On \({\mathcal B}_{p,k}\)-boundedness and compactness of linear pseudo- differential operators (Q910629)

Summary: Boundedness and compactness arguments in the Hörmander spaces \({\mathcal B}_{p,k}\) for linear pseudo-differential operators L(X,D) are considered. The symbol L(x,\(\xi)\) of L(X,D) is assumed to obey appropriate temperate criteria, which guarantee that L(X,D) maps the Schwartz class \({\mathcal S}\) into itself and that the formal transpose \(L'(X,D): {\mathcal S}\to {\mathcal S}\) exists. A characterization for the boundedness of the operator \(L'(X,D): {\mathcal B}_{1,kk^{\sim}}\to {\mathcal B}_{1,k}\) is obtained. A sufficient condition for the boundedness of the operator \(L'(X,D)* {\mathcal B}_{p,kk^{\sim}}\to {\mathcal B}_{p,k}\) with \(p\in [1,\infty)\) is established as well. Finally, the compactness of the continuous extension of \(L'(X,D): {\mathcal B}_{p,kk^{\sim}}(G)\to {\mathcal B}_{p,k}\) is studied, where G is an open bounded set in \({\mathbb{R}}^ n\) and where \({\mathcal B}_{p,kk^{\sim}}(G)\) is (essentially) the completion of \(C^{\infty}_ 0(G)\) with respect to the \({\mathcal B}_{p,kk^{\sim}}\)-norm.