# Local vanishing mean oscillation

Almaz Butaev  (A.B.) Department of Mathematical Sciences, P.O. Box 210025,
University of Cincinnati, Cincinnati, OH 45221–0025, U.S.A
<EMAIL_ADDRESS>and  Galia Dafni  (G.D.) Concordia University,
Department of Mathematics and Statistics, Montréal, QC H3G 1M8, Canada
<EMAIL_ADDRESS>
###### Abstract.

We consider various notions of vanishing mean oscillation on a (possibly
unbounded) domain $\Omega\subset{{\mathbb{R}}^{n}}$, and prove an analogue of
Sarason’s theorem, giving sufficient conditions for the density of bounded
Lipschitz functions in the nonhomogeneous space ${\rm vmo}(\Omega)$. We also
study ${\rm cmo}(\Omega)$, the closure in ${{\rm bmo}(\Omega)}$ of the
continuous functions with compact support in $\Omega$. Using these
approximation results, we prove that there is a bounded extension from ${\rm
vmo}(\Omega)$ and ${\rm cmo}(\Omega)$ to the corresponding spaces on
${{\mathbb{R}}^{n}}$, if and only if $\Omega$ is a locally uniform domain.

###### Key words and phrases:

vanishing mean oscillation, Lipschitz functions, approximation, extension
domain, $\ed$-domain, locally uniform domain

###### 2020 Mathematics Subject Classification:

42B35, 46E35, 41A30

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

##  1\. Introduction

In the theory of function spaces, it is useful to define “local” versions of
certain spaces. For function spaces on ${{\mathbb{R}}^{n}}$, this can refer to
functions defined globally which belong to a certain space when restricted to
compact sets (such as ${L^{p}_{\rm loc}}({{\mathbb{R}}^{n}})$) or
alternatively to spaces defined on a given domain
$\Omega\subset{{\mathbb{R}}^{n}}$. In the latter case one can then use “local”
to refer to behavior away from the boundary ${\partial\Omega}$. When
considering behavior up to the boundary, the notions of “vanishing” at the
boundary, approximation by smooth functions, and extension to
${{\mathbb{R}}^{n}}$ are interrelated and intimately connected with the
geometry of the domain (for example, in the case of Sobolev spaces and
Triebel-Lizorkin spaces, see [5, 15, 16, 17, 20, 23]).

In the case of functions of bounded mean oscillation, there are various
notions of “local” and “vanishing” in the literature. The original definition
of bounded mean oscillation by John and Nirenberg [13] was on a fixed cube
$Q_{0}\subset{{\mathbb{R}}^{n}}$: $f\in{\rm BMO}(Q_{0})$ if

$\sup\limits_{Q\subset Q_{0}}\fint_{Q}|f(x)-f_{Q}|dx<\infty,$

where the supremum is over all parallel subcubes of $Q_{0}$, $|Q|$ denotes
Lebesgue measure, and $f_{Q}:=\fint_{Q}f:=|Q|^{-1}\int_{Q}f$ is the average of
$f$ on $Q$. This can be extended to give a definition of ${\rm BMO}(\Omega)$
on any domain (or even open set) $\Omega\subset{{\mathbb{R}}^{n}}$ by taking
the supremum over all cubes $Q\subset\Omega$ with sides parallel to the axes.
When $\Omega$ is connected, this supremum defines a norm modulo constants, and
${\rm BMO}(\Omega)$ is a Banach space.

A more refined measure of the mean oscillation of a function $f\in{L^{1}_{\rm
loc}}({{\mathbb{R}}^{n}})$ is given by the modulus of mean oscillation

(1) $\omega(f,t):=\sup\limits_{\begin{subarray}{c}\ell(Q)<t\\\
Q\subset{{\mathbb{R}}^{n}}\end{subarray}}\fint_{Q}|f(x)-f_{Q}|dx,\quad t>0,$

where $\ell(Q)$ denotes the sidelength of $Q$. We then have $\|f\|_{{\rm
BMO}({{\mathbb{R}}^{n}})}:=\sup_{t>0}\omega(f,t)$. A local version of BMO can
be defined by fixing a finite $T>0$ and considering locally integrable
functions with $\omega(f,T)<\infty$. The set of such functions does not depend
on the choice of $T$ and is strictly larger than ${\rm
BMO}({{\mathbb{R}}^{n}})$ since it contains, for example, all uniformly
continuous functions.

Another version of BMO which is called “local” is the space ${\rm
bmo}({{\mathbb{R}}^{n}})$ (not to be confused with what is known as “little”
BMO and has the same notation), introduced by Goldberg [12] as the dual of the
local Hardy space $h^{1}({{\mathbb{R}}^{n}})$ and consisting of locally
integrable functions $f$ satisfying

$\|f\|_{{\rm bmo}({{\mathbb{R}}^{n}})}:=\omega(f,1)+\sup_{\ell(Q)\geq
1}|f|_{Q}<\infty,$

where once more the supremum is taken over all cubes
$Q\subset{{\mathbb{R}}^{n}}$ with sides parallel to the axes. Here again the
scale $1$ can be replaced by any finite $T$ without changing the collection of
functions in ${\rm bmo}({{\mathbb{R}}^{n}})$, only affecting the norm, and we
can also restrict the second term to the supremum of the averages of $|f|$ on
cubes whose sidelength is exactly equal to $1$. As sets of functions, ${\rm
bmo}({{\mathbb{R}}^{n}})$ is strictly smaller than ${\rm
BMO}({{\mathbb{R}}^{n}})$ (it does not contain $\log|x|$, for example) and
should be considered as a nonhomogeneous version of BMO, not taken modulo
constants. Both $h^{1}$ and ${\rm bmo}$ are part of the scale of
nonhomogeneous Triebel-Lizorkin spaces - see [28, Theorem 1.7.1].

The notion of vanishing mean oscillation was introduced by Sarason [21]. The
space ${\rm VMO}({{\mathbb{R}}^{n}})$ can be defined using either one of the
two characterizations in the following theorem, which was proved in [21] for
the case $n=1$.

######  Theorem A (Sarason).

For $f\in{\rm BMO}({{\mathbb{R}}^{n}})$,

(2) $\displaystyle{\lim_{t\rightarrow 0^{+}}\omega(f,t)=0}$

if and only if $f\in\overline{{\rm UC}({{\mathbb{R}}^{n}})\cap{\rm
BMO}({{\mathbb{R}}^{n}})}$, the closure of the uniformly continuous functions
in ${\rm BMO}$.

A smaller space which is sometimes also called ${\rm VMO}({{\mathbb{R}}^{n}})$
(see [10]), and serves as a predual to the Hardy space
$H^{1}({{\mathbb{R}}^{n}})$, is the closure in ${\rm BMO}({{\mathbb{R}}^{n}})$
of the continuous functions with compact support (or equivalently the
${C^{\infty}}$ functions with compact support). We will denote this space by
${\rm CMO}({{\mathbb{R}}^{n}})$ for “continuous mean oscillation”, following
Neri [19]. As stated in [19] and proved by Uchiyama in [29], in addition to
(2), functions in ${\rm CMO}({{\mathbb{R}}^{n}})$ also satisfy vanishing mean
oscillation conditions as the size of the cube increases to $\infty$ and as
the cube itself goes to $\infty$. Recently subspaces between ${\rm
CMO}({{\mathbb{R}}^{n}})$ and ${\rm VMO}({{\mathbb{R}}^{n}})$ were considered
in [26, 27].

The nonhomogeneous versions of the spaces ${\rm VMO}({{\mathbb{R}}^{n}})$ and
${\rm CMO}({{\mathbb{R}}^{n}})$, denoted ${\rm vmo}({{\mathbb{R}}^{n}})$ and
${\rm cmo}({{\mathbb{R}}^{n}})$, are the corresponding subspaces of ${\rm
bmo}({{\mathbb{R}}^{n}})$, and the vanishing mean oscillation conditions
characterizing the latter were given in [2, 11] (see Proposition 1).
Bourdaud’s paper [2] contains extensive coverage of the properties of ${\rm
BMO}({{\mathbb{R}}^{n}})$ and ${\rm bmo}({{\mathbb{R}}^{n}})$ (treating it
modulo constants as a subspace of ${\rm BMO}$) as well as their vanishing
subspaces.

The focus of our work are the versions of these spaces on a domain
$\Omega\subset{{\mathbb{R}}^{n}}$, and the corresponding approximation and
extension results. The definition of the modulus of oscillation can be adapted
by restricting the cubes to lie inside the domain, namely

(3) ${\omega_{\Omega}}(f,t):=\sup\limits_{\begin{subarray}{c}\ell(Q)<t\\\
Q\subset\Omega\end{subarray}}\fint_{Q}|f(x)-f_{Q}|dx,\quad t>0,$

and ${\rm BMO}(\Omega)$ defined to consist of those $f\in{L^{1}_{\rm
loc}}(\Omega)$ with $\sup_{t>0}{\omega_{\Omega}}(f,t)<\infty$. The question of
the definition of ${\rm VMO}(\Omega)$ is more delicate: for which domains does
a version of Sarason’s theorem hold? For a bounded domain $\Omega$, Brezis and
Nirenberg [4] give many results on ${\rm VMO}(\Omega)$, including a strong
version of Sarason’s result, which they attribute to Jones, not only
identifying the subspace of ${\rm BMO}(\Omega)$ consisting of functions with
vanishing mean oscillation with the closure of the uniformly continuous
functions, but also with the closure of the continuous, or smooth, functions
with compact support in $\Omega$.

Jones [14] showed that there is a bounded linear extension from ${\rm
BMO}(\Omega)$ to ${\rm BMO}({{\mathbb{R}}^{n}})$ if and only if $\Omega$ is a
uniform domain. In [7], we composed Jones’ extension operator with an
averaging operator on the complement of ${\overline{\Omega}}$ to obtain an
operator simultaneously extending ${\rm BMO}$, ${\rm VMO}$ and ${\rm CMO}$, as
well as Lipschitz functions, on a uniform domain $\Omega$, and also
characterized such domains in terms of the existence of a bounded extension
from ${\rm CMO}(\Omega)$ to ${\rm BMO}({{\mathbb{R}}^{n}})$. As a corollary,
we obtained a version of Sarason’s theorem for these domains.

Turning to the nonhomogeneous case, in [8] we proved a version of Jones’
theorem for ${\rm bmo}$, identifying the extension domains with locally
uniform domains, which we in turn showed are equivalent to the $\ed$-domains
used in Jones’ extension results for Sobolev spaces in [17]. To define ${\rm
bmo}$ on $\Omega$ we need to fix a scale $\lambda$. We say $f\in{{\rm
bmo}_{\lambda}(\Omega)}$ if $f$ is integrable on every cube $Q\subset\Omega$
and

(4) $\|f\|_{{\rm
bmo}_{\lambda}(\Omega)}:={\omega_{\Omega}}(f,\lambda)+\sup_{Q\subset\Omega,\ell(Q)\geq\lambda}|f|_{Q}<\infty.$

Due to the extension theorem in [8], if $\Omega$ is an $\ed$-domain, there is
a natural scale $\lambda_{\epsilon,\delta}$ such that membership in ${{\rm
bmo}_{\lambda}(\Omega)}$ is independent of $\lambda$ provided
$\lambda\leq\lambda_{\epsilon,\delta}$, and we can define ${{\rm
bmo}(\Omega)}$ to be ${\rm bmo}_{\lambda_{\epsilon,\delta}}(\Omega)$. As
pointed out above, in the case $\Omega={{\mathbb{R}}^{n}}$, one can take
$\lambda$ to be any finite positive number.

In the special case when $\Omega$ is a bounded domain, ${\rm BMO}(\Omega)$ and
${{\rm bmo}_{\lambda}(\Omega)}$ coincide for $\lambda$ sufficiently small,
provided we consider them both modulo constants or fix the average on some
large cube in $\Omega$ to be zero, say. In such a case Jones’ extension in
[14] will vanish on all cubes sufficiently far away from $\Omega$. In [6], we
gave an extension operator for ${\rm VMO}(\Omega)$ on a bounded uniform domain
which does not use averaging and preserves the property of the Jones extension
that the values of the extended function on a Whitney cube of
${{\mathbb{R}}^{n}}\setminus{\overline{\Omega}}$ are completely determined by
the average of the original function on a matching cube inside $\Omega$.

Turning to unbounded domains, in the present paper we prove an extension
theorem for the vanishing mean oscillation subspaces of ${\rm bmo}(\Omega)$,
or equivalently a nonhomogeneous version of the VMO extension theorem in [7].
Unlike in [7], where we used the extension result to prove the analogue of
Sarason’s theorem, here we first prove the approximation theorem and then
obtain the extension as a corollary. It is natural to state the approximation
in terms of Lipschitz functions since, as shown in [7], the averaging process
used by Sarason yields this strong form of uniform continuity, and it is used
for both the approximation and the extension arguments. As in the setting of
metric measure spaces, Lipschitz functions are the “smooth” functions in this
context.

######  Theorem 1.

Suppose $\Omega\subset{{\mathbb{R}}^{n}}$ is a domain and $\lambda$ is such
that for all $0<\lambda^{\prime}\leq\lambda$, ${\rm
bmo}_{\lambda^{\prime}}(\Omega)={{\rm bmo}_{\lambda}(\Omega)}$ as sets. Then

$\\{f\in{{\rm bmo}_{\lambda}(\Omega)}:\lim_{t\rightarrow
0^{+}}{\omega_{\Omega}}(f,t)=0\\}=\overline{{\rm
UC}(\Omega)\cap{L^{\infty}}(\Omega)}=\overline{{\rm
Lip}_{b}(\Omega)}=\overline{{\rm Lip}_{b,0}(\Omega)},$

where the closures are in the ${{\rm bmo}_{\lambda}(\Omega)}$ norm, ${\rm
Lip}_{b}(\Omega)$ denotes the bounded Lipschitz functions in $\Omega$,
equipped with the nonhomogeneous norm

$\|f\|_{{\rm Lip}_{b}}:=\|f\|_{\infty}+\sup_{x\neq
y}\frac{|f(x)-f(y)|}{|x-y|},$

and ${\rm Lip}_{b,0}(\Omega)$ consists of those $f\in{\rm Lip}_{b}(\Omega)$
with ${\rm dist}({\rm supp}(f),{\partial\Omega})>0$.

As pointed out above, for an $\ed$-domain $\Omega$, the hypothesis of the
theorem holds for $\lambda=\lambda_{\epsilon,\delta}$, and as a result we can
not only unambiguously define ${{\rm bmo}(\Omega)}$ but also ${\rm
vmo}(\Omega)$. We define ${\rm cmo}(\Omega)$ to be the closure in ${{\rm
bmo}(\Omega)}$ of $C_{c}(\Omega)$, the continuous functions with compact
support in $\Omega$. This can be identified (see Proposition 4) with the space
of functions in ${{\rm vmo}(\Omega)}$ which vanish at infinity. Note that both
${\rm Lip}_{b}(\Omega)$ and $C_{c}(\Omega)$ are continuously embedded in
${{\rm bmo}_{\lambda}(\Omega)}$ since $\|f\|_{{\rm
bmo}_{\lambda}(\Omega)}\leq\|f\|_{\infty}$.

Using these definitions, we can state our extension result.

######  Theorem 2.

Let $\Omega\subset{{\mathbb{R}}^{n}}$ be an $\ed$-domain. Then there exists a
linear extension operator $T$ such that

  * (i)

$T:{{\rm bmo}(\Omega)}\to{\rm bmo}({{\mathbb{R}}^{n}})$ is bounded;

  * (ii)

$T:{\rm vmo}(\Omega)\to{\rm vmo}({{\mathbb{R}}^{n}})$ is bounded;

  * (iii)

$T:{\rm cmo}(\Omega)\to{\rm cmo}({{\mathbb{R}}^{n}})$ is bounded;

  * (iv)

$T:{\rm Lip}_{b}(\Omega)\to{\rm Lip}_{b}({{\mathbb{R}}^{n}})$ is bounded.

Boundedness in (i)-(iv) refers to the ${\rm bmo}$ norm while in (v) the
boundedness is with respect to the norm $\|f\|_{{\rm Lip}_{b}}$.

Note that while Theorem 1 holds under weaker assumptions on the domain (see
Section 5 for examples which are not $\ed$ domains), it is not possible to
weaken the assumptions in Theorem 2, as the converse holds. This follows from
the results in [8] by observing that the functions used in the proof of
Theorem 3.1 there, constructed from the quasihyperbolic metric, are continuous
with compact support in $\Omega$. Thus we can state the converse under the
weakest hypotheses.

######  Theorem 3 ([8]).

If $\Omega\subset{{\mathbb{R}}^{n}}$ is a domain and for some $\lambda>0$
there is an extension operator $T:C_{c}(\Omega)\to{\rm
BMO}({{\mathbb{R}}^{n}})$ such that for some $C$ and all $f\in C_{c}(\Omega)$,

$\|Tf\|_{{\rm BMO}({{\mathbb{R}}^{n}})}\leq C\|f\|_{{\rm
bmo}_{\lambda}(\Omega)},$

then $\Omega$ is an $\ed$-domain.

We start, in Section 2, with some results on approximation by smooth and
compactly supported functions in ${\rm bmo}({{\mathbb{R}}^{n}})$, and clarify
the notion of “vanishing at infinity” in this context, answering a question
posed by Bourdaud in [2]. Section 3 addresses the same questions on a domain
$\Omega$, culminating in the proof of Theorem 1 in Subsection 3.2. The proof
of Theorem 2 can be found in Section 4. Finally, Section 5 provides some
examples and counterexamples to illustrate the results.

##  2\. Approximation and vanishing mean oscillation in ${{\mathbb{R}}^{n}}$

The approximation of functions in ${\rm VMO}({{\mathbb{R}}^{n}})$ by Lipschitz
functions can be proved by the same technique use by Sarason on
${\mathbb{R}}$, namely first taking the averages of the function on a
sufficiently fine grid of cubes, and then smoothing out the resulting step
function by convolution with a mollifier. Even when convolving with the
normalized characteristic function of a ball, this produces a Lipschitz
function (see [7]). Averaging functions of vanishing mean oscillation in order
to get smooth functions is a standard technique, and can be used to show that
if the modulus of oscillation is sufficiently rapidly decreasing, the
functions themselves are smooth, as originally shown by Campanato [9], Meyers
[18] and Spanne [24] (see also [22]). For ${\rm BMO}$ functions without
vanishing mean oscillation, on a bounded domain, averaging results in
Lipschitz up to a log factor, as shown in [3, Lemma B.9].

For $f\in{\rm bmo}({{\mathbb{R}}^{n}})$, the functions resulting from this
process, for a given grid size, are bounded, as we can control the average of
$f$ on a cube $Q$ with $\ell(Q)<1$ by taking a chain $Q=Q_{0}\subset
Q_{1}\subset\ldots Q_{k}$ with $\ell(Q_{i})=2\ell(Q_{i-1})$, $1\leq i\leq
k-1$, and $1=\ell(Q_{k})\leq 2\ell(Q_{k-1})<2$, and using the standard
estimate

(5) $|f|_{Q}\leq\sum_{i=1}^{k}||f|_{Q_{i-1}}-|f|_{Q_{i}}|+|f|_{Q_{k}}\leq
2^{n+1}k\sup_{1\leq i\leq
k}\fint_{Q_{i}}|f-f_{Q_{i}}|+|f|_{Q_{k}}\lesssim\log\Big{(}\frac{2}{\ell(Q)}\Big{)}\|f\|_{\rm
bmo}.$

Here and throughout the paper we use $a\lesssim b$ to denote the existence of
a constant $C$ (usually only depending on the dimension) such that $a\leq Cb$.

This gives us the result of Bourdaud [2, Théorème 1] identifying ${\rm
vmo}({{\mathbb{R}}^{n}})$ with the closure of the bounded uniformly continuous
functions in ${\rm bmo}({{\mathbb{R}}^{n}})$.

######  Theorem B ([2]).

A function $f\in{\rm bmo}({{\mathbb{R}}^{n}})$ satisfies (2) if and only if it
can be approximated in the ${\rm bmo}$ norm by bounded uniformly continuous
functions.

Bourdaud also proves (see [2, Théorème 4]) that functions in ${\rm
vmo}({{\mathbb{R}}^{n}})$ can be approximated by $C^{\infty}$ functions, but
such functions do not have uniform (on ${{\mathbb{R}}^{n}}$) bounds on their
derivatives. What is possible, as noted above, is to strengthen uniform
continuity to Lipschitz continuity.

One can get more smoothness when there is vanishing at infinity. The following
characterization of ${\rm cmo}({{\mathbb{R}}^{n}})$ captured this notion in
the ${\rm bmo}$ sense.

######  Proposition 1 ([2, 11]).

For $f\in{\rm bmo}({{\mathbb{R}}^{n}})$, the following are equivalent:

  1. (1)

$f\in{\rm cmo}({{\mathbb{R}}^{n}})$, the closure in ${\rm
bmo}({{\mathbb{R}}^{n}})$ of $C_{c}({{\mathbb{R}}^{n}})$;

  2. (2)

$f$ satisfies (2) together with
$\displaystyle{\lim_{\beta\rightarrow\infty}\sup\\{|f|_{Q}:{\rm
dist}(Q,0)>\beta,\ell(Q)\geq 1\\}=0}$;

  3. (3)

$f$ satisfies (2) together with

$\limsup_{x\rightarrow\infty}\sup\left\\{\fint_{Q}|f-f_{Q}|:\mbox{center of
}Q=x,\ell(Q)\leq 1\right\\}=0=\limsup_{x\rightarrow\infty}|f|_{Q_{0}+x},$

where $Q_{0}=[0,1]^{n}$.

The equivalence of the first two conditions was proved by the second author in
[11], independently of the work of Bourdaud [2], who proved the equivalence of
the first and third condition. Bourdaud defines ${\rm
cmo}({{\mathbb{R}}^{n}})$ as the closure in ${\rm bmo}({{\mathbb{R}}^{n}})$ of
${\mathcal{D}}({{\mathbb{R}}^{n}})$, the set of smooth functions with compact
support, so the identifications above include the approximation by Lipschitz
functions of compact support. Note that the fact that the vanishing at
infinity of the averages over large cubes in the second condition is
sufficient to give the vanishing at infinity of the oscillation on small cubes
in the third condition can be seen by combining Sarason’s VMO condition (2)
with the standard log estimate (5), but may fail when not working on all of
${{\mathbb{R}}^{n}}$ (see Example 2).

Bourdaud, in the same paper [2, p. 1217 (1)], poses as a question for further
study the characterization of the notion of vanishing at infinity for
functions in ${\rm bmo}({{\mathbb{R}}^{n}})$. We answer this question by
giving an analogue of Proposition 1 for functions which do not necessarily
satisfy the VMO condition (2) .

######  Definition 1.

We denote by ${{\rm bmo}_{c}}({{\mathbb{R}}^{n}})$ the subspace of functions
$f$ in ${\rm bmo}({{\mathbb{R}}^{n}})$ which have compact support. We say
$f\in{\rm bmo}({{\mathbb{R}}^{n}})$ vanishes at infinity if

(6) $\lim_{R\rightarrow\infty}\|f\|_{{\rm
bmo}({{\mathbb{R}}^{n}}\setminus\overline{B(0,R)})}=0,$

where the ${\rm bmo}$ norm is taken here in the sense of Definition 4 with
$\lambda=1$.

######  Proposition 2.

For $f\in{\rm bmo}({{\mathbb{R}}^{n}})$, the following are equivalent:

  1. (i)

$f$ is in the closure of ${{\rm bmo}_{c}}({{\mathbb{R}}^{n}})$ in ${\rm
bmo}({{\mathbb{R}}^{n}})$;

  2. (ii)

$f$ vanishes at infinity;

  3. (iii)

$\displaystyle{\lim_{\beta\rightarrow\infty}\gamma(f,\beta)=0}$, where

$\gamma(f,\beta):=\sup_{{\rm
dist}(Q,0)>\beta,\ell(Q)<1}\fint_{Q}|f-f_{Q}|+\sup_{{\rm
dist}(Q,0)>\beta,\ell(Q)=1}|f|_{Q};$

and

  4. (iv)

$\lim_{\beta\rightarrow\infty}\left(\sup_{{\rm
dist}(Q,0)>\beta,\ell(Q)<1}\fint_{Q}|f-f_{Q}|+\sup_{{\rm
dist}(Q,0)>\beta,\ell(Q)\leq 1}|f|_{Q}\ell(Q)\right)=0.$

###### Proof.

Condition (6) is satisfied for any function of compact support, and is
preserved when taking limits in the $\|\cdot\|_{{\rm
bmo}({{\mathbb{R}}^{n}})}$ norm, so (i) $\implies$ (ii). Moreover,
$\gamma(f,\beta)\leq\|f\|_{{\rm
bmo}({{\mathbb{R}}^{n}}\setminus\overline{B(0,\beta)})}$ so (ii) $\implies$
(iii).

To show (iii) $\implies$ (iv), it suffices to bound $|f|_{Q}\ell(Q)$ for a
cube $Q$ with ${\rm dist}(Q,0)>\beta$, and $\ell(Q)<1$. Proceeding as in (5)
with $Q\subset Q_{k}\subset{{\mathbb{R}}^{n}}\setminus\overline{B(0,\beta)}$
and $\ell(Q_{k})=1$, we have

$|f|_{Q}\lesssim\log(2/\ell(Q))\;\gamma(f,\beta)\lesssim\ell(Q)^{-1}\gamma(f,\beta).$

This shows the quantity in parenthesis in (iv) is controlled by
$\gamma(f,\beta)$.

We will now complete the circle of equivalences by proving that (iv)
$\implies$ (i). Assume $f$ satisfies condition (iv) (and hence also (iii)).
First note that if $P$ is a cube whose sidelength is greater than $1$, then
taking a cube $P^{\prime}\supset P$ with $\ell(P^{\prime})$ an integer and
$\frac{\ell(P^{\prime})}{\ell(P)}<2$, and writing $P^{\prime}$ as the union of
cubes $Q_{i}$ of sidelength $1$ and pairwise disjoint interiors, we have

$\displaystyle\fint_{P}|f|$ $\displaystyle\leq$
$\displaystyle\frac{1}{|P|}\left(\sum_{{\rm
dist}(Q_{i},0)\leq\beta}\int_{P\cap Q_{i}}|f|+\sum_{{\rm
dist}(Q_{i},0)>\beta}|Q_{i}||f|_{Q_{i}}\right)$ $\displaystyle\leq$
$\displaystyle\frac{1}{|P|}\int_{P\cap B(0,\beta+\sqrt{n})}|f|+2^{n}\sup_{{\rm
dist}(Q,0)>\beta,\ell(Q)=1}|f|_{Q}.$

Thus (iii) implies the seemingly stronger vanishing condition for averages
over large cubes,

(7) $\lim_{\beta\rightarrow\infty}\left[\sup_{{\rm
dist}(Q,0)>\beta,\ell(Q)\geq 1}|f|_{Q}\
+\sup_{\ell(Q)\geq\beta}|f|_{Q}\right]=0,$

and in particular (iii) $\implies$ (ii).

Fix $k\in{\mathbb{N}}$ and let $f_{k}=\psi_{k}f$, where $\psi_{k}$ is a
Lipschitz function which is equal to $1$ on $B(0,k)$ and to $0$ outside
$B(0,2k)$, with $0\leq\psi_{k}\leq 1$ and $\|\psi_{k}\|_{\rm Lip}\leq k^{-1}$.
Since the support of $f_{k}$ is compact, we just need to show
$\|f-f_{k}\|_{{\rm bmo}({{\mathbb{R}}^{n}})}\rightarrow 0$ as
$k\rightarrow\infty$.

Write $g_{k}=1-\psi_{k}$ so $f-f_{k}=fg_{k}$. For any cube $Q$,
$|fg_{k}|_{Q}\leq|f|_{Q}$, so by (iii) and (7) we know the averages of
$fg_{k}$ decay as the size of the cube increases. Hence we may assume that
${\rm diam}(Q)<k/2$; from this, knowing $fg_{k}=0$ on $B(0,k)$, we can
restrict to cubes with ${\rm dist}(Q,0)\geq k/2$, and therefore, again by (7),
to $\ell(Q)<1$.

In order to bound the oscillation of $fg_{k}$ over such cubes $Q$, setting
$c_{Q}=f_{Q}(g_{k})_{Q}$, we have

$\displaystyle\fint_{Q}|fg_{k}-c_{Q}|$ $\displaystyle\leq$
$\displaystyle\|g_{k}\|_{\infty}\fint_{Q}|f-f_{Q}|+2|f_{Q}|\fint_{Q}|g_{k}-(g_{k})_{Q}|$
$\displaystyle\leq$ $\displaystyle\fint_{Q}|f-f_{Q}|+2|f_{Q}|\;{\rm
diam}(Q_{k})\;\|g_{k}\|_{\rm Lip}$ $\displaystyle\lesssim$
$\displaystyle\sup_{{\rm dist}(Q,0)\geq
k/2,\ell(Q)<1}\left(\fint_{Q}|f-f_{Q}|+|f_{Q}|\ell(Q)k^{-1}\right).$

By (iv), this supremum will go to zero as $k\rightarrow\infty$. ∎

In [1, Proposition 3.3], estimate (2) above and the logarithmic estimate (5)
are used to bound the ${\rm bmo}$ norm of the product of a function in ${\rm
bmo}({{\mathbb{R}}^{n}})$ with a bounded function in ${\rm
lmo}({{\mathbb{R}}^{n}})$. However, as can be seen in the proof, the
logarithmic estimate (5) itself is not necessary to prove the approximation
(i), and may not hold in an arbitrary domain (see Examples 1 and 2).

##  3\. Approximation and vanishing mean oscillation on a domain

Our goal is to study the approximation by Lipschitz functions and functions of
compact support and prove Theorem 1, which is the analogue of Theorems A and
B, as well as analogues of Propositions 1 and 2 in the nonhomogeneous space
${\rm bmo}$ on a domain $\Omega$. Such results are not only of interest in
themselves but will lead to a simple proof of the extension theorem, Theorem
2.

###  3.1. Approximation in ${{\rm bmo}_{\lambda}(\Omega)}$

We start with the case where we do not assume any Sarason-type vanishing mean
oscillation condition. We will need the following notation and terminology.

######  Definition 2.

We denote by ${{\rm bmo}_{\lambda,c}(\Omega)}$ the subspace of functions $f$
in ${{\rm bmo}_{\lambda}(\Omega)}$ which have compact support in $\Omega$. We
say $f\in{{\rm bmo}_{\lambda}(\Omega)}$ vanishes at infinity if

(9) $\lim_{R\rightarrow\infty}\|f\|_{{{\rm
bmo}_{\lambda}}(\Omega\setminus\overline{B(0,R)})}=0.$

We say $f\in{{\rm bmo}_{\lambda}(\Omega)}$ vanishes at the boundary if

(10) $\lim_{t\rightarrow 0}\|f\|_{{{\rm
bmo}_{\lambda}}(\Omega\setminus\mathring{\Omega}_{t})}=0$

where we define

$\mathring{\Omega}_{t}:=\\{x\in\Omega:{d_{\Omega}}(x)\geq
t\\},\quad{d_{\Omega}}(x):={\rm dist}(x,{\partial\Omega}).$

To avoid confusion, it should be pointed out that in [8], $\lambda$ was
replaced by $\lambda/4$ in the definition of $\mathring{\Omega}_{\lambda}$,
while in [2] the notation $\Omega_{t}$ was used completely differently, for
the set ${{\mathbb{R}}^{n}}\setminus\overline{B(0,t)}$.

Note that for an arbitrary domain $\Omega$ and $\lambda>0$, if $f\in{{\rm
bmo}_{\lambda}(\Omega)}$ with ${\rm dist}({\rm supp}(f),{\partial\Omega}))>0$
then $\|f\|_{{{\rm bmo}_{\lambda}}(\Omega\setminus\mathring{\Omega}_{t})}=0$
for all sufficiently small $t$. Since $\|f\|_{{{\rm
bmo}_{\lambda}}(\Omega\setminus\mathring{\Omega}_{t})}\leq\|f\|_{{\rm
bmo}_{\lambda}(\Omega)}$, we have that the limit of such functions vanishes at
the boundary.

Similarly, if $f\in{{\rm bmo}_{\lambda}(\Omega)}$ has compact support in
${\overline{\Omega}}$ (we use the notation ${{\rm
bmo}_{\lambda,c}({\overline{\Omega}})}$ to denote such functions) then $f$
vanishes outside a bounded set and $\|f\|_{{{\rm
bmo}_{\lambda}}(\Omega\setminus\overline{B(0,R)})}=0$ for all sufficiently
large $R$. This again is dominated by the ${{\rm bmo}_{\lambda}(\Omega)}$
norm, so the limit of such functions vanishes at infinity.

In order to get the reverse implications, we need to assume, for functions in
${{\rm bmo}_{\lambda}(\Omega)}$, some control of the averages over small cubes
in terms of the sidelength of the cubes. This is accomplished by the following
two lemmas, stated and proved for general domains.

######  Lemma 1.

Let $\Omega$ be a domain, $\lambda>0$ and $f\in{{\rm bmo}_{\lambda}(\Omega)}$.
Suppose $f$ vanishes at infinity and

(11) $\sup_{Q\subset\Omega,{\rm
dist}(Q,0)>\beta,\ell(Q)<\lambda}|f_{Q}|\ell(Q)<\infty$

for $\beta$ sufficiently large. Then $f$ is in the closure of ${{\rm
bmo}_{\lambda,c}({\overline{\Omega}})}$ in ${{\rm bmo}_{\lambda}(\Omega)}$.

###### Proof.

We proceed as in the proof of Proposition 2. The vanishing at infinity
immediately implies
$\displaystyle{\lim_{\beta\rightarrow\infty}\gamma_{\Omega}(f,\beta)=0}$,
where

$\gamma_{\Omega}(f,\beta):=\sup_{Q\subset\Omega,{\rm
dist}(Q,0)>\beta,\ell(Q)<\lambda}\fint_{Q}|f-f_{Q}|+\sup_{Q\subset\Omega,{\rm
dist}(Q,0)>\beta,\ell(Q)=\lambda}|f|_{Q}.$

By restricting the arguments to cubes lying inside $\Omega$, we get from this
an analogue of (7), namely

(12) $\lim_{\beta\rightarrow\infty}\left[\sup_{Q\subset\Omega,{\rm
dist}(Q,0)>\beta,\ell(Q)\geq\lambda}|f|_{Q}\
+\sup_{Q\subset\Omega,\ell(Q)\geq\beta}|f|_{Q}\right]=0.$

Then we define the functions $f_{k}=f\psi_{k}$, noting that the supports of
these functions are bounded and are therefore compact subsets of
${\overline{\Omega}}$. We continue with the arguments in the proof of
Proposition 2, restricted to cubes $Q\subset\Omega$. In the final step, we
apply (2) to estimate the oscillation of $f-f_{k}$ on cubes $Q\subset\Omega$
with ${\rm dist}(Q,0)\geq k/2$ and $\ell(Q)<\lambda$. As in the final step of
that proof, all that is needed for this is the vanishing of
$\gamma_{\Omega}(f,\beta)$ and (11). ∎

To obtain the analogous result for the vanishing at the boundary, we need to
refine our hypothesis.

######  Lemma 2.

Let $\Omega$ be a domain, $\lambda>0$ and $f\in{{\rm bmo}_{\lambda}(\Omega)}$.
Suppose $f$ vanishes at the boundary and there exists a function
$\varphi_{\lambda}:(0,\lambda/2)\rightarrow(0,\infty)$ with
$\varphi_{\lambda}(t)=1$ for $t\geq\lambda/4$, $t\varphi_{\lambda}(t)$
monotone increasing,
$\int_{0}^{\lambda/4}\frac{dt}{t\varphi_{\lambda}(t)}=\infty$, and such that

(13) $\sup_{\begin{subarray}{c}2Q\subset\Omega\\\ {\rm
dist}(Q,{\partial\Omega})<\lambda/4\end{subarray}}|f_{Q}|\lesssim\varphi_{\lambda}(\ell(Q)).$

Then there exists a sequence $\\{f_{j}\\}$ converging to $f$ in ${{\rm
bmo}_{\lambda}(\Omega)}$, with ${\rm dist}({\rm
supp}(f_{j}),{\partial\Omega}))>0$ for each $j$.

###### Proof.

Set $f_{j}=fh_{j}$, where $h_{j}$ are modified versions of the auxiliary
functions introduced in the proof of [4, Theorem 1]:

$h_{j}(x):=\Big{(}1-\frac{1}{j}\int_{{d_{\Omega}}(x)}^{\lambda/4}\frac{dt}{t\varphi_{\lambda}(t)}\Big{)}_{+}.$

Here the notation $F_{+}$ denotes $\max(F,0)$. For
$x\in\mathring{\Omega}_{\lambda/4}$ we have that $h_{j}(x)=1$, giving
$f_{j}(x)=f(x)$. Moreover, by the hypothesis on $\varphi_{\lambda}$, there is
a sequence of positive numbers $\alpha_{j}$ converging to zero such that
${d_{\Omega}}(x)\leq\alpha_{j}\iff\int_{{d_{\Omega}}(x)}^{\lambda/4}\frac{dt}{t\varphi_{\lambda}(t)}\geq
j\iff h_{j}(x)=0$, so ${\rm dist}({\rm
supp}(f_{j}),{\partial\Omega}))\geq\alpha_{j}$. That $f_{j}\in{{\rm
bmo}_{\lambda}(\Omega)}$ and $f_{j}\rightarrow f$ will follow by estimating
$\|f-f_{j}\|_{{\rm bmo}_{\lambda}(\Omega)}$.

Let $g=f-f_{j}$. To estimate $\|g\|_{{\rm bmo}_{\lambda}(\Omega)}$, we use a
“local-to-global” property - see [4, Theorem A1.1] applied to cubes (balls in
the $\ell^{\infty}$ norm) and [8, Lemma 3.5]:

$\|g\|_{{\rm
bmo}_{\lambda}(\Omega)}\lesssim\sup_{2Q\subset\Omega}\fint_{Q}|g-g_{Q}|+\|g\|_{{L^{\infty}}(\mathring{\Omega}_{\lambda/4})}.$

Since $g=0$ on $\mathring{\Omega}_{\lambda/4}$, it remains to bound the
oscillation of $g$ over cubes $Q$ with $2Q\subset\Omega$ and ${\rm
dist}(Q,{\partial\Omega})<\lambda/4$, which means $\ell(Q)<\lambda/2$.

As in the proof of Proposition 2, we can control the oscillation of
$g=f(1-h_{j})$ over $Q$ by

(14)
$\|1-h_{j}\|_{{L^{\infty}}(Q)}\fint_{Q}|f-f_{Q}|+|f_{Q}|\fint_{Q}|h_{j}-(h_{j})_{Q}|.$

For a given $\eta>0$, take $t_{0}$ sufficiently small so that $\|f\|_{{{\rm
bmo}_{\lambda}}(\Omega\setminus\mathring{\Omega}_{t_{0}})}<\eta/2$. Then if
$Q\cap\mathring{\Omega}_{t_{0}}=\varnothing$ we have
$\fint_{Q}|f(x)-f_{Q}|dx<\eta/2$ and $\|1-h_{j}\|_{{L^{\infty}}(Q)}\leq 1$.

When $Q\cap\mathring{\Omega}_{t_{0}}$ is nonempty, the condition
$2Q\subset\Omega$ forces ${\rm dist}(Q,{\partial\Omega})\geq
t_{0}/(1+\sqrt{n})$. Hence for $j$ sufficiently large so that
$\alpha_{j}<t_{0}/(1+\sqrt{n})$, we have

$|1-h_{j}(x)|=\frac{1}{j}\int_{{d_{\Omega}}(x)}^{\lambda/4}\frac{dt}{t\varphi_{\lambda}(t)}\leq\frac{1}{j}\int_{\frac{t_{0}}{1+\sqrt{n}}}^{\lambda/4}\frac{dt}{t\varphi_{\lambda}(t)},\quad\forall\;x\in
Q.$

Since the integral on the right depends only on $\lambda$ and $t_{0}$,
bounding the oscillation of $f$ on $Q$ by $\|f\|_{{\rm
bmo}_{\lambda}(\Omega)}$, we can make the first term in (14) smaller than
$\eta/2$ by taking $j$ sufficiently large.

For the second term in (14), letting
$\Phi_{\lambda}(x)=\int_{{d_{\Omega}}(x)}^{\lambda/4}\frac{dt}{t\varphi_{\lambda}(t)}$
and noting that truncations do not increase oscillation and that
${d_{\Omega}}$ is a $1$-Lipschitz function, we can proceed as in the proof of
[4, Lemma 4]:

(15) $\displaystyle|f_{Q}|\fint_{Q}|h_{j}(x)-(h_{j})_{Q}|dx$
$\displaystyle\leq$
$\displaystyle\frac{|f_{Q}|}{j}\fint_{Q}\fint_{Q}|\Phi_{\lambda}(x)-\Phi_{\lambda}(y)|dxdy$
$\displaystyle\lesssim$
$\displaystyle\frac{\varphi_{\lambda}(\ell(Q))}{j}\frac{{\rm
diam}(Q)}{\inf_{x\in Q}{d_{\Omega}}(x)\varphi_{\lambda}({d_{\Omega}}(x))}$
$\displaystyle\lesssim$
$\displaystyle\frac{1}{j}\frac{\ell(Q)\varphi_{\lambda}(\ell(Q))}{{\rm
dist}(Q,{\partial\Omega})\varphi_{\lambda}({\rm dist}(Q,{\partial\Omega}))}$
$\displaystyle\lesssim$ $\displaystyle\frac{1}{j}.$

Here we have used (13) and the fact that $2Q\subset\Omega$ implies
$\ell(Q)\leq{\rm dist}(Q,{\partial\Omega})$, as well as the fact that
$t\varphi_{\lambda}(t)$ is an increasing function. Thus the second term in
(14) can be made small for $j$ sufficiently large. ∎

The choice of $\varphi_{\lambda}(t)=\frac{1}{t}$, corresponding to the bound
in (11), does not satisfy the conditions in Lemma 2. At the other extreme,
putting $\varphi_{\lambda}(t)=1$ in the hypotheses of Lemma 2 corresponds to
$f$ being a bounded function, which is the special case shown in the proof of
Theorem 1 in [4] for a bounded domain, and in Lemma 3 of [7] for a general
domain. In these results the vanishing at the boundary follows immediately
from assuming the Sarason-type VMO condition restricted to $\Omega$, as will
be seen in Section 3.2.

We now introduce certain geometric conditions on the domain which will allow
us to apply Lemmas 1 and 2 to all functions in ${{\rm
bmo}_{\lambda}(\Omega)}$. Let us recall the definition of an $\ed$-domain by
Jones [17].

######  Definition 3.

Given $\epsilon\in(0,1]$ and $\delta>0$, an $\ed$-domain is a domain $\Omega$
such that every pair of points $x,y$ in $\Omega$ with $|x-y|<\delta$ may be
joined by a rectifiable curve $\gamma$ lying in $\Omega$, with

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

and such that for any point $z$ on $\gamma$,

${\rm dist}(z,{\partial\Omega})\geq\epsilon\frac{|z-x||z-y|}{|x-y|}.$

For such domains we have the following analogue of (5), which can be shown
directly by a combination of Proposition 4.9 and Lemma 4.4 in [8], or by
extending $f$ to ${{\rm bmo}_{\lambda}}({{\mathbb{R}}^{n}})$ and then using
(5).

######  Lemma 3 ([8]).

Let $\Omega$ be an $\ed$-domain and
$0<\lambda\leq\lambda_{\epsilon,\delta}:=\frac{\epsilon^{2}\delta}{320n(1+\sqrt{n}\epsilon)}$.
If $f\in{{\rm bmo}_{\lambda}(\Omega)}$ and $Q\subset\Omega$ is a cube with
sidelength $\ell(Q)<\lambda$, then

(16)
$|f|_{Q}\lesssim\left(1+\log\left(\frac{\lambda}{\ell(Q)}\right)\right)\|f\|_{{{\rm
bmo}_{\lambda}(\Omega)}}.$

If $\lambda_{1}<\lambda_{2}\leq\lambda_{\epsilon,\delta}$, then (16) gives

(17) $\|f\|_{{\rm bmo}_{\lambda_{2}}(\Omega)}\leq 2\|f\|_{{\rm
bmo}_{\lambda_{1}}(\Omega)}\leq
C\log\frac{\lambda_{2}}{\lambda_{1}}\|f\|_{{\rm bmo}_{\lambda_{2}}(\Omega)}.$

This justifies using the notation ${{\rm bmo}(\Omega)}$ for ${{\rm
bmo}_{\lambda}(\Omega)}$, where we fix $\lambda=\lambda_{\epsilon,\delta}$ for
the norm. The same estimates also show that the notions of vanishing at
infinity and vanishing at the boundary are independent of the choice of
$\lambda$. Moreover, ${{\rm bmo}_{\lambda,c}(\Omega)}$ can be denoted by
${{{\rm bmo}_{c}}(\Omega)}$ independently of $\lambda$.

With the help of the three lemmas, we can prove an analogue of Proposition 2.

######  Proposition 3.

Let $\Omega$ be an $\ed$-domain and $f\in{{\rm bmo}(\Omega)}$. Then the
following are equivalent:

  1. (i)

there exists a sequence $\\{f_{j}\\}$ converging to $f$ in ${{\rm
bmo}(\Omega)}$, with ${\rm dist}({\rm supp}(f_{j}),{\partial\Omega}))>0$ for
each $j$;

  2. (ii)

$f$ vanishes at the boundary;

Moreover, we have the equivalence of the following two conditions:

  1. (a)

$f$ is in the closure of ${{{\rm bmo}_{c}}(\Omega)}$ in ${{\rm bmo}(\Omega)}$;

  2. (b)

$f$ vanishes at the boundary and vanishes at infinity.

###### Proof.

As was pointed out above, the implication in one direction of each pair holds
in any domain, so it remains to show the implications (ii) $\implies$ (i) and
(b) $\implies$ (a).

Since $\Omega$ is an $\ed$ domain and $f\in{{\rm bmo}(\Omega)}$, by Lemma 3,
both (11) and (13) with

$\varphi_{\lambda}(t)=1+\log_{+}\frac{\lambda}{4t},\quad\lambda=\lambda_{\epsilon,\delta},$

hold for $f$ and also for $|f|$. Assuming $f$ vanishes at the boundary, we
obtain from Lemma 2 the approximation by functions $f_{j}=fh_{j}$ in ${{\rm
bmo}(\Omega)}$ which are supported away from the boundary.

If in addition $f$ vanishes at infinity, we apply the proof of Lemma 1 to
$f_{j}$ and obtain the sequence of functions
$f_{j,k}=f_{j}\psi_{k}=fh_{j}\psi_{k}$ having compact support in $\Omega$. We
need to show that by choosing $j$ and $k$ sufficiently large, we can make
$\|f_{j,k}-f\|_{{\rm bmo}(\Omega)}$ small. This means estimating
$\|f_{j,k}-f_{j}\|_{{\rm bmo}(\Omega)}$.

From the proof of Lemma 2 we know that $0\leq h_{j}\leq 1$ so for every
$Q\subset\Omega$, $|f_{j}|_{Q}\leq|f|_{Q}$. As $f$ vanishes at infinity, this
means that (12) holds with $f$ replaced by $f_{j}$. Thus it suffices to bound
the oscillation of $f_{j,k}-f_{j}$ on cubes $Q\subset\Omega$ with ${\rm
dist}(Q,0)\geq k/2$ and $\ell(Q)<\lambda$. From (2), we can control this
oscillation by

$\sup_{Q\subset\Omega,{\rm dist}(Q,0)\geq
k/2,\ell(Q)<\lambda}\left(\fint_{Q}|f_{j}-(f_{j})_{Q}|+|(f_{j})_{Q}|\ell(Q)k^{-1}\right),$

which in turn, by (11), (13), (14) and (15), can be controlled by

$\sup_{Q\subset\Omega,{\rm dist}(Q,0)\geq
k/2,\ell(Q)<\lambda}\fint_{Q}|f-f_{Q}|+j^{-1}+k^{-1}.$

It is then possible to choose $j$ and $k$ sufficient large to make both this
quantity and $\|f_{j}-f\|_{{\rm bmo}(\Omega)}$ small. ∎

###  3.2. Approximation in ${{\rm vmo}_{\lambda}(\Omega)}$

We now consider functions in ${{\rm bmo}_{\lambda}(\Omega)}$ which also
satisfy a Sarason-type VMO condition, namely

(18) $\lim_{t\rightarrow 0^{+}}{\omega_{\Omega}}(f,t)=0,$

where ${\omega_{\Omega}}(f,t)$ is defined in (3).

The following two lemmas will lead to the proof of Theorem 1.

######  Lemma 4.

Let $\Omega$ be a domain, $\lambda>0$. If $f$ is a bounded function satisfying
(18) then $f$ can be approximated in ${{\rm bmo}_{\lambda}(\Omega)}$ by
bounded Lipschitz functions supported away from ${\partial\Omega}$.

###### Proof.

This is the special case discussed after the proof of Lemma 2, where the lemma
can be applied with $\varphi_{\lambda}(t)=1$ and tells us that we may
approximate $f$ in ${{\rm bmo}_{\lambda}(\Omega)}$ by functions supported away
from ${\partial\Omega}$. Since the approximations are products of $f$ with
bounded functions $h_{j}$, they are bounded. To further approximate such
functions by Lipschitz functions supported away from ${\partial\Omega}$ can be
accomplished by the averaging process described at the beginning of Section 2,
and whose details can be found in Section 2 of [7]. As the functions are
supported away from ${\partial\Omega}$, the grid can be chosen fine enough so
that the resulting Lipschitz functions are also supported away from the
boundary. Moreover, while the approximation in [7] is in the ${\rm BMO}$ norm,
it can be seen from the calculations following [7, Equation (9)] that for
large cubes, what is estimated are the averages of the error function, so the
approximation is actually in the ${{\rm bmo}_{\lambda}(\Omega)}$ norm. This
argument is also referred to in the proof of [7, Proposition 3]. However,
unlike in that case where $f$ is assumed to have compact support, or the
domain is bounded, as in [4], here we cannot immediately obtain smooth
functions with uniform bounds by convolving with a smooth mollifier, since we
do not necessarily have control of the $L^{1}$ norm of $f$. ∎

Based on the previous lemma, we will be able to prove Theorem 1 if we can
approximate $f$ in the ${{\rm bmo}_{\lambda}(\Omega)}$ norm by bounded
functions. The standard way to do this is by truncations, namely setting
$f^{t}=\max(\min(f,t),-t)$ and letting $t\rightarrow\infty$. In the case of a
bounded domain, or when $f$ has compact support, the convergence of $f^{t}$ to
$f$ in $L^{1}$ together with (18) gives the convergence in ${\rm BMO}(\Omega)$
(see [4, Lemma A1.4]) and also in ${{\rm bmo}_{\lambda}(\Omega)}$.

For unbounded domains and general $f\in{{\rm bmo}_{\lambda}(\Omega)}$, we need
to find a criterion for approximation by bounded functions. This is
accomplished by the following lemma, which assumes some uniform control of the
averages of $f$ on cubes of a given size. Since we are assuming (18), we can
get by with a weaker assumption than those in Lemmas 1 and 2.

######  Lemma 5.

Let $\Omega$ be a domain, $\lambda>0$ and $f\in{{\rm bmo}_{\lambda}(\Omega)}$.
Suppose $f$ satisfies (18) and for every $\ell>0$,

$C_{\ell}=\sup_{2Q\subset\Omega,\ell(Q)\geq\ell}|f|_{Q}<\infty.$

Then $f$ can be approximated in ${{\rm bmo}_{\lambda}(\Omega)}$ by bounded
functions.

###### Proof.

For a positive $\ell<\lambda/8\sqrt{n}$ and $x\in\Omega$, let
$\ell(x)=\min(\frac{{d_{\Omega}}(x)}{2\sqrt{n}},\ell)$, take $Q_{x}$ to be the
cube centered at $x$ with sidelength $\ell(x)$, and set

$\tilde{f}(x)=\fint_{Q_{x}}f,\quad
g(x)=\max(\min(\tilde{f}(x),C_{\ell}),-C_{\ell}).$

As $g$ is a truncation, $|g|\leq C_{\ell}$. Since $2Q_{x}\subset\Omega$, by
the definition of $\ell(x)$ we have $|\tilde{f}(x)|\leq C_{\ell}$ whenever
${d_{\Omega}}(x)\geq 2\ell\sqrt{n}$, i.e. when
$x\in\mathring{\Omega}_{2\ell\sqrt{n}}$, so $g=\tilde{f}$ on
$\mathring{\Omega}_{2\ell\sqrt{n}}$.

To estimate the ${{\rm bmo}_{\lambda}(\Omega)}$ norm of $h:=f-g$, we again use
the local-to-global property ([8, Lemma 3.5]):

(19) $\|h\|_{{\rm
bmo}_{\lambda}(\Omega)}\lesssim\sup_{2Q\subset\Omega}\fint_{B}|h-h_{Q}|+\sup_{2Q\subset\Omega,\ell(Q)\geq\lambda/2}|h|_{Q}.$

Note that cubes over which the supremum in the second term on the right is
taken satisfy ${\rm dist}(Q,{\partial\Omega})\geq\ell(Q)/2\geq\lambda/4$, so
to bound this term it suffices to control the averages $|h|_{Q}$ for
$Q\subset\mathring{\Omega}_{\lambda/4}$. By the choice of $\ell$,
$\mathring{\Omega}_{\lambda/4}\subset\mathring{\Omega}_{2\ell\sqrt{n}}$.

So let us first consider a cube $Q\subset\mathring{\Omega}_{2\ell\sqrt{n}}$.
For $x\in Q$, $g(x)=\tilde{f}(x)=f_{Q_{x}}$ with $\ell(Q_{x})=\ell$. Suppose
$\ell(Q)\geq\ell$. Then for $z\in Q$,

(20) $2^{-n}\ell^{n}\leq|Q_{z}\cap Q|\leq\ell^{n},$

and therefore

$\fint_{Q_{z}\cap Q}|h|\leq\frac{2^{n}}{\ell^{n}}\int_{Q_{z}\cap
Q}|f(x)-f_{Q_{x}}|dx\leq\frac{2^{n}}{\ell^{2n}}\int_{Q_{z}}\int_{Q_{x}}|f(x)-f(y)|dydx\leq
2^{1+3n}\fint_{2Q_{z}}|f-f_{2Q_{z}}|,$

where we have used the fact that $x\in Q_{z}\implies Q_{x}\subset 2Q_{z}$.
Since $2Q_{z}\subset\Omega$, the right-hand-side is bounded by
$2^{1+3n}{\omega_{\Omega}}(f,2\ell)$. From this, applying (20) again, we get
that

$\fint_{Q}|h|\leq\fint_{Q}\left\\{\int_{z\in Q_{x}\cap
Q}\frac{2^{n}dz}{|Q_{z}\cap Q|}\right\\}|h(x)|dx\ =2^{n}\fint_{Q}\fint_{x\in
Q_{z}\cap Q}|h(x)|dxdz\lesssim{\omega_{\Omega}}(f,2\ell).$

Thus ${\omega_{\Omega}}(f,2\ell)$ controls the second term in (19), as well as
bounding the oscillation of $h$ over cubes in
$\mathring{\Omega}_{2\ell\sqrt{n}}$ with $\ell(Q)\geq\ell$.

It now remains to estimate the oscillation of $h$ over cubes $Q$ with
$2Q\subset\Omega$ and such that either
$Q\not\subset\mathring{\Omega}_{2\ell\sqrt{n}}$ or $\ell(Q)<\ell$. In the
first case we also have $\ell(Q)\leq 2{\rm
dist}(Q,{\partial\Omega})<4\sqrt{n}\ell$ so we will deal with the two
simultaneously.

Since $|h-h_{Q}|\leq|f-f_{Q}|+|g-g_{Q}|$ and truncation reduces oscillation,
we have

$\fint_{Q}|h-h_{Q}|\leq{\omega_{\Omega}}(f,4\sqrt{n}\ell)+\fint_{Q}|\tilde{f}(x)-\tilde{f}_{Q}|dx\leq{\omega_{\Omega}}(f,4\sqrt{n}\ell)+\fint_{Q}\fint_{Q}|f_{Q_{x}}-f_{Q_{y}}|dydx.$

For $x\in Q$, we take $Q^{\prime}_{x}$ to be the cube centered at $x$ of
sidelength $d=\frac{{\rm dist}(Q,{\partial\Omega})}{2\sqrt{n}}$, noting that

$d=\min\Big{(}\frac{{\rm
dist}(Q,{\partial\Omega})}{2\sqrt{n}},\ell\Big{)}\leq\ell(x)\leq\frac{{\rm
dist}(Q,{\partial\Omega})+{\rm diam}(Q)}{2\sqrt{n}}\leq cd,\quad
c:=1+2\sqrt{n}.$

Then

$|f_{Q^{\prime}_{x}}-f_{Q_{z}}|\leq\fint_{Q^{\prime}_{x}}|f-f_{Q_{x}}|\leq
c^{n}\fint_{Q_{x}}|f-f_{Q_{x}}|\leq c^{n}\omega(f,4\sqrt{n}\ell).$

The triangle inequality then gives us

$\fint_{Q}\fint_{Q}|f_{Q_{z}}-f_{Q_{y}}|dydx\leq
2c^{n}\omega(f,4\sqrt{n}\ell)+\fint_{Q}\fint_{Q}|f_{Q^{\prime}_{x}}-f_{Q^{\prime}_{y}}|dydx.$

Finally, writing the averages as convolution with
$d^{-n}\chi_{[-d/2,d/2]^{n}}$, we can bound the second term on the right-hand-
side by

$\fint_{Q}\fint_{Q}\fint_{[-d/2,d/2]^{n}}|f(x-z)-f(y-z)|dzdydx=\fint_{[-d/2,d/2]^{n}}\fint_{Q-z}\fint_{Q-z}|f(u)-f(v)|dudvdz.$

This quantity is also bounded by $\omega(f,4\sqrt{n}\ell)$, since for
$z\in[-d/2,d/2]^{n}$, the choice of $d$ gives ${\rm
dist}(Q-z,{\partial\Omega})\geq{\rm dist}(Q,{\partial\Omega})-d/2>0$ so
$Q-z\subset\Omega$.

In summary, for the given $\ell$, we have shown the existence of a bounded
function $g$ such that $\|f-g\|_{{\rm
bmo}_{\lambda}(\Omega)}\lesssim\omega(f,4\sqrt{n}\ell)$, where the constants
depend only on $n$. By choice of $\ell$, we can make this as small as we like.
∎

###### Proof of Theorem 1.

On any domain $\Omega$ and for any $\lambda>0$, if $f$ can be approximated in
${{\rm bmo}_{\lambda}(\Omega)}$ by uniformly continuous functions, then (18)
holds. For the converse inclusions we show the strongest, namely that any
function in ${{\rm bmo}_{\lambda}(\Omega)}$, with $\lambda$ as in the
hypotheses of the theorem, can be approximated by bounded Lipschitz functions
supported away from the boundary. This follows from Lemmas 4 and 5 by noting
that for any $\ell\leq\lambda$, $C_{\ell}\leq\|f\|_{{\rm
bmo}_{\ell}(\Omega)}<\infty$. ∎

As a corollary of Theorem 1, Lemma 3 and Proposition 3, we get the following.

######  Proposition 4.

Let $\Omega$ be an $\ed$-domain and $f\in{{\rm bmo}(\Omega)}$. Then $f$ can be
approximated in ${{\rm bmo}(\Omega)}$ by bounded Lipschitz functions supported
away from ${\partial\Omega}$ if and only if (18) holds.

In addition, the following are equivalent:

  1. (i)

$f$ is in ${\rm cmo}(\Omega)$, the closure of $C_{c}(\Omega)$ in ${{\rm
bmo}(\Omega)}$;

  2. (ii)

$f$ is in the closure of $C_{c}^{\infty}(\Omega)$ in ${{\rm bmo}(\Omega)}$;

  3. (iii)

$f$ vanishes at infinity and (18) holds.

###### Proof.

We only need to show the implication (iii) $\implies$ (ii), as convolution
with a smooth mollifier gives the approximation, in the $L^{\infty}$ norm, of
a function in $C_{c}(\Omega)$ by functions in $C_{c}^{\infty}(\Omega)$. If $f$
vanishes at infinity and (18) holds, then $f$ vanishes at the boundary and we
can follow the proof of Proposition 3, starting with the sequence
$\\{f_{j}\\}$ of bounded Lipschitz functions supported away from
${\partial\Omega}$ which approximates $f$, and multiplying by the cut-off
functions $\psi_{k}$ to make them of compact support in $\Omega$. ∎

##  4\. Proof of Theorem 2: extension from a locally uniform domain

In this section we assume $\Omega$ is an $\ed$ domain and fix
$\lambda\leq\lambda_{\epsilon,\delta}$, where
$\lambda_{\epsilon,\delta}:=\frac{\epsilon^{2}\delta}{320n(1+\sqrt{n}\epsilon)}$.
As in [8], for $f\in{{\rm bmo}_{\lambda}(\Omega)}$ we define a function
${T_{\lambda}}f:{{\mathbb{R}}^{n}}\to{\mathbb{R}}$ by

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

Here $E^{\prime}$ denotes the Whitney decomposition of $\Omega^{\prime}$, the
complement of ${\overline{\Omega}}$ (see [25, Section VI.1] for the definition
and properties of Whitney cubes), and we have fixed, for each $Q\in
E^{\prime}$ with $\ell(Q)\leq\lambda$, a choice of matching cube $Q^{*}\in E$
with $\ell(Q)\leq\ell(Q^{*})\leq 4\ell(Q)$ and ${\rm dist}(Q,Q^{*})\leq
C\ell(Q)$. The existence of such a cube is guaranteed by [17, Lemma 2.4],
since $\lambda_{\epsilon,\delta}\leq\epsilon\delta/(16n)$.

By [17, Lemma 2.3], ${\partial\Omega}$ has measure zero, so it suffices to
extend $f$ to $\Omega^{\prime}$. We now proceed to average the step function
${T_{\lambda}}f$ on $\Omega^{\prime}$, as in [7]. Set

$\widetilde{{T_{\lambda}}}f(x)=\left\\{\begin{array}[]{cc}f(x),&x\in\Omega,\\\
A({T_{\lambda}}f)(x)&x\in\Omega^{\prime},\end{array}\right.$

where the averaging operator $A$, applied applied to $\phi={T_{\lambda}}f$ on
$\Omega^{\prime}$, is defined by

$A(\phi)(x):=\fint_{B(x,R(x))}\phi(y)dy,\quad x\in\Omega^{\prime}.$

Here

(22) $R(x)=c_{n}{d_{\Omega}}(x)$

where for $x\in\Omega^{\prime}$, with an abuse of notation we denote
${d_{\Omega}}(x):={\rm dist}(x,{\partial\Omega})$, just like for points inside
$\Omega$. As in [7], we choose the constant $c_{n}$ sufficiently small so that
for $x\in\Omega^{\prime}$, the collection

${\mathcal{N}}(x):=\\{Q\in E^{\prime}:Q\cap B(x,R(x))\neq\varnothing\\}$

consists exactly of the Whitney cube containing $x$ and the Whitney cubes
adjacent to it, and moreover that at the center $x_{Q}$ of a cube $Q\in
E^{\prime}$, $A(\phi)(x_{Q})=\phi(x_{Q})$.

Having proved in [8] the boundedness of the map $f\rightarrow{T_{\lambda}}f$
from ${{\rm bmo}_{\lambda}(\Omega)}$ to ${{\rm
bmo}_{\lambda}}({{\mathbb{R}}^{n}})$, we now note that it also applies to the
map $f\rightarrow\widetilde{{T_{\lambda}}}f$. As in [7], this follows from the
fact that the difference between ${T_{\lambda}}f$ and its averaging
$\widetilde{{T_{\lambda}}}f=A({T_{\lambda}}f)$ on a Whitney cube in
$\Omega^{\prime}$ is bounded by a constant times $\|{T_{\lambda}}f\|_{{\rm
BMO}({{\mathbb{R}}^{n}})}$, since by the choice of $c_{n}$, the averaging
takes places only on adjacent Whitney cubes.

Having part (i) of Theorem 2, we proceed to part (iv), which will give us the
other two parts by approximation.

###  4.1. Boundedness on ${\rm Lip}_{b}(\Omega)$

The ideas are similar to those in Section 3.3 of [7], with two main
differences. First, we are dealing with bounded functions, which makes things
easier. On the other hand, we need to deal with the “jump” in the definition
of ${T_{\lambda}}$, namely the separate definitions for cubes near and far
away from the boundary.

Starting with a bounded Lipschitz function $f$ on $\Omega$, by definition
$\|\widetilde{{T_{\lambda}}}f\|_{\infty}=\|f\|_{\infty}$. If we can show that
$\widetilde{{T_{\lambda}}}f$ is Lipschitz on $\Omega^{\prime}$ with
$\|\widetilde{{T_{\lambda}}}f\|_{{\rm
Lip}_{b}(\Omega^{\prime})}\lesssim\|f\|_{{\rm Lip}_{b}(\Omega)}$, then the
same argument as in Section 3.3 of [7], i.e. extending to ${\partial\Omega}$
by uniform continuity, will give us the desired Lipschitz continuity on all of
${{\mathbb{R}}^{n}}$ and prove part (iv) of Theorem 2.

As in [7, Lemma 1 and Corollary 3], the key is the inherent local Lipschitz
nature of the averaging itself, which gives, for
$x_{1},x_{2}\in\Omega^{\prime}$,

(23)
$|\widetilde{{T_{\lambda}}}f(x_{1})-\widetilde{{T_{\lambda}}}f(x_{2})|\leq
C\sup_{Q,Q^{\prime}\in{\mathcal{N}}(x_{1})\cup{\mathcal{N}}(x_{2})}\left|({T_{\lambda}}f)_{Q}-({T_{\lambda}}f)_{Q^{\prime}}\right|\min\left(\frac{|x_{1}-x_{2}|}{\min_{i=1,2}{d_{\Omega}}(x_{i})},1\right).$

Thus if both $x_{i}$ are sufficiently far from ${\partial\Omega}$, say
contained in Whitney cubes of sidelength at least $\lambda/40\sqrt{n}$, then

$|\widetilde{{T_{\lambda}}}f(x_{1})-\widetilde{{T_{\lambda}}}f(x_{2})|\leq
C_{\lambda,n}\|{T_{\lambda}}f\|_{\infty}|x_{1}-x_{2}|\leq
C_{\lambda,n}\|f\|_{\infty}|x_{1}-x_{2}|.$

On the other hand, if one of the points, say $x_{1}$, belongs to a Whitney
cube of sidelength less than $\lambda/40\sqrt{n}$, while the other, $x_{2}$,
belongs to a cube of sidelength greater than $\lambda/4$, then by the triangle
inequality and the properties of Whitney cubes,

$|x_{1}-x_{2}|\geq{d_{\Omega}}(x_{2})-{d_{\Omega}}(x_{1})\geq\lambda/4-(5\sqrt{n}\lambda/40\sqrt{n})=\lambda/8,$

and the trivial $L^{\infty}$ bound (not using (23)) gives

$|\widetilde{{T_{\lambda}}}f(x_{1})-\widetilde{{T_{\lambda}}}f(x_{2})|\leq
C_{\lambda}\|\widetilde{{T_{\lambda}}}f\|_{\infty}|x_{1}-x_{2}|=C_{\lambda}\|f\|_{\infty}|x_{1}-x_{2}|.$

Thus it remains to consider the case when both points lie in Whitney cubes,
say $Q_{1},Q_{2}$, respectively, of sidelength at most $\lambda/4$. Recalling
that the Whitney cubes in ${\mathcal{N}}(x_{i})$ are either $Q_{i}$ or
adjacent to it, by the properties of adjacent Whitney cubes, any cubes
$Q,Q^{\prime}$ in ${\mathcal{N}}(x_{1})\cup{\mathcal{N}}(x_{2})$, have
sidelength bounded by $\lambda$, so

$|\widetilde{{T_{\lambda}}}f(x_{1})-\widetilde{{T_{\lambda}}}f(x_{2})|\leq
C\sup_{Q,Q^{\prime}\in{\mathcal{N}}(x_{1})\cup{\mathcal{N}}(x_{2})}\left|f_{Q^{*}}-f_{(Q^{\prime})^{*}}\right|\min\left(\frac{|x_{1}-x_{2}|}{\min_{i=1,2}{d_{\Omega}}(x_{i})},1\right).$

In the special case when $Q_{1}$ and $Q_{2}$ are adjacent (which includes the
case $Q_{1}=Q_{2}$), since $\lambda<\epsilon\delta/16n$, we can apply [17,
Lemma 2.8] to get that the shortest Whitney chain connecting $Q_{1}^{*}$ and
$Q_{2}^{*}$ has length $m$ bounded by a constant. By transitivity, this also
applies to any two cubes $Q^{*},(Q^{\prime})^{*}$ which are matching to cubes
$Q,Q^{\prime}\in{\mathcal{N}}(x_{1})\cup{\mathcal{N}}(x_{2})$. Along such a
chain, the difference of averages of $f$ on two adjacent cubes is bounded by
$\|f\|_{{\rm Lip}(\Omega)}$ times the sum of the diameters of the two cubes.
Again by the properties of Whitney cubes, the largest cube along such a chain
has diameter at most $4^{m}$ times the smallest. This gives (see also [7,
Lemma 4] for the general case of a function in ${\rm BMO}(\Omega)$ on any
domain $\Omega$) that

$\left|f_{Q^{*}}-f_{(Q^{\prime})^{*}}\right|\lesssim\|f\|_{{\rm
Lip}(\Omega)}\ell(Q^{*}).$

Since in this special case the sidelengths of
$Q\in{\mathcal{N}}(x_{1})\cup{\mathcal{N}}(x_{2})$ and a matching cube $Q^{*}$
are comparable to $\ell(Q_{i})$, which are in turn comparable to
${d_{\Omega}}(x_{i})$, we get that

$|\widetilde{{T_{\lambda}}}f(x_{1})-\widetilde{{T_{\lambda}}}f(x_{2})|\lesssim\|f\|_{{\rm
Lip}(\Omega)}|x_{1}-x_{2}|.$

Finally, when $Q_{1}$ and $Q_{2}$ are not adjacent, we follow the argument in
Section 3.3 of [7]. In this case $|x_{1}-x_{2}|$ must be at least as large as
the sidelength of the smallest Whitney cube adjacent to either $Q_{1}$ or
$Q_{2}$, which means $|x_{1}-x_{2}|\gtrsim\max_{i=1,2}\ell(Q_{i})$, so the
estimate is achieved by showing that

$\left|f_{Q_{1}^{*}}-f_{Q_{2}^{*}}\right|\lesssim\|f\|_{{\rm
Lip}(\Omega)}\max_{i=1,2}\ell(Q_{i})\lesssim\|f\|_{{\rm
Lip}(\Omega)}|x_{1}-x_{2}|,$

and noting that the left-hand-side is just the difference of values of
$\widetilde{{T_{\lambda}}}f$ at the center points of the $Q_{i}$, by the
choice of $c_{n}$ in (22). The comparison of the value of
$\widetilde{{T_{\lambda}}}f(x_{i})$ to the value of
$\widetilde{{T_{\lambda}}}f$ at the center of $Q_{i}$ can be bound by the same
quantity using the case of adjacent (in this case identical) cubes above.

It should be noted that while Lipschitz extensions do not need any restriction
on the domain, for this particular extension we have had to use some of the
geometric properties of $\ed$ domains, namely the “reflection” lemmas from
[17], in particular Lemmas 2.5 and 2.8.

###  4.2. The ${\rm vmo}$ and ${\rm cmo}$ extensions

We follow the argument in [7, Sections 3.4 and 3.5]. Here Proposition 4 gives
us, in the case of $f\in{\rm vmo}(\Omega)$, the approximation by bounded
Lipschitz functions in ${{\rm bmo}(\Omega)}$, and part (ii) of Theorem 2
follows from parts (i), (iv) and Theorem B.

For ${\rm cmo}(\Omega)$, we just need to add to this argument the fact that
the extension maps functions of compact support in $\Omega$ to functions of
compact support in ${{\mathbb{R}}^{n}}$. This is shown in [7, Section 3.5] in
the homogeneous case (where compact support means the function is constant
outside a compact set), but the case here is simpler since the extension
${T_{\lambda}}f$ is zero, by definition, on Whitney cubes sufficiently far
from ${\partial\Omega}$ and therefore, by the local nature of the averaging
process (i.e. the choice of $c_{n}$ in (22)), so is
$\widetilde{{T_{\lambda}}}f$. More specifically, it will vanish on any $Q\in
E^{\prime}$ all of whose neighbors have sidelength greater than $\lambda$,
which means it is guaranteed to vanish if $\ell(Q)>4\lambda$.

Suppose $f\in{{\rm bmo}(\Omega)}$ is supported in $B(0,R)$. By the argument
above, we only need to study $\widetilde{{T_{\lambda}}}f$ on Whitney cubes
$Q\in E^{\prime}$ with $\ell(Q)\leq 4\lambda$. Since
$4\lambda<\epsilon\delta/16n$, we can apply [17, Lemmas 2.4 and 2.5] to such
cubes to conclude that their matching cubes $Q^{*}$ must lie within a distance
comparable to $\ell(Q)$. Thus ${T_{\lambda}}f=0$ on $Q$ whenever ${\rm
dist}(Q,0)\geq R+C_{\epsilon,n,\lambda}$. Again by the local nature of the
averaging process, this gives $\widetilde{{T_{\lambda}}}f=0$ on any Whitney
cube in $\Omega^{\prime}$ such that all its neighbors lie in the complement of
$B(0,R+C_{\epsilon,n,\lambda})$. Since we are only looking at cubes of
sidelength bounded by $4\lambda$, this shows that $\widetilde{{T_{\lambda}}}f$
is supported in $B(0,R^{\prime})$ for some $R^{\prime}>R$.

##  5\. Examples

We close the paper with a couple of examples to illustrate Theorem 1 and some
of the results in Section 3.

######  Example 1.

In ${\mathbb{R}}^{2}$, our domain $\Omega$ consists of the left half-plane
$\\{(x,y):x<0\\}$ connected to infinitely many disjoint strips lying in the
right half-plane, parallel to the positive $x$ axis. The $n$th strip $S_{n}$
has vertical width $1/n$ and horizontal length $L_{n}$. If we take a cube of
sidelength $\ell$, there are only finitely many strips $S_{n}$ which can
contain this cube, namely those with $n<\ell^{-1}$.

If $f$ is in ${{\rm bmo}_{\lambda}(\Omega)}$ and $\ell<1/n<\lambda$, then
starting from a cube $Q$ of sidelength $\ell$ in $S_{n}$, to reach a cube of
sidelength $\lambda$ we have to get to the left half-plane, which means going
through a chain of Whitney cubes in $S_{n}$ of sidelength $\gtrsim\ell$ and
hence whose length $m\lesssim L_{n}\ell^{-1}$. The standard telescoping sum
argument then gives the bound $|f|_{Q}\lesssim L_{n}\ell^{-1}\|f\|_{{\rm
bmo}_{\lambda}(\Omega)}$. Since $n$ is bounded by $\ell^{-1}$, we get that for
$0<\lambda^{\prime}<\lambda$,

$\|f\|_{{\rm bmo}_{\lambda^{\prime}}(\Omega)}\lesssim\|f\|_{{\rm
bmo}_{\lambda}(\Omega)}+\sup\\{|f|_{Q}:\lambda^{\prime}\leq\ell(Q)<\lambda\\}\lesssim\|f\|_{{\rm
bmo}_{\lambda}(\Omega)}\Big{(}1+\max_{n<\frac{1}{\lambda^{\prime}}}\frac{L_{n}}{\lambda^{\prime}}\Big{)}<\infty.$

By Theorem 1, any $f\in{{\rm bmo}_{\lambda}(\Omega)}$ which satisfies the
vanishing mean oscillation condition $\displaystyle{\lim_{t\rightarrow
0^{+}}{\omega_{\Omega}}(f,t)=0}$ can therefore be approximated in ${{\rm
bmo}_{\lambda}(\Omega)}$ by bounded Lipschitz functions.

On the other hand, such a domain $\Omega$ can only be an $\ed$ domain if the
lengths $L_{n}$ go to zero sufficiently fast as $n\rightarrow\infty$, in order
to guarantee that when two points are at some distance
$\delta^{\prime}\leq\delta$, they can belong to $S_{n}$ only for sufficiently
small $n$, depending on $\delta^{\prime}$, and such $S_{n}$ are wide enough
for them to be joined by the appropriate $\epsilon$ cigar. If $L_{n}$ do not
go to zero, $\Omega$ will not be $\ed$.

How fast do $L_{n}$ have to go to zero for $\Omega$ to be $\ed$? Consider the
function $f(x,y)$ on $\Omega$ which is $0$ on the left half-plane and is equal
to $nx$ on $S_{n}$. Since all cubes contained in the strip $S_{n}$ are of
sidelength at most $1/n$, the mean oscillation of $f$ is bounded by $1$.
Moreover, since only finitely many of the strips contain cubes of sidelength
at least $\lambda$, and $0\leq f\leq nL_{n}$ on $S_{n}$, we have

$\sup\\{|f|_{Q}:\ell(Q)\geq\lambda\\}\lesssim\max_{n<\frac{1}{\lambda}}nL_{n}<\infty.$

If we could extend $f$ to a function in ${\rm bmo}({{\mathbb{R}}^{n}})$ then
its averages would have to satisfy the logarithmic estimate (5), so for each
$n$, taking a cube $Q$ of sidelength $\frac{1}{2n}$ at the rightmost tip of
$S_{n}$, we would need to have

$nL_{n}\approx|f|_{Q}\lesssim\|f\|_{{{\rm
bmo}_{\lambda}(\Omega)}}\log\Big{(}\frac{2}{\ell(Q)}\Big{)}=\|f\|_{{{\rm
bmo}_{\lambda}(\Omega)}}\log(4n).$

Thus $L_{n}$ must be ${\mathcal{O}}(\frac{\log n}{n})$ as
$n\rightarrow\infty$.

######  Example 2.

We now give a variation on Example 1 which does not satisfy the hypotheses of
Theorem 1. For each integer $n\geq 1$, instead of one strip we attach to the
left half-plane $n$ disjoint strips in the right half-plane, $S_{n,j},1\leq
j\leq n$, with vertical width $\frac{1}{j}$ and horizontal length $n$. This
means that for any $\ell<1$, there will be cubes of size $\ell$ in infinitely
many strips, further and further away from the left half-plane.

For this domain, fixing any $\lambda>0$, we want to show there is a function
$f$ in ${{\rm bmo}_{\lambda}(\Omega)}$ which is not in ${\rm
bmo}_{\lambda^{\prime}}(\Omega)$ for some $\lambda^{\prime}<\lambda$. Define
$f$ to be $0$ in the left half-plane and on any strip which contains cubes of
size $\lambda$. On all the other strips $S_{n,j}$, define $f(x,y)=c_{j}x$,
$c_{j}\neq 0$. Then $f$ is Lipschitz with constant $c_{j}$ on $S_{n,j}$, which
contains only cubes of sidelength bounded by $\frac{1}{j}$, and therefore has
vanishing mean oscillation provided $\frac{c_{j}}{j}\rightarrow 0$ as
$j\rightarrow\infty$. It is also in ${{\rm bmo}_{\lambda}(\Omega)}$ because it
is zero on cubes of sidelength $\lambda$ or greater contained in $\Omega$.
However, for $\lambda^{\prime}<\frac{1}{j}<\lambda$, there will be cubes $Q$
of size $\lambda^{\prime}$ in all $S_{n,j}$ with $n\geq j$, for which the
averages $|f|_{Q}\approx nc_{j}$. Thus $f\not\in{\rm
bmo}_{\lambda^{\prime}}(\Omega)$.

Note that while $f$ satisfies the vanishing mean oscillation condition
$\displaystyle{\lim_{t\rightarrow 0^{+}}{\omega_{\Omega}}(f,t)=0}$, and its
averages over all cubes of side $\lambda$ or larger are zero, it does not
vanish at infinity in the sense Definition 2, as can be seen by looking at
cubes of sidelength approximately $1$ in $S_{n,1}$, for arbitrary large $n$,
whose distance from the origin is going to infinity but on which the
oscillation of $f$ is approximately $c_{1}$. Thus the implication $2\implies
3$ in Proposition 1 does not hold in a general domain.

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