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Zeitschrift für Analysis und ihre Anwendungen


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Volume 8, Issue 5, 1989, pp. 389–406
DOI: 10.4171/ZAA/361

Published online: 1989-10-31

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

Jouko Tervo[1]

(1) University of Kuopio, Finland

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.

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Tervo Jouko: Some Properties of Pseudo-Differential Operators with Amplitude $Q(x, y, \xi$). Z. Anal. Anwend. 8 (1989), 389-406. doi: 10.4171/ZAA/361