Fundamental solution for a degenerate hyperbolic operator in Gevrey classes (Q1328875)

The authors consider degenerate hyperbolic operators modelled on \[ L = D^ 2_ t - t^{2p} x^{2q} D^ 2_ x + at^ r x^ sD_ x. \] The Cauchy problem for \(t = 0\) is well posed in the Gevrey classes \(G^ t\) for suitable indices \(t\) depending on \(p,q,r,s\). A fundamental solution \(E(t,s)\) is here constructed, of the form \[ E(t,s) = \sum_ \pm I_ \pm E_ \pm + E_ 0 \] where \(I_ \pm\) are classical Fourier integral operators, and \(E_ \pm\), \(E_ 0\) are infinite order pseudo- differential operators acting on Gevrey ultra-distributions. By using preceding results of \textit{K. Shinkai} [Commun. Partial Differ. Equations 7, 581-607 (1982; Zbl 0501.35050)], a precise geometric bound is deduced for the Gevrey wave front set of the solutions. The proofs are based on a general calculus for infinite order operators and on a reduction to a perfectly diagonalized system.