# Sharp Hölder Regularity for Nirenberg’s Complex Frobenius Theorem

Liding Yao

###### Abstract

Nirenberg’s famous complex Frobenius theorem gives necessary and sufficient
conditions on a locally integrable structure for when the manifold is locally
diffeomorphic to $\mathbb{R}^{r}\times\mathbb{C}^{m}\times\mathbb{R}^{N-r-2m}$
through a coordinate chart $F$ in such a way that the structure is locally
spanned by $F^{*}\frac{\partial}{\partial
t^{1}},\dots,F^{*}\frac{\partial}{\partial
t^{r}},F^{*}\frac{\partial}{\partial
z^{1}},\dots,F^{*}\frac{\partial}{\partial z^{m}}$, where we have given
$\mathbb{R}^{r}\times\mathbb{C}^{m}\times\mathbb{R}^{N-r-2m}$ coordinates
$(t,z,s)$. In this paper, we give the optimal Hölder-Zygmund regularity for
the coordinate charts which achieve this realization. Namely, if the structure
has Hölder-Zygmund regularity of order $\alpha>1$, then the coordinate chart
$F$ that maps to $\mathbb{R}^{r}\times\mathbb{C}^{m}\times\mathbb{R}^{N-r-2m}$
may be taken to have Hölder-Zygmund regularity of order $\alpha$, and this is
sharp. Furthermore, we can choose this $F$ in such a way that the vector
fields $F^{*}\frac{\partial}{\partial
t^{1}},\dots,F^{*}\frac{\partial}{\partial
t^{r}},F^{*}\frac{\partial}{\partial
z^{1}},\dots,F^{*}\frac{\partial}{\partial z^{m}}$ on the original manifold
have Hölder-Zygmund regularity of order $\alpha-\varepsilon$ for every
$\varepsilon>0$, and we give an example to show that the regularity for
$F^{*}\frac{\partial}{\partial z}$ is optimal.

Keywords — Complex Frobenius structures, Newlander-Nirenberg theorem, Bi-
parameter Hölder spaces, Malgrange’s factorizations.

###### Contents

  1. 1 Introduction
    1. 1.1 A first glimpse to the main results
    2. 1.2 Historical remarks
    3. 1.3 Idea of the proof
    4. 1.4 Notations and organizations
  2. 2 The Bi-parameter Hölder-Zygmund Structures
    1. 2.1 Hölder-Zygmund spaces of single and bi-parameters
    2. 2.2 A proposition for bi-parameter products
    3. 2.3 Compositions and inverse mapping with parameter
  3. 3 The Vector Fields and Ordinary Differential Equations
    1. 3.1 Review of tangent subbundles and involutivity
    2. 3.2 Review of autonomous ODE flows
    3. 3.3 The bi-layer Frobenius theorem and a holomorphic analogy
  4. 4 Some Elliptic PDEs
    1. 4.1 Existence of a particular PDE
    2. 4.2 Real-analyticity of a particular PDE
  5. 5 Malgrange’s Factorization with Parameters
    1. 5.1 Overview of the key estimate
    2. 5.2 Some transition lemmas
    3. 5.3 Proof of Theorem 5.1
  6. 6 The Rough Complex Frobenius Theorem: Statements and Proofs
  7. 7 The Sharpness of Coordinates and Coordinate Vector Fields
    1. 7.1 Sharpness of $F^{\prime\prime}$, $F^{\prime\prime\prime}$ and $F^{*}\frac{\partial}{\partial s}$: some elementary examples
    2. 7.2 Sharpness of $F^{*}\frac{\partial}{\partial z}$: a $\bar{\partial}$-equation with parameter
  8. A Some Results with non $C^{1}$ Parameters
  9. B Holomorphic Extension of Laplacian
    1. B.1 Holomorphic Hölder spaces and extension of harmonic functions
    2. B.2 Construction and Hölder regularity for real inverse Laplacian $\mathbf{P}$
    3. B.3 Construction of the holomorphic extension $\tilde{\mathbf{P}}$
    4. B.4 Hölder-Zygmund regularities for $\tilde{\mathbf{P}}$

##  1 Introduction

For $\alpha\in(0,\infty)$, let $\mathscr{C}^{\alpha}$ be the class of Hölder-
Zygmund functions111For non-integer exponents, the Hölder-Zygmund space agrees
with the classical Hölder space. When $\alpha\in\mathbb{Z}_{+}$ we have
$C^{\alpha-1,1}\subsetneq\mathscr{C}^{\alpha}$. See Remark 2.3. of order
$\alpha$ (See Definitions 1.8). Let
$\mathscr{C}^{\alpha-}=\bigcap_{\varepsilon>0}\mathscr{C}^{\alpha-\varepsilon}$.
And we denote $\mathscr{C}^{\infty}=C^{\infty}$ as the class of smooth
functions.

Given a smooth manifold $\mathfrak{M}$, a $\mathscr{C}^{\alpha}$-complex
Frobenius structure on $\mathfrak{M}$ is a $\mathscr{C}^{\alpha}$-complex
tangent subbundle222For (real or complex) linear spaces $V,W$, we use $V\leq
W$ as saying that $V$ is a (real or complex) subspace of $W$. And we use
$\mathcal{S}\leq\mathbb{C}T\mathfrak{M}$ as saying $\mathcal{S}$ is a complex
subbundle of $\mathbb{C}T\mathfrak{M}$.
$\mathcal{S}\leq\mathbb{C}T\mathfrak{M}$, such that
$\mathcal{S}+\bar{\mathcal{S}}$ is also a complex tangential subbundle, and
both $\mathcal{S}$, $\mathcal{S}+\bar{\mathcal{S}}$ satisfy the involutivity
condition, namely

$\forall X,Y\in
C^{1}_{\mathrm{loc}}(\mathfrak{M};\mathcal{S})\Rightarrow[X,Y]\in
C^{0}_{\mathrm{loc}}(\mathfrak{M};\mathcal{S}),\quad\text{and }\forall X,Y\in
C^{1}_{\mathrm{loc}}(\mathfrak{M};\mathcal{S}+\bar{\mathcal{S}})\Rightarrow[X,Y]\in
C^{0}_{\mathrm{loc}}(\mathfrak{M};\mathcal{S}+\bar{\mathcal{S}}).$

See Definition 3.10.

The complex Frobenius theorem was first introduced by Louis Nirenberg. It is a
generalization of the real Frobenius theorem and the Newlander-Nirenberg
theorem.

######  Proposition 1.0 (L. Nirenberg [Nir57]).

Let $r,m,N$ be nonnegative integers satisfying $r+2m\leq N$. Let
$\mathfrak{M}$ be an $N$-dimensional smooth manifold and let $\mathcal{S}$ be
a smooth complex Frobenius structure on $\mathfrak{M}$, such that
$\operatorname{rank}(\mathcal{S}\cap\bar{\mathcal{S}})=r$ and
$\operatorname{rank}\mathcal{S}=r+m$. Then for any $p\in\mathfrak{M}$ there is
a smooth coordinate system
$F=(F^{\prime},F^{\prime\prime},F^{\prime\prime\prime}):U\subseteq\mathfrak{M}\to\mathbb{R}^{r}_{t}\times\mathbb{C}^{m}_{z}\times\mathbb{R}^{N-r-2m}_{s}$
near $p$, such that $\mathcal{S}|_{U}$ is spanned by the smooth vector fields
$F^{*}\frac{\partial}{\partial t^{1}},\dots,F^{*}\frac{\partial}{\partial
t^{r}},F^{*}\frac{\partial}{\partial
z^{1}},\dots,F^{*}\frac{\partial}{\partial z^{m}}$.

Here $(t,z,s)=(t^{1},\dots,t^{r},z^{1},\dots,z^{m},s^{1},\dots,s^{N-r-2m})$ is
the standard coordinate system for
$\mathbb{R}^{r}\times\mathbb{C}^{m}\times\mathbb{R}^{N-r-2m}$. All the ranks
and spans are in the sense of complex subspaces, not as real subspaces.

###  1.1 A first glimpse to the main results

In this paper we find such a coordinate chart $F$ with optimal Hölder-Zygmund
regularity, when the complex Frobenius structure $\mathcal{S}$ has low
smoothness. To obtain the best results, we compute the regularities
$F^{\prime}$, $F^{\prime\prime}$, $F^{\prime\prime\prime}$ separately, instead
of merely giving a single regularity result for
$F=(F^{\prime},F^{\prime\prime},F^{\prime\prime\prime})$.

We say a complex tangent subbundle $\mathcal{S}\leq\mathbb{C}T\mathfrak{M}$ is
$\mathscr{C}^{\alpha}$, if locally it is spanned by
$\mathscr{C}^{\alpha}$-vector fields (Definitions 3.5).

######  Theorem 1.1.

Let $\alpha\in(1,\infty)$, let $\mathfrak{M}$ be an $N$-dimensional smooth
manifold, and let $\mathcal{S}$ be a $\mathscr{C}^{\alpha}$-complex Frobenius
structure such that $\operatorname{rank}(\mathcal{S}\cap\bar{\mathcal{S}})=r$
and $\operatorname{rank}\mathcal{S}=r+m$.

Then for any $p\in\mathfrak{M}$ there is a $\mathscr{C}^{\alpha}$-coordinate
system
$F=(F^{\prime},F^{\prime\prime},F^{\prime\prime\prime}):U\subseteq\mathfrak{M}\to\mathbb{R}^{r}_{t}\times\mathbb{C}^{m}_{z}\times\mathbb{R}^{N-r-2m}_{s}$
near $p$, such that $\mathcal{S}|_{U}$ is spanned by
$F^{*}\frac{\partial}{\partial t^{1}},\dots,F^{*}\frac{\partial}{\partial
t^{r}},F^{*}\frac{\partial}{\partial
z^{1}},\dots,F^{*}\frac{\partial}{\partial z^{m}}$.

In fact $F^{\prime}\in\mathscr{C}^{\infty}$,
$F^{\prime\prime}\in\mathscr{C}^{\alpha}$,
$F^{\prime\prime\prime}\in\mathscr{C}^{\alpha}$. Moreover
$F^{*}\frac{\partial}{\partial t^{1}},\dots,F^{*}\frac{\partial}{\partial
t^{r}},F^{*}\frac{\partial}{\partial
z^{1}},\dots,F^{*}\frac{\partial}{\partial z^{m}}$ are
$\mathscr{C}^{\alpha-}$-complex vector fields in $U$.

In this theorem $F^{\prime}\in\mathscr{C}^{\infty}$,
$F^{\prime\prime}\in\mathscr{C}^{\alpha}$ and $F^{*}\frac{\partial}{\partial
z}\in\mathscr{C}^{\alpha-}$ are all optimal, in the sense that given
$\alpha>1$, we have examples of complex Frobenius structures $\mathcal{S}$,
such that if $F$ is a $C^{1}$-chart representing $\mathcal{S}$, then
$F^{\prime\prime}\notin\mathscr{C}^{\alpha+\varepsilon}$ for any
$\varepsilon>0$,
$F^{\prime\prime\prime}\notin\mathscr{C}^{\alpha+\varepsilon}$ for any
$\varepsilon>0$, and $F^{*}\frac{\partial}{\partial
z}\notin\mathscr{C}^{\alpha}$, respectively. See Section 7. As a corollary,
the regularity statement of our coordinate chart $F\in\mathscr{C}^{\alpha}$ is
optimal.

The subbundle $\mathcal{S}+\bar{\mathcal{S}}$ and
$\mathcal{S}\cap\bar{\mathcal{S}}$ (if they have constant ranks, see Remark
3.13 (ii)) are always at least as regular as the subbundle $\mathcal{S}$.
However, it is possible that $\mathcal{S}+\bar{\mathcal{S}}$ and
$\mathcal{S}\cap\bar{\mathcal{S}}$ have better regularity than $\mathcal{S}$.
See Remark 3.13 (iii). When $\mathcal{S}\cap\bar{\mathcal{S}}$ is more regular
than $\mathcal{S}$, we have an improvement of Theorem 1.1:

######  Theorem 1.2.

Let $\alpha,\beta\in(1,\infty)$ satisfy $\alpha+1\leq\beta$ and let
$\mathfrak{M}$ be an $N$-dimensional smooth manifold. Let $\mathcal{S}$ be a
$\mathscr{C}^{\alpha}$-complex Frobenius structure, such that
$\operatorname{rank}\mathcal{S}=r+m$ and $\mathcal{S}\cap\bar{\mathcal{S}}$ is
a $\mathscr{C}^{\beta}$-complex tangential subbundle with complex rank $r$.

Then for any $p\in\mathfrak{M}$ there is a $\mathscr{C}^{\alpha}$-coordinate
system
$F:U\subseteq\mathfrak{M}\to\mathbb{R}^{r}\times\mathbb{C}^{m}\times\mathbb{R}^{N-r-2m}$
near $p$, such that in addition to the consequences in Theorem 1.1, we have
that $F^{*}\frac{\partial}{\partial t^{1}},\dots,F^{*}\frac{\partial}{\partial
t^{r}}$ are $\mathscr{C}^{\beta}$-vector fields on $U$.

There is another characterization for representing the complex Frobenius
structure using the inverse of coordinate charts, the so-called “regular
parameterization”. Let333We use $f^{\mathrm{Inv}}$ instead of $f^{-1}$ to
refer the inverse map of $f$, if it is bijective.
$\Phi=F^{\mathrm{Inv}}:F(U)\subseteq\mathbb{R}^{N}\to\mathfrak{M}$ be a such
parameterization. For $F$ we work with the regularities of $F^{\prime}$,
$F^{\prime\prime}$ and $F^{\prime\prime\prime}$, while for $\Phi$ we work with
that of $[t\mapsto\Phi(t,z_{0},s_{0})]$, $[z\mapsto\Phi(t_{0},z,s_{0})]$ and
$[s\mapsto\Phi(t_{0},z_{0},s)]$, for any fixed $t_{0},z_{0},s_{0}$.

For a general version of Theorems 1.1 and 1.2, which includes the estimates
for $\Phi$, see Theorems 6.1 and 6.2.

To see that $F^{*}\frac{\partial}{\partial z}\in\mathscr{C}^{\alpha-}$ is
sharp; in Section 7.2 we construct an example that meets the following:

######  Theorem 1.3.

Let $\alpha>1$. There is a $\mathscr{C}^{\alpha}$-complex Frobenius structure
$\mathcal{S}$ on $\mathbb{R}^{3}$, such that
$\operatorname{rank}\mathcal{S}=1$,
$\operatorname{rank}(\mathcal{S}+\bar{\mathcal{S}})=2$, and there does not
exist a $C^{1}$-coordinate chart
$F:U\subseteq\mathbb{R}^{3}\to\mathbb{C}^{1}_{z}\times\mathbb{R}^{1}_{s}$ near
$0\in\mathbb{R}^{3}$ such that $F^{*}\frac{\partial}{\partial z}$ spans
$\mathcal{S}|_{U}$ and $F^{*}\frac{\partial}{\partial
z}\in\mathscr{C}^{\alpha}(U;\mathbb{C}^{3})$.

See Theorem 7.12 for a general version.

###  1.2 Historical remarks

There are many previous discussions of complex Frobenius theorem in the
nonsmooth setting. Nijenhuis & Woolf [NW63] studied Newlander-Nirenberg
theorem with nonsmooth parameters, which is a special case of nonsmooth
complex Frobenius theorem.

Later Hill & Taylor [HT07] studied nonsmooth complex Frobenius structures.
They assumed the complex Frobenius structure to be at least $C^{1,1}$ in order
to achieve the condition called “Hypothesis V” in their paper.

Recently, Gong [Gon20] proved a better estimate for complex Frobenius
structure. He showed that for a non-integer $\alpha>1$ and a
$\mathscr{C}^{\alpha}$-complex Frobenius structure
$\mathcal{S}\leq\mathbb{C}T\mathfrak{M}$, locally one can find a
$\mathscr{C}^{\alpha-}$-coordinate chart
$F:U\subseteq\mathfrak{M}\to\mathbb{R}^{r}\times\mathbb{C}^{m}\times\mathbb{R}^{N-r-2m}$
such that $F^{*}\frac{\partial}{\partial
t^{1}},\dots,F^{*}\frac{\partial}{\partial
t^{r}},F^{*}\frac{\partial}{\partial
z^{1}},\dots,F^{*}\frac{\partial}{\partial z^{m}}$ span $\mathcal{S}$. Though
he did not have the sharp result, the statement $F\in\mathscr{C}^{\alpha-}$ is
endpoint sharp (up to a loss of $\varepsilon>0$ derivatives). The proof
technique in his paper is quite different for ours. He used homotopy formulae
with Nash-Moser iterations.

Our result is stronger because we have not only the optimal regularity of the
coordinate chart that $F\in\mathscr{C}^{\alpha}$, but also the optimal
regularity for other ingredients like the components of the coordinates and
the coordinate vector fields. See Section 7.

As the origin of the complex Frobenius theorem, the sharp estimate to
Newlander-Nirenberg is due to Malgrange [Mal69]. He proved that for $\alpha>1$
one can always obtain a $\mathscr{C}^{\alpha+1}$ local complex coordinate
system for a $\mathscr{C}^{\alpha}$ integrable almost complex structure. His
argument also works for $\frac{1}{2}<\alpha\leq 1$, see [HT03]. We remark that
we cannot replace $(\mathscr{C}^{\alpha},\mathscr{C}^{\alpha+1})$ by
$(C^{k},C^{k+1})$, see [Yao21].

Later Street used Malgrange’s method and obtained a sharp estimate for
elliptic structure.

######  Proposition 1.4 ([Str20], Sharp estimate for elliptic structures).

Let $\alpha>1$, let $\mathfrak{M}$ be a $\mathscr{C}^{\alpha+1}$-manifold, and
let $r,m$ be integers satisfying $\dim\mathfrak{M}=r+2m$. Endow
$\mathbb{R}^{r}\times\mathbb{C}^{m}$ with standard coordinate system
$(t,z)=(t^{1},\dots,t^{r},z^{1},\dots,z^{m})$.

Let $\mathcal{S}\leq\mathbb{C}T\mathfrak{M}$ be a
$\mathscr{C}^{\alpha}$-elliptic structure of rank $r+m$. Then for any
$p\in\mathfrak{M}$ there exists a $\mathscr{C}^{\alpha+1}$-coordinate system
$F:U\subseteq\mathfrak{M}\to\mathbb{R}^{r}\times\mathbb{C}^{m}$ near $p$, such
that $\mathcal{S}|_{U}$ is spanned by $F^{*}\frac{\partial}{\partial
t^{1}},\dots,F^{*}\frac{\partial}{\partial
t^{r}},F^{*}\frac{\partial}{\partial
z^{1}},\dots,F^{*}\frac{\partial}{\partial z^{m}}$.

Note that in Proposition 1.4 we have $F^{*}\frac{\partial}{\partial
t^{1}},\dots,F^{*}\frac{\partial}{\partial
t^{r}},F^{*}\frac{\partial}{\partial
z^{1}},\dots,F^{*}\frac{\partial}{\partial z^{m}}\in\mathscr{C}^{\alpha}$
because $F\in\mathscr{C}^{\alpha+1}$. However in a more general setting
(Theorem 1.1), due to the influence of the parameters, we only have
$F^{*}\frac{\partial}{\partial t^{1}},\dots,F^{*}\frac{\partial}{\partial
t^{r}}$, $F^{*}\frac{\partial}{\partial
z^{1}},\dots,F^{*}\frac{\partial}{\partial z^{m}}\in\mathscr{C}^{\alpha-}$.
Both results for $F$ and $F^{*}\frac{\partial}{\partial
z^{1}},\dots,F^{*}\frac{\partial}{\partial z^{m}}$ are sharp.

###  1.3 Idea of the proof

Let $\mathfrak{M}$ be a (smooth) manifold. In this section we use the
following terminologies:

######  Definition 1.5.

We call a $C^{1}$ complex tangent subbundle
$\mathcal{S}\subseteq\mathbb{C}T\mathfrak{M}$ is involutive, if for every
$C^{1}$-vector fields $X,Y$ that are sections of $\mathcal{S}$, the Lie
bracket $[X,Y]$ is a $C^{0}$-section of $\mathcal{S}$. (See also Definition
3.7.)

  * •

An elliptic structure on $\mathfrak{M}$ is an involutive complex subbundle
$\mathcal{S}\subseteq\mathbb{C}T\mathfrak{M}$ such that
$\mathbb{C}T\mathfrak{M}=\mathcal{S}+\bar{\mathcal{S}}$.

  * •

A complex structure444In this paper we use “complex structure” to refer the
almost complex structure satisfying involutivity condition. See also [BCH08,
Section I.8]. on $\mathfrak{M}$ is an involutive complex subbundle
$\mathcal{S}\subseteq\mathbb{C}T\mathfrak{M}$ such that
$\mathbb{C}T\mathfrak{M}=\mathcal{S}\oplus\bar{\mathcal{S}}$.

  * •

An essentially real structure is a complex subbundle
$\mathcal{S}\subseteq\mathbb{C}T\mathfrak{M}$ such that
$\mathcal{S}=\bar{\mathcal{S}}$.

Here $\bar{\mathcal{S}}=\\{(p,u-iv):p\in\mathfrak{M},\ u,v\in
T_{p}\mathfrak{M},\ u+iv\in\mathcal{S}_{p}\\}$ is the complex conjugate
subbundle of $\mathcal{S}$.

To prove Theorem 1.1, notice that $\mathcal{S}+\bar{\mathcal{S}}$ is an
essentially real structure. By applying the real Frobenius theorem, we get a
foliation of $\mathscr{C}^{\alpha+1}$-manifolds, such that by restricted to
each leaf $\mathcal{S}$ becomes an elliptic structure. The theorem is proved
once we have the estimate of the coordinate system and the coordinate vector
fields for elliptic structures with parameters:

######  Theorem 1.6.

Let $\mathfrak{M}$ and $\mathfrak{N}$ be two smooth manifolds such that
$\dim\mathfrak{M}=r+2m$ and $\dim\mathfrak{N}=q$. Let
$\mathcal{S}\leq(\mathbb{C}T\mathfrak{M})\times\mathfrak{N}$ be a rank-$(r+m)$
$\mathscr{C}^{\alpha}$-involutive subbundle such that
$\mathcal{S}+\bar{\mathcal{S}}=(\mathbb{C}T\mathfrak{M})\times\mathfrak{N}$.
Then for any $(p_{0},q_{0})\in\mathfrak{M}\times\mathfrak{N}$ there is a
$\mathscr{C}^{\alpha}$-coordinate chart $F:U\times
V\subseteq\mathfrak{M}\times\mathfrak{N}\to\mathbb{R}^{r}_{t}\times\mathbb{C}^{m}_{z}\times\mathbb{R}^{q}_{s}$
such that $\mathcal{S}|_{U\times V}$ is spanned by
$\mathscr{C}^{\alpha-}$-vector fields $F^{*}\frac{\partial}{\partial
t^{1}},\dots,F^{*}\frac{\partial}{\partial
t^{r}},F^{*}\frac{\partial}{\partial
z^{1}},\dots,F^{*}\frac{\partial}{\partial z^{m}}$.

See Theorem 5.1 for a more general version, where we consider the case of
mixed regularity $\mathcal{S}\in\mathscr{C}^{\alpha,\beta}$ with
$0<\beta\leq\alpha+1$ (see Definitions 2.10, 2.15, 3.2, 3.6 and 3.7). Theorem
5.1 implies Proposition 1.4 which can be viewed as the “parameter free” case.
See also Remarks 5.2, 5.3 and 5.4.

The idea is based on Malgrange’s factorization method for the sharp Newlander-
Nirenberg theorem, whose sketch can be found in, for example [BCH08, Page 47]
and [HT03, Section 3]. Malgrange’s idea is to find an intermediate coordinate
change $H$ such that the generators for $H_{*}\mathcal{S}$ satisfy an elliptic
PDE system. Using the PDE we can show that $H_{*}\mathcal{S}$ is a
$\mathscr{C}^{\alpha}$-family of real-analytic elliptic structures. Finally we
construct another coordinate change $G$ using holomorphic Frobenius theorem,
now with parameter, so that $G_{*}H_{*}\mathcal{S}$ is the span of
$\frac{\partial}{\partial t^{1}},\dots,\frac{\partial}{\partial
t^{r}},\frac{\partial}{\partial z^{1}},\dots,\frac{\partial}{\partial z^{m}}$.
Thus $F=G\circ H$ is the desired coordinate chart.

See Section 5.1 for a detailed sketch to the proof of Theorem 1.6.

Note that in Theorem 1.6 we work on the elliptic structures rather than the
complex structures. In this way we do not have to pick generators for
$\mathcal{S}$ that satisfy the “Hypothesis V” in [HT07].

For the proof of Theorem 1.2, we follow Nirenberg’s original proof by applying
real Frobenius theorem twice on the essentially real structures
$\mathcal{S}\cap\bar{\mathcal{S}}$ and $\mathcal{S}+\bar{\mathcal{S}}$, see
[BCH08, the proof of Theorem I.12.1]. This ends up with a family of complex
structures. We complete the proof by applying Theorem 1.6 with the special
case $r=0$.

To get the optimal Hölder-Zygmund regularity during this reduction, we combine
the two real Frobenius theorems into one step. We call it the bi-layer
Frobenius theorem.

######  Theorem 1.7.

Let $\mathfrak{M}$ be a smooth manifold with dimension $(r+m+q)$. Let
$\alpha,\beta>1$ and let $\mathcal{V}\leq\mathcal{E}\leq T\mathfrak{M}$ be two
involutive real subbundles such that $\mathcal{V}\in\mathscr{C}^{\alpha}$,
$\mathcal{E}\in\mathscr{C}^{\beta}$, $\operatorname{rank}\mathcal{V}=r$ and
$\operatorname{rank}\mathcal{E}=r+m$.

Then for any $p\in\mathfrak{M}$ there is a
$\mathscr{C}^{\min(\alpha,\beta)}$-coordinate system
$F=(F^{\prime},F^{\prime\prime},F^{\prime\prime\prime}):U\subseteq\mathfrak{M}\to\mathbb{R}^{r}_{t}\times\mathbb{R}^{m}_{x}\times\mathbb{R}^{q}_{s}$
near $p$, such that $\mathcal{V}|_{U}$ is spanned by
$F^{*}\frac{\partial}{\partial t^{1}},\dots,F^{*}\frac{\partial}{\partial
t^{r}}$ and $\mathcal{E}|_{U}$ is spanned by $F^{*}\frac{\partial}{\partial
t^{1}},\dots,F^{*}\frac{\partial}{\partial
t^{r}},F^{*}\frac{\partial}{\partial
x^{1}},\dots,F^{*}\frac{\partial}{\partial x^{m}}$.

Moreover for their regularities we have $F^{\prime}\in\mathscr{C}^{\infty}$,
$F^{\prime\prime}\in\mathscr{C}^{\alpha}$,
$F^{\prime\prime\prime}\in\mathscr{C}^{\beta}$, $F^{*}\frac{\partial}{\partial
t^{1}},\dots,F^{*}\frac{\partial}{\partial t^{r}}\in\mathscr{C}^{\alpha}$ and
$F^{*}\frac{\partial}{\partial x^{1}},\dots,F^{*}\frac{\partial}{\partial
x^{m}}\in\mathscr{C}^{\min(\alpha-1,\beta)}$.

See Theorem 3.27 for a more general version. In Theorem 1.2 we take
$\alpha+1\geq\beta$, $\mathcal{V}=\mathcal{S}\cap T\mathfrak{M}$ and
$\mathcal{E}=(\mathcal{S}+\bar{\mathcal{S}})\cap T\mathfrak{M}$.

Unlike the sharp Newlander-Nirenberg theorem or Street’s result where
$F^{*}\frac{\partial}{\partial z^{1}},\dots,F^{*}\frac{\partial}{\partial
z^{m}}$ are $\mathscr{C}^{\alpha}$-vector fields, in our result we show that
$F^{*}\frac{\partial}{\partial z^{j}}\in\mathscr{C}^{\alpha-}$ is optimal.

One reason for the regularity loss comes from solving elliptic pdes. Say we
have a $\mathscr{C}^{\alpha}$-subbundle for some $\alpha>0$. In the first step
of Malgrange’s proof of the sharp Newlander-Nirenberg theorem we need to solve
an elliptic pde system for $f=f(x)$ that has the following form:

$\Delta f=\phi(f,\nabla f,\nabla^{2}f,a,\nabla a).$ (1.1)

Here $a=a(x)\in\mathscr{C}^{\alpha}$ is the vector-valued coefficient that
comes from the subbundle, and $\phi$ is a rational function which is defined
and small when $f$ and $a$ are both small.

Since Laplace’s equation gains two derivatives in Hölder-Zygmund spaces, we
can find a $f\in\mathscr{C}^{\alpha+1}$ solving (1.1). This is why we can find
a $\mathscr{C}^{\alpha+1}$-coordinate system for a
$\mathscr{C}^{\alpha}$-integrable almost complex structure. In particular the
coordinate vector fields are $\mathscr{C}^{\alpha}$ because $\nabla
f\in\mathscr{C}^{\alpha}$.

When we do estimate on the complex Frobenius structure, we need to solve a pde
system that has similar expression to (1.1) but with a parameter: we need
$f=f(x,s)$ for the system $\Delta_{x}f=\phi(f,\nabla_{x}f,a,\nabla_{x}a)$ with
$\mathscr{C}^{\alpha}$-coefficient $a=a(x,s)$.

We need to introduce some suitable function spaces that capture the property
that the pde system “gains 2 derivatives on $x$-variable”. Note that by
comparing to $a(x,s)$, the function $\phi(f,\nabla_{x}f,a,\nabla_{x}a)$ lose
one derivative in $x$, and by taking inverse Laplacian, $f$ should gain 1
derivative in $x$. Thus we can ask what is the good function space
$\mathscr{X}=\mathscr{X}_{x,s}$ to work on, such that if
$a\in\mathscr{X}_{x,s}$ then $\nabla_{x}f\in\mathscr{X}_{x,s}$.

Unfortunately the space $\mathscr{C}^{\alpha}_{x,s}$ does not satisfy our
need. When $a\in\mathscr{C}^{\alpha}_{x,s}$ we may not find a solution $f$
such that $\nabla f\in\mathscr{C}^{\alpha}_{x,s}$. This is roughly because the
$x$-variable Riesz transform (say $x\in\mathbb{R}^{n}$ and $s\in
V\subseteq\mathbb{R}^{m}$)

$R_{j}f(x,s):=\frac{\Gamma(\frac{n+1}{2})}{\pi^{\frac{n+1}{2}}}\lim\limits_{\varepsilon\to
0}\int_{\mathbb{R}^{n}\backslash
B^{n}(0,\varepsilon)}\frac{y_{j}-x_{j}}{|y-x|^{n+1}}f(y,s)dy,\quad
j=1,\dots,n,$

does not map $\mathscr{C}^{\alpha}(\mathbb{R}^{n}\times V)$ into itself. That
is why $F^{*}\frac{\partial}{\partial z^{j}}\in\mathscr{C}^{\alpha-}$ is the
best one we can get.

To see that why
$R_{j}:\mathscr{C}^{\alpha}_{x,s}\not\to\mathscr{C}^{\alpha}_{x,s}$ holds, we
consider the bi-parameter decomposition
$\mathscr{C}^{\alpha}_{x,s}=\mathscr{C}^{\alpha}_{x}L^{\infty}_{s}\cap
L^{\infty}_{x}\mathscr{C}^{\alpha}_{s}$ (see Lemma 2.12): we have
$\|f\|_{\mathscr{C}^{\alpha}_{x,s}}\approx\sup_{s}\|f(\cdot,s)\|_{\mathscr{C}^{\alpha}_{x}}+\sup_{x}\|f(x,\cdot)\|_{\mathscr{C}^{\alpha}_{s}}$.
We see that
$R_{j}:\mathscr{C}^{\alpha}_{x}L^{\infty}_{s}\to\mathscr{C}^{\alpha}_{x}L^{\infty}_{s}$
is still bounded because Riesz transforms are bounded in
$\mathscr{C}^{\alpha}$-spaces. However
$R_{j}:L^{\infty}_{x}\mathscr{C}^{\alpha}_{s}\not\to
L^{\infty}_{x}\mathscr{C}^{\alpha}_{s}$ is not bounded because Riesz
transforms are unbounded on $L^{\infty}$-spaces.

To construct the example of complex Frobenius structure such that
$F^{*}\frac{\partial}{\partial z}\notin\mathscr{C}^{\alpha}$, we follow a
similar but more sophisticated construction to [Yao21]. See Propositions 7.3
and 7.9 in Section 7.2.

###  1.4 Notations and organizations

We use distributions for generalized functions, rather than tangential
subbundles on manifolds.

Given a bijection $f:U\to V$, we use $f^{\mathrm{Inv}}:V\to U$ as the inverse
function of $f$. We do not use the notion $f^{-1}$ in order to reduce
confusion when $(\cdot)^{-1}$ is used for inverting a matrix.

We use $A\Subset B$ to denote that $A$ is a relatively compact subset of $B$.

By $A\lesssim B$ we mean that $A\leq CB$ where $C$ is a constant independent
of $A,B$. We use $A\approx B$ for “$A\lesssim B$ and $B\lesssim A$”. And we
use $A\lesssim_{x}B$ to emphasize the constant dependence on quantity $x$.

We use $\mathbb{B}^{n}=B^{n}(0,1)$ as the unit ball in $\mathbb{R}^{n}$ and
$r\mathbb{B}^{n}=B^{n}(0,r)$ for $r>0$. We use a complex cone
$\mathbb{H}^{n}:=\\{x+iy\in\mathbb{C}^{n}:x\in\mathbb{B}^{n},4|y|<1-|x|\\}\subseteq\mathbb{C}^{n}$.

For functions $f:U\to\mathbb{R}$, $g:V\to\mathbb{R}$, we use the tensor
notation $f\otimes g:U\times V\to\mathbb{R}$ as $(f\otimes g)(x,y):=f(x)g(y)$.

For the index of Hölder regularity, we use the convention
$\infty=\infty+1=\infty-1=\infty+=\infty-=\frac{\infty}{2}$.

We denote by $\mathscr{C}^{\alpha}$ the class of Hölder-Zygmund functions:

######  Definition 1.8.

Let $\alpha\in(0,\infty]$, let $\Omega\subset\mathbb{R}^{n}$ be a convex open
subset and let $X=\mathbb{R}^{m}$ or $\mathbb{C}^{m}$. The space
$\mathscr{C}^{\alpha}(\Omega;X)$ of bounded ($X$ vector-valued) Hölder-Zygmund
functions on $\Omega$, is given recursively as follows:

  1. (i)

For $0<\alpha<1$: $\mathscr{C}^{\alpha}(\Omega;X)$ consists of those
continuous functions $f:\Omega\to X$ such that:

$\textstyle\|f\|_{\mathscr{C}^{\alpha}(\Omega;X)}:=\sup_{x\in\Omega}|f(x)|_{X}+\sup_{\begin{subarray}{c}x,y\in\Omega;x\neq
y\end{subarray}}|x-y|^{-\alpha}|f(x)-f(y)|_{X}<\infty.$

  2. (ii)

For $\alpha=1$: $\mathscr{C}^{1}(\Omega;X)$ consists of those continuous
functions $f:\Omega\to X$ such that:

$\textstyle\|f\|_{\mathscr{C}^{1}(\Omega;X)}:=\sup_{x\in\Omega}|f(x)|_{X}+\sup_{\begin{subarray}{c}x,y\in\Omega;x\neq
y\end{subarray}}|x-y|^{-1}|\frac{f(x)+f(y)}{2}-f(\frac{x+y}{2})|_{X}<\infty.$

  3. (iii)

For $\alpha>1$: $\mathscr{C}^{\alpha}(\Omega;X)$ consists of
$f\in\mathscr{C}^{\alpha-1}(\Omega;X)$ such that the following norm is finite:

$\textstyle\|f\|_{\mathscr{C}^{\alpha}(\Omega;X)}:=\|f\|_{\mathscr{C}^{\alpha-1}(\Omega;X)}+\sum_{j=1}^{n}\|\partial_{j}f\|_{\mathscr{C}^{\alpha-1}(\Omega;X)}.$

  4. (iv)

We use
$\mathscr{C}^{\infty}(\Omega;X):=\bigcap_{\beta>0}\mathscr{C}^{\beta}(\Omega;X)$
and
$\mathscr{C}^{\alpha-}(\Omega;X):=\bigcap_{\beta<\alpha}\mathscr{C}^{\beta}(\Omega;X)$,
both endowed with the standard limit Fréchet topologies. That is, we say
$f_{k}\to f_{0}$ in $\mathscr{C}^{\alpha-}$ if $f_{k}\to f_{0}$ in
$\mathscr{C}^{\beta}$ for all $\beta<\alpha$.

  5. (v)

We use
$\mathscr{C}^{\alpha+}(\Omega):=\bigcup_{\beta>\alpha}\mathscr{C}^{\beta}(\Omega)$,
whose topology is used in the paper.

We use $\mathscr{C}^{\beta}(\Omega)=\mathscr{C}^{\beta}(\Omega;\mathbb{R})$
for $\beta\in\\{\alpha,\alpha-,\alpha+:\alpha>0\\}\cup\\{\infty\\}$.

We use $\|f\|_{\mathscr{C}^{\alpha}(\Omega)}$ either when
$f:\Omega\to\mathbb{R}$ is a function or when the codomain of $f$ is clear.
And we use $\|f\|_{\mathscr{C}^{\alpha}}$ when both the domain and the
codomain are clear.

We define $\mathscr{C}^{\alpha}(\Omega;X)$ for $\alpha\leq 0$ in Definition
2.4 using the Besov space $\mathscr{B}_{\infty\infty}^{\alpha}$.

Let $U\subseteq\mathbb{R}^{m}_{x}$ and $V\subseteq\mathbb{R}^{n}_{s}$ be two
convex open subsets. For $\alpha,\beta\in(0,\infty]$, we denote by
$\mathscr{C}^{\alpha,\beta}(U,V)$ the space of functions on $U\times V$ that
is bounded $\mathscr{C}^{\alpha}$ in $x$ and bounded $\mathscr{C}^{\beta}$ in
$s$ separately, see Definition 2.10. For $\alpha,\beta\in(-\infty,\infty]$ we
denote by $\mathscr{C}^{\alpha}\mathscr{C}^{\beta}(U,V)$ the space of
functions on $U\times V$ that is bounded $\mathscr{C}^{\alpha}$ in $x$ and
bounded $\mathscr{C}^{\beta}$ in $s$ simultaneously, see Definition 2.6. In
most of the cases we work on the spaces like
$\mathscr{C}^{\alpha}L^{\infty}\cap\mathscr{C}^{\beta}\mathscr{C}^{\gamma}(U,V)$.

Given a coordinate chart $x=(x^{1},\dots,x^{n})$, we write
$dx:=[dx^{1},\dots,dx^{n}]$ as a row vector and $\frac{\partial}{\partial
x}:=\begin{bmatrix}\partial_{x^{1}}\\\ \vdots\\\
\partial_{x^{n}}\end{bmatrix}$ as a column vector. We may use $X$ sometimes as
a single vector field, and sometimes as a (column) collection of vector fields
$X=[X_{1},\dots,X_{r}]^{\top}$. We use these conventions when there is no
ambiguity.

In a mixed real and complex domain
$\Omega\subseteq\mathbb{R}^{r}\times\mathbb{C}^{m}$ with standard (real and
complex) coordinate system $(t,z)=(t^{1},\dots,t^{r},z^{1},\dots,z^{m})$, for
a function $f$ on $\Omega$, we use $\nabla f,\nabla_{z}f,\partial_{z}f$ as
functions taking values in column vectors, which have rows $r+2m$, $2m$ and
$m$ respectively as follows,

$\begin{gathered}\textstyle\nabla
f=\nabla_{t,z}f=\partial_{t,z,\bar{z}}f=\big{[}\frac{\partial f}{\partial
t^{1}},\dots,\frac{\partial f}{\partial t^{r}},\frac{\partial f}{\partial
z^{1}},\dots,\frac{\partial f}{\partial z^{m}},\frac{\partial
f}{\partial\bar{z}^{1}},\dots,\frac{\partial
f}{\partial\bar{z}^{m}}\big{]}^{\top},\\\
\textstyle\nabla_{z}f=\partial_{z,\bar{z}}f=\big{[}\frac{\partial f}{\partial
z^{1}},\dots,\frac{\partial f}{\partial z^{m}},\frac{\partial
f}{\partial\bar{z}^{1}},\dots,\frac{\partial
f}{\partial\bar{z}^{m}}\big{]}^{\top},\qquad\partial_{z}f=\frac{\partial
f}{\partial z}=\big{[}\frac{\partial f}{\partial z^{1}},\dots,\frac{\partial
f}{\partial z^{m}}\big{]}^{\top}.\end{gathered}$ (1.2)

For a map $F:\mathfrak{M}\to\mathbb{R}^{r}\times\mathbb{C}^{m}$, we usually
use the notation $F=(F^{\prime},F^{\prime\prime})$ where
$F^{\prime}:\mathfrak{M}\to\mathbb{R}^{r}$ and
$F^{\prime\prime}:\mathfrak{M}\to\mathbb{C}^{m}$ are the corresponding
components of $F$.

The paper is organized as follows:

In Section 2 we give discussions of Hölder-Zygmund spaces in the bi-parameter
setting. We prove some results for multiplications and compositions of bi-
parameter Hölder-Zygmund spaces in Sections 2.2 and 2.3 respectively. These
results will be used in Section 4.

In Section 3 we review the involutive structures and ODE flows. We prove the
bi-layer Frobenius theorem Theorem 1.7 (see Theorem 3.27) and a version of
holomorphic Frobenius theorem (Proposition 3.31) in Section 3.3. Proposition
3.31 will be used in the last step of Malgrange’s factorization in Section
5.1.

In Section 4 we prove two results for the nonlinear elliptic PDE system with
parameters: one for existence (Section 4.1) and one for interior real-
analyticity (Section 4.2).

In Section 5 we state and prove the general version of the key result Theorem
1.6 (Theorem 5.1). We first sketch the idea using Malgrange’s method in
Section 5.1 and then prove them in Section 5.2. To get a desired coordinate
chart, we first pick some good generators for the given family of elliptic
structures. Then we use the existence result in Section 4.1 to construct an
coordinate change such that in the new coordinates the generators satisfy some
extra conditions. By the real-analyticity result in Section 4.2 such
generators admit a holomorphic extension to a complex domain. Finally, we use
the holomorphic Frobenius theorem to complete the proof of Theorem 1.6. We
finish the proof of Theorem 5.1 in Section 5.3.

In Section 6 we complete the proof of Theorems 1.1 and 1.2 (see Theorems 6.1
and 6.2).

In Section 7 we explain which parts of the results in the main theorems are
sharp. For the sharpness of $F^{*}\frac{\partial}{\partial
z}\in\mathscr{C}^{\alpha-}$ we give a complete construction of the complex
Frobenius structure (Propositions 7.3 and 7.9) and give the proof of Theorem
1.3 in Section 7.2 (see Theorems 7.11 and 7.12).

In Section B we include a result of the holomorphic extension for a right
inverse Laplacian from $\mathbb{B}^{n}\subset\mathbb{R}^{n}$ to
$\mathbb{H}^{n}\subset\mathbb{C}^{n}$ (Proposition 4.8) based on [Mor58].

### Acknowledgement

I would like to express my great appreciation for my advisor Prof. Brian
Street for his help, not only giving the idea of the project but also for his
essential guidance throughout the project. I would also like to thanks to
Prof. Xianghong Gong for the invaluable advise for the paper.

##  2 The Bi-parameter Hölder-Zygmund Structures

In this section we set up some bi-parameter Hölder-Zygmund spaces that are
used to discuss the Hölder regularity of some geometric objects. The results
in this section are mostly for technical purposes.

###  2.1 Hölder-Zygmund spaces of single and bi-parameters

Recall the classical (single-parameter) Hölder-Zygmund spaces in Definition
1.8. In applications we require the Littlewood-Paley characterizations for
Hölder-Zygmund spaces.

######  Definition 2.1.

A (Fourier) dyadic resolution of unity for $\mathbb{R}^{n}$ is a sequence of
real-valued Schwartz functions $\phi=(\phi_{j})_{j=0}^{\infty}$ on
$\mathbb{R}^{n}$ such that

  1. (a)

The Fourier transform
$\hat{\phi}_{0}(\xi)=\int_{\mathbb{R}^{n}}\phi(x)e^{-2\pi ix\xi}dx$ satisfies
$\operatorname{supp}\hat{\phi}_{0}\subset\\{|\xi|<2\\}$ and
$\hat{\phi}_{0}\big{|}_{\\{|\xi|<1\\}}\equiv 1$.

  2. (b)

For $j\geq 1$,
$\phi_{j}(x):=2^{nj}\phi_{0}(2^{j}x)-2^{n(j-1)}\phi_{0}(2^{j-1}x)$ for
$x\in\mathbb{R}^{n}$.

Immediately we see that

$\operatorname{supp}\phi_{0}\subset\\{|\xi|<2\\},\quad\operatorname{supp}\phi_{j}\subset\\{2^{j-1}<|\xi|<2^{j+1}\\},\quad
j\geq 1.$ (2.1)

######  Lemma 2.2.

Let $\alpha>0$ and let $\phi=(\phi_{j})_{j=0}^{\infty}$ be a dyadic
resolution.

  1. (i)

$f\in\mathscr{C}^{\alpha}(\mathbb{R}^{n})$ if and only if $\phi_{j}\ast f\in
C^{0}(\mathbb{R}^{n})$ for every $j\geq 0$ and $\sup_{j\geq
0}2^{j\alpha}\|\phi_{j}\ast f\|_{L^{\infty}(\mathbb{R}^{n})}<\infty$. Moreover
there is a $C=C(\phi,\alpha)>0$ that does not depend on $f$ such that

$\textstyle C^{-1}\|f\|_{\mathscr{C}^{\alpha}(\mathbb{R}^{n})}\leq\sup_{j\geq
0}2^{j\alpha}\|\phi_{j}\ast f\|_{L^{\infty}(\mathbb{R}^{n})}\leq
C\|f\|_{\mathscr{C}^{\alpha}(\mathbb{R}^{n})},\quad\forall
f\in\mathscr{C}^{\alpha}(\mathbb{R}^{n}).$

  2. (ii)

Let $\Omega\subset\mathbb{R}^{n}$ be a bounded convex open set with smooth
boundary. Then $\mathscr{C}^{\alpha}(\Omega)$ has an equivalent norm

$f\mapsto\inf\\{\|\tilde{f}\|_{\mathscr{C}^{\alpha}(\mathbb{R}^{n})}:\tilde{f}\in\mathscr{C}^{\alpha}(\mathbb{R}^{n}),\tilde{f}\big{|}_{\Omega}=f\\}.$

  3. (iii)

Let $\Omega\subseteq\mathbb{R}^{n}$ be either the whole space or a bounded
convex open set with smooth boundary. Suppose $0<\alpha<2$, then
$\mathscr{C}^{\alpha}(\Omega)$ has an equivalent norm

$\textstyle f\mapsto\sup_{x\in\Omega}|f(x)|+\sup_{x,y\in\Omega;x\neq
y}\big{|}\frac{f(x)+f(y)}{2}-f\big{(}\frac{x+y}{2}\big{)}\big{|}\cdot|x-y|^{-\alpha}.$

See [Tri10, Chapters 2.2.2, 2.5.7 and 3.4.2] for the proof.

######  Remark 2.3.

Using Lemma 2.2 and Definition 1.8 we know that for positive noninteger value,
the Hölder-Zygmund spaces coincide with the classical Hölder spaces, that is
$\mathscr{C}^{m+s}(\Omega)=C^{m,s}(\Omega)$ for $m\geq 0$, $0<s<1$. However
for integer cases $C^{m}(\Omega)\subsetneq
C^{m-1,1}(\Omega)\subsetneq\mathscr{C}^{m}(\Omega)$.

Following the notations in [Tri10, Definition 2.3.1/2(i)] for $\alpha>0$ the
Hölder-Zygmund spaces $\mathscr{C}^{\alpha}(\Omega)$ are indeed the Besov
spaces $\mathscr{B}_{\infty\infty}^{\alpha}(\Omega)$. Thus it is natural to
define $\mathscr{C}^{\alpha}$-spaces for $\alpha\leq 0$.

######  Definition 2.4.

Let $\alpha\leq 0$, let $X=\mathbb{R}^{m}$ or $\mathbb{C}^{m}$, and let
$(\phi_{j})_{j=0}^{\infty}$ be a dyadic resolution of unity in
$\mathbb{R}^{n}$. We define
$\mathscr{C}^{\alpha}:=\mathscr{B}_{\infty\infty}^{\alpha}$. More precisely,
$\mathscr{C}^{\alpha}(\mathbb{R}^{n};X)$ is the space of all temper
distributions $f\in\mathscr{S}^{\prime}(\mathbb{R}^{n};X)$ such that

$\textstyle\|f\|_{\mathscr{C}^{\alpha}(\mathbb{R}^{n};X)}:=\sup_{j\geq
0}2^{j\alpha}\|\phi_{j}\ast f\|_{L^{\infty}(\mathbb{R}^{n};X)}<\infty.$

Let $\Omega\subset\mathbb{R}^{n}$ be a bounded convex domain, we define
$\mathscr{C}^{\alpha}(\Omega):=\\{\tilde{f}|_{\Omega}:\tilde{f}\in\mathscr{C}^{\alpha}(\mathbb{R}^{n})\\}$
with norm

$\|f\|_{\mathscr{C}^{\alpha}(\Omega;X)}:=\inf\\{\|\tilde{f}\|_{\mathscr{C}^{\alpha}(\mathbb{R}^{n};X)}:\tilde{f}|_{\Omega}=f\\}.$

The $\mathscr{C}^{\alpha}$-norms depend on $\phi=(\phi_{j})_{j=0}^{\infty}$,
while the spaces themselves are not. See [Tri10, Chapter 2.3.2].

We use Littlewood-Paley decomposition along different variables to define bi-
parameter Zygmund structures.

######  Notation 2.5.

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 $\mathbb{R}^{q}$.

######  Definition 2.6.

Let $\alpha,\beta\in\mathbb{R}$. 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
$\phi=(\phi_{j})_{j=0}^{\infty}\subset\mathscr{S}(\mathbb{R}^{n})$ and
$\psi=(\psi_{k})_{k=0}^{\infty}\subset\mathscr{S}(\mathbb{R}^{q})$ be dyadic
resolutions for $\mathbb{R}^{n}$ and $\mathbb{R}^{q}$.

  1. (i)

We define
$\mathscr{C}^{\alpha}_{x}L^{\infty}_{s}(\mathbb{R}^{n},\mathbb{R}^{q})=L^{\infty}_{s}\mathscr{C}^{\alpha}_{x}(\mathbb{R}^{q},\mathbb{R}^{n})$
to be the set of all
$f\in\mathscr{S}^{\prime}(\mathbb{R}^{n}\times\mathbb{R}^{q})$ such that
$\phi_{j}\ast_{x}f\in L^{\infty}(\mathbb{R}^{n}\times\mathbb{R}^{q})$ for
every $j\geq 0$ and

$\textstyle\|f\|_{\mathscr{C}^{\alpha}_{x}L^{\infty}_{s}(\mathbb{R}^{n},\mathbb{R}^{q})}=\|f\|_{\mathscr{C}^{\alpha}_{x}L^{\infty}_{s}(\mathbb{R}^{n},\mathbb{R}^{q})_{\phi}}:=\sup_{j\geq
0}2^{j\alpha}\|\phi_{j}\ast_{x}f\|_{L^{\infty}(\mathbb{R}^{n}\times\mathbb{R}^{q})}<\infty.$

  2. (ii)

We define
$\mathscr{C}^{\alpha}_{x}\mathscr{C}^{\beta}_{s}(\mathbb{R}^{m},\mathbb{R}^{q})$
to be the set of all
$f\in\mathscr{S}^{\prime}(\mathbb{R}^{n}\times\mathbb{R}^{q})$ such that

$\textstyle\|f\|_{\mathscr{C}^{\alpha}_{x}\mathscr{C}^{\beta}_{s}(\mathbb{R}^{n},\mathbb{R}^{q})}=\|f\|_{\mathscr{C}^{\alpha}_{x}\mathscr{C}^{\beta}_{s}(\mathbb{R}^{n},\mathbb{R}^{q})_{\phi,\psi}}:=\sup_{j,k\geq
0}2^{j\alpha+k\beta}\|\phi_{j}\ast_{x}(\psi_{k}\ast_{s}f)\|_{L^{\infty}(\mathbb{R}^{n}\times\mathbb{R}^{q})}<\infty.$

  3. (iii)

For
$\mathscr{X}\in\\{\mathscr{C}^{\alpha}_{x}L^{\infty}_{s},\mathscr{C}^{\alpha}_{x}\mathscr{C}^{\beta}_{s}\\}$,
let $U\subset\mathbb{R}^{m}$ and $V\subset\mathbb{R}^{n}$ be two open sets,
define $\mathscr{X}(U_{x},V_{s})$ by

$\textstyle\mathscr{X}(U,V):=\\{\tilde{f}|_{U\times
V}:\tilde{f}\in\mathscr{X}(\mathbb{R}^{n},\mathbb{R}^{q})\\},\quad\|f\|_{\mathscr{X}(U,V)}:=\inf_{\tilde{f}|_{U\times
V}=f}\|\tilde{f}\|_{\mathscr{X}(\mathbb{R}^{n},\mathbb{R}^{q})}.$

Let $\mathscr{X},\mathscr{Y}$ be the function classes in (i) or (ii), we have
convention

$\mathscr{X}\cap\mathscr{Y}(U,V):=\mathscr{X}(U,V)\cap\mathscr{Y}(U,V),\quad\|f\|_{\mathscr{X}\cap\mathscr{Y}(U,V)}:=\|f\|_{\mathscr{X}}+\|f\|_{\mathscr{Y}}.$

For vector valued functions, we use
$\mathscr{X}\cap\mathscr{Y}(U,V;\mathbb{R}^{m})$ as the spaces of all vector
valued functions (or distributions) $f=(f^{1},\dots,f^{m}):U\times
V\to\mathbb{R}^{m}$ such that $f^{j}\in\mathscr{X}\cap\mathscr{Y}(U,V)$ for
all $j=1,\dots,m$. And we use

$\textstyle\|f\|_{\mathscr{X}\cap\mathscr{Y}(U,V;\mathbb{R}^{m})}=\sup_{1\leq
j\leq m}\|f^{j}\|_{\mathscr{X}\cap\mathscr{Y}(U,V)}.$

We may omit the variables, the domains and the codomains in the norms when
they are clear.

######  Remark 2.7.

  1. (i)

By Lemma 2.2 (i) and [Tri10, Chapter 2.3.2], different choices of $(\phi_{j})$
and $(\psi_{k})$ result to the equivalent norms. We would use the norm
notation $\|f\|_{\mathscr{X}(U,V)}$ through fixing a choice of $(\phi,\psi)$
implicitly.

  2. (ii)

We have
$\mathscr{C}^{\alpha}_{x}\mathscr{C}^{\beta}_{s}(U_{x},V_{s})=\mathscr{C}^{\beta}_{s}\mathscr{C}^{\alpha}_{x}(V_{s},U_{x})$.
This is due to the fact that

$\phi_{j}\ast_{x}(\psi_{k}\ast_{s}f)=\psi_{k}\ast_{s}(\phi_{j}\ast_{x}f)=(\phi_{j}\otimes\psi_{k})\ast
f,\quad\text{where }(\phi_{j}\otimes\psi_{k})(x,s)=\phi_{j}(x)\psi_{k}(s).$
(2.2)

  3. (iii)

When $\alpha>0$ and when $U,V$ are smooth domains, we have
$\mathscr{C}^{\alpha}\mathscr{C}^{\beta}(U,V)=\mathscr{C}^{\alpha}(U;\mathscr{C}^{\beta}(V))$
in the sense that we take $X=\mathscr{C}^{\beta}(V)$ in Definition 1.8. We
leave the proof to readers.

  4. (iv)

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 $L^{\infty}(U;X)$ is the set of all $X$-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. See for
example [HvNVW16, Remark 1.2.16].

  5. (v)

In the definition
$f\in\mathscr{C}^{\alpha}_{x}L^{\infty}_{s}\cap\mathscr{C}^{\beta}_{x}\mathscr{C}^{\gamma}_{s}(U,V)$
means there are extensions
$\tilde{f}_{1}\in\mathscr{C}^{\alpha}_{x}L^{\infty}_{s}(\mathbb{R}^{n},\mathbb{R}^{q})$
and
$\tilde{f}_{2}\in\mathscr{C}^{\beta}_{x}\mathscr{C}^{\gamma}_{s}(\mathbb{R}^{n},\mathbb{R}^{q})$
such that $\tilde{f}_{1}|_{U\times V}=\tilde{f}_{2}|_{U\times V}=f$. When $U$
and $V$ are smooth domains, by Lemma 2.8 below we can choose a suitable
extension $\tilde{f}_{1}=\tilde{f}_{2}$ by taking
$\tilde{f}=E_{x}E_{s}f=E_{s}E_{x}f$ that extends
$\mathscr{C}^{\alpha}_{x}L^{\infty}_{s}$ and
$L^{\infty}_{x}\mathscr{C}^{\beta}_{s}$ simultaneously.

  6. (vi)

Let $\alpha_{0},\alpha_{1},\beta_{0},\beta_{1}\in\mathbb{R}$. 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 convex domains,

$\mathscr{C}^{\alpha_{0}}\mathscr{C}^{\beta_{0}}\cap\mathscr{C}^{\alpha_{1}}\mathscr{C}^{\beta_{1}}(U,V)\subseteq\mathscr{C}^{\alpha_{\theta}}\mathscr{C}^{\beta_{\theta}}(U,V),\quad\alpha_{\theta}:=(1-\theta)\alpha_{0}+\theta\alpha_{1},\
\beta_{\theta}:=(1-\theta)\beta_{0}+\theta\beta_{1}.$ (2.3)

######  Lemma 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)

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

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

$\tilde{E}:\mathscr{X}\mathscr{Y}(U,V)\to\mathscr{X}\mathscr{Y}(\mathbb{R}^{n},\mathbb{R}^{q}),\quad\mathscr{X},\mathscr{Y}\in\\{L^{\infty},\mathscr{C}^{\alpha}:-R<\alpha<R\\}.$
(2.4)

###### Proof.

The construction $E_{x}$ and $E_{s}$ for (a) can be found in [Tri10, Theorems
2.9.2, 2.9.4 (i) and 3.3.4 (i)], which are both $L^{\infty}$ and
$\mathscr{C}^{\alpha}$-bounded for $-R<\alpha<R$.

For continuous functions $u\in C^{0}(U)$, $v\in C^{0}(V)$ we have equality
$E_{x}E_{s}(u\otimes v)=E_{x}u\otimes E_{s}v=E_{s}E_{x}(u\otimes v)$. Thus by
taking finite linear combinations of simple tensors and taking $C^{0}$-limit,
we see that $E_{x}E_{s}=E_{s}E_{x}:C^{0}(U\times V)\to
L^{\infty}(\mathbb{R}^{n}\times\mathbb{R}^{q})$ is an extension operator for
$U\times V$.

Let $\varepsilon>0$ and let $(\phi_{j}),(\psi_{k})$ be dyadic resolutions for
$\mathbb{R}^{n}_{x}$, $\mathbb{R}^{q}_{s}$ respectively. Clearly for
$\tilde{f}\in\mathscr{C}^{-R+\varepsilon}\mathscr{C}^{-R+\varepsilon}(\mathbb{R}^{n},\mathbb{R}^{q})$,
the double sum $\sum_{j,k=0}^{\infty}(\phi_{j}\otimes\psi_{k})\ast\tilde{f}$
converges to $\tilde{f}$ in
$\mathscr{C}^{-R+\frac{\varepsilon}{2}}\mathscr{C}^{-R+\frac{\varepsilon}{2}}(\mathbb{R}^{n},\mathbb{R}^{q})$.
Since $((\phi_{j}\otimes\psi_{k})\ast\tilde{f})|_{U\times V}\in C^{0}(U\times
V)$ for each $j,k$, by Definition 2.6 (iii) we see that
$E_{x}E_{s}f=E_{s}E_{x}f\in\mathscr{C}^{-R+\frac{\varepsilon}{2}}\mathscr{C}^{-R+\frac{\varepsilon}{2}}(\mathbb{R}^{n},\mathbb{R}^{q})$
whenever
$f\in\mathscr{C}^{-R+\varepsilon}\mathscr{C}^{-R+\varepsilon}(\mathbb{R}^{n},\mathbb{R}^{q})$.
Therefore
$E_{x}E_{s},E_{s}E_{x}\mathscr{C}^{(-R)+}\mathscr{C}^{(-R)+}(U,V)\to\mathscr{C}^{(-R)+}\mathscr{C}^{(-R)+}(\mathbb{R}^{n},\mathbb{R}^{q})$
are both defined and equal. This proves (b).

Note that the same argument also shows that

$\phi_{j}\ast_{x}(E_{s}g)(x,s)=E_{s}(\phi_{j}\ast_{x}g)(x,s),\quad\text{for
}(x,s)\in\mathbb{R}^{n}\times\mathbb{R}^{q},\quad j\geq 0,\quad
g\in\mathscr{C}^{(-R)+}\mathscr{C}^{(-R)+}(U,\mathbb{R}^{q}).$ (2.5)

To prove (2.4), we only prove the case $\mathscr{C}^{\alpha}L^{\infty}$ for
$-R<\alpha<R$. The argument for the rest of the cases are similar and we omit
the details.

Using the boundedness
$E_{x}:\mathscr{C}^{\alpha}(U)\to\mathscr{C}^{\alpha}(\mathbb{R}^{n})$ and
$E_{s}:L^{\infty}(V)\to L^{\infty}(\mathbb{R}^{q})$ we have

$\displaystyle\|\tilde{E}(\tilde{f}|_{U\times
V})\|_{\mathscr{C}^{\alpha}L^{\infty}(\mathbb{R}^{n},\mathbb{R}^{q})}=\sup_{j\geq
0}2^{j\alpha}\|\phi_{j}\ast_{x}(E_{s}E_{x}(\tilde{f}|_{U\times
V}))\|_{L^{\infty}(\mathbb{R}^{n}\times\mathbb{R}^{q})}=\sup_{j\geq
0}2^{j\alpha}\|E_{s}(\phi_{j}\ast_{x}E_{x}(\tilde{f}|_{U\times
V}))\|_{L^{\infty}(\mathbb{R}^{n}\times\mathbb{R}^{q})}$ $\displaystyle\leq$
$\displaystyle\|E_{s}\|_{L^{\infty}}\sup_{j\geq
0}2^{j\alpha}\mathop{\operatorname{essup}}_{s\in
V}\|\phi_{j}\ast_{x}E_{x}(\tilde{f}(\cdot,s)|_{U})\|_{L^{\infty}(\mathbb{R}^{n})}\leq\|E_{s}\|_{L^{\infty}}\|E_{x}\|_{\mathscr{C}^{\alpha}}\mathop{\operatorname{essup}}_{s\in\mathbb{R}^{q}}\|\tilde{f}(\cdot,s)\|_{\mathscr{C}^{\alpha}(U)}\lesssim_{\tilde{E}}\|\tilde{f}\|_{\mathscr{C}^{\alpha}L^{\infty}(\mathbb{R}^{n},\mathbb{R}^{q})}.$

For a function $f\in\mathscr{C}^{\alpha}L^{\infty}(U,V)$, taking the infimum
of all extensions
$\tilde{f}\in\mathscr{C}^{\alpha}L^{\infty}(\mathbb{R}^{n},\mathbb{R}^{q})$
over $f$, we get
$\|\tilde{E}f\|_{\mathscr{C}^{\alpha}L^{\infty}(\mathbb{R}^{n},\mathbb{R}^{q})}\lesssim\|f\|_{\mathscr{C}^{\alpha}L^{\infty}(U,V)}$
from the above inequality. ∎

More generally, for a linear operator $T=T_{x}$ acting on functions in
$U\subseteq\mathbb{R}^{n}_{x}$, we have natural extension
$\tilde{T}=T\otimes\operatorname{id}_{s}$ acting on functions in $U\times
V\subseteq\mathbb{R}^{n}_{x}\times\mathbb{R}^{q}_{s}$.

######  Lemma 2.9.

Let $R_{1}<R_{2}$ and $r\in\mathbb{R}$. Let $U\subseteq\mathbb{R}^{n}$ and
$V\subseteq\mathbb{R}^{q}$ be either the total spaces or bounded smooth
domains. Suppose $T$ is a linear operator on functions of $U$ such that
$T:\mathscr{C}^{\alpha}(U)\to\mathscr{C}^{\alpha+r}(U)$ for all
$R_{1}<\alpha<R_{2}$. Then $(\tilde{T}f)(x,s):=T(f(\cdot,s))(x)$ defines a
linear operator that has boundedness
$\tilde{T}:\mathscr{C}^{\alpha}\mathscr{X}(U,V)\to\mathscr{C}^{\alpha}\mathscr{X}(U,V)$
for all $R_{1}<\alpha<R_{2}$ and
$\mathscr{X}\in\\{L^{\infty},\mathscr{C}^{\beta}:\beta\in\mathbb{R}\\}$.

###### Proof.

By passing to an extension $\tilde{E}=E_{x}E_{s}$ given in Lemma 2.8, it
suffices to consider the case $U=\mathbb{R}^{n}$ and $V=\mathbb{R}^{q}$. We
only prove the case $\mathscr{X}=\mathscr{C}^{\beta}$ since the case
$\mathscr{X}=L^{\infty}$ is similar but simpler.

Clearly for $u\in\mathscr{C}^{R_{1}+}(\mathbb{R}^{n})$ and $v\in
C^{0}(\mathbb{R}^{q})$ we have $\tilde{T}(u\otimes v)=Tu\otimes v$ and
$\psi_{k}\ast_{s}\tilde{T}(u\otimes v)=Tu\otimes(\psi_{k}\ast v)$ for all
$k\geq 0$. Taking finite linear combinations of simple tensors and then taking
limit similar to the proof of Lemma 2.8 (b), we get
$\tilde{T}:\mathscr{C}^{R_{1}+\varepsilon}\mathscr{C}^{-R}(\mathbb{R}^{n},\mathbb{R}^{q})\to\mathscr{C}^{R_{1}+r+\frac{\varepsilon}{2}}\mathscr{C}^{-R-\varepsilon}(\mathbb{R}^{n},\mathbb{R}^{q})$
is defined and bounded for all $R>0$ and $0<\varepsilon<R_{2}-R_{1}$, and we
have similar to (2.5) that

$\psi_{k}\ast_{s}(\tilde{T}f)=\tilde{T}(\psi_{k}\ast_{s}f),\quad k\geq 0,\quad
f\in\mathscr{C}^{R_{1}+}\mathscr{C}^{-R}(\mathbb{R}^{n},\mathbb{R}^{q}).$

Therefore, for $R_{1}<\alpha<R_{2}$ and $\beta\in\mathbb{R}$,

$\displaystyle\|\tilde{T}f\|_{\mathscr{C}^{\alpha+r}\mathscr{C}^{\beta}(\mathbb{R}^{n},\mathbb{R}^{q})}=\sup_{j,k\geq
0}2^{j(\alpha+r)+k\beta}\|(\phi_{j}\otimes\psi_{k})\ast\tilde{T}f\|_{L^{\infty}(\mathbb{R}^{n}\times\mathbb{R}^{q})}=\sup_{j,k\geq
0}2^{j(\alpha+r)+k\beta}\|\phi_{j}\ast_{x}\tilde{T}(\psi_{k}\ast_{s}f)\|_{L^{\infty}(\mathbb{R}^{n}\times\mathbb{R}^{q})}$
$\displaystyle\leq$
$\displaystyle\|\tilde{T}\|_{\mathscr{C}^{\alpha}\to\mathscr{C}^{\alpha+r}}\sup_{j,k\geq
0}2^{j(\alpha+r)+k\beta}\mathop{\operatorname{essup}}_{s\in\mathbb{R}^{q}}\|\phi_{j}\ast((\psi_{k}\ast_{s}f)(\cdot,s))\|_{L^{\infty}(\mathbb{R}^{n})}=\|\tilde{T}\|_{\mathscr{C}^{\alpha}\to\mathscr{C}^{\alpha+r}}\|f\|_{\mathscr{C}^{\alpha}\mathscr{C}^{\beta}(\mathbb{R}^{n},\mathbb{R}^{q})}.$

This completes the proof. ∎

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

######  Definition 2.10.

Let $\alpha,\beta>0$, and let $U\subseteq\mathbb{R}^{n}$,
$V\subseteq\mathbb{R}^{q}$ be two convex open sets. We define
$\mathscr{C}^{\alpha,\beta}_{x,s}(U,V)$ to be the set of all continuous
functions $f:U\times V\to\mathbb{R}$ satisfying

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

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 2.11.

  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 2.12 (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 Zygmund spaces and the version on manifold
analogously. See Definition 2.15 (iv) and Definition 3.2.

Also see Definition 2.15 when $\alpha,\beta$ are not real numbers.

######  Lemma 2.12.

Let $\alpha,\beta>0$, and let $U\subset\mathbb{R}^{m}$,
$V\subset\mathbb{R}^{n}$ be two bounded convex open sets with smooth
boundaries.

  1. (i)

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

  2. (ii)

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

###### Proof.

(i): By Lemma 2.8 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

$\textstyle\sup_{x\in
U}\|f(x,\cdot)\|_{\mathscr{C}^{\beta}(V)}\geq\mathop{\operatorname{essup}}_{x\in
U}\|f(x,\cdot)\|_{\mathscr{C}^{\beta}(V)}=\|E_{s}f\|_{L^{\infty}\mathscr{C}^{\beta}(U,\mathbb{R}^{q})}\approx\|\tilde{E}f\|_{L^{\infty}\mathscr{C}^{\beta}(\mathbb{R}^{n},\mathbb{R}^{q})}\approx\|f\|_{L^{\infty}\mathscr{C}^{\beta}(U,V)}.$
(2.6)

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 (2.6)
is indeed an equality. By Remark 2.7 (vi) we have
$\mathscr{C}^{\alpha}L^{\infty}\cap
L^{\infty}\mathscr{C}^{\beta}(U,V)\subset\mathscr{C}^{\alpha}\mathscr{C}^{0}\cap\mathscr{C}^{0}\mathscr{C}^{\beta}(U,V)\subset\mathscr{C}^{\frac{\alpha}{2}}\mathscr{C}^{\frac{\beta}{2}}(U,V)\subset
C^{0}(U,V)$. 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
2.2 (i) and Remark 2.7 (ii),

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

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

######  Remark 2.13.

Let $\alpha,\beta>0$ and let $U,V$ be two bounded smooth domains. Since
$\mathscr{C}^{0}\subset L^{\infty}\subset\mathscr{C}^{1}$, 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 2.14 (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,

$[\mathscr{C}^{\alpha_{0},\beta_{0}}(U,V),\mathscr{C}^{\alpha_{1},\beta_{1}}(U,V)]_{\theta}=\mathscr{C}^{(1-\theta)\alpha_{0}+\theta\alpha_{1},(1-\theta)\beta_{0}+\theta\beta_{1}}(U,V),\quad\text{for
all }\alpha_{0},\alpha_{1},\beta_{0},\beta_{1}>0,\quad\theta\in[0,1].$ (2.7)

We also leave the proof of (2.7) to readers.

Here are more definitions.

######  Definition 2.15.

Let $\alpha,\beta>0$ and let $U\in\mathbb{R}^{n}$, $V\in\mathbb{R}^{q}$ be two
open convex 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)

$\mathscr{C}^{\alpha,\beta-}(U,V)=\mathscr{C}^{\beta-,\alpha}(U,V):=\bigcap_{0<\gamma<\beta}\mathscr{C}^{\alpha,\gamma}(U,V)$.

  3. (iii)

$\mathscr{C}^{\infty,\beta}(U,V):=\bigcap_{\gamma>0}\mathscr{C}^{\gamma,\beta}(U,V)$
and
$\mathscr{C}^{\infty}\mathscr{C}^{\beta}(U,V):=\bigcap_{\gamma>0}\mathscr{C}^{\gamma}\mathscr{C}^{\beta}(U,V)$.

  4. (iv)

Let $\gamma>0$ and let $W\subset\mathbb{R}^{m}$ be as well an open convex set.
We define $\mathscr{C}^{\alpha,\beta,\gamma}(U,V,W)$ to be the set of all
continuous functions $f:U\times V\times W\to\mathbb{R}$ satisfying

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

######  Remark 2.16.

By Lemma 2.12 (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 2.6 and 2.10.

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

######  Example 2.17.

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)
$\operatorname{supp}\hat{\psi}_{k}=\operatorname{supp}2^{k-1}\hat{\psi}_{1}\subsetneq(-2^{k+1},-2^{k-1})\cup(2^{k-1},2^{k+1})$
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 2.18.

Let $\alpha,\beta>0$ 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 2.7 (vi) 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 2.19.

Let $\beta>0$, and let $U\subset\mathbb{R}^{n}_{x}$ and
$V\subset\mathbb{R}^{q}_{s}$ be two bounded 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.

###### 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.

Let $E_{s}:\mathscr{C}^{\beta}(V)\to\mathscr{C}^{\beta}(\mathbb{R}^{q})$ be an
$s$-variable extension as in Lemma 2.8. Clearly
$\nabla_{x}^{k}E_{s}f\in\mathscr{C}^{-k-1}\mathscr{C}^{\beta}(U,\mathbb{R}^{n};\mathbb{R}^{2^{k}})$
for every $k\geq 0$.

Let $(\psi_{j})_{j=0}^{\infty}$ be a dyadic resolution for
$\mathbb{R}^{q}_{s}$. By Lemma B.1 (iv) we have

$2^{j\beta}|\psi_{j}\ast_{s}E_{s}\nabla_{x}^{k}f(x,s)|\lesssim_{n,k}(1-|x|)^{-k-1}2^{j\beta}\|\psi_{j}\ast_{s}E_{s}\nabla_{x}^{k}f\|_{\mathscr{C}^{-k-1}(\mathbb{B}^{n})}\lesssim_{\beta,E}\|\nabla_{x}^{k}f\|_{\mathscr{C}^{-k-1}\mathscr{C}^{\beta}(\mathbb{B}^{n},V)}.$

Taking supremum in $s\in\mathbb{R}^{q}$ and $j\geq 0$, by Lemmas 2.2 (i) we
get

$\sup_{|\nu|\leq
k}\|\partial_{x}^{\nu}f\|_{L^{\infty}\mathscr{C}^{\beta}(r\mathbb{B}^{n},V)}\lesssim_{V,n,k,\beta}(1-r)^{-k-1}\|f\|_{\mathscr{C}^{-1}\mathscr{C}^{\beta}(\mathbb{B}^{n},V)},\quad\forall
0<r<1.$

Therefore
$f\in\mathscr{C}^{k}_{\mathrm{loc}}\mathscr{C}^{\beta}(\mathbb{B}^{n},V)$ for
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 A proposition for bi-parameter products

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

Recall in Definition 2.6, we fix two dyadic resolutions
$(\phi_{j})_{j=0}^{\infty}\subset\mathscr{S}(\mathbb{R}^{m})$ and
$(\psi_{k})_{k=0}^{\infty}\subset\mathscr{S}(\mathbb{R}^{n})$ to define the
$\mathscr{C}^{\alpha}L^{\infty}$-norms and the
$\mathscr{C}^{\alpha}\mathscr{C}^{\beta}$-norms.

######  Proposition 2.20.

Let $\alpha,\beta>0$ and $\eta,\theta\in[0,1]$ satisfy $\eta+\theta<\alpha$.
There is a $C=C_{\alpha,\beta,\eta,\theta,\phi,\psi}>0$ 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
$g\in\mathscr{C}^{\alpha-\theta}L^{\infty}\cap\mathscr{C}^{-\eta-\theta}\mathscr{C}^{\beta}(\mathbb{R}^{n},\mathbb{R}^{q})$.

We postpone the proof to the end of this subsection.

For applications, we use the cases $(\eta,\theta)=(1,0)$ and
$(\eta,\theta)=(0,0)$ to prove Proposition 4.3 that the left hand side of
(4.3) behave well. And we use $(\eta,\theta)=(0,1)$ to prove Proposition 2.27
which is required in Proposition 4.3 (iii).

######  Corollary 2.21.

Let $\alpha,\beta>0$ and $\eta,\theta\in[0,1]$ satisfy $\eta+\theta<\alpha$.
Let $U\subset\mathbb{R}^{m}$ and $V\subset\mathbb{R}^{n}$ be two bounded open
convex subsets with smooth boundaries. Then the product operation is bilinear
bounded as the following:

$[(f,g)\to
fg]:\mathscr{C}^{\alpha}L^{\infty}\cap\mathscr{C}^{-\eta}\mathscr{C}^{\beta}(U,V)\times\mathscr{C}^{\alpha-\theta}L^{\infty}\cap\mathscr{C}^{-\eta-\theta}\mathscr{C}^{\beta}(U,V)\to\mathscr{C}^{\alpha-\theta}L^{\infty}\cap\mathscr{C}^{-\eta-\theta}\mathscr{C}^{\beta}(U,V).$
(2.8)

In particular it is bilinearly bounded between the following spaces: for
$\alpha>1$,

  1. (i)

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

  2. (ii)

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

  3. (iii)

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

###### Proof.

Indeed (i), (ii) and (iii) are the cases where $(\eta,\theta)=(0,1)$,
$(\eta,\theta)=(0,0)$ and $(\eta,\theta)=(1,0)$ in (2.8) respectively.

By Lemma 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 2.20,

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

So we obtain (2.8). ∎

######  Corollary 2.22.

Let $\alpha,\beta>0$ and $\eta\in[0,1]$ satisfy $\eta<\alpha$. Let $m,n,q\geq
0$ and let $U\subset\mathbb{R}^{n}$, $V\subset\mathbb{R}^{q}$ be bounded
convex open sets with smooth boundaries. Then there is a
$c_{0}=c_{0}(U,V,m,n,q,\alpha,\beta,\eta)>0$, such that for any matrix-valued
function $A:U\times V\to\mathbb{C}^{m\times m}$ satisfying
$\|A-I_{m}\|_{\mathscr{C}^{\alpha}L^{\infty}\cap\mathscr{C}^{-\eta}\mathscr{C}^{\beta}(U,V;\mathbb{C}^{m\times
m})}<c_{0}$, we have:

  * •

$A(x,s)\in\mathbb{C}^{m\times m}$ is an invertible matrix for every $(x,s)\in
U\times V$.

  * •

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

Moreover, the map of inverting matrices

$[A\mapsto
A^{-1}]:\\{A\in\mathscr{C}^{\alpha}L^{\infty}\cap\mathscr{C}^{-\eta}\mathscr{C}^{\beta}(U,V;\mathbb{C}^{m\times
m}):\|A-I_{m}\|_{\mathscr{C}^{\alpha}L^{\infty}\cap\mathscr{C}^{-\eta}\mathscr{C}^{\beta}}<c_{0}\\}\to\mathscr{C}^{\alpha}L^{\infty}\cap\mathscr{C}^{-\eta}\mathscr{C}^{\beta}(U,V;\mathbb{C}^{m\times
m}),$

is real-analytic and in particular Fréchet differentiable.

###### Proof.

Applying Corollary 2.21 to the components of the matrix product
$(A-I_{m})^{k}=(A-I_{m})^{k-1}(A-I_{m})$ recursively in $k$, with the
parameter $\theta=0$, we know there is a $C=C(U,V,m,n,q,\alpha,\beta,\eta)>0$
such that

$\displaystyle\|(A-I_{m})^{k}\|_{\mathscr{C}^{\alpha}L^{\infty}\cap\mathscr{C}^{-\eta}\mathscr{C}^{\beta}}\leq
C^{k-1}\|A-I_{m}\|_{\mathscr{C}^{\alpha}L^{\infty}\cap\mathscr{C}^{-\eta}\mathscr{C}^{\beta}}^{k},\quad\forall\
k\geq 1,\
A\in\mathscr{C}^{\alpha}L^{\infty}\cap\mathscr{C}^{-\eta}\mathscr{C}^{\beta}(U,V;\mathbb{C}^{m\times
m}).$

Now take $c_{0}=\frac{1}{2}C^{-1}$, we see that the sum
$\sum_{k=0}^{\infty}(I_{m}-A)^{k}$ is norm convergent in
$\mathscr{C}^{\alpha}L^{\infty}\cap\mathscr{C}^{-\eta}\mathscr{C}^{\beta}(U,V;\mathbb{C}^{m\times
m})$ whose limit equals to $(I_{m}-(I_{m}-A))^{-1}=A^{-1}$. So $A(x,s)$ is an
invertible matrix for all $(x,s)\in U\times V$.

That
$\mathscr{C}^{\alpha}L^{\infty}\cap\mathscr{C}^{-\eta}\mathscr{C}^{\beta}$ is
closed under products also implies that $[A\mapsto
A^{-1}=I+\sum_{k=1}^{\infty}(I-A)^{k}]$ is a real-analytic map in the open set
$\\{A:\|A-I_{m}\|_{\mathscr{C}^{\alpha}L^{\infty}\cap\mathscr{C}^{-\eta}\mathscr{C}^{\beta}}<c_{0}\\}$
with respect to the norm-topology. ∎

######  Remark 2.23.

By taking $\eta=0$ and forgetting the $s$-variable, we get the following
classical result:

For any $\alpha>0$ and bounded convex smooth open $U\subset\mathbb{R}^{n}$,
there is a $c_{0}^{\prime}(U,m,\alpha)>0$, such that

$\forall A\in\mathscr{C}^{\alpha}(U;\mathbb{C}^{m\times
m}),\quad\|A-I_{m}\|_{\mathscr{C}^{\alpha}(U;\mathbb{C}^{m\times
m})}<c_{0}^{\prime}\quad\Rightarrow\quad\|A^{-1}-I_{m}\|_{\mathscr{C}^{\alpha}(U;\mathbb{C}^{m\times
m})}\leq 2\|A-I_{m}\|_{\mathscr{C}^{\alpha}(U;\mathbb{C}^{m\times m})}.$

The proof of Proposition 2.20 use paraproduct decompositions. The single
parameter case is the following:

######  Lemma 2.24 (The paraproduct decomposition).

Let $(\phi_{j})_{j=0}^{\infty}\subset\mathscr{S}(\mathbb{R}^{n})$ be a dyadic
resolution. For bounded continuous functions $u,v$ defined on
$\mathbb{R}^{n}$, consider the following decomposition

$uv=\sum_{j=0}^{\infty}\sum_{j^{\prime}=0}^{j-3}(\phi_{j}\ast
u)(\phi_{j^{\prime}}\ast
v)+\sum_{j^{\prime}=0}^{\infty}\sum_{j=0}^{j^{\prime}-3}(\phi_{j}\ast
u)(\phi_{j^{\prime}}\ast v)+\sum_{|j-j^{\prime}|\leq 2}(\phi_{j}\ast
u)(\phi_{j^{\prime}}\ast v).$ (2.9)

Then for every $l\geq 0$,

$\phi_{l}\ast(uv)=\phi_{l}\ast\bigg{(}\sum_{j=l-2}^{l+2}\sum_{j^{\prime}=0}^{j-3}(\phi_{j}\ast
u)(\phi_{j^{\prime}}\ast
v)+\sum_{j^{\prime}=l-2}^{l+2}\sum_{j=0}^{j^{\prime}-3}(\phi_{j}\ast
u)(\phi_{j^{\prime}}\ast v)+\sum_{\begin{subarray}{c}j,j^{\prime}\geq l-3\\\
|j-j^{\prime}|\leq 2\end{subarray}}(\phi_{j}\ast u)(\phi_{j^{\prime}}\ast
v)\bigg{)}.$ (2.10)

######  Remark 2.25.

In [BCD11, Chapter 2.8.1, (2.41)] (2.9) is denoted by
$uv=T(u,v)+T(v,u)+R(u,v)$ where $T$ is the paraproduct operator associated
with $\phi$.

###### Proof.

(2.10) can be obtained by investigating the Fourier support condition in
Definition 2.1. Note that $\phi_{l}\ast\left((\phi_{j}\ast
u)(\phi_{j^{\prime}}\ast v)\right)\neq 0$ only when
$\operatorname{supp}\hat{\phi}_{l}\cap(\operatorname{supp}\hat{\phi}_{j}+\operatorname{supp}\hat{\phi}_{j^{\prime}})\neq\varnothing$,
and by (2.1) we have on the following:

$\operatorname{supp}\hat{\phi}_{l}\subset\begin{cases}\\{2^{l-1}<|\xi|<2^{l+1}\\},&l\geq
1,\\\
\\{|\xi|<2\\},&l=0;\end{cases}\quad\operatorname{supp}\hat{\phi}_{j}+\operatorname{supp}\hat{\phi}_{j^{\prime}}\subset\begin{cases}\\{2^{\max(j,j^{\prime})-2}<|\xi|<2^{\max(j,j^{\prime})+2}\\},&|j-j^{\prime}|\geq
3,\\\ \\{|\xi|<2^{\max(j,j^{\prime})+2}\\},&|j-j^{\prime}|\leq 2.\end{cases}$

For details, see [Tri10, Proof of Theorem 2.8.2], [Gra14, Chapter 4.4] or
[BCD11, Proof of Theorem 2.52]. Note that their choice of decompositions (2.9)
may have slight differences. ∎

######  Lemma 2.26.

Let $\alpha>0$ and $\beta\in(-\alpha,\alpha]$. Let $U\subset\mathbb{R}^{n}$ be
a bounded convex open set with smooth boundary.

  1. (i)

There is a $C=C(U,\alpha,\beta)\geq 0$ such that
$\|fg\|_{\mathscr{C}^{\beta}(U)}\leq
C\|f\|_{\mathscr{C}^{\alpha}(U)}\|g\|_{\mathscr{C}^{\beta}(U)}$ for all
$f\in\mathscr{C}^{\alpha}(U)$, $g\in\mathscr{C}^{\beta}(U)$.

  2. (ii)

There is a $C=C(U,\alpha)\geq 0$ such that
$\|fg\|_{\mathscr{C}^{\alpha}(U)}\leq
C(\|f\|_{\mathscr{C}^{\alpha}(U)}\|g\|_{L^{\infty}(U)}+\|f\|_{L^{\infty}(U)}\|g\|_{\mathscr{C}^{\alpha}(U)})$
for all $f,g\in\mathscr{C}^{\alpha}(U)$.

###### Proof.

See [Tri10, Theorem 3.3.2(ii)] for (i) and [BCD11, Corollary 2.86] for (ii).
Note that by Lemma 2.2 and Definition 2.4 Hölder-Zygmund spaces are a special
case to Besov spaces
$\mathscr{C}^{\alpha}(U)=\mathscr{B}_{\infty\infty}^{\alpha}(U)$ for all
$\alpha\in\mathbb{R}$. ∎

We now prove Proposition 2.20.

###### Proof of Proposition 2.20.

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. Note that by assumption $f,g$ are both
$L^{\infty}$-functions, so their product is already defined in
$L^{\infty}(U\times V)$. What we need is to show that $fg$ lays in the desired
function class.

Since $\alpha-\theta>0$, by Lemma 2.26 (i) for almost every
$s\in\mathbb{R}^{q}$,
$\|f(\cdot,s)g(\cdot,s)\|_{\mathscr{C}^{\alpha-\theta}(\mathbb{R}^{n})}\lesssim_{\alpha-\theta}\|f(\cdot,s)\|_{\mathscr{C}^{\alpha}}\|g(\cdot,s)\|_{\mathscr{C}^{\alpha-\theta}}$,
with the implied constant being independent of $f$ and $g$. So by taking
essential supremum of $s$, we get

$\displaystyle\|fg\|_{\mathscr{C}^{\alpha-\theta}L^{\infty}(\mathbb{R}^{n},\mathbb{R}^{q})}=$
$\displaystyle\mathop{\operatorname{essup}}_{s\in\mathbb{R}^{q}}\|f(\cdot,s)g(\cdot,s)\|_{\mathscr{C}^{\alpha-\theta}(\mathbb{R}^{n})}\lesssim\|f\|_{\mathscr{C}^{\alpha}L^{\infty}(\mathbb{R}^{n},\mathbb{R}^{q})}\|g\|_{\mathscr{C}^{\alpha-\theta}L^{\infty}(\mathbb{R}^{n},\mathbb{R}^{q})}$
(2.11) $\displaystyle\leq$
$\displaystyle\|f\|_{\mathscr{C}^{\alpha}L^{\infty}\cap\mathscr{C}^{-\eta}\mathscr{C}^{\beta}(\mathbb{R}^{n},\mathbb{R}^{q})}\|g\|_{\mathscr{C}^{\alpha-\theta}L^{\infty}\cap\mathscr{C}^{-\eta-\theta}\mathscr{C}^{\beta}(\mathbb{R}^{n},\mathbb{R}^{q})}.$

Our goal is to show the bilinear control
$\|fg\|_{\mathscr{C}^{-\eta-\theta}\mathscr{C}^{\beta}}\lesssim\|f\|_{\mathscr{C}^{\alpha}L^{\infty}\cap\mathscr{C}^{-\eta}\mathscr{C}^{\beta}}\|g\|_{\mathscr{C}^{\alpha-\theta}L^{\infty}\cap\mathscr{C}^{-\eta-\theta}\mathscr{C}^{\beta}}$.

Before we move into the computations, we first upgrade (2.10) to the bi-
parameter case.

For bounded functions $f$ and $g$ defined on
$\mathbb{R}^{n}\times\mathbb{R}^{q}$, we denote

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

We can 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:

$\displaystyle fg=\sum_{\mu,\nu=1}^{3}S^{\mu\nu}=$
$\displaystyle\sum\limits_{j=3}^{\infty}\sum\limits_{j^{\prime}=0}^{j-3}\sum\limits_{k=3}^{\infty}\sum\limits_{k^{\prime}=0}^{k-3}f_{jk}g_{j^{\prime}k^{\prime}}+\sum\limits_{j^{\prime}=3}^{\infty}\sum\limits_{j=0}^{j^{\prime}-3}\sum\limits_{k=3}^{\infty}\sum\limits_{k^{\prime}=0}^{k-3}f_{jk}g_{j^{\prime}k^{\prime}}+\sum\limits_{|j-j^{\prime}|\leq
2}\sum\limits_{k=3}^{\infty}\sum\limits_{k^{\prime}=0}^{k-3}f_{jk}g_{j^{\prime}k^{\prime}}$
(2.12)
$\displaystyle+\sum\limits_{j=3}^{\infty}\sum\limits_{j^{\prime}=0}^{j-3}\sum\limits_{k^{\prime}=3}^{\infty}\sum\limits_{k=0}^{k^{\prime}-3}f_{jk}g_{j^{\prime}k^{\prime}}+\sum\limits_{j^{\prime}=3}^{\infty}\sum\limits_{j=0}^{j^{\prime}-3}\sum\limits_{k^{\prime}=3}^{\infty}\sum\limits_{k=0}^{k^{\prime}-3}f_{jk}g_{j^{\prime}k^{\prime}}+\sum\limits_{|j-j^{\prime}|\leq
2}\sum\limits_{k=3}^{\infty}\sum\limits_{k=0}^{k^{\prime}-3}f_{jk}g_{j^{\prime}k^{\prime}}$
$\displaystyle+\sum\limits_{j=3}^{\infty}\sum\limits_{j^{\prime}=0}^{j-3}\sum\limits_{|k-k^{\prime}|\leq
2}f_{jk}g_{j^{\prime}k^{\prime}}+\sum\limits_{j^{\prime}=3}^{\infty}\sum\limits_{j=0}^{j^{\prime}-3}\sum\limits_{|k-k^{\prime}|\leq
2}f_{jk}g_{j^{\prime}k^{\prime}}+\sum\limits_{|j-j^{\prime}|\leq
2}\sum\limits_{|k-k^{\prime}|\leq 2}f_{jk}g_{j^{\prime}k^{\prime}}.$

Here $S^{\mu\nu}$, $1\leq\mu,\nu\leq 3$, is the $\mu$-th row $\nu$-th column
term in (2.12). We denote

$S^{\mu\nu}_{lq}:=\|(\phi_{j}\otimes\psi_{k})\ast
S^{\mu\nu}\|_{L^{\infty}(\mathbb{R}^{n}\times\mathbb{R}^{q})},\quad l,q\geq
0.$

Applying Lemma 2.24 on $x$-variable and $s$-variable separately we get

$(fg)_{lq}=(\phi_{l}\otimes\psi_{q})\ast\Bigg{(}\sum\limits_{j=l-2}^{l+2}\sum\limits_{j^{\prime}=0}^{j-3}+\sum\limits_{j^{\prime}=l-2}^{l+2}\sum\limits_{j=0}^{j^{\prime}-3}+\sum\limits_{\begin{subarray}{c}|j-j^{\prime}|\leq
2\\\ j,j^{\prime}\geq
l-3\end{subarray}}\Bigg{)}\Bigg{(}\sum\limits_{k=q-2}^{q+2}\sum\limits_{k^{\prime}=0}^{k-3}+\sum\limits_{k^{\prime}=q-2}^{q+2}\sum\limits_{k=0}^{k^{\prime}-3}+\sum\limits_{\begin{subarray}{c}|k-k^{\prime}|\leq
2\\\ k,k^{\prime}\geq
q-3\end{subarray}}\Bigg{)}(f_{jk}g_{j^{\prime}k^{\prime}}).$ (2.13)

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

$\displaystyle|f_{jk}|\lesssim_{\phi,\psi}a_{jk}\cdot\|f\|_{\mathscr{C}^{\alpha}L^{\infty}\cap\mathscr{C}^{-\eta}\mathscr{C}^{\beta}},\quad|g_{jk}|\lesssim_{\phi,\psi}b_{jk}\cdot\|g\|_{\mathscr{C}^{\alpha-\theta}L^{\infty}\cap\mathscr{C}^{-\eta-\theta}\mathscr{C}^{\beta}},$
$\displaystyle\text{where }a_{jk}:=\begin{cases}2^{-j\alpha},&j\geq\frac{\beta
k}{\alpha+\eta}\\\ 2^{j\eta-k\beta},&j\leq\frac{\beta
k}{\alpha+\eta},\end{cases}\quad
b_{jk}:=\begin{cases}2^{-j(\alpha-\theta)},&j\geq\frac{\beta
k}{\alpha+\eta}\\\ 2^{j(\eta+\theta)-k\beta},&j\leq\frac{\beta
k}{\alpha+\eta},\end{cases}\quad\text{for }j,k\geq 0.$ (2.14)

For convenience we can normalize $f$ and $g$ so that
$\|f\|_{\mathscr{C}^{\alpha}L^{\infty}\cap\mathscr{C}^{-\eta}\mathscr{C}^{\beta}}=\|g\|_{\mathscr{C}^{\alpha-\theta}L^{\infty}\cap\mathscr{C}^{-\eta-\theta}\mathscr{C}^{\beta}}=1$.

Note that when $l\geq\frac{\beta q}{\alpha+\eta}$, by (2.11) we have

$|(fg)_{lq}|\lesssim
2^{-l(\alpha-\theta)}\|fg\|_{\mathscr{C}^{\alpha-\theta}\mathscr{C}^{0}}\lesssim
2^{-l(\alpha-\theta)}\|f\|_{\mathscr{C}^{\alpha}L^{\infty}\cap\mathscr{C}^{-\eta}\mathscr{C}^{\beta}}\|g\|_{\mathscr{C}^{\alpha-\theta}L^{\infty}\cap\mathscr{C}^{-\eta-\theta}\mathscr{C}^{\beta}}=2^{-l(\alpha-\theta)}\leq
2^{l(\eta+\theta)-q\beta}.$

So we only need to show that

$\textstyle\sup_{x\in\mathbb{R}^{n},s\in\mathbb{R}^{q}}|(fg)_{lq}(x,s)|\leq\sum_{\mu,\nu=1}^{3}S^{\mu\nu}_{lq}\lesssim_{\phi,\psi,\alpha,\beta,\theta,\eta}2^{l(\eta+\theta)-q\beta},\qquad\text{for
}l\leq\tfrac{\beta q}{\alpha+\eta}.$ (2.15)

We prove the control by estimating $S^{\mu\nu}_{lq}$ one by one. We use the
following facts for geometric sums,

$\sum_{j=u}^{v}2^{j\gamma}\leq\begin{cases}C_{\gamma}2^{u\gamma},&\gamma<0,\\\
v-u,&\gamma=0,\\\
C_{\gamma}2^{v\gamma},&\gamma>0,\end{cases}\quad\Rightarrow\quad\sum_{j=0}^{v}2^{j\gamma}\leq
C_{\gamma}\min(v2^{v\gamma},1),\quad\text{for }u,v\geq 0.$ (2.16)

For the first term $S^{11}_{lq}$, by (2.14) we have
$S^{11}_{lq}\leq\sum_{j=l-2}^{l+2}\sum_{j^{\prime}=0}^{j-3}\sum_{k=q-2}^{q+2}\sum_{k^{\prime}=0}^{k-3}a_{jk}b_{j^{\prime}k^{\prime}}$.
By (2.16) we know $\sum_{j=l-2}^{l+2}\sum_{k=q-2}^{q+2}a_{jk}\leq
C_{\alpha,\eta,\theta}\cdot a_{lq}$, so we only need to consider the sums for
$j^{\prime},k^{\prime}$ and omit $j,k$, as the following

$\displaystyle\sum_{j=l-2}^{l+2}\sum_{j^{\prime}=0}^{j-3}\sum_{k=q-2}^{q+2}\sum_{k^{\prime}=0}^{k-3}a_{jk}b_{j^{\prime}k^{\prime}}\lesssim
a_{lq}\sum_{j^{\prime}=0}^{l}\sum_{k^{\prime}=0}^{q}b_{j^{\prime}k^{\prime}}\leq
2^{l\eta-q\beta}\sum_{j^{\prime}=0}^{l}\Big{(}\sum_{k^{\prime}\leq\frac{\alpha+\eta}{\beta}j^{\prime}}2^{-j^{\prime}(\alpha-\theta)}+\sum_{\frac{\alpha+\eta}{\beta}j^{\prime}\leq
k^{\prime}\leq q}2^{j^{\prime}(\eta+\theta)-k^{\prime}\beta}\Big{)}$ (2.17)
$\displaystyle\lesssim
2^{l\eta-q\beta}\sum_{j^{\prime}=0}^{l}(j^{\prime}2^{-j^{\prime}(\alpha-\theta)}+2^{j^{\prime}(\eta+\theta)-j^{\prime}(\alpha+\eta)})\lesssim
2^{l\eta-q\beta}\leq 2^{l(\eta+\theta)-q\beta}.$

The deductions for the rest of the terms are similar, we sketch them as below.
Note that the assumption $\alpha-\eta-\theta>0$ implies
$\frac{2\alpha-\theta}{\alpha+\eta}>1$:

$\displaystyle S_{lq}^{12}\lesssim$ $\displaystyle\sum_{j\leq
l}\sum_{k^{\prime}\leq q}a_{jq}b_{lk^{\prime}}\leq\sum_{j\leq
l}2^{j\eta-q\beta}\Big{(}\sum_{k^{\prime}\leq\frac{\alpha+\eta}{\beta}l}2^{-l(\alpha-\theta)}+\sum_{\frac{\alpha+\eta}{\beta}l\leq
k^{\prime}\leq q}2^{l(\eta+\theta)-k^{\prime}\beta}\Big{)}$ (2.18)
$\displaystyle\lesssim
l2^{l\eta-q\beta}(l2^{-l(\alpha-\theta)}+2^{l(\eta+\theta)-l(\alpha+\eta)})=2^{l(\eta+\theta)-q\beta}(l^{2}2^{-l\alpha}+l2^{-l\alpha})\lesssim
2^{l(\eta+\theta)-q\beta}.$ $\displaystyle S_{lq}^{13}\lesssim$
$\displaystyle\sum_{l^{\prime}\geq l}\sum_{k^{\prime}\leq
q}a_{l^{\prime}q}b_{l^{\prime}k^{\prime}}\leq\sum_{l^{\prime}\leq\frac{\beta
q}{\alpha+\eta}}2^{l^{\prime}\eta-q\beta}\Big{(}\sum_{k^{\prime}\leq\frac{\alpha+\eta}{\beta}l^{\prime}}2^{-l^{\prime}(\alpha-\theta)}+\sum_{\frac{\alpha+\eta}{\beta}l^{\prime}\leq
k^{\prime}\leq
q}2^{l^{\prime}(\eta+\theta)-k^{\prime}\beta}\Big{)}+\sum_{l^{\prime}\geq\frac{\beta
q}{\alpha+\eta}}2^{-l^{\prime}\alpha}\sum_{k^{\prime}\leq
q}2^{-l^{\prime}(\alpha-\theta)}$
$\displaystyle\lesssim\sum_{l^{\prime}\leq\frac{\beta
q}{\alpha+\eta}}2^{l^{\prime}\eta-q\beta}(l^{\prime}2^{-l^{\prime}(\alpha-\theta)}+2^{-l^{\prime}(\alpha-\theta)})+\sum_{l^{\prime}\geq\frac{\beta
q}{\alpha+\eta}}2^{-l^{\prime}(2\alpha-\theta)}\lesssim
2^{-q\beta}\sum_{l^{\prime}\leq\frac{\beta
q}{\alpha+\eta}}l^{\prime}2^{-l^{\prime}(\alpha-\eta-\theta)}+2^{-\frac{2\alpha-\theta}{\alpha+\eta}\beta
q}\lesssim 2^{-q\beta}.$ $\displaystyle S_{lq}^{21}\lesssim$
$\displaystyle\sum_{j^{\prime}\leq l}\sum_{k\leq
q}a_{lk}b_{j^{\prime}q}\leq\Big{(}\sum_{k\leq\frac{\alpha+\eta}{\beta}l}2^{-l\alpha}+\sum_{\frac{\alpha+\eta}{\beta}l\leq
k\leq q}2^{l\eta-k\beta}\Big{)}\sum_{j^{\prime}\leq
l}2^{j^{\prime}(\eta+\theta)-q\beta}\lesssim(l2^{-l\alpha}+2^{-l\alpha})\cdot
l2^{l(\eta+\theta)-q\beta}\lesssim 2^{l(\eta+\theta)-q\beta}.$ $\displaystyle
S_{lq}^{22}\lesssim$ $\displaystyle\sum_{j\leq l}\sum_{k\leq
q}a_{jk}b_{lq}\leq\sum_{j,k=0}^{\infty}a_{jk}\cdot
2^{l(\eta+\theta)-q\beta}=\sum_{k=0}^{\infty}\Big{(}\sum_{j\geq\frac{\beta
k}{\alpha+\eta}}2^{-j\alpha}+\sum_{j\leq\frac{\beta
k}{\alpha+\eta}}2^{j\eta-k\beta}\Big{)}2^{l(\eta+\theta)-q\beta}\lesssim
2^{l(\eta+\theta)-q\beta}.$ $\displaystyle S_{lq}^{23}\lesssim$
$\displaystyle\sum_{l^{\prime}\geq l}\sum_{k\leq
q}a_{l^{\prime}k}b_{l^{\prime}q}\leq\sum_{l^{\prime}\leq\frac{\beta
q}{\alpha+\eta}}\Big{(}\sum_{k\leq\frac{\alpha+\eta}{\beta}l^{\prime}}2^{-l^{\prime}\alpha}+\sum_{\frac{\alpha+\eta}{\beta}l^{\prime}\leq
k\leq
q}2^{l^{\prime}\eta-k\beta}\Big{)}2^{l^{\prime}(\eta+\theta)-q\beta}+\sum_{l^{\prime}\geq\frac{\beta
q}{\alpha+\eta}}\sum_{k\leq
q}2^{-l^{\prime}\alpha}2^{-l^{\prime}(\alpha-\theta)}$
$\displaystyle\lesssim\sum_{l^{\prime}\leq\frac{\beta
q}{\alpha+\eta}}(l^{\prime}2^{-l^{\prime}\alpha}+2^{-l^{\prime}\alpha})\cdot
2^{l^{\prime}(\eta+\theta)-q\beta}+\sum_{l^{\prime}\geq\frac{\beta
q}{\alpha+\eta}}2^{-l^{\prime}(2\alpha-\theta)}\lesssim
2^{-q\beta}\sum_{l^{\prime}\leq\frac{\beta
q}{\alpha+\eta}}l^{\prime}2^{-l^{\prime}(\alpha-\eta-\theta)}+2^{-\frac{2\alpha-\theta}{\alpha+\eta}\beta
q}\lesssim 2^{-q\beta}.$ $\displaystyle S_{lq}^{31}\lesssim$
$\displaystyle\sum_{j^{\prime}\leq l}\sum_{q^{\prime}\geq
q}a_{lq^{\prime}}b_{j^{\prime}q^{\prime}}\leq\sum_{j^{\prime}\leq
l}\sum_{q^{\prime}\geq
q}2^{l\eta-q^{\prime}\beta}2^{j^{\prime}(\eta+\theta)-q^{\prime}\beta}\lesssim
l2^{l(2\eta+\theta)}\sum_{q^{\prime}\geq q}2^{-2q^{\prime}\beta}\lesssim
2^{l(\eta+\theta)}2^{-q\beta}(l2^{l\eta-q\beta})\overset{l\leq\frac{\beta
q}{\alpha+\eta}}{\lesssim}2^{l(\eta+\theta)-q\beta}.$ $\displaystyle
S_{lq}^{32}\lesssim$ $\displaystyle\sum_{j\leq l}\sum_{q^{\prime}\geq
q}a_{jq^{\prime}}b_{lq^{\prime}}\leq\sum_{j\leq l}\sum_{q^{\prime}\geq
q}2^{j\eta-q^{\prime}\beta}2^{l(\eta+\theta)-q^{\prime}\beta}\lesssim
l2^{l(2\eta+\theta)}\sum_{q^{\prime}\geq q}2^{-2q^{\prime}\beta}\lesssim
2^{l(\eta+\theta)}2^{-q\beta}(l2^{l\eta-q\beta})\lesssim
2^{l(\eta+\theta)-q\beta}.$ $\displaystyle S_{lq}^{33}\lesssim$
$\displaystyle\sum_{l^{\prime}\geq l}\sum_{q^{\prime}\geq
q}a_{l^{\prime}q^{\prime}}b_{l^{\prime}q^{\prime}}\leq\sum_{q^{\prime}\geq
q}\Big{(}\sum_{l\leq l^{\prime}\leq\frac{\beta
q^{\prime}}{\alpha+\eta}}2^{l^{\prime}\eta-q^{\prime}\beta}2^{l^{\prime}(\eta+\theta)-q^{\prime}\beta}+\sum_{l^{\prime}\geq\frac{\beta
q^{\prime}}{\alpha+\eta}}2^{-l^{\prime}\alpha}2^{-l^{\prime}(\alpha-\theta)}\Big{)}$
$\displaystyle\lesssim\sum_{q^{\prime}\geq
q}(q^{\prime}2^{q^{\prime}(\frac{2\eta+\theta}{\alpha+\eta}\beta-2\beta)}+2^{-\frac{2\alpha-\theta}{\alpha+\eta}\beta
q^{\prime}})=\sum_{q^{\prime}\geq
q}(q^{\prime}2^{-\frac{2\alpha-\theta}{\alpha+\eta}\beta
q^{\prime}}+2^{-\frac{2\alpha-\theta}{\alpha+\eta}\beta q^{\prime}})\lesssim
q2^{-\frac{2\alpha-\theta}{\alpha+\eta}\beta q}\lesssim 2^{-q\beta}\leq
2^{l(\eta+\theta)-q\beta}.$

We conclude that $S^{\mu\nu}_{lq}\lesssim 2^{l(\eta+\theta)-q\beta}$ for all
non-negatives $l\leq\frac{\beta q}{\alpha+\eta}$ and all $1\leq\mu,\nu\leq 3$.
Thus (2.15) holds and we have finished the proof. ∎

###  2.3 Compositions and inverse mapping with parameter

In this subsection we work on the concept of compositions between parameter
dependent maps. We are going to prove an auxiliary result, Proposition 2.27,
that is used in Proposition 4.3 (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 $s\in\mathbb{R}^{q}$.

######  Proposition 2.27.

Let $\alpha>1$ and $\beta\in(0,\alpha+1]$. Then for any $\varepsilon>0$ there
exists a $\delta=\delta(n,q,\alpha,\beta,\varepsilon)>0$ such that the
following is true:

Suppose
$F\in\mathscr{C}^{\alpha+1}L^{\infty}\cap\mathscr{C}^{1}\mathscr{C}^{\beta}(\mathbb{B}^{n},\mathbb{B}^{q};\mathbb{R}^{n})$
and
$g\in\mathscr{C}^{\alpha}L^{\infty}\cap\mathscr{C}^{0}\mathscr{C}^{\beta}(\mathbb{B}^{n},\mathbb{B}^{q})$
satisfy

$(F-\mathcal{I})|_{(\partial\mathbb{B}^{n})\times\mathbb{B}^{q}}\equiv
0\quad\text{and}\quad\|F-\mathcal{I}\|_{\mathscr{C}^{\alpha+1}L^{\infty}\cap\mathscr{C}^{1}\mathscr{C}^{\beta}(\mathbb{B}^{n},\mathbb{B}^{q};\mathbb{R}^{n})}+\|g\|_{\mathscr{C}^{\alpha}L^{\infty}\cap\mathscr{C}^{0}\mathscr{C}^{\beta}(\mathbb{B}^{n},\mathbb{B}^{q})}<\delta.$
(2.19)

Then $\tilde{F}(x,s):=(F(x,s),s)$ is bijective on
$\mathbb{B}^{n}\times\mathbb{B}^{q}$. Moreover, by setting
$\tilde{\Phi}(y,s):=\tilde{F}^{\mathrm{Inv}}(y,s)$ for
$(y,s)\in\mathbb{B}^{n}\times\mathbb{B}^{q}$, we have
$g\circ\tilde{\Phi}\in\mathscr{C}^{\alpha}L^{\infty}\cap\mathscr{C}^{-1}\mathscr{C}^{\beta}(\mathbb{B}^{n},\mathbb{B}^{q})$
and

$\|g\circ\tilde{\Phi}\|_{\mathscr{C}^{\alpha}L^{\infty}\cap\mathscr{C}^{-1}\mathscr{C}^{\beta}(\mathbb{B}^{n},\mathbb{B}^{q})}<\varepsilon.$
(2.20)

We postpone the proof to the end of this subsection.

######  Remark 2.28.

In Proposition 2.27 we may have
$g\circ\tilde{\Phi}\in\mathscr{C}^{\alpha}L^{\infty}\cap\mathscr{C}^{0}\mathscr{C}^{\beta}(\mathbb{B}^{n},\mathbb{B}^{q})$.
But in applications, the Proposition 4.7, our result in Proposition 2.27 is
enough.

The proof is a combination of quantitative controls for composition and
inverse mapping. Recall the quantitative results for the one-parameter cases:

######  Lemma 2.29.

Let $m,n\geq 1$ be integers. Let $\alpha>0$.

  1. (i)

Let $M>0$ and $\beta>0$, such that $\max(\alpha,\beta)>1$. There is a
$K_{1}=K_{1}(m,n,\alpha,\beta,M)>0$ satisfying the following:

Suppose $g\in\mathscr{C}^{\alpha}(\mathbb{B}^{n};\mathbb{R}^{m})$ satisfies
$g(\mathbb{B}^{n})\subseteq\mathbb{B}^{m}$ and
$\|g\|_{\mathscr{C}^{\alpha}(\mathbb{B}^{n};\mathbb{R}^{m})}\leq M$. Then for
every $f\in\mathscr{C}^{\beta}(\mathbb{B}^{m})$, we have $f\circ
g\in\mathscr{C}^{\min(\alpha,\beta)}(\mathbb{B}^{n})$ with $\|f\circ
g\|_{\mathscr{C}^{\min(\alpha,\beta)}(\mathbb{B}^{n})}\leq
K_{1}\|f\|_{\mathscr{C}^{\beta}(\mathbb{B}^{m})}$.

  2. (ii)

There is a $K_{2}(n,\alpha)>0$ satisfying the following:

Suppose $F\in\mathscr{C}^{\alpha+1}(\mathbb{B}^{n};\mathbb{R}^{n})$ satisfy
$\|F-\operatorname{id}\|_{\mathscr{C}^{\alpha+1}(\mathbb{B}^{n};\mathbb{R}^{n})}<K_{2}^{-1}$
and $F|_{\partial\mathbb{B}^{n}}\equiv\operatorname{id}$. Then $F$ is a
$\mathscr{C}^{\alpha+1}$-diffeomorphism on $\mathbb{B}^{n}$, and for the
inverse map $F^{\mathrm{Inv}}:\mathbb{B}^{n}\to\mathbb{B}^{n}$ we have
$\|F^{\mathrm{Inv}}\|_{\mathscr{C}^{\alpha+1}(\mathbb{B}^{n};\mathbb{R}^{n})}<K_{2}$.

###### Proof.

For the proof of (ii), see [SY21, Proposition 4.10].

For (i), see [Str21, Lemma 5.8] for the case $\alpha>1$. We prove the case
$0<\alpha\leq 1<\beta$ below.

By Lemma 2.2 (iii) we have $\|f\circ
g\|_{\mathscr{C}^{\alpha}(\mathbb{B}^{n})}\approx_{n,\alpha}\|f\circ
g\|_{C^{0}(\mathbb{B}^{n})}+\sup\limits_{x_{0}\neq
x_{1}}|\tfrac{f(g(x_{0}))+f(g(x_{1}))}{2}-f(g(\tfrac{x_{0}+x_{1}}{2}))|\cdot|x_{0}-x_{1}|^{-\alpha}$
and
$|\tfrac{f(g(x_{0}))+f(g(x_{1}))}{2}-f(\tfrac{g(x_{0})+g(x_{1})}{2})|\lesssim_{m,\beta}\|f\|_{\mathscr{C}^{\min(\beta,\frac{3}{2})}}|g(x_{0})-g(x_{1})|^{\min(\beta,\frac{3}{2})}$
for $x_{0},x_{1}\in\mathbb{B}^{n}$. Therefore

$\displaystyle|\tfrac{f(g(x_{0}))+f(g(x_{1}))}{2}-f(g(\tfrac{x_{0}+x_{1}}{2}))|\leq|\tfrac{f(g(x_{0}))+f(g(x_{1}))}{2}-f(\tfrac{g(x_{0})+g(x_{1})}{2})|+|f(\tfrac{g(x_{0})+g(x_{1})}{2})-f(g(\tfrac{x_{0}+x_{1}}{2}))|$
$\displaystyle\lesssim$
${}_{m,\beta}\|f\|_{\mathscr{C}^{\min(\beta,\frac{3}{2})}}|g(x_{0})-g(x_{1})|^{\min(\beta,\frac{3}{2})}+\|f\|_{C^{0,1}}|\tfrac{g(x_{0})+g(x_{1})}{2}-g(\tfrac{x_{0}+x_{1}}{2})|$
$\displaystyle\lesssim$
${}_{n,\alpha}\|f\|_{\mathscr{C}^{\min(\beta,\frac{3}{2})}}\|g\|_{\mathscr{C}^{\alpha/\min(\beta,\frac{3}{2})}}^{\min(\beta,\frac{3}{2})}|x_{0}-x_{1}|^{\alpha}+\|f\|_{C^{0,1}}\|g\|_{\mathscr{C}^{\alpha}}|x_{0}-x_{1}|^{\beta}\lesssim_{M}\|f\|_{\mathscr{C}^{\beta}}|x_{0}-x_{1}|^{\alpha}.$

In particular we can choose
$K_{1}(m,n,\alpha,\beta,M)\approx_{m,n,\alpha,\beta}(1+M)^{\min(\beta,\frac{3}{2})}$.
∎

######  Lemma 2.30.

Let $n,q\in\mathbb{Z}_{+}$, $\alpha>0$ 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 $K_{3}=K_{3}(n,q,\alpha,\beta)>1$ that
satisfies the following:

Let
$F\in\mathscr{C}^{\alpha+1,\beta}(\mathbb{B}^{n},\mathbb{B}^{q};\mathbb{R}^{n})$
satisfies
$\|F-\mathcal{I}\|_{\mathscr{C}^{\alpha+1,\beta}(\mathbb{B}^{n},\mathbb{B}^{q};\mathbb{R}^{n})}<K_{3}^{-1}$
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)

Define $\Phi(y,s):=F(\cdot,s)^{\mathrm{Inv}}(y)$ for $y\in\mathbb{B}^{n}$,
$s\in\mathbb{B}^{q}$. Then
$\Phi\in\mathscr{C}^{\alpha+1,\beta}(\mathbb{B}^{n},\mathbb{B}^{q};\mathbb{R}^{n})$
and satisfies

$\|\Phi\|_{\mathscr{C}^{\alpha+1,\beta}(\mathbb{B}^{n},\mathbb{B}^{q};\mathbb{R}^{n})}\leq
K_{3}.$ (2.21)

###### Proof.

We let $K_{3}$ be a large constant which may change from line to line.

Let $K_{2}=K_{2}(n,\alpha)>0$ be the constant in Lemma 2.29 (ii). So we can
choose $C_{1}>0$ such that

$\|F-\mathcal{I}\|_{\mathscr{C}^{\alpha+1,\beta}(\mathbb{B}^{n},\mathbb{B}^{q};\mathbb{R}^{n})}<C_{1}^{-1}\quad\Rightarrow\quad\|F(\cdot,s)-\operatorname{id}_{\mathbb{R}^{n}}\|_{\mathscr{C}^{\alpha+1}(\mathbb{B}^{n};\mathbb{R}^{n})}<K_{2}^{-1},\
\forall s\in\mathbb{B}^{q}.$

By assumption
$(F(\cdot,s)-\mathcal{I}(\cdot,s))|_{\partial\mathbb{B}^{n}}=(F(\cdot,s)-\operatorname{id}_{\mathbb{R}^{n}})|_{\partial\mathbb{B}^{n}}\equiv
0$, so by Lemma 2.29 (ii) $F(\cdot,s):\mathbb{B}^{n}\to\mathbb{B}^{n}$ is a
$\mathscr{C}^{\alpha+1}$-diffeomorphism for each $s\in\mathbb{B}^{n}$ and we
have

$\textstyle\|F-\mathcal{I}\|_{\mathscr{C}^{\alpha+1,\beta}(\mathbb{B}^{n},\mathbb{B}^{q};\mathbb{R}^{n})}<C_{1}^{-1}\quad\Rightarrow\quad\sup_{s\in\mathbb{B}^{q}}\|F(\cdot,s)^{\mathrm{Inv}}\|_{\mathscr{C}^{\alpha+1}(\mathbb{B}^{n};\mathbb{R}^{n})}<K_{2}.$
(2.22)

In particular $\Phi(y,s)$ is pointwise defined for
$(y,s)\in\mathbb{B}^{n}\times\mathbb{B}^{q}$. This finishes the proof of (i).

Let $c^{\prime}_{0}=c^{\prime}_{0}(\mathbb{B}^{n},n,\alpha)\in(0,1)$ be the
constant in Remark 2.23, we have, for every $s\in\mathbb{B}^{q}$,

$\|\nabla_{x}F(\cdot,s)-I_{n}\|_{\mathscr{C}^{\alpha}(\mathbb{B}^{n};\mathbb{R}^{n\times
n})}<c^{\prime}_{0}\ \Rightarrow\
\|(\nabla_{x}F(\cdot,s))^{-1}-I_{n}\|_{\mathscr{C}^{\alpha}(\mathbb{B}^{n};\mathbb{R}^{n\times
n})}<2c^{\prime}_{0}.$ (2.23)

Thus we can find a $C_{2}\geq\max(2c^{\prime}_{0}{}^{-1},C_{1},K_{2},1)$, such
that

$\|F-\mathcal{I}\|_{\mathscr{C}^{\alpha+1,\beta}(\mathbb{B}^{n},\mathbb{B}^{q};\mathbb{R}^{n})}<C_{2}^{-1}\
\Rightarrow\
\sup_{x\in\mathbb{B}^{n},s\in\mathbb{B}^{q}}|(\nabla_{x}F(x,s))^{-1}|\leq\sup_{s\in\mathbb{B}^{q}}\|(\nabla_{x}F(\cdot,s))^{-1}\|_{\mathscr{C}^{\alpha}(\mathbb{B}^{n};\mathbb{R}^{n\times
n})}<C_{2}.$ (2.24)

In particular
$\inf_{x\in\mathbb{B}^{n},s\in\mathbb{B}^{q}}|\det(\nabla_{x}F(x,s))^{-1}|>C_{2}^{-n}$.
Thus when $0<\beta<\min(\frac{2\alpha+1}{\alpha+1},\frac{3}{2})$ (in
particular when $0<\beta\leq\frac{3\alpha+1}{2\alpha+1}$), we get
$\|\Phi\|_{\mathscr{C}^{\alpha+1,\beta}}\lesssim_{n,q,\alpha,\beta}1$ by
applying Proposition A.1.

It remains to prove the case $\beta>\frac{3\alpha+1}{2\alpha+1}$. By chain
rule, we have

$0_{q\times
n}=\nabla_{s}(F(\Phi(y,s),s))=\big{(}(\nabla_{x}F)(\Phi(y,s),s)\big{)}\cdot\nabla_{s}\Phi(y,s)+(\nabla_{s}F)(\Phi(y,s),s).$

Therefore

$\nabla_{s}\Phi(y,s)=-((\nabla_{x}F)^{-1}\cdot\nabla_{s}F)\circ(\Phi(y,s),s).$
(2.25)

What we need is to show
$\|\nabla_{s}\Phi(y,\cdot)\|_{\mathscr{C}^{\beta-1}(\mathbb{B}^{q};\mathbb{R}^{q\times
n})}\lesssim_{n,q,\alpha,\beta}1$. We will proceed by induction on
$k=\lceil\beta-\frac{3\alpha+1}{2\alpha+1}\rceil$. The base case $k=1$ has
done since we have obtained $K_{3}(n,q,\alpha,\beta)>0$ for
$\beta\leq\frac{3\alpha+1}{2\alpha+1}$ from the above discussion.

Suppose the case $k\geq 1$ is done. Let $\beta$ satisfies
$\lceil\beta-\frac{3\alpha+1}{2\alpha+1}\rceil=k+1$. We denote
$\gamma(\beta):=\max(\beta-1,\frac{3\alpha+1}{2\alpha+1})>1$. By induction
hypothesis and
$\|\Phi\|_{\mathscr{C}^{\gamma(\beta)}_{(x,s)}}\lesssim\|\Phi\|_{\mathscr{C}^{\alpha+1,\gamma(\beta)}_{x,s}}$
there is a $C_{3}=C_{3}(n,q,\alpha,\beta)>0$ such that

$\|F-\mathcal{I}\|_{\mathscr{C}^{\alpha+1,\gamma(\beta)}}\leq
K_{3}(n,q,\alpha,\gamma(\beta))^{-1}\quad\Rightarrow\quad\|\Phi\|_{\mathscr{C}^{\gamma(\beta)}(\mathbb{B}^{n}\times\mathbb{B}^{q};\mathbb{R}^{n})}\leq
C_{3}.$ (2.26)

Applying Lemma 2.29 (i) to (2.25), taking $f$ to be the components of
$(\nabla_{x}F)^{-1}\cdot\nabla_{s}F\in\mathscr{C}^{\beta-1}(\mathbb{B}^{n}\times\mathbb{B}^{q};\mathbb{R}^{q\times
n})$, we have, when the assumption of (2.19) is satisfied,

$\displaystyle\|\nabla_{s}\Phi\|_{\mathscr{C}^{\beta-1}(\mathbb{B}^{n},\mathbb{B}^{q})}\lesssim$
$\displaystyle\sup_{x,s}\|((\nabla_{x}F)^{-1}\cdot\nabla_{s}F)(\Phi(y,\cdot),s)\|_{\mathscr{C}^{\beta-1}(\mathbb{B}^{q})}+\sup_{x,s}\|((\nabla_{x}F)^{-1}\cdot\nabla_{s}F)(\Phi(y,s),\cdot)\|_{\mathscr{C}^{\beta-1}(\mathbb{B}^{q})}$
(2.27) $\displaystyle\leq$ $\displaystyle
K_{1}(n,q,\gamma(\beta),\beta-1,C_{3})\sup_{s\in\mathbb{B}^{q}}\|(\nabla_{x}F)^{-1}\nabla_{s}F(\cdot,s)\|_{\mathscr{C}^{\beta-1}(\mathbb{B}^{n})}$
$\displaystyle+K_{1}(q,q,\gamma(\beta),\beta-1,\|\operatorname{id}_{\mathbb{B}^{q}}\|_{\mathscr{C}^{\gamma(\beta)}})\sup_{x\in\mathbb{B}^{n}}\|(\nabla_{x}F)^{-1}\nabla_{s}F(x,\cdot)\|_{\mathscr{C}^{\beta-1}(\mathbb{B}^{q})}.$

Similar to the argument of (2.24), by considering the constants
$c^{\prime}_{0}(\mathbb{B}^{n},n,\beta-1)$ and
$c^{\prime}_{0}(\mathbb{B}^{q},n,\beta-1)$ in Remark 2.23, we can find a
$C_{4}=C_{4}(n,q,\alpha,\beta)>\max(C_{2},C_{3},K_{3}(n,q,\alpha,\gamma(\beta)))$,
such that

$\|F-\mathcal{I}\|_{\mathscr{C}^{\alpha+1,\beta}(\mathbb{B}^{n},\mathbb{B}^{q};\mathbb{R}^{n})}<C_{4}^{-1}\
\Rightarrow\
\sup_{s\in\mathbb{B}^{q}}\|(\nabla_{x}F(\cdot,s))^{-1}\|_{\mathscr{C}^{\beta-1}(\mathbb{B}^{n};\mathbb{R}^{n\times
n})}+\sup_{x\in\mathbb{B}^{n}}\|(\nabla_{x}F(x,\cdot))^{-1}\|_{\mathscr{C}^{\beta-1}(\mathbb{B}^{q};\mathbb{R}^{n\times
n})}<C_{4}.$

Thus
$\|(\nabla_{x}F)^{-1}\nabla_{s}F\|_{\mathscr{C}^{\beta-1}(\mathbb{B}^{n}\times\mathbb{B}^{q};\mathbb{R}^{q\times
n})}\lesssim_{n,q,\alpha,\beta}1$. Combining this with (2.27) we see that
$\sup\limits_{y\in\mathbb{B}^{n}}\|\Phi(y,\cdot)\|_{\mathscr{C}^{\beta}(\mathbb{B}^{q};\mathbb{R}^{n})}\lesssim_{n,q,\alpha,\beta}1$.
Thus we get a $K_{3}(n,q,\alpha,\beta)$ for such $\beta$, finishing the
induction and hence the whole proof. ∎

To prove Proposition 2.27 we also need an auxiliary result for Laplacian.

######  Lemma 2.31.

Let $\alpha>-1$. Let $U\subset\mathbb{R}^{n}_{x}$ and
$V\subset\mathbb{R}^{q}_{s}$ be two bounded convex open sets with smooth
boundaries. Then the $x$-variable Laplacian
$\Delta_{x}=\sum_{j=1}^{n}\partial_{x^{j}}^{2}$ has a bounded linear inverse
$\mathcal{P}$ as the following map

$\Delta_{x}:\\{f\in\mathscr{C}^{\alpha+1}_{x}\mathscr{X}_{s}(U,V):f|_{(\partial
U)\times
V}=0\\}\to\mathscr{C}^{\alpha-1}_{x}\mathscr{X}_{s}(U,V),\quad\mathscr{X}\in\\{L^{\infty},\mathscr{C}^{\beta}:\beta\in\mathbb{R}\\}.$

In particular
$\mathcal{P}:\mathscr{C}^{\alpha-1}\mathscr{X}(U,V)\to\mathscr{C}^{\alpha+1}\mathscr{X}(U,V)$
is bounded linear.

###### Proof.

Indeed, for any $v\in\mathscr{C}^{(-2)+}(U)$ there is a unique continuous
$u\in C^{0}(\overline{U})$ satisfying $\Delta u=v$ and $u\big{|}_{\partial
U}=0$. Moreover the solution operator $[v\mapsto
u]:\mathscr{C}^{\mu-1}(U)\to\mathscr{C}^{\mu+1}(U)$ is bounded linear for all
$\mu>-1$. See [FR95, Theorem 15]. One can also see [GT01, Theorems 6.19 and
8.33] for the cases where $\mu$ is positive non-integer.

We can denote the above solution operator as $\mathcal{P}_{0}$. So
$\mathcal{P}_{0}:\mathscr{C}^{\alpha-1}(U)\to\mathscr{C}^{\alpha+1}(U)$ is
bounded linear. Therefore by Lemma 2.9,
$\mathcal{P}f(x,s):=\mathcal{P}_{0}(f(\cdot,s))(x)$ gives the bounded operator
$\mathcal{P}:\mathscr{C}^{\alpha-1}\mathscr{X}(U,V)\to\mathscr{C}^{\alpha+1}\mathscr{X}(U,V)$.
∎

###### Proof of Proposition 2.27.

We have
$\|F-\mathcal{I}\|_{\mathscr{C}^{\alpha+1,\beta}(\mathbb{B}^{n},\mathbb{B}^{q};\mathbb{R}^{n})}\lesssim\|F-\mathcal{I}\|_{\mathscr{C}^{\alpha+1}L^{\infty}\cap\mathscr{C}^{1}\mathscr{C}^{\beta}(\mathbb{B}^{n},\mathbb{B}^{q};\mathbb{R}^{n})}$
by Remark 2.13. Therefore there is a $\delta_{1}>0$ such that if (2.19) is
satisfied with $\delta=\delta_{1}$, then
$\|F-\mathcal{I}\|_{\mathscr{C}^{\alpha+1,\beta}(\mathbb{B}^{n},\mathbb{B}^{q};\mathbb{R}^{n})}\leq
K_{3}^{-1}$, wh ere $K_{3}=K_{3}(n,q,\alpha,\beta)>0$ is the constant in Lemma
2.30. Thus by Lemma 2.30 (i) $\Phi(y,s)=F(\cdot,s)^{\mathrm{Inv}}(y)$ is well-
defined and $\|\Phi\|_{\mathscr{C}^{\alpha+1,\beta}}\leq K_{3}$ holds.

Applying Lemma 2.29 (i) along with its constant $K_{1}$, we have for every
$f\in\mathscr{C}^{\alpha+1,\beta}(\mathbb{B}^{n},\mathbb{B}^{q})$,

$\displaystyle\textstyle\sup_{s\in\mathbb{B}^{q}}\|f\circ(\Phi(\cdot,s),s)\|_{\mathscr{C}^{\alpha+1}(\mathbb{B}^{n})}$
$\displaystyle\textstyle\leq
K_{1}(n,n,\alpha+1,\alpha+1,K_{3})\cdot\sup_{s\in\mathbb{B}^{q}}\|f(\cdot,s)\|_{\mathscr{C}^{\alpha+1}(\mathbb{B}^{n})};$
$\displaystyle\textstyle\sup_{y\in\mathbb{B}^{n},s\in\mathbb{B}^{q}}\|f\circ(\Phi(y,\cdot),s)\|_{\mathscr{C}^{\beta}(\mathbb{B}^{n})}$
$\displaystyle\textstyle\leq
K_{1}(n,q,\alpha+1,\beta,K_{3})\cdot\sup_{s\in\mathbb{B}^{q}}\|f(\cdot,s)\|_{\mathscr{C}^{\alpha+1}(\mathbb{B}^{n})};$
$\displaystyle\textstyle\sup_{y\in\mathbb{B}^{n},s\in\mathbb{B}^{q}}\|f\circ(\Phi(y,s),\cdot)\|_{\mathscr{C}^{\beta}(\mathbb{B}^{n})}$
$\displaystyle\textstyle\leq
K_{1}(q,q,\alpha+1,\beta,\|\operatorname{id}_{\mathbb{B}^{q}}\|_{\mathscr{C}^{\alpha+1}})\cdot\sup_{x\in\mathbb{B}^{n}}\|f(x,\cdot)\|_{\mathscr{C}^{\beta}(\mathbb{B}^{n})}.$

Therefore we can find a $C_{1}=C_{1}(n,q,\alpha,\beta)>0$ such that

$\|f\circ\tilde{\Phi}\|_{\mathscr{C}^{\alpha+1,\beta}(\mathbb{B}^{n},\mathbb{B}^{q})}\leq
C_{1}\|f\|_{\mathscr{C}^{\alpha+1,\beta}(\mathbb{B}^{n},\mathbb{B}^{q})},\quad\forall
f\in\mathscr{C}^{\alpha+1,\beta}(\mathbb{B}^{n},\mathbb{B}^{q}).$ (2.28)

In particular by taking $f$ to be the components of $F-\mathcal{I}$, we have
$\|\mathcal{I}-\Phi\|_{\mathscr{C}^{\alpha+1,\beta}(\mathbb{B}^{n},\mathbb{B}^{q};\mathbb{R}^{n})}\lesssim\|F-\mathcal{I}\|_{\mathscr{C}^{\alpha+1,\beta}(\mathbb{B}^{n},\mathbb{B}^{q};\mathbb{R}^{n})}$.
By
$\mathscr{C}^{\alpha+1,\beta}\subset\mathscr{C}^{\alpha+1}L^{\infty}\cap\mathscr{C}^{0}\mathscr{C}^{\beta}$
in Remark 2.13 and that $\nabla_{y}\mathcal{I}(y,s)=I_{n}$, we know
$\|\nabla_{y}\Phi-
I_{n}\|_{\mathscr{C}^{\alpha}_{y}L^{\infty}_{s}\cap\mathscr{C}^{-1}_{y}\mathscr{C}^{\beta}_{s}}\lesssim\|F-\mathcal{I}\|_{\mathscr{C}^{\alpha+1,\beta}_{x,s}}$.
Thus, we can find a $C_{2}=C_{2}(n,q,\alpha,\beta)>0$ such that

$\|\nabla_{y}\Phi-
I_{n}\|_{\mathscr{C}^{\alpha}L^{\infty}\cap\mathscr{C}^{-1}\mathscr{C}^{\beta}(\mathbb{B}^{n},\mathbb{B}^{q};\mathbb{R}^{n\times
n})}\leq
C_{2}\|F-\mathcal{I}\|_{\mathscr{C}^{\alpha+1,\beta}(\mathbb{B}^{n},\mathbb{B}^{q};\mathbb{R}^{n})},\text{
when }\eqref{Eqn::Hold::CompThm::Assum}\text{ is satisfied with
}\delta=\delta_{1}.$

By Corollary 2.22 (with $\eta=1$), we see that if
$C_{2}\|F-\mathcal{I}\|_{\mathscr{C}^{\alpha+1,\beta}(\mathbb{B}^{n},\mathbb{B}^{q};\mathbb{R}^{n})}$
is small enough, then
$(\nabla_{y}\Phi)^{-1}\in\mathscr{C}^{\alpha}L^{\infty}\cap\mathscr{C}^{-1}\mathscr{C}^{\beta}(\mathbb{B}^{n},\mathbb{B}^{q};\mathbb{R}^{n\times
n})$. Therefore we can find a $\delta_{2}\in(0,\delta_{1})$ such that

$\|(\nabla_{y}\Phi)^{-1}\|_{\mathscr{C}^{\alpha}L^{\infty}\cap\mathscr{C}^{-1}\mathscr{C}^{\beta}(\mathbb{B}^{n},\mathbb{B}^{q};\mathbb{R}^{n\times
n})}\leq\delta_{2}^{-1},\text{ when }\eqref{Eqn::Hold::CompThm::Assum}\text{
is satisfied with }\delta=\delta_{2}.$ (2.29)

Now for
$g\in\mathscr{C}^{\alpha}L^{\infty}\cap\mathscr{C}^{0}\mathscr{C}^{\beta}(\mathbb{B}^{n},\mathbb{B}^{q})$,
we take $G_{j}=\partial_{x^{j}}\mathcal{P}g$ where
$\mathcal{P}:\mathscr{C}^{\alpha}L^{\infty}\cap\mathscr{C}^{0}\mathscr{C}^{\beta}(\mathbb{B}^{n},\mathbb{B}^{q})\to\mathscr{C}^{\alpha+2}L^{\infty}\cap\mathscr{C}^{2}\mathscr{C}^{\beta}(\mathbb{B}^{n},\mathbb{B}^{q})$
is the operator in Lemma 2.31. Thus
$G=(G_{1},\dots,G_{n})\in\mathscr{C}^{\alpha+1}L^{\infty}\cap\mathscr{C}^{1}\mathscr{C}^{\beta}(\mathbb{B}^{n},\mathbb{B}^{q};\mathbb{R}^{n})$
and

$g\circ\tilde{\Phi}=(\operatorname{div}_{x}G)\circ\tilde{\Phi},\qquad(\nabla_{x}G)\circ\tilde{\Phi}=\nabla_{y}(G\circ\tilde{\Phi})\cdot(\nabla_{y}\Phi)^{-1}.$

By Corollary 2.21 (iii) we have

$\displaystyle\|g\circ\tilde{\Phi}\|_{\mathscr{C}^{\alpha}L^{\infty}\cap\mathscr{C}^{-1}\mathscr{C}^{\beta}(\mathbb{B}^{n},\mathbb{B}^{q})}$
$\displaystyle\lesssim\|\nabla_{y}(G\circ\tilde{\Phi})\|_{\mathscr{C}^{\alpha}L^{\infty}\cap\mathscr{C}^{-1}\mathscr{C}^{\beta}(\mathbb{B}^{n},\mathbb{B}^{q};\mathbb{R}^{n\times
n})}\|(\nabla_{y}\Phi)^{-1}\|_{\mathscr{C}^{\alpha}L^{\infty}\cap\mathscr{C}^{-1}\mathscr{C}^{\beta}(\mathbb{B}^{n},\mathbb{B}^{q};\mathbb{R}^{n\times
n})}.$

Therefore by (2.29), (2.28) and Remark 2.13, we have: when (2.19) is satisfied
with $\delta=\delta_{2}$,

$\displaystyle\|g\circ\tilde{\Phi}\|_{\mathscr{C}^{\alpha}L^{\infty}\cap\mathscr{C}^{-1}\mathscr{C}^{\beta}(\mathbb{B}^{n},\mathbb{B}^{q})}$
$\displaystyle\lesssim\|G\circ\tilde{\Phi}\|_{\mathscr{C}^{\alpha}L^{\infty}\cap\mathscr{C}^{0}\mathscr{C}^{\beta}(\mathbb{B}^{n},\mathbb{B}^{q};\mathbb{R}^{n})}\lesssim\|G\circ\tilde{\Phi}\|_{\mathscr{C}^{\alpha+1,\beta}(\mathbb{B}^{n},\mathbb{B}^{q};\mathbb{R}^{n})}$
$\displaystyle\lesssim\|G\|_{\mathscr{C}^{\alpha+1,\beta}(\mathbb{B}^{n},\mathbb{B}^{q};\mathbb{R}^{n})}\lesssim\|G\|_{\mathscr{C}^{\alpha+1}L^{\infty}\cap\mathscr{C}^{1}\mathscr{C}^{\beta}(\mathbb{B}^{n},\mathbb{B}^{q};\mathbb{R}^{n})}\lesssim\|g\|_{\mathscr{C}^{\alpha}L^{\infty}\cap\mathscr{C}^{0}\mathscr{C}^{\beta}(\mathbb{B}^{n},\mathbb{B}^{q})}.$

In other words, there is a $C_{3}=C_{3}(n,q,\alpha,\beta)>0$, such that

$\|g\circ\tilde{\Phi}\|_{\mathscr{C}^{\alpha}L^{\infty}\cap\mathscr{C}^{-1}\mathscr{C}^{\beta}(\mathbb{B}^{n},\mathbb{B}^{q})}\leq
C_{3}\|g\|_{\mathscr{C}^{\alpha}L^{\infty}\cap\mathscr{C}^{0}\mathscr{C}^{\beta}(\mathbb{B}^{n},\mathbb{B}^{q})},\quad\forall\
g\in\mathscr{C}^{\alpha}L^{\infty}\cap\mathscr{C}^{0}\mathscr{C}^{\beta}(\mathbb{B}^{n},\mathbb{B}^{q}).$

Now for every $\varepsilon>0$, take
$\delta=\min(C_{3}^{-1}\varepsilon,\delta_{2})$, we get (2.20) and complete
the proof. ∎

In applications, we also need a local version of the composition estimate.

######  Lemma 2.32.

Let $\alpha,\beta,\gamma,\delta>0$. Let $U_{1}\subseteq\mathbb{R}^{n}_{x}$,
$V_{1}\subseteq\mathbb{R}^{p}_{t}$, $U_{2}\subseteq\mathbb{R}^{q}_{y}$ and
$V_{2}\subseteq\mathbb{R}^{r}_{s}$ be four open sets.

  1. (i)

Suppose $\max(\alpha,\gamma)>1$ and $\max(\beta,\gamma)>1$. Let
$g\in\mathscr{C}^{\alpha,\beta}_{\mathrm{loc}}(U_{1},V_{1};U_{2})$ and
$f\in\mathscr{C}^{\gamma,\delta}_{\mathrm{loc}}(U_{2},V_{2})$. We define
$h:U_{1}\times V_{1}\times V_{2}\to\mathbb{R}$ as $h(x,t,s):=f(g(x,t),s)$.
Then
$h\in\mathscr{C}^{\min(\alpha,\gamma),\min(\beta,\gamma),\delta}_{\mathrm{loc}}(U_{1},V_{1},V_{2})$.

  2. (ii)

Let
$A\in\mathscr{C}^{\alpha,\beta}_{\mathrm{loc}}(U_{1},V_{1};\mathbb{C}^{m\times
m})$ be a matrix function such that $A(x,s)\in\mathbb{R}^{n\times n}$ is
invertible for every $(x,s)$. Then
$A^{-1}\in\mathscr{C}^{\alpha,\beta}_{\mathrm{loc}}(U_{1},V_{1};\mathbb{C}^{m\times
m})$ as well.

  3. (iii)

Let
$F\in\mathscr{C}^{\alpha+1,\beta}_{\mathrm{loc}}(U_{1},V_{1};\mathbb{R}^{n})$
and $(x_{0},t_{0})\in U_{1}\times V_{1}$. Suppose
$\nabla_{x}F(x_{0},t_{0})\in\mathbb{R}^{n\times n}$ is an invertible matrix.
Then there are neighborhoods $U_{1}^{\prime}\subseteq U_{1}$ of $x_{0}$,
$V_{1}^{\prime}\subseteq V_{1}$ of $t_{0}$ and
$\Omega_{1}\subseteq\mathbb{R}^{n}$ of $F(x_{0})$ such that

     * •

For each $t\in V^{\prime}_{1}$, $F(\cdot,t):U^{\prime}_{1}\to
F(U^{\prime}_{1},t)\subseteq\mathbb{R}^{n}$ is bijective and
$F(U^{\prime}_{1},t)\supseteq\Omega_{1}$.

     * •

Set $\Phi(x,t):=F(\cdot,t)^{\mathrm{Inv}}(x)$, then
$\Phi\in\mathscr{C}^{\alpha+1,\min(\alpha+1,\beta)}_{\mathrm{loc}}(\Omega_{1},V^{\prime}_{1};U^{\prime}_{1})$.

In particular for the map $\tilde{F}(x,t):=(F(x,t),t))$ we have that
$\tilde{F}:U_{1}^{\prime}\times
V_{1}^{\prime}\to\tilde{F}(U^{\prime}_{1}\times V^{\prime}_{1})$ is
homeomorphism.

###### Proof.

By passing to local and taking scaling, we can assume
$U_{1},V_{1},U_{2},V_{2}$ are all unit balls, and every maps have finite
norms.

For (i), what we need to show is that $\sup_{t\in V_{1},s\in
V_{2}}\|h(\cdot,t,s)\|_{\mathscr{C}^{\min(\alpha+1,\gamma)}(U_{1})}$,
$\sup_{x\in U_{1},s\in
V_{2}}\|h(x,\cdot,s)\|_{\mathscr{C}^{\min(\beta,\gamma)}(V_{1})}$ and
$\sup_{x\in U_{1},t\in V_{1}}\|h(x,y,\cdot)\|_{\mathscr{C}^{\delta}(V_{2})}$
are all finite.

Clearly $h\in\mathscr{C}^{\delta}_{s}L^{\infty}_{x,t}$ because
$\sup_{x,t}\|h(x,t,\cdot)\|_{\mathscr{C}^{\delta}(V_{2})}\leq\sup_{y\in
U_{2}}\|f(y,\cdot)\|_{\mathscr{C}^{\delta}(V_{2})}$.

By Lemma 2.29 (i) (since we assume $V_{1}$ and $U_{2}$ are balls) we have a
$C>0$ that depends on $U_{2},V_{1},\beta,\gamma$ and the upper bound of
$\sup_{s^{\prime}\in
V_{2}}\|g(\cdot,s^{\prime})\|_{\mathscr{C}^{\alpha}(U_{1})}$ but not on $t\in
V_{1}$ or $s\in V_{2}$, such that

$\|h(\cdot,t,s)\|_{\mathscr{C}^{\min(\alpha,\gamma)}(U_{1})}\leq
C\|f(\cdot,t)\|_{\mathscr{C}^{\gamma}(U_{2})}.$

Taking supremum over $t\in V_{1}$ and $s\in V_{2}$ we get $\sup_{t\in
V_{1},s\in
V_{2}}\|h(\cdot,t,s)\|_{\mathscr{C}^{\min(\alpha,\gamma)}(U_{1})}<\infty$.

Switching the role of $x$ and $t$ we get $\sup\limits_{x\in U_{1},s\in
V_{2}}\|h(x,\cdot,s)\|_{\mathscr{C}^{\min(\beta,\gamma)}(V_{2})}<\infty$,
finishing the proof of (i).

For (ii), by assumption $\det A:U_{1}\times V_{1}\to\mathbb{C}$ is non-
vanishing in the domain. By continuity and by shrinking $U_{1},V_{1}$ we can
assume that $\inf_{U\times V}|\det A|>c$ for some $c>0$. By [Str21, Lemma 5.7]
(where we take $X=\nabla$ in the reference), we get $\sup_{x}\|\det
A(x,\cdot)^{-1}\|_{\mathscr{C}^{\beta}_{s}}<\infty$ and $\sup_{s}\|\det
A(\cdot,s)^{-1}\|_{\mathscr{C}^{\alpha}_{x}}<\infty$. So $(\det
A)^{-1}\in\mathscr{C}^{\alpha,\beta}_{x,s}$.

By Cramer’s rule, the entries of $A^{-1}$ are the products of $(\det A)^{-1}$
and the entries of $A$, thus $A^{-1}\in\mathscr{C}^{\alpha,\beta}_{x,s}$.

For (iii), apply the Inverse Function Theorem on $F(\cdot,t_{0})$, we can find
a neighborhood $\tilde{U}_{1}\subseteq U_{1}$ of $x_{0}$ such that
$F(\cdot,t_{0}):\tilde{U}_{1}\to F(\tilde{U}_{1},t_{0})$ is a
$\mathscr{C}^{\alpha+1}$-diffeomorphism. Shrinking $\tilde{U}_{1}$ if
necessary, we can assume
$F(\cdot,t_{0}),F(\cdot,t_{0})^{\mathrm{Inv}}\in\mathscr{C}^{\alpha+1}(\tilde{U}_{1};\mathbb{R}^{n})$
both have bounded $C^{1}$-norm.

By (2.3) we have
$F\in\mathscr{C}^{\alpha+1}\mathscr{C}^{0}\cap\mathscr{C}^{0}\mathscr{C}^{\beta}\subset\mathscr{C}^{\frac{\alpha}{2}+1}_{x}\mathscr{C}^{\frac{\alpha\beta}{2(\alpha+1)}}_{s}$,
thus $[t\mapsto F(\cdot,t)]:V_{1}\to C^{1}(\tilde{U}_{1};\mathbb{R}^{n})$ is a
continuous map (in fact Hölder) with respect to the $C^{1}$-norm topology.
Since the set of bounded $C^{1}$-embedding map in
$C^{1}(\tilde{U}_{1};\mathbb{R}^{n})$ is open, we can now find a neighborhood
$\tilde{V}_{1}\subseteq V_{1}$ of $t_{0}$ such that $F(\cdot,t)$ is
$C^{1}$-embedding for all $t\in\tilde{V}_{1}$. By Inverse Function Theorem
again, the inverse map of $F(\cdot,t):\tilde{U}_{1}\to F(\tilde{U}_{1},t)$ is
also $\mathscr{C}^{\alpha+1}$.

Take $U^{\prime}_{1}:=\tilde{U}_{1}$ and take $\Omega_{1}$ such that
$F(x_{0},t_{0})\in\Omega_{1}\Subset F(U^{\prime}_{1},t_{0})$. By continuity
there is a neighborhood $V^{\prime}_{1}\subseteq\tilde{V}_{1}$ of $t_{0}$ such
that $\Omega_{1}\Subset F(U^{\prime}_{1},t)$ for all $t\in V^{\prime}_{1}$ as
well. Now we get $U^{\prime}_{1},V^{\prime}_{1},\Omega_{1}$ as desired and
$\Phi:\Omega_{1}\times V^{\prime}_{1}\to U^{\prime}_{1}$ is pointwise defined.

Since $F(\cdot,t)$ is $C^{1}$-embedding which is homeomorphic to its image,
the result $\sup_{t\in
V^{\prime}_{1}}\|\Phi(\cdot,t)\|_{\mathscr{C}^{\alpha+1}(\Omega_{1};\mathbb{R}^{n})}<\infty$
then follows from [Str21, Lemma 5.9].

When $\beta\leq 1$, the result
$\sup_{x\in\Omega_{1}}\|\Phi(x,\cdot)\|_{\mathscr{C}^{\beta}(V_{1};\mathbb{R}^{n})}<\infty$
follows from Proposition A.1 via a scaling. When $\beta>1$, applying Inverse
Function Theorem to the $\mathscr{C}^{\min(\alpha+1,\beta)}$-map $[(x,t)\in
U^{\prime}_{1}\times V^{\prime}_{1}\mapsto(F(x,t),t)]$, we see that its
inverse $[(x,t)\in\Omega_{1}\times V^{\prime}_{1}\mapsto(\Phi(x,t),t)]$ is
also $\mathscr{C}^{\min(\alpha+1,\beta)}$, which means
$\sup_{x\in\Omega_{1}}\|\Phi(x,\cdot)\|_{\mathscr{C}^{\min(\alpha+1,\beta)}(V_{1};\mathbb{R}^{n})}<\infty$.

Therefore either case we have
$\Phi\in\mathscr{C}^{\alpha+1}_{x}L^{\infty}_{t}\cap
L^{\infty}_{x}\mathscr{C}^{\min(\alpha+1,\beta)}_{t}=\mathscr{C}^{\alpha+1,\min(\alpha+1,\beta)}_{x,t}$.

Now $\tilde{F}$ and $\tilde{F}^{\mathrm{Inv}}$ are both
$\mathscr{C}^{\alpha+1,\beta}\subset C^{0}$ so $\tilde{F}$ is homeomorphic to
its image. ∎

##  3 The Vector Fields and Ordinary Differential Equations

###  3.1 Review of tangent subbundles and involutivity

In this part we give some conventions of notations and terminologies.

For $\kappa>1$, we can define $\mathscr{C}^{\alpha}$-maps between two
$\mathscr{C}^{\kappa}$-manifolds, since we have compositions
$\mathscr{C}^{\alpha}_{\mathrm{loc}}\circ\mathscr{C}^{\kappa}_{\mathrm{loc}}\subseteq\mathscr{C}^{\alpha}_{\mathrm{loc}}$
and
$\mathscr{C}^{\kappa}_{\mathrm{loc}}\circ\mathscr{C}^{\alpha}_{\mathrm{loc}}\subseteq\mathscr{C}^{\alpha}_{\mathrm{loc}}$.

######  Notation 3.1.

Let $\kappa>1$, $\alpha\in(0,\kappa]$, and let $\mathfrak{M},\mathfrak{N}$ be
two $\mathscr{C}^{\kappa}$-differential manifolds. We use
$\mathscr{C}^{\alpha}_{\mathrm{loc}}(\mathfrak{M};\mathfrak{N})$ as the set of
all (locally) $\mathscr{C}^{\alpha}$-maps from $\mathfrak{M}$ to
$\mathfrak{N}$.

We can define the mixed regularity $\mathscr{C}^{\alpha,\beta}$ on product
manifolds.

######  Definition 3.2.

Let $\kappa>1$ and $\alpha,\beta\in(0,\kappa]$. Let
$\mathfrak{M}^{m},\mathfrak{N}^{n},\mathfrak{P}^{q}$ be three
$\mathscr{C}^{\kappa}$-manifolds.

We say a continuous map $f:\mathfrak{M}\times\mathfrak{N}\to\mathfrak{P}$ is
$\mathscr{C}^{\alpha,\beta}_{\mathrm{loc}}$, denoted by
$f\in\mathscr{C}^{\alpha,\beta}_{\mathrm{loc}}(\mathfrak{M},\mathfrak{N};\mathfrak{P})$,
if for every coordinate charts $\phi:U\subseteq\mathfrak{M}\to\mathbb{R}^{m}$,
$\psi:V\subseteq\mathfrak{N}\to\mathbb{R}^{n}$ and
$\rho:W\subseteq\mathfrak{P}\to\mathbb{R}^{q}$ that satisfy $f(U\times
V)\subseteq W$, we have

$\rho\circ
f\circ(\phi^{\mathrm{Inv}},\psi^{\mathrm{Inv}})\in\mathscr{C}^{\alpha,\beta}_{\mathrm{loc}}(\phi(U),\psi(V);\mathbb{R}^{q}).$
(3.1)

######  Remark 3.3.

  1. (i)

The definition still has symmetry for indices:
$\mathscr{C}^{\alpha,\beta}_{\mathrm{loc}}(\mathfrak{M},\mathfrak{N};\mathfrak{P})=\mathscr{C}^{\beta,\alpha}_{\mathrm{loc}}(\mathfrak{N},\mathfrak{M};\mathfrak{P})$.

  2. (ii)

To check $f\in\mathscr{C}^{\alpha,\beta}_{\mathrm{loc}}$ it suffices to show
(3.1) is valid on a coordinate cover $\\{(\phi_{i},\psi_{i},\rho_{i})\\}_{i\in
I}$ for $\mathfrak{M}\times\mathfrak{N}\times
f(\mathfrak{M}\times\mathfrak{N})\subseteq\mathfrak{M}\times\mathfrak{N}\times\mathfrak{P}$
such that $f(U_{i}\times V_{i})\subseteq W_{i}$ for each $i\in I$. The proof
can be done using Lemma 2.32 (i). Such cover always exists because of the
continuity of $f$. We leave the proof to readers.

  3. (iii)

In applications we only consider differentiable maps localized near a fixed
point. By passing to a precompact coordinate chart near a point, everything is
reduced to the case of Euclidean space, and by shrinking the domains every
functions or vector fields would have bounded Hölder-Zygmund regularities.

Definition 3.2 is applied for labeling component-wise regularity for vector
fields and parameterizations with parameters.

######  Definition 3.4.

Let $1<\alpha\leq\kappa$ and let $\mathfrak{M}$ be a $m$-dimensional
$\mathscr{C}^{\kappa}$-manifold. We say
$\Phi:\Omega\subseteq\mathbb{R}^{m}\to\mathfrak{M}$ is a
$\mathscr{C}^{\alpha}$-regular parameterization, if $\Phi$ is injective and
$\Phi^{\mathrm{Inv}}:\Phi(\Omega)\subseteq\mathfrak{M}\to\mathbb{R}^{m}$ is a
$\mathscr{C}^{\alpha}$-coordinate chart.

When $\kappa>2$, the (real) tangent bundle $T\mathfrak{M}$ of a
$\mathscr{C}^{\kappa}$-manifold $\mathfrak{M}$ has a natural
$\mathscr{C}^{\kappa-1}$-differential structure. Its complexification, the
complex tangent bundle
$\mathbb{C}T\mathfrak{M}:=T\mathfrak{M}\otimes\mathbb{C}=T\mathfrak{M}\oplus
iT\mathfrak{M}$ is also a $\mathscr{C}^{\kappa-1}$-differential manifold. So
we can define (real or complex) $\mathscr{C}^{\alpha}$-vector fields when
$\alpha\in(0,\kappa-1]$.

Similarly, the cotangent bundle $T^{*}\mathfrak{M}$ and the complex cotangent
bundle $\mathbb{C}T^{*}\mathfrak{M}$ are also
$\mathscr{C}^{\kappa-1}$-manifolds.

######  Definition 3.5.

Let $\kappa>2$, $0<\alpha\leq\kappa$ and let $\mathfrak{M}$ be a
$\mathscr{C}^{\kappa}$-manifold. A $\mathscr{C}^{\alpha}$-complex tangent
subbundle of rank $r$ is a subset
$\mathcal{S}\subseteq\mathbb{C}T\mathfrak{M}$ satisfying that for any
$p\in\mathfrak{M}$,

  * •

$\mathcal{S}_{p}:=\mathcal{S}\cap\mathbb{C}T_{p}\mathfrak{M}$ is a
$r$-dimensional complex vector subspace.

  * •

There is an open neighborhood $U\subseteq\mathfrak{M}$ of $p$ and
$\mathscr{C}^{\alpha}$-complex vector fields $X_{1},\dots,X_{r}$ on $U$, such
that $X_{1}|_{q},\dots,X_{r}|_{q}$ form a complex linear basis for
$\mathcal{S}_{q}$ for all $q\in U$.

We denote it as $\mathcal{S}\leq\mathbb{C}T\mathfrak{M}$. And we say
$(X_{1},\dots,X_{r})$ form a $\mathscr{C}^{\alpha}$-local basis for
$\mathcal{V}$ (or $\mathcal{S}$) on $U$.

In the paper we do not deal with abstract vector bundle, we will use
“subbundle” as the abbreviation of (complex) tangent subbundle.

The real tangent subbundles can be defined in the same way through replacing
every $\mathbb{C}T\mathfrak{M}$ by $T\mathfrak{M}$. But we focus mostly on
complex subbundles since every real subbundle $\mathcal{V}\leq T\mathfrak{M}$
can be identified as a complex subbundle
$\mathcal{V}\otimes\mathbb{C}\leq\mathbb{C}T\mathfrak{M}$.

Alternatively a rank $r$ $\mathscr{C}^{\alpha}$-subbundle
$\mathcal{S}\leq\mathbb{C}T\mathfrak{M}$ can be equivalently defined as a
$\mathscr{C}^{\alpha}$-section of the complex Grassmannian bundle
$\mathbf{Gr}_{\mathbb{C}}(r,\mathbb{C}T\mathfrak{M})=\coprod_{p\in\mathfrak{M}}\\{\text{rank
}r\text{ complex spaces of }\mathbb{C}T_{p}\mathfrak{M}\\}$. For their
equivalence we leave the proof to readers.

In Theorem 5.1 we consider tangent subbundles with mixed regularities.

######  Definition 3.6.

Let $0<\alpha,\beta\leq\kappa$ and let $\mathfrak{M},\mathfrak{N}$ be two
$\mathscr{C}^{\kappa}$-manifolds. A $\mathscr{C}^{\alpha,\beta}$-complex
tangent subbundle of rank $r$ is a
$\mathscr{C}^{\min(\alpha,\beta)}$-subbundle
$\mathcal{S}\leq\mathbb{C}T(\mathfrak{M}\times\mathfrak{N})$ such that, for
any $(u_{0},v_{0})\in\mathfrak{M}\times\mathfrak{N}$ there is a neighborhood
$U\times V\subseteq\mathfrak{M}\times\mathfrak{N}$ of $(u_{0},v_{0})$ and
complex vector fields
$X_{1},\dots,X_{r}\in\mathscr{C}^{\alpha,\beta}_{\mathrm{loc}}(U,V;\mathbb{C}T(\mathfrak{M}\times\mathfrak{N}))$,
such that $X_{1}|_{(u,v)},\dots,X_{r}|_{(u,v)}$ form a complex linear basis
for $\mathcal{S}_{(u,v)}$ for all $(u,v)\in U\times V$.

Equivalently this is saying
$\mathcal{S}\in\mathscr{C}^{\alpha,\beta}_{\mathrm{loc}}(\mathfrak{M},\mathfrak{N};\mathbf{Gr}_{\mathbb{C}}(r,\mathbb{C}T(\mathfrak{M}\times\mathfrak{N})))$
as a section of the Grassmannian bundle.

Recall the notion of involutive structure in Definition 1.5. Let
$x=(x^{1},\dots,x^{n})$ be a local coordinate system, for (at least $C^{1}$)
vector fields $X=:\sum_{j=1}^{n}X^{j}\frac{\partial}{\partial x^{j}}$ and
$Y=:\sum_{k=1}^{n}Y^{k}\frac{\partial}{\partial x^{k}}$, we recall that

$[X,Y]=\sum_{j,k=1}^{n}\Big{(}X^{j}\frac{\partial Y^{k}}{\partial
x^{j}}-Y^{j}\frac{\partial X^{k}}{\partial
x^{j}}\Big{)}\frac{\partial}{\partial x^{k}}.$ (3.2)

For completeness we give definition of the involutivity on subbundles with
mixed regularities.

######  Definition 3.7.

Let $\kappa>2$, $\alpha\in(1,\kappa]$ and $\beta\in(0,\kappa]$. Let
$\mathfrak{M},\mathfrak{N}$ be two $\mathscr{C}^{\kappa}$-manifolds and
$\mathcal{S}$ be a $\mathscr{C}^{\alpha,\beta}$-complex subbundle such that
$\mathcal{S}\leq(\mathbb{C}T\mathfrak{M})\times\mathfrak{N}$. We say
$\mathcal{S}$ is involutive, if for every $C^{0}$-sections
$X,Y:\mathfrak{M}\times\mathfrak{N}\to\mathcal{S}$ of $\mathcal{S}$ that are
$C^{1}_{\mathrm{loc}}$ in $\mathfrak{M}$, the commutator $[X,Y]$ is a
$C^{0}$-section of $\mathcal{S}$.

######  Remark 3.8.

Definition 3.7 does not require $X$ or $Y$ to be differentiable on
$\mathfrak{N}$, since when
$\mathcal{S}\leq(\mathbb{C}T\mathfrak{M})\times\mathfrak{N}$ the
differentiations in (3.2) are only taken on $\mathfrak{M}$ but not on
$\mathfrak{N}$. This is critical for the case $\beta\leq 1$.

An important fact for involutive structures is that we can pick a collection
of good basis.

######  Lemma 3.9 (Canonical local basis).

Let $\alpha,\beta>0$, $\kappa\geq\max(\alpha,\beta)+1$. Let $\mathfrak{M}$ and
$\mathfrak{N}$ be two $\mathscr{C}^{\kappa}$-manifolds. Let $\mathcal{S}$ be a
$\mathscr{C}^{\alpha,\beta}$-complex subbundle over
$\mathfrak{M}\times\mathfrak{N}$ such that
$\mathcal{S}\leq(\mathbb{C}T\mathfrak{M})\times\mathfrak{N}$.

Let $p\in\mathfrak{M}$ and $\lambda_{0}\in\mathfrak{N}$. Then there are
numbers $r,m,q\geq 0$ and a (mixed real and complex)
$\mathscr{C}^{\kappa}$-coordinate system
$(t,z,s)=(t^{1},\dots,t^{r},z^{1},\dots,z^{m},s^{1},\dots,s^{q}):\tilde{U}\subseteq\mathfrak{M}\to\mathbb{R}^{r}\times\mathbb{C}^{m}\times\mathbb{R}^{q}$
near $p$, such that

$\mathcal{S}_{(p,\lambda_{0})}\oplus\operatorname{Span}(\partial_{\bar{z}^{1}}|_{p},\dots,\partial_{\bar{z}^{m}}|_{p},\partial_{s^{1}}|_{p},\dots,\partial_{s^{q}}|_{p})=\mathbb{C}T_{p}\mathfrak{M}$
(3.3)

Moreover there are a neighborhood $U\subseteq\tilde{U}$ of $p$, a neighborhood
$V\subseteq\mathfrak{N}$ of $\lambda_{0}$, and a
$\mathscr{C}^{\min(\alpha,\beta)}$-local basis
$X=[X_{1},\dots,X_{r+m}]^{\top}$ for $\mathcal{S}$ on $U\times V$, such that
the collection vectors $X^{\prime}=[X_{1},\dots,X_{r}]^{\top}$ and
$X^{\prime\prime}=[X_{r+1},\dots,X_{r+m}]^{\top}$ are of the form

$\begin{pmatrix}X^{\prime}\\\
X^{\prime\prime}\end{pmatrix}=\begin{pmatrix}I_{r}&&A^{\prime}&B^{\prime}\\\
&I_{m}&A^{\prime\prime}&B^{\prime\prime}\end{pmatrix}\begin{pmatrix}\partial/\partial
t\\\ \partial/\partial z\\\ \partial/\partial{\bar{z}}\\\ \partial/\partial
s\end{pmatrix},$ (3.4)

where
$A^{\prime}\in\mathscr{C}^{\alpha,\beta}_{\mathrm{loc}}(U,V;\mathbb{C}^{r\times
m})$,
$A^{\prime\prime}\in\mathscr{C}^{\alpha,\beta}_{\mathrm{loc}}(U,V;\mathbb{C}^{m\times
m})$,
$B^{\prime}\in\mathscr{C}^{\alpha,\beta}_{\mathrm{loc}}(U,V;\mathbb{C}^{r\times
q})$ and
$B^{\prime\prime}\in\mathscr{C}^{\alpha,\beta}_{\mathrm{loc}}(U,V;\mathbb{C}^{m\times
q})$. In addition,

  1. (i)

Such $X_{1},\dots,X_{r+m}$ are uniquely determined by the coordinate system
$(t,z,s)$.

  2. (ii)

If $\mathcal{S}$ is involutive, then $X_{1},\dots,X_{r+m}$ are pairwise
commutative.

  3. (iii)

Suppose $\mathcal{S}=\bar{\mathcal{S}}$, then we must have $m=0$,
$A^{\prime}=0$, $A^{\prime\prime}=0$, $B^{\prime\prime}=0$ and $B^{\prime}$ is
real-valued.

###### Proof.

Clearly $\mathcal{S}_{(p,\lambda_{0})}\cap\bar{\mathcal{S}}_{(p,\lambda_{0})}$
and $\mathcal{S}_{(p,\lambda_{0})}+\bar{\mathcal{S}}_{(p,\lambda_{0})}$ are
both complex linear subspaces of $\mathbb{C}T_{p}\mathfrak{M}$. We have
$r:=\operatorname{rank}(\mathcal{S}_{(p,\lambda_{0})}\cap\bar{\mathcal{S}}_{(p,\lambda_{0})})$,
$m:=\operatorname{rank}\mathcal{S}_{(p,\lambda_{0})}-r$ and
$q:=\dim\mathfrak{M}-\operatorname{rank}(\mathcal{S}_{(p,\lambda_{0})}+\bar{\mathcal{S}}_{(p,\lambda_{0})})$.
Note that $\dim\mathfrak{M}=r+2m+q$. See [BCH08, Lemma I.8.5] for details of
(3.3).

The formula (3.4) follows from the standard linear algebra argument. See
[Lun92, Lemma 1] or [BCH08, Page 18]. Note that by Lemma 2.32 (ii), if