Characterization of Non-Smooth Pseudodifferential Operators

Abstract

Smooth pseudodifferential operators on ${\mathbb{R}}^n$ can be characterized by their mapping properties between $L^p-$Sobolev spaces due to Beals and Ueberberg. In applications such a characterization would also be useful in the non-smooth case, for example to show the regularity of solutions of a partial differential equation. Therefore, we will show that every linear operator $P$, which satisfies some specific continuity assumptions, is a non-smooth pseudodifferential operator of the symbol-class $C^{\tau }{S}_{1,0}^m({\mathbb{R}}^n\times {\mathbb{R}}^n)$. The main new difficulties are the limited mapping properties of pseudodifferential operators with non-smooth symbols.

arXiv comment42 pages