Let Φ be an N-function. We show that a function u∈LΦ(ℝn) belongs to the Orlicz-Sobolev space W1,Φ(ℝn) if and only if it satisfies the (generalized) Φ-Poincaré inequality. Under more restrictive assumptions on Φ, an analog of the result holds in a general metric measure space setting.

1. Introduction

Let X=(X,d,μ) be a metric measure space, Ω⊂X open, and Φ a Young function. A pair (u,g) of measurable functions, u∈Lloc1(Ω) and g≥0, satisfy the Φ-Poincaré inequality in Ω, if there is a constant τ≥1 such that⨍B|u-uB|dμ≤rBΦ-1(⨍τBΦ(g)dμ),
for every ball B=B(x,rB) such that τB⊂Ω. Here, uB=⨍Budμ=μ(B)-1∫Budμ and τB=B(x,τrB). If Φ(t)=tp, then (1.1) reduces to the familiar p-Poincaré inequality. The Φ-Poincaré inequality was introduced in [1] and further studied in [2–5].

In the euclidean setting, it is well known that u∈Wloc1,1(Ω) satisfies the 1-Poincaré inequality⨍B|u-uB|dμ≤CnrB⨍B|∇u|dμ,
for every ball B⊂Ω. Thus, by Jensen’s inequality, (1.1) holds with τ=1 and g=Cn|∇u|. Our first result, Theorem 1.1 below, says that also the converse holds: if u∈LΦ(Ω) and there exists g∈LΦ(Ω) such that (1.1) holds (for the normalized pair), then u belongs to the Sobolev class W1,Φ(Ω). More generally, we show that u∈W1,Φ(Ω) if and only if the number‖u‖Aτ1,Φ(Ω)=supB∈Bτ(Ω)‖∑B∈B(rB-1⨍B|u-uB|dμ)χB‖LΦ(Ω),
whereBτ(Ω)={{Bi}:ballsτBiaredisjointandcontainedinΩ},
is finite. Note that ∥u∥Aτ1,Φ(Ω)≤1 if and only if there is a functional ν:{B⊂Ω:Bisaball}→[0,∞) such that ∑iν(Bi)≤1,
whenever the balls Bi are disjoint, and that the generalized Φ-Poincaré inequality⨍B|u-uB|dμ≤rBΦ-1(ν(τB)μ(B))
holds whenever τB⊂Ω. In particular, if a pair (u/∥g∥LΦ(Ω),g/∥g∥LΦ(Ω)) satisfies the Φ-Poincaré inequality in Ω, then‖u‖Aτ1,Φ(Ω)≤‖g‖LΦ(Ω).
The spaces Aτ1,Φ(Ω)={u∈Lloc1(Ω):∥u∥Aτ1,Φ(Ω)<∞}, for Φ(t)=tp, were studied in [6].

Theorem 1.1.

Suppose that Φ is an N-function, Ω⊂ℝn is open, and u∈LΦ(Ω), then the following conditions are equivalent:

u∈W1,Φ(Ω),

there exists g∈LΦ(Ω) such that the pair (u/∥g∥LΦ(Ω),g/∥g∥LΦ(Ω)) satisfies the Φ-Poincaré inequality in Ω,

u∈Aτ1,Φ(Ω) for some τ≥1.

Moreover, if a functional ν satisfies (1.5) and (1.6), one has that
|∇u(x)|≤C(Cd,τ)limsupr→0Φ-1(ν(B(x,r))μ(B(x,r))),
for a.e. x∈Ω. In particular, (b) implies that |∇u(x)|≤C(n)g(x) for a.e. x∈Ω.

Notice that Φ(t)=t is not an N-function. In this case, (a) and (b) are still equivalent, but (a) and (c) are not. In fact, it was shown in [6] that Aτ1,1(Ω) coincides with BV(Ω), the space of functions of bounded variation. The equivalence of (a) and (b) in the case Φ(t)=tp, p>1 was proved in [7] and in the case p=1 in [8]. A different proof of the case p≥1 was provided in [9]. The case where both Φ and its conjugate are doubling can be found in [5]. The equivalence of (a) and (c) in the case Φ(t)=tp, p>1, was proved in [6]. The proof in [6] relies on a reflexivity argument which does not extend to the present setting. Our proof is a modification of the proof from [9].

The rest of our results are partial analogs of Theorem 1.1 in a general metric measure space setting. Let μ be a Borel regular outer measure satisfying 0<μ(U)<∞, whenever U is nonempty, open, and bounded. Suppose further that μ is doubling, that is, there exists a constant Cd such thatμ(2B)≤Cdμ(B),
whenever B=B(x,r) is a ball and 2B=B(x,2r).

Our substitute for the usual Sobolev class W1,Φ is based on upper gradients. We call a Borel function g:X→[0,∞] an upper gradient of a function u:X→ℝ¯ if|u(γ(0))-u(γ(l))|≤∫γgds,
for all rectifiable curves γ:[0,l]→X. Further, g as above is called a Φ-weak upper gradient if (1.10) holds for all curves γ except for a family of Φ-modulus zero, see Section 2.2 below. The concept of an upper gradient was introduced in [10]; also see [7]. The Sobolev space N1,Φ(X) consists of all functions in LΦ(X) that have a Φ-weak upper gradient that belongs to LΦ(X).

Theorem 1.2.

Suppose that Φ is a doubling Young function, Ω⊂X is open, u,g∈LΦ(Ω), and that the pair (u/∥g∥LΦ(Ω),g/∥g∥LΦ(Ω)) satisfies the Φ-Poincaré inequality in Ω, then a representative of u has a Φ-weak upper gradient gu∈LΦ(Ω) such that gu(x)≤C(Cd)g(x) for a.e. x∈Ω.

In the case Φ(t)=tp, p≥1, the result was essentially proved in [8], see [11].

If both Φ and its conjugate are doubling, then a generalization of the proof of [6, Theorem 1.1 (2)] yields the following.

Theorem 1.3.

Let Ω⊂X be an open set, and let Φ be a doubling Young function whose conjugate is doubling, then a representative of u∈Aτ1,Φ(Ω)∩LΦ(Ω) has a Φ-weak upper gradient g with ∥g∥LΦ(Ω)≤C(Cd,τ)∥u∥Aτ1,Φ(Ω). Moreover, for a functional ν satisfying (1.5) and (1.6), one has that
g(x)≤C(Cd,τ)limsupr→0Φ-1(ν(B(x,r))μ(B(x,r))),
for a.e. x∈Ω.

We say that a space X supports the Φ-Poincaré inequality if there exist constants CP and τ such that⨍B|u-uB|dμ≤CPrBΦ-1(⨍τBΦ(g)dμ),
whenever B⊂X is a ball, u∈Lloc1(X), and g is a Φ-weak upper gradient of u. The spaces supporting the Φ-Poincaré inequality include Riemannian manifolds with nonnegative Ricci curvature, Carnot groups, and general Carnot—Carathéodory spaces associated with a system of vector fields satisfying Hörmander’s condition; see [11, 12] and the references therein.

Theorem 1.4.

Suppose that Φ is a doubling Young function, X supports the Φ-Poincaré inequality, Ω⊂X is open, and u∈LΦ(Ω), then the following conditions are equivalent.

u∈N1,Φ(Ω).

There exists g∈LΦ(Ω) such that the pair (u/∥g∥LΦ(Ω),g/∥g∥LΦ(Ω)) satisfies the Φ-Poincaré inequality in Ω.

If also the conjugate of Φ is doubling, then (a) and (b) are equivalent to

u∈Aτ1,Φ(Ω) for some τ≥1.

2. Preliminaries

Throughout this paper, C will denote a positive constant whose value is not necessarily the same at each occurrence. By writing C=C(λ1,…,λn), we indicate that the constant depends only on λ1,…,λn.

2.1. Young Functions and Orlicz Spaces

In this subsection, we recall the basic facts about Young functions and Orlicz spaces. An exhaustive treatment of the subject is [13]. In the case of N-functions, good expositions are also [14] and [15, Chapter 8].

A function Φ:[0,∞)→[0,∞] is called a Young function if it has the formΦ(t)=∫0tϕ(s)ds,
where ϕ:[0,∞)→[0,∞] is an increasing, left-continuous function, which is neither identically zero nor identically infinite on (0,∞). If, in addition, 0<Φ(t)<∞ for t>0, limt→0Φ(t)/t=0, and limt→∞Φ(t)/t=∞, then Φ is called an N-function.

A Young function is convex and, in particular, satisfiesΦ(ɛt)≤ɛΦ(t),
for 0<ɛ≤1 and 0≤t<∞.

If Φ is a real-valued Young function and μ(X)<∞, then Jensen’s inequalityΦ(⨍Xudμ)≤⨍XΦ(u)dμ
holds for 0≤u∈L1(X).

The right-continuous generalized inverse of a Young function Φ isΦ-1(t)=inf{s:Φ(s)>t}.
We have thatΦ(Φ-1(t))≤t≤Φ-1(Φ(t)),
for t≥0.

The conjugate of a Young function Φ is the Young function defined byΦ̂(t)=sup{ts-Φ(s):s>0},
for t≥0.

The conjugate of an N-function is an N-function.

Let Φ be a Young function. The Orlicz space LΦ(X) is the set of all measurable functions u for which there exists λ>0 such that∫XΦ(|u(x)|λ)dμ(x)<∞.
The Luxemburg norm of u∈LΦ(X) is‖u‖LΦ(X)=inf{λ>0:∫XΦ(|u(x)|λ)dμ(x)≤1}.
If ∥u∥LΦ(X)≠0, we have that∫XΦ(|u(x)|‖u‖LΦ(X))dμ(x)≤1.
The following generalized Hölder inequality holds for Luxemburg norms:∫Xu(x)v(x)dμ(x)≤2‖u‖LΦ(X)‖v‖LΦ̂(X).

A Young function Φ is doubling if there exists a constant CΦ≥1 such thatΦ(2t)≤CΦΦ(t),
for t≥0. Notice that a doubling Young function is realvalued and Φ(x)=0 if and only if x=0.

Lemma 2.1.

Let Φ be a doubling Young function.

The space C0(X) of bounded, boundedly supported continuous functions is dense in LΦ(X).

The modular convergence and the norm convergence are equivalent, that is,
‖fj-f‖LΦ(X)⟶0
if and only if
∫XΦ(|fj-f|)dμ⟶0.

If Φ is doubling, simple functions are dense in LΦ(X) [13, Chapter III, Corollary 5]. Hence, the proof of (1.1) is the same as in the Lp-case; see, for example, [11, Theorem 4.2]. For the proof of (1.3), see [13, Chapter III, Theorem 12].

If Φ is doubling, then (LΦ(X))*=LΦ̂(X), see [13, Chapter IV, Corollary 9]. So, if both Φ and Φ̂ are doubling, LΦ(X) is reflexive. Thus, every bounded sequence in LΦ(X) admits a weakly converging subsequence. In the proof of Theorem 1.2, we need to extract a weakly converging subsequence also when Φ̂ is not doubling. For this, we need the following lemma.

Lemma 2.2.

Suppose that Φ is a doubling Young function and that {gi}⊂LΦ(X) satisfies
supi‖gi‖LΦ(X)<∞,limμ(A)→0supi∫AΦ(|gi|)dμ=0,
then there exists a subsequence (gij) of (gi) and g∈LΦ(X) such that gij→g weakly in LΦ(X).

Proof.

Since Φ is doubling, the dual of LΦ(X) is LΦ̂(X). By [13, page 144, Corollary 2], a sequence {gi} has a weakly converging subsequence if for each h∈LΦ̂(X),
supi|∫Xgihdμ|<∞,limμ(A)→0supi∫A|gih|dμ=0.
By the Hölder inequality and Lemma 2.1(2), these follow from (2.14).

2.2. Sobolev Spaces

The Φ-modulus of a curve family Γ isModΦ(Γ)=inf‖g‖LΦ(X),
where the infimum is taken over all Borel functions g:X→[0,∞] satisfying∫γgds≥1,
for all locally rectifiable curves γ∈Γ.

The Sobolev space N1,Φ(X), consisting of the functions u∈LΦ(X) having a Φ-weak upper gradient g∈LΦ(X), was introduced by Shanmugalingam [16], when Φ(t)=tp, and extended to the Orlicz case by Tuominen [1]. The space N1,Φ(X) is a Banach space with the norm‖u‖N1,Φ(X)=‖u‖LΦ(X)+inf‖g‖LΦ(X),
where the infimum is taken over Φ-weak upper gradients g∈LΦ(X) of u.

Lemma 2.3.

Suppose that ui→u∈LΦ(X) and gi→g∈LΦ(X) weakly in LΦ(X) and that gi is a Φ-weak upper gradient of ui, then g is a Φ-weak upper gradient of a representative of u. Moreover, g(x)≤limsupi→∞gi(x) for a.e. x∈X.

Proof.

By Mazur’s lemma ([17, Page 120, Theorem 2]), there is a sequence (g̃i) of convex combinations
g̃i=∑j=iniλi,jgj,
where λi,j≥0 and ∑j=1niλi,j=1, such that g̃i→g in LΦ(X). Hence, a subsequence of (g̃i) converges pointwise a.e., which implies that g(x)≤limsupi→∞g̃i(x)≤limsupi→∞gi(x) for a.e. x∈X. The fact that g is a Φ-weak upper gradient of a representative of u was proved in [1, Theorem 4.17].

If Φ is doubling and Ω⊂ℝn is an open set, then N1,Φ(Ω) is isomorphic to W1,Φ(Ω) [1, Theorem 6.19]. As usual, W1,Φ(Ω) is the space of functions u∈LΦ(Ω) having weak partial derivatives in LΦ(Ω). A function vi∈Lloc1(Ω) is a weak partial derivative of u (with respect to xi) if∫u∂φ∂xi=-∫viφ,
for all φ∈C0∞(Ω).

Lemma 2.4.

Let Φ be an N-function. Suppose that the functional ∂u/∂xi:C0∞(Ω)→ℝ,
∂u∂xi[φ]:=-∫u∂φ∂xi,
is bounded with respect to the norm ∥·∥LΦ̂(Ω), then u has a weak partial derivative vi such that ∥vi∥LΦ(Ω)≤∥∂u/∂xi∥.

Proof.

Denote by EΦ̂(Ω) the closure of the space of bounded, boundedly supported functions in LΦ̂(Ω). By [15, Theorem 8.21(d)], C0∞(Ω) is dense in EΦ̂(Ω). Thus, ∂u/∂xi extends to a continuous linear functional on EΦ̂(Ω). By [15, Theorem 8.19], the dual of EΦ̂(Ω) is isomorphic to LΦ(Ω); there exists vi∈LΦ(Ω) such that
∂u∂xi[φ]=∫φvi,
for φ∈EΦ̂(Ω). Moreover, ∥vi∥LΦ(Ω)≤∥∂u/∂xi∥. The claim follows.

2.3. Lipschitz Functions

A function u:X→ℝ is L-Lipschitz if |u(x)-u(y)|≤Ld(x,y) for all x,y∈X. The (upper) pointwise Lipschitz constant of a locally Lipschitz function u isLipu(x)=limsupr→0r-1supd(x,y)≤r|u(x)-u(y)|.
It is well known that Lipu is an upper gradient of u; see, for example, [18].

3. Proofs

The proof of Theorem 1.1 is a modification of the proof of case p>1 of [9, Lemma 6].

Proof of Theorem <xref ref-type="statement" rid="thm1.1">1.1</xref>.

As noted in the introduction, (a) ⇒ (b) ⇒ (c). Let us show that (c) ⇒ (a). Fix u∈Aτ1,Φ(Ω). We will show that the functional ∂u/∂xi:C0∞(Ω)→ℝ,
∂u∂xi[φ]:=-∫u∂φ∂xi,
satisfies
|∂u∂xi[φ]|≤C(n,τ)‖u‖Aτ1,Φ(Ω)‖φ‖LΦ̂(suppφ).

Choose 0≤ψ∈C0∞(B(0,1)) such that ∫ψ=1, and let ψɛ(x)=ɛ-nψ(x/ɛ) for ɛ>0, Then∂u∂xi[φ]=-limɛ→0∫(u*ψɛ)∂φ∂xi=limɛ→0∫(u*∂ψɛ∂xi)φ.
By the Hölder inequality,
|∂u∂xi[φ]|≤2liminfɛ→0‖u*∂ψɛ∂xi‖LΦ(suppφ)‖φ‖LΦ̂(suppφ).
Since ∫∂ψɛ/∂xi=0, we have that
(u*∂ψɛ∂xi)(x)=((u-uB(x,ɛ))*∂ψɛ∂xi)(x).
Thus,
|u*∂ψɛ∂xi|(x)≤C(n)ɛ-1⨍B(x,ɛ)|u(y)-uB(x,ɛ)|dy.
Let K=suppφ and let 0<ɛ<d(K,Ωc)/3τ. Cover K with balls B(xj,2ɛ), xj∈K, such that the balls B(xj,ɛ) are disjoint. If x∈B(xj,2ɛ), then B(x,ɛ)⊂B(xj,3ɛ) and (3.6) implies that
|u*∂ψɛ∂xi|(x)≤C(n)(3ɛ)-1⨍B(xj,3ɛ)|u(y)-uB(xj,3ɛ)|dy.
Thus,
|u*∂ψɛ∂xi|≤C(n)∑j(3ɛ)-1⨍B(xj,3ɛ)|u(y)-uB(xj,3ɛ)|dyχB(xj,3ɛ).
Since the balls B(xj,ɛ) are disjoint, it follows that the family ℬ={B(xj,3ɛ)} can be divided into k=C(n,τ) subfamilies ℬ1,…ℬk such that each of the families τℬj={τB:B∈ℬj} consists of disjoint balls. Hence,
‖u*∂ψɛ∂xi‖LΦ(Ω)≤C(n)∑j=1k‖∑B∈BjrB-1⨍B|u-uB|dμχB‖LΦ(Ω)≤C(n,τ)‖u‖Aτ1,Φ(Ω).
This, combined with (3.4), yields (3.2). By Lemma 2.4, u has a weak partial derivative vi∈LΦ(Ω).

Since |vi(x)|≤limsupɛ→0|(∂/∂xi)(u*ψɛ)(x)| for a.e. x∈Ω and (∂/∂xi)(u*ψɛ)(x)=u*(∂ψɛ/∂xi)(x) for small ɛ, it follows that|vi(x)|≤limsupɛ→0|u*∂ψɛ∂xi(x)|,
for a.e. x∈Ω. Let ν be a functional satisfying (1.5) and ((1.6), then, by (3.6),
|u*∂ψɛ∂xi|(x)≤C(n)Φ-1(ν(B(x,τɛ))μ(B(x,ɛ)))≤C(n,τ)Φ-1(ν(B(x,τɛ))μ(B(x,τɛ))).
Thus, (1.8) holds for a.e. x∈Ω. If condition (b) is satisfied, we have
|u*∂ψɛ∂xi|(x)≤C(n)‖g‖LΦ(Ω)Φ-1(⨍B(x,τɛ)Φ(g(y)‖g‖LΦ(Ω))dy),
which implies that |vi(x)|≤C(n)g(x) for a.e. x∈Ω. This completes the proof.

The proofs of Theorems 1.2 and 1.3 are based on approximation by Lipschitz convolutions. The same technique was employed in [6–8]. The proof of Theorem 1.3 is a generalization of the proof of [6, Theorem 1.1]. Using a partition of unity and averages of u on balls, we construct a sequence of locally Lipschitz functions uj so that uj→u in LΦ(Ω) and that‖Lipuj‖LΦ(Ω)≤C‖u‖Aτ1,Φ(Ω).
Since LΦ(Ω) is reflexive, a subsequence of (Lipuj) converges weakly, and the claim follows from Lemma 2.3.

Under the assumptions of Theorem 1.2, LΦ(Ω) may not be reflexive, but using the Φ-Poincaré inequality, we can show that (Lipuj) is uniformly Φ-integrable. The existence of a weakly converging subsequence then follows from Lemma 2.2. A similar argument was used in [8].

We need a couple of standard lemmas. For the proofs, see [19, Theorem III.1.3] and [20, Lemmas 2.9 and 2.16].

Lemma 3.1.

Let Ω⊂X be open. Given ɛ>0, λ≥1, there is a cover {Bi=B(xi,ri)} of Ω with the following properties:

ri≤ɛ for all i,

λBi⊂Ω for all i,

if λBi meets λBj, then ri≤2rj,

each ball λBi meets at most C=C(Cd,λ) balls λBj.

A collection {Bi} as above is called an (ɛ,λ)-covering of Ω. Clearly, an (ɛ,λ)-cover is an (ɛ′,λ′)-cover provided ɛ′≥ɛ and λ′≤λ.

Lemma 3.2.

Let Ω⊂X be open, and let ℬ={Bi=B(xi,ri)} be an (∞,2)-cover of Ω, then there is a collection {φi} of functions Ω→ℝ such that

each φi is C(Cd)ri-1-Lipschitz,

0≤φi≤1 for all i,

φi(x)=0 for x∈X∖2Bi for all i,

∑iφi(x)=1 for all x∈Ω.

A collection {φi} as above is called a partition of unity with respect to ℬ.

Let ℬ={Bi} be as in the lemma above, and let {φi} be a partition of unity with respect to ℬ. For a locally integrable function u on Ω, defineuB(x)=∑iuBiφi(x).
The following lemma describes the most important properties of uℬ.

Lemma 3.3.

(1) The function uℬ is locally Lipschitz. Moreover, for each x∈Bi,
LipuB(x)≤C(Cd)rBi-1⨍5Bi|u-u5Bi|dμ.

(2) Let Φ be a doubling Young function, and let u∈LΦ(Ω). If ℬk is an (ɛk,2)-cover of Ω and ɛk→0 as k→∞, then uℬk→u in LΦ(Ω).

Proof.

(1) See the proof of [6, Lemma 5.3(1)].

(2) We begin by showing that, for every w∈LΦ(Ω),‖wB‖LΦ(Ω)≤C(Cd)‖w‖LΦ(Ω).
We may assume that ∥w∥LΦ(Ω)=1. By Jensen’s inequality, Φ(|wℬ|)≤(Φ(|w|))ℬ. Hence, by the properties of the functions φi,
∫ΩΦ(|wB|)dμ≤∫Ω(Φ(|w|))Bdμ≤∑i∫Ω(Φ(|w|))Biφidμ≤∑i∫2BiΦ(|w|)Bidμ≤Cd∑i∫BiΦ(|w|)dμ=Cd∫ΩΦ(|w|)∑iχBidμ≤C(Cd)∫ΩΦ(|w|)dμ≤C(Cd).
Thus, by (2.2), we obtain (3.16).

Let u∈LΦ(Ω) and ɛ>0. By Lemma 2.1(1), there exists v∈C0(Ω) such that ∥u-v∥LΦ(Ω)<ɛ. Then, by (3.16), we obtain‖uB-vB‖LΦ(Ω)=‖(u-v)B‖LΦ(Ω)≤C(Cd)‖u-v‖LΦ(Ω)<C(Cd)ɛ,
and so
‖uB-u‖LΦ(Ω)≤‖uB-vB‖LΦ(Ω)+‖vB-v‖LΦ(Ω)+‖v-u‖LΦ(Ω)<‖vB-v‖LΦ(Ω)+C(Cd)ɛ.
Therefore, it suffices to show that ∥vℬk-v∥LΦ(Ω)→0 as ɛk→0. Now, |vℬk-v|≤2sup|v|, and for all x, we have that
|vBk(x)-v(x)|≤∑2Bi∋x⨍Bi|v(y)-v(x)|dμ(y)≤C(Cd)⨍B(x,5ɛk)|v(y)-v(x)|dμ(y),
which converges to 0 as ɛk→0 by the continuity of v. Thus, by the dominated convergence theorem,
∫ΩΦ(|vBk-v|)dμ⟶0,
and so, by Lemma 2.1(2), ∥vℬk-v∥LΦ(Ω)→0.

Proof of Theorem <xref ref-type="statement" rid="thm1.3">1.3</xref>.

Let u∈Aτ1,Φ(Ω)∩LΦ(Ω). For j∈ℕ, let ℬj be a (j-1,5τ)-cover (and hence also a (j-1,2)-cover) of Ω, then, by Lemma 3.3(2), uj:=uℬj→u in LΦ(Ω). Let us show that
‖Lipuj‖LΦ(Ω)≤C(Cd,τ)‖u‖Aτ1,Φ(Ω).
By Lemma 3.3(1),
Lipuj≤C(Cd)∑B∈BjrB-1⨍5B|u-u5B|dμχB.
It follows from Lemma 3.1(4) that ℬj can be divided into k=C(Cd,τ) subfamilies ℬj,1,…,ℬj,k so that each of the families 5τℬj,l consists of disjoint balls. Since the families 5ℬj,1,…,5ℬj,k belong to ℬτ(Ω), we have that
‖Lipuj‖LΦ(Ω)≤C(Cd)∑l=1k‖∑B∈Bj,lrB-1⨍5B|u-u5B|dμχB‖LΦ(Ω)≤C(Cd)∑l=1k‖∑B∈5Bj,lrB-1⨍B|u-uB|dμχB‖LΦ(Ω)≤C(Cd,τ)‖u‖Aτ1,Φ(Ω).

Since Φ and Φ̂ are doubling, LΦ(Ω) is reflexive. Thus, the bounded sequence (Lipuj) has a subsequence that converges weakly to some g∈LΦ(Ω). By Lemma 2.3, g is a Φ-weak upper gradient of a representative of u. As a weak limit, g satisfies‖g‖LΦ(Ω)≤limsupj→∞‖Lipuj‖LΦ(Ω)≤C(Cd,τ)‖u‖Aτ1,Φ(Ω).
Let ν be a functional satisfying (1.5) and (1.6). Using Lemma 3.3(1), we obtain
Lipuj(x)≤C(Cd)(10j-1)-1⨍B(x,10j-1)|u-uB(x,10j-1)|dμ≤C(Cd)Φ-1(ν(B(x,10τj-1))μ(B(x,10j-1)))≤C(Cd,τ)Φ-1(ν(B(x,10τj-1))μ(B(x,10τj-1))).
Since, by Lemma 2.3, g(x)≤limsupj→∞Lipuj(x) for a.e. x∈Ω, the pointwise inequality (1.11) follows.

Proof of Theorem <xref ref-type="statement" rid="thm1.2">1.2</xref>.

We may assume that ∥g∥LΦ(Ω)=1. Define the functions uj as in the proof of Theorem 1.3. By (3.22) and (1.7), we have that
‖Lipuj‖LΦ(Ω)≤C(Cd,τ).
Let us show that
limμ(E)→0supj∫EΦ(Lipuj)dμ=0.
By Lemma 3.3(1) and by the Φ-Poincaré inequality,
Lipuj≤C(Cd)∑B∈BjrB-1⨍5B|u-u5B|dμχB≤C(Cd)∑B∈BjΦ-1(⨍5τBΦ(g)dμ)χB.
Thus,
∫EΦ(Lipuj)dμ≤C(Cd,CΦ)∑B∈Bjμ(E∩B)μ(5τB)∫5τBΦ(g)dμ.
Since Bj can be divided into k=C(Cd,τ) subfamilies ℬj,1,…,ℬj,k so that each of the families 5τℬj,l consists of disjoint balls, it suffices to show that, for 1≤l≤k,
limμ(E)→0∑B∈Bj,lμ(E∩B)μ(5τB)∫5τBΦ(g)dμ=0.
Fix ɛ>0. Then, there exists δ>0 such that ∫AΦ(g)<ɛ whenever μ(A)<δ. Denote by ℬ the family of those balls B in ℬj,l for which
μ(E∩B)μ(5τB)<ɛ.
Also, let ℬ′=ℬj,l∖ℬ. Now, if μ(E)<ɛδ, we have that μ(⋃B∈ℬ′5τB)≤ɛ-1μ(E)<δ. Thus,
∑B∈Bj,lμ(E∩B)μ(5τB)∫5τBΦ(g)dμ=∑B∈Bμ(E∩B)μ(5τB)∫5τBΦ(g)dμ+∑B∈B′μ(E∩B)μ(5τB)∫5τBΦ(g)dμ≤ɛ∫ΩΦ(g)dμ+∫⋃B∈B′5τBΦ(g)dμ≤2ɛ.
This completes the proof of (3.28).

By Lemma 2.2, a subsequence of (Lipuj) converges weakly to some gu∈LΦ(Ω), which, by Lemma 2.3, is a Φ-weak upper gradient of a representative of u. Moreover, gu(x)≤limsupj→∞Lipuj(x) for a.e. x∈Ω. It follows from Lemma 3.3(1) and from the Φ-Poincaré inequality thatLipuj(x)≤C(Cd)Φ-1(⨍B(x,10τj-1)Φ(g)dμ).
Thus, gu(x)≤C(Cd)g(x) for a.e. x∈Ω.

Acknowledgments

An earlier version of this paper was a part of the authors Ph.D. thesis written under the supervision of Professor Pekka Koskela. The research was supported by Vilho, Yrjö and Kalle Väisälä Foundation.

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