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Product (α1, α2)-modulation spaces

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Abstract

We study fundamental properties of product (α1, α2)-modulation spaces built by (α1, α2)-coverings of ℝn1 × ℝn2. Precisely we prove embedding theorems between these spaces with different parameters and other classical spaces. Furthermore, we specify their duals. The characterization of product modulation spaces via the short time Fourier transform is also obtained. Families of tight frames are constructed and discrete representations in terms of corresponding sequence spaces are derived. Fourier multipliers are studied and as applications we extract lifting properties and the identification of our spaces with (fractional) Sobolev spaces with mixed smoothness.

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Acknowledgements

The first author was supported by University of Cyprus and New Function Spaces in Harmonic Analysis and Their Applications in Statistics (Individual Grant). The first author expresses her gratitude to Professor George Kyriazis, for his fruitful discussions, his encouragement and for the problems that he brought to her attention. The authors thank Professor Hans G. Feichtinger for his several suggestions and remarks that improved significantly the manuscript. Finally, the authors feel grateful to the three anonymous referees, for their detailed reports.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, National University of Ireland, Maynooth, W23 F2K8, Ireland

    Galatia Cleanthous

  2. Faculty of Engineering, Mathematics and Science, Trinity College of Dublin, Dublin, D02 PN40, Ireland

    Athanasios G. Georgiadis

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  1. Galatia Cleanthous
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  2. Athanasios G. Georgiadis
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Correspondence to Athanasios G. Georgiadis.

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Dedicated to the Memory of Michel Marias

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Cleanthous, G., Georgiadis, A.G. Product (α1, α2)-modulation spaces. Sci. China Math. 65, 1599–1640 (2022). https://doi.org/10.1007/s11425-021-1923-7

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