Full characterization of the embedding relations between α-modulation spaces

Guo, Weichao; Fan, Dashan; Zhao, Guoping

doi:10.1007/s11425-016-9151-1

Cited by 21 publications

(27 citation statements)

References 27 publications

(32 reference statements)

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“…Decomposition spaces, originally introduced by Feichtinger and Gröbner [16], provide a unified generalisation of modulation and Besov spaces and were used to introduce the α-modulation spaces [25], which have recently received great attention [36,55,32,3,19,31,33,13].…”

Section: Decomposition Spaces and Their Relation To Frames And Sparsitymentioning

confidence: 99%

“…2) In Subsection 6.5, we shall see that the wave packet smoothness spaces W p,q s (α, α) are identical to the α-modulation spaces M s,α p,q (R 2 ) introduced in Gröbner's PhD thesis [25] and studied further in [3,19,55,33,36,32,61]. Therefore, Theorem 6.5 can be seen as a generalisation of the characterisation of the embeddings between α-modulation spaces, which were first studied in [25,33] and fully understood in [60,32,61].…”

Section: Proof Of B) Here Again There Existsmentioning

confidence: 99%

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Design and properties of wave packet smoothness spaces

Bytchenkoff

Voigtlaender

2020

Journal de Mathématiques Pures et Appliquées

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We introduce a family of quasi-Banach spaces -which we call wave packet smoothness spacesthat includes those function spaces which can be characterised by the sparsity of their expansions in Gabor frames, wave atoms, and many other frame constructions. We construct Banach frames for and atomic decompositions of the wave packet smoothness spaces and study their embeddings in each other and in a few more classical function spaces such as Besov and Sobolev spaces. RésuméNous introduisons une famille d'espaces affines quasi-normés complets -que nous appellerons espaces de paquets d'ondelettes réguliers -qui incluent de nombreux espaces de fonctions caractérisés par leurs transformées, comme celle de Gabor ou en ondelettes, clairsemées. Nous construisons des cadres de Banach et des décompositions atomiques pour ces espaces etétudions leurs inclusions l'un dans l'autre ainsi que dans quelques espaces de fonctions devenus classiques tels que ceux de Sobolev ou de Besov.(2.1)To ensure that the decomposition space D(Q, L p , q w ) is indeed a well-defined quasi-Banach space, certain conditions must be imposed on the covering Q, the partition of unity (ϕ i ) i∈I and the weight w = (w i ) i∈I . These conditions and elementary properties of the decomposition spaces D(Q, L p , q w ) will be reminded in Section 2.1.An attractive feature of decomposition spaces is the recently developed theory of structured Banach frame decompositions of decomposition spaces [63], which shows that there is a close connexion between the existence of a sparse expansion of a given function f in terms of a given frame, and the membership of f in a certain decomposition space, which depends on the frame under consideration.This theory will be formally introduced in Section 2.2; here, we outline the underlying intuition. We first note that most frame constructions used in harmonic analysis have two crucial properties:• The frame is a generalised shift-invariant system (see [34,59,11,21,43] for more about these systems), i.e., it is of the form Ψ = (L x ψ j ) j∈I,x∈Γ j for suitable generators (ψ j ) j∈I and certain lattices Γ j = δB j Z d where the matrices B j ∈ GL(R d ) are determined by the frame construction and δ > 0 globally stands for the sampling density.

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Section: Decomposition Spaces and Their Relation To Frames And Sparsitymentioning

confidence: 99%

Section: Proof Of B) Here Again There Existsmentioning

confidence: 99%

Design and properties of wave packet smoothness spaces

Bytchenkoff

Voigtlaender

2020

Journal de Mathématiques Pures et Appliquées

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“…Finally, we recall a disjoint property for the α-covering, which will be used in the proof Lemma 4.1. [14]). Let α ∈ [0, 1) be arbitrary.…”

Section: Denote Bymentioning

confidence: 99%

Sharp estimates of unimodular Fourier multipliers on Wiener amalgam spaces

Guo

Zhao

2020

Journal of Functional Analysis

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We study the boundedness on the Wiener amalgam spaces W p,q s of Fourier multipliers with symbols of the type e iµ(ξ) , for some real-valued functions µ(ξ) whose prototype is |ξ| β with β ∈ (0, 2]. Under some suitable assumptions on µ, we give the characterization of W p,q s → W p,q boundedness of e iµ(D) , for arbitrary pairs of 0 < p, q ≤ ∞. Our results are an essential improvement of the previous known results, for both sides of sufficiency and necessity, even for the special case µ(ξ) = |ξ| β with 1 < β < 2.(t, x) ∈ R × R n , the formal solution is given by u(t, x) = e it|D| β u 0 (x), where e it|D| β is just the unimodular Fourier multiplier defined by e it|D| β f = F −1 (e it|ξ| β F f ). Note that e it|D| β u 0 is the free solution of the Schrödinger equation when β = 2, and the free solution of the wave equation when β = 1. Hence, in order to study the dispersive equation, it is of great interest to study the boundedness of unimodular Fourier multipliers on several function spaces.2000 Mathematics Subject Classification. 42B15, 42B35.

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“…One can find some basic properties about -modulation spaces in [7,8]. Among many features of the -modulation spaces, an interesting subject is the inclusion between -modulation and function spaces, have been concerned by many authors to this topic, see [8][9][10][11]. As applications, -modulation spaces are quite recently applied to the field of partial differential equations.…”

Section: Introductionmentioning

confidence: 99%