Nonregular Pseudo-Differential Operators

Abstract

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