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

This is the authors' abstract. We don't add key points for this paper.

We deduce one-parameter group properties for pseudo-differential operators [Formula: see text], where [Formula: see text] belongs to the class [Formula: see text] of certain Gevrey symbols. We use this to show that there are pseudo-differential operators [Formula: see text] and [Formula: see text] which are inverses to each other, where [Formula: see text] and [Formula: see text]. We apply these results to deduce lifting property for modulation spaces and construct explicit isomorphisms between them. For each weight functions [Formula: see text] moderated by GRS submultiplicative weights, we prove that the Toeplitz operator (or localization operator) [Formula: see text] is an isomorphism from [Formula: see text] to [Formula: see text] for every [Formula: see text].