Boundary values of cohomology classes as hyperfunctions

Abstract

This work defines the boundary value of a cohomology class of degree q (0 ≤ q ≤ m -1) valued in the sheaf of germs of holomorphic functions, in a wedge whose edge lies on an arbitrary m-dimensional totally real (smooth) submanifold X of Cm and whose directing cone has its singular homology in dimension q generated by one q-cycle c ⊂ Sm - 1. The boundary value is defined as a hyperfunction in X. After “microlocalizing” about the cycle, it is equivalent to use the Dolbeaurt or the Čech realizations of the cohomology. Provided the cone generated by c bounds a convex cone the boundary value map is injective. The corresponding spaces of (germs of) hyperfunctions are characterized by their hypo-analytic wave-front set. Locally, every hyperfunction solution of the wave equation in Rm (m ≥ 3) is the boundary value of a cohomology class of degre m- 2 in cones about deleted hyperplanes that do not intersect the light cone. © 1995 Academic Press Limited.