On the ranges of realizations in distribution spaces (Q1188376)

Summary: The paper deals with the closedness of ranges of the surjectivity of realizations related to linear partial differential operators \(L(x,D)\). A characterization (with the help of a certain coercivity condition) of the surjectivity of the maximal realization \(L_{p,k,G}^{'\#}\) in \(B_{p,k}(G)=\{u\in D'(G)| u=f_{u| G}\) for some \(f_ u\in B_{p,k}\}\) is established. Here \(B_{p,k}\) \((p\in(0,1)\), \(k\in K)\) is the Hörmander space. Furthermore, the closedness of the range \(R(\Lambda^ \sim_{p,k}(G))\) corresponding to the minimal realization \(\Lambda^ \sim_{p,k}(G)\) in local Hörmander spaces \(B^{loc}_{p,k}(G)=\{u\in D'(G)|\Psi u\in B_{p,k}\) for any \(\Psi\in C_ 0^ \infty(G)\}\) is considered.