Research Article | Open Access
Weighted Variable Sobolev Spaces and Capacity
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.
In 1991 Kováčik and Rákosník  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  and Lemma 2.3 in ). The boundedness of the maximal operator was an open problem in for a long time. It was first proved by Diening  over bounded domains, under the assumption that is locally log-Hölder 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, 3–5]. Sobolev capacity for constant exponent spaces has found a great number of uses, see Maz′ja , Evans and Gariepy , and Heinonen et al. . Also Kilpeläinen  introduced weighted Sobolev capacity and discussed the role of capacity in the pointwise definition of functions in Sobolev spaces involving weights of Muckenhoupt’s -class. Variable Sobolev capacity was introduced in the spaces by Harjulehto et al. . They generalized the Sobolev capacity to the variable exponent case. Our purpose is to generalize some results of [9–12] 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 . 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 [13–15]. 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 Muckenhoupt’s 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 . Also Miller showed that the Schwartz class is dense in for and [17, Lemma 2.1].
Hästö 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 .
We say that satisfies the local log-Hölder continuity condition if for all . If for some , and all , then we say satisfies the log-Hölder decay condition (at infinity). We denote by the class of variable exponents which are log-Hölder continuous, that is, which satisfy the local log-Hölder continuity condition and the log-Hölder 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 Hölder’s inequality for variable exponent Lebesgue spaces , 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 .
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 .
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 , 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 .(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 . 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, Lemma 4.1]. 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 .
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 . Since , then we have for almost everywhere in by . Since , 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 .
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 Theorem 4.7 in , 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  the proof is completed.
Now we show that every quasicontinuous function satisfies a weak type capacity inequality; the proofs follow the ideas by .
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, Theorem 2.2]. 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
Proposition 3.15. Let . If , then there is a such that
Theorem 3.16. Let . If 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