# Improving the Regularity of Vector Fields

Brian Street111The first author was partially supported by National Science
Foundation Grant No. 1764265. and Liding Yao222Corresponding Author.
<EMAIL_ADDRESS>
###### Abstract

Let $\alpha>0$, $\beta>\alpha$, and let $X_{1},\ldots,X_{q}$ be
$\mathscr{C}^{\alpha}_{\mathrm{loc}}$ vector fields on a
$\mathscr{C}^{\alpha+1}$ manifold which span the tangent space at every point,
where $\mathscr{C}^{s}$ denotes the Zygmund-Hölder space of order $s$. We give
necessary and sufficient conditions for when there is a
$\mathscr{C}^{\beta+1}$ structure on the manifold, compatible with its
$\mathscr{C}^{\alpha+1}$ structure, with respect to which $X_{1},\ldots,X_{q}$
are $\mathscr{C}^{\beta}_{\mathrm{loc}}$. This strengthens previous results of
the first author which dealt with the setting $\alpha>1$,
$\beta>\max\\{\alpha,2\\}$.

##  1 Introduction

Fix $\alpha>0$ and let $X_{1},\ldots,X_{q}$ be
$\mathscr{C}^{\alpha}_{\mathrm{loc}}$ vector fields on a
$\mathscr{C}^{\alpha+1}$ manifold $\mathfrak{M}$ of dimension $n$, which span
the tangent space at every point, where $\mathscr{C}^{\alpha}$ denotes the
Zygmund-Hölder space of order $\alpha$.333For $m\in\mathbb{N}$ and
$\alpha\in(0,1)$, $\mathscr{C}^{m+\alpha}$ coincides with the usual Hölder
space $C^{m,\alpha}$. For $\alpha\in\\{0,1\\}$, these spaces differ:
$C^{m+1,0}\subsetneq C^{m,1}\subsetneq\mathscr{C}^{m+1}$. In this paper, we
investigate the following:

######  Question 1.

Fix $\beta\in[\alpha,\infty)$. When is there a $\mathscr{C}^{\beta+1}$
manifold structure on $\mathfrak{M}$, compatible with its
$\mathscr{C}^{\alpha+1}$ structure, respect to which $X_{1},\ldots,X_{q}$ are
$\mathscr{C}^{\beta}_{\mathrm{loc}}$?

Question 1 is local in nature, so we focus instead on the following local
version:

######  Question 2.

Fix $\beta\in[\alpha,\infty)$ and $x\in\mathfrak{M}$. When is there a
neighborhood $U\subseteq\mathfrak{M}$ of $x$ and a
$\mathscr{C}_{\mathrm{loc}}^{\alpha+1}$ diffeomorphism
$\Phi:\mathbb{B}^{n}\xrightarrow{\sim}U$, such that
$\Phi^{*}X_{1},\ldots,\Phi^{*}X_{q}$ are $\mathscr{C}^{\beta}$ vector fields
on $\mathbb{B}^{n}$? Here, $\mathbb{B}^{n}$ denotes the open unit ball in
$\mathbb{R}^{n}$.

We give necessary and sufficient conditions on $X_{1},\ldots,X_{q}$ for when
Question 2 has an affirmative answer; and therefore give necessary and
sufficient conditions for when Question 1 has an affirmative answer.

When $\alpha>1$ and $\beta>2$, Questions 1 and 2 were completely answered in
work of the first author and Stovall [SS18, Str21, Str20a]; which also proved
stronger quantitative results (see Section 1.3 for our distinction between
quantitative and qualitative results). In this paper, by focusing only on the
qualitative Questions 1 and 2 we are able to prove results for all $\alpha>0$,
$\beta\geq\alpha$, and the proof is simpler. Our methods can also be used to
improve the quantitative results of [SS18, Str21, Str20a]: see Section 9.

###  1.1 Informal Statement of Results

Much of the difficulty in this paper comes from working with $\alpha$ and
$\beta$ small. In this section, we informally describe the results without
worrying about such difficulties.

We begin with the case when $\beta=\infty$ and $X_{1},\ldots,X_{q}$ are
$C^{1}_{\mathrm{loc}}$ vector fields on a $C^{2}$ manifold $\mathfrak{M}$ of
dimension $n$, which span the tangent space at every point. This is a special
cases of results in [Str21].

Because $X_{1},\ldots,X_{q}$ are $C^{1}$ vector fields which span the tangent
space at every point, we may write

$[X_{i},X_{j}]=\sum_{k=1}^{q}c_{i,j}^{k}X_{k},\quad
c_{i,j}^{k}:\mathfrak{M}\rightarrow\mathbb{R}\text{ is continuous}.$

######  Theorem 1.1 ([Str21]).

Fix $x\in\mathfrak{M}$. The following are equivalent:

  1. (i)

There is a neighborhood $U\subseteq\mathfrak{M}$ of $x$ and a
$C^{2}_{\mathrm{loc}}$ diffeomoprhism
$\Phi:\mathbb{B}^{n}\xrightarrow{\sim}U$, such that
$\Phi^{*}X_{1},\ldots,\Phi^{*}X_{q}$ are $C^{\infty}_{\mathrm{loc}}$ vector
fields on $\mathbb{B}^{n}$.

  2. (ii)

The functions $c_{i,j}^{k}$ can be chosen so that the following holds. There
is a neighborhood $V\subseteq\mathfrak{M}$ of $x$ such that for every
$L\in\mathbb{N}$ and every list $l_{1},\ldots,l_{L}\in\\{1,\ldots,q\\}$, the
functions $X_{l_{1}}X_{l_{2}}\cdots
X_{l_{L}}c_{i,j}^{k}\big{|}_{V}:V\rightarrow\mathbb{R}$ are continuous. We
will write this condition as $c_{i,j}^{k}\big{|}_{V}\in C^{\infty}_{X}(V)$.

In this paper, $X_{1},\ldots,X_{q}$ are $\mathscr{C}^{\alpha}_{\mathrm{loc}}$
vector fields on a $\mathscr{C}^{\alpha+1}$ manifold, $\mathfrak{M}$.
Informally, we wish our main result to say that the following are equivalent:

  1. (a)

There is a neighborhood $U\subseteq\mathfrak{M}$ of $x$ and a
$\mathscr{C}^{\alpha+1}_{\mathrm{loc}}$ diffeomoprhism
$\Phi:\mathbb{B}^{n}\xrightarrow{\sim}U$, such that
$\Phi^{*}X_{1},\ldots,\Phi^{*}X_{q}$ are $\mathscr{C}^{\beta}_{\mathrm{loc}}$
vector fields on $\mathbb{B}^{n}$.

  2. (b)

The $c_{i,j}^{k}$ can be chosen such that
$c_{j,k}^{l}\big{|}_{V}\in\mathscr{C}^{\beta-1}_{X}(V)$ for some neighborhood
$V\subseteq\mathfrak{M}$ of $x$, where $\mathscr{C}^{\beta-1}_{X}(V)$ is an
appropriate Zygmund-Hölder space with respect to the vector fields
$X_{1},\ldots,X_{q}$.

When $\alpha>1$ and $\beta>2$, this equivalence was proved in [Str21]; the
main result of this paper gives an extension of this result to all
$\beta\geq\alpha>0$. Unfortunately, for $\alpha>0$ small, the commutator of
$[X_{j},X_{k}]$ does not even immediately make sense and many of the usual
operations on $\mathfrak{M}$ do not make immediate sense. Thus, much of this
paper is devoted to making sense of conditions similar to b in such low
regularity. As we will see, this is a bit easier when $\alpha>1/2$ and so our
results take a different form depending on whether $\alpha\in(0,1/2]$ or
$\alpha>1/2$.

###  1.2 Relation to Results of DeTurck and Kazdan

The results in this paper may be reminiscent of the celebrated results of
DeTurck and Kazdan [DK81] regarding a coordinate system in which a Riemannian
metric tensor has optimal regularity. It seems that there are no direct
implications between our results and their results; however there are many
similarities. We present this in more detail in Section 8; there the following
ideas are discussed.

DeTurck and Kazdan showed that a Riemannian metric tensor has optimal
regularity in harmonic coordinates [DK81, Lemma 1.2]. Analogously, we present
a natural Riemannian metric associated to vector fields $X_{1},\ldots,X_{n}$
(which form a basis of the tangent space at every point) such that
$X_{1},\ldots,X_{n}$ have optimal regularity in harmonic coordinates with
respect to this metric.

DeTurck and Kazdan also showed that a Riemannian metric tensor may not have
optimal regularity in geodesic normal coordinates [DK81, Example 2.3].
Analogously, we show that vector fields may not have optimal regularity in
canonical coordinates of the first kind.

However, the heart of our main result is not just to provide a coordinate
system in which vector fields have optimal regularity. Instead, we provide a
test to determine what that optimal regularity is. This test can be carried
out in any coordinate system and does not require solving any differential
equations.

Both this work and [DK81] use methods introduced by Malgrange [Mal69].

######  Remark 1.2.

It may be somewhat unexpected that vector fields may not have optimal
regularity in canonical coordinates of the first kind. Indeed, there is a long
history of writing vector fields in these coordinates because they provide a
coordinate system in which the vector fields are often particularly easy to
study. In the theory of Lie groups this is classical (see, for example,
[Che46, page 115]). Outside of the setting of Lie groups, canonical
coordinates have been used in the quantitative study of sub-Riemannian
geometry, beginning with the work of Nagel, Stein, and Wainger [NSW85], and
later used by Tao and Wright [TW03], the first author [Str11], Montanari and
Morbidelli [MM12], and the first author and Stovall [SS18], among others. In
[Str21], the first author moved beyond canonical coordinates to strengthen
these theories. We see now that this is necessary: sharp results like the ones
in this paper and in [Str21] cannot be obtained using canonical coordinates.

######  Remark 1.3.

If $X$ is a nonzero $C^{1}$ vector field on a one dimensional manifold
$\mathfrak{M}$, then the canonical coordinate system with respect to $X$, near
the point $x_{0}$, is the map $\Phi_{x_{0}}(t)=e^{tX}x_{0}$. Since
$\Phi_{x_{0}}^{*}X=\partial_{t}$, we see that canonical coordinate system do
provide optimal regularity in this simple setting. However, once we move to
two dimensions, with two vector fields, Lemma 8.6 shows that canonical
coordinates may not give the optimal regularity.

###  1.3 Qualitative versus Quantitative

Most of the results in this paper are qualitative in the following sense. We
give necessary and sufficient conditions so that a map $\Phi$ as in Question 2
exists. If one traces through the proof, the $\mathscr{C}^{\beta}$ norms of
the coefficients of $\Phi^{*}X_{1},\ldots,\Phi^{*}X_{q}$ depend on (among
other things) quantities like:

  * •

Upper bounds for $\mathscr{C}^{\alpha}$ norms of the coefficients of
$X_{1},\ldots,X_{q}$ in some fixed coordinate system near $x$.

  * •

A lower bound, $>0$, for the quantity:

$\max_{j_{1},\ldots,j_{n}\in\\{1,\ldots
q\\}}\left|\det\left(X_{j_{1}}(x)|\ldots|X_{j_{n}}(x)\right)\right|,$ (1.1)

where in the above expression $X_{1},\ldots,X_{q}$ are written as column
vectors in the same fixed coordinate system near $x$.

Unfortunately, both above quantities depend on the choice of the original
coordinate system. Thus, if the vector fields are given in a coordinate system
where the above upper and lower bounds are bad, the estimates our proof gives
are bad; even if there exists a different (unknown) choice of coordinate
system where the above estimates are better. Thus, while our results are
qualitatively optimal (we give necessary and sufficient conditions for each
$\beta$), the estimates which follow from our proofs may be far from optimal
unless one happens to know a good coordinate system in which to write the
vector fields in the first place.

In the papers [SS18, Str21, Str20a], the first author and Stovall give such
estimates on $\Phi^{*}X_{1},\ldots,\Phi^{*}X_{q}$ in terms of quantities which
are invariant under arbitrary $C^{2}$ diffeomorphisms (we call such estimates
quantitative). Thus, they do not depend on any choice of coordinate system.
This is useful for questions from partial differential equations and harmonic
analysis, where $\Phi$ can be used as a scaling map. Such scaling maps
originated in the smooth setting in the foundational work of Nagel, Stein, and
Wainger [NSW85] and were later worked on by Tao and Wright [TW03], the first
author [Str11], and in the above-mentioned series of papers by the first
author and Stovall [SS18, Str21, Str20a]. Similar scaling in a non-smooth
setting was studied by Montanari and Morbidelli [MM12], though they do not
address questions like the ones in this paper.

In Section 9, we use the main methods of this paper, combined with the methods
of [SS18, Str21] to improve the main quantitative result of [Str21] (and also
the main quantitative result of [Str20b]).

##  2 Function Spaces

To state our main result, we need to introduce several function spaces related
to the classical Zygmund-Hölder spaces. Because we are working in low
regularity, some care is needed in the definitions.

###  2.1 Classical Zygmund-Hölder spaces

In this section, we describe the classical Zygmund-Hölder spaces, and the
corresponding spaces on a manifold; see Section 4 for proofs of the results
stated here.

In what follows, $U$ will either be equal to $\mathbb{R}^{n}$ or equal to an
bounded open set with smooth boundary in $\mathbb{R}^{n}$–we will usually be
interested in the case when $U$ is either $\mathbb{R}^{n}$ or an open ball in
$\mathbb{R}^{n}$. We define the Zygmund-Hölder space
$\mathscr{C}^{s}(U):=\mathscr{B}^{s}_{\infty,\infty}(U)$, where
$\mathscr{B}^{s}_{\infty,\infty}$ denotes the classical Besov space (see,
[Tri10, Section 2.3] for $\mathscr{B}^{s}_{\infty,\infty}(U)$ when
$U=\mathbb{R}^{n}$, and [Tri92, Chapter 5] or [Tri06, Chapter 1.11] when $U$
is an bounded smooth domain).444Many results concerning $\mathscr{C}^{s}(U)$,
where $U$ is a bounded smooth domain, follow from the corresponding results
concerning $\mathscr{C}^{s}(\mathbb{R}^{n})$ via the theory described in
[Tri92, Chapter 5]. We similarly define the vector valued space
$\mathscr{C}^{s}(U;\mathbb{R}^{m})$. The space $\mathscr{C}^{s}(U)$ has some
particularly concrete characterizations:

######  Remark 2.1.

For $U=\mathbb{R}^{n}$ or $U$ an bounded open set with smooth boundary in
$\mathbb{R}^{n}$, we have

  1. (i)

$s\in(0,2)$: $\mathscr{C}^{s}(U)$ consists of those continuous functions
$f:U\rightarrow\mathbb{R}$ such that the following norm is finite

$\sup_{x\in U}|f(x)|+\sup_{\begin{subarray}{c}x\in U\\\
h\in\mathbb{R}^{n},h\neq 0\\\ x+h,x+2h\in
U\end{subarray}}|h|^{-s}|f(x+2h)-2f(x+h)+f(x)|.$

Moreover, the above expression gives a norm equivalent to
$\|f\|_{\mathscr{C}^{s}(U)}$. See [Tri10, Theorem 2.5.7 (ii)] and [Tri10,
(3.4.2/6)].

  2. (ii)

$s\in(0,1)$: $\mathscr{C}^{s}(U)$ consists of those continuous functions
$f:U\rightarrow\mathbb{R}$ such that the following norm is finite

$\sup_{x\in U}|f(x)|+\sup_{\begin{subarray}{c}x,y\in U\\\ x\neq
y\end{subarray}}|x-y|^{-s}|f(x)-f(y)|.$

Moreover, the above expression gives a norm equivalent to
$\|f\|_{\mathscr{C}^{s}(U)}$. See [Tri10, Remark 2.2.2/3] and [Tri10,
(3.4.2/6)].

  3. (iii)

$s\in(1,\infty]$: $\mathscr{C}^{s}(U)$ consists of those continuous
$f:U\rightarrow\mathbb{R}^{n}$, such that
$f,\partial_{x_{j}}f\in\mathscr{C}^{s-1}(U)$, $1\leq j\leq n$. We have
$\|f\|_{\mathscr{C}^{s}(U)}\approx\|f\|_{\mathscr{C}^{s-1}(U)}+\sum_{j=1}^{n}\|\partial_{x_{j}}f\|_{\mathscr{C}^{s-1}(U)}$.
See [Tri10, Theorem 2.5.7 (ii)] and [Tri10, Theorem 3.3.5(i)].

  4. (iv)

$s\in(-\infty,0]$: $\mathscr{C}^{s}(U)$ consists of those distributions
$f\in\mathscr{D}^{\prime}(U)$ such that
$f=g_{0}+\sum_{j=1}^{n}\partial_{x_{j}}g_{j}$ for some
$g_{0},\ldots,g_{n}\in\mathscr{C}^{s+1}(U)$. We have
$\|f\|_{\mathscr{C}^{s}(U)}\approx\inf\sum_{j=0}^{n}\|g_{j}\|_{\mathscr{C}^{s+1}(U)}$,
where the infimum is taken over all such choices of $g_{0},\ldots,g_{n}$. When
$U=\mathbb{R}^{n}$, this can be seen by letting
$g_{0}=(I+\bigtriangleup)^{-1}f$ and
$g_{j}=-\partial_{x_{j}}(I+\bigtriangleup)^{-1}f$, for $j=1,\ldots n$. See
[Tri10, Theorem 2.3.8].

  5. (v)

When $m\in\mathbb{N}$ and $r\in(0,1)$, then $\mathscr{C}^{m+r}(U)=C^{m,r}(U)$,
with equivalence of norms. See [Tri06, Theorem 1.118 (i)]. However, when
$r\in\\{0,1\\}$ these spaces differ.

######  Lemma 2.2.

Let $r,s\in\mathbb{R}$ with $r+s>0$, $r\geq s$. The product map $(f,g)\mapsto
fg$ can be defined as a continuous map
$\mathscr{C}^{r}(U)\times\mathscr{C}^{s}(U)\rightarrow\mathscr{C}^{s}(U)$.

###### Proof.

This is a special case of [Tri10, Theorem 2.8.2(i)] when $U=\mathbb{R}^{n}$
and [Tri10, Theorem 3.3.2(ii)] when $U$ is bounded open set with smooth
boundary. ∎

######  Definition 2.3.

Let $U\subseteq\mathbb{R}^{n}$ be an open set. For $s\in\mathbb{R}$, we define
$\mathscr{C}^{s}(U;TU)$, to be the space of vector fields (with distribution
coefficients) $Y=\sum_{j=1}^{n}a_{j}\partial_{x_{j}}$, where
$a_{j}\in\mathscr{C}^{s}(U)$. We identify $Y$ with the distribution
$(a_{1},\ldots,a_{n})\in\mathscr{C}^{s}(U;\mathbb{R}^{n})$ and define

$\|Y\|_{\mathscr{C}^{s}(U)}:=\|(a_{1},\ldots,a_{n})\|_{\mathscr{C}^{s}(U;\mathbb{R}^{n})}.$

######  Definition 2.4.

Let $U\subseteq\mathbb{R}^{n}$ be an open set. For $s\in\mathbb{R}$ and
$k\in\mathbb{N}$, we define
$\mathscr{C}^{s}\left(U;{\textstyle\bigwedge}^{k}T^{*}U\right)$, to be the
space of $k$-forms (with distribution coefficients) $\omega=\sum_{1\leq
i_{1}<i_{2}<\cdots<i_{k}\leq
n}\omega_{i_{1},\ldots,i_{k}}dx_{1}\wedge\cdots\wedge dx_{k}$, where
$\omega_{i_{1},\ldots,i_{k}}\in\mathscr{C}^{s}(U)$. We identify $\omega$ with
the distribution $(\omega_{i_{1},\ldots,i_{k}})_{1\leq
i_{1}<i_{2}<\ldots<i_{k}\leq n}$, and define

$\|\omega\|_{\mathscr{C}^{s}(U)}:=\|(\omega_{i_{1},\ldots,i_{k}})\|_{\mathscr{C}^{s}(U;\mathbb{R}^{Q_{n,k}})},$

where $Q_{n,k}=\dim{\textstyle\bigwedge}^{k}\mathbb{R}^{n}$.

######  Definition 2.5.

Let $U\subseteq\mathbb{R}^{n}$ be an open set. For $s\in\mathbb{R}$, we define
$\mathscr{C}^{s}_{\mathrm{loc}}(U)$ to be the space of distributions
$f\in\mathscr{D}^{\prime}(U)$, such that for every $x\in U$, there exists an
open ball $U^{\prime}\subseteq U$ containing $x$ with
$f\big{|}_{U^{\prime}}\in\mathscr{C}^{s}(U^{\prime})$. We similarly define
$\mathscr{C}^{s}_{\mathrm{loc}}(U;TU)$ and
$\mathscr{C}^{s}_{\mathrm{loc}}\left(U;{\textstyle\bigwedge}^{k}T^{*}U\right)$.

For $\alpha>0$, we define $\mathscr{C}^{\alpha+1}$ manifolds in the usual way:
the transition functions are assumed to be
$\mathscr{C}^{\alpha+1}_{\mathrm{loc}}$ (see [Str21, Section 5.4] for some
comments on this). Such manifold are, in particular, $C^{1}$ manifolds, and so
it makes sense to talk about, for example, vector fields on such manifolds.

######  Remark 2.6.

On a $\mathscr{C}^{\alpha+1}$ manifold $\mathfrak{M}$, it makes sense to talk
about functions in $\mathscr{C}^{s}_{\mathrm{loc}}(\mathfrak{M})$ for
$s\in(-\alpha,\alpha+1]$, vector fields in
$\mathscr{C}^{s}_{\mathrm{loc}}(\mathfrak{M};T\mathfrak{M})$ for
$s\in(-\alpha,\alpha]$, and $k$-forms in
$\mathscr{C}^{s}_{\mathrm{loc}}\left(\mathfrak{M};{\textstyle\bigwedge}^{k}T^{*}\mathfrak{M}\right)$
for $s\in(-\alpha,\alpha]$. See Lemma 4.1 and Definition 4.2.

Let $\iota_{Y}$ denote the interior product with respect to the vector field
$Y$ and let $\mathrm{Lie}_{Y}$ denote the Lie derivative with respect to $Y$.

######  Proposition 2.7.

Let $\mathfrak{M}$ be a $\mathscr{C}^{\alpha+1}$ manifold for some $\alpha>0$.

  1. (i)

For $\beta\in(-\alpha,\alpha]$, the map $(Y,\omega)\mapsto\iota_{Y}\omega$ is
a continuous map

$\mathscr{C}^{\alpha}_{\mathrm{loc}}(\mathfrak{M};T\mathfrak{M})\times\mathscr{C}^{\beta}_{\mathrm{loc}}\left(\mathfrak{M};{\textstyle\bigwedge}^{k}T^{*}\mathfrak{M}\right)\rightarrow\mathscr{C}_{\mathrm{loc}}^{\beta}\left(\mathfrak{M};{\textstyle\bigwedge}^{k-1}T^{*}\mathfrak{M}\right).$

  2. (ii)

For $\beta\in(-\alpha,\alpha]$, the map $(\eta,\omega)\mapsto\eta\wedge\omega$
is a continuous map

$\mathscr{C}^{\alpha}_{\mathrm{loc}}\left(\mathfrak{M};{\textstyle\bigwedge}^{l}T^{*}\mathfrak{M}\right)\times\mathscr{C}^{\beta}_{\mathrm{loc}}\left(\mathfrak{M};{\textstyle\bigwedge}^{k}T^{*}\mathfrak{M}\right)\rightarrow\mathscr{C}_{\mathrm{loc}}^{\beta}\left(\mathfrak{M};{\textstyle\bigwedge}^{k+l}T^{*}\mathfrak{M}\right).$

  3. (iii)

If $\alpha>1/2$ and $\beta\in(-\alpha+1,\alpha]$, the map $\omega\mapsto
d\omega$ is continuous

$\mathscr{C}^{\beta}_{\mathrm{loc}}\left(\mathfrak{M};{\textstyle\bigwedge}^{k}T^{*}\mathfrak{M}\right)\rightarrow\mathscr{C}^{\beta-1}_{\mathrm{loc}}\left(\mathfrak{M};{\textstyle\bigwedge}^{k+1}T^{*}\mathfrak{M}\right).$

  4. (iv)

For $\beta\in(-\alpha+1,\alpha+1]$, the map

$(Y,f)\mapsto Yf=:\mathrm{Lie}_{Y}f$

is continuous

$\mathscr{C}^{\alpha}_{\mathrm{loc}}(\mathfrak{M};T\mathfrak{M})\times\mathscr{C}^{\beta}_{\mathrm{loc}}(\mathfrak{M})\rightarrow\mathscr{C}^{\beta-1}_{\mathrm{loc}}(\mathfrak{M}).$

  5. (v)

If $\alpha>1/2$, then for $\beta\in(-\alpha+1,\alpha]$, the map

$(Y,Z)\mapsto[Y,Z]=:\mathrm{Lie}_{Y}Z$

is continuous

$\mathscr{C}^{\alpha}_{\mathrm{loc}}(\mathfrak{M};T\mathfrak{M})\times\mathscr{C}^{\beta}_{\mathrm{loc}}(\mathfrak{M};T\mathfrak{M})\rightarrow\mathscr{C}^{\beta-1}_{\mathrm{loc}}(\mathfrak{M};T\mathfrak{M}).$

  6. (vi)

If $\alpha>1/2$, then for $\beta\in(-\alpha+1,\alpha]$, the map

$(Y,\omega)\mapsto d\iota_{Y}\omega+\iota_{Y}d\omega=:\mathrm{Lie}_{Y}\omega$

is continuous

$\mathscr{C}^{\alpha}_{\mathrm{loc}}(\mathfrak{M};T\mathfrak{M})\times\mathscr{C}^{\beta}_{\mathrm{loc}}(\mathfrak{M};T\mathfrak{M})\rightarrow\mathscr{C}^{\beta-1}_{\mathrm{loc}}(\mathfrak{M};T\mathfrak{M}).$

As can be seen in Section 1.1, our main results are in terms of the
commutators of vector fields: i.e., the Lie derivative of one vector field
with respect to another. Proposition 2.7 (v) shows that such Lie derivatives,
$\mathrm{Lie}_{Y}$, only make sense when
$Y\in\mathscr{C}^{\alpha}_{\mathrm{loc}}(\mathfrak{M};T\mathfrak{M})$ for
$\alpha>1/2$. Because of this, when $\alpha>1/2$, the characterizations in our
main result can be made somewhat simpler. However, it is still possible to
make sense of some of these ideas when $\alpha\in(0,1/2]$, as we now make
precise.

######  Proposition 2.8.

Let $\alpha>0$ and $\beta,\gamma\in(-\alpha,\alpha+1]$. Let
$U,V\subseteq\mathbb{R}^{n}$ be open and let $F:U\xrightarrow{\sim}V$ be a
$\mathscr{C}^{\alpha+1}_{\mathrm{loc}}$ diffeomorphism. Fix a $k$-form
$\theta\in\mathscr{C}^{\gamma}_{\mathrm{loc}}\left(U;{\textstyle\bigwedge}^{k}T^{*}U\right)$.
Then, the following are equivalent:

  1. (i)

$d\theta\in\mathscr{C}^{\beta-1}_{\mathrm{loc}}\left(U;{\textstyle\bigwedge}^{k+1}T^{*}U\right)$.

  2. (ii)

$d(F_{*}\theta)\in\mathscr{C}^{\beta-1}_{\mathrm{loc}}\left(V;{\textstyle\bigwedge}^{k+1}T^{*}V\right)$.

Moreover, in this case, for all $p\in U$, there is a neighborhood
$V^{\prime}\subseteq V$ of $F(p)$ and
$\tau\in\mathscr{C}^{\beta}_{\mathrm{loc}}\left(V^{\prime};{\textstyle\bigwedge}^{k}T^{*}V^{\prime}\right)$
such that $d(F_{*}\theta)\big{|}_{V^{\prime}}=d\tau$.

For the remainder of this section, let $\mathfrak{M}$ be a
$\mathscr{C}^{\alpha+1}$ manifold for some $\alpha>0$.

######  Definition 2.9.

Let $\gamma\in(-\alpha,\alpha]$, $\beta\in[\gamma,\alpha+1]$, and
$\theta\in\mathscr{C}^{\gamma}_{\mathrm{loc}}\left(\mathfrak{M};{\textstyle\bigwedge}^{k}T^{*}\mathfrak{M}\right)$.
We say $d\theta$ has regularity
$\mathscr{C}^{\beta-1}_{\mathrm{loc}}(\mathfrak{M})$, if for any
$p\in\mathfrak{M}$, there is a $\mathscr{C}^{\alpha+1}_{\mathrm{loc}}$
coordinate system $F:V\xrightarrow{\sim}U$, where $V$ is a neighborhood of $p$
and $U\subseteq\mathbb{R}^{n}$ is open, such that
$dF_{*}\theta\in\mathscr{C}^{\beta-1}_{\mathrm{loc}}\left(U;{\textstyle\bigwedge}^{k+1}T^{*}U\right)$.

Proposition 2.8 shows that Definition 2.9 is well-defined: it does not depend
on the choice of the coordinate system $F$. However, we do not define the form
$d\theta$ itself: we only define its regularity. Indeed, if
$\beta-1\leq-\alpha$, the space
$\mathscr{C}^{\beta-1}_{\mathrm{loc}}\left(\mathfrak{M};{\textstyle\bigwedge}^{k+1}T^{*}\mathfrak{M}\right)$
is not well-defined. However, when $\beta-1>-\alpha$, the form $d\theta$ is
well-defined, as the next result shows.

######  Lemma 2.10.

Let $\gamma\in(-\alpha,\alpha]$ and $\beta\in(-\alpha+1,\alpha+1]$. Suppose
$\theta\in\mathscr{C}^{\gamma}_{\mathrm{loc}}\left(\mathfrak{M};{\textstyle\bigwedge}^{k}T^{*}\mathfrak{M}\right)$
is such that $d\theta$ has regularity
$\mathscr{C}^{\beta-1}_{\mathrm{loc}}(\mathfrak{M})$. Then, $d\theta$ is given
by a well-defined form in
$\mathscr{C}^{\beta-1}_{\mathrm{loc}}\left(\mathfrak{M};{\textstyle\bigwedge}^{k+1}T^{*}\mathfrak{M}\right)$.
I.e., there is a unique form
$\tau\in\mathscr{C}^{\beta-1}_{\mathrm{loc}}\left(\mathfrak{M};{\textstyle\bigwedge}^{k+1}T^{*}\mathfrak{M}\right)$
such that in every coordinate system $F:V\xrightarrow{\sim}U$, where
$V\subseteq\mathfrak{M}$ and $U\subseteq\mathbb{R}^{n}$ are open, we have
$F_{*}\tau=d(F_{*}\theta)$. Furthermore, this form $\tau$ is closed in the
sense that in every such coordinate system, we have $dF_{*}\tau=0$.

######  Definition 2.11.

For $\gamma\in(-\alpha,\alpha]$, $\beta\in(-\alpha+1,\alpha+1]$, and
$\theta\in\mathscr{C}^{\gamma}_{\mathrm{loc}}\left(\mathfrak{M};{\textstyle\bigwedge}^{k}T^{*}\mathfrak{M}\right)$,
we write
$d\theta\in\mathscr{C}^{\beta-1}_{\mathrm{loc}}\left(\mathfrak{M};{\textstyle\bigwedge}^{k+1}T^{*}\mathfrak{M}\right)$
to mean $d\theta$ has regularity
$\mathscr{C}^{\beta-1}_{\mathrm{loc}}(\mathfrak{M})$, and we identify
$d\theta$ with the unique closed form in
$\mathscr{C}^{\beta-1}_{\mathrm{loc}}\left(\mathfrak{M};{\textstyle\bigwedge}^{k+1}T^{*}\mathfrak{M}\right)$
given in Lemma 2.10.

######  Convention 2.12.

For $\beta\in\mathbb{R}$, we say
$\theta\in\mathscr{C}^{\beta^{+}}_{\mathrm{loc}}\left(\mathfrak{M};{\textstyle\bigwedge}^{k}T^{*}\mathfrak{M}\right)$,
if
$\theta\in\mathscr{C}^{\beta+\varepsilon}_{\mathrm{loc}}\left(\mathfrak{M};{\textstyle\bigwedge}^{k}T^{*}\mathfrak{M}\right)$
for some $\varepsilon>0$.

###  2.2 Zygmund-Hölder spaces with respect to vector fields

Let $\mathfrak{M}$ be a $\mathscr{C}^{\alpha+1}$ manifold for some $\alpha>0$,
and let
$X_{1},\ldots,X_{q}\in\mathscr{C}^{\alpha}_{\mathrm{loc}}(\mathfrak{M};T\mathfrak{M})$
be $\mathscr{C}^{\alpha}_{\mathrm{loc}}$ vector fields which span the tangent
space to $\mathfrak{M}$ at every point. Since $\mathfrak{M}$ is only a
$\mathscr{C}^{\alpha+1}$ manifold, it does not make sense to talk about
whether a function on $\mathfrak{M}$ has regularity higher than
$\mathscr{C}^{\alpha+1}_{\mathrm{loc}}$. However, we can make sense of higher
regularity with respect to the vector fields $X_{1},\ldots,X_{q}$.

######  Definition 2.13 ($\mathscr{C}^{\beta}_{X,\mathrm{loc}}$-functions).

For $\beta>-\alpha$, we let
$\mathscr{C}^{\beta}_{X,\mathrm{loc}}(\mathfrak{M})$ be the space of those
functions $\mathfrak{M}\rightarrow\mathbb{R}$ defined recursively by:

  * •

If $\beta\in(-\alpha,1]$,
$\mathscr{C}^{\beta}_{X,\mathrm{loc}}(\mathfrak{M}):=\mathscr{C}^{\beta}_{\mathrm{loc}}(\mathfrak{M})$.

  * •

If $\beta>1$, $\mathscr{C}^{\beta}_{X,\mathrm{loc}}(\mathfrak{M})$ consists of
those $f\in\mathscr{C}^{\beta-1}_{X,\mathrm{loc}}(\mathfrak{M})\bigcap
C^{1}_{\mathrm{loc}}(\mathfrak{M})$ such that
$\mathrm{Lie}_{X_{j}}f=X_{j}f\in\mathscr{C}^{\beta-1}_{X,\mathrm{loc}}(\mathfrak{M})$,
for $1\leq j\leq q$.

We can make a similar definition for vector fields and forms, so long as
$\alpha>1/2$:

######  Definition 2.14 ($\mathscr{C}^{\beta}_{X,\mathrm{loc}}$-vector
fields).

Suppose $\alpha>1/2$. For $\beta>-\alpha$, we let
$\mathscr{C}^{\beta}_{X,\mathrm{loc}}(\mathfrak{M};T\mathfrak{M})$ be be the
space of those vector fields on $\mathfrak{M}$ defined recursively by:

  * •

If $\beta\in(-\alpha,\frac{1}{2}]$,
$\mathscr{C}^{\beta}_{X,\mathrm{loc}}(\mathfrak{M};T\mathfrak{M})=\mathscr{C}^{\beta}_{\mathrm{loc}}(\mathfrak{M};T\mathfrak{M})$.

  * •

If $\beta>\frac{1}{2}$,
$\mathscr{C}^{\beta}_{X,\mathrm{loc}}(\mathfrak{M};T\mathfrak{M})$ consists of
those
$Y\in\mathscr{C}^{\beta-1}_{X,\mathrm{loc}}(\mathfrak{M};T\mathfrak{M})\bigcap\mathscr{C}^{\frac{1}{2}}_{\mathrm{loc}}(\mathfrak{M};T\mathfrak{M})$
such that
$\mathrm{Lie}_{X_{j}}Y=[X_{j},Y]\in\mathscr{C}^{\beta-1}_{X,\mathrm{loc}}(\mathfrak{M};T\mathfrak{M})$,
for $1\leq j\leq q$.

######  Definition 2.15 ($\mathscr{C}^{\beta}_{X,\mathrm{loc}}$-forms).

Suppose $\alpha>1/2$ and $k\geq 1$. For $\beta>-\alpha$, we let
$\mathscr{C}^{\beta}_{X,\mathrm{loc}}\left(\mathfrak{M};{\textstyle\bigwedge}^{k}T^{*}\mathfrak{M}\right)$
be the space of those $k$-forms on $\mathfrak{M}$ defined recursively by:

  * •

If $\beta\in(-\alpha,\frac{1}{2}]$,
$\mathscr{C}^{\beta}_{X,\mathrm{loc}}\left(\mathfrak{M};{\textstyle\bigwedge}^{k}T^{*}\mathfrak{M}\right)=\mathscr{C}^{\beta}_{\mathrm{loc}}\left(\mathfrak{M};{\textstyle\bigwedge}^{k}T^{*}\mathfrak{M}\right)$.

  * •

If $\beta>\frac{1}{2}$,
$\mathscr{C}^{\beta}_{X,\mathrm{loc}}\left(\mathfrak{M};{\textstyle\bigwedge}^{k}T^{*}\mathfrak{M}\right)$
consists of those
$\theta\in\mathscr{C}^{\beta-1}_{X,\mathrm{loc}}\left(\mathfrak{M};{\textstyle\bigwedge}^{k}T^{*}\mathfrak{M}\right)\bigcap\mathscr{C}^{\frac{1}{2}}_{\mathrm{loc}}\left(\mathfrak{M};{\textstyle\bigwedge}^{k}T^{*}\mathfrak{M}\right)$
such that
$\mathrm{Lie}_{X_{j}}\theta=(d\iota_{X_{j}}+\iota_{X_{j}}d)\theta\in\mathscr{C}^{\beta-1}_{X,\mathrm{loc}}\left(\mathfrak{M};{\textstyle\bigwedge}^{k}T^{*}\mathfrak{M}\right)$,
for $1\leq j\leq q$.

When $\alpha\in(0,1/2]$, we cannot use Definition 2.15. However, we can make
an appropriate analog of Definition 2.9 with respect to the vector fields
$X_{1},\ldots,X_{q}$:

######  Definition 2.16 ($\mathscr{C}^{\beta}_{X,\mathrm{loc}}$ for
differentials).

Suppose $\alpha>0$. Let $\beta>-\alpha$, $1\leq k\leq n-1$ and let
$\theta\in\mathscr{C}^{(-\alpha)^{+}}_{\mathrm{loc}}\left(\mathfrak{M};{\textstyle\bigwedge}^{k}T^{*}\mathfrak{M}\right)$.
We say $d\theta$ has regularity
$\mathscr{C}^{\beta-1}_{X,\mathrm{loc}}(\mathfrak{M})$ if:

  * •

If $\beta\in(-\alpha,1]$, we assume $d\theta$ has regularity
$\mathscr{C}^{\beta-1}_{\mathrm{loc}}(\mathfrak{M})$.

  * •

If $\beta\in(1,2]$, we assume
$d\theta\in\mathscr{C}^{0^{+}}_{\mathrm{loc}}\left(\mathfrak{M};{\textstyle\bigwedge}^{k+1}T^{*}\mathfrak{M}\right)$
and $\mathrm{Lie}_{X_{j}}d\theta=d\iota_{X_{j}}d\theta$ has regularity
$\mathscr{C}^{\beta-2}_{\mathrm{loc}}(\mathfrak{M})$, for $1\leq j\leq q$.

  * •

If $\beta>2$, we assume $d\theta$ has regularity
$\mathscr{C}^{\beta-2}_{X,\mathrm{loc}}(\mathfrak{M})$, and
$\mathrm{Lie}_{X_{j}}d\theta=d\iota_{X_{j}}d\theta$ has regularity
$\mathscr{C}^{\beta-2}_{X,\mathrm{loc}}(\mathfrak{M})$, for $1\leq j\leq q$.

######  Remark 2.17.

Note that if $\beta\geq 0$, Definitions 2.13 and 2.16 do not depend on
$\alpha$. Similarly, when $\beta\geq-\frac{1}{2}$, Definitions 2.14 and 2.15
do not depend on $\alpha$.

##  3 The Main Result

######  Theorem 3.1.

Let $\alpha,\beta>0$, and let $X_{1},\ldots,X_{q}$ be
$\mathscr{C}^{\alpha}_{\mathrm{loc}}$ vector fields on a
$\mathscr{C}^{\alpha+1}$ manifold $\mathfrak{M}$ of dimension $n$, which span
the tangent space at every point. Fix a point $p\in\mathfrak{M}$, and re-order
$X_{1},\ldots,X_{q}$ so that $X_{1}(p),\ldots,X_{n}(p)$ form a basis for
$T_{p}\mathfrak{M}$. Let $\lambda^{1},\ldots,\lambda^{n}$ be the dual basis
for $X_{1},\ldots,X_{n}$, defined on a neighborhood of $p$. Then, the
following are equivalent:

  1. (a)

There is a neighborhood $U\subseteq\mathfrak{M}$ of $p$ and a
$\mathscr{C}^{\alpha+1}_{\mathrm{loc}}$ diffeomorphism
$\Phi:\mathbb{B}^{n}\xrightarrow{\sim}U$, such that $\Phi(0)=p$ and
$\Phi^{*}X_{1},\ldots,\Phi^{*}X_{q}\in\mathscr{C}^{\beta}(\mathbb{B}^{n};T\mathbb{B}^{n})$.

  2. (b)

There is a neighborhood $U$ of $p$ such that for $1\leq j\leq n$,
$d\lambda^{j}$ has regularity $\mathscr{C}^{\beta-1}_{X,\mathrm{loc}}(U)$, and
for $1\leq j\leq n$, $n+1\leq k\leq q$,
$\langle\lambda^{j},X_{k}\rangle\in\mathscr{C}^{\beta}_{X,\mathrm{loc}}(U)$.

If $\alpha>1/2$, then in addition we have the following equivalent conditions:

  1. 3.

There is a neighborhood $U$ of $p$ such that for $1\leq j\leq n$,
$\lambda^{j}\in\mathscr{C}^{\beta}_{X,\mathrm{loc}}\left(U,T^{*}U\right)$, and
for $1\leq j\leq n$, $n+1\leq k\leq q$,
$\langle\lambda^{j},X_{k}\rangle\in\mathscr{C}^{\beta}_{X,\mathrm{loc}}(U)$.

  2. 4.

There is a neighborhood $U$ of $p$ such that for $1\leq j\leq q$,
$X_{j}\in\mathscr{C}^{\beta}_{X,\mathrm{loc}}\left(U;TU\right)$.

######  Remark 3.2.

(a) $\Leftrightarrow$ 4 is the conclusion alluded to in Section 1.1.

##  4 Classical Function Spaces, Revisited

In this section, we prove the basic results we require about Zygmund-Hölder
spaces. In particular, we prove the results from Section 2. We begin by
discussing the main result we need to help understand the various objects
under consideration on $\mathscr{C}^{\alpha+1}$-manifolds.

######  Lemma 4.1.

Fix $\alpha>0$, let $U,V\subseteq\mathbb{R}^{n}$ be open sets, and let
$\Phi:U\xrightarrow{\sim}V$ be a $\mathscr{C}^{\alpha+1}_{\mathrm{loc}}$
diffeomorphism.

  1. (i)

For $\beta\in(-\alpha,\alpha+1]$ and
$f\in\mathscr{C}^{\beta}_{\mathrm{loc}}(V)$, $f\circ\Phi$ is defined as a
distribution and we have $f\circ\Phi\in\mathscr{C}^{\beta}_{\mathrm{loc}}(U)$.

  2. (ii)

For $\beta\in(-\alpha,\alpha]$ and
$X\in\mathscr{C}^{\beta}_{\mathrm{loc}}(V;TV)$, $\Phi^{*}X$ is defined and
$\Phi^{*}X\in\mathscr{C}^{\beta}_{\mathrm{loc}}(U;TU)$.

  3. (iii)

For $\beta\in(-\alpha,\alpha]$ and
$\omega\in\mathscr{C}^{\beta}_{\mathrm{loc}}\left(V;{\textstyle\bigwedge}^{k}T^{*}V\right)$,
then $\Phi^{*}\omega$ is defined and
$\Phi^{*}\omega\in\mathscr{C}^{\beta}_{\mathrm{loc}}\left(U;{\textstyle\bigwedge}^{k}T^{*}U\right)$

###### Proof.

We give $U$ coordinates $x^{1},\ldots,x^{n}$ and $V$ coordinates
$y^{1},\ldots,y^{n}$. We begin with (i). For $\beta>0$, $f$ is a continuous
function and the regularity of $f\circ\Phi$ is classical. See [Str21, Lemma
5.15] for a discussion of this classical case. For nonpositive $\beta$, we
proceed by induction. We prove the result for
$\beta\in(-l+1,-l]\cap(-\alpha,\alpha+1]$, for $l=-1,0,1,2,\ldots$. The base
case, $l=-1$, follows from the above discussion for $\beta>0$. For
$l\in\mathbb{N}$, we assume the result for $l-1$ and prove it for $l$.

Fix a point $y_{0}\in V$, and let $B_{y_{0}}\subseteq V$ be an open ball
centered at $y_{0}$ with $\overline{B_{y_{0}}}\subseteq V$. By Remark 2.1
(iv), we may write $f=g_{0}+\sum_{j=1}^{n}\partial_{y^{j}}g_{j}$, where
$g_{0},\ldots,g_{n}\in\mathscr{C}^{\beta+1}(B_{y_{0}})$. Letting
$\Psi=(\psi_{1},\ldots\psi_{n}):=\Phi^{-1}$, we have

$(\partial_{y^{j}}g_{j})\circ\Phi=\sum_{k=1}^{n}\left(\partial_{x^{j}}(g_{j}\circ\Phi)\right)\left(\left(\partial_{y^{j}}\psi_{k}\right)\circ\Phi\right).$

By the already proved case $\beta=\alpha$, we have
$\left(\partial_{y^{k}}\psi_{k}\right)\circ\Phi\in\mathscr{C}^{\alpha}_{\mathrm{loc}}(\Psi(B_{y_{0}}))$
and by the inductive hypothesis
$\partial_{x^{k}}(g_{j}\circ\Phi)\in\mathscr{C}^{\beta}_{\mathrm{loc}}(\Psi(B_{y_{0}}))$.
Also, by the inductive hypothesis
$g_{0}\circ\Phi\in\mathscr{C}^{\beta+1}_{\mathrm{loc}}(\Psi(B_{y_{0}}))\subseteq\mathscr{C}^{\beta}_{\mathrm{loc}}(\Psi(B_{y_{0}}))$.
Using Lemma 2.2, we conclude
$f\circ\Phi=g_{0}\circ\Phi+\sum(\partial_{y^{j}}g_{j})\circ\Phi\in\mathscr{C}^{\beta}_{\mathrm{loc}}(\Psi(B_{y_{0}}))$.
It is easy to see that the distribution obtained in this way does not depend
on the choice of $g_{0},\ldots,g_{n}$ with
$f=g_{0}+\sum_{j=1}^{n}\partial_{y^{j}}g_{j}$. Since $y_{0}\in V$ was
arbitrary, this completes the proof of (i).

For (ii), write $X=\sum_{j=1}^{n}a_{j}\partial_{y^{j}}$, where
$a_{j}\in\mathscr{C}^{\beta}_{\mathrm{loc}}(V)$. Then, if
$\Psi=(\psi_{1},\ldots,\psi_{n})=\Phi^{-1}$,

$\Phi^{*}X=\sum_{k=1}^{n}\sum_{j=1}^{n}\left(a_{j}\circ\Phi\right)\left((\partial_{y^{j}}\psi_{k})\circ\Phi\right)\partial_{x^{k}}.$

By (i), $a_{j}\circ\Phi\in\mathscr{C}^{\beta}_{\mathrm{loc}}(U)$, and
$(\partial_{x^{j}}\phi_{k})\circ\Phi\in\mathscr{C}^{\alpha}_{\mathrm{loc}}(U)$.
Since $\beta\in(-\alpha,\alpha]$, by hypothesis, Lemma 2.2 implies
$\Phi^{*}X\in\mathscr{C}^{\beta}_{\mathrm{loc}}(U;TU)$.

The proof of (iii) is very similar to the proof of (ii), and follows easily by
combining (i) with Lemma 2.2. We leave the details to the reader. ∎

Lemma 4.1 establishes Remark 2.6: On a $\mathscr{C}^{\alpha+1}$ manifold
$\mathfrak{M}$, it makes sense to talk about functions in
$\mathscr{C}^{s}_{\mathrm{loc}}(\mathfrak{M})$ for $s\in(-\alpha,\alpha+1]$,
vector fields in $\mathscr{C}^{s}_{\mathrm{loc}}(\mathfrak{M};T\mathfrak{M})$
for $s\in(-\alpha,\alpha]$, and $k$-forms in
$\mathscr{C}^{s}_{\mathrm{loc}}\left(\mathfrak{M};{\textstyle\bigwedge}^{k}T^{*}\mathfrak{M}\right)$
for $s\in(-\alpha,\alpha]$. This is because these properties are invariant
under $\mathscr{C}^{\alpha+1}_{\mathrm{loc}}$ diffeomorphisms. By a similar
proof, one can show that the more general concept of a
$\mathscr{C}^{\beta}_{\mathrm{loc}}$ tensor is well-defined for
$\beta\in(-\alpha,\alpha]$, though the only tensors we use in this paper are
vector fields and forms.

For completeness we put the definition of functions, vector fields and
differential forms on manifold as below, which is the obvious analog of the
standard definitions (see, for example, [Hör03, Definition 6.3.3]):

######  Definition 4.2.

Let $\alpha>0$, $\beta\in(-\alpha,\alpha]$ and $\gamma\in(-\alpha,\alpha+1]$.
Let $\mathfrak{M}$ be a $n$-dimensional $\mathscr{C}^{\alpha+1}$-manifold
equipped with the maximal $\mathscr{C}^{\alpha+1}$-atlas
$\mathscr{A}=\\{\phi:U_{\phi}\subseteq\mathfrak{M}\to\mathbb{R}^{n}\\}$.
Namely, each $\phi\in\mathscr{A}$ is a homeomorphism
$\phi:U_{\phi}\xrightarrow{\sim}\phi(U_{\phi})\subseteq\mathbb{R}^{n}$;
$\phi\circ\psi^{-1}\in\mathscr{C}^{\alpha+1}_{\mathrm{loc}}(\psi(U_{\psi}\cap
U_{\phi});\mathbb{R}^{n})$ whenever $\phi,\psi\in\mathscr{A}$ satisfy
$U_{\psi}\cap U_{\phi}\neq\varnothing$; and $\mathscr{A}$ is maximal with
these properties.

  * •

A $\mathscr{C}^{\gamma}_{\mathrm{loc}}$-function is a collection
$f=\\{f_{\phi}\in\mathscr{C}^{\gamma}_{\mathrm{loc}}(\phi(U_{\phi}))\\}_{\phi\in\mathscr{A}}$,
such that

$f_{\phi}=f_{\psi}\circ(\psi\circ\phi^{-1}),\quad\text{on }\phi(U_{\phi}\cap
U_{\psi})\text{ whenever }U_{\phi}\cap U_{\psi}\neq\varnothing.$

  * •

A $\mathscr{C}^{\beta}_{\mathrm{loc}}$-vector field is a collection
$X=\\{X_{\phi}\in\mathscr{C}^{\beta}_{\mathrm{loc}}(\phi(U_{\phi});T\mathbb{R}^{n})\\}_{\phi\in\mathscr{A}}$,
such that

$X_{\phi}=(\psi\circ\phi^{-1})^{*}X_{\psi},\quad\text{on }\phi(U_{\phi}\cap
U_{\psi})\text{ whenever }U_{\phi}\cap U_{\psi}\neq\varnothing.$ (4.1)

  * •

Let $1\leq k\leq n$. A $\mathscr{C}^{\beta}_{\mathrm{loc}}$ $k$-form is a
collection
$\omega=\left\\{\omega_{\phi}\in\mathscr{C}^{\beta}_{\mathrm{loc}}\left(\phi(U_{\phi});{\textstyle\bigwedge}^{k}T^{*}\mathbb{R}^{n}\right)\right\\}_{\phi\in\mathscr{A}}$,
such that

$\omega_{\psi}=(\psi\circ\phi^{-1})^{*}\omega_{\phi},\quad\text{on
}\phi(U_{\phi}\cap U_{\psi})\text{ whenever }U_{\phi}\cap
U_{\psi}\neq\varnothing.$ (4.2)

######  Remark 4.3.

By Lemma 4.1 we are able to pullback functions, vector fields, and
differential forms using $\mathscr{C}^{\alpha+1}$-transition maps. Thus, the
above objects are well-defined.

To prove Proposition 2.7 we use the following:

######  Lemma 4.4.

Let $\alpha>0$, let $U,V\subseteq\mathbb{R}^{n}$ be two open sets and let
$\Phi:U\xrightarrow{\sim}V$ be a $\mathscr{C}^{\alpha+1}$-diffeomorphism.

  1. (i)

For $\beta\in(-\alpha,\alpha]$, if
$Y\in\mathscr{C}^{\alpha}_{\mathrm{loc}}(V;TV)$ and
$\omega\in\mathscr{C}^{\beta}_{\mathrm{loc}}\left(V;{\textstyle\bigwedge}^{k}T^{*}V\right)$
then $\Phi^{*}(\iota_{Y}\omega)=\iota_{\Phi^{*}Y}\Phi^{*}\omega$, and their
common value is in
$\mathscr{C}^{\beta}_{\mathrm{loc}}\left(U;{\textstyle\bigwedge}^{k-1}T^{*}U\right)$.

  2. (ii)

For $\beta\in(-\alpha,\alpha]$, if
$\sigma\in\mathscr{C}^{\alpha}_{\mathrm{loc}}\left(V;{\textstyle\bigwedge}^{l}T^{*}V\right)$
and
$\omega\in\mathscr{C}^{\beta}_{\mathrm{loc}}\left(V;{\textstyle\bigwedge}^{k}T^{*}V\right)$
then $\Phi^{*}(\sigma\wedge\omega)=\Phi^{*}\sigma\wedge\Phi^{*}\omega$, and
their common value is in
$\mathscr{C}^{\beta}_{\mathrm{loc}}\left(U;{\textstyle\bigwedge}^{k+l}T^{*}U\right)$.

  3. (iii)

For $\alpha>\frac{1}{2}$ and $\beta\in(1-\alpha,\alpha]$, if
$\omega\in\mathscr{C}^{\beta}_{\mathrm{loc}}\left(V;{\textstyle\bigwedge}^{k}T^{*}V\right)$
then $\Phi^{*}d\omega=d\Phi^{*}\omega$, and their common value is in
$\mathscr{C}^{\beta-1}_{\mathrm{loc}}\left(U;{\textstyle\bigwedge}^{k+1}T^{*}U\right)$.

  4. (iv)

For $\beta\in(1-\alpha,\alpha+1]$, if
$Y\in\mathscr{C}^{\alpha}_{\mathrm{loc}}(V;TV)$ and
$f\in\mathscr{C}^{\beta}_{\mathrm{loc}}(V)$ then
$\Phi^{*}(Yf)=(\Phi^{*}Y)(\Phi^{*}f)$, and their common value is in
$\mathscr{C}^{\beta-1}_{\mathrm{loc}}(U)$.

  5. (v)

If $\alpha>\frac{1}{2}$ and $\beta\in(1-\alpha,\alpha]$, if
$Y\in\mathscr{C}^{\alpha}_{\mathrm{loc}}(V;TV)$ and
$Z\in\mathscr{C}^{\beta}_{\mathrm{loc}}(V;TV)$ then
$\Phi^{*}[Y,Z]=[\Phi^{*}Y,\Phi^{*}Z]$ , and their common value is in
$\mathscr{C}^{\beta-1}_{\mathrm{loc}}(U;TU)$.

  6. (vi)

If $\alpha>\frac{1}{2}$ and $\beta\in(1-\alpha,\alpha]$, if
$Y\in\mathscr{C}^{\alpha}_{\mathrm{loc}}(V;TV)$ and
$\omega\in\mathscr{C}^{\beta}_{\mathrm{loc}}\left(V;{\textstyle\bigwedge}^{k}T^{*}V\right)$
then $\Phi^{*}\mathrm{Lie}_{Y}\omega=\mathrm{Lie}_{\Phi^{*}Y}\Phi^{*}\omega$,
and their common value is in
$\mathscr{C}^{\beta-1}_{\mathrm{loc}}\left(U;{\textstyle\bigwedge}^{k}T^{*}U\right)$.

###### Proof.

The formal computations are standard in differential geometry. What we need to
be careful is that the products and compositions are defined, due to the low
regularity of the objects involved.

We only prove (i) and (iii), since the arguments for the others are similar.
We endow $V\subseteq\mathbb{R}^{n}$ with the standard coordinate system
$(x^{1},\dots,x^{n})$, and write $\Phi=:(\phi^{1},\dots,\phi^{n})$ where
$\phi^{j}\in\mathscr{C}^{\alpha+1}_{\mathrm{loc}}(U)$.

(i): We write $Y=\sum_{i=1}^{n}a^{i}\frac{\partial}{\partial x^{i}}$ and
$\omega_{j_{1}\dots j_{k}}:=\omega(\frac{\partial}{\partial
x^{j_{1}}},\dots,\frac{\partial}{\partial x^{j_{k}}})$ for $1\leq
j_{1},\dots,j_{k}\leq n$. By Lemma 2.2, $a^{i}\omega_{ij_{1}\dots
j_{k-1}}\in\mathscr{C}^{\beta}_{\mathrm{loc}}(V)$ and therefore

$\iota_{Y}\omega=\frac{1}{(k-1)!}\sum_{i=1}^{n}\sum_{j_{1},\dots,j_{k-1}=1}^{n}a^{i}\omega_{ij_{1}\dots
j_{k-1}}dx^{j_{1}}\wedge\dots\wedge
dx^{j_{k-1}}\in\mathscr{C}^{\beta}_{\mathrm{loc}}\left(U;{\textstyle\bigwedge}^{k-1}T^{*}U\right).$
(4.3)

Note that $\frac{\partial}{\partial\phi^{i}}=\Phi^{*}\frac{\partial}{\partial
x^{i}}$, $i=1,\dots,n$, are $\mathscr{C}^{\alpha}$-vector fields on $U$ that
satisfy555We write $\delta_{i}^{j}$ for the Kronecker delta functions (see
(5.6)).
$d\phi^{j}(\frac{\partial}{\partial\phi^{i}})=\frac{\partial\phi^{j}}{\partial\phi^{i}}=\delta_{i}^{j}$
for $1\leq i,j\leq n$. By Lemma 4.1 (i), $a^{i}\circ\Phi,\omega_{ij_{1}\dots
j_{k-1}}\circ\Phi,(a^{i}\omega_{ij_{1}\dots
j_{k-1}})\circ\Phi\in\mathscr{C}^{\beta}_{\mathrm{loc}}(U)$ are all defined.
Therefore we have the following, where all the products and compositions are
defined.

$\displaystyle\iota_{\Phi^{*}Y}(\Phi^{*}\omega)=\frac{1}{k!}\sum_{j_{0},\dots,j_{k-1}=1}^{n}\iota_{\sum_{i=1}^{n}(a^{i}\circ\Phi)\frac{\partial}{\partial\phi^{i}}}\left((\omega_{j_{0}\dots
j_{k-1}}\circ\Phi)\cdot d\phi^{j_{0}}\wedge\dots\wedge d\phi^{j_{k-1}}\right)$
$\displaystyle=\frac{1}{k!}\sum_{i=1}^{n}\sum_{j_{0},j_{1},\dots,j_{k-1}=1}^{n}\sum_{\rho=0}^{k-1}(a^{i}\circ\Phi)\cdot(\omega_{j_{0}\dots\widehat{j_{\rho}}\dots
j_{k}}\circ\Phi)\cdot(-1)^{\rho}\frac{\partial\phi^{j_{\rho}}}{\partial\phi^{i}}d\phi^{j_{0}}\wedge\dots\wedge
d\phi^{j_{\rho-1}}\wedge d\phi^{j_{\rho+1}}\wedge\dots\wedge d\phi^{j_{k-1}}$
$\displaystyle=\frac{1}{(k-1)!}\sum_{i=1}^{n}\sum_{j_{1},\dots,j_{k-1}=1}^{n}((a^{i}\omega_{j_{1}\dots
j_{k-1}})\circ\Phi)\cdot d\phi^{j_{1}}\wedge\dots\wedge
d\phi^{j_{k-1}}=\Phi^{*}(\iota_{Y}\omega).$

The equality holds in
$\mathscr{C}^{\beta}_{\mathrm{loc}}\left(V;{\textstyle\bigwedge}^{k-1}T^{*}V\right)$,
completing the proof.

(iii): By passing to linear combinations it suffices to consider the form
$\omega=fdx^{i_{1}}\wedge\dots\wedge dx^{i_{k}}$ where
$f\in\mathscr{C}^{\beta}_{\mathrm{loc}}(V)$.

By Lemma 4.1, $f\circ\Phi\in\mathscr{C}^{\beta}_{\mathrm{loc}}(U)$ and
$\frac{\partial f}{\partial
x^{j}}\circ\Phi\in\mathscr{C}^{\beta-1}_{\mathrm{loc}}(U)$. Since we also have
$d\phi^{i_{1}},\dots,d\phi^{i_{k}}\in\mathscr{C}^{\alpha}_{\mathrm{loc}}(U)$,
Lemma 2.2 shows that all below products are defined. We have,

$\Phi^{*}d(fdx^{i_{1}}\wedge\dots\wedge
dx^{i_{k}})=\Phi^{*}\sum_{j=1}^{n}\frac{\partial f}{\partial
x^{j}}dx^{j}\wedge dx^{i_{1}}\wedge\dots\wedge
dx^{i_{k}}=\sum_{j=1}^{n}\Big{(}\frac{\partial f}{\partial
x^{j}}\circ\Phi\Big{)}d\phi^{j}\wedge d\phi^{i_{1}}\wedge\dots\wedge
d\phi^{i_{k}},$ (4.4) $\begin{split}&d\Phi^{*}(fdx^{i_{1}}\wedge\dots\wedge
dx^{i_{k}})=d(f\circ\Phi)\wedge d\phi^{i_{1}}\wedge\dots\wedge
d\phi^{i_{k}}=\sum_{l=1}^{n}\frac{\partial(f\circ\Phi)}{\partial x^{l}}\wedge
d\phi^{i_{1}}\wedge\dots\wedge d\phi^{i_{k}}\\\
&=\sum_{l,j=1}^{n}\Big{(}\frac{\partial f}{\partial
x^{j}}\circ\Phi\Big{)}\frac{\partial\phi^{j}}{\partial x^{l}}dx^{l}\wedge
d\phi^{i_{1}}\wedge\dots\wedge d\phi^{i_{k}}.\end{split}$ (4.5)

Since $\sum_{l=1}^{n}\frac{\partial\phi^{j}}{\partial x^{l}}dx^{l}=d\phi^{j}$,
we have that (4.4) and (4.5) are equal, completing the proof. ∎

###### Proof of Proposition 2.7.

We only prove (i), since the arguments for the other parts are similar.

By Definition 4.2, we can write
$Y=\\{Y_{\phi}\in\mathscr{C}^{\alpha}_{\mathrm{loc}}(\phi(U_{\phi});T\mathbb{R}^{n})\\}_{\phi\in\mathscr{A}}$
and
$\omega=\left\\{\omega_{\phi}\in\mathscr{C}^{\beta}_{\mathrm{loc}}\left(\phi(U_{\phi});{\textstyle\bigwedge}^{k}T^{*}\mathbb{R}^{n}\right)\right\\}_{\phi\in\mathscr{A}}$,
where $\mathscr{A}$ is the maximal $\mathscr{C}^{\alpha+1}$-atlas for
$\mathfrak{M}$.

For each $\phi\in\mathscr{A}$, by applying Lemma 2.2 on the coordinate
components of $Y_{\phi}$ and $\omega_{\phi}$, we see that the map
$(Y_{\phi},\omega_{\phi})\mapsto\iota_{Y_{\phi}}\omega_{\phi}$ is a continuous
map
$\mathscr{C}^{\alpha}_{\mathrm{loc}}(\phi(U_{\phi});T\mathbb{R}^{n})\times\mathscr{C}^{\beta}_{\mathrm{loc}}\left(\phi(U_{\phi});{\textstyle\bigwedge}^{k}T^{*}\mathbb{R}^{n}\right)\to\mathscr{C}^{\beta}_{\mathrm{loc}}\left(\phi(U_{\phi});{\textstyle\bigwedge}^{k-1}T^{*}\mathbb{R}^{n}\right)$.

By (4.1) and (4.2) we have $(\psi\circ\phi^{-1})^{*}Y_{\psi}=Y_{\phi}$ and
$(\psi\circ\phi^{-1})^{*}\omega_{\psi}=\omega_{\phi}$ on $\phi(U_{\phi}\cap
U_{\psi})$. By Lemma 4.4 (i) we see that

$(\psi\circ\phi^{-1})^{*}(\iota_{Y_{\psi}}\omega_{\psi})=\iota_{(\psi\circ\phi^{-1})^{*}Y_{\psi}}\left((\psi\circ\phi^{-1})^{*}\omega_{\psi}\right)=\iota_{Y_{\phi}}\omega_{\phi},\quad\text{on
}\phi(U_{\phi}\cap U_{\psi}),\text{ whenever }U_{\phi}\cap
U_{\psi}\neq\varnothing.$

Therefore $\\{\iota_{Y_{\phi}}\omega_{\phi}\\}_{\phi\in\mathscr{A}}$ is a
collection of $(k-1)$-forms satisfying (4.2), and therefore defines a
$(k-1)$-form on $\mathfrak{M}$, which is denoted by $\iota_{Y}\omega$.

Finally, the continuity of $(Y,\omega)\mapsto\iota_{Y}\omega$ comes from the
fact that $(Y_{\phi},\omega_{\phi})\mapsto\iota_{Y_{\phi}}\omega_{\phi}$ is
continuous for each $\phi\in\mathscr{A}$. ∎

We now turn to the proofs of Proposition 2.8 and Lemma 2.10. For these, we
require several standard objects and results.

######  Notation 4.5.

We use the co-differential, $\vartheta=\vartheta_{\mathbb{R}^{n}}$, which is a
linear operator taking $k$ forms to $k-1$ forms, satisfying for $1\leq
i_{1}<i_{2}<\cdots<i_{k}\leq n$,

$\vartheta\left(fdx^{i_{1}}\wedge\cdots\wedge
dx^{i_{k}}\right)=\sum_{l=1}^{k}\frac{\partial f}{\partial
x^{i_{l}}}(-1)^{l}dx^{i_{1}}\wedge\cdots\wedge dx^{i_{l-1}}\wedge
dx^{i_{l+1}}\wedge\cdots\wedge dx^{i_{k}}.$

In particular on 1-forms, $-\vartheta$ is the divergence operator, namely

$\text{For
}\theta=\sum_{i=1}^{n}\theta_{i}dx^{i},\quad\vartheta\theta=-\sum_{i=1}^{n}\frac{\partial\theta_{i}}{\partial
x^{i}}.$

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

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

We will make use of the classical Newtonian potential. Let

$\mathcal{G}(x):=\begin{cases}-\frac{|x|}{2},&n=1,\\\
-\frac{1}{2\pi}\log|x|,&n=2,\\\ |\mathbb{S}^{n-1}|^{-1}|x|^{2-n},&n\geq
3.\end{cases}$ (4.6)

######  Lemma 4.6.

Let $\omega=\sum_{1\leq i_{1}<\cdots<i_{k}\leq
n}\omega_{i_{1},\ldots,i_{k}}dx^{i_{k}}\wedge\cdots\wedge dx^{i_{k}}$ be a
$k$-form where each $\omega_{i_{1},\ldots,i_{k}}$ is a compactly supported
distribution on $\mathbb{R}^{n}$. Then, $\sigma:=\mathcal{G}*\omega$ is a
distribution on $\mathbb{R}^{n}$ satisfying $\bigtriangleup\sigma=\omega$.
Moreover, if for some open set $U\subseteq\mathbb{R}^{n}$ and
$\beta\in\mathbb{R}$ we have
$\omega\big{|}_{U}\in\mathscr{C}_{\mathrm{loc}}^{\beta}\left(U;{\textstyle\bigwedge}^{k}T^{*}U\right)$,
then
$\sigma\big{|}_{U}\in\mathscr{C}_{\mathrm{loc}}^{\beta+2}\left(U;{\textstyle\bigwedge}^{k}T^{*}U\right)$.

###### Proof.

The convolution $\mathcal{G}*\omega$ makes sense because
$\omega_{i_{1},\ldots,i_{k}}$ is a compactly supported distribution and
$\mathcal{G}$ is a distribution. Since $\mathcal{G}$ is well-known to be a
fundamental solution for the Laplacian $\bigtriangleup$, we have
$\bigtriangleup(\mathcal{G}\ast\omega)=(\bigtriangleup\mathcal{G})*\omega=\omega$.
Since $\bigtriangleup\sigma=\omega$, the classical interior regularity for
elliptic equations shows that if
$\omega\big{|}_{U}\in\mathscr{C}_{\mathrm{loc}}^{\beta}\left(U;{\textstyle\bigwedge}^{k}T^{*}U\right)$,
then
$\sigma\big{|}_{U}\in\mathscr{C}_{\mathrm{loc}}^{\beta+2}(U;{\textstyle\bigwedge}^{k}T^{*}U)$;
see, for example, [Tay11, Proposition 4.1]. ∎

######  Lemma 4.7.

Let $0\leq k\leq n$, $\gamma\in\mathbb{R}$ and let $U\subset\mathbb{R}^{n}$ be
a bounded open set, then there is a $C_{U,\gamma}>0$ such that
$\|\mathcal{G}\ast\omega\|_{\mathscr{C}^{\gamma+2}(U;\wedge^{k}T^{*}U)}\leq
C_{U,\gamma}\|\omega\|_{\mathscr{C}^{\gamma}(U;\wedge^{k}T^{*}U)}$ for all
compactly supported $k$-forms
$\omega\in\mathscr{C}_{c}^{\gamma}\left(U;{\textstyle\bigwedge}^{k}T^{*}U\right)$.

###### Proof.

Set $\mathscr{X}$ to be the completion of
$\mathscr{C}_{c}^{\gamma}\left(U;{\textstyle\bigwedge}^{k}T^{*}U\right)$ under
the $\mathscr{C}^{\gamma}$-norm. Thus, $\mathscr{X}$ is a closed subspace of
$\mathscr{C}^{\gamma}\left(\mathbb{R}^{n};{\textstyle\bigwedge}^{k}T^{*}\mathbb{R}^{n}\right)$
and $\|\omega\|_{\mathscr{X}}=\|\omega\|_{\mathscr{C}^{\gamma}}$ for all
$\omega\in\mathscr{X}$.

When $\omega\in\mathscr{X}$, we have that
$\operatorname{supp}\omega\subseteq\overline{U}$ so
$\omega\in\mathscr{C}^{\gamma}_{c}\left(\mathbb{R}^{n};{\textstyle\bigwedge}^{k}T^{*}\mathbb{R}^{n}\right)$.
By Lemma 4.6,
$\mathcal{G}\ast\omega\in\mathscr{C}^{\gamma+2}_{\mathrm{loc}}\left(\mathbb{R}^{n};{\textstyle\bigwedge}^{k}T^{*}\mathbb{R}^{n}\right)$
is well-defined. By restricting it to $U$ we get
$(\mathcal{G}\ast\omega)\big{|}_{U}\in\mathscr{C}^{\gamma+2}\left(U;{\textstyle\bigwedge}^{k}T^{*}U\right)$.

By the closed graph theorem we have
$\|\mathcal{G}\ast\omega\|_{\mathscr{C}^{\gamma+2}(U;\wedge^{k}T^{*}U)}\leq
C\|\omega\|_{\mathscr{X}}=C\|\omega\|_{\mathscr{C}^{\gamma}(U;\wedge^{k}T^{*}U)}$
for some $C$ that does not depend on $\omega$.

Therefore, for the same constant $C$ we have
$\|\mathcal{G}\ast\omega\|_{\mathscr{C}^{\gamma+2}(U;\wedge^{k}T^{*}U)}\leq
C\|\omega\|_{\mathscr{C}^{\gamma}(U;\wedge^{k}T^{*}U)}$ for all
$\omega\in\mathscr{C}_{c}^{\gamma}\left(U;{\textstyle\bigwedge}^{k}T^{*}U\right)$.
∎

######  Lemma 4.8.

Let $0\leq k\leq n$, $\gamma\in\mathbb{R}$, $\beta>\gamma-1$ and let
$U^{\prime}\Subset U\subseteq\mathbb{R}^{n}$ be two open sets.666Here, and in
the rest of the paper, $A\Subset B$ denotes that $A$ is a relatively compact
subset of $B$. Suppose
$\theta\in\mathscr{C}^{\gamma}_{\mathrm{loc}}\left(U,{\textstyle\bigwedge}^{k}T^{*}U\right)$
satisfies
$d\theta\in\mathscr{C}^{\beta-1}_{\mathrm{loc}}\left(U,{\textstyle\bigwedge}^{k+1}T^{*}U\right)$.
Then, there exist
$\rho\in\mathscr{C}^{\beta}_{\mathrm{loc}}\left(U^{\prime};{\textstyle\bigwedge}^{k}T^{*}U\right)$
and
$\xi\in\mathscr{C}^{\gamma+1}_{\mathrm{loc}}\left(U^{\prime};{\textstyle\bigwedge}^{k-1}T^{*}U^{\prime}\right)$
such that

$\theta\big{|}_{U^{\prime}}=\rho+d\xi.$

Note that the case $\beta\leq\gamma-1$ holds automatically if we pick
$\rho:=\theta$ and $\xi:=0$.

###### Proof.

Let $\chi\in C^{\infty}_{c}(U)$ satisfy $\chi\equiv 1$ on a neighborhood over
$\overline{U^{\prime}}$. Define

$\widehat{\rho}:=\mathcal{G}*\vartheta
d(\chi\theta),\quad\widehat{\xi}:=\mathcal{G}*\vartheta(\chi\theta),\qquad\rho:=\widehat{\rho}\big{|}_{U^{\prime}},\quad\xi:=\widehat{\xi}\big{|}_{U^{\prime}}.$

Since $\chi\theta\in\mathscr{C}^{\gamma}$, Lemma 4.6 shows
$\widehat{\rho}\in\mathscr{C}^{\gamma}_{\mathrm{loc}}\left(\mathbb{R}^{n};{\textstyle\bigwedge}^{k}T^{*}\mathbb{R}^{n}\right)$
and
$\widehat{\xi}\in\mathscr{C}^{\gamma+1}_{\mathrm{loc}}\left(\mathbb{R}^{n};{\textstyle\bigwedge}^{k-1}T^{*}\mathbb{R}^{n}\right)$.
Thus,
$\xi\in\mathscr{C}^{\gamma+1}_{\mathrm{loc}}\left(U^{\prime};{\textstyle\bigwedge}^{k-1}T^{*}U^{\prime}\right)$.
Also, $\bigtriangleup\widehat{\rho}\big{|}_{U^{\prime}}=\vartheta
d(\chi\theta)\big{|}_{U^{\prime}}=\vartheta
d\theta\big{|}_{U^{\prime}}\in\mathscr{C}_{\mathrm{loc}}^{\beta-2}$, by
hypothesis. Thus, by the interior regularity of elliptic PDEs (see [Tay11,
Proposition 4.1]), we have
$\rho=\widehat{\rho}\big{|}_{U^{\prime}}\in\mathscr{C}^{\beta}_{\mathrm{loc}}$.
We also have,

$\theta\big{|}_{U^{\prime}}=\bigtriangleup(\mathcal{G}*(\chi\theta))\big{|}_{U^{\prime}}=(\vartheta
d+d\vartheta)(\mathcal{G}*(\chi\theta))\big{|}_{U^{\prime}}=\mathcal{G}*\vartheta
d(\chi\theta)\big{|}_{U^{\prime}}+d\mathcal{G}*\vartheta(\chi\theta)\big{|}_{U^{\prime}}=\widehat{\rho}\big{|}_{U^{\prime}}+d\widehat{\xi}\big{|}_{U^{\prime}}=\rho+d\xi,$

as desired. ∎

Finally, we require paraproduct decompositions. Let
$\psi_{0}\in\mathscr{S}(\mathbb{R}^{n})$ be a Schwartz function whose Fourier
transform, $\hat{\psi}_{0}(\xi)=\int\psi_{0}(x)e^{2\pi ix\cdot\xi}\>dx$,
satisfies $\operatorname{supp}\hat{\psi}_{0}\subseteq\\{\xi:|\xi|<8/3\\}$ and
$\hat{\psi}_{0}(\xi)\equiv 1$ for $|\xi|\leq 3/2$. Set

$\psi_{j}(x):=\begin{cases}2^{nj}\psi_{0}(2^{j}x)&j>0,\\\
0&j\leq-1.\end{cases}$

Associated to $\psi_{0}$, we define two bilinear operators each taking a $k$
form $\sigma$ and an $l$-form $\omega$ and outputting a $(k+l)$-form,

$\mathfrak{P}(\sigma,\omega):=\sum_{j=0}^{\infty}\left((\psi_{j}-\psi_{j-1})*\sigma\right)\wedge(\psi_{j-2}*\omega),\quad\mathfrak{R}(\sigma,\omega):=\sum_{|j-k|\leq
1}\left((\psi_{j}-\psi_{j-1})*\sigma\right)\wedge\left((\psi_{k}-\psi_{k-1})*\omega\right).$

######  Lemma 4.9.

We have the following properties of $\mathfrak{P}$ and $\mathfrak{R}$. Fix
$k,l\in\\{0,\ldots n\\}$.

  1. (i)

For $\alpha\in\mathbb{R}$, $\mathfrak{P}$ defines a continuous bilinear map
$\mathfrak{P}:\mathscr{C}^{\alpha}\left(\mathbb{R}^{n};{\textstyle\bigwedge}^{k}T^{*}\mathbb{R}^{n}\right)\times
L^{\infty}\left(\mathbb{R}^{n};{\textstyle\bigwedge}^{l}T^{*}\mathbb{R}^{n}\right)\rightarrow\mathscr{C}^{\alpha}\left(\mathbb{R}^{n};{\textstyle\bigwedge}^{k+l}T^{*}\mathbb{R}^{n}\right)$.

  2. (ii)

For $\alpha\in\mathbb{R}$ and $\beta<0$, $\mathfrak{P}$ defines a continuous
bilinear map
$\mathfrak{P}:\mathscr{C}^{\alpha}\left(\mathbb{R}^{n};{\textstyle\bigwedge}^{k}T^{*}\mathbb{R}^{n}\right)\times\mathscr{C}^{\beta}\left(\mathbb{R}^{n};{\textstyle\bigwedge}^{l}T^{*}\mathbb{R}^{n}\right)\rightarrow\mathscr{C}^{\alpha+\beta}\left(\mathbb{R}^{n};{\textstyle\bigwedge}^{k+l}T^{*}\mathbb{R}^{n}\right)$.

  3. (iii)

For $\alpha,\beta\in\mathbb{R}$ with $\alpha+\beta>0$, $\mathfrak{R}$ defines
a continuous bilinear map
$\mathfrak{R}:\mathscr{C}^{\alpha}\left(\mathbb{R}^{n};{\textstyle\bigwedge}^{k}T^{*}\mathbb{R}^{n}\right)\times\mathscr{C}^{\beta}\left(\mathbb{R}^{n};{\textstyle\bigwedge}^{l}T^{*}\mathbb{R}^{n}\right)\rightarrow\mathscr{C}^{\alpha+\beta}\left(\mathbb{R}^{n};{\textstyle\bigwedge}^{k+l}T^{*}\mathbb{R}^{n}\right)$.

  4. (iv)

$\sigma\wedge\omega=\mathfrak{P}(\sigma,\omega)+(-1)^{kl}\mathfrak{P}(\omega,\sigma)+\mathfrak{R}(\omega,\sigma)$
holds for
$\sigma\in\mathscr{C}^{\alpha}\left(\mathbb{R}^{n};{\textstyle\bigwedge}^{k}T^{*}\mathbb{R}^{n}\right)$
and
$\omega\in\mathscr{C}^{\beta}\left(\mathbb{R}^{n};{\textstyle\bigwedge}^{l}T^{*}\mathbb{R}^{n}\right)$,
where $\alpha+\beta>0$.

  5. (v)

$\mathfrak{P}$ satisfies
$d\mathfrak{P}(\sigma,\omega)=\mathfrak{P}(d\sigma,\omega)+(-1)^{k}\mathfrak{P}(\sigma,d\omega)$,
for $k$-forms $\sigma$ and $l$-forms $\omega$.

###### Proof.

For $0$ forms, (i) and (ii) can be found in [BCD11, Theorem 2.82] and (iii)
can be found in [BCD11, Theorem 2.85]. By passing to their coordinate
components we obtain the results for arbitrary forms. (iv) follows easily from
the fact that
$\sum_{j=0}^{N}(\psi_{j}-\psi_{j-1})*\omega=\psi_{N}*\omega\xrightarrow{N\rightarrow\infty}\omega$.
(v) follows directly from the definitions. ∎

###### Proof of Proposition 2.8.

Let
$\theta\in\mathscr{C}^{\gamma}_{\mathrm{loc}}\left(U;{\textstyle\bigwedge}^{k}T^{*}U\right)$
be such that
$d\theta\in\mathscr{C}^{\beta-1}_{\mathrm{loc}}\left(U;{\textstyle\bigwedge}^{k}T^{*}U\right)$.
We will show, for any point $p\in U$, there is a neighborhood
$V^{\prime}\subseteq V$ of $F(p)$ and
$\tau\in\mathscr{C}^{\beta}_{\mathrm{loc}}\left(V^{\prime};{\textstyle\bigwedge}^{k}T^{*}V^{\prime}\right)$,
with $d(F_{*}\theta)\big{|}_{V^{\prime}}=d\tau$. This will prove
(i)$\Rightarrow$(ii) and the existence of $\tau$ as claimed in the
proposition. The reverse implication follows by reversing the roles of
$F_{*}\theta$ and $\theta$.

Let $U^{\prime}\Subset U$ be an open neighborhood of $p$. By Lemma 4.8, there
are
$\rho\in\mathscr{C}^{\beta}_{\mathrm{loc}}\left(U^{\prime},{\textstyle\bigwedge}^{k}T^{*}U^{\prime}\right)$
and
$\xi\in\mathscr{C}^{\gamma+1}_{\mathrm{loc}}\left(U^{\prime},{\textstyle\bigwedge}^{k-1}T^{*}U^{\prime}\right)$
such that $\theta\big{|}_{U^{\prime}}=\rho+d\xi$. By Lemma 4.4 (iii),
$dF_{*}d\xi=d^{2}F_{*}\xi=0$, so

$dF_{*}\theta\big{|}_{F(U^{\prime})}=dF_{*}\rho\big{|}_{F(U^{\prime})}+dF_{*}d\xi\big{|}_{F(U^{\prime})}=dF_{*}\rho\big{|}_{F(U^{\prime})}.$
(4.7)

Since
$\rho\in\mathscr{C}^{\beta}_{\mathrm{loc}}\left(U^{\prime},{\textstyle\bigwedge}^{k}T^{*}U^{\prime}\right)$
and $F$ is a $\mathscr{C}^{\alpha+1}_{\mathrm{loc}}$ diffeomorphism, Lemma 4.1
shows
$F_{*}\rho\in\mathscr{C}^{\min\\{\alpha,\beta\\}}_{\mathrm{loc}}\left(F(U^{\prime}),{\textstyle\bigwedge}^{k}T^{*}\mathbb{R}^{n}\right)$.
If $\beta\leq\alpha$, then we have
$dF_{*}\rho\in\mathscr{C}^{\beta-1}_{\mathrm{loc}}$, completing the proof with
$V^{\prime}:=F(U^{\prime})$ and $\tau:=F_{*}\rho$.

However, if $\beta>\alpha$, this does not imply the desired result. To show
$dF_{*}\rho\in\mathscr{C}^{\beta-1}_{\mathrm{loc}}$ near $F(p)$, we construct
a new $k$-form $\tau\in\mathscr{C}^{\beta}$ such that $dF_{*}\rho=d\tau$ near
$F(p)$. The construction requires paraproducts.

Let $U^{\prime\prime}\Subset U^{\prime}$ be a smaller open neighborhood of
$p$. We claim that there exist $\tilde{\rho}_{i_{1}\cdots
i_{k}}\in\mathscr{C}^{\beta}(\mathbb{R}^{n})$ and $\tilde{\mu}^{i_{1}\cdots
i_{k}}\in\mathscr{C}^{\alpha+1}\left(\mathbb{R}^{n};{\textstyle\bigwedge}^{k-1}T^{*}\mathbb{R}^{n}\right)$,
with compact supports and such that

$F_{*}\rho\big{|}_{F(U^{\prime\prime})}=\sum_{1\leq i_{1}<\cdots<i_{k}\leq
n}\tilde{\rho}_{i_{1}\ldots i_{k}}d\tilde{\mu}^{i_{1}\ldots
i_{k}}\big{|}_{F(U^{\prime\prime})}.$ (4.8)

Write $\rho=\sum_{1\leq i_{1}<\cdots<i_{k}\leq n}\rho_{i_{1}\ldots
i_{k}}dx^{i_{1}}\wedge\cdots\wedge dx^{i_{k}}$ and denote
$\Phi:=F^{-1}:V\xrightarrow{\sim}U$. Take $\chi_{1}\in
C_{c}^{\infty}(F(U^{\prime}))$ such that
$\chi_{1}\big{|}_{F(U^{\prime\prime})}\equiv 1$. Define
$(\tilde{\phi}^{1},\ldots,\tilde{\phi}^{n}):=\chi_{1}\Phi$, so that each
$\tilde{\phi}^{j}$ is compactly supported and
$\tilde{\phi}^{j}\in\mathscr{C}^{\alpha+1}(\mathbb{R}^{n})$. Let
$\tilde{\rho}_{i_{1}\ldots
i_{k}}:=\chi_{1}(\rho_{i_{1}\ldots_{k}}\circ\Phi)\in\mathscr{C}^{\beta}(\mathbb{R}^{n})$.
We have $\rho_{i_{1}\dots
i_{k}}\circ\Phi\big{|}_{F(U^{\prime\prime})}=\tilde{\rho}_{i_{1}\dots
i_{k}}\big{|}_{F(U^{\prime\prime})}$ and
$\phi^{j}\big{|}_{F(U^{\prime\prime})}=\tilde{\phi}^{j}\big{|}_{F(U^{\prime\prime})}$,
so

$F_{*}\rho\big{|}_{F(U^{\prime\prime})}=\sum_{1\leq i_{1}<\cdots<i_{k}\leq
n}\tilde{\rho}_{i_{1}\ldots i_{k}}d\tilde{\phi}^{i_{1}}\wedge\cdots\wedge
d\tilde{\phi}^{i_{k}}\big{|}_{F(U^{\prime\prime})}.$ (4.9)

Using Lemma 4.6, we set $\mu^{i_{1}\cdots
i_{k}}:=\mathcal{G}*\vartheta(d\tilde{\phi}^{i_{1}}\wedge\cdots\wedge
d\tilde{\phi}^{i_{k}})\in\mathscr{C}_{\mathrm{loc}}^{\alpha+1}\left(\mathbb{R}^{n};{\textstyle\bigwedge}^{k-1}T^{*}\mathbb{R}^{n}\right)$.
Since $d\tilde{\phi}^{i_{1}}\wedge\cdots\wedge d\tilde{\phi}^{i_{k}}$ is
closed, we have

$\begin{split}&d\mu^{i_{1}\cdots
i_{k}}=d\vartheta\mathcal{G}*(d\tilde{\phi}^{i_{1}}\wedge\cdots\wedge
d\tilde{\phi}^{i_{k}})+\vartheta\mathcal{G}*d(d\tilde{\phi}^{i_{1}}\wedge\cdots\wedge
d\tilde{\phi}^{i_{k}})\\\
&=\bigtriangleup\mathcal{G}*(d\tilde{\phi}^{i_{1}}\wedge\cdots\wedge
d\tilde{\phi}^{i_{k}})=d\tilde{\phi}^{i_{1}}\wedge\cdots\wedge
d\tilde{\phi}^{i_{k}}.\end{split}$ (4.10)

Setting $\tilde{\mu}^{i_{1}\cdots i_{k}}:=\chi_{1}\mu^{i_{1}\cdots
i_{k}}\in\mathscr{C}^{\alpha+1}\left(\mathbb{R}^{n};{\textstyle\bigwedge}^{k-1}T^{*}\mathbb{R}^{n}\right)$,
(4.8) follows by combining (4.9) and (4.10).

Define

$\tau:=\sum_{1\leq i_{1}<\cdots<i_{k}\leq
n}\mathfrak{P}(\tilde{\rho}_{i_{1}\ldots i_{k}},d\tilde{\mu}^{i_{1}\ldots
i_{k}})+(-1)^{k}\mathfrak{P}(\tilde{\mu}^{i_{1}\ldots,i_{k}},d\tilde{\rho}_{i_{1}\ldots
i_{k}})+\mathfrak{R}(\tilde{\rho}_{i_{1}\ldots
i_{k}},d\tilde{\mu}^{i_{1}\ldots i_{k}}).$ (4.11)

We will show
$\tau\in\mathscr{C}^{\beta}\left(\mathbb{R}^{n};{\textstyle\bigwedge}^{k}T^{*}\mathbb{R}^{n}\right)$
and
$d\tau\big{|}_{F(U^{\prime\prime})}=dF_{*}\rho\big{|}_{F(U^{\prime\prime})}$;
this will complete the proof with $V^{\prime}:=F(U^{\prime\prime})$ and we
have used (4.7).

We turn to showing
$d\tau\big{|}_{F(U^{\prime\prime})}=dF_{*}\rho\big{|}_{F(U^{\prime\prime})}$.
Since $\tilde{\mu}^{i_{1}\ldots i_{k}}$ is a $(k-1)$-form, Lemma 4.9 (v) shows
$d\mathfrak{P}(\tilde{\mu}^{i_{1}\ldots i_{k}},\tilde{\rho}_{i_{1}\ldots
i_{k}})=\mathfrak{P}(d\tilde{\mu}^{i_{1}\ldots
i_{k}},\tilde{\rho}_{i_{1}\ldots
i_{k}})+(-1)^{k-1}\mathfrak{P}(\tilde{\mu}^{i_{1}\ldots
i_{k}},d\tilde{\rho}_{i_{1}\ldots i_{k}})$. Applying differential to both
sides of this equation and using $d^{2}=0$, we obtain

$d\mathfrak{P}(d\tilde{\mu}^{i_{1}\ldots i_{k}},\tilde{\rho}_{i_{1}\ldots
i_{k}})=(-1)^{k}d\mathfrak{P}(\tilde{\mu}^{i_{1}\ldots
i_{k}},d\tilde{\rho}_{i_{1}\ldots i_{k}}).$ (4.12)

Using Lemma 4.9 (iv) in the case $l=0$, (4.12), and (4.8), we have

$\begin{split}&d\tau\big{|}_{F(U^{\prime\prime})}=\sum_{1\leq
i_{1}<\cdots<i_{k}\leq n}d\mathfrak{P}(\tilde{\rho}_{i_{1}\ldots
i_{k}},d\tilde{\mu}^{i_{1}\ldots
i_{k}})+(-1)^{k}d\mathfrak{P}(\tilde{\mu}^{i_{1}\ldots
i_{k}},d\tilde{\rho}_{i_{1}\ldots
i_{k}})+d\mathfrak{R}(\tilde{\rho}_{i_{1}\ldots
i_{k}},d\tilde{\mu}^{i_{1}\ldots i_{k}})\big{|}_{F(U^{\prime\prime})}\\\
&=\sum_{1\leq i_{1}<\cdots<i_{k}\leq n}d\mathfrak{P}(\tilde{\rho}_{i_{1}\ldots
i_{k}},d\tilde{\mu}^{i_{1}\ldots
i_{k}})+d\mathfrak{P}(d\tilde{\mu}^{i_{1}\ldots
i_{k}},\tilde{\rho}_{i_{1}\ldots
i_{k}})+d\mathfrak{R}(\tilde{\rho}_{i_{1}\ldots
i_{k}},d\tilde{\mu}^{i_{1}\ldots i_{k}})\big{|}_{F(U^{\prime\prime})}\\\
&=\sum_{1\leq i_{1}<\cdots<i_{k}\leq n}d\left(\tilde{\rho}_{i_{1}\ldots
i_{k}}d\tilde{\mu}^{i_{1}\ldots i_{k}}\right)\big{|}_{F(U^{\prime\prime})}\\\
&=dF_{*}\rho\big{|}_{F(U^{\prime\prime})},\end{split}$

as desired.

Finally, we show
$\tau\in\mathscr{C}^{\beta}\left(\mathbb{R}^{n};{\textstyle\bigwedge}^{k}T^{*}\mathbb{R}^{n}\right)$,
which will complete the proof. Using that $\tilde{\rho}_{i_{1}\ldots
i_{k}}\in\mathscr{C}^{\beta}(\mathbb{R}^{n};T^{*}\mathbb{R}^{n})$ and
$d\tilde{\mu}^{i_{1}\ldots
i_{k}}\in\mathscr{C}^{\alpha}\left(\mathbb{R}^{n};{\textstyle\bigwedge}^{k}T^{*}\mathbb{R}^{n}\right)\subset
L^{\infty}$, Lemma 4.9 (i) shows $\mathfrak{P}(\tilde{\rho}_{i_{1}\ldots
i_{k}},d\tilde{\mu}^{i_{1}\ldots
i_{k}})\in\mathscr{C}^{\beta}\left(\mathbb{R}^{n};{\textstyle\bigwedge}^{k}T^{*}\mathbb{R}^{n}\right)$
and Lemma 4.9 (iii) shows $\mathfrak{R}(\tilde{\rho}_{i_{1}\ldots
i_{k}},d\tilde{\mu}^{i_{1}\ldots
i_{k}})\in\mathscr{C}^{\alpha+\beta}\left(\mathbb{R}^{n};{\textstyle\bigwedge}^{k}T^{*}\mathbb{R}^{n}\right)\subsetneq\mathscr{C}^{\beta}$.

Thus, the proof will be complete once we show
$\mathfrak{P}(\tilde{\mu}^{i_{1}\ldots i_{k}},d\tilde{\rho}_{i_{1}\ldots
i_{k}})\in\mathscr{C}^{\beta}\left(\mathbb{R}^{n};{\textstyle\bigwedge}^{k}T^{*}\mathbb{R}^{n}\right)$.
If $\beta>1$, then $d\tilde{\rho}_{i_{1}\ldots i_{k}}\in L^{\infty}$. Using
that $\tilde{\mu}^{i_{1}\ldots i_{k}}\in\mathscr{C}^{\alpha+1}$, Lemma 4.9 (i)
then implies $\mathfrak{P}(\tilde{\mu}^{i_{1}\ldots
i_{k}},d\tilde{\rho}_{i_{1}\ldots
i_{k}})\in\mathscr{C}^{\alpha+1}\subseteq\mathscr{C}^{\beta}$. If $\beta\leq
1$, then $d\tilde{\rho}_{i_{1}\ldots
i_{k}}\in\mathscr{C}^{\beta}\subsetneq\mathscr{C}^{\beta-1-\alpha}$. Since
$\beta-1-\alpha<0$ and $\tilde{\mu}^{i_{1}\ldots
i_{k}}\in\mathscr{C}^{\alpha+1}$, Lemma 4.9 (ii) shows
$\mathfrak{P}(\tilde{\mu}^{i_{1}\ldots i_{k}},d\tilde{\rho}_{i_{1}\ldots
i_{k}})\in\mathscr{C}^{\alpha+1+\beta-1-\alpha}=\mathscr{C}^{\beta}$. This
completes the proof. ∎

###### Proof of Lemma 2.10.

Let $\mathscr{A}=\\{\phi:U_{\phi}\subseteq\mathfrak{M}\to\mathbb{R}^{n}\\}$ be
the $\mathscr{C}^{\alpha+1}$-atlas of $\mathfrak{M}$. By Definition 4.2,
$\theta\in\mathscr{C}^{\gamma}_{\mathrm{loc}}\left(\mathfrak{M};{\textstyle\bigwedge}^{k}T^{*}\mathfrak{M}\right)$
is the collection
$\Big{\\{}\theta_{\phi}\in\mathscr{C}^{\gamma}_{\mathrm{loc}}\left(U_{\phi};{\textstyle\bigwedge}^{k}T^{*}U_{\phi}\right)\Big{\\}}_{\phi\in\mathscr{A}}$
which satisfies $(\phi\circ\psi)^{*}\theta_{\psi}\big{|}_{\phi(U_{\phi}\cap
U_{\psi})}=\theta_{\phi}\big{|}_{\phi(U_{\phi}\cap U_{\psi})}$ whenever
$U_{\phi}\cap U_{\psi}\neq\varnothing$.

We claim that $\big{\\{}d\theta_{\phi}\big{\\}}_{\phi\in\mathscr{A}}$ defines
a $\mathscr{C}^{\beta-1}_{\mathrm{loc}}$ $(k+1)$-form on $\mathfrak{M}$ (see
Definition 4.2). Namely, we claim

$(\phi\circ\psi^{-1})^{*}d\theta_{\psi}\big{|}_{\phi(U_{\phi}\cap
U_{\psi})}=d\theta_{\phi}\big{|}_{\phi(U_{\phi}\cap U_{\psi})},\quad\text{
whenever }\phi,\psi\in\mathscr{A}\text{ satisfy }U_{\phi}\cap
U_{\psi}\neq\varnothing,$ (4.13)

and their common value on $\phi(U_{\phi}\cap U_{\psi})$ is
$\mathscr{C}^{\beta-1}_{\mathrm{loc}}$.

Indeed, once (4.13) is shown, then
$\tau=\\{d\theta_{\phi}\\}_{\phi\in\mathscr{A}}$ is the desired $(k+1)$-form.
To see that this $\tau$ is closed in the sense of the statement of the lemma,
note that if $F\in\mathscr{A}$ is a $\mathscr{C}^{\alpha+1}$-coordinate chart
on $\mathfrak{M}$, then $F_{*}\tau=d\theta_{F}$ and therefore
$dF_{*}\tau=d^{2}\theta_{F}=0$.

First, we claim that $d\theta_{\phi}$ is $\mathscr{C}^{\beta}_{\mathrm{loc}}$
for every $\phi\in\mathscr{A}$. The assumption that $d\theta$ has regularity
$\mathscr{C}^{\beta-1}_{\mathrm{loc}}(\mathfrak{M})$ (see Definition 2.9) says
that we can find a covering of coordinate charts
$\\{\phi_{j}:U_{\phi_{j}}\subseteq\mathfrak{M}\to\mathbb{R}^{n}\\}_{j\in
I}\subseteq\mathscr{A}$ (that is $\bigcup_{j}U_{\phi_{j}}=\mathfrak{M}$) such
that
$d(\phi_{j})_{*}\theta\in\mathscr{C}^{\beta-1}_{\mathrm{loc}}\left(U_{\phi_{j}};{\textstyle\bigwedge}^{k+1}T^{*}(U_{\phi_{j}})\right)$,
i.e.
$d\theta_{\phi_{j}}\in\mathscr{C}^{\beta-1}_{\mathrm{loc}}\left(U_{\phi_{j}};{\textstyle\bigwedge}^{k+1}T^{*}(U_{\phi_{j}})\right)$
for each $j\in I$.

Let $\psi\in\mathscr{A}$. For each $p\in U_{\psi}\subseteq\mathfrak{M}$, we
can find a $j_{0}\in I$ such that $p\in U_{\phi_{j_{0}}}$. By Proposition 2.8
we see that
$d(\psi\circ\phi_{j_{0}})^{*}\theta_{\phi_{j_{0}}}\big{|}_{\psi(U_{\psi}\cap
U_{\phi_{j_{0}}})}\in\mathscr{C}^{\beta-1}_{\mathrm{loc}}\left(\psi(U_{\psi}\cap
U_{\phi_{j_{0}}});{\textstyle\bigwedge}^{k+1}T^{*}\mathbb{R}^{n}\right)$. By
(4.2), $d\theta_{\psi}\big{|}_{\psi(U_{\psi}\cap
U_{\phi_{j_{0}}})}=d(\psi\circ\phi_{j_{0}})^{*}\theta_{\phi_{j_{0}}}\big{|}_{\psi(U_{\psi}\cap
U_{\phi_{j_{0}}})}$ so $d\theta_{\psi}$ is
$\mathscr{C}^{\beta-1}_{\mathrm{loc}}$ near $p\in U_{\psi}$. Since $p$ is
arbitrary, we know
$d\theta_{\psi}\in\mathscr{C}^{\beta-1}_{\mathrm{loc}}\left(\psi(U_{\psi});{\textstyle\bigwedge}^{k+1}T^{*}\mathbb{R}^{n}\right)$.
Therefore, $\\{d\theta_{\phi}\\}$ is a collection of
$\mathscr{C}^{\beta-1}_{\mathrm{loc}}$-forms.

We turn to proving (4.13). Let $p\in U_{\phi}$, and let $V\Subset U_{\phi}$ be
a neighborhood of $p$. We will show:

$(\phi\circ\psi^{-1})^{*}d\theta_{\psi}\big{|}_{\phi(V\cap
U_{\psi})}=d\theta_{\phi}\big{|}_{\phi(V\cap U_{\psi})}\text{ whenever
}\psi\in\mathscr{A}\text{ satisfies }V\cap U_{\psi}\neq\varnothing.$ (4.14)

Since $\phi(V)\Subset\phi(U_{\phi})$, by Lemma 4.8 there exists a
$\rho\in\mathscr{C}^{\beta}_{\mathrm{loc}}\left(\phi(V);{\textstyle\bigwedge}^{k}T^{*}\mathbb{R}^{n}\right)$
and
$\xi\in\mathscr{C}^{\gamma+1}_{\mathrm{loc}}\left(\phi(V);{\textstyle\bigwedge}^{k-1}T^{*}\mathbb{R}^{n}\right)$
such that $\theta_{\phi}\big{|}_{\phi(V)}=\rho+d\xi$, and therefore
$d\theta_{\phi}\big{|}_{\phi(V)}=d\rho$.

By Lemma 4.4 (iii), since $\beta>1-\alpha$ and $\gamma+1>1-\alpha$,

$d(\psi\circ\phi^{-1})^{*}\rho=(\psi\circ\phi^{-1})^{*}d\rho,\quad
d(\psi\circ\phi^{-1})^{*}\xi=(\psi\circ\phi^{-1})^{*}d\xi,\qquad\text{on
}\psi(V\cap U_{\psi}).$

Therefore on $\psi(V\cap U_{\psi})$,

$(\psi\circ\phi^{-1})^{*}d\theta_{\phi}=(\psi\circ\phi^{-1})^{*}d\rho=d(\psi\circ\phi^{-1})^{*}\rho=d(\psi\circ\phi^{-1})^{*}\theta_{\phi}-d(\psi\circ\phi^{-1})^{*}d\xi=d\theta_{\psi}-d^{2}(\psi\circ\phi^{-1})^{*}\xi=d\theta_{\psi}.$

This proves (4.14). Since $p\in U_{\phi}$ was arbitrary, (4.13) follows,
completing the proof. ∎

In the proof of Theorem 3.1, we need a version of Proposition 2.8 on 1-forms
where we keep track of various estimates. We are concerned with the case when
$F$ is a $\mathscr{C}^{\alpha+1}$-diffeomorphism on $\mathbb{B}^{n}$ and is
close to the identity map, and our 1-form $\theta$ defined on $\mathbb{B}^{n}$
is such that
$\|\theta\|_{\mathscr{C}^{\alpha}}+\|d\theta\|_{\mathscr{C}^{\beta-1}}$ is
small.

######  Proposition 4.10.

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

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

  1. (i)

Let
$\Phi=(\phi^{1},\dots,\phi^{n}):\mathbb{B}^{n}\xrightarrow{\sim}\mathbb{B}^{n}$
be the inverse map of $F$. Then,

$\|g\circ\Phi\|_{\mathscr{C}^{\beta}(\mathbb{B}^{n})}\leq
C_{0}\|g\|_{\mathscr{C}^{\beta}(\mathbb{B}^{n})},\quad\forall
g\in\mathscr{C}^{\beta}(\mathbb{B}^{n}).$ (4.15)

In particular
$\|\Phi\|_{\mathscr{C}^{\alpha+1}(\mathbb{B}^{n};\mathbb{R}^{n})}\leq C_{0}$.

  2. (ii)

$\|\nabla\Phi-I_{n}\|_{\mathscr{C}^{\alpha}(\mathbb{B}^{n};\mathbb{M}^{n\times
n})}\leq C_{0}\|R\|_{\mathscr{C}^{\alpha+1}(\mathbb{B}^{n};\mathbb{R}^{n})}$.

  3. (iii)

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

$\|d(F_{*}\theta)\|_{\mathscr{C}^{\beta-1}\left(\mathbb{B}^{n};\wedge^{2}T^{*}\mathbb{B}^{n}\right)}\leq
C_{0}\|d\theta\|_{\mathscr{C}^{\beta-1}\left(\mathbb{B}^{n};\wedge^{2}T^{*}\mathbb{B}^{n}\right)}.$
(4.16)

In the proof of Proposition 4.10 we need to follow convention for matrix-
valued functions.

######  Convention 4.11.

For matrix-valued map $A=(a_{i}^{j}):\mathbb{B}^{n}\to\mathbb{M}^{n\times n}$,
we use the matrix norm

$|A(x)|_{\mathbb{M}^{n\times
n}}:=\sup\limits_{v\in\mathbb{R}^{n}\backslash\\{0\\}}\frac{|A(x)\cdot
v|}{|v|},\quad\|A\|_{C^{0}(\mathbb{B}^{n};\mathbb{M}^{n\times
n})}=\sup\limits_{x\in\mathbb{B}^{n}}|A(x)|_{\mathbb{M}^{n\times n}}.$ (4.17)

For Zygmund-Hölder norms of $A$ we use the component-wise norm; namely,
$\|A\|_{\mathscr{C}^{\alpha}(\mathbb{B}^{n};\mathbb{M}^{n\times
n})}:=\|(a_{1}^{1},\dots,a_{n}^{n})\|_{\mathscr{C}^{\alpha}(\mathbb{B}^{n};\mathbb{R}^{n^{2}})}$.

######  Remark 4.12.

Let $\alpha>0$, and let $U\subseteq\mathbb{R}^{n}$ be an open set. It follows
from Lemma 2.2 that there is a $\tilde{C}_{U,\alpha}>0$, such that

$\|AB\|_{\mathscr{C}^{\alpha}(U;\mathbb{M}^{n\times
n})}\leq\tilde{C}_{U,\alpha}\|A\|_{\mathscr{C}^{\alpha}(U;\mathbb{M}^{n\times
n})}\|B\|_{\mathscr{C}^{\alpha}(U;\mathbb{M}^{n\times n})},\quad\forall
A,B\in\mathscr{C}^{\alpha}(U;\mathbb{M}^{n\times n}).$ (4.18)

######  Lemma 4.13.

Let $\alpha>0$, and let $B\subset\mathbb{R}^{n}$ be an open ball. There is a
$\tilde{c}_{B,\alpha}>0$, such that if
$\|A\|_{\mathscr{C}^{\alpha}(B;\mathbb{M}^{n\times n})}<\tilde{c}_{B,\alpha}$,
then $I+A(x)$ is an invertible matrix for every $x\in B$. In addition, the map

$A\mapsto(I+A)^{-1}:\\{M\in\mathscr{C}^{\alpha}(B;\mathbb{M}^{n\times
n}):\|M\|_{\mathscr{C}^{\alpha}}<\tilde{c}_{B,\alpha}\\}\to\mathscr{C}^{\alpha}(B;\mathbb{M}^{n\times
n}),$

is continuous and satisfies
$\|(I+A)^{-1}-I\|_{\mathscr{C}^{\alpha}(B;\mathbb{M}^{n\times n})}\leq
2\|A\|_{\mathscr{C}^{\alpha}(B;\mathbb{M}^{n\times n})}$.

###### Proof.

We take
$\tilde{c}_{B,\alpha}=\frac{1}{2}\min\\{\tilde{C}_{B,\alpha}^{-1},1/2\\}<\frac{1}{2}$
where $\tilde{C}_{B,\alpha}$ is in (4.18).

When $\|A\|_{\mathscr{C}^{\alpha}}<\tilde{c}_{B,\alpha}$, we have
$\|A^{k}\|_{\mathscr{C}^{\alpha}}\leq\tilde{C}_{B,\alpha}\|A\|_{\mathscr{C}^{\alpha}}\|A^{k-1}\|_{\mathscr{C}^{\alpha}}\leq\frac{1}{2}\|A^{k-1}\|_{\mathscr{C}^{\alpha}}$
for all $k\in\mathbb{Z}_{+}$. Therefore, for such $A$ we have
$\|A^{k}\|_{\mathscr{C}^{\alpha}}\leq 2^{1-k}\|A\|_{\mathscr{C}^{\alpha}}\leq
2^{-k}$.

We know $(I+A)^{-1}=\sum_{k=0}^{\infty}(-1)^{k}A^{k}$ whenever the right hand
side absolutely converges. This power series has $\mathscr{C}^{\alpha}$-norm
convergent radius larger than $\tilde{c}_{B,\alpha}$ and is continuous in the
domain $\\{\|A\|_{\mathscr{C}^{\alpha}}<\tilde{c}_{B,\alpha}\\}$.

Finally we have

$\|(I+A)^{-1}-I\|_{\mathscr{C}^{\alpha}}=\Big{\|}\sum_{k=1}^{\infty}(-1)^{k}A^{k}\Big{\|}_{\mathscr{C}^{\alpha}}\leq\sum_{k=1}^{\infty}\|A^{k}\|_{\mathscr{C}^{\alpha}}\leq\sum_{k=1}^{\infty}2^{1-k}\|A\|_{\mathscr{C}^{\alpha}}\leq
2\|A\|_{\mathscr{C}^{\alpha}}.$

∎

###### Proof of Proposition 4.10.

We let $C_{0}$ be a large constant which may change from line to line. In
particular, we will choose $C_{0}$ large enough such that
$\|R\|_{\mathscr{C}^{\alpha+1}}\leq C_{0}^{-1}$ implies
$\|R\|_{C^{0}}+\|\nabla R\|_{C^{0}}\leq\frac{1}{4}$ and $\|\nabla
R\|_{\mathscr{C}^{\alpha}}\leq\tilde{c}_{\mathbb{B}^{n},\alpha}$, where
$\tilde{c}_{\mathbb{B}^{n},\alpha}$ is in Lemma 4.13.

By Lemma 4.13, $\nabla F(x)=I+\nabla R(x)$ is an invertible matrix for every
$x\in\mathbb{B}^{n}$, and we have

$\|(\nabla F)^{-1}-I\|_{\mathscr{C}^{\alpha}}=\|(I+\nabla
R)^{-1}-I\|_{\mathscr{C}^{\alpha}}\leq 2\|\nabla R\|_{\mathscr{C}^{\alpha}}.$
(4.19)

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

$|F(x_{1})-F(x_{2})|\geq|x_{1}-x_{2}|-|R(x_{1})-R(x_{2})|\geq\textstyle\frac{3}{4}|x_{1}-x_{2}|.$
(4.20)

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

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

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

$\|\Phi\|_{\mathscr{C}^{\alpha+1}(\mathbb{B}^{n};\mathbb{R}^{n})}\leq C_{1}.$
(4.21)

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

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

By (4.20), $|F(x_{1})-F(x_{2})|\geq\frac{3}{4}|x_{1}-x_{2}|$ implies
$\sup\limits_{x\in\mathbb{B}^{n}}|(\nabla F(x))^{-1}|\leq\frac{4}{3}$, so
$\|(\nabla F)^{-1}\|_{C^{0}(\mathbb{B}^{n};\mathbb{M}^{n\times
n})}\leq\frac{4}{3}$, which does not depend on $R$ as well. This establishes
(4.21).

By [Str21, Lemma 5.8], we know for every $\tilde{C}_{1}>0$ there is a
$C_{2}=C_{2}(\alpha,\beta,\tilde{C}_{1})>0$ such that
$\|g\circ\Phi\|_{\mathscr{C}^{\beta}(\mathbb{B}^{n})}\leq
C_{2}\|g\|_{\mathscr{C}^{\beta}(\mathbb{B}^{n})}$ holds when
$\|\Phi\|_{\mathscr{C}^{\alpha+1}(\mathbb{B}^{n};\mathbb{R}^{n})}\leq\tilde{C}_{1}$.
By possibly increasing $C_{0}$ so that $C_{0}\geq C_{2}(\alpha,\beta,C_{1})$,
we obtain $\|g\circ\Phi\|_{\mathscr{C}^{\beta}(\mathbb{B}^{n})}\leq
C_{0}\|g\|_{\mathscr{C}^{\beta}(\mathbb{B}^{n})}$, which is (4.15).

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

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

$\|\nabla\Phi-I\|_{\mathscr{C}^{\alpha}(\mathbb{B}^{n};\mathbb{M}^{n\times
n})}=\|((\nabla F)^{-1}-I)\circ\Phi\|_{\mathscr{C}^{\alpha}}\leq
C_{0}\|(\nabla F)^{-1}-I\|_{\mathscr{C}^{\alpha}}\leq 2C_{0}\|\nabla
R\|_{\mathscr{C}^{\alpha}(\mathbb{B}^{n};\mathbb{M}^{n\times n})},$

and we obtain (ii) by replacing $C_{0}$ with $2C_{0}$.

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

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

Similar to the proof of Lemma 4.8, we define a 1-form $\rho$ and a function
$\xi$ by (see (4.6))

$\rho=\sum_{i=1}^{n}\rho_{i}dx^{i}:=\mathcal{G}\ast\vartheta
d\theta,\quad\xi:=\mathcal{G}\ast\vartheta\theta.$

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

By Lemma 4.6, we have
$\rho\in\mathscr{C}^{\beta}_{\mathrm{loc}}\left(\mathbb{R}^{n};T^{*}\mathbb{R}^{n}\right)$,
$\xi\in\mathscr{C}^{\alpha+1}_{\mathrm{loc}}(\mathbb{R}^{n})$, and
$\rho+d\xi=\mathcal{G}\ast(\vartheta d+d\vartheta)\theta=\theta$. Moreover, by
Lemma 4.7 with $\gamma=\beta-2$ and the assumption
$\operatorname{supp}\vartheta
d\theta\subseteq\operatorname{supp}\theta\subseteq\frac{3}{4}\mathbb{B}^{n}\Subset\mathbb{B}^{n}$,
we have

$\|\rho\|_{\mathscr{C}^{\beta}(\mathbb{B}^{n};T^{*}\mathbb{B}^{n})}\lesssim_{\beta}\|\vartheta
d\theta\|_{\mathscr{C}^{\beta-2}}\lesssim_{\beta}\|d\theta\|_{\mathscr{C}^{\beta-1}}.$
(4.22)

By Lemma 4.4 (iii), $d(F_{*}d\xi)=d^{2}F_{*}\xi=0$ on $\mathbb{B}^{n}$, so we
have that

$d(F_{*}\theta)=d\left(F_{*}\rho\big{|}_{\mathbb{B}^{n}}\right)+d\left(F_{*}d\xi\big{|}_{\mathbb{B}^{n}}\right)=d\left(F_{*}\rho\big{|}_{\mathbb{B}^{n}}\right)=d\left(\sum_{i=1}^{n}(\rho_{i}\circ\Phi)d\phi^{i}\right),\quad\text{on
}\mathbb{B}^{n}.$ (4.23)

Thus, to prove (4.16), by (4.22) it suffices to show

$\|d(F_{*}\theta)\|_{\mathscr{C}^{\beta-1}(\frac{3}{4}\mathbb{B}^{n};\wedge^{2}T^{*}\mathbb{R}^{n})}\lesssim\|\rho\|_{\mathscr{C}^{\beta}\left(\mathbb{B}^{n};T^{*}\mathbb{B}^{n}\right)}.$
(4.24)

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

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

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

$\sum_{i=1}^{n}(\tilde{\rho}_{i}d\tilde{\phi}^{i})\big{|}_{\frac{3}{4}\mathbb{B}^{n}}=\sum_{i=1}^{n}\left((\rho_{i}\circ\Phi)d\phi^{i}\right)\big{|}_{\frac{3}{4}\mathbb{B}^{n}}=(F_{*}\rho)\big{|}_{\frac{3}{4}\mathbb{B}^{n}}.$
(4.25)

By (4.25) and (4.23) we have

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

By Lemma 2.2 we have
$\|\tilde{\rho}_{i}\|_{\mathscr{C}^{\beta}(\mathbb{R}^{n})}\lesssim_{\beta}\|\chi\|_{\mathscr{C}^{\beta}(\mathbb{B}^{n})}\|\rho_{i}\circ\Phi\|_{\mathscr{C}^{\beta}(\mathbb{B}^{n})}$
and
$\|\tilde{\phi}^{i}\|_{\mathscr{C}^{\alpha+1}(\mathbb{R}^{n})}\lesssim_{\alpha}\|\chi\|_{\mathscr{C}^{\alpha+1}(\mathbb{B}^{n})}\|\Phi\|_{\mathscr{C}^{\alpha+1}(\mathbb{B}^{n})}$,
by (4.15) we have
$\|\rho_{i}\circ\Phi\|_{\mathscr{C}^{\beta}(\mathbb{B}^{n})}\lesssim\|\rho_{i}\|_{\mathscr{C}^{\beta}(\mathbb{B}^{n})}$
and $\|\Phi\|_{\mathscr{C}^{\alpha+1}(\mathbb{B}^{n};\mathbb{R}^{n})}\lesssim
1$. Combining them we get

$\|\tilde{\rho}_{i}\|_{\mathscr{C}^{\beta}(\mathbb{R}^{n})}\lesssim_{\alpha,\beta}\|\rho_{i}\|_{\mathscr{C}^{\beta}(\mathbb{B}^{n})},\quad\|\tilde{\phi}^{i}\|_{\mathscr{C}^{\alpha+1}(\mathbb{R}^{n})}\lesssim_{\alpha}1,\quad
1\leq i\leq n.$ (4.26)

Thus, to obtain (4.24) and complete the proof, it suffices to show

$\|d(\tilde{\rho}_{i}d\tilde{\phi}^{i})\|_{\mathscr{C}^{\beta-1}\left(\mathbb{R}^{n};\wedge^{2}T^{*}\mathbb{R}^{n}\right)}\lesssim_{\alpha,\beta}\|\tilde{\rho}_{i}\|_{\mathscr{C}^{\beta}(\mathbb{B}^{n})},\quad
1\leq i\leq n.$ (4.27)

Similar to (4.11), we define 1-form $\tau_{i}$ on $\mathbb{R}^{n}$ by

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

By Lemma 4.9 (iv) and (v), we have
$\tilde{\rho}_{i}d\tilde{\phi}^{i}=\mathfrak{P}(\tilde{\rho}_{i},d\tilde{\phi}^{i})+\mathfrak{P}(d\tilde{\phi}^{i},\tilde{\rho}_{i})+\mathfrak{R}(\tilde{\rho}_{i},d\tilde{\phi}^{i})$
and
$d\mathfrak{P}(\tilde{\phi}^{i},\tilde{\rho}_{i})=\mathfrak{P}(d\tilde{\phi}^{i},\tilde{\rho}_{i})+\mathfrak{P}(\tilde{\phi}^{i},d\tilde{\rho}_{i})$.
Therefore for $1\leq i\leq n$,

$d\tau_{i}=d\mathfrak{P}(\tilde{\rho}_{i},d\tilde{\phi}^{i})-d\mathfrak{P}(\tilde{\phi}^{i},d\tilde{\rho}_{i})+d\mathfrak{R}(\tilde{\rho}_{i},d\tilde{\phi}^{i})=d\mathfrak{P}(\tilde{\rho}_{i},d\tilde{\phi}^{i})+d\mathfrak{P}(d\tilde{\phi}^{i},\tilde{\rho}_{i})+d\mathfrak{R}(\tilde{\rho}_{i},d\tilde{\phi}^{i})=d(\tilde{\rho}_{i}d\tilde{\phi}^{i}),\quad\text{on
}\mathbb{R}^{n}.$ (4.28)

We claim

$\|\tau_{i}\|_{\mathscr{C}^{\beta}(\mathbb{R}^{n};T^{*}\mathbb{R}^{n})}\lesssim\|\tilde{\rho}_{i}\|_{\mathscr{C}^{\beta}(\mathbb{R}^{n};T^{*}\mathbb{R}^{n})},\quad
1\leq i\leq n.$ (4.29)

By Lemma 4.9 (i) and (iii), along with the fact that
$\|\tilde{\phi}^{i}\|_{\mathscr{C}^{\alpha+1}}\lesssim 1$, we get
$\|\mathfrak{P}(\tilde{\rho}_{i},d\tilde{\phi}^{i})\|_{\mathscr{C}^{\beta}}\lesssim\|\tilde{\rho}_{i}\|_{\mathscr{C}^{\beta}}$
and
$\|\mathfrak{R}(\tilde{\rho}_{i},d\tilde{\phi}^{i})\|_{\mathscr{C}^{\beta}}\lesssim\|\mathfrak{R}(\tilde{\rho}_{i},d\tilde{\phi}^{i})\|_{\mathscr{C}^{\alpha+\beta}}\lesssim\|\tilde{\rho}_{i}\|_{\mathscr{C}^{\beta}}$.

To complete the proof of (4.29), we need to show

$\|\mathfrak{P}(\tilde{\phi}^{i},d\tilde{\rho}_{i})\|_{\mathscr{C}^{\beta}(\mathbb{R}^{n};T^{*}\mathbb{R}^{n})}\lesssim_{\alpha,\beta}\|\tilde{\rho}_{i}\|_{\mathscr{C}^{\beta}(\mathbb{R}^{n};T^{*}\mathbb{R}^{n})},\quad
1\leq i\leq n.$ (4.30)

We separate the proof of (4.30) into two cases: $\beta>1$ and $0<\beta\leq 1$.

For the case $\beta>1$, (4.30) follows from Lemma 4.9 (i) with (4.26) that
$\tilde{\phi}^{i}\in\mathscr{C}^{\alpha+1}\subseteq\mathscr{C}^{\beta}$ and
$d\tilde{\rho}_{i}\in\mathscr{C}^{\beta-1}\subsetneq L^{\infty}$. For the case
$\beta\leq 1$, our assumption $0<\alpha\leq\beta$ implies that
$\alpha\in(0,1]$. So $\beta-\alpha-1<\beta-1\leq 0$ and we have
$d\tilde{\rho}_{i}\in\mathscr{C}^{\beta-1}\subsetneq\mathscr{C}^{\beta-\alpha-1}$.
By Lemma 4.9 (ii) along with $\tilde{\phi}^{i}\in\mathscr{C}^{\alpha+1}$ from
(4.26), we get
$\mathfrak{P}(\tilde{\phi}^{i},d\tilde{\rho}_{i})\in\mathscr{C}^{(\alpha+1)+(\beta-\alpha-1)}=\mathscr{C}^{\beta}$,
which gives (4.30) and establishes (4.29).

Using (4.28) and (4.29), we see
$\|d(\tilde{\rho}_{i}d\tilde{\phi}^{i})\|_{\mathscr{C}^{\beta-1}(\mathbb{R}^{n};\wedge^{2}T^{*}\mathbb{R}^{n})}=\|d\tau_{i}\|_{\mathscr{C}^{\beta-1}}\lesssim\|\tau_{i}\|_{\mathscr{C}^{\beta}}\lesssim\|\tilde{\rho}_{i}\|_{\mathscr{C}^{\beta}}$,
establishing (4.27) and completing the proof (by possibly increasing $C_{0}$).
∎

##  5 The Key Estimate

Let $\alpha>0$ and $\beta\in[\alpha,\alpha+1]$. Suppose
$\lambda^{1},\ldots,\lambda^{n}$ are $\mathscr{C}^{\alpha}$ $1$-forms on an
open set $U\subseteq\mathbb{R}^{n}$ which span the cotangent space at every
point of $U$. If we know that
$d\lambda^{j}\in\mathscr{C}^{\beta-1}_{\mathrm{loc}}$ for each $j$, it is not
necessarily true that $\lambda^{j}\in\mathscr{C}^{\beta}_{\mathrm{loc}}$.
However, it is a consequence of our main result (Theorem 3.1) that near each
point, one can always change coordinates so that the forms are in
$\mathscr{C}^{\beta}_{\mathrm{loc}}$ (see also Corollary 5.17 and Remark
5.18).

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

######  Theorem 5.1.

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

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

$\sum_{i=1}^{n}(\|\lambda^{i}-dx^{i}\|_{\mathscr{C}^{\alpha}(\mathbb{B}^{n};T^{*}\mathbb{B}^{n})}+\|d\lambda^{i}\|_{\mathscr{C}^{\beta-1}(\mathbb{B}^{n};\wedge^{2}T^{*}\mathbb{B}^{n})})\leq
c.$ (5.1)

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

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

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

The proof of Theorem 5.1 is inspired by Malgrange’s proof of the Newlander-
Nirenberg Theorem [Mal69].

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

$\lambda^{i}=dx^{i}+\sum_{j=1}^{n}a_{j}^{i}dx^{j},\quad\eta^{i}=dy^{i}+\sum_{j=1}^{n}b_{j}^{i}dy^{j},\quad
i=1,\dots,n,$ (5.3)

and define coefficient matrices

$A:=(a_{j}^{i})_{n\times n}=\begin{pmatrix}a_{1}^{1}&\cdots&a_{n}^{1}\\\
\vdots&\ddots&\vdots\\\ a_{1}^{n}&\cdots&a_{n}^{n}\end{pmatrix},\quad
B:=(b_{j}^{i})_{n\times n}=\begin{pmatrix}b_{1}^{1}&\cdots&b_{n}^{1}\\\
\vdots&\ddots&\vdots\\\ b_{1}^{n}&\cdots&b_{n}^{n}\end{pmatrix}.$ (5.4)

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

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

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

$\begin{split}&g=\sum_{i,j=1}^{n}g_{ij}dx^{i}dx^{j}:=\sum_{i,j,k=1}^{n}(\delta_{i}^{k}+a_{i}^{k})(\delta_{j}^{k}+a_{j}^{k})dx^{i}dx^{j},\\\
&h=\sum_{i,j=1}^{n}h_{ij}dy^{i}dy^{j}:=\sum_{i,j,k=1}^{n}(\delta_{i}^{k}+b_{i}^{k})(\delta_{j}^{k}+b_{j}^{k})dy^{i}dy^{j}.\end{split}$
(5.5)

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

$\delta_{i}^{j}=\delta_{ij}=\delta^{ij}=\begin{cases}1,&i=j,\\\ 0,&i\neq
j.\end{cases}$ (5.6)

We use the following notations from classical Riemannian differential
geometry:

$\begin{split}&g^{ij}:=g(dx^{i},dx^{j}),\quad\sqrt{\det
g}:=\left|\frac{\lambda^{1}\wedge\dots\wedge\lambda^{n}}{dx^{1}\wedge\dots\wedge
dx^{n}}\right|,\\\ &h^{ij}:=h(dy^{i},dy^{j}),\quad\sqrt{\det
h}:=\left|\frac{\eta^{1}\wedge\dots\wedge\eta^{n}}{dy^{1}\wedge\dots\wedge
dy^{n}}\right|.\end{split}$ (5.7)

######  Remark 5.2.

  1. (a)

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

  2. (b)

Using matrix notations in (5.3) we have $(h_{ij})_{n\times
n}=(I+B)^{\top}(I+B)$ and so we know $(h^{ij})_{n\times
n}=(h_{ij})^{-1}=(I+B)^{-1}\left((I+B)^{-1}\right)^{\top}$ and $\sqrt{\det
h}=\det(I+B)$. Similarly, we have $(g_{ij})=(I+A)^{\top}(I+A)=(g^{ij})^{-1}$
and $\sqrt{\det g}=\det(I+A)$. More importantly, we have the following lemma.

######  Lemma 5.3.

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

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

are norm continuous maps, with

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

######  Remark 5.4.

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

###### Proof.

By Lemma 2.2, the space
$\mathscr{C}^{\gamma}(\mathbb{B}^{n};\mathbb{M}^{n\times n})$ is closed under
matrix multiplication. By Remark 5.2 (b) $\sqrt{\det h}=\det(I+B)$ is a
polynomial in the components of $B$, so in particular is a norm continuous
function on $\mathscr{C}^{\gamma}(\mathbb{B}^{n};\mathbb{M}^{n\times n})$.
Note that $\det(I+0)=1$ so we have $\|\sqrt{\det
h}-1\|_{\mathscr{C}^{\gamma}(\mathbb{B}^{n})}\lesssim_{\gamma}\|B\|_{\mathscr{C}^{\gamma}(\mathbb{B}^{n};\mathbb{M}^{n\times
n})}$ when the right hand side is small.

By Lemma 4.13, by choosing $c_{n,\gamma}<\tilde{c}_{\mathbb{B}^{n},\gamma}$
where $\tilde{c}_{\mathbb{B}^{n},\gamma}$ is the constant in Lemma 4.13, we
see that the map $B\mapsto(I+B)^{-1}$ is $\mathscr{C}^{\gamma}$-norm
continuous with
$\|(I+B)^{-1}-I\|_{\mathscr{C}^{\gamma}(\mathbb{B}^{n};\mathbb{M}^{n\times
n})}\leq 2\|B\|_{\mathscr{C}^{\gamma}(\mathbb{B}^{n};\mathbb{M}^{n\times
n})}$.

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

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

######  Convention 5.5.

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

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

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

######  Lemma 5.6.

Let $\alpha>0$, and let $\lambda^{1},\dots,\lambda^{n}$ be
$\mathscr{C}^{\alpha}$ 1-forms defined on $\mathbb{B}^{n}$ that span the
cotangent space at every point, with $(a_{i}^{j})$, $g$, $(g^{ij})$, and
$\sqrt{\det g}$ given in (5.4), (5.5), and (5.7).

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

$\sum_{i,j=1}^{n}\frac{\partial}{\partial x^{j}}\Big{(}\sqrt{\det g}\cdot
g^{ij}\frac{\partial R^{k}}{\partial
x^{i}}\Big{)}=\sum_{i,j=1}^{n}\frac{\partial}{\partial x^{j}}\big{(}\sqrt{\det
g}\cdot g^{ij}a_{i}^{k}\big{)},\quad\text{in }\mathbb{B}^{n}_{x},\quad
k=1,\dots,n.$ (5.8)

Then for the pushforward 1-forms $\eta^{k}=F_{*}\lambda^{k}$, $k=1,\dots,n$,
the coefficients $(b_{i}^{j})$ defined in (5.4) satisfy

$\sum_{i,j=1}^{n}\frac{\partial}{\partial y^{i}}\left(\sqrt{\det
h}h^{ij}b_{j}^{k}\right)=0,\quad\text{in }\mathbb{B}^{n}_{y},\quad
k=1,\dots,n.$ (5.9)

###### Proof.

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

$C_{c}^{1}(\mathbb{B}_{x}^{n})=\\{v\circ F:v\in
C_{c}^{1}(\mathbb{B}_{y}^{n})\\}.$ (5.10)

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

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

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

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

Using (5.8), we get $\langle
dF^{k}-\lambda^{k},du\rangle_{\mathbb{B}^{n}_{x};g}=0$ for all $u\in
C_{c}^{1}(\mathbb{B}^{n})$. By (5.10) we have $\langle
dF^{k}-\lambda^{k},d(v\circ F)\rangle_{\mathbb{B}^{n}_{x};g}=0$ for all $v\in
C_{c}^{1}(\mathbb{B}^{n}_{y})$.

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

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

Integrating by parts, we obtain (5.9). ∎

We will choose a coordinate chart $F$ so that (5.8) is satisfied, and
therefore (5.9) will be satisfied as well.

To prove Theorem 5.1 we will prove the following:

  * •

There exists a $R\in\mathscr{C}^{\alpha+1}(\mathbb{B}^{n};\mathbb{R}^{n})$
satisfying (5.8) with boundary condition
$R\big{|}_{\partial\mathbb{B}^{n}}=0$. Moreover, we can choose $R$ with
$\|R\|_{\mathscr{C}^{\alpha+1}}\lesssim_{\alpha,\beta}\|A\|_{\mathscr{C}^{\alpha}}$.
Thus, by taking $c>0$ small, we may take $\|R\|_{\mathscr{C}^{\alpha+1}}$
small.

  * •

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

  * •

Using that
$B\in\mathscr{C}^{\alpha}(\partial\mathbb{B}^{n};\mathbb{M}^{n\times n})$
satisfies (5.9), if
$\|B\|_{\mathscr{C}^{\beta}(\partial\mathbb{B}^{n})}+\|d\eta\|_{\mathscr{C}^{\beta-1}}$
is small, we will show
$B\in\mathscr{C}^{\beta}(\mathbb{B}^{n};\mathbb{M}^{n\times n})$ and
$\|B\|_{\mathscr{C}^{\beta}}\lesssim_{\alpha,\beta}\|A\|_{\mathscr{C}^{\alpha}}$.

The last step above requires the Zygmund-Hölder well-posedness for the
Dirichlet problem.

######  Lemma 5.7 (The Dirichlet Problem).

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

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

See [FR95, Theorem 15] for a proof of Lemma 5.7.

######  Definition 5.8 (Dirichlet solution on ball).

Let $\gamma>0$ and $f\in\mathscr{C}^{\gamma-2}(\mathbb{B}^{n})$, we write
$\mathfrak{D}(f)$ for the unique solution
$u\in\mathscr{C}^{\gamma}(\mathbb{B}^{n})$ such that $\bigtriangleup u=f$ and
$u\big{|}_{\partial\mathbb{B}^{n}}=0$. For a $\mathscr{C}^{\gamma-2}$-vector
valued function $g=(g_{1},\dots,g_{m})$ on $\mathbb{B}^{n}$, we let
$\mathfrak{D}(g):=(\mathfrak{D}(g_{1}),\dots,\mathfrak{D}(g_{m}))$.

######  Remark 5.9.

In Lemma 5.7, for $\gamma>0$, $\mathscr{C}^{\gamma}(\partial\mathbb{B}^{n})$
is the Zymgund-Hölder space on the sphere
$\mathbb{S}^{n-1}=\partial\mathbb{B}^{n}$. One can use Definition 4.2 to
define $\mathscr{C}^{\gamma}$-functions on it. Note that the sphere is a
compact manifold, and therefore
$\mathscr{C}^{\gamma}_{\mathrm{loc}}(\mathbb{S}^{n-1})=\mathscr{C}^{\gamma}(\mathbb{S}^{n-1})$.
The norm can be defined using any finite atlas, and the equivalence class of
the norm does not depend on the choice of atlas. Moreover we have

$\|f\|_{\mathscr{C}^{\gamma}(\partial\mathbb{B}^{n})}\approx_{\gamma}\inf\\{\|\tilde{f}\|_{\mathscr{C}^{\gamma}(\mathbb{B}^{n})}:\tilde{f}\big{|}_{\partial\mathbb{B}^{n}}=f\\},$
(5.11)

since the trace operator
$(\cdot)|_{\partial\mathbb{B}^{n}}:\mathscr{C}^{\gamma}(\mathbb{B}^{n})\to\mathscr{C}^{\gamma}(\partial\mathbb{B}^{n})$
is continuous and surjective; see [Tri10, Theorem 2.7.2]. In fact given a
function $f\in\mathscr{C}^{\gamma}(\mathbb{S}^{n-1})$, we can take
$\tilde{f}(x)=f(\frac{x}{|x|})\chi(|x|)$ where $\chi\in
C_{c}^{\infty}(-\frac{1}{2},2)$ such that $\chi(1)=1$, then we have
$\tilde{f}\big{|}_{\mathbb{S}^{n-1}}=f$ and
$\|\tilde{f}\|_{\mathscr{C}^{\gamma}(\mathbb{B}^{n})}\approx_{\gamma}\|f\|_{\mathscr{C}^{\gamma}(\mathbb{S}^{n-1})}$.

###  5.2 The existence proposition

In this section, we show that there exists a
$\mathscr{C}^{\alpha+1}$-diffeomorphism $F=\mathrm{id}+R$ solving (5.8) and
which satisfies good estimates.

######  Proposition 5.10.

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

  * -

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

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

  1. (i)

$R$ solves the equation

$\sum_{i,j=1}^{n}\frac{\partial}{\partial x^{j}}\Big{(}\sqrt{\det g}\cdot
g^{ij}\frac{\partial R^{k}}{\partial
x^{i}}\Big{)}=\sum_{i,j=1}^{n}\frac{\partial}{\partial x^{j}}\big{(}\sqrt{\det
g}\cdot g^{ij}a_{i}^{k}\big{)},\quad\text{in }\mathbb{B}^{n}_{x},\quad
k=1,\dots,n.$ (5.8)

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

$\textstyle\|R\|_{\mathscr{C}^{\alpha+1}(\mathbb{B}^{n};\mathbb{R}^{n})}+\|\nabla
R\|_{\mathscr{C}^{\beta}(\mathbb{B}^{n}\backslash\frac{3}{4}\mathbb{B}^{n};\mathbb{R}^{n})}\leq
c_{1}^{-1}\|A\|_{\mathscr{C}^{\alpha}}.$ (5.12)

  2. (ii)

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

  3. (iii)

Let $\Phi=F^{-1}:\mathbb{B}^{n}_{y}\to\mathbb{B}^{n}_{x}$ be its inverse map,
then

$\|\nabla\Phi-I\|_{\mathscr{C}^{\alpha}(\mathbb{B}^{n};\mathbb{M}^{n\times
n})}+\|\nabla\Phi-I\|_{\mathscr{C}^{\beta}(\partial\mathbb{B}^{n};\mathbb{M}^{n\times
n})}\leq c_{1}^{-1}\|A\|_{\mathscr{C}^{\alpha}}.$ (5.13)

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

######  Remark 5.11.

The map $F$ in Proposition 5.10 is uniquely determined by $A$. This is due to
the well-posedness of the Dirichlet problem for the second order elliptic
equations, since $R$ satisfies (5.8) with
$R\big{|}_{\partial\mathbb{B}^{n}}=0$.

######  Remark 5.12.

As we will see in the proof of Theorem 5.1, the map $F$ from Proposition 5.10
is the map of the same name in Theorem 5.1.

###### Proof.

We let $c_{1}$ be a small constant which may change from line to line. Note
that if (5.12) and (5.13) are valid for some $\tilde{c}_{1}$, then they are
also valid for any $0<c_{1}\leq\tilde{c}_{1}$.

First pick $c_{1}<\tilde{c}_{\mathbb{B}^{n},\alpha}$ where
$\tilde{c}_{\mathbb{B}^{n},\alpha}$ is the constant in Lemma 4.13. By Lemma
4.13, the assumption
$\|A\|_{\mathscr{C}^{\alpha}}<c_{1}(<\tilde{c}_{\mathbb{B}^{n},\alpha})$
implies that $I+A$ is invertible at every point and
$(I+A)^{-1}\in\mathscr{C}^{\alpha}(\mathbb{B}^{n};\mathbb{M}^{n\times n})$.
Therefore, $g$ given in (5.5) is indeed a $\mathscr{C}^{\alpha}$-Riemannian
metric.

By the assumption $\operatorname{supp}A\subsetneq\frac{1}{2}\mathbb{B}^{n}$,
we have $\sqrt{\det g}\cdot
g^{ij}\big{|}_{\mathbb{B}^{n}\backslash\frac{1}{2}\mathbb{B}^{n}}=\delta^{ij}$.
Since $(\sqrt{\det g}g^{ij}(x))_{n\times n}$ is an invertible matrix for
$x\in\frac{1}{2}\mathbb{B}^{n}$, the second order operator
$\sum_{i,j=1}^{n}\partial_{x^{j}}(\sqrt{\det g}\cdot g^{ij}\partial_{x^{i}})$
is uniformly elliptic on $\mathbb{B}^{n}$. Classical existence theorems (for
example, [GT01, Theorem 8.3]) show that for each $k=1,\dots,n$ there
exists777Here $H^{1}(\mathbb{B}^{n})$ stands for the classical $L^{2}$-Sobolev
space of order $1$, and $H^{-1}(\mathbb{B}^{n})=H_{0}^{1}(\mathbb{B}^{n})^{*}$
is the $L^{2}$-Sobolev space of order $-1$. a $R^{k}\in H^{1}(\mathbb{B}^{n})$
that satisfies (5.8) with Dirichlet boundary condition
$R^{k}\big{|}_{\partial\mathbb{B}^{n}}=0$, since
$\sum_{i,j=1}^{n}\frac{\partial}{\partial x^{j}}\left(\sqrt{\det
g}g^{ij}a_{i}^{k}\right)\in\mathscr{C}^{\alpha-1}(\mathbb{B}^{n})\subset
H^{-1}(\mathbb{B}^{n})$. By a classical regularity estimate (see [GT01,
Theorem 8.34] or [FR95, Theorem 15]), we know
$R^{k}\in\mathscr{C}^{\alpha+1}(\mathbb{B}^{n};\mathbb{R}^{n})$.

To show $\|R\|_{\mathscr{C}^{\alpha+1}}\lesssim\|A\|_{\mathscr{C}^{\alpha}}$,
we write (5.8) as

$\bigtriangleup R^{k}=\sum_{i,j=1}^{n}\frac{\partial}{\partial
x^{j}}\left(\left(\delta^{ij}-\sqrt{\det g}g^{ij}\right)\frac{\partial
R^{k}}{\partial x^{i}}\right)+\sum_{i,j=1}^{n}\frac{\partial}{\partial
x^{j}}\left(\sqrt{\det g}g^{ij}a_{i}^{k}\right),\quad\text{in
}\mathbb{B}^{n}_{x},\quad k=1,\dots,n.$ (5.14)

By Remark 5.4 (see also Lemma 5.3), we see that $\|\delta^{ij}-\sqrt{\det
g}g^{ij}\|_{\mathscr{C}^{\alpha}(\mathbb{B}^{n})}\lesssim_{\alpha}\|A\|_{\mathscr{C}^{\alpha}}$.
Therefore we have

$\|\bigtriangleup
R\|_{\mathscr{C}^{\alpha-1}}\lesssim_{\alpha}\sum_{i,j,k=1}^{n}\left(\|\delta^{ij}-\sqrt{\det
g}g^{ij}\|_{\mathscr{C}^{\alpha}}\|\partial_{x^{i}}R^{k}\|_{\mathscr{C}^{\alpha}}+\|\sqrt{\det
g}g^{ij}a_{i}^{k}\|_{\mathscr{C}^{\alpha}}\right)\lesssim\|A\|_{\mathscr{C}^{\alpha}}\|R\|_{\mathscr{C}^{\alpha+1}}+\|A\|_{\mathscr{C}^{\alpha}},$
(5.15)

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

The assumption $R\big{|}_{\partial\mathbb{B}^{n}}=0$ implies that
$R=\mathfrak{D}(\bigtriangleup R)$, where $\mathfrak{D}$ is the zero Dirichlet
boundary solution operator given in Definition 5.8. Since, by Lemma 5.7,
$\mathfrak{D}:\mathscr{C}^{\alpha-1}(\mathbb{B}^{n})\to\mathscr{C}^{\alpha+1}(\mathbb{B}^{n})$
is bounded, (5.15) implies

$\|R\|_{\mathscr{C}^{\alpha+1}(\mathbb{B}^{n};\mathbb{R}^{n})}\leq\tilde{C}_{1}\|A\|_{\mathscr{C}^{\alpha}(\mathbb{B}^{n};\mathbb{M}^{n\times
n})}\|R\|_{\mathscr{C}^{\alpha+1}(\mathbb{B}^{n};\mathbb{R}^{n})}+\tilde{C}_{1}\|A\|_{\mathscr{C}^{\alpha}(\mathbb{B}^{n};\mathbb{M}^{n\times
n})},$ (5.16)

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

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

$\textstyle\|R\|_{\mathscr{C}^{\alpha+1}(\mathbb{B}^{n};\mathbb{R}^{n})}\leq\frac{3}{2}\tilde{C}_{1}\|A\|_{\mathscr{C}^{\alpha}}\leq\frac{1}{2}c_{1}^{-1}\|A\|_{\mathscr{C}^{\alpha}},\quad\text{when
}\|A\|_{\mathscr{C}^{\alpha}}<c_{1}\leq(3\tilde{C}_{1})^{-1}.$ (5.17)

This is part of the estimate in (5.12).

Next we show
$\|R\|_{\mathscr{C}^{\beta+1}(\mathbb{B}^{n}\backslash\frac{3}{4}\mathbb{B}^{n})}\lesssim\|A\|_{\mathscr{C}^{\alpha}}$.
Note that by the support assumption
$\operatorname{supp}A\subsetneq\frac{1}{2}\mathbb{B}^{n}$ we have $\sqrt{\det
g}g^{ij}\big{|}_{\mathbb{B}^{n}\backslash\frac{1}{2}\mathbb{B}^{n}}=\delta^{ij}$
and $a_{i}^{k}\big{|}_{\mathbb{B}^{n}\backslash\frac{1}{2}\mathbb{B}^{n}}=0$,
so the right hand side of (5.14) is zero in
$\mathbb{B}^{n}\backslash\frac{1}{2}\mathbb{B}^{n}$. Therefore each $R^{k}$
are harmonic functions in the domain
$\mathbb{B}^{n}\backslash\frac{1}{2}\mathbb{B}^{n}$.

The estimate
$\|R\|_{\mathscr{C}^{\alpha+1}(\mathbb{B}^{n})}\lesssim_{\alpha}\|A\|_{\mathscr{C}^{\alpha}}$
implies
$\|R\|_{\mathscr{C}^{\alpha+1}(\partial(\frac{1}{2}\mathbb{B}^{n}))}\lesssim_{\alpha}\|A\|_{\mathscr{C}^{\alpha}}$
since the trace map
$\mathscr{C}^{\alpha+1}(\mathbb{B}^{n})\to\mathscr{C}^{\alpha+1}(\partial(\frac{1}{2}\mathbb{B}^{n}))$
is bounded (see Remark 5.9). By classical interior estimates of harmonic
functions (for example, [GT01, Theorem 2.10]) since $\bigtriangleup
R\big{|}_{\mathbb{B}^{n}\backslash\frac{1}{2}\mathbb{B}^{n}}=0$, we have

$\|R\|_{\mathscr{C}^{\beta+1}(\partial\frac{3}{4}\mathbb{B}^{n})}\lesssim\|R\|_{C^{\lceil\beta\rceil+1}(\partial\frac{3}{4}\mathbb{B}^{n})}\lesssim\|R\|_{C^{0}(\mathbb{B}^{n}\backslash\frac{2}{3}\mathbb{B}^{n})}\lesssim\|A\|_{\mathscr{C}^{\alpha}}.$
(5.18)

Therefore, along with the fact that $R\big{|}_{\partial\mathbb{B}^{n}}=0$,
$R\big{|}_{\partial(\mathbb{B}^{n}\backslash\frac{3}{4}\mathbb{B}^{n})}=R\big{|}_{\partial\frac{3}{4}\mathbb{B}^{n}}\cup
R\big{|}_{\partial\mathbb{B}^{n}}$ has $\mathscr{C}^{\beta+1}$ norm bounded by
a constant times $\|A\|_{\mathscr{C}^{\alpha}}$. By classical regularity
estimates of harmonic functions (also see Lemma 5.7) on
$\mathbb{B}^{n}\backslash\frac{3}{4}\mathbb{B}^{n}$ we know

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

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

$\|\nabla
R\|_{\mathscr{C}^{\beta}(\mathbb{B}^{n}\backslash\frac{3}{4}\mathbb{B}^{n})}\leq\tilde{C}_{2}\|A\|_{\mathscr{C}^{\alpha}}.$
(5.19)

Taking $c_{1}<\frac{1}{2}\tilde{C}_{2}^{-1}$ we have $\|\nabla
R\|_{\mathscr{C}^{\beta}(\mathbb{B}^{n}\backslash\frac{3}{4}\mathbb{B}^{n})}\leq\frac{1}{2}c_{1}^{-1}\|A\|_{\mathscr{C}^{\alpha}}$.
Combining this with (5.17), completes the proof of (5.12).

We can take $c_{1}>0$ possibly smaller so that
$c_{1}<\frac{1}{3}(C_{0}\tilde{C}_{1})^{-1}$, where $C_{0}$ is the constant in
Proposition 4.10 and $\tilde{C}_{1}$ is the constant in (5.16). By (5.17) we
know $\|R\|_{\mathscr{C}^{\alpha+1}}\leq C_{0}^{-1}$. So by Proposition 4.10,
the map $F=\mathrm{id}+R$ has $\mathscr{C}^{\alpha+1}$-inverse. We conclude
$\Phi=F^{-1}\in\mathscr{C}^{\alpha+1}(\mathbb{B}^{n};\mathbb{R}^{n})$.

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

Finally we prove (iii). Note that by Proposition 4.10 (i), (5.13) gives
$\|\Phi\|_{\mathscr{C}^{\alpha+1}(\mathbb{B}^{n};\mathbb{R}^{n})}\lesssim 1$,
which is $\|\Phi\|_{\mathscr{C}^{\alpha+1}}\leq c_{1}^{-1}$ by choosing
$c_{1}$ small.

By Proposition 4.10 (ii) and using that
$c_{1}<\frac{1}{3}(C_{0}\tilde{C}_{1})^{-1}$, we get
$\|\nabla\Phi-I\|_{\mathscr{C}^{\alpha}}\leq
C_{0}\|R\|_{\mathscr{C}^{\alpha+1}}\leq
C_{0}\tilde{C}_{1}\|A\|_{\mathscr{C}^{\alpha}}\leq\frac{1}{2}c_{1}^{-1}\|A\|_{\mathscr{C}^{\alpha}}$,
which proves half of (5.13).

To show the second half of (5.13), we need to show
$\|\nabla\Phi-I\|_{\mathscr{C}^{\beta}(\partial\mathbb{B}^{n})}\lesssim\|A\|_{\mathscr{C}^{\alpha}}$.

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

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

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

We shrink $c_{1}>0$ so that
$c_{1}<\tilde{c}_{\mathbb{B}^{n},\beta}\cdot(\tilde{C}_{\mathbb{B}^{n},\beta}\cdot\tilde{C}_{2}\|\chi\|_{\mathscr{C}^{\beta}})^{-1}$,
where $\tilde{C}_{\mathbb{B}^{n},\beta}$ is in (4.18), $\tilde{C}_{2}$ is in
(5.19), and $\tilde{c}_{\mathbb{B}^{n},\beta}$ is in Lemma 4.13. Then the
assumption $\|A\|_{\mathscr{C}^{\alpha}}<c_{1}$ implies

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

Therefore we can apply Lemma 4.13 to $\chi\nabla
R\in\mathscr{C}^{\beta}(\mathbb{B}^{n};\mathbb{M}^{n\times n})$ to obtain
$\|(I+\chi\nabla R)^{-1}-I\|_{\mathscr{C}^{\beta}}\leq 2\|\chi\nabla
R\|_{\mathscr{C}^{\beta}}$. Hence, along with (5.11),

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

So by possibly shrinking $c_{1}>0$, we get
$\|\nabla\Phi-I\|_{\mathscr{C}^{\beta}(\partial\mathbb{B}^{n})}\leq\frac{1}{2}c_{1}^{-1}\|A\|_{\mathscr{C}^{\alpha}}$,
which completes the second half of (5.13). ∎

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

######  Lemma 5.13.

Let $\alpha>0$ and let $\beta\in[\alpha,\alpha+1]$. There is a
$c_{2}=c_{2}(n,\alpha,\beta)>0$ such that the following holds. Let
$A\in\mathscr{C}^{\alpha}_{c}(\frac{1}{2}\mathbb{B}^{n},\mathbb{M}^{n\times
n})$ be the coefficient matrix for $\lambda^{1},\dots,\lambda^{n}$ (see (5.3))
satisfying the assumptions of Proposition 5.10 and also satisfying

  1. (a)

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

  2. (b)

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

Suppose $\Phi=F^{-1}:\mathbb{B}^{n}_{y}\to\mathbb{B}^{n}_{x}$ satisfies the
conclusions of Proposition 5.10. Then, for the 1-forms
$\eta^{k}=\Phi^{*}\lambda^{k}$ ($k=1,\dots,n$) with coefficient matrix
$B=(b^{i}_{j})_{n\times n}$ (see (5.3)), we have:

  1. (i)

$B$ satisfies the PDE system (5.9).

  2. (ii)

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

  3. (iii)

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

###### Proof.

Part i is obtained in Lemma 5.6.

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

$\eta^{k}=\Phi^{*}\Big{(}dx^{k}+\sum_{i=1}^{n}a_{i}^{k}dx^{i}\Big{)}=d\phi^{k}+\sum_{i=1}^{n}(a_{i}^{k}\circ\Phi)d\phi^{i},\quad
b_{j}^{k}=\frac{\partial(\phi^{k}-y^{k})}{\partial
y^{j}}+\sum_{i=1}^{n}(a_{i}^{k}\circ\Phi)\frac{\partial\phi^{i}}{\partial
y^{j}},\quad 1\leq j,k\leq n.$ (5.20)

From (5.20) we know that
$\|B\|_{\mathscr{C}^{\alpha}}\lesssim\|\nabla\Phi-I\|_{\mathscr{C}^{\alpha}}+\|A\circ\Phi\|_{\mathscr{C}^{\alpha}}\|\nabla\Phi\|_{\mathscr{C}^{\alpha}}$.
By Proposition 5.10 (iii) we know $\|\Phi\|_{\mathscr{C}^{\alpha+1}}\lesssim
1$ and
$\|\nabla\Phi-I\|_{\mathscr{C}^{\alpha}}\lesssim\|A\|_{\mathscr{C}^{\alpha}}$.
By Proposition 4.10 (i) we get
$\|A\circ\Phi\|_{\mathscr{C}^{\alpha}}\lesssim\|A\|_{\mathscr{C}^{\alpha}}$.
Combining these we get

$\|B\|_{\mathscr{C}^{\alpha}}\lesssim\|\nabla\Phi-I\|_{\mathscr{C}^{\alpha}}+\|A\circ\Phi\|_{\mathscr{C}^{\alpha}}\|\nabla\Phi\|_{\mathscr{C}^{\alpha}}\lesssim\|A\|_{\mathscr{C}^{\alpha}}.$
(5.21)

Since we have $A\equiv 0$ outside $\frac{1}{2}\mathbb{B}^{n}$ in particular
$A\big{|}_{\partial\mathbb{B}^{n}}=0$, it follows that $\eta^{k}=d\phi^{k}$ on
$\partial\mathbb{B}^{n}$. Therefore
$\|B\|_{\mathscr{C}^{\beta}(\partial\mathbb{B}^{n})}=\|\nabla\Phi-I\|_{\mathscr{C}^{\beta}(\partial\mathbb{B}^{n})}$.
So by Proposition 5.10 (iii)

$\|B\|_{\mathscr{C}^{\beta}(\partial\mathbb{B}^{n})}=\|\nabla\Phi-I\|_{\mathscr{C}^{\beta}(\partial\mathbb{B}^{n})}\lesssim\|A\|_{\mathscr{C}^{\alpha}}$
(5.22)

By choosing $c_{2}>0$ small, (5.21) and (5.22) complete the proof of ii.

Finally, for iii, we apply Proposition 4.10 (iii) with $\theta=\lambda^{k}$,
for each $k=1,\dots,n$. Since $d(F_{*}\theta)=d(F_{*}\lambda^{k})=d\eta^{k}$,
by (4.16) we get
$\|d\eta^{k}\|_{\mathscr{C}^{\beta-1}(\mathbb{B}^{n};\wedge^{2}T^{*}\mathbb{B}^{n})}\lesssim\|d\lambda^{k}\|_{\mathscr{C}^{\beta-1}}$.
Taking $c_{2}$ smaller, we complete the proof. ∎

###  5.3 The regularity proposition

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

######  Proposition 5.14.

Let $\alpha>0$ and $\beta\in[\alpha,\alpha+1]$. There is a
$c_{3}=c_{3}(n,\alpha,\beta)>0$, such that if
$\eta^{1},\dots,\eta^{n}\in\mathscr{C}^{\alpha}(\mathbb{B}^{n};T^{*}\mathbb{B}^{n})$
with coefficient matrix
$B\in\mathscr{C}^{\alpha}(\mathbb{B}^{n};\mathbb{M}^{n\times n})$ (see (5.3))
such that $B$ solves the PDE (5.9),
$B\big{|}_{\partial\mathbb{B}^{n}}\in\mathscr{C}^{\beta}(\partial\mathbb{B}^{n};\mathbb{M}^{n\times
n})$ with

$\|B\|_{\mathscr{C}^{\alpha}(\mathbb{B}^{n};\mathbb{M}^{n\times
n})}+\|B\|_{\mathscr{C}^{\beta}(\partial\mathbb{B}^{n};\mathbb{M}^{n\times
n})}+\sum_{l=1}^{n}\|d\eta^{l}\|_{\mathscr{C}^{\beta-1}\left(\mathbb{B}^{n};\wedge^{2}T^{*}\mathbb{B}^{n}\right)}<c_{3},$
(5.23)

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

$\|B\|_{\mathscr{C}^{\beta}(\mathbb{B}^{n};\mathbb{M}^{n\times n})}\leq
c_{3}^{-1}\Big{(}\|B\|_{\mathscr{C}^{\alpha}(\mathbb{B}^{n};\mathbb{M}^{n\times
n})}+\|B\|_{\mathscr{C}^{\beta}(\partial\mathbb{B}^{n};\mathbb{M}^{n\times
n})}+\sum_{l=1}^{n}\|d\eta^{l}\|_{\mathscr{C}^{\beta-1}(\mathbb{B}^{n};\wedge^{2}T^{*}\mathbb{B}^{n})}\Big{)}.$
(5.24)

###### Proof.

We can write $B=\mathfrak{D}(\bigtriangleup B)+(B-\mathfrak{D}(\bigtriangleup
B))$, where $\mathfrak{D}$ is defined in Definition 5.8 and is the zero
Dirichlet boundary solution operator to the Laplacian equation on the unit
ball.

Note that $B-\mathfrak{D}(\bigtriangleup B)$ is the harmonic function whose
boundary value equals to $B\big{|}_{\partial\mathbb{B}^{n}}$ (which might not
be zero). By Lemma 5.7 using the assumption
$B\big{|}_{\partial\mathbb{B}^{n}}\in\mathscr{C}^{\beta}(\partial\mathbb{B}^{n};\mathbb{M}^{n\times
n})$, we get $B-\mathfrak{D}(\bigtriangleup
B)\in\mathscr{C}^{\beta}(\mathbb{B}^{n};\mathbb{M}^{n\times n})$ and

$\|B-\mathfrak{D}(\bigtriangleup
B)\|_{\mathscr{C}^{\beta}(\mathbb{B}^{n})}\lesssim\|B\|_{\mathscr{C}^{\beta}(\partial\mathbb{B}^{n})}.$
(5.25)

We can rewrite (5.9) as

$-\sum_{i=1}^{n}\frac{\partial}{\partial
y^{i}}b_{i}^{k}=\sum_{i,j=1}^{n}\frac{\partial}{\partial
y^{i}}\left(\big{(}\sqrt{\det
h}h^{ij}-\delta^{ij}\big{)}b_{j}^{k}\right),\quad\text{in
}\mathbb{B}^{n}_{y},\quad k=1,\dots,n.$ (5.26)

The left hand side of (5.26) is $\vartheta_{\mathbb{R}^{n}}\eta^{k}$. By Lemma
5.3, the right hand side of (5.26) is the derivatives of rational functions of
the components of $B$, which vanish to second order at $B=0$. More precisely,
using Lemma 5.3, we can rewrite (5.26) as

$\vartheta_{\mathbb{R}^{n}}\eta^{k}=-\sum_{i=1}^{n}\frac{\partial}{\partial
y^{i}}b_{i}^{k}=\sum_{i=1}^{n}\frac{\partial}{\partial
y^{i}}\mathcal{R}_{i}^{k}(B),\quad\text{in }\mathbb{B}^{n}_{y},\quad
k=1,\dots,n.$ (5.27)

Here $\mathcal{R}_{i}^{k}$ are rational functions (see Lemma 5.3) defined in a
neighborhood of origin in $\mathbb{M}^{n\times n}$ with
$|\mathcal{R}_{i}^{k}(u)|\lesssim|u|_{\mathbb{M}^{n\times n}}^{2}$ for
suitably small matrices $u\in\mathbb{M}^{n\times n}$, and we have

$|\mathcal{R}_{i}^{k}(u_{1})-\mathcal{R}_{i}^{k}(u_{2})|\lesssim(|u_{1}|_{\mathbb{M}^{n\times
n}}+|u_{2}|_{\mathbb{M}^{n\times n}})|u_{1}-u_{2}|_{\mathbb{M}^{n\times
n}},\quad\text{when }u_{1},u_{2}\in\mathbb{M}^{n\times n}\text{ small}.$
(5.28)

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

$\mathcal{R}_{i}^{k}(u)=\sum_{r=2}^{\infty}\sum_{j_{1},\dots,j_{r},l_{1},\dots,l_{r}=1}^{n}a_{i,r;l_{1}\dots
l_{r}}^{k;j_{1}\dots j_{r}}u_{j_{1}}^{l_{1}}\dots
u_{j_{r}}^{l_{r}},\quad\text{converging when }|u|_{\mathbb{M}^{n\times
n}}\text{ is small.}$ (5.29)

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

By Lemma 2.2, we can replace $u\in\mathbb{M}^{n\times n}$ in (5.29) by
$f\in\mathscr{C}^{\gamma}(\mathbb{B}^{n};\mathbb{M}^{n\times n})$, and as in
(5.28), for $\gamma>0$ there is a $\tilde{C}_{\mathcal{R},\gamma}>0$, such
that when $\|f_{1}\|_{\mathscr{C}^{\gamma}(\mathbb{B}^{n};\mathbb{M}^{n\times
n})}+\|f_{2}\|_{\mathscr{C}^{\gamma}(\mathbb{B}^{n};\mathbb{M}^{n\times
n})}\leq\tilde{C}_{\mathcal{R},\gamma}^{-1}$,

$\|\mathcal{R}_{i}^{k}(f_{1})-\mathcal{R}_{i}^{k}(f_{2})\|_{\mathscr{C}^{\gamma}(\mathbb{B}^{n})}\leq\tilde{C}_{\mathcal{R},\gamma}(\|f_{1}\|_{\mathscr{C}^{\gamma}(\mathbb{B}^{n};\mathbb{M}^{n\times
n})}+\|f_{2}\|_{\mathscr{C}^{\gamma}(\mathbb{B}^{n};\mathbb{M}^{n\times
n})})\|f_{1}-f_{2}\|_{\mathscr{C}^{\gamma}(\mathbb{B}^{n};\mathbb{M}^{n\times
n})},\quad 1\leq i,k\leq n.$ (5.30)

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

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

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

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

$\bigtriangleup
b_{j}^{k}=\left\langle\bigtriangleup\eta^{k},\frac{\partial}{\partial
y^{j}}\right\rangle=\sum_{i,j=1}^{n}\frac{\partial^{2}}{\partial y^{j}\partial
y^{i}}\mathcal{R}_{i}^{k}(B)+\left\langle\vartheta_{\mathbb{R}^{n}}d\eta^{k},\frac{\partial}{\partial
y^{j}}\right\rangle,\quad\text{in }\mathbb{B}^{n}_{y},\quad k=1,\dots,n.$
(5.31)

Let
$\tilde{\xi}_{0}:=\min(\tilde{C}_{\mathcal{R},\alpha}^{-1},\tilde{C}_{\mathcal{R},\beta}^{-1})$
where $\tilde{C}_{\mathcal{R},\gamma}$ is the constant in (5.30). Let
$\xi=\xi_{B}\in(0,\tilde{\xi}_{0}]$ to be determined. We define metric spaces
$\mathscr{X}_{\gamma,\xi}$ and an operator
$\mathcal{T}_{B}:\mathscr{X}_{\alpha,\tilde{\xi}_{0}}\to\mathscr{C}^{\alpha}(\mathbb{B}^{n};\mathbb{M}^{n\times
n})$ by

$\begin{gathered}\mathscr{X}_{\gamma,\xi}:=\\{f\in\mathscr{C}^{\gamma}(\mathbb{B}^{n};\mathbb{M}^{n\times
n}):\|f\|_{\mathscr{C}^{\gamma}}\leq\xi\\}\subset\mathscr{C}^{\gamma}(\mathbb{B}^{n};\mathbb{M}^{n\times
n}),\quad\text{for }\gamma\in\\{\alpha,\beta\\}\text{ and
}\xi\in(0,\tilde{\xi}_{0}].\\\
\mathcal{T}_{B}[f]^{k}_{j}:=b_{j}^{k}-\mathfrak{D}(\bigtriangleup
b_{j}^{k})+\Big{\langle}\mathfrak{D}(\vartheta_{\mathbb{R}^{n}}d\eta^{k}),\frac{\partial}{\partial
y^{j}}\Big{\rangle}+\sum_{i=1}^{n}\mathfrak{D}\Big{(}\frac{\partial^{2}\mathcal{R}_{i}^{k}(f)}{\partial
y^{j}\partial y^{i}}\Big{)},\quad 1\leq j,k\leq n.\end{gathered}$ (5.32)

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

Note that from (5.32) and (5.31) we have $B=\mathcal{T}_{B}[B]$. Our goal is
to show that when $c_{3}$ and $\xi$ are both suitably small, we have
$B\in\mathscr{X}_{\alpha,\xi}$ and that $\mathcal{T}_{B}$ is a contraction
mapping on both $\mathscr{X}_{\alpha,\xi}$ and $\mathscr{X}_{\beta,\xi}$, thus
by uniqueness of the fixed point we conclude that $B$ is a