Continuity properties for modulation spaces, with applications to pseudo-differential calculus—I

Abstract

Let M p, q denote the modulation space with parameters p, q∈[1,∞]. If 1/ p 1+1/ p 2=1+1/ p 0 and 1/ q 1+1/ q 2=1/ q 0, then it is proved that M p 1,q 1 ∗M p 2,q 2 ⊂M p 0,q 0 . The result is used to get inclusions between modulation spaces, Besov spaces and Schatten classes in calculus of Ψdo (pseudo-differential operators), and to extend the definition of Toeplitz operators. We also discuss continuity of ambiguity functions and Ψdo in the framework of modulation spaces.