Embeddings of Decomposition Spaces

• Autore: Voigtlaender Felix
• Pubblicazione: Providence : , : American Mathematical Society, , 2023; ©2023
• Edizione: 1st ed.
• Descrizione fisica: 1 online resource (268 pages)
• Disciplina: 518/.2; 518.2
• Soggetto topico: Decomposition (Mathematics); Harmonic analysis; Harmonic analysis on Euclidean spaces -- Harmonic analysis in several variables -- Function spaces arising in harmonic analysis; Functional analysis -- Linear function spaces and their duals -- Banach spaces of continuous, differentiable or analytic functions; Functional analysis -- Linear function spaces and their duals -- Sobolev spaces and other spaces of ``smooth'' functions, embedding theorems, trace theorems
• Classificazione: 42B3546E1546E35
• Nota di contenuto: Cover -- Title page -- Chapter 1. Introduction -- 1.1. Motivation and related work -- 1.2. Notation -- A comment on the writing style and the length of the paper -- 1.3. Overview of new results -- 1.4. Structure of the paper -- Acknowledgments -- Chapter 2. Different classes of coverings and their relations -- 2.1. Admissible and (semi/almost)-structured coverings -- 2.2. Relations between coverings -- Chapter 3. (Fourier-side) decomposition spaces -- 3.1. Convolution relations for ^{ }, ∈(0,1) -- 3.2. Definition of decomposition spaces -- 3.3. Well-definedness of decomposition spaces -- 3.4. Completeness of decomposition spaces -- Chapter 4. Nested sequence spaces -- Chapter 5. Sufficient conditions for embeddings -- Chapter 6. Necessary conditions for embeddings -- 6.1. Elementary necessary conditions -- 6.2. Coincidence of decomposition spaces -- 6.3. Improved necessary conditions -- 6.4. Further necessary conditions in case of ₁= ₂ -- 6.5. Complete characterizations for relatively moderate coverings -- Chapter 7. An overview of the derived embedding results -- 7.1. A collection of readily applicable embedding results -- 7.2. Embeddings between decomposition spaces: A user's guide -- Chapter 8. Decomposition spaces as spaces of tempered distributions -- Chapter 9. Applications -- 9.1. Embeddings of -modulation spaces -- 9.2. Embeddings between -modulation spaces and Besov spaces -- 9.3. Embeddings between homogeneous and inhomogeneous Besov spaces -- Bibliography -- Back Cover.
• Sommario/riassunto: "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 [equation] is determined by a covering [equation] of the frequency domain, an integrability exponent p, and a sequence space [equation]. Given these ingredients, the decomposition space norm of a distribution g is defined as [equation] is a suitable partition of unity for Q. We establish readily verifiable criteria which ensure the existence of a continuous inclusion ("an embedding") [equation], mostly concentrating on the case where [equation]. Under suitable assumptions on Q, P, we will see that the relevant sufficient conditions are [equation] and finiteness of a nested norm of the form [equation]. Like the sets Ij, the exponents t, s and the weights [omega], [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 p1, p2, our criteria yield a complete characterization for the existence of the embedding. The same holds for arbitrary values of p1, p2 under more strict assumptions on the coverings. We also prove a rigidity result, namely that--[equation]--two decomposition spaces [equation] and [equation] can only coincide if their "ingredients" are equivalent, that is, if [equation] and [equation] and if the coverings Q,P and the weights w, v are equivalent in a suitable sense. The resulting embedding theory is illustrated by applications to [omega]-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"--
• Titolo autorizzato: Embeddings of Decomposition Spaces
• ISBN: 1-4704-7542-1
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 9910915800203321
• Lo trovi qui: Univ. Federico II
• Serie: Memoirs of the American Mathematical Society Series