Smooth pseudodifferential operators on 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 τ S m 1,0 (R n × R n ). The main new difficulties are the limited mapping properties of pseudodifferential operators with non-smooth symbols.In 1977 Beals [4] proved a characterization of smooth pseudodifferential operators, for example of the Hörmander class S m ρ,δ (R n × R n ) with 0 ≤ δ ≤ ρ ≤ 1 and δ < 1. Later Ueberberg [22] generalized this characterization for L p −Sobolev spaces for pseudodifferential operators of the Hörmander class S m ρ,δ (R n × R n ) with 0 ≤ δ ≤ ρ ≤ 1 and δ < 1. In the literature there are some other characterizations in the smooth case, e.g.[9], [13] or [19]. But the most important one for this section is that one of Ueberberg, cf. [22]. It is based on the method for characterizing algebras of pseudodifferential operators developed by Beals [4], [5], Coifman, Meyer [6] and Cordes [7], [8]. Since non-smooth pseudodifferential operators are used in the regularity theory for partial differential equations, such a characterization is also useful in the non-smooth case. We use the main ideas of the characterization of Ueberberg in the smooth case, cf. [22], in order to derive a characterization for non-smooth pseudodifferential operators of the symbol-class p ∈ C τ * S m ρ,0 (R n × R n ; M) with ρ ∈ {0, 1}. Here the Hölder-Zygmund space C τ * (R n ), τ > 0, is defined by the symbol-class Cm ,s S 0 0,0 (R n × R n ; M − 1). Subsection 4.2 is devoted to the symbol reduction of non-smooth double symbols to non-smooth single symbols. Details for the third tool are proved in Subsection 4.3. There a family (T ε ) ε∈(0,1] fulfilling the following three properties is constructed: T ε : S ′ (R n ) → S(R n ) is continuous for all ε ∈ (0, 1] and converges pointwise if ε → 0. Moreover, all iterated commutators of T ε are uniformly bounded with respect to ε as maps from L q (R n ) to L q (R n ).In Section 5 we illustrate the usefulness of such a characterization: We show that the composition P Q of two non-smooth pseudodifferential operators P and Q is a non-smooth pseudodifferential operator again if Q is smooth enough. This is done by means of the characterization of non-smooth pseudodifferential operators.Section 3 is devoted to some properties of pseudodifferential operators with single symbols, cf. Subsection 3.
