s)\end{pmatrix}\text{ for }(\tau,w)\in\mathbb{B}^{r+2m},s\in\mathbb{B}^{q}.$

So we have
$\|A\|_{\mathscr{C}^{\alpha,\beta}(\mathbb{B}^{r+2m},\mathbb{B}^{q})}=\|\tilde{A}(\lambda\cdot)\|_{\mathscr{C}^{\alpha,\beta}(\lambda_{0}\mathbb{B}^{r+2m},\lambda_{0}\mathbb{B}^{q})}<\delta$
as desired. ∎

Next we show that (4.1.5) and (4.1.6) are associated. Here we endow
$\mathbb{R}^{r}\times\mathbb{C}^{m}$ with two standard coordinate systems
$(\tau,w)=(\tau^{1},\dots,\tau^{r},w^{1},\dots,w^{m})$ and
$(\sigma,\zeta)=(\sigma^{1},\dots,\sigma^{r},\zeta^{1},\dots,\zeta^{m})$, and
we still use $s=(s^{1},\dots,s^{q})$ as the standard coordinate system for
$\mathbb{R}^{q}$.

######  Lemma 4.1.19.

Let $\varepsilon,\delta>0$,
$A:\mathbb{B}^{r+2m}_{\tau,w}\times\mathbb{B}^{q}_{s}\to\mathbb{C}^{(r+m)\times
m}$ and
$H:\mathbb{B}^{r+2m}_{\tau,w}\times\mathbb{B}^{q}_{s}\to\mathbb{B}^{r+2m}_{\sigma,\zeta}\times\mathbb{B}^{q}_{s}$
be as in (the assumptions and consequence) of Proposition 4.1.6. Then,

  1. (i)

If $L_{*}\mathcal{S}$ has a local basis with the form (4.1.3), then $(H\circ
L)_{*}\mathcal{S}$ has a local basis with the form (4.1.4).

  2. (ii)

Suppose $\mathcal{S}$ is involutive and $H^{\prime\prime}$ is a solution to
(4.1.6), then the coefficient matrix $B=[B^{\prime},B^{\prime\prime}]^{\top}$
in (4.1.4) is a solution to (4.1.7).

###### Proof.

Following Convention 4.1.4, by assumption
$H(\tau,w,s)=(\tau,H^{\prime\prime}(\tau,w,s),s)$, we have

$H^{*}(d\sigma,d\zeta,d\bar{\zeta},ds)=(d\tau,dw,d\bar{w},ds)\begin{pmatrix}I&H^{\prime\prime}_{\tau}&\bar{H}^{\prime\prime}_{\tau}\\\
&H^{\prime\prime}_{w}&\bar{H}^{\prime\prime}_{w}\\\
&H^{\prime\prime}_{\bar{w}}&\bar{H}^{\prime\prime}_{\bar{w}}\\\
&H^{\prime\prime}_{s}&\bar{H}^{\prime\prime}_{s}&I\end{pmatrix}\ \Rightarrow\
\begin{pmatrix}\frac{\partial}{\partial\tau}\\\ \frac{\partial}{\partial w}\\\
\frac{\partial}{\partial\bar{w}}\end{pmatrix}=\begin{pmatrix}I&H^{\prime\prime}_{\tau}&\bar{H}^{\prime\prime}_{\tau}\\\
&H^{\prime\prime}_{w}&\bar{H}^{\prime\prime}_{w}\\\
&H^{\prime\prime}_{\bar{w}}&\bar{H}^{\prime\prime}_{\bar{w}}\end{pmatrix}\cdot\begin{pmatrix}H^{*}\frac{\partial}{\partial\sigma}\\\
H^{*}\frac{\partial}{\partial\zeta}\\\
H^{*}\frac{\partial}{\partial\bar{\zeta}}\end{pmatrix}.$ (4.1.37)

Therefore taking pushforward of (4.1.3) under $H$ we have,

$H_{*}X=H_{*}\left(\begin{pmatrix}I&&A^{\prime}\\\
&I&A^{\prime\prime}\end{pmatrix}\begin{pmatrix}I&H^{\prime\prime}_{\tau}&\bar{H}^{\prime\prime}_{\tau}\\\
&H^{\prime\prime}_{w}&\bar{H}^{\prime\prime}_{w}\\\
&H^{\prime\prime}_{\bar{w}}&\bar{H}^{\prime\prime}_{\bar{w}}\end{pmatrix}\right)\begin{pmatrix}\frac{\partial}{\partial\sigma}\\\
\frac{\partial}{\partial\zeta}\\\
\frac{\partial}{\partial\bar{\zeta}}\end{pmatrix}=\left(\begin{pmatrix}I&H^{\prime\prime}_{\tau}+A^{\prime}H^{\prime\prime}_{\bar{w}}&\bar{H}^{\prime\prime}_{\tau}+A^{\prime}\bar{H}^{\prime\prime}_{\bar{w}}\\\
&H^{\prime\prime}_{w}+A^{\prime\prime}\bar{H}^{\prime\prime}_{\bar{w}}&\bar{H}^{\prime\prime}_{w}+A^{\prime\prime}\bar{H}^{\prime\prime}_{\bar{w}}\end{pmatrix}\circ
H^{\mathrm{Inv}}\right)\cdot\begin{pmatrix}\frac{\partial}{\partial\sigma}\\\
\frac{\partial}{\partial\zeta}\\\
\frac{\partial}{\partial\bar{\zeta}}\end{pmatrix}.$

The matrix
$\begin{pmatrix}I&H^{\prime\prime}_{\tau}+A^{\prime}H^{\prime\prime}_{\bar{w}}\\\
&H^{\prime\prime}_{w}+A^{\prime\prime}\bar{H}^{\prime\prime}_{\bar{w}}\end{pmatrix}$
is invertible (see Lemma 4.1.7 (i)), we see that
$H_{*}(L_{*}\mathcal{S})=(H\circ L)_{*}\mathcal{S}$ has a local basis

$H_{*}\left(\begin{pmatrix}I&H^{\prime\prime}_{\tau}+A^{\prime}H^{\prime\prime}_{\bar{w}}\\\
&H^{\prime\prime}_{w}+A^{\prime\prime}\bar{H}^{\prime\prime}_{\bar{w}}\end{pmatrix}^{-1}\begin{pmatrix}X^{\prime}\\\
X^{\prime\prime}\end{pmatrix}\right)=\left(\left(I_{r+m}+\begin{pmatrix}I&H^{\prime\prime}_{\tau}+A^{\prime}H^{\prime\prime}_{\bar{w}}\\\
&H^{\prime\prime}_{w}+A^{\prime\prime}H^{\prime\prime}_{\bar{w}}\end{pmatrix}^{-1}\begin{pmatrix}\bar{H}^{\prime\prime}_{\tau}+A^{\prime}\bar{H}^{\prime\prime}_{\bar{w}}\\\
\bar{H}^{\prime\prime}_{w}+A^{\prime\prime}\bar{H}^{\prime\prime}_{\bar{w}}\end{pmatrix}\right)\circ
H^{\mathrm{Inv}}\right)\cdot\begin{pmatrix}\frac{\partial}{\partial\sigma}\\\
\frac{\partial}{\partial\zeta}\\\
\frac{\partial}{\partial\bar{\zeta}}\end{pmatrix}.$

This gives (4.1.4) and finishes the proof of (i).

Recall the notations $B^{\prime}=(b_{j}^{l})_{\begin{subarray}{c}1\leq j\leq
r;1\leq l\leq m\end{subarray}}$ and
$B^{\prime\prime}=(b_{r+k}^{l})_{\begin{subarray}{c}1\leq k,l\leq
m\end{subarray}}$. By (4.1.4) we have,

$b_{j}^{l}\circ
H=\left(\begin{pmatrix}I&H^{\prime\prime}_{\tau}+A^{\prime}H^{\prime\prime}_{\bar{w}}\\\
&H^{\prime\prime}_{w}+A^{\prime\prime}H^{\prime\prime}_{\bar{w}}\end{pmatrix}^{-1}\\!\\!\begin{pmatrix}\bar{H}^{\prime\prime}_{\tau}+A^{\prime}\bar{H}^{\prime\prime}_{\bar{w}}\\\
\bar{H}^{\prime\prime}_{w}+A^{\prime\prime}\bar{H}^{\prime\prime}_{\bar{w}}\end{pmatrix}\right)_{j}^{l},\quad
1\leq j\leq m+r,\quad 1\leq l\leq m.$ (4.1.38)

By (4.1.37) we have, for $1\leq j\leq r$ and $1\leq k\leq m$,

$\left.\frac{\partial}{\partial\sigma^{j}}\right|_{H(\tau,w,s)}=\left(\begin{pmatrix}I&H^{\prime\prime}_{\tau}&\bar{H}^{\prime\prime}_{\tau}\\\
&H^{\prime\prime}_{w}&\bar{H}^{\prime\prime}_{w}\\\
&H^{\prime\prime}_{\bar{w}}&\bar{H}^{\prime\prime}_{\bar{w}}\end{pmatrix}^{-1}\;\begin{pmatrix}\frac{\partial}{\partial\tau}\\\
\frac{\partial}{\partial w}\\\
\frac{\partial}{\partial\bar{w}}\end{pmatrix}\right)_{j},\
\left.\frac{\partial}{\partial\bar{\zeta}^{k}}\right|_{H(\tau,w,s)}=\left(\begin{pmatrix}I&H^{\prime\prime}_{\tau}&\bar{H}^{\prime\prime}_{\tau}\\\
&H^{\prime\prime}_{w}&\bar{H}^{\prime\prime}_{w}\\\
&H^{\prime\prime}_{\bar{w}}&\bar{H}^{\prime\prime}_{\bar{w}}\end{pmatrix}^{-1}\;\begin{pmatrix}\frac{\partial}{\partial\tau}\\\
\frac{\partial}{\partial w}\\\
\frac{\partial}{\partial\bar{w}}\end{pmatrix}\right)_{r+m+k}.$ (4.1.39)

Plugging (4.1.38) and (4.1.39) into (4.1.6) we get
$\sum_{j=1}^{r}\frac{\partial
b_{j}^{l}}{\partial\sigma^{j}}+\sum_{k=1}^{m}\frac{\partial
b_{r+k}^{l}}{\partial\bar{\zeta}^{k}}=0$ for $1\leq l\leq m$, which is the
last row of (4.1.7).

Since $\mathcal{S}$ is involutive, $(H\circ L)_{*}\mathcal{S}$ is also
involutive, by Lemma 2.3.9 (iii), $T_{1},\dots,T_{r},Z_{1},\dots,Z_{m}$ given
in (4.1.4) are pairwise commutative. Writing out the condition
$[T_{j},T_{j^{\prime}}]=0$ $(1\leq j<j^{\prime}\leq r)$, $[T_{j},Z_{k}]=0$
$(1\leq j\leq r,1\leq k\leq m)$ and $[Z_{k},Z_{k^{\prime}}]=0$ $(1\leq
k<k^{\prime}\leq m)$ in the coordinate components we get the first three rows
of (4.1.7), finishing the proof of (ii). ∎

Now we have the PDE system (4.1.7). We show that it satisfies the assumption
of Proposition 4.1.12.

######  Lemma 4.1.20.

Suppose
$B:\mathbb{B}^{r+2m}_{\sigma,\zeta}\times\mathbb{B}^{q}_{s}\to\mathbb{C}^{(r+m)\times
m}$ solves (4.1.7), then
$f(\sigma,\xi,\eta,s):=B(\sigma,\frac{1}{2}(\xi+i\eta),s)$ is defined on
$\mathbb{B}^{r+2m}_{\sigma,\xi,\eta}\times\mathbb{B}^{q}_{s}$ and $f$ solves a
PDE with the form
$Lf=\Delta_{\sigma,\xi,\eta}\Theta(f,\nabla_{\sigma,\xi,\eta}f)$ where $L$ and
$\Theta$ satisfy the assumptions of Proposition 4.1.12.

###### Proof.

We can write (4.1.7) as the form
$DB=\tilde{\Theta}(B,\nabla_{\sigma,\xi,\eta}B)$ where the coefficients of $D$
and $\Theta$ take value in $\mathbb{C}^{m\times\frac{(r+m)(r+m-1)}{2}+m}$.

By direct computation (see also [Str20b, Lemma B.5]) we have
$D^{*}D=\sum_{j=1}^{r}\frac{\partial^{2}}{\partial\sigma^{j}\partial\sigma^{j}}+\sum_{k=1}^{m}\frac{\partial^{2}}{\partial\zeta^{k}\partial\bar{\zeta}^{k}}=\Delta_{\sigma}+\Box_{\zeta}$,
acting on the components of the vector value functions.

Using the real coordinates $\xi=\operatorname{Re}\zeta$,
$\eta=\operatorname{Im}\zeta$, we see that
$\Box_{\zeta}=\frac{1}{4}\sum_{k=1}^{m}(\frac{\partial^{2}}{\partial\xi^{k}\partial\xi^{k}}+\frac{\partial^{2}}{\partial\eta^{k}\partial\eta^{k}})=\frac{1}{4}\Delta_{\xi,\eta}$.
So for $f(\sigma,\xi,\eta,s)=B(\sigma,\frac{\xi+i\eta}{2},s)$ we have
$(D^{*}DB)(\sigma,\frac{\xi+i\eta}{2},s)=(\Delta_{\sigma}+\Delta_{\xi,\eta})f(\sigma,\xi,\eta,s)$
and
$\nabla_{\xi,\eta}f(\sigma,\xi,\eta,s)=\frac{1}{2}(\nabla_{\xi,\eta}B)(\sigma,\frac{\xi+i\eta}{2},s)$.
Therefore $f$ satisfies
$\Delta_{\sigma,\xi,\eta}f=D^{*}\tilde{\Theta}(f,(\nabla_{\sigma}f,2\nabla_{\xi,\eta}f))$.
Taking $L:=D^{*}$ and
$\Theta(f,\nabla_{\sigma,\xi,\eta}f):=\tilde{\Theta}(f,(\nabla_{\sigma}f,2\nabla_{\xi,\eta}f))$
we finish the proof. ∎

Finally we find the map $G^{\prime\prime}$ in (4.1.2) using Proposition
2.3.45. We work on
$\frac{1}{2}\mathbb{B}^{r+2m}\subset\mathbb{R}^{r+2m}_{\sigma,\xi,\eta}$ and
its “complex extension”
$\frac{1}{2}\mathbb{H}^{r+2m}\subset\mathbb{C}^{r+2m}$.

We endow $\mathbb{C}^{r+2m}$ with standard complex coordinates
$(\boldsymbol{\sigma},\boldsymbol{\xi},\boldsymbol{\eta})=(\boldsymbol{\sigma}^{1},\dots,\boldsymbol{\sigma}^{r},\boldsymbol{\xi}^{1},\dots,\boldsymbol{\xi}^{m},\boldsymbol{\eta}^{1},\dots,\boldsymbol{\eta}^{m})$
in the way that $\sigma=\operatorname{Re}\boldsymbol{\sigma}$,
$\xi=\operatorname{Re}\boldsymbol{\xi}$ and
$\eta=\operatorname{Re}\boldsymbol{\eta}$. We denote
$\frac{\partial}{\partial\boldsymbol{\zeta}}:=\frac{1}{2}(\frac{\partial}{\partial\boldsymbol{\xi}}-i\frac{\partial}{\partial\boldsymbol{\eta}})=[\frac{1}{2}(\frac{\partial}{\partial\boldsymbol{\xi}^{1}}-i\frac{\partial}{\partial\boldsymbol{\eta}^{1}}),\dots,\frac{1}{2}(\frac{\partial}{\partial\boldsymbol{\xi}^{m}}-i\frac{\partial}{\partial\boldsymbol{\eta}^{m}})]^{\top}$.

Note that in this place, we temporarily forget the original complex structure
of $\mathbb{C}^{m}_{\zeta}$. The objects
$\boldsymbol{\zeta}=\boldsymbol{\xi}+i\boldsymbol{\eta}$ and
$\frac{\partial}{\partial\boldsymbol{\zeta}}$ are just conventions.

In this setting, $T_{1},\dots,T_{r},Z_{1},\dots,Z_{m}$ given in (4.1.4) are
complex vector fields defined on
$\frac{1}{2}\mathbb{B}^{r+2m}\times\frac{1}{2}\mathbb{B}^{q}\subset\mathbb{R}^{r+2m}_{\sigma,\xi,\eta}\times\mathbb{R}^{q}_{s}$.

######  Lemma 4.1.21.

Let $\tilde{\beta}\in\\{\gamma,\gamma-:\gamma\in\mathbb{R}_{+}\\}$. Suppose
$T_{1},\dots,T_{r},Z_{1},\dots,Z_{m}$ in (4.1.4) are commutative, and the
coefficient matrix function
$B:\frac{1}{2}\mathbb{B}^{r+2m}_{\sigma,\xi,\eta}\times\frac{1}{2}\mathbb{B}^{q}_{s}\to\mathbb{C}^{(r+m)\times
m}$ has a $(\sigma,\xi,\eta)$-variable holomorphic extension
$\mathfrak{B}\in\mathscr{C}^{\infty}_{\mathrm{loc}}\mathscr{C}^{\tilde{\beta}}(\frac{1}{2}\mathbb{H}^{r+2m},\frac{1}{2}\mathbb{B}^{q};\mathbb{C}^{(r+m)\times
m})$. Then for any
$(\sigma_{0},\xi_{0},\eta_{0},s_{0})\in\frac{1}{2}\mathbb{B}^{r+2m}\times\frac{1}{2}\mathbb{B}^{q}$
there is a neighborhood $U_{1}\times
V_{1}\subset\frac{1}{2}\mathbb{B}^{r+2m}\times\frac{1}{2}\mathbb{B}^{q}$ of
$(\sigma_{0},\xi_{0},\eta_{0},s_{0})$ and a map
$G^{\prime\prime}\in\mathscr{C}^{\infty}\mathscr{C}^{\tilde{\beta}}(U_{1},V_{1};\mathbb{C}^{m})$
such that $G^{\prime\prime}(\sigma_{0},\xi_{0},\eta_{0},s_{0})=0$ and

$(\operatorname{Span}(T_{1}(\cdot,s),\dots,T_{r}(\cdot,s),Z_{1}(\cdot,s),\dots,Z_{m}(\cdot,s)))^{\bot}|_{U_{1}}=\operatorname{Span}(dG^{\prime\prime}(\cdot,s)^{1},\dots,dG^{\prime\prime}(\cdot,s)^{m})\leq\mathbb{C}T^{*}U_{1},\quad\forall\
s\in V_{1}.$

Recall the notation of dual bundle in (2.3.6).

###### Proof.

We define collections of vector fields
$\mathfrak{T}=[\mathfrak{T}_{1},\dots,\mathfrak{T}_{r}]^{\top}$ and
$\mathfrak{Z}=[\mathfrak{Z}_{1},\dots,\mathfrak{Z}_{m}]^{\top}$ as

$\begin{pmatrix}\mathfrak{T}\\\
\mathfrak{Z}\end{pmatrix}:=\begin{pmatrix}I_{r}&&\mathfrak{B}^{\prime}\\\
&I_{m}&\mathfrak{B}^{\prime\prime}\end{pmatrix}\begin{pmatrix}\partial_{\boldsymbol{\sigma}}\\\
\partial_{\boldsymbol{\zeta}}\\\
\partial_{\bar{\boldsymbol{\zeta}}}\end{pmatrix}.$

Here $\mathfrak{B}^{\prime}$ is the upper $r\times m$ block of $\mathfrak{B}$,
and $\mathfrak{B}^{\prime\prime}$ is the lower $m\times m$ block of
$\mathfrak{B}$.

Thus $(T_{1},\dots,T_{r},Z_{1},\dots,Z_{m})$ is the “real domain restriction”
of
$(\mathfrak{T}_{1},\dots,\mathfrak{T}_{r},\mathfrak{Z}_{1},\dots,\mathfrak{Z}_{m})$
in the sense of Proposition 2.3.45. We see that the coefficients of
$[\mathfrak{T}_{j},\mathfrak{T}_{j^{\prime}}]$,
$[\mathfrak{T}_{j},\mathfrak{Z}_{k}]$ and
$[\mathfrak{Z}_{k},\mathfrak{Z}_{k^{\prime}}]$ are the holomorphic extension
those of $[T_{j},T_{j^{\prime}}]$, $[T_{j},Z_{k}]$ and
$[Z_{k},Z_{k^{\prime}}]$.

By assumption $T_{1},\dots,T_{r},Z_{1},\dots,Z_{m}$ are commutative, so
$[T_{j},T_{j^{\prime}}]=[T_{j},Z_{k}]=[Z_{k},Z_{k^{\prime}}]=0$. Since a
holomorphic function in $\mathbb{H}^{r+2m}$ that vanishes in
$\mathbb{B}^{r+2m}$ must be a zero function, we know
$[\mathfrak{T}_{j},\mathfrak{T}_{j^{\prime}}]=[\mathfrak{T}_{j},\mathfrak{Z}_{k}]=[\mathfrak{Z}_{k},\mathfrak{Z}_{k^{\prime}}]=0$
as well. Thus
$\mathfrak{T}_{1},\dots,\mathfrak{T}_{r},\mathfrak{Z}_{1},\dots,\mathfrak{Z}_{m}$
are commutative.

Now the assumptions of Proposition 2.3.45 are satisfied. By Proposition 2.3.45
we can find a neighborhood $\tilde{U}_{1}\times
V_{1}\subseteq\frac{1}{2}\mathbb{H}^{r+2m}\times\frac{1}{2}\mathbb{B}^{q}$ of
$(\sigma_{0},\xi_{0},\eta_{0},s_{0})$ and a map
$\widetilde{G}^{\prime\prime}\in\mathscr{C}^{\infty}\mathscr{C}^{\tilde{\beta}}(\tilde{U}_{1},V_{1};\mathbb{C}^{m})$
such that $\widetilde{G}^{\prime\prime}(\sigma_{0},\xi_{0},\eta_{0},s_{0})=0$
and for $U_{1}:=\tilde{U}_{1}\cap\mathbb{R}^{r+2m}$ we have for each $s\in
V_{1}$,

$(\operatorname{Span}(T_{1}(\cdot,s),\dots,T_{r}(\cdot,s),Z_{1}(\cdot,s),\dots,Z_{m}(\cdot,s)))^{\bot}|_{U_{1}}=\operatorname{Span}(d(\widetilde{G}^{\prime\prime}(\cdot,s)^{1}|_{U_{1}}),\dots,d(\widetilde{G}^{\prime\prime}(\cdot,s)^{m}|_{U_{1}}))\leq\mathbb{C}T^{*}U_{1}.$

Clearly $\widetilde{G}^{\prime\prime}|_{U_{1}\times
V_{1}}\in\mathscr{C}^{\infty}\mathscr{C}^{\tilde{\beta}}(U_{1},V_{1};\mathbb{C}^{m})$
holds. Thus, $G^{\prime\prime}:=\widetilde{G}^{\prime\prime}|_{U_{1}\times
V_{1}}$ is as desired. ∎

####  4.1.6 Proof of Theorem 4.1

In the proof we often consider the local bases of cotangent subbundles.

######  Convention 4.1.22.

Let $M$ and $N$ be two smooth manifolds, let
$\mathcal{T}\leq(\mathbb{C}TM)\times N$ be a continuous subbundle. Let
$\mu=(\mu^{1},\dots,\mu^{n}):M\to\mathbb{R}^{n}$ and
$\nu=(\nu^{1},\dots,\nu^{q}):N\to\mathbb{R}^{q}$ be two smooth coordinate
charts and let $f=(f^{1},\dots,f^{n-r}):M\times N\to\mathbb{C}^{n-r}$ be a
continuous map which is $C^{1}$ in $M$. We say $\mathcal{T}^{\bot}$ is spanned
by $df^{1},\dots,df^{n-r},d\nu^{1},\dots,d\nu^{q}$, denoted by
$\mathcal{T}^{\bot}=\operatorname{Span}(df,d\nu)$, if

$\operatorname{Span}\Big{(}\sum_{j=1}^{n}\frac{\partial
f^{i}}{\partial\mu^{j}}(x,s)d\mu^{j}|_{x}\Big{)}_{i=1}^{r}=\\{\theta\in
T^{*}_{x}M:\langle\theta,X\rangle=0,\ \forall
X\in\mathcal{T}_{(x,s)}\\},\quad\forall\ u\in M,\ s\in N.$ (4.1.40)

######  Remark 4.1.23.

When $\tilde{\beta}>1$ and when $f\in\mathscr{C}^{\tilde{\beta}}$, (4.1.40)
coincides with the classical notion
$\mathcal{T}^{\bot}=\operatorname{Span}(df,d\nu)$ since
$\operatorname{Span}(df,d\nu)=\operatorname{Span}(\frac{\partial
f}{\partial\mu}d\mu+\frac{\partial
f}{\partial\nu}d\nu,d\nu)=\operatorname{Span}(\frac{\partial
f}{\partial\mu}d\mu,d\nu)$.

The convention is used specifically for the case $\tilde{\beta}\leq 1$, since
when $f\in\mathscr{C}^{\alpha+1,\tilde{\beta}}$, the collection of
differentials $df$ is no longer pointwise defined, while $\frac{\partial
f}{\partial\mu}\cdot d\mu\equiv df\pmod{d\nu}$ and the left hand side is still
pointwise.

We now begin the proof. On the space
$\mathbb{R}^{r}_{\sigma}\times\mathbb{C}^{m}_{\zeta}$, we write
$\xi=\operatorname{Re}\zeta$ and $\eta=\operatorname{Im}\zeta$. Recall the
notation $\beta^{\sim\alpha}$ and $\beta^{\wedge\alpha}$ in (4.1.1).

Proof of Theorem 4.1. By Lemma 4.1.20 and Proposition 4.1.12, along with a
scaling, we can find a $\varepsilon_{0}=\varepsilon_{0}(r,m,q,\alpha)>0$ such
that,

  1. (B.1)

If
$B\in\mathscr{C}^{\alpha}L^{\infty}\cap\mathscr{C}^{-1}\mathscr{C}^{\beta^{\sim\alpha}}(\mathbb{B}^{r+2m}_{\sigma,\eta,\xi},\mathbb{B}^{q}_{s};\mathbb{C}^{(r+m)\times
m})$ satisfies (4.1.7) with

$\|B\|_{\mathscr{C}^{\alpha}L^{\infty}(\mathbb{B}^{r+2m},\mathbb{B}^{q})}<\varepsilon_{0},$
(4.1.41)

then $B(\sigma,\frac{\xi+i\eta}{2},s)$ has a $(\sigma,\xi,\eta)$-holomorphic
extension
$\mathfrak{B}(\boldsymbol{\sigma},\frac{1}{2}\boldsymbol{\xi},\frac{1}{2}\boldsymbol{\eta},s)\in\mathscr{C}^{\infty}_{\mathrm{loc}}\mathscr{C}^{\beta^{\sim\alpha}}(\mathbb{H}^{r+2m}_{\boldsymbol{\sigma},\boldsymbol{\xi},\boldsymbol{\eta}},\mathbb{B}^{q}_{s};\mathbb{C}^{(r+m)\times
m})$.

For such $\varepsilon_{0}>0$, we can find a $\delta_{0}>0$ such that the
results of Proposition 4.1.6 are satisfied with $\varepsilon=\varepsilon_{0}$,
$\delta=\delta_{0}$.

For such $\delta_{0}>0$, by Lemma 4.1.18 there is a smooth coordinate chart
$L:=((L^{\prime},L^{\prime\prime}),L^{\prime\prime\prime}):U_{0}\times
V_{0}\subseteq\mathfrak{M}\times\mathfrak{N}\xrightarrow{\sim}\mathbb{B}^{r+2m}_{\tau,w}\times\mathbb{B}^{q}_{s}$
near $(u_{0},v_{0})$ such that,

  1. (L.1)

$(L^{\prime},L^{\prime\prime}):U_{0}\subseteq\mathfrak{M}\xrightarrow{\sim}\mathbb{B}^{r+2m}_{\tau,w}$
and
$L^{\prime\prime\prime}:V_{0}\subseteq\mathfrak{N}\xrightarrow{\sim}\mathbb{B}^{q}_{s}$
are both smooth coordinate charts, and $L(u_{0},v_{0})=(0,0)$.

  2. (L.2)

$L_{*}\mathcal{S}$ has a $\mathscr{C}^{\alpha,\beta}_{(\tau,w),s}$-local basis
with form (4.1.3), such that:

     * •

When $\beta\in\mathbb{R}_{+}$, (4.1.33) is satisfied with $\delta=\delta_{0}$,
i.e.
$\|A\|_{\mathscr{C}^{\alpha,\beta}(\mathbb{B}^{r+2m},\mathbb{B}^{q};\mathbb{C}^{(r+m)\times
m})}<\delta_{0}$.

     * •

When $\beta\in\mathbb{R}_{+}^{-}(=\\{\gamma-:\gamma\in\mathbb{R}_{+}\\})$,
$\|A\|_{\mathscr{C}^{\alpha}L^{\infty}(\mathbb{B}^{r+2m},\mathbb{B}^{q};\mathbb{C}^{(r+m)\times
m})}<\delta_{0}$.

By Corollary 4.1.8 if $\beta\in\mathbb{R}_{+}$ and $\alpha\neq 1$, and
Proposition 4.1.9 if $\beta\in\mathbb{R}_{+}^{-}$ or $\alpha=1$, we can find a
$\mathscr{C}^{\alpha+1,\beta^{\sim\alpha}}$-map
$H:\mathbb{B}^{r+2m}_{\tau,w}\times\mathbb{B}^{q}_{s}\to\mathbb{R}^{r}_{\sigma}\times\mathbb{C}^{m}_{\zeta}\times\mathbb{B}^{q}_{s}$
that has the form $H(\tau,w,s)=(\tau,H^{\prime\prime}(\tau,w,s),s)$ such that

  1. (H.1)

$H:\mathbb{B}^{r+2m}_{\tau,w}\times\mathbb{B}^{q}_{s}\to\mathbb{B}^{r+2m}_{\sigma,\zeta}\times\mathbb{B}^{q}_{s}$
is homeomorphism, and
$H,H^{\mathrm{Inv}}\in\mathscr{C}^{\alpha+1,\beta}(\mathbb{B}^{r+2m},\mathbb{B}^{q};\mathbb{R}^{r}\times\mathbb{C}^{m})$.

  2. (H.2)

$H(\frac{1}{4}\mathbb{B}^{r+2m}\times\frac{1}{4}\mathbb{B}^{q})\subseteq\frac{1}{2}\mathbb{B}^{r+2m}\times\frac{1}{2}\mathbb{B}^{q}$.

  3. (H.3)

$H^{\prime\prime}\in\mathscr{C}^{\alpha+1,\beta^{\sim\alpha}}(\mathbb{B}^{r+2m},\mathbb{B}^{q};\mathbb{C}^{m})$
and
$\nabla_{\tau,w}H^{\prime\prime}\in\mathscr{C}^{\alpha,\beta^{\wedge\alpha}}(\mathbb{B}^{r+2m},\mathbb{B}^{q};\mathbb{C}^{(r+2m)\times
m})$.

  4. (H.4)

$H^{\prime\prime}$ solves (4.1.6).

  5. (H.5)

When $\beta\in\mathbb{R}_{+}$ and $\alpha\neq 1$, (4.1.17) is satisfied with
$\varepsilon=\varepsilon_{0}$.

When $\beta\in\mathbb{R}_{+}^{-}$ or $\alpha=1$, (4.1.19) is satisfied with
$\varepsilon=\varepsilon_{0}$.

Here (H.1) shows that $(H\circ L)_{*}\mathcal{S}$ is defined on
$\mathbb{B}^{r+2m}_{\sigma,\zeta}\times\mathbb{B}^{q}_{s}$.

By Lemma 4.1.19 (ii), (H.4) implies that $B$ solves (4.1.7). By comparing
(4.1.9) and (4.1.4) we see that $B=\Lambda[A,H^{\prime\prime}]\circ
H^{\mathrm{Inv}}$, so (H.5) implies (4.1.41) with
$\varepsilon=\varepsilon_{0}$. Thus we get the result (B.1): a matrix-valued
function
$\mathfrak{B}\in\mathscr{C}^{\infty}_{\mathrm{loc}}\mathscr{C}^{\beta^{\sim\alpha}}(\frac{1}{2}\mathbb{H}^{r+2m},\frac{1}{2}\mathbb{B}^{q};\mathbb{C}^{(r+m)\times
m})$.

Applying Lemma 4.1.21 on $H\circ
L(u_{0},v_{0})=H(0,0)\in\frac{1}{2}\mathbb{B}^{r+2m}\times\frac{1}{2}\mathbb{B}^{q}$,
we can find a neighborhood $U_{1}\times
V_{1}\subseteq\frac{1}{2}\mathbb{B}^{r+2m}\times\frac{1}{2}\mathbb{B}^{q}$ of
$H(0,0)$ and a $G^{\prime\prime}:U_{1}\times V_{1}\to\mathbb{C}^{m}$ such that

  1. (G.1)

$G^{\prime\prime}\in\mathscr{C}^{\infty}\mathscr{C}^{\beta^{\sim\alpha}}(U_{1},V_{1};\mathbb{C}^{m})$
and $G^{\prime\prime}(H(0,0))=0$. In particular
$G^{\prime\prime}\in\mathscr{C}^{\infty,\beta^{\sim\alpha}}_{(\sigma,\zeta),s}$
and
$\nabla_{\sigma,\zeta}G^{\prime\prime}\in\mathscr{C}^{\infty,\beta^{\sim\alpha}}_{(\sigma,\zeta),s}$.

  2. (G.2)

For each $s\in V_{1}$,
$\operatorname{Span}d(G^{\prime\prime}(\cdot,s))=\operatorname{Span}(T(\cdot,s),Z(\cdot,s))|_{U_{1}}^{\bot}=(H\circ
L)_{*}\mathcal{S}|_{U_{1}\times\\{s\\}}^{\bot}$. Equivalently we have
$\operatorname{Span}(dG^{\prime\prime},ds)=(H\circ
L)_{*}\mathcal{S}|_{U_{1}\times V_{1}}^{\bot}$ in the sense of Convention
4.1.22.

Therefore, $(\operatorname{Span}d(G^{\prime\prime}\circ
H),ds)|_{H^{\mathrm{Inv}}(U_{1}\times
V_{1})}=L_{*}\mathcal{S}|_{H^{\mathrm{Inv}}(U_{1}\times V_{1})}$ in the sense
of Convention 4.1.22, which means

$\operatorname{Span}(d(G^{\prime\prime}\circ H\circ
L),dL^{\prime\prime\prime})|_{(H\circ L)^{\mathrm{Inv}}(U_{1}\times
V_{1})}=\mathcal{S}|_{(H\circ L)^{\mathrm{Inv}}(U_{1}\times
V_{1})}^{\bot}\text{ in the sense of Convention
\ref{Conv::EllipticPara::CoSpan}}.$ (4.1.42)

Next, we claim that

  1. (L.3)

$\big{(}dL^{\prime},d(\operatorname{Re}G^{\prime\prime}\circ H\circ
L(\cdot,v_{0})),d(\operatorname{Im}G^{\prime\prime}\circ H\circ
L(\cdot,v_{0}))\big{)}$ is a collection of $(r+2m)$-real differentials that
are linearly independent at $u_{0}\in\mathfrak{M}$.

Indeed, we have $\operatorname{Span}(d(\overline{G^{\prime\prime}}\circ
H),ds)=L_{*}\bar{\mathcal{S}}^{\bot}$ in the sense of Convention 4.1.22, which
means for each $s\in V_{1}$,

$\operatorname{Span}(d(\operatorname{Re}G^{\prime\prime}\circ
H(\cdot,s)),d(\operatorname{Im}G^{\prime\prime}\circ
H(\cdot,s))^{\bot}=\operatorname{Span}(d(G^{\prime\prime}\circ
H(\cdot,s)),d(\overline{G^{\prime\prime}}\circ
H(\cdot,s)))^{\bot}=L_{*}(\mathcal{S}\cap\bar{\mathcal{S}})|_{H^{\mathrm{Inv}}(U_{1}\times\\{s\\})},$

has rank $r=\dim\mathfrak{M}-(r+2m)$. Therefore,
$(d(\operatorname{Re}G^{\prime\prime}\circ
H(\cdot,s),d(\operatorname{Im}G^{\prime\prime}\circ H(\cdot,s))$ is a linearly
independent collection in the domain for all $s\in V_{1}$.

On the other hand, by (4.1.3) we have
$L_{*}\mathcal{S}=\operatorname{Span}(\frac{\partial}{\partial\tau}+A^{\prime}\frac{\partial}{\partial
w},\frac{\partial}{\partial
w}+A^{\prime\prime}\frac{\partial}{\partial\bar{w}})$, so
$L_{*}(\mathcal{S}\cap\bar{\mathcal{S}})=\operatorname{Span}(\frac{\partial}{\partial\tau}+\frac{1}{2}A^{\prime}\frac{\partial}{\partial
w}+\frac{1}{2}\bar{A}^{\prime}\frac{\partial}{\partial\bar{w}}$, which means
$L_{*}(\mathcal{S}\cap\bar{\mathcal{S}})^{\bot}\cap\operatorname{Span}d\tau=\\{0\\}$.
We conclude that $(d\tau,d(\operatorname{Re}G^{\prime\prime}\circ
H(\cdot,s)),d(\operatorname{Im}G^{\prime\prime}\circ H(\cdot,s))$ is a
linearly independent collection in the domain for all $s\in V_{1}$. Since
$dL^{\prime}=L^{*}d\tau$ and $dL^{\prime\prime\prime}=L^{*}ds$, taking
compositions with $L$ we obtain (L.3).

Now endow $\mathbb{R}^{r}\times\mathbb{C}^{m}\times\mathbb{R}^{q}$ with
standard coordinates
$(t,z,s)=(t^{1},\dots,t^{r},z^{1},\dots,z^{m},s^{1},\dots,s^{q})$. We take

$F=(F^{\prime},F^{\prime\prime},F^{\prime\prime\prime}):=(L^{\prime},\overline{G^{\prime\prime}}\circ
H\circ L,L^{\prime\prime\prime}):(H\circ L)^{\mathrm{Inv}}(U_{1}\times
V_{1})\subseteq\mathfrak{M}\times\mathfrak{N}\to\mathbb{R}^{r}_{t}\times\mathbb{C}^{m}_{z}\times\mathbb{R}^{q}_{s}.$
(4.1.43)

By (L.1), (H.3), (G.1) and Lemma 2.2.38 (i) we have

  1. (F.1)

$F^{\prime}(u_{0})=0$, $F^{\prime\prime}(u_{0},v_{0})=0$,
$F^{\prime\prime\prime}(v_{0})=0$. And for every product neighborhood $U\times
V\subseteq(H\circ L)^{\mathrm{Inv}}(U_{1}\times V_{1})$ of $(u_{0},v_{0})$, we
have $F^{\prime}\in\mathscr{C}^{\infty}_{\mathrm{loc}}(U;\mathbb{R}^{r})$,
$F^{\prime\prime}\in\mathscr{C}^{\alpha+1,\beta^{\sim\alpha}}_{\mathrm{loc}}(U,V;\mathbb{C}^{m})$
and
$F^{\prime\prime\prime}\in\mathscr{C}^{\infty}_{\mathrm{loc}}(V;\mathbb{R}^{q})$.

In particular (1) is satisfied, since $L^{\prime\prime\prime}$ is already a
smooth coordinate chart.

By (F.1)
$(F^{\prime},F^{\prime\prime})\in\mathscr{C}^{\alpha+1,\beta^{\sim\alpha}}(U,V;\mathbb{R}^{r}\times\mathbb{C}^{m})$
for every $U\times V\subseteq(H\circ L)^{\mathrm{Inv}}(U_{1}\times V_{1})$. By
(L.3), $(dF^{\prime},dF^{\prime\prime}(\cdot,v_{0}))$ has full rank $r+2m$ in
the domain. Therefore, by Lemma 2.2.38 (iii), we can find a small enough
neighborhood $U\times V\subseteq(H\circ L)^{\mathrm{Inv}}(U_{1}\times V_{1})$
of $(u_{0},v_{0})$, such that $F:U\times V\to F(U\times
V)\subseteq\mathbb{R}^{r}_{t}\times\mathbb{C}^{m}_{z}\times\mathbb{R}^{q}_{s}$
is homeomorphism, and

  1. (F.2)

$F^{\mathrm{Inv}}\in\mathscr{C}^{\alpha+1,\beta}_{(t,z),s}(\Omega^{\prime}\times\Omega^{\prime\prime},\Omega^{\prime\prime\prime};\mathfrak{M}\times\mathfrak{N})$
whenever
$\Omega^{\prime}_{t}\times\Omega^{\prime\prime}_{z}\times\Omega^{\prime\prime\prime}_{s}\subseteq
F(U\times V)$.

Define $\Phi:=F^{\mathrm{Inv}}:\Omega\to U\times V$ where
$\Omega=\Omega^{\prime}_{s}\times\Omega^{\prime\prime}_{z}\times\Omega^{\prime\prime\prime}_{s}$
is an arbitrary product neighborhood of $(0,0,0)$ in the assumption of Theorem
4.1. We are going to show that $F$ and $\Phi$ are as desired.

Firstly, by (F.1), we get
$\Phi^{\prime\prime}=(L^{\prime\prime\prime})^{\mathrm{Inv}}=(F^{\prime\prime\prime})^{\mathrm{Inv}}$
is a smooth parameterization, which is the result (5).

By (F.1) and (F.2) we know $F\in\mathscr{C}^{\beta}$ and
$\Phi\in\mathscr{C}^{\beta}$. Thus (9) holds and $F$, $\Phi$ are both
homeomorphic to their images respectively. Since
$F(\cdot,v)\in\mathscr{C}^{\alpha+1}$,
$\Phi(\cdot,s)\in\mathscr{C}^{\alpha+1}$ for each $v$ and $s$, we know
$(F^{\prime},F^{\prime\prime}(\cdot,v))$ is a $\mathscr{C}^{\alpha+1}$-chart
and $\Phi^{\prime}(\cdot,s)$ is a $\mathscr{C}^{\alpha+1}$-parameterization,
which gives (2) and (6).

To prove (3) and (7), we use
$\widehat{U}:=(L^{\prime},L^{\prime\prime})(U)\subseteq\mathbb{R}^{r}_{\tau}\times\mathbb{C}^{m}_{w}$,
$\widehat{V}:=L^{\prime\prime\prime}(V)\subseteq\mathbb{R}^{q}_{s}$,
$\widehat{F}^{\prime\prime}:=F^{\prime\prime}\circ L^{\mathrm{Inv}}$ and
$\widehat{\Phi}:=L\circ\Phi$. Since $(L^{\prime},L^{\prime\prime})$ and
$L^{\prime\prime\prime}$ are smooth charts, it is equivalent but more
convenient to work on the chart
$\widehat{F}:\widehat{U}\times\widehat{V}\to\mathbb{R}^{r}_{t}\times\mathbb{C}^{m}_{z}\times\mathbb{R}^{q}_{s}$
and the parameterization
$\widehat{\Phi}:\Omega\to\widehat{U}\times\widehat{V}$: it suffices to show
the following

$\begin{gathered}\widehat{F}^{\prime\prime}\in\mathscr{C}^{\alpha+1,\beta^{\sim\alpha}}_{(\tau,w),s}(\widehat{U},\widehat{V};\mathbb{C}^{m}),\quad\nabla_{\tau,w}\widehat{F}^{\prime\prime}\in\mathscr{C}^{\alpha,\beta^{\wedge\alpha}}(\widehat{U},\widehat{V};\mathbb{C}^{(r+2m)\times
m}),\\\
\widehat{\Phi}\in\mathscr{C}^{\alpha+1,\beta^{\sim\alpha}}_{(t,z),s}(\Omega^{\prime}\times\Omega^{\prime\prime},\Omega^{\prime\prime\prime};\widehat{U}\times\widehat{V}),\quad\nabla_{t,z}\widehat{\Phi}\in\mathscr{C}^{\alpha,\beta^{\wedge\alpha}}(\Omega^{\prime}\times\Omega^{\prime\prime},\Omega^{\prime\prime\prime};\mathbb{C}^{(r+2m)\times(r+2m)}).\end{gathered}$
(4.1.44)

Here for every vector field
$X\in\mathscr{C}^{\infty}_{\mathrm{loc}}(\mathfrak{M};T\mathfrak{M})$,
$(L^{\prime},L^{\prime\prime})_{*}X$ must be the smooth linear combinations of
$\frac{\partial}{\partial\tau},\frac{\partial}{\partial
w},\frac{\partial}{\partial\bar{w}}$ on $\widehat{U}$. Thus
$\nabla_{\tau,w}\widehat{F}^{\prime\prime}\in\mathscr{C}^{\alpha,\beta^{\wedge\alpha}}_{(\tau,w),s}$
implies
$XF^{\prime\prime}\in\mathscr{C}^{\alpha,\beta^{\wedge\alpha}}_{\mathrm{loc}}$.

Proof of (4.1.44): By (4.1.43) we have
$\widehat{F}^{\prime\prime}(\tau,w,s)=\overline{G^{\prime\prime}}(H(\tau,w,s))=\overline{G^{\prime\prime}}(\tau,H^{\prime\prime}(\tau,w,s),s)$.
By (H.3) and (G.1) we have
$G^{\prime\prime}\in\mathscr{C}^{\infty,\beta^{\sim\alpha}}_{(\sigma,\zeta),s}$,
$H\in\mathscr{C}^{\alpha+1,\beta^{\sim\alpha}}_{(\tau,w),s}$,
$\nabla_{\sigma,\zeta}G^{\prime\prime}\in\mathscr{C}^{\infty,\beta^{\sim\alpha}}_{(\sigma,\zeta),s}$
and
$\nabla_{\tau,w}H\in\mathscr{C}^{\alpha,\beta^{\wedge\alpha}}_{(\tau,w),s}$.
Therefore, by Lemma 2.2.38 (i) we get
$\widehat{F}^{\prime\prime}\in\mathscr{C}^{\alpha+1,\beta^{\sim\alpha}}_{(\tau,w),s}$
and
$\nabla_{\tau,w}\widehat{F}^{\prime\prime}=((\nabla_{\sigma,\zeta}\overline{G^{\prime\prime}})\circ
H)\cdot\nabla_{\tau,w}H\in\mathscr{C}^{\alpha+1,\beta^{\sim\alpha}}_{(\tau,w),s}\cdot\mathscr{C}^{\alpha,\beta^{\wedge\alpha}}_{(\tau,w),s}\subseteq\mathscr{C}^{\alpha,\beta^{\wedge\alpha}}_{(\tau,w),s}$.

Since
$F\circ(L^{\mathrm{Inv}})(\tau,w,s)=(\tau,\widehat{F}^{\prime\prime}(\tau,w,s),s)$,
we see that $\widehat{\Phi}$ has the form

$\widehat{\Phi}(t,z,s)=(t,\widehat{\Psi}(t,z,s),s),\quad\text{where
}\widehat{\Psi}:\Omega^{\prime}_{t}\times\Omega^{\prime\prime}_{z}\times\Omega^{\prime\prime\prime}_{s}\to\mathbb{C}^{m}_{w},\quad\widehat{\Psi}(t,z,s):=\widehat{F}^{\prime\prime}(t,\cdot,s)^{\mathrm{Inv}}(z).$

Since
$\widehat{F}^{\prime\prime}\in\mathscr{C}^{\alpha+1,\beta^{\sim\alpha}}_{(\tau,w),s}$,
applying Lemma 2.2.38 (iii) to the map
$[(\tau,w),s)\mapsto(\tau,\widehat{F}^{\prime\prime}(\tau,w,s))]$ we get
$\widehat{\Psi}\in\mathscr{C}^{\alpha+1,\beta^{\sim\alpha}}_{(t,z),s}$.

Using the chain rules on
$z=\widehat{F}^{\prime\prime}(\widehat{\Phi}(t,z,s))=\widehat{F}^{\prime\prime}(t,\widehat{\Psi}(t,z,s),s)$
we have (recall $\nabla_{w}=[\frac{\partial}{\partial
w},\frac{\partial}{\partial\bar{w}}]^{\top}$),

$\displaystyle I_{2m}$
$\displaystyle=((\nabla_{w}\widehat{F}^{\prime\prime})\circ\widehat{\Phi})\cdot\nabla_{z}\widehat{\Psi},$
$\displaystyle 0_{r\times m}$
$\displaystyle=\nabla_{\tau}\widehat{F}^{\prime\prime}\circ\widehat{\Phi}+((\nabla_{w}\widehat{F}^{\prime\prime})\circ\widehat{\Phi})\cdot\nabla_{t}\widehat{\Psi};$
$\displaystyle\Longrightarrow\quad\nabla_{z}\widehat{\Psi}$
$\displaystyle=(\nabla_{w}\widehat{F}^{\prime\prime})^{-1}\circ\widehat{\Phi},$
$\displaystyle\nabla_{t}\widehat{\Psi}$
$\displaystyle=((\nabla_{w}\widehat{F}^{\prime\prime})^{-1}\cdot\nabla_{\tau}\widehat{F}^{\prime\prime})\circ\widehat{\Phi}.$

Since
$\nabla_{\tau,w}\widehat{F}^{\prime\prime}\in\mathscr{C}^{\alpha,\beta^{\wedge\alpha}}_{(\tau,w),s}$,
by Lemma 2.2.38 (ii) we have
$(\nabla_{w}\widehat{F}^{\prime\prime})^{-1}\in\mathscr{C}^{\alpha,\beta^{\wedge\alpha}}_{(\tau,w),s}$
thus
$(\nabla_{w}\widehat{F}^{\prime\prime})^{-1}\cdot\nabla_{\tau}\widehat{F}^{\prime\prime}\in\mathscr{C}^{\alpha,\beta^{\wedge\alpha}}_{(\tau,w),s}$.
Applying Lemma 2.2.38 (i) with
$\widehat{\Phi}\in\mathscr{C}^{\alpha+1,\beta^{\sim\alpha}}_{(t,z),s}$, we get
$\nabla_{t,z}\widehat{\Psi}\in\mathscr{C}^{\alpha,\beta^{\wedge\alpha}}_{(t,z),s}$.
Therefore
$\nabla_{t,z}\widehat{\Phi}\in\mathscr{C}^{\alpha,\beta^{\wedge\alpha}}_{(t,z),s}$
finishing the proof of (4.1.44) and hence the proof of (3) and (7).

Now by (F.1) and (7) we see that
$F\in\mathscr{C}^{\alpha+1,\beta^{\sim\alpha}}_{u,v}$ and
$\nabla_{t,z}\Phi\in\mathscr{C}^{\alpha,\beta^{\wedge\alpha}}_{(t,z),s}$. By
Lemma 2.2.38 (i) we see that $\frac{\partial\Phi}{\partial t^{j}}\circ
F\in\mathscr{C}^{\alpha,\beta^{\wedge\alpha}}_{\mathrm{loc}}(U,V;T\mathfrak{M})$
and $\frac{\partial\Phi}{\partial z^{k}}\circ
F\in\mathscr{C}^{\alpha,\beta^{\wedge\alpha}}_{\mathrm{loc}}(U,V;\mathbb{C}T\mathfrak{M})$
for $1\leq j\leq r$ and $1\leq k\leq m$. Thus $F^{*}\frac{\partial}{\partial
t^{j}}=\frac{\partial\Phi}{\partial t^{j}}\circ F$,
$F^{*}\frac{\partial}{\partial z^{k}}=\frac{\partial\Phi}{\partial z^{k}}\circ
F$ are both well-defined vector fields and have regularity
$\mathscr{C}^{\alpha}\cap\mathscr{C}^{\beta^{\wedge\alpha}}=\mathscr{C}^{\min(\alpha,\beta^{\wedge\alpha})}$.

By definition, $dF^{\prime}=F^{*}dt$, $dF^{\prime\prime\prime}=F^{*}ds$,
$dF^{\prime\prime}=F^{*}dz$ and $d\overline{F^{\prime\prime}}=F^{*}d\bar{z}$.
So by (4.1.42) and (4.1.43), in the sense of Convention 4.1.22,

$\textstyle\operatorname{Span}(F^{*}\frac{\partial}{\partial
t},F^{*}\frac{\partial}{\partial
z})=\operatorname{Span}(F^{*}d\bar{z},F^{*}ds)^{\bot}=\operatorname{Span}(d\overline{F^{\prime\prime}},dF^{\prime\prime\prime})^{\bot}=\mathcal{S}|_{U\times
V}.$

This proves (4). Immediately (8) follows. Now the whole proof is complete.∎

###  4.2 Improving the Regularity of Rough 1-forms

Let $\alpha>0$ and $\beta\in[\alpha,\alpha+1]$. Suppose
$\lambda^{1},\ldots,\lambda^{n}$ are $\mathscr{C}^{\alpha}$ $1$-forms on an
open set $U\subseteq\mathbb{R}^{n}$ which span the cotangent space at every
point in $U$. If we know that
$d\lambda^{j}\in\mathscr{C}^{\beta-1}_{\mathrm{loc}}$ for each $j$, it is not
necessarily true that $\lambda^{j}\in\mathscr{C}^{\beta}_{\mathrm{loc}}$.
However, one can always change coordinates so that the forms are in
$\mathscr{C}^{\beta}_{\mathrm{loc}}$ (see also Corollary 4.2.13).

The next result is a special case of this idea, where we present an intital
setting where we may find a $\mathscr{C}^{\alpha+1}$-diffemorphism such that
$F_{*}\lambda^{j}\in\mathscr{C}^{\beta}_{\mathrm{loc}}$.

######  Theorem 4.2 (Improving regularity of rough 1-forms).

Let $\alpha>0$ and $\beta\in[\alpha,\alpha+1]$. Let $x=(x^{1},\dots,x^{n})$
and $y=(y^{1},\dots,y^{n})$ be two coordinate systems for $\mathbb{R}^{n}$.
There exists $c=c(n,\alpha,\beta)>0$ such that the following holds.

Suppose
$\lambda^{i}\in\mathscr{C}^{\alpha}(\mathbb{B}^{n};T^{*}\mathbb{B}^{n})$,
$i=1,\dots,n$ are 1-forms on $\mathbb{B}^{n}$ such that
$\operatorname{supp}(\lambda^{i}-dx^{i})\subsetneq\frac{1}{2}\mathbb{B}^{n}$
for each $i$, and

$\sum_{i=1}^{n}(\|\lambda^{i}-dx^{i}\|_{\mathscr{C}^{\alpha}(\mathbb{B}^{n};T^{*}\mathbb{B}^{n})}+\|d\lambda^{i}\|_{\mathscr{C}^{\beta-1}(\mathbb{B}^{n};\wedge^{2}T^{*}\mathbb{B}^{n})})\leq
c.$ (4.2.1)

Then, there exists a $\mathscr{C}^{\alpha+1}$-diffeomorphism
$F:\mathbb{B}^{n}_{x}\xrightarrow{\sim}\mathbb{B}^{n}_{y}$, such that
$B^{n}(F(0),\frac{1}{6})\subseteq
F(\frac{1}{3}\mathbb{B}^{n})\cap\frac{3}{4}\mathbb{B}^{n}$,
$F_{*}\lambda^{i}\in\mathscr{C}^{\beta}(\mathbb{B}^{n};T^{*}\mathbb{B}^{n})$
for $i=1,\dots,n$, and moreover

$\displaystyle\|F-\operatorname{id}\|_{\mathscr{C}^{\alpha+1}(\mathbb{B}^{n};\mathbb{R}^{n})}+\sum_{i=1}^{n}\|F_{*}\lambda^{i}-dy^{i}\|_{\mathscr{C}^{\beta}\left(\mathbb{B}^{n};T^{*}\mathbb{B}^{n}\right)}\leq
c^{-1}\sum_{i=1}^{n}\left(\|\lambda^{i}-dx^{i}\|_{\mathscr{C}^{\alpha}\left(\mathbb{B}^{n};T^{*}\mathbb{B}^{n}\right)}+\|d\lambda^{i}\|_{\mathscr{C}^{\beta-1}\left(\mathbb{B}^{n};\wedge^{2}T^{*}\mathbb{B}^{n}\right)}\right).$
(4.2.2)

######  Remark 4.2.1.

Notice that if $\tilde{c}(n,\alpha,\beta)>0$ is a small constant that
satisfies the results in Theorem 4.2 with $c=\tilde{c}$, then any constant
$c<\tilde{c}$ would satisfies the results as well. Therefore by shrinking $c$
if possible, we can make the left hand side of (4.2.1) bounded by, say
$\frac{1}{12}$.

####  4.2.1 Outline of the proof: the dual Malgrange method

The proof of Theorem 4.2 is inspired by Malgrange’s proof of the Newlander-
Nirenberg Theorem [Mal69].

Let $x=(x^{1},\dots,x^{n})$ and $y=(y^{1},\dots,y^{n})$ be two coordinate
systems on the unit ball $\mathbb{B}^{n}\subset\mathbb{R}^{n}$. In this
section, we write 1-forms $\lambda^{1},\dots,\lambda^{n}$,
$\eta^{1},\dots,\eta^{n}$ as

$\lambda^{i}=dx^{i}+\sum_{j=1}^{n}a_{j}^{i}dx^{j},\quad\eta^{i}=dy^{i}+\sum_{j=1}^{n}b_{j}^{i}dy^{j},\quad
i=1,\dots,n,$ (4.2.3)

and define coefficient matrices

$A:=(a_{j}^{i})_{n\times n}=\begin{pmatrix}a_{1}^{1}&\cdots&a_{n}^{1}\\\
\vdots&\ddots&\vdots\\\ a_{1}^{n}&\cdots&a_{n}^{n}\end{pmatrix},\quad
B:=(b_{j}^{i})_{n\times n}=\begin{pmatrix}b_{1}^{1}&\cdots&b_{n}^{1}\\\
\vdots&\ddots&\vdots\\\ b_{1}^{n}&\cdots&b_{n}^{n}\end{pmatrix}.$ (4.2.4)

In this section, $\lambda^{1},\ldots,\lambda^{n}$ are given
$\mathscr{C}^{\alpha}$ $1$-forms on $\mathbb{B}^{n}\subset\mathbb{R}^{n}$
which span the cotangent space at every point. And
$\eta^{i}:=F_{*}\lambda^{i}$ are the push-forward $1$-forms by the unknown
$\mathscr{C}^{\alpha+1}$-diffeomorphism
$F:\mathbb{B}^{n}\xrightarrow{\sim}\mathbb{B}^{n}$, which we are solving for.
Thus, $\eta^{1},\ldots,\eta^{n}$ are also $\mathscr{C}^{\alpha}$ $1$-forms
defined on $\mathbb{B}^{n}$ which span the cotangent space at every point.

As in Malgrange’s work [Mal69], the main idea is to choose $F$ so that the
matrix $B$ satisfies a nonlinear elliptic PDE. That
$\eta=F_{*}\lambda\in\mathscr{C}^{\beta}$ will follow from the classical
interior regularity of elliptic PDEs. We will show such an $F$ exists by
showing that it suffices for $F$ to satisfy a different nonlinear elliptic
PDE, whose solution is guaranteed by classical elliptic theory.

Given collections $(\lambda^{1},\dots,\lambda^{n})$ and
$(\eta^{1},\dots,\eta^{n})$ of 1-forms on $\mathbb{B}^{1}$, as above, that
both span their co-tangent spaces at every point, we define Riemannian metrics
$g$ and $h$ by

$\begin{split}&g=\sum_{i,j=1}^{n}g_{ij}dx^{i}dx^{j}:=\sum_{i,j,k=1}^{n}(\delta_{i}^{k}+a_{i}^{k})(\delta_{j}^{k}+a_{j}^{k})dx^{i}dx^{j},\\\
&h=\sum_{i,j=1}^{n}h_{ij}dy^{i}dy^{j}:=\sum_{i,j,k=1}^{n}(\delta_{i}^{k}+b_{i}^{k})(\delta_{j}^{k}+b_{j}^{k})dy^{i}dy^{j}.\end{split}$
(4.2.5)

Here, $\delta_{i}^{j}$, $\delta_{ij}$, $\delta^{ij}$ are the Kronecker delta
functions:

$\delta_{i}^{j}=\delta_{ij}=\delta^{ij}=\begin{cases}1,&i=j,\\\ 0,&i\neq
j.\end{cases}$ (4.2.6)

We use the following notations from classical Riemannian differential
geometry:

$\begin{split}&g^{ij}:=g(dx^{i},dx^{j}),\quad\sqrt{\det
g}:=\left|\frac{\lambda^{1}\wedge\dots\wedge\lambda^{n}}{dx^{1}\wedge\dots\wedge
dx^{n}}\right|,\\\ &h^{ij}:=h(dy^{i},dy^{j}),\quad\sqrt{\det
h}:=\left|\frac{\eta^{1}\wedge\dots\wedge\eta^{n}}{dy^{1}\wedge\dots\wedge
dy^{n}}\right|.\end{split}$ (4.2.7)

######  Remark 4.2.2.

  1. (a)

We can write $h=\sum_{i=1}^{n}\eta^{i}\cdot\eta^{i}$. It is non-degenerate
since $\eta^{1},\dots,\eta^{n}$ span the cotangent space at every point.
Moreover $\eta^{1},\dots,\eta^{n}$ form an orthogonal basis with respect to
this metric $h$. Similar remarks hold for
$g=\sum_{i=1}^{n}\lambda^{i}\cdot\lambda^{i}$.

  2. (b)

Using matrix notations in (4.2.3) we have $(h_{ij})_{n\times
n}=(I+B)^{\top}(I+B)$ and so we know $(h^{ij})_{n\times
n}=(h_{ij})^{-1}=(I+B)^{-1}\left((I+B)^{-1}\right)^{\top}$ and $\sqrt{\det
h}=\det(I+B)$. Similarly, we have $(g_{ij})=(I+A)^{\top}(I+A)=(g^{ij})^{-1}$
and $\sqrt{\det g}=\det(I+A)$. More importantly, we have the following lemma.

######  Lemma 4.2.3.

Let $B$, $h^{ij}$ and $\sqrt{\det h}$ be as above. Then $h^{ij}$ and
$\sqrt{\det h}$ are rational functions of the components of $B$. Moreover for
every $\gamma>0$, there is a $c_{n,\gamma}>0$ such that

$h^{ij},\sqrt{\det
h}:\left\\{B\in\mathscr{C}^{\gamma}(\mathbb{B}^{n};\mathbb{R}^{n\times
n}):\|B\|_{\mathscr{C}^{\gamma}}<c_{n,\gamma}\right\\}\to\mathscr{C}^{\gamma}(\mathbb{B}^{n}),\quad
1\leq i,j\leq n,$

are norm continuous maps, with

$\sum_{i,j=1}^{n}\|h^{ij}-\delta^{ij}\|_{\mathscr{C}^{\gamma}(\mathbb{B}^{n})}\leq
c_{n,\gamma}^{-1}\|B\|_{\mathscr{C}^{\gamma}(\mathbb{B}^{n};\mathbb{R}^{n\times
n})},\quad\|\sqrt{\det h}-1\|_{\mathscr{C}^{\gamma}(\mathbb{B}^{n})}\leq
c_{n,\gamma}^{-1}\|B\|_{\mathscr{C}^{\gamma}(\mathbb{B}^{n};\mathbb{R}^{n\times
n})}.$

######  Remark 4.2.4.

The same results hold for $g^{ij}$ and $\sqrt{\det g}$. Namely,
$\|g^{ij}-\delta^{ij}\|_{\mathscr{C}^{\alpha}}+\|\sqrt{\det
g}-1\|_{\mathscr{C}^{\alpha}}\leq
c_{n,\alpha}^{-1}\|A\|_{\mathscr{C}^{\alpha}}$ holds with the same constant
$c_{n,\alpha}>0$.

###### Proof.

By Lemma 2.1.17, the space
$\mathscr{C}^{\gamma}(\mathbb{B}^{n};\mathbb{R}^{n\times n})$ is closed under
matrix multiplication. By Remark 4.2.2 (b) $\sqrt{\det h}=\det(I+B)$ is a
polynomial in the components of $B$, so in particular is a norm continuous
function on $\mathscr{C}^{\gamma}(\mathbb{B}^{n};\mathbb{R}^{n\times n})$.
Note that $\det(I+0)=1$ so we have $\|\sqrt{\det
h}-1\|_{\mathscr{C}^{\gamma}(\mathbb{B}^{n})}\lesssim_{\gamma}\|B\|_{\mathscr{C}^{\gamma}(\mathbb{B}^{n};\mathbb{R}^{n\times
n})}$ when the right hand side is small.

By Lemma 2.2.30 (v), choosing
$c_{n,\gamma}<\tilde{c}(\mathbb{B}^{n},\alpha,n)$ where $\tilde{c}$ is the
constant in Lemma 2.2.30 (v), we see that the map $B\mapsto(I+B)^{-1}$ is
$\mathscr{C}^{\gamma}$-norm continuous with
$\|(I+B)^{-1}-I\|_{\mathscr{C}^{\gamma}(\mathbb{B}^{n};\mathbb{R}^{n\times
n})}\leq 2\|B\|_{\mathscr{C}^{\gamma}(\mathbb{B}^{n};\mathbb{R}^{n\times
n})}$.

Thus, in the domain $\|B\|_{\mathscr{C}^{\gamma}}<c_{n,\gamma}$, the map
$B\mapsto(I+B)^{-1}\left((I+B)^{-1}\right)^{\top}$ is also continuous and
satisfies
$\|(I+B)^{-1}\left((I+B)^{-1}\right)^{\top}-I\|_{\mathscr{C}^{\gamma}}\lesssim\|(I+B)^{\top}(I+B)-I\|_{\mathscr{C}^{\gamma}}\lesssim\|B\|_{\mathscr{C}^{\gamma}}$.
It follows with $(h^{ij})_{n\times n}=(h_{ij})^{-1}_{n\times
n}=(I+B)^{-1}\left((I+B)^{-1}\right)^{\top}$, we have that $h^{ij}$ are norm
continuous and
$\|h^{ij}-\delta^{ij}\|_{\mathscr{C}^{\gamma}}\lesssim\|B\|_{\mathscr{C}^{\gamma}}$.

By possibly shrinking $c_{n,\gamma}$ we get $\|\sqrt{\det
h}-1\|_{\mathscr{C}^{\gamma}}\leq
c_{n,\gamma}^{-1}\|B\|_{\mathscr{C}^{\gamma}}$ and
$\sum_{i,j=1}^{n}\|h^{ij}-\delta^{ij}\|_{\mathscr{C}^{\gamma}}\leq
c_{n,\gamma}^{-1}\|B\|_{\mathscr{C}^{\gamma}}$. ∎

######  Convention 4.2.5.

Given a Riemannian metric $g$, we use the co-differential $\vartheta_{g}$ as
the adjoint of differential with respect to $h$. That is, for any $k$-form
$\phi$ and any compactly supported $(k-1)$-form $\psi$,

$(\vartheta_{g}\phi,\psi)_{g}:=(\phi,d\psi)_{g}=\int g(\phi,d\psi)\
d\operatorname{Vol}_{g},$

where $d\operatorname{Vol}_{g}$ is the Riemannian volume density induced by
$g$. In local coordinates, $d\operatorname{Vol}_{g}=\sqrt{\det g}\
d\operatorname{Vol}_{\mathbb{R}^{n}}$, where
$d\operatorname{Vol}_{\mathbb{R}^{n}}$ is the usual Lebesgue density on
$\mathbb{R}^{n}$. We write $\vartheta_{\mathbb{R}^{n}}=\vartheta$ for the
usual co-differential with respect to the flat metric
$\sum_{i,j=1}^{n}\delta_{ij}dy^{i}dy^{j}$ on $\mathbb{R}^{n}$.

######  Lemma 4.2.6.

Let $\alpha>0$, and let $\lambda^{1},\dots,\lambda^{n}$ be
$\mathscr{C}^{\alpha}$ 1-forms defined on $\mathbb{B}^{n}$ that span the
cotangent space at every point, with $(a_{i}^{j})$, $g$, $(g^{ij})$, and
$\sqrt{\det g}$ given in (4.2.4), (4.2.5), and (4.2.7).

Suppose $F=\operatorname{id}+R:\mathbb{B}^{n}_{x}\to\mathbb{B}^{n}_{y}$ is a
$\mathscr{C}^{\alpha+1}$-diffeomorphism that satisfies

$\sum_{i,j=1}^{n}\frac{\partial}{\partial x^{j}}\Big{(}\sqrt{\det g}\cdot
g^{ij}\frac{\partial R^{k}}{\partial
x^{i}}\Big{)}=\sum_{i,j=1}^{n}\frac{\partial}{\partial x^{j}}\big{(}\sqrt{\det
g}\cdot g^{ij}a_{i}^{k}\big{)},\quad\text{in }\mathbb{B}^{n}_{x},\quad
k=1,\dots,n.$ (4.2.8)

Then for the pushforward 1-forms $\eta^{k}=F_{*}\lambda^{k}$, $k=1,\dots,n$,
the coefficients $(b_{i}^{j})$ defined in (4.2.4) satisfy

$\sum_{i,j=1}^{n}\frac{\partial}{\partial y^{i}}\left(\sqrt{\det
h}h^{ij}b_{j}^{k}\right)=0,\quad\text{in }\mathbb{B}^{n}_{y},\quad
k=1,\dots,n.$ (4.2.9)

###### Proof.

Note that the composition of a $C^{1}$-function and a
$\mathscr{C}^{\alpha+1}$-diffeomorphism is still $C^{1}$, and being compactly
supported is preserved under homeomorphism, so we have

$C_{c}^{1}(\mathbb{B}_{x}^{n})=\\{v\circ F:v\in
C_{c}^{1}(\mathbb{B}_{y}^{n})\\}.$ (4.2.10)

By assumption $R\in\mathscr{C}^{\alpha+1}(\mathbb{B}^{n};\mathbb{R}^{n})$, and
$a_{i}^{k},g^{ij},\sqrt{\det g}\in\mathscr{C}^{\alpha}(\mathbb{B}^{n})$, so
(4.2.8) makes sense in
$\mathscr{C}^{\alpha-1}_{\mathrm{loc}}(\mathbb{B}^{n}_{x})\subsetneq
C_{c}^{1}(\mathbb{B}^{n}_{x})^{\prime}$ and the equality can be viewed as
elements of the dual of $C_{c}^{1}(\mathbb{B}^{n})$.

For any $u\in C_{c}^{1}(\mathbb{B}^{n})$, integrating by parts, we obtain

$\displaystyle 0$
$\displaystyle=\bigg{\langle}\sum_{i,j=1}^{n}\frac{\partial}{\partial
x^{j}}\bigg{(}\sqrt{\det g}\cdot g^{ij}\Big{(}\frac{\partial R^{k}}{\partial
x^{i}}-a_{i}^{k}\Big{)}\bigg{)},u\bigg{\rangle}_{\mathbb{B}^{n}_{x}}=-\int_{\mathbb{B}^{n}}\sum_{i,j=1}^{n}g^{ij}\Big{(}\frac{\partial
R^{k}}{\partial x^{i}}-a_{i}^{k}\Big{)}\frac{\partial u}{\partial
x^{j}}\sqrt{\det g}dx$ $\displaystyle=-\langle
dR^{k}-(\lambda^{k}-dx^{k}),du\rangle_{\mathbb{B}^{n}_{x};g}=-\langle
dF^{k}-\lambda^{k},du\rangle_{\mathbb{B}_{x}^{n};g}$

Here $\langle\cdot,\cdot\rangle_{\mathbb{B}_{x}^{n};g}$ are the dual pairs for
linear functionals and test functions induced by $g$. Namely, for $u,v\in
C^{0}(\mathbb{B}^{n})$ and $\phi,\psi\in
C^{0}(\mathbb{B}^{n};T^{*}\mathbb{B}^{n})$, $\langle
u,v\rangle_{\mathbb{B}^{n};g}=\int_{\mathbb{B}^{n}}uv\sqrt{\det g}dx$ and
$\langle\phi,\psi\rangle_{\mathbb{B}^{n};g}=\int_{\mathbb{B}^{n}}g(\phi,\psi)\sqrt{\det
g}dx$.

Using (4.2.8), we get $\langle
dF^{k}-\lambda^{k},du\rangle_{\mathbb{B}^{n}_{x};g}=0$ for all $u\in
C_{c}^{1}(\mathbb{B}^{n})$. By (4.2.10) we have $\langle
dF^{k}-\lambda^{k},d(v\circ F)\rangle_{\mathbb{B}^{n}_{x};g}=0$ for all $v\in
C_{c}^{1}(\mathbb{B}^{n}_{y})$.

Note that
$F_{*}(g(\phi,\psi))=(F_{*}g)(F_{*}\phi,F_{*}\psi)=h(F_{*}\phi,F_{*}\psi)$ for
all $\phi,\psi\in C^{0}_{\mathrm{loc}}(\mathbb{B}^{n};T^{*}\mathbb{B}^{n})$,
and $F_{*}\sqrt{\det g}=\sqrt{\det F_{*}g}=\sqrt{\det h}$, so we have, for
every $v\in C_{c}^{1}(\mathbb{B}^{n}_{y})$,

$0=\langle dF^{k}-\lambda^{k},d(v\circ
F)\rangle_{\mathbb{B}^{n}_{x};g}=\langle
F_{*}(dF^{k}-\lambda^{k}),F_{*}F^{*}dv\rangle_{\mathbb{B}^{n}_{y};h}=\langle
dy^{k}-\eta^{k},dv\rangle_{\mathbb{B}^{n}_{y};h}=\int_{\mathbb{B}^{n}}\sum_{i,j=1}^{n}h^{ij}b_{j}^{k}\frac{\partial
v}{\partial y^{i}}\sqrt{\det h}dy.$

Integrating by parts, we obtain (4.2.9). ∎

We will choose a coordinate chart $F$ so that (4.2.8) is satisfied, and
therefore (4.2.9) will be satisfied as well.

To prove Theorem 4.2 we will prove the following:

  * •

There exists a $R\in\mathscr{C}^{\alpha+1}(\mathbb{B}^{n};\mathbb{R}^{n})$
satisfying (4.2.8) with boundary condition
$R\big{|}_{\partial\mathbb{B}^{n}}=0$. Moreover, we can choose $R$ with
$\|R\|_{\mathscr{C}^{\alpha+1}}\lesssim_{\alpha,\beta}\|A\|_{\mathscr{C}^{\alpha}}$.
Thus, by taking $c>0$ small, we may take $\|R\|_{\mathscr{C}^{\alpha+1}}$
small.

  * •

When $\|R\|_{\mathscr{C}^{\alpha+1}}$ is small, $F=\operatorname{id}+R$ is a
$\mathscr{C}^{\alpha+1}$-diffeomorphism of $\mathbb{B}^{n}$. And under the
assumption $\operatorname{supp}A\subsetneq\frac{1}{2}\mathbb{B}^{n}$, we have
$\|B\|_{\mathscr{C}^{\beta}(\partial\mathbb{B}^{n})}\lesssim_{\alpha,\beta}\|A\|_{\mathscr{C}^{\alpha}}$
and
$\|d\eta\|_{\mathscr{C}^{\beta-1}}\lesssim_{\alpha,\beta}\|d\lambda\|_{\mathscr{C}^{\beta-1}}$.
In particular, by taking $c>0$ small, we may take
$\|B\|_{\mathscr{C}^{\beta}(\partial\mathbb{B}^{n})}+\|d\eta\|_{\mathscr{C}^{\beta-1}}$
small.

  * •

Using that
$B\in\mathscr{C}^{\alpha}(\partial\mathbb{B}^{n};\mathbb{R}^{n\times n})$
satisfies (4.2.9), if
$\|B\|_{\mathscr{C}^{\beta}(\partial\mathbb{B}^{n})}+\|d\eta\|_{\mathscr{C}^{\beta-1}}$
is small, we will show
$B\in\mathscr{C}^{\beta}(\mathbb{B}^{n};\mathbb{R}^{n\times n})$ and
$\|B\|_{\mathscr{C}^{\beta}}\lesssim_{\alpha,\beta}\|A\|_{\mathscr{C}^{\alpha}}$.

####  4.2.2 The existence proposition

In this section, we show that there exists a
$\mathscr{C}^{\alpha+1}$-diffeomorphism $F=\operatorname{id}+R$ solving
(4.2.8) and which satisfies good estimates.

######  Proposition 4.2.7.

Let $\alpha>0$ and let $\beta\in[\alpha,\alpha+1]$. There is a
$c_{1}=c_{1}(n,\alpha,\beta)\in(0,1)$ such that, if $A=(a_{j}^{k})_{n\times
n}:\mathbb{B}^{n}\to\mathbb{R}^{n\times n}$ satisfies

  * -

$A\in\mathscr{C}^{\alpha}_{c}(\frac{1}{2}\mathbb{B}^{n};\mathbb{R}^{n\times
n})$ and $\|A\|_{\mathscr{C}^{\alpha}}<c_{1}$,

then the matrix $(I+A)(x)$ is invertible for every $x\in\mathbb{B}^{n}$, and
there is a $\mathscr{C}^{\alpha+1}$-map $F=\operatorname{id}+R$ on
$\mathbb{B}^{n}$ such that

  1. (i)

$R$ solves the equation

$\sum_{i,j=1}^{n}\frac{\partial}{\partial x^{j}}\Big{(}\sqrt{\det g}\cdot
g^{ij}\frac{\partial R^{k}}{\partial
x^{i}}\Big{)}=\sum_{i,j=1}^{n}\frac{\partial}{\partial x^{j}}\big{(}\sqrt{\det
g}\cdot g^{ij}a_{i}^{k}\big{)},\quad\text{in }\mathbb{B}^{n}_{x},\quad
k=1,\dots,n.$ (4.2.8)

with boundary condition $R\big{|}_{\partial\mathbb{B}^{n}}=0$, and we have

$\textstyle\|R\|_{\mathscr{C}^{\alpha+1}(\mathbb{B}^{n};\mathbb{R}^{n})}+\|\nabla
R\|_{\mathscr{C}^{\beta}(\mathbb{B}^{n}\backslash\frac{3}{4}\mathbb{B}^{n};\mathbb{R}^{n})}\leq
c_{1}^{-1}\|A\|_{\mathscr{C}^{\alpha}}.$ (4.2.11)

  2. (ii)

$F:\mathbb{B}^{n}_{x}\to\mathbb{B}^{n}_{y}$ is a
$\mathscr{C}^{\alpha+1}$-diffeomorphism such that
$B^{n}(F(0),\frac{1}{6})\subseteq
F(\frac{1}{3}\mathbb{B}^{n})\cap\frac{3}{4}\mathbb{B}^{n}$.

  3. (iii)

Let $\Phi=F^{\mathrm{Inv}}:\mathbb{B}^{n}_{y}\to\mathbb{B}^{n}_{x}$ be its
inverse map, then

$\|\nabla\Phi-I\|_{\mathscr{C}^{\alpha}(\mathbb{B}^{n};\mathbb{R}^{n\times
n})}+\|\nabla\Phi-I\|_{\mathscr{C}^{\beta}(\partial\mathbb{B}^{n};\mathbb{R}^{n\times
n})}\leq c_{1}^{-1}\|A\|_{\mathscr{C}^{\alpha}}.$ (4.2.12)

In particular
$\|\Phi\|_{\mathscr{C}^{\alpha+1}(\mathbb{B}^{n};\mathbb{R}^{n})}\leq
c_{1}^{-1}$.

######  Remark 4.2.8.

The map $F$ in Proposition 4.2.7 is uniquely determined by $A$. This is due to
the well-posedness of the Dirichlet problem for the second order elliptic
equations, since $R$ satisfies (4.2.8) with
$R\big{|}_{\partial\mathbb{B}^{n}}=0$.

######  Remark 4.2.9.

As we will see in the proof of Theorem 4.2, the map $F$ from Proposition 4.2.7
is the map of the same name in Theorem 4.2.

###### Proof.

We let $c_{1}$ be a small constant which may change from line to line. Note
that if (4.2.11) and (4.2.12) are valid for some $\tilde{c}_{1}$, then they
are also valid for any $0<c_{1}\leq\tilde{c}_{1}$.

First pick $c_{1}<\tilde{c}(\mathbb{B}^{n},\alpha,n)$ where $\tilde{c}$ is the
constant in Lemma 2.2.30 (v). By Lemma 2.2.30 (v), the assumption
$\|A\|_{\mathscr{C}^{\alpha}}<c_{1}(<\tilde{c}_{\mathbb{B}^{n},\alpha,n})$
implies that $I+A$ is invertible at every point and
$(I+A)^{-1}\in\mathscr{C}^{\alpha}(\mathbb{B}^{n};\mathbb{R}^{n\times n})$.
Therefore, $g$ given in (4.2.5) is indeed a $\mathscr{C}^{\alpha}$-Riemannian
metric.

By the assumption $\operatorname{supp}A\subsetneq\frac{1}{2}\mathbb{B}^{n}$,
we have $\sqrt{\det g}\cdot
g^{ij}\big{|}_{\mathbb{B}^{n}\backslash\frac{1}{2}\mathbb{B}^{n}}=\delta^{ij}$.
Since $(\sqrt{\det g}g^{ij}(x))_{n\times n}$ is an invertible matrix for
$x\in\frac{1}{2}\mathbb{B}^{n}$, the second order operator
$\sum_{i,j=1}^{n}\partial_{x^{j}}(\sqrt{\det g}\cdot g^{ij}\partial_{x^{i}})$
is uniformly elliptic on $\mathbb{B}^{n}$. Classical existence theorems (for
example, [GT01, Theorem 8.3]) show that for each $k=1,\dots,n$ there
exists222Here $H^{1}(\mathbb{B}^{n})$ stands for the classical $L^{2}$-Sobolev
space of order $1$, and $H^{-1}(\mathbb{B}^{n})=H_{0}^{1}(\mathbb{B}^{n})^{*}$
is the $L^{2}$-Sobolev space of order $-1$. a $R^{k}\in H^{1}(\mathbb{B}^{n})$
that satisfies (4.2.8) with Dirichlet boundary condition
$R^{k}\big{|}_{\partial\mathbb{B}^{n}}=0$, since
$\sum_{i,j=1}^{n}\frac{\partial}{\partial x^{j}}\left(\sqrt{\det
g}g^{ij}a_{i}^{k}\right)\in\mathscr{C}^{\alpha-1}(\mathbb{B}^{n})\subset
H^{-1}(\mathbb{B}^{n})$. By a classical regularity estimate (see [GT01,
Theorem 8.34] or [FR95, Theorem 15]), we know
$R^{k}\in\mathscr{C}^{\alpha+1}(\mathbb{B}^{n};\mathbb{R}^{n})$.

To show $\|R\|_{\mathscr{C}^{\alpha+1}}\lesssim\|A\|_{\mathscr{C}^{\alpha}}$,
we write (4.2.8) as

$\Delta R^{k}=\sum_{i,j=1}^{n}\frac{\partial}{\partial
x^{j}}\left(\left(\delta^{ij}-\sqrt{\det g}g^{ij}\right)\frac{\partial
R^{k}}{\partial x^{i}}\right)+\sum_{i,j=1}^{n}\frac{\partial}{\partial
x^{j}}\left(\sqrt{\det g}g^{ij}a_{i}^{k}\right),\quad\text{in
}\mathbb{B}^{n}_{x},\quad k=1,\dots,n.$ (4.2.13)

By Remark 4.2.4 (see also Lemma 4.2.3), we see that $\|\delta^{ij}-\sqrt{\det
g}g^{ij}\|_{\mathscr{C}^{\alpha}(\mathbb{B}^{n})}\lesssim_{\alpha}\|A\|_{\mathscr{C}^{\alpha}}$.
Therefore

$\|\Delta
R\|_{\mathscr{C}^{\alpha-1}}\lesssim_{\alpha}\sum_{i,j,k=1}^{n}\left(\|\delta^{ij}-\sqrt{\det
g}g^{ij}\|_{\mathscr{C}^{\alpha}}\|\partial_{x^{i}}R^{k}\|_{\mathscr{C}^{\alpha}}+\|\sqrt{\det
g}g^{ij}a_{i}^{k}\|_{\mathscr{C}^{\alpha}}\right)\lesssim\|A\|_{\mathscr{C}^{\alpha}}\|R\|_{\mathscr{C}^{\alpha+1}}+\|A\|_{\mathscr{C}^{\alpha}},$
(4.2.14)

where the implicit constants depend only on $n$ and $\alpha$ but not $A$ or
$R$.

The assumption $R\big{|}_{\partial\mathbb{B}^{n}}=0$ implies that
$R=\mathcal{P}_{0}^{\mathbb{B}^{n}}\Delta R$, where
$\mathcal{P}_{0}^{\mathbb{B}^{n}}$ is the zero Dirichlet boundary solution
operator given in Notation 2.1.43. Since, by Lemma 2.1.41,
$\mathcal{P}_{0}^{\mathbb{B}^{n}}:\mathscr{C}^{\alpha-1}(\mathbb{B}^{n})\to\mathscr{C}^{\alpha+1}(\mathbb{B}^{n})$
is bounded, (4.2.14) implies

$\|R\|_{\mathscr{C}^{\alpha+1}(\mathbb{B}^{n};\mathbb{R}^{n})}\leq\tilde{C}_{1}\|A\|_{\mathscr{C}^{\alpha}(\mathbb{B}^{n};\mathbb{R}^{n\times
n})}\|R\|_{\mathscr{C}^{\alpha+1}(\mathbb{B}^{n};\mathbb{R}^{n})}+\tilde{C}_{1}\|A\|_{\mathscr{C}^{\alpha}(\mathbb{B}^{n};\mathbb{R}^{n\times
n})},$ (4.2.15)

where $\tilde{C}_{1}=\tilde{C}_{1}(n,\alpha)>1$ is a constant depending only
on $n$ and $\alpha$ but not $A$ or $R$.

Choosing $c_{1}$ small enough so that $c_{1}\tilde{C}_{1}\leq\frac{1}{3}$,
then we get
$\|R\|_{\mathscr{C}^{\alpha+1}}\leq\frac{1}{3}\|R\|_{\mathscr{C}^{\alpha+1}}+\tilde{C}_{1}\|A\|_{\mathscr{C}^{\alpha}}$
when $A$ satisfies the assumption $\|A\|_{\mathscr{C}^{\alpha}}<c_{1}$.
Therefore

$\textstyle\|R\|_{\mathscr{C}^{\alpha+1}(\mathbb{B}^{n};\mathbb{R}^{n})}\leq\frac{3}{2}\tilde{C}_{1}\|A\|_{\mathscr{C}^{\alpha}}\leq\frac{1}{2}c_{1}^{-1}\|A\|_{\mathscr{C}^{\alpha}},\quad\text{when
}\|A\|_{\mathscr{C}^{\alpha}}<c_{1}\leq(3\tilde{C}_{1})^{-1}.$ (4.2.16)

This is part of the estimate in (4.2.11).

Next we show
$\|R\|_{\mathscr{C}^{\beta+1}(\mathbb{B}^{n}\backslash\frac{3}{4}\mathbb{B}^{n})}\lesssim\|A\|_{\mathscr{C}^{\alpha}}$.
Note that by the support assumption
$\operatorname{supp}A\subsetneq\frac{1}{2}\mathbb{B}^{n}$ we have $\sqrt{\det
g}g^{ij}\big{|}_{\mathbb{B}^{n}\backslash\frac{1}{2}\mathbb{B}^{n}}=\delta^{ij}$
and $a_{i}^{k}\big{|}_{\mathbb{B}^{n}\backslash\frac{1}{2}\mathbb{B}^{n}}=0$,
so the right hand side of (4.2.13) is zero in
$\mathbb{B}^{n}\backslash\frac{1}{2}\mathbb{B}^{n}$. Therefore each $R^{k}$ is
a harmonic functions in the domain
$\mathbb{B}^{n}\backslash\frac{1}{2}\mathbb{B}^{n}$.

The estimate
$\|R\|_{\mathscr{C}^{\alpha+1}(\mathbb{B}^{n})}\lesssim_{\alpha}\|A\|_{\mathscr{C}^{\alpha}}$
implies
$\|R\|_{\mathscr{C}^{\alpha+1}(\partial(\frac{1}{2}\mathbb{B}^{n}))}\lesssim_{\alpha}\|A\|_{\mathscr{C}^{\alpha}}$
since the trace map
$\mathscr{C}^{\alpha+1}(\mathbb{B}^{n})\to\mathscr{C}^{\alpha+1}(\partial(\frac{1}{2}\mathbb{B}^{n}))$
is bounded (see Remark 2.1.42). By classical interior estimates of harmonic
functions (for example, [GT01, Theorem 2.10]) since $\Delta
R\big{|}_{\mathbb{B}^{n}\backslash\frac{1}{2}\mathbb{B}^{n}}=0$, we have

$\|R\|_{\mathscr{C}^{\beta+1}(\partial\frac{3}{4}\mathbb{B}^{n})}\lesssim\|R\|_{C^{\lceil\beta\rceil+1}(\partial\frac{3}{4}\mathbb{B}^{n})}\lesssim\|R\|_{C^{0}(\mathbb{B}^{n}\backslash\frac{2}{3}\mathbb{B}^{n})}\lesssim\|A\|_{\mathscr{C}^{\alpha}}.$
(4.2.17)

Therefore, along with the fact that $R\big{|}_{\partial\mathbb{B}^{n}}=0$,
$R\big{|}_{\partial(\mathbb{B}^{n}\backslash\frac{3}{4}\mathbb{B}^{n})}=R\big{|}_{\partial\frac{3}{4}\mathbb{B}^{n}}\cup
R\big{|}_{\partial\mathbb{B}^{n}}$ has $\mathscr{C}^{\beta+1}$ norm bounded by
a constant times $\|A\|_{\mathscr{C}^{\alpha}}$. By classical regularity
estimates of harmonic functions (also see Lemma 2.1.41) on
$\mathbb{B}^{n}\backslash\frac{3}{4}\mathbb{B}^{n}$ we know

$\|R\|_{\mathscr{C}^{\beta+1}(\mathbb{B}^{n}\backslash\frac{3}{4}\mathbb{B}^{n})}\lesssim_{\beta}\|R\|_{\mathscr{C}^{\beta+1}(\partial(\mathbb{B}^{n}\backslash\frac{3}{4}\mathbb{B}^{n}))}\lesssim_{\alpha,\beta}\|A\|_{\mathscr{C}^{\alpha}}.$

In particular, there is a $\tilde{C}_{2}=\tilde{C}_{2}(n,\alpha,\beta)>0$ that
depends on neither $A$ nor $R$ such that

$\|\nabla
R\|_{\mathscr{C}^{\beta}(\mathbb{B}^{n}\backslash\frac{3}{4}\mathbb{B}^{n})}\leq\tilde{C}_{2}\|A\|_{\mathscr{C}^{\alpha}}.$
(4.2.18)

Taking $c_{1}<\frac{1}{2}\tilde{C}_{2}^{-1}$ we have $\|\nabla
R\|_{\mathscr{C}^{\beta}(\mathbb{B}^{n}\backslash\frac{3}{4}\mathbb{B}^{n})}\leq\frac{1}{2}c_{1}^{-1}\|A\|_{\mathscr{C}^{\alpha}}$.
Combining this with (4.2.16), completes the proof of (4.2.11).

We can take $c_{1}>0$ possibly smaller so that
$c_{1}<\frac{1}{3}(K_{2}\tilde{C}_{1})^{-1}$, where
$K_{2}=K_{2}(n,\alpha,\beta)$ is the constant in Proposition 2.1.44 and
$\tilde{C}_{1}$ is the constant in (4.2.15). By (4.2.16) we know
$\|R\|_{\mathscr{C}^{\alpha+1}}\leq K_{2}^{-1}$. So by Proposition 2.1.44, the
map $F=\operatorname{id}+R$ has $\mathscr{C}^{\alpha+1}$-inverse. We conclude
$\Phi=F^{\mathrm{Inv}}\in\mathscr{C}^{\alpha+1}(\mathbb{B}^{n};\mathbb{R}^{n})$.

Since $\|R\|_{C^{1}(\mathbb{B}^{n};\mathbb{R}^{n\times
n})}\lesssim_{\alpha}\|R\|_{\mathscr{C}^{\alpha+1}(\mathbb{B}^{n};\mathbb{R}^{n})}$,
by possibly shrinking $c_{1}$ we can ensure $\|R\|_{C^{0}}+\|\nabla
R\|_{C^{0}}\leq\frac{1}{2}$. So $F(0)=R(0)\in B^{n}(0,\frac{1}{2})$, which
implies $B^{n}(F(0),\frac{1}{6})\subseteq
B^{n}(0,\frac{1}{2}+\frac{1}{6})\subset\frac{3}{4}\mathbb{B}^{n}$, and
$|F(x_{1})-F(x_{2})|\geq|x_{1}-x_{2}|-|R(x_{1})-R(x_{2})|\geq\frac{1}{2}|x_{1}-x_{2}|$
for $x_{1},x_{2}\in\mathbb{B}^{n}$. Thus, if $|F(x)-F(0)|<\frac{1}{6}$ then
$|x-0|<\frac{1}{3}$; i.e., $B^{n}(F(0),\frac{1}{6})\subseteq
F(\frac{1}{3}\mathbb{B}^{n})$. So $B^{n}(F(0),\frac{1}{6})\subseteq
F(\frac{1}{3}\mathbb{B}^{n})\cap\frac{3}{4}\mathbb{B}^{n}$ finishing the proof
of (ii).

Finally we prove (iii). Note that by Proposition 2.1.44 (i), (4.2.12) gives
$\|\Phi\|_{\mathscr{C}^{\alpha+1}(\mathbb{B}^{n};\mathbb{R}^{n})}\lesssim 1$
and , which is $\|\Phi\|_{\mathscr{C}^{\alpha+1}}\leq c_{1}^{-1}$ by choosing
$c_{1}$ small.

By Proposition 2.1.44 (i) and using that
$c_{1}<\frac{1}{3}(C_{0}\tilde{C}_{1})^{-1}$, we get
$\|\nabla\Phi-I\|_{\mathscr{C}^{\alpha}}\leq
K_{2}\|R\|_{\mathscr{C}^{\alpha+1}}\leq
K_{2}\tilde{C}_{1}\|A\|_{\mathscr{C}^{\alpha}}\leq\frac{1}{2}c_{1}^{-1}\|A\|_{\mathscr{C}^{\alpha}}$,
which proves half of (4.2.12).

To show the second half of (4.2.12), we need to show
$\|\nabla\Phi-I\|_{\mathscr{C}^{\beta}(\partial\mathbb{B}^{n})}\lesssim\|A\|_{\mathscr{C}^{\alpha}}$.

The assumption $R\big{|}_{\partial\mathbb{B}^{n}}=0$ implies
$F\big{|}_{\partial\mathbb{B}^{n}}=\operatorname{id}\big{|}_{\partial\mathbb{B}^{n}}=\Phi\big{|}_{\partial\mathbb{B}^{n}}$
and therefore

$(\nabla\Phi-I)\big{|}_{\partial\mathbb{B}^{n}}=((\nabla\Phi)\circ\Phi^{\mathrm{Inv}})\big{|}_{\partial\mathbb{B}^{n}}-I=(\nabla
F)^{-1}\big{|}_{\partial\mathbb{B}^{n}}-I.$

Fix $\chi\in
C_{c}^{\infty}(2\mathbb{B}^{n}\backslash\frac{3}{4}\mathbb{B}^{n})$ such that
$\chi\equiv 1$ in a neighborhood of $\partial\mathbb{B}^{n}$, so $\nabla
R(x)=\chi(x)\nabla R(x)$ for $x\in\mathbb{B}^{n}$ near
$\partial\mathbb{B}^{n}$.

We shrink $c_{1}>0$ so that
$c_{1}<\tilde{c}_{\mathbb{B}^{n},\beta,n}\cdot(2\tilde{C}_{2}\|\chi\|_{\mathscr{C}^{\beta}})^{-1}$,
where $\tilde{C}_{2}$ is in (4.2.18), and $\tilde{c}_{\mathbb{B}^{n},\beta,n}$
is in Lemma 2.2.30 (v). Then by Lemma 2.1.17 (ii) the assumption
$\|A\|_{\mathscr{C}^{\alpha}}<c_{1}$ implies

$\|\chi\nabla R\|_{\mathscr{C}^{\beta}(\mathbb{B}^{n})}\leq
2\|\chi\|_{\mathscr{C}^{\beta}_{c}(\mathbb{B}^{n}\backslash\frac{3}{4}\mathbb{B}^{n})}\|\nabla
R\|_{\mathscr{C}^{\beta}(\mathbb{B}^{n}\backslash\frac{3}{4}\mathbb{B}^{n})}\leq
2\tilde{C}_{2}\|A\|_{\mathscr{C}^{\alpha}_{c}(\frac{1}{2}\mathbb{B}^{n})}<\tilde{c}_{\mathbb{B}^{n},\beta}.$

Therefore we can apply Lemma 2.2.30 (v) to $\chi\nabla
R\in\mathscr{C}^{\beta}(\mathbb{B}^{n};\mathbb{R}^{n\times n})$ to obtain
$\|(I+\chi\nabla R)^{-1}-I\|_{\mathscr{C}^{\beta}}\leq 2\|\chi\nabla
R\|_{\mathscr{C}^{\beta}}$. Hence, along with (2.1.27),

$\|(I+\nabla
R)^{-1}-I\|_{\mathscr{C}^{\beta}(\partial\mathbb{B}^{n})}\lesssim\|(I+\chi\nabla
R)^{-1}-I\|_{\mathscr{C}^{\beta}(\mathbb{B}^{n})}\lesssim\|\chi\nabla
R\|_{\mathscr{C}^{\beta}(\mathbb{B}^{n})}\lesssim_{\chi}\|R\|_{\mathscr{C}^{\beta+1}(\mathbb{B}^{n}\backslash\frac{3}{4}\mathbb{B}^{n})}\lesssim\|A\|_{\mathscr{C}^{\alpha}}.$

So by possibly shrinking $c_{1}>0$, we get
$\|\nabla\Phi-I\|_{\mathscr{C}^{\beta}(\partial\mathbb{B}^{n})}\leq\frac{1}{2}c_{1}^{-1}\|A\|_{\mathscr{C}^{\alpha}}$,
which completes the second half of (4.2.12). ∎

We now have pushforward 1-forms
$\eta^{1}=F_{*}\lambda^{1},\dots,\eta^{n}=F_{*}\lambda^{n}$. Their norms admit
some control, as the next lemma shows.

######  Lemma 4.2.10.

Let $\alpha>0$ and let $\beta\in[\alpha,\alpha+1]$. There is a
$c_{2}=c_{2}(n,\alpha,\beta)>0$ such that the following holds. Let
$A\in\mathscr{C}^{\alpha}_{c}(\frac{1}{2}\mathbb{B}^{n},\mathbb{R}^{n\times
n})$ be the coefficient matrix for $\lambda^{1},\dots,\lambda^{n}$ (see
(4.2.3)) satisfying the assumptions of Proposition 4.2.7 and also satisfying

  1. (a)

$\|A\|_{\mathscr{C}^{\alpha}(\mathbb{B}^{n};\mathbb{R}^{n\times n})}<c_{2}$.

  2. (b)

For $k=1,\dots,n$,
$d\lambda^{k}\in\mathscr{C}^{\beta-1}\left(\mathbb{B}^{n};\wedge^{2}T^{*}\mathbb{B}^{n}\right)$
with
$\sum_{k=1}^{n}\|d\lambda^{k}\|_{\mathscr{C}^{\beta-1}(\mathbb{B}^{n};\wedge^{2}T^{*}\mathbb{B}^{n})}<c_{2}$.

Suppose $\Phi=F^{\mathrm{Inv}}:\mathbb{B}^{n}_{y}\to\mathbb{B}^{n}_{x}$
satisfies the conclusions of Proposition 4.2.7. Then, for the 1-forms
$\eta^{k}=\Phi^{*}\lambda^{k}$ ($k=1,\dots,n$) with coefficient matrix
$B=(b^{i}_{j})_{n\times n}$ (see (4.2.3)), we have:

  1. (i)

$B$ satisfies the PDE system (4.2.9).

  2. (ii)

$\|B\|_{\mathscr{C}^{\alpha}(\mathbb{B}^{n};\mathbb{R}^{n\times
n})}+\|B\|_{\mathscr{C}^{\beta}(\partial\mathbb{B}^{n};\mathbb{R}^{n\times
n})}<c_{2}^{-1}\|A\|_{\mathscr{C}^{\alpha}}$.

  3. (iii)

$d\eta^{k}\in\mathscr{C}^{\beta-1}\left(\mathbb{B}^{n};\wedge^{2}T^{*}\mathbb{B}^{n}\right)$
for $k=1,\dots,n$, with
$\|d\eta^{k}\|_{\mathscr{C}^{\beta-1}(\mathbb{B}^{n};\wedge^{2}T^{*}\mathbb{B}^{n})}<c_{2}^{-1}\|d\lambda^{k}\|_{\mathscr{C}^{\beta-1}}$.

###### Proof.

Part (i) is obtained in Lemma 4.2.6.

For part (ii), write $\Phi=(\phi^{1},\dots,\phi^{n})$, where
$\phi^{k}\in\mathscr{C}^{\alpha+1}(\mathbb{B}^{n})$, $k=1,\dots,n$. Therefore

$\eta^{k}=\Phi^{*}\Big{(}dx^{k}+\sum_{i=1}^{n}a_{i}^{k}dx^{i}\Big{)}=d\phi^{k}+\sum_{i=1}^{n}(a_{i}^{k}\circ\Phi)d\phi^{i},\quad
b_{j}^{k}=\frac{\partial(\phi^{k}-y^{k})}{\partial
y^{j}}+\sum_{i=1}^{n}(a_{i}^{k}\circ\Phi)\frac{\partial\phi^{i}}{\partial
y^{j}},\quad 1\leq j,k\leq n.$ (4.2.19)

From (4.2.19) we know that
$\|B\|_{\mathscr{C}^{\alpha}}\lesssim\|\nabla\Phi-I\|_{\mathscr{C}^{\alpha}}+\|A\circ\Phi\|_{\mathscr{C}^{\alpha}}\|\nabla\Phi\|_{\mathscr{C}^{\alpha}}$.
By Proposition 4.2.7 (iii) we know $\|\Phi\|_{\mathscr{C}^{\alpha+1}}\lesssim
1$ and
$\|\nabla\Phi-I\|_{\mathscr{C}^{\alpha}}\lesssim\|A\|_{\mathscr{C}^{\alpha}}$.
By Proposition 2.1.30 (i) we get
$\|A\circ\Phi\|_{\mathscr{C}^{\alpha}}\lesssim\|A\|_{\mathscr{C}^{\alpha}}$.
Combining these we get

$\|B\|_{\mathscr{C}^{\alpha}}\lesssim\|\nabla\Phi-I\|_{\mathscr{C}^{\alpha}}+\|A\circ\Phi\|_{\mathscr{C}^{\alpha}}\|\nabla\Phi\|_{\mathscr{C}^{\alpha}}\lesssim\|A\|_{\mathscr{C}^{\alpha}}.$
(4.2.20)

Since we have $A\equiv 0$ outside $\frac{1}{2}\mathbb{B}^{n}$ in particular
$A\big{|}_{\partial\mathbb{B}^{n}}=0$, it follows that $\eta^{k}=d\phi^{k}$ on
$\partial\mathbb{B}^{n}$. Therefore
$\|B\|_{\mathscr{C}^{\beta}(\partial\mathbb{B}^{n})}=\|\nabla\Phi-I\|_{\mathscr{C}^{\beta}(\partial\mathbb{B}^{n})}$.
So by Proposition 4.2.7 (iii)

$\|B\|_{\mathscr{C}^{\beta}(\partial\mathbb{B}^{n})}=\|\nabla\Phi-I\|_{\mathscr{C}^{\beta}(\partial\mathbb{B}^{n})}\lesssim\|A\|_{\mathscr{C}^{\alpha}}$
(4.2.21)

By choosing $c_{2}>0$ small, (4.2.20) and (4.2.21) complete the proof of (ii).

Finally, for (iii), we apply Proposition 2.1.44 (ii) with
$\lambda=\lambda^{k}$, for each $k=1,\dots,n$. Since
$d(F_{*}\lambda^{k})=d\eta^{k}$, by (2.1.28) we get
$\|d\eta^{k}\|_{\mathscr{C}^{\beta-1}(\mathbb{B}^{n};\wedge^{2}T^{*}\mathbb{B}^{n})}\lesssim\|d\lambda^{k}\|_{\mathscr{C}^{\beta-1}}$.
Taking $c_{2}$ smaller, we complete the proof. ∎

####  4.2.3 The regularity proposition

In this part, we show that the 1-forms $\eta^{1},\dots,\eta^{n}$ are indeed
$\mathscr{C}^{\beta}$, by using the interior regularity theory for elliptic
PDEs.

######  Proposition 4.2.11.

Let $\alpha>0$ and $\beta\in[\alpha,\alpha+1]$. There is a
$c_{3}=c_{3}(n,\alpha,\beta)>0$, such that if
$\eta^{1},\dots,\eta^{n}\in\mathscr{C}^{\alpha}(\mathbb{B}^{n};T^{*}\mathbb{B}^{n})$
with coefficient matrix
$B\in\mathscr{C}^{\alpha}(\mathbb{B}^{n};\mathbb{R}^{n\times n})$ (see
(4.2.3)) such that $B$ solves the PDE (4.2.9),
$B\big{|}_{\partial\mathbb{B}^{n}}\in\mathscr{C}^{\beta}(\partial\mathbb{B}^{n};\mathbb{R}^{n\times
n})$ with

$\|B\|_{\mathscr{C}^{\alpha}(\mathbb{B}^{n};\mathbb{R}^{n\times
n})}+\|B\|_{\mathscr{C}^{\beta}(\partial\mathbb{B}^{n};\mathbb{R}^{n\times
n})}+\sum_{l=1}^{n}\|d\eta^{l}\|_{\mathscr{C}^{\beta-1}\left(\mathbb{B}^{n};\wedge^{2}T^{*}\mathbb{B}^{n}\right)}<c_{3},$
(4.2.22)

then $B\in\mathscr{C}^{\beta}(\mathbb{B}^{n};\mathbb{R}^{n\times n})$.
Moreover

$\|B\|_{\mathscr{C}^{\beta}(\mathbb{B}^{n};\mathbb{R}^{n\times n})}\leq
c_{3}^{-1}\Big{(}\|B\|_{\mathscr{C}^{\alpha}(\mathbb{B}^{n};\mathbb{R}^{n\times
n})}+\|B\|_{\mathscr{C}^{\beta}(\partial\mathbb{B}^{n};\mathbb{R}^{n\times
n})}+\sum_{l=1}^{n}\|d\eta^{l}\|_{\mathscr{C}^{\beta-1}(\mathbb{B}^{n};\wedge^{2}T^{*}\mathbb{B}^{n})}\Big{)}.$
(4.2.23)

###### Proof.

We can write $B=\mathcal{P}_{0}\Delta B+(B-\mathcal{P}_{0}\Delta B)$, where
$\mathcal{P}_{0}=\mathcal{P}_{0}^{\mathbb{B}^{n}}$ is given in Lemma 2.2.35,
which is the zero Dirichlet boundary solution operator on the unit ball.

Note that $B-\mathcal{P}_{0}\Delta B$ is the harmonic function whose boundary
value equals to $B\big{|}_{\partial\mathbb{B}^{n}}$ (which might not be zero).
By Lemma 2.1.41 using the assumption
$B\big{|}_{\partial\mathbb{B}^{n}}\in\mathscr{C}^{\beta}(\partial\mathbb{B}^{n};\mathbb{R}^{n\times
n})$, we get $B-\mathcal{P}_{0}\Delta
B\in\mathscr{C}^{\beta}(\mathbb{B}^{n};\mathbb{R}^{n\times n})$ and

$\|B-\mathcal{P}_{0}\Delta
B\|_{\mathscr{C}^{\beta}(\mathbb{B}^{n})}\lesssim\|B\|_{\mathscr{C}^{\beta}(\partial\mathbb{B}^{n})}.$
(4.2.24)

We can rewrite (4.2.9) as

$-\sum_{i=1}^{n}\frac{\partial}{\partial
y^{i}}b_{i}^{k}=\sum_{i,j=1}^{n}\frac{\partial}{\partial
y^{i}}\left(\big{(}\sqrt{\det
h}h^{ij}-\delta^{ij}\big{)}b_{j}^{k}\right),\quad\text{in
}\mathbb{B}^{n}_{y},\quad k=1,\dots,n.$ (4.2.25)

The left hand side of (4.2.25) is $\vartheta_{\mathbb{R}^{n}}\eta^{k}$. By
Lemma 4.2.3, the right hand side of (4.2.25) is the derivatives of rational
functions of the components of $B$, which vanish to second order at $B=0$.
More precisely, using Lemma 4.2.3, we can rewrite (4.2.25) as

$\vartheta_{\mathbb{R}^{n}}\eta^{k}=-\sum_{i=1}^{n}\frac{\partial}{\partial
y^{i}}b_{i}^{k}=-\sum_{i=1}^{n}\frac{\partial}{\partial
y^{i}}\mathcal{R}_{i}^{k}(B),\quad\text{in }\mathbb{B}^{n}_{y},\quad
k=1,\dots,n.$ (4.2.26)

Here $\mathcal{R}_{i}^{k}$ are rational functions (see Lemma 4.2.3) defined in
a neighborhood of origin in $\mathbb{R}^{n\times n}$ with
$|\mathcal{R}_{i}^{k}(u)|\lesssim|u|_{\mathbb{R}^{n\times n}}^{2}$ for
suitably small matrices $u\in\mathbb{R}^{n\times n}$, and we have

$|\mathcal{R}_{i}^{k}(u_{1})-\mathcal{R}_{i}^{k}(u_{2})|\lesssim(|u_{1}|_{\mathbb{R}^{n\times
n}}+|u_{2}|_{\mathbb{R}^{n\times n}})|u_{1}-u_{2}|_{\mathbb{R}^{n\times
n}},\quad\text{when }u_{1},u_{2}\in\mathbb{R}^{n\times n}\text{ small}.$
(4.2.27)

We can pass this fact from matrices to matrix-valued functions. Indeed,
$\mathcal{R}_{i}^{k}$ has convergent power expansion in a neighborhood of $0$
as

$\mathcal{R}_{i}^{k}(u)=\sum_{r=2}^{\infty}\sum_{j_{1},\dots,j_{r},l_{1},\dots,l_{r}=1}^{n}a_{i,r;l_{1}\dots
l_{r}}^{k;j_{1}\dots j_{r}}u_{j_{1}}^{l_{1}}\dots
u_{j_{r}}^{l_{r}},\quad\text{converging when }|u|_{\mathbb{R}^{n\times
n}}\text{ is small.}$ (4.2.28)

Here $a_{i,r;l_{1}\dots l_{r}}^{k;j_{1}\dots j_{r}}\in\mathbb{R}$. The power
expansion starts at $r=2$ since the zero and the first order terms all vanish.

By Lemma 2.1.17, we can replace $u\in\mathbb{R}^{n\times n}$ in (4.2.28) by
$f\in\mathscr{C}^{\gamma}(\mathbb{B}^{n};\mathbb{R}^{n\times n})$, and as in
(4.2.27), for $\gamma>0$ there is a $\tilde{C}_{\mathcal{R},\gamma}>0$, such
that when $\|f_{1}\|_{\mathscr{C}^{\gamma}(\mathbb{B}^{n};\mathbb{R}^{n\times
n})}+\|f_{2}\|_{\mathscr{C}^{\gamma}(\mathbb{B}^{n};\mathbb{R}^{n\times
n})}\leq\tilde{C}_{\mathcal{R},\gamma}^{-1}$,

$\|\mathcal{R}_{i}^{k}(f_{1})-\mathcal{R}_{i}^{k}(f_{2})\|_{\mathscr{C}^{\gamma}(\mathbb{B}^{n})}\leq\tilde{C}_{\mathcal{R},\gamma}(\|f_{1}\|_{\mathscr{C}^{\gamma}(\mathbb{B}^{n};\mathbb{R}^{n\times
n})}+\|f_{2}\|_{\mathscr{C}^{\gamma}(\mathbb{B}^{n};\mathbb{R}^{n\times
n})})\|f_{1}-f_{2}\|_{\mathscr{C}^{\gamma}(\mathbb{B}^{n};\mathbb{R}^{n\times
n})},\quad 1\leq i,k\leq n.$ (4.2.29)

Using the fact that
$-\Delta\eta^{k}=d\vartheta_{\mathbb{R}^{n}}\eta^{k}+\vartheta_{\mathbb{R}^{n}}d\eta^{k}$,
we further have

$\Delta\eta^{k}=d\sum_{i=1}^{n}\frac{\partial}{\partial
y^{i}}\mathcal{R}_{i}^{k}(B)-\vartheta_{\mathbb{R}^{n}}d\eta^{k}=\sum_{i,j=1}^{n}\frac{\partial^{2}}{\partial
y^{j}\partial
y^{i}}\mathcal{R}_{i}^{k}(B)dy^{j}-\sum_{j=1}^{n}\left\langle\vartheta_{\mathbb{R}^{n}}d\eta^{k},\frac{\partial}{\partial
y^{j}}\right\rangle dy^{j}.$

Here $\langle\cdot,\cdot\rangle$ denotes the pairing between $1$ forms and
vector fields.

On the other hand,
$\Delta\eta^{k}=\sum_{j=1}^{n}\Delta(\delta_{j}^{k}+b_{j}^{k})dy^{j}=\sum_{j=1}^{n}\Delta
b_{j}^{k}dy^{j}$, therefore

$\Delta b_{j}^{k}=\left\langle\Delta\eta^{k},\frac{\partial}{\partial
y^{j}}\right\rangle=\sum_{i,j=1}^{n}\frac{\partial^{2}}{\partial y^{j}\partial
y^{i}}\mathcal{R}_{i}^{k}(B)-\left\langle\vartheta_{\mathbb{R}^{n}}d\eta^{k},\frac{\partial}{\partial
y^{j}}\right\rangle,\quad\text{in }\mathbb{B}^{n}_{y},\quad k=1,\dots,n.$
(4.2.30)

Let
$\tilde{\xi}_{0}:=\min(\tilde{C}_{\mathcal{R},\alpha}^{-1},\tilde{C}_{\mathcal{R},\beta}^{-1})$
where $\tilde{C}_{\mathcal{R},\gamma}$ is the constant in (4.2.29). Let
$\xi=\xi_{B}\in(0,\tilde{\xi}_{0}]$ to be determined. We define metric spaces
$\mathscr{X}_{\gamma,\xi}$ and an operator
$\mathcal{T}_{B}:\mathscr{X}_{\alpha,\tilde{\xi}_{0}}\to\mathscr{C}^{\alpha}(\mathbb{B}^{n};\mathbb{R}^{n\times
n})$ by

$\begin{gathered}\mathscr{X}_{\gamma,\xi}:=\\{f\in\mathscr{C}^{\gamma}(\mathbb{B}^{n};\mathbb{R}^{n\times
n}):\|f\|_{\mathscr{C}^{\gamma}}\leq\xi\\}\subset\mathscr{C}^{\gamma}(\mathbb{B}^{n};\mathbb{R}^{n\times
n}),\quad\text{for }\gamma\in\\{\alpha,\beta\\}\text{ and
}\xi\in(0,\tilde{\xi}_{0}].\\\
\mathcal{T}_{B}[f]^{k}_{j}:=b_{j}^{k}-\mathcal{P}_{0}\Delta
b_{j}^{k}-\Big{\langle}\mathcal{P}_{0}(\vartheta_{\mathbb{R}^{n}}d\eta^{k}),\frac{\partial}{\partial
y^{j}}\Big{\rangle}+\sum_{i=1}^{n}\mathcal{P}_{0}\Big{(}\frac{\partial^{2}\mathcal{R}_{i}^{k}(f)}{\partial
y^{j}\partial y^{i}}\Big{)},\quad 1\leq j,k\leq n.\end{gathered}$ (4.2.31)

We endow $\mathscr{X}_{\gamma,\xi}$ with the metric induced by the norm
$\|\cdot\|_{\mathscr{C}^{\gamma}}$, which makes $\mathscr{X}_{\gamma,\xi}$ a
complete metric space.

Note that from (4.2.31) and (4.2.30) we have $B=\mathcal{T}_{B}[B]$. Our goal
is to show that when $c_{3}$ and $\xi$ are both suitably small, we have
$B\in\mathscr{X}_{\alpha,\xi}$ and that $\mathcal{T}_{B}$ is a contraction
mapping on both $\mathscr{X}_{\alpha,\xi}$ and $\mathscr{X}_{\beta,\xi}$, thus
by uniqueness of the fixed point we conclude that $B$ is a
$\mathscr{C}^{\beta}$-matrix and $\|B\|_{\mathscr{C}^{\beta}}<\xi$.

By Lemma 2.1.41,
$\mathcal{P}_{0}:\mathscr{C}^{\gamma-2}(\mathbb{B}^{n};\mathbb{R}^{n\times
n})\to\mathscr{C}^{\gamma}(\mathbb{B}^{n};\mathbb{R}^{n\times n})$ is bounded
for $\gamma\in\\{\alpha,\beta\\}$. By (4.2.29), we know for every
$f_{1},f_{2}\in\mathscr{X}_{\gamma,\xi}$,

$\|\mathcal{T}_{B}[f_{1}]-\mathcal{T}_{B}[f_{2}]\|_{\mathscr{C}^{\gamma}}\leq\|\mathcal{P}_{0}\|_{\mathscr{C}^{\gamma-2}\to\mathscr{C}^{\gamma}}\|\nabla^{2}\|_{\mathscr{C}^{\gamma}\to\mathscr{C}^{\gamma-2}}\sum_{k=1}^{n}\|\mathcal{R}_{k}^{l}(f_{1})-\mathcal{R}_{k}^{l}(f_{2})\|_{\mathscr{C}^{\gamma}}\leq\xi
C^{\prime}_{\mathcal{R},\gamma}\|f_{1}-f_{2}\|_{\mathscr{C}^{\gamma}}.$
(4.2.32)

where $C^{\prime}_{\mathcal{R},\gamma}>1$ is a constant that only depends on
$n,(\mathcal{R}_{j}^{k}),\gamma$ but not on $B,\xi,f_{1},f_{2}$.

On the other hand $\mathcal{T}_{B}[0]_{j}^{k}=b_{j}^{k}-\mathcal{P}_{0}\Delta
b_{j}^{k}-\big{\langle}\mathcal{P}_{0}(\vartheta_{\mathbb{R}^{n}}d\eta^{k}),\frac{\partial}{\partial
y^{j}}\big{\rangle}$. By (4.2.24), for $\gamma\in\\{\alpha,\beta\\}$,

$\|\mathcal{T}_{B}[0]\|_{\mathscr{C}^{\gamma}}\leq\|\mathcal{T}_{B}[0]\|_{\mathscr{C}^{\beta}}\leq\|B-\mathcal{P}_{0}\Delta
B\|_{\mathscr{C}^{\beta}(\mathbb{B}^{n})}+\sum_{k=1}^{n}\|\mathcal{P}_{0}(\vartheta_{\mathbb{R}^{n}}d\eta^{k})\|_{\mathscr{C}^{\beta}}\lesssim\|B\|_{\mathscr{C}^{\beta}(\partial\mathbb{B}^{n})}+\sum_{k=1}^{n}\|d\eta^{k}\|_{\mathscr{C}^{\beta-1}}.$
(4.2.33)

So by possibly increasing $C^{\prime}_{\mathcal{R},\gamma}$, we have, for
$f_{1}\in\mathscr{X}_{\gamma,\xi}$, using (4.2.32) and (4.2.33),

$\|\mathcal{T}_{B}[f_{1}]\|_{\mathscr{C}^{\gamma}}\leq\|\mathcal{T}_{B}[f_{1}]-\mathcal{T}_{B}[0]\|_{\mathscr{C}^{\gamma}}+\|\mathcal{T}_{B}[0]\|_{\mathscr{C}^{\gamma}}\leq
C^{\prime}_{\mathcal{R},\gamma}\Big{(}\|B\|_{\mathscr{C}^{\beta}(\partial\mathbb{B}^{n})}+\sum_{l=1}^{n}\|d\eta^{l}\|_{\mathscr{C}^{\beta-1}}+\xi\|f_{1}\|_{\mathscr{C}^{\gamma}}\Big{)}.$
(4.2.34)

Take $c_{3}>0$ satisfying
$c_{3}<\frac{1}{4}\max(1,C^{\prime}_{\mathcal{R},\alpha},C^{\prime}_{\mathcal{R},\beta})^{-2}$,
and take

$\xi=\xi_{B}:=2\max(C^{\prime}_{\mathcal{R},\alpha},C^{\prime}_{\mathcal{R},\beta})\left(\|B\|_{\mathscr{C}^{\alpha}(\mathbb{B}^{n};\mathbb{R}^{n\times
n})}+\|B\|_{\mathscr{C}^{\beta}(\partial\mathbb{B}^{n})}+\sum_{k=1}^{n}\|d\eta^{k}\|_{\mathscr{C}^{\beta-1}}\right).$
(4.2.35)

By the assumption (4.2.22),
$\xi_{B}\leq\frac{1}{2}\max(C^{\prime}_{\mathcal{R},\alpha},C^{\prime}_{\mathcal{R},\beta})^{-1}<\tilde{\xi}_{0}$,
so $\mathcal{T}_{B}$ is defined on $\mathscr{X}_{\alpha,\tilde{\xi}_{0}}$ and
by (4.2.34) $\mathcal{T}_{B}$ maps $\mathscr{X}_{\gamma,\xi_{B}}$ into
$\mathscr{X}_{\gamma,\xi_{B}}$ for $\gamma\in\\{\alpha,\beta\\}$.

Since $\xi_{B}C^{\prime}_{\mathcal{R},\gamma}<\frac{1}{2}$ for
$\gamma\in\\{\alpha,\beta\\}$, using (4.2.32), $\mathcal{T}_{B}$ is a
contraction mapping on the domain $\mathscr{X}_{\gamma,\xi_{B}}$, for
$\gamma\in\\{\alpha,\beta\\}$.

Note that $\xi_{B}\geq\|B\|_{\mathscr{C}^{\alpha}}$, and so
$B\in\\{f\in\mathscr{C}^{\alpha}(\mathbb{B}^{n};\mathbb{R}^{n\times
n}):\|f\|_{\mathscr{C}^{\alpha}}\leq\xi_{B}\\}=\mathscr{X}_{\alpha,\xi_{B}}$.
Therefore, $B$ is a fixed point for $\mathcal{T}_{B}$ in
$\mathscr{X}_{\alpha,\xi_{B}}$, which is unique since $\mathcal{T}_{B}$ is a
contraction mapping on $\mathscr{X}_{\alpha,\xi_{B}}$.

On the other hand $\mathcal{T}_{B}$ also has a unique fixed point in
$\mathscr{X}_{\beta,\xi_{B}}\subsetneq\mathscr{X}_{\alpha,\xi_{B}}$.
Therefore, by uniqueness,
$B\in\mathscr{X}_{\beta,\xi_{B}}=\\{f\in\mathscr{C}^{\beta}(\mathbb{B}^{n};\mathbb{R}^{n\times
n}):\|f\|_{\mathscr{C}^{\beta}}\leq\xi_{B}\\}$. In particular,
$\|B\|_{\mathscr{C}^{\beta}}\leq\xi_{B}$. Thus by (4.2.35),

$\|B\|_{\mathscr{C}^{\beta}(\mathbb{B}^{n})}\leq\xi_{B}\lesssim_{\alpha,\beta}\|B\|_{\mathscr{C}^{\alpha}(\mathbb{B}^{n})}+\|B\|_{\mathscr{C}^{\beta}(\partial\mathbb{B}^{n})}+\sum_{l=1}^{n}\|d\eta^{l}\|_{\mathscr{C}^{\beta-1}}.$

Thus, we have established (4.2.23) which completes the proof. ∎

####  4.2.4 The proof of Theorem 4.2 and a qualitative regularity improving

Using Propositions 4.2.7 and 4.2.11 we can prove Theorem 4.2.

###### Proof of Theorem 4.2.

Let $c_{1},c_{2},c_{3}>0$ be the small constants in Proposition 4.2.7, Lemma
4.2.10, and Proposition 4.2.11. We take
$c=\frac{1}{2n^{2}}\min(c_{1},c_{2}c_{3})$ in the assumption of Theorem 4.2.

Let $F$, $R=F-\operatorname{id}$, $\Phi=F^{\mathrm{Inv}}$, $A$, $B$ and
$\eta^{i}=F_{*}\lambda^{i}$ be as in Proposition 4.2.7. Recall $\eta^{i}$ and
$B=(b_{i}^{j})$ are given in (4.2.3) and (4.2.4).

When the assumption (4.2.1) is satisfied, by Proposition 4.2.7 (ii) we have
that $F$ is $\mathscr{C}^{\alpha+1}$-diffeomorphism and satisfies
$B^{n}(F(0),\frac{1}{6})\subseteq
F(\frac{1}{3}\mathbb{B}^{n})\cap\frac{3}{4}\mathbb{B}^{n}$. And by (4.2.11),
we have
$\|F-\operatorname{id}\|_{\mathscr{C}^{\alpha+1}}=\|R\|_{\mathscr{C}^{\alpha+1}}\leq
c_{1}^{-1}\|A\|_{\mathscr{C}^{\alpha}}\leq\frac{1}{2n}c^{-1}\sum_{i=1}^{n}\|\lambda^{i}-dx^{i}\|_{\mathscr{C}^{\alpha}}$.
This implies half of the estimate (4.2.2).

By Lemma 4.2.10 (ii) and (iii), we have
$\|B\|_{\mathscr{C}^{\alpha}(\mathbb{B}^{n})}+\|B\|_{\mathscr{C}^{\beta}(\partial\mathbb{B}^{n})}<c_{2}^{-1}\|A\|_{\mathscr{C}^{\alpha}}$
and
$\|d\eta^{k}\|_{\mathscr{C}^{\beta-1}}<c_{2}^{-1}\|d\lambda^{k}\|_{\mathscr{C}^{\beta-1}}$,
$k=1,\dots,n$.

Thus,
$\|B\|_{\mathscr{C}^{\alpha}(\mathbb{B}^{n})}+\|B\|_{\mathscr{C}^{\beta}(\partial\mathbb{B}^{n})}+\sum_{k=1}^{n}\|d\eta^{k}\|_{\mathscr{C}^{\beta-1}}<c_{2}^{-1}\|A\|_{\mathscr{C}^{\alpha}}+c_{2}^{-1}\sum_{k=1}^{n}\|d\lambda^{k}\|_{\mathscr{C}^{\beta-1}}<2c_{2}^{-1}c<c_{3}$.
By (4.2.23) in Proposition 4.2.11, we get

$\displaystyle\sum_{k=1}^{n}\|\eta^{k}-dy^{k}\|_{\mathscr{C}^{\beta}}\leq
n\|B\|_{\mathscr{C}^{\beta}(\mathbb{B}^{n})}\leq
nc_{3}^{-1}\left(\|B\|_{\mathscr{C}^{\alpha}}+\|B\|_{\mathscr{C}^{\beta}(\partial\mathbb{B}^{n})}+\sum_{k=1}^{n}\|d\eta^{k}\|_{\mathscr{C}^{\beta-1}}\right)$
$\displaystyle\leq nc_{3}^{-1}\cdot
c_{2}^{-1}\left(\|A\|_{\mathscr{C}^{\alpha}}+\sum_{k=1}^{n}\|d\lambda^{k}\|_{\mathscr{C}^{\beta-1}}\right)\leq
n^{2}(c_{2}c_{3})^{-1}\sum_{k=1}^{n}\left(\|\lambda^{k}-dx^{k}\|_{\mathscr{C}^{\alpha}}+\|d\lambda^{k}\|_{\mathscr{C}^{\beta-1}}\right).$

This gives the second half of the estimate (4.2.2) since
$n^{2}(c_{2}c_{3})^{-1}\leq\frac{1}{2}c^{-1}$. ∎

In Theorem 4.2, we assumed (4.2.1) which is a smallness assumption. When
(4.2.1) is not satisfied, we may use a scaling argument to transfer to a
setting where it is satisfied, as the next result shows.

######  Proposition 4.2.12 (The scaling argument).

Let $\alpha>0$, $\beta\in[\alpha,\alpha+1]$ and let $\mu_{0},\tilde{c},M>0$.
There exists a
$\kappa_{0}=\kappa_{0}(n,\alpha,\beta,\mu_{0},\tilde{c},M)\in(0,\mu_{0}]$ that
satisfies the following:

Suppose
$\xi^{1},\dots,\xi^{n}\in\mathscr{C}^{\alpha}(\mu_{0}\mathbb{B}^{n};T^{*}\mathbb{R}^{n})$
such that $\xi^{i}\big{|}_{0}=dx^{i}\big{|}_{0}$ for $i=1,\dots,n$ and
$d\xi^{1},\dots,d\xi^{n}\in\mathscr{C}^{\beta-1}\left(\mu_{0}\mathbb{B}^{n};\wedge^{2}T^{*}\mathbb{R}^{n}\right)$
with estimate

$\sum_{i=1}^{n}\|\xi^{i}\|_{\mathscr{C}^{\alpha}(\mu_{0}\mathbb{B}^{n};T^{*}\mathbb{R}^{n})}+\|d\xi^{i}\|_{\mathscr{C}^{\beta-1}(\mu_{0}\mathbb{B}^{n};\wedge^{2}T^{*}\mathbb{R}^{n})}<M.$
(4.2.36)

Then there are 1-forms
$\lambda^{1},\dots,\lambda^{n}\in\mathscr{C}^{\alpha}(\mathbb{B}^{n};T^{*}\mathbb{B}^{n})$
such that

  1. (i)

$\lambda^{i}\big{|}_{\frac{1}{3}\mathbb{B}^{n}}=\frac{1}{\kappa_{0}}\cdot(\phi_{\kappa_{0}}^{*}\xi^{i})\big{|}_{\frac{1}{3}\mathbb{B}^{n}}$,
$i=1,\dots,n$, where $\phi_{\kappa_{0}}(x):=\kappa_{0}\cdot x$ is the scaling
map.

  2. (ii)

$\lambda^{1},\dots,\lambda^{n}$ satisfy the assumptions of Theorem 4.2 with
the constant $c=\tilde{c}$. That is,

     * •

$\lambda^{1},\dots,\lambda^{n}$ span the cotangent space at every point in
$\mathbb{B}^{n}$.

     * •

$\operatorname{supp}(\lambda^{i}-dx^{i})\subsetneq\frac{1}{2}\mathbb{B}^{n}$.

     * •

$\sum_{i=1}^{n}(\|\lambda^{i}-dx^{i}\|_{\mathscr{C}^{\alpha}}+\|d\lambda^{i}\|_{\mathscr{C}^{\beta-1}})\leq\tilde{c}$.

###### Proof.

First we construct 1-forms
$\rho^{1},\dots,\rho^{n}\in\mathscr{C}^{\beta}(\mu_{0}\mathbb{B}^{n};T^{*}\mathbb{R}^{n})$
such that for $i=1,\dots,n$,

  1. (a)

$\rho^{i}\big{|}_{0}=0$ and
$d\rho^{i}\big{|}_{\frac{\mu_{0}}{2}\mathbb{B}^{n}}=d\xi^{i}\big{|}_{\frac{\mu_{0}}{2}\mathbb{B}^{n}}$.

  2. (b)

There is a $C_{0}=C_{0}(n,\alpha,\beta,\mu_{0})>0$ that does not depend on
$\xi^{i}$, such that

$\|\rho^{i}\|_{\mathscr{C}^{\beta}(\mu_{0}\mathbb{B}^{n};T^{*}\mathbb{R}^{n})}\leq
C_{0}(\|\xi^{i}\|_{\mathscr{C}^{\alpha}(\mu_{0}\mathbb{B}^{n};T^{*}\mathbb{R}^{n})}+\|d\xi^{i}\|_{\mathscr{C}^{\beta-1}(\mu_{0}\mathbb{B}^{n};\wedge^{2}T^{*}\mathbb{R}^{n})}).$
(4.2.37)

Take a $\chi_{0}\in C_{c}^{\infty}(\mu_{0}\mathbb{B}^{n})$ such that
$\chi_{0}\big{|}_{\frac{\mu_{0}}{2}\mathbb{B}^{n}}\equiv 1$. Define

$\tilde{\rho}^{i}:=-\mathbf{\Gamma}\ast\vartheta
d(\chi_{0}\xi^{i})=-\mathbf{\Gamma}\ast\vartheta(\chi_{0}\cdot
d\xi^{i}+d\chi_{0}\wedge\xi^{i}),\quad\rho^{i}:=\tilde{\rho}^{i}-(\tilde{\rho}^{i}\big{|}_{0}),\quad
i=1,\dots,n.$ (4.2.38)

Recall $\vartheta$ is the codifferential from Notation 2.1.38, and
$\mathbf{\Gamma}$ is the fundamental solution of Laplacian as in (2.1.26). The
convolution is defined in $\mathbb{R}^{n}$ using Lemma 2.1.40, since the
support $\operatorname{supp}\vartheta
d(\chi_{0}\xi^{i})\subseteq\operatorname{supp}\chi_{0}\Subset\mu_{0}\mathbb{B}^{n}$
is compact.

Clearly $\rho^{i}\big{|}_{0}=0$. Similar to the proof of Proposition 2.1.44,
since $\chi_{0}\big{|}_{\frac{\mu_{0}}{2}\mathbb{B}^{n}}\equiv 1$, we have

$d\xi^{i}\big{|}_{\frac{\mu_{0}}{2}\mathbb{B}^{n}}=d(\chi_{0}\xi^{i})\big{|}_{\frac{\mu_{0}}{2}\mathbb{B}^{n}}=-(\vartheta
d+d\vartheta)(\mathbf{\Gamma}\ast
d(\chi_{0}\xi^{i}))\big{|}_{\frac{\mu_{0}}{2}\mathbb{B}^{n}}=-(\mathbf{\Gamma}\ast
d\vartheta
d(\chi_{0}\xi^{i}))\big{|}_{\frac{\mu_{0}}{2}\mathbb{B}^{n}}=d\tilde{\rho}^{i}\big{|}_{\frac{\mu_{0}}{2}\mathbb{B}^{n}}=d\rho^{i}\big{|}_{\frac{\mu_{0}}{2}\mathbb{B}^{n}}.$

So condition (a) is satisfied.

By Lemma 2.1.40 we have, for every $\mu>0$,

$\|\mathcal{G}\ast\vartheta\omega\|_{\mathscr{C}^{\beta}(\mu\mathbb{B}^{n};\wedge^{2}T^{*}\mathbb{R}^{n})}\lesssim_{\beta,\mu}\|\omega\|_{\mathscr{C}^{\beta-1}(\mu\mathbb{B}^{n};T^{*}\mathbb{R}^{n})},\quad\forall\omega\in\mathscr{C}^{\beta-1}_{c}(\mu\mathbb{B}^{n};T^{*}\mathbb{R}^{n}).$
(4.2.39)

Take $\omega=\vartheta(\chi_{0}\cdot d\xi^{i}+d\chi_{0}\wedge\xi^{i})$,
$\mu=\mu_{0}$ in (4.2.39) and by Lemma 2.1.17, we have

$\displaystyle\|\rho^{i}\|_{\mathscr{C}^{\beta}(\mu_{0}\mathbb{B}^{n};T^{*}\mathbb{R}^{n})}\leq\|\tilde{\rho}^{i}\|_{\mathscr{C}^{\beta}(\mu_{0}\mathbb{B}^{n};T^{*}\mathbb{R}^{n})}+|\tilde{\rho}^{i}(0)|\leq
2\|\tilde{\rho}^{i}\|_{\mathscr{C}^{\beta}(\mu_{0}\mathbb{B}^{n};T^{*}\mathbb{R}^{n})}$
(4.2.40) $\displaystyle\lesssim_{\beta,\mu_{0}}\|\vartheta(\chi_{0}\cdot
d\xi^{i}+d\chi_{0}\wedge\xi^{i})\|_{\mathscr{C}^{\beta-2}(\mu_{0}\mathbb{B}^{n};\wedge^{2}T^{*}\mathbb{R}^{n})}\lesssim_{\beta,\mu_{0}}\|\chi_{0}\|_{\mathscr{C}^{\alpha+1}}\|d\xi^{i}\|_{\mathscr{C}^{\beta-1}}+\|d\chi_{0}\|_{\mathscr{C}^{\alpha}}\|\xi^{i}\|_{\mathscr{C}^{\beta-1}}$
$\displaystyle\lesssim_{\alpha,\beta,\mu_{0},\chi_{0}}\|d\xi^{i}\|_{\mathscr{C}^{\beta-1}(\mu_{0}\mathbb{B}^{n};\wedge^{2}T^{*}\mathbb{R}^{n})}+\|\xi^{i}\|_{\mathscr{C}^{\alpha}(\mu_{0}\mathbb{B}^{n};T^{*}\mathbb{R}^{n})}.$

(4.2.40) gives us the $C_{0}$ for condition (b). This complete the proof of
(a) and (b) and we get $\rho^{1},\dots,\rho^{n}$ as desired.

Fix $\chi_{1}\in C_{c}^{\infty}(\frac{1}{2}\mathbb{B}^{n})$ such that
$\chi_{1}\big{|}_{\frac{1}{3}\mathbb{B}^{n}}\equiv 1$. For $\kappa>0$, let
$\phi_{\kappa}(x):=\kappa\cdot x$, so $\phi_{\kappa}$ maps
$\frac{1}{2}\mathbb{B}^{n}$ into $\frac{\mu_{0}}{2}\mathbb{B}^{n}$ when
$\kappa\in(0,\mu_{0}]$. For $\kappa\in(0,\mu_{0}]$, we define 1-forms
$\lambda_{\kappa}^{1},\dots,\lambda_{\kappa}^{n}$ and
$\tau_{\kappa}^{1},\dots,\tau_{\kappa}^{n}$ by

$\lambda_{\kappa}^{i}:=dx^{i}+\tfrac{1}{\kappa}\chi_{1}\cdot\phi_{\kappa}^{*}(\xi^{i}-dx^{i}),\quad\tau_{\kappa}^{i}:=\tfrac{1}{\kappa}\chi_{1}\cdot(\phi_{\kappa}^{*}\rho^{i})+\tfrac{1}{\kappa}\mathcal{G}\ast\vartheta\left(d\chi_{1}\wedge\phi_{\kappa}^{*}(\xi^{i}-dx^{i})\right),\quad
i=1,\dots,n.$ (4.2.41)

Since $\xi^{i}\in\mathscr{C}^{\alpha}$ and $\rho^{i}\in\mathscr{C}^{\beta}$,
we have
$\lambda_{\kappa}^{i}\in\mathscr{C}^{\alpha}(\mathbb{B}^{n};T^{*}\mathbb{R}^{n})$,
$\tau_{\kappa}^{i}\in\mathscr{C}^{\beta}(\mathbb{B}^{n};T^{*}\mathbb{R}^{n})$
(by (4.2.39)) and
$\operatorname{supp}(\lambda_{\kappa}^{i}-dx^{i})\subseteq\operatorname{supp}\chi_{1}\Subset\frac{1}{2}\mathbb{B}^{n}$.
And since $\chi_{1}\big{|}_{\frac{1}{3}\mathbb{B}^{n}}\equiv 1$ and
$\phi_{\kappa}^{*}dx=\kappa dx$,

$\lambda_{\kappa}^{i}\big{|}_{\frac{1}{3}\mathbb{B}^{n}}=dx^{i}+\tfrac{1}{\kappa}\cdot\phi_{\kappa}^{*}(\xi^{i}-dx^{i})\big{|}_{\frac{1}{3}\mathbb{B}^{n}}=\tfrac{1}{\kappa}(\phi_{\kappa}^{*}\xi^{i})\big{|}_{\frac{1}{3}\mathbb{B}^{n}}.$
(4.2.42)

We write $\xi^{i}$ and $\rho^{i}$, $i=1,\dots,n$ as

$\xi^{i}=dx^{i}+\sum_{j=1}^{n}a^{i}_{j}(x)dx^{j},\quad\rho^{i}=\sum_{j=1}^{n}b^{i}_{j}(x)dx^{j},\quad\text{where
}a^{i}_{j}\in\mathscr{C}^{\alpha}(\mu_{0}\mathbb{B}^{n}),\
b^{i}_{j}\in\mathscr{C}^{\beta}(\mu_{0}\mathbb{B}^{n}).$ (4.2.43)

By assumption $\xi^{i}\big{|}_{0}=dx^{i}\big{|}_{0}$ and
$\rho^{i}\big{|}_{0}=0$ for $i=1,\dots,n$, so $a^{i}_{j}(0)=b^{i}_{j}(0)=0$
for all $1\leq i,j\leq n$. And we have

$\lambda_{\kappa}^{i}=dx^{i}+\sum_{j=1}^{n}\chi_{1}(x)a^{i}_{j}(\kappa
x)dx^{j},\quad\tau_{\kappa}^{i}=\sum_{j=1}^{n}\Big{(}\chi_{1}(x)b^{i}_{j}(\kappa
x)dx^{j}+\mathcal{G}\ast\vartheta\big{(}a^{i}_{j}(\kappa x)d\chi_{1}\wedge
dx^{j}\big{)}\Big{)},\quad i=1,\dots,n.$ (4.2.44)

Since
$\phi_{\kappa}(\frac{1}{2}\mathbb{B}^{n})\subseteq\frac{\mu_{0}}{2}\mathbb{B}^{n}$
and $\operatorname{supp}\chi_{1}\Subset\frac{1}{2}\mathbb{B}^{n}$, by
condition (a) we have
$\chi_{1}\cdot\phi_{\kappa}^{*}d\rho^{i}=\chi_{1}\cdot\phi_{\kappa}^{*}d\xi^{i}$
for $i=1,\dots,n$. It follows that $d\lambda_{\kappa}^{i}=d\tau_{\kappa}^{i}$
for $i=1,\dots,n$; indeed,

$d\lambda_{\kappa}^{i}-d\tau_{\kappa}^{i}=\tfrac{1}{\kappa}d\chi_{1}\wedge\phi_{\kappa}^{*}(\xi^{i}-dx^{i})+\tfrac{1}{\kappa}\chi_{1}\cdot\phi_{\kappa}^{*}d\xi^{i}-\tfrac{1}{\kappa}d\chi_{1}\wedge\phi_{\kappa}^{*}(\xi^{i}-dx^{i})-\tfrac{1}{\kappa}\chi_{1}\cdot\phi_{\kappa}^{*}d\rho^{i}=\tfrac{1}{\kappa}\chi_{1}\cdot\phi_{\kappa}^{*}(d\xi^{i}-d\rho^{i})=0.$
(4.2.45)

Applying Lemmas 2.1.17 and 2.1.12 to $a^{i}_{j}$ we have

$\|\chi_{1}\cdot
a^{i}_{j}(\kappa\cdot)\|_{\mathscr{C}^{\alpha}(\mathbb{B}^{n})}\leq
2\|\chi_{1}\|_{\mathscr{C}^{\alpha}}\|a^{i}_{j}(\kappa\cdot)\|_{\mathscr{C}^{\alpha}(\mathbb{B}^{n})}\lesssim_{\alpha,\mu_{0}}\kappa^{\min(\alpha,\frac{1}{2})}\|\chi_{1}\|_{\mathscr{C}^{\alpha}}\|a^{i}_{j}\|_{\mathscr{C}^{\alpha}(\mu_{0}\mathbb{B}^{n})},\quad\forall\kappa\in(0,\mu_{0}].$
(4.2.46)

Using (4.2.44) and (4.2.46), we deduce that

$\sum_{i=1}^{n}\|\lambda_{\kappa}^{i}-dx^{i}\|_{\mathscr{C}^{\alpha}}\leq\sum_{i,j=1}^{n}\|\chi_{1}(x)a^{i}_{j}(\kappa
x)\|_{\mathscr{C}^{\alpha}}\lesssim_{\alpha,\mu_{0}}\kappa^{\min(\alpha,\frac{1}{2})}\|\chi_{1}\|_{\mathscr{C}^{\alpha}}\sum_{i,j=1}^{n}\|a^{i}_{j}\|_{\mathscr{C}^{\alpha}(\mu_{0}\mathbb{B}^{n})}.$
(4.2.47)

By (4.2.45) we have $d\lambda_{\kappa}^{i}=d\tau_{\kappa}^{i}$, and therefore

$\|d\lambda_{\kappa}^{i}\|_{\mathscr{C}^{\beta-1}(\mathbb{B}^{n};\wedge^{2}T^{*}\mathbb{B}^{n})}=\|d\tau_{\kappa}^{i}\|_{\mathscr{C}^{\beta-1}(\mathbb{B}^{n};\wedge^{2}T^{*}\mathbb{B}^{n})}\lesssim_{\beta}\|\tau_{\kappa}^{i}\|_{\mathscr{C}^{\beta}(\mathbb{B}^{n};T^{*}\mathbb{B}^{n})},\quad
i=1,\dots,n.$ (4.2.48)

Applying Lemma 2.1.12 to $a^{i}_{j}$ and $b^{i}_{j}$ we get that for
$0<\kappa<\mu_{0}$,

$\displaystyle\|\chi_{1}\cdot b^{i}_{j}(\kappa
x)\|_{\mathscr{C}^{\beta}(\mathbb{B}^{n})}\lesssim_{\beta}\|\chi_{1}\|_{\mathscr{C}^{\beta}}\|b^{i}_{j}(\kappa
x)\|_{\mathscr{C}^{\beta}(\mathbb{B}^{n})}\lesssim_{\beta}\kappa^{\min(\beta,\frac{1}{2})}\|\chi_{1}\|_{\mathscr{C}^{\beta}}\|b^{i}_{j}\|_{\mathscr{C}^{\beta}}\leq\kappa^{\min(\alpha,\frac{1}{2})}\|\chi_{1}\|_{\mathscr{C}^{\beta}}\|b^{i}_{j}\|_{\mathscr{C}^{\beta}(\mu_{0}\mathbb{B}^{n})}.$
(4.2.49) $\displaystyle\|a^{i}_{j}(\kappa
x)d\chi_{1}\|_{\mathscr{C}^{\alpha}(\mathbb{B}^{n};T^{*}\mathbb{R}^{n})}\lesssim_{\alpha}\|a^{i}_{j}(\kappa
x)\|_{\mathscr{C}^{\alpha}(\mathbb{B}^{n})}\|\chi_{1}\|_{\mathscr{C}^{\alpha+1}}\lesssim_{\alpha}\kappa^{\min(\alpha,\frac{1}{2})}\|\chi_{1}\|_{\mathscr{C}^{\alpha+1}}\|a^{i}_{j}\|_{\mathscr{C}^{\alpha}(\mu_{0}\mathbb{B}^{n})}.$
(4.2.50)

Letting $\omega=a_{j}^{i}(\kappa x)d\chi_{1}\wedge
dx^{j}\in\mathscr{C}^{\alpha}(\mathbb{B}^{n};T^{*}\mathbb{B}^{n})\subseteq\mathscr{C}^{\beta-1}(\mathbb{B}^{n};T^{*}\mathbb{B}^{n})$
and $\mu=1$ in (4.2.39), we see that

$\displaystyle\sum_{i=1}^{n}\|d\lambda_{\kappa}^{i}\|_{\mathscr{C}^{\beta-1}(\mathbb{B}^{n};\wedge^{2}T^{*}\mathbb{B}^{n})}=\sum_{i=1}^{n}\|d\tau_{\kappa}^{i}\|_{\mathscr{C}^{\beta-1}(\mathbb{B}^{n};\wedge^{2}T^{*}\mathbb{B}^{n})}\lesssim_{\beta}\sum_{i=1}^{n}\|\tau_{\kappa}^{i}\|_{\mathscr{C}^{\beta}(\mathbb{B}^{n};T^{*}\mathbb{B}^{n})}$
by (4.2.48) (4.2.51)
$\displaystyle\leq\sum_{i,j=1}^{n}\big{(}\|\chi_{1}(x)b_{j}^{i}(\kappa
x)dx^{j}\|_{\mathscr{C}^{\beta}}+\big{\|}\mathcal{G}\ast\vartheta\big{(}a^{i}_{j}(\kappa
x)d\chi_{1}\wedge dx^{j}\big{)}\big{\|}_{\mathscr{C}^{\beta}}\big{)}$ by
(4.2.44)
$\displaystyle\lesssim_{\beta}\sum_{i,j=1}^{n}\big{(}\|\chi_{1}(x)b_{j}^{i}(\kappa
x)\|_{\mathscr{C}^{\beta}(\mathbb{B}^{n})}+\|a^{i}_{j}(\kappa
x)d\chi_{1}\wedge
dx^{j}\big{\|}_{\mathscr{C}^{\beta-1}(\mathbb{B}^{n};\wedge^{2}T^{*}\mathbb{B}^{n})}\big{)}$
by (4.2.39)
$\displaystyle\lesssim_{\alpha,\beta}\|\chi_{1}\|_{\mathscr{C}^{\alpha+1}}\sum_{i,j=1}^{n}\big{(}\|b_{j}^{i}(\kappa
x)\|_{\mathscr{C}^{\beta}(\mathbb{B}^{n})}+\|a^{i}_{j}(\kappa
x)d\chi_{1}\big{\|}_{\mathscr{C}^{\alpha}(\mathbb{B}^{n};T^{*}\mathbb{B}^{n})}\big{)}$
$\displaystyle\text{since }\alpha\geq\beta-1$
$\displaystyle\lesssim_{\alpha,\beta}\kappa^{\min(\alpha,\frac{1}{2})}\|\chi_{1}\|_{\mathscr{C}^{\alpha+1}}\sum_{i,j=1}^{n}\Big{(}\|a^{i}_{j}\|_{\mathscr{C}^{\alpha}(\mu_{0}\mathbb{B}^{n})}+\|b^{i}_{j}\|_{\mathscr{C}^{\beta}(\mu_{0}\mathbb{B}^{n})}\Big{)}$
$\displaystyle\text{by \eqref{Eqn::Rough1Form::ScalingTmp2.1} and
\eqref{Eqn::Rough1Form::ScalingTmp2.2}}.$

Note that $\chi_{1}$ is a fixed cut-off function whose $\mathscr{C}^{\alpha}$
and $\mathscr{C}^{\alpha+1}$-norms depend only on $n,\alpha$. So combining
(4.2.47) and (4.2.51) we have

$\sum_{i=1}^{n}(\|\lambda_{\kappa}^{i}-dx^{i}\|_{\mathscr{C}^{\alpha}}+\|d\lambda_{\kappa}^{i}\|_{\mathscr{C}^{\beta-1}})\lesssim_{\alpha,\beta,\mu_{0}}\kappa^{\min(\alpha,\frac{1}{2})}\sum_{i,j=1}^{n}\big{(}\|a^{i}_{j}\|_{\mathscr{C}^{\alpha}(\mu_{0}\mathbb{B}^{n};T^{*}\mathbb{R}^{n})}+\|b^{i}_{j}\|_{\mathscr{C}^{\beta}(\mu_{0}\mathbb{B}^{n};\wedge^{2}T^{*}\mathbb{R}^{n})}\big{)}.$
(4.2.52)

By (4.2.43) we have
$\sum_{i,j=1}^{n}\|a^{i}_{j}\|_{\mathscr{C}^{\alpha}}\lesssim
1+\sum_{i=1}^{n}\|\xi^{i}\|_{\mathscr{C}^{\alpha}}$ and
$\sum_{i,j=1}^{n}\|b^{i}_{j}\|_{\mathscr{C}^{\beta}}\lesssim\sum_{i=1}^{n}\|\rho^{i}\|_{\mathscr{C}^{\beta}}$.
And combining (4.2.52) with (4.2.37), we can find a
$C_{1}=C_{1}(n,\alpha,\beta,\mu_{0})>0$ that does not depend on the other
quantities, such that

$\sum_{i=1}^{n}\|\lambda_{\kappa}^{i}-dx^{i}\|_{\mathscr{C}^{\alpha}}+\|d\lambda_{\kappa}^{i}\|_{\mathscr{C}^{\beta-1}}\leq
C_{1}\cdot\kappa^{\min(\alpha,\frac{1}{2})}\sum_{i=1}^{n}\Big{(}1+\|\xi^{i}\|_{\mathscr{C}^{\alpha}(\mu_{0}\mathbb{B}^{n};T^{*}\mathbb{R}^{n})}+\|d\xi^{i}\|_{\mathscr{C}^{\beta-1}(\mu_{0}\mathbb{B}^{n};\wedge^{2}T^{*}\mathbb{R}^{n})}\Big{)},\quad\forall\kappa\in(0,\mu_{0}].$
(4.2.53)

Now applying assumption (4.2.36) to (4.2.53) we have

$\sum_{i=1}^{n}\|\lambda_{\kappa}^{i}-dx^{i}\|_{\mathscr{C}^{\alpha}}+\|d\lambda_{\kappa}^{i}\|_{\mathscr{C}^{\beta-1}}\leq\kappa^{\min(\alpha,\frac{1}{2})}C_{1}\cdot(M+n).$

Since $\|(\langle\lambda_{\kappa}^{i}-dx^{i},\frac{\partial}{\partial
x^{j}}\rangle)_{n\times n}\|_{C^{0}(\mathbb{B}^{n};\mathbb{R}^{n\times
n})}\lesssim_{\alpha}\sum_{i=1}^{n}\|\lambda_{\kappa}^{i}-dx^{i}\|_{\mathscr{C}^{\alpha}(\mathbb{B}^{n})}$,
we can find a $\tilde{c}^{\prime}=\tilde{c}^{\prime}(n,\alpha)>0$ such that

$\sum_{i=1}^{n}\|\lambda_{\kappa}^{i}-dx^{i}\|_{\mathscr{C}^{\alpha}}\leq\tilde{c}^{\prime}\quad\text{implies}\quad\textstyle\|(\langle\lambda_{\kappa}^{i}-dx^{i},\frac{\partial}{\partial
x^{j}}\rangle)_{n\times n}\|_{C^{0}(\mathbb{B}^{n};\mathbb{R}^{n\times
n})}\leq\frac{1}{2}.$

In particular $\left(\langle\lambda_{\kappa}^{i},\frac{\partial}{\partial
x^{j}}\rangle\right)_{n\times
n}=I+\left(\langle\lambda_{\kappa}^{i}-dx^{i},\frac{\partial}{\partial
x^{j}}\rangle\right)_{n\times n}$ is invertible at every point in
$\mathbb{B}^{n}$, which means
$(\lambda_{\kappa}^{1},\dots,\lambda_{\kappa}^{n})$ span the tangent space at
every point in $\mathbb{B}^{n}$.

We take $\kappa_{0}=\kappa_{0}(n,\alpha,\beta,\mu_{0},M,\tilde{c})>0$ such
that

$0<\kappa_{0}<\mu_{0}\quad\text{and}\quad\kappa_{0}^{\min(\alpha,\frac{1}{2})}C_{1}\cdot(M+n)\leq\min(\tilde{c},\tilde{c}^{\prime}).$

Take $\lambda^{i}=\lambda_{\kappa_{0}}^{i}$ for $i=1,\dots,n$. We have
$\sum_{i=1}^{n}\|\lambda^{i}-dx^{i}\|_{\mathscr{C}^{\alpha}}+\|d\lambda^{i}\|_{\mathscr{C}^{\beta-1}}\leq\min(\tilde{c},\tilde{c}^{\prime})$.
By our assumption on $\tilde{c}^{\prime}$, $\lambda^{1},\dots,\lambda^{n}$
span the tangent space at every point in $\mathbb{B}^{n}$. Note that by
(4.2.44)
$\operatorname{supp}(\lambda_{\kappa_{0}}^{i}-dx^{i})\subseteq\operatorname{supp}\chi_{1}\Subset\frac{1}{2}\mathbb{B}^{n}$.
This shows conclusion (ii) is satisfied.

By (4.2.42) we get the conclusion (i), finishing the proof. ∎

We can give a sufficient and necessary condition for some 1-forms to have
better regularity in some special coordinate system:

######  Corollary 4.2.13.

Let $\alpha>0$ and let $\beta\in[\alpha,\alpha+1]$ be such that
$\alpha+\beta>1$. Let $\lambda^{1},\dots,\lambda^{n}$ be
$\mathscr{C}^{\alpha}$ 1-forms on a smooth manifold $\mathfrak{M}$ of
dimension $n$, which span the cotangent space at every point. Then following
are equivalent:

  1. (1)

For every $p\in\mathfrak{M}$ there exist a neighborhood
$U\subseteq\mathfrak{M}$ of $p$ and a
$\mathscr{C}^{\alpha+1}_{\mathrm{loc}}$-diffeomorphism
$\Phi:\mathbb{B}^{n}\xrightarrow{\sim}U\subseteq\mathfrak{M}$ such that
$\Phi(0)=p$ and
$\Phi^{*}\lambda^{1},\dots,\Phi^{*}\lambda^{n}\in\mathscr{C}^{\beta}(\mathbb{B}^{n};T^{*}\mathbb{B}^{n})$.

  2. (2)

$d\lambda^{1},\dots,d\lambda^{n}\in\mathscr{C}^{\beta-1}_{\mathrm{loc}}(\mathfrak{M};\wedge^{2}T^{*}\mathfrak{M})$.
(see Definition 3.1.1 (iii)).

######  Remark 4.2.14.

  1. (i)

When $X_{1},\dots,X_{n}$ are $\mathscr{C}^{\alpha}$ vector fields that span
the cotangent space at every point, we can apply Corollary 4.2.13 by taking
$(\lambda^{1},\dots,\lambda^{n})$ to be the dual basis of
$(X_{1},\dots,X_{n})$, and we see that
$\Phi^{*}\lambda^{1},\dots,\Phi^{*}\lambda^{n}$ are $\mathscr{C}^{\beta}$ if
and only if $\Phi^{*}X_{1},\dots,\Phi^{*}X_{n}$ are $\mathscr{C}^{\beta}$.

  2. (ii)

When $\beta>\alpha+1$, the result is still true if we replace
$d\lambda^{1},\dots,d\lambda^{n}\in\mathscr{C}^{\beta-1}_{\mathrm{loc}}$ by
$d\lambda^{1},\dots,d\lambda^{n}\in\mathscr{C}^{\beta-1}_{X,\mathrm{loc}}$ in
(2). See [SY22, Definition 2.15].

  3. (iii)

The assumption $\alpha+\beta>1$ can be removed. When $\alpha+\beta\leq 1$, in
particular $0<\alpha\leq\beta<1$, we cannot pullback a
$\mathscr{C}^{\beta-1}$-form under a $\mathscr{C}^{\alpha+1}$-map. However the
statement
$d\lambda^{1},\dots,d\lambda^{n}\in\mathscr{C}^{\beta-1}_{\mathrm{loc}}(\mathfrak{M};\wedge^{2}T^{*}\mathfrak{M})$
still make sense because for a 1-form $\xi\in\mathscr{C}^{0+}$ and a
$\mathscr{C}^{\alpha+1}$-diffeomorphism $\phi$, we have
$d\xi\in\mathscr{C}^{\beta-1}_{\mathrm{loc}}$ if and only if
$d(\phi^{*}\xi)\in\mathscr{C}^{\beta-1}_{\mathrm{loc}}$, see [SY22,
Proposition 2.8].

###### Proof.

By passing to a local coordinate system, we can assume $\mathfrak{M}$ to be an
open subset of $\mathbb{R}^{n}$.

(1)$\Rightarrow$(2): For such $\Phi$, we have
$d(\Phi^{*}\lambda^{i})\in\mathscr{C}^{\beta-1}_{\mathrm{loc}}(\mathbb{B}^{n};T^{*}\mathbb{B}^{n})$
for $i=1,\dots,n$. By Lemma 2.1.17 (i) along with direct computation (see
[SY22, Lemma 4.4]) we see that $\Phi^{*}d\lambda^{i}=d(\Phi^{*}\lambda^{i})$.
So $\Phi^{\mathrm{Inv}}:U\subseteq\mathfrak{M}\to\mathbb{R}^{n}$ is the
desired coordinate chart that shows
$d\lambda^{1},\dots,d\lambda^{n}\in\mathscr{C}^{\beta-1}_{\mathrm{loc}}$.

(2)$\Rightarrow$(1): Let $p\in\mathfrak{M}$.

By passing to local coordinate system and applying an invertible linear
transformation we can find a $\mu_{0}>0$ and a
$\mathscr{C}^{\alpha+1}$-coordinate chart
$F_{0}:U_{0}\subseteq\mathfrak{M}\xrightarrow{\sim}\mu_{0}\mathbb{B}^{n}_{x}$
such that

  * •

$F_{0}(p)=0$.

  * •

$((F_{0})_{*}\lambda^{i})\big{|}_{0}=dx^{i}\big{|}_{0}$ for $i=1,\dots,n$.

Take $c>0$ be the small constant in Theorem 4.2. By Proposition 4.2.12 with
$\tilde{c}=c$, we can find a $\kappa_{0}\in(0,\mu_{0}]$ and 1-forms
$\tilde{\lambda}^{1},\dots,\tilde{\lambda}^{n}\in\mathscr{C}^{\alpha}(\mathbb{B}^{n};T^{*}\mathbb{B}^{n})$
such that for scaling map
$\phi_{\kappa_{0}}:\mathbb{B}^{n}\to\mu_{0}\mathbb{B}^{n}$,
$\phi_{\kappa_{0}}(x)=\kappa_{0}x$, we have

  1. (a)

$\tilde{\lambda}^{1},\dots,\tilde{\lambda}^{n}$ span the cotangent space at
every point.

  2. (b)

$\operatorname{supp}(\tilde{\lambda}^{i}-dx^{i})\subsetneq\frac{1}{2}\mathbb{B}^{n}$
for $i=1,\dots,n$.

  3. (c)

$(\tilde{\lambda}^{1},\dots,\tilde{\lambda}^{n})\big{|}_{\frac{1}{3}\mathbb{B}^{n}}=\frac{1}{\kappa_{0}}\cdot((F_{0}^{-1}\circ\phi_{\kappa_{0}})^{*}\lambda^{1},\dots,(F_{0}^{-1}\circ\phi_{\kappa_{0}})^{*}\lambda^{n})\big{|}_{\frac{1}{3}\mathbb{B}^{n}}$.

  4. (d)

$\sum_{i=1}^{n}\|\tilde{\lambda}^{i}-dx^{i}\|_{\mathscr{C}^{\alpha}}+\|d\tilde{\lambda}^{i}\|_{\mathscr{C}^{\beta-1}}<c$.

We set $F_{1}:=\phi_{\kappa_{0}}^{\mathrm{Inv}}$.

Applying Theorem 4.2 to $\tilde{\lambda}^{1},\dots,\tilde{\lambda}^{n}$ we
obtain a $\mathscr{C}^{\alpha+1}$-chart
$F_{2}:\mathbb{B}^{n}\xrightarrow{\sim}\mathbb{B}^{n}$ such that
$F_{2}(\frac{1}{3}\mathbb{B}^{n})\supseteq B^{n}(F_{2}(0),\frac{1}{6})$,
$\|F_{2}-\operatorname{id}\|_{\mathscr{C}^{\alpha+1}}<c$ and
$(F_{2})_{*}\tilde{\lambda}^{1},\dots,(F_{2})_{*}\tilde{\lambda}^{n}\in\mathscr{C}^{\beta}(\mathbb{B}^{n};T^{*}\mathbb{B}^{n})$.

By (c), $\left((F_{2}\circ F_{1}\circ
F_{0})_{*}\lambda^{i}\right)\big{|}_{F_{2}(\frac{1}{3}\mathbb{B}^{n})}={\tfrac{1}{\kappa_{0}}}\cdot(F_{2})_{*}\tilde{\lambda}^{i}\big{|}_{F_{2}(\frac{1}{3}\mathbb{B}^{n})}\in\mathscr{C}^{\beta}$
for $i=1,\dots,n$. We can take an affine linear transformation
$F_{3}:\mathbb{R}^{n}\to\mathbb{R}^{n}$ such that $F_{3}(F_{2}(0))=0$ and
$F_{3}(B^{n}(F_{2}(0),\frac{1}{6}))\supseteq\mathbb{B}^{n}$.

Now we have $F_{0}(U_{0})\supseteq\mu_{0}\mathbb{B}^{n}$,
$F_{1}(\mu_{0}\mathbb{B}^{n})\supseteq\mathbb{B}^{n}$,
$F_{2}(\frac{1}{3}\mathbb{B}^{n})\supseteq B^{n}(F_{2}(0),\frac{1}{6})$ and
$F_{3}(B^{n}(F_{2}(0),\frac{1}{6})\supseteq\mathbb{B}^{n}$. Take
$\Phi:=(F_{3}\circ F_{2}\circ F_{1}\circ
F_{0})^{\mathrm{Inv}}:\mathbb{B}^{n}\to\mathfrak{M}$. Since
$F_{0},F_{1},F_{2},F_{3}$ are all $\mathscr{C}^{\alpha+1}$-diffeomorphism onto
their images, we know know
$\Phi:\mathbb{B}^{n}\xrightarrow{\sim}\Phi(\mathbb{B}^{n})\subseteq\mathfrak{M}$
is a $\mathscr{C}^{\alpha+1}$ diffeomorphism. Moreover, we have $\Phi(0)=p$
because $F_{0}(p)=0$, $F_{1}(0)=0$, $F_{3}(F_{2}(0))=0$. Thus, $\Phi$ is the
diffeomorphism we desire with $U:=\Phi(\mathbb{B}^{n})$, completing the proof.
∎

####  4.2.5 Remarks in harmonic coordinates

Given a non-smooth Riemannian metric $g$ on a manifold, $\mathfrak{M}$,
DeTurck and Kazdan showed that $g$ has optimal regularity in harmonic
coordinates [DK81, Lemma 1.2] (in the Zygmund-Hölder sense), but may not have
optimal regularity in geodesic normal coordinates [DK81, Example 2.3] (in
fact, the regularity of $g$ in geodesic normal coordinates may be two
derivatives worse than the regularity in harmonic coordinates).

We present analogous results for vector fields. Let $X_{1},\dots,X_{n}$ be
$C^{1}_{\mathrm{loc}}$-vector fields on a $C^{2}$-manifold $\mathfrak{M}$ that
form a local basis for the tangent space at every point. In Section 5.3.4 we
give an example showing that the canonical coordinate system with respect to
$X_{1},\dots,X_{n}$ may not have optimal regularity

In Section 4.2.1 we defined a Riemannian metric
$g=\sum_{i=1}^{n}\lambda^{i}\cdot\lambda^{i}$ where
$(\lambda^{1},\dots,\lambda^{n})$ is the dual basis of $(X_{1},\dots,X_{n})$
(see Remark 4.2.2). With respect to this metric, $X_{1},\dots,X_{n}$ form an
orthonormal basis at every point. Since $X_{1},\dots,X_{n}\in C^{1}$, we can
talk about the metric Laplacian $\Delta_{g}$ with respect to $g$.

######  Proposition 4.2.15.

In harmonic coordinates with respect to $g$, $X_{1},\dots,X_{n}$ have optimal
regularity.

More precisely, let $X_{1},\dots,X_{n}$ and $g$ be as above, and let
$\beta>1$. Suppose there is a $\mathscr{C}^{\beta+1}$-atlas $\mathfrak{A}$
which is compatible with the $C^{2}$-atlas of $\mathfrak{M}$, such that
$X_{1},\dots,X_{n}$ are $\mathscr{C}^{\beta}$ on $\mathfrak{A}$, let
$\psi:U\subseteq\mathfrak{M}\to V\subseteq\mathbb{R}^{n}$ be a harmonic
coordinate chart333Such a harmonic coordinate chart $\psi$ always exists
locally when $\beta>1$, see also [DK81, Lemma 1.2]., then
$\psi_{*}X_{1},\dots,\psi_{*}X_{n}\in\mathscr{C}^{\beta}_{\mathrm{loc}}(V;\mathbb{R}^{n})$.

###### Proof.

It suffices to show that $\psi$ is a
$\mathscr{C}_{\mathrm{loc}}^{\beta+1}$-map with respect to
$(\mathfrak{M},\mathfrak{A})$. Once this is done, using that
$X_{1},\dots,X_{n}$ are $\mathscr{C}^{\beta}_{\mathrm{loc}}$ with respect to
$(\mathfrak{M},\mathfrak{A})$, we get that
$\psi_{*}X_{1},\dots,\psi_{*}X_{n}\in\mathscr{C}^{\beta}_{\mathrm{loc}}$.

Since the statement $\psi\in\mathscr{C}_{\mathrm{loc}}^{\beta+1}(U;V)$ is
local, we may without loss of generality, shrink $U$. By doing so, the
hypotheses of the proposition imply that there is a
$\mathscr{C}^{\beta+1}$-coordinate chart
$x=(x^{1},\dots,x^{n}):U\subseteq\mathfrak{M}\to\mathbb{R}^{n}$ on $U$
(respect to $\mathfrak{A}$). In this coordinate chart we can write
$\Delta_{g}=\frac{1}{\sqrt{\det g}}\sum_{i,j=1}^{n}\frac{\partial}{\partial
x^{i}}(\sqrt{\det g}g^{ij}\frac{\partial}{\partial x^{j}})$ where $g^{ij}$ and
$\sqrt{\det g}$ are as in (4.2.7).

By assumption $\Delta_{g}\psi^{k}=0$ for $k=1,\dots,n$. Note that on
$(x^{1},\dots,x^{n})$, $\Delta_{g}$ is a second order divergent form elliptic
operator with $\mathscr{C}^{\beta}$-coefficients. By a classical elliptic
estimates (for example, [Tay11, Proposition 4.1]) we have that $\psi^{k}$ are
all $\mathscr{C}^{\beta+1}_{\mathrm{loc}}$ with respect to $\mathfrak{A}$,
completing the proof. ∎

######  Remark 4.2.16.

In fact the coordinate chart we construct in Proposition 4.2.7 (also see
(4.2.8)) is closely related to harmonic coordinates.

######  Remark 4.2.17.

While Proposition 4.2.15 shows that $X_{1},\dots,X_{n}$ have optimal
regularity with respect to harmonic coordinates, this fact along does not give
a practical test for what the optimal regularity is. Corollary 4.2.13, on the
other hand, provides such a test.

######  Remark 4.2.18.

Proposition 4.2.15 shows that harmonic coordinates induces a
$\mathscr{C}^{\beta+1}$-atlas with respect to which $X_{1},\dots,X_{n}$ are
$\mathscr{C}^{\beta}_{\mathrm{loc}}$. It is possible that the harmonic
coordinates induces some $\mathscr{C}^{\gamma+1}$-atlas for some
$\gamma>\beta$ while $X_{1},\dots,X_{n}$ are only $\mathscr{C}^{\beta}$ with
respect to this atlas; see Example 4.2.19 below.

######  Example 4.2.19.

Endow $\mathbb{R}^{2}$ with standard coordinates $(x,y)$, and let $\theta\in
C^{1}(\mathbb{R}^{2})$ be a function which is not smooth. Set
$X:=\cos(\theta(x,y))\frac{\partial}{\partial
x}+\sin(\theta(x,y))\frac{\partial}{\partial y}$ and
$Y:=-\sin(\theta(x,y))\frac{\partial}{\partial
x}+\cos(\theta(x,y))\frac{\partial}{\partial x}$. The corresponding metric is
$g=dx^{2}+dy^{2}$ since $X,Y$ form an orthonormal basis with respect to the
standard Euclidean metric, thus
$\Delta_{g}=\Delta=\partial_{x}^{2}+\partial_{y}^{2}$.

Therefore the singleton $\\{(x,y):\mathbb{R}^{2}\to\mathbb{R}^{2}\\}$ is an
atlas of harmonic coordinates for $\mathbb{R}^{2}$, and since harmonic
functions are real-analytic, we know the collection of harmonic coordinates
with respect to $\Delta_{g}$ defines an real-analytic structure for
$\mathbb{R}^{2}$ (which coincides with the standard real-analytic structure).
Even though the differential structure induced by the harmonic coordinates is
real analytic, $X$ and $Y$ cannot be smooth under any coordinate system (since
they are not smooth with respect to these harmonic coordinates).

##  Chapter 5 The Theorems and The Proofs

###  5.1 The Real Frobenius Theorems

####  5.1.1 History remark for rough involutivity

For vector fields $X$ and $Y$ which are not $C^{1}$, the commutator $[X,Y]$
may not be defined pointwise. Because of this, any Frobenius type theorem for
subbundles of less than $C^{1}$-regularity requires a suitable notion of
involutivity. This leads to the following:

######  Question 5.1.1.

Let $\mathcal{V}$ be a tangent subbundle which is not $C^{1}$, and let vector
fields $X,Y$ be sections of $\mathcal{V}$.

  * •

What is the good definition of the Lie bracket $[X,Y]$?

  * •

With this definition how can we say whether $[X,Y]$ is a section of
$\mathcal{V}$?

When $X$ and $Y$ are Lipschitz vector fields, the Lie bracket $[X,Y]$ makes
sense as a vector field with $L^{\infty}$-coefficients. And we can talk about
sections and involutivity almost everywhere. In [Sim96], Simić gives the
following definition:

######  Definition 5.1.2 (Almost everywhere characterization).

Let $\mathfrak{M}$ be a $C^{1,1}$-manifold, let $\mathcal{V}$ be a Lipschitz
subbundle of $T\mathfrak{M}$, and let $X,Y$ be two Lipschitz sections of
$\mathcal{V}$. We say $[X,Y]$ is an almost everywhere section of
$\mathcal{V}$, if as a vector field with $L^{\infty}$-coefficients,
$[X,Y](p)\in\mathcal{V}_{p}$ holds for almost every $p\in\mathfrak{M}$.

We say $\mathcal{V}$ is almost everywhere involutive, if for any Lipschitz
sections $X,Y$ of $\mathcal{V}$, the Lie bracket $[X,Y]$ is an almost
everywhere section of $\mathcal{V}$.

Later Rampazzo [Ram07] gave another characterizations using set-valued Lie
brackets:

######  Definition 5.1.3 (Set valued characterization).

Let $\mathfrak{M}$ be a $C^{1,1}$-manifold, let $\mathcal{V}$ be a Lipschitz
subbundle of $T\mathfrak{M}$, and let $X,Y$ be two Lipschitz sections of
$\mathcal{V}$. The set-valued Lie bracket $[X,Y]_{set}\subseteq\mathcal{V}$ is
defined as the convex hall
$[X,Y]_{set}(p):=\operatorname{conv}([X,Y]_{set}^{+}(p))\subseteq
T_{p}\mathfrak{M}$, $p\in\mathfrak{M}$, where

$[X,Y]_{set}^{+}:=\overline{\\{(q,[X,Y](q))\in T\mathfrak{M}:X,Y\text{ are
differentiable at }q\\}}\subseteq T\mathfrak{M}.$

And we say $\mathcal{V}$ is set-valuedly involutive, if
$[X,Y]_{set}\subseteq\mathcal{V}$ for every Lipschitz sections $X,Y$ of
$\mathcal{V}$.

Definitions 5.1.2 and 5.1.3 are equivalent, see [Ram07, Section 2] and [Ram07,
Theorem 4.11 (I) and (III)].

An advantage of set valued bracket is that it is defined pointwise everywhere.
Note that a Lipschitz function is differentiable almost everywhere, so
$[X,Y]_{set}(p)\neq\varnothing$ for all $p\in M$.

######  Proposition 5.1.4 (Lipschitz Frobenius theorem, [Sim96, Ram07]).

Let $\mathfrak{M}$ be a $n$-dimensional $C^{1,1}$ manifold, and let
$\mathcal{V}\leq T\mathfrak{M}$ be a Lipschitz tangent subbundle of rank $r$.
If $\mathcal{V}$ is either almost everywhere involutive or set-valuedly
involutive, then for any $p\in\mathfrak{M}$, there is a neighborhood
$\Omega\subseteq\mathbb{R}^{r}_{t}\times\mathbb{R}^{n-r}_{s}$ of $0$ and a bi-
Lipschitz parameterization $\Phi(t,s):\Omega\to\mathfrak{M}$ satisfying the
following:

  1. (i)

$\Phi(0)=p$.

  2. (ii)

$\frac{\partial\Phi}{\partial t^{1}},\dots,\frac{\partial\Phi}{\partial
t^{r}}:\Omega\to T\mathfrak{M}$ are Lipschitz continuous maps.

  3. (iii)

For any $(t,s)\in\Omega$, $\frac{\partial\Phi}{\partial
t^{1}}(t,s),\dots,\frac{\partial\Phi}{\partial t^{r}}(t,s)\in
T_{\Phi(t,s)}\mathfrak{M}$ form a basis of the linear subspace
$\mathcal{V}_{\Phi(t,s)}$.

Both [Sim96] and [Ram07] only deal with Lipschitz vector fields and Lipschitz
subbundles. What happen if they are non-Lipschitz? Indeed, $[X,Y]$ is not
necessarily $L^{1}_{\mathrm{loc}}$ so we cannot use an almost everywhere
characterization. A non-Lipschitz section can be nowhere differentiable so we
cannot use set-valued bracket either.

To resolve this problem, we use generalized functions. When the vector fields
$X,Y$ are log-Lipschitz, or more generally when $X,Y$ are Hölder
$\mathscr{C}^{\alpha}$ for some $\frac{1}{2}<\alpha<1$, by Corollary 2.1.19,
$[X,Y]$ is a vector field with $\mathscr{C}^{\alpha-1}$-coefficients, see
Lemma 2.1.34 and Definition 3.1.1.

By imitating Definition  ‣ 1.1 we can give a definition for the distributional
sections of $\mathcal{V}$ as follows:

######  Definition (Distributional sections of subbundles).

Let $\alpha>0$ and $\beta>-\alpha$. Let $\mathcal{V}$ be a
$\mathscr{C}^{\alpha}$ tangent subbundle of a manifold $\mathfrak{M}$. We say
a $\mathscr{C}^{\beta}$-vector field
$Z=\sum_{i=1}^{n}Z^{i}\frac{\partial}{\partial x^{i}}$ is a distributional
section of $\mathcal{V}$, if for any log-Lipschitz section $\theta$ of
$\mathcal{V}^{\bot}$, written as $\theta=\sum_{j=1}^{n}\theta_{j}dx^{j}$, the
generalized function111Using Lemma 2.1.18 (ii), the right hand side of (5.1.1)
makes sense as a distribution in
$\mathscr{C}^{\beta}_{\mathrm{loc}}(\mathfrak{M})$. Also see Definition 3.1.1.

$\langle\theta,Z\rangle=\sum_{j=1}^{n}Z^{j}\theta_{j}\in\mathscr{D}^{\prime}(\mathfrak{M}),$
(5.1.1)

is identically zero in the sense of distributions.

One can see that this is the characterization in Proposition 3.1.10 (S.4). See
also Remark 3.1.6 (c).

The characterization of distributional sections and distributional
involutivity recover the above characterizations when $\mathcal{V}$ is
Lipschitz. Indeed, for Lipschitz vector fields $X,Y$ that are sections of
$\mathcal{V}$, $[X,Y]=0$ lays in $\mathcal{V}$ almost everywhere if and only
if $\langle\theta,[X,Y]\rangle=0$, so Definitions  ‣ 1.1 and 5.1.2 are
equivalent.

Recently in [LTW19], a version of involutivity was introduced using
approximations by smooth functions. [LTW19] works on differential forms (see
[LTW19, Definition 1.12]). An equivalent characterization using vector fields
is as follows:

######  Definition 5.1.5.

Let $\mathcal{V}$ be a continuous rank-$r$ tangent subbundle over
$\mathbb{R}^{n}$. We say $\mathcal{V}$ is strongly asymptotically involutive,
if for any $p\in\mathbb{R}^{n}$ there is a neighborhood
$U\subseteq\mathbb{R}^{n}$ of $p$, and sequence of $C^{1}$-vector fields
$\\{X_{1}^{\nu},\dots,X^{\nu}_{r}\\}_{\nu=1}^{\infty}$ on $U$, such that

  1. (i)

For each $j=1,\dots,r$, $\\{X_{j}^{\nu}\\}_{\nu=1}^{\infty}$ uniformly
converges to a continuous vector field $X_{j}$, such that for each $q\in U$,
$X_{1}(q),\dots,X_{r}(q)\in\mathbb{R}^{n}$ form a linear basis of
$\mathcal{V}_{q}$.

  2. (ii)

There exists $\\{c_{jk}^{l\nu}\\}_{\nu=1}^{\infty}\subset C^{0}(\bar{U})$ for
$1\leq j,k,l\leq r$, such that

$\exists t_{0}>0,\ \lim\limits_{\nu\to\infty}\bigg{(}\max\limits_{1\leq
j,k\leq
r}\Big{\|}[X_{j}^{\nu},X_{k}^{\nu}]-\sum_{l=1}^{r}c_{jk}^{l\nu}X_{l}^{\nu}\Big{\|}_{C^{0}(U;\mathbb{R}^{n})}\bigg{)}\cdot\exp\big{(}t_{0}\cdot\max\limits_{1\leq
l\leq r}\|\nabla X_{l}^{\nu}\|_{C^{0}(U;\mathbb{R}^{n\times n})}\big{)}=0.$
(5.1.2)

In [LTW19] it is shown that if $\mathcal{V}$ is strongly asymptotically
involutive, then locally there exists a topological parameterization
$\Phi(t,s)\in C^{1}_{t}(C^{0}_{s})$ such that
$\Phi_{*}\frac{\partial}{\partial
t^{1}},\dots,\Phi_{*}\frac{\partial}{\partial t^{r}}$ span $\mathcal{V}$.
However, given a continuous subbundle, we do not know an algorithm to check
whether such an approximation exists. In contrast, Definition  ‣ 1.1 can be
directly checked using coordinate systems.

In the log-Lipschitz setting we prove that distributional involutivity implies
strongly asymptotically involutivity. See Corollary 2.1.50 and Remark 2.1.51.

####  5.1.2 The Frobenius theorem for log-Lipschitz subbundles: proof of
Theorem 1

For application to the proof of Theorems 4 and 5, we give a refinement of
estimates in the results of Theorem 1.

######  Theorem 5.1.

Keep the assumptions of Theorem 1, by shrinking the neighborhood $\Omega$ that
has the form
$\Omega=\Omega^{\prime}\times\Omega^{\prime\prime}\subseteq\mathbb{R}^{r}_{t}\times\mathbb{R}^{n-r}_{s}$
and write the inverse map $F:=\Phi^{\mathrm{Inv}}$ as
$F=(F^{\prime},F^{\prime\prime})$ where
$F^{\prime}:\Phi(\Omega)\to\mathbb{R}^{r}$ and
$F^{\prime\prime}:\Phi(\Omega)\to\mathbb{R}^{n-r}$, then $F^{\prime}\in
C^{1,1}_{\mathrm{loc}}(\Phi(\Omega);\mathbb{R}^{r})$. Moreover

  1. (5)

For any $\varepsilon>0$ there is an open subset
$\Omega^{\prime}_{\varepsilon}\subseteq\Omega^{\prime}$ of $0$, such that
$\Phi\in\mathscr{C}_{\mathrm{loc}}^{1+\mathrm{LogL},1-\varepsilon}(\Omega^{\prime}_{\varepsilon},\Omega^{\prime\prime};\mathfrak{M})$
and $\frac{\partial\Phi}{\partial t^{1}},\dots,\frac{\partial\Phi}{\partial
t^{r}}\in\mathscr{C}_{\mathrm{loc}}^{\mathrm{LogL},1-\varepsilon}(\Omega^{\prime}_{\varepsilon},\Omega^{\prime\prime};T\mathfrak{M})$.
Moreover by taking
$\Omega_{\varepsilon}=\Omega^{\prime}_{\varepsilon}\times\Omega^{\prime\prime}$,
and
$F^{\prime\prime}\in\mathscr{C}^{1-\varepsilon}_{\mathrm{loc}}(\Phi(\Omega_{\varepsilon});\mathbb{R}^{n-r})$.

  2. (5’)

Suppose $\mathcal{V}\in\mathscr{C}^{1}$. Then for the same
$\Omega^{\prime}_{\varepsilon}\subseteq\Omega^{\prime}$ in (5), automatically
$\Phi\in\mathscr{C}_{\mathrm{loc}}^{2,1-\varepsilon}(\Omega^{\prime}_{\varepsilon},\Omega^{\prime\prime};\mathfrak{M})$
and $\frac{\partial\Phi}{\partial t^{1}},\dots,\frac{\partial\Phi}{\partial
t^{r}}\in\mathscr{C}_{\mathrm{loc}}^{1,1-\varepsilon}(\Omega^{\prime}_{\varepsilon},\Omega^{\prime\prime};T\mathfrak{M})$.

  3. (5”)

Suppose $\mathcal{V}$ is little log-Lipschitz. Then automatically
$\Phi\in\mathscr{C}^{1+\mathrm{LogL},1-}(\Omega^{\prime},\Omega^{\prime\prime};\mathfrak{M})$,
$\frac{\partial\Phi}{\partial t^{1}},\dots,\frac{\partial\Phi}{\partial
t^{r}}\in\mathscr{C}^{\mathrm{LogL},1-}(\Omega^{\prime},\Omega^{\prime\prime};T\mathfrak{M})$.
and
$F^{\prime\prime}\in\mathscr{C}^{1-}_{\mathrm{loc}}(\Phi(\Omega);\mathbb{R}^{n-r})$.

  4. (5’’’)

Let
$\alpha\in\\{\mathrm{Lip}\\}\cup(1,\infty]\cup\\{k+\mathrm{Lip},k+\mathrm{LogL}:k=1,2,3,\dots\\}$.
Suppose $\mathfrak{M}$ is smooth and $\mathcal{V}\in\mathscr{C}^{\alpha}$ for
some $\alpha>1$, then automatically
$\Phi\in\mathscr{C}_{\mathrm{loc}}^{\alpha+1,\alpha}(\Omega^{\prime},\Omega^{\prime\prime};\mathfrak{M})$,
$\frac{\partial\Phi}{\partial t^{1}},\dots,\frac{\partial\Phi}{\partial
t^{r}}\in\mathscr{C}_{\mathrm{loc}}^{\alpha}(\Omega;T\mathfrak{M})$,
$F^{\prime}\in\mathscr{C}^{\infty}_{\mathrm{loc}}(\Phi(\Omega);\mathbb{R}^{r})$
and
$F^{\prime\prime}\in\mathscr{C}^{\alpha}_{\mathrm{loc}}(\Phi(\Omega);\mathbb{R}^{n-r})$.

Note that (5), (5”) and (5’’’) in Theorem 5.1 are stronger statement than
(4), (4’) and (4”) in Theorem 1, respectively.

###### Proof of Theorems 1 and 5.1.

By Remark 3.1.2 and by passing to a local coordinate chart of the manifold
$\mathfrak{M}$, we can assume that $\mathfrak{M}\subseteq\mathbb{R}^{n}$ is an
open subset with the base point $p=0\in\mathbb{R}^{n}$.

By Lemma 2.3.9 we can find a neighborhood $U\subseteq\mathfrak{M}$ of $p=0$, a
linear coordinate system $(x,y)=(x^{1},\dots,x^{r},y^{1},\dots,y^{n-r})$ for
$\mathbb{R}^{n}$, and commutative log-Lipschitz vector fields
$X_{1},\dots,X_{r}$ defined on $U$ that span $\mathcal{V}$ at every point in
$U$ and has the form (2.3.4), i.e. for some
$a_{j}^{k}\in\mathscr{C}^{\mathrm{LogL}}(U)$,

$X_{j}=\frac{\partial}{\partial
x^{j}}+\sum_{k=1}^{n-r}a_{j}^{k}\frac{\partial}{\partial y^{k}},\quad 1\leq
j\leq r.$ (5.1.3)

Let $\Omega^{\prime\prime}\subseteq\mathbb{R}^{n-r}$ be an open neighborhood
of $0\in\mathbb{R}^{n-r}$ such that
$\\{(0,s)\in\mathbb{R}^{r}\times\mathbb{R}^{n-r}:v\in\Omega^{\prime\prime}\\}\Subset
U$. We can take a small enough number $\tau_{0}>0$ such that the following map
$\Phi(t,s):B^{r}(0,\tau_{0})\times\Omega^{\prime\prime}\subset\mathbb{R}^{r}\times\mathbb{R}^{n-r}\to
U$ is defined:

$\Phi(t,s):=\Phi(t^{1},\dots,t^{r},s):=e^{t^{1}X_{1}}\dots
e^{t^{r}X_{r}}(0,s),\quad|t|<\tau_{0},\quad s\in\Omega^{\prime\prime}.$
(5.1.4)

Clearly $\Phi(0,0)=0$ which gives (1).

Applying Lemma 2.3.31 with $S=\\{0\\}\times\Omega^{\prime\prime}$, by choosing
$\tau_{0}>0$ small enough, $\Phi$ is well-defined, differentiable in $t$, and
satisfies

$\frac{\partial}{\partial t^{j}}\Phi(t,s)=X_{j}(\Phi(t,s)),\quad 1\leq j\leq
r,\quad|t|<\tau_{0},\quad s\in\Omega^{\prime\prime}.$

Since $X_{1},\dots,X_{r}$ are linearly independent and span $\mathcal{V}$ in
the domain, we see that $\frac{\partial\Phi}{\partial
t^{1}}(t,s),\dots,\frac{\partial\Phi}{\partial t^{r}}(t,s)$ spans
$\mathcal{V}_{\Phi(t,s)}$ for $(t,s)$ in the domain
$B^{r}(0,\tau_{0})\times\Omega^{\prime\prime}$. This gives (3).

Next we prove that $\Phi$ is homeomorphism onto its image by constructing the
inverse map. We still use $(x,y)$ as the standard coordinate system for
$\mathfrak{M}\subseteq\mathbb{R}^{n}$. Thus we have the natural coordinate
maps $x^{j}(q^{1},\dots,q^{n})=q^{j}$ for $1\leq j\leq r$ and
$y^{k}(q^{1},\dots,q^{n})=q^{r+k}$ for $1\leq k\leq n-r$ and
$q\in\mathbb{R}^{n}$. Define $F^{\prime}(q):=x(q)$ and
$F^{\prime\prime}=(\lambda^{1},\dots\lambda^{n-r}):\Phi(\Omega)\subseteq\mathbb{R}^{n}_{q}\to\mathbb{R}^{n-r}$
as

$\lambda^{j}(q):=y^{j}(e^{-q^{r}X_{r}}\dots e^{-q^{1}X_{1}}(q)),\quad 1\leq
j\leq n-r.$ (5.1.5)

We are going to show
$\Phi\circ(F^{\prime},F^{\prime\prime})=(F^{\prime},F^{\prime\prime})\circ\Phi=\operatorname{id}$.
This implies $F=(F^{\prime},F^{\prime\prime})=\Phi^{\mathrm{Inv}}$ and thus
$\Phi$ is homeomorphic onto its image.

Indeed, since $X_{j}$ are of the form (5.1.3), we see that $\Phi$ can be
written as

$\Phi(t,s)=(t,\phi(t,s)),\quad\text{where
}\phi:B^{r}(0,\tau_{0})\times\Omega^{\prime\prime}\to\mathbb{R}^{n-r}.$

Also we have

$e^{-q^{r}X_{r}}\dots e^{-q^{1}X_{1}}(q)=(0,y(e^{-q^{r}X_{r}}\dots
e^{-q^{1}X_{1}}(q))),\quad x(e^{-q^{r}X_{r}}\dots e^{-q^{1}X_{1}}(q))=0,\quad
q\in\Phi(B^{r}(0,\tau_{0})\times\Omega^{\prime\prime}).$ (5.1.6)

Therefore for $(t,s)\in B^{r}(0,\tau_{0})\times\Omega^{\prime\prime}$, using
$\Phi(t,s)=(t,\phi(t,s))$ we have

$\displaystyle(F^{\prime},F^{\prime\prime})(\Phi(t,s))=$
$\displaystyle(x(t,\phi(t,s)),y(e^{-t^{r}X_{r}}\dots
e^{-t^{1}X_{1}}(\Phi(t,s))))$ (5.1.7) $\displaystyle=$
$\displaystyle(t,y(e^{-t^{r}X_{r}}\dots e^{-t^{1}X_{1}}e^{t^{1}X_{1}}\dots
e^{t^{r}X_{r}}(0,s)))$ $\displaystyle=$ $\displaystyle(t,y(0,s))=(t,s).$

For $q\in\Phi(B^{r}(0,\tau_{0})\times\Omega^{\prime\prime})$, by (5.1.6) we
have

$\displaystyle\Phi((F^{\prime}(q),F^{\prime\prime}(q))=$
$\displaystyle\Phi(x(q),y(e^{-q^{r}X_{r}}\dots e^{-q^{1}X_{1}}(q)))$ (5.1.8)
$\displaystyle=$ $\displaystyle e^{q^{1}X_{1}}\dots
e^{q^{r}X_{r}}(0,y(e^{-q^{r}X_{r}}\dots e^{-q^{1}X_{1}}(q)))$ $\displaystyle=$
$\displaystyle e^{q^{1}X_{1}}\dots e^{q^{r}X_{r}}(e^{-q^{r}X_{r}}\dots
e^{-q^{1}X_{1}}(q))=q.$

We get $\Phi\circ(F^{\prime},F^{\prime\prime})=\operatorname{id}$ and
$(F^{\prime},F^{\prime\prime})\circ\Phi=\operatorname{id}$. Since
$B^{r}(0,\tau_{0})\times\Omega^{\prime\prime}$ is open, we know
$\Phi(B^{r}(0,\tau_{0})\times\Omega^{\prime\prime})=(x,\lambda)^{-1}(B^{r}(0,\tau_{0})\times\Omega^{\prime\prime})$
is also open. This completes the proof of (2) whenever $\Omega$ is an open
neighborhood of $0$ satisfying $\Omega\subseteq
B^{r}(0,\tau_{0})\times\Omega^{\prime\prime}$.

Now $\Phi^{\mathrm{Inv}}=(F^{\prime},F^{\prime\prime})$ and it remains to
prove the additional results in Theorem 5.1, which implies (4) and (4’) and
(4”).

Since $F^{\prime}(q)=x(q)$ arise from the coordinate chart, the regularity of
$F^{\prime}$ is the same as the regularity assumption for $\mathfrak{M}$.

By Lemma 2.3.9 (ii), since the log-Lipschitz vector fields $X_{1},\dots,X_{r}$
in (5.1.3) are uniquely determined by $\mathcal{V}$ and the coordinate system
$(x^{1},\dots,x^{r},y^{1},\dots,y^{n-r})$, we see that

  * •

If $\mathcal{V}$ is $\mathscr{C}^{\alpha}$ where
$\alpha\in\\{\mathrm{logl},\mathrm{Lip}\\}\cup[1,\infty]\cup\\{k+\mathrm{LogL},k+\mathrm{Lip}:k=1,2,3,\dots\\}$,
then $X_{1},\dots,X_{r}$ are automatically $\mathscr{C}^{\alpha}$ as well.

By Corollary 2.3.42 we see that

  * •

For every $\varepsilon>0$ there exists $\tau_{\varepsilon}\in(0,\tau_{0})$
such that
$\Phi\in\mathscr{C}^{1+\mathrm{LogL},1-\frac{\varepsilon}{2}}(B^{r}(0,\tau_{\varepsilon}),\Omega^{\prime\prime};U)$
and
$F^{\prime\prime}\in\mathscr{C}^{1-\frac{\varepsilon}{2}}(\Phi(B^{r}(0,\tau_{\varepsilon})\times\Omega^{\prime\prime});\mathbb{R}^{n-r})$.
In particular by Lemma 2.2.38 (i), $\frac{\partial\Phi}{\partial
t^{j}}=X_{j}\circ\Phi\in\mathscr{C}^{\mathrm{LogL},1-\varepsilon}(B^{r}(0,\tau_{\varepsilon}),\Omega^{\prime\prime};U)$
for $1\leq j\leq r$.

  * •

If $X_{1},\dots,X_{r}\in\mathscr{C}^{1}$, then automatically
$\Phi\in\mathscr{C}^{2}L^{\infty}(B^{r}(0,\tau_{0}),\Omega^{\prime\prime};U)$
and $\frac{\partial\Phi}{\partial
t^{j}}\in\mathscr{C}^{1}L^{\infty}(B^{r}(0,\tau_{0}),\Omega^{\prime\prime};U)$
for $1\leq j\leq r$. In particular
$\Phi\in\mathscr{C}^{2,1-\frac{\varepsilon}{2}}(B^{r}(0,\tau_{\varepsilon}),\Omega^{\prime\prime};U)$
and $\frac{\partial\Phi}{\partial t^{1}},\dots,\frac{\partial\Phi}{\partial
t^{r}}\in\mathscr{C}^{1,1-\varepsilon}(B^{r}(0,\tau_{\varepsilon}),\Omega^{\prime\prime};U)$.

  * •

If $X_{1},\dots,X_{r}\in\mathscr{C}^{\mathrm{logl}}$, then automatically
$\Phi\in\mathscr{C}^{1+\mathrm{LogL},1-}(B^{r}(0,\tau_{0}),\Omega^{\prime\prime};U)$
and
$F^{\prime\prime}\in\mathscr{C}^{1-}(\Phi(B^{r}(0,\tau_{0})\times\Omega^{\prime\prime});\mathbb{R}^{n-r})$.
In particular $\frac{\partial\Phi}{\partial
t^{j}}=X_{j}\circ\Phi\in\mathscr{C}^{\mathrm{LogL},1-}(B^{r}(0,\tau_{0}),\Omega^{\prime\prime};U)$
for $1\leq j\leq r$.

By Lemmas 2.3.41 and 2.2.38 (i) and Corollary 2.1.31 (i) we see that

  * •

If $X_{1},\dots,X_{r}$ are all $\mathscr{C}^{\alpha}$ where
$\alpha\in\\{\mathrm{Lip}\\}\cup(1,\infty]\cup\\{k+\mathrm{LogL},k+\mathrm{Lip}:k=1,2,3,\dots\\}$,
then
$\Phi\in\mathscr{C}^{\alpha+1,\alpha}(B^{r}(0,\tau_{0}),\Omega^{\prime\prime};U)$
and
$F^{\prime\prime}\in\mathscr{C}^{\alpha}(\Phi(B^{r}(0,\tau_{0})\times\Omega^{\prime\prime});\mathbb{R}^{n-r})$.
In particular $\frac{\partial\Phi}{\partial
t^{j}}=X_{j}\circ\Phi\in\mathscr{C}^{\alpha}(B^{r}(0,\tau_{0})\times\Omega^{\prime\prime};U)$
for $1\leq j\leq r$.

Take $\Omega^{\prime}:=B^{r}(0,\tau_{0})$ and
$\Omega^{\prime}_{\varepsilon}:=B^{r}(0,\tau_{\varepsilon})$. The above
results give (5), (5’), (5”) and (5’’’) respectively, finishing the whole
proof. ∎

####  5.1.3 A PDE counterpart of Theorem 1

We can interpret Theorem 1 in terms of the first-order PDE system following
from [Ram07, Section 5]:

######  Theorem 5.2.

Let $\mathfrak{M}$ be a $n$-dimensional $C^{1,1}$-manifold. Let
$L_{1},\dots,L_{r}$ be some first order differential operators on
$\mathfrak{M}$ with log-Lipschitz coefficients, such that

$L_{j}L_{k}-L_{k}L_{j}=\sum_{l=1}^{r}c_{jk}^{l}\cdot L_{l},$ (5.1.9)

for some
$c_{jk}^{l}\in\mathscr{C}^{\mathrm{LogL}-1}_{\mathrm{loc}}(\mathfrak{M})$,
$1\leq j,k,l\leq r$, in the sense of distributions.

Suppose there is a $(n-r)$-dimensional $C^{1}$-submanifold
$\mathfrak{S}\subset\mathfrak{M}$ such that

$\operatorname{Span}(L_{1}|_{q}\dots,L_{r}|_{q})\oplus
T_{q}\mathfrak{S}=T_{q}\mathfrak{M},\qquad\forall q\in\mathfrak{S}.$ (5.1.10)

Let $0<\beta\leq 1$ and
$h\in\mathscr{C}^{\beta}_{\mathrm{loc}}(\mathfrak{S})$. Then for any
$p\in\mathfrak{S}$ and any $0<\varepsilon<\beta$, there is a neighborhood
$U\subseteq\mathfrak{M}$ of $p$, such that there exists a unique solution
$f\in\mathscr{C}^{\beta-\varepsilon}_{\mathrm{loc}}(U)$ to the Cauchy problem

$\begin{cases}L_{j}f=0,&1\leq j\leq r,\\\ f|_{\mathfrak{S}\cap
U}=h.\end{cases}$ (5.1.11)

Here $L_{j}f=0$ holds in the sense of distributions.

Applying Lemma 2.1.18 (ii) with
$L_{j}\in\mathscr{C}^{1-}_{\mathrm{loc}}(U;T\mathfrak{M})$ and
$df\in\mathscr{C}^{\beta-1-\varepsilon}_{\mathrm{loc}}(U;T^{*}\mathfrak{M})$,
we know that $L_{j}f\in C^{\beta-1-\varepsilon}_{\mathrm{loc}}(U)$ is defined
as a distribution.

If in addition $L_{1},\dots,L_{r}$ have little log-Lipschitz coefficients, we
can show that $f$ given (5.1.11) is indeed $\mathscr{C}^{\beta-}$. We omit the
proof to reader.

The proof is based on the construction of $\Phi$ in (5.1.4), along with the
following proposition on inverse functions.

######  Lemma 5.1.6.

Let $(x,y)=(x^{1},\dots,x^{r},y^{1},\dots,y^{n-r})$ be the standard coordinate
system of $\mathbb{R}^{r}\times\mathbb{R}^{n-r}$. Let
$U\subseteq\mathbb{R}^{n}$ be an open neighborhood of $0$. Let
$X_{1},\dots,X_{r}$ be commutative log-Lipschitz vector fields on $U$ that
have the form $X_{j}=\frac{\partial}{\partial
x^{j}}+\sum_{k=1}^{n-r}a_{j}^{k}\frac{\partial}{\partial y^{k}}$ for $1\leq
j\leq r$.

Let $\Gamma=(\Gamma^{1},\dots,\Gamma^{n-r}):\mathbb{R}^{n-r}\to\mathbb{R}^{r}$
be a $C^{1}$-map such that $\Gamma(0)=0$ and $\nabla\Gamma(0)=0$. There is a
neighborhood $U_{0}^{\prime\prime}\subseteq\mathbb{R}^{n-r}$ such that

$e^{\Gamma(s)\cdot X}(0,\mu(s))=(\Gamma(s),s),$

defines a unique map $\mu:U_{0}^{\prime\prime}\to\mathbb{R}^{n-r}$. Moreover,

  * •

$\mu$ is hemeomorphism onto its image. In particular
$\mu(U_{0}^{\prime\prime})\subseteq\mathbb{R}^{n-r}$ is open.

  * •

For any $\varepsilon>0$ there is a smaller neighborhood
$U_{\varepsilon}^{\prime\prime}\subseteq U_{0}^{\prime\prime}$ such that
$\mu\in\mathscr{C}^{1-\varepsilon}(U_{\varepsilon}^{\prime\prime};\mathbb{R}^{n-r})$
and
$\mu^{\mathrm{Inv}}\in\mathscr{C}^{1-\varepsilon}(\mu(U_{\varepsilon}^{\prime\prime});\mathbb{R}^{n-r})$.

###### Proof.

By assumption $X_{j}=\frac{\partial}{\partial
x^{j}}+\sum_{k=1}^{n-r}a_{j}^{k}\frac{\partial}{\partial y^{k}}$, we get
$\mu(s)=y(e^{-\Gamma(s)\cdot X}(\Gamma(s),s))$. Thus $\mu$ is uniquely defined
and is a continuous map in the domain and satisfies $\mu(0)=0$. By Lemma
2.3.31 (since $X_{1},\dots,X_{r}$ are commutative) we see that for every
$0<\varepsilon<1$ there is a neighborhood
$\tilde{U}_{\varepsilon}^{\prime\prime}$ of $0$ such that
$\mu\in\mathscr{C}^{1-\varepsilon}(\tilde{U}_{\varepsilon}^{\prime\prime};\mathbb{R}^{n-r})$.

It remains to show that for every $0<\varepsilon<1$ there are a
$c_{\varepsilon}>0$ and a neighborhood
$U^{\prime\prime}_{\varepsilon}\subseteq\tilde{U}_{\varepsilon}^{\prime\prime}$
of $0$, such that

$|e^{-\Gamma(s_{1})\cdot X}(\Gamma(s_{1}),s_{1})-e^{-\Gamma(s_{2})\cdot
X}(\Gamma(s_{2}),s_{2})|\geq
c|s_{1}-s_{2}|^{\frac{1}{1-\varepsilon}},\quad\forall s_{1},s_{2}\in