Research Article | Open Access

# Weighted Variable Sobolev Spaces and Capacity

**Academic Editor:**V. M. Kokilashvili

#### Abstract

We define weighted variable Sobolev capacity and discuss properties of capacity in the space . We investigate the role of capacity in the pointwise definition of functions in this space if the Hardy-Littlewood maximal operator is bounded on the space . Also it is shown the relation between the Sobolev capacity and Bessel capacity.

#### 1. Introduction

In 1991 Kovรกฤik and Rรกkosnรญk [1] introduced the variable exponent Lebesgue space and Sobolev space in higher dimensional Euclidean spaces. The spaces and have many common properties. A crucial difference between and the classical Lebesgue spaces is that is not invariant under translation in general (Example 2.9 in [1] and Lemmaโโ2.3 in [2]). The boundedness of the maximal operator was an open problem in for a long time. It was first proved by Diening [2] over bounded domains, under the assumption that is locally log-Hรถlder continuous, that is, He later extended the result to unbounded domains by supposing, in addition, that the exponent is constant outside a large ball. After this paper, many interesting and important papers appeared in nonweighted and weighted variable exponent spaces. For more details and historical background, see [1, 3โ5]. Sobolev capacity for constant exponent spaces has found a great number of uses, see Mazโฒja [6], Evans and Gariepy [7], and Heinonen et al. [8]. Also Kilpelรคinen [9] introduced weighted Sobolev capacity and discussed the role of capacity in the pointwise definition of functions in Sobolev spaces involving weights of Muckenhouptโs -class. Variable Sobolev capacity was introduced in the spaces by Harjulehto et al. [10]. They generalized the Sobolev capacity to the variable exponent case. Our purpose is to generalize some results of [9โ12] to the weighted variable exponent case.

#### 2. Definition and Preliminary Results

We study weighted variable Lebesgue and Sobolev spaces in the -dimensional Euclidean space , . Throughout this paper all sets and functions are Lebesgue measurable. The Lebesgue measure and the characteristic function of a subset will be denoted by and , respectively. The space consists of all (classes of) measurable functions on such that for any compact subset . It is a topological vector space with the family of seminorms . A Banach function space (shortly BF-space) on is a Banach space of measurable functions which is continuously embedded into , that is, for any compact subset there exists some constant such that for all . We denote it by . The class is defined as set of infinitely differentiable functions with compact support in . For a measurable function (called a variable exponent on ), we put For every measurable functions on we define the function The function is convex modular; that is, , if and only if , and is convex. The variable exponent Lebesgue spaces (or generalized Lebesgue spaces) is defined as the set of all measurable functions on such that for some , equipped with the Luxemburg norm If , then if and only if . The set is a Banach space with the norm . If is a constant function, then the norm coincides with the usual Lebesgue norm [1]. In this paper we assume that .

A positive, measurable, and locally integrable function is called a weight function. The weighted modular is defined by The weighted variable exponent Lebesgue space consists of all measurable functions on for which . The relations between the modular and as follows: see [13โ15]. Moreover, if , then we have , since one easily sees that and .

The Schwartz class consists of all infinitely differentiable and rapidly decreasing functions in . Then and any derivative die out faster than reciprocal of any polynomial at infinity. That is, if and only if for any and there is a constant such that In particular, for , Also it is well known that .

For and we denote an open ball with center and radius by . For , we denote the (centered) Hardy-Littlewood maximal operator of by where the supremum is taken over all balls .

Let . A weight satisfies Muckenhouptโs condition, or , if there exist positive constants and such that, for every ball , or The infimum over the constants and is called the and , respectively. Also it is known that . Let . Then Muckenhoupt proved that if and only if the Hardy-Littlewood maximal operator is bounded on [16]. Also Miller showed that the Schwartz class is dense in for and [17, Lemma 2.1].

Hรคstรถ and Diening defined the class to consist of those weights for which where denotes the set of all balls in , and . Note that this class is ordinary Muckenhoupt class if is a constant function [13].

We say that satisfies the local log-Hรถlder continuity condition if for all . If for some , and all , then we say satisfies the log-Hรถlder decay condition (at infinity). We denote by the class of variable exponents which are log-Hรถlder continuous, that is, which satisfy the local log-Hรถlder continuity condition and the log-Hรถlder decay condition.

Let , and . If , then there exists a constant depending on the characteristics of and such that [13, Lemma 3.1]. As a result of this Lemma we have for and .

Let and . Then if and only if [13, Theorem 1.1].

We use the notation that is, the maximal operator is bounded on . Hence we can find a sufficient condition for .

Proposition 2.1. *Let be a weight function and . If , then .*

*Proof. *Suppose that , and let be a compact set. For , by using Hรถlderโs inequality for variable exponent Lebesgue spaces [1], then there exists a such that
It is known that if and only if for . Since , then we have
If we use (2.17) and (2.18), then the proof is completed.

*Definition 2.2 (Mollifiers). *Let be a nonnegative, radial, decreasing function belonging to and having the properties:(i) if ,(ii).

Let . If the function is nonnegative, belongs to , and satisfies(i) if and(ii), then is called a mollifier and we define the convolution by

The following proposition was proved in [18, Proposition 2.7].

Proposition 2.3. *Let be a mollifier and . Then
*

Proposition 2.4. *If and , then in as .*

*Proof. *Let and be given. If is continuous, then the assertion is trivial. By Proposition 2.3, we have
and we have for all . It can be proved that the class of continuous functions with compact support is dense in the space . Then there is a function such that
Also it is well known that if , then for all . It is easily seen that uniformly on compact sets as . Hence we have
where , compact. Hence as and we write
Finally by using (2.22) and (2.24),
The proof is complete.

As a direct consequence of Proposition 2.4 there follows.

Corollary 2.5. *Let . The class is dense in .*

This result was proved without the assumption that the maximal operator is bounded in by Kokilashvili and Samko [19].

*Remark 2.6. *Let and . Then every function in has distributional derivatives by Proposition 2.1.

#### 3. Weighted Variable Sobolev Spaces

Let , and . We define the weighted variable Sobolev spaces by equipped with the norm where is a multiindex, , and . It can be shown that is a reflexive Banach space. Throughout this paper, we will always assume that and .

The space is defined by The function is defined as . The norm .

The Bessel kernel order , is defined by Let . The weighted variable Bessel potential space is, for , defined by and is equipped with the norm If we put and .

Let . If , then . Indeed, since and is radial, we have , [20, page 62]. The assertion thus follows from boundedness of maximal function in .

The unweighted variable Bessel potential space was firstly studied by Almeida and Samko in [21].

Lemma 3.1. *Let , , and . Then*(i)* is dense in , ,*(ii)*The Schwartz class is dense in , .*

*Proof. *(i) By Proposition 2.4 the proof is complete.

(ii) Let . The class is dense in by Corollary 2.5. It remains only to show that . Let . Then there exist and such that
Also since and , then
It is known that for . Also the fact that the Muckenhoupt weights with constant are integrable with some power weight. Then
see [22, Lemma 1]. If we use (3.9) in (3.8), then the Schwartz class is dense in .

Let and . Then there is a function such that . By density of in we can find a sequence converging to in . Since the mapping maps onto [20], the functions , , belong to . Moreover,
and the assertion follows.

The following Theorem can be proved similarly in [12, Theorem 3.1].

Theorem 3.2. *Let and . Then and the corresponding norms are equivalent.*

*Remark 3.3. *The equivalence of the spaces and fails when or .

For , we denote The Sobolev -capacity of is defined by In case , we set . The -capacity has the following properties.(i). (ii)If , then .(iii)If is a subset of , then (iv)If and are subsets of , then (v)If are compact, then

Note that the assertion (v) above is not true in general for noncompact sets [9].(vi) If are subsets of , then (vii) If for , then

For the proof of these properties see [8, 10]. Hence the Sobolev capacity is an outer measure. A set function which satisfies the capacity properties (i), (ii), (v), and (vi) is called Choquet capacity; see [23]. Therefore we have the following result.

Corollary 3.4. *The set function , , is a Choquet capacity. In particular, all Suslin sets are capacitable, that is,
*

Lemma 3.5. *Let for . Then every measurable set satisfies .*

*Proof. *If , then there is an open set such that in and hence
We obtain the claim by taking the infimum on .

*Definition 3.6 (Bessel Capacity). *Let , . Define that the -Bessel capacity in is the number
where the infimum is taken over all such that on . Since is nonnegative we can assume that .

Theorem 3.7. * is an outer capacity defined on all subsets of .*

*Proof. *It is known that(i);
(ii)if , then ;(iii)if for , then
by [12, Lemmaโโ4.1]. We will show that
for any . Let be arbitrary. Obviously We assume that . If there must exist a test function (measurable and nonnegative) for , call it , such that on , and
Let . Since is lower semicontinuous in , is an open set and since on , . Therefore is a test function for and we have
This proves the theorem as .

Now we give relationship between the capacities and [12].

Lemma 3.8. *Let and . Then
**
Here and are positive constants independent of .*

Proposition 3.9. *Let .*(i)*If , then and for almost everywhere in .*(ii)*Let . Then and there exists a constant such that the inequality
holds for all .*

*Proof. *(i) By Proposition 2.1 we have and . Sinceโโ, then we have for almost everywhere in by [24]. Sinceโโ, and , then . Hence .

(ii) Let . By using definition of , we have
and . Therefore we have
and . Since , then . Hence we write
by (3.28). If we set , then
This completes the proof.

Proposition 3.10. *Let . Then for every and every we have
*

*Proof. *Since is lower semicontinuous, the set is open for every . By Proposition 3.9 we can take as a test function for the capacity. Then we have

We say that a property holds -quasi everywhere if it holds except in a set of capacity zero. A function is -quasicontinuous in if for each there exists an open set with such that restricted to is continuous. The following proof of theorem is quite similar to Theorem 4.7 in [11].

Theorem 3.11. *Let . If , then the limit
**
exists -quasi everywhere in . The function is the -quasicontinuous representative of .*

*Proof. *Since the class is dense in by Lemma 3.1, then we can choose a sequence such that
For we denote
By using Proposition 3.10 and the subadditivity of we have
and . If we follow the proof of Theoremโโ4.7 in [11], then this proves the theorem.

Corollary 3.12. *Let . If and is quasicontinuous, then we have
**-quasi everywhere in .*

*Proof. *By using the Theorem in [25] the proof is completed.

Now we show that every quasicontinuous function satisfies a weak type capacity inequality; the proofs follow the ideas by [10].

Lemma 3.13. *Let and . If is a nonnegative -quasicontinuous function such that on . Then for every there exists a function such that .*

*Proof. *Let , and let be an open set such that is continuous in and . By definition of there exists a such that . If we set , then it is easy to show that by [10, Theoremโโ2.2]. Since the function is continuous and the set is open, then the set
is open, contains , and on , thus . It is known that for and , and . Hence we obtain and
This completes the proof as .

Theorem 3.14. *Let . If is a -quasicontinuous function and , then
*

*Proof. *By [10, Theoremโโ2.2], and . By Lemma 3.13, there is a sequence such that
Hence we have by [10, Lemmaโโ2.6] that
By definition of , we write
Therefore

Proposition 3.15. *Let . If , then there is a such that
*

*Proof. *For , we take . Choose such that and by Theorem 3.2. Then
and . Also it is known that if , then by [12, Lemmaโโ4.1]. Therefore we have

The following Theorem is obtained directly from Lemma 3.8 and Theorem 3.11.

Theorem 3.16. *Let . Ifโโ and is quasicontinuous, then the limit
**
exists -quasi everywhere in .*

The following proposition can be proved similarly as in [12, Proposition 5.1].

Proposition 3.17. *Let . Every is quasicontinuous. That is, for every , there exists a set , , so that restricted to is continuous.*

Proposition 3.18. *Let . Then for all and we have
*

*Proof. *We first note that by definition of -capacity, is a test function for the Bessel capacity. Hence

Proposition 3.19. *Let . If and
**
then .*

*Proof. *By Proposition 3.18, we write as .

Proposition 3.20. *Let . If , then
**
for -q.e. .*

*Proof. *Let be the characteristic function for the unit ball , and define for , , . Then
As , for every . This implies that, for fixed , for a.e. . It was shown that