Pseudo-differential operators of infinite order on \({\mathcal D}_{L^2}^{\{\sigma\}}({\mathcal D}_{L^2}^{\{\sigma\}'})\) and applications to the Cauchy problem for some elementary operators (Q810773)

The author studies the well-posedness in a space \(D_{L^ 2}^{\{\sigma \}'}\) of ultradistributions for a Cauchy problem of operators of the form \(P=\partial_ t+a(t,x,D_ x)\), where the symbol of the pseudodifferential operator \(a(t,x,\xi)\in C^{\infty}\) belongs to \(S^ p_{1,0}\) with \(p\in [0,1)\), analytic in \(\xi\) and in some Gevrey class with respect to x. He proves that the (forward) Cauchy problem with the initial datum at \(t=s\) is well-posed if a condition \[ \liminf_{\rho \to +\infty}\rho^{-1/\sigma}\int^{t}_{s}Re a(t',x,\rho \eta)dt'\geq 0,\quad \forall t\geq s \] holds uniformly with respect to \((x,\eta)\in R^ n\times S_{n-1}\). It is known by \textit{S. Mizohata} [On the Cauchy problem (Acad. Press, Bejing 1985; Zbl 0616.35002)] and \textit{K. Taniguchi} [Math. Jap. 30, 719-741 (1985; Zbl 0584.35104)] that the Cauchy problem is well-posed in the case where \(\sigma\leq 1/p\) (since the condition is empty in this case). This sufficient condition is motivated by the result concerning a necessary condition for an analogous Cauchy problem studied by \textit{S. Mizohata} [Proc. Taniguchi Int. Symp., Katata and Kyoto, 1984, 193-233 (1986; Zbl 0665.35006)].