Characterization of Non-Smooth Pseudodifferential Operators

TL;DR

This paper extends the characterization of pseudodifferential operators to the non-smooth case, showing that operators satisfying certain continuity conditions are indeed non-smooth pseudodifferential operators with specific symbol classes.

Contribution

It provides a new characterization of non-smooth pseudodifferential operators based on their continuity properties, generalizing classical results to less regular symbols.

Findings

Operators with specific continuity assumptions are non-smooth pseudodifferential operators.

The characterization applies to symbols in the class $C^{ au} S^m_{1,0}$.

Addresses difficulties due to limited mapping properties of non-smooth 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.