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Complex manifolds --- Embedding theorems

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Let X be an abstract not necessarily compact orientable CR manifold of dimension 2n-1, ngeqslant 2, and let L^k be the k-th tensor power of a CR complex line bundle L over X. Given qin {0,1,ldots ,n-1}, let Box ^{(q)}_{b,k} be the Gaffney extension of Kohn Laplacian for (0,q) forms with values in L^k. For lambda geq 0, let Pi ^{(q)}_{k,leq lambda} :=E((-infty ,lambda ]), where E denotes the spectral measure of Box ^{(q)}_{b,k}. In this work, the author proves that Pi ^{(q)}_{k,leq k^{-N_0}}F^*_k, F_kPi ^{(q)}_{k,leq k^{-N_0}}F^*_k, N_0geq 1, admit asymptotic expansions with respect to k on the non-degenerate part of the characteristic manifold of Box ^{(q)}_{b,k}, where F_k is some kind of microlocal cut-off function. Moreover, we show that F_kPi ^{(q)}_{k,leq 0}F^*_k admits a full asymptotic expansion with respect to k if Box ^{(q)}_{b,k} has small spectral gap property with respect to F_k and Pi^{(q)}_{k,leq 0} is k-negligible away the diagonal with respect to F_k. By using these asymptotics, the authors establish almost Kodaira embedding theorems on CR manifolds and Kodaira embedding theorems on CR manifolds with transversal CR S^1 action.

Embedding theorems. --- CR submanifolds. --- Manifolds (Mathematics) --- Embeddings (Mathematics) --- Kernel functions. --- Asymptotic expansions.

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Partial differential equations --- 51 --- Differential equations, Elliptic --- -Embedding theorems --- Imbedding theorems --- Theorems, Embedding --- Theorems, Imbedding --- Embeddings (Mathematics) --- Elliptic differential equations --- Elliptic partial differential equations --- Linear elliptic differential equations --- Differential equations, Linear --- Differential equations, Partial --- Mathematics --- Numerical solutions --- 51 Mathematics --- Embedding theorems --- Equations aux derivees partielles elliptiques --- Espaces de sobolev --- Methodes variationnelles --- Problemes aux limites

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51 --- Differential equations, Partial --- -Embedding theorems --- Sobolev spaces --- Spectral theory (Mathematics) --- Functional analysis --- Hilbert space --- Measure theory --- Transformations (Mathematics) --- Spaces, Sobolev --- Function spaces --- Imbedding theorems --- Theorems, Embedding --- Theorems, Imbedding --- Embeddings (Mathematics) --- Partial differential equations --- Mathematics --- Numerical solutions --- 51 Mathematics --- Embedding theorems --- Numerical analysis --- Espaces fonctionnels --- Function spaces. --- Espaces de sobolev

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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 [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"--

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.