Liftings for ultra-modulation spaces, and one-parameter groups of Gevrey-type pseudo-differential operators. (English) Zbl 1442.35573
Summary: We deduce one-parameter group properties for pseudo-differential operators \(\text{Op}(a)\), where \(a\) belongs to the class \(\Gamma_\ast^{(\omega_0)}\) of certain Gevrey symbols. We use this to show that there are pseudo-differential operators \(\text{Op}(a)\) and \(\text{Op}(b)\) which are inverses to each other, where \(a \in \Gamma_\ast^{(\omega_0)}\) and \(b \in \Gamma_\ast^{(1 / \omega_0)}\). We apply these results to deduce lifting property for modulation spaces and construct explicit isomorphisms between them. For each weight functions \(\omega, \omega_0\) moderated by GRS submultiplicative weights, we prove that the Toeplitz operator (or localization operator) \(\text{Tp}(\omega_0)\) is an isomorphism from \(M_{(\omega)}^{p, q}\) to \(M_{(\omega / \omega_0)}^{p, q}\) for every \(p, q \in (0, \infty]\).
MSC:
| 35S05 | Pseudodifferential operators as generalizations of partial differential operators |
| 47D06 | One-parameter semigroups and linear evolution equations |
| 46B03 | Isomorphic theory (including renorming) of Banach spaces |
| 42B35 | Function spaces arising in harmonic analysis |
| 47B35 | Toeplitz operators, Hankel operators, Wiener-Hopf operators |
| 46F05 | Topological linear spaces of test functions, distributions and ultradistributions |
| 46G15 | Functional analytic lifting theory |
| 47L80 | Algebras of specific types of operators (Toeplitz, integral, pseudodifferential, etc.) |
Keywords:
Gelfand-Shilov spaces; Banach function spaces; ultra-distributions; confinement of symbols; comparable symbols; symbol-valued evolution equations; Toeplitz operatorsReferences:
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