# Musielak-Orlicz-Sobolev embeddings: Necessary and Sufficient Conditions

Ankur Pandey and Nijjwal Karak

###### Abstract.

In this paper we study the necessary and sufficient conditions on domain for
Musielak-Orlicz-Sobolev embedding of the space
$W^{1,\Phi(\cdot,\cdot)}(\Omega)$ where
$\Phi(x,t):=t^{p(x)}{(\log(e+t))}^{q(x)}$.

Both the authors gratefully acknowledge support from DST-SERB
(SRG/2021/000118). The first author also acknowledges Council of Scientific
and Industrial Research (CSIR) for awarding Junior Research Fellowship
(E-certificate no. 22D/23J05002).

Keywords: Orlicz spaces, Orlicz-Sobolev spaces, Musielak-Orlicz spaces,
Musielak-Orlicz-Sobolev spaces, Variable exponent Sobolev spaces.
2020 Mathematics Subject Classification: 46E35, 46E30.

##  1\. Introduction

We assume throughout the paper that $\Omega$ is an open subset of
$\mathbb{R}^{n}$ and the variable exponents p and q are continuous functions
defined on $\Omega$ or $\mathbb{R}^{n}$, satisfying

  1. (p1)

$1\leq p^{-}:=inf_{x\in\mathbb{R}^{n}}p(x)\leq
sup_{x\in\mathbb{R}^{n}}p(x)=:p^{+}<\infty$

  2. (q1)

$-\infty<q^{-}:=inf_{x\in\mathbb{R}^{n}}q(x)\leq
sup_{x\in\mathbb{R}^{n}}q(x)=:q^{+}<\infty$.

The following two conditions on $p$ and $q$ will also be used which, in
literature, are known as the log-Hölder continuous and the log-log-Hölder
continuous respectively:

  1. (p2)

$|p(x)-p(y)|\leq\frac{C}{\log(e+1/|x-y|)}$ whenever $x\in\mathbb{R}^{n}$ and
$y\in\mathbb{R}^{n}$

  2. (q2)

$|q(x)-q(y)|\leq\frac{C}{\log(e+(e+1/|x-y|))}$ whenever $x\in\mathbb{R}^{n}$
and $y\in\mathbb{R}^{n}$.

For the variable exponent Sobolev space $W^{1,p(\cdot)}(\Omega),$ the Sobolev-
type (continuous) embedding $W^{1,p(\cdot)}\hookrightarrow
L^{p^{*}(\cdot)}(\Omega)$ was established in [Die04] for bounded domains with
locally Lipschitz boundary, with the condition $(p2)$ on the exponent $p.$

For Musielak-Orlicz-Sobolev spaces, Sobolev-type embedding have been studied
in [AC24, Fan12, HMOS10, MOS18]. In this paper, we concentrate on the class of
functions $\Phi(x,t):=t^{p(x)}{(\log(e+t))}^{q(x)}$. For this class of
functions $\Phi(x,t)$ the following embedding was established in [HMOS10] for
the space $W^{1,\Phi(\cdot,\cdot)}_{0}(\Omega)$.

######  Theorem 1.1 ([HMOS10]).

Let p satisfy (p1), (p2) and q satisfy (q1), (q2). If $p^{+}<n$ then for every
$u\in W^{1,\Phi(\cdot,\cdot)}_{0}(\Omega),$

$||u||_{L^{\Psi(\cdot,\cdot)}(\Omega)}\leq
C||u||_{W^{1,\Phi(\cdot,\cdot)}(\Omega)},$

where $\Phi(x,t):=t^{p(x)}{(\log(e+t))}^{q(x)}$ and
$\Psi(x,t):=t^{p^{*}(x)}{(\log(e+t))}^{q(x)p^{*}(x)/p(x)}.$ Here $p^{*}(x)$
denotes the Sobolev conjugate of $p(x),$ that is $1/p^{*}(x)=1/p(x)-1/n$.

Here we establish the embedding for $W^{1,\Phi(\cdot,\cdot)}(\Omega)$ for
bounded domains $\Omega$ with Lipschitz boundary.

######  Theorem 1.2.

Let $\Omega$ be an open and bounded set with Lipschitz boundary so that
$\operatorname{diam}(\Omega)>0$. Suppose that the exponent
$p:\Omega\rightarrow[1,\infty)$ is log-Hölder continuous with $1\leq p^{-}\leq
p^{+}<n$ and the exponent $q:\Omega\rightarrow(-\infty,\infty)$ is log-log-
Hölder continuous with $p^{+}+q^{+}\geq 1$. Consider
$\Phi(x,t):=t^{p(x)}(\log(e+t))^{q(x)}$ and
$\Psi(x,t):=t^{p^{*}(x)}(\log(e+t))^{q(x)p^{*}(x)/p(x)},$ where
$1/p^{*}(x)=1/p(x)-1/n$. Then there exists a constant $C$ such that whenever
$u\in W^{1,\Phi(\cdot,\cdot)}(\Omega),$

(1.1) $||u||_{L^{\Psi(\cdot,\cdot)}(\Omega)}\leq
C||u||_{W^{1,\Phi(\cdot,\cdot)}(\Omega)}.$

For the necessary part, it was shown in [GKP21, GKP23] that $\Omega$ must
satisfy the measure density condition to have the embedding
$W^{1,p(\cdot)}(\Omega)\hookrightarrow L^{p^{*}}(\Omega),$ if $p$ satisfies
the log-Hölder condition. Note that $\Omega$ satisfies measure density
condition if there exists a constant $c>0$ such that for every $x$ in
$\bar{\Omega}$ and each $R$ in $]0,1/2],$ one has $|B_{R}(x)\cap\Omega|\geq
cR^{n}.$ This condition was first appeared as a necessary condition for
Sobolev embedding in [HaKT08]. Recently, a weaker version of the measure
density condition, namely log-measure density condition, has appeared in
[HK22] as a necessary condition of certain Orlicz-Sobolev embedding and also
in [GKP23] as a necessary condition of Sobolev-type embedding of
$W^{1,p(\cdot)}(\Omega)$ if $p$ is log-log-Hölder continuous on $\Omega.$

######  Definition 1.3.

A subset $\Omega$ of $\mathbb{R}^{n}$ is said to satisfy the log $s$-measure
density condition if there exist two positive constants $c$ and $\alpha$ such
that for every $x$ in $\bar{\Omega}$ and each $R$ in $]0,1/2]$ one has

$cR^{s}(\log(\frac{1}{R}))^{-\alpha}\leq|B_{R}(x)\cap\Omega|.$

If $s=n$, one says that $\Omega$ satisfies the log-measure density condition.

Here we prove that if the embedding holds, then $\Omega$ satisfies log-measure
density condition.

######  Theorem 1.4.

Let $\Omega$ be an open subset of $\mathbb{R}^{n},$
$\Phi(x,t):=t^{p(x)}{(\log(e+t))}^{q(x)}$ and
$\Psi(x,t):=t^{p^{*}(x)}{(\log(e+t))}^{q(x)p^{*}(x)/p(x)}$ with
${p^{-}+q^{-}}\geq 1,$ where $p^{*}(x)$ denotes the Sobolev conjugate of
$p(x),$ that is, $1/p^{*}(x)=1/p(x)-1/n$. Suppose that

  1. 1.

The exponent $p(\cdot)$ is log-Hölder continuous with $p^{+}<n$,

  2. 2.

$W^{1,\Phi(\cdot,\cdot)}(\Omega)\hookrightarrow
L^{\Psi(\cdot,\cdot)}(\Omega).$

Then $\Omega$ satisfies the log-measure density condition.

Note that, we do not require the log-log-Hölder continuity of $q$ here, unlike
Theorem 1.2. On the other hand, the condition ${p^{-}+q^{-}}\geq 1$ implies
the condition ${p^{+}+q^{+}}\geq 1$ of Theorem 1.2, both of which are trivial
if $q^{-}\geq 0.$

##  2\. Notations and Preliminary Results

######  Definition 2.1.

A function $f:(0,\infty)\rightarrow\mathbb{R}$ is almost increasing if there
exists a constant $a\geq 1$ such that $f(s)\leq af(t)$ for all $0<s<t.$
Similarly, a function $f:(0,\infty)\rightarrow\mathbb{R}$ is almost decreasing
if there exists a constant $b\geq 1$ such that $f(s)\geq bf(t)$ for all
$0<s<t.$

######  Definition 2.2.

Let $f:(0,\infty)\rightarrow\mathbb{R}$ and $p,q>0$.We say that f satisfies

  1. (i)

$(Inc)_{p}$ if $f(t)/t^{p}$ is increasing;

  2. (ii)

$(aInc)_{p}$ if $f(t)/t^{p}$ is almost increasing;

  3. (iii)

$(Deq)_{q}$ if $f(t)/t^{q}$ is decreasing;

  4. (iv)

$(aDeq)_{q}$ if $f(t)/t^{q}$ is almost decreasing.

######  Definition 2.3.

Let $(\Omega,\Sigma,\mu)$ be a $\sigma$-finite, complete measure space. A
function $\Phi:\Omega\times[0,\infty)\rightarrow[0,\infty]$ is said to be a
(generalized) $\Phi$-prefunction on $(\Omega,\Sigma,\mu)$ if
$x\rightarrow\Phi(x,|f(x)|)$ is measurable for every $f\in L^{0}(\Omega,\mu)$
and $\Phi(x,.)$ is a $\Phi$-prefunction for $\mu$-almost every $x\in\Omega$.
We say that the $\Phi$-prefunction $\Phi$ is a (generalized weak)
$\Phi$-function if it satisfies $(aInc)_{1}$. The sets of generalized weak
$\Phi$-function is denoted by $\Phi_{w}(\Omega,\mu)$.

######  Definition 2.4.

We say that $\Phi\in\Phi_{w}(\Omega,\mu)$ satisfies $(A0)$ if there exits a
constant $\beta\in(0,1]$ such that

$\beta\leq\Phi^{-1}(x,1)\leq{1/\beta}$ for $\mu$-almost every $x\in\Omega$.

######  Definition 2.5.

Let $\Phi\in\Phi_{w}(\Omega,\mu)$. We say that $\Phi$ satisfies (A1) if there
exists $\beta\in(0,1)$ such that

$\beta\Phi^{-1}(x,t)\leq\Phi^{-1}(y,t)$

for every $t\in[1,\frac{1}{|B|}]$, almost every $x,y\in B\cap\Omega$ and every
ball $B$ with $|B|\leq 1$.

######  Definition 2.6.

We say that $\Phi\in\Phi_{w}(\Omega,\mu)$ satisfies $(A2)$ if for every $s>0$
there exists $\beta\in(0,1]$ and $h\in L^{1}(\Omega)\cap L^{\infty}(\Omega)$
such that

$\beta\Phi^{-1}(x,t)\leq\Phi^{-1}(y,t)$

for almost every $x,y\in\Omega$ and every $t\in[h(x)+h(y),s]$.

Here we prove an elementary result regarding a class of functions satisfying
the above conditions.

######  Lemma 2.7.

If $p(\cdot)$ satisfies $(p1),(p2),$ and $q(\cdot)$ satisfies $(q1),(q2)$ then
the function $\Phi(x,t):=t^{p(x)}{(\log(e+t))}^{q(x)}$ with $q(x)\geq 0$
satisfies $(A0),(A1),(A2)$ and $(Dec)_{p^{+}+q^{+}}$.

###### Proof.

It is easy to verify the existence of constants $c_{1}$ and $c_{2}$ such that

$\frac{c_{2}t^{\frac{1}{p(x)}}}{(\log(e+t))^{\frac{q(x)}{p(x)}}}\leq\Phi^{-1}(x,t)\leq\frac{c_{1}t^{\frac{1}{p(x)}}}{(\log(e+t))^{\frac{q(x)}{p(x)}}}$

and hence

$\frac{c_{2}}{2^{\frac{q^{+}}{p^{-}}}}\leq\frac{c_{2}}{(\log(e+1))^{\frac{q(x)}{p(x)}}}\leq\Phi^{-1}(x,1)\leq\frac{c_{1}}{(\log(e+1))^{\frac{q(x)}{p(x)}}}\leq
c_{1}.$

Now we can choose $c_{1}\geq 1$ so that $c_{1}c_{2}\geq
2^{\frac{q^{+}}{p^{-}}}$. Then (A0) follows by choosing
$\beta=\frac{1}{c_{1}}.$

To show condition (A1), by symmetry, we may assume that $p(x)<p(y)$. If
$t\in[1,\frac{1}{|B|}]$, then

$\displaystyle\frac{\Phi^{-1}(x,t)}{\Phi^{-1}(y,t)}$ $\displaystyle\leq$
$\displaystyle
c_{1}t^{\frac{1}{p(x)}-\frac{1}{p(y)}}(\log(e+t))^{\frac{q(y)}{p(y)}-\frac{q(x)}{p(x)}}\leq
c_{1}t^{\frac{1}{p(x)}-\frac{1}{p(y)}}(\log(e+t))^{\frac{q(y)-q(x)}{p^{-}}}$
$\displaystyle\leq$ $\displaystyle
c_{1}|B|^{\frac{-c}{\log(e+\frac{1}{|x-y|})}}(\log(e+\frac{1}{|B|}))^{\frac{C}{(p^{-})\log(e+(e+\frac{1}{|x-y|}))}}$
$\displaystyle\leq$ $\displaystyle
c_{1}e^{\frac{cn\log{\frac{1}{|x-y|}}}{\log(e+\frac{1}{|x-y|})}}{e^{\frac{c_{0}\log(e+(e+\frac{1}{|x-y|^{n}}))}{\log(e+(e+\frac{1}{|x-y|}))}}}$
$\displaystyle\leq$ $\displaystyle c_{1}e^{cn}e^{\max\\{c_{0},\frac{c_{0}\ln
n}{\ln(\ln(e+1))}\\}}.$

This yields that $\beta\Phi^{-1}(x,t)\leq\Phi^{-1}(y,t)$ where
$\frac{1}{\beta}=c_{1}e^{cn}e^{\max\\{c_{0},\frac{c_{0}\ln
n}{\ln(\ln(e+1))}\\}}$ and hence (A1) follows.

To show condition (A2), first note that the function $p(\cdot)$ satisfies
Nekvinda’s decay condition, that is, there exist $c_{1}\in(0,1)$ and
$p_{\infty}\in[1,\infty]$ such that

$\int_{(p(x)\not=p_{\infty})}c_{1}^{\frac{1}{|\frac{1}{p_{\infty}}-\frac{1}{p(x)}|}}dx<\infty$.

When $p(x)<p_{\infty},$ it follows from Young’s inequality that

$\displaystyle(c_{1}s)^{p(x)}$ $\displaystyle\leq$ $\displaystyle
s^{p_{\infty}}+c_{1}^{\frac{1}{|\frac{1}{p_{\infty}}-\frac{1}{p(x)}|}}$
$\displaystyle(c_{1}s)^{p(x)}{(\log(e+s))}^{q(x)}$ $\displaystyle\leq$
$\displaystyle
s^{p_{\infty}}{(\log(e+s))}^{q(x)}+c_{1}^{\frac{1}{|\frac{1}{p_{\infty}}-\frac{1}{p(x)}|}}{(\log(e+s))}^{q(x)}$
$\displaystyle\leq$ $\displaystyle
s^{p_{\infty}}(\log(e+s))^{q_{+}}+c_{1}^{\frac{1}{|\frac{1}{p_{\infty}}-\frac{1}{p(x)}|}}{(\log(e+1))}^{q^{+}},$

and the opposite case follows analogously. Hence there holds

$\Phi(x,c_{1}s)\leq\Phi_{\infty}(s)+h(x)$,

where

$h(x):=c_{1}^{\frac{1}{|\frac{1}{p_{\infty}}-\frac{1}{p(x)}|}}{(\log(e+1))}^{q^{+}}.$

So by Lemma 4.2.7 of [HH19], (A2) follows (see also [HHSa24]).

Finally, the condition $(Dec)_{p^{+}+q^{+}}$ follows easily, since we have for
$0\leq s\leq t,$

$\frac{\Phi(x,t)}{t^{p^{+}+q^{+}}}=t^{p(x)-p^{+}-q^{+}}{(\log(e+t))}^{q(x)}\leq
s^{p(x)-p^{+}-q^{+}}{(\log(e+s))}^{q(x)}=\frac{\Phi(x,s)}{s^{p^{+}+q^{+}}}$,

where we have used the fact that the function $t^{Q(x)}{(\log(e+t))}^{q(x)}$
is decreasing when $Q(x)+q^{+}\leq 0.$ ∎

######  Definition 2.8.

Let $\Phi\in\Phi_{w}(\Omega,\mu)$ and let $\rho_{\Phi}$ be given by

$\rho_{\Phi}(f):=\int_{\Omega}\Phi(x,|f(x)|)d\mu(x)$

for all $f\in L^{0}(\Omega,\mu)$. The function $\rho_{\Phi}$ is called a
modular. The set

$L^{\Phi(\cdot,\cdot)}(\Omega,\mu)$ := $\\{f\in
L^{0}(\Omega,\mu):\rho_{\Phi}(\lambda f)<\infty$ for some $\lambda>0\\}$

is called a generalized Orlicz space or Musielak-Orlicz (M-O) space.

######  Definition 2.9.

Let $\Phi\in\Phi_{w}(\Omega,\mu)$. The function $u\in
L^{\Phi(\cdot,\cdot)}\cap L^{1}_{loc}(\Omega)$ belongs to Musielak-Orlicz-
Sobolev space $W^{1,\Phi(\cdot,\cdot)}(\Omega)$ if its weak partial
derivatives $\delta_{\alpha}u$ exist and belong to
$L^{\Phi(\cdot,\cdot)}(\Omega)$ for all $|\alpha|\leq 1$. We define a
semimodular on $W^{1,\Phi(\cdot,\cdot)}(\Omega)$ by

$\rho_{W^{1,\Phi(\cdot,\cdot)}(\Omega)}(u):=\sum_{0\leq|\alpha|\leq
1}\rho_{\Phi}(\delta_{\alpha}u)$.

It induces a (quasi-) norm

$||u||_{W^{1,\Phi(\cdot,\cdot)}(\Omega)}:=\inf\\{\lambda>0:\rho_{W^{1,\Phi(\cdot,\cdot)}(\Omega)}\left(\frac{u}{\lambda}\right)\leq
1\\}$.

Here we prove three lemmas to estimate the norm of the characteristic function
of a measurable set, considering three different sets of values of $q.$

######  Lemma 2.10.

Let $\Phi:\Omega\times[0,\infty)\rightarrow[0,\infty)$ be given by
$\Phi(x,t):=t^{p(x)}(\log(e+t))^{q(x)}$ with $q(x)\geq 0$ for all $x$ and
$A\subset\Omega$ is a measurable set. Then

(2.1)
$\min\\{|A|^{\frac{1}{p_{A}^{+}}},|A|^{\frac{1}{p_{A}^{-}}}\\}\leq\|1_{A}\|_{L^{\Phi(\cdot,\cdot)}(\Omega)}\leq\max\\{|A|^{\frac{1}{p_{A}^{+}}}(\log(e+\frac{1}{|A|}))^{q_{A}^{+}},|A|^{\frac{1}{p_{A}^{-}}}(\log(1+e))^{q_{A}^{+}}\\},$

###### Proof.

We start with the proof of the second inequality of (2.1). Let $u>|A|$ and
assume first that $u\leq 1$. Then

$\displaystyle\bigintsss_{A}\Phi\bigg{(}x,\frac{1}{u^{\frac{1}{p_{A}^{+}}}(\log(e+\frac{1}{u}))^{q_{A}^{+}}}\bigg{)}dx$
$\displaystyle=$
$\displaystyle\bigints_{A}\frac{\bigg{(}\log\Big{(}e+\frac{1}{u^{\frac{1}{p_{A}^{+}}}\big{(}\log(e+\frac{1}{u})\big{)}^{q_{A}^{+}}}\Big{)}\bigg{)}^{q(x)}}{u^{\frac{p(x)}{p_{A}^{+}}}(\log(e+\frac{1}{u}))^{p(x)q_{A}^{+}}}dx$
$\displaystyle\leq$
$\displaystyle\frac{|A|\bigg{(}\log\Big{(}e+\frac{1}{u^{\frac{1}{p_{A}^{+}}}(\log(e+\frac{1}{u}))^{q_{A}^{+}}}\Big{)}\bigg{)}^{q_{A}^{+}}}{u(\log(e+\frac{1}{u}))^{q_{A}^{+}}}$
$\displaystyle<$ $\displaystyle 1,$

where in the final inequality we have used the fact that
$u^{{\frac{1}{p_{A}^{+}}}-1}(\log(e+\frac{1}{u}))^{q_{A}^{+}}\geq 1$. Hence we
have $\|1_{A}\|_{L^{\Phi(\cdot,\cdot)}(\Omega)}\leq
u^{\frac{1}{p_{A}^{+}}}(\log(e+\frac{1}{u}))^{q_{A}^{+}}$. If $u>1$, we can
similarly show that $\|1_{A}\|_{L^{\Phi(\cdot,\cdot)}(\Omega)}\leq
u^{\frac{1}{p_{A}^{-}}}(\log(1+e))^{q_{A}^{+}}$. The second inequality follows
as $u\rightarrow|A|^{+}$.

Let us then prove the first inequality of (2.1). Let $u<|A|$ and assume first
that $u\leq 1$. Then

(2.2)
$\bigintsss_{A}\Phi\Big{(}x,\frac{1}{u^{\frac{1}{p_{A}^{-}}}}\Big{)}dx=\bigintss_{A}\frac{\Big{(}\log\big{(}e+\frac{1}{u^{\frac{1}{p_{A}^{-}}}}\big{)}\Big{)}^{q(x)}}{u^{\frac{p(x)}{p_{A}^{-}}}}dx\geq\frac{|A|}{u}>1.$

Hence we get
$u^{\frac{1}{p_{A}^{-}}}\leq\|1_{A}\|_{L^{\Phi(\cdot,\cdot)}(\Omega)}$. If
$u>1$, we can similarly show that
$u^{\frac{1}{p_{A}^{+}}}\leq\|1_{A}\|_{L^{\Phi(\cdot,\cdot)}(\Omega)}$. The
first inequality follows as $u\rightarrow|A|^{-}$. ∎

######  Lemma 2.11.

Let $\Phi:\Omega\times[0,\infty)\rightarrow[0,\infty)$ be given by
$\Phi(x,t):=t^{p(x)}(\log(e+t))^{q(x)}$ with $q(x)<0$ for all $x$. Assume that
$p_{A}^{-}+q_{A}^{-}\geq 1$ and $A\subset\Omega$ is a measurable set with
$|A|<\frac{1}{2}$. Then

(2.3)
$|A|^{\frac{1}{p_{A}^{-}}}(\log(e+\frac{1}{|A|}))^{\frac{q_{A}^{-}}{p_{A}^{-}}}\leq\|1_{A}\|_{L^{\Phi(\cdot,\cdot)}(\Omega)}\leq\max\\{|A|^{\frac{1}{p_{A}^{+}}},|A|^{\frac{1}{p_{A}^{-}}}\\},$

###### Proof.

We start with the proof of the second inequality of (2.3). Let $u>|A|$ and
assume that $u\leq 1$. Then we have

(2.4)
$\bigintsss_{A}\Phi\big{(}x,\frac{1}{u^{\frac{1}{p_{A}^{+}}}}\big{)}dx=\bigints_{A}\frac{\big{(}\log(e+\frac{1}{u^{\frac{1}{p_{A}^{+}}}})\big{)}^{q(x)}}{u^{\frac{p(x)}{p_{A}^{+}}}}dx\leq\frac{|A|}{u}<1,$

and hence
$u^{\frac{1}{p_{A}^{+}}}\geq\|1_{A}\|_{L^{\Phi(\cdot,\cdot)}(\Omega)}$. If
$u>1$, we can similarly show that
$u^{\frac{1}{p_{A}^{-}}}\geq\|1_{A}\|_{L^{\Phi(\cdot,\cdot)}(\Omega)}$. The
second inequality follows as $u\rightarrow|A|^{+}$.

Let us then prove the first inequality of (2.3). Let $u<|A|$. Then

$\displaystyle\bigintss_{A}\Phi\bigg{(}x,\frac{1}{u^{\frac{1}{p_{A}^{-}}}(\log(e+\frac{1}{u}))^{\frac{q_{A}^{-}}{p_{A}^{-}}}}\bigg{)}dx$
$\displaystyle=$
$\displaystyle\bigints_{A}\frac{\bigg{(}\log(e+\frac{1}{u^{\frac{1}{p_{A}^{-}}}\big{(}\log(e+\frac{1}{u})\big{)}^{\frac{q_{A}^{-}}{p_{A}^{-}}}})\bigg{)}^{q(x)}}{u^{\frac{p(x)}{p_{A}^{-}}}(\log(e+\frac{1}{u}))^{p(x)\frac{q_{A}^{-}}{p_{A}^{-}}}}dx$
$\displaystyle\geq$
$\displaystyle\frac{|A|\bigg{(}\log(e+\frac{1}{u^{\frac{1}{p_{A}^{-}}}\big{(}\log(e+\frac{1}{u})\big{)}^{\frac{q_{A}^{-}}{p_{A}^{-}}}})\bigg{)}^{q_{A}^{-}}}{u\big{(}\log(e+\frac{1}{u})\big{)}^{q_{A}^{-}}}$
$\displaystyle>$ $\displaystyle 1,$

where we have used the fact that
$u^{\frac{1}{p_{A}^{-}}-1}(\log(e+\frac{1}{u}))^{\frac{q_{A}^{-}}{p_{A}^{-}}}>1$
under the assumption $p_{A}^{-}+q_{A}^{-}\geq 1$ which follows from the
increasing property of the function $\Phi(t)=\frac{t^{r}}{(\log(e+t))^{m}}$
for all $t>2$ when $r\geq m,$ $m>0.$ Therefore we have
$u^{\frac{1}{p_{A}^{-}}}(\log(e+\frac{1}{u}))^{\frac{q_{A}^{-}}{p_{A}^{-}}}\leq||1_{A}||_{L^{\Phi(\cdot,\cdot)}(\Omega)}.$
The first inequality follows as $u\rightarrow|A|^{-}$. ∎

######  Lemma 2.12.

Let $\Phi:\Omega\times[0,\infty)\rightarrow[0,\infty)$ be given by
$\Phi(x,t):=t^{p(x)}(\log(e+t))^{q(x)}$ where $q(x)<0$ for some $x$ and
$q(x)\geq 0$ for some $x$. Assume that $p_{A}^{-}+q_{A}^{-}\geq 1$ and
$A\subset\Omega$ is a measurable set with $|A|<\frac{1}{2}$. Then, there exist
constants $b_{1}>0$, $b_{2}>0$ such that

(2.5)
$b_{1}|A|^{\frac{1}{p_{A}^{-}}}\big{(}\log(e+\frac{1}{|A|})\big{)}^{\frac{q_{A}^{-}}{p_{A}^{-}}}\leq\|1_{A}\|_{L^{\Phi(\cdot,\cdot)}(\Omega)}\leq
b_{2}\max\\{|A|^{\frac{1}{p_{A}^{+}}}\big{(}\log(e+\frac{1}{|A|})\big{)}^{q_{A}^{+}},|A|^{\frac{1}{p_{A}^{-}}}\big{(}\log(1+e)\big{)}^{q_{A}^{+}}\\},$

###### Proof.

We start with the proof of the second inequality of (2.5). Let $u>|A|$ and
assume first that $u\leq 1$. Then

$\displaystyle\bigintsss_{A}\Phi\Big{(}x,\frac{1}{2u^{\frac{1}{p_{A}^{+}}}(\log\big{(}e+\frac{1}{2u}\big{)}\big{)}^{q_{A}^{+}}}\Big{)}dx$
$\displaystyle=$
$\displaystyle\bigintss_{A}\frac{\bigg{(}\log\Big{(}e+\frac{1}{2u^{\frac{1}{p_{A}^{+}}}\big{(}\log\big{(}e+\frac{1}{2u}\big{)}\big{)}^{q_{A}^{+}}}\Big{)}\bigg{)}^{q(x)}}{2^{p(x)}u^{\frac{p(x)}{p_{A}^{+}}}\big{(}\log(e+\frac{1}{2u})\big{)}^{p(x)q_{A}^{+}}}dx$
$\displaystyle=$ $\displaystyle\bigintss_{A\cap(x:q(x)\geq
0)}\frac{\bigg{(}\log\Big{(}e+\frac{1}{2u^{\frac{1}{p_{A}^{+}}}\big{(}\log\big{(}e+\frac{1}{2u}\big{)}\big{)}^{q_{A}^{+}}}\Big{)}\bigg{)}^{q(x)}}{2^{p(x)}u^{\frac{p(x)}{p_{A}^{+}}}\bigg{(}\log\big{(}e+\frac{1}{2u}\big{)}\bigg{)}^{p(x)q_{A}^{+}}}dx$
$\displaystyle+$
$\displaystyle\bigintss_{A\cap(x:q(x)<0)}\frac{\bigg{(}\log\Big{(}e+\frac{1}{2u^{\frac{1}{p_{A}^{+}}}\big{(}\log\big{(}e+\frac{1}{2u}\big{)}\big{)}^{q_{A}^{+}}}\Big{)}\bigg{)}^{q(x)}}{2^{p(x)}u^{\frac{p(x)}{p_{A}^{+}}}\big{(}\log\big{(}e+\frac{1}{2u}\big{)}\big{)}^{p(x)q_{A}^{+}}}dx$
$\displaystyle\leq$
$\displaystyle\frac{1}{2}\frac{|A|\bigg{(}\log\Big{(}e+\frac{1}{2u^{\frac{1}{p_{A}^{+}}}\big{(}\log\big{(}e+\frac{1}{2u}\big{)}\big{)}^{q_{A}^{+}}}\Big{)}\bigg{)}^{q_{A}^{+}}}{u\big{(}\log\big{(}e+\frac{1}{2u}\big{)}\big{)}^{q_{A}^{+}}}+\frac{|A|}{2u}$
$\displaystyle<$ $\displaystyle 1,$

where in the final inequality we have used the fact that
$u^{{\frac{1}{p_{A}^{+}}}-1}(\log(e+\frac{1}{u}))^{q_{A}^{+}}\geq 1$. Hence we
have $\|1_{A}\|_{L^{\Phi(\cdot,\cdot)}(\Omega)}\leq
2u^{\frac{1}{p_{A}^{+}}}(\log(e+\frac{1}{2u}))^{q_{A}^{+}}$. If $u>1$, we can
similarly show that $\|1_{A}\|_{L^{\Phi(\cdot,\cdot)}(\Omega)}\leq
u^{\frac{1}{p_{A}^{-}}}(\log(1+e))^{q_{A}^{+}}$. The second inequality follows
as $u\rightarrow|A|^{+}$.
Let us then prove the first inequality of (2.5). Let $u<|A|$. Then

$\displaystyle\bigintss_{A}\Phi\bigg{(}x,\frac{1}{2^{\frac{1}{{p_{A}}^{+}}}u^{\frac{1}{p_{A}^{-}}}\big{(}\log(e+\frac{1}{u2^{1/{p_{A}}^{+}}})\big{)}^{\frac{q_{A}^{-}}{p_{A}^{-}}}}\bigg{)}dx$
$\displaystyle=$
$\displaystyle\bigints_{A}\frac{\bigg{(}\log(e+\frac{1}{2^{\frac{1}{{p_{A}}^{+}}}u^{\frac{1}{p_{A}^{-}}}\big{(}\log(e+\frac{1}{u2^{1/{p_{A}}^{+}}})\big{)}^{\frac{q_{A}^{-}}{p_{A}^{-}}}})\bigg{)}^{q(x)}}{2^{\frac{p(x)}{{p_{A}}^{+}}}u^{\frac{p(x)}{p_{A}^{-}}}(\log(e+\frac{1}{u2^{1/{p_{A}}^{+}}}))^{p(x)\frac{q_{A}^{-}}{p_{A}^{-}}}}dx$
$\displaystyle=$ $\displaystyle\bigintss_{A\cap(x:q(x)\geq
0)}\frac{\bigg{(}\log(e+\frac{1}{2^{\frac{1}{{p_{A}}^{+}}}u^{\frac{1}{p_{A}^{-}}}\big{(}\log(e+\frac{1}{u2^{1/{p_{A}}^{+}}})\big{)}^{\frac{q_{A}^{-}}{p_{A}^{-}}}})\bigg{)}^{q(x)}}{2^{\frac{p(x)}{{p_{A}}^{+}}}u^{\frac{p(x)}{p_{A}^{-}}}\big{(}\log(e+\frac{1}{u2^{1/{p_{A}}^{+}}})\big{)}^{p(x)\frac{q_{A}^{-}}{p_{A}^{-}}}}dx$
$\displaystyle+$
$\displaystyle\bigintss_{A\cap(x:q(x)<0)}\frac{\bigg{(}\log(e+\frac{1}{2^{\frac{1}{{p_{A}}^{+}}}u^{\frac{1}{p_{A}^{-}}}\big{(}\log(e+\frac{1}{u2^{1/{p_{A}}^{+}}})\big{)}^{\frac{q_{A}^{-}}{p_{A}^{-}}}})\bigg{)}^{q(x)}}{2^{\frac{p(x)}{{p_{A}}^{+}}}u^{\frac{p(x)}{p_{A}^{-}}}\big{(}\log(e+\frac{1}{u2^{1/{p_{A}}^{+}}})\big{)}^{p(x)\frac{q_{A}^{-}}{p_{A}^{-}}}}dx$
$\displaystyle\geq$
$\displaystyle\frac{|A|}{2u}+\frac{|A|\bigg{(}\log(e+\frac{1}{2^{\frac{1}{{p_{A}}^{+}}}u^{\frac{1}{p_{A}^{-}}}\big{(}\log(e+\frac{1}{u2^{1/{p_{A}}^{+}}})\big{)}^{\frac{q_{A}^{-}}{p_{A}^{-}}}})\bigg{)}^{q_{A}^{-}}}{2u\big{(}\log(e+\frac{1}{u2^{1/{p_{A}}^{+}}})\big{)}^{q_{A}^{-}}}$
$\displaystyle>$ $\displaystyle 1,$

where in the final inequality we have used the fact that
$u^{\frac{1}{p_{A}^{-}}-1}(\log(e+\frac{1}{u2^{1/{p_{A}}^{+}}}))^{\frac{q_{A}^{-}}{p_{A}^{-}}}>1$
when $p_{A}^{-}+q_{A}^{-}\geq 1.$ Therefore we have
$2^{\frac{1}{{p_{A}}^{+}}}u^{\frac{1}{p_{A}^{-}}}(\log(e+\frac{1}{u2^{1/{p_{A}}^{+}}}))^{\frac{q_{A}^{-}}{p_{A}^{-}}}\leq\|1_{A}\|_{L^{\Phi(\cdot,\cdot)}(\Omega)}.$
The first inequality follows as $u\rightarrow|A|^{-}$. ∎

In the following lemma, we extend the exponent $p$ from $\Omega$ to
$\mathbb{R}^{n}$ preserving the modulus of continuity as well as upper and
lower bounds, using the technique of Edmunds and Rákosník [ER00, Theorem 4.1]
which was originally introduced by Hestenes [Hes41]. The same technique was
also used in [Die04]. We recall the proof here for the convenience of the
readers. Also note that, here we will use the lemma only to extend the
exponent functions.

######  Lemma 2.13.

Let $\Omega\subset\mathbb{R}^{n}$ be an open, bounded set with Lipschitz
boundary. Let $p:\Omega\rightarrow(-\infty,\infty)$ satisfy the uniform
continuity condition

$|p(x)-p(y)|\leq\rho(|x-y|)$ for all $x,y\in\Omega$

where $\rho$ is concave for $t\geq 0$ and $\rho(t)\rightarrow 0$ for
$t\rightarrow 0^{+}$. Then there exists an extension $p_{1}$ on
$\mathbb{R}^{n}$ of $p$ and a constant $C>0$, such that

$|p_{1}(x)-p_{1}(y)|\leq\rho(C|x-y|)$ for all $x,y\in\Omega$

Moreover, there holds $p_{1}^{-}=p^{-}$ and $p_{1}^{+}=p^{+}$.

###### Proof.

Let ${V_{j}},j=1,...,k,$ be the covering of the boundary $\partial\Omega$
which corresponds to the local description of $\partial\Omega$. More
precisely, for each $j=1,...,k,$ there is a local coordinate system
$(x^{\prime},x_{n})$ such that

$V_{j}=\\{(x^{\prime},x_{n}):|x_{i}|<\delta,i=1,...,n-1,{a_{j}}(x^{\prime})-\beta<x_{n}<{a_{j}}(x^{\prime})+\beta\\},$
$V_{j}\cap\Omega=\\{x\in
V_{j}:{a_{j}}(x^{\prime})<x_{n}<{a_{j}}(x^{\prime})+\beta\\}$

and

$\\{x\in\bar{V_{j}}:x_{n}<{a}(x^{\prime})\\}\cap\bar{\Omega}=\varnothing,$

where $\beta,\delta$ are some fixed positive numbers and ${a_{j}}\in
C^{0,1}((-\delta,\delta)^{n-1})$ are the functions describing the boundary.
Define the mappings

$T_{j}:(-\delta,\delta)^{n-1}\times(-\beta,\beta)\rightarrow\mathbb{R}^{n},\quad
j=1,...,n,$

by

$T_{j}(x^{\prime},x_{n})=(x^{\prime},x_{n}+a_{j}(x^{\prime})).$

Then the $T_{j}$ are bi-Lipschitz mappings. To these flattened domains
${{T_{j}}^{-1}}(V_{j})$, define the reflection operator

$Ef(x)=\begin{cases}f(x^{\prime},x_{n})\quad for\quad x_{n}\geq 0,\\\
f(x^{\prime},x_{n})\quad for\quad x_{n}<0\end{cases}$

and the functions ${p_{1}}_{j}$ on $V_{j}\cup\Omega$ by

${p_{1}}_{j}(x)=\begin{cases}p(x)\quad for\quad x\in\Omega,\\\
Er_{j}({{T_{J}}^{-1}}(x)))\quad for\quad x\in V_{j}/\Omega,\end{cases}$

where $r_{j}:=p\circ T_{j}.$ Note that since $E,T_{j},$ and ${T_{j}}^{-1}$ are
Lipschitz there exists $C>0$ such that

$|{{p_{1}}_{j}}(x)-{{p_{1}}_{j}}(y)|\leq\rho(C|x-y|)$ for all $x,y\in\Omega$

Then extend the functions ${p_{1}}_{j}$ on $\Omega$ to $\tilde{{p}_{1}}_{j}$
on $\mathbb{R}^{n}$ preserving their upper and lower bounds. Note that this
extension is possible due to McShane [McS34, Theorem 2 and Corollary 2] and
the fact that $\rho$ is concave with $\rho(t)\rightarrow 0$ for $t\rightarrow
0^{+}$. Define $p_{1}:\mathbb{R}^{n}\rightarrow(-\infty,\infty)$ by

${p_{1}}(x):=\min\limits_{{j=1,...,k}}{\tilde{{p}_{1}}_{j}}(x)$ for
$x\in\mathbb{R}^{n}$

Thus there holds

$|p_{1}(x)-p_{1}(y)|\leq\rho(C|x-y|)$ for all $x,y\in\Omega$

This proves the theorem. ∎

######  Lemma 2.14.

Let $\Omega\subset\mathbb{R}^{n}$ be an open, bounded set with Lipschitz
boundary. Suppose that $p$ satisfies $(p1),$ $(p2)$ and $q$ satisfies $(q1)$
and $(q2).$ Then there exists an extension $p_{1}$ on $\mathbb{R}^{n}$ of $p$
with $p_{1}^{-}=p^{-},$ $p_{1}^{+}=p^{+}$ and an extension $q_{1}$ on
$\mathbb{R}^{n}$ of $q$ with $q_{1}^{-}=q^{-}$ and $q_{1}^{+}=q^{+}$, which
satisfies the same local uniform continuity conditions (with possibly
different constants).

###### Proof.

Since the mapping $\rho:t\rightarrow C/\log(e+1/t)$ is concave for $t\geq 0$
and $\rho(t)\rightarrow 0$ for $t\rightarrow 0^{+}$ and $p$ satisfies
uniformly the local continuity condition such that

$|p(x)-p(y)|\leq\rho(|x-y|)$ for all $x,y\in\Omega$,

Due to Lemma 2.13 it follows that there exists an extension $p_{1}$ on
$\mathbb{R}^{n}$ of $p$, which possesses all the desired properties. The proof
for the extension of $q$ is similar. ∎

######  Theorem 2.15 ([Juu23]).

Let $\Omega$ be an $(\epsilon,\delta)$ -domain with $rad(\Omega)>0.$ Suppose
that $\Phi\in\Phi_{w}(\Omega,\mu)$ satisfies $(A0),(A1),(A2)$ and $(aDec)_{q}$
with $q\geq 1.$ Let $\Psi\in\Phi_{w}(\mathbb{R}^{n},\mu)$ be the extension of
$\Phi$ which also satisfies $(A0),(A1),(A2)$ and $(aDec)_{q}$ with $q\geq 1.$
Then there exists an operator
$\Lambda:W^{1,\Phi(\cdot,\cdot)}(\Omega)\hookrightarrow
W^{1,\Psi(\cdot,\cdot)}(\Omega)$ and a constant $B$ such that

$||\Lambda u||_{W^{1,\Psi(\cdot,\cdot)}(\mathbb{R}^{n})}\leq
B||u||_{W^{1,\Phi(\cdot,\cdot)}(\Omega)},$

for every $u\in W^{1,\Phi(\cdot,\cdot)}(\Omega).$

##  3\. Main Results

Proof of Theorem 1.2 By Lemma 2.14, we obtain an extension $p_{1}$ on
$\mathbb{R}^{n}$ of $p$ and an extension $q_{1}$ on $\mathbb{R}^{n}$ of $q.$
Consider $\Phi_{1}(x,t):=t^{p_{1}(x)}\ {(\log(e+t))}^{q_{1}(x)}$ and
${\Psi_{1}}(x,t):=t^{{p_{1}}^{*}(x)}(\log(e+t))^{{q_{1}}(x){p_{1}}^{*}(x)/{p_{1}}(x)}.$
Note that $p_{1}$ satisfies the conditions $(p1)$ and $(p2)$ whereas $q_{1}$
satisfies the conditions $(q1)$ and $(q2),$ and therefore by Lemma 2.7,
$\Phi_{1}$ satisfies $(A0),(A1),(A2)$ and $(Dec)_{p_{1}^{+}+q_{1}^{+}}$. Since
${p^{+}+q^{+}}\geq 1,$ we get, by Theorem 2.15, a linear extension operator
$\mathcal{E}:W^{1,\Phi(\cdot,\cdot)}(\Omega)\rightarrow
W^{1,\Phi_{1}(\cdot,\cdot)}(\mathbb{R}^{n})$ and a constant $c_{1}$ such that

(3.1) $||v||_{W^{1,\Phi_{1}(\cdot,\cdot)}(\mathbb{R}^{n})}\leq
c_{1}||u||_{W^{1,\Phi(\cdot,\cdot)}(\Omega)}$

and $v|_{\Omega}=u$ for all $u\in W^{1,\Phi_{1}(\cdot,\cdot)}(\Omega),$ where
$\mathcal{E}u=:v.$ On the other hand, using Theorem 1.1, we get a constant
$c_{2}$ such that

(3.2) $||v||_{L^{\Psi_{1}(\cdot,\cdot)}(\mathbb{R}^{n})}\leq
c_{2}||v||_{W^{1,\Phi_{1}(\cdot,\cdot)}(\mathbb{R}^{n})}$

for all $v\in W^{1,\Phi_{1}(\cdot,\cdot)}_{0}(\mathbb{R}^{n}).$ Also, since
$\Phi_{1}$ satisfies $(A0),(A1),(A2)$ and $(Dec)_{p_{1}^{+}+q_{1}^{+}}$, by
Theorem 6.4.4 of [HH19] we have
$W^{1,\Phi_{1}(\cdot,\cdot)}_{0}(\mathbb{R}^{n})=W^{1,\Phi_{1}(\cdot,\cdot)}(\mathbb{R}^{n})$.
Hence the inequality (3.2) holds for all $v\in
W^{1,\Phi_{1}(\cdot,\cdot)}(\mathbb{R}^{n}).$ Finally, we use the inequalities
(3.1), (3.2) and the facts that $\Phi_{1},$ $\Psi_{1}$ are the extensions of
$\Phi$ and $\Psi$ respectively, we obtain, for all $u\in
W^{1,\Phi(\cdot,\cdot)}(\Omega),$

$\displaystyle||u||_{L^{\Psi(\cdot,\cdot)}(\Omega)}=||v||_{L^{\Psi(\cdot,\cdot)}(\Omega)}\leq||v||_{L^{\Psi_{1}(\cdot,\cdot)}(\mathbb{R}^{n})}\leq
c_{2}||v||_{W^{1,\Phi_{1}(\cdot,\cdot)}(\mathbb{R}^{n})}$ $\displaystyle\leq$
$\displaystyle C||u||_{W^{1,\Phi(\cdot,\cdot)}(\Omega)},$

where $C=c_{1}c_{2}.$ ∎

######  Remark 3.1.

Note that in the above proof, we are using Lemma 2.14 only to extend the
functions $p$ and $q$ which satisfy the same local uniform continuity
condition and not using the boundedness of the linear extension operator. We
do not know, if we can avoid the lemma and prove the extension of the
functions $p$ and $q$ in more general domains.

Proof of Theorem 1.4 For a fixed $x$ in $\bar{\Omega}$ define
$A_{R}:=B_{R}(x)\cap\Omega$. It is enough to consider the case when
$|A_{R}|\leq 1$, otherwise $|A_{R}|\geq 1\geq R^{n}$ whenever $R\leq 1$ and
there is nothing to prove. Moreover, it is enough to consider $R\leq r_{0}$
for some $0<r_{0}\leq 1/4.$ For such an $R$, denote by $\tilde{R}\leq R$ the
smallest real number such that

$|A_{\tilde{R}}|=\frac{1}{2}|A_{R}|$

To prove Theorem 1.4, we need following Lemma:

######  Lemma 3.2.

If we have the same assumptions as in Theorem 1.4, then there exist positive
constants $C_{1}$, $C_{2}$, $C_{3}$ such that for all $x$ in $\bar{\Omega}$
and every $R$ in $]0,1]$ we have

$R-\tilde{R}\leq
C_{1}|A_{R}|^{\frac{1}{n}+\frac{1}{p_{A_{R}}^{+}}-\frac{1}{p_{A_{R}}^{-}}}(\log(e+\frac{1}{|A_{R}|}))^{q_{A_{R}}^{+}}$

when $q(x)\geq 0$ for all $x$,

$R-\tilde{R}\leq
C_{2}|A_{R}|^{\frac{1}{n}+\frac{1}{p_{A_{R}}^{+}}-\frac{1}{p_{A_{R}}^{-}}}(\log(e+\frac{1}{|A_{R}|}))^{Q_{A_{R}}}$

when $q(x)<0$ for all $x$, and

$R-\tilde{R}\leq
C_{3}|A_{R}|^{\frac{1}{n}+\frac{1}{p_{A_{R}}^{+}}-\frac{1}{p_{A_{R}}^{-}}}(\log(e+\frac{1}{|A_{R}|}))^{X_{A_{R}}}$

when $q(x)<0$ for some $x$ and $q(x)\geq 0$ for some $x$.

###### Proof.

Since $W^{1,\Phi(\cdot,\cdot)}(\Omega)\hookrightarrow
L^{\Psi(\cdot,\cdot)}(\Omega),$ there exists a constant $c_{1}>0$ such that
whenever $u\in W^{1,\Phi(\cdot,\cdot)}(\Omega)$ one has the inequality

(3.3) $||u||_{L^{\Psi(\cdot,\cdot)}(\Omega)}\leq
c_{1}||u||_{W^{1,\Phi(\cdot,\cdot)}(\Omega)}.$

For a fixed $x\in\bar{\Omega}$ let $u(y):=\phi(y-x),$ where $y\in\Omega$ and
$\phi$ is a cut-off function so that

  1. (1)

$\phi:\mathbb{R}^{n}\rightarrow[0,1]$,

  2. (2)

$\hbox{spt}\ \phi\subset B_{R}(0)$,

  3. (3)

$\phi|_{B_{\tilde{R}}(0)}=1$, and

  4. (4)

$|\nabla\phi|\leq\tilde{c}/(R-\tilde{R})$ for some constant $\tilde{c}$.

Note that we have the inequalities

$\|1_{B_{\tilde{R}}}\|_{L^{\Psi(\cdot,\cdot)(\Omega)}}\leq\|u\|_{L^{\Psi(\cdot,\cdot)(\Omega)}},\quad\quad\|u\|_{L^{\Phi(\cdot,\cdot)(\Omega)}}\leq\|1_{B_{R}}\|_{L^{\Phi(\cdot,\cdot)(\Omega)}}$

and

$\|\nabla
u\|_{L^{\Phi(\cdot,\cdot)(\Omega)}}\leq\frac{\tilde{c}}{R-\tilde{R}}\|1_{B_{R}\thicksim
B_{\tilde{R}}}\|_{L^{\Phi(\cdot,\cdot)(\Omega)}}\leq\frac{\tilde{c}}{R-\tilde{R}}\
\|1_{B_{R}}\|_{L^{\Phi(\cdot,\cdot)(\Omega)}}.$

Use these inequalities in inequality (3.3) to obtain

$\displaystyle\|1_{B_{\tilde{R}}}\|_{L^{\Psi(\cdot,\cdot)(\Omega)}}$
$\displaystyle\leq$ $\displaystyle
c_{1}(\|1_{B_{R}}\|_{L^{\Phi(\cdot,\cdot)(\Omega)}}+\frac{\tilde{c}}{R-\tilde{R}}\
\|1_{B_{R}}\|_{L^{\Phi(\cdot,\cdot)(\Omega)}})$ $\displaystyle\leq$
$\displaystyle\frac{2c_{1}\max\\{1,\tilde{c}\\}}{R-\tilde{R}}\|1_{B_{R}}\|_{L^{\Phi(\cdot,\cdot)(\Omega)}}$

and hence

$\displaystyle
R-\tilde{R}\leq{c_{2}}\frac{\|1_{B_{R}}\|_{L^{\Phi(\cdot,\cdot)(\Omega)}}}{\|1_{B_{\tilde{R}}}\|_{L^{\Psi(\cdot,\cdot)(\Omega)}}},$

where $c_{2}:=2c_{1}\max\\{1,\tilde{c}\\}.$

Case-1 ($q(x)\geq 0$ for all $x\in\Omega$): Using the norm estimates in Lemma
2.10, we get

$\displaystyle R-\tilde{R}$ $\displaystyle\leq$ $\displaystyle c_{2}\
\frac{|A_{R}|^{\frac{1}{p_{A_{R}}^{+}}}(\log(e+\frac{1}{|A_{R}|}))^{q_{A_{R}}^{+}}}{|A_{\tilde{R}}|^{\frac{1}{p*_{A_{R}^{-}}}}}$
$\displaystyle=$ $\displaystyle c_{2}\
\frac{|A_{R}|^{\frac{1}{p_{A_{R}}^{+}}}(\log(e+\frac{1}{|A_{R}|}))^{q_{A_{R}}^{+}}}{|A_{\tilde{R}}|^{\frac{1}{p_{A_{R}}^{-}}-\frac{1}{n}}}$
$\displaystyle=$ $\displaystyle c_{2}\
2^{\frac{1}{p_{A_{R}}^{-}}-\frac{1}{n}}|A_{R}|^{\frac{1}{p_{A_{R}}^{+}}-\frac{1}{p_{A_{R}}^{-}}+\frac{1}{n}}(\log(e+\frac{1}{|A_{R}|}))^{q_{A_{R}}^{+}}$
$\displaystyle\leq$ $\displaystyle c_{2}\ 2^{\frac{1}{p^{-}}-\frac{1}{n}}\
|A_{R}|^{\frac{1}{p_{A_{R}}^{+}}-\frac{1}{p_{A_{R}}^{-}}+\frac{1}{n}}(\log(e+\frac{1}{|A_{R}|}))^{q_{A_{R}}^{+}}$

as claimed.

Case-2 ($q(x)\leq 0$ for all $x\in\Omega$): Using the norm estimates in Lemma
2.11, we get

$\displaystyle R-\tilde{R}$ $\displaystyle\leq$ $\displaystyle c_{2}\
\frac{|A_{R}|^{\frac{1}{p_{A_{R}}^{+}}}}{|A_{\tilde{R}}|^{\frac{1}{p*_{A_{R}}^{-}}}(\log(e+\frac{1}{|A_{\tilde{R}}|}))^{\frac{q_{A_{R}}^{-}p*_{A_{R}}^{+}}{p*_{A_{R}}^{-}p_{A_{R}}^{+}}}}$
$\displaystyle=$ $\displaystyle c_{2}\
\frac{|A_{R}|^{\frac{1}{p_{A_{R}}^{+}}}}{|A_{\tilde{R}}|^{\frac{1}{p_{A_{R}}^{-}}-\frac{1}{n}}(\log(e+\frac{1}{|A_{\tilde{R}}|}))^{-Q_{A_{R}}}}$
$\displaystyle\equiv$ $\displaystyle c_{2}\
2^{\frac{1}{p_{A_{R}}^{-}}-\frac{1}{n}}|A_{R}|^{\frac{1}{p_{A_{R}}^{+}}-\frac{1}{p_{A_{R}}^{-}}+\frac{1}{n}}(\log(e+\frac{1}{|A_{R}|}))^{Q_{A_{R}}}$
$\displaystyle\leq$ $\displaystyle c_{2}\ 2^{\frac{1}{p^{-}}-\frac{1}{n}}\
|A_{R}|^{\frac{1}{p_{A_{R}}^{+}}-\frac{1}{p_{A_{R}}^{-}}+\frac{1}{n}}(\log(e+\frac{1}{|A_{R}|}))^{Q_{A_{R}}}$

where
$Q_{A_{R}}=\frac{-q_{A_{R}}^{-}(n-p_{A_{R}}^{-})}{p_{A_{R}}^{-}(n-p_{A_{R}}^{+})}\geq
0.$
Case-3 ($q(x)<0$ for some $x\in\Omega$ and $q(x)\geq 0$ for some
$x\in\Omega$): Using the norm estimates in Lemma 2.12, we get

$\displaystyle R-\tilde{R}$ $\displaystyle\leq$ $\displaystyle c_{2}^{\prime}\
\frac{|A_{R}|^{\frac{1}{p_{A_{R}}^{+}}}(\log(e+\frac{1}{|A_{R}|}))^{q_{A_{R}}^{+}}}{|A_{\tilde{R}}|^{\frac{1}{p*_{A_{R}}^{-}}}(\log(e+\frac{1}{|A_{\tilde{R}}|}))^{\frac{q_{A_{R}}^{-}p*_{A_{R}}^{+}}{p*_{A_{R}}^{-}p_{A_{R}}^{+}}}}$
$\displaystyle=$ $\displaystyle c_{2}^{\prime}\
\frac{|A_{R}|^{\frac{1}{p_{A_{R}}^{+}}}(\log(e+\frac{1}{|A_{R}|}))^{q_{A_{R}}^{+}}}{|A_{\tilde{R}}|^{\frac{1}{p_{A_{R}}^{-}}-\frac{1}{n}}(\log(e+\frac{1}{|A_{\tilde{R}}|}))^{-Q_{A_{R}}}}$
$\displaystyle\equiv$ $\displaystyle c_{2}^{\prime}\
2^{\frac{1}{p_{A_{R}}^{-}}-\frac{1}{n}}|A_{R}|^{\frac{1}{p_{A_{R}}^{+}}-\frac{1}{p_{A_{R}}^{-}}+\frac{1}{n}}(\log(e+\frac{1}{|A_{R}|}))^{T_{A_{R}}}$
$\displaystyle\leq$ $\displaystyle c_{2}^{\prime}\
2^{\frac{1}{p^{-}}-\frac{1}{n}}\
|A_{R}|^{\frac{1}{p_{A_{R}}^{+}}-\frac{1}{p_{A_{R}}^{-}}+\frac{1}{n}}(\log(e+\frac{1}{|A_{R}|}))^{T_{A_{R}}}$
$\displaystyle\leq$ $\displaystyle c_{2}^{\prime}\
2^{\frac{1}{p^{-}}-\frac{1}{n}}\
|A_{R}|^{\frac{1}{p_{A_{R}}^{+}}-\frac{1}{p_{A_{R}}^{-}}+\frac{1}{n}}(\log(e+\frac{1}{|A_{R}|}))^{S_{A_{R}}}$

where $c_{2}^{\prime}=\frac{c_{2}b_{2}}{b_{1}}$,
$T_{A_{R}}=q_{A_{R}}^{+}+Q_{A_{R}}=q_{A_{R}}^{+}-\frac{q_{A_{R}}^{-}(n-p_{A_{R}}^{-})}{p_{A_{R}}^{-}(n-p_{A_{R}}^{+})}$
and $S_{A_{R}}=\max\\{q_{A_{R}}^{+},Q_{A_{R}},T_{A_{R}}\\}$
∎

To continue the proof of Theorem 1.4, construct the sequence $\\{R_{i}\\}$ by
setting $R_{0}:=R$, and then define $R_{i+1}:=\tilde{R_{i}}$ inductively for
$i\geq 0$. It follows that

$|A_{R_{i}}|=\frac{1}{2^{i}}\ |A_{R}|,$

with $\lim_{i\rightarrow\infty}R_{i}=0$.

Case-1 ($q(x)\geq 0$ for all $x\in\Omega$): Using Lemma 3.2 one obtains

$\displaystyle R_{i}-R_{i+1}$ $\displaystyle\leq$ $\displaystyle
C_{1}|A_{R_{i}}|^{\frac{1}{n}+\frac{1}{p_{A_{R_{i}}}^{+}}-\frac{1}{p_{A_{R_{i}}}^{-}}}(\log(e+\frac{1}{|A_{R_{i}}|}))^{q_{A_{R_{i}}}^{+}}$
$\displaystyle\leq$ $\displaystyle
C_{1}|A_{R_{i}}|^{\frac{1}{n}+\frac{1}{p_{A_{R}}^{+}}-\frac{1}{p_{A_{R}}^{-}}}(\log(e+\frac{1}{|A_{R_{i}}|}))^{q_{A_{R}}^{+}}$
$\displaystyle=$ $\displaystyle
C_{1}\frac{|A_{R}|^{\eta_{R}}}{2^{i\eta_{R}}}(\log(e+\frac{2^{i}}{|A_{R}|}))^{q_{A_{R}}^{+}},$

where $\eta_{R}:=\frac{1}{n}+\frac{1}{p_{A_{R}}^{+}}-\frac{1}{p_{A_{R}}^{-}}.$
Since we have $(\log(e+\frac{2^{i}}{|A_{R}|}))^{q_{A_{R}}^{+}}\leq
i^{q_{A_{R}}^{+}}(\log(e+\frac{2}{|A_{R}|}))^{q_{A_{R}}^{+}}\leq
2^{q^{+}}i^{q_{A_{R}}^{+}}(\log(e+\frac{1}{|A_{R}|}))^{q_{A_{R}}^{+}}$ for
$i\geq 1,$ we obtain

$\displaystyle R_{i}-R_{i+1}$ $\displaystyle\leq$ $\displaystyle
c_{3}\frac{i^{q_{A_{R}}^{+}}|A_{R}|^{\eta_{R}}}{2^{i\eta_{R}}}(\log(e+\frac{1}{|A_{R}|}))^{q_{A_{R}}^{+}},$

where $c_{3}={C_{1}}2^{q^{+}}.$

Note that $\eta_{R}\geq\eta:=\frac{1}{n}+\frac{1}{p^{+}}-\frac{1}{p^{-}}>0$
and by integral test
$\sum_{i=1}^{\infty}{i^{q_{A_{R}}^{+}}}{2^{-i\eta_{R}}}\leq\frac{(q_{A_{R}}^{+})!}{(\eta_{R}\ln
2)^{(q_{A_{R}}^{+}+1)}}$ and hence

$\displaystyle R=\sum_{i=0}^{\infty}(R_{i}-R_{i+1})$ $\displaystyle\leq$
$\displaystyle
c_{3}|A_{R}|^{\eta_{R}}(\log(e+\frac{1}{|A_{R}|}))^{q_{A_{R}}^{+}}\Big{(}1+\sum_{i=1}^{\infty}{i^{q_{A_{R}}^{+}}}{2^{-i\eta_{R}}}\Big{)}$
$\displaystyle\leq$ $\displaystyle
c_{3}|A_{R}|^{\eta_{R}}(\log(e+\frac{1}{|A_{R}|}))^{q_{A_{R}}^{+}}\Big{(}\frac{(q_{A_{R}}^{+})!}{(\eta_{R}\ln
2)^{(q_{A_{R}}^{+}+1)}}+1\Big{)}$ $\displaystyle\leq$ $\displaystyle
c_{3}|A_{R}|^{\eta_{R}}(\log(e+\frac{1}{|A_{R}|}))^{q_{A_{R}}^{+}}\Big{(}\frac{(q_{A_{R}}^{+})!}{(\eta\ln
2)^{(q_{A_{R}}^{+}+1)}}+1\Big{)}$ $\displaystyle\leq$ $\displaystyle
c_{3}|A_{R}|^{\eta_{R}}(\log(e+\frac{1}{|A_{R}|}))^{q_{A_{R}}^{+}}\Big{(}\frac{(q^{+})!}{({\min(1,\eta)}\ln
2)^{(q^{+}+1)}}+1\Big{)}$

Moreover, since
$c_{4}:=1/\max\Big{\\{}1,\Big{(}\frac{c_{3}(q^{+})!}{({\min(1,\eta)}\ln
2)^{(q^{+}+1)}}+c_{3}\Big{)}\Big{\\}}\leq 1$ one has

(3.4)
$\displaystyle|A_{R}|(\log(e+\frac{1}{|A_{R}|}))^{\frac{q_{A_{R}}^{+}}{\eta_{R}}}\geq
c_{4}^{1/\eta_{R}}R^{1/\eta_{R}}\geq
c_{4}^{1/\eta}R^{1/\eta_{R}}=c_{4}^{1/\eta}R^{n}R^{\beta_{R}/\eta_{R}},$

where $\beta_{R}:=1-n\eta_{R}$.

Now, we would like to find a constant $\tilde{\eta}>0$, independent of $x$ and
$R$, such that
$\frac{1}{s}+\frac{1}{p_{A_{R}}^{+}}-\frac{1}{p_{A_{R}}^{-}}=:\eta_{R}\geq\tilde{\eta}>0$
for all $R\leq r_{0}.$ Towards this end, the log-Hölder continuity of $p$
gives, for any $z$ and $y$ in $A_{R},$

$|\frac{1}{p(z)}-\frac{1}{p(y)}|\leq\frac{C_{\text{log}}}{\log(e+1/|z-y|)},$

and taking the supremum over all pairs of points in $A_{R}$ one gets

(3.5)
$\frac{1}{p_{A_{R}}^{-}}-\frac{1}{p_{A_{R}}^{+}}\leq\frac{C_{\text{log}}}{\log(1/(2R))}.$

Suppose now that for some $R\leq 1/4$ we have that $\eta_{R}\leq 0$. Then
(3.5) gives
$\frac{1}{s}\leq\frac{C_{\text{log}}}{\log\left(\frac{1}{2R}\right)}$, which
further implies $R\geq\frac{1}{2}e^{-sC_{\text{log}}}$.

Hence we have the following conclusion:

  * •

If $\frac{1}{2}e^{-sC_{\text{log}}}>\frac{1}{4}$, then there is no
$R\leq\frac{1}{4}$ for which $\eta_{R}\leq 0$.

  * •

If $\frac{1}{2}e^{-sC_{\text{log}}}\leq\frac{1}{4}$, then $\eta_{R}\leq 0$
implies $R\geq\frac{1}{2}e^{-sC_{\text{log}}}.$

Therefore if we choose
$r_{0}=\frac{1}{2}\min\\{\frac{1}{4},\frac{1}{2}e^{-sC_{\text{log}}}\\}$, then
$\eta_{r_{0}}>0$, and also

(3.6) $\frac{1}{s}>\frac{C_{\text{log}}}{\log(1/(2r_{0}))}.$

But $\eta_{r_{0}}$ may depend on the point $x$ fixed at the beginning of the
proof. To obtain the required $\tilde{\eta}$, we apply again log-Hölder
continuity of $1/p$ on $A_{r_{0}}$, to obtain

(3.7)
$\frac{1}{p_{A_{r_{0}}}^{-}}-\frac{1}{p_{A_{r_{0}}}^{+}}\leq\frac{C_{\text{log}}}{\log(1/(2r_{0}))},$

and (3.6) together with (3.7) give

$\eta_{r_{0}}=\frac{1}{s}+\frac{1}{p_{A_{r_{0}}}^{+}}-\frac{1}{p_{A_{r_{0}}}^{-}}\geq\frac{1}{s}-\frac{C_{\text{log}}}{\log(1/(2r_{0}))}>0.$

Choosing $\tilde{\eta}:=\frac{1}{s}-\frac{C_{\text{log}}}{\log(1/(2r_{0}))},$
we get that $\eta_{R}\geq\eta_{r_{0}}\geq\tilde{\eta}>0$ for all $R\leq
r_{0}$. This is our desired $\tilde{\eta}.$

Therefore, from equation (3.4) one sees that if a positive lower bound for
$R^{\beta_{R}/\eta_{R}}$ is provided, the proof of Theorem 1.4 is finished. To
achieve such a lower bound, we see that from the log-Hölder continuity of $p,$

$|p(z)-p(y)|\leq\frac{C_{\text{log}}}{\log(e+1/|z-y|)};$

taking the supremum over pairs of points in $A_{R}$ one gets

$p_{A_{R}}^{+}-p_{A_{R}}^{-}\leq\frac{C_{\text{log}}}{\log(1/(2R))},$

or

$\log\left(1/(2R)^{p_{A_{R}}^{+}-p_{A_{R}}^{-}}\right)\leq C_{\text{log}},$

therefore

(3.8)
$R^{p_{A_{R}}^{+}-p_{A_{R}}^{-}}\geq\frac{e^{-C_{\text{log}}}}{2^{p_{A_{R}}^{+}-p_{A_{R}}^{-}}}\geq\frac{e^{-C_{\text{log}}}}{2^{(p^{+}-p^{-})}}.$

But

$R^{\frac{\beta_{R}}{\eta_{R}}}\geq
R^{\frac{\beta_{R}}{\eta}}=R^{\frac{n(p_{A_{R}}^{+}-p_{A_{R}}^{-})}{\eta
p_{A_{R}}^{+}p_{A_{R}}^{-}}}\geq(R^{p_{A_{R}}^{+}-p_{A_{R}}^{-}})^{n/\eta(p^{-})^{2}},$

hence using (3.8) the required bound

$R^{\frac{\beta_{R}}{\eta_{R}}}\geq\left(\frac{e^{-C_{\text{log}}}}{2^{(p^{+}-p^{-})}}\right)^{n/\eta(p^{-})^{2}}=:c_{5}>0.$

Taking $f(t)=t(\log(e+\frac{1}{t}))^{\frac{q_{A_{R}}^{+}}{\eta_{R}}}$, we see
that (3.4) becomes $f(|A_{R}|)\geq cR^{n},$ where $c:=c_{4}^{1/\eta}c_{5}$ and
hence $|A_{R}|\geq f^{-1}(cR^{n})$ which further implies that
$cR^{n}(\log(e+\frac{1}{R}))^{\frac{-q^{+}}{\eta}}\leq
cR^{n}(\log(e+\frac{1}{R}))^{\frac{-q_{A_{R}}^{+}}{\eta_{R}}}\leq|B_{R}(x)\cap\Omega|$.
So, $\Omega$ satisfies the log-measure density condition.

Case-2 ($q(x)\leq 0$ for all $x\in\Omega$): Using Lemma 3.2 one obtains

$\displaystyle R_{i}-R_{i+1}$ $\displaystyle\leq$ $\displaystyle
C_{2}|A_{R_{i}}|^{\frac{1}{n}+\frac{1}{p_{A_{R_{i}}}^{+}}-\frac{1}{p_{A_{R_{i}}}^{-}}}(\log(e+\frac{1}{|A_{R_{i}}|}))^{Q_{A_{R}}}$
$\displaystyle\leq$ $\displaystyle
C_{2}|A_{R_{i}}|^{\frac{1}{n}+\frac{1}{p_{A_{R}}^{+}}-\frac{1}{p_{A_{R}}^{-}}}(\log(e+\frac{1}{|A_{R}|}))^{Q_{A_{R}}}$
$\displaystyle=$ $\displaystyle
C_{2}\frac{|A_{R}|^{\eta_{R}}}{2^{i\eta_{R}}}(\log(e+\frac{2^{i}}{|A_{R}|}))^{Q_{A_{R}}}$
$\displaystyle\leq$ $\displaystyle
c_{6}\frac{2^{Q}i^{Q_{A_{R}}}|A_{R}|^{\eta_{R}}}{2^{i\eta_{R}}}(\log(e+\frac{1}{|A_{R}|}))^{Q_{A_{R}}},$

where $\eta_{R}:=\frac{1}{n}+\frac{1}{p_{A_{R}}^{+}}-\frac{1}{p_{A_{R}}^{-}}$
and $c_{6}=C_{2}(2)^{Q}.$ Note that
$\eta_{R}\geq\eta:=\frac{1}{n}+\frac{1}{p^{+}}-\frac{1}{p^{-}}>0$ and hence

$\displaystyle R=\sum_{i=0}^{\infty}(R_{i}-R_{i+1})$ $\displaystyle\leq$
$\displaystyle
c_{6}|A_{R}|^{\eta_{R}}(\log(e+\frac{1}{|A_{R}|}))^{Q_{A_{R}}}\Big{(}1+\sum_{i=1}^{\infty}{i^{Q_{A_{R}}}}{2^{-i\eta_{R}}}\Big{)}$
$\displaystyle\leq$ $\displaystyle
c_{6}|A_{R}|^{\eta_{R}}(\log(e+\frac{1}{|A_{R}|}))^{Q_{A_{R}}}\Big{(}1+\frac{(Q_{A_{R}}!)}{(\eta_{R}\ln
2)^{({Q_{A_{R}}}+1)}}\Big{)}$ $\displaystyle\leq$ $\displaystyle
c_{6}|A_{R}|^{\eta_{R}}(\log(e+\frac{1}{|A_{R}|}))^{Q_{A_{R}}}\Big{(}1+\frac{(Q_{A_{R}})!}{(\eta\ln
2)^{(Q_{A_{R}}+1)}}\Big{)}$ $\displaystyle\leq$ $\displaystyle
c_{6}|A_{R}|^{\eta_{R}}(\log(e+\frac{1}{|A_{R}|}))^{Q_{A_{R}}}\Big{(}1+\frac{(Q)!}{(\min(1,\eta)\ln
2)^{(Q+1)}}\Big{)}$

where $Q=\frac{-q^{-}(n-p^{-})}{p^{-}(n-p^{+})}\geq Q_{A_{R}}\geq 0.$
Moreover, since
$c_{7}:=1/\max\Big{\\{}\Big{(}1,c_{6}+\frac{c_{6}(Q)!}{(\min(1,\eta)\ln
2)^{(Q+1)}}\Big{)}\Big{\\}}\leq 1$ one has

(3.9)
$\displaystyle|A_{R}|(\log(e+\frac{1}{|A_{R}|}))^{\frac{Q_{A_{R}}}{\eta_{R}}}\geq
c_{4}^{1/\eta_{R}}R^{1/\eta_{R}}\geq
c_{7}^{1/\eta}R^{1/\eta_{R}}=c_{7}^{1/\eta}R^{n}R^{\beta_{R}/\eta_{R}},$

where $\beta_{R}:=1-n\eta_{R}$. Now we can proceed similarly as in case-1 to
obtain $cR^{n}(\log(e+\frac{1}{R}))^{\frac{-Q}{\eta}}\leq
cR^{n}(\log(e+\frac{1}{R}))^{\frac{-Q_{A_{R}}}{\eta_{R}}}\leq|B_{R}(x)\cap\Omega|$.
So, $\Omega$ satisfies the log-measure density condition.

Case-3 ($q(x)<0$ for some $x\in\Omega$ and $q(x)\geq 0$ for some
$x\in\Omega$): Using Lemma 3.2 one obtains

$\displaystyle R_{i}-R_{i+1}$ $\displaystyle\leq$ $\displaystyle
C_{3}|A_{R_{i}}|^{\frac{1}{n}+\frac{1}{p_{A_{R_{i}}}^{+}}-\frac{1}{p_{A_{R_{i}}}^{-}}}(\log(e+\frac{1}{|A_{R_{i}}|}))^{S_{A_{R}}}$
$\displaystyle\leq$ $\displaystyle
C_{3}|A_{R_{i}}|^{\frac{1}{n}+\frac{1}{p_{A_{R}}^{+}}-\frac{1}{p_{A_{R}}^{-}}}(\log(e+\frac{1}{|A_{R}|}))^{S_{A_{R}}}$
$\displaystyle=$ $\displaystyle
C_{3}\frac{|A_{R}|^{\eta_{R}}}{2^{i\eta_{R}}}(\log(e+\frac{2^{i}}{|A_{R}|}))^{S_{A_{R}}}$
$\displaystyle\leq$ $\displaystyle
c_{8}\frac{2^{S}i^{S_{A_{R}}}|A_{R}|^{\eta_{R}}}{2^{i\eta_{R}}}(\log(e+\frac{1}{|A_{R}|}))^{S_{A_{R}}},$

where $\eta_{R}:=\frac{1}{n}+\frac{1}{p_{A_{R}}^{+}}-\frac{1}{p_{A_{R}}^{-}}$
and $c_{8}=C_{3}(2)^{S}.$ Note that
$\eta_{R}\geq\eta:=\frac{1}{n}+\frac{1}{p^{+}}-\frac{1}{p^{-}}>0$ and hence

$\displaystyle R=\sum_{i=0}^{\infty}(R_{i}-R_{i+1})$ $\displaystyle\leq$
$\displaystyle
c_{8}|A_{R}|^{\eta_{R}}(\log(e+\frac{1}{|A_{R}|}))^{S_{A_{R}}}\Big{(}1+\sum_{i=1}^{\infty}{i^{S_{A_{R}}}}{2^{-i\eta_{R}}}\Big{)}$
$\displaystyle\leq$ $\displaystyle
c_{8}|A_{R}|^{\eta_{R}}(\log(e+\frac{1}{|A_{R}|}))^{S_{A_{R}}}\Big{(}1+\frac{(S_{A_{R}}!)}{(\eta_{R}\ln
2)^{({S_{A_{R}}}+1)}}\Big{)}$ $\displaystyle\leq$ $\displaystyle
c_{8}|A_{R}|^{\eta_{R}}(\log(e+\frac{1}{|A_{R}|}))^{S_{A_{R}}}\Big{(}1+\frac{(S_{A_{R}})!}{(\eta\ln
2)^{(S_{A_{R}}+1)}}\Big{)}$ $\displaystyle\leq$ $\displaystyle
c_{8}|A_{R}|^{\eta_{R}}(\log(e+\frac{1}{|A_{R}|}))^{S_{A_{R}}}\Big{(}1+\frac{(S)!}{(\min(1,\eta)\ln
2)^{(S+1)}}\Big{)}$

where $S=\max\\{q^{+},Q,T\\}\geq S_{A_{R}}\geq 0$ and
$T=q^{+}-\frac{q^{-}(n-p^{-})}{p^{-}(n-p^{+})}\geq T_{A_{R}}.$
Moreover, since
$c_{9}:=1/\max\Big{\\{}\Big{(}1,c_{8}+\frac{c_{8}(S)!}{(\min(1,\eta)\ln
2)^{(S+1)}}\Big{)}\Big{\\}}\leq 1$ one has

(3.10)
$\displaystyle|A_{R}|(\log(e+\frac{1}{|A_{R}|}))^{\frac{S_{A_{R}}}{\eta_{R}}}\geq
c_{9}^{1/\eta_{R}}R^{1/\eta_{R}}\geq
c_{9}^{1/\eta}R^{1/\eta_{R}}=c_{9}^{1/\eta}R^{n}R^{\beta_{R}/\eta_{R}},$

where $\beta_{R}:=1-n\eta_{R}$. Now we can proceed similarly as in case-1 to
obtain $cR^{n}(\log(e+\frac{1}{R}))^{\frac{-S}{\eta}}\leq
cR^{n}(\log(e+\frac{1}{R}))^{\frac{-S_{A_{R}}}{\eta_{R}}}\leq|B_{R}(x)\cap\Omega|$.
So, $\Omega$ satisfies the log-measure density condition. ∎

## References

  * [AC24] Lars Diening Andrea Cianchi.  Sobolev embeddings in musielak-orlicz spaces.  Advances in Mathematics, 447, 2024.
  * [Die04] Lars Diening.  Riesz potential and Sobolev embeddings on generalized Lebesgue and Sobolev spaces $L^{p(\cdot)}$ and $W^{k,p(\cdot)}$.  Math. Nachr., 268:31–43, 2004.
  * [ER00] David E. Edmunds and Jiri Rákosnik.  Sobolev embeddings with variable exponent.  Studia Math., 143:267–293, 2000.
  * [Fan12] Xianling Fan.  An imbedding theorem for Musielak-Sobolev spaces.  Nonlinear Anal., 75(4):1959–1971, 2012.
  * [GKP21] Przemysł aw Górka, Nijjwal Karak, and Daniel J. Pons.  Variable exponent Sobolev spaces and regularity of domains.  J. Geom. Anal., 31(7):7304–7319, 2021.
  * [GKP23] Przemysł aw Górka, Nijjwal Karak, and Daniel J. Pons.  Variable exponent sobolev spaces and regularity of domains-ii.  Rev Mat Complut, 2023.
  * [HaKT08] Piotr Hajł asz, Pekka Koskela, and Heli Tuominen.  Sobolev embeddings, extensions and measure density condition.  J. Funct. Anal., 254(5):1217–1234, 2008.
  * [Hes41] Magnus R. Hestenes.  Extension of the range of a differentiable function.  Duke Math. J., 8:183–192, 1941.
  * [HH19] Petteri Harjulehto and Peter Hästö.  Orlicz spaces and generalized Orlicz spaces, volume 2236 of Lecture Notes in Mathematics.  Springer, Cham, 2019.
  * [HHSa24] Petteri Harjulehto, Peter Hästö, and Artur Sł abuszewski.  A revised condition for harmonic analysis in generalized orlicz spaces on unbounded domains.  Math. Nachr., 2024.
  * [HK22] Toni Heikkinen and Nijjwal Karak.  Orlicz-Sobolev embeddings, extensions and Orlicz-Poincaré inequalities.  J. Funct. Anal., 282(2):Paper No. 109292, 53, 2022.
  * [HMOS10] Peter Hästö, Yoshihiro Mizuta, Takao Ohno, and Tetsu Shimomura.  Sobolev inequalities for Orlicz spaces of two variable exponents.  Glasg. Math. J., 52(2):227–240, 2010.
  * [Juu23] Jonne Juusti.  Extension in generalized Orlicz-Sobolev spaces.  J. Math. Anal. Appl., 522(1):Paper No. 126941, 12, 2023.
  * [McS34] Edward James McShane.  Extension of range of functions.  Bull. Amer. Math. Soc., 40:837–842, 1934.
  * [MOS18] Yoshihiro Mizuta, Takao Ohno, and Tetsu Shimomura.  Sobolev inequalities for musielak–orlicz spaces.  manuscripta math., 155:209–227, 2018.

Ankur Pandey
Department of Mathematics,
Birla Institute of Technology and Science-Pilani, Hyderabad Campus,
Hyderabad-500078, India
<EMAIL_ADDRESS><EMAIL_ADDRESS>

Nijjwal Karak
Department of Mathematics,
Birla Institute of Technology and Science-Pilani, Hyderabad Campus,
Hyderabad-500078, India
<EMAIL_ADDRESS><EMAIL_ADDRESS>