Some properties of pseudo-differential operators with amplitude Q(x,y,\(\xi\) ) (Q909148)

The paper considers linear pseudo-differential operators Q with amplitude \(Q(\cdot,\cdot,\cdot)\in C^{\infty}(G\times G\times {\mathbb{R}}^ n)\), where G is an open set in \({\mathbb{R}}^ n\). A criterion, which guarantees that Q maps continuously \(C^{\infty}_ 0(G)\) into \(C^{\infty}(G)\) and that the continuous formal transpose \(Q': C^{\infty}_ 0(G)\to C^{\infty}(G)\) of Q exists, is verified. Furthermore, a condition, under which the extension \({\mathbb{Q}}\) of Q from \(E'(G)\) to \(D'(G)\) is pseudolocal, is expressed. Also, the decomposition \(Q=Q+R_{\ell}\) where \(\tilde Q\) is properly supported (as pseudolocal) and the extension of \(R_{\ell}\) maps \(E'(G)\) into \(C^{\infty}(G)\), is considered.