Iterates and hypoellipticity of partial differential operators on non-quasianalytic classes
The author studies the problem of the iterates, cf. [\textit{H. Komatsu}, Proc. Japan Acad. 36, 90--93 (1960; Zbl 0144.35505); \textit{T. Kotake} and \textit{M. S. Narasimhan}, Bull. Soc. Math. Fr. 90, 449--471 (1962; Zbl 0104.32503)], in the frame of the ultradifferentiable classes of \textit{R. W. Braun}, \textit{R. Meise} and \textit{B. A. Taylor} [Result. Math. 17, No. 3--4, 206--237 (1990; Zbl 0735.46022)] defined in terms of a weight function \(\lambda\). Operators under consideration here are partial differential operators with constant coefficients \(P(D)\). A detailed study is given for the space of the \(\lambda\)-ultradifferentiable vectors, i.e. \(C^\infty\) functions \(f\) such that \(P^N(D)f\), \(N= 1,2,\dots\), can be estimated in terms of \(N\) and \(\lambda\). In particular, under an additional assumption on the weight \(\lambda\), the author proves that the space of the \(\lambda\)-ultradifferentiable vectors coincides with the space of the \(\lambda\)-ultradifferentiable functions if and only if \(P(D)\) is elliptic.
scientific article; zbMATH DE number 5808429
scientific article