# The reduced Bergman kernel and its properties

Sahil Gehlawat*, Aakanksha Jain† and Amar Deep Sarkar‡ SG: Harish-Chandra
Research Institute, A CI of Homi Bhabha National Institute, Chhatnag Road,
Jhunsi, Prayagraj - 211019, India<EMAIL_ADDRESS><EMAIL_ADDRESS>AJ: Department of Mathematics, Indian Institute of
Science, Bangalore 560 012, India<EMAIL_ADDRESS>ADS: Indian Institute
of Science Education and Research Kolkata, India<EMAIL_ADDRESS>
###### Abstract.

In this article, we study some properties of the $n$-th order weighted reduced
Bergman kernels for planar domains, $n\geq 1$. Specifically, we look at
Ramadanov type theorems, localization, and boundary behaviour of the weighted
reduced Bergman kernel and its higher-order counterparts. We also give a
transformation formula for these kernels under biholomorphisms.

###### Key words and phrases:

reduced Bergman kernel, Ramadanov theorem, localization, boundary behaviour,
transformation formula

###### 1991 Mathematics Subject Classification:

Primary: 30H20, 46E22; Secondary: 30C40

*The author is supported by the postdoctoral fellowship of Harish-Chandra Research Institute, Prayagraj (Allahabad).

$\dagger$The author is supported by the PMRF Ph.D. fellowship.

$\ddagger$The author is supported by the postdoctoral fellowship of Indian
Institute of Science Education and Research Kolkata.

##  1\. Introduction

Recall that the Bergman space associated with a domain $D\subset\mathbb{C}$
consists of square-integrable holomorphic functions on it. An important space
that has a close relationship with this space is the collection of all
holomorphic functions whose derivatives are square-integrable with respect to
the area measure. This space can be associated with a closed subspace of the
Bergman space, which we call the reduced Bergman space. The reduced Bergman
space is a reproducing kernel Hilbert space with its reproducing kernel called
the reduced Bergman kernel; see below for the definitions.

In a previous work (see [3]), we proved a transformation formula for the
weighted reduced Bergman kernels under proper holomorphic maps between bounded
planar domains. In continuation of our previous effort, this note is a step
forward in creating a dictionary between the Bergman kernel and the reduced
Bergman kernel. This note consists of two parts.

In the first part, we prove Ramadanov type theorem for the weighted reduced
Bergman kernel. We also prove Ramadanov type theorem for the higher-order
weighted reduced Bergman kernels (see Definition 1.2).

The second part of this article is about the local analysis of these $n$-th
order weighted reduced Bergman kernels near a boundary point. In particular,
we study localization of the $n$-th order weighted reduced Bergman kernels.
Using this localization, we make some observations about the boundary
asymptotics of these kernels. Similar ideas have been used to study the
boundary behaviour of the weighted Bergman kernels, see [6]. In a related work
by the third author, similar boundary behaviour was observed for the class of
finitely connected domains bounded by Jordan curves for the span metric,
closely connected to the reduced Bergman kernel, see [13].

We also prove a transformation formula for these $n$-th order reduced Bergman
kernels under biholomorphisms, that we will use to get the boundary asymptotes
for these kernels; which is interesting in its own right.

For a domain $D\subset\mathbb{C}$, we will be working with admissible weights
due to Pasternak-Winiarski. The space of admissible weights on $D$ is denoted
by $AW(D)$.

######  Definition 1.1.

(See, [8], [9]) Let $D\subset\mathbb{C}$ be a domain and $\mu$ be a positive
measurable real-valued function on $D$. The weight $\mu$ is called an
admissible weight on $D$ if

  1. (i)

the weighted Bergman space $A^{2}_{\mu}(D)$ is a closed subspace of
$L^{2}_{\mu}(D)$ (and hence a Hilbert space), and

  2. (ii)

the evaluation functional $A^{2}_{\mu}(D)\ni f\mapsto f(z)\in\mathbb{C}$ is
continuous for every $z\in D$.

M. Sakai ([11], [12]) defined the $n$-th order weighted reduced Bergman kernel
in the following way:

######  Definition 1.2.

Let $D\subset\mathbb{C}$ be a domain, $\mu\in AW(D)$, $\zeta\in D$, and $n$ be
a positive integer. Define

$AD^{\mu}(D,\zeta^{n})=\left\\{f\in\mathcal{O}(D):f(\zeta)=f^{\prime}(\zeta)=\cdots=f^{(n-1)}(\zeta)=0,\,\int_{D}\lvert
f^{\prime}(z)\rvert^{2}\mu(z)dA(z)<\infty\right\\}.$

This is a Hilbert space with respect to the inner product

$\langle
f,g\rangle_{AD^{\mu}(D,\zeta^{n})}=\int_{D}f^{\prime}(z)\,\overline{g^{\prime}(z)}\mu(z)\,dA(z),\quad
f,g\in AD^{\mu}(D,\zeta^{n}).$

The linear functional defined by $AD^{\mu}(D,\zeta^{n})\ni f\mapsto
f^{(n)}(\zeta)\in\mathbb{C},$ is continuous. By Riesz representation theorem,
there exists a unique function $M_{D,\mu,n}(\cdot,\zeta)\in
AD^{\mu}(D,\zeta^{n})$ such that $f^{(n)}(\zeta)=\langle
f,M_{D,\mu,n}(\cdot,\zeta)\rangle_{AD^{\mu}(D,\zeta^{n})}$ for every $f\in
AD^{\mu}(D,\zeta^{n})$. Define

$\tilde{K}_{D,\mu,n}(z,\zeta)=\frac{\partial}{\partial
z}M_{D,\mu,n}(z,\zeta),\quad z,\zeta\in D.$

The kernel $\tilde{K}_{D,\mu,n}$ is called the $n$-th order weighted reduced
Bergman kernel of $D$ with respect to the weight $\mu$. Putting $n=1$ gives us
the weighted reduced Bergman kernel $\tilde{K}_{D,\mu}$ of $D$ with respect to
the weight $\mu$. So,

$f^{(n)}(\zeta)=\int_{D}f^{\prime}(z)\,\overline{\tilde{K}_{D,\mu,n}(z,\zeta)}\,\mu(z)\,dA(z)\quad\text{for
}f\in AD^{\mu}(D,\zeta^{n}).$

We now compute the $n$-th order reduced Bergman kernels for the unit disc
$\mathbb{D}$.

######  Example 1.

Let $\zeta\in\mathbb{D}$ and $f\in AD(\mathbb{D},\zeta^{n})$ for
$n\in\mathbb{Z}^{+}$, that is $f^{(k)}(\zeta)=0$ for all $0\leq k\leq n-1$.
For $g\in AD(\mathbb{D},0^{n})$ and $0<r<1$, the Cauchy integral formula gives
us

$\displaystyle g^{(n)}(0)$ $\displaystyle=$ $\displaystyle\frac{(n-1)!}{2\pi
i}\int_{|\xi|=r}\frac{g^{\prime}(\xi)}{\xi^{n}}\,d\xi=\frac{(n-1)!}{2\pi
i}\int_{0}^{2\pi}\frac{g^{\prime}(re^{it})}{(re^{it})^{n}}re^{it}i\,dt$
$\displaystyle=$
$\displaystyle\frac{(n-1)!}{2\pi}\int_{0}^{2\pi}\frac{g^{\prime}(re^{it})}{r^{n-1}}\overline{(e^{it})^{n-1}}\,dt.$

Multiplying both sides by $r^{2n-1}$ and integrating with respect to parameter
$r$, we get

$\int_{0}^{1}g^{(n)}(0)r^{2n-1}dr=\frac{(n-1)!}{2\pi}\int_{0}^{1}\int_{0}^{2\pi}g^{\prime}(re^{it})\overline{(re^{it})^{n-1}}\,r\,dr\,dt.$

By change of variables on the right hand side of the above equation, we get

$g^{(n)}(0)=\frac{n!}{\pi}\int_{\mathbb{D}}g^{\prime}(\xi)\overline{(\xi)^{n-1}}\,dA(\xi).$
(1)

Let $\tilde{K}_{n}(\cdot,\cdot)$ denote the $n$-th order reduced Bergman
kernel of $\mathbb{D}$. For all $\zeta\in\mathbb{D}$, and $f\in
AD(\mathbb{D},\zeta^{n})$, we have

$f^{(n)}(\zeta)=\int_{\mathbb{D}}f^{\prime}(\xi)\overline{\tilde{K}_{n}(\xi,\zeta)}\,dA(\xi).$
(2)

Let $\phi_{\zeta}:\mathbb{D}\rightarrow\mathbb{D}$ be the automorphism of unit
disc given by $\phi_{\zeta}(z)=\frac{\zeta-z}{1-z\overline{\zeta}}$. Note that
$\phi_{\zeta}(0)=\zeta$, $\phi_{\zeta}(\zeta)=0$, and
$\phi_{\zeta}\circ\phi_{\zeta}(z)=z$ for all $z\in\mathbb{D}$. Consider the
holomorphic function $f\circ\phi_{\zeta}$. Observe that $f\circ\phi_{\zeta}\in
AD(\mathbb{D},0^{n})$ and
$(f\circ\phi_{\zeta})^{(n)}(0)=f^{(n)}(\zeta)(\phi_{\zeta}^{\prime}(0))^{n}$.
On substituting $g=f\circ\phi_{\zeta}$ in equation (1), we get

$\displaystyle f^{(n)}(\zeta)(\phi_{\zeta}^{\prime}(0))^{n}$ $\displaystyle=$
$\displaystyle\frac{n!}{\pi}\int_{\mathbb{D}}(f\circ\phi_{\zeta})^{\prime}(\chi)\overline{(\chi)^{n-1}}\,dA(\chi)$
$\displaystyle=$
$\displaystyle\frac{n!}{\pi}\int_{\mathbb{D}}f^{\prime}(\phi_{\zeta}(\chi))\phi_{\zeta}^{\prime}(\chi)\overline{(\chi)^{n-1}}\,dA(\chi).$

Now by doing change of variables $\xi=\phi_{\zeta}(\chi)$, we get
$\chi=\phi_{\zeta}(\xi)$, and
$\phi_{\zeta}^{\prime}(\chi)=\frac{1}{\phi_{\zeta}^{\prime}(\xi)}$. Therefore

$\displaystyle f^{(n)}(\zeta)(\phi_{\zeta}^{\prime}(0))^{n}$ $\displaystyle=$
$\displaystyle\frac{n!}{\pi}\int_{\mathbb{D}}f^{\prime}(\xi)(\phi_{\zeta}^{\prime}(\xi))^{-1}\overline{(\phi_{\zeta}(\xi))^{n-1}}\,|\phi_{\zeta}^{\prime}(\xi)|^{2}\,dA(\xi)$
$\displaystyle=$
$\displaystyle\frac{n!}{\pi}\int_{\mathbb{D}}f^{\prime}(\xi)\overline{(\phi_{\zeta}(\xi))^{n-1}}\,\overline{\phi_{\zeta}^{\prime}(\xi)}\,dA(\xi).$

Therefore we get

$f^{(n)}(\zeta)=\int_{\mathbb{D}}f^{\prime}(\xi)\frac{n!\overline{(\phi_{\zeta}(\xi))^{n-1}}\,\overline{\phi_{\zeta}^{\prime}(\xi)}}{\pi(\phi_{\zeta}^{\prime}(0))^{n}}\,dA(\xi).$
(3)

Comparing the equations (2) and (3), we get

$\tilde{K}_{n}(\xi,\zeta)=\frac{n!}{\pi}\frac{(\phi_{\zeta}(\xi))^{n-1}\,\phi_{\zeta}^{\prime}(\xi)}{\overline{(\phi_{\zeta}^{\prime}(0))^{n}}}.$

We can check that
$\phi_{\zeta}^{\prime}(\xi)=\frac{|\zeta|^{2}-1}{(1-\overline{\zeta}\xi)^{2}}$,
which gives $\phi_{\zeta}^{\prime}(0)=|\zeta|^{2}-1$. Therefore,

$\tilde{K}_{n}(\xi,\zeta)=\frac{n!}{\pi}\frac{(\zeta-\xi)^{n-1}}{(1-\overline{\zeta}\xi)^{n-1}}\frac{|\zeta|^{2}-1}{(1-\overline{\zeta}\xi)^{2}}\frac{1}{(|\zeta|^{2}-1)^{n}}.$

Thus, for $n\geq 1$, the $n$-th order reduced Bergman kernel is given by

$\tilde{K}_{n}(\xi,\zeta)=\frac{n!}{\pi}\frac{(\xi-\zeta)^{n-1}}{(1-\overline{\zeta}\xi)^{n+1}(1-|\zeta|^{2})^{n-1}}.$
(4)

The following is an equivalent definition of the $1$-st order weighted reduced
Bergman kernel. We shall use both definitions interchangeably.

######  Definition 1.3.

Let $D$ be a domain in $\mathbb{C}$ and $\mu\in\text{AW}(D)$. The weighted
reduced Bergman space of $D$ is the space of all the square-integrable
holomorphic functions on $D$ whose primitive exists on $D$, i.e.,

$\mathcal{D}^{\mu}(D)=\left\\{f\in\mathcal{O}(D):f=g^{\prime}\ \text{for
some}\ g\in\mathcal{O}(D)\,\,\text{and}\,\,\int_{D}\lvert
f(z)\rvert^{2}\mu(z)dA(z)<\infty\right\\}.$

This is a Hilbert space with respect to the inner product

$\langle f,g\rangle:=\int_{D}f(z)\overline{g(z)}\mu(z)dA(z).$

For every $\zeta\in D$, the evaluation functional

$f\mapsto f(\zeta),\quad\quad f\in\mathcal{D}^{\mu}(D)$

is a bounded linear functional, and therefore $\mathcal{D}^{\mu}(D)$ is a
reproducing kernel Hilbert space. The reproducing kernel of
$\mathcal{D}^{\mu}(D)$, denoted by $\tilde{K}_{D,\mu}(\cdot,\cdot)$, is called
the ($1$-st order) weighted reduced Bergman kernel of $D$ with respect to the
weight $\mu$. It satisfies the reproducing property:

$f(\zeta)=\int_{D}f(z)\overline{\tilde{K}_{D,\mu}(z,\zeta)}\mu(z)dA(z)$

for all $f\in\mathcal{D}^{\mu}(D)$ and $\zeta\in D$.

######  Remark 1.4.

It will easily follow from the above definition that $\tilde{K}_{D,\mu}$ is
holomorphic in the first variable, anti-holomorphic in the second variable,
and that $\tilde{K}_{D,\mu}\in C^{\infty}(D\times D)$.

It is known (see [1, p. 26], [2, p. 476]) that for a domain
$D\subset\mathbb{C}$, $\mu\in AW(D)$, and $n\geq 2$,

$\tilde{K}_{D,\mu,n}(z,\zeta)=\frac{(-1)^{n-1}}{J_{n-2}}\det\left(\begin{matrix}\tilde{K}_{0,\bar{0}}(z,\zeta)&\ldots&\tilde{K}_{0,\overline{n-1}}(z,\zeta)\\\
\tilde{K}_{0,\bar{0}}&\ldots&\tilde{K}_{0,\overline{n-1}}\\\
\tilde{K}_{1,\bar{0}}&\ldots&\tilde{K}_{1,\overline{n-1}}\\\ \vdots&&\vdots\\\
\tilde{K}_{n-2,\bar{0}}&\ldots&\tilde{K}_{n-2,\overline{n-1}}\end{matrix}\right),$
(5)

where $J_{n}=\det\left(\tilde{K}_{j\bar{k}}\right)_{j,k=0}^{n}$ and

$\tilde{K}_{j\bar{k}}(z,\zeta)=\frac{\partial^{j+k}}{\partial
z^{j}\partial\bar{\zeta}^{k}}\tilde{K}_{D,\mu}(z,\zeta),\quad\tilde{K}_{j\bar{k}}\equiv\tilde{K}_{j\bar{k}}(\zeta,\zeta).$

Here $J_{n}>0$ for all $\zeta\notin N_{D}(\mu):=\\{z\in
D:\tilde{K}_{D,\mu}(z,z)=0\\}$. Thus, $\tilde{K}_{D,\mu,n}\in
C^{\infty}(D\times(D\setminus N_{D}(\mu))$. If $\mu\in L^{1}(D)$, then
$N_{D}(\mu)=\emptyset$, and therefore $\tilde{K}_{D,\mu,n}\in
C^{\infty}(D\times D)$.

Notation:  For a non-negative integer $p$, we denote
$\tilde{K}^{(p)}_{D,\mu,n}(z,\zeta):=\frac{\partial^{p}}{\partial
z^{p}}\tilde{K}_{D,\mu,n}(z,\zeta)$.

######  Remark 1.5.

Let $D\subset\mathbb{C}$ be a bounded domain, $\mu\in L^{1}(D)$ be an
admissible weight on $D$, $\zeta\in D$ and $n$ be a positive integer.

  1. 1.

Note that,
$(M_{D,\mu,n}(\cdot,\zeta))^{(n)}(\zeta)=\|M_{D,\mu,n}(\cdot,\zeta)\|^{2}_{AD^{\mu}(D,\zeta^{n})}.$
Thus, $(M_{D,\mu,n}(\cdot,\zeta))^{(n)}(\zeta)=0$ would imply that
$M_{D,\mu,n}(\cdot,\zeta)\equiv 0$ and hence $f^{(n)}(\zeta)=0$ for all $f\in
AD^{\mu}(D,\zeta^{n})$. This is not true since $D$ is bounded and $\mu\in
L^{1}(D)$. Hence, $(M_{D,\mu,n}(\cdot,\zeta))^{(n)}(\zeta)>0$.

  2. 2.

Consider the extremal problem $\min\\{\|f\|_{AD^{\mu}(D,\zeta^{n})}:f\in
AD^{\mu}(D,\zeta^{n}),\,f^{(n)}(\zeta)=1\\}.$ Then

$g(z)=\frac{M_{D,\mu,n}(z,\zeta)}{\tilde{K}^{(n-1)}_{D,\mu,n}(\zeta,\zeta)},\quad
z\in D$

is the unique function solving the extremal problem with
$\|g\|_{AD^{\mu}(D,\zeta^{n})}=\sqrt{\tilde{K}_{D,\mu,n}^{(n-1)}(\zeta,\zeta)}$.

  3. 3.

Consider the extremal problem $\max\\{|f^{(n)}(\zeta)|:f\in
AD^{\mu}(D,\zeta^{n}),\,\|f\|_{AD^{\mu}(D,\zeta^{n})}=1\\}$. Then

$h(z)=\frac{M_{D,\mu,n}(z,\zeta)}{\sqrt{\tilde{K}^{(n-1)}_{D,\mu,n}(\zeta,\zeta)}},\quad
z\in D$

is the unique function solving the extremal problem such that
$h^{(n)}(\zeta)>0$. Here,
$h^{(n)}(\zeta)=\sqrt{\tilde{K}_{D,\mu,n}^{(n-1)}(\zeta,\zeta)}$.

Ramadanov [10] showed that for a sequence of domains $D_{j}\subset\mathbb{C}$
such that $D_{j}\subset D_{j+1}$ for all $j\in\mathbb{Z}^{+}$, and
$D:=\bigcup_{j=1}^{\infty}D_{j}$, the Bergman kernel $K_{j}(\cdot,\cdot)$
corresponding to the domain $D_{j}$ converges uniformly on compacts of
$D\times D$ to the Bergman kernel $K(\cdot,\cdot)$ corresponding to the domain
$D$. Over the years, different versions of Ramadanov type theorems have been
proved for the Bergman kernels and the weighted Bergman kernels (see, for
instance, [7], [14]), for different types of convergence of the sequence of
domains. Let us prove some of these Ramadanov type theorems for the $n$-th
order weighted reduced Bergman kernels.

######  Theorem 1.6.

Let $\\{D_{i}\\}_{i=1}^{\infty}$ be a sequence of domains in $\mathbb{C}$ with
$\mu_{i}\in AW(D_{i})$.

  1. (1)

Set $D:=\cup_{i}D_{i}$ and let $\mu\in AW(D)$. Assume that for any
$i\in\mathbb{Z}^{+}$, there exists a $j=j(i)\in\mathbb{Z}^{+}$ such that
$D_{i}\subset D_{k}$ and $\mu_{i}(z)\leq\mu_{k}(z)\leq\mu(z)$ for all $k\geq
j(i)$ and $z\in D_{i}$. If $\mu_{i}\longrightarrow\mu$ pointwise a.e. on $D$
then

    1. (A)

we have

$\lim\limits_{i\rightarrow\infty}\tilde{K}_{D_{i},\mu_{i}}=\tilde{K}_{D,\mu}$

locally uniformly on $D\times D$.

    2. (B)

for $n>1$ and $w\in D$,

$\lim\limits_{i\rightarrow\infty}\tilde{K}_{D_{i},\mu_{i},n}(\cdot,w)=\tilde{K}_{D,\mu,n}(\cdot,w)$

locally uniformly on $D$. In particular, $\tilde{K}_{D_{i},\mu_{i},n}$
converges to $\tilde{K}_{D,\mu,n}$ pointwise on $D\times D$. Furthermore, all
the derivatives of $\tilde{K}_{D_{i},\mu_{i},n}$ converge locally uniformly to
the respective derivatives of $\tilde{K}_{D,\mu,n}$ on $D\times(D\setminus
N_{D}(\mu))$.

  2. (2)

Suppose that $D=\cap_{i}D_{i}$ is a domain and $\mu\in AW(D)$ be such that
$\mu(z)\leq\mu_{i}(z)$ for all $i\in\mathbb{Z}^{+}$ and $z\in D$. Assume that
$\mu_{i}\longrightarrow\mu$ pointwise a.e. on $D$ and that for every fixed
$t\in D$, we have

$\lim\limits_{i\rightarrow\infty}\tilde{K}_{D_{i},\mu_{i}}(t,t)=\tilde{K}_{D,\mu}(t,t).$

Then,

    1. (A)

we have

$\lim\limits_{i\rightarrow\infty}\tilde{K}_{D_{i},\mu_{i}}=\tilde{K}_{D,\mu}$

locally uniformly on $D\times D$.

    2. (B)

for $n>1$, all the derivatives of $\tilde{K}_{D_{i},\mu_{i},n}$ converge
locally uniformly to the respective derivatives of $\tilde{K}_{D,\mu,n}$ on
$D\times(D\setminus N_{D}(\mu))$.

######  Remark 1.7.

If $\mu\in L^{1}(D)$, we have uniform convergence on all the compact subsets
of $D\times D$.

We have the following corollary about the convergence of the corresponding
kernel functions $M_{D_{i},\mu_{i},n}$ of the domains $D_{i}$.

######  Corollary 1.8.

Let $(D_{i},\mu_{i},D,\mu)$ for $i\in\mathbb{Z}^{+}$ be as in Theorem 1.6
$(1)$, then for $n\geq 1$ and for all $\xi\in D$, the $n$-th order kernel
function $M_{D_{i},\mu_{i},n}(\cdot,\xi)$ converges locally uniformly on the
domain $D$ to the kernel function $M_{D,\mu,n}(\cdot,\xi)$.

If $(D_{i},\mu_{i},D,\mu)$ are as in Theorem 1.6 $(2)$, then

  1. (1)

for all $\xi\in D$, the $1$-st order kernel function
$M_{D_{i},\mu_{i}}(\cdot,\xi)$ converges locally uniformly on the domain $D$
to the kernel function $M_{D,\mu}(\cdot,\xi)$.

  2. (2)

for $n>1$ and for all $\xi\in D\setminus N_{D}(\mu)$, the $n$-th order kernel
function $M_{D_{i},\mu_{i},n}(\cdot,\xi)$ converges locally uniformly on the
domain $D$ to the kernel function $M_{D,\mu,n}(\cdot,\xi)$.

Moving ahead to the second part of this note, we study the boundary behaviour
of $\tilde{K}^{(n-1)}_{D,\mu,n}(z,z)$ as $z$ approaches $\partial D$. We will
start by studying two main ingredients, namely, the localization of
$\tilde{K}^{(n-1)}_{D,\mu,n}(z,z)$ and the transformation formula for
$\tilde{K}^{(n-1)}_{D,\mu,n}$ under biholomorphisms.

For a domain $D\subset\mathbb{C}$ and a point $p\in\partial D$, let
$\mathcal{A}(D,p)$ denote the collection of all admissible weights $\mu$ on
$D$ that are in $L^{\infty}(D)$ and extend continuously to $p$ with
$\mu(p)>0$.

######  Definition 1.9.

Let $D\subset\mathbb{C}$ be a domain. A boundary point $p\in\partial D$ is
called a holomorphic local peak point of $D$ if there is a neighborhood $W$ of
$p$ in $\mathbb{C}$ and a function $h$ such that

  1. (i)

$h$ is continuous on $\overline{D}\cap W$,

  2. (ii)

$h$ is holomorphic on $D\cap W$,

  3. (iii)

$|h(z)|<1$ for all $z\in(\overline{D}\cap W)\setminus\\{p\\}$ and $h(p)=1$.

######  Theorem 1.10.

Let $D$ be a bounded domain in $\mathbb{C}$, $p\in\partial D$,
$\mu\in\mathcal{A}(D,p)$ and $n$ be a positive integer. There exists a
neighborhood $U$ of $p$ in $\mathbb{C}$ such that $D\cap U$ is connected and
$\mu\geq c$ a.e. on $D\cap U$ for some constant $c>0$. Hence, the $n$-th order
weighted reduced Bergman kernel $\tilde{K}_{D\cap U,\mu,n}$ of $D\cap U$ with
respect to the weight $\mu|_{D\cap U}$ is well-defined.

Suppose that $p$ is a holomorphic local peak point, and there exists a
neighborhood $N_{p}$ of $p$ in $\mathbb{C}$ such that $D\cap N_{p}$ is simply
connected. Then,

$\lim_{\zeta\rightarrow p}\frac{\tilde{K}^{(n-1)}_{D\cap
U,\mu,n}(\zeta,\zeta)}{\tilde{K}^{(n-1)}_{D,\mu,n}(\zeta,\zeta)}=1.$

Let us see an example where we study boundary behaviour of these kernel
functions explicitly using the above localization before studying it for
general domains.

######  Example 2.

Consider the domain
$D=\mathbb{D}\setminus\cup_{i=1}^{N}\overline{D(q_{i},r_{i})}$ for
$N\in\mathbb{Z}^{+}$, such that
$\overline{D(q_{i},r_{i})}\cap\overline{D(q_{j},r_{j})}=\emptyset$, if $i\neq
j$, where $q_{i}\in\mathbb{D}$, and
$\overline{D(q_{i},r_{i})}\subset\mathbb{D}$ for all $1\leq i\leq N$. We want
to study the behaviour of the kernel functions on the diagonal near the
boundary of $D$. Let $p\in\partial\mathbb{D}$, i.e., $|p|=1$, and observe that
for a sufficiently small neighourhood $U$ of $p$, we have $D\cap
U=\mathbb{D}\cap U$. Therefore, the above localization theorem 1.10 tells us
that

$\lim_{z\to
p}{\frac{\tilde{K}_{D,n}^{(n-1)}(z,z)}{\tilde{K}_{\mathbb{D},n}^{(n-1)}(z,z)}}=1.$
(6)

Now using the Leibniz rule to differentiate the equation (4), with respect to
the variable $z$ upto $(n-1)$ times, we get

$\tilde{K}_{\mathbb{D},n}^{(n-1)}(z,w)=\frac{1}{\pi(1-|w|^{2})^{n-1}}\sum_{k=0}^{n-1}{\binom{n-1}{k}^{2}(n+k)!\,(n-k-1)!\,\frac{\overline{w}^{k}(z-w)^{k}}{(1-\overline{w}z)^{n+k+1}}}.$

Taking $z=w$, all the terms in the above summation vanishes except for $k=0$,
so we get

$\tilde{K}_{\mathbb{D},n}^{(n-1)}(z,z)=\frac{n!\,(n-1)!}{\pi(1-|z|^{2})^{2n}}.$

Using this in the equation (6), we get

$\lim_{z\to
p}\left((1-|z|^{2})^{2n}\,\tilde{K}_{D,n}^{(n-1)}(z,z)\right)=\frac{n!\,(n-1)!}{\pi}.$
(7)

This is true for every finitely connected domain $D\subset\mathbb{D}$ such
that $\partial\mathbb{D}\subset\partial D$, and for every
$p\in\partial\mathbb{D}$.

Now we would like to study these kernel functions near other boundary
components. As we can see $\partial
D=\partial\mathbb{D}\cup\left(\cup_{i=1}^{N}\partial D(q_{i},r_{i})\right)$.
Take a point $p_{i}\in\partial D(q_{i},r_{i})=\\{|z-q_{i}|=r_{i}\\}$. Consider
the univalent holomorphic map $f:D\rightarrow\mathbb{D}$ given by
$f(z)=\frac{r_{i}}{z-q_{i}}$. One can easily see that
$\tilde{D}:=f(D)\subset\mathbb{D}$ is an $(N+1)-$connected domain with
$\partial\mathbb{D}\subset\partial\tilde{D}$ (since
$f(\\{|z-q_{i}|=r_{i}\\})=\\{|w|=1\\}$). Therefore equation (7) holds true for
the domain $\tilde{D}$. Now using the transformation formula proved in Theorem
1.11, we get

$\tilde{K}_{D,n}^{(n-1)}(z,z)=|f^{\prime}(z)|^{2n}\,\tilde{K}_{\tilde{D},n}^{(n-1)}\left(\frac{r_{i}}{z-q_{i}},\frac{r_{i}}{z-q_{i}}\right).$

Note that $f^{\prime}(z)=\frac{-r_{i}}{(z-q_{i})^{2}}$. Now multiplying both
sides by $\left(1-\lvert\frac{r_{i}}{z-q_{i}}\rvert^{2}\right)^{2n}$, and
taking the limit $z\to p_{i}$, we get

$\lim_{z\to
p_{i}}{\left(1-\left|\frac{r_{i}}{z-q_{i}}\right|^{2}\right)^{2n}\tilde{K}_{D,n}^{(n-1)}(z,z)}\\\
=\,\lim_{z\to
p_{i}}\frac{r^{2n}}{|z-q_{i}|^{4n}}\left(1-\left|\frac{r_{i}}{z-q_{i}}\right|^{2}\right)^{2n}\tilde{K}_{\tilde{D},n}^{(n-1)}\left(\frac{r_{i}}{z-q_{i}},\frac{r_{i}}{z-q_{i}}\right).$

Using the change of variables $\tilde{z}=\frac{r_{i}}{z-q_{i}}$, with
$\tilde{p}_{i}=\frac{r_{i}}{p_{i}-q_{i}}\in\partial\mathbb{D}$, we get

$\lim_{z\to
p_{i}}\frac{r_{i}^{2n}}{|z-q_{i}|^{4n}}\left(1-\left|\frac{r_{i}}{z-q_{i}}\right|^{2}\right)^{2n}\tilde{K}_{\tilde{D},n}^{(n-1)}\left(\frac{r_{i}}{z-q_{i}},\frac{r_{i}}{z-q_{i}}\right)\\\
=\,\frac{1}{r_{i}^{2n}}\lim_{\tilde{z}\to\tilde{p}_{i}}|\tilde{z}|^{4n}\,(1-|\tilde{z}|^{2})^{2n}\,\tilde{K}_{\tilde{D},n}^{(n-1)}(\tilde{z},\tilde{z}).$

Now using the fact that $|\tilde{p}_{i}|=1$, and equation (7) for the domain
$\tilde{D}$ gives us

$\lim_{z\to
p_{i}}{\left(1-\left|\frac{r_{i}}{z-q_{i}}\right|^{2}\right)^{2n}\tilde{K}_{D,n}^{(n-1)}(z,z)}=\frac{n!\,(n-1)!}{\pi
r_{i}^{2n}}.$ (8)

######  Theorem 1.11.

Let $D_{1}$, $D_{2}$ be domains in $\mathbb{C}$ and $n$ be a positive integer.
Let $\nu$ be an admissible weight on $D_{2}$ and $f:D_{1}\rightarrow D_{2}$ be
a biholomorphism. Then,

$(\overline{f^{\prime}(\zeta)})^{n}\,M_{D_{2},\nu,n}(f(z),f(\zeta))=M_{D_{1},\nu\circ
f,n}(z,\zeta),\quad z,\zeta\in D_{1}.$

Consequently, we have

$(f^{\prime}(z))^{n}\,\tilde{K}^{(n-1)}_{D_{2},\nu,n}(f(z),f(\zeta))\,(\overline{f^{\prime}(\zeta)})^{n}=\tilde{K}_{D_{1},\nu\circ
f,n}^{(n-1)}(z,\zeta),\quad z,\zeta\in D_{1}.$

######  Theorem 1.12.

Let $D\subset\mathbb{C}$ be a bounded domain and $p\in\partial D$. Assume that
$\partial D$ is $C^{2}$-smooth near $p$. If $\psi$ is a $C^{2}$-defining
function for $D$ near $p$ with $\frac{\partial\psi}{\partial z}(p)=1$, then
for a positive integer $n$

$\tilde{K}_{D,n}^{(n-1)}(z,z)\sim\frac{n!(n-1)!}{\pi}\frac{1}{(\psi(z))^{2n}}\approx\frac{1}{(\delta(z))^{2n}}\quad\text{as}\quad
z\rightarrow p.$

Here, $\delta(z)$ denotes the Euclidean distance of $z$ from $\partial D$. For
functions $f$ and $g$ on $D$, the notation $f\sim\lambda g$ as $z\rightarrow
p$ means that $f(z)/g(z)\rightarrow\lambda$ as $z\rightarrow p$, and $f\approx
g$ as $z\rightarrow p$ means that $f/g$ is bounded above and below by positive
constants in some neighborhood of $p$ in $\overline{D}$.

######  Theorem 1.13.

Let $D\subset\mathbb{C}$ be a bounded domain, $p\in\partial D$ and
$\nu\in\mathcal{A}(D,p)$. Assume that $\partial D$ is $C^{2}$-smooth near $p$.
Then

$\tilde{K}_{D,\nu,n}^{(n-1)}(z,z)\sim\frac{1}{\nu(p)}\,\tilde{K}_{D,n}^{(n-1)}(z,z)$

as $z\rightarrow p$.

Acknowledgement. The authors would like to thank Kaushal Verma for all the
helpful discussions and suggestions.

##  2\. Ramadanov Type Theorems

We will use the following characterization for the kernel functions.

######  Lemma 2.1.

Let $D\subset\mathbb{C}$ be a domain, $\mu\in AW(D)$ and $n\in\mathbb{Z}^{+}$.
For $w\in D$, let $S^{w}_{D,\mu,n}\subset AD^{\mu}(D,w^{n})$ denote the set of
all functions $f$ such that

$f^{(n)}(w)\geq 0\quad\text{and}\quad\lVert
f\rVert_{AD^{\mu}(D,w^{n})}\leq\sqrt{f^{(n)}(w)}.$

Then the $n$-th order weighted kernel function $M_{D,\mu,n}(\cdot,w)$ is
uniquely characterized by the properties:

  1. (i)

$M_{D,\mu,n}(\cdot,w)\in S_{D,\mu,n}^{w}$;

  2. (ii)

if $f\in S^{w}_{D,\mu,n}$ and
$f^{(n)}(w)\geq\tilde{K}^{(n-1)}_{D,\mu,n}(w,w)$, then $f(\cdot)\equiv
M_{D,\mu,n}(\cdot,w)$.

###### Proof.

Let $\varphi_{1},\varphi_{2}\in AD^{\mu}(D,w^{n})$ satisfy $(i)$ and $(ii)$.
Either $\varphi_{1}^{(n)}(w)\geq\varphi_{2}^{(n)}(w)$ or
$\varphi_{2}^{(n)}(w)\geq\varphi_{1}^{(n)}(w)$. In both the cases, we have
$\varphi_{1}\equiv\varphi_{2}$. Therefore, the function satisfying both $(i)$
and $(ii)$ is unique. We shall show that $M_{D,\mu,n}(\cdot,w)$ satisfies the
two properties. Since

$(M_{D,\mu,n}(\cdot,w))^{(n)}(w)=\tilde{K}_{D,\mu,n}^{(n-1)}(w,w)\geq
0\quad\text{and}\quad\lVert
M_{D,\mu,n}(\cdot,w)\rVert_{AD^{\mu}(D,w^{n})}=\sqrt{\tilde{K}^{(n-1)}_{D,\mu,n}(w,w)},$

property $(i)$ holds. Let $f\in S^{w}_{D,\mu,n}$ such that
$f^{(n)}(w)\geq\tilde{K}_{D,\mu,n}^{(n-1)}(w,w)$. If $f^{(n)}(w)=0$ then
$\tilde{K}^{(n-1)}_{D,\mu,n}(w,w)=0$ and therefore $\lVert
f\rVert_{AD^{\mu}(D,w^{n})}=\lVert
M_{D,\mu,n}(\cdot,w)\rVert_{AD^{\mu}(D,w^{n})}=0$. Hence, $f$ and
$M_{D,\mu,n}(\cdot,w)$ are constant functions on $D$ that vanish at $w\in D$.
Thus,

$f(\cdot)\equiv M_{D,\mu,n}(\cdot,w)\equiv 0.$

So, assume that $f^{(n)}(w)>0$. If $\tilde{K}_{D,\mu,n}^{(n-1)}(w,w)=0$ then
by the similar argument as above $M_{D,\mu,n}(\cdot,w)\equiv 0$ and therefore
$g^{(n)}(w)=\langle g,M_{D,\mu,n}(\cdot,w)\rangle_{AD^{\mu}(D,w^{n})}=0$ for
all $g\in AD^{\mu}(D,w^{n})$. This is a contradiction as $f^{(n)}(w)\neq 0$.
Therefore, $\tilde{K}^{(n-1)}_{D,\mu,n}(w,w)>0$.

Note that the function $f(\cdot)/f^{(n)}(w)$ belongs to the set $\\{h\in
AD^{\mu}(D,w^{n}):h^{(n)}(w)=1\\}$. Observe that

$\left\lVert\frac{f(\cdot)}{f^{(n)}(w)}\right\rVert_{AD^{\mu}(D,w^{n})}\leq\frac{\sqrt{f^{(n)}(w)}}{f^{(n)}(w)}=\frac{1}{\sqrt{f^{(n)}(w)}}\leq\frac{1}{\sqrt{\tilde{K}^{(n-1)}_{D,\mu,n}(w,w)}}=\left\lVert\frac{M_{D,\mu,n}(\cdot,w)}{\tilde{K}^{(n-1)}_{D,\mu,n}(w,w)}\right\rVert_{AD^{\mu}(D,w^{n})}$

By the uniqueness of the minimal problem: $\min\\{\lVert
h\rVert_{AD^{\mu}(D,w^{n})}:h\in AD^{\mu}(D,w^{n}),h^{(n)}(w)=1\\}$, we
therefore have

$\frac{f(\cdot)}{f^{(n)}(w)}\equiv\frac{M_{D,\mu,n}(\cdot,w)}{\tilde{K}^{(n-1)}_{D,\mu,n}(w,w)}\quad\text{and}\quad\frac{1}{\sqrt{f^{(n)}(w)}}=\frac{1}{\sqrt{\tilde{K}^{(n-1)}_{D,\mu,n}(w,w)}}.$

So, $f(\cdot)\equiv M_{D,\mu,n}(\cdot,w)$. ∎

In a similar manner, we can characterize the $1$-st order reduced Bergman
kernel as follows.

######  Lemma 2.2.

Let $D\subset\mathbb{C}$ be a domain, and $\mu$ be an admissible weight on
$D$. For $w\in D$, let $\tilde{S}_{D,\mu}^{w}\subset\mathcal{D}^{\mu}(D)$
denote the set of all functions $f$ such that $f(w)\geq 0$ and $\lVert
f\rVert_{L^{2}_{\mu}(D)}\leq\sqrt{f(w)}$. Then the $1$-st order weighted
reduced Bergman kernel function $\tilde{K}_{D,\mu}(\cdot,w)$ is uniquely
characterized by the properties:

  1. (i)

$\tilde{K}_{D,\mu}(\cdot,w)\in\tilde{S}^{w}_{D,\mu}$;

  2. (ii)

if $f\in\tilde{S}_{D,\mu}^{w}$ and $f(w)\geq\tilde{K}_{D,\mu}(w,w)$, then
$f(\cdot)\equiv\tilde{K}_{D,\mu}(\cdot,w)$.

###### Proof.

Similar to the proof of Lemma 2.1. ∎

###### Proof of Theorem 1.6 $(1)$.

We will start by proving the monotonicity property:
$\tilde{K}_{D_{i},\mu_{i},n}^{(n-1)}(z,z)\geq\tilde{K}_{D_{k},\mu_{k},n}^{(n-1)}(z,z)$
for all $z\in D_{i}$ and $k\geq j(i)$.

Fix $z\in D_{i}$ and $k\geq j(i)$. If
$\tilde{K}_{D_{k},\mu_{k},n}^{(n-1)}(z,z)=0$ then there is nothing to prove.
Therefore, assume that $\tilde{K}^{(n-1)}_{D_{k},\mu_{k},n}(z,z)>0$. Observe
that

$\int_{D_{i}}\left\lvert\frac{(M_{D_{k},\mu_{k},n}(\cdot,z))^{\prime}(\zeta)}{\tilde{K}_{D_{k},\mu_{k},n}^{(n-1)}(z,z)}\right\rvert^{2}\mu_{i}(\zeta)dA(\zeta)\leq\int_{D_{i}}\left\lvert\frac{(M_{D_{k},\mu_{k},n}(\cdot,z))^{\prime}(\zeta)}{\tilde{K}_{D_{k},\mu_{k},n}^{(n-1)}(z,z)}\right\rvert^{2}\mu_{k}(\zeta)dA(\zeta)\\\
\leq\int_{D_{k}}\left\lvert\frac{(M_{D_{k},\mu_{k},n}(\cdot,z))^{\prime}(\zeta)}{\tilde{K}_{D_{k},\mu_{k},n}^{(n-1)}(z,z)}\right\rvert^{2}\mu_{k}(\zeta)dA(\zeta)=\frac{1}{\tilde{K}_{D_{k},\mu_{k},n}^{(n-1)}(z,z)}<\infty.$

Therefore
$M_{D_{k},\mu_{k},n}(\cdot,z)/\tilde{K}^{(n-1)}_{D_{k},\mu_{k},n}(z,z)$
belongs to the set $\\{h\in AD^{\mu_{i}}(D_{i},z^{n}):h^{(n)}(z)=1\\}$. Note
that $\tilde{K}^{(n-1)}_{D_{i},\mu_{i},n}(z,z)>0$ as we have a function in
$AD^{\mu_{i}}(D_{i},z^{n})$ whose $n$-th derivative does not vanish at $z$.

Thus, by the minimality, we have

$\frac{1}{\tilde{K}^{(n-1)}_{D_{i},\mu_{i},n}(z,z)}=\int_{D_{i}}\left\lvert\frac{(M_{D_{i},\mu_{i},n}(\cdot,z))^{\prime}(\zeta)}{\tilde{K}_{D_{i},\mu_{i},n}^{(n-1)}(z,z)}\right\rvert^{2}\mu_{i}(\zeta)\,dA(\zeta)\\\
\leq\int_{D_{i}}\left\lvert\frac{(M_{D_{k},\mu_{k},n}(\cdot,z))^{\prime}(\zeta)}{\tilde{K}_{D_{k},\mu_{k},n}^{(n-1)}(z,z)}\right\rvert^{2}\mu_{i}(\zeta)\,dA(\zeta)\leq\frac{1}{\tilde{K}^{(n-1)}_{D_{k},\mu_{k},n}(z,z)}.$

Hence,
$\tilde{K}_{D_{i},\mu_{i},n}^{(n-1)}(z,z)\geq\tilde{K}^{(n-1)}_{D_{k},\mu_{k},n}(z,z)$.
Since $D_{i}\subset D$ for all $i\in\mathbb{Z}^{+}$ and $\mu_{i}(z)\leq\mu(z)$
for all $z\in D_{i}$, it can be proved in a similar manner that
$\tilde{K}_{D_{i},\mu_{i},n}^{(n-1)}(z,z)\geq\tilde{K}_{D,\mu,n}^{(n-1)}(z,z)$
for all $i\in\mathbb{Z}^{+}$ and $z\in D_{i}$.

Now we will show that $\\{\tilde{K}_{D_{k},\mu_{k}}\\}$ and
$\\{\tilde{K}_{D_{k},\mu_{k},n}(\cdot,w)\\}$ are normal families for any $w\in
D$.

Let $K\subset D$ be a compact subset. There exists an $i\in\mathbb{Z}^{+}$
such that $K\subset D_{i}$, and therefore $K\subset D_{i}\subset D_{k}$ for
all $k\geq j(i)$. Choose $i\in\mathbb{Z}^{+}$ so that $w\in D_{i}$. For
$z,\zeta\in K$ and $k\geq j(i)$, we have by monotonicity that

$\lvert\tilde{K}_{D_{k},\mu_{k}}(z,\zeta)\rvert\leq\sqrt{\tilde{K}_{D_{k},\mu_{k}}(z,z)}\sqrt{\tilde{K}_{D_{k},\mu_{k}}(\zeta,\zeta)}\leq\sqrt{\tilde{K}_{D_{i},\mu_{i}}(z,z)}\sqrt{\tilde{K}_{D_{i},\mu_{i}}(\zeta,\zeta)}\leq
M$

and

$\displaystyle|\tilde{K}_{D_{k},\mu_{k},n}(z,w)|$ $\displaystyle=$
$\displaystyle|(M_{D_{k},\mu_{k},n}(\cdot,w))^{\prime}(z)|=|\langle(M_{D_{k},\mu_{k},n}(\cdot,w))^{\prime},\tilde{K}_{D_{k},\mu_{k}}(\cdot,z)\rangle_{L^{2}_{\mu_{k}}(D_{k})}|$
$\displaystyle\leq$
$\displaystyle\|(M_{D_{k},\mu_{k},n}(\cdot,w))^{\prime}\|_{L^{2}_{\mu_{k}}(D_{k})}\,\|\tilde{K}_{D_{k},\mu_{k}}(\cdot,z)\|_{L^{2}_{\mu_{k}}(D_{k})}$
$\displaystyle=$
$\displaystyle\sqrt{\tilde{K}^{(n-1)}_{D_{k},\mu_{k},n}(w,w)}\,\sqrt{\tilde{K}_{D_{k},\mu_{k}}(z,z)}\leq\sqrt{\tilde{K}^{(n-1)}_{D_{i},\mu_{i},n}(w,w)}\,\sqrt{\tilde{K}_{D_{i},\mu_{i}}(z,z)}\leq
C,$

where

$M=\sup_{t\in K}\lvert\tilde{K}_{D_{i},\mu_{i}}(t,t)\rvert\geq
0\quad\text{and}\quad C=\sup\limits_{t\in
K}\sqrt{\tilde{K}_{D_{i},\mu_{i}}(t,t)}\,\sqrt{\tilde{K}^{(n-1)}_{D_{i},\mu_{i},n}(w,w)}\geq
0.$

By Montel’s theorem, $\\{\tilde{K}_{D_{k},\mu_{k}}\\}_{k=1}^{\infty}$ and
$\\{\tilde{K}_{D_{k},\mu_{k},n}(\cdot,w)\\}_{k=1}^{\infty}$ are normal
families. We will show that for $n\geq 1$ and $w\in D$, every convergent
subsequence of $\\{\tilde{K}_{D_{k},\mu_{k},n}(\cdot,w)\\}_{k=1}^{\infty}$
converges to $\tilde{K}_{D,\mu,n}(\cdot,w)$. This will imply that

$\lim\limits_{k\rightarrow\infty}\tilde{K}_{D_{k},\mu_{k}}=\tilde{K}_{D,\mu}\quad\text{and}\quad\lim\limits_{k\rightarrow\infty}\tilde{K}_{D_{k},\mu_{k},n}(\cdot,w)=\tilde{K}_{D,\mu,n}(\cdot,w)$

locally uniformly on $D\times D$ and $D$ respectively.

So, without loss of generality, assume that

$\lim\limits_{k\rightarrow\infty}\tilde{K}_{D_{k},\mu_{k},n}(\cdot,w)=\tilde{K}$

locally uniformly on $D$ for some holomorphic function $\tilde{K}$. Since
$\tilde{K}^{(p)}_{D_{k},\mu_{k},n}(w,w)=0$ for all $0\leq p\leq n-2$ and large
$k$, we have $\tilde{K}^{(p)}(w)=0$ for all $0\leq p\leq n-2$. Let
$\gamma\subset\subset D$ be a closed curve. Then $\gamma\subset D_{k}$ for
large enough $k$. Since the holomorphic functions
$\tilde{K}_{D_{k},\mu_{k},n}(\cdot,w)$ have a primitive on $D_{k}$ and
converge locally uniformly to $\tilde{K}$ on $D$,

$\int_{\gamma}\tilde{K}(z)dz=\int_{\gamma}\lim_{k\rightarrow\infty}\tilde{K}_{D_{k},\mu_{k},n}(z,w)dz=\lim_{k\rightarrow\infty}\int_{\gamma}\tilde{K}_{D_{k},\mu_{k},n}(z,w)dz=0.$

Thus, there exists a holomorphic function $M$ on $D$ such that
$M^{\prime}=\tilde{K}$. Choose $M$ so that $M(w)=0$. Therefore,

$M(w)=M^{\prime}(w)=\cdots=M^{(n-1)}(w)=0.$

Let $K\subset D$ be a compact subset. Then $K\subset D_{k}$ for large enough
$k$. By Fatou’s lemma,

$\displaystyle\int_{K}|M^{\prime}(z)|^{2}\mu(z)dA(z)$ $\displaystyle=$
$\displaystyle\int_{K}|\tilde{K}(z)|^{2}\mu(z)dA(z)\leq\liminf_{k\rightarrow\infty}\int_{K}|\tilde{K}_{D_{k},\mu_{k},n}(z,w)|^{2}\mu_{k}(z)dA(z)$
$\displaystyle\leq$
$\displaystyle\liminf_{k\rightarrow\infty}\int_{D_{k}}|\tilde{K}_{D_{k},\mu_{k},n}(z,w)|^{2}\mu_{k}(z)dA(z)=\liminf_{k\rightarrow\infty}\tilde{K}^{(n-1)}_{D_{k},\mu_{k}}(w,w)$
$\displaystyle=$ $\displaystyle\tilde{K}^{(n-1)}(w)=M^{(n)}(w).$

Since compact $K\subset D$ was arbitrary, we have

$\|M^{\prime}\|_{L^{2}_{\mu}(D)}^{2}=\int_{D}|M^{\prime}(z)|^{2}\mu(z)\,dA(z)\leq
M^{(n)}(w)<\infty.$

Therefore, $M\in S_{D,\mu,n}^{w}$. Since
$\tilde{K}^{(n-1)}_{D_{i},\mu_{i},n}(w,w)\geq\tilde{K}^{(n-1)}_{D,\mu,n}(w,w)$
for all $i\in\mathbb{Z}^{+}$, we have that
$M^{(n)}(w)=\tilde{K}^{(n-1)}(w)\geq\tilde{K}^{(n-1)}_{D,\mu,n}(w,w)$. Hence,
Lemma 2.1 implies that $M(\cdot)\equiv M_{D,\mu,n}(\cdot,w)$ and therefore
$\tilde{K}=\tilde{K}_{D,\mu,n}(\cdot,w)$.

It now follows from the equation (5) that all the derivatives of
$\tilde{K}_{D_{i},\mu_{i},n}$ are the fractions where the numerator is a
polynomial in the derivatives of $\tilde{K}_{D_{i},\mu_{i}}$ and the
denominator is some power of $J_{n-2}$ (corresponding to the domain $D_{i}$
and weight $\mu_{i}$) which is non-zero on $D_{i}\times(D_{i}\setminus
N_{D_{i}}(\mu_{i}))$ for all $i\in\mathbb{Z}^{+}$.

If $K\subset D$ is compact such that $\tilde{K}_{D,\mu}(z,z)\neq 0$ for all
$z\in K$, then $\tilde{K}_{D_{i},\mu_{i}}(z,z)\neq 0$ for all large $i$ and
$z\in K$. Thus, any compact set in $D\times(D\setminus N_{D}(\mu))$ is
eventually contained in $D_{i}\times(D_{i}\setminus N_{D_{i}}(\mu_{i}))$.

Since $\tilde{K}_{D_{i},\mu_{i}}$ are holomorphic in the first and anti-
holomorphic in the second variable, all of its derivatives converge locally
uniformly to the respective derivatives of $\tilde{K}_{D,\mu}$. Therefore, for
$n>1$, all the derivatives of $\tilde{K}_{D_{i},\mu_{i},n}$ converge locally
uniformly to the respective derivatives of $\tilde{K}_{D,\mu,n}$ on
$D\times(D\setminus N_{D}(\mu))$. ∎

###### Proof of Theorem 1.6 $(2)$.

We will first show part (A). Let $K\subset D$ be a compact subset. For $z,w\in
K$ and $k\in\mathbb{Z}^{+}$, we have by monotonicity that

$\lvert\tilde{K}_{D_{k},\mu_{k}}(z,w)\rvert\leq\sqrt{\tilde{K}_{D_{k},\mu_{k}}(z,z)}\sqrt{\tilde{K}_{D_{k},\mu_{k}}(w,w)}\leq\sqrt{\tilde{K}_{D,\mu}(z,z)}\sqrt{\tilde{K}_{D,\mu}(w,w)}\leq
M,$

where $M=\sup_{t\in K}\lvert\tilde{K}_{D,\mu}(t,t)\rvert>0$. By Montel’s
theorem, $\\{K_{D_{k},\mu_{k}}\\}_{k=1}^{\infty}$ is a normal family and
therefore has a subsequence that converges locally uniformly on $D\times D$.
We will show that every such subsequence must converge to $\tilde{K}_{D,\mu}$.
So, without loss of generality, assume that

$\lim\limits_{k\rightarrow\infty}\tilde{K}_{D_{k},\mu_{k}}=\tilde{K}$

locally uniformly on $D\times D$, for some $\tilde{K}$. Fix $w\in D$. Again,
let $K\subset D$ be a compact subset. By Fatou’s lemma,

$\displaystyle\int_{K}|\tilde{K}(z,w)|^{2}\mu(z)\,dA(z)$ $\displaystyle\leq$
$\displaystyle\liminf_{k\rightarrow\infty}\int_{K}|\tilde{K}_{D_{k},\mu_{k}}(z,w)|^{2}\mu_{k}(z)\,dA(z)$
$\displaystyle\leq$
$\displaystyle\liminf_{k\rightarrow\infty}\int_{D_{k}}|\tilde{K}_{D_{k},\mu_{k}}(z,w)|^{2}\mu_{k}(z)\,dA(z)$
$\displaystyle=$
$\displaystyle\liminf_{k\rightarrow\infty}\tilde{K}_{D_{k},\mu_{k}}(w,w)=\tilde{K}_{D,\mu}(w,w)=\tilde{K}(w,w).$

Since compact $K\subset D$ was arbitrary, we have

$\|\tilde{K}(\cdot,w)\|_{L^{2}_{\mu}(D)}^{2}=\int_{D}|\tilde{K}(z,w)|^{2}\mu(z)\,dA(z)\leq\tilde{K}_{D,\mu}(w,w)<\infty.$

Let $\gamma\subset\subset D$ be a closed curve. Then $\gamma\subset D_{k}$ for
large enough $k$. Since the holomorphic functions
$\tilde{K}_{D_{k},\mu_{k}}(\cdot,w)$ have a primitive on $D_{k}$ and converge
locally uniformly to $\tilde{K}(\cdot,w)$ on $D$,

$\int_{\gamma}\tilde{K}(z,w)dz=\int_{\gamma}\lim_{k\rightarrow\infty}\tilde{K}_{D_{k},\mu_{k}}(z,w)dz=\lim_{k\rightarrow\infty}\int_{\gamma}\tilde{K}_{D_{k},\mu_{k}}(z,w)dz=0.$

Thus, $\tilde{K}(\cdot,w)$ has a primitive on $D$. Therefore,
$\tilde{K}(\cdot,w)\in\mathcal{D}^{\mu}(D)$. Thus, the function
$\tilde{K}(\cdot,w)\in\tilde{S}_{D,\mu}^{w}$ and
$\tilde{K}(w,w)=\tilde{K}_{D,\mu}(w,w)$. Hence, Lemma 2.2 implies that
$\tilde{K}(\cdot,w)\equiv\tilde{K}_{D,\mu}(\cdot,w)$. Since $w\in D$ was
arbitrary, we conclude that $\tilde{K}\equiv\tilde{K}_{D,\mu}$.

Part (B) follows from the determinant formula (5) as in the proof of Theorem
1.6 $(1)$. ∎

##  3\. Boundary behaviour

###### Proof of Theorem 1.10.

Since $\mu$ extends continuously to $p$ with $\mu(p)>0$, there exists a
neighborhood $U$ of $p$ in $\mathbb{C}$ such that $D\cap U$ is connected and
$\mu\geq c$ a.e. on $D\cap U$ for some constant $c>0$. So, $(\mu|_{D\cap
U})^{-1}\in L^{1}(D\cap U)$. Therefore, $\mu|_{D\cap U}$ is an admissible
weight on $D\cap U$ and hence $\tilde{K}_{D\cap U,\mu,n}$ is well-defined.

As $D\cap U\subset D$, the monotonicity of the solution of the extremal
problems in Remark 1.5 implies that $\tilde{K}^{(n-1)}_{D\cap
U,\mu,n}(\zeta,\zeta)\geq\tilde{K}^{(n-1)}_{D,\mu,n}(\zeta,\zeta)$ for all
$\zeta\in D\cap U$. Therefore,

$1\leq\liminf_{\zeta\rightarrow p}\frac{\tilde{K}^{(n-1)}_{D\cap
U,\mu,n}(\zeta,\zeta)}{\tilde{K}^{(n-1)}_{D,\mu,n}(\zeta,\zeta)}.$

It is now enough to show that

$\limsup_{\zeta\rightarrow p}\frac{\tilde{K}^{(n-1)}_{D\cap
U,\mu,n}(\zeta,\zeta)}{\tilde{K}^{(n-1)}_{D,\mu,n}(\zeta,\zeta)}\leq 1.$ (9)

Since $p$ is a holomorphic local peak point, choose $h$ and $W$ as in the
Definition 1.9. For any neighborhood $\hat{U}\subset U$ of $p$, monotonicity
gives that

$\limsup_{\zeta\rightarrow p}\frac{\tilde{K}^{(n-1)}_{D\cap
U,\mu,n}(\zeta,\zeta)}{\tilde{K}^{(n-1)}_{D,\mu,n}(\zeta,\zeta)}\leq\limsup_{\zeta\rightarrow
p}\frac{\tilde{K}^{(n-1)}_{D\cap\hat{U},\mu,n}(\zeta,\zeta)}{\tilde{K}^{(n-1)}_{D,\mu,n}(\zeta,\zeta)}.$

Therefore, we may assume that $U\subset(N_{p}\cap W)$ without loss of
generality. Choose an open neighborhood $U_{0}\subset\subset U$ of $p$ such
that $h\neq 0$ on $\bar{D}\cap U_{0}$. Then there exists a constant
$a\in(0,1)$ such that $|h|\leq a$ on $\overline{(U\setminus U_{0})\cap D}$.
Choose a cut-off function $\chi:\mathbb{C}\longrightarrow[0,1]$ such that
$\chi\equiv 1$ on some neighborhood $\hat{U}_{0}$ of $\overline{U}_{0}$ and
$supp\,\chi\subset U$. Fix $\zeta\in D\cap U_{0}$ and put

$f=\frac{M_{D\cap U,\mu,n}(\cdot,\zeta)}{\sqrt{(M_{D\cap
U,\mu,n}(\cdot,\zeta))^{(n)}(\zeta)}}.$

Then, $f\in AD^{\mu}(D\cap U,\zeta^{n})$ such that $\lVert
f\rVert_{AD^{\mu}(D\cap U,\zeta^{n})}=1$ and $\lvert
f^{(n)}(\zeta)\rvert^{2}=\tilde{K}^{(n-1)}_{D\cap U,\mu,n}(\zeta,\zeta)$.

Without loss of generality, assume that $p$ lies on the outer boundary curve
of $D$, i.e., the boundary of the unbounded component of $\mathbb{C}\setminus
D$. Let $\hat{D}$ be the domain obtained by filling all the bounded components
of $\mathbb{C}\setminus D$. Then, $\hat{D}$ is a simply connected domain. For
$k\geq 1$, define

$\alpha_{k}=\begin{cases}\bar{\partial}(\chi
f^{\prime}h^{k})=f^{\prime}h^{k}\frac{\partial\chi}{\partial\bar{z}}\,d\bar{z}&,\text{
on }D\cap U=\hat{D}\cap U\\\ 0&,\text{ on }\hat{D}\setminus U\end{cases}.$

Then, $\alpha_{k}$ is a smooth $\bar{\partial}$-closed $(0,1)$-form on
$\hat{D}$. Put $\phi(z)=(2n+2)\log\lvert z-\zeta\rvert$ for $z\in\hat{D}$. The
function $\phi$ is a subharmonic function on $\hat{D}$. Also,

$\displaystyle\int_{\hat{D}}\lvert\alpha_{k}\rvert^{2}(z)\,\exp(-\phi(z))\,dA(z)$
$\displaystyle=$ $\displaystyle\int_{D\cap U}\lvert
f^{\prime}(z)\rvert^{2}\lvert
h(z)\rvert^{2k}\left\lvert\frac{\partial\chi}{\partial\bar{z}}(z)\right\rvert^{2}\lvert
z-\zeta\rvert^{-(2n+2)}\,dA(z)$ $\displaystyle\leq$ $\displaystyle
C_{1}\int_{D\cap(U\setminus\hat{U}_{0})}\lvert f^{\prime}(z)\rvert^{2}\lvert
h(z)\rvert^{2k}\,dA(z)$ $\displaystyle\leq$ $\displaystyle
C_{1}\,a^{2k}\frac{1}{c}\int_{D\cap U}\lvert
f^{\prime}(z)\rvert^{2}\mu(z)\,dA(z)=C_{2}\,a^{2k}<\infty.$

where $C_{1}>0$ is chosen independent of $\zeta\in D\cap U_{0}$ and
$C_{2}=C_{1}/c$. Therefore, $\alpha_{k}$ belongs to
$L^{2}_{(0,1)}(\hat{D},\exp(-\phi))$, the Hilbert space of $(0,1)$-forms that
are square-integrable with respect to the weight function $\exp(-\phi)$. By
Hormander’s theory (see [5]), there exists a smooth function $g_{k}$ on
$\hat{D}$ such that $\bar{\partial}g_{k}=\alpha_{k}$ and

$\int_{\hat{D}}\lvert g_{k}(z)\rvert^{2}(1+\lvert
z\rvert^{2})^{-2}\exp(-\phi(z))\,dA(z)\leq C_{2}\,a^{2k}.$

Since $D$ is bounded, $\hat{D}$ is also bounded. Therefore, we have

$\int_{\hat{D}}\lvert g_{k}(z)\rvert^{2}\lvert
z-\zeta\rvert^{-(2n+2)}\,dA(z)\leq C_{3}\,a^{2k}<\infty,$

where $C_{3}=C_{2}\,\sup_{\hat{D}}(1+\lvert z\rvert^{2})^{2}>0$. Since the
above integral is finite, we must have that
$g_{k}(\zeta)=g_{k}^{\prime}(\zeta)=\cdots=g_{k}^{(n-1)}(\zeta)=0$. As $\mu\in
L^{\infty}(D)$, there exists a constant $M>0$ so that

$\displaystyle\int_{\hat{D}}\lvert g_{k}(z)\rvert^{2}\mu(z)dA(z)$
$\displaystyle\leq$ $\displaystyle M\sup_{\hat{D}}\lvert
z-\zeta\rvert^{2n+2}\int_{\hat{D}}\lvert g_{k}(z)\rvert^{2}\lvert
z-\zeta\rvert^{-(2n+2)}dA(z)$ $\displaystyle\leq$ $\displaystyle
M(diamD)^{2n+2}\,C_{3}\,a^{2k}<\infty.$

Therefore, $g_{k}\in L^{2}_{\mu}(\hat{D})$ and $\lVert
g_{k}\rVert_{L^{2}_{\mu}(\hat{D})}\leq C_{0}\,a^{k}$, where
$C_{0}=\sqrt{M(diamD)^{2n+2}C_{3}}>0$. Put, $\hat{F}_{k}=\chi
f^{\prime}h^{k}-g_{k}$. Then, $\hat{F}_{k}\in\mathcal{O}(\hat{D})$. We also
have
$\hat{F}_{k}(\zeta)={\hat{F}_{k}}^{\prime}(\zeta)=(\hat{F}_{k})^{(n-2)}(\zeta)=0$
and $(\hat{F}_{k})^{(n-1)}(\zeta)=f^{(n)}(\zeta)(h(\zeta))^{k}$. Since
$\hat{D}$ is simply connected, $\hat{F}_{k}$ has a primitive on $\hat{D}$.
Choose $F_{k}\in\mathcal{O}(D)$ such that $F_{k}(\zeta)=0$ and
$F_{k}^{\prime}=\hat{F}_{k}$. Since
$F_{k}(\zeta)=F_{k}^{\prime}(\zeta)=\cdots=F_{k}^{(n-1)}(\zeta)=0$ and

$\lVert{F_{k}}^{\prime}\rVert_{L^{2}_{\mu}(D)}=\lVert\hat{F}_{k}\rVert_{L^{2}_{\mu}(D)}\leq\lVert
f^{\prime}\rVert_{L^{2}_{\mu}(D\cap U)}+\lVert
g_{k}\rVert_{L^{2}_{\mu}(D)}\leq 1+C_{0}\,a^{k},$

we have $F_{k}\in AD^{\mu}(D,\zeta^{n})$ and
$\|F_{k}\|_{AD^{\mu}(D,\zeta^{n})}\leq 1+C_{0}\,a^{k}$. Hence,

$\displaystyle\lvert h(\zeta)\rvert^{2k}\,\tilde{K}^{(n-1)}_{D\cap
U,\mu,n}(\zeta,\zeta)$ $\displaystyle=$ $\displaystyle\lvert
h(\zeta)\rvert^{2k}\,\lvert
f^{(n)}(\zeta)\rvert^{2}=|F_{k}^{(n)}(\zeta)\rvert^{2}$ $\displaystyle=$
$\displaystyle\left\lvert\langle
F_{k},M_{D,\mu,n}(\cdot,\zeta)\rangle_{AD^{\mu}(D,\zeta^{n})}\right\rvert^{2}$
$\displaystyle\leq$ $\displaystyle\lVert
F_{k}\rVert_{AD^{\mu}(D,\zeta^{n})}^{2}\,\lVert
M_{D,\mu,n}(\cdot,\zeta)\rVert_{AD^{\mu}(D,\zeta^{n})}^{2}\leq(1+C_{0}\,a^{k})^{2}\,\tilde{K}^{(n-1)}_{D,\mu,n}(\zeta,\zeta).$

Therefore,

$\frac{\tilde{K}^{(n-1)}_{D\cap
U,\mu,n}(\zeta,\zeta)}{\tilde{K}^{(n-1)}_{D,\mu,n}(\zeta,\zeta)}\leq\frac{(1+C_{0}\,a^{k})^{2}}{\lvert
h(\zeta)\rvert^{2k}}.$

Since $C_{0}$ is independent of $\zeta$, we have

$\limsup_{\zeta\rightarrow p}\frac{\tilde{K}^{(n-1)}_{D\cap
U,\mu,n}(\zeta,\zeta)}{\tilde{K}^{(n-1)}_{D,\mu,n}(\zeta,\zeta)}\leq(1+C_{0}\,a^{k})^{2}.$

Finally, since $a\in(0,1)$ and $C_{0}$ is independent of $k$, taking the limit
$k\rightarrow\infty$ gives (9). ∎

###### Proof of Theorem 1.11.

Let $\\{\varphi_{n}\\}$ be a Cauchy sequence in $A^{2}_{\nu\circ f}(D_{1})$.
Then, applying the change of variables formula

$\int_{D_{1}}|\varphi_{n}-\varphi_{m}|^{2}(z)\,(\nu\circ
f)(z)\,dA(z)=\int_{D_{2}}|(\varphi_{n}\circ f^{-1})-(\varphi_{m}\circ
f^{-1})|^{2}(z)\,\nu(z)\,|(f^{-1})^{\prime}(z)|^{2}\,dA(z)$

implies that $\\{(\varphi_{n}\circ f^{-1})\,(f^{-1})^{\prime}\\}$ is a Cauchy
sequence in $A^{2}_{\nu}(D_{2})$. Therefore, there exists a function
$\varphi\in A^{2}_{\nu}(D_{2})$ such that $\|(\varphi_{n}\circ
f^{-1})\,(f^{-1})^{\prime}-\varphi\|_{L^{2}_{\nu}(D_{2})}\rightarrow 0$ as
$n\rightarrow\infty$. It can now be checked using change of variables formula,
as above, that $(\varphi\circ f)\,f^{\prime}\in A^{2}_{\nu\circ f}(D_{1})$ and
$\|\varphi_{n}-(\varphi\circ f)\,f^{\prime}\|_{L^{2}_{\nu\circ
f}(D_{1})}\rightarrow 0$ and $n\rightarrow\infty$. Thus, $A^{2}_{\nu\circ
f}(D_{1})$ is a closed subspace of $L^{2}_{\nu\circ f}(D_{1})$, and hence a
Hilbert space. Now for $z\in D_{1}$, the evaluation functional
$A^{2}_{\nu}(D_{2})\ni h\mapsto h(f(z))$ is continuous, i.e., there exists a
constant $C_{z}>0$ such that

$|h(f(z))|\leq C_{z}\,\|h\|_{L^{2}_{\nu}(D_{2})}\quad\text{for all }h\in
A^{2}_{\nu}(D_{2}).$

For every $g\in A^{2}_{\nu\circ f}(D_{1})$, the function $(g\circ
f^{-1})\,(f^{-1})^{\prime}\in A^{2}_{\nu}(D_{2})$. Therefore,

$\displaystyle|g(z)|$ $\displaystyle=$
$\displaystyle|f^{\prime}(z)|\,|g(z)\,(f^{-1})^{\prime}(f(z))|\,\leq\,|f^{\prime}(z)|\,C_{z}\,\|(g\circ
f^{-1})\,(f^{-1})^{\prime}\|_{L^{2}_{\nu}(D_{2})}$ $\displaystyle=$
$\displaystyle|f^{\prime}(z)|\,C_{z}\,\|g\|_{L^{2}_{\nu\circ f}(D_{1})}.$

Hence, the evaluation functional $A^{2}_{\nu\circ f}(D_{1})\ni g\mapsto g(z)$
is continuous. So, $\nu\circ f$ is an admissible weight on $D_{1}$ and
therefore $M_{D_{1},n,\nu\circ f}$ is well-defined.

Let $\zeta\in D_{1}$ be arbitrary. Let $\varphi\in
AD^{\nu}(D_{2},(f(\zeta))^{n})$ be arbitrary. Note that $(\varphi\circ
f)(\zeta)=(\varphi\circ f)^{\prime}(\zeta)=\cdots=(\varphi\circ
f)^{(n-1)}(\zeta)=0$. Also,

$\displaystyle\int_{D_{1}}|(\varphi\circ f)^{\prime}(z)|^{2}(\nu\circ
f)(z)\,dA(z)$ $\displaystyle=$
$\displaystyle\int_{D_{2}}|\varphi^{\prime}(w)|^{2}\,|f^{\prime}(f^{-1}(w))|^{2}\,|(f^{-1})^{\prime}(w)|^{2}\nu(w)\,dA(w)$
$\displaystyle=$
$\displaystyle\int_{D_{2}}|\varphi^{\prime}(w)|^{2}\nu(w)\,dA(w).$

Therefore, $\varphi\circ f\in AD^{\nu\circ f}(D_{1},\zeta^{n})$. So,

$\displaystyle\varphi^{(n)}(f(\zeta))(f^{\prime}(\zeta))^{n}$ $\displaystyle=$
$\displaystyle(\varphi\circ f)^{(n)}(\zeta)=\int_{D_{1}}(\varphi\circ
f)^{\prime}(z)\,\overline{(M_{D_{1},\nu\circ
f,n}(\cdot,\zeta))^{\prime}(z)}(\nu\circ f)(z)\,dA(z)$ $\displaystyle=$
$\displaystyle\int_{D_{2}}\varphi^{\prime}(w)\,f^{\prime}(f^{-1}(w))\,\overline{(M_{D_{1},\nu\circ
f,n}(\cdot,\zeta))^{\prime}(f^{-1}(w))}\,\nu(w)\,|(f^{-1})^{\prime}(w)|^{2}\,dA(w)$
$\displaystyle=$
$\displaystyle\int_{D_{2}}\varphi^{\prime}(w)\,\overline{(M_{D_{1},\nu\circ
f,n}(\cdot,\zeta))^{\prime}(f^{-1}(w))\cdot(f^{-1})^{\prime}(w)}\,\nu(w)\,dA(w)$
$\displaystyle=$
$\displaystyle\int_{D_{2}}\varphi^{\prime}(w)\,\overline{(M_{D_{1},\nu\circ
f,n}(\cdot,\zeta)\circ f^{-1})^{\prime}(w)}\,\nu(w)\,dA(w).$

Since $(M_{D_{1},\nu\circ f,n}(\cdot,\zeta)\circ f^{-1})\in
AD^{\nu}(D_{2},(f(\zeta))^{n})$, it follows by the uniqueness of the
reproducing kernel of $AD^{\nu}(D_{2},(f(\zeta))^{n})$ that
$M_{D_{2},\nu,n}(w,f(\zeta))=(\overline{f^{\prime}(\zeta)})^{-n}\,M_{D_{1},\nu\circ
f,n}(f^{-1}(w),\zeta)$ for all $w\in D_{2}$ and $\zeta\in D_{1}$. That is,

$(\overline{f^{\prime}(\zeta)})^{n}\,M_{D_{2},\nu,n}(f(z),f(\zeta))=M_{D_{1},\nu\circ
f,n}(z,\zeta)\quad\text{for all }z,\zeta\in D_{1}.$

This can be differentiated $n$ times to obtain the other transformation
formula. ∎

###### Proof of Theorem 1.12.

We will use the scaling principle in the case when $D$ is simply connected. So
let us quickly recall the scaling principle:

Let $\psi$ be a $C^{2}$ defining function for $D$ near $p$ such that
$\frac{\partial\psi}{\partial z}(p)=1$. Let $U$ be a neighborhood of $p$ in
$\mathbb{C}$ where $\psi$ is defined. Choose a sequence $p_{j}$ in $D\cap U$
converging to the boundary point $p$. For $j\in\mathbb{Z}^{+}$, define the
affine maps

$T_{j}(z)=\frac{z-p_{j}}{-\psi(p_{j})},\quad\quad z\in\mathbb{C}.$

Let $D_{j}:=T_{j}(D)$. Note that $0\in D_{j}$ for every $j$ as
$T_{j}(p_{j})=0$. Let $K\subset\mathbb{C}$ be compact set. Since
$\psi(p_{j})\rightarrow 0$ as $j\rightarrow\infty$, $\\{T_{j}(U)\\}$ is an
eventually increasing family of open sets that exhaust $\mathbb{C}$ and hence
$K\subset T_{j}(U)$ for all large $j$. Taking the Taylor series expansion of
$\psi$ near $z=p_{j}$, the functions

$\displaystyle\psi\circ T_{j}^{-1}(z)$ $\displaystyle=$
$\displaystyle\psi(p_{j}+z(-\psi(p_{j})))$ $\displaystyle=$
$\displaystyle\psi(p_{j})+2Re\left(\frac{\partial\psi}{\partial
z}(p_{j})z\right)(-\psi(p_{j}))+\psi(p_{j})^{2}o(1)$

are therefore well-defined on $K$ for all large $j$. Let $\psi_{j}$ be the
defining functions of $D_{j}$ defined near $T_{j}(p)\in\partial D_{j}$ on
$T_{j}(U)$, given by

$\displaystyle\psi_{j}(z)$ $\displaystyle=$
$\displaystyle\frac{1}{(-\psi(p_{j}))}\psi\circ T_{j}^{-1}(z)$
$\displaystyle=$ $\displaystyle-1+2Re\left(\frac{\partial\psi}{\partial
z}(p_{j})z\right)+(-\psi(p_{j}))o(1).$

It is easy to see that $\psi_{j}$ converges to

$\psi_{\infty}(z)=-1+2Re\left(\frac{\partial\psi}{\partial
z}(p)z\right)=-1+2Rez.$

uniformly on all the compact subsets of $\mathbb{C}$. Let $\mathcal{H}$ denote
the half-space defined by $\psi_{\infty}$, i.e.

$\mathcal{H}=\left\\{z:-1+2Rez<0\right\\}.$

Since $\psi_{j}$ converges to $\psi_{\infty}$ uniformly on compacts, it can be
shown that the closure of the domains $T_{j}(D\cap U)$ (and therefore
$T_{j}(D)=D_{j}$) converge to the closure of the half-space $\mathcal{H}$ in
the Hausdorff sense. Therefore, every compact $K\subset\mathcal{H}$ is
eventually contained in $D_{j}$, and every compact
$K\subset\mathbb{C}\setminus\overline{\mathcal{H}}$ is eventually contained in
$\mathbb{C}\setminus\overline{D_{j}}$.

We want to study the behaviour of the sequence
$\tilde{K}^{(n-1)}_{D,n}(p_{j},p_{j})$ as $j\rightarrow\infty$. Applying
Theorem (1.11) on $T_{j}:D\longrightarrow D_{j}$ gives

$\tilde{K}_{D,n}^{(n-1)}(z,\zeta)=({T_{j}}^{\prime}(z))^{n}\,\tilde{K}^{(n-1)}_{D_{j},n}(T_{j}(z),T_{j}(\zeta))\,(\overline{{T_{j}}^{\prime}(\zeta)})^{n}\quad
z,\zeta\in D.$

Since $T_{j}^{\prime}\equiv-1/\psi(p_{j})$ and $T_{j}(p_{j})=0$, we obtain

$\tilde{K}_{D,n}^{(n-1)}(p_{j},p_{j})=\frac{1}{(\psi(p_{j}))^{2n}}\,\tilde{K}_{D_{j},n}^{(n-1)}(0,0),\quad
j\in\mathbb{Z}^{+}.$ (10)

Therefore, it is enough to study the behaviour of
$\tilde{K}_{D_{j},n}^{(n-1)}(0,0)$ as $j\rightarrow\infty$. This is called the
scaling principle where a boundary problem has been converted to an interior
problem at the cost of varying the domains (for details, see [4]).

###### Case 1..

Let $D$ be simply connected. Since $T_{j}$’s are affine maps, the domains
$D_{j}$’s are also simply connected. Let
$F_{j}:D_{j}\longrightarrow\mathcal{H}$ be the Riemann maps such that
$F_{j}(0)=0$ and $F_{j}^{\prime}(0)>0$. Applying Theorem (1.11) on the Riemann
maps $F_{j}$ gives

$\tilde{K}_{D_{j},n}^{(n-1)}(z,w)=(F_{j}^{\prime}(z))^{n}\,\tilde{K}_{\mathcal{H},n}^{(n-1)}(F_{j}(z),F_{j}(w))\,(\overline{F_{j}^{\prime}(w)})^{n},\quad\quad
z,w\in D_{j}.$ (11)

The problem is therefore reduced to studying the Riemann maps $F_{j}$.

######  Lemma 3.1.

The Riemann maps $F_{j}$ converge to the identity map $i_{\mathcal{H}}$
locally uniformly on $\mathcal{H}$.

###### Proof.

Since every compact $K\subset\mathbb{C}\setminus\overline{\mathcal{H}}$ is
eventually contained in $\mathbb{C}\setminus\overline{D_{j}}$, we observe that
both $\\{F_{j}\\}$ and $\\{F_{j}^{-1}\\}$ omit atleast two values in
$\mathbb{C}$ and fixes $0$. Therefore, $\\{F_{j}\\}$ and $\\{F_{j}^{-1}\\}$
are normal families of holomorphic functions, i.e. there exists a subsequence
$\\{j_{k}\\}$ of the sequence of natural numbers such that both
$\\{F_{j_{k}}\\}$ and $\\{F_{j_{k}}^{-1}\\}$ converge locally uniformly on
$\mathcal{H}$ to some holomorphic functions
$F,\,G:\mathcal{H}\rightarrow\overline{\mathcal{H}}$ respectively. Since
$F(0)=0=G(0)$, the Open mapping theorem implies that
$F,\,G:\mathcal{H}\longrightarrow\mathcal{H}$.

Now, $F_{j_{k}}\circ F_{j_{k}}^{-1}\equiv id_{\mathcal{H}}$ and
$F_{j_{k}}^{-1}\circ F_{j_{k}}\equiv id_{D_{j}}$. Since $F_{j_{k}}\circ
F_{j_{k}}^{-1}$ converges to $F\circ G$ locally uniformly on $\mathcal{H}$, we
have $F\circ G\equiv id_{\mathcal{H}}$. Since any compact
$K\subset\mathcal{H}$ is eventually contained in $D_{j}$, the sequence
$F_{j_{k}}^{-1}\circ F_{j_{k}}$ converges to $G\circ F$ uniformly on $K$. So,
$(G\circ F)|_{K}\equiv id_{K}$ and therefore $G\circ F\equiv
id_{\mathcal{H}}$. Thus, $F$ is an automorphism of $\mathcal{H}$ such that
$F(0)=0$ and $F^{\prime}(0)>0$.

Let $\varphi:\mathcal{H}\rightarrow\mathbb{D}$ be the Riemann map such that
$\varphi(0)=0$ and $\varphi^{\prime}(0)>0$. Then, $\phi=\varphi\circ
F\circ\varphi^{-1}$ is an automorphism of $\mathbb{D}$ such that $\phi(0)=0$
and $\phi^{\prime}(0)>0$. So, $\phi\equiv id_{\mathbb{D}}$ and therefore
$F\equiv id_{\mathcal{H}}$. Hence, the Riemann maps $F_{j}$ converge to the
identity map $id_{\mathcal{H}}$ locally uniformly on $\mathcal{H}$. ∎

Applying Theorem 1.11 on a Riemann map
$\varphi:\mathcal{H}\rightarrow\mathbb{D}$ gives that for $z,w\in\mathcal{H}$,

$\tilde{K}^{(n-1)}_{\mathcal{H},n}(z,w)=(\varphi^{\prime}(z))^{n}\,\tilde{K}^{(n-1)}_{\mathbb{D},n}(\varphi(z),\varphi(w))\,\overline{(\varphi^{\prime}(w))^{n}}.$

Since $\tilde{K}^{(n-1)}_{\mathbb{D},n}$ is continuous on
$\mathbb{D}\times\mathbb{D}$ (see Remark 1.5), the kernel function
$\tilde{K}_{\mathcal{H},n}^{(n-1)}$ is continuous on
$\mathcal{H}\times\mathcal{H}$. Therefore, we conclude from equation (11) and
Lemma 3.1 that

$\lim_{j\rightarrow\infty}\tilde{K}_{D_{j},n}^{(n-1)}=\tilde{K}_{\mathcal{H},n}^{(n-1)}$

locally uniformly on $\mathcal{H}\times\mathcal{H}$. Thus, it follows from
equation (10) that

${\psi(p_{j})}^{2n}\,\tilde{K}_{D,n}^{(n-1)}(p_{j},p_{j})\rightarrow\tilde{K}_{\mathcal{H},n}^{(n-1)}(0,0)\quad\text{as
}j\rightarrow\infty.$

By a previous calculation, for $z,\zeta\in\mathbb{D}$

$\tilde{K}_{\mathbb{D},n}(z,\zeta)=\frac{n!}{\pi}\frac{(z-\zeta)^{n-1}}{(1-z\overline{\zeta})^{n+1}(1-|\zeta|^{2})^{n-1}}.$

Upon differentiating $n-1$ times with respect to $z$, we obtain

$\displaystyle\tilde{K}_{\mathbb{D},n}^{(n-1)}(z,\zeta)$ $\displaystyle=$
$\displaystyle\frac{n!}{\pi}\frac{1}{(1-|\zeta|^{2})^{n-1}}\sum_{k=0}^{n-1}\binom{n-1}{k}\frac{(n-1)!}{k!}(z-\zeta)^{k}\frac{(n+k)!}{n!}\frac{\overline{\zeta}^{k}}{(1-z\overline{\zeta})^{n+k+1}}$
$\displaystyle=$
$\displaystyle\frac{1}{\pi}\frac{1}{(1-|\zeta|^{2})^{n-1}}\sum_{k=0}^{n-1}\binom{n-1}{k}^{2}(n+k)!(n-k-1)!\,\frac{\overline{\zeta}^{k}(z-\zeta)^{k}}{(1-z\overline{\zeta})^{n+k+1}}.$

Therefore,

$\displaystyle\tilde{K}_{\mathbb{D},n}^{(n-1)}(z,z)$ $\displaystyle=$
$\displaystyle\frac{1}{\pi}\frac{1}{(1-|z|^{2})^{n-1}}\,n!(n-1)!\,\frac{1}{(1-|z|^{2})^{n+1}}$
$\displaystyle=$
$\displaystyle\frac{n!(n-1)!}{\pi}\frac{1}{(1-|z|^{2})^{2n}}.$

Applying Theorem 1.11 on the biholomorphism
$f:\mathcal{H}\longrightarrow\mathbb{D}$ defined by

$f(z)=\frac{2z+1}{-2z+3}$

gives that

$\tilde{K}^{(n-1)}_{\mathcal{H},n}(0,0)=|f^{\prime}(0)|^{2n}\,\tilde{K}^{(n-1)}_{\mathbb{D},n}(f(0),f(0))=\left(\frac{8}{9}\right)^{2n}\,\frac{n!(n-1)!}{\pi}\left(\frac{9}{8}\right)^{2n}=\frac{n!(n-1)!}{\pi}.$

Hence, we have proved that

$\tilde{K}_{D,n}^{(n-1)}(z,z)\sim\frac{n!(n-1)!}{\pi}\frac{1}{(\psi(z))^{2n}}\quad\text{
as }z\rightarrow p.$ (12)

Let $\delta(z)=dist(z,\partial D)$. Since $\frac{\partial\psi}{\partial
z}(p)=1$, $(-\psi(z))\sim 2\,\delta(z)$ as $z\rightarrow p$. Therefore,

$\tilde{K}_{D,n}^{(n-1)}(z,z)\sim\frac{n!(n-1)!}{\pi\,2^{2n}}\frac{1}{(\delta(z))^{2n}}\quad\text{
as }z\rightarrow p.$ (13)

In particular, we have proved that

$\tilde{K}_{D,n}^{(n-1)}(z,z)\approx\frac{1}{(\delta(z))^{2n}}\quad\text{ as
}z\rightarrow p.$

This proves the theorem when $D$ is simply connected. ∎

###### Case 2..

Let $D$ be a bounded domain in $\mathbb{C}$ and $p\in\partial\mathbb{D}$.
Assume that $\partial D$ is $C^{2}$-smooth near $p$. Choose a neighborhood $U$
of $p$ such that $D\cap U$ is simply connected.

Since $D\cap U$ is simply connected and $\partial(D\cap U)$ is $C^{2}$-smooth
near $p$, we have by Case (1) that

$\tilde{K}_{D\cap
U,n}^{(n-1)}(z,z)\sim\frac{n!(n-1)!}{\pi}\frac{1}{(\psi(z))^{2n}}\approx\frac{1}{(\delta(z))^{2n}}\quad\text{
as }z\rightarrow p.$

By Theorem 1.10, we have

$\lim_{z\rightarrow p}\frac{\tilde{K}_{D\cap
U,n}^{(n-1)}(z,z)}{\tilde{K}_{D,n}^{(n-1)}(z,z)}=1.$

Therefore,

$\tilde{K}_{D,n}^{(n-1)}(z,z)\sim\frac{n!(n-1)!}{\pi}\frac{1}{(\psi(z))^{2n}}\approx\frac{1}{(\delta(z))^{2n}}$

as $z\rightarrow p$. ∎

This completes the proof of the theorem. ∎

###### Proof of Theorem 1.13.

Since $\nu$ is continuous at $p$, for $0<\epsilon<\nu(p)$, there exits
$\delta>0$ such that whenever $|z-p|<\delta$, we have

$\nu(p)-\epsilon<\nu(z)<\nu(p)+\epsilon.$

Set $N_{p}:=B(p,\delta)\cap D$. Now, for all $z\in D$, define
$\nu_{+}(z)=\nu(p)-\epsilon$ and $\nu_{-}(z)=\nu(p)+\epsilon$. So, we get for
all $z\in N_{p}$,

$\frac{\tilde{K}_{N_{p},\nu_{-},n}^{(n-1)}(z,z)}{\tilde{K}_{D,\nu,n}^{(n-1)}(z,z)}\leq\frac{\tilde{K}_{N_{p},\nu,n}^{(n-1)}(z,z)}{\tilde{K}_{D,\nu,n}^{(n-1)}(z,z)}\leq\frac{\tilde{K}_{N_{p},\nu_{+},n}^{(n-1)}(z,z)}{\tilde{K}_{D,\nu,n}^{(n-1)}(z,z)}.$

Note that by definition, it can seen that
$\tilde{K}_{N_{p},\nu_{-},n}^{(n-1)}(z,z)=\frac{\tilde{K}_{N_{p},n}^{(n-1)}(z,z)}{\nu(p)+\epsilon}$
and
$\tilde{K}_{N_{p},\nu_{+},n}^{(n-1)}(z,z)=\frac{\tilde{K}_{N_{p},n}^{(n-1)}(z,z)}{\nu(p)-\epsilon}$.
Applying the localization for the weighted $n$-th order reduced Bergman
kernel, we get

$\lim_{z\to
p}\frac{\tilde{K}_{N_{p},\nu,n}^{(n-1)}(z,z)}{\tilde{K}_{D,\nu,n}^{(n-1)}(z,z)}=1.$

Therefore, we obtain from the inequality above,

$\limsup_{z\to
p}\frac{\tilde{K}_{N_{p},n}^{(n-1)}(z,z)/(\nu(p)+\epsilon)}{\tilde{K}_{D,\nu,n}^{(n-1)}(z,z)}\leq
1\,\,\text{and}\,\,\liminf_{z\to
p}\frac{\tilde{K}_{N_{p},n}^{(n-1)}(z,z)/(\nu(p)-\epsilon)}{\tilde{K}_{D,\nu,n}^{(n-1)}(z,z)}\geq
1.$

So

$\limsup_{z\to
p}\frac{\psi(z)^{2n}\tilde{K}_{N_{p},n}^{(n-1)}(z,z)/(\nu(p)+\epsilon)}{\psi(z)^{2n}\tilde{K}_{D,\nu,n}^{(n-1)}(z,z)}\leq
1\,\,\text{and}\,\,\liminf_{z\to
p}\frac{\psi(z)^{2n}\tilde{K}_{N_{p},n}^{(n-1)}(z,z)/(\nu(p)-\epsilon)}{\psi(z)^{2n}\tilde{K}_{D,\nu,n}^{(n-1)}(z,z)}\geq
1.$

Now, using the boundary behaviour of the non-weighted $n$-th order reduced
Bergman kernel

$\tilde{K}_{N_{p},n}^{(n-1)}(z,z)\sim\frac{n!(n-1)!}{\pi}\frac{1}{(\psi(z))^{2n}}\quad\text{
as }z\rightarrow p,$

we get

$\liminf_{z\to
p}\tilde{K}_{D,\nu,n}^{(n-1)}(z,z)\geq\frac{1}{\nu(p)+\epsilon}\frac{n!(n-1)!}{\pi}\frac{1}{(\psi(z))^{2n}},$

and

$\limsup_{z\to
p}\tilde{K}_{D,\nu,n}^{(n-1)}(z,z)\leq\frac{1}{\nu(p)-\epsilon}\frac{n!(n-1)!}{\pi}\frac{1}{(\psi(z))^{2n}}.$

Since $\epsilon$ is arbitrarily small, we have

$\tilde{K}_{D,\nu,n}^{(n-1)}(z,z)\sim\frac{1}{\nu(p)}\frac{n!(n-1)!}{\pi}\frac{1}{(\psi(z))^{2n}}\quad\text{
as }z\rightarrow p.$

This proves the required result. ∎

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# Revolutionizing MRI Data Processing Using FSL: Preliminary Findings with the
Fugaku Supercomputer

Tianxiang Lyu  Graduate School of Medicine, Juntendo University  Wataru Uchida
Graduate School of Medicine, Juntendo University  Faculty of Health Data
Science, Juntendo University  Zhe Sun  Graduate School of Medicine, Juntendo
University  Faculty of Health Data Science, Juntendo University  Christina
Andica  Graduate School of Medicine, Juntendo University  Faculty of Health
Data Science, Juntendo University  Keita Tokuda  Graduate School of Medicine,
Juntendo University  Faculty of Health Data Science, Juntendo University  Rui
Zou  Graduate School of Medicine, Juntendo University  Jie Mao  Graduate
School of Medicine, Juntendo University  Keigo Shimoji  Graduate School of
Medicine, Juntendo University  Department of Radiology, Juntendo University
Koji Kamagata  Graduate School of Medicine, Juntendo University  Department of
Radiology, Juntendo University  Mitsuhisa Sato  Graduate School of Medicine,
Juntendo University  Faculty of Health Data Science, Juntendo University
Ryutaro Himeno  Graduate School of Medicine, Juntendo University  Faculty of
Health Data Science, Juntendo University  Shigeki Aoki  Graduate School of
Medicine, Juntendo University  Faculty of Health Data Science, Juntendo
University  Department of Radiology, Juntendo University

##  1 Abstract

The amount of Magnetic resonance imaging data has grown tremendously recently,
creating an urgent need to accelerate data processing, which requires
substantial computational resources and time. In this preliminary study, we
applied FMRIB Software Library commands on T1-weighted and diffusion-weighted
images of a single young adult using the Fugaku supercomputer. The tensor-
based measurements and subcortical structure segmentations performed on Fugaku
supercomputer were highly consistent with those from conventional systems,
demonstrating its reliability and significantly reduced processing time.

##  2 Introduction

Magnetic resonance imaging (MRI) has revolutionized neuroscience research,
enabling comprehensive analyses of brain structure, function, and connectivity
in both healthy individuals and those with neurological or psychiatric
conditions. To advance our understanding of the brain and its disorders,
large-scale neuroimaging datasets like the Human Connectome Project (HCP)[2],
the UK Biobank (UKB)[7], the Adolescent Brain Cognitive Development (ABCD)[1],
and Brain/MINDS-beyond have been established. However, the sheer volume of
data in these datasets poses a significant computational challenge, hindering
research progress and limiting the scope of analysis.[6]

Traditional methods for processing brain MRI data often rely on resources
available at the university laboratory level, which are insufficient to handle
the massive scale of these datasets. The processing time required for
thousands of brain images becomes impractical, leading to delays in research
projects and potentially limiting the insights that can be derived from these
valuable datasets. Supercomputers offer a promising solution by enabling high-
speed, parallel processing of large datasets[5], but deploying complex MRI
data processing pipelines on these systems remains a challenge due to issues
such as software dependencies, hardware-software compatibility, and the
complexity of composite pipelines involving multiple software packages.

Previous attempts to implement MRI data processing pipelines on supercomputers
have been limited in scope. While studies have successfully implemented the
FreeSurfer pipeline[5], which accepts T1-weighted images as input, on the
supercomputer Fugaku, there are no known studies that have implemented the
FMRIB Software Library (FSL) pipeline, which can accept both T1-weighted and
diffusion-weighted images as input. The ability to process diverse types of
MRI data on a supercomputer would dramatically increase the options for brain
image analysis and accelerate research progress.

To run FSL on Fugaku supercomputer, The main difficulty is complication. FSL
official do not provide a correct configuration setting for Fugaku
supercomputer. The reason should be obivous that FSL offcial have no chance
accessing Fugaku before, and all codes in FSL have been compiled and run for
many years on Intel-based platform only, therefore it is hard to launch on
Fugaku supercomputer. Therefore, our main efforts is to modify the source
codes and configure files of FSL, in order to make it able to be compiled and
run on Fugaku supercomputer. The main challenges are shown in the following:

  * •

Configuration files were written 10 years ago, where the system could not
guess build type.

  * •

The software dependencies on Fugaku are not compatible with FSL.

  * •

Some source codes are written with Intel-based CPU SIMD instrunctions only.

This study aims to address these challenges by demonstrating the feasibility
of utilizing the FSL pipeline on the Fugaku supercomputer system. By
showcasing the successful integration of FSL onto a leading-edge
supercomputer, we seek to pave the way for efficient and scalable analysis of
large-scale neuroimaging datasets. The unique capabilities of the Fugaku
supercomputer, such as its high-performance computing power and ability to
handle complex workflows, make it particularly suitable for this research. The
successful implementation of the FSL pipeline on Fugaku has the potential to
revolutionize brain research by enabling researchers to process and analyze
vast amounts of diverse MRI data efficiently, ultimately leading to new
insights into brain structure, function, and disorders.

##  3 Method

###  3.1 Study Participant

The MRI data of a 26-year-old female subject analyzed in this study was
randomly selected from the S1200 dataset of the WU-Minn Human Connectome
Project (HCP) consortium,[10] which consists of high-quality neuroimaging data
acquired from a large cohort of healthy young adults. Upon further
examination, the subject was found to have normal cognitive function as
indicated by a Mini-Mental State Examination score of 30, no vascular risk
factors (normal body mass index, blood pressure, lipid, and glucose profiles),
and no history of neurological or psychological diseases, alcohol abuse, or
drug abuse.

###  3.2 Data Acquisition

Imaging data from the HCP database were acquired using an HCP-customized
Siemens 3T Connectome Skyra magnet.[11] Diffusion-weighted images were
acquired using a spin-echo echo-planar imaging sequence with the following
parameters: repetition time (TR) = 5520 $ms$, echo time (TE) = 89.5 $ms$, flip
angle = 78°, field of view (FOV) = 210 × 180 $mm$, matrix size = 144 × 124,
and voxel size = 1.5 × 1.5 × 1.5 $mm^{3}$. Diffusion weighting was applied
along 90 diffusion-encoding directions at b = 1000, 2000, and 3000 $s/mm^{2}$,
with an additional six b = 0 $s/mm^{2}$ (non-diffusion-weighted) images
acquired. From the available diffusion-weighted data, only the volumes with b
= 0 and 1000 s/mm2 were extracted and utilized for analysis in this study,
because the current work focused exclusively on diffusion tensor imaging (DTI)
analyses. At higher b-values (2000 and 3000 s/mm2), the diffusion signal
becomes increasingly influenced by non-Gaussian motion effects, violating the
Gaussian diffusion assumption underlying the DTI model. T1-weighted images
were acquired using a 3D Magnetization-Prepared Rapid Gradient-Echo sequence
with the following parameters: TR = 2400 $ms$, TE = 2.14 $ms$, flip angle =
8°, FOV = 224 × 224 $mm$, matrix size = 320 × 320, slice thickness = 0.7 $mm$,
and voxel size = 0.7 × 0.7 × 0.7 $mm^{3}$. All imaging data were processed and
reconstructed using the HCP’s minimal preprocessing pipelines,[3] which
included gradient nonlinearity correction, motion correction, and registration
to the MNI anatomical template. On visual examination, T1 and diffusion-
weighted images of the subject assessed in this study were free from artifacts
and structural abnormalities.

###  3.3 Processing Pipelines

The analysis pipeline was implemented using tools from FSL version 6.0,[4]
which provides a comprehensive library of brain image analysis techniques
including segmentation, registration, and mathematical operations. The
specific steps shown in Figure 1 were as follows:

  1. 1.

Estimation of DTI parameters from diffusion-weighted images using DTIFIT,
including Fractional Anisotropy (FA) [unitless], Mean Diffusivity (MD)
[$\times 10^{-3}$ mm2/s], Axial Diffusivity (AD) [$\times 10^{-3}$ mm2/s], and
Radial Diffusivity (RD) [$\times 10^{-3}$ mm2/s].

  2. 2.

Segmentation of subcortical structures from T1-weighted images using FMRIB’s
Integrated Registration and Segmentation Tool (FIRST).[8] Subcortical
structures to be evaluated included accumbens, amygdala, brainstem, caudate,
hippocampus, pallidum, putamen, and thalamus.

  3. 3.

Linear registration of the T1-weighted image to diffusion space using FMRIB’s
Linear Image Registration Tool (FLIRT) (Jenkinson and Smith, 2001) and
epi_reg. The masks of subcortical regions obtained from FIRST were then
transformed to diffusion space using nearest neighbor interpolation.

  4. 4.

Computation of average DTI parameter values within each subcortical region
using fslstats.

Figure 1: Processing pipeline set on supercomputers

###  3.4 System Configuration

The environment used in this study is solely FMRIB Software Library (FSL)
version 6.0.5, and the platforms are shown as the following. Because it is the

The analysis results of version 7.2.0 were not examined by Haddad et al.21;
thus, we present new test–retest results.

####  3.4.1 Fugaku Supercomputer

The supercomputer Fugaku is a homogeneous, CPU-based distributed machine with
158,976 nodes. Based on the ARM v8.2A architecture, the CPU of the system,
A64FX, has 32 GB HBM2 memory and 48 compute cores, with total memory bandwidth
1024 GB/s. 12 compute cores form one Core Memory Group (CMG). On the ALUs
side, there is a couple of 512-bit wide SIMD units inside each compute core,
where the whole CPU performs 3.072 TF/s for double-precision, 6.144 TF/s for
single-precision, and 12.288 TF/s for half-precision. As to cache, Each
compute core contains 64 KB of L1 data cache, and 32 MB L2 cache is shared by
all. For interconnect, low latency and high throughput are achieved by a
6-dimensional mesh/torus network, with a 6.8 GB/s link bandwidth and 40.8 GB/s
of injection bandwidth per compute node.

  * •

Operating System: Red Hat Enterprise Linux 8 McKernel

  * •

Arch: Armv8.2-A SVE 512 bit

  * •

Chip: Fujitsu A64FX

  * •

CPU: 48 compute cores and 2-4 assistant cores with HBM2 32 GiB, 1024 GB/s

  * •

Memory: 4.85 PiB HBM2

To run FSL on Fugaku supercomputer, The main difficulty is handle the
compatibility of software environment. However, FSL official do not provide a
correct configuration setting for Fugaku supercomputer. Therefore, it is
critical to build up an environment with usable configuration. Our current
software environment is shown as the following:

  * •

GNU Compiler: 8.5.0

  * •

OpenBLAS: 0.3.21

  * •

Lapack: 3.10.8

####  3.4.2 Conventional node

To evaluate the reproducibility of our brain image analysis software
implemented on a supercomputer, we used a MacBook Pro as the comparison node.
The specifications of the MacBook Pro are as follows:

  * •

Operating System: macOS 14.2

  * •

Model: MacBook Pro (Model Identifier: MacBookPro18,4)

  * •

Chip: Apple M1 Max

  * •

CPU: 10-core (8 performance cores and 2 efficiency cores)

  * •

Memory: 64 GB LPDDR5

These specifications provided a baseline for comparing the performance and
reproducibility of the analysis software when run on a high-performance
supercomputer versus a conventional node. Because there is official
configuration of FSL on macOS, the software environment is managed by FSL
install script.

###  3.5 Data Parallelism Strategies

In this section, we will talk about the strategies of data parallel processing
on supercomputers, which is designed for processing large numbers of brain
image with high efficiency, considering not only the CPU architecture but also
the available computing resource of the whole system.

####  3.5.1 NUMA-Aware Optimizations

HBM memory offer higher memory bandwidth, but there are several NUMAs in one
CPU. Data transfer inside one NUMA can achieve highest memory bandwidth with
lowest response latency. Therefore, if memory size is available, one
processing pipelines should be performed in one NUMA in order to achieve best
performance. Also, most of processing libraries are not supported with multi-
threading parallelization, which makes it much more important to execute
several pipelines on one CPU, in order to utilize multiple computing cores.
Inside one computing node, GNU Parallel library is used to execute pipelines
in parallelism. And the maximal subjects processed on one compute node is
related to the memory consumption of executed pipelines.

####  3.5.2 Bulk Jobs

There are hundreds to thousands of computing node in supercomputers, offering
the promise of expressive speedups over single workstation. One of the main
technical contribution of our paper is to utilize massive computing nodes
using bulk jobs for scheduling. Bulk jobs decomposes a whole data processing
into many subtasks, wherer there is no dependency or communication required
between the execution of the subtasks. Then all subtask will be submitted to
the waiting list of the workload manager. where each subtask will be executed
with available computing resource. Such an parallel strategy can be readily
deployed on leading-edge supercomputers and experimental evidence shows that
such parallelization introduces more flexibility and efficiency of the
computing resource on leading-edge supercomputers.

Figure 2: Data Parallelism Processing Framework

###  3.6 Verification of pipeline

We verified the reproducibility of the pipeline implementation on the
supercomputer Fugaku escribed in the previous section via comparing the
results from the conventional node. First, to evaluate the reproducibility of
voxel-wise DTI parameters, we assessed the correlation between DTI measures
derived from the supercomputer and from the conventional node for each voxel.
The $R^{2}$ was used as an index for evaluating the correlation. $R^{2}$ was
defined as

$R^{2}(y,\hat{y})=1-\frac{\sum_{i=1}^{n}(y_{i}-\hat{y}i)^{2}}{\sum{i=1}^{n}(y_{i}-\bar{y})^{2}}$

where $\bar{y}=\frac{1}{n}\sum_{i=1}^{n}y_{i}$ and
$\sum_{i=1}^{n}(y_{i}-\hat{y}i)^{2}=\sum{i=1}^{n}\epsilon_{i}^{2}$.

In addition, to compare the accuracy of DTI parameters estimation in local
brain structures, we calculated the average DTI parameters for each deep gray
matter structure defined in the diffusion-weighted imaging space and evaluated
the correlation of the estimation results between general node and
supercomputer as well as the voxel-wise approach.

Figure 3: Comparison between Fugaku and Conventional Node

Next, to confirm the segmentation accuracy of deep gray matter using FIRST, we
used the dice index of ROIs defined by pair of two nodes. The Dice similarity
coefficient between two sets A and B is defined as:

$\mathrm{dice}(A,B)=\frac{2\times|A\cap B|}{|A|+|B|}$

where $|A|$, $|B|$denotes the number of elements of node A and B,
respectively, and $|A\cap B|$ represents the cardinality of the intersection
of pair of two nodes. In simple terms, the Dice index is calculated as twice
the number of elements in the intersection divided by the total number of
elements in both sets.

(a) FA

(b) MD

(c) AD

(d) RD

Figure 4: Scatter Plot of DTIFIT Values

##  4 Results

###  4.1 Diffusion Tensor Imaging

The voxel-wise correlation of DTI parameters between the supercomputer and
conventional node was evaluated using the coefficient of determination
($R^{2}$). The scatter plots in Figure 2B show the voxel-wise correlation for
each DTI parameter, with $R^{2}$ values of 1.0 for FA, 1.0 for MD, 1.0 for AD,
and 1.0 for RD. These high $R^{2}$ values indicate an excellent agreement
between the DTI results obtained from the supercomputer and the conventional
node.

Figure 5: Structure and Dice Coefficient

For the region-wise evaluation of DTI parameters in deep gray matter
structures, scatter plots were created to assess the correlation between the
supercomputer and conventional node results (Figure 4). The $R^{2}$ values
were 1.0 for FA, 0.99 for MD, 0.99 for AD, and 0.99 for RD, demonstrating a
high degree of consistency in the regional DTI parameter estimation between
the conventional node and supercomputers including the Fugaku.

(a) FA

(b) MD

(c) AD

(d) RD

Figure 6: Scatter Plot of DTIFIT Values

###  4.2 Subcortical Sturctures Segmentation

The deep gray matter segmentation results obtained using FIRST on the
supercomputer were compared to those from the conventional node by calculating
the dice index for each structure. As shown in Table 1, the dice indices
ranged from 0.897 for the accumbents to 0.993 for the putamen and thalamus.
These values indicate a high degree of overlap between the segmentation
results from the two platforms.

##  5 Discussion

This study aimed to implement the FMRIB Software Library (FSL), a well-
established brain diffusion MRI analysis pipeline, on the Fugaku supercomputer
for the first time. The successful execution of FSL commands such as DTIFIT,
FIRST, and FLIRT on brain MRI data of healthy young adults demonstrates the
feasibility and potential of utilizing supercomputers for advanced brain image
analysis.

The implementation of FSL in the Fugaku supercomputer has increased the
possibility of significantly reducing the time required to analyze and process
large numbers of brain images based on large databases compared to
conventional methods. This speed-up enables the analysis of larger sample
sizes and more complex datasets in realistic timeframes, increasing the
possibility of obtaining robust and reliable research results from large-scale
studies based on multi-center collaborative databases, which cannot be
achieved through small-scale, single-institution studies.

Despite the different CPU architectures of Fugaku (ARM-based) and common
commercially available workstations (Intel-based), the results obtained from
both systems showed a nearly perfect correlation and high overlap in
fractional anisotropy (FA), mean diffusivity (MD), axial diffusivity (AD),
radial diffusivity (RD), and subcortical structure volumes. This finding
strongly supports the reliability and validity of using supercomputers for
brain image analysis.

While some small-area variability was observed in the structural segmentation
by FIRST, the degree of this variability was minimal[12]. Due to the limited
sample size of this study, the potential reasons for these discrepancies and
their impact on the overall results remain uncertain[9]. Future studies with
larger sample sizes are needed to investigate and address these variations.

The successful implementation of FSL on Fugaku has significant implications
for democratizing brain imaging research. By making large-scale neuroimaging
studies more accessible and efficient, especially for smaller laboratories
with limited computational resources, this advancement paves the way for
researchers to conduct brain imaging studies based on large databases. This
democratization of brain imaging analysis research will facilitate
collaboration among researchers.

Compared to previous attempts to use supercomputers for brain imaging
analysis, the findings of this study are highly innovative, as we have
developed a pipeline that can handle both T1-weighted images and diffusion MRI
as input images. This advancement expands the scope of brain imaging research
that can be conducted using supercomputers.

The main limitation of this preliminary study is the single-subject design,
which limits the generalizability of the results. Future validation in a
larger number of subjects is necessary. Additionally, only the FSL pipeline
was implemented in this study. Future attempts should be made to implement and
validate other brain imaging analysis pipelines such as MRtrix3 and ANTs.

In conclusion, this study successfully demonstrates the feasibility and
potential of brain image analysis using FSL on the Fugaku supercomputer. The
results showed high consistency with conventional processing methods and
highlighted the significance of supercomputing in reducing processing time,
enabling large-scale studies, and democratizing brain imaging research. Future
research should focus on validating these findings in larger samples and
integrating other widely used brain image processing pipelines into the Fugaku
supercomputer.

##  6 Acknowledgement

This work was supported by JSPS KAKENHI Grant Grant-in-Aid for Scientific
Research(B) Number 24K03005.

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