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differential-operators
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Jannes Bantje
differential-operators
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834eb326
Commit
834eb326
authored
Feb 18, 2019
by
Jannes Bantje
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explicit mention of order 1 case
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contents/diffop.tex
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834eb326
...
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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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