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

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