}\lambda=\sum_{i=1}^{n}\lambda_{i}dx^{i},\quad\vartheta\lambda=-\sum_{i=1}^{n}\frac{\partial\lambda_{i}}{\partial
x^{i}}.$

For any form $\omega$, we have
$d(\vartheta\omega)+\vartheta(d\omega)=-\Delta\omega$, where
$\Delta=\sum_{i=1}^{n}\partial_{x^{i}}^{2}$ is the classical Laplacian acting
on the components of $\omega$; in this setting $\Delta$ is called the
(negative) Hodge Laplacian.

######  Notation 2.1.39.

We use $\mathbf{\Gamma}$ as the Newtonian potential in $\mathbb{R}^{n}$,
namely

$\mathbf{\Gamma}(x):=\begin{cases}\frac{|x|}{2},&n=1,\\\
\frac{1}{2\pi}\log|x|,&n=2,\\\ -|\mathbb{S}^{n-1}|^{-1}|x|^{2-n},&n\geq
3.\end{cases}$ (2.1.26)

We will often convolve functions with $k$-forms. Formally, if
$\omega=\sum_{1\leq i_{1}<\cdots<i_{k}\leq
n}\omega_{i_{1},\ldots,i_{k}}dx^{i_{1}}\wedge\cdots\wedge dx^{i_{k}}$ is a
$k$-form, and $\phi$ is a function, we set $\phi*\omega=\sum_{1\leq
i_{1}<\cdots<i_{k}\leq
n}(\phi*\omega_{i_{1},\ldots,i_{k}})dx^{i_{1}}\wedge\cdots\wedge dx^{i_{k}}$.

######  Lemma 2.1.40.

Let $U\subset\mathbb{R}^{n}$ be a bounded open subset, then for every
$\alpha\in\mathbb{R}$ the map $[f\mapsto(\mathbf{\Gamma}\ast
f)|_{U}]:\mathscr{C}^{\alpha}_{c}(U;\wedge^{k}T^{*}U)\to\mathscr{C}^{\alpha+2}(U;\wedge^{k}T^{*}U)$
is continuous for all $0\leq k\leq n$. Moreover there is a $C=C(U,\alpha)>0$
such that $\|\mathbf{\Gamma}\ast f\|_{\mathscr{C}^{\alpha+2}(U)}\leq
C\|f\|_{\mathscr{C}^{\alpha}(U)}$ for all
$f\in\mathscr{C}^{\alpha}_{c}(U;\wedge^{k}T^{*}U)$.

###### Proof.

By passing to the coordinate components it suffices to prove the case $k=0$.

Let $\chi\in\mathscr{S}(\mathbb{R}^{n})$ whose Fourier transform is
$C_{c}^{\infty}$. We write $\mathbf{\Gamma}_{0}:=\chi\ast\mathbf{\Gamma}$ and
$\mathbf{\Gamma}_{\infty}:=(\delta_{0}-\chi)\ast\mathbf{\Gamma}$. Since
$\mathbf{\Gamma}$ is well-known to be a fundamental solution for the
Laplacian, we have the Fourier transform
$((I-\Delta)\mathbf{\Gamma}_{\infty})^{\wedge}(\xi)=(1-\hat{\chi}(\xi))(4\pi^{2}|\xi|^{-2}+1)$,
which is a bounded smooth function. Therefore by Hörmander-Mikhlin multiplier
theorem (see [Tri10, Theorem 2.3.7]),
$[f\mapsto(I-\Delta)\mathbf{\Gamma}_{\infty}\ast
f]:\mathscr{B}_{\infty\infty}^{\alpha}(\mathbb{R}^{n})\to\mathscr{B}_{\infty\infty}^{\alpha}(\mathbb{R}^{n})$.
By [Tri10, Theorem 2.3.8] we get
$(I-\Delta)^{-2}:\mathscr{B}_{\infty\infty}^{\alpha}(\mathbb{R}^{n})\to\mathscr{B}_{\infty\infty}^{\alpha+2}(\mathbb{R}^{n})$.

Recall from Lemma 2.1.4 and Definition 2.1.6 that
$\mathscr{C}^{\alpha}(\mathbb{R}^{n})=\mathscr{B}_{\infty\infty}^{\alpha}(\mathbb{R}^{n})$,
so $[f\mapsto\mathbf{\Gamma}_{\infty}\ast
f]:\mathscr{C}^{\alpha}_{c}(U)\subset\mathscr{C}^{\alpha}(\mathbb{R}^{n})\to\mathscr{C}^{\alpha+2}(\mathbb{R}^{n})$
is bounded linear.

On the other hand
$\operatorname{supp}\hat{\mathbf{\Gamma}}_{0}\subseteq\operatorname{supp}\hat{\chi}$
is compact, thus
$\mathbf{\Gamma}_{0}\in\mathscr{C}^{\infty}_{\mathrm{loc}}(\mathbb{R}^{n})$.
Clearly $[f\mapsto\mathbf{\Gamma}_{0}\ast
f]:\mathscr{E}(\mathbb{R}^{n})\to\mathscr{C}^{\infty}_{\mathrm{loc}}(\mathbb{R}^{n})$
is continuous and thus $[f\mapsto\mathbf{\Gamma}_{0}\ast
f]:\mathscr{C}^{\alpha}_{c}(U)\to\mathscr{C}^{\alpha+2}_{\mathrm{loc}}(\mathbb{R}^{n})$
is also bounded linear.

Now $[f\mapsto\mathbf{\Gamma}\ast
f]:\mathscr{C}^{\alpha}_{c}(U)\to\mathscr{C}^{\alpha+2}_{\mathrm{loc}}(\mathbb{R}^{n})$
is bounded linear, taking restriction to $U$ we get the continuity.

Let $\tilde{U}\subset\mathbb{R}^{n}$ be a larger bounded open subset such that
$U\Subset\tilde{U}$ is precompact. We know the Banach subspace
$\\{f\in\mathscr{C}^{\alpha}(\mathbb{R}^{n}):f|_{U^{c}}\equiv
0\\}\subset\mathscr{C}^{\alpha}(\mathbb{R}^{n})$ is a bounded subset of
$\mathscr{C}^{\alpha}_{c}(\tilde{U})$. Since $[f\mapsto(\mathbf{\Gamma}\ast
f)|_{\tilde{U}}]:\mathscr{C}^{\alpha}_{c}(\tilde{U})\to\mathscr{C}^{\alpha+2}(\tilde{U})$
is continuous, we know the following map is bounded:

$[f\mapsto(\mathbf{\Gamma}\ast
f)|_{U}]:\\{f\in\mathscr{C}^{\alpha}(\mathbb{R}^{n}):f|_{U^{c}}\equiv
0\\}\hookrightarrow\mathscr{C}^{\alpha}_{c}(\tilde{U})\to\mathscr{C}^{\alpha+2}(\tilde{U})\xrightarrow{(-)|_{U}}\mathscr{C}^{\alpha+2}(U).$

Thus such constant $C(U,\alpha)>0$ exists. ∎

We also consider the Dirichlet problem of the Laplacian equation:

######  Lemma 2.1.41 (The Dirichlet Problem).

Let $\alpha>0$, and let $U$ be a bounded domain with smooth boundary. Then for
$f\in\mathscr{C}^{\alpha-2}(U)$ and $g\in\mathscr{C}^{\alpha}(\partial U)$
there is a unique $u\in\mathscr{C}^{\alpha}(U)$ such that $\Delta u=f$ and
$u\big{|}_{\partial U}=g$.

Moreover, the solution map $(f,g)\mapsto u$ is continuous linear map
$\mathscr{C}^{\alpha-2}(U)\oplus\mathscr{C}^{\alpha}(\partial
U)\rightarrow\mathscr{C}^{\alpha}(U)$.

See [FR95, Theorem 15] for a proof of Lemma 2.1.41.

######  Remark 2.1.42.

In Lemma 2.1.41, for $\alpha>0$, $\mathscr{C}^{\alpha}(\partial U)$ is the
Zymgund-Hölder space on manifold. One can use Definition 3.1.1 to define
$\mathscr{C}^{\alpha}$-functions on it. Note that the sphere is a compact
manifold, and therefore $\mathscr{C}^{\alpha}_{\mathrm{loc}}(\partial
U)=\mathscr{C}^{\alpha}(\partial U)$. The norm can be defined using any finite
atlas, and the equivalence class of the norm does not depend on the choice of
atlas. Moreover we have

$\|f\|_{\mathscr{C}^{\alpha}(\partial
U)}\approx_{\alpha}\inf\\{\|\tilde{f}\|_{\mathscr{C}^{\alpha}(U)}:\tilde{f}\big{|}_{\partial
U}=f\\},$ (2.1.27)

since the trace operator $(\cdot)|_{\partial
U}:\mathscr{C}^{\alpha}(U)\to\mathscr{C}^{\alpha}(\partial U)$ is continuous
and surjective; see [Tri10, Theorem 2.7.2].

In the case of unit ball $\mathbb{B}^{n}$ and its boundary
$\mathbb{S}^{n-1}=\partial\mathbb{B}^{n}$, given a function
$f\in\mathscr{C}^{\alpha}(\mathbb{S}^{n-1})$, we can take
$\tilde{f}(x)=f(\frac{x}{|x|})\chi(|x|)$ where $\chi\in
C_{c}^{\infty}(-\frac{1}{2},2)$ such that $\chi(1)=1$, then we have
$\tilde{f}\big{|}_{\mathbb{S}^{n-1}}=f$ and
$\|\tilde{f}\|_{\mathscr{C}^{\alpha}(\mathbb{B}^{n})}\approx_{\alpha}\|f\|_{\mathscr{C}^{\alpha}(\mathbb{S}^{n-1})}$.

######  Notation 2.1.43.

Let $U\subset\mathbb{R}^{n}$ be a bounded smooth domain. We denote
$\mathcal{P}_{0}=\mathcal{P}_{0}^{U}$ to be the Laplacian solution operator
with zero Dirichlet boundary. Namely $\mathcal{P}_{0}$ is the map
$(f,0)\mapsto u$ in Lemma 2.1.41.

In Propositions 4.2.7 and 4.2.7 we need the following quantitative version
inverse function theorem and the pushforwards of the exact forms.

######  Proposition 2.1.44.

Let $\alpha\in(0,\infty)$ and $\beta\in[\alpha,\alpha+1]$ be two real numbers,
then there is a constant $K_{2}=K_{2}(n,\alpha,\beta)>1$ satisfying the
following:

Suppose $R\in\mathscr{C}^{\alpha+1}(\mathbb{B}^{n};\mathbb{R}^{n})$ satisfies
$R\big{|}_{\partial\mathbb{B}^{n}}=0$ and $\|R\|_{\mathscr{C}^{\alpha+1}}\leq
K_{2}^{-1}$, then the map
$F:=\operatorname{id}+R:\mathbb{B}^{n}\to\mathbb{R}^{n}$ is a
$\mathscr{C}^{\alpha+1}$-diffeomorphism of $\mathbb{B}^{n}$. Moreover,

  1. (i)

Let
$\Phi=(\phi^{1},\dots,\phi^{n}):\mathbb{B}^{n}\xrightarrow{\sim}\mathbb{B}^{n}$
be the inverse map of $F$. Then
$\|\Phi\|_{\mathscr{C}^{\alpha+1}(\mathbb{B}^{n};\mathbb{R}^{n})}\leq K_{2}$
and $\|\nabla\Phi-
I_{n}\|_{\mathscr{C}^{\alpha}(\mathbb{B}^{n};\mathbb{R}^{n\times n})}\leq
K_{2}\|R\|_{\mathscr{C}^{\alpha+1}(\mathbb{B}^{n};\mathbb{R}^{n})}$.

  2. (ii)

If $\lambda\in\mathscr{C}^{\alpha}_{c}(\mathbb{B}^{n};T^{*}\mathbb{B}^{n})$
satisfies $\operatorname{supp}\lambda\subsetneq\frac{1}{2}\mathbb{B}^{n}$ and
$d\lambda\in\mathscr{C}^{\beta-1}(\mathbb{B}^{n};\wedge^{2}T^{*}\mathbb{B}^{n})$,
then $\operatorname{supp}F_{*}\lambda\subsetneq\frac{3}{4}\mathbb{B}^{n}$ and

$\|d(F_{*}\lambda)\|_{\mathscr{C}^{\beta-1}(\mathbb{B}^{n};\wedge^{2}T^{*}\mathbb{B}^{n})}\leq
K_{2}\|d\lambda\|_{\mathscr{C}^{\beta-1}(\mathbb{B}^{n};\wedge^{2}T^{*}\mathbb{B}^{n})}.$
(2.1.28)

###### Proof.

We let $K_{2}$ be a large constant which may change from line to line. In
particular, we will choose $K_{2}$ large enough such that
$\|R\|_{\mathscr{C}^{\alpha+1}}\leq K_{2}^{-1}$ implies
$\|R\|_{C^{0}}+\|\nabla R\|_{C^{0}}\leq\frac{1}{4}$ and $\|\nabla
R\|_{\mathscr{C}^{\alpha}}\leq c_{0}(\mathbb{B}^{n},\alpha,n)$, where $c_{0}$
is in Lemma 2.1.28 (i).

By Lemma 2.1.28 (i), $\nabla F(x)=I+\nabla R(x)$ is an invertible matrix for
every $x\in\mathbb{B}^{n}$, and we have

$\|(\nabla F)^{-1}-I\|_{\mathscr{C}^{\alpha}}=\|(I+\nabla
R)^{-1}-I\|_{\mathscr{C}^{\alpha}}\leq 2\|\nabla R\|_{\mathscr{C}^{\alpha}}.$
(2.1.29)

Since $\|\nabla R\|_{C^{0}}\leq\frac{1}{4}$, we have
$|R(x_{1})-R(x_{2})|\leq\|\nabla
R\|_{C^{0}}|x_{1}-x_{2}|\leq\frac{1}{4}|x_{1}-x_{2}|$, which implies

$|F(x_{1})-F(x_{2})|\geq|x_{1}-x_{2}|-|R(x_{1})-R(x_{2})|\geq\textstyle\frac{3}{4}|x_{1}-x_{2}|.$
(2.1.30)

This implies $F$ is injective. By the Inverse Function Theorem, we know
$F:\mathbb{B}^{n}\xrightarrow{\sim}F(\mathbb{B}^{n})$ is a
$\mathscr{C}^{\alpha+1}$-diffeomorphism.

The assumption $R\big{|}_{\partial\mathbb{B}^{n}}=0$ gives
$F(\partial\mathbb{B}^{n})=\partial\mathbb{B}^{n}$. Since $F(\mathbb{B}^{n})$
is contractible and
$\overline{F(\mathbb{B}^{n})}\supset\partial\mathbb{B}^{n}$, we get that
$F(\mathbb{B}^{n})=\mathbb{B}^{n}$. We conclude $F$ is a
$\mathscr{C}^{\alpha+1}$-diffeomorphism on $\mathbb{B}^{n}$.

Proof of (i): First, we claim that there is a $C_{1}(n,\alpha)>0$, which does
not depend on $R$, such that whenever $R$ satisfies the assumptions of the
proposition, we have

$\|\Phi\|_{\mathscr{C}^{\alpha+1}(\mathbb{B}^{n};\mathbb{R}^{n})}\leq C_{1}.$
(2.1.31)

In particular $\|\Phi\|_{\mathscr{C}^{\alpha+1}}\leq K_{2}$ holds by taking
$K_{2}\geq C_{1}$.

Since $\Phi$ is the inverse map of $F$, by [Str21, Lemma 5.9] or by induction
on $\alpha$ with chain rule, we know
$\|\Phi\|_{\mathscr{C}^{\alpha+1}(\mathbb{B}^{n};\mathbb{R}^{n})}$ only
depends on $n$, $\alpha$,
$\|F\|_{\mathscr{C}^{\alpha+1}(\mathbb{B}^{n};\mathbb{R}^{n})}$, and
$\|(\nabla F)^{-1}\|_{C^{0}(\mathbb{B}^{n};\mathbb{R}^{n\times n})}$. We will
show that $\|F\|_{\mathscr{C}^{\alpha+1}(\mathbb{B}^{n};\mathbb{R}^{n})}$ and
$\|(\nabla F)^{-1}\|_{C^{0}(\mathbb{B}^{n};\mathbb{R}^{n\times n})}$ have
bounds that do not depend on $R$.

We have
$\|F\|_{\mathscr{C}^{\alpha+1}(\mathbb{B}^{n};\mathbb{R}^{n})}\leq\|\operatorname{id}\|_{\mathscr{C}^{\alpha+1}(\mathbb{B}^{n})}+\|R\|_{\mathscr{C}^{\alpha+1}(\mathbb{B}^{n};\mathbb{R}^{n})}\leq\|\operatorname{id}\|_{\mathscr{C}^{\alpha+1}(\mathbb{B}^{n})}+\frac{1}{4}$.
The right hand side of this inequality does not depend on $R$.

By (2.1.30), $|F(x_{1})-F(x_{2})|\geq\frac{3}{4}|x_{1}-x_{2}|$ implies
$\sup\limits_{x\in\mathbb{B}^{n}}|(\nabla F(x))^{-1}|\leq\frac{4}{3}$, so
$\|(\nabla F)^{-1}\|_{C^{0}(\mathbb{B}^{n};\mathbb{R}^{n\times
n})}\leq\frac{4}{3}$, which does not depend on $R$ as well. This establishes
(2.1.31).

Note that the identity matrix $I$ can be viewed as a constant function defined
on the unit ball. Since $\Phi$ is a $\mathscr{C}^{\alpha+1}$-diffeomorphism on
$\mathbb{B}^{n}$, we get the equality $I=I\circ\Phi$ as a matrix function on
$\mathbb{B}^{n}$.

By the chain rule $I=\nabla(F\circ\Phi)=((\nabla F)\circ\Phi)\cdot\nabla\Phi$,
so $\nabla\Phi-I=(\nabla F)^{-1}\circ\Phi-I=((\nabla F)^{-1}-I)\circ\Phi$.

By Proposition 2.1.30 (i), we know
$\|G\circ\Phi\|_{\mathscr{C}^{\alpha}(\mathbb{B}^{n};\mathbb{R}^{n\times
n})}\leq
K_{1}(n^{2},n,\alpha,\alpha,C_{1})\|G\|_{\mathscr{C}^{\alpha}(\mathbb{B}^{n};\mathbb{R}^{n\times
n})}$ holds when (2.1.31) is satisfied, where $K_{1}$ is the constant in
Proposition 2.1.30. Therefore by taking $G=(\nabla F)^{-1}-I$, we see that

$\|\nabla\Phi-I\|_{\mathscr{C}^{\alpha}(\mathbb{B}^{n};\mathbb{R}^{n\times
n})}=\|((\nabla F)^{-1}-I)\circ\Phi\|_{\mathscr{C}^{\alpha}}\leq
K_{1}\|(\nabla F)^{-1}-I\|_{\mathscr{C}^{\alpha}}\leq 2K_{1}\|\nabla
R\|_{\mathscr{C}^{\alpha}(\mathbb{B}^{n};\mathbb{R}^{n\times n})}\leq
2K_{1}\|R\|_{\mathscr{C}^{\alpha+1}(\mathbb{B}^{n};\mathbb{R}^{n})},$

and we obtain $\|\nabla\Phi-I\|_{\mathscr{C}^{\alpha}}\leq
K_{2}\|R\|_{\mathscr{C}^{\alpha+1}}$ by replacing $K_{2}$ with
$\max(K_{2},2K_{1})$.

Proof of (ii): Let
$\lambda\in\mathscr{C}_{c}^{\alpha}(\mathbb{B}^{n};T^{*}\mathbb{B}^{n})$ be as
in the assumption of (ii). In particular,
$\operatorname{supp}\lambda\subsetneq\frac{1}{2}\mathbb{B}^{n}$.

By the assumption $\|R\|_{C^{0}}\leq\frac{1}{4}$, we have
$F(\frac{1}{2}\mathbb{B}^{n})\subseteq\frac{1}{2}\mathbb{B}^{n}+\frac{1}{4}\mathbb{B}^{n}=\frac{3}{4}\mathbb{B}^{n}$,
so
$\operatorname{supp}F_{*}\lambda=F(\operatorname{supp}\lambda)\subseteq\frac{3}{4}\mathbb{B}^{n}$.

We define a 1-form $\rho$ and a function $\xi$ by

$\rho=\sum_{i=1}^{n}\rho_{i}dx^{i}:=-\mathbf{\Gamma}\ast\vartheta
d\lambda,\quad\xi:=-\mathbf{\Gamma}\ast\vartheta\lambda.$

$\rho$ and $\xi$ are globally defined in $\mathbb{R}^{n}$ because $\lambda$ is
compactly supported in $\frac{1}{2}\mathbb{B}^{n}$.

By Lemma 2.1.40, we have
$\rho\in\mathscr{C}^{\beta}_{\mathrm{loc}}(\mathbb{R}^{n};T^{*}\mathbb{R}^{n})$,
$\xi\in\mathscr{C}^{\alpha+1}_{\mathrm{loc}}(\mathbb{R}^{n})$ and

$\|\rho\|_{\mathscr{C}^{\beta}(\mathbb{B}^{n};T^{*}\mathbb{B}^{n})}\lesssim_{\beta}\|\vartheta
d\lambda\|_{\mathscr{C}^{\beta-2}}\lesssim_{\beta}\|d\lambda\|_{\mathscr{C}^{\beta-1}},\quad\|\xi\|_{\mathscr{C}^{\alpha}(\mathbb{B}^{n})}\lesssim_{\alpha}\|\vartheta\lambda\|_{\mathscr{C}^{\alpha-1}}\lesssim_{\alpha}\|\lambda\|_{\mathscr{C}^{\alpha}}.$
(2.1.32)

Since convolution commutes with $d$ and $\vartheta$, by construction
$\lambda-\rho=-\mathbf{\Gamma}\ast(d\vartheta+\vartheta
d)\lambda+\mathbf{\Gamma}\ast\vartheta\lambda=-d(\mathbf{\Gamma}\ast\vartheta
f)=d\xi$. By direct computation along with Lemma 2.1.17 $F_{*}df=d(f\circ
F^{\mathrm{Inv}})=dF_{*}f$ for all
$f\in\mathscr{C}^{\alpha+1}_{\mathrm{loc}}(\mathbb{R}^{n})\subset
C^{1}_{\mathrm{loc}}(\mathbb{R}^{n})$, therefore

$d(F_{*}\lambda)=d(F_{*}\rho)+d(F_{*}d\xi)=d(F_{*}\rho)+d^{2}(\xi\circ
F^{\mathrm{Inv}})=d(F_{*}\rho)=d\Big{(}\sum_{i=1}^{n}(\rho_{i}\circ\Phi)d\phi^{i}\Big{)},\quad\text{on
}\mathbb{B}^{n}.$ (2.1.33)

Thus, to prove (2.1.28), by (2.1.32) and the fact that
$\operatorname{supp}dF_{*}\lambda\subseteq\frac{3}{4}\mathbb{B}^{n}$ it
suffices to show

$\|d(F_{*}\rho)\|_{\mathscr{C}^{\beta-1}(\frac{3}{4}\mathbb{B}^{n};\wedge^{2}T^{*}\mathbb{R}^{n})}\lesssim\|\rho\|_{\mathscr{C}^{\beta}(\mathbb{B}^{n};T^{*}\mathbb{B}^{n})}.$
(2.1.34)

Fix $\chi\in C_{c}^{\infty}(\mathbb{B}^{n})$ such that
$\chi|_{\frac{3}{4}\mathbb{B}^{n}}\equiv 1$. For each $1\leq i\leq n$, set

$\tilde{\rho}_{i}:=\chi(\rho_{i}\circ\Phi),\quad\tilde{\phi}^{i}:=\chi\phi^{i}.$

So $\tilde{\rho}_{i}\in\mathscr{C}^{\beta}_{c}(\mathbb{B}^{n})$ and
$d\tilde{\phi}^{i}\in\mathscr{C}^{\alpha}_{c}(\mathbb{B}^{n};T^{*}\mathbb{B}^{n})$
are globally defined 1-forms for each $i$, such that

$\sum_{i=1}^{n}(\tilde{\rho}_{i}d\tilde{\phi}^{i})\big{|}_{\frac{3}{4}\mathbb{B}^{n}}=\sum_{i=1}^{n}\left((\rho_{i}\circ\Phi)d\phi^{i}\right)\big{|}_{\frac{3}{4}\mathbb{B}^{n}}=(F_{*}\rho)\big{|}_{\frac{3}{4}\mathbb{B}^{n}}.$
(2.1.35)

By (2.1.35) and (2.1.33) we have

$\|d(F_{*}\lambda)\|_{\mathscr{C}^{\beta-1}(\frac{3}{4}\mathbb{B}^{n};\wedge^{2}T^{*}\mathbb{R}^{n})}=\Big{\|}\sum_{i=1}^{n}d(\tilde{\rho}_{i}d\tilde{\phi}^{i})\Big{\|}_{\mathscr{C}^{\beta-1}(\frac{3}{4}\mathbb{B}^{n};\wedge^{2}T^{*}\mathbb{R}^{n})}\leq\sum_{i=1}^{n}\|d(\tilde{\rho}_{i}d\tilde{\phi}^{i})\|_{\mathscr{C}^{\beta-1}(\mathbb{R}^{n};\wedge^{2}T^{*}\mathbb{R}^{n})}.$

By Lemma 2.1.17 we have
$\|\tilde{\rho}_{i}\|_{\mathscr{C}^{\beta}(\mathbb{R}^{n})}\lesssim_{\beta}\|\chi\|_{\mathscr{C}^{\beta}(\mathbb{B}^{n})}\|\rho_{i}\circ\Phi\|_{\mathscr{C}^{\beta}(\mathbb{B}^{n})}$
and
$\|\tilde{\phi}^{i}\|_{\mathscr{C}^{\alpha+1}(\mathbb{R}^{n})}\lesssim_{\alpha}\|\chi\|_{\mathscr{C}^{\alpha+1}(\mathbb{B}^{n})}\|\Phi\|_{\mathscr{C}^{\alpha+1}(\mathbb{B}^{n})}$,
by the result (i) and Proposition 2.1.30 (i) we have
$\|\Phi\|_{\mathscr{C}^{\alpha+1}(\mathbb{B}^{n};\mathbb{R}^{n})}\lesssim 1$
and
$\|\rho_{i}\circ\Phi\|_{\mathscr{C}^{\beta}(\mathbb{B}^{n})}\lesssim\|\rho_{i}\|_{\mathscr{C}^{\beta}(\mathbb{B}^{n})}$.
Combining them we get

$\|\tilde{\rho}_{i}\|_{\mathscr{C}^{\beta}(\mathbb{R}^{n})}\lesssim_{\alpha,\beta}\|\rho_{i}\|_{\mathscr{C}^{\beta}(\mathbb{B}^{n})},\quad\|\tilde{\phi}^{i}\|_{\mathscr{C}^{\alpha+1}(\mathbb{R}^{n})}\lesssim_{\alpha}1,\quad
1\leq i\leq n.$ (2.1.36)

Thus, to obtain (2.1.34) and complete the proof, it suffices to show

$\|d(\tilde{\rho}_{i}d\tilde{\phi}^{i})\|_{\mathscr{C}^{\beta-1}(\mathbb{R}^{n};\wedge^{2}T^{*}\mathbb{R}^{n})}\lesssim_{\alpha,\beta}\|\tilde{\rho}_{i}\|_{\mathscr{C}^{\beta}(\mathbb{B}^{n})},\quad
1\leq i\leq n.$ (2.1.37)

To prove (2.1.37) we define 1-form $\tau_{i}$ on $\mathbb{R}^{n}$ by

$\tau_{i}:=\mathfrak{P}(\tilde{\rho}_{i},d\tilde{\phi}^{i})-\mathfrak{P}(\tilde{\phi}^{i},d\tilde{\rho}_{i})+\mathfrak{R}(\tilde{\rho}_{i},d\tilde{\phi}^{i}),\quad
1\leq i\leq n.$

By Lemma 2.1.13, we have
$\tilde{\rho}_{i}d\tilde{\phi}^{i}=\mathfrak{P}(\tilde{\rho}_{i},d\tilde{\phi}^{i})+\mathfrak{P}(d\tilde{\phi}^{i},\tilde{\rho}_{i})+\mathfrak{R}(\tilde{\rho}_{i},d\tilde{\phi}^{i})$.
Since as vector-valued functions
$d\mathfrak{P}(\tilde{\phi}^{i},\tilde{\rho}_{i})=\mathfrak{P}(d\tilde{\phi}^{i},\tilde{\rho}_{i})+\mathfrak{P}(\tilde{\phi}^{i},d\tilde{\rho}_{i})$
holds for $1\leq i\leq n$ on $\mathbb{R}^{n}$, we have

$d\tau_{i}=d\mathfrak{P}(\tilde{\rho}_{i},d\tilde{\phi}^{i})-d\mathfrak{P}(\tilde{\phi}^{i},d\tilde{\rho}_{i})+d\mathfrak{R}(\tilde{\rho}_{i},d\tilde{\phi}^{i})=d\mathfrak{P}(\tilde{\rho}_{i},d\tilde{\phi}^{i})+d\mathfrak{P}(d\tilde{\phi}^{i},\tilde{\rho}_{i})+d\mathfrak{R}(\tilde{\rho}_{i},d\tilde{\phi}^{i})=d(\tilde{\rho}_{i}d\tilde{\phi}^{i}).$
(2.1.38)

We claim that

$\|\tau_{i}\|_{\mathscr{C}^{\beta}(\mathbb{R}^{n};T^{*}\mathbb{R}^{n})}\lesssim\|\tilde{\rho}_{i}\|_{\mathscr{C}^{\beta}(\mathbb{R}^{n};T^{*}\mathbb{R}^{n})},\quad
1\leq i\leq n.$ (2.1.39)

By Lemma 2.1.13, along with the fact that
$\|\tilde{\phi}^{i}\|_{\mathscr{C}^{\alpha+1}}\lesssim 1$, we get
$\|\mathfrak{P}(\tilde{\rho}_{i},d\tilde{\phi}^{i})\|_{\mathscr{C}^{\beta}}\lesssim\|\tilde{\rho}_{i}\|_{\mathscr{C}^{\beta}}$
and
$\|\mathfrak{R}(\tilde{\rho}_{i},d\tilde{\phi}^{i})\|_{\mathscr{C}^{\beta}}\lesssim\|\mathfrak{R}(\tilde{\rho}_{i},d\tilde{\phi}^{i})\|_{\mathscr{C}^{\alpha+\beta}}\lesssim\|\tilde{\rho}_{i}\|_{\mathscr{C}^{\beta}}$.

To complete the proof of (2.1.39), we need to show

$\|\mathfrak{P}(\tilde{\phi}^{i},d\tilde{\rho}_{i})\|_{\mathscr{C}^{\beta}(\mathbb{R}^{n};T^{*}\mathbb{R}^{n})}\lesssim_{\alpha,\beta}\|\tilde{\rho}_{i}\|_{\mathscr{C}^{\beta}(\mathbb{R}^{n};T^{*}\mathbb{R}^{n})},\quad
1\leq i\leq n.$ (2.1.40)

We separate the proof of (2.1.40) into two cases: $\beta>1$ and $0<\beta\leq
1$. We apply Lemma 2.1.16.

For the case $\beta>1$, (2.1.40) follows from (2.1.18) with (2.1.36) that
$\tilde{\phi}^{i}\in\mathscr{C}^{\alpha+1}\subseteq\mathscr{C}^{\beta}$ and
$d\tilde{\rho}_{i}\in\mathscr{C}^{\beta-1}\subsetneq L^{\infty}$. For the case
$\beta\leq 1$, our assumption $0<\alpha\leq\beta$ implies that
$\alpha\in(0,1]$. So $\beta-\alpha-1<\beta-1\leq 0$ and we have
$d\tilde{\rho}_{i}\in\mathscr{C}^{\beta-1}\subsetneq\mathscr{C}^{\beta-\alpha-1}$.
By (2.1.19) along with $\tilde{\phi}^{i}\in\mathscr{C}^{\alpha+1}$ from
(2.1.36), we get
$\mathfrak{P}(\tilde{\phi}^{i},d\tilde{\rho}_{i})\in\mathscr{C}^{(\alpha+1)+(\beta-\alpha-1)}=\mathscr{C}^{\beta}$,
which gives (2.1.40) and establishes (2.1.39).

Using (2.1.38) and (2.1.39), we see
$\|d(\tilde{\rho}_{i}d\tilde{\phi}^{i})\|_{\mathscr{C}^{\beta-1}(\mathbb{R}^{n};\wedge^{2}T^{*}\mathbb{R}^{n})}=\|d\tau_{i}\|_{\mathscr{C}^{\beta-1}}\lesssim\|\tau_{i}\|_{\mathscr{C}^{\beta}}\lesssim\|\tilde{\rho}_{i}\|_{\mathscr{C}^{\beta}}$,
establishing (2.1.37) and completing the proof (by possibly increasing
$K_{2}$). ∎

In Section 4.1.3, we need the following existence of second order elliptic
operators with zero Dirichlet boundary:

######  Proposition 2.1.45.

Let $m,n\geq 1$ and $\alpha\in(\frac{1}{2},1]$. Let $U\subseteq\mathbb{R}^{n}$
be a bounded smooth domain. Then there is a $K_{3}=K_{3}(U,m,\alpha)>0$ that
satisfies the following:

Let $\displaystyle
L_{B}:=\sum_{j,k=1}^{n}B^{1}_{jk}\partial_{j}(B^{2}_{jk}\partial_{k})+\sum_{l=1}^{n}B^{3}_{l}\partial_{l}$
be a second order differential operator on $\mathbb{C}^{m}$-valued functions,
with $B^{1}_{jk},B^{2}_{jk}\in\mathscr{C}^{\alpha}(U;\mathbb{C}^{m\times m})$
and $B^{3}_{l}\in\mathscr{C}^{\alpha-1}(U;\mathbb{C}^{m\times m})$. Suppose

$\|B\|_{\alpha}=\|B\|_{\alpha;\phi,E}:=\sum_{j,k=1}^{n}\sum_{\nu=1}^{2}\|B^{\nu}_{jk}\|_{\mathscr{C}^{\alpha}(U;\mathbb{C}^{m\times
m})}+\sum_{l=1}^{n}\|B^{3}_{l}\|_{\mathscr{C}^{\alpha-1}(U;\mathbb{C}^{m\times
m})}<K_{3}^{-1}.$ (2.1.41)

Then the following map is (both side) invertible,

$\Delta+L_{B}:\\{F\in\mathscr{C}^{\beta+1}(U;\mathbb{C}^{m}):F|_{\partial
U}\equiv
0\\}\to\mathscr{C}^{\beta-1}(U;\mathbb{C}^{m}),\quad\forall\beta\in(1-\alpha,\alpha].$

The difficult part is to show the invertibility when $\beta$ closed to
$1-\alpha$, whereas $K_{3}$ cannot depend on $\beta$.

In this subsection only, we use Littlewood-Paley characterizations to define
$\mathscr{C}^{\beta}$-norms. More precisely:

######  Definition 2.1.46.

Let $\phi=(\phi_{j})_{j=0}^{\infty}$ be a dyadic resolution (see Definition
2.1.2) of $\mathbb{R}^{n}$ and let $E$ be a fixed extension operator for $U$
such that $E:\mathscr{C}^{\beta}(U)\to\mathscr{C}^{\beta}(\mathbb{R}^{n})$ is
bounded linear for all $-1\leq\beta\leq 2$. Let $\mathscr{X}$ be a finite
dimensional vector space, for
$\tilde{f}\in\mathscr{C}^{\beta}(\mathbb{R}^{n};\mathscr{X})$ and
$f\in\mathscr{C}^{\beta}(U;\mathscr{X})$ we use the following norms

$\|\tilde{f}\|_{\mathscr{C}^{\beta}_{\phi}(\mathscr{X})}:=\sup_{j\geq
0}2^{j\beta}\|\phi_{j}\ast\tilde{f}\|_{L^{\infty}(\mathbb{R}^{n};\mathscr{X})},\quad\|f\|_{\mathscr{C}^{\beta}_{\phi,E}(\mathscr{X})}:=\|Ef\|_{\mathscr{C}^{\beta}_{\phi}(\mathscr{X})}.$
(2.1.42)

By Lemma 2.1.5 such $E$ exists. By Lemma 2.1.4 and Definition 2.1.6,
$\|\cdot\|_{\mathscr{C}^{\beta}_{\phi}(\mathscr{X})}$ is an equivalent norm of
$\mathscr{C}^{\beta}(\mathbb{R}^{n};\mathscr{X})$ for all $-1\leq\beta\leq 2$.

Clearly for every $-1\leq\beta_{0}\leq\beta_{1}\leq 2$,

$\|\tilde{f}\|_{\mathscr{C}^{\beta_{0}}_{\phi}}\leq\|\tilde{f}\|_{\mathscr{C}^{\beta_{1}}_{\phi}},\quad\|f\|_{\mathscr{C}^{\beta_{0}}_{\phi,E}}\leq\|f\|_{\mathscr{C}^{\beta_{1}}_{\phi,E}},\quad\forall\tilde{f}\in\mathscr{C}^{\beta_{1}}(\mathbb{R}^{n}),\quad
f\in\mathscr{C}^{\beta_{1}}(U).$

Note that both inequalities have no extra constant.

For $\beta\in(0,2]$, we see that for every
$\tilde{f}\in\mathscr{C}^{\beta}(\mathbb{R}^{n})$,

$\textstyle\|\tilde{f}\|_{L^{\infty}}\leq\sum_{j=0}^{\infty}\|\phi_{j}\ast\tilde{f}\|_{L^{\infty}}\leq\|\tilde{f}\|_{\mathscr{C}^{\beta}_{\phi}}\sum_{j=0}^{\infty}2^{-j\beta}=\|\tilde{f}\|_{\mathscr{C}^{\beta}_{\phi}}\tfrac{1}{1-2^{-\beta}}\leq\frac{3}{\beta}\|\tilde{f}\|_{\mathscr{C}^{\beta}_{\phi}}.$
(2.1.43)

Thus
$\|f\|_{L^{\infty}}\leq\frac{3}{\beta}\|f\|_{\mathscr{C}^{\beta}_{\phi,E}}$ as
well.

Moreover by [Tri10, Theorem 2.4.7(i)],
$\\{\mathscr{C}^{\beta}(\mathbb{R}^{n}),\|\cdot\|_{\mathscr{C}^{\beta}_{\phi}}:-1\leq\beta\leq
2\\}$ forms an exact complex interpolation family, see [Tri95, Definition
1.2.2/2]. In particular, for any
$\beta_{0},\beta_{1},\gamma_{0},\gamma_{1}\in[-1,2]$ and for any linear
operator $\mathcal{T}$ such that
$\mathcal{T}:\mathscr{C}^{\beta_{j}}(\mathbb{R}^{n})\to\mathscr{C}^{\gamma_{j}}(\mathbb{R}^{n})$
is bounded for $j=0,1$, then
$\mathcal{T}:\mathscr{C}^{(1-\theta)\beta_{0}+\theta\beta_{1}}(\mathbb{R}^{n})\to\mathscr{C}^{(1-\theta)\gamma_{0}+\theta\gamma_{1}}(\mathbb{R}^{n})$
is also bounded for all $0<\theta<1$, with operator norm

$\|\mathcal{T}\|_{\mathscr{C}^{(1-\theta)\beta_{0}+\theta\beta_{1}}_{\phi}\to\mathscr{C}^{(1-\theta)\gamma_{0}+\theta\gamma_{1}}_{\phi}}\leq\|\mathcal{T}\|_{\mathscr{C}^{\beta_{0}}_{\phi}\to\mathscr{C}^{\gamma_{0}}_{\phi}}^{1-\theta}\|\mathcal{T}\|_{\mathscr{C}^{\beta_{1}}_{\phi}\to\mathscr{C}^{\gamma_{1}}_{\phi}}^{\theta}.$
(2.1.44)

There is no extra constant in (2.1.44) as well.

Applying Lemma 2.2.35 via scaling,
$\Delta:\\{f\in\mathscr{C}^{\beta+1}(U):f|_{\partial
U}\\}\to\mathscr{C}^{\beta-1}(U)$ is invertible for all $0\leq\beta\leq 1$.
Recall that we denote its inverse by
$\mathcal{P}_{0}:\mathscr{C}^{\beta-1}(U)\to\mathscr{C}^{\beta+1}(U)$ in
Notation 2.1.43.

By (2.1.42)
$\|\mathcal{P}_{0}\|_{\mathscr{C}^{\beta-1}_{\phi,E}\to\mathscr{C}^{\beta+1}_{\phi,E}}=\|\tilde{f}\mapsto
E\mathcal{P}_{0}(\tilde{f}|_{U})\|_{\mathscr{C}^{\beta-1}_{\phi}\to\mathscr{C}^{\beta+1}_{\phi}}$.
Therefore by (2.1.44), there is a $C_{0}=C_{0}(U,m,\phi,E)>0$ such that

$\|\mathcal{P}_{0}\|_{\mathscr{C}^{\beta-1}_{\phi,E}\to\mathscr{C}^{\beta+1}_{\phi,E}}\leq
C_{0},\quad\forall 0\leq\beta\leq 1.$ (2.1.45)

We need an improvement of Lemma 2.1.16.

######  Lemma 2.1.47 ([BCD11, Theorems 2.82 and 2.85]).

Let $\mathfrak{P}$ and $\mathfrak{R}$ be the paraproduct operators in
Definition 2.1.14 associated with the same $(\phi_{j})_{j=0}^{\infty}$ in
Definition 2.1.46. There is $C_{2}=C_{2}(n,\phi)>0$ such that

$\displaystyle\|\mathfrak{P}(f,g)\|_{\mathscr{C}^{\beta}_{\phi}}$
$\displaystyle\leq C_{2}\|f\|_{\mathscr{C}^{\beta}_{\phi}}\|g\|_{L^{\infty}},$
$\displaystyle\forall\beta\in[-1,2],\quad
f\in\mathscr{C}^{\beta}(\mathbb{R}^{n}),\ g\in L^{\infty}(\mathbb{R}^{n});$
(2.1.46)
$\displaystyle\|\mathfrak{P}(f,g)\|_{\mathscr{C}^{\beta+\gamma}_{\phi}}$
$\displaystyle\leq\frac{C_{2}}{-\gamma}\|f\|_{\mathscr{C}^{\beta}_{\phi}}\|g\|_{\mathscr{C}^{\gamma}_{\phi}},$
$\displaystyle\forall\beta\in[-1,2],\quad\gamma\in[-1,0),\
f\in\mathscr{C}^{\beta}(\mathbb{R}^{n}),\
g\in\mathscr{C}^{\gamma}(\mathbb{R}^{n});$ (2.1.47)
$\displaystyle\|\mathfrak{R}(f,g)\|_{\mathscr{C}^{\beta+\gamma}_{\phi}}$
$\displaystyle\leq\frac{C_{2}\|f\|_{\mathscr{C}^{\beta}_{\phi}}\|g\|_{\mathscr{C}^{\gamma}_{\phi}}}{\beta+\gamma},$
$\displaystyle\forall\beta,\gamma\in[-1,2],\ \beta+\gamma>0,\
f\in\mathscr{C}^{\beta}(\mathbb{R}^{n}),\
g\in\mathscr{C}^{\gamma}(\mathbb{R}^{n}).$ (2.1.48)

###### Proof of Proposition 2.1.45.

In the proof we use $\mathscr{C}^{\gamma}_{\phi}$-norm from Definition 2.1.46
instead of the classical $\mathscr{C}^{\gamma}$-norm in Definition 1.2.1.

Recall in Notation 2.1.43
$\mathcal{P}_{0}:\mathscr{C}^{\beta-1}(U;\mathbb{C}^{m})\to\mathscr{C}^{\beta+1}(U;\mathbb{C}^{m})$
is the inverse operator to $\Delta$ with zero Dirichlet boundary, for all
$\beta\in(1-\alpha,\alpha]$. Thus if $\Delta+L_{B}$ is invertible, then
$(\Delta+L_{B})^{-1}=(\operatorname{id}+\mathcal{P}_{0}L_{B})^{-1}\mathcal{P}_{0}$,
and $\operatorname{id}+\mathcal{P}_{0}L_{B}$ is an isomorphism on
$\\{F\in\mathscr{C}^{\beta+1}(U;\mathbb{C}^{m}):F|_{\partial U}\equiv 0\\}$.

By (2.1.45) $\mathcal{P}_{0}$ is uniformly bounded. Therefore to prove
(2.1.49) it suffices to show $\operatorname{id}+\mathcal{P}_{0}L_{B}$ is an
isomorphism on $\mathscr{C}^{\beta+1}(U;\mathbb{C}^{m})$ when (2.1.41) is
satisfied.

We also assume that $K_{3}$ can depend on $\phi$ and $E$ in the proof. The
extra dependence can be eliminated when we switch the
$\|\cdot\|_{\mathscr{C}^{\alpha}_{\phi,E}}$ norm back to the
$\|\cdot\|_{\mathscr{C}^{\alpha}(U)}$ norm.

We claim that, by taking $K_{3}$ suitably large, we have

$\|(\operatorname{id}+\mathcal{P}_{0}L_{B})^{-1}\|_{\mathscr{C}^{\beta-1}_{\phi,E}(\mathbb{C}^{m})\to\mathscr{C}^{\beta+1}_{\phi,E}(\mathbb{C}^{m})}\leq
K_{3}\cdot(\beta+\alpha-1)^{-1},\quad\forall\beta\in(1-\alpha,\alpha].$
(2.1.49)

For $F=(F^{1},\dots,F^{m})$, we write

$(L_{B}F)^{i}=\sum_{j,k=1}^{n}\sum_{u,v=1}^{m}B^{1,iu}_{jk}\partial_{j}(B^{2,uv}_{jk}\partial_{k}F^{v})+\sum_{l=1}^{n}\sum_{u=1}^{m}B^{3,iu}_{l}\partial_{l}F^{u},\quad
i=1,\dots,m.$

We take decomposition $L_{B}=L_{B,1}+L_{B,2}$ where for $i=1,\dots,m$,

$\displaystyle(L_{B,1}F)^{i}:=$
$\displaystyle\sum_{j,k=1}^{n}\sum_{u,v=1}^{m}\left(\mathfrak{P}\big{(}EB^{1,iu}_{jk},E\partial_{j}(B^{2,uv}_{jk}\partial_{k}F^{v})\big{)}+\mathfrak{P}\big{(}E\partial_{j}(B^{2,uv}_{jk}\partial_{k}F^{v}),EB^{1,iu}_{jk}\big{)}\right)\Big{|}_{U}$
$\displaystyle+\sum_{l=1}^{n}\sum_{u=1}^{m}\left(\mathfrak{P}(EB^{3,iu}_{l},E\partial_{l}F^{u})+\mathfrak{P}(E\partial_{l}F^{u},EB^{3,iu}_{l})\right)\Big{|}_{U};$
$\displaystyle(L_{B,2}F)^{i}:=$
$\displaystyle\sum_{j,k=1}^{n}\sum_{u,v=1}^{m}\mathfrak{R}\big{(}EB^{1,iu}_{jk},E\partial_{j}(B^{2,uv}_{jk}\partial_{k}F^{v})\big{)}\Big{|}_{U}+\sum_{l=1}^{n}\sum_{u=1}^{m}\mathfrak{R}(EB^{3,iu}_{l},E\partial_{l}F^{u})\Big{|}_{U}.$

By Lemma 2.1.47
$\|B^{2,uv}_{jk}\partial_{k}F^{v}\|_{\mathscr{C}^{\beta}_{\phi,E}}$ is
uniformly bounded in $\alpha\in(\frac{1}{2},1]$ and
$\beta\in(1-\alpha,\alpha]$:

$\displaystyle\|B^{2,uv}_{jk}\partial_{k}F^{v}\|_{\mathscr{C}^{\beta}_{\phi,E}}=\|\mathfrak{P}(EB^{2,uv}_{jk},E\partial_{k}F^{v})+\mathfrak{P}(E\partial_{k}F^{v},EB^{2,uv}_{jk})+\mathfrak{R}(EB^{2,uv}_{jk},E\partial_{k}F^{v})\|_{\mathscr{C}^{\beta}_{\phi}}$
$\displaystyle\leq$ $\displaystyle
C_{2}\min\big{(}\|EB^{2,uv}_{jk}\|_{\mathscr{C}^{\beta}_{\phi}}\|E\partial_{k}F^{v}\|_{L^{\infty}},\tfrac{1}{\alpha-\beta}\|EB^{2,uv}_{jk}\|_{\mathscr{C}^{\alpha}_{\phi}}\|E\partial_{k}F^{v}\|_{\mathscr{C}^{\beta-\alpha}_{\phi}}\big{)}$
$\displaystyle+C_{2}\|EB^{2,uv}_{jk}\|_{L^{\infty}}\|E\partial_{k}F^{v}\|_{\mathscr{C}^{\beta}_{\phi}}+\tfrac{C_{2}}{\alpha}\|EB^{2,uv}_{jk}\|_{\mathscr{C}^{\alpha}_{\phi}}\|E\partial_{k}F^{v}\|_{\mathscr{C}^{\beta-\alpha}_{\phi}}$
(by Lemma 2.1.47) $\displaystyle\leq$ $\displaystyle
C_{2}(\min(\tfrac{3}{\beta},\tfrac{1}{\alpha-\beta})+1+\tfrac{1}{\alpha})\|EB^{2,uv}_{jk}\|_{\mathscr{C}^{\alpha}_{\phi}}\|E\partial_{k}F^{v}\|_{\mathscr{C}^{\beta}_{\phi}}\leq
11C_{2}\|B^{2,uv}_{jk}\|_{\mathscr{C}^{\alpha}_{\phi,E}}\|\partial_{k}F^{v}\|_{\mathscr{C}^{\beta}_{\phi,E}}.$
(by (2.1.43))

Similarly
$L_{B,1}:\mathscr{C}^{\beta+1}(U;\mathbb{C}^{m})\to\mathscr{C}^{\beta-1}(U;\mathbb{C}^{m})$
is uniformly bounded in $\alpha\in(\frac{1}{2},1],\beta\in(1-\alpha,\alpha]$:

$\displaystyle\|L_{B,1}F\|_{\mathscr{C}^{\beta-1}_{\phi}}\lesssim$
${}_{E}\tfrac{1}{\alpha+1-\beta}\|EB^{1}\|_{\mathscr{C}^{\alpha}_{\phi}}\|E\nabla(B^{2}\otimes\nabla
F)\|_{\mathscr{C}^{\beta-1-\alpha}_{\phi}}+\|EB^{1}\|_{L^{\infty}}\|E\nabla(B^{2}\otimes\nabla
F)\|_{\mathscr{C}^{\beta-1}_{\phi}}$
$\displaystyle+\min\big{(}\|EB^{3}\|_{\mathscr{C}^{\beta-1}_{\phi}}\|E\nabla
F\|_{L^{\infty}},\tfrac{1}{\alpha-\beta}\|EB^{3}\|_{\mathscr{C}^{\alpha-1}_{\phi}}\|E\nabla
F\|_{\mathscr{C}^{\beta-\alpha}}\big{)}+\|EB^{3}\|_{\mathscr{C}^{-1}_{\phi}}\|E\nabla
F\|_{\mathscr{C}^{\beta}_{\phi}}$ $\displaystyle\leq$
$\displaystyle\big{(}\tfrac{1}{\alpha+1-\beta}+\tfrac{1}{\alpha}\big{)}\|B_{1}\|_{\mathscr{C}^{\alpha}_{\phi,E}}\|\nabla(B^{2}\otimes\nabla
F)\|_{\mathscr{C}^{\beta-1}_{\phi,E}}+\big{(}\min(\tfrac{1}{\beta},\tfrac{1}{\alpha-\beta})+1\big{)}\|B^{3}\|_{\mathscr{C}^{\alpha-1}_{\phi,E}}\|\nabla
F\|_{\mathscr{C}^{\beta}_{\phi,E}}$ $\displaystyle\lesssim$
${}_{U,m}\|B\|_{\alpha}(1+\|B\|_{\alpha})\|F\|_{\mathscr{C}^{\beta+1}_{\phi,E}}.$

Here we use $\frac{1}{\alpha+1-\beta}<\frac{1}{3/2-1}=2$ and
$\min(\frac{1}{\beta},\frac{1}{\alpha-\beta})\leq\frac{2}{\alpha}<4$.

$L_{B,2}:\mathscr{C}^{\beta+1}(U;\mathbb{C}^{m})\to\mathscr{C}^{\alpha-1}(U;\mathbb{C}^{m})$
is not uniformly bounded but gains a few derivatives: we have
$\alpha+\beta>1$,

$\displaystyle\|L_{B,2}F\|_{\mathscr{C}^{\alpha-1}_{\phi,E}}$
$\displaystyle\leq\|L_{B,2}F\|_{\mathscr{C}^{\alpha+\beta-1}_{\phi,E}}\lesssim_{U,m,E}\tfrac{1}{\alpha+\beta-1}\big{(}\|B^{1}\|_{\mathscr{C}^{\alpha}_{\phi,E}}\|\nabla(B^{2}\otimes\nabla
F)\|_{\mathscr{C}^{\beta-1}_{\phi,E}}+\|B^{3}\|_{\mathscr{C}^{\alpha-1}_{\phi,E}}\|\nabla
F\|_{\mathscr{C}^{\beta}_{\phi,E}}\big{)}$
$\displaystyle\lesssim_{U,m}(\alpha+\beta-1)^{-1}\|B\|_{\alpha}(1+\|B\|_{\alpha})\|F\|_{\mathscr{C}^{\beta+1}_{\phi,E}}.$

Combining (2.1.45), there is a $C_{3}=C_{3}(U,m,\phi,E)>1$ such that for every
$\alpha\in(\frac{1}{2},1]$, $\beta\in(1-\alpha,\alpha]$ and
$F\in\mathscr{C}^{\beta+1}(U;\mathbb{C}^{m})$,

$\|\mathcal{P}_{0}L_{B,1}F\|_{\mathscr{C}^{\beta+1}_{\phi,E}}\leq
C_{3}\|B\|_{\alpha}(1+\|B\|_{\alpha})\|F\|_{\mathscr{C}^{\beta+1}_{\phi,E}},\quad\|\mathcal{P}_{0}L_{B,2}F\|_{\mathscr{C}^{\alpha+1}_{\phi,E}}\leq\frac{C_{3}\|B\|_{\alpha}(1+\|B\|_{\alpha})}{\alpha+\beta-1}\|F\|_{\mathscr{C}^{\beta+1}_{\phi,E}}.$
(2.1.50)

Take $K_{3}:=\frac{8C_{3}}{2\alpha-1}$, we see that
$K_{3}=K_{3}(U,m,\phi,E,\alpha)\in(0,1)$. When (2.1.41) is satisfied, (2.1.50)
implies that
$\|\mathcal{P}_{0}L_{B,1}\|_{\mathscr{C}^{\beta+1}_{\phi,E}\to\mathscr{C}^{\beta+1}_{\phi,E}}\leq\frac{1}{4}$
and
$\|\mathcal{P}_{0}L_{B,2}\|_{\mathscr{C}^{\beta+1}_{\phi,E}\to\mathscr{C}^{\beta+1}_{\phi,E}}\leq\frac{1}{4}\cdot\frac{2\alpha-1}{\alpha+\beta-1}$.
In particular
$\|\mathcal{P}_{0}L_{B}\|_{\mathscr{C}^{\alpha+1}_{\phi,E}\to\mathscr{C}^{\alpha+1}_{\phi,E}}\leq\frac{1}{4}+\frac{1}{4}<\frac{1}{2}$
is a contraction mapping.

Therefore for every $\alpha\in(\frac{1}{2},1]$ and
$\beta\in(1-\alpha,\alpha]$, using power series
$(\operatorname{id}+T)^{-1}=\sum_{k=0}^{\infty}(-1)^{k}T^{k}$, provided that
the right hand side converges, taking
$T\in\\{\mathcal{P}_{0}L_{B,1},(\operatorname{id}+\mathcal{P}_{0}L_{B,1})^{-1}\mathcal{P}_{0}L_{B,2}\\}$
we get

$\displaystyle\|(\operatorname{id}+\mathcal{P}_{0}L_{B,1})^{-1}\|_{\mathscr{C}^{\beta+1}_{\phi,E}\to\mathscr{C}^{\beta+1}_{\phi,E}}$
$\displaystyle\textstyle\leq\sum_{k=0}^{\infty}\|\mathcal{P}_{0}L_{B}\|_{\mathscr{C}^{\beta+1}_{\phi,E}\to\mathscr{C}^{\beta+1}_{\phi,E}}^{k}\leq\sum_{k=0}^{\infty}4^{-k}\leq\frac{4}{3}.$
$\displaystyle\|(\operatorname{id}+(\operatorname{id}+\mathcal{P}_{0}L_{B,1})^{-1}\mathcal{P}_{0}L_{B,2})^{-1}\|_{\mathscr{C}^{\alpha+1}_{\phi,E}\to\mathscr{C}^{\alpha+1}_{\phi,E}}$
$\displaystyle\textstyle\leq\sum\limits_{k=0}^{\infty}\|(\operatorname{id}+\mathcal{P}_{0}L_{B,1})^{-1}\|_{\mathscr{C}^{\alpha+1}_{\phi,E}\to\mathscr{C}^{\alpha+1}_{\phi,E}}^{k}\|\mathcal{P}_{0}L_{B,2}\|_{\mathscr{C}^{\alpha+1}_{\phi,E}\to\mathscr{C}^{\alpha+1}_{\phi,E}}^{k}\leq\sum\limits_{k=0}^{\infty}3^{-k}\leq\frac{3}{2}.$

Finally for the $\mathscr{C}^{\beta+1}$-boundedness of
$(\operatorname{id}+\mathcal{P}_{0}L_{B})^{-1}$, we use

$\displaystyle(\operatorname{id}+\mathcal{P}_{0}L_{B})^{-1}=$
$\displaystyle(\operatorname{id}+\mathcal{P}_{0}L_{B,1}+\mathcal{P}_{0}L_{B,2})^{-1}=(\operatorname{id}+(\operatorname{id}+\mathcal{P}_{0}L_{B,1})^{-1}\mathcal{P}_{0}L_{B,2})^{-1}(\operatorname{id}+\mathcal{P}_{0}L_{B,1})^{-1}$
$\displaystyle=$
$\displaystyle(\operatorname{id}+\mathcal{P}_{0}L_{B,1})^{-1}-((\operatorname{id}+\mathcal{P}_{0}L_{B,1})^{-1}\mathcal{P}_{0}L_{B,2})(\operatorname{id}+(\operatorname{id}+\mathcal{P}_{0}L_{B,1})^{-1}\mathcal{P}_{0}L_{B,2})^{-1}(\operatorname{id}+\mathcal{P}_{0}L_{B,1})^{-1}.$

Therefore

$\displaystyle\|(\operatorname{id}+\mathcal{P}_{0}L_{B})^{-1}\|_{\mathscr{C}^{\beta+1}_{\phi,E}\to\mathscr{C}^{\beta+1}_{\phi,E}}\leq$
$\displaystyle\|(\operatorname{id}+\mathcal{P}_{0}L_{B,1})^{-1}\|_{\mathscr{C}^{\beta+1}_{\phi,E}\to\mathscr{C}^{\beta+1}_{\phi,E}}\big{(}1+\|\mathcal{P}_{0}L_{B,2}\|_{\mathscr{C}^{\beta+1}_{\phi,E}\to\mathscr{C}^{\alpha+1}_{\phi,E}}$
$\displaystyle\times\|(\operatorname{id}+(\operatorname{id}+\mathcal{P}_{0}L_{B,1})^{-1}\mathcal{P}_{0}L_{B,2})^{-1}\|_{\mathscr{C}^{\alpha+1}_{\phi,E}\to\mathscr{C}^{\alpha+1}_{\phi,E}}\|(\operatorname{id}+\mathcal{P}_{0}L_{B,1})^{-1}\|_{\mathscr{C}^{\alpha+1}_{\phi,E}\to\mathscr{C}^{\alpha+1}_{\phi,E}}\big{)}$
$\displaystyle\leq$
$\displaystyle\tfrac{4}{3}\times\big{(}1+\tfrac{2\alpha-1}{4(\alpha+\beta-1)}+\tfrac{3}{2}\cdot\tfrac{4}{3}\big{)}\leq\tfrac{13}{3}(\alpha+\beta-1)^{-1}.$

Since $K_{3}=\frac{8C_{3}}{2\alpha-1}>\frac{3}{13}$, we get (2.1.45) and
finish the proof. ∎

####  2.1.4 An approximation theorem on vanishing products

Recall the Littlewood-Paley operators
$\mathsf{P}_{\sigma}=\phi_{\sigma}\ast(-)$ and
$\mathsf{S}_{\sigma}=\sum_{j=0}^{\sigma}\phi_{\sigma}\ast(-)=2^{n\sigma}\phi_{0}(2^{\sigma}\cdot)\ast(-)$
in Definition 2.1.2, where $\phi=(\phi_{j})_{j=0}^{\infty}$ is a dyadic
resolution.

The log-Lipschitz Frobenius theorems (Theorems 1 and 2) rely heavily on the
following:

######  Theorem 2.1.48.

Let $\alpha\in(0,1)$ and $\beta\in(0,\alpha)$. Let $m,n\geq 1$. There is a
$C=C_{n,m,\phi,\alpha,\beta}>0$ such that, if
$\mathbf{f}\in\mathscr{C}^{\alpha}(\mathbb{R}^{n};\mathbb{R}^{m})$ and
$\mathbf{g}\in\mathscr{C}^{-\beta}(\mathbb{R}^{n};\mathbb{R}^{m})$ satisfy
$\mathbf{f}\cdot\mathbf{g}=0$ in the sense of distributions, then

$\|(\mathsf{S}_{\sigma}\mathbf{f})\cdot(\mathsf{S}_{\sigma}\mathbf{g})\|_{C^{0}(\mathbb{R}^{n})}\leq
C\cdot
2^{-(\alpha-\beta)\sigma}\|\mathbf{f}\|_{\mathscr{C}^{\alpha}(\mathbb{R}^{n};\mathbb{R}^{m})}\|\mathbf{g}\|_{\mathscr{C}^{-\beta}(\mathbb{R}^{n};\mathbb{R}^{m})},\qquad\forall\sigma\geq
1.$ (2.1.51)

By Lemma 2.1.17 (i), we know the equality $\mathbf{f}\cdot\mathbf{g}=0$ holds
in $\mathscr{C}^{-\beta}(\mathbb{R}^{n})$. Note that we already have the
convergence
$\lim\limits_{\sigma\to\infty}(\mathsf{S}_{\sigma}\mathbf{f})\cdot(\mathsf{S}_{\sigma}\mathbf{g})=0$
in $\mathscr{C}^{-\beta-\delta}(\mathbb{R}^{n})$, for every $\delta>0$.
Theorem 2.1.48 says that this sequence converges uniformly with quantitative
speed, which is better than the convergence in some spaces of distributions.

In the proof we use the following modified paraproduct formula, which can be
deduced using (2.1.3) and (2.1.17):

$\mathsf{P}_{l}(\mathbf{f}\cdot\mathbf{g})=\mathsf{P}_{l}\sum_{j=l-2}^{l+2}(\mathsf{P}_{j}\mathbf{f})\cdot(\mathsf{S}_{j-6}\mathbf{g})+\mathsf{P}_{l}\sum_{k=l-2}^{l+2}(\mathsf{S}_{k-6}\mathbf{f})\cdot(\mathsf{P}_{k}\mathbf{g})+\mathsf{P}_{l}\sum_{j,k\geq
l-2;|j-k|\leq 5}(\mathsf{P}_{j}\mathbf{f})\cdot(\mathsf{P}_{k}\mathbf{g}).$
(2.1.52)

We leave the deduction to readers.

###### Proof of Theorem 2.1.48.

We set
$\|\mathbf{f}\|_{\mathscr{C}^{\alpha}}=\|\mathbf{g}\|_{\mathscr{C}^{-\beta}}=1$
without loss of generality.

Clearly
$\|2^{n\sigma}\phi_{0}(2^{\sigma}\cdot)\|_{L^{1}}=\|\phi_{0}\|_{L^{1}}$ for
all $\sigma\geq 0$, and
$\|\phi_{\sigma}\|_{L^{1}}=\|2^{n(\sigma-1)}\phi_{1}(2^{\sigma-1}\cdot)\|_{L^{1}}=\|\phi_{1}\|_{L^{1}}$
for all $\sigma\geq 1$. Therefore we have the following uniform bounds,

$\|\mathsf{S}_{\sigma}\|_{C^{0}\to
C^{0}}\leq\|2^{n\sigma}\phi_{0}(2^{\sigma}\cdot)\|_{L^{1}}=\|\phi_{0}\|_{L^{1}},\quad\|\mathsf{P}_{\sigma}\|_{C^{0}\to
C^{0}}\leq\|\phi_{\sigma}\|_{L^{1}}\leq\|\phi_{0}\|_{L^{1}}+\|\phi_{1}\|_{L^{1}},\quad\forall\sigma\geq
0.$ (2.1.53)

By (2.1.2), $\mathsf{S}_{\sigma}f$ and $\mathsf{S}_{\sigma}g$ both have
Fourier supports in $\\{|\xi|<2^{\sigma+1}\\}$. Thus when $l\geq\sigma+3$,
$\operatorname{supp}(\mathsf{S}_{\sigma}f\cdot\mathsf{S}_{\sigma}g)^{\wedge}\subseteq\\{|\xi|<2^{\sigma+2}\\}$
is disjoint from
$\operatorname{supp}\hat{\phi}_{l}\subseteq\\{|\xi|>2^{l-1}\\}$, which means
$\mathsf{P}_{l}(\mathsf{S}_{\sigma}f\cdot\mathsf{S}_{\sigma}g)=0$. Therefore
we have

$\mathsf{S}_{\sigma}\mathbf{f}\cdot\mathsf{S}_{\sigma}\mathbf{g}=\sum_{l=0}^{\infty}\mathsf{P}_{l}(\mathsf{S}_{\sigma}\mathbf{f}\cdot\mathsf{S}_{\sigma}\mathbf{g})=\sum_{l=0}^{\sigma+2}\mathsf{P}_{l}(\mathsf{S}_{\sigma}\mathbf{f}\cdot\mathsf{S}_{\sigma}\mathbf{g})=\mathsf{S}_{\sigma-4}(\mathsf{S}_{\sigma}\mathbf{f}\cdot\mathsf{S}_{\sigma}\mathbf{g})+\sum_{l=\sigma-3}^{\sigma+2}\mathsf{P}_{l}(\mathsf{S}_{\sigma}\mathbf{f}\cdot\mathsf{S}_{\sigma}\mathbf{g}).$
(2.1.54)

Our goal is to prove
$\|\mathsf{S}_{\sigma-4}(\mathsf{S}_{\sigma}\mathbf{f}\cdot\mathsf{S}_{\sigma}\mathbf{g})\|_{C^{0}(\mathbb{R}^{n})}\lesssim
2^{-(\alpha-\beta)\sigma}$ and
$\sum_{l=\sigma-3}^{\sigma+2}\|\mathsf{P}_{l}(\mathsf{S}_{\sigma}\mathbf{f}\cdot\mathsf{S}_{\sigma}\mathbf{g})\|_{C^{0}(\mathbb{R}^{n})}\lesssim
2^{-(\alpha-\beta)\sigma}$, where the implicit constants depend only on
$m,n,\alpha,\beta$ and $\phi$.

By (2.1.52) and the assumption $\mathbf{f}\cdot\mathbf{g}=0$, we know,

$\mathsf{P}_{l}\sum_{j=l-2}^{l+2}(\mathsf{P}_{j}\mathbf{f})\cdot(\mathsf{S}_{j-6}\mathbf{g})+\mathsf{P}_{l}\sum_{k=l-2}^{l+2}(\mathsf{S}_{k-6}\mathbf{f})\cdot(\mathsf{P}_{k}\mathbf{g})+\mathsf{P}_{l}\sum_{j,k\geq
l-2;|j-k|\leq
5}(\mathsf{P}_{j}\mathbf{f})\cdot(\mathsf{P}_{k}\mathbf{g})=0,\quad
l=0,1,2,\dots.$ (2.1.55)

For $\sigma,l\geq 0$, we have decomposition

$\displaystyle\mathsf{P}_{l}(\mathsf{S}_{\sigma}\mathbf{f}\cdot\mathsf{S}_{\sigma}\mathbf{g})=\mathsf{P}_{l}(\mathsf{S}_{\sigma}\mathbf{f}\cdot\mathsf{S}_{\sigma}\mathbf{g})-\mathsf{P}_{l}(\mathbf{f}\cdot\mathbf{g})=$
$\displaystyle\mathsf{P}_{l}\sum_{j=l-2}^{l+2}\big{(}(\mathsf{P}_{j}\mathsf{S}_{\sigma}\mathbf{f})\cdot(\mathsf{S}_{j-6}\mathsf{S}_{\sigma}\mathbf{g})-\mathsf{P}_{j}\mathbf{f}\cdot\mathsf{S}_{j-6}\mathbf{g}\big{)}$
(2.1.56)
$\displaystyle+\mathsf{P}_{l}\sum_{k=l-2}^{l+2}\big{(}(\mathsf{S}_{k-6}\mathsf{S}_{\sigma}\mathbf{f})\cdot(\mathsf{P}_{k}\mathsf{S}_{\sigma}\mathbf{g})-\mathsf{S}_{k-6}\mathbf{f}\cdot\mathsf{P}_{k}\mathbf{g}\big{)}$
(2.1.57) $\displaystyle+\mathsf{P}_{l}\sum_{j,k\geq l-2;|j-k|\leq
5}\big{(}(\mathsf{P}_{j}\mathsf{S}_{\sigma}\mathbf{f})\cdot(\mathsf{P}_{k}\mathsf{S}_{\sigma}\mathbf{g})-\mathsf{P}_{j}\mathbf{f}\cdot\mathsf{P}_{k}\mathbf{g}\big{)}.$
(2.1.58)

When $l\leq\sigma-4$ and $|j-l|\leq 2$, we have $j\leq\sigma-2$. By (2.1.3) we
have $\mathsf{P}_{j}\mathsf{S}_{\sigma}=\mathsf{P}_{j}$ and
$\mathsf{S}_{j-6}\mathsf{S}_{\sigma}=\mathsf{S}_{j-6}$, so in this case
(2.1.56) is zero. Similarly (2.1.57) vanishes as well. Thus for
$\mathsf{S}_{\sigma-4}(\mathsf{S}_{\sigma}\mathbf{f}\cdot\mathsf{S}_{\sigma}\mathbf{g})$
in (2.1.54) we have

$\displaystyle\mathsf{S}_{\sigma-4}(\mathsf{S}_{\sigma}\mathbf{f}\cdot\mathsf{S}_{\sigma}\mathbf{g})=$
$\displaystyle\sum_{l=0}^{\sigma-4}\mathsf{P}_{l}\sum_{j,k\geq l-2;|j-k|\leq
5}\big{(}(\mathsf{P}_{j}\mathsf{S}_{\sigma}\mathbf{f}\cdot\mathsf{P}_{k}\mathsf{S}_{\sigma}\mathbf{g})-\mathsf{P}_{j}\mathbf{f}\cdot\mathsf{P}_{k}\mathbf{g}\big{)}$
$\displaystyle=$
$\displaystyle\sum_{l=0}^{\sigma-4}\mathsf{P}_{l}\sum_{j,k\geq\sigma-2;|j-k|\leq
5}\big{(}(\mathsf{P}_{j}\mathsf{S}_{\sigma}\mathbf{f})\cdot(\mathsf{P}_{k}\mathsf{S}_{\sigma}\mathbf{g})-\mathsf{P}_{j}\mathbf{f}\cdot\mathsf{P}_{k}\mathbf{g}\big{)}$
$\displaystyle=$
$\displaystyle\mathsf{S}_{\sigma-4}\sum_{j,k\geq\sigma-2;|j-k|\leq
5}\big{(}(\mathsf{P}_{j}\mathsf{S}_{\sigma}\mathbf{f})\cdot(\mathsf{P}_{k}\mathsf{S}_{\sigma}\mathbf{g})-\mathsf{P}_{j}\mathbf{f}\cdot\mathsf{P}_{k}\mathbf{g}\big{)}.$

Using (2.1.53) and Lemma 2.1.4 (i) we have an estimate for (2.1.58): for every
$l\geq 0$,

$\displaystyle\sum_{\begin{subarray}{c}j,k\geq l-2\\\ |j-k|\leq
5\end{subarray}}\big{\|}(\mathsf{P}_{j}\mathsf{S}_{\sigma}\mathbf{f})\cdot(\mathsf{P}_{k}\mathsf{S}_{\sigma}\mathbf{g})-\mathsf{P}_{j}\mathbf{f}\cdot\mathsf{P}_{k}\mathbf{g}\big{\|}_{C^{0}}\leq\sum_{\begin{subarray}{c}j,k\geq
l-2\\\ |j-k|\leq 5\end{subarray}}\big{(}\|\mathsf{S}_{\sigma}\|_{C^{0}\to
C^{0}}^{2}+1\big{)}\|\mathsf{P}_{j}\mathbf{f}\|_{C^{0}}\|\mathsf{P}_{k}\mathbf{g}\|_{C^{0}}$
(2.1.59) $\displaystyle\lesssim$
$\displaystyle{}_{\phi}\sum_{\begin{subarray}{c}j,k\geq l-2\\\ |j-k|\leq
5\end{subarray}}\|\mathsf{P}_{j}\mathbf{f}\|_{C^{0}}\|\mathsf{P}_{k}\mathbf{g}\|_{C^{0}}\lesssim_{\phi,\alpha,\beta}\sum_{\begin{subarray}{c}j,k\geq
l-2\\\ |j-k|\leq 5\end{subarray}}2^{-j\alpha}2^{k\beta}\lesssim\sum_{q\geq
l}2^{-q\alpha}2^{q\beta}\lesssim 2^{-l(\alpha-\beta)}.$

Taking $l=\sigma$ in (2.1.59) and using (2.1.53) again we get

$\displaystyle\|\mathsf{S}_{\sigma-4}(\mathsf{S}_{\sigma}\mathbf{f}\cdot\mathsf{S}_{\sigma}\mathbf{g})\|_{C^{0}}\leq\|\mathsf{S}_{\sigma-4}\|_{C^{0}\to
C^{0}}\sum_{\begin{subarray}{c}j,k\geq l-2\\\ |j-k|\leq
5\end{subarray}}\big{\|}(\mathsf{P}_{j}\mathsf{S}_{\sigma}\mathbf{f})\cdot(\mathsf{P}_{k}\mathsf{S}_{\sigma}\mathbf{g})-\mathsf{P}_{j}\mathbf{f}\cdot\mathsf{P}_{k}\mathbf{g}\big{\|}_{C^{0}}\lesssim
2^{-\sigma(\alpha-\beta)}.$ (2.1.60)

This gives the first estimate
$\|\mathsf{S}_{\sigma-4}(\mathsf{S}_{\sigma}\mathbf{f}\cdot\mathsf{S}_{\sigma}\mathbf{g})\|_{C^{0}}\lesssim
2^{-\sigma(\alpha-\beta)}$ for (2.1.54).

Next we prove
$\|\mathsf{P}_{l}(\mathsf{S}_{\sigma}\mathbf{f}\cdot\mathsf{S}_{\sigma}\mathbf{g})\|_{C^{0}(\mathbb{R}^{n})}\lesssim
2^{-\sigma(\alpha-\beta)}$ when $\sigma-3\leq l\leq\sigma+2$. In this case
(2.1.56) and (2.1.57) do not vanish.

The term (2.1.56) is still good: by (2.1.53) and Lemma 2.1.4 (i) we have

$\displaystyle\Big{\|}\sum_{j=l-2}^{l+2}\big{(}(\mathsf{P}_{j}\mathsf{S}_{\sigma}\mathbf{f})(\mathsf{S}_{j-6}\mathsf{S}_{\sigma}\mathbf{g})-\mathsf{P}_{j}\mathbf{f}\cdot\mathsf{S}_{j-6}\mathbf{g}\big{)}\Big{\|}_{C^{0}}$
$\displaystyle\lesssim\sum_{j=l-2}^{l+2}\big{(}\|\mathsf{S}_{\sigma}\|_{C^{0}\to
C^{0}}^{2}+1\big{)}\|\mathsf{P}_{j}\mathbf{f}\|_{C^{0}}\|\mathsf{S}_{j-6}\mathbf{g}\|_{C^{0}}$
(2.1.61) $\displaystyle\lesssim\sum_{j=l-2}^{l+2}2^{-j\alpha}2^{j\beta}\approx
2^{-l(\alpha-\beta)}.$

Combining (2.1.61) and (2.1.59) we have

$\displaystyle\Big{\|}\mathsf{P}_{l}\sum_{j=l-2}^{l+2}(\mathsf{P}_{j}\mathbf{f})\cdot(\mathsf{S}_{j-6}\mathbf{g})+\mathsf{P}_{l}\sum_{j,k\geq
l-2;|j-k|\leq
5}(\mathsf{P}_{j}\mathbf{f})\cdot(\mathsf{P}_{k}\mathbf{g})\Big{\|}_{C^{0}}$
(2.1.62) $\displaystyle\lesssim$ $\displaystyle\|\mathsf{P}_{l}\|_{C^{0}\to
C^{0}}\Big{(}\sum_{j=l-2}^{l+2}2^{-j\alpha}2^{j\beta}+\sum_{j,k\geq
l-2;|j-k|\leq 5}2^{-j\alpha}2^{k\beta}\Big{)}\approx 2^{-l(\alpha-\beta)}.$

By assumption $\sigma-3\leq l\leq\sigma+2$, so the right hand side above is
bounded by $2^{-\sigma(\alpha-\beta)}$.

It remains to estimate (2.1.57). Note that for $|k-l|\leq 2$ and
$l\leq\sigma+2$ we have $k\leq\sigma+4$. By (2.1.3) we have
$\mathsf{S}_{k-6}\mathsf{S}_{\sigma}=\mathsf{S}_{j-6}$. Therefore we can
rewrite the summand in (2.1.57) as,

$\textstyle(\mathsf{S}_{k-6}\mathsf{S}_{\sigma}\mathbf{f})\cdot(\mathsf{P}_{k}\mathsf{S}_{\sigma}\mathbf{g})-\mathsf{S}_{k-6}\mathbf{f}\cdot\mathsf{P}_{k}\mathbf{g}=\mathsf{S}_{k-6}\mathbf{f}\cdot(\mathsf{P}_{k}\mathsf{S}_{\sigma}\mathbf{g}-\mathsf{P}_{k}\mathbf{g})=\mathsf{S}_{k-6}\mathbf{f}\cdot\big{(}\mathsf{P}_{k}\sum_{\mu=\sigma+1}^{\sigma+5}\mathsf{P}_{\mu}\mathbf{g}\big{)}.$
(2.1.63)

Here for the last equality above, we use the fact
$\mathsf{P}_{k}(\mathsf{S}_{\sigma}-\operatorname{id})=\mathsf{P}_{k}\sum_{\mu=\sigma+1}^{\infty}\mathsf{P}_{\mu}=\mathsf{P}_{k}\sum_{\mu=\sigma+1}^{\sigma+5}\mathsf{P}_{\mu}$.
We define

$\textstyle\psi_{\sigma}:=\sum_{\mu=\sigma+1}^{\sigma+5}\phi_{\mu},\quad\sigma\geq
0.$

Thus
$\mathsf{P}_{k}\sum_{\mu=\sigma+1}^{\sigma+5}\mathsf{P}_{\mu}\mathbf{g}=\mathsf{P}_{k}(\psi_{\sigma}\ast\mathbf{g})=\psi_{\sigma}\ast\mathsf{P}_{k}\mathbf{g}$.
Therefore for $x\in\mathbb{R}^{n}$,

$\displaystyle\mathsf{S}_{k-6}\mathbf{f}\cdot(\psi_{\sigma}\ast\mathsf{P}_{k}\mathbf{g})(x)=\int_{\mathbb{R}^{n}}(\mathsf{S}_{k-6}\mathbf{f})(x)\cdot(\mathsf{P}_{k}\mathbf{g})(x-y)\psi_{\sigma}(y)dy$
(2.1.64) $\displaystyle=$
$\displaystyle\big{(}\psi_{\sigma}\ast(\mathsf{S}_{k-6}\mathbf{f}\cdot\mathsf{P}_{k}\mathbf{g})\big{)}(x)+\int_{\mathbb{R}^{n}}\psi_{\sigma}(y)\big{(}(\mathsf{S}_{k-6}\mathbf{f})(x)-(\mathsf{S}_{k-6}\mathbf{f})(x-y)\big{)}\cdot(\mathsf{P}_{k}\mathbf{g})(x-y)dy$
$\displaystyle=$
$\displaystyle\big{(}\psi_{\sigma}\ast(\mathsf{S}_{k-6}\mathbf{f}\cdot\mathsf{P}_{k}\mathbf{g})\big{)}(x)+\int_{\mathbb{R}^{n}}\Big{(}\int_{0}^{1}y\psi_{\sigma}(y)\cdot(\nabla\mathsf{S}_{k-6}\mathbf{f})(x-ty)dt\Big{)}\cdot(\mathsf{P}_{k}\mathbf{g})(x-y)dy.$

By scaling we have
$\|x\phi_{\mu}(x)\|_{L^{1}_{x}}=2^{1-\mu}\|x\phi_{1}(x)\|_{L^{1}_{x}}$ and
$\|\phi_{\mu}\|_{L^{1}}=\|\phi_{1}\|_{L^{1}}$ for all $\mu\geq 1$, thus

$\|x\psi_{\sigma}(x)\|_{L^{1}_{x}}=2^{-\sigma}\|x\psi_{0}(x)\|_{L^{1}_{x}}\approx_{\phi}2^{-\sigma},\quad\|\psi_{\sigma}\|_{L^{1}}\approx_{\phi}1,\quad\forall\sigma\geq
0.$ (2.1.65)

Combining (2.1.65) with (2.1.55) and (2.1.64), for $\sigma-3\leq
l\leq\sigma+2$ we have

$\displaystyle\Big{\|}\mathsf{P}_{l}\sum_{k=l-2}^{l+2}\big{(}(\mathsf{S}_{k-6}\mathsf{S}_{\sigma}\mathbf{f})\cdot(\mathsf{P}_{k}\mathsf{S}_{\sigma}\mathbf{g})-\mathsf{S}_{k-6}\mathbf{f}\cdot\mathsf{P}_{k}\mathbf{g}\big{)}\Big{\|}_{C^{0}}$
$\displaystyle\leq$
$\displaystyle\Big{\|}\mathsf{P}_{l}\sum_{k=l-2}^{l+2}\psi_{\sigma}\ast(\mathsf{S}_{k-6}\mathbf{f}\cdot\mathsf{P}_{k}\mathbf{g})\Big{\|}_{C^{0}}+\|\mathsf{P}_{l}\|_{C^{0}\to
C^{0}}\sum_{k=l-2}^{l+2}\int_{\mathbb{R}^{n}}|y\psi_{\sigma}(y)|\|\nabla\mathsf{S}_{k-6}\mathbf{f}\|_{C^{0}}\|\mathsf{P}_{k}\mathbf{g}\|_{C^{0}}dy$
$\displaystyle\leq$
$\displaystyle\Big{\|}\psi_{\sigma}\ast\mathsf{P}_{l}\Big{(}\sum_{j=l-2}^{l+2}\mathsf{P}_{j}\mathbf{f}\cdot\mathsf{S}_{j-6}\mathbf{g}+\sum_{\begin{subarray}{c}|j-k|\leq
5\\\ j,k\geq
l-2\end{subarray}}\mathsf{P}_{j}\mathbf{f}\cdot\mathsf{P}_{k}\mathbf{g}\Big{)}\Big{\|}_{C^{0}}+\|\phi_{l}\|_{L^{1}}\sum_{k=l-2}^{l+2}\|y\psi_{\sigma}\|_{L^{1}_{y}}\|\mathsf{S}_{k-6}\nabla\mathbf{f}\|_{C^{0}}\|\mathsf{P}_{k}\mathbf{g}\|_{C^{0}}$
$\displaystyle\text{by }\eqref{Eqn::Hold::PfApThm::Para0}$
$\displaystyle\lesssim$
$\displaystyle\|\psi_{\sigma}\|_{L^{1}}2^{-\sigma(\alpha-\beta)}+\|x\mapsto
x\psi_{\sigma}(x)\|_{L^{1}}\sum_{k=l-2}^{l+2}\sum_{\mu=0}^{k-6}\|\mathsf{P}_{\mu}\nabla\mathbf{f}\|_{C^{0}}\|\mathsf{P}_{k}\mathbf{g}\|_{C^{0}}$
$\displaystyle\text{by }\eqref{Eqn::Hold::PfApThm::Term0C0}$
$\displaystyle\lesssim$
$\displaystyle\|\psi_{\sigma}\|_{L^{1}}2^{-\sigma(\alpha-\beta)}+2^{1-\sigma}\|x\mapsto
x\psi_{1}(x)\|_{L^{1}}\sum_{\mu=0}^{\sigma}\|\nabla\mathbf{f}\|_{\mathscr{C}^{\alpha-1}}2^{\mu(1-\alpha)}\|\mathbf{g}\|_{\mathscr{C}^{-\beta}}2^{l\beta}$
$\displaystyle\text{by }\eqref{Eqn::Hold::HoldChar::LPHoldChar}$
$\displaystyle\lesssim$ $\displaystyle
2^{-\sigma(\alpha-\beta)}+2^{-\sigma}2^{\sigma(1-\alpha)}2^{\sigma\beta}\lesssim
2^{-\sigma(\alpha-\beta)}.$ $\displaystyle\text{by
}\eqref{Eqn::Hold::PfApThm::PsiNorm}$

Here for the last inequality we use the fact
$\|\nabla\mathbf{f}\|_{\mathscr{C}^{\alpha-1}}\lesssim\|\mathbf{f}\|_{\mathscr{C}^{\alpha}}=1$
from Remark 2.1.7 (i).

Combining the above estimate with (2.1.59) and (2.1.61) we get
$\|\mathsf{P}_{l}(\mathsf{S}_{\sigma}\mathbf{f}\cdot\mathsf{S}_{\sigma}\mathbf{g})\|_{C^{0}(\mathbb{R}^{n})}\lesssim
2^{-\sigma(\alpha-\beta)}$ when $\sigma-3\leq l\leq\sigma+2$, which is the
second estimate for (2.1.54).

We therefore have
$\|\mathsf{S}_{\sigma}\mathbf{f}\cdot\mathsf{S}_{\sigma}\mathbf{g}\|_{C^{0}}\lesssim
2^{-\sigma(\alpha-\beta)}$ and finish the proof. ∎

In application we take $\mathbf{f}$ and $\mathbf{g}$ in Theorem 2.1.48 to be
the coefficients of vector fields and their derivatives. The theorem give a
systematical way of approximating vector fields.

######  Proposition 2.1.49.

Let $m\geq 1$ and $0<\alpha,\beta<1$ be satisfy
$\alpha>\max(\frac{1}{2},\beta)$. Let $\Omega\subseteq\mathbb{R}^{n}$ be an
open set, Let $\Omega^{\prime}\Subset\Omega$ be a precompact open subset and
let $\chi\in C_{c}^{\infty}(\Omega)$ satisfies $\chi|_{\Omega^{\prime}}\equiv
1$.

Let
$X_{1},\dots,X_{m}\in\mathscr{C}^{\alpha}_{\mathrm{loc}}(\Omega;\mathbb{R}^{n})$
be vector fields on $\Omega$. Assuming there are distributions
$(c_{ij}^{k})_{i,j,k=1}^{m}\subset\mathscr{C}^{-\beta}_{\mathrm{loc}}(\Omega)$
such that

$[X_{i},X_{j}]=\sum_{k=1}^{m}c_{ij}^{k}X_{k},\quad 1\leq i,j\leq m.$ (2.1.66)

For $\sigma=0,1,2,\dots$, we define

$X^{\sigma}_{i}:=\mathsf{S}_{\sigma}(\chi X_{i})=(\mathsf{S}_{\sigma}(\chi
a_{i}^{1}),\dots,\mathsf{S}_{\sigma}(\chi a_{i}^{n})),\quad
c_{ij}^{k\sigma}:=\mathsf{S}_{\sigma}(\chi c_{ij}^{k}),\quad 1\leq i,j,k\leq
m.$ (2.1.67)

Then there is a constant constant $C$ that does not depend on $\sigma$ (but
can depend on $X_{i},c_{ij}^{k},\chi,\phi$), such that

$\Big{\|}[X^{\sigma}_{i},X^{\sigma}_{j}]-\sum_{k=1}^{m}c_{ij}^{k\sigma}X_{k}^{\sigma}\Big{\|}_{C^{0}(\Omega^{\prime};\mathbb{R}^{n})}\leq
C2^{-\sigma\min(\alpha-\beta,2\alpha-1)},\quad\forall\sigma\geq 0.$ (2.1.68)

Recall from Lemmas 2.1.18 (ii) and Corollary 2.1.19,
$c_{ij}^{k}X_{k}\in\mathscr{C}^{-\beta}_{\mathrm{loc}}(\Omega;\mathbb{R}^{n})$
and
$[X_{i},X_{j}]\in\mathscr{C}^{\alpha-1}_{\mathrm{loc}}(\Omega;\mathbb{R}^{n})$
are defined as distributions.

###### Proof.

We can assume $\beta\geq 1-\alpha$, hence the right hand side of (2.1.68)
becomes $C2^{-\sigma(\alpha-\beta)}$. By computation,

$[\chi X_{i},\chi X_{j}]-(X_{i}\chi)\cdot\chi X_{j}+(X_{j}\chi)\cdot\chi
X_{i}-\sum_{k=1}^{m}\chi c_{ij}^{k}\cdot\chi X_{k}=0\quad\text{on
}\mathbb{R}^{n},\quad 1\leq i,j\leq m.$ (2.1.69)

By writing $X_{i}=:\sum_{l=1}^{n}a^{l}\frac{\partial}{\partial x^{l}}$ where
$a^{l}\in\mathscr{C}^{\alpha}_{\mathrm{loc}}(\Omega)$, (2.1.69) becomes

$\sum_{q=1}^{n}\Big{(}\chi a_{i}^{q}\frac{\partial(\chi a_{j}^{l})}{\partial
x^{q}}-\chi a_{j}^{q}\frac{\partial(\chi a_{i}^{l})}{\partial
x^{q}}\Big{)}+\sum_{q=1}^{n}\Big{(}\chi a_{i}^{l}\cdot
a_{j}^{q}\frac{\partial\chi}{\partial x^{q}}-\chi a_{j}^{l}\cdot
a_{i}^{q}\frac{\partial\chi}{\partial x^{q}}\Big{)}-\sum_{k=1}^{m}\chi
c_{ij}^{k}\cdot\chi a_{k}^{l}=0,\ 1\leq i,j\leq m,\ 1\leq l\leq n.$ (2.1.70)

We see that the above equality holds in
$\mathscr{C}^{\min(\alpha-1,-\beta)}(\mathbb{R}^{n})=\mathscr{C}^{-\beta}(\mathbb{R}^{n})$.

In order to apply Theorem 2.1.48, we define a function
$\mathbf{f}_{ij}^{l}\in\mathscr{C}^{\alpha}(\mathbb{R}^{n};\mathbb{R}^{2n+m+1})$
and a distribution
$\mathbf{g}_{ij}^{l}\in\mathscr{C}^{-\beta}(\mathbb{R}^{n};\mathbb{R}^{2n+m+1})$
for $1\leq i,j\leq m$ and $1\leq l\leq n$, as

$\begin{array}[]{ccccccccccccccc}\@add@centering\mathbf{f}_{ij}^{l}:=\big{(}&\chi
a_{i}^{1}&,\dots,&\chi a_{i}^{n}&,&-\chi a_{j}^{1}&,\dots,&-\chi
a_{j}^{n}&,&-\chi a_{1}^{l}&,\dots,&-\chi
a_{m}^{l}&,&\textstyle\sum_{q=1}^{n}\big{(}\chi
a_{i}^{l}a_{j}^{q}\partial_{q}\chi-\chi
a_{j}^{l}a_{i}^{q}\partial_{q}\chi\big{)}&\big{)},\\\
\mathbf{g}_{ij}^{l}:=\big{(}&\partial_{1}(\chi
a_{j}^{l})&,\dots,&\partial_{n}(\chi a_{j}^{l})&,&\partial_{1}(\chi
a_{i}^{l})&,\dots,&\partial_{n}(\chi a_{i}^{l})&,&\chi c_{ij}^{1}&,\dots,&\chi
c_{ij}^{m}&,&\centering 1&\big{)}.\end{array}$

The condition (2.1.70) is now the same as saying
$\mathbf{f}_{ij}^{l}\cdot\mathbf{g}_{ij}^{l}=0\in\mathscr{C}^{-\beta}(\mathbb{R}^{n})$
for all $1\leq i,j\leq m$, $1\leq l\leq n$.

By Theorem 2.1.48 we have
$\|\mathsf{S}_{\sigma}\mathbf{f}_{ij}^{l}\cdot\mathsf{S}_{\sigma}\mathbf{g}_{ij}^{l}\|_{C^{0}}\lesssim
2^{-\sigma(\alpha-\beta)}$ for all $i,j,l$ and $\sigma$, in other words,

$\bigg{\|}[X_{i}^{\sigma},X_{j}^{\sigma}]+\mathsf{S}_{\sigma}(X_{j}\chi)\cdot
X_{i}^{\sigma}-\mathsf{S}_{\sigma}(X_{i}\chi)\cdot
X_{j}^{\sigma}-\sum_{k=1}^{m}c_{ij}^{k\sigma}X_{k}^{\sigma}\bigg{\|}_{C^{0}(\mathbb{R}^{n})}\lesssim
2^{-\sigma(\alpha-\beta)},\quad 1\leq i,j\leq m,\quad\sigma\geq 0.$

To prove (2.1.68) it remains to show that
$\|\mathsf{S}_{\sigma}(X_{j}\chi)\cdot
X_{i}^{\sigma}-\mathsf{S}_{\sigma}(X_{i}\chi)\cdot
X_{j}^{\sigma}\|_{C^{0}(\Omega^{\prime})}\lesssim 2^{-\sigma(\alpha-\beta)}$
for all $1\leq i,j\leq m$. Indeed, by assumption
$\chi|_{\Omega^{\prime}}\equiv 1$, so $X_{i}\chi|_{\Omega^{\prime}}\equiv 0$.
Hence for $1\leq i\leq m$,

$\displaystyle\|\mathsf{S}_{\sigma}(X_{i}\chi)\|_{C^{0}(\Omega^{\prime})}=$
$\displaystyle\|\mathsf{S}_{\sigma}(X_{i}\chi)-X_{i}\chi\|_{C^{0}(\Omega^{\prime})}\leq\|\mathsf{S}_{\sigma}(X_{i}\chi)-X_{i}\chi\|_{C^{0}(\mathbb{R}^{n})}\leq\sum_{k=\sigma+1}^{\infty}\|\mathsf{P}_{\sigma}(X_{i}\chi)\|_{C^{0}(\mathbb{R}^{n})}$
(2.1.71) $\displaystyle\lesssim$
$\displaystyle\|X_{i}\chi\|_{\mathscr{C}^{\alpha}(\mathbb{R}^{n})}2^{-\sigma\alpha}\lesssim_{\alpha,X,\chi}2^{-\sigma(\alpha-\beta)}.$

Therefore

$\displaystyle\sum_{i,j=1}^{m}\|\mathsf{S}_{\sigma}(X_{j}\chi)\cdot
X_{i}^{\sigma}-\mathsf{S}_{\sigma}(X_{i}\chi)\cdot
X_{j}^{\sigma}\|_{C^{0}(\Omega^{\prime};\mathbb{R}^{n})}$
$\displaystyle\leq\sum_{i,j=1}^{m}\|\mathsf{S}_{\sigma}(X_{j}\chi)\|_{C^{0}(\Omega^{\prime})}\|X_{i}^{\sigma}\|_{C^{0}(\Omega;\mathbb{R}^{n})}$
$\displaystyle\lesssim 2^{-\sigma(\alpha-\beta)}\sup_{1\leq i\leq m}\|\chi
X_{i}\|_{C^{0}(\mathbb{R}^{n})}\lesssim 2^{-\sigma(\alpha-\beta)}.$

This completes the proof. ∎

As a direct application to Proposition 2.1.49, we see that for log-Lipschitz
subbundles, the distributional involutivity (see Definition  ‣ 1.1) implies
the strongly asymptotically involutivity (see Definition 5.1.5).

######  Corollary 2.1.50.

Following the assumptions of Proposition 2.1.49, and assuming
$X_{1},\dots,X_{m}\in\mathscr{C}^{\mathrm{LogL}}_{\mathrm{loc}}(\Omega;\mathbb{R}^{n})$,
then
$\lim_{\nu\to\infty}\|X_{i}^{\nu}-X_{i}\|_{C^{0}(\Omega^{\prime};\mathbb{R}^{n})}=0$
for all $1\leq i\leq m$, and there is a $t_{0}>0$ such that

$\lim\limits_{\nu\to\infty}\max_{1\leq i,j\leq
m}\Big{\|}[X^{\nu}_{i},X^{\nu}_{j}]-\sum_{k=1}^{m}c_{ij}^{k\nu}X_{k}^{\nu}\Big{\|}_{C^{0}(\Omega^{\prime};\mathbb{R}^{n})}\exp\Big{(}t_{0}\cdot\max_{1\leq
l\leq m}\|\nabla X_{l}^{\nu}\|_{C^{0}(\Omega^{\prime};\mathbb{R}^{n\times
n})}\Big{)}=0.$ (2.1.72)

In particular, for the case of two commutative vector fields ($m=2$ and
$c_{ij}^{k}\equiv 0$), namely $[X_{1},X_{2}]=0$ in $\Omega$, we have

$\exists t_{0}>0,\
\lim\limits_{\nu\to\infty}\|[X_{1}^{\nu},X_{2}^{\nu}]\|_{C^{0}(\Omega^{\prime};\mathbb{R}^{n})}e^{t_{0}\left(\|\nabla
X_{1}^{\nu}\|_{C^{0}(\Omega^{\prime};\mathbb{R}^{n\times n})}+\|\nabla
X_{2}^{\nu}\|_{C^{0}(\Omega^{\prime};\mathbb{R}^{n\times n})}\right)}=0.$
(2.1.73)

######  Remark 2.1.51.

For a distributional involutive log-Lipschitz subbundle $\mathcal{V}\leq
T\mathfrak{M}$, by Lemma 2.3.9 (iii) locally we can find a log-Lipschitz basis
$(X_{1},\dots,X_{r})$ for $\mathcal{V}$ ($r=\operatorname{rank}\mathcal{V}$)
such that $[X_{i},X_{j}]=0$ for all $1\leq i,j\leq r$. So by (2.1.72) with
$c_{ij}^{k}=c_{ij}^{k\nu}\equiv 0$, we see that the condition (5.1.2) is
satisfied (see Section 5.1.1). We conclude that a log-Lipschitz involutive
subbundle is always strongly asymptotic involutive (see Definition 5.1.5).

###### Proof of Corollary 2.1.50.

By assumption $(X_{i}^{\nu}-X_{i})|_{\Omega^{\prime}}=(X_{i}^{\nu}-\chi
X_{i})|_{\Omega^{\prime}}=(\mathsf{S}_{\nu}(\chi X_{i})-\chi
X_{i})|_{\Omega^{\prime}}$ for all $1\leq i\leq m$. Thus similar to (2.1.71),

$\|X_{i}^{\nu}-X_{i}\|_{C^{0}(\Omega^{\prime};\mathbb{R}^{n})}\leq\|\mathsf{S}_{\nu}(\chi
X_{i})-\chi
X_{i}\|_{C^{0}(\Omega^{\prime};\mathbb{R}^{n})}\leq\sum_{\nu+1}^{\infty}\|\mathsf{P}_{\nu}(\chi
X_{i})\|_{C^{0}(\Omega^{\prime};\mathbb{R}^{n})}\lesssim_{\alpha}\|\chi
X_{i}\|_{C^{0,\alpha}(\mathbb{R}^{n};\mathbb{R}^{n})}2^{-\nu\alpha}.$

Let $\nu\to\infty$ we see that
$\lim_{\nu\to\infty}\|X_{i}^{\nu}-X_{i}\|_{C^{0}(\Omega^{\prime};\mathbb{R}^{n})}=0$.

By Proposition 2.1.49 there is a $C_{1}>0$ and a $\delta>0$ (in fact
$\delta=\min(\alpha-\beta,2\alpha-1)$) such that

$\Big{\|}[X^{\nu}_{i},X^{\nu}_{j}]-\sum_{k=1}^{m}c_{ij}^{k\nu}X_{k}^{\nu}\Big{\|}_{C^{0}(\Omega^{\prime};\mathbb{R}^{n})}\leq
C_{1}2^{-\delta\nu},\quad\forall\nu=1,2,3,\dots.$

By Lemma 2.1.22 (ii) we can find a $C_{2}>0$ such that

$\|\nabla X_{i}^{\nu}\|_{C^{0}(\Omega^{\prime};\mathbb{R}^{n})}\leq\|\nabla
X_{i}^{\nu}\|_{C^{0}(\mathbb{R}^{m};\mathbb{R}^{n})}\leq
C_{2}\nu,\quad\forall\nu=1,2,3,\dots.$

Therefore for $t_{0}>0$ we have

$\Big{\|}[X^{\nu}_{i},X^{\nu}_{j}]-\sum_{k=1}^{m}c_{ij}^{k\nu}X_{k}^{\nu}\Big{\|}_{C^{0}(\Omega^{\prime};\mathbb{R}^{n})}e^{t_{0}\cdot\max_{1\leq
l\leq m}\|\nabla X_{l}^{\nu}\|_{C^{0}(\Omega^{\prime};\mathbb{R}^{n\times
n})}}\leq C_{1}2^{-\delta\nu}e^{t_{0}C_{2}\nu},\quad\nu=1,2,3,\dots.$ (2.1.74)

Taking $t_{0}=(2C_{2})^{-1}\delta\log 2$, the right hand side of (2.1.74) is
$C_{1}2^{-\frac{\delta}{2}\nu}$, which goes to $0$ as $\nu\to\infty$.
Therefore we get (2.1.72) for such $t_{0}$.

Replacing $t_{0}$ by $\frac{t_{0}}{2}$ we get (2.1.73). ∎

###  2.2 The Bi-parameter Hölder-Zygmund Spaces

In this section we set up some bi-parameter Hölder-Zygmund spaces and prove
some results of products and compositions.

####  2.2.1 Basic definitions

We define bi-parameter Hölder-Zygmund structures using vector valued
formulation.

######  Definition 2.2.1.

Let $\alpha\in\mathbb{R}$ and $\beta>0$. Let $x=(x^{1},\dots,x^{n})$ and
$s=(s^{1},\dots,s^{q})$ be the coordinate system of $\mathbb{R}^{n}$ and
$\mathbb{R}^{q}$ respectively. Let $U\subseteq\mathbb{R}^{n}$ and
$V\subseteq\mathbb{R}^{q}$ be two open subsets. Let $\mathscr{X}$ be a Banach
space.

  1. (i)

We define
$\mathscr{C}^{\alpha}_{x}L^{\infty}_{s}(U,V;\mathscr{X})=L^{\infty}_{s}\mathscr{C}^{\alpha}_{x}(V,U;\mathscr{X})$
to be the set of all $f\in\mathscr{D}^{\prime}(U\times V)$ such that
$x\mapsto\langle
f(x,\cdot),u\rangle_{\mathscr{D}^{\prime},C_{c}^{\infty}(V)}\in\mathscr{C}^{\alpha}(U)$
for all $u\in C_{c}^{\infty}(V)$ and

$\|f\|_{\mathscr{C}^{\alpha}L^{\infty}(U,V;\mathscr{X})}:=\sup_{u\in
C_{c}^{\infty}(V);\|u\|_{L^{1}(V)}=1}\|x\mapsto\langle
f(x,\cdot),u\rangle_{L^{\infty}(V),L^{1}(V)}\|_{\mathscr{C}^{\alpha}_{x}(U;\mathscr{X})}<\infty.$
(2.2.1)

  2. (ii)

We define
$\mathscr{C}^{\alpha}_{x}\mathscr{C}^{\beta}_{s}(U,V;\mathscr{X}):=\mathscr{C}^{\beta}(V;\mathscr{C}^{\alpha}(U;\mathscr{X}))$
with the same norm given in Definition 1.2.1.

  3. (iii)

For
$\mathscr{Y},\mathscr{Z}\in\\{\mathscr{C}^{\alpha}L^{\infty},\mathscr{C}^{\alpha}\mathscr{C}^{\beta}:\alpha\in\mathbb{R},\beta>0\\}$,
we define
$\mathscr{Y}\cap\mathscr{Z}(U,V;\mathscr{X}):=\mathscr{Y}(U,V;\mathscr{X})\cap\mathscr{Z}(U,V;\mathscr{X})$
and
$\mathscr{Y}+\mathscr{Z}(U,V;\mathscr{X}):=\mathscr{Y}(U,V;\mathscr{X})+\mathscr{Z}(U,V;\mathscr{X})$
with norms

$\displaystyle\|f\|_{\mathscr{Y}\cap\mathscr{Z}(U,V;\mathscr{X})}$
$\displaystyle:=\|f\|_{\mathscr{Y}(U,V;\mathscr{X})}+\|f\|_{\mathscr{Z}(U,V;\mathscr{X})},$
$\displaystyle\|f\|_{\mathscr{Y}+\mathscr{Z}(U,V;\mathscr{X})}$
$\displaystyle:=\inf\\{\|f_{1}\|_{\mathscr{Y}(U,V;\mathscr{X})}+\|f_{2}\|_{\mathscr{Z}(U,V;\mathscr{X})}:f=f_{1}+f_{2}\\}.$

######  Remark 2.2.2.

  1. (i)

The $\mathscr{C}^{\alpha}\mathscr{C}^{\beta}$ spaces are symmetric in the
sense that
$\mathscr{C}^{\alpha}\mathscr{C}^{\beta}(U,V)=\mathscr{C}^{\beta}\mathscr{C}^{\alpha}(V,U)$
when $\alpha>0$. This follows from Definition 1.2.1 since we have
$\mathscr{C}^{\beta}(V;\mathscr{C}^{\alpha}(U))=\mathscr{C}^{\alpha}(U;\mathscr{C}^{\beta}(V))$.

  2. (ii)

When $\alpha>0$, the pair $\langle
f(x,\cdot),u\rangle_{\mathscr{D}^{\prime},C_{c}^{\infty}(V)}$ can be defined
pointwise for $x\in U$. When $\alpha\leq 0$, the map $[x\mapsto\langle
f(x,\cdot),u\rangle_{\mathscr{D}^{\prime},C_{c}^{\infty}(V)}]$ can be realized
as a distribution on $U$ in the way that $v\in C_{c}^{\infty}(U)\mapsto\langle
f,v\otimes u\rangle_{\mathscr{D}^{\prime},C_{c}^{\infty}(U\times V)}$, where
$v\otimes u(x,s)=v(x)u(s)$.

  3. (iii)

(2.2.1) is a rigorous way of saying $\mathop{\operatorname{essup}}_{s\in
V}\|f(\cdot,s)\|_{\mathscr{C}^{\alpha}(U)}<\infty$, see Lemma 2.2.6.

  4. (iv)

For $\alpha,\gamma\leq 0$, the definition
$f\in\mathscr{C}^{\alpha}L^{\infty}\cap\mathscr{C}^{\gamma}\mathscr{C}^{\beta}(U,V)$
means there are extensions
$\tilde{f}_{1}\in\mathscr{C}^{\alpha}L^{\infty}(U,\mathbb{R}^{q})$ and
$\tilde{f}_{2}\in\mathscr{C}^{\gamma}\mathscr{C}^{\beta}(U,\mathbb{R}^{q})$
such that $\tilde{f}_{1}|_{U\times V}=\tilde{f}_{2}|_{U\times V}=f$. When $U$
is a smooth domains, by the existence of common extension operator we can
choose $\tilde{f}_{1}=\tilde{f}_{2}:=E_{x}f$ where
$E_{x}:\mathscr{C}^{\delta}(U)\to\mathscr{C}^{\delta}(\mathbb{R}^{n})$ is
extension operator which is bounded for $\delta\in\\{\alpha,\gamma\\}$.

Let $U,V$ be to arbitrary open sets, we use
$\mathcal{L}(L^{1}(V),\mathscr{C}^{\alpha}(U))$ as the space of all bounded
linear operators $L^{1}(V)\to\mathscr{C}^{\alpha}(U)$ endowed with the
standard operator norm. This is in fact the same as
$\mathscr{C}^{\alpha}L^{\infty}(U,V)$, see Lemma 2.2.3.

Let $T:\mathscr{C}^{\alpha_{1}}(U_{1})\to\mathscr{C}^{\alpha_{2}}(U_{1})$ be a
bounded linear operator, by Lemma 2.1.3 and Definition 2.2.1 (ii), the
operator $\tilde{T}f(x,s):=T(f(\cdot,s))(x)$ defines a bounded linear operator
$\tilde{T}:\mathscr{C}^{\alpha_{1}}\mathscr{C}^{\beta}(U_{1},V)\to\mathscr{C}^{\alpha_{2}}\mathscr{C}^{\beta}(U_{1},V)$
for $\beta>0$ with the same operator norms. This is also true if we replace
$\mathscr{C}^{\beta}$ by $L^{\infty}$.

To summary, for $\mathscr{C}^{\alpha}L^{\infty}$-spaces we have the following:

######  Lemma 2.2.3.

Let $\alpha_{1},\alpha_{2}\in\mathbb{R}$ and let $U_{1},U_{2},V$ be arbitrary
open sets.

  1. (i)

There is a canonical correspondence
$\mathscr{C}^{\alpha_{1}}L^{\infty}(U_{1},V)\cong\mathcal{L}(L^{1}(V),\mathscr{C}^{\alpha_{1}}(U_{1}))$
with the respective norms are the same.

  2. (ii)

Let $T:\mathscr{C}^{\alpha_{1}}(U_{1})\to\mathscr{C}^{\alpha_{2}}(U_{2})$ be a
bounded linear operator, then there is a unique operator
$\tilde{T}:\mathscr{C}^{\alpha_{1}}(U_{1},V)\to\mathscr{C}^{\alpha_{2}}(U_{2},V)$
such that

$[y\in
U_{2}\mapsto\langle(\tilde{T}f)(y,\cdot),u\rangle_{\mathscr{D}^{\prime},C_{c}^{\infty}(V)}]=T[x\in
U_{1}\mapsto\langle
f(x,\cdot),u\rangle_{\mathscr{D}^{\prime},C_{c}^{\infty}(V)}],\quad\forall
f\in\mathscr{C}^{\alpha_{1}}(U_{1}),\quad u\in C_{c}^{\infty}(V).$ (2.2.2)

Moreover $\tilde{T}$ and $T$ have the same operator norms.

###### Proof.

Let $F:L^{1}(V)\to\mathscr{C}^{\alpha_{1}}(U_{1})$ be a bounded linear
operator, clearly
$F:C_{c}^{\infty}(V)\to\mathscr{C}^{\alpha_{1}}(U_{1})\subset\mathscr{D}^{\prime}(U_{1})$
is bounded. Thus by the Schwartz Kernel Theorem $F$ corresponds to a
distribution $f\in\mathscr{D}^{\prime}(U_{1}\times V)$ such that $\langle
F(u),v\rangle_{\mathscr{D}^{\prime},C_{c}^{\infty}(U)}=\langle f,u\otimes
v\rangle_{\mathscr{D}^{\prime},C_{c}^{\infty}(U\times V)}$. Since
$\|x\mapsto\langle
f(x,\cdot),u\rangle_{\mathscr{D}^{\prime},C_{c}^{\infty}(V)}\|_{\mathscr{C}^{\alpha_{1}}(U_{1})}=\|F(u)\|_{\mathscr{C}^{\alpha_{1}}(U_{1})}$,
taking supremum over $u\in C_{c}^{\infty}$ we get
$\|f\|_{\mathscr{C}^{\alpha_{1}}L^{\infty}}\leq\|F\|_{L^{1}\to\mathscr{C}^{{\alpha_{1}}}}$.
In particular such $F$ gives a
$f\in\mathscr{C}^{\alpha_{1}}L^{\infty}(U_{1},V)$

Conversely, for a $f\in\mathscr{C}^{\alpha_{1}}L^{\infty}(U_{1},V)$, we have a
map $F:C_{c}^{\infty}(V)\to\mathscr{C}^{\alpha_{1}}(U_{1})$ given by
$F(u)=\langle f,u\rangle_{\mathscr{D}^{\prime},C_{c}^{\infty}(V)}$. By
(2.2.1),
$\|F(u)\|_{\mathscr{C}^{\alpha_{1}}(U_{1})}\leq\|f\|_{\mathscr{C}^{\alpha_{1}}L^{\infty}(U_{1},V)}\|u\|_{L^{1}(V)}$.
Since $C_{c}^{\infty}(V)\subset L^{1}(V)$ is dense, $F$ uniquely extends to a
bounded linear map $F:L^{1}(V)\to\mathscr{C}^{\alpha_{1}}(U_{1})$ with
$\|F\|_{L^{1}\to\mathscr{C}^{\alpha_{1}}}\leq\|f\|_{\mathscr{C}^{\alpha_{1}}L^{\infty}}$.
This finishes the proof of (i).

Now the right hand side of (2.2.2) defines a bounded linear map
$\mathcal{L}(L^{1}(V),\mathscr{C}^{\alpha_{1}}(U_{1}))\to\mathcal{L}(L^{1}(V),\mathscr{C}^{\alpha_{2}}(U_{2}))$,
which has the same operator norm to $T$. By (i) this descends to a unique
bounded linear map
$\mathscr{C}^{\alpha_{1}}(U_{1},V)\to\mathscr{C}^{\alpha_{2}}(U_{2},V)$, which
is our $\tilde{T}$. This finishes the proof of (ii). ∎

######  Notation 2.2.4.

For functions $f(x,s)$ and $\rho(x)$, we use $\ast_{x}$ as the convolution
acting on $x$-variable:

$\rho\ast_{x}f(x,s)=(\rho\ast f(\cdot,s))(x).$

Note that if
$f\in\mathscr{S}^{\prime}(\mathbb{R}^{n}_{x}\times\mathbb{R}^{q}_{s})$ and
$\rho\in\mathscr{S}(\mathbb{R}^{n})$, then $\rho\ast_{x}f$ is a well-defined
tempered distribution in $\mathbb{R}^{n}_{x}\times\mathbb{R}^{q}_{s}$, since
$\rho\ast_{x}f=(\rho\otimes\delta_{0^{q}})\ast f$. Here $\delta_{0^{q}}$ is
the Direc delta measure of the origin in $\mathbb{R}^{q}$.

######  Remark 2.2.5.

In the thesis we use a different formulation of bi-parameter spaces from
[Yao22, Section 2.1]. These two definitions coincide by the Lemma 2.2.6 below.

Definition 2.2.1 (ii) is better than the characterization in Lemma 2.2.6 (iii)
because of Lemma 2.1.3: for an operator
$T:\mathscr{C}^{\alpha_{1}}(U_{1})\to\mathscr{C}^{\alpha_{2}}(U_{2})$ and its
lift
$\tilde{T}:\mathscr{C}^{\alpha_{1}}\mathscr{C}^{\beta}(U_{1},V)\to\mathscr{C}^{\alpha_{2}}\mathscr{C}^{\beta}(U_{1},V)$,
we have $\|\tilde{T}\|=\|T\|$ by Lemma 2.1.3. However we do not know the
relation between $\|\tilde{T}\|$ and $\|T\|$ if we use (2.2.3) to define
$\|\tilde{T}\|$.

######  Lemma 2.2.6.

Let $\phi=(\phi_{j})_{j=0}^{\infty}\subset\mathscr{S}(\mathbb{R}^{n})$ and
$\psi=(\psi_{k})_{k=0}^{\infty}\subset\mathscr{S}(\mathbb{R}^{q})$ be two
dyadic resolutions. Let $\alpha\in\mathbb{R}$ and $\beta>0$. Let
$U\subseteq\mathbb{R}^{n}$ and $V\subseteq\mathbb{R}^{q}$ be two bounded
smooth domains.

  1. (i)

$f\in\mathscr{C}^{\alpha}L^{\infty}(\mathbb{R}^{n},V)$ if and only if
$\phi_{j}\ast_{x}f\in L^{\infty}(\mathbb{R}^{n}\times V)$ for all $j\geq 0$
and $\sup_{j\geq 0}2^{j\alpha}\|\phi_{j}\ast_{x}f\|_{L^{\infty}}<\infty$.
Moreover $f\mapsto\sup_{j\geq 0}2^{j\alpha}\|\phi_{j}\ast_{x}f\|_{L^{\infty}}$
is an equivalent norm for $\mathscr{C}^{\alpha}L^{\infty}(\mathbb{R}^{n},V)$.

  2. (ii)

$f\in\mathscr{C}^{\alpha}\mathscr{C}^{\beta}(\mathbb{R}^{n},\mathbb{R}^{q})$
has the following equivalent norm:

$f\mapsto\sup_{j,k\geq 0}2^{j\alpha+k\beta}\|(\phi_{j}\otimes\psi_{k})\ast
f\|_{L^{\infty}(\mathbb{R}^{n}\times\mathbb{R}^{q})},\quad\text{where
}\phi_{j}\otimes\psi_{k}(x,s)=\phi_{j}(x)\psi_{k}(s).$

  3. (iii)

Let $\mathscr{X}\in\\{L^{\infty},\mathscr{C}^{\beta}\\}$.
$\mathscr{C}^{\alpha}\mathscr{X}(U,V)$ has equivalent norm

$f\mapsto\inf\\{\|\tilde{f}\|_{\mathscr{C}^{\alpha}\mathscr{X}(\mathbb{R}^{n},\mathbb{R}^{q})}:\tilde{f}|_{U\times
V}=f\\}.$ (2.2.3)

  4. (iv)

For $\alpha>0$,
$\mathscr{C}^{\alpha}L^{\infty}(U,V)=\mathscr{C}^{\alpha}(U;L^{\infty}(V))$
and the corresponding norms are equivalent. And the common space equals to the
following

$\\{f\in L^{\infty}(U\times V):f(\cdot,s)\in\mathscr{C}^{\alpha}(U)\text{ a.e.
}s\in V\text{ with }\mathop{\operatorname{essup}}_{s\in
V}\|f(\cdot,s)\|_{\mathscr{C}^{\alpha}(U)}<\infty\\}.$ (2.2.4)

######  Remark 2.2.7.

We postpone the proof of Lemma 2.2.6 after the proof of Lemma 2.2.8. Thus from
then on we can view
$f\in\mathscr{C}^{\alpha}L^{\infty}\cap\mathscr{C}^{\gamma}\mathscr{C}^{\beta}(U,V)$
as a concrete function/distribution on product space $U\times V$ satisfying
the corresponding Littlewood-Paley characterization, rather than a vector-
valued map. Thus by taking coordinate components components, we see that Lemma
2.2.6 is also true if we replace $\mathscr{C}^{\alpha}\mathscr{X}(U,V)$
($\mathscr{X}\in\\{L^{\infty},\mathscr{C}^{\beta}$) by
$\mathscr{C}^{\alpha}\mathscr{X}(U,V;\mathbb{R}^{m})$ or
$\mathscr{C}^{\alpha}\mathscr{X}(U,V;\mathbb{C}^{m})$.

On the domain case we need the following result of extension operators.

######  Lemma 2.2.8 (Common extensions).

Let $R>0$, and let $U\subset\mathbb{R}^{n}$, $V\subset\mathbb{R}^{q}$ be two
bounded domain with smooth boundaries. Then there are extension operators
$E_{x}:L^{\infty}(U)\to L^{\infty}(\mathbb{R}^{n})$ and
$E_{s}:L^{\infty}(V)\to L^{\infty}(\mathbb{R}^{q})$ such that

  1. (a)

$E_{x}:\mathscr{C}^{\alpha}(U)\to\mathscr{C}^{\alpha}(\mathbb{R}^{n})$ and
$E_{s}:\mathscr{C}^{\alpha}(V)\to\mathscr{C}^{\alpha}(\mathbb{R}^{q})$ are
both bounded linear for all $-R<\alpha<R$.

  2. (b)

$E_{x}E_{s},E_{s}E_{x}:\mathscr{C}^{(-R)+}L^{\infty}(U,V)\to\mathscr{C}^{(-R)+}L^{\infty}(\mathbb{R}^{n},\mathbb{R}^{q})$
are well-defined and equal.

Moreover $\tilde{E}:=E_{x}E_{s}$ is an extension operator for $U\times V$ that
has boundedness

$\tilde{E}:\mathscr{C}^{\alpha}\mathscr{X}(U,V)\to\mathscr{C}^{\alpha}\mathscr{X}(\mathbb{R}^{n},\mathbb{R}^{q}),\quad-R<\alpha<R,\quad\mathscr{X}\in\\{L^{\infty},\mathscr{C}^{\beta}:0<\beta<R\\}.$
(2.2.5)

###### Proof.

We define $E_{x}$ and $E_{s}$ as in (2.1.11) and (2.1.12) by taking $\Omega=U$
and $\Omega=V$ respectively and taking $M=2\lceil R\rceil$. By [Tri10,
Theorems 2.9.2, 2.9.4 (i) and 3.3.4 (i)] $E_{x}$ and $E_{s}$ are both
$L^{\infty}$ and $\mathscr{C}^{\alpha}$-bounded for $-R<\alpha<R$. The
property (b) follows immediately from the expression themselves.

One can see that for the adjoint of $E_{s}$ we have
$E_{s}^{*}:L^{1}(\mathbb{R}^{q})\to L^{1}(V)$ with
$\|E_{s}^{*}\|_{L^{1}(\mathbb{R}^{q})\to L^{1}(V)}=\|E_{s}\|_{L^{\infty}(V)\to
L^{\infty}(\mathbb{R}^{q})}\leq\sum_{j=1}^{M}|a_{j}|$ where $(a_{j})$ are in
(2.1.11). So for $U^{\prime}\in\\{U,\mathbb{R}^{n}\\}$,

$\|x\mapsto\langle
E_{s}f(x,\cdot),u\rangle_{L^{\infty},L^{1}(\mathbb{R}^{q})}\|_{\mathscr{C}^{\alpha}(U^{\prime})}=\|x\mapsto\langle
f(x,\cdot),E_{s}^{*}u\rangle_{L^{\infty},L^{1}(V)}\|_{\mathscr{C}^{\alpha}(U^{\prime})}\leq\sum_{j=1}^{M}|a_{j}|\|f\|_{\mathscr{C}^{\alpha}L^{\infty}(U^{\prime},V)}\|u\|_{L^{1}(\mathbb{R}^{q})}.$

Therefore we have the boundedness
$E_{s}:\mathscr{C}^{\alpha}L^{\infty}(U^{\prime},V)\to\mathscr{C}^{\alpha}L^{\infty}(U^{\prime},\mathbb{R}^{n})$.

For $V^{\prime}\in\\{V,\mathbb{R}^{q}\\}$, by Lemma 2.1.3
$E_{x}:\mathscr{C}^{\alpha}(U)\to\mathscr{C}^{\alpha}(\mathbb{R}^{n})$
descends to
$E_{x}:\mathscr{C}^{\beta}(V^{\prime};\mathscr{C}^{\alpha}(U))\to\mathscr{C}^{\beta}(V^{\prime};\mathscr{C}^{\alpha}(\mathbb{R}^{n}))$,
i.e.
$E_{x}:\mathscr{C}^{\alpha}\mathscr{C}^{\beta}(U,V^{\prime})\to\mathscr{C}^{\alpha}\mathscr{C}^{\beta}(\mathbb{R}^{n},V^{\prime})$.
By Definition 2.2.1 (i) we have
$E_{x}:\mathscr{C}^{\alpha}L^{\infty}(U,V^{\prime})\to\mathscr{C}^{\alpha}L^{\infty}(\mathbb{R}^{n},V^{\prime})$
immediately. Therefore (2.2.5) is done by the following:

$E_{s}E_{x}:\mathscr{C}^{\alpha}\mathscr{X}(U,V)\xrightarrow{E_{x}}\mathscr{C}^{\alpha}\mathscr{X}(\mathbb{R}^{n},V)\xrightarrow{E_{s}}\mathscr{C}^{\alpha}\mathscr{X}(\mathbb{R}^{n},\mathbb{R}^{q}).\qed$

###### Proof of Lemma 2.2.6.

By the Schwartz Kernel Theorem (see [Trè06, Theorem 51.7])
$\mathscr{C}^{\alpha}L^{\infty}(\mathbb{R}^{n},V)$ can be identified as a
subspace of $\mathscr{D}^{\prime}(\mathbb{R}^{n}\times V)$. Thus we can view
the element $f:L^{1}(V)\to\mathscr{C}^{\alpha}(\mathbb{R}^{n})$ as a
distribution on $\mathbb{R}^{n}\times V$.

(i): By Lemma 2.1.4 (i) (if $\alpha>0$) and Definition 2.1.6 (if $\alpha\leq
0$) we have $\|f(\cdot,s)\|_{\mathscr{C}^{\alpha}(\mathbb{R}^{n})}\approx
2^{j\alpha}\|\phi_{j}\ast_{x}f(\cdot,s)\|_{L^{\infty}}$ for every $s\in V$.
Since $(L^{1})^{*}=L^{\infty}$, for a $g\in L^{\infty}(\mathbb{R}^{n}\times
V)$ we have $\|g(x,\cdot)\|_{L^{\infty}(V)}=\sup_{\|u\|_{L^{1}}=1}\langle
g(x,\cdot),u\rangle_{L^{\infty},L^{1}}$. Therefore

$\|f\|_{\mathscr{C}^{\alpha}L^{\infty}(\mathbb{R}^{n},V)}\approx\sup_{\|u\|_{L^{1}}=1}\sup_{j\geq
0}2^{j\alpha}\sup_{x\in\mathbb{R}^{n}}\langle\phi_{j}\ast_{x}f(x,\cdot),u\rangle_{L^{\infty},L^{1}(V)}=\sup_{j\geq
0}2^{j\alpha}\|\phi_{j}\ast_{x}f\|_{L^{\infty}(\mathbb{R}^{n}\times V)}.$

Here we use that fact that $\phi_{j}\ast_{x}f$ is continuous along
$x$-variable, which means the sup and the essential sup of $x$ are the same.

(ii): Since $\beta>0$, by Lemma 2.1.4 (i) again we have

$\|f\|_{\mathscr{C}^{\alpha}\mathscr{C}^{\beta}(\mathbb{R}^{n},\mathbb{R}^{q})}\approx\sup_{k\geq
0}\sup_{s\in\mathbb{R}^{q}}2^{k\beta}\|\psi_{k}\ast_{s}f(\cdot,s)\|_{\mathscr{C}^{\alpha}(\mathbb{R}^{n})}\approx\sup_{j,k\geq
0}2^{j\alpha+k\beta}\|\psi_{k}\ast_{s}(\phi_{j}\ast_{x}f)\|_{L^{\infty}(\mathbb{R}^{n}\times\mathbb{R}^{q})}.$

This completes the proof since

$\psi_{k}\ast_{s}(\phi_{j}\ast_{x}f)=\phi_{j}\ast_{x}(\psi_{k}\ast_{s}f)=(\phi_{j}\otimes\psi_{k})\ast
f.$ (2.2.6)

The proof of (iii) follows immediately from Lemma 2.2.8.

Finally to prove (iv), clearly by definition, the space
$\mathscr{C}^{\alpha}(U;L^{\infty}(V))$ and (2.2.4) are equal. By (i) and
Lemma 2.1.4 2.1.4 we see that
$\mathscr{C}^{\alpha}L^{\infty}(\mathbb{R}^{n},V)=\mathscr{C}^{\alpha}(\mathbb{R}^{n};L^{\infty}(V))=\\{f\in
L^{\infty}(\mathbb{R}^{n}\times V):\mathop{\operatorname{essup}}_{s\in
V}\|f(\cdot,s)\|<\infty\\}$ with equivalent norms. Using (iii) we can replace
$\mathbb{R}^{n}$ by bounded smooth domain $U$. ∎

######  Remark 2.2.9.

Now $f\in L^{\infty}\mathscr{C}^{\alpha}(U,V)$ is saying that
$f(\cdot,s)\in\mathscr{C}^{\alpha}(U)$ almost every $s\in V$ and is
essentially bounded. However $L^{\infty}\mathscr{C}^{\alpha}(U,V)$ does not
coincide with the vector-valued space $L^{\infty}(V;\mathscr{C}^{\alpha}(U))$.
In classical definitions $L^{\infty}(U;\mathscr{X})$ is the set of all
$\mathscr{X}$-valued strong measurable function (quotiented by almost
everywhere) which is essentially bounded. Such functions in
$L^{\infty}(U;\mathscr{C}^{\alpha}(V))$ must be almost everywhere separable
valued. But $\mathscr{C}^{\alpha}(U)$ is not a separable space. So
$L^{\infty}(U;\mathscr{C}^{\alpha}(V))\subsetneq
L^{\infty}\mathscr{C}^{\alpha}(U,V)$ and the inclusion is strict, and moreover
$L^{\infty}\mathscr{C}^{\alpha}(U,V)\not\subset
L^{1}(V;\mathscr{C}^{\alpha}(U))$. See for example [HvNVW16, Remark 1.2.16].

######  Remark 2.2.10.

Let $\alpha_{0},\alpha_{1}\in\mathbb{R}$ and $\beta_{0},\beta_{1}>0$. Clearly
$\min(2^{j\alpha_{0}+k\beta_{0}},2^{j\alpha_{1}+k\beta_{1}})\leq
2^{j((1-\theta)\alpha_{0}+\theta\alpha_{1})+k((1-\theta)\beta_{0}+\theta\beta_{1})}$
for $\theta\in[0,1]$. Thus we have inclusion: for $U,V$ either the total space
or bounded smooth domains,

$\mathscr{C}^{\alpha_{0}}\mathscr{C}^{\beta_{0}}\cap\mathscr{C}^{\alpha_{1}}\mathscr{C}^{\beta_{1}}(U,V)\subset\mathscr{C}^{\alpha_{\theta}}\mathscr{C}^{\beta_{\theta}}(U,V),\quad\alpha_{\theta}:=(1-\theta)\alpha_{0}+\theta\alpha_{1},\
\beta_{\theta}:=(1-\theta)\beta_{0}+\theta\beta_{1}.$ (2.2.7)

In particular
$\mathscr{C}^{\alpha}L^{\infty}\cap\mathscr{C}^{0}\mathscr{C}^{\beta}(U,V)\subset\mathscr{C}^{(1-\theta)\alpha}\mathscr{C}^{\theta\beta}(U,V)$
for all $\alpha,\beta>0$.

As a corollary to Lemma 2.2.6, the paraproduct operators can be adapted to bi-
parameter Hölder spaces.

######  Lemma 2.2.11.

Let $(s^{1},\dots,s^{q})$ be the standard coordinate system for
$\mathbb{R}^{q}$ and let $\phi=(\phi_{j})_{j=0}^{\infty}$ be a dyadic
resolution for $\mathbb{R}^{q}$.

$U,V,W$ be three open sets and let
$\Pi:\mathscr{C}^{\alpha_{1}}(U)\times\mathscr{C}^{\alpha_{2}}(V)\to\mathscr{C}^{\alpha_{3}}(W)$
be a bounded bilinear operator where
$\alpha_{1},\alpha_{2},\alpha_{3}\in\mathbb{R}$. Then the $s$-variable
paraproduct operators

$\mathfrak{P}^{\Pi}(f,g):=\sum_{k=0}^{\infty}\sum_{k^{\prime}=0}^{k-3}\Pi(\phi_{k}\ast_{s}f,\phi_{k^{\prime}}\ast_{s}g),\quad\mathfrak{R}^{\Pi}(f,g):=\sum_{|k-k^{\prime}|\leq
2}\Pi(\phi_{k}\ast_{s}f,\phi_{k^{\prime}}\ast_{s}g),$

define bilinearly bounded maps
$\mathfrak{P}^{\Pi},\mathfrak{R}^{\Pi}:\mathscr{C}^{\alpha_{1}}\mathscr{C}^{\beta}(U,\mathbb{R}^{q})\times\mathscr{C}^{\alpha_{2}}L^{\infty}(V,\mathbb{R}^{q})\to\mathscr{C}^{\alpha_{3}}\mathscr{C}^{\beta}(W,\mathbb{R}^{q})$,
for every $\beta>0$.

###### Proof.

Direct computation using Lemmas 2.1.13 and 2.1.16. ∎

Next we define the mixed Hölder-Zygmund space
$\mathscr{C}^{\alpha,\beta}_{x,s}$ as the following

######  Definition 2.2.12.

Let $\alpha,\beta\in\mathbb{R}_{\mathbb{E}}^{+}$ be two generalized indices.
Let $U\subseteq\mathbb{R}^{n}$ and $V\subseteq\mathbb{R}^{q}$ be two open
sets, and let $\mathscr{X}$ be a Banach space. We define
$\mathscr{C}^{\alpha,\beta}_{x,s}(U,V;\mathscr{X})$ to be the set of all
continuous functions $f:U\times V\to\mathscr{X}$ such that $\\{f(x,\cdot):x\in
U\\}\subset\mathscr{C}^{\beta}(V;\mathscr{X})$ and $\\{f(\cdot,s):s\in
V\\}\subset\mathscr{C}^{\alpha}(U;\mathscr{X})$ are two bounded sets with
respect to the topologies of $\mathscr{C}^{\beta}(V;\mathscr{X})$ and
$\mathscr{C}^{\alpha}(U;\mathscr{X})$ respectively.

When $\alpha,\beta\in\mathbb{R}_{+}$, we use the norm,

$\textstyle\|f\|_{\mathscr{C}^{\alpha,\beta}_{x,s}(U,V;\mathscr{X})}:=\sup_{x\in
U}\|f(x,\cdot)\|_{\mathscr{C}^{\beta}(V;\mathscr{X})}+\sup_{s\in
V}\|f(\cdot,s)\|_{\mathscr{C}^{\alpha}(U;\mathscr{X})}.$

We use abbreviation $f\in\mathscr{C}^{\alpha,\beta}_{x,s}$ when the domain is
clear. We use $f\in\mathscr{C}^{\alpha,\beta}$ when the variables are also
clear.

######  Remark 2.2.13.

  1. (i)

The norm definition is symmetric and we have
$\mathscr{C}^{\alpha,\beta}_{x,s}=\mathscr{C}^{\beta,\alpha}_{s,x}$. But in
applications we mostly refer the left variable as space variable, and the
right variable as parameter. For the notation $\mathscr{C}^{\alpha,\beta}$, we
usually use $\alpha\geq\beta$.

  2. (ii)

When $\alpha=\beta$, we have
$\mathscr{C}^{\alpha,\alpha}=\mathscr{C}^{\alpha}$. See Lemma 2.2.15 (ii)
below. So our $\mathscr{C}^{\alpha,\beta}$-spaces is a generalization to the
single variable cases.

  3. (iii)

We can define three variables mixed Hölder-Zygmund spaces and the version on
manifold analogously. See Definition 2.2.18 (ii) and Definition 2.3.2.

For $\gamma\in\mathbb{R}_{+}$, the spaces $\mathscr{C}^{\gamma-,\beta}$ and
$\mathscr{C}^{\gamma+,\beta}$ are just for convenient purpose:

######  Lemma 2.2.14.

Let $\gamma\in\mathbb{R}_{+}$ and $\beta\in\mathbb{R}_{\mathbb{E}}^{+}$. Let
$U\subseteq\mathbb{R}^{n}$ and $V\subseteq\mathbb{R}^{q}$ be two open subsets.
Then $f\in\mathscr{C}^{\gamma-,\beta}(U,V)$ if and only if
$f\in\mathscr{C}^{\gamma-\varepsilon,\beta}(U,V)$ for all
$0<\varepsilon<\gamma$.

Thus the discussion of $\mathscr{C}^{\alpha,\beta}$-regularities for positive
$\alpha,\beta\in\mathbb{R}_{\mathbb{E}}^{+}$ can be reduced to the case where
$\alpha,\beta\in(0,\infty)\cup\\{k+\mathrm{LogL},k+\mathrm{Lip}:k=0,1,2,\dots\\}$.

###### Proof.

$f\in\mathscr{C}^{\gamma-,\beta}(U,V)$ if and only if $\\{f(\cdot,s):s\in
V\\}\subset\mathscr{C}^{\gamma-}(U)$ is bounded, if and only if
$\\{f(\cdot,s):s\in V\\}\subset\mathscr{C}^{\gamma-\varepsilon}(U)$ is bounded
for all $\varepsilon>0$, if and only if
$f\in\mathscr{C}^{\gamma-\varepsilon,\beta}(U,V)$ for all $\varepsilon>0$. ∎

######  Lemma 2.2.15.

Let $\alpha,\beta\in(0,\infty)$, and let $U\subset\mathbb{R}^{m}$,
$V\subset\mathbb{R}^{n}$ be two bounded smooth domains. Then

  1. (i)

$\mathscr{C}^{\alpha,\beta}(U,V)=\mathscr{C}^{\alpha}L^{\infty}(U,V)\cap\mathscr{C}^{\beta}L^{\infty}(V,U)$
with equivalent norms.

  2. (ii)

$\mathscr{C}^{\alpha,\alpha}(U,V)=\mathscr{C}^{\alpha}(U\times V)$ with
equivalent norms.

###### Proof.

(i): By Lemma 2.2.8 we have an extension operator
$\tilde{E}:=E_{x}E_{s}=E_{s}E_{x}:L^{\infty}\mathscr{C}^{\beta}(U,V)\to
L^{\infty}\mathscr{C}^{\beta}(\mathbb{R}^{n},\mathbb{R}^{q})$, thus by Lemma
2.2.6 (iii),

$\textstyle\sup_{x\in
U}\|f(x,\cdot)\|_{\mathscr{C}^{\beta}(V)}\geq\mathop{\operatorname{essup}}_{x\in
U}\|f(x,\cdot)\|_{\mathscr{C}^{\beta}(V)}=\|E_{s}f\|_{\mathscr{C}^{\beta}L^{\infty}(\mathbb{R}^{q},U)}\approx\|\tilde{E}f\|_{\mathscr{C}^{\beta}L^{\infty}(\mathbb{R}^{n},\mathbb{R}^{q})}\approx\|f\|_{L^{\infty}\mathscr{C}^{\beta}(U,V)}.$
(2.2.8)

Similarly $\sup_{s\in
V}\|f(\cdot,s)\|_{\mathscr{C}^{\alpha}(U)}\gtrsim\|f\|_{\mathscr{C}^{\alpha}L^{\infty}(U,V)}$.
Therefore
$\|f\|_{\mathscr{C}^{\alpha,\beta}(U,V)}\gtrsim\|f\|_{\mathscr{C}^{\alpha}L^{\infty}\cap
L^{\infty}\mathscr{C}^{\beta}(U,V)}$.

For the converse direction, it suffices to show the first inequality in
(2.2.8) is indeed an equality. By Remark 2.2.10 we have
$\mathscr{C}^{\alpha}L^{\infty}\cap
L^{\infty}\mathscr{C}^{\beta}(U,V)\subset\mathscr{C}^{\frac{\alpha}{2}}\mathscr{C}^{\frac{\beta}{2}}(U,V)\subset
C^{0}(U,V)$. Thus $f\in\mathscr{C}^{\alpha}L^{\infty}\cap
L^{\infty}\mathscr{C}^{\beta}(U,V)$ can be pointwise defined, and the
essential supremum is indeed the supremum. This completes the proof.

(ii): Clearly
$\|f(x,\cdot)\|_{\mathscr{C}^{\alpha}(V)}\leq\|f\|_{\mathscr{C}^{\alpha}(U\times
V)}$ and
$\|f(\cdot,s)\|_{\mathscr{C}^{\alpha}(U)}\leq\|f\|_{\mathscr{C}^{\alpha}(U\times
V)}$ for each $x\in U$ and $s\in V$. Therefore
$\|f\|_{\mathscr{C}^{\alpha,\alpha}(U,V)}\leq
2\|f\|_{\mathscr{C}^{\alpha}(U\times V)}$.

For the converse, we can choose $(\phi_{j})_{j=0}^{\infty}$ and
$(\psi_{k})_{k=0}^{\infty}$ as dyadic resolutions for $\mathbb{R}^{n}_{x}$ and
$\mathbb{R}^{q}_{s}$ respectively, such that
$\operatorname{supp}\hat{\phi}_{0}\subset B^{n}(0,\sqrt{2})$ and
$\operatorname{supp}\hat{\psi}_{0}\subset B^{q}(0,\sqrt{2})$ additionally.
Define $(\pi_{l})_{l=0}^{\infty}$ as $\pi_{0}=\phi_{0}\otimes\psi_{0}$ and
$\pi_{l}=2^{l(n+q)}\pi_{0}(2^{l}\cdot)-2^{(l-1)(n+q)}\pi_{0}(2^{l-1}\cdot)$
for $l\geq 1$. We see that $(\pi_{l})_{l}$ is a dyadic resolutions for
$\mathbb{R}^{n+q}$ and

$\textstyle\pi_{l}=\sum_{\max(j,k)=l}\phi_{j}\otimes\psi_{k}=\phi_{l}\otimes\delta_{0\in\mathbb{R}^{q}}+\delta_{0\in\mathbb{R}^{n}}\otimes\psi_{l}-\sum_{\min(j,k)=l}\phi_{j}\otimes\psi_{k},\quad\forall\
l\geq 0.$

Therefore for $\tilde{f}\in\mathscr{C}^{\alpha}(\mathbb{R}^{n+q})$, by Lemma
2.1.4 (i) and (2.2.6),

$\displaystyle\textstyle\|\pi_{l}\ast\tilde{f}\|_{L^{\infty}}\leq\|\phi_{l}\ast_{x}\tilde{f}\|_{L^{\infty}}+\|\psi_{l}\ast_{s}\tilde{f}\|_{L^{\infty}}+\sum_{j=l}^{\infty}\|\psi_{l}\ast_{s}(\phi_{j}\ast_{x}\tilde{f})\|_{L^{\infty}}+\sum_{k=l+1}^{\infty}\|\phi_{l}\ast_{x}(\psi_{k}\ast_{s}\tilde{f})\|_{L^{\infty}}$
$\displaystyle\lesssim$
${}_{\alpha}\|\tilde{f}\|_{\mathscr{C}^{\alpha}_{x}L^{\infty}_{s}}2^{-l\alpha}+\|\tilde{f}\|_{L^{\infty}_{x}\mathscr{C}^{\alpha}_{s}}2^{-l\alpha}+\|\psi_{l}\|_{L^{1}}\|\tilde{f}\|_{\mathscr{C}^{\alpha}_{x}L^{\infty}_{s}}\sum_{j=l}^{\infty}2^{-j\alpha}+\|\phi_{l}\|_{L^{1}}\|\tilde{f}\|_{L^{\infty}_{x}\mathscr{C}^{\alpha}_{s}}\sum_{k=l+1}^{\infty}2^{-k\alpha}\lesssim\|\tilde{f}\|_{\mathscr{C}^{\alpha}_{x}L^{\infty}_{s}\cap
L^{\infty}_{x}\mathscr{C}^{\alpha}_{s}}2^{-l\alpha}.$

Thus
$\|\tilde{f}\|_{\mathscr{C}^{\alpha}(\mathbb{R}^{n}\times\mathbb{R}^{q})}\lesssim_{\alpha}\|\tilde{f}\|_{\mathscr{C}^{\alpha}L^{\infty}\cap
L^{\infty}\mathscr{C}^{\alpha}(\mathbb{R}^{n},\mathbb{R}^{q})}$. Restricting
to the domain $U\times V$ we finish the proof. ∎

######  Remark 2.2.16.

Let $\alpha,\beta>0$ and let $U,V$ be two bounded smooth domains. Since
$\mathscr{C}^{0}\subset L^{\infty}\subset\mathscr{C}^{1}$, using Lemma 2.2.6
we have
$\mathscr{C}^{\alpha}L^{\infty}\cap\mathscr{C}^{1}\mathscr{C}^{\beta}(U,V)\subset\mathscr{C}^{\alpha,\beta}(U,V)\subset\mathscr{C}^{\alpha}L^{\infty}\cap\mathscr{C}^{0}\mathscr{C}^{\beta}(U,V)$.

######  Remark 2.2.17 (Comparison of different working spaces).

The bi-parameter structures used in [Gon20, Section 2] and [NW63, Section 4.4]
are the $\mathcal{C}^{\alpha,\beta}$-spaces, which are similar to our
$\mathscr{C}^{\alpha,\beta}$-spaces. We recall them here:

Let $k\geq l\geq 0$ be integers, let $r,s\in[0,1)$ and let
$U\subseteq\mathbb{R}^{n}_{x}$, $V\subseteq\mathbb{R}^{q}_{s}$ be two open
(bounded smooth) subsets. The space $\mathcal{C}^{k+r,l+s}(U,V)$ is the set of
all continuous functions $f:U\times V\to\mathbb{R}$ whose norm below is
finite:

$\textstyle\|f\|_{\mathcal{C}^{k+r,l+s}(U,V)}:=\sum_{j=0}^{l}\sum_{i=0}^{k-j}\big{(}\sup_{s\in
V}\|\nabla_{x}^{i}\nabla_{s}^{j}f(\cdot,s)\|_{C^{0,r}(U)}+\sup_{x\in
U}\|\nabla_{x}^{i}\nabla_{s}^{j}f(x,\cdot)\|_{C^{0,s}(V)}\big{)}.$

Here $C^{0,\gamma}$ is the standard Hölder space $(0\leq\gamma<1)$.

One can see that $\mathcal{C}^{k+r,l+s}(U,V)=\mathscr{C}^{k+r,l+s}(U,V)$ when
$r,s\neq 0$ and
$\mathcal{C}^{k+r,l+s}(U,V)\subsetneq\mathscr{C}^{k+r,l+s}(U,V)$ when $r$ or
$s=0$. We leave the proof to readers.

The $\mathscr{C}^{\alpha,\beta}$-spaces is better than the
$\mathcal{C}^{\alpha,\beta}$-spaces in some sense. On one hand
$\mathcal{C}^{\alpha,\beta}$ are only defined for
$\lfloor\alpha\rfloor\geq\lfloor\beta\rfloor$, while
$\mathscr{C}^{\alpha,\beta}$ are defined for all positive $\alpha,\beta$. On
the other hand, the discussions of $\mathcal{C}^{\alpha,\beta}$ are always
separated between the integer cases and non-integer cases, while for
$\mathscr{C}^{\alpha,\beta}$-spaces the discussion can be unified as we have
complex interpolations,

$[\mathscr{C}^{\alpha_{0},\beta_{0}}(U,V),\mathscr{C}^{\alpha_{1},\beta_{1}}(U,V)]_{\theta}=\mathscr{C}^{(1-\theta)\alpha_{0}+\theta\alpha_{1},(1-\theta)\beta_{0}+\theta\beta_{1}}(U,V),\quad\text{for
all }\alpha_{0},\alpha_{1},\beta_{0},\beta_{1}>0,\quad\theta\in[0,1].$ (2.2.9)

We also leave the proof of (2.2.9) to readers.

Here are more definitions.

######  Definition 2.2.18.

Let $\alpha,\beta\in\mathbb{R}_{\mathbb{E}}^{+}$. Let
$U\subseteq\mathbb{R}^{n}$, $V\subseteq\mathbb{R}^{q}$ be two open sets. We
define the following spaces:

  1. (i)

$\mathscr{C}^{\alpha,\beta}_{\mathrm{loc}}(U,V):=\\{f\in
C^{0}_{\mathrm{loc}}(U\times
V):f\in\mathscr{C}^{\alpha,\beta}(U^{\prime},V^{\prime}),\ \forall
U^{\prime}\Subset U,\ V^{\prime}\Subset V\\}$.

  2. (ii)

Let $\gamma\in\mathbb{R}_{\mathbb{E}}^{+}$. Let $W\subseteq\mathbb{R}^{m}$ be
an open convex set. We define $\mathscr{C}^{\alpha,\beta,\gamma}(U,V,W)$ to be
the set of all continuous functions $f:U\times V\times W\to\mathbb{R}$ such
that $\\{f(\cdot,s,t):s\in V,t\in W\\}\subset\mathscr{C}^{\alpha}(U)$,
$\\{f(x,\cdot,t):x\in U,t\in W\\}\subset\mathscr{C}^{\beta}(V)$ and
$\\{f(x,s,\cdot):x\in U,s\in V\\}\subset\mathscr{C}^{\gamma}(W)$ are all
bounded subsets with respect to their topologies.

When $\alpha,\beta,\gamma\in(0,\infty)$ are numbers, we use the norm

$\displaystyle\|f\|_{\mathscr{C}^{\alpha,\beta,\gamma}_{x,s,t}(U,V,W)}:=$
$\displaystyle\sup_{x\in U,s\in
V}\|f(x,s,\cdot)\|_{\mathscr{C}^{\gamma}(W)}+\sup_{x\in U,t\in
W}\|f(x,\cdot,t)\|_{\mathscr{C}^{\beta}(V)}+\sup\limits_{s\in V,t\in
W}\|f(\cdot,s,t)\|_{\mathscr{C}^{\alpha}(U)}.$

######  Remark 2.2.19.

By Lemma 2.2.15 (i) we can say that
$\mathscr{C}^{\alpha,\beta,\gamma}_{x,s,t}=\mathscr{C}^{\alpha}_{x}L^{\infty}_{s,t}\cap\mathscr{C}^{\beta}_{s}L^{\infty}_{x,t}\cap\mathscr{C}^{\gamma}_{t}L^{\infty}_{x,s}=\mathscr{C}^{\alpha}_{x}L^{\infty}_{s,t}\cap
L^{\infty}_{x}\mathscr{C}^{\beta,\gamma}_{s,t}$, where
$\mathscr{C}^{\alpha}_{x}L^{\infty}_{s,t}\cap\mathscr{C}^{\beta}_{s}L^{\infty}_{x,t}\cap\mathscr{C}^{\gamma}_{t}L^{\infty}_{x,s}$
and $\mathscr{C}^{\alpha}_{x}L^{\infty}_{s,t}\cap
L^{\infty}_{x}\mathscr{C}^{\beta,\gamma}_{s,t}$ are defined analogously by
Definitions 2.2.1 and 2.2.12.

For $0<\alpha<\infty$, $0<\beta<\infty$, that
$f\in\mathscr{C}^{\alpha+1,\beta}_{x,s}$ does not imply
$\nabla_{x}f\in\mathscr{C}^{\alpha,\beta}_{x,s}$.

######  Example 2.2.20.

Let $\alpha,\beta>0$. We consider the following
$f:\mathbb{R}_{x}\times\mathbb{R}_{s}\to\mathbb{C}$:

$f(x,s):=\sum_{l=1}^{\infty}2^{-l(\alpha+1)}\exp(i2^{l}x+i2^{\frac{\alpha+1}{\beta}l}s).\quad\text{
We have }\frac{\partial f}{\partial
x}(x,s)=\sum_{l=1}^{\infty}i2^{-l\alpha}\exp(i2^{l}x+i2^{\frac{\alpha+1}{\beta}l}s).$

One can check that $f\in\mathscr{C}^{\alpha+1}_{x}L^{\infty}_{s}$ and $f\in
L^{\infty}_{x}\mathscr{C}^{\beta}_{s}$. So
$f\in\mathscr{C}^{\alpha+1,\beta}_{x,s}$.

Let $\psi=(\psi_{k})_{k=0}^{\infty}$ be a dyadic resolution. By (2.1.2)
$\operatorname{supp}\hat{\psi}_{k}=\operatorname{supp}2^{k-1}\hat{\psi}_{1}\subsetneq(-2^{k+1},-2^{k-1})\cup(2^{k-1},2^{k+1})$
for $k\geq 1$, so there is an $\varepsilon_{0}>0$ such that
$\operatorname{supp}\hat{\psi}_{k}\subseteq
2^{k}\cdot\big{(}[-2+\varepsilon_{0},-\frac{1}{2}-\varepsilon_{0}]\cup[\frac{1}{2}+\varepsilon_{0},2-\varepsilon_{0}]\big{)}$.
Note that there are infinitely many integer pairs $(k,l)$ such that
$|k-\frac{\alpha+1}{\beta}l|<\frac{\varepsilon_{0}}{2}$, and for such a pair
$(\phi_{k}\ast_{s}f)(x,s)=i2^{-l\alpha}\exp(i2^{l}x+i2^{\frac{\alpha+1}{\beta}l}s)$.

So $\|\psi_{k}\ast_{s}f\|_{L^{\infty}(\mathbb{R}^{2})}\approx
2^{-k\frac{\alpha\beta}{\alpha+1}}$ holds for infinite many $k\geq 0$, which
means $f\notin
L^{\infty}_{x}\mathscr{C}^{\frac{\alpha\beta}{\alpha+1}+\varepsilon}_{s}$ for
any $\varepsilon>0$.

In general, for $0<\alpha,\beta<\infty$ and a function
$f\in\mathscr{C}^{\alpha+1,\beta}_{x,s}$, the best we can guarantee is
$\nabla_{x}f\in\mathscr{C}^{\alpha,\frac{\alpha\beta}{\alpha+1}}_{x,s}$, where
the index $\frac{\alpha\beta}{\alpha+1}$ is optimal. For applications we need
the following endpoint optimal result.

######  Lemma 2.2.21.

Let $\alpha,\beta\in(0,\infty)$ and let $U\subseteq\mathbb{R}^{n}_{x}$,
$V\subseteq\mathbb{R}^{q}_{s}$ be two bounded convex smooth domains. Then
$\nabla_{x}:\mathscr{C}^{\alpha+1,\beta}(U,V)\to\mathscr{C}^{\alpha,\frac{\alpha\beta}{\alpha+1}-}(U,V;\mathbb{R}^{n})$
is bounded. In particular, if $f\in\mathscr{C}^{\infty,\beta}(U,V)$, then
$\nabla_{x}f\in\mathscr{C}^{\infty,\beta-}(U,V;\mathbb{R}^{n})$.

###### Proof.

Clearly
$\nabla_{x}:\mathscr{C}^{\alpha+1,\beta}_{x,s}\subset\mathscr{C}^{\alpha+1}_{x}L^{\infty}_{s}\cap\mathscr{C}^{0}_{x}\mathscr{C}^{\beta}_{s}\to\mathscr{C}^{\alpha}_{x}L^{\infty}_{s}\cap\mathscr{C}^{-1}_{x}\mathscr{C}^{\beta}_{s}$
is bounded. By Remark 2.2.10 we have embedding
$\mathscr{C}^{\alpha}_{x}L^{\infty}_{s}\cap\mathscr{C}^{-1}_{x}\mathscr{C}^{\beta}_{s}\subset\mathscr{C}^{(1-\theta)\alpha-\theta}_{x}\mathscr{C}^{\theta\beta}_{s}$
for all $\theta\in(0,1)$. Thus
$\mathscr{C}^{\alpha}_{x}L^{\infty}_{s}\cap\mathscr{C}^{-1}_{x}\mathscr{C}^{\beta}_{s}\subset\bigcap_{0<\theta<\frac{\alpha}{\alpha+1}}\mathscr{C}^{(1-\theta)\alpha-\theta}_{x}\mathscr{C}^{\theta\beta}_{s}\subset
L^{\infty}_{x}\mathscr{C}^{\frac{\alpha\beta}{\alpha+1}-}_{s}$. This proves
the boundedness
$\nabla_{x}:\mathscr{C}^{\alpha+1,\beta}_{x,s}\to\mathscr{C}^{\alpha,\frac{\alpha\beta}{\alpha+1}-}_{x,s}$.

Let $\alpha\to+\infty$ we get
$\nabla_{x}:\mathscr{C}^{\infty,\beta}_{x,s}\to\mathscr{C}^{\infty,\beta-}_{x,s}$.
∎

On the other hand, if $f(x,s)$ is, say a harmonic function in $x$, then the
assumption $f\in\mathscr{C}^{\alpha+1,\beta}_{x,s}$ can imply
$\nabla_{x}f\in\mathscr{C}^{\alpha,\beta}_{x,s}$. In fact we have
$f\in\mathscr{C}^{\infty}_{x}\mathscr{C}^{\beta}_{s}$ by the following.

######  Lemma 2.2.22.

Let $\beta\in\mathbb{R}_{\mathbb{E}}^{+}$, and let
$U\subset\mathbb{R}^{n}_{x}$ and $V\subset\mathbb{R}^{q}_{s}$ be two bounded
convex open sets with smooth boundaries. Suppose
$f\in\mathscr{C}^{-1}\mathscr{C}^{\beta}(U,V)$ satisfies $\Delta_{x}f=0$. Then
$f\in\mathscr{C}^{\infty}_{\mathrm{loc}}\mathscr{C}^{\beta}(U,V)$. In
particular $f\in\mathscr{C}^{\beta+1,\beta}_{\mathrm{loc}}(U,V)$ and
$\nabla_{x}f\in\mathscr{C}^{\beta}_{\mathrm{loc}}(U\times V;\mathbb{R}^{n})$.

As a corollary, let $\tilde{U}\subseteq\mathbb{C}^{m}_{z}$ be an open set. If
$g\in\mathscr{C}^{\beta}_{\mathrm{loc}}(\tilde{U}\times V;\mathbb{C})$ is
holomorphic in $z$, then
$\partial_{z}g\in\mathscr{C}^{\beta}_{\mathrm{loc}}(\tilde{U}\times
V;\mathbb{C}^{m})$ holds.

Here for $\beta\in\mathbb{R}_{\mathbb{E}}^{+}\backslash\mathbb{R}_{+}$, we use
the space
$\mathscr{C}^{\alpha}\mathscr{C}^{\beta}(U,V):=\mathscr{C}^{\beta}(V;\mathscr{C}^{\alpha}(U))$
and
$\mathscr{C}^{\alpha}_{\mathrm{loc}}\mathscr{C}^{\beta}(U,V):=\mathscr{C}^{\beta}(V;\mathscr{C}^{\alpha}_{\mathrm{loc}}(U))$
adapted from Definition 2.2.1 (ii).

###### Proof.

Since the result is local in $x\in U$, it is enough to consider the case where
$U=\mathbb{B}^{n}$ is the unit ball. By Definition 1.2.1 (iv) it suffices to
consider the case
$\beta\in\mathbb{R}_{+}\cup\\{k+\mathrm{LogL},k+\mathrm{Lip}:k=0,1,2,\dots\\}$.

Applying the spherical average formula, (or by Lemma 2.4.4 (iv)) we have

$|\nabla^{k}h(x)|\lesssim_{n,k}(1-|x|)^{-k-1}\|\nabla^{k}h\|_{\mathscr{C}^{-k-1}(\mathbb{B}^{n})}\lesssim_{k}(1-|x|)^{-k-1}\|h\|_{\mathscr{C}^{-1}(\mathbb{B}^{n})},\quad\forall
h\in\mathscr{C}^{-1}(\mathbb{B}^{n}),\ \Delta h=0.$

When $\beta\in(0,1)$, for $s_{1},s_{2}\in V$, by taking
$h_{s_{1},s_{2}}(x)=\frac{f(x,s_{1})-f(x,s_{2})}{|s_{1}-s_{2}|^{\beta}}$ we
see that for every $\Delta_{x}f=0$,

$\|f\|_{\mathscr{C}^{k}\mathscr{C}^{\beta}(r\mathbb{B}^{n},V)}=\sup_{s_{1},s_{2}\in
V}\|h_{s_{1},s_{2}}\|_{\mathscr{C}^{k}(r\mathbb{B}^{n})}\lesssim_{k}(1-r)^{-k-1}\sup_{s_{1},s_{2}\in
V}\|h_{s_{1},s_{2}}\|_{\mathscr{C}^{-1}(\mathbb{B}^{n})}=(1-r)^{-k-1}\|f\|_{\mathscr{C}^{-1}\mathscr{C}^{\beta}(\mathbb{B}^{n},V)}.$

Similarly arguments show that
$\|f\|_{\mathscr{C}^{k}\mathscr{C}^{\beta}(r\mathbb{B}^{n},V)}\lesssim_{k}(1-r)^{-k-1}\|f\|_{\mathscr{C}^{-1}\mathscr{C}^{\beta}(\mathbb{B}^{n},V)}$
for
$\beta\in[1,\infty)\cup\\{k+\mathrm{LogL},k+\mathrm{Lip}:k=0,1,2,\dots\\}$,
since all $\mathscr{C}^{\beta}$-norms are defined by a supremum over some
linear functionals.

Therefore $f\in\mathscr{C}^{k}\mathscr{C}^{\beta}(r\mathbb{B}^{n},V)$ for all
$r<1$ and all $k$, which means
$f\in\mathscr{C}^{\infty}_{\mathrm{loc}}\mathscr{C}^{\beta}(\mathbb{B}^{n},V)$.
The results $f\in\mathscr{C}^{\beta+1,\beta}_{\mathrm{loc}}$ and
$\nabla_{x}f\in\mathscr{C}^{\beta}_{\mathrm{loc}}$ follow immediately.

When $g(z,s)$ is holomorphic in $z$, we know $g$ is complex harmonic in
$z$-variable. So we reduce it to the real case by considering
$\tilde{U}\subseteq\mathbb{R}^{2m}$. ∎

####  2.2.2 Propositions on bi-parameter products

In this subsection we compute the mixed regularity of products using bi-
parameter paraproducts.

As a motivation, in Section 4.1.2 we need to solve a non-linear elliptic pde
system to find a suitable coordinate change that has best possible component-
wise regularity. We start with a coefficient function
$A\in\mathscr{C}^{\alpha,\beta}(U,V)$ and we want to find a solution
$H\in\mathscr{X}^{1}(U,V)$ in a suitable bi-parameter Hölder space
$\mathscr{X}^{1}$. To show the existence of $H$ we need spaces
$\mathscr{X}^{1}$, $\mathscr{X}^{0}$ and $\mathscr{X}^{-1}$ that are as small
as possible, and satisfy the following (see Proposition 4.1.6):

  * •

$\nabla_{x}:\mathscr{X}^{1}(U,V)\to\mathscr{X}^{0}(U,V;\mathbb{R}^{n})$,
$\nabla_{x}:\mathscr{X}^{0}(U,V)\to\mathscr{X}^{-1}(U,V;\mathbb{R}^{n})$ are
bounded; and $\Delta_{x}:\mathscr{X}^{1}(U,V)\to\mathscr{X}^{-1}(U,V)$ is
right invertible.

  * •

We have product maps
$\mathscr{X}^{0}(U,V)\times\mathscr{X}^{-1}(U,V)\to\mathscr{X}^{-1}(U,V)$ and
$\mathscr{X}^{0}(U,V)\times\mathscr{X}^{0}(U,V)\to\mathscr{X}^{0}(U,V)$.

  * •

$\mathscr{X}^{0}(U,V)\supseteq\mathscr{C}^{\alpha,\beta}(U,V)$.

In application $\mathscr{X}^{1}$ labels the mixed regularity for the
coordinate change, and $\mathscr{X}^{0}$ labels the mixed regularity for the
pushforward vector fields which are the generators for a given subbundle.

We shall see that for $k=1,0,-1$, we can pick
$\mathscr{X}^{k}=\mathscr{C}^{\alpha+k}_{x}L^{\infty}_{s}\cap\mathscr{C}^{k}_{x}\mathscr{C}^{\beta}_{s}$
when $\alpha>1$, and
$\mathscr{X}^{k}=\mathscr{C}^{\alpha+k}_{x}L^{\infty}_{s}\cap\mathscr{C}^{k}_{x}\mathscr{C}^{\beta}_{s}+\mathscr{C}^{\alpha+k}_{x}\mathscr{C}^{(2-1/\alpha)\beta}_{s}$
when $\frac{1}{2}<\alpha<1$. Note that
$\mathscr{C}^{\alpha+k}_{x}\mathscr{C}^{(2-1/\alpha)\beta}_{s}\subseteq\mathscr{C}^{k}_{x}\mathscr{C}^{\beta}_{s}$
when $\alpha>1$.

Firstly we extend Lemma 2.1.17 (i) to the bi-parameter case.

######  Lemma 2.2.23.

Let $U\subseteq\mathbb{R}^{n}$ and $V\subseteq\mathbb{R}^{q}$ be either the
total spaces or bounded smooth domains. The product map $[(f,g)\mapsto
fg]:\mathscr{C}^{\alpha}L^{\infty}(U,V)\times\mathscr{C}^{\beta}L^{\infty}(U,V)\to\mathscr{C}^{\min(\alpha,\beta)}L^{\infty}(U,V)$
is bounded for all $\alpha,\beta\in\mathbb{R}$ such that $\alpha+\beta>0$.

###### Proof.

By Lemma 2.2.6 (iii) it suffices to prove $U=\mathbb{R}^{n}$ and
$V=\mathbb{R}^{q}$. By symmetry we can assume $\beta\leq\alpha$.

Let $(\phi_{j})_{j=0}^{\infty}$ be a dyadic resolution for $\mathbb{R}^{n}$.
By Lemma 2.2.6 (i) it suffices to prove
$\|\phi_{j}\ast_{x}(fg)\|_{L^{\infty}(\mathbb{R}^{n}\times\mathbb{R}^{q})}\lesssim
2^{-j\beta}\|f\|_{\mathscr{C}^{\alpha}L^{\infty}}\|g\|_{\mathscr{C}^{\beta}L^{\infty}}$.
Indeed by Lemma 2.1.13,

$\displaystyle\|\phi_{l}\ast_{x}(fg)\|_{L^{\infty}}$
$\displaystyle\leq\|\phi_{l}\|_{L^{1}}\bigg{(}\sum_{j=l-2}^{l+2}\sum_{j^{\prime}=0}^{j-3}+\sum_{j^{\prime}=l-2}^{l+2}\sum_{j=0}^{j^{\prime}-3}+\sum_{\begin{subarray}{c}j,j^{\prime}\geq
l-3\\\ |j-j^{\prime}|\leq
2\end{subarray}}\bigg{)}\|\phi_{j}\ast_{x}f\|_{L^{\infty}}\|\phi_{j^{\prime}}\ast_{x}g\|_{L^{\infty}}$
$\displaystyle\lesssim\|f\|_{\mathscr{C}^{\alpha}L^{\infty}}\|g\|_{\mathscr{C}^{\beta}L^{\infty}}\bigg{(}\sum_{j=l-2}^{l+2}\sum_{j^{\prime}=0}^{j-3}+\sum_{j^{\prime}=l-2}^{l+2}\sum_{j=0}^{j^{\prime}-3}+\sum_{\begin{subarray}{c}j,j^{\prime}\geq
l-3\\\ |j-j^{\prime}|\leq 2\end{subarray}}\bigg{)}2^{-j\alpha-j\beta}\approx
2^{-j\beta}\|f\|_{\mathscr{C}^{\alpha}L^{\infty}}\|g\|_{\mathscr{C}^{\beta}L^{\infty}}.$

Here we use the assumption $\alpha+\beta>0$ and $\beta\leq\alpha$. ∎

In this subsection, we fix
$(\phi_{j})_{j=0}^{\infty}\subset\mathscr{S}(\mathbb{R}^{n})$ and
$(\psi_{k})_{k=0}^{\infty}\subset\mathscr{S}(\mathbb{R}^{q})$ as two dyadic
resolutions for their spaces. For bounded functions $f$ and $g$ defined on
$\mathbb{R}^{n}\times\mathbb{R}^{q}$, we denote

$f_{jk}:=(\phi_{j}\otimes\psi_{k})\ast f,\quad
g_{jk}:=(\phi_{j}\otimes\psi_{k})\ast g,\quad j,k\geq 0.$

We write $\displaystyle fg=\bigg{(}\sum\limits_{j^{\prime}+3\leq
j}+\sum\limits_{j+3\leq j^{\prime}}+\sum\limits_{|j-j^{\prime}|\leq
2}\bigg{)}\bigg{(}\sum\limits_{k^{\prime}+3\leq k}+\sum\limits_{k+3\leq
k^{\prime}}+\sum\limits_{|k-k^{\prime}|\leq
2}\bigg{)}(f_{jk}g_{j^{\prime}k^{\prime}})$ as $3\times 3$ sums:

$\displaystyle fg=\sum_{\mu,\nu=1}^{3}S^{\mu\nu}=$
$\displaystyle\sum\limits_{j=3}^{\infty}\sum\limits_{j^{\prime}=0}^{j-3}\sum\limits_{k=3}^{\infty}\sum\limits_{k^{\prime}=0}^{k-3}f_{jk}g_{j^{\prime}k^{\prime}}+\sum\limits_{j=3}^{\infty}\sum\limits_{j^{\prime}=0}^{j-3}\sum\limits_{k^{\prime}=3}^{\infty}\sum\limits_{k=0}^{k^{\prime}-3}f_{jk}g_{j^{\prime}k^{\prime}}+\sum\limits_{j=3}^{\infty}\sum\limits_{j^{\prime}=0}^{j-3}\sum\limits_{|k-k^{\prime}|\leq
2}f_{jk}g_{j^{\prime}k^{\prime}}$ (2.2.10)
$\displaystyle+\sum\limits_{j^{\prime}=3}^{\infty}\sum\limits_{j=0}^{j^{\prime}-3}\sum\limits_{k=3}^{\infty}\sum\limits_{k^{\prime}=0}^{k-3}f_{jk}g_{j^{\prime}k^{\prime}}+\sum\limits_{j^{\prime}=3}^{\infty}\sum\limits_{j=0}^{j^{\prime}-3}\sum\limits_{k^{\prime}=3}^{\infty}\sum\limits_{k=0}^{k^{\prime}-3}f_{jk}g_{j^{\prime}k^{\prime}}+\sum\limits_{j^{\prime}=3}^{\infty}\sum\limits_{j=0}^{j^{\prime}-3}\sum\limits_{|k-k^{\prime}|\leq
2}f_{jk}g_{j^{\prime}k^{\prime}}$
$\displaystyle+\sum\limits_{|j-j^{\prime}|\leq
2}\sum\limits_{k=3}^{\infty}\sum\limits_{k^{\prime}=0}^{k-3}f_{jk}g_{j^{\prime}k^{\prime}}+\sum\limits_{|j-j^{\prime}|\leq
2}\sum\limits_{k=3}^{\infty}\sum\limits_{k=0}^{k^{\prime}-3}f_{jk}g_{j^{\prime}k^{\prime}}+\sum\limits_{|j-j^{\prime}|\leq
2}\sum\limits_{|k-k^{\prime}|\leq 2}f_{jk}g_{j^{\prime}k^{\prime}}.$

Here $S^{\mu\nu}(f,g)$, $1\leq\mu,\nu\leq 3$, is defined to be the $\mu$-th
row $\nu$-th column term in (2.2.10).

The discussion on $\alpha>1$ follows from the following

######  Proposition 2.2.24.

Let $\alpha,\beta\in(0,\infty)$, $\eta\in(-\alpha,\alpha]$ and
$\theta\in[0,\alpha-\eta)$. There is $C=C_{\alpha,\beta,\eta,\theta}>0$ such
that

$\|fg\|_{\mathscr{C}^{\alpha-\theta}L^{\infty}\cap\mathscr{C}^{-\eta-\theta}\mathscr{C}^{\beta}(\mathbb{R}^{n},\mathbb{R}^{q})}\leq
C\|f\|_{\mathscr{C}^{\alpha}L^{\infty}\cap\mathscr{C}^{-\eta}\mathscr{C}^{\beta}(\mathbb{R}^{n},\mathbb{R}^{q})}\|g\|_{\mathscr{C}^{\alpha-\theta}L^{\infty}\cap\mathscr{C}^{-\eta-\theta}\mathscr{C}^{\beta}(\mathbb{R}^{n},\mathbb{R}^{q})},$

for all
$f\in\mathscr{C}^{\alpha}L^{\infty}\cap\mathscr{C}^{-\eta}\mathscr{C}^{\beta}(\mathbb{R}^{n},\mathbb{R}^{q})$
and
$g\in\mathscr{C}^{\alpha-\theta}L^{\infty}\cap\mathscr{C}^{-\eta-\theta}\mathscr{C}^{\beta}(\mathbb{R}^{n},\mathbb{R}^{q})$.

For applications, we use the cases $(\eta,\theta)=(1,0)$ and
$(\eta,\theta)=(0,0)$ to prove Corollary 4.1.8 that the left hand side of
(4.1.6) behave well. And we use $(\eta,\theta)=(0,1)$ to prove Proposition
2.2.36 which is required in Corollary 4.1.8 (iii).

###### Proof.

By Lemma 2.2.23, since $\alpha-\theta>\eta\geq-\alpha$, we have boundedness
$[(f,g)\mapsto
fg]:\mathscr{C}^{\alpha}L^{\infty}\cap\mathscr{C}^{-\eta}\mathscr{C}^{\beta}(\mathbb{R}^{n},\mathbb{R}^{q})\times\mathscr{C}^{\alpha-\theta}L^{\infty}\cap\mathscr{C}^{-\eta-\theta}\mathscr{C}^{\beta}(\mathbb{R}^{n},\mathbb{R}^{q})\to\mathscr{C}^{\alpha-\theta}L^{\infty}\cap\mathscr{C}^{-\eta-\theta}\mathscr{C}^{\beta}(\mathbb{R}^{n},\mathbb{R}^{q})$.
Our goal is to show the bilinear control
$\|fg\|_{\mathscr{C}^{-\eta-\theta}\mathscr{C}^{\beta}}\lesssim\|f\|_{\mathscr{C}^{\alpha}L^{\infty}\cap\mathscr{C}^{-\eta}\mathscr{C}^{\beta}}\|g\|_{\mathscr{C}^{\alpha-\theta}L^{\infty}\cap\mathscr{C}^{-\eta-\theta}\mathscr{C}^{\beta}}$.

By assumption $\alpha>\max(0,\eta+\theta)$ and $\eta>-\alpha$. Using Lemma
2.1.16 we have the boundedness:

$\mathfrak{P}_{x}:\mathscr{C}^{-\eta}(\mathbb{R}^{n})\times\mathscr{C}^{\alpha-\theta}(\mathbb{R}^{n})\to\begin{cases}\mathscr{C}^{-\eta}(\mathbb{R}^{n})&0\leq\theta<\alpha\\\
\mathscr{C}^{(-\eta)-\varepsilon}(\mathbb{R}^{n})(\forall\varepsilon>0)&\alpha=\theta\\\
\mathscr{C}^{\alpha-\eta-\theta}(\mathbb{R}^{n})&\alpha<\theta\leq\alpha-\eta\end{cases}\quad\subseteq\mathscr{C}^{-\eta-\theta}(\mathbb{R}^{n}).$

Similarly argument shows that for all $\alpha,\beta,\eta,\theta$ that satisfy
the assumption:

$\displaystyle\mathfrak{P}_{x},\mathfrak{R}_{x}\text{ are bounded
}\mathscr{C}^{\alpha-\theta}\times\mathscr{C}^{-\eta}\to\mathscr{C}^{-\eta-\theta},\
\mathscr{C}^{\alpha}\times\mathscr{C}^{-\eta-\theta}\to\mathscr{C}^{-\eta-\theta},\
\mathscr{C}^{-\eta-\theta}\times\mathscr{C}^{\alpha}\to\mathscr{C}^{-\eta-\theta}.$

Thus applying Lemma 2.2.11 with
$\Pi(u,v)\in\\{\mathfrak{P}_{x}(u,v),\mathfrak{P}_{x}(v,u),\mathfrak{R}_{x}(u,v)\\}$
in the lemma, we get:

$\displaystyle S^{11},S^{13},S^{21},S^{23},S^{31},S^{33}$
$\displaystyle:\mathscr{C}^{-\eta}\mathscr{C}^{\beta}(\mathbb{R}^{n},\mathbb{R}^{q})\times\mathscr{C}^{\alpha-\theta}L^{\infty}(\mathbb{R}^{n},\mathbb{R}^{q})\to\mathscr{C}^{-\eta-\theta}\mathscr{C}^{\beta}(\mathbb{R}^{n},\mathbb{R}^{q});$
$\displaystyle S^{12},S^{13},S^{22},S^{23},S^{32},S^{33}$
$\displaystyle:\mathscr{C}^{\alpha}L^{\infty}(\mathbb{R}^{n},\mathbb{R}^{q})\times\mathscr{C}^{-\eta-\theta}\mathscr{C}^{\beta}(\mathbb{R}^{n},\mathbb{R}^{q})\to\mathscr{C}^{-\eta-\theta}\mathscr{C}^{\beta}(\mathbb{R}^{n},\mathbb{R}^{q}).$

Therefore
$\|S^{\mu\nu}(f,g)\|_{\mathscr{C}^{-\eta-\theta}\mathscr{C}^{\beta}}\lesssim\|f\|_{\mathscr{C}^{\alpha-\theta}L^{\infty}\cap\mathscr{C}^{-\eta-\theta}\mathscr{C}^{\beta}}\|g\|_{\mathscr{C}^{\alpha-\theta}L^{\infty}\cap\mathscr{C}^{-\eta-\theta}\mathscr{C}^{\beta}}$
for all $\mu,\nu\in\\{1,2,3\\}$, finishing the proof. ∎

######  Remark 2.2.25.

The proof above is shorter than the proof in [Yao22, Proposition 2.30].

######  Corollary 2.2.26.

Let $\alpha,\beta,\eta,\theta$ satisfy the assumption of Proposition 2.2.24.
Let $U\subset\mathbb{R}^{m}$ and $V\subset\mathbb{R}^{n}$ be two bounded open
convex subsets with smooth boundaries. Then the following product is
bilinearly bounded:

$[(f,g)\to
fg]:\mathscr{C}^{\alpha}L^{\infty}\cap\mathscr{C}^{-\eta}\mathscr{C}^{\beta}(U,V)\times\mathscr{C}^{\alpha-\theta}L^{\infty}\cap\mathscr{C}^{-\eta-\theta}\mathscr{C}^{\beta}(U,V)\to\mathscr{C}^{\alpha-\theta}L^{\infty}\cap\mathscr{C}^{-\eta-\theta}\mathscr{C}^{\beta}(U,V).$
(2.2.11)

In particular it is bilinearly bounded between the following spaces:

  1. (i)

$\mathscr{C}^{\alpha}L^{\infty}\cap\mathscr{C}^{0}\mathscr{C}^{\beta}(U,V)\times\mathscr{C}^{\alpha-1}L^{\infty}\cap\mathscr{C}^{-1}\mathscr{C}^{\beta}(U,V)\to\mathscr{C}^{\alpha-1}L^{\infty}\cap\mathscr{C}^{-1}\mathscr{C}^{\beta}(U,V)$
for $\alpha>1$.

  2. (ii)

$\mathscr{C}^{\alpha}L^{\infty}\cap\mathscr{C}^{0}\mathscr{C}^{\beta}(U,V)\times\mathscr{C}^{\alpha}L^{\infty}\cap\mathscr{C}^{0}\mathscr{C}^{\beta}(U,V)\to\mathscr{C}^{\alpha}L^{\infty}\cap\mathscr{C}^{0}\mathscr{C}^{\beta}(U,V)$
for $\alpha>0$.

  3. (iii)

$\mathscr{C}^{\alpha}L^{\infty}\cap\mathscr{C}^{-1}\mathscr{C}^{\beta}(U,V)\times\mathscr{C}^{\alpha}L^{\infty}\cap\mathscr{C}^{-1}\mathscr{C}^{\beta}(U,V)\to\mathscr{C}^{\alpha}L^{\infty}\cap\mathscr{C}^{-1}\mathscr{C}^{\beta}(U,V)$
for $\alpha>1$.

###### Proof.

Indeed (i), (ii) and (iii) are the cases where $(\eta,\theta)=(0,1)$,
$(\eta,\theta)=(0,0)$ and $(\eta,\theta)=(1,0)$ in (2.2.11) respectively.

By Lemma 2.2.8 there is an extension operator such that
$\tilde{E}:\mathscr{X}(U,V)\to\mathscr{X}(\mathbb{R}^{n},\mathbb{R}^{q})$ and
$\tilde{E}:\mathscr{Y}(U,V)\to\mathscr{Y}(\mathbb{R}^{n},\mathbb{R}^{q})$.
Here we use
$\mathscr{X}=\mathscr{C}^{\alpha}L^{\infty}\cap\mathscr{C}^{-\eta}\mathscr{C}^{\beta}$
and
$\mathscr{Y}=\mathscr{C}^{\alpha-\theta}L^{\infty}\cap\mathscr{C}^{-\eta-\theta}\mathscr{C}^{\beta}$.
Applying Proposition 2.2.24,

$\|fg\|_{\mathscr{Y}(U,V)}\leq\|\tilde{E}f\tilde{E}g\|_{\mathscr{Y}(\mathbb{R}^{n},\mathbb{R}^{q})}\lesssim\|\tilde{E}f\|_{\mathscr{X}(\mathbb{R}^{n},\mathbb{R}^{q})}\|\tilde{E}g\|_{\mathscr{Y}(\mathbb{R}^{n},\mathbb{R}^{q})}\lesssim\|f\|_{\mathscr{X}(U,V)}\|g\|_{\mathscr{Y}(U,V)}.$

So we obtain (2.2.11). ∎

For $\frac{1}{2}<\alpha<1$, the bi-parameter spaces are unfortunately a bit
convoluted.

######  Proposition 2.2.27.

Let $\frac{1}{2}<\alpha<1$ and $\beta>0$. Let $U\subseteq\mathbb{R}^{n}$ and
$V\subseteq\mathbb{R}^{q}$ be either the total spaces or bounded smooth
domains. Then the product map

$\mathscr{C}^{\alpha}L^{\infty}\cap\mathscr{C}^{0}\mathscr{C}^{\beta}+\mathscr{C}^{\alpha}\mathscr{C}^{(2-\frac{1}{\alpha})\beta}(U,V)\times\mathscr{C}^{\alpha-1}L^{\infty}\cap(\mathscr{C}^{-1}\mathscr{C}^{\beta}+\mathscr{C}^{\alpha-1}\mathscr{C}^{(2-\frac{1}{\alpha})\beta}(U,V)\to\mathscr{C}^{\alpha-1}L^{\infty}\cap\mathscr{C}^{-1}\mathscr{C}^{\beta}+\mathscr{C}^{\alpha-1}\mathscr{C}^{(2-\frac{1}{\alpha})\beta}(U,V)$

is bounded bilinear.

######  Remark 2.2.28.

  1. (i)

There is no need to put parentheses on the spaces: we have
$\mathscr{C}^{\alpha}L^{\infty}\cap(\mathscr{C}^{0}\mathscr{C}^{\beta}+\mathscr{C}^{\alpha}\mathscr{C}^{(2-\frac{1}{\alpha})\beta})=(\mathscr{C}^{\alpha}L^{\infty}\cap\mathscr{C}^{0}\mathscr{C}^{\beta})+\mathscr{C}^{\alpha}\mathscr{C}^{(2-\frac{1}{\alpha})\beta}$
because
$\mathscr{C}^{\alpha}\mathscr{C}^{(2-\frac{1}{\alpha})\beta}\subset\mathscr{C}^{\alpha}L^{\infty}$.
Later it more convenient to consider
$\mathscr{C}^{\alpha}L^{\infty}\cap(\mathscr{C}^{0}\mathscr{C}^{\beta}+\mathscr{C}^{\alpha}\mathscr{C}^{(2-\frac{1}{\alpha})\beta})$.

  2. (ii)

For $\frac{1}{2}<\alpha<1$, by Remarks 2.2.10 and 2.2.16
$\mathscr{C}^{\alpha,\beta}\subset\mathscr{C}^{\alpha}L^{\infty}\cap\mathscr{C}^{\varepsilon+1-\alpha}\mathscr{C}^{(2-\frac{1+\varepsilon}{\alpha})\beta}$
holds for all $0<\varepsilon<2\alpha-1$. Applying Proposition 2.2.24 with
$\eta=\alpha-1-\varepsilon$ and $\theta=1$, we have the product map

$\mathscr{C}^{\alpha}L^{\infty}\cap\mathscr{C}^{\varepsilon+1-\alpha}\mathscr{C}^{(2-\frac{1+\varepsilon}{\alpha})\beta}\times\mathscr{C}^{\alpha-1}L^{\infty}\cap\mathscr{C}^{\varepsilon-\alpha}\mathscr{C}^{(2-\frac{1+\varepsilon}{\alpha})\beta}\to\mathscr{C}^{\alpha-1}L^{\infty}\cap\mathscr{C}^{\varepsilon-\alpha}\mathscr{C}^{(2-\frac{1+\varepsilon}{\alpha})\beta},\quad\forall
0<\varepsilon<2\alpha-1.$

Thus for $f,g\in\mathscr{C}^{\alpha,\beta}(\mathbb{R}^{n},\mathbb{R}^{q})$ we
get $(-\Delta)^{-\frac{1}{2}}(f\cdot\nabla
g)\in\mathscr{C}^{\alpha,(2-\frac{1}{\alpha})\beta-}$ which is endpoint
optimal. By introducing the term
$\mathscr{C}^{\alpha}\mathscr{C}^{(2-\frac{1}{\alpha})\beta}$ in Proposition
2.2.27, we get $(-\Delta)^{-\frac{1}{2}}(f\cdot\nabla
g)\in\mathscr{C}^{\alpha,(2-\frac{1}{\alpha})\beta}$, which is sharp. See
Example 2.2.29.

###### Proof of Proposition 2.2.27.

Firstly by Lemma 2.2.11, we see that
$\|fg\|_{\mathscr{C}^{\alpha-1}L^{\infty}}\lesssim\|f\|_{\mathscr{C}^{\alpha}L^{\infty}}\|g\|_{\mathscr{C}^{\alpha-1}L^{\infty}}$.

It is more convenient to introduce
$\mathfrak{P}^{\prime}(u,v):=\mathfrak{P}(v,u)$, thus we have
$uv=\mathfrak{P}(u,v)+\mathfrak{P}^{\prime}(u,v)+\mathfrak{R}(u,v)$.

By Lemma 2.1.16 $\mathfrak{P}_{x},\mathfrak{P}_{x}^{\prime}$ are both bounded
as $\mathscr{C}^{\alpha}\times\mathscr{C}^{-1}\to\mathscr{C}^{-1}$ and
$\mathscr{C}^{\alpha-1}\times\mathscr{C}^{0}\to\mathscr{C}^{-1}$. Therefore by
Lemma 2.2.11, again with
$\Pi(u,v)\in\\{\mathfrak{P}_{x}(u,v),\mathfrak{P}_{x}(v,u),\mathfrak{R}_{x}(u,v)\\}$
in the lemma,

$\displaystyle S^{11},S^{13},S^{21},S^{23}$
$\displaystyle:\mathscr{C}^{0}\mathscr{C}^{\beta}(\mathbb{R}^{n},\mathbb{R}^{q})\times\mathscr{C}^{\alpha-1}L^{\infty}(\mathbb{R}^{n},\mathbb{R}^{q})\to\mathscr{C}^{-1}\mathscr{C}^{\beta}(\mathbb{R}^{n},\mathbb{R}^{q});$
$\displaystyle S^{12},S^{13},S^{22},S^{23}$
$\displaystyle:\mathscr{C}^{\alpha}L^{\infty}(\mathbb{R}^{n},\mathbb{R}^{q})\times\mathscr{C}^{-1}\mathscr{C}^{\beta}(\mathbb{R}^{n},\mathbb{R}^{q})\to\mathscr{C}^{-1}\mathscr{C}^{\beta}(\mathbb{R}^{n},\mathbb{R}^{q}).$

Similarly, the boundedness
$\mathfrak{P}_{x},\mathfrak{P}_{x}^{\prime},\mathfrak{R}_{x}:\mathscr{C}^{\alpha}\times\mathscr{C}^{\alpha-1}\to\mathscr{C}^{\alpha-1}$
gives

$\displaystyle S^{11},S^{13},S^{21},S^{23},S^{31},S^{33}$
$\displaystyle:\mathscr{C}^{\alpha}\mathscr{C}^{(2-\frac{1}{\alpha})\beta}(\mathbb{R}^{n},\mathbb{R}^{q})\times\mathscr{C}^{\alpha-1}L^{\infty}(\mathbb{R}^{n},\mathbb{R}^{q})\to\mathscr{C}^{\alpha-1}\mathscr{C}^{(2-\frac{1}{\alpha})\beta}(\mathbb{R}^{n},\mathbb{R}^{q});$
$\displaystyle S^{12},S^{13},S^{22},S^{23},S^{32},S^{33}$
$\displaystyle:\mathscr{C}^{\alpha}L^{\infty}(\mathbb{R}^{n},\mathbb{R}^{q})\times\mathscr{C}^{\alpha-1}\mathscr{C}^{(2-\frac{1}{\alpha})\beta}(\mathbb{R}^{n},\mathbb{R}^{q})\to\mathscr{C}^{\alpha-1}\mathscr{C}^{(2-\frac{1}{\alpha})\beta}(\mathbb{R}^{n},\mathbb{R}^{q}).$

It remains prove the following:

$S^{31},S^{32},S^{33}:\mathscr{C}^{\alpha}L^{\infty}\cap\mathscr{C}^{0}\mathscr{C}^{\beta}\times\mathscr{C}^{\alpha-1}L^{\infty}\cap\mathscr{C}^{-1}\mathscr{C}^{\beta}\to\mathscr{C}^{0}\mathscr{C}^{(2-1/\alpha)\beta}(\subset\mathscr{C}^{\alpha-1}\mathscr{C}^{(2-1/\alpha)\beta}).$
(2.2.12)

The assumption $f\in\mathscr{C}^{\alpha}L^{\infty}$ implies that
$\|f_{jk}\|_{L^{\infty}}\lesssim_{\phi,\psi}\|f\|_{\mathscr{C}^{\alpha}L^{\infty}}2^{-j\alpha}$,
and the assumption $f\in\mathscr{C}^{0}\mathscr{C}^{\beta}$ implies that
$\|f_{jk}\|_{L^{\infty}}\lesssim_{\phi,\psi}\|f\|_{\mathscr{C}^{0}\mathscr{C}^{\beta}}2^{j-k\beta}$.
Similar control works for $g$. To conclude this, we have

$\displaystyle\|f_{jk}\|_{L^{\infty}}\lesssim\min(2^{-j\alpha},2^{-k\beta})\|f\|_{\mathscr{C}^{\alpha}L^{\infty}\cap\mathscr{C}^{0}\mathscr{C}^{\beta}},\quad\|g_{jk}\|_{L^{\infty}}\lesssim\min(2^{j(1-\alpha)},2^{j-k\beta})\|g\|_{\mathscr{C}^{\alpha-1}L^{\infty}\cap\mathscr{C}^{-1}\mathscr{C}^{\beta}}.$
(2.2.13)

For convenience we normalize $f$ and $g$ so that
$\|f\|_{\mathscr{C}^{\alpha}L^{\infty}\cap\mathscr{C}^{0}\mathscr{C}^{\beta}}=\|g\|_{\mathscr{C}^{\alpha-\theta}L^{\infty}\cap\mathscr{C}^{-\eta-\theta}\mathscr{C}^{\beta}}=1$.
Therefore what we need is to show for $\nu=1,2,3$,

$\textstyle S^{3\nu}_{lm}:=\|(\phi_{l}\otimes\psi_{m})\ast
S^{3\nu}(f,g)\|_{L^{\infty}}\lesssim_{\phi,\psi,\alpha,\beta}2^{-m(2-\frac{1}{\alpha})\beta},\text{
for }l,m\geq 0.$ (2.2.14)

Applying Lemma 2.1.13 on $x$-variable and $s$-variable separately we get

$(\phi_{l}\otimes\psi_{m})\ast(S^{31}+S^{32}+S^{33})(f,g)=\sum\limits_{\begin{subarray}{c}|j-j^{\prime}|\leq
2\\\ j,j^{\prime}\geq
l-3\end{subarray}}\Bigg{(}\sum\limits_{k=m-2}^{m+2}\sum\limits_{k^{\prime}=0}^{k-3}+\sum\limits_{k^{\prime}=m-2}^{m+2}\sum\limits_{k=0}^{k^{\prime}-3}+\sum\limits_{\begin{subarray}{c}|k-k^{\prime}|\leq
2\\\ k,k^{\prime}\geq
m-3\end{subarray}}\Bigg{)}(f_{jk}g_{j^{\prime}k^{\prime}}).$ (2.2.15)

Notice that $f\in\mathscr{C}^{\alpha}L^{\infty}$ and
$g\in\mathscr{C}^{\alpha-1}L^{\infty}$ give the following improvement of
(2.2.13):

$\displaystyle\textstyle\big{\|}\sum_{k=1}^{m}f_{jk}\big{\|}_{L^{\infty}}$
$\displaystyle=\|2^{mq}\psi_{0}(2^{m}\cdot)\ast_{s}(\phi_{j}\ast_{x}f)\|_{L^{\infty}}\lesssim\|\phi_{j}\ast_{x}f\|_{L^{\infty}}\lesssim
2^{-j\alpha}\|f\|_{\mathscr{C}^{\alpha}L^{\infty}};$
$\displaystyle\textstyle\big{\|}\sum_{k=1}^{m}g_{jk}\big{\|}_{L^{\infty}}$
$\displaystyle=\|2^{mq}\psi_{0}(2^{m}\cdot)\ast_{s}(\phi_{j}\ast_{x}g)\|_{L^{\infty}}\lesssim\|\phi_{j}\ast_{x}g\|_{L^{\infty}}\lesssim
2^{j(1-\alpha)}\|g\|_{\mathscr{C}^{\alpha-1}L^{\infty}}.$

Therefore for $l,m\geq 0$,

$\displaystyle S^{31}_{lm}\leq$
$\displaystyle\sum_{\begin{subarray}{c}j,j^{\prime}\geq l-3\\\
|j-j^{\prime}|\leq
l-3\end{subarray}}\sum_{k=m-2}^{m+2}\|f_{jk}\|_{L^{\infty}}\bigg{\|}\sum_{k^{\prime}=0}^{k-3}g_{j^{\prime}k^{\prime}}\bigg{\|}_{L^{\infty}}\lesssim\sum_{\begin{subarray}{c}j,j^{\prime}\geq
l-3\\\ |j-j^{\prime}|\leq
l-3\end{subarray}}\sum_{k=m-2}^{m+2}\min(2^{-j\alpha},2^{-k\beta})2^{j^{\prime}(1-\alpha)}$
$\displaystyle\approx$ $\displaystyle\sum_{l^{\prime}\geq
l}\min(2^{-l^{\prime}\alpha},2^{-m\beta})2^{l^{\prime}(1-\alpha)}=\sum_{l\leq
l^{\prime}\leq
m\beta/\alpha}2^{l^{\prime}(1-\alpha)-m\beta}+\sum_{l^{\prime}>m\beta/\alpha}2^{-l^{\prime}(2\alpha-1)}\lesssim
2^{-m(2-\frac{1}{\alpha})\beta};$ $\displaystyle S^{32}_{lm}\leq$
$\displaystyle\sum_{\begin{subarray}{c}j,j^{\prime}\geq l-3\\\
|j-j^{\prime}|\leq
l-3\end{subarray}}\sum_{k^{\prime}=m-2}^{m+2}\|g_{j^{\prime}k^{\prime}}\|_{L^{\infty}}\bigg{\|}\sum_{k=0}^{k^{\prime}-3}f_{jk}\bigg{\|}_{L^{\infty}}\lesssim\sum_{\begin{subarray}{c}j,j^{\prime}\geq
l-3\\\ |j-j^{\prime}|\leq
l-3\end{subarray}}\sum_{k^{\prime}=m-2}^{m+2}\min(2^{j^{\prime}(1-\alpha)},2^{j^{\prime}-k^{\prime}\beta})2^{-j\alpha}$
$\displaystyle\approx$ $\displaystyle\sum_{l^{\prime}\geq
l}\min(2^{l^{\prime}(1-\alpha)},2^{l^{\prime}-m\beta})2^{-l^{\prime}\alpha}\lesssim
2^{-m(2-\frac{1}{\alpha})\beta};$ $\displaystyle S^{33}_{lm}\leq$
$\displaystyle\sum_{\begin{subarray}{c}j,j^{\prime}\geq l-3\\\
|j-j^{\prime}|\leq l-3\end{subarray}}\sum_{\begin{subarray}{c}k,k^{\prime}\geq
m-3\\\ |k-k^{\prime}|\leq
2\end{subarray}}\|f_{jk}\|_{L^{\infty}}\|g_{j^{\prime}k^{\prime}}\|_{L^{\infty}}\lesssim\sum_{\begin{subarray}{c}j,j^{\prime}\geq
l-3\\\ |j-j^{\prime}|\leq
l-3\end{subarray}}\sum_{\begin{subarray}{c}k,k^{\prime}\geq m-3\\\
|k-k^{\prime}|\leq
2\end{subarray}}\min(2^{-j\alpha},2^{-k\beta})\min(2^{j^{\prime}(1-\alpha)},2^{j^{\prime}-k^{\prime}\beta})$
$\displaystyle\approx$ $\displaystyle\sum_{l^{\prime}\geq
l}\sum_{m^{\prime}\geq
m}\min(2^{-l^{\prime}(2\alpha-1)},2^{l^{\prime}-2m^{\prime}\beta})=\sum_{m^{\prime}\geq
m}\big{(}\sum_{l\leq l^{\prime}\leq
m^{\prime}\beta/\alpha}2^{l^{\prime}-2m^{\prime}\beta}+\sum_{l^{\prime}>m^{\prime}\beta/\alpha}2^{-l^{\prime}(2\alpha-1)}\bigg{)}$
$\displaystyle\lesssim$ $\displaystyle\sum_{m^{\prime}\geq
m}2^{-m^{\prime}(2-1/\alpha)\beta}\lesssim 2^{-m(2-\frac{1}{\alpha})\beta}.$

We get (2.2.14) hence (2.2.12), finishing the proof. ∎

We should point out that when $\frac{1}{2}<\alpha<1$, the regularity loss in
parameter is inevitable.

######  Example 2.2.29.

For $\alpha,\beta>0$, $\eta>-\alpha$ and $0<\theta<2\alpha$. Let

$f(x,s):=\sum_{j=1}^{\infty}2^{-\frac{j\alpha\beta}{\alpha-\eta}}e^{2\pi
i(2^{\frac{j\beta}{\alpha-\eta}}x+2^{j}s)}\sum_{k=1}^{j-1}2^{\frac{\theta\beta}{\alpha-\eta}(k-j)}e^{2\pi
i2^{k}s},\quad
g(x,s):=\sum_{j=1}^{\infty}2^{-\frac{j(\alpha-\theta)\beta}{\alpha-\eta}}e^{2\pi
i(2^{\frac{j\beta}{\alpha-\eta}}x+2^{j}s)}.$

One can see that
$f\in\mathscr{C}^{\alpha}L^{\infty}\cap\bigcap_{\lambda\in\mathbb{R}}\mathscr{C}^{(1-\lambda)\alpha}\mathscr{C}^{\lambda\frac{\alpha\beta}{\alpha-\eta}}(\mathbb{R},\mathbb{R})$
and
$g\in\mathscr{C}^{\alpha-\theta}L^{\infty}\cap\bigcap_{\lambda\in\mathbb{R}}\mathscr{C}^{(1-\lambda)(\alpha-\theta)}\mathscr{C}^{\lambda\frac{(\alpha-\theta)\beta}{\alpha-\eta}}(\mathbb{R},\mathbb{R})$.
Taking $\lambda=\frac{\alpha-\eta}{\alpha}$ and
$\lambda=\frac{\alpha-\eta}{\alpha-\theta}$ respectively we get
$f\in\mathscr{C}^{\alpha}L^{\infty}\cap\mathscr{C}^{-\eta}\mathscr{C}^{\beta}$
and
$g\in\mathscr{C}^{\alpha-\theta}L^{\infty}\cap\mathscr{C}^{-\eta-\theta}\mathscr{C}^{\beta}$.

Let $(\phi_{j})_{j=0}^{\infty}\subset\mathscr{S}(\mathbb{R})$ be a dyadic
resolution, we have

$(\phi_{0}\ast_{s}(fg))(x,s)=\sum_{j=0}^{\infty}\sum_{k=0}^{j-1}2^{-j(2\alpha-\theta)\frac{\beta}{\alpha-\eta}+\frac{\theta\beta(k-j)}{\alpha-\eta}}e^{2\pi
i2^{k}s}=\sum_{k=0}^{\infty}2^{-k(2\alpha-\theta)\frac{\beta}{\alpha-\eta}}e^{2\pi
i2^{k}s}\frac{1}{2^{\frac{2\alpha\beta}{\alpha-\eta}}-1}.$

Note that the assumption $\theta\leq 2\alpha$ is necessary otherwise the sum
diverges.

Thus $\phi_{0}\ast_{s}(fg)\in
L^{\infty}\mathscr{C}^{(2\alpha-\theta)\frac{\beta}{\alpha-\eta}}(\mathbb{R},\mathbb{R})$
but $fg\notin\mathscr{C}^{-M}\mathscr{C}^{\gamma}(\mathbb{R},\mathbb{R})$ for
all $M>0$ and $\gamma>(2\alpha-\theta)\frac{\beta}{\alpha-\eta}$.

Therefore when $\eta+\theta>\alpha$ we see that
$\frac{2\alpha-\theta}{\alpha-\eta}<1$, thus
$fg\notin\mathscr{C}^{-M}\mathscr{C}^{\beta}(\mathbb{R},\mathbb{R})$ for all
$M>0$. This example shows that in order for Proposition 2.2.24 to be true we
must assume $\eta+\theta\leq\alpha$.

One can find a more delicate example to show that the case
$\eta+\theta=\alpha$ also fails. We leave the detail to readers.

As a special case, when $\eta=0$ and $\theta=1$, we see that
$fg\notin\mathscr{C}^{-M}\mathscr{C}^{(2-1/\alpha)\beta+}(\mathbb{R},\mathbb{R})$
for all $M>0$. Thus the index $(2-1/\alpha)\beta$ in Proposition 2.2.27 cannot
be improved.

In application we need Corollary 2.2.26 and Proposition 2.2.27 on matrix-
valued functions,

######  Lemma 2.2.30.

For any $\alpha,\beta>0$, $0\leq\eta<\alpha$ and any bounded smooth domains
$U\subset\mathbb{R}^{n}$, $V\subset\mathbb{R}^{q}$, there exists a
$\tilde{c}=\tilde{c}(U,V,\alpha,\beta,\eta,m)>0$ such that

  1. (i)

$\|A^{-1}-I_{m}\|_{\mathscr{C}^{\alpha}L^{\infty}}\leq
2\|A-I_{m}\|_{\mathscr{C}^{\alpha}L^{\infty}}$, for all
$A\in\mathscr{C}^{\alpha}L^{\infty}(U,V;\mathbb{C}^{m\times m})$ such that
$\|A-I_{m}\|_{\mathscr{C}^{\alpha}L^{\infty}}\leq\tilde{c}$.

  2. (ii)

$\|A^{-1}-I_{m}\|_{\mathscr{C}^{\alpha}\mathscr{C}^{\beta}}\leq
2\|A-I_{m}\|_{\mathscr{C}^{\alpha}\mathscr{C}^{\beta}}$, for all
$A\in\mathscr{C}^{\alpha}\mathscr{C}^{\beta}(U,V;\mathbb{C}^{m\times m})$ such
that $\|A-I_{m}\|_{\mathscr{C}^{\alpha}\mathscr{C}^{\beta}}\leq\tilde{c}$.

  3. (iii)

$\|A^{-1}-I_{m}\|_{\mathscr{C}^{\alpha,\beta}}\leq
2\|A-I_{m}\|_{\mathscr{C}^{\alpha,\beta}}$, for all
$A\in\mathscr{C}^{\alpha,\beta}(U,V;\mathbb{C}^{m\times m})$ such that
$\|A-I_{m}\|_{\mathscr{C}^{\alpha,\beta}}\leq\tilde{c}$.

  4. (iv)

$\|A^{-1}-I_{m}\|_{\mathscr{C}^{\alpha}L^{\infty}\cap\mathscr{C}^{-\eta}\mathscr{C}^{\beta}}\leq
2\|A-I_{m}\|_{\mathscr{C}^{\alpha}L^{\infty}\cap\mathscr{C}^{-\eta}\mathscr{C}^{\beta}}$,
for all
$A\in\mathscr{C}^{\alpha}L^{\infty}\cap\mathscr{C}^{-\eta}\mathscr{C}^{\beta}(U,V;\mathbb{C}^{m\times
m})$ such that
$\|A-I_{m}\|_{\mathscr{C}^{\alpha}L^{\infty}\cap\mathscr{C}^{-\eta}\mathscr{C}^{\beta}}\leq\tilde{c}$.

  5. (v)

$\|A^{-1}-I_{m}\|_{\mathscr{C}^{\alpha}}\leq
2\|A-I_{m}\|_{\mathscr{C}^{\alpha}}$, for all
$A\in\mathscr{C}^{\alpha}(U;\mathbb{C}^{m\times m})$ with
$\|A-I_{m}\|_{\mathscr{C}^{\alpha}}\leq\tilde{c}$. Here
$\tilde{c}=\tilde{c}(U,\alpha,m)>0$.

###### Proof.

These are immediate from Lemmas 2.1.28 and 2.1.17, where (iv) comes from
Corollary 2.2.26. ∎

####  2.2.3 Compositions and inverse mapping with parameters

In this subsection we work on the concept of compositions between parameter
dependent maps. We are going to prove an auxiliary result, Proposition 2.2.36,
that is used in Corollary 4.1.8 (iii).

We recall that for a bijection $f:U_{1}\to U_{2}$, we denote its inverse map
by $f^{\mathrm{Inv}}:U_{2}\to U_{1}$.

In this subsection, for maps $F(x,s):U\times
V\subseteq\mathbb{R}^{n}\times\mathbb{R}^{q}\to\mathbb{R}^{n}$, usually we
would denote by $\tilde{F}$ the map $\tilde{F}(x,s):=(F(x,s),s)$. Note that if
$F(\cdot,s)$ has inverse for every $s$, then in this notation we have

$\tilde{F}^{\mathrm{Inv}}(y,s)=(F(\cdot,s)^{\mathrm{Inv}}(y),s).$

Let $x=(x^{1},\dots,x^{n})$ and $y=(y^{1},\dots,y^{n})$ be two standard
coordinate systems of $\mathbb{R}^{n}$, and let $s=(s^{1},\dots,s^{q})$ be the
standard coordinate system of $\mathbb{R}^{q}$. We let $\mathcal{I}(x,s):=x$
for $x\in\mathbb{R}^{n}$ and $s\in\mathbb{R}^{q}$.

The quantitative composition estimate Proposition 2.1.30 can be generalized to
the case with parameters.

######  Lemma 2.2.31.

Let $m,n,q\geq 1$ be integers. Let $\alpha,\beta,\gamma\in\mathbb{R}_{+}$
satisfy $\max(\alpha,\beta)>1$ and $\max(\beta,\gamma)>1$. For any $M>0$ there
is a $K_{4}(m,n,q,\alpha,\beta,\gamma,M)>0$ satisfying the following:

Suppose
$g\in\mathscr{C}^{\alpha,\gamma}(\mathbb{B}^{n},\mathbb{B}^{q};\mathbb{R}^{m})$
satisfies $g(\mathbb{B}^{n},s)\subseteq\mathbb{B}^{m}$ for all
$s\in\mathbb{B}^{q}$ and
$\|g\|_{\mathscr{C}^{\alpha,\gamma}(\mathbb{B}^{n};\mathbb{R}^{m})}\leq M$.
Then for every
$f\in\mathscr{C}^{\beta,\gamma}(\mathbb{B}^{m},\mathbb{B}^{q})$, the function
$h(x,s):=f(g(x,s),s)$ satisfies
$h\in\mathscr{C}^{\min(\alpha,\beta),\min(\beta,\gamma)}(\mathbb{B}^{n},\mathbb{B}^{q})$
with
$\|h\|_{\mathscr{C}^{\min(\alpha,\beta),\min(\beta,\gamma)}(\mathbb{B}^{n},\mathbb{B}^{q})}\leq
K_{4}\|f\|_{\mathscr{C}^{\beta,\gamma}(\mathbb{B}^{m},\mathbb{B}^{q})}$.

###### Proof.

Applying Proposition 2.1.30 (i), when
$\|g\|_{\mathscr{C}^{\alpha,\gamma}}=\sup_{s}\|g(\cdot,s)\|_{\mathscr{C}^{\alpha}}+\sup_{x}\|g(x,\cdot)\|_{\mathscr{C}^{\gamma}}<M$
we have,

$\displaystyle\|f(g(\cdot,s),s)\|_{\mathscr{C}^{\min(\alpha,\beta)}(\mathbb{B}^{n})}\leq$
$\displaystyle
K_{1}(m,n,\alpha,\beta,M)\|f(\cdot,s)\|_{\mathscr{C}^{\beta}(\mathbb{B}^{m})}$
$\displaystyle\forall s\in\mathbb{B}^{q};$
$\displaystyle\|f(g(x,\cdot),s)\|_{\mathscr{C}^{\min(\beta,\gamma)}(\mathbb{B}^{q})}\leq$
$\displaystyle
K_{1}(m,q,\gamma,\beta,M)\|f(\cdot,s)\|_{\mathscr{C}^{\beta}(\mathbb{B}^{m})}$
$\displaystyle\forall x\in\mathbb{B}^{n},s\in\mathbb{B}^{q};$
$\displaystyle\|f(g(x,s),\cdot)\|_{\mathscr{C}^{\min(\beta,\gamma)}(\mathbb{B}^{q})}\leq$
$\displaystyle\textstyle
C(q,\beta,\gamma)\sup_{y\in\mathbb{B}^{m}}\|f(y,\cdot)\|_{\mathscr{C}^{\gamma}(\mathbb{B}^{q})}$
$\displaystyle\forall x\in\mathbb{B}^{n},s\in\mathbb{B}^{q}.$

Here $C(q,\beta,\gamma)>0$ is the constant such that
$\|u\|_{\mathscr{C}^{\min(\beta,\gamma)}(\mathbb{B}^{q})}\leq
C\|u\|_{\mathscr{C}^{\gamma}(\mathbb{B}^{q})}$. Therefore by Lemma 2.2.15
(ii),

$\displaystyle\|h\|_{\mathscr{C}^{\min(\alpha,\beta),\min(\alpha,\gamma)}}\lesssim_{q,\beta}\sup_{s\in\mathbb{B}^{q}}\|h(\cdot,s)\|_{\mathscr{C}^{\min(\alpha,\beta)}}+\sup_{x\in\mathbb{B}^{n},s\in\mathbb{B}^{q}}(\|f(g(x,\cdot),s)\|_{\mathscr{C}^{\min(\beta,\gamma)}}+\|f(g(x,s),\cdot)\|_{\mathscr{C}^{\min(\beta,\gamma)}})$
$\displaystyle\lesssim$
${}_{m,n,q,\alpha,\beta,\gamma,M}\sup_{s\in\mathbb{B}^{q}}\|f(\cdot,s)\|_{\mathscr{C}^{\beta}(\mathbb{B}^{m})}+\sup_{s\in\mathbb{B}^{q}}\|f(\cdot,s)\|_{\mathscr{C}^{\beta}(\mathbb{B}^{m})}+\sup_{y\in\mathbb{B}^{m}}(\|f(y,\cdot)\|_{\mathscr{C}^{\gamma}(\mathbb{B}^{q})}\lesssim\|f\|_{\mathscr{C}^{\beta,\gamma}(\mathbb{B}^{m},\mathbb{B}^{q})}.$

By keeping track of the constant above we obtain $K_{4}$. ∎

The quantitative inverse function theorem, Proposition 2.1.44 (i), can also be
generalized to the case with parameters.

######  Proposition 2.2.32.

Let $n,q\geq 1$, $\alpha>0$ and $\beta\in(0,\alpha+1]$. Write
$\mathcal{I}:\mathbb{R}^{n}\times\mathbb{R}^{q}\to\mathbb{R}^{n}$ as
$\mathcal{I}(x,s):=x$. There is a $K_{5}=K_{5}(n,q,\alpha,\beta)>1$ that
satisfies the following:

Let
$F\in\mathscr{C}^{\alpha+1,\beta}(\mathbb{B}^{n},\mathbb{B}^{q};\mathbb{R}^{n})$
satisfies
$\|F-\mathcal{I}\|_{\mathscr{C}^{\alpha+1,\beta}(\mathbb{B}^{n},\mathbb{B}^{q};\mathbb{R}^{n})}<K_{5}^{-1}$
and $(F-\mathcal{I})|_{(\partial\mathbb{B}^{n})\times\mathbb{B}^{q}}\equiv 0$.
Then

  1. (i)

For each $s\in\mathbb{B}^{q}$, $F(\cdot,s):\mathbb{B}^{n}\to\mathbb{B}^{n}$ is
bijective.

  2. (ii)