Hyperfunctions with Real Analytic Parameters and Continuation of Solutions of Systems of Partial Differential Equations

The purpose of this paper is to introduce a "sheaf" of hyperfunctions with Cω-parameters at the boundary, to study its behavior under the trace morphism, and to state some criteria on extension of solutions of systems of P.D.E. across singular sets of codimension ≥ 1. Let M be a Cω-manifold, X a complexification of M, Ω an open set with Cω-boundary N = ∂Ω, Y a complexilication of N, V a closed conic regular involutive submanifold of T*MX\ T*YX transversal to N × MT*MX with regular involutjve intersection. Let be a X-module for which Y is non-characteristic. We introduce a complex aM X = a, VM X whose 0th cohomology consists of those hyperfunctions on M which depend real analytically on the variables transversal to the leaves of V. We also introduce a new complex aΩ X = a, VΩ X whose main feature is that traces on N of H0( a, VΩ X)-so1utions of belong to H0( a, V′N Y)). (Here V′ = ρ −1(V) with T*Y ← ρY × XT*X → T*X.) We are then able to state in Section 3 several principles on continuation of hyperfunction solutions with Cω-parameters (resp. real analytic solutions) to systems across a subset S contained in a non-characteristic boundary N. The method (inspired by [Kan2]) consists in proving that under some hypotheses on non-microcharacteristicity of N, ΓΩ(H0( aM X))-solution f of belongs automatically to H0( aΩ X) (Ω = Ω± denoting the components of M \ N), and therefore their traces γ ± (f) on N satisfy SSγ ± (f) ∩ V′ = Ø. The extendability of f across S is then an immediate consequence of the propagation 0 for γ+ (f) − γ−(f) from N \ S up to N.