A generalization of the conjugate Hardy spaces

Qi’an Guan; Zheng Yuan

doi:10.1017/nmj.2026.10120

A generalization of the conjugate Hardy $H^2$ spaces

Part of: Spaces and algebras of analytic functions Analytic continuation Genetics and population dynamics Holomorphic functions of several complex variables

Published online by Cambridge University Press: 03 August 2026

Qi’an Guan

and

Zheng Yuan

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Qi’an Guan: Affiliation:
School of Mathematical Sciences, Peking University , China
Zheng Yuan*: Affiliation:
State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science , Chinese Academy of Sciences, China
*: Corresponding author: Zheng Yuan; Email: yuanzheng@amss.ac.cn

Article contents

Abstract
Introduction
Preparations
Proofs of Theorems and
Proof of Theorem
Proof of Corollary
Funding statement
References

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Abstract

In this article,we consider a generalization of the conjugate Hardy $H^2$ spaces and give some properties of the minimal norm of the generalization and some relations between the norm of the generalization and the minimal $L^2$ integrals. As applications, we give some monotonicity results for the conjugate Hardy $H^2$ kernels and the Bergman kernels on planar regions, and some relations between the conjugate Hardy $H^2$ kernels and the Bergman kernels on planar regions.

Keywords

Ohsawa–Saitoh–Hardy H2 spaces conjugate Hardy H2 spaces minimal L2 integrals conjugate Hardy H2 kernels Bergman kernels

MSC classification

Primary: 30H10: Hardy spaces 32A10: Holomorphic functions

Secondary: 32D15: Continuation of analytic objects 30H20: Bergman spaces, Fock spaces

Information

Type: Article
Information: Nagoya Mathematical Journal , Volume 261 , 2026 , e40

DOI: https://doi.org/10.1017/nmj.2026.10120 [Opens in a new window]
Creative Commons: This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0), which permits unrestricted re-use, distribution and reproduction, provided the original article is properly cited.
Copyright: © The Author(s), 2026. Published by Cambridge University Press on behalf of Foundation Nagoya Mathematical Journal

1. Introduction

The Hardy space $H^2(\Delta )$ consists of the holomorphic functions f on the unit disc $\Delta \subset \mathbb {C}$ such that $\int _{0}^{2\pi }|f(re^{i\theta })|^2d\theta $ is bounded for $0<r<1$ , which was introduced into analysis by Hardy [Reference Hardy14]. Parreau [Reference Parreau16] and Rudin [Reference Rudin17] independently generalized the Hardy space to arbitrary planar domains in a conformally invariant way.

Let D be a planar region bounded by finite analytic Jordan curves. Let $z_0\in D$ , and $G_{D}(\cdot ,z_0)$ be the Green function on D. In [Reference Saitoh18], Saitoh proved that

$$ \begin{align*}\int_D|f'(z)|^2\le\frac{1}{2\pi}\int_{\partial D}|f'(z)|^2\left(\frac{\partial G_D(z,z_0)}{\partial v_z}\right)^{-1}|dz|\end{align*} $$

for any holomorphic function f on $\overline {D}$ , where $\partial /\partial v_z$ denotes the derivative along the outer normal unit vector $v_z$ . The conjugate Hardy $H^2$ space $H^{2}_{(c)}(D)$ (see [Reference Saitoh19]) is defined as the set of all holomorphic functions $f(z)$ on D such that the subharmonic functions $|f(z)|^2$ have harmonic majorants $U(z)$ :

$$ \begin{align*}|f(z)|^2\le U(z)\,\, \text{on}\,\,D.\end{align*} $$

Each function $f(z)\in H^{2}_{(c)}(D)$ has Fatou’s nontangential boundary value a.e. on $\partial D$ belonging to $L^2(\partial D)$ (see [Reference Duren2]). We recall the norm on $H^{2}_{(c)}(D)$ (see [Reference Saitoh19]):

$$ \begin{align*}\|f\|_{H^{2}_{(c)}(D)}:=\left(\frac{1}{2\pi}\int_{\partial D}|f(z)|^2\left(\frac{\partial G_D(z,z_0)}{\partial v_z}\right)^{-1}|dz|\right)^{\frac{1}{2}}\end{align*} $$

StartLayout 1st Row StartMetric f EndMetric Subscript upper H Sub Subscript left parenthesis c right parenthesis Sub Superscript 2 Subscript left parenthesis upper D right parenthesis Baseline colon equals left parenthesis StartFraction 1 Over 2 pi EndFraction integral Underscript partial differential upper D Endscripts StartAbsoluteValue f left parenthesis z right parenthesis EndAbsoluteValue squared left parenthesis StartFraction partial differential upper G Subscript upper D Baseline left parenthesis z comma z 0 right parenthesis Over partial differential v Subscript z Baseline EndFraction right parenthesis Superscript negative 1 Baseline StartAbsoluteValue d z EndAbsoluteValue right parenthesis Superscript one half EndLayout

for any $f\in H^{2}_{(c)}(D)$ .

The conjugate Hardy $H^2$ kernel (see [Reference Saitoh19]) was defined by

$$ \begin{align*} \hat{K}_{D}(z_0):=\frac{1}{\inf\left\{\|f\|^2_{H^2_{(c)}(D)}:f\in H^2_{(c)}(D)\,\&\,f(z_0)=1\right\}}. \end{align*} $$

StartLayout 1st Row ModifyingAbove upper K With caret Subscript upper D Baseline left parenthesis z 0 right parenthesis colon equals StartFraction 1 Over inf left brace StartMetric f EndMetric Subscript upper H Sub Subscript left parenthesis c right parenthesis Sub Superscript 2 Subscript left parenthesis upper D right parenthesis Superscript 2 Baseline colon f element of upper H Subscript left parenthesis c right parenthesis Superscript 2 Baseline left parenthesis upper D right parenthesis ampersand f left parenthesis z 0 right parenthesis equals 1 right brace EndFraction period EndLayout

Saitoh posed the following conjecture for the conjugate Hardy $H^2$ kernel (see [Reference Saitoh19], [Reference Yamada21]).

Conjecture 1.1. If D is not simply connected, then $\hat K_{D}(z_0)>\pi B_D(z_0)$ , where $B_D(z_0)$ is the Bergman kernel on D.

In [Reference Guan6], Guan gave an affirmative answer to the above conjecture by using a concavity property for minimal $L^2$ integrals (see [Reference Berndtsson and Lempert1], [Reference Guan5]) and a solution of the equality part of Suita conjecture (see [Reference Guan and Zhou13]). Some recent results about general versions of Saitoh’s conjecture can be referred to [Reference Guan and Yuan10]–[Reference Guan and Yuan12].

In [Reference Ohsawa15], Ohsawa gave an extension theorem involving the Hardy $H^2$ space as a limiting case for the weighted Bergman spaces on bonded pseudoconvex domains in $\mathbb {C}^n$ .

Inspired by Saitoh’s work in [Reference Saitoh19] and Ohsawa’s work in [Reference Ohsawa15], we consider a generalization of the conjugate Hardy $H^2$ spaces as a limiting case on complex manifolds in this article.

Let M be an n-dimensional complex manifold, and let $K_{M}$ be the canonical line bundle on M. Let $\psi <0$ be a plurisubharmonic function on M. Let $\rho \ge 0$ be a weight, which is a Lebesgue measurable function on M.

Definition 1.2. The generalization of the conjugate Hardy $H^2$ space is defined as the set of all $f\in H^0(M,\mathcal {O}(K_M))$ such that

$$ \begin{align*}\liminf_{t\rightarrow0+0}\frac{1}{t}\int_{\{-t<\psi<0\}}|f|^2\rho<+\infty,\end{align*} $$ limit inf Underscript t right arrow 0 plus 0 Endscripts StartFraction 1 Over t EndFraction integral Underscript left brace negative t less than psi less than 0 right brace Endscripts StartAbsoluteValue f EndAbsoluteValue squared rho less than plus infinity comma

which is called Ohsawa–Saitoh–Hardy space and denoted by $OSH^2_{\rho }(M,\psi )$ . We define a norm on the space $OSH^2_{\rho }(M,\psi )$ as follows:

$$ \begin{align*}\|f\|_{OSH^2_{\rho}(M,\psi)}:=\left(\liminf_{t\rightarrow0+0}\frac{1}{t}\int_{\{-t<\psi<0\}}|f|^2\rho\right)^{\frac{1}{2}}\end{align*} $$

for any $f\in OSH^2_{\rho }(M,\psi )$ , where $|f|^2:=\sqrt {-1}^{n^2}f\wedge \bar f$ .

If $M=D$ is a planar region bounded by finite analytic Jordan curves, $\psi =2G_{D}(\cdot ,z_0)$ and $\rho \equiv 1$ , the Ohsawa–Saitoh–Hardy space $OSH^2(M,\psi )$ is the space $\{fdz:f\in H^2_{(c)}(D)\}$ and $\|fdz\|^2_{OSH^2(M,\psi )}=2\pi \|f\|^2_{H^2_{(c)}(D)}$ for any $f\in H^2_{(c)}(D)$ (for the proof, see Proposition 2.15). Thus, the Ohsawa–Saitoh–Hardy space $OSH^2_{\rho }(M,\psi )$ is a generalization of the conjugate Hardy $H^2$ spaces.

Let X and Z be closed subsets of M, and assume that $(M,X,Z)$ satisfies the following conditions:

$I.\, X$ is a closed subset of M and X is locally negligible with respect to $L^2$ holomorphic functions, that is, for any local coordinated neighborhood $U\subset M$ and for any $L^2$ holomorphic function f on $U\backslash X$ , there exists an $L^2$ holomorphic function $\tilde {f}$ on U such that $\tilde {f}|_{U\backslash X}=f$ with the same $L^2$ norm.

$ II.\, Z$ is an analytic subset of M and $M\backslash (X\cup Z)$ is a weakly pseudoconvex Kähler manifold.

Let us give an example for the above $(M,X,Z)$ : $M:=\{(z,w)\in \mathbb {C}^2:|z|\le 1,0<|w|< 2\}\cup \{(z,w)\in \mathbb {C}^2:1<|z|<2,|w|<2\}$ , $X:=\emptyset $ and $Z:=\{(z,w)\in M:w=0\}$ . Then $M\backslash (X\cup Z)$ is a weakly pseudoconvex Kähler manifold, but M is not.

Let $\varphi $ be a Lebesgue measurable function on M, such that $\psi +\varphi $ is a plurisubharmonic function on M.

Definition 1.3. We call a positive measurable function c on $(0,+\infty )$ in class $P_{0,M}$ if the following two statements hold:

$(1)\, c(t)e^{-t}$ is decreasing with respect to t;

$(2)$ there is a closed subset E of M such that $E\subset Z\cap \{\psi (z)=-\infty \}$ and for any compact subset $K\subset M\backslash E$ , $e^{-\varphi }c(-\psi )$ has a positive lower bound on K.

It is clear that $c\equiv 1\in P_{0,M}$ when $\varphi $ is plurisubharmonic, and if $\psi $ is an analytic weight and $\{\psi (z)=-\infty \}=Z$ , then any c that satisfies condition $(1)$ belongs to $P_{0,M}$ .

Let $Z_0$ be a subset of $\{\psi =-\infty \}$ such that $Z_0 \cap Supp(\mathcal {O}/\mathcal {I}(\varphi +\psi ))\neq \emptyset $ . Let $U \supset Z_0$ be an open subset of M, and let f be a holomorphic $(n,0)$ form on U. Let $\mathcal {F}_{z_0} \supset \mathcal {I}(\varphi +\psi )_{z_0}$ be an ideal of $\mathcal {O}_{z_0}$ for any $z_{0}\in Z_0$ . Let $c\in P_{0,M}$ such that $\int _0^{+\infty }c(t)e^{-t}dt<+\infty $ .

Denote

$$ \begin{align*}\nonumber \begin{aligned} \inf\Bigg\{\int_{ \{ \psi<-t\}}|\tilde{f}|^2e^{-\varphi}c(-\psi): &\tilde{f}\in H^0(\{\psi<-t\},\mathcal{O}(K_M)) \\ &\&\, (\tilde{f}-f)\in H^0(Z_0 ,(\mathcal{O} (K_M) \otimes \mathcal{F})|_{Z_0})\Bigg\} \end{aligned} \end{align*} $$

StartLayout 1st Row StartLayout 1st Row 1st Column inf left brace integral Underscript left brace psi less than negative t right brace Endscripts StartAbsoluteValue f overtilde EndAbsoluteValue squared e Superscript negative phi Baseline c left parenthesis negative psi right parenthesis colon 2nd Column f overtilde element of upper H Superscript 0 Baseline left parenthesis left brace psi less than negative t right brace comma script upper O left parenthesis upper K Subscript upper M Baseline right parenthesis right parenthesis 2nd Row 1st Column Blank 2nd Column ampersand left parenthesis f overtilde minus f right parenthesis element of upper H Superscript 0 Baseline left parenthesis upper Z 0 comma left parenthesis script upper O left parenthesis upper K Subscript upper M Baseline right parenthesis circled times script upper F right parenthesis vertical bar Subscript upper Z 0 Baseline right parenthesis right brace EndLayout EndLayout

by $G(t)$ , where $t\in [0,+\infty )$ and $(\tilde {f}-f)\in H^0(Z_0 ,(\mathcal {O} (K_M) \otimes \mathcal {F})|_{Z_0})$ means $(\tilde {f}-f,z_0)\in (\mathcal {O}(K_M)\otimes \mathcal {F})_{z_0}$ for all $z_0\in Z_0$ .

Assume that there exists $t \in [0,+\infty )$ satisfying that $G(t)<+\infty $ . Theorem 2.5 shows that $G(h^{-1}(r))$ is concave on $[0,\int _0^{+\infty }c(t)e^{-t}]$ , then the right-hand derivative $G^{\prime }_{+}(t):=\lim _{t_1\rightarrow 0+0}\frac {G(t+t_1)-G(t)}{t_1}$ exists and is finite for any $t\ge 0$ .

Let $\rho =e^{-\varphi }c(-\psi )$ on M. Denote by $M_t:=\{\psi <-t\}$ and $\psi _t=\psi +t$ for any $t\ge 0$ . Let $F_t$ be the unique holomorphic $(n,0)$ form on $M_t$ (see Lemma 2.6) satisfying that $(F_t-f)\in H^0(Z_0 ,(\mathcal {O} (K_M) \otimes \mathcal {F})|_{Z_0})$ and $G(t)=\int _{ \{ \psi <-t\}}|F_t|^2e^{-\varphi }c(-\psi )$ . By definitions, we have $F_t\in OSH_{\rho }^2(M_t,\psi _t)$ and

(1.1) $$ \begin{align} \|F_t\|_{OSH_{\rho}^2(M_t,\psi_t)}^2=\liminf_{B\rightarrow0+0}\frac{\int_{\{-t-B<\psi<-t\}}|F_t|^2e^{-\varphi}c(-\psi)}{B}\leq-G^{\prime}_{+}(t). \end{align} $$

Denote by $c_+(t):=\lim _{s\rightarrow t+0}c(s)$ . The concavity of $G(h^{-1}(r))$ shows that $\frac {e^{t}}{c_+(t)}G^{\prime }_+(t)$ is decreasing on $[0,+\infty )$ . We give some properties for $\|F_t\|_{OSH_{\rho }^2(M_t,\psi _t)}^2$ as follows.

Theorem 1.4. The following three statements hold:

$(1)\, \frac {e^{t}}{c_+(t)}\|F_t\|_{OSH_{\rho }^2(M_t,\psi _t)}^2$ is increasing on $[0,+\infty )$ ;

$(2)\, \|F_t\|_{OSH_{\rho }^2(M_t,\psi _t)}^2=-G^{\prime }_{+}(t)$ holds for a.e. $t\in (0,+\infty )$ ;

$(3)$ when functions $G^{\prime }_+$ and c are both continuous at $t\in (0,+\infty )$ , equality

$$ \begin{align*}\|F_t\|_{OSH_{\rho}^2(M_t,\psi_t)}^2=-G^{\prime}_{+}(t)\end{align*} $$

holds.

Denote by

$$ \begin{align*}\nonumber \begin{aligned} H(t):=\inf\Bigg\{\|\tilde f\|_{OSH_{\rho}^2(M_t,\psi_t)}^2: \tilde{f}\in H^0(\{\psi<-t\},\mathcal{O}(K_M)) \\ \&\, (\tilde{f}-f)\in H^0(Z_0 ,(\mathcal{O} (K_M) \otimes \mathcal{F})|_{Z_0})\Bigg\} \end{aligned} \end{align*} $$

StartLayout 1st Row StartLayout 1st Row upper H left parenthesis t right parenthesis colon equals inf left brace StartMetric f overtilde EndMetric Subscript upper O upper S upper H Sub Subscript rho Sub Superscript 2 Subscript left parenthesis upper M Sub Subscript t Subscript comma psi Sub Subscript t Subscript right parenthesis Superscript 2 Baseline colon f overtilde element of upper H Superscript 0 Baseline left parenthesis left brace psi less than negative t right brace comma script upper O left parenthesis upper K Subscript upper M Baseline right parenthesis right parenthesis 2nd Row ampersand left parenthesis f overtilde minus f right parenthesis element of upper H Superscript 0 Baseline left parenthesis upper Z 0 comma left parenthesis script upper O left parenthesis upper K Subscript upper M Baseline right parenthesis circled times script upper F right parenthesis vertical bar Subscript upper Z 0 Baseline right parenthesis right brace EndLayout EndLayout

for any $t\ge 0$ .

Following from inequality (1.1), we have

(1.2) $$ \begin{align} H(t)\le -G^{\prime}_{+}(t) \end{align} $$

for any $t\ge 0$ . We give a necessary and sufficient condition for $H(t)=-G^{\prime }_{+}(t)$ for a.e. $t\in (0,+\infty )$ .

Theorem 1.5. The following two statements are equivalent:

$(1)\ H(t)=-G^{\prime }_{+}(t)$ holds for a.e. $t\in (0,+\infty )$ ;

$(2)$ there exists $F\in H^0(M,\mathcal {O}(K_M))$ such that $(F-f)\in H^0(Z_0 ,(\mathcal {O} (K_M) \otimes \mathcal {F})|_{Z_0})$ , $G(t)=\int _{\{\psi <-t\}}|F|^2e^{-\varphi }c(-\psi )$ for any $t\in [0,+\infty ),$ and $H(t)=\|F\|_{OSH_{\rho }^2(M_t,\psi _t)}^2$ for a.e. $t\in (0,+\infty )$ .

Denote by

$$ \begin{align*} \begin{aligned} \mathcal{H}^2(M_t,\rho):=\Bigg\{\tilde f:\tilde{f}\in H^0(\{\psi<-t\},\mathcal{O}(K_M)),\,\int_{\{\psi<-t\}}|\tilde f|^2e^{-\varphi}c(-\psi)<+\infty\Bigg\} \end{aligned} \end{align*} $$

for $t\ge 0$ .

Assume that $H(t)$ is Lebesgue measurable on $(0,+\infty )$ , then it is clear that $\int _0^{+\infty }H(t)dt\le G(0)$ . Denote by $\hat H(t):=\int _t^{+\infty }H(s)ds$ . We give a concavity property for $\hat H(t)$ in the following theorem.

Theorem 1.6. Assume that $OSH_{\rho }^2(M_t,\psi _t)\subset \mathcal {H}^2(M_t,\rho )$ for a.e. $t_0\in (0,+\infty )$ . Then $\hat H(h^{-1}(r))$ is concave with respect to $r\in [0,\int _0^{+\infty }c(t)e^{-t}dt]$ , where $h(t):=\int _t^{+\infty }c(s)e^{-s}ds$ for $t\ge 0$ .

1.1. Applications: On planar regions

In this section, we consider the case that $M=D$ is a planar region bounded by finite analytic Jordan curves.

Let $z_0\in D$ , and $G_{D}(\cdot ,z_0)$ be the Green function on D. Denote the set of all critical values of $G_{D}(\cdot ,z_0)$ by $N\subset (-\infty ,0)$ . $N\subset \subset (-\infty ,0)$ is a discrete set (see Lemma 2.14). Denote by

$$ \begin{align*}D_t:=\{z\in D:2G_D(z,z_0)<-t\}\end{align*} $$

for any $t\ge 0$ . For any $t\in [0,+\infty )\backslash -2N$ , $D_t$ is a planar region bounded by finite analytic Jordan curves, thus the conjugate Hardy $H^2$ space $H^{2}_{(c)}(D_t)$ is well defined.

We recall the conjugate Hardy $H^2$ kernel (see [Reference Saitoh19])

(1.3) $$ \begin{align} \hat K_{D_t}(z_0):=\frac{1}{\inf\left\{\|f\|^2_{H^2_{(c)}(D_t)}:f\in H^2_{(c)}(D_t)\,\&\,f(z_0)=1\right\}},\end{align} $$

StartLayout 1st Row ModifyingAbove upper K With caret Subscript upper D Sub Subscript t Subscript Baseline left parenthesis z 0 right parenthesis colon equals StartFraction 1 Over inf left brace StartMetric f EndMetric Subscript upper H Sub Subscript left parenthesis c right parenthesis Sub Superscript 2 Subscript left parenthesis upper D Sub Subscript t Subscript right parenthesis Superscript 2 Baseline colon f element of upper H Subscript left parenthesis c right parenthesis Superscript 2 Baseline left parenthesis upper D Subscript t Baseline right parenthesis ampersand f left parenthesis z 0 right parenthesis equals 1 right brace EndFraction comma EndLayout

where $t\in [0,+\infty )\backslash -2N$ , $\|f\|^2_{H^2_{(c)}(D_t)}:=\frac {1}{2\pi }\int _{\partial D_t}|f(z)|^2\left (\frac {\partial G_D(z,z_0)}{\partial v_z}\right )^{-1}|dz|$ and $\partial /\partial v_z$ denotes the derivative along the outer normal unit vector $v_z$ . Following from Proposition 2.15, for $t\in -2N$ , we define that

$$ \begin{align*}H^2_{(c)}(D_t):=\left\{f\in\mathcal{O}(D_t):\|f\|^2_{H^2_{(c)}(D_t)}:=\liminf_{B\rightarrow0+0}\frac{\int_{\{-t-B<2G_D(\cdot,z_0)<-t\}}|f|^2d\lambda_D}{B\pi}<+\infty\right\},\end{align*} $$

StartLayout 1st Row upper H Subscript left parenthesis c right parenthesis Superscript 2 Baseline left parenthesis upper D Subscript t Baseline right parenthesis colon equals StartSet f element of script upper O left parenthesis upper D Subscript t Baseline right parenthesis colon StartMetric f EndMetric Subscript upper H Sub Subscript left parenthesis c right parenthesis Sub Superscript 2 Subscript left parenthesis upper D Sub Subscript t Subscript right parenthesis Superscript 2 Baseline colon equals limit inf Underscript upper B right arrow 0 plus 0 Endscripts StartFraction integral Underscript left brace negative t minus upper B less than 2 upper G Subscript upper D Baseline left parenthesis dot comma z 0 right parenthesis less than negative t right brace Endscripts StartAbsoluteValue f EndAbsoluteValue squared d lamda Subscript upper D Baseline Over upper B pi EndFraction less than plus infinity EndSet comma EndLayout

and $\hat K_{D_t}(z_0)$ is defined as equality (1.3), where $d\lambda _D$ is the Lebesgue measure on D.

Denote that $B_{D_t}(\cdot ,\cdot )$ is the Bergman kernel on $D_t$ . Then we have

$$ \begin{align*}\frac{2}{B_{D_t}(z_0,z_0)}=\inf\left\{\int_{D_t}|\tilde f|^2:\tilde f\in H^0(D_t,\mathcal{O}(K_D))\,\&\,\tilde f(z_0)=dz\right\}.\end{align*} $$

Note that $\frac {B_{D_t}(\cdot ,z_0)}{B_{D_t}(z_0,z_0)}$ is a holomorphic function on $D_t$ satisfying

$$ \begin{align*}\int_{D_t}\left|\frac{B_{D_t}(\cdot,z_0)}{B_{D_t}(z_0,z_0)}\right|{}^2=\inf\left\{\int_{D_t}|\tilde f|^2d\lambda_D:\tilde f\in \mathcal{O}(D_t)\,\&\,\tilde f(z_0)=1\right\},\end{align*} $$

then Theorem 1.4 implies the following result.

Corollary 1.7. The following two statements hold:

$(1)\, \frac {\|B_{D_t}(\cdot ,z_0)\|_{H^2_{(c)}(D_t)}^2}{|B_{D_t}(z_0,z_0)|^2}e^t$ is an increasing function on $[0,+\infty )$ ;

$(2)$ equality

$$ \begin{align*}\pi\|B_{D_t}(\cdot,z_0)\|_{H^2_{(c)}(D_t)}^2=\frac{d}{dt}B_{D_t}(z_0,z_0)\end{align*} $$

holds a.e. on $(0,+\infty )$ .

Inequality (1.2) implies that $\frac {\pi }{\hat K_{D_t}(z_0)}\le -\frac {d}{dt}\frac {1}{B_{D_t}(z_0,z_0)}.$ The following corollary gives a monotonicity of $\hat K_{D_t}(z_0)e^{-t}$ and a characterization for $\frac {\pi }{\hat K_{D_t}(z_0)}=-\frac {d}{dt}\frac {1}{B_{D_t}(z_0,z_0)}$ a.e. on $(0,+\infty )$ .

Corollary 1.8. $\hat K_{D_t}(z_0)e^{-t}$ is a decreasing function on $[0,+\infty )$ . Moreover, the following three statements are equivalent:

$(1)\, \frac {\pi }{\hat K_{D_t}(z_0)}=-\frac {d}{dt}\frac {1}{B_{D_t}(z_0,z_0)}$ a.e. on $(0,+\infty )$ ;

$(2)\, \hat K_{D_t}(z_0)e^{-t}$ is a constant function on $[0,+\infty )$ ;

$(3)\, D$ is simply connected.

The following remark shows that Corollary 1.8 can deduce the solution of Conjecture 1.1 (Saitoh’s conjecture for conjugate Hardy $H^2$ kernels).

Remark 1.9. As $\hat K_{D_t}(z_0)e^{-t}$ is a decreasing function on $[0,+\infty )$ and inequality $\frac {\pi }{\hat K_{D_t}(z_0)}\le -\frac {d}{dt}\frac {1}{B_{D_t}(z_0,z_0)}$ holds for any $t\ge 0$ , we have

$$ \begin{align*} \nonumber \begin{aligned} \frac{\pi}{\hat K_{D}(z_0)}=&\int_{0}^{{+\infty}}\frac{\pi e^{-t}}{\hat K_{D}(z_0)}dt \le\int_{0}^{{+\infty}}\frac{\pi}{\hat K_{D_t}(z_0)}dt\\ \le& \int_0^{+\infty}-\frac{d}{dt}\frac{1}{B_{D_t}(z_0,z_0)}dt\le \frac{1}{B_{D}(z_0,z_0)}, \end{aligned} \end{align*} $$

StartLayout 1st Row StartLayout 1st Row 1st Column StartFraction pi Over ModifyingAbove upper K With caret Subscript upper D Baseline left parenthesis z 0 right parenthesis EndFraction equals 2nd Column integral Subscript 0 Superscript plus infinity Baseline StartFraction pi e Superscript negative t Baseline Over ModifyingAbove upper K With caret Subscript upper D Baseline left parenthesis z 0 right parenthesis EndFraction d t less than or equals integral Subscript 0 Superscript plus infinity Baseline StartFraction pi Over ModifyingAbove upper K With caret Subscript upper D Sub Subscript t Subscript Baseline left parenthesis z 0 right parenthesis EndFraction d t 2nd Row 1st Column less than or equals 2nd Column integral Subscript 0 Superscript plus infinity Baseline minus StartFraction d Over d t EndFraction StartFraction 1 Over upper B Subscript upper D Sub Subscript t Subscript Baseline left parenthesis z 0 comma z 0 right parenthesis EndFraction d t less than or equals StartFraction 1 Over upper B Subscript upper D Baseline left parenthesis z 0 comma z 0 right parenthesis EndFraction comma EndLayout EndLayout

which shows that $\hat K_{D}(z_0)\ge \pi B_{D}(z_0,z_0)$ . If the equality $\hat K_{D}(z_0)=\pi B_{D}(z_0,z_0)$ holds, then $\hat K_{D_t}(z_0)e^{-t}$ is a constant function on $[0,+\infty )$ . Thus, using the characterization in Corollary 1.8, Conjecture 1.1 has been proved.

2. Preparations

In this section, we do some preparations.

2.1. Minimal $L^2$ integrals and Ohsawa–Saitoh–Hardy space

In this section, we recall and give some results on minimal $L^2$ integrals $G(t)$ and Ohsawa–Saitoh–Hardy space $OSH_{\rho }^2(M,\psi )$ .

We firstly introduce a property of coherent analytic sheaves and a convergence property of holomorphic $(n,0)$ forms.

Lemma 2.1 (See [Reference Grauert and Remmert4]).

Let N be a submodule of $\mathcal O_{\mathbb C^n,o}^q$ , $1\leq q\leq \infty $ , and let $f_j\in \mathcal O_{\mathbb C^n}(U)^q$ be a sequence of q-tuples holomorphic in an open neighborhood U of the origin o. Assume that the $f_j$ converge uniformly in U toward a q-tuples $f\in \mathcal O_{\mathbb C^n}(U)^q$ , assume furthermore that all germs $(f_{j},o)$ belong to N. Then $(f,o)\in N$ .

Lemma 2.2 (See [Reference Guan and Yuan9]).

Let M be a complex manifold. Let S be an analytic subset of M. Let $\{g_j\}_{j=1,2,\ldots }$ be a sequence of nonnegative Lebesgue measurable functions on M, which satisfies that $g_j$ are almost everywhere convergent to g on M when $j\rightarrow +\infty $ , where g is a nonnegative Lebesgue measurable function on M. Assume that for any compact subset K of $M\backslash S$ , there exist $s_K\in (0,+\infty )$ and $C_K\in (0,+\infty )$ such that

$$ \begin{align*}\int_{K}{g_j}^{-s_K}dV_M\leq C_K\end{align*} $$

for any j, where $dV_M$ is a continuous volume form on M.

Let $\{F_j\}_{j=1,2,\ldots }$ be a sequence of holomorphic $(n,0)$ forms on M. Assume that $\liminf _{j\rightarrow +\infty }\int _{M}|F_j|^2g_j\leq C$ , where C is a positive constant. Then there exists a subsequence $\{F_{j_l}\}_{l=1,2,\ldots }$ , which satisfies that $\{F_{j_l}\}$ is uniformly convergent to a holomorphic $(n,0)$ form F on M on any compact subset of M when $l\rightarrow +\infty $ , such that

$$ \begin{align*}\int_{M}|F|^2g\leq C.\end{align*} $$

In the following, we follow the notations and assumptions in Theorem 1.4.

Lemma 2.3 (See [Reference Guan, Mi and Yuan7], [Reference Guan, Mi and Yuan8]).

Let $B \in (0, +\infty )$ and $t_0>t_1 > 0$ be arbitrarily given. Let F be a holomorphic $(n,0)$ form on $\{\psi < -t_0\}$ such that

$$ \begin{align*} \int_{\{\psi<-t_0\}} {|F|}^2e^{-\varphi}c(-\psi)<+\infty. \nonumber \end{align*} $$ integral Underscript left brace psi less than minus t 0 right brace Endscripts StartAbsoluteValue upper F EndAbsoluteValue squared e Superscript negative phi Baseline c left parenthesis negative psi right parenthesis less than plus infinity period

Then there exists a holomorphic $(n,0)$ form $\tilde {F}$ on $\{\psi <-t_1\}$ such that

$$ \begin{align*}\nonumber \int_{\{\psi<-t_1\}}|\tilde{F}-(1-b_{t_0,B}(\psi))F|^2e^{-\varphi-\psi+v_{t_0,B}(\psi)}c(-v_{t_0,B}(\psi)) \le C\int_{t_1}^{t_0+B}c(t)e^{-t}dt, \end{align*} $$

where $C=\int _M \frac {1}{B} \mathbb {I}_{\{-t_0-B< \psi < -t_0\}} {|F|}^2 e^{{-}\varphi -\psi }$ , $b_{t_0,B}(t)=\int ^{t}_{-\infty }\frac {1}{B} \mathbb {I}_{\{-t_0-B< s < -t_0\}}ds,$ and $v_{t_0,B}(t)=\int ^{t}_{-t_0}b_{t_0,B}(s)ds-t_0$ .

The following lemma gives a characterization for the minimal $L^2$ integral $G(t)$ being equal to $0$ .

Lemma 2.4 (See [Reference Guan, Mi and Yuan7], [Reference Guan, Mi and Yuan8]).

$f\in H^0(Z_0,(\mathcal {O}(K_{M})\otimes \mathcal {F})|_{Z_0})\Leftrightarrow G(t)=0$ .

We recall a concavity property for the minimal $L^2$ integral $G(t)$ .

Theorem 2.5 (See [Reference Guan, Mi and Yuan7], [Reference Guan, Mi and Yuan8]).

$G(h^{-1}(r))$ is concave with respect to $r\in (0,\int _{0}^{+\infty }c(t)e^{-t}dt)$ , $\lim \limits _{t\to 0+0}G(t)=G(0),$ and $\lim \limits _{t \to +\infty }G(t)=0$ .

The following lemma shows the existence and uniqueness of the minimal holomorphic form $F_t$ .

Lemma 2.6 (See [Reference Guan, Mi and Yuan7], [Reference Guan, Mi and Yuan8]).

Assume that $G(t)<+\infty $ for some $t\in [T,+\infty )$ . Then there exists a unique holomorphic $(n,0)$ form $F_{t}$ on $\{\psi <-t\}$ satisfying $(F_{t}-f)\in H^{0}(Z_0,(\mathcal {O}(K_{M})\otimes \mathcal {F})|_{Z_0})$ and $\int _{\{\psi <-t\}}|F_{t}|^{2}e^{-\varphi }c(-\psi )=G(t)$ . Furthermore, for any holomorphic $(n,0)$ form $\hat {F}$ on $\{\psi <-t\}$ satisfying $(\hat {F}-f)\in H^{0}(Z_0,(\mathcal {O}(K_{M})\otimes \mathcal {F})|_{Z_0})$ and $\int _{\{\psi <-t\}}|\hat {F}|^{2}e^{-\varphi }c(-\psi )<+\infty $ , we have the following equality:

$$ \begin{align*}\int_{\{\psi<-t\}}|F_{t}|^{2}e^{-\varphi}c(-\psi)+\int_{\{\psi<-t\}}|\hat{F}-F_{t}|^{2}e^{-\varphi}c(-\psi) = \int_{\{\psi<-t\}}|\hat{F}|^{2}e^{-\varphi}c(-\psi). \end{align*} $$

Let $F_t$ be the unique holomorphic $(n,0)$ form on $M_t$ satisfying that $(F_t-f)\in H^0(Z_0 ,(\mathcal {O} (K_M) \otimes \mathcal {F})|_{Z_0})$ and $G(t)=\int _{ \{ \psi <-t\}}|F_t|^2e^{-\varphi }c(-\psi )$ . By definitions, we have $F_t\in OSH_{\rho }^2(M_t,\psi _t)$ and

(2.1) $$ \begin{align} \|F_t\|_{OSH_{\rho}^2(M_t,\psi_t)}^2=\liminf_{B\rightarrow0+0}\frac{\int_{\{-t-B<\psi<-t\}}|F_t|^2e^{-\varphi}c(-\psi)}{B}\leq-G^{\prime}_{+}(t). \end{align} $$

Following the concavity of $G(h^{-1}(r))$ , we have

(2.2) $$ \begin{align} G(t_1)\le G(t_0)+\frac{\int_{t_1}^{t_0}c(s)e^{-s}ds}{c_+(t_0)e^{-t_0}}(-G^{\prime}_{+}(t_0)) \end{align} $$

for any $0\le t_1<t_0<+\infty $ . The following proposition shows that inequality (2.2) also holds when replacing $-G^{\prime }_{+}(t)$ by $\|F_t\|_{OSH_{\rho }^2(M_t,\psi _t)}^2$ .

Proposition 2.7. For any $t_0>t_1\ge 0$ , inequality

$$ \begin{align*} \nonumber G(t_1)\le G(t_0)+\frac{\int_{t_1}^{t_0}c(s)e^{-s}ds}{c_+(t_0)e^{-t_0}}\|F_{t_0}\|_{OSH_{\rho}^2(M_{t_0},\psi_t)}^2 \end{align*} $$

holds, where $c_+(t)=\lim _{s\rightarrow t+0}c(s)$ .

Proof. Lemma 2.3 shows that for any $B>0$ , there exists a holomorphic $(n,0)$ form $\tilde {F}_{B}$ on $\{\psi <-t_1\}$ , such that $(\tilde {F}_{B}-F_{t_{0}})\in H^{0}(Z_0,(\mathcal {O}(K_M)\otimes \mathcal {I}(\varphi +\psi ))|_{Z_0}) \subseteq H^{0}(Z_0,(\mathcal {O}(K_{M})\otimes \mathcal {F})|_{Z_0})$ and

(2.3) $$ \begin{align} \begin{aligned} &\int_{\{\psi<-t_1\}}|\tilde{F}_{B}-(1-b_{t_{0},B}(\psi))F_{t_{0}}|^{2}e^{-\varphi-\psi+v_{t_0,B}(\psi)}c(-v_{t_0,B}(\psi)) \\\leq& \int_{t_1}^{t_{0}+B}c(t)e^{-t}dt \int_{\{\psi<-t_1\}}\frac{1}{B}(\mathbb{I}_{\{-t_{0}-B<\psi<-t_{0}\}})|F_{t_{0}}|^{2}e^{-\varphi-\psi} \\\leq& \frac{e^{t_{0}+B}\int_{t_1}^{t_{0}+B}c(t)e^{-t}dt}{\inf_{t\in(t_{0},t_{0}+B)}c(t)} \int_{\{\psi<-t_1\}}\frac{1}{B}(\mathbb{I}_{\{-t_{0}-B<\psi<-t_{0}\}})|F_{t_{0}}|^{2}e^{-\varphi}c(-\psi). \end{aligned} \end{align} $$

StartLayout 1st Row StartLayout 1st Row 1st Column Blank 2nd Column integral Underscript left brace psi less than minus t 1 right brace Endscripts StartAbsoluteValue upper F overtilde Subscript upper B Baseline minus left parenthesis 1 minus b Subscript t 0 comma upper B Baseline left parenthesis psi right parenthesis right parenthesis upper F Subscript t 0 Baseline EndAbsoluteValue squared e Superscript negative phi minus psi plus v Super Subscript t 0 comma upper B Superscript left parenthesis psi right parenthesis Baseline c left parenthesis minus v Subscript t 0 comma upper B Baseline left parenthesis psi right parenthesis right parenthesis 2nd Row 1st Column less than or equals 2nd Column integral Subscript t 1 Superscript t 0 plus upper B Baseline c left parenthesis t right parenthesis e Superscript negative t d t integral Underscript left brace psi less than minus t 1 right brace Endscripts StartFraction 1 Over upper B EndFraction left parenthesis double struck upper I Subscript left brace minus t 0 minus upper B less than psi less than minus t 0 right brace Baseline right parenthesis StartAbsoluteValue upper F Subscript t 0 Baseline EndAbsoluteValue squared e Superscript negative phi minus psi 3rd Row 1st Column less than or equals 2nd Column StartFraction e Superscript t 0 plus upper B Baseline integral Subscript t 1 Superscript t 0 plus upper B Baseline c left parenthesis t right parenthesis e Superscript negative t Baseline d t Over inf Underscript t element of left parenthesis t 0 comma t 0 plus upper B right parenthesis Endscripts c left parenthesis t right parenthesis EndFraction integral Underscript left brace psi less than minus t 1 right brace Endscripts StartFraction 1 Over upper B EndFraction left parenthesis double struck upper I Subscript left brace minus t 0 minus upper B less than psi less than minus t 0 right brace Baseline right parenthesis StartAbsoluteValue upper F Subscript t 0 Baseline EndAbsoluteValue squared e Superscript negative phi Baseline c left parenthesis negative psi right parenthesis period EndLayout EndLayout

As $t\leq v_{t_0,B}(t)$ , the decreasing property of $c(t)e^{-t}$ shows that

$$ \begin{align*}e^{-\psi+v_{t_0,B}(\psi)}c(-v_{t_0,B}(\psi))\geq c(-\psi).\end{align*} $$

Inequalities (2.3) and (2.1) imply that

(2.4) $$ \begin{align} \begin{aligned} &\liminf_{B\rightarrow0+0}\int_{\{\psi<-t_1\}}|\tilde{F}_{B}-(1-b_{t_{0},B}(\psi))F_{t_{0}}|^{2}e^{-\varphi}c(-\psi)\\ &\quad \le \liminf_{B\rightarrow0+0}\int_{\{\psi<-t_1\}}|\tilde{F}_{B}-(1-b_{t_{0},B}(\psi))F_{t_{0}}|^{2}e^{-\varphi-\psi+v_{t_0,B}(\psi)}c(-v_{t_0,B}(\psi))\\ &\quad \le \frac{\int_{t_1}^{t_0}c(s)e^{-s}ds}{c_+(t_0)e^{-t_0}}\|F_{t_0}\|_{OSH_{\rho}^2(M_{t_0},\psi_t)}^2\\ &\quad <+\infty. \end{aligned} \end{align} $$

StartLayout 1st Row StartLayout 1st Row 1st Column Blank 2nd Column limit inf Underscript upper B right arrow 0 plus 0 Endscripts integral Underscript left brace psi less than minus t 1 right brace Endscripts StartAbsoluteValue upper F overtilde Subscript upper B Baseline minus left parenthesis 1 minus b Subscript t 0 comma upper B Baseline left parenthesis psi right parenthesis right parenthesis upper F Subscript t 0 Baseline EndAbsoluteValue squared e Superscript negative phi Baseline c left parenthesis negative psi right parenthesis 2nd Row 1st Column Blank 2nd Column less than or equals limit inf Underscript upper B right arrow 0 plus 0 Endscripts integral Underscript left brace psi less than minus t 1 right brace Endscripts StartAbsoluteValue upper F overtilde Subscript upper B Baseline minus left parenthesis 1 minus b Subscript t 0 comma upper B Baseline left parenthesis psi right parenthesis right parenthesis upper F Subscript t 0 Baseline EndAbsoluteValue squared e Superscript negative phi minus psi plus v Super Subscript t 0 comma upper B Superscript left parenthesis psi right parenthesis Baseline c left parenthesis minus v Subscript t 0 comma upper B Baseline left parenthesis psi right parenthesis right parenthesis 3rd Row 1st Column Blank 2nd Column less than or equals StartFraction integral Subscript t 1 Superscript t 0 Baseline c left parenthesis s right parenthesis e Superscript negative s Baseline d s Over c Subscript plus Baseline left parenthesis t 0 right parenthesis e Superscript minus t 0 Baseline EndFraction StartMetric upper F Subscript t 0 Baseline EndMetric Subscript upper O upper S upper H Sub Subscript rho Sub Superscript 2 Subscript left parenthesis upper M Sub Subscript t 0 Subscript comma psi Sub Subscript t Subscript right parenthesis Superscript 2 Baseline 4th Row 1st Column Blank 2nd Column less than plus infinity period EndLayout EndLayout

Note that

$$ \begin{align*} \nonumber \begin{aligned} &\left(\int_{\{\psi<-t_1\}}|\tilde{F}_{B}-(1-b_{t_{0},B}(\psi))F_{t_{0}}|^{2}e^{-\varphi}c(-\psi)\right)^{\frac{1}{2}} \\ \geq\ &\left(\int_{\{\psi<-t_1\}}|\tilde{F}_{B}|^{2}e^{-\varphi}c(-\psi)\right)^{\frac{1}{2}}-\left(\int_{\{\psi<-t_1\}}|(1-b_{t_{0},B}(\psi))F_{t_{0}}|^{2}e^{-\varphi}c(-\psi)\right)^{\frac{1}{2}}\\ \ge\ &\left(\int_{\{\psi<-t_1\}}|\tilde{F}_{B}|^{2}e^{-\varphi}c(-\psi)\right)^{\frac{1}{2}}-\left(\int_{\{\psi<-t_0\}}|F_{t_{0}}|^{2}e^{-\varphi}c(-\psi)\right)^{\frac{1}{2}}. \end{aligned} \end{align*} $$

StartLayout 1st Row StartLayout 1st Row 1st Column Blank 2nd Column left parenthesis integral Underscript left brace psi less than minus t 1 right brace Endscripts StartAbsoluteValue upper F overtilde Subscript upper B Baseline minus left parenthesis 1 minus b Subscript t 0 comma upper B Baseline left parenthesis psi right parenthesis right parenthesis upper F Subscript t 0 Baseline EndAbsoluteValue squared e Superscript negative phi Baseline c left parenthesis negative psi right parenthesis right parenthesis Superscript one half 2nd Row 1st Column greater than or equals 2nd Column left parenthesis integral Underscript left brace psi less than minus t 1 right brace Endscripts StartAbsoluteValue upper F overtilde Subscript upper B Baseline EndAbsoluteValue squared e Superscript negative phi Baseline c left parenthesis negative psi right parenthesis right parenthesis Superscript one half Baseline minus left parenthesis integral Underscript left brace psi less than minus t 1 right brace Endscripts StartAbsoluteValue left parenthesis 1 minus b Subscript t 0 comma upper B Baseline left parenthesis psi right parenthesis right parenthesis upper F Subscript t 0 Baseline EndAbsoluteValue squared e Superscript negative phi Baseline c left parenthesis negative psi right parenthesis right parenthesis Superscript one half 3rd Row 1st Column greater than or equals 2nd Column left parenthesis integral Underscript left brace psi less than minus t 1 right brace Endscripts StartAbsoluteValue upper F overtilde Subscript upper B Baseline EndAbsoluteValue squared e Superscript negative phi Baseline c left parenthesis negative psi right parenthesis right parenthesis Superscript one half Baseline minus left parenthesis integral Underscript left brace psi less than minus t 0 right brace Endscripts StartAbsoluteValue upper F Subscript t 0 Baseline EndAbsoluteValue squared e Superscript negative phi Baseline c left parenthesis negative psi right parenthesis right parenthesis Superscript one half Baseline period EndLayout EndLayout

Then it follows from inequality (2.4) that

$$ \begin{align*} \nonumber \begin{aligned} &\liminf_{B\rightarrow0+0}\left(\int_{\{\psi<-t_1\}}|\tilde{F}_{B}|^{2}e^{-\varphi}c(-\psi)\right)^{\frac{1}{2}} \\ &\quad \leq\frac{\int_{t_1}^{t_0}c(s)e^{-s}ds}{c_+(t_0)e^{-t_0}}\|F_{t_0}\|_{OSH_{\rho}^2(M_{t_0},\psi_t)}^2+\left(\int_{\{\psi<-t_0\}}|F_{t_{0}}|^{2}e^{-\varphi}c(-\psi)\right)^{\frac{1}{2}}. \end{aligned} \end{align*} $$

StartLayout 1st Row StartLayout 1st Row 1st Column Blank 2nd Column limit inf Underscript upper B right arrow 0 plus 0 Endscripts left parenthesis integral Underscript left brace psi less than minus t 1 right brace Endscripts StartAbsoluteValue upper F overtilde Subscript upper B Baseline EndAbsoluteValue squared e Superscript negative phi Baseline c left parenthesis negative psi right parenthesis right parenthesis Superscript one half 2nd Row 1st Column Blank 2nd Column less than or equals StartFraction integral Subscript t 1 Superscript t 0 Baseline c left parenthesis s right parenthesis e Superscript negative s Baseline d s Over c Subscript plus Baseline left parenthesis t 0 right parenthesis e Superscript minus t 0 Baseline EndFraction StartMetric upper F Subscript t 0 Baseline EndMetric Subscript upper O upper S upper H Sub Subscript rho Sub Superscript 2 Subscript left parenthesis upper M Sub Subscript t 0 Subscript comma psi Sub Subscript t Subscript right parenthesis Superscript 2 Baseline plus left parenthesis integral Underscript left brace psi less than minus t 0 right brace Endscripts StartAbsoluteValue upper F Subscript t 0 Baseline EndAbsoluteValue squared e Superscript negative phi Baseline c left parenthesis negative psi right parenthesis right parenthesis Superscript one half Baseline period EndLayout EndLayout

As $c\in P_{0,M}$ , it follows from Lemma 2.2 that there exists a subsequence of $\{\tilde {F}_B\}$ , denoted by $\{\tilde {F}_{B_k}\}_{k\in \mathbb {N}^+}$ ( $B_k\rightarrow 0$ when $k\rightarrow +\infty $ ), which is uniformly convergent to a holomorphic $(n,0)$ form $F_1$ on $\{\psi <-t_1\}$ on any compact subset of $\{\psi <-t_1\}$ when $k\rightarrow +\infty $ , such that

$$ \begin{align*}\int_{\{\psi<-t_1\}}|F_1|^2e^{-\varphi}c(-\psi)\le\liminf_{B\rightarrow0+0}\int_{\{\psi<-t_1\}}|\tilde{F}_{B}|^{2}e^{-\varphi}c(-\psi)<+\infty.\end{align*} $$

As $(\tilde {F}_B-F_{t_{0}})\in H^{0}(Z_0,(\mathcal {O}(K_{M})\otimes \mathcal {F})|_{Z_0})$ for any $B>0$ , we have $(F_1-F_{t_{0}})\in H^{0}(Z_0,(\mathcal {O}(K_{M})\otimes \mathcal {F})|_{Z_0})$ by Lemma 2.1. Note that

$$ \begin{align*} \lim_{k\rightarrow+\infty}b_{t_0,B_k}(t)=\lim_{k\rightarrow+\infty}\int_{-\infty}^{t}\frac{1}{B_k}\mathbb{I}_{\{-t_0-B_k<s<-t_0\}}ds=\left\{ \begin{array}{lcl} 0 & \text{if}& x\in(-\infty,-t_0)\\ 1 & \text{if}& x\in[-t_0,+\infty) \end{array} \right. \end{align*} $$

and

$$ \begin{align*} \lim_{j\rightarrow+\infty}v_{t_0,B_k}(t)=\lim_{j\rightarrow+\infty}\int_{-t_0}^{t}b_{t_0,B_k}ds-t_0=\left\{ \begin{array}{lcl} -t_0 & \text{if}& x\in(-\infty,-t_0)\\ t & \text{if}& x\in[-t_0,+\infty) \end{array}. \right. \end{align*} $$

Following from inequality (2.4) and Fatou’s lemma, we have

$$ \begin{align*} \nonumber \begin{aligned} &\int_{\{\psi<-t_0\}}|F_1-F_{t_0}|^2e^{-\varphi}c(-\psi)+\int_{\{-t_0\le\psi<-t_1\}}|F_1|^2e^{-\varphi}c(-\psi)\\ &\quad =\int_{\{\psi<-t_1\}}\lim_{k\rightarrow+\infty}|\tilde{F}_{B}-(1-b_{t_{0},B}(\psi))F_{t_{0}}|^{2}e^{-\varphi}c(-\psi) \\ &\quad \le\liminf_{B\rightarrow0+0}\int_{\{\psi<-t_1\}}|\tilde{F}_{B}-(1-b_{t_{0},B}(\psi))F_{t_{0}}|^{2}e^{-\varphi}c(-\psi)\\ &\quad \le \frac{\int_{t_1}^{t_0}c(s)e^{-s}ds}{c_+(t_0)e^{-t_0}}\|F_{t_0}\|_{OSH_{\rho}^2(M_{t_0},\psi_t)}^2. \end{aligned} \end{align*} $$

StartLayout 1st Row StartLayout 1st Row 1st Column Blank 2nd Column integral Underscript left brace psi less than minus t 0 right brace Endscripts StartAbsoluteValue upper F 1 minus upper F Subscript t 0 Baseline EndAbsoluteValue squared e Superscript negative phi Baseline c left parenthesis negative psi right parenthesis plus integral Underscript left brace minus t 0 less than or equals psi less than minus t 1 right brace Endscripts StartAbsoluteValue upper F 1 EndAbsoluteValue squared e Superscript negative phi Baseline c left parenthesis negative psi right parenthesis 2nd Row 1st Column Blank 2nd Column equals integral Underscript left brace psi less than minus t 1 right brace Endscripts limit Underscript k right arrow plus infinity Endscripts StartAbsoluteValue upper F overtilde Subscript upper B Baseline minus left parenthesis 1 minus b Subscript t 0 comma upper B Baseline left parenthesis psi right parenthesis right parenthesis upper F Subscript t 0 Baseline EndAbsoluteValue squared e Superscript negative phi Baseline c left parenthesis negative psi right parenthesis 3rd Row 1st Column Blank 2nd Column less than or equals limit inf Underscript upper B right arrow 0 plus 0 Endscripts integral Underscript left brace psi less than minus t 1 right brace Endscripts StartAbsoluteValue upper F overtilde Subscript upper B Baseline minus left parenthesis 1 minus b Subscript t 0 comma upper B Baseline left parenthesis psi right parenthesis right parenthesis upper F Subscript t 0 Baseline EndAbsoluteValue squared e Superscript negative phi Baseline c left parenthesis negative psi right parenthesis 4th Row 1st Column Blank 2nd Column less than or equals StartFraction integral Subscript t 1 Superscript t 0 Baseline c left parenthesis s right parenthesis e Superscript negative s Baseline d s Over c Subscript plus Baseline left parenthesis t 0 right parenthesis e Superscript minus t 0 Baseline EndFraction StartMetric upper F Subscript t 0 Baseline EndMetric Subscript upper O upper S upper H Sub Subscript rho Sub Superscript 2 Subscript left parenthesis upper M Sub Subscript t 0 Subscript comma psi Sub Subscript t Subscript right parenthesis Superscript 2 Baseline period EndLayout EndLayout

Using Lemma 2.6, we get that

$$ \begin{align*} \nonumber \begin{aligned} G(t_1)&\leq\int_{\{\psi<-t_1\}}|F_1|^2e^{-\varphi}c(-\psi)\\ &=\int_{\{-t_0\le\psi<-t_1\}}|F_1|^2e^{-\varphi}c(-\psi)+\int_{\{\psi<-t_0\}}|F_1-F_{t_0}|^2e^{-\varphi}c(-\psi)+G(t_0)\\ &\le \frac{\int_{t_1}^{t_0}c(s)e^{-s}ds}{c_+(t_0)e^{-t_0}}\|F_{t_0}\|_{OSH_{\rho}^2(M_{t_0},\psi_t)}^2+G(t_0). \end{aligned} \end{align*} $$

StartLayout 1st Row StartLayout 1st Row 1st Column upper G left parenthesis t 1 right parenthesis 2nd Column less than or equals integral Underscript left brace psi less than minus t 1 right brace Endscripts StartAbsoluteValue upper F 1 EndAbsoluteValue squared e Superscript negative phi Baseline c left parenthesis negative psi right parenthesis 2nd Row 1st Column Blank 2nd Column equals integral Underscript left brace minus t 0 less than or equals psi less than minus t 1 right brace Endscripts StartAbsoluteValue upper F 1 EndAbsoluteValue squared e Superscript negative phi Baseline c left parenthesis negative psi right parenthesis plus integral Underscript left brace psi less than minus t 0 right brace Endscripts StartAbsoluteValue upper F 1 minus upper F Subscript t 0 Baseline EndAbsoluteValue squared e Superscript negative phi Baseline c left parenthesis negative psi right parenthesis plus upper G left parenthesis t 0 right parenthesis 3rd Row 1st Column Blank 2nd Column less than or equals StartFraction integral Subscript t 1 Superscript t 0 Baseline c left parenthesis s right parenthesis e Superscript negative s Baseline d s Over c Subscript plus Baseline left parenthesis t 0 right parenthesis e Superscript minus t 0 Baseline EndFraction StartMetric upper F Subscript t 0 Baseline EndMetric Subscript upper O upper S upper H Sub Subscript rho Sub Superscript 2 Subscript left parenthesis upper M Sub Subscript t 0 Subscript comma psi Sub Subscript t Subscript right parenthesis Superscript 2 Baseline plus upper G left parenthesis t 0 right parenthesis period EndLayout EndLayout

Thus, Proposition 2.7 holds.

Recall that

for any $t\ge 0$ , then we have $H(t)\le -G_+'(t)$ . Following the concavity of $G(h^{-1}(r))$ , we have

(2.5) $$ \begin{align} \int_{t_1}^{t_0}-G^{\prime}_+(t)dt=G(t_1)-G(t_0)\le\frac{\int_{t_1}^{t_0}c(s)e^{-s}ds}{c_+(t_0)e^{-t_0}}(-G^{\prime}_{+}(t_0)) \end{align} $$

for any $0\le t_1<t_0<+\infty $ . The following proposition shows that when replacing $-G_+'(t)$ by $H(t)$ , inequality (2.5) also holds under some assumptions.

Proposition 2.8. Let $t_0>0$ . Assume that $OSH_{\rho }^2(M_{t_0},\psi _{t_0})\subset \mathcal {H}^2(M_{t_0},\rho )$ and $H(t)$ is Lebesgue measurable on $(0,+\infty )$ . Then we have

$$ \begin{align*} \nonumber \frac{\int_{t_1}^{t_0}H(t)dt}{\int_{t_1}^{t_0}c(t)e^{-t}dt}\le \frac{e^{t_0}}{c_+(t_0)}H(t_0), \end{align*} $$

StartLayout 1st Row StartFraction integral Subscript t 1 Superscript t 0 Baseline upper H left parenthesis t right parenthesis d t Over integral Subscript t 1 Superscript t 0 Baseline c left parenthesis t right parenthesis e Superscript negative t Baseline d t EndFraction less than or equals StartFraction e Superscript t 0 Baseline Over c Subscript plus Baseline left parenthesis t 0 right parenthesis EndFraction upper H left parenthesis t 0 right parenthesis comma EndLayout

where $t_1\in [0,t_0)$ .

Proof. Without loss of generality, we assume $H(t_0)<+\infty $ . By definition of $H(t_0)$ , there exists $\{f_j\}_{j\in \mathbb {Z}_{>0}}\subset OSH_{\rho }^2(M_{t_0},\psi _{t_0})\subset \mathcal {H}^2(M_{t_0},\rho )$ such that $(f_j-f)\in H^0(Z_0,(\mathcal {O}(K_M)\otimes \mathcal {F})|_{Z_0})$ for any j and

(2.6) $$ \begin{align} H(t_0)=\lim_{j\rightarrow+\infty}\|f_j\|^2_{OSH_{\rho}^2(M_{t_0},\psi_{t_0})}. \end{align} $$

Lemma 2.3 shows that for any $B>0$ and $j>0$ , there exists a holomorphic $(n,0)$ form $\tilde {F}_{j,B}$ on $\{\psi <-t_1\}$ , such that $(\tilde {F}_{j,B}-f_j)\in H^{0}(Z_0,(\mathcal {O}(K_M)\otimes \mathcal {I}(\varphi +\psi ))|_{Z_0}) \subseteq H^{0}(Z_0,(\mathcal {O}(K_{M})\otimes \mathcal {F})|_{Z_0})$ and

(2.7) $$ \begin{align} \begin{aligned} &\int_{\{\psi<-t_1\}}|\tilde{F}_{j,B}-(1-b_{t_{0},B}(\psi))f_j|^{2}e^{-\varphi-\psi+v_{t_0,B}(\psi)}c(-v_{t_0,B}(\psi)) \\ &\quad \leq \int_{t_1}^{t_{0}+B}c(t)e^{-t}dt \int_{\{\psi<-t_1\}}\frac{1}{B}(\mathbb{I}_{\{-t_{0}-B<\psi<-t_{0}\}})|f_j|^{2}e^{-\varphi-\psi} \\ &\quad \leq \frac{e^{t_{0}+B}\int_{t_1}^{t_{0}+B}c(t)e^{-t}dt}{\inf_{t\in(t_{0},t_{0}+B)}c(t)} \int_{\{\psi<-t_1\}}\frac{1}{B}(\mathbb{I}_{\{-t_{0}-B<\psi<-t_{0}\}})|f_j|^{2}e^{-\varphi}c(-\psi). \end{aligned} \end{align} $$

As $H(t)\le \liminf _{\epsilon \rightarrow 0+0}\frac {1}{\epsilon }\int _{\{-t-\epsilon <\psi <-t\}}|\tilde F_{j,B}|^2e^{-\varphi }c(-\psi )$ and $v_{t_0,B}(t)=t$ for $t\in [-t_0,-t_1)$ , inequality (2.7) implies that

(2.8) $$ \begin{align} \begin{aligned} \int_{t_1}^{t_0}H(t)dt\le &\int_{\{-t_0\le\psi<-t_1\}}|\tilde F_{j,B}|^2e^{-\varphi}c(-\psi)\\ \le&\int_{\{\psi<-t_1\}}|\tilde{F}_{j,B}-(1-b_{t_{0},B}(\psi))f_j|^{2}e^{-\varphi-\psi+v_{t_0,B}(\psi)}c(-v_{t_0,B}(\psi))\\ \le &\frac{e^{t_{0}+B}\int_{t_1}^{t_{0}+B}c(t)e^{-t}dt}{\inf_{t\in(t_{0},t_{0}+B)}c(t)} \int_{\{\psi<-t_1\}}\frac{1}{B}(\mathbb{I}_{\{-t_{0}-B<\psi<-t_{0}\}})|f_j|^{2}e^{-\varphi}c(-\psi). \end{aligned} \end{align} $$

StartLayout 1st Row StartLayout 1st Row 1st Column integral Subscript t 1 Superscript t 0 Baseline upper H left parenthesis t right parenthesis d t less than or equals 2nd Column integral Underscript left brace minus t 0 less than or equals psi less than minus t 1 right brace Endscripts StartAbsoluteValue upper F overtilde Subscript j comma upper B Baseline EndAbsoluteValue squared e Superscript negative phi Baseline c left parenthesis negative psi right parenthesis 2nd Row 1st Column less than or equals 2nd Column integral Underscript left brace psi less than minus t 1 right brace Endscripts StartAbsoluteValue upper F overtilde Subscript j comma upper B Baseline minus left parenthesis 1 minus b Subscript t 0 comma upper B Baseline left parenthesis psi right parenthesis right parenthesis f Subscript j Baseline EndAbsoluteValue squared e Superscript negative phi minus psi plus v Super Subscript t 0 comma upper B Superscript left parenthesis psi right parenthesis Baseline c left parenthesis minus v Subscript t 0 comma upper B Baseline left parenthesis psi right parenthesis right parenthesis 3rd Row 1st Column less than or equals 2nd Column StartFraction e Superscript t 0 plus upper B Baseline integral Subscript t 1 Superscript t 0 plus upper B Baseline c left parenthesis t right parenthesis e Superscript negative t Baseline d t Over inf Underscript t element of left parenthesis t 0 comma t 0 plus upper B right parenthesis Endscripts c left parenthesis t right parenthesis EndFraction integral Underscript left brace psi less than minus t 1 right brace Endscripts StartFraction 1 Over upper B EndFraction left parenthesis double struck upper I Subscript left brace minus t 0 minus upper B less than psi less than minus t 0 right brace Baseline right parenthesis StartAbsoluteValue f Subscript j Baseline EndAbsoluteValue squared e Superscript negative phi Baseline c left parenthesis negative psi right parenthesis period EndLayout EndLayout

Letting $B\rightarrow 0+0$ and $j\rightarrow +\infty $ , equality (2.6) and inequality (2.8) show

$$ \begin{align*}\frac{\int_{t_1}^{t_0}H(t)dt}{\int_{t_1}^{t_0}c(t)e^{-t}dt}\le \frac{e^{t_0}}{c_+(t_0)}H(t_0).\end{align*} $$

Thus, Proposition 2.8 holds.

2.2. Conjugate Hardy $H^2$ spaces on planar regions

In this section, we recall and give some results on the conjugate Hardy $H^2$ spaces on planar regulars.

Let D be a planar regular region bounded by finite analytic Jordan curves (see [Reference Saitoh19], [Reference Yamada21]). Let $z_0\in D$ , and $G_{D}(\cdot ,z_0)$ be the Green function on D. The definition of $H^{2}_{(c)}(D)$ (see [Reference Saitoh19]) can be referred to Section 1.1. The following lemma gives some properties for $H^2_{(c)}(D)$ .

Lemma 2.9 [Reference Rudin17].

$\mathrm{(a)}$ If $f\in H^2_{(c)}(D)$ , there is a function $f_*$ on $\partial D$ such that f has nontangential boundary value $f_*$ almost everywhere on $\partial D$ . And the map $\gamma :f\longmapsto f_*\in L^2(\partial D)$ satisfies that

$$ \begin{align*}u_f(z_0)=\frac{1}{2\pi}\int_{D}|\gamma(f)|^2\frac{\partial G_{D}(z,z_0)}{\partial v_z}|dz|\end{align*} $$

holds for any $f\in H^2_{(c)}(D)$ , where $u_f$ is the least harmonic majorant of $|f|^2$ .

$\mathrm{(b)}$ For any $g\in L^2(\partial D)$ , $g\in \gamma (H_{(c)}^2(D))$ if and only if

$$ \begin{align*}\int_{\partial D}g(z)\phi(z)dz=0\end{align*} $$

holds for any holomorphic function $\phi $ on a neighborhood of $\overline D$ .

$\mathrm{(c)}$ The inverse of $\gamma $ is given by

$$ \begin{align*} \nonumber f(w)=\frac{1}{2\pi\sqrt{-1}}\int_{\partial D}\frac{f_*(z)}{z-w}dz \end{align*} $$ StartLayout 1st Row f left parenthesis w right parenthesis equals StartFraction 1 Over 2 pi StartRoot negative 1 EndRoot EndFraction integral Underscript partial differential upper D Endscripts StartFraction f Subscript asterisk Baseline left parenthesis z right parenthesis Over z minus w EndFraction d z EndLayout

for any $z\in D$ .

Each function $f(z)\in H_{(c)}^2(D)$ has Fatou’s nontangential boundary value a.e. on $\partial D$ belonging to $L^2(\partial D)$ , then $f\in H_{(c)}^2(D)$ can be seen as a function on $\overline D$ without misunderstanding.

We recall a basic formula, which will be used in the subsequent discussion.

Lemma 2.10 (See [Reference Guan and Yuan11]).

Let $\psi \in C^1(U)$ satisfy $\psi |_{\partial D}=0$ and $|d\psi |\not =0$ on $\partial D$ , where U is a neighborhood of $\partial D$ . Then $\frac {\partial \psi }{\partial v_z}=\left (\left (\frac {\partial \psi }{\partial x}\right )^2+\left (\frac {\partial \psi }{\partial y}\right )^2\right )^{\frac {1}{2}}$ on $\partial D$ , where $\partial /\partial v_z$ denotes the derivative along the outer normal unit vector $v_z$ and $z=(x,y)$ .

We recall the following coarea formula.

Lemma 2.11 (See [Reference Federer3]).

Suppose that $\Omega $ is an open set in $\mathbb {R}^n$ and $u\in C^1(\Omega )$ . Then for any $g\in L^1(\Omega )$ ,

$$ \begin{align*}\int_{\Omega}g(x)|\nabla u(x)|dx=\int_{\mathbb{R}}\left(\int_{u^{-1}(t)}g(x)dH_{n-1}(x)\right)dt,\end{align*} $$

where $H_{n-1}$ is the $(n-1)$ -dimensional Hausdorff measure.

Denote by

$$ \begin{align*}D_t:=\{z\in D:2G_D(z,z_0)<-t\}\end{align*} $$

for any $t\ge 0$ . The following two lemmas will be used in the proof of Proposition 2.15.

Lemma 2.12 (See [Reference Guan and Yuan12]).

Let $\varphi $ be a positive Lebesgue measurable function on $U\cap \overline D$ satisfying that $\lim _{z\rightarrow \tilde z}\varphi (z)=\varphi (\tilde z)$ for any $\tilde z\in \partial D$ . Then

$$ \begin{align*} \nonumber \int_{\partial D}|f|^2\varphi|dz|=\lim_{t\rightarrow0+0}\int_{\partial D_t}|f|^2\varphi|dz| \end{align*} $$ integral Underscript partial differential upper D Endscripts StartAbsoluteValue f EndAbsoluteValue squared phi StartAbsoluteValue d z EndAbsoluteValue equals limit Underscript t right arrow 0 plus 0 Endscripts integral Underscript partial differential upper D Subscript t Baseline Endscripts StartAbsoluteValue f EndAbsoluteValue squared phi StartAbsoluteValue d z EndAbsoluteValue

holds for any $f\in H_{(c)}^2(D)$ .

Lemma 2.13 (See [Reference Guan and Yuan10]).

Let f be a holomorphic function on D. Assume that

$$ \begin{align*} \nonumber \liminf_{r\rightarrow1-0}\frac{\int_{\{z\in D:2G_{D}(z,z_0)\ge\log r\}}|f(z)|^2d\lambda_D}{1-r}<+\infty, \end{align*} $$

then we have $f\in H_{(c)}^{2}(D)$ , where $d\lambda _D$ is the Lebesgue measure on D.

Denote the set of all critical values of $G_{D}(\cdot ,z_0)$ by $N\subset (-\infty ,0)$ .

Lemma 2.14. $N\subset \subset (-\infty ,0)$ is a discrete set.

Proof. As D is a planar region bounded by finite analytic Jordan curves, $G_{D}(\cdot ,z_0)$ can be extended to a harmonic function on $U\cup D\backslash \{z_0\}$ and $|\nabla G_{D}(\cdot ,z_0)|\not =0$ on $\partial D$ , where U is a neighborhood of $\partial D$ . Note that $G_{D}(\cdot ,z_0)-\log |z-z_0|$ is harmonic near $z_0$ . There exists a holomorphic function $\tilde {f}_{z_0}$ near $z_0$ such that $\tilde {f}^{\prime }_{z_0}(z_0)\not =0$ and $\log |\tilde {f}_{z_0}|=G_{D}(\cdot ,z_0)$ near o. Hence, we have $N\subset \subset (-\infty ,0)$ . Since $G_{D}(\cdot ,z_0)$ is harmonic on $D\backslash \{z_0\}$ , the set of all critical points $G_{D}(\cdot ,z_0)$ is a discrete subset of $D\backslash \{z_0\}$ . Thus, we obtain that $N\subset \subset (-\infty ,0)$ is a discrete set.

For any $t\in [0,+\infty )\backslash -2N$ , $D_t$ is a planar region bounded by finite analytic Jordan curves. The following proposition shows that for planar region $D_t$ and trivial weight $\rho \equiv 1$ , the Ohsawa–Saitoh–Hardy space is just the conjugate Hardy $H^2$ space.

Proposition 2.15. Let $t\in [0,+\infty )\backslash -2N$ . Then

$$ \begin{align*}OSH^2(D_t,2G_{D}(\cdot,z_0)+t)=\{fdz:f\in H^2_{(c)}(D_t)\},\end{align*} $$

and

$$ \begin{align*}\|fdz\|^2_{OSH^2(D_t,2G_{D}(\cdot,z_0)+t)}=2\pi\|f\|_{H^2_{(c)}(D_t)}^2\end{align*} $$

for any $f\in H^2_{(c)}(D_t),$ where $\|f\|_{H^2_{(c)}(D)}:=\left (\frac {1}{2\pi }\int _{\partial D}|f(z)|^2\left (\frac {\partial G_{D}(z,z_0)}{\partial v_z} \right )^{-1}|dz|\right )^{\frac {1}{2}}$ .

Proof. As D is a planar regular region bounded by finite analytic Jordan curves and $G_D(\cdot ,z_0)$ is harmonic on $D\backslash \{z_0\}$ , it suffices to prove the case $t=0$ . Let f be any holomorphic function on D.

If $fdz\in OSH^2(D,2G_{D}(\cdot ,z_0))$ , then we have

$$ \begin{align*} \nonumber \begin{aligned} \liminf_{B\rightarrow 0+0}\frac{1}{B}\int_{\{-B\le\psi<0\}}|f|^2d\lambda_D &=\liminf_{B\rightarrow 0+0}\frac{1}{2B}\int_{\{-B<\psi<0\}}|fdz|^2 \\ &=\frac{1}{2}\|f\|^2_{OSH^2(D,\psi)}\\ &<+\infty, \end{aligned} \end{align*} $$

StartLayout 1st Row 1st Column limit inf Underscript upper B right arrow 0 plus 0 Endscripts StartFraction 1 Over upper B EndFraction integral Underscript left brace negative upper B less than or equals psi less than 0 right brace Endscripts StartAbsoluteValue f EndAbsoluteValue squared d lamda Subscript upper D 2nd Column equals limit inf Underscript upper B right arrow 0 plus 0 Endscripts StartFraction 1 Over 2 upper B EndFraction integral Underscript left brace negative upper B less than psi less than 0 right brace Endscripts StartAbsoluteValue f d z EndAbsoluteValue squared 2nd Row 1st Column Blank 2nd Column equals one half StartMetric f EndMetric Subscript upper O upper S upper H squared left parenthesis upper D comma psi right parenthesis Superscript 2 Baseline 3rd Row 1st Column Blank 2nd Column less than plus infinity comma EndLayout

where $d\lambda _D$ is the Lebesgue measure on D and $\psi =2G_{D}(z,z_0)$ , which implies that $f\in H^2_{(c)}(D)$ by Lemma 2.13.

In the following, assume that $f\in H^2_{(c)}(D)$ . Using Lemmas 2.11 and 2.12, we have

$$ \begin{align*} \nonumber \begin{aligned} \lim_{B\rightarrow 0+0}\frac{1}{B}\int_{\{-B<\psi<0\}}|f|^2d\lambda_D&=\lim_{B\rightarrow 0+0}\frac{\int_{0}^{B}\left(\int_{\{\psi=-t\}}|f|^2|\nabla \psi|^{-1}|dz|\right)dt}{B}\\ &=\frac{1}{2}\int_{\partial D}|f(z)|^2\left(\frac{\partial G_D(z,z_0)}{\partial v_z}\right)^{-1}|dz|, \end{aligned} \end{align*} $$

StartLayout 1st Row StartLayout 1st Row 1st Column limit Underscript upper B right arrow 0 plus 0 Endscripts StartFraction 1 Over upper B EndFraction integral Underscript left brace negative upper B less than psi less than 0 right brace Endscripts StartAbsoluteValue f EndAbsoluteValue squared d lamda Subscript upper D 2nd Column equals limit Underscript upper B right arrow 0 plus 0 Endscripts StartFraction integral Subscript 0 Superscript upper B Baseline left parenthesis integral Underscript left brace psi equals negative t right brace Endscripts StartAbsoluteValue f EndAbsoluteValue squared StartAbsoluteValue nabla psi EndAbsoluteValue Superscript negative 1 Baseline StartAbsoluteValue d z EndAbsoluteValue right parenthesis d t Over upper B EndFraction 2nd Row 1st Column Blank 2nd Column equals one half integral Underscript partial differential upper D Endscripts StartAbsoluteValue f left parenthesis z right parenthesis EndAbsoluteValue squared left parenthesis StartFraction partial differential upper G Subscript upper D Baseline left parenthesis z comma z 0 right parenthesis Over partial differential v Subscript z Baseline EndFraction right parenthesis Superscript negative 1 Baseline StartAbsoluteValue d z EndAbsoluteValue comma EndLayout EndLayout

which implies that $f\in OSH^2(D,2G_{D}(z,z_0))$ and

$$ \begin{align*}2\pi\|f\|_{H^2_{(c)}(D)}^2=\int_{\partial D}|f(z)|^2\left(\frac{\partial G_D(z,z_0)}{\partial v_z}\right)^{-1}|dz|=\|fdz\|^2_{OSH^2(D,2G_{D}(z,z_0))}.\end{align*} $$

Thus, Proposition 2.15 holds.

According to the above proposition, for $t\in -2N$ , we define that

$$ \begin{align*}H^2_{(c)}(D_t):=\{f\in\mathcal{O}(D_t):fdz\in OSH^2(D_t,2G_{D}(\cdot,z_0)+t)\}\end{align*} $$

and

$$ \begin{align*}\|f\|^2_{H^2_{(c)}(D_t)}:=\frac{1}{2\pi}\|fdz\|^2_{OSH^2(D_t,2G_{D}(z,z_0)+t)}\end{align*} $$

for any $f\in H^2_{(c)}(D_t)$ . For any $t\ge 0$ , the conjugate Hardy $H^2$ kernel is defined by

$$ \begin{align*}\hat K_{D_t}(z_0):=\frac{1}{\inf\left\{\|f\|^2_{H^2_{(c)}(D_t)}:f\in H^2_{(c)}(D_t)\,\&\,f(z_0)=1\right\}}.\end{align*} $$

We recall an existence and uniqueness property for $\hat K_{D_t}(z_0)$ .

Lemma 2.16 (See [Reference Guan and Yuan12]).

For any $t\in [0,+\infty )\backslash -2N,$ there exists a unique holomorphic function $f_t\in H^2_{(c)}(D_t)$ such that $f_t(z_0)=1$ and $\|f_t\|^2_{H_{(c)}^2(D_t)}=\frac {1}{\hat K_{D_t}(z_0)}$ .

The following lemma will be used in the proof of Lemma 2.18.

Lemma 2.17. Let $\rho _1\in C([-1,1]\times [0,1])$ be a positive function. Let $\{t_j\}_{j\in \mathbb {Z}_{\ge 1}}\in (0,1)$ be a decreasing sequence satisfying $\lim _{j\rightarrow +\infty }t_j=0$ , and let $f_{t_j}\in L^2[0,1]$ for any $j\ge 1$ , which satisfies that

(2.9) $$ \begin{align} T:=\lim_{j\rightarrow+\infty}\int_0^1|f_{t_j}(x)|^2\rho_1(t_j,x)dx<+\infty. \end{align} $$ upper T colon equals limit Underscript j right arrow plus infinity Endscripts integral Subscript 0 Superscript 1 Baseline StartAbsoluteValue f Subscript t Sub Subscript j Subscript Baseline left parenthesis x right parenthesis EndAbsoluteValue squared rho 1 left parenthesis t Subscript j Baseline comma x right parenthesis d x less than plus infinity period

Then there exists a subsequence of $\{f_{t_j}\}$ (denoted also by $\{f_{t_j}\}$ ), which weakly converges to $f_0\in L^2[0,1]$ and satisfies that

$$ \begin{align*}\int_0^1|f_0|^2\rho_1(0,x)dx\le T\end{align*} $$ integral Subscript 0 Superscript 1 Baseline StartAbsoluteValue f 0 EndAbsoluteValue squared rho 1 left parenthesis 0 comma x right parenthesis d x less than or equals upper T

and

$$ \begin{align*}\int_0^1f_0(x)\rho_2(0,x)dx=\lim_{j\rightarrow+\infty}\int_0^1f_{t_j}(x)\rho_2(t_j,x)dx\end{align*} $$

for any $\rho _2\in C([-1,1]\times [0,1]).$

Proof. As $\rho _1>0$ and $\rho _1\in C([-1,1]\times [0,1])$ , inequality (2.9) implies

$$ \begin{align*}\lim_{j\rightarrow+\infty}\int_0^1|f_{t_j}(x)|^2dx<+\infty.\end{align*} $$ limit Underscript j right arrow plus infinity Endscripts integral Subscript 0 Superscript 1 Baseline StartAbsoluteValue f Subscript t Sub Subscript j Subscript Baseline left parenthesis x right parenthesis EndAbsoluteValue squared d x less than plus infinity period

Since $L^2[0,1]$ is a Hilbert space, there exists a subsequence of $\{f_{t_j}\}$ denoted also by $\{f_{t_j}\}$ such that $\{f_{t_j}\}$ weakly converges to $f_0\in L^2[0,1]$ and $\{\rho _1(t_j,\cdot )^{\frac {1}{2}}f_{t_j}\}$ weakly converges to $\tilde f_0\in L^2[0,1]$ .

For any $\rho _2\in C([-1,1]\times [0,1])$ , we have

(2.10) $$ \begin{align} \begin{aligned} &\lim_{j\rightarrow+\infty}\int_0^1f_{t_j}\rho_2(t_j,x)dx\\ &\quad =\lim_{j\rightarrow+\infty}\int_0^1f_{t_j}(x)\left(\rho_2(t_j,x)-\rho_2(0,x)\right)dx+\lim_{j\rightarrow+\infty}\int_0^1f_{t_j}(x)\rho_2(0,x)dx\\ &\quad =\int_0^1f_{0}(x)\rho_2(0,x)dx. \end{aligned} \end{align} $$

StartLayout 1st Row StartLayout 1st Row 1st Column Blank 2nd Column limit Underscript j right arrow plus infinity Endscripts integral Subscript 0 Superscript 1 Baseline f Subscript t Sub Subscript j Baseline rho 2 left parenthesis t Subscript j Baseline comma x right parenthesis d x 2nd Row 1st Column Blank 2nd Column equals limit Underscript j right arrow plus infinity Endscripts integral Subscript 0 Superscript 1 Baseline f Subscript t Sub Subscript j Baseline left parenthesis x right parenthesis left parenthesis rho 2 left parenthesis t Subscript j Baseline comma x right parenthesis minus rho 2 left parenthesis 0 comma x right parenthesis right parenthesis d x plus limit Underscript j right arrow plus infinity Endscripts integral Subscript 0 Superscript 1 Baseline f Subscript t Sub Subscript j Baseline left parenthesis x right parenthesis rho 2 left parenthesis 0 comma x right parenthesis d x 3rd Row 1st Column Blank 2nd Column equals integral Subscript 0 Superscript 1 Baseline f 0 left parenthesis x right parenthesis rho 2 left parenthesis 0 comma x right parenthesis d x period EndLayout EndLayout

For any $g\in C[0,1]$ , equality (2.10) deduces that

$$ \begin{align*} \nonumber \lim_{j\rightarrow+\infty}\int_0^1f_{t_j}(x)\rho_1(t_j,x)^{\frac{1}{2}}g(x)dx =\int_0^1f_{0}(x)\rho_1(0,x)^{\frac{1}{2}}g(x)dx, \end{align*} $$

which shows that $\tilde f_0=f_{0}(x)\rho _1(0,x)^{\frac {1}{2}}$ and

$$ \begin{align*}\int_0^1|f_0|^2\rho_1(0,x)dx\le\lim_{j\rightarrow+\infty}\int_0^1|f_{t_j}(x)|^2\rho_1(t_j,x)dx.\end{align*} $$

Thus, Lemma 2.17 holds.

The following lemma shows that $\hat K_{D_t}(z_0)$ is right continuous.

Lemma 2.18. For any $t_0\ge 0$ , we have $\lim _{t\rightarrow t_0+0}\hat K_{D_t}(z_0)=\hat K_{D_{t_0}}(z_0).$

Proof. Firstly, we consider the case $t_0\in [0,+\infty )\backslash -2N$ .

For any $t_0\in [0,+\infty )\backslash -2N,$ let $f_{t_0}$ be the unique holomorphic function such that $f_{t_0}(z_0)=1$ and

(2.11) $$ \begin{align} \|f_{t_0}\|^2_{H_{(c)}^2(D_t)}=\frac{1}{\hat K_{D_{t_0}}(z_0)} \end{align} $$

by Lemma 2.16. Following from Lemma 2.12, we have

(2.12) $$ \begin{align} \begin{aligned} \|f_{t_0}\|^2_{H_{(c)}^2(D_{t_0})}=\frac{1}{2\pi}\lim_{t\rightarrow t_0+0}\int_{\{2G_D(\cdot,z_0)=-t\}}|f_{t_0}|^2|\nabla G_D(\cdot,z_0)|^{-1}|dz|. \end{aligned} \end{align} $$

As $\frac {1}{2\pi }\int _{\{2G_D(\cdot ,z_0)=-t\}}|f_{t_0}|^2|\nabla G_D(\cdot ,z_0)|^{-1}|dz|\ge \frac {1}{\hat K_{D_t}(z_0)}$ , it follows from equalities (2.11) and (2.12) that

(2.13) $$ \begin{align} \liminf_{t\rightarrow t_0+0}\hat K_{D_t}(z_0)\ge\hat K_{D_{t_0}}(z_0). \end{align} $$

Then, we will prove $\limsup _{t\rightarrow t_0+0}\hat K_{D_t}(z_0)\le \hat K_{D_{t_0}}(z_0).$ It suffices to consider the case $\limsup _{t\rightarrow t_0+0}\hat K_{D_t}(z_0)>0$ . There exists a decreasing sequence $\{t_j\}_{j\in \mathbb {Z}_{\ge 1}}$ such that $\lim _{j\rightarrow +\infty }t_j=t_0$ and

(2.14) $$ \begin{align} \limsup_{t\rightarrow t_0+0}\hat K_{D_t}(z_0)=\lim_{j\rightarrow+\infty}\hat K_{D_{t_j}}(z_0). \end{align} $$

Without loss of generality, assume that $|\nabla G_D(\cdot ,z_0)|\not =0$ on $\{-t_1<2G_D(\cdot ,z_0)<-t_0\}$ . For any $j\ge 1$ , it follows from Lemma 2.9 that

$$ \begin{align*}\int_{\partial D_{t_j}}\frac{f_{t_j}(z)}{z-w}dz=f_{t_j}(w)\end{align*} $$ StartLayout 1st Row integral Underscript partial differential upper D Subscript t Sub Subscript j Baseline Endscripts StartFraction f Subscript t Sub Subscript j Subscript Baseline left parenthesis z right parenthesis Over z minus w EndFraction d z equals f Subscript t Sub Subscript j Baseline left parenthesis w right parenthesis EndLayout

for any $w\in D_{t_j}$ and

$$ \begin{align*}\int_{\partial D_{t_j}}f_{t_j}(z)\phi(z)dz=0\end{align*} $$ integral Underscript partial differential upper D Subscript t Sub Subscript j Baseline Endscripts f Subscript t Sub Subscript j Baseline left parenthesis z right parenthesis phi left parenthesis z right parenthesis d z equals 0

for any holomorphic function $\phi $ on a neighborhood of $\overline D_{t_j}$ . Note that

$$ \begin{align*}\lim_{j\rightarrow+\infty}\frac{1}{2\pi}\int_{\partial D_{t_j}}|f_{t_j}|^2\left(\frac{\partial G_D(z,z_0)}{\partial v_z} \right)^{-1}|dz|=\lim_{j\rightarrow+\infty}\frac{1}{\hat K_{D_j}(z_0)}<+\infty.\end{align*} $$

Dividing $\{z\in D:-t_1<2G_D(z,z_0)<-t_0\}$ into finite small domains and using Lemma 2.17, there exist a subsequence of $\{f_{t_j}\}$ (denoted also by $\{f_{t_j}\}$ ) and an element $f_0\in L^2(\partial D_{t_0})$ satisfying that

$$ \begin{align*}\int_{\partial D_{t_0}}\frac{f_{0}(z)}{z-w}dz&=\lim_{j\rightarrow+\infty}\int_{\partial D_{t_j}}\frac{f_{t_j}(z)}{z-w}dz=\lim_{j\rightarrow+\infty}f_{t_j}(w),\\\int_{\partial D_{t_j}}f_{0}(z)\phi(z)dz&=\lim_{j\rightarrow+\infty}\int_{\partial D_{t_j}}f_{t_j}(z)\phi(z)dz=0\end{align*} $$

StartLayout 1st Row 1st Column integral Underscript partial differential upper D Subscript t 0 Endscripts StartFraction f 0 left parenthesis z right parenthesis Over z minus w EndFraction d z 2nd Column equals limit Underscript j right arrow plus infinity Endscripts integral Underscript partial differential upper D Subscript t Sub Subscript j Subscript Baseline Endscripts StartFraction f Subscript t Sub Subscript j Subscript Baseline left parenthesis z right parenthesis Over z minus w EndFraction d z equals limit Underscript j right arrow plus infinity Endscripts f Subscript t Sub Subscript j Subscript Baseline left parenthesis w right parenthesis comma 2nd Row 1st Column integral Underscript partial differential upper D Subscript t Sub Subscript j Endscripts f 0 left parenthesis z right parenthesis phi left parenthesis z right parenthesis d z 2nd Column equals limit Underscript j right arrow plus infinity Endscripts integral Underscript partial differential upper D Subscript t Sub Subscript j Baseline Endscripts f Subscript t Sub Subscript j Baseline left parenthesis z right parenthesis phi left parenthesis z right parenthesis d z equals 0 EndLayout

for any holomorphic function $\phi $ on a neighborhood of $\overline D_{t_0}$ and

(2.15) $$ \begin{align} \frac{1}{2\pi}\int_{\partial D_{t_0}}|f_{0}|^2\left(\frac{\partial G_D(z,z_0)}{\partial v_z} \right)^{-1}|dz|\le\lim_{j\rightarrow+\infty}\frac{1}{2\pi}\int_{\partial D_{t_j}}|f_{t_j}|^2\left(\frac{\partial G_D(z,z_0)}{\partial v_z} \right)^{-1}|dz|, \end{align} $$

which shows that there exists an element in $H^2_{(c)}(D_{t_0})$ (denoted also by $f_0$ ) whose nontangential boundary limit is $f_0$ on $\partial D_{t_0}$ by Lemma 2.9. Then, we have

$$ \begin{align*}f_0(z_0)=\lim_{j\rightarrow+\infty}f_{t_j}(z_0)=1.\end{align*} $$

Equalities (2.14) and (2.15) show that

$$ \begin{align*}\limsup_{t\rightarrow t_0+0}\hat K_{D_t}(z_0)\le \frac{1}{\frac{1}{2\pi}\int_{\partial D_{t_0}}|f_{0}|^2\left(\frac{\partial G_D(z,z_0)}{\partial v_z} \right)^{-1}|dz|}\le\hat K_{D_{t_0}}(z_0).\end{align*} $$

StartLayout 1st Row limit sup Underscript t right arrow t 0 plus 0 Endscripts ModifyingAbove upper K With caret Subscript upper D Sub Subscript t Subscript Baseline left parenthesis z 0 right parenthesis less than or equals StartStartFraction 1 OverOver StartFraction 1 Over 2 pi EndFraction integral Underscript partial differential upper D Subscript t 0 Baseline Endscripts StartAbsoluteValue f 0 EndAbsoluteValue squared left parenthesis StartFraction partial differential upper G Subscript upper D Baseline left parenthesis z comma z 0 right parenthesis Over partial differential v Subscript z Baseline EndFraction right parenthesis Superscript negative 1 Baseline StartAbsoluteValue d z EndAbsoluteValue EndEndFraction less than or equals ModifyingAbove upper K With caret Subscript upper D Sub Subscript t 0 Subscript Baseline left parenthesis z 0 right parenthesis period EndLayout

Combining equality (2.13), we have $\lim _{t\rightarrow t_0+0}\hat K_{D_t}(z_0)=\hat K_{D_{t_0}}(z_0)$ .

Now, we consider the case $t_0\in -2N$ .

Note that $G_D(\cdot ,z_0)$ is harmonic on $D\backslash \{z_0\}$ . Using Riemann mapping theorem repeatedly, we obtain a biholomorphic mapping $\mu : \tilde D\rightarrow D_{t_0}$ , where $\tilde D$ is a planar region bounded by finite analytic Jordan curves and $\tilde z_0\in \tilde D$ satisfying $\mu (\tilde z_0)=z_0$ . It is clear that $G_{\tilde D}(\cdot ,\tilde z_0)=G_D(\cdot ,z_0)\circ \mu +\frac {t_0}{2}$ . Denote by

$$ \begin{align*}\tilde K(t):=\frac{1}{\inf\left\{\|\tilde f\|^2_{H_{(c)}^2(\tilde D)}:\tilde f\in H^2_{(c)}(\tilde D)\,\&\,\tilde f(\tilde z_0)=1\right\}}.\end{align*} $$

StartLayout 1st Row ModifyingAbove upper K With tilde left parenthesis t right parenthesis colon equals StartFraction 1 Over inf left brace StartMetric ModifyingAbove f With tilde EndMetric Subscript upper H Sub Subscript left parenthesis c right parenthesis Sub Superscript 2 Subscript left parenthesis upper D overtilde right parenthesis Superscript 2 Baseline colon ModifyingAbove f With tilde element of upper H Subscript left parenthesis c right parenthesis Superscript 2 Baseline left parenthesis upper D overtilde right parenthesis ampersand ModifyingAbove f With tilde left parenthesis z overtilde Subscript 0 Baseline right parenthesis equals 1 right brace EndFraction period EndLayout

Then we have

(2.16) $$ \begin{align} \begin{aligned} \frac{\pi}{\hat K_{D_{t_0+t}}(z_0)}&=\inf_{\tilde f\in\mathcal{O}(D_{t_0+t})\,\&\,\tilde f( z_0)=1}\liminf_{B\rightarrow0+0}\frac{1}{B}\int_{\{-t-t_0-B<2G_D(\cdot,z_0)<-t-t_0\}}|\tilde f|^2d\lambda_{D} \\&=\inf_{\tilde f\in\mathcal{O}(D_{t_0+t})\,\&\,\tilde f( z_0)=1}\liminf_{B\rightarrow0+0}\frac{1}{B}\int_{\{-t-B<2G_D(\cdot,z_0)\circ\mu+t_0<-t\}}|\tilde f\circ\mu|^2|\mu'|^2d\lambda_{\tilde D}\\ &=\inf_{\tilde f\in\mathcal{O}(\tilde D_t)\,\&\,\tilde f(\tilde z_0)=\mu'(\tilde z_0)}\liminf_{B\rightarrow0+0}\frac{1}{B}\int_{\{-t-B<2G_{\tilde D}(w,\tilde z_0)<-t\}}|\tilde f|^2d\lambda_{\tilde D}\\ &=|\mu'(\tilde z_0)|^2\frac{\pi}{\tilde K(t)} \end{aligned} \end{align} $$

StartLayout 1st Row StartLayout 1st Row 1st Column StartFraction pi Over ModifyingAbove upper K With caret Subscript upper D Sub Subscript t 0 plus t Subscript Baseline left parenthesis z 0 right parenthesis EndFraction 2nd Column equals inf Underscript ModifyingAbove f With tilde element of script upper O left parenthesis upper D Subscript t 0 plus t Baseline right parenthesis ampersand ModifyingAbove f With tilde left parenthesis z 0 right parenthesis equals 1 Endscripts limit inf Underscript upper B right arrow 0 plus 0 Endscripts StartFraction 1 Over upper B EndFraction integral Underscript left brace negative t minus t 0 minus upper B less than 2 upper G Subscript upper D Baseline left parenthesis dot comma z 0 right parenthesis less than negative t minus t 0 right brace Endscripts StartAbsoluteValue ModifyingAbove f With tilde EndAbsoluteValue squared d lamda Subscript upper D Baseline 2nd Row 1st Column Blank 2nd Column equals inf Underscript ModifyingAbove f With tilde element of script upper O left parenthesis upper D Subscript t 0 plus t Baseline right parenthesis ampersand ModifyingAbove f With tilde left parenthesis z 0 right parenthesis equals 1 Endscripts limit inf Underscript upper B right arrow 0 plus 0 Endscripts StartFraction 1 Over upper B EndFraction integral Underscript left brace negative t minus upper B less than 2 upper G Subscript upper D Baseline left parenthesis dot comma z 0 right parenthesis ring mu plus t 0 less than negative t right brace Endscripts StartAbsoluteValue ModifyingAbove f With tilde ring mu EndAbsoluteValue squared StartAbsoluteValue mu Superscript prime Baseline EndAbsoluteValue squared d lamda Subscript upper D overtilde Baseline 3rd Row 1st Column Blank 2nd Column equals inf Underscript ModifyingAbove f With tilde element of script upper O left parenthesis upper D overtilde Subscript t Baseline right parenthesis ampersand ModifyingAbove f With tilde left parenthesis z overtilde Subscript 0 Baseline right parenthesis equals mu prime left parenthesis z overtilde Subscript 0 Baseline right parenthesis Endscripts limit inf Underscript upper B right arrow 0 plus 0 Endscripts StartFraction 1 Over upper B EndFraction integral Underscript left brace negative t minus upper B less than 2 upper G Subscript upper D overtilde Baseline left parenthesis w comma z overtilde Subscript 0 Baseline right parenthesis less than negative t right brace Endscripts StartAbsoluteValue ModifyingAbove f With tilde EndAbsoluteValue squared d lamda Subscript upper D overtilde Baseline 4th Row 1st Column Blank 2nd Column equals StartAbsoluteValue mu prime left parenthesis z overtilde Subscript 0 Baseline right parenthesis EndAbsoluteValue squared StartFraction pi Over ModifyingAbove upper K With tilde left parenthesis t right parenthesis EndFraction EndLayout EndLayout

for any $t\ge 0$ by Proposition 2.15, where $\tilde D_t:=\{w\in \tilde D:2G_{\tilde D}(w,\tilde z_0)<-t\}=\mu ^{-1}(\{z\in D:2G_D(z,z_0)<-t_0-t\})$ , and $d\lambda _{\tilde D}$ and $d\lambda _D$ are the Lebesgue measures on $\tilde D$ and $D,$ respectively.

As $\tilde D$ is a planar region bounded by finite analytic Jordan curves, we get that

$$ \begin{align*}\lim_{t\rightarrow0+0}\tilde K(t)=\tilde K(0)\end{align*} $$

by above discussions. Combining equality (2.16), we obtain that

$$ \begin{align*}\lim_{t\rightarrow t_0+0}\hat K_{D_t}(z_0)=\hat K_{D_{t_0}}(z_0).\end{align*} $$

Thus, Lemma 2.18 has been proved.

Remark 2.19. Following the discussion in the above proof, we know that Lemma 2.16 holds for $t\in -2N$ .

It follows from Propositions 2.8 and 2.15 that

(2.17) $$ \begin{align} \frac{\int_{t_1}^{t_0}\frac{1}{\hat K_{D_t}(z_0)}dt}{e^{-t_1}-e^{-t_0}}\le \frac{e^{t_0}}{\hat K_{D_{t_0}}(z_0)} \end{align} $$

StartLayout 1st Row StartFraction integral Subscript t 1 Superscript t 0 Baseline StartFraction 1 Over ModifyingAbove upper K With caret Subscript upper D Sub Subscript t Subscript Baseline left parenthesis z 0 right parenthesis EndFraction d t Over e Superscript minus t 1 Baseline minus e Superscript minus t 0 Baseline EndFraction less than or equals StartFraction e Superscript t 0 Baseline Over ModifyingAbove upper K With caret Subscript upper D Sub Subscript t 0 Subscript Baseline left parenthesis z 0 right parenthesis EndFraction EndLayout

holds for any $0\le t_1<t_0<+\infty $ . The following proposition gives a necessary condition for inequality (2.17) to become an equality.

Proposition 2.20. If inequality (2.17) becomes an equality, then there exists a holomorphic function F on $\{2G_D(\cdot ,z_0)<-t_1\}$ such that $\frac {1}{\hat K_{D_t}(z_0)}=\|F\|_{H^2_{(c)}(D_t)}^2$ for a.e. $t\in (t_1,t_0)$ and $F(z_0)=1$ .

Proof. Let $f_{t_0}$ be the holomorphic function on $\{2G_D(\cdot ,z_0)<-t_0\}$ such that $f_{t_0}(z_0)=1$ and

(2.18) $$ \begin{align} \frac{1}{\hat K_{D_{t _0}}(z_0)}=\|f_{t_0}\|_{H^2_{(c)}(D_{t_0})}^2. \end{align} $$

Lemma 2.3 shows that, for any $B>0$ , there exists a holomorphic function $\tilde {F}_{B}$ on $\{2G_D(\cdot ,z_0)<-t_1\}$ such that $\tilde F_{B}(z_0)=1$ and

(2.19) $$ \begin{align} \begin{aligned} &\int_{\{\psi<-t_1\}}|\tilde{F}_{B}-(1-b_{t_{0},B}(\psi))f_{t_0}|^{2}d\lambda_D\\ &\quad \le\int_{\{\psi<-t_1\}}|\tilde{F}_{B}-(1-b_{t_{0},B}(\psi))f_{t_0}|^{2}e^{-\psi+v_{t_0,B}(\psi)}d\lambda_D \\ &\quad \leq\left(e^{-t_1}-e^{-t_0-B}\right) \int_{\{\psi<-t_1\}}\frac{1}{B}\mathbb{I}_{\{-t_{0}-B<\psi<-t_{0}\}}|f_{t_0}|^{2}e^{-\psi}d\lambda_D, \end{aligned} \end{align} $$

where $\psi :=2G_D(\cdot ,z_0)$ . Note that $\liminf _{B\rightarrow 0+0}\frac {1}{B}\mathbb {I}_{\{-t_{0}-B<\psi <-t_{0}\}}|f_{t_0}|^{2}e^{-\psi }d\lambda _D<+\infty $ and $\liminf _{B\rightarrow 0+0}\int _{\{\psi <-t_1\}}|(1-b_{t_{0},B}(\psi ))f_{t_0}|^{2}d\lambda _D\le \int _{\{\psi <-t_0\}}|f_{t_0}|^2d\lambda _D<+\infty $ , then inequality (2.19) implies that

$$ \begin{align*}\liminf_{B\rightarrow0+0}\int_{\{\psi<-t_1\}}|\tilde F_B|^2d\lambda_D<+\infty.\end{align*} $$ limit inf Underscript upper B right arrow 0 plus 0 Endscripts integral Underscript left brace psi less than minus t 1 right brace Endscripts StartAbsoluteValue upper F overtilde Subscript upper B Baseline EndAbsoluteValue squared d lamda Subscript upper D Baseline less than plus infinity period

There is a subsequence of $\{\tilde F_B\}_{B>0}$ denoted by $\{\tilde F_{B_j}\}_{j\in \mathbb {Z}_{\ge 1}}$ (when $j\rightarrow +\infty $ , $B_j\rightarrow 0$ ), which uniformly convergent to a holomorphic function F on $\{\psi <-t_1\}$ on any compact subset of $\{\psi <-t_1\}$ . Then we have $F(z_0)=1$ , and it follows from inequality (2.19), Fatou’s lemma, and Proposition 2.15 that

(2.20) $$ \begin{align} \begin{aligned} &\int_{\{-t_0\le\psi<-t_1\}}|F|^2d\lambda_D+\int_{\{\psi<-t_0\}}|F-f_{t_0}|^2d\lambda_D\\ &\quad \le\liminf_{j\rightarrow+\infty}\int_{\{\psi<-t_1\}}|\tilde{F}_{B_j}-(1-b_{t_{0},B_j}(\psi))f_{t_0}|^{2}d\lambda_D\\ &\quad \le\left(e^{-t_1}-e^{-t_0}\right)e^{t_0} \liminf_{j\rightarrow+\infty}\int_{\{-t_{0}-B_j<\psi<-t_{0}\}}\frac{1}{B_j}|f_{t_0}|^{2}d\lambda_D\\ &\quad =\left(e^{-t_1}-e^{-t_0}\right)e^{t_0}\pi \|f_{t_0}\|_{H^2_{(c)}(D_t)}^2. \end{aligned} \end{align} $$

StartLayout 1st Row 1st Column Blank 2nd Column integral Underscript left brace minus t 0 less than or equals psi less than minus t 1 right brace Endscripts StartAbsoluteValue upper F EndAbsoluteValue squared d lamda Subscript upper D plus integral Underscript left brace psi less than minus t 0 right brace Endscripts StartAbsoluteValue upper F minus f Subscript t 0 Baseline EndAbsoluteValue squared d lamda Subscript upper D 2nd Row 1st Column Blank 2nd Column less than or equals limit inf Underscript j right arrow plus infinity Endscripts integral Underscript left brace psi less than minus t 1 right brace Endscripts StartAbsoluteValue upper F overtilde Subscript upper B Sub Subscript j Subscript Baseline minus left parenthesis 1 minus b Subscript t 0 comma upper B Sub Subscript j Subscript Baseline left parenthesis psi right parenthesis right parenthesis f Subscript t 0 Baseline EndAbsoluteValue squared d lamda Subscript upper D Baseline 3rd Row 1st Column Blank 2nd Column less than or equals left parenthesis e Superscript minus t 1 Baseline minus e Superscript minus t 0 Baseline right parenthesis e Superscript t 0 Baseline limit inf Underscript j right arrow plus infinity Endscripts integral Underscript left brace minus t 0 minus upper B Subscript j Baseline less than psi less than minus t 0 right brace Endscripts StartFraction 1 Over upper B Subscript j Baseline EndFraction StartAbsoluteValue f Subscript t 0 Baseline EndAbsoluteValue squared d lamda Subscript upper D Baseline 4th Row 1st Column Blank 2nd Column equals left parenthesis e Superscript minus t 1 Baseline minus e Superscript minus t 0 Baseline right parenthesis e Superscript t 0 Baseline pi StartMetric f Subscript t 0 Baseline EndMetric Subscript upper H Sub Subscript left parenthesis c right parenthesis Sub Superscript 2 Subscript left parenthesis upper D Sub Subscript t Subscript right parenthesis Superscript 2 Baseline period EndLayout

By Proposition 2.15, we have $\frac {1}{\hat K_{D_t}(z_0)}\le \frac {1}{\pi }\liminf _{\epsilon \rightarrow 0+0}\frac {1}{\epsilon }\int _{\{-t-\epsilon <\psi <-t\}}| F|^2d\lambda _D$ , hence

$$ \begin{align*}\int_{t_1}^{t_0}\frac{1}{\hat K_{D_t}(z_0)}dt\le \frac{1}{\pi}\int_{\{-t_0\le\psi<-t_1\}}|F|^2d\lambda_D\end{align*} $$

and the inequality becomes an equality if and only if

$$ \begin{align*}\frac{1}{\hat K_{D_t}(z_0)}= \frac{1}{\pi}\liminf_{\epsilon\rightarrow0+0}\frac{1}{\epsilon}\int_{\{-t-\epsilon<\psi<-t\}}| F|^2d\lambda_D\end{align*} $$

holds a.e. on $(t_1,t_0)$ . As $\frac {\int _{t_1}^{t_0}\frac {1}{\hat K_{D_t}(z_0)}dt}{e^{-t_1}-e^{-t_0}}= \frac {e^{t_0}}{\hat K_{D_{t_0}}(z_0)}$ , it follows from equality (2.18), inequality (2.20), and the above characterization that

$$ \begin{align*}\frac{1}{\hat K_{D_t}(z_0)}= \frac{1}{\pi}\liminf_{\epsilon\rightarrow0+0}\frac{1}{\epsilon}\int_{\{-t-\epsilon<\psi<-t\}}| F|^2d\lambda_D=\|F\|^2_{H^2_{(c)}(D_t)}\end{align*} $$

holds a.e. on $(t_1,t_0)$ .

Thus, Proposition 2.20 holds.

3. Proofs of Theorems 1.4 and 1.5

In this section, we prove Theorems 1.4 and 1.5.

3.1. Proof of Theorem 1.4.

Firstly, we prove statement $(1)$ .

Proposition 2.7 shows that

$$ \begin{align*}\frac{\|F_{t_0}\|_{OSH_{\rho}^2(M_{t_0},\psi_{t_0})}^2}{e^{-t_0}c_+(t_0)}\ge\frac{G(t_1)-G(t_0)}{\int_{t_1}^{t_0}c(t)e^{-t}dt}\end{align*} $$

StartLayout 1st Row StartFraction StartMetric upper F Subscript t 0 Baseline EndMetric Subscript upper O upper S upper H Sub Subscript rho Sub Superscript 2 Subscript left parenthesis upper M Sub Subscript t 0 Subscript comma psi Sub Subscript t 0 Subscript right parenthesis Superscript 2 Baseline Over e Superscript minus t 0 Baseline c Subscript plus Baseline left parenthesis t 0 right parenthesis EndFraction greater than or equals StartFraction upper G left parenthesis t 1 right parenthesis minus upper G left parenthesis t 0 right parenthesis Over integral Subscript t 1 Superscript t 0 Baseline c left parenthesis t right parenthesis e Superscript negative t Baseline d t EndFraction EndLayout

for any $t_0>t_1\ge 0$ . Following from the concavity of $G(h^{-1}(r))$ and inequality (2.1), we have

(3.1) $$ \begin{align} \frac{-G_+'(t_0)}{e^{-t_0}c_+(t_0)}\ge \frac{\|F_{t_0}\|_{OSH_{\rho}^2(M_{t_0},\psi_{t_0})}^2}{e^{-t_0}c_+(t_0)}\ge \frac{G(t_1)-G(t_0)}{\int_{t_1}^{t_0}c(t)e^{-t}dt}\ge \frac{-G_+'(t_1)}{e^{-t_1}c_+(t_1)}, \end{align} $$

StartLayout 1st Row StartFraction minus upper G prime Subscript plus Baseline left parenthesis t 0 right parenthesis Over e Superscript minus t 0 Baseline c Subscript plus Baseline left parenthesis t 0 right parenthesis EndFraction greater than or equals StartFraction StartMetric upper F Subscript t 0 Baseline EndMetric Subscript upper O upper S upper H Sub Subscript rho Sub Superscript 2 Subscript left parenthesis upper M Sub Subscript t 0 Subscript comma psi Sub Subscript t 0 Subscript right parenthesis Superscript 2 Baseline Over e Superscript minus t 0 Baseline c Subscript plus Baseline left parenthesis t 0 right parenthesis EndFraction greater than or equals StartFraction upper G left parenthesis t 1 right parenthesis minus upper G left parenthesis t 0 right parenthesis Over integral Subscript t 1 Superscript t 0 Baseline c left parenthesis t right parenthesis e Superscript negative t Baseline d t EndFraction greater than or equals StartFraction minus upper G prime Subscript plus Baseline left parenthesis t 1 right parenthesis Over e Superscript minus t 1 Baseline c Subscript plus Baseline left parenthesis t 1 right parenthesis EndFraction comma EndLayout

which implies that $\frac {\|F_{t}\|_{OSH_{\rho }^2(M_{t},\psi _{t})}^2}{e^{-t_0}c_+(t)}$ is increasing on $[0,+\infty )$ .

As $G(h^{-1}(r))$ is concave, it is clear that statement $(3)$ can deduce statement $(2)$ . Thus, it suffices to prove statement $(3)$ .

Note that $\lim _{t_1\rightarrow t_0-0}c_+(t_1)=c_-(t_0)$ , where $c_-(t_0)=\lim _{t\rightarrow t_0-0}c(t)$ . Thus, $\frac {-G_+'(t_0)}{e^{-t_0}c_+(t_0)}=\frac {\|F_{t_0}\|_{OSH_{\rho }^2(M_{t_0},\psi _{t_0})}^2}{e^{-t_0}c_+(t_0)}$ , that is, $-G_+'(t_0)=\|F_{t_0}\|_{OSH_{\rho }^2(M_{t_0},\psi _{t_0})}^2$ if the functions $G^{\prime }_+$ and c are both continuous at $t_0$ according to inequality (3.1). As $c(t)e^{-t}$ is decreasing on $(0,+\infty )$ and $G(h^{-1}(r))$ is concave on $[0,\int _0^{+\infty }c(t)e^{-t}]$ , we know that functions $G^{\prime }_+$ and c are both continuous a.e. on $(0,+\infty )$ . Thus, Theorem 1.4 holds.

3.2. Proof of Theorem 1.5.

In this section, we prove Theorem 1.5.

Step 1. $(1)\Rightarrow (2)$ . Denote by

$$ \begin{align*}I(t):=\int_{\{\psi<-t\}}|F_0|^2e^{-\varphi}c(-\psi),\end{align*} $$ upper I left parenthesis t right parenthesis colon equals integral Underscript left brace psi less than negative t right brace Endscripts StartAbsoluteValue upper F 0 EndAbsoluteValue squared e Superscript negative phi Baseline c left parenthesis negative psi right parenthesis comma

where $F_0$ is the unique holomorphic $(n,0)$ form on M satisfying that $(F_0-f)\in H^0(Z_0 ,(\mathcal {O} (K_M) \otimes \mathcal {F})|_{Z_0})$ and $G(0)=\int _{ M}|F_0|^2e^{-\varphi }c(-\psi )$ . Note that $I(t)$ is decreasing on $[0,+\infty )$ and $\lim _{t\rightarrow +\infty }I(t)=0$ . Then $I'(t)$ exists a.e. on $(0,+\infty )$ and

$$ \begin{align*} \nonumber -I'(t)=\lim_{B\rightarrow0+0}\frac{1}{B}\int_{\{-t-B\le\psi<-t\}}|F_0|^2e^{-\varphi}c(-\psi)\ge H(t). \end{align*} $$

Thus, combining with statement $(1)$ in Theorem 1.5, we have

(3.2) $$ \begin{align} G(0)=I(0)\ge\int_0^{+\infty}-I'(t)dt\ge \int_0^{+\infty}H(t)dt=\int_0^{+\infty}-G_+'(t)dt=G(0), \end{align} $$

where the last “=” holds by the absolute continuity of $G(t)$ . Inequality (3.2) implies that $-I'(t)=H(t)$ a.e. on $(0,+\infty )$ and $I(0)=\int _0^{+\infty }-I'(t)dt$ , which show that

$$ \begin{align*} \nonumber \begin{aligned} \int_{\{\psi<-t\}}|F_0|^2e^{-\varphi}c(-\psi) =&I(t) =\int_t^{+\infty}-I'(s)ds =G(t) =\int_{\{\psi<-t\}}|F_t|^2e^{-\varphi}c(-\psi) \end{aligned} \end{align*} $$

for any $t\ge 0$ . Following from the uniqueness of $F_t$ , we have

$$ \begin{align*}F_0|_{\{\psi<-t\}}=F_t\end{align*} $$

for any $t\ge 0$ . Thus, we have $H(t)=-G^{\prime }_+(t)=\|F_0\|_{OSH_{\rho }^2(M_{t},\psi _{t})}^2$ for a.e. $t\in (0,+\infty )$ .

Step 2. $(2)\Rightarrow (1)$

As $G(t)=\int _{\{\psi <-t\}}|F|^2e^{-\varphi }c(-\psi )$ for any $t\ge 0$ , then $G_+'(t)=\|F\|_{OSH_{\rho }^2(M_{t},\psi _{t})}^2$ for any $t\in (0,+\infty )$ . Combining $H(t)=\|F\|_{OSH_{\rho }^2(M_{t},\psi _{t})}^2$ for a.e. $t\in (0,+\infty )$ , we have

$$ \begin{align*}H(t)=-G^{\prime}_+(t)\end{align*} $$

for a.e. $t\in (0,+\infty )$ .

Thus, Theorem 1.5 holds.

4. Proof of Theorem 1.6

In this section, we prove Theorem 1.6.

Following from the monotonicity of $c(t)e^{-t}$ , we know that $c_+(t)=c(t)$ a.e. on $(0,+\infty )$ . Note that $OSH_{\rho }^2(M_{t_0},\psi _{t_0})\subset \mathcal {H}^2(M_{t_0},\rho )$ holds for a.e. $t_0\in (0,+\infty )$ in Theorem 1.6, then it follows from Proposition 2.8 that inequality

$$ \begin{align*}\frac{\int_{t_1}^{t_0}H(t)dt}{\int_{t_1}^{t_0}c(t)e^{-t}dt}\le \frac{e^{t_0}}{c(t_0)}H(t_0)\end{align*} $$

holds for a.e. $t_0\in (0,+\infty )$ . Thus, Theorem 1.6 holds by the following lemma and $\int _0^{+\infty }H(t)dt\le G(0)<+\infty $ .

Lemma 4.1. Let $a(t)$ and $b(t)$ be nonnegative Lebesgue measurable functions on $(0,+\infty )$ , which satisfy that $\int _0^{+\infty }a(t)dt<+\infty $ , $\int _0^{+\infty }b(t)dt<+\infty $ , $b(t)>0$ a.e. on $(0,+\infty )$ and

(4.1) $$ \begin{align} \frac{\int_{t_1}^{t_0}a(t)dt}{\int_{t_1}^{t_0}b(t)dt}\le \frac{a(t_0)}{b(t_0)} \end{align} $$ StartLayout 1st Row StartFraction integral Subscript t 1 Superscript t 0 Baseline a left parenthesis t right parenthesis d t Over integral Subscript t 1 Superscript t 0 Baseline b left parenthesis t right parenthesis d t EndFraction less than or equals StartFraction a left parenthesis t 0 right parenthesis Over b left parenthesis t 0 right parenthesis EndFraction EndLayout

StartLayout 1st Row StartFraction integral Subscript t 1 Superscript t 0 Baseline a left parenthesis t right parenthesis d t Over integral Subscript t 1 Superscript t 0 Baseline b left parenthesis t right parenthesis d t EndFraction less than or equals StartFraction a left parenthesis t 0 right parenthesis Over b left parenthesis t 0 right parenthesis EndFraction EndLayout

holds for a.e. $t_0\in (0,+\infty )$ and any $t_1\in [0,t_0)$ . Then $\hat a(\hat b^{-1}(r))$ is concave on $(0,\int _0^{+\infty }b(t)dt]$ , where $\hat a(t)=\int _t^{+\infty }a(s)ds$ and $\hat b(t)=\int _t^{+\infty }b(s)ds$ for $t\ge 0$ .

Proof. As $\hat a'(t)=a(t)$ and $\hat b'(t)=b(t)$ a.e. on $(0,+\infty )$ , inequality (4.1) implies that

$$ \begin{align*} \nonumber \frac{\hat a(t_1)-\hat a(t_0)}{\hat b(t_1)-\hat b(t_0)}\le \frac{\hat a'(t_0)}{\hat b'(t_0)} \end{align*} $$

holds for a.e. $t_0\in (0,+\infty )$ and any $t_1\in [0,t_0)$ . Let $r_1=\hat b(t_1)$ and $r_0=\hat b(t_0)$ , and denote by $g(r):=\hat a(\hat b(r))$ , then we have

(4.2) $$ \begin{align} \frac{g(r_1)-g(r_0)}{r_1-r_0}\le g'(r_0) \end{align} $$

holds for a.e. $r_0\in (0,\int _0^{+\infty }b(t)dt)$ and any $r_1\in (r_0,\int _0^{+\infty }b(t)dt]$ .

We prove the concavity of $g(r)$ by contradiction: if not, there exists $0< r_2<r_3<r_4\le \int _0^{+\infty }b(t)dt$ such that

$$ \begin{align*} \nonumber \frac{g(r_3)-g(r_2)}{r_3-r_2}<\frac{g(r_4)-g(r_2)}{r_4-r_2}<\frac{g(r_4)-g(r_3)}{r_4-r_3}. \end{align*} $$

Consider

$$ \begin{align*}\tilde g(r)=g(r)-g(r_4)-\frac{g(r_4)-g(r_2)}{r_4-r_2}(r-r_4)\end{align*} $$

on $(0,\int _0^{+\infty }b(t)dt]$ . As $g(r)$ is continuous on $(0,\int _0^{+\infty }b(t)dt]$ , then $\tilde g(r)$ is continuous on $(0,\int _0^{+\infty }b(t)dt]$ . Note that $\tilde {g}(r_2)=\tilde {g}(r_4)=0$ and $\tilde {g}(r_3)<0$ . It follows from the continuity of $\tilde {g}(r)$ that there exists $r_5\in (r_2,r_4)$ such that

$$ \begin{align*}\tilde{g}(r_5)=\inf_{r\in[r_2,r_4]}\tilde{g}(r)<0.\end{align*} $$

It is clear that there exists small enough $\epsilon>0$ such that

$$ \begin{align*}\frac{\tilde{g}(r)-\tilde{g}(r_4)}{r-r_4}>0\end{align*} $$

for any $r\in (r_5-\epsilon ,r_5+\epsilon )\subset (r_2,r_4)$ , which implies that

$$ \begin{align*}g'(r)\ge \frac{{g}(r)-{g}(r_4)}{r-r_4}>\frac{g(r_4)-g(r_2)}{r_4-r_2}\end{align*} $$

for a.e. $r\in (r_5-\epsilon ,r_5+\epsilon )$ by inequality (4.2). As g is increasing on $(0,\int _0^{+\infty }b(t)dt]$ , we have

$$ \begin{align*} \nonumber g(r_5)-g(r_5-\epsilon)\ge\int_{r_5-\epsilon}^{r_5}g'(r)dr>\epsilon \frac{g(r_4)-g(r_2)}{r_4-r_2}, \end{align*} $$

that is, $\tilde g(r_5)>\tilde g(r_5-\epsilon )$ , which contradicts to $\tilde {g}(r_5)=\inf _{r\in [r_2,r_4]}\tilde {g}(r)$ .

Thus, Lemma 4.1 holds.

5. Proof of Corollary 1.8

In this section, we prove Corollary 1.8 in three steps.

Step 1. $\hat K_{D_t}(z_0)e^{-t}$ is decreasing.

For any $t\ge 0,$ note that for any holomorphic f on $D_t$ , if there exists $B>0$ such that $\int _{\{-B-t<2G_D(\cdot ,z_0)<-t\}}|f|^2d\lambda _D<+\infty ,$ then $\int _{D_t}|f|^2d\lambda _D<+\infty $ .

It follows from Proposition 2.15 that

$$ \begin{align*} \frac{1}{\hat K_{D_t}(z_0)}&=\inf\left\{\|\tilde f\|^2_{H^2_{(c)}(D_t)}:\tilde f\in \mathcal{O}(D_t)\,\&\,\tilde f(z_0)=1 \right\}\\ &=\frac{1}{\pi}\inf\left\{\liminf_{B\rightarrow0+0}\frac{1}{B}\int_{\{-B-t<2G_D(\cdot,z_0)<-t\}}|f|^2d\lambda_D:\tilde f\in \mathcal{O}(D_t)\,\&\,\tilde f(z_0)=1 \right\}. \end{align*} $$

StartLayout 1st Row 1st Column StartFraction 1 Over ModifyingAbove upper K With caret Subscript upper D Sub Subscript t Subscript Baseline left parenthesis z 0 right parenthesis EndFraction 2nd Column equals inf left brace StartMetric ModifyingAbove f With tilde EndMetric Subscript upper H Sub Subscript left parenthesis c right parenthesis Sub Superscript 2 Subscript left parenthesis upper D Sub Subscript t Subscript right parenthesis Superscript 2 Baseline colon ModifyingAbove f With tilde element of script upper O left parenthesis upper D Subscript t Baseline right parenthesis ampersand ModifyingAbove f With tilde left parenthesis z 0 right parenthesis equals 1 right brace 2nd Row 1st Column Blank 2nd Column equals StartFraction 1 Over pi EndFraction inf left brace limit inf Underscript upper B right arrow 0 plus 0 Endscripts StartFraction 1 Over upper B EndFraction integral Underscript left brace negative upper B minus t less than 2 upper G Subscript upper D Baseline left parenthesis dot comma z 0 right parenthesis less than negative t right brace Endscripts StartAbsoluteValue f EndAbsoluteValue squared d lamda Subscript upper D Baseline colon ModifyingAbove f With tilde element of script upper O left parenthesis upper D Subscript t Baseline right parenthesis ampersand ModifyingAbove f With tilde left parenthesis z 0 right parenthesis equals 1 right brace period EndLayout

Lemma 2.18 implies that $\frac {1}{\hat K_{D_t}(z_0)}$ is a Lebesgue measurable function with respect to $t\in [0,+\infty )$ . Then Theorem 1.6 tells us that $\int _{-\log r}^{+\infty }\frac {1}{\hat K_{D_t}(z_0)}dt$ is concave with respect to $r\in [0,\int _0^{+\infty }c(t)e^{-t}dt]$ . By Lemma 2.18, $\frac {1}{\hat K_{D_t}(z_0)}=\lim _{t_1\rightarrow t+0}\frac {1}{\hat K_{D_{t_1}}(z_0)}$ . Thus, we get that $\hat K_{D_t}(z_0)e^{-t}$ is decreasing with respect to $t\in [0,+\infty )$ .

Step 2. $(3)\Rightarrow (1)$ and $(2)$ .

Assume that D is simply connected, then there is a biholomorphic map $p:\Delta \rightarrow D$ such that $p(o)=z_0$ . Following from Lemma 2.10, we have

$$ \begin{align*} \begin{aligned} \frac{\partial G_{\Delta_t}(\cdot,o)}{\partial v_w}&=\frac{\partial (G_{D_t}(\cdot,z_0)\circ p)}{\partial v_w}=|\nabla (G_{D_t}(\cdot,z_0)\circ p)|\\ &=|\nabla G_{D_t}(\cdot,z_0)\circ p\cdot\nabla p|=\frac{\partial G_{D_t}(\cdot,z_0)}{\partial v_z}\circ p\cdot |p'|, \end{aligned} \end{align*} $$

StartLayout 1st Row StartLayout 1st Row 1st Column StartFraction partial differential upper G Subscript normal upper Delta Sub Subscript t Subscript Baseline left parenthesis dot comma o right parenthesis Over partial differential v Subscript w Baseline EndFraction 2nd Column equals StartFraction partial differential left parenthesis upper G Subscript upper D Sub Subscript t Subscript Baseline left parenthesis dot comma z 0 right parenthesis ring p right parenthesis Over partial differential v Subscript w Baseline EndFraction equals StartAbsoluteValue nabla left parenthesis upper G Subscript upper D Sub Subscript t Subscript Baseline left parenthesis dot comma z 0 right parenthesis ring p right parenthesis EndAbsoluteValue 2nd Row 1st Column Blank 2nd Column equals StartAbsoluteValue nabla upper G Subscript upper D Sub Subscript t Subscript Baseline left parenthesis dot comma z 0 right parenthesis ring p dot nabla p EndAbsoluteValue equals StartFraction partial differential upper G Subscript upper D Sub Subscript t Subscript Baseline left parenthesis dot comma z 0 right parenthesis Over partial differential v Subscript z Baseline EndFraction ring p dot StartAbsoluteValue p Superscript prime Baseline EndAbsoluteValue comma EndLayout EndLayout

where $\Delta _t:=\{z\in \mathbb {C}:2\log |z|<-t\}$ for any $t\ge 0$ . We have

$$ \begin{align*} \nonumber \begin{aligned} &\frac{1}{\hat K_{D_t}(z_0)}\\ =\ &\inf\left\{\frac{1}{2\pi}\int_{\partial D_t}|\tilde f(z)|^2\left(\frac{\partial G_{D_t}(\cdot,z_0)}{\partial v_z} \right)^{-1}|dz|:\tilde f(z)\in H^2_{(c)}(D_t)\,\&\, \tilde f(z_0)=1\right\}\\ =\ &\inf\left\{\frac{1}{2\pi}\int_{\partial \Delta_t}|\tilde f(w)|^2\left(\frac{\partial G_{\Delta_t}(\cdot,o)}{\partial v_w} \right)^{-1}|p'(w)|^2|dw|:\tilde f(w)\in H^2_{(c)}(\Delta_t)\,\&\, \tilde f(o)=1\right\}\\ =\ &\frac{|p'(o)|^2}{\hat K_{\Delta_t}(o)} \end{aligned} \end{align*} $$

StartLayout 1st Row StartLayout 1st Row 1st Column Blank 2nd Column StartFraction 1 Over ModifyingAbove upper K With caret Subscript upper D Sub Subscript t Subscript Baseline left parenthesis z 0 right parenthesis EndFraction 2nd Row 1st Column equals 2nd Column inf left brace StartFraction 1 Over 2 pi EndFraction integral Underscript partial differential upper D Subscript t Baseline Endscripts StartAbsoluteValue ModifyingAbove f With tilde left parenthesis z right parenthesis EndAbsoluteValue squared left parenthesis StartFraction partial differential upper G Subscript upper D Sub Subscript t Subscript Baseline left parenthesis dot comma z 0 right parenthesis Over partial differential v Subscript z Baseline EndFraction right parenthesis Superscript negative 1 Baseline StartAbsoluteValue d z EndAbsoluteValue colon ModifyingAbove f With tilde left parenthesis z right parenthesis element of upper H Subscript left parenthesis c right parenthesis Superscript 2 Baseline left parenthesis upper D Subscript t Baseline right parenthesis ampersand ModifyingAbove f With tilde left parenthesis z 0 right parenthesis equals 1 right brace 3rd Row 1st Column equals 2nd Column inf left brace StartFraction 1 Over 2 pi EndFraction integral Underscript partial differential normal upper Delta Subscript t Baseline Endscripts StartAbsoluteValue ModifyingAbove f With tilde left parenthesis w right parenthesis EndAbsoluteValue squared left parenthesis StartFraction partial differential upper G Subscript normal upper Delta Sub Subscript t Subscript Baseline left parenthesis dot comma o right parenthesis Over partial differential v Subscript w Baseline EndFraction right parenthesis Superscript negative 1 Baseline StartAbsoluteValue p prime left parenthesis w right parenthesis EndAbsoluteValue squared StartAbsoluteValue d w EndAbsoluteValue colon ModifyingAbove f With tilde left parenthesis w right parenthesis element of upper H Subscript left parenthesis c right parenthesis Superscript 2 Baseline left parenthesis normal upper Delta Subscript t Baseline right parenthesis ampersand ModifyingAbove f With tilde left parenthesis o right parenthesis equals 1 right brace 4th Row 1st Column equals 2nd Column StartFraction StartAbsoluteValue p prime left parenthesis o right parenthesis EndAbsoluteValue squared Over ModifyingAbove upper K With caret Subscript normal upper Delta Sub Subscript t Subscript Baseline left parenthesis o right parenthesis EndFraction EndLayout EndLayout

and

$$ \begin{align*} \begin{aligned} \frac{1}{B_{D_t}(z_0,z_0)} &=\inf\left\{\int_{D_t}|\tilde f|^2d\lambda_{D} :\tilde f\in\mathcal{O}(D_t)\,\&\,\tilde f(z_0)=1 \right\}\\ &=\inf\left\{\int_{D_t}|\tilde f|^2|p'|^2d\lambda_{\Delta} :\tilde f\in\mathcal{O}(\Delta_t)\,\&\,\tilde f(o)=1 \right\}\\ &=\frac{|p'(o)|^2}{B_{\Delta_t}(o,o)}, \end{aligned} \end{align*} $$

StartLayout 1st Row StartLayout 1st Row 1st Column StartFraction 1 Over upper B Subscript upper D Sub Subscript t Subscript Baseline left parenthesis z 0 comma z 0 right parenthesis EndFraction 2nd Column equals inf left brace integral Underscript upper D Subscript t Baseline Endscripts StartAbsoluteValue ModifyingAbove f With tilde EndAbsoluteValue squared d lamda Subscript upper D Baseline colon ModifyingAbove f With tilde element of script upper O left parenthesis upper D Subscript t Baseline right parenthesis ampersand ModifyingAbove f With tilde left parenthesis z 0 right parenthesis equals 1 right brace 2nd Row 1st Column Blank 2nd Column equals inf left brace integral Underscript upper D Subscript t Baseline Endscripts StartAbsoluteValue ModifyingAbove f With tilde EndAbsoluteValue squared StartAbsoluteValue p Superscript prime Baseline EndAbsoluteValue squared d lamda Subscript normal upper Delta Baseline colon ModifyingAbove f With tilde element of script upper O left parenthesis normal upper Delta Subscript t Baseline right parenthesis ampersand ModifyingAbove f With tilde left parenthesis o right parenthesis equals 1 right brace 3rd Row 1st Column Blank 2nd Column equals StartFraction StartAbsoluteValue p prime left parenthesis o right parenthesis EndAbsoluteValue squared Over upper B Subscript normal upper Delta Sub Subscript t Subscript Baseline left parenthesis o comma o right parenthesis EndFraction comma EndLayout EndLayout

where $d\lambda _D$ and $d\lambda _{\Delta }$ are the Lebesgue measures on D and $\Delta ,$ respectively. It is clear that $\hat K_{\Delta _t}(o)=e^{t}$ and $B_{\Delta _t}(o,o)=\frac { e^{t}}{\pi }$ , thus statements $(1)$ and $(2)$ hold.

Step 3. $(2)\Rightarrow (1)$

Assume that $\hat {K}_{D_t}(z_0)e^{-t}=c$ is a constant function on $[0,+\infty )$ . Hence, we have $\frac {\int _{t_1}^{t_0}\frac {\pi }{\hat {K}_{D_t}(z_0)}dt}{e^{-t_1}-e^{-t_0}}=\frac {\pi }{c}=e^{t_0}\frac {\pi }{\hat {K}_{D_{t_0}}(z_0)}$ for any $0\le t_1<t_0<+\infty $ . By Proposition 2.20, there exists a holomorphic function F on D such that

$$ \begin{align*}\frac{1}{\hat{K}_{D_t}(z_0)}=\frac{1}{2\pi}\int_{\partial D_t}|F|^2\left(\frac{\partial G_{D}(\cdot,z_0)}{\partial v_z}\right)^{-1}|dz|\end{align*} $$

a.e. on $[0,+\infty )$ , and $F(z_0)=1$ .

For any $f\in \mathcal {O}(\{2G_D(\cdot ,z_0)<-t\})$ satisfying $\int _{\{2G_D(\cdot ,z_0)<-t\}}|f|^2<+\infty $ and $\tilde f(z_0)=1$ , we have

$$ \begin{align*}\frac{1}{2\pi}\int_{\partial D_t}|\tilde f|^2\left(\frac{\partial G_{D}(\cdot,z_0)}{\partial v_z}\right)^{-1}|dz|\ge\frac{1}{\hat{K}_{D_t}(z_0)},\end{align*} $$

which implies that

$$ \begin{align*} \nonumber \begin{aligned} \int_{D_t}|\tilde f|^2d\lambda_D&=\int_{-\infty}^{-t}\int_{D_s}|\tilde f|^2\left(\frac{\partial 2G_{D}(\cdot,z_0)}{\partial v_z}\right)^{-1}|dz|ds\\ &\ge \int_{-\infty}^{-t}\frac{\pi}{\hat K_{D_s}(z_0)} ds\\ &=\int_{-\infty}^{-t}\int_{D_s}|F|^2\left(\frac{\partial 2G_{D}(\cdot,z_0)}{\partial v_z}\right)^{-1}|dz|ds\\ &= \int_{D_t}|F|^2d\lambda_D \end{aligned} \end{align*} $$

StartLayout 1st Row StartLayout 1st Row 1st Column integral Underscript upper D Subscript t Endscripts StartAbsoluteValue f overtilde EndAbsoluteValue squared d lamda Subscript upper D 2nd Column equals integral Subscript negative infinity Superscript negative t Baseline integral Underscript upper D Subscript s Baseline Endscripts StartAbsoluteValue f overtilde EndAbsoluteValue squared left parenthesis StartFraction partial differential 2 upper G Subscript upper D Baseline left parenthesis dot comma z 0 right parenthesis Over partial differential v Subscript z Baseline EndFraction right parenthesis Superscript negative 1 Baseline StartAbsoluteValue d z EndAbsoluteValue d s 2nd Row 1st Column Blank 2nd Column greater than or equals integral Subscript negative infinity Superscript negative t Baseline StartFraction pi Over ModifyingAbove upper K With caret Subscript upper D Sub Subscript s Subscript Baseline left parenthesis z 0 right parenthesis EndFraction d s 3rd Row 1st Column Blank 2nd Column equals integral Subscript negative infinity Superscript negative t Baseline integral Underscript upper D Subscript s Baseline Endscripts StartAbsoluteValue upper F EndAbsoluteValue squared left parenthesis StartFraction partial differential 2 upper G Subscript upper D Baseline left parenthesis dot comma z 0 right parenthesis Over partial differential v Subscript z Baseline EndFraction right parenthesis Superscript negative 1 Baseline StartAbsoluteValue d z EndAbsoluteValue d s 4th Row 1st Column Blank 2nd Column equals integral Underscript upper D Subscript t Baseline Endscripts StartAbsoluteValue upper F EndAbsoluteValue squared d lamda Subscript upper D Baseline EndLayout EndLayout

according to Lemma 2.11. Then we have

$$ \begin{align*} \nonumber \begin{aligned} &\int_{\{2G_{D}(\cdot,z_0)<-t\}}|F|^2d\lambda_D\\ &\quad =\inf\left\{\int_{\{2G_D(\cdot,z_0)<-t\}}|\tilde f|^2d\lambda_D:\mathcal{O}(\{2G_D(\cdot,z_0)<-t\})\,\&\,\tilde f(z_0)=1\right\} \end{aligned} \end{align*} $$

StartLayout 1st Row StartLayout 1st Row 1st Column Blank 2nd Column integral Underscript left brace 2 upper G Subscript upper D Baseline left parenthesis dot comma z 0 right parenthesis less than negative t right brace Endscripts StartAbsoluteValue upper F EndAbsoluteValue squared d lamda Subscript upper D 2nd Row 1st Column Blank 2nd Column equals inf left brace integral Underscript left brace 2 upper G Subscript upper D Baseline left parenthesis dot comma z 0 right parenthesis less than negative t right brace Endscripts StartAbsoluteValue ModifyingAbove f With tilde EndAbsoluteValue squared d lamda Subscript upper D Baseline colon script upper O left parenthesis left brace 2 upper G Subscript upper D Baseline left parenthesis dot comma z 0 right parenthesis less than negative t right brace right parenthesis ampersand ModifyingAbove f With tilde left parenthesis z 0 right parenthesis equals 1 right brace EndLayout EndLayout

for any $t\ge 0$ , which deduces that statement $(1)$ holds by Theorem 1.5.

Step 4. $(1)\Rightarrow (3)$

Assume that statement $(1)$ holds, then there exists a holomorphic function F on D such that

$$ \begin{align*} \nonumber\begin{aligned} G(t)&:=\inf\left\{\int_{\{2G_D(\cdot,z_0)<-t\}}|\tilde f|^2d\lambda_D:\mathcal{O}(\{2G_D(\cdot,z_0)<-t\})\,\&\,\tilde f(z_0)=1\right\}\\ &=\int_{\{2G_{D}(\cdot,z_0)<-t\}}|F|^2 \end{aligned} \end{align*} $$

StartLayout 1st Row StartLayout 1st Row 1st Column upper G left parenthesis t right parenthesis 2nd Column colon equals inf left brace integral Underscript left brace 2 upper G Subscript upper D Baseline left parenthesis dot comma z 0 right parenthesis less than negative t right brace Endscripts StartAbsoluteValue ModifyingAbove f With tilde EndAbsoluteValue squared d lamda Subscript upper D Baseline colon script upper O left parenthesis left brace 2 upper G Subscript upper D Baseline left parenthesis dot comma z 0 right parenthesis less than negative t right brace right parenthesis ampersand ModifyingAbove f With tilde left parenthesis z 0 right parenthesis equals 1 right brace 2nd Row 1st Column Blank 2nd Column equals integral Underscript left brace 2 upper G Subscript upper D Baseline left parenthesis dot comma z 0 right parenthesis less than negative t right brace Endscripts StartAbsoluteValue upper F EndAbsoluteValue squared EndLayout EndLayout

for any $t\ge 0$ and $F(z_0)=1$ according to Theorem 1.5.

Let $p:\Delta \rightarrow D$ be the universal covering map, then there is a holomorphic function $f_{z_0}$ on $\Delta $ such that $|f_{z_0}|=p^*\left (e^{G_D(\cdot ,z_0)}\right )$ . Take $t_0$ large enough such that $D_{t_0}$ is simply connected. By the uniqueness property of the minimal function, we have $F=c\frac {dp_*(f_{z_0})}{dz}$ on $D_{t_0}$ , where c is a constant and $p_*(f_{z_0})$ is well-defined on $D_{t_0}$ . By the uniqueness property of holomorphic form, we have $p^*(Fdz)=cdf_{z_0}$ on $\Delta $ . Hence, we get that $p_*(f_{z_0})$ is single-valued on D, which shows that D is simply connected (see [Reference Suita20]).

Thus, Corollary 1.8 holds.

Acknowledgements

The authors would like to thank Dr. Shijie Bao and Dr. Zhitong Mi for checking the manuscript and providing some useful suggestions. The authors would like to thank the anonymous reviewers’ helpful comments.

Funding statement

The first named author was supported by the National Key R&D Program of China (Grant No. 2021YFA1003100), NSFC-12425101, and NSFC-11825101. The second named author was supported by NSFC-12501106 and China Postdoctoral Science Foundation (Grant Nos. BX20230402 and 2023M743719).

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Article contents

A generalization of the conjugate Hardy $H^2$ spaces

Abstract

Keywords

MSC classification

Information

1. Introduction

1.1. Applications: On planar regions

2. Preparations

2.1. Minimal $L^2$ integrals and Ohsawa–Saitoh–Hardy space

Lemma 2.1 (See [Reference Grauert and Remmert4]).

Lemma 2.2 (See [Reference Guan and Yuan9]).

Lemma 2.3 (See [Reference Guan, Mi and Yuan7], [Reference Guan, Mi and Yuan8]).

Lemma 2.4 (See [Reference Guan, Mi and Yuan7], [Reference Guan, Mi and Yuan8]).

Theorem 2.5 (See [Reference Guan, Mi and Yuan7], [Reference Guan, Mi and Yuan8]).

Lemma 2.6 (See [Reference Guan, Mi and Yuan7], [Reference Guan, Mi and Yuan8]).

2.2. Conjugate Hardy $H^2$ spaces on planar regions

Lemma 2.9 [Reference Rudin17].

Lemma 2.10 (See [Reference Guan and Yuan11]).

Lemma 2.11 (See [Reference Federer3]).

Lemma 2.12 (See [Reference Guan and Yuan12]).

Lemma 2.13 (See [Reference Guan and Yuan10]).

Lemma 2.16 (See [Reference Guan and Yuan12]).

3. Proofs of Theorems 1.4 and 1.5

3.1. Proof of Theorem 1.4.

3.2. Proof of Theorem 1.5.

4. Proof of Theorem 1.6

5. Proof of Corollary 1.8

Acknowledgements

Funding statement

References

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