Liess, Otto; Rodino, Luigi Inhomogeneous Gevrey classes and related pseudodifferential operators. (English) Zbl 0557.35131 Boll. Unione Mat. Ital., VI. Ser., C, Anal. Funz. Appl. 3, 233-323 (1984). The purpose of this paper is to develop the theory of pseudodifferential operators which are available to the investigation of the hypoellipticity in Gevrey classes or inhomogeneous Gevrey classes. The paper is composed of four parts. The first part is devoted to the investigation of the inhomogeneous Gevrey classes \(G_{\phi}\), which is defined through the weight function \(\phi\) (\(\xi)\). Such classes have been considered previously, starting from another definition. The authors use here a definition which is well adopted to the study of operators with variable coefficients. In the second part they give definitions of pseudo-differential operators acting on the function classes defined in the first part, and develop a symbolic calculus. They adopt the definition of symbol classes, similar to the ones of R. Beals. In the third part they give propositions related to the pseudo-local property of the pseudodifferential operators in the sense of the above Gevrey classes, and some fundamental properties of pseudodifferential operators. In the last part they construct parametrices for some operators (see, for example, Theorem 4.3.1) and give some results on the hypoellipticity in the inhomogeneous spaces. Reviewer: M.Nagase Cited in 4 ReviewsCited in 19 Documents MSC: 35S05 Pseudodifferential operators as generalizations of partial differential operators 65H10 Numerical computation of solutions to systems of equations 35A30 Geometric theory, characteristics, transformations in context of PDEs Keywords:hypoellipticity; Gevrey classes; symbolic calculus; pseudo-local property; parametrices × Cite Format Result Cite Review PDF