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Smooth tame Fréchet algebras and Lie groups of pseudodifferential operators. (English) Zbl 0763.47022

In order to use pseudodifferential operators in Nash-Moser methodology for the local production of smooth solutions to nonlinear partial differential equations it is necessary to have a smooth time Lie group structure in the sense of Hamilton for a subgroup of invertible elements of some large pseudodifferential algebra and a tame Fréchet space topology in this algebra.
The author introduces and studies these structures for the Cordes type algebra \(\text{OPS}_{gs}^{0,0}\), which consists of global zero-order pseudodifferential operators whose symbols are smooth functions on \(\mathbb{R}^{2n}\) and remain bounded after an arbitrary application of some phase space differential operators.

MSC:

47G30 Pseudodifferential operators
35S50 Paradifferential operators as generalizations of partial differential operators in context of PDEs
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