# Strongly symmetric homeomorphisms on the real line with uniform continuity

Huaying Wei  Department of Mathematics and Statistics, Jiangsu Normal
University Xuzhou 221116, PR China<EMAIL_ADDRESS>and  Katsuhiko Matsuzaki
Department of Mathematics, School of Education, Waseda University Shinjuku,
Tokyo 169-8050, Japan<EMAIL_ADDRESS>
###### Abstract.

We investigate strongly symmetric homeomorphisms of the real line which appear
in harmonic analysis aspects of quasiconformal Teichmüller theory. An element
in this class can be characterized by a property that it can be extended
quasiconformally to the upper half-plane so that its complex dilatation
induces a vanishing Carleson measure. However, differently from the case on
the unit circle, strongly symmetric homeomorphisms on the real line are not
preserved under either the composition or the inversion. In this paper, we
present the difference and the relation between these two cases. In
particular, we show that if uniform continuity is assumed for strongly
symmetric homeomorphisms of the real line, then they are preserved by those
operations. We also show that the barycentric extension of uniformly
continuous one induces a vanishing Carleson measure and so do the composition
and the inverse of those quasiconformal homeomorphisms of the upper half-
plane.

###### Key words and phrases:

quasiconformal, asymptotically conformal, strongly quasisymmetric, symmetric
homeomorphism, BMO function, VMO function, Carleson measure,
$A_{\infty}$-weight, barycentric extension, Beurling–Ahlfors extension

###### 2020 Mathematics Subject Classification:

Primary 30C62, 42A45; Secondary 30H35, 26A46, 37E10

Research supported by Japan Society for the Promotion of Science (KAKENHI
18H01125 and 21F20027).

##  1\. Introduction and statement of the main results

The universal Teichmüller space and its subspaces are regarded as the spaces
consisting of quasiconformal mappings on the complex plane. By introducing
various particular properties to these mappings from view points of complex
analysis and harmonic analysis, we can study those concepts through such
subspaces reflecting their properties. For instance, studies on Teichmüller
spaces of integrable complex dilatations with Weil–Petersson metrics are in
Cui [9], Takhtajan and Teo [33], and Shen [30], those of BMO and VMO functions
are in Astala and Zinsmeister [3] and Shen and Wei [31], and those of
$C^{1+\alpha}$-diffeomorphisms are in [23].

In the conformally invariant formulation, Teichmüller spaces defined on the
upper half-plane $\mathbb{U}$ are the same as those defined on the unit disk
$\mathbb{D}$. However, if we consider subspaces of the universal Teichmüller
space by imposing certain conditions on quasiconformal and quasisymmetric
mappings, the theory can differ greatly depending on whether the conditions
are placed on the compact set (the unit circle $\mathbb{S}$) or on the non-
compact set (the real line $\mathbb{R}$).

In this paper, we study the class of strongly symmetric homeomorphisms in the
non-compact setting. A sense-preserving homeomorphism $h$ of $\mathbb{R}$ is
called strongly symmetric if $h$ is locally absolutely continuous,
$h^{\prime}$ is an $A_{\infty}$-weight, and $\log h^{\prime}$ is a VMO
function. We denote the set of all strongly symmetric homeomorphisms on
$\mathbb{R}$ by ${\rm SS}(\mathbb{R})$. The set ${\rm SS}(\mathbb{S})$ of
those on $\mathbb{S}$, which defines the original VMO Teichmüller space as in
[31], is a natural counterpart to $\rm SS(\mathbb{R})$.

The situation on $\mathbb{R}$ is more complicated than that on $\mathbb{S}$
because one has to worry about behavior at $\infty$. Typically, we have found
a phenomenon that $\rm SS(\mathbb{R})$ does not constitute a group by the
composition of mappings in [37] whereas $\rm SS(\mathbb{S})$ is a group. This
causes a trouble in the theory of Teichmüller spaces. The VMO Teichmüller
space $T_{v}(\mathbb{R})$ is defined as the set of all equivalence classes of
${\rm SS}(\mathbb{R})$ by affine transformations, and its complex analytic
structure is studied in [30] and [38]. On the contrary, homogeneity of
Teichmüller space is important to consider the group of automorphisms of the
Teichmüller space and also to introduce an invariant metric with respect to
the analytic structure; $T_{v}(\mathbb{R})$ lacks this nature.

In this paper, we consider a condition under which ${\rm SS}(\mathbb{R})$ is
preserved by the composition, and prove the following result.

######  Theorem 1.

If $g,h\in{\rm SS}(\mathbb{R})$ and $h^{-1}$ is uniformly continuous on
$\mathbb{R}$, then $g\circ h^{-1}\in{\rm SS}(\mathbb{R})$.

Let ${\rm SS}_{\rm uc}(\mathbb{R})$ denote a subset of ${\rm SS}(\mathbb{R})$
consisting of all elements $h$ such that both $h$ and $h^{-1}$ are uniformly
continuous. Then, ${\rm SS}_{\rm uc}(\mathbb{R})$ becomes a group by Theorem
1. Every element $h\in{\rm SS}_{\rm uc}(\mathbb{R})$ acts on
$T_{v}(\mathbb{R})$ as an automorphism that maps the equivalence class of $h$
to the origin of $T_{v}(\mathbb{R})$ (see Section 3). Hence, ${\rm SS}_{\rm
uc}(\mathbb{R})$ is embedded into the group ${\rm Aut}(T_{v}(\mathbb{R}))$ of
biholomorphic automorphisms of $T_{v}(\mathbb{R})$, which plays the role of
the Teichmüller modular group of $T_{v}(\mathbb{R})$. In order to see that the
action of $h$ is biholomorphic, the following investigation on quasiconformal
extension is necessary.

The complex dilatation $\mu_{F}$ of a quasiconformal homeomorphism $F$ is
defined by $\mu_{F}=F_{\bar{z}}/F_{z}$. It satisfies $\|\mu_{F}\|_{\infty}<1$.
Let $\mathcal{M}(\mathbb{U})$ be the set of all measurable functions $\mu$ on
$\mathbb{U}$ such that $\|\mu\|_{\infty}<1$ and $|\mu(z)|^{2}dxdy/y$ is a
Carleson measure on $\mathbb{U}$. In addition, if $|\mu(z)|^{2}dxdy/y$ is a
vanishing Carleson measure, then the subset of all such $\mu$ is denoted by
$\mathcal{M}_{0}(\mathbb{U})$.

The following chain rule of complex dilatations is obtained by refinement of
the argument in Cui and Zinsmeister [10, Lemma 10] who showed the first
statement in the case that $G$ is the identity map.

######  Theorem 2.

Let $G$ and $H$ be quasiconformal homeomorphisms of $\mathbb{U}$ onto itself,
and assume that $H$ is bi-Lipschitz with respect to the hyperbolic metric on
$\mathbb{U}$. Then, $(1)$ $\mu_{G\circ H^{-1}}$ belongs to
$\mathcal{M}(\mathbb{U})$ if $\mu_{G},\,\mu_{H}\in\mathcal{M}(\mathbb{U})$;
$(2)$ $\mu_{G\circ H^{-1}}$ belongs to $\mathcal{M}_{0}(\mathbb{U})$ if
$\mu_{G},\,\mu_{H}\in\mathcal{M}_{0}(\mathbb{U})$ and in addition if the
boundary extension $h^{-1}$ of $H^{-1}$ to $\mathbb{R}$ is uniformly
continuous.

We remark that if the uniform continuity is dropped then statement (2) is no
longer valid due to the lack of group structure of $\rm SS(\mathbb{R})$. The
relation between $\mathcal{M}_{0}(\mathbb{U})$ and $\rm SS(\mathbb{R})$ will
be given in Proposition 8.

By statement (2) of Theorem 2, we can define a biholomorphic automorphism of
$\mathcal{M}_{0}(\mathbb{U})$ induced by some quasiconformal extension of
$h\in{\rm SS}_{\rm uc}(\mathbb{R})$. Suppose that $h$ extends to a bi-
Lipschitz quasiconformal homeomorphism $H$ of $\mathbb{U}$ such that
$\mu_{H}\in\mathcal{M}_{0}(\mathbb{U})$, which is known to be always the case
independently of Theorem 3 below (see Proposition 16). Then, by representing
any element of $\mathcal{M}_{0}(\mathbb{U})$ by $\mu_{G}$ for a quasiconformal
homeomorphism $G$ of $\mathbb{U}$, we have the right translation
$r_{H}:\mathcal{M}_{0}(\mathbb{U})\to\mathcal{M}_{0}(\mathbb{U})$ by the
correspondence $\mu_{G}\mapsto\mu_{G\circ H^{-1}}$. Standard arguments show
that $r_{H}$ is biholomorphic, and moreover, this action is projected down to
$T_{v}(\mathbb{R})$ under the Teichmüller projection to induce a biholomorphic
automorphism $R_{h}:T_{v}(\mathbb{R})\to T_{v}(\mathbb{R})$ well defined by
$h\in{\rm SS}_{\rm uc}(\mathbb{R})$ (Theorem 17).

There are several ways to extend quasisymmetric homeomorphisms of $\mathbb{S}$
and $\mathbb{R}$ to quasiconformal homeomorphisms. The classical one is due to
Beurling and Ahlfors [4], and its variants and modified versions are also
introduced by Semmes [28] and by Fefferman, Kenig and Pipher [14]. Including
the barycentric extension introduced by Douady and Earle [12], all of them
have a property that the extension map is a bi-Lipschitz diffeomorphism with
respect to the hyperbolic metric. In addition, the conformal naturality of the
barycentric extension is useful in the theory of quasiconformal mappings, in
particular, when we consider a Möbius group action on the Teichmüller space.
However, it is so far unknown whether the complex dilatation of the
barycentric extension $e(h)$ of a strongly symmetric homeomorphism $h$ of
$\mathbb{R}$ induces a vanishing Carleson measure on $\mathbb{U}$. In this
paper we prove it does if a strongly symmetric homeomorphism $h$ and its
inverse $h^{-1}$ are uniformly continuous on $\mathbb{R}$, as stated in the
following result.

######  Theorem 3.

If $h\in{\rm SS}_{\rm uc}(\mathbb{R})$, then
$\mu_{e(h)}\in\mathcal{M}_{0}(\mathbb{U})$.

This result is a consequence of Theorem 2, and implies that a biholomorphic
automorphism $R_{h}$ of $T_{v}(\mathbb{R})$ for $h\in{\rm SS}_{\rm
uc}(\mathbb{R})$ is lifted canonically to the biholomorphic automorphism
$r_{H}$ of $\mathcal{M}_{0}(\mathbb{U})$ by the barycentric extension
$H=e(h)$.

We end this introduction (Section 1) with showing the organization of the rest
of this paper. (Section 2): Definitions and review of basic results are given.
These are concerning strongly symmetric and quasisymmetric homeomorphisms, BMO
and VMO functions, the Muckenhoupt weights, the Carleson measures, the spaces
$\mathcal{M}(\mathbb{U})$ and $\mathcal{M}_{0}(\mathbb{U})$ of Beltrami
coefficients, and their Teichmüller spaces. (Section 3): Under the assumption
of uniform continuity, the group structure is considered. The proof of Theorem
1 is given. We also prove a similar result in Theorem 11 for symmetric
homeomorphisms of $\mathbb{R}$, a vanishing class of quasisymmetric
homeomorphisms, since it has the property parallel to Theorem 1. (Section 4):
The composition of quasiconformal homeomorphisms whose complex dilatations
satisfy the Carleson measure condition is considered. The uniform continuity
condition is applied in the case for the vanishing Carleson measure condition.
The proof of Theorem 2 is given. (Section 5): The barycentric extension
defined on $\mathbb{R}$ is considered. The proof of Theorem 3 is given. We
also give another proof of Theorem 1 based on Theorem 2. (Section 6):
Comparisons between strongly symmetric homeomorphisms on $\mathbb{R}$ and
those on $\mathbb{S}$ under the conjugation by the Cayley transformation are
addressed.

##  2\. Preliminaries

###  2.1. Quasisymmetric homeomorphisms

As background knowledge, the definitions of quasisymmetric homeomorphisms and
the universal Teichmüller space are given. Including the concept of
quasiconformal mapping, a basic reference of those is [1].

######  Definition.

An increasing homeomorphism $h$ of the real line $\mathbb{R}$ onto itself is
said to be quasisymmetric if there exists a constant $M\geq 1$ such that

$\frac{1}{M}\leq\frac{h(x+t)-h(x)}{h(x)-h(x-t)}\leq M$

for all $x\in\mathbb{R}$ and $t>0$. The least possible value of such $M$ is
called the quasisymmetry constant of $h$.

If we define a measure $m_{h}$ by $m_{h}(E)=|h(E)|$ for a measurable subset
$E\subset\mathbb{R}$ with respect to the Lebesgue measure $|\cdot|$, then the
boundedness of the quasisymmetry quotient of $h$ is equivalent to that $m_{h}$
is a doubling measure, i.e., there is some constant $M^{\prime}\geq 1$ such
that $m_{h}(2I)\leq M^{\prime}m_{h}(I)$ for every bounded closed interval
$I=[x-t,x+t]$ and its double $2I=[x-2t,x+2t]$.

Beurling and Ahlfors [4] proved the following theorem.

######  Proposition 4.

An increasing homeomorphism $h$ of the real line $\mathbb{R}$ onto itself is
quasisymmetric if and only if there exists some quasiconformal homeomorphism
of the upper half-plane $\mathbb{U}$ onto itself that is continuously
extendable to the boundary map $h$.

This quasiconformal extension is explicitly written in terms of $h$, and is
called the Beurling–Ahlfors extension in the literature. Later, Douady and
Earle [12] gave a quasiconformal extension of a quasisymmetric homeomorphism,
called the barycentric extension, in a conformally natural way. We will
explain this extension in Section 4.

######  Definition.

Let $\rm QS(\mathbb{R})$ denote the group of all quasisymmetric homeomorphisms
of $\mathbb{R}$. The universal Teichmüller space $T$ is defined as the group
$\rm QS(\mathbb{R})$ modulo the left action of the group $\rm Aff(\mathbb{R})$
of all real affine mappings $z\mapsto az+b,\,a>0,b\in\mathbb{R}$, i.e.,
$T={\rm Aff}(\mathbb{R})\backslash{\rm QS}(\mathbb{R})$.

See monographs [21, 24] for comprehensive introduction on Teichmüller spaces.

Let $M(\mathbb{U})$ denote the open unit ball of the Banach space
$L^{\infty}(\mathbb{U})$ of essentially bounded measurable functions on
$\mathbb{U}$. An element in $M(\mathbb{U})$ is called a Beltrami coefficient.
By the measurable Riemann mapping theorem, a Beltrami coefficient $\mu\in
M(\mathbb{U})$ determines uniquely a quasiconformal homeomorphism $F$ on
$\mathbb{U}$ with its complex dilatation $\mu_{F}=F_{\bar{z}}/F_{z}$ equal to
$\mu$ up to post composition with conformal mappings. See [1].

By Proposition 4 with the measurable Riemann mapping theorem, the universal
Teichmüller space $T$ can be also defined as the set of all equivalence
classes $[\mu]$ of $\mu\in M(\mathbb{U})$, where $\mu,\mu^{\prime}\in
M(\mathbb{U})$ are equivalent if they produce quasiconformal homeomorphisms of
$\mathbb{U}$ onto itself having the same boundary extension to $\mathbb{R}$.
We call the quotient map $\pi:M(\mathbb{U})\to T$ the Teichmüller projection.

###  2.2. BMO, VMO, and $A_{\infty}$-weights

The functions in $\rm BMO(\mathbb{R})$ are characterized by the boundedness of
their mean oscillations over intervals. The functions in $\rm VMO(\mathbb{R})$
are those with additional property that their mean oscillations over small
intervals are small. To be precise:

######  Definition.

A locally integrable function $u$ on $\mathbb{R}$ belongs to BMO if

$\|u\|_{\rm
BMO(\mathbb{R})}=\sup_{I\subset\mathbb{R}}\frac{1}{|I|}\int_{I}|u(x)-u_{I}|dx<\infty,$

where the supremum is taken over all bounded intervals $I$ on $\mathbb{R}$ and
$u_{I}$ denotes the integral mean of $u$ over $I$. The set of all BMO
functions on $\mathbb{R}$ is denoted by ${\rm BMO}(\mathbb{R})$. This is
regarded as a Banach space with the BMO-norm $\|\cdot\|_{\rm BMO(\mathbb{R})}$
by ignoring the difference of constant functions. It is said that $u\in{\rm
BMO}(\mathbb{R})$ belongs to VMO if

$\lim_{|I|\to 0}\frac{1}{|I|}\int_{I}|u(x)-u_{I}|dx=0,$

and the set of all such functions is denoted by ${\rm VMO}(\mathbb{R})$.

The spaces $\rm BMO(\mathbb{S})$ and $\rm VMO(\mathbb{S})$ can be defined in
the same way.

Sarason [26] proved that $\rm VMO(\mathbb{R})$ is the closure of ${\rm
BMO}(\mathbb{R})\cap{\rm UC}(\mathbb{R})$ in the BMO-norm, where ${\rm
UC}(\mathbb{R})$ denotes the set of uniformly continuous functions on
$\mathbb{R}$. In particular, $\rm VMO(\mathbb{R})$ is a closed subspace of
${\rm BMO}(\mathbb{R})$.

######  Remark.

The space ${\rm VMO}(\mathbb{S})$ is the closure of $C^{\infty}(\mathbb{S})$
in the BMO-norm. Differently from the case of $\mathbb{S}$, however, it was
shown by Martell, Mitrea et al. [22, Theorem 1.8, Corollary 1.7] that
$L^{\infty}(\mathbb{R})\cap{\rm UC}(\mathbb{R})$ is not dense in ${\rm
VMO}(\mathbb{R})$, while ${\rm BMO}(\mathbb{R})\cap C^{\alpha}(\mathbb{R})$ is
dense in $\rm VMO(\mathbb{R})$ in the BMO-norm. Here, $C^{\alpha}(\mathbb{R})$
denotes the set of Hölder continuous functions on $\mathbb{R}$ of order
$\alpha\in(0,1)$.

BMO functions satisfy the following John–Nirenberg inequality (see [18,
Section VI.2], [32, Section IV.1.3]).

######  Proposition 5.

There exists two universal positive constants $C_{1},C_{2}>0$ such that for
any BMO function $u$ on $\mathbb{R}$ $($or on $\mathbb{S}$$)$, any bounded
closed interval $J\subset I$ on any interval $I\subset\mathbb{R}$ $($or
$I\subset\mathbb{S}$$)$, the inequality

$\frac{1}{|J|}|\\{z\in J:|u(x)-u_{J}|>t\\}|\leq C_{1}{\rm
exp}\left(\frac{-C_{2}t}{\|u\|_{{\rm BMO}(I)}}\right)$

holds for all $t>0$, where $\|u\|_{{\rm BMO}(I)}$ is the BMO-norm of $u$ on
$I$.

The exponentials of BMO functions are closely related to the Muckenhoupt
weights. There are several equivalent definitions of $A_{\infty}$-weights (see
[7]), and the following is one of them.

######  Definition.

A locally integrable non-negative measurable function $\omega\geq 0$ on
$\mathbb{R}$ is called a weight. We say that $\omega$ is an
$A_{\infty}$-weight if there exist two positive constants $C$ and $\alpha$
such that

$\frac{\int_{E}\omega(x)dx}{\int_{I}\omega(x)dx}\leq
C\left(\frac{|E|}{|I|}\right)^{\alpha}$

whenever $I\subset\mathbb{R}$ is a bounded closed interval and $E\subset I$ a
measurable subset.

A weight $\omega$ is called doubling if the measure $\omega(x)dx$ is doubling.
If we use the above definition, it is easy to see that an $A_{\infty}$-weight
is doubling. But, the converse is not. Fefferman and Muckenhoupt [15] provided
an example of a weight that satisfies the doubling condition but not
$A_{\infty}$.

If $\omega$ is an $A_{\infty}$-weight, then $\log\omega$ is a BMO function.
Conversely, if $\log\omega$ is a real-valued BMO function, then
$\omega^{\delta}$ is an $A_{\infty}$-weight for some small $\delta>0$. This is
a consequence from the John–Nirenberg inequality, but $\omega$ itself need not
be even locally integrable, and thus need not be an $A_{\infty}$-weight (see
[16, p.409]). However, if $\log\omega$ is a real-valued VMO function on the
unit circle $\mathbb{S}$, then by the John–Nirenberg inequality, $\omega$ is
an $A_{\infty}$-weight on $\mathbb{S}$ (see [16, p.474]). Namely, the local
BMO norm of a VMO function $\log\omega$ can be made so small on a small
interval that $\omega$ is a local $A_{\infty}$-weight on this interval. Then,
this is in fact an $A_{\infty}$-weight on $\mathbb{S}$ by its compactness.

###  2.3. Beltrami coefficients inducing Carleson measures

We define the spaces of Beltrami coefficients characterizing particular
quasiconformal homeomorphisms considered in our research.

######  Definition.

Let $\lambda$ be a positive Borel measure on the upper half-plane
$\mathbb{U}$. We say that $\lambda$ is a Carleson measure if

$\|\lambda\|_{c}=\sup_{I\subset\mathbb{R}}\frac{\lambda(I\times(0,|I|])}{|I|}<\infty,$

where the supremum is taken over all bounded closed interval
$I\subset\mathbb{R}$ and $I\times(0,|I|]\subset\mathbb{U}$ is a Carleson box.
The set of all Carleson measures on $\mathbb{U}$ is denoted by ${\rm
CM}(\mathbb{U})$. A Carleson measure $\lambda\in{\rm CM}(\mathbb{U})$ is
called vanishing if

$\lim_{|I|\to 0}\frac{\lambda(I\times(0,|I|])}{|I|}=0.$

The set of all vanishing Carleson measures on $\mathbb{U}$ is denoted by ${\rm
CM}_{0}(\mathbb{U})$.

######  Definition.

Let $\mathcal{L}(\mathbb{U})$ be the Banach space of all essentially bounded
measurable functions $\mu$ on $\mathbb{U}$ such that $\lambda_{\mu}\in{\rm
CM}(\mathbb{U})$ for $d\lambda_{\mu}(z)=|\mu(z)|^{2}\rho_{\mathbb{U}}(z)dxdy$.
Here, $\rho_{\mathbb{U}}$ is the hyperbolic density on $\mathbb{U}$. The norm
of $\mathcal{L}(\mathbb{U})$ is given by
$\|\mu\|_{\infty}+\|\lambda_{\mu}\|_{c}^{1/2}$. Let
$\mathcal{L}_{0}(\mathbb{U})$ be the subspace of $\mathcal{L}(\mathbb{U})$
consisting of all elements $\mu$ such that $\lambda_{\mu}\in{\rm
CM}_{0}(\mathbb{U})$. Moreover, we set the corresponding spaces of Beltrami
coefficients as
$\mathcal{M}(\mathbb{U})=M(\mathbb{U})\cap\mathcal{L}(\mathbb{U})$, and
$\mathcal{M}_{0}(\mathbb{U})=M(\mathbb{U})\cap\mathcal{L}_{0}(\mathbb{U})$.

On the unit disk $\mathbb{D}$, the corresponding spaces of Carleson measures
$\rm CM(\mathbb{D})$ and vanishing Carleson measures $\rm CM_{0}(\mathbb{D})$
are defined in the same way (see [18, p.231]), and so are
$\mathcal{M}(\mathbb{D})$ and $\mathcal{M}_{0}(\mathbb{D})$.

###  2.4. Strongly quasisymmetric homeomorphism

The notion of strongly quasisymmetric homeomorphisms was introduced by Semmes
[28]. This subclass is much related with and also has wide application to some
important problems in real and harmonic analysis (see [11]).

######  Definition.

An increasing homeomorphism $h$ of $\mathbb{R}$ onto itself is said to be
strongly quasisymmetric if $h$ is locally absolutely continuous and
$h^{\prime}$ belongs to the class of $A_{\infty}$-weights. Let ${\rm
SQS}(\mathbb{R})$ denote the set of all strongly quasisymmetric homeomorphisms
of $\mathbb{R}$ onto itself.

In particular, if $h\in\rm SQS(\mathbb{R})$ then $\log h^{\prime}$ belongs to
$\rm BMO(\mathbb{R})$. By the definition of $A_{\infty}$-weight, we see that
${\rm SQS}(\mathbb{R})$ is preserved under the composition of elements. In
addition, by [7, Lemma 5], the inverse operation also preserves ${\rm
SQS}(\mathbb{R})$. Thus, ${\rm SQS}(\mathbb{R})$ is a group. Moreover, since
an $A_{\infty}$-weight defines a doubling measure, $\rm SQS(\mathbb{R})$ is a
subgroup of $\rm QS(\mathbb{R})$.

######  Definition.

We say that a strongly quasisymmetric homeomorphism $h\in\rm SQS(\mathbb{R})$
is strongly symmetric if $\log h^{\prime}$ belongs to $\rm VMO(\mathbb{R})$.
Let $\rm SS(\mathbb{R})$ denote the set of all strongly symmetric
homeomorphisms of $\mathbb{R}$.

On the unit circle $\mathbb{S}$, strongly quasisymmetric and symmetric
homeomorphisms are defined similarly. The sets of those are denoted by ${\rm
SQS}(\mathbb{S})$ and ${\rm SS}(\mathbb{S})$ respectively. These classes were
investigated in [13, 31, 35, 40] during their study of BMO theory on
Teichmüller spaces. In particular, it is known that ${\rm SS}(\mathbb{S})$ is
a subgroup of ${\rm SQS}(\mathbb{S})$.

Concerning the quasiconformal extension of strongly quasisymmetric
homeomorphisms to $\mathbb{U}$, and conversely the boundary extension of
quasiconformal homeomorphisms with complex dilatations in
$\mathcal{M}(\mathbb{U})$, the results in Fefferman, Kenig and Pipher [14,
Theorems 2.3 and 4.2] imply the following claim adapted to our purpose.

######  Proposition 6.

An increasing homeomorphism $h$ of $\mathbb{R}$ onto itself belongs to ${\rm
SQS}(\mathbb{R})$ if and only if $h$ continuously extends to some
quasiconformal homeomorphism of $\mathbb{U}$ onto itself whose complex
dilatation belongs to $\mathcal{M}(\mathbb{U})$.

Precisely, a variant of the Beurling–Ahlfors extension by the heat kernel
introduced in [14] gives an appropriate extension (quasiconformal
diffeomorphism) for strongly quasisymmetric homeomorphisms, while a variant of
the Beurling–Ahlfors extension constructed by Semmes [28] is also valid in
this case under the assumption that $\log h^{\prime}$ has small BMO norm.

Due to the conformal invariance of Carleson measures (see [18, Lemma VI.3.3]),
the corresponding statement to Proposition 6 for ${\rm SQS}(\mathbb{S})$ and
$\mathcal{M}(\mathbb{D})$ holds true. However, for strongly symmetric
homeomorphisms, the situation is different.

In [31, Theorem 4.1], the corresponding claim for strongly symmetric
homeomorphisms on $\mathbb{S}$ is proved though this only asserts the
existence of desired quasiconformal extension. Later, under this existence
result, it is shown that the barycentric extension is in fact such an
extension (see Remark in Section 5).

######  Proposition 7.

A sense-preserving homeomorphism $\varphi$ of $\mathbb{S}$ onto itself belongs
to $\rm SS(\mathbb{S})$ if and only if $\varphi$ continuously extends to some
quasiconformal homeomorphism of $\mathbb{D}$ onto itself whose complex
dilatation belongs to $\mathcal{M}_{0}(\mathbb{D})$.

This result uses a fact that $C^{\infty}(\mathbb{S})$ is dense in ${\rm
VMO}(\mathbb{S})$, and does not directly imply the corresponding statement for
${\rm SS}(\mathbb{R})$ and $\mathcal{M}_{0}(\mathbb{U})$. Nevertheless, the
result itself is satisfied and we obtain the following characterization of
strongly symmetric homeomorphisms on $\mathbb{R}$ by their quasiconformal
extensions.

######  Proposition 8.

An increasing homeomorphism $h$ of $\mathbb{R}$ onto itself belongs to ${\rm
SS}(\mathbb{R})$ if and only if $h$ continuously extends to some
quasiconformal homeomorphism of $\mathbb{U}$ onto itself whose complex
dilatation belongs to $\mathcal{M}_{0}(\mathbb{U})$.

Indeed, it was proved in [38, Theorem 4.1] that the variant of the
Beurling–Ahlfors extension by the heat kernel yields such a quasiconformal
extension. The fact that the above boundary extension is strongly symmetric
was obtained in [29, Theorem 2.2].

######  Definition.

The quotient space $T_{b}={\rm Aff}(\mathbb{R})\backslash{\rm
SQS}(\mathbb{R})$ is called the BMO Teichmüller space. This can be also
defined by $T_{b}=\pi(\mathcal{M}(\mathbb{U}))$.

This space was introduced by Astala and Zinsmeister [3]. By the conformal
invariance, we can also define this by
$T_{b}=\mbox{M\"{o}b}(\mathbb{S})\backslash{\rm SQS}(\mathbb{S})$, where
$\mbox{M\"{o}b}(\mathbb{S})$ is the group of Möbius transformations keeping
$\mathbb{S}$ invariant.

######  Definition.

The quotient space $T_{v}=\mbox{M\"{o}b}(\mathbb{S})\backslash{\rm
SS}(\mathbb{S})$ is called the VMO Teichmüller space, and
$T_{v}(\mathbb{R})={\rm Aff}(\mathbb{R})\backslash{\rm SS}(\mathbb{R})$ the
VMO Teichmüller space on the real line. These can be also defined by
$T_{v}=\pi(\mathcal{M}_{0}(\mathbb{D}))$ and
$T_{v}(\mathbb{R})=\pi(\mathcal{M}_{0}(\mathbb{U}))$.

These two VMO Teichmüller spaces are different. Considering all of them on
$\mathbb{R}$ by taking the conjugate, we have the strict inclusion relations
$T_{v}\subset T_{v}(\mathbb{R})\subset T_{b}$ (see Theorem 19). The VMO
Teichmüller space on the real line was introduced by Shen [29]. The
Teichmüller spaces $T_{b}$ and $T_{v}$ possess the group structure inherited
from those of ${\rm SQS}(\mathbb{S})$ and ${\rm SS}(\mathbb{S})$.

##  3\. Uniform continuity: Proofs of Theorem 1 and a theorem for symmetric
homeomorphisms

In this section, we give the real-variable proof of Theorem 1 for strongly
symmetric homeomorphisms. We also show that symmetric homeomorphisms defined
below have a very similar property to that in Theorem 1.

It is known that each strongly quasisymmetric homeomorphism $h$ induces a
bounded linear isomorphism $P_{h}:\rm BMO(\mathbb{R})\to BMO(\mathbb{R})$ by
$P_{h}(u)=u\circ h$ (see Jones [20]). We note that the operator $P_{h}$ does
not necessarily maps $\rm VMO(\mathbb{R})$ into itself. For example, there
exist strongly symmetric homeomorphisms $g$ and $h$ (see Section 6 for
specific constructions) such that $\log(g\circ h)^{\prime}=\log
g^{\prime}\circ h+\log h^{\prime}\notin\rm VMO(\mathbb{R})$. Since $\log
h^{\prime}\in\rm VMO(\mathbb{R})$, we see that $P_{h}(\log g^{\prime})=\log
g^{\prime}\circ h\notin\rm VMO(\mathbb{R})$.

However, under the uniform continuity of $h$, the operator $P_{h}$ maps $\rm
VMO(\mathbb{R})$ properly. Namely, we have the following:

######  Proposition 9.

Let $h\in{\rm SQS}(\mathbb{R})$ such that $h$ is uniformly continuous on
$\mathbb{R}$. Then, the operator $P_{h}$ maps $\rm VMO(\mathbb{R})$ into $\rm
VMO(\mathbb{R})$.

###### Proof.

We follow the proof of Anderson, Becker and Lesley [2, Lemma]. To see that
this is composed by real analytic arguments, we show their proof.

Let $v=P_{h}(u)$ for any $u\in{\rm BMO}(\mathbb{R})$, which is also in $\rm
BMO(\mathbb{R})$. For every bounded interval $I\subset\mathbb{R}$, we have

$\int_{I}|v(x)-v_{I}|dx\leq\int_{I}|v(x)-u_{h(I)}|dx+\int_{I}|v_{I}-u_{h(I)}|dx\leq
2\int_{I}|v(x)-u_{h(I)}|dx.$

Here, for $E_{t}=\\{y\in h(I):|u(y)-u_{h(I)}|>t\\}$, Proposition 5 implies
that

$\frac{|E_{t}|}{|h(I)|}\leq C_{1}{\rm exp}\left(\frac{-C_{2}t}{\|u\|_{{\rm
BMO}(h(I))}}\right).$

Since the inverse $h^{-1}$ also belongs to ${\rm SQS}(\mathbb{R})$, there are
positive constants $C$ and $\alpha$ for the $A_{\infty}$-weight
$(h^{-1})^{\prime}$ such that

$\frac{|h^{-1}(E_{t})|}{|I|}\leq
C\left(\frac{|E_{t}|}{|h(I)|}\right)^{\alpha}.$

We consider the distribution function $\lambda(t)=|h^{-1}(E_{t})|$, where

$h^{-1}(E_{t})=\\{x\in I:|u\circ h(x)-u_{h(I)}|>t\\}.$

Then, we have

$\displaystyle\quad\int_{I}|v(x)-u_{h(I)}|dx=\int_{I}|u\circ
h(x)-u_{h(I)}|dx=\int_{0}^{\infty}\lambda(t)dt$ $\displaystyle\leq
CC_{1}^{\alpha}|I|\int_{0}^{\infty}{\rm exp}\left(\frac{-\alpha
C_{2}t}{\|u\|_{{\rm BMO}(h(I))}}\right)dt=\frac{CC_{1}^{\alpha}}{\alpha
C_{2}}|I|\|u\|_{{\rm BMO}(h(I))}.$

Therefore, we conclude that

$\frac{1}{|I|}\int_{I}|P_{h}(u)(x)-(P_{h}(u))_{I}|dx\leq\frac{2CC_{1}^{\alpha}}{\alpha
C_{2}}\|u\|_{{\rm BMO}(h(I))}.$

Since $h$ is uniformly continuous on $\mathbb{R}$, we have $|h(I)|\to 0$ as
$|I|\to 0$. Hence, the BMO norm $\|u\|_{{\rm BMO}(h(I))}$ tends uniformly to
$0$ as $|I|\to 0$ owning to $u\in\rm VMO(\mathbb{R})$. This implies that
$P_{h}(u)\in\rm VMO(\mathbb{R})$. ∎

Theorem 1 follows from this proposition.

###### Proof of Theorem 1.

Since $\rm SQS(\mathbb{R})$ is a group, we have $g\circ h^{-1}\in\rm
SQS(\mathbb{R})$ by the condition $g,h\in\rm SS(\mathbb{R})\subset\rm
SQS(\mathbb{R})$. Noting that

$\log(g\circ h^{-1})^{\prime}=(\log g^{\prime}-\log h^{\prime})\circ
h^{-1}=P_{h^{-1}}(\log g^{\prime}-\log h^{\prime}),$

we conclude by Proposition 9 that $\log(g\circ h^{-1})^{\prime}\in\rm
VMO(\mathbb{R})$, and thus $g\circ h^{-1}\in\rm SS(\mathbb{R})$. ∎

Theorem 1 implies the following as well:

######  Corollary 10.

The following statements hold:

  1. (1)

If $h\in\rm SS(\mathbb{R})$ and $h^{-1}$ is uniformly continuous on
$\mathbb{R}$, then $h^{-1}\in\rm SS(\mathbb{R})$;

  2. (2)

If $g,h\in\rm SS(\mathbb{R})$ and $h,h^{-1}$ are uniformly continuous on
$\mathbb{R}$, then $g\circ h\in\rm SS(\mathbb{R})$;

  3. (3)

The set of elements $h\in\rm SS(\mathbb{R})$ such that both $h$ and $h^{-1}$
are uniformly continuous on $\mathbb{R}$ is a subgroup of ${\rm
SQS}(\mathbb{R})$.

######  Definition.

The subgroup of ${\rm SQS}(\mathbb{R})$ consisting of all elements $h\in\rm
SS(\mathbb{R})$ such that both $h$ and $h^{-1}$ are uniformly continuous on
$\mathbb{R}$ is denoted by ${\rm SS}_{\rm uc}(\mathbb{R})$.

As mentioned in Section 1, ${\rm SS}_{\rm uc}(\mathbb{R})$ acts on the VMO
Teichmüller space $T_{v}(\mathbb{R})={\rm Aff}(\mathbb{R})\backslash{\rm
SS}(\mathbb{R})$ as a group of its automorphisms. For any $h\in{\rm SS}_{\rm
uc}(\mathbb{R})$, this action is defined by $h_{*}([g])=[g\circ h^{-1}]$ for
every $[g]\in T_{v}(\mathbb{R})$, where $[g]$ denotes the equivalence class
represented by $g\in{\rm SS}(\mathbb{R})$.

Next, we consider a similar problem for symmetric homeomorphisms of
$\mathbb{R}$.

######  Definition.

A quasisymmetric homeomorphism $h$ of $\mathbb{R}$ is said to be symmetric if

$\lim_{t\to 0}\,\frac{h(x+t)-h(x)}{h(x)-h(x-t)}=1$

uniformly for all $x\in\mathbb{R}$. Let $\rm S(\mathbb{R})$ denote the subset
of $\rm QS(\mathbb{R})$ consisting of all symmetric homeomorphisms of
$\mathbb{R}$.

It is known that $h$ is symmetric if and only if $h$ can be extended to an
asymptotically conformal homeomorphism $H$ of the upper half-plane
$\mathbb{U}$ onto itself (see [6, 17]). Here, we say that $H$ is
asymptotically conformal if its complex dilatation $\mu_{H}=H_{\bar{z}}/H_{z}$
satisfies that

${\rm
ess}\\!\\!\\!\\!\\!\\!\\!\sup_{0<y<t\qquad}\\!\\!\\!\\!\\!\\!\\!|\mu_{H}(x+iy)|\to
0\quad(t\to 0).$

In fact, the Beurling–Ahlfors extension of $h$ is asymptotically conformal
when $h$ is symmetric.

The class $\rm S(\mathbb{R})$ was first studied by Carleson [6] when he
discussed absolute continuity of quasisymmetric homeomorphisms. It was
investigated in depth later by Gardiner and Sullivan [17] in their study of
the asymptotic Teichmüller space
$T_{0}=\mbox{M\"{o}b}(\mathbb{S})\backslash{\rm S}(\mathbb{S})$ by using ${\rm
S}(\mathbb{S})$ similarly defined on the unit circle $\mathbb{S}$. Recently,
Hu, Wu and Shen [19] introduced the Teichmüller space $T_{0}(\mathbb{R})={\rm
Aff}(\mathbb{R})\backslash\rm S(\mathbb{R})$ on the real line. This has been
further generalized in [36].

The inclusion relation $\rm SS(\mathbb{R})\subset\rm S(\mathbb{R})$ is seen
from the characterization of VMO functions in Sarason [26, Theorem 2]. In more
detail, by the John–Nirenberg inequality, the local $A_{2}$-constant for the
exponential of a VMO function tends to $1$ when the interval gets small. This
can be applied to show the above inclusion. See [29, Lemma 3.3].

In [37], we constructed counter-examples for showing that the class $\rm
S(\mathbb{R})$ does not constitute a group under the composition. To be
precise, we have proved that neither the composition nor the inverse preserves
this class. However, we have the following result similar to Theorem 1 for
strongly symmetric homeomorphisms. This can be regarded as its prototype in
the non-compact setting.

######  Theorem 11.

If $g,h\in\rm S(\mathbb{R})$ and $h^{-1}$ is uniformly continuous on
$\mathbb{R}$, then $g\circ h^{-1}\in\rm S(\mathbb{R})$.

###### Proof.

Suppose that $g\circ h^{-1}$ is not symmetric. Then, for some $\delta>0$,
there are consecutive bounded closed intervals $J_{n}$ and $J_{n}^{\prime}$ in
$\mathbb{R}$ such that $|J_{n}|=|J_{n}^{\prime}|\to 0$ $(n\to\infty)$ and

$\max\left\\{\frac{|g\circ h^{-1}(J_{n})|}{|g\circ
h^{-1}(J_{n}^{\prime})|},\frac{|g\circ h^{-1}(J_{n}^{\prime})|}{|g\circ
h^{-1}(J_{n})|}\right\\}\geq 1+\delta.$

Without loss of generality, we may assume that $|g\circ
h^{-1}(J_{n})|\geq(1+\delta)|g\circ h^{-1}(J_{n}^{\prime})|$. Let
$I_{n}=h^{-1}(J_{n})$ and $I_{n}^{\prime}=h^{-1}(J_{n}^{\prime})$. Since
$h^{-1}$ is uniformly continuous, we have $|I_{n}|\to 0$ and
$|I_{n}^{\prime}|\to 0$. By the symmetry of $g$ with
$|g(I_{n})|\geq(1+\delta)|g(I_{n}^{\prime})|$, we see that there exists
$\varepsilon>0$ such that $|I_{n}|\geq(1+\varepsilon)|I_{n}^{\prime}|$ for all
sufficiently large $n$. We choose $\widetilde{I}_{n}\subset I_{n}$ so that
$\widetilde{I}_{n}$ and $I_{n}^{\prime}$ are consecutive intervals in
$\mathbb{R}$ with $|\widetilde{I}_{n}|=|I_{n}^{\prime}|$. Then,
$|I_{n}|\geq(1+\varepsilon)|\widetilde{I}_{n}|$ but

$\lim_{n\to\infty}\frac{|h(I_{n})|}{|h(\widetilde{I}_{n})|}=\lim_{n\to\infty}\frac{|h(I_{n})|}{|h(I^{\prime}_{n})|}=\frac{|J_{n}|}{|J^{\prime}_{n}|}=1.$

This contradicts that $h$ is a symmetric homeomorphism. ∎

As an immediate consequence from Theorem 11, we have:

######  Corollary 12.

The following statements hold:

  1. (1)

If $h\in\rm S(\mathbb{R})$ and $h^{-1}$ is uniformly continuous on
$\mathbb{R}$, then $h^{-1}\in\rm S(\mathbb{R})$;

  2. (2)

If $g,h\in\rm S(\mathbb{R})$ and $h,h^{-1}$ are uniformly continuous on
$\mathbb{R}$, then $g\circ h\in\rm S(\mathbb{R})$;

  3. (3)

The set of elements $h\in\rm S(\mathbb{R})$ such that both $h$ and $h^{-1}$
are uniformly continuous on $\mathbb{R}$ is a subgroup of ${\rm
QS}(\mathbb{R})$.

Thus, the group of those $h\in{\rm S}(\mathbb{R})$ with both $h$ and $h^{-1}$
being uniformly continuous acts on the Teichmüller space $T_{0}(\mathbb{R})$.

##  4\. The chain rule of complex dilatations: Proof of Theorem 2

The idea of the proof of Theorem 2 originally appeared in Cui and Zinsmeister
[10, Lemma 10] (and in Semmes [28, lemma 4.8] for a different statement), and
we supply necessary ingredients for it to make our proof more understandable.
These include stability of quasi-geodesics, the Carleson embedding theorem,
and certain properties of $A_{\infty}$-weights in the Muckenhoupt theory.

First, we prepare two lemmas for the proof of this theorem. For every
$z=(x,y)\in\mathbb{U}$, we define a closed interval $I_{z}\subset\mathbb{R}$
by $I_{z}=[x-y,x+y]$. Conversely, for a closed interval
$I=[a,b]\subset\mathbb{R}$, we define a point $q(I)\in\mathbb{U}$ by
$q(I)=(\frac{a+b}{2},\frac{b-a}{2})$. Let $\gamma_{a,b}$ denote the hyperbolic
geodesic line in the hyperbolic plane $\mathbb{U}$ joining two points $a$ and
$b$ on the boundary at infinity ${\mathbb{R}}\cup\\{\infty\\}$. Then,
$q([a,b])$ is the intersection of $\gamma_{a,b}$ and
$\gamma_{(a+b)/2,\infty}$.

Hereafter, we use a convenient notation $A\asymp B$, which means that there is
a constant $C\geq 1$ satisfying that $C^{-1}A\leq B\leq CA$ uniformly with
respect to certain circumstances obvious from the context.

######  Lemma 13.

Let $F:\mathbb{U}\to\mathbb{U}$ be a bi-Lipschitz quasiconformal homeomorphism
with respect to the hyperbolic metric that extends to a quasisymmetric
homeomorphism $f:\mathbb{R}\to\mathbb{R}$ with $f(\infty)=\infty$. Then, there
is a constant $C\geq 1$ depending only on the bi-Lipschitz constant
$L=L(F)\geq 1$ of $F$ such that

$C^{-1}\frac{|f(I_{z})|}{|I_{z}|}\leq\frac{{\rm Im}\,F(z)}{{\rm Im}\,z}\leq
C\frac{|f(I_{z})|}{|I_{z}|}$

for every $z\in\mathbb{U}$.

###### Proof.

For $I_{z}=[a,b]$, the point $z$ is the intersection of the geodesic lines
$\gamma_{a,b}$ and $\gamma_{(a+b)/2,\infty}$. Then, $F(z)$ is the intersection
of the quasi-geodesic lines $F(\gamma_{a,b})$ and
$F(\gamma_{(a+b)/2,\infty})$. By the stability of quasi-geodesics (see [5,
p.401] and [8, p.41]), we see that $F(\gamma_{a,b})$ is within a bounded
hyperbolic distance of $\gamma_{f(a),f(b)}$ and that
$F(\gamma_{(a+b)/2,\infty})$ is within a bounded hyperbolic distance of
$\gamma_{f((a+b)/2),\infty}$, where the bounds depend only on the bi-Lipschitz
constant $L$. This shows that the hyperbolic distance between the
intersections $F(z)$ and $\gamma_{f(a),f(b)}\cap\gamma_{f((a+b)/2),\infty}$ is
bounded from above by a constant depending only on $L$.

On the other hand, by the quasisymmetry of $f$ on $\mathbb{R}$, there exists
some constant $M\geq 1$ which also depends only on $L$ such that

$M^{-1}\leq\frac{f(b)-f((a+b)/2)}{f((a+b)/2)-f(a)}\leq M.$

This shows that the hyperbolic distance between the intersections
$\gamma_{f(a),f(b)}\cap\gamma_{f((a+b)/2),\infty}$ and
$\gamma_{f(a),f(b)}\cap\gamma_{(f(a)+f(b))/2,\infty}=q(f(I_{z}))$ is bounded
from above by a constant depending only on $L$.

Combining the boundedness from above of the two hyperbolic distances mentioned
above, we conclude that the hyperbolic distance between $F(z)$ and
$q(f(I_{z}))$ is bounded from above by a constant depending only on $L$. By
the formula of the hyperbolic distance $d_{H}$ on the upper half-plane
$\mathbb{U}$, for any two points $z,w\in\mathbb{U}$, it holds that

$\sinh\left(\frac{d_{H}(z,w)}{2}\right)=\frac{|z-w|}{2({\rm Im}\,z\,{\rm
Im}\,w)^{\frac{1}{2}}}\geq\frac{1}{2}\left|\left(\frac{{\rm Im}\,z}{{\rm
Im}\,w}\right)^{\frac{1}{2}}-\left(\frac{{\rm Im}\,w}{{\rm
Im}\,z}\right)^{\frac{1}{2}}\right|.$

Thus,

$C^{-1}{\rm Im}\,F(z)\leq{\rm Im}\,q(f(I_{z}))\leq C{\rm Im}\,F(z)$

for some constant $C\geq 1$ depending only on $L$. Consequently, we have

$C^{-1}\frac{{\rm Im}\,F(z)}{{\rm
Im}\,z}\leq\frac{|f(I_{z})|}{|I_{z}|}=\frac{{\rm Im}\,q(f(I_{z}))}{{\rm
Im}\,z}\leq C\frac{{\rm Im}\,F(z)}{{\rm Im}\,z},$

which is the required inequalities. ∎

######  Lemma 14.

Let $F:\mathbb{U}\to\mathbb{U}$ and $f:\mathbb{R}\to\mathbb{R}$ be as in Lemma
13. There is a constant $\alpha\geq 1$ depending only on the bi-Lipschitz
constant $L=L(F)$ such that for every bounded closed interval
$I\subset\mathbb{R}$, the image $F(Q_{I})$ of the Carleson box
$Q_{I}=I\times(0,|I|]\subset\mathbb{U}$ is contained in the Carleson box
$Q_{\alpha f(I)}$ associated with the interval $\alpha f(I)\subset\mathbb{R}$
with the same center as $f(I)$ and with length $|\alpha f(I)|=\alpha|f(I)|$.

###### Proof.

We take any point $z$ on the upper side $I\times\\{|I|\\}$ of the Carleson box
$Q_{I}$ and consider $F(z)$. Lemma 13 shows that

${\rm Im}\,F(z)\asymp\frac{{\rm
Im}\,z\cdot|f(I_{z})|}{|I_{z}|}=\frac{1}{2}|f(I_{z})|$

is satisfied with the comparability constant $C\geq 1$ depending only on $L$.
Since $I\subset I_{z}\subset 3I$, the quasisymmetry of $f$ with a constant
$M\geq 1$ depending only on $L$ implies that $|f(I_{z})|\leq(2M+1)|f(I)|$.
Hence, ${\rm Im}\,F(z)$ is bounded by $C(M+\frac{1}{2})|f(I)|$.

By the stability of quasi-geodesics, the images of the left and the right
sides of $Q_{I}$ under $F$ are within a bounded hyperbolic distance of the
hyperbolic geodesic lines towards $\infty$ from the left and the right end
points of the interval $f(I)$, respectively. Combining this with the above
estimate on the image of the upper side of $Q_{I}$, we can find the required
constant $\alpha\geq 1$ depending only on $L$ such that $F(Q_{I})\subset
Q_{\alpha f(I)}$. ∎

###### Proof of Theorem 2.

Statement (1) for the case of $G={\rm id}$ was proved by Cui and Zinsmeister
[10, Lemma 10]. We will give the proof of the other statements. Our argument
also gives a detailed exposition of their proof simultaneously. We remark in
advance that in virtue of Proposition 6 we can use the strongly quasisymmetric
properties of the boundary extension $h$ of $H$ to $\mathbb{R}$ in the proof.

Let $I\subset\mathbb{R}$ be a bounded closed interval and
$Q_{I}=I\times(0,|I|]$ the associated Carleson box. Then, by change of
variables $\zeta=H(z)$, we obtain that

$\begin{split}\mathcal{I}&=\iint_{Q_{I}}|\mu_{G\circ
H^{-1}}(\zeta)|^{2}\rho_{\mathbb{U}}(\zeta)d\xi d\eta\\\
&=\iint_{H^{-1}(Q_{I})}\left|\frac{\mu(z)-\nu(z)}{1-\overline{\nu(z)}\mu(z)}\right|^{2}\rho_{\mathbb{U}}(H(z))J_{H}(z)dxdy,\\\
\end{split}$

where $J_{H}$ denotes the Jacobian of $H$. We set $J=h^{-1}(I)$, where $h$ is
the boundary extension of $H$ to $\mathbb{R}$. By Lemma 14, there is a
constant $\alpha\geq 1$ depending only on the bi-Lipschitz constant
$L=L(H)=L(H^{-1})\geq 1$ such that $H^{-1}(Q_{I})\subset Q_{\alpha J}$, where
$\alpha J$ is the interval with the same center as $J$, but with length
$\alpha|J|$. Then, for some constant $K>0$ depending only on
$\|\mu\|_{\infty}$ and $\|\nu\|_{\infty}$, we have that

$\mathcal{I}\leq K\iint_{Q_{\alpha
J}}|\mu(z)-\nu(z)|^{2}\rho_{\mathbb{U}}(z)\frac{{\rm Im}\,z}{{\rm
Im}\,H(z)}J_{H}(z)dxdy.$

Since $H$ is bi-Lipschitz with respect to the hyperbolic metric, we see that

$\left(\frac{{\rm Im}\,z}{{\rm Im}\,H(z)}\right)^{2}J_{H}(z)\asymp 1.$

Moreover, Lemma 13 implies that

$\frac{{\rm Im}\,z}{{\rm Im}\,H(z)}\asymp\frac{|I_{z}|}{|h(I_{z})|}.$

Both comparabilities as above are given by constants depending only on the bi-
Lipschitz constant $L$. Hence, there is a constant $\widetilde{L}\geq 1$
depending only on $L$ such that

$\widetilde{L}^{-1}\frac{|h(I_{z})|}{|I_{z}|}\leq\frac{{\rm Im}\,z}{{\rm
Im}\,H(z)}J_{H}(z)\leq\widetilde{L}\,\frac{|h(I_{z})|}{|I_{z}|}$

for every $z\in\mathbb{U}$. By setting
$\widetilde{\omega}(z)=|h(I_{z})|/|I_{z}|$, we have

$\mathcal{I}\leq K\widetilde{L}\iint_{Q_{\alpha
J}}\widetilde{\omega}(z)|\mu(z)-\nu(z)|^{2}\rho_{\mathbb{U}}(z)dxdy.$

Now we apply the following Carleson embedding theorem (see [27, Theorem
1.57]). It says that assuming $\lambda\in{\rm CM}(\mathbb{U})$ is a Carleson
measure on the upper half-plane $\mathbb{U}$ and $F:\mathbb{U}\to[0,\infty)$
is a non-negative Borel measurable function in general, we have

$\iint_{\mathbb{U}}F(z)d\lambda(z)\leq
A\|\lambda\|_{c}\int_{\mathbb{R}}F^{\ast}(t)dt,$

where $F^{\ast}(t)=\sup_{z\in\Gamma(t)}F(z)$ denotes the non-tangential
maximal function of $F$ at $t\in\mathbb{R}$ with a cone
$\Gamma(t)=\\{z=(x,y)\in\mathbb{U}\mid|x-t|\leq y\\}$, and $A>0$ is an
absolute constant.

By the assumption $\mu,\nu\in\mathcal{M}(\mathbb{U})$, the measure
$\lambda_{\mu-\nu}$ defined by

$d\lambda_{\mu-\nu}=|\mu(z)-\nu(z)|^{2}\rho_{\mathbb{U}}(z)dxdy$

belongs to $\rm CM(\mathbb{U})$. Then, we obtain that

$\begin{split}\mathcal{I}&\leq
K\widetilde{L}\iint_{\mathbb{U}}\widetilde{\omega}(z)1_{Q_{\alpha
J}}(z)|\mu(z)-\nu(z)|^{2}\rho_{\mathbb{U}}(z)dxdy\\\ &\leq
AK\widetilde{L}\,\|\lambda_{\mu-\nu}1_{Q_{\alpha
J}}\|_{c}\int_{\mathbb{R}}(\widetilde{\omega}1_{Q_{\alpha
J}})^{\ast}(t)dt=C(J)\int_{3\alpha J}(\widetilde{\omega}1_{Q_{\alpha
J}})^{\ast}(t)dt,\\\ \end{split}$

where the constant $C(J)=AK\widetilde{L}\,\|\lambda_{\mu-\nu}1_{Q_{\alpha
J}}\|_{c}>0$ depends also on the interval $J$, but is bounded due to
$\lambda_{\mu-\nu}\in\rm CM(\mathbb{U})$.

The boundary extension $h$ of $H$ to $\mathbb{R}$ is strongly quasisymmetric
by Proposition 6. We consider the $A_{\infty}$-weight $\omega=h^{\prime}$ on
$\mathbb{R}$ and set $\varphi=\omega 1_{3\alpha J}$. Let

$M\varphi(t)=\sup_{t\in I}\frac{1}{|I|}\int_{I}\varphi(s)ds$

denote the Hardy–Littlewood (uncentered) maximal function of $\varphi$. Then,
we can show that $(\widetilde{\omega}1_{Q_{\alpha J}})^{\ast}(t)\leq
M\varphi(t)$ for any $t\in 3\alpha J$. Indeed, $t\in I_{z}\subset 3\alpha J$
for every $z\in\Gamma(t)\cap Q_{\alpha J}$, and hence

$\displaystyle(\widetilde{\omega}1_{Q_{\alpha
J}})^{\ast}(t)=\sup_{z\in\Gamma(t)\cap Q_{\alpha J}}\widetilde{\omega}(z)$
$\displaystyle=\sup_{z\in\Gamma(t)\cap Q_{\alpha
J}}\frac{|h(I_{z})|}{|I_{z}|}$ $\displaystyle\leq\sup_{t\in I_{z}\subset
3\alpha J}\frac{1}{|I_{z}|}\int_{I_{z}}\omega(t)dt\leq M\varphi(t).$

Therefore,

$\mathcal{I}\leq C(J)\int_{3\alpha J}M\varphi(t)dt.$

By the reverse Hölder inequality in the Muckenhoupt theory (see [7, Lemma 5]),
there exist $C>0$ and $p>1$ such that for any bounded closed interval
$I\subset\mathbb{R}$, we have

$\frac{1}{|I|}\int_{I}\omega(t)^{p}dt\leq
C\left(\frac{1}{|I|}\int_{I}\omega(t)dt\right)^{p}.$

It follows that

$\begin{split}\mathcal{I}&\leq C(J)|3\alpha
J|^{\frac{1}{p^{\prime}}}\left(\int_{3\alpha
J}(M\varphi)^{p}\right)^{\frac{1}{p}}\leq C_{1}(J)|3\alpha
J|^{\frac{1}{p^{\prime}}}\left(\int_{\mathbb{R}}\varphi^{p}\right)^{\frac{1}{p}}\\\
&=C_{1}(J)|3\alpha J|^{\frac{1}{p^{\prime}}}\left(\int_{3\alpha
J}\omega^{p}\right)^{\frac{1}{p}}\leq C_{2}(J)\int_{3\alpha
J}\omega=C_{2}(J)|h(3\alpha J)|\\\ \end{split}$

for $1/p+1/p^{\prime}=1$, where $C_{1}(J)$ and $C_{2}(J)$ are some constant
multiples of $C(J)$. We have used the Hölder inequality in the first
inequality, the strong $L^{p}$-estimate for maximal functions (see [7, Theorem
I], [18, Theorem 4.3]) in the second inequality, and the reverse Hölder
inequality mentioned above in the last inequality.

Finally, we estimate $|h(3\alpha J)|$. Let $M\geq 1$ be the quasisymmetry
constant of $h$. Then,

$|h(3\alpha J)|\leq\frac{M^{3\alpha}-1}{M-1}|h(J)|=\widetilde{M}|I|,$

where the constant involving $M$ and $\alpha$ is replaced with
$\widetilde{M}\geq 1$. This yields an estimate

$\frac{\mathcal{I}}{|I|}=\frac{1}{|I|}\iint_{Q_{I}}|\mu_{G\circ
H^{-1}}(\zeta)|^{2}\rho_{\mathbb{U}}(\zeta)d\xi
d\eta\leq\widetilde{M}C_{2}(J).$

Since $C_{2}(J)$ is bounded independent of $J$, we see that $\mu_{G\circ
H^{-1}}$ belongs to $\mathcal{M}(\mathbb{U})$. This proves statement (1). We
further assume that $\mu,\nu\in\mathcal{M}_{0}(\mathbb{U})$. Then,
$\lambda_{\mu-\nu}\in\rm CM_{0}(\mathbb{U})$. Since

$C_{2}(J)\asymp\|\lambda_{\mu-\nu}1_{Q_{\alpha J}}\|_{c},$

this tends uniformly to $0$ as $|J|\to 0$. Moreover, by the assumption that
$h^{-1}$ is uniformly continuous on $\mathbb{R}$, we see that $|J|\to 0$
uniformly as $|I|\to 0$. This shows that $\mu_{G\circ H^{-1}}$ belongs to
$\mathcal{M}_{0}(\mathbb{U})$, which proves statement (2). ∎

##  5\. The barycentric extension: Proof of Theorem 3

In this section, after introducing the barycentric extension, we prove Theorem
3, and provide a complex-variable proof for Theorem 1. We also show the fact
that the group ${\rm SS}_{\rm uc}$ acts on the Teichmüller space
$T_{v}(\mathbb{R})$ as biholomorphic automorphisms.

We first recall the barycentric extension for a homeomorphism of the unit
circle $\mathbb{S}$ introduced by Douady and Earle [12], and then we translate
it to the setting of the real line $\mathbb{R}$.

For $\varphi\in{\rm QS}(\mathbb{S})$, the average of $\varphi$ taken at
$w\in\mathbb{D}$ is defined by

$\xi_{\varphi}(w)=\frac{1}{2\pi}\int_{\mathbb{S}}\gamma_{w}(\varphi(\zeta))|d\zeta|$

where the Möbius transformation

$\gamma_{w}(z)=\frac{z-w}{1-\bar{w}z}\in\mbox{\rm{M\"{o}b}}(\mathbb{D})$

maps $w$ to the origin $0$. The barycenter of $\varphi$ is the unique point
$w_{0}\in\mathbb{D}$ such that $\xi_{\varphi}(w_{0})=0$. The value of the
barycentric extension $e(\varphi)$ at the origin $0$ is defined to be the
barycenter $w_{0}=w_{0}(\varphi)$, that is, $e(\varphi)(0)=w_{0}(\varphi)$.
For an arbitrary point $z\in\mathbb{D}$, the barycentric extension
$e(\varphi)$ is defined by

${e(\varphi)(z)=e(\varphi\circ\gamma_{z}^{-1})(0)},$

which satisfies the conformal naturality such that

${e(\gamma_{1}\circ\varphi\circ\gamma_{2})=e(\gamma_{1})\circ e(\varphi)\circ
e(\gamma_{2})}$

for any $\gamma_{1},\gamma_{2}\in\mbox{\rm{M\"{o}b}}(\mathbb{S})$. Moreover,
$e(\varphi)$ is a quasiconformal diffeomorphism of $\mathbb{D}$ onto itself
and is even bi-Lipschitz under the hyperbolic metric on $\mathbb{D}$.

We fix a Cayley transformation $T:\mathbb{U}\to\mathbb{D}$ given by
$w=T(z)=(z-i)/(z+i)$. For a quasisymmetric homeomorphism $h\in{\rm
QS}(\mathbb{R})$, we set $\varphi=T\circ h\circ T^{-1}$, and define
$e(h)=T^{-1}\circ e(\varphi)\circ T$, where $e(\varphi)$ is the barycentric
extension of $\varphi\in{\rm QS}(\mathbb{S})$. We also call $e(h)$ the
barycentric extension of $h$. The complex dilatation of $e(h)$ is denoted by
$\mu_{e(h)}$. The barycentric extension $e(h)$ on $\mathbb{U}$ also satisfies
the conformal naturality and bi-Lipschitz continuity as in the case of
$\mathbb{D}$.

Suppose that $\varphi\in{\rm QS}(\mathbb{S})$ has a quasiconformal extension
to $\mathbb{D}$ with complex dilatation $\mu$. We denote by
$e^{-1}(\varphi^{-1})$ the inverse mapping of the barycentric extension
$e(\varphi^{-1})$. By checking the proof of [34, Corollary 3.4], we see that
$\mu_{e^{-1}(\varphi^{-1})}\in\mathcal{M}(\mathbb{D})$ if
$\mu\in\mathcal{M}(\mathbb{D})$, and
$\mu_{e^{-1}(\varphi^{-1})}\in\mathcal{M}_{0}(\mathbb{D})$ if
$\mu\in\mathcal{M}_{0}(\mathbb{D})$. Furthermore, the results on the
barycentric extension $e(\varphi)$ itself were deduced from this by using [3,
Lemma 10] or [28, Lemme 4.8] (which are generalized to Theorem 2). Namely,
$\mu_{e(\varphi)}\in{\mathcal{M}}(\mathbb{D})$ if
$\mu\in{\mathcal{M}}(\mathbb{D})$, and
$\mu_{e(\varphi)}\in{\mathcal{M}}_{0}(\mathbb{D})$ if
$\mu\in{\mathcal{M}}_{0}(\mathbb{D})$. This claim is also true for
${\mathcal{M}}(\mathbb{U})$ on the upper half-plane $\mathbb{U}$ by the
conformal invariance, but is not known to be true for
${\mathcal{M}}_{0}(\mathbb{U})$. The former result on
$\mu_{e^{-1}(\varphi^{-1})}$ was translated to the setting on $\mathbb{U}$ in
[30, Theorem 4.3] as follows.

######  Lemma 15.

Let $H$ be a quasiconformal homeomorphism of $\mathbb{U}$ onto itself whose
complex dilatation is in ${\mathcal{M}}_{0}(\mathbb{U})$. Then, for the
boundary extension $h$ of $H$ to $\mathbb{R}$, the complex dilatation
$\mu_{e^{-1}(h^{-1})}$ is also in ${\mathcal{M}}_{0}(\mathbb{U})$.

This lemma implies that once we obtain such a quasiconformal homeomorphism
$H$, we can replace it with the bi-Lipschitz diffeomorphism given by means of
the barycentric extension with the same boundary extension $h$ as $H$ and with
its complex dilatation in the same class ${\mathcal{M}}_{0}(\mathbb{U})$ as
$H$. For a given $h\in{\rm SS}(\mathbb{R})$, the existence of the appropriate
quasiconformal extension $H$ is guaranteed by Proposition 8. Thus, we can
prepare the following claim for the proof of Theorem 3.

######  Proposition 16.

Any $h\in{\rm SS}(\mathbb{R})$ extends continuously to a bi-Lipschitz
diffeomorphism $e^{-1}(h^{-1})$ of $\mathbb{U}$ onto itself whose complex
dilatation belongs to ${\mathcal{M}}_{0}(\mathbb{U})$.

######  Remark.

On the unit disk $\mathbb{D}$, the barycentric extension $e(\varphi)$
satisfies $\mu_{e(\varphi)}\in{\mathcal{M}}(\mathbb{D})$ for $\varphi\in{\rm
SQS}(\mathbb{S})$ and $\mu_{e(\varphi)}\in{\mathcal{M}}_{0}(\mathbb{D})$ for
$\varphi\in{\rm SS}(\mathbb{S})$. These facts follow from the above arguments
combined with Proposition 6 (applied to $\mathbb{D}$) and Proposition 7, which
originally appeared in [10] and [34].

###### Proof of Theorem 3.

Since $h\in\rm SS(\mathbb{R})$ and $h^{-1}$ is uniformly continuous on
$\mathbb{R}$, we conclude by Corollary 10 that $h^{-1}\in\rm SS(\mathbb{R})$.
Then by Proposition 16, we have
$\mu_{e^{-1}(h)}\in\mathcal{M}_{0}(\mathbb{U})$. Since $e^{-1}(h)$ is bi-
Lipschitz on $\mathbb{U}$ and $h$ is uniformly continuous on $\mathbb{R}$,
Theorem 2 implies that $\mu_{e(h)}\in\mathcal{M}_{0}(\mathbb{U})$. This
completes the proof. ∎

The complex-variable proof of Theorem 1 is also instructive. We give the
argument by using the results on quasiconformal extensions in Theorem 2,
Proposition 8, and Proposition 16.

###### Another proof of Theorem 1.

For $g,h\in\rm SS(\mathbb{R})$, we set $G=e^{-1}(g^{-1})$ and
$H=e^{-1}(h^{-1})$ (only for $H$, this particular construction is necessary).
By Proposition 16, we have $\mu_{G},\,\mu_{H}\in\mathcal{M}_{0}(\mathbb{U})$.
Since $H=e^{-1}(h^{-1})$ is bi-Lipschitz with respect to the hyperbolic metric
on $\mathbb{U}$ and $h^{-1}$ is uniformly continuous on $\mathbb{R}$ by
assumption, we conclude by Theorem 2 that $\mu_{G\circ
H^{-1}}\in\mathcal{M}_{0}(\mathbb{U})$. Then, the boundary extension $g\circ
h^{-1}$ of $G\circ H^{-1}$ belongs to ${\rm SS}(\mathbb{R})$ by Proposition 8.
∎

Finally, we mention the action of the group ${\rm SS}_{\rm uc}(\mathbb{R})$ on
$T_{v}(\mathbb{R})$. For each $h\in{\rm SS}_{\rm uc}(\mathbb{R})$, this can be
simply defined by $h_{*}([g])=[g\circ h^{-1}]$ for $[g]\in T_{v}(\mathbb{R})$,
but to see that this gives a biholomorphic automorphism of
$T_{v}(\mathbb{R})$, we have to extend $h$ quasiconformally to $\mathbb{U}$
and consider its action on ${\mathcal{M}}_{0}(\mathbb{U})$.

######  Theorem 17.

For any $h\in{\rm SS}_{\rm uc}(\mathbb{R})$, let $H=e(h)$. For any
$\mu\in{\mathcal{M}}_{0}(\mathbb{U})$, let $G$ be a quasiconformal
homeomorphism of $\mathbb{U}$ onto itself whose complex dilatation is $\mu$.
Then, the correspondence $\mu\mapsto\mu_{G\circ H^{-1}}$ defines a
biholomorphic automorphism
$r_{H}:{\mathcal{M}}_{0}(\mathbb{U})\to{\mathcal{M}}_{0}(\mathbb{U})$.
Moreover, this map descends down to a biholomorphic automorphism
$R_{h}:T_{v}(\mathbb{R})\to T_{v}(\mathbb{R})$ that coincides with $h_{*}$ and
satisfies $\pi\circ r_{H}=R_{h}\circ\pi$ for the Teichmüller projection
$\pi:{\mathcal{M}}_{0}(\mathbb{U})\to T_{v}(\mathbb{R})$.

Indeed, in the proof of Theorem 2, we obtain that

$\|\lambda_{\mu_{G\circ
H^{-1}}}\|_{c}\lesssim\|\lambda_{\mu_{G}-\mu_{H}}\|_{c}\lesssim\|\lambda_{\mu_{G}}\|_{c}+\|\lambda_{\mu_{H}}\|_{c},$

from which we see that $r_{H}$ is locally bounded. Then, the remaining
arguments for showing holomorphy are carried out in a standard way. See [31,
Remark 5.1] and [39, Proposition 3.1]. The properties of the inverse maps are
clear by $(r_{H})^{-1}=r_{H^{-1}}$ and $(R_{h})^{-1}=R_{h^{-1}}$.

##  6\. Comparisons of $\rm SS(\mathbb{R})$ and $\rm SS(\mathbb{S})$

The conformal invariance of strongly quasisymmetric homeomorphisms is well
understood. However, this is not the case for strongly symmetric
homeomorphisms. The problem comes from the uniformity of vanishing quantities
related to VMO and vanishing Carleson measures. In this section, we will
clarify the relationship between strongly symmetric homeomorphisms on
$\mathbb{R}$ and those on $\mathbb{S}$ along with some observations which may
be of independent interests.

We switch the definition of Carleson measure to measuring the intersection of
a disk with $\mathbb{U}$ or $\mathbb{D}$ instead of measuring a Carleson box
or sector. More precisely, a positive Borel measure $\lambda$ on $\mathbb{D}$
(similarly on $\mathbb{U}$) is a Carleson measure if

$\sup_{\Delta(\xi,r)}\frac{\lambda(\Delta(\xi,r)\cap\mathbb{D})}{r}<\infty,$

where the supremum is taken over all closed disks $\Delta(\xi,r)$ with center
$\xi\in\mathbb{S}$ and radius $r\in(0,2)$. A vanishing Carleson measure
$\lambda\in\rm CM_{0}(\mathbb{D})$ is defined by verifying a uniform vanishing
limit of the above quantity as $r\to 0$. These definitions are equivalent to
the previous ones.

We begin with considering the correspondence between $\rm CM_{0}(\mathbb{U})$
and $\rm CM_{0}(\mathbb{D})$ under the Cayley transformation
$T:\mathbb{U}\to\mathbb{D}$ defined by $T(z)=(z-i)/(z+i)$. The following is a
basic fact.

######  Lemma 18.

If $\lambda\in\rm CM_{0}(\mathbb{D})$, then the pull-back measure
$T^{*}\lambda$ on $\mathbb{U}$ satisfying
$d(T^{*}\lambda)=|T^{\prime}|^{-1}d\lambda\circ T$ belongs to $\rm
CM_{0}(\mathbb{U})$.

###### Proof.

Let $\Delta(x,r)$ denote a closed disk with center $x\in\mathbb{R}$ and radius
$r>0$. Then,

$\displaystyle\quad\
\frac{1}{r}\iint_{\Delta(x,r)\cap\mathbb{U}}d(T^{*}\lambda)(z)=\frac{1}{r}\iint_{\Delta(x,r)\cap\mathbb{U}}|T^{\prime}(z)|^{-1}d\lambda\circ
T(z)$ $\displaystyle=\frac{1}{{\rm
rad}(T(\Delta(x,r)))}\iint_{T(\Delta(x,r))\cap\mathbb{D}}\frac{{\rm
rad}(T(\Delta(x,r)))}{{\rm
rad}(\Delta(x,r))}|(T^{-1})^{\prime}(w)|d\lambda(w).$

Here, ${\rm rad}(T(\Delta(x,r)))|(T^{-1})^{\prime}(w)|/{\rm rad}(\Delta(x,r))$
is uniformly bounded by some constant $C>0$ on $T(\Delta(x,r))$ for all
sufficiently small $r>0$.

Since $\lambda\in\rm CM_{0}(\mathbb{D})$, for every $\varepsilon>0$, there is
some $\delta>0$ such that if ${\rm rad}(T(\Delta(x,r)))<2\delta$ then

$\frac{C}{{\rm
rad}(T(\Delta(x,r)))}\iint_{T(\Delta(x,r))\cap\mathbb{D}}d\lambda(w)<\varepsilon.$

Moreover, ${\rm rad}(T(\Delta(x,r)))\leq 2\,{\rm rad}(\Delta(x,r))$ by
$|T^{\prime}(x)|\leq 2$ for all $x\in\mathbb{R}$. Hence, if $r<\delta$ then

$\frac{1}{r}\iint_{\Delta(x,r)\cap\mathbb{U}}d(T^{*}\lambda)(z)<\varepsilon$

for all $x\in\mathbb{R}$. Therefore, we have $T^{*}\lambda\in\rm
CM_{0}(\mathbb{U})$. ∎

The following is the main result in this section. This also compares $T_{v}$
with $T_{v}(\mathbb{R})$.

######  Theorem 19.

If $\varphi\in\rm SS(\mathbb{S})$, then $h=T^{-1}\circ\varphi\circ T\in\rm
SS(\mathbb{R})$. The converse does not necessarily hold. In other words, there
exists $h\in\rm SS(\mathbb{R})$ such that $\varphi=T\circ h\circ
T^{-1}\notin\rm SS(\mathbb{S})$.

######  Remark.

We point out that there is another way of constructing a strongly symmetric
homeomorphism on $\mathbb{R}$ from that on $\mathbb{S}$. This is done by
taking a lift against the universal covering projection
$\mathbb{R}\to\mathbb{S}$ defined by $x\mapsto e^{ix}$. Namely, for each
sense-preserving homeomorphism $\varphi$ of $\mathbb{S}$ onto itself, there
exists a strictly increasing homeomorphism $\hat{h}$ of $\mathbb{R}$ onto
itself that satisfies $\varphi(e^{ix})=e^{i\hat{h}(x)}$. Then,
$\hat{h}(x+2\pi)-\hat{h}(x)\equiv 2\pi$ and
$\hat{h}^{\prime}(x)=|\varphi^{\prime}(e^{ix})|$. If $\varphi\in\rm
SS(\mathbb{S})$, we have by Partyka [25, Lemma 2.2] that $\hat{h}\in{\rm
SS}(\mathbb{R})$.

###### Proof of Theorem 19.

By Proposition 7, any $\varphi\in\rm SS(\mathbb{S})$ extends to a
quasiconformal homeomorphism $\Phi$ of $\mathbb{D}$ onto itself whose complex
dilatation $\mu_{\Phi}$ induces a vanishing Carleson measure
$\lambda_{\mu_{\Phi}}\in{\rm CM}_{0}(\mathbb{D})$. Then,
$H=T^{-1}\circ\Phi\circ T$ is a quasiconformal extension of $h$ to
$\mathbb{U}$ whose complex dilatation $\mu_{H}$ induces a measure
$\lambda_{\mu_{H}}$ on $\mathbb{U}$ such that

$\begin{split}d\lambda_{\mu_{H}}(z)&=|\mu_{H}(z)|^{2}\rho_{\mathbb{U}}(z)dxdy\\\
&=|\mu_{\Phi}(T(z))|^{2}\rho_{\mathbb{D}}(T(z))|T^{\prime}(z)|dxdy\\\
&=|T^{\prime}(z)|^{-1}d\lambda_{\mu_{\Phi}}\circ
T(z)=d(T^{*}\lambda_{\mu_{\Phi}})(z).\\\ \end{split}$

It follows from Lemma 18 that $\lambda_{\mu_{H}}\in\rm CM_{0}(\mathbb{U})$,
and thus $h\in\rm SS(\mathbb{R})$ by Proposition 8.

In order to prove the second assertion, we first recall two functions $h$ and
$g$ constructed in [37] (the roles of $h$ and $g$ are exchanged here).

The function $h$ is simply defined as follows:

$h(x)=\begin{cases}(x+1)^{2}-1,&x\geq 0\\\ -(x-1)^{2}+1,&x\leq 0.\end{cases}$

To construct the function $g$, we consider a function $g_{1}(x)=x^{2}/24$ on
the interval $[1,12]\subset\mathbb{R}$. We draw the graph of $y=g_{1}(x)$ on
the $xy$-plane and its $\pi$-rotating copy around the point $O=(1,g_{1}(1))$.
The union of these two curves is denoted by $\mathcal{G}_{1}$. Its end points
are $E=(12,g_{1}(12))$ and the antipodal point $E^{\prime}$ on the copy. We
move $\mathcal{G}_{1}$ by parallel translation so that $E^{\prime}$ coincides
with the origin $(0,0)$ of the $xy$-plane. In the positive direction, we put
each $\mathcal{G}_{1}$ from one to another so that $E^{\prime}$ coincides with
$E$. The resulting curve that is a graph on $\\{x\geq 0\\}$ is denoted by
$\mathcal{G}_{+}$. We also make its $\pi$-rotating copy around the origin
$(0,0)$, which is denoted by $\mathcal{G}_{-}$. Then, we set
$\mathcal{G}=\mathcal{G}_{+}\cup\mathcal{G}_{-}$. This curve $\mathcal{G}$ on
the $xy$-plane defines a function $y=g(x)$ for $x\in\mathbb{R}$ that has
$\mathcal{G}$ as its graph.

We have shown in [37, Corollary 5.6] that $g,h\in\rm SS(\mathbb{R})$ but
$g\circ h\notin\rm SS(\mathbb{R})$. Suppose that both $T\circ g\circ T^{-1}$
and $T\circ h\circ T^{-1}$ are in $\rm SS(\mathbb{S})$. Since $\rm
SS(\mathbb{S})$ is a group, we have that $T\circ g\circ h\circ T^{-1}$ is in
$\rm SS(\mathbb{S})$. Then, we conclude by the above argument that $g\circ
h\in\rm SS(\mathbb{R})$. This is a contradiction, and thus either $T\circ
g\circ T^{-1}$ or $T\circ h\circ T^{-1}$ is not in $\rm SS(\mathbb{S})$. ∎

######  Remark.

As an immediate consequence of Theorem 19, we see that there exists
$\tilde{\lambda}\in\rm CM_{0}(\mathbb{U})$ such that the push-forward measure
$T_{*}{\tilde{\lambda}}$ on $\mathbb{D}$ satisfying
$d(T_{*}{\tilde{\lambda}})=d((T^{-1})^{*}{\tilde{\lambda}})=|(T^{-1})^{\prime}|^{-1}d\tilde{\lambda}\circ
T^{-1}$ is not in $\rm CM_{0}(\mathbb{D})$.

Next, we consider the correspondence between VMO functions on $\mathbb{R}$ and
on $\mathbb{S}$. Similar results to Theorem 19 can be obtained; the
boundedness of $|T^{\prime}(x)|$ also transforms $\rm VMO(\mathbb{S})$ into
$\rm VMO(\mathbb{R})$ under the Cayley transformation $T$.

######  Proposition 20.

If $v\in\rm VMO(\mathbb{S})$, then $u=v\circ T\in\rm VMO(\mathbb{R})$. The
converse does not necessarily hold. In other words, there exists $u\in\rm
VMO(\mathbb{R})$ such that $v=u\circ T^{-1}\notin\rm VMO(\mathbb{S})$.

###### Proof.

It is easy to see that

$\frac{1}{|I|}\int_{I}|u(x)-u_{I}|dx\leq\frac{2}{|I|}\int_{I}|u(x)-c|dx$

for any bounded closed interval $I\subset\mathbb{R}$ and any $c\in\mathbb{R}$.
For $c=v_{J}$, $J=T(I)$ and $x=T^{-1}(\xi)$, the right side term in the above
inequality turns out to be

$\frac{2}{|J|}\int_{J}|v(\xi)-v_{J}|\frac{|J|}{|I|}|(T^{-1})^{\prime}(\xi)||d\xi|.$

Here, $|J||(T^{-1})^{\prime}(\xi)|/|I|$ is bounded from above by an absolute
constant $C>0$ for all sufficiently small intervals $I$. We see that $|J|\to
0$ as $|I|\to 0$ by $|T^{\prime}(z)|\leq 2$. Hence,

$\frac{1}{|I|}\int_{I}|u(x)-u_{I}|dx\leq\frac{2C}{|J|}\int_{J}|v(\xi)-v_{J}||d\xi|\to
0$

as $|I|\to 0$ by the condition $v\in\rm VMO(\mathbb{S})$. This implies that
$u\in\rm VMO(\mathbb{R})$.

For the converse direction, we set $v(\xi)=\log|1-\xi|$. This does not belong
to $\rm VMO(\mathbb{S})$. However,

$u(x)=v\circ T(x)=-\log|x+i|+\log 2$

belongs to $\rm VMO(\mathbb{R})$ because it is in $\rm BMO(\mathbb{R})$ and is
uniformly continuous on $\mathbb{R}$ (see [26]). ∎

######  Acknowledgments.

The authors would like to thank the referee for a very careful reading of the
manuscript and for several suggestions that greatly improve the presentation
of the paper.

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# Neutron polarisation transfer, $C_{x^{\prime}}^{n}$, in $\pi^{+}$
photoproduction off the proton

M. Bashkanov<EMAIL_ADDRESS>D.P. Watts  S.J.D. Kay  S. Abt  P.
Achenbach  P. Adlarson  F. Afzal  Z. Ahmed  C.S. Akondi  J.R.M. Annand  R.
Beck  M. Biroth  N. Borisov  A. Braghieri  W.J. Briscoe  F. Cividini  C.
Collicott  S. Costanza  A. Denig  E.J. Downie  P. Drexler  S. Fegan  A. Fix
S. Gardner  D. Ghosal  D.I. Glazier  I. Gorodnov  W. Gradl  D. Gurevich  L.
Heijkenskjöld  D. Hornidge  G.M. Huber  V.L. Kashevarov  M. Korolija  B.
Krusche  A. Lazarev  K. Livingston  S. Lutterer  I.J.D. MacGregor  D.M. Manley
P.P. Martel  R. Miskimen  E. Mornacchi  C. Mullen  A. Neganov  A. Neiser  M.
Ostrick  P.B. Otte  D. Paudyal  P. Pedroni  T. Rostomyan  V. Sokhoyan  K.
Spieker  O. Steffen  I.I. Strakovsky  T. Strub  I. Supek  A. Thiel  M. Thiel
A. Thomas  Yu.A. Usov  S. Wagner  N.K. Walford  J. Wettig  M. Wolfes  N.
Zachariou  Department of Physics, University of York, Heslington, York, Y010
5DD, UK  University of Regina, Regina, SK S4S-0A2 Canada  Kent State
University, Kent, Ohio 44242, USA  SUPA School of Physics and Astronomy,
University of Glasgow, Glasgow, G12 8QQ, UK  Department of Physics, University
of Basel, Ch-4056 Basel, Switzerland  Institut für Kernphysik, University of
Mainz, D-55099 Mainz, Germany  Helmholtz-Institut für Strahlen- und
Kernphysik, University Bonn, D-53115 Bonn, Germany  Joint Institute for
Nuclear Research, 141980 Dubna, Russia  INFN Sezione di Pavia, I-27100 Pavia,
Pavia, Italy  Center for Nuclear Studies, The George Washington University,
Washington, DC 20052, USA  Department of Astronomy and Physics, Saint Mary’s
University, E4L1E6 Halifax, Canada  Dipartimento di Fisica, Università di
Pavia, I-27100 Pavia, Italy  II. Physikalisches Institut, University of
Giessen, D-35392 Giessen, Germany  Institute for Nuclear Research, RU-125047
Moscow, Russia  Mount Allison University, Sackville, New Brunswick E4L1E6,
Canada  Rudjer Boskovic Institute, HR-10000 Zagreb, Croatia  University of
Massachusetts, Amherst, Massachusetts 01003, USA  University of California Los
Angeles, Los Angeles, California 90095-1547, USA  Racah Institute of Physics,
Hebrew University of Jerusalem, Jerusalem 91904, Israel  Department of Physics
and Astronomy, Rutgers University, Piscataway, New Jersey, 08854-8019
Jefferson Lab, 12000 Jefferson Ave., Newport News, VA 23606, USA  Tomsk
Polytechnic University, 634034 Tomsk, Russia

###### Abstract

We report a first measurement of the double-polarisation observable,
$C_{x^{\prime}}$, in $\pi^{+}$ photoproduction off the proton. The
$C_{x^{\prime}}$ double-polarisation observable represents the transfer of
polarisation from a circularly polarised photon beam to the recoiling neutron.
The MAMI circularly polarised photon beam impinged on a liquid deuterium
target cell, with reaction products detected in the Crystal Ball calorimeter.
Ancillary apparatus surrounding the target provided tracking, particle
identification and determination of recoil nucleon polarisation. The
$C_{x^{\prime}}$ observable is determined for photon energies 800-1400 MeV,
providing new constraints on models aiming to elucidate the spectrum and
properties of nucleon resonances. This is the first determination of any
polarisation observable from the beam-recoil group of observables for this
reaction, providing a valuable constraint and systematic check of the current
solutions of partial wave analysis based theoretical models.

###### keywords:

photoproduction

††journal: Physics Letters B

##  1 Introduction

Photoinduced reactions on proton and neutron targets have played a key role in
progressing our knowledge of the excited nucleon spectrum the past decade [1],
catalysed by quality nucleon photoproduction data from MAMI, JLAB, ELSA and
other facilities [2]. These have provided a step change in the number of
measured observables, statistical accuracy, and kinematic coverage.

Pion photoproduction is the simplest photoinduced reaction on the nucleon. The
reaction can be described theoretically with four complex amplitudes, which
can be fully constrained by kinematically complete measurements of a chosen
set of eight observables taken from the cross section, single polarisation
observables (where the polarisation of either photon beam, target or recoiling
nucleon is determined) and double-polarisation observables formed from
simultaneous determination of two of the above polarisation quantities. Recent
work has indicated that the properties of the different partial waves in the
reaction may converge with fewer measurements than the mathematically complete
eight, as discussed in Ref. [3].

Previous double-polarisation measurements for $n\pi^{+}$ photoproduction are
limited to the beam-target group of observables. Measurements of the $G$
observable (linearly polarised beam and transversely polarised target) [6],
the $E$ observable (circularly polarised beam and longitudinally polarised
target [7]) and more limited data sets for $H$ (circularly polarised beam and
transversely polarised target) have recently been obtained[10, 20]. These
double-polarisation data, combined with the cross section and single
polarisation observables ($\Sigma$, $T$ and $P$) [6, 8] comprise the current
world data base.

The lack of any double-polarisation measurements from the beam-recoil group
for this channel mean the mathematically "complete" constraint has not been
achieved. This lack of previous data reflects the challenges in measurement of
recoil nucleon polarisation, requiring a secondary rescattering of the
ejectile nucleon in a spin-analysing polarimeter material. Such measurements
are only feasible at high photon beam intensities111To achieve such
measurements around four orders of magnitude larger statistics are required
than for a typical beam-target measurements as only $\sim$2% of the ejected
events can be analysed with a practical thickness of nucleon scattering
medium. . The new $C_{x^{\prime}}$ data presented here therefore provide an
important cross check of convergence in the model predictions. Observables
from the beam-recoil group give different sensitivities to the underlying
reaction amplitudes, so even if determined with less precision than the beam-
target measurements which dominate the database, they have the potential to
provide important information on the road to convergence in the model
predictions. Some constraint on systematics in both the reaction modelling and
the fitted database can also be achieved as $C_{x^{\prime}}$ is obtained with
very different systematics to the beam-target double-polarisation data.222For
example, correlated systematics could potentially arise as all the beam-target
observables employed common methodologies in determining the degree of linear
beam polarisation and MAMI/ELSA had common polarised target systems.

The photoproduction of $p\pi^{0}$ is the sister reaction to $n\pi^{+}$. For
the $p\pi^{0}$ reaction the database is more complete. The differential cross-
section, $\Sigma,P,T,G,H,C_{x^{\prime}}$ observables have been determined over
the full energy range of the new data and $E,O_{x^{\prime}},O_{z^{\prime}}$
for part of the range. Such $p\pi^{0}$ data are simultaneously fitted by
theoretical models along with the $n\pi^{+}$ data. In combination,
sensitivities to the isospin of the contributing nucleon resonances and
backgrounds can be achieved.

The leading models to interpret pion photoproduction data, and infer
information on the nucleon resonance spectrum, are the SAID [8] and BnGa [9]
frameworks, which are both based on partial wave analysis (PWA) methods fitted
to the data. The latest iterations of these models (SAID MA19 and BnGa BG2019)
experienced substantial change due to the inclusion of new data on the double-
polarisation observable G, recently measured at CLAS [6].

In this work we make a first determination of the $C_{x^{\prime}}$ observable
for the $p(\gamma^{\odot},\vec{n}\pi^{+})$ reaction. This is the first
measurement of any observable from the beam-recoil double polarisation group
for this reaction and the results are compared to nearly converged predictions
from the leading partial wave analysis based theoretical models. The
$C_{x^{\prime}}$ observable is extracted using a bootstrap statistical
technique which gives access to statistical and systematic errors in a robust
way. We compare the new data to the SAID [8] and BnGa [9] models, which are
fitted to a database currently unconstrained by measurements of beam-recoil
observables in this reaction channel.

##  2 Experimental details

The measurement employed a new, large acceptance, neutron polarimeter [11]
within the Crystal Ball detector at the A2@MAMI [12] facility during a 300
hour beamtime. A 1557 MeV longitudinally polarised electron beam impinged on
either a thin amorphous (cobalt-iron alloy) or crystalline (diamond) radiator,
producing circularly (alloy) or elliptically (diamond) polarised
bremsstrahlung photons. The electron helicity was regularly flipped to produce
a photon beam with equal amounts of both circular photon polarisations. As
linear photon beam polarisation is not used to extract $C_{x^{\prime}}^{n}$,
equal flux from the two elliptical polarisation settings were combined to
increase the circularly polarised photon yield333 Separate treatment of the
events with two elliptical photon polarisations and with pure circular photon
polarisation, gave consistent results within the achievable statistical
accuracy.. The photons were energy-tagged ($\Delta E\sim~{}2$ MeV) by the
Glasgow-Mainz Tagger [13] and impinged on a 10 cm long liquid deuterium target
cell. Reaction products were detected by the Crystal Ball (CB) [14], a highly
segmented NaI(Tl) photon calorimeter covering nearly 96% of $4\pi$ steradians.
For this experiment, a new bespoke 24 element, 7 cm diameter and 30 cm long
plastic scintillator barrel (PID-POL) [15] surrounded the target, with a
smaller diameter than the earlier PID detector [15], but which provided
similar particle identification capabilities. A 2.6 cm thick cylinder of
analysing material (graphite) for nucleon polarimetry was placed around PID-
POL, covering polar angles $12^{\circ}<\Theta<150^{\circ}$ and occupying the
space between PID-POL and the Multi Wire Proportional Chamber (MWPC) [16]. The
MWPC provided charged particle tracking for particles passing out of the
graphite into the CB. At forward angles, an additional 2.6 cm thick graphite
disk covered the range $2<\Theta<12^{\circ}$ [15, 17, 18]. A GEANT4
visualisation of the experimental setup can be seen in Fig. 1.

Figure 1: Crystal Ball setup during the polarimeter beamtime. The cryogenic
target (red cell) is surrounded by the PID barrel (blue), the graphite
polarimeter (grey), the MWPC (blue/green) and the Crystal Ball (white).

The cryogenic deuterium target provided a source of weakly bound protons and
neutrons. The $d(\gamma,\pi^{+}\vec{n})n_{spec}$ events of interest consist of
a primary charged pion track and a reconstructed neutron, which undergoes a
$(n,p)$ charge-exchange reaction in the graphite to produce a secondary
proton; the spectator neutron is not detected. The secondary proton gives
signals in the MWPC and CB. The primary $\pi^{+}$ was identified using the
correlation between the energy deposits in the PID and CB using $\Delta E-E$
analysis [15] along with an associated charged track in the MWPC. The
intercept of the primary $\pi^{+}$ track with the photon beamline allowed
determination of the production vertex, and hence permitted the yield
originating from the target cell windows to be removed. Neutron 12C$(n,p)$
charge exchange candidates required an absence of a PID-POL signal on the
reconstructed neutron path, while having an associated track in the MWPC and
signal in the CB from the scattered secondary proton. The incident neutron
angle ($\Theta_{n}$) was determined from reaction kinematics using
$E_{\gamma}$ and the production vertex coordinates. A distance of closest
approach condition was imposed to ensure a crossing of the (reconstructed)
neutron track and the secondary proton candidate track (measured with MWPC and
CB). Once candidate pion and neutron tracks were identified, a kinematic fit
was employed to increase the purity of the data sample and improve the
determination of the reaction kinematics (see Ref [19] for details).

##  3 Determination of $C_{x^{\prime}}^{n}$

The transferred neutron polarisation ($C_{x^{\prime}}^{n}$) was determined
through analysis of the neutron-spin dependent 12C$(n,p)$ reactions occurring
in the graphite polarimeter. The spin-orbit component of the nucleon-nucleon
interaction results in a $\phi$-anisotropy in the produced yield of secondary
protons (see Ref. [20] for details). We followed the same procedure to
determine $C_{x^{\prime}}^{n}$, as described in Ref. [18], where the same
observable was extracted for the deuteron photodisintegration reaction. The
$C_{x^{\prime}}^{n}$ extraction employs a combination of log-likelihood and
bootstrap techniques, as discussed below. The likelihood was defined as

$L_{i}=c_{i}\left[1+C^{n}_{x^{\prime}}\cdot P_{\gamma,i}^{\odot}\cdot
A_{y,i}\cdot\sin(\phi^{scat}_{i})\right]A,$ (1)

where $c_{i}$ is a normalisation coefficient, $A$ is the detector acceptance,
$C^{n}_{x^{\prime}}$ is the spin transfer observable of interest,
$P_{\gamma,i}^{\odot}$ is the degree of photon circular polarisation,
$A_{y,i}$ is the neutron analysing power444$P_{\gamma,i}^{\odot}$ is photon
energy dependent and $A_{y}$ depends on the ejected neutron energy and
scattered proton polar angle. Both variables are evaluated on an event-by-
event basis. and $\phi^{scat}$ is the $\phi$ scattering angle in the primed
frame, see Fig. 2. The log-likelihood function that was maximised to obtain
the observable of interest is given by

$\log L=b+\sum_{i}\log\left[1+C^{n}_{x^{\prime}}\cdot
P_{\gamma,i}^{\odot}\cdot A_{y,i}\cdot\sin(\phi^{scat}_{i})\right],$ (2)

where the constant $b$ is an observable-independent constant, which absorbs
the normalisation coefficient and detector acceptance. The summation $(i)$
runs over all events. To reduce systematic dependencies, the events were only
retained if $A_{y}(np)$ was above 0.1, and the proton scattering angle was in
the range $\Theta^{scat}_{p}\in 15~{}-~{}45^{\circ}$, where
$\Theta^{scat}_{p}$ is the polar angle of the scattered proton relative to the
direction of the neutron. The above cuts restrict the kinetic energy of the
neutrons to be above 200 MeV, as well as their polar angle, $\Theta_{n}$ , to
be larger than $\sim 40^{\circ}$, allowing them to be well within the
acceptance of the polarimeter. Events meeting the conditions required a photon
energy larger than 800 MeV and neutron centre-of-mass angle close to
$90^{\circ}$.

Figure 2: Kinematics of the reaction in the centre-of-mass system. The z-axis
is oriented along the photon beam, the y-axis is vertically upwards in the
laboratory; the $z^{\prime}$-axis is oriented along the ejectile neutron
direction, and the $y^{\prime}$-axis is perpendicular to the reaction plane.

The extraction of $C_{x^{\prime}}^{n}$ is simpler than the previous extraction
of $P_{y}$ due to cancellations in the acceptance.555The determination of
$P_{y}$ is obtained from a ($\cos(\phi^{scat}_{i})$ dependence, which is
sensitive to any systematics between forward and backward angle scattering
[17]. In contrast, $C_{x^{\prime}}^{n}$ has a ($\sin(\phi^{scat}_{i})$
dependence (left-right asymmetry) for which the cylindrically symmetrical A2
setup has minimal acceptance/efficiency effects. A regular ($\sim 1~{}s$)
flipping of the photon helicity allowed us to reduce systematic effects
associated with a temporal variation of the detector acceptance to negligible
values. In addition, in the determination of the $C_{x^{\prime}}^{n}$
observable, the acceptance effects completely factor-out in the likelihood
extraction (unlike the extraction of $P_{y}$). The use of an unbinned (in
$\phi$ and $E_{\gamma}$) likelihood method eliminates bin-size related
systematics arising from fits of trigonometric functions to binned
$\phi^{scat}$-asymmetries.

The spin transfer observable,
$C_{x^{\prime}}^{n}=f(\Theta_{N}^{*},E_{\gamma})$, was determined in
$10^{\circ}$ neutron centre-of-mass (CM) angle bins using a likelihood-
extraction in which a smooth energy dependence is assumed. To get robust error
evaluations within the likelihood method we employed a bootstrap procedure
[21]. This method involves randomly selecting $N$ events out of our sample of
$N$ events allowing repetitions666For example, in the case of the $80^{\circ}$
bin it is 9200 random events out of a 9200 event sample.. For each
combination, a likelihood fit is carried out to extract $C_{x^{\prime}}^{n}$.
Each fit produces a smooth function $C_{x^{\prime}}^{n}=f(E_{\gamma})$ for the
given $\Theta_{n}$ bin. Multiple repetitions of this procedure enable
determination of the most likely $C_{x^{\prime}}$ and its associated errors
$C_{x^{\prime}}^{n}$. These are presented in Fig. 3 (middle). Additional
studies of systematic effects were obtained by relaxing the analysis cuts. The
systematic errors are extracted from the resulting variations in the extracted
values of $C_{x^{\prime}}$. The magnitude of the systematic errors in each bin
are influenced by the achievable statistical accuracy in the bin. The
extracted systematic errors are also shown in Fig. 3 (bottom)777The dataset
corresponds to kinematical regions where $C_{x^{\prime}}^{n}$ statistical
errors are less than 0.5 or systematical errors are smaller than 0.8..

Figure 3: Two-dimensional $C_{x^{\prime}}^{n}$ dependence as a function of
neutron centre-of-mass angle,$\Theta^{*}_{n}$ and photon energy $E_{\gamma}$
(top), statistical (middle) and systematical (bottom) uncertainties for this
distribution. Systematic uncertainty contour lines at 0.4 are also shown on
the top plot as dash-dot lines.

##  4  Results

Our $C_{x^{\prime}}^{n}$ results, along with associated statistical and
estimated systematic errors, are presented in Fig. 3 as a function of photon
energy and neutron CM polar angle. The new data cover photon energy bins from
800 MeV up to 1400 MeV and CM angles for the neutron of 70-120∘. The magnitude
of $C_{x^{\prime}}^{n}$ is seen to be small over much of the sampled phase
space, albeit with indications of localised regions of both high and low
$C_{x^{\prime}}^{n}$. The statistical and systematic uncertainties are both
largely uniform over much of the phase space, but with expected deterioration
close to the edge of the polarimeter acceptance.

Example comparisons of the new data with recent solutions of the SAID [8] and
BnGa [6] partial wave fits are shown in Fig. 4 and 5. In Fig. 4 the data are
presented as a function of photon energy for a fixed angular bin of 80∘. For
photon energies up to 1200 MeV the data are well described by all SAID and
BnGa solutions (see figure caption) within the achievable experimental
errors888The structure seen in theory curves around $E_{\gamma}\sim 700$ MeV
originates from the $\eta$-meson production threshold.. Above this, the data
may suggest higher $C_{x^{\prime}}^{n}$ than both model predictions - albeit
in regions where the estimated systematic uncertainties are large.

The angular distribution of $C_{x^{\prime}}^{n}$ at 1100 MeV is presented in
Fig. 5 (top), compared to the model predictions. The observed zero crossing at
around $90^{\circ}$ is broadly consistent with all model predictions, however
the second crossing around $108^{\circ}$ is in better agreement with the
latest BnGa solution. The angular distribution of $C_{x^{\prime}}^{n}$ at 1200
MeV, Fig. 5 (bottom), shows a broadly similar shape to the lower energy bin,
but with indications of regions of higher positive $C_{x^{\prime}}^{n}$. The
divergence between the models is larger for this higher photon energy bin. The
most recent SAID and BnGa fits (red solid and red-dashed respectively) include
the large new G- dataset [6] for $n\pi^{+}$, but it’s inclusion results in
both models giving a poorer description of this independent
$C_{x^{\prime}}^{n}$ variable. This suggests constraint from a different
double-polarisation group, even with poorer coverage and accuracy, has the
potential to provide new information. Future fits including the new data may
improve convergence between the models for this photon energy region.

The statistical and systematic uncertainties in $C_{x^{\prime}}^{n}$
extraction could be significantly reduced in the future due to recent upgrades
in the MAMI beam intensity. Accurate beam-recoil data appears important even
when, as is the case here, an extensive database of beam-target double-
polarisation observables has already been obtained.

Figure 4: The neutron spin transfer observable $C_{x^{\prime}}^{n}$ for the
$75^{\circ}-85^{\circ}$, neutron CM angles as a function of photon beam
energy. The dashed area represents the data together with its statistical
uncertainty, while the hatched area shows the systematic uncertainty (the data
and uncertainties are a 1D slice of Fig. 3).The solid lines are the SAID PWA
solutions, SAID-2019(red) [8], SAID-2012(blue) solution [22]; the dashed lines
are the BnGa solutions, BnGa-19(red) [6], BnGa-2014-1(blue),
BnGa-2011-1(black).

Figure 5: The neutron spin transfer observable $C_{x^{\prime}}^{n}$ for the
1100 MeV (top) and 1200 MeV (bottom) beam photon energies as a function of
neutron CM angle ($5^{\circ}$ bin spacing). The dashed area represents the
data together with its statistical uncertainty, while the hatched area shows
the systematic uncertainty. The theory lines are the same as in Fig. 4.

##  5  Summary

This work provides the first measurement of a beam-recoil double-polarisation
observable, $C_{x^{\prime}}^{n}$, for $\pi^{+}$ photoproduction off the
proton, obtained for photon energies 800-1300 MeV. The measurement provides an
important check of the leading partial wave analysis based models which fit
the world data on photo-meson production from the nucleon to extract the
nucleon excitation spectrum. The general trends of the new
$C_{x^{\prime}}^{n}$ data are reproduced by the two leading model predictions
(SAID and BnGa), although deviations between the predictions remain in this
photon energy region. The inclusion of the most recent beam-target double-
polarisation data in the fitted database, resulted in a poorer description of
the new $C_{x^{\prime}}^{n}$ data. This suggests beam-recoil observables give
sensitivities to reaction amplitudes that are not currently well constrained
by world data and are an important addition to the database on the path
towards model convergence in the extracted amplitudes.

##  6 Acknowledgement

In accordance with UK data management policy, all data are available for
downloading from PURE [23]. This work has been supported by the U.K. STFC
(ST/V002570/1, ST/L00478X/2, ST/V001035/1, ST/P004385/2, ST/T002077/1,
ST/L005824/1, 57071/1, 50727/1 ) grants, the Deutsche Forschungsgemeinschaft
(SFB443, SFB/TR16, and SFB1044), DFG-RFBR (Grant No. 09-02-91330),
Schweizerischer Nationalfonds (Contracts No. 200020-175807, No. 200020-156983,
No. 132799, No. 121781, No. 117601), the U.S. Department of Energy (Offices of
Science and Nuclear Physics, Awards No. DE-SC0014323, DEFG02-99-ER41110, No.
DE-FG02-88ER40415, No. DEFG02-01-ER41194) and National Science Foundation
(Grants NSF OISE-1358175; PHY-1039130, PHY-1714833, No. IIA-1358175,
PHY-2012940), INFN (Italy), and NSERC of Canada (Grant No. FRN-
SAPPJ2015-00023). This infrastructure is part of a project that has received
funding from the European Union’s Horizon 2020 research and innovation
programme under grant agreement No 824093.

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