Mixed-norm 𝛼-modulation spaces

Cleanthous, Galatia; Georgiadis, Athanasios

doi:10.1090/tran/8023

Mixed-norm $\alpha$-modulation spaces
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by Galatia Cleanthous and Athanasios G. Georgiadis PDF

Trans. Amer. Math. Soc. 373 (2020), 3323-3356 Request permission

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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