### 数学代写|复分析作业代写Complex function代考|KMA152

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## 数学代写|复分析作业代写Complex function代考|The Oka–Weil Theorem and Its Generalizations

The analogue of Runge’s theorem (see Theorems 2 and 4) on Stein manifolds and Stein spaces is the following theorem due to K. Oka [132] and A. Weil [171]. All complex spaces are assumed to be reduced.

Theorem 18 (The Oka-Weil Theorem) If $X$ is a Stein space and $K$ is a compact $\mathscr{O}(X)$-convex subset of $X$, then every holomorphic function in an open neighborhood of $K$ can be approximated uniformly on $K$ by functions in $\mathscr{O}(X)$.

Proof Two proofs of this result are available in the literature. The original one, due to $\mathrm{K}$. Oka and A. Weil, proceeds as follows. A compact $\mathscr{O}(X)$-convex subset $K$ in a Stein space $X$ admits a basis of open Stein neighborhoods of the form
$$P=\left{x \in X:\left|h_1(x)\right|<1, \ldots,\left|h_N(x)\right|<1\right}$$
with $h_1, \ldots, h_N \in \mathscr{O}(X)$. We may assume that the function $f \in \mathscr{O}(K)$ to be approximated is holomorphic on $P$. By adding more functions if necessary, we can ensure that the map $h=\left(h_1, \ldots, h_N\right): X \rightarrow \mathbb{C}^N$ embeds $P$ onto a closed complex subvariety $A=h(P)$ of the unit polydisc $\mathbb{D}^N \subset \mathbb{C}^N$. Hence, there is a function $g \in \mathscr{O}(A)$ such that $g \circ h=f$ on $P$. By the Oka-Cartan extension theorem [62, Corollary 2.6.3], $g$ extends to a holomorphic function $G$ on $\mathbb{D}^N$. Expanding $G$ into a power series and precomposing its Taylor polynomials by $h$ gives a sequence of holomorphic functions on $X$ converging to $f$ uniformly on $K$.

Another approach uses the method of L. Hörmander for solving the $\bar{\partial}$-equation with $L^2$-estimates (see $[94,96]$ ). We consider the case $X=\mathbb{C}^n$; the general case reduces to this one by standard methods of Oka-Cartan theory. Assume that $f$ is a holomorphic function in a neighborhood $U \subset \mathbb{C}^n$ of $K$. Choose a pair of neighborhoods $W \Subset V \Subset U$ of $K$ and a smooth function $\chi: \mathbb{C}^n \rightarrow[0,1]$ such that $\chi=1$ on $\bar{V}$ and $\operatorname{supp}(\chi) \subset U$. By choosing $W \supset K$ small enough, there is a nonnegative plurisubharmonic function $\rho \geq 0$ on $\mathbb{C}^n$ that vanishes on $W$ and satisfies $\rho \geq c>0$ on $U \backslash V$. Note that the smooth $(0,1)$-form
$$\alpha=\bar{\partial}(\chi f)=f \bar{\partial} \chi=\sum_{i=1}^n \alpha_i d \bar{z}_i$$
is supported in $U \backslash V$. Hörmander’s theory for the $\bar{\partial}$-complex (see [96, Theorem 4.4.2]) furnishes for any $t>0$ a smooth function $h_t$ on $\mathbb{C}^n$ satisfying $\bar{\partial} h_t=\alpha \quad$ and $\quad \int_{\mathbb{C}^n} \frac{\left|h_t\right|^2}{\left(1+|z|^2\right)^2} e^{-t \rho} d \lambda \leq \int_{\mathbb{C}^n} \sum_{i=1}^n\left|\alpha_i\right|^2 e^{-t \rho} d \lambda$

## 数学代写|复分析作业代写Complex function代考|Mergelyan’s Theorem in Higher Dimensions

As we have seen in Sections 2-4, the Mergelyan approximation theory in the complex plane and on Riemann surfaces was a highly developed subject around mid twentieth century. Around the same time, it became clear that the situation is much more complicated in higher dimensions. For example, in $1955 \mathrm{~J}$. Wermer [173] constructed an arc in $\mathbb{C}^3$ which fails to have the Mergelyan property. This suggests that, in several variables, one has to be much more restrictive about the sêts on which one considers Mergeelyan type anpproximation problems.

There are two lines of investigations in the literature: approximation on submanifolds of $\mathbb{C}^n$ of various degrees of smoothness and approximation on closures of bounded pseudoconvex domains. In neither category the problem is completely understood, and even with these restrictions, the situation is substantially more complicated than in dimension one. For example, R. Basener (1973), [14] (generalizing a result of B. Cole (1968), [39]) showed that Bishop’s peak point criterium does not suffice even for smooth polynomially convex submanifolds of $\mathbb{C}^n$. Even more surprisingly, it was shown by K. Diederich and J. E. Fornæss in 1976 [42] that there exist bounded pseudoconvex domains with smooth boundaries in $\mathbb{C}^2$ on which the Mergelyan property fails. The picture for curves is more complete; see G. Stolzenberg [153], H. Alexander [5], and P. Gauthier and E. Zeron [80].

In this section we outline the developments starting around the 1960s, give proofs in some detail in the cases of totally real manifolds and strongly pseudoconvex domains, and provide some new results on combinations of such sets.

Definition 4 Let $(X, J)$ be a complex manifold, and let $M \subset X$ be a $\mathscr{C}^1$ submanifold.
(a) $M$ is totally real at a point $p \in M$ if $T_p M \cap J T_p M={0}$. If $M$ is totally real at all points, we say that $M$ is a totally real submanifold of $X$.
(b) $M$ is a stratified totally real submanifold of $X$ if $M=\bigcup_{i=1}^l M_i$, with $M_i \subset M_{i+1}$ locally closed sers, such that $M_1$ and $M_{i+1} \backslash M_i$ are torally real submanifolds.

We now introduce suitable types of sets for Mergelyan approximation. The following notion is a generalization of the one for Riemann surfaces in Definition 3 . Recall that a compact set $S$ in a complex manifold $X$ is a Stein compact if $S$ admits a basis of open Stein neighborhoods in $X$.

# 复分析代写

## 数学代写|复分析作业代写Complex function代考|The Oka–Weil Theorem and Its Generalizations

$$\alpha=\bar{\partial}(\chi f)=f \bar{\partial} \chi=\sum_{i=1}^n \alpha_i d \bar{z}i$$ 支持 $U \backslash V$. 霍曼德的理论 $\bar{\partial}$-complex（见[96，定理 4.4.2]） 提供任何 $t>0$ 平滑函数 $h_t$ 在 $\mathbb{C}^n$ 令人满意 $\bar{\partial} h_t=\alpha \quad$ 和 $\quad \int{\mathbb{C}^n} \frac{\left|h_t\right|^2}{\left(1+|z|^2\right)^2} e^{-t \rho} d \lambda \leq \int_{\mathbb{C}^n} \sum_{i=1}^n\left|\alpha_i\right|^2 e^{-t \rho} d \lambda$

## 数学代写|复分析作业代写Complex function代考|Mergelyan’s Theorem in Higher Dimensions

(一种) $M$ 在某一点上是完全真实的 $p \in M$ 如果 $T_p M \cap J T_p M=0$. 如果 $M$ 在所有点上都是完全真实 的，我们说 $M$ 是一个完全真实的子流形 $X$.
(乙) $M$ 是一个分层的完全真实的子流形 $X$ 如果 $M=\bigcup_{i=1}^l M_i ＼mathrm{~ ， 和 ~} M_i \subset M_{i+1}$ 本地封闭的 sers， 这样 $M_1$ 和 $M_{i+1} \backslash M_i$ 是真正的子流形。

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