# Fuglede’s theorem in generalized Orlicz–Sobolev spaces

Jonne Juusti  Department of Mathematics and Statistics, FI-20014 University of
Turku, Finland<EMAIL_ADDRESS>
###### Abstract.

In this paper, we show that Orlicz–Sobolev spaces $W^{1,\varphi}(\Omega)$ can
be characterized with the ACL- and ACC-characterizations. ACL stands for
absolutely continuous on lines and ACC for absolutely continuous on curves.
Our results hold under the assumptions that $C^{1}(\Omega)$ functions are
dense in $W^{1,\varphi}(\Omega)$, and $\varphi(x,\beta)\geqslant 1$ for some
$\beta>0$ and almost every $x\in\Omega$. The results are new even in the
special cases of Orlicz and double phase growth.

###### Key words and phrases:

Generalized Orlicz space, Orlicz–Sobolev space, ACC, ACL, modulus

###### 2010 Mathematics Subject Classification:

46E35

##  1\. Introduction

In this paper, we study the ACL- and ACC-characterizations of Orlicz–Sobolev
spaces $W^{1,\varphi}(\Omega)$, where $\varphi$ has generalized Orlicz growth
and $\Omega\subset\mathbb{R}^{n}$ is an open set. ACL stands for absolutely
continuous on lines and ACC for absolutely continuous on curves. Special cases
of Orlicz growth include the constant exponent case $\varphi(x,t)=t^{p}$, the
Orlicz case $\varphi(x,t)=\varphi(t)$, the variable exponent case
$\varphi(x,t)=t^{p(x)}$, and the double phase case
$\varphi(x,t)=t^{p}+a(x)t^{q}$. Generalized Orlicz and Orlicz–Sobolev spaces
on $\mathbb{R}^{n}$ have been recently studied for example in [4, 3, 11], and
in a more general setting in [1, 10].

The ACL-characterization of the classical constant exponent Sobolev spaces was
given by Nikodym in [9]. It states that a function $u\in L^{p}(\Omega)$
belongs to $W^{1,p}(\Omega)$ if and only if it has representative $\tilde{u}$
that is absolutely continuous on almost every line segment parallel to the
coordinate axes and the classical partial derivatives of $\tilde{u}$ belong to
$L^{p}(\Omega)$. Moreover the classical partial derivatives are equal to the
weak partial derivatives. In [5], Fuglede gave a finer version of this
characterization, namely, the ACC-characterization. The ACC-characterization
states that a function $u\in L^{p}(\Omega)$ belongs to $W^{1,p}(\Omega)$ if
and only if it has representative $\tilde{u}$ that is absolutely continuous on
every rectifiable curve outside a family of zero $p$-modulus and the
(classical) partial derivatives $\tilde{u}$ belong to $L^{p}(\Omega)$.

In [7], it was shown that variable exponent Sobolev space
$W^{1,p(\cdot)}(\Omega)$ also has the ACL- and ACC-characterizations, if the
exponent satisfies suitable conditions and $C^{1}(\Omega)$ functions are
dense. In Section 8 of [10], it was shown that the results hold in the space
$W^{1,\varphi}(\mathbb{R}^{n})$, if $C^{1}(\mathbb{R}^{n})$-functions are
dense and $\varphi$ satisfies certain conditions. In this paper, we generalize
the results even further. We show that the results hold for the space
$W^{1,\varphi}(\Omega)$, and we do so using fewer assumptions than in [7] or
[10]. There are two assumptions we need to make: First that $C^{1}(\Omega)$
functions are dense in $W^{1,\varphi}(\Omega)$. And second, that
$\varphi(x,\beta)\geqslant 1$ for some $\beta>0$ and almost every
$x\in\Omega$. To best of our knowledge, the results are new even in the
special cases of Orlicz and double phase growth.

We base our approach on [7], but make some modifications to both make the
results more general and simplify some of the results. One difference is that
we use a slightly different definition for the modulus of a curve family. Our
definition of is based on the norm, while the definition in [7] is based on
the modular. The reason for defining the modulus differently has to do with
the fact that modular convergence is a weaker concept than norm convergence.
Another difference with [7] is that we do not use the theory of capacities.
This has two advantages: First, the use of capacities would force us to make
some extra assumptions on $\varphi$. Second, we can prove our results directly
in $W^{1,\varphi}(\Omega)$, for any $\Omega\subset\mathbb{R}^{n}$, whereas in
[7] the results are first proven in the case $\Omega=\mathbb{R}^{n}$, and this
case is then used to prove the results for $\Omega\subset\mathbb{R}^{n}$.

The structure of this paper is as follows: Section 2 covers preliminaries
about generalized Orlicz and Orlicz–Sobolev spaces. In section 3 we define and
discuss the modulus of a curve family. In section 4 we prove two lemmas, which
we will need in order to prove our main results. In section 5 we prove our
main results, the ACL- and ACC-characterizations of $W^{1,\varphi}(\Omega)$.

##  2\. Preliminaries

Throughout this paper, we assume that $\Omega\subset\mathbb{R}^{n}$ is an open
set. The following definitions are as in [6], which we use as a general
reference to background theory in generalized Orlicz spaces.

######  Definition 2.1.

We say that $\varphi:\Omega\times[0,\infty)\to[0,\infty]$ is a weak
$\Phi$-function, and write $\varphi\in\Phi_{w}(\Omega)$, if the following
conditions hold

  * •

For every measurable $f:\Omega\to[-\infty,\infty]$ the function
$x\mapsto\varphi(x,|f|)$ is measurable, and for every $x\in\Omega$ the
function $t\mapsto\varphi(x,t)$ is non-decreasing.

  * •

$\displaystyle\varphi(x,0)=\lim_{t\to 0^{+}}\varphi(x,t)=0$ and
$\displaystyle\lim_{t\to\infty}\varphi(x,t)=\infty$ for every $x\in\Omega$.

  * •

The function $t\mapsto\frac{\varphi(x,t)}{t}$ is $L$-almost increasing for
$t>0$ uniformly in $\Omega$. "Uniformly" means that $L$ is independent of $x$.

If $\varphi\in\Phi_{w}(\Omega)$ is additionally convex and left-continuous,
then $\varphi$ is a convex $\Phi$-function, and we write
$\varphi\in\Phi_{c}(\Omega)$.

Two functions $\varphi$ and $\psi$ are equivalent, $\varphi\simeq\psi$, if
there exists $L\geqslant 1$ such that
$\psi(x,\frac{t}{L})\leqslant\varphi(x,t)\leqslant\psi(x,Lt)$ for every
$x\in\Omega$ and every $t>0$. Equivalent $\Phi$-functions give rise to the
same space with comparable norms.

We define the left-inverse of $\varphi$ by setting

$\varphi^{-1}(x,\tau):=\inf\\{t\geqslant 0:\varphi(x,t)\geqslant\tau\\}.$

### Assumptions

Let us write $\varphi^{+}_{B}(t):=\sup_{x\in B}\varphi(x,t)$ and
$\varphi^{-}_{B}(t):=\inf_{x\in B}\varphi(x,t)$; and abbreviate
$\varphi^{\pm}:=\varphi^{\pm}_{\Omega}$. We state some assumptions for later
reference.

  * (A0)

There exists $\beta\in(0,1)$ such that $\varphi(x,\beta)\leqslant
1\leqslant\varphi(x,1/\beta)$ for almost every $x$.

  * (A1)

There exists $\beta\in(0,1)$ such that, for every ball $B$ and a.e. $x,y\in
B\cap\Omega$,

$\beta\varphi^{-1}(x,t)\leqslant\varphi^{-1}(y,t)\quad\text{when}\quad
t\in\bigg{[}1,\frac{1}{|B|}\bigg{]}.$

  * (A2)

For every $s>0$ there exist $\beta\in(0,1]$ and $h\in L^{1}(\Omega)\cap
L^{\infty}(\Omega)$ such that

$\beta\varphi^{-1}(x,t)\leqslant\varphi^{-1}(y,t)$

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

  * (aInc)p

There exist $L\geqslant 1$ such that $t\mapsto\frac{\varphi(x,t)}{t^{p}}$ is
$L$-almost increasing in $(0,\infty)$.

  * (aDec)q

There exist $L\geqslant 1$ such that $t\mapsto\frac{\varphi(x,t)}{t^{q}}$ is
$L$-almost decreasing in $(0,\infty)$.

We say that $\varphi$ satisfies (aInc), if it satisfies (aInc)p for some
$p>1$. Similarly, $\varphi$ satisfies (aDec), if it satisfies (aDec)q for some
$q>1$. We write (Inc) if the ratio is increasing rather than just almost
increasing, similarly for (Dec). See [6, Table 7.1] for an interpretation of
the assumptions in some special cases.

### Generalized Orlicz spaces

We recall some definitions. We denote by $L^{0}(\Omega)$ the set of measurable
functions in $\Omega$.

######  Definition 2.2.

Let $\varphi\in\Phi_{w}(\Omega)$ and define the modular $\varrho_{\varphi}$
for $f\in L^{0}(\Omega)$ by

$\varrho_{\varphi}(f):=\int_{\Omega}\varphi(x,|f(x)|)\,dx.$

The _generalized Orlicz space_ , also called Musielak–Orlicz space, is defined
as the set

$L^{\varphi}(\Omega):=\big{\\{}f\in L^{0}(\Omega)\colon\lim_{\lambda\to
0^{+}}\varrho_{\varphi}(\lambda f)=0\big{\\}}$

equipped with the (Luxemburg) norm

(2.3)
$\|f\|_{L^{\varphi}(\Omega)}:=\inf\Big{\\{}\lambda>0\colon\varrho_{\varphi}\Big{(}\frac{f}{\lambda}\Big{)}\leqslant
1\Big{\\}}.$

If the set is clear from the context we abbreviate
$\|f\|_{L^{\varphi}(\Omega)}$ by $\|f\|_{\varphi}$.

The following lemma is a direct consequence of the proof of [6, Theorem
3.3.7].

######  Lemma 2.4.

If $(f_{i})$ is a Cauchy sequence in $L^{\varphi}(\Omega)$ such that the
pointwise limit $f(x):=\lim_{i\to\infty}f_{i}(x)$ ($\pm\infty$ allowed) exists
for almost every $x\in\Omega$, then $f$ is the limit of $(f_{i})$ in
$L^{\varphi}(\Omega)$.

Hölder’s inequality holds in generalized Orlicz spaces with constant $2$,
without restrictions on the $\Phi_{w}$-function [6, Lemma 3.2.13]:

$\int_{\Omega}|f|\,|g|\,dx\leqslant 2\|f\|_{\varphi}\|g\|_{\varphi^{*}}.$

######  Definition 2.5.

A function $u\in L^{\varphi}(\Omega)$ belongs to the Orlicz–Sobolev space
$W^{1,\varphi}(\Omega)$ if its weak partial derivatives
$\partial_{1}u,\ldots,\partial_{n}u$ exist and belong to the space
$L^{\varphi}(\Omega)$. For $u\in W^{1,\varphi}(\Omega)$, we define the norm

$\|u\|_{W^{1,\varphi}(\Omega)}:=\|u\|_{\varphi}+\|\nabla u\|_{\varphi}.$

Here $\|\nabla u\|_{\varphi}$ is short for $\big{\|}|\nabla
u|\big{\|}_{\varphi}$. Again, if $\Omega$ is clear from the context, we
abbreviate $\|u\|_{W^{1,\varphi}(\Omega)}$ by $\|u\|_{1,\varphi}$.

Many of our results need the assumption that $C^{1}(\Omega)$-functions are
dense in $W^{1,\varphi}(\Omega)$. A suuficient condition is given by [6,
Theorem 6.4.7], which states that $C^{\infty}(\Omega)$-functions are dense in
$W^{1,\varphi}(\Omega)$, if $\varphi$ satisfies (A0), (A1), (A2) and (aDec).
By [6, Lemma 4.2.3], (A2) can be omitted, if $\Omega$ is bounded.

##  3\. Modulus of a family of curves

By a curve, we mean any continuous function $\gamma:I\to\mathbb{R}^{n}$, where
$I=[a,b]$ is a closed interval. If a curve $\gamma$ is rectifiable, we may
assume that $I=[0,\ell(\gamma)]$, where $\ell(\gamma)$ denotes the length of
$\gamma$. We denote the image of $\gamma$ by $\operatorname{Im}(\gamma)$, and
by $\Gamma_{rect}(\Omega)$ we denote the family of all rectifiable curves
$\gamma$ such that $\operatorname{Im}(\gamma)\subset\Omega$. Let
$\Gamma\subset\Gamma_{rect}(\Omega)$. We say that a Borel function
$u:\Omega\to[0,\infty]$ is _$\Gamma$ -admissible_, if

$\int_{\gamma}u\,ds\geqslant 1$

for all $\gamma\in\Gamma$, where $ds$ denotes the integral with respect to
curve length. We denote the set of all $\Gamma$-admissible functions by
$F_{adm}(\Gamma)$.

######  Definition 3.1.

Let $\Gamma\subset\Gamma_{rect}(\Omega)$. Let $\varphi\in\Phi_{w}(\Omega)$. We
define the _$\varphi$ -modulus_ of $\Gamma$ by

$\operatorname{M_{\varphi}}(\Gamma):=\inf_{u\in
F_{adm}(\Gamma)}\|u\|_{\varphi}.$

If $F_{adm}(\Gamma)=\emptyset$, we set
$\operatorname{M_{\varphi}}(\Gamma):=\infty$. A family of curves $\Gamma$ is
_exceptional_ , if $\operatorname{M_{\varphi}}(\Gamma)=0$.

The definition above is as in [8]. The following lemma gives some useful
properties of the modulus. Items (a) and (b) are items (a) and (c) of [8,
Lemma 4.5], and item (c) is a consequence of [8, Proposition 4.8]. To use the
lemma, we must check that $L^{\varphi}(\Omega)$ satisfies conditions (P0),
(P1), (P2) and (RF) stated at the beginning of section 2 in [8]. The
conditions (P0), (P1) and (P2) are easy to check. For (RF) to hold, there must
exists $c\geqslant 1$ such that

$\Big{\|}\sum_{i=1}^{\infty}u_{i}\Big{\|}_{\varphi}\leqslant\sum_{i=1}^{\infty}c^{i}\|u_{i}\|_{\varphi}$

holds for non-negative $u_{i}\in L^{\varphi}(\Omega)$. This is an easy
consequence of [6, Lemma 3.2.5], which states that there exists $c\geqslant 1$
such that

$\Big{\|}\sum_{i=1}^{\infty}u_{i}\Big{\|}_{\varphi}\leqslant
c\sum_{i=1}^{\infty}\|u_{i}\|_{\varphi}.$

######  Lemma 3.2.

Let $\varphi\in\Phi_{w}(\Omega)$, then the $\varphi$-modulus has the following
properties:

  * (a)

if $\Gamma_{1}\subset\Gamma_{2}$, then
$\operatorname{M_{\varphi}}(\Gamma_{1})\leqslant\operatorname{M_{\varphi}}(\Gamma_{2})$,

  * (b)

if $\operatorname{M_{\varphi}}(\Gamma_{i})=0$ for every $i\in\mathbb{N}$, then
$\operatorname{M_{\varphi}}(\bigcup_{i=1}^{\infty}\Gamma_{i})=0$.

  * (c)

$\operatorname{M_{\varphi}}(\Gamma)=0$ if and only if there exists a non-
negative Borel function $u\in L^{\varphi}(\Omega)$ such that
$\int_{\gamma}u\,ds=\infty$ for every $\gamma\in\Gamma$.

In [5], the $L^{p}$-modulus was originally defined by

$\operatorname{M}_{p}(\Gamma):=\inf_{u\in
F_{adm}(\Gamma)}\int_{\Omega}u^{p}\,dx.$

This differs from Definition 3.1 in that the infimum is taken over the
modulars of admissible functions instead of their norms. A similar approach
was taken in the variable exponent case in [7]. Following the original
approach, we could have defined the modulus by

$\operatorname{\widetilde{M}_{\varphi}}(\Gamma):=\inf_{u\in
F_{adm}(\Gamma)}\int_{\Omega}\varphi(x,u(x))\,dx.$

In the case $\varphi(x,t)=t^{p}$, where $1\leqslant p<\infty$, we have
$\operatorname{\widetilde{M}_{\varphi}}(\Gamma)=\operatorname{M_{\varphi}}(\Gamma)^{p}$.
Thus in this special case $\operatorname{\widetilde{M}_{\varphi}}(\Gamma)=0$
if and only if $\operatorname{M_{\varphi}}(\Gamma)=0$. Since we are only
interested in whether a family of curves is exceptional or not, in this case
it does not matter whether we use $\operatorname{M_{\varphi}}$ or
$\operatorname{\widetilde{M}_{\varphi}}$.

In the general case, the situation is somewhat more complicated. Let
$\varphi\in\Phi_{w}(\Omega)$. By [6, Corollary 3.2.8], if $\|u\|_{\varphi}<1$,
then $\varrho_{\varphi}(u)\lesssim\|u\|_{\varphi}$. Thus
$\operatorname{M_{\varphi}}(\Gamma)=0$ implies
$\operatorname{\widetilde{M}_{\varphi}}(\Gamma)=0$. The converse implication
does not necessarily hold, as the next example shows, which is the main reason
for using norms instead of modulars in Definition 3.1.

######  Example 3.3.

Define $\varphi\in\Phi_{w}(\mathbb{R}^{2})$ by

$\varphi(x,t):=\left\\{\begin{array}[]{ll}0&\text{ if }t\leqslant 1,\\\
t-1&\text{ if }t>1.\end{array}\right.$

For $y\in[0,1]$, let $\gamma_{y}:[0,1]\to\mathbb{R}^{2},z\mapsto(y,z)$, and
let $\Gamma:=\\{\gamma_{y}:y\in[0,1]\\}$. Let $u=1$ everywhere. Then

$\int_{\gamma}u(s)\,ds=1$

for every $\gamma\in\Gamma$, and therefore $u\in F_{adm}(\Gamma)$. Since
$\varphi(x,u(x))=0$ for every $x\in\mathbb{R}^{2}$, we have
$\varrho_{\varphi}(u)=0$, and thus
$\operatorname{\widetilde{M}_{\varphi}}(\Gamma)=0$.

To show that $\operatorname{M_{\varphi}}(\Gamma)>0$, suppose on the contrary,
that $\operatorname{M_{\varphi}}(\Gamma)=0$. Then by Lemma 3.2 (c) there
exists some $v\in L^{\varphi}(\mathbb{R}^{2})$ such that
$\int_{\gamma}v\,ds=\infty$ for every $\gamma\in\Gamma$. Thus

$\int_{[0,1]}v(y,z)\,dz=\int_{\gamma_{y}}v\,ds=\infty$

for every $y\in[0,1]$. Let $\lambda>0$. Since $\varphi(x,t)\geqslant t-1$ for
every $x\in\mathbb{R}^{2}$ and every $t\geqslant 0$, Fubini’s theorem implies
that

$\int_{\mathbb{R}^{2}}\varphi(x,\lambda
v(x))\,dx\geqslant\int_{[0,1]}\int_{[0,1]}\lambda
v(y,z)-1\,dz\,dy=\infty-\int_{[0,1]}\int_{[0,1]}1\,dz\,dy=\infty$

Since $\lambda>0$ was arbitrary, it follows by (2.3) that
$\|v\|_{\varphi}=\infty$. But this is impossible, since $v\in
L^{\varphi}(\mathbb{R}^{2})$. Thus the assumption that
$\operatorname{M_{\varphi}}(\Gamma)=0$ must be wrong and
$\operatorname{M_{\varphi}}(\Gamma)>0$.

Note that if $\varphi\in\Phi_{w}(\Omega)$ satisfies (aDec)q for $1\leqslant
q<\infty$, then, by [6, Lemma 3.2.9] (since $\varphi$ satisfies (aInc)1 by
definition) we have

$\|u\|_{\varphi}\lesssim\max\\{\varrho_{\varphi}(u),\varrho_{\varphi}(u)^{\frac{1}{q}}\\}.$

Thus, if $\varphi$ satisfies (aDec), then
$\operatorname{\widetilde{M}_{\varphi}}(\Gamma)=0$ if and only if
$\operatorname{M_{\varphi}}(\Gamma)=0$.

##  4\. Fuglede’s Lemma

######  Lemma 4.1 (Fuglede’s lemma).

Let $\varphi\in\Phi_{w}(\Omega)$, and let $(u_{i})$ be a sequence of non-
negative Borel functions converging to zero in $L^{\varphi}(\Omega)$. Then
there exists a subsequence $(u_{i_{k}})$ and an exceptional set
$\Gamma\subset\Gamma_{rect}(\Omega)$ such that for all $\gamma\notin\Gamma$ we
have

$\lim_{k\to\infty}\int_{\gamma}u_{i_{k}}\,ds=0.$

###### Proof.

Let $(v_{k}):=(u_{i_{k}})$ be a subsequence of $(u_{i})$, such that

$\|v_{k}\|_{\varphi}\leqslant 2^{-k}.$

Let $\Gamma\subset\Gamma_{rect}(\Omega)$ be the family of curves $\gamma$,
such that $\int_{\gamma}v_{k}\,ds\nrightarrow 0$ as $k\to\infty$. For every
$j\in\mathbb{N}$, let

$w_{j}:=\sum_{k=1}^{j}v_{k}.$

Since every $v_{k}$ is a non-negative Borel function, every $w_{j}$ is also a
non-negative Borel function. Since the sequence $(w_{j}(x))$ is increasing for
every $x\in\Omega$, the limit $w(x):=\lim_{j\to\infty}w_{j}(x)$ (possibly
$\infty$) exists. By [6, Corollary 3.2.5], if $j<m$, then

$\|w_{m}-w_{j}\|_{\varphi}=\Big{\|}\sum_{k=j+1}^{m}v_{k}\Big{\|}_{\varphi}\lesssim\sum_{k=j+1}^{m}\|v_{k}\|_{\varphi}\leqslant\sum_{k=j+1}^{m}2^{-k}<2^{-j},$

which implies that $(w_{j})$ is a Cauchy sequence in $L^{\varphi}(\Omega)$. By
Lemma 2.4, $w$ is the limit of $(w_{j})$ in $L^{\varphi}(\Omega)$, which
implies that $w\in L^{\varphi}(\Omega)$, and therefore
$\|w\|_{\varphi}<\infty$.

Suppose now that $\gamma\in\Gamma$. Then

$\int_{\gamma}w\,ds=\sum_{k=1}^{\infty}\int_{\gamma}v_{k}\,ds=\infty,$

because $\sum_{k=1}^{\infty}\int_{\gamma}v_{k}\,ds<\infty$ would imply that
$\lim_{k\to\infty}\int_{\gamma}v_{k}\,ds=0$. Thus $w/m$ is $\Gamma$-admissible
for every $m\in\mathbb{N}$. Since
$\lim_{m\to\infty}\|w/m\|_{\varphi}=\lim_{m\to\infty}\|w\|_{\varphi}/m=0$, we
have $\operatorname{M_{\varphi}}(\Gamma)=0$. ∎

Let $E\subset\Omega$. We denote by $\Gamma_{E}$ the set of all curves
$\gamma\in\Gamma_{rect}(\Omega)$, such that the
$E\cap\operatorname{Im}(\gamma)$ is nonempty.

The next lemma is, in a sense, a combination of [7, Lemma 3.1] and [2, Lemma
5.1]. The former of the aforementioned lemmas states that if
$C^{1}(\mathbb{R}^{n})$ functions are dense in the variable exponent Sobolev
space $W^{1,p(\cdot)}(\mathbb{R}^{n})$ and $1<p^{-}\leqslant p^{+}<\infty$,
then $\Gamma_{E}$ is exceptional whenever $E\subset\mathbb{R}^{n}$ is of
capacity zero. The latter states that if $\varphi\in\Phi_{w}(\mathbb{R}^{n})$
satisfies (aInc) and (aDec), then for every Cauchy sequence in
$C(\mathbb{R}^{n})\cap W^{1,\varphi}(\mathbb{R}^{n})$ there exists a
subsequence which converges pointwise outside a set of zero capacity. The
beginning of the proof of our lemma is similar to [2, Lemma 5.1], but then we
use the ideas from [7, Lemma 3.1] and modify the proof to replace convergence
outside a set of capacity zero by convergence outside a set $E$, such that
$\Gamma_{E}$ is exceptional. The reason that we do not simply prove a direct
generalization of [7, Lemma 3.1] and then use [2, Lemma 5.1] is, that our
proof avoids the use of capacities. This has two advantages: First, we can
drop the assumptions (aInc) and (aDec). And second, our new result works in
$W^{1,\varphi}(\Omega)$ for any $\Omega\subset\mathbb{R}^{n}$, while in [7,
Lemma 3.1] and [2, Lemma 5.1] we have $\Omega=\mathbb{R}^{n}$.

######  Lemma 4.2.

Let $\varphi\in\Phi_{w}(\Omega)$ and let $(u_{i})$ be a Cauchy sequence of
functions in $C^{1}(\Omega)\cap W^{1,\varphi}(\Omega)$. Then there exists a
set $E$ and a subsequence $(u_{i_{k}})$ such that
$\operatorname{M_{\varphi}}(\Gamma_{E})=|E|=0$ and $(u_{i_{k}})$ converges
pointwise everywhere outside $E$.

###### Proof.

By [6, Lemma 3.3.6] there exists a subsequence of $(u_{i})$ that converges
pointwise almost everywhere. Thus we can choose a subsequence
$(v_{k}):=(u_{i_{k}})$, such that $(v_{k})$ converges pointwise almost
everywhere, and

$\|v_{k+1}-v_{k}\|_{1,\varphi}<4^{-k}$

for every $k\in\mathbb{N}$. For every $k\in\mathbb{N}$, let
$f_{k}:=2^{k}(v_{k+1}-v_{k})\in C^{1}(\Omega)\cap W^{1,\varphi}(\Omega)$. For
every $j\in\mathbb{N}$, let

$g_{j}:=\sum_{k=1}^{j}|f_{k}|\quad\text{and}\quad h_{j}:=\sum_{k=1}^{j}|\nabla
f_{k}|.$

Since the sequences $(g_{j}(x))$ and $(h_{j}(x))$ are increasing for every
$x\in\Omega$, the limits $g(x):=\lim_{j\to\infty}g_{j}(x)$ and
$h(x):=\lim_{j\to\infty}h_{j}(x)$ (possibly $\infty$) exist. Since the
functions $g_{j}$ are continuous, $g$ is a Borel function. If $j<m$, then by
[6, Corollary 3.2.5]

$\|g_{m}-g_{j}\|_{\varphi}\lesssim\sum_{k=j+1}^{m}\|f_{k}\|_{\varphi}\leqslant\sum_{k=j+1}^{\infty}\|f_{k}\|_{1,\varphi}<\sum_{k=j+1}^{\infty}2^{-k}=2^{-j},$

which implies that $(g_{j})$ is a Cauchy sequence in $L^{\varphi}(\Omega)$. By
Lemma 2.4, $g$ is the limit of $(g_{j})$ in $L^{\varphi}(\Omega)$. Similarly,
since

$\|h_{m}-h_{j}\|_{\varphi}\lesssim\sum_{k=j+1}^{m}\|\nabla
f_{k}\|_{\varphi}\leqslant\sum_{k=j+1}^{\infty}\|f_{k}\|_{1,\varphi}<2^{-j},$

we find that $h$ is the limit of $h_{j}$ in $L^{\varphi}(\Omega)$.

Since $f_{k}\in C^{1}(\Omega)$, for any $k\in\mathbb{N}$ we have

$\big{|}|f_{k}(x)|-|f_{k}(y)|\big{|}\leqslant|f_{k}(x)-f_{k}(y)|\leqslant\int_{\gamma}|\nabla
f_{k}|\,ds$

for every $x,y\in\Omega$ and any $\gamma\in\Gamma_{rect}(\Omega)$ containing
$x$ and $y$. Thus for every $j\in\mathbb{N}$ we have

(4.3)
$|g_{j}(x)-g_{j}(y)|\leqslant\sum_{k=1}^{j}\big{|}|f_{k}(x)|-|f_{k}(y)|\big{|}\leqslant\sum_{k=1}^{j}\int_{\gamma}|\nabla
f_{k}|\,ds=\int_{\gamma}h_{j}\,ds,$

for every $x,y\in\Omega$ and any $\gamma\in\Gamma_{rect}(\Omega)$ containing
$x$ and $y$.

Denote by $E$ the set of points $x\in\Omega$ such that the sequence
$(v_{k}(x))$ does not converge. Since $(v_{k})$ converges pointwise almost
everywhere, we have $|E|=0$. It is easy to see that if $x\in E$, then
$x\in\\{|f_{k}|>1\\}$ for infinitely many $k\in\mathbb{N}$, and therefore
$g(x)=\infty$. Thus

$E\subset E_{\infty}:=\\{x\in\Omega:g(x)=\infty\\},$

and $\Gamma_{E}\subset\Gamma_{E_{\infty}}$. Next we construct a set
$\Gamma\subset\Gamma_{rect}(\Omega)$ such that
$\Gamma_{E_{\infty}}\subset\Gamma$ and $\operatorname{M_{\varphi}}(\Gamma)=0$.
It then follows by Lemma 3.2 (a) that
$\operatorname{M_{\varphi}}(\Gamma_{E})=\operatorname{M_{\varphi}}(\Gamma_{E_{\infty}})=0$.

By Lemma 4.1, considering a subsequence if necessary, we find an exceptional
set $\Gamma_{1}\subset\Gamma_{rect}(\Omega)$ such that

$\lim_{j\to\infty}\int_{\gamma}h-h_{j}\,ds=0$

for every $\gamma\in\Gamma_{rect}(\Omega)\setminus\Gamma_{1}$. Let

$\Gamma_{2}:=\Big{\\{}\gamma\in\Gamma_{rect}(\Omega):\int_{\gamma}g\,ds=\infty\Big{\\}}\quad\text{and}\quad\Gamma_{3}:=\Big{\\{}\gamma\in\Gamma_{rect}(\Omega):\int_{\gamma}h\,ds=\infty\Big{\\}}.$

For every $m\in\mathbb{N}$, the function $g/m$ is $\Gamma_{2}$ admissible,
hence $\operatorname{M_{\varphi}}(\Gamma_{2})\leqslant\|g\|_{\varphi}/m$. Thus
it follows that $\operatorname{M_{\varphi}}(\Gamma_{2})=0$. Similarly, we see
that $\operatorname{M_{\varphi}}(\Gamma_{3})=0$. Let
$\Gamma:=\Gamma_{1}\cup\Gamma_{2}\cup\Gamma_{3}$. By Lemma 3.2 (b)
$\operatorname{M_{\varphi}}(\Gamma)=0$.

It remains to show that $\Gamma_{E_{\infty}}\subset\Gamma$. Suppose that
$\gamma\in\Gamma_{rect}(\Omega)\setminus\Gamma$. Since
$\gamma\notin\Gamma_{2}$, there must exist some
$y\in\operatorname{Im}(\gamma)$ with $g(y)<\infty$. By (4.3), for any
$x\in\operatorname{Im}(\gamma)$ and any $j\in\mathbb{N}$ we have

$g_{j}(x)\leqslant g_{j}(y)+|g_{j}(x)-g_{j}(y)|\leqslant
g_{j}(y)+\int_{\gamma}h_{j}\,ds.$

Since $\gamma\notin\Gamma_{1}$, it follows that

$\lim_{j\to\infty}\int_{\gamma}h_{j}\,ds=\int_{\gamma}h\,ds,$

where the right-hand side is finite because $\gamma\notin\Gamma_{3}$. Thus we
have

$g(x)=\lim_{j\to\infty}g_{j}(x)\leqslant\lim_{j\to\infty}\Big{(}g_{j}(y)+\int_{\gamma}h_{j}\,ds\Big{)}=g(y)+\int_{\gamma}h\,ds<\infty.$

Since $x\in\operatorname{Im}(\gamma)$ was arbitrary, it follows that
$\gamma\notin\Gamma_{E_{\infty}}$. And since $\gamma\notin\Gamma$ was
arbitrary, it follows that $\Gamma_{E_{\infty}}\subset\Gamma$. ∎

##  5\. Fuglede’s Theorem

We begin this section by defining some notations. Let $k\in\\{1,2,\dots,n\\}$.
If $z\in\mathbb{R}$ and $y=(y_{1},y_{2},\dots,y_{n-1})\in\mathbb{R}^{n-1}$ we
define

$(y,z)_{k}:=(y_{1},\dots,y_{k-1},z,y_{k},\dots,y_{n-1})\in\mathbb{R}^{n}.$

For every $x=(x_{1},x_{2},\dots,x_{n})\in\mathbb{R}^{n}$, we write
$\tilde{x}_{k}:=(x_{1},\dots,x_{k-1},x_{k+1},\dots,x_{n})\in\mathbb{R}^{n-1}$.
With these notations, we have $x=(\tilde{x}_{k},x_{k})_{k}$. We define
$\widetilde{\Omega}_{k}\subset\mathbb{R}^{n-1}$ by

$\widetilde{\Omega}_{k}:=\\{\tilde{x}_{k}:x\in\Omega\\}=\\{y\in\mathbb{R}^{n-1}:(y,z)_{k}\in\Omega\text{
for some }z\in\mathbb{R}\\}.$

The set $\widetilde{\Omega}_{k}$ is, in a sense, the orthogonal projection of
$\Omega$ into the space $\\{x\in\mathbb{R}^{n}:x_{k}=0\\}$, but strictly
speaking this is not true, since a projection is a function
$P:\mathbb{R}^{n}\to\mathbb{R}^{n}$, but
$\widetilde{\Omega}_{k}\subset\mathbb{R}^{n-1}$. For every
$y\in\widetilde{\Omega}_{k}$, we let $Z_{k}(y)\subset\mathbb{R}$ be the set of
points $z$, such that $(y,z)_{k}\in\Omega$. Note that
$\Omega=\\{(y,z)_{k}:y\in\widetilde{\Omega}_{k}\text{ and }z\in Z_{k}(y)\\}$.

Since we will be using Lebesgue measures with different dimensions
simultaneously, we will use subscripts to differentiate them, i.e.
$m$-dimensional measure will be denoted by $|\cdot|_{m}$.

######  Definition 5.1.

We say that $u:\Omega\to\mathbb{R}$ is _absolutely continuous on lines_ ,
$u\in ACL(\Omega)$, if it is absolutely continuous on almost every line
segment in $\Omega$ parallel to the coordinate axes. More formally, let
$k\in\\{1,2,\dots,n\\}$ and let $E_{k}\subset\widetilde{\Omega}_{k}$ be the
set of points $y$ such that the function

$f_{y}:Z_{k}(y)\to[-\infty,\infty],\,f_{y}(z)=u((y,z)_{k})$

is absolutely continuous on every compact interval $[a,b]\subset Z_{k}(y)$.
Then $u\in ACL(\Omega)$ if and only if $|\widetilde{\Omega}_{k}\setminus
E_{k}|_{n-1}=0$ for every $k$.

Let $u\in ACL(\Omega)$. Absolute continuity implies that the classical partial
derivative $\partial_{k}u$ of $u\in ACL(\Omega)$ exist for every $x\in\Omega$
such that $\tilde{x}_{k}\in E_{k}$. Since $|\widetilde{\Omega}_{k}\setminus
E_{k}|_{n-1}=0$, it follows by Fubini’s theorem that $\partial_{k}u$ exists
for almost every $x\in\Omega$. Another application of Fubini’s theorem shows
that the classical partial derivative is equal to the weak partial derivative,
see [12, Theorem 2.1.4]. Since the partial derivatives exist almost
everywhere, it follows that the gradient $\nabla u$ exists almost everywhere.
A function $u\in ACL(\Omega)$ is said to belong to $ACL^{\varphi}(\Omega)$, if
$|\nabla u|\in L^{\varphi}(\Omega)$.

The following lemma follows immediately from the definitions of
$L^{\varphi}(\Omega)$, $ACL^{\varphi}(\Omega)$ and $W^{1,\varphi}(\Omega)$.

######  Lemma 5.2.

If $\varphi\in\Phi_{w}(\Omega)$, then $ACL^{\varphi}(\Omega)\cap
L^{\varphi}(\Omega)\subset W^{1,\varphi}(\Omega)$.

######  Definition 5.3.

For any $u:\Omega\to R$, let $\Gamma_{NAC}(u)\subset\Gamma_{rect}(\Omega)$ be
the family of curves $\gamma:[0,\ell(\gamma)]\to\Omega$ such that
$u\circ\gamma$ is not absolutely continuous on $[0,\ell(\gamma)]$. If
$\operatorname{M_{\varphi}}(\Gamma_{NAC}(u))=0$, then we say that $u$ is
_absolutely continuous on curves_ , $u\in ACC(\Omega)$.

In the next lemma, we show that $ACC(\Omega)$ is a subset of $ACL(\Omega)$, if
$\varphi$ satisfies a suitable condition.

######  Lemma 5.4.

Let $\varphi\in\Phi_{w}(\Omega)$ and assume that $\varphi$ satisfies the
following condition:

(5.5) $\text{there exist }\beta>0\text{ such that }\varphi(x,\beta)\geqslant
1\text{ for almost every }x\in\Omega.$

Then

$ACC(\Omega)\subset ACL(\Omega).$

######  Remark 5.6.

Note that (A0) implies (5.5), but not the other way around, since we do not
assume that $\varphi(x,1/\beta)\leqslant 1$. We also note (5.5) is equivalent
to

(5.7) $\text{there exist }\beta>0\text{ and }\delta>0\text{ such that
}\varphi(x,\beta)\geqslant\delta\text{ for almost every }x\in\Omega.$

It is clear that (5.5) is just a special case of (5.7) with $\delta=1$. It is
also clear that (5.7) implies (5.5), if $\delta>1$. Suppose then, that
$\varphi$ satisfies (5.7) with $0<\delta<1$. Then

(5.8) $\frac{\delta}{\beta}\leqslant\frac{\varphi(x,\beta)}{\beta}$

for almost every $x\in\Omega$. By (aInc)1 (which $\varphi$ satisfies by
definition of $\Phi_{w}$), there exist a constant $a\geqslant 1$ such that

(5.9) $\frac{\varphi(x,\beta)}{\beta}\leqslant a\frac{\varphi(x,t)}{t}$

for almost every $x\in\Omega$ and every $t\geqslant\beta$. Choosing
$t:=a\beta/\delta>\beta$, it follows from (5.8) and (5.9) that
$\varphi(x,a\beta/\delta)\geqslant 1$ for almost every $x\in\Omega$, and
therefore $\varphi$ satisfies (5.5). Thus the choice $\delta=1$ in (5.5) has
no special meaning, except for making notations simpler by getting rid of
$\delta$.

###### Proof of Lemma 5.4.

Let $u\in ACC(\Omega)$, and let $k\in\\{1,\dots,n\\}$ and let
$E_{k}\subset\mathbb{R}^{n-1}$ be as in Definition 5.3. By Lemma 3.2, there
exists a non-negative Borel function $v\in L^{\varphi}(\Omega)$ such that
$\int_{\gamma}v\,ds=\infty$ for every $\gamma\in\Gamma_{NAC}(u)$. For every
$y\in\widetilde{\Omega}_{k}\setminus E_{k}$, let $I(y)\subset Z_{k}(y)$ be
some compact interval such that $v$ is not absolutely continuous on $I(y)$,
and let $\gamma_{y}:[0,|I(y)|_{1}]\to\Omega$ be a parametrization of $I(y)$.
Since $\gamma_{y}\in\Gamma_{NAC}(u)$, it follows that
$\int_{I(y)}v((y,z)_{k})\,dz=\int_{\gamma_{y}}v(s)\,ds=\infty$.

From (5.8) (with $\delta=1$) and (5.9) we get

$\varphi(x,t)\geqslant\frac{t}{a\beta}$

for almost every $x\in\Omega$ and every $t\geqslant\beta$. Since
$\varphi(x,t)\geqslant 0$, it follows that

(5.10) $\varphi(x,t)\geqslant\frac{t}{a\beta}-\frac{1}{a}$

for almost every $x\in\Omega$ and every $t\geqslant 0$. Let
$\lambda>\|v\|_{\varphi}$. By (2.3) and Fubini’s theorem we have

(5.11) $\displaystyle
1\geqslant\int_{\Omega}\varphi\bigg{(}x,\frac{v(x)}{\lambda}\bigg{)}\,dx=\int_{\widetilde{\Omega}_{k}}\int_{Z_{k}(y)}\varphi\bigg{(}(y,z)_{k},\frac{v((y,z)_{k})}{\lambda}\bigg{)}\,dz\,dy$
$\displaystyle\geqslant\int_{\widetilde{\Omega}_{k}\setminus
E_{k}}\int_{I(y)}\varphi\bigg{(}(y,z)_{k},\frac{v((y,z)_{k})}{\lambda}\bigg{)}\,dz\,dy.$

By (5.10) we have

$\int_{I(y)}\varphi\bigg{(}(y,z)_{k},\frac{v((y,z)_{k})}{\lambda}\bigg{)}\,dz\geqslant\int_{I(y)}\frac{v((y,z)_{k})}{a\beta\lambda}\,dz-\int_{I(y)}\frac{1}{a}\,dz.$

Since $\int_{I(y)}v((y,z)_{k})\,dz=\infty$, the first integral on the right-
hand side is infinite, and since $I(y)$ is compact, the second integral is
finite. Thus

$\int_{I(y)}\varphi\bigg{(}(y,z)_{k},\frac{v((y,z)_{k})}{\lambda}\bigg{)}\,dz=\infty.$

Inserting this into (5.11), we get

$1\geqslant\int_{\widetilde{\Omega}_{k}\setminus
E_{k}}\int_{I(y)}\varphi\bigg{(}(y,z)_{k},\frac{v((y,z)_{k})}{\lambda}\bigg{)}\,dz\,dy\\\
=\int_{\widetilde{\Omega}_{k}\setminus E_{k}}\infty\,dy.$

This is possible only if $|\widetilde{\Omega}_{k}\setminus E_{k}|_{n-1}=0$.
Thus $u\in ACL(\Omega)$. ∎

The next example shows that the assumption (5.5) in the preceding lemma is not
redundant.

######  Example 5.12.

Let $\Omega=\mathbb{R}^{2}$. For $x=(y,z)\in\mathbb{R}^{2}$, let

$\varphi(x,t):=\left\\{\begin{array}[]{ll}t&\text{if }y=0,\\\ 0&\text{if
}y\neq 0\text{ and }t\leqslant|y|^{-1},\\\ t&\text{if }y\neq 0\text{ and
}t>|y|^{-1}.\end{array}\right.$

It easily follows from [6, Theorem 2.5.4] that
$\varphi\in\Phi_{w}(\mathbb{R}^{2})$. Define $u:\mathbb{R}^{2}\to\mathbb{R}$
by

$u(y,z):=\left\\{\begin{array}[]{ll}0&\text{if }y<0,\\\ 1&\text{if }y=0,\\\
2&\text{if }y>0.\end{array}\right.$

It is trivial that $u\notin ACL(\mathbb{R}^{2})$. It is however the case that
$u\in ACC(\mathbb{R}^{2})$.

It is easy to see, that $\Gamma_{NAC}(u)=\Gamma_{E}$, where
$E:=\\{(y,z)\in\mathbb{R}^{2}:y=0\\}$. Define $v:\mathbb{R}^{2}\to[0,\infty]$
by

$v(y,z):=\left\\{\begin{array}[]{ll}\infty&\text{if }y=0,\\\ |y|^{-1}&\text{if
}y\neq 0.\end{array}\right.$

Since the set

$\\{(y,z)\in\mathbb{R}^{2}:v(y,z)>r\\}=\\{(y,z)\in\mathbb{R}^{2}:|y|<r^{-1}\\}$

is open for every $r\in\mathbb{R}$, it follows that $v$ is a Borel function.
Fix $\gamma\in\Gamma_{E}$. For every $a\in[0,\ell(\gamma)]$, we write
$(y_{a},z_{a}):=\gamma(a)$. Now, there exists some $b\in[0,\ell(\gamma)]$ with
$y_{b}=0$. Since $\gamma$ is parametrized by arc-length, we have

$|y_{a}|=|y_{a}-y_{b}|\leqslant|\gamma(a)-\gamma(b)|\leqslant|a-b|$

for every $a\in[0,\ell(\gamma)]$. If $a\neq b$, then
$v(\gamma(a))\geqslant|a-b|^{-1}$, since if $y_{a}=0$, then
$v(\gamma(a))=\infty$, and if $y_{a}\neq 0$, then
$v(\gamma(a))=|y_{a}|^{-1}\geqslant|a-b|^{-1}$. Thus

$\int_{\gamma}v\,ds=\int_{0}^{b}v(\gamma(a))\,da+\int_{b}^{\ell(\gamma)}v(\gamma(a))\,da\geqslant\int_{0}^{b}\frac{1}{|a-b|}\,da+\int_{b}^{\ell(\gamma)}\frac{1}{|a-b|}\,da=\infty.$

Since this holds for all $\gamma\in\Gamma_{E}$, by Lemma 3.2 (c), to show that
$\operatorname{M_{\varphi}}(\Gamma_{E})=0$, it suffices to show that $v\in
L^{\varphi}(\mathbb{R}^{2})$. If $x=(y,z)$ and $y\neq 0$, then
$\varphi(x,v(x))=\varphi(x,|y|^{-1})=0$. Thus $\varphi(x,v(x))=0$ almost
everywhere, and $\varrho_{\varphi}(v)=0$. By (2.3), it follows that
$\|v\|_{\varphi}\leqslant 1$, and therefore $v\in
L^{\varphi}(\mathbb{R}^{2})$.

We know that $\nabla u$ exists for every $u\in ACL(\Omega)$. Thus, if
$\varphi$ satisfies (5.5), then Lemma 5.4 implies that $\nabla u$ exists for
every $u\in ACC(\Omega)$. We say that $u\in ACC^{\varphi}(\Omega)$, if $u\in
ACC(\Omega)$ and $\nabla u\in L^{\varphi}(\Omega)$.

######  Theorem 5.13 (Fuglede’s theorem).

Let $\varphi\in\Phi_{w}(\Omega)$ satisfy (5.5). If $C^{1}(\Omega)$-functions
are dense in $W^{1,\varphi}(\Omega)$, then $u\in W^{1,\varphi}(\Omega)$ if and
only if $u\in L^{\varphi}(\Omega)$ and it has a representative that belongs to
$ACC^{\varphi}(\Omega)$. In short

$ACC^{\varphi}(\Omega)\cap L^{\varphi}(\Omega)=W^{1,\varphi}(\Omega).$

###### Proof.

By Lemmas 5.2 and 5.4, we have

$ACC^{\varphi}(\Omega)\cap L^{\varphi}(\Omega)\subset
ACL^{\varphi}(\Omega)\cap L^{\varphi}(\Omega)\subset W^{1,\varphi}(\Omega).$

Thus it suffices to show that $W^{1,\varphi}(\Omega)\subset
ACC^{\varphi}(\Omega)$. Since $|\nabla u|\in L^{\varphi}(\Omega)$ whenever
$u\in W^{1,\varphi}(\Omega)$, we only have to show that
$W^{1,\varphi}(\Omega)\subset ACC(\Omega)$.

Suppose that $u\in W^{1,\varphi}(\Omega)$. Let $(u_{i})$ be a sequence of
functions in $C^{1}(\Omega)\cap W^{1,\varphi}(\Omega)$ converging to $u$ in
$W^{1,\varphi}(\Omega)$. By Lemma 4.2, passing to a subsequence if necessary,
we may assume that $(u_{i})$ converges pointwise everywhere, except in a set
$E$ with $\operatorname{M_{\varphi}}(\Gamma_{E})=|E|_{n}=0$. Let
$\tilde{u}(x):=\liminf_{i\to\infty}u_{i}(x)$ for every $x\in\Omega$. Since the
functions $u_{i}$ are continuous, it follows that $\tilde{u}$ is a Borel
function. Since $u_{i}(x)$ converges for every $x\in\Omega\setminus E$, it
follows that $\tilde{u}(x)=\lim_{i\to\infty}u_{i}(x)$ for $x\in\Omega\setminus
E$. By Lemma 2.4, $u_{i}\to\tilde{u}$ in $L^{\varphi}(\Omega)$, and it follows
that $\tilde{u}=u$ almost everywhere.

Since $u_{i}\to u$ in $W^{1,\varphi}(\Omega)$ we may assume, considering a
subsequence if necessary, that

$\|\nabla u_{i+1}-\nabla u_{i}||_{\varphi}<2^{-i}$

for every $i\in\mathbb{N}$. Since

$u_{i}=u_{1}+\sum_{j=1}^{i-1}(u_{j+1}-u_{j}),$

we have $|\nabla u_{i}|\leqslant g_{i}$ for every $i\in\mathbb{N}$, where

$g_{i}=|\nabla u_{1}|+\sum_{j=i}^{i-1}|\nabla u_{j+1}-\nabla u_{j}|.$

Since the sequence $(g_{i}(x))$ is increasing for every $x\in\Omega$, the
limit $g(x):=\lim_{i\to\infty}g_{i}(x)$ (possibly $\infty$) exists. Since the
functions $g_{i}$ are continuous, $g$ is a Borel function. For every $m>n$ we
have

$\|g_{m}-g_{n}\|_{\varphi}=\Big{\|}\sum_{j=n}^{m-1}|\nabla u_{j+1}-\nabla
u_{j}|\Big{\|}_{\varphi}\leqslant\sum_{j=n}^{\infty}\|\nabla u_{j+1}-\nabla
u_{j}\|_{\varphi}<\sum_{j=n}^{\infty}2^{-i}<2^{-n+1},$

i.e. $(g_{i})$ is a Cauchy sequence in $L^{\varphi}(\Omega)$. Lemma 2.4
implies that $g_{i}\to g$ in $L^{\varphi}(\Omega)$.

Let

$\Gamma_{1}:=\Big{\\{}\gamma\in\Gamma_{rect}(\Omega):\int_{\gamma}g\,ds=\infty\Big{\\}}.$

Since $g/j$ is $\Gamma_{1}$-admissible for every $j\in\mathbb{N}$, we find
that $\operatorname{M_{\varphi}}(\Gamma_{1})=0$. By Lemma 4.1, passing to a
subsequence if necessary, we find an exceptional set
$\Gamma_{2}\subset\Gamma_{rect}(\Omega)$, such that

$\lim_{i\to\infty}\int_{\gamma}g-g_{i}\,ds=0$

for every $\gamma\in\Gamma_{rect}(\Omega)\setminus\Gamma_{2}$. The set
$\Gamma_{2}$ has the following property: if
$\gamma\in\Gamma_{rect}(\Omega)\setminus\Gamma_{2}$ and $0\leqslant a\leqslant
b\leqslant\ell(\gamma)$, then
$\gamma|_{[a,b]}\in\Gamma_{rect}(\Omega)\setminus\Gamma_{2}$. The reason is
that, since $g-g_{i}\geqslant 0$, we have

$\int_{\gamma}g-g_{i}\,ds\geqslant\int_{\gamma|_{[a,b]}}g-g_{i}\,ds\geqslant
0,$

and since the first term tends to zero, the middle term must also tend to
zero. Let $\Gamma:=\Gamma_{1}\cup\Gamma_{2}\cup\Gamma_{E}$. By Lemma 3.2 (b)
$\operatorname{M_{\varphi}}(\Gamma)=0$.

We complete the proof by showing that $\tilde{u}\circ\gamma$ is absolutely
continuous for every $\gamma\in\Gamma_{rect}(\Omega)\setminus\Gamma$. Let
$k\in\mathbb{N}$ and for $j\in\\{1,2,\dots,k\\}$, let
$(a_{j},b_{j})\subset[0,\ell(\gamma)]$ be disjoint intervals. Since
$\operatorname{Im}(\gamma)$ does not intersect $E$, and $u_{i}\in
C^{1}(\Omega)$ for every $i$, we have

$\sum_{j=1}^{k}|\tilde{u}(\gamma(b_{j}))-\tilde{u}(\gamma(a_{j}))|=\lim_{i\to\infty}\sum_{j=1}^{k}|u_{i}(\gamma(b_{j}))-u_{i}(\gamma(a_{j}))|\leqslant\limsup_{i\to\infty}\sum_{j=1}^{k}\int_{\gamma|_{[a_{j},b_{j}]}}|\nabla
u_{i}|\,ds.$

Using first the fact that $|\nabla u_{i}|\leqslant g_{i}$, and then the fact
that $\gamma|_{[a_{j},b_{j}]}\notin\Gamma_{2}$, we get

$\limsup_{i\to\infty}\sum_{j=1}^{k}\int_{\gamma|_{[a_{j},b_{j}]}}|\nabla
u_{i}|\,ds\leqslant\limsup_{i\to\infty}\sum_{j=1}^{k}\int_{\gamma|_{[a_{j},b_{j}]}}g_{i}\,ds=\sum_{j=1}^{k}\int_{\gamma|_{[a_{j},b_{j}]}}g\,ds.$

Thus

$\sum_{j=1}^{k}|\tilde{u}(\gamma(b_{j}))-\tilde{u}(\gamma(a_{j}))|\leqslant\sum_{j=1}^{k}\int_{\gamma|_{[a_{j},b_{j}]}}g\,ds$

Since $\gamma\notin\Gamma_{1}$, we have $g\circ\gamma\in
L^{1}[0,\ell(\gamma)]$, which together with the inequality above implies that
$\tilde{u}\circ\gamma$ is absolutely continuous on $[0,\ell(\gamma)]$. ∎

We can combine Theorem 5.13 with Lemmas 5.2 and 5.4 to get the following
corollary:

######  Corollary 5.14.

Let $\varphi\in\Phi_{w}(\Omega)$ satisfy (5.5). If $C^{1}(\Omega)$-functions
are dense in $W^{1,\varphi}(\Omega)$, then

$ACC^{\varphi}(\Omega)\cap L^{\varphi}(\Omega)=ACL^{\varphi}(\Omega)\cap
L^{\varphi}(\Omega)=W^{1,\varphi}(\Omega).$

As was noted at the end of Section 2, $C^{\infty}(\Omega)$ functions are dense
in $W^{1,\varphi}(\Omega)$ if $\varphi$ satisfies (A0), (A1), (A2) and (aDec).
By Remark 5.6, (A0) implies (5.5). Thus Corollary 5.14 also holds with
assumptions (A0), (A1), (A2) and (aDec), instead of (5.5) and density of
$C^{1}(\Omega)$-functions.

## References

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BppAttack | 88.49 | 18.36 | 67.93 | 50.14 | 93.20 | 13.53 | 75.46 | 54.04 | 93.30 | 30.29 | 63.73 | 50.00 | 88.47 | 32.62 | 58.89 | 48.50 | 90.97 | 2.72 | 82.06 | 59.20 | 93.23 | 33.03 | 61.39 | 50.00 | 91.47 | 21.62 | 69.37 | 50.00 | 92.24 | 2.57 | 87.20 | 59.53
Input | 89.73 | 38.96 | 49.76 | 53.08 | 92.03 | 26.13 | 51.19 | 60.59 | 93.05 | 56.61 | 39.22 | 50.00 | 90.26 | 9.89 | 59.54 | 67.88 | 89.21 | 33.53 | 52.99 | 55.53 | 93.05 | 58.94 | 37.43 | 50.00 | 91.94 | 47.32 | 45.53 | 50.00 | 93.16 | 46.93 | 48.42 | 50.19
LC | 91.84 | 76.53 | 22.69 | 49.11 | 92.72 | 20.68 | 73.96 | 74.84 | 93.58 | 71.38 | 27.93 | 49.98 | 89.18 | 11.91 | 77.11 | 77.45 | 93.59 | 71.31 | 28.04 | 49.98 | 93.66 | 55.57 | 42.56 | 57.84 | 92.05 | 1.28 | 91.44 | 84.20 | 92.63 | 29.68 | 65.80 | 70.29
LF | 91.41 | 13.17 | 77.67 | 48.93 | 93.42 | 12.48 | 81.00 | 50.06 | 93.27 | 9.63 | 83.07 | 51.40 | 90.40 | 3.57 | 84.83 | 53.00 | 93.48 | 12.47 | 81.21 | 50.09 | 93.57 | 10.51 | 82.41 | 51.11 | 89.68 | 4.57 | 83.62 | 52.14 | 92.86 | 18.39 | 75.06 | 49.66
SIG | 92.29 | 60.33 | 25.07 | 49.28 | 93.53 | 37.71 | 43.07 | 51.68 | 93.64 | 38.99 | 44.12 | 51.09 | 87.48 | 13.70 | 45.68 | 60.66 | 93.59 | 40.93 | 41.98 | 50.10 | 93.75 | 37.60 | 45.49 | 51.83 | 89.34 | 15.96 | 42.92 | 60.46 | 93.21 | 41.34 | 41.50 | 49.74
SSBA | 90.43 | 4.72 | 85.84 | 48.27 | 93.72 | 1.60 | 90.28 | 49.93 | 93.86 | 1.41 | 90.84 | 50.09 | 89.23 | 1.66 | 84.68 | 47.67 | 93.74 | 1.80 | 90.14 | 49.92 | 93.83 | 1.43 | 90.76 | 50.06 | 93.89 | 1.62 | 90.64 | 50.00 | 93.39 | 1.83 | 90.08 | 49.75
TrojanNN | 91.91 | 98.04 | 1.93 | 48.97 | 92.70 | 96.50 | 3.36 | 50.14 | 93.85 | 98.19 | 1.79 | 49.89 | 87.90 | 13.41 | 70.46 | 89.28 | 93.94 | 97.94 | 2.01 | 50.04 | 93.83 | 96.32 | 3.60 | 50.79 | 93.74 | 99.11 | 0.89 | 49.84 | 92.60 | 93.84 | 5.64 | 51.42
WaNet | 91.19 | 1.09 | 89.81 | 49.50 | 91.88 | 1.06 | 91.00 | 49.85 | 93.49 | 0.74 | 92.92 | 50.02 | 91.73 | 0.82 | 90.87 | 49.78 | 91.84 | 1.07 | 90.69 | 49.83 | 93.44 | 0.76 | 92.93 | 50.01 | 90.71 | 0.94 | 89.81 | 49.26 | 93.04 | 1.02 | 91.57 | 50.00
VGG19-BN | BadNets | 91.84 | 1.31 | 91.03 | 49.84 | 90.50 | 0.70 | 90.41 | 49.30 | 90.15 | 1.36 | 89.87 | 49.00 | 88.42 | 1.26 | 87.82 | 48.14 | 92.17 | 1.03 | 91.82 | 50.00 | 88.75 | 1.70 | 88.21 | 48.30 | 89.33 | 0.86 | 89.26 | 48.64 | 10.11 | 0.07 | 11.10 | 9.42
Blended | 91.45 | 58.23 | 36.19 | 49.56 | 90.65 | 29.68 | 58.40 | 58.58 | 90.65 | 17.59 | 66.38 | 64.63 | 86.83 | 28.47 | 57.10 | 57.28 | 92.23 | 49.76 | 44.39 | 49.94 | 89.01 | 30.39 | 55.62 | 57.41 | 92.34 | 48.53 | 45.57 | 50.00 | 10.00 | 100.00 | 0.00 | 8.83
BppAttack | 89.19 | 2.51 | 83.11 | 49.25 | 91.92 | 1.57 | 86.16 | 50.36 | 91.78 | 1.79 | 85.60 | 50.24 | 90.77 | 4.36 | 83.20 | 50.00 | 88.75 | 2.51 | 82.22 | 49.03 | 92.01 | 2.12 | 85.31 | 50.08 | 90.69 | 2.28 | 84.60 | 50.00 | 19.75 | 7.40 | 17.70 | 14.53
Input | 88.00 | 74.71 | 20.91 | 49.36 | 91.45 | 60.72 | 32.99 | 50.00 | 91.39 | 65.26 | 29.59 | 50.00 | 88.73 | 10.66 | 30.43 | 68.59 | 89.23 | 72.10 | 24.11 | 49.98 | 91.02 | 70.97 | 24.80 | 50.00 | 91.10 | 6.79 | 55.04 | 70.79 | 90.21 | 43.31 | 45.21 | 52.53
LC | 89.06 | 4.10 | 83.96 | 63.09 | 90.56 | 17.13 | 73.64 | 57.32 | 90.16 | 21.46 | 71.40 | 54.96 | 90.36 | 27.61 | 62.51 | 51.98 | 92.00 | 36.17 | 59.56 | 49.80 | 89.31 | 38.78 | 56.07 | 48.46 | 92.39 | 33.60 | 61.70 | 50.00 | 89.59 | 10.11 | 79.31 | 60.35
LF | 90.53 | 1.70 | 90.08 | 49.12 | 89.48 | 0.90 | 89.51 | 48.60 | 90.29 | 1.23 | 90.12 | 49.00 | 89.18 | 0.46 | 89.64 | 48.62 | 92.16 | 0.86 | 92.06 | 49.94 | 89.21 | 1.27 | 89.00 | 48.46 | 90.38 | 0.80 | 90.46 | 49.05 | 10.00 | 100.00 | 0.00 | 8.86
SIG | 90.13 | 33.89 | 44.77 | 48.96 | 90.37 | 21.38 | 53.33 | 54.61 | 90.56 | 17.33 | 54.68 | 56.73 | 89.54 | 26.57 | 46.36 | 51.60 | 92.11 | 35.09 | 45.28 | 49.96 | 88.33 | 44.89 | 38.49 | 48.06 | 92.20 | 32.43 | 48.06 | 50.00 | 89.35 | 33.59 | 46.42 | 48.57
SSBA | 90.57 | 1.30 | 88.11 | 49.30 | 90.78 | 1.63 | 87.69 | 49.32 | 90.63 | 1.97 | 86.70 | 49.25 | 85.89 | 4.11 | 80.04 | 46.88 | 91.86 | 1.88 | 88.60 | 49.86 | 88.75 | 2.81 | 84.90 | 48.31 | 90.24 | 1.31 | 86.99 | 49.13 | 10.00 | 100.00 | 0.00 | 8.93
TrojanNN | 90.57 | 99.58 | 0.41 | 49.13 | 90.46 | 95.49 | 3.97 | 50.80 | 90.19 | 98.40 | 1.57 | 49.21 | 88.84 | 17.33 | 49.63 | 89.06 | 92.22 | 99.09 | 0.89 | 49.96 | 88.25 | 72.92 | 21.56 | 60.98 | 92.30 | 98.92 | 1.06 | 50.00 | 10.72 | 0.13 | 11.67 | 58.60
WaNet | 88.15 | 3.02 | 84.60 | 49.00 | 91.81 | 1.27 | 89.63 | 50.65 | 91.87 | 1.19 | 90.24 | 50.69 | 89.96 | 0.97 | 87.57 | 50.70 | 88.94 | 3.10 | 85.31 | 49.39 | 91.41 | 1.00 | 89.92 | 50.78 | 90.16 | 2.57 | 87.10 | 50.00 | 10.00 | 100.00 | 0.00 | 9.92
ConvNeXt_tiny | BadNets | 92.16 | 1.31 | 91.33 | 49.93 | 88.87 | 1.03 | 88.37 | 48.28 | 89.68 | 1.16 | 89.14 | 48.69 | 89.99 | 1.14 | 89.52 | 48.84 | 92.14 | 0.96 | 91.66 | 49.92 | 87.61 | 1.57 | 87.16 | 47.65 | 92.30 | 0.80 | 91.97 | 50.00 | 86.55 | 1.52 | 85.99 | 47.12
Blended | 91.97 | 61.08 | 35.31 | 51.36 | 89.67 | 19.59 | 65.62 | 70.95 | 89.71 | 26.63 | 59.88 | 67.45 | 89.26 | 36.56 | 51.96 | 62.26 | 92.12 | 61.91 | 34.91 | 51.01 | 85.99 | 19.59 | 58.86 | 69.11 | 92.17 | 64.00 | 33.11 | 50.00 | 74.04 | 6.27 | 58.97 | 69.80
BppAttack | 87.01 | 2.38 | 83.24 | 49.76 | 87.88 | 1.87 | 82.61 | 50.34 | 88.59 | 1.81 | 83.91 | 50.37 | 87.92 | 1.61 | 82.84 | 50.47 | 87.45 | 2.66 | 83.40 | 49.90 | 88.51 | 2.86 | 82.78 | 50.00 | 88.62 | 1.93 | 84.28 | 50.31 | 89.09 | 1.92 | 85.96 | 50.32
Input | 86.66 | 39.48 | 49.93 | 49.54 | 88.72 | 17.17 | 66.42 | 50.00 | 89.86 | 19.20 | 65.48 | 50.00 | 87.96 | 9.48 | 66.48 | 52.99 | 87.38 | 35.39 | 54.17 | 49.90 | 89.34 | 18.96 | 65.53 | 50.00 | 87.54 | 35.73 | 53.50 | 49.98 | 93.20 | 35.31 | 56.96 | 50.00
LC | 91.69 | 3.41 | 88.77 | 50.27 | 89.02 | 1.66 | 87.47 | 49.81 | 89.97 | 2.88 | 87.43 | 49.67 | 89.51 | 4.26 | 85.77 | 48.76 | 91.84 | 4.58 | 88.04 | 49.87 | 87.07 | 2.13 | 84.48 | 48.60 | 89.74 | 2.10 | 87.71 | 49.95 | 92.99 | 2.09 | 90.67 | 51.13
LF | 91.67 | 0.79 | 91.44 | 49.68 | 88.83 | 1.12 | 88.76 | 48.26 | 89.62 | 1.37 | 89.21 | 48.66 | 90.00 | 0.60 | 90.02 | 48.93 | 92.23 | 0.81 | 91.98 | 49.96 | 84.83 | 2.86 | 83.89 | 46.26 | 92.31 | 0.76 | 92.10 | 50.00 | 91.58 | 0.77 | 91.34 | 49.64
SIG | 91.80 | 18.10 | 65.27 | 52.28 | 88.82 | 10.82 | 65.82 | 54.43 | 89.39 | 6.40 | 70.64 | 56.93 | 89.69 | 7.70 | 67.66 | 56.43 | 92.18 | 22.56 | 63.19 | 50.25 | 85.86 | 22.29 | 58.56 | 47.22 | 92.50 | 23.39 | 62.66 | 50.00 | 92.83 | 19.94 | 63.56 | 51.72
SSBA | 92.17 | 1.09 | 89.71 | 49.88 | 88.78 | 1.17 | 86.42 | 48.19 | 89.67 | 1.23 | 87.34 | 48.64 | 90.44 | 0.78 | 87.69 | 49.16 | 92.23 | 1.07 | 89.77 | 49.92 | 87.24 | 1.18 | 84.76 | 47.42 | 92.40 | 1.06 | 90.00 | 50.00 | 75.61 | 3.36 | 72.10 | 41.60
TrojanNN | 91.98 | 95.81 | 4.02 | 49.84 | 89.13 | 49.20 | 43.99 | 71.48 | 89.42 | 69.31 | 27.41 | 61.57 | 90.44 | 39.06 | 53.49 | 77.21 | 92.28 | 95.66 | 4.16 | 49.99 | 87.29 | 43.98 | 44.61 | 73.17 | 89.56 | 67.86 | 28.67 | 62.37 | 83.05 | 15.88 | 61.86 | 85.10
WaNet | 87.11 | 1.19 | 84.40 | 49.54 | 88.51 | 1.07 | 87.40 | 50.00 | 88.72 | 1.45 | 87.10 | 50.00 | 88.30 | 1.06 | 87.08 | 50.00 | 88.02 | 0.92 | 86.03 | 50.00 | 88.47 | 1.41 | 87.19 | 50.00 | 88.06 | 0.95 | 86.17 | 50.00 | 84.24 | 1.49 | 82.86 | 48.10
ViT_b_16 | BadNets | NaN | NaN | NaN | 1.81 | 94.41 | 0.34 | 94.76 | 48.84 | 96.61 | 0.34 | 96.54 | 49.94 | 27.31 | 10.78 | 24.07 | 15.30 | NaN | NaN | NaN | 1.81 | 26.23 | 12.02 | 22.64 | 14.76 | 96.72 | 0.33 | 96.56 | 50.00 | 76.08 | 2.67 | 74.49 | 39.68
Blended | NaN | NaN | NaN | 40.20 | 94.57 | 55.90 | 38.00 | 59.53 | 96.78 | 76.68 | 22.32 | 50.21 | 27.45 | 19.03 | 20.59 | 44.41 | NaN | NaN | NaN | 40.20 | 26.92 | 16.96 | 20.77 | 45.18 | 96.63 | 78.38 | 20.76 | 49.97 | 96.15 | 72.87 | 26.03 | 51.84
BppAttack | NaN | NaN | NaN | 21.90 | 93.96 | 29.70 | 64.08 | 52.78 | 93.96 | 37.24 | 58.19 | 50.00 | 79.11 | 3.79 | 67.20 | 59.56 | NaN | NaN | NaN | 21.90 | 78.27 | 8.98 | 58.83 | 56.54 | 91.46 | 35.26 | 58.06 | 50.00 | 66.79 | 5.73 | 61.17 | 52.43
Input | 90.47 | 58.82 | 36.99 | 49.41 | 94.09 | 20.02 | 65.02 | 54.61 | 94.47 | 39.77 | 56.37 | 50.00 | 82.10 | 5.81 | 66.48 | 56.94 | 89.19 | 68.98 | 26.94 | 48.77 | 76.95 | 22.69 | 48.53 | 45.92 | 94.57 | 8.53 | 80.58 | 60.35 | 96.49 | 56.83 | 42.51 | 50.00
LC | 96.26 | 9.88 | 87.20 | 51.64 | 94.75 | 35.44 | 52.91 | 49.10 | 96.40 | 13.56 | 84.03 | 49.92 | 26.07 | 14.19 | 22.91 | 14.75 | 95.99 | 14.56 | 81.78 | 49.72 | 28.51 | 14.34 | 24.91 | 15.98 | 96.35 | 12.42 | 84.90 | 50.42 | 96.16 | 6.23 | 90.42 | 53.42
LF | 96.27 | 0.40 | 96.17 | 49.90 | 94.98 | 0.22 | 95.19 | 49.31 | 96.50 | 0.32 | 96.44 | 50.01 | 27.83 | 10.28 | 26.06 | 15.68 | 95.79 | 0.81 | 95.64 | 49.66 | 24.20 | 6.70 | 23.11 | 13.86 | 96.41 | 0.37 | 96.29 | 49.96 | 56.83 | 5.91 | 56.26 | 30.18
SIG | 96.38 | 10.59 | 78.00 | 49.89 | 94.67 | 3.22 | 74.23 | 51.75 | 96.67 | 8.38 | 80.06 | 50.14 | 27.02 | 7.51 | 27.62 | 15.78 | 95.87 | 13.22 | 76.18 | 49.63 | 26.27 | 12.04 | 23.64 | 14.83 | 96.61 | 8.67 | 80.47 | 50.00 | 96.23 | 5.23 | 82.00 | 51.53
SSBA | 96.40 | 1.58 | 93.98 | 49.86 | 93.85 | 1.11 | 91.30 | 48.77 | 96.58 | 1.14 | 94.28 | 50.12 | 26.67 | 10.00 | 25.20 | 15.00 | 95.88 | 2.11 | 93.13 | 49.60 | 27.88 | 11.87 | 25.46 | 15.60 | 96.68 | 1.48 | 94.18 | 50.00 | 72.44 | 3.49 | 67.46 | 37.88
TrojanNN | 96.54 | 89.28 | 10.69 | 49.89 | 94.53 | 54.24 | 27.30 | 66.33 | 96.66 | 84.98 | 14.96 | 52.03 | 28.12 | 13.61 | 22.44 | 53.44 | 96.17 | 93.96 | 6.02 | 49.70 | 25.22 | 10.21 | 22.43 | 53.69 | 96.53 | 69.43 | 30.28 | 59.74 | 88.35 | 75.10 | 21.78 | 52.81
WaNet | 91.56 | 0.75 | 91.16 | 49.83 | 92.96 | 0.77 | 92.44 | 50.00 | 93.77 | 0.66 | 93.27 | 50.02 | 85.72 | 1.81 | 84.84 | 46.91 | 90.68 | 0.90 | 90.34 | 49.39 | 77.14 | 3.04 | 75.74 | 42.62 | 93.91 | 0.56 | 93.51 | 50.07 | 73.83 | 2.67 | 73.05 | 40.96

##  IV Acknowledgement

This work is supported by the National Natural Science Foundation of China
under grant No.62076213, Shenzhen Science and Technology Program under grant
No.RCYX20210609103057050, and the university development fund of the Chinese
University of Hong Kong, Shenzhen under grant No.01001810. Chao Shen is
supported by the National Key Research and Development Program of China
(2020AAA0107702), National Natural Science Foundation of China (U21B2018,
62161160337, 62132011), Shaanxi Province Key Industry Innovation Program
(2021ZDLGY01-02).