# Locally uniform domains and extension of bmo functions

Almaz Butaev A.B. was partially supported by the Natural Sciences and
Engineering Research Council (NSERC) of Canada. Galia Dafni G.D. was partially
supported by the Natural Sciences and Engineering Research Council (NSERC) of
Canada, and the Centre de recherches mathématiques (CRM).

###### Abstract

We prove that for a domain $\Omega\subset{\mathbb{R}}^{n}$, being
${(\epsilon,\delta)}$ in the sense of Jones is equivalent to being an
extension domain for $\textup{bmo}(\Omega)$, the nonhonomogeneous version of
the space of function of bounded mean oscillation on $\Omega$. In the process
we demonstrate that these conditions are equivalent to local versions of two
other conditions characterizing uniform domains, one involving the presence of
length cigars between nearby points and the other a local version of the
quasi-hyperbolic uniform condition. Our results show that the definition of
$\textup{bmo}(\Omega)$ is closely connected to the geometry of the domain.

##  1 Introduction

Let $\Omega$ be a domain (open connected set) in ${\mathbb{R}}^{n}$ and
consider a space $X$ of functions on $\Omega$. We are interested in the
relationship between the geometry of the domain $\Omega$ and the existence of
a bounded linear extension operator $T:X(\Omega)\to X({\mathbb{R}}^{n})$. Can
we characterize the domain in terms of such extensions, namely give conditions
on $\Omega$ which are necessary and sufficient for the existence of the
operator $T$? If such a map exists, we say that the domain is an extension
domain for the given function space $X$. We refer to [4] and [5] for a
comprehensive account of extension problems for different function spaces $X$.

Although extension domains have been widely studied, the problem of their
complete characterization is still open in general, even for the classical
Sobolev spaces $W^{s,p}$ (see however [18]). When $X=\textup{BMO}$, a complete
description of the extension domains has been given by Jones [13]. The space
$\textup{BMO}(\Omega)$ consists of functions $f$ which are locally integrable
in $\Omega$ and have bounded mean oscillation in $\Omega$, that is,

$\|f\|_{\textup{BMO}(\Omega)}:=\sup_{Q\subset\Omega}\fint_{Q}|f(x)-f_{Q}|dx<\infty,$
(1)

where here and in what follows $Q$ denotes a cube with sides parallel to the
axes, $|Q|$ is its measure and $f_{Q}=\fint_{Q}f:=\frac{1}{|Q|}\int_{Q}fdx$ is
the mean of $f$ over $Q$. The seminorm $\|f\|_{\textup{BMO}(\Omega)}$ defines
a norm modulo constants.

Jones gave a geometric condition on the domain which was necessary and
sufficient for the existence of a bounded extension map from
$\textup{BMO}(\Omega)$ to $\textup{BMO}({\mathbb{R}}^{n})$. This condition was
shown in [11] to be equivalent to $\Omega$ being a uniform domain, previously
defined in [16] (see also [20]): there are constants $a,b>0$ such that any
$x,y\in\Omega$ are connected by a rectifiable curve
$\gamma=\gamma(x,y)\subset\Omega$ with

$\begin{cases}s(\gamma(x,y))\leq a|x-y|\;\mbox{ and}\\\
\min(s(\gamma(x,z)),s(\gamma(z,y)))\leq
b\;\textup{dist}(z,{\partial\Omega})\quad\mbox{for all points $z$ along the
curve $\gamma$.}\end{cases}$ (2)

Here $s(\gamma(p,q))$ denotes the arclength of the curve between the points
$p$ and $q$.

In [14], Jones introduced local versions of uniform domains called
${(\epsilon,\delta)}$ domains (uniform domains are $(\epsilon,\infty)$ see
Definition 2.4). He showed that such domains are extension domains for the
Sobolev spaces $W^{s,p}$, $1\leq p\leq\infty$, $s\in\mathbb{N}$ and this
result is sharp in dimension $2$: if a finitely connected open set is an
extension domain for all Sobolev spaces, then it is an $(\epsilon,\delta)$
domain. Christ in [6] extended these results to certain spaces of fractional
smoothness which had been simultaneously studied by DeVore and Sharpley in [8]
and later in [9]; there followed many further extensions and variations - for
a recent example see [15].

We summarize the results of Jones [13, 14], and Christ [6] using the notation
in [6], where a maximal function $N_{\alpha}$ which originates in the work of
Calderón is used to define spaces

$\mathcal{E}_{\alpha}^{p}(\Omega)=\\{f\in
L^{1}_{loc}(\Omega):\|f\|_{\mathcal{E}_{\alpha}^{p}}:=\|N_{\alpha}f\|_{L^{p}(\Omega)}<\infty\\}$

and

$\mathcal{R}_{\alpha}^{p}(\Omega)=\\{f\in
L^{1}_{loc}(\Omega):\|f\|_{\mathcal{R}_{\alpha}^{p}}:=\|f\|_{L^{p}(\Omega)}+\|N_{\alpha}f\|_{L^{p}(\Omega)}<\infty\\}.$

When $\alpha$ is a positive integer and $1<p\leq\infty$, these are the
homogeneous and nonhomogeneous Sobolev spaces, while for $\alpha=0$,
$N_{\alpha}$ is the sharp function and therefore
$\mathcal{E}^{\infty}_{0}(\Omega)=\textup{BMO}(\Omega)$.

Homogeneous Case

Sufficiency: if $\Omega\subset{\mathbb{R}}^{n}$ is an $(\epsilon,\infty)$
domain then $\Omega$ is an extension domain for $\mathcal{E}_{\alpha}^{p}$
when $1<p<\infty$ and $\alpha>0$, or $p=\infty$ and $\alpha=0$.

Necessity: if a finitely connected domain $\Omega\subset\mathbb{R}^{2}$ is an
extension domain for $\mathcal{E}_{\alpha}^{p}$, where $\alpha p=2$ and
$0<\alpha\leq 1$, then $\Omega$ is an $(\epsilon,\infty)$ domain. Furthermore,
if a domain $\Omega\subset{\mathbb{R}}^{n}$ is an extension domain for
$\mathcal{E}^{\infty}_{0}$ then $\Omega$ is an $(\epsilon,\infty)$ domain.

Nonhomogeneous Case

Sufficiency: if $\Omega\subset{\mathbb{R}}^{n}$ is an ${(\epsilon,\delta)}$
domain then $\Omega$ is an extension domain for $\mathcal{R}_{\alpha}^{p}$
with $1<p<\infty$ and $\alpha>0$.

Necessity: if a finitely connected domain $\Omega\subset\mathbb{R}^{2}$ is an
extension domain for $\mathcal{R}_{\alpha}^{p}$, where $\alpha p=2$ and
$0<\alpha\leq 1$, then $\Omega$ is an ${(\epsilon,\delta)}$ domain.

Comparing the two cases, the following questions arise: does the sufficiency
in the nonhomogeneous case extend to $p=\infty$ and $\alpha=0$, and what is
the space corresponding to $\mathcal{R}_{0}^{\infty}(\Omega)$? Does the
necessity in the nonhomogeneous case extend to any domain $\Omega$ in any
dimension, when $\alpha=0$ and $p=\infty$?

We answer these questions by identifying ${(\epsilon,\delta)}$ domains as the
extension domains for bmo, known alternatively as local, localized or
nonhomogeneous BMO. The space $\textup{bmo}({\mathbb{R}}^{n})$ was introduced
by Goldberg [12] and consists of locally integrable functions $f$ satisfying

$\|f\|_{\textup{bmo}({\mathbb{R}}^{n})}:=\sup_{\ell(Q)<1}\fint_{Q}|f(x)-f_{Q}|dx+\sup_{\ell(Q)\geq
1}|f|_{Q}<\infty,$

where the supremum is taken over all cubes $Q\subset{\mathbb{R}}^{n}$, and
$\ell(Q)$ denotes the sidelength of $Q$. Goldberg’s choice of the constant $1$
in the definition can be replaced by a constant $\lambda$ without changing the
space, only the norm.

On a domain $\Omega$, the space corresponding to the endpoint $\alpha=0$ in
the nonhomogeneous case can be identified with $\textup{bmo}(\Omega)$ (see
Triebel [19], Section 1.7.2). However, when $\Omega$ is unbounded, it turns
out that it is not sufficient to consider averages over balls or cubes of size
$1$; the choice of constant, or scale, in the definition of
$\textup{bmo}(\Omega)$ is important.

Fix $\lambda>0$. We say $f\in{\textup{bmo}_{\lambda}(\Omega)}$ if $f$ is
integrable on every cube $Q\subset\Omega$ and

$\|f\|_{\textup{bmo}_{\lambda}(\Omega)}:=\sup_{Q\subset\Omega,\ell(Q)<\lambda}\fint_{Q}|f(x)-f_{Q}|dx+\sup_{Q\subset\Omega,\ell(Q)\geq\lambda}|f|_{Q}<\infty.$
(3)

If the domain does not contain any cube of sidelength $\lambda$ or larger, we
have ${\textup{bmo}_{\lambda}(\Omega)}={\textup{BMO}(\Omega)}$ with the same
norms, namely we are back in the homogeneous case. Thus we need to choose
$\lambda$ sufficiently small for a cube of sidelength $\lambda$ to be
contained in $\Omega$, but this is not enough. For example, consider the
Lipschitz domain $\Omega\subset\mathbb{R}^{2}$ lying between the $x$-axis and
the curve $y=\max(1,1-x)$. Clearly $\Omega$ contains arbitrarily large cubes,
but choosing $\lambda$ too large, for example $\lambda>1$ in this example,
results in the existence of functions belonging to
${\textup{bmo}_{\lambda}(\Omega)}$, such as $f(x,y)=\max(x,0)$, but not having
an extension to $\textup{bmo}(\mathbb{R}^{2})$. Thus the correct scale for the
definition of bmo on a domain is intimately connected with the geometry.

We now state the main result of this paper, using some terminology adapted
from [21], as well as two distances ${j_{\Omega}}$ and ${k_{\Omega}}$ defined
on $\Omega\Omega$, where ${k_{\Omega}}$ is the quasi-hyperbolic metric,
introduced by Gehring and Palka [10] (see Section 2).

######  Theorem 1.1.

Let $\Omega\subset{\mathbb{R}}^{n}$ be a domain. Then the following are
equivalent

  * (i)

There are positive constants $\epsilon$ and $\delta$ such that $\Omega$ is an
${(\epsilon,\delta)}$ domain.

  * (ii)

$\Omega$ is a locally uniform domain, i.e. there are positive constants $a$,
$b$ and $\delta$ such that for any $x,y\in\Omega$ with $|x-y|<\delta$, there
is a rectifiable curve $\gamma(x,y)\subset\Omega$ for which (2) holds.

  * (iii)

$\Omega$ is strongly locally quasi-hyperbolically uniform, i.e. there are
positive constants $c$, $d$ and $\delta$ such that for any $x,y\in\Omega$ with
$|x-y|<\delta$,

${k_{\Omega}}(z,w)\leq c{j_{\Omega}}(z,w)+d$

for all points $z,w$ along the quasi-hyperbolic geodesic between $x$ and $y$.

  * (iv)

There is a positive constant $\lambda_{0}$ such that for each
$\lambda\leq\lambda_{0}$, we have a bounded linear extension operator
${T_{\lambda}}:{\textup{bmo}_{\lambda}(\Omega)}\to\textup{bmo}({\mathbb{R}}^{n})$,
with the constant in the bound depending only on $\lambda_{0}$.

In particular, if these conditions hold, we see that for each
$\lambda\leq\lambda_{0}$, the set ${\textup{bmo}_{\lambda}(\Omega)}$ consists
precisely of restrictions to $\Omega$ of elements of
$\textup{bmo}({\mathbb{R}}^{n})$, hence all these sets are the same and we can
define $\textup{bmo}(\Omega)$ to be ${\textup{bmo}_{\lambda}(\Omega)}$ with
$\lambda=\lambda_{0}$.

The theorem follows from the following implications:

$\text{(iii)}\implies\text{(ii)}\implies\text{(i)}\implies\text{(iv)}\implies\text{(iii)}.$

The fact that condition (iii) implies (ii) has already been established by
Theorem 2 in [11]. Implication $\text{(ii)}\implies\text{(i)}$ is easy (see
Proposition 2.5). The major work in this paper is to show the implications
$\text{(i)}\implies\text{(iv)}\implies\text{(iii)}$.

In Section 2 we will introduce the necessary notation and definitions. Apart
from domains satisfying conditions (i)-(iii), we define weakly locally quasi-
hyperbolically uniform domains as domains satisfying a weaker version of (iii)
and show that ${(\epsilon,\delta)}$ domains are weakly locally quasi-
hyperbolically uniform domains (Proposition 2.6). In Section 3 we establish
some estimates for $\textup{bmo}_{\lambda}$ functions on weakly locally quasi-
hyperbolically uniform domains and on ${(\epsilon,\delta)}$ domains. Based on
these estimates and certain lemmas of Jones from [14], in Section 4 we
construct the extension operator, thus showing that (i) $\implies$ (iv). The
extension is a local version of Jones’ extension in [13], in the sense that it
is defined by using averages of the function on “reflected” Whitney cubes near
the boundary, while, like the extension in [14], it is zero far away from the
boundary. Showing that such an extension is in
$\textup{bmo}({\mathbb{R}}^{n})$ relies on the control of the averages of a
function in ${\textup{bmo}_{\lambda}(\Omega)}$ on large cubes, and the
logarithmic growth (in sidelength) that it imposes on the average over any
cube (see Proposition 3.8).

Finally, in Section 5, we show that (iv) $\implies$ (iii). More precisely, if
there is a bounded linear extension from ${\textup{bmo}_{\lambda}(\Omega)}$ to
$\textup{BMO}({\mathbb{R}}^{n})$, we show that $\Omega$ is strongly locally
quasi-hyperbolically uniform. A key tool is the use of truncation to create
functions in ${\textup{bmo}_{\lambda}(\Omega)}$ based on the quasi-hyperbolic
distance ${k_{\Omega}}$, which is in $\textup{BMO}(\Omega)$.

Note that the equivalence of conditions (i),(ii) and (iii), while known in the
case of uniform domains, is not found in the literature in the local case.
Other variants of these conditions can be found in [1], where the notion of
“semi-uniform” is introduced, and in [2], where the ${(\epsilon,\delta)}$
condition is further localized (note that the condition that is called
“locally uniform” there is stated in terms of what are known as “distance
cigars”, not “length cigars” as in (2), and is therefore immediately
equivalent to ${(\epsilon,\delta)}$ \- see [20], Section 2.4). It would be
interesting to have a proof of the equivalence of (i), (ii) and (iii) without
going through the bmo extension. Furthermore, each of these three conditions
is known to imply the weak local quasi-hyperbolically uniform condition, but
it is unknown whether this condition is indeed weaker.

##  2 The geometric setting

In what follows, $\Omega$ will always denote a domain (open and connected set)
in ${\mathbb{R}}^{n}$. We denote by $d_{\Omega}(x)$ the distance to the
boundary, from either inside or outside the domain:

${d_{\Omega}}(x):=\textup{dist}(x,{\partial\Omega}).$

For a rectifiable curve $\gamma$, we denote its arc length by $s(\gamma)$. If
$p,q$ are two points along the curve $\gamma$, we denote by $\gamma(p,q)$ the
portion of the curve between $p$ and $q$,.

###  2.1 The quasi-hyperbolic distance

The quasi-hyperbolic metric, introduced by Gehring and Palka [10], is defined
on $\Omega\times\Omega$ by

${k_{\Omega}}(x,y):=\inf_{\gamma(x,y)\subset\Omega}\int_{\gamma(x,y)}\frac{ds}{\textup{dist}(z,{\partial\Omega})},$
(4)

where the infimum is taken over all rectifiable curves connecting $x$ and $y$,
and the integral is taken with respect to arc length $ds$. Gehring and Osgood
showed (see Lemma 1 in [11]) that between any two points $x,y\in\Omega$ there
is always a curve minimizing ${k_{\Omega}}$, called a quasi-hyperbolic
geodesic, namely a curve $\gamma=\gamma(x,y)$ for which

$\int_{\gamma}\frac{ds}{\textup{dist}(z,{\partial\Omega})}={k_{\Omega}}(x,y).$

Adapting the terminology in [21] to the local case, we state the following
definition.

######  Definition 2.1.

For $x,y\in\Omega$, define

${j_{\Omega}}(x,y):=\frac{1}{2}\log\left[\left(1+\frac{|x-y|}{\textup{dist}(x,{\partial\Omega})}\right)\left(1+\frac{|x-y|}{\textup{dist}(y,{\partial\Omega})}\right)\right].$
(5)

A domain $\Omega$ will be called weakly locally quasi-hyperbolically uniform
if there are positive constants $c$, $d$ and $\delta$ such that for any
$z,w\in\Omega$ with $|z-w|<\delta$,

${k_{\Omega}}(z,w)\leq c{j_{\Omega}}(z,w)+d.$ (6)

A domain $\Omega$ will be called strongly locally quasi-hyperbolically uniform
if there are positive constants $c$, $d$ and $\delta$ such that for any
$x,y\in\Omega$ with $|x-y|<\delta$, (6) holds for all points $z,w$ along the
quasi-hyperbolic geodesic between $x$ and $y$.

The quasi-hyperbolically uniform conditions in the non-local case, which can
be considered a version of Jones’ $d_{1}-d_{2}$ inequality in [13], were shown
by Gehring and Osgood [11] to be equivalent to the domain being uniform. As it
applies to a given curve, Theorem 1 in [11] shows that the local uniformity
condition (ii) of Theorem 1.1 implies that $\Omega$ is weakly locally quasi-
hyperbolically uniform. Conversely, Theorem 2 in the same paper shows that the
strong local quasi-hyperbolically uniform condition (iii) implies condition
(ii).

We will also need to look at the quasi-hyperbolic distance to the interior of
the domain, meaning the set of points at a certain minimum distance from the
boundary.

######  Definition 2.2.

Given domain $\Omega$ and $\lambda>0$ we denote by
$\mathring{\Omega}_{\lambda}$ the set
$\\{x\in\Omega:{d_{\Omega}}(x)\geq\lambda\\}$ and define, for $x\in\Omega$,

${k_{\Omega}}(x,\mathring{\Omega}_{\lambda}):=\inf_{p\in\mathring{\Omega}_{\lambda}}{k_{\Omega}}(x,p).$

######  Lemma 2.3.

Let $\lambda>0$. For each $x\in\Omega\setminus\mathring{\Omega}_{\lambda}$
there exists a point $x^{\prime}\in\mathring{\Omega}_{\lambda}$ and a curve
$\gamma(x,x^{\prime})$ such that

${k_{\Omega}}(x,\mathring{\Omega}_{\lambda})={k_{\Omega}}(x,x^{\prime})=\int_{\gamma(x,x^{\prime})}\frac{ds}{{d_{\Omega}}(z)}.$

###### Proof.

We will only show that
${k_{\Omega}}(x,\mathring{\Omega}_{\lambda})={k_{\Omega}}(x,x^{\prime})$ for
some $x^{\prime}\in\mathring{\Omega}_{\lambda}$ as the existence of a geodesic
$\gamma(x,x^{\prime})$ is proved by Lemma 1 in [11].

First of all, note that $\mathring{\Omega}_{\lambda}$ is a closed set and
${k_{\Omega}}(x,\cdot)$ is a continuous function, so if
$\mathring{\Omega}_{\lambda}$ is bounded then
$\inf\limits_{y\in\mathring{\Omega}_{\lambda}}{k_{\Omega}}(x,y)$ is attained
in at some point in $\mathring{\Omega}_{\lambda}$.

For unbounded $\mathring{\Omega}_{\lambda}$, let $y_{0}$ be any fixed point in
$\mathring{\Omega}_{\lambda}$ and
$R=\max(\lambda\cdot{k_{\Omega}}(x,y_{0}),|x-y_{0}|)$. We claim that

$\inf_{y\in\mathring{\Omega}_{\lambda}}{k_{\Omega}}(x,y)=\inf_{y^{\prime}\in\mathring{\Omega}_{\lambda}\cap
B_{R}(x)}{k_{\Omega}}(x,y^{\prime}),$

where $B_{R}(x)=\\{z\in{\mathbb{R}}^{n}:|x-z|\leq R\\}$. As the latter infimum
is attained at some point in $\mathring{\Omega}_{\lambda}\cap B_{R}(x)$, all
we need to finish the proof is to show that if
$y\in\mathring{\Omega}_{\lambda}$ with $|x-y|>R$ then

${k_{\Omega}}(x,y)\geq\inf_{y^{\prime}\in\mathring{\Omega}_{\lambda}\cap
B_{R}(x)}{k_{\Omega}}(x,y^{\prime}).$

Let $\gamma(x,y)$ be a quasi-hyperbolic geodesic between $x$ and $y$. If there
is a $z\in\gamma(x,y)$ such that $z\in\mathring{\Omega}_{\lambda}\cap
B_{R}(x)$, then

${k_{\Omega}}(x,y)\geq{k_{\Omega}}(x,z)\geq\inf_{y^{\prime}\in\mathring{\Omega}_{\lambda}\cap
B_{R}(x)}{k_{\Omega}}(x,y^{\prime}).$

Otherwise, $\gamma(x,y)\cap
B_{R}(x)\cap\mathring{\Omega}_{\lambda}=\varnothing$ and by our choice of $R$

${k_{\Omega}}(x,y)\geq\int_{\gamma\cap
B_{R}(x)}\frac{ds}{{d_{\Omega}}(z)}\geq\frac{s(\gamma\cap
B_{R}(x))}{\lambda}\geq\frac{R}{\lambda}\geq{k_{\Omega}}(x,y_{0})\geq\inf_{y^{\prime}\in\mathring{\Omega}_{\lambda}\cap
B_{R}(x)}{k_{\Omega}}(x,y^{\prime}).$

∎

###  2.2 ${(\epsilon,\delta)}$-domains

######  Definition 2.4 (Jones [14]).

Given $\epsilon,\delta>0$, a domain $\Omega$ is called an
${(\epsilon,\delta)}$-domain if for any $x,y\in\Omega$ with $|x-y|<\delta$,
there is a rectifiable curve $\gamma\subset\Omega$ joining $x$ to $y$ such
that

$s(\gamma)\leq\epsilon^{-1}|x-y|,$ (7)

and

${d_{\Omega}}(z)\geq\epsilon\frac{|z-x||z-y|}{|x-y|}\quad\forall z\in\gamma.$
(8)

Note that if $\Omega$ is an ${(\epsilon,\delta)}$-domain then it is
$(\epsilon^{\prime},\delta)$-domain and $(\epsilon,\delta^{\prime})$-domain
for any $\epsilon^{\prime}\leq\epsilon$ and $\delta^{\prime}\leq\delta$. For
this reason, we will always assume that $0<\epsilon,\delta<1$. Moreover, we
will also assume that $\delta\leq\textup{diam}(\Omega)$ since any
${(\epsilon,\delta)}$-domain $\Omega$ with $\delta>\textup{diam}(\Omega)$ is
automatically an $(\epsilon,\infty)$ domain. An $(\epsilon,\infty)$ domain is
also called a uniform domain.

As seen from the work of Gehring and Osgood [11], for uniform domains
($\delta=\infty$), the conditions in Definition 2.4 can be replaced by the
equivalent condition that for some constants $a,b>0$, for any $x,y\in\Omega$
there exists a rectifiable curve $\gamma(x,y)\subset\Omega$ such that (2)
holds. Theorem 1.1 shows that the equivalence between these conditions also
holds locally. One direction is easy.

######  Proposition 2.5.

Let $\Omega\subset{\mathbb{R}}^{n}$ be a domain and $\delta>0$. Then $\Omega$
is an ${(\epsilon,\delta)}$ domain for some $\epsilon>0$ if there are
constants $a,b>0$ such that for all $x,y\in\Omega$ with $|x-y|<\delta$, there
exists a rectifiable curve $\gamma(x,y)\subset\Omega$ for which (2) hold.

###### Proof.

The implication holds for an individual curve. If for $x,y\in\Omega$ there
exists a rectifiable curve $\gamma(x,y)\subset\Omega$ satisfying (2), then (7)
is satisfied with $\epsilon\leq a^{-1}$, and for all points $z$ along
$\gamma$,

$\frac{|z-x||z-y|}{|x-y|}\leq
a\frac{s(\gamma(z,x))s(\gamma(z,y))}{s(\gamma)}\leq
a\min(s(\gamma(z,x)),s(\gamma(z,y))\leq ab\;{d_{\Omega}}(z),$

so (8) is satisfied with $\epsilon\leq(ab)^{-1}$. ∎

The other direction of this equivalence will only follow at the end of the
paper, after we have shown the full circle of implications in Theorem 1.1.

The following Proposition shows that ${(\epsilon,\delta)}$ domains are weakly
locally quasi-hyperbolically uniform.

######  Proposition 2.6.

Suppose that $\Omega$ an ${(\epsilon,\delta)}$-domain. Then for $x,y\in\Omega$
with $|x-y|<\delta$, there exists a curve
$\gamma^{\prime}=\gamma^{\prime}(x,y)\subset\Omega$ such that

$\int_{\gamma^{\prime}}\frac{ds}{d_{\Omega}(z)}\leq
C_{\epsilon}({j_{\Omega}}(x,y)+1)$

In particular, we have

${k_{\Omega}}(x,y)\leq C_{\epsilon}({j_{\Omega}}(x,y)+1)$

and $\Omega$ is weakly locally quasi-hyperbolically uniform.

We will need the following simple result.

######  Lemma 2.7.

Let $\epsilon,K,C_{0},C_{1}$ be positive constants and $z_{1},z_{2},z$ be
points in $\Omega$ with

$|z_{1}-z_{2}|\leq C_{0}K,\quad d_{\Omega}(z_{i})\geq C_{1}K,\quad i=1,2,$

and

$d_{\Omega}(z)\geq\epsilon\frac{|z-z_{1}||z-z_{2}|}{|z_{1}-z_{2}|}.$

Then

$d_{\Omega}(z)\geq CK,$

where $C=\min(\frac{C_{1}}{2},\frac{C_{1}^{2}\epsilon}{4C_{0}})$.

###### Proof.

If for $i=1$ or $2$ we have $|z-z_{i}|\leq d_{\Omega}(z_{i})/2$, then the
triangle inequality gives

$d_{\Omega}(z)\geq d_{\Omega}(z_{i})-|z-z_{i}|\geq\frac{C_{1}}{2}K.$

Thus we may assume $|z-z_{i}|\geq d_{\Omega}(z_{i})/2$ for both $i=1$ and $2$.
Then

$d_{\Omega}(z)\geq\epsilon\frac{|z-z_{1}||z-z_{2}|}{|z_{1}-z_{2}|}\geq\epsilon\frac{d_{\Omega}(z_{1})d_{\Omega}(z_{2})}{4C_{0}K}\geq\frac{\epsilon
C_{1}^{2}}{4C_{0}}K.$

∎

###### Proof of Proposition 2.6.

Let $x,y\in\Omega$, $d:=|x-y|<\delta$ and $\gamma=\gamma(x,y)$ be the curve
satisfying (7) and (8). We partition $\gamma$ into three portions
$\gamma(x,\tilde{x})$, $\gamma(\tilde{x},\tilde{y})$ and $\gamma(\tilde{y},y)$
by choosing a point $\tilde{x}$ on $\gamma$ with $|\tilde{x}-x|=d/2$ and a
point $\tilde{y}$ on $\gamma(\tilde{x},y)$ such that $|\tilde{y}-y|=d/2$, with
the condition that the portion $\gamma(\tilde{x},\tilde{y})$ lie outside the
balls $B(x,d/2)$, and $B(y,d/2)$.

Claim 1:

$\int_{\gamma(\tilde{x},\tilde{y})}\frac{ds}{d_{\Omega}(z)}\leq
C\epsilon^{-2}.$

Indeed, by construction each $z\in\gamma(\tilde{x},\tilde{y})$ satisfies
$\min(|z-x|,|z-y|)\geq d/2$. So by conditions (7) and (8) we have

$\int_{\gamma(\tilde{x},\tilde{y})}\frac{ds}{d_{\Omega}(z)}\leq\epsilon^{-1}\int_{\gamma(\tilde{x},\tilde{y})}\frac{|x-y|ds}{|z-x||z-y|}\leq
4\epsilon^{-1}\frac{s(\gamma(\tilde{x},\tilde{y}))}{d}\leq
4\epsilon^{-1}\frac{s(\gamma)}{d}\leq 4\epsilon^{-2}.$

Claim 2: There exist curves
$\tilde{\gamma}_{1},\tilde{\gamma}_{2}\subset\Omega$, where
$\tilde{\gamma}_{1}=\tilde{\gamma}_{1}(x,\tilde{x})$ and
$\tilde{\gamma}_{2}=\tilde{\gamma}_{2}(\tilde{y},y)$ such that

$\int_{\tilde{\gamma}_{1}}\frac{ds}{d_{\Omega}(z)}\lesssim
1+\log\left(\frac{d}{d_{\Omega}(x)}\right),\quad\int_{\tilde{\gamma}_{2}}\frac{ds}{d_{\Omega}(z)}\lesssim
1+\log\left(\frac{d}{d_{\Omega}(y)}\right).$

We will prove the first inequality as the proof for the second is identical.

If $d_{\Omega}(x)>d$ then $|\tilde{x}-x|<d_{\Omega}(x)/2$ and we can take
$\tilde{\gamma}_{1}$ to be the straight line from $x$ to $\tilde{x}$ inside
the ball $B(x,d_{\Omega}(x)/2)$. Then for every
$z\in\tilde{\gamma}_{1}(x,\tilde{x})$,

$d_{\Omega}(z)\geq d_{\Omega}(x)/2$

and therefore

$\int_{\tilde{\gamma}_{1}}\frac{ds}{d_{\Omega}(z)}\leq
2\frac{|\tilde{x}-x|}{d_{\Omega}(x)}<1.$

Therefore, we will assume $d_{\Omega}(x)\leq d$. Let $N$ be the integer for
which $2^{N-1}\leq\frac{d}{d_{\Omega}(x)}<2^{N}$. Take a sequence of point
$x=z_{N+1},z_{N},\ldots,z_{1}=\tilde{x}$ along the curve $\gamma(x,\tilde{x})$
with $|z_{k}-x|=2^{-k}d$. Note that

$|z_{k}-z_{k+1}|\leq|z_{k}-x|+|x-z_{k+1}|\leq 2^{1-k}d.$ (9)

For $k=1,\ldots,N$, since $|z_{k}-y|\geq|x-y|-|z_{k}-x|=d-2^{-k}d\geq d/2$,
condition (8) gives

$d_{\Omega}(z_{k})\geq\epsilon\frac{|z_{k}-x||z_{k}-y|}{d}\geq\frac{\epsilon}{2}|z_{k}-x|=\frac{\epsilon}{2}2^{-k}d.$
(10)

For $k=N+1$, our choice of $N$ guarantees that

$d_{\Omega}(z_{N+1})=d_{\Omega}(x)\geq 2^{-(N+1)}d.$ (11)

Since $\Omega$ is an ${(\epsilon,\delta)}$ domain and $|z_{k}-z_{k+1}|\leq
d<\delta$, there exist curves
$\alpha_{k}=\alpha_{k}(z_{k+1},z_{k})\subset\Omega$, $k=1,\ldots,N$, that
satisfy

$s(\alpha_{k})\leq\epsilon^{-1}|z_{k}-z_{k+1}|\leq\epsilon^{-1}2^{1-k}d$ (12)

and

$d_{\Omega}(z)\geq\epsilon\frac{|z-z_{k}||z-z_{k+1}|}{|z_{k}-z_{k+1}|}\text{
for all }z\in\alpha_{k}.$ (13)

In view of (9), (10), (11) and (13), we can apply Lemma 2.7 with $z_{1}$ and
$z_{2}$ replaced by $z_{k}$ and $z_{k+1}$, $K=2^{-k}d$, $C_{0}=2$ and
$C_{1}=\frac{\epsilon}{2}$ to get that

$d_{\Omega}(z)\geq C2^{-k}d\quad\text{ for all }z\in\alpha_{k},\quad
k=1,\ldots N,$

where $C=\epsilon^{3}/32$. Combining with (12), this gives

$\int\limits_{\alpha_{k}}\frac{ds}{d_{\Omega}(z)}\leq\frac{1}{16\epsilon^{4}}.$

Then setting $\tilde{\gamma}_{1}=\tilde{\gamma}_{1}(x,\tilde{x})$ to be the
curve composed of $\alpha_{N},\alpha_{N-1},\ldots\alpha_{1}$, we get, by the
choice of $N$, that

$\int\limits_{\tilde{\gamma}_{1}}\frac{ds}{d_{\Omega}(z)}=\sum_{k=1}^{N}\int\limits_{\alpha_{k}}\frac{ds}{d_{\Omega}(z)}\leq\frac{N}{16\epsilon^{4}}\leq\frac{1}{16\epsilon^{4}}\left(1+\log\left(\frac{d}{d_{\Omega}(x)}\right)\right),$

which proves the claim.

Finally, the curve $\gamma^{\prime}$ composed of $\tilde{\gamma}_{1}$,
$\gamma(\tilde{x},\tilde{y})$ and $\tilde{\gamma}_{2}$ gives the desired bound
on $\int_{\gamma^{\prime}}\frac{ds}{d_{\Omega}(z)}$, with a constant
$C_{\epsilon}$ which is a multiple of $\epsilon^{-4}$. ∎

###  2.3 Whitney cubes

Along with $\Omega$ we consider the interior of its complement
$\Omega^{\prime}=({\mathbb{R}}^{n}\setminus\Omega)^{\circ}$. By $E$ and
$E^{\prime}$ we denote the Whitney decompositions of $\Omega$ and
$\Omega^{\prime}$, respectively. We recall that the Whitney decomposition of
an open set $O\subsetneq{\mathbb{R}}^{n}$ is a collection of closed dyadic
cubes $\\{Q_{I}\\}$ such that $O=\cup_{i}Q_{i}$ and

$Q^{\circ}_{i}\cap Q_{j}^{\circ}=\emptyset,\text{ if }i\neq j,$ (14)
$1\leq\frac{\textup{dist}(Q_{j},\partial\Omega)}{\ell(Q_{j})}\leq 4\sqrt{n},$
(15) $\frac{1}{4}\leq\frac{\ell(Q_{j})}{\ell(Q_{i})}\leq 4,\text{ if
}Q_{i}\cap Q_{j}\neq\emptyset.$ (16)

Cubes in the Whitney decomposition of $O$ will be called Whitney cubes of $O$.
Any two (closed) Whitney cubes $Q_{1},Q_{2}$ such that $Q_{1}\cap
Q_{2}\neq\emptyset$ will be called adjacent cubes. Note that this sometimes
includes the case when $Q_{1}=Q_{2}$. As in [13], we will use the term Whitney
chain for a finite sequence of Whitney cubes with each cube adjacent to the
preceding one.

When $\Omega$ is an ${(\epsilon,\delta)}$ domain, the Whitney decompositions
$E$ and $E^{\prime}$ cover ${\mathbb{R}}^{n}$ up to a set of measure zero.
This follows from Lemma 2.3 in [14], stating that the boundary
${\partial\Omega}$ of an ${(\epsilon,\delta)}$ domain has zero $n$-dimensional
Lebesgue measure, although, as Jones notes, it could have positive Hausdorff
measure for dimensions arbitrary close to $n$. Thus to extend a function from
$\Omega$ to ${\mathbb{R}}^{n}$ we only need to define it on cubes in
$E^{\prime}$.

The following lemma allows us to view cubes in $E^{\prime}$ which are close to
the boundary as reflections of corresponding cubes in $E$.

######  Lemma 2.8 (Lemma 2.4 in Jones [14]).

Let $\Omega$ be an ${(\epsilon,\delta)}$-domain. If $Q\in E^{\prime}$ and
$\ell(Q)\leq\epsilon\delta/(16n)$, then there exists $Q^{*}\in E$ such that

$1\leq\frac{\ell(Q^{*})}{\ell(Q)}\leq 4$ (17)

and

$\textup{dist}(Q^{*},Q)\leq C_{\epsilon,n}\ell(Q),$ (18)

where $C_{\epsilon,n}=5\sqrt{n}+8n\cdot\epsilon^{-2}$.

Any such $Q^{*}\in E$ associated to a cube $Q$ as in the lemma will be called
a matching cube of $Q$, keeping in mind that the choice of $Q^{*}$ is not
unique.

######  Lemma 2.9.

Let $\Omega$ be an ${(\epsilon,\delta)}$-domain. If $Q$ is a dyadic cube in
${\mathbb{R}}^{n}$ with $\ell(Q)<\delta$, then there is $z\in Q^{\circ}$ such
that

${d_{\Omega}}(z)\geq\epsilon\ell(Q)/32.$

###### Proof.

Let $c$ be the center of $Q$, $r=\ell(Q)/8$ and consider the open annulus

$A=\\{x\in{\mathbb{R}}^{n}:r<|x-c|<2r\\}\subset Q^{\circ}.$

If $A\cap\Omega=\emptyset$ , namely $A\subset\Omega^{\prime}$, then $A$
contains points of $Q$ of distance at least $r/2=\ell(Q)/16$ from
${\partial\Omega}$. Thus we may assume that there is an $x\in\Omega\cap A$.
Applying the same reasoning to $A\setminus\overline{B(x,2r)}$, which also
contains a ball of radius $r/2$, we may assume there is a $y\in\Omega\cap A$
such that $|x-y|>2r.$

Since $|x-y|\leq\textup{diam}(A)=4r<\delta$, there is a curve
$\gamma\subset\Omega$ connecting $x$ and $y$ and satisfying conditions (7) and
(8). Let $z$ be a point on this curve such that $|x-z|=r$. Then by the choice
of $x,y$,

$|z-r|\leq|x-c|+|x-z|<3r<\ell(Q)/2,$

which means that $z\in Q^{\circ}$. Moreover, by $\eqref{John_cond}$ and since
$|y-z|\geq|y-x|-|x-z|>r$,

${d_{\Omega}}(z)\geq\frac{\epsilon|x-z||y-z|}{|x-y|}\geq\frac{\epsilon
r}{4}=\frac{\epsilon\ell(Q)}{32}.$

∎

######  Lemma 2.10.

Let $\Omega$ be an ${(\epsilon,\delta)}$-domain. If $Q\subset{\mathbb{R}}^{n}$
is a dyadic cube with $\ell(Q)<\delta$, then there is $Q_{0}\in E\cup
E^{\prime}$ with $Q_{0}\supset Q$ or $Q\supset Q_{0}$ and

$\ell(Q_{0})\ \geq\frac{\epsilon}{160\sqrt{n}}\ell(Q).$

###### Proof.

Apply Lemma 2.9 to the given cube $Q$ to get $z\in Q^{\circ}$ such that
${d_{\Omega}}(z)\geq\epsilon\ell(Q)/32$. Since $z\not\in{\partial\Omega}$,
there is a cube $Q_{0}$ in $E\cup E^{\prime}$ containing $z$. As $Q$ and
$Q_{0}$ are dyadic cubes with intersecting interiors, either $Q_{0}\supset Q$
or $Q\supset Q_{0}$. There is nothing to prove in the former case.

In the latter case, note that (15) implies

$1\leq\frac{{d_{\Omega}}(z)}{\ell(Q_{0})}\leq\frac{\textup{dist}(Q_{0},\partial\Omega)+\textup{diam}(Q_{0})}{\ell(Q_{0})}\leq
5\sqrt{n}$

and therefore

$\ell(Q_{0})\geq\frac{{d_{\Omega}}(z)}{5\sqrt{n}}\geq\frac{\epsilon\ell(Q)}{160\sqrt{n}}.$

∎

As mentioned in the introduction, one of the key issues is how big of a cube
can be fit inside the domain, and for this reason we are looking for Whitney
cubes of a certain minimal size. We can apply the lemma above to obtain the
following result, which is reminiscent of the notion of plumpness in [20].

######  Proposition 2.11.

If $\Omega$ is an ${(\epsilon,\delta)}$-domain and $x\in\Omega$, then the
Whitney decomposition $E$ contains a cube $S$ of sidelength at least
$\epsilon\delta/(320n)$, and whose distance from $x$ is less than
$\delta(\epsilon^{-1}+\sqrt{n})$.

###### Proof.

Let $x\in\Omega$ and take any dyadic cube $Q$ containing $x$ whose sidelength
satisfies $\delta/2\leq\ell(Q)<\delta$. By Lemma 2.10, there is a cube
$Q_{0}\in E\cup E^{\prime}$ such that $Q_{0}\supset Q$, or $Q\supset Q_{0}$
and $\ell(Q_{0})\geq\epsilon\ell(Q)/(160\sqrt{n})$. In the first case we must
have $Q_{0}\in E$ and $\textup{dist}(Q_{0},x)=0$, so we let $S=Q_{0}$.

In the second case, it is possible that $Q_{0}\in E^{\prime}$, in which case,
provided $\ell(Q_{0})\leq\epsilon\delta/(16n)$, can apply Lemma 2.8 to $Q_{0}$
to get a matching cube $Q_{0}^{*}\in E$ with $\ell(Q_{0}^{*})=\ell(Q_{0})$,
and let $S=Q_{0}^{*}$. If our original choice of $Q_{0}$ happens to be too
large, going along the straight line from $Q_{0}\subset\Omega^{\prime}$ to
$x\in\Omega$ which lies in $Q$, we obtain a chain of Whitney cubes in
$E^{\prime}$ whose sidelengths tend to zero as they approach
${\partial\Omega}$. By (16), the sidelengths of consecutive cubes in the chain
vary by a factor of at most $4$, so we can replace our choice by another cube
$Q_{0}$ in $E^{\prime}$ with sidelength in
$(\epsilon\delta/(64n),\epsilon\delta/(16n))$, and take its matching cube as
our desired cube $S$ in $E$. Our $S$ will have
$\ell(S)\geq\min(\epsilon\ell(Q)/(160\sqrt{n}),\epsilon\delta/(64n))\geq\epsilon\delta/(320n)$,
and since our chosen cube $Q_{0}$ in $E^{\prime}$ intersects our original cube
$Q$, by (18) we will have that

$\textup{dist}(S,x)\leq\textup{dist}(S,Q_{0})+\textup{diam}(Q_{0})+\textup{dist}(Q_{0},x)\leq(C_{\epsilon,n}+\sqrt{n})\ell(Q_{0})+\textup{diam}(Q)<\delta(\epsilon^{-1}+\sqrt{n}).$

∎

##  3 Bounded mean oscillation in $\Omega$

In this section we look at functions in $\textup{BMO}(\Omega)$ and
${\textup{bmo}_{\lambda}(\Omega)}$ and their properties, first assuming that
$\Omega$ is any domain in ${\mathbb{R}}^{n}$, and then specializing to weakly
locally quasi-hyperbolically uniform and ${(\epsilon,\delta)}$ domains
$\Omega$.

###  3.1 Local-to-global property and the quasi-hyperbolic distance

A key connection between the geometry of the domain and the space
${\textup{BMO}(\Omega)}$, defined by condition (1), is via the quasi-
hyperbolic distance ${k_{\Omega}}$. Jones [13] states that ${k_{\Omega}}$ is
equivalent to the distance he calls $d_{1}$, which defines a function in
$\textup{BMO}(\Omega)$, but the fact that ${k_{\Omega}}$ is itself in
$\textup{BMO}(\Omega)$ can be shown directly, as follows.

######  Lemma 3.1.

For any fixed $a\in\Omega$, the function $f(x)={k_{\Omega}}(a,x)$ is an
element of ${\textup{BMO}(\Omega)}$ and

$\|f\|_{{\textup{BMO}(\Omega)}}\leq c,$

where $c$ depends only on the dimension $n$.

###### Proof.

We will use the following result, sometimes known as a “local-to-global”
property, due to Reimann and Rychener [17] (also attributed to Jones - see
Theorem A1.1 and Corollary A1.1 in [3]): there is a constant $C$ that depends
only on the dimension $n$ such that for $f\in{L^{1}_{\text{loc}}}(\Omega)$,

$\|f\|_{{\textup{BMO}(\Omega)}}\leq
C\sup\limits_{2B\subset\Omega}\fint_{B}|f(x)-f_{B}|dx.$ (19)

Here the supremum is taken over all balls $B$ whose doubles $2B$ are contained
in $\Omega$ (the notation $cB$ denotes the concentric ball of $c$ times the
radius). Since

$\fint_{B}|f(x)-f_{B}|dx\leq\frac{1}{|B|^{2}}\int_{B}\int_{B}|f(x)-f(y)|dxdy,$

it is enough to establish that

$\sup\limits_{2B\subset\Omega}\frac{1}{|B|^{2}}\int_{B}\int_{B}|{k_{\Omega}}(a,x)-{k_{\Omega}}(a,y)|dxdy\leq
C^{\prime}$

for some $C^{\prime}$ depending at most on $n$.

Suppose $x,y\in B\subset\Omega$. Since ${k_{\Omega}}$ is a distance, the
triangle inequality gives

$|{k_{\Omega}}(a,x)-{k_{\Omega}}(a,y)|\leq{k_{\Omega}}(x,y)\leq\int_{x}^{y}\frac{ds}{{d_{\Omega}}(z)}\leq\frac{\textup{diam}(B)}{\textup{dist}(B,{\partial\Omega})},$
(20)

where the integral on the right is over the line segment $z=(1-t)x+ty,$ $0\leq
t\leq 1$. Since $2B\subset\Omega$ means
$\textup{dist}(B,{\partial\Omega})\geq\textup{diam}(B)/2$, we get
$|{k_{\Omega}}(a,x)-{k_{\Omega}}(a,y)|\leq 2$. ∎

######  Remark 3.2.

If we replace $B$ in (20) by a Whitney cube $Q\in E$, we see from (15) that

$\sup_{x,y\in Q}|{k_{\Omega}}(a,x)-{k_{\Omega}}(a,y)|\leq\sqrt{n}.$

Moreover, this is true (with a larger constant) if $x$ and $y$ are in adjacent
Whitney cubes. From this we can deduce the observation, in [13], that the
quasi-hyperbolic distance ${k_{\Omega}}(x,y)$ is equivalent to the distance
$d_{1}$, defined by Jones [13], between the Whitney cubes containing $x$ and
$y$, namely the length of a shortest Whitney chain between them.

We now establish the analogue of (19) for ${\textup{bmo}_{\lambda}(\Omega)}$.

######  Lemma 3.3.

Let $f\in{\textup{BMO}(\Omega)}$. Then

$\|f\|_{\textup{bmo}_{\lambda}(\Omega)}\lesssim\sup_{2B\subset\Omega}\fint_{B}|f(x)-f_{B}|dx+\sup_{2B\subset\Omega,\textup{diam}(B)\geq\lambda/2}|f|_{B}<\infty.$
(21)

In particular, if $f\in L^{\infty}(\mathring{\Omega}_{\lambda/4})$, then

$\|f\|_{{\textup{bmo}_{\lambda}(\Omega)}}\lesssim\|f\|_{{\textup{BMO}(\Omega)}}+\|f\|_{L^{\infty}(\mathring{\Omega}_{\lambda/4})}.$

Here and below, we use the notation $u\lesssim v$ when there exists a constant
$c$ such that $u\leq cv$.

###### Proof.

The first term on the right-hand-side of (21) controls
$\|f\|_{{\textup{BMO}(\Omega)}}$ by (19). To control the averages over large
cubes, let $Q\subset\Omega$ be any cube such that $\ell(Q)\geq\lambda$. We
take $B$ to be the ball co-centered with $Q$ with
$\textup{diam}(B)=\ell(Q)/2$. Then $2B\subset Q\subset\Omega$, and

$|f|_{Q}\leq\fint_{Q}|f(x)-f_{B}|+|f|_{B}\leq\fint_{Q}\fint_{B}|f(x)-f(y)|dxdy+|f|_{B}\leq\frac{|Q|}{|B|}\fint_{Q}\fint_{Q}|f(x)-f(y)|dxdy+|f|_{B}.$

The first term on the right-hand-side is controlled, up to a constant
depending only on $n$, by $\|f\|_{{\textup{BMO}(\Omega)}}$, and therefore,
applying (19) again, the right-hand-side is controlled by the right-hand-side
of (21).

Finally, note that if $2B\subset\Omega$ and $\textup{diam}(B)\geq\lambda/2$,
then $\textup{dist}(B,{\partial\Omega})\geq\lambda/4$, i.e.
$B\subset\mathring{\Omega}_{\lambda/4}$. ∎

###  3.2 Averages of $\textup{bmo}_{\lambda}$ functions

One of the ways of understanding the bounded mean oscillation condition is by
looking at differences of averages of a BMO function on dyadic cubes. When the
cubes are adjacent, we have the following special case of Lemma 2.2 in [13]:

######  Lemma 3.4 ([13]).

Let $\Omega$ be any domain and $Q_{1},Q_{2}\in E$ be two adjacent Whitney
cubes of $\Omega$. Then

$|f_{Q_{1}}-f_{Q_{2}}|\lesssim\|f\|_{{\textup{BMO}(\Omega)}}.$

In the nonhomogeneous case, we also have absolute control of the averages when
the cubes are large. The following lemma, which is the
$\textup{bmo}({\mathbb{R}}^{n})$ analogue of Lemma 2.3 in [13], allows us to
measure the bmo norm only on dyadic cubes, provided we have control of the
differences of averages over adjacent cubes.

######  Lemma 3.5.

Let $\mathfrak{D}({\mathbb{R}}^{n})$ be the collection of dyadic cubes in
${\mathbb{R}}^{n}$. Given $f\in{L^{1}_{\text{loc}}}({\mathbb{R}}^{n})$, denote

$a_{f}=\sup\limits_{Q\in\mathfrak{D}({\mathbb{R}}^{n})}\fint_{Q}|f(x)-f_{Q}|dx,$
$b_{f}=\sup\limits_{\begin{subarray}{c}Q_{1},Q_{2}\in\mathfrak{D}({\mathbb{R}}^{n})\\\
\ell(Q_{1})=\ell(Q_{2})\\\ Q_{1}\cap
Q_{2}\neq\emptyset\end{subarray}}|f_{Q_{1}}-f_{Q_{2}}|,$

and

$c_{f}=\sup_{Q\in\mathfrak{D}({\mathbb{R}}^{n}):\ell(Q)\geq\lambda/16\sqrt{n}}|f|_{Q}.$

Then

$\|f\|_{\textup{bmo}_{\lambda}({\mathbb{R}}^{n})}\lesssim a_{f}+b_{f}+c_{f}.$

###### Proof.

The inequality

$\sup_{Q\subset\Omega}\fint_{Q}|f(x)-f_{Q}|dx\lesssim a_{f}+b_{f}$

is Lemma 2.3 in [13]. More precisely, its proof shows that if
$Q\subset\Omega$, $\\{Q_{i}\\}$ is the Whitney decomposition of the interior
of $Q$ and $Q_{0}$ is the cube in this decomposition which contains the center
of $Q$, then

$\fint_{Q}|f(x)-f_{Q_{0}}|dx\lesssim a_{f}+b_{f}.$

When $\ell(Q)\geq\lambda$, the sidelength of this chosen cube satisfies
$\ell(Q_{0})\geq\lambda/(16\sqrt{n})$ and

$|f|_{Q}\leq\fint_{Q}|f(x)-f_{Q_{0}}|dx+|f|_{Q_{0}}\lesssim
a_{f}+b_{f}+c_{f}.$

∎

In addition to having control of the averages of a
${\textup{bmo}_{\lambda}(\Omega)}$ function when $\ell(Q)\geq\lambda$, our
goal in this section is to estimate the growth of such averages when $\ell(Q)$
is small. When working on ${\mathbb{R}}^{n}$, or on a metric measure spaces
with doubling (see [7]), it is not difficult to show that the averages of a
bmo function on cubes or balls grow at most logarithmically as the sidelength
shrinks. To get a similar result in a domain, we need to exploit the geometry.

######  Lemma 3.6.

Let $\Omega$ be any domain and $x,y\in\Omega$ be connected by a rectifiable
curve $\gamma$. Let $\\{Q_{i}\\}_{i=1}^{m}\subset E$ be the Whitney cubes
covering $\gamma$. Then

$m\lesssim\int_{\gamma}\frac{ds}{{d_{\Omega}}(z)}+1.$

In particular, if $\Omega$ is weakly locally quasi-hyperbolically uniform,
then for $x,y\in\Omega$ with $|x-y|<\delta$, there exists a sequence of
pairwise adjacent Whitney cubes $\\{Q_{i}\\}_{i=1}^{m}$ such that $x\in
Q_{1}$, $y\in Q_{m}$ and

$m\leq C_{\epsilon}({j_{\Omega}}(x,y)+1).$ (22)

###### Proof.

Fixing $\alpha\in(0,1)$, we have

$\int_{\gamma}\frac{ds}{{d_{\Omega}}(z)}\geq\sum_{i}\frac{s(\gamma\cap
Q_{i})}{\textup{dist}(Q_{i},{\partial\Omega})}\geq\sum_{i}\frac{s(\gamma\cap
Q_{i})}{\ell(Q_{i})4\sqrt{n}}\gtrsim\;\\#\\{i:s(\gamma\cap
Q_{i})\geq\alpha\ell(Q_{i})\\}.$

Let $\mathcal{A}$ be the collection of those cubes $Q_{i}$ with $s(\gamma\cap
Q_{i})\geq\alpha\ell(Q_{i})$. For each $Q_{i}$, either one of the endpoints of
$\gamma$ lies in an adjacent cube or $\gamma$ exits the set
$\mathcal{N}(Q_{i})$ consisting of $Q_{i}$ and all its neighboring cubes,
meaning that

$s(\gamma\cap\mathcal{N}(Q_{i}))\geq\textup{dist}(Q_{i},\partial\mathcal{N}(Q_{i}))\geq\ell(Q_{i})/4.$

As the number of cubes in $\mathcal{N}(Q_{i})$ is bounded by some $D_{n}$, and
the size of the cubes in $\mathcal{N}(Q_{i})$ is bounded by $4\ell(Q_{i})$, if
all those cubes do not belong to $\mathcal{A}$ then

$\ell(Q_{i})/4<4D_{n}\alpha\ell(Q_{i}),$

which is impossible if we choose $\alpha\leq 1/(16D_{n})$. Noting that there
can be at most $2D_{n}$ cubes $Q_{i}$ for which $x$ or $y$ lie in an adjacent
cube, we get that

$m\leq
D_{n}(\\#\mathcal{A}+2)\lesssim\int_{\gamma}\frac{ds}{{d_{\Omega}}(z)}+1.$

This proves the first part of the theorem.

For the second part, we apply the above to the quasi-hyperbolic geodesic
$\gamma^{\prime}(x,y)$ and use (6). ∎

As a consequence of this lemma we obtain a local version of Lemma 2.1 in [13]
which will prove very useful in what follows.

######  Corollary 3.7.

Assume $\Omega$ is weakly locally quasi-hyperbolically uniform and let
$Q_{1},Q_{2}\in E$ be such that $\ell(Q_{1})\leq\ell(Q_{2})$. If
$\textup{dist}(Q_{1},Q_{2})<\delta$, then

$|f_{Q_{1}}-f_{Q_{2}}|\lesssim\|f\|_{\textup{BMO}(\Omega)}\left(1+\log_{+}\left(\frac{\textup{dist}(Q_{1},Q_{2})+\ell(Q_{2})}{\ell(Q_{1})}\right)\right)$

with a constant depending on $\epsilon$. Here $\log_{+}(x):=\max(\log x,0)$.

###### Proof.

We apply Lemma 3.6 to points $x\in Q_{1}$, $y\in Q_{2}$ with $|x-y|<\delta$ to
get a curve between $x$ and $y$ for which we can use estimate (22) to bound
the length $m$ of the chain of Whitney cubes along this curve by
$C_{\epsilon}({j_{\Omega}}(x,y)+1)$. Note that
$|x-y|\leq\textup{dist}(Q_{1},Q_{2})+\textup{diam}(Q_{1})+\textup{diam}(Q_{2})$,
so the hypothesis and the properties of Whitney cubes, we have

${j_{\Omega}}(x,y)=\frac{1}{2}\log\left[\left(1+\frac{|x-y|}{\textup{dist}(x,{\partial\Omega})}\right)\left(1+\frac{|x-y|}{\textup{dist}(y,{\partial\Omega})}\right)\right]\lesssim
1+\log_{+}\left(\frac{\textup{dist}(Q_{1},Q_{2})+\ell(Q_{2})}{\ell(Q_{1})}\right),$

The conclusion follows by applying Lemma 3.4 to adjacent cubes along this
chain. ∎

We now come to the desired logarithmic growth estimate on the averages of
${\textup{bmo}_{\lambda}(\Omega)}$ functions on Whitney cubes, which will
prove very useful in what follows.

######  Proposition 3.8.

Let $\Omega$ be an ${(\epsilon,\delta)}$ domain. If
$\lambda\leq\frac{\epsilon^{2}\delta}{320n(1+\sqrt{n}\epsilon)}$, then for any
$f\in{\textup{bmo}_{\lambda}(\Omega)}$ and $Q\in E$,

$|f_{Q}|\lesssim\left(1+\log_{+}\left(\frac{\lambda}{\ell(Q)}\right)\right)\|f\|_{{\textup{bmo}_{\lambda}(\Omega)}}.$

###### Proof.

There is nothing to prove if $\ell(Q)\geq\lambda$, so we assume that
$\ell(Q)\leq\lambda$.

Given a chain of Whitney cubes $\\{Q_{i}\\}_{i=1}^{m}$ of length $m$, starting
at $Q_{1}=Q$, we can write

$|f_{Q}|\leq\sum_{i=1}^{m-1}|f_{Q_{i}}-f_{Q_{i+1}}|+|f_{Q_{m}}|\leq
m\|f\|_{{\textup{BMO}(\Omega)}}+|f|_{Q_{m}}$

due to Lemma 3.4. Thus it remain to bound $m$ and $|f|_{Q_{m}}$.

Let $x\in Q$. Since $\Omega$ is ${(\epsilon,\delta)}$, it is
$(\epsilon,\delta^{\prime})$ for $\delta^{\prime}<\delta$. Applying
Proposition 2.11 with $\delta^{\prime}$ instead of $\delta$, where
$\delta^{\prime}\leq\delta(\epsilon^{-1}+\sqrt{n})^{-1}$, we get the existence
of a Whitney cube $S\in E$ whose sidelength is at least
$\epsilon\delta^{\prime}/(320n)$, and whose distance from $x$ is less than
$\delta^{\prime}(\epsilon^{-1}+\sqrt{n})\leq\delta$. Taking a point $y\in S$
with $|x-y|<\delta$, we can use Lemma 3.6 to get the chain of cubes
$\\{Q_{i}\\}_{i=1}^{m}$ with $Q_{m}=S$ and

$m\lesssim\log\left[\left(1+\frac{|x-y|}{\textup{dist}(x,{\partial\Omega})}\right)\left(1+\frac{|x-y|}{\textup{dist}(y,{\partial\Omega})}\right)\right]+1\lesssim\log\left[\left(1+\frac{|x-y|}{\ell(Q)}\right)\left(1+\frac{|x-y|}{\ell(Q_{m})}\right)\right]+1,$

by the properties of Whitney cubes. Now since

$\lambda\leq\frac{\epsilon^{2}\delta}{320n(1+\sqrt{n}\epsilon)}=\frac{\epsilon\delta}{320n(\epsilon^{-1}+\sqrt{n})},$

we can take $\delta^{\prime}:=320n\epsilon^{-1}\lambda$, which satisfies the
conditions above, and therefore

$|x-y|<\delta^{\prime}(\epsilon^{-1}+\sqrt{n})=320n\epsilon^{-1}(\epsilon^{-1}+\sqrt{n})\lambda=C_{n,\epsilon}\lambda.$

Moreover, $\ell(Q_{m})\geq\epsilon\delta^{\prime}/(320n)=\lambda$, giving us
that $|f|_{Q_{m}}\leq\|f\|_{\textup{bmo}_{\lambda}(\Omega)}$ and

$m\lesssim\log\left[\left(1+\frac{C_{n,\epsilon}\lambda}{\ell(Q)}\right)\left(1+\frac{C_{n,\epsilon}\lambda}{\lambda}\right)\right]+1\lesssim
1+\log_{+}\left(\frac{\lambda}{\ell(Q)}\right).$

∎

##  4 The ${\textup{bmo}_{\lambda}(\Omega)}$ extension

In this section we assume $\Omega$ is an ${(\epsilon,\delta)}$ domain and fix
$\lambda\leq\lambda_{\epsilon,\delta}$, where
$\lambda_{\epsilon,\delta}:=\frac{\epsilon^{2}\delta}{320n(1+\sqrt{n}\epsilon)}$.
For any $f\in{\textup{bmo}_{\lambda}(\Omega)}$, define a function
${T_{\lambda}}f:{\mathbb{R}}^{n}\to\mathbb{R}$ by

${T_{\lambda}}f(x)=\begin{cases}f(x)&\textup{if }x\in\Omega;\\\
f_{Q^{*}}&\textup{if }x\in Q\in E^{\prime}:\ell(Q)\leq\lambda;\\\
0&\textup{otherwise}.\end{cases}$ (23)

For this definition we have fixed a choice of $Q^{*}$, as in Lemma 2.8, for
each $Q\in E^{\prime}$ with $\ell(Q)\leq\lambda$, since
$\lambda_{\epsilon,\delta}\leq\epsilon\delta/(16n)$. The operator
$f\mapsto{T_{\lambda}}f$ is clearly linear. We will show that
${T_{\lambda}}f\in\textup{bmo}({\mathbb{R}}^{n})$, thus proving the
implication (i) $\implies$ (iv) in Theorem 1.1.

The following lemma shows that the estimates on the averages of
${\textup{bmo}_{\lambda}(\Omega)}$ functions given by Corollary 3.7 and
Proposition 3.8 also hold for ${T_{\lambda}}f$, a fact which will be essential
to our proof.

######  Lemma 4.1.

Let $Q\in E\cup E^{\prime}$. Then

$|({T_{\lambda}}f)_{Q}|\lesssim\left(1+\log_{+}\left(\frac{\lambda}{\ell(Q)}\right)\right)\|f\|_{{\textup{bmo}_{\lambda}(\Omega)}}.$

Moreover, if $Q_{1},Q_{2}\in E\cup E^{\prime}$ with
$\ell(Q_{1})\leq\ell(Q_{2})$ and $\textup{dist}(Q_{1},Q_{2})<\delta$, then

$|({T_{\lambda}}f)_{Q_{1}}-({T_{\lambda}}f)_{Q_{2}}|\lesssim\left(1+\log_{+}\left(\frac{\min(\textup{dist}(Q_{1},Q_{2})+\ell(Q_{2}),\lambda)}{\ell(Q_{1})}\right)\right)\|f\|_{{\textup{bmo}_{\lambda}(\Omega)}}.$

The constants in both inequalities depend on $\epsilon$.

###### Proof.

The first inequality follows from the definition of ${T_{\lambda}}$,
Proposition 3.8 and Lemma 2.8, noting that the matching cube $Q^{*}$ satisfies
$\ell(Q^{*})\geq\ell(Q)$.

To establish the second inequality, we may thus assume
$\textup{dist}(Q_{1},Q_{2})+\ell(Q_{2})<\lambda$. Then both cubes have
sidelength bounded by $\lambda$, so by the definition of ${T_{\lambda}}$ and
Lemma 2.8 we can write

$|({T_{\lambda}}f)_{Q_{1}}-({T_{\lambda}}f)_{Q_{2}}|=|f_{S_{1}}-f_{S_{2}}|$

for $S_{1},S_{2}\in E$ with
$\frac{1}{4}\leq\frac{\ell(S_{i})}{\ell(Q_{i})}\leq 4$ (possibly
$S_{i}=Q_{i}$) and $\textup{dist}(S_{i},Q_{i})\leq C_{\epsilon,n}\ell(Q_{i})$,
where $C_{\epsilon,n}=5\sqrt{n}+8n\cdot\epsilon^{-2}$. Thus

$\displaystyle\textup{dist}(S_{1},S_{2})$ $\displaystyle\leq$ $\displaystyle
C_{\epsilon,n}\ell(Q_{1})+\textup{diam}(Q_{1})+\textup{dist}(Q_{1},Q_{2})+\textup{diam}(Q_{2})+C_{\epsilon,n}\ell(Q_{2})$
$\displaystyle\leq$ $\displaystyle
2(C_{\epsilon,n}+\sqrt{n})\ell(Q_{2})+\textup{dist}(Q_{1},Q_{2})$
$\displaystyle<$ $\displaystyle 2(6\sqrt{n}+8n\cdot\epsilon^{-2})\lambda$
$\displaystyle\leq$ $\displaystyle
2(6\sqrt{n}+8n\cdot\epsilon^{-2})\frac{\epsilon^{2}\delta}{320n(1+\sqrt{n}\epsilon)}<\delta,$

and the result follows from Corollary 3.7. ∎

To prove that ${T_{\lambda}}f\in\textup{bmo}({\mathbb{R}}^{n})$, we will use
Lemma 3.5. Note that when it comes to estimating the oscillation of
${T_{\lambda}}f$ on a cube $Q$, if $Q\subset\Omega$ then we already control
the oscillation by the ${\textup{bmo}_{\lambda}(\Omega)}$ norm, while if $Q$
is contained inside a Whitney cube in $E^{\prime}$ then ${T_{\lambda}}f$ is
constant on $Q$ and the oscillation in zero. The following lemma (essentially
Lemma 2.11 in [13]) will help us to estimate the oscillation when we are not
in either of these cases.

######  Lemma 4.2.

Let $Q$ be a dyadic cube in ${\mathbb{R}}^{n}$ with $\ell(Q)<\delta$. Assume
$Q\not\subset\Omega$ and $Q$ is not contained inside any Whitney cube in
$E^{\prime}$. Then there exists a Whitney cube $Q_{0}\in E\cup E^{\prime}$
such that $Q_{0}\subset Q$, $|Q_{0}|\gtrsim\epsilon^{n}|Q|$ and

$\fint_{Q}|{T_{\lambda}}f(x)-({T_{\lambda}}f)_{Q_{0}}|\leq
C_{\epsilon}\|f\|_{{\textup{bmo}_{\lambda}(\Omega)}}.$

###### Proof.

We follow the proof of Lemma 2.11 in [13]. By Lemma 2.10, there is a cube
$Q_{0}\in E\cup E^{\prime}$ with $Q_{0}\supset Q$ or $Q\supset Q_{0}$ and
$\ell(Q_{0})\geq\frac{\epsilon}{160\sqrt{n}}\ell(Q)$. The assumptions on $Q$
rule out the first case, so we have $2^{-k}\ell(Q)\leq\ell(Q_{0})<\ell(Q)$,
where $k$ is the positive integer for which
$2^{-k}\leq\frac{\epsilon}{160\sqrt{n}}<2^{-k+1}$.

Partition $Q$ into $2^{kn}$ dyadic cubes $\\{Q^{1}_{j}\\}$ of sidelength
$2^{-k}\ell(Q)$. If there exists some Whitney cube $S^{1}_{j}\in E\cup
E^{\prime}$ with $Q^{1}_{j}\subset S^{1}_{j}$, we say that $Q^{1}_{j}$ belongs
to $F_{1}$. Since at least one of the $Q^{1}_{j}$ is contained in $Q_{0}$,
$F_{1}\neq\emptyset$, and therefore, denoting by $R_{1}$ the union of all
subcubes not in $F_{1}$, we have

$|R^{1}|=|Q|-\sum_{Q^{1}_{j}\in
F_{1}}|Q^{1}_{j}|\leq(1-2^{-kn})|Q|\leq(1-\epsilon_{1}^{n})|Q|,$

where we set $\epsilon_{1}:=\frac{\epsilon}{320\sqrt{n}}$.

Partitioning each of cubes $Q^{1}_{j}\not\in F_{1}$ further into $2^{kn}$
dyadic cubes of sidelength $2^{-k}\ell(Q^{1}_{j})=2^{-2k}\ell(Q)$, we denote
all of those cubes by $\\{Q^{2}_{j^{\prime}}\\}$. By Lemma 2.10 applied to
$Q^{1}_{j}\not\in F_{1}$, we again have that at least one of the subcubes
$Q^{2}_{j^{\prime}}$ is contained in a Whitney cube $S^{2}_{j^{\prime}}\in
E\cup E^{\prime}$, and furthermore $S^{2}_{j^{\prime}}\subset Q^{1}_{j}$. We
collect all the subcubes $Q^{2}_{j^{\prime}}$ which lie in Whitney cubes into
a collection $F_{2}$, and denote the union of the remaining subcubes by
$R_{2}$. Then again

$|R^{2}|\leq(1-\epsilon_{1}^{n})^{2}|Q|.$

We continue this process recursively. For each $N\in\mathbb{N}$, at the $N$th
stage we have

$Q=\bigcup_{i=1}^{N}\bigcup_{Q^{i}_{j}\in F_{i}}Q^{i}_{j}\cup R_{N},$

where the union is over a finite collection of pairwise disjoint cubes and the
remainder set satisfies

$|R_{N}|\leq(1-\epsilon_{1}^{n})^{N}|Q|.$ (24)

Letting $N\rightarrow 0$, we have that
$\chi_{Q}=\displaystyle{\sum_{i=1}^{\infty}\sum_{Q^{i}_{j}\in
F_{i}}\chi_{Q^{i}_{j}}}$ almost everywhere and by the monotone convergence
theorem

$\displaystyle\fint_{Q}|{T_{\lambda}}f(x)-({T_{\lambda}}f)_{Q_{0}}|dx$
$\displaystyle=$ $\displaystyle\sum_{i=1}^{\infty}\sum_{Q^{i}_{j}\in
F_{i}}\fint_{Q}|{T_{\lambda}}f(x)-({T_{\lambda}}f)_{Q_{0}}|dx$
$\displaystyle\leq$ $\displaystyle\sum_{i=1}^{\infty}\sum_{Q^{i}_{j}\in
F_{i}}\frac{|Q^{i}_{j}|}{|Q|}\left(\fint_{Q^{i}_{j}}|{T_{\lambda}}f(x)-({T_{\lambda}}f)_{Q^{i}_{j}}|dx+|({T_{\lambda}}f)_{Q^{i}_{j}}-({T_{\lambda}}f)_{Q_{0}}|\right).$

Since each $Q^{i}_{j}$ is contained in a Whitney cube in $E\cup E^{\prime}$,
either $Q^{i}_{j}\subset\Omega$ or ${T_{\lambda}}f$ is constant on
$Q^{i}_{j}$, so

$\fint_{Q^{i}_{j}}|{T_{\lambda}}f(x)-({T_{\lambda}}f)_{Q^{i}_{j}}|dx\leq\|f\|_{\textup{BMO}(\Omega)}\leq
2\|f\|_{{\textup{bmo}_{\lambda}(\Omega)}}.$ (25)

Moreover, by the selection of $Q^{i}_{j}$, we know that the Whitney cube
containing $Q^{i}_{j}$, denoted by $S^{i}_{j}$, must have $\ell(S^{i}_{j})\leq
2^{-k(i-1)}\ell(Q)=2^{k}\ell(Q^{i}_{j})$. If $S^{i}_{j}\in E^{\prime}$ then
${T_{\lambda}}f$ is constant on $S^{i}_{j}$ and
$({T_{\lambda}}f)_{S^{i}_{j}}=({T_{\lambda}}f)_{Q^{i}_{j}}$, while if
$S^{i}_{j}\in E$ then

$|({T_{\lambda}}f)_{Q^{i}_{j}}-({T_{\lambda}}f)_{S^{i}_{j}}|\leq\fint_{Q^{i}_{j}}|{T_{\lambda}}f(x)-({T_{\lambda}}f)_{S^{i}_{j}}|dx\leq
2^{kn}\fint_{S^{i}_{j}}|{T_{\lambda}}f(x)-({T_{\lambda}}f)_{S^{i}_{j}}|dx\leq
C_{\epsilon}\|f\|_{{\textup{bmo}_{\lambda}(\Omega)}}.$

Recalling that $Q_{0}\subset Q$ with $Q_{0}\in E\cap E^{\prime}$ and
$\ell(Q_{0})\geq 2^{-k}\ell(Q)\geq\ell(Q^{i}_{j})$, we apply Lemma 4.1 to
$S^{i}_{j}$ and $Q_{0}$ to get

$\displaystyle|({T_{\lambda}}f)_{Q^{i}_{j}}-({T_{\lambda}}f)_{Q_{0}}|$
$\displaystyle\leq$ $\displaystyle
C_{\epsilon}\|f\|_{{\textup{bmo}_{\lambda}(\Omega)}}+|({T_{\lambda}}f)_{S^{i}_{j}}-({T_{\lambda}}f)_{Q_{0}}|$
$\displaystyle\lesssim$
$\displaystyle\|f\|_{{\textup{bmo}_{\lambda}(\Omega)}}+\left(1+\log_{+}\left(\frac{\textup{dist}(S^{i}_{j},Q_{0})+\ell(Q_{0})}{\ell(S^{i}_{j})}\right)\right)$
$\displaystyle\lesssim$
$\displaystyle\|f\|_{{\textup{bmo}_{\lambda}(\Omega)}}+\log_{+}\left(\frac{\textup{diam}(Q)}{2^{-k(i-1)}\ell(Q)}\right)\leq
C_{\epsilon}\;i\|f\|_{{\textup{bmo}_{\lambda}(\Omega)}}$

Finally, combining the previous estimate with (25), noting that
$\sum_{j}|Q^{i}_{j}|\leq|R_{i-1}|$, and using (24), we have

$\fint_{Q}|{T_{\lambda}}f(x)-({T_{\lambda}}f)_{Q_{0}}|dx\lesssim\sum_{i=1}^{\infty}\sum_{Q^{i}_{j}\in
F_{i}}\frac{|Q^{i}_{j}|}{|Q|}i\|f\|_{{\textup{bmo}_{\lambda}(\Omega)}}\lesssim\sum_{i=1}^{\infty}(1-\epsilon_{1}^{n})^{i-1}i\|f\|_{{\textup{bmo}_{\lambda}(\Omega)}}\lesssim\|f\|_{{\textup{bmo}_{\lambda}(\Omega)}}$

with a constant depending on $\epsilon$. ∎

For $Q$ as in the Lemma, we can now conclude that
$\int_{Q}|{T_{\lambda}}f|\lesssim|({T_{\lambda}}f)_{Q^{1}}|+\|f\|_{\textup{bmo}_{\lambda}(\Omega)}<\infty$.
Thus

######  Corollary 4.3.

The function ${T_{\lambda}}f$ is locally integrable on ${\mathbb{R}}^{n}$. In
particular,
${\textup{bmo}_{\lambda}(\Omega)}\subset{L^{1}_{\text{loc}}}(\Omega)$.

Note that the conclusion is not immediate for an arbitrary domain, since in
the definition of ${\textup{bmo}_{\lambda}(\Omega)}$ by (3) we only assumed
$f$ was integrable on cubes $Q\subset\Omega$.

######  Lemma 4.4.

There are constants depending on $\epsilon$ such that

$a_{f}=\sup\limits_{\begin{subarray}{c}Q\in\mathfrak{D}({\mathbb{R}}^{n})\\\
\ell(Q)\leq\lambda\end{subarray}}\fint_{Q}|{T_{\lambda}}(x)-({T_{\lambda}}f)_{Q}|dx\lesssim\|f\|_{{\textup{bmo}_{\lambda}(\Omega)}},$
$b_{f}=\sup\limits_{\begin{subarray}{c}Q_{1},Q_{2}\in\mathfrak{D}({\mathbb{R}}^{n})\\\
\ell(Q_{1})=\ell(Q_{2})\\\ Q_{1},Q_{2}\text{ are
adjacent}\end{subarray}}|({T_{\lambda}}f)_{Q_{1}}-({T_{\lambda}}f)_{Q_{2}}|\leq\|f\|_{{\textup{bmo}_{\lambda}(\Omega)}}$

and

$c_{f}=\sup\limits_{\begin{subarray}{c}Q\in\mathfrak{D}({\mathbb{R}}^{n})\\\
\ell(Q)\geq\lambda/16\sqrt{n}\end{subarray}}|{T_{\lambda}}f|_{Q}\lesssim\|f\|_{{\textup{bmo}_{\lambda}(\Omega)}}.$

###### Proof.

Let us first consider a dyadic cube $Q\subset{\mathbb{R}}^{n}$ with
$\ell(Q)\leq\lambda$. As previously observed, if $Q\subset\Omega$ or $Q\subset
S\in E^{\prime}$ then, by definition, the oscillation of ${T_{\lambda}}$ on
$Q$ is bounded by $\|f\|_{\textup{bmo}_{\lambda}(\Omega)}$ or zero. Otherwise,
as $\ell(Q)<\delta$, we have a Whitney subcube $Q_{0}\in E\cup E^{\prime}$ for
which Lemma 4.2 gives

$\fint_{Q}|{T_{\lambda}}f(x)-({T_{\lambda}}f)_{Q}|dx\leq
2\fint_{Q}|{T_{\lambda}}f(x)-({T_{\lambda}}f)_{Q_{0}}|dx\lesssim\|f\|_{{\textup{bmo}_{\lambda}(\Omega)}}.$
(26)

If in addition $\ell(Q)\geq\frac{\lambda}{16\sqrt{n}}$, then we also have, by
applying Lemma 4.1 to $Q_{0}$,

$|Tf_{\lambda}|_{Q}\leq\fint_{Q}|{T_{\lambda}}f(x)-({T_{\lambda}}f)_{Q_{0}}|dx+|({T_{\lambda}}f)_{Q_{0}}|\lesssim(1+\log_{+}(16\sqrt{n}))\|f\|_{{\textup{bmo}_{\lambda}(\Omega)}}.$

If $Q\subset{\mathbb{R}}^{n}$ is dyadic with $\ell(Q)>\lambda$, then we can
partition $Q$ into $2^{k}$ equal dyadic subcubes ${Q_{i}}$ of sidelength
$2^{-k}\ell(Q)$, where $k\in\mathbb{N}$ is such that
$2^{-k}\leq\lambda/\ell(Q)<2^{1-k}$, and write the mean $|Tf_{\lambda}|_{Q}$
as the average of the means taken over the $Q_{i}$. Since
$\ell(Q_{i})\leq\lambda<\delta$, we can apply Lemma 4.2 to each of the
$Q_{i}$, and denote by $Q_{i,0}$ the corresponding Whitney cubes. Then
applying Lemma 4.1 to the $Q_{i,0}$, and using the fact that
$\ell(Q_{i})=2^{-k}\ell(Q)\geq\lambda/2$, we have

$\displaystyle|Tf_{\lambda}|_{Q}$ $\displaystyle\leq$
$\displaystyle\sum_{i}\frac{|Q_{i}|}{|Q|}\cdot\left(\fint_{Q_{i}}|{T_{\lambda}}f(x)-({T_{\lambda}}f)_{Q_{i,0}}|dx+|({T_{\lambda}}f)_{Q_{i,0}}|\right)$
$\displaystyle\lesssim$
$\displaystyle\|f\|_{{\textup{bmo}_{\lambda}(\Omega)}}+\left(1+\log_{+}\left(\frac{\lambda}{2^{-k}\ell(Q)}\right)\right)\|f\|_{{\textup{bmo}_{\lambda}(\Omega)}}\lesssim\|f\|_{{\textup{bmo}_{\lambda}(\Omega)}}.$

Finally, consider adjacent dyadic cubes $Q_{1},Q_{2}\subset{\mathbb{R}}^{n}$
of equal sidelength. If this sidelength is at least $\lambda$, then we can
apply the previous estimate to get

$|{T_{\lambda}}f_{Q_{1}}-({T_{\lambda}}f)_{Q_{2}}|\lesssim\|f\|_{{\textup{bmo}_{\lambda}(\Omega)}}.$

Hence, we assume that $\ell(Q_{1})=\ell(Q_{2})<\lambda$ and again, as
$\lambda<\delta$, apply Lemma 4.2 to get Whitney cubes $Q_{1,0}\subset Q_{1}$
and $Q_{2,0}\subset Q_{2}$ such that

$|({T_{\lambda}}f)_{Q_{i}}-({T_{\lambda}}f)_{Q_{i,0}}|\leq\fint_{Q_{i}}|{T_{\lambda}}f(x)-({T_{\lambda}}f)_{Q_{i,0}}|\lesssim\|f\|_{{\textup{bmo}_{\lambda}(\Omega)}},\quad
i=1,2.$

Moreover, by Lemma 4.1,

$|({T_{\lambda}}f)_{Q_{1,0}}-({T_{\lambda}}f)_{Q_{2,0}}|\lesssim\left(1+\log_{+}\left(\frac{\textup{dist}(Q_{1,0},Q_{2,0})}{\ell(Q_{1,0})}+1\right)\right)\|f\|_{{\textup{bmo}_{\lambda}(\Omega)}}\lesssim\|f\|_{{\textup{bmo}_{\lambda}(\Omega)}}$

since
$\textup{dist}(Q_{1,0},Q_{2,0})\leq\textup{dist}(Q_{1},Q_{2})+\textup{diam}(Q_{1})+\textup{diam}(Q_{2})=2\sqrt{n}\ell(Q_{1})$
and $\ell(Q_{1,0})\gtrsim\epsilon\ell(Q_{1})$. Hence,

$|({T_{\lambda}}f)_{Q_{1}}-({T_{\lambda}}f)_{Q_{2}}|\leq\sum_{i=1,2}|({T_{\lambda}}f)_{Q_{i}}-({T_{\lambda}}f)_{Q_{i,0}}|+|({T_{\lambda}}f)_{Q_{1,0}}-({T_{\lambda}}f)_{Q_{2,0}}|\lesssim\|f\|_{{\textup{bmo}_{\lambda}(\Omega)}}.$

∎

By Lemmas 3.5 and 4.4, we have that
${T_{\lambda}}f\in\textup{bmo}_{\lambda}({\mathbb{R}}^{n})=\textup{bmo}({\mathbb{R}}^{n})$.
Note that the constants in the above estimates depended on $\epsilon$ but not
on the choice of $\lambda$, as long as $\lambda\leq\lambda_{\epsilon,\delta}$.

##  5 Proof of implication (iv) $\implies$ (iii)

In this section we assume that $\Omega$ is a $\textup{bmo}_{\lambda}$
extension domain, or, specifically, the weaker assumption that there is a
bounded linear extension from ${\textup{bmo}_{\lambda}(\Omega)}$ to
$\textup{BMO}({\mathbb{R}}^{n})$. Our goal is to show that $\Omega$ satisfies
the strong local quasi-hyperbolic uniformity condition (iii): there exists
$\delta,c,d>0$, depending only on $\Omega$ and $\lambda$, such that for all
$x,y\in\Omega$ with $|x-y|<\delta$, we have

${k_{\Omega}}(z,w)\leq c{j_{\Omega}}(z,w)+d$ (27)

for all points $z,w$ on a quasi-hyperbolic geodesic connecting $x$ to $y$.

We start by noting that if

$|x-y|<\max({d_{\Omega}}(x),{d_{\Omega}}(y))/2,$ (28)

then for any $u$ on the line segment $[x,y]$,

$|x-y|\leq\max({d_{\Omega}}(x),{d_{\Omega}}(y))/2\leq{d_{\Omega}}(u).$

Therefore, for $z,w$ on the quasi-hyperbolic geodesic $\gamma(x,y)$

${k_{\Omega}}(z,w)\leq{k_{\Omega}}(x,y)\leq\int_{[x,y]}\frac{ds}{{d_{\Omega}}(u)}\leq\frac{2|x-y|}{\max({d_{\Omega}}(x),{d_{\Omega}}(y))}\leq
1.$

Hence for $x,y$ satisfying (28), (27) holds with $c=0$, $d=1$ and any
$\delta>0$. In particular, (28) holds whenever
$x\in\mathring{\Omega}_{\lambda/4}$ or $y\in\mathring{\Omega}_{\lambda/4}$ and
$|x-y|<\lambda/8$.

Thus we may restrict ourselves to $\delta<\lambda/8$ and $(x,y)$ in the set

$E_{\lambda}:=\left\\{(x,y)\in\Omega\times\Omega:\frac{\max({d_{\Omega}}(x),{d_{\Omega}}(y))}{2}\leq|x-y|<\frac{\lambda}{8}\right\\}\subset(\Omega\setminus\mathring{\Omega}_{\lambda/4})\times(\Omega\setminus\mathring{\Omega}_{\lambda/4})$

It is useful to introduce the following function on $\Omega\times\Omega$:

$\eta_{\lambda}(x,y):={k_{\Omega}}(x,\mathring{\Omega}_{\lambda/4})+{k_{\Omega}}(y,\mathring{\Omega}_{\lambda/4}).$

Assuming that $\Omega$ is a $\textup{bmo}_{\lambda}$ extension domain, we will
establish the following three facts regarding ${k_{\Omega}},{j_{\Omega}}$ and
$\eta_{\lambda}$

Claim 1: There exist $c,d>0$ such that for all $x,y\in\Omega$ with
${k_{\Omega}}(x,y)\leq\eta_{\lambda}(x,y)$, (27) holds for all
$z,w\in\gamma(x,y)$, where $\gamma(x,y)$ is a quasi-hyperbolic geodesic.

Claim 2: For each $C>0$ there exists $\rho>0$ such that all $x,y\in\Omega$
satisfying

${j_{\Omega}}(x,y)\leq
C,\quad\quad{d_{\Omega}}(x)<\rho\quad\text{and}\quad{d_{\Omega}}(y)<\rho,$

we have

${k_{\Omega}}(x,y)\leq\eta_{\lambda}(x,y).$

Claim 3: Let $x,y\in E_{\lambda}$ and
$\gamma_{1}(x,x^{\prime}),\gamma_{2}(y,y^{\prime})$ be quasi-hyperbolic
geodesics from $x$ and $y$ to $\mathring{\Omega}_{\lambda/4}$, as in Lemma
2.3. Then there exist points $\tilde{x}\in\gamma_{1}$ and
$\tilde{y}\in\gamma_{2}$ such that

${d_{\Omega}}(\tilde{x})={d_{\Omega}}(\tilde{y})=2|x-y|$ (29)

and

${j_{\Omega}}(\tilde{x},\tilde{y})\leq C,$ (30)

where $C$ depends on $\Omega$ and $\lambda$ only.

Assuming these claims we will complete the proof in the following subsection.
The remaining subsections are devoted to the proofs of Claims 1,2 and 3.

###  5.1 Proof of implication (iv) $\implies$ (iii) assuming Claims $1,2$ and
$3$

We want to show that there is a small enough $\delta>0$ so that $|x-y|<\delta$
implies ${k_{\Omega}}(x,y)\leq\eta(x,y)$ for all $(x,y)\in E_{\lambda}$. By
Claim 1 this is enough to complete the proof.

First of all, let us fix $C>0$ in the statement of Claim 3 whose value is
determined by $\Omega$ and $\lambda$. By Claim 2, there is a $\rho>0$
associated to this value $C$ so that the conclusion of Claim 2 holds. Set
$\delta=\rho/2$.

Then for each $(x,y)\in E_{\lambda}$ with $|x-y|<\delta$, consider the points
$\tilde{x},\tilde{y}$ on the geodesics from $x$ and $y$ to
$\mathring{\Omega}_{\lambda/4}$ satisfying (29) and (30). Note that in this
case ${d_{\Omega}}(\tilde{x})={d_{\Omega}}(\tilde{y})<\rho$ and
${j_{\Omega}}(\tilde{x},\tilde{y})\leq C$. Thus, evoking Claim 2, we get

${k_{\Omega}}(\tilde{x},\tilde{y})\leq\eta(\tilde{x},\tilde{y}).$

Finally, as $\tilde{x}$ and $\tilde{y}$ lie on the quasi-hyperbolic geodesics
to $\mathring{\Omega}_{\lambda/4}$, the last inequality implies

$\displaystyle{k_{\Omega}}(x,y)$ $\displaystyle\leq$
$\displaystyle{k_{\Omega}}(x,\tilde{x})+{k_{\Omega}}(\tilde{x},\tilde{y})+{k_{\Omega}}(\tilde{y},y)$
$\displaystyle\leq$
$\displaystyle{k_{\Omega}}(x,\tilde{x})+{k_{\Omega}}(\tilde{x},\mathring{\Omega}_{\lambda/4})+{k_{\Omega}}(\tilde{y},\mathring{\Omega}_{\lambda/4})+{k_{\Omega}}(\tilde{y},y)$
$\displaystyle=$
$\displaystyle{k_{\Omega}}(x,\mathring{\Omega}_{\lambda/4})+{k_{\Omega}}(y,\mathring{\Omega}_{\lambda/4})=\eta_{\lambda}(x,y).$

###  5.2 Proofs of Claims 1 and 2

The following lemma forms a basis for both claims.

######  Lemma 5.1.

Let domain $\Omega$ be such that any $f\in{\textup{bmo}_{\lambda}(\Omega)}$
can be extended to some $F\in\textup{BMO}({\mathbb{R}}^{n})$ with
$\|F\|_{\textup{BMO}({\mathbb{R}}^{n})}\leq
C\|f\|_{{\textup{bmo}_{\lambda}(\Omega)}}$. Then there exists $C^{\prime}$
depending only on $C$ and $n$ such that for all $z_{1},z_{2}\in\Omega$ and
$R_{1},R_{2}>0$ with

$R_{i}\leq\min({k_{\Omega}}(z_{i},\mathring{\Omega}_{\lambda/4}),{k_{\Omega}}(z_{1},z_{2})),\quad
i=1,2$

we have

$R_{1}+R_{2}\leq C^{\prime}({j_{\Omega}}(z_{1},z_{2})+1).$

###### Proof.

Put

$f_{1}(x)=\max(R_{1}-{k_{\Omega}}(z_{1},x),0),\qquad
f_{2}(x)=\max(R_{2}-{k_{\Omega}}(z_{2},x),0).$

Lemma 3.1 and the fact that truncation of a BMO function is still in BMO show
that both $f_{i}\in\textup{BMO}(\Omega)$. Moreover, as
$R_{i}\leq{k_{\Omega}}(z_{i},\mathring{\Omega}_{\lambda/4})$

$f_{i}=0\mbox{ on }\mathring{\Omega}_{\lambda/4},\ i=1,2.$

Let $f=f_{1}-f_{2}$. Thus by Lemma 3.3, $f\in{\textup{bmo}_{\lambda}(\Omega)}$
with $\|f\|_{\textup{bmo}_{\lambda}(\Omega)}<c$ for some $c$ depending on $n$
only.

Furthermore, $f_{1}(z_{2})=f_{2}(z_{1})=0$ as
$R_{i}\leq{k_{\Omega}}(z_{1},z_{2})$ for both $i=1,2$. Thus
$f(z_{1})-f(z_{2})=R_{1}+R_{2}$. Let $Q_{i}$ be the Whitney cubes containing
$z_{i}$. By Remark 3.2,

$R_{1}+R_{2}\leq|f_{Q_{1}}-f_{Q_{2}}|+\sup_{y\in
Q_{1}}|f(z_{1})-f(y)|+\sup_{y\in
Q_{2}}|f(z_{2})-f(y)|\leq|f_{Q_{1}}-f_{Q_{2}}|+2\sqrt{n}.$ (31)

It remains to estimate $|f_{Q_{1}}-f_{Q_{2}}|$. As
$f\in{\textup{bmo}_{\lambda}(\Omega)}$ and $\Omega$ is assumed to be an
extension domain, we may apply Lemma 2.1 in [13] to a
$\textup{BMO}({\mathbb{R}}^{n})$ extension of $f$ and obtain

$|f_{Q_{1}}-f_{Q_{2}}|\leq
C^{\prime\prime}d_{2}(Q_{1},Q_{2}):=C^{\prime\prime}\left(\left|\log\frac{l(Q_{1})}{l(Q_{2})}\right|+\log\left(2+\frac{\textrm{dist}(Q_{1},Q_{2})}{l(Q_{1})+l(Q_{2})}\right)\right),$

where $C^{\prime\prime}$ depends only on extension constant $C$ and $n$. By
the properties of Whitney cubes, the right-hand-side is bounded by a constant
times the following quantity, which is labeled by
$\overline{d}_{2}(x_{1},x_{2})$ in [13]:

$\left|\log\frac{{d_{\Omega}}(x_{1})}{{d_{\Omega}}(x_{2})}\right|+\log\left(2+\frac{|x_{1}-x_{2}|}{{d_{\Omega}}(x_{1})+{d_{\Omega}}(x_{2})}\right).$

Finally, since ${d_{\Omega}}$ is Lipschitz with constant $1$, we can bound
this quantity by a constant multiple of

${j_{\Omega}}(x_{1},x_{2})=\frac{1}{2}\log\left(1+\frac{|x_{1}-x_{2}|}{{d_{\Omega}}(x_{1})}\right)\left(1+\frac{|x_{1}-x_{2}|}{{d_{\Omega}}(x_{2})}\right).$

Combining the last three estimates with (31) gives
${k_{\Omega}}(x_{1},x_{2})\leq C^{\prime}{j_{\Omega}}(x_{1},x_{2})+2\sqrt{n}$.

∎

Now Claim 1 follows from the above lemma as

######  Corollary 5.2.

If $\Omega$ is a $\textup{bmo}_{\lambda}$ extension domain, then there exist
$c,d>0$ depending on $\Omega$ and $\lambda$ such that the following is true:
for all $x_{1},x_{2}\in\Omega$ with

${k_{\Omega}}(x_{1},x_{2})\leq\eta_{\lambda}(x_{1},x_{2}),$ (32)

we have

${k_{\Omega}}(z,w)\leq c{j_{\Omega}}(z,w)+d$

for all $z,w$ lying on a quasi-hyperbolic geodesic connecting $x_{1}$ and
$x_{2}$.

###### Proof.

First of all, note that if ${k_{\Omega}}(x_{1},x_{2})\leq\eta(x_{1},x_{2})$,
then for all $z,w$ lying on a quasi-hyperbolic geodesic connecting $x_{1}$ and
$x_{2}$ we have

${k_{\Omega}}(z,w)\leq\eta(z,w).$

This shows that it is enough to establish the fact for $z=x_{1}$ and
$w=x_{2}$.

There is nothing to prove when $\eta_{\lambda}(x_{1},x_{2})=0$, so we will
assume the contrary and set

$\theta:=\frac{{k_{\Omega}}(x_{2},\mathring{\Omega}_{\lambda/4})}{\eta_{\lambda}(x_{1},x_{2})}=\frac{{k_{\Omega}}(x_{2},\mathring{\Omega}_{\lambda/4})}{{k_{\Omega}}(x_{1},\mathring{\Omega}_{\lambda/4})+{k_{\Omega}}(x_{2},\mathring{\Omega}_{\lambda/4})}.$

Then put

$R_{1}=(1-\theta){k_{\Omega}}(x_{1},x_{2}),\qquad
R_{2}=\theta{k_{\Omega}}(x_{1},x_{2}).$

As ${k_{\Omega}}(x_{1},x_{2})\leq\eta_{\lambda}(x_{1},x_{2})$, the conditions
of Lemma 5.1 are satisfied and thus

${k_{\Omega}}(x_{1},x_{2})=R_{1}+R_{2}\leq c{j_{\Omega}}(x_{1},x_{2})+d.$

∎

######  Corollary 5.3.

Let $\Omega$ be a $\textup{bmo}_{\lambda}$ extension domain. Then for all
$x_{1},x_{2}\in\Omega$ such that

$\eta_{\lambda}(x_{1},x_{2})\leq{k_{\Omega}}(x_{1},x_{2})$

we have

$\eta_{\lambda}(x_{1},x_{2})\leq c{j_{\Omega}}(x_{1},x_{2})+d,$

where $c,d$ depend only on $\Omega$ and $\lambda$.

###### Proof.

As $\eta_{\lambda}(x_{1},x_{2})\leq{k_{\Omega}}(x_{1},x_{2})$, we can evoke
Lemma 5.1 with $R_{i}={k_{\Omega}}(x_{i},\mathring{\Omega}_{\lambda/4})$ to
obtain

$\eta_{\lambda}(x_{1},x_{2})=R_{1}+R_{2}\leq c{j_{\Omega}}+d.$

∎

Now we prove the contrapositive form of Claim 2.

######  Lemma 5.4.

Let $\Omega$ be a $\textup{bmo}_{\lambda}$ extension domain. Then for any
$C>0$ there exists $\rho=\rho(\Omega,\lambda,C)>0$ such that for all
$x_{1},x_{2}\in\Omega$ with

$\eta(x_{1},x_{2})\leq{k_{\Omega}}(x_{1},x_{2})\quad\text{ and
}\quad{j_{\Omega}}(x_{1},x_{2})<C$

we have

${d_{\Omega}}(x_{i})\geq\rho,\quad i=1,2.$

###### Proof.

If $x_{1},x_{2}$ satisfies the above conditions, Corollary 5.3 implies that
for $C^{\prime}=cC+d$ where $c,d$ depend on $\Omega$ and $\lambda$, we have

${k_{\Omega}}(x_{i},\mathring{\Omega}_{\lambda/4})\leq C^{\prime},\quad
i=1,2.$

By Lemma 2.3 there is $x^{\prime}_{i}$ in $\mathring{\Omega}_{\lambda/4}$ such
that
${k_{\Omega}}(x_{i},\mathring{\Omega}_{\lambda/4})={k_{\Omega}}(x_{i},x^{\prime}_{i})$.
Then by Lemma 2.1 in [10] we have

$C^{\prime}\geq{k_{\Omega}}(x_{i},\mathring{\Omega}_{\lambda/4})={k_{\Omega}}(x_{i},x^{\prime}_{i})\geq\ln\frac{{d_{\Omega}}(x^{\prime}_{i})}{{d_{\Omega}}(x_{i})}.$

Hence,

${d_{\Omega}}(x_{i})\geq{d_{\Omega}}(x^{\prime}_{i})e^{-C^{\prime}}\geq\lambda
e^{-C^{\prime}}/4=:\rho,\quad i=1,2.$

∎

###  5.3 Proof of Claim 3

By definition $E_{\lambda}$ contains only $(x,y)$ with
$\max({d_{\Omega}}(x),{d_{\Omega}}(y))\leq 2|x-y|\leq\lambda/4$, while
$x^{\prime},y^{\prime}\in\mathring{\Omega}_{\lambda/4}$ implies
$\min({d_{\Omega}}(x^{\prime}),{d_{\Omega}}(y^{\prime}))\geq\lambda/4$. As
${d_{\Omega}}$ is continuous there are points
$\tilde{x}\in\gamma_{1}(x,x^{\prime})$ and
$\tilde{y}\in\gamma_{1}(y,y^{\prime})$ such that
${d_{\Omega}}(\tilde{x})={d_{\Omega}}(\tilde{y})=2|x-y|$. We choose
$\tilde{x}$ and $\tilde{y}$ so that $s(\gamma_{1}(x,\tilde{x}))$ and
$s(\gamma_{2}(y,\tilde{y}))$ are minimal.

As (29) holds by the choice of $\tilde{x}$ and $\tilde{y}$, it remains to show
(30). Since by the triangle inequality

$|\tilde{x}-\tilde{y}|\leq|x-y|+|x-\tilde{x}|+|y-\tilde{y}|\leq|x-y|+s(\gamma_{1}(x,\tilde{x}))+s(\gamma_{2}(y,\tilde{y})),$

it suffices to show that for some constant $C>0$ depending on $\Omega$ and
$\lambda$ only, $s(\gamma_{1}(x,\tilde{x}))\leq C|x-y|$ and
$s(\gamma_{2}(y,\tilde{y}))\leq C|x-y|$. This will establish that

${j_{\Omega}}(\tilde{x},\tilde{y}):=\frac{1}{2}\log\left[\left(1+\frac{|\tilde{x}-\tilde{y}|}{{d_{\Omega}}(\tilde{x})}\right)\left(1+\frac{|\tilde{x}-\tilde{y}|}{{d_{\Omega}}(\tilde{y})}\right)\right]\leq\log\left(1+\frac{C|x-y|}{2|x-y|}\right)<C.$

By symmetry, it is enough to show only

$s(\gamma_{1}(x,\tilde{x}))\leq C|x-y|$

assuming ${d_{\Omega}}(x)<2|x-y|$ (otherwise $\tilde{x}=x$ and there is
nothing to prove).

Let $N\geq 0$ be the integer for which

$2^{N}<\frac{2|x-y|}{{d_{\Omega}}(x)}\leq 2^{N+1}.$

Next, we define a sequence of points $w_{0},w_{1},\dots w_{N+1}$ by setting
$w_{0}=x$, $w_{N+1}=\tilde{x}$ and, if $1\leq j\leq N$, $w_{j}$ to be the
first point along $\gamma_{1}(x,\tilde{x})$ for which
${d_{\Omega}}(w_{j})=2^{j}{d_{\Omega}}(x)$. Then by definition,

${d_{\Omega}}(w_{N})\leq{d_{\Omega}}(w_{N+1})\leq
2{d_{\Omega}}(w_{N})\quad\text{and}\quad{d_{\Omega}}(w_{j+1})=2{d_{\Omega}}(w_{j})\quad
j=0,\dots,N.$ (33)

Furthermore, for $j=0,\dots,N+1$

${k_{\Omega}}(w_{j-1},w_{j})=\int_{\gamma_{1}(w_{j-1},w_{j})}\frac{ds}{{d_{\Omega}}(z)}\geq\frac{s(\gamma_{1}(w_{j-1},w_{j}))}{\max_{z\in\gamma_{1}(w_{j-1},w_{j})}{d_{\Omega}}(z)}=\frac{s(\gamma_{1}(w_{j-1},w_{j}))}{{d_{\Omega}}(w_{j})}.$
(34)

As $\gamma_{1}(x,x^{\prime})$ is a quasi-hyperbolic geodesic, we have

${k_{\Omega}}(w_{j-1},w_{j})\leq{k_{\Omega}}(w_{j-1},\mathring{\Omega}_{\lambda/4})\leq\eta_{\lambda}(w_{j-1},w_{j}),\qquad
j=1,\dots,N+1.$

This shows that we can apply Corollary 5.2 to $w_{j-1},w_{j}$ to obtain

${k_{\Omega}}(w_{j-1},w_{j})\leq c{j_{\Omega}}(w_{j-1},w_{j})+d,$

where $c,d$ depend on $\Omega$ and $\lambda$. By (33) this gives

${k_{\Omega}}(w_{j-1},w_{j})\leq
c\log\left(1+\frac{|w_{j-1}-w_{j}|}{{d_{\Omega}}(w_{j})}\right)+d+1.$

Combining it with (34) we get

$\frac{s(\gamma_{1}(w_{j-1},w_{j}))}{{d_{\Omega}}(w_{j})}\leq
C^{\prime}\left(\log\left(1+\frac{s(\gamma_{1}(w_{j-1},w_{j}))}{{d_{\Omega}}(w_{j})}\right)+1\right)$

with $C^{\prime}=\max(c,d+1)$. But $u(\log(1+u)+1))^{-1}\rightarrow\infty$ as
$u\rightarrow\infty$, which means

$\frac{s(\gamma_{1}(w_{j-1},w_{j}))}{{d_{\Omega}}(w_{j})}\leq M$

for some constant $M$ depending on $C^{\prime}$, and hence only on $\Omega$
and $\lambda$. This gives our desired inequality

$s(\gamma_{1}(x,\tilde{x}))=\sum_{j=1}^{N+1}s(\gamma_{1}(w_{j-1},w_{j}))\leq
M\sum_{j=1}^{N+1}{d_{\Omega}}(w_{j})=M\left(\sum_{j=0}^{N}{d_{\Omega}}(w_{j})+{d_{\Omega}}(\tilde{x})\right)\leq$
$\leq M(2^{N+1}{d_{\Omega}}(x)+{d_{\Omega}}(\tilde{x}))\leq C|x-y|,$

where $C=6M$.

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