Fuglede’s theorem in generalized Orlicz–Sobolev spaces

Abstract

In this paper, we show that Orlicz–Sobolev spaces $W^{1,\varphi }(\varOmega )$ 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 $C^1(\varOmega )$ functions are dense in $W^{1,\varphi }(\varOmega )$, and $\varphi (x,\beta ) \ge 1$ for some $\beta > 0$ and almost every $x \in \varOmega$. The results are new even in the special cases of Orlicz and double phase growth.

1 Introduction

In this paper, we study the ACL- and ACC-characterizations of Orlicz–Sobolev spaces $W^{1,\varphi }(\varOmega )$, where $\varphi$ has generalized Orlicz growth and $\varOmega \subset {\mathbb {R}}^n$ 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, 5, 13], and in a more general setting in [1, 12]. ACC-characterization has been used for example in [9] to study properties of capacities in the variable exponent case.

The ACL-characterization of the classical constant exponent Sobolev spaces was given by Nikodym [11]. It states that a function $u \in L^p(\varOmega )$ belongs to $W^{1,p}(\varOmega )$ 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 $L^p(\varOmega )$. Moreover the classical partial derivatives are equal to the weak partial derivatives. Fuglede [6] gave a finer version of this characterization, namely, the ACC-characterization. The ACC-characterization states that a function $u \in L^p(\varOmega )$ belongs to $W^{1,p}(\varOmega )$ 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 $L^p(\varOmega )$.

In [8], it was shown that variable exponent Sobolev space $W^{1,p(\cdot )}(\varOmega )$ also has the ACL- and ACC-characterizations, if the exponent satisfies suitable conditions and $C^1(\varOmega )$ functions are dense. In Section 8 of [12], 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 $W^{1,\varphi }(\varOmega )$, and we do so using fewer assumptions than in [8] or [12]. There are two assumptions we need to make: First that $C^1(\varOmega )$ functions are dense in $W^{1,\varphi }(\varOmega )$. And second, that $\varphi (x,\beta ) \ge 1$ for some $\beta > 0$ and almost every $x \in \varOmega$. 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 [8], 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 [8] 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 [8] 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 $W^{1,\varphi }(\varOmega )$, for any $\varOmega \subset {\mathbb {R}}^n$, whereas in [8] the results are first proven in the case $\varOmega = {\mathbb {R}}^n$, and this case is then used to prove the results for $\varOmega \subset {\mathbb {R}}^n$.

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

Let us say a few words about why one might be interested in studying ACL- or ACC-characterizations. One reason is that ACL-functions have classical partial derivatives almost everywhere, and ACC-functions are a subclass of ACL-functions under the assumptions we use. ACL- and ACC-functions also have some nice closure properties, for example the product and the maximum of two ACC-funtions is an ACC-function, and the composition of an ACC-function with a Lipschitz function is an ACC-function, and similar results hold for ACL. Another reason for studying ACC-characterization in particular is that the theory can be applied in a more general setting. In a general metric space, the concept of direction does not really make sense, so the concept of an ACL-functions cannot be used. But the concept of an ACC-function can still be defined, and has been used in the study of Newtonian spaces on general metric spaces, see [3, 10] for example.

2 Preliminaries

Throughout this paper, we assume that $\varOmega \subset {\mathbb {R}}^n$ is an open set. The following definitions are as in [7], which we use as a general reference to background theory in generalized Orlicz spaces.

Definition 2.1

We say that $\varphi : \varOmega \times [0, \infty ) \rightarrow [0, \infty ]$ is a weak $\varPhi$-function, and write $\varphi \in \varPhi _w(\varOmega )$, if the following conditions hold

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

• $\varphi (x, 0) = \lim _{t \rightarrow 0^+} \varphi (x,t) =0$ and $\lim _{t \rightarrow \infty }\varphi (x,t)=\infty$ for every $x\in \varOmega$.

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

If $\varphi \in \varPhi _w(\varOmega )$ is additionally convex and left-continuous, then $\varphi$ is a convex $\varPhi$-function, and we write $\varphi \in \varPhi _c(\varOmega )$.

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

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

$$\begin{aligned} \varphi ^{-1}(x,\tau ) := \inf \{t \ge 0 : \varphi (x,t) \ge \tau \}. \end{aligned}$$

2.1 Assumptions

We state some assumptions for later reference.

(A0):

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

(A1):

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

$$\begin{aligned} \beta \varphi ^{-1}(x, t) \le \varphi ^{-1} (y, t) \quad \text {when}\quad t \in \left[ 1, \frac{1}{|B|}\right] . \end{aligned}$$

(A2):

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

$$\begin{aligned} \beta \varphi ^{-1}(x,t) \le \varphi ^{-1}(y,t) \end{aligned}$$

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

(aInc)$_p$:

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

(aDec)$_q$:

There exist $L\ge 1$ 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 [7, Table 7.1] for an interpretation of the assumptions in some special cases.

2.2 Generalized Orlicz spaces

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

Definition 2.2

Let $\varphi \in \varPhi _w(\varOmega )$ and define the modular $\varrho _\varphi$ for $f\in L^0(\varOmega )$ by

$$\begin{aligned} \varrho _\varphi (f) := \int _\varOmega \varphi (x, |f(x)|)\,dx. \end{aligned}$$

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

$$\begin{aligned} L^\varphi (\varOmega ) := \big \{f \in L^0(\varOmega ) :\lim _{\lambda \rightarrow 0^+} \varrho _\varphi (\lambda f) = 0\big \} \end{aligned}$$

equipped with the (Luxemburg) norm

$$\begin{aligned} \Vert f\Vert _{L^\varphi (\varOmega )} := \inf \left\{ \lambda >0 :\varrho _\varphi \left( \frac{f}{\lambda } \right) \le 1\right\} . \end{aligned}$$

(2.1)

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

The following lemma is a direct consequence of the proof of [7, Theorem 3.3.7].

Lemma 2.3

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

Definition 2.4

A function $u \in L^\varphi (\varOmega )$ belongs to the Orlicz–Sobolev space $W^{1, \varphi }(\varOmega )$ if its weak partial derivatives $\partial _1 u, \ldots , \partial _n u$ exist and belong to the space $L^{\varphi }(\varOmega )$. For $u \in W^{1,\varphi }(\varOmega )$, we define the norm

$$\begin{aligned} \Vert u \Vert _{W^{1,\varphi }(\varOmega )} := \Vert u \Vert _{\varphi } + \Vert \nabla u \Vert _{\varphi }. \end{aligned}$$

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

Many of our results need the assumption that $C^1(\varOmega )$-functions are dense in $W^{1,\varphi }(\varOmega )$. A sufficient condition is given by [7, Theorem 6.4.7], which states that $C^\infty (\varOmega )$-functions are dense in $W^{1,\varphi }(\varOmega )$, if $\varphi$ satisfies (A0), (A1), (A2) and (aDec). By [7, Lemma 4.2.3], (A2) can be omitted, if $\varOmega$ is bounded.

3 Modulus of a family of curves

By a curve, we mean any continuous function $\gamma : I \rightarrow {\mathbb {R}}^n$, 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 ${\text {Im}}(\gamma )$, and by $\varGamma _{{\textit{rect}}}(\varOmega )$ we denote the family of all rectifiable curves $\gamma$ such that ${\text {Im}}(\gamma ) \subset \varOmega$. Let $\varGamma \subset \varGamma _{{\textit{rect}}}(\varOmega )$. We say that a Borel function $u: \varOmega \rightarrow [0,\infty ]$ is $\varGamma$-admissible, if

$$\begin{aligned} \int _\gamma u \,ds \ge 1 \end{aligned}$$

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

Definition 3.1

Let $\varGamma \subset \varGamma _{{\textit{rect}}}(\varOmega )$. Let $\varphi \in \varPhi _w(\varOmega )$. We define the $\varphi$-modulus of $\varGamma$ by

$$\begin{aligned} {\text {M}}_{\varphi }(\varGamma ) := \inf _{u\in F_{{\textit{adm}}}(\varGamma )} \Vert u\Vert _\varphi . \end{aligned}$$

If $F_{{\textit{adm}}}(\varGamma ) = \emptyset$, we set ${\text {M}}_{\varphi }(\varGamma ) := \infty$. A family of curves $\varGamma$ is exceptional, if ${\text {M}}_{\varphi }(\varGamma ) = 0$.

The definition above is as in [10]. The following lemma gives some useful properties of the modulus. Items (a) and (b) are items (a) and (c) of [10, Lemma 4.5], and item (c) is a consequence of [10, Proposition 4.8]. To use the lemma, we must check that $L^{\varphi }(\varOmega )$ satisfies conditions (P0), (P1), (P2) and (RF) stated at the beginning of section 2 in [10]. The conditions (P0), (P1) and (P2) are easy to check. For (RF) to hold, there must exists $c \ge 1$ such that

$$\begin{aligned} \left\| \sum _{i=1}^\infty u_i \right\| _\varphi \le \sum _{i=1}^\infty c^{i}\Vert u_i\Vert _\varphi \end{aligned}$$

holds for non-negative $u_i \in L^\varphi (\varOmega )$. This is an easy consequence of [7, Lemma 3.2.5], which states that there exists $c \ge 1$ such that

$$\begin{aligned} \left\| \sum _{i=1}^\infty u_i \right\| _\varphi \le c\sum _{i=1}^\infty \Vert u_i\Vert _\varphi . \end{aligned}$$

Lemma 3.2

Let $\varphi \in \varPhi _w(\varOmega )$, then the $\varphi$-modulus has the following properties:

• (a) if $\varGamma _1 \subset \varGamma _2$, then ${\text {M}}_{\varphi }(\varGamma _1) \le {\text {M}}_{\varphi }(\varGamma _2)$,

• (b) if ${\text {M}}_{\varphi }(\varGamma _i) = 0$ for every $i \in {\mathbb {N}}$, then ${\text {M}}_{\varphi }(\bigcup _{i=1}^\infty \varGamma _i) = 0$.

• (c) ${\text {M}}_{\varphi }(\varGamma ) = 0$ if and only if there exists a non-negative Borel function $u \in L^\varphi (\varOmega )$ such that $\int _\gamma u \,ds = \infty$ for every $\gamma \in \varGamma$.

In [6], the $L^p$-modulus was originally defined by

$$\begin{aligned} {\text {M}}_p(\varGamma ) := \inf _{u \in F_{{\textit{adm}}}(\varGamma )} \int _\varOmega u^p \,dx. \end{aligned}$$

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 [8]. Following the original approach, we could have defined the modulus by

$$\begin{aligned} {\widetilde{{\text {M}}}_{\varphi }}(\varGamma ) := \inf _{u\in F_{{\textit{adm}}}(\varGamma )} \int _{\varOmega } \varphi (x, u(x)) \,dx. \end{aligned}$$

In the case $\varphi (x,t) = t^p$, where $1\le p<\infty$, we have ${\widetilde{{\text {M}}}_{\varphi }}(\varGamma ) = {\text {M}}_{\varphi }(\varGamma )^p$. Thus in this special case ${\widetilde{{\text {M}}}_{\varphi }}(\varGamma ) = 0$ if and only if ${\text {M}}_{\varphi }(\varGamma ) = 0$. 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 ${\text {M}}_{\varphi }$ or ${\widetilde{{\text {M}}}_{\varphi }}$.

In the general case, the situation is somewhat more complicated. Let $\varphi \in \varPhi _w(\varOmega )$. By [7, Corollary 3.2.8], if $\Vert u\Vert _\varphi < 1$, then $\varrho _\varphi (u) \lesssim \Vert u\Vert _\varphi$. Thus ${\text {M}}_{\varphi }(\varGamma ) = 0$ implies ${\widetilde{{\text {M}}}_{\varphi }}(\varGamma ) = 0$. 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 $\varphi \in \varPhi _w({\mathbb {R}}^2)$ by

$$\begin{aligned} \varphi (x,t) := \left\{ \begin{array}{ll} 0 &{}\quad \text {if}\;t\le 1, \\ t-1 &{} \quad \text {if}\;t>1. \end{array}\right. \end{aligned}$$

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

$$\begin{aligned} \int _\gamma u(s) \,ds = 1 \end{aligned}$$

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

To show that ${\text {M}}_{\varphi }(\varGamma ) > 0$, suppose on the contrary, that ${\text {M}}_{\varphi }(\varGamma ) = 0$. 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 $\gamma \in \varGamma$. Thus

$$\begin{aligned} \int _{[0,1]} v(y,z) \,dz = \int _{\gamma _y} v \,ds = \infty \end{aligned}$$

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

$$\begin{aligned} \int _{{\mathbb {R}}^2} \varphi (x,\lambda v(x)) \,dx \ge \int _{[0,1]}\int _{[0,1]} \lambda v(y,z) - 1 \,dz\,dy = \infty -\int _{[0,1]}\int _{[0,1]} 1 \,dz\,dy = \infty \end{aligned}$$

Since $\lambda > 0$ was arbitrary, it follows by (2.1) that $\Vert v\Vert _\varphi = \infty$. But this is impossible, since $v \in L^\varphi ({\mathbb {R}}^2)$. Thus the assumption that ${\text {M}}_{\varphi }(\varGamma ) = 0$ must be wrong and ${\text {M}}_{\varphi }(\varGamma ) > 0$.

Note that if $\varphi \in \varPhi _w(\varOmega )$ satisfies (aDec)$_q$ for $1\le q<\infty$, then, by [7, Lemma 3.2.9] (since $\varphi$ satisfies (aInc)$_1$ by definition) we have

$$\begin{aligned} \Vert u\Vert _\varphi \lesssim \max \{\varrho _\varphi (u),\varrho _\varphi (u)^{\frac{1}{q}}\}. \end{aligned}$$

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

4 Fuglede’s lemma

Lemma 4.1

(Fuglede’s lemma) Let $\varphi \in \varPhi _w(\varOmega )$, and let $(u_i)$ be a sequence of non-negative Borel functions converging to zero in $L^\varphi (\varOmega )$. Then there exists a subsequence $(u_{i_k})$ and an exceptional set $\varGamma \subset \varGamma _{{\textit{rect}}}(\varOmega )$ such that for all $\gamma \notin \varGamma$ we have

$$\begin{aligned} \lim _{k\rightarrow \infty } \int _\gamma u_{i_k} \,ds = 0. \end{aligned}$$

Proof

Let $(v_k) := (u_{i_k})$ be a subsequence of $(u_i)$, such that

$$\begin{aligned} \Vert v_k\Vert _\varphi \le 2^{-k}. \end{aligned}$$

Let $\varGamma \subset \varGamma _{{\textit{rect}}}(\varOmega )$ be the family of curves $\gamma$, such that $\int _\gamma v_k \,ds \nrightarrow 0$ as $k \rightarrow \infty$. For every $j \in {\mathbb {N}}$, let

$$\begin{aligned} w_j := \sum _{k=1}^j v_k. \end{aligned}$$

Since every $v_k$ is a non-negative Borel function, it follows that every $w_j$ is also a non-negative Borel function. And since the sequence $(w_j(x))$ is increasing for every $x\in \varOmega$, it follows that the limit $w(x) := \lim _{j\rightarrow \infty }w_j(x)$ (possibly $\infty$) exists. By [7, Corollary 3.2.5], if $j<m$, then

$$\begin{aligned} \Vert w_m-w_j\Vert _\varphi =\left\| \sum _{k=j+1}^m v_k \right\| _\varphi \le \sum _{k=j+1}^m \Vert v_k \Vert _\varphi \le \sum _{k=j+1}^m 2^{-k} < 2^{-j}, \end{aligned}$$

which implies that $(w_j)$ is a Cauchy sequence in $L^\varphi (\varOmega )$. By Lemma 2.3, w is the limit of $(w_j)$ in $L^\varphi (\varOmega )$, which implies that $w \in L^\varphi (\varOmega )$, and therefore $\Vert w \Vert _\varphi < \infty$.

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

$$\begin{aligned} \int _\gamma w \,ds = \sum _{k=1}^\infty \int _\gamma v_k \,ds = \infty , \end{aligned}$$

because $\sum _{k=1}^\infty \int _\gamma v_k \,ds < \infty$ would imply that $\lim _{k\rightarrow \infty }\int _\gamma v_k \,ds = 0$. Thus w/m is $\varGamma$-admissible for every $m\in {\mathbb {N}}$. Since $\lim _{m\rightarrow \infty } \Vert w/m\Vert _\varphi = \lim _{m\rightarrow \infty } \Vert w\Vert _\varphi / m = 0$, we have ${\text {M}}_{\varphi }(\varGamma ) = 0$. $\square$

Let $E \subset \varOmega$. We denote by $\varGamma _E$ the set of all curves $\gamma \in \varGamma _{{\textit{rect}}}(\varOmega )$, such that the $E \cap {\text {Im}}(\gamma )$ is nonempty.

The next lemma is, in a sense, a combination of [8, 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 $1< p^{-}\le p^+ < \infty$, then $\varGamma _E$ is exceptional whenever $E \subset {\mathbb {R}}^n$ is of capacity zero. The latter states that if $\varphi \in \varPhi _w({\mathbb {R}}^n)$ 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 [8, Lemma 3.1] and modify the proof to replace convergence outside a set of capacity zero by convergence outside a set E, such that $\varGamma _E$ is exceptional. The reason that we do not simply prove a direct generalization of [8, 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 $W^{1,\varphi }(\varOmega )$ for any $\varOmega \subset {\mathbb {R}}^n$, while in [8, Lemma 3.1] and [2, Lemma 5.1] we have $\varOmega = {\mathbb {R}}^n$.

Lemma 4.2

Let $\varphi \in \varPhi _w(\varOmega )$ and let $(u_i)$ be a Cauchy sequence of functions in $C^1(\varOmega )\cap W^{1,\varphi }(\varOmega )$. Then there exists a set E and a subsequence $(u_{i_k})$ such that ${\text {M}}_{\varphi }(\varGamma _E) = |E| = 0$ and $(u_{i_k})$ converges pointwise everywhere outside E.

Proof

By [7, 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

$$\begin{aligned} \Vert v_{k+1} - v_k \Vert _{1,\varphi } < 4^{-k} \end{aligned}$$

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

$$\begin{aligned} g_j := \sum _{k=1}^j |f_k| \quad \text {and}\quad h_j := \sum _{k=1}^j |\nabla f_k|. \end{aligned}$$

Since the sequences $(g_j(x))$ and $(h_j(x))$ are increasing for every $x \in \varOmega$, the limits $g(x) := \lim _{j\rightarrow \infty } g_j(x)$ and $h(x) :=\lim _{j\rightarrow \infty } h_j(x)$ (possibly $\infty$) exist. Since the functions $g_j$ are continuous, g is a Borel function. If $j<m$, then by [7, Corollary 3.2.5]

$$\begin{aligned} \Vert g_m -g_j\Vert _{\varphi } \lesssim \sum _{k=j+1}^m \Vert f_k\Vert _{\varphi } \le \sum _{k=j+1}^\infty \Vert f_k\Vert _{1,\varphi } < \sum _{k=j+1}^\infty 2^{-k} = 2^{-j}, \end{aligned}$$

which implies that $(g_j)$ is a Cauchy sequence in $L^{\varphi }(\varOmega )$. By Lemma 2.3, g is the limit of $(g_j)$ in $L^\varphi (\varOmega )$. Similarly, since

$$\begin{aligned} \Vert h_m-h_j\Vert _\varphi \lesssim \sum _{k=j+1}^m \Vert \nabla f_k\Vert _{\varphi } \le \sum _{k=j+1}^\infty \Vert f_k\Vert _{1,\varphi } < 2^{-j}, \end{aligned}$$

we find that h is the limit of $h_j$ in $L^\varphi (\varOmega )$.

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

$$\begin{aligned} \big | |f_k(x)| - |f_k(y)|\big | \le |f_k(x) - f_k(y)| \le \int _\gamma |\nabla f_k| \,ds \end{aligned}$$

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

$$\begin{aligned} |g_j(x)-g_j(y)| \le \sum _{k=1}^j \big | |f_k(x)| - |f_k(y)|\big | \le \sum _{k=1}^j \int _\gamma |\nabla f_k| \,ds = \int _\gamma h_j \,ds, \end{aligned}$$

(4.1)

for every $x,y\in \varOmega$ and any $\gamma \in \varGamma _{{\textit{rect}}}(\varOmega )$ containing x and y.

Denote by E the set of points $x \in \varOmega$ 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

$$\begin{aligned} E \subset E_\infty := \{x \in \varOmega : g(x) = \infty \}, \end{aligned}$$

and $\varGamma _E \subset \varGamma _{E_\infty }$. Next we construct a set $\varGamma \subset \varGamma _{{\textit{rect}}}(\varOmega )$ such that $\varGamma _{E_\infty } \subset \varGamma$ and ${\text {M}}_{\varphi }(\varGamma ) = 0$. It then follows by Lemma 3.2(a) that ${\text {M}}_{\varphi }(\varGamma _E) = {\text {M}}_{\varphi }(\varGamma _{E_\infty }) = 0$.

By Lemma 4.1, considering a subsequence if necessary, we find an exceptional set $\varGamma _1 \subset \varGamma _{{\textit{rect}}}(\varOmega )$ such that

$$\begin{aligned} \lim _{j\rightarrow \infty } \int _\gamma h - h_j \,ds = 0 \end{aligned}$$

for every $\gamma \in \varGamma _{{\textit{rect}}}(\varOmega ) {\setminus } \varGamma _1$. Let

$$\begin{aligned} \varGamma _2 := \left\{ \gamma \in \varGamma _{{\textit{rect}}}(\varOmega ) : \int _\gamma g \,ds = \infty \right\} \quad \text {and} \quad \varGamma _3 := \left\{ \gamma \in \varGamma _{{\textit{rect}}}(\varOmega ) : \int _\gamma h \,ds = \infty \right\} . \end{aligned}$$

For every $m \in {\mathbb {N}}$, the function g/m is $\varGamma _2$ admissible, hence ${\text {M}}_{\varphi }(\varGamma _2) \le \Vert g \Vert _\varphi / m$. Thus it follows that ${\text {M}}_{\varphi }(\varGamma _2) = 0$. Similarly, we see that ${\text {M}}_{\varphi }(\varGamma _3) = 0$. Let $\varGamma := \varGamma _1 \cup \varGamma _2 \cup \varGamma _3$. By Lemma 3.2(b) ${\text {M}}_{\varphi }(\varGamma ) = 0$.

It remains to show that $\varGamma _{E_\infty } \subset \varGamma$. Suppose that $\gamma \in \varGamma _{{\textit{rect}}}(\varOmega ){\setminus }\varGamma$. Since $\gamma \notin \varGamma _2$, there must exist some $y \in {\text {Im}}(\gamma )$ with $g(y) < \infty$. By (4.1), for any $x \in {\text {Im}}(\gamma )$ and any $j \in {\mathbb {N}}$ we have

$$\begin{aligned} g_j(x) \le g_j(y) + |g_j(x) - g_j(y)| \le g_j(y) + \int _\gamma h_j \,ds. \end{aligned}$$

Since $\gamma \notin \varGamma _1$, it follows that

$$\begin{aligned} \lim _{j\rightarrow \infty } \int _\gamma h_j \,ds = \int _\gamma h \,ds, \end{aligned}$$

where the right-hand side is finite because $\gamma \notin \varGamma _3$. Thus we have

$$\begin{aligned} g(x) = \lim _{j\rightarrow \infty } g_j(x) \le \lim _{j\rightarrow \infty } \left( g_j(y) + \int _\gamma h_j \,ds \right) = g(y) + \int _\gamma h \,ds < \infty . \end{aligned}$$

Since $x \in {\text {Im}}(\gamma )$ was arbitrary, it follows that $\gamma \notin \varGamma _{E_\infty }$. And since $\gamma \notin \varGamma$ was arbitrary, it follows that $\varGamma _{E_\infty } \subset \varGamma$. $\square$

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

$$\begin{aligned} (y,z)_k := (y_1,\dots ,y_{k-1},z,y_{k},\dots ,y_{n-1}) \in {\mathbb {R}}^n. \end{aligned}$$

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 ${\widetilde{\varOmega }}_k\subset {\mathbb {R}}^{n-1}$ by

$$\begin{aligned} {\widetilde{\varOmega }}_k:= \{ {\tilde{x}}_k : x \in \varOmega \} = \{ y \in {\mathbb {R}}^{n-1} : (y,z)_k \in \varOmega \text { for some } z \in {\mathbb {R}}\}. \end{aligned}$$

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

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 $u : \varOmega \rightarrow {\mathbb {R}}$ is absolutely continuous on lines, $u \in {\textit{ACL}}(\varOmega )$, if it is absolutely continuous on almost every line segment in $\varOmega$ parallel to the coordinate axes. More formally, let $k \in \{1,2,\dots ,n\}$ and let $E_k \subset {\widetilde{\varOmega }}_k$ be the set of points y such that the function

$$\begin{aligned} f_y: Z_k(y) \rightarrow [-\infty ,\infty ],\, f_y(z) = u((y,z)_k) \end{aligned}$$

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

Let $u \in {\textit{ACL}}(\varOmega )$. Absolute continuity implies that the classical partial derivative $\partial _k u$ of $u \in {\textit{ACL}}(\varOmega )$ exist for every $x \in \varOmega$ such that ${\tilde{x}}_k \in E_k$. Since $|{\widetilde{\varOmega }}_k{\setminus } E_k|_{n-1} = 0$, it follows by Fubini’s theorem that $\partial _k u$ exists for almost every $x \in \varOmega$. Another application of Fubini’s theorem shows that the classical partial derivative is equal to the weak partial derivative, see [14, Theorem 2.1.4]. Since the partial derivatives exist almost everywhere, it follows that the gradient $\nabla u$ exists almost everywhere. A function $u \in {\textit{ACL}}(\varOmega )$ is said to belong to ${\textit{ACL}}^\varphi (\varOmega )$, if $|\nabla u| \in L^\varphi (\varOmega )$.

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

Lemma 5.2

If $\varphi \in \varPhi _w(\varOmega )$, then ${\textit{ACL}}^\varphi (\varOmega ) \cap L^\varphi (\varOmega ) \subset W^{1,\varphi }(\varOmega )$.

Definition 5.3

For any $u : \varOmega \rightarrow R$, we define $\varGamma _{{\textit{NAC}}}(u) \subset \varGamma _{{\textit{rect}}}(\varOmega )$ as the family of curves $\gamma : [0,\ell (\gamma )] \rightarrow \varOmega$ such that $u \circ \gamma$ is not absolutely continuous on $[0,\ell (\gamma )]$. If ${\text {M}}_{\varphi }(\varGamma _{{\textit{NAC}}}(u)) = 0$, then we say that u is absolutely continuous on curves, $u \in {\textit{ACC}}(\varOmega )$.

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

Lemma 5.4

Let $\varphi \in \varPhi _w(\varOmega )$ and assume that $\varphi$ satisfies the following condition:

$$\begin{aligned} \text {there exist }\beta > 0 \text { such that }\varphi (x,\beta )\ge 1\text { for almost every }x \in \varOmega . \end{aligned}$$

(5.1)

Then

$$\begin{aligned} {\textit{ACC}}(\varOmega ) \subset {\textit{ACL}}(\varOmega ). \end{aligned}$$

Remark 5.5

Note that (A0) implies (5.1), but not the other way around, since we do not assume that $\varphi (x,1/\beta ) \le 1$. We also note (5.1) is equivalent to

$$\begin{aligned} \text {there exist }\beta> 0 \text { and }\delta > 0\text { such that }\varphi (x,\beta )\ge \delta \text { for almost every }x \in \varOmega . \end{aligned}$$

(5.2)

It is clear that (5.1) is just a special case of (5.2) with $\delta = 1$. It is also clear that (5.2) implies (5.1), if $\delta > 1$. Suppose then, that $\varphi$ satisfies (5.2) with $0< \delta < 1$. Then

$$\begin{aligned} \frac{\delta }{\beta } \le \frac{\varphi (x,\beta )}{\beta } \end{aligned}$$

(5.3)

for almost every $x \in \varOmega$. By (aInc)$_1$ (which $\varphi$ satisfies by definition of $\varPhi _w$), there exist a constant $a \ge 1$ such that

$$\begin{aligned} \frac{\varphi (x,\beta )}{\beta } \le a\frac{\varphi (x,t)}{t} \end{aligned}$$

(5.4)

for almost every $x \in \varOmega$ and every $t \ge \beta$. Choosing $t := a\beta / \delta > \beta$, it follows from (5.3) and (5.4) that $\varphi (x,a\beta /\delta ) \ge 1$ for almost every $x \in \varOmega$, and therefore $\varphi$ satisfies (5.1). Thus the choice $\delta = 1$ in (5.1) has no special meaning, except for making notations simpler by getting rid of $\delta$.

Proof of Lemma 5.4

Let $u \in {\textit{ACC}}(\varOmega )$, and let $k \in \{1,\dots ,n\}$ and let $E_k \subset {\mathbb {R}}^{n-1}$ be as in Definition 5.1. By Lemma 3.2, there exists a non-negative Borel function $v \in L^\varphi (\varOmega )$ such that $\int _\gamma v \,ds = \infty$ for every $\gamma \in \varGamma _{{\textit{NAC}}}(u)$. For every $y \in {\widetilde{\varOmega }}_k{\setminus } E_k$, let $I(y) \subset Z_k(y)$ be some compact interval such that v is not absolutely continuous on I(y), and let $\gamma _y : [0,|I(y)|_1] \rightarrow \varOmega$ be a parametrization of I(y). Since $\gamma _y \in \varGamma _{{\textit{NAC}}}(u)$, it follows that $\int _{I(y)} v((y,z)_k) \, dz = \int _{\gamma _y} v(s) \,ds = \infty$.

From (5.3) (with $\delta = 1$) and (5.4) we get

$$\begin{aligned} \varphi (x,t) \ge \frac{t}{a\beta } \end{aligned}$$

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

$$\begin{aligned} \varphi (x,t) \ge \frac{t}{a\beta } - \frac{1}{a} \end{aligned}$$

(5.5)

for almost every $x \in \varOmega$ and every $t \ge 0$. Let $\lambda > \Vert v \Vert _\varphi$. By (2.1) and Fubini’s theorem we have

$$\begin{aligned} \begin{aligned} 1&\ge \int _\varOmega \varphi \left( x, \frac{v(x)}{\lambda }\right) \,dx = \int _{{\widetilde{\varOmega }}_k}\int _{Z_k(y)} \varphi \left( (y,z)_k,\frac{v((y,z)_k)}{\lambda }\right) \,dz\,dy \\&\ge \int _{{\widetilde{\varOmega }}_k{\setminus } E_k} \int _{I(y)} \varphi \left( (y,z)_k, \frac{v((y,z)_k)}{\lambda }\right) \,dz\,dy. \end{aligned} \end{aligned}$$

(5.6)

By (5.5) we have

$$\begin{aligned} \int _{I(y)} \varphi \left( (y,z)_k, \frac{v((y,z)_k)}{\lambda }\right) \,dz \ge \int _{I(y)} \frac{v((y,z)_k)}{a\beta \lambda } \,dz - \int _{I(y)} \frac{1}{a} \, dz. \end{aligned}$$

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

$$\begin{aligned} \int _{I(y)} \varphi \left( (y,z)_k, \frac{v((y,z)_k)}{\lambda }\right) \,dz = \infty . \end{aligned}$$

Inserting this into (5.6), we get

$$\begin{aligned} 1&\ge \int _{{\widetilde{\varOmega }}_k{\setminus } E_k} \int _{I(y)} \varphi \left( (y,z)_k, \frac{v((y,z)_k)}{\lambda }\right) \,dz\,dy \\&= \int _{{\widetilde{\varOmega }}_k{\setminus } E_k} \infty \,dy. \end{aligned}$$

This is possible only if $|{\widetilde{\varOmega }}_k{\setminus } E_k|_{n-1} = 0$. Thus $u \in {\textit{ACL}}(\varOmega )$. $\square$

The next example shows that the assumption (5.1) in the preceding lemma is not redundant.

Example 5.6

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

$$\begin{aligned} \varphi (x,t) := \left\{ \begin{array}{ll} t &{}\quad \text {if}\;y = 0, \\ 0 &{}\quad \text {if}\;y \ne 0\;\text {and}\;t \le |y|^{-1}, \\ t &{} \quad \text {if}\;y \ne 0\;\text {and}\;t > |y|^{-1}. \end{array}\right. \end{aligned}$$

It easily follows from [7, Theorem 2.5.4] that $\varphi \in \varPhi _w({\mathbb {R}}^2)$. Define $u : {\mathbb {R}}^2 \rightarrow {\mathbb {R}}$ by

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

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

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

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

Since the set

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

is open for every $r \in {\mathbb {R}}$, it follows that v is a Borel function. Fix $\gamma \in \varGamma _E$. 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

$$\begin{aligned} |y_a| = |y_a - y_b| \le |\gamma (a) - \gamma (b)| \le |a - b| \end{aligned}$$

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

$$\begin{aligned} \int _\gamma v \,ds = \int _0^b v(\gamma (a)) \,da + \int _b^{\ell (\gamma )} v(\gamma (a)) \,da \ge \int _0^b \frac{1}{|a-b|} \,da + \int _b^{\ell (\gamma )} \frac{1}{|a-b|} \,da = \infty . \end{aligned}$$

Since this holds for all $\gamma \in \varGamma _E$, by Lemma 3.2(c), to show that ${\text {M}}_{\varphi }(\varGamma _E) = 0$, it suffices to show that $v \in L^\varphi ({\mathbb {R}}^2)$. If $x = (y,z)$ and $y \ne 0$, then $\varphi (x,v(x)) = \varphi (x, |y|^{-1}) = 0$. Thus $\varphi (x,v(x)) = 0$ almost everywhere, and $\varrho _\varphi (v) = 0$. By (2.1), it follows that $\Vert v\Vert _\varphi \le 1$, and therefore $v \in L^\varphi ({\mathbb {R}}^2)$.

We know that $\nabla u$ exists for every $u \in {\textit{ACL}}(\varOmega )$. Thus, if $\varphi$ satisfies (5.1), then Lemma 5.4 implies that $\nabla u$ exists for every $u \in {\textit{ACC}}(\varOmega )$. We say that $u \in {\textit{ACC}}^\varphi (\varOmega )$, if $u \in {\textit{ACC}}(\varOmega )$ and $\nabla u \in L^\varphi (\varOmega )$.

Theorem 5.7

(Fuglede’s theorem) Let $\varphi \in \varPhi _w(\varOmega )$ satisfy (5.1). If $C^1(\varOmega )$-functions are dense in $W^{1,\varphi }(\varOmega )$, then $u \in W^{1,\varphi }(\varOmega )$ if and only if $u \in L^\varphi (\varOmega )$ and it has a representative that belongs to ${\textit{ACC}}^\varphi (\varOmega )$. In short

$$\begin{aligned} {\textit{ACC}}^\varphi (\varOmega ) \cap L^\varphi (\varOmega ) = W^{1,\varphi }(\varOmega ). \end{aligned}$$

Proof

By Lemmas 5.2 and 5.4 , we have

$$\begin{aligned} {\textit{ACC}}^\varphi (\varOmega ) \cap L^\varphi (\varOmega ) \subset {\textit{ACL}}^\varphi (\varOmega ) \cap L^\varphi (\varOmega ) \subset W^{1,\varphi }(\varOmega ). \end{aligned}$$

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

Suppose that $u \in W^{1,\varphi }(\varOmega )$. Let $(u_i)$ be a sequence of functions in $C^1(\varOmega ) \cap W^{1,\varphi }(\varOmega )$ converging to u in $W^{1,\varphi }(\varOmega )$. 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 ${\text {M}}_{\varphi }(\varGamma _E) =|E|_n = 0$. Let ${\tilde{u}}(x) := \liminf _{i\rightarrow \infty } u_i(x)$ for every $x \in \varOmega$. Since the functions $u_i$ are continuous, it follows that ${\tilde{u}}$ is a Borel function. Since $u_i(x)$ converges for every $x\in \varOmega {\setminus } E$, it follows that ${\tilde{u}}(x) = \lim _{i\rightarrow \infty }u_i(x)$ for $x\in \varOmega {\setminus } E$. By Lemma 2.3, $u_i \rightarrow {\tilde{u}}$ in $L^\varphi (\varOmega )$, and it follows that ${\tilde{u}} = u$ almost everywhere.

Since $u_i \rightarrow u$ in $W^{1,\varphi }(\varOmega )$ we may assume, considering a subsequence if necessary, that

$$\begin{aligned} \Vert \nabla u_{i+1} - \nabla u_i||_\varphi < 2^{-i} \end{aligned}$$

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

$$\begin{aligned} u_i = u_1 + \sum _{j=1}^{i-1} (u_{j+1} - u_j), \end{aligned}$$

we have $|\nabla u_i| \le g_i$ for every $i \in {\mathbb {N}}$, where

$$\begin{aligned} g_i = |\nabla u_1| + \sum _{j=i}^{i-1}|\nabla u_{j+1} - \nabla u_j|. \end{aligned}$$

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

$$\begin{aligned} \Vert g_m - g_n\Vert _\varphi = \left\| \sum _{j=n}^{m-1} |\nabla u_{j+1} - \nabla u_j|\right\| _\varphi \lesssim \sum _{j=n}^\infty \Vert \nabla u_{j+1} - \nabla u_j\Vert _\varphi< \sum _{j=n}^\infty 2^{-i} < 2^{-n+1}, \end{aligned}$$

i.e. $(g_i)$ is a Cauchy sequence in $L^\varphi (\varOmega )$. Lemma 2.3 implies that $g_i \rightarrow g$ in $L^\varphi (\varOmega )$.

Let

$$\begin{aligned} \varGamma _1 := \left\{ \gamma \in \varGamma _{{\textit{rect}}}(\varOmega ) : \int _\gamma g \,ds = \infty \right\} . \end{aligned}$$

Since g/j is $\varGamma _1$-admissible for every $j\in {\mathbb {N}}$, we find that ${\text {M}}_{\varphi }(\varGamma _1) = 0$. By Lemma 4.1, passing to a subsequence if necessary, we find an exceptional set $\varGamma _2 \subset \varGamma _{{\textit{rect}}}(\varOmega )$, such that

$$\begin{aligned} \lim _{i\rightarrow \infty } \int _\gamma g - g_i \,ds = 0 \end{aligned}$$

for every $\gamma \in \varGamma _{{\textit{rect}}}(\varOmega ) {\setminus } \varGamma _2$. The set $\varGamma _2$ has the following property: if $\gamma \in \varGamma _{{\textit{rect}}}(\varOmega ) {\setminus } \varGamma _2$ and $0\le a\le b\le \ell (\gamma )$, then $\gamma |_{[a,b]} \in \varGamma _{{\textit{rect}}}(\varOmega ) {\setminus } \varGamma _2$. The reason is that, since $g-g_i\ge 0$, we have

$$\begin{aligned} \int _\gamma g - g_i \,ds \ge \int _{\gamma |_{[a,b]}} g - g_i \,ds \ge 0, \end{aligned}$$

and since the first term tends to zero, the middle term must also tend to zero. Let $\varGamma := \varGamma _1 \cup \varGamma _2 \cup \varGamma _E$. By Lemma 3.2(b) ${\text {M}}_{\varphi }(\varGamma ) = 0$.

We complete the proof by showing that ${\tilde{u}} \circ \gamma$ is absolutely continuous for every $\gamma \in \varGamma _{{\textit{rect}}}(\varOmega ) {\setminus } \varGamma$. 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 ${\text {Im}}(\gamma )$ does not intersect E, and $u_i \in C^1(\varOmega )$ for every i, we have

$$\begin{aligned} \sum _{j=1}^k |{\tilde{u}}(\gamma (b_j)) - {\tilde{u}}(\gamma (a_j))|= & {} \lim _{i\rightarrow \infty } \sum _{j=1}^k |u_i(\gamma (b_j)) - u_i(\gamma (a_j))|\\\le & {} \limsup _{i\rightarrow \infty } \sum _{j=1}^k \int _{\gamma |_{[a_j,b_j]}} |\nabla u_i| \,ds. \end{aligned}$$

Using first the fact that $|\nabla u_i| \le g_i$, and then the fact that $\gamma |_{[a_j,b_j]} \notin \varGamma _2$, we get

$$\begin{aligned} \limsup _{i\rightarrow \infty } \sum _{j=1}^k \int _{\gamma |_{[a_j,b_j]}} |\nabla u_i| \,ds \le \limsup _{i\rightarrow \infty } \sum _{j=1}^k \int _{\gamma |_{[a_j,b_j]}} g_i \,ds = \sum _{j=1}^k\int _{\gamma |_{[a_j,b_j]}} g \,ds. \end{aligned}$$

Thus

$$\begin{aligned} \sum _{j=1}^k |{\tilde{u}}(\gamma (b_j)) - {\tilde{u}}(\gamma (a_j))| \le \sum _{j=1}^k\int _{\gamma |_{[a_j,b_j]}} g \,ds \end{aligned}$$

Since $\gamma \notin \varGamma _1$, 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 $[0,\ell (\gamma )]$. $\square$

We can combine Theorem 5.7 with Lemmas 5.2 and 5.4 to get the following corollary:

Corollary 5.8

Let $\varphi \in \varPhi _w(\varOmega )$ satisfy (5.1). If $C^1(\varOmega )$-functions are dense in $W^{1,\varphi }(\varOmega )$, then

$$\begin{aligned} {\textit{ACC}}^\varphi (\varOmega ) \cap L^\varphi (\varOmega ) = {\textit{ACL}}^\varphi (\varOmega ) \cap L^\varphi (\varOmega ) = W^{1,\varphi }(\varOmega ). \end{aligned}$$

As was noted at the end of Sect. 2, $C^\infty (\varOmega )$ functions are dense in $W^{1,\varphi }(\varOmega )$ if $\varphi$ satisfies (A0), (A1), (A2) and (aDec). By Remark 5.5, (A0) implies (5.1). Thus Corollary 5.8 also holds with assumptions (A0), (A1), (A2) and (aDec), instead of (5.1) and density of $C^1(\varOmega )$-functions.