Gevrey class solutions of hypoelliptic equations (Q5942138)

The authors consider a linear partial differential operator \(P(x,D)\) hypoelliptic of constant strength for \(x\in\Omega \subset\mathbb{R}^n\), and assume that the symbol \[ P(x,\eta)= \sum_{\alpha\in N}c_\alpha (x)\eta^\alpha \] is multi-quasi-elliptic: \[ 1+\bigl |P(x,\eta)\bigr |\geq c\sum_{\alpha\in N}|\eta^\alpha |, \] where the Newton polyhedron \(N\) is convex and complete. Basing on \(N\), one may define the multi-anisotropic Gevrey classes \(G^N\); if the coefficients \(c_\alpha(x)\) are constant or analytic then \(P(x,D)\) is \(G\)-hypoelliptic [see, for example, \textit{L. Zanghirati}, Boll. Unione Mat. Ital., Suppl. 1, 177-195 (1980; Zbl 0447.35023)]. The new interesting result of the present paper is to obtain \(G^N\)-hypoellipticity under minimal assumptions on the coefficients \(c_\alpha(x)\); namely the authors assume \(c_\alpha (x)\in G^M\), where \(M\) is the complementary polyhedron of \(N\), according to a suitable definition.