Abstract

Let Φ be an N-function. We show that a function belongs to the Orlicz-Sobolev space 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 be a metric measure space, open, and a Young function. A pair of measurable functions, and , satisfy the -Poincaré inequality in , if there is a constant such that for every ball such that . Here, and . If , then (1.1) reduces to the familiar -Poincaré inequality. The -Poincaré inequality was introduced in [1] and further studied in [25].

In the euclidean setting, it is well known that satisfies the 1-Poincaré inequality for every ball . Thus, by Jensen’s inequality, (1.1) holds with and . Our first result, Theorem 1.1 below, says that also the converse holds: if and there exists such that (1.1) holds (for the normalized pair), then belongs to the Sobolev class . More generally, we show that if and only if the number where is finite. Note that if and only if there is a functional such that whenever the balls are disjoint, and that the generalized -Poincaré inequality holds whenever . In particular, if a pair satisfies the -Poincaré inequality in , then The spaces , for , were studied in [6].

Theorem 1.1. Suppose that is an -function, is open, and , then the following conditions are equivalent:(a),(b)there exists such that the pair satisfies the -Poincaré inequality in ,(c) for some . Moreover, if a functional satisfies (1.5) and (1.6), one has that for a.e. . In particular, (b) implies that for a.e. .

Notice that is not an -function. In this case, (a) and (b) are still equivalent, but (a) and (c) are not. In fact, it was shown in [6] that coincides with , the space of functions of bounded variation. The equivalence of (a) and (b) in the case , was proved in [7] and in the case in [8]. A different proof of the case was provided in [9]. The case where both and its conjugate are doubling can be found in [5]. The equivalence of (a) and (c) in the case , , 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 be a Borel regular outer measure satisfying , whenever is nonempty, open, and bounded. Suppose further that is doubling, that is, there exists a constant such that whenever is a ball and .

Our substitute for the usual Sobolev class is based on upper gradients. We call a Borel function an upper gradient of a function if for all rectifiable curves . Further, as above is called a -weak upper gradient if (1.10) holds for all curves except for a family of -modulus zero, see Section 2.2 below. The concept of an upper gradient was introduced in [10]; also see [7]. The Sobolev space consists of all functions in that have a -weak upper gradient that belongs to .

Theorem 1.2. Suppose that is a doubling Young function, is open, , and that the pair satisfies the -Poincaré inequality in , then a representative of has a -weak upper gradient such that for a.e. .

In the case , , the result was essentially proved in [8], see [11].

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

Theorem 1.3. Let be an open set, and let be a doubling Young function whose conjugate is doubling, then a representative of has a -weak upper gradient with . Moreover, for a functional satisfying (1.5) and (1.6), one has that for a.e. .

We say that a space supports the -Poincaré inequality if there exist constants and such that whenever is a ball, , and is a -weak upper gradient of . The spaces supporting the -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 is a doubling Young function, supports the -Poincaré inequality, is open, and , then the following conditions are equivalent.(a).(b)There exists such that the pair satisfies the -Poincaré inequality in .If also the conjugate of is doubling, then (a) and (b) are equivalent to(c) for some .

2. Preliminaries

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

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 -functions, good expositions are also [14] and [15, Chapter 8].

A function is called a Young function if it has the form where is an increasing, left-continuous function, which is neither identically zero nor identically infinite on . If, in addition, for , , and , then is called an -function.

A Young function is convex and, in particular, satisfies for and .

If is a real-valued Young function and , then Jensen’s inequality holds for .

The right-continuous generalized inverse of a Young function is We have that for .

The conjugate of a Young function is the Young function defined by for .

The conjugate of an -function is an -function.

Let be a Young function. The Orlicz space is the set of all measurable functions for which there exists such that The Luxemburg norm of is If , we have that The following generalized Hölder inequality holds for Luxemburg norms:

A Young function is doubling if there exists a constant such that for . Notice that a doubling Young function is realvalued and if and only if .

Lemma 2.1. Let be a doubling Young function.(1)The space of bounded, boundedly supported continuous functions is dense in .(2)The modular convergence and the norm convergence are equivalent, that is, if and only if

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

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

Lemma 2.2. Suppose that is a doubling Young function and that satisfies then there exists a subsequence of and such that weakly in .

Proof. Since is doubling, the dual of is . By [13, page 144, Corollary 2], a sequence has a weakly converging subsequence if for each , By the Hölder inequality and Lemma 2.1, these follow from (2.14).

2.2. Sobolev Spaces

The -modulus of a curve family is where the infimum is taken over all Borel functions satisfying for all locally rectifiable curves .

The Sobolev space , consisting of the functions having a -weak upper gradient , was introduced by Shanmugalingam [16], when , and extended to the Orlicz case by Tuominen [1]. The space is a Banach space with the norm where the infimum is taken over -weak upper gradients of .

Lemma 2.3. Suppose that and weakly in and that is a -weak upper gradient of , then is a -weak upper gradient of a representative of . Moreover, for a.e. .

Proof. By Mazur’s lemma ([17, Page 120, Theorem 2]), there is a sequence of convex combinations where and , such that in . Hence, a subsequence of converges pointwise a.e., which implies that for a.e. . The fact that is a -weak upper gradient of a representative of was proved in [1, Theorem 4.17].

If is doubling and is an open set, then is isomorphic to [1, Theorem 6.19]. As usual, is the space of functions having weak partial derivatives in . A function is a weak partial derivative of (with respect to ) if for all .

Lemma 2.4. Let be an -function. Suppose that the functional , is bounded with respect to the norm , then has a weak partial derivative such that .

Proof. Denote by the closure of the space of bounded, boundedly supported functions in . By [15, Theorem ], is dense in . Thus, extends to a continuous linear functional on . By [15, Theorem 8.19], the dual of is isomorphic to ; there exists such that for . Moreover, . The claim follows.

2.3. Lipschitz Functions

A function is -Lipschitz if for all . The (upper) pointwise Lipschitz constant of a locally Lipschitz function is It is well known that is an upper gradient of ; see, for example, [18].

3. Proofs

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

Proof of Theorem 1.1. As noted in the introduction, (a) (b) (c). Let us show that (c) (a). Fix . We will show that the functional , satisfies
Choose such that , and let for , Then By the Hölder inequality, Since , we have that Thus, Let and let . Cover with balls , , such that the balls are disjoint. If , then and (3.6) implies that Thus, Since the balls are disjoint, it follows that the family can be divided into subfamilies such that each of the families consists of disjoint balls. Hence, This, combined with (3.4), yields (3.2). By Lemma 2.4, has a weak partial derivative .
Since for a.e. and for small , it follows that for a.e. . Let be a functional satisfying (1.5) and ((1.6), then, by (3.6), Thus, (1.8) holds for a.e. . If condition (b) is satisfied, we have which implies that for a.e. . 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 on balls, we construct a sequence of locally Lipschitz functions so that in and that Since is reflexive, a subsequence of converges weakly, and the claim follows from Lemma 2.3.

Under the assumptions of Theorem 1.2, may not be reflexive, but using the -Poincaré inequality, we can show that is uniformly -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 be open. Given , , there is a cover of with the following properties:(1) for all ,(2) for all ,(3)if meets , then ,(4)each ball meets at most balls .

A collection as above is called an -covering of . Clearly, an -cover is an -cover provided and .

Lemma 3.2. Let be open, and let be an -cover of , then there is a collection of functions such that(1)each is -Lipschitz,(2) for all ,(3) for for all ,(4) for all .A collection as above is called a partition of unity with respect to .

Let be as in the lemma above, and let be a partition of unity with respect to . For a locally integrable function on , define The following lemma describes the most important properties of .

Lemma 3.3. The function is locally Lipschitz. Moreover, for each ,
Let be a doubling Young function, and let . If is an -cover of and as , then in .

Proof. See the proof of [6, Lemma ].
We begin by showing that, for every , We may assume that . By Jensen’s inequality, . Hence, by the properties of the functions , Thus, by (2.2), we obtain (3.16).
Let and . By Lemma 2.1, there exists such that . Then, by (3.16), we obtain and so Therefore, it suffices to show that as . Now, , and for all , we have that which converges to 0 as by the continuity of . Thus, by the dominated convergence theorem, and so, by Lemma 2.1, .

Proof of Theorem 1.3. Let . For , let be a -cover (and hence also a -cover) of , then, by Lemma 3.3, in . Let us show that By Lemma 3.3, It follows from Lemma 3.1 that can be divided into subfamilies so that each of the families consists of disjoint balls. Since the families belong to , we have that
Since and are doubling, is reflexive. Thus, the bounded sequence has a subsequence that converges weakly to some . By Lemma 2.3, is a -weak upper gradient of a representative of . As a weak limit, satisfies Let be a functional satisfying (1.5) and (1.6). Using Lemma 3.3, we obtain Since, by Lemma 2.3, for a.e. , the pointwise inequality (1.11) follows.

Proof of Theorem 1.2. We may assume that . Define the functions as in the proof of Theorem 1.3. By (3.22) and (1.7), we have that Let us show that By Lemma 3.3 and by the -Poincaré inequality, Thus, Since can be divided into subfamilies so that each of the families consists of disjoint balls, it suffices to show that, for , Fix . Then, there exists such that whenever . Denote by the family of those balls in for which Also, let . Now, if , we have that . Thus, This completes the proof of (3.28).
By Lemma 2.2, a subsequence of converges weakly to some , which, by Lemma 2.3, is a -weak upper gradient of a representative of . Moreover, for a.e. . It follows from Lemma 3.3 and from the -Poincaré inequality that Thus, for a.e. .

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.