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Embeddings of Decomposition Spaces - Felix Voigtlaender

Embeddings of Decomposition Spaces

Buch | Softcover
253 Seiten
2023
American Mathematical Society (Verlag)
978-1-4704-5990-1 (ISBN)
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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? In a nutshell, in order to apply the embedding results presented in this article, no knowledge of Fourier analysis is required.
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, Lp, Y ) is determined by a covering Q = (Qi)i?I of the frequency domain, an integrability exponent p, and a sequence space Y ? CI . Given these ingredients, the decomposition space norm of a distribution g is defined as g D(Q,Lp,Y ) = F?1 (?i · g ) Lp i?I Y , where (?i)i?I is a suitable partition of unity for Q.

We establish readily verifiable criteria which ensure the existence of a continuous inclusion ("an embedding") D(Q, Lp1 , Y ) ? D(P, Lp2 , Z), mostly concentrating on the case where Y = q1 w (I) and Z = q2 v (J). Under suitable assumptions on Q,P, we will see that the relevant sufficient conditions are p1 ? p2 and finiteness of a nested norm of the form (?i?j · vj/wi)i?Ij t j?J s , with Ij = {i ? I : Qi ? Pj = ?} for j ? J .

Like the sets Ij , the exponents t, s and the weights ?, ? 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.

Felix Voigtlaender, Catholic University of Eichstatt-Ingolstadt, Germany.

Erscheinungsdatum
Reihe/Serie Memoirs of the American Mathematical Society ; Volume: 287 Number: 1426
Verlagsort Providence
Sprache englisch
Maße 178 x 254 mm
Themenwelt Mathematik / Informatik Mathematik Analysis
ISBN-10 1-4704-5990-6 / 1470459906
ISBN-13 978-1-4704-5990-1 / 9781470459901
Zustand Neuware
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