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 \(C^{\tau } S^m_{1,0}(\mathbb {R}^n \times \mathbb {R}^n)\). The main new difficulties are the limited mapping properties of pseudodifferential operators with non-smooth symbols.