Edge-of-the-wedge type theorems for hyperfunction solutions
Let \(M\) be a real analytic manifold and \(N\) its closed submanifold of codimension \(d\geq 1\). We take \(X\) (resp., \(Y)\) as a complexification of \(M\) (resp., \(N\) in \(X) \), and assume here that \({\mathcal M}\) is a coherent module over the sheaf of ring \({\mathcal D}_X\) of holomorphic differential operators on \(X\) and that \(Y\) is noncharacteristic for \({\mathcal M}\). The author examines the very important problem of receiving sufficient conditions on the module \({\mathcal M}\) for the vanishing of the local cohomologies of the hyperfunction solution complex of \( {\mathcal M}\): \[ H^j \mu_NR\Hom_{{\mathcal D}_X} ({\mathcal M}, {\mathcal B}_M) \simeq 0 \quad \text{for} \quad j<d, \] where \(\mu_N\) is the Sato's microlocalization functor along \(N\). This problem generalizes such classical facts as the Holmgren's theorem and the abstract edge-of-the-wedge theorem of Martineau and Kashiwara, and, on the other hand, many concrete results can be deduced from this type of vanishing theorems on cohomologies. Note that M. Kashiwara and T. Kawai earlier discussed the analogous question with the additional supposition of the ellipticity of \({\mathcal M}\), but in the present paper this result of Kashiwara-Kawai is extended with the help of the theory of bimicrolocalization to the cases where the solutions are not necessarily real analytic. So, in this paper a class of microlocally hyperbolic systems is introduced, and the author proves that for such systems a local Bochner-type extension theorem on hyperfunction solutions holds. Besides, the case of partially elliptic systems of Bony-Schapira type is considered. All this by combining of the author's approach with the ideas of other mathematicians (N. Tose, J.-M. Delort etc.), allows to get both the well-known results, generalizing essentially some of them, and many examples of systems that cannot be treated by Kashiwara-Kawai's theory. Also it should be marked that this paper of K. Takeuchi contains an extensive bibliography.
scientific article; zbMATH DE number 1054078
scientific article