Second microlocalization at the boundary and microhyperbolicity
Let \(M=R^ n\), \(\Omega =\{x\in M|\) \(x_ 1>0\}\). Let \(P=P(x,D)\) be a differential operator with analytic coefficients defined on M and assume that \(\{x_ 1=0\}\) is noncharacteristic for P. The authors generalize the notion of microhyperbolicity at the boundary and call it \(\Omega\)-V- hyperbolicity for an involutive submanifold V of \(T^*_ MX\). It is proved that, if P is \(\Omega\)-V-hyperbolic for some V, P is \(\Omega\)- regular in the sense of \textit{K. Kataoka} [J. Fac. Sci., Univ. Tokyo, Sect. I A 28, 31-56 (1981; Zbl 0459.35098)] and \textit{P. Schapira} [C. R. Acad. Sci., Paris, Sér. I 302, 383-386 (1986; Zbl 0592.58052)], i.e., the micro-analytic regularity of every hyperfunction solution to P defined on \(\Omega\) propagates up to the boundary. This result comprises as special cases \(\Omega\)-regularity of microhyperbolic operators [see \textit{K. Kataoka} (loc. cit.), \textit{P. Schapira}, Publ. Res. Inst. Math. Sci, Kyoto Univ. 12, Suppl., 441-453 (1977; Zbl 0378.35065)] and that of non-micro-characteristic operators [see \textit{P. Schapira}, Sém. Goulaouic-Schwartz 1976-1977, Equat. Deriv. Part. Anal. Fonct., Exp. No.IX, 10 p. (1977; Zbl 0361.35062)].
scientific article; zbMATH DE number 4171455
scientific article