LEADER 06055nam 22005653 450 001 9910915800203321 005 20240723140218.0 010 $a1-4704-7542-1 035 $a(MiAaPQ)EBC30671913 035 $a(Au-PeEL)EBL30671913 035 $a(PPN)272106690 035 $a(CKB)27902414000041 035 $a(EXLCZ)9927902414000041 100 $a20230804d2023 uy 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aEmbeddings of Decomposition Spaces 205 $a1st ed. 210 1$aProvidence :$cAmerican Mathematical Society,$d2023. 210 4$dİ2023. 215 $a1 online resource (268 pages) 225 1 $aMemoirs of the American Mathematical Society Series ;$vv.287 311 08$aPrint version: Voigtlaender, Felix Embeddings of Decomposition Spaces Providence : American Mathematical Society,c2023 9781470459901 327 $aCover -- 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 ?= ? -- 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. 330 $a"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"--$cProvided by publisher. 410 0$aMemoirs of the American Mathematical Society Series 606 $aDecomposition (Mathematics) 606 $aHarmonic analysis 606 $aHarmonic analysis on Euclidean spaces -- Harmonic analysis in several variables -- Function spaces arising in harmonic analysis$2msc 606 $aFunctional analysis -- Linear function spaces and their duals -- Banach spaces of continuous, differentiable or analytic functions$2msc 606 $aFunctional analysis -- Linear function spaces and their duals -- Sobolev spaces and other spaces of ``smooth'' functions, embedding theorems, trace theorems$2msc 615 0$aDecomposition (Mathematics) 615 0$aHarmonic analysis. 615 7$aHarmonic analysis on Euclidean spaces -- Harmonic analysis in several variables -- Function spaces arising in harmonic analysis. 615 7$aFunctional analysis -- Linear function spaces and their duals -- Banach spaces of continuous, differentiable or analytic functions. 615 7$aFunctional analysis -- Linear function spaces and their duals -- Sobolev spaces and other spaces of ``smooth'' functions, embedding theorems, trace theorems. 676 $a518/.2 676 $a518.2 686 $a42B35$a46E15$a46E35$2msc 700 $aVoigtlaender$b Felix$01779869 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910915800203321 996 $aEmbeddings of Decomposition Spaces$94303467 997 $aUNINA