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

### Jouko Tervo

University of Kuopio, Finland

## 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.

## Cite this article

Jouko Tervo, On $\mathcal B_{p,k}$-Boundedness and Compactness of Linear Pseudo-Differential Operators. Z. Anal. Anwend. 7 (1988), no. 1, pp. 41–56

DOI 10.4171/ZAA/281