Compactness Properties for Modulation Spaces

Abstract

We prove that if ω 1 and ω 2 are moderate weights and B is a suitable (quasi-)Banach function space, then a necessary and sufficient condition for the embedding i :between two modulation spaces to be compact is that the quotient ω 2 /ω 1 vanishes at infinity. Moreover we show, that the boundedness of ω 2 /ω 1 a necessary and sufficient condition for the previous embedding to be continuous. Germany

Continuity of Gevrey–Hörmander pseudo-differential operators on modulation spaces

“…M(ω, B) is the same as in Definition 1.5, and let φ ∈ Finally we recall the following result on completeness for M(ω, B). We refer to [34] for a proof of the first assertion and [22] for the second one.…”

Liftings for ultra-modulation spaces, and one-parameter groups of Gevrey-type pseudo-differential operators

“…We omit the proof since it follows by similar arguments as in the proof of Proposition 11.3.2 in [21]. (See also [36] for topological aspects of M (ω, B).) Proposition 1.20.…”

Uncertainty relations for a non-canonical phase-space noncommutative algebra

“…(2) weighted L 2 -spaces: If m(x, ω) = m(x) = 1 + |x| 2 s/2 with s ∈ R, then In the sequel, we shall need the following compact embedding theorem for modulation spaces, which was proved by Boggiatto and Toft [16] (see also [68]): where we wrote collectively z = (x, ω) ∈ R 2d .…”

“…( In the sequel, we shall need the following compact embedding theorem for modulation spaces, which was proved by Boggiatto and Toft [16] (see also [68]):…”