Fuglede’s theorem in generalized Orlicz–Sobolev spaces

In this paper, we show that Orlicz–Sobolev spaces W1,φ(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W^{1,\varphi }(\varOmega )$$\end{document} 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 C1(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^1(\varOmega )$$\end{document} functions are dense in W1,φ(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W^{1,\varphi }(\varOmega )$$\end{document}, and φ(x,β)≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi (x,\beta ) \ge 1$$\end{document} for some β>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta > 0$$\end{document} and almost every x∈Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x \in \varOmega $$\end{document}. The results are new even in the special cases of Orlicz and double phase growth.


INTRODUCTION
In this paper, we study the ACL-and ACC-characterizations of Orlicz-Sobolev spaces W 1,ϕ (Ω), where ϕ has generalized Orlicz growth and Ω ⊂ 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 ϕ(x, t) = t p , the Orlicz case ϕ(x, t) = ϕ(t), the variable exponent case ϕ(x, t) = t p(x) , and the double phase case ϕ(x, t) = t p + a(x)t q .Generalized Orlicz and Orlicz-Sobolev spaces on 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 ∈ L p (Ω) belongs to W 1,p (Ω) if and only if it has representative ũ that is absolutely continuous on almost every line segment parallel to the coordinate axes and the classical partial derivatives of ũ belong to L p (Ω). 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 ACCcharacterization states that a function u ∈ L p (Ω) belongs to W 1,p (Ω) if and only if it has representative ũ that is absolutely continuous on every rectifiable curve outside a family of zero p-modulus and the (classical) partial derivatives ũ belong to L p (Ω).
In [7], it was shown that variable exponent Sobolev space W 1,p(•) (Ω) also has the ACLand ACC-characterizations, if the exponent satisfies suitable conditions and C 1 (Ω) functions are dense.In Section 8 of [10], it was shown that the results hold in the space W 1,ϕ (R n ), if C 1 (R n )-functions are dense and ϕ satisfies certain conditions.In this paper, we generalize the results even further.We show that the results hold for the space W 1,ϕ (Ω), and we do so using fewer assumptions than in [7] or [10].There are two assumptions we need to make: First that C 1 (Ω) functions are dense in W 1,ϕ (Ω).And second, that ϕ(x, β) 1 for some β > 0 and almost every x ∈ Ω.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 ϕ.Second, we can prove our results directly in W 1,ϕ (Ω), for any Ω ⊂ R n , whereas in [7] the results are first proven in the case Ω = R n , and this case is then used to prove the results for Ω ⊂ 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,ϕ (Ω).

PRELIMINARIES
Throughout this paper, we assume that Ω ⊂ 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.
Two functions ϕ and ψ are equivalent, ϕ ≃ ψ, if there exists L 1 such that ψ(x, t L ) ϕ(x, t) ψ(x, Lt) for every x ∈ Ω and every t > 0. Equivalent Φ-functions give rise to the same space with comparable norms.
Generalized Orlicz spaces.We recall some definitions.We denote by L 0 (Ω) the set of measurable functions in Ω. Definition 2.2.Let ϕ ∈ Φ w (Ω) and define the modular ̺ ϕ for f ∈ L 0 (Ω) by The generalized Orlicz space, also called Musielak-Orlicz space, is defined as the set If the set is clear from the context we abbreviate f L ϕ (Ω) by f ϕ .
The following lemma is a direct consequence of the proof of [6,Theorem 3.3.7].

MODULUS OF A FAMILY OF CURVES
By a curve, we mean any continuous function γ : If a curve γ is rectifiable, we may assume that I = [0, ℓ(γ)], where ℓ(γ) denotes the length of γ.We denote the image of γ by Im(γ), and by Γ rect (Ω) we denote the family of all rectifiable curves γ such that Im(γ) ⊂ Ω.Let Γ ⊂ Γ rect (Ω).We say that a Borel function for all γ ∈ Γ, where ds denotes the integral with respect to curve length.We denote the set of all Γ-admissible functions by F adm (Γ).
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 ϕ (Ω) 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 1 such that . This is an easy consequence of [6, Lemma 3.2.5],which states that there exists c 1 such that In [5], the L p -modulus was originally defined by 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 ˆΩ ϕ(x, u(x)) dx.
In the case ϕ(x, t) = t p , where 1 p < ∞, we have M ϕ (Γ) = M ϕ (Γ) p .Thus in this special case M ϕ (Γ) = 0 if and only if M ϕ (Γ) = 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 M ϕ or M ϕ .
In the general case, the situation is somewhat more complicated.Let 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.

FUGLEDE'S LEMMA
Lemma 4.1 (Fuglede's lemma).Let ϕ ∈ Φ w (Ω), and let (u i ) be a sequence of non-negative Borel functions converging to zero in L ϕ (Ω).Then there exists a subsequence (u i k ) and an exceptional set Γ ⊂ Γ rect (Ω) such that for all γ / ∈ Γ we have Let Γ ⊂ Γ rect (Ω) be the family of curves γ, such that ´γ v k ds 0 as k → ∞.For every j ∈ N, let 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 ∈ Ω, the limit w(x) := lim j→∞ w j (x) (possibly ∞) exists.By [6, Corollary 3.2.5],if j < m, then which implies that (w j ) is a Cauchy sequence in L ϕ (Ω).By Lemma 2.4, w is the limit of (w j ) in L ϕ (Ω), which implies that w ∈ L ϕ (Ω), and therefore w ϕ < ∞.
Let E ⊂ Ω.We denote by Γ E the set of all curves γ ∈ Γ rect (Ω), such that the E ∩ Im(γ) 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 (R n ) functions are dense in the variable exponent Sobolev space W 1,p(•) (R n ) and 1 < p − p + < ∞, then Γ E is exceptional whenever E ⊂ R n is of capacity zero.The latter states that if ϕ ∈ Φ w (R n ) satisfies (aInc) and (aDec), then for every Cauchy sequence in C(R n )∩W 1,ϕ (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 Γ 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,ϕ (Ω) for any Ω ⊂ R n , while in [7, Lemma 3.1] and [2, Lemma 5.1] we have Ω = R n .Lemma 4.2.Let ϕ ∈ Φ w (Ω) and let (u i ) be a Cauchy sequence of functions in C 1 (Ω) ∩ W 1,ϕ (Ω).Then there exists a set E and a subsequence (u i k ) such that M ϕ (Γ 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 Since the sequences (g j (x)) and (h j (x)) are increasing for every x ∈ Ω, the limits g(x) := lim j→∞ g j (x) and h(x) := lim j→∞ h j (x) (possibly ∞) exist.Since the functions g j are continuous, g is a Borel function.If j < m, then by [6, Corollary 3.2.5] which implies that (g j ) is a Cauchy sequence in L ϕ (Ω).By Lemma 2.4, g is the limit of (g j ) in L ϕ (Ω).Similarly, since we find that h is the limit of h j in L ϕ (Ω).
Since f k ∈ C 1 (Ω), for any k ∈ N we have for every x, y ∈ Ω and any γ ∈ Γ rect (Ω) containing x and y.Thus for every j ∈ N we have for every x, y ∈ Ω and any γ ∈ Γ rect (Ω) containing x and y.
Denote by E the set of points x ∈ Ω 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 ∈ E, then x ∈ {|f k | > 1} for infinitely many k ∈ N, and therefore g(x) = ∞.Thus and By Lemma 4.1, considering a subsequence if necessary, we find an exceptional set and It remains to show that Γ E∞ ⊂ Γ. Suppose that γ ∈ Γ rect (Ω) \ Γ.Since γ / ∈ Γ 2 , there must exist some y ∈ Im(γ) with g(y) < ∞.By (4.3), for any x ∈ Im(γ) and any j ∈ N we have g j (x) g j (y) + |g j (x) − g j (y)| g j (y) + ˆγ h j ds.

For every
The set Ω k is, in a sense, the orthogonal projection of Ω into the space {x ∈ R n : x k = 0}, but strictly speaking this is not true, since a projection is a function P : R n → R n , but 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 |•| m .Definition 5.1.We say that u : Ω → R is absolutely continuous on lines, u ∈ ACL(Ω), if it is absolutely continuous on almost every line segment in Ω parallel to the coordinate axes.More formally, let k ∈ {1, 2, . . ., n} and let E k ⊂ Ω k be the set of points y such that the function Let u ∈ ACL(Ω).Absolute continuity implies that the classical partial derivative it follows by Fubini's theorem that ∂ k u exists for almost every x ∈ Ω.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 ∇u exists almost everywhere.A function u ∈ ACL(Ω) is said to belong to The following lemma follows immediately from the definitions of L ϕ (Ω), ACL ϕ (Ω) and Definition 5.3.For any u : Ω → R, let Γ N AC (u) ⊂ Γ rect (Ω) be the family of curves γ : [0, ℓ(γ)] → Ω such that u • γ is not absolutely continuous on [0, ℓ(γ)].If M ϕ (Γ N AC (u)) = 0, then we say that u is absolutely continuous on curves, u ∈ ACC(Ω).
In the next lemma, we show that ACC(Ω) is a subset of ACL(Ω), if ϕ satisfies a suitable condition.
Proof of Lemma 5.4  Since ´I(y) v((y, z) k ) dz = ∞, the first integral on the right-hand side is infinite, and since I(y) is compact, the second integral is finite.Thus Inserting this into (5.11),we get The next example shows that the assumption (5.5) in the preceding lemma is not redundant.
Suppose that u ∈ W 1,ϕ (Ω).Let (u i ) be a sequence of functions in C 1 (Ω) ∩ W 1,ϕ (Ω) converging to u in W 1,ϕ (Ω).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 M ϕ (Γ E ) = |E| n = 0. Let ũ(x) := lim inf i→∞ u i (x) for every x ∈ Ω.Since the functions u i are continuous, it follows that ũ is a Borel function.Since u i (x) converges for every x ∈ Ω \ E, it follows that ũ(x) = lim i→∞ u i (x) for x ∈ Ω \ E. By Lemma 2.4, u i → ũ in L ϕ (Ω), and it follows that ũ = u almost everywhere.
Since u i → u in W 1,ϕ (Ω) we may assume, considering a subsequence if necessary, that for every i ∈ N. Since we have |∇u i | g i for every i ∈ N, where Since the sequence (g i (x)) is increasing for every x ∈ Ω, the limit g(x) := lim i→∞ g i (x) (possibly ∞) exists.Since the functions g i are continuous, g is a Borel function.For every m > n we have i.e. (g i ) is a Cauchy sequence in L ϕ (Ω).Lemma 2.4 implies that g i → g in L ϕ (Ω).Let Γ 1 := γ ∈ Γ rect (Ω) : ˆγ g ds = ∞ .
Since g/j is Γ 1 -admissible for every j ∈ N, we find that M ϕ (Γ 1 ) = 0.By Lemma 4.1, passing to a subsequence if necessary, we find an exceptional set Γ 2 ⊂ Γ rect (Ω), such that We complete the proof by showing that ũ • γ is absolutely continuous for every γ ∈ Γ rect (Ω) \ Γ.Let k ∈ N and for j ∈ {1, 2, . . ., k}, let (a j , b j ) ⊂ [0, ℓ(γ)] be disjoint intervals.Since Im(γ) does not intersect E, and u i ∈ C 1 (Ω) for every i, we have We can combine Theorem 5.13 with Lemmas 5.2 and 5.4 to get the following corollary: For every j ∈ N, let g j := j k=1 |f k | and h j := j k=1 |∇f k |.