Proof. Note that \(\left(\xi\cdot \frac{(z-z_0)^{\alpha_0}}{\xi_{\alpha_0}}\right)(z_0)=1\), and \(\left\|\frac{(z-z_0)^{\alpha_0}}{\xi_{\alpha_0}}\right\|_p<+\infty\) as \(\Omega\) is bounded. Thus, \(m_{\xi,p}(z_0)<+\infty\).

Let \(\{f_j\}\) be a sequence of elements in \(A^p(\Omega)\) such that \((\xi\cdot f_j)(z_0)=1\) for every \(j\) and \(\|f_j\|_p\rightarrow m_{\xi,p}(z_0)\) as \(j\rightarrow +\infty\). Then 3 shows that \(\{f_j\}\) forms a normal family on \(\Omega\). Therefore, there exists a subsequence \(\{f_{j_k}\}\) of \(\{f_j\}\) such that \(\{f_{j_k}\}\) converges uniformly to a function \(f_0\in\mathcal{O}(\Omega)\) on every compact subset of \(\Omega\). It follows from Lemma 14 that \((\xi\cdot f_0)(z_0)=1\). In addition, Fatou’s Lemma implies \[\int_{\Omega}|f_0|^p\le \lim_{k\rightarrow +\infty}\int_{\Omega}|f_{j_k}|^p=m_{\xi,p}(z_0)^p.\] Therefore, \(\|f_0\|_p=m_{\xi,p}(z_0)\). ◻

Proof. Suppose \(f_1, f_2\) are two elements in \(A^p(\Omega)\) which satisfy \((\xi\cdot f_j)(z_0)=1\) and \(m_{\xi,p}(z_0)=\|f_j\|_p\) for \(j=1,2\). On the one hand, we have \(\left(\xi\cdot \frac{f_1+f_2}{2}\right)(z_0)=1\), which implies \(\left\|\frac{f_1+f_2}{2}\right\|_p\ge m_{\xi,p}(z_0)\) by the definition of \(m_{\xi,p}(z_0)\). On the other hand, since \(\|f_1\|_p=\|f_2\|_p=m_{\xi,p}(z_0)\), we can see \[\left\|\frac{f_1+f_2}{2}\right\|^p_p=\int_{\Omega} \left|\frac{f_1+f_2}{2}\right|^p \le \int_{\Omega}\frac{|f_1|^p+|f_2|^p}{2}=m_{\xi,p}(z_0),\] according to the convexity of the function \(x^p\) on \((0,+\infty)\). Therefore, \(\left\|\frac{f_1+f_2}{2}\right\|_p=m_{\xi,p}(z_0)\). Lemma 12 shows that \(f_1=f_2\). ◻

Proof. Let \(f_0\) be an element of \(A^p(\Omega)\) which satisfies \((\xi\cdot f_0)(z)=1\) and \(m_{\xi,p}(z)=\|f_0\|_p\). Since \(K_{\xi,p}(z)=m_{\xi,p}(z)^{-p}=\|f_0\|_p^{-p}=1/\int_{\Omega}|f_0|^p\), we have \[K_{\xi,p}(z)= \frac{|(\xi\cdot f_0)(z)|^p}{\int_{\Omega}|f_0|^p}\le \sup_{f\in A^p(\Omega)}\frac{|(\xi\cdot f)(z)|^p}{\int_{\Omega}|f|^p}.\]

On the other hand, let \(f\in A^p(\Omega)\) be an element such that \((\xi\cdot f)(z)\neq 0\). Then \[\left(\xi\cdot\frac{f}{(\xi\cdot f)(z)}\right)(z)=1.\] Then according to the definition of \(m_{\xi,p}(z)\), we have \[m_{\xi,p}(z)\le \left\| \frac{f}{(\xi\cdot f)(z)} \right\|_p=\frac{1}{|(\xi\cdot f)(z)|}\|f\|_p,\] which implies that \[\frac{|(\xi\cdot f)(z)|^p}{\int_{\Omega}|f|^p}=\frac{|(\xi\cdot f)(z)|^p}{\|f\|_p^p}\le m_{\xi,p}(z)^{-p}=K_{\xi,p}(z).\] Since \(f\in A^p(\Omega)\) is arbitrarily chosen with \((\xi\cdot f)(z)\neq 0\), we get \[\sup_{f\in A^p(\Omega)}\frac{|(\xi\cdot f)(z)|^p}{\int_{\Omega}|f|^p}\le K_{\xi,p}(z).\] Therefore, Proposition 18 is proved. ◻

Proof of Proposition 1. Let \(r\mathrel{\vcenter{:}}= \inf_{z\in K}{\text{dist}(z,\partial \Omega)}\), and \[U\mathrel{\vcenter{:}}= \Big\{z\in\Omega\colon \text{dist}(z,K)\le\frac{r}{2}\Big\}.\] Then for any \(z\in K\), \(\overline{B(z,\frac{r}{2})}\subset U\) and for any \(z\in U\), \(B(z,\frac{r}{2})\subset \Omega\).

Since \(\log|f|\) is a plurisubharmonic function for any holomorphic function \(f\), \[|f(z)|^p\le\frac{1}{\frac{\pi^n}{n!}\left(\frac{r}{2}\right)^{2n}}\int_{B(z,\frac{r}{2})}|f|^p\le \frac{1}{\frac{\pi^n}{n!}\left(\frac{r}{2}\right)^{2n}}\|f\|_{L^p(\Omega)}^p\] holds for any \(p>0\) and \(z\in U\). Therefore, \[\begin{align} \label{b1} \sup_{U}|f|\le (n!)^\frac{1}{p}\left(\frac{4}{\pi r^2}\right)^{\frac{n}{p}}\|f\|_{L^p(\Omega)}. \end{align}\tag{4}\] For any \(z=(z_1,\ldots,z_n)\in K\), Note that \[\overline{B\left(z_1,\frac{r}{2\sqrt{n}}\right)}\times\cdots\times \overline{B\left(z_n,\frac{r}{2\sqrt{n}}\right)}\subset \overline{B\left(z,\frac{r}{2}\right)}\subset U.\] It follows from Cauchy’s estimate that \[\begin{align} \label{b2} \left|\frac{f^{(\alpha)}(z)}{\alpha!}\right|\le \left(\frac{2\sqrt{n}}{r}\right)^{|\alpha|}\sup_{U}|f| \end{align}\tag{5}\] for any \(\alpha\in \mathbb{N}^n\) and \(z\in K\). Then the inequalities 4 and 5 imply that \[\begin{align} \label{b3} \left|\frac{f^{(\alpha)}(z)}{\alpha!}\right|&\le \left(\frac{2\sqrt{n}}{r}\right)^{|\alpha|}(n!)^\frac{1}{p}\left(\frac{4}{\pi r^2}\right)^{\frac{n}{p}}\|f\|_{L^p(\Omega)} \end{align}\tag{6}\] for any \(\alpha\in \mathbb{N}^n\) and \(z\in K\).

The definition of \(\ell_1^{(n)}\) implies that \[\sum_{\alpha\in\mathbb{N}^n}|\xi_{\alpha}|\left(\frac{2\sqrt{n}}{r}\right)^{|\alpha|}<C_p'\] for a positive constant \(C_p'\). Set \(C_{K,p}=C_p'\cdot (n!)^\frac{1}{p}\left(\frac{4}{\pi r^2}\right)^{\frac{n}{p}}\), which is independent of \(f\), and we get that \[|(\xi\cdot f)(z)|\le C_{K,p}\|f\|_{L^p(\Omega)}\] holds for all \(z\in K\). Proposition 1 is proved. ◻

Proof. Let \(R>0\) be the diameter of \(\Omega\). Recall that \(\xi_{\alpha_0}\neq 0\) for an \(\alpha_0\in\mathbb{N}^n\) by assumption. For any \(z_0\in\Omega\), we have \((\xi\cdot (z-z_0)^{\alpha_0})(z_0)=\xi_{\alpha_0}\). Then Proposition 18 implies that \[\frac{|\xi_{\alpha_0}|^p}{\int_{B(o,R)}|z^{\alpha_0}|^p}\le \frac{|\xi_{\alpha_0}|^p}{\int_{\Omega}|(z-z_0)^{\alpha_0}|^p}\le K_{\xi,p}(z_0).\] Note that \(\frac{|\xi_{\alpha_0}|^p}{\int_{B(o,R)}|z^{\alpha_0}|^p}\) is a positive constant independent of the choice of \(z_0\). Therefore, \[\begin{align} \label{align3} \frac{|\xi_{\alpha_0}|^p}{\int_{B(o,R)}|z^{\alpha_0}|^p}\le K_{\xi,p}(z) \end{align}\tag{7}\] for all \(z\in\Omega\), i.e., \(K_{\xi,p}(z)\) is uniformly bounded from below by a positive constant.

On the other hand, for a compact subset \(K\) of \(\Omega\), Proposition 18 and Proposition 1 imply that \[\begin{align} \label{align4} K_{\xi,p}(z)\le C \end{align}\tag{8}\] for all \(z\in K\), where \(C\) is a positive constant depending on \(p\) and \(\text{dist}(K,\partial \Omega)\). ◻

Proof. Let \(K\) be a compact subset of \(\Omega\). Choose an open set \(U\) such that \(K\subset U\Subset \Omega\). For any \(z_0\in K\), By Proposition 15, there exists some \(f_0\in A^p(\Omega)\) such that \((\xi\cdot f_{0})(z_0)=\big(K_{\xi,p}(z_0)\big)^{1/p}\) and \(\|f_0\|_p=1\).

Proposition 1 implies that \[\sup_{z\in \overline{U}} |(\xi\cdot f_0)(z)|\le C_p\] for a constant \(C_p>0\) independent of \(z_0\) and \(f_0\). Since \((\xi\cdot f_0)\) is a holomorphic function by Lemma 13, we get from Cauchy’s estimate that \((\xi\cdot f_0)^{(\alpha)}\) is uniformly bounded on \(K\) for a fixed \(\alpha\in\mathbb{N}^n\). In particular, there exists a constant \(C_K>0\) independent of \(z_0\) and \(f_0\) such that \[\begin{align} (K_{\xi,p}(z_0))^{\frac{1}{p}}=|(\xi\cdot f_0)(z_0)|&\le |(\xi\cdot f_0)(z)|+C_K|z-z_0|\\ &\le (K_{\xi,p}(z))^{\frac{1}{p}}+C_K|z-z_0| \end{align}\] for any \(z\in K\). As \(K_{\xi,p}(z)\) is uniformly bounded on \(K\) by 8 and larger than a positive constant by 7 , we conclude that \(K_{\xi,p}(z)\) is Lipschitz continuous on \(K\), which finishes the proof. ◻

Proof. For any \(z_0\in\Omega\), we can choose \(j_0\) such that \(z_0\in \Omega_{j_0}\) whenever \(j\ge j_0\).

By Proposition 15, we can choose an \(f_j\in A^{p}(\Omega)\) for each \(j\ge j_0\) such that \[(\xi\cdot f_j)(z_0)=K_{\xi,\Omega_j,p}(z_0)^{1/p} \quad \text{and} \quad \|f_j\|_{L^p(\Omega_j)}=1.\] It follows from 10 that \(\lim_{j\rightarrow +\infty}K_{\xi,\Omega_j,p}(z_0)\) exists.

According to Proposition 1 and Montel’s theorem, we can extract a subsequence \((j_k)\) of \((j)\) such that \(\{f_{j_k}\}\) converges locally uniformly to a holomorophic function \(f_0\) on \(\Omega\). Lemma 14 implies that \[(\xi\cdot f_0)(z_0)=\lim_{k\rightarrow +\infty}(\xi\cdot f_{j_k})(z_0)=\lim_{k\rightarrow +\infty}K_{\xi,\Omega_{j_k},p}(z_0)^{1/p}=\lim_{j\rightarrow +\infty}K_{\xi,\Omega_j,p}(z_0)^{1/p}.\] On the other hand, Fatou’s Lemma induces \[\|f_0\|^p_{L^p(\Omega)}=\int_{\Omega}|f_0|^p\le \liminf_{k\rightarrow +\infty}\int_{\Omega_{j_k}}|f_{j_k}|^p=1.\] Therefore, Proposition 18 implies \[K_{\xi,\Omega,p}(z_0)\ge |(\xi\cdot f_0)(z_0)|^p=\lim_{j\rightarrow +\infty}K_{\xi,\Omega_j,p}(z_0).\] Combining this inequality with 10 , we get \[K_{\xi,\Omega,p}(z_0)=\lim_{j\rightarrow +\infty}K_{\xi,\Omega_j,p}(z_0).\]

Given arbitrary \(\epsilon>0\), we choose \(j_1\) such that \[K_{\xi,\Omega_{j_1},p}(z_0)\le K_{\xi,\Omega,p}(z_0)+\frac{\epsilon}{3}.\] Choose a neighborhood \(U_0\) of \(z_0\) such that \[|K_{\xi,\Omega_{j_1},p}(z)-K_{\xi,\Omega_{j_1},p}(z_0)|<\frac{\epsilon}{3}\] and \[|K_{\xi,\Omega,p}(z)-K_{\xi,\Omega,p}(z_0)|<\frac{\epsilon}{3}\] for any \(z\in U_0\). Then \[K_{\xi,\Omega_{j_1},p}(z)\le K_{\xi,\Omega,p}(z)+\epsilon\] for any \(z\in U_0\).

Since \(z_0\) is arbitrarily chosen and \(K_{\xi,\Omega_{j},p}(z)\) is decreasing in \(j\), we conclude that \[\lim_{j\rightarrow +\infty}K_{\xi,\Omega_j,p}(z)=K_{\xi,\Omega,p}(z)\] locally uniformly on \(\Omega\). ◻

Proof. By a change of coordinates, we can assume that \(z_0=(o',o'')\), where \(o'\in\Omega_1\) and \(o''\in\Omega_2\).

Denote the coordinate of \(\Omega_0\) by \((z',z'')\), where \(z'\) is the coordinate of \(\Omega_1\) and \(z''\) is the coordinate of \(\Omega_2\). Let \(f_1\in A^p(\Omega_1)\) and \(f_2\in A^p(\Omega_2)\) satisfying that \[(\xi^{(1)}\cdot f_1)(o')=1, \quad (\xi^{(2)}\cdot f_2)(o'')=1,\] and \[\|f_1\|_{L^p(\Omega_1)}=m_{\xi^{(1)},\Omega_1,p}(o'),\quad \|f_2\|_{L^p(\Omega_2)}=m_{\xi^{(2)},\Omega_2,p}(o'').\] Suppose that we have the following Taylor’s expansions \[f_1(z')=\sum_{\alpha\in\mathbb{N}^{m_1}}a_{\alpha}(z')^{\alpha}\] near \(o'\) and \[f_2(z'')=\sum_{\beta\in\mathbb{N}^{m_2}}b_{\beta}(z'')^{\beta}\] near \(o''\). Let \(f(z',z'')\mathrel{\vcenter{:}}= f_1(z')f(z'')\), which is a holomorphic function on \(\Omega_0\) such that \[f(z',z'')=\sum_{\alpha\in\mathbb{N}^{m_1},\,\beta\in\mathbb{N}^{m_2}}a_{\alpha}b_{\beta}(z')^{\alpha}(z'')^{\beta}\] near \((o',o'')\). Therefore, \[\begin{align} (\xi^{(0)}\cdot f)(o',o'')&=\sum_{\alpha\in\mathbb{N}^{m_1},\beta\in\mathbb{N}^{m_2}}a_{\alpha}b_{\beta}\xi^{(1)}_{\alpha}\xi^{(2)}_{\beta}\notag\\ &=\left(\sum_{\alpha\in\mathbb{N}^{m_1}}a_{\alpha}\xi^{(1)}_{\alpha}\right)\cdot\left(\sum_{\beta\in\mathbb{N}^{m_2}}b_{\beta}\xi^{(2)}_{\beta}\right)=1.\notag \end{align}\] Fubini’s theorem shows that \[\int_{\Omega_0}|f|^p=\int_{\Omega_1}|f_1|^p\int_{\Omega_2}|f_2|^p=m_{\xi^{(1)},\Omega_1,p}(z_1)^pm_{\xi^{(2)},\Omega_2,p}(z_2)^p,\] yielding that \[m_{\xi^{(0)},\Omega_0,p}(o',o'')\le m_{\xi^{(1)},\Omega_1,p}(o')\cdot m_{\xi^{(2)},\Omega_2,p}(o'').\]

Next, we prove \[\begin{align} \label{a2} m_{\xi^{(0)},\Omega_0,p}(o',o'')\ge m_{\xi^{(1)},\Omega_1,p}(o')\cdot m_{\xi^{(2)},\Omega_2,p}(o''). \end{align}\tag{11}\] For any \(h\in A^p(\Omega_0)\), set \[g(z'')=(\xi^{(1)}\cdot h(\cdot,z''))(o').\] By the definition, it can be seen that \(g(z'')\) is a holomorphic function on \(\Omega_2\). Suppose \[h(z',z'')=\sum_{\alpha\in\mathbb{N}^{m_1},\beta\in\mathbb{N}^{m_2}}a_{\alpha,\beta}(z')^{\alpha}(z'')^{\beta}\] near \((o',o'')\). Then \[g(z'')=\sum_{\beta\in\mathbb{N}^{m_2}}\left(\sum_{\alpha\in\mathbb{N}^{m_1}}a_{\alpha,\beta}\xi^{(1)}_{\alpha}\right)(z'')^{\beta},\] near \(o''\), which implies that \[(\xi^{(2)}\cdot g)(o'')=\sum_{\alpha\in\mathbb{N}^{m_1},\beta\in\mathbb{N}^{m_2}}a_{\alpha,\beta}\xi^{(1)}_{\alpha}\xi^{(2)}_{\beta}=(\xi^{(0)}\cdot h)(o',o'').\] According to Proposition 18, we have \[|(\xi^{(2)}\cdot g)(o'')|^p\le K_{\xi^{(2)},\Omega_2,p}(o'')\int_{\Omega_2}|g|^p,\] and \[|g(z'')|^p=|(\xi^{(1)}\cdot h(\cdot,z''))(o')|^p\le K_{\xi^{(1)},\Omega_1,p}(o')\int_{\Omega_1}|h(\cdot,z'')|^p,\] for any \(z''\in\Omega_2\). Hence, \[|(\xi^{(0)}\cdot h)(o',o'')|^p\le K_{\xi^{(1)},\Omega_1,p}(o')\cdot K_{\xi^{(2)},\Omega_2,p}(o'')\int_{\Omega_0}|h|^p\] for any \(h\in A^p(\Omega_0)\), which implies that \[K_{\Omega_0,\xi^{(0)},p}(o',o'')\le K_{\xi^{(1)},\Omega_1,p}(o')\cdot K_{\xi^{(2)},\Omega_2,p}(o''),\] and then 11 follows from \(K_{\xi,p}=m_{\xi,p}^{-p}\) for any \(p\) and \(\xi\).

Let \(m(z,z_0)=m_{\xi^{(1)},\Omega_1,p}(w_1,z_1)\cdot m_{\xi^{(2)},\Omega_2,p}(w_2,z_2)\) for \(p\ge 1\). Since \[(\xi^{(0)}\cdot m(\cdot,z_0))(z_1,z_2)=1\] and \[\|m(\cdot,z_0)\|_{L^p(\Omega_0)}=m_{\xi^{(1)},\Omega_1,p}(z_1)\cdot m_{\xi^{(2)},\Omega_2,p}(z_2),\] We get that ?? holds by Proposition 16. ◻

Proof. For \(t\in \Delta\mathrel{\vcenter{:}}= \{z\in\mathbb{C}\colon|z|<1\}\), set \(f_t\mathrel{\vcenter{:}}= tf+m_{\xi,p}(\cdot,z_0)\). Note that \(f_t\) is a holomorphic function on \(\Omega\) satisfying \((\xi\cdot f_t)(z_0)=1\) for any such \(t\).

For any \(t\in\Delta\), rewrite \[|f_t|^p=\left(|m_{\xi,p}(\cdot,z_0)|^2+tf\overline{m_{\xi,p}(\cdot,z_0)}+\overline{tf}m_{\xi,p}(\cdot,z_0)+tf\overline{tf}\right)^\frac{p}{2}.\] Therefore, \[\frac{\partial |f_t|^p}{\partial t}=\frac{p}{2}|f_t|^{p-2}(f\overline{m_{\xi,p}(\cdot,z_0)}+f\overline{tf})=\frac{p}{2}|f_t|^{p-2}f\overline{f_t}\] outside a zero measure set \(f_t^{-1}(0)\subset\Omega\). We get \[\left|\frac{\partial |f_t|^p}{\partial t}\right|\le\frac{p}{2}(|f|+|m_{\xi,p}(\cdot,z_0)|)^{p-1}|f|\] for every \(t\in\Delta\). Note that Hölder inequality implies \[\begin{align} \label{align7} \int_{\Omega}(|f|+|m_{\xi,p}(\cdot,z_0)|)^{p-1}|f|\le \||f|+|m_{\xi,p}(\cdot,z_0)|\|_p^{p-1}\|f\|_p<+\infty \end{align}\tag{12}\] when \(p>1\), and inequality 12 obviously holds for \(p=1\). Since \(\|f_t\|_p\) takes its minimal value at \(t=0\) by the definition of \(m_{\xi,p}(\cdot,z_0)\), we conclude from 12 and the dominated convergence theorem that \[0=\frac{\partial \|f_t\|_p^p}{\partial t}\bigg|_{t=0}=\int_{\Omega}\left(\frac{\partial |f_t|^p}{\partial t}\right)\bigg|_{t=0}=\frac{p}{2}\int_{\Omega}|m_{\xi,p}(\cdot,z_0)|^{p-2}\overline{m_{\xi,p}(\cdot,z_0)}f.\] ◻

Proof of Theorem 2. Recall that \((\xi\cdot (z-z_0)^{\alpha_0})(z_0)=\xi_{\alpha_0}\neq0\). For any \(f\in A^p(\Omega)\), \[f(z)-\frac{(\xi\cdot f)(z_0)}{\xi_{\alpha_0}}(z-z_0)^{\alpha_0}\in A^p(\Omega),\] and \[\left(\xi\cdot\Big(f(z)-\frac{(\xi\cdot f)(z_0)}{\xi_{\alpha_0}}(z-z_0)^{\alpha_0}\Big)\right)(z_0)=0.\] Lemma 23 implies that \[\int_{\Omega}|m_{\xi,p}(\cdot,z_0)|^{p-2}\overline{m_{\xi,p}(\cdot,z_0)}\left(f-\frac{(\xi\cdot f)(z_0)}{\xi_{\alpha_0}}(\cdot-z_0)^{\alpha_0}\right)=0,\] which can be rewritten as \[\begin{align} \label{align8} \int_{\Omega}|m_{\xi,p}(\cdot,z_0)|^{p-2}\overline{m_{\xi,p}(\cdot,z_0)}f=\frac{(\xi\cdot f)(z_0)}{\xi_{\alpha_0}}\int_{\Omega}|m_{\xi,p}(\cdot,z_0)|^{p-2}\overline{m_{\xi,p}(\cdot,z_0)}(\cdot-z_0)^{\alpha_0}. \end{align}\tag{13}\] Choose \(f=m_{\xi,p}(\cdot,z_0)\). Since \((\xi\cdot m_{\xi,p}(\cdot,z_0))(z_0)=1\), we then get \[\begin{align} \label{align9} m_{\xi,p}(z_0)^p=\frac{1}{\xi_{\alpha_0}}\int_{\Omega}|m_{\xi,p}(\cdot,z_0)|^{p-2}\overline{m_{\xi,p}(\cdot,z_0)}(\cdot-z_0)^{\alpha_0}. \end{align}\tag{14}\] Combining 13 and 14 , we get Theorem 2. ◻

Proof. For any \(z\neq w\in K\), \[\begin{align} &\left|\frac{H_{\xi,p}(z,w)}{z-w}\right|\notag\\ \le&\;\left|\frac{K_{\xi,p}(z)(\xi\cdot m_{\xi,p}(w,w)-\xi\cdot m_{\xi,p}(w,z))-K_{\xi,p}(w)(\xi\cdot m_{\xi,p}(z,w)-\xi\cdot m_{\xi,p}(z,z))}{z-w}\right| \notag\\ \le&\; K_{\xi,p}(z)\frac{|(\xi\cdot m_{\xi,p}(w,w)-\xi\cdot m_{\xi,p}(w,z))-(\xi\cdot m_{\xi,p}(z,w)-\xi\cdot m_{\xi,p}(z,z))|}{|z-w|}\notag\\ &+\frac{ |K_{\xi,p}(z)- K_{\xi,p}(w)|}{|z-w|}|\xi\cdot m_{\xi,p}(z,w)-\xi\cdot m_{\xi,p}(z,z)|. \notag \end{align}\]

For any \(\gamma>0\), inequality 6 implies that \[\begin{align} &\frac{|(\xi\cdot m_{\xi,p}(w,w)-\xi\cdot m_{\xi,p}(w,z))-(\xi\cdot m_{\xi,p}(z,w)-\xi\cdot m_{\xi,p}(z,z))|}{|z-w|}\notag\\ \le&\; C_{\gamma}'\|\xi\cdot m_{\xi,p}(\cdot,z)-\xi\cdot m_{\xi,p}(\cdot,w)\|_{L^{\gamma}(U)}\notag \end{align}\] for a constant \(C_{\gamma}'\) and the sub mean-value formula shows that \[|\xi\cdot m_{\xi,p}(z,z)-\xi\cdot m_{\xi,p}(z,w)|\le C_{\gamma}''\|\xi\cdot m_{\xi,p}(\cdot,z)-\xi\cdot m_{\xi,p}(\cdot,w)\|_{L^{\gamma}(U)}\] for a constant \(C_{\gamma}''\).

The inequality 8 and Proposition 20 imply that \(K_{\xi,p}(z)\) and \[\frac{ |K_{\xi,p}(z)- K_{\xi,p}(w)|}{|z-w|}\] are bounded by a constant \(C'\). Then ?? can be proved by combining the above results.

Note that \[\|\xi\cdot m_{\xi,p}(\cdot,z)-\xi\cdot m_{\xi,p}(\cdot,w)\|_{L^{\gamma}(U)}\le \mathrm{vol}(U)^{\frac{1}{\gamma}}\sup_{U} |\xi\cdot m_{\xi,p}(\cdot,z)-\xi\cdot m_{\xi,p}(\cdot,w)|.\] We get ?? . ◻

Proof. Since \[\begin{align} \label{al9} \int_{\Omega}|m_{\xi,p}(\cdot,w_{j})|^p=\frac{1}{K_{\xi,p}(w_{j})}\le \frac{\int_{B(o,R)}|z^{\alpha_0}|^p}{|\xi_{\alpha_0}|^p} \end{align}\tag{15}\] by 7 , we know from the sub mean-value formula that \(\big(m_{\xi,p}(\cdot,w_{j})\big)_{j\ge 1}\) forms a normal family. There exists a subsequence \(j_k\) such that \(m_{\xi,p}(\cdot,w_{j_k})\) converge locally uniformly to a holomorphic function \(m_0(\,\cdot\,)\) on \(\Omega\). Fatou’s lemma shows that \[\int_{\Omega}|m_0|^p\le \liminf_{k\rightarrow +\infty}\int_{\Omega}|m_{\xi,p}(\cdot,w_{j_k})|^p=\liminf_{k\rightarrow +\infty} m_{\xi,p}(w_{j_k})^p=m_{\xi,p}(w_0)^p.\]

We can assume that \(w_{j_k}\in K\) for a compact subset \(K\) of \(\Omega\). Inequalities 6 and 15 show \[\frac{|m_{\xi,p}(w_{j_k},w_{j_k})-m_{\xi,p}(w_0,w_{j_k})|}{|w_{j_k}-w_0|}\le C\] for a constant \(C\), which implies that \(m_{\xi,p}(w_0,w_{j_k})\rightarrow 1\) and hence \(m_0(w_0)=1\). We deduce from Proposition 16 that \(m_0(\cdot)=m_{\xi,p}(\cdot,w_0)\), Lemma 27 has been proved. ◻

Proof of Theorem 3. Note that 7 and 8 imply that \[\frac{1}{C_1}\le K_{\xi,p}(w)\le C_1\] for any \(w\in K\), where \(C_1\) is a constant.

First, we assume \(1<p\le 2\). Hölder’s inequality and ?? show that \[\begin{align} &\int_{\Omega}|m_{\xi,p}(\cdot,w_1)-m_{\xi,p}(\cdot,w_2)|^p\notag\\ =&\;\int_{\Omega}|m_{\xi,p}(\cdot,w_1)-m_{\xi,p}(\cdot,w_2)|^p|m(\cdot,w_1)|^{\frac{(p-2)p}{2}}|m(\cdot,w_1)|^{\frac{(2-p)p}{2}}\notag\\ \le&\;\left(\int_{\Omega}|m_{\xi,p}(\cdot,w_1)-m_{\xi,p}(\cdot,w_2)|^2|m(\cdot,w_1)|^{p-2} \right)^{\frac{p}{2}} \left(\int_{\Omega}|m(\cdot,w_1)|^p\right)^{1-\frac{p}{2}} \notag\\ \le&\; C_2 H_{\xi,p}(w_1,w_2)^{\frac{p}{2}},\notag \end{align}\] where \(C_2\) is a constant. Let \(U\) be an open set such that \(K\subset U\Subset \Omega\). Combining the above results with ?? , we conclude that \[\label{eq4} \begin{align} \|m_{\xi,p}(\cdot,w_1)&-m_{\xi,p}(\cdot,w_2)\|_{L^p(\Omega)}\le C_2^{\frac{1}{p}} H_{\xi,p}(w_1,w_2)^{\frac{1}{2}}\\ &\le C_2^{\frac{1}{p}}C_{\gamma}^{\frac{1}{2}}|w_1-w_2|^{\frac{1}{2}}\cdot\|\xi\cdot m_{\xi,p}(\cdot,w_1)-\xi\cdot m_{\xi,p}(\cdot,w_2)\|_{L^p(U)}^{\frac{1}{2}}. \end{align}\tag{16}\]

It follows from Proposition 1 and inequality 16 that \[\begin{align} &\|m_{\xi,p}(\cdot,w_1)-m_{\xi,p}(\cdot,w_2)\|_{L^p(\Omega)}\notag\\ \le& \;C_2^{\frac{1}{p}}C_{\gamma}^{\frac{1}{2}}|w_1-w_2|^{\frac{1}{2}}\|\xi\cdot m_{\xi,p}(\cdot,w_1)-\xi\cdot m_{\xi,p}(\cdot,w_2)\|_{L^p(U)}^{\frac{1}{2}}\notag\\ \le& \;C_3|w_1-w_2|^{\frac{1}{2}}\|m_{\xi,p}(\cdot,w_1)-m_{\xi,p}(\cdot,w_2)\|_{L^p(\Omega)}^{\frac{1}{2}}\notag \end{align}\] for a constant \(C_3\), which implies that \[\label{ali3} \|m_{\xi,p}(\cdot,w_1)-m_{\xi,p}(\cdot,w_2)\|_{L^p(\Omega)}\le C_3^2|w_1-w_2|.\tag{17}\] Then ?? follows from the sub mean-value formula.

Proposition 1 and inequality 17 imply that \[\sup_{K}|\xi\cdot m_{\xi,p}(\cdot,w_1)-\xi\cdot m_{\xi,p}(\cdot,w_2)|\le C_3'|w_1-w_2|\] for a constant \(C_3'\). Thus, ?? holds.

Now we assume \(p>2\). By Lemma 27, \(\big(m_{\xi,p}(\cdot,w)\big)_{w\in K}\) forms a compact family of holomorphic functions in the topology of locally uniform convergence. Demailly–Kollár’s theorem on the semi-continuity property of the complex singularity exponent ([23]) implies that \[\label{ali2} \int_{U_1}|m_{\xi,p}(\cdot,w)|^{-c}\le M\tag{18}\] for all \(w\in K\), where \(U_1\) is an open subset of \(\Omega\) such that \(K\subset U_1\), and \(c>0\), \(M>0\) are constants.

Denote \(\beta=\frac{2c}{p-2+c}<2\). Then Hölder inequality, inequalities 18 and ?? indicate that \[\begin{align} &\int_{U_1}|m_{\xi,p}(\cdot,w_1)-m_{\xi,p}(\cdot,w_2)|^\beta\\ =&\;\int_{U_1}|m_{\xi,p}(\cdot,w_1)-m_{\xi,p}(\cdot,w_2)|^\beta|m_{\xi,p}(\cdot,w_1)|^{\frac{(p-2)\beta}{2}}|m_{\xi,p}(\cdot,w_1)|^{\frac{(2-p)\beta}{2}}\\ \le&\; \left\{\int_{U_1}|m_{\xi,p}(\cdot,w_1)|^{p-2}|m_{\xi,p}(\cdot,w_1)-m_{\xi,p}(\cdot,w_2)|^2\right\}^{\frac{\beta}{2}}\cdot\left\{\int_{U_1}|m_{\xi,p}(\cdot,w_1)|^{-c}\right\}^{1-\frac{\beta}{2}}\\ \le&\; C_4 H_{\xi,p}(w_1,w_2)^{\frac{\beta}{2}} \end{align}\] for a constant \(C_4\). Take an open set \(U_2\) such that \[K\subset U_2\Subset U_1.\] Combining the above results with ?? and Proposition 1, we conclude that \[\begin{align} \|m_{\xi,p}(\cdot,w_1)&-m_{\xi,p}(\cdot,w_2)\|_{L^\beta(U_1)}\le C_4^{\frac{1}{\beta}}H_{\xi,p}(w_1,w_2)^{\frac{1}{2}}\\ &\le C_4^{\frac{1}{\beta}}C_{\beta}^{\frac{1}{2}}|w_1-w_2|^{\frac{1}{2}}\|\xi\cdot m_{\xi,p}(\cdot,w_1)-\xi\cdot m_{\xi,p}(\cdot,w_2)\|_{L^\beta(U_2)}^{\frac{1}{2}}\\ &\le C_5|w_1-w_2|^{\frac{1}{2}}\|m_{\xi,p}(\cdot,w_1)-m_{\xi,p}(\cdot,w_2)\|_{L^\beta(U_1)}^{\frac{1}{2}} \end{align}\] for a constant \(C_5\), which implies that \[\begin{align} \label{ali6} \|m_{\xi,p}(\cdot,w_1)-m_{\xi,p}(\cdot,w_2)\|_{L^{\beta}(U_1)}\le C_5^2|w_1-w_2|. \end{align}\tag{19}\] Thus, we get ?? by the sub-mean value formula.

Proposition 1 and 19 imply that \[\sup_{K}|\xi\cdot m_{\xi,p}(\cdot,w_1)-\xi\cdot m_{\xi,p}(\cdot,w_2)|\le C_5'|w_1-w_2|\] for a constant \(C_5'\), which gives ?? . ◻

Proof of Theorem 4. Let \(U_1\) be an open set such that \(K\subset U_1\Subset \Omega\backslash\{m_{\xi,1}(\cdot,w_1)=0\}\). It follows from 7 that \(m_{\xi,1}(w)^p=\int_{\Omega}|m_{\xi,1}(\cdot,w)|^p\) is uniformly bounded. Note that \(m_{\xi,1}(\cdot,w_1)\) is a non-vanishing continuous function on \(\overline{U_1}\). Hence, there exists a constant \(C_1\) such that \[\label{al10} \frac{1}{C_1}\le |m_{\xi,1}(z,w_1)|\le |m_{\xi,1}(z,w_1)|+|m_{\xi,1}(z,w_2)|\le C_1\tag{20}\] for any \(z,w_2\in U_1\).

According to ?? and 20 , we have \[\begin{align} \int_{U_1}\left|\mathrm{Im}\,\left\{\frac{m_{\xi,1}(\cdot,w_2)}{m_{\xi,1}(\cdot,w_1)}\right\}\right|^2&=\int_{U_1}\left|\mathrm{Im}\,\left\{\frac{m_{\xi,1}(\cdot,w_2)\overline{m_{\xi,1}(\cdot,w_1)}}{|m_{\xi,1}(\cdot,w_1)|^2}\right\}\right|^2 \notag\\ &\le C_1^5\int_{U_1}\frac{|\mathrm{Im}\,(\overline{m_{\xi,1}(\cdot,w_1)}m_{\xi,1}(\cdot,w_2))|^2}{(|m_{\xi,1}(\cdot,w_1)|+|m_{\xi,1}(\cdot,w_2)|)^3}\\ &\le C_2H_{\xi,1}(w_1,w_2)\notag \end{align}\] for a constant \(C_2\). Let \(U_2, U_3, U_4\) be domains such that \(K\Subset U_4 \Subset U_3\Subset U_2\Subset U_1\). It follows from Proposition 1, ?? and the sub mean-value formula that \[\begin{align} \sup_{U_2}\left|\mathrm{Im}\,\left\{\frac{m_{\xi,1}(\cdot,w_2)}{m_{\xi,1}(\cdot,w_1)}\right\}\right|&\le C_3\left\{\int_{U_1}\left|\mathrm{Im}\,\left\{\frac{m_{\xi,1}(\cdot,w_2)}{m_{\xi,1}(\cdot,w_1)}\right\}\right|^2\right\}^{\frac{1}{2}}\\ &\le C_3C_2^{\frac{1}{2}}H_{\xi,1}(w_1,w_2)^{\frac{1}{2}}\notag\\ &\le C_3C_2^{\frac{1}{2}}C'^{\frac{1}{2}}|z-w|^{\frac{1}{2}}\cdot\sup_{U_4} |\xi\cdot m_{\xi,p}(\cdot,w_1)-\xi\cdot m_{\xi,p}(\cdot,w_2)|^{\frac{1}{2}}\\ &\le C_4|z-w|^{\frac{1}{2}}\cdot\sup_{U_3} |m_{\xi,p}(\cdot,w_1)-m_{\xi,p}(\cdot,w_2)|^{\frac{1}{2}} \end{align}\] for a constant \(C_4\).

Recall that the Borel–Carathéodory inequality (see e.g., [9]) shows \[\sup_{\Delta_r}|f|\le \frac{2r}{R-r}\sup_{\Delta_R}\mathrm{Im}\,f+\frac{R+r}{R-r}|f(0)|\] for any holomorophic function \(f\) on \(\Delta_R\mathrel{\vcenter{:}}= \{z\in\mathbb{C}\colon|z|<R\}\) and \(0<r<R\). Let \(r=\frac{1}{2}R\), and it follows that \[\label{ali4} \sup_{\Delta_{\frac{R}{2}}}|f|\le 2\sup_{\Delta_R}\mathrm{Im}\,f+3|f(0)|.\tag{21}\]

Let \(h\mathrel{\vcenter{:}}= \frac{m_{\xi,1}(\cdot,w_2)}{m_{\xi,1}(\cdot,w_1)}-1\). Since \(m_{\xi,1}(w)=\int_{\Omega}|m_{\xi,1}(\cdot,w)|\) is locally Lipschitz continuous by Proposition 20, it follows from the sub mean-value formula that \[|h(w_1)|\le C_5|w_1-w_2|\] for a constant \(C_5\). Since \(\{B_{\frac{1}{2}\mathrm{dist}(w,\partial U_2 )}(w):w\in \overline{U_3}\}\) form an open covering of \(U_3\), a finite chain of balls lying in \(B_{\frac{1}{2}\mathrm{dist}(w,\partial U_2 )}(w)\) for some \(w\in \overline{U_3}\) connect \(w_1\) to any point in \(U_3\). Applying (21 ) successively on each ball, we conclude that \[\sup_{U_3} |h|\le C_6|w_1-w_2|+C_7|w_1-w_2|^{\frac{1}{2}}\cdot\sup_{U_3} |m_{\xi,p}(\cdot,w_1)-m_{\xi,p}(\cdot,w_2)|^{\frac{1}{2}}\] for two constants \(C_6\) and \(C_7\), which implies that \[\begin{align} &\sup_{U_3} |m_{\xi,p}(\cdot,w_1)-m_{\xi,p}(\cdot,w_2)|\notag\\ &\le C_1C_6|w_1-w_2|+C_1C_7|w_1-w_2|^{\frac{1}{2}}\cdot\sup_{U_3} |m_{\xi,p}(\cdot,w_1)-m_{\xi,p}(\cdot,w_2)|^{\frac{1}{2}}\notag \end{align}\] by 20 . As a result, \[\begin{align} \label{ali7} \sup_{U_3} |m_{\xi,p}(\cdot,w_1)-m_{\xi,p}(\cdot,w_2)|\le C_8|w_1-w_2| \end{align}\tag{22}\] for a constant \(C_8\).

Proposition 1 and 22 imply that \[\sup_{K}|\xi\cdot m_{\xi,p}(\cdot,w_1)-\xi\cdot m_{\xi,p}(\cdot,w_2)|\le C_8'|w_1-w_2|\] for a constant \(C_8'\). Theorem 4 is proved. ◻

Proof of Theorem 5.. Let \(U_2\Subset U_1\Subset\Omega\). Then \[h\mathrel{\vcenter{:}}= \frac{\xi\cdot K_{\xi,p}(\cdot,z)-\xi\cdot K_{\xi,p}(\cdot,w)}{z-w}\] is a uniformly bounded holomorphic function on \(U_1\) for every \(z\neq w\in U_2\) by Theorem 3. Cauchy’s estimate implies that \[\sup_{U_2}\left|\frac{\partial h}{\partial x_j}\right|\le C_1\] for a constant \(C_1\). Note that \[\int_{\Omega}|K_{\xi,p}(\cdot,z)|^p=|K_{\xi,p}(z)|^{p-1}\] is uniformly bounded for any \(z\in U_1\) by 8 . It follows from Proposition 1 and Cauchy’s estimate that \[\Bigg|\frac{\partial (\xi\cdot K_{\xi,p}(\cdot,z))}{\partial x_j}\bigg|_z-\frac{\partial (\xi\cdot K_{\xi,p}(\cdot,z))}{\partial x_j}\bigg|_w \Bigg| \le C_2|z-w|\] for a constant \(C_2\) and any \(z,w\in U_2\). As a result, we get \[\label{ali8} \begin{align} & \Bigg|\frac{\partial (\xi\cdot K_{\xi,p}(\cdot,z))}{\partial x_j}\Big|_z-\frac{\partial (\xi\cdot K_{\xi,p}(\cdot,w))}{\partial x_j}\Big|_w\Bigg|\\ \le&\; \Bigg|\frac{\partial (\xi\cdot K_{\xi,p}(\cdot,z))}{\partial x_j}\Big|_z-\frac{\partial (\xi\cdot K_{\xi,p}(\cdot,z))}{\partial x_j}\Big|_w\Bigg|+\left|\frac{\partial h}{\partial x_j}(w)\right||z-w|\\ \le&\; (C_1+C_2)|z-w| \end{align}\tag{23}\] for any \(z,w\in U_2\).

Let \(e_1,\ldots,e_{2n}\) be the standard basis in \(\mathbb{R}^{2n}=\mathbb{C}^n\). Then \[\begin{align} &K_{\xi,p}(z+te_j)^{-1}-K_{\xi,p}(z)^{-1}\notag\\ =&\;m_{\xi,p}(z+te_j)^p-m_{\xi,p}(z)^p\notag\\ \le&\; p\mathop{\mathrm{Re}}\int_{\Omega}|m_{\xi,p}(\cdot,z+te_j)|^{p-2}\overline{m_{\xi,p}(\cdot,z+te_j)}(m_{\xi,p}(\cdot,z+te_j)-m_{\xi,p}(\cdot,z))\notag \end{align}\] by ?? . Note that \(\xi\cdot m_{\xi,p}(z,z)=\xi\cdot m_{\xi,p}(z+te_j,z+te_j)=1\). We get from Theorem 2, inequality 7 and Proposition 20 that \[\begin{align} &K_{\xi,p}(z+te_j)^{-1}-K_{\xi,p}(z)^{-1}\notag\\ \le&\;\frac{p}{K_{\xi,p}(z+te_j)}\mathop{\mathrm{Re}}\{\xi\cdot m_{\xi,p}(z+te_j,z+te_j)-\xi\cdot m_{\xi,p}(z+te_j,z)\}\notag\\ =&\;-\frac{p}{K_{\xi,p}(z+te_j)}\mathop{\mathrm{Re}}\{\xi\cdot m_{\xi,p}(z+te_j,z)-\xi\cdot m_{\xi,p}(z,z)\}\notag\\ \le&\;-\frac{pt}{K_{\xi,p}(z)}\mathop{\mathrm{Re}}\frac{\partial (\xi\cdot m_{\xi,p}(\cdot,z))}{\partial x_j}\bigg|_z+C_3|t|^2\notag \end{align}\] for a constant \(C_3\) as \(t\) tends to \(0\).

Similarly, we have \[\begin{align} &K_{\xi,p}(z)^{-1}-K_{\xi,p}(z+te_j)^{-1}\notag\\ \le&\; \frac{p}{K_{\xi,p}(z)}\cdot\mathop{\mathrm{Re}}\{\xi\cdot m_{\xi,p}(z,z)-\xi\cdot m_{\xi,p}(z,z+te_j)\}\notag\\ =&\;-\frac{p}{K_{\xi,p}(z+te_j)}\cdot\mathop{\mathrm{Re}}\{\xi\cdot m_{\xi,p}(z,z+te_j)-\xi\cdot m_{\xi,p}(z+te_j,z+te_j)\}\notag\\ \le&\; \frac{pt}{K_{\xi,p}(z)}\cdot\mathop{\mathrm{Re}}\frac{\partial (\xi\cdot m_{\xi,p}(\cdot,z+te_j))}{\partial x_j}\bigg|_z+C_4|t|^2\notag \end{align}\] for a constant \(C_4\) as \(t\) tends to \(0\). Combining the above results with 23 , we conclude that \[\frac{\partial K_{\xi,p}^{-1}}{\partial x_j}(z)=-\frac{p}{K_{\xi,p}(z)}\cdot\mathop{\mathrm{Re}}\frac{\partial (\xi\cdot m_{\xi,p}(\cdot,z))}{\partial x_j}\bigg|_z,\] which implies that \[\frac{\partial K_{\xi,p}}{\partial x_j}(z)=-K_{\xi,p}(z)^2\frac{\partial K_{\xi,p}^{-1}}{\partial x_j}(z)=p\mathop{\mathrm{Re}}\frac{\partial (\xi\cdot K_{\xi,p}(\cdot,z))}{\partial x_j}\bigg|_z.\] Combining with 23 , we get \(K_{\xi,p}\in C_{\mathop{\mathrm{loc}}}^{1,1}(\Omega)\). ◻

Proof. It suffices to prove that for any \(z\in\Omega\) , there exists a positive constant \(c\) such that \[\liminf_{r\rightarrow 0+}\frac{1}{r^2} \left(\frac{1}{2\pi}\int_0^{2\pi}\log K_{\xi,p}(z+re^{i\theta }X)d\theta-\log K_{\xi,p}(z)\right)>c|X|^2\] for any \(X\in \mathbb{R}^n\). By a translation, we can assume that \(z=o\).

Arbitrarily choose \(h\in A^p(\Omega)\) satisfying \((\xi\cdot h)(o)=0\). Set \(h_t\mathrel{\vcenter{:}}= m_{\xi,p}(\cdot,o)+th\), where \(t\in\mathbb{C}\). Proof of Lemma 23 implies that \[\frac{\partial \|h_t\|_p^p}{\partial t}(0)=\frac{\partial \|h_t\|_p^p}{\partial \bar{t}}(0)=0.\] Since \[\begin{align} &\frac{\partial^2 |h_t|^p}{\partial t^2}=\frac{p(p-2)}{4}|h_t|^{p-4}(\bar{h}_th)^2,\notag\\ &\frac{\partial^2 |h_t|^p}{\partial t\partial \bar{t}}=\frac{p^2}{4}|h_t|^{p-2}|h|^2\notag \end{align}\] by direct calculations, \[\begin{align} &\left|\frac{\partial^2 |h_t|^p}{\partial t^2}\right|\le\frac{p(p-2)}{4}(|m_{\xi,p}(\cdot,o)|+|h|)^{p-2}|h|^2,\notag\\ &\left|\frac{\partial^2 |h_t|^p}{\partial t\partial \bar{t}}\right|=\frac{p^2}{4}(|m_{\xi,p}(\cdot,o)|+|h|)^{p-2}|h|^2\notag \end{align}\] for \(|t|\le 1\). Hölder’s inequality implies that \[\int_{\Omega}(|m_{\xi,p}(\cdot,o)|+|h|)^{p-2}|h|^2\le\|h\|_p^2\left\{\int_{\Omega}(|m_{\xi,p}(\cdot,o)|+|h|)^p\right\}^{1-\frac{2}{p}}\eqqcolon c<\infty.\] Therefore the dominated convergence theorem shows that \[\begin{align} \label{alig2} \left|\frac{\partial^2 \|h_t\|_p^p}{\partial t^2}(0)\right|\le \frac{p(p-2)}{4}c, \quad \left|\frac{\partial^2 \|h_t\|_p^p}{\partial t\partial \bar{t}}(0)\right|\le \frac{p^2}{4}c. \end{align}\tag{24}\]

Denote the coordinate by \(z\mathrel{\vcenter{:}}= (z_1,\ldots,z_n)\). For any \(0\neq X\in\mathbb{R}^n\), set \[h^X\mathrel{\vcenter{:}}= \left(\sum_{j=1}^n\frac{X_jz_j}{((\alpha_0)_j+1)\|X\|}\right)z^{\alpha_0}.\] By the assumption that \(\xi_{\alpha}=0\) for all \(\alpha>\alpha_0\) with \(|\alpha|=|\alpha_0|+1\), we see \((\xi\cdot h^X)(o)=0\). Since the \(L^p\) norm of \(h^X\) is uniformly bounded by a constant independent of \(X\), it follows from 24 that \[\begin{align} \label{alig3} \|h^X_t\|_p^p\le \|m_{\xi,p}(\cdot,o)\|_p^p\big(1+C_1|t|^2+o(|t|^2)\big) \end{align}\tag{25}\] for a constant \(C_1>0\).

At \(re^{i\theta}X=(re^{i\theta}X_1,\ldots,re^{i\theta}X_n)\), \[\begin{align} h^X=re^{i\theta}\|X\|(z-re^{i\theta}X)^{\alpha_0}&+\left(\sum_{j=1}^n\frac{X_j(z_j-re^{i\theta}X_j)}{((\alpha_0)_j+1)\|X\|}\right)(z-re^{i\theta}X)^{\alpha_0}\\ &+\sum_{\alpha<\alpha_0}a_{\alpha}(z-re^{i\theta}X)^{\alpha}, \end{align}\] where \(a_{\alpha}=O(r^2)\) for any \(\alpha<\alpha_0\). Therefore \[(\xi\cdot h^X)(re^{i\theta}X)=re^{i\theta}\|X\|\xi_{\alpha_0}+O(r^2).\]

Since \(\xi\cdot m_{\xi,p}(\cdot,o)\) is a holomorphic function satisfying that \((\xi\cdot m_{\xi,p}(\cdot,o))(o)=1\), \((\xi\cdot m_{\xi,p}(\cdot,o))(re^{i\theta}X)=1+O(r)\).

It then follows that \[(\xi\cdot h^X_t)(re^{i\theta}X)=\left\{(\xi\cdot m_{\xi,p}(\cdot,o))(re^{i\theta}X)\right\}(1+tre^{i\theta}\|X\|\xi_{\alpha_0}+O(|t|r^2)),\] which implies that \[\begin{align} \label{alig4} |(\xi\cdot h^X_t)(re^{i\theta}X)|^p=\left|(\xi\cdot m_{\xi,p}(\cdot,o))(re^{i\theta}X)\right|^p|1+ptre^{i\theta}\|X\|\xi_{\alpha_0}+O(|t|r^2)|. \end{align}\tag{26}\]

It follows from Proposition 18, inequality 25 and equality 26 that \[\begin{align} \label{alig5} K_{\xi,p}(re^{i\theta}X)\ge\frac{\left|(\xi\cdot m_{\xi,p}(\cdot,o))(re^{i\theta}X)\right|^p}{\|m_{\xi,p}(\cdot,o)\|_p^p}\cdot\frac{|1+ptre^{i\theta}\|X\|\xi_{\alpha_0}+O(|t|r^2)|}{1+C_1|t|^2+o(|t|^2)}. \end{align}\tag{27}\]

Let \(t=\frac{pr\|X\|\bar{\xi}_{\alpha_0}}{2C_1}e^{-i\theta}\). Then inequality 27 reduces to

\[K_{\xi,p}(re^{i\theta }X)\ge \frac{\left|(\xi\cdot m_{\xi,p}(\cdot,o))(re^{i\theta}X)\right|^p}{\|m_{\xi,p}(\cdot,o)\|_p^p}\cdot\frac{1+\frac{p^2r^2\|X\|^2|\xi_{\alpha_0}|^2}{2C_1}+O(|r|^3)}{1+\frac{p^2r^2\|X\|^2|\xi_{\alpha_0}|^2}{4C_1}+o(|r|^2)}.\] Consequently, \[\begin{align} &\liminf_{r\rightarrow 0+}\frac{1}{r^2} \left(\frac{1}{2\pi}\int_0^{2\pi}\log K_{\xi,p}(re^{i\theta }X)d\theta-\log K_{\xi,p}(o)\right)\\ \ge&\; \liminf_{r\rightarrow 0+}\frac{1}{r^2} \left(\frac{1}{2\pi}\int_0^{2\pi}\log \frac{\left|(\xi\cdot m_{\xi,p}(\cdot,o))(re^{i\theta}X)\right|^p}{\|m_{\xi,p}(\cdot,o)\|_p^p}d\theta-\log K_{\xi,p}(o)\right)+\frac{p^2\|X\|^2|\xi_{\alpha_0}|^2}{4C_1}\\ \ge&\; \frac{p^2\|X\|^2|\xi_{\alpha_0}|^2}{4C_1}, \end{align}\] where the second inequality follows from the plurisubharmonic property of \[\log \frac{\left|(\xi\cdot m_{\xi,p}(\cdot,o))\right|^p}{\|m_{\xi,p}(\cdot,o)\|_p^p}.\]

The proof of Proposition 28 is done. ◻

Proof of Theorem 6. Let \(L\subset\mathbb{C}^n\) be a complex line such that \(\Omega\cap L\neq\emptyset\). We can assume that \(L=\{z_2=\cdots=z_n=0\}\) by a unitary transformation.

For \(z^0=(z_1^0,0,\ldots,0)\in\partial\Omega\cap L\), since the \(L^p(\Omega\cap L)\) norm of the function \(\frac{1}{z_1-z^0_1}\) is bounded by a constant \(c_1\), it follows from Lemma 29 that there exists a holomorphic function \(F\) on \(\Omega\) whose \(L^p(\Omega)\) norm is bounded by a constant \(c_2\) such that \(F|_{\Omega\cap L}=\frac{1}{z_1-z^0_1}\).

Choose any \(z'=(z'_1,0,\ldots,0)\in \Omega\cap L\). Since \((z-z')^{\alpha_0}\) is uniformly bounded on \(\Omega\), it can be obtained that \(G\mathrel{\vcenter{:}}= F\cdot(z-z')^{\alpha_0}\) is a holomorphic function on \(\Omega\) whose \(L^p(\Omega)\) norm is bounded by a constant \(c_3\).

Note that \((\partial^{\alpha_0}G)|_{z'}=\alpha_0 !\cdot F(z')\) and \((\partial^{\alpha}G)|_{z'}=0\) for any \(\alpha\not \ge\alpha_0\), which implies that \[(\xi\cdot G)(z')=F(z')=\frac{1}{z'_1-z_1^0}.\] It follows from Proposition 18 that \[K_{\xi,p}(z')\ge \frac{1}{c_3^p \cdot |z'-z_0|^p}.\] Since \(L\) and \(z'\in L\) are arbitrarily chosen, the proof of Theorem 6 is done. ◻

Proof. By 28 , there exists a constant \(C_1\) such that for any \(z_0\in\Omega\), there exists \(F\in A^2(\Omega)\) satisfying that \[\int_{\Omega}|F|^2\le C_1\delta(z_0)^2|F(z_0)|^2 .\] Since \((z-z_0)^{\alpha_0}\) is uniformly bounded in \(\Omega\), if we let \(G=F\cdot (z-z_0)^{\alpha_0}\), then \[\label{al12} \int_{\Omega}|G|^2\le C_2\delta(z_0)^2|F(z_0)|^2\tag{29}\] for a constant \(C_2\). As \((\partial^{\alpha_0}G)|_{z_0}=\alpha_0 !\cdot F(z_0)\) and \((\partial^{\alpha}G)|_{z_0}=0\) for any \(\alpha\not \ge\alpha_0\), we have \((\xi\cdot G)(z_0)=F(z_0)\). Then it follows from Proposition 18 and 29 that \[K_{\xi,2}(z_0)\ge\frac{|(\xi\cdot(F\cdot(z-z_0)^{\alpha_0}))(z_0)|^2}{\int_{\Omega}|F\cdot (z-z_0)^{\alpha_0}|^2}\ge \frac{1}{C_2\delta(z_0)^2}.\] Proposition 30 is proved. ◻

Proof. It is clear that \(\kappa_{\eta,D}^{\mathbf{E}_0,p}(z)\le \kappa_{\xi,D}^p(z)\) for any \(\xi\in\mathbf{S}_{\eta}\). Then we may assume \(\kappa_{\eta,D}^{\mathbf{E}_0,p}(z)<+\infty\).

We prove the theorem in three steps.

Step 1: Suppose \(D\) is bounded, and we prove \(\kappa_{\eta,D}^{\mathbf{E}_0,p}(z)=\min_{\xi\in\mathbf{S}_{\eta}}\kappa_{\xi,D}^p(z)\).

In case \(D\) is bounded, any polynomial on \(\mathbb{C}^n\) belongs to \(A^p(D)\). Then one can see \(\mathscr{A}^p_{\mathbf{E}_0}(D)\) is a non-empty closed subspace of the Banach space \(A^p(D)\), and for any \(f\in\mathscr{A}^p_{\mathbf{E}_0}(D)\), \[|(\eta\cdot f)(z)|\le \kappa_{\eta,D}^{\mathbf{E}_0,p}(z)\cdot \|f\|_{p,D}.\] By the Hahn–Banach Theorem, the continuous linear functional \(\Lambda_{\eta}\colon f\mapsto (\eta\cdot f)(z)\) on \(\mathscr{A}^p_{\mathbf{E}_0}(D)\) can be extended to a continuous linear functional \(\Lambda\) on \(A^p(D)\), with \(\Lambda|_{\mathscr{A}^p_{\mathbf{E}_0}(D)}=\Lambda_{\eta}\), and \[|\Lambda(f)|\le \kappa_{\eta,D}^{\mathbf{E}_0,p}(z)\cdot \|f\|_{p,D}, \quad \forall\, f\in A^p(D).\]

Next, we prove that there exists some \(\xi\in\mathbf{S}_{\eta}\) such that \(\Lambda_{\xi}=\Lambda\), where \(\Lambda_{\xi}\colon f\mapsto (\xi\cdot f)(z)\). This will imply \(\xi\in\mathbf{S}_{\eta}\) and \(\|\Lambda_{\xi}\|\le \kappa_{\eta,D}^{\mathbf{E}_0,p}(z)\), which yields \(\kappa^p_{\xi,D}(z)=\kappa_{\eta,D}^{\mathbf{E}_0,p}(z)\). Let \(f\in A^p(D)\). By Taylor’s expansion, one can uniquely decompose \(f\) into \(f=f_0+f_1\), where \[\begin{align} \begin{aligned} \left\{\alpha \colon D^{\alpha}f_0(z)\neq 0\right\}&\subset \mathbf{E}_0, \\ \left\{\alpha \colon D^{\alpha}f_1(z)\neq 0\right\}&\subset\mathbf{E}_1. \end{aligned} \end{align}\] Since \(\mathbf{E}_0\) is a finite set, \(f_0\) is a polynomial, which belongs to \(A^p(D)\). Then \(f_1\in \mathscr{A}^p_{\mathbf{E}_0}(D)\), and \(\Lambda(f_1)=\Lambda_{\eta}(f_1)\). We get \[\Lambda(f)=\Lambda(f_0)+\Lambda(f_1)=\Lambda(f_0)+\Lambda_{\eta}(f_1),\] where \[\Lambda(f_0)=\sum_{\alpha\in\mathbf{E}_0}\Lambda(z^{\alpha})\cdot\frac{D^{\alpha}f(z)}{\alpha!}, \quad \forall\, f\in A^p(D).\] Let \(\xi=(\xi_{\alpha})\in\ell_1^{(n)}\), where \[\xi_{\alpha}= \begin{cases} \Lambda(z^{\alpha}), \;& \alpha\in\mathbf{E}_0,\\ \eta_{\alpha}, \;& \alpha\in\mathbf{E}_1.\\ \end{cases}\] Then \(\xi\in\mathbf{S}_{\eta}\) satisfying \(\Lambda_{\xi}=\Lambda\), which implies that \(\kappa_{\eta,D}^{\mathbf{E}_0,p}(z)=\min_{\xi\in\mathbf{S}_{\eta}}\kappa_{\xi,D}^p(z)\).

Step 2: For a general domain \(D\) in \(\mathbb{C}^n\), we prove \(\kappa_{\eta,D}^{\mathbf{E}_0,p}(z)=\inf_{\xi\in\mathbf{S}_{\eta}}\kappa_{\xi,D}^p(z)\).

Let \(D_1\subset D_2\subset\cdots\subset D_j\subset\cdots\) be an increasing sequence of bounded domains in \(\mathbb{C}^n\) such that \(D=\bigcup_{j=1}^{\infty}D_j\) and \(z\in D_1\). By Step 1, \[\kappa_{\eta,D_j}^{\mathbf{E}_0,p}(z)=\min_{\xi\in\mathbf{S}_{\eta}}\kappa_{\xi,D_j}^p(z), \quad \forall\, j\ge 1.\] By Montel’s Theorem, we also have (e.g. see the proof of Proposition 21) \[\lim_{j\to \infty}\kappa_{\eta,D_j}^{\mathbf{E}_0,p}(z)=\kappa_{\eta,D}^{\mathbf{E}_0,p}(z),\] and \[\lim_{j\to \infty}\kappa_{\xi,D_j}^p(z)=\kappa_{\xi,D_j}^p(z), \quad \forall\, \xi\in\mathbf{S}_{\eta}.\] Consequently, \(\kappa_{\eta,D}^{\mathbf{E}_0,p}(z)=\inf_{\xi\in\mathbf{S}_{\eta}}\kappa_{\xi,D}^p(z)\).

Step 3: We prove \(\inf_{\xi\in\mathbf{S}_{\eta}}\kappa_{\xi,D}^p(z)=\min_{\xi\in\mathbf{S}_{\eta}}\kappa_{\xi,D}^p(z)\).

Note that \(\mathbf{S}_{\eta}\) is a linear space of finite dimension (\(\dim\mathbf{S}_{\eta}=|\mathbf{E}_0|\)). Proposition 20 shows that the function \(\xi\mapsto \kappa_{\xi,D}^p(z)\) is a continuous function on \(\mathbf{S}_{\eta}\). For any \(\alpha\in\mathbf{E}_0\), we may assume \[\big\{f\in A^p(D) \colon D^{\alpha}f(z)\neq 0\big\}\neq\emptyset.\] Otherwise, the value of \(\xi_{\alpha}\) (\(\alpha\)-th position of \(\xi\)) does not affect \(\kappa^p_{\xi,D}(z)\). Then now, one can find that \[\lim_{|\xi_{\alpha}|\to+\infty}\kappa_{\xi,D}^p(z)=+\infty,\] where the limit is taken when the values of the other positions of \(\xi\) are fixed except the \(\alpha\)-th position. It follows that \(\inf_{\xi\in\mathbf{S}_{\eta}}\kappa_{\xi,D}^p(z)=\inf_{\xi\in \mathscr{K}}\kappa_{\xi,D}^p(z)\), where \(\mathscr{K}\) is a compact subset of \(\mathbf{S}_{\eta}\). Thus, the continuity of \(\kappa_{\xi,D}^p(z)\) in \(\xi\) implies that \(\inf_{\xi\in\mathbf{S}_{\eta}}\kappa_{\xi,D}^p(z)=\min_{\xi\in\mathbf{S}_{\eta}}\kappa_{\xi,D}^p(z)\) holds.

The proof is complete. ◻

An alternative proof of Theorem 7 for \(p=2\). We may assume \(\inf_{\xi\in\mathbf{S}_H}K_{\xi,\Omega}(z)>0\), otherwise the result is trivial. According to (2 ), we only need to prove that there exists some \(\xi_H\in\mathbf{S}_H\) such that \(K^H_{\Omega}(z)=K_{\xi_H,\Omega}(z)\).

Using Lemma 32, we can get a complete orthonormal basis \(\{\sigma_{\alpha}\}_{\alpha\in\mathbf{E}}\) of \(A^2(\Omega)\) satisfying the conditions in Lemma 32. For any \(f\in A^2(\Omega)\), if we write \[f=\sum_{\alpha\in\mathbf{E}}a_{\alpha}\sigma_{\alpha},\] where \(a_{\alpha}\in\mathbb{C}\), then \[\label{eq-Dbetafz} (D^{\beta}f)(z)=\sum_{\alpha\in\mathbf{E}}a_{\alpha}(D^{\beta}\sigma_{\alpha})(z)=\sum_{\alpha\preceq\beta,\, \alpha\in\mathbf{E}}a_{\alpha}(D^{\beta}\sigma_{\alpha})(z).\tag{30}\] Now for any \(\xi=(\xi_{\beta})\in\mathbf{S}_H\), we can write \[(\xi\cdot f)(z)=\sum_{|\beta|\le k}\frac{\xi_{\beta}}{\beta!}(D^{\beta}f)(z).\] Combining with (30 ), we have \[\begin{align} \begin{aligned} (\xi\cdot f)(z)&=\sum_{|\beta|\le k}\frac{\xi_{\beta}}{\beta!}\left(\sum_{\alpha\preceq\beta,\, \alpha\in\mathbf{E}}a_{\alpha}(D^{\beta}\sigma_{\alpha})(z)\right)\\ &=\sum_{|\alpha|\le k,\, \alpha\in\mathbf{E}}\left(\sum_{\beta\succeq\alpha, |\beta|\le k}\xi_{\beta}\frac{(D^{\beta}\sigma_{\alpha})(z)}{\beta!}\right)a_{\alpha}. \end{aligned} \end{align}\] Set \[\label{calpha} c_{\alpha}\mathrel{\vcenter{:}}= \sum_{\beta\succeq\alpha,\, |\beta|\le k}\xi_{\beta}\frac{(D^{\beta}\sigma_{\alpha})(z)}{\beta!}\tag{31}\] for each \(\alpha\in\mathbf{E}\) with \(|\alpha|\le k\). Then using the Cauchy–Schwarz inequality, we can compute that \[\begin{align} \begin{aligned} K_{\xi,\Omega}(z)&\mathrel{\vcenter{:}}=\sup_{f\in A^2(\Omega)}\frac{|(\xi\cdot f)(z)|^2}{\int_{\Omega}|f|^2}\\ &=\sup_{\{a_{\alpha}\}_{\alpha\in\mathbf{E}}\in l^2}\frac{\left|\sum_{|\alpha|\le k, \alpha\in\mathbf{E}}c_{\alpha}a_{\alpha}\right|^2}{\sum_{\alpha\in\mathbf{E}}|a_{\alpha}|^2}\\ &=\sum_{|\alpha|\le k,\, \alpha\in\mathbf{E}}|c_{\alpha}|^2, \end{aligned} \end{align}\] where the supremum is achieved if and only if \[f=C\sum_{\alpha\le k,\, \alpha\in\mathbf{E}}\bar{c}_{\alpha}\sigma_{\alpha},\] for some \(C\in\mathbb{C}\setminus\{0\}\). Note that \[(D^{\beta}\sigma_{\alpha})(z)=0, \quad \forall\, \alpha\in\mathbf{E}, \;\beta\in\mathbb{N}^n, \;\beta\prec\alpha.\] If we let \[\label{eq-calpha} c_{\alpha}=0, \quad \forall\, \alpha\in\mathbf{E}, \;|\alpha|<k,\tag{32}\] then \[K_{\xi,\Omega}(z)=\frac{|(\xi\cdot f_0)(z)|^2}{\int_{\Omega}|f_0|^2},\] for some \(f_0\in A^2(\Omega)\) with \(\|f_0\|_{L^2(\Omega)}=1\) and \(f_0^{(j)}(z)=0\), \(j=0,\ldots,k-1\). Since \(\xi\in\mathbf{S}_H\), we also have \[(\xi\cdot f_0)(z)=P_H(f_0)(z).\] Now it is clear that \[K_{\xi, \Omega}(z)=K^H_{\Omega}(z)\] for every \(\xi=(\xi_{\beta})\in\mathbf{S}_H\) such that (32 ) holds. Now we only need to prove the existence of such \(\xi\). Actually, the equation (32 ) is equivalent to the following linear equation system: \[\sum_{\beta\succeq\alpha, \, |\beta|\le k}\xi_{\beta}\frac{(D^{\beta}\sigma_{\alpha})(z)}{\beta!}=0, \quad \forall\, \alpha\in\mathbf{E}, \;|\alpha|\le k-1.\] This linear equation system must have solutions since \[(D^{\alpha}\sigma_{\alpha})(z)\neq 0, \quad \forall\, \alpha\in\mathbf{E}.\] In summary, we get \[K^H_{\Omega}(z)=\inf_{\xi\in\mathbf{S}_H}K_{\xi,\Omega}(z),\] and the infimum can be achieved, i.e., \[K^H_{\Omega}(z)=\inf_{\xi\in\mathbf{S}_H}K_{\xi,\Omega}(z)=\min_{\xi\in\mathbf{S}_H}K_{\xi,\Omega}(z).\] ◻

Proof. The proof is almost as the same as the proof of Proposition 33 in [13], except using the optimal \(L^p\) extension theorem rather than the optimal \(L^2\) extension theorem.

Proposition 21 indicates that \(\log K_{\xi,\Omega_{\zeta},p}(z)\) is upper-semicontinuous on \(\Omega\), and we have got the plurisubharmonicity of \(\log K_{\xi,\Omega_{\zeta},p}(z)\) in \(z\) (see Proposition 18). In order to get the mean value inequality of \(\log K_{\xi,\Omega_{\zeta},p}(z)\) in \(\zeta\), we need to apply the optimal \(L^p\) extension theorem and Guan–Zhou method.

Let \((z_0,\zeta_0)\in\Omega\) and \(f\in A^p(\Omega_{\zeta_0})\) such that \[K_{\xi,\Omega_{\zeta_0},p}(z_0)=\frac{|(\xi\cdot f)(z_0)|^p}{\int_{\Omega_{\zeta_0}}|f|^p}.\] For any \(r>0\) with \(\Delta(\zeta_0,r)\subseteq\omega\), we prove the sub-mean value inequality of \(\log K_{\xi,\Omega_{\zeta},p}(z_0)\) in \(\zeta\in\Delta(\zeta_0,r)\). We simply assume \(\omega=\Delta(\zeta_0,r)\). Then according to Lemma 35, there exists \(F\in A^p(\Omega)\) such that \(F|_{\Omega_{\zeta_0}}=f\) and \[\frac{1}{\pi r^2}\int_{\Omega}|F|^p\le\int_{\Omega_{\zeta_0}}|f|^p.\] Consequently, \[\begin{align} \begin{aligned} \log\left(\int_{\Omega_{\zeta_0}}|f|^p\right)&\ge\log\left(\frac{1}{\pi r^2}\int_{\Omega}|F|^p\right)\\ &= \log\left(\frac{1}{\pi r^2}\int_{\zeta\in\Delta(\zeta_0,r)}\int_{\Omega_{\zeta}}|F_{\zeta}|^p\right)\\ &\ge \frac{1}{\pi r^2}\int_{\zeta\in\Delta(\zeta_0,r)}\log\left(\int_{\Omega_{\zeta}}|F_{\zeta}|^p\right)\\ &\ge \frac{1}{\pi r^2}\int_{\zeta\in\Delta(\zeta_0,r)}\bigg(\log|(\xi\cdot F_{\zeta})(z_0)|^p-\log K_{\xi,\Omega_{\zeta},p}(z_0)\bigg), \end{aligned} \end{align}\] where \(F_{\zeta}(z)\mathrel{\vcenter{:}}= F(z,\zeta)\). Since \(\xi\in\ell_1^{(n)}\), we can verify that \((\xi\cdot F_{\zeta})(z_0)\) is holomorphic in \(\zeta\) (see also [13], [18]), yielding that \(\log|(\xi\cdot F_{\zeta})(z_0)|^p\) is subharmonic in \(\zeta\). Eventually, we get that \[\log K_{\xi,\Omega_{\zeta_0},p}(z_0)\le\frac{1}{\pi r^2}\int_{\zeta\in\Delta(\zeta_0,r)}\log K_{\xi,\Omega_{\zeta},p}(z_0),\] which is the sub-mean value inequality of \(\log K_{\xi,\Omega_{\zeta},p}(z_0)\) in \(\zeta\) on \(\Delta(\zeta_0,r)\), and we get the desired result. ◻

Proof. Note that the domain \[\big\{(z,\zeta)\in D\times\mathbb{C}\colon \varphi(z)-\mathop{\mathrm{Re}}\zeta<0\big\}\] is pseudoconvex in \(\mathbb{C}^{n+1}\). Then Proposition 37 implies that \(\log K_{\xi,\{\varphi<\mathop{\mathrm{Re}}\zeta\},p}(o)\) is plurisubharmonic in \(\zeta\). Thus, since \(\log K_{\xi,\{\varphi<\mathop{\mathrm{Re}}\zeta\},p}(o)\) only depends on \(\mathop{\mathrm{Re}}\zeta\), we get that \(\log K_{\xi,\{\varphi<t\},p}(o)\) is convex in \(t\in (-\infty,0]\). ◻

Proof. We choose some sufficiently small \(r_0>0\) and a real constant \(M\) such that \[G_{D}(z,o)\le \log|z|+M, \quad \forall\, |z|<r_0.\] It follows that \[\mathbb{B}^n(o,e^{a-M})\subset \{G_{D}(\cdot,o)<a\}\] for sufficiently negative \(a\). Thus, according to the estimate (9 ) on balls, we deduce \[K_{\xi,\{G_{D}(\cdot,o)<a\},p}(o)\le K_{\xi,\mathbb{B}^n(o,e^{a-M}),p}(o)\le C e^{-(2n+pk)a},\] for some positive constant \(C\) independent of \(a\). ◻

Proof of Theorem 8. First, Proposition 37 verifies the convexity of the function \[(-\infty,0] \ni a \to \log K_{\xi,\{G<a\},p}(o).\] Here we denote the pluricomplex Green function \(G_{\Omega}(\cdot,o)\) by \(G\) for short. It follows that \(e^{(2n+pk)a}K_{\xi,\{G<a\},p}(o)\) is also convex in \(a\in (-\infty,a]\). In addition, since \(e^{(2n+pk)a}K_{\xi,\{G<a\},p}(o)\) is bounded above by Lemma 38, we obtain that the function \(e^{(2n+pk)a}K_{\xi,\{G<a\},p}(o)\) is non-decreasing in \(a\in (-\infty,0]\).

Next, we prove the inequality \[\limsup_{a\to -\infty} K^{H,p}_{\Omega_a}(o)\ge K_{I_{\Omega}(o)}^{H,p}(o).\] Like what is done in [20], we can approximate the pseudoconvex domain \(\Omega\) by bounded hyperconvex domains, which allows us to assume that \(\Omega\) is a bounded hyperconvex domain. Then as in [20] for the case \(p=2\), the convergence in the sense of Hausdorff (cf. [27]): \(\Omega_a\to I_{\Omega}(o)\) for the bounded hyperconvex domain \(\Omega\) implies the desired inequality.

Now for every \(\xi\in\mathbf{S}_H\), we obtain \[\begin{align} \begin{aligned} \lim_{a\to -\infty}e^{2n+pk}K_{\xi,\{G<a\},p}(o)&\ge \limsup_{a\to -\infty} e^{2n+pk}K^{H,p}_{\{G<a\}}(o)\\ &=\limsup_{a\to -\infty} K^{H,p}_{\Omega_a}(o)\ge K_{I_{\Omega}(o)}^{H,p}(o). \end{aligned} \end{align}\]

The proof is complete. ◻

Proof of Corollary 9. If the space \(A^p(I_{\Omega}(o))\) is nontrivial, then there exists a homogeneous \(H\) such that \(K^{H,p}_{I_{\Omega}(o)}(o)>0\). Hence, choosing any \(\xi\in\mathbf{S}_H\), Theorem 8 indicates that \(K_{\xi,\Omega,p}(o)>0\), yielding that \(A^p(\Omega)\) is nontrivial. ◻

Proof of Corollary 10. According to Theorem 7 and Theorem 8, we have that \[K^{H,p}_{\{G<a\}}(o)=\inf_{\xi\in\mathbf{S}_H}K_{\xi,\{G<a\},p}(o),\] and \[(-\infty,0]\ni a \to e^{(2n+pk)a}K_{\xi,\{G<a\},p}(o)\] is non-decreasing. It follows that \[e^{(2n+pk)a}K^{H,p}_{\{G<a\}}(o)=\inf_{\xi\in\mathbf{S}_H}e^{(2n+pk)a}K_{\xi,\{G<a\},p}(o)\] is also non-decreasing in \(a\in (-\infty,0]\). Since \[e^{(2n+pk)a}K^{H,p}_{\{G<a\}}(o)=K^{H,p}_{\Omega_a}(o),\] we obtain that \(K^{H,p}_{\Omega_a}(o)\) is non-decreasing in \(a\in (-\infty,0]\). ◻

Proof of Corollary 11. Combining Theorem 7 and Theorem 8, for \(p\in [1,2]\), we have \[K^{H,p}_{\Omega}(o)=\inf_{\xi\in\mathbf{S}_H}K_{\xi,\Omega,p}(o)\ge K_{I_{\Omega}(o)}^{H,p}(o).\]

If the space \(A^p(I_{\Omega}(o))\) is infinitely dimensional, then there exists a sequence of homogeneous polynomials \(\{H_j\}_{j\ge 1}\) of degree \(k_j\), where \(k_j\) increasingly tends to \(+\infty\), such that \(K^{H_j,p}_{I_{\Omega}(o)}>0\) for each \(j\). Since \(K^{H_j,p}_{\Omega}(o)\ge K_{I_{\Omega}(o)}^{H_j,p}(o)\), we have \(K^{H_j,p}_{\Omega}(o)>0\) for each \(j\), which implies \(A^p(\Omega)\) is infinitely dimensional. ◻