JournalszaaVol. 7, No. 1pp. 41–56

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

  • Jouko Tervo

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

Abstract

Boundedness and compactness arguments in the Hörmander spaces Bp,k\mathcal B_{p,k} for linear pseudo-differential operators L(X,D)L(X, D) are considered. The symbol L(x,ξ)L(x, \xi) of L(X,D)L(X, D) is assumed to obey appropriate temperate criteria, which guarantee that L(X,D)L(X, D) maps the Schwartz class S\mathcal S into itself and,that the formal transpose L(X,D):SSL’(X, D): \mathcal S \to \mathcal S exists. A characterization for the boundedness of the operator L(X,D):B1,kk~B1,kL’(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):Bp,kkBp,kL’(X, D): \mathcal B_{p,kk} \sim \to \mathcal B_{p,k} with p[1,]p \in [1, \infty] is established as well. Finally, the compactness of the continuous extension of L(X,D):Bp,kk(G)Bp,kL’(X, D): \mathcal B_{p,kk} \sim (G) \to \mathcal B_{p,k} is studied, where GG is an open bounded set in Rn\mathbb R^n and where Bp,kk(G)\mathcal B_{p,kk} \sim (G) is (essentially) the completion of C0(G)C_0^{\infty} (G) with respect to the Bp,kk\mathcal B_{p,kk} \sim-norm.

Cite this article

Jouko Tervo, On Bp,k\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