JournalszaaVol. 15, No. 1pp. 109–148

Nonregular Pseudo-Differential Operators

  • Jürgen Marschall

    Universität der Bundeswehr München, Neubiberg, Germany
Nonregular Pseudo-Differential Operators cover
Download PDF

Abstract

We study the boundedness properties of pseudo-differential operators a(x,D)a(x, D) and their adjoints a(x,D)a(x, D)* with symbols in a certain vector-valued Besov space on Besov spaces Bp,qsB^s_{p,q} and Triebel spaces Fp,qs(0<p,q<)F^s_{p,q} (0 < p,q < \infty). Applications are given to multiplication properties of Besov and Triebel spaces. We show that our results are best possible for both pseudo-differential estimates and multiplication. Denoting by (,)(\cdot,\cdot) the duality between Besov and between Triebel spaces we derive general conditions under which (a(x,D)f,g)=(fa(x,D)g)(a(x, D)f, g) = (f a(x, D)*g) holds. This requires a precise definition of a(x,D)fa(x, D)f and a(x,D)fa(x, D)*f for fFp,qsf \in F^s_{p,q} and fBp,qsf \in B^s_{p,q}.

Cite this article

Jürgen Marschall, Nonregular Pseudo-Differential Operators. Z. Anal. Anwend. 15 (1996), no. 1, pp. 109–148

DOI 10.4171/ZAA/691