Nonregular Pseudo-Differential Operators

Abstract

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 $f \in B^s\_{p,q}$.