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Analytic-Gevrey hypoellipticity for a class of pseudo-differential operators with multiple characteristics. (English) Zbl 0704.35035

The paper concerns pseudo-differential operators of the form \(P=A^ m+lower\) order terms, \(m\geq 2\), where A is classically analytic of complex valued principal type. The authors prove that the (non) hypoellipticity of A implies the (non) hypoellipticity of P in the analytic class and in the Gevrey classes \({\mathbb{G}}^ s\) for \(1<s<m/(m- 1)\), independently of the lower order terms.
The lower order terms may have influence when arguing in \(C^{\infty}\) or in \({\mathbb{G}}^ s\) for \(s\geq m/(m-1)\), as shown in the second part of the paper by means of model operators in \({\mathbb{R}}^ 2\) of the type \[ P=(D_ x+ix^{2k}D_ y)^ m+\sum c_{\alpha \beta \gamma}x^{\alpha}D^{\beta}_ xD^{\gamma}_ y \] where the sum is finite with \(\beta +\gamma \leq m-1\); such operators P are always analytically hypoelliptic, but not \(C^{\infty}\) hypoelliptic under suitable assumptions on \(c_{\alpha \beta \gamma}\).
Reviewer: L.Rodino

MSC:

35H10 Hypoelliptic equations
35S05 Pseudodifferential operators as generalizations of partial differential operators
35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
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