Some Properties of Pseudo-Differential Operators with Amplitude $Q(x,y,\xi$)

Abstract

The paper considers linear pseudo-differential operators $Q$ with amplitude $Q(\cdot ,\cdot ,\cdot )\in C^{infty}(g\times G\times {\mathbb{R}}^n$), where G is an open set in ${\mathbb{R}}^n$. A criterion, which guarantees that $Q$ maps continuously $C_0^{infty}(G)$ into $C^{infty}(G)$ and that the continuous formal transpose $Q’:C_0^{infty}(G)\to C^{infty}$ of $Q$ exists, is verified. Furthermnore, a condition, under which the extension $\mathbb{Q}$ of $Q$ from $E’(G)$ to $D’(G)$ is pseudolocal, is expressed. Also, the decomposition $Q=\bar{Q}+R_1$ where $Q$ is properly supported (and pseudolocal) and $R_1$ maps $E’(G)$ into $C_{infty}(G)$, is considered.