On hypoellipticity for a class of pseudo-differential operators (Q5939525)

The author proves a result of hypoellipticity. The hypotheses of the main theorem are very technical, and we cannot describe them here; the author gives a series of interesting examples, such as NEWLINE\[NEWLINEa(x)(-\Delta)^M +(-\Delta)^NNEWLINE\]NEWLINE and NEWLINE\[NEWLINE\sum^n_{k=1} a_k(x)D^2_{x_k}+1,NEWLINE\]NEWLINE which turn out to be hypoelliptic under suitable assumptions on \(a,a_k,M,N\). Basic tool in the proof is an a priori estimate in weighted Sobolev spaces, cf. [\textit{K. Kajitani} and \textit{S. Wakabayashi} Bull. Sci. Math., II. Sér., 115, No. 4, 397-449 (1991; Zbl 0758.35097)].