Edge-of-the-wedge theorem for elliptic systems (Q1915414)

Let \(M\) be a real analytic manifold of dimension \(n\), \(N\subset M\) is a real submanifold of \(M\), and \(X\) and \(Y\) are their complexifications. \({\mathcal P}_X\) and \({\mathcal B}_M\) denote the sheaf of rings of linear holomorphic differential operators on \(X\) and the sheaf of Sato's hyperfunctions correspondingly. In the present paper an alternative proof is given of the Kashiwara-Kawai theorem for left coherent elliptic \({\mathcal P}_X\)-modules \({\mathcal M}\) for which \(Y\) is non characteristic, i.e. that \(H^j\mu_N(\text{R}{\mathcal H}\text{om}_{{\mathcal P_X}}({\mathcal M},^*)) = 0\), where \(^*={\mathcal A}_m\) or \({\mathcal B}_M\), \(\mu_N\) is the Sato microlocalization functor, and \(j<\text{cod}_MN\). It is also shown that the result fails to be true in higher dimensions for the sheaf \({\mathcal A}_M\) without the ellipticity requirement.