A weak extension theorem for inhomogeneous differential equations (Q2716420)

The author proves some extension theorems of Bochner's type. Namely, letting \(M\) be a smooth manifold, \(Z\) a point of \(M\), \(P\) a differential operator on \(M\) with smooth coefficients, given \(f\in D'(M)\), one assumes \(Pu= f\) in \(M-Z\) for some \(u\in D'(M-Z)\). Then, by taking \(f\) in suitable classes and assuming the singularity of \(u\) at \(Z\) is sufficiently weak, one concludes that \(u\) can be extended to a solution of \(Pu=f\) in the whole \(M\). The functional setting is here given by \(L^p\) and Besov spaces. Several interresting examples are proposed at the end of the paper.