Proof. If d is finite, the claim is clearly true. Hence, it remains to prove it for the case $d=\infty .$ In this case, X is noncompact. Let $\{K_{i}\}_{i\in \mathbb {N}}$ be an increasing sequence of compact subsets of X such that $X=\cup _{i\in \mathbb {N}} K_{i}$ . We can take each $K_{i}$ to be the closure of a relatively compact open subset $U_{i}$ of X. Then to prove this proposition, we only need to show that for each i, $\psi ^{S}_{\eta }$ is almost surely a holomorphic section of L on $U_{i}$ .

Let K be a compact subset of X, and let U be an open relatively compact neighborhood of K. Similarly to estimate (2.4), there exists a constant $C_{U}>0$ such that for $s\in H^{0}_{(2)}(X,L)$ ,

(2.9) $$ \begin{align} \sup_{x\in K} |s(x)|_{h_{L}}\leq C_{U}\|s\|_{\mathcal{L}^{2}(\overline{U},L)}. \end{align} $$

By Equation (2.7), we have

(2.10) $$ \begin{align} \sum_{j=1}^{d}\|S_{j}\|^{2}_{\mathcal{L}^{2}(\overline{U},L)}=\int_{x\in\overline{U}}P(x,x)\mathrm{dV}(x)<+\infty. \end{align} $$

For $j\in \mathbb {N}_{>0}$ , $x\in X$ we consider the $L_{x}$ -valued random variable $X_{j}(x)=\eta _{j}S_{j}(x)$ . Since $\eta _{j}$ is centered, we infer

(2.11) $$ \begin{align} \mathbb{E}[X_{j}(x)]=0\in L_{x}. \end{align} $$

It is then consistent to define the variance as

$$ \begin{align*} \mathrm{Var}(X_{j}(x))=\mathbb{E}[|X_{j}(x)|_{h_{L}}^{2}], \end{align*} $$

and we can compute

(2.12) $$ \begin{align} \mathrm{Var}(X_{j}(x))=\mathbb{E}[|X_{j}(x)|_{h_{L}}^{2}]= \mathrm{Var}(\eta_{j})|S_{j}(x)|^{2}_{h_{L}}= \mathrm{Var}(\eta_{1})|S_{j}(x)|^{2}_{h_{L}}. \end{align} $$

We next prove that for any $k\in \mathbb {N}, N\in \mathbb {N}_{>0}$ and for $r>0$ , we have

(2.13) $$ \begin{align} \mathbb{P}\Big(\sup_{\ell=1,\ldots, N}\sup_{x\in K}\big|\sum_{j=1}^{\ell}X_{k+j}(x)\big|_{h_{L}}>r\Big)<\frac{C^{2}_{U}\mathrm{Var}(\eta_{1})}{r^{2}}\sum_{j=1}^{N}\|S_{k+j}\|^{2}_{\mathcal{L}^{2}(\overline{U},L)}. \end{align} $$

For this purpose, define the stochastic process

(2.14) $$ \begin{align} Y_{\ell}=\big\|\sum_{j=1}^{\ell}X_{k+j}\big\|^2_{\mathcal{L}^{2}(\overline{U},L)}, \quad \ell =1, \ldots, N, \end{align} $$

and observe that by virtue of Equation (2.9), we have

(2.15) $$ \begin{align} \sup_{x\in K}\big|\sum_{j=1}^{\ell}X_{k+j}(x)\big|_{h_{L}}\leq C_{U}Y_{\ell}^{\frac12}. \end{align} $$

As a consequence, we have

(2.16) $$ \begin{align} \mathbb{P}\Big(\sup_{\ell=1,\ldots, N}\sup_{x\in K}\big|\sum_{j=1}^{\ell}X_{k+j}(x)\big|_{h_{L}}>r\Big) \le \mathbb{P}\Big(\sup_{\ell=1,\ldots, N} C_{U}^2 Y_{\ell} >r^2\Big). \end{align} $$

Now, the process $(Y_{\ell })$ , $\ell =1, \ldots , N,$ is a submartingale with respect to the filtration $(\mathcal F_\ell ),$ where $\mathcal F_\ell = \sigma \big (\langle X_{k+i},X_{k+j} \rangle _{\mathcal {L}^{2}(\overline {U},L)},\ i,j =1, \ldots , \ell \big )$ . Therefore, Doob’s submartingale inequality (see, e.g., [Reference Klenke47, Lemma 11.1]) yields

(2.17) $$ \begin{align} \mathbb{P}\Big(\sup_{\ell=1, \ldots, N} Y_{\ell}> \frac{r^2}{C_{U}^2}\Big) \le C_{U}^2 \frac{\mathbb{E}[Y_N]}{r^2}, \end{align} $$

which immediately entails Equation (2.13). Now, letting $N\rightarrow +\infty $ in Equation (2.13), we get

(2.18) $$ \begin{align} \mathbb{P}\Big(\sup_{\ell\in\mathbb{N}_{>0}}\sup_{x\in K}|\sum_{j=1}^{\ell}X_{k+j}(x)|_{h_{L}}>r \Big)\leq \frac{C^{2}_{U}\mathrm{Var}(\eta_{1})}{r^{2}}\sum_{j=1}^{+\infty}\|S_{k+j}\|^{2}_{\mathcal{L}^{2}(\overline{U},L)}. \end{align} $$

Then taking the limit of Equation (2.18) as $k\rightarrow \infty $ , and using Equation (2.10), we infer

(2.19) $$ \begin{align} \mathbb{P}\Big(\limsup_{k\rightarrow+\infty}\sup_{\ell\in\mathbb{N}_{>0}}\sup_{x\in K}|\sum_{j=1}^{\ell}X_{k+j}(x)|_{h_{L}}>r\Big)=0. \end{align} $$

Therefore, a union bound along the sequence of $r = \frac 1N$ immediately supplies us with

(2.20) $$ \begin{align} \mathbb{P}\Big(\limsup_{k\rightarrow+\infty}\sup_{\ell\in\mathbb{N}_{>0}}\sup_{x\in K}|\sum_{j=1}^{\ell}X_{k+j}(x)|_{h_{L}}>0\Big)=0. \end{align} $$

If we take V to be a relatively compact open subset of X, and take $K=\overline {V}$ , then by Equation (2.20), the sum $\sum _{j=1}^{\infty } X_{j}$ is almost surely uniformly convergent on K so that it almost surely defines a holomorphic section on V. This completes the proof of our proposition.

Proof. It is sufficient to find a sequence $\eta '=\{\eta ^{\prime }_{j}\}_{j=1}^{d}$ of i.i.d. standard complex Gaussian random variables such that a.s. $\psi ^{S'}_{\eta }=\psi ^{S}_{\eta '}$ . Let $\ell ^{2}(\mathbb {C})$ denote the Hilbert space of $\ell ^{2}$ -summable complex sequences. If $u=(u_{j})_{j\in \mathbb {N}}\in \ell ^{2}(\mathbb {C})$ , set

(2.21) $$ \begin{align} (\eta,u)_{\ell^{2}}=\sum_{j\in\mathbb{N}} \eta_{j}\bar{u}_{j}. \end{align} $$

By Kolmogorov’s three-series theorem (cf. [Reference Varadhan77]), the series in definition (2.21) is almost surely convergent so that $(\eta ,u)_{\ell ^{2}}$ is a well-defined random variable. By the property of Gaussian random variables, we conclude that $(\eta ,u)_{\ell ^{2}}$ is a centered complex Gaussian random variable with variance $|u|^{2}_{\ell ^{2}}$ . In particular, if $|u|_{\ell ^{2}}=1$ , then $(\eta ,u)_{\ell ^{2}}$ has the same distribution as $\eta _1$ . Moreover, if nonzero $u,v\in \ell ^{2}$ is such that $(u,v)_{\ell ^{2}}=0$ , then $(\eta ,u)_{\ell ^{2}}$ and $(\eta ,v)_{\ell ^{2}}$ are independent. Take $(a_{ij}\in \mathbb {C})_{i,j\in \mathbb {N}}$ such that for each i,

(2.22) $$ \begin{align} S^{\prime}_{i}=\sum_{j\in\mathbb{N}}a_{ij}S_{j}. \end{align} $$

For $j\in \mathbb {N}$ , set $b_{j}=(\bar {a}_{ij})_{i\in \mathbb {N}}$ . Then $b_{j}\in \ell ^{2}(\mathbb {C})$ is with norm $1$ , moreover, if $j\neq j'$ , then $(b_{j},b_{j'})_{\ell ^{2}}=0$ . Now, define

(2.23) $$ \begin{align} \eta^{\prime}_{j}=(\eta,b_j)_{\ell^{2}}. \end{align} $$

Then $\eta '=(\eta ^{\prime }_{j})_{j\in \mathbb {N}}$ is a sequence of i.i.d. centered Gaussian random variables with the same distribution as $\eta $ . By definition, we get that almost surely, $\psi ^{S'}_{\eta }=\psi ^{S}_{\eta '}$ . Therefore, $\psi ^{S'}_{\eta }$ and $\psi ^{S}_{\eta }$ have the same distribution.

Proof. The event that $\psi ^{S}_{\eta }$ is $\mathcal {L}^{2}$ -integrable is equivalent to the event $\{\sum _{j=1}^{\infty } |\eta _{j}|^{2}< \infty \}$ . But, for example, by the law of large numbers, we infer

(2.26) $$ \begin{align} \mathbb{P}\Big(\sum_{j=1}^{\infty} |\eta_{j}|^{2}< \infty\Big)=0, \end{align} $$

and the statement of the lemma follows.

Proof. Let $e_{L}:U\to L$ be a local holomorphic frame of L. Let $\varphi \in \mathscr {C}^\infty (U)$ be the local weight of $h_L$ with respect to $e_L$ , that is, $|e_L|^2_{h_L}=e^{-2\varphi }$ on U. We consider an orthonormal basis $\{S_j\}_{j=1}^d$ of $H^0_{(2)}(X,L)$ and write

(2.30) $$ \begin{align} S_{j}(x)=f_{j}(x)e_{L}(x), \:\:x\in U,\:\:f_j\in\mathscr{O}(U), \end{align} $$

where $f_{j}$ are nontrivial holomorphic functions on U. Then $P(x,x)=\sum _{j=1}^d|S_j|^2_{h_L}=\sum _{j=1}^d|f_j|^2e^{-2\varphi }$ on U, hence

(2.31) $$ \begin{align} \log P(x,x)=\log\Big(\sum_{j=1}^d|f_j|^2\Big)-2\varphi\,. \end{align} $$

The series $\sum _{j=1}^d|f_j|^2$ converges locally uniformly on U, thus $\log \!\big (\sum _{j=1}^d|f_j|^2\big )$ is a plurisubharmonic function that is not identically $-\infty $ , hence locally integrable.

Proof of Theorem 1.1.

We fix a test form $\varphi \in \Omega ^{n-1,n-1}_{0}(X)$ and evaluate $\mathbb {E}[\langle [\operatorname {\mathrm {Div}}(\psi ^{S}_{\eta })], \varphi \rangle ]$ . By applying the Lelong–Poincaré formula (2.29) to $\psi ^{S}_{\eta }$ , we get

(2.34) $$ \begin{align} \begin{aligned} \langle [\operatorname{\mathrm{Div}}(\psi^{S}_{\eta})],\varphi\rangle &= \int_{X}\big(\frac{\sqrt{-1}}{2\pi}\partial\bar{\partial} \log|\psi^{S}_{\eta}|^{2}_{h_{L}}+ c_{1}(L,h_{L})\big)\wedge \varphi\\ &=\int_{X}c_{1}(L,h_{L})\wedge\varphi+ \frac{\sqrt{-1}}{2\pi}\int_{X} \log|\psi^{S}_{\eta}|_{h_{L}}^{2} \partial\bar{\partial}\varphi\\ &=\int_{X}c_{1}(L,h_{L})\wedge\varphi+\frac{\sqrt{-1}}{2\pi} \int_{X\backslash \text{Bl}(X,L)}\log|\psi^{S}_{\eta}|_{h_{L}}^{2} \partial\bar{\partial}\varphi. \end{aligned} \end{align} $$

The last identity follows from that $\log |\psi ^{S}_{\eta }|_{h_{L}}^{2}$ is almost surely locally integrable by Proposition 2.1 and Remark 2.4 (b), and that $\text {Bl}(X,L)$ has Lebesgue measure zero in X. For $x\in X\backslash \text {Bl}(X,L)$ , we have $P(x,x)\neq 0$ , and let $e_{L}(x)$ be a unit vector of L at x, define

(2.35) $$ \begin{align} b(x)=\big(P(x,x)^{-1/2}S_{j}(x)/e_{L}(x)\big)_{j\in\mathbb{N}}\in \ell^{2}(\mathbb{C}). \end{align} $$

We have $|b(x)|_{\ell ^{2}}=1$ . Note that

(2.36) $$ \begin{align} P(x,x)^{-1/2}\psi^{S}_{\eta}=(\eta,\overline{b(x)})_{\ell^{2}}\,e_{L}(x). \end{align} $$

Then

(2.37) $$ \begin{align} \mathbb{E}[\log|P(x,x)^{-1/2}\psi^{S}_{\eta}|_{h_{L}}^{2}]=\mathbb{E}[\log\big|(\eta,\overline{b(x)})_{\ell^{2}}\big|^{2}]=\mathbb{E}[\log|\eta_{1}|^{2}]. \end{align} $$

Note that $\mathbb {E}[|\log |\eta _{1}|^{2}|]<\infty $ . By Lemma 2.6, $\log P(x,x)$ is locally integrable on X, then we can apply the Fubini’s theorem to the following integrals so that

(2.38) $$ \begin{align} \begin{aligned} \mathbb{E}\big[\int_{X\backslash \text{Bl}(X,L)} \log|P(x,x)^{-1/2}\psi^{S}_{\eta}|_{h_{L}}^{2} \partial\bar{\partial} \varphi\big] &=\int_{X\backslash \text{Bl}(X,L)} \mathbb{E}\big[\log|P(x,x)^{-1/2}\psi^{S}_{\eta}|_{h_{L}}^{2}\big] \partial\bar{\partial} \varphi\\ &=\mathbb{E}[\log|\eta_{1}|^{2}]\int_{X}\partial\bar{\partial} \varphi=0. \end{aligned} \end{align} $$

Then by formula (2.34), we get

(2.39) $$ \begin{align} \begin{aligned} \mathbb{E}\left[\langle[\operatorname{\mathrm{Div}}(\psi^{S}_{\eta})],\varphi\rangle \right] &=\int_{X}c_{1}(L,h_{L})\wedge\varphi+ \frac{\sqrt{-1}}{2\pi} \int_{X\backslash\text{Bl}(X,L)}\log{P(x,x)}\cdot \partial\bar{\partial}\varphi\\ &=\big\langle c_{1}(L,h_{L})+\frac{\sqrt{-1}}{2\pi}\partial\bar{\partial}\log(P(x,x)), \varphi\big\rangle. \end{aligned} \end{align} $$

This completes the proof.

Proof. It is clear that the second part (ii) of this theorem follows directly from the expansion (3.6). We only need to prove the first part (i). By Theorem 1.1 for $L^{p}$ ,

(3.10) $$ \begin{align} \mathbb{E}[[\operatorname{\mathrm{Div}}(\psi^{S_{p}}_{\eta})]]=\gamma(L^{p},h_{p}), \end{align} $$

where $\gamma (L^{p}, h_{p})$ is the corresponding Fubini–Study current defined via formula (1.4). To obtain the expansion (3.6), it is enough to study the expansion of $\frac {\sqrt {-1}}{2\pi p}\partial \overline {\partial }\log P_{p}(x,x)$ in p.

For this purpose, we apply the expansion (3.3) to $P_{p}(x,x)$ . Note that $\boldsymbol {b}_{0}$ is uniformly bounded from below by a positive constant. Then for a given compact subset K of X, and $k,\ell \in \mathbb {N}$ , we take the expansion (3.3) but with $\mathscr {C}^{\ell +2}(K)$ -norm. We conclude that, for all sufficiently large p, we have $P_{p}(x,x)>0$ , and

(3.11) $$ \begin{align} \big|\sum_{r=1}^{k}\boldsymbol{b}_{r}(x)p^{-r}+C_{k,\ell+2,K} p^{-k-1}\big|\leq \frac{1}{2}\boldsymbol{b}_{0}(x). \end{align} $$

Then $\frac {\sqrt {-1}}{2\pi p}\partial \overline {\partial }\log P_{p}(x,x)$ is a smooth $(1,1)$ -form on K, and the expansion (3.6) is the consequence of the following identity and the Taylor expansion of $\log (1+x)$ with $|x|<1$ ,

(3.12) $$ \begin{align} \begin{aligned} \log P_{p}(x,x)&=n\log p+\log\big(\sum_{r=0}^{k}\boldsymbol{b}_{r}(x)p^{-r}+R_{k}(p)\big)\\ &=n\log p+\log \boldsymbol{b}_{0}(x)+\log\big(1+\sum_{r=1}^{k}p^{-r}\frac{\boldsymbol{b}_{r}(x)}{\boldsymbol{b}_{0}(x)}+R_{k}(p)\big), \end{aligned} \end{align} $$

where $R_{k}(p)=\mathcal {O}(p^{-k-1})$ denotes the remainder term. In particular, we get formulas (3.7).

Proof. To prove this theorem, we mainly follow the arguments from proof of [Reference Ma and Marinescu56, Theorem 5.3.3], and the possibility of infinite dimension does not lead to complications in this setting. Fix a nontrivial test form $\varphi \in \Omega ^{n-1,n-1}_{0}(X)$ . Note that from the proof of Theorem 3.1, we have the convergence

(3.20) $$ \begin{align} \lim_{p\rightarrow\infty}\Big\langle \frac{1}{p}\gamma(L^{p},h_{p}),\varphi\Big\rangle=\langle c_{1}(L,h_{L}),\varphi\rangle. \end{align} $$

Defining the random variable

(3.21) $$ \begin{align} Y_{p}=\frac{1}{p}\Big\langle [\operatorname{\mathrm{Div}}(\psi^{S_{p}}_{\eta})]-\gamma(L^{p},h_{p}),\varphi\Big\rangle, \end{align} $$

statement (3.19) is equivalent to proving the almost sure convergence

(3.22) $$ \begin{align} Y_{p}\rightarrow 0. \end{align} $$

For any $x\in \operatorname {\mathrm {supp}} \varphi $ , let $e_{L}(x)$ denote a unit vector of $(L_{x},h_{L,x})$ . Set

(3.23) $$ \begin{align} b_{p}(x)=(P_{p}(x,x)^{-1/2}S^{p}_{j}(x)/e_{L}^{\otimes p}(x))\in\ell^{2}(\mathbb{C}). \end{align} $$

Then $\eta \cdot b_{p}(x)$ is a standard complex Gaussian variable. The covariance matrix of the Gaussian vector $(\eta \cdot b_{p}(x),\eta \cdot b_{p}(y))$ depends smoothly on $(x,y)\in \operatorname {\mathrm {supp}}\varphi \times \operatorname {\mathrm {supp}}\varphi $ .

For $v=(v_{1},v_{2})\in \mathbb {C}^{2}$ with $\|v\|=1$ , we consider the integral

(3.24) $$ \begin{align} \rho(v):=\frac{1}{4\pi^{2}}\int_{\mathbb{C}^{2}}e^{-\frac{1}{2}(|z_{1}|^{2}+|z_{2}|^{2})}\big|\log|z_{1}|\cdot\log|v_{1}z_{1}+v_{2}z_{2}|\big|\mathrm{dV}(z). \end{align} $$

The computations in [Reference Ma and Marinescu56, Eqs. (5.3.13) to (5.3.15)] then show that

(3.25) $$ \begin{align} C:=\sup_{v\in\mathbb{C}^{2},\|v\|=1 }\rho(v)<\infty, \end{align} $$

so for $x,y\in \operatorname {\mathrm {supp}}\varphi $ we have

(3.26) $$ \begin{align} \mathbb{E}\Big[\Big|\log\big|P_{p}(x,x)^{-1/2}\sum_{j}\eta_{j}S^{p}_{j}(x) \big|_{h_{p}}\log\big|P_{p}(y,y)^{-1/2} \sum_{j}\eta_{j}S^{p}_{j}(y)\big|_{h_{p}}\Big|\Big]\leq C. \end{align} $$

Note that

(3.27) $$ \begin{align} \mathbb{E}[|Y_{p}|^{2}]=\frac{1}{p^{2}}\mathbb{E}\big[\left|\langle [\operatorname{\mathrm{Div}}(\psi^{S_{p}}_{\eta})],\varphi\rangle\right|{}^{2}\big]-\frac{1}{p^{2}}\big|\langle \gamma(L^{p},h_{p}),\varphi\rangle\big|^{2}. \end{align} $$

Then by relations (2.29) and (3.26) and the Fubini–Tonelli theorem we infer that

(3.28) $$ \begin{align} \begin{aligned} \mathbb{E}[|Y_{p}|^{2}] & =\frac{1}{\pi^{2}p^{2}}\int_{X\times X} \big(\partial\overline{\partial}\varphi(x)\big) \big(\overline{\partial\overline{\partial}\varphi(y)}\big)\\ &\quad \mathbb{E}\Big[\log\big|P_{p}(x,x)^{-1/2}\sum_{j}\eta_{j}S^{p}_{j}(x) \big|_{h_{p}}\log\big|P_{p}(y,y)^{-1/2} \sum_{j}\eta_{j}S^{p}_{j}(y)\big|_{h_{p}}\Big]. \end{aligned} \end{align} $$

By estimate (3.26), we conclude $\mathbb {E}[|Y_{p}|^{2}]=\mathcal {O}(\frac {1}{p^{2}})$ . Hence, $\mathbb {E}[\sum _{p\geq 1}|Y_{p}|^{2}]=\sum _{p\geq 1}\mathbb {E}[|Y_{p}|^{2}]<\infty $ , thus $Y_{p}\rightarrow 0$ almost surely.

Proof. Let’s assume at first that $\Theta $ is a Kähler form with $\int _{X} c_{1}(L,h_{L})\wedge \Theta ^{n-1}<\infty $ . For all $\varphi \in \Omega ^{n-1,n-1}_{0}(X)$ , $s_{p}\in H^{0}(X,L^{p})$ , due to the positivity of the current $[\operatorname {\mathrm {Div}}(s_{p})]$ , we have

(3.30) $$ \begin{align} |\langle[\operatorname{\mathrm{Div}}(s_{p})],\varphi\rangle|\leq |\varphi|_{\mathscr{C}^{0}(X)}\langle[\operatorname{\mathrm{Div}}(s_{p})], \Theta^{n-1}\rangle=p|\varphi|_{\mathscr{C}^{0}(X)}\int_{X}c_{1}(L,h_{L})\wedge\Theta^{n-1}. \end{align} $$

The last identity follows from the Poncaré–Lelong formula. We may set $C=\int _{X}c_{1}(L,h_{L})\wedge \Theta ^{n-1}<\infty $ , then

(3.31) $$ \begin{align} \frac{1}{p}|\langle[\operatorname{\mathrm{Div}}(s_{p})],\varphi\rangle|\leq C|\varphi|_{\mathscr{C}^{0}(X)}. \end{align} $$

By considering a countable $\mathscr {C}^{0}$ -dense family of $\varphi $ ’s in $\Omega ^{n-1,n-1}_{0}(X)$ , and applying Theorem 3.7, we get the equality (3.29).

If $\int _{X} c_{1}(L,h_{L})^{n}<\infty $ , we use $c_{1}(L,h_{L})$ instead of $\Theta $ in the above arguments to get the equality (3.29).

Proof of Theorem 1.4.

The Poincaré–Lelong formula (2.29) shows that

(3.35) $$ \begin{align} \frac{\sqrt{-1}}{\pi}\partial\overline{\partial} \log{|\psi^{S_{p}}_{\eta}|_{h_{p}}}=[\operatorname{\mathrm{Div}}(\psi^{S_{p}}_{\eta})]-pc_{1}(L,h_L) \end{align} $$

as an identity of $(1,1)$ -currents on $X.$ Now, fix $\varphi \in \Omega ^{n-1,n-1}_{0}(X)$ , and fix a relatively compact open subset $U\subset X$ such that $\mathrm {supp}\ \varphi \subset U$ . Then

(3.36) $$ \begin{align} \begin{aligned} \Big(\frac{1}{p}[\operatorname{\mathrm{Div}}(s_{p})],\varphi\Big) -\int_{X}c_{1}(L,h_L)\wedge \varphi=\frac{\sqrt{-1}}{p\pi}\int_{X} \log{|\psi^{S_{p}}_{\eta}|_{h_{p}}}\,\partial\overline{\partial}\varphi. \end{aligned} \end{align} $$

Since $\varphi $ has a compact support in U, so has $\partial \overline {\partial }\varphi $ . Set

(3.37) $$ \begin{align} S_{\varphi}=\max_{x\in U}\left|\frac{\sqrt{-1}\partial \overline{\partial}\varphi(x)}{\mathrm{dV}(x)}\right|. \end{align} $$

We can and we may assume that $S_{\varphi }>0$ . Then

(3.38) $$ \begin{align} \begin{aligned} \left|\frac{\sqrt{-1}}{p\pi}\int_{X}\log{|\psi^{S_{p}}_{\eta}|_{h_{p}}}\, \partial\overline{\partial}\varphi\right|\leq \frac{S_{\varphi}}{p\pi}\int_{U} \big|\log{|\psi^{S_{p}}_{\eta}(x)|_{h_{p}}}\big|\,\mathrm{dV}(x). \end{aligned} \end{align} $$

Applying Proposition 3.11 to right-hand side of inequality (3.38) we get estimate (1.10).

Proof of Theorem 1.5.

Estimate (1.14) is a direct consequence of estimate (1.13) by taking $\delta =n\operatorname {\mathrm {Vol}}^{L}_{2n}(U)$ . Hence, it is sufficient to prove estimate (1.13). For this purpose, let $\chi _{U}$ denote the characteristic function of U on X. Let $\delta>0$ be arbitrary, and take $\psi _{1}$ , $\psi _{2}\in \mathcal {C}^{\infty }_{0}(X,\mathbb {R})$ such that $0\leq \psi _{1}\leq \chi _{U}\leq \psi _{2}\leq 1,$ and

(3.39) $$ \begin{align} \int_{X} \psi_{1}\frac{c_{1}(L,h_{L})^{n}}{n!}\geq \operatorname{\mathrm{Vol}}^{L}_{2n}(U)-\delta, \quad \int_{X} \psi_{2}\frac{c_{1}(L,h_{L})^{n}}{n!}\leq \operatorname{\mathrm{Vol}}^{L}_{2n}(U)+\delta\,. \end{align} $$

Note that the existence of such functions is guaranteed by the assumption that $\partial U$ has measure $0$ with respect to $\mathrm {dV}$ , hence also to $\frac {1}{n!}c_{1}(L,h_{L})^{n}$ . For $j\in \{1,2\},$ set $\varphi _{j}=\frac {1}{(n-1)!}{\psi _{j}}c_{1}(L,h_{L})^{n-1}$ . By applying Theorem 1.4 to $\varphi _{j}$ separately, we get exactly estimate (1.13).

Proof. The basic idea of the proof is that the local sup-norm of a holomorphic function is bounded by its local $\mathcal {L}^{2}$ -norm as in estimate (2.9). We fix $\delta>0$ and let $r>0$ be sufficiently small so that we can choose a finite set of points $\{x_{j}\}_{j=1}^{\ell }\subset U$ such that the geodesic open balls $B^{X}(x_{j},r)$ , $j=1,\ldots , \ell $ form an open covering of $\overline {U}$ . Since r is sufficiently small, then we can assume that each larger ball $B^{X}(x_{j},2r)$ lies in a complex chart (hence viewed as an open subset of $\mathbb {C}^{n}$ ), and that for each j, we can fix a local holomorphic frame $e_{L,j}$ of L on a neighborhood of $B^{X}(x_{j}, 2r)$ with $\sup _{x\in B^{X}(x_{j},2r)} |e_{L,j}(x)|_{h_{L}}=1$ . Set

(3.42) $$ \begin{align} \nu=\min\big\{ \inf_{x\in B^{X}(x_{j},2r)}|e_{L,j}(x)|_{h_{L}} \,:\,j=1 , \ldots, \ell\big\}. \end{align} $$

It is clear that $0<\nu \leq 1$ . By fixing r small enough, we can and do assume that

(3.43) $$ \begin{align} -\log{\nu}\leq \frac{\delta}{4}\,\cdot \end{align} $$

As in estimate (2.9), since U is relatively compact, there exists a constant $C>0$ such that for each $j=1,\ldots , \ell $ , if f is a holomorphic function on a neighborhood of $B^{X}(x_{j},2r)$ , then

(3.44) $$ \begin{align} \sup_{x\in B^{X}(x_{j},r)}|f(x)|\leq C\|f\|_{\mathcal{L}^{2}(B^{X}(x_{j},2r))}, \end{align} $$

where the volume form $\mathrm {dV}(x)$ on X is used in the norm $\|\cdot \|_{\mathcal {L}^{2}(B^{X}(x_{j},2r))}$ . Note that the choices of $x_{j}$ , r, $\ell $ and the constants $\nu $ , C are independent of the tensor power p. Set $\widetilde {U}=\cup _{j} B^{X}(x_{j},2r)\supset U$ . For $p\in \mathbb {N}, s_{p}\in H^{0}(X,L^{p})$ , on each $B^{X}(x_{j},2r)$ , we write

(3.45) $$ \begin{align} s_{p}|_{B^{X}(x_{j},2r)}=f_{j}e_{L,j}^{\otimes p}, \end{align} $$

where $f_{j}$ is a holomorphic function on the chart in $\mathbb {C}^{n}$ corresponding to $B^{X}(x_{j},2r)$ . Then we have

(3.46) $$ \begin{align} \begin{aligned} \mathcal{M}^{U}_{p}(s_{p})=\sup_{x\in U}|s_{p}(x)|_{h_{p}}&\leq \max_{j} \sup_{x\in B^{X}(x_{j},r)}|f_{j}(x)|\\ &\leq C\max_{j} \{\|f_{j}\|_{\mathcal{L}^{2}(B^{X}(x_{j},2r))}\}\\ &\leq \frac{C}{\nu^{p}}\max_{j}\{\|s_{p}\|_{\mathcal{L}^{2}(B^{X}(x_{j},2r),L^{p})}\}\\ &\leq \frac{C}{\nu^{p}}\|s_{p}\|_{\mathcal{L}^{2}(\widetilde{U},L^{p})}. \end{aligned} \end{align} $$

The next step is to estimate the quantity $\mathbb {E}[\|\psi ^{S_{p}}_{\eta }\|^{2p^{n}}_{\mathcal {L}^{2}(\widetilde {U},L^{p})}]$ for $p\geq 2$ . Applying Hölder’s inequality with $\frac {1}{p^{n}}+\frac {p^{n}-1}{p^{n}}=1$ , we get

(3.47) $$ \begin{align} \mathbb{E}\big[\|\psi^{S_{p}}_{\eta}\|^{2p^{n}}_{\mathcal{L}^{2}(\widetilde{U},L^{p})}\big] \leq \operatorname{\mathrm{Vol}}(\widetilde{U})^{p^{n}-1}\mathbb{E}\Big[\int_{\widetilde{U}} |\psi^{S_{p}}_{\eta}(x)|^{2p^{n}}_{h_{p}}(x)\mathrm{dV}\Big]. \end{align} $$

As in estimate (3.44), on a neighborhood of $B^{X}(x_{j}, 2r)$ , write

(3.48) $$ \begin{align} S^{p}_{i}=f^{p}_{i}e_{L,i}^{\otimes p}. \end{align} $$

If $x\in B^{X}(x_{j},2r)$ , set

(3.49) $$ \begin{align} F_{j}(x)=\sum_{i=1}^{d_{p}} \eta_{i} f^{p}_{i}(x). \end{align} $$

Then $F_{j}(x)$ is a complex Gaussian random variable with (total) variance $\sum _{i=1}^{d_{p}}|f^{p}_{i}(x)|^{2}$ . By our assumption on the local frame $e_{L,j}$ , we get

(3.50) $$ \begin{align} \sum_{i=1}^{d_{p}}|f^{p}_{i}(x)|^{2}\leq \frac{1}{\nu^{2p}}P_{p}(x,x). \end{align} $$

Then we have

(3.51) $$ \begin{align} \mathbb{E}\big[|F_{j}(x)|^{2p^{n}}\big]=p^{n}! \Big(\sum_{i=1}^{d_{p}}|f^{p}_{i}(x)|^{2}\Big)^{p^{n}}. \end{align} $$

As a consequence, we get that for $x\in \widetilde {U}$ ,

(3.52) $$ \begin{align} \begin{aligned} \mathbb{E}\big[|\psi^{S_{p}}_{\eta}(x)|_{h_{p}}^{2p^{n}}\big]\leq\frac{p^{n}!}{\nu^{2p^{n+1}}}(P_{p}(x,x))^{p^{n}}. \end{aligned} \end{align} $$

Since we are in the context of $\sigma $ -finite measures and the integrands are nonnegative, Tonelli’s theorem applies so that

(3.53) $$ \begin{align} \mathbb{E}\Big[\int_{\widetilde{U}}|\psi^{S_{p}}_{\eta}(x)|^{2p^{n}}_{h_{p}}\mathrm{dV}(x)\Big]\leq \frac{p^{n}!}{\nu^{2p^{n+1}}}\int_{\widetilde{U}} (P_{p}(x,x))^{p^{n}}\mathrm{dV}(x). \end{align} $$

Moreover, by the on-diagonal estimate for the Bergman kernel on a given compact subset, there exists a constant $C_{\widetilde {U}}>0$ (independent of p) such that for $p\in \mathbb {N}$ , $x\in \widetilde {U}$ ,

(3.54) $$ \begin{align} P_{p}(x,x)\leq C_{\widetilde{U}}p^{n}. \end{align} $$

Combining estimate (3.47) with the above inequalities, we infer that

(3.55) $$ \begin{align} \mathbb{E}\Big[\|\psi^{S_{p}}_{\eta}\|^{2p^{n}}_{\mathcal{L}^{2}(\widetilde{U},L^{p})} \Big]\leq \big(C_{\widetilde{U}}\operatorname{\mathrm{Vol}}(\widetilde{U})\big)^{p^{n}}\frac{p^{n}!}{\nu^{2p^{n+1}}}(p^{n})^{p^{n}}. \end{align} $$

By applying estimate (3.46) to $\psi ^{S_{p}}_{\eta }$ , we get

(3.56) $$ \begin{align} \begin{aligned} \mathbb{E}\big[\mathcal{M}^{U}_{p}(\psi^{S_{p}}_{\eta})^{2p^{n}}\big]&\leq \Big(\frac{C}{\nu^{p}}\Big)^{2p^{n}}\mathbb{E}\big[\|\psi^{S_{p}}_{\eta}\|^{2p^{n}}_{\mathcal{L}^{2}(\widetilde{U},L^{p})}\big] \leq \frac{(\widetilde{C}p^{n})^{2p^{n}}}{\nu^{4p^{n+1}}}, \end{aligned} \end{align} $$

where $C>0$ , $\widetilde {C}>0$ are constants independent of p. Then estimate (3.41) follows from Chebyshev’s inequality and the inequality $\frac {1}{\nu }\leq e^{\frac {\delta }{4}}$ from (3.43).

Proof. For $x\in X$ , we fix some $\lambda _{x}\in L_{x}$ with $|\lambda _{x}|_{h}=1$ and set

(3.68) $$ \begin{align} \xi_{x}=\frac{\langle \lambda_{x}^{\otimes p}, \psi^{S_{p}}_{\eta}(x)\rangle_{h_{p}}}{\sqrt{P_{p}(x,x)}}\,\cdot \end{align} $$

Then $\xi _{x}$ is a complex Gaussian random variable. Moreover, for any two points $x,y\in X$ , we have

(3.69) $$ \begin{align} \big|\mathbb{E}[\xi_{x}\overline{\xi}_{y}]\big| =N_{p}(x,y). \end{align} $$

Then by the asymptotics (3.66), using arguments similar to those used in [Reference Shiffman, Zelditch and Zrebiec70, Subsection 3.2] or the proof of [Reference Drewitz, Liu and Marinescu27, Theorem 1.13], we can prove a more general version of estimate (3.67) as follows: for a sequence of positive numbers $\{\lambda _{p}\}_{p\in \mathbb {N}}$ with $\lambda _p\rightarrow 0$ , we have

(3.70) $$ \begin{align} \mathbb{P}\big(\mathcal{M}^{U}_{p}(\psi^{S_{p}}_{\eta})\leq \lambda_{p}\big)\leq e^{Cp^{n}\log \lambda_{p} +C'p^{n}\log p},\quad p\gg 0. \end{align} $$

Then, for any $\delta>0$ , choosing $\lambda _{p}=e^{-\delta p}$ in estimate (3.70), we recover (3.67).

Proof of Proposition 3.11.

The proof of Proposition 3.11 follows by combining from the arguments in [Reference Shiffman, Zelditch and Zrebiec70, Subsection 4.1] with Corollary 3.16. Here, we just sketch the proof.

For $t>0$ , set

(3.72) $$ \begin{align} \log^{+}{t}=\max\{\log{t},0\},\; \log^{-}{t}:=\log^{+}(1/t)=\max\{-\log{t},0\}. \end{align} $$

Then

(3.73) $$ \begin{align} |\log{t}|=\log^{+}{t}+\log^{-}{t}. \end{align} $$

Let U be a relatively compact nonempty open subset in $X.$ Then for any nonzero holomorphic section $s_{p}\in H^{0}(X,L^{p})$ , we have that $\big |\log |s_{p}|_{h_{p}}\big |$ is integrable on $\overline {U}$ with respect to $\mathrm {dV}$ . We now start with showing that

(3.74) $$ \begin{align} \mathbb{P}\left( \int_{U}\log^{+}{|\psi^{S_{p}}_{\eta}(x)|_{h_{p}}}\,\mathrm{dV}(x)\geq \frac{\delta}{2} p\right)\leq e^{-C_{U,\delta}p^{n+1}}. \end{align} $$

For this purpose, observe that on U we have

(3.75) $$ \begin{align} \log^{+}{|\psi^{S_{p}}_{\eta}|_{h_{p}}}\leq \big|\log\mathcal{M}^{U}_{p}(\psi^{S_{p}}_{\eta})\big|, \end{align} $$

which then supplies us with

(3.76) $$ \begin{align} \mathbb{P}\left( \int_{U}\log^{+}{|\psi^{S_{p}}_{\eta}(x)|_{h_{p}}}\,\mathrm{dV}(x)\geq \frac{\delta}{2} p\right) \leq \mathbb{P}\left( \left|\log\mathcal{M}^{U}_{p}(\psi^{S_{p}}_{\eta})\right|\geq \frac{\delta}{2 \operatorname{\mathrm{Vol}}(U)} p\right), \end{align} $$

where $\operatorname {\mathrm {Vol}}(U)$ denotes the volume of U with respect to $\mathrm {dV}$ . In combination with Corollary 3.16, this immediately implies estimate (3.74). The next step is to prove that

(3.77) $$ \begin{align} \mathbb{P}\left( \int_{U}\log^{-}{|\psi^{S_{p}}_{\eta}(x)|_{h_{p}}}\,\mathrm{dV}(x)\geq \frac{\delta}{2} p\right)\leq e^{-C_{U,\delta}\,p^{n+1}}. \end{align} $$

Suppose that U contains an annulus $B(2,3):=\{z\in \mathbb {C}^{n}:2<|z|<3\}$ (possibly after rescaling of coordinates) and the line bundle L on $B(4):=\{|z|<4\}$ (still contained in U) has a holomorphic local frame $e_{L}$ . We will mainly work on this annulus $B(2,3)$ instead of a coordinate disc. The reason is that, as in [Reference Shiffman, Zelditch and Zrebiec70, p. 1991 in Subsection 4.1], we will need the two-sided positive bounds for the Poisson kernel of the disc and such bounds will hold on the annulus $B(2,3)$ .

Set $\alpha (x)= \log |e_{L}(x)|^{2}_{h_L}$ . We can then write

(3.78) $$ \begin{align} \psi^{S_{p}}_{\eta}=F_{p}e^{\otimes p}_{L}, \end{align} $$

where $F_{p}$ is a random holomorphic function on $B(4)$ . Then

(3.79) $$ \begin{align} \log|\psi^{S_{p}}_{\eta}|_{h_{p}}=\log|F_{p}|+\frac{p}{2}\alpha. \end{align} $$

In the following estimates, each $K_{i}$ , $i \in \mathbb {N},$ denotes a sufficiently large positive constant. Then by relations (3.73) and (3.76), we have

(3.80) $$ \begin{align} \mathbb{P}\left( \int_{B(2,3)}\log^{+}{|F_{p}|}\,\mathrm{dV}\geq K_{1} p \right)\leq e^{-C_{U,K_{1}}\, p^{n+1}}. \end{align} $$

Using the Poisson kernel and the submean inequality for $\log (|F_{p}|)$ (see [Reference Shiffman, Zelditch and Zrebiec70, Subsection 4.1, pp. 1991]), we can improve estimate (3.80) to get

(3.81) $$ \begin{align} \mathbb{P}\left( \int_{B(2,3)}\big|\log{|F_{p}|}\big|\,\mathrm{dV}\geq K_{2} p \right)\leq e^{-C_{U,K_{2}}\, p^{n+1}}. \end{align} $$

From this point, we proceed as in [Reference Shiffman, Zelditch and Zrebiec70, Section 4.1, p. 1992]. For $\delta \in (0,\frac {1}{2}]$ , we fix a grid in the polar coordinate system of $B(2,3)$ so that, by enlarging a bit the grid cells, we obtain an open covering $\{U_j\}_{j=1}^q$ of $B(2,3)$ consisting of small boxes of diameters $\simeq \delta ^{2n+2}$ . Then for a finite set of points $\{z_{j}\}_{j=1}^{q}$ with $z_j\in U_j$ and for all $s_{p}$ , $p\in \mathbb {N}$ , we have

(3.82) $$ \begin{align} \begin{aligned} &-\int_{B(2,3)} \log|s_{p}|_{h_{p}}\mathrm{dV}\\ &\quad \leq -\sum_{j=1}^{q} \mu_{j} \log|s_{p}|_{h_{p}}(z_{j})+K_{3}\delta\int_{B(2,3)} \big|\log{|f_{p}|}\big|\,\mathrm{dV}+p\delta K_{3} \sup_{z\in B(2,3)}|d\alpha(z)|_{g^{T^{\ast}X}}, \end{aligned} \end{align} $$

where the constant $K_{3}$ does not depend on $\delta $ , but the quantities q and $\mu _{j}>0$ only depend on $\delta $ , and we have $\sum _{j=1}^{q}\mu _{j}\simeq 1$ . Applying the above inequality to $\psi ^{S_{p}}_{\eta }$ and $F_{p}$ , we can use Corollary 3.16 for each $U_j$ : except an event of probability $\leq e^{-c_jp^{n+1}}$ , we can always choose $z_j\in U_j$ such that $\log |\psi ^{S_{p}}_{\eta }|_{h_{p}}(z_{j})> -\delta p$ . Then combining estimate (3.81) with estimate (3.82), we infer that

(3.83) $$ \begin{align} \mathbb{P}\left( -\int_{B(2,3)}\log{|\psi^{S_{p}}_{\eta}|_{h_{p}}}\,\mathrm{dV}\geq K_{4}\delta p\right)\leq e^{-C_{U,\delta}p^{n+1}},\;\forall\;p\gg 0. \end{align} $$

Noting that $\log ^{-}=-\log +\log ^{+}$ and that a finite set of annuli of the form $B(2,3)$ covers U, we can infer estimate (3.77) from estimates (3.76) and (3.83). This completes our proof.

Proof. For $U=\varnothing $ the statement is trivial, so assume U nonempty is as in the assumptions. Fix a relatively compact open neighborhood $U'$ of $\overline {U}$ and define the strictly positive quantity

(3.88) $$ \begin{align} \widetilde{M}:=\min_{z\in \overline{U'}} e^{-\frac{|z|^{2}}{2}} \in (0, 1). \end{align} $$

Let the constants K and $C'$ be the ones in estimate (3.86) for the constant $c=-2\log {\widetilde {M}}>0$ and for $U'$ . For $p\in \mathbb {N}$ , write $S^{p}_{0}\equiv p^{n/2}$ for the unit constant section in $H^{0}_{(2)}(\mathbb {C}^{n},L^{p})$ corresponding to $\alpha =(0,\ldots ,0)\in \mathbb {N}^{n}$ . Then

(3.89) $$ \begin{align} \min_{z\in \overline{U'}} |S^{p}_{0}(z)|_{h_{p}}=p^{n/2}\widetilde{M}^{p}. \end{align} $$

Defining the random holomorphic sections

(3.90) $$ \begin{align} \begin{aligned} \psi^{S_{p}}_{\eta,\mathrm{I}}(z):=\sum_{ \alpha \in I^{*}(Kp)}\eta_{\alpha} S^{p}_{\alpha}(z)\,, \quad \psi^{S_{p}}_{\eta,\mathrm{II}}(z):=\sum_{\alpha \notin I(Kp)}\eta_{\alpha} S^{p}_{\alpha}(z), \end{aligned} \end{align} $$

we can decompose

(3.91) $$ \begin{align} \psi^{S_{p}}_{\eta}=\eta_{0} S^{p}_{0} + \psi^{S_{p}}_{\eta,\mathrm{I}} + \psi^{S_{p}}_{\eta,\mathrm{II}}. \end{align} $$

Note that the three random sections on the right-hand side of equality (3.91) are independent from each other. In the rest of the proof, we will regard the above sections as being holomorphic functions on $\mathbb {C}^{n}$ , and let $|\cdot |$ denote the standard modulus on $\mathbb {C}$ (instead of considering the norm $|\cdot |_{h_{p}}$ on line bundle). Applying estimate (2.9) to the function $\psi ^{S_{p}}_{\eta ,\mathrm {II}}$ and using the estimate (3.86), we arrive at the upper bound

(3.92) $$ \begin{align} \begin{aligned} \mathbb{E}\Big[\sup_{z\in U}\Big|\psi^{S_{p}}_{\eta,\mathrm{II}}(z)\Big|^{2}\Big]&\leq C_{U'}\sigma^{2}\int_{U'}\sum_{\alpha \notin I(Kp)}|S^{p}_{\alpha}(z)|^{2}\mathrm{dV}(z)\\ &\leq \widetilde{C}_{U'}\operatorname{\mathrm{Vol}}(U')\sigma^{2} \widetilde{M}^{-2p} e^{-cp}\\ &= \widetilde{C}_{U'}\operatorname{\mathrm{Vol}}(U')\sigma^{2} =:\widetilde{C}', \end{aligned} \end{align} $$

where the last equality follows from our choice $c=-2\log {\widetilde {M}}$ . For any $\lambda>0,$ as a consequence of Chebyshev’s inequality in combination with estimate (3.92), we have

(3.93) $$ \begin{align} \mathbb{P}\Big( \sup_{z\in U}\Big|\psi^{S_{p}}_{\eta,\mathrm{II}}(z)\Big| < \lambda\Big)\geq 1-\frac{\widetilde{C}'}{\lambda^{2}}. \end{align} $$

We define the good event

(3.94) $$ \begin{align} \Omega_{p}=\left\{|\eta_{0}|\geq 1\,;\, |\eta_{\alpha}|\leq \frac{1}{3\sqrt{q_{p}-1}}\widetilde{M}^{p}, \alpha \in I^{\ast}(Kp)\,;\, \sup_{z\in U}\left|\psi^{S_{p}}_{\eta,\mathrm{II}}(z)\right| < \frac{1}{3}p^{n/2} \right\}. \end{align} $$

For all sufficiently large $p\in \mathbb {N}$ , we have

(3.95) $$ \begin{align} \begin{aligned} \mathbb{P}(\Omega_{p})&=\mathbb{P}(|\eta_{0}|\geq 1) \cdot \mathbb{P}\left(\sup_{z\in U}\left|\psi^{S_{p}}_{\eta,\mathrm{II}}(z)\right| < \frac{1}{3}p^{n/2} \right)\\ &\qquad\cdot\mathbb{P}\Bigg(|\eta_{\alpha}|\leq \frac{1}{3\sqrt{q_{p}-1}}\widetilde{M}^{p}, \alpha \in I^{\ast}(Kp)\Bigg)\\ &\geq e^{-1}\Big(1-\frac{9\widetilde{C}'}{ p^{n}}\Big)\cdot \Pi_{\alpha \in I^{\ast}(Kp)}\left(\frac{1}{18(q_{p}-1)}\widetilde{M}^{2p}\right). \end{aligned} \end{align} $$

Then by formula (3.85), there exists $C^{\prime }_{U}>0$ such that for $p\gg 1$ ,

(3.96) $$ \begin{align} \mathbb{P}(\Omega_{p})\geq e^{-C^{\prime}_{U}p^{n+1}}. \end{align} $$

Our lemma then follows once we show the inclusion

(3.97) $$ \begin{align} \Omega_{p}\subset \left\{\operatorname{\mathrm{Div}}(\psi^{S_{p}}_{\eta})\cap U=\varnothing\right\}. \end{align} $$

Indeed, if $|\eta _{\alpha }|\leq \frac {1}{3\sqrt {q_{p}-1}}\widetilde {M}^{p}$ , $\alpha \in I^{\ast }(Kp)$ , then for $z\in U$ ,

(3.98) $$ \begin{align} \begin{aligned} \big|\psi^{S_{p}}_{\eta,\mathrm{I}}(z)\big|^{2}&\leq \Big(\sum_{\alpha\in I^{\ast}(pK)}|\eta_{\alpha}|^{2}\Big)\Big(\sum_{\alpha\in I^{\ast}(pK)}|S^{p}_{\alpha}(z)|^{2}\Big)\\ &\leq \frac{1}{\widetilde{M}^{2p}}\Big(\sum_{\alpha\in I^{\ast}(pK)}|\eta_{\alpha}|^{2}\Big)P_{p}(z,z)\\ &\leq \frac{1}{9}p^{n}. \end{aligned} \end{align} $$

As a consequence, on $\Omega _{p}$ and for $z\in U$ , we get

(3.99) $$ \begin{align} \begin{aligned} \big|\psi^{S_{p}}_{\eta,\mathrm{I}}(z) + \psi^{S_{p}}_{\eta,\mathrm{II}}(z)\big|&\leq \big|\psi^{S_{p}}_{\eta,\mathrm{I}}(z)\big|+ \big|\psi^{S_{p}}_{\eta,\mathrm{II}}(z)\big| \leq \frac{1}{3}p^{n/2}+\frac{1}{3}p^{n/2}\\ &< p^{n/2}\leq \left|\eta_{0}S^{p}_{0}(z)\right|. \end{aligned} \end{align} $$

The above strict inequality implies that the inclusion (3.97) holds. This finishes the proof of the lemma.