Journal of Function Spaces

Journal of Function Spaces / 2012 / Article

Research Article | Open Access

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

Ismail Aydin, "Weighted Variable Sobolev Spaces and Capacity", Journal of Function Spaces, vol. 2012, Article ID 132690, 17 pages, 2012. 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(𝑦)≀𝐢𝑔𝛼(𝑦) for 0<π‘Ÿβ‰€1 and π‘¦βˆˆβ„π‘› [26, page 161]. By Proposition 3.19, the integrand in (3.53) is dominated by a constant times 𝑔𝛼(𝑦)|𝑓(π‘₯βˆ’π‘¦)|, which is 𝐿1(ℝ𝑛) function for 𝐡𝛼,𝑝(β‹…),𝑀-q.e. π‘₯βˆˆβ„π‘›. If we use the Lebesgue’s dominated convergence theorem, then the proof is completed.

Acknowledgments

The author would like to thank Petteri Harjulehto for his significant suggestions, comments, and corrections to the original version of this paper. He is also grateful to Stefan Samko for his attention to a gap in the proof of Lemma 3.1. He also thanks the referee for carefully reading the paper and useful comments.

References

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