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Remarks on the Sibony functions and pseudometrics

  • Marek JarnickiEmail author
  • Peter Pflug
Open Access
Article
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Abstract

We discuss some basic properties of the Sibony functions and pseudometrics.

Keywords

Sibony function Sibony pseudometric 

Mathematics Subject Classification

32F45 

1 Introduction

Let \(G\subset {\mathbb {C}}^n\) be a domain. For \(a\in G\) let
$$\begin{aligned} {\mathcal {M}}_G(a):&=\{|f|: f\in {\mathcal {O}}(G,{\mathbb {D}}),\;f(a) =0\},\\ {\mathcal {S}}^{(p)}_G(a):&=\{\root p \of {u}:\;\; u:G\longrightarrow [0,1): \log u\in \mathcal {PSH}(G),\\&\qquad u\in \mathcal C^p(\{a\}),\;\exists {C>0}: u(z)\le C\Vert z-a\Vert ^p,\;z\in G\},\quad p\in {\mathbb {N}},\\ {\mathcal {K}}_G(a):&=\{u:\;\;u:G\longrightarrow [0,1): \log u\in \mathcal {PSH}(G),\\&\qquad \exists {C>0}: u(z)\le C\Vert z-a\Vert ,\;z\in G\}, \end{aligned}$$
where \({\mathbb {D}}\subset {\mathbb {C}}\) stands for the unit disc, \({\mathcal {O}}(G,{\mathbb {D}})\), resp. \(\mathcal {PSH}(G)\) denote the set of all holomorphic functions on G having values in \({\mathbb {D}}\), resp. the set of all plurisubharmonic functions on G, and “\(u\in \mathcal C^p(\{a\})\)” means that u is of class \(\mathcal C^p\) in a neighborhood of a (cf. [1, § 4.2]). Note that \({\mathcal {S}}^{(1)}_G(a)\) is different from \({\mathcal {K}}_G(a)\) (see Remark 2.1(c)). Put
$$\begin{aligned} {\mathcal {S}}_G(a):= & {} {\mathcal {S}}^{(2)}_G(a)=\{\sqrt{u}:\;\; u:G\longrightarrow [0,1): \log u\in \mathcal {PSH}(G),\\&u\in \mathcal C^2(\{a\}),\;u(0)=0\}. \end{aligned}$$
Obviously, \({\mathcal {M}}_G(a)\subset {\mathcal {S}}_G(a)\subset {\mathcal {K}}_G(a)\) and \({\mathcal {S}}^{(p)}_G(a)\subset {\mathcal {K}}_G(a)\), \(p\in {\mathbb {N}}\). If \({\mathcal {F}}\in \{{\mathcal {M}}, {\mathcal {S}}^{(p)}, {\mathcal {K}}\}\), then we define:
$$\begin{aligned} d_G^{\mathcal {F}}(a,z):&=\sup \{v(z):\; v\in {\mathcal {F}}_G(a)\},\quad a, z\in G,\\ \delta _G^{\mathcal {F}}(a;X):&=\sup \Big \{\limsup _{\lambda \rightarrow 0}\frac{v(a+\lambda X)}{|\lambda |}: v\in {\mathcal {F}}_G(a)\Big \},\quad a\in G,\;X\in {\mathbb {C}}^n. \end{aligned}$$
For \({\mathcal {F}}\in \{{\mathcal {M}}, {\mathcal {S}}, {\mathcal {K}}\}\) the families \((d^{\mathcal {F}}_G)_G\) and \((\delta ^{\mathcal {F}}_G)_G\) are holomorphically contractible, i.e.

\(\bullet \)   \(d^{\mathcal {F}}_{{\mathbb {D}}}(0,t)=t\), \(t\in [0,1)\),    \(\delta ^{\mathcal {F}}_{{\mathbb {D}}}(0;1)=1\);

\(\bullet \)   for any domains \(G\subset {\mathbb {C}}^n\), \(D\subset {\mathbb {C}}^m\) and for any holomorphic mapping \(F:G\longrightarrow D\) we have
$$\begin{aligned} d_D^{\mathcal {F}}(F(a),F(z))\le & {} d_G^{\mathcal {F}}(a,z),\quad a,z\in G, \end{aligned}$$
(1.1)
$$\begin{aligned} \delta _D^{\mathcal {F}}(F(a);F'(a)(X))\le & {} \delta _G^{\mathcal {F}}(a;X),\quad a\in G,\;X\in {\mathbb {C}}^n. \end{aligned}$$
(1.2)
In particular, the families \((d^{\mathcal {F}}_G)_G\) and \((\delta ^{\mathcal {F}}_G)_G\) are invariant under biholomorphic mappings.
If \({\mathcal {F}}={\mathcal {M}}\), then we get the Möbius pseudodistance\(\varvec{m}_G:=d_G^{\mathcal {M}}\) and the Carathéodory–Reiffen pseudometric\({\varvec{\gamma }}_G:=\delta _G^{\mathcal {M}}\). It is known that
$$\begin{aligned} {\varvec{\gamma }}_G(a,z)=\lim _{\lambda \rightarrow 0}\frac{\varvec{m}_G(a,a+\lambda X)}{|\lambda |}=\max \{|f'(z)(X)|: f\in {\mathcal {O}}(G,{\mathbb {D}}),\;f(a) =0\}.\nonumber \\ \end{aligned}$$
(1.3)
If \({\mathcal {F}}={\mathcal {S}}\), then we get the Sibony function\(\varvec{s}_G:=d_G^{\mathcal {S}}\) and the Sibony pseudometric\(\varvec{S}_G:=\delta _G^{\mathcal {S}}\). It is known that
$$\begin{aligned} \varvec{S}_G(a;X)=\sup \{\sqrt{{\mathcal {L}}u(a;X)}: u\in {\mathcal {S}}_G(a)\}, \end{aligned}$$
where \({\mathcal {L}}u(a;X):=\sum _{j,k=1}^n\frac{\partial ^2u}{\partial z_j\partial \overline{z}_k}(a)X_j{{\overline{X}}}_k\) is the Levi form (cf. [1, Proposition 4.2.16]). In particular, \(\varvec{S}_G(a;\cdot )\) is a \({\mathbb {C}}\)-seminorm.
If \({\mathcal {F}}={\mathcal {K}}\), then we get the pluricomplex Green function\(\varvec{g}_G:=d_G^{\mathcal {K}}\) and the Azukawa pseudometric\(\varvec{A}_G:=\delta _G^{\mathcal {K}}\). It is known that \(\varvec{g}_G(a,\cdot )\in {\mathcal {K}}_G(a)\), \(\log \varvec{A}_G(a;\cdot )\in \mathcal {PSH}({\mathbb {C}}^n)\), and
$$\begin{aligned} \varvec{A}_G(a;X)=\limsup _{\lambda \rightarrow 0}\frac{\varvec{g}_G(a,a+\lambda X)}{|\lambda |}\quad \text {(cf. [JP 2013, Lemma 4.2.3]).} \end{aligned}$$
(1.4)
If \({\mathcal {F}}={\mathcal {S}}^{(p)}\), \(p\ne 2\), then we get the higher order Sibony function\(\varvec{s}^{(p)}_G:=d^{{\mathcal {S}}^{(p)}}_G\) and the higher order Sibony pseudometric\(\varvec{S}^{(p)}_G:=\delta ^{{\mathcal {S}}^{(p)}}_G\).

Recall that many properties of complex domains/manifolds are encoded in \(\varvec{m}_G\), \({\varvec{\gamma }}_G\), \(\varvec{g}_G\), and \(\varvec{A}_G\). Therefore, these functions have been important tools to investigate various geometric problems in several complex analysis. Their study over the last years has led to the point that their basic properties are now well understood. To extent the family of invariant functions higher order functions were introduced and studied, e.g. the higher Carathéodory-Reiffen pseudometrics. But in contrast to that little is known on properties of \(\varvec{S}_G\) and almost nothing on \(\varvec{s}^{(p)}_G\), \(p\in {\mathbb {N}}\), and \(\varvec{S}^{(p)}_G\), \(p\ne 2\).

The main aim of this note is to show that the basic properties of \(\varvec{s}^{(p)}_G\) and \(\varvec{S}^{(p)}_G\) differ essentially from the corresponding properties of \(\varvec{m}_G\), \(\varvec{g}_G\), \({\varvec{\gamma }}_G\), and \(\varvec{A}_G\). Surprisingly, as we will see, many properties of the so far studied invariant functions fail to hold.

2 Holomorphic contractibility

Remark 2.1

  1. (a)

    \({\mathcal {S}}^{(p)}_G(a)=\{\root p \of {u}:\;u:G\longrightarrow [0,1): \log u\in \mathcal {PSH}(G),\;u\in \mathcal C^p(\{a\}),\;{\text {ord}}_au\ge p\}\), where \({\text {ord}}_au\) denotes the order of zero of u at a.

     
  2. (b)
    In view of the Taylor formula, we have
    $$\begin{aligned} \varvec{S}^{(p)}_G(a;X)=\sup \big \{(\tfrac{1}{p!}|u^{(p)}(a)(X)|)^{1/p}: \root p \of {u}\in {\mathcal {S}}^{(p)}_G(a)\big \},\quad a\in G,\;X\in {\mathbb {C}}^n, \end{aligned}$$
    where \(u^{(p)}(a):{\mathbb {C}}^n\longrightarrow {\mathbb {R}}\) stands for the p-th Fréchet differential of u at a.
     
  3. (c)

    In view of (b), we get \(\varvec{S}^{(p)}_G(a;\cdot )\equiv 0\) for p odd. In particular, \(\varvec{S}^{(1)}_{\mathbb {D}}(0;1)=0<1=\varvec{A}_{\mathbb {D}}(0;1)\).

     
  4. (d)

    \(\varvec{s}^{(p)}_G\le \varvec{g}_G\), \(\varvec{S}^{(p)}_G\le \varvec{A}_G\). In particular, \(\varvec{s}^{(p)}_{{\mathbb {D}}}(0,\lambda )\le \varvec{g}_{{\mathbb {D}}}(0,\lambda )=|\lambda |\), \(\varvec{S}^{(p)}_{{\mathbb {D}}}(0;1)\le \varvec{A}_{{\mathbb {D}}}(0;1)=1\).

     
  5. (e)

    If \(\varvec{g}^{p+\varepsilon }_G(a,\cdot )\in \mathcal C^p(\{a\})\) for \(0<\varepsilon \ll 1\), then \(\varvec{g}^{1+\varepsilon /p}_G(a,\cdot )\in {\mathcal {S}}^{(p)}_G(a)\). Consequently, \(\varvec{s}^{(p)}_G(a,\cdot )=\varvec{g}_G(a,\cdot )\). In particular, \(\varvec{s}^{(p)}_{{\mathbb {D}}}(0,\lambda )=|\lambda |\), \(\lambda \in {\mathbb {D}}\).

     
  6. (f)

    If \(\varvec{g}^{2p}_G(a,\cdot )\in \mathcal C^{2p}(\{a\})\), then \(\varvec{S}^{(2p)}_G(a;\cdot )=\varvec{A}_G(a;\cdot )\). In particular, \(\varvec{S}^{(2p)}_{\mathbb {D}}(0;1)=1\).

     
  7. (g)

    If \(F:G\longrightarrow D\) is holomorphic, then \(v\circ F\in {\mathcal {S}}^{(p)}_G(a)\) for every \(v\in {\mathcal {S}}^{(p)}_D(F(a))\). Consequently, the family \((\varvec{s}^{(p)}_G)_G\) (resp. \((\varvec{S}^{(p)}_G)_G\)) satisfies (1.1) (resp. (1.2)).

     
  8. (h)

    The families \((\varvec{s}^{(p)}_G)_G\) and \((\varvec{S}^{(2p)}_G)_G\) are holomorphically contractible. They will be the main objects of our investigation in the sequel.

     
  9. (i)

    \(\varvec{m}_G\le \varvec{s}^{(p)}_G\le \varvec{g}_G\) and \({\varvec{\gamma }}_G\le \varvec{S}^{(2p)}_G\le \varvec{A}_G\).

     

3 Upper semicontinuity

It is known that for \({\mathcal {F}}\in \{{\mathcal {M}}, {\mathcal {K}}\}\) the functions \(G\times G\ni (z,w)\longmapsto d_G^{\mathcal {F}}(z,w)\) and \(G\times {\mathbb {C}}^n\ni (z,X)\longmapsto \delta _G^{\mathcal {F}}(z;X)\) are upper semicontinuous (cf. [1, Propositions 2.6.1, 2.7.1(c), 4.2.10(g,k)]). We will prove that in general the functions \(\varvec{s}^{(p)}_G(\cdot ,z^0)\) and \(\varvec{S}^{(2p)}_G(\cdot ;X^0)\) are not upper semicontinuous (Examples 3.13.3).

Recall that \(\varvec{S}_G(a;\cdot )\) is a seminorm and therefore it is continuous. We do not know whether the functions \(\varvec{s}_G(a,\cdot )\), \(p\in {\mathbb {N}}\), and \(\varvec{S}^{(2p)}_G(a;\cdot )\), \(p\ge 2\), are upper semicontinuous.

Example 3.1

(cf. [1, Example 4.2.18]) Let
$$\begin{aligned} G:=\{(z_1,z_2,z_3)\in {\mathbb {C}}^3: |z_1|e^{\varphi (z_2,z_3)}<1\} \end{aligned}$$
with
$$\begin{aligned} \varphi (\xi ,\eta ):=\sum _{k=1}^\infty \lambda _k\log \Big (\frac{|\xi -a_k|^2+|\eta |}{k}\Big ), \quad (\xi ,\eta )\in {\mathbb {C}}^2, \end{aligned}$$
where \((a_k)^\infty _{k=1}\subset {\mathbb {D}}{\setminus }\{0\}\) is a dense subset of \({\mathbb {D}}\) and \((\lambda _k)_{k=1}^\infty \subset (0,1]\) are chosen so that \(\varphi (0,0)>-\infty \) and \(\varphi \in \mathcal C^\infty ({\mathbb {C}}\times {\mathbb {C}}_*)\), where \({\mathbb {C}}_*:={\mathbb {C}}{\setminus }\{0\}\). Note that G is a pseudoconvex Hartogs domain.
Let \(c_t:=(0,0,t)\in G\), \(t>0\), \(z^0:=(b,0,0)\in G\) with \(b\ne 0\), and let \(X^0:=(1,0,0)\). We will show that
$$\begin{aligned} \varvec{s}^{(p)}_G(0,z^0)= & {} 0<|b|e^{\varphi (0,0)}\le \varvec{s}^{(p)}_G(c_t,z^0),\\ \varvec{S}^{(2p)}_G(0;X^0)= & {} 0<e^{\varphi (0,0)}\le \varvec{S}^{(2p)}_G(c_t;X^0),\quad 0<t\ll 1, \end{aligned}$$
which shows that the functions \(\varvec{s}^{(p)}_G(\cdot ,z^0)\) and \(\varvec{S}^{(2p)}_G(\cdot ;X^0)\) are not upper semicontinuous at 0.

Indeed, the function \(G\ni (z_1,z_2,z_3)\overset{v}{\longmapsto }(|z_1|e^{\varphi (z_2,z_3)})^{1+\varepsilon /p}\) belongs to \({\mathcal {S}}^{(p)}_G(c_t)\) for all \(\varepsilon >0\) and \(t>0\). Hence, \(\varvec{s}^{(p)}_G(c_t,z^0)\ge |b|e^{\varphi (0,0)}>0\). Analogously, the function \(G\ni (z_1,z_2,z_3)\overset{v}{\longmapsto }|z_1|e^{\varphi (z_2,z_3)}\) belongs to \({\mathcal {S}}^{(2p)}_G(c_t)\) for all \(t>0\). Hence, \(\varvec{S}^{(2p)}_G(c_t;X^0)\ge \limsup _{\lambda \rightarrow 0}\frac{v(c_t+\lambda X^0)}{|\lambda |}=e^{\varphi (0,t)}\ge e^{\varphi (0,0)}>0\).

On the other hand, let \(\root p \of {u}\in {\mathcal {S}}^{(p)}_G(0)\) (resp.  \(\root 2p \of {u}\in {\mathcal {S}}^{(2p)}_G(0)\)). Since \({\mathbb {C}}\times \{a_k\}\times \{0\}\subset G\), we get \(u(z_1,a_k,0)={\text {const}}(k)\), \(z_1\in {\mathbb {C}}\), \(k\in {\mathbb {N}}\). Since \(\{0\}\times {\mathbb {C}}\times \{0\}\subset G\), we get \(u(0,z_2,0)={\text {const}}=u(0)=0\), \(z_2\in {\mathbb {C}}\). Thus, \(u(z_1,a_k,0)=0\), \(z_1\in {\mathbb {C}}\), \(k\in {\mathbb {N}}\). Since \(u\in \mathcal C^p(\{0\})\) (resp. \(u\in \mathcal C^{2p}(\{0\})\)), we conclude that \(u=0\) in \(U\times \{0\}\), where U is a neighborhood of (0, 0). Since \(\log u\in \mathcal {PSH}(G)\), we get \(u(z_1,z_2,0)=0\) for all \((z_1,z_2,0)\in G\). Consequently, \(\varvec{s}^{(p)}_G(0,z^0)=0\) (resp. \(\varvec{S}^{(2p)}_G(0;X^0)=0\)).

Example 3.2

In view of Example 3.1, one could expect that perhaps the families \((\varvec{s}^{(p)*}_G)_G\) and/or \((\varvec{S}^{(2p)*}_G)_G\) are holomorphically contractible, where \(\varvec{s}^{(p)*}_G:=(\varvec{s}^{(p)}_G)^*\), \(\varvec{S}^{(2p)*}_G:=(\varvec{S}^{(2p)}_G)^*\), and \({}^*\) denotes the upper semicontinuous regularization. We will prove that unfortunately they are not holomorphically contractible.

Keep the notation from Example 3.1. Let
$$\begin{aligned} D:=\{(z_1,z_2)\in {\mathbb {C}}^2: (z_1,z_2,0)\in G\},\quad D\ni (z_1,z_2)\overset{F}{\longmapsto }(z_1,z_2,0)\in G. \end{aligned}$$
Then \(\varvec{s}^{(p)*}_G(0,z^0)\ge \limsup _{t\rightarrow 0+}\varvec{s}^{(p)}_G(c_t,z^0)\ge |b|e^{\varphi (0,0)}>0\) and \(\varvec{S}^{(2p)*}_G(0;X^0)\ge \limsup _{t\rightarrow 0+}\varvec{S}^{(2p)}_G(c_t;X^0)\ge e^{\varphi (0,0)}>0\).
On the other hand, let \(w^0\in D\cap ({\mathbb {C}}\times {\mathbb {D}})\) and let \(\root p \of {u}\in {\mathcal {S}}^{(p)}_D(\{w^0\})\) (resp.  \(\root 2p \of {u}\in {\mathcal {S}}^{(2p)}_D(\{w^0\})\)). Since \({\mathbb {C}}\times \{a_k\}\subset D\), we get \(u(z_1,a_k)={\text {const}}(k)\), \(z_1\in {\mathbb {C}}\), \(k\in {\mathbb {N}}\). Since \(\{0\}\times {\mathbb {C}}\subset D\), we get \(u(0,z_2)={\text {const}}=u(0,0)\), \(z_2\in {\mathbb {C}}\). Thus, \(u(z_1,a_k)={\text {const}}\), \(z_1\in {\mathbb {C}}\), \(k\in {\mathbb {N}}\). Since \(u\in \mathcal C^p(\{w^0\})\) (resp. \(u\in \mathcal C^{2p}(\{w^0\})\)), we conclude that \(u=0\) in \(U\times \{0\}\), where U is a neighborhood of \(w^0\). Hence, since \(\log u\in \mathcal {PSH}(G)\), we get \(u(z_1,z_2)=0\) for all \((z_1,z_2)\in D\). Consequently, \(\varvec{s}^{(p)}_D=0\) on \((D\cap ({\mathbb {C}}\times {\mathbb {D}}))\times D\) (resp. \(\varvec{S}^{(2p)}_D=0\) on \((D\cap ({\mathbb {C}}\times {\mathbb {D}}))\times {\mathbb {C}}^2\)). In particular, \(\varvec{s}^{(p)*}_D(0,(b,0))=0\) (resp. \(\varvec{S}^{(2p)*}_D(0;(1,0))=0\)) and therefore
$$\begin{aligned} \varvec{s}^{(p)*}_G(F(0,0),F(b,0))&>0=\varvec{s}^{(p)*}_D((0,0),(b,0)),\\ \varvec{S}^{(2p)*}_G(F(0,0);F'(0,0)(1,0))&>0=\varvec{S}^{(2p)*}_D((0,0);(1,0)). \end{aligned}$$

Example 3.3

For \(n\ge 2\) and \(\alpha =(\alpha _1,\dots ,\alpha _n)\in {\mathbb {R}}^n{\setminus }\{0\}\) let
$$\begin{aligned} \varvec{D}_\alpha :=\{z\in {\mathbb {C}}^n(\alpha ): |z^\alpha |:=|z_1|^{\alpha _1}\cdots |z_n|^{\alpha _n}<1\}, \end{aligned}$$
where \({\mathbb {C}}^n(\alpha ):=\{(z_1,\dots ,z_n)\in {\mathbb {C}}^n: \forall {j\in \{1,\dots ,n\}}: (\alpha _j<0 \Longrightarrow z_j\ne 0)\}\). Note that \(\varvec{D}_\alpha \) is a pseudoconvex Reinhardt domain. For \(a=(a_1,\dots ,a_n)\in \varvec{D}_\alpha \) define
$$\begin{aligned} \Xi (a):= & {} \{j\in \{1,\dots ,n\}: \alpha _j>0,\;a_j=0\},\\ r(a):= & {} \left\{ \begin{array}{ll} 1 &{} \text { if } \sigma (a)=0 \\ \sum \nolimits _{j\in \Xi (a)}\alpha _j &{} \text { if } \sigma (a)\ge 1\end{array}\right. ,\quad \sigma (a):=\#\Xi (a),\quad \mu (a):=\min \{\alpha _j: j\in \Xi (a)\}. \end{aligned}$$
Note that if \(\sigma (a)=1\), then \(r(a)=\mu (a)\).

The following results are known (cf. [1, §§ 6.2, 6.3], and [2, Theorem 1]).

\(\bullet \)   If \(\alpha _1,\dots ,\alpha _n\in {\mathbb {Z}}\) are relatively prime, then
$$\begin{aligned} \varvec{m}_{\varvec{D}_\alpha }(a,z)= & {} \varvec{m}_{{\mathbb {D}}}(a^\alpha ,z^\alpha ),\quad \varvec{g}_{\varvec{D}_\alpha }(a,z)=\big (\varvec{m}_{{\mathbb {D}}}(a^\alpha ,z^\alpha )\big )^{1/r},\\ \varvec{A}_{\varvec{D}_\alpha }(a;X)= & {} \Big ({\varvec{\gamma }}_{\mathbb {D}}(a^\alpha ; \tfrac{1}{r!}\prod _{j\notin \Xi (a)}a_j^{\alpha _j}\cdot \prod _{j\in \Xi (a)}X_j^{\alpha _j})\Big )^{1/r},\quad r=r(a),\\ \varvec{s}_{\varvec{D}_\alpha }(a,z)= & {} {\left\{ \begin{array}{ll}\varvec{m}_{{\mathbb {D}}}(a^\alpha ,z^\alpha ) &{} \text { if }\sigma (a)=0\\ |z^\alpha |^{1/\mu (a)} &{} \text { if }\sigma (a)\ge 1\end{array}\right. },\\ \varvec{S}_{\varvec{D}_\alpha }(a;X)= & {} {\left\{ \begin{array}{ll}\varvec{A}_{\varvec{D}_\alpha }(a;X) &{} \text { if }\sigma (a)\le 1\\ 0 &{} \text { if }\sigma (a)\ge 2\end{array}\right. },\quad a,z\in \varvec{D}_\alpha ,\; X\in {\mathbb {C}}^n.\\ \end{aligned}$$
\(\bullet \)   If \(\alpha \notin {\mathbb {R}}\cdot {\mathbb {Z}}^n\), then
$$\begin{aligned}&\varvec{m}_{\varvec{D}_\alpha }\equiv 0,\quad \varvec{g}_{\varvec{D}_\alpha }(a,z)={\left\{ \begin{array}{ll} 0 &{} \text { if }\sigma (a)=0\\ |z^\alpha |^{1/r} &{} \text { if }\sigma (a)\ge 1 \end{array}\right. },\\&\varvec{A}_{\varvec{D}_\alpha }(a;X)={\left\{ \begin{array}{ll}0 &{} \text { if }\sigma (a)=0\\ \big (\prod _{j\notin \Xi (a)}|a_j|^{\alpha _j}\cdot \prod _{j\in \Xi (a)}|X_j|^{\alpha _j}\big )^{1/r} &{} \text { if }\sigma (a)\ge 1 \end{array}\right. },\quad r=r(a),\\&\varvec{s}_{\varvec{D}_\alpha }(a,z)={\left\{ \begin{array}{ll}0 &{} \text { if }\sigma (a)=0\\ |z^\alpha |^{1/\mu (a)} &{} \text { if }\sigma (a)\ge 1\end{array}\right. },\\&\varvec{S}_{\varvec{D}_\alpha }(a;X)={\left\{ \begin{array}{ll}\varvec{A}_{\varvec{D}_\alpha }(a;X) &{} \text { if }\sigma (a)=1\\ 0 &{} \text { if }\sigma (a)\ne 1\end{array}\right. },\quad a, z\in \varvec{D}_\alpha ,\; X\in {\mathbb {C}}^n. \end{aligned}$$
In particular, if \(n=3\) and \(\alpha =(1,2,2)\), then \(\varvec{s}_{\varvec{D}_\alpha }((0,0,0),z)=|z^\alpha |\) and \(\varvec{s}_{\varvec{D}_\alpha }((1/k,0,0),z)=|z^\alpha |^{1/2}\), \(k\in {\mathbb {N}}\). Thus, the function \(\varvec{s}_{\varvec{D}_\alpha }(\cdot , z^0)\) is not upper semicontinuous at (0, 0, 0) for all \(z^0=(z^0_1,z^0_2,z^0_3)\in \varvec{D}_\alpha \) with \(z^0_1z^0_2z^0_3\ne 0\).

Notice that using the above effective formulas one may easily construct many other counterexamples.

Example 3.4

Keep the notation from Example 3.3. Assume that \(\alpha _1,\dots ,\alpha _n\in {\mathbb {R}}_*:={\mathbb {R}}{\setminus }\{0\}\), \(a_1\cdots a_s\ne 0\), \(a_{s+1}=\dots =a_n=0\), \(s:=n-\sigma (a)\). In particular, \(\alpha _{s+1},\dots ,\alpha _n>0\).

First observe that if \(\sigma (a)\le 1\), then \(\varvec{g}^{p+\varepsilon }_{\varvec{D}_\alpha }(a,\cdot )\in \mathcal C^p(\{a\})\) and consequently \(\varvec{s}^{(p)}_{\varvec{D}_\alpha }(a,\cdot )=\varvec{g}_{\varvec{D}_\alpha }(a,\cdot )\) (Remark 2.1(e)). Similarly, if \(\sigma (a)\le 1\), then \(\varvec{g}^{2p}_{\varvec{D}_\alpha }(a,\cdot )\in \mathcal C^\infty (\{a\})\) and consequently \(\varvec{S}^{(2p)}_{\varvec{D}_\alpha }(a;\cdot )=\varvec{A}_{\varvec{D}_\alpha }(a;\cdot )\) (Remark 2.1(f)). Problems start when \(\sigma (a)\ge 2\). We do not know effective formulas for \(\varvec{s}^{(p)}_{\varvec{D}_\alpha }(a,\cdot )\), \(p\ne 2\), and \(\varvec{S}^{(2p)}_{\varvec{D}_\alpha }(a;\cdot )\), \(p\ge 2\). To illustrate problems we discuss some particular cases.

\(\bullet \)   Assume that \(\sigma (a)\ge 1\) and \(k_j:=\frac{p\alpha _j}{r(a)}\in {\mathbb {N}}\), \(j=s+1,\dots ,n\). Then
$$\begin{aligned} \varvec{g}^{2p}_{\varvec{D}_\alpha }(a,z)=\prod _{j=1}^s|z_j|^{2p\alpha _j/r(a)}\cdot \prod _{j=s+1}^n|z_j|^{2k_j} \end{aligned}$$
and consequently \(\varvec{g}^{2p}_{\varvec{D}_\alpha }(a,\cdot )\in \mathcal C^\infty (\{a\})\) which gives \(\varvec{S}^{(2p)}_{\varvec{D}_\alpha }(a;\cdot )=\varvec{A}_{\varvec{D}_\alpha }(a;\cdot )\). For example, if \(n=2\), \(\alpha =(1,1)\), \(a=(0,0)\), then \(\varvec{S}^{(4k)}_{\varvec{D}_\alpha }(a;X)=|X_1X_2|^{1/2}\), \(k\in {\mathbb {N}}\).

\(\bullet \)   Assume that \(\sigma (a)\ge 1\) and there exists a \(j_0\in \{s+1,\dots ,n\}\) such that \(\frac{2p\alpha _{j_0}}{r(a)}\notin {\mathbb {N}}\). Then \(\varvec{S}^{(2p)}_{\varvec{D}_\alpha }(a;\cdot )\equiv 0\).

Indeed, we may assume that \(j_0=n\). Let \(r:=r(a)\) and let \(k\in {\mathbb {N}}_0\) be such that \(k<\frac{2p\alpha _n}{r}<k+1\). In view of Remark 2.1(b), we have to prove that \(u^{(2p)}(a)\equiv 0\) for all \(\root 2p \of {u}\in {\mathcal {S}}^{(2p)}_{\varvec{D}_\alpha }(a)\). Fix such a u and suppose that \(u^{(2p)}(a)(X^0)\ne 0\) for some \(X^0\ne 0\). We have
$$\begin{aligned}&\Big (\frac{1}{(2p)!}|u^{(2p)}(a)(X)|\Big )^{1/2p}\le \varvec{S}^{(2p)}_{\varvec{D}_\alpha }(a;X)\\&\quad \le \varvec{A}_{\varvec{D}_\alpha }(a;X)=\Big (\prod _{j=1}^s|a_j|^{\alpha _j}\cdot \prod _{j=s+1}^n|X_j|^{\alpha _j}\Big )^{1/r}. \end{aligned}$$
Write \(u^{(2p)}(a)(X_1^0,\dots ,X_{n-1}^0,tX_n^0)=A_dt^d+\dots +A_0\), \(t\in {\mathbb {R}}\), with \(A_d\ne 0\). We have \(|A_dt^d+\dots +A_0|\le {\text {const}}|t|^{2p\alpha _n/r}\), \(t\in {\mathbb {R}}\). Taking \(t\longrightarrow \infty \) we get \(d\le k\). On the other hand, taking \(t\longrightarrow 0\) we get \(A_d=0\); a contradiction.
For example let \(n=2\), \(\alpha =(q,1)\), \(a=(0,0)\), where
$$\begin{aligned} 0<q\notin \{\tfrac{2p-k}{k}: k=1,\dots ,2p-1\}\cap \{\tfrac{k}{2p-k}: k=1,\dots ,2p-1\}. \end{aligned}$$
Then \(\varvec{S}^{(2p)}_{\varvec{D}_\alpha }(a;\cdot )\equiv 0\).

\(\bullet \)   As a consequence, we conclude that for every \(s\in \{0,\dots ,n-2\}\) there exists a set \(C_s\) dense in \({\mathbb {R}}_*^s\times {\mathbb {R}}_{>0}^{n-s}\) (\({\mathbb {R}}_{>0}:=(0,+\infty )\)) such that for any \(\alpha \in C_s\), \(a\in \varvec{D}_\alpha \cap ({\mathbb {C}}_*^s\times \{0\}^{n-s})\), and \(p\in {\mathbb {N}}\) we have \(\varvec{S}^{(2p)}_{\varvec{D}_\alpha }(a;\cdot )\equiv 0\).

Indeed, we may put
$$\begin{aligned} C_s:=({\mathbb {R}}_*^s\times {\mathbb {R}}_{>0}^{n-s}){\setminus }\bigcup _{\begin{array}{c} p,k\in {\mathbb {N}}:\; k<2p\\ j\in \{s+1,\dots ,n\} \end{array}}\{\alpha \in {\mathbb {R}}^n: 2p\alpha _j=k(\alpha _{s+1}+\dots +\alpha _n)\}. \end{aligned}$$

Now we turn to discuss a special case where \(G\subset {\mathbb {C}}^n\) is a complete n-circled domain (Example 3.5).

Example 3.5

Let \(G\subset {\mathbb {C}}^n\) be a complete n-circled domain, i.e. for any \(z=(z_1,\dots ,z_n)\in G\) and \(\lambda =(\lambda _1,\dots ,\lambda _n)\in {{\overline{{\mathbb {D}}}}}^n\), the point \(\lambda \cdot z:=(\lambda _1z_1,\dots ,\lambda _nz_n)\) belongs to G.
  1. (a)

    Since \(\varvec{s}^{(p)}_G(0,\cdot )\le \varvec{g}_G(0,\cdot )\) and the Green function is upper semicontinuous, the function \(\varvec{s}^{(p)}_G(0,\cdot )\) is continuous at 0.

     
  2. (b)

    The function \(\varvec{s}^{(p)}_G(0,\cdot )\) is upper semicontinuous in the domain \(G{\setminus }\varvec{V}_0\), where \(\varvec{V}_0:=\{(z_1,\dots ,z_n)\in {\mathbb {C}}^n: z_1\cdots z_n=0\}\).

    Indeed, let \(M:=\{a\in G: \varvec{s}^{(p)}_G(0,\cdot ) \text { is not upper semicontinuous at }a\}\). Since \(\varvec{s}^{(p)}_G(0,\cdot )\) is invariant under n-rotations (i.e. under mappings \(G\ni z\longmapsto \lambda \cdot z\in G\), \(\lambda \in {\mathbb {T}}^n\), where \({\mathbb {T}}:=\partial {\mathbb {D}}\)), the set M is also invariant under n-rotations. It is known that M is pluripolar, i.e. there exists a \(v\in \mathcal {PSH}({\mathbb {C}}^n)\), \(v\not \equiv -\infty \), such that \(M\subset v^{-1}(-\infty )\) (cf. [3, Theorem 4.7.6]). Suppose that \(a=(a_1,\dots ,a_n)\in M{\setminus }\varvec{V}_0\). Then \(v(\lambda \cdot a)=-\infty \) for all \(\lambda \in {\mathbb {T}}^n\). Consequently, by the maximum principle for plurisubharmonic functions, \(v(z_1,\dots ,z_n)=-\infty \) for all \(|z_j|\le |a_j|\), \(j=1,\dots ,n\). Hence, \(v\equiv -\infty \); a contradiction.

     
  3. (c)

    Let \(a=(0,\dots ,0,a_{s+1},\dots ,a_n)=:(0,b)\in G\cap \varvec{V}_0\), \(1\le s\le n-1\), \(a_{s+1}\cdots a_n\ne 0\). Define \(D:=\{\zeta \in {\mathbb {C}}^{n-s}: (\underset{s\times }{\underbrace{0,\dots ,0}},\zeta )\in G\}\). Note that D is a complete \((n-s)\)-circled domain with \(b\in D\). Let \({\mathfrak {h}}_D\) denote the Minkowski functional of D (\({\mathfrak {h}}_D(\zeta ):=\inf \{1/t: t>0,\;t\zeta \in D\}\), \(\zeta \in {\mathbb {C}}^{n-s}\)). Observe that \({\mathfrak {h}}_D\) is continuous (because D is \((n-s)\)-circled).

     
Assume that \(\varvec{s}^{(p)}_D(0,b)={\mathfrak {h}}_D(b)\). Then the function \(\varvec{s}^{(p)}_G(0,\cdot )\) is upper semicontinuous at a.
Indeed, let \(0<R<1\) and \(k>0\) be such that \({\mathfrak {h}}_D(b)<R\) and \(\Vert b\Vert <k\). Note that \(\{\zeta \in D: {\mathfrak {h}}_D(\zeta )<R,\;\Vert \zeta \Vert <k\}\subset \subset D\). Consequently, there exists an \(\varepsilon >0\) such that \(U:=\{(z',z'')\in {\mathbb {C}}^n: \Vert z'\Vert<\varepsilon ,\;{\mathfrak {h}}_D(z'')<R,\;\Vert z''\Vert <k\}\subset G\). Then for \(z=(z',z'')\in U\) we have
$$\begin{aligned} \varvec{s}^{(p)}_G(0,z)\le \varvec{s}^{(p)}_U(0,z)\le \varvec{g}_U(0,z)\le \max \Big \{\frac{\Vert z'\Vert }{\varepsilon }, \frac{{\mathfrak {h}}_D(z'')}{R}, \frac{\Vert z''\Vert }{k}\Big \}. \end{aligned}$$
Hence,
$$\begin{aligned} \limsup _{z\rightarrow a}\varvec{s}^{(p)}_G(0,z)\le \limsup _{(z',z'')\rightarrow (0,b)} \max \Big \{\frac{\Vert z'\Vert }{\varepsilon }, \frac{{\mathfrak {h}}_D(z'')}{R}, \frac{\Vert z''\Vert }{k}\Big \}=\max \Big \{\frac{{\mathfrak {h}}_D(b)}{R}, \frac{\Vert b\Vert }{k}\Big \}. \end{aligned}$$
Letting \(R\longrightarrow 1\) and \(k\longrightarrow +\infty \) we get \(\limsup _{z\rightarrow a}\varvec{s}^{(p)}_G(0,z)\le {\mathfrak {h}}_D(b)\).
On the other side, since the projection \({\mathbb {C}}^s\times {\mathbb {C}}^{n-s}\ni (z',z'')\longmapsto z''\in D\) is well-defined, we get \({\mathfrak {h}}_D(b)=\varvec{s}^{(p)}_D(0,b)\le \varvec{s}^{(p)}_G(0,a)\).
  1. (d)

    Observe that \(\varvec{s}^{(p)}_D(0,b)={\mathfrak {h}}_D(b)\) in the case where D is convex. If \(s=n-1\), then D is either a disc or the whole \({\mathbb {C}}\). Thus, if \(s=n-1\), then the function \(\varvec{s}^{(p)}_G(0,\cdot )\) is upper semicontinuous at each point \(a\in \varvec{V}_0\) of the form \(a=(0,\dots ,0,a_j,0,\dots ,0)\in G\).

     
  2. (e)

    Consequently, if \(n=2\), then the function \(\varvec{s}^{(p)}_G(0,\cdot )\) is globally upper semicontinuous.

     
  3. (f)

    If G is bounded, then the function \(\varvec{s}^{(p)}_G(0,\cdot )\) is globally upper semicontinuous.

     
Indeed, we proceed by induction on \(n\ge 2\). The case \(n=2\) is solved in (e). Suppose the result is true for \(n-1\ge 2\). Let \(a=(a_1,\dots ,a_n)\in G\cap \varvec{V}_0\) (see Example 3.5 (b)). We may assume that \(a_{n-1}\ne 0\), \(a_n=0\). Define \(D:=\{z'\in {\mathbb {C}}^{n-1}: (z',0)\in G\}\); D is a bounded complete \((n-1)\)-circled domain. Thus, by the inductive assumption, \(\varvec{s}^{(p)}_D(0,\cdot )\) is upper semicontinuous. Since G is bounded, for every \(0<r<1\) with \(a\in rG\) there exists an \(\varepsilon >0\) such that \((rD)\times {\mathbb {D}}(\varepsilon )\subset \subset G\). Suppose that \(G\subset {\mathbb {D}}^n(R)\) and let \(\eta >0\) be such that \(rR|\frac{z_n}{z_{n-1}}|<\varepsilon \) for \(z\in U:=\{z=(z',z_n)\in a+{\mathbb {D}}^n(\eta ): z'\in rD\}\). For \(z\in U\) consider the holomorphic mapping \(F_z:rD\longrightarrow G\), \(F_z(w):=(w,w_{n-1}\frac{z_n}{z_{n-1}})\). We have \(\varvec{s}^{(p)}_G(0,F_z(w))\le \varvec{s}^{(p)}_{rD}(0,w)=\varvec{s}^{(p)}_D(0,w/r)\). In particular, \(\varvec{s}^{(p)}_G(0,z)=\varvec{s}^{(p)}_G(0,F_z(z'))\le \varvec{s}^{(p)}_D(0,z'/r)\). Thus, \(\limsup _{z\rightarrow a}\varvec{s}^{(p)}_G(0,z)\le \limsup _{z\rightarrow a}\varvec{s}^{(p)}_D(0,z'/r)=\varvec{s}^{(p)}_D(0,a'/r)\). Letting \(r\longrightarrow 1-\) (and using once again the upper semicontinuity of \(\varvec{s}^{(p)}_D(0,\cdot )\) we get \(\limsup _{z\rightarrow a}\varvec{s}^{(p)}_G(0,z)\le \varvec{s}^{(p)}_D(0,a')\le \varvec{s}^{(p)}_G(0,a)\) (cf. Example 3.5 (c)).

Note that if \(n\ge 3\) and D is unbounded, then it is not known whether the function \(\varvec{s}^{(p)}_G(0,\cdot )\) is globally upper semicontinuous.

4 Increasing domains property

Let \((G_k)_{k=1}^\infty \) be sequence of domains in \({\mathbb {C}}^n\) such that \(G_k\nearrow G\), i.e. \(G_k\subset G_{k+1}\), \(k\in {\mathbb {N}}\), and let \(G=\bigcup _{k=1}^\infty G_k\). It is known that if \({\mathcal {F}}\in \{{\mathcal {M}}, {\mathcal {K}}\}\), then \(d_{G_k}^{\mathcal {F}}\searrow d_G^{\mathcal {F}}\) and \(\delta _{G_k}^{\mathcal {F}}\searrow \delta _G^{\mathcal {F}}\) (cf. [1, Propositions 2.7.1(a), 4.2.10(a)]). We will show that this is not true for \({\mathcal {F}}={\mathcal {S}}^{(p)}\).

Example 4.1

Let
$$\begin{aligned} \varphi _k(\lambda ):=\sum _{s=2}^k\frac{1}{s^2}\log \Big |\lambda -\frac{1}{s}\Big |,\quad k\ge 2,\quad \varphi (\lambda ):=\sum _{s=2}^\infty \frac{1}{s^2}\log \Big |\lambda -\frac{1}{s}\Big |,\quad |\lambda |<\frac{1}{2}. \end{aligned}$$
Observe that \(\varphi _k\in \mathcal {PSH}\) and \(\varphi _k\searrow \varphi \). Moreover, \(\varphi _k\in \mathcal C^\infty (\tfrac{1}{k}{\mathbb {D}})\). Define
$$\begin{aligned} G_k:= & {} \{(z_1,z_2)\in {\mathbb {C}}^2: |z_1|<1/2,\;|z_2|e^{\varphi _k(z_1)}<1\},\\ G:= & {} \{(z_1,z_2)\in {\mathbb {C}}^2: |z_1|<1/2,\;|z_2|e^{\varphi (z_1)}<1\}. \end{aligned}$$
Note that \(G_k\) is a Hartogs domain in \({\mathbb {C}}^2\), \(k\ge 2\), and \(G_k\nearrow G\). For each \(k\ge 2\) the function \(G_k\ni (z_1,z_2)\longmapsto (|z_2|e^{\varphi _k(z_1)})^{1+\varepsilon /p}\) belongs to \({\mathcal {S}}^{(p)}_{G_k}((0,0))\), \(\varepsilon >0\). Hence, \(\varvec{s}^{(p)}_{G_k}((0,0),(0,z_2))\ge |z_2|e^{\varphi _k(0)}\ge |z_2|e^{\varphi (0)}\) for \(|z_2|<e^{-\varphi (0)}\).

Analogously, since the function \(G_k\ni (z_1,z_2)\longmapsto |z_2|e^{\varphi _k(z_1)}\) belongs to \({\mathcal {S}}^{(2p)}_{G_k}((0,0))\), we get \(\varvec{S}^{(2p)}_{G_k}((0,0);(0,X_2))\ge |X_2|e^{\varphi (0)}\) for \(X_2\in {\mathbb {C}}\) and \(k\ge 2\).

Now let \(\root p \of {u}\in {\mathcal {S}}^{(p)}_G((0,0))\). Since \(\{1/s\}\times {\mathbb {C}}\subset G\), the Liouville type theorem for subharmonic functions gives \(u(1/s,z_2)={\text {const}}(s)=:c_s\), \(s\ge 2\), \(z_2\in {\mathbb {C}}\). Since \(u(0,0)=0\), we conclude that \(c_s\longrightarrow 0\). Since u is continuous near (0, 0), we get \(u(0,z_2)=\lim _{s\rightarrow +\infty }u(1/s,z_2)=\lim _{s\rightarrow +\infty }c_s=0\), \(|z_2|\ll 1\). Hence, since \(\log u\in \mathcal {PSH}(G)\), we have \(u(0,z_2)=0\) for all \(|z_2|<e^{\varphi (0)}\). Consequently, \(\varvec{s}^{(p)}_G((0,0),(0,z_2))=0\), \(|z_2|<e^{\varphi (0)}\), and \(\varvec{S}^{(p)}_G((0,0);(0,X_2))=0\), \(X_2\in {\mathbb {C}}\).

5 Relations between \((\varvec{m}_G, \varvec{s}_G, \varvec{g}_G)\) and \(({\varvec{\gamma }}_G, \varvec{S}_G, \varvec{A}_G\))

We will discuss the following two problems. Find a pseudoconvex domain \(G\subset {\mathbb {C}}^n\), \(a\in G\), and \(z^0\in G\) (resp. \(X_0\in {\mathbb {C}}^n\)) such that
$$\begin{aligned}&\varvec{m}_G(a,z^0)<\varvec{s}_G(a,z^0)<\varvec{g}_G(a, z^0)\\&\quad \text {(resp. } {\varvec{\gamma }}_G(a;X_0)<\varvec{S}_G(a;X_0)<\varvec{A}_G(a;X_0)\text {)}. \end{aligned}$$

Example 5.1

If \(\alpha _1,\dots ,\alpha _n\in {\mathbb {Z}}\) are relatively prime, \(\sigma (a)\ge 2\), and \(\mu (a)\ge 2\), then the domain \(G=\varvec{D}_\alpha \) (cf. Example 3.3) is an example of a pseudoconvex domain (unfortunately, unbounded) such that \(\varvec{m}_G(a,\cdot )<\varvec{s}_G(a,\cdot )<\varvec{g}_G(a,\cdot )\) on \(\varvec{D}_\alpha {\setminus }\varvec{V}_0\). It is not known whether there exists a bounded pseudoconvex domain with this property.

Example 5.2

Let \(G\subset {\mathbb {C}}^n\) be a balanced domain (i.e. \({{\overline{{\mathbb {D}}}}}\cdot G=G\)) and let \({\mathfrak {h}}_G(z)\) be the Minkowski functional of G. It is known that \(G=\{z\in {\mathbb {C}}^n: {\mathfrak {h}}_G(z)<1\}\). Moreover, \(\varvec{g}_G(0,\cdot )={\mathfrak {h}}_G\) in G\(\Longleftrightarrow \)\(\varvec{A}_G(0;\cdot )\equiv {\mathfrak {h}}_G\)\(\Longleftrightarrow \)G is pseudoconvex \(\Longleftrightarrow \)\(\log {\mathfrak {h}}_G\in \mathcal {PSH}({\mathbb {C}}^n)\) (cf. [1, Proposition 4.2.10(b)]).

Let \(\widehat{G}\) be the convex envelope of G. It is known that \(\widehat{G}\) is also balanced and \({\mathfrak {h}}_{\widehat{G}}=\sup \{q: q:{\mathbb {C}}^n\longrightarrow [0,+\infty )\) is a \({\mathbb {C}}\)-seminorm with \(q\le {\mathfrak {h}}_G\}\). Moreover (cf. [1, Proposition 2.3.1(d)]), \({\varvec{\gamma }}_G(0;\cdot )\equiv {\mathfrak {h}}_{\widehat{G}}\). Thus, if G is pseudoconvex, then \({\varvec{\gamma }}_G(0;\cdot )={\mathfrak {h}}_{\widehat{G}}\ge \varvec{S}_G(0;\cdot )\) and hence \({\varvec{\gamma }}_G(0;\cdot )\equiv \varvec{S}_G(0;\cdot )\equiv {\mathfrak {h}}_{\widehat{G}}\le {\mathfrak {h}}_G\equiv \varvec{A}_G(0;\cdot )\). Consequently, we get the following result.

If G is a balanced pseudoconvex non-convex domain, then
$$\begin{aligned} {\mathfrak {h}}_{\widehat{G}}\equiv {\varvec{\gamma }}_G(0;\cdot )\equiv \varvec{S}_G(0;\cdot )\genfrac{}{}{0.0pt}0{\le }{\not \equiv }\varvec{A}_G(0;\cdot )\equiv {\mathfrak {h}}_G. \end{aligned}$$

In particular, the result solves the problem formulated in Example 4.2.17 from [1].

Example 5.3

Keep the notation from Example 3.1. Then
$$\begin{aligned} {\varvec{\gamma }}_G(c_t;X_0)<\varvec{S}^{(2p)}_G(c_t;X_0)=\varvec{A}_G(c_t;X_0)=e^{\varphi (0,t)},\quad p\in {\mathbb {N}},\; 0<t\ll 1. \end{aligned}$$
Indeed, the function \(G\ni (z_1,z_2,z_3)\overset{v}{\longmapsto }|z_1|e^{\varphi (z_2,z_3)}\) is of the class \({\mathcal {S}}^{(2p)}_G(c_t)\), which gives
$$\begin{aligned} \varvec{S}^{(2p)}_G(c_t;X_0)\ge \limsup _{\lambda \rightarrow 0}\frac{v(c_t+\lambda X_0)}{|\lambda |}=e^{\varphi (0,t)}>0,\quad t>0. \end{aligned}$$
Observe that the mapping \(e^{-\varphi (0,t)}{\mathbb {D}}\ni \lambda \overset{F}{\longmapsto }(\lambda ,0,t)\in G\) is well-defined. Hence, using the holomorphic contractibility, we get
$$\begin{aligned} \varvec{A}_G(c_t;X_0)=\varvec{A}_G(F(0);F'(0)(X_0))\le \varvec{A}_{e^{-\varphi (0,t)}{\mathbb {D}}}(0;1)=e^{\varphi (0,t)}. \end{aligned}$$
Thus, \(\varvec{S}^{(2p)}_G(c_t;X^0)=\varvec{A}_G(c_t;X^0)=e^{\varphi (0,t)}\ge e^{\varphi (0,0)}>0\), \(t>0\).

Now, to get the result it suffices to show that \({\varvec{\gamma }}_G((0,0,0);X^0)=0\) and then use the continuity of \({\varvec{\gamma }}_G(\cdot ;X^0)\). For this, let \(f\in {\mathcal {O}}(G,{\mathbb {D}})\) such that \(f(0,0,0)=0\). Since \(\{0\}\times {\mathbb {C}}^2\subset G\), the Liouville theorem implies that \(f(0,\cdot ,\cdot )={\text {const}}\). Since \(f(0,0,0)=0\), we get \(f(0,\cdot ,\cdot )\equiv 0\). Since \({\mathbb {C}}\times \{a_k\}\times \{0\}\subset G\), we get \(f(\cdot ,a_k,0)={\text {const}}(k)\). Thus, \(f(\cdot ,a_k,0)\equiv 0\). Since the sequence \((a_k)_{k=1}^\infty \) is dense in \({\mathbb {D}}\), we conclude that \(f=0\) on \(({\mathbb {C}}\times {\mathbb {D}}\times \{0\})\cap G\). Thus, \(f(z_1,0,0)=0\) provided that \(|z_1|<e^{-\varphi (0,0)}\). Hence, \(f'(0,0,0)(X^0)=0\) and so \({\varvec{\gamma }}_G((0,0,0);X^0)=0\).

6 Derivative

Recall that for \({\mathcal {F}}\in \{{\mathcal {M}}, {\mathcal {K}}\}\) we have \(\delta ^{\mathcal {F}}_G(a;X)=\limsup _{\lambda \rightarrow 0}\frac{d^{\mathcal {F}}_G(a,a+\lambda X)}{|\lambda |},\quad a\in G,\;X\in {\mathbb {C}}^n\) (cf. (1.3) and (1.4)). It is an open problem whether
$$\begin{aligned} \varvec{S}^{(2p)}_G(a;X)=\limsup _{\lambda \rightarrow 0}\frac{\varvec{s}^{(2p)}_G(a,a+\lambda X)}{|\lambda |},\quad a\in G,\;X\in {\mathbb {C}}^n. \end{aligned}$$
Observe that
$$\begin{aligned} \varvec{S}^{(2p)}_G(a;X)= & {} \sup \Big \{\limsup _{\lambda \rightarrow 0}\frac{v(a+\lambda X)}{|\lambda |}: v\in {\mathcal {S}}^{(2p)}_G(a)\Big \}\\\le & {} \limsup _{\lambda \rightarrow 0}\frac{\varvec{s}^{(2p)}_G(a,a+\lambda X)}{|\lambda |}\le \limsup _{\lambda \rightarrow 0}\frac{\varvec{g}_G(a,a+\lambda X)}{|\lambda |}=\varvec{A}_G(a;X), \end{aligned}$$
so the problem is trivial if \(\varvec{S}^{(2p)}_G(a;X)=\varvec{A}_G(a;X)\).

Notes

References

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    Jarnicki, M., Pflug, P.: Invariant Distances and Metrics in Complex Analysis, 2nd extended edition, de Gruyter Expositions in Mathematics 9, Walter de Gruyter, xvii+861 pp (2013)Google Scholar
  2. 2.
    Jarnicki, M., Pflug, P.: A remark on the Sibony function. J. Math. Anal. Appl. 461, 1374–1377 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Klimek, M.: Pluripotential Theory. Oxford University Press, Oxford (1991)zbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Institute of Mathematics, Faculty of Mathematics and Computer ScienceJagiellonian UniversityKrakówPoland
  2. 2.Institut für MathematikCarl von Ossietzky Universität OldenburgOldenburgGermany

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