Journal of Function Spaces

Journal of Function Spaces / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 132690 | 17 pages | https://doi.org/10.1155/2012/132690

Weighted Variable Sobolev Spaces and Capacity

Academic Editor: V. M. Kokilashvili
Received07 Jul 2010
Accepted21 Nov 2010
Published03 Jan 2012

Abstract

We define weighted variable Sobolev capacity and discuss properties of capacity in the space π‘Š1,𝑝(β‹…)(ℝ𝑛,𝑀). 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 π‘Š1,𝑝(β‹…)(ℝ𝑛,𝑀). Also it is shown the relation between the Sobolev capacity and Bessel capacity.

1. Introduction

In 1991 KovÑčik and RΓ‘kosnΓ­k [1] introduced the variable exponent Lebesgue space 𝐿𝑝(β‹…)(ℝ𝑛) and Sobolev space π‘Šπ‘˜,𝑝(β‹…)(ℝ𝑛) in higher dimensional Euclidean spaces. The spaces 𝐿𝑝(β‹…)(ℝ𝑛) and 𝐿𝑝(ℝ𝑛) have many common properties. A crucial difference between 𝐿𝑝(β‹…)(ℝ𝑛) and the classical Lebesgue spaces 𝐿𝑝(ℝ𝑛) is that 𝐿𝑝(β‹…)(ℝ𝑛) is not invariant under translation in general (Example 2.9 in [1] and Lemma  2.3 in [2]). The boundedness of the maximal operator was an open problem in 𝐿𝑝(β‹…)(ℝ𝑛) for a long time. It was first proved by Diening [2] over bounded domains, under the assumption that 𝑝(β‹…) is locally log-HΓΆlder continuous, that is, ||||≀𝐢𝑝(π‘₯)βˆ’π‘(𝑦)||||||||≀1βˆ’lnπ‘₯βˆ’π‘¦,π‘₯,π‘¦βˆˆΞ©,π‘₯βˆ’π‘¦2.(1.1) He later extended the result to unbounded domains by supposing, in addition, that the exponent 𝑝(β‹…) 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 Mazβ€²ja [6], Evans and Gariepy [7], and Heinonen et al. [8]. Also KilpelΓ€inen [9] introduced weighted Sobolev capacity and discussed the role of capacity in the pointwise definition of functions in Sobolev spaces involving weights of Muckenhoupt’s 𝐴𝑝-class. Variable Sobolev capacity was introduced in the spaces π‘Š1,𝑝(β‹…)(ℝ𝑛) by Harjulehto et al. [10]. 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 𝑛-dimensional Euclidean space ℝ𝑛, 𝑛β‰₯2. Throughout this paper all sets and functions are Lebesgue measurable. The Lebesgue measure and the characteristic function of a subset π΄βŠ‚β„π‘› will be denoted by πœ‡(𝐴)=|𝐴| and πœ’π΄, respectively. The space 𝐿1loc(ℝ𝑛) consists of all (classes of) measurable functions 𝑓 on ℝ𝑛 such that π‘“πœ’πΎβˆˆπΏ1(ℝ𝑛) for any compact subset πΎβŠ‚β„π‘›. It is a topological vector space with the family of seminorms π‘“β†¦β€–π‘“πœ’πΎβ€–πΏ1. A Banach function space (shortly BF-space) on ℝ𝑛 is a Banach space (𝐡,‖⋅‖𝐡) of measurable functions which is continuously embedded into 𝐿1loc(ℝ𝑛), that is, for any compact subset πΎβŠ‚β„π‘› there exists some constant 𝐢𝐾>0 such that β€–π‘“πœ’πΎβ€–πΏ1≀𝐢𝐾‖𝑓‖𝐡 for all π‘“βˆˆπ΅. We denote it by 𝐡β†ͺ𝐿1loc(ℝ𝑛). The class 𝐢∞0(ℝ𝑛) is defined as set of infinitely differentiable functions with compact support in ℝ𝑛. For a measurable function π‘βˆΆβ„π‘›β†’[1,∞) (called a variable exponent on ℝ𝑛), we putπ‘βˆ’=essinfπ‘₯βˆˆβ„π‘›π‘(π‘₯),𝑝+=esssupπ‘₯βˆˆβ„π‘›π‘(π‘₯).(2.1) For every measurable functions 𝑓 on ℝ𝑛 we define the functionπœšπ‘(β‹…)(ξ€œπ‘“)=ℝ𝑛||||𝑓(π‘₯)𝑝(π‘₯)𝑑π‘₯.(2.2) The function πœšπ‘(β‹…) is convex modular; that is, πœšπ‘(β‹…)(𝑓)β‰₯0, πœšπ‘(β‹…)(𝑓)=0 if and only if 𝑓=0, πœšπ‘(β‹…)(βˆ’π‘“)=πœšπ‘(β‹…)(𝑓) and πœšπ‘(β‹…) is convex. The variable exponent Lebesgue spaces (or generalized Lebesgue spaces) 𝐿𝑝(β‹…)(ℝ𝑛) is defined as the set of all measurable functions 𝑓 on ℝ𝑛 such that πœšπ‘(β‹…)(πœ†π‘“)<∞ for some πœ†>0, equipped with the Luxemburg norm‖𝑓‖𝑝(β‹…)ξ‚»=infπœ†>0βˆΆπœšπ‘ξ‚΅π‘“πœ†ξ‚Άξ‚Όβ‰€1.(2.3) If 𝑝+<∞, then π‘“βˆˆπΏπ‘(β‹…)(ℝ𝑛) if and only if πœšπ‘(β‹…)(𝑓)<∞. The set 𝐿𝑝(β‹…)(ℝ𝑛) is a Banach space with the norm ‖⋅‖𝑝(β‹…). If 𝑝(π‘₯)=𝑝 is a constant function, then the norm ‖⋅‖𝑝(β‹…) coincides with the usual Lebesgue norm ‖⋅‖𝑝 [1]. In this paper we assume that 𝑝+<∞.

A positive, measurable, and locally integrable function π‘€βˆΆβ„π‘›β†’(0,∞) is called a weight function. The weighted modular is defined byπœšπ‘(β‹…),𝑀(ξ€œπ‘“)=ℝ𝑛||||𝑓(π‘₯)𝑝(π‘₯)𝑀(π‘₯)𝑑π‘₯.(2.4) The weighted variable exponent Lebesgue space 𝐿𝑝(β‹…)(ℝ𝑛,𝑀) consists of all measurable functions 𝑓 on ℝ𝑛 for which ‖𝑓‖𝑝(β‹…),𝑀=‖𝑓𝑀1/𝑝(β‹…)‖𝑝(β‹…)<∞. The relations between the modular πœšπ‘(β‹…),𝑀(β‹…) and ‖⋅‖𝑝(β‹…),𝑀 as follows:ξ‚†πœšmin𝑝(β‹…),𝑀(𝑓)1/π‘βˆ’,πœšπ‘(β‹…),𝑀(𝑓)1/𝑝+≀‖𝑓‖𝑝(β‹…),π‘€ξ‚†πœšβ‰€max𝑝(β‹…),𝑀(𝑓)1/π‘βˆ’,πœšπ‘(β‹…),𝑀(𝑓)1/𝑝+min‖𝑓‖𝑝+𝑝(β‹…),𝑀,β€–π‘“β€–π‘βˆ’π‘(β‹…),π‘€ξ‚‡β‰€πœšπ‘(β‹…),𝑀(𝑓)≀max‖𝑓‖𝑝+𝑝(β‹…),𝑀,β€–π‘“β€–π‘βˆ’π‘(β‹…),𝑀,(2.5) see [13–15]. Moreover, if 0<𝐢≀𝑀, then we have 𝐿𝑝(β‹…)(ℝ𝑛,𝑀)β†ͺ𝐿𝑝(β‹…)(ℝ𝑛), since one easily sees thatπΆξ€œβ„π‘›||||𝑓(π‘₯)𝑝(π‘₯)ξ€œπ‘‘π‘₯≀ℝ𝑛||||𝑓(π‘₯)𝑝(π‘₯)𝑀(π‘₯)𝑑π‘₯(2.6) and 𝐢‖𝑓‖𝑝(β‹…)≀‖𝑓‖𝑝(β‹…),𝑀.

The Schwartz class 𝑆=𝑆(ℝ𝑛) consists of all infinitely differentiable and rapidly decreasing functions in ℝ𝑛. Then 𝑓 and any derivative 𝐷𝛽𝑓 die out faster than reciprocal of any polynomial at infinity. That is, π‘“βˆˆπ‘† if and only if for any 𝛽 and π‘˜>0 there is a constant 𝐢=𝐢(𝛽,π‘˜) such that||𝐷𝛽||≀𝐢𝑓(π‘₯)(1+|π‘₯|)π‘˜.(2.7) In particular, for 𝛽=0,||||≀𝐢𝑓(π‘₯)(1+|π‘₯|)π‘˜.(2.8) Also it is well known that 𝐢∞0(ℝ𝑛)βŠ‚π‘†.

For π‘₯βˆˆβ„π‘› and π‘Ÿ>0 we denote an open ball with center π‘₯ and radius π‘Ÿ by 𝐡(π‘₯,π‘Ÿ). For π‘“βˆˆπΏ1loc(ℝ𝑛), we denote the (centered) Hardy-Littlewood maximal operator 𝑀𝑓 of 𝑓 by𝑀𝑓(π‘₯)=supπ‘Ÿ>01||||ξ€œπ΅(π‘₯,π‘Ÿ)𝐡(π‘₯,π‘Ÿ)||||𝑓(𝑦)𝑑𝑦,(2.9) where the supremum is taken over all balls 𝐡(π‘₯,π‘Ÿ).

Let 1≀𝑝<∞. A weight 𝑀 satisfies Muckenhoupt’s 𝐴𝑝(ℝ𝑛)=𝐴𝑝 condition, or π‘€βˆˆπ΄π‘, if there exist positive constants 𝐢 and 𝑐 such that, for every ball π΅βŠ‚β„π‘›,ξ‚΅1||𝐡||ξ€œπ΅1𝑀𝑑π‘₯ξ‚Άξ‚΅||𝐡||ξ€œπ΅π‘€βˆ’1/(π‘βˆ’1)𝑑π‘₯π‘βˆ’1≀𝐢,1≀𝑝<∞,(2.10) orξ‚΅1||𝐡||ξ€œπ΅π‘€π‘‘π‘₯ξ‚Άξ‚΅esssup𝐡1𝑀≀𝑐,𝑝=1.(2.11) The infimum over the constants 𝐢 and 𝑐 is called the 𝐴𝑝 and 𝐴1, respectively. Also it is known that 𝐴∞=⋃1≀𝑝<βˆžπ΄π‘. Let 1<𝑝<∞. Then Muckenhoupt proved that π‘€βˆˆπ΄π‘ if and only if the Hardy-Littlewood maximal operator is bounded on 𝐿𝑝(ℝ𝑛,𝑀) [16]. Also Miller showed that the Schwartz class 𝑆 is dense in 𝐿𝑝(ℝ𝑛,𝑀) for 1<𝑝<∞ and π‘€βˆˆπ΄π‘ [17, Lemma 2.1].

HΓ€stΓΆ and Diening defined the class 𝐴𝑝(β‹…) to consist of those weights 𝑀 for which‖𝑀‖𝐴𝑝(β‹…)∢=supπ΅βˆˆβ„¬||𝐡||βˆ’π‘π΅β€–π‘€β€–πΏ1(𝐡)β€–β€–β€–1𝑀‖‖‖𝐿𝑝′(β‹…)/𝑝(β‹…)(𝐡)<∞,(2.12) where ℬ denotes the set of all balls in ℝ𝑛, π‘π΅βˆ«=((1/|𝐡|)𝐡(1/𝑝(π‘₯))𝑑π‘₯)βˆ’1 and 1/𝑝(β‹…)+1/π‘ξ…ž(β‹…)=1. Note that this class is ordinary Muckenhoupt class 𝐴𝑝 if 𝑝 is a constant function [13].

We say that 𝑝(β‹…) satisfies the local log-HΓΆlder continuity condition if||||≀𝐢𝑝(π‘₯)βˆ’π‘(𝑦)ξ€·||||ξ€Έlog𝑒+1/π‘₯βˆ’π‘¦(2.13) for all π‘₯,π‘¦βˆˆβ„π‘›. If||𝑝(π‘₯)βˆ’π‘βˆž||≀𝐢log(𝑒+|π‘₯|)(2.14) for some π‘βˆž>1, 𝐢>0 and all π‘₯βˆˆβ„π‘›, then we say 𝑝(β‹…) satisfies the log-HΓΆlder decay condition (at infinity). We denote by 𝑃log(ℝ𝑛) the class of variable exponents which are log-HΓΆlder continuous, that is, which satisfy the local log-HΓΆlder continuity condition and the log-HΓΆlder decay condition.

Let 𝑝,π‘žβˆˆπ‘ƒlog(ℝ𝑛), 1<π‘βˆ’β‰€π‘+<∞ and 1<π‘žβˆ’β‰€π‘ž+<∞. If π‘žβ‰€π‘, then there exists a constant 𝐢>0 depending on the characteristics of 𝑝 and π‘ž such that ‖𝑀‖𝐴𝑝(β‹…)β‰€πΆβ€–π‘€β€–π΄π‘ž(β‹…) [13, Lemma 3.1]. As a result of this Lemma we have𝐴1βŠ‚π΄π‘βˆ’βŠ‚π΄π‘(β‹…)βŠ‚π΄π‘+βŠ‚π΄βˆž(2.15) for π‘βˆˆπ‘ƒlog(ℝ𝑛) and 1<π‘βˆ’β‰€π‘+<∞.

Let π‘βˆˆπ‘ƒlog(ℝ𝑛) and 1<π‘βˆ’β‰€π‘+<∞. Then π‘€βˆΆπΏπ‘(β‹…)(ℝ𝑛,𝑀)β†ͺ𝐿𝑝(β‹…)(ℝ𝑛,𝑀) if and only if π‘€βˆˆπ΄π‘(β‹…) [13, Theorem 1.1].

We use the notation𝒫(ℝ𝑛𝑝)=(β‹…)∢1<π‘βˆ’β‰€π‘(π‘₯)≀𝑝+<∞,‖𝑀𝑓‖𝑝(β‹…),𝑀≀𝐢‖𝑓‖𝑝(β‹…),𝑀,(2.16) that is, the maximal operator 𝑀 is bounded on 𝐿𝑝(β‹…)(ℝ𝑛,𝑀). Hence we can find a sufficient condition for 𝑝(β‹…)βˆˆπ’«(ℝ𝑛).

Proposition 2.1. Let 𝑀 be a weight function and 1<π‘βˆ’β‰€π‘(π‘₯)≀𝑝+<∞. If π‘€βˆ’1/(𝑝(β‹…)βˆ’1)∈𝐿1loc(ℝ𝑛), then 𝐿𝑝(β‹…)(ℝ𝑛,𝑀)β†ͺ𝐿1loc(ℝ𝑛).

Proof. Suppose that π‘“βˆˆπΏπ‘(β‹…)(ℝ𝑛,𝑀), and let πΎβŠ‚β„π‘› be a compact set. For 1/𝑝(β‹…)+1/π‘ž(β‹…)=1, by using HΓΆlder’s inequality for variable exponent Lebesgue spaces [1], then there exists a 𝐴𝐾>0 such that ξ€œπΎ||||𝑓(π‘₯)𝑑π‘₯≀𝐴𝐾‖‖𝑓𝑀1/𝑝(β‹…)‖‖𝑝(β‹…),πΎβ€–β€–π‘€βˆ’1/𝑝(β‹…)β€–β€–π‘ž(β‹…),𝐾≀𝐴𝐾‖‖𝑓𝑀1/𝑝(β‹…)‖‖𝑝(β‹…)β€–β€–π‘€βˆ’1/𝑝(β‹…)β€–β€–π‘ž(β‹…),𝐾.(2.17) It is known that β€–π‘€βˆ’1/𝑝(β‹…)β€–π‘ž(β‹…),𝐾<∞ if and only if πœšπ‘ž(β‹…),𝐾(π‘€βˆ’1/𝑝(β‹…))<∞ for π‘ž+<∞. Since π‘€βˆ’1/(𝑝(β‹…)βˆ’1)∈𝐿1loc(ℝ𝑛), then we have πœšπ‘ž(β‹…),πΎξ€·π‘€βˆ’1/𝑝(β‹…)ξ€Έ=ξ€œπΎπ‘€(π‘₯)βˆ’π‘ž(π‘₯)/𝑝(π‘₯)ξ€œπ‘‘π‘₯=𝐾𝑀(π‘₯)βˆ’1/(𝑝(π‘₯)βˆ’1)𝑑π‘₯=𝐡𝐾<∞.(2.18) If we use (2.17) and (2.18), then the proof is completed.

Definition 2.2 (Mollifiers). Let πœ‘βˆΆβ„π‘›β†’β„ be a nonnegative, radial, decreasing function belonging to 𝐢∞0(ℝ𝑛) and having the properties:(i)πœ‘(π‘₯)=0 if |π‘₯|β‰₯1,(ii)βˆ«β„π‘›πœ‘(π‘₯)𝑑π‘₯=1.
Let πœ€>0. If the function πœ‘πœ€(π‘₯)=πœ€βˆ’π‘›πœ‘(π‘₯/πœ€) is nonnegative, belongs to 𝐢∞0(ℝ𝑛), and satisfies(i)πœ‘πœ€(π‘₯)=0 if |π‘₯|β‰₯πœ€ and(ii)βˆ«β„π‘›πœ‘πœ€(π‘₯)𝑑π‘₯=1, then πœ‘πœ€ is called a mollifier and we define the convolution by πœ‘πœ€ξ€œβˆ—π‘“(π‘₯)=β„π‘›πœ‘πœ€(π‘₯βˆ’π‘¦)𝑓(𝑦)𝑑𝑦.(2.19)

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

Proposition 2.3. Let πœ‘πœ€ be a mollifier and π‘“βˆˆπΏ1loc(ℝ𝑛). Then supπœ€>0||πœ‘πœ€||βˆ—π‘“(π‘₯)≀𝑀𝑓(π‘₯).(2.20)

Proposition 2.4. If 𝑝(β‹…)βˆˆπ’«(ℝ𝑛) and π‘“βˆˆπΏπ‘(β‹…)(ℝ𝑛,𝑀), then πœ‘πœ€βˆ—π‘“β†’π‘“ in 𝐿𝑝(β‹…)(ℝ𝑛,𝑀) as πœ€β†’0+.

Proof. Let π‘“βˆˆπΏπ‘(β‹…)(ℝ𝑛,𝑀) and πœ€>0 be given. If 𝑓 is continuous, then the assertion is trivial. By Proposition 2.3, we have β€–β€–πœ‘πœ€β€–β€–βˆ—π‘“π‘(β‹…),𝑀≀‖𝑀𝑓‖𝑝(β‹…),𝑀≀𝐢‖𝑓‖𝑝(β‹…),𝑀(2.21) and we have πœ‘πœ€βˆ—π‘“βˆˆπΏπ‘(β‹…)(ℝ𝑛,𝑀) for all πœ€>0. It can be proved that the class 𝐢0(ℝ𝑛) of continuous functions with compact support is dense in the space 𝐿𝑝(β‹…)(ℝ𝑛,𝑀). Then there is a function π‘”βˆˆπΆ0(ℝ𝑛) such that β€–π‘“βˆ’π‘”β€–π‘(β‹…),𝑀<πœ€.(2.22) Also it is well known that if π‘”βˆˆπΆ0(ℝ𝑛), then πœ‘πœ€βˆ—π‘”βˆˆπΆβˆž0(ℝ𝑛) for all πœ€>0. It is easily seen that πœ‘πœ€βˆ—π‘”β†’π‘” uniformly on compact sets as πœ€β†’0+. Hence we have ||πœ‘πœ€||βˆ—π‘”(π‘₯)βˆ’π‘”(π‘₯)𝑝(π‘₯)𝜚⟢0,𝑝(β‹…),π‘€ξ€·πœ‘πœ€ξ€Έ=ξ€œβˆ—π‘”βˆ’π‘”πΎ||πœ‘πœ€||βˆ—π‘”(π‘₯)βˆ’π‘”(π‘₯)𝑝(π‘₯)𝑀(π‘₯)𝑑π‘₯β‰€πœ€π‘βˆ’ξ€œπΎπ‘€(π‘₯)𝑑π‘₯,(2.23) where supp(πœ‘πœ€βˆ—π‘”)βˆͺsuppπ‘”βŠ‚πΎ, πΎβŠ‚β„π‘› compact. Hence πœšπ‘(β‹…),𝑀(πœ‘πœ€βˆ—π‘”βˆ’π‘”)β†’0 as πœ€β†’0+ and we write β€–β€–πœ‘πœ€β€–β€–βˆ—π‘”βˆ’π‘”π‘(β‹…),𝑀<πœ€.(2.24) Finally by using (2.22) and (2.24), β€–β€–π‘“βˆ’πœ‘πœ€β€–β€–βˆ—π‘“π‘(β‹…),π‘€β‰€β€–π‘“βˆ’π‘”β€–π‘(β‹…),𝑀+β€–β€–π‘”βˆ’πœ‘πœ€β€–β€–βˆ—π‘”π‘(β‹…),𝑀+β€–β€–πœ‘πœ€βˆ—π‘”βˆ’πœ‘πœ€β€–β€–βˆ—π‘“π‘(β‹…),𝑀<(𝐢+2)πœ€.(2.25) The proof is complete.

As a direct consequence of Proposition 2.4 there follows.

Corollary 2.5. Let 𝑝(β‹…)βˆˆπ’«(ℝ𝑛). The class 𝐢∞0(ℝ𝑛) is dense in 𝐿𝑝(β‹…)(ℝ𝑛,𝑀).

This result was proved without the assumption that the maximal operator is bounded in 𝐿𝑝(β‹…)(ℝ𝑛,𝑀) by Kokilashvili and Samko [19].

Remark 2.6. Let 1<π‘βˆ’β‰€π‘(π‘₯)≀𝑝+<∞ and π‘€βˆ’1/(𝑝(β‹…)βˆ’1)∈𝐿1loc(ℝ𝑛). Then every function in 𝐿𝑝(β‹…)(ℝ𝑛,𝑀) has distributional derivatives by Proposition 2.1.

3. Weighted Variable Sobolev Spaces

Let 1<π‘βˆ’β‰€π‘(π‘₯)≀𝑝+<∞, π‘€βˆ’1/(𝑝(β‹…)βˆ’1)∈𝐿1loc(ℝ𝑛) and π‘˜βˆˆβ„•. We define the weighted variable Sobolev spaces π‘Šπ‘˜,𝑝(β‹…)(ℝ𝑛,𝑀) byπ‘Šπ‘˜,𝑝(β‹…)(ℝ𝑛,𝑀)=π‘“βˆˆπΏπ‘(β‹…)(ℝ𝑛,𝑀)βˆΆπ·π›Όπ‘“βˆˆπΏπ‘(β‹…)(ℝ𝑛,𝑀),0≀|𝛼|β‰€π‘˜(3.1) equipped with the normβ€–π‘“β€–π‘˜,𝑝(β‹…),𝑀=0≀|𝛼|β‰€π‘˜β€–π·π›Όπ‘“β€–π‘(β‹…),𝑀(3.2) where π›Όβˆˆβ„•π‘›0 is a multiindex, |𝛼|=𝛼1+𝛼2+β‹―+𝛼𝑛, and 𝐷𝛼=πœ•|𝛼|/(πœ•π›Ό1π‘₯1β‹―πœ•π›Όπ‘›π‘₯𝑛). It can be shown that π‘Šπ‘˜,𝑝(β‹…)(ℝ𝑛,𝑀) is a reflexive Banach space. Throughout this paper, we will always assume that 1<π‘βˆ’β‰€π‘(π‘₯)≀𝑝+<∞ and π‘€βˆ’1/(𝑝(β‹…)βˆ’1)∈𝐿1loc(ℝ𝑛).

The space π‘Š1,𝑝(β‹…)(ℝ𝑛,𝑀) is defined byπ‘Š1,𝑝(β‹…)(ℝ𝑛,𝑀)=π‘“βˆˆπΏπ‘(β‹…)(ℝ𝑛||||,𝑀)βˆΆβˆ‡π‘“βˆˆπΏπ‘(β‹…)(ℝ𝑛,𝑀).(3.3) The function 𝜚1,𝑝(β‹…),π‘€βˆΆπ‘Š1,𝑝(β‹…)(ℝ𝑛,𝑀)β†’[0,∞) is defined as 𝜚1,𝑝(β‹…),𝑀(𝑓)=πœšπ‘(β‹…),𝑀(𝑓)+πœšπ‘(β‹…),𝑀(βˆ‡π‘“). The norm ‖𝑓‖1,𝑝(β‹…),𝑀=‖𝑓‖𝑝(β‹…),𝑀+β€–βˆ‡π‘“β€–π‘(β‹…),𝑀.

The Bessel kernel 𝑔𝛼 order 𝛼,𝛼>0, is defined byπ‘”π›Όπœ‹(π‘₯)=𝑛/2ξ€œΞ“(𝛼/2)∞0π‘’βˆ’π‘ βˆ’(πœ‹2|π‘₯|2)/𝑠𝑠(π›Όβˆ’π‘›)/2𝑑𝑠𝑠,π‘₯βˆˆβ„π‘›.(3.4) Let 𝛼β‰₯0. The weighted variable Bessel potential space ℒ𝛼,𝑝(β‹…)(ℝ𝑛,𝑀) is, for 𝛼>0, defined byℒ𝛼,𝑝(β‹…)(ℝ𝑛,𝑀)∢=β„Ž=π‘”π›Όβˆ—π‘“;π‘“βˆˆπΏπ‘(β‹…)(ℝ𝑛,𝑀),(3.5) and is equipped with the normβ€–β„Žβ€–π›Ό;𝑝(β‹…),𝑀=‖𝑓‖𝑝(β‹…),𝑀.(3.6) If 𝛼=0 we put 𝑔0βˆ—π‘“=𝑓 and β„’0,𝑝(β‹…)(ℝ𝑛,𝑀)=𝐿𝑝(β‹…)(ℝ𝑛,𝑀).

Let 𝑝(β‹…)βˆˆπ’«(ℝ𝑛). If π‘“βˆˆπΏπ‘(β‹…)(ℝ𝑛,𝑀), then π‘”π›Όβˆ—π‘“βˆˆπΏπ‘(β‹…)(ℝ𝑛,𝑀). Indeed, since π‘”π›ΌβˆˆπΏ1(ℝ𝑛) and 𝑔𝛼 is radial, we have (π‘”π›Όβˆ—π‘“)(π‘₯)≀𝑀𝑓(π‘₯), π‘₯βˆˆβ„π‘› [20, page 62]. The assertion thus follows from boundedness of maximal function in 𝐿𝑝(β‹…)(ℝ𝑛,𝑀).

The unweighted variable Bessel potential space ℒ𝛼,𝑝(β‹…)(ℝ𝑛) was firstly studied by Almeida and Samko in [21].

Lemma 3.1. Let 𝑝(β‹…)βˆˆπ‘ƒlog(ℝ𝑛), 1<π‘βˆ’β‰€π‘+<∞, and π‘€βˆˆπ΄π‘(β‹…). Then(i)𝐢∞0(ℝ𝑛) is dense in π‘Šπ‘˜,𝑝(β‹…)(ℝ𝑛,𝑀), π‘˜βˆˆβ„•,(ii)The Schwartz class 𝑆 is dense in ℒ𝛼,𝑝(β‹…)(ℝ𝑛,𝑀), 𝛼β‰₯0.

Proof. (i) By Proposition 2.4 the proof is complete.
(ii) Let 𝛼=0. The class 𝐢∞0(ℝ𝑛) is dense in 𝐿𝑝(β‹…)(ℝ𝑛,𝑀) by Corollary 2.5. It remains only to show that π‘†βŠ‚πΏπ‘(β‹…)(ℝ𝑛,𝑀). Let π‘“βˆˆπ‘†. Then there exist 𝐢=𝐢(π‘Ÿ)>0 and π‘Ÿ>0 such that ||||≀𝐢𝑓(π‘₯)(1+|π‘₯|)π‘Ÿ.(3.7) Also since π‘Ÿπ‘(π‘₯)β‰₯π‘Ÿ and (1+|π‘₯|)π‘Ÿβ‰₯1, then πœšπ‘(β‹…),𝑀(ξ€œπ‘“)=ℝ𝑛||||𝑓(π‘₯)𝑝(π‘₯)𝐢𝑀(π‘₯)𝑑π‘₯≀maxπ‘βˆ’,𝐢𝑝+ξ‚‡ξ€œβ„π‘›π‘€(π‘₯)(1+|π‘₯|)π‘Ÿπ‘(π‘₯)𝐢𝑑π‘₯≀maxπ‘βˆ’,𝐢𝑝+ξ‚‡ξ€œβ„π‘›π‘€(π‘₯)(1+|π‘₯|)π‘Ÿπ‘‘π‘₯.(3.8) It is known that 𝐴𝑝(β‹…)βŠ‚π΄π‘+ for 1<𝑝+<∞. Also the fact that the Muckenhoupt weights with constant 𝑝+ are integrable with some power weight. Then ξ€œβ„π‘›π‘€(π‘₯)(1+|π‘₯|)π‘Ÿπ‘‘π‘₯<∞,(3.9) see [22, Lemma 1]. If we use (3.9) in (3.8), then the Schwartz class 𝑆 is dense in 𝐿𝑝(β‹…)(ℝ𝑛,𝑀).
Let 𝛼>0 and β„Žβˆˆβ„’π›Ό,𝑝(β‹…)(ℝ𝑛,𝑀). Then there is a function π‘“βˆˆπΏπ‘(β‹…)(ℝ𝑛,𝑀) such that β„Ž=π‘”π›Όβˆ—π‘“. By density of 𝐢∞0(ℝ𝑛) in π‘“βˆˆπΏπ‘(β‹…)(ℝ𝑛,𝑀) we can find a sequence (𝑓𝑗)π‘—βˆˆβ„•βŠ‚πΆβˆž0(ℝ𝑛)βŠ‚π‘† converging to 𝑓 in 𝐿𝑝(β‹…)(ℝ𝑛,𝑀). Since the mapping π‘“β†¦π‘”π›Όβˆ—π‘“ maps 𝑆 onto 𝑆 [20], the functions β„Žπ‘—=π‘”π›Όβˆ—π‘“π‘—, π‘—βˆˆβ„•, belong to 𝑆. Moreover, β€–β€–β„Žβˆ’β„Žπ‘—β€–β€–π›Ό;𝑝(β‹…),𝑀=β€–β€–π‘“βˆ’π‘“π‘—β€–β€–π‘(β‹…),π‘€βŸΆ0asπ‘—βŸΆβˆž(3.10) and the assertion follows.

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

Theorem 3.2. Let 𝑝(β‹…)βˆˆπ’«(ℝ𝑛) and π‘˜βˆˆβ„•. Then β„’π‘˜,𝑝(β‹…)(ℝ𝑛,𝑀)=π‘Šπ‘˜,𝑝(β‹…)(ℝ𝑛,𝑀) and the corresponding norms are equivalent.

Remark 3.3. The equivalence of the spaces β„’π‘˜,𝑝(β‹…)(ℝ𝑛,𝑀) and π‘Šπ‘˜,𝑝(β‹…)(ℝ𝑛,𝑀) fails when 𝑝=1 or 𝑝=∞.

For πΈβŠ‚β„π‘›, we denote𝑆𝑝(β‹…),𝑀(𝐸)=π‘“βˆˆπ‘Š1,𝑝(β‹…)(ℝ𝑛,𝑀)βˆΆπ‘“β‰₯1inopensetcontaining𝐸.(3.11) The Sobolev (𝑝(β‹…),𝑀)-capacity of 𝐸 is defined by𝐢𝑝(β‹…),𝑀(𝐸)=infπ‘“βˆˆπ‘†π‘(β‹…),𝑀(𝐸)𝜚1,𝑝(β‹…),𝑀(𝑓)=infπ‘“βˆˆπ‘†π‘(β‹…),𝑀(𝐸)ξ€œβ„π‘›ξ‚€||||𝑓(π‘₯)𝑝(π‘₯)+||||βˆ‡π‘“(π‘₯)𝑝(π‘₯)𝑀(π‘₯)𝑑π‘₯.(3.12) In case 𝑆𝑝(β‹…),𝑀(𝐸)=βˆ…, we set 𝐢𝑝(β‹…),𝑀(𝐸)=∞. The 𝐢𝑝(β‹…),𝑀-capacity has the following properties.(i)𝐢𝑝(β‹…),𝑀(βˆ…)=0. (ii)If 𝐸1βŠ‚πΈ2, then 𝐢𝑝(β‹…),𝑀(𝐸1)≀𝐢𝑝(β‹…),𝑀(𝐸2).(iii)If 𝐸 is a subset of ℝ𝑛, then 𝐢𝑝(β‹…),𝑀𝐢(𝐸)=inf𝑝(β‹…),𝑀(π‘ˆ)βˆΆπΈβŠ‚π‘ˆ,π‘ˆopen.(3.13)(iv)If 𝐸1 and 𝐸2 are subsets of ℝ𝑛, then 𝐢𝑝(β‹…),𝑀𝐸1βˆͺ𝐸2ξ€Έ+𝐢𝑝(β‹…),𝑀𝐸1∩𝐸2≀𝐢𝑝(β‹…),𝑀𝐸1ξ€Έ+𝐢𝑝(β‹…),𝑀𝐸2ξ€Έ.(3.14)(v)If 𝐾1βŠƒπΎ2βŠƒβ‹― are compact, then limπ‘–β†’βˆžπΆπ‘(β‹…),𝑀𝐾𝑖=𝐢𝑝(β‹…),π‘€ξƒ©βˆžξ™π‘–=1𝐾𝑖ξƒͺ.(3.15)

Note that the assertion (v) above is not true in general for noncompact sets [9].(vi) If 𝐸1βŠ‚πΈ2βŠ‚β‹― are subsets of ℝ𝑛, then limπ‘–β†’βˆžπΆπ‘(β‹…),𝑀𝐸𝑖=𝐢𝑝(β‹…),π‘€ξƒ©βˆžξšπ‘–=1𝐸𝑖ξƒͺ.(3.16)(vii) If πΈπ‘–βŠ‚β„π‘› for 𝑖=1,2,…, then 𝐢𝑝(β‹…),π‘€ξƒ©βˆžξšπ‘–=1𝐸𝑖ξƒͺβ‰€βˆžξ“π‘–=1𝐢𝑝(β‹…),𝑀𝐸𝑖.(3.17)

For the proof of these properties see [8, 10]. Hence the Sobolev 𝐢𝑝(β‹…),𝑀 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 𝐸↦𝐢𝑝(β‹…),𝑀(𝐸), πΈβŠ‚β„π‘›, is a Choquet capacity. In particular, all Suslin sets πΈβŠ‚β„π‘› are capacitable, that is, 𝐢𝑝(β‹…),𝑀(𝐸)=infπ‘ˆπΈβŠ‚π‘ˆopen𝐢𝑝(β‹…),𝑀(π‘ˆ)=supπΎπΎβŠ‚πΈcompact𝐢𝑝(β‹…),𝑀(𝐾).(3.18)

Lemma 3.5. Let 𝑀(π‘₯)β‰₯1 for π‘₯βˆˆβ„π‘›. Then every measurable set πΈβŠ‚β„π‘› satisfies |𝐸|≀𝐢𝑝(β‹…),𝑀(𝐸).

Proof. If π‘“βˆˆπ‘†π‘(β‹…),𝑀(𝐸), then there is an open set πΈβŠ‚π‘ˆ such that 𝑓β‰₯1 in π‘ˆ and hence ||𝐸||≀||π‘ˆ||β‰€ξ€œβ„π‘›||||𝑓(π‘₯)𝑝(π‘₯)ξ€œπ‘€(π‘₯)𝑑π‘₯≀ℝ𝑛||||𝑓(π‘₯)𝑝(π‘₯)+||||βˆ‡π‘“(π‘₯)𝑝(π‘₯)𝑀(π‘₯)𝑑π‘₯.(3.19) We obtain the claim by taking the infimum on 𝑆𝑝(β‹…),𝑀(𝐸).

Definition 3.6 (Bessel Capacity). Let πΈβŠ‚β„π‘›, 𝛼>0. Define that the (𝛼,𝑝(β‹…),𝑀)-Bessel capacity in ℒ𝛼,𝑝(β‹…)(ℝ𝑛,𝑀) is the number 𝐡𝛼,𝑝(β‹…),𝑀(𝐸)=infπœšπ‘(β‹…),𝑀(𝑓),(3.20) where the infimum is taken over all π‘“βˆˆπΏπ‘(β‹…)(ℝ𝑛,𝑀) such that π‘”π›Όβˆ—π‘“β‰₯1 on 𝐸. Since 𝑔𝛼 is nonnegative we can assume that 𝑓β‰₯0.

Theorem 3.7. 𝐡𝛼,𝑝(β‹…),𝑀 is an outer capacity defined on all subsets of ℝ𝑛.

Proof. It is known that(i)𝐡𝛼,𝑝(β‹…),𝑀(βˆ…)=0; (ii)if 𝐸1βŠ‚πΈ2, then 𝐡𝛼,𝑝(β‹…),𝑀(𝐸1)≀𝐡𝛼,𝑝(β‹…),𝑀(𝐸2);(iii)if πΈπ‘–βŠ‚β„π‘› for 𝑖=1,2,…, then 𝐡𝛼,𝑝(β‹…),π‘€ξƒ©βˆžξšπ‘–=1𝐸𝑖ξƒͺβ‰€βˆžξ“π‘–=1𝐡𝛼,𝑝(β‹…),𝑀𝐸𝑖(3.21)by [12, Lemma  4.1]. We will show that 𝐡𝛼,𝑝(β‹…),𝑀(𝐸)=infπΊπΈβŠ‚πΊopen𝐡𝛼,𝑝(β‹…),𝑀(𝐺).(3.22) for any πΈβŠ‚β„π‘›. Let πΈβŠ‚β„π‘› be arbitrary. Obviously 𝐡𝛼,𝑝(β‹…),𝑀(𝐸)≀infπΊπΈβŠ‚πΊopen𝐡𝛼,𝑝(β‹…),𝑀(𝐺). We assume that 𝐡𝛼,𝑝(β‹…),𝑀(𝐸)<∞. If 0<πœ€<1 there must exist a test function (measurable and nonnegative) for 𝐡𝛼,𝑝(β‹…),𝑀(𝐸), call it 𝑓, such that π‘”π›Όβˆ—π‘“β‰₯1 on 𝐸, and πœšπ‘(β‹…),𝑀(𝑓)<𝐡𝛼,𝑝(β‹…),𝑀(𝐸)+πœ€.(3.23) Let 𝐺={π‘₯βˆˆβ„π‘›βˆΆπ‘”π›Όβˆ—π‘“>1βˆ’πœ€}. Since π‘”π›Όβˆ—π‘“ is lower semicontinuous in π‘₯, 𝐺 is an open set and since π‘”π›Όβˆ—π‘“>1βˆ’πœ€ on 𝐸, πΊβŠƒπΈ. Therefore (1βˆ’πœ€)βˆ’1𝑓 is a test function for 𝐡𝛼,𝑝(β‹…),𝑀(𝐺) and we have 𝐡𝛼,𝑝(β‹…),𝑀(𝐺)β‰€πœšπ‘(β‹…),𝑀𝑓1βˆ’πœ€β‰€(1βˆ’πœ€)βˆ’π‘+πœšπ‘(β‹…),𝑀(𝑓)<(1βˆ’πœ€)βˆ’π‘+𝐡𝛼,𝑝(β‹…),𝑀(𝐸)+πœ€.(3.24) This proves the theorem as πœ€β†’0+.

Now we give relationship between the capacities 𝐡𝛼,𝑝(β‹…),𝑀 and 𝐢𝑝(β‹…),𝑀 [12].

Lemma 3.8. Let 𝑝(β‹…)βˆˆπ’«(ℝ𝑛) and πΈβŠ‚β„π‘›. Then 𝐡1,𝑝(β‹…),𝑀𝐢(𝐸)≀𝑐max𝑝(β‹…),𝑀(𝐸)π‘βˆ’/𝑝+,𝐢𝑝(β‹…),𝑀(𝐸)𝑝+/π‘βˆ’ξ‚‡,𝐢𝑝(β‹…),𝑀𝐡(𝐸)≀𝐢max1,𝑝(β‹…),𝑀(𝐸)π‘βˆ’/𝑝+,𝐡1,𝑝(β‹…),𝑀(𝐸)𝑝+/π‘βˆ’ξ‚‡.(3.25) Here 𝑐 and 𝐢 are positive constants independent of 𝐸.

Proposition 3.9. Let 𝑝(β‹…)βˆˆπ’«(ℝ𝑛).(i)If π‘“βˆˆπ‘Š1,𝑝(β‹…)(ℝ𝑛,𝑀), then π‘€π‘“βˆˆπ‘Š1,𝑝(β‹…)(ℝ𝑛,𝑀) and |βˆ‡π‘€π‘“(π‘₯)|≀𝑀|βˆ‡π‘“(π‘₯)| for almost everywhere in ℝ𝑛.(ii)Let 1≀𝑠<∞. Then 𝑠𝑝(β‹…)βˆˆπ’«(ℝ𝑛) and there exists a constant 𝐢>0 such that the inequality ‖𝑀𝑓‖1,𝑠𝑝(β‹…),𝑀≀𝐢‖𝑓‖1,𝑠𝑝(β‹…),𝑀(3.26) holds for all π‘“βˆˆπ‘Š1,𝑠𝑝(β‹…)(ℝ𝑛,𝑀).

Proof. (i) By Proposition 2.1 we have 𝐿𝑝(β‹…)(ℝ𝑛,𝑀)β†ͺ𝐿𝑝(β‹…)loc(ℝ𝑛,𝑀)β†ͺ𝐿1loc(ℝ𝑛) and π‘Š1,𝑝(β‹…)(ℝ𝑛,𝑀)β†ͺπ‘Š1,𝑝(β‹…)loc(ℝ𝑛,𝑀)β†ͺπ‘Š1,1loc(ℝ𝑛). Sinceβ€‰β€‰π‘“βˆˆπ‘Š1,1loc(ℝ𝑛), then we have |βˆ‡π‘€π‘“(π‘₯)|≀𝑀|βˆ‡π‘“(π‘₯)| for almost everywhere in ℝ𝑛 by [24]. Since  𝑓, |βˆ‡π‘“|βˆˆπΏπ‘(β‹…)(ℝ𝑛,𝑀) and 𝑝(β‹…)βˆˆπ’«(ℝ𝑛), then 𝑀𝑓,|βˆ‡π‘€π‘“|βˆˆπΏπ‘(β‹…)(ℝ𝑛,𝑀). Hence π‘€π‘“βˆˆπ‘Š1,𝑝(β‹…)(ℝ𝑛,𝑀).
(ii) Let π‘“βˆˆπΏπ‘ π‘(β‹…)(ℝ𝑛,𝑀). By using definition of ‖⋅‖𝑝(β‹…),𝑀, we have ‖𝑓‖𝑠𝑝(β‹…),𝑀=β€–β€–||𝑓||𝑠‖‖1/𝑠𝑝(β‹…),𝑀(3.27) and |𝑓|π‘ βˆˆπΏπ‘(β‹…)(ℝ𝑛,𝑀). Therefore we have ‖𝑀𝑓‖𝑠𝑝(β‹…),𝑀=β€–(𝑀𝑓)𝑠‖1/𝑠𝑝(β‹…),𝑀≀‖‖𝑀||𝑓||𝑠‖‖1/𝑠𝑝(β‹…),𝑀‖‖||𝑓||≀𝑐𝑠‖‖1/𝑠𝑝(β‹…),𝑀=𝑐‖𝑓‖𝑠𝑝(β‹…),𝑀(3.28) and 𝑠𝑝(β‹…)βˆˆπ’«(ℝ𝑛). Since π‘“βˆˆπ‘Š1,𝑠𝑝(β‹…)(ℝ𝑛,𝑀), then 𝑓,|βˆ‡π‘“|βˆˆπΏπ‘ π‘(β‹…)(ℝ𝑛,𝑀). Hence we write ‖𝑀𝑓‖1,𝑠𝑝(β‹…),𝑀=β€–(𝑀𝑓)𝑠‖1/𝑠𝑝(β‹…),𝑀+β€–β€–||||βˆ‡π‘€π‘“π‘ β€–β€–1/𝑠𝑝(β‹…),𝑀≀‖(𝑀𝑓)𝑠‖1/𝑠𝑝(β‹…),𝑀+‖‖𝑀||||ξ€Έβˆ‡π‘“π‘ β€–β€–1/𝑠𝑝(β‹…),𝑀≀‖‖𝑀||𝑓||𝑠‖‖1/𝑠𝑝(β‹…),𝑀+‖‖𝑀||||βˆ‡π‘“π‘ ξ€Έβ€–β€–1/𝑠𝑝(β‹…),𝑀≀𝐢1β€–β€–||𝑓||𝑠‖‖1/𝑠𝑝(β‹…),𝑀+𝐢2β€–β€–||||βˆ‡π‘“π‘ β€–β€–1/𝑠𝑝(β‹…),𝑀.(3.29) by (3.28). If we set 𝐢=max{𝐢1,𝐢2}, then ‖𝑀𝑓‖1,𝑠𝑝(β‹…),𝑀≀𝐢‖𝑓‖1,𝑠𝑝(β‹…),𝑀.(3.30) This completes the proof.

Proposition 3.10. Let 𝑝(β‹…)βˆˆπ’«(ℝ𝑛). Then for every πœ†>0 and every π‘“βˆˆπ‘Š1,𝑝(β‹…)(ℝ𝑛,𝑀) we have 𝐢𝑝(β‹…),𝑀({π‘₯βˆˆβ„π‘›ξƒ―β€–β€–β€–π‘“βˆΆπ‘€π‘“(π‘₯)>πœ†})≀𝑐maxπœ†β€–β€–β€–π‘+1,𝑝(β‹…),𝑀,β€–β€–β€–π‘“πœ†β€–β€–β€–π‘βˆ’1,𝑝(β‹…),𝑀.(3.31)

Proof. Since 𝑀𝑓 is lower semicontinuous, the set {π‘₯βˆˆβ„π‘›βˆΆπ‘€π‘“(π‘₯)>πœ†} is open for every πœ†>0. By Proposition 3.9 we can take (𝑀𝑓)/πœ†=𝑀(𝑓/πœ†) as a test function for the capacity. Then we have 𝐢𝑝(β‹…),𝑀({π‘₯βˆˆβ„π‘›βˆΆπ‘€π‘“(π‘₯)>πœ†})β‰€πœš1,𝑝(β‹…),π‘€ξ‚΅π‘€π‘“πœ†ξ‚Άξƒ―β€–β€–β€–π‘€π‘“β‰€maxπœ†β€–β€–β€–π‘+1,𝑝(β‹…),𝑀,β€–β€–β€–π‘€π‘“πœ†β€–β€–β€–π‘βˆ’1,𝑝(β‹…),𝑀‖‖‖𝑓≀𝑐maxπœ†β€–β€–β€–π‘+1,𝑝(β‹…),𝑀,β€–β€–β€–π‘“πœ†β€–β€–β€–π‘βˆ’1,𝑝(β‹…),𝑀.(3.32)

We say that a property holds (𝑝(β‹…),𝑀)-quasi everywhere if it holds except in a set of capacity zero. A function 𝑓 is (𝑝(β‹…),𝑀)-quasicontinuous in ℝ𝑛 if for each πœ€>0 there exists an open set 𝐸 with 𝐢𝑝(β‹…),𝑀(𝐸)<πœ€ such that 𝑓 restricted to ℝ𝑛⧡𝐸 is continuous. The following proof of theorem is quite similar to Theorem 4.7 in [11].

Theorem 3.11. Let 𝑝(β‹…)βˆˆπ’«(ℝ𝑛). If π‘“βˆˆπ‘Š1,𝑝(β‹…)(ℝ𝑛,𝑀), then the limit π‘“βˆ—(π‘₯)=limπ‘Ÿβ†’01||||ξ€œπ΅(π‘₯,π‘Ÿ)𝐡(π‘₯,π‘Ÿ)𝑓(𝑦)𝑑𝑦(3.33) exists (𝑝(β‹…),𝑀)-quasi everywhere in ℝ𝑛. The function π‘“βˆ— is the (𝑝(β‹…),𝑀)-quasicontinuous representative of 𝑓.

Proof. Since the class 𝐢∞0(ℝ𝑛) is dense in π‘Š1,𝑝(β‹…)(ℝ𝑛,𝑀) by Lemma 3.1, then we can choose a sequence (𝑓𝑖) such that β€–β€–π‘“βˆ’π‘“π‘–β€–β€–1,𝑝(β‹…),𝑀≀2βˆ’2𝑖.(3.34) For 𝑖=1,2,… we denote 𝐴𝑖=ξ€½π‘₯βˆˆβ„π‘›ξ€·βˆΆπ‘€π‘“βˆ’π‘“π‘–ξ€Έ(π‘₯)>2βˆ’π‘–ξ€Ύ,𝐡𝑖=βˆžξšπ‘—=𝑖𝐴𝑗,𝐸=βˆžξ™π‘—=1𝐡𝑗.(3.35) By using Proposition 3.10 and the subadditivity of 𝐢𝑝(β‹…),𝑀 we have 𝐢𝑝(β‹…),π‘€ξ€·π΄π‘–ξ€ΈβŽ§βŽͺ⎨βŽͺβŽ©β€–β€–β€–β€–π‘€ξ€·β‰€π‘maxπ‘“βˆ’π‘“π‘–ξ€Έ2βˆ’π‘–β€–β€–β€–β€–π‘+1,𝑝(β‹…),𝑀,β€–β€–β€–β€–π‘€ξ€·π‘“βˆ’π‘“π‘–ξ€Έ2βˆ’π‘–β€–β€–β€–β€–π‘βˆ’1,𝑝(β‹…),π‘€βŽ«βŽͺ⎬βŽͺβŽ­ξƒ―ξ‚΅1=𝑐max2βˆ’π‘–ξ‚Άπ‘+β€–β€–π‘€ξ€·π‘“βˆ’π‘“π‘–ξ€Έβ€–β€–π‘+1,𝑝(β‹…),𝑀,ξ‚΅12βˆ’π‘–ξ‚Άπ‘βˆ’β€–β€–π‘€ξ€·π‘“βˆ’π‘“π‘–ξ€Έβ€–β€–π‘βˆ’1,𝑝(β‹…),𝑀1≀𝑐max2βˆ’π‘–ξ‚Άπ‘+ξ€·2βˆ’2𝑖𝑝+,ξ‚΅12βˆ’π‘–ξ‚Άπ‘βˆ’ξ€·2βˆ’2π‘–ξ€Έπ‘βˆ’ξƒ°β‰€π‘2βˆ’π‘–,(3.36)𝐢𝑝(β‹…),𝑀(𝐡𝑖)≀𝑐21βˆ’π‘– and 𝐢𝑝(β‹…),𝑀(𝐸)=0. If we follow the proof of Theorem  4.7 in [11], then this proves the theorem.

Corollary 3.12. Let 𝑝(β‹…)βˆˆπ’«(ℝ𝑛). If π‘“βˆˆπ‘Š1,𝑝(β‹…)(ℝ𝑛,𝑀) and 𝑓 is quasicontinuous, then we have 𝑓(π‘₯)=limπ‘Ÿβ†’01||||ξ€œπ΅(π‘₯,π‘Ÿ)𝐡(π‘₯,π‘Ÿ)𝑓(𝑦)𝑑𝑦(3.37)(𝑝(β‹…),𝑀)-quasi everywhere in ℝ𝑛.

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 𝑝+<∞ and πΈβŠ‚β„π‘›. If π‘’βˆˆπ‘Š1,𝑝(β‹…)(ℝ𝑛,𝑀) is a nonnegative (𝑝(β‹…),𝑀)-quasicontinuous function such that 𝑒β‰₯1 on 𝐸. Then for every πœ€>0 there exists a function β„Žβˆˆπ‘†π‘(β‹…),𝑀(𝐸) such that 𝜚1,𝑝(β‹…),𝑀(π‘’βˆ’β„Ž)<πœ€.

Proof. Let 0<𝛿<1, and let π‘‰βŠ‚β„π‘› be an open set such that 𝑒 is continuous in ℝ𝑛⧡𝑉 and 𝐢𝑝(β‹…),𝑀(𝑉)<𝛿. By definition of 𝐢𝑝(β‹…),𝑀 there exists a π‘£βˆˆπ‘†π‘(β‹…),𝑀(𝐸) such that 𝜚1,𝑝(β‹…),𝑀(𝑣)<𝛿. If we set β„Ž=(1+𝛿)𝑒+|𝑣|, then it is easy to show that β„Žβˆˆπ‘Š1,𝑝(β‹…)(ℝ𝑛,𝑀) by [10, Theorem  2.2]. Since the function 𝑒 is continuous and the set 𝑉 is open, then the set 𝐺={π‘₯βˆˆβ„π‘›β§΅π‘‰βˆΆπ‘’(π‘₯)>1}βˆͺ𝑉(3.38) is open, contains 𝐸, and β„Žβ‰₯1 on 𝐺, thus β„Žβˆˆπ‘†π‘(β‹…),𝑀(𝐸). It is known that for 1≀𝑝(β‹…)≀𝑝+<∞ and π‘Ž,𝑏β‰₯0, (π‘Ž+𝑏)𝑝(β‹…)≀2𝑝+βˆ’1(π‘Žπ‘(β‹…)+𝑏𝑝(β‹…)) and |βˆ‡|𝑣||=|βˆ‡π‘£|. Hence we obtain ||𝑣|+𝛿𝑒|𝑝(β‹…)≀2𝑝+βˆ’1(|𝑣|𝑝(β‹…)+|𝛿𝑒|𝑝(β‹…)) and 𝜚1,𝑝(β‹…),𝑀(ξ€œπ‘’βˆ’β„Ž)=ℝ𝑛|||||𝑣|+𝛿𝑒𝑝(π‘₯)+||||βˆ‡(|𝑣|+𝛿𝑒)𝑝(π‘₯)𝑀(π‘₯)𝑑π‘₯≀2𝑝+βˆ’1ξ€œβ„π‘›ξ‚€|𝑣|𝑝(β‹…)+||||𝛿𝑒𝑝(β‹…)+||βˆ‡||||||𝑣(π‘₯)𝑝(π‘₯)+||||π›Ώβˆ‡π‘’(π‘₯)𝑝(π‘₯)𝑀(π‘₯)𝑑π‘₯≀2𝑝+βˆ’1ξ€·πœš1,𝑝(β‹…),𝑀(𝑣)+π›Ώπ‘βˆ’πœš1,𝑝(β‹…),𝑀(𝑒)<2𝑝+βˆ’1𝛿+π›Ώπ‘βˆ’πœš1,𝑝(β‹…),𝑀.(𝑒)(3.39) This completes the proof as 𝛿→0.

Theorem 3.14. Let 𝑝+<∞. If π‘’βˆˆπ‘Š1,𝑝(β‹…)(ℝ𝑛,𝑀) is a (𝑝(β‹…),𝑀)-quasicontinuous function and πœ†>0, then 𝐢𝑝(β‹…),𝑀π‘₯βˆˆβ„π‘›βˆΆ||||β‰€ξ€œπ‘’(π‘₯)>πœ†ξ€Ύξ€Έβ„π‘›ξ‚΅|||𝑒(π‘₯)πœ†|||𝑝(π‘₯)+|||βˆ‡π‘’(π‘₯)πœ†|||𝑝(π‘₯)𝑀(π‘₯)𝑑π‘₯.(3.40)

Proof. By [10, Theorem  2.2], |𝑒|βˆˆπ‘Š1,𝑝(β‹…)(ℝ𝑛,𝑀) and |βˆ‡|𝑒||=|βˆ‡π‘’|. By Lemma 3.13, there is a sequence β„Žπ‘—βˆˆπ‘†π‘(β‹…),𝑀({π‘₯βˆˆβ„π‘›βˆΆ|𝑒(π‘₯)|/πœ†>1}) such that 𝜚1,𝑝(β‹…),𝑀|𝑒|πœ†βˆ’β„Žπ‘—ξ‚ΆβŸΆ0asπ‘—βŸΆβˆž.(3.41) Hence we have by [10, Lemma  2.6] that 𝜚1,𝑝(β‹…),π‘€ξ€·β„Žπ‘—ξ€ΈβŸΆπœš1,𝑝(β‹…),𝑀|𝑒|πœ†ξ‚Άasπ‘—βŸΆβˆž.(3.42) By definition of 𝐢𝑝(β‹…),𝑀, we write 𝐢𝑝(β‹…),𝑀π‘₯βˆˆβ„π‘›βˆΆ||𝑒||(π‘₯)>πœ†ξ€Ύξ€Έβ‰€πœš1,𝑝(β‹…),π‘€ξ€·β„Žπ‘—ξ€Έ.(3.43) Therefore 𝐢𝑝(β‹…),𝑀π‘₯βˆˆβ„π‘›βˆΆ||||𝑒(π‘₯)>πœ†ξ€Ύξ€Έβ‰€πœš1,𝑝(β‹…),𝑀|𝑒|πœ†ξ‚Άasπ‘—βŸΆβˆž.(3.44)

Proposition 3.15. Let 𝑝(β‹…)βˆˆπ’«(ℝ𝑛). If π‘’βˆˆπ‘Š1,𝑝(β‹…)(ℝ𝑛,𝑀), then there is a 𝐢>0 such that 𝐡1,𝑝(β‹…),𝑀({π‘₯βˆˆβ„π‘›ξ‚»β€–β€–β€–π‘’βˆΆπ‘€π‘’(π‘₯)β‰₯πœ†})≀𝐢maxπœ†β€–β€–β€–π‘+1,𝑝(β‹…),𝑀,β€–β€–β€–π‘’πœ†β€–β€–β€–π‘βˆ’1,𝑝(β‹…),𝑀.(3.45)

Proof. For π‘Ÿ>0, we take β„Ž=|𝐡(0,π‘Ÿ)|βˆ’1πœ’π΅(0,π‘Ÿ). Choose π‘“βˆˆπΏπ‘(β‹…)(ℝ𝑛,𝑀) such that 𝑒=𝑔1βˆ—π‘“ and ‖𝑓‖𝑝(β‹…),π‘€β‰ˆβ€–π‘’β€–1,𝑝(β‹…),𝑀 by Theorem 3.2. Then 1||||ξ€œπ΅(π‘₯,π‘Ÿ)𝐡(π‘₯,π‘Ÿ)||||1𝑒(𝑦)𝑑𝑦=||||ξ€œπ΅(0,π‘Ÿ)𝐡(0,π‘Ÿ)πœ’π΅(0,π‘Ÿ)(||||𝑔π‘₯βˆ’π‘¦)𝑒(𝑦)𝑑𝑦=(β„Žβˆ—|𝑒|)(π‘₯)β‰€β„Žβˆ—1βˆ—||𝑓||=𝑔(π‘₯)1βˆ—ξ€·||𝑓||ξ€·π‘”β„Žβˆ—ξ€Έξ€Έ(π‘₯)≀1ξ€Έβˆ—π‘€π‘“(π‘₯)(3.46) and 𝑀𝑒(π‘₯)≀(𝑔1βˆ—π‘€π‘“)(π‘₯). Also it is known that if 𝐸1βŠ‚πΈ2, then 𝐡1,𝑝(β‹…),𝑀(𝐸1)≀𝐡1,𝑝(β‹…),𝑀(𝐸2) by [12, Lemma  4.1]. Therefore we have 𝐡1,𝑝(β‹…),𝑀({π‘₯βˆˆβ„π‘›βˆΆπ‘€π‘’(π‘₯)β‰₯πœ†})≀𝐡1,𝑝(β‹…),𝑀π‘₯βˆˆβ„π‘›βˆΆξ€·π‘”1ξ€Έβˆ—π‘€π‘“(π‘₯)β‰₯πœ†ξ€Ύξ€Έβ‰€πœšπ‘(β‹…),π‘€ξ‚΅π‘€π‘“πœ†ξ‚Άξƒ―β€–β€–β€–π‘€π‘“β‰€maxπœ†β€–β€–β€–π‘+𝑝(β‹…),𝑀,β€–β€–β€–π‘€π‘“πœ†β€–β€–β€–π‘βˆ’π‘(β‹…),𝑀‖‖‖𝑓≀𝑐maxπœ†β€–β€–β€–π‘+𝑝(β‹…),𝑀,β€–β€–β€–π‘“πœ†β€–β€–β€–π‘βˆ’π‘(β‹…),𝑀‖‖‖𝑒≀𝐢maxπœ†β€–β€–β€–π‘+1,𝑝(β‹…),𝑀,β€–β€–β€–π‘’πœ†β€–β€–β€–π‘βˆ’1,𝑝(β‹…),𝑀.(3.47)

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

Theorem 3.16. Let 𝑝(β‹…)βˆˆπ’«(ℝ𝑛). Ifβ€‰β€‰π‘’βˆˆπ‘Š1,𝑝(β‹…)(ℝ𝑛,𝑀) and 𝑒 is quasicontinuous, then the limit 𝑒(π‘₯)=limπ‘Ÿβ†’01||||ξ€œπ΅(π‘₯,π‘Ÿ)𝐡(π‘₯,π‘Ÿ)𝑒(𝑦)𝑑𝑦(3.48) exists (1,𝑝(β‹…),𝑀)-quasi everywhere in ℝ𝑛.

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

Proposition 3.17. Let 𝑝(β‹…)βˆˆπ’«(ℝ𝑛). Every π‘’βˆˆβ„’1,𝑝(β‹…)(ℝ𝑛,𝑀) is quasicontinuous. That is, for every πœ€>0, there exists a set πΉβŠ‚β„π‘›, 𝐡1,𝑝(β‹…),𝑀(𝐹)β‰€πœ€, so that 𝑒 restricted to ℝ𝑛⧡𝐹 is continuous.

Proposition 3.18. Let 1<π‘βˆ’β‰€π‘+<∞. Then for all π‘“βˆˆπΏπ‘(β‹…)(ℝ𝑛,𝑀) and 0<πœ†<∞ we have 𝐡𝛼,𝑝(β‹…),𝑀π‘₯βˆˆβ„π‘›βˆΆξ€·π‘”π›Όξ€Έξƒ―β€–β€–β€–π‘“βˆ—π‘“(π‘₯)β‰₯πœ†ξ€Ύξ€Έβ‰€maxπœ†β€–β€–β€–π‘+𝑝(β‹…),𝑀,β€–β€–β€–π‘“πœ†β€–β€–β€–π‘βˆ’π‘(β‹…),𝑀.(3.49)

Proof. We first note that by definition of 𝐡1,𝑝(β‹…),𝑀-capacity, πœ†βˆ’1𝑓 is a test function for the Bessel capacity. Hence 𝐡𝛼,𝑝(β‹…),𝑀π‘₯βˆˆβ„π‘›βˆΆξ€·π‘”π›Όξ€Έβˆ—π‘“(π‘₯)β‰₯πœ†ξ€Ύξ€Έβ‰€πœšπ‘(β‹…),π‘€ξ‚΅π‘“πœ†ξ‚Άξƒ―β€–β€–β€–π‘“β‰€maxπœ†β€–β€–β€–π‘+𝑝(β‹…),𝑀,β€–β€–β€–π‘“πœ†β€–β€–β€–π‘βˆ’π‘(β‹…),𝑀.(3.50)

Proposition 3.19. Let 1<π‘βˆ’β‰€π‘+<∞. If π‘“βˆˆπΏπ‘(β‹…)(ℝ𝑛,𝑀) and 𝐸=π‘₯βˆˆβ„π‘›βˆΆξ€·π‘”π›Όξ€Έξ€Ύβˆ—π‘“(π‘₯)=∞,(3.51) then 𝐡𝛼,𝑝(β‹…),𝑀(𝐸)=0.

Proof. By Proposition 3.18, we write 𝐡𝛼,𝑝(β‹…),𝑀(𝐸)=0 as πœ†β†’βˆž.

Proposition 3.20. Let 1<π‘βˆ’β‰€π‘+<∞. If π‘“βˆˆπΏπ‘(β‹…)(ℝ𝑛,𝑀), then limπ‘Ÿβ†’01||||ξ€œπ΅(π‘₯,π‘Ÿ)𝐡(π‘₯,π‘Ÿ)𝑔𝛼(ξ€·π‘”βˆ—π‘“π‘¦)𝑑𝑦=𝛼(βˆ—π‘“π‘₯)(3.52) for 𝐡𝛼,𝑝(β‹…),𝑀-q.e. π‘₯βˆˆβ„π‘›.

Proof. Let πœ’ be the characteristic function for the unit ball 𝐡(0,1), and define for π‘Ÿ>0, πœ’π‘Ÿ(π‘₯)=(1/|𝐡(0,1)|)πœ’(π‘₯/π‘Ÿ), π‘₯βˆˆβ„π‘›. Then 1||||ξ€œπ΅(π‘₯,π‘Ÿ)𝐡(π‘₯,π‘Ÿ)𝑔𝛼(βˆ—π‘“π‘¦)𝑑𝑦=πœ’π‘Ÿβˆ—ξ€·π‘”π›Όξ€Έ(ξ€·πœ’βˆ—π‘“π‘₯)=π‘Ÿβˆ—π‘”π›Όξ€Έ=ξ€œβˆ—π‘“(π‘₯)𝐡(π‘₯,π‘Ÿ)πœ’π‘Ÿβˆ—π‘”π›Ό(𝑦)𝑓(π‘₯βˆ’π‘¦)𝑑𝑦.(3.53) As π‘Ÿβ†’0, πœ’π‘Ÿβˆ—π‘”π›Ό(𝑦)→𝑔𝛼(𝑦) for every π‘¦βˆˆβ„π‘›. This implies that, for fixed π‘₯βˆˆβ„π‘›, πœ’π‘Ÿβˆ—π‘”π›Ό(𝑦)𝑓(π‘₯βˆ’π‘¦)→𝑔𝛼(𝑦)𝑓(π‘₯βˆ’π‘¦) for a.e. π‘¦βˆˆβ„π‘›. It was shown that πœ’π‘Ÿβˆ—π‘”π›Ό