# Characterizations of Orlicz-Sobolev Spaces by Means of Generalized Orlicz-Poincaré Inequalities

Authors: Toni Heikkinen
Journal: Journal of Function Spaces and Applications
 (2012)

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## Abstract

Let Φ be anN-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.


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## Body

## 1. Introduction

LetX=(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(1.1)⨍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 thatu∈Wloc1,1(Ω) satisfies the 1-Poincaré inequality(1.2)⨍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(1.3)‖u‖Aτ1,Φ(Ω)=sup⁡B∈Bτ(Ω)‖∑B∈B(rB-1⨍B|u-uB|dμ)χB‖LΦ(Ω),
where(1.4)Bτ(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 (1.5)∑iν(Bi)≤1,
whenever the balls Bi are disjoint, and that the generalized Φ-Poincaré inequality(1.6)⨍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(1.7)‖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 thatΦ is an N-function, Ω⊂ℝn is open, and u∈LΦ(Ω), then the following conditions are equivalent:(a)
u∈W1,Φ(Ω),(b)
there existsg∈LΦ(Ω) such that the pair (u/∥g∥LΦ(Ω),g/∥g∥LΦ(Ω)) satisfies the Φ-Poincaré inequality in Ω,(c)
u∈Aτ1,Φ(Ω) for some τ≥1. Moreover, if a functional ν satisfies (1.5) and (1.6), one has that
(1.8)|∇u(x)|≤C(Cd,τ)limsup⁡r→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 thatΦ(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 Theorem1.1 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 that(1.9)μ(2B)≤Cdμ(B),
whenever B=B(x,r) is a ball and 2B=B(x,2r).Our substitute for the usual Sobolev classW1,Φ is based on upper gradients. We call a Borel function g:X→[0,∞] an upper gradient of a function u:X→ℝ¯ if(1.10)|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 thatΦ 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 caseΦ(t)=tp, p≥1, the result was essentially proved in [8], see [11].If bothΦ and its conjugate are doubling, then a generalization of the proof of [6, Theorem 1.1 (2)] yields the following.Theorem 1.3.
LetΩ⊂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
(1.11)g(x)≤C(Cd,τ)limsup⁡r→0Φ-1(ν(B(x,r))μ(B(x,r))),
for a.e. x∈Ω.We say that a spaceX supports the Φ-Poincaré inequality if there exist constants CP and τ such that(1.12)⨍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 thatΦ 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.(a)
u∈N1,Φ(Ω).(b)
There existsg∈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(c)
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 functionΦ:[0,∞)→[0,∞] is called a Young function if it has the form(2.1)Φ(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, satisfies(2.2)Φ(ɛt)≤ɛΦ(t),
for 0<ɛ≤1 and 0≤t<∞.IfΦ is a real-valued Young function and μ(X)<∞, then Jensen’s inequality(2.3)Φ(⨍Xudμ)≤⨍XΦ(u)dμ
holds for 0≤u∈L1(X).The right-continuous generalized inverse of a Young functionΦ is(2.4)Φ-1(t)=inf⁡{s:Φ(s)>t}.
We have that(2.5)Φ(Φ-1(t))≤t≤Φ-1(Φ(t)),
for t≥0.The conjugate of a Young functionΦ is the Young function defined by(2.6)Φ̂(t)=sup⁡{ts-Φ(s):s>0},
for t≥0.The conjugate of anN-function is an N-function.LetΦ 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(2.7)∫XΦ(|u(x)|λ)dμ(x)<∞.
The Luxemburg norm of u∈LΦ(X) is(2.8)‖u‖LΦ(X)=inf⁡{λ>0:∫XΦ(|u(x)|λ)dμ(x)≤1}.
If ∥u∥LΦ(X)≠0, we have that(2.9)∫XΦ(|u(x)|‖u‖LΦ(X))dμ(x)≤1.
The following generalized Hölder inequality holds for Luxemburg norms:(2.10)∫Xu(x)v(x)dμ(x)≤2‖u‖LΦ(X)‖v‖LΦ̂(X).A Young functionΦ is doubling if there exists a constant CΦ≥1 such that(2.11)Φ(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.
LetΦ be a doubling Young function.(1)
The spaceC0(X) of bounded, boundedly supported continuous functions is dense in LΦ(X).(2)
The modular convergence and the norm convergence are equivalent, that is,(2.12)‖fj-f‖LΦ(X)⟶0
if and only if
(2.13)∫XΦ(|fj-f|)dμ⟶0.IfΦ 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].IfΦ 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 thatΦ is a doubling Young function and that {gi}⊂LΦ(X) satisfies
(2.14)sup⁡i‖gi‖LΦ(X)<∞,lim⁡μ(A)→0sup⁡i∫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.
SinceΦ 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),
(2.15)sup⁡i|∫Xgihdμ|<∞,lim⁡μ(A)→0sup⁡i∫A|gih|dμ=0.
By the Hölder inequality and Lemma 2.1(2), these follow from (2.14).


### 2.2. Sobolev Spaces

TheΦ-modulus of a curve family Γ is(2.16)ModΦ(Γ)=inf⁡⁡‖g‖LΦ(X),
where the infimum is taken over all Borel functions g:X→[0,∞] satisfying(2.17)∫γgds≥1,
for all locally rectifiable curves γ∈Γ.The Sobolev spaceN1,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(2.18)‖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 thatui→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
(2.19)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].IfΦ 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(2.20)∫u∂φ∂xi=-∫viφ,
for all φ∈C0∞(Ω).Lemma 2.4.
LetΦ be an N-function. Suppose that the functional ∂u/∂xi:C0∞(Ω)→ℝ,
(2.21)∂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 byEΦ̂(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
(2.22)∂u∂xi[φ]=∫φvi,
for φ∈EΦ̂(Ω). Moreover, ∥vi∥LΦ(Ω)≤∥∂u/∂xi∥. The claim follows.


### 2.3. Lipschitz Functions

A functionu: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 is(2.23)Lipu(x)=limsup⁡r→0r-1sup⁡d(x,y)≤r|u(x)-u(y)|.
It is well known that Lipu is an upper gradient of u; see, for example, [18].


## 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 functionΦ:[0,∞)→[0,∞] is called a Young function if it has the form(2.1)Φ(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, satisfies(2.2)Φ(ɛt)≤ɛΦ(t),
for 0<ɛ≤1 and 0≤t<∞.IfΦ is a real-valued Young function and μ(X)<∞, then Jensen’s inequality(2.3)Φ(⨍Xudμ)≤⨍XΦ(u)dμ
holds for 0≤u∈L1(X).The right-continuous generalized inverse of a Young functionΦ is(2.4)Φ-1(t)=inf⁡{s:Φ(s)>t}.
We have that(2.5)Φ(Φ-1(t))≤t≤Φ-1(Φ(t)),
for t≥0.The conjugate of a Young functionΦ is the Young function defined by(2.6)Φ̂(t)=sup⁡{ts-Φ(s):s>0},
for t≥0.The conjugate of anN-function is an N-function.LetΦ 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(2.7)∫XΦ(|u(x)|λ)dμ(x)<∞.
The Luxemburg norm of u∈LΦ(X) is(2.8)‖u‖LΦ(X)=inf⁡{λ>0:∫XΦ(|u(x)|λ)dμ(x)≤1}.
If ∥u∥LΦ(X)≠0, we have that(2.9)∫XΦ(|u(x)|‖u‖LΦ(X))dμ(x)≤1.
The following generalized Hölder inequality holds for Luxemburg norms:(2.10)∫Xu(x)v(x)dμ(x)≤2‖u‖LΦ(X)‖v‖LΦ̂(X).A Young functionΦ is doubling if there exists a constant CΦ≥1 such that(2.11)Φ(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.
LetΦ be a doubling Young function.(1)
The spaceC0(X) of bounded, boundedly supported continuous functions is dense in LΦ(X).(2)
The modular convergence and the norm convergence are equivalent, that is,(2.12)‖fj-f‖LΦ(X)⟶0
if and only if
(2.13)∫XΦ(|fj-f|)dμ⟶0.IfΦ 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].IfΦ 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 thatΦ is a doubling Young function and that {gi}⊂LΦ(X) satisfies
(2.14)sup⁡i‖gi‖LΦ(X)<∞,lim⁡μ(A)→0sup⁡i∫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.
SinceΦ 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),
(2.15)sup⁡i|∫Xgihdμ|<∞,lim⁡μ(A)→0sup⁡i∫A|gih|dμ=0.
By the Hölder inequality and Lemma 2.1(2), these follow from (2.14).


## 2.2. Sobolev Spaces

TheΦ-modulus of a curve family Γ is(2.16)ModΦ(Γ)=inf⁡⁡‖g‖LΦ(X),
where the infimum is taken over all Borel functions g:X→[0,∞] satisfying(2.17)∫γgds≥1,
for all locally rectifiable curves γ∈Γ.The Sobolev spaceN1,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(2.18)‖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 thatui→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
(2.19)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].IfΦ 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(2.20)∫u∂φ∂xi=-∫viφ,
for all φ∈C0∞(Ω).Lemma 2.4.
LetΦ be an N-function. Suppose that the functional ∂u/∂xi:C0∞(Ω)→ℝ,
(2.21)∂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 byEΦ̂(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
(2.22)∂u∂xi[φ]=∫φvi,
for φ∈EΦ̂(Ω). Moreover, ∥vi∥LΦ(Ω)≤∥∂u/∂xi∥. The claim follows.


## 2.3. Lipschitz Functions

A functionu: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 is(2.23)Lipu(x)=limsup⁡r→0r-1sup⁡d(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 Theorem1.1 is a modification of the proof of case p>1 of [9, Lemma 6].Proof of Theorem1.1.
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,
(3.1)∂u∂xi[φ]:=-∫u∂φ∂xi,
satisfies
(3.2)|∂u∂xi[φ]|≤C(n,τ)‖u‖Aτ1,Φ(Ω)‖φ‖LΦ̂(supp⁡φ).
Choose0≤ψ∈C0∞(B(0,1)) such that ∫ψ=1, and let ψɛ(x)=ɛ-nψ(x/ɛ) for ɛ>0, Then(3.3)∂u∂xi[φ]=-lim⁡ɛ→0∫(u*ψɛ)∂φ∂xi=lim⁡ɛ→0∫(u*∂ψɛ∂xi)φ.
By the Hölder inequality,
(3.4)|∂u∂xi[φ]|≤2liminf⁡ɛ→0‖u*∂ψɛ∂xi‖LΦ(supp⁡φ)‖φ‖LΦ̂(supp⁡φ).
Since ∫∂ψɛ/∂xi=0, we have that
(3.5)(u*∂ψɛ∂xi)(x)=((u-uB(x,ɛ))*∂ψɛ∂xi)(x).
Thus,
(3.6)|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
(3.7)|u*∂ψɛ∂xi|(x)≤C(n)(3ɛ)-1⨍B(xj,3ɛ)|u(y)-uB(xj,3ɛ)|dy.
Thus,
(3.8)|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,
(3.9)‖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(3.10)|vi(x)|≤limsup⁡ɛ→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),
(3.11)|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
(3.12)|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 Theorems1.2 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(3.13)‖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 Theorem1.2, 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.
LetΩ⊂X be open. Given ɛ>0, λ≥1, there is a cover {Bi=B(xi,ri)} of Ω with the following properties:(1)
ri≤ɛ for all i,(2)
λBi⊂Ω for all i,(3)
ifλBi meets λBj, then ri≤2rj,(4)
each ballλBi 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.
LetΩ⊂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(1)
eachφi is C(Cd)ri-1-Lipschitz,(2)
0≤φi≤1 for all i,(3)
φi(x)=0 for x∈X∖2Bi for all i,(4)
∑iφi(x)=1 for all x∈Ω.A collection {φi} as above is called a partition of unity with respect to ℬ.Letℬ={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, define(3.14)uB(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,
(3.15)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),(3.16)‖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,
(3.17)∫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).
Letu∈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(3.18)‖uB-vB‖LΦ(Ω)=‖(u-v)B‖LΦ(Ω)≤C(Cd)‖u-v‖LΦ(Ω)<C(Cd)ɛ,
and so
(3.19)‖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
(3.20)|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,
(3.21)∫ΩΦ(|vBk-v|)dμ⟶0,
and so, by Lemma 2.1(2), ∥vℬk-v∥LΦ(Ω)→0.Proof of Theorem1.3.
Letu∈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
(3.22)‖Lipuj‖LΦ(Ω)≤C(Cd,τ)‖u‖Aτ1,Φ(Ω).
By Lemma 3.3(1),
(3.23)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
(3.24)‖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).
SinceΦ 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(3.25)‖g‖LΦ(Ω)≤limsup⁡j→∞‖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
(3.26)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 Theorem1.2.
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
(3.27)‖Lipuj‖LΦ(Ω)≤C(Cd,τ).
Let us show that
(3.28)lim⁡μ(E)→0sup⁡j∫EΦ(Lipuj)dμ=0.
By Lemma 3.3(1) and by the Φ-Poincaré inequality,
(3.29)Lipuj≤C(Cd)∑B∈BjrB-1⨍5B|u-u5B|dμχB≤C(Cd)∑B∈BjΦ-1(⨍5τBΦ(g)dμ)χB.
Thus,
(3.30)∫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,
(3.31)lim⁡μ(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
(3.32)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,
(3.33)∑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 Lemma2.2, 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 that(3.34)Lipuj(x)≤C(Cd)Φ-1(⨍B(x,10τj-1)Φ(g)dμ).
Thus, gu(x)≤C(Cd)g(x) for a.e. x∈Ω.



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*Source: 426067-2012-02-13.xml*