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Opérateurs hypoelliptiques dans des espaces de Gevrey. (French) Zbl 0551.35085

In this paper the author proves the following result. Let P(x,D) be a differential operator of order m in an open subset X of \({\mathbb{R}}^ n\) with \(G^ r(X)\) \((r>1)\) coefficients, where \(G^ r(X)\) means the r-th Gevrey class in X. Assume that there exist \(\rho\),\(\delta\), \(0\leq \delta <\rho \leq 1\), such that the estimates \[ (1)\quad | D_ x^{\alpha}\partial^{\beta}_{\xi}P(x,\xi)| \leq C_ KA_ k^{| \alpha |}(\alpha !)^ r<\xi >^{\delta | \alpha | -\rho | \beta |},\quad (2)\quad | P^{-1}(x,\xi)| \leq C_ K(1+| \xi |)^{\tilde m}K \] hold if \(x\in K\), \(| \xi | \geq R_ K\), for each compact subset K of \(\Omega\). Then the following holds if \(d=r/(\rho -\delta)\) and \(\omega\) is an open subset of X: If \(u\in G'{}^ d(X)\) (the space of Gevrey ultradistributions of order d in X) and \(Pu\in G^ s(\omega)\) for some \(s\geq d\), then \(u\in G^ s(\omega)\). This result is proved by a construction of a parametrix based on a variation on the calculus of analytic pseudodifferential operators of Boutet de Monvel and Krée.
Reviewer: P.Godin

MSC:

35S05 Pseudodifferential operators as generalizations of partial differential operators
65H10 Numerical computation of solutions to systems of equations
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