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

Abstract

We deduce one-parameter group properties for pseudo-differential operators Op(a), where a belongs to the class Γ (ω 0 ) * of certain Gevrey symbols. We use this to show that there are pseudo-differential operators Op(a) and Op(b) which are inverses to each others, where a ∈ ΓWe apply these results to deduce lifting property for modulation spaces and construct explicit isomorpisms between them. For each weight functions ω, ω 0 moderated by GRS submultiplicative weights, we prove that the Toeplitz operator (or localization operator) Tp(ω 0 ) is an isomorphism from M p,q(ω) onto M p,q(ω/ω 0 ) for every p, q ∈ (0, ∞].

The Weyl product on quasi-Banach modulation spaces

“…where Op A (a) is the A-indexed pseudodifferential operator with symbol a. This family of calculi contains the Weyl quantization as the special case A = 1 2 I. The sufficient conditions and the necessary conditions that we find extend results [7,23] where the same problem was studied for the narrower range of Lebesgue The Weyl product on quasi-Banach modulation spaces parameters [1, ∞].…”

“…We consider a family of pseudodifferential calculi parameterized by the real [3,36]. Let s ≥ 1 2 , let a ∈ S s (R 2d ) and let A ∈ M(d, R) be fixed. The pseudodifferential operator Op A (a) is the linear and continuous operator…”

The Weyl product on quasi-Banach modulation spaces

“…Denote the Gelfand-Shilov space of order 1/2 by S 1/2 , and the weighted modulation space with Lebesgue parameters p, q > 0 and with weight ω by M p,q (ω) . Then the map (a 1 , a 2 ) → a 1 #a 2 from S 1/2 (R 2d ) × S 1/2 (R 2d ) to S 1/2 (R 2d ) extends uniquely to a continuous map from M p 1 ,q 1 (ω 1 ) (R 2d ) × M p 2 ,q 2 (ω 2 ) (R 2d ) to M p 0 ,q 0 (ω 0 ) (R 2d ), and a 1 #a 2 M p 0 ,q 0 (ω 0 )…”

Compactness Properties for Modulation Spaces

“…Because of Proposition 1.5 (1) we are allowed to be rather imprecise concerning the choice of φ ∈ M r (v) \0 in (1.11). For instance let C > 0 be a constant and Ω be a subset of Σ ′ 1 .…”

“…Let ω = e −2| · |/h 0 , ω 0 = ω * e −| · | 2 /2 and let v = 1/ω 0 . Then [1,Proposition 1.6] shows that |D β ω 0 | h |β| β! e −2| · |/h 0 for every h > 0.…”