Weighted Variable Sobolev Spaces and Capacity

Abstract

We define weighted variable Sobolev capacity and discuss properties of capacity in the space 𝑊^1,𝑝(⋅)(ℝ^𝑛,𝑤). 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 𝑊^1,𝑝(⋅)(ℝ^𝑛,𝑤). Also it is shown the relation between the Sobolev capacity and Bessel capacity.

1. Introduction

In 1991 Kováčik and Rákosník [1] introduced the variable exponent Lebesgue space 𝐿^𝑝(⋅)(ℝ^𝑛) and Sobolev space 𝑊^{𝑘,𝑝(⋅)}(ℝ^𝑛) in higher dimensional Euclidean spaces. The spaces 𝐿^𝑝(⋅)(ℝ^𝑛) and 𝐿^𝑝(ℝ^𝑛) have many common properties. A crucial difference between 𝐿^𝑝(⋅)(ℝ^𝑛) and the classical Lebesgue spaces 𝐿^𝑝(ℝ^𝑛) is that 𝐿^𝑝(⋅)(ℝ^𝑛) 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 𝐿^𝑝(⋅)(ℝ^𝑛) for a long time. It was first proved by Diening [2] over bounded domains, under the assumption that 𝑝(⋅) is locally log-Hölder continuous, that is, ||𝑝(𝑥)−𝑝(𝑦)||≤𝐶−ln||𝑥−𝑦||,𝑥,𝑦∈Ω,||𝑥−𝑦||≤12.(1.1) He later extended the result to unbounded domains by supposing, in addition, that the exponent 𝑝(⋅) 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 [6], Evans and Gariepy [7], and Heinonen et al. [8]. Also Kilpeläinen [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 𝐴_𝑝-class. Variable Sobolev capacity was introduced in the spaces 𝑊^1,𝑝(⋅)(ℝ^𝑛) 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 ℝ^𝑛, 𝑛≥2. Throughout this paper all sets and functions are Lebesgue measurable. The Lebesgue measure and the characteristic function of a subset 𝐴⊂ℝ^𝑛 will be denoted by 𝜇(𝐴)=|𝐴| and 𝜒_𝐴, respectively. The space 𝐿^1loc(ℝ^𝑛) consists of all (classes of) measurable functions 𝑓 on ℝ^𝑛 such that 𝑓𝜒_𝐾∈𝐿¹(ℝ^𝑛) for any compact subset 𝐾⊂ℝ^𝑛. It is a topological vector space with the family of seminorms 𝑓↦‖𝑓𝜒_𝐾‖_𝐿1. A Banach function space (shortly BF-space) on ℝ^𝑛 is a Banach space (𝐵,‖⋅‖_𝐵) of measurable functions which is continuously embedded into 𝐿^1loc(ℝ^𝑛), that is, for any compact subset 𝐾⊂ℝ^𝑛 there exists some constant 𝐶_𝐾>0 such that ‖𝑓𝜒_𝐾‖_𝐿1≤𝐶_𝐾‖𝑓‖_𝐵 for all 𝑓∈𝐵. We denote it by 𝐵↪𝐿^1loc(ℝ^𝑛). The class 𝐶^∞0(ℝ^𝑛) is defined as set of infinitely differentiable functions with compact support in ℝ^𝑛. For a measurable function 𝑝∶ℝ^𝑛→[1,∞) (called a variable exponent on ℝ^𝑛), we put𝑝⁻=essinf_{𝑥∈ℝ𝑛}𝑝(𝑥),𝑝⁺=esssup_{𝑥∈ℝ𝑛}𝑝(𝑥).(2.1) For every measurable functions 𝑓 on ℝ^𝑛 we define the function𝜚_𝑝(⋅)(𝑓)=_ℝ𝑛||𝑓(𝑥)||^𝑝(𝑥)𝑑𝑥.(2.2) The function 𝜚_𝑝(⋅) is convex modular; that is, 𝜚_𝑝(⋅)(𝑓)≥0, 𝜚_𝑝(⋅)(𝑓)=0 if and only if 𝑓=0, 𝜚_𝑝(⋅)(−𝑓)=𝜚_𝑝(⋅)(𝑓) and 𝜚_𝑝(⋅) is convex. The variable exponent Lebesgue spaces (or generalized Lebesgue spaces) 𝐿^𝑝(⋅)(ℝ^𝑛) is defined as the set of all measurable functions 𝑓 on ℝ^𝑛 such that 𝜚_𝑝(⋅)(𝜆𝑓)<∞ for some 𝜆>0, equipped with the Luxemburg norm‖𝑓‖_𝑝(⋅)=inf𝜆>0∶𝜚_𝑝𝑓𝜆≤1.(2.3) If 𝑝⁺<∞, then 𝑓∈𝐿^𝑝(⋅)(ℝ^𝑛) if and only if 𝜚_𝑝(⋅)(𝑓)<∞. The set 𝐿^𝑝(⋅)(ℝ^𝑛) is a Banach space with the norm ‖⋅‖_𝑝(⋅). If 𝑝(𝑥)=𝑝 is a constant function, then the norm ‖⋅‖_𝑝(⋅) coincides with the usual Lebesgue norm ‖⋅‖_𝑝 [1]. In this paper we assume that 𝑝⁺<∞.

A positive, measurable, and locally integrable function 𝑤∶ℝ^𝑛→(0,∞) is called a weight function. The weighted modular is defined by𝜚_{𝑝(⋅),𝑤}(𝑓)=_ℝ𝑛||𝑓(𝑥)||^𝑝(𝑥)𝑤(𝑥)𝑑𝑥.(2.4) The weighted variable exponent Lebesgue space 𝐿^𝑝(⋅)(ℝ^𝑛,𝑤) consists of all measurable functions 𝑓 on ℝ^𝑛 for which ‖𝑓‖_{𝑝(⋅),𝑤}=‖𝑓𝑤^1/𝑝(⋅)‖_𝑝(⋅)<∞. The relations between the modular 𝜚_{𝑝(⋅),𝑤}(⋅) and ‖⋅‖_{𝑝(⋅),𝑤} as follows:minm𝜚in^{𝑝(⋅),𝑤}(‖𝑓𝑓)‖^1/𝑝−,_{𝑝+𝑝(⋅),𝑤}𝜚_{𝑝(⋅),𝑤}(𝑓)^1/𝑝+≤‖𝑓‖^{𝑝(⋅),𝑤}≤max𝜚_{𝑝(⋅),𝑤}(𝑓)^1/𝑝−,𝜚_{𝑝(⋅),𝑤}(𝑓)^1/𝑝+,‖𝑓‖^𝑝−_{𝑝(⋅),𝑤}≤𝜚^{𝑝(⋅),𝑤}(𝑓)≤max‖𝑓‖^𝑝+_{𝑝(⋅),𝑤},‖𝑓‖^𝑝−_{𝑝(⋅),𝑤},(2.5) see [13–15]. Moreover, if 0<𝐶≤𝑤, then we have 𝐿^𝑝(⋅)(ℝ^𝑛,𝑤)↪𝐿^𝑝(⋅)(ℝ^𝑛), since one easily sees that𝐶_ℝ𝑛||𝑓(𝑥)||^𝑝(𝑥)𝑑𝑥≤_ℝ𝑛||𝑓(𝑥)||^𝑝(𝑥)𝑤(𝑥)𝑑𝑥(2.6) and 𝐶‖𝑓‖_𝑝(⋅)≤‖𝑓‖_{𝑝(⋅),𝑤}.

The Schwartz class 𝑆=𝑆(ℝ^𝑛) consists of all infinitely differentiable and rapidly decreasing functions in ℝ^𝑛. Then 𝑓 and any derivative 𝐷^𝛽𝑓 die out faster than reciprocal of any polynomial at infinity. That is, 𝑓∈𝑆 if and only if for any 𝛽 and 𝑘>0 there is a constant 𝐶=𝐶(𝛽,𝑘) such that||𝐷^𝛽𝑓(𝑥)||≤𝐶(1+|𝑥|)^𝑘.(2.7) In particular, for 𝛽=0,||𝑓(𝑥)||≤𝐶(1+|𝑥|)^𝑘.(2.8) Also it is well known that 𝐶^∞0(ℝ^𝑛)⊂𝑆.

For 𝑥∈ℝ^𝑛 and 𝑟>0 we denote an open ball with center 𝑥 and radius 𝑟 by 𝐵(𝑥,𝑟). For 𝑓∈𝐿^1loc(ℝ^𝑛), we denote the (centered) Hardy-Littlewood maximal operator 𝑀𝑓 of 𝑓 by𝑀𝑓(𝑥)=sup_𝑟>01||𝐵(𝑥,𝑟)||_{𝐵(𝑥,𝑟)}||𝑓(𝑦)||𝑑𝑦,(2.9) where the supremum is taken over all balls 𝐵(𝑥,𝑟).

Let 1≤𝑝<∞. A weight 𝑤 satisfies Muckenhoupt’s 𝐴_𝑝(ℝ^𝑛)=𝐴_𝑝 condition, or 𝑤∈𝐴_𝑝, if there exist positive constants 𝐶 and 𝑐 such that, for every ball 𝐵⊂ℝ^𝑛,||𝐵1||_𝐵𝑤𝑑𝑥1||𝐵||_𝐵𝑤^{−1/(𝑝−1)}𝑑𝑥^𝑝−1≤𝐶,1≤𝑝<∞,(2.10) or||𝐵1||_𝐵𝑤𝑑𝑥esssup_𝐵1𝑤≤𝑐,𝑝=1.(2.11) The infimum over the constants 𝐶 and 𝑐 is called the 𝐴_𝑝 and 𝐴₁, respectively. Also it is known that 𝐴_∞=⋃_1≤𝑝<∞𝐴_𝑝. Let 1<𝑝<∞. Then Muckenhoupt proved that 𝑤∈𝐴_𝑝 if and only if the Hardy-Littlewood maximal operator is bounded on 𝐿^𝑝(ℝ^𝑛,𝑤) [16]. Also Miller showed that the Schwartz class 𝑆 is dense in 𝐿^𝑝(ℝ^𝑛,𝑤) for 1<𝑝<∞ and 𝑤∈𝐴_𝑝 [17, Lemma 2.1].

Hästö and Diening defined the class 𝐴_𝑝(⋅) to consist of those weights 𝑤 for which‖𝑤‖_{𝐴𝑝(⋅)}∶=sup_𝐵∈ℬ||𝐵||^−𝑝𝐵‖𝑤‖_𝐿¹_(𝐵)‖‖‖1𝑤‖‖‖_𝐿^{𝑝′(⋅)/𝑝(⋅)}_(𝐵)<∞,(2.12) where ℬ denotes the set of all balls in ℝ^𝑛, 𝑝_𝐵=((1/|𝐵|)∫_𝐵(1/𝑝(𝑥))𝑑𝑥)⁻¹ and 1/𝑝(⋅)+1/𝑝^(⋅)=1. Note that this class is ordinary Muckenhoupt class 𝐴_𝑝 if 𝑝 is a constant function [13].

We say that 𝑝(⋅) satisfies the local log-Hölder continuity condition if||𝑝(𝑥)−𝑝(𝑦)||≤𝐶log𝑒+1/||𝑥−𝑦||(2.13) for all 𝑥,𝑦∈ℝ^𝑛. If||𝑝(𝑥)−𝑝_∞||≤𝐶log(𝑒+|𝑥|)(2.14) for some 𝑝_∞>1, 𝐶>0 and all 𝑥∈ℝ^𝑛, then we say 𝑝(⋅) satisfies the log-Hölder decay condition (at infinity). We denote by 𝑃^log(ℝ^𝑛) 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 𝑝,𝑞∈𝑃^log(ℝ^𝑛), 1<𝑝⁻≤𝑝⁺<∞ and 1<𝑞⁻≤𝑞⁺<∞. If 𝑞≤𝑝, then there exists a constant 𝐶>0 depending on the characteristics of 𝑝 and 𝑞 such that ‖𝑤‖_{𝐴𝑝(⋅)}≤𝐶‖𝑤‖_{𝐴𝑞(⋅)} [13, Lemma 3.1]. As a result of this Lemma we have𝐴₁⊂𝐴_𝑝−⊂𝐴_𝑝(⋅)⊂𝐴_𝑝+⊂𝐴_∞(2.15) for 𝑝∈𝑃^log(ℝ^𝑛) and 1<𝑝⁻≤𝑝⁺<∞.

Let 𝑝∈𝑃^log(ℝ^𝑛) and 1<𝑝⁻≤𝑝⁺<∞. Then 𝑀∶𝐿^𝑝(⋅)(ℝ^𝑛,𝑤)↪𝐿^𝑝(⋅)(ℝ^𝑛,𝑤) if and only if 𝑤∈𝐴_𝑝(⋅) [13, Theorem 1.1].

We use the notation𝒫(ℝ^𝑛)=𝑝(⋅)∶1<𝑝⁻≤𝑝(𝑥)≤𝑝⁺<∞,‖𝑀𝑓‖_{𝑝(⋅),𝑤}≤𝐶‖𝑓‖_{𝑝(⋅),𝑤},(2.16) that is, the maximal operator 𝑀 is bounded on 𝐿^𝑝(⋅)(ℝ^𝑛,𝑤). Hence we can find a sufficient condition for 𝑝(⋅)∈𝒫(ℝ^𝑛).

Proposition 2.1. Let 𝑤 be a weight function and 1<𝑝⁻≤𝑝(𝑥)≤𝑝⁺<∞. If 𝑤^{−1/(𝑝(⋅)−1)}∈𝐿^1loc(ℝ^𝑛), then 𝐿^𝑝(⋅)(ℝ^𝑛,𝑤)↪𝐿^1loc(ℝ^𝑛).

Proof. Suppose that 𝑓∈𝐿^𝑝(⋅)(ℝ^𝑛,𝑤), and let 𝐾⊂ℝ^𝑛 be a compact set. For 1/𝑝(⋅)+1/𝑞(⋅)=1, by using Hölder’s inequality for variable exponent Lebesgue spaces [1], then there exists a 𝐴_𝐾>0 such that _𝐾||𝑓(𝑥)||𝑑𝑥≤𝐴_𝐾‖‖𝑓𝑤^1/𝑝(⋅)‖‖_{𝑝(⋅),𝐾}‖‖𝑤^{−1/𝑝(⋅)}‖‖_{𝑞(⋅),𝐾}≤𝐴_𝐾‖‖𝑓𝑤^1/𝑝(⋅)‖‖_𝑝(⋅)‖‖𝑤^{−1/𝑝(⋅)}‖‖_{𝑞(⋅),𝐾}.(2.17) It is known that ‖𝑤^{−1/𝑝(⋅)}‖_{𝑞(⋅),𝐾}<∞ if and only if 𝜚_{𝑞(⋅),𝐾}(𝑤^{−1/𝑝(⋅)})<∞ for 𝑞⁺<∞. Since 𝑤^{−1/(𝑝(⋅)−1)}∈𝐿^1loc(ℝ^𝑛), then we have 𝜚_{𝑞(⋅),𝐾}𝑤^{−1/𝑝(⋅)}=_𝐾𝑤(𝑥)^{−𝑞(𝑥)/𝑝(𝑥)}𝑑𝑥=_𝐾𝑤(𝑥)^{−1/(𝑝(𝑥)−1)}𝑑𝑥=𝐵_𝐾<∞.(2.18) If we use (2.17) and (2.18), then the proof is completed.

Definition 2.2 (Mollifiers). Let 𝜑∶ℝ^𝑛→ℝ be a nonnegative, radial, decreasing function belonging to 𝐶^∞0(ℝ^𝑛) and having the properties:(i)𝜑(𝑥)=0 if |𝑥|≥1,(ii)∫_ℝ𝑛𝜑(𝑥)𝑑𝑥=1.
Let 𝜀>0. If the function 𝜑_𝜀(𝑥)=𝜀^−𝑛𝜑(𝑥/𝜀) is nonnegative, belongs to 𝐶^∞0(ℝ^𝑛), and satisfies(i)𝜑_𝜀(𝑥)=0 if |𝑥|≥𝜀 and(ii)∫_ℝ𝑛𝜑_𝜀(𝑥)𝑑𝑥=1, then 𝜑_𝜀 is called a mollifier and we define the convolution by 𝜑_𝜀∗𝑓(𝑥)=_ℝ𝑛𝜑_𝜀(𝑥−𝑦)𝑓(𝑦)𝑑𝑦.(2.19)

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

Proposition 2.3. Let 𝜑_𝜀 be a mollifier and 𝑓∈𝐿^1loc(ℝ^𝑛). Then sup_𝜀>0||𝜑_𝜀∗𝑓(𝑥)||≤𝑀𝑓(𝑥).(2.20)

Proposition 2.4. If 𝑝(⋅)∈𝒫(ℝ^𝑛) and 𝑓∈𝐿^𝑝(⋅)(ℝ^𝑛,𝑤), then 𝜑_𝜀∗𝑓→𝑓 in 𝐿^𝑝(⋅)(ℝ^𝑛,𝑤) as 𝜀→0⁺.

Proof. Let 𝑓∈𝐿^𝑝(⋅)(ℝ^𝑛,𝑤) and 𝜀>0 be given. If 𝑓 is continuous, then the assertion is trivial. By Proposition 2.3, we have ‖‖𝜑_𝜀∗𝑓‖‖_{𝑝(⋅),𝑤}≤‖𝑀𝑓‖_{𝑝(⋅),𝑤}≤𝐶‖𝑓‖_{𝑝(⋅),𝑤}(2.21) and we have 𝜑_𝜀∗𝑓∈𝐿^𝑝(⋅)(ℝ^𝑛,𝑤) for all 𝜀>0. It can be proved that the class 𝐶₀(ℝ^𝑛) of continuous functions with compact support is dense in the space 𝐿^𝑝(⋅)(ℝ^𝑛,𝑤). Then there is a function 𝑔∈𝐶₀(ℝ^𝑛) such that ‖𝑓−𝑔‖_{𝑝(⋅),𝑤}<𝜀.(2.22) Also it is well known that if 𝑔∈𝐶₀(ℝ^𝑛), then 𝜑_𝜀∗𝑔∈𝐶^∞0(ℝ^𝑛) for all 𝜀>0. It is easily seen that 𝜑_𝜀∗𝑔→𝑔 uniformly on compact sets as 𝜀→0⁺. Hence we have 𝜚||𝜑_𝜀∗𝑔(𝑥)−𝑔(𝑥)||^𝑝(𝑥)⟶0,_{𝑝(⋅),𝑤}𝜑_𝜀∗𝑔−𝑔≤=𝜀_𝐾𝑝−||𝜑_𝜀∗𝑔(𝑥)−𝑔(𝑥)||^𝑝(𝑥)𝑤(𝑥)𝑑𝑥_𝐾𝑤(𝑥)𝑑𝑥,(2.23) where supp(𝜑_𝜀∗𝑔)∪supp𝑔⊂𝐾, 𝐾⊂ℝ^𝑛 compact. Hence 𝜚_{𝑝(⋅),𝑤}(𝜑_𝜀∗𝑔−𝑔)→0 as 𝜀→0⁺ and we write ‖‖𝜑_𝜀∗𝑔−𝑔‖‖_{𝑝(⋅),𝑤}<𝜀.(2.24) Finally by using (2.22) and (2.24), ‖‖𝑓−𝜑_𝜀∗𝑓‖‖_{𝑝(⋅),𝑤}≤‖𝑓−𝑔‖_{𝑝(⋅),𝑤}+‖‖𝑔−𝜑_𝜀∗𝑔‖‖_{𝑝(⋅),𝑤}+‖‖𝜑_𝜀∗𝑔−𝜑_𝜀∗𝑓‖‖_{𝑝(⋅),𝑤}<(𝐶+2)𝜀.(2.25) The proof is complete.

As a direct consequence of Proposition 2.4 there follows.

Corollary 2.5. Let 𝑝(⋅)∈𝒫(ℝ^𝑛). The class 𝐶^∞0(ℝ^𝑛) is dense in 𝐿^𝑝(⋅)(ℝ^𝑛,𝑤).

This result was proved without the assumption that the maximal operator is bounded in 𝐿^𝑝(⋅)(ℝ^𝑛,𝑤) by Kokilashvili and Samko [19].

Remark 2.6. Let 1<𝑝⁻≤𝑝(𝑥)≤𝑝⁺<∞ and 𝑤^{−1/(𝑝(⋅)−1)}∈𝐿^1loc(ℝ^𝑛). Then every function in 𝐿^𝑝(⋅)(ℝ^𝑛,𝑤) has distributional derivatives by Proposition 2.1.

3. Weighted Variable Sobolev Spaces

Let 1<𝑝⁻≤𝑝(𝑥)≤𝑝⁺<∞, 𝑤^{−1/(𝑝(⋅)−1)}∈𝐿^1loc(ℝ^𝑛) and 𝑘∈ℕ. We define the weighted variable Sobolev spaces 𝑊^{𝑘,𝑝(⋅)}(ℝ^𝑛,𝑤) by𝑊^{𝑘,𝑝(⋅)}(ℝ^𝑛,𝑤)=𝑓∈𝐿^𝑝(⋅)(ℝ^𝑛,𝑤)∶𝐷^𝛼𝑓∈𝐿^𝑝(⋅)(ℝ^𝑛,𝑤),0≤|𝛼|≤𝑘(3.1) equipped with the norm‖𝑓‖_{𝑘,𝑝(⋅),𝑤}=_{0≤|𝛼|≤𝑘}‖𝐷^𝛼𝑓‖_{𝑝(⋅),𝑤}(3.2) where 𝛼∈ℕ^𝑛0 is a multiindex, |𝛼|=𝛼₁+𝛼₂+⋯+𝛼_𝑛, and 𝐷^𝛼=𝜕^|𝛼|/(𝜕^𝛼1_𝑥1⋯𝜕^𝛼𝑛_𝑥𝑛). It can be shown that 𝑊^{𝑘,𝑝(⋅)}(ℝ^𝑛,𝑤) is a reflexive Banach space. Throughout this paper, we will always assume that 1<𝑝⁻≤𝑝(𝑥)≤𝑝⁺<∞ and 𝑤^{−1/(𝑝(⋅)−1)}∈𝐿^1loc(ℝ^𝑛).

The space 𝑊^1,𝑝(⋅)(ℝ^𝑛,𝑤) is defined by𝑊^1,𝑝(⋅)(ℝ^𝑛,𝑤)=𝑓∈𝐿^𝑝(⋅)(ℝ^𝑛,𝑤)∶||∇𝑓||∈𝐿^𝑝(⋅)(ℝ^𝑛,𝑤).(3.3) The function 𝜚_{1,𝑝(⋅),𝑤}∶𝑊^1,𝑝(⋅)(ℝ^𝑛,𝑤)→[0,∞) is defined as 𝜚_{1,𝑝(⋅),𝑤}(𝑓)=𝜚_{𝑝(⋅),𝑤}(𝑓)+𝜚_{𝑝(⋅),𝑤}(∇𝑓). The norm ‖𝑓‖_{1,𝑝(⋅),𝑤}=‖𝑓‖_{𝑝(⋅),𝑤}+‖∇𝑓‖_{𝑝(⋅),𝑤}.

The Bessel kernel 𝑔_𝛼 order 𝛼,𝛼>0, is defined by𝑔_𝛼(𝑥)=𝜋^𝑛/2Γ(𝛼/2)^∞0𝑒^{−𝑠−(𝜋2|𝑥|2)/𝑠}𝑠^{(𝛼−𝑛)/2}𝑑𝑠𝑠,𝑥∈ℝ^𝑛.(3.4) Let 𝛼≥0. The weighted variable Bessel potential space ℒ^{𝛼,𝑝(⋅)}(ℝ^𝑛,𝑤) is, for 𝛼>0, defined byℒ^{𝛼,𝑝(⋅)}(ℝ^𝑛,𝑤)∶=ℎ=𝑔_𝛼∗𝑓;𝑓∈𝐿^𝑝(⋅)(ℝ^𝑛,𝑤),(3.5) and is equipped with the norm‖ℎ‖_{𝛼;𝑝(⋅),𝑤}=‖𝑓‖_{𝑝(⋅),𝑤}.(3.6) If 𝛼=0 we put 𝑔₀∗𝑓=𝑓 and ℒ^0,𝑝(⋅)(ℝ^𝑛,𝑤)=𝐿^𝑝(⋅)(ℝ^𝑛,𝑤).

Let 𝑝(⋅)∈𝒫(ℝ^𝑛). If 𝑓∈𝐿^𝑝(⋅)(ℝ^𝑛,𝑤), then 𝑔_𝛼∗𝑓∈𝐿^𝑝(⋅)(ℝ^𝑛,𝑤). Indeed, since 𝑔_𝛼∈𝐿¹(ℝ^𝑛) and 𝑔_𝛼 is radial, we have (𝑔_𝛼∗𝑓)(𝑥)≤𝑀𝑓(𝑥), 𝑥∈ℝ^𝑛 [20, page 62]. The assertion thus follows from boundedness of maximal function in 𝐿^𝑝(⋅)(ℝ^𝑛,𝑤).

The unweighted variable Bessel potential space ℒ^{𝛼,𝑝(⋅)}(ℝ^𝑛) was firstly studied by Almeida and Samko in [21].

Lemma 3.1. Let 𝑝(⋅)∈𝑃^log(ℝ^𝑛), 1<𝑝⁻≤𝑝⁺<∞, and 𝑤∈𝐴_𝑝(⋅). Then(i)𝐶^∞0(ℝ^𝑛) is dense in 𝑊^{𝑘,𝑝(⋅)}(ℝ^𝑛,𝑤), 𝑘∈ℕ,(ii)The Schwartz class 𝑆 is dense in ℒ^{𝛼,𝑝(⋅)}(ℝ^𝑛,𝑤), 𝛼≥0.

Proof. (i) By Proposition 2.4 the proof is complete.
(ii) Let 𝛼=0. The class 𝐶^∞0(ℝ^𝑛) is dense in 𝐿^𝑝(⋅)(ℝ^𝑛,𝑤) by Corollary 2.5. It remains only to show that 𝑆⊂𝐿^𝑝(⋅)(ℝ^𝑛,𝑤). Let 𝑓∈𝑆. Then there exist 𝐶=𝐶(𝑟)>0 and 𝑟>0 such that ||𝑓(𝑥)||≤𝐶(1+|𝑥|)^𝑟.(3.7) Also since 𝑟𝑝(𝑥)≥𝑟 and (1+|𝑥|)^𝑟≥1, then 𝜚_{𝑝(⋅),𝑤}(𝑓)=≤ma_ℝ𝑛x||𝑓(𝑥𝐶)||^𝑝(𝑥)_𝑝−𝑤(𝑥)𝑑𝑥,𝐶^𝑝+_ℝ𝑛𝑤(𝑥)(1+|𝑥|)^{𝑟𝑝(𝑥)}≤max𝐶_𝑝−𝑑𝑥,𝐶^𝑝+_ℝ𝑛𝑤(𝑥)(1+|𝑥|)^𝑟𝑑𝑥.(3.8) It is known that 𝐴_𝑝(⋅)⊂𝐴_𝑝+ for 1<𝑝⁺<∞. Also the fact that the Muckenhoupt weights with constant 𝑝⁺ are integrable with some power weight. Then _ℝ𝑛𝑤(𝑥)(1+|𝑥|)^𝑟𝑑𝑥<∞,(3.9) see [22, Lemma 1]. If we use (3.9) in (3.8), then the Schwartz class 𝑆 is dense in 𝐿^𝑝(⋅)(ℝ^𝑛,𝑤).
Let 𝛼>0 and ℎ∈ℒ^{𝛼,𝑝(⋅)}(ℝ^𝑛,𝑤). Then there is a function 𝑓∈𝐿^𝑝(⋅)(ℝ^𝑛,𝑤) such that ℎ=𝑔_𝛼∗𝑓. By density of 𝐶^∞0(ℝ^𝑛) in 𝑓∈𝐿^𝑝(⋅)(ℝ^𝑛,𝑤) we can find a sequence (𝑓_𝑗)_𝑗∈ℕ⊂𝐶^∞0(ℝ^𝑛)⊂𝑆 converging to 𝑓 in 𝐿^𝑝(⋅)(ℝ^𝑛,𝑤). Since the mapping 𝑓↦𝑔_𝛼∗𝑓 maps 𝑆 onto 𝑆 [20], the functions ℎ_𝑗=𝑔_𝛼∗𝑓_𝑗, 𝑗∈ℕ, belong to 𝑆. Moreover, ‖‖ℎ−ℎ_𝑗‖‖_{𝛼;𝑝(⋅),𝑤}=‖‖𝑓−𝑓_𝑗‖‖_{𝑝(⋅),𝑤}⟶0as𝑗⟶∞(3.10) and the assertion follows.

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

Theorem 3.2. Let 𝑝(⋅)∈𝒫(ℝ^𝑛) and 𝑘∈ℕ. Then ℒ^{𝑘,𝑝(⋅)}(ℝ^𝑛,𝑤)=𝑊^{𝑘,𝑝(⋅)}(ℝ^𝑛,𝑤) and the corresponding norms are equivalent.

Remark 3.3. The equivalence of the spaces ℒ^{𝑘,𝑝(⋅)}(ℝ^𝑛,𝑤) and 𝑊^{𝑘,𝑝(⋅)}(ℝ^𝑛,𝑤) fails when 𝑝=1 or 𝑝=∞.

For 𝐸⊂ℝ^𝑛, we denote𝑆_{𝑝(⋅),𝑤}(𝐸)=𝑓∈𝑊^1,𝑝(⋅)(ℝ^𝑛,𝑤)∶𝑓≥1inopensetcontaining𝐸.(3.11) The Sobolev (𝑝(⋅),𝑤)-capacity of 𝐸 is defined by𝐶_{𝑝(⋅),𝑤}(𝐸)=inf_{𝑓∈𝑆𝑝(⋅),𝑤(𝐸)}𝜚_{1,𝑝(⋅),𝑤}(𝑓)=inf_{𝑓∈𝑆𝑝(⋅),𝑤(𝐸)}_ℝ𝑛||𝑓(𝑥)||^𝑝(𝑥)+||∇𝑓(𝑥)||^𝑝(𝑥)𝑤(𝑥)𝑑𝑥.(3.12) In case 𝑆_{𝑝(⋅),𝑤}(𝐸)=∅, we set 𝐶_{𝑝(⋅),𝑤}(𝐸)=∞. The 𝐶_{𝑝(⋅),𝑤}-capacity has the following properties.(i)𝐶_{𝑝(⋅),𝑤}(∅)=0. (ii)If 𝐸₁⊂𝐸₂, then 𝐶_{𝑝(⋅),𝑤}(𝐸₁)≤𝐶_{𝑝(⋅),𝑤}(𝐸₂).(iii)If 𝐸 is a subset of ℝ^𝑛, then 𝐶_{𝑝(⋅),𝑤}(𝐸)=inf𝐶_{𝑝(⋅),𝑤}(𝑈)∶𝐸⊂𝑈,𝑈open.(3.13)(iv)If 𝐸₁ and 𝐸₂ are subsets of ℝ^𝑛, then 𝐶_{𝑝(⋅),𝑤}𝐸₁∪𝐸₂+𝐶_{𝑝(⋅),𝑤}𝐸₁∩𝐸₂≤𝐶_{𝑝(⋅),𝑤}𝐸₁+𝐶_{𝑝(⋅),𝑤}𝐸₂.(3.14)(v)If 𝐾₁⊃𝐾₂⊃⋯ are compact, then lim_𝑖→∞𝐶_{𝑝(⋅),𝑤}𝐾_𝑖=𝐶_{𝑝(⋅),𝑤}^∞_𝑖=1𝐾_𝑖.(3.15)

Note that the assertion (v) above is not true in general for noncompact sets [9].(vi) If 𝐸₁⊂𝐸₂⊂⋯ are subsets of ℝ^𝑛, then lim_𝑖→∞𝐶_{𝑝(⋅),𝑤}𝐸_𝑖=𝐶_{𝑝(⋅),𝑤}^∞_𝑖=1𝐸_𝑖.(3.16)(vii) If 𝐸_𝑖⊂ℝ^𝑛 for 𝑖=1,2,…, then 𝐶_{𝑝(⋅),𝑤}^∞_𝑖=1𝐸_𝑖≤^∞_𝑖=1𝐶_{𝑝(⋅),𝑤}𝐸_𝑖.(3.17)

For the proof of these properties see [8, 10]. Hence the Sobolev 𝐶_{𝑝(⋅),𝑤} 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↦𝐶_{𝑝(⋅),𝑤}(𝐸), 𝐸⊂ℝ^𝑛, is a Choquet capacity. In particular, all Suslin sets 𝐸⊂ℝ^𝑛 are capacitable, that is, 𝐶_{𝑝(⋅),𝑤}(𝐸)=inf_{𝑈𝐸⊂𝑈open}𝐶_{𝑝(⋅),𝑤}(𝑈)=sup_{𝐾𝐾⊂𝐸compact}𝐶_{𝑝(⋅),𝑤}(𝐾).(3.18)

Lemma 3.5. Let 𝑤(𝑥)≥1 for 𝑥∈ℝ^𝑛. Then every measurable set 𝐸⊂ℝ^𝑛 satisfies |𝐸|≤𝐶_{𝑝(⋅),𝑤}(𝐸).

Proof. If 𝑓∈𝑆_{𝑝(⋅),𝑤}(𝐸), then there is an open set 𝐸⊂𝑈 such that 𝑓≥1 in 𝑈 and hence ||𝐸||≤||𝑈||≤_ℝ𝑛||𝑓(𝑥)||^𝑝(𝑥)𝑤(𝑥)𝑑𝑥≤_ℝ𝑛||𝑓(𝑥)||^𝑝(𝑥)+||∇𝑓(𝑥)||^𝑝(𝑥)𝑤(𝑥)𝑑𝑥.(3.19) We obtain the claim by taking the infimum on 𝑆_{𝑝(⋅),𝑤}(𝐸).

Definition 3.6 (Bessel Capacity). Let 𝐸⊂ℝ^𝑛, 𝛼>0. Define that the (𝛼,𝑝(⋅),𝑤)-Bessel capacity in ℒ^{𝛼,𝑝(⋅)}(ℝ^𝑛,𝑤) is the number 𝐵_{𝛼,𝑝(⋅),𝑤}(𝐸)=inf𝜚_{𝑝(⋅),𝑤}(𝑓),(3.20) where the infimum is taken over all 𝑓∈𝐿^𝑝(⋅)(ℝ^𝑛,𝑤) such that 𝑔_𝛼∗𝑓≥1 on 𝐸. Since 𝑔_𝛼 is nonnegative we can assume that 𝑓≥0.

Theorem 3.7. 𝐵_{𝛼,𝑝(⋅),𝑤} is an outer capacity defined on all subsets of ℝ^𝑛.

Proof. It is known that(i)𝐵_{𝛼,𝑝(⋅),𝑤}(∅)=0; (ii)if 𝐸₁⊂𝐸₂, then 𝐵_{𝛼,𝑝(⋅),𝑤}(𝐸₁)≤𝐵_{𝛼,𝑝(⋅),𝑤}(𝐸₂);(iii)if 𝐸_𝑖⊂ℝ^𝑛 for 𝑖=1,2,…, then 𝐵_{𝛼,𝑝(⋅),𝑤}^∞_𝑖=1𝐸_𝑖≤^∞_𝑖=1𝐵_{𝛼,𝑝(⋅),𝑤}𝐸_𝑖(3.21)by [12, Lemma  4.1]. We will show that 𝐵_{𝛼,𝑝(⋅),𝑤}(𝐸)=inf_{𝐺𝐸⊂𝐺open}𝐵_{𝛼,𝑝(⋅),𝑤}(𝐺).(3.22) for any 𝐸⊂ℝ^𝑛. Let 𝐸⊂ℝ^𝑛 be arbitrary. Obviously 𝐵_{𝛼,𝑝(⋅),𝑤}(𝐸)≤inf_{𝐺𝐸⊂𝐺open}𝐵_{𝛼,𝑝(⋅),𝑤}(𝐺). We assume that 𝐵_{𝛼,𝑝(⋅),𝑤}(𝐸)<∞. If 0<𝜀<1 there must exist a test function (measurable and nonnegative) for 𝐵_{𝛼,𝑝(⋅),𝑤}(𝐸), call it 𝑓, such that 𝑔_𝛼∗𝑓≥1 on 𝐸, and 𝜚_{𝑝(⋅),𝑤}(𝑓)<𝐵_{𝛼,𝑝(⋅),𝑤}(𝐸)+𝜀.(3.23) Let 𝐺={𝑥∈ℝ^𝑛∶𝑔_𝛼∗𝑓>1−𝜀}. Since 𝑔_𝛼∗𝑓 is lower semicontinuous in 𝑥, 𝐺 is an open set and since 𝑔_𝛼∗𝑓>1−𝜀 on 𝐸, 𝐺⊃𝐸. Therefore (1−𝜀)⁻¹𝑓 is a test function for 𝐵_{𝛼,𝑝(⋅),𝑤}(𝐺) and we have 𝐵_{𝛼,𝑝(⋅),𝑤}(𝐺)≤𝜚_{𝑝(⋅),𝑤}1𝑓−𝜀≤(1−𝜀)^−𝑝+𝜚_{𝑝(⋅),𝑤}(𝑓)<(1−𝜀)^−𝑝+𝐵_{𝛼,𝑝(⋅),𝑤}(𝐸)+𝜀.(3.24) This proves the theorem as 𝜀→0⁺.

Now we give relationship between the capacities 𝐵_{𝛼,𝑝(⋅),𝑤} and 𝐶_{𝑝(⋅),𝑤} [12].

Lemma 3.8. Let 𝑝(⋅)∈𝒫(ℝ^𝑛) and 𝐸⊂ℝ^𝑛. Then 𝐵𝐶^{1,𝑝(⋅),𝑤}_{𝑝(⋅),𝑤}(𝐸)≤𝑐max𝐶_{𝑝(⋅),𝑤}(𝐸)^{𝑝−/𝑝+},𝐶_{𝑝(⋅),𝑤}(𝐸)^{𝑝+/𝑝−},(𝐸)≤𝐶max𝐵_{1,𝑝(⋅),𝑤}(𝐸)^{𝑝−/𝑝+},𝐵_{1,𝑝(⋅),𝑤}(𝐸)^{𝑝+/𝑝−}.(3.25) Here 𝑐 and 𝐶 are positive constants independent of 𝐸.

Proposition 3.9. Let 𝑝(⋅)∈𝒫(ℝ^𝑛).(i)If 𝑓∈𝑊^1,𝑝(⋅)(ℝ^𝑛,𝑤), then 𝑀𝑓∈𝑊^1,𝑝(⋅)(ℝ^𝑛,𝑤) and |∇𝑀𝑓(𝑥)|≤𝑀|∇𝑓(𝑥)| for almost everywhere in ℝ^𝑛.(ii)Let 1≤𝑠<∞. Then 𝑠𝑝(⋅)∈𝒫(ℝ^𝑛) and there exists a constant 𝐶>0 such that the inequality ‖𝑀𝑓‖_{1,𝑠𝑝(⋅),𝑤}≤𝐶‖𝑓‖_{1,𝑠𝑝(⋅),𝑤}(3.26) holds for all 𝑓∈𝑊^{1,𝑠𝑝(⋅)}(ℝ^𝑛,𝑤).

Proof. (i) By Proposition 2.1 we have 𝐿^𝑝(⋅)(ℝ^𝑛,𝑤)↪𝐿^𝑝(⋅)loc(ℝ^𝑛,𝑤)↪𝐿^1loc(ℝ^𝑛) and 𝑊^1,𝑝(⋅)(ℝ^𝑛,𝑤)↪𝑊^{1,𝑝(⋅)loc}(ℝ^𝑛,𝑤)↪𝑊^1,1loc(ℝ^𝑛). Since  𝑓∈𝑊^1,1loc(ℝ^𝑛), then we have |∇𝑀𝑓(𝑥)|≤𝑀|∇𝑓(𝑥)| for almost everywhere in ℝ^𝑛 by [24]. Since  𝑓, |∇𝑓|∈𝐿^𝑝(⋅)(ℝ^𝑛,𝑤) and 𝑝(⋅)∈𝒫(ℝ^𝑛), then 𝑀𝑓,|∇𝑀𝑓|∈𝐿^𝑝(⋅)(ℝ^𝑛,𝑤). Hence 𝑀𝑓∈𝑊^1,𝑝(⋅)(ℝ^𝑛,𝑤).
(ii) Let 𝑓∈𝐿^{𝑠𝑝(⋅)}(ℝ^𝑛,𝑤). By using definition of ‖⋅‖_{𝑝(⋅),𝑤}, we have ‖𝑓‖_{𝑠𝑝(⋅),𝑤}=‖‖||𝑓||^𝑠‖‖^{1/𝑠𝑝(⋅),𝑤}(3.27) and |𝑓|^𝑠∈𝐿^𝑝(⋅)(ℝ^𝑛,𝑤). Therefore we have ‖𝑀𝑓‖_{𝑠𝑝(⋅),𝑤}=‖(𝑀𝑓)^𝑠‖^{1/𝑠𝑝(⋅),𝑤}≤‖‖𝑀||𝑓||^𝑠‖‖^{1/𝑠𝑝(⋅),𝑤}≤𝑐‖‖||𝑓||^𝑠‖‖^{1/𝑠𝑝(⋅),𝑤}=𝑐‖𝑓‖_{𝑠𝑝(⋅),𝑤}(3.28) and 𝑠𝑝(⋅)∈𝒫(ℝ^𝑛). Since 𝑓∈𝑊^{1,𝑠𝑝(⋅)}(ℝ^𝑛,𝑤), then 𝑓,|∇𝑓|∈𝐿^{𝑠𝑝(⋅)}(ℝ^𝑛,𝑤). Hence we write ‖𝑀𝑓‖_{1,𝑠𝑝(⋅),𝑤}=‖(𝑀𝑓)^𝑠‖^{1/𝑠𝑝(⋅),𝑤}+‖‖||∇𝑀𝑓||^𝑠‖‖^{1/𝑠𝑝(⋅),𝑤}≤‖(𝑀𝑓)^𝑠‖^{1/𝑠𝑝(⋅),𝑤}+‖‖𝑀||∇𝑓||^𝑠‖‖^{1/𝑠𝑝(⋅),𝑤}≤‖‖𝑀||𝑓||^𝑠‖‖^{1/𝑠𝑝(⋅),𝑤}+‖‖𝑀||∇𝑓||^𝑠‖‖^{1/𝑠𝑝(⋅),𝑤}≤𝐶₁‖‖||𝑓||^𝑠‖‖^{1/𝑠𝑝(⋅),𝑤}+𝐶₂‖‖||∇𝑓||^𝑠‖‖^{1/𝑠𝑝(⋅),𝑤}.(3.29) by (3.28). If we set 𝐶=max{𝐶₁,𝐶₂}, then ‖𝑀𝑓‖_{1,𝑠𝑝(⋅),𝑤}≤𝐶‖𝑓‖_{1,𝑠𝑝(⋅),𝑤}.(3.30) This completes the proof.

Proposition 3.10. Let 𝑝(⋅)∈𝒫(ℝ^𝑛). Then for every 𝜆>0 and every 𝑓∈𝑊^1,𝑝(⋅)(ℝ^𝑛,𝑤) we have 𝐶_{𝑝(⋅),𝑤}({𝑥∈ℝ^𝑛∶𝑀𝑓(𝑥)>𝜆})≤𝑐max‖‖‖𝑓𝜆‖‖‖^𝑝+_{1,𝑝(⋅),𝑤},‖‖‖𝑓𝜆‖‖‖^𝑝−_{1,𝑝(⋅),𝑤}.(3.31)

Proof. Since 𝑀𝑓 is lower semicontinuous, the set {𝑥∈ℝ^𝑛∶𝑀𝑓(𝑥)>𝜆} is open for every 𝜆>0. By Proposition 3.9 we can take (𝑀𝑓)/𝜆=𝑀(𝑓/𝜆) as a test function for the capacity. Then we have 𝐶_{𝑝(⋅),𝑤}({𝑥∈ℝ^𝑛∶𝑀𝑓(𝑥)>𝜆})≤𝜚_{1,𝑝(⋅),𝑤}≤max‖‖‖𝑀𝑀𝑓𝜆𝑓𝜆‖‖‖^𝑝+_{1,𝑝(⋅),𝑤},‖‖‖𝑀𝑓≤𝑐max‖‖‖𝑓𝜆‖‖‖^𝑝−_{1,𝑝(⋅),𝑤}𝜆‖‖‖^𝑝+_{1,𝑝(⋅),𝑤},‖‖‖𝑓𝜆‖‖‖^𝑝−_{1,𝑝(⋅),𝑤}.(3.32)

We say that a property holds (𝑝(⋅),𝑤)-quasi everywhere if it holds except in a set of capacity zero. A function 𝑓 is (𝑝(⋅),𝑤)-quasicontinuous in ℝ^𝑛 if for each 𝜀>0 there exists an open set 𝐸 with 𝐶_{𝑝(⋅),𝑤}(𝐸)<𝜀 such that 𝑓 restricted to ℝ^𝑛⧵𝐸 is continuous. The following proof of theorem is quite similar to Theorem 4.7 in [11].

Theorem 3.11. Let 𝑝(⋅)∈𝒫(ℝ^𝑛). If 𝑓∈𝑊^1,𝑝(⋅)(ℝ^𝑛,𝑤), then the limit 𝑓^∗(𝑥)=lim_𝑟→01||𝐵(𝑥,𝑟)||_{𝐵(𝑥,𝑟)}𝑓(𝑦)𝑑𝑦(3.33) exists (𝑝(⋅),𝑤)-quasi everywhere in ℝ^𝑛. The function 𝑓^∗ is the (𝑝(⋅),𝑤)-quasicontinuous representative of 𝑓.

Proof. Since the class 𝐶^∞0(ℝ^𝑛) is dense in 𝑊^1,𝑝(⋅)(ℝ^𝑛,𝑤) by Lemma 3.1, then we can choose a sequence (𝑓_𝑖) such that ‖‖𝑓−𝑓_𝑖‖‖_{1,𝑝(⋅),𝑤}≤2^−2𝑖.(3.34) For 𝑖=1,2,… we denote 𝐴_𝑖=𝑥∈ℝ^𝑛∶𝑀𝑓−𝑓_𝑖(𝑥)>2^−𝑖,𝐵_𝑖=^∞_𝑗=𝑖𝐴_𝑗,𝐸=^∞_𝑗=1𝐵_𝑗.(3.35) By using Proposition 3.10 and the subadditivity of 𝐶_{𝑝(⋅),𝑤} we have 𝐶_{𝑝(⋅),𝑤}𝐴_𝑖≤𝑐max⎧⎪⎨⎪⎩‖‖‖‖𝑀𝑓−𝑓^𝑖2^−𝑖‖‖‖‖^𝑝+_{1,𝑝(⋅),𝑤},‖‖‖‖𝑀𝑓−𝑓^𝑖2^−𝑖=𝑐max1‖‖‖‖^𝑝−_{1,𝑝(⋅),𝑤}⎫⎪⎬⎪⎭2^−𝑖^𝑝+‖‖𝑀𝑓−𝑓_𝑖‖‖^𝑝+_{1,𝑝(⋅),𝑤},12^−𝑖^𝑝−‖‖𝑀𝑓−𝑓_𝑖‖‖^𝑝−_{1,𝑝(⋅),𝑤}≤𝑐max12^−𝑖^𝑝+2^−2𝑖^𝑝+,12^−𝑖^𝑝−2^−2𝑖^𝑝−≤𝑐2^−𝑖,(3.36)𝐶_{𝑝(⋅),𝑤}(𝐵_𝑖)≤𝑐2^1−𝑖 and 𝐶_{𝑝(⋅),𝑤}(𝐸)=0. If we follow the proof of Theorem  4.7 in [11], then this proves the theorem.

Corollary 3.12. Let 𝑝(⋅)∈𝒫(ℝ^𝑛). If 𝑓∈𝑊^1,𝑝(⋅)(ℝ^𝑛,𝑤) and 𝑓 is quasicontinuous, then we have 𝑓(𝑥)=lim_𝑟→01||𝐵(𝑥,𝑟)||_{𝐵(𝑥,𝑟)}𝑓(𝑦)𝑑𝑦(3.37)(𝑝(⋅),𝑤)-quasi everywhere in ℝ^𝑛.

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 𝐸⊂ℝ^𝑛. If 𝑢∈𝑊^1,𝑝(⋅)(ℝ^𝑛,𝑤) is a nonnegative (𝑝(⋅),𝑤)-quasicontinuous function such that 𝑢≥1 on 𝐸. Then for every 𝜀>0 there exists a function ℎ∈𝑆_{𝑝(⋅),𝑤}(𝐸) such that 𝜚_{1,𝑝(⋅),𝑤}(𝑢−ℎ)<𝜀.

Proof. Let 0<𝛿<1, and let 𝑉⊂ℝ^𝑛 be an open set such that 𝑢 is continuous in ℝ^𝑛⧵𝑉 and 𝐶_{𝑝(⋅),𝑤}(𝑉)<𝛿. By definition of 𝐶_{𝑝(⋅),𝑤} there exists a 𝑣∈𝑆_{𝑝(⋅),𝑤}(𝐸) such that 𝜚_{1,𝑝(⋅),𝑤}(𝑣)<𝛿. If we set ℎ=(1+𝛿)𝑢+|𝑣|, then it is easy to show that ℎ∈𝑊^1,𝑝(⋅)(ℝ^𝑛,𝑤) by [10, Theorem  2.2]. Since the function 𝑢 is continuous and the set 𝑉 is open, then the set 𝐺={𝑥∈ℝ^𝑛⧵𝑉∶𝑢(𝑥)>1}∪𝑉(3.38) is open, contains 𝐸, and ℎ≥1 on 𝐺, thus ℎ∈𝑆_{𝑝(⋅),𝑤}(𝐸). It is known that for 1≤𝑝(⋅)≤𝑝⁺<∞ and 𝑎,𝑏≥0, (𝑎+𝑏)^𝑝(⋅)≤2^𝑝+−1(𝑎^𝑝(⋅)+𝑏^𝑝(⋅)) and |∇|𝑣||=|∇𝑣|. Hence we obtain ||𝑣|+𝛿𝑢|^𝑝(⋅)≤2^𝑝+−1(|𝑣|^𝑝(⋅)+|𝛿𝑢|^𝑝(⋅)) and 𝜚_{1,𝑝(⋅),𝑤}(𝑢−ℎ)=≤2_ℝ𝑛_𝑝+−1|||𝑣|+𝛿𝑢||^𝑝(𝑥)+||∇(|𝑣|+𝛿𝑢)||^𝑝(𝑥)𝑤(𝑥)𝑑𝑥_ℝ𝑛|𝑣|^𝑝(⋅)≤2_𝑝+−1+||𝛿𝑢||^𝑝(⋅)+||∇||𝑣(𝑥)||||^𝑝(𝑥)+||𝛿∇𝑢(𝑥)||^𝑝(𝑥)𝑤(𝑥)𝑑𝑥𝜚_{1,𝑝(⋅),𝑤}(𝑣)+𝛿^𝑝−𝜚_{1,𝑝(⋅),𝑤}(𝑢)<2^𝑝+−1𝛿+𝛿^𝑝−𝜚_{1,𝑝(⋅),𝑤}(𝑢).(3.39) This completes the proof as 𝛿→0.

Theorem 3.14. Let 𝑝⁺<∞. If 𝑢∈𝑊^1,𝑝(⋅)(ℝ^𝑛,𝑤) is a (𝑝(⋅),𝑤)-quasicontinuous function and 𝜆>0, then 𝐶_{𝑝(⋅),𝑤}𝑥∈ℝ^𝑛∶||𝑢(𝑥)||>𝜆≤_ℝ𝑛|||𝑢(𝜆𝑥)|||^𝑝(𝑥)+|||∇𝑢(𝑥)𝜆|||^𝑝(𝑥)𝑤(𝑥)𝑑𝑥.(3.40)

Proof. By [10, Theorem  2.2], |𝑢|∈𝑊^1,𝑝(⋅)(ℝ^𝑛,𝑤) and |∇|𝑢||=|∇𝑢|. By Lemma 3.13, there is a sequence ℎ_𝑗∈𝑆_{𝑝(⋅),𝑤}({𝑥∈ℝ^𝑛∶|𝑢(𝑥)|/𝜆>1}) such that 𝜚_{1,𝑝(⋅),𝑤}|𝜆𝑢|−ℎ^𝑗⟶0as𝑗⟶∞.(3.41) Hence we have by [10, Lemma  2.6] that 𝜚_{1,𝑝(⋅),𝑤}ℎ_𝑗⟶𝜚_{1,𝑝(⋅),𝑤}|𝜆𝑢|as𝑗⟶∞.(3.42) By definition of 𝐶_{𝑝(⋅),𝑤}, we write 𝐶_{𝑝(⋅),𝑤}𝑥∈ℝ^𝑛∶||𝑢(𝑥)||>𝜆≤𝜚_{1,𝑝(⋅),𝑤}ℎ_𝑗.(3.43) Therefore 𝐶_{𝑝(⋅),𝑤}𝑥∈ℝ^𝑛∶||𝑢(𝑥)||>𝜆≤𝜚_{1,𝑝(⋅),𝑤}|𝜆𝑢|as𝑗⟶∞.(3.44)

Proposition 3.15. Let 𝑝(⋅)∈𝒫(ℝ^𝑛). If 𝑢∈𝑊^1,𝑝(⋅)(ℝ^𝑛,𝑤), then there is a 𝐶>0 such that 𝐵_{1,𝑝(⋅),𝑤}({𝑥∈ℝ^𝑛∶𝑀𝑢(𝑥)≥𝜆})≤𝐶max‖‖‖𝑢𝜆‖‖‖^𝑝+_{1,𝑝(⋅),𝑤},‖‖‖𝑢𝜆‖‖‖^𝑝−_{1,𝑝(⋅),𝑤}.(3.45)

Proof. For 𝑟>0, we take ℎ=|𝐵(0,𝑟)|⁻¹𝜒_𝐵(0,𝑟). Choose 𝑓∈𝐿^𝑝(⋅)(ℝ^𝑛,𝑤) such that 𝑢=𝑔₁∗𝑓 and ‖𝑓‖_{𝑝(⋅),𝑤}≈‖𝑢‖_{1,𝑝(⋅),𝑤} by Theorem 3.2. Then 1||𝐵(𝑥,𝑟)||_{𝐵(𝑥,𝑟)}||𝑢(𝑦)||𝑑𝑦=1||𝐵(0,𝑟)||_𝐵(0,𝑟)=(ℎ∗|𝑢|)(𝑥)≤𝜒ℎ∗_𝐵(0,𝑟)𝑔(₁𝑥−𝑦)||𝑢(𝑦)||𝑑𝑦=𝑔∗||𝑓||(𝑥)₁∗ℎ∗||𝑓||(𝑥)≤𝑔₁∗𝑀𝑓(𝑥)(3.46) and 𝑀𝑢(𝑥)≤(𝑔₁∗𝑀𝑓)(𝑥). Also it is known that if 𝐸₁⊂𝐸₂, then 𝐵_{1,𝑝(⋅),𝑤}(𝐸₁)≤𝐵_{1,𝑝(⋅),𝑤}(𝐸₂) by [12, Lemma  4.1]. Therefore we have 𝐵_{1,𝑝(⋅),𝑤}({𝑥∈ℝ^𝑛∶𝑀𝑢(𝑥)≥𝜆})≤≤𝜚𝐵_{1,𝑝(⋅),𝑤𝑝(⋅),𝑤}𝑥∈ℝ^𝑛∶𝑔₁∗𝑀𝑓(𝑥)≥𝜆≤max𝑀‖‖‖𝑀𝑓𝜆𝑓𝜆‖‖‖^𝑝+_{𝑝(⋅),𝑤},‖‖‖𝑀𝑓≤𝑐max‖‖‖𝑓𝜆‖‖‖^𝑝−_{𝑝(⋅),𝑤}𝜆‖‖‖^𝑝+_{𝑝(⋅),𝑤},‖‖‖𝑓≤𝐶max‖‖‖𝑢𝜆‖‖‖^𝑝−_{𝑝(⋅),𝑤}𝜆‖‖‖^𝑝+_{1,𝑝(⋅),𝑤},‖‖‖𝑢𝜆‖‖‖^𝑝−_{1,𝑝(⋅),𝑤}.(3.47)

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

Theorem 3.16. Let 𝑝(⋅)∈𝒫(ℝ^𝑛). If  𝑢∈𝑊^1,𝑝(⋅)(ℝ^𝑛,𝑤) and 𝑢 is quasicontinuous, then the limit 𝑢(𝑥)=lim_𝑟→01||𝐵(𝑥,𝑟)||_{𝐵(𝑥,𝑟)}𝑢(𝑦)𝑑𝑦(3.48) exists (1,𝑝(⋅),𝑤)-quasi everywhere in ℝ^𝑛.

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

Proposition 3.17. Let 𝑝(⋅)∈𝒫(ℝ^𝑛). Every 𝑢∈ℒ^1,𝑝(⋅)(ℝ^𝑛,𝑤) is quasicontinuous. That is, for every 𝜀>0, there exists a set 𝐹⊂ℝ^𝑛, 𝐵_{1,𝑝(⋅),𝑤}(𝐹)≤𝜀, so that 𝑢 restricted to ℝ^𝑛⧵𝐹 is continuous.

Proposition 3.18. Let 1<𝑝⁻≤𝑝⁺<∞. Then for all 𝑓∈𝐿^𝑝(⋅)(ℝ^𝑛,𝑤) and 0<𝜆<∞ we have 𝐵_{𝛼,𝑝(⋅),𝑤}𝑥∈ℝ^𝑛∶𝑔_𝛼∗𝑓(𝑥)≥𝜆≤max‖‖‖𝑓𝜆‖‖‖^𝑝+_{𝑝(⋅),𝑤},‖‖‖𝑓𝜆‖‖‖^𝑝−_{𝑝(⋅),𝑤}.(3.49)

Proof. We first note that by definition of 𝐵_{1,𝑝(⋅),𝑤}-capacity, 𝜆⁻¹𝑓 is a test function for the Bessel capacity. Hence 𝐵_{𝛼,𝑝(⋅),𝑤}𝑥∈ℝ^𝑛∶𝑔_𝛼∗𝑓(𝑥)≥𝜆≤𝜚_{𝑝(⋅),𝑤}≤max𝑓‖‖‖𝜆𝑓𝜆‖‖‖^𝑝+_{𝑝(⋅),𝑤},‖‖‖𝑓𝜆‖‖‖^𝑝−_{𝑝(⋅),𝑤}.(3.50)

Proposition 3.19. Let 1<𝑝⁻≤𝑝⁺<∞. If 𝑓∈𝐿^𝑝(⋅)(ℝ^𝑛,𝑤) and 𝐸=𝑥∈ℝ^𝑛∶𝑔_𝛼∗𝑓(𝑥)=∞,(3.51) then 𝐵_{𝛼,𝑝(⋅),𝑤}(𝐸)=0.

Proof. By Proposition 3.18, we write 𝐵_{𝛼,𝑝(⋅),𝑤}(𝐸)=0 as 𝜆→∞.

Proposition 3.20. Let 1<𝑝⁻≤𝑝⁺<∞. If 𝑓∈𝐿^𝑝(⋅)(ℝ^𝑛,𝑤), then lim_𝑟→01||𝐵(𝑥,𝑟)||_{𝐵(𝑥,𝑟)}𝑔_𝛼∗𝑓(𝑦)𝑑𝑦=𝑔_𝛼∗𝑓(𝑥)(3.52) for 𝐵_{𝛼,𝑝(⋅),𝑤}-q.e. 𝑥∈ℝ^𝑛.

Proof. Let 𝜒 be the characteristic function for the unit ball 𝐵(0,1), and define for 𝑟>0, 𝜒_𝑟(𝑥)=(1/|𝐵(0,1)|)𝜒(𝑥/𝑟), 𝑥∈ℝ^𝑛. Then 1||𝐵(𝑥,𝑟)||_{𝐵(𝑥,𝑟)}𝑔_𝛼∗𝑓(𝑦)𝑑𝑦==𝜒_𝑟∗𝑔_𝛼∗𝑓(𝑥)=𝜒_𝑟∗𝑔_𝛼∗𝑓(𝑥)_{𝐵(𝑥,𝑟)}𝜒_𝑟∗𝑔_𝛼(𝑦)𝑓(𝑥−𝑦)𝑑𝑦.(3.53) As 𝑟→0, 𝜒_𝑟∗𝑔_𝛼(𝑦)→𝑔_𝛼(𝑦) for every 𝑦∈ℝ^𝑛. This implies that, for fixed 𝑥∈ℝ^𝑛, 𝜒_𝑟∗𝑔_𝛼(𝑦)𝑓(𝑥−𝑦)→𝑔_𝛼(𝑦)𝑓(𝑥−𝑦) for a.e. 𝑦∈ℝ^𝑛. It was shown that 𝜒_𝑟∗𝑔_𝛼(𝑦)≤𝐶𝑔_𝛼(𝑦) for 0<𝑟≤1 and 𝑦∈ℝ^𝑛 [26, page 161]. By Proposition 3.19, the integrand in (3.53) is dominated by a constant times 𝑔_𝛼(𝑦)|𝑓(𝑥−𝑦)|, which is 𝐿¹(ℝ^𝑛) function for 𝐵_{𝛼,𝑝(⋅),𝑤}-q.e. 𝑥∈ℝ^𝑛. 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.