we use Fourier expansion ... • 293 1 vote 1 answer 287 views ### L^p norm of Littlewood-Paley block of a smooth function I'm working on a thesis concerning the Littlewood-Paley decomposition of distributions and the use of paraproducts. I'm referring to the book of Bahouri, Chemin, Danchin and during a proof (page 88 ... • 1,548 6 votes 1 answer 850 views ### \mathcal{C}^{\alpha} Besov spaces: Definition I'm reading an article for my future thesis (I'm a third-year undergraduate) where the authors define the generalized Holder Spaces as a special class of Besov Spaces. Define \chi,\tilde{\chi}\in C_{... • 1,548 3 votes 0 answers 106 views ### Fractional Sobolev space on Union of Sets I have a question on fractional Sobolev Hilbert space H^s with fractional s: I have a bounded Lipschitz domain \Omega\subseteq\mathbb{R}^d that is the union of two Lipschitz domains \Omega_1,\... 6 votes 1 answer 1k views ### Interpretation of Besov Space parameters I've been reading about Besov spaces (my reference thus far has been "Mathematical foundations of infinite-dimensional statistical models" (Nickl & Gine), and I've been struggling a bit with the ... • 10.8k 1 vote 1 answer 106 views ### On the Besov space I want to prove this:$$ 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\... • 700 0 votes 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 ... • 1,035 3 votes 1 answer 2k views ### 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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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 ...
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{...