Journal of Function Spaces

Journal of Function Spaces / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 426067 | 15 pages | https://doi.org/10.1155/2012/426067

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

Academic Editor: Olli Martio
Received04 Nov 2009
Accepted27 Feb 2010
Published13 Feb 2012

Abstract

Let Φ be an N-function. We show that a function 𝑢𝐿Φ(𝑛) belongs to the Orlicz-Sobolev space 𝑊1,Φ(𝑛) 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, Ω𝑋 open, and Φ a Young function. A pair (𝑢,𝑔) of measurable functions, 𝑢𝐿1loc(Ω) and 𝑔0, satisfy the Φ-Poincaré inequality in Ω, if there is a constant 𝜏1 such that𝐵||𝑢𝑢𝐵||𝑑𝜇𝑟𝐵Φ1𝜏𝐵Φ(𝑔)𝑑𝜇,(1.1) for every ball 𝐵=𝐵(𝑥,𝑟𝐵) such that 𝜏𝐵Ω. Here, 𝑢𝐵=𝐵𝑢𝑑𝜇=𝜇(𝐵)1𝐵𝑢𝑑𝜇 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 [25].

In the euclidean setting, it is well known that 𝑢𝑊1,1loc(Ω) satisfies the 1-Poincaré inequality𝐵||𝑢𝑢𝐵||𝑑𝜇𝐶𝑛𝑟𝐵𝐵||||𝑢𝑑𝜇,(1.2) for every ball 𝐵Ω. 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 𝑢𝐿Φ(Ω) and there exists 𝑔𝐿Φ(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 𝑢𝑊1,Φ(Ω) if and only if the number𝑢𝐴𝜏1,Φ(Ω)=sup𝜏(Ω)𝐵𝑟1𝐵||𝑢𝑢𝐵||𝜒𝑑𝜇𝐵𝐿Φ(Ω),(1.3) where𝜏𝐵(Ω)=𝑖balls𝜏𝐵𝑖aredisjointandcontainedinΩ,(1.4) is finite. Note that 𝑢𝐴𝜏1,Φ(Ω)1 if and only if there is a functional 𝜈{𝐵Ω𝐵isaball}[0,) such that 𝑖𝜈𝐵𝑖1,(1.5) whenever the balls 𝐵𝑖 are disjoint, and that the generalized Φ-Poincaré inequality𝐵||𝑢𝑢𝐵||𝑑𝜇𝑟𝐵Φ1𝜈(𝜏𝐵)𝜇(𝐵)(1.6) holds whenever 𝜏𝐵Ω. In particular, if a pair (𝑢/𝑔𝐿Φ(Ω),𝑔/𝑔𝐿Φ(Ω)) satisfies the Φ-Poincaré inequality in Ω, then𝑢𝐴𝜏1,Φ(Ω)𝑔𝐿Φ(Ω).(1.7) The spaces 𝐴𝜏1,Φ(Ω)={𝑢𝐿1loc(Ω)𝑢𝐴𝜏1,Φ(Ω)<}, for Φ(𝑡)=𝑡𝑝, were studied in [6].

Theorem 1.1. Suppose that Φ is an 𝑁-function, Ω𝑛 is open, and 𝑢𝐿Φ(Ω), then the following conditions are equivalent:(a)𝑢𝑊1,Φ(Ω),(b)there exists 𝑔𝐿Φ(Ω) such that the pair (𝑢/𝑔𝐿Φ(Ω),𝑔/𝑔𝐿Φ(Ω)) satisfies the Φ-Poincaré inequality in Ω,(c)𝑢𝐴𝜏1,Φ(Ω) for some 𝜏1. Moreover, if a functional 𝜈 satisfies (1.5) and (1.6), one has that ||||𝐶𝑢(𝑥)𝐶𝑑,𝜏limsup𝑟0Φ1𝜈(𝐵(𝑥,𝑟)),𝜇(𝐵(𝑥,𝑟))(1.8) for a.e. 𝑥Ω. In particular, (b) implies that |𝑢(𝑥)|𝐶(𝑛)𝑔(𝑥) for a.e. 𝑥Ω.

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 𝑝1 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),(1.9) 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 𝑔𝑋[0,] an upper gradient of a function 𝑢𝑋 if||||𝑢(𝛾(0))𝑢(𝛾(𝑙))𝛾𝑔𝑑𝑠,(1.10) 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, Ω𝑋 is open, 𝑢,𝑔𝐿Φ(Ω), and that the pair (𝑢/𝑔𝐿Φ(Ω),𝑔/𝑔𝐿Φ(Ω)) satisfies the Φ-Poincaré inequality in Ω, then a representative of 𝑢 has a Φ-weak upper gradient 𝑔𝑢𝐿Φ(Ω) such that 𝑔𝑢(𝑥)𝐶(𝐶𝑑)𝑔(𝑥) for a.e. 𝑥Ω.

In the case Φ(𝑡)=𝑡𝑝, 𝑝1, 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 Ω𝑋 be an open set, and let Φ be a doubling Young function whose conjugate is doubling, then a representative of 𝑢𝐴𝜏1,Φ(Ω)𝐿Φ(Ω) has a Φ-weak upper gradient 𝑔 with 𝑔𝐿Φ(Ω)𝐶(𝐶𝑑,𝜏)𝑢𝐴𝜏1,Φ(Ω). Moreover, for a functional 𝜈 satisfying (1.5) and (1.6), one has that 𝐶𝑔(𝑥)𝐶𝑑,𝜏limsup𝑟0Φ1𝜈(𝐵(𝑥,𝑟)),𝜇(𝐵(𝑥,𝑟))(1.11) for a.e. 𝑥Ω.

We say that a space 𝑋 supports the Φ-Poincaré inequality if there exist constants 𝐶𝑃 and 𝜏 such that𝐵||𝑢𝑢𝐵||𝑑𝜇𝐶𝑃𝑟𝐵Φ1𝜏𝐵Φ(𝑔)𝑑𝜇,(1.12) whenever 𝐵𝑋 is a ball, 𝑢𝐿1loc(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, Ω𝑋 is open, and 𝑢𝐿Φ(Ω), then the following conditions are equivalent.(a)𝑢𝑁1,Φ(Ω).(b)There exists 𝑔𝐿Φ(Ω) such that the pair (𝑢/𝑔𝐿Φ(Ω),𝑔/𝑔𝐿Φ(Ω)) satisfies the Φ-Poincaré inequality in Ω.If also the conjugate of Φ is doubling, then (a) and (b) are equivalent to(c)𝑢𝐴𝜏1,Φ(Ω) for some 𝜏1.

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)=𝑡0𝜙(𝑠)𝑑𝑠,(2.1) 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, lim𝑡0Φ(𝑡)/𝑡=0, and lim𝑡Φ(𝑡)/𝑡=, then Φ is called an 𝑁-function.

A Young function is convex and, in particular, satisfiesΦ(𝜀𝑡)𝜀Φ(𝑡),(2.2) for 0<𝜀1 and 0𝑡<.

If Φ is a real-valued Young function and 𝜇(𝑋)<, then Jensen’s inequalityΦ𝑋𝑢𝑑𝜇𝑋Φ(𝑢)𝑑𝜇(2.3) holds for 0𝑢𝐿1(𝑋).

The right-continuous generalized inverse of a Young function Φ isΦ1(𝑡)=inf{𝑠Φ(𝑠)>𝑡}.(2.4) We have thatΦΦ1(𝑡)𝑡Φ1(Φ(𝑡)),(2.5) for 𝑡0.

The conjugate of a Young function Φ is the Young function defined byΦ(𝑡)=sup{𝑡𝑠Φ(𝑠)𝑠>0},(2.6) for 𝑡0.

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 that𝑋Φ||||𝑢(𝑥)𝜆𝑑𝜇(𝑥)<.(2.7) The Luxemburg norm of 𝑢𝐿Φ(𝑋) is𝑢𝐿Φ(𝑋)=inf𝜆>0𝑋Φ||||𝑢(𝑥)𝜆.𝑑𝜇(𝑥)1(2.8) If 𝑢𝐿Φ(𝑋)0, we have that𝑋Φ||||𝑢(𝑥)𝑢𝐿Φ(𝑋)𝑑𝜇(𝑥)1.(2.9) The following generalized Hölder inequality holds for Luxemburg norms:𝑋𝑢(𝑥)𝑣(𝑥)𝑑𝜇(𝑥)2𝑢𝐿Φ(𝑋)𝑣𝐿Φ(𝑋).(2.10)

A Young function Φ is doubling if there exists a constant 𝐶Φ1 such thatΦ(2𝑡)𝐶ΦΦ(𝑡),(2.11) for 𝑡0. 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.(1)The space 𝐶0(𝑋) of bounded, boundedly supported continuous functions is dense in 𝐿Φ(𝑋).(2)The modular convergence and the norm convergence are equivalent, that is, 𝑓𝑗𝑓𝐿Φ(𝑋)0(2.12) if and only if 𝑋Φ||𝑓𝑗||𝑓𝑑𝜇0.(2.13)

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))Φ=𝐿(𝑋), see [13, Chapter IV, Corollary 9]. So, if both Φ and Φ 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 Φ 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 sup𝑖𝑔𝑖𝐿Φ(𝑋)<,lim𝜇(𝐴)0sup𝑖𝐴Φ||𝑔𝑖||𝑑𝜇=0,(2.14) then there exists a subsequence (𝑔𝑖𝑗) of (𝑔𝑖) and 𝑔𝐿Φ(𝑋) such that 𝑔𝑖𝑗𝑔 weakly in 𝐿Φ(𝑋).

Proof. Since Φ is doubling, the dual of 𝐿Φ(𝑋) is 𝐿Φ(𝑋). By [13, page 144, Corollary 2], a sequence {𝑔𝑖} has a weakly converging subsequence if for each Φ𝐿(𝑋), sup𝑖||||𝑋𝑔𝑖||||𝑑𝜇<,lim𝜇(𝐴)0sup𝑖𝐴||𝑔𝑖||𝑑𝜇=0.(2.15) 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)=inf𝑔𝐿Φ(𝑋),(2.16) where the infimum is taken over all Borel functions 𝑔𝑋[0,] satisfying𝛾𝑔𝑑𝑠1,(2.17) for all locally rectifiable curves 𝛾Γ.

The Sobolev space 𝑁1,Φ(𝑋), consisting of the functions 𝑢𝐿Φ(𝑋) having a Φ-weak upper gradient 𝑔𝐿Φ(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 norm𝑢𝑁1,Φ(𝑋)=𝑢𝐿Φ(𝑋)+inf𝑔𝐿Φ(𝑋),(2.18) where the infimum is taken over Φ-weak upper gradients 𝑔𝐿Φ(𝑋) of 𝑢.

Lemma 2.3. Suppose that 𝑢𝑖𝑢𝐿Φ(𝑋) and 𝑔𝑖𝑔𝐿Φ(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)limsup𝑖𝑔𝑖(𝑥) for a.e. 𝑥𝑋.

Proof. By Mazur’s lemma ([17, Page 120, Theorem 2]), there is a sequence (̃𝑔𝑖) of convex combinations ̃𝑔𝑖=𝑛𝑖𝑗=𝑖𝜆𝑖,𝑗𝑔𝑗,(2.19) where 𝜆𝑖,𝑗0 and 𝑛𝑖𝑗=1𝜆𝑖,𝑗=1, such that ̃𝑔𝑖𝑔 in 𝐿Φ(𝑋). Hence, a subsequence of (̃𝑔𝑖) converges pointwise a.e., which implies that 𝑔(𝑥)limsup𝑖̃𝑔𝑖(𝑥)limsup𝑖𝑔𝑖(𝑥) for a.e. 𝑥𝑋. 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 Ω𝑛 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 𝑢𝐿Φ(Ω) having weak partial derivatives in 𝐿Φ(Ω). A function 𝑣𝑖𝐿1loc(Ω) is a weak partial derivative of 𝑢 (with respect to 𝑥𝑖) if𝑢𝜕𝜑𝜕𝑥𝑖𝑣=𝑖𝜑,(2.20) for all 𝜑𝐶0(Ω).

Lemma 2.4. Let Φ be an 𝑁-function. Suppose that the functional 𝜕𝑢/𝜕𝑥𝑖𝐶0(Ω), 𝜕𝑢𝜕𝑥𝑖[𝜑]𝑢=𝜕𝜑𝜕𝑥𝑖,(2.21) is bounded with respect to the norm 𝐿Φ(Ω), then 𝑢 has a weak partial derivative 𝑣𝑖 such that 𝑣𝑖𝐿Φ(Ω)𝜕𝑢/𝜕𝑥𝑖.

Proof. Denote by 𝐸Φ(Ω) the closure of the space of bounded, boundedly supported functions in 𝐿Φ(Ω). By [15, Theorem 8.21(d)], 𝐶0(Ω) is dense in 𝐸Φ(Ω). Thus, 𝜕𝑢/𝜕𝑥𝑖 extends to a continuous linear functional on 𝐸Φ(Ω). By [15, Theorem 8.19], the dual of 𝐸Φ(Ω) is isomorphic to 𝐿Φ(Ω); there exists 𝑣𝑖𝐿Φ(Ω) such that 𝜕𝑢𝜕𝑥𝑖[𝜑]=𝜑𝑣𝑖,(2.22) for Φ𝜑𝐸(Ω). Moreover, 𝑣𝑖𝐿Φ(Ω)𝜕𝑢/𝜕𝑥𝑖. The claim follows.

2.3. Lipschitz Functions

A function 𝑢𝑋 is 𝐿-Lipschitz if |𝑢(𝑥)𝑢(𝑦)|𝐿𝑑(𝑥,𝑦) for all 𝑥,𝑦𝑋. The (upper) pointwise Lipschitz constant of a locally Lipschitz function 𝑢 isLip𝑢(𝑥)=limsup𝑟0𝑟1sup𝑑(𝑥,𝑦)𝑟||||.𝑢(𝑥)𝑢(𝑦)(2.23) 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 1.1. As noted in the introduction, (a) (b) (c). Let us show that (c) (a). Fix 𝑢𝐴𝜏1,Φ(Ω). We will show that the functional 𝜕𝑢/𝜕𝑥𝑖𝐶0(Ω), 𝜕𝑢𝜕𝑥𝑖[𝜑]𝑢=𝜕𝜑𝜕𝑥𝑖,(3.1) satisfies ||||𝜕𝑢𝜕𝑥𝑖[𝜑]||||𝐶(𝑛,𝜏)𝑢𝐴𝜏1,Φ(Ω)𝜑𝐿Φ(supp𝜑).(3.2)
Choose 0𝜓𝐶0(𝐵(0,1)) such that 𝜓=1, and let 𝜓𝜀(𝑥)=𝜀𝑛𝜓(𝑥/𝜀) for 𝜀>0, Then𝜕𝑢𝜕𝑥𝑖[𝜑]=lim𝜀0𝑢𝜓𝜀𝜕𝜑𝜕𝑥𝑖=lim𝜀0𝑢𝜕𝜓𝜀𝜕𝑥𝑖𝜑.(3.3) By the Hölder inequality, ||||𝜕𝑢𝜕𝑥𝑖[𝜑]||||2liminf𝜀0𝑢𝜕𝜓𝜀𝜕𝑥𝑖𝐿Φ(supp𝜑)𝜑𝐿Φ(supp𝜑).(3.4) Since 𝜕𝜓𝜀/𝜕𝑥𝑖=0, we have that 𝑢𝜕𝜓𝜀𝜕𝑥𝑖(𝑥)=𝑢𝑢𝐵(𝑥,𝜀)𝜕𝜓𝜀𝜕𝑥𝑖(𝑥).(3.5) Thus, ||||𝑢𝜕𝜓𝜀𝜕𝑥𝑖||||(𝑥)𝐶(𝑛)𝜀1𝐵(𝑥,𝜀)||𝑢(𝑦)𝑢𝐵(𝑥,𝜀)||𝑑𝑦.(3.6) Let 𝐾=supp𝜑 and let 0<𝜀<𝑑(𝐾,Ω𝑐)/3𝜏. Cover 𝐾 with balls 𝐵(𝑥𝑗,2𝜀), 𝑥𝑗𝐾, such that the balls 𝐵(𝑥𝑗,𝜀) are disjoint. If 𝑥𝐵(𝑥𝑗,2𝜀), then 𝐵(𝑥,𝜀)𝐵(𝑥𝑗,3𝜀) and (3.6) implies that ||||𝑢𝜕𝜓𝜀𝜕𝑥𝑖||||(𝑥)𝐶(𝑛)(3𝜀)1𝐵(𝑥𝑗,3𝜀)|||𝑢(𝑦)𝑢𝐵(𝑥𝑗,3𝜀)|||𝑑𝑦.(3.7) Thus, ||||𝑢𝜕𝜓𝜀𝜕𝑥𝑖||||𝐶(𝑛)𝑗(3𝜀)1𝐵(𝑥𝑗,3𝜀)|||𝑢(𝑦)𝑢𝐵(𝑥𝑗,3𝜀)|||𝑑𝑦𝜒𝐵(𝑥𝑗,3𝜀).(3.8) Since the balls 𝐵(𝑥𝑗,𝜀) are disjoint, it follows that the family ={𝐵(𝑥𝑗,3𝜀)} can be divided into 𝑘=𝐶(𝑛,𝜏) subfamilies 1,𝑘 such that each of the families 𝜏𝑗={𝜏𝐵𝐵𝑗} consists of disjoint balls. Hence, 𝑢𝜕𝜓𝜀𝜕𝑥𝑖𝐿Φ(Ω)𝐶(𝑛)𝑘𝑗=1𝐵𝑗𝑟𝐵1𝐵||𝑢𝑢𝐵||𝑑𝜇𝜒𝐵𝐿Φ(Ω)𝐶(𝑛,𝜏)𝑢𝐴𝜏1,Φ(Ω).(3.9) This, combined with (3.4), yields (3.2). By Lemma 2.4, 𝑢 has a weak partial derivative 𝑣𝑖𝐿Φ(Ω).
Since |𝑣𝑖(𝑥)|limsup𝜀0|(𝜕/𝜕𝑥𝑖)(𝑢𝜓𝜀)(𝑥)| for a.e. 𝑥Ω and (𝜕/𝜕𝑥𝑖)(𝑢𝜓𝜀)(𝑥)=𝑢(𝜕𝜓𝜀/𝜕𝑥𝑖)(𝑥) for small 𝜀, it follows that||𝑣𝑖||(𝑥)limsup𝜀0||||𝑢𝜕𝜓𝜀𝜕𝑥𝑖||||(𝑥),(3.10) for a.e. 𝑥Ω. Let 𝜈 be a functional satisfying (1.5) and ((1.6), then, by (3.6), ||||𝑢𝜕𝜓𝜀𝜕𝑥𝑖||||(𝑥)𝐶(𝑛)Φ1𝜈(𝐵(𝑥,𝜏𝜀))𝜇(𝐵(𝑥,𝜀))𝐶(𝑛,𝜏)Φ1𝜈(𝐵(𝑥,𝜏𝜀)).𝜇(𝐵(𝑥,𝜏𝜀))(3.11) Thus, (1.8) holds for a.e. 𝑥Ω. If condition (b) is satisfied, we have ||||𝑢𝜕𝜓𝜀𝜕𝑥𝑖||||(𝑥)𝐶(𝑛)𝑔𝐿Φ(Ω)Φ1𝐵(𝑥,𝜏𝜀)Φ𝑔(𝑦)𝑔𝐿Φ(Ω),𝑑𝑦(3.12) which implies that |𝑣𝑖(𝑥)|𝐶(𝑛)𝑔(𝑥) for a.e. 𝑥Ω. 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 [68]. 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 𝑢𝑗𝑢 in 𝐿Φ(Ω) and thatLip𝑢𝑗𝐿Φ(Ω)𝐶𝑢𝐴𝜏1,Φ(Ω).(3.13) 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 Ω𝑋 be open. Given 𝜀>0, 𝜆1, there is a cover {𝐵𝑖=𝐵(𝑥𝑖,𝑟𝑖)} of Ω with the following properties:(1)𝑟𝑖𝜀 for all 𝑖,(2)𝜆𝐵𝑖Ω for all 𝑖,(3)if 𝜆𝐵𝑖 meets 𝜆𝐵𝑗, then 𝑟𝑖2𝑟𝑗,(4)each ball 𝜆𝐵𝑖 meets at most 𝐶=𝐶(𝐶𝑑,𝜆) balls 𝜆𝐵𝑗.

A collection {𝐵𝑖} as above is called an (𝜀,𝜆)-covering of Ω. Clearly, an (𝜀,𝜆)-cover is an (𝜀,𝜆)-cover provided 𝜀𝜀 and 𝜆𝜆.

Lemma 3.2. Let Ω𝑋 be open, and let ={𝐵𝑖=𝐵(𝑥𝑖,𝑟𝑖)} be an (,2)-cover of Ω, then there is a collection {𝜑𝑖} of functions Ω such that(1)each 𝜑𝑖 is 𝐶(𝐶𝑑)𝑟𝑖1-Lipschitz,(2)0𝜑𝑖1 for all 𝑖,(3)𝜑𝑖(𝑥)=0 for 𝑥𝑋2𝐵𝑖 for all 𝑖,(4)𝑖𝜑𝑖(𝑥)=1 for all 𝑥Ω.A collection {𝜑𝑖} as above is called a partition of unity with respect to .

Let ={𝐵𝑖} be as in the lemma above, and let {𝜑𝑖} be a partition of unity with respect to . For a locally integrable function 𝑢 on Ω, define𝑢(𝑥)=𝑖𝑢𝐵𝑖𝜑𝑖(𝑥).(3.14) The following lemma describes the most important properties of 𝑢.

Lemma 3.3. (1) The function 𝑢 is locally Lipschitz. Moreover, for each 𝑥𝐵𝑖, Lip𝑢(𝐶𝑥)𝐶𝑑𝑟𝐵1𝑖5𝐵𝑖||𝑢𝑢5𝐵𝑖||𝑑𝜇.(3.15)
(2) Let Φ be a doubling Young function, and let 𝑢𝐿Φ(Ω). If 𝑘 is an (𝜀𝑘,2)-cover of Ω and 𝜀𝑘0 as 𝑘, then 𝑢𝑘𝑢 in 𝐿Φ(Ω).

Proof. (1) See the proof of [6, Lemma 5.3(1)].
(2) We begin by showing that, for every 𝑤𝐿Φ(Ω),𝑤𝐿Φ(Ω)𝐶𝐶𝑑𝑤𝐿Φ(Ω).(3.16) We may assume that 𝑤𝐿Φ(Ω)=1. By Jensen’s inequality, Φ(|𝑤|)(Φ(|𝑤|)). Hence, by the properties of the functions 𝜑𝑖, ΩΦ||𝑤||𝑑𝜇Ω(Φ(|𝑤|))𝑑𝜇𝑖Ω(Φ(|𝑤|))𝐵𝑖𝜑𝑖𝑑𝜇𝑖2𝐵𝑖Φ(|𝑤|)𝐵𝑖𝑑𝜇𝐶𝑑𝑖𝐵𝑖Φ(|𝑤|)𝑑𝜇=𝐶𝑑ΩΦ(|𝑤|)𝑖𝜒𝐵𝑖𝐶𝑑𝜇𝐶𝑑Ω𝐶Φ(|𝑤|)𝑑𝜇𝐶𝑑.(3.17) Thus, by (2.2), we obtain (3.16).
Let 𝑢𝐿Φ(Ω) and 𝜀>0. By Lemma 2.1(1), there exists 𝑣𝐶0(Ω) such that 𝑢𝑣𝐿Φ(Ω)<𝜀. Then, by (3.16), we obtain𝑢𝑣𝐿Φ(Ω)=(𝑢𝑣)𝐿Φ(Ω)𝐶𝐶𝑑𝑢𝑣𝐿Φ(Ω)𝐶<𝐶𝑑𝜀,(3.18) and so 𝑢𝑢𝐿Φ(Ω)𝑢𝑣𝐿Φ(Ω)+𝑣𝑣𝐿Φ(Ω)+𝑣𝑢𝐿Φ(Ω)<𝑣𝑣𝐿Φ(Ω)𝐶+𝐶𝑑𝜀.(3.19) Therefore, it suffices to show that 𝑣𝑘𝑣𝐿Φ(Ω)0 as 𝜀𝑘0. Now, |𝑣𝑘𝑣|2sup|𝑣|, and for all 𝑥, we have that ||𝑣𝑘(||𝑥)𝑣(𝑥)2𝐵𝑖𝑥𝐵𝑖||||𝐶𝑣(𝑦)𝑣(𝑥)𝑑𝜇(𝑦)𝐶𝑑𝐵(𝑥,5𝜀𝑘)||||𝑣(𝑦)𝑣(𝑥)𝑑𝜇(𝑦),(3.20) which converges to 0 as 𝜀𝑘0 by the continuity of 𝑣. Thus, by the dominated convergence theorem, ΩΦ||𝑣𝑘||𝑣𝑑𝜇0,(3.21) and so, by Lemma 2.1(2), 𝑣𝑘𝑣𝐿Φ(Ω)0.

Proof of Theorem 1.3. Let 𝑢𝐴𝜏1,Φ(Ω)𝐿Φ(Ω). For 𝑗, let 𝑗 be a (𝑗1,5𝜏)-cover (and hence also a (𝑗1,2)-cover) of Ω, then, by Lemma 3.3(2), 𝑢𝑗=𝑢𝑗𝑢 in 𝐿Φ(Ω). Let us show that Lip𝑢𝑗𝐿Φ(Ω)𝐶𝐶𝑑,𝜏𝑢𝐴𝜏1,Φ(Ω).(3.22) By Lemma 3.3(1), Lip𝑢𝑗𝐶𝐶𝑑𝐵𝑗𝑟𝐵15𝐵||𝑢𝑢5𝐵||𝑑𝜇𝜒𝐵.(3.23) It follows from Lemma 3.1(4) that 𝑗 can be divided into 𝑘=𝐶(𝐶𝑑,𝜏) subfamilies 𝑗,1,,𝑗,𝑘 so that each of the families 5𝜏𝑗,𝑙 consists of disjoint balls. Since the families 5𝑗,1,,5𝑗,𝑘 belong to 𝜏(Ω), we have that Lip𝑢𝑗𝐿Φ(Ω)𝐶𝐶𝑑𝑘𝑙=1𝐵𝑗,𝑙𝑟𝐵15𝐵||𝑢𝑢5𝐵||𝑑𝜇𝜒𝐵𝐿Φ(Ω)𝐶𝐶𝑑𝑘𝑙=1𝐵5𝑗,𝑙𝑟𝐵1𝐵||𝑢𝑢𝐵||𝑑𝜇𝜒𝐵𝐿Φ(Ω)𝐶𝐶𝑑,𝜏𝑢𝐴𝜏1,Φ(Ω).(3.24)
Since Φ and Φ are doubling, 𝐿Φ(Ω) is reflexive. Thus, the bounded sequence (Lip𝑢𝑗) has a subsequence that converges weakly to some 𝑔𝐿Φ(Ω). By Lemma 2.3, 𝑔 is a Φ-weak upper gradient of a representative of 𝑢. As a weak limit, 𝑔 satisfies𝑔𝐿Φ(Ω)limsup𝑗Lip𝑢𝑗𝐿Φ(Ω)𝐶𝐶𝑑,𝜏𝑢𝐴𝜏1,Φ(Ω).(3.25) Let 𝜈 be a functional satisfying (1.5) and (1.6). Using Lemma 3.3(1), we obtain Lip𝑢𝑗(𝐶𝑥)𝐶𝑑10𝑗11𝐵(𝑥,10𝑗1)||𝑢𝑢𝐵(𝑥,10𝑗1)||𝐶𝑑𝜇𝐶𝑑Φ1𝜈𝐵𝑥,10𝜏𝑗1𝜇𝐵𝑥,10𝑗1𝐶𝐶𝑑Φ,𝜏1𝜈𝐵𝑥,10𝜏𝑗1𝜇𝐵𝑥,10𝜏𝑗1.(3.26) Since, by Lemma 2.3, 𝑔(𝑥)limsup𝑗Lip𝑢𝑗(𝑥) for a.e. 𝑥Ω, the pointwise inequality (1.11) follows.

Proof of Theorem 1.2. We may assume that 𝑔𝐿Φ(Ω)=1. Define the functions 𝑢𝑗 as in the proof of Theorem 1.3. By (3.22) and (1.7), we have that Lip𝑢𝑗𝐿Φ(Ω)𝐶𝐶𝑑.,𝜏(3.27) Let us show that lim𝜇(𝐸)0sup𝑗𝐸ΦLip𝑢𝑗𝑑𝜇=0.(3.28) By Lemma 3.3(1) and by the Φ-Poincaré inequality, Lip𝑢𝑗𝐶𝐶𝑑𝐵𝑗𝑟𝐵15𝐵||𝑢𝑢5𝐵||𝑑𝜇𝜒𝐵𝐶𝐶𝑑𝐵𝑗Φ15𝜏𝐵𝜒Φ(𝑔)𝑑𝜇𝐵.(3.29) Thus, 𝐸ΦLip𝑢𝑗𝐶𝑑𝜇𝐶𝑑,𝐶Φ𝐵𝑗𝜇(𝐸𝐵)𝜇(5𝜏𝐵)5𝜏𝐵Φ(𝑔)𝑑𝜇.(3.30) Since 𝐵𝑗 can be divided into 𝑘=𝐶(𝐶𝑑,𝜏) subfamilies 𝑗,1,,𝑗,𝑘 so that each of the families 5𝜏𝑗,𝑙 consists of disjoint balls, it suffices to show that, for 1𝑙𝑘, lim𝜇(𝐸)0𝐵𝑗,𝑙𝜇(𝐸𝐵)𝜇(5𝜏𝐵)5𝜏𝐵Φ(𝑔)𝑑𝜇=0.(3.31) Fix 𝜀>0. Then, there exists 𝛿>0 such that 𝐴Φ(𝑔)<𝜀 whenever 𝜇(𝐴)<𝛿. Denote by the family of those balls 𝐵 in 𝑗,𝑙 for which 𝜇(𝐸𝐵)𝜇(5𝜏𝐵)<𝜀.(3.32) Also, let =𝑗,𝑙. Now, if 𝜇(𝐸)<𝜀𝛿, we have that 𝜇(𝐵5𝜏𝐵)𝜀1𝜇(𝐸)<𝛿. Thus, 𝐵𝑗,𝑙𝜇(𝐸𝐵)𝜇(5𝜏𝐵)5𝜏𝐵Φ(𝑔)𝑑𝜇=𝐵𝜇(𝐸𝐵)𝜇(5𝜏𝐵)5𝜏𝐵Φ(𝑔)𝑑𝜇+𝐵𝜇(𝐸𝐵)𝜇(5𝜏𝐵)5𝜏𝐵Φ(𝑔)𝑑𝜇𝜀ΩΦ(𝑔)𝑑𝜇+𝐵5𝜏𝐵Φ(𝑔)𝑑𝜇2𝜀.(3.33) This completes the proof of (3.28).
By Lemma 2.2, a subsequence of (Lip𝑢𝑗) converges weakly to some 𝑔𝑢𝐿Φ(Ω), which, by Lemma 2.3, is a Φ-weak upper gradient of a representative of 𝑢. Moreover, 𝑔𝑢(𝑥)limsup𝑗Lip𝑢𝑗(𝑥) for a.e. 𝑥Ω. It follows from Lemma 3.3(1) and from the Φ-Poincaré inequality thatLip𝑢𝑗𝐶(𝑥)𝐶𝑑Φ1𝐵(𝑥,10𝜏𝑗1).Φ(𝑔)𝑑𝜇(3.34) Thus, 𝑔𝑢(𝑥)𝐶(𝐶𝑑)𝑔(𝑥) for a.e. 𝑥Ω.

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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Copyright © 2012 Toni Heikkinen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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