Commit 834eb326 authored by Jannes Bantje's avatar Jannes Bantje

explicit mention of order 1 case

parent 62955af5
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......@@ -209,6 +209,25 @@ This allows us, to combine the symbols of differential operators into a space:
\]
\end{definition}
All of this becomes a lot clearer, if one considers operators of order $1$.
Let $D$ be such an operator and $g \in C^\infty_c(U)$, which we view as a multiplication operator on $C^\infty_c(M,E)$.
The commutator $\benbrace*{D,g}$ acts on $u \in C^\infty_c(U,E)$ via
\[
\enbrace[\big]{\benbrace*{D,g}u}(x) = \sum_{i=1}^{d} a_i(x) \diff{g}{x_i}(x) u(x)
\]
In other words $\benbrace*{D,g}$ is acting via the following element of $C_c^\infty(U,\End(E))$
\[
x \mapsto \sum_{i=1}^{d} a_i(x) \diff{g}{x_i}(x)
\]
The computation in the proof of \cref{lem:symbol_welldefined} now shows:
\begin{lemma}
If $D$ is an operator of order $1$, the symbol can be computed as
\[
\sigma_1(D)(\mathd g) u = i \,\benbrace*{D,g} u
\]
\end{lemma}
The symbol gives raise to an extremly important class of differential operators:
\begin{definition}
......
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