Resolutions by hyperfunctions of sheaves of solutions of differential equations with constant coefficients

Recently R. HARVEY [-4] and G. BENGEL [1] proved the existence and regularity theorems for hyperfunction solutions of single differential equations with constant coefficients. We extend their results to systems of differential equations with constant coefficients. Since the sheaf of hyperfunctions is flabby, we have therefore a flabby resolution of the sheaf of hyperfunction solutions of the homogeneous equation. If the system is elliptic, it is shown that any hyperfunction solution is analytic. Thus we obtain a concrete flabby resolution of the sheaf of regular solutions in terms of hyperfunctions. We apply this resolution first to the Cauchy-Riemann system and to single elliptic operators, and give a proof of the vanishing cohomology theorem by MALGRANGE [-10] for coherent analytic sheaves and G.BENGEL'S results [1] on P-functionals. Then applying the resolution to the exterior differentiation, we obtain a kind of the Alexander-Pontrjagin duality theorem and hence a purely analytic proof of the Jordan-Brouwer theorem.