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Pseudodifferential operators of infinite order and Gevrey classes. (English) Zbl 0601.35110

Let \(\Omega\) be an open set in \({\mathbb{R}}^ n\) and let \(\theta\),p,\(\delta\) be real numbers such that \(\theta >1\), \(0\leq \delta <\rho \leq 1\), \(\theta\rho\geq 1\). Then let \(S_{\rho,\delta}^{\infty,\theta}(\Omega)\) be a space of all functions \(p\in C^{\infty}(\Omega \times {\mathbb{R}}^ n)\) satisfying the following condition: for every compact subset \(K\subset \Omega\) there exist constant \(C>0\) and \(B\geq 0\) and for every \(\epsilon >0\) there exists a constant \(c_{\epsilon}\) such that \[ \sup_{x\in K}| D^{\alpha}_{\xi} D_ x^{\beta}p(x,\xi)| \leq c_{\epsilon} C^{| \alpha +\beta |} \alpha ! \beta !^{\theta (\rho -\delta)}(1+| \xi |)^{-\rho | \alpha | +\delta | \beta |} \exp (\epsilon | \xi |)^{1/\theta} \] for every \(\alpha,\beta \in {\mathbb{Z}}_+^ n\) and for every \(\xi \in {\mathbb{R}}^ n\) such that \(| \xi | \geq B| \alpha |^{\theta}\). The paper studies the operator \[ Pu(x)=(2\pi)^{-n}\int \exp (i<x,\xi >)p(x,\xi)\hat u(\xi)d\xi,\quad u\in G_ 0^{(\theta)}(\Omega),\quad (p\in S^{\infty,\theta}_{\rho,\delta}(\Omega)) \] it appears that the operator P is of infinite order, acts on spaces of Gevrey functions \(G_ 0^{(\theta)}(\Omega)\) and their duals and is Gevrey-pseudolocal. For the corresponding symbols the classical symbolic calculus is developed and the existence of a parametrix is proved. This makes it possible to obtain a result of propagation of Gevrey regularity for operators with ”hypoellipticity condition”.
Reviewer: J.Kostarčuk

MSC:

35S05 Pseudodifferential operators as generalizations of partial differential operators
35S15 Boundary value problems for PDEs with pseudodifferential operators
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