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$α$-Modulation Spaces (I)

First, we consider some fundamental properties including dual spaces, complex interpolations of $α$-modulation spaces $M^{s,α}_{p,q}$ with $0<p,q \le \infty$. Next, necessary and sufficient conditions for the scaling property and the inclusions between $α_1$-modulation and $α_2$-modulation spaces are obtained. Finally, we give some criteria for $α$-modulation spaces constituting multiplication algebra. As a by-product, we show that there exists an $α$-modulation space which is not an interpolation space between modulation and Besov spaces.