# A bridge connecting convex analysis and complex analysis and
$L^{2}$-estimate of $d$ and $\bar{\partial}$

Fusheng Deng  Fusheng Deng: School of Mathematical Sciences, University of
Chinese Academy of Sciences
Beijing 100049, P. R. China<EMAIL_ADDRESS>,  Jinjin Hu  Jinjin Hu:
School of Mathematical Sciences, University of Chinese Academy of Sciences
Beijing 100049, P. R. China<EMAIL_ADDRESS>,  Weiwen Jiang
Weiwen Jiang: Institute of Mathematics, Academy of Mathematics and Systems
Science, Chinese Academy of Sciences, Beijing 100190, P. R. China
<EMAIL_ADDRESS>and  Xiangsen Qin  Xiangsen Qin: School of Mathematical
Sciences, University of Chinese Academy of Sciences
Beijing 100049, P. R. China<EMAIL_ADDRESS>
###### Abstract.

We propose a way to connect complex analysis and convex analysis. As
applications, we derive some results about $L^{2}$-estimate for $d$-equation
and prove some curvature positivity related to convex analysis from well known
$L^{2}$-estimate for $\bar{\partial}$-equation or the results we prove in
complex analysis.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Notations and conventions
    2. 2.2 Regularized max function
    3. 2.3 Curvature positivity of Hermitian holomorphic vector bundles
    4. 2.4 $L^{2}$-estimates in complex analysis
    5. 2.5 Curvature positivity of real vector bundles
    6. 2.6 Some basic knowledge from measure theory
  3. 3 The proof of Theorem 1.1
  4. 4 The proof of Theorem 1.2
  5. 5 The proof of Theorem 1.3
  6. 6 The proof of Theorem 1.4
  7. 7 The proof of Theorem 1.6

##  1\. Introduction

It is well known that complex analysis and convex analysis are related or
analogous in some aspects. For example, a function $\varphi(z)$ on
$\mathbb{C}^{n}$ that is independent of the imaginary part of $z$ is
plurisubharmonic if and only if it is convex when viewed as a function on
$\mathbb{R}^{n}$, and a tube domain of the form
$D+i\mathbb{R}^{n}\subset\mathbb{C}^{n}$ is pseudoconvex if and only if its
base $D$ is a convex domain in $\mathbb{R}^{n}$. In the present paper, we
propose another principle that connecting complex analysis and convex analysis
in a certain way. As applications, we show that some important (both old and
new) results in convex analysis can be deduced from analogous results in
complex analysis.

The principle can be described as follows. Starting from a domain (connected
open set) $D\subset\mathbb{R}^{n}$, one can form a tube domain
$T_{D}:=D+i\mathbb{R}^{n}$ in $\mathbb{C}^{n}$. Now the domain $T_{D}$ admits
the obvious symmetry $\mathbb{R}^{n}$ given by translations along the
imaginary part. But the trouble is that it is difficult to make use of the
symmetry since $\mathbb{R}^{n}$ is noncompact as a Lie group. So we take a
step forward by considering the domain

$R_{D}:=\exp T_{D}=\\{(e^{z_{1}},\cdots,e^{z_{n}})|\ (z_{1},\cdots,z_{n})\in
T_{D}\\},$

which is a Reinhardt domain in $\mathbb{C}^{n}$ that has no intersection with
the coordinate axes. Now the key point is that translation symmetry of $T_{D}$
induces rotation symmetry of $R_{D}$ given by the compact Lie group
$\mathbb{T}^{n}$, the torus group of dimension $n$. Through this way, we get a
one to one correspondence between domains in $\mathbb{R}^{n}$ and Reinhardt
domains in $(\mathbb{C}^{*})^{n}$. It follows that, in principle, problems
about $D$ can be translated to analogous rotation-invariant problems about
$R_{D}$, and vice versa.

This principle has been implicitly used in [8] and [11] for the study of
positivity of direct image sheaves, and has applied in [9], [10] deducing some
important results about real analysis from analogous results from complex
analysis.

The above principle has a lot of other applications. In this paper, we give
some typical examples to show the power of it.

The first example is the following

######  Theorem 1.1.

Let $D\subset\mathbb{R}_{x}^{n}$ be a convex domain,
$V\subset\mathbb{C}_{\tau}^{m}$ be a pseudoconvex domain, and let
$W_{z}:=D+i\mathbb{R}_{y}^{n}\subset\mathbb{C}_{z}^{n}.$ Let
$\varphi(\tau,z),\ \psi(\tau,z)$ be plurisubharmonic functions on $V\times W,$
which are independent of $\operatorname{Im}(z)$ (the imaginary part of $z$)
for any $z\in W$. We assume that $\psi$ is $C^{2}$ and is strictly
plurisubharmonic with respect to $\tau$. Then for any $\bar{\partial}$-closed
measurable $(0,1)$-form $f:=\sum^{n}_{j=1}f_{j}d\bar{\tau}_{j}$ in $V$, we can
solve the equation $\bar{\partial}u=f$ in $V$ with the estimate

$\int_{V\times D}|u|^{2}e^{-\varphi-\psi}d\lambda\leq\int_{V\times
D}\sum^{n}_{j,k=1}\psi^{j\bar{k}}f_{j}\bar{f}_{k}e^{-\varphi-\psi}d\lambda,$

provided the right hand side is finite, where
$\left(\psi^{j\bar{k}}\right)_{n\times
n}:=\left(\frac{\partial^{2}\psi}{\partial\tau_{j}\partial\bar{\tau}_{k}}\right)^{-1}_{n\times
n}$.

Theorem 1.1 is a slight generalization of [1, Proposition 2.3], which plays an
essential role in Berndtsson’s proof of a Prékopa-type inequality for
plurisubharmonic functions on tube domains. Berndtsson’s original proof
involves taking limit of balls in $\mathbb{R}^{n}$ and an infinite dimensional
version of the Arzelà-Ascoli Theorem. Our proof of Theorem 1.1, based on the
principle mentioned above, will be very simple and different. Following
Berndtsson’s method, Inayama proved an analogue of Theorem 1.1 for
$L^{2}$-extension of holomorphic functions.

######  Theorem 1.2 ([15, Theorem 1.1]).

For any $r>0,$ let $\Delta:=\\{\tau\in\mathbb{C}|\ |\tau|<r\\}.$ Let
$D\subset\mathbb{R}_{x}^{n}$ be a bounded convex domain, $\varphi(\tau,z)$ be
a plurisubharmonic function on
$\Delta_{\tau}\times(D_{x}+i\mathbb{R}^{n}_{y}),$ which is independent of
$\operatorname{Im}(z)$. Then there exists a holomorphic function $f$ on
$\Delta$ satisfying $f(0)=1$ and

$\int_{\Delta\times D}|f(\tau)|^{2}e^{-\varphi(\tau,x)}d\lambda(\tau,x)\leq\pi
r^{2}\int_{D}e^{-\varphi(0,x)}d\lambda(x),$

provided the right hand side is finite.

We will show that, combing the optimal $L^{2}$-extension theorems for
holomorphic functions in [4], [13], Theorem 1.2 can be also deduced from the
above principle directly. Of course, the disc $\Delta$ in Theorem 1.2 can be
replaced by general bounded domains in $\mathbb{C}^{k}$, with the form of the
estimate modified accordingly. But to clarify the main idea of the proof, we
only consider the case $k=1$.

We now turn to the discussion of $L^{2}$-estimate for the $d$-operator on
convex domains. In complex analysis, the $L^{2}$-estimate of
$\bar{\partial}$-equation on general pseudoconvex domains was established by
Hörmander in 1965 in his fundamental work [14]. Hörmander’s result was
generalized to much more general context by Demailly and other authors in the
early eighties (see e.g. [7]). Surprisingly, one had not seen analogous work
about the $L^{2}$-estimate for the $d$-operator until a decade later when
Brascamp and Lieb proved a parallel result in [5] on $\mathbb{R}^{n}$. The
$L^{2}$-estimate for the $d$-operator on general convex domains was proved
only recently in [16] for line bundles. For positively curved vector bundles
on the whole $\mathbb{R}^{n}$, the $L^{2}$-estimate for the $d$-operator
parallel to Brascamp-Lieb’s result is given in [6] in 2019. The main
motivation to [5], [6] is to prove the Brunn-Minkowski type result in convex
analysis for Riemannian vector bundles with positive curvature. As indicated
in Hörmander’s work, generalizing an estimate for the $d$-operator from the
whole $\mathbb{R}^{n}$ to general convex domains may require a highly
nontrivial effort to prove some integral inequalities for certain forms that
are not smooth. In the present paper, we generalize the result in [6] from
$\mathbb{R}^{n}$ to general convex domains very easily by applying well known
results about the $L^{2}$-estimate of $\bar{\partial}$ and the above proposed
principle.

######  Theorem 1.3.

Let $D\subset\mathbb{R}^{n}$ be a convex domain, and let $(E,g)$ be a Nakano
positive Riemannian vector bundle of finite rank $r$ over $D$, then for any
$d$-closed $f\in L^{2}(D,\Lambda^{1}T^{*}D\otimes E),$ there exists $u\in
L^{2}(D,E)$ with $du=f$ and satisfying the estimate

$\int_{D}|u|_{g}^{2}d\lambda\leq\int_{D}\langle(\Theta^{(E,g)})^{-1}f,f\rangle_{g}d\lambda.$

In the above theorem,

$\Theta^{(E,g)}:=\left(\theta_{jk}^{(E,g)}\right)_{n\times n},\
\theta_{jk}^{(E,g)}:=-\frac{\partial}{\partial
x^{k}}\left(g^{-1}\frac{\partial g}{\partial x^{j}}\right)$

is the curvature operator of $(E,g)$. We will give the definition of the
involved Nakano positivity of $(E,g)$ later (see Definition 2.2).

Theorem 1.3 can be generalized to closed $E$-valued forms of higher degree via
the same argument. We omit the exact formulation for the general context here.
The main idea in the proof of Theorem 1.3 is as follows. We transform from $D$
to $R_{D}$ and then solve the analogous $\bar{\partial}$-equation on $R_{D}$
with the $L^{2}$-estimate known in complex analysis. Since everything on
$R_{D}$ that we are handling is invariant under the natural torus action on
$R_{D}$, the uniqueness of the minimal solution implies that the minimal
solution of the $\bar{\partial}$-equation is also invariant under the torus
action. It follows that the minimal solution on $R_{D}$ induces a solution of
the $d$-equation on $D$ with the desired $L^{2}$-estimate.

Now we consider the curvature strict positivity of the direct image bundles
associated to a strictly pseudoconvex family of bounded domains. We consider
the families that all fibers are circular domains or Reinhardt domains. Recall
that a domain $D\subset\mathbb{C}^{m}$ is called a circular domain if it is
invariant under the action of $\mathbb{S}^{1}$ on $\mathbb{C}^{m}$ given by

$e^{i\theta}\cdot(z_{1},\cdots,z_{m}):=(e^{i\theta}z_{1},\cdots,e^{i\theta}z_{m}),\
\theta\in\mathbb{R},$

and is called a Reinhardt domain if it is invariant under the action of the
torus group $\mathbb{T}^{m}$ on $\mathbb{C}^{m}$ given by

$(e^{i\theta_{1}},\cdots,e^{i\theta_{m}})\cdot(z_{1},\cdots,z_{m}):=(e^{i\theta_{1}}z_{1},\cdots,e^{i\theta_{m}}z_{m}),\
\theta_{i}\in\mathbb{R}.$

######  Definition 1.1.

Let $U\subset\mathbb{C}^{n}$ be a domain, and let
$p\colon\mathbb{C}^{n}\times\mathbb{C}^{m}\rightarrow\mathbb{C}^{n}$ be the
natural projection.

  * (1)

A _family of bounded domains_ (of dimension $m$ over $U$) is a domain
$\Omega\subset U\times\mathbb{C}^{m}$ such that $p(\Omega)=U$ and all fibers
$\Omega_{t}:=p^{-1}(t)\subset\mathbb{C}^{m}\ (t\in U)$ are bounded.

  * (2)

A family of bounded domains $\Omega$ over $U$ has _ $C^{2}$ boundary _if there
exists a $C^{2}$-function $\rho$ defined on $U\times\mathbb{C}^{m}$ such that
$\Omega=\\{(t,z)\in U\times\mathbb{C}^{m}|\ \rho(t,z)<0\\}$ and
$d(\rho|_{\Omega_{t}})\neq 0$ on $\partial\Omega_{t}$ for all $t\in U$. Such a
function $\rho$ is called a _defining function_ of $\Omega$.

  * (3)

A family of bounded domains $\Omega\subset U\times\mathbb{C}^{m}$ is called
_strictly pseudoconvex_ if it admits a $C^{2}$-boundary defining function that
is strictly plurisubharmonic on some neighborhood of $\overline{\Omega}$ in
$U\times\mathbb{C}^{m}$.

For any $k\in\mathbb{N}$, let $\rm\rm P^{k}$ be the space of homogenous
polynomials on $\mathbb{C}^{m}$ of degree $k$.

######  Theorem 1.4.

Let $\Omega\subset U\times\mathbb{C}^{m}$ be a strictly pseudoconvex family of
bounded domains over $U\subset\mathbb{C}^{n}$, and let $(F,h^{F})$ be a Nakano
semi-positive trivial Hermitian vector bundle of finite rank $r$ over some
neighborhood of $\overline{\Omega}$ in $U\times\mathbb{C}^{m}$. We assume that
all fibers $\Omega_{t}\ (t\in U)$ are (connected) circular domains in
$\mathbb{C}^{m}$ containing the origin and $h^{F}_{(t,z)}$ is
$\mathbb{S}^{1}$-invariant with respect to $z$. Let $\\{e_{1},\cdots,e_{r}\\}$
be the canonical holomorphic frame of $F$. For any $k\in\mathbb{N},\ t\in U$,
set

$E^{k}_{t}:=\\{f=\sum_{\lambda=1}^{r}f_{\lambda}\otimes e_{\lambda}|\
f_{\lambda}\in\rm\rm P^{k}\text{ for all }\lambda\\}$

with inner product $h_{t}$ given by

$h_{t}(f,g):=\int_{\Omega_{t}}\sum_{\lambda,\mu=1}^{r}f_{\lambda}(z)\overline{g_{\mu}(z)}h^{F}_{(t,z)}(e_{\lambda},e_{\mu})d\lambda(z),\
\forall f,g\in E^{k}_{t}.$

We set $E^{k}:=\cup_{t\in U}E^{k}_{t}$ and view it as a (trivial) holomorphic
vector bundle over $U$ in a natural way, then the curvature of $(E^{k},h)$ is
strictly positive in the sense of Nakano.

Considering strictly pseudoconvex family of Reinhardt domains, we have the
following parallel result, whose proof is similar to that of Theorem 1.4, so
we omit it.

######  Theorem 1.5.

Let $\Omega\subset U\times\mathbb{C}^{m}$ be a strictly pseudoconvex family of
bounded domains over a domain $U\subset\mathbb{C}^{n},$ and let $(F,h^{F})$ be
a Nakano positive trivial Hermitian vector bundle of finite rank $r$ defined
on some neighborhood of $\overline{\Omega}$. We assume that all fibers
$\Omega_{t}\ (t\in U)$ are (connected) Reinhardt domains and $h^{F}_{(t,z)}$
is $\mathbb{T}^{n}$-invariant with respect to $z$. Let
$\\{e_{1},\cdots,e_{r}\\}$ be the canonical holomorphic frame of $F$. For any
$k_{1},\cdots,k_{m}\in\mathbb{N},\ t\in U$, set

$E^{k_{1},\cdots,k_{m}}_{t}:=\\{f=\sum_{\lambda=1}^{r}f_{\lambda}\otimes
e_{\lambda}|\ f_{\lambda}\in\mathbb{C}z_{1}^{k_{1}}\cdots z_{m}^{k_{m}}\text{
for all }\lambda\\}$

with inner product $h_{t}$ given by

$h_{t}(f,g):=\int_{\Omega_{t}}\sum_{\lambda,\mu=1}^{r}f_{\lambda}(z)\overline{g_{\mu}(z)}h^{F}_{(t,z)}(e_{\lambda},e_{\mu})d\lambda(z),\
\forall f,g\in E^{k_{1},\cdots,k_{m}}_{t}.$

We set

$E^{k_{1},\cdots,k_{m}}:=\mathop{\cup}_{t\in U}E^{k_{1},\cdots,k_{m}}_{t}$

and view it as a (trivial) holomorphic vector bundle over $U$ in a natural
way, then the curvature of $(E^{k_{1},\cdots,k_{m}},h)$ is strictly positive
in the sense of Nakano.

In Theorem 1.4 and Theorem 1.5, the vector bundle $F$ is assumed to be trivial
just for simplicity of presentation.

We should remark that the strict positivity of the curvature of the direct
images in Theorem 1.4 and Theorem 1.5 comes from the strict pseudoconvexity of
the families, but not from the curvature positivity of $(F,h^{F})$. Similar
considerations have appeared in [19] and [9] for direct images of Hermitian
line bundles. The proofs of Theorem 1.4 and Theorem 1.5 rely on a development
of the method in [9]. Our argument depends on Deng-Ning-Wang-Zhou’s integral
characterization of Nakano positivity in [12] and is different from the method
in [19].

Assume that $D\subset\mathbb{C}^{m}$ is a bounded circular domain containing
the origin and $\varphi$ is an $\mathbb{S}^{1}$-invariant continuous function
on $\overline{D}$. Let $A^{2}(D,\varphi)$ be the space of $L^{2}$ holomorphic
functions on $D$ with weight $e^{-\varphi}$. Then ${\rm P}^{k}\subset
A^{2}(D,\varphi)$ for $k\geq 0$, and ${\rm P}^{k}\bot{\rm P}^{l}$ provided
that $k\neq l$. Moreover, we have the decomposition

$A^{2}(D,\varphi)=\overline{\mathop{\bigoplus}_{k}}{\rm P}^{k},$

where $\overline{\bigoplus}$ means taking orthogonal direct sum first and then
taking closure. In terms of representation theory, $\mathbb{S}^{1}$ has a
natural unitary representation on $A^{2}(D,\varphi)$ and $\rm P^{k}$ is the
submodule corresponding to the irreducible character $\alpha\mapsto\alpha^{k}$
of $\mathbb{S}^{1}$. For Reinhardt domains, we have similar decompositions by
considering representation of the torus group $\mathbb{T}^{m}$. In this
meaning, Theorem 1.4 and Theorem 1.5 can be generalized to any compact group
actions. The case of the trivial group action (without group action) is
carried out in detail by the last author [17].

If we just consider curvature positivity but not strict positivity of the
direct image, Theorem 1.4 and Theorem 1.5 are due to Deng-Zhang-Zhou [11] for
the line bundle case, based on the fundamental result of Berndtsson in [3].

Using our principle and Theorem 1.5, we can get the strict curvature
positivity of the direct images of Riemannian vector bundles over a strict
convex family of bounded domains in $\mathbb{R}^{m}$.

Recall that a $C^{2}$ function defined on a domain in $\mathbb{R}^{n}$ is
called strictly convex if its real Hessian is positive definite everywhere.

######  Definition 1.2.

Let $U\subset\mathbb{R}^{n}$ be a domain, and let
$p\colon\mathbb{R}^{n}\times\mathbb{R}^{m}\rightarrow\mathbb{R}^{n}$ be the
natural projection.

  * (1)

A _family of bounded domains_ (of dimension $m$ over $U$) is a domain
$D\subset U\times\mathbb{R}^{m}$ such that $p(D)=U$ and all fibers
$D_{t}:=p^{-1}(t)(t\in U)$ are bounded.

  * (2)

A family of bounded domains $D$ over $U$ has _ $C^{2}$ boundary _if there
exists a $C^{2}$-function $\rho$ defined on $U\times\mathbb{R}^{m}$ such that
$D=\\{(t,x)\in U\times\mathbb{R}^{m}|\ \rho(t,x)<0\\}$ and
$d(\rho|_{D_{t}})\neq 0$ on $\partial D_{t}$ for all $t\in U$. Such a function
$\rho$ is called a _defining function_ of $D$.

  * (3)

A family of bounded domains $D\subset U\times\mathbb{C}^{m}$ is called
_strictly convex_ if it admits a $C^{2}$-boundary defining function that is
strictly convex on some neighborhood of $\overline{D}$.

######  Theorem 1.6.

Let $D\subset U_{0}\times\mathbb{R}^{m}$ be a strictly convex family of
bounded domains (with connected fibers) over a domain
$U_{0}\subset\mathbb{R}^{n},$ and let $(F,g^{F})$ be a trivial vector bundle
of finite rank $r$ defined on some neighborhood of $\overline{D}$. Let
$\\{e_{1},\cdots,e_{r}\\}$ be the canonical frame of $F$. For any $t\in
U_{0}$, set $E_{t}:=\mathbb{R}^{r}$, with an inner product $g_{t}^{E}$ given
by

$g_{t}^{E}(u,v):=\int_{D_{t}}\sum_{\lambda,\mu=1}^{r}u_{\lambda}v_{\mu}g^{F}_{(t,x)}(e_{\lambda},e_{\mu})d\lambda(x)$

for all

$u:=(u_{1},\cdots,u_{r})\in E_{t},\ v:=(v_{1},\cdots,v_{r})\in E_{t}.$

We set $E:=\cup_{t\in U_{0}}E_{t}$ and view it as a Riemannian (trivial)
vector bundle over $U_{0}$, then the curvature of $(E,g^{E})$ is strictly
positive in the sense of Nakano if $(F,g^{F})$ is Nakano positive on some
neighborhood of $\overline{D}$.

The curvature positivity of $(E,g^{E})$ is originally proved in [18] in the
case that $D=\mathbb{R}^{n}\times\mathbb{R}^{m}$ is a product, based on the
idea of [3] and Fourier transform, and is proved by different methods in [6]
and [8]. Here we are interested in the curvature strict positivity of
$(E,g^{E})$ and on deriving it from Theorem 1.5 following the principle
proposed in the beginning of this section.

### Acknowledgements

This research is supported by National Key R&D Program of China (No.
2021YFA1003100), NSFC grants (No. 11871451, 12071310), and the Fundamental
Research Funds for the Central Universities.

##  2\. Preliminaries

In this section, we fix some notations, conventions and collect some knowledge
that are needed in our discussions.

###  2.1. Notations and conventions

Our convection for $\mathbb{N}$ is that $\mathbb{N}:=\\{0,1,2,3,\cdots\\}$,
and set $\mathbb{C}^{*}:=\\{z\in\mathbb{C}|\ z\neq 0\\}.$ For any complex
number $z\in\mathbb{C}$, let $\operatorname{Re}(z)$ (resp.
$\operatorname{Im}(z)$) denote the real part (rep. imaginary part) of $z$. For
any $z:=(z_{1},\cdots,z_{n})\in\mathbb{C}^{n}$, set

$\operatorname{Re}(z):=(\operatorname{Re}(z_{1}),\cdots,\operatorname{Re}(z_{n})),\
\operatorname{Im}(z):=(\operatorname{Im}(z_{1}),\cdots,\operatorname{Im}(z_{n})).$

For any subset $A\subset\mathbb{R}^{n}$, let $\overline{A}$ denote the closure
of $A$ in $\mathbb{R}^{n}$, and let $\partial A:=\overline{A}\setminus A$. Any
neighborhood of a subset $A\subset\mathbb{R}^{n}$ is open by default. We say
$U$ is a domain of $\mathbb{R}^{n}$ if it is a connected open subset of
$\mathbb{R}^{n}$.
For any $k\in\mathbb{N}\cup\\{\infty\\}$. If $U\subset\mathbb{R}^{n}$ is an
open subset, then we let $C^{k}(U)$ be the space of complex valued functions
which are of class $C^{k}$. More generally, if $A\subset\mathbb{R}^{n}$ is an
arbitrary subset, let $C^{k}(A)$ denote the space of complex valued functions
which is of class $C^{k}$ in an open neighborhood of $A$. We write
$A\subset\subset U$ if $\overline{A}$ is a compact subset of $U$. A real
valued function $\varphi\in C^{2}(U)$ is strictly convex if its real Hessian
$\left(\varphi_{jk}\right)_{1\leq j,k\leq n}$ is positive definite for any
$x\in U$, where

$\varphi_{jk}:=\frac{\partial^{2}\varphi}{\partial x_{j}\partial x_{k}},\
1\leq j,k\leq n,$

and we let $\left(\varphi^{jk}\right)_{1\leq j,k\leq n}$ be the inverse matrix
of its real Hessian. We recall that an open subset $U\subset\mathbb{R}^{n}$ is
strictly convex if it has $C^{2}$-boundary, and it has a $C^{2}$-boundary
defining function which is strictly convex in a neighborhood of
$\overline{U}$.
If $U\subset\mathbb{C}^{n}$ is a open subset, let $\mathcal{O}(U)$ denote the
space of holomorphic functions in $U$. We recall an upper semi-continuous
function $f\colon U\rightarrow[-\infty,\infty)$ is plurisubharmonic if its
restriction to every complex line in $U$ is subharmonic, i.e. satisfying the
submean value inequality. A real valued function $\varphi\in C^{2}(U)$ is
strictly plurisubharmonic if its complex Hessian
$\left(\varphi_{j\bar{k}}\right)_{1\leq j,k\leq n}$ is positive definite for
any $z\in U$, where

$\varphi_{j\bar{k}}:=\frac{\partial^{2}\varphi}{\partial
z_{j}\partial\bar{z}_{k}},\ 1\leq j,k\leq n,$

and we let $\left(\varphi^{j\bar{k}}\right)_{1\leq j,k\leq n}$ be the inverse
matrix of its complex Hessian. We recall that $U$ is pseudoconvex if it has a
smooth plurisubharmonic function $p$ such that its sublevel sets are
relatively compact in $U$, i.e.

$\\{z\in U|\ p(z)\leq c\\}\subset\subset U$

for any $c\in\mathbb{R}$. $U$ is strictly pseudoconvex if it has
$C^{2}$-boundary, and it has a $C^{2}$-boundary defining function which is
strictly plurisubahrmonic in a neighborhood of $\overline{U}$.
Let $d\lambda$ denote the Lebesgue measure on $\mathbb{R}^{n}$. Note that all
measurable sets, measurable maps, measurable differential forms will be
Lebesgue measurable. If $A\subset\mathbb{R}^{n}$ is a measurable subset, we
also write $|A|$ for the measure of $A.$ If an open subset
$U\subset\mathbb{R}^{n}$ has $C^{1}$-boundary, let $dS$ denote the surface
measure on $\partial U$.
For any open subset $U\subset\mathbb{R}^{n}$, let $L^{2}(U)$ denote the space
of $L^{2}$-integrable complex valued functions on $U$. For any $f\in
L^{2}(U)$, let $\|f\|_{L^{2}(U)}$ denote the $L^{2}$-norm of $f$.
For any open subset $U\subset\mathbb{R}^{n}$ (resp. $U\subset\mathbb{C}^{n}$),
and for any $k\in\mathbb{N}$ (resp. $p,q\in\mathbb{N}$), let
$\Lambda^{k}T^{*}U$ (resp. $\Lambda^{p,q}T^{*}U)$ denote the bundle of smooth,
complex valued $k$-forms (resp. $(p,q)$-forms) on $U$.
If $(X,\omega)$ is a Kähler manifold, then we write $d\lambda(\omega)$ for the
Riemannian volume form determined by $\omega$.
A metric on a vector bundle will be of class $C^{2}$, all sections of a vector
bundle will be global unless stated otherwise. Let $(E,h^{E})$ be a Riemannian
smooth (resp. Hermitian holomorphic) vector bundle of finite rank $r$ over an
open subset $U\subset\mathbb{R}^{n}$ (resp. $U\subset\mathbb{C}^{n}$), and let
$e_{1},\cdots,e_{r}$ be a local smooth (resp, holomorphic) frame of $E$. For
any $x\in U$, let $h^{E}_{x}$ denote the inner product on the fiber $E_{x}$ of
$x$. For any $1\leq\alpha,\beta\leq r,$ set

$h^{E}_{\alpha\bar{\beta}}:=h^{E}(e_{\alpha},e_{\beta}),$

and let $\left((h^{E})^{\alpha\bar{\beta}}\right)_{1\leq\alpha,\beta\leq n}$
be the inverse matrix of
$\left(h^{E}_{\alpha\bar{\beta}}\right)_{1\leq\alpha,\beta\leq r}.$ Let $u,v$
be any measurable sections of $E$, write

$u=\sum_{\lambda=1}^{r}u_{\lambda}e_{\lambda},\
v=\sum_{\mu=1}^{r}v_{\mu}e_{\mu},$

then we let

$|u|^{2}:=\sum_{\lambda=1}^{r}|u_{\lambda}|^{2},\ \langle
u,v\rangle_{h^{E}}:=h^{E}(u,v)=\sum_{\lambda,\mu=1}^{r}u_{\lambda}\overline{v_{\mu}}h^{E}(e_{\lambda},e_{\mu}),$
$|u|_{h}^{2}:=\langle u,u\rangle_{h},\ (u,v)_{h}:=\int_{U}\langle
u,v\rangle_{h^{E}}d\lambda,\ \|u\|_{h^{E}}^{2}:=(u,u)_{h^{E}}.$

Let $L^{2}(U,E)$ be the space of measurable $E$-valued sections which are
$L^{2}$-integrable on $U$, i.e.

$L^{2}(U,E):=\\{u\text{ is a measurable }E\text{-valued section of }E|\
\|u\|_{h^{E}}^{2}<\infty\\}.$

For any $k\in\mathbb{N}\cup\\{\infty\\}$, let $C^{k}(U,E)$ be the space of
$E$-valued sections which are of class $C^{k}$. We also let $C_{c}^{k}(U,E)$
be the space of elements of $C^{k}(U,E)$ which has compact support.

###  2.2. Regularized max function

Let $\psi\in C^{\infty}(\mathbb{R})$ be a nonnegative even function with
support in $[-1,1]$ such that

$\int_{\mathbb{R}}\psi(x)dx=1.$

######  Lemma 2.1 ([7, Lemma 5.18, Chapter I]).

For any $\eta:=(\eta_{1},\eta_{2})\in(0,+\infty)\times(0,+\infty)$, the
function $\max\nolimits_{\eta}\colon\mathbb{R}^{2}\rightarrow\mathbb{R}$
defined by

$(t_{1},t_{2})\mapsto\int_{\mathbb{R}^{2}}\max\\{t_{1}+x_{1},t_{2}+x_{2}\\}\frac{1}{\eta_{1}\eta_{2}}\psi\left(\frac{x_{1}}{\eta_{1}}\right)\psi\left(\frac{x_{2}}{\eta_{2}}\right)dx_{1}dx_{2}$

has the following properties

  * (1)

$\max_{\eta}\\{t_{1},t_{2}\\}$ is non decreasing in all variables, smooth and
convex on $\mathbb{R}^{2}$;

  * (2)

$\max\\{t_{1},t_{2}\\}\leq\max_{\eta}\\{t_{1},t_{2}\\}\leq\max\\{t_{1}+\eta_{1},t_{2}+\eta_{2}\\}$;

  * (3)

If $u_{1},u_{2}$ are plurisubharmonic functions, then
$\max_{\eta}\\{u_{1},u_{2}\\}$ is also plurisubharmonic.

###  2.3. Curvature positivity of Hermitian holomorphic vector bundles

Let $U\subset\mathbb{C}_{t}^{n}$ be a domain, and let $(E,h^{E})$ be a
Hermitian holomorphic trivial vector bundle over $U$ of finite rank $r$. The
Chern connection is now given by a collection of differential operators
$\\{D_{t_{j}}^{E}\\}_{1\leq j\leq n}$, which satisfies

$\partial_{t_{j}}h^{E}(u,v)=h^{E}(D_{t_{j}}^{E}u,v)+h^{E}(u,\bar{\partial}_{t_{j}}v),\
\forall u,v\in C^{2}(U,E),$

where $\partial_{t_{j}}:=\frac{\partial}{\partial t_{j}}$ and
$\bar{\partial}_{t_{j}}:=\frac{\partial}{\partial\bar{t}_{j}}$ for any $1\leq
j\leq n.$ For any $1\leq j,k\leq n,$ let
$\Theta^{(E,h^{E})}_{jk}:=[D_{t_{j}}^{E},\bar{\partial}_{t_{k}}]$ (Lie
bracket), then the Chern curvature of $(E,h^{E})$ is given by

$\Theta^{(E,h^{E})}=\sum_{j,k=1}^{n}\Theta^{(E,h^{E})}_{jk}dt_{j}\wedge
d\bar{t}_{k}.$

######  Definition 2.1.

The curvature of $(E,h^{E})$ is said to be positive (resp. strictly positive)
in the sense of Nakano if for any nonzero $n$-tuple $(u_{1},\cdots,u_{n})\\\
\in C^{\infty}(U,E),$ we have

$\sum_{j,k=1}^{n}h^{E}\left(\Theta_{jk}^{(E,h^{E})}u_{j},u_{k}\right)\geq 0\
(\text{resp.}>0).$

We need another simple lemma in the proof of Theorem 1.4.

######  Lemma 2.2 (see [7, Theorem (14.5), Chapter V]).

Let $(F,h)$ be a Hermitian holomorphic vector bundle over $X$, and let $E,G$
be two holomorphic subbundles of $F$ such that $F=E\oplus G$ and $E$ is
orthogonal to $G$, then the curvature of these bundles satisfies

$\Theta^{(F,h)}=\Theta^{(E,h)}\oplus\Theta^{(G,h)}.$

Although the above lemma is originally stated for finite rank bundles, it can
be easily generalized to infinite rank bundles.

###  2.4. $L^{2}$-estimates in complex analysis

In this subsection, we collect some well known $L^{2}$-estimates in complex
analysis that will be used later.

######  Lemma 2.3 ([2, Lemma 1.6.4]).

Let $\Omega\subset\mathbb{C}^{n}$ be a pseudoconvex domain, let $\varphi$ be a
plurisubharmonic function on $\Omega,$ and let $\psi\in C^{2}(\Omega)$ be a
strictly plurisubharmonic function. Then for any $\bar{\partial}$-closed
measurable $(0,1)$-form $f:=\sum_{j=1}^{n}f_{j}d\bar{z}_{j}$ on $\Omega$, we
can solve $\bar{\partial}u=f$ with the estimate

$\int_{\Omega}|u|^{2}e^{-\varphi-\psi}d\lambda\leq\int_{\Omega}\sum_{j,k=1}^{n}\psi^{j\bar{k}}f_{j}\bar{f_{k}}e^{-\varphi-\psi}d\lambda,$

provided the right hand side is finite.

######  Lemma 2.4 ([4, Theorem 1]).

Let $0\in\Delta\subset\mathbb{C}$ be a bounded domain, let
$g_{\Delta}(\cdot,z)$ be the (negative) Green function for $\Delta$ with pole
at $z,$ and let $\Omega\subset\Delta\times\mathbb{C}^{n}$ be a pseudoconvex
domain. Then for any plurisubharmonic function $\varphi$ on $\Omega,$
holomorphic function $f$ on $H:=\\{(z_{1},\cdots,z_{n+1})\in\Omega|\
z_{1}=0\\},$ there exists $F\in\mathcal{O}(\Omega)$ such that
$F|_{\Omega^{\prime}}=f$ and

$\int_{\Omega}|F|^{2}e^{-\varphi}d\lambda\leq\frac{\pi}{(c_{\Delta}(0))^{2}}\int_{H}|f|^{2}e^{-\varphi}d\lambda,$

provided the right hand side is finite, where

$c_{\Delta}(0):=\exp(\lim_{z\rightarrow 0}(g_{\Delta}(z,0)-\log|z|)).$

A fundamental result about the $L^{2}$-estimate of $\bar{\partial}$ for a
Hermitian holomorphic vector bundle is the following, which is due to
Hörmander and Demailly.

######  Lemma 2.5 ([7, Theorem 6.1, Chapter VIII]).

Let $X$ be a complete Kähler manifold with a Kähler metric $\omega$ which is
not necessarily complete. Let $(E,h^{E})$ be a holomorphic vector bundle over
$X$ with a Nakano positive Hermitian metric $h^{E}$, and define the operator
$A:=i\Theta^{(E,h^{E})}\wedge\Lambda_{\omega}$ acts on
$\Lambda^{n,q}T^{*}X\otimes E$ ($\Lambda_{\omega}$ is the adjoint operator of
the $E$-valued Lefschetz operator, $q\geq 1$). Then for any $v\in
L^{2}(X,\Lambda^{n,q}T^{*}X\otimes E)$ satisfying $\bar{\partial}v=0$, there
exists $u\in L^{2}(X,\Lambda^{n,q-1}T^{*}X\otimes E)$ such that
$\bar{\partial}u=v$ and

$\int_{X}|u|^{2}_{h^{E}}d\lambda(\omega)\leq\int_{X}\langle
A^{-1}v,v\rangle_{h^{E}}d\lambda(\omega),$

provided the right hand side is finite.

The following result of Deng-Ning-Wang-Zhou shows that the converse of the
above Lemma also holds, and hence gives an equivalent integral form
characterization of the curvature positivity of Hermitian holomorphic vector
bundles.

######  Lemma 2.6 ([12, Theorem 1.1]).

Let $U\subset\mathbb{C}^{n}$ be a bounded domain with the standard Kähler
metric $\omega:=i\sum_{j=1}^{n}dz_{j}\wedge d\bar{z}_{j}$, $(E,h)$ be a
Hermitian holomorphic vector bundle over $U$ with smooth Hermitian metric $h$,
and let $\theta\in C^{0}(U,\wedge^{1,1}T^{*}U\otimes\operatorname{End}(E))$
with $\theta^{*}=\theta$. If for any strictly plurisubharmonic function $\psi$
on $U$ and $v\in C_{c}^{\infty}(U,\wedge^{n,1}T^{*}U\otimes E)$ with
$\bar{\partial}f=0$ and
$i\partial\bar{\partial}\psi\otimes\operatorname{Id}_{E}+\theta>0$ on
$\operatorname{supp}(v)$ (support of $v$), there is a measurable section $u$
of $\wedge^{n,0}T^{*}U\otimes E$, satisfying $\bar{\partial}u=v$ and

(2.1) $\int_{U}|u|_{h}^{2}e^{-\psi}d\lambda(z)\leq\int_{U}\langle
B_{i\partial\bar{\partial}\psi,\theta}^{-1}v,v\rangle_{h}e^{-\psi}d\lambda(z),$

provided that the right hand side is finite, then $i\Theta^{E}\geq\theta$ in
the sense of Nakano, where

$B_{i\partial\bar{\partial}\psi,\theta}:=[i\partial\bar{\partial}\psi\otimes\operatorname{Id}_{E}+\theta,\Lambda_{\omega}].$

Let us also note that the following simple lemma.

######  Lemma 2.7 ([7, Theorem 5.2, Chapter VIII]).

Let $\Omega\subset\mathbb{C}^{n}$ be a pseudoconvex domain, then $\Omega$ is a
complete Kähler manifold.

###  2.5. Curvature positivity of real vector bundles

In this subsection, we give the definition of curvature posivity of a real
vector bundle, which is similar to Definition 2.1.

######  Definition 2.2 ([18, Definition 2]).

Let $g$ be a $C^{2}$ Riemannian metric on the trivial vector bundle
$E:=D\times\mathbb{R}^{r}$ over an open subset $D\subset\mathbb{R}^{n}$, and
let

$\Theta^{(E,g)}_{jk}:=-\frac{\partial}{\partial
x^{k}}\left(g^{-1}\frac{\partial g}{\partial x^{j}}\right),\ 1\leq j,k\leq n,$

where differentiation should be interpreted elementwise. The curvature of
$(E,g)$ is said to be positive (resp. strictly positive) in the sense of
Nakano if for any nonzero $n$-tuple $(u_{1},\cdots,u_{n})\in
C^{\infty}(U,\mathbb{R}^{n}),$ we have

$\sum_{j,k=1}^{n}g(\Theta_{jk}^{(E,g)}u_{j},u_{k})\geq 0\ (\text{resp.}>0).$

###  2.6. Some basic knowledge from measure theory

In this subsection, we give some basic knowledge from measure theory that will
be used several times when we consider applications to convex analysis.

######  Definition 2.3.

Let $U\subset\mathbb{R}^{m},\ V\subset\mathbb{R}^{n}$ be two measurable
subsets, and let $\phi\colon U\rightarrow V$ be a measurable map. For any
$A\subset V$ which is measurable, define $\mu(A):=|\phi^{-1}(A)|$, then $\mu$
is a measure on $V$, which is called the pushforward measure of the Lebesgue
measure on $U$ via $\phi$.

######  Lemma 2.8.

Let $U\subset\mathbb{R}^{m},\ V\subset\mathbb{R}^{n}$ be two measurable
subsets, let $\phi\colon U\rightarrow V$ be a (Lebesgue) measurable map, and
let $\mu$ be the pushforward measure of the Lebesgue measure on $U$ via
$\phi$, then for any measurable function $f\colon
V\rightarrow[-\infty,\infty]$, we have

$\int_{U}fd\mu=\int_{\phi^{-1}(U)}f\circ\phi d\lambda$

provided one side of the above exists.

######  Lemma 2.9.

Let $U\subset\mathbb{R}^{n}$ be an open subset, $\phi\colon
U\rightarrow\mathbb{R}^{n}$ be a $C^{1}$ map, then for any integrable function
$f$ on $\phi(U)$, we know

$\int_{\phi(U)}f(y)d\lambda(y)=\int_{U}f(\phi(x))J_{\phi}(x)d\lambda(x),$

where $J_{\phi}:=|\det(D(\phi))|$ is the absolute value of the Jacobian of
$\phi$.

Combining the above two lemmas, we have the following

######  Corollary 2.10.

Let $U\subset\mathbb{R}^{n}$ be an open subset, $\phi\colon
U\rightarrow\mathbb{R}^{n}$ be a $C^{1}$ map which is injective and has non-
vanishing Jacobian on $U$, and let $\mu$ be the pushforward measure of the
Lebesgue measure on $U$ via $\phi$, then the Nadon-Nikodym derivative of $\mu$
with respect to the Lebesgue measure $\lambda$ on $V$ is

$\frac{d\mu}{d\lambda}=J_{\phi}^{-1}\circ\phi^{-1}.$

##  3\. The proof of Theorem 1.1

In this section, we give the proof of Theorem 1.1. For convenience, we restate
it here.

######  Theorem 3.1 (= Theorem 1.1).

Let $D\subset\mathbb{R}_{x}^{n}$ be a convex domain,
$V\subset\mathbb{C}_{\tau}^{m}$ be a pseudoconvex domain, and let
$W_{z}:=D_{x}+i\mathbb{R}_{y}^{n}\subset\mathbb{C}_{z}^{n}.$ Let
$\varphi(\tau,z),\ \psi(\tau,z)$ be plurisubharmonic functions on $V\times W,$
which are independent of $\operatorname{Im}(z)$ for any $z\in W$. We assume
that $\psi$ is a $C^{2}$, strictly plurisubharmonic function with respect to
$\tau$. Then for any $\bar{\partial}$-closed measurable $(0,1)$-form
$f:=\sum^{n}_{j=1}f_{j}d\bar{\tau}_{j}$ in $V$, we can solve the equation
$\bar{\partial}u=f$ in $V$ with the estimate

$\int_{V\times D}|u|^{2}e^{-\varphi-\psi}d\lambda\leq\int_{V\times
D}\sum^{n}_{j,k=1}\psi^{j\bar{k}}f_{j}\bar{f}_{k}e^{-\varphi-\psi}d\lambda,$

provided the right hand side is finite, where
$\left(\psi^{j\bar{k}}\right)_{n\times
n}:=\left(\frac{\partial^{2}\psi}{\partial\tau_{j}\partial\bar{\tau}_{k}}\right)^{-1}_{n\times
n}$.

###### Proof..

Let $\Omega:=V\times W.$ By the usual approximation technique, we may assume
$\psi$ is of class $C^{2}$ on $\Omega$. Consider the map

$\operatorname{id}\times\phi\colon\mathbb{C}_{\tau}^{m}\times\mathbb{C}_{z}^{n}\rightarrow\mathbb{C}_{\tau}^{m}\times(\mathbb{C}_{w}^{*})^{n},\
(\tau,z_{1},\cdots,z_{n})\rightarrow(\tau,e^{z_{1}},\cdots,e^{z_{n}}),$

and let $\Omega^{\prime}:=(\operatorname{id}\times\phi)(\Omega),$ then
$\Omega^{\prime}$ is a pseudoconvex domain. It is clear that $\psi$ induces a
$C^{2}$, plurisubharmonic function $\psi^{\prime}$ on $\Omega^{\prime}$ by

$\psi^{\prime}(\tau,e^{z_{1}},\cdots,e^{z_{n}})=\psi(\tau,z_{1},\cdots,z_{n}),$

and $\psi^{\prime}$ is strictly plurisubharmonic with respect to $\tau.$
Similarly, $\varphi$ also induces a plurisubharmonic function
$\varphi^{\prime}$ on $\Omega^{\prime}$. We may regard $f$ as a
$\bar{\partial}$-closed $(0,1)$-form on $\Omega^{\prime}$. For any measurable
subset $U\subset(\mathbb{C}_{w}^{*})^{n},$ define

$\mu(U):=\left|\\{z\in\mathbb{C}^{n}|\ \phi(z)\in U,\
\operatorname{Im}(z_{1}),\operatorname{Im}(z_{2}),\cdots,\operatorname{Im}(z_{n})\in(0,2\pi)\\}\right|,$

then $\mu$ is a measure on $(\mathbb{C}_{w}^{*})^{n},$ and by Corollary 2.10,
we know

$d\mu(w)=\frac{1}{|w_{1}|^{2}\cdots|w_{n}|^{2}}d\lambda(w).$

By Lemma 2.8, we have

(3.1)
$\int_{\Omega^{\prime}}\sum_{j,k=1}^{n}(\psi^{\prime})^{j\bar{k}}f_{j}\bar{f}_{k}e^{-\varphi^{\prime}-\psi^{\prime}}d\mu=(2\pi)^{n}\int_{V\times
D}\sum_{j,k=1}^{n}\psi^{j\bar{k}}f_{j}\bar{f}_{k}e^{-\varphi-\psi}d\lambda<\infty.$

Let
$\varphi^{\prime\prime}(w):=\varphi^{\prime}(w)+\sum_{j=1}^{n}\log|w_{j}|^{2}$
for any $w\in\Omega^{\prime}$, then $\varphi^{\prime\prime}$ is a
plurisubharmonic function on $\Omega^{\prime}$, and

$e^{-\varphi^{\prime}}d\mu(w)=e^{-\varphi^{\prime\prime}}d\lambda(w),\ \forall
w\in(\mathbb{C}^{*})^{n}.$

By Lemma 2.3, we can solve $\bar{\partial}_{(\tau,w)}u=f$ on $\Omega^{\prime}$
with the estimate

$\int_{\Omega^{\prime}}|u|^{2}e^{-\varphi^{\prime\prime}-\psi^{\prime}}d\lambda\leq\int_{\Omega^{\prime}}\sum_{j,k=1}^{n}(\psi^{\prime})^{j\bar{k}}f_{j}\bar{f}_{k}e^{-\varphi^{\prime\prime}-\psi^{\prime\prime}}d\lambda,$

i.e.

(3.2)
$\int_{\Omega^{\prime}}|u|^{2}e^{-\varphi^{\prime}-\psi^{\prime}}d\mu\leq\int_{\Omega^{\prime}}\sum_{j,k=1}^{n}(\psi^{\prime})^{j\bar{k}}f_{j}\bar{f}_{k}e^{-\varphi^{\prime}-\psi^{\prime}}d\mu,$

Take $u$ to be minimal, then $u$ is independent of $\operatorname{Im}(w)$ by
the uniqueness of minimal solution. As $f$ is independent of $w$, we know $u$
is holomorphic in $w$, so $u$ is independent of $w$. Thus, we can regard $u$
as a function on $V$, then we have $\bar{\partial}u=f$. By Lemma 2.8 again, we
have

(3.3)
$\int_{\Omega^{\prime}}|u|^{2}e^{-\varphi^{\prime}-\psi^{\prime}}d\mu=(2\pi)^{n}\int_{V\times
D}|u|^{2}e^{-\varphi-\psi}d\lambda.$

Combining Formulas (3.1), (3.2), (3.3), we complete the proof. ∎

##  4\. The proof of Theorem 1.2

In this section, we give and prove a more general version of Theorem 1.2.

######  Theorem 4.1.

Let $D\subset\mathbb{R}_{x}^{n}$ be a convex domain,
$0\in\Delta\subset\mathbb{C}_{\tau}$ be a bounded domain, and let
$\varphi(\tau,z)$ be a plurisubharmonic function on
$\Delta_{\tau}\times(D_{x}+i\mathbb{R}^{n}_{y}),$ which is independent of
$\operatorname{Im}(z)$. Then there exists a holomorphic function $f$ on
$\Delta$ satisfying $f(0)=1$ and

$\int_{\Delta\times
D}|f(\tau)|^{2}e^{-\varphi(\tau,x)}d\lambda(\tau,x)\leq\frac{\pi}{(c_{\Delta}(0))^{2}}\int_{D}e^{-\varphi(0,x)}d\lambda(x),$

provided the right hand side is finite.

###### Proof..

Let $W_{z}:=D_{x}+i\mathbb{R}^{n}_{y}\subset\mathbb{C}_{z}^{n},\
\Omega:=\Delta\times W.$ Consider the map

$\operatorname{id}\times\phi\colon\mathbb{C}_{\tau}^{m}\times\mathbb{C}_{z}^{n}\rightarrow\mathbb{C}_{\tau}^{m}\times(\mathbb{C}_{w}^{*})^{n},\
(\tau,z_{1},\cdots,z_{n})\rightarrow(\tau,e^{z_{1}},\cdots,e^{z_{n}}),$

and let $\Omega^{\prime}:=(\operatorname{id}\times\phi)(\Omega),$ then
$\Omega^{\prime}$ is a pseudoconvex domain. It is clear that $\varphi$ induces
a $C^{2}$, plurisubharmonic function $\varphi^{\prime}$ on $\Omega^{\prime}$
by

$\varphi^{\prime}(\tau,e^{z_{1}},\cdots,e^{z_{n}})=\varphi(\tau,z_{1},\cdots,z_{n}).$

Similar as in the proof of Theorem 3.1, we may get a measure on
$(\mathbb{C}_{w}^{*})^{n}$ via $\phi$ which satisfies

$d\mu(w)=\frac{1}{|w_{1}|^{2}\cdots|w_{n}|^{2}}d\lambda(w),\ \forall
w\in(\mathbb{C}^{*})^{n}.$

Let
$\varphi^{\prime\prime}(w):=\varphi^{\prime}(w)+\sum_{j=1}^{n}\log|w_{j}|^{2}$
for any $w\in\Omega^{\prime}$, then $\varphi^{\prime\prime}$ is a
plurisubharmonic function on $\Omega^{\prime}$, and

$e^{-\varphi^{\prime}}d\mu(w)=e^{-\varphi^{\prime\prime}}d\lambda(w).$

Let $H:=\Omega^{\prime}\cap\\{\tau=0\\}=\\{0\\}\times\pi(W),$ then by Lemma
2.8, we have

(4.1)
$\int_{H}e^{-\varphi^{\prime\prime}(0,w)}d\lambda(w)=\int_{H}e^{-\varphi^{\prime}(0,w)}d\mu(w)=(2\pi)^{n}\int_{D}e^{-\varphi(0,x)}d\lambda(x)<\infty$

By Lemma 2.4, we know there is a holomorphic function $f$ on $\Omega^{\prime}$
such that $f|_{H}=1$ and

(4.2)
$\int_{\Omega^{\prime}}|f(\tau,w)|^{2}e^{-\varphi^{\prime\prime}(\tau,w)}d\lambda(\tau)d\lambda(w)\leq\frac{\pi}{(c_{\Delta}(0))^{2}}\int_{H}e^{-\varphi^{\prime\prime}(0,w)}d\lambda(w),$

i.e.

(4.3)
$\int_{\Omega^{\prime}}|f(\tau,w)|^{2}e^{-\varphi^{\prime}(\tau,w)}d\lambda(\tau)d\mu(w)\leq\frac{\pi}{(c_{\Delta}(0))^{2}}\int_{H}e^{-\varphi^{\prime}(0,w)}d\mu(w).$

Take $f$ to be minimal, then we know $f$ is independent of the argument of
$w$, and then $f$ is independent of $w$ as $f$ is holomorphic in $w.$ We may
thus regard $f$ as a holomorphic function on $\Delta,$ then $f(0)=1.$ By Lemma
2.8 again, we have

(4.4)
$\int_{\Omega^{\prime}}|f(\tau,w)|^{2}e^{-\varphi^{\prime}(\tau,w)}d\lambda(\tau)d\mu(w)=(2\pi)^{n}\int_{\Delta\times
D}|f(\tau)|^{2}e^{-\varphi(\tau,x)}d\lambda(\tau,x).$

Therefore, the proof is complete by combining Formulas (4.1), (4.3), (4.4). ∎

##  5\. The proof of Theorem 1.3

To prove Theorem 1.3, let us firstly recall the complexification of a real
inner product. Let $(V,g)$ be a finite dimensional real inner product space,
and let $V_{\mathbb{C}}:=V\otimes_{\mathbb{R}}\mathbb{C}$ be the
complexification of $V$, then we can extend $g$ to a Hermitian inner product
$h$ on $V_{\mathbb{C}}$ by setting

$h(x\otimes z,y\otimes w)=z\bar{w}g(x,y),\ \forall x,y\in V,\ \forall
z,w\in\mathbb{C}.$

When we define a Hermitian inner product $h$ as above, we always say $g$
extends to a Hermitian inner product $h$ for brevity.
Now we give the proof of Theorem 1.3.

######  Theorem 5.1 (= Theorem 1.3).

Let $D\subset\mathbb{R}^{n}$ be a convex domain, and let $(E,g)$ be a Nakano
positive trivial vector bundle of finite rank $r$ over $D$, then for any
$d$-closed $f\in L^{2}(D,\Lambda^{1}T^{*}D\otimes E),$ there exists $u\in
L^{2}(D,E)$ such that

$du=f,\
\int_{D}|u|_{g}^{2}d\lambda\leq\int_{D}\langle(\Theta^{(E,g)})^{-1}f,f\rangle_{g}d\lambda,$

provided the right hand side is finite.

###### Proof..

Let $\Omega_{z}:=D_{x}+i\mathbb{R}_{y}^{n}\subset\mathbb{C}_{z}^{n}.$ Consider
the map

$\phi\colon\mathbb{C}_{z}^{n}\rightarrow(\mathbb{C}_{w}^{*})^{n},\
(z_{1},\cdots,z_{n})\mapsto(e^{z_{1}},\cdots,e^{z_{n}}),$

and let $\Omega^{\prime}:=\phi(\Omega),$ then $\Omega^{\prime}$ is a complete
Kähler manifold with the canonical Kähler metric $\omega$ by Lemma 2.7.
Similar as in the proof of Theorem 3.1, we may get a measure on
$(\mathbb{C}_{w}^{*})^{n}$ via $\phi$ which satisfies

$d\mu(w)=\frac{1}{|w_{1}|^{2}\cdots|w_{n}|^{2}}d\lambda(w),\ \forall
w\in(\mathbb{C}^{*})^{n}.$

Let $E^{\prime}:=\Omega^{\prime}\times\mathbb{C}^{r}$ be the trivial vector
bundle over $\Omega^{\prime}.$ We extend $g$ to a Hermitian metric $h$ on the
trivial vector bundle $\Omega\times\mathbb{C}^{r}$ over $\Omega$ such that $h$
is independent of $\operatorname{Im}(z)$ for any $z\in\Omega$, then $h$
induces a Hermitian metric $h^{\prime}$ on $E^{\prime}$ via $\phi.$ Similar to
the proof of Theorem 3.1, we know $f$ induces a $\bar{\partial}$-closed
$(0,1)$-form $f^{\prime}\in
L^{2}(\Omega^{\prime},\Lambda^{0,1}T^{*}\Omega^{\prime}\otimes E^{\prime}).$
For any $w\in\Omega^{\prime}$, we set

$(h^{\prime\prime})_{w}:=\frac{1}{|w_{1}|^{2}\cdots|w_{n}|^{2}}(h^{\prime})_{w}.$

Since $(E,g)$ is Nakano positive, then

$A_{E^{\prime}}:=[i\Theta^{(E^{\prime},h^{\prime\prime})},\Lambda_{\omega}]\geq
0\text{ on }\Lambda^{0,1}T^{*}\Omega^{\prime}\otimes E^{\prime}$

by a simple computation. By Lemma 2.8, we have

(5.1) $\displaystyle\int_{\Omega^{\prime}}\langle
A^{-1}_{E^{\prime}}f^{\prime},f^{\prime}\rangle_{h^{\prime\prime}}d\lambda$
$\displaystyle=\int_{\Omega^{\prime}}\langle
A^{-1}_{E^{\prime}}f^{\prime},f^{\prime}\rangle_{h^{\prime}}d\mu=(2\pi)^{n}\int_{D}\langle(\Theta^{(E,g)})^{-1}f,f\rangle_{g}d\lambda$
$\displaystyle<\infty.$

By Lemma 2.5, we may solve $\bar{\partial}u^{\prime}=f^{\prime}$ with the
estimate

(5.2)
$\int_{\Omega^{\prime}}|u^{\prime}|^{2}_{h^{\prime\prime}}d\lambda\leq\int_{\Omega^{\prime}}\langle
A_{E^{\prime}}^{-1}f^{\prime},f^{\prime}\rangle_{h^{\prime\prime}}d\lambda.$

Take $u^{\prime}$ to be minimal, then $u^{\prime}$ is independent of
$\operatorname{Im}(w)$. For any $x\in D$, define $u(x):=u^{\prime}(\pi(x)),$
then $du=f$. By Lemma 2.8 again, we know

(5.3)
$\int_{\Omega^{\prime}}|u^{\prime}|^{2}_{h^{\prime}}d\mu=(2\pi)^{n}\int_{D}|u|_{g}^{2}d\lambda.$

Thus, the desired result follows from Formulas (5.1), (5.2), (5.3). ∎

##  6\. The proof of Theorem 1.4

In this section, we will give the proof of Theorem 1.4. Let us first fix some
notations and make some preparations.
In this section, we always let $j,k=1,\cdots,n$ represent the indices of the
components of $t=(t_{1},\cdots,t_{n})\in U$, $p,q,s,a=1,\cdots,m$ represent
the indices of the components of $z=(z_{1},\cdots,z_{m})\in\mathbb{C}^{m}$,
$\lambda,\mu,\alpha,\beta,\gamma=1,\cdots,r$ represent the indices of the
components of $F$. To simplify the notation, we always write $dz$ for
$dz_{1}\wedge\cdots\wedge dz_{m}$.
Write

$D_{t_{j}}^{F}e_{\lambda}=\sum_{\mu}\Gamma_{j\lambda}^{\mu}dt_{j}\otimes
e_{\mu},\
D_{z_{p}}^{F}e_{\lambda}=\sum_{\mu}\Gamma_{p\lambda}^{\mu}dz_{p}\otimes
e_{\mu},$

and we also adopt the following notations

$\left(\begin{array}[]{cc}(A_{pq\lambda}^{\alpha})&(A_{pj\lambda}^{\alpha})\\\
(A_{jp\lambda}^{\alpha})&(A_{jk\lambda}^{\alpha})\end{array}\right):=\left(\begin{array}[]{cc}\left(-\frac{\partial\Gamma_{p\lambda}^{\alpha}}{\partial\bar{z}_{q}}\right)&\left(-\frac{\partial\Gamma_{p\lambda}^{\alpha}}{\partial\bar{t}_{j}}\right)\\\
\left(-\frac{\partial\Gamma_{j\lambda}}{\partial\bar{z}_{p}}\right)&\left(-\frac{\partial\Gamma_{j\lambda}^{\alpha}}{\partial\bar{t}_{k}}\right)\end{array}\right),\
\ h^{F}_{\lambda\mu}:=h_{(t,z)}^{F}(e_{\lambda},e_{\mu}),$
$\left(\begin{array}[]{cc}(A_{pq\lambda\mu})&(A_{pj\lambda\mu})\\\
(A_{jp\mu\lambda})&(A_{jk\lambda\mu})\end{array}\right):=\left(\begin{array}[]{cc}\left(-\frac{\partial\Gamma_{p\lambda}^{\alpha}}{\partial\bar{z}_{q}}h^{F}_{\alpha\mu}\right)&\left(-\frac{\partial\Gamma_{p\lambda}^{\alpha}}{\partial\bar{t}_{j}}h^{F}_{\alpha\mu}\right)\\\
\left(-\frac{\partial\Gamma_{p\lambda}^{\alpha}}{\partial\bar{t}_{j}}h^{F}_{\alpha\mu}\right)^{*}&\left(-\frac{\partial\Gamma_{j\lambda}^{\alpha}}{\partial\bar{t}_{k}}h^{F}_{\alpha\mu}\right)\end{array}\right),$

where ∗ represents the conjugate of the transpose of a matrix. It is clear
that

$\Theta_{jk}^{F}=[D_{t_{j}}^{F},\bar{\partial}_{t_{k}}]=\sum_{\lambda,\alpha}A_{jk\lambda}^{\alpha}e_{\lambda}^{*}\otimes
e_{\alpha},$

where $e_{1}^{*},\cdots,e_{r}^{*}$ is the dual basis of $e_{1},\cdots,e_{r}$.
To prove Theorem 1.4, we need the following key lemma, which is a
generalization of [3, Formula (3.1)].

######  Lemma 6.1.

Let $U\subset\mathbb{C}^{n}$ be a pseudoconvex domain, and let
$D\subset\mathbb{C}^{m}$ be a pseudoconvex bounded domain. let $(F,h^{F})$ be
a Nakano positive trivial vector bundle of finite rank $r$ defined on some
neighborhood of the closure of $\Omega:=U\times D$. Let
$\\{e_{1},\cdots,e_{r}\\}$ be the canonical holomorphic frame of $(F,h^{F})$.
For any $t\in U$, set

$E_{t}:=\\{f=\sum_{\lambda=1}^{r}f_{\lambda}\otimes e_{\lambda}|\
f_{\lambda}\in\mathcal{O}(D)\text{ for all }\lambda,\
\|f\|^{2}_{t}<\infty\\},$

where

$\|f\|^{2}_{t}:=\int_{D}\sum_{\lambda,\mu=1}^{r}f_{\lambda}(z)\overline{f_{\mu}(z)}h^{F}_{(t,z)}(e_{\lambda},e_{\mu})d\lambda(z),$

and set $E:=\cup_{t\in U}E_{t}$, then $E$ is a holomorphic vector bundle (of
infinite rank) over $U$ with a Hermitian metric $h$ given by

$h_{t}^{E}(f,g):=\int_{D}\sum_{\lambda,\mu=1}^{r}f_{\lambda}(z)\overline{g_{\mu}(z)}h^{F}_{(t,z)}(e_{\lambda},e_{\mu})d\lambda(z),\
\forall f,g\in E_{t}.$

Then for any smooth sections $u_{j}:=\sum_{\lambda}u_{j\lambda}dz\otimes
e_{\lambda}(1\leq j\leq n)$ of $E$, we have

$\displaystyle\sum_{j,k}h^{E}\left(\Theta^{E}_{jk}u_{j},u_{k}\right)$
$\displaystyle\geq\int_{D}\sum_{j,k,\beta,\gamma}\left(A_{jk\beta\gamma}-\sum_{p,q,\lambda,\mu}B_{pq\lambda\mu}A_{jq\beta\mu}A_{pk\lambda\gamma}\right)u_{j\beta}\overline{u_{k\gamma}}d\lambda$
$\displaystyle=:\int_{D}\sum_{j,k,\beta,\gamma}H(h^{F})_{jk\beta\gamma}u_{j\beta}\overline{u_{k\gamma}}d\lambda,$

where $B$ is uniquely determined by $A$ and satisfies

$\sum_{q,\beta}A_{pq\lambda\beta}B_{sq\alpha\beta}=\delta_{ps}\delta_{\lambda\alpha}.$

###### Proof..

For any $t\in U$, set

$G_{t}:=\\{u:=\sum_{\lambda}u_{\lambda}dz\otimes e_{\lambda}|\ u_{\lambda}\in
L^{2}(D)\text{ for all }\lambda\\}$

with an inner product $h^{G}_{t}$ given by

$h_{t}^{G}(u,v):=\int_{\Omega_{t}}\sum_{\lambda,\mu}u_{\lambda}(z)\overline{v_{\mu}(z)}h^{F}_{(t,z)}(e_{\lambda},e_{\mu})d\lambda(z),\
\forall u,v\in G_{t},$

then $G:=\cup_{t\in U}G_{t}$ can be viewed as a Hermitian holomorphic
(trivial) vector bundle over $U.$ Let $\pi\colon G\rightarrow E$ be the
fiberwise orthogonal projection, and let $\pi_{\perp}$ be the orthogonal
projection on the orthogonal complement of $E$ in $G.$ Put

$w:=\pi_{\perp}(\sum_{j}D^{F}_{t_{j}}u_{j}),\
\|w\|_{h^{F}}^{2}:=\int_{D}h^{F}(w,w)d\lambda,$

then we have

$D^{E}=\pi\circ D^{G}$

and

$\displaystyle\quad\sum_{j,k}h^{E}\left(\Theta^{E}_{jk}u_{j},u_{k}\right)$
$\displaystyle=\sum_{j,k}h^{G}\left(\Theta^{G}_{jk}u_{j},u_{k}\right)-h^{G}\left(\pi_{\perp}\sum_{j}D^{G}_{t_{j}}u_{j},\pi_{\perp}\sum_{j}D^{G}_{t_{j}}u_{j}\right)$
$\displaystyle=\sum_{j,k}\int_{D}h^{F}\left(\Theta^{F}_{jk}u_{j},u_{k}\right)d\lambda-\left\|w\right\|_{h^{F}}^{2}$

for any smooth sections $u_{1},\cdots,u_{n}$ of $E$. For fixed $t\in U$, $w$
solves the $\bar{\partial}_{z}$-equation

$\bar{\partial}_{z}w=\bar{\partial}_{z}\left(\pi_{\perp}\sum_{j}D^{F}_{t_{j}}u_{j}\right)=\bar{\partial}_{z}\left(\sum_{j}D^{F}_{t_{j}}u_{j}\right).$

We will use Lemma 2.5 to give an estimate of $\|w\|_{h^{F}}^{2}$. By Lemma
2.7, we know $D$ is a complete Kähler manifold. Let $\omega$ be the standard
Kähler metric on $D$, and set $A:=[i\Theta^{(F_{t},h^{F})},\Lambda_{\omega}]$,
then we know $A\geq 0$ as $(F,h^{F})$ is Nakano strictly positive. Set
$v:=(-1)^{n}\bar{\partial}_{z}w\in L^{2}(D,\wedge^{n,1}T^{*}D\otimes F_{t}),$
then $v$ satisfies $\bar{\partial}_{z}v=0.$ By Lemma 2.5, we have

$\|w\|_{h^{F}}^{2}\leq\int_{D}\langle
A^{-1}v,v\rangle_{h^{F}}d\lambda(\omega),$

then

$\sum_{j,k}h^{E}\left(\Theta^{E}_{jk}u_{j},u_{k}\right)\geq\sum_{j,k}\int_{D}h^{F}\left(\Theta^{F}_{jk}u_{j},u_{k}\right)d\lambda-\int_{D}\langle
A^{-1}v,v\rangle_{h^{F}}d\lambda.$

Now we need to compute the right hand side the above inequality. To do so, we
need some simple but long computation.
It is clear that

$\sum_{j,k}h^{F}\left(\Theta_{jk}^{F}u_{j},u_{k}\right)=\sum_{j,k,\lambda,\mu}A_{jk\lambda}^{\alpha}u_{j\lambda}\overline{u_{k\mu}}h^{F}_{\alpha\mu}=\sum_{j,k,\lambda,\mu}A_{jk\lambda\mu}u_{j\lambda}\overline{u_{k\mu}},$

and

$\displaystyle v$
$\displaystyle=\bar{\partial}_{z}((-1)^{n}\sum_{j}D_{t_{j}}^{F}u_{j})=\sum_{j,\lambda,\mu}u_{j\lambda}\frac{\partial\Gamma_{j\lambda}^{\mu}}{\partial\bar{z}_{p}}dz\wedge
d\bar{z}_{p}\otimes e_{\mu}$
$\displaystyle=\sum_{j,\lambda,\mu}u_{j\mu}A_{jp\mu}^{\lambda}dz\wedge
d\bar{z}_{p}\otimes e_{\lambda}.$

Hence it suffices to prove that

$\langle
A^{-1}v,v\rangle_{h^{F}}=\sum_{j,k,p,q,\lambda,\mu,\beta,\gamma}B_{pq\lambda\mu}A_{jq\beta\mu}A_{pk\lambda\gamma}u_{j\beta}\overline{u_{k\gamma}}$

for some $B$ to be determined later.
For any

$g:=\sum_{p,\lambda}g_{p\lambda}dz\wedge d\bar{z}_{p}\otimes e_{\lambda}\in
L^{2}(D,\wedge^{n,1}T^{*}D\otimes F_{t}),$

we have

$\displaystyle Ag$
$\displaystyle=i\Theta^{(F_{t},h^{F})}(\Lambda_{\omega}g)=\left(i\sum_{p,q,\lambda,\mu}A_{pq\lambda}^{\mu}dz_{p}\wedge
d\bar{z}_{q}\otimes e_{\lambda}^{*}\otimes e_{\mu}\right)(\Lambda_{\omega}g)$
$\displaystyle=\sum_{p,q,j,\lambda,\mu}A_{pq\lambda}^{\mu}g_{p\lambda}dz\wedge
d\bar{z}_{q}\otimes e_{\mu},$

where $e_{1}^{*},\cdots,e_{r}^{*}$ is the dual frame of $e_{1},\cdots,e_{r}.$
Note that we may view $A$ as a linear map $T$ which satisfies

$T(d\bar{z}_{p}\otimes
e_{\lambda})=\sum_{q,\mu}A_{pq\lambda}^{\mu}d\bar{z}_{q}\otimes e_{\mu}.$

Since $(F,h^{F})$ is Nakano strictly positive, then for any nonzero matrices
$u_{p\lambda}$, we have

$\sum_{p,q,\lambda,\mu}A_{pq\lambda\mu}u_{p\lambda}\overline{u_{q\mu}}>0,$

and then $T$ is invertible. Let $T^{-1}$ be the inverse map of $T$, and write

$T^{-1}(d\bar{z}_{q}\otimes
e_{\mu})=\sum_{p,\lambda}B_{pq\lambda}^{\mu}d\bar{z}_{p}\otimes e_{\lambda},$

then we have

$d\bar{z}_{p}\otimes
e_{\lambda}=T^{-1}(\sum_{q,\mu}A_{pq\lambda}^{\mu}d\bar{z}_{q}\otimes
e_{\mu})=\sum_{q,s,\alpha,\mu}A_{pq\lambda}^{\mu}B_{sq\alpha}^{\mu}d\bar{z}_{s}\otimes
e_{\alpha},$

i.e.,

$\sum_{q,\mu}A_{pq\lambda}^{\mu}B_{sq\alpha}^{\mu}=\delta_{ps}\delta_{\lambda\alpha}.$

Let

$B_{sq\alpha\beta}:=\sum_{\mu}B_{sq\alpha}^{\mu}(h^{F})^{\beta\mu},$

where $\left((h^{F})^{\beta\mu}\right)_{1\leq\beta,\mu\leq r}$ is the inverse
matrix of $\left(h_{\beta\mu}^{F}\right)_{1\leq\beta,\mu\leq r}$, then we have

$\sum_{q,\beta}A_{pq\lambda\beta}B_{sq\alpha\beta}=\sum_{q,\mu}A_{pq\lambda}^{\mu}B_{sq\alpha}^{\mu}=\delta_{ps}\delta_{\lambda\alpha}.$

Define linear maps

$S_{1}(d\bar{z}_{p}\otimes
e_{\lambda})=\sum_{q,\mu}A_{pq\lambda\mu}d\bar{z}_{q}\otimes e_{\mu},\
S_{2}(d\bar{t}_{j}\otimes
e_{\lambda})=\sum_{q,\mu}A_{jq\lambda\mu}d\bar{z}_{q}\otimes e_{\mu},$
$S_{3}(d\bar{z}_{p}\otimes
e_{\lambda})=\sum_{k,\mu}A_{pk\lambda\mu}d\bar{t}_{k}\otimes e_{\mu},\
S_{4}(d\bar{t}_{j}\otimes
e_{\lambda})=\sum_{k,\mu}A_{jk\lambda\mu}d\bar{t}_{k}\otimes e_{\mu},$
$B(d\bar{z}_{q}\otimes
e_{\mu})=\sum_{p,\lambda}B_{pq\lambda\mu}d\bar{z}_{p}\otimes e_{\lambda},$

then we have

$BS_{1}(d\bar{z}_{p}\otimes
e_{\lambda})=B\left(\sum_{q,\mu}A_{pq\lambda\mu}d\bar{z}_{q}\otimes
e_{\mu}\right)=\sum_{q,s,\mu,\alpha}A_{pq\lambda\mu}B_{sq\alpha\mu}d\bar{z}_{s}\otimes
e_{\alpha}=d\bar{z}_{p}\otimes e_{\lambda},$

so $B=S_{1}^{-1}$.
Let

$v_{q\lambda}:=\sum_{j,\mu}u_{j\mu}A_{jp\mu}^{\lambda},$

then we know

$A^{-1}v=\sum_{p,q,\lambda,\mu}B_{pq\lambda}^{\mu}v_{q\mu}dz_{1}\wedge\cdots\wedge
dz_{n}\wedge d\bar{z}_{p}\otimes e_{\lambda},$

and

$\displaystyle\langle A^{-1}v,v\rangle_{h^{F}}$
$\displaystyle=\sum_{p,q,\lambda,\mu,\alpha}B_{pq\lambda}^{\mu}v_{q\mu}\overline{v_{p\alpha}}h^{F}_{\lambda\alpha}=\sum_{p,q,j,k,\lambda,\mu,\beta,\gamma}B_{pq\lambda}^{\mu}A_{jq\beta}^{\mu}\overline{A_{kp\gamma\lambda}}u_{j\beta}\overline{u_{k\gamma}}$
$\displaystyle=\sum_{j,k,p,q,\lambda,\mu,\beta,\gamma}B_{pq\lambda\mu}A_{jq\beta\mu}A_{pk\lambda\gamma}u_{j\beta}\overline{u_{k\gamma}}.$

Then the proof is complete. ∎

Now we shall prove Theorem 1.4.

######  Theorem 6.2 (= Theorem 1.4).

Let $\Omega\subset U\times\mathbb{C}^{m}$ be a strictly pseudoconvex family of
bounded domains over $U\subset\mathbb{C}^{n}$, and let $(F,h^{F})$ be a Nakano
positive trivial Hermitian vector bundle of finite rank $r$ over some
neighborhood of $\overline{\Omega}$ in $U\times\mathbb{C}^{m}$. We assume that
all fibers $\Omega_{t}\ (t\in U)$ are (connected) circular domains in
$\mathbb{C}^{m}$ containing the origin and $h^{F}_{(t,z)}$ is
$\mathbb{S}^{1}$-invariant with respect to $z$. Let $\\{e_{1},\cdots,e_{r}\\}$
be the canonical holomorphic frame of $F$. For any $k\in\mathbb{N},\ t\in U$,
set

$E^{k}_{t}:=\\{f=\sum_{\lambda=1}^{r}f_{\lambda}\otimes e_{\lambda}|\
f_{\lambda}\in\rm P^{k}\text{ for all }\lambda\\}$

with inner product $h_{t}$ given by

$h_{t}(f,g):=\int_{\Omega_{t}}\sum_{\lambda,\mu=1}^{r}f_{\lambda}(z)\overline{g_{\mu}(z)}h^{F}_{(t,z)}(e_{\lambda},e_{\mu})d\lambda(z),\
\forall f,g\in E^{k}_{t}.$

We set $E^{k}:=\cup_{t\in U}E^{k}_{t}$ and view it as a (trivial) holomorphic
vector bundle over $U$ in a natural way, then the curvature of $(E^{k},h)$ is
strictly positive in the sense of Nakano.

###### Proof..

Since $\Omega$ is strictly pseudoconvex with $C^{2}$-boundary, there is a
defining function $\rho$ which is $C^{2}$-smooth and strictly plurisubharmonic
on some neighborhood of $\overline{\Omega}$ and $\mathbb{S}^{1}$-invariant
with respect to $z$. For any fixed $t_{0}\in U$ and any $q>0$, let
$D:=\\{z\in\mathbb{C}^{n}|\ \rho(t_{0},z)<q\\}$, then there exists a
neighborhood $U^{\prime}\subset\subset U$ of $t_{0}$ and $0<q<1$ such that

  * (1)

$\rho$ is defined, $C^{2}$-smooth and strictly plurisubharmonic on some
neighborhood of the closure of $U^{\prime}\times D$;

  * (2)

$(F,h^{F})$ is defined and Nakano positive on some neighborhood of the closure
of $U^{\prime}\times D$;

  * (3)

let $p\colon\mathbb{C}^{n}\times\mathbb{C}^{m}\rightarrow\mathbb{C}^{n}$ be
the natural projection, then $p^{-1}(U^{\prime})\cap\Omega\subset
U^{\prime}\times D$;

  * (4)

$D$ is a pseudoconvex circular domain;

  * (5)

$\Omega_{t_{0}}\subset\subset D.$

Since the result to be proved is local, we may assume

$U=U^{\prime},\ \Omega\subset U\times D=:\tilde{\Omega},\
\Omega_{t_{0}}\subset\subset D.$

For any $N\in\mathbb{N},\ \epsilon>0$, we set

$h^{F}_{N}:=h^{F}e^{-N\max\nolimits_{\left(\frac{1}{N^{3}},\frac{1}{N^{3}}\right)}\\{0,\rho\\}},\
h^{F}_{N,\epsilon}:=h_{N}^{F}e^{-\epsilon|t|^{2}-\epsilon|z|^{2}},$

where $\max\nolimits_{(-,-)}$ is the regularized max function defined as in
Lemma 2.1. It is clear that $(F,h_{N,\epsilon}^{F})$ is Nakano strictly
positive on some neighborhood of the closure of $\tilde{\Omega}$.
Let $E$ be the trivial vector bundle over $U$ as in Lemma 6.1. For any $t\in
U,\ f,g\in E_{t}$, set

$\left(h_{N}^{E}\right)_{t}(f,g):=\int_{\Omega_{t}}\sum_{\lambda,\mu=1}^{r}f_{\lambda}\overline{g_{\mu}}\left(h_{N}^{F}\right)_{(t,z)}(e_{\lambda},e_{\mu})d\lambda(z),$

then $h_{N}^{E}$ is a Hermitian metric on $E$. We may similarly define
$h_{N,\epsilon}^{E}$ according to $h_{N,\epsilon}^{F}.$ It is clear that
$E^{k}$ is a holomorphic subbundle of $E$ (the metric on $E^{k}$ is similar to
that on $E$). Similar to the proof of [9, Lemma 3.1], by Lemma 6.1, we know
for any sections $u_{1},\cdots,u_{n}$ of $E^{k}$, we have

(6.1)
$\sum_{j,l}\left.h_{N,\epsilon}^{E}\left(\Theta_{jl}^{(E^{k},h_{N,\epsilon}^{E})}u_{j},u_{l}\right)\right|_{t_{0}}\geq\int_{D}\sum_{j,l,\beta,\gamma}H(h^{F}_{N,\epsilon})_{jl\beta\gamma}(t_{0},z)u_{j\beta}\overline{u_{l\gamma}}d\lambda(z).$

In fact, since $D$ is a circular domain containing the origin, any
$f\in\mathcal{O}(D)$ can be represented as a series

$f=\sum_{j=0}^{+\infty}f_{j}$

that is convergent locally uniformly on $D$, where each $f_{j}$ is a
homogenous polynomial of degree $j$. For any $\mathbb{S}^{1}$-invariant
continuous bounded function $\psi$ on $D$, and any homogenous polynomials
$g_{j},g_{l}$ of degree $j$ and $l$ respectively, we have

$\int_{D}g_{j}\bar{g}_{l}\psi d\lambda=0,\ \forall j\neq l.$

It follows that, for any $t\in U$, an element $f$ in the orthogonal complement
$(E^{k})_{t}^{\bot}$ of $E_{t}^{k}$ in $E_{t}$ has the form

$f=\sum_{j\geq 0,j\neq k}f_{k},$

where each $f_{j}$ is a homogeneous polynomial of degree $j$. Hence
$(E^{k})_{t}^{\bot}$ as a vector space is independent of the choice of the
hermitian metric $h^{F}$ and is also a holomorphic subbundle of $E$. By Lemma
6.1 and Lemma 2.2, we know Inequality (6.4) holds.
For any $N\in\mathbb{N},\ s>\frac{1}{N^{2}}$, set

$d(z,\Omega_{t_{0}}):=\inf_{w\in\Omega_{t_{0}}}|z-w|,\
\Omega_{t_{0},s}^{N}:=\\{z\in\mathbb{C}^{m}|\
\frac{1}{N^{2}}<d(z,\Omega_{t_{0}})<s\\},$

Fix an $s$ such that $\Omega_{t_{0},s}^{N}\subset\subset D$ for all $N>>1$. It
is clear that

$\max\nolimits_{(\frac{1}{N^{3}},\frac{1}{N^{3}})}\\{0,\rho\\}=\rho\text{ on
}\Omega^{N}_{t_{0},s}$

for all $N>>1$, then we have

(6.2)
$\int_{\Omega_{t_{0},s}}N\sum_{j,l,\beta,\gamma}H(h_{N,\epsilon}^{F})_{jl\beta\gamma}u_{j\beta}\overline{u_{l\gamma}}d\lambda\geq\int_{\Omega_{t_{0},s}}\frac{N}{2}\sum_{j,l,\beta}H(\rho)_{jl}u_{j\beta}\overline{u_{l\beta}}e^{-N\rho}d\lambda$

on $\Omega^{N}_{t_{0},s}$ for all $N>>1$, where
$H(\rho)_{jl}=\rho_{jl}-\sum_{p,q}\rho^{pq}\rho_{jq}\rho_{pl}$. The remaining
is similar to the proof of Theorem 4.2 of [9]. We will first prove
$i\Theta^{(E^{k},h_{N}^{E})}\geq\theta$ in the sense of Nakano for some
$\theta$, then by Lemma 2.6, we get $i\Theta_{(E^{k},h)}\geq\theta$ whenever
$N\rightarrow\infty$, which will completes the proof. The details is stated as
follows.
By the assumption and the tube neighborhood theorem, it is clear that there
are constants $\delta_{0},\delta_{1}>0$ such that (in the following, whenever
we say a number is a constant, we mean that it is a number that does not
depend on the sections of $E$ and $N,\epsilon,s$)

$\displaystyle\int_{\Omega_{t_{0},s}}N\sum_{j,l,\beta}H(\rho)_{jl}u_{j\beta}\overline{u_{l\beta}}e^{-N\rho}d\lambda\geq\delta_{0}\int_{\Omega_{t_{0},s}}N\sum_{j,\beta}|u_{j\beta}|^{2}e^{-N\rho}d\lambda$
$\displaystyle\geq$
$\displaystyle\delta_{1}\int_{\zeta\in\partial\Omega_{t_{0}}}dS(\zeta)\int^{s}_{1/N^{2}}N\sum_{j,\beta}|u_{j\beta}(\zeta+\tau\mathbf{n}_{\zeta})|^{2}e^{-N\rho(\zeta+\tau\mathbf{n}_{\zeta})}d\tau$
$\displaystyle\geq$
$\displaystyle\delta_{1}\sum_{j,\beta}\int_{\zeta\in\partial\Omega_{t_{0}}}dS(\zeta)\inf_{1/N^{2}\leq\tau\leq
s}|u_{j\beta}(\zeta+\tau\mathbf{n}_{\zeta})|^{2}\int^{s}_{1/N^{2}}Ne^{-N\rho(\zeta+\tau\mathbf{n}_{\zeta})}d\tau$
$\displaystyle\geq$
$\displaystyle\delta_{1}\sum_{j,\beta}\int_{\zeta\in\partial\Omega_{t_{0}}}dS(\zeta)\inf_{0\leq\tau\leq
s}|u_{j\beta}(\zeta+\tau\mathbf{n}_{\zeta})|^{2}\int^{s}_{1/N^{2}}Ne^{-NT\tau}d\tau,$

where $\mathbf{n}_{\zeta}$ is the unit outward normal vector of
$\partial\Omega_{t_{0}}$ at $\zeta$, and $T>0$ is a constant such that
$\rho(\zeta+\tau\mathbf{n}_{\zeta})\leq T\tau$ for all
$\zeta\in\partial\Omega_{t_{0}}$ and $0\leq\tau\leq s$. Since

$\lim_{N\rightarrow\infty}\int^{s}_{1/N^{2}}Ne^{-NT\tau}d\tau=\frac{1}{T}>0,$

then for $N$ sufficiently large, we have

(6.3) $\int^{s}_{1/N^{2}}Ne^{-NT\tau}d\tau\geq\frac{1}{2T}.$

Combining Inequalities (6.4), (6.2) and (6.3), we know there is a constant
$\delta_{4}>0$ such that, for any sections $u_{1},\cdots,u_{n}$ of $E^{k}$, we
have

(6.4)
$\sum_{j,l}\left.h_{N,\epsilon}^{E}\left(\Theta_{jl}^{(E^{k},h_{N,\epsilon}^{E})}u_{j},u_{l}\right)\right|_{t_{0}}\geq\delta_{4}\sum_{j,\beta}\int_{\partial\Omega_{t_{0}}}\inf_{0\leq\tau\leq
s}|u_{j\beta}(\zeta+\tau\mathbf{n}_{\zeta})|^{2}dS(\zeta).$

Let $\epsilon\rightarrow 0+$, we get

(6.5)
$\displaystyle\sum_{j,l}\left.h^{E}_{N}\left(\Theta_{jl}^{(E^{k},h^{E}_{N})}u_{j},u_{l}\right)\right|_{t_{0}}\geq\delta_{4}\sum_{j,\beta}\int_{\partial\Omega_{t_{0}}}\inf_{0\leq\tau\leq
s}|u_{j\beta}(\zeta+\tau\mathbf{n}_{\zeta})|^{2}dS$

for $N$ sufficiently large.
We claim that there is a constant $\delta_{5}>0$ such that, for any sections
$u_{1},\cdots,u_{n}$ of $E^{k}$, we have

(6.6) $\sum_{j,\beta}\int_{\partial\Omega_{t_{0}}}\inf_{0\leq\tau\leq
s}|u_{j\beta}(\zeta+\tau\mathbf{n}_{\zeta})|^{2}dS\geq\delta_{5}\sum_{j}h^{E}(u_{j},u_{j}).$

Without loss of generality, we only need to sections $u_{1},\cdots,u_{n}$ of
$E^{k}$ such that

$\sum_{j,\beta}\int_{\Omega_{t_{0}}}|u_{j\beta}|^{2}d\lambda=1.$

Since each $u_{j\beta}$ is a homogenous polynomial of degree $k$ and
$\Omega_{t_{0}}$ contains the origin, we may choose a constant $M_{1}>0$ and a
large ball $B$ with $\overline{D}\subset B$ such that
$\sum_{j,\beta}\int_{B}|u_{j\beta}|^{2}d\lambda(z)\leq M_{1}$. By Cauchy’s
Inequality for holomorphic functions, there is a constant $M_{2}>0$ such that
$\sum_{j}|du_{j}^{2}|\leq M_{2}$ on $D$, then it is obvious that

(6.7) $\sum_{j,\beta}\inf_{0\leq\tau\leq
s}|u_{j\beta}(\zeta+\tau\mathbf{n}_{\zeta})|^{2}\geq\sum_{j,\beta}|u_{j\beta}(\zeta)|^{2}-nrsM_{2}$

for all $\zeta\in\partial\Omega_{t_{0}}$. By Corollary 1.7 of [10], there is a
constant $\delta_{6}>0$ such that

$\sum_{j,\beta}\int_{\partial\Omega_{t_{0}}}|u_{j\beta}(\zeta)|^{2}dS\geq\delta_{6}\sum_{j,\beta}\int_{\Omega_{t_{0}}}|u_{j\beta}|^{2}d\lambda=\delta_{6}.$

Choose $N>>1$ enough large and $s>0$ enough small, then by continuity, we know
there exists a constant $\delta_{7}>0$ such that

$\sum_{j,\beta}\int_{\Omega_{t_{0}}}|u_{j\beta}|^{2}d\lambda-
nrsM_{2}\geq\delta_{7}\sum_{j}h^{E}(u_{j},u_{j}),$

which completes the proof of Inequality (6.6).
By Inequality (6.5) and Inequality (6.7), there is a constant $\delta_{8}>0$
such that for $N$ sufficiently large and all sections $u_{1},\cdots,u_{n}$ of
$E^{k}$, we have

(6.8)
$\sum_{j,l}\left.h^{E}_{N}\left(\Theta_{jl}^{(E^{k},h^{E}_{N})}u_{j},u_{l}\right)\right|_{t_{0}}\geq\delta_{8}\sum_{j}h^{E}(u_{j},u_{j}).$

In another word, if we take (where we will shrink $U$ if necessary to gurantee
$\delta_{8}$ is a continuous function of $t_{0}$)

$\theta:=i\delta_{8}\sum_{j}dt_{j}\wedge
d\bar{t}_{j}\otimes\operatorname{Id}_{E^{k}}\in
C^{0}(U,\wedge^{1,1}T^{*}_{U}\otimes\operatorname{End}(E^{k})),$

then we have $i\Theta^{(E^{k},h^{E}_{N})}\geq\theta$ in the sense of Nakano.
Now we prove $i\Theta^{(E^{k},h^{E})}\geq\theta$ in the sense of Nakano. Let
$\psi$ be a strictly plurisubharmonic function on $U$, and $f\in
C_{c}^{\infty}(U,\wedge^{n,1}T^{*}_{U}\otimes E^{k})$ satisfies
$\bar{\partial}f=0$ and

$\int_{U}h^{E}\left(B_{i\partial\bar{\partial}\psi,\theta}^{-1}f,f\right)e^{-\psi}d\lambda<+\infty,$

where $B_{i\partial\bar{\partial}\psi,\theta}$ is given as in Lemma 2.6. Then
there exists a constant $M_{3}>0$ such that

$\int_{U}h_{N}^{E}\left(B_{i\partial\bar{\partial}\psi,\theta}^{-1}f,f\right)e^{-\psi}d\lambda\leq
M_{3},\ \forall N>>1.$

By Lemma 2.5, for all $N>>1$, there are measurable sections $u_{N}$ of
$\wedge^{n,0}T^{*}_{U}\otimes E^{k}$ on $U$, such that $\bar{\partial}u_{N}=f$
and

$\int_{U}h_{N}^{E}\left(u_{N},u_{N}\right)e^{-\psi}d\lambda_{t}\leq\int_{U}h_{N}^{E}\left(B_{i\partial\bar{\partial}\psi,\theta}^{-1}f,f\right)e^{-\psi}d\lambda_{t}\leq
M_{3}.$

Since $h^{E}_{N}=h^{E}$ on $\Omega$, we have

$\int_{U}h^{E}(u_{N},u_{N})e^{-\psi}d\lambda\leq\int_{U}h_{N}^{E}\left(u_{N},u_{N}\right)e^{-\psi}d\lambda\leq
M_{3},\ \forall N>>1.$

Since every bounded sequence in a Hilbert space has a weakly convergent
subsequence, by taking a subsequence if necessary, we may assume $u_{N}$
converges weakly to $u$, which is a measurable section of
$\wedge^{n,0}T^{*}_{U}\otimes E^{k}$. Moreover we have $\bar{\partial}u=f$ in
the sense of distribution and

$\int_{U}h^{E}(u,u)e^{-\psi}d\lambda\leq\liminf_{N\rightarrow\infty}\int_{U}h^{E}(u_{N},u_{N})e^{-\psi}d\lambda\leq
M_{3}.$

By Lebesgue’s Dominated Convergence Theorem, we know

$\lim_{N\rightarrow\infty}\int_{U}h_{N}^{E}\left(B_{i\partial\bar{\partial}\psi,\theta}^{-1}f,f\right)e^{-\psi}d\lambda=\int_{U}h^{E}\left(B_{i\partial\bar{\partial}\psi,\theta}^{-1}f,f\right)e^{-\psi}d\lambda,$

so we get

$\int_{U}h^{E}(u,u)e^{-\psi}d\lambda\leq\leq\int_{U}h^{E}\left(B_{i\partial\bar{\partial}\psi,\theta}^{-1}f,f\right)e^{-\psi}d\lambda.$

It follows from Lemma 2.6 that $i\Theta^{(E^{k},h)}\geq\theta$ in the sense of
Nakano. In particular, the curvature of $(E^{k},h)$ is strictly positive in
the sense of Nakano. ∎

##  7\. The proof of Theorem 1.6

In this section, we give the proof of Theorem 1.6.

######  Theorem 7.1 (= Theorem 1.6).

Let $D\subset U_{0}\times\mathbb{R}^{m}$ be a strictly convex family of
bounded domains (with connected fibers) over a domain
$U_{0}\subset\mathbb{R}^{n},$ and let $(F,g^{F})$ be a trivial vector bundle
of finite rank $r$ defined on some neighborhood of $\overline{D}$. Let
$\\{e_{1},\cdots,e_{r}\\}$ be the canonical frame of $F$. For any $t\in
U_{0}$, set $E_{t}:=\mathbb{R}^{r}$, with an inner product $g_{t}^{E}$ given
by

$g_{t}^{E}(u,v):=\int_{D_{t}}\sum_{\lambda,\mu=1}^{r}u_{\lambda}v_{\mu}g^{F}_{(t,x)}(e_{\lambda},e_{\mu})d\lambda(x)$

for all

$u:=(u_{1},\cdots,u_{r})\in E_{t},\ v:=(v_{1},\cdots,v_{r})\in E_{t}.$

We set $E:=\cup_{t\in U_{0}}E_{t}$ and view it as a Riemann (trivial) vector
bundle over $U_{0}$, then the curvature of $(E,g^{E})$ is strictly positive in
the sense of Nakano if $(F,g^{F})$ is Nakano positive on some neighborhood of
$\overline{D}$.

###### Proof..

Consider the map

$\phi\colon\mathbb{C}_{z}^{m}\rightarrow(\mathbb{C}_{w}^{*})^{m},\
(z_{1},\cdots,z_{m})\mapsto(e^{z_{1}},\cdots,e^{z_{m}}).$

Similar as in the proof of Theorem 3.1, we may get a measure on
$(\mathbb{C}_{w}^{*})^{n}$ via $\phi$ which satisfies

$d\mu(w)=\frac{1}{|w_{1}|^{2}\cdots|w_{n}|^{2}}d\lambda(w),\ \forall
w\in(\mathbb{C}^{*})^{n}.$

Set $U:=U_{0}+i\mathbb{R}^{n}\subset\mathbb{C}_{\tau}^{n}$. For any $\tau\in
U$, set

$\Omega_{\tau}:=D_{\operatorname{Re}(\tau)}+i\mathbb{R}^{m}\subset\mathbb{C}_{z}^{m},\
(\Omega^{\prime})_{\tau}:=\phi(D_{\operatorname{Re}(\tau)}+i\mathbb{R}^{m})\subset(\mathbb{C}_{w}^{*})^{m},$

then both $\Omega:=\cup_{\tau\in U}\Omega_{\tau}$ and
$\Omega^{\prime}:=\cup_{\tau\in U}(\Omega^{\prime})_{\tau}$ are strictly
pseudoconvex family of domains over $U$. It is clear that each $\Omega_{\tau}$
is a (connected) Reinhardt domain for each $\tau\in U$. We extend $g^{E}$ to a
Hermitian metric $h^{E^{\prime}}$ on the trivial vector bundle
$E^{\prime}:=U\times\mathbb{C}^{r}$ over $U$ such that $h^{E^{\prime}}$ is
independent of $\operatorname{Im}(\tau)$ for any $\tau\in U$. We also extend
$g^{F}$ to a Hermitian metric $h^{\prime}$ on the trivial vector bundle
$F^{\prime}:=\Omega\times\mathbb{C}^{r}$ over $\Omega$ such that $h^{\prime}$
is independent of $\operatorname{Im}(\tau),\ \operatorname{Im}(z)$ for any
$(\tau,z)\in\Omega.$ $h^{\prime\prime}$ clearly induces a Hermitian metric
$h^{\prime\prime}$ on the trivial vector bundle
$F^{\prime\prime}:=\Omega^{\prime}\times\mathbb{C}^{r}$. It is obvious that
$(F^{\prime\prime},h^{\prime\prime})$ is Nakano positive on some neighborhood
of the closure of $\Omega^{\prime}$ by assumption. Let

$h_{(\tau,w)}:=\frac{h^{\prime\prime}_{(\tau,w)}}{|w_{1}|^{2}\cdots|w_{m}|^{2}}\text{
for any }w\in\Omega_{\tau}\text{ and any }\tau\in U,$

then $(F^{\prime\prime},h)$ is again Nakano positive on some neighborhood of
the closure of $\Omega^{\prime}$, and $h_{(\tau,w)}$ is
$\mathbb{T}^{n}$-invariant with respect to $w$ for any $w\in\Omega_{\tau}$ and
any $\tau\in U$. Let $\\{e_{1}^{\prime\prime},\cdots,e_{r}^{\prime\prime}\\}$
be the canonical holomorphic frame of $F^{\prime\prime}$.
For any $\tau\in U$, by Lemma 2.8, we have

$h_{\tau}^{E^{\prime}}(u,v)=\frac{1}{(2\pi)^{m}}\int_{(\Omega^{\prime})_{\tau}}\sum_{\lambda,\mu=1}^{r}u_{\lambda}\overline{v_{\mu}}h_{(\tau,w)}(e_{\lambda}^{\prime\prime},e_{\mu}^{\prime\prime})d\lambda(w),\
\forall u,v\in(E^{\prime})_{\tau}.$

By Theorem 1.5, we know $(E^{\prime},h^{E^{\prime}})$ is Nakano strictly
positive on $U$. Therefore, $(E,g^{E})$ is also strictly positive in the sense
of Nakano. ∎

## References

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