We study the boundedness properties of pseudo-differential operators and their adjoints with symbols in a certain vector-valued Besov space on Besov spaces and Triebel spaces . 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 the duality between Besov and between Triebel spaces we derive general conditions under which holds. This requires a precise definition of and for and .
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Jürgen Marschall, Nonregular Pseudo-Differential Operators. Z. Anal. Anwend. 15 (1996), no. 1, pp. 109–148DOI 10.4171/ZAA/691