Weighted embedding theorems for radial Besov and Triebel-Lizorkin spaces

Pablo L. De Nápoli; Irene Drelichman; Nicolas Saintier

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Displaying similar documents to “Weighted embedding theorems for radial Besov and Triebel-Lizorkin spaces”

Compactness criteria in function spaces

Monika Dörfler, Hans G. Feichtinger, Karlheinz Gröchenig (2002)

Colloquium Mathematicae

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The classical criterion for compactness in Banach spaces of functions can be reformulated into a simple tightness condition in the time-frequency domain. This description preserves more explicitly the symmetry between time and frequency than the classical conditions. The result is first stated and proved for $L ² (ℝ^{d})$ , and then generalized to coorbit spaces. As special cases, we obtain new characterizations of compactness in Besov-Triebel-Lizorkin, modulation and Bargmann-Fock spaces. ...

Asymptotic behaviour of Besov norms via wavelet type basic expansions

Anna Kamont (2016)

Annales Polonici Mathematici

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J. Bourgain, H. Brezis and P. Mironescu [in: J. L. Menaldi et al. (eds.), Optimal Control and Partial Differential Equations, IOS Press, Amsterdam, 2001, 439-455] proved the following asymptotic formula: if $Ω \subset ℝ^{d}$ is a smooth bounded domain, 1 ≤ p < ∞ and $f \in W^{1, p} (Ω)$ , then $l i m_{s ↗ 1} (1 - s) \int_{Ω} \int_{Ω} {(| f (x) - f (y) |}^{p} {) / (| | x - y | |}^{d + s p}) d x d y = K \int_{Ω} {| \nabla f (x) |}^{p} d x$ , where K is a constant depending only on p and d. The double integral on the left-hand side of the above formula is an equivalent seminorm in the Besov space $B_{p}^{s, p} (Ω)$ . The purpose of this paper is to obtain analogous asymptotic formulae...

Local means and wavelets in function spaces

Hans Triebel (2008)

Banach Center Publications

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The paper deals with local means and wavelet bases in weighted and unweighted function spaces of type $B_{p q}^{s}$ and $F_{p q}^{s}$ on ℝⁿ and on ⁿ.

Unconditional biorthogonal wavelet bases in $L^{p} (ℝ^{d})$

Waldemar Pompe (2002)

Colloquium Mathematicae

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We prove that a biorthogonal wavelet basis yields an unconditional basis in all spaces $L^{p} (ℝ^{d})$ with 1 < p < ∞, provided the biorthogonal wavelet set functions satisfy weak decay conditions. The biorthogonal wavelet set is associated with an arbitrary dilation matrix in any dimension.

Inequivalence of Wavelet Systems in $L ₁ (ℝ^{d})$ and $B V (ℝ^{d})$

Paweł Bechler (2005)

Bulletin of the Polish Academy of Sciences. Mathematics

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Theorems stating sufficient conditions for the inequivalence of the d-variate Haar wavelet system and another wavelet system in the spaces $L ₁ (ℝ^{d})$ and $B V (ℝ^{d})$ are proved. These results are used to show that the Strömberg wavelet system and the system of continuous Daubechies wavelets with minimal supports are not equivalent to the Haar system in these spaces. A theorem stating that some systems of smooth Daubechies wavelets are not equivalent to the Haar system in $L ₁ (ℝ^{d})$ is also shown.

Good-λ inequalities for wavelets of compact support

Sarah V. Cook (2004)

Colloquium Mathematicae

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For a wavelet ψ of compact support, we define a square function $S_{w}$ and a maximal function NΛ. We then obtain the $L_{p}$ equivalence of these functions for 0 < p < ∞. We show this equivalence by using good-λ inequalities.

Refinement type equations: sources and results

Rafał Kapica, Janusz Morawiec (2013)

Banach Center Publications

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It has been proved recently that the two-direction refinement equation of the form $f (x) = \sum_{n \in} c_{n, 1} f (k x - n) + \sum_{n \in ℤ} c_{n, - 1} f (- k x - n)$ can be used in wavelet theory for constructing two-direction wavelets, biorthogonal wavelets, wavelet packages, wavelet frames and others. The two-direction refinement equation generalizes the classical refinement equation $f (x) = \sum_{n \in ℤ} c ₙ f (k x - n)$ , which has been used in many areas of mathematics with important applications. The following continuous extension of the classical refinement equation $f (x) = \int_{ℝ} c (y) f (k x - y) d y$ has also various interesting...

Haar wavelets on the Lebesgue spaces of local fields of positive characteristic

Biswaranjan Behera (2014)

Colloquium Mathematicae

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We construct the Haar wavelets on a local field K of positive characteristic and show that the Haar wavelet system forms an unconditional basis for $L^{p} (K)$ , 1 < p < ∞. We also prove that this system, normalized in $L^{p} (K)$ , is a democratic basis of $L^{p} (K)$ . This also proves that the Haar system is a greedy basis of $L^{p} (K)$ for 1 < p < ∞.

The characterization of radial subspaces of Besov and Lizorkin-Triebel spaces by differences

Winfried Sickel, Leszek Skrzypczak, Jan Vybíral (2014)

Banach Center Publications

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We continue our earlier investigations of radial subspaces of Besov and Lizorkin-Triebel spaces on $ℝ^{d}$ . This time we study characterizations of these subspaces by differences.

Sharp embedding results for spaces of smooth functions with power weights

Martin Meyries, Mark Veraar (2012)

Studia Mathematica

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We consider function spaces of Besov, Triebel-Lizorkin, Bessel-potential and Sobolev type on $ℝ^{d}$ , equipped with power weights $w (x) = {| x |}^{γ}$ , γ > -d. We prove two-weight Sobolev embeddings for these spaces. Moreover, we precisely characterize for which parameters the embeddings hold. The proofs are presented in such a way that they also hold for vector-valued functions.

Decomposition systems for function spaces

G. Kyriazis (2003)

Studia Mathematica

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Let $Θ : = θ_{I}^{e} : e \in E, I \in D$ be a decomposition system for $L ₂ (ℝ^{d})$ indexed over D, the set of dyadic cubes in $ℝ^{d}$ , and a finite set E, and let $Θ ̃ : = Θ ̃_{I}^{e} : e \in E, I \in D$ be the corresponding dual functionals. That is, for every $f \in L ₂ (ℝ^{d})$ , $f = \sum_{e \in E} \sum_{I \in D} ⟨ f, Θ ̃_{I}^{e} ⟩ θ_{I}^{e}$ . We study sufficient conditions on Θ,Θ̃ so that they constitute a decomposition system for Triebel-Lizorkin and Besov spaces. Moreover, these conditions allow us to characterize the membership of a distribution f in these spaces by the size of the coefficients $⟨ f, Θ ̃_{I}^{e} ⟩$ , e ∈ E, I ∈ D. Typical examples of such decomposition...

Function spaces with dominating mixed smoothness

Jan Vybiral

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We study several techniques which are well known in the case of Besov and Triebel-Lizorkin spaces and extend them to spaces with dominating mixed smoothness. We use the ideas of Triebel to prove three important decomposition theorems. We deal with so-called atomic, subatomic and wavelet decompositions. All these theorems have much in common. Roughly speaking, they say that a function f belongs to some function space (say $S_{p, q}^{r ̅} A$ ) if, and only if, it can be decomposed as $f (x) = \sum_{ν} \sum_{m} λ_{ν m} a_{ν m} (x)$ , convergence in S’, with...

Jacobi decomposition of weighted Triebel-Lizorkin and Besov spaces

George Kyriazis, Pencho Petrushev, Yuan Xu (2008)

Studia Mathematica

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The Littlewood-Paley theory is extended to weighted spaces of distributions on [-1,1] with Jacobi weights $w (t) = {(1 - t)}^{α} {(1 + t)}^{β}$ . Almost exponentially localized polynomial elements (needlets) $φ_{ξ}$ , $ψ_{ξ}$ are constructed and, in complete analogy with the classical case on ℝⁿ, it is shown that weighted Triebel-Lizorkin and Besov spaces can be characterized by the size of the needlet coefficients $⟨ f, φ_{ξ} ⟩$ in respective sequence spaces.

Existence of solutions to the (rot,div)-system in $L_{p}$ -weighted spaces

Wojciech M. Zajączkowski (2010)

Applicationes Mathematicae

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The existence of solutions to the elliptic problem rot v = w, div v = 0 in a bounded domain Ω ⊂ ℝ³, ${v \cdot n ̅ |}_{S} = 0$ , S = ∂Ω in weighted $L_{p}$ -Sobolev spaces is proved. It is assumed that an axis L crosses Ω and the weight is a negative power function of the distance to the axis. The main part of the proof is devoted to examining solutions of the problem in a neighbourhood of L. The existence in Ω follows from the technique of regularization.

Embeddings of Besov-Morrey spaces on bounded domains

Dorothee D. Haroske, Leszek Skrzypczak (2013)

Studia Mathematica

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We study embeddings of spaces of Besov-Morrey type, $i d_{Ω} :_{p ₁, u ₁, q ₁}^{s ₁} (Ω) ↪_{p ₂, u ₂, q ₂}^{s ₂} (Ω)$ , where $Ω \subset ℝ^{d}$ is a bounded domain, and obtain necessary and sufficient conditions for the continuity and compactness of $i d_{Ω}$ . This continues our earlier studies relating to the case of $ℝ^{d}$ . Moreover, we also characterise embeddings into the scale of $L_{p}$ spaces or into the space of bounded continuous functions.

A note on integer translates of a square integrable function on ℝ

Maciej Paluszyński (2010)

Colloquium Mathematicae

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We consider the subspace of L²(ℝ) spanned by the integer shifts of one function ψ, and formulate a condition on the family ${ψ (\cdot - n)}_{n = - \infty}^{\infty}$ , which is equivalent to the weight function $\sum_{n = - \infty}^{\infty} | ψ ̂ (\cdot + n) | ²$ being > 0 a.e.

The Properties of the Weighted Space $H_{2, α}^{k} (Ω)$ and Weighted Set $W_{2, α}^{k} (Ω, δ)$

V.A. Rukavishnikov, E.V. Matveeva, E.I. Rukavishnikova (2018)

Communications in Mathematics

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We study the properties of the weighted space $H_{2, α}^{k} (Ω)$ and weighted set $W_{2, α}^{k} (Ω, δ)$ for boundary value problem with singularity.

Estimation of a smoothness parameter by spline wavelets

Magdalena Meller, Natalia Jarzębkowska (2013)

Applicationes Mathematicae

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We consider the smoothness parameter of a function f ∈ L²(ℝ) in terms of Besov spaces $B_{2, \infty}^{s} (ℝ)$ , $s * (f) = s u p s > 0 : f \in B_{2, \infty}^{s} (ℝ)$ . The existing results on estimation of smoothness [K. Dziedziul, M. Kucharska and B. Wolnik, J. Nonparametric Statist. 23 (2011)] employ the Haar basis and are limited to the case 0 < s*(f) < 1/2. Using p-regular (p ≥ 1) spline wavelets with exponential decay we extend them to density functions with 0 < s*(f) < p+1/2. Applying the Franklin-Strömberg wavelet p = 1, we prove that the...

Lipschitz continuity in Muckenhoupt 𝓐₁ weighted function spaces

Dorothee D. Haroske (2011)

Banach Center Publications

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We study continuity envelopes of function spaces $B_{p, q}^{s} (ℝ ⁿ, w)$ and $F_{p, q}^{s} (ℝ ⁿ, w)$ where the weight belongs to the Muckenhoupt class ₁. This essentially extends partial forerunners in [13, 14]. We also indicate some applications of these results.

Embeddings of doubling weighted Besov spaces

Dorothee D. Haroske, Philipp Skandera (2014)

Banach Center Publications

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We study continuous embeddings of Besov spaces of type $B_{p, q}^{s} (ℝ ⁿ, w)$ , where s ∈ ℝ, 0 < p < ∞, 0 < q ≤ ∞, and the weight w is doubling. This approach generalises recent results about embeddings of Muckenhoupt weighted Besov spaces. Our main argument relies on appropriate atomic decomposition techniques of such weighted spaces; here we benefit from earlier results by Bownik. In addition, we discuss some other related weight classes briefly and compare corresponding results.

Polar wavelets and associated Littlewood-Paley theory

Epperson Jay, Frazier Michael

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Abstract We develop an almost orthogonal wavelet-type expansion in ℝ² which is adapted to polar coordinates. We start by defining a product Fourier-Hankel transform f̂ and proving a sampling formula for f such that f̂ is compactly supported. For general f, the sampling formula and a partition of unity lead to an identity of the form $f = \sum_{μ, k, m} ⟨ f, φ_{μ k m} ⟩ ψ_{μ k m}$ , in which each function $φ_{μ k m}$ and $ψ_{μ k m}$ is concentrated near a certain annular sector, has compactly supported product Fourier-Hankel transform, and is smooth...

Parallel implementation of Wavelet-Galerkin method

Finěk, Václav, Šimůnková, Martina

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We present here some details of our implementation of Wavelet-Galerkin method for Poisson equation in C language parallelized by POSIX threads library and show its performance in dimensions $d \in {3, 4, 5}$ .

Solutions to the equation div u = f in weighted Sobolev spaces

Katrin Schumacher (2008)

Banach Center Publications

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We consider the problem div u = f in a bounded Lipschitz domain Ω, where f with $\int_{Ω} f = 0$ is given. It is shown that the solution u, constructed as in Bogovski’s approach in [1], fulfills estimates in the weighted Sobolev spaces $W_{w}^{k, q} (Ω)$ , where the weight function w is in the class of Muckenhoupt weights $A_{q}$ .

On Clifford-type structures

Wiesław Królikowski

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Polyanalytic Besov spaces and approximation by dilatations

Ali Abkar (2024)

Czechoslovak Mathematical Journal

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Using partial derivatives $\partial f / \partial z$ and $\partial f / \partial \bar{z}$ , we introduce Besov spaces of polyanalytic functions in the open unit disk, as well as in the upper half-plane. We then prove that the dilatations of functions in certain weighted polyanalytic Besov spaces converge to the same functions in norm. When restricted to the open unit disk, we prove that each polyanalytic function of degree $q$ can be approximated in norm by polyanalytic polynomials of degree at most $q$ .

Complex interpolation of function spaces with general weights

Douadi Drihem (2023)

Commentationes Mathematicae Universitatis Carolinae

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We present the complex interpolation of Besov and Triebel–Lizorkin spaces with generalized smoothness. In some particular cases these function spaces are just weighted Besov and Triebel–Lizorkin spaces. As a corollary of our results, we obtain the complex interpolation between the weighted Triebel–Lizorkin spaces ${\dot{F}}_{p_{0}, q_{0}}^{s_{0}} (ω_{0})$ and ${\dot{F}}_{\infty, q_{1}}^{s_{1}} (ω_{1})$ with suitable assumptions on the parameters $s_{0}, s_{1}, p_{0}, q_{0}$ and $q_{1}$ , and the pair of weights $(ω_{0}, ω_{1})$ .

The linear bound in A₂ for Calderón-Zygmund operators: a survey

Michael Lacey (2011)

Banach Center Publications

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For an L²-bounded Calderón-Zygmund Operator T acting on $L ² (ℝ^{d})$ , and a weight w ∈ A₂, the norm of T on L²(w) is dominated by $C_{T} {| | w | |}_{A ₂}$ . The recent theorem completes a line of investigation initiated by Hunt-Muckenhoupt-Wheeden in 1973 (MR0312139), has been established in different levels of generality by a number of authors over the last few years. It has a subtle proof, whose full implications will unfold over the next few years. This sharp estimate requires that the A₂ character of the weight can...