On ${\mathcal{B}}_{p,k}$-Boundedness and Compactness of Linear Pseudo-Differential Operators

Abstract

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 $L’(X,D):{\mathcal{B}}_{1,kk}\tilde{\to }{\mathcal{B}}_{1,k}$ 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 $C_0^{\infty }(G)$ with respect to the ${\mathcal{B}}_{p,kk}\sim$-norm.