MIXED-NORM α-MODULATION SPACES

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 ((p) over right arrow -BAPU). We prove that the definition is independent of the (p) over right arrow -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.