α-modulation spaces (I) scaling, embedding and algebraic properties

Abstract

First, we consider some fundamental properties including dual spaces, complex interpolations of α-modulation spaces M_p,q^s,α with 0 < p, q ≤ ∞. Next, necessary and sufficient conditions for the scaling property and the inclusions between α₁-modulation and α₂-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. In a subsequent paper, we will give some applications of α-modulation spaces to nonlinear dispersive wave equations.