A generalization of Bochner's extension theorem and its application (Q1810340)

Let \(M\) be a smooth paracompact real manifold and \(Z\) a closed and smooth submanifold of \(M\) of co-dimension \(d.\) Let \(P\) be a differential operator on \(M\) with smooth coefficients. Finally, let \(u\) be a distribution on \(M\) which satisfies \(Pu=f\) outside \(Z.\) The authors prove that, if \(f\) is microlocally integrable and \(u\) satisfies a microlocal \(p\)-integrability condition with \(m\leq d(1-1/p),\) then \(u\) is locally extendable to \(Z.\) An application to semilinear differential equations is also proposed.