Embeddings of decomposition spaces. (English) Zbl 1525.42001
Memoirs of the American Mathematical Society 1426. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-5990-1/pbk; 978-1-4704-7542-0/ebook). vi, 255 p. (2023).
It is well known that decomposition spaces were first introduced by H. G. Feichtinger and P. Gröbner [Math. Nachr. 123, 97–120 (1985; Zbl 0586.46030)] to give a unifying framework for Besov spaces, modulation spaces, and certain Wiener amalgam spaces. Moreover, many smoothness spaces in harmonic analysis are decomposition spaces.
In this monograph, the author systematically develops a general and easy-to-use framework for proving and disproving the existence of embeddings between decomposition spaces. To be precise, the author establishes criteria that ensure the existence of embeddings between two decomposition spaces. It is worth pointing out that the sufficient criteria given in this monograph are sharp. Moreover, the embedding results for \(\alpha\)-modulation and Besov spaces obtained in the existing literature are special cases of the embedding theory given in this monograph. The embeddings theory developed in this monograph will have a wide range of potential applications in harmonic analysis and partial differential equations.
In this monograph, the author systematically develops a general and easy-to-use framework for proving and disproving the existence of embeddings between decomposition spaces. To be precise, the author establishes criteria that ensure the existence of embeddings between two decomposition spaces. It is worth pointing out that the sufficient criteria given in this monograph are sharp. Moreover, the embedding results for \(\alpha\)-modulation and Besov spaces obtained in the existing literature are special cases of the embedding theory given in this monograph. The embeddings theory developed in this monograph will have a wide range of potential applications in harmonic analysis and partial differential equations.
Reviewer: Sibei Yang (Lanzhou)
MSC:
| 42-02 | Research exposition (monographs, survey articles) pertaining to harmonic analysis on Euclidean spaces |
| 46-02 | Research exposition (monographs, survey articles) pertaining to functional analysis |
| 42B35 | Function spaces arising in harmonic analysis |
| 46E15 | Banach spaces of continuous, differentiable or analytic functions |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 42C15 | General harmonic expansions, frames |
| 42C40 | Nontrigonometric harmonic analysis involving wavelets and other special systems |
| 22D10 | Unitary representations of locally compact groups |
| 46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |
| 47B25 | Linear symmetric and selfadjoint operators (unbounded) |
| 41A17 | Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities) |
| 46B15 | Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces |
| 42B15 | Multipliers for harmonic analysis in several variables |
Keywords:
function spaces; smoothness spaces; decomposition spaces; embeddings; frequency coverings; Besov spaces; \( \alpha \)-modulation spaces; coorbit spacesCitations:
Zbl 0586.46030References:
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