### 数学代写|复分析作业代写Complex function代考|MATH307

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• Foundations of Data Science 数据科学基础

## 数学代写|复分析作业代写Complex function代考|Approximation on Totally Real Submanifolds

In this section we present an optimal $\mathscr{C}^k$-approximation result on totally real submanifolds. With essentially no extra effort we get approximation results on stratified totally real manifolds and on admissible sets (see Theorems 20 and 21).
There is a long history on approximation on totally real submanifolds, starting with J. Wermer [173] on curves and R. O. Wells [172] on real analytic manifolds. The first general result on approximation on totally real manifolds with various degrees of smoothness is due to L. Hörmander and J. Wermer [97]. Their work is based on $L^2$-methods for solving the $\bar{\partial}$-equation, and the passage from $L^2$ to $\mathscr{C}^k$-estimates led to a gap between the order $m$ of smoothness of the manifold $M$ on which the approximation takes place, and the order $k$ of the norm of the Banach space $\mathscr{C}^k(M)$ in which the approximation takes place. Subsequently, several authors worked on decreasing the gap between $m$ and $k$, introducing more precise integral kernel methods for solving $\bar{\partial}$. The optimal result with $m=k$ was eventually obtained by M. Range and Y.-T. Siu [139]. Subsequent improvements were made by F. Forstnerič, E. Løw, and N. Øvrelid [66] in 2001. They developed Henkin-type kernels adapted to this situation and obtained optimal results on approximation of $\bar{\partial}$-flat functions in tubes around totally real manifolds by holomorphic functions. In 2009 , B. Berndtsson [18] used $L^2$-theory to give a new approach to uniform approximation by holomorphic functions on compact zero sets of strongly plurisubharmonic functions. A novel byproduct of his method is that, in the case of polynomial approximation, one gets a bound on the degree of the approximating polynomial in terms of the closeness of the approximation.

We will not go into the details of the $L^2$ or the integral kernel approaches, but will instead present a method based on convolution with the Gaussian kernel which originates in the proof of Weierstrass’s Theorem 1 on approximating continuous functions on $\mathbb{R}$ by holomorphic polynomials. This approach is perhaps the most elementary one, and is particularly well suited for proving Runge-Mergelyan type approximation results with optimal regularity on (strongly) admissible sets. It seems that the first modern application of this method was made in 1981 by $\mathrm{S}$. Baouendi and F. Treves [12] to obtain local approximation of Cauchy-Riemann (CR) functions on CR submanifolds. The use of this method on totally real manifolds was developed further by P. Manne [118] in 1993 to obtain Carleman approximation on totally real submanifolds (see also [119]).

## 数学代写|复分析作业代写Complex function代考|Approximation on Strongly Pseudoconvex Domains

As we have seen, proofs of the Mergelyan theorem in one complex variable depend heavily on integral representations of holomorphic or $\bar{\partial}$-flat functions. The single most important reason why the one-dimensional proofs work so well is that the Cauchy-Green kernel (4) provides a solution to the inhomogeneous $\bar{\partial}$-equation which is uniformly bounded on all of $\mathbb{C}$ in terms of sup-norm of the data and the area of its support (see (6)). This allows uniform approximation of functions in $\mathscr{A}(K)$ on any compact set $K \subset \mathbb{C}$ with not too rough boundary by functions in $\mathscr{O}(K)$ (see Vitushkin’s Theorem 7). Nothing like that holds in several variables, and the question of uniform approximability is highly sensitive to the shape of the boundary even for smoothly bounded domains.

The idea of developing holomorphic integral kernels for domains in $\mathbb{C}^n$ with comparable properties to those of the Cauchy kernel in one variable was promoted by H. Grauert already around 1960; however, it took almost a decade to be realized. The first such constructions were given in 1969 by G. M. Henkin [92] and E. Ramírez de Arellano [138] for the class of strongly pseudoconvex domains. These kernels provide solution operators for the $\bar{\partial}$-equation which are bounded in the $\mathscr{C}^k$ norms and improve the regularity by $1 / 2$. We state here a special case of their results for $(0,1)$-forms, but in a more precise form which can be found in the works by I. Lieb and M. Range [112, Theorem 1], I. Lieb and J. Michel [111], and [62, Theorem 2.7.3]. A brief historical review of the kernel method is given in [66, pp. 392-393]. Given a domain $\Omega \subset \mathbb{C}^n$, we denote by $\mathscr{C}_{(0,1)}^k(\bar{\Omega})$ the space of all differential $(0,1)$-forms of class $\mathscr{C}^k$ on $\bar{\Omega}$.

Theorem 23 If $\Omega$ is a bounded strongly pseudoconvex Stein domain with boundary of class $\mathscr{C}^k$ for some $k \in{2,3, \ldots}$ in a complex manifold $X$, there exists a bounded linear operator $T: \mathscr{\zeta}{(0,1)}^0(\bar{\Omega}) \rightarrow \mathscr{L}^0(\bar{\Omega})$ satisfying the following properties: (i) If $f \in \mathscr{C}{0,1}^0(\bar{\Omega}) \cap \mathscr{C}{0,1}^1(\Omega)$ and $\bar{\partial} f=0$, then $\bar{\partial}(T f)=f$. (ii) If $f \in \mathscr{C}{0,1}^0(\bar{\Omega}) \cap \mathscr{C}{0,1}^r(\Omega)$ for some $r \in{1, \ldots, k}$ then $$|T f|{\mathscr{G} l, 1 / 2(\bar{\Omega})} \leq C_{l, \Omega}|f|_{\mathscr{C}{0,1}(\bar{\Omega})}, \quad l=0,1, \ldots, r .$$ Moreover, the constants $C{l, \Omega}$ may be chosen uniformly for all domains sufficiently $\mathscr{C}^k$ close to $\bar{\Omega}$.

# 复分析代写

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