Geometric Properties of a General Kohn-Nirenberg Domain in \(\mathbb{C}^{n}\)

1. Introduction

Consider a domain $\Omega \in {\mathbb{C}}^n$ with smooth boundary $\partial \Omega$. As the boundary point is a strongly pseudoconvex point $p\in \partial \Omega$, we can find a local system of holomorphic coordinates. Hakim [1] and Pflug[2] showed that every strongly pseudoconvex point of $\partial \Omega$ is a peak point. But this property fails for weakly pseudoconvex boundary point in general. Kohn and Nirenberg have an example that is defined with the boundary point $(0,0)\in \partial \Omega$. The representing example [3] is $\Omega =\{z={(z_1,z_2)}^T\in {\mathbb{C}}^2:r\left(z\right):=Re{z}_2+{\left|z_1\right|}^8+{\displaystyle \frac{15}{7}}{\left|z_1\right|}^{2}Re{z}_1^6<0\}$, which is a pseudoconvex domain with point in the boundary that does not admit any peak function, supporting surface and the boundary can not be convexifiable by any local holomorphic coordinates[4,5,6]. The existence of supporting functions and smooth peak functions and the properties of convexifiability have been done by Pflug [2], by Kola$\breve{r}$[7], by J.Han [8], D.Zhao [9] and by J. Byun and H. R. Cho [10]. In [11,12], Taeyong Ahn etc. provide a tool to construct global holomorphic peaks from local holomorphic supporting functions for a class of unbounded domains in ${\mathbb{C}}^n$. But it is still an open question whether any Kohn-Nirenberg domain is biholomorphic to a bounded domain. In order to better understand the properties of the Kohn-Nirenberg domain, in [13], Simone Calamai provides some new examples of Kohn-Nirenberg domain that develop some properties and theories about convexifiability in ${\mathbb{C}}^2$.

Let ${\mathcal{O}}^{\alpha }(\Omega )(0\le \alpha \le \infty )$ denote the space of functions holomorphic on $\Omega$ and of class $C^{\alpha }$ on $\overline{\Omega }$. Recall a point $\xi \in \partial \Omega$ is a peak point to $\Omega$ if there is a function $f\in {\mathcal{O}}^{\alpha }(\Omega )$ satisfying $f\left(\xi \right)=1$ and $\left|f\right(z\left)\right|<1$ for all $z\in \overline{\Omega }\setminus \left\{\xi \right\}$. We call f a peak function. A holomorphic supporting surface for $\Omega$ at $\xi$ is a complex manifold M of co-dimension 1 with the property: there exists a neighborhood $N\left(\xi \right)$ of $\xi$ such that $\overline{\Omega }\cap N\left(\xi \right)\cap M=\left\{\xi \right\}$. In [9], D. Zhao etc. considered a general moditication of the Kohn-Nirenberg domain near the origin in ${\mathbb{C}}^n$, namely, the domain ${\Omega }_K=\{(z_1,\dots ,z_n)\in {\mathbb{C}}^n:Re{z}_n+{\sum }_{j=1}^{n-1}({\left|z_j\right|}^p+K_j{\left|z_j\right|}^{p-q}Re\left(z_j^q\right))<0\}$, where $K=(K_1,...,K_{n-1})\in {\mathbb{R}}^{n-1},p,q\in {\mathbb{Z}}^{+}$ and $p-q-2\ge 0$. They proved the following sufficient condition.

Theorem 1 

([9]). Given the above domain ${\Omega }_K$ with $\left|K_j\right|\le {\displaystyle \frac{p^2}{p^2-q^2}}(1\le j\le n-1)$, then

1. 

${\Omega }_K$ is a pseudoconvex domain.

2. 

If $q\mid p$ or $q\nmid p$ but $\left|K_j\right|<1(1\le j\le n-1)$, there exists a $C^{\infty }$-peak function and a supporting surface at the origin $0\in \partial {\Omega }_K$.

3. 

If $q\nmid p$ and ${\sum }_{j=1}^{n-1}\left|K_j\right|>n-1$, there does not exist any $C^{\infty }$-peak function and supporting surface at $0\in \partial {\Omega }_K$.

In this paper, based on the domain ${\Omega }_K$, we define a general Kohn-Nirenberg type domain as follows.

$$\begin{array}{cc}\hfill {\Omega }_{K,L}& =\{(z_1,\dots ,z_n)\in {\mathbb{C}}^n:Re{z}_n+g{\left|z_n\right|}^2+{\sum }_{j=1}^{n-1}({\left|z_j\right|}^p+K_j{\left|z_j\right|}^{p-q}Re\left(z_j^q\right)+\hfill \\ \hfill & L_j{\left|z_j\right|}^{p-2q}Im\left(z_j^{2q}\right))<0\}\hfill \end{array}$$

where $K=(K_1,K_2,\dots ,K_{n-1})\in {\mathbb{R}}^{n-1},L=(L_1,L_2,\dots ,L_{n-1})\in {\mathbb{R}}^{n-1},g\ge 0,p\in {\mathbb{Z}}^{+},q\in {\mathbb{Z}}^{+}$ and $p-2q-1\ge 0$.

We see that ${\Omega }_{K,L}$ is a modificationn of ${\Omega }_K$. If we do not consider the term $g{\left|z_n\right|}^2$, the general modified domain ${\Omega }_{K,L}$ is a weighted-bumped domain [12], denoted by ${\Omega }_{WB}$. If we consider this term, the Theorem 4.6 [11] has a argument that there exists global holomorphic supporting function when $g>0$ and a bound point of ${\Omega }_{WB}$ admits a local holomorphic supporting function. Thus ${\Omega }_{K,L}$ keeps the main features. It will be interesting to study whether the existence of supporting surface and peak function at the origin in [9] can be generalized to the domain ${\Omega }_{K,L}$.

For the domain ${\Omega }_{K,L}$, we study the existence of the holomorphic peak function, supporting surface at the boundary points. The main result of this article is the following theorem.

Theorem 2 

Let ${\Omega }_{K,L}$ be the above domain with $\left|K_j\right|+2\left|L_j\right|\le {\displaystyle \frac{p^2}{p^2-q^2}}(1\le j\le n-1)$, we have

1. 

${\Omega }_{K,L}$ is a peseudoconvex domain.

2. 

If $q\mid p$ or $q\nmid p$ and $\left|K_j\right|+\left|L_j\right|<1(1\le j\le n-1)$, there exists holomorphic peak function and supporting surface at the origin $0\in \partial {\Omega }_{K,L}$.

3. 

If $q\nmid p$ and ${\sum }_{j=1}^{n-1}\left|K_j\right|>n-1$, there does not exist any supporting surface at $0\in \partial {\Omega }_{K,L}$.

The structure of this article is as follows. In Sect. Section 2, we give some basic definitions for Kohn-Nirenberg domain. In Sect. Section 3, we provide the proof of theorem 2 about (1)-(3).

2. Basic Definitions and Lemmas

Let $\Omega =\{(z_1,\dots ,z_n)\in {\mathbb{C}}^n:r\left(z\right)<0\}$ be a domain in ${\mathbb{C}}^n$ with smooth boundary, its defining function is $r\left(z\right)$. Let ${\mathcal{O}}^{\alpha }(\Omega )(0\le \alpha \le \infty )$ be the space of holomorphic functions on ${\Omega }_{K,L}$ and $C^{\alpha }$-continuous on ${\overline{\Omega }}_{K,L}$. For a point $\xi \in \partial \Omega$ and a vector $t=(t_1,\dots ,t_n)\in {\mathbb{C}}^n$, we write $\partial r_{\xi }\left(t\right)={\sum }_{j=1}^n{\displaystyle \frac{\partial r}{\partial z_j}}\left(\xi \right)t_j$. The Levi form of $r\left(z\right)$ at $\xi$ applied to $t\in {\mathbb{C}}^n$ is $L(\xi ,t)={\sum }_{j,k=1}^n{\displaystyle \frac{{\partial }^{2}r}{\partial z_j\partial {\overline{z}}_k}}\left(\xi \right)t_j{\overline{t}}_k$. $\xi$ is called a pseudoconvex point if $L(\xi ,t)={\sum }_{j,k=1}^n{\displaystyle \frac{{\partial }^{2}r}{\partial z_j\partial {\overline{z}}_k}}\left(\xi \right)t_j{\overline{t}}_k\ge 0$ for all $t\in T_{\xi }\partial \Omega =\{t\in {\mathbb{C}}^n:\partial r_{\xi }\left(t\right)={\sum }_{j=1}^n{\displaystyle \frac{\partial r}{\partial z_j}}\left(\xi \right)t_j=0\}$, where $T_{\xi }\partial \Omega =\{t\in {\mathbb{C}}^n:\partial r_{\xi }\left(t\right)=0\}$ is the corresponding complex tangent space. If the Levi form is positive at boundary point $\xi$, i.e.$L(\xi ,t)>0$, we call $\xi$ a strong pseudoconvex point. If all the boundary points are (strong) pseudoconvex points, the domain $\Omega$ is called a (strong) pseudoconvex domain.

Lemma 1 

([9]). For any real number K with $\left|K\right|\le {\displaystyle \frac{m^2}{m^2-1}}(1<m\in {\mathbb{Z}}^{+})$, there exist constants $b\in \mathbb{R}$ and $C>0$ such that

$$1+Kcost+bcos\left(mt\right)>C,\mathrm{for}\mathrm{all}t\in [0,2\pi )$$

where $b=b\left(m\right)$ and $C=C\left(m\right)$ depend only on m .

Lemma 2 

If $\left|K\right|+2\left|L\right|\le {\displaystyle \frac{m^2}{m^2-1}}(1<m\in {\mathbb{Z}}^{+})$, there exist constants $b\in \mathbb{R}$ and $C>0$ such that

$$1+Kcos\left(t\right)+Lsin\left(2t\right)+bcos\left(mt\right)>C,\mathrm{for}\forall t\in [0,2\pi )$$

Proof. 

If $\left|K\right|+2\left|L\right|\le {\displaystyle \frac{m^2}{m^2-1}}$, then $|K+2Lsin\left(t\right)|\le \left|K\right|+2\left|L\right|\le {\displaystyle \frac{m^2}{m^2-1}}$. From Lemma 1 and $Kcos\left(t\right)+Lsin\left(2t\right)=(K+2Lsin(t\left)\right)cos\left(t\right)$, we have constants $b\in \mathbb{R}$ and $C>0$ such that

$$1+Kcos\left(t\right)+Lsin\left(2t\right)+bcos\left(mt\right)>C,\mathrm{for}t\in [0,2\pi )$$

where $b=b\left(m\right)$ and $C=C\left(m\right)$ depend only on m.

Set the domain ${\Omega }_{K,L}=\{(z_1,\dots ,z_n)\in {\mathbb{C}}^n:r\left(z\right)<0\}$, its defining function $r\left(z\right)$ on ${\mathbb{C}}^n$ is as follows

$$r\left(z\right)=Re{z}_n+g{\left|z_n\right|}^2+{\sum }_{j=1}^{n-1}({\left|z_j\right|}^p+K_j{\left|z_j\right|}^{p-q}Re{z}_j^q+L_j{\left|z_j\right|}^{p-2q}Im{z}_j^{2q})$$

where $K=(K_1,...,K_{n-1})\in {\mathbb{R}}^{n-1},L=(L_1,...,L_{n-1})\in {\mathbb{R}}^{n-1},g\ge 0,p\in {\mathbb{Z}}^{+},q\in {\mathbb{Z}}^{+}$ and $p-2q-1\ge 0$. □

Lemma 3 

If $(p^2-q^2)\left|K_j\right|+(p^2-4q^2)\left|L_j\right|\le p^2(1\le j\le n-1)(g\ge 0)$, then ${\Omega }_{K,L}$ is a pseudoconvex domain.

Proof. 

For boundary point $\xi =(z_1,\dots ,z_n)\in \partial \Omega$ and tangent vector $t=(t_1,\dots ,t_n)$, we compute the Levi form and get

$$\begin{array}{cc}\hfill r_{z_j}& ={\displaystyle \frac{1}{2}}p{z}_j^{{\displaystyle \frac{1}{2}}p-1}{\left({\overline{z}}_j\right)}^{{\displaystyle \frac{1}{2}}p}+{\displaystyle \frac{K_j(p+q)}{4}}{z}_j^{{\displaystyle \frac{1}{2}}p+{\displaystyle \frac{1}{2}}q-1}{\left({\overline{z}}_j\right)}^{{\displaystyle \frac{1}{2}}p-{\displaystyle \frac{1}{2}}q}+{\displaystyle \frac{K_j(p-q)}{4}}{z}_j^{{\displaystyle \frac{1}{2}}p-{\displaystyle \frac{1}{2}}q-1}{\left({\overline{z}}_j\right)}^{{\displaystyle \frac{1}{2}}p+{\displaystyle \frac{1}{2}}q}+\hfill \\ & {\displaystyle \frac{L_j(p+2q)}{4i}}{z}_j^{{\displaystyle \frac{1}{2}}p+q-1}{\left({\overline{z}}_j\right)}^{{\displaystyle \frac{1}{2}}p-q}-{\displaystyle \frac{L_j(p-2q)}{4i}}{z}_j^{{\displaystyle \frac{1}{2}}p-q-1}{\left({\overline{z}}_j\right)}^{{\displaystyle \frac{1}{2}}p+q}\hfill \end{array}$$

where $i=\sqrt{-1}$. $r_{z_j{\overline{z}}_k}=0(j\ne k)$ and $r_{z_n{\overline{z}}_n}=g$.

For $1\le j\le n-1$, there is

$$\begin{array}{cc}\hfill r_{z_j{\overline{z}}_j}& ={\displaystyle \frac{1}{4}}{p}^2{\left(z_j\right)}^{{\displaystyle \frac{1}{2}}p-1}+{\displaystyle \frac{K_j(p+q)(p-q)}{8}}{\left(z_j\right)}^{{\displaystyle \frac{1}{2}}p+{\displaystyle \frac{1}{2}}q-1}{\left({\overline{z}}_j\right)}^{{\displaystyle \frac{1}{2}}p-{\displaystyle \frac{1}{2}}q-1}+{\displaystyle \frac{K_j(p-q)}{8}}(p+q)\hfill \\ \hfill & {\left(z_j\right)}^{{\displaystyle \frac{1}{2}}p-{\displaystyle \frac{1}{2}}q-1}{\left({\overline{z}}_j\right)}^{{\displaystyle \frac{1}{2}}p+{\displaystyle \frac{1}{2}}q-1}+{\displaystyle \frac{L_j(p+2q)(p-2q)}{8i}}{\left(z_j\right)}^{{\displaystyle \frac{1}{2}}p+q-1}{\left({\overline{z}}_j\right)}^{{\displaystyle \frac{1}{2}}p-q-1}-\hfill \\ \hfill & {\displaystyle \frac{L_j(p-2q)(p+2q)}{8i}}{\left(z_j\right)}^{{\displaystyle \frac{1}{2}}p-q-1}{\left({\overline{z}}_j\right)}^{{\displaystyle \frac{1}{2}}p+q-1}\hfill \\ \hfill & ={\displaystyle \frac{1}{4}}\left\{p^2\mid z_j{\mid }^{p-2}+K_j(p^2-q^2){\left(z_j\right)}^{{\displaystyle \frac{1}{2}}p-{\displaystyle \frac{1}{2}}q-1}{\left({\overline{z}}_j\right)}^{{\displaystyle \frac{1}{2}}p-{\displaystyle \frac{1}{2}}q-1}\left[{\displaystyle \frac{{\left(z_j\right)}^q+{\left({\overline{z}}_j\right)}^q}{2}}\right]+L_j\right.\hfill \\ \hfill & (p^2-4q^2){\left(z_j\right)}^{{\displaystyle \frac{1}{2}}p-q-1}{\left({\overline{z}}_j\right)}^{{\displaystyle \frac{1}{2}}p-q-1}\left.\left[{\displaystyle \frac{{\left(z_j\right)}^{2q}-{\left({\overline{z}}_j\right)}^{2q}}{2i}}\right]\right\}\hfill \\ \hfill & ={\displaystyle \frac{1}{4}}\left\{p^2{\left|z_j\right|}^{p-2}+(p^2-q^2)K_j{\left|z_j\right|}^{p-q-2}Re{\left(z_j\right)}^q+(p^2-4q^2)L_j{\left|z_j\right|}^{p-2q-2}\right.\left.Im{\left(z_j\right)}^{2q}\right\}\hfill \end{array}$$

If $(p^2-q^2)\left|K_j\right|+(p^2-4q^2)\left|L_j\right|\le p^2$ and $g\ge 0$, we have the Levi form

$$\begin{array}{cc}\hfill L(\xi ,t)& ={\sum }_{j,k=1}^n{\displaystyle \frac{{\partial }^{2}r}{\partial z_j\partial {\overline{z}}_k}}\left(\xi \right)t_j{\overline{t}}_k\hfill \\ & ={\sum }_{j=1}^{n-1}{r}_{z_j{\overline{z}}_j}{\left|t_j\right|}^2+g{\left|t_n\right|}^2\hfill \\ & \ge {\displaystyle \frac{1}{4}}{\sum }_{j=1}^{n-1}(p^2-(p^2-q^2))\left|K_j\right|-(p^2-4q^2)\left|L_j\right|){\left|z_j\right|}^{p-2}{\left|t_j\right|}^2+g{\left|t_n\right|}^2\hfill \\ & \ge 0.\hfill \end{array}$$

Thus the Levi form $L(\xi ,t)$ is semi-positive definite, which proves that ${\Omega }_{K,L}$ is pseudoconvex. □

3. Holomorphic Peak Function and Supporting Surface of the General Modified Domain ${\Omega }_{K,L}$

Here we prove the main Theorem 2 about (1)-(3).

Proof 

(Proof of Theorem 2 about (1)). Let ${\Omega }_{K,L}=\{z\in {\mathbb{C}}^n:Re{z}_n+g{\left|z_n\right|}^2+{\sum }_{j=1}^{n-1}({\left|z_j\right|}^p+K_j{\left|z_j\right|}^{p-q}$

$Re{z}_j^q+L_j{\left|z_j\right|}^{p-2q}Im{z}_j^{2q})<0\}$. Suppose $|K_j|+2\left|L_j\right|\le {\displaystyle \frac{p^2}{p^2-q^2}}(1\le j\le n-1,g\ge 0)$, then $(p^2-q^2)\left|K_j\right|+(p^2-4q^2)\left|L_j\right|\le (p^2-q^2)\left|K_j\right|+2(p^2-q^2)\left|L_j\right|\le p^2(1\le j\le n-1,g\ge 0)$. So Lemma 3 implies that ${\Omega }_{K,L}$ is pseudoconvex. □

Remark 1. 

Let ${\Omega }_{K,L}$ be the above domain. If $\left|K_j\right|+2\left|L_j\right|\le {\displaystyle \frac{p^2}{p^2-q^2}}(1\le j\le n-1,g\ge 0)$, it is easy to see the origin is a weakly pseudoconvex (not strong pseudoconvex ) boundary point of ${\Omega }_{K,L}$.

Proof 

(Proof of Theorem 2 about (2)). Let ${\Omega }_{K,L}$ be the above domain with $\left|K_j\right|+2\left|L_j\right|\le {\displaystyle \frac{p^2}{p^2-q^2}}(1\le j\le n-1,g\ge 0)$.

For $r\left(z\right)=Re{z}_n+g{\left|z_n\right|}^2+{\sum }_{j=1}^{n-1}({\left|z_j\right|}^p+K_j{\left|z_j\right|}^{p-q}Re{z}_j^q+L_j{\left|z_j\right|}^{p-2q}Im{z}_j^{2q})$, we consider two cases.

• The case $q\mid p(p=mq)$. Let $z_j=r_j{e}^{i{\theta }_j}(1\le j\le n-1)$ and $z_n=u+iv$. In polar coordinate system, we have $r(z_1,z_2,\dots ,z_n)=u+g{u}^2+g{v}^2+{\sum }_{j=1}^{n-1}[1+K_{j}cos\left(q{\theta }_j\right)+L_{j}sin\left(2q{\theta }_j\right)]r_j^p$. Then we consider the coordinate transformation $z_1^{*}=z_1;z_2^{*}=z_2;\cdots ;$
  $z_{n-1}^{*}=z_{n-1};z_n^{*}=z_n-{\sum }_{j=1}^{n-1}{b}_j{z}_j^p$, and $b_j\in \mathbb{R}$. In the new coordinate system, after dropping the stars we have

  $$\begin{array}{cc}\hfill r(z_1,z_2,\dots ,z_n)& =u+g{(u+{\sum }_{j=1}^{n-1}{b}_{j}cos\left(p{\theta }_j\right)r_j^p)}^2+g{(v+{\sum }_{j=1}^{n-1}{b}_{j}sin\left(p{\theta }_j\right)r_j^p)}^2+\hfill \\ \hfill & {\sum }_{j=1}^{n-1}[1+K_{j}cos\left(q{\theta }_j\right)+L_{j}sin\left(2q{\theta }_j\right)+b_{j}cos\left(p{\theta }_j\right)]r_j^p\hfill \end{array}$$
  Note that $\left|K_j\right|+2\left|L_j\right|\le {\displaystyle \frac{p^2}{p^2-q^2}}={\displaystyle \frac{m^2}{m^2-1}}(p=mq)$. Lemma 2 implies that there exists $C>0$ such that

  $${\sum }_{j=1}^{n-1}[1+K_{j}cos\left(q{\theta }_j\right)+L_{j}sin\left(2q{\theta }_j\right)+b_{j}cos\left(p{\theta }_j\right)]r_j^p>C{\sum }_{j=1}^{n-1}{r}_j^p$$
  The point 0 belongs to the set $\{(z_1,z_2,\dots ,0):z_j\in \mathbb{C},1\le j\le n-1\}\cap N\left(0\right)\cap {\overline{\Omega }}_{K,L}$, where $N\left(0\right)$ is a neighborhood of 0. For all $0\ne z\in \{(z_1,z_2,...,0):z_j\in \mathbb{C},1\le j\le n-1\}\cap N\left(0\right)\cap {\overline{\Omega }}_{K,L}$, there exists j such that $z_j\ne 0$. We have

  $$\begin{array}{cc}\hfill r(z_1,z_2,\dots ,0)& =g\left\{{\left({\sum }_{j=1}^{n-1}{b}_{j}cos\left(p{\theta }_j\right)r_j^p\right)}^2+{\sum }_{j=1}^{n-1}{b}_{j}sin\left(p{\theta }_j\right)r_j^p{)}^2\right\}+0+{\sum }_{j=1}^{n-1}[1+\hfill \\ \hfill & K_{j}cos\left(q{\theta }_j\right)+L_{j}sin\left(2q{\theta }_j\right)+b_{j}cos\left(p{\theta }_j\right)]r_j^p>C{\sum }_{j=1}^{n-1}{r}_j^p>0\hfill \end{array}$$
  This is a contradiction with the definition $r(z_1,z_2,\dots ,z_n)\le 0$ and implies that

  $$\{(z_1,z_2,...,0):z_j\in \mathbb{C},1\le j\le n-1\}\cap N\left(0\right)\cap {\overline{\Omega }}_{K,L}=\left\{0\right\}.$$
  Thus in the new coordinates, the complex manifold $\{(z_1,z_2,\dots ,0):z_j\in \mathbb{C},1\le j\le n-1\}$ is a holomorphic supporting surface at the origin $0\in \partial {\Omega }_{K,L}$. The holomorphic supporting function is $f(z_1,z_2,\dots ,z_n)=z_n$ at 0. And the corresponding holomorphic peak function is $h(z_1,z_2,\dots ,z_n)=e^{z_n+z_n^2}$ for the origin 0.
  In fact, it is obvious that $h\left(0\right)=1$. Put $z_n=u+iv$. For $\forall z\in {\overline{\Omega }}_{K,L}-\left\{0\right\}$, we have $r(z_1,z_2,\dots ,z_n)\le 0$, i.e.

  $$\begin{array}{cc}\hfill & u+g{(u+{\sum }_{j=1}^{n-1}{b}_{j}cos\left(p{\theta }_j\right)r_j^p)}^2+g{(v+{\sum }_{j=1}^{n-1}{b}_{j}sin\left(p{\theta }_j\right)r_j^p)}^2+\hfill \\ & {\sum }_{j=1}^{n-1}[1+K_{j}cos\left(q{\theta }_j\right)+L_{j}sin\left(2q{\theta }_j\right)+b_{j}cos\left(p{\theta }_j\right)]r_j^p\le 0\hfill \end{array}$$
  Thus

  $$u\le -{\sum }_{j=1}^{n-1}[1+K_{j}cos\left(q{\theta }_j\right)+L_{j}sin\left(2q{\theta }_j\right)+b_{j}cos\left(p{\theta }_j\right)]r_j^p\le -C{\sum }_{j=1}^{n-1}{r}_j^p\le 0.$$
  If $u=0$, then $v<0$ and $\left|h(z_1,z_2,\dots ,z_n)\right|=e^{u+u^2-v^2}=e^{-v^2}<1$.
  If $u<0$, then $\left|h(z_1,z_2,\dots ,z_n)\right|=e^{u+u^2-v^2}=e^{-v^2}<1$.
  So, the function $h(z_1,z_2,\dots ,z_n)$ is a local peak function at $0\in \partial {\Omega }_{K,L}$. Further, Hakim and Sibony [1,2] show that there is a global peak function with the same regularity as $h(z_1,z_2,\dots ,z_n)$.
• The case $q\nmid p$ and $\left|K_j\right|+\left|L_j\right|<1(1\le j\le n-1)$.There exists holomorphic peak function and supporting surface at the origin. Note that ${\sum }_{j=1}^{n-1}({\left|z_j\right|}^p+K_j{\left|z_j\right|}^{p-q}Re\left(z_j^q\right)+L_j{\left|z_j\right|}^{p-2q}Im\left(z_j^{2q}\right))\ge {\sum }_{j=1}^{n-1}[1-(\left|K_j\right|+\left|L_j\right|)]{\left|z_j\right|}^p$, similar to case (I), we have the complex manifold $\{(z_1,z_2,...,0):z_j\in \mathbb{C},1\le j\le n-1\}$, which is also a holomorphic supporting surface at the origin. At the same time, $h(z_1,z_2,...,z_n)=e^{z_n+z_n^2}$ is a local peak function at $0\in \partial {\Omega }_{K,L}$. In [1,2], Hakim and Sibony show that there is a global peak function with the same regularity as h.

□

Proof 

(Proof of Theorem 2 about (3)). Assume that there exists supporting surface at the origin $0\in \partial {\Omega }_{K,L}$. The support surface M as a complex manifold of co-dimension 1 implies that there are an open neighbourhood $N\left(0\right)\subset {\mathbb{C}}^n$ and holomorphic function f on $N\left(0\right)$ such that

• $M\cap N\left(0\right)=\{z\in N(0):f(z)=0\}$;
• $rank({\displaystyle \frac{\partial f}{\partial z_1}},...,{\displaystyle \frac{\partial f}{\partial z_n}})=1$.

We shall study two different cases.

• The case ${\displaystyle \frac{\partial f}{\partial z_n}}=0$, there is some j such that ${\displaystyle \frac{\partial f}{\partial z_j}}\ne 0$. The implicit function theorem implies that

  $$M=\{(z_1,\dots ,z_n)\in {\mathbb{C}}^n:z_j=\varphi (z_1,\dots ,z_{j-1},z_j,\dots ,z_n)\}$$
  Now let $z_1=\cdots =z_{j-1}=z_{j+1}=\cdots =z_n=-\epsilon$, then

  $$\begin{array}{cc}\hfill r\left(z\right)& =-\epsilon +g{\epsilon }^2+(n-2){\epsilon }^p+{(-1)}^p{\sum }_{l\ne j}{K}_l{\left|\varphi (-\epsilon ,\dots ,-\epsilon )\right|}^p+K_j{\left|\varphi (-\epsilon ,\dots ,-\epsilon )\right|}^{p-q}\hfill \\ \hfill & Re{\left[\varphi (-\epsilon ,\dots ,-\epsilon )\right]}^q+L_j{\left|\varphi (-\epsilon ,\dots ,-\epsilon )\right|}^{p-2q}Re{\left[\varphi (-\epsilon ,\dots ,-\epsilon )\right]}^{2q}\hfill \\ & =-\epsilon +O\left({\epsilon }^2\right)<0.\hfill \end{array}$$
  If $\epsilon$ is small, then $M\cap {\overline{\Omega }}_{K,L}\ne \left\{0\right\}$ in every small neighborhood of 0. Therefore, we have a contradiction with M as a support surface.
• The case ${\displaystyle \frac{\partial f}{\partial z_n}}\ne 0$, the implicit function theorem implies that

  $$M=\{(z_1,...,z_n)\in {\mathbb{C}}^n:z_n=\varphi (z_1,...,z_{n-1})\}$$

We shall divide this into three different cases.

• When $\varphi (z_1,\dots ,z_{n-1})={\sum }_{s=t}^{\infty }{F}_s(z_1,\dots ,z_{n-1}),t\ge p+1$, where $F_s$ is the sum of those terms $C_{{\alpha }_1,\dots ,{\alpha }_{n-1}}{z}_1^{{\alpha }_1}\cdots z_{n-1}^{{\alpha }_{n-1}}$ in the power series for which ${\alpha }_1+\cdots +{\alpha }_{n-1}=s$.
  We let $z_j=\epsilon e^{i{\displaystyle \frac{(\chi \left(K_j\right)+1)}{q}}\pi },1\le j\le n-1$, where $\chi \left(K_j\right)$ is defined by

  $$\begin{array}{c}\hfill \chi \left(K_j\right)=\left\{\begin{array}{c}0K_j>0\\ 1K_j<0\end{array}\right.(1\le j\le n-1).\end{array}$$
  And $z_n=\varphi (z_1,...,z_{n-1})$. If $\epsilon$ is sufficiently small,

  $$\begin{array}{cc}\hfill r\left(z\right)& =Re\left[\varphi (\epsilon e^{i{\displaystyle \frac{\chi \left(K_1\right)+1}{q}}\pi },\dots ,\epsilon e^{i{\displaystyle \frac{\chi \left(K_{n-1}\right)+1}{q}}\pi })\right]+g{\left|\varphi (·)\right|}^2+(n-1){\epsilon }^p-{\epsilon }^p{\sum }_{j=1}^{n-1}\left|K_j\right|\hfill \\ & =[(n-1)-{\sum }_{j=1}^{n-1}\left|K_j\right|]{\epsilon }^p+O\left({\epsilon }^{p+1}\right)<0.\hfill \end{array}$$
  Hence M is not a supporting surface. It is a contradiction.
• When $\varphi (z_1,\dots ,z_{n-1})={\sum }_{s=t}^{\infty }{F}_s(z_1,\dots ,z_{n-1}),1\le t\le p-1$, $F_t(z_1,\dots ,z_{n-1})\ne 0$. We can suppose $F_t(t_1,\dots ,t_{n-1})={\lambda }_1\ne 0$ and choose $\theta$ such that ${\lambda }_1{e}^{i\theta t}<0$.
  Let $z_1=t_1\epsilon e^{i\theta },z_2=t_2\epsilon e^{i\theta },\dots ,z_{n-1}=t_{n-1}\epsilon e^{i\theta }$, it is easy to see that

  $$F_t(z_1,\dots ,z_{n-1})=F_t(t_1,\dots ,t_{n-1}){\epsilon }^t{e}^{i\theta }={\lambda }_1{e}^{it\theta }{\epsilon }^t=-\left|{\lambda }_1\right|{\epsilon }^t.$$
  Then

  $$\begin{array}{cc}\hfill r\left(z\right)& =Re\left[\varphi (z_1,\dots ,z_{n-1})\right]+g{\left|\varphi (z_1,\dots ,z_{n-1})\right|}^2+{\epsilon }^p{\sum }_{j=1}^{n-1}{\left|t_j\right|}^p+{\epsilon }^{p}cosq\theta \hfill \\ \hfill & \left({\sum }_{j=1}^{n-1}{K}_j{\left|t_j\right|}^p\right)+{\epsilon }^{p}sin2q\theta \left({\sum }_{j=1}^{n-1}{L}_j{\left|t_j\right|}^p\right)\hfill \\ & =-\left|{\lambda }_1\right|{\epsilon }^t+O\left({\epsilon }^{t+1}\right)<0\hfill \end{array}$$

  if $\epsilon$ is sufficiently small. Hence M is not a supporting surface. It is a contradiction.
• Then the only remaining case is when

  $$\varphi (z_1,\dots ,z_{n-1})={\sum }_{s=p}^{\infty }{F}_s(z_1,\dots ,z_{n-1}).$$
  Let $z_j^l=\epsilon e^{i{\displaystyle \frac{\chi \left(K_j\right)+l}{q}}\pi }(1\le j\le n-1)$, where $\chi \left(K_j\right)$ is defined by

  $$\begin{array}{c}\hfill \chi \left(K_j\right)=\left\{\begin{array}{c}0K_j>0\\ 1K_j<0\end{array}\right.\end{array}$$

  and $z_n^l=\varphi (z_1^l,\dots ,z_{n-1}^l)$. Then

  $$\begin{array}{cc}\hfill r\left(z^l\right)& =Re\left[\varphi (z_1^l,\dots ,z_{n-1}^l)\right]+g{\left|\varphi (z_1^l,\dots ,z_{n-1}^l)\right|}^2+(n-1){\epsilon }^p+{\epsilon }^{p}cosl\pi \left({\sum }_{j=1}^{n-1}\left|K_j\right|\right)\hfill \\ & ={\epsilon }^p[Re\left(\lambda e^{i{\displaystyle \frac{p\pi }{q}}l}\right)+O\left(\epsilon \right)+(n-1)+cosl\pi \left({\sum }_{j=1}^{n-1}\left|K_j\right|\right)],\hfill \end{array}$$

  where $\lambda =F_p(e^{i{\displaystyle \frac{\chi \left(K_1\right)}{q}}\pi },...,e^{i{\displaystyle \frac{\chi \left(K_{n-1}\right)}{q}}\pi })$. We take $l=1,3,...,2q-1$, then

  $$\begin{array}{cc}\hfill \sum _{l=1,3,\dots ,2q-1}r\left(z^l\right)& ={\epsilon }^p\left\{Re\lambda \left(\sum _{l=1,3,\dots ,2q-1}{e}^{i{\displaystyle \frac{p\pi }{q}}l}\right)+O\left(\epsilon \right)+q(n-1)+\sum _{l=1,3,\dots ,2q-1}cosl\pi \right.\hfill \\ \hfill & \left.\left({\sum }_{j=1}^{n-1}\left|K_j\right|\right)\right\}\hfill \end{array}$$
  Since l takes odd integers, we obtain

  $$\sum _{l=1,3,\dots ,2q-1}cosl\pi =-q.$$
  If $q\nmid p$, we have ${\sum }_{l=1,3,\dots ,2q-1}{e}^{i{\displaystyle \frac{p\pi }{q}}l}=0$. Therefore, when $\epsilon$ is small,

  $$\begin{array}{cc}\hfill \sum _{l=1,3,\dots ,2q-1}r\left(z^l\right)& ={\epsilon }^p\left\{O\left(\epsilon \right)+q[(n-1)-\sum _{j=1}^{n-1}\left|K_j\right|]\right\}<0.\hfill \end{array}$$
  Since $M\cap {\overline{\Omega }}_{K,L}=\left\{0\right\}$ and $z^l\in M$, it follows that $r\left(z^l\right)>0$ for all l.
  Thus

  $$\sum _{l=1,3,\dots ,2q-1}r\left(z^l\right)>0.$$
  Hence we get a contradiction. This completes the proof.

□

Conjecture. According to the result of Theorem 2 about (3), we can conjecture that ${\Omega }_{K,L}$ has no peak functions at 0, under the same condition.