# On local existence of solutions for nonlinear systems of Cauchy-Riemann
operator of any order in the plane

Yifei Pan

###### Abstract

We prove a general local existence theorem for nonlinear systems of Cauchy-
Riemann operator of any order in one complex variable with initial values at a
given point, which is a counterpart of local existence of ODE.

##  1 Introduction

000MSC 2010: 35G20 (Primary); 32G05, 30G20 (Secondary)

In this paper we consider local existence of solutions for nonlinear partial
differential systems of Cauchy-Riemann operator in dimension two from point
view of complex analysis.

First we introduce some simple notations. Let $D$ denote the closed disk
$\\{z\in\mathbb{C}\mid|z|\leq R\\}$. Let $m$ be a positive integer. For each
$k$: $1\leq k\leq m$, we denote $\mathcal{D}^{k}v$ as a vector in
$\mathbb{C}^{2^{k}}$ with entries $\partial^{\mu}\bar{\partial}^{\nu}v$ where
$\mu+\nu=k$ where $v$ is a complex valued function on $D$. Here we use the
standard complex derivatives for partial derivatives as
$\partial=\frac{1}{2}(\frac{\partial}{\partial x}-i\frac{\partial}{\partial
y}),\mbox{ }\bar{\partial}=\frac{1}{2}(\frac{\partial}{\partial
x}+i\frac{\partial}{\partial y}),$ where $z=x+iy$. Let $u$ be a map from $D$
to $\mathbb{C}^{n}$ with $u=(u^{1},...,u^{n})$. We denote $\mathcal{D}^{k}u$
as $(\mathcal{D}^{k}u^{1},...,\mathcal{D}^{k}u^{n})$, which is a vector in
$\mathbb{C}^{2^{k}n}$.

Let $\Omega$ be the set

$\Omega=D\times\mathbb{C}^{n}\times\mathbb{C}^{2n}\times\cdot\cdot\cdot\times\mathbb{C}^{2^{m-1}n},$

with coordinates as $(z,\eta_{0},\eta_{1},...,\eta_{m-1})$.

The following is the main result of this paper, which can be considered as an
analogue for systems of ordinary differential equations with initial
conditions.

######  Theorem 1.1.

Let $a:\Omega\to\mathbb{C}^{n}$ be any function of class $C^{1+\alpha}$. Let
$\mu,\nu$ be nonnegative integers such that $\mu+\nu=m\geq 1$, and
$c_{i,\bar{j}}$ be any vector in $\mathbb{C}^{n}$ for $i+j\leq m-1$. Then the
system of Cauchy-Riemann operators with initial values at the origin:

$\displaystyle\partial^{\mu}\bar{\partial}^{\nu}u(z)$ $\displaystyle=$
$\displaystyle a(z,u,\mathcal{D}^{1}u,...,\mathcal{D}^{m-1}u)$
$\displaystyle\partial^{i}\bar{\partial}^{j}u(0)$ $\displaystyle=$
$\displaystyle c_{i,\bar{j}},\mbox{ for }i+j\leq m-1$

has (infinitely many) solutions $u:D\to\mathbb{C}^{n}$ of class $C^{m+\alpha}$
in $D$ for sufficiently small values of $R$.

As a special case we state the following

######  Corollary 1.2.

Let $a:D\times\mathbb{C}^{n}\to\mathbb{C}^{n}$ be any function of class
$C^{1+\alpha}$. Let $c$ be any vector in $\mathbb{C}^{n}$. Then the following
nonlinear first odrer partial differential system of Cauchy-Riemnn operator

$\displaystyle\bar{\partial}u(z)$ $\displaystyle=$ $\displaystyle a(z,u(z))$
$\displaystyle u(0)$ $\displaystyle=$ $\displaystyle c$

has (infinitely many) solutions $u:D\to\mathbb{C}^{n}$ of class $C^{1+\alpha}$
in $D$ for sufficiently small values of $R$.

In order to prove Theorem 1.1, it suffices to prove the case of zero initial
values. The general case of initial values can be converted easily. The whole
paper is devoted to give a proof for the case of zero initial values.

Finally, The main idea and method have been applied to higher dimensions of
elliptic operators [PZ], PY].

###  1.1 Examples of no solutions-Mizohata equation

After a famous example of Lewy [L] in $\mathbb{R}^{3}$, Mizohata [M]
considered, in $\mathbb{R}^{2}$, the following equation

$\frac{\partial u}{\partial x}+ix\frac{\partial u}{\partial y}=F(x,y).$

It was proved in [M] that there is a smooth function $F$ for which the above
equation has no solution near the origin. Converting the equation to complex
one, one has the following equation

$\displaystyle\bar{\partial}u=\frac{1}{1+\mbox{Re}z}F(z,\bar{z})-\frac{1-\mbox{Re}z}{1+\mbox{Re}z}\partial
u.$ (1)

According to the notation as Theorem 1.1, we have

$a(z,\eta_{0},\eta)=\frac{1}{1+\mbox{Re}z}F(z,\bar{z})-\frac{1-\mbox{Re}z}{1+\mbox{Re}z}\eta,$

whence

$\partial_{\eta}(0)=-1,\bar{\partial}_{\eta}a(0)=0.$

Therefore the condition (1) is not met for Theorem 1.1. Taking
$\partial^{\mu}\bar{\partial}^{\nu-1}$ on both side, we have

$\partial^{\mu}\bar{\partial}^{\nu}u=-\frac{1-\mbox{Re}z}{1+\mbox{Re}z}\partial^{\mu+1}\bar{\partial}^{\nu-1}u+\cdot\cdot\cdot+\partial^{\mu}\bar{\partial}^{\nu-1}\left\\{\frac{1}{1+\mbox{Re}z}F(z,\bar{z})\right\\}.$

This is a differential equation of any order which has no solutions locally at
the origin.

###  1.2 Osserman’s theorem

We remark that it is classical that the equation $\Delta u=e^{2u}$ has no
solutions in the whole plane $\mathbb{C}$ due to Ahlfors [A], and local
existence puts constrain on the radius of existence due to Osserman [O].
Therefore, the vanishing condition in Theorem B can’t be dropped, and the
existence of radius in Theorem A can’t be arbitrarily large. In fact, solve a
solution $u$ by Theorem A, of

$\Delta u=e^{2u},u(0)=a,\nabla u(0)=b$

for $|z|\leq R$. Then by [O], we have $R\leq 2e^{-a}$, which goes to zero if
$a\to\infty$.

Also we can solve the following

$\bar{\partial}u=u^{2}$

specifically to get

$u(z)=\frac{1}{\varphi(z)+\bar{z}}$

where $\varphi(z)$ is an entire function. From this, we can see the radius of
existence of the solution can be any radius $R$. On the other hand, we can
solve

$\bar{\partial}u=e^{u}$

to get

$u(z)=\ln(\varphi(z)+\bar{z})$

where we can define a branch cut so that $u$ is well-defined.

##  2 Function spaces and their norms

###  2.1 Hölder space

Let $C^{\alpha}(D)$ be the set of all functions $f$ on $D$ for which

$H_{\alpha}[f]=\sup\left\\{{\frac{|f(z)-f(z^{\prime})|}{|z-z^{\prime}|^{\alpha}}\bigg{|}z,z^{\prime}\in\mathrm{D}}\right\\}$

is finite. Let $C^{k}(D)$ be the set of all function $f$ on $D$ whose
$k^{\textup{th}}$ order partial derivatives exist and are continuous, $k$ an
integer, $k\geq 0$. $C^{k+\alpha}(D)$ is the set of all functions $f$ on $D$
whose $k^{\textup{th}}$ order partial derivatives exist and belong to
$C^{\alpha}(D)$.

The symbol $|f|$ or $|f|_{\mathrm{D}}$ denotes $\textup{sup}_{z\in D}|f(z)|$.
For $f\in C^{\alpha}(D)$ we define

$\|f\|=|f|+(2R)^{\alpha}H_{\alpha}[f].$

The set of $n$-tuples $f=(f_{1},...,f_{n})$ of functions (vector functions or
maps) of $C^{\alpha}(D)$ is denoted by $[C^{\alpha}(D)]^{n}$, and
$H_{\alpha}[f]$ is defined as the maximum of $H_{\alpha}[f_{i}](i=1,..,n)$. In
a similar fashion we define $|f|_{A}=\sup_{z\in A}|f(z)|$ for functions and
vector functions, and write $|f|$ when the domain is understood. Finally, in
this paper throughout, the norm of $\mathbb{C}^{N}$ is taken as
$|v|=\max|v_{j}|$.

The following lemma is well-known; for a proof see ([NW], 7.1b).

######  Lemma 2.1.

The function $\|\cdot\cdot\cdot\|$ defined on $C^{\alpha}(D)$ is a norm, with
respect to which $C^{\alpha}(D)$ is a Banach algebra: $\|fg\|\leq\|f\|\|g\|$.

The following simple lemma is to be used multiple times throughout the paper.

######  Lemma 2.2.

If $f\in C^{k+\alpha}(D)$, then

$|f(z^{\prime})-\sum_{l=0}^{k}\frac{1}{l!}\sum_{i+j=l}\partial^{i}\bar{\partial}^{j}f(z)(z^{\prime}-z)^{i}{(\bar{z}^{\prime}-\bar{z})}^{j}|\leq\bigg{\\{}\sum_{i+j=k}H_{\alpha}[\partial^{i}\bar{\partial}^{j}f]\bigg{\\}}|z^{\prime}-z|^{k+\alpha}.$

###### Proof.

Expanding at $z$, we have the formula

$\displaystyle
f(z^{\prime})-\sum_{l=0}^{k-1}\frac{1}{l!}\sum_{i+j=l}\partial^{i}\bar{\partial}^{j}f(z)(z^{\prime}-z)^{i}{(\bar{z}^{\prime}-\bar{z})}^{j}$
$\displaystyle=$
$\displaystyle\int_{0}^{1}\int_{0}^{t_{k-1}}\cdot\cdot\cdot\int_{0}^{t_{1}}\left\\{\frac{d^{k}}{dt^{k}}f(tz^{\prime}+(1-t)z)\right\\}dtdt_{1}\cdot\cdot\cdot
dt_{k-1}$ $\displaystyle=$
$\displaystyle\int_{0}^{1}\int_{0}^{t_{k-1}}\cdot\cdot\cdot\int_{0}^{t_{1}}\left\\{\sum_{i+j=k}\partial^{i}\bar{\partial}^{j}f(tz^{\prime}+(1-t)z)(z^{\prime}-z)^{i}{(\bar{z}^{\prime}-\bar{z})}^{j}\right\\}dtdt_{1}\cdot\cdot\cdot
dt_{k-1}.$

Hence, by subtracting kth term
$\frac{1}{k!}\sum_{i+j=k}\partial^{i}\bar{\partial}^{j}f(z)(z^{\prime}-z)^{i}{(\bar{z}^{\prime}-\bar{z})}^{j}$
from both sides, we have

$f(z^{\prime})-\sum_{l=0}^{k}\frac{1}{l!}\sum_{i+j=l}\partial^{i}\bar{\partial}^{j}f(z)(z^{\prime}-z)^{i}{(\bar{z}^{\prime}-\bar{z})}^{j}$
$=\int_{0}^{1}\int_{0}^{t_{k-1}}\cdot\cdot\cdot\int_{0}^{t_{1}}\left\\{\sum_{i+j=k}\\{\partial^{i}\bar{\partial}^{j}f(tz^{\prime}+(1-t)z)-\partial^{i}\bar{\partial}^{j}f(z)\\}(z^{\prime}-z)^{i}{(\bar{z}^{\prime}-\bar{z})}^{j}\right\\}dtdt_{1}\cdot\cdot\cdot
dt_{k-1}$

Thus we have,

$|f(z^{\prime})-\sum_{l=0}^{k}\frac{1}{l!}\sum_{i+j=l}\partial^{i}\bar{\partial}^{j}f(z)(z^{\prime}-z)^{i}{(\bar{z}^{\prime}-\bar{z})}^{j}|$
$\leq\sum_{i+j=k}H_{\alpha}[\partial^{i}\bar{\partial}^{j}f]|z^{\prime}-z|^{k+\alpha}.$

This completes the proof. ∎

###  2.2 Function spaces with vanishing order at the origin

Our idea of solving differential equations of order $m$ is to look for
solutions that vanish up to $m-1$ order at the origin; this way the norm
estimate of the function space to be considered later is made possible using
only $m$th order derivatives. The general case of initial values can be
converted to the special case of zero initial values. We denote for $k\geq 1$,
$C_{0}^{k+\alpha}(D)$ the set of all functions in $C^{k+\alpha}(D)$ whose
derivatives vanish up to order $k-1$ at the origin. Specifically

$C_{0}^{k+\alpha}(D)=\\{f\in
C^{k+\alpha}(D)\big{|}\partial^{i}\bar{\partial}^{j}f(0)=0,i+j\leq k-1\\}.$

We now define functions $\|\cdot\cdot\cdot\|^{(k)}$ on $C^{k+\alpha}(D)$
inductively. On $C^{1+\alpha}(D)$ we define, following [NW],
$\|f\|^{(1)}=\max\\{\|\partial f\|,\|\bar{\partial}f\|\\}.$ For $k\geq 2$, we
define $\|f\|^{(k)}=\max\\{\|\partial
f\|^{(k-1)},\|\bar{\partial}f\|^{(k-1)}\\}.$ Obviously, we have the definition
$\|\cdot\cdot\cdot\|^{(k)}$ in terms of $\|\cdot\cdot\cdot\|$:
$\|f\|^{(k)}=\max_{i+j=k}\\{\|\partial^{i}\bar{\partial}^{j}f\|\\}.$ We point
out that the function $\|\cdot\cdot\cdot\|^{(k)}$ on $C^{k+\alpha}(D)$ is not
a norm since $\|f\|^{(k)}=0$ if and only if $f$ is a polynomial of degree at
most $k-1$. However it becomes norm when restricted to the subspace
$C_{0}^{k+\alpha}(D)$, which is to be proved below. In this paper, we denote
$C$ as a constant independent of $R$ varying from line to line. First we
obtain some useful estimates.

######  Lemma 2.3.

If $f\in C_{0}^{k+\alpha}(D)$, then $\|f\|\leq CR^{k}\|f\|^{(k)}.$

###### Proof.

Let $f\in C_{0}^{k+\alpha}(D)$, then

$\displaystyle f(z)$ $\displaystyle=$
$\displaystyle\int_{0}^{1}\int_{0}^{t_{k-1}}\cdot\cdot\cdot\int_{0}^{t_{1}}\left\\{\frac{d^{k}}{dt^{k}}f(tz)\right\\}dtdt_{1}\cdot\cdot\cdot
dt_{k-1}$ $\displaystyle=$
$\displaystyle\int_{0}^{1}\int_{0}^{t_{k-1}}\cdot\cdot\cdot\int_{0}^{t_{1}}\left\\{\sum_{i+j=k}\partial^{i}\bar{\partial}^{j}f(tz)z^{i}\bar{z}^{j}\right\\}dtdt_{1}\cdot\cdot\cdot
dt_{k-1}$ $\displaystyle=$
$\displaystyle\sum_{i+j=k}\left\\{\int_{0}^{1}\int_{0}^{t_{k-1}}\cdot\cdot\cdot\int_{0}^{t_{1}}\partial^{i}\bar{\partial}^{j}f(tz)dtdt_{1}\cdot\cdot\cdot
dt_{k-1}\right\\}z^{i}\bar{z}^{j}.$

Applying norm inequality, we obtain

$\displaystyle\|f\|$ $\displaystyle\leq$
$\displaystyle\sum_{i+j=k}\frac{1}{k!}\|\partial^{i}\bar{\partial}^{j}f\|\|z^{i}\bar{z}^{j}\|$
$\displaystyle\leq$
$\displaystyle\sum_{i+j=k}\frac{1}{k!}\|\partial^{i}\bar{\partial}^{j}f\|\|z\|^{k}\leq
CR^{k}\|f\|^{(k)},$

where we have used $\|z\|=3R$, which is easily verified. ∎

######  Lemma 2.4.

If $f\in C_{0}^{m+\alpha}(D)$, then, for $l\leq m$,

$\|f\|^{(l)}\leq CR^{m-l}\|f\|^{(m)}.$

###### Proof.

Let $f\in C_{0}^{m+\alpha}(D)$. If $i+j=l$, then
$\partial^{i}\bar{\partial}^{j}f\in C_{0}^{m-l+\alpha}(D)$. By Lemma 2.3, we
have

$\|\partial^{i}\bar{\partial}^{j}f\|\leq
CR^{m-l}\|\partial^{i}\bar{\partial}^{j}f\|^{(m-l)}\leq CR^{m-l}\|f\|^{(m)}.$

∎

######  Lemma 2.5.

The function space $C_{0}^{k+\alpha}(D)$ equipped with the function
$\|\cdot\cdot\cdot\|^{(k)}$ is a Banach space.

###### Proof.

First define $|||f|||=\sum_{i+j=0}^{k}\|\partial^{i}\bar{\partial}^{j}f\|$. It
is well known that $C^{k+\alpha}(D)$ equipped with the norm $|||\cdot|||$ is a
Banach space. By Lemma 2.4 the norm $\|\cdot\cdot\cdot\|^{(k)}$ for
$C_{0}^{k+\alpha}(D)$ is equivalent to the norm $|||\cdot|||$ of
$C^{k+\alpha}(D)$. Note that $C_{0}^{k+\alpha}(D)$ is a closed subspace of
$C^{k+\alpha}(D)$ and is therefore a Banch space with norm
$\|\cdot\cdot\cdot\|^{(k)}$. ∎

##  3 Cauchy-Green operator and high order derivative formula

###  3.1 Basic definitions and properties

The operators are defined for integrable functions on $D$ and $C=\\{|z|=R\\}$
as follows:

$\displaystyle Tf(z)$ $\displaystyle=$ $\displaystyle\frac{-1}{2\pi
i}\int_{D}\frac{f(\zeta)d\bar{\zeta}\wedge d\zeta}{\zeta-z},\mbox{
}^{2}Tf(z)=\frac{-1}{2\pi
i}\int_{D}\frac{f(\zeta)-f(z)}{(\zeta-z)^{2}}d\bar{\zeta}\wedge d\zeta$
$\displaystyle Sf(z)$ $\displaystyle=$ $\displaystyle\frac{1}{2\pi
i}\int_{C}\frac{f(\zeta)d\zeta}{\zeta-z},\hskip
28.45274ptS_{b}f(z)=\frac{1}{2\pi
i}\int_{C}\frac{f(\zeta)d\bar{\zeta}}{\zeta-z},$ .

Using polar coordinates, we see that $Tf$ is defined for continuous $f$, and
${}^{2}Tf$ for $C^{\alpha}(D)$. The operator $S$ is the familiar Cauchy
integral. We also define related operators:
$\overline{T},\overline{S},\overline{S}_{b}$, and ${}^{2}\overline{T}$ as
follows: $\overline{T}(f)=\overline{T(\bar{f})}$,
$\overline{S}(f)=\overline{S(\bar{f})}$,
$\overline{S}_{b}(f)=\overline{S_{b}(\bar{f})}$ and
${}^{2}\overline{T}(f)=\overline{{}^{2}T(\bar{f})}$ . Importantly, we will use
operators of $T^{\mu}\overline{T}^{\nu}$, which is a composition of
$T,\overline{T}$. Here we assume
$T^{0}=\mathrm{Id},\overline{T}^{0}=\mathrm{Id}$ and etc. We note here that
$S_{b}$ in this paper is the same as $\overline{S}$ in [NW]. The following
estimate holds:

$\displaystyle|Tf|\leq 4R|f|.$

More generally, if $\triangle$ is a bounded domain, then $T_{\triangle}f$ is
defined for continuous $f$ on $\triangle$ by

$T_{\triangle}f(z)=\frac{-1}{2\pi
i}\int_{\triangle}\frac{f(\zeta)d\bar{\zeta}\wedge d\zeta}{\zeta-z},\mbox{
}S_{\triangle}f(z)=\frac{1}{2\pi
i}\int_{\partial\triangle}\frac{f(\zeta)d\zeta}{\zeta-z}.$

We have

$|T_{\triangle}f|\leq 2\mathrm{diam}(\triangle)|f|_{\triangle}.$

The fundamental property between operators $T,S$ is the following ([NW],
6.1a).

######  Lemma 3.1.

If $f\in C^{1}(D)$, then

$T\bar{\partial}f=f-Sf\mbox{ on }\mbox{}\mathrm{Int}(D),$
$\overline{T}\partial f=f-\overline{S}f\mbox{ on }\mbox{}\mathrm{Int}(D).$

######  Lemma 3.2.

If $f\in C^{m+\alpha}(D)$ and $\mu+\nu=m$, then

$\mathrm{T}^{\nu}\overline{T}^{\mu}(\partial^{\mu}\bar{\partial}^{\nu}f)=f-\sum_{j=0}^{\nu-1}T^{j}(S(\bar{\partial}^{j}f))-\sum_{j=0}^{\mu-1}T^{\nu}\overline{T}^{j}(\overline{S}(\partial^{j}\bar{\partial}^{\nu}f))\mbox{
on }\mathrm{Int}(D)$

if $\mu,\nu\geq 1$; otherwise

$T^{m}(\bar{\partial}^{m}g)=g-\sum_{j=0}^{m-1}T^{j}(S(\bar{\partial}^{j}g))$
$\overline{T}^{m}(\partial^{m}g)=g-\sum_{j=0}^{m-1}\overline{T}^{j}(\overline{S}(\partial^{j}g)).$

###### Proof.

Just apply Lemma 3.1 repeatedly. ∎

###  3.2 A new high order integral operator

For $k\geq 0$, we define a new operator ${}^{k+2}Tf$ on $C^{k+\alpha}(D)$ by

${}^{k+2}Tf(z)=\frac{-(k+1)!}{2\pi
i}\int_{D}\frac{f(\zeta)-P_{k}(\zeta,z)}{(\zeta-z)^{k+2}}d\bar{\zeta}\wedge
d\zeta,$

where $P_{k}(\zeta,z)$ is the Taylor expansion of $f$ at $z$ of degree $k$.
Namely

$P_{k}(\zeta,z)=\sum_{l=0}^{k}\frac{1}{l!}\sum_{i+j=l}\partial^{i}\bar{\partial}^{j}f(z)(\zeta-z)^{i}{(\bar{\zeta}-\bar{z})}^{j}.$

We also define ${}^{k+2}\overline{T}f=\overline{{}^{k+2}T(\bar{f})}$. More
generally, if $\Delta$ is a bounded domain, then ${}^{k+2}T_{\Delta}$ is
defined as

${}^{k+2}T_{\Delta}f(z)=\frac{-(k+1)!}{2\pi
i}\int_{\Delta}\frac{f(\zeta)-P_{k}(\zeta,z)}{(\zeta-z)^{k+2}}d\bar{\zeta}\wedge
d\zeta.$

Note the case $k=0$ was defined in [NW]. The following is well-known and
classical(see [NW], [V]).

######  Lemma 3.3.

Let $f\in C^{\alpha}(D)$. Then $Tf\in C^{1+\alpha}(D),^{2}Tf\in
C^{\alpha}(D)$. Moreover

$\bar{\partial}Tf=f,\mbox{ }\partial Tf=^{2}Tf,$ $H_{\alpha}[^{2}Tf]\leq
CH_{\alpha}[f],$

If $f\in C^{k+\alpha}(D)(k\geq 0)$, then $Tf\in f\in C^{k+1+\alpha}(D)$.

We prove the following extension of Lemma 3.3 for the operator ${}^{k+2}Tf$,
which is one of key tools in the proof of the results that follow.

######  Theorem 3.4.

Let $f\in C^{k+\alpha}(D)(k\geq 0)$. Then

$\partial^{i}\bar{\partial}^{j}Tf=\partial^{i}\bar{\partial}^{j-1}f,$

if $j\geq 1,i+j\leq k+1;$ otherwise

$\partial^{k+1}Tf=^{k+2}Tf.$

Before giving a proof of Theorem 3.4, we need two lemma on compution of
integrals.

######  Lemma 3.5.

If $\varphi(\zeta)$ is holomorphic in $\mathbb{C}$, then

$\int_{\triangle}\frac{\overline{\phi(\zeta)}}{\zeta-w}d\bar{\zeta}\wedge
d\zeta$

is anti-holomorphic in $w\in\mathrm{Int}(\triangle)$, where
$\triangle=\\{|\zeta-z_{0}|\leq r\\}.$

###### Proof.

Writing $\phi(\zeta)=\sum a_{l}(\zeta-z_{0})^{l}$, we only have to prove the
lemma for $\phi(\zeta)=(\zeta-z_{0})^{l}$. Indeed,

$\displaystyle\int_{\triangle}\frac{(\bar{\zeta}-\bar{z}_{0})^{l}}{\zeta-w}d\bar{\zeta}\wedge
d\zeta$ $\displaystyle=$ $\displaystyle{\frac{-2\pi
i}{(l+1)}}T_{\triangle}(\bar{\partial}(\bar{\zeta}-\bar{z}_{0})^{l+1})(w)$
$\displaystyle=$ $\displaystyle{\frac{-2\pi
i}{(l+1)}}\\{(\bar{w}-\bar{z}_{0})^{l+1}-S_{\triangle}((\bar{\zeta}-\bar{z}_{0})^{l+1})(w)\\},$

where we have

$\displaystyle S_{\triangle}((\bar{\zeta}-\bar{z}_{0})^{l+1})(w)$
$\displaystyle=$ $\displaystyle\frac{1}{2\pi i}\int_{|\zeta-
z_{0}|=r}\frac{(\bar{\zeta}-\bar{z}_{0})^{l+1}}{\zeta-w}d\zeta$
$\displaystyle=$ $\displaystyle\frac{r^{2(l+1)}}{2\pi i}\int_{|\zeta-
z_{0}|=r}\frac{1}{(\zeta-z_{0})^{l+1}(\zeta-w)}d\zeta$ $\displaystyle=$
$\displaystyle 0,$

where in the last equality, we have used the residue theorem to get zero
integral. ∎

######  Corollary 3.6.

It holds for $l\geq 0$,

$\int_{\triangle}\frac{(\bar{\zeta}-\bar{z}_{0})^{l}}{\zeta-w}d\bar{\zeta}\wedge
d\zeta={\frac{-2\pi i}{(l+1)}}(\bar{w}-\bar{z}_{0})^{l+1}$

where $\triangle=\\{|\zeta-z_{0}|\leq r\\}.$

######  Lemma 3.7.

If $l\geq 1$, then

$\int_{\triangle}\frac{(\bar{\zeta}-\bar{z})^{l}}{(\zeta-z)^{l+1}}d\bar{\zeta}\wedge
d\zeta=0$

for $z\in\mathrm{Int(\triangle)}.$

###### Proof.

We notice that $\frac{(\bar{\zeta}-\bar{z})^{l}}{(\zeta-z)^{l+1}}$ is
integrable in $\triangle$ using polar coordinate at $z$. Let $\epsilon(z)$ be
the disk of radius $\epsilon$ and center at $z$ so that
$\epsilon(z)\subset\triangle$. Now we have

$\displaystyle\int_{\triangle}\frac{(\bar{\zeta}-\bar{z})^{l}}{(\zeta-z)^{l+1}}d\bar{\zeta}\wedge
d\zeta$ $\displaystyle=$
$\displaystyle\int_{\triangle\setminus\epsilon(z)}\frac{(\bar{\zeta}-\bar{z})^{l}}{(\zeta-z)^{l+1}}d\bar{\zeta}\wedge
d\zeta+\int_{\epsilon(z)}\frac{(\bar{\zeta}-\bar{z})^{l}}{(\zeta-z)^{l+1}}d\bar{\zeta}\wedge
d\zeta$ $\displaystyle=$
$\displaystyle\frac{1}{l!}\frac{d^{l}}{dw^{l}}\int_{\triangle\setminus\epsilon(z)}\frac{(\bar{\zeta}-\bar{z})^{l}}{\zeta-w}d\bar{\zeta}\wedge
d\zeta\bigg{|}_{w=z}+\int_{\epsilon(z)}\frac{(\bar{\zeta}-\bar{z})^{l}}{(\zeta-z)^{l+1}}d\bar{\zeta}\wedge
d\zeta$ $\displaystyle=$ $\displaystyle I_{1}|_{w=z}+I_{2}$

We have

$I_{1}=\frac{1}{l!}\frac{d^{l}}{dw^{l}}\left\\{\int_{\triangle}\frac{(\bar{\zeta}-\bar{z})^{l}}{\zeta-w}d\bar{\zeta}\wedge
d\zeta-\int_{\epsilon(z)}\frac{(\bar{\zeta}-\bar{z})^{l}}{\zeta-w}d\bar{\zeta}\wedge
d\zeta\right\\}$

where $w\in\epsilon(z)$. By Lemma 3.5, $I_{1}=0$. $\lim_{\epsilon\to
0}I_{2}=0$ is obvious. This proof is complete. ∎

Now we are ready to give proof of Therem 3.4.

###### Proof.

It suffices to show that if $f\in C^{k+\alpha}(D)$, then ${}^{k+1}Tf$ has a
total differential, which means we must show that there are numbers $A$ and
$B$, depending on $z$, such that

${}^{k+1}Tf(z)-^{k+1}Tf(z^{\prime})=A(z-z^{\prime})+B(\bar{z}-\bar{z}^{\prime})+\varepsilon(z,z^{\prime})\cdot|z-z^{\prime}|$

where $\varepsilon(z,z^{\prime})\to 0$ as $z^{\prime}\to z$. It is claimed
that $A=^{k+2}Tf(z)$ and $B=\partial^{k}f(z)$. To this end, denote the left
side in the above by $I$, and restrict $z^{\prime}$ so that
$\rho=|z-z^{\prime}|<\frac{1}{2}(R-|z|)$. Let $\delta$ be the disk of radius
$\rho$ and center $(z+z^{\prime})/2$. Notice $\delta\subset D$. Then

$I=^{k+1}T_{\delta}f(z)-^{k+1}T_{\delta}f(z^{\prime})-\frac{k!}{2\pi
i}\int_{D\setminus\delta}\bigg{\\{}\frac{f(\zeta)-P_{k-1}(\zeta,z)}{(\zeta-z)^{k+1}}-\frac{f(\zeta)-P_{k-1}(\zeta,z^{\prime})}{(\zeta-z^{\prime})^{k+1}}\bigg{\\}}d\bar{\zeta}\wedge
d\zeta.$

We write by Lemma 2.2 that

$f(\zeta)=P_{k-1}(\zeta,z)+\frac{1}{k!}\sum_{i+j=k}\partial^{i}\bar{\partial}^{j}f(z)(\zeta-z)^{i}(\bar{\zeta}-\bar{z})^{j}+E_{k}(\zeta,z)$

where

$|E_{k}(\zeta,z)|\leq\frac{1}{k!}\bigg{\\{}\sum_{i+j=k}H_{\alpha}[\partial^{i}\bar{\partial}^{j}f]\bigg{\\}}|\zeta-z|^{k+\alpha}.$

We now consider the terms in order. First we need the following, using Lemma
3.6 and Corollary 3.5,

${}^{k+1}T_{\delta}f(z)$ $\displaystyle=$ $\displaystyle\frac{-k!}{2\pi
i}\int_{\delta}\frac{f(\zeta)-P_{k-1}(\zeta,z)}{(\zeta-z)^{k+1}}d\bar{\zeta}\wedge
d\zeta$ $\displaystyle=$ $\displaystyle\frac{-1}{2\pi
i}\sum_{i+j=k}\partial^{i}\bar{\partial}^{j}f(z)\int_{\delta}\frac{(\zeta-z)^{i}(\bar{\zeta}-\bar{z})^{j}}{(\zeta-z)^{k+1}}d\bar{\zeta}\wedge
d\zeta$ $\displaystyle+$ $\displaystyle\frac{-k!}{2\pi
i}\int_{\delta}\frac{E_{k}(\zeta,z)}{(\zeta-z)^{k+1}}d\bar{\zeta}\wedge
d\zeta$ $\displaystyle=$ $\displaystyle\partial^{k}f(z)\frac{-1}{2\pi
i}\int_{\delta}\frac{1}{(\zeta-z)}d\bar{\zeta}\wedge d\zeta+\frac{-k}{2\pi
i}\int_{\delta}\frac{E_{k}(\zeta,z)}{(\zeta-z)^{k+1}}d\bar{\zeta}\wedge
d\zeta$ $\displaystyle=$
$\displaystyle\partial^{k}f(z)(\bar{z}-\frac{\bar{z}+\bar{z}^{\prime}}{2})+\frac{-k}{2\pi
i}\int_{\delta}\frac{E_{k}(\zeta,z)}{(\zeta-z)^{k+1}}d\bar{\zeta}\wedge
d\zeta$ $\displaystyle=$
$\displaystyle\frac{1}{2}\partial^{k}f(z)(\bar{z}-\bar{z}^{\prime})+I_{3}.$

Similarly we have

${}^{k+1}T_{\delta}f(z^{\prime})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\partial^{k}f(z^{\prime})(\bar{z}^{\prime}-\bar{z})+\frac{-k}{2\pi
i}\int_{\delta}\frac{E_{k}(\zeta,z^{\prime})}{(\zeta-z^{\prime})^{k+1}}d\bar{\zeta}\wedge
d\zeta$ $\displaystyle=$
$\displaystyle\frac{1}{2}\partial^{k}f(z^{\prime})(\bar{z}^{\prime}-\bar{z})+I_{3}^{\prime}.$

Now we estimate $I_{3},I_{3}^{\prime}$. It follows

$|I_{3}|=\bigg{|}\frac{-k}{2\pi
i}\int_{\delta}\frac{E_{k}(\zeta,z)}{(\zeta-z)^{k+1}}d\bar{\zeta}\wedge
d\zeta\bigg{|}\leq
C\sum_{i+j=k}H_{\alpha}[\partial^{i}\bar{\partial}^{j}f]\int_{0}^{2\pi}\int_{0}^{r_{1}(\theta)}\frac{r^{k+\alpha}2rdrd\theta}{r^{k+1}}$

where $r_{1}(\theta)$ is the distance from $z$ to the boundary of $\delta$ in
direction $\theta$; note $r_{1}(\theta)\leq 2\rho$. Continuing the
computation, we have

$|I_{3}|\leq
C\sum_{i+j=k}H_{\alpha}[\partial^{i}\bar{\partial}^{j}f](2\rho)^{1+\alpha}$

which approaches $0$ as $\rho\to 0$. The same estimate holds true for
$I_{3}^{\prime}$. Thus we have

$\displaystyle|^{k+1}T_{\delta}f(z)-^{k+1}T_{\delta}f(z^{\prime})-\partial^{k}f(z)(\bar{z}-\bar{z}^{\prime})|$
$\displaystyle\leq$
$\displaystyle|I_{3}|+|I_{3}^{\prime}|+\bigg{|}\frac{1}{2}\partial^{k}f(z)(\bar{z}-\bar{z}^{\prime})-\frac{1}{2}\partial^{k}f(z^{\prime})(\bar{z}^{\prime}-\bar{z})-\partial^{k}f(z)(\bar{z}-\bar{z}^{\prime})\bigg{|}$
$\displaystyle\leq$
$\displaystyle|I_{3}|+|I_{3}^{\prime}|+\frac{1}{2}H_{\alpha}[\partial^{k}f]|z-z^{\prime}|^{1+\alpha}.$

On the other hand, to continue to estimate, we have

$\displaystyle\int_{D\setminus\delta}\bigg{\\{}\frac{f(\zeta)-P_{k-1}(\zeta,z)}{(\zeta-z)^{k+1}}-\frac{f(\zeta)-P_{k-1}(\zeta,z^{\prime})}{(\zeta-z^{\prime})^{k+1}}\bigg{\\}}d\bar{\zeta}\wedge
d\zeta$ $\displaystyle=$
$\displaystyle\int_{D\setminus\delta}f(\zeta)\bigg{(}\frac{1}{(\zeta-z)^{k+1}}-\frac{1}{(\zeta-z^{\prime})^{k+1}}\bigg{)}d\bar{\zeta}\wedge
d\zeta-\int_{D\setminus\delta}\frac{P_{k-1}(\zeta,z)}{(\zeta-z)^{k+1}}d\bar{\zeta}\wedge
d\zeta$ $\displaystyle+$
$\displaystyle\int_{D\setminus\delta}\frac{P_{k-1}(\zeta,z^{\prime})}{(\zeta-z^{\prime})^{k+1}}d\bar{\zeta}\wedge
d\zeta$ $\displaystyle=$ $\displaystyle I_{4}+I_{5}+I_{6}.$

We need the following lemma to proceed.

######  Lemma 3.8.

If $m\geq 2,n\geq 0$, then

$\int_{D\setminus\delta}\frac{(\bar{\zeta}-\bar{z})^{n}}{(\zeta-z)^{m}}d\bar{\zeta}\wedge
d\zeta=\int_{D\setminus\delta}\frac{(\bar{\zeta}-\bar{z}^{\prime})^{n}}{(\zeta-z^{\prime})^{m}}d\bar{\zeta}\wedge
d\zeta=0.$

Actually more is true:

$\int_{D\setminus\delta}\frac{(\bar{\zeta}-\bar{s})^{n}}{(\zeta-s)^{m}}d\bar{\zeta}\wedge
d\zeta=0$

for any $s\in\delta$.

###### Proof.

We have

$\int_{D\setminus\delta}\frac{(\bar{\zeta}-\bar{z})^{n}}{(\zeta-z)^{m}}d\bar{\zeta}\wedge
d\zeta=\frac{1}{(m-1)!}\frac{d^{m-1}}{dw^{m-1}}\int_{D\setminus\delta}\frac{(\bar{\zeta}-\bar{z})^{n}}{\zeta-w}d\bar{\zeta}\wedge
d\zeta\bigg{|}_{w=z},$
$=\frac{1}{(m-1)!}\frac{d^{m-1}}{dw^{m-1}}\\{\int_{D}\frac{(\bar{\zeta}-\bar{z})^{n}}{\zeta-w}d\bar{\zeta}\wedge
d\zeta-\int_{\delta}\frac{(\bar{\zeta}-\bar{z})^{n}}{\zeta-w}d\bar{\zeta}\wedge
d\zeta\\}\bigg{|}_{w=z},$

which is zero by applying Lemma 3.4. ∎

By Lemma 3.8, we have

$I_{5}=I_{6}=0.$

Now we only have to estimate $I_{4}$. Indeed,

$I_{4}=-(k+1)\int_{z^{\prime}}^{z}\bigg{(}\int_{D\setminus\delta}\frac{f(\zeta)}{(\zeta-w)^{k+2}}d\bar{\zeta}\wedge
d\zeta\bigg{)}dw$

where the integration with respect to $w$ is along the straight line segment
from $z^{\prime}$ to $z$. Now it suffices to estimate

$\displaystyle\int_{D\setminus\delta}\frac{f(\zeta)}{(\zeta-w)^{k+2}}d\bar{\zeta}\wedge
d\zeta-\int_{D}\frac{f(\zeta)-P_{k}(\zeta,z)}{(\zeta-z)^{k+2}}d\bar{\zeta}\wedge
d\zeta$ $\displaystyle=$
$\displaystyle\int_{D\setminus\delta}\frac{f(\zeta)}{(\zeta-w)^{k+2}}d\bar{\zeta}\wedge
d\zeta-\int_{D\setminus\delta}\frac{f(\zeta)-P_{k}(\zeta,z)}{(\zeta-z)^{k+2}}d\bar{\zeta}\wedge
d\zeta$ $\displaystyle-$
$\displaystyle\int_{\delta}\frac{f(\zeta)-P_{k}(\zeta,z)}{(\zeta-z)^{k+2}}d\bar{\zeta}\wedge
d\zeta$ $\displaystyle=$
$\displaystyle\int_{D\setminus\delta}f(\zeta)\bigg{\\{}\frac{1}{(\zeta-w)^{k+2}}-\frac{1}{(\zeta-z)^{k+2}}\bigg{\\}}d\bar{\zeta}\wedge
d\zeta-\int_{\delta}\frac{f(\zeta)-P_{k}(\zeta,z)}{(\zeta-z)^{k+2}}d\bar{\zeta}\wedge
d\zeta$ $\displaystyle=$ $\displaystyle I_{7}+I_{8}.$

Here we have used Lemma 3.8 to conclude

$\int_{D\setminus\delta}\frac{P_{k}(\zeta,z)}{(\zeta-z)^{k+2}}d\bar{\zeta}\wedge
d\zeta=0.$

Now we have

$\displaystyle I_{7}$ $\displaystyle=$
$\displaystyle\int_{D\setminus\delta}f(\zeta)\bigg{\\{}\frac{1}{(\zeta-w)^{k+2}}-\frac{1}{(\zeta-z)^{k+2}}\bigg{\\}}d\bar{\zeta}\wedge
d\zeta$ $\displaystyle=$
$\displaystyle(k+2)\int_{D\setminus\delta}f(\zeta)\int_{w}^{z}\frac{ds}{(\zeta-s)^{k+3}}d\bar{\zeta}\wedge
d\zeta$ $\displaystyle=$
$\displaystyle(k+2)\int_{w}^{z}ds\int_{D\setminus\delta}\frac{f(\zeta)-P_{k}(\zeta,s)}{(\zeta-s)^{k+3}}d\bar{\zeta}\wedge
d\zeta$

where we have used that

$\int_{D\setminus\delta}\frac{P_{k}(\zeta,s)}{(\zeta-s)^{k+3}}d\bar{\zeta}\wedge
d\zeta=0,$

for $s\in\delta$ by Lemma 3.8. Now

$\displaystyle|I_{7}|$ $\displaystyle\leq$ $\displaystyle
C|w-z|\bigg{|}\int_{D\setminus\delta}\frac{f(\zeta)-P_{k}(\zeta,s)}{(\zeta-s)^{k+3}}d\bar{\zeta}\wedge
d\zeta\bigg{|}$ $\displaystyle\leq$ $\displaystyle
C|w-z|\sum_{i+j=k}H_{\alpha}[\partial^{i}\bar{\partial}^{j}f]2\pi\int_{\rho/2}^{2R}\frac{r^{k+\alpha}2rdr}{r^{k+3}}$
$\displaystyle=$ $\displaystyle
C|w-z|\sum_{i+j=k}H_{\alpha}[\partial^{i}\bar{\partial}^{j}f]2\pi\frac{1}{1-\alpha}((\rho/2)^{\alpha-1}-(2R)^{\alpha-1})$
$\displaystyle\leq$ $\displaystyle
C\sum_{i+j=k}H_{\alpha}[\partial^{i}\bar{\partial}^{j}f]\rho^{\alpha}.$

In the last inequality we have used that $|w-z|\leq|z-z^{\prime}|=\rho.$ We
have thus shown that

${}^{k+1}Tf(z)-^{k+1}Tf(z^{\prime})=^{k+2}Tf(z)(z-z^{\prime})+\partial^{k}f(z)(\bar{z}-\bar{z}^{\prime})+O(|z-z^{\prime}|^{1+\alpha})$

as $z^{\prime}\to z$. Thus ${}^{k+1}Tf$ has a total differential and
$\partial(^{k+1}Tf)=^{k+2}Tf$. ∎

In what follows, we will show that ${}^{k+2}Tf\in C^{\alpha}(D)$ if $f\in
C^{k+\alpha}(D)$.

######  Lemma 3.9.

Let $f\in C^{k+\alpha}(D)$. Then

$|^{k+2}Tf|\leq
CR^{\alpha}\sum_{i+j=k}H_{\alpha}[\partial^{i}\bar{\partial}^{j}f].$

###### Proof.

We compute, using Lemma 2.2,

$\displaystyle|^{k+2}Tf(z)|$ $\displaystyle=$
$\displaystyle\frac{(k+1)!}{2\pi}\bigg{|}\int_{D}\frac{f(\zeta)-P_{k}(\zeta,z)}{(\zeta-z)^{k+2}}d\bar{\zeta}\wedge
d\zeta\bigg{|},$ $\displaystyle\leq$ $\displaystyle
C\sum_{i+j=k}H_{\alpha}[\partial^{i}\bar{\partial}^{j}f]\int_{0}^{2\pi}\int_{0}^{2R}\frac{r^{k+\alpha}2rdrd\theta}{r^{k+2}}$
$\displaystyle\leq$ $\displaystyle
CR^{\alpha}\sum_{i+j=k}H_{\alpha}[\partial^{i}\bar{\partial}^{j}f].$

∎

The following is well known [NW, K]

######  Lemma 3.10.

If $f\in C^{1+\alpha}(D)$, then

${}^{2}Tf=T(\partial f)-S_{b}(f).$

######  Lemma 3.11.

If $f\in C^{k+\alpha}(D)$, then for $k\geq 1$

${}^{k+2}Tf=^{2}T(\partial^{k}f)-\sum_{i=1}^{k}\partial^{i}S_{b}(\partial^{k-i}f).$

###### Proof.

By Lemma 3.10, we have

${}^{2}Tf=T(\partial f)-S_{b}(f).$

Taking derivative, we have

$\partial^{2}Tf=\partial T(\partial f)-\partial S_{b}(f),$

which is equivalent to

${}^{3}Tf=^{2}T(\partial f)-\partial S_{b}(f).$

A mathematical induction finishes the proof. ∎

######  Lemma 3.12.

If $f\in C^{k+\alpha}(D)$, then

$H_{\alpha}[\partial^{k}S_{b}f]\leq
C\sum_{i+j=k}H_{\alpha}[\partial^{i}\bar{\partial}^{j}f].$

###### Proof.

We notice that

$\partial^{k}S_{b}f(z)=\frac{k!}{2\pi
i}\int_{C}\frac{f(\zeta)}{(\zeta-z)^{k+1}}d\bar{\zeta},$

and $S_{b}f(z)$ is holomorphic in $\mathrm{Int}(D)$. First we need the
following

$\int_{C}\frac{P_{k}(\zeta,w)}{(\zeta-w)^{k+2}}d\bar{\zeta}=0,$

for $w\in\mathrm{Int}(D)$, and where $P_{k}(\zeta,w)$ is the Taylor expansion
of $f(\zeta)$ of order $k$ at $w$. Indeed,

$\int_{C}\frac{P_{k}(\zeta,w)}{(\zeta-w)^{k+2}}d\bar{\zeta}=\sum_{l=0}^{k}\frac{1}{l!}\sum_{i+j=l}\partial^{i}\bar{\partial}^{j}f(w)\int_{C}\frac{(\zeta-w)^{i}(\bar{\zeta}-\bar{w})^{j}}{(\zeta-w)^{k+2}}d\bar{\zeta}.$

It suffices to prove the following

$\int_{C}\frac{(\bar{\zeta}-\bar{w})^{n}}{(\zeta-w)^{m+2}}d\bar{\zeta}=0$

for $m,n\geq 0$. This is equivalent to prove that

$\int_{C}\frac{\bar{\zeta}^{n}}{(\zeta-w)^{m+2}}d\bar{\zeta}=0$

for $m,n\geq 0$. This is equivalent to prove that

$\int_{C}\frac{1}{\zeta^{n+2}(\zeta-w)^{m+2}}d\zeta=0,$

which is true by the residue theorem. Now we are ready to estimate.

$\displaystyle\frac{2\pi
i}{k!}(\partial^{k}S_{b}f(z)-\partial^{k}S_{b}f(z^{\prime}))$ $\displaystyle=$
$\displaystyle\int_{C}f(\zeta)\bigg{[}\frac{1}{(\zeta-z)^{k+1}}-\frac{1}{(\zeta-z^{\prime})^{k+1}}\bigg{]}d\bar{\zeta}$
$\displaystyle=$
$\displaystyle\frac{1}{k+1}\int_{C}f(\zeta)\int_{z}^{z^{\prime}}\frac{dw}{(\zeta-w)^{k+2}}d\bar{\zeta}$
$\displaystyle=$
$\displaystyle\frac{1}{k+1}\int_{z}^{z^{\prime}}dw\int_{C}\frac{f(\zeta)}{(\zeta-w)^{k+2}}d\bar{\zeta}$
$\displaystyle=$
$\displaystyle\frac{1}{k+1}\int_{z}^{z^{\prime}}dw\int_{C}\frac{f(\zeta)-P_{k}(\zeta,w)}{(\zeta-w)^{k+2}}d\bar{\zeta}.$

In order to apply estimates in [NW], we convert integral to be on unit circle.
Setting $w=R\omega$ and $\zeta=R\eta$, this becomes

$\frac{1}{(k+1)R^{k}}\int_{z/R}^{z^{\prime}/R}d\omega\int_{|\eta|=1}\frac{f(R\eta)-P_{k}(R\eta,R\omega)}{(\eta-\omega)^{k+2}}d\bar{\eta}$

where the line integral is taken on the shorter segment of the circle through
$z/R$ and $z^{\prime}/R$ and orthogonal to the unit circle(see [NW] (6.2a)).
Further, setting $\tau=(\eta-\omega)/(1-\bar{\omega}\eta)$, we have

$\eta=\frac{\tau+\omega}{1+\bar{\omega}\tau},\mbox{
}\eta-\omega=\frac{\tau(1-\omega\bar{\omega})}{1+\tau\bar{\omega}},\mbox{
}d\eta=\frac{(1-\omega\bar{\omega})}{(1+\tau\bar{\omega})^{2}}d\tau.$

Hence

$\displaystyle\frac{2\pi}{k!}$
$\displaystyle|\partial^{k}S_{b}f(z)-\partial^{k}S_{b}f(z^{\prime})|$
$\displaystyle\leq$
$\displaystyle\frac{C}{(R^{k}}\int_{z/R}^{z^{\prime}/R}|d\omega|\int_{|\eta|=1}\bigg{|}\frac{f(R\eta)-P_{k}(R\eta,R\omega)}{(\eta-\omega)^{k+2}}\bigg{|}|d\bar{\eta}|$
$\displaystyle\leq$
$\displaystyle\frac{C}{R^{k}}\int_{z/R}^{z^{\prime}/R}|d\omega|\int_{|\eta|=1}\frac{|f(R\frac{\tau+\omega}{1+\bar{\omega}\tau})-P_{k}(R\frac{\tau+\omega}{1+\bar{\omega}\tau},R\omega)||1+\tau\bar{\omega}|^{k}}{(1-\omega\bar{\omega})^{k+1}}|d\eta|$
$\displaystyle\leq$
$\displaystyle\frac{C}{R^{k}}\sum_{i+j=k}H_{\alpha}[\partial^{i}\bar{\partial}^{j}f]\int_{z/R}^{z^{\prime}/R}|d\omega|\int_{|\eta|=1}\frac{|R\frac{\tau+\omega}{1+\bar{\omega}\tau}-R\omega|^{k+\alpha}|1+\tau\bar{\omega}|^{k}}{(1-\omega\bar{\omega})^{k+1}}|d\eta|$
$\displaystyle\leq$ $\displaystyle
CR^{\alpha}\sum_{i+j=k}H_{\alpha}[\partial^{i}\bar{\partial}^{j}f]\int_{z/R}^{z^{\prime}/R}\frac{|d\omega|}{(1-\omega\bar{\omega})^{1-\alpha}}\int_{|\tau|=1}\frac{|d\tau|}{|1+\bar{\omega}\tau|^{\alpha}}.$

According to [NW](6.2a and p473), one has

$\int_{|\tau|=1}\frac{|d\tau|}{|1+\bar{\omega}\tau|^{\alpha}}\leq\frac{4\pi}{1-\alpha},$
$\int_{z/R}^{z^{\prime}/R}\frac{|d\omega|}{(1-\omega\bar{\omega})^{1-\alpha}}\leq\frac{2}{\alpha}|z-z^{\prime}|^{\alpha}\frac{1}{R^{\alpha}}.$

Hence

$H_{\alpha}[\partial^{k}S_{b}f]\leq
C\sum_{i+j=k}H_{\alpha}[\partial^{i}\bar{\partial}^{j}f].$

∎

The following is the key result to be used to prove existence of solutions in
what follows.

######  Theorem 3.13.

If $f\in C^{k+\alpha}(D)$, then

$H_{\alpha}[^{k+2}Tf]\leq
C\sum_{i+j=k}H_{\alpha}[\partial^{i}\bar{\partial}^{j}f],$
$H_{\alpha}[^{k+2}\overline{T}f]\leq
C\sum_{i+j=k}H_{\alpha}[\partial^{i}\bar{\partial}^{j}f].$

###### Proof.

By Lemma 3.11, and applying Lemma 3.12, we have

${}^{k+2}Tf$ $\displaystyle=$
${}^{2}T(\partial^{k}f)-\sum_{i=1}^{k}\partial^{i}S_{b}(\partial^{k-i}f).$
$\displaystyle H_{\alpha}[^{k+2}Tf]$ $\displaystyle\leq$ $\displaystyle
H_{\alpha}[^{2}T(\partial^{k}f)]+\sum_{i=1}^{k}H_{\alpha}[\partial^{i}S_{b}(\partial^{k-i}f)]$
$\displaystyle\leq$ $\displaystyle
CH_{\alpha}[\partial^{k}f]+C\left\\{\sum_{i=1}^{k}\frac{1}{i}\sum_{p+q=i}H_{\alpha}[\partial^{p}\bar{\partial}^{q}\partial^{k-i}f]\right\\}$
$\displaystyle\leq$ $\displaystyle
C\sum_{i+j=k}H_{\alpha}[\partial^{i}\bar{\partial}^{j}f].$

∎

######  Lemma 3.14.

If $h\in C^{k+\alpha}(D)$ $(k\geq 0)$, then

$\|Th\|^{(k+1)}\leq C\|h\|^{(k)},$ $\|\overline{T}h\|^{(k+1)}\leq
C\|h\|^{(k)}.$

###### Proof.

Let $i+j=k+1$. If $j\geq 1$, then
$\partial^{i}\bar{\partial}^{j}Th=\partial^{i}\bar{\partial}^{j-1}h$.
Otherwise, $\partial^{k+1}Th=^{k+2}Th$. We have by Lemma 3.9, 3.13

$\displaystyle\|^{k+2}Th\|$ $\displaystyle=$
$\displaystyle|^{k+2}Th|+(2R)^{\alpha}H_{\alpha}[^{k+2}Th]$
$\displaystyle\leq$ $\displaystyle
CR^{\alpha}\sum_{i+j=k}H_{\alpha}[\partial^{i}\bar{\partial}^{j}h]+(2R)^{\alpha}C\sum_{i+j=k}H_{\alpha}[\partial^{i}\bar{\partial}^{j}h]$
$\displaystyle\leq$
$\displaystyle(CR^{\alpha}+(2R)^{\alpha}C)\sum_{i+j=k}H_{\alpha}[\partial^{i}\bar{\partial}^{j}h]$
$\displaystyle\leq$ $\displaystyle C\|h\|^{(k)}.$

In the last inequality we have used that
$H_{\alpha}[\partial^{i}\bar{\partial}^{j}h]\leq(2R)^{-\alpha}\|h\|^{(k)}$ for
$i+j=k$. The proof is now complete. ∎

The following is the main result of this section.

######  Theorem 3.15.

If $h\in C^{\alpha}(D)$ and $\mu+\nu=m$ , then

$\|T^{\nu}\overline{T}^{\mu}h\|^{(m)}\leq C\|h\|.$

###### Proof.

Since, if $\nu\geq 1$,

$\|T^{\nu}\overline{T}^{\mu}h\|^{(m)}=\|T(T^{\nu-1}\overline{T}^{\mu}h)\|^{(m-1+1)},$

which is less than, by invoking Lemma 3.14

$C\|T^{\nu-1}\overline{T}^{\mu}h\|^{(m-1)},$

which is less than, by repeating the argument, $C\|h\|.$ This gives the proof
of the theorem. ∎

##  4 A High order integral operator

In order to prove the existence of solutions of system with arbitrary initial
values we will first solve the equation with zero initial values at the
origin. Namely, we are to solve the system of equations

$\displaystyle\partial^{\mu}\bar{\partial}^{\nu}u(z)$ $\displaystyle=$
$\displaystyle a(z,u(z),\mathcal{D}^{1}u(z),...,\mathcal{D}^{m-1}u(z))$ (2)

with zero initial values at the origin:

$\displaystyle\mathcal{D}^{k}u(0)$ $\displaystyle=$ $\displaystyle 0,\mbox{
for }0\leq k\leq m-1$ (3)

Every solution $u$ of (1) satisfies the following integral equation, by Lemma
3.2,

$\displaystyle
u(z)=\psi(z)+T^{\nu}\overline{T}^{\mu}a(\zeta,u,\mathcal{D}^{1}u,...,\mathcal{D}^{m-1}u)(z),$
(4)

where $\partial^{\mu}\bar{\partial}^{\nu}\psi(z)=0$. In what follows, we will
introduce function spaces that are Banach with norms to reflect the initial
values condition, and will modify the equation (3) to fit the defined Banach
space and to apply fixed point theorem to produce the needed solutions. In
doing so, we will take a full advantage of two parameters: the radius $R$ of
$D$ and the radius $\gamma$ of a closed ball in the Banach space where the
solutions are sought from.

###  4.1 Iteration procedures

For the sake of generality we consider a Banach space $\mathbf{B}(R)$
depending on a positive parameter $R$. $R$ will be less than some fixed
positive quantity. The norm on $\mathbf{B}(R)$ is denoted by
$\|\cdot\cdot\cdot\|$; it may also depend on $R$. The following deals with a
map $\mathbf{\Theta}:\mathbf{A}(R,\gamma)\to\mathbf{B}(R)$, where
$\mathbf{A}(R,\gamma)(\gamma>0)$ is a closed subset of $\mathbf{B}(R)$ defined
as $\mathbf{A}(R,\gamma)=\\{\Omega|\|\Omega\|\leq\gamma\\}$. Here we actually
have a family of such maps with two parameters $R,\gamma$.

######  Lemma 4.1.

Let $\mathbf{\Theta}:\mathbf{A}(R,\gamma)\to\mathbf{B}(R)$, and and let
$\delta(R,\gamma)$ and $\eta(R,\gamma)$ exist such that for all
$\Omega,\Omega^{\prime}\in\mathbf{A}(R,\gamma)$

$\|\mathbf{\Theta}(\Omega^{\prime})-\mathbf{\Theta}(\Omega)\|\leq\delta(R,\gamma)\|\Omega^{\prime}-\Omega\|,$
$\|\mathbf{\Theta}(\Omega)\|\leq\eta(R,\gamma).$

If there exist $R_{0}>0,\gamma_{0}>0$ such that $\delta(R,\gamma_{0})\leq 3/4$
and $\eta(R,\gamma_{0})\leq\frac{\gamma_{0}}{2}$ for $R\leq R_{0}$. Let
$\psi\in\mathbf{B}(R)$ be such that $\|\psi\|\leq\frac{\gamma_{0}}{2}$. Then
the following equation

$\Omega=\psi+\mathbf{\Theta}(\Omega)$

has a unique solution in $\mathbf{A}(R,\gamma_{0})$ for $R\leq R_{0}$. The
solution $\Omega$ is the limit of the sequence

$\displaystyle\Omega_{N+1}=\psi+\mathbf{\Theta}(\Omega_{N})\mbox{ for
}N=1,2,...$

where $\Omega_{1}\in\mathbf{A}(R,\gamma_{0})$.

###  4.2 Integral equations on Banach spaces

Here we fix the Banach space to work on. Let $\mathbf{B}(R)$ be
$[C^{m+\alpha}_{0}(D)]^{n}$, which is, by Lemma 2.6 a Banach space with norm
$||\cdot\cdot\cdot||^{(m)}$ defined as follows: if
$f=(f_{1},...,f_{n})\in\mathbf{B}(R)$, then $\|f\|^{(m)}=\max_{1\leq i\leq
n}\|f_{i}\|^{(m)}$. Let $\gamma$ be a positive number, and we consider

$\mathbf{A}(R,\gamma)=\\{f\in\mathbf{B}(R)|\|f\|^{(m)}\leq\gamma\\}.$

We note that $\mathbf{A}(R,\gamma)$ is a closed subset of $\mathbf{B}(R)$. Now
we assume $u=(u^{1},...,u^{n})$ and $a=(a^{1},...,a^{n})$. According to (1) ,
we consider equations

$u^{i}=\psi^{i}+T^{\nu}\overline{T}^{\mu}a^{i}(z,u,\mathcal{D}^{1}u,...,\mathcal{D}^{m-1}u)$

where $\psi=(\psi^{1},...,\psi^{n})$ is such that
$\partial^{\mu}\bar{\partial}^{\nu}\partial\psi=0$. To simplify the notation
we define

$\omega^{i}(f)=T^{\nu}\overline{T}^{\mu}a^{i}(z,f,\mathcal{D}^{1}f,...,\mathcal{D}^{m-1}f)$

for all $f\in\mathbf{B}(R)$ and $i=1,2,...,n$. To assure the zero initial
satisfied, we consider

$\displaystyle\mathbf{\Theta}^{i}(f)(\zeta)$ $\displaystyle=$
$\displaystyle\omega^{i}(f)(\zeta)-\sum_{p=0}^{m-1}\frac{1}{p!}\sum_{k+l=p}[\partial^{k}\bar{\partial}^{l}\omega^{i}(f)](0)\zeta^{k}\bar{\zeta}^{l}.$
(5)

We first notice that $T,\overline{T}$ map $C^{\alpha}(\mathrm{D})\to
C^{1+\alpha}(D)$, and it follows, by the construction (1),
$\mathbf{\Theta}^{i}(f)(\zeta)\in C^{m+\alpha}_{0}(D)$ for
$f\in\mathbf{B}(R)$. Thus we define a map from $\mathbf{B}(R)$ to
$\mathbf{B}(R)$.

$\displaystyle\mathbf{\Theta}$ $\displaystyle:$
$\displaystyle\mathbf{B}(R)\rightarrow\mathbf{B}(R)$
$\displaystyle\mathbf{\Theta}(f)$ $\displaystyle=$
$\displaystyle(\mathbf{\Theta}^{1}(f),...,\mathbf{\Theta}^{n}(f)).$ (6)

Here we recall again $\|\mathbf{\Theta}(f)\|^{(m)}=\max_{1\leq i\leq
n}\|\mathbf{\Theta}^{i}(f)\|^{(m)}$ and recall that for $f\in
C^{m+\alpha}_{0}(D)$,
$\|f\|^{(m)}=\max_{k+l=m}\\{\|\partial^{k}\bar{\partial}^{l}f\|\\}$. In order
to apply iteration procedure of Lemma 4.1, we will first estimate

$\|\mathbf{\Theta}^{i}(f)-\mathbf{\Theta}^{i}(g)\|^{(m)}=\|\mathbf{\omega}^{i}(f)-\mathbf{\omega}^{i}(g)\|^{(m)}$

for all $f,g\in\mathbf{A}(R,\gamma)$ in terms of $\|f-g\|^{(m)}$. and next we
will want to estimate
$\|\mathbf{\Theta}^{i}(f)\|^{(m)}=\|\mathbf{\omega}^{i}(f)\|^{(m)}$ in terms
of a constant.

####  4.2.1 Estimate of
$\|\mathbf{\omega}^{i}(f)-\mathbf{\omega}^{i}(g)\|^{(m)}$

In this section we always have functions $f,g$ to be in
$\mathbf{A}(R,\gamma)$. We first begin with the estimate by Theorem 3.15:

$\displaystyle\|\mathbf{\omega}^{i}(f)-\mathbf{\omega}^{i}(g)\|^{(m)}$ (7)
$\displaystyle=$
$\displaystyle\|T^{\nu}\overline{T}^{\mu}(a^{i}(z,f,\mathcal{D}^{1}f,...,\mathcal{D}^{m-1}f)-a^{i}(z,g,\mathcal{D}^{1}g,...,\mathcal{D}^{m-1}g))\|^{(m)}$
$\displaystyle\leq$ $\displaystyle
C\|a^{i}(z,f,\mathcal{D}^{1}f,...,\mathcal{D}^{m-1}f)-a^{i}(z,g,\mathcal{D}^{1}g,...,\mathcal{D}^{m-1}g)\|,$

In order to estimate (5), we first estimate the norm $|\cdot\cdot\cdot|$. In
fact, we have

$\displaystyle
a^{i}(\zeta,\mathcal{D}^{0}f(\zeta),\mathcal{D}^{1}f(\zeta),...,\mathcal{D}^{m-1}f(\zeta))-a^{i}(\zeta,\mathcal{D}^{0}g(\zeta),\mathcal{D}^{1}g(\zeta),...,\mathcal{D}^{m-1}g(\zeta))$
(8) $\displaystyle=$
$\displaystyle\int_{0}^{1}\frac{d}{dt}a^{i}(\zeta,tf(\zeta)+(1-t)g(\zeta),...,t\mathcal{D}^{m-1}f(\zeta)+(1-t)\mathcal{D}^{m-1}g(\zeta))dt$
$\displaystyle=$
$\displaystyle\sum_{j=0}^{n}\sum_{p=0}^{m-1}\sum_{k+l=p}A^{p,j}_{k,l}\partial^{k}\bar{\partial}^{l}(f_{j}-g_{j})+\bar{A}^{p,j}_{k,l}\overline{\partial^{k}\bar{\partial}^{l}(f_{j}-g_{j})},$

where

$\displaystyle A^{p,j}_{k,l}=\int_{0}^{1}\frac{\partial
a^{i}}{\partial\eta_{p,j}^{k,l}}(\zeta,\mathcal{W}^{0},...,\mathcal{W}^{m-1})dt,\mbox{
}\bar{A}^{p,j}_{k,l}=\int_{0}^{1}\frac{\partial
a^{i}}{\bar{\partial}\eta_{p,j}^{k,l}}(\zeta,\mathcal{W}^{0},...,\mathcal{W}^{m-1})dt$
(9)

where we have used the shorten notations:

$\mathcal{W}^{k}=t\mathcal{D}^{k}f(\zeta)+(1-t)\mathcal{D}^{k}g(\zeta)$

for $k=0,1,...,m-1$, and $\eta_{p,j}^{k,l}$ is a variable in $\eta_{p}$.
Therefore, taking norm $\|\cdot\cdot\cdot\|$ on (6) we have, using Lemma 2.4,

$\displaystyle\|a^{i}(\zeta,\mathcal{D}^{0}f,\mathcal{D}^{1}f,...,\mathcal{D}^{m-1}f)-a^{i}(\zeta,\mathcal{D}^{0}g,\mathcal{D}^{1}g,...,\mathcal{D}^{m-1}g)\|$
(10) $\displaystyle\leq$
$\displaystyle\sum_{j=0}^{n}\sum_{p=0}^{m-1}\sum_{k+l=p}\|A^{p,j}_{k,l}\partial^{k}\bar{\partial}^{l}(f_{j}-g_{j})\|+\|\bar{A}^{p,j}_{k,l}\overline{\partial^{k}\bar{\partial}^{l}(f_{j}-g_{j})}\|$
$\displaystyle\leq$
$\displaystyle\sum_{j=0}^{n}\sum_{p=0}^{m-1}\sum_{k+l=p}\|A^{p,j}_{k,l}\|\|\partial^{k}\bar{\partial}^{l}(f_{j}-g_{j})\|+\|\bar{A}^{p,j}_{k,l}\|\|\overline{\partial^{k}\bar{\partial}^{l}(f_{j}-g_{j})}\|$
$\displaystyle\leq$
$\displaystyle\sum_{j=0}^{n}\sum_{p=0}^{m-1}\sum_{k+l=p}\\{\|A^{p,j}_{k,l}\|+\|\bar{A}^{p,j}_{k,l}\|\\}\|{\partial^{k}\bar{\partial}^{l}(f-g)}\|$
$\displaystyle\leq$ $\displaystyle
C\sum_{j=0}^{n}\sum_{p=0}^{m-1}\sum_{k+l=p}\\{\|A^{p,j}_{k,l}\|+\|\bar{A}^{p,j}_{k,l}\|\\}R^{m-p}\|f-g\|^{(m)}.$

In order to estimate the quantities $\|A^{p,j}_{k,l}\|,\bar{A}^{p,j}_{k,l}\|$
in (8), we need to study the ranges of $\mathcal{W}^{k},k=0,1,...,m-1$ for
$f,g\in\mathbf{A}(R,\gamma)$. The following is what we need.

######  Lemma 4.2.

If $f,g\in\mathbf{A}(R,\gamma)$, then

$\displaystyle|\mathcal{W}^{k}|$ $\displaystyle\leq$ $\displaystyle
CR^{m-k}\gamma,\mbox{ }k=0,1,...,m-1.$

###### Proof.

If $f$ is a function in $C_{0}^{m+\alpha}(D)$, then by Lemma 2.4 we have, if
$i+j=k\leq m-1$,

$\|\partial^{i}\bar{\partial}^{j}f\|\leq CR^{m-k}\|f\|^{(m)}\leq
CR^{m-k}\gamma.$

Particularly, it implies

$|\partial^{i}\bar{\partial}^{j}f|\leq CR^{m-k}\gamma,$

whence, by the definition of norm, we have

$|\mathcal{W}^{k}|\leq CR^{m-k}\gamma$

for $k=0,1,...,m-1$. ∎

To continue on the estimates, we need a lemma on Lipschitz properties of
$C_{0}^{m+\alpha}(D)$.

######  Lemma 4.3.

Let $f\in C_{0}^{m+\alpha}(D)$. Then, for $\zeta^{\prime},\zeta\in D$, and
$i+j=l\leq m-1$, we have

$|\partial^{i}\bar{\partial}^{j}f(\zeta^{\prime})-\partial^{i}\bar{\partial}^{j}f(\zeta)|\leq
CR^{m-l-1}\|f\|^{(m)}|\zeta^{\prime}-\zeta|.$

###### Proof.

We have

$\displaystyle\partial^{i}\bar{\partial}^{j}f(\zeta^{\prime})-\partial^{i}\bar{\partial}^{j}f(\zeta)$
$\displaystyle=$
$\displaystyle\int_{0}^{1}\frac{d}{dt}\\{\partial^{i}\bar{\partial}^{j}f(t\zeta^{\prime}+(1-t)\zeta)\\}dt$
$\displaystyle=$
$\displaystyle\int_{0}^{1}\partial^{i+1}\bar{\partial}^{j}f(t\zeta^{\prime}+(1-t)\zeta)(\zeta^{\prime}-\zeta)+\partial^{i}\bar{\partial}^{j+1}f(t\zeta^{\prime}+(1-t)\zeta)\overline{(\zeta^{\prime}-\zeta)}dt,$

whence

$\displaystyle|\partial^{i}\bar{\partial}^{j}f(\zeta^{\prime})-\partial^{i}\bar{\partial}^{j}f(\zeta)|$
$\displaystyle\leq$
$\displaystyle(|\partial^{i+1}\bar{\partial}^{j}f|+|\partial^{i}\bar{\partial}^{j+1}f|)|\zeta^{\prime}-\zeta|$
$\displaystyle\leq$ $\displaystyle 2\|f\|^{(l+1)}|\zeta^{\prime}-\zeta|$
$\displaystyle\leq$ $\displaystyle CR^{m-l-1}\|f\|^{(m)}$

where in the last inequality we have used Lemma 2.4. ∎

In the following we will use a compact set in $\Omega$ defined as follows:

$E(R,\gamma)=D\times\Pi_{k=0}^{m-1}\\{z\in\mathbb{C}^{n2^{k}}||z|\leq
CR^{m-k}\gamma\\}.$

Now we first define some constants, whose usefulness will be self-evident
during proofs below and later sections.

$\displaystyle A(R,\gamma)$ $\displaystyle=$
$\displaystyle\max\bigg{\\{}\bigg{|}\frac{\partial
a^{i}}{\partial\eta_{p,j}^{k,l}}\bigg{|}_{E(R,\gamma)},\bigg{|}\frac{\partial
a^{i}}{\bar{\partial}\eta_{p,j}^{k,l}}\bigg{|}_{E(R,\gamma)}\bigg{|},\bigg{|}\frac{\partial
a^{i}}{\partial\zeta}\bigg{|}_{E(R,\gamma)},\bigg{|}\frac{\partial
a^{i}}{\partial\bar{\zeta}}\bigg{|}_{E(R,\gamma)}\bigg{\\}}$ $\displaystyle
H_{\alpha}^{A}[R,\gamma]$ $\displaystyle=$
$\displaystyle\max\bigg{\\{}H_{\alpha}\bigg{[}\frac{\partial
a^{i}}{\partial\eta_{p,j}^{k,l}}\bigg{]}_{E(R,\gamma)},H_{\alpha}\bigg{[}\frac{\partial
a^{i}}{\partial\bar{\eta}_{p,j}^{k,l}}\bigg{]}_{E(R,\gamma)},\bigg{|}\frac{\partial
a^{i}}{\partial\zeta}\bigg{|}_{E(R,\gamma)},\bigg{|}\frac{\partial
a^{i}}{\partial\bar{\zeta}}\bigg{|}_{E(R,\gamma)}\bigg{\\}}$

where max is taken over $k+l=p;0\leq p\leq m-1;i,j=1,..n$. Let $h$ be a
function defined on open set $\Omega$ in $\mathbb{C}^{N}$. The Hölder constant
of $h$ is defined

$H_{\alpha}[h]=\sup\bigg{\\{}\frac{|h(u)-h(v)|}{|u-v|^{\alpha}}\bigg{|}u,v\in\Omega\bigg{\\}}.$

Now if we restrict $h$ to be on one variable:
$g(\zeta)=h(u_{1},\cdot\cdot\cdot,u_{i},\zeta,u_{i+1},\cdot\cdot\cdot,u_{N})$,
then $H_{\alpha}[g]\leq H_{\alpha}[h]$. This fact is used below.

It is obvious from (7) that

$\displaystyle\left|A^{p,j}_{k,l}\right|\leq A(R,\gamma),\mbox{
}\left|\bar{A}^{p,j}_{k,l}\right|\leq A(R,\gamma).$ (11)

Now we estimate $H_{\alpha}[A^{p,j}_{k,l}]$ by inserting terms,

$\displaystyle A^{p,j}_{k,l}(\zeta^{\prime})-A^{p,j}_{k,l}(\zeta)$
$\displaystyle=$ $\displaystyle\int_{0}^{1}\\{\frac{\partial
a^{i}}{\partial\eta_{p,j}^{k,l}}(\zeta^{\prime},\mathcal{W}^{0}(\zeta^{\prime}),...,\mathcal{W}^{m-2}(\zeta^{\prime}),\mathcal{W}^{m-1}(\zeta^{\prime}))-\frac{\partial
a^{i}}{\partial\eta_{p,j}^{k,l}}(\zeta,\mathcal{W}^{0}(\zeta),...,\mathcal{W}^{m-2}(\zeta),\mathcal{W}^{m-1}(\zeta))\\}dt$
$\displaystyle=$ $\displaystyle\int_{0}^{1}\\{\frac{\partial
a^{i}}{\partial\eta_{p,j}^{k,l}}(\zeta^{\prime},\mathcal{W}^{0}(\zeta^{\prime}),...,\mathcal{W}^{m-2}(\zeta^{\prime}),\mathcal{W}^{m-1}(\zeta^{\prime}))-\frac{\partial
a^{i}}{\partial\eta_{p,j}^{k,l}}(\zeta,\mathcal{W}^{0}(\zeta^{\prime}),...,\mathcal{W}^{m-2}(\zeta^{\prime}),\mathcal{W}^{m-1}(\zeta^{\prime}))\\}dt$
$\displaystyle+$ $\displaystyle\int_{0}^{1}\\{\frac{\partial
a^{i}}{\partial\eta_{p,j}^{k,l}}(\zeta,\mathcal{W}^{0}(\zeta^{\prime}),...,\mathcal{W}^{m-2}(\zeta^{\prime}),\mathcal{W}^{m-1}(\zeta^{\prime}))-\frac{\partial
a^{i}}{\partial\eta_{p,j}^{k,l}}(\zeta,\mathcal{W}^{0}(\zeta),\mathcal{W}^{1}(\zeta^{\prime}),...,\mathcal{W}^{m-2}(\zeta^{\prime}),\mathcal{W}^{m-1}(\zeta^{\prime}))\\}dt$
$\displaystyle+$ $\displaystyle\cdot\cdot\cdot$ $\displaystyle+$
$\displaystyle\int_{0}^{1}\\{\frac{\partial
a^{i}}{\partial\eta_{p,j}^{k,l}}(\zeta,\mathcal{W}^{0}(\zeta),...,\mathcal{W}^{m-2}(\zeta),\mathcal{W}^{m-1}(\zeta^{\prime}))-\frac{\partial
a^{i}}{\partial\eta_{p,j}^{k,l}}(\zeta,\mathcal{W}^{0}(\zeta),\mathcal{W}^{1}(\zeta),...,\mathcal{W}^{m-2}(\zeta),\mathcal{W}^{m-1}(\zeta))\\}dt,$

whence, since $a$ is of $C^{1+\alpha}$ and by Lemma 4.3,

$\displaystyle|A^{p,j}_{k,l}(\zeta^{\prime})-A^{p,j}_{k,l}(\zeta)|$
$\displaystyle\leq$ $\displaystyle
H_{\alpha}^{A}[R,\gamma]\bigg{\\{}|\zeta^{\prime}-\zeta|^{\alpha}+\sum_{l=0}^{m-1}\sum_{i+j=l}(|\partial^{i}\bar{\partial}^{j}f(\zeta^{\prime})-\partial^{i}\bar{\partial}^{j}f(\zeta)|+|\partial^{i}\bar{\partial}^{j}g(\zeta^{\prime})-\partial^{i}\bar{\partial}^{j}g(\zeta)|)^{\alpha}\bigg{\\}}$
$\displaystyle\leq$ $\displaystyle
H_{\alpha}^{A}[R,\gamma]\bigg{\\{}|\zeta^{\prime}-\zeta|^{\alpha}+C\sum_{l=0}^{m-1}\sum_{i+j=l}(R^{m-l-1}\gamma)^{\alpha}|\zeta^{\prime}-\zeta|^{\alpha}\bigg{\\}}$

Therefore, from (8) we have

$\displaystyle H_{\alpha}[A^{p,j}_{k,l}]$ $\displaystyle\leq$ $\displaystyle
H_{\alpha}^{A}[R,\gamma]C(R,\gamma),$ (13)

where $C(R,\gamma)=1+C\sum_{l=0}^{m-1}(R^{m-l-1}\gamma)^{\alpha}$. By (10),
(11), we have

$\displaystyle\|A^{p,j}_{k,l}\|$ $\displaystyle=$
$\displaystyle|A^{p,j}_{k,l}|+(2R)^{\alpha}H_{\alpha}[A^{p,j}_{k,l}]$ (14)
$\displaystyle\leq$ $\displaystyle
A(R,\gamma)+(2R)^{\alpha}H_{\alpha}^{A}[R,\gamma]C(R,\gamma)$

Of course, $\|\bar{A}^{p,j}_{k,l}\|$ satisfies the same estimate. Putting
(6),(8),(11), and (12) together we have thus shown that

$\displaystyle\|a^{i}(\zeta,\mathcal{D}^{0}f,\mathcal{D}^{1}f,...,\mathcal{D}^{m-1}f)-a^{i}(\zeta,\mathcal{D}^{0}g,\mathcal{D}^{1}g,...,\mathcal{D}^{m-1}g)\|$
$\displaystyle\leq$ $\displaystyle\delta(R,\gamma)\|f-g\|^{(m)}$
$\displaystyle\|\mathbf{\omega}^{i}(f)-\mathbf{\omega}^{i}(g)\|^{(m)}$
$\displaystyle\leq$ $\displaystyle\delta(R,\gamma)|\|f-g\|^{(m)}$ (15)

where

$\displaystyle\delta(R,\gamma)=C(\sum_{p=0}^{m-1}R^{m-p})(A(R,\gamma)+(2R)^{\alpha}H_{\alpha}^{A}[R,\gamma]C(R,\gamma)).$
(16)

####  4.2.2 Estimate of $\|\mathbf{\Theta}^{i}(f)\|^{(m)}$

We begin with

$\displaystyle
a^{i}(\zeta,\mathcal{D}^{0}f(\zeta),\mathcal{D}^{1}f(\zeta),...,\mathcal{D}^{m-1}f(\zeta))-a^{i}(0,...,0)$
$\displaystyle=$
$\displaystyle\int_{0}^{1}\frac{d}{dt}a^{i}(t\zeta,tf(\zeta),...,t\mathcal{D}^{m-1}f(\zeta))dt$
$\displaystyle=$ $\displaystyle
C^{i}\zeta+\bar{C}^{i}\bar{\zeta}+\sum_{j=0}^{n}\sum_{p=0}^{m-1}\sum_{k+l=p}B^{p,j}_{k,l}\partial^{k}\bar{\partial}^{l}(f_{j})+\bar{B}^{p,j}_{k,l}\overline{\partial^{k}\bar{\partial}^{l}(f_{j})}$

where we have

$B^{i}=\int_{0}^{1}\frac{\partial
a^{i}}{\partial\zeta}(t\zeta,tf(\zeta),...,t\mathcal{D}^{m-1}f(\zeta))dt,\bar{B}^{i}=\int_{0}^{1}\frac{\partial
a^{i}}{\partial\bar{\zeta}}(t\zeta,tf(\zeta),...,t\mathcal{D}^{m-1}f(\zeta))dt,$
$B^{p,j}_{k,l}=\int_{0}^{1}\frac{\partial
a^{i}}{\partial\eta_{p,j}^{k,l}}(t\zeta,tf(\zeta),...,t\mathcal{D}^{m-1}f(\zeta))dt,\bar{B}^{p,j}_{k,l}=\int_{0}^{1}\frac{\partial
a^{i}}{\bar{\partial}\eta_{p,j}^{k,l}}(t\zeta,tf(\zeta),...,t\mathcal{D}^{m-1}f(\zeta))dt,$

It follows by Theorem 3.15

$\displaystyle\|\omega^{i}(f)\|^{(m)}$ $\displaystyle\leq$ $\displaystyle
C\|a^{i}(f)\|.$

Following the above estimate, we have

$\displaystyle\|a^{i}(f)\|\leq C(|a^{i}(0)|+A(R,\gamma)R+\delta(R,\gamma)).$
(17)

##  5 Proof of Theorem 1.1

###  5.1 Case of zero initial values

First, we note from (14) that for each $\gamma$, $\lim_{\mathrm{R}\to
0}\delta(R,\gamma)=0$. Let
$\eta(R,\gamma)=C|a(0)|+CA(R,\gamma)R+C\delta(R,\gamma)$ from (16). Now we
choose $\gamma_{0}$ so large that $\frac{\gamma_{0}}{4}>C|a(0)|$. Then we
choose $R$ sufficiently small that
$CA(R,\gamma)R+C\delta(R,\gamma_{0})\leq\frac{\gamma_{0}}{4}$. As a result, we
have

$\eta(R,\gamma_{0})\leq\frac{\gamma_{0}}{2}.$

If necessary, we choose $R$ further so that

$\delta(R,\gamma_{0})<3/4.$

Let $\psi(z)$ be a homogenous polynomial map of degree $m$ such that
$\|\psi\|\leq\frac{\gamma_{0}}{2}$. Then we can apply Lemma 4.1 to

$u=\psi+\mathbf{\Theta}(u)$

on $A(R,\gamma_{0})$ to get the desired unique solution that vanish up to
order $m-1$ at the origin. $\psi$ be arbitrary, we have obtained infintely
many solutions.

###  5.2 General initial values

To get the general case, we consider a new system, $p(z)=(p^{1},...,p^{n})$
with $\partial^{i}\bar{\partial}^{j}p(0)=c_{i,\bar{j}}$.

$\partial^{\mu}\bar{\partial}^{\nu}\tilde{u}=a(z,\tilde{u}+p(z),\mathcal{D}^{1}\tilde{u}+\mathcal{D}^{1}p(z),...,\mathcal{D}^{m-1}\tilde{u}+\mathcal{D}^{m-1}p(z)).$

This can be written as

$\partial^{\mu}\bar{\partial}^{\nu}\tilde{u}=b(z,\tilde{u},\mathcal{D}^{1}\tilde{u},...,\mathcal{D}^{m-1}\tilde{u}).$

We can solve $\tilde{u}$ as just proved so that $\tilde{u}$ vanishes up to
order $m-1$ at the origin. Then $u=\tilde{u}+p(z)$ solves the original
equation with desired property of derivatives at the origin since
$\partial^{\mu}\bar{\partial}^{\nu}\tilde{u}=\partial^{\mu}\bar{\partial}^{\nu}u$.
The proof is complete.

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Department of Mathematical Sciences

Indiana University - Purdue University Fort Wayne

Fort Wayne, IN 46805-1499, USA.

<EMAIL_ADDRESS>


The homology $H_*(\mathbb{Z}S^D(M))$ of the simplicial set $S^D(M)$ is called the singular homology of $M$, denoted $H_*(M; \mathbb{Z})$. The singular homology $H_*(M, N; \mathbb{Z})$ of a relative space $(M, N)$ is defined by

Under the same notation as in Example <ref>,
the smooth map $\widetilde{\Theta} : (k(W), k(\partial W)) \to (k(S)\cup_{k(f)} k(W), k(S))$ described before Proposition <ref> induces an isomorphism on the singular
homology and then on the singular de Rham cohomology.

In order to prove the result,
we use the excision theorem on the singular homology twice. We define the composite $\overline{r}\circ l :  (X, A) :=(k(S)\cup_{k(f)} k(W), k(S)\cup_{k(f)}k(\partial W \times [0, \e))) \to (k(S)\cup_{k(f)} k(W), k(S))$ of pairs of diffological spaces with maps $l$ and $\overline{r}$ in Theorem <ref> and Lemma <ref>.
Then, the composite
is a homotopy equivalence with the inclusion $i_1$ as the inverse. In fact, we have
$(\overline{r}\circ l)\circ i_1 = id$ and $i_1\circ (\overline{r}\circ l) \simeq id$. The second homotopy is induced by the homotopy
$\overline{H}$ in the proof of Theorem <ref>. Since $l$ is smooth, it follows from Lemma <ref> that
$U:= k(S)\cup_{k(f)}k(\partial W \times [0, \delta))$ is an open subset of $k(S)\cup_{k(f)} k(W)$, where $\delta = \frac{\e}{2}$.
The excision axiom for the singular homology of a diffeological space (<cit.>) enables us to deduce that the inclusion

\[j : (X- U, A- U)=(W', \partial W \times [\delta, \e)) \to (X, A)
\]
is an isomorphism on the singular homology.

Define an increasing smooth function  $r_1 :  [0, \e) \to [0, \e)$ which satisfies the condition that $r_1(t) = \delta$ for
$0 \leq t \leq \delta$ and $r_1(t) = t$ for $\delta' < t <\e$  for some $\delta'$ with
$\delta < \delta' <  \e$. We may have a smooth map $\widetilde{1\times r_1} : W \to W'$ extending the map $1\times r_1 : \partial W\times [0, \e) \to
\partial W\times [\delta, \e)$.  With the smooth map, we obtain a commutative square

\[
\xymatrix@C25pt@R20pt{
(W', \partial W\times [\delta, \e))& (W' - V', \partial W\times [\delta, \e) - V') \ar[l]^-{excision} \\
(W,  \partial W\times [0, \e)) \ar[u]^-{\widetilde{1\times r_1}} & (W - V, \partial W\times [0, \e) - V),  \ar[l]^-{excision}  \ar[u]_{(\widetilde{1\times r_1)}|}
\]

where $V = \partial W \times [0, \delta')$ and $V = \partial W \times [\delta, \delta')$.
It is readily seen that the restriction $(\widetilde{1\times r_1})|$ is the identity map between the manifolds. Therefore, we see that  the map
$\widetilde{1\times r_1}$ induces the isomorphism on the relative singular homology.  Moreover, it follows from the proof of Theorem <ref>
that the restriction
$(\overline{r}\circ l)| : (W,  \partial W\times [0, \e)) \to (W,  \partial W)$ is a smooth homotopy eqivalence with the inclusion  $i_2$ as a homotopy inverse. Fourthermore, we have

\[
\widetilde{\Theta} = (\overline{r} \circ l) \circ i_1 \circ (\widetilde{1\times r_1}) \circ i_2.
\]

It turns out that the map $\widetilde{\Theta}$ induces an isomorphism on the relative singular homology. The result <cit.> implies that
the $\widetilde{\Theta}$ gives rise to an isomorphism on the singular de Rham cohomology; see Remark <ref>.

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This necessitates  ....

This revised version contains a brief review of diffeology.


# The future of human-centric eXplainable Artificial Intelligence (XAI) is not
post-hoc explanations

Vinitra Swamy<EMAIL_ADDRESS>
Jibril Frej<EMAIL_ADDRESS>
Tanja Käser<EMAIL_ADDRESS>
EPFL, Switzerland

###### Abstract

Explainable Artificial Intelligence (XAI) plays a crucial role in enabling
human understanding and trust in deep learning systems. As models get larger,
more ubiquitous, and pervasive in aspects of daily life, explainability is
necessary to minimize adverse effects of model mistakes. Unfortunately,
current approaches in human-centric XAI (e.g. predictive tasks in healthcare,
education, or personalized ads) tend to rely on a single post-hoc explainer,
whereas recent work has identified systematic disagreement between post-hoc
explainers when applied to the same instances of underlying black-box models.
In this paper, we therefore present a call for action to address the
limitations of current state-of-the-art explainers. We propose a shift from
post-hoc explainability to designing interpretable neural network
architectures. We identify five needs of human-centric XAI (real-time,
accurate, actionable, human-interpretable, and consistent) and propose two
schemes for interpretable-by-design neural network workflows (adaptive routing
with InterpretCC and temporal diagnostics with I2MD). We postulate that the
future of human-centric XAI is neither in explaining black-boxes nor in
reverting to traditional, interpretable models, but in neural networks that
are intrinsically interpretable.

##  1 Introduction

The rise of neural networks is accompanied by a severe disadvantage: the lack
of transparency of their decisions. Deep models are often considered black-
boxes, producing highly accurate results while providing little insight into
how they arrive at those conclusions. This disadvantage is especially relevant
in human-centric domains where model decisions have large, real-world impacts
(?, ?).

The goal of eXplainable AI (XAI) is to circumvent this failing by either
producing interpretations for black-box model decisions or making the model’s
decision-making process transparent. As illustrated in Figure 1, model
explanations range from local (single point) to global granularity (entire
sample). Moreover, explainability can be integrated into the modeling pipeline
at three stages:

  1. 1.

Intrinsic explainability: traditional ML models (e.g., decision trees)
explicitly define the decision pathway.

  2. 2.

In-hoc explainability: interpreting the model gradients at inference or
customizing training protocols for additional information; e.g., Grad-CAM uses
backpropagation to highlight important regions of an input image (?).

  3. 3.

Post-hoc explainability: after the decision is made, an explainer is fit on
top of the black-box model to interpret the results.

Figure 1: Explainability can be intrinsic (by design), in-hoc (e.g., gradient
methods), or post-hoc (e.g., LIME, SHAP). Furthermore, the granularity of
model explanations ranges from local (single user, a group of users) to global
(entire sample).

In human-centric domains, researchers and practitioners tend to use either
intrinsically interpretable traditional ML models (?, ?) or apply a single
post-hoc explainer (?, ?). Unfortunately, recent research shows that post-hoc
explanations might be unfaithful to the true model (?), inconsistent (?), or
method-dependent (?, ?, ?). Furthermore, evaluating the quality of the
provided explanations is a challenge, since there is often no ground truth (?,
?).

In this paper, we therefore present a call-to-action to address the
limitations of current state-of-the-art explainability methods. While previous
work (?) has made a strong argument for moving away from black-box models and
using inherent interpretability (i.e. traditional ML models) for impactful
decisions, we suggest exploring strategies to make deep learning approaches
intrinsically interpretable, guaranteeing transparency, robustness, and
trustworthiness. We believe that human-centric domains should profit from both
explainability and the recent advances in state-of-the-art machine learning
methods, including large language models (LLMs).

In the following, we define five needs of human-centric XAI: real-time,
accurate, actionable, human-interpretable, and consistent. We discuss the
limitations of current XAI methods, their inability to meet the requirements
for human-centric XAI, and two ideas towards inherently interpretable deep
learning workflows. We hope this paper will serve as a guideline for achieving
consistency and reliability in human-centric XAI systems.

##  2 Requirements for Human-Centric eXplainable AI

Neural networks have an enormous potential for impacting human life, from
areas like personalized healthcare or educational tutoring to smart farming
and finance. We define human-centric as any application where a human directly
uses model predictions in decision-making. In light of the specific challenges
in human-centric domains (?), we define five requirements that explanations
should fulfill.

  1. 1.

Real-Time: Explanations should be provided in real-time or with minimal delay
to support timely decision-making (in the scale of seconds, not tens of
minutes), e.g., ? (?).

  2. 2.

Accurate explanations with certainty: Explanations need to be accurate,
reflecting the neural network’s decision-making process or at least
accompanied by a level of confidence (?, ?).

  3. 3.

Actionable: Explanations should provide actionable insights, empowering model
deployers to take appropriate actions or make informed interventions (?).

  4. 4.

Human interpretable: Explanations should be understandable to a broad audience
beyond computer scientists (?, ?). We believe that LLMs will be crucial in
improving the understandability of explanations.

  5. 5.

Consistent: Explanations should be consistent across similar instances or
contexts, ensuring reliability and predictability in the decision-making
process. In a time series of interactive predictions, the explanations should
not drastically differ (?).

##  3 Explainers of Today: State-of-the-Art and Limitations

Research and adoption of neural network explainability in human-centric areas
has surged over the last eight years. In-hoc methods like layer relevance
propagation (?) or concept-activation vectors (?) have shown success in
student success prediction (?) or identifying skin conditions (?), but require
specific model architectures or access to model weights. Intrinsic explainers
like neural additive models (?) have shown aptitude for personalized
treatments of COVID-19 patients, but require developer effort and could affect
model performance. Post-hoc approaches are most commonly favored, as there is
no impact on model accuracy and no additional effort required during training.
Local, instance-specific post-hoc techniques such as LIME (?), SHAP (?), or
counterfactuals (?), have been effectively utilized for tasks like predicting
ICU mortality (?), non-invasive ventilation for ALS patients (?), credit risk
(?), or loan repayment (?).

Post-hoc approaches, while popular, are accompanied by weaknesses in real-
world settings. The computational time is often in the tens of minutes; not
real-time enough for users, students, or patients to make a decision based on
the explanation alongside a prediction. In most cases, there is no measurement
of confidence in a generated post-hoc explanation. The actionability and
human-interpretability of the explanation are solely based on the input
format. As human-centric tasks often use tabular or time series data, the
subsequent explanations are often not concise, actionable or interpretable
easily beyond the scope of a data scientist’s knowledge (?). Recent research
on explanation user design has shown that humans across healthcare, law,
finance, education, and e-commerce, among others, prefer hybrid text and
visual explanations (?), a format not easily provided by current post-hoc
libraries. Lastly, the consistency of the explanations is not inherently
measured. Several explainability methods could produce vastly different
explanations with different random seeds or at different time steps (?).

Furthermore, post-hoc explanations are difficult to evaluate. Current metrics
(e.g. saliency, faithfulness) aim to quantify the quality of an explanation in
comparison to expert-generated ground truth (?). However, accurate
explanations need to be true to the model internals, not human perceptions. In
this light, the most trustworthy metrics measure the prediction gap (e.g. PIU,
PGU), removing features that are considered important by the explanation and
seeing how the prediction changes (?). This approach is still time-consuming
and imperfect, as it fails to account for cross-feature dependencies. Recent
literature (?, ?, ?) has examined the results of over $50$ explainability
methods with diverse datasets ranging from criminal justice to healthcare to
education through a variety of metrics (rank agreement, Jenson-Shannon
distance) and demonstrated strong, systematic disagreement across methods.
Validating explanations through human experts can also be difficult:
explanations are subjective, and most can be justified. ? (?), ? (?), and ?
(?) have conducted user studies to examine trust in explainers, measuring data
scientist and human expert preference of explanations. Results indicate that
while humans generally find explanations helpful, no method is recognized as
most trustworthy. As further shown by ? (?), most preferred explanations align
with the prior beliefs of validators.

We anticipate that the state-of-the-art in AI will continue to prefer large,
pretrained deep models over traditional interpretable models for the
foreseeable future; the capabilities and ease-of-use of neural networks
outweigh any black-box drawbacks. Our goal is therefore to identify a way to
use deep learning in an interpretable workflow.

##  4 Intrinsically Interpretable Deep Learning Design

In human-centric applications, there is no margin for error; It is crucial to
prioritize designs that are intrinsically interpretable as opposed to
imperfect approximations of importance. We present two ideas towards
intrinsically interpretable deep learning workflows (InterpretCC and I2MD),
targeting both local and global explanability.

Figure 2: Proposed architecture of adaptive routing with Intepretable
Conditional Computation (InterpretCC, left). A discriminator layer adaptively
selects feature groupings as important, then sends truncated feature sets to
expert sub-networks. Example of global model benchmarks with Interpretable
Iterative Model Diagnostics (I2MD, right). Knowledge graphs are extracted from
a language model at iterative stages of training and compared over time with
diagnostic benchmarks.

###  4.1 InterpretCC: Interpretable Conditional Computation

InterpretCC aims to guarantee an explanation’s accuracy to model behavior with
$100\%$ certainty, while maintaining performance by input point adaptivity.
This approach is inspired by conditional computation in neural networks (?) to
speed up neural network computation. While it is similar to feature grouping
interpretability approaches like ? (?, ?), it differentiates by its focus on
human-specified feature groupings and an adaptive choice of expert
subnetworks. Extensive experiments of InterpretCC have been performed in ?
(?).

The simplest implementation of InterpretCC is a feature gating model that
learns a dynamic feature mask and enforces sparsity regularization on the
input features. For each point, the goal is to choose a minimal predictive
feature set. While it might seem that model accuracy will be compromised by
this approach, the adaptivity has potential to improve performance by reducing
noise. This idea can be expanded to an interpretable mixture-of-experts model
by dynamically activating expert sub-networks (Figure 2). Instead of
restricting the features individually, we can group features together
meaningfully (either by human specification or automated approaches). Expert
sub-networks are trained only using their subset of features and either
activated or ignored for each point.

InterpretCC optimizes the interpretability-accuracy trade-off: easy-to-
classify instances use less features and therefore have high interpretability
while difficult-to-classify points use more features and do not trade accuracy
for interpretability. The advantages of InterpretCC are multifold, as
explanations are 1) real-time (a prediction is provided simultaneously with
the explanation) 2) accurate (the model only uses specific features or feature
groups), 3) consistent (the same learned experts will be activated for each
point), and 4) human interpretable (sparse explanations with human-specified
groupings of features). InterpretCC’s actionability depends on the
actionability of the user-specified features (e.g., in the breast cancer
setting of ? (?) with grouped cell nuclei, explanations are not actionable).

###  4.2 I2MD: Interpretable Iterative Model Diagnostics

Current deep learning performance metrics (accuracy, F1 score) paint a starkly
incomplete global picture of model strengths and weaknesses. I2MD seeks to
address this gap by examining the differential diagnostics of iterative model
snapshots during training to build a detailed understanding of model
abilities. Experimental evaluation of this approach in ? (?, ?) demonstrates
how language models can be interpreted by comparing knowledge graphs extracted
during various stages of training (Figure 2). This enables understanding of
which skills the model learns at what time. During training or fine-tuning,
tailored I2MD datasets can be created to target extracted model weaknesses;
this results in a more performant model earlier in the training process and
closes the loop, integrating XAI results back into the modeling pipeline.
While the idea of iterative temporal diagnostics has been extensively
discussed for usability (?), its role in interpretability has not been
explored yet.

The I2MD approach provides explanations that are consistent (a model snapshot
will extract the same diagnostic explanations every time) and actionable
(granular benchmarking allows developers to correct their models with custom
datasets). However, it is not real-time, as extracting diagnostics from model
snapshots is time-consuming in the training process. I2MD’s human
interpretability depends on the choice and granularity of diagnostics.
Likewise, accuracy depends on the breadth of the diagnostics chosen and does
not have a measure of certainty. A narrow iterative benchmark might not fully
capture model weaknesses, while an overly broad iterative benchmark might not
be easily understandable, illustrating the interpretability-accuracy tradeoff.

##  5 Conclusion

The evolving landscape of machine learning models, characterized by the
ubiquity of LLMs, transformers, and other advanced techniques, necessitates a
departure from the traditional approach of explaining black-box models.
Instead, there is a growing need to incorporate interpretability as an
inherent feature of model design. In this work, we have discussed five needs
of human-centric XAI and have shown that the current state-of-the-art is not
meeting these needs. We have also presented two initial ideas towards
intrinsic interpretable design for neural networks and discussed their
applications towards the five needs of human-centric XAI. As researchers,
model developers, and practitioners, we must move away from imperfect, post-
hoc XAI estimation and towards guaranteed interpretability with less friction
and higher adoption in deep learning workflows.

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