Structured, compactly supported Banach frame decompositions of decomposition spaces

Voigtlaender, Felix

doi:10.4064/dm804-5-2021

Cited by 4 publications

(2 citation statements)

References 72 publications

(237 reference statements)

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“…The isotropic Besov spaces are known [14] to have orthonormal bases made of wavelets [7]. Herein we shall use the innovative approach reported in [15,16] to construct Banach frames and sets of atoms for the anisotropic homoand inhomogeneous Besov spaces introduced in [4] as decomposition spaces [8]. Similar results, although formulated and achieved rather differently, were reported in [4] and [3].…”

Section: Introductionmentioning

confidence: 68%

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Construction of Banach frames and atomic decompositions of anisotropic Besov spaces

Bytchenkoff¹

2020

Preprint

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Section: Introductionmentioning

confidence: 68%

“…This is a slightly reformulated statement of Theorem 2.11 in [5]. The proof of this theorem in its more general and simplified formulations can be found in [15] and [16] respectively. Theorem 3.7.…”

Section: Construction Of Banach Frames For and Atomic Decompositions ...mentioning

confidence: 96%

Construction of Banach frames and atomic decompositions of anisotropic Besov spaces

Bytchenkoff¹

2020

Preprint

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A framework of Besov–Triebel–Lizorkin type spaces via ball quasi-Banach function sequence spaces I: real-variable characterizations

Sun,

Yang,

Yuan

2024

Math. Ann.

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Embeddings of Decomposition Spaces

Voigtlaender¹

2023

Memoirs of the AMS

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Many smoothness spaces in harmonic analysis are decomposition spaces. In this paper we ask: Given two such spaces, is there an embedding between the two? A decomposition space D ( Q , L p , Y ) \mathcal {D}(\mathcal {Q}, L^p, Y) is determined by a covering Q = ( Q i ) i ∈ I \mathcal {Q} = (Q_i)_{i \in I} of the frequency domain, an integrability exponent p p , and a sequence space Y ⊂ C I {Y \subset \mathbb {C}^I} . Given these ingredients, the decomposition space norm of a distribution g g is defined as ‖ g ‖ D ( Q , L p , Y ) = ‖ ( ‖ F − 1 ( φ i ⋅ g ^ ) ‖ L p ) i ∈ I ‖ Y , { \left \Vert g \right \Vert _{\mathcal {D}(\mathcal {Q}, L^p, Y)} = \left \Vert \left ( \left \Vert \mathcal {F}^{-1} \left ( \varphi _{i} \cdot \widehat {g} \right ) \right \Vert _{L^{p}} \right )_{i \in I} \right \Vert _{Y} } , where ( φ i ) i ∈ I (\varphi _i)_{i \in I} is a suitable partition of unity for Q \mathcal {Q} . We establish readily verifiable criteria which ensure the existence of a continuous inclusion (“an embedding”) D ( Q , L p 1 , Y ) ↪ D ( P , L p 2 , Z ) \mathcal {D}(\mathcal {Q},L^{p_1},Y) \hookrightarrow \mathcal {D}(\mathcal {P},L^{p_2},Z) , mostly concentrating on the case where Y = ℓ w q 1 ( I ) Y = \ell _{w}^{q_{1}} ( I ) and Z = ℓ v q 2 ( J ) Z = \ell _{v}^{q_{2}} (J) . Under suitable assumptions on Q , P \mathcal {Q}, \mathcal {P} , we will see that the relevant sufficient conditions are p 1 ≤ p 2 p_{1} \leq p_{2} and finiteness of a nested norm of the form \[ ‖ ( ‖ ( α i β j ⋅ v j / w i ) i ∈ I j ‖ ℓ t ) j ∈ J ‖ ℓ s , with I j = { i ∈ I : Q i ∩ P j ≠ ∅ } for j ∈ J . \left \Vert \left ( \left \Vert \left ( \alpha _{i} \beta _{j} \cdot v_{j} / w_{i} \right )_{ i \in I_{j}} \right \Vert _{\ell ^{t}} \right )_{j\in J} \right \Vert _{\ell ^{s}} \, , \quad \text {with} \quad I_{j} = \left \{ i \in I \,:\, Q_{i} \cap P_{j} \neq \varnothing \right \} \quad \text {for } j \in J \,. \] Like the sets I j I_j , the exponents t , s t, s and the weights α , β \alpha , \beta only depend on the quantities used to define the decomposition spaces. In a nutshell, in order to apply the embedding results presented in this article, no knowledge of Fourier analysis is required; instead, one only has to study the geometric properties of the involved coverings, so that one can decide the finiteness of certain sequence space norms defined in terms of the coverings. These sufficient criteria are quite sharp: For almost arbitrary coverings and certain ranges of p 1 , p 2 p_1, p_2 , our criteria yield a complete characterization for the existence of the embedding. The same holds for arbitrary values of p 1 , p 2 p_1, p_2 under more strict assumptions on the coverings. We also prove a rigidity result, namely that—for ( p 1 , q 1 ) ≠ ( 2 , 2 ) (p_{1}, q_{1}) \neq (2, 2) —two decomposition spaces D ( Q , L p 1 , ℓ w q 1 ) \mathcal {D}(\mathcal {Q}, L^{p_1}, \ell _w^{q_1}) and D ( P , L p 2 , ℓ v q 2 ) \mathcal {D}(\mathcal {P}, L^{p_2}, \ell _v^{q_2}) can only coincide if their “ingredients” are equivalent, that is, if p 1 = p 2 p_{1} = p_{2} and q 1 = q 2 q_{1} = q_{2} and if the coverings Q , P \mathcal {Q}, \mathcal {P} and the weights w , v w, v are equivalent in a suitable sense. The resulting embedding theory is illustrated by applications to α \alpha -modulation and Besov spaces. All known embedding results for these spaces are special cases of our approach; often, we improve considerably upon the state of the art.

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Structured, compactly supported Banach frame decompositions of decomposition spaces

Abstract: 7. Compactly supported Banach frames and atomic decompositions for α-modulation spaces 8. Compactly supported Banach frames and atomic decompositions for inhomogeneous Besov spaces .

Cited by 4 publications

References 72 publications

Construction of Banach frames and atomic decompositions of anisotropic Besov spaces

Construction of Banach frames and atomic decompositions of anisotropic Besov spaces

A framework of Besov–Triebel–Lizorkin type spaces via ball quasi-Banach function sequence spaces I: real-variable characterizations

Embeddings of Decomposition Spaces

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