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Zeitschrift für Analysis und ihre Anwendungen

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Volume 15, Issue 1, 1996, pp. 109–148
DOI: 10.4171/ZAA/691

Published online: 1996-03-31

Nonregular Pseudo-Differential Operators

Jürgen Marschall[1]

(1) Universität der Bundeswehr München, Neubiberg, Germany

We study the boundedness properties of pseudo-differential operators $a(x, D)$ and their adjoints $a(x, D)*$ with symbols in a certain vector-valued Besov space on Besov spaces $B^s_{p,q}$ and Triebel spaces $F^s_{p,q} (0 < p,q < \infty)$. 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 $(a(x, D)f, g) = (f a(x, D)*g)$ holds. This requires a precise definition of $a(x, D)f$ and $a(x, D)*f$ for $f \in F^s_{p,q}$ and $f \in B^s_{p,q}$.

Keywords: Pseudo-differential operators, function spaces, nonregular symbols, paramultiplication, pointwise multiplication

Marschall Jürgen: Nonregular Pseudo-Differential Operators. Z. Anal. Anwend. 15 (1996), 109-148. doi: 10.4171/ZAA/691