  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.
