Sharpness of Complex Interpolation on $$\alpha $$ α -Modulation Spaces

Guo, Weichao; Fan, Dashan; Wu, Huoxiong; Zhao, Guoping

doi:10.1007/s00041-015-9424-z

Cited by 22 publications

(19 citation statements)

References 19 publications

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“…In [16], the authors mention the fact that the α -modulation space cannot be obtained by interpolation between modulation spaces and Besov spaces . Thus, it shows that our proofs for in Theorems 3.1 and 3.3 are meaningful.…”

Section: Main Results and Proofsmentioning

confidence: 94%

Strongly singular integrals along curves on α-modulation spaces

Wang

2017

J Inequal Appl

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Section: Main Results and Proofsmentioning

confidence: 94%

Strongly singular integrals along curves on α-modulation spaces

Wang

2017

J Inequal Appl

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“…Combing with the estimates of term I and II, we have |V φψj (x, ξ)| x −L uniformly for all j ∈ N. By the definition of W p,∞ , [11,13,19]). Now, the desired conclusion follows by an interpolation argument among (4.1), (4.11) and (4.12).…”

Section: )mentioning

confidence: 88%

“…On the other hand, by the duality of (4.11), we obtain by the known inclusion between modulation and Besov space (see [11,13,19] Proof. We only state the proof for p = ∞.…”

Section: (Iimentioning

confidence: 96%

“…One basic but important problem is how to characterize the inclusion relations between theses two kinds of spaces. A lot of attentions have been paid to this topic, for example, one can see [19,21,26] for the inclusion relations between modulation and Besov spaces, [11,13,20] for the more general inclusion relations under the frame of α-modulation space, [16] for the inclusion relations between local Hardy spaces h p and modulation spaces, and [17] for the inclusion relation between Sobolev and modulation spaces, etc. In particular, Cunanana, Kobayashib and Sugimotoa [3] recently give some sufficient and necessary conditions for the inclusion relations between W s p,q and L p in the range of 1 p, q ∞, they also give a corollary for the inclusion relations between W s p,q and B p,q (the Besov space B 0 p,q , see (2.16) for the definition).…”

Section: Introductionmentioning

confidence: 99%

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Characterization of inclusion relations between Wiener amalgam and some classical spaces

Guo¹,

Wu²,

Yang³

et al. 2017

Journal of Functional Analysis

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In this paper, we establish the sharp conditions for the inclusion relations between Besov spaces Bp,q and Wiener amalgam spaces W s p,q . We also obtain the optimal inclusion relations between local hardy spaces h p and Wiener amalgam spaces W s p,q , which completely improve and extend the main results obtained by Cunanana, Kobayashib and Sugimotoa in [J. Funct. Anal. 268 (2015), 239-254]. In addition, we establish some mild characterizations of inclusion relations between Triebel-Lizorkin and Wiener amalgam spaces, which relates some modern inequalities to classical inequalities.2000 Mathematics Subject Classification. 42B35.

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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%

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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Sharpness of Complex Interpolation on $$\alpha $$ α -Modulation Spaces

Cited by 22 publications

References 19 publications

Strongly singular integrals along curves on α-modulation spaces

Strongly singular integrals along curves on α-modulation spaces

Characterization of inclusion relations between Wiener amalgam and some classical spaces

Design and properties of wave packet smoothness spaces

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