Schapira, Pierre Front d’onde analytique au bord. II. (Analytic wave front set at the boundary. II). (French) Zbl 0638.58027 Sémin., Équations Dériv. Partielles 1985-1986, Exposé No. 13, 12 p. (1986). [For part I, cf. C. R. Acad. Sci., Paris, Sér. I 302, 383-386 (1986; Zbl 0592.58052).] Let M be a real analytic manifold, X a complexification of M, \(\Omega\) an open subset of M. Using the functor \(\mu\) hom we construct a sheaf \(C_{\Omega | X}\) on \(T^*X\), which extends the sheaf \(C_ M\) of Sato’s microfunctions above \(\Omega\), and permits us to define a new analytic wave front set \(SS_{\Omega}(u)\), when \(u\in B_ M(\Omega)\) is a hyperfunction on \(\Omega\). If \({\mathcal M}\) is a system of microdifferential equations and \(u\in Ext^ j_{E_ X}({\mathcal M},C_ M)\) we also introduce \(SS_{\Omega}^{{\mathcal M},j}(u)\), a closed subset of \(T^*X.\) Let now N be a submanifold of M of codimension d, Y the complexification of N in X, and let \(\omega\) be an open subset of N with \({\bar \omega}\subset {\bar \Omega}\). Under a topological assumption on \(\omega\) and \(\Omega\) we show how to define for any sheaf F on M a morphism (in the derived category of sheaves on M) of “boundary values”, \(b: R\Gamma_{\Omega}(F)_{| N}\to R\Gamma_{\omega}(F)\otimes \omega_{N| M}[d],\) and we apply this construction to the case where \(F=R Hom_{D_ X}({\mathcal M},B_ M)\). If Y is non-characteristic for \({\mathcal M}\), and \({\mathcal M}_ Y\) denotes the induced system by \({\mathcal M}\) on Y, we obtain a morphism \(R Hom_{D_ X}(M,\Gamma_{\Omega}(B_ M))_{| N}\to R Hom_{D_ Y}({\mathcal M}_ Y,\Gamma_{\omega}(B_ N))\) and we prove that if \(\rho\) and \({\bar \omega}\) denote the natural maps from \(Y\times T^*X\) to \(T^*Y\) and \(T^*X\) respectively, one has the inclusion \[ SS_{\omega}^{{\mathcal M}_{Y,j}}(b(u))\quad \subset \quad \rho {\bar \omega}^{-1} SS_{\Omega}^{{\mathcal M},j}(u), \] with equality when \(d=1\), \(\omega =N\). Finally we define the boundary value morphism in case Y is characteristic, but \({\mathcal M}\) is “regular” along Y. Cited in 5 ReviewsCited in 13 Documents MSC: 58J47 Propagation of singularities; initial value problems on manifolds Keywords:complexification; analytic wave front set; microdifferential equations Citations:Zbl 0592.58052 × Cite Format Result Cite Review PDF Full Text: Numdam EuDML