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'$\bar{\partial}$ operator'
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| Title of object: |
$\bar{\partial}$ operator |
| Canonical Name: |
BarpartialOperator |
| Type: |
Definition |
| Created on: |
2005-04-05 20:22:45 |
| Modified on: |
2005-04-15 19:37:14 |
| Classification: |
msc:30E99, msc:32A99 |
| Defines: |
$\partial$ operator |
| Synonyms: |
$\bar{\partial}$ operator=d bar operator
$\bar{\partial}$ operator=d-bar operator |
Revision comment (for changes between this and next version):
| consistent with upper/lower indicies |
Preamble:
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\usepackage{amsmath}
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%\usepackage{psfrag}
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\newtheorem*{rmk}{Remark} |
Content:
Let $G \subset {\mathbb{C}}^n$ be a domain and let
$f \colon G \to {\mathbb{C}}$ be a $C^1$
function (continuously differentiable)
$(z^1,\ldots,z^n) \mapsto f(z^1,\ldots,z^n)$ where $z^j = x^j + i y^j$.
We can think of $G$ as a subset of ${\mathbb{R}}^{2n}$.
We therefore
have the following partial derivatives for all $1 \leq j \leq n$,
\begin{align*}
\frac{\partial f}{\partial z^j} & :=
\frac{1}{2} \left(
\frac{\partial f}{\partial x^j} - i \frac{\partial f}{\partial y^j}
\right) ,
\\
\frac{\partial f}{\partial \bar{z}^j} & :=
\frac{1}{2} \left(
\frac{\partial f}{\partial x^j} + i \frac{\partial f}{\partial y^j}
\right) .
\end{align*}
Now let $d$ be the standard exterior derivative on
${\mathbb{R}}^{2n}$ and the $dx^j$ and $dy^j$ the standard basis of cotangent
vectors. Then if we define
\begin{align*}
dz^j & := dx^j + i dy^j , \\
d\bar{z}^j & := dx^j - i dy^j ,
\end{align*}
then we can define two new operators acting on $C^1$ functions on $G$
giving 1-forms by
\begin{align*}
\partial f & := \sum_{j=1}^n \frac{\partial f}{\partial z^j} dz^j , \\
\bar{\partial} f & := \sum_{j=1}^n \frac{\partial f}{\partial \bar{z}^j}
d\bar{z}^j .
\end{align*}
By direct calculation we immediately see that
\begin{equation*}
df = \partial f + \bar{\partial} f .
\end{equation*}
Similarly we now define $\partial$ and $\bar{\partial}$
on arbitrary differential form
$\omega = \sum_{\alpha,\beta} f_{\alpha,\beta} dz^\alpha \wedge
d\bar{z}^\beta$, where $\alpha$ and $\beta$ range over all multi-indices with
elements less then $n$, where if $\alpha = (\alpha_1,\ldots,\alpha_k)$
then $dz^\alpha = dz_{\alpha_1} \wedge \ldots \wedge dz^{\alpha_k}$,
and $f_{\alpha,\beta}$ is a $C^1$, complex valued function
on $G$.
\begin{align*}
\partial \omega
& :=
\sum_{\alpha,\beta} \frac{\partial f_{\alpha,\beta}}{\partial z^j} dz^j
\wedge
dz^\alpha \wedge d\bar{z}^\beta
, \\
\bar{\partial} \omega
& :=
\sum_{\alpha,\beta} \frac{\partial f_{\alpha,\beta}}{\partial \bar{z}^j}
d\bar{z}^j
\wedge
dz^\alpha \wedge d\bar{z}^\beta .
\end{align*}
Again a direct calculation shows that $d = \partial + \bar{\partial}$.
The Cauchy-Riemann equations are then given by
\begin{equation*}
\bar{\partial} f = 0
\end{equation*}
That is, $f$ is holomorphic if and only if it satisfies the above equations.
Note that this only applies to functions. If $\bar{\partial}\omega = 0$
for a differential form, then the coefficients in the standard basis
need not be holomorphic.
\begin{prop}
$\bar{\partial}$ and $\partial$ satisfy the following properties
\begin{itemize}
\item $\bar{\partial}$ and $\partial$ are linear,
\item $\bar{\partial}^2 = \bar{\partial} \bar{\partial} = 0$ and $\partial^2 = \partial \partial = 0$,
\item $\bar{\partial} \partial - \partial \bar{\partial} = 0$.
\end{itemize}
\end{prop}
While $\bar{\partial} u = 0$ is our condition for $u$ to be a
holomorphic function it turns out that it is more important to solve the inhomogeneous
$\bar{\partial}u = f$ equation, as that allows us to construct holomorphic
objects from nonholomorphic ones.
\begin{thebibliography}{9}
\bibitem{Hormander:several}
Lars H\"ormander.
{\em \PMlinkescapetext{An Introduction to Complex Analysis in Several
Variables}},
North-Holland Publishing Company, New York, New York, 1973.
\bibitem{Krantz:several}
Steven~G.\@ Krantz.
{\em \PMlinkescapetext{Function Theory of Several Complex Variables}},
AMS Chelsea Publishing, Providence, Rhode Island, 1992.
\end{thebibliography} |
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