1.1 Background

The Ohsawa–Takegoshi $L^{2}$ extension theorem concerning the extension problem with $L^{2}$ estimates of holomorphic sections of a Hermitian holomorphic bundle over complex manifolds under certain geometric conditions is one of the most important results in several complex variables and complex geometry. After it was established in [Reference Ohsawa15], many versions of the Ohsawa–Takegoshi type extension theorems have been obtained such as the generalizations given in a series of papers on the extension of $L^{2}$ holomorphic functions by Ohsawa himself [Reference Ohsawa15–Reference Ohsawa17], Manivel–Demailly $L^{2}$ extension theorem [Reference Demailly, Dolbeault, Iordan, Henkin, Skoda and Trépreau3, Reference Manivel13], the $L^{2}$ extension theorems by Siu [Reference Siu, Noguchi, Fujimoto, Kajiwara and Ohsawa21] and Berndtsson [Reference Berndtsson1], the $L^{2}$ extension theorem with gain by McNeal and Varolin [Reference McNeal and Varolin14] and an $L^{2}$ extension theorem of Demailly et al. [Reference Demailly, Hacon and Păun4], and so on. We remark here that recently Guan and Zhou have established in a unified way an optimal version of $L^{2}$ extension theorem [Reference Guan and Zhou6, Reference Guan and Zhou7, Reference Guan and Zhou8] which gives all optimal estimates of the above theorems.

Also various results have been established by using the Ohsawa–Takegoshi extension theorem as well as its optimal version, such as the invariance of plurigenera on projective manifolds [Reference Siu22, Reference Siu23], Demailly’s strong openness conjecture [Reference Guan and Zhou9], the Suita conjecture, L-conjecture, Berndtsson’s theorem [Reference Berndtsson2] on the log-plurisubharmonicity of the relative Bergman kernel (see [Reference Guan and Zhou8]), and so on. We remark here that the last one can be directly derived by the optimal version of $L^{2}$ extension theorem. Also as an application of the above mentioned Guan-Zhou’s unified result about the Ohsawa–Takegoshi extension theorem with optimal estimates, a complete solution of the Suita conjecture [Reference Guan and Zhou8] has been obtained by Guan-Zhou.

In [Reference Ohsawa17], Ohsawa posed the following question about $L^{2}$ extension.

Question. Given a subharmonic function $\psi $ on $\mathbb {C}$ such that

$$ \begin{align*} \int_{\mathbb{C}}e^{-\psi}\bigg(=\int_{\mathbb{C}}e^{-\psi}dxdy\bigg)<\infty, \end{align*} $$

and any subharmonic function $\varphi $ on $\mathbb {C}$ , does there exist a holomorphic function f on $\mathbb {C}$ such that

$$ \begin{align*} f(0)=1 \end{align*} $$

and

$$ \begin{align*} \int_{\mathbb{C}}|f(z)|^{2}e^{-\varphi(z)-\psi(z)}\leq e^{-\varphi(0)}\int_{\mathbb{C}}e^{-\psi} \ ? \end{align*} $$

In the present paper, we give an answer to this question by using Guan-Zhou’s optimal $L^{2}$ extension theorem.

1.2 Notions and notations

In order to state the optimal $L^{2}$ extension theorem which we’ll use, we recall some symbols and notions in [Reference Ohsawa15] and [Reference Ohsawa16]. For the fundamentals on $L^{2}$ method, the reader is referred to [Reference Hörmander11, Reference Hörmander12].

Let M be a complex n-dimensional manifold, and S be a closed complex subvariety of M. Let $dV_{M}$ be a continuous volume form on M. We consider a class of upper-semi-continuous function $\Psi $ from M to the interval $[-\infty ,A)$ , where $A\in (-\infty ,+\infty ]$ , such that

$(1)$ $\Psi ^{-1}(-\infty )\supset S$ , and $\Psi ^{-1}(-\infty )$ is a closed subset of M;

$(2)$ If S is l-dimensional around a point $x\in S_{reg}$ ( $S_{reg}$ is the regular part of S), there exists a local coordinate $(z_{1},\ldots ,z_{n})$ on a neighborhood U of x such that $z_{l+1}=\cdots =z_{n}=0$ on $S\cap U$ and

$$ \begin{align*} \mathrm{sup}_{U\setminus S}|\Psi(z)-(n-l)\log\sum_{l+1}^{n}|z_{j}|^{2}|<\infty. \end{align*} $$

The set of such polar functions $\Psi $ will be denoted by $\#_{A}(S)$ and the subset of plurisubharmonic functions $\Psi $ in $\#_{A}(S)$ will be denoted by $\Delta _{A}(S)$ .

For each $\Psi \in \#_{A}(S)$ , one can associate a positive measure $dV_{M}[\Psi ]$ on $S_{reg}$ as the minimum element of the partially ordered set of positive measures $d\mu $ satisfying

$$ \begin{align*} \int_{S_{l}}fd\mu\geq\limsup_{t\to\infty}\frac{2(n-l)} {\sigma_{2n-2l-1}}\int_{M}fe^{-\Psi}\mathbb{I}_{\{-1-t<\Psi<-t\}}dV_{M} \end{align*} $$

for any nonnegative continuous function f with supp $ f\Subset M$ , where $\mathbb {I}_{\{-1-t<\Psi <-t\}}$ is the characteristic function of the set $\{-1-t<\Psi <-t\}$ . Here we denote by $S_{l}$ the l-dimensional component of $S_{reg}$ , denote by $\sigma _{m}$ the volume of the unit sphere in $\mathbb {R}^{m+1}$ .

Let $\omega $ be a Kähler metric on $M\setminus (X\cup S)$ , where X is a closed subset of M such that $S_{sing}\subset X$ ( $S_{sing}$ is the singular part of S). We can also define a measure $dV_{\omega }[\Psi ]$ on $S\setminus X$ as the minimum element of the partially ordered set of positive measures $d{\mu '}$ satisfying

$$ \begin{align*} \int_{S_{l}}fd{\mu'}\geq\limsup_{t\to\infty}\frac{2(n-l)} {\sigma_{2n-2l-1}}\int_{M\setminus (X\cup S)}fe^{-\Psi}\mathbb{I}_{\{-1-t<\Psi<-t\}}dV_{\omega} \end{align*} $$

for any nonnegative continuous function f with supp $(f)\Subset M\setminus X$ . As

$$ \begin{align*} \{-1-t<\Psi<-t\} \cap \mathrm{supp}(f)\Subset M\setminus (X\cup S), \end{align*} $$

the right-hand side of the above inequality is well-defined.

Let u be a continuous section of $K_{M}\otimes E$ , where E is a holomorphic vector bundle equipped with a continuous metric h on M. We define

$$ \begin{align*} |u|^{2}_{h}|_{V}:=\frac{c_{n}h(e,e)v\wedge\bar{v}}{dV_{M}}, \end{align*} $$

and

$$ \begin{align*} |u|^{2}_{h,\omega}|_{V}:=\frac{c_{n}h(e,e)v\wedge\bar{v}}{dV_{\omega}}, \end{align*} $$

where $u|_{V}=v\otimes e$ for an open set $V\subset M\setminus (X\cup S)$ , v is a continuous section of $K_{M}|_{V}$ , e is a continuous section of $E|_{V}$ and $c_{n}=i^{n^{2}}$ such that $c_{n} v\wedge \bar {v}$ is positive (especially, we define

$$ \begin{align*} |u|^{2}|_{V}:=\frac{c_{n}u\wedge\bar{u}}{dV_{M}}, \end{align*} $$

when u is a continuous section of $K_{M}$ ). It is clear that $|u|^{2}_{h}$ is independent of the choices of V. In fact, one may see the relationship between $dV_{\omega }[\Psi ]$ and $dV_{M}[\Psi ]$ (respectively, $dV_{\omega }$ and $dV_{M}$ ). It is clear that

$$ \begin{align*}\int_{M\setminus(X\cup S)}f|u|^{2}_{h,\omega}dV_{\omega}[\Psi]& =\int_{M\setminus(X\cup S)}f|u|^{2}_{h}dV_{M}[\Psi],\\\Bigg(\text{resp.} \int_{M\setminus(X\cup S)}f|u|^{2}_{h,\omega}dV_{\omega} &=\int_{M\setminus(X\cup S)}f|u|^{2}_{h}dV_{M},\Bigg),\end{align*} $$

where f is a continuous function with compact support on $M\setminus X$ .

It is clear that $|u|^{2}_{h}$ is independent of the choices of U, while $|u|^{2}_{h}dV_{M}$ is independent of the choices of $dV_{M}$ (respectively, $|u|^{2}_{h}dV_{M}[\Psi ]$ is independent of the choices of $dV_{M}$ ).

1.3 The optimal $L^{2}$ extension theorem

In [Reference Guan and Zhou8], Guan and Zhou found an Ohsawa–Takegoshi type $L^{2}$ extension theorem with an optimal $L^{2}$ estimate, which is stated as follows.

Let $c_{A}(t)$ be a positive function in $C^{\infty }((-A,+\infty ))$ $(A\in (-\infty ,+\infty ])$ , satisfying

$$ \begin{align*} \int_{-A}^{\infty}c_{A}(t)e^{-t}dt<\infty \end{align*} $$

and

(1.1) $$ \begin{align} (\int_{-A}^{t}c_{A}(t_{1})e^{-t_{1}}dt_{1})^{2}>c_{A}(t)e^{-t} \int_{-A}^{t}\int_{-A}^{t_{2}}c_{A}(t_{1})e^{-t_{1}}dt_{1}dt_{2}, \end{align} $$

for any $t\in (-A,+\infty )$ , that is,

$$ \begin{align*} F(t)=\log\bigg(\int_{-A}^{t}\int_{-A}^{t_{2}}c_{A}(t_{1})e^{-t_{1}}dt_{1}dt_{2}\bigg) \end{align*} $$

is strictly concave for any $t\in (-A,+\infty )$ .

An easy example of such functions $c_{A}(t)$ is when $c_{A}(t)e^{-t}$ is decreasing with respect to t.

Theorem 1.5. Let $(M,S)$ be an almost Stein pair, and $\Psi $ be a plurisubharmonic function in $\Delta _{A}(S)\cap C^{\infty }(M\setminus (S\cup X))$ (X is as in the definition of almost Stein pair). Let h be a smooth metric on a holomorphic vector bundle E on M with rank r, such that $he^{-\Psi }$ is semi-positive in the sense of Nakano on $M\setminus (S\cup X)$ (when E is a line bundle, h can be chosen as a semipositive singular metric). Then there exists a uniform constant $\mathbf {C}=1$ , which is optimal, such that, for any holomorphic section f of $K_{M}\otimes E|_{S}$ on S satisfying condition that

$$ \begin{align*} \sum_{k=1}^{n}\frac{\pi^{k}}{k!}\int_{S_{n-k}}|f|^{2}_{h}dV_{M}[\Psi]<\infty, \end{align*} $$

there exists a holomorphic section F of $K_{M}\otimes E$ on M satisfying $F = f$ on $ S$ and

$$ \begin{align*} \int_{M}c_{A}(-\Psi)|F|^{2}_{h}dV_{M} \leq\mathbf{C}\int_{-A}^{\infty}c_{A}(t)e^{-t}dt\sum_{k=1}^{n}\frac{\pi^{k}}{k!}\int_{S_{n-k}}|f|^{2}_{h}dV_{M}[\Psi]. \end{align*} $$

As we have mentioned, various results have been derived from the above theorem. A survey about the optimal $L^{2}$ extension and its applications is given in [Reference Zhou24].

Especially, when we take M to be a pseudoconvex domain D in $\mathbb {C}^{n}$ with coordinates $(z_{1},z_{2},\ldots ,z_{n})$ , H be the intersection of the set $\{z_{n-l+1}=\cdots =z_{n}=0\}$ with D, let $dV_{D}$ and $dV_{H}$ be the Lebesgue measures on D and $H,$ respectively, $\Psi (z)=l\log (|z_{n-l+1}|^{2}+\cdots +|z_{n}|^{2})$ , then $dV_{D}[l\log (|z_{n-l+1}|^{2}+\cdots +|z_{n}|^{2})] = dV_{H}$ . For any positive function $c_{A}(t)$ that satisfies (1.1), Theorem 1.5 degenerates to the following.

Theorem 1.6. Let $D\subset \mathbb {C}^{n}$ be a pseudoconvex domain, let l be an integer between $1$ and n, let $\varphi $ be a plurisubharmonic function on $D,$ and let $H=D\cap \{z_{n}=\cdots =z_{n-l+1}=0\}$ . Then for any holomorphic function f on H satisfying

$$ \begin{align*} \int_{H}|f|^{2}e^{-\varphi}dV_{H}<\infty, \end{align*} $$

there exists a holomorphic function F on D satisfying $F=f$ on H and

$$ \begin{align*} \int_{D}c_{A}(-l\log(|z_{n}|^{2}+\cdots+|z_{n-l+1}|^{2}))|F|^{2}e^{-\varphi}dV_{D} \leq\frac{\pi^{l}}{l!}\int_{-A}^{\infty}c_{A}(t)e^{-t}dt\int_{H}|f|^{2}e^{-\varphi}dV_{H}. \end{align*} $$

By using Theorem 1.6, we obtain the following

Theorem 1.7. If $D\subset \mathbb {C}^{n}$ be a pseudoconvex domain, l be an integer between $1$ and n, $H=D\cap \{z_{n-l+1}=\cdots =z_{n}=0\}$ , $\psi (z)=\rho ((\sum _{j=n-l+1}^{n}|z_{j}|^{2})^{l/2})$ where $\rho $ is a radial subharmonic function on $\mathbb {C}$ . Then for every plurisubharmonic function $\varphi $ on D and every holomorphic function f on H such that

$$ \begin{align*} \int_{H}|f|^{2} e^{-\varphi}dV_{H}<\infty, \end{align*} $$

there exists a holomorphic function F on D such that

$$ \begin{align*} F=f \end{align*} $$

on H and

(1.2) $$ \begin{align} \int_{D}|F|^{2} e^{-\varphi-\psi}dV_{D}\leqslant 2{\sigma_{l}}\int_{0}^{e^{\frac{A}{2}}}e^{-\rho(t)}tdt \int_{H}|f|^{2} e^{-\varphi}dV_{H}, \end{align} $$

where $A=\mathrm {sup}_{z\in D} l\log (\sum _{j=n-l+1}^{n}|z_{j}|^{2})$ .

When $n=1$ , as a corollary of Theorem 1.7, we conclude that the question of Ohsawa holds under a natural condition.

Corollary 1.9. If $\psi $ is a subharmonic function on $\mathbb {C}$ such that $i\partial \bar \partial \psi $ is rotational invariant and satisfies

$$ \begin{align*} \int_{\mathbb{C}}e^{-\psi}<\infty, \end{align*} $$

then for every subharmonic function $\varphi $ on $\mathbb {C}$ , there exists a holomorphic function f on $\mathbb {C}$ such that

$$ \begin{align*} f(0)=1 \end{align*} $$

and

$$ \begin{align*} \int_{\mathbb{C}}|f|^{2} e^{-\varphi-\psi}\leqslant e^{-\varphi(0)}\int_{\mathbb{C}} e^{-\psi}. \end{align*} $$

However, without the condition that $i\partial \bar \partial \psi $ is rotational invariant, the above question of Ohsawa fails in general. Recently, Guan gave a counterexample in [Reference Guan5] by showing that there exist subharmonic functions $\psi $ and $\varphi $ on $\mathbb {C}$ such that the Bergman space $A^{2}(\mathbb {C},e^{-\varphi -\psi })$ with respect to $\varphi +\psi $ contains only zero. In the last section, we will give another counterexample with nontrivial $A^{2}(\mathbb {C},e^{-\varphi -\psi })$ .