# operator

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 $z^{j}=x^{j}+iy^{j}$. We can think of $G$ as a subset of ${\mathbb{R}}^{2n}$. We therefore have the following partial derivatives for all $1\leq j\leq n$,

 $\displaystyle\frac{\partial f}{\partial z^{j}}$ $\displaystyle:=\frac{1}{2}\left(\frac{\partial f}{\partial x^{j}}-i\frac{% \partial f}{\partial y^{j}}\right),$ $\displaystyle\frac{\partial f}{\partial\bar{z}^{j}}$ $\displaystyle:=\frac{1}{2}\left(\frac{\partial f}{\partial x^{j}}+i\frac{% \partial f}{\partial y^{j}}\right).$

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

 $\displaystyle dz^{j}$ $\displaystyle:=dx^{j}+idy^{j},$ $\displaystyle d\bar{z}^{j}$ $\displaystyle:=dx^{j}-idy^{j},$

then we can define two new operators acting on $C^{1}$ functions on $G$ giving 1-forms by

 $\displaystyle\partial f$ $\displaystyle:=\sum_{j=1}^{n}\frac{\partial f}{\partial z^{j}}dz^{j},$ $\displaystyle\bar{\partial}f$ $\displaystyle:=\sum_{j=1}^{n}\frac{\partial f}{\partial\bar{z}^{j}}d\bar{z}^{j}.$

By direct calculation we immediately see that

 $df=\partial f+\bar{\partial}f.$

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 $G$.

 $\displaystyle\partial\omega$ $\displaystyle:=\sum_{\alpha,\beta}\frac{\partial f_{\alpha,\beta}}{\partial z^% {j}}dz^{j}\wedge dz^{\alpha}\wedge d\bar{z}^{\beta},$ $\displaystyle\bar{\partial}\omega$ $\displaystyle:=\sum_{\alpha,\beta}\frac{\partial f_{\alpha,\beta}}{\partial% \bar{z}^{j}}d\bar{z}^{j}\wedge dz^{\alpha}\wedge d\bar{z}^{\beta}.$

Again a direct calculation shows that $d=\partial+\bar{\partial}$.

The Cauchy-Riemann equations are then given by

 $\bar{\partial}f=0$

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.

###### Proposition.

$\bar{\partial}$ and $\partial$ satisfy the following properties

• $\bar{\partial}$ and $\partial$ are linear,

• $\bar{\partial}^{2}=\bar{\partial}\bar{\partial}=0$ and $\partial^{2}=\partial\partial=0$,

• $\bar{\partial}\partial-\partial\bar{\partial}=0$.

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.

## References

• 1 Lars Hörmander. , North-Holland Publishing Company, New York, New York, 1973.
• 2 Steven G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.
Title operator barpartialOperator 2013-03-22 15:10:39 2013-03-22 15:10:39 jirka (4157) jirka (4157) 7 jirka (4157) Definition msc 30E99 msc 32A99 d bar operator d-bar operator $\partial$ operator