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Transactions of the American Mathematical Society

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Mixed-norm $ \alpha$-modulation spaces


Authors: Galatia Cleanthous and Athanasios G. Georgiadis
Journal: Trans. Amer. Math. Soc. 373 (2020), 3323-3356
MSC (2010): Primary 42B35, 46E35, 42B25, 42B15; Secondary 46F10
DOI: https://doi.org/10.1090/tran/8023
Published electronically: February 19, 2020
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Abstract: Mixed-norm $ \alpha $-modulation spaces are introduced and their properties are explored. Precisely we define $ \alpha $-modulation spaces with mixed Lebesgue norms, based on mixed-norm bounded admissible partitions of unity ($ \vec {p}$-BAPU). We prove that the definition is independent of the $ \vec {p}$-BAPU and that the lifting property holds for the newly introduced spaces. We provide embedding theorems between the new spaces, we find their duals, and we study the boundedness of Fourier multipliers under general mixed-norm conditions. In addition we obtain a discrete decomposition based on frame systems whose construction combines the operations used to construct Gabor and wavelet systems. Some properties of mixed Lebesgue spaces, of independent interest, are proved as well.


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Additional Information

Galatia Cleanthous
Affiliation: School of Mathematics, Statistics and Physics, Newcastle University, Newcastle upon Tyne, United Kingdom
Address at time of publication: Faculty of Engineering, Mathematics and Science, Trinity College of Dublin, Dublin, Ireland
Email: cleanthg@tcd.ie

Athanasios G. Georgiadis
Affiliation: Department of Mathematics and Statistics, University of Cyprus, 1678 Nicosia, Cyprus
Address at time of publication: Faculty of Engineering, Mathematics and Science, Trinity College of Dublin, Dublin, Ireland
Email: georgiaa@tcd.ie

DOI: https://doi.org/10.1090/tran/8023
Keywords: $\alpha$-modulation spaces, Besov spaces, mixed-norms, equivalent norms, embeddings, dual spaces, Fourier multipliers, frames, discrete decomposition, sequence spaces.
Received by editor(s): April 20, 2018
Received by editor(s) in revised form: March 28, 2019, May 28, 2019, and August 18, 2019
Published electronically: February 19, 2020
Article copyright: © Copyright 2020 American Mathematical Society