&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 4.5 (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 (5.3) 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 (5.3) we have,

$b_{j}^{l}\circ
H=\left(\begin{pmatrix}I&H^{\prime\prime}_{\tau}+A^{\prime}H^{\prime\prime}_{\bar{w}}\\\
&H^{\prime\prime}_{w}+A^{\prime\prime}H^{\prime\prime}_{\bar{w}}\end{pmatrix}^{-1}\\!\\!\begin{pmatrix}\bar{H}^{\prime\prime}_{\tau}+A^{\prime}\bar{H}^{\prime\prime}_{\bar{w}}\\\
\bar{H}^{\prime\prime}_{w}+A^{\prime\prime}\bar{H}^{\prime\prime}_{\bar{w}}\end{pmatrix}\right)_{j}^{l},\quad
1\leq j\leq m+r,\quad 1\leq l\leq m.$ (5.11)

By (5.10) we have, for $1\leq j\leq r$ and $1\leq k\leq m$,

$\left.\frac{\partial}{\partial\sigma^{j}}\right|_{H(\tau,w,s)}=\left(\begin{pmatrix}I&H^{\prime\prime}_{\tau}&\bar{H}^{\prime\prime}_{\tau}\\\
&H^{\prime\prime}_{w}&\bar{H}^{\prime\prime}_{w}\\\
&H^{\prime\prime}_{\bar{w}}&\bar{H}^{\prime\prime}_{\bar{w}}\end{pmatrix}^{-1}\;\begin{pmatrix}\frac{\partial}{\partial\tau}\\\
\frac{\partial}{\partial w}\\\
\frac{\partial}{\partial\bar{w}}\end{pmatrix}\right)_{j},\
\left.\frac{\partial}{\partial\bar{\zeta}^{k}}\right|_{H(\tau,w,s)}=\left(\begin{pmatrix}I&H^{\prime\prime}_{\tau}&\bar{H}^{\prime\prime}_{\tau}\\\
&H^{\prime\prime}_{w}&\bar{H}^{\prime\prime}_{w}\\\
&H^{\prime\prime}_{\bar{w}}&\bar{H}^{\prime\prime}_{\bar{w}}\end{pmatrix}^{-1}\;\begin{pmatrix}\frac{\partial}{\partial\tau}\\\
\frac{\partial}{\partial w}\\\
\frac{\partial}{\partial\bar{w}}\end{pmatrix}\right)_{r+m+k}.$ (5.12)

Plugging (5.11) and (5.12) into (4.3) 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 (5.5).

Since $\mathcal{S}$ is involutive, $(H\circ L)_{*}\mathcal{S}$ is also
involutive, by Lemma 3.9 (ii), $T_{1},\dots,T_{r},Z_{1},\dots,Z_{m}$ given in
(5.3) 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 (5.5), finishing the proof of (ii). ∎

Now we have the PDE system (5.5). We show that it satisfies the assumption of
Proposition 4.7.

######  Lemma 5.7.

Suppose
$B:\mathbb{B}^{r+2m}_{\sigma,\zeta}\times\mathbb{B}^{q}_{s}\to\mathbb{C}^{(r+m)\times
m}$ solves (5.5), 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 4.7.

###### Proof.

We can write (5.5) 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 [Str20, 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 (5.1) using Proposition 3.31. 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 (5.3) 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 5.8.

Suppose $T_{1},\dots,T_{r},Z_{1},\dots,Z_{m}$ in (5.3) 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
$\mathfrak{B}\in\mathscr{C}^{\infty}_{\mathrm{loc}}\mathscr{C}^{\beta}(\frac{1}{2}\mathbb{H}^{r+2m},\frac{1}{2}\mathbb{B}^{q};\mathbb{C}^{(r+m)\times
m})$. 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
$G^{\prime\prime}\in\mathscr{C}^{\infty}\mathscr{C}^{\beta}(U_{1},V_{1};\mathbb{C}^{m})$
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 (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 3.31. 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 3.31 are satisfied. By Proposition 3.31 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
$\widetilde{G}^{\prime\prime}\in\mathscr{C}^{\infty}\mathscr{C}^{\beta}(\tilde{U}_{1},V_{1};\mathbb{C}^{m})$
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 $\widetilde{G}^{\prime\prime}|_{U_{1}\times
V_{1}}\in\mathscr{C}^{\infty}\mathscr{C}^{\beta}(U_{1},V_{1};\mathbb{C}^{m})$
holds. Thus, $G^{\prime\prime}:=\widetilde{G}^{\prime\prime}|_{U_{1}\times
V_{1}}$ is as desired. ∎

###  5.3 Proof of Theorem 5.1

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

######  Convention 5.9.

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

$\operatorname{Span}\Big{(}\sum_{j=1}^{n}\frac{\partial
f^{i}}{\partial\mu^{j}}(x,s)d\mu^{j}|_{x}\Big{)}_{i=1}^{r}=\\{\theta\in
T^{*}_{x}M:\langle\theta,X\rangle=0,\ \forall
X\in\mathcal{T}_{(x,s)}\\},\quad\forall\ u\in M,\ s\in N.$ (5.13)

######  Remark 5.10.

When $\beta>1$ and when $f\in\mathscr{C}^{\beta}$, (5.13) 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 $\beta\leq 1$, since when
$f\in\mathscr{C}^{\alpha+1,\beta}$, 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 $\eta=\operatorname{Im}\zeta$.

By Lemma 5.7 and Proposition 4.7, along with a scaling, we can find a
$\varepsilon_{0}=\varepsilon_{0}(r,m,q,\alpha)>0$ such that

  1. (B.1)

If
$B\in\mathscr{C}^{\alpha}L^{\infty}\cap\mathscr{C}^{-1}\mathscr{C}^{\beta}(\mathbb{B}^{r+2m}_{\sigma,\eta,\xi},\mathbb{B}^{q}_{s};\mathbb{C}^{(r+m)\times
m})$ satisfies (5.5) with

$\|B\|_{\mathscr{C}^{\alpha}L^{\infty}\cap\mathscr{C}^{-1}\mathscr{C}^{\beta}(\mathbb{B}^{r+2m},\mathbb{B}^{q})}<\varepsilon_{0},$
(5.14)

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

For such $\varepsilon_{0}>0$, we can find a $\delta_{0}>0$ such that the
results of Proposition 4.3 are satisfied with $\varepsilon=\varepsilon_{0}$,
$\delta=\delta_{0}$.

For such $\delta_{0}>0$, by Lemma 5.5 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 (5.2), where (5.6) is satisfied with $\delta=\delta_{0}$.

Now
$\|A\|_{\mathscr{C}^{\alpha,\beta}(\mathbb{B}^{r+2m},\mathbb{B}^{q};\mathbb{C}^{(r+m)\times
m})}<\delta_{0}$ from (5.6). By Proposition 4.3 and Corollary 4.4 we can find
a $\mathscr{C}^{\alpha+1,\beta}$-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)

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

  4. (H.4)

$H^{\prime\prime}$ solves (4.3).

  5. (H.5)

(4.4) 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 5.6 (ii), (H.4) implies that $B$ solves (5.5). By comparing (4.2) and
(5.3) we see that $B=\Lambda[A,H^{\prime\prime}]\circ H^{\mathrm{Inv}}$, so
(H.5) implies (5.14) with $\varepsilon=\varepsilon_{0}$. Thus we get the
result (B.1): a matrix-valued function
$\mathfrak{B}\in\mathscr{C}^{\infty}_{\mathrm{loc}}\mathscr{C}^{\beta}(\frac{1}{2}\mathbb{H}^{r+2m},\frac{1}{2}\mathbb{B}^{q};\mathbb{C}^{(r+m)\times
m})$.

Applying Lemma 5.8 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)

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

  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 5.9.

Therefore, $(\operatorname{Span}d(G^{\prime\prime}\circ
H),ds)|_{H^{-1}(U_{1}\times V_{1})}=L_{*}\mathcal{S}|_{H^{-1}(U_{1}\times
V_{1})}$ in the sense of Convention 5.9, which means

$\operatorname{Span}(d(G^{\prime\prime}\circ H\circ
L),dL^{\prime\prime\prime})|_{(H\circ L)^{-1}(U_{1}\times
V_{1})}=\mathcal{S}|_{(H\circ L)^{-1}(U_{1}\times V_{1})}^{\bot}\text{ in the
sense of Convention \ref{Conv::Key::CoSpan}}.$ (5.15)

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 5.9, which
means for each $s\in V_{1}$,

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

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 (5.2) 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

$F=(F^{\prime},F^{\prime\prime},F^{\prime\prime\prime}):=(L^{\prime},\overline{G^{\prime\prime}}\circ
H\circ L,L^{\prime\prime\prime}):(H\circ L)^{-1}(U_{1}\times
V_{1})\subseteq\mathfrak{M}\times\mathfrak{N}\to\mathbb{R}^{r}_{t}\times\mathbb{C}^{m}_{z}\times\mathbb{R}^{q}_{s}.$
(5.16)

By (L.1), (H.3), (G.1) and Lemma 2.32 (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 $U\times
V\subseteq(H\circ L)^{-1}(U_{1}\times V_{1})$ of $(u_{0},v_{0})$, we have
$F^{\prime}\in\mathscr{C}^{\infty}_{\mathrm{loc}}(U;\mathbb{R}^{r})$,
$F^{\prime\prime}\in\mathscr{C}^{\alpha+1,\beta}_{\mathrm{loc}}(U,V;\mathbb{C}^{m})$
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)
$(F^{\prime},F^{\prime\prime})\in\mathscr{C}^{\alpha+1,\beta}(U,V;\mathbb{R}^{r}\times\mathbb{C}^{m})$
for every $U\times V\subseteq(H\circ L)^{-1}(U_{1}\times V_{1})$. By (L.3),
$(dF^{\prime},dF^{\prime\prime}(\cdot,v_{0}))$ has full rank $r+2m$ in the
domain. Therefore, by Lemma 2.32 (iii), we can find a small enough
neighborhood $U\times V\subseteq(H\circ L)^{-1}(U_{1}\times V_{1})$ 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
5.1. 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 $F\in\mathscr{C}^{\beta}$,
$\Phi\in\mathscr{C}^{\beta}$. 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
$\widehat{U}:=(L^{\prime},L^{\prime\prime})(U)\subseteq\mathbb{R}^{r}_{\tau}\times\mathbb{C}^{m}_{w}$,
$\widehat{V}:=L^{\prime\prime\prime}(V)\subseteq\mathbb{R}^{q}_{s}$,
$\widehat{F}^{\prime\prime}:=F^{\prime\prime}\circ L^{-1}$ 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

$\begin{gathered}\widehat{F}^{\prime\prime}\in\mathscr{C}^{\alpha+1,\beta}_{(\tau,w),s}(\widehat{U},\widehat{V};\mathbb{C}^{m}),\quad\nabla_{\tau,w}\widehat{F}^{\prime\prime}\in\mathscr{C}^{\alpha,\beta-}(\widehat{U},\widehat{V};\mathbb{C}^{(r+2m)\times
m}),\\\
\widehat{\Phi}\in\mathscr{C}^{\alpha+1,\beta}_{(t,z),s}(\Omega^{\prime}\times\Omega^{\prime\prime},\Omega^{\prime\prime\prime};\widehat{U}\times\widehat{V}),\quad\nabla_{t,z}\widehat{\Phi}\in\mathscr{C}^{\alpha,\beta-}(\Omega^{\prime}\times\Omega^{\prime\prime},\Omega^{\prime\prime\prime};\mathbb{C}^{(r+2m)\times(r+2m)}).\end{gathered}$
(5.17)

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
$\nabla_{\tau,w}\widehat{F}^{\prime\prime}\in\mathscr{C}^{\alpha,\beta-}_{(\tau,w),s}$
implies $XF^{\prime\prime}\in\mathscr{C}^{\alpha,\beta-}_{\mathrm{loc}}$.

Proof of (5.17): By (5.16) 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
$G^{\prime\prime}\in\mathscr{C}^{\infty,\beta}_{(\sigma,\zeta),s}$,
$H\in\mathscr{C}^{\alpha+1,\beta}_{(\tau,w),s}$,
$\nabla_{\sigma,\zeta}G^{\prime\prime}\in\mathscr{C}^{\infty,\beta}_{(\sigma,\zeta),s}$
and $\nabla_{\tau,w}H\in\mathscr{C}^{\alpha,\beta-}_{(\tau,w),s}$. Therefore,
by Lemma 2.32 (i) we get
$\widehat{F}^{\prime\prime}\in\mathscr{C}^{\alpha+1,\beta}_{(\tau,w),s}$ and
$\nabla_{\tau,w}\widehat{F}^{\prime\prime}=((\nabla_{\sigma,\zeta}\overline{G^{\prime\prime}})\circ
H)\cdot\nabla_{\tau,w}H\in\mathscr{C}^{\alpha+1,\beta}_{(\tau,w),s}\cdot\mathscr{C}^{\alpha,\beta-}_{(\tau,w),s}\subseteq\mathscr{C}^{\alpha,\beta-}_{(\tau,w),s}$.

Since $F\circ L^{-1}(\tau,w,s)=(\tau,\widehat{F}^{\prime\prime}(\tau,w,s),s)$,
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
$\widehat{F}^{\prime\prime}\in\mathscr{C}^{\alpha+1,\beta}_{(\tau,w),s}$,
applying Lemma 2.32 (iii) to the map
$[(\tau,w),s)\mapsto(\tau,\widehat{F}^{\prime\prime}(\tau,w,s))]$ we get
$\widehat{\Psi}\in\mathscr{C}^{\alpha+1,\beta}_{(t,z),s}$.

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
$\nabla_{\tau,w}\widehat{F}^{\prime\prime}\in\mathscr{C}^{\alpha,\beta-}_{(\tau,w),s}$,
by Lemma 2.32 (ii) we have
$(\nabla_{w}\widehat{F}^{\prime\prime})^{-1}\in\mathscr{C}^{\alpha,\beta-}_{(\tau,w),s}$
thus
$(\nabla_{w}\widehat{F}^{\prime\prime})^{-1}\cdot\nabla_{\tau}\widehat{F}^{\prime\prime}\in\mathscr{C}^{\alpha,\beta-}_{(\tau,w),s}$.
Applying Lemma 2.32 (i) with
$\widehat{\Phi}\in\mathscr{C}^{\alpha+1,\beta}_{(t,z),s}$, we get
$\nabla_{t,z}\widehat{\Psi}\in\mathscr{C}^{\alpha,\beta-}_{(t,z),s}$.
Therefore $\nabla_{t,z}\widehat{\Phi}\in\mathscr{C}^{\alpha,\beta-}_{(t,z),s}$
finishing the proof of (5.17) and hence the proof of (3) and (7).

Now by (F.1) and (7) we see that $F\in\mathscr{C}^{\alpha+1,\beta}_{u,v}$ and
$\nabla_{t,z}\Phi\in\mathscr{C}^{\alpha,\beta-}_{(t,z),s}$. By Lemma 2.32 (i)
we see that $\frac{\partial\Phi}{\partial t^{j}}\circ
F\in\mathscr{C}^{\alpha,\beta-}_{\mathrm{loc}}(U,V;T\mathfrak{M})$ and
$\frac{\partial\Phi}{\partial z^{k}}\circ
F\in\mathscr{C}^{\alpha,\beta-}_{\mathrm{loc}}(U,V;\mathbb{C}T\mathfrak{M})$
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
$\mathscr{C}^{\alpha}\cap\mathscr{C}^{\beta-}=\mathscr{C}^{\min(\alpha,\beta-)}$.

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 (5.15) and (5.16), in the sense of Convention 5.9,

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

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

##  6 The Rough Complex Frobenius Theorem: Statements and Proofs

Now we give more general statements for Theorems 1.1 and 1.2 with proofs.

Recall Definitions 2.10, 2.15 and 3.2 for the
$\mathscr{C}^{\alpha,\beta}$-spaces, Definition 3.5 for a local basis, and
(1.2) for the column and matrix convention of Jacobian matrices.

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

The refinement of Theorem 1.1 is the following.

######  Theorem 6.1.

Let $\alpha,\beta,\kappa\in(1,\infty]$ satisfy $\alpha\leq\beta\leq\kappa-1$,
and let $r,m,q$ be nonnegative integers. Let $\mathfrak{M}$ be a
$(r+2m+q)$-dimensional $\mathscr{C}^{\kappa}$-manifold.

Let $\mathcal{S}$ be a $\mathscr{C}^{\alpha}$-complex Frobenius structure over
$\mathfrak{M}$ that has complex rank $r+m$, such that
$\mathcal{S}+\bar{\mathcal{S}}$ is a $\mathscr{C}^{\beta}$-subbundle and has
complex rank $r+2m$.

Then for any $p_{0}\in\mathfrak{M}$ there is an open neighborhood
$\Omega=\Omega^{\prime}\times\Omega^{\prime\prime}\times\Omega^{\prime\prime\prime}\subseteq\mathbb{R}^{r}_{t}\times\mathbb{C}^{m}_{z}\times\mathbb{R}^{q}_{s}$
of $(0,0,0)$ and a map $\Phi:\Omega\to\mathfrak{M}$ such that:

  1. (1)

$\Phi:\Omega\to\Phi(\Omega)$ is a $\mathscr{C}^{\alpha}$-diffeomorphism and
$\Phi(0)=p_{0}$.

  2. (2)

$\Phi\in\mathscr{C}^{\alpha+1,\alpha}_{\mathrm{loc}}(\Omega^{\prime}\times\Omega^{\prime\prime},\Omega^{\prime\prime\prime};\mathfrak{M})$.

  3. (3)

$\frac{\partial\Phi}{\partial t^{1}},\cdots,\frac{\partial\Phi}{\partial
t^{r}}\in\mathscr{C}^{\alpha,\alpha-}_{\mathrm{loc}}(\Omega^{\prime}\times\Omega^{\prime\prime},\Omega^{\prime\prime\prime};T\mathfrak{M})$,
and $\frac{\partial\Phi}{\partial z^{1}},\dots,\frac{\partial\Phi}{\partial
z^{m}},\frac{\partial\Phi}{\partial\bar{z}^{1}},\dots,\frac{\partial\Phi}{\partial\bar{z}^{m}}\in\mathscr{C}^{\alpha,\alpha-}_{\mathrm{loc}}(\Omega^{\prime}\times\Omega^{\prime\prime},\Omega^{\prime\prime\prime};\mathbb{C}T\mathfrak{M})$.

  4. (4)

For any $u\in\Omega$, the complex subspace $\mathcal{S}_{\Phi(u)}$ is spanned
by $\frac{\partial\Phi}{\partial t^{1}}(u),\cdots,\frac{\partial\Phi}{\partial
t^{r}}(u),\frac{\partial\Phi}{\partial
z^{1}}(u),\dots,\frac{\partial\Phi}{\partial
z^{m}}(u)\in\mathbb{C}T_{\Phi(u)}\mathfrak{M}$.

For the coordinate side, set $U:=\Phi(\Omega)\subseteq\mathfrak{M}$ and let
$F=(F^{\prime},F^{\prime\prime},F^{\prime\prime\prime}):U\to\mathbb{R}^{r}\times\mathbb{C}^{m}\times\mathbb{R}^{q}$
be the inverse map of $\Phi$. Then:

  1. (5)

$F$ is a $\mathscr{C}^{\alpha}$-coordinate chart of $\mathfrak{M}$ near
$p_{0}$ and $F(p_{0})=(0,0,0)$.

  2. (6)

$F^{\prime}\in\mathscr{C}^{\kappa}_{\mathrm{loc}}(U;\mathbb{R}^{r})$,
$F^{\prime\prime}\in\mathscr{C}^{\alpha}_{\mathrm{loc}}(U;\mathbb{C}^{m})$ and
$F^{\prime\prime\prime}\in\mathscr{C}^{\beta}_{\mathrm{loc}}(U;\mathbb{R}^{q})$.

  3. (7)

$F^{*}\frac{\partial}{\partial t^{1}},\dots,F^{*}\frac{\partial}{\partial
t^{r}},F^{*}\frac{\partial}{\partial
z^{1}},\dots,F^{*}\frac{\partial}{\partial z^{m}}$ are
$\mathscr{C}^{\alpha-}$-vector fields defined on $U$.

  4. (8)

$\mathcal{S}|_{U}$ has a local basis $(F^{*}\frac{\partial}{\partial
t^{1}},\dots,F^{*}\frac{\partial}{\partial
t^{r}},F^{*}\frac{\partial}{\partial
z^{1}},\dots,F^{*}\frac{\partial}{\partial z^{m}})$.

In particular,

  1. (9)

$\mathcal{S}^{\bot}|_{U}$ has a local basis
$(F^{*}d\bar{z}^{1},\dots,F^{*}d\bar{z}^{m},F^{*}ds^{1},\dots,F^{*}ds^{q})$.

  2. (10)

$(\mathcal{S}\cap\bar{\mathcal{S}})|_{U}$ has a local basis
$(F^{*}\frac{\partial}{\partial t^{1}},\dots,F^{*}\frac{\partial}{\partial
t^{r}})$.

  3. (11)

$(\mathcal{S}+\bar{\mathcal{S}})|_{U}$ has a local basis
$(F^{*}\frac{\partial}{\partial t^{1}},\dots,F^{*}\frac{\partial}{\partial
t^{r}},F^{*}\frac{\partial}{\partial
z^{1}},\dots,F^{*}\frac{\partial}{\partial
z^{m}},F^{*}\frac{\partial}{\partial\bar{z}^{1}},\dots,F^{*}\frac{\partial}{\partial\bar{z}^{m}})$.

The refinement of Theorem 1.2 is the following

######  Theorem 6.2.

Let $\alpha,\beta,\kappa>1$, $r,m,q\geq 0$, $\mathfrak{M}$ and
$\mathcal{S}\leq\mathbb{C}T\mathfrak{M}$ be the same as in Theorem 6.1.

Suppose in addition to the assumptions of Theorem 6.1, there is a
$\gamma\in(1,\infty]$ such that $\alpha+1\leq\gamma\leq\kappa-1$ and
$\mathcal{S}\cap\bar{\mathcal{S}}$ is a $\mathscr{C}^{\gamma}$-subbundle.

Then for any $p_{0}\in\mathfrak{M}$ there are an open neighborhood
$\Omega=\Omega^{\prime}\times\Omega^{\prime\prime}\times\Omega^{\prime\prime\prime}\subseteq\mathbb{R}^{r}_{t}\times\mathbb{C}^{m}_{z}\times\mathbb{R}^{q}_{s}$
of $(0,0,0)$, and a $\mathscr{C}^{\alpha}$-map $\Phi:\Omega\to\mathfrak{M}$,
such that in additional to all the consequences in Theorem 6.1,

  1. (12)

$\Phi\in\mathscr{C}^{\gamma+1,\alpha+1,\alpha}_{\mathrm{loc}}(\Omega^{\prime},\Omega^{\prime\prime},\Omega^{\prime\prime\prime};\mathfrak{M})$.

  2. (13)

$\frac{\partial\Phi}{\partial t^{1}},\cdots,\frac{\partial\Phi}{\partial
t^{r}}\in\mathscr{C}^{\gamma,\alpha+1,\alpha}_{\mathrm{loc}}(\Omega^{\prime},\Omega^{\prime\prime},\Omega^{\prime\prime\prime};T\mathfrak{M})$.

  3. (14)

For $F=\Phi^{\mathrm{Inv}}$, the vector fields $F^{*}\frac{\partial}{\partial
t^{1}},\dots,F^{*}\frac{\partial}{\partial t^{r}}$ defined on $U=\Phi(\Omega)$
are all $\mathscr{C}^{\gamma}$.

Some of the results are directly implied by others, we mark them as below.

######  Remark 6.3.

  1. (i)

The case $\alpha=\infty$ (which implies $\beta=\gamma=\kappa=\infty$) is the
classical result [Nir57] (see Proposition 1.0).

  2. (ii)

(7) is implied by (3) and (5).

We have $F^{*}\frac{\partial}{\partial t^{j}}=\frac{\partial\Phi}{\partial
t^{j}}\circ F$ and $F^{*}\frac{\partial}{\partial
z^{k}}=\frac{\partial\Phi}{\partial z^{k}}\circ F$ for $1\leq j\leq r$ and
$1\leq k\leq m$. (3) shows that $\frac{\partial\Phi}{\partial
t^{j}},\frac{\partial\Phi}{\partial z^{k}}\in\mathscr{C}^{\alpha-}$ and (5)
shows that $F\in\mathscr{C}^{\alpha}$. Taking compositions we get
$\frac{\partial\Phi}{\partial t^{j}}\circ F,\frac{\partial\Phi}{\partial
z^{k}}\circ F\in\mathscr{C}^{\alpha-}$. ∎

  1. (iii)

(4) and (8) are equivalent.

This is because $F^{*}\frac{\partial}{\partial
t^{j}}|_{p}=\frac{\partial\Phi}{\partial t^{j}}(F(p))$ and
$F^{*}\frac{\partial}{\partial z^{k}}|_{p}=\frac{\partial\Phi}{\partial
z^{k}}(F(p))$ for $p\in U$, $1\leq j\leq r$ and $1\leq k\leq m$. ∎

  1. (iv)

(9), (10) and (11) are all implied by (8).

Since $(F^{*}dt,F^{*}dz,F^{*}d\bar{z},F^{*}ds)$ is the dual basis of
$(F^{*}\frac{\partial}{\partial t},F^{*}\frac{\partial}{\partial
z},F^{*}\frac{\partial}{\partial\bar{z}},F^{*}\frac{\partial}{\partial s})$,
and by (8) $F^{*}\frac{\partial}{\partial t},F^{*}\frac{\partial}{\partial z}$
span $\mathcal{S}|_{U}$. Taking the dual bundle, we get that
$F^{*}d\bar{z},F^{*}ds$ span $\mathcal{S}^{\bot}|_{U}$.

For the complex conjugate of (8), $\bar{\mathcal{S}}|_{U}$ has local basis
$F^{*}\frac{\partial}{\partial t},F^{*}\frac{\partial}{\partial\bar{z}}$.
Taking intersection we get that $\mathcal{S}\cap\bar{\mathcal{S}}|_{U}$ is
spanned by $F^{*}\frac{\partial}{\partial t}$. Taking union we get that
$\mathcal{S}+\bar{\mathcal{S}}|_{U}$ is spanned by
$F^{*}\frac{\partial}{\partial t},F^{*}\frac{\partial}{\partial
z},F^{*}\frac{\partial}{\partial\bar{z}}$. ∎

The proofs of Theorems 6.1 and 6.2 are combinations of the real Frobenius
theorem, the Theorem 3.27 or Corollary 3.29, and the estimate of the
parameterized elliptic structures, the Theorem 5.1.

###### Proof of Theorem 6.1.

By Remark 6.3 (i) only the case $\alpha<\infty$ needs a proof.

We fix the point $p_{0}\in\mathfrak{M}$. First we apply real Frobenius
theorem, the Corollary 3.23 $\mathcal{S}+\bar{\mathcal{S}}$. We endow
$\mathbb{R}^{r+2m}\times\mathbb{R}^{q}$ with standard coordinate system
$(\nu,\lambda)=(\nu^{1},\dots,\nu^{r+2m},\lambda^{1},\dots,\lambda^{q})$.

Since $\mathcal{S}+\bar{\mathcal{S}}$ is $\mathscr{C}^{\beta}$ and has rank
$r+2m$, we can find a neighborhood
$\Omega_{1}=\Omega^{\prime}_{1}\times\Omega^{\prime\prime\prime}_{1}\subseteq\mathbb{R}^{r+2m}_{\nu}\times\mathbb{R}^{q}_{\lambda}$
of $(0,0)$ and a $\mathscr{C}^{\beta}$-parameterization
$\Phi_{1}:\Omega_{1}\to\mathfrak{M}$, such that by taking
$U_{1}:=\Phi_{1}(\Omega_{1})$ and
$F_{1}=(F_{1}^{\prime},F_{1}^{\prime\prime\prime}):U_{1}\to\mathbb{R}^{r+2m}\times\mathbb{R}^{q}$,
$F_{1}:=\Phi_{1}^{\mathrm{Inv}}$, we have

  1. (S1.1)

$\Phi_{1}(0,0)=p_{0}$.

  2. (S1.2)

$\Phi_{1}\in\mathscr{C}^{\beta+1,\beta}_{\nu,\lambda,\mathrm{loc}}(\Omega_{1}^{\prime},\Omega_{1}^{\prime\prime\prime};\mathfrak{M})$
and
$\nabla_{\nu}\Phi_{1}\in\mathscr{C}^{\beta}_{\mathrm{loc}}(\Omega_{1};T\mathfrak{M}\otimes\mathbb{R}^{r+2m})$.

  3. (S1.3)

$F_{1}^{\prime}\in\mathscr{C}^{\kappa}_{\mathrm{loc}}(U_{1};\mathbb{R}^{r+2m})$
and
$F_{1}^{\prime\prime\prime}\in\mathscr{C}^{\beta}_{\mathrm{loc}}(U_{1};\mathbb{R}^{q})$.

  4. (S1.4)

$(\mathcal{S}+\bar{\mathcal{S}})|_{U_{1}}$ has a $\mathscr{C}^{\beta}$-local
basis
$(F_{1}^{*}\frac{\partial}{\partial\nu^{1}},\dots,F_{1}^{*}\frac{\partial}{\partial\nu^{r+2m}})$.

Now $\Phi_{1}^{*}\mathcal{S}=(F_{1})_{*}\mathcal{S}\leq\mathbb{C}T\Omega_{1}$.
By (S1.4),
$\Phi_{1}^{*}\mathcal{S}\leq\Phi_{1}^{*}(\mathcal{S}+\bar{\mathcal{S}})=\operatorname{Span}(\frac{\partial}{\partial\nu^{1}},\dots,\frac{\partial}{\partial\nu^{r+2m}})|_{\Omega_{1}}=(\mathbb{C}T\Omega_{1}^{\prime})\times\Omega_{1}^{\prime\prime\prime}$.

Since
$F_{1}^{*}\frac{\partial}{\partial\nu^{1}},\dots,F_{1}^{*}\frac{\partial}{\partial\nu^{r+2m}}\in\mathscr{C}^{\beta}\subseteq\mathscr{C}^{\alpha}$,
by Lemma 3.17 $\Phi_{1}^{*}\mathcal{S}$ is a $\mathscr{C}^{\alpha}$-subbundle
of $(\mathbb{C}T\Omega_{1}^{\prime})\times\Omega_{1}^{\prime\prime\prime}$.

Now Theorem 5.1 applies to
$\Phi_{1}^{*}\mathcal{S}\leq(\mathbb{C}T\Omega_{1}^{\prime})\times\Omega_{1}^{\prime\prime\prime}$.
We endow $\mathbb{R}^{r}\times\mathbb{C}^{m}\times\mathbb{R}^{q}$ with
standard real and complex coordinate system
$(t,z,s)=(t^{1},\dots,t^{r},z^{1},\dots,z^{m},s^{1},\dots,s^{q})$. By Theorem
5.1, we can find a neighborhood
$\Omega=\Omega^{\prime}\times\Omega^{\prime\prime}\times\Omega^{\prime\prime\prime}\subseteq\mathbb{R}^{r}_{t}\times\mathbb{C}^{m}_{z}\times\mathbb{R}^{q}_{s}$
of $(0,0,0)$ and a $\mathscr{C}^{\alpha}$-parameterization
$\Phi_{2}=(\Phi_{2}^{\prime},\Phi_{2}^{\prime\prime\prime}):\Omega\to\Omega_{1}^{\prime}\times\Omega_{1}^{\prime\prime\prime}$,
such that by taking $U_{2}:=\Phi_{2}(\Omega)$ and
$F_{2}=(F_{2}^{\prime},F_{2}^{\prime\prime},F_{2}^{\prime\prime\prime}):=\Phi_{2}^{\mathrm{Inv}}:U_{2}\to\mathbb{R}^{r}\times\mathbb{C}^{m}\times\mathbb{R}^{q}$,

  1. (S1.5)

$\Phi_{2}(0,0,0)=(0,0)$.

  2. (S1.6)

$\Phi_{2}^{\prime\prime\prime}=\Phi_{2}^{\prime\prime\prime}(s):\Omega^{\prime\prime\prime}\subseteq\mathbb{R}^{q}_{s}\to\Omega^{\prime\prime\prime}_{1}$
is a smooth parameterization that does not depend on $(t,z)$.

  3. (S1.7)

$\Phi_{2}^{\prime}\in\mathscr{C}^{\alpha+1,\alpha}_{(t,z),s}(\Omega^{\prime}\times\Omega^{\prime\prime},\Omega^{\prime\prime\prime};\Omega_{1}^{\prime})$
and
$\nabla_{t,z}\Phi_{2}^{\prime}\in\mathscr{C}^{\alpha,\alpha-}(\Omega^{\prime}\times\Omega^{\prime\prime},\Omega^{\prime\prime\prime};\mathbb{C}^{(r+2m)\times(r+2m)})$.

  4. (S1.8)

$F_{2}^{\prime}=F_{2}^{\prime}(\nu)\in C^{\infty}$ does not depend on
$\lambda$; and
$F_{2}^{\prime\prime\prime}=F_{2}^{\prime\prime\prime}(\lambda)\in C^{\infty}$
does not depend on $\nu$.

  5. (S1.9)

$\Phi_{1}^{*}\mathcal{S}|_{U_{2}}$ has a $\mathscr{C}^{\alpha-}$-local basis
$(F_{2}^{*}\frac{\partial}{\partial
t^{1}},\dots,F_{2}^{*}\frac{\partial}{\partial
t^{r}},F_{2}^{*}\frac{\partial}{\partial
z^{1}},\dots,F_{2}^{*}\frac{\partial}{\partial z^{m}})$.

We now define $\Phi:=\Phi_{1}\circ\Phi_{2}:\Omega\to\mathfrak{M}$,
$U:=\Phi(\Omega)$ and
$F=(F^{\prime},F^{\prime\prime},F^{\prime\prime\prime}):=\Phi^{\mathrm{Inv}}$.

Immediately by (S1.1) and (S1.5) we have (1) and (5).

By (S1.4) and (S1.9) we have
$\mathcal{S}=\Phi_{1}^{*}\operatorname{Span}(F_{2}^{*}\frac{\partial}{\partial
t},F_{2}^{*}\frac{\partial}{\partial
z})=\operatorname{Span}(F_{1}^{*}F_{2}^{*}\frac{\partial}{\partial
t},F_{1}^{*}F_{2}^{*}\frac{\partial}{\partial
z})=\operatorname{Span}(F^{*}\frac{\partial}{\partial
t},F^{*}\frac{\partial}{\partial z})$, which is (8). By Remark 6.3 (iii) we
get (4).

By (S1.6) we have
$\Phi(t,z,s)=\Phi_{1}(\Phi_{2}^{\prime}(t,z,s),\Phi_{2}^{\prime\prime\prime}(s))$.
By chain rule we have

$\nabla_{t,z}\Phi(t,z,s)=((\nabla_{\nu}\Phi_{1})\circ\Phi_{2})(t,z,s))\cdot\nabla_{t,z}\Phi_{2}^{\prime}(t,z,s).$

Thus (2) and (3) follow by applying Lemma 2.32 (i) with (S1.2) and (S1.7).

Clearly $F^{\prime\prime}=F_{2}^{\prime\prime}\circ
F_{1}\in\mathscr{C}^{\alpha}$. By (S1.3) and (S1.8) we have
$F^{\prime}=F_{2}^{\prime}\circ F_{1}^{\prime}\in\mathscr{C}^{\kappa}$ and
$F^{\prime\prime\prime}=F_{2}^{\prime\prime\prime}\circ
F_{1}^{\prime\prime\prime}\in\mathscr{C}^{\beta}$, giving the result (6).

The rest of the results in Theorem 6.1 follow from Remark 6.3. ∎

###### Proof of Theorem 6.2.

We use a construction that is slightly different to the proof of Theorem 6.1.
Instead of applying Corollary 3.29, we apply Theorem 3.27 in the first step.

We endow $\mathbb{R}^{r}\times\mathbb{R}^{2m}\times\mathbb{R}^{q}$ with
standard coordinate system
$(\mu,\nu,\lambda)=(\mu^{1},\dots,\mu^{r},\nu^{1},\dots,\nu^{2m},\lambda^{1},\dots,\lambda^{q})$.

Since $\mathcal{S}+\bar{\mathcal{S}}\in\mathscr{C}^{\beta}$ has rank $r+2m$
and $\mathcal{S}\cap\bar{\mathcal{S}}\in\mathscr{C}^{\gamma}$ has rank $r$, we
can find a neighborhood
$\Omega_{1}=\Omega^{\prime}_{1}\times\Omega^{\prime\prime}_{1}\times\Omega^{\prime\prime\prime}_{1}\subseteq\mathbb{R}^{r}_{\mu}\times\mathbb{R}^{2m}_{\nu}\times\mathbb{R}^{q}_{\lambda}$
of $(0,0,0)$ and a $\mathscr{C}^{\min(\beta,\gamma)}$-parameterization
$\Phi_{1}:\Omega_{1}\to\mathfrak{M}$, such that by taking
$U_{1}:=\Phi_{1}(\Omega_{1})$ and
$F_{1}=(F_{1}^{\prime},F_{1}^{\prime\prime},F_{1}^{\prime\prime\prime}):U_{1}\to\mathbb{R}^{r}\times\mathbb{R}^{2m}\times\mathbb{R}^{q}$
as $F_{1}:=\Phi_{1}^{\mathrm{Inv}}$, we have

  1. (S2.1)

$\Phi_{1}(0,0,0)=p_{0}$,

  2. (S2.2)

$\Phi_{1}\in\mathscr{C}^{\gamma+1,\min(\gamma,\beta+1),\min(\gamma,\beta)}_{\mu,\nu,\lambda,\mathrm{loc}}(\Omega_{1}^{\prime},\Omega_{1}^{\prime\prime},\Omega_{1}^{\prime\prime\prime};\mathfrak{M})$
and
$\nabla_{\mu}\Phi_{1}\in\mathscr{C}^{\gamma,\min(\gamma,\beta+1),\min(\gamma,\beta)}_{\mathrm{loc}}(\Omega_{1}^{\prime},\Omega_{1}^{\prime\prime},\Omega_{1}^{\prime\prime\prime};T\mathfrak{M}\otimes\mathbb{R}^{r})$.

  3. (S2.3)

$F_{1}^{\prime}\in\mathscr{C}^{\kappa}_{\mathrm{loc}}(U_{1};\mathbb{R}^{r})$,
$F_{1}^{\prime\prime}\in\mathscr{C}^{\gamma}_{\mathrm{loc}}(U_{1};\mathbb{R}^{2m})$
and
$F_{1}^{\prime\prime\prime}\in\mathscr{C}^{\beta}_{\mathrm{loc}}(U_{1};\mathbb{R}^{q})$.

  4. (S2.4)

On $U_{1}$,
$F_{1}^{*}\frac{\partial}{\partial\mu^{1}},\dots,F_{1}^{*}\frac{\partial}{\partial\mu^{r}}$
are $\mathscr{C}^{\gamma}$-vector fields and
$F_{1}^{*}\frac{\partial}{\partial\nu^{1}},\dots,F_{1}^{*}\frac{\partial}{\partial\nu^{2m}}$
are $\mathscr{C}^{\min(\gamma-1,\beta)}$-vector fields.

  5. (S2.5)

$(\mathcal{S}\cap\bar{\mathcal{S}})|_{U_{1}}=\operatorname{Span}(F_{1}^{*}\frac{\partial}{\partial\mu^{1}},\dots,F_{1}^{*}\frac{\partial}{\partial\mu^{r}})$,
and
$(\mathcal{S}+\bar{\mathcal{S}})|_{U_{1}}=\operatorname{Span}(F_{1}^{*}\frac{\partial}{\partial\mu^{1}},\dots,F_{1}^{*}\frac{\partial}{\partial\mu^{r}},F_{1}^{*}\frac{\partial}{\partial\nu^{1}},\dots,F_{1}^{*}\frac{\partial}{\partial\nu^{2m}})$.

By (S2.4) and (S2.5), $\mathcal{S}$ is contained in the span of
$F_{1}^{*}\frac{\partial}{\partial\mu^{1}},\dots,F_{1}^{*}\frac{\partial}{\partial\mu^{r}},F_{1}^{*}\frac{\partial}{\partial\nu^{1}},\dots,F_{1}^{*}\frac{\partial}{\partial\nu^{2m}}\in\mathscr{C}^{\min(\beta,\gamma-1)}$.
By assumption $\min(\gamma-1,\beta)\geq\alpha$, using Lemma 3.17 we see that
$\Phi_{1}^{*}\mathcal{S}\leq\mathbb{C}T\Omega_{1}$ is a
$\mathscr{C}^{\alpha}$-subbundle

By the standard linear algebra argument we can find a linear complex
coordinate system $w=(w^{1},\dots,w^{m})$ on $\mathbb{R}^{2m}_{\nu}$ such that

$\textstyle(\Phi_{1}^{*}\mathcal{S})_{(0,0)}=\operatorname{Span}\big{(}\frac{\partial}{\partial\mu^{1}}|_{(0,0)},\dots,\frac{\partial}{\partial\mu^{r}}|_{(0,0)},\frac{\partial}{\partial
w^{1}}|_{(0,0)},\dots,\frac{\partial}{\partial
w^{m}}|_{(0,0)}\big{)}\leq\mathbb{C}T_{(0,0)}(\mathbb{R}^{r}_{\mu}\times\mathbb{C}^{m}_{w}).$

In particular
$(\Phi_{1}^{*}\mathcal{S})_{(0,0)}\oplus\operatorname{Span}\big{(}\frac{\partial}{\partial\bar{w}^{1}}|_{(0,0)},\dots,\frac{\partial}{\partial\bar{w}^{m}}|_{(0,0)},\frac{\partial}{\partial\lambda^{1}}|_{(0,0)},\dots,\frac{\partial}{\partial\lambda^{q}}|_{(0,0)})=\mathbb{C}T_{(0,0)}(\mathbb{R}^{r}_{\mu}\times\mathbb{C}^{m}_{w})$.
Applying Lemma 3.9, we can find a neighborhood $W=W^{\prime}\times
W^{\prime\prime}\times
W^{\prime\prime\prime}\subseteq\Omega_{1}^{\prime}\times\Omega_{1}^{\prime\prime}\times\Omega_{1}^{\prime\prime\prime}$
of $(0,0,0)\in\Omega$ such that $\Phi_{1}^{*}\mathcal{S}$ has the following
local basis on $W$,

$X=\begin{pmatrix}X^{\prime}\\\
X^{\prime\prime}\end{pmatrix}\begin{pmatrix}I&&A^{\prime}&B^{\prime}\\\
&I&A^{\prime\prime}&B^{\prime\prime}\end{pmatrix}\begin{pmatrix}\partial_{\mu}\\\
\partial_{w}\\\ \partial_{\bar{w}}\\\ \partial_{\lambda}\end{pmatrix}.$

where $A^{\prime},A^{\prime\prime},B^{\prime},B^{\prime\prime}$ are
$\mathscr{C}^{\alpha}$-matrix functions defined in $W$.

Since
$\Phi_{1}^{*}\mathcal{S}\leq\mathbb{C}T(\Omega_{1}^{\prime}\times\Omega_{1}^{\prime\prime})\times\Omega_{1}^{\prime\prime\prime}$,
we have $B^{\prime}\equiv 0$ and $B^{\prime\prime}\equiv 0$. Since
$\frac{\partial}{\partial\mu^{1}},\dots,\frac{\partial}{\partial\mu^{r}}$ are
sections of
$\Phi_{1}^{*}(\mathcal{S}\cap\bar{\mathcal{S}})\subset\Phi_{1}^{*}\mathcal{S}$,
by Lemma 3.9 (i) we have $A^{\prime}=0$, i.e.
$X_{j}=\frac{\partial}{\partial\mu^{j}}$ for $1\leq j\leq r$.

By Lemma 3.9 (ii), since $\Phi_{1}^{*}\mathcal{S}$ is involutive, we see that
$[X_{j},X_{k}]=0$ for $1\leq j\leq r<k\leq r+m$. So
$\frac{\partial}{\partial\mu^{j}}A^{\prime\prime}(\mu,w,s)=0$ for $1\leq j\leq
r$ in the domain, which means
$A^{\prime\prime}(\mu,w,s)=A^{\prime\prime}(w,s)$.

Define a complex tangent subbundle
$\mathcal{T}\leq\mathbb{C}T(W^{\prime\prime}\times W^{\prime\prime\prime})$ as
$\mathcal{T}:=\operatorname{Span}(\frac{\partial}{\partial
w}+A^{\prime\prime}(w,s)\frac{\partial}{\partial\bar{w}})$. We see that
$\mathcal{T}$ is a $\mathscr{C}^{\alpha}$-involutive subbundle of rank $m$ and
satisfies

$\mathcal{T}\cap\bar{\mathcal{T}}=0,\quad\mathcal{T}+\bar{\mathcal{T}}=(\mathbb{C}TW^{\prime\prime})\times
W^{\prime\prime\prime},\quad(\Phi_{1}^{*}\mathcal{S})|_{W}=(\mathbb{C}TW^{\prime})\otimes_{\mathbb{C}}\mathcal{T}.$
(6.1)

Endow $\mathbb{C}^{m}\times\mathbb{R}^{q}$ with standard coordinate system
$(z,s)=(z^{1},\dots,z^{m},s^{1},\dots,s^{q})$. By Theorem 5.1 (by taking $r=0$
in this theorem), we can find a neighborhood
$\Omega_{2}=\Omega_{2}^{\prime\prime}\times\Omega_{2}^{\prime\prime\prime}\subseteq\mathbb{C}^{m}_{z}\times\mathbb{R}^{q}_{s}$
of $(0,0)$ and a $\mathscr{C}^{\alpha}$-parameterization
$\Phi_{2}=(\Phi_{2}^{\prime\prime},\Phi_{2}^{\prime\prime\prime}):\Omega_{2}\to
W^{\prime\prime}\times W^{\prime\prime\prime}$, such that by taking
$U_{2}:=\Phi_{2}(\Omega_{2})$ and
$F_{2}=(F_{2}^{\prime\prime},F_{2}^{\prime\prime\prime}):U_{2}\to\mathbb{C}^{m}\times\mathbb{R}^{q}$,
$F_{2}:=\Phi_{2}^{\mathrm{Inv}}$, we have

  1. (S2.6)

$\Phi_{2}(0,0,0)=(0,0)$.

  2. (S2.7)

$\Phi_{2}^{\prime\prime\prime}=\Phi_{2}^{\prime\prime\prime}(s):\Omega_{2}^{\prime\prime\prime}\subseteq\mathbb{R}^{q}_{s}\to\Omega_{1}^{\prime\prime\prime}$
is a smooth parameterization that does not depend on $z$.

  3. (S2.8)

$\Phi_{2}^{\prime\prime}\in\mathscr{C}^{\alpha+1,\alpha}_{z,s}(\Omega_{2}^{\prime\prime},\Omega_{2}^{\prime\prime\prime};W^{\prime\prime})$
and
$\nabla_{z}\Phi_{2}^{\prime\prime}\in\mathscr{C}^{\alpha,\alpha-}(\Omega_{2}^{\prime\prime},\Omega_{2}^{\prime\prime\prime};\mathbb{C}^{2m\times
2m})$

  4. (S2.9)

$F_{2}^{\prime\prime\prime}=F_{2}^{\prime\prime\prime}(\lambda)\in C^{\infty}$
does not depend on $(\mu,\nu)$; and
$F_{2}^{\prime\prime}\in\mathscr{C}^{\alpha}(U_{2};\mathbb{C}^{m})$.

  5. (S2.10)

$\mathcal{T}|_{U_{2}}$ has a $\mathscr{C}^{\alpha-}$-local basis
$(F_{2}^{*}\frac{\partial}{\partial
z^{1}},\dots,F_{2}^{*}\frac{\partial}{\partial z^{m}})$.

Define $\Omega^{\prime}:=W^{\prime}$,
$\Omega^{\prime\prime}:=\Omega_{2}^{\prime\prime}$,
$\Omega^{\prime\prime\prime}:=\Omega_{2}^{\prime\prime\prime}$ and
$\Phi:\Omega\to\mathfrak{M}$ as

$\Phi(t,z,s):=\Phi_{1}(t,\Phi_{2}(t,z))=\Phi_{1}(t,\Phi^{\prime\prime}_{2}(z,s),\Phi_{2}^{\prime\prime\prime}(s)),\quad
t\in\Omega^{\prime},z\in\Omega^{\prime\prime},s\in\Omega^{\prime\prime\prime}.$
(6.2)

We define $U:=\Phi(\Omega)$ and
$F=(F^{\prime},F^{\prime\prime},F^{\prime\prime\prime}):=\Phi^{\mathrm{Inv}}$.
Immediately by (S2.1) and (S2.6) we have (1) and (5).

Taking the chain rules on (6.2) we have

$\nabla_{t}\Phi(t,z,s)=\nabla_{\mu}\Phi_{1}(t,\Phi_{2}^{\prime\prime}(z,s),\Phi_{2}^{\prime\prime\prime}(s)),\quad\nabla_{z}\Phi(t,z,s)=(\nabla_{\nu}\Phi_{1})(t,\Phi_{2}^{\prime\prime}(z,s),\Phi_{2}^{\prime\prime\prime}(s))\cdot\nabla_{z}\Phi_{2}(z,s).$
(6.3)

Based on (S2.2) and (S2.8), applying Lemma 2.32 (i) to (6.2) and (6.3), we get
(12), (3) and (13). Note that (12) is stronger than (2).

By (6.2), (S2.3) and (S2.9) we see that
$F=(F^{\prime},F^{\prime\prime},F^{\prime\prime\prime})$ satisfies

$F^{\prime}=F_{1}^{\prime}=\operatorname{id}_{\mathbb{R}^{r}}\circ
F_{1}^{\prime}\in\mathscr{C}^{\kappa}_{\mathrm{loc}}(U;\mathbb{R}^{r}),\quad
F^{\prime\prime}=F_{2}^{\prime\prime}\circ
F_{1}^{\prime\prime}\in\mathscr{C}^{\alpha}_{\mathrm{loc}}(U;\mathbb{C}^{m}),\quad
F^{\prime\prime\prime}=F_{2}^{\prime\prime\prime}\circ
F_{1}^{\prime\prime\prime}\in\mathscr{C}^{\beta}_{\mathrm{loc}}(U;\mathbb{R}^{q}).$
(6.4)

This gives (6).

By (6.3), we get $\frac{\partial}{\partial
t^{j}}\Phi(t,z,s)=\frac{\partial\Phi_{1}}{\partial\mu^{j}}(t,\Phi_{2}^{\prime\prime}(z,s),\Phi_{2}^{\prime\prime\prime}(s))=(\Phi_{1})_{*}\frac{\partial}{\partial\mu^{j}}|_{\Phi(t,z,s)}$
for $1\leq j\leq r$. So $F^{*}\frac{\partial}{\partial
t^{j}}=F_{1}^{*}\frac{\partial}{\partial\mu^{j}}\in\mathscr{C}^{\gamma}$ for
$1\leq j\leq r$, finishing the proof of (14).

By (6.1) and (S2.10) we see that
$\Phi_{1}^{*}\mathcal{S}=\operatorname{Span}(\frac{\partial}{\partial\mu},F_{2}^{*}\frac{\partial}{\partial
z})$, so by writing
$\operatorname{id}_{\mathbb{R}^{r}}:\mathbb{R}^{r}_{\mu}\to\mathbb{R}^{r}_{t}$
as the identity we have
$\Phi_{1}^{*}\mathcal{S}=\operatorname{Span}(\operatorname{id}_{\mathbb{R}^{r}}^{*}\frac{\partial}{\partial
t},F_{2}^{*}\frac{\partial}{\partial z})$. Using (6.4) we have
$F=(\operatorname{id}_{\mathbb{R}^{r}},F_{2})\circ F_{1}$, taking pushforward
of $\Phi_{1}$ we get
$\mathcal{S}|_{U}=F_{1}^{*}\operatorname{Span}(\operatorname{id}_{\mathbb{R}^{r}}^{*}\frac{\partial}{\partial
t},F_{2}^{*}\frac{\partial}{\partial
z})|_{U}=\operatorname{Span}(F^{*}\frac{\partial}{\partial
t},F^{*}\frac{\partial}{\partial z})$, finishing the proof of (8).

The rest of the results follow from Remark 6.3 as well. ∎

##  7 The Sharpness of Coordinates and Coordinate Vector Fields

Fixed four numbers $\alpha,\beta,\gamma,\kappa>1$ such that
$\alpha\leq\min(\beta,\gamma)$ and $\kappa\geq\max(\beta,\gamma)+1$. Consider
all possible complex Frobenius structures
$\mathcal{S}\leq\mathbb{C}T\mathfrak{M}$ among all
$\mathscr{C}^{\kappa}$-manifolds $\mathfrak{M}$ such that
$\mathcal{S}\in\mathscr{C}^{\alpha}$,
$\mathcal{S}+\bar{\mathcal{S}}\in\mathscr{C}^{\beta}$ and
$\mathcal{S}\cap\bar{\mathcal{S}}\in\mathscr{C}^{\gamma}$.

Fix a point $p\in\mathfrak{M}$, let
$F=(F^{\prime},F^{\prime\prime},F^{\prime\prime\prime}):U\subseteq\mathfrak{M}\to\mathbb{R}^{r}_{t}\times\mathbb{C}^{m}_{z}\times\mathbb{R}^{n}_{s}$
be any desired $C^{1}$-coordinate chart near a fixed point $p$ that represents
$\mathcal{S}$, that is, $F^{*}\frac{\partial}{\partial
t^{1}},\dots,F^{*}\frac{\partial}{\partial
t^{r}},F^{*}\frac{\partial}{\partial
z^{1}},\dots,F^{*}\frac{\partial}{\partial z^{m}}$ span $\mathcal{S}$ in $U$.

Recall the notation $\mathscr{C}^{\mu+}=\bigcup_{\nu>\mu}\mathscr{C}^{\nu}$
for $\mu\in\mathbb{R}$.

We list the best Hölder-Zygmund regularities the six objects: $F^{\prime}$,
$F^{\prime\prime}$, $F^{\prime\prime\prime}$, $F^{*}\frac{\partial}{\partial
t}$, $F^{*}\frac{\partial}{\partial z}$, $F^{*}\frac{\partial}{\partial s}$,
as follows.

  * •

$F^{\prime}\notin\mathscr{C}^{\kappa+}$: $\mathfrak{M}$ is
$\mathscr{C}^{\kappa}$, its coordinate functions are at most
$\mathscr{C}^{\kappa}$, so $F^{\prime}\in\mathscr{C}^{\kappa}$ is optimal.

  * •

$F^{\prime\prime\prime}\notin\mathscr{C}^{\beta+}$: This is due to the
sharpness of real Frobenius theorem that
$\mathcal{S}+\bar{\mathcal{S}}\in\mathscr{C}^{\beta}$ only guarantees
$F^{\prime\prime\prime}$ to be $\mathscr{C}^{\beta}$. See Proposition 7.1
below.

  * •

$F^{*}\frac{\partial}{\partial t}\notin\mathscr{C}^{\gamma+}$:
$F^{*}\frac{\partial}{\partial t}$ spans $\mathcal{S}\cap\bar{\mathcal{S}}$
which is $\mathscr{C}^{\gamma}$. If $F^{*}\frac{\partial}{\partial
t}\in\mathscr{C}^{\gamma+}$ then
$\mathcal{S}\cap\bar{\mathcal{S}}\in\mathscr{C}^{\gamma+}$ as well.

  * •

$F^{\prime\prime}\notin\mathscr{C}^{\alpha+}$ and
$F^{*}\frac{\partial}{\partial s}\notin\mathscr{C}^{(\alpha-1)+}$: The
argument is similar to the sharpness of real Frobenius theorem. See
Proposition 7.2 below.

  * •

$F^{*}\frac{\partial}{\partial z}\notin\mathscr{C}^{\alpha}$: This is the
counter intuitive part in our results. See Section 7.2 for the complete
deduction and Proposition 7.3, Theorem 7.11 for the proof.

When $\gamma\geq\alpha+1$ (that $\mathcal{S}\cap\bar{\mathcal{S}}$ is at least
1 order more regular than $\mathcal{S}$), by Theorem 6.2, all six regularity
estimates above are sharp.

When $\alpha\leq\gamma<\alpha+1$, from Theorem 1.1 we only have
$F^{*}\frac{\partial}{\partial t}\in\mathscr{C}^{\alpha-}$, the rest of five
estimates are still sharp.

When $\alpha\leq\gamma<\alpha+1$ and $\gamma>2$, using Theorem 6.1 we can find
a $\mathscr{C}^{\alpha}$-coordinate chart
$F=(F^{\prime},F^{\prime\prime},F^{\prime\prime\prime})$ to
$\mathbb{R}^{r}_{t}\times\mathbb{C}^{m}_{z}\times\mathbb{R}^{n}_{s}$ such that
$F^{\prime\prime}\in\mathscr{C}^{\alpha}$, $F^{*}\frac{\partial}{\partial
z}\in\mathscr{C}^{\alpha-}$ but $F^{*}\frac{\partial}{\partial
t}\in\mathscr{C}^{\alpha-}$. If we use Theorem 6.2, then we can get
$F^{*}\frac{\partial}{\partial t}\in\mathscr{C}^{\gamma}$ which is sharp, but
neither $F^{\prime\prime}\in\mathscr{C}^{\gamma-1}$,
$F^{*}\frac{\partial}{\partial z}\in\mathscr{C}^{(\gamma-1)-}$ nor
$F^{*}\frac{\partial}{\partial s}\in\mathscr{C}^{\gamma-2}$ is sharp.

We do not know whether we can choose suitable coordinate chart $F$ so that
$F^{*}\frac{\partial}{\partial t}\in\mathscr{C}^{\gamma}$ and
$F^{\prime\prime}\in\mathscr{C}^{\alpha}$ holds in the same time.

###  7.1 Sharpness of $F^{\prime\prime}$, $F^{\prime\prime\prime}$ and
$F^{*}\frac{\partial}{\partial s}$: some elementary examples

In this subsection we use the conventions $f_{u}=\partial_{u}f$,
$f_{v}=\partial_{v}f$, etc.

######  Proposition 7.1.

Let $\beta>1$, and let $(u,v)$ be the standard coordinate system for
$\mathbb{R}^{2}$. We define $\mathcal{V}\leq T\mathbb{R}^{2}$ as

$\textstyle\mathcal{V}:=\operatorname{Span}_{\mathbb{R}}T,\quad
T:=\frac{\partial}{\partial u}+f(v)\frac{\partial}{\partial
v},\quad\text{where }f(v):=\max(0,v)^{\beta}.$

Suppose
$F=(F^{\prime},F^{\prime\prime\prime}):U\subseteq\mathbb{R}^{2}_{u,v}\to\mathbb{R}_{x}\times\mathbb{R}_{s}$
is a $C^{1}$-coordinate chart near $(0,0)$ such that
$F^{*}\frac{\partial}{\partial x}$ spans $\mathcal{V}|_{U}$, then
$\partial_{v}F^{\prime\prime\prime}\notin\mathscr{C}^{(\beta-1)+}_{\mathrm{loc}}(U)$.
In particular
$F^{\prime\prime\prime}\notin\mathscr{C}^{\beta+}_{\mathrm{loc}}(U;\mathbb{R}^{2})$.

Note that $\mathcal{V}$ has rank 1 so is automatically involutive.

###### Proof.

By direct computation

$e^{tT}(u,v)=\begin{cases}\big{(}u+t,\frac{v}{\left(1-(\beta-1)v^{\beta-1}t\right)^{\frac{1}{\beta-1}}}\big{)}&\text{for
}v>0\text{ and }-\infty<t<(\beta-1)v^{1-\beta},\\\ (u+t,v)&\text{for }v\leq
0\text{ and }t\in\mathbb{R}.\end{cases}$

First we show that for every $\delta\neq 0$, the function

$g^{\delta}:I_{\delta}\to\mathbb{R},\quad
g^{\delta}(v):=\begin{cases}\frac{v}{\left(1-(\beta-1)\delta
v^{\beta-1}\right)^{\frac{1}{\beta-1}}},&v>0,\\\ v,&v\leq
0,\end{cases}\quad\text{where
}I_{\delta}=\begin{cases}\left(-\infty,(\frac{\delta}{\beta-1})^{\frac{1}{1-\beta}}\right),&\delta>0,\\\
\mathbb{R},&\delta<0,\end{cases}$ (7.1)

is not $\mathscr{C}^{\beta+}$ near $v=0$. Note that $e^{uT}(0,v)=(u,g^{u}(v))$
for every $u$.

Indeed by Taylor’s expansion,

$\frac{v}{\left(1-(\beta-1)\delta
v^{\beta-1}\right)^{\frac{1}{\beta-1}}}=v+\delta
v^{\beta}+\sum_{k=2}^{\infty}{-\frac{1}{\beta-1}\choose
k}\left(-(\beta-1)\delta\right)^{k}\cdot v^{k\beta-k+1},\quad\text{converging
for }0\leq v<(\tfrac{\delta}{\beta-1})^{\frac{1}{1-\beta}}.$

Hence
$g^{\delta}(v)=v+\max(0,v)^{\beta}+\sum_{k=2}^{\infty}{-\frac{1}{\beta-1}\choose
k}\left(-(\beta-1)\delta\right)^{k}\cdot\max(0,v)^{k\beta-k+1}$ holds near
$v=0$.

Now near $v=0$ $\sum_{k=2}^{\infty}{-\frac{1}{\beta-1}\choose
k}\left(-(\beta-1)\delta\right)^{k}\cdot\max(0,v)^{k\beta-k+1}$ is a
$\mathscr{C}^{2\beta-1}_{\mathrm{loc}}\subsetneq\mathscr{C}^{\beta+}_{\mathrm{loc}}$-function,
but $\max(0,v)^{\beta}\notin\mathscr{C}^{\beta+}$. So
$g^{\delta}\notin\mathscr{C}^{\beta+}$ near $v=0$. Therefore the function
$(u,v)\mapsto g^{u}(v)$ is not $\mathscr{C}^{\beta+}$ near $(u,v)=(0,0)$.

By assumption $F^{*}\frac{\partial}{\partial x}$ spans $\mathcal{V}|_{U}$. So
$\mathcal{V}^{\bot}|_{U}=\operatorname{Span}dF^{\prime\prime}$, which means
that we have a transport equation

$\textstyle
TF^{\prime\prime\prime}(u,v)=0\quad\text{i.e.}\quad\partial_{u}F^{\prime\prime\prime}(u,v)=-f(v)\cdot\partial_{v}F^{\prime\prime\prime}(u,v),\quad\text{for
all }(u,v)\in U.$ (7.2)

Since $F$ is a $C^{1}$-chart, we have
$(F_{u}^{\prime\prime\prime}(0,0),F_{v}^{\prime\prime\prime}(0,0))\neq(0,0)$.
By (7.2) and $f(0)=0$, we have $F_{u}^{\prime\prime\prime}(0,0)=0$. Thus
$F_{v}^{\prime\prime\prime}(0,0)\neq 0$.

Therefore $h(v):=F^{\prime\prime\prime}(0,v)$ is a $C^{1}$-function that has
non-vanishing derivatives in a neighborhood of $v=0$. By the Inverse Function
Theorem, on a smaller neighborhood of $v=0$, $h$ is a $C^{1}$-diffeomorphism
onto its image.

Suppose by contrast $F^{\prime\prime\prime}_{v}\in\mathscr{C}^{(\beta-1)+}$ in
a neighborhood of $(u,v)=(0,0)$. By (7.2) since
$f\in\mathscr{C}^{\beta}_{\mathrm{loc}}\subset\mathscr{C}^{(\beta-1)+}$, we
have $F^{\prime\prime\prime}_{u}\in\mathscr{C}^{(\beta-1)+}$ as well, which
means $F^{\prime\prime\prime}\in\mathscr{C}^{\beta+}$ near $(u,v)=(0,0)$.
Since $h(v)=F^{\prime\prime\prime}(0,v)$, we have $h\in\mathscr{C}^{\beta+}$
in the domain and by the Inverse Function Theorem
$h^{\mathrm{Inv}}\in\mathscr{C}^{\beta+}$.

So $h^{\mathrm{Inv}}\circ F^{\prime\prime}(u,v)$ is a
$\mathscr{C}^{\beta+}$-function defined in a neighborhood of $(u,v)=(0,0)$. By
(7.2) and (7.1),
$F^{\prime\prime\prime}(u,v)=F^{\prime\prime\prime}(e^{-uT}(u,v))=F^{\prime\prime\prime}(0,g^{-u}(v))$
holds a neighborhood of $0$. So $h^{\mathrm{Inv}}\circ
F^{\prime\prime\prime}(u,v)=g^{-u}(v)$, contradicting to the fact that
$(u,v)\mapsto g^{u}(v)$ is not $\mathscr{C}^{\beta+}$ near $(0,0)$. This
concludes that $F_{v}^{\prime\prime\prime}\notin\mathscr{C}^{(\beta-1)+}$ near
$(u,v)=(0,0)$. Therefore $F^{\prime\prime\prime}\notin\mathscr{C}^{\beta+}$
and we finish the proof. ∎

######  Proposition 7.2.

Endow $\mathbb{R}^{3}$ with standard coordinate system $(u,v,\theta)$. Let
$\alpha>0$. We define $\mathcal{S}\leq\mathbb{C}T\mathbb{R}^{3}$ as

$\mathcal{S}:=\operatorname{Span}_{\mathbb{C}}Z,\quad
Z:=\partial_{u}+ie^{f(\theta)}\partial_{v}\quad\text{where
}f(\theta):=\max(0,\theta)^{\alpha}.$

Suppose
$F=(F^{\prime\prime},F^{\prime\prime\prime}):U\subseteq\mathbb{R}^{3}_{u,v,\theta}\to\mathbb{C}^{1}_{z}\times\mathbb{R}^{1}_{s}$
is a $C^{1}$-coordinate chart near $0\in\mathbb{R}^{3}$ such that that
$F(0)=(0,0)$ and $F^{*}\frac{\partial}{\partial z}$ spans $\mathcal{S}|_{U}$.
Then both $F^{\prime\prime}\notin\mathscr{C}^{\alpha+}$ and
$F^{*}\frac{\partial}{\partial s}\notin\mathscr{C}^{(\alpha-1)+}$ near
$(0,0,0)$.

By assumption $Z\in\mathscr{C}^{\alpha}$, $\mathcal{S}$ is involutive since it
has rank 1, and
$\mathcal{S}+\bar{\mathcal{S}}=\operatorname{Span}(\frac{\partial}{\partial
u},\frac{\partial}{\partial v})$ is involutive as well. So $\mathcal{S}$ is a
$\mathscr{C}^{\alpha}$-complex Frobenius structure.

###### Proof.

By assumption
$\mathcal{S}+\bar{\mathcal{S}}|_{U}=\operatorname{Span}(\partial_{u},\partial_{v})|_{U}=\operatorname{Span}(F^{*}\frac{\partial}{\partial
z},F^{*}\frac{\partial}{\partial\bar{z}})=(\operatorname{Span}F^{*}ds)^{\bot}=(\operatorname{Span}dF^{\prime\prime\prime})^{\bot}$,
so $\partial_{u}F^{\prime\prime\prime}=\partial_{v}F^{\prime\prime\prime}=0$
which means
$F^{\prime\prime\prime}(u,v,\theta)=F^{\prime\prime\prime}(\theta)$. So
$dF^{\prime\prime\prime}=F^{\prime\prime\prime}_{\theta}d\theta$ and
$F^{\prime\prime\prime}_{\theta}(\theta)$ is non-vanishing in a neighborhood
of $\theta=0$.

Now
$\mathcal{S}^{\bot}|_{U}=(\operatorname{Span}Z)^{\bot}|_{U}=\operatorname{Span}(F^{*}d\bar{z},F^{*}ds)=\operatorname{Span}(d\bar{F}^{\prime\prime},dF^{\prime\prime\prime})$,
so

$Z\bar{F}^{\prime\prime}=\bar{F}^{\prime\prime}_{u}+ie^{f(\theta)}\bar{F}^{\prime\prime}_{v}=0,\quad\Rightarrow\quad
F^{\prime\prime}_{v}=-ie^{-f(\theta)}F^{\prime\prime}_{u},\quad\text{in }U.$

Here we use $F^{\prime\prime}_{u}=\partial_{u}F^{\prime\prime}$,
$F^{\prime\prime}_{v}=\partial_{v}F^{\prime\prime}$ and
$F^{\prime\prime}_{\theta}=\partial_{\theta}F^{\prime\prime}$. So

$\begin{pmatrix}F^{*}dz\\\ F^{*}d\bar{z}\\\
F^{*}ds\end{pmatrix}=\begin{pmatrix}F^{\prime\prime}_{u}&-F^{\prime\prime}_{u}&F^{\prime\prime}_{\theta}\\\
\bar{F}^{\prime\prime}_{u}&\bar{F}^{\prime\prime}_{u}&\bar{F}^{\prime\prime}_{\theta}\\\
&&F^{\prime\prime\prime}_{\theta}\end{pmatrix}\begin{pmatrix}du\\\
-ie^{-f(\theta)}dv\\\ d\theta\end{pmatrix}\ \Rightarrow\
\begin{pmatrix}\partial_{u}\\\ ie^{f(\theta)}\partial_{v}\\\
\partial_{\theta}\end{pmatrix}=\begin{pmatrix}F^{\prime\prime}_{u}&\bar{F}^{\prime\prime}_{u}&\\\
-F^{\prime\prime}_{u}&\bar{F}^{\prime\prime}_{u}\\\
F^{\prime\prime}_{\theta}&\bar{F}^{\prime\prime}_{\theta}&F^{\prime\prime\prime}_{\theta}\end{pmatrix}\begin{pmatrix}F^{*}\partial_{z}\\\
F^{*}\partial_{\bar{z}}\\\ F^{*}\partial_{s}\end{pmatrix}.$ (7.3)

Applying Cramer’s rule for inverse matrices to the right hand equation of
(7.3), we have

$F^{*}\frac{\partial}{\partial
s}=\frac{1}{F^{\prime\prime\prime}_{\theta}(\theta)}\bigg{(}\frac{\partial}{\partial\theta}-\frac{\bar{F}^{\prime\prime}_{\theta}}{\bar{F}^{\prime\prime}_{u}}\Big{(}\frac{\partial}{\partial
u}+ie^{f(\theta)}\frac{\partial}{\partial
v}\Big{)}-\frac{F^{\prime\prime}_{\theta}}{F^{\prime\prime}_{u}}\Big{(}\frac{\partial}{\partial
u}-ie^{f(\theta)}\frac{\partial}{\partial v}\Big{)}\bigg{)},\quad\text{in }U.$
(7.4)

In particular we see that $F^{\prime\prime}_{u}$ and
$F^{\prime\prime}_{v}=-ie^{-f}F^{\prime\prime}_{u}$ are both non-vanishing in
the domain $U$.

For fixed $\theta\in\mathbb{R}$, $\partial_{u}+ie^{f(\theta)}\partial_{v}$ is
scaled $\overline{\partial}$-operator on $\mathbb{R}^{2}_{u,v}$ that
annihilates $F^{\prime\prime}(\cdot,\theta_{0})$. Therefore by a scaling of
the Cauchy Integral Formula, for every $\tilde{V}^{\prime}\times
V^{\prime\prime}\Subset U$, $(u,v)\in\tilde{V}^{\prime}$, $\theta\in
V^{\prime\prime}$ and $k\geq 0$,

$F^{\prime\prime}(u,v,\theta)=\frac{1}{2\pi i}\int_{\partial
V^{\prime}}\frac{F^{\prime\prime}(\xi,\eta,\theta)(d\xi+ie^{-f(\theta)}d\eta)}{(\xi-u)+ie^{-f(\theta)}(\eta-v)},\quad\frac{\partial^{k}F^{\prime\prime}(u,v,\theta)}{\partial
u^{k}}=\frac{k!}{2\pi i}\int_{\partial
V^{\prime}}\frac{F^{\prime\prime}(\xi,\eta,\theta)(d\xi+ie^{-f(\theta)}d\eta)}{((\xi-u)+ie^{-f(\theta)}(\eta-v))^{k+1}}.$

Since $F^{\prime\prime}\in C^{0}(\tilde{V}^{\prime}\times
V^{\prime\prime};\mathbb{C})$, for every precompact subset
$V^{\prime}\Subset\tilde{V}^{\prime}$ we have

$\|\partial_{u}^{k}F^{\prime\prime}\|_{L^{\infty}(V^{\prime}\times
V^{\prime\prime};\mathbb{C}^{2^{k}})}\leq
k!\operatorname{dist}(V^{\prime},\partial\tilde{V}^{\prime})^{-k-1}\|F^{\prime\prime}\|_{L^{\infty}(\tilde{V}^{\prime}\times
V^{\prime\prime};\mathbb{C})}<\infty,\quad\forall k\geq 0.$

Using $F_{v}^{\prime\prime}=-ie^{-f(\theta)}F_{u}^{\prime\prime}$, we see that
$\|\nabla_{u,v}^{k}F^{\prime\prime}\|_{L^{\infty}(V^{\prime}\times
V^{\prime\prime};\mathbb{C}^{2^{k}})}<\infty$ for every $V^{\prime}\times
V^{\prime\prime}\Subset U$, which means
$F^{\prime\prime}\in\mathscr{C}^{\infty}_{u,v}L^{\infty}_{\theta}(V^{\prime},V^{\prime\prime};\mathbb{C})$.

Let $\mu>0$ be such that
$F^{\prime\prime}\in\mathscr{C}^{\mu}_{\mathrm{loc}}(U)$. Since
$F^{\prime\prime}\in\mathscr{C}^{\infty}_{u,v}L^{\infty}_{\theta}(V^{\prime},V^{\prime\prime};\mathbb{C})$
for every $V^{\prime}\times V^{\prime\prime}\Subset U$, we have:

  1. (7.$\mu$)

For every $\mu>0$, if
$F^{\prime\prime}\in\mathscr{C}^{\mu}_{\mathrm{loc}}(U)$, then
$F^{\prime\prime}\in\mathscr{C}^{\infty,\mu-}_{(u,v),s}(V^{\prime},V^{\prime\prime};\mathbb{C})$
holds for all $V^{\prime}\times V^{\prime\prime}\Subset U$.

Now suppose by contrast that
$F^{\prime\prime}\in\mathscr{C}^{\alpha+}_{\mathrm{loc}}(U;\mathbb{C})$. Say
$F^{\prime\prime}\in\mathscr{C}^{\alpha+2\varepsilon}_{\mathrm{loc}}(U;\mathbb{C})$
for some $\varepsilon>0$. Take $\mu=\alpha+\varepsilon$ in (7.$\mu$), we have
$F^{\prime\prime}_{u},F^{\prime\prime}_{v}\in\mathscr{C}^{\alpha+\varepsilon}(V^{\prime}\times
V^{\prime\prime};\mathbb{C})$ for every precompact neighborhood
$V^{\prime}\times V^{\prime\prime}\Subset U$ of
$0\in\mathbb{R}^{2}_{u,v}\times\mathbb{R}^{1}_{\theta}$.

Since $F^{\prime\prime}_{u},F^{\prime\prime}_{v}$ are both non-vanishing, we
have
$-ie^{f(\theta)}=F^{\prime\prime}_{v}/F_{u}^{\prime\prime}\in\mathscr{C}^{\alpha+\varepsilon}$
near $(0,0,0)$. However $-ie^{f}\notin\mathscr{C}^{\alpha+}$ near $\theta=0$
since $f\notin\mathscr{C}^{\alpha+}$ near $\theta=0$. Contradiction!

This concludes the proof of $F^{\prime\prime}\notin\mathscr{C}^{\alpha+}$ near
$(0,0,0)$.

To show $F^{*}\frac{\partial}{\partial s}\notin\mathscr{C}^{(\alpha-1)+}$ near
$(0,0,0)$, we can assume $F^{\prime\prime\prime}\in\mathscr{C}^{\alpha+}$,
since otherwise
$F^{\prime\prime\prime}_{\theta}\notin\mathscr{C}^{(\alpha-1)+}$ in a
neighborhood of $\theta=0$ and by (7.4) the
$\frac{\partial}{\partial\theta}$-component of $F^{*}\frac{\partial}{\partial
s}$ is not $\mathscr{C}^{(\alpha-1)+}$ already.

Let $\mu>0$ be the largest number such that
$F^{\prime\prime}\in\mathscr{C}^{\mu-}$ near $(0,0,0)$. By assumption
$F^{\prime\prime}\in C^{1}$ so $\mu\geq 1$. Since
$F^{\prime\prime}\notin\mathscr{C}^{\alpha+}$ near $(0,0,0)$, we see that
$\mu\leq\alpha$.

By (7.$\mu$), we know
$F^{\prime\prime}_{u},F^{\prime\prime}_{v}\in\mathscr{C}^{\mu-\varepsilon}$
near $(0,0,0)$ for every $\varepsilon>0$, in particular
$F^{\prime\prime}_{u},F^{\prime\prime}_{v}\in\mathscr{C}^{\mu-\frac{1}{2}}$
near $(0,0,0)$.

We see that $F^{\prime\prime}_{\theta}\notin\mathscr{C}^{(\mu-1)+}$ near
$(0,0,0)$. Suppose otherwise, i.e.
$F^{\prime\prime}_{\theta}\in\mathscr{C}^{\mu-1+\varepsilon}$ for some
$0<\varepsilon<\frac{1}{2}$. Since
$F^{\prime\prime}_{u},F^{\prime\prime}_{v}\in\mathscr{C}^{\mu-\frac{1}{2}}$,
we have $F^{\prime\prime}\in\mathscr{C}^{\mu+\varepsilon}$. This contradicts
to the maximality of $\mu$.

Now we have
$F^{\prime\prime}_{\theta}/F^{\prime\prime}_{u}\notin\mathscr{C}^{(\mu-1)+}$
for such $\mu\geq 1$, in particular
$F^{\prime\prime}_{\theta}/F^{\prime\prime}_{u}\notin\mathscr{C}^{(\alpha-1)+}$.
Since $Z=\partial_{u}+ie^{f}\partial_{v}$ and
$\bar{Z}=\partial_{u}-ie^{f}\partial_{v}$ are both
$\mathscr{C}^{\alpha}$-vector fields in $\mathbb{R}^{3}$ and we already
assumed $F^{\prime\prime\prime}_{\theta}\in\mathscr{C}^{(\alpha-1)+}$, using
(7.4) again we conclude that $F^{*}\frac{\partial}{\partial
s}\notin\mathscr{C}^{(\alpha-1)+}$ near $(0,0,0)$. This completes the whole
proof. ∎

###  7.2 Sharpness of $F^{*}\frac{\partial}{\partial z}$: a
$\bar{\partial}$-equation with parameter

In this subsection we will construct a rank-1 complex Frobenius structure
$\mathcal{S}$ on $\mathbb{C}^{1}\times\mathbb{R}^{1}$, such that
$F^{*}\frac{\partial}{\partial z}\notin\mathscr{C}^{\alpha}$ whenever $F$ is a
coordinate chart representing $\mathcal{S}$ near the origin.

We sketch the idea to the proof of Theorem 1.3 here. It is adapted from
[Yao21], where we prove that there is a $C^{k}$-integrable almost complex
structure that does not admit a $C^{k+1}$-coordinate chart.

Endow $\mathbb{C}^{1}\times\mathbb{R}^{1}$ with standard coordinate system
$(w,\theta)$. We consider a subbundle
$\mathcal{S}\leq\mathbb{C}T(\mathbb{C}^{1}\times\mathbb{R}^{1})$ as

$\mathcal{S}:=\operatorname{Span}(\partial_{w}+a(w,\theta)\partial_{\bar{w}}),\quad\text{where
}a\in\mathscr{C}^{\alpha}(\mathbb{C}^{1}\times\mathbb{R}^{1};\mathbb{C})\text{
such that }\|a\|_{L^{\infty}}<1.$

Clearly $\mathcal{S}$ is involutive and
$\mathcal{S}+\bar{\mathcal{S}}=\operatorname{Span}(\partial_{w},\partial_{\bar{w}})=(\mathbb{C}T\mathbb{C}_{w}^{1})\times\mathbb{R}^{1}_{\theta}$
since $\operatorname{rank}\mathcal{S}=1$ and $\sup_{w,\theta}|a(w,\theta)|<1$.
So $\mathcal{S}$ is a $\mathscr{C}^{\alpha}$-complex Frobenius structure.

Step 1 (Proposition 7.3): We show that, the existence of a coordinate chart
$F$ near $(0,0)$ such that $F^{*}\frac{\partial}{\partial
z}\in\mathscr{C}^{\alpha}$ spans $\mathcal{S}$, is equivalent to the existence
of $f\in\mathscr{C}^{\alpha}$ near $(w,\theta)=(0,0)$ that solves

$(\partial_{w}+a(w,\theta)\partial_{\bar{w}})f=-\partial_{\bar{w}}a,\quad\text{or
equivalently}\quad f_{w}+(af)_{\bar{w}}-a_{\bar{w}}f=-a_{\bar{w}}.$ (7.5)

Step 2 (Proposition 7.9): We construct an $a(w,\theta)$ such that $a$ is
smooth away from $\\{w=0\\}=\\{0\\}\times\mathbb{R}^{1}_{\theta}$, and the
complex Hilbert transform (also known as the Beurling transform)
$\partial_{w}^{-1}\partial_{\bar{w}}a\notin\mathscr{C}^{\alpha}$ near $w=0$.

Step 3 (Theorem 7.11): We show that there is no $f\in\mathscr{C}^{\alpha}$
defined near $(w,\theta)=(0,0)$ that solves (7.5). This can be done by showing
that:

  * •

For every $f$ that solves the homogeneous equation
$(\partial_{w}+a\partial_{\bar{w}})f=0$ with $f(0,\theta)=0$, we must have
$f\in\mathscr{C}^{\alpha}$.

  * •

There exists a non-$\mathscr{C}^{\alpha}$ function $f$ such that
$f(0,\theta)=0$ and $f$ solves (7.5).

Thus, by the superposition principle, every solution to (7.5) cannot be
$\mathscr{C}^{\alpha}$. By Step 1 we complete the proof.

To make the result more general, we consider the case of mixed regularity,
where $a\in\mathscr{C}^{\alpha,\beta}_{w,\theta}$ for arbitrary
$\alpha>\frac{1}{2}$ and $\beta>0$ (recall again from Lemma 2.12 that
$\mathscr{C}^{\alpha}_{(w,\theta)}=\mathscr{C}^{\alpha,\alpha}_{w,\theta}$).
See Theorem 7.12.

We now start the proof. We use the notation $f_{w}=\partial_{w}f$ etc.

######  Proposition 7.3.

Let $\alpha>\frac{1}{2},\beta>0$, let
$a\in\mathscr{C}^{\alpha,\beta}(\mathbb{C}^{1}_{w},\mathbb{R}^{1}_{\theta};\mathbb{C})$
be such that $\sup|a|<1$, and let
$\mathcal{S}=\operatorname{Span}(\partial_{w}+a(w,\theta)\partial_{\bar{w}})$
be a $\mathscr{C}^{\min(\alpha,\beta)}$-complex Frobenius structure with rank
$1$. The following (A) and (B) are equivalent:

  1. (A)

There is a continuous map $F:U_{1}\times
V_{1}\subseteq\mathbb{C}^{1}_{w}\times\mathbb{R}^{1}_{\theta}\to\mathbb{C}^{1}_{z}\times\mathbb{R}^{1}_{s}$
near $(0,0)$ such that

    1. (A.1)

$F$ is homeomorphic to its image, $\nabla_{z}F\in C^{0}(U_{1}\times
V_{1};\mathbb{C}^{2\times 2})$ and $\nabla_{w}(F^{\mathrm{Inv}})\in
C^{0}(U_{1}\times V_{1};\mathbb{C}^{2\times 2})$.

    2. (A.2)

$F^{*}\frac{\partial}{\partial
z}\in\mathscr{C}^{\alpha,\beta}(U_{1},V_{1};\mathbb{C}^{3})$ and
$F^{*}\frac{\partial}{\partial z}$ spans $\mathcal{S}|_{U_{1}\times V_{1}}$.

  2. (B)

There is a function $f\in\mathscr{C}^{\alpha,\beta}(U_{2},V_{2};\mathbb{C})$
defined in a neighborhood $U_{2}\times
V_{2}\subseteq\mathbb{C}^{1}_{w}\times\mathbb{R}^{1}_{\theta}$ of $(0,0)$ that
solves (7.5).

The results are also true for $\alpha$ or $\beta=\infty$, without any change
of the proof.

For the application to Theorem 1.3, we only need the contraposition of (A)
$\Rightarrow$ (B).

###### Proof.

(A) $\Rightarrow$ (B): Write $F=:(F^{\prime},F^{\prime\prime})$ and
$\Phi=(\Phi^{\prime},\Phi^{\prime\prime}):=F^{\mathrm{Inv}}:F(U_{1}\times
V_{1})\subseteq\mathbb{C}^{1}_{z}\times\mathbb{R}^{1}_{s}\to\mathbb{C}^{1}_{w}\times\mathbb{R}^{1}_{\theta}$.
By assumption
$\nabla_{z}\Phi=(\nabla_{z}\Phi^{\prime},\nabla_{z}\Phi^{\prime\prime})\in
C^{0}(F(U_{1}\times V_{1});\mathbb{C}^{2\times 2})$, so
$F^{*}\frac{\partial}{\partial
z}=\frac{\partial(\operatorname{Re}\Phi^{\prime},\operatorname{Im}\Phi^{\prime},\Phi^{\prime\prime})}{\partial
z}\circ F$ and
$F^{*}\frac{\partial}{\partial\bar{z}}=\frac{\partial(\operatorname{Re}\Phi^{\prime},\operatorname{Im}\Phi^{\prime},\Phi^{\prime\prime})}{\partial\bar{z}}\circ
F$ are well-defined continuous vector fields on
$\mathbb{R}^{3}_{(\operatorname{Re}w,\operatorname{Im}w,\theta)}$.

By assumption $(\mathcal{S}+\bar{\mathcal{S}})|_{U_{1}\times
V_{1}}=\operatorname{Span}(\partial_{w},\partial_{\bar{w}})$ is spanned by
$F^{*}\frac{\partial}{\partial z}$ and $F^{*}\frac{\partial}{\partial z}$, so
$\partial_{w}F^{\prime\prime}=\partial_{\bar{w}}F^{\prime\prime}=0$ in the
sense of distributions, which means
$F^{\prime\prime}(w,\theta)=F^{\prime\prime}(\theta)$.

By assumption $d\bar{F}^{\prime}=F^{*}d\bar{z}$ is continuous in $w$ and is
annihilated by $\operatorname{Span}F^{*}\frac{\partial}{\partial
z}=\operatorname{Span}(\partial_{w}+a\partial_{\bar{w}})$. Therefore

$(\partial_{w}+a\partial_{\bar{w}})\bar{F}^{\prime}=\bar{F}^{\prime}_{w}+a\bar{F}^{\prime}_{\bar{w}}=0,\quad\text{i.e.}\quad
F^{\prime}_{\bar{w}}=-\bar{a}F^{\prime}_{w}.$ (7.6)

So for the transition matrix (cf. (7.3)) we have

$\begin{pmatrix}F^{*}dz\\\ F^{*}d\bar{z}\\\
F^{*}ds\end{pmatrix}=\begin{pmatrix}F^{\prime}_{w}&-\bar{a}F^{\prime}_{w}&F^{\prime}_{\theta}\\\
-a\bar{F}^{\prime}_{\bar{w}}&\bar{F}^{\prime}_{\bar{w}}&\bar{F}^{\prime}_{\theta}\\\
&&F^{\prime\prime}_{\theta}\end{pmatrix}\begin{pmatrix}dw\\\ d\bar{w}\\\
d\theta\end{pmatrix}\quad\Rightarrow\quad\begin{pmatrix}\partial_{w}\\\
\partial_{\bar{w}}\end{pmatrix}=\begin{pmatrix}F^{\prime}_{w}&-a\bar{F}^{\prime}_{\bar{w}}\\\
-\bar{a}F^{\prime}_{w}&\bar{F}^{\prime}_{\bar{w}}\end{pmatrix}\begin{pmatrix}F^{*}\partial_{z}\\\
F^{*}\partial_{\bar{z}}\end{pmatrix}.$ (7.7)

Applying the Cramer’s rule of inverse matrices to the right hand matrix in
(7.7), we have

$F^{*}\frac{\partial}{\partial
z}=\frac{\bar{F}^{\prime}_{\bar{w}}}{(1-|a|^{2})|F^{\prime}_{w}|^{2}}(\partial_{w}+a\partial_{\bar{w}})=\frac{\partial_{w}+a\partial_{\bar{w}}}{(1-|a|^{2})F^{\prime}_{w}}\quad\text{in}\quad
U_{1}\times V_{1}.$ (7.8)

Since $F^{*}\frac{\partial}{\partial
z}\in\mathscr{C}^{\alpha,\beta}(U_{1},V_{1};\mathbb{C}^{3})$ and
$a\in\mathscr{C}^{\alpha,\beta}(U_{1},V_{1};\mathbb{C})$, we see that
$F^{\prime}_{w}\in\mathscr{C}^{\alpha,\beta}_{\mathrm{loc}}(U_{1},V_{1};\mathbb{C})$
and $F^{\prime}_{w}\not\equiv 0$ in $U_{1}$.

Take $U_{2}\times V_{2}\subseteq U_{1}\times V_{1}$ as a neighborhood of
$(0,0)$ such that $\sup_{(w,\theta)\in U_{2}\times
V_{2}}|\frac{F^{\prime}_{w}(w,\theta)}{F^{\prime}_{w}(0,\theta)}-1|<1$. We
define

$f(w,\theta):=\log\frac{\bar{F}^{\prime}_{\bar{w}}(w,\theta)}{\bar{F}^{\prime}_{\bar{w}}(0,\theta)}=\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n}\Big{(}\frac{\bar{F}^{\prime}_{\bar{w}}(w,\theta)}{\bar{F}^{\prime}_{\bar{w}}(0,\theta)}-1\Big{)}^{n}.$

Since
$\big{[}(w,\theta)\mapsto\frac{F^{\prime}_{w}(w,\theta)}{F^{\prime}_{w}(0,\theta)}\big{]}\in\mathscr{C}^{\alpha,\beta}(U_{2},V_{2};\mathbb{C})$,
we see that $f\in\mathscr{C}^{\alpha,\beta}(U_{2},V_{2};\mathbb{C})$ as well.
So (7.5) follows:

$f_{w}+af_{\bar{w}}=\frac{\bar{F}^{\prime}_{w\bar{w}}}{\bar{F}^{\prime}_{\bar{w}}}+\frac{a\bar{F}^{\prime}_{\bar{w}\bar{w}}}{\bar{F}^{\prime}_{\bar{w}}}=\frac{(\bar{F}^{\prime}_{w}+a\bar{F}^{\prime}_{\bar{w}})_{\bar{w}}}{\bar{F}^{\prime}_{\bar{w}}}-\frac{a_{\bar{w}}\bar{F}^{\prime}_{\bar{w}}}{\bar{F}^{\prime}_{\bar{w}}}=-a_{\bar{w}}.$
(7.9)

(B) $\Rightarrow$ (A): This is the reverse of the above argument. By shrinking
$U_{2}\times V_{2}$ we assume $U_{2}$ and $V_{2}$ are both convex
neighborhoods.

Suppose $f\in\mathscr{C}^{\alpha,\beta}(U_{2},V_{2};\mathbb{C})$ is a solution
to (7.5). Take $\mu:=e^{f}$, we know
$\mu\in\mathscr{C}^{\alpha,\beta}(U_{2},V_{2};\mathbb{C})$ satisfies

$(a\mu)_{\bar{w}}=a_{\bar{w}}e^{f}+af_{\bar{w}}e^{f}=-f_{w}e^{f}=-\mu_{w},\quad\text{in}\quad
U_{2}\times V_{2}.$ (7.10)

Therefore $\mu(\cdot,\theta)(d\bar{w}-a(\cdot,\theta)dw)$ is a closed 1-form
in $U_{2}\subseteq\mathbb{C}^{1}$, for each $\theta\in\mathbb{R}$. We define
$F^{\prime}:U_{2}\times V_{2}\to\mathbb{C}$ as

$F^{\prime}(w,\theta):=\int_{0}^{1}(w\cdot\bar{\mu}(tw,\theta)+\bar{w}\cdot\bar{a}(tw,\theta)\bar{\mu}(tw,\theta))dt.$
(7.11)

We see that
$F^{\prime}\in\mathscr{C}^{\alpha,\beta}_{\mathrm{loc}}(U_{1},V_{1};\mathbb{C})$
and satisfies $d\bar{F}^{\prime}=\mu d\bar{w}-a\mu
dw+\bar{F}^{\prime}_{\theta}d\theta$. In other words

$\nabla_{w}(F^{\prime},\bar{F}^{\prime})=\begin{pmatrix}F^{\prime}_{w}&\bar{F}^{\prime}_{w}\\\
F^{\prime}_{\bar{w}}&\bar{F}^{\prime}_{\bar{w}}\end{pmatrix}=\begin{pmatrix}\bar{\mu}&-a\mu\\\
-\bar{a}\bar{\mu}&\mu\end{pmatrix}\quad\Rightarrow\quad\det\big{(}\nabla_{w}(F^{\prime},\bar{F}^{\prime})\big{)}=|\mu|^{2}(1-|a|^{2})>0.$

Since $a,\mu\in\mathscr{C}^{\alpha,\beta}$, we see that
$F^{\prime}\in\mathscr{C}^{\alpha+1,\beta}$ and the matrix function
$\nabla_{w}(F^{\prime},\bar{F}^{\prime})\in\mathscr{C}^{\alpha,\beta}$ is
invertible at $(0,0)$. Applying Lemma 2.32 (iii) to
$(\operatorname{Re}F^{\prime},\operatorname{Im}F^{\prime})\in\mathscr{C}^{\alpha+1,\beta}(U_{2},V_{2};\mathbb{R}^{2})$,
we can find a neighborhood $U_{1}\times V_{1}\subseteq U_{2}\times V_{2}$ such
that $F^{\prime}(\cdot,\theta):U_{1}\to\mathbb{C}^{1}_{z}$ is injective for
each $\theta\in V_{1}$ and its inverse map
$\Phi^{\prime}(z,s)=F^{\prime}(\cdot,s)^{\mathrm{Inv}}(z)$ is
$\mathscr{C}^{\alpha+1,\beta}_{z,s}$ near $(F^{\prime}(0,0),0)=(0,0)$.

Take $F(w,\theta):=(F^{\prime}(w,\theta),\theta)$ and
$\Phi:=F^{\mathrm{Inv}}$, we see that $\Phi(z,s)=(\Phi^{\prime}(z,s),s)$.
Therefore $F:U_{1}\times V_{1}\to\mathbb{C}^{1}\times\mathbb{R}^{1}$ is
homeomorphic to its image (since $F,\Phi$ are both
$\mathscr{C}^{\alpha+1,\beta}\subset C^{0}$), and
$\nabla_{z}(F^{\mathrm{Inv}})=(\nabla_{z}\Phi^{\prime},0)$ is a continuous map
on $F(U_{1}\times V_{1})$. This completes the proof of (A)(A.1).

Now for each $\theta\in V_{1}$,
$F^{\prime}(\cdot,\theta):U_{1}\to\mathbb{C}_{z}$ is a
$\mathscr{C}^{\alpha+1}$-coordinate chart satisfying
$d(\bar{F}^{\prime}(\cdot,\theta))=\mu(\cdot,\theta)d\bar{w}-a\mu(\cdot,\theta)dw$
in $U_{1}$, so $(\partial_{w}+a\partial_{\bar{w}})F^{\prime}=0$, which means
$F^{\prime}(\cdot,\theta)^{*}d\bar{z}=d\bar{F}^{\prime}(\cdot,\theta)$ spans
$\mathcal{S}|_{U_{1}\times\\{\theta\\}}=\operatorname{Span}(\partial_{w}+a(\cdot,\theta)\partial_{\bar{w}})$.
Taking a dual we get
$\mathcal{S}|_{U_{1}\times\\{\theta\\}}=\operatorname{Span}F^{\prime}(\cdot,\theta)^{*}\frac{\partial}{\partial
z}$ for each $\theta\in V_{1}$.

Now $F(w,\theta)=(F^{\prime}(w,\theta),\theta)$ and
$\Phi(z,s)=(\Phi^{\prime}(z,s),s)$, so $((\nabla_{z}\Phi)\circ
F)(w,\theta)=((\nabla_{z}\Phi^{\prime}(\cdot,\theta))\circ
F^{\prime}(\cdot,\theta))(w)$ hence $F^{*}\frac{\partial}{\partial
z}|_{(w,\theta)}=F^{\prime}(\cdot,\theta)^{*}\frac{\partial}{\partial z}|_{w}$
spans $\mathcal{S}_{(w,\theta)}$ for each $w\in U_{1}$ and $\theta\in V_{2}$,
finishing the proof of (A)(A.2). ∎

######  Remark 7.4.

We do not know how to relax the assumption $\alpha>\frac{1}{2}$ to $\alpha>0$.
The difficulty comes from (7.9) and (7.10) in the proof, where the
intermediate calculations need a product of a
$\mathscr{C}^{\alpha-1}$-function and a $\mathscr{C}^{\alpha}$-function. On
the other hand, the PDE (7.5) can still be defined when
$0<\alpha\leq\frac{1}{2}$, by requiring $a$ to stay in the space, for example,
$\mathscr{C}^{\alpha,\beta}(\mathbb{C}^{1}_{w},\mathbb{R}^{1}_{\theta};\mathbb{C})\cap
L^{\infty}(\mathbb{R}^{1}_{\theta};W^{1,2}(\mathbb{C}^{1}_{w};\mathbb{C}))$
(see Definition 7.5 and Proposition 7.9 (i)).

For Step 2, the construction of $a(w,\theta)$ requires the following.

######  Definition 7.5.

Let $k\geq 0$ be an integer, $\beta>0$, $1\leq p<\infty$. Let
$U\subseteq\mathbb{R}^{n}_{x}$ be an open set and let
$(\psi_{j})_{j=0}^{\infty}$ be a dyadic resolution of $\mathbb{R}^{q}_{s}$. We
denote by $\mathscr{C}^{\beta}_{s}(\mathbb{R}^{q};W^{k,p}_{x}(U))$ the space
of all distributions $f\in\mathscr{D}^{\prime}(\mathbb{R}^{q}_{s}\times
U_{x})$ such that $\psi_{j}\ast_{s}f(\cdot,s)\in W^{k,p}(U)$ for every
$s\in\mathbb{R}$ (recall Notation 2.5 for $\psi_{j}\ast_{s}f$), with

$\|f\|_{\mathscr{C}^{\beta}(\mathbb{R}^{q};W^{k,p}(U))}=\|f\|_{\mathscr{C}^{\beta}(W^{k,p})}:=\sup_{j\geq
0,s\in\mathbb{R}^{q}}2^{j\beta}\|(\psi_{j}\ast_{s}f)(\cdot,s)\|_{W^{k,p}(U)}<\infty.$

We denote by $L^{\infty}(\mathbb{R}^{q};W^{k,p}(U))$ the space of all
$L^{1}_{\mathrm{loc}}$-function $f:\mathbb{R}^{q}\times U\to\mathbb{R}$ such
that $f(\cdot,s)\in W^{k,p}(U)$ for almost every $s\in\mathbb{R}^{q}$ with
$\|f\|_{L^{\infty}(W^{k,p})}:=\mathop{\operatorname{essup}}_{s\in\mathbb{R}^{q}}\|f(\cdot,s)\|_{W^{k,p}(U)}<\infty$.

One can check that $\mathscr{C}^{\beta}(\mathbb{R}^{q};W^{k,p}(U))$ is the
same as putting $X=W^{k,p}(U)$ in Definition 1.8 (see also Remark 2.7). In
particular the space does not depend on the choice of
$(\psi_{j})_{j=0}^{\infty}$. We leave their proof to readers since we are
allowed to fix a choice of $(\psi_{j})_{j}$ in the later discussion.

When $U$ is a smooth domain, the space $L^{\infty}(\mathbb{R}^{q};W^{k,p}(U))$
coincides with the classical vector-valued $L^{\infty}$-space (cf. Remark 2.7
(iv)) since $W^{k,p}(U)$ is a separable space.

######  Lemma 7.6.

Let $\beta>0$ and $1<p<\infty$. Let $U\subset\mathbb{R}^{n}$ be a bounded
smooth domain.

  1. (i)

The product map $[(f,g)\mapsto
fg]:\mathscr{C}^{\beta}(\mathbb{R}^{q};L^{p}(U))\times
L^{\infty}\mathscr{C}^{\beta}(U,\mathbb{R}^{q})\to\mathscr{C}^{\beta}(\mathbb{R}^{q};L^{p}(U))$
is bounded bilinear.

  2. (ii)

When $\beta<1$, the product map $[(f,g)\mapsto fg]:(L^{\infty}\cap
W^{1,2})(U)\times\mathscr{C}^{\beta-1}(U)\to\mathscr{C}^{\beta-1}(U)$ is
bounded bilinear.

###### Proof.

(i) can follow from the boundedness of the product map $L^{p}(U)\times
L^{\infty}(U)\to L^{p}(U)$ and that $\mathscr{C}^{\beta}(\mathbb{R}^{q})$ is
closed under multiplications. More precisely, by Lemma 2.24 we have the
following uniformly in $l\geq 0$ and $s\in\mathbb{R}^{q}$:

$\displaystyle\|(\psi_{l}\ast_{s}(fg))(\cdot,s)\|_{L^{p}(U)}\leq\|\psi_{l}\|_{L^{1}}\big{(}\sum_{j=l-2}^{l+2}\sum_{j^{\prime}=0}^{j-3}+\sum_{j=l-2}^{l+2}\sum_{j^{\prime}=0}^{j-3}+\sum_{\begin{subarray}{c}j,j^{\prime}\geq
l-3\\\ |j-j^{\prime}|\leq
2\end{subarray}}\big{)}\|(\psi_{j}\ast_{s}f)(\cdot,s)\|_{L^{p}(U)}\|(\psi_{j^{\prime}}\ast_{s}g)(\cdot,s)\|_{L^{\infty}(U)}$
$\displaystyle\lesssim$
${}_{\psi}\|f\|_{\mathscr{C}^{\beta}_{s}(L^{p}_{x})}\|g\|_{L^{\infty}_{x}\mathscr{C}^{\beta}_{s}}\big{(}\sum_{|j-l|\geq
2}\sum_{j^{\prime}\geq j-3}+\sum_{|j^{\prime}-l|\geq 2}\sum_{j\geq
j^{\prime}-3}+\sum_{\begin{subarray}{c}j,j^{\prime}\geq l-3\\\
|j-j^{\prime}|\leq
2\end{subarray}}\big{)}2^{-j\beta}2^{-j^{\prime}\beta}\lesssim_{\beta}\|f\|_{\mathscr{C}^{\beta}_{s}(L^{p}_{x})}\|g\|_{L^{\infty}_{x}\mathscr{C}^{\beta}_{s}}2^{-l\beta}.$

Proof of (ii) follows from [BCD11, Chapter 2.8.1], where in the reference we
use Bony’s decomposition $uv=T(u,v)+T(v,u)+R(u,v)$ for distributions $u,v$ on
$\mathbb{R}^{2}$ (see [BCD11, (2.41)] and Remark 2.25). By [BCD11, Theorem
2.82] we have
$T:L^{\infty}(\mathbb{R}^{2})\times\mathscr{C}^{\beta-1}(\mathbb{R}^{2})\to\mathscr{C}^{\beta-1}(\mathbb{R}^{2})$
and $T:\mathscr{C}^{\beta-1}(\mathbb{R}^{2})\times
L^{\infty}(\mathbb{R}^{2})\to\mathscr{C}^{\beta-1}(\mathbb{R}^{2})$. By
[BCD11, Theorem 2.85] we have
$R:W^{1,2}(\mathbb{R}^{2})\times\mathscr{C}^{\beta-1}(\mathbb{R}^{2})\to\mathscr{B}_{22}^{\beta}(\mathbb{R}^{2})$.
Here $\mathscr{B}_{22}^{\beta}(\mathbb{R}^{2})=H^{\beta}(\mathbb{R}^{2})$ is
the fractional $L^{2}$-Sobolev space (see for example [BCD11, Definition 2.68]
or [Tri10, Definition 2.3.1/2(i)]). We have Sobolev embedding
$\mathscr{B}_{22}^{\beta}(\mathbb{R}^{2})\hookrightarrow\mathscr{C}^{\beta-1}(\mathbb{R}^{2})$,
whose proof can be found in for example [Tri10, Chapter 2.7.1]666We recall the
correspondence of Besov space
$\mathscr{C}^{\beta}(\mathbb{R}^{q})=\mathscr{B}_{\infty\infty}^{\beta}(\mathbb{R}^{q})$
and Triebel-Lizorkin space
$L^{p}(\mathbb{R}^{n})=\mathscr{F}_{p2}^{0}(\mathbb{R}^{n})$ from [Tri10,
Theorems 2.5.7 and 2.5.12].. Thus
$R:W^{1,2}(\mathbb{R}^{2})\times\mathscr{C}^{\beta-1}(\mathbb{R}^{2})\to\mathscr{C}^{\beta-1}(\mathbb{R}^{2})$
is bounded.

Therefore $[(u,v)\mapsto uv]:(L^{\infty}\cap
W^{1,2})(\mathbb{R}^{2})\times\mathscr{C}^{\beta-1}(\mathbb{R}^{2})\to\mathscr{C}^{\beta-1}(\mathbb{R}^{2})$
is bilinearly bounded. Taking restrictions to $U$ we complete the proof. ∎

######  Lemma 7.7.

Let $\mathcal{P}_{0}$ and $\mathcal{P}$ be the zero Dirichlet boundary
solution operators in Lemma 2.31 with
$U=\mathbb{B}^{2}_{w}\subset\mathbb{C}^{1}$ and $V=\mathbb{R}^{1}_{\theta}$ in
the lemma. Define $\mathcal{T}_{0}:=4\partial_{\bar{w}}\mathcal{P}_{0}$ and
$\mathcal{T}:=4\partial_{\bar{w}}\mathcal{P}$. Then

  1. (i)

$\partial_{w}\mathcal{T}f=f$ for all $f\in
L^{\infty}(\mathbb{B}^{2},\mathbb{R}^{1};\mathbb{C})$ and $\mathcal{T}$ has
the following boundedness: for $\alpha>0$, $1<p<\infty$ and
$\mathscr{X}\in\\{L^{\infty},\mathscr{C}^{\gamma}:\gamma>0\\}$,

$\displaystyle\mathcal{T}:\mathscr{C}^{\alpha-1}\mathscr{X}(\mathbb{B}^{2},\mathbb{R}^{1};\mathbb{C})\to\mathscr{C}^{\alpha}\mathscr{X}(\mathbb{B}^{2},\mathbb{R}^{1};\mathbb{C})\quad\text{and}\quad\mathcal{T}:\mathscr{C}^{\alpha}(\mathbb{R}^{1};L^{p}(\mathbb{B}^{2}))\to\mathscr{C}^{\alpha}(\mathbb{R}^{1};W^{1,p}(\mathbb{B}^{2})).$
(7.12)

  2. (ii)

Moreover, if $f\in
L^{\infty}\mathscr{C}^{\beta}(\mathbb{B}^{2},\mathbb{R}^{1};\mathbb{C})$
satisfies $f|_{\\{|w|>\frac{1}{2}\\}}\equiv 0$, then

$f-\mathcal{T}\partial_{w}f\in\mathscr{C}^{\infty}\mathscr{C}^{\beta}(\mathbb{B}^{2},\mathbb{R}^{1};\mathbb{C}).$
(7.13)

###### Proof.

That $\partial_{w}\mathcal{T}=\operatorname{id}$ and
$\mathcal{T}:\mathscr{C}^{\alpha-1}_{w}\mathscr{X}_{\theta}\to\mathscr{C}^{\alpha}_{w}\mathscr{X}_{\theta}$
are the immediate consequences to Lemma 2.31 since
$\Delta_{\operatorname{Re}w,\operatorname{Im}w}=4\frac{\partial^{2}}{\partial
w\partial\bar{w}}$.

We have boundedness $\mathcal{P}_{0}:L^{p}(\mathbb{B}^{2})\to
W^{2,p}(\mathbb{B}^{2})$ by [GT01, Theorem 9.13], thus
$\mathcal{T}_{0}:L^{p}(\mathbb{B}^{2};\mathbb{C})\to
W^{1,p}(\mathbb{B}^{2};\mathbb{C})$. We get
$\mathcal{T}:\mathscr{C}^{\beta}(L^{p})\to\mathscr{C}^{\beta}(W^{1,p})$ by the
same reason to Lemma 2.9. Thus we prove (i).

To prove (ii), we need to show that
$\operatorname{id}-\mathcal{T}_{0}\partial_{w}:L^{\infty}(\frac{1}{2}\mathbb{B}^{2};\mathbb{C})\to\mathscr{C}^{M}(\mathbb{B}^{2};\mathbb{C})$
is bounded for all $M>0$.

Let $g\in L^{\infty}(\frac{1}{2}\mathbb{B}^{2};\mathbb{C})$ with zero
extension outside $\frac{1}{2}\mathbb{B}^{2}$. By formula of Green functions
(see for example [Eva10, Theorem 2.12 and Section 2.2.c]) we have for
$w\in\mathbb{B}^{2}$,

$g(w)-\mathcal{T}_{0}\partial_{w}g(w)=g(w)-\int_{\frac{1}{2}\mathbb{B}^{2}}4g(z)\partial^{2}_{z\bar{w}}\big{(}\mathbf{\Gamma}(z-w)-\mathbf{\Gamma}\big{(}|z|w-\tfrac{z}{|z|}\big{)}\big{)}dxdy=\int_{\frac{1}{2}\mathbb{B}^{2}}4g(z)\partial^{2}_{z\bar{w}}\mathbf{\Gamma}\big{(}|z|w-\tfrac{z}{|z|}\big{)}dxdy.$

Here $z=x+iy\in\mathbb{B}^{2}$ and $\mathbf{\Gamma}(z)=\frac{1}{2\pi}\log|z|$
(which is the $n=2$ case in (B.1)). The first equality follows from
integration by part, the second equality follows from the fact that
$\partial_{z\bar{w}}\mathbf{\Gamma}(z-w)=\partial_{z\bar{z}}\mathbf{\Gamma}(z-w)=\frac{1}{4}\Delta_{x,y}\mathbf{\Gamma}(z-w)=\frac{\delta_{w}(z)}{4}$.

Clearly $(z,w)\mapsto\mathbf{\Gamma}(|z|w-z/|z|)$ is a smooth function on
$(z,w)\in r\mathbb{B}^{2}\times\mathbb{B}^{2}$ for any $r<1$. Thus for any
integer $M\geq 0$

$\|\nabla^{M}(g-\mathcal{T}_{0}\partial_{w}g)\|_{L^{\infty}(\mathbb{B}^{2};\mathbb{C}^{2^{k}})}\leq\|g\|_{L^{\infty}(\frac{1}{2}\mathbb{B}^{2};\mathbb{C})}\sup_{|w|<\frac{1}{2}}\int_{\mathbb{B}^{2}}\big{|}\nabla^{M+2}_{z,w}\mathbf{\Gamma}\big{(}|z|w-\tfrac{z}{|z|}\big{)}\big{|}dxdy\lesssim_{M}\|g\|_{L^{\infty}(\frac{1}{2}\mathbb{B}^{2};\mathbb{C})}.$

We conclude that
$\operatorname{id}-\mathcal{T}_{0}\partial_{w}:L^{\infty}(\frac{1}{2}\mathbb{B}^{2};\mathbb{C})\to\mathscr{C}^{M}(\mathbb{B}^{2};\mathbb{C})$
is bounded for all $M>0$. By Lemma 2.9, we get the boundedness
$\operatorname{id}-\mathcal{T}\partial_{w}:L^{\infty}\mathscr{C}^{\beta}(\frac{1}{2}\mathbb{B}^{2},\mathbb{R}^{1};\mathbb{C})\to\mathscr{C}^{M}\mathscr{C}^{\beta}(\mathbb{B}^{2},\mathbb{R}^{1};\mathbb{C})$.
The proof is complete. ∎

######  Lemma 7.8.

Let $\beta>0$ and let $(c_{k})_{k=1}^{\infty}\subset\mathbb{C}$ be a sequence.
Define $g(\theta):=\sum_{k=1}^{\infty}c_{k}e^{2\pi i\cdot
2^{k}\theta}2^{-\beta\theta}$, then

  1. (i)

$\|g\|_{\mathscr{C}^{\beta}(\mathbb{R};\mathbb{C})}\approx\sup_{k\geq
0}|c_{k}|$ with the implied constant depending only on $\beta$.

  2. (ii)

If $\varlimsup_{k\to\infty}|c_{k}|=\infty$, then
$g\notin\mathscr{C}^{\beta}(I;\mathbb{C})$ for every interval
$I\subseteq\mathbb{R}$.

###### Proof.

Let $\\{\psi_{k}\\}_{k=0}^{\infty}$ be a dyadic resolution for $\mathbb{R}$.
Recall from (2.1)
$\operatorname{supp}\hat{\psi}_{k}\subset(-2^{k-1},-2^{k+1})\cup(2^{k-1},2^{k+1})$
holds for $k\geq 1$. Since the Fourier support of $e^{2\pi i2^{k}\theta}$ is
$\\{2^{k}\\}$, we have $\psi_{k}\ast g(\theta)=c_{k}2^{-\beta\theta}e^{2\pi
i\cdot 2^{k}\theta}$ for all $k\geq 1$. Therefore by Lemma 2.2 (i), we get the
result (i):

$\textstyle\|g\|_{\mathscr{C}^{\beta}(\mathbb{R})}\approx_{\psi,\beta}\sup_{k\geq
0}2^{k\beta}\|\psi_{k}\ast g\|_{L^{\infty}}=\sup_{k\geq 1}|c_{k}|.$

Now suppose $\varlimsup_{k\to\infty}|c_{k}|=+\infty$, so
$\|g\|_{\mathscr{C}^{\beta}(\mathbb{R})}\approx_{\psi,\beta}=\sup|c_{k}|=\infty$.

For a given interval $I\subseteq\mathbb{R}$, $I$ contains a dyadic
subinterval, say $[n2^{-k_{0}},(n+1)2^{-k_{0}}]$ where $n\in\mathbb{Z}$ and
$k_{0}\in\mathbb{Z}_{+}$. Take
$g^{k_{0}}(\theta):=\sum_{k=k_{0}+1}^{\infty}c_{k}2^{2\pi
i2^{k}\theta}2^{-k\beta}$, so

$\textstyle g(\theta)-g^{k_{0}}(\theta)=\sum_{k=1}^{k_{0}}c_{k}e^{2\pi
i2^{k}\theta}2^{-\beta\theta},$

must have finite $\mathscr{C}^{\beta}(\mathbb{R})$-norm (though whose bound
depends on $k_{0}$).

Since $g^{k_{0}}$ is $2^{-k_{0}}$-periodic,
$\|g\|_{\mathscr{C}^{\beta}(\mathbb{R})}=\infty$ and
$\|g-g^{k_{0}}\|_{\mathscr{C}^{\beta}(\mathbb{R})}<\infty$, we know
$\|g^{k_{0}}\|_{\mathscr{C}^{\beta}(n2^{-k_{0}},(n+1)2^{-k_{0}})}=\infty$.
Therefore, we complete the proof of (ii) by:

$\textstyle\|g\|_{\mathscr{C}^{\beta}(I)}\geq\|g\|_{\mathscr{C}^{\beta}([n2^{-k_{0}},(n+1)2^{-k_{0}}])}\geq\|g^{k_{0}}\|_{\mathscr{C}^{\beta}([n2^{-k_{0}},(n+1)2^{-k_{0}}])}-\|g-g^{k_{0}}\|_{\mathscr{C}^{\beta}(\mathbb{R})}=\infty.\qed$

We now construct our $\mathscr{C}^{\alpha,\beta}$-function $a(w,\theta)$ and
finish the Step 2:

######  Proposition 7.9.

Let $\alpha,\beta>0$. There is a function $A(w,\theta)$ defined in
$\mathbb{C}^{1}_{w}\times\mathbb{R}^{1}_{\theta}$, that satisfies the
following properties:

  1. (i)

$A\in\mathscr{C}^{\alpha,\beta}(\mathbb{C}^{1},\mathbb{R}^{1};\mathbb{C})\cap\mathscr{C}^{\beta}(\mathbb{R}^{1};W^{1,2}(\mathbb{C}^{1};\mathbb{C}))$
and $[(w,\theta)\mapsto
w\partial_{\bar{w}}A(w,\theta)]\in\mathscr{C}^{\alpha,\beta}(\mathbb{C}^{1},\mathbb{R}^{1};\mathbb{C})$.

  2. (ii)

$[(w,\theta)\mapsto
w^{-1}A(w,\theta)]\in\mathscr{C}^{\frac{\beta}{2}}(\mathbb{R}^{1};L^{p}(\mathbb{C}^{1};\mathbb{C}))$
for all $1<p<\frac{4}{2-\min(\alpha,2)}$. In particular
$w^{-1}A\in\mathscr{C}^{\frac{\beta}{2}}(\mathbb{R}^{1};L^{\frac{\alpha+4}{2}})$.

  3. (iii)

$\mathcal{T}[\partial_{\bar{w}}A]\notin
L^{\infty}_{w}\mathscr{C}^{\beta}_{\theta}$ near $(w,\theta)=(0,0)$, where
$\mathcal{T}$ be the inverse $\partial_{w}$-operator in Lemma 7.7.

The construction is essentially the parameter version of [Yao21].

######  Remark 7.10.

By taking some simple modifications from the construction below (see (7.15)),
one can also prove Proposition 7.9 for the case $\alpha=\infty$,
$\beta<\infty$. We leave the details to readers.

###### Proof of Proposition 7.9.

First we find a function $b\in C_{c}^{0}(\mathbb{B}^{2};\mathbb{C})$ such
that,

  1. (b.1)

$b\in C^{\infty}_{\mathrm{loc}}(\mathbb{B}^{2}\backslash\\{0\\};\mathbb{C})$,
$b\in W^{1,2}(\mathbb{B}^{2};\mathbb{C})$ and $wb\in
C_{c}^{1}(\mathbb{B}^{2};\mathbb{C})$.

  2. (b.2)

$\mathcal{T}_{0}\partial_{\bar{w}}b\notin L^{\infty}$ near $w=0$. Here
$\mathcal{T}_{0}$ be the inverse $\partial_{w}$-operator in Lemma 7.7.

Indeed, we take a function $b$ such that

$b(w):=\partial_{w}\big{(}\bar{w}(-\log|w|)^{\frac{1}{3}}\big{)}=-\tfrac{1}{6}(-\log|w|)^{-\frac{2}{3}}\tfrac{|w|^{2}}{w^{2}},\quad\text{when
}|w|<\tfrac{1}{2}.$ (7.14)

And $b$ is smoothly cutoffed outside $|w|<\frac{1}{2}$.

Immediately we get $b\in
C^{\infty}_{\mathrm{loc}}(\mathbb{B}^{2}\backslash\\{0\\};\mathbb{C})$ and
$wb(w)\in C^{1}_{c}$. And clearly we have $|\nabla
b(w)|\lesssim|w|^{-1}(-\log|w|)^{-\frac{2}{3}}$, so $b\in W^{1,2}$, which
gives (b.1).

To prove (b.2), since
$\partial_{w}\mathcal{T}_{0}\partial_{\bar{w}}b=\partial_{\bar{w}}b$ it
suffices to show that if $U$ is a neighborhood of $0$ and
$f:U\subseteq\frac{1}{2}\mathbb{B}^{2}\to\mathbb{C}$ is a function satisfies
$\partial_{w}f=\partial_{\bar{w}}b$, then $f\notin L^{\infty}$ near $0$.

Indeed by (7.14) in a neighborhood of $0$ we have
$f_{w}=(\bar{w}(-\log|w|)^{\frac{1}{2}})_{w\bar{w}}$, so
$f-(\bar{w}(-\log|w|)^{\frac{1}{2}})_{\bar{w}}$ is annihilated by
$\partial_{w}$ which is an anti-holmorphic function. Therefore
$f-(\bar{w}(-\log|w|)^{\frac{1}{2}})_{\bar{w}}$ is a bounded function near
$w=0$. On the other hand

$(\bar{w}(-\log|w|)^{\frac{1}{3}})_{\bar{w}}=(-\log|w|)^{\frac{1}{3}}+O(1)\quad\text{as
}w\to 0,$

is not bounded near $0$. We conclude that $f\notin L^{\infty}$ near $0$.

Let $\rho_{0}\in\mathscr{S}(\mathbb{C}^{1}_{w})$ be a Schwartz function whose
Fourier transform satisfies $\hat{\rho}_{0}\in C_{c}^{\infty}(\mathbb{B}^{2})$
and $\hat{\rho}_{0}|_{\frac{1}{2}\mathbb{B}^{2}}\equiv 1$. Define functions
$(\rho_{k},c_{k})_{k=0}^{\infty}$ and $A(w,\theta)$ as

$\rho_{k}(w):=2^{n\frac{\beta}{\alpha}k}\rho_{0}(2^{\frac{\beta}{\alpha}k}w),\quad
c_{k}(w):=\rho_{k}\ast b(w)-\rho_{k}\ast b(0),\quad k\geq 0;\quad
A(w,\theta):=\sum_{j=0}^{\infty}c_{j}(w)e^{2\pi i2^{j}\theta}2^{-j\beta}.$
(7.15)

Thus, we have
$wc_{k}=\int_{\mathbb{C}}(w-z)\rho_{k}(w-z)b(z)d\operatorname{vol}(z)=(w\rho_{k})\ast
b$ for $k\geq 0$. By scaling

$\|\rho_{k}\|_{L^{1}}=\|\rho_{0}\|_{L^{1}},\quad\|\nabla\rho_{k}\|_{L^{1}}=2^{\frac{\beta}{\alpha}k}\|\nabla\rho_{0}\|_{L^{1}},\quad\|w\nabla\rho_{k}\|_{L^{1}}=\|w\nabla\rho_{0}\|_{L^{1}}\quad\text{for
each }k\geq 1.$ (7.16)

Therefore,

$\begin{gathered}\|c_{k}\|_{L^{\infty}\cap W^{1,2}}\leq 2\|\rho_{k}\ast
b\|_{L^{\infty}\cap W^{1,2}}\lesssim\|\rho_{k}\|_{L^{1}}\|b\|_{L^{\infty}\cap
W^{1,2}}\lesssim 1;\\\
\|w\partial_{\bar{w}}c_{k}\|_{L^{\infty}}=\|\partial_{\bar{w}}(w\rho_{k})\ast
b\|_{L^{\infty}}\leq\|w\partial_{\bar{w}}\rho_{k}\|_{L^{1}}\|b\|_{L^{\infty}}\lesssim
1.\end{gathered}$ (7.17)

Let $(\phi_{j})_{j=0}^{\infty}$ and $(\psi_{k})_{k=0}^{\infty}$ be two dyadic
resolutions for $\mathbb{C}^{1}_{w}$ and $\mathbb{R}^{1}_{\theta}$
respectively (recall Definition 2.1). Recall from (2.1) $\hat{\phi}_{0}\in
C_{c}^{\infty}(2\mathbb{B}^{2})$, $\hat{\psi}_{0}\in
C_{c}^{\infty}(2\mathbb{B}^{1})$, $\operatorname{supp}\hat{\phi}_{j}\subset
2^{j+1}\mathbb{B}^{2}\backslash 2^{j-1}\mathbb{B}^{2}$ and
$\operatorname{supp}\hat{\psi}_{j}\subset 2^{j+1}\mathbb{B}^{1}\backslash
2^{j-1}\mathbb{B}^{1}$ for $j\geq 1$. Since
$\operatorname{supp}\hat{\rho}_{0}\subset\mathbb{B}^{2}$, by scaling we have
$\operatorname{supp}\hat{\rho}_{k}\subset
2^{\frac{\beta}{\alpha}k}\mathbb{B}^{2}$. Thus,

$\phi_{j}\ast_{w}A(w,\theta)=\sum_{k\geq\max(0,\frac{\alpha}{\beta}(j-1))}(\phi_{j}\ast
c_{k})(w)e^{2\pi
i2^{k}\theta}2^{-k\beta},\quad\psi_{j}\ast_{\theta}A(w,\theta)=c_{j}(w)e^{2\pi
i2^{j}\theta}2^{-j\beta},\quad j\geq 0.$ (7.18)

By (7.17) and (7.18) we get $A\in\mathscr{C}^{\alpha}L^{\infty}\cap
L^{\infty}\mathscr{C}^{\beta}(\mathbb{C}^{1},\mathbb{R}^{1};\mathbb{C})\cap\mathscr{C}^{\beta}(\mathbb{R}^{1};W^{1,2}(\mathbb{C}^{1};\mathbb{C}))$,
since for $j\geq 0$,

$\begin{gathered}\|\phi_{j}\ast_{w}A\|_{L^{\infty}(\mathbb{C}^{1}\times\mathbb{R}^{1};\mathbb{C})}\leq\sum_{k\geq\max(0,\frac{\alpha}{\beta}(j-1))}\|\phi_{j}\|_{L^{1}(\mathbb{C}^{1})}\|c_{k}\|_{L^{\infty}(\mathbb{C}^{1}\times\mathbb{R}^{1};\mathbb{C})}2^{-k\beta}\lesssim_{\phi,c,\alpha,\beta}2^{-j\alpha};\\\
\sup_{\theta\in\mathbb{R}^{1}}\|\psi_{j}\ast_{\theta}A(\cdot,\theta)\|_{L^{\infty}\cap
W^{1,2}(\mathbb{C}^{1};\mathbb{C})}\leq\|\psi_{j}\|_{L^{1}(\mathbb{R})}\|c_{j}\|_{L^{\infty}\cap
W^{1,2}(\mathbb{C}^{1};\mathbb{C})}2^{-j\beta}\lesssim_{\psi,c,\beta}2^{-j\beta}.\end{gathered}$

Taking supremum over $j\geq 0$ we get (i).

Using (7.16) and (7.17) we also have $\|c_{k}\|_{C^{1}}\lesssim
2^{\frac{\beta}{\alpha}k}$ and $\|w\nabla c_{k}\|_{C^{0}}\lesssim 1$. Thus

$|\nabla
c_{k}(w)|\lesssim\min(|w|^{-1},2^{\frac{\beta}{\alpha}k})\leq|w|^{\min(\frac{\alpha}{2},1)-1}2^{k\frac{\beta}{2}\cdot\min(\frac{\alpha}{2},1)}\leq|w|^{\min(\frac{\alpha}{2},1)-1}2^{k\frac{\beta}{2}},\quad\text{for
all}\quad k\geq 0,\quad w\in\mathbb{C}^{1}.$

By (7.15) we have $c_{k}(0,\theta)\equiv 0$ for all $\theta\in\mathbb{R}$.
Thus by (7.18),

$|\psi_{k}\ast_{\theta}(w^{-1}A)(w,\theta)|=|w^{-1}c_{k}(w)|2^{-k\beta}\leq|\nabla
c_{k}(w)|2^{-k\beta}\leq|w|^{\min(\frac{\alpha}{2},1)-1}2^{-k\frac{\beta}{2}},\quad
k\geq 0,\quad w\in\mathbb{C}^{2},\quad\theta\in\mathbb{R}^{1}.$

On the other hand $|w|^{-\gamma}\in L^{\frac{2}{\gamma}-}(\mathbb{B}^{2})$
holds for all $0<\gamma<2$. Thus for
$1<p<\frac{2}{1-\min(\alpha/2,1)}=\frac{4}{2-\min(\alpha,2)}$ we have
$\sup_{k\geq
0;\theta\in\mathbb{R}^{1}}2^{k\frac{\beta}{2}}\|\psi_{k}\ast_{\theta}(w^{-1}A)(\cdot,\theta)\|_{L^{p}(\mathbb{C}^{1};\mathbb{C})}<\infty$.

Clearly $2<\frac{\alpha+4}{2}<\frac{4}{2-\min(\alpha,2)}$, we finish the proof
of (ii).

To prove (iii), since $\mathcal{T}$ only acts on $w$-variable, by (7.15) we
have

$\mathcal{T}A_{\bar{w}}(w,\theta)=\sum_{k=1}^{\infty}2^{-k\alpha}e^{2\pi
i2^{k}\theta}\mathcal{T}_{0}\partial_{\bar{w}}c_{k}(w)=\sum_{k=1}^{\infty}2^{-k\alpha}e^{2\pi
i2^{k}\theta}\mathcal{T}_{0}(\rho_{k}\ast\partial_{\bar{w}}b)(w),\quad
w\in\mathbb{C}^{1},\ \theta\in\mathbb{R}.$

Now
$\lim\limits_{k\to\infty}\rho_{k}\ast(\partial_{\bar{w}}b)=\partial_{\bar{w}}b$
holds (as $L^{2}$-functions), and by (b.2)
$\mathcal{T}_{0}\partial_{\bar{w}}b\notin L^{\infty}$ near $w=0$. So for any
neighborhood $U\subseteq\mathbb{C}^{1}$ of $w=0$, we have
$\varlimsup_{k\to\infty}\|\mathcal{T}_{0}\partial_{w}c_{k}\|_{L^{\infty}(U;\mathbb{C})}=+\infty$.
Therefore by Lemma 7.8 (ii), for any interval $I\subseteq\mathbb{R}$
containing $\theta=0$, we have
$\|\mathcal{T}\partial_{\bar{w}}A\|_{L^{\infty}\mathscr{C}^{\alpha}(U,I;\mathbb{C})}\gtrsim\varlimsup_{k\to\infty}\|\mathcal{T}_{0}\partial_{\bar{w}}b\|_{L^{\infty}(U;\mathbb{C})}=+\infty$.

So $\mathcal{T}\partial_{\bar{w}}A\notin
L^{\infty}_{w}\mathscr{C}^{\beta}_{\theta}$ near $(0,0)$, finishing the proof
of (iii). ∎

Now we prove Step 3, which is the non-existence of
$f\in\mathscr{C}^{\alpha,\beta}$ to (7.5). Cf. [Yao21, Proposition 7]. Here we
consider all $\alpha,\beta>0$.

######  Theorem 7.11 ($\bar{\partial}$-equation for sharpness of
$F^{*}\frac{\partial}{\partial z}\notin\mathscr{C}^{\alpha}$).

Let $\alpha,\beta>0$, let $A$ be as in Proposition 7.9. Then there is a
constant $\delta=\delta(\alpha,\beta,A)>0$, such that for any neighborhood
$U_{2}\times V_{2}\subset\mathbb{C}^{1}_{w}\times\mathbb{R}^{1}_{\theta}$ of
$(0,0)$ there is no $f\in\mathscr{C}^{\alpha,\beta}(U_{2},V_{2};\mathbb{C})$
solving (7.5) with $a(w,\theta):=\delta\cdot A(w,\theta)$.

This result contains the largest possible range for $\alpha$ and $\beta$. The
proof can be simplified if $\alpha>1$.

###### Proof.

Let
$\mathcal{T}:\mathscr{C}^{\alpha-1}L^{\infty}(\mathbb{B}^{2},\mathbb{R}^{1};\mathbb{C})\to\mathscr{C}^{\alpha}L^{\infty}(\mathbb{B}^{2},\mathbb{R}^{1};\mathbb{C})$
be the inverse $\partial_{w}$-operator as in Lemma 7.7. Recall that
$\mathcal{T}:\mathscr{C}^{\beta}(\mathbb{R}^{1};L^{p}(\mathbb{B}^{2};\mathbb{C}))\to\mathscr{C}^{\beta}(\mathbb{R}^{1};W^{1,p}(\mathbb{B}^{2};\mathbb{C}))$
is also bounded for $1<p<\infty$.

We define a linear operator $\mathfrak{T}=\mathfrak{T}_{A}$ on functions in
$\mathbb{B}^{2}\times\mathbb{C}^{1}$ as

$\mathfrak{T}[\varphi]:=\mathcal{T}(A\cdot\partial_{\bar{w}}\varphi),\quad\varphi\in\mathscr{C}^{\alpha}L^{\infty}(\mathbb{B}^{2},\mathbb{R}^{1};\mathbb{C}).$
(7.19)

By Lemma 2.26 (i), Lemma 7.6 (i) and (ii),

$\displaystyle\|A\partial_{\bar{w}}\varphi\|_{\mathscr{C}^{\alpha-1}L^{\infty}(\mathbb{B}^{2},\mathbb{R}^{1};\mathbb{C})}$
$\displaystyle\lesssim_{\alpha}\|A\|_{\mathscr{C}^{\alpha}L^{\infty}}\|\partial_{\bar{w}}\varphi\|_{\mathscr{C}^{\alpha-1}L^{\infty}}\lesssim_{A}\|\varphi\|_{\mathscr{C}^{\alpha}L^{\infty}(\mathbb{B}^{2},\mathbb{R}^{1};\mathbb{C})},$
$\displaystyle\text{provided }\alpha>\tfrac{1}{2};$
$\displaystyle\|A\partial_{\bar{w}}\varphi\|_{\mathscr{C}^{\alpha-1}L^{\infty}(\mathbb{B}^{2},\mathbb{R}^{1};\mathbb{C})}$
$\displaystyle\lesssim_{\alpha}\|A\|_{L^{\infty}(\mathbb{R}^{1};L^{\infty}\cap
W^{1,2})}\|\partial_{\bar{w}}\varphi\|_{\mathscr{C}^{\alpha-1}L^{\infty}}\lesssim_{A}\|\varphi\|_{\mathscr{C}^{\alpha}L^{\infty}(\mathbb{B}^{2},\mathbb{R}^{1};\mathbb{C})},$
$\displaystyle\text{provided }0<\alpha<1;$
$\displaystyle\|A\partial_{\bar{w}}\varphi\|_{\mathscr{C}^{\beta}(\mathbb{R}^{1};L^{p}(\mathbb{B}^{2};\mathbb{C}))}$
$\displaystyle\lesssim_{\beta}\|A\|_{L^{\infty}\mathscr{C}^{\beta}}\|\partial_{\bar{w}}\varphi\|_{\mathscr{C}^{\beta}(\mathbb{R}^{1};L^{p}(\mathbb{B}^{2};\mathbb{C}))}\lesssim_{A,p}\|\varphi\|_{\mathscr{C}^{\beta}(\mathbb{R}^{1};W^{1,p}(\mathbb{B}^{2};\mathbb{C}))},$
$\displaystyle\text{for all }1<p<\infty.$

Taking $p=p_{\alpha}:=4+16/\alpha$, we conclude that $\mathfrak{T}$ is a
bounded endomorphism on both
$\mathscr{C}^{\alpha}L^{\infty}(\mathbb{B}^{2},\mathbb{R}^{1};\mathbb{C})$ and
$\mathscr{C}^{\alpha}L^{\infty}(\mathbb{B}^{2},\mathbb{R}^{1};\mathbb{C})\cap\mathscr{C}^{\beta}(\mathbb{R}^{1};W^{1,4+\frac{16}{\alpha}}(\mathbb{B}^{2};\mathbb{C}))$.

We now take $\delta=\delta(\alpha,\beta,A)>0$ small such that

$\delta\big{(}\|\mathfrak{T}\|_{\mathscr{C}^{\alpha}L^{\infty}(\mathbb{B}^{2},\mathbb{R}^{1};\mathbb{C})}+\|\mathfrak{T}\|_{\mathscr{C}^{\alpha}L^{\infty}(\mathbb{B}^{2},\mathbb{R}^{1};\mathbb{C})\cap\mathscr{C}^{\beta}(\mathbb{R}^{1};W^{1,4+16/\alpha}(\mathbb{B}^{2};\mathbb{C}))}\big{)}\leq\tfrac{1}{2}.$
(7.20)

Note that $\mathfrak{T}$ is now contraction map in these two spaces.

We claim that $a(w,\theta)=\delta A(w,\theta)$ is as desired.

Suppose by contradiction there is a $U_{2}\times V_{2}\ni(0,0)$ and a
$f\in\mathscr{C}^{\alpha,\beta}(U_{2},V_{2};\mathbb{C})$ solving (7.5). Take
$0<\varepsilon_{0}<\frac{1}{2}$ such that
$\varepsilon_{0}\mathbb{B}^{2}\subseteq U_{2}$ and
$(-\varepsilon_{0},\varepsilon_{0})\subseteq V_{2}$. We define $\tilde{\chi}$
and $\chi$ such that

$\tilde{\chi}\in
C_{c}^{\infty}(-\varepsilon_{0},\varepsilon_{0}),\quad\tilde{\chi}|_{(-\frac{1}{2}\varepsilon_{0},\frac{1}{2}\varepsilon_{0})}\equiv
1,\quad\chi\in
C_{c}^{\infty}(\varepsilon_{0}\mathbb{B}^{2}\times(-\varepsilon_{0},\varepsilon_{0})),\quad\chi(w,\theta):=\tilde{\chi}(|w|)\tilde{\chi}(\theta).$
(7.21)

Therefore $\chi\in C_{c}^{\infty}(U_{2})$ equals to 1 in the neighborhood
$\frac{\varepsilon_{0}}{2}\mathbb{B}^{2}\times(-\frac{\varepsilon_{0}}{2},\frac{\varepsilon_{0}}{2})$
of $(0,0)$, and $\nabla_{w}\chi$ vanishes in the neighborhood
$\frac{\varepsilon_{0}}{2}\mathbb{B}^{2}\times\mathbb{R}^{1}$ of
$\\{w=0\\}=\\{0\\}_{w}\times\mathbb{R}^{1}_{\theta}$.

Define $g,h\in\mathscr{C}^{\alpha,\beta}_{c}(U_{2},V_{2};\mathbb{C})$ as
$g:=\chi f$ and $h(w,\theta):=wg(w,\theta)$, so

$g_{w}+a\cdot g_{\bar{w}}=(\chi_{w}+a\chi_{\bar{w}})f-\chi a_{\bar{w}},\quad
h_{w}+a\cdot h_{\bar{w}}=g+w(\chi_{w}+a\chi_{\bar{w}})f-\chi
wa_{\bar{w}},\quad\text{in }\mathbb{B}^{2}_{w}\times\mathbb{R}^{1}_{\theta}.$
(7.22)

Claim 1:
$h\in\mathscr{C}^{\beta}(\mathbb{R}^{1};W^{1,4+\frac{16}{\alpha}}(\mathbb{B}^{2};\mathbb{C}))$
and therefore
$h_{\bar{w}}\in\mathscr{C}^{\beta}(\mathbb{R}^{1};L^{4+\frac{16}{\alpha}}(\mathbb{B}^{2};\mathbb{C}))$.

Rewriting the second equation of (7.22) we have

$h=-\mathcal{T}(ah_{\bar{w}})+(h-\mathcal{T}h_{w})+\mathcal{T}(g+w(\chi_{w}+a\chi_{\bar{w}})f-\chi
wa_{\bar{w}}).$

In other words $h=\varphi$ is a solution to the following affine linear
equation:

$\varphi=-\delta\mathfrak{T}[\varphi]+(h-\mathcal{T}h_{w})+\mathcal{T}(g+w(\chi_{w}+a\chi_{\bar{w}})f-\chi
wa_{\bar{w}}).$ (7.23)

By assumption
$g\in\mathscr{C}^{\alpha,\beta}(\mathbb{B}^{2},\mathbb{R}^{1};\mathbb{C})$,
$f\nabla_{w}\chi\in\mathscr{C}^{\alpha,\beta}(\mathbb{B}^{2},\mathbb{R}^{1};\mathbb{C}^{2})$.
By Proposition 7.9 (i)
$wa_{\bar{w}}\in\mathscr{C}^{\alpha,\beta}(\mathbb{B}^{2},\mathbb{R}^{1};\mathbb{C})$.
Therefore by (7.12) (using
$L^{\infty}_{w}\mathscr{C}^{\beta}_{\theta}\subset\mathscr{C}^{\beta}_{\theta}(L^{4+\frac{16}{\alpha}}_{w})$)
we get

$\mathcal{T}(g+w(\chi_{w}+a\chi_{\bar{w}})f-\chi
wa_{\bar{w}})\in\mathcal{T}(\mathscr{C}^{\alpha,\beta}(\mathbb{B}^{2},\mathbb{R}^{1};\mathbb{C}))\subset\mathscr{C}^{\alpha}L^{\infty}(\mathbb{B}^{2},\mathbb{R}^{1};\mathbb{C})\cap\mathscr{C}^{\beta}(\mathbb{R}^{1};W^{1,4+\frac{16}{\alpha}}(\mathbb{B}^{2};\mathbb{C})).$
(7.24)

By assumption
$h\in\mathscr{C}^{\alpha,\beta}(\mathbb{B}^{2},\mathbb{R}^{1};\mathbb{C})\subset
L^{\infty}\mathscr{C}^{\beta}(\mathbb{B}^{2},\mathbb{R}^{1};\mathbb{C})$ is
supported in $\frac{1}{2}\mathbb{B}^{2}_{w}\times\mathbb{R}^{1}_{\theta}$, so
by (7.13)
$h-\mathcal{T}h_{w}\in\mathscr{C}^{\infty}\mathscr{C}^{\beta}(\mathbb{B}^{2},\mathbb{R}^{1};\mathbb{C})$.
In particular

$h-\mathcal{T}h_{w}\in\mathscr{C}^{\alpha}L^{\infty}(\mathbb{B}^{2},\mathbb{R}^{1};\mathbb{C})\cap\mathscr{C}^{\beta}(\mathbb{R}^{1};W^{1,4+\frac{16}{\alpha}}(\mathbb{B}^{2};\mathbb{C})).$
(7.25)

Using (7.24), (7.25) and the assumption (7.20),
$\big{[}\varphi\mapsto-\delta\mathfrak{T}[\varphi]+(h-\mathcal{T}h_{w})+\mathcal{T}(g+w(\chi_{w}+a\chi_{\bar{w}})f-\chi
wa_{\bar{w}})\big{]}$ is a contraction map in both
$\mathscr{C}^{\alpha}L^{\infty}(\mathbb{B}^{2},\mathbb{R}^{1};\mathbb{C})$ and
$\mathscr{C}^{\alpha}L^{\infty}(\mathbb{B}^{2},\mathbb{R}^{1};\mathbb{C})\cap\mathscr{C}^{\beta}(\mathbb{R}^{1};W^{1,4+\frac{16}{\alpha}}(\mathbb{B}^{2};\mathbb{C}))$.
Since (7.23) has a unique fixed point in both spaces, and
$h\in\mathscr{C}^{\alpha}L^{\infty}(\mathbb{B}^{2},\mathbb{R}^{1};\mathbb{C})$
is a fixed point to (7.23), we conclude that
$h\in\mathscr{C}^{\beta}(\mathbb{R}^{1};W^{1,4+\frac{16}{\alpha}}(\mathbb{B}^{2};\mathbb{C}))$.
Thus
$h_{\bar{w}}\in\mathscr{C}^{\beta}(\mathbb{R}^{1};L^{4+\frac{16}{\alpha}}(\mathbb{B}^{2};\mathbb{C}))$.
This finishes the proof of Claim 1.

Claim 2: $a\cdot
g_{\bar{w}}\in\mathscr{C}^{\varepsilon-1}_{w}\mathscr{C}^{\beta}_{\theta}(\mathbb{B}^{2},\mathbb{R}^{1};\mathbb{C})$
for some $\varepsilon>0$, and therefore $\mathcal{T}(a\cdot g_{\bar{w}})\in
L^{\infty}\mathscr{C}^{\beta}(\mathbb{B}^{2},\mathbb{R}^{1};\mathbb{C})$.

Let $(\psi_{j})_{j=0}^{\infty}$ be a dyadic resolution for
$\mathbb{R}^{1}_{\theta}$. By Lemma 2.24, for $l\geq 0$, on
$\mathbb{C}^{1}\times\mathbb{R}^{1}$,

$\displaystyle\psi_{l}\ast_{\theta}(ag_{\bar{w}})$
$\displaystyle=\psi_{l}\ast_{\theta}\Big{(}\sum_{j=l-2}^{l+2}\sum_{j^{\prime}=0}^{j-3}+\sum_{j=l-2}^{l+2}\sum_{j^{\prime}=0}^{j-3}+\sum_{\begin{subarray}{c}j,j^{\prime}\geq
l-3;|j-j^{\prime}|\leq
2\end{subarray}}\Big{)}(\psi_{j}\ast_{\theta}a)\cdot(\psi_{j^{\prime}}\ast_{\theta}g_{\bar{w}})=:P_{l}^{1}+P_{l}^{2}+P_{l}^{3}.$

Our goal is to show that for $\varepsilon=\frac{\alpha}{8+2\alpha}>0$,
$\sup_{l\geq
0;\theta\in\mathbb{R}^{1}}2^{l\beta}\|P_{l}^{\nu}(\cdot,\theta)\|_{\mathscr{C}^{\varepsilon-1}(\mathbb{C}^{1};\mathbb{C})}<\infty$
for $\nu=1,2,3$.

By assumption $a\in\mathscr{C}^{\beta}(\mathbb{R}^{1};L^{\infty}\cap
W^{1,2}(\mathbb{C}^{1};\mathbb{C}))$ and
$g_{\bar{w}}\in\mathscr{C}^{\alpha-1}L^{\infty}\cap\mathscr{C}^{-1}\mathscr{C}^{\beta}(\mathbb{C}^{1},\mathbb{R}^{1};\mathbb{C})\subset\mathscr{C}^{\frac{\alpha}{2}-1}\mathscr{C}^{\frac{\beta}{2}}(\mathbb{C}^{1},\mathbb{R}^{1};\mathbb{C})$,
so we have
$\sup_{\theta\in\mathbb{R}^{1}}\|\psi_{j}\ast_{\theta}a(\cdot,\theta)\|_{L^{\infty}\cap
W^{1,2}}\lesssim 2^{-j\beta}$ and
$\sup_{\theta\in\mathbb{R}^{1}}\|\psi_{j^{\prime}}\ast_{\theta}g_{\bar{w}}(\cdot,\theta)\|_{\mathscr{C}^{\frac{\alpha}{2}-1}}\lesssim
2^{-j^{\prime}\frac{\beta}{2}}$. Thus by Lemma 7.6 (ii), we have for each
$\theta\in\mathbb{R}^{1}$,

$\displaystyle\|P^{1}_{l}(\cdot,\theta)\|_{\mathscr{C}^{\frac{\alpha}{2}-1}}$
$\displaystyle\leq\|\psi_{l}\|_{L^{1}}\sum_{j=l-2}^{l+2}\sum_{j^{\prime}=0}^{j-3}\|\psi_{j}\ast_{\theta}a(\cdot,\theta)\|_{L^{\infty}\cap
W^{1,2}}\|\psi_{j^{\prime}}\ast_{\theta}g_{\bar{w}}(\cdot,\theta)\|_{\mathscr{C}^{\frac{\alpha}{2}-1}}\lesssim_{a,g}\sum_{j=l-2}^{l+2}\sum_{j^{\prime}=0}^{j-3}2^{-j\beta}2^{-j^{\prime}\frac{\beta}{2}}\lesssim
2^{-l\beta};$
$\displaystyle\|P^{3}_{l}(\cdot,\theta)\|_{\mathscr{C}^{\frac{\alpha}{2}-1}}$
$\displaystyle\leq\|\psi_{l}\|_{L^{1}}\sum_{\begin{subarray}{c}j,j^{\prime}\geq
l-3\\\ |j-j^{\prime}|\leq
2\end{subarray}}\|\psi_{j}\ast_{\theta}a(\cdot,\theta)\|_{L^{\infty}\cap
W^{1,2}}\|\psi_{j^{\prime}}\ast_{\theta}g_{\bar{w}}(\cdot,\theta)\|_{\mathscr{C}^{\frac{\alpha}{2}-1}}\lesssim_{a,g}\sum_{\begin{subarray}{c}j,j^{\prime}\geq
l-3\\\ |j-j^{\prime}|\leq
2\end{subarray}}2^{-j\beta}2^{-j^{\prime}\frac{\beta}{2}}\lesssim
2^{-l\frac{3}{2}\beta}.$

Taking supremum over $\theta\in\mathbb{R}^{1}$ we get $\sup_{l\geq
0;\theta\in\mathbb{R}^{1}}2^{l\beta}(\|P^{1}_{l}(\cdot,\theta)\|_{\mathscr{C}^{\frac{\alpha}{2}-1}}+\|P^{3}_{l}(\cdot,\theta)\|_{\mathscr{C}^{\frac{\alpha}{2}-1}})<\infty$,
bounding the first and the third term.

For the second sum we use
$(\psi_{j}\ast_{\theta}a)\cdot(\psi_{j^{\prime}}\ast_{\theta}g_{\bar{w}})=(\psi_{j}\ast_{\theta}(w^{-1}a))\cdot(\psi_{j^{\prime}}\ast_{\theta}wg_{\bar{w}})=(\psi_{j}\ast_{\theta}(w^{-1}a))\cdot(\psi_{j^{\prime}}\ast_{\theta}h_{\bar{w}})$.
By Proposition 7.9 (ii)
$\sup_{\theta\in\mathbb{R}^{1}}\|\psi_{j}\ast_{\theta}(w^{-1}a)(\cdot,\theta)\|_{L^{\frac{\alpha+4}{2}}}\lesssim
2^{-j\frac{\beta}{2}}$, by Claim 1
$\sup_{\theta\in\mathbb{R}^{1}}\|\psi_{j^{\prime}}\ast_{\theta}h_{\bar{w}}(\cdot,\theta)\|_{L^{4+\frac{16}{\alpha}}}\lesssim
2^{-j^{\prime}\beta}$. And since the product map $L^{\frac{\alpha+4}{2}}\times
L^{4+\frac{16}{\alpha}}\to L^{\frac{4(\alpha+4)}{\alpha+8}}$ is bounded, we
have

$\displaystyle\|P^{2}_{l}(\cdot,\theta)\|_{L^{\frac{4(\alpha+4)}{\alpha+8}}}$
$\displaystyle\leq\|\psi_{l}\|_{L^{1}}\sum_{j^{\prime}=l-2}^{l+2}\sum_{j=0}^{j^{\prime}-3}\|\psi_{j}\ast_{\theta}(w^{-1}a)(\cdot,\theta)\|_{L^{\frac{\alpha+4}{2}}}\|\psi_{j^{\prime}}\ast_{\theta}h_{\bar{w}}(\cdot,\theta)\|_{L^{4+\frac{16}{\alpha}}}\lesssim\sum_{j^{\prime}=l-2}^{l+2}\sum_{j=0}^{j^{\prime}-3}2^{-j\frac{\beta}{2}}2^{-j^{\prime}\beta}\lesssim
2^{-l\beta}.$

By Sobolev embedding
$L^{\frac{4(\alpha+4)}{\alpha+8}}(\mathbb{C}^{1})\hookrightarrow\mathscr{C}^{\frac{\alpha}{8+2\alpha}-1}(\mathbb{C}^{1})$
(also see [Tri10, Chapter 2.7.1]), we conclude that $\sup_{l\geq
0;\theta\in\mathbb{R}^{1}}2^{l\beta}\|P^{2}_{l}(\cdot,\theta)\|_{\mathscr{C}^{\frac{\alpha}{8+2\alpha}-1}(\mathbb{C}^{1};\mathbb{C})}<\infty$.

Since $\frac{\alpha}{8+2\alpha}<\frac{\alpha}{2}$, we conclude that
$ag_{\bar{w}}\in\mathscr{C}^{\frac{\alpha}{8+2\alpha}-1}\mathscr{C}^{\beta}(\mathbb{C}^{1},\mathbb{R}^{1};\mathbb{C})$.
By Lemma 7.7 (i) we get $\mathcal{T}(a\cdot
g_{\bar{w}})\in\mathscr{C}^{\frac{\alpha}{8+2\alpha}}\mathscr{C}^{\beta}(\mathbb{B}^{2},\mathbb{R}^{1};\mathbb{C})\subset
L^{\infty}\mathscr{C}^{\beta}(\mathbb{B}^{2},\mathbb{R}^{1};\mathbb{C})$,
finishing the proof of Claim 2.

Final Step: We use $\mathcal{T}(a\cdot g_{\bar{w}})\in
L^{\infty}_{w}\mathscr{C}^{\beta}_{\theta}$ to obtain a contradiction.

Now rewriting the first equation in (7.22) we have

$g=-\mathcal{T}(ag_{\bar{w}})+(g-\mathcal{T}g_{w})+\mathcal{T}((\chi_{w}+a\chi_{\bar{w}})f)-\mathcal{T}((\chi-1)a_{\bar{w}})-\mathcal{T}(a_{\bar{w}}).$
(7.26)

By Claim 2, we have $\mathcal{T}(ag_{\bar{w}})\in
L^{\infty}\mathscr{C}^{\beta}(\mathbb{B}^{2},\mathbb{R}^{1};\mathbb{C})$. By
assumption $f\in\mathscr{C}^{\alpha,\beta}_{w,\theta}$, we have
$(\chi_{w}+a\chi_{\bar{w}})f\in\mathscr{C}^{\alpha,\beta}(\mathbb{B}^{2},\mathbb{R}^{1};\mathbb{C})$,
so
$\mathcal{T}((\chi_{w}+a\chi_{\bar{w}})f)\in\mathscr{C}^{1}\mathscr{C}^{\beta}(\mathbb{B}^{2},\mathbb{R}^{1};\mathbb{C})\subset
L^{\infty}\mathscr{C}^{\beta}(\mathbb{B}^{2},\mathbb{R}^{1};\mathbb{C})$ as
well.

Since $g\in\mathscr{C}^{\alpha,\beta}_{w,\theta}\subset
L^{\infty}_{w}\mathscr{C}^{\beta}_{\theta}$ is supported in
$\\{|w|<\frac{1}{2}\\}\times\mathbb{R}^{1}$, by (7.13) $g-\mathcal{T}g_{w}\in
L^{\infty}\mathscr{C}^{\beta}(\mathbb{B}^{2},\mathbb{R}^{1};\mathbb{C})$ as
well.

Let $\mu(\theta):=\tilde{\chi}(2\theta)$, we have
$\mu|_{(-\frac{1}{4}\varepsilon_{0},\frac{1}{4}\varepsilon_{0})}\equiv 1$.
Thus by (7.21), for every
$(w,\theta)\in\frac{\varepsilon_{0}}{2}\mathbb{B}^{2}\times(-\frac{\varepsilon_{0}}{4},\frac{\varepsilon_{0}}{4})$,
using
$\mu(\chi-1)|_{\frac{\varepsilon_{0}}{2}\mathbb{B}^{2}\times(-\frac{\varepsilon_{0}}{4},\frac{\varepsilon_{0}}{4})}=0$
we have

$\mathcal{T}((\chi-1)a_{\bar{w}})(w,\theta)=\mu(\theta)\mathcal{T}((\chi-1)a_{\bar{w}})(w,\theta)=\mathcal{T}(\mu(\chi-1)a_{\bar{w}})(w,\theta)=\big{(}\mathcal{T}(\mu(\chi-1)a_{\bar{w}})-\mu(\chi-1)a_{\bar{w}}\big{)}(w,\theta).$

By assumption $\mu(\chi-1)a_{\bar{w}}$ is supported in
$\\{\frac{\varepsilon_{0}}{2}<|w|<\varepsilon_{0}\\}\times(-\frac{\varepsilon_{0}}{4},\frac{\varepsilon_{0}}{4})$,
so by Proposition 7.9 (i) we have $\mu(\chi-1)a_{\bar{w}}\in
L^{\infty}\mathscr{C}^{\beta}(\mathbb{B}^{2},\mathbb{R}^{1};\mathbb{C})$.
Using Lemma 7.7 (ii) we get $\mathcal{T}((\chi-1)a_{\bar{w}})\in
L^{\infty}\mathscr{C}^{\beta}(\mathbb{B}^{2},\mathbb{R}^{1};\mathbb{C})$ as
well.

However, by Proposition 7.9 (iii) we have $\mathcal{T}(a_{\bar{w}})\notin
L^{\infty}_{w}\mathscr{C}^{\beta}_{\theta}$ near $(0,0)$. Therefore near
$(w,\theta)=(0,0)$, the left hand side of (7.26) is in
$L^{\infty}_{w}\mathscr{C}^{\beta}_{\theta}$ while the right hand side is not,
this contradicts to the assumption
$f\in\mathscr{C}^{\alpha,\beta}(U_{2},V_{2};\mathbb{C})$ and finishes the
proof. ∎

To summarize this subsection, what we obtained is the following:

######  Theorem 7.12.

Let $\alpha>\frac{1}{2}$ and $\beta>0$, let
$a\in\mathscr{C}^{\alpha,\beta}(\mathbb{B}^{2}_{w},\mathbb{R}^{1}_{\theta};\mathbb{C})$
be as in Theorem 7.11 (see (7.15)). Let
$\mathcal{S}:=\operatorname{Span}(\partial_{w}+a(w,\theta)\partial_{\bar{w}})$
be a rank 1 complex Frobenius structure in
$\mathbb{C}^{1}\times\mathbb{R}^{1}$.

Then for any neighborhood $U\times
V\subseteq\mathbb{C}^{1}_{w}\times\mathbb{R}^{1}_{\theta}$ of $(0,0)$ there is
no continuous map $F:U\times V\to\mathbb{C}^{1}_{z}\times\mathbb{R}^{1}_{s}$
such that $F$ is homeomorphic to its image, $\nabla_{w}F$ and
$\nabla_{z}(F^{\mathrm{Inv}})$ are both continuous, and
$F^{*}\frac{\partial}{\partial z}$ spans $\mathcal{S}|_{U\times V}$.

In particular we have the result for Theorem 1.3: if $\alpha=\beta>1$, then
there is no $C^{1}$-coordinate chart $F:U\times
V\subseteq\mathbb{C}^{1}_{z}\times\mathbb{R}^{1}_{s}$ near $(0,0)$ such that
$F^{*}\frac{\partial}{\partial z}$ spans $\mathcal{S}|_{U\times V}$.

###### Proof.

Suppose, by contrast, such continuous map $F$ exists. Then by Proposition 7.3
(A) $\Rightarrow$ (B) there are a neighborhood $U_{2}\times V_{2}\subseteq
U\times V$ of $(0,0)$ and a function
$f\in\mathscr{C}^{\alpha,\beta}(U_{2},V_{2};\mathbb{C})$ that solves the PDE
$f_{w}+af_{\bar{w}}=-a_{\bar{w}}$. However by Theorem 7.11 such $f$ would not
exist, contradiction.

When $\alpha=\beta>1$, a $C^{1}$-coordinate chart $F$ implies $\nabla_{w}F\in
C^{0}$ and $\nabla_{z}(F^{\mathrm{Inv}})\in C^{0}$ automatically, so the
assumptions of Proposition 7.3 (A) are satisfied. By Proposition 7.3 (A)
$\Rightarrow$ (B) and Theorem 7.11 the proof follows. ∎

##  Appendix A Some Results with non $C^{1}$ Parameters

######  Proposition A.1.

Let $\alpha>0$ and $0<\beta<\min(\frac{2\alpha+1}{\alpha+1},\frac{3}{2})$. Let
$F\in\mathscr{C}^{\alpha+1,\beta}(\mathbb{B}^{n}_{x},\mathbb{B}^{q}_{s};\mathbb{R}^{n}_{y})$
be a map such that $F(\cdot,s)$ is a $\mathscr{C}^{\alpha+1}$-diffeomorphism
onto its image and $F(\mathbb{B}^{n},s)\supseteq\mathbb{B}^{n}$ for each
$s\in\mathbb{B}^{q}$. Then $\Phi(y,s):=F(\cdot,s)^{\mathrm{Inv}}(y)$ satisfies
$\Phi\in\mathscr{C}^{\alpha+1,\beta}(\mathbb{B}^{n}_{x},\mathbb{B}^{q}_{s};\mathbb{R}^{n}_{y})$.

Moreover, for any $M>1$ there is a $K_{4}=K_{4}(n,q,\alpha,\beta,M)>0$ that
does not depend on $F$, such that if
$\|F\|_{\mathscr{C}^{\alpha+1,\beta}(\mathbb{B}^{n},\mathbb{B}^{q};\mathbb{R}^{n})}<M$
and
$\inf_{x\in\mathbb{B}^{n},s\in\mathbb{B}^{q}}|\det(\nabla_{x}F(x,s))^{-1}|>M^{-1}$
then
$\|\Phi\|_{\mathscr{C}^{\alpha+1,\beta}(\mathbb{B}^{n},\mathbb{B}^{q};\mathbb{R}^{n})}<K_{4}$.

The result is indeed true for all $0<\beta\leq\alpha+1$. When $\beta>1$, the
proof is simpler by applying the Inverse Function Theorem on the
$\mathscr{C}^{\beta}$-map $(x,s)\mapsto(F(x,s),s)$. In our case, we deal with
the difference quotiences directly.

###### Proof.

It suffices to prove the quantitative result: by assumption,
$\|F(\cdot,s)\|_{\mathscr{C}^{\alpha+1}}<M$ and
$\inf_{x\in\mathbb{B}^{n}}|\det(\nabla_{x}F(x,s))^{-1}|>M^{-1}$, for each
$s\in\mathbb{B}^{q}$. Applying [Str21, Lemma 5.9] we get
$\|\Phi(\cdot,s)\|_{\mathscr{C}^{\alpha+1}(\mathbb{B}^{n};\mathbb{R}^{n})}\lesssim_{n,\alpha,M}1$.

Thus to prove the proposition, it suffices to prove
$\|\Phi(y,\cdot)\|_{\mathscr{C}^{\beta}(\mathbb{B}^{q};\mathbb{R}^{n})}\lesssim_{n,q,\alpha,\beta,M}1$
uniformly in $y\in\mathbb{B}^{n}$. Without loss of generality, we assume
$\alpha<1$ from now on, which means $\frac{2\alpha+1}{\alpha+1}<\frac{3}{2}$.

Let $y\in\mathbb{B}^{n}$ and $s_{0},s_{1}\in\mathbb{B}^{q}$. For $j=0,1$,
applying mean value theorem to $F(\cdot,s_{j})$,

$\Phi(y,s_{0})-\Phi(y,s_{1})=\nabla_{x}F\big{(}(1-\theta_{j})\Phi(y,s_{0})+\theta_{j}\Phi(y,s_{1}),s_{j}\big{)}^{-1}\cdot\big{(}F(\Phi(y,s_{0}),s_{j})-F(\Phi(y,s_{1}),s_{j}))\big{)},\text{
for some }\theta_{j}\in(0,1).$ (A.1)

Now
$F\in\mathscr{C}^{\alpha+1,\beta}(\mathbb{B}^{n},\mathbb{B}^{q};\mathbb{R}^{n})$.
By Lemma 2.18, we have
$\nabla_{x}F\in\mathscr{C}^{\alpha,\frac{\alpha\beta}{2\alpha+1}}(\mathbb{B}^{n},\mathbb{B}^{q};\mathbb{R}^{n\times
n})$ with
$\|\nabla_{x}F\|_{\mathscr{C}^{\alpha,\frac{\alpha\beta}{2\alpha+1}}_{x,s}}\lesssim_{n,q,\alpha,\beta}\|F\|_{\mathscr{C}^{\alpha+1,\beta}_{x,s}}\leq
M$. By [Str21, Lemma 5.7] and the cofactor representation of
$(\nabla_{x}F)^{-1}$, we can find a $C_{1}=C_{1}(n,q,\alpha,\beta,M)>0$ such
that

$\sup_{s\in\mathbb{B}^{q}}\|\nabla_{x}F(\cdot,s)^{-1}\|_{\mathscr{C}^{\alpha}(\mathbb{B}^{n};\mathbb{R}^{n\times
n})}+\sup_{x\in\mathbb{B}^{n}}\|\nabla_{x}F(x,\cdot)^{-1}\|_{\mathscr{C}^{\frac{\alpha\beta}{2\alpha+1}}(\mathbb{B}^{q};\mathbb{R}^{n\times
n})}<C_{1},$ (A.2)

when the assumptions
$\|F\|_{\mathscr{C}^{\alpha+1,\beta}(\mathbb{B}^{n},\mathbb{B}^{q};\mathbb{R}^{n})}<M$
and $\inf_{x\in\mathbb{B}^{n}}|\det(\nabla_{x}F(x,s))^{-1}|>M^{-1}$ are
satisfied.

We separate the discussions for $\beta<1$ and $\beta>1$.

Case $\beta\in(0,1)$: It suffices to show
$\sup_{y,s_{0},s_{1}}|\Phi(y,s_{0})-\Phi(y,s_{1})|\lesssim_{n,q,\alpha,\beta,M}|s_{0}-s_{1}|^{\beta}$.

We have $y=F(\Phi(y,s_{0}),s_{0})=F(\Phi(y,s_{1}),s_{1})$ since
$\Phi(\cdot,s_{j})=F(\cdot,s_{j})^{\mathrm{Inv}}$. Therefore

$\displaystyle\textstyle|\Phi(y,s_{0})-\Phi(y,s_{1})|\leq\big{(}\sup_{\mathbb{B}^{n}\times\mathbb{B}^{q}}|(\nabla_{x}F)^{-1}|\big{)}\cdot|F(\Phi(y,s_{0}),s_{0})-F(\Phi(y,s_{1}),s_{0}))|$
$\displaystyle\leq$ $\displaystyle\textstyle
C_{1}|F(\Phi(y,s_{1}),s_{1})-F(\Phi(y,s_{1}),s_{0}))|\leq
C_{1}\sup_{x\in\mathbb{B}^{n}}\|F(x,\cdot)\|_{\mathscr{C}^{\beta}(\mathbb{B}^{q})}|s_{0}-s_{1}|^{\beta}\leq
C_{1}M|s_{0}-s_{1}|^{\beta}.$

This complete the proof of $0<\beta<1$.

Case $\beta\in(1,\frac{2\alpha+1}{\alpha+1}]$: It suffices to show
$\sup_{y,s_{0},s_{1}}|\Phi(y,s_{0})+\Phi(y,s_{1})-2\Phi(y,\frac{s_{0}+s_{1}}{2})|\lesssim_{n,q,\alpha,\beta,M}|s_{0}-s_{1}|^{\beta}$.

Fix a $\tilde{y}\in\mathbb{B}^{n}$. For $s_{0},s_{1}\in\mathbb{B}^{q}$ we
denote $s_{\frac{1}{2}}:=\frac{s_{0}+s_{1}}{2}$,
$x_{i}:=\Phi(\tilde{y},s_{i})$ for $i=0,\frac{1}{2},1$.

Take $\varepsilon=1-\frac{\alpha+1}{2\alpha+1}\beta$, so
$0<\varepsilon<1-\frac{\beta}{\alpha+1}$. Since
$\|F\|_{\mathscr{C}^{\alpha+1,1-\varepsilon}_{x,s}}\lesssim_{\varepsilon}\|F\|_{\mathscr{C}^{\alpha+1,\beta}_{x,s}}\leq
M$ we have $\|F\|_{\mathscr{C}^{\alpha+1,1-\varepsilon}_{x,s}}\leq\tilde{M}$
for some $\tilde{M}=\tilde{M}(n,q,\alpha,M)>0$. Let
$C_{2}:=K_{4}(n,q,\alpha,1-\varepsilon,\tilde{M})$. By the proven case
$\beta=1-\varepsilon$ above we have
$\sup_{x\in\mathbb{B}^{n}}\|\Phi(x,\cdot)\|_{\mathscr{C}^{1-\varepsilon}(\mathbb{B}^{q};\mathbb{R}^{n})}<C_{2}$.
In particular $|x_{i}-x_{j}|=|\Phi(\tilde{y},s_{i})-\Phi(\tilde{y},s_{j})|\leq
C_{2}\cdot|s_{i}-s_{j}|^{1-\varepsilon}$.

By the mean value theorem we can find $\xi_{0}$ and $\xi_{1}$ contained in the
triangle with vertices $x_{0},x_{\frac{1}{2}},x_{1}$ such that,

$F(x_{0},s_{0})-F(x_{\frac{1}{2}},s_{0})=\nabla_{x}F(\xi_{0},s_{0})\cdot(x_{0}-x_{\frac{1}{2}}),\quad
F(x_{1},s_{1})-F(x_{\frac{1}{2}},s_{1})=\nabla_{x}F(\xi_{1},s_{1})\cdot(x_{1}-x_{\frac{1}{2}}).$

Clearly $|\xi_{0}-\xi_{1}|\leq\max_{i,j=0,\frac{1}{2},1}|x_{i}-x_{j}|$.
Therefore by (A.2) and the property $|x_{i}-x_{j}|\leq
C_{2}\cdot|s_{i}-s_{j}|^{\beta-1+\varepsilon}$,

$|\nabla_{x}F(\xi_{0},s_{0})^{-1}-\nabla_{x}F(\xi_{1},s_{1})^{-1}|\leq
C_{1}(|\xi_{0}-\xi_{1}|^{\alpha}+|s_{0}-s_{1}|^{\frac{\alpha\beta}{2\alpha+1}})\leq
C_{1}(C_{2}^{\alpha}+1)|s_{0}-s_{1}|^{\beta-1+\varepsilon}.$ (A.3)

Here we use the fact that
$|s_{0}-s_{1}|^{\frac{\alpha\beta}{2\alpha+1}}=|s_{0}-s_{1}|^{\beta-1+\varepsilon}$
and $|\xi_{0}-\xi_{1}|^{\alpha}\leq
C_{2}^{\alpha}|s_{0}-s_{1}|^{\alpha(1-\varepsilon)}\leq
C_{2}^{\alpha}|s_{0}-s_{1}|^{\beta-1+\varepsilon}$.

Therefore using $\tilde{y}=F(x_{j},s_{j})$ for $j=0,\frac{1}{2},1$, we have

$\displaystyle x_{0}+x_{1}-2x_{\frac{1}{2}}$
$\displaystyle=\nabla_{x}F(\xi_{0},s_{0})^{-1}\cdot(F(x_{0},s_{0})-F(x_{\frac{1}{2}},s_{0}))+\nabla_{x}F(\xi_{1},s_{1})^{-1}\cdot(F(x_{1},s_{1})-F(x_{\frac{1}{2}},s_{1}))$
$\displaystyle=\nabla_{x}F(\xi_{0},s_{0})^{-1}\cdot(2\tilde{y}-F(x_{\frac{1}{2}},s_{0})-F(x_{\frac{1}{2}},s_{1}))+\big{(}\nabla_{x}F(\xi_{0},s_{0})^{-1}-\nabla_{x}F(\xi_{1},s_{1})^{-1}\big{)}\cdot(\tilde{y}-F(x_{\frac{1}{2}},s_{1})).$

Applying (A.2) and (A.3) to the above equality:

$\displaystyle|x_{0}+x_{1}-2x_{\frac{1}{2}}|\leq
C_{1}|2F(x_{\frac{1}{2}},s_{\frac{1}{2}})-F(x_{\frac{1}{2}},s_{0})-F(x_{\frac{1}{2}},s_{1})|+C_{1}(C_{2}^{\alpha}+1)|s_{0}-s_{1}|^{\beta-1+\varepsilon}\cdot|F(x_{\frac{1}{2}},s_{\frac{1}{2}})-F(x_{\frac{1}{2}},s_{1})|$
$\displaystyle\lesssim$
${}_{q,\beta}C_{1}\|F\|_{L^{\infty}\mathscr{C}^{\beta}}|s_{0}-s_{1}|^{\beta}+C_{1}(C_{2}^{\alpha}+1)\|F\|_{L^{\infty}\mathscr{C}^{1-\varepsilon}}|s_{0}-s_{1}|^{\beta-1+\varepsilon+1-\varepsilon}\leq
C_{1}(M+(C_{2}^{\alpha}+1)\tilde{M})|s_{0}-s_{1}|^{\beta}.$

Thus
$\sup_{y\in\mathbb{B}^{n}}\|\Phi(y,\cdot)\|_{\mathscr{C}^{\beta}(\mathbb{B}^{q})}\lesssim_{n,q,\alpha,\beta,M}1$
and we complete the proof. ∎

######  Lemma A.2.

Let $\alpha>1$, $\beta\in(0,\alpha]$, and let
$U_{0}\subseteq\mathbb{R}^{n}_{x}$, $V_{0}\subseteq\mathbb{R}^{q}_{s}$ be two
open sets. Let
$X\in\mathscr{C}^{\alpha,\beta}_{\mathrm{loc}}(U_{0},V_{0};\mathbb{R}^{n})$ be
a vector field on $U_{0}$ with parameter on $V_{0}$.

Then for any $U_{1}\times V_{1}\Subset U_{0}\times V_{0}$ there is a
$\varepsilon_{0}>0$ and a unique map
$\Phi(t,x,s):(-\varepsilon_{0},\varepsilon_{0})\times U_{1}\times V_{1}\to
U_{0}\times V_{0}$ such that $\Phi(\cdot,\cdot,s)$ is $C^{1}$ for each $s\in
V_{1}$; $\frac{\partial\Phi}{\partial t}(t,x,s)=X(\Phi(t,x,s),s)$ and
$\Phi(0,x,s)=x$ in the domain. Moreover
$\Phi\in\mathscr{C}^{\alpha+1,\alpha,\beta}((-\varepsilon_{0},\varepsilon_{0}),U_{1},V_{1};U_{0})$.

###### Proof.

Take $\varepsilon_{0}<\frac{1}{2}\operatorname{dist}(U_{1},\partial U_{0})$.
For each $s\in V_{1}$, $X(\cdot,s)$ is a $\mathscr{C}^{\alpha}$-vector field.
By Lemma 3.22,
$\exp_{X(\cdot,s)}\in\mathscr{C}^{\alpha+1,\alpha}((-\varepsilon_{0},\varepsilon_{0}),U_{1};U_{0})$
is the unique $C^{1}$-map $\phi(\cdot,s)$ such that $\frac{\partial}{\partial
t}\phi(\cdot,s)=X(\cdot,s)\circ\phi(\cdot,s)$ and $\phi(0,x,s)=x$. Thus, our
map $\Phi(t,x,s)$ is pointwisely defined and $C^{1}$ in $(t,x)$. By the same
argument in the proof of Lemma 3.22 we see that
$\sup_{s}\|\Phi(\cdot,s)\|_{\mathscr{C}^{\alpha+1,\alpha}}<\infty$.

When $\beta>1$, $(X(x,s),s)$ is a $\mathscr{C}^{\beta}$-vector field on
$U_{0}\times V_{0}$ and we see that $(\Phi(t,x,s),s)$ is its ODE flow, so by
Lemma 3.22 we have $\Phi\in\mathscr{C}^{\beta+1,\beta}_{t,(x,s)}$. Since we
already have
$\sup_{s}\|\Phi(\cdot,s)\|_{\mathscr{C}^{\alpha+1,\alpha}}<\infty$, we get
$\Phi\in\mathscr{C}^{\alpha+1,\alpha,\beta}_{t,x,s}$.

When $\beta<1$, by shrinking $U_{0}\times V_{0}$ if necessary, we can assume
$\|X\|_{\mathscr{C}^{\alpha,\beta}(U_{0},V_{0})}<\infty$. We fix $x\in U_{1}$
and $s_{0},s_{1}\in V_{1}$. Define $\psi_{j}(t):=\Phi(t,x,s_{j})$ for $j=0,1$
and $0\leq t<\varepsilon_{0}$, we see that

$\displaystyle\textstyle|\psi_{0}(t)-\psi_{1}(t)|\leq\int_{0}^{t}|X(\psi_{0}(\tau),s_{0})-X(\psi_{1}(\tau),s_{1})|d\tau$