On solvability of linear partial differential equations in local spaces \(B^{loc}_{p,k}(G)\) (Q1113002)

Let \(L(x,D)=\sum_{| \sigma | \leq r}a_{\sigma}(x)D^{\sigma}\) be a linear partial differential operator with \(C^{\infty}(G)\)- coefficients \(a_{\sigma}\), where G is an open subset in \({\mathbb{R}}^ n\). Denote by \(\Lambda^{\sim}_{p,k}(G)\) the minimal closed realization of L(x,D) in the local Hörmander space \({\mathcal B}^{loc}_{p,k}(G)\). The closedness of the range \(R(\Lambda^{\sim}_{p,k}(G))\) and of the range \(R(\Lambda^{\sim +}_{p,k}(G))\) of the dual operator \(\Lambda^{\sim +}_{p,k}(G)\) is considered. Among other things, one shows necessary and sufficient conditions for the closedness of \(R(\Lambda^{\sim}_{p,k}(G))\). The surjectivity of \(\Lambda^{\sim}_{p,k}(G)\) is also characterized. As an application a sufficient condition for the closedness of \(R(\Lambda^{\sim}_{2,1}(B(0,R)))\), where B(0,R) is the open ball in \({\mathbb{R}}^ n\), is established, when certain a priori estimates for the formal transpose L'(x,D) of L(x,D) hold.