---
abstract: 'We study $c$-completeness on domains in ${{\mathbb C}}^n$. We reprove Sibony/Selby result on completeness on the complex plane. We also give a characterization of $c$-completeness in Reinhardt domains.'
address: 'Faculty of Mathematics and Computer Science, Jagiellonian University, Łojasiewicza 6, 30-348 Kraków, Poland'
author:
- Armen Edigarian
title: 'On Carathéodory completeness in ${{\mathbb C}}^n$'
---

[^1]

Introduction
============

Let $X$ be a complex manifold. Put $$c_X(z,w)=\sup\{\rho(f(z),f(w)): f\in{{\mathcal O}}(X,{{\mathbb D}})\},$$ where $\rho$ denotes the hyperbolic (Poincaré) distance in the unit disc ${{\mathbb D}}\subset{{\mathbb C}}$ (see e.g. [@J-P], Chapter I) and ${{\mathcal O}}(X,{{\mathbb D}})$ denotes the set of all holomorphic mappings $X\to{{\mathbb D}}$. We call $c_X$ the Carathéodory pseudodistance (see e.g. [@J-P], Chapter II).

Note that, in general, $c_X$ is not a distance (e.g., $c_{{{\mathbb C}}^n}\equiv0$). We say that a manifold $X$ is $c$-hyperbolic if $c_X$ is a distance, i.e., for any $z,w\in X$ there exists a bounded holomorphic function $f$ on $X$ such that $f(z)\not=f(w)$. For $c$-hyperbolic manifold it is natural to study a subclass of manifolds for which the metric space $(X,c_X)$ is complete. We call this type of manifolds $c$-complete. Moreover, we say that a manifold $X$ is $c$-finitely compact if for any $x_0\in X$ and for any $r>0$ we have $\{x\in X: c_X(x_0,x)<r\}\Subset X$. Note that any $c$-finitely compact domain is $c$-complete. Whether the inverse implication holds for domains in ${{\mathbb C}}^n$ is still an open question.

The following Conjecture is open for more then 30 years

Let $D\subset{{\mathbb C}}^n$ be a bounded pseudoconvex domain with $C^\infty$ boundary. Then $D$ is $c$-complete.

In our opinion, one of the main problem is lack of good conjectures, which “approximate” to the above one. Our aim is to give such a conjecture and verify it for some class of domains.

First recall the following result, proved by P. Pflug

\[thm:1:pflug\] Let $D\subset{{\mathbb C}}^n$ be a $c$-hyperbolic domain. Then the following conditions are equivalent:

1.  $D$ is $c$-finitely compact;

2.  for any sequence $\{z_n\}_{n=1}^\infty\subset D$ without accumulation points (w.r.t. usual topology in $D$) there exists an $f\in{{\mathcal O}}(D,{{\mathbb D}})$ with $\sup_{n}|f(z_n)|=1$.

Following Kosiński and Zwonek [@KZ], for a domain $D\subset{{\mathbb C}}^n$ we say that $\zeta\in\partial D$ is a weak peak point if there exists $f\in{{\mathcal O}}(D)\cap C(D\cup\{\zeta\})$ such that $|f|<1$ on $D$ and $f(\zeta)=1$.

The main result of the paper is to show

\[thm:1\] Let $D\subset{{\mathbb C}}^n$ be a $c$-hyperbolic pseudoconvex domain. Assume that

-   $n=1$, or

-   $D$ is a Reinhardt domain.

Then the following conditions are equivalent:

1.  $D$ is $c$-complete.

2.  $D$ is $c$-finitely compact.

3.  any $\zeta\in\partial D$ is a weak peak point.

Theorem \[thm:1\] in case $n=1$ is the result proved by N. Sibony [@Sib] and M.A. Selby [@Sel]. However, their proof heavily depends on Melnikov’s theorem (see e.g. [@GG]) which is used to show the existence of a specific measure and, in our opinion, are strictly one-dimensional. We show that the measure one can get from a generalization of Edwards’ theorem.

Note that in Theorem \[thm:1\] the equivalence $(1)\Longleftrightarrow (2)$ is well-known. So, our result states that these conditions are equivalent to condition $(3)$.

The above result suggests that c-finitely compactness is equivalent to the existence of weak peak functions.

Non-compact version of Edwards’ Theorem
=======================================

This part of the paper is motivated by [@GPP] and we use methods of that paper.

Let $X$ be a topological space and let $C(X)$ (resp. $C_{{{\mathbb C}}}(X)$) be the set of all real-valued (resp. complex-valued) continuous functions on $X$. We say that $S\subset C(X)$ is a convex cone if $S$ contains constant functions and $\alpha f+\beta g\in S$ for any $f,g\in S$ and any $\alpha,\beta\ge0$. Fix $x\in X$. For any ${\varphi}\in C(X)$ we consider its envelope defined as $$\Phi_x({\varphi})=\sup\{\psi(x): \psi\in S, \psi\le{\varphi}\}.$$ Note that $\Phi_x:C(X)\to[-\infty,+\infty)$ is a positive superlinear operator, i.e.,

1.  $\Phi_x(c{\varphi})=c\Phi_x({\varphi})$ for any ${\varphi}\in C(X)$ and any $c\ge0$;

2.  $\Phi_x({\varphi}_1+{\varphi}_2)\ge \Phi_x({\varphi}_1)+\Phi_x({\varphi}_2)$ for any ${\varphi}_1,{\varphi}_2\in C(X)$;

3.  $\Phi_x({\varphi})\ge0$ for any ${\varphi}\in C(X)$ such that ${\varphi}\ge0$.

We have the following result (which follows from a version of Hahn-Banach theorem, see e.g., [@GPP]).

Let $X$ be a topological space and let $x\in X$ be a fixed point. Assume that $S\subset C(X)$ is a convex cone. Then $$\Phi_x({\varphi})=\min\{L({\varphi}): L\text{ is a positive linear operator on }C(X), L\ge\Phi_x\}.$$

There is an extensive literature on the study of positive linear functionals on $C(X)$ (see e.g., [@Hewitt], [@BP], and [@Res])[^2]. However, we are interested on very special ones, related to compact spaces. Let $X$ be a normal topological space. We say that $X$ is of $GPP$-type if for any positive linear functional $\Lambda:C(X)\to{{\mathbb R}}$ there exists a compact set $K\subset X$ such that $\Lambda({\varphi})=0$ for any ${\varphi}\in C(X)$ with ${\varphi}=0$ on $K$. According to [@GPP] any locally compact $\sigma$-compact Hausdorff space is of $GPP$-type. We have another example.

Let $D\subset{{\mathbb C}}^n$ be a domain and let $K\subset\partial D$ be a compact set. Then $X=D\cup K$ is of $GPP$-type.

Take sequences $R_m,r_m$ such that $R_m>r_m>R_{m+1}$ and $R_m\to 0$ (e.g., $R_m=\frac{1}{3^m}$ and $r_m=\frac{2}{3^{m+1}}$). Consider functions $\chi_m\in C^\infty({{\mathbb R}})$ such that $0\le \chi_m\le 1$ having the following properties: $$\chi_1(t)=
\begin{cases}
1& t\ge R_1\\
0& t\le r_1
\end{cases}$$ and for any $m\ge2$ $$\chi_m(t)=
\begin{cases}
1-\sum_{j=1}^{m-1}\chi_j(t) & t\ge R_m\\
0& t\le r_m
\end{cases}.$$ Note that $\sum_{m=1}^\infty \chi_m(t)=1$ for $t>0$. Moreover, $\chi_m(t)=0$ for $t\ge R_{m-1}$ and $t\le r_{m}$.

Put $A_1=\{x\in X: \operatorname{dist}(x,K)\ge r_1\}$ and $A_m=\{x\in X: r_m\le \operatorname{dist}(x,K)\le R_{m-1}\}$, $m\ge2$. Note that $A_m$, $m\ge2$, are (relatively) closed sets in $D$ and that $\chi_m(\operatorname{dist}(x,K))=0$ for $x\in{{\mathbb C}}^n\setminus A_m$. Moreover, $\cup_{m=1}^\infty A_m=D$. We want to show that for any $m\ge1$ there exists a compact set $K_m\subset A_m$ with the following property: $\Lambda({\varphi})=0$ whenever ${\varphi}\in C(X)$, ${\varphi}\ge0$, and ${\varphi}=0$ on $(X\setminus A_m)\cup K_m$.

Fix $m\ge1$. For any integer $j\ge1$ we put $$K_{mj}=\{x\in A_m: \operatorname{dist}(x,\partial D)\ge\frac{1}{j}, \|x\|\le j\}.$$ Note that $K_{mj}$ are compact sets. We also have $\cup_{j=1}^\infty K_{mj}=A_m$ for any $m\ge1$. Let us show that there exists a $j=j(m)$ such that $\Lambda({\varphi})=0$ whenever ${\varphi}\in C(X)$, ${\varphi}\ge0$, ${\varphi}=0$ on $(D\setminus A_{m})\cup K_{mj}$. Indeed, assume that for any $j\ge1$ there exists ${\varphi}_{j}\in C(X)$ such that ${\varphi}_j\ge0$ on $X$, ${\varphi}_j=0$ on $(X\setminus A_{m})\cup K_{mj}$, and $\Lambda({\varphi}_j)\not=0$. Without loss of generality we may assume that $\Lambda({\varphi}_j)=1$. Consider ${\varphi}=\sum_{j=1}^\infty {\varphi}_j$. Note that ${\varphi}\in C(X)$. Moreover, $\Lambda({\varphi})\ge \sum_{j=1}^N \Lambda({\varphi}_j)=N$ for any $N\in{{\mathbb N}}$. Hence, $\Lambda({\varphi})=+\infty$. A contradiction.

Consider a set $$L=\cup_{j=1}^\infty K_{mj(m)}\cup K.$$ Note that $L$ is a compact set. It suffices to show that $\Lambda({\varphi})=0$ for any ${\varphi}\in C(X)$ such that ${\varphi}=0$ on $L$. First let us shot the mentioned property for ${\varphi}\ge0$. So, fix a function ${\varphi}\in C(X)$ such that ${\varphi}\ge0$ and ${\varphi}=0$ on $L$. Fix $\epsilon>0$. Since ${\varphi}=0$ on $K$, there exists a $\delta>0$ such that ${\varphi}<\epsilon$ on $\{z\in D: \operatorname{dist}(z,K)\le\delta\}$. We have $${\varphi}=\sum_{m=0}^\infty {\varphi}\psi_m=\sum_{m=0}^{m_0}{\varphi}\psi_m+\sum_{m=m_0+1}^\infty{\varphi}\psi_m.$$ Put $\tilde{\varphi}=\sum_{m=m_0+1}^\infty{\varphi}\psi_m$. For sufficiently big $m_0$ we have $\tilde{\varphi}=0$ on $\{z\in D: \operatorname{dist}(z,K)>\delta\}$. And, therefore, $0\le\tilde{\varphi}\le\epsilon$ on $D$. So, $$\Lambda({\varphi})=\sum_{m=0}^{m_0}\Lambda({\varphi}\psi_m)+\Lambda(\tilde{\varphi})=\Lambda(\tilde{\varphi})\le\epsilon\Lambda(1).$$ Since $\epsilon>0$ was arbitrary, we get $\Lambda({\varphi})=0$.

If ${\varphi}$ is not necessary positive, then take ${\varphi}={\varphi}_+-{\varphi}_-$, where ${\varphi}_+=\max\{{\varphi},0\}$ and ${\varphi}_-={\varphi}_+-{\varphi}$. Then $L({\varphi})=L({\varphi}_+)-L({\varphi}_-)=0$.

We need also the following version of the Riesz representation theorem.

Let $X$ be a normal topological space and let $L:C(X)\to{{\mathbb R}}$ be a positive linear functional. Assume that there exists a compact set $K\subset X$ such that $L({\varphi})=0$ whenever ${\varphi}\in C(X)$, ${\varphi}=0$ on $K$. Then there exists a Borel finite measure $\mu$ with support in $K$ such that $$L({\varphi})=\int{\varphi}(x)d\mu(x)\quad\text{ for any }{\varphi}\in C(X).$$

Let us define a positive linear operator $\widetilde L:C(K)\to{{\mathbb R}}$. Fix ${\varphi}\in C(K)$. From the normality of $X$ there exists $\widetilde{\varphi}\in C(X)$ such that $\widetilde{\varphi}={\varphi}$ on $K$. We put $\widetilde L({\varphi})=L(\widetilde{\varphi})$. Now we use the classical Riesz representation theorem for the operator $\widetilde L$ and get $\mu$.

For each convex cone $S$ and a point $x\in X$ we associate two sets:

1.  $J_x^{S}(X)$ - the set of all *Jensen measures* with barycenter at $x$ which consists of all Borel probability measures $\mu$ with compact support such that $\psi(x)\le\int \psi d\mu$ for any $\psi\in X$;

2.  $R_x^{S}(X)$ - the set of all *representing measures* with barycenter at $x$ which consists of all Borel probability measures $\mu$ with compact support such that $\psi(x)=\int \psi d\mu$ for any $\psi\in X$;

Note that $R_x^{S}(X)\subset J_x^{S}(X)$.

In 1965 Edwards [@E] proved the following result:

Let $X$ be a compact topological space and let $S\subset C(X)$ be a convex cone. Assume that ${\varphi}$ is a lower semicontinuous function on $X$. Then $$\Phi_x({\varphi})=\min\Big\{\int {\varphi}d\mu: \mu\in J_{x}^S(X)\Big\}.$$

In 2013 Gogus, Perkins, and Poletsky [@GPP] proved the following non-compact version of Edwards’ theorem

\[thm:GPP\] Let $X$ be a locally compact $\sigma$-compact Hausdorff space and let $S\subset C(X)$ be a convex cone. Assume ${\varphi}$ be a continuous function on $X$. Then $\Phi_x({\varphi})\equiv-\infty$ or $$\Phi_x({\varphi})=\min\Big\{\int {\varphi}d\mu: \mu\in J_{x}^S(X)\Big\}.$$

From the above results we get

Let $X$ be a normal topological space of $GPP$-type and let $S\subset C(X)$ be a convex cone. If ${\varphi}\in C(X)$ then for any $x\in X$ we have $\Phi_x({\varphi})=-\infty$ or $$\Phi_x({\varphi})=\inf\Big\{\int{\varphi}d\mu: \mu\in J_{x}^{{{\mathcal S}}}\Big\}\quad\text{ for any }x\in X.$$

We have the following important corollary, which is a non-compact counterpart of Theorems II.11.1 and II.11.3 in [@Gam].

\[cor:10\] Let $D\subset{{\mathbb C}}^n$ be a domain and let $\zeta\in\partial D$. We put $X=D\cup\{\zeta\}$ and $S=\{\Re(f): f\in A(X)\}$, where $A(X)=H^\infty(D)\cap C_{{{\mathbb C}}}(X)$. Then the following conditions are equivalent:

1.  $R_{\zeta}^S(X)=\{\delta_\zeta\}$;

2.  there exists a function $f\in A(X)$ such that $f(\zeta)=1$ and $|f|<1$ on $D$, i.e., $\zeta$ is a peak point for $A(X)$;

3.  there exist a $c\in(0,1)$ and an $M\ge1$ such that for any $r>0$ there exists an $f\in A(X)$ with

    1.  $\|f\|_{X}\le M$;

    2.  $f(\zeta)=1$;

    3.  $|f|\le c$ on $X\setminus{{\mathbb D}}(\zeta,r)$.

Carathéodory completeness on the complex plane
==============================================

Before we state main results of this section, let us recall some notions and results from one-dimensional analysis.

Let $D\subset{{\mathbb C}}$ be a domain and let $\zeta\in\partial D$. Recall that if $$\limsup_{r\to0}\frac{{{\mathcal L}}({{\mathbb D}}(\zeta,r)\setminus D)}{r^2}>0$$ then $\zeta$ is a peak point for $A(\overline{D})={{\mathcal O}}(D)\cap C(\overline{D})$. Here, ${{\mathcal L}}$ denotes the Lebesgue measure on the complex plane. Essentially, its Curtis’s Criterion and the relation between analytic capacity and the Lebesgue measure (see e.g. [@Gam], Corollary VIII.4.2).

So, if $\zeta$ is not a peak point for $A(\overline{D})$ then $$\label{eq:5}
\lim_{r\to0}\frac{{{\mathcal L}}({{\mathbb D}}(\zeta,r)\cap D)}{\pi r^2}=1.$$

Let ${{\mathcal M}}$ denotes the set of all positive probability measure in ${{\mathbb C}}$ with compact support and let $\mu\in{{\mathcal M}}$. We define its Newton potential as $M(\xi)=\int\frac{1}{|w-\xi|}d\mu(w)$. The following result is a corollary of Fubini’s theorem (see e.g. [@Sto], Lemma 26.16).

\[prop:11\] For any $\zeta\in{{\mathbb C}}$ we have $$\lim_{r\to0}\frac{1}{\pi r^2}\int_{{{\mathbb D}}(\zeta,r)}|w-\zeta|\cdot M(w)d{{\mathcal L}}(w)=\mu(\{\zeta\}).$$ In particular, if $\mu(\{\zeta\})=0$ then for any $\epsilon>0$ the set $$\Pi(\epsilon)=\{w\in{{\mathbb C}}: |w-\zeta|\cdot M(w)>\epsilon\}$$ has the property $$\lim_{r\to0}\frac{{{\mathcal L}}(\Pi(\epsilon)\cap{{\mathbb D}}(\zeta,r))}{\pi r^2}=0.$$

As a Corollary of Proposition \[prop:11\] and we get the following

\[cor:12\] Let $D\subset{{\mathbb C}}$ be a domain, let $\zeta\in\partial D$, and let $\mu$ be a Borel measure with compact support in $D\cup\{\zeta\}$ such that $\mu(\{\zeta\})=0$. Assume that $\zeta$ is not a peak point for $A(\overline D)$. Then $$\lim_{r\to0}\frac{{{\mathcal L}}(D\cap{{\mathbb D}}(\zeta,r)\setminus \Pi(\epsilon))}{\pi r^2}=1.$$

The main result of this section is the following.

\[thm:2\] Let $D\subset{{\mathbb C}}^n$ be a domain. Consider the following conditions:

1.  \[3\] for any $\zeta\in\partial D$ there exist no a Borel probability measure $\mu$ with compact support in $D\cup\{\zeta\}$ such that $\mu\not=\delta_{\zeta}$ and $$|f(\zeta)|\le\int_{D\cup\{\zeta\}} |f(w)| d\mu(w)\quad\text{ for any }f\in A(D\cup\{\zeta\}).$$

2.  \[3a\] for any $\zeta\in\partial D$ there exist no a Borel probability measure $\mu$ with compact support in $D\cup\{\zeta\}$ such that $\mu\not=\delta_{\zeta}$ and $$f(\zeta)=\int_{D\cup\{\zeta\}} f(w)d\mu(w)\quad\text{ for any }f\in A(D\cup\{\zeta\}).$$

3.  \[4\] there exists an $f\in A(D\cup\{\zeta\})$ such that $f(\zeta)=1$ and $|f|<1$ on $D$.

4.  \[2\] $D$ is $c$-finitely compact.

5.  \[1\] $D$ is $c$-complete.

Then $\eqref{3}\implies\eqref{3a}\implies\eqref{4}\implies\eqref{2}\implies\eqref{1}$. Moreover, if $n=1$ then $\eqref{1}\implies\eqref{3}$ and, therefore, all the above conditions are equivalent.

Note that the implications $\eqref{3}\implies\eqref{3a}$ and $\eqref{4}\implies\eqref{2}\implies\eqref{1}$ are immediate. The implication $\eqref{3a}\implies\eqref{4}$ follows from Corollary \[cor:10\].

Assume that $n=1$. Let us prove $\eqref{1}\implies\eqref{3}$. Assume that there exists a positive probability measure $\mu$ such that $\mu(D)=1$ and $$|f(\zeta)|\le\int_D |f|d\mu\quad\text{ for any }f\in A(D\cup\{\zeta\}).$$ Fix $f\in A(D\cup\{\zeta\})$. Then there exists a sequence $f_n\in H^\infty(D)$ such that $\|f_n\|_{D}\le 17\|f\|_{D}$, $f_n$ extends to be analytic in a neighborhood of $\zeta$ and $f_n$ converges uniformly to $f$ on any set of type $D\setminus{{\mathbb D}}(\zeta,\epsilon)$, where $\epsilon>0$.

For any $\eta\in D$ we put $$ g\_n(z)=. $$ Note that $g_n\in H^\infty(D\cup\{\zeta\})$. Then $$|g_n(\zeta)|\le\int_{D}|g_n(w)|d\mu(w)\le 2\|f_n\|_{\infty} M(\eta)\le 34\|f\|_{\infty} M(\eta)$$ and, therefore, $$|f_n(\zeta)-f_n(\eta)|\le |\zeta-\eta|\cdot 2\|f_n\|_{\infty} M(\eta)\le |\zeta-\eta|\cdot 34\|f\|_{\infty} M(\eta).$$ For any $\eta_1,\eta_2\in D$ we have $$|f(\eta_1)-f(\eta_2)|\le 34\|f\|_{\infty}\cdot\big(|\zeta-\eta_1|\cdot  M(\eta_1)+|\zeta-\eta_2|\cdot M(\eta_2)\big).$$ According to Corollary \[cor:12\] we can take a sequence $\{\eta_\nu\}$ such that $\eta_\nu\to\zeta$ and $|\zeta-\eta_\nu|\cdot  M(\eta_\nu)\le\frac{1}{2^\nu}$. Then $\{\eta_\nu\}$ is a c-Cauchy sequence. A contradiction.

The Reinhardt domains
=====================

A domain $D\subset{{\mathbb C}}^n$ is called *Reinhardt* if $(\lambda_1 z_1,\dots,\lambda_n z_n)\in D$ for all points $z=(z_1,\dots,z_n)\in D$ and any $|\lambda_1|=\dots=|\lambda_n|=1$. Let us denote $$V_j=\{z\in{{\mathbb C}}^n: z_j=0\}, \quad j=1,\dots,n.$$

Recall the following result of W. Zwonek (see [@Zwo1], Theorem 2).

\[thm:14\] Let $D$ be a hyperbolic pseudoconvex Reinhardt domain. Then the following conditions are equivalent:

1.  $D$ is c-finitely compact;

2.  $D$ is c-complete;

3.  $D$ is bounded and for any $j\in\{1,\dots,n\}$ $$\label{eq:3:1}
    \text{if }\overline{D}\cap V_j\not=\varnothing,\text{ then } D\cap V_j\not=\varnothing.$$

All we have to prove is that condition $(3)$ in Theorem \[thm:14\] implies that any $\zeta\in\partial D$ is a weak peak point. Without loss of generality we may assume that

(i) $D\subset{{\mathbb D}}^n$;

(ii) $D\cap V_j\not=\varnothing$, $j=1,\dots,m$.

(iii) $\overline{D}\cap V_j=\varnothing$, $j=m+1,\dots,n$.

Fix a boundary point $\zeta=(\zeta_1,\dots,\zeta_n)\in\partial D$. First assume that $\zeta_1\cdot\dots\cdot\zeta_n\not=0$. Then there exist $\xi_1,\dots,\xi_n\in{{\mathbb R}}$ such that ${\varphi}(z)<{\varphi}(\zeta)$ for any $z\in D$, where ${\varphi}(z)=|z_1|^{\xi_1}\cdot\dots\cdot|z_n|^{\xi_n}$. Note that $\xi_1,\dots,\xi_m\ge0$ and that there exists a $\delta_0>0$ such that for any $z\in D$ we have $|z_j|\ge \delta_0$ for $j=m+1,\dots,n$. Without loss of generality we may assume that $\xi_1,\dots,\xi_\ell>0$ and $\xi_{\ell+1}=\dots=\xi_m=0$.

Fix a point $\eta\in D$ such that $\eta_1\cdot\ldots\cdot \eta_n\not=0$. Put $R=\frac{{\varphi}(\eta)}{{\varphi}(\zeta)}<1$. We want to show that there exists a sequence $g_k\in{{\mathcal O}}(D)\cap C(D\cup\{\zeta\})$ such that

1.  $\|g_k\|_{D}\to1$ when $k\to\infty$;

2.  $g_k(\zeta)=1$ for any $k\ge1$;

3.  $|g_k(\eta)|\le R$ for any $k\ge1$.

Fix $\epsilon>0$. There exist $\beta_{1},\dots,\beta_{n}\in{{\mathbb R}}$ and $q\in{{\mathbb N}}$, $q\ge2$, such that $\operatorname{sign}\beta_j=\operatorname{sign}\xi_j$ and $$|q\xi_{j}-\beta_{j}|\le \epsilon\quad\text{ for any } j=1,\dots,n.$$ Put $f(z)=z_1^{\beta_{1}}\cdot\dots\cdot z_n^{\beta_{n}}$ and $g(z)=\frac{f(z)}{f(\zeta)}$. We have $$|f(z)|={\varphi}(z)^{q}|z_1|^{\beta_{1}-\xi_1 q}\cdot\dots\cdot |z_n|^{\beta_{n}-\xi_n q}.$$ Hence, $$|g(z)|=\left(\frac{{\varphi}(z)}{{\varphi}(\zeta)}\right)^{q}\cdot
\left(\frac{|z_1|}{|\zeta_1|}\right)^{\beta_{1}-\xi_1 q}
\cdot\ldots\cdot
\left(\frac{|z_n|}{|\zeta_n|}\right)^{\beta_{n}-\xi_n q}$$ and, in particular, $$|g(z)|\le\left(\frac{|z_1|}{|\zeta_1|}\right)^{\beta_{1}-\xi_1 q}
\cdot\ldots\cdot
\left(\frac{|z_n|}{|\zeta_n|}\right)^{\beta_{n}-\xi_n q}.$$

$1^o$ Let us show that $|g(\eta)|<R$. Indeed, $$|g(\eta)|\le R^q\cdot
\left(\prod_{j=1}^n \max\Big\{\frac{|\eta_j|}{|\zeta_j|}, \frac{|\zeta_j|}{|\eta_j|}\Big\}\right)^{\epsilon}.$$ For sufficiently small $\epsilon>0$ we have $|g(\eta)|<R$.

$2^o$ Let us now estimate $\|g\|_{D}$. There exists $\delta_1>$ such that for any $z\in G$ with $\min\{|z_1|,\dots,|z_\ell|\}\le\delta_1$ we have $$|z_1|^{\beta_1}\cdot\ldots\cdot |z_n|^{\beta_n}\le{\varphi}(\zeta)^{q}.$$ Note that $$|z_1|^{\beta_1}\cdot\ldots\cdot |z_n|^{\beta_n}\le
|z_1|^{\beta_1}\cdot\ldots\cdot |z_\ell|^{\beta_\ell}\cdot \delta_0^{\sum_{j=m+1}^n \beta_j}\le
\delta_1^{q\min\{\zeta_1,\dots,\zeta_\ell\}-\epsilon}\cdot \delta_0^{q\sum_{j=m+1}^n (\zeta_j-\frac{\epsilon}{q})}.$$ For sufficiently small $\delta_1>0$ we have $$\delta_1^{\min\{\zeta_1,\dots,\zeta_\ell\}-\frac{\epsilon}{2}}\cdot \delta_0^{\sum_{j=m+1}^n (\zeta_j-\frac{\epsilon}{2})}
\le{\varphi}(\zeta).$$ Fix such a $\delta_1>0$. Take $z\in D$ such that $\min\{|z_1|,\dots,|z_\ell|\}\le\delta_1$. Then $|f(z)|\le{\varphi}(\zeta)^q$. Therefore, $$|g(z)|\le \prod_{j=1}^n |\zeta_j|^{q\zeta_j-\beta_j}\le
\left(\frac{1}{\prod_{j=1}^n |\zeta_j|}\right)^\epsilon.$$ Take $z\in D$ such that $|z_1|,\dots,|z_\ell|\ge\delta_1$. Then $$|g(z)|\le \left(\prod_{j=1}^n \max\Big\{\frac{\delta_1}{|\zeta_j|}, \frac{|\zeta_j|}{\delta_1}\Big\}\right)^{\epsilon}.$$ All in all, we have $\|g\|_{D}\to1$ when $\epsilon\to0$.

In case, $\zeta_1\cdot\ldots\zeta_n=0$, consider a projection $\pi:{{\mathbb C}}^n\to{{\mathbb C}}^{n-1}$. Take $\widetilde D=\pi(D)$.

For the construction of a peak function we need the following simple, however, useful result.

Let $\Omega\subset{{\mathbb C}}^n$ be a domain and let $\zeta\in\partial\Omega$, $\eta\in\Omega$, $r\in[0,1)$ be fixed. Assume that for any $\epsilon>0$ there exists $f_\epsilon\in{{\mathcal O}}(\Omega)\cap C(\Omega\cup\{\zeta\})$ such that

1.  $|f_\epsilon|<1+\epsilon$ on $\Omega$;

2.  $f_\epsilon(\zeta)=1$;

3.  $|f_\epsilon(\eta)|<r$.

Then there exists $F\in{{\mathcal O}}(\Omega)\cap C(\Omega\cup\{\zeta\})$ such that

1.  $|F|<1$ on $\Omega$;

2.  $F(\zeta)=1$.

It is easy to check

Let $\epsilon>0$. Then $$\Re\left(\frac{1+z}{1-z}\right)>\frac{1}{\epsilon}\Longleftrightarrow |z-\frac{1}{1+\epsilon}|<\frac{\epsilon}{1+\epsilon}.$$

Without loss of generality we may assume that $r=0$. Indeed, take a sequence $$g_\epsilon(\lambda)=a(1+\epsilon)\cdot\frac{f_\epsilon(\lambda)-f_\epsilon(\eta)}
{(1+\epsilon)^2-\overline{f_\epsilon(\eta)}f_\epsilon(\lambda)},$$ where $$a=\frac{1}{1+\epsilon}\cdot\frac{(1+\epsilon)^2-\overline{f_\epsilon(\eta)}}{1-f_\epsilon(\eta)}.$$ Note that $g_\epsilon(\eta)=0$, $g_\epsilon(\zeta)=1$, and $$|g_\epsilon(\lambda)|\le |a|=
\left|\frac{1}{1+\epsilon}\cdot\frac{(1+\epsilon)^2-f_\epsilon(\eta)}{1-f_\epsilon(\eta)}\right|\le
\frac{1}{1+\epsilon}\left(1+\frac{2\epsilon+\epsilon^2}{1-r}\right)\to1\quad\text{when }\epsilon\to0.$$

So, we assume that $r=0$. Put $\epsilon_k=\frac{1}{4^k}$. Then there exists $f_k=f_{\epsilon_k}\in{{\mathcal O}}(\Omega)\cap C(\Omega\cup\{\zeta\})$ such that

1.  $|f_k|<1+\epsilon_k$ on $\Omega$;

2.  $f_k(\zeta)=1$;

3.  $f_k(\eta)=0$.

Put $U_k=\{z\in\Omega\cup\{\zeta\}: |f_k(z)-1|<\epsilon_k\}$ and $$h(z)=\sum_{k=1}^\infty \frac{1}{2^k}\cdot \frac{1+\tilde f_k(z)}{1-\tilde f_k(z)},$$ where $\tilde f_k=\frac{f_k}{1+\epsilon_k}$. Note that $\Re h(z)\ge \frac{1}{2^k}\cdot \Re\left(\frac{1+\tilde f_k(z)}{1-\tilde f_k(z)}\right)\ge 2^k$ for $z\in U_k$. Put $F=\frac{h-1}{h+1}$. Then $|F-1|\le\frac{2}{|h+1|}\le\frac{2}{|\Re h+1|}\le \frac{2}{2^k+1}$ on $U_k$.

Using Sibony’s ideas we show that there exists a domain $D\subset{{\mathbb C}}^2$ and a boundary point such that a weak peak function exists, however, peak function does not exist.

Fix an irrational number $\alpha>0$. Let $D\subset{{\mathbb C}}^2$ be a domain and let $(z_0,w_0)\in\partial D$. Assume that there exists a neighborhood $U\subset{{\mathbb C}}^2$ of $(z_0,w_0)$ such that $$D\cap U=\{(z,w)\in{{\mathbb C}}^2: |z|\cdot |w|^\alpha<1\}\cap U.$$ We want to show that there does not exist a holomorphic function $f\in{{\mathcal O}}(D)\cap C(D\cup U)$ such that $|f|<1$ on $D$ and $f(z_0,w_0)=1$.

Indeed, assume that such a function exists. There exists a neighborhood $V\subset{{\mathbb C}}$ of the origin such that $(z_0e^{-\alpha\lambda},w_0e^{\lambda})\in U$ whenever $\lambda\in V$. For sufficiently big $n\in{{\mathbb N}}$ consider functions $$\psi_n(\lambda)= f(z_0e^{-\alpha\lambda},(1-\frac1n)w_0e^{\lambda}).$$ Note that $\psi_n:V\to D\cap U$ is a holomorphic mapping. Hence, there exists a subsequence $\{n_k\}$ such that $\psi_{n_k}$ tends locally uniformly on $V$ to a holomorphic mapping $\psi:V\to{{\mathbb C}}^2$. It is easy to see (use continuity of $f$) that $$\psi(\lambda)=f(z_0e^{-\alpha\lambda},w_0e^{\lambda}).$$ So, we get that $\psi:V\to \overline{D\cap U}$ is a holomorphic mapping such that $|\psi|\le1$ and $\psi(0)=1$. Hence, $\psi\equiv1$. Since $\alpha$ is irrational, we get $\{(z_0e^{-\alpha\lambda},w_0e^{\lambda}): \lambda\in V\}$ is dense in a neighborhood of $(z_0,w_0)$. From the continuity of $f$ we get that $f=1$ on a relatively open subset $\partial D$ containing $(z_0,w_0)$. Then a function $f(z_0,\lambda w_0)=1$ on the open subset of the unit circle containing $1$. Hence, $f(z_0,\lambda w_0)=1$ everywhere. A contradiction.

[10]{}

P. Berti, P. Rigo, *Integral representation of linear functionals on spaces of unbounded functions*, PAMS 128 (2000), 3251–3258.

E. Bishop, *A minimal boundary for function algebras*, Pacific J. Math. 9 (1959), 629–642.

Edwards D. A., *Choquet boundary theory for certain spaces of lower semicontinuous functions, in Function Algebras,* Proc. Internat. Sympos. on Function Algebras, Chicago, (1966), 300–309.

T.W. Gamelin, *Uniform algebras*, Prentice-Hall, 1969.

T.W. Gamelin, J. Garnett, *Distinguished homomorphisms and fiber algebras*, Trans. Amer. Math. Soc (1970), 455–474.

N. Gogus, T. Perkins, E. Poletsky, *Non-compact versions of Edwards’ Theorem*, Positivity 17 (2013), 459–473.

M. Jarnicki & P. Pflug, *Invariant distances and metrics in complex analysis*, De Gruyter Expositions in Mathematics 9, 2nd ext. edition, 2013.

E. Hewitt, *Linear functionals on spaces of continuous functions*, Fund. Math. 37 (1950), 161–189.

Ł. Kosiński, W. Zwonek, *Proper holomorphic mappings vs. peak points and Shilov boundary*, Ann. Polon. Math. 107 (2013), 97–108.

E. A. Poletsky, *Plurisubharmonic functions as solutions of variational problems,* Proc. Sympos. Pure Math. [**52**]{} (1991), Part 1, 163–171.

F. Wikström, *Jensen measures and boundary values of plurisubharmonic functions,* Ark. Mat. [**39**]{}(2001), 181–200.

P. Ressel, *Integral representations on convex semigroups*, Math. Scand. 61 (1987), 93–111.

W. Rudin, *Real and Complex Analysis*, McGraw-Hill, 1987.

M.A. Selby, *On completeness with respect to the Carathéodory metric*, Canad. Math. Bull. 17 (1974), 261–263.

N. Sibony, *Prolongement de fonctions holomorphes bornées et metrique de Carathéodory*, Invent. Math. 29 (1975), 205–230.

E.L. Stout, *The theory of uniform algebras*, Bogden & Quigley, 1971.

W. Zwonek, *On Caratheodory completeness of pseudoconvex Reinhardt domains*, PAMS 128 (2000), 857–864.

[^1]: Research partially supported by the National Science Center of Poland (NCN) grant UMO-2011/03/B/ST1/04758.

[^2]: The author is thankful to Piotr Niemiec and Jan Stochel for the references on linear functionals.
