The EMS Publishing House is now EMS Press and has its new home at ems.press.

Please find all EMS Press journals and articles on the new platform.

Zeitschrift für Analysis und ihre Anwendungen


Full-Text PDF (1187 KB) | Metadata | Table of Contents | ZAA summary
Volume 7, Issue 1, 1988, pp. 41–56
DOI: 10.4171/ZAA/281

Published online: 1988-02-29

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

Jouko Tervo[1]

(1) University of Kuopio, Finland

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.

No keywords available for this article.

Tervo Jouko: On $\mathcal B_{p,k}$-Boundedness and Compactness of Linear Pseudo-Differential Operators. Z. Anal. Anwend. 7 (1988), 41-56. doi: 10.4171/ZAA/281