Second Microlocalization at the Boundary and Microhyperbolicity

The purpose of this paper is to construct the "sheaf" of 2-hyperfunctions at the boundary along an involutive submanifold and to generalize the notion of microhyperbolicity at the boundary. Let M be a real analytic manifold, X a complexification of M, and let Q be an open subset of M with Cw-boundary N. Let Fbe a conic involutive submanifold of T^X which intersects transversally to N x T%f X with regular involutive intersection. Then we define the complex of ^j-Modules $2$x °f 2-hyperfunctions at the boundary along V, which appears to be a useful tool in studying noncharacteristic boundary value problems. Remark that the complex ^Q\X was first introduced by P. Schapira [S 3] for the microlocal study of boundary value problems. Next we introduce the notion of O-F-hyperboli city of a system M of microdifferential equations and prove that it implies "propagation of zeros up to the boundary" of cohomology groups of the complex Rjf#mSx(Jt, ^2Q\x}- This implies in particular "D-regularity" of M in the sense of [S 3].