Research Article
Capacities in Generalized Orlicz Spaces
Debangana Baruah,1 Petteri Harjulehto,1 and Peter Hästö 1,2
1Department of Mathematics and Statistics, 20014, University of Turku, Finland
2Department of Mathematics, 90014, University of Oulu, Finland
Correspondence should be addressed to Peter Ha¨sto¨; peter.hasto@oulu.fi
Received 28 June 2018; Accepted 9 September 2018; Published 1 October 2018
Academic Editor: Alberto Fiorenza
Copyright © 2018 Debangana Baruah et al. This is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly
cited.
In this paper basic properties of both Sobolev and relative capacities are studied in generalized Orlicz spaces. The capacities
are compared with each other and the Hausdorff measure. As an application, the existence of quasicontinuous representative of
generalized Orlicz functions is proved.
1. Introduction
In the calculus of variations one studies existence and pro-
perties of solutions to minimization problems such as
min
𝑢∈𝑊1,1
∫𝐹 (𝑥, |∇𝑢|) 𝑑𝑥. (1)
Classical techniques, by De Giorgi and Moser, cover both the
linear case and the𝑝-growth case, where𝐹(𝑡) ≈ 𝑡𝑝. Marcellini
[1] developed the theory of (𝑝, 𝑞)-growth, which is based on
the growth assumption 𝑡𝑝 − 1 ≲ 𝐹(𝑥, 𝑡) ≲ 𝑡𝑞 + 1, 𝑞 > 𝑝.
Zhikov [2] studied such minimizers as models of anisotropic
materials and also observed that they exhibit the so-called
Lavrentiev phenomenon whereby minimizers do not have
improved regularity and may even be discontinuous.
In the variable exponent case, 𝐹(𝑥, 𝑡) ≈ 𝑡𝑝(𝑥), the change
in the anisotropy (growth rate) is gradual owing to the conti-
nuity of 𝑝. For instance, in electrorheological fluid dynamics,
where the anisotropy depends on the smooth electrical field,
this is a reasonable assumption [3]. In other situations, such
as composite materials, a more clear-cut transition is better.
To this end, Baroni, Colombo, and Mingione [4–7] have
developed a regularity theory of the double phase functional𝐹(𝑥, 𝑡) = 𝑡𝑝 + 𝑎(𝑥)𝑡𝑞, 𝑞 > 𝑝, which has the property that
the growth rate changes abruptly from 𝑝 to 𝑞 in the sets{𝑎 = 0} and {𝑎 > 0} (see also [8–12]). Recently, we were
able to generalize their first step, showing Ho¨lder continuity,
to the general Φ-growth case [13]. Also other results have
recently been obtained for partial differential equations with
generalized Orlicz growth, cf. [13–16].
A different approach to differential equations is based
on (nonlinear) potential theory. The foundation of nonlin-
ear potential theory includes general notions of a Radon
measure, a capacity and generalized functions. The sets of
capacity zero are the exceptional sets for representatives of
the function. In this paper we give basic properties of both
Sobolev capacity and relative capacity in the generalized
Orlicz setting. We follow the general framework of [17] and
its previous adaption to the variable exponent setting [18,
Chapter 10]. The results can be applied, e.g., in the study of
boundary behavior of solutions to PDE.
We consider general Φ-functions which need not be
convex. Many of the proofs in this paper follow a standard
pattern, since they do not depend on the exact form of the
integrand used in the definition of capacity.Wehave therefore
omitted or abbreviated several proofs (e.g., Section 6). Nev-
ertheless, it is necessary to check these basic building blocks
in order to proceed with constructing the theory, since the
results are not covered by earlier results, merely similar.There
are also some proofs which are new, namely, Theorem 9,
Example 13, and Proposition 20. Furthermore, this general
setting clarifies the necessity of various assumptions for
different properties (e.g., Theorem 8). In particular, we find
that the relative capacity is Choquet also for nonconvexΦ-function, whereas convexity is needed for the Sobolev
capacity.
Hindawi
Journal of Function Spaces
Volume 2018, Article ID 8459874, 10 pages
https://doi.org/10.1155/2018/8459874
2 Journal of Function Spaces
It should be noted that Ohno and Shimomura [19] have
recently studied (Sobolev) capacity in the generalized Orlicz
case. However, they consider the capacity in ametric measure
space setting with Hajłasz gradients. These results therefore
work in the Euclidean setting only when the maximal
operator is bounded, since the Hajłasz gradient corresponds
to𝑀(|∇𝑓|).
The outline of the paper is as follows. We start by
introducing our notation and basic definitions.Thenwe study
Sobolev capacity and compare it with the Hausdorff measure.
Next we derive existence of a quasicontinuous representative
of generalized Orlicz function. Finally, we study relative
capacity and compare it with the Sobolev capacity.
2. Preliminaries
We study spaces of functions defined inR𝑛 or opens setsΩ ⊂
R𝑛.
A real-valued function is 𝐿-almost increasing, 𝐿 ⩾ 1, if𝐿𝑓(𝑠) ⩾ 𝑓(𝑡) for 𝑠 > 𝑡. So a 1-almost increasing function is
increasing. 𝐿-almost decreasing is defined analogously.
Definition 1. We say that 𝜑 : Ω × [0,∞) 󳨀→ [0,∞] is a weakΦ-function, and write 𝜑 ∈ Φ𝑤(Ω), if the following conditions
hold:
(i) For every 𝑡 ∈ [0,∞) the function 𝑥 󳨃󳨀→ 𝜑(𝑥, 𝑡) is
measurable and for every 𝑥 ∈ Ω the function 𝑡 󳨃󳨀→𝜑(𝑥, 𝑡) is non-decreasing and left-continuous.
(ii) 𝜑(𝑥, 0) = lim𝑡󳨀→0+𝜑(𝑥, 𝑡) = 0 and lim𝑡󳨀→∞𝜑(𝑥, 𝑡) =∞ for every 𝑥 ∈ Ω.
(iii) There exists 𝐿 ⩾ 1 such that 𝑡 󳨃󳨀→ 𝜑(𝑥, 𝑡)/𝑡 is 𝐿-almost
increasing in (0,∞), for every 𝑥 ∈ Ω.
If 𝜑 ∈ Φ𝑤(Ω) is convex with respect to the second variable,
then it is called a convexΦ-function, and we write 𝜑 ∈ Φ𝑐(Ω).
If there exists 𝛽 ∈ (0, 1] such that 𝜑(𝑥, 𝛽) ⩽ 1 and𝜑(𝑥, 1/𝛽) ⩾ 1 for all 𝑥 ∈ Ω, then we say that (A0) holds.
Further, we say that 𝜑 ∈ Φ𝑤(Ω) satisfies
(aInc)𝑝 if there exists 𝐿 ⩾ 1 such that 𝑡 󳨃󳨀→ 𝜑(𝑥, 𝑡)/𝑡𝑝 is 𝐿-
almost increasing in (0,∞) for every 𝑥 ∈ Ω,(aInc) if there exist 𝑝 > 1 such that (aInc)𝑝 holds,(aDec)𝑞 if there exists 𝐿 ⩾ 1 such that 𝑡 󳨃󳨀→ 𝜑(𝑥, 𝑡)/𝑡𝑞 is 𝐿-
almost decreasing in (0,∞) for every 𝑥 ∈ Ω,(aDec) if there exist 𝑞 > 1 such that (aDec)𝑞 holds.
The corresponding conditions with 𝐿 = 1 are denoted by
(Inc) and (Dec). Note that the definition of weak Φ-function
includes assumption (aInc)1.
Definition 2. Let 𝜑 ∈ Φ𝑤(Ω) and define the modular 󰜚𝜑(⋅) for𝑓 ∈ 𝐿0(Ω) by
󰜚𝜑(⋅) (𝑓) fl ∫
Ω
𝜑 (𝑥, 󵄨󵄨󵄨󵄨𝑓 (𝑥)󵄨󵄨󵄨󵄨) 𝑑𝑥. (2)
The generalized Orlicz space, also called Musielak–Orlicz
space, is defined as the set
𝐿𝜑(⋅) (Ω) fl {𝑓 ∈ 𝐿0 (Ω) : lim
𝜆󳨀→0+
󰜚𝜑(⋅) (𝜆𝑓) = 0} (3)
equipped with the (Luxemburg) norm
󵄩󵄩󵄩󵄩𝑓󵄩󵄩󵄩󵄩𝜑(⋅) fl inf {𝜆 > 0 : 󰜚𝜑(⋅) (𝑓𝜆) ⩽ 1} . (4)
For 𝜑 ∈ Φ𝑐(Ω), 𝐿𝜑(⋅)(Ω) is a Banach space [18, Theorem
2.3.13.].
Definition 3. A function 𝑓 ∈ 𝐿𝜑(⋅)(Ω) belongs to Sobolev
space 𝑊1,𝜑(⋅)(Ω), if its weak partial derivatives 𝜕1𝑓, . . . , 𝜕𝑛𝑓
exist and belong to 𝐿𝜑(⋅)(Ω), that is,
𝑊1,𝜑(⋅) (Ω) fl {𝑓 ∈ 𝑊1,1𝑙𝑜𝑐 (Ω) : 𝑓, 󵄨󵄨󵄨󵄨∇𝑓󵄨󵄨󵄨󵄨 ∈ 𝐿𝜑(⋅) (Ω)} . (5)
We define a semimodular on𝑊1,𝜑(⋅)(Ω) by
󰜚1,𝜑(⋅) (𝑓) fl 󰜚𝜑(⋅) (𝑓) + 󰜚𝜑(⋅) (󵄨󵄨󵄨󵄨∇𝑓󵄨󵄨󵄨󵄨) . (6)
which induces a quasinorm by ‖𝑓‖1,𝜑(⋅) fl inf{𝜆 > 0 :󰜚1,𝜑(⋅)(𝑓/𝜆) ⩽ 1}.
Lemma 4. Let 𝜑 ∈ Φ𝑐(Ω) satisfy (aInc) and (aDec), and let𝑓𝑗, 𝑔𝑗 ∈ 𝑊1,𝜑(⋅)(Ω) for 𝑗 = 1, 2, . . . Assume further that the
sequence (󰜚1,𝜑(⋅)(𝑔𝑗))∞𝑗=1 is bounded. If 󰜚1,𝜑(⋅)(𝑓𝑗 − 𝑔𝑗) 󳨀→ 0 as𝑗 󳨀→ ∞, then󵄨󵄨󵄨󵄨󵄨󰜚1,𝜑(⋅) (𝑓𝑗) − 󰜚1,𝜑(⋅) (𝑔𝑗)󵄨󵄨󵄨󵄨󵄨 󳨀→ 0 𝑎𝑠 𝑗 󳨀→ ∞. (7)
Proof. Since 𝜑 is convex and satisfies (aDec) it satisfies (Dec)
for some possible larger exponent 𝑞 by Lemma 2.6 of [13]. Let𝜆 ∈ (0, 1). By convexity and (Dec) we obtain
𝜑 (𝑥, 󵄨󵄨󵄨󵄨󵄨𝑓𝑗󵄨󵄨󵄨󵄨󵄨) = 𝜑 (𝑥, 󵄨󵄨󵄨󵄨󵄨𝑓𝑗 − 𝑔𝑗 + 𝑔𝑗󵄨󵄨󵄨󵄨󵄨)
⩽ 𝜑(𝑥, 𝜆𝜆 󵄨󵄨󵄨󵄨󵄨𝑓𝑗 − 𝑔𝑗󵄨󵄨󵄨󵄨󵄨 + 1 − 𝜆1 − 𝜆 󵄨󵄨󵄨󵄨󵄨𝑔𝑗󵄨󵄨󵄨󵄨󵄨)
⩽ 𝜆𝜑(𝑥, 1𝜆 󵄨󵄨󵄨󵄨󵄨𝑓𝑗 − 𝑔𝑗󵄨󵄨󵄨󵄨󵄨)
+ (1 − 𝜆) 𝜑 (𝑥, 11 − 𝜆 󵄨󵄨󵄨󵄨󵄨𝑔𝑗󵄨󵄨󵄨󵄨󵄨)
⩽ 𝜆1−𝑞𝜑 (𝑥, 󵄨󵄨󵄨󵄨󵄨𝑓𝑗 − 𝑔𝑗󵄨󵄨󵄨󵄨󵄨)
+ (1 − 𝜆)1−𝑞 𝜑 (𝑥, 󵄨󵄨󵄨󵄨󵄨𝑔𝑗󵄨󵄨󵄨󵄨󵄨) .
(8)
Next we subtract 𝜑(𝑥, 𝑔𝑗) from both sides and integrate overΩ. Hence
󰜚𝜑(⋅) (𝑓𝑗) − 󰜚𝜑(⋅) (𝑔𝑗) ⩽ 𝜆1−𝑞󰜚𝜑(⋅) (𝑓𝑗 − 𝑔𝑗)
+ ((1 − 𝜆)1−𝑞 − 1) 󰜚𝜑(⋅) (𝑔𝑗) . (9)
Note that the choice 𝜆 = 1/2 implies that (󰜚𝜑(⋅)(𝑓𝑗)) is a
bounded sequence.
Journal of Function Spaces 3
Swapping 𝑓𝑗 and 𝑔𝑗 gives a similar inequality, and,
combining the inequalities, we find that󵄨󵄨󵄨󵄨󵄨󰜚𝜑(⋅) (𝑓𝑗) − 󰜚𝜑(⋅) (𝑔𝑗)󵄨󵄨󵄨󵄨󵄨
⩽ 𝜆1−𝑞󰜚𝜑(⋅) (𝑓𝑗 − 𝑔𝑗) 𝑑𝑥
+ ((1 − 𝜆)1−𝑞 − 1) (󰜚𝜑(⋅) (𝑓𝑗) + 󰜚𝜑(⋅) (𝑔𝑗)) .
(10)
The same argument can be applied also to the weak gradient.
Hence󵄨󵄨󵄨󵄨󵄨󰜚1,𝜑(⋅) (𝑓𝑗) − 󰜚1,𝜑(⋅) (𝑔𝑗)󵄨󵄨󵄨󵄨󵄨
⩽ 𝜆1−𝑞󰜚1,𝜑(⋅) (𝑓𝑗 − 𝑔𝑗)
+ ((1 − 𝜆)1−𝑞 − 1) (󰜚1,𝜑(⋅) (𝑓𝑗) + 󰜚1,𝜑(⋅) (𝑔𝑗)) .
(11)
Let 𝜀 > 0 be given. Since 󰜚1,𝜑(⋅)(𝑓𝑗) + 󰜚1,𝜑(⋅)(𝑔𝑗) ⩽ 2𝑐, we
can choose 𝜆 so small that
((1 − 𝜆)1−𝑞 − 1) (󰜚1,𝜑(⋅) (𝑓𝑗) + 󰜚1,𝜑(⋅) (𝑔𝑗)) ⩽ 𝜀2 . (12)
We can then choose 𝑗0 so large that 𝜆1−𝑞󰜚1,𝜑(⋅)(𝑓𝑗 − 𝑔𝑗) ⩽ 𝜀/2
when 𝑗 ⩾ 𝑗0 and it follows that |󰜚1,𝜑(⋅)(𝑓𝑗)−󰜚1,𝜑(⋅)(𝑔𝑗)| ⩽ 𝜀.
The first claim of the following lemma has been proved in
[16, Lemma 4.4]. The proof for the second claim is similar.
Lemma 5. Let 𝜑 ∈ Φ𝑤(R𝑛) satisfy (A0) and let 𝐸 ⊂ R𝑛 be
bounded.
(a) If 𝜑 satisfies (aInc)𝑝, then 𝐿𝜑(⋅)(𝐸) 󳨅→ 𝐿𝑝(𝐸).
(b) If 𝜑 satisfies (aDec)𝑞, then 𝐿𝑞(𝐸) 󳨅→ 𝐿𝜑(⋅)(𝐸).
Lemma 6. Let Ω ⊂ R𝑛. Let 𝜑 ∈ Φ𝑤(Ω) satisfy (aInc)𝑝 and
(aDec)𝑞. Then
󵄩󵄩󵄩󵄩𝑓󵄩󵄩󵄩󵄩𝜑(⋅) ⩽ max {(𝐿󰜚𝜑(⋅) (𝑓))1/𝑞 , (𝐿󰜚𝜑(⋅) (𝑓))1/𝑝} (13)
where 𝐿 is the maximum of the constants from (aInc) and
(aDec).
Proof. If 󰜚𝜑(⋅)(𝑓) = 0, then ‖𝑓‖𝜑(⋅) = 0 and the claim holds.
Let 󰜚𝜑(⋅)(𝑓) > 0 and assume first that 𝐿󰜚𝜑(⋅)(𝑓) ⩽ 1.Then
(aDec) gives that
𝜑(𝑥, 󵄨󵄨󵄨󵄨𝑓 (𝑥)󵄨󵄨󵄨󵄨(𝐿󰜚𝜑(⋅) (𝑓))1/𝑞)
⩽ 𝐿 (𝐿󰜚𝜑(⋅) (𝑓))−1 𝜑 (𝑥, 󵄨󵄨󵄨󵄨𝑓 (𝑥)󵄨󵄨󵄨󵄨)
= (󰜚𝜑(⋅) (𝑓))−1 𝜑 (𝑥, 󵄨󵄨󵄨󵄨𝑓 (𝑥)󵄨󵄨󵄨󵄨) .
(14)
Integrating over Ω, we find that 󰜚𝜑(⋅)(𝑓/(𝐿󰜚𝜑(⋅)(𝑓))1/𝑞)) ⩽ 1,
which yields ‖𝑓‖𝜑(⋅) ⩽ (𝐿󰜚𝜑(⋅)(𝑓))1/𝑞. If 𝐿󰜚𝜑(⋅)(𝑓) > 1 we simi-
larly use (aInc) to conclude that ‖𝑓‖𝜑(⋅) ⩽ (𝐿󰜚𝜑(⋅)(𝑓))1/𝑝. The
claim follows from these two cases.
3. Sobolev Capacity
We define a set of test-functions for the capacity of the set 𝐸
by
𝑆1,𝜑(⋅) (𝐸) fl {𝑓 ∈ 𝑊1,𝜑(⋅) (R𝑛) : 𝑓
⩾ 1 in an open set containing 𝐸 and 𝑓 ⩾ 0} . (15)
The generalized Orlicz 𝜑(⋅)-capacity of 𝐸 is defined by
𝐶𝜑(⋅) (𝐸) fl inf
𝑓∈𝑆1,𝜑(⋅)(𝐸)
∫
R𝑛
𝜑 (𝑥, 𝑓) + 𝜑 (𝑥, 󵄨󵄨󵄨󵄨∇𝑓󵄨󵄨󵄨󵄨) 𝑑𝑥. (16)
We now prove the following properties for the set func-
tion 𝐸 󳨃󳨀→ 𝐶𝜑(⋅)(𝐸). We emphasize that these do not require
the convexity of 𝜑.
Proposition 7. Let 𝜑 ∈ Φ𝑤(R𝑛).
(S1) 𝐶𝜑(⋅)(0) = 0.
(S2) If 𝐸1 ⊂ 𝐸2 ⊂ R𝑛, then 𝐶𝜑(⋅)(𝐸1) ⩽ 𝐶𝜑(⋅)(𝐸2).
(S3) If 𝐸 ⊂ R𝑛, then 𝐶𝜑(⋅)(𝐸) = inf𝑈⊃𝐸 𝑜𝑝𝑒𝑛𝐶𝜑(⋅)(𝑈).
(S4) If 𝐸1, 𝐸2 ⊂ R𝑛, then 𝐶𝜑(⋅)(𝐸1 ∪ 𝐸2) + 𝐶𝜑(⋅)(𝐸1 ∩ 𝐸2) ⩽𝐶𝜑(⋅)(𝐸1) + 𝐶𝜑(⋅)(𝐸2).
(S5) If𝐾1 ⊃ 𝐾2 ⊃ . . . are compact, then lim𝑖󳨀→∞𝐶𝜑(⋅)(𝐾𝑖) =𝐶𝜑(⋅)(⋂∞𝑖=1𝐾𝑖).
Proof. (S1) follows from testing with 𝑓 = 0. Since a test-
function for𝐸2 is also a test-function for𝐸1, (S2) follows from
the infimum in the definition of capacity.
If 𝑈 ⊃ 𝐸, then 𝐶𝜑(⋅)(𝐸) ⩽ 𝐶𝜑(⋅)(𝑈) by (S2). Thus,
𝐶𝜑(⋅) (𝐸) ⩽ inf
𝑈⊃𝐸 open
𝐶𝜑(⋅) (𝑈) . (17)
For the opposite inequality, let 𝜀 > 0 and choose𝑓 ∈ 𝑆1,𝜑(⋅)(𝐸)
such that
󰜚1,𝜑(⋅) (𝑓) ⩽ 𝐶𝜑(⋅) (𝐸) + 𝜀. (18)
Denote 𝑉 fl int{𝑓 ⩾ 1}. Then 𝑓 ∈ 𝑆1,𝜑(⋅)(𝑉), and so
inf
𝑈⊃𝐸 open
𝐶𝜑(⋅) (𝑈) ⩽ 𝐶𝜑(⋅) (𝑉) ⩽ 󰜚1,𝜑(⋅) (𝑓)
⩽ 𝐶𝜑(⋅) (𝐸) + 𝜀. (19)
This implies (S3) as 𝜀 󳨀→ 0+.
Let 𝜀 > 0. Choose 𝑓𝑗 ∈ 𝑆1,𝜑(⋅)(𝐸𝑗) such that
󰜚1,𝜑(⋅) (𝑓𝑗) ⩽ 𝐶𝜑(⋅) (𝐸𝑗) + 𝜀 for 𝑗 ∈ {1, 2} . (20)
Since max{𝑓1, 𝑓2} ⩾ 1 in an open set containing 𝐸1 ∪ 𝐸2,
it follows that max{𝑓1, 𝑓2} ∈ 𝑆1,𝜑(⋅)(𝐸1 ∪ 𝐸2). Similarly,
min{𝑓1, 𝑓2} ∈ 𝑆1,𝜑(⋅)(𝐸1 ∩ 𝐸2).
4 Journal of Function Spaces
The lattice property of Sobolev functions [18, Proposition
8.1.9] implies that ∇max{𝑓1, 𝑓2} = ∇𝑓1 and ∇min{𝑓1, 𝑓2} =∇𝑓2 almost everywhere in 𝐴 = {𝑓1 ⩾ 𝑓2}. Therefore
∫
𝐴
𝜑 (𝑥, 󵄨󵄨󵄨󵄨∇max {𝑓1, 𝑓2}󵄨󵄨󵄨󵄨) 𝑑𝑥
+ ∫
𝐴
𝜑 (𝑥, 󵄨󵄨󵄨󵄨∇min {𝑓1, 𝑓2}󵄨󵄨󵄨󵄨) 𝑑𝑥
= ∫
𝐴
𝜑 (𝑥, 󵄨󵄨󵄨󵄨∇𝑓1󵄨󵄨󵄨󵄨) 𝑑𝑥 + ∫
𝐴
𝜑 (𝑥, 󵄨󵄨󵄨󵄨∇𝑓2󵄨󵄨󵄨󵄨) 𝑑𝑥.
(21)
The same argument also gives the equality in the complement
of 𝐴, i.e. {𝑓1 < 𝑓2}. Hence
󰜚𝜑(⋅) (∇max {𝑓1, 𝑓2}) + 󰜚𝜑(⋅) (∇min {𝑓1, 𝑓2})
= 󰜚𝜑(⋅) (∇𝑓1) + 󰜚𝜑(⋅) (∇𝑓2) (22)
almost everywhere. An analogous results holds without the
derivative ∇.
Since max{𝑓1, 𝑓2} ∈ 𝑆1,𝜑(⋅)(𝐸1 ∪ 𝐸2), it follows from the
definition that 𝐶𝜑(⋅)(𝐸1 ∪ 𝐸2) ⩽ 󰜚1,𝜑(⋅)(max{𝑓1, 𝑓2}), and
similarly 𝐶𝜑(⋅)(𝐸1∩𝐸2) ⩽ 󰜚1,𝜑(⋅)(min{𝑓1, 𝑓2}). Combining this
with the conclusion of the previous paragraph, we find that
𝐶𝜑(⋅) (𝐸1 ∪ 𝐸2) + 𝐶𝜑(⋅) (𝐸1 ∩ 𝐸2)
⩽ 󰜚1,𝜑(⋅) (max {𝑓1, 𝑓2}) + 󰜚1,𝜑(⋅) (min {𝑓1, 𝑓2})
= 󰜚1,𝜑(⋅) (𝑓1) + 󰜚1,𝜑(⋅) (𝑓2)
⩽ 𝐶𝜑(⋅) (𝐸1) + 𝐶𝜑(⋅) (𝐸2) + 2𝜀.
(23)
(S4) follows from this as 𝜀 󳨀→ 0+.
It remains to prove (S5). Since ⋂∞𝑖=1𝐾𝑖 ⊂ 𝐾𝑗 for any 𝑗,
the “⩾”-inequality follows from (S2). To prove the opposite
inequality, we choose an open 𝑈 ⊃ ⋂∞𝑖=1𝐾𝑖. Since ⋂∞𝑖=1𝐾𝑖 is
compact and (𝐾𝑗) is decreasing, there is a positive integer 𝑘
such that, 𝐾𝑖 ⊂ 𝑈 for all 𝑖 ⩾ 𝑘. Then, again by (S2), we have
lim
𝑖󳨀→∞
𝐶𝜑(⋅) (𝐾𝑖) ⩽ 𝐶𝜑(⋅) (𝑈) . (24)
Taking the infimum of this inequality over such sets 𝑈, we
obtain the claim by (S3).
Notice that we need convexity for the next property.
Theorem 8. Let 𝜑 ∈ Φ𝑐(R𝑛) satisfy (aInc) and (aDec) and𝐸1 ⊂ 𝐸2 ⊂ ⋅ ⋅ ⋅ ⊂ R𝑛. Then
(S6) lim𝑖󳨀→∞𝐶𝜑(⋅)(𝐸𝑖) = 𝐶𝜑(⋅)(⋃∞𝑖=1 𝐸𝑖).
Proof. Let us denote𝐸 fl ⋃∞𝑖=1 𝐸𝑖. By (S2), lim𝑖󳨀→∞𝐶𝜑(⋅)(𝐸𝑖) ⩽𝐶𝜑(⋅)(𝐸).
Now to prove the opposite inequality, wemay assume that
lim𝑖󳨀→∞𝐶𝜑(⋅)(𝐸𝑖) < ∞. Let 𝑓𝑖 ∈ 𝑆1,𝜑(⋅)(𝐸𝑖) and 󰜚1,𝜑(⋅)(𝑓𝑖) ⩽𝐶𝜑(⋅)(𝐸𝑖) + 2−𝑖 for every 𝑖 ∈ N.
The space 𝐿𝜑(⋅)(R𝑛) is uniformly convex and reflexive [20,
Theorem 1.3]. The same holds for 𝑊1,𝜑(⋅), which is a closed
subspace of (𝐿𝜑(⋅))𝑛+1. By reflexivity, the bounded sequence(𝑓𝑖) has a subsequence which converges weakly to a function𝑓 ∈ 𝑊1,𝜑(⋅)(R𝑛). It follows from the Banach-Saks theorem
that
1𝑚
𝑚∑
𝑖=1
𝑓𝑖 󳨀→ 𝑓 in 𝑊1,𝜑(⋅) (R𝑛) as 𝑚 󳨀→ ∞. (25)
Let 𝑔𝑗 fl (1/𝑗(𝑗 − 1))∑𝑗2𝑖=𝑗+1 𝑓𝑖. Then
1𝑗2
𝑗2∑
𝑖=1
𝑓𝑖 − 𝑔𝑗 = ( 1𝑗2 − 1𝑗 (𝑗 − 1))
𝑗2∑
𝑖=1
𝑓𝑖
+ 1𝑗 (𝑗 − 1)
𝑗∑
𝑖=1
𝑓𝑖
(26)
from which we get, by the triangle inequality, that
󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩
1𝑗2
𝑗2∑
𝑖=1
𝑓𝑖 − 𝑔𝑗
󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩1,𝜑(⋅) ⩽
1𝑗 − 1
󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩
1𝑗2
𝑗2∑
𝑖=1
𝑓𝑖
󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩1,𝜑(⋅)
+ 1𝑗 − 1
󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩
1𝑗
𝑗∑
𝑖=1
𝑓𝑖
󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩1,𝜑(⋅) 󳨀→ 0
(27)
as 𝑗 󳨀→ ∞, so that 𝑔𝑗 󳨀→ 𝑓 in𝑊1,𝜑(⋅)(R𝑛).
By the convexity of the modular and the choice of 𝑔𝑗, we
obtain that
󰜚1,𝜑(⋅) (𝑔𝑗) ⩽ 1𝑗 (𝑗 − 1)
𝑗2∑
𝑖=𝑗+1
󰜚1,𝜑(⋅) (𝑓𝑖) ⩽ sup
𝑖⩾𝑗+1
󰜚1,𝜑(⋅) (𝑓𝑖)
⩽ sup
𝑖⩾𝑗+1
(𝐶𝜑(⋅) (𝐸𝑖) + 2−𝑖) .
(28)
Now 𝐶𝜑(⋅)(𝐸𝑖) increases in 𝑖 whereas 2−𝑖 decreases. Hence
󰜚1,𝜑(⋅) (𝑔𝑗) ⩽ lim
𝑖󳨀→∞
𝐶𝜑(⋅) (𝐸𝑖) + 2−𝑗. (29)
By considering a subsequence if necessary, we may
assume that ‖𝑔𝑗+1−𝑔𝑗‖1,𝜑(⋅) ⩽ 2−𝑗.Then ℎ𝑗 fl 𝑔𝑗+∑∞𝑖=𝑗 |𝑔𝑖+1−𝑔𝑖| ∈ 𝑊1,𝜑(⋅)(R𝑛). By definition of a test-function, there exists
an open set 𝑈𝑖 ⊃ 𝐸𝑖 such that 𝑓𝑖 ⩾ 1 in 𝑈𝑖. As 𝐸𝑗 is an
increasing sequence, it follows that 𝐸𝑗 ⊂ 𝑈𝑖 for every 𝑗 ⩽ 𝑖. In⋂𝑗2𝑖=𝑗+1𝑈𝑗, 𝑓𝑖 ⩾ 1 for all relevant 𝑖, so that 𝑔𝑗 ⩾ 1 in the same
(open) set. Since ℎ𝑗 ⩾ sup𝑖⩾𝑗𝑔𝑗, we have
𝐸𝑖 ⊂ int {𝑔𝑖 ⩾ 1} ⊂ int {ℎ𝑗 ⩾ 1} . (30)
Taking the union over 𝑖 of the previous inclusion, we find thatℎ𝑗 ⩾ 1 in the open set ⋃∞𝑖=𝑗 int{𝑔𝑗 ⩾ 1} ⊃ 𝐸, so that ℎ𝑗 ∈𝑆1,𝜑(⋅)(𝐸). Hence
𝐶𝜑(⋅) (𝐸) ⩽ 󰜚1,𝜑(⋅) (ℎ𝑗) for 𝑗 = 1, 2, . . . (31)
Journal of Function Spaces 5
Furthermore,
󵄩󵄩󵄩󵄩󵄩ℎ𝑗 − 𝑔𝑗󵄩󵄩󵄩󵄩󵄩1,𝜑(⋅) ⩽ ∞∑
𝑖=𝑗
󵄩󵄩󵄩󵄩𝑔𝑖+1 − 𝑔𝑖󵄩󵄩󵄩󵄩1,𝜑(⋅) ⩽ ∞∑
𝑖=𝑗
2−𝑖 = 2−𝑗+1 (32)
and, hence, by [18, Corollary 2.1.15], 󰜚1,𝜑(⋅)(ℎ𝑗 − 𝑔𝑗) ⩽ ‖ℎ𝑗 −𝑔𝑗‖1,𝜑(⋅) 󳨀→ 0 as 𝑗 󳨀→ ∞.Then, using Lemma 4, we also have|󰜚1,𝜑(⋅)(ℎ𝑗) − 󰜚1,𝜑(⋅)(𝑔𝑗)| 󳨀→ 0 as 𝑗 󳨀→ ∞, which we apply in
(31) to obtain
𝐶𝜑(⋅) (𝐸) ⩽ lim
𝑗󳨀→∞
󰜚1,𝜑(⋅) (ℎ𝑗) = lim
𝑗󳨀→∞
󰜚1,𝜑(⋅) (𝑔𝑗) (33)
and further (29) implies
𝐶𝜑(⋅) (𝐸) ⩽ lim
𝑗󳨀→∞
󰜚1,𝜑(⋅) (𝑔𝑗)
⩽ lim
𝑗󳨀→∞
( lim
𝑖󳨀→∞
𝐶𝜑(⋅) (𝐸𝑖) + 2−𝑗)
= lim
𝑖󳨀→∞
𝐶𝜑(⋅) (𝐸𝑖) .
(34)
In the next result we use a trick to get back to weak Φ-
functions despite an application of (S6).
Theorem 9. Let 𝜑 ∈ Φ𝑤(R𝑛) satisfy (aInc) and (aDec) and𝐸1, 𝐸2, . . . ⊂ R𝑛. Then
(S7) 𝐶𝜑(⋅)(⋃∞𝑖=1 𝐸𝑖) ⩽ ∑∞𝑖=1 𝐶𝜑(⋅)(𝐸𝑖).
Proof. Let 𝜓 ∈ Φ𝑐(R𝑛) with 𝜑 ≃ 𝜓 [21, Lemma 3.1]. Since 𝜑
satisfies (aDec) 𝜑 ≃ 𝜓 yields 𝜑 ≈ 𝜓. Denote 𝐹𝑚𝑘 fl ∪𝑘𝑖=𝑚𝐸𝑖 for𝑘 = 𝑚,𝑚 + 1, . . . By induction on (S4), we obtain that
𝐶𝜑(⋅) (𝐹𝑚𝑘 ) = 𝐶𝜑(⋅)( 𝑘⋃
𝑖=𝑚
𝐸𝑖) ⩽ 𝑘∑
𝑖=𝑚
𝐶𝜑(⋅) (𝐸𝑖)
⩽ ∞∑
𝑖=𝑚
𝐶𝜑(⋅) (𝐸𝑖) .
(35)
The same inequality holds also for𝜓. Now, using (S6) for (𝐹𝑘),
we have
𝐶𝜓(⋅)( ∞⋃
𝑘=𝑚
𝐸𝑘) = 𝐶𝜓(⋅)( ∞⋃
𝑘=𝑚
𝐹𝑚𝑘 ) = lim
𝑘󳨀→∞
𝐶𝜓(⋅) (𝐹𝑚𝑘 )
⩽ ∞∑
𝑖=𝑚
𝐶𝜓(⋅) (𝐸𝑖)
(36)
Furthermore, for 𝑘 = 1, 2, . . ., by (S4) and 𝜑 ≈ 𝜓,
𝐶𝜑(⋅)(∞⋃
𝑗=1
𝐸𝑗) ⩽ 𝐶𝜑(⋅) (𝐹1𝑘) + 𝐶𝜑(⋅)( ∞⋃
𝑗=𝑘+1
𝐸𝑗)
⩽ 𝐶𝜑(⋅) (𝐹1𝑘) + 𝑐𝐶𝜓(⋅)( ∞⋃
𝑗=𝑘+1
𝐸𝑗) .
(37)
By the two estimates in the previous paragraph,we obtain that
𝐶𝜑(⋅)(∞⋃
𝑗=1
𝐸𝑗) ⩽ ∞∑
𝑖=1
𝐶𝜑(⋅) (𝐸𝑖) + 𝑐∞∑
𝑖=𝑘
𝐶𝜓(⋅) (𝐸𝑖) . (38)
Since the sum is finite (otherwise, there is nothing to prove),
the second term on the right hand side tends to zero as 𝑘 󳨀→∞. This gives the claim.
Remark 10. A set function satisfying the properties (S1), (S2),
and (S7) is called an outer measure. This holds if 𝜑 ∈ Φ𝑤(Ω) is
satisfies (aInc) and (aDec). If 𝜑 is convex and satisfies (aInc)
and (aDec), then it is a Choquet capacity, [22], i.e., it satisfies
(S1), (S2), (S5), and (S6). Then, for every Borel set 𝐸 ⊂ Ω,
𝐶𝜑(⋅) (𝐸)
= sup {𝐶𝜑(⋅) (𝐾) : 𝐾 is compact and 𝐾 ⊂ 𝐸} . (39)
4. Sobolev Capacity and Hausdorff Measure
In this section, we discuss simple relations between the
generalized Orlicz capacity and the Lebesgue and Hausdorff
measures.
Lemma 11. Let 𝜑 ∈ Φ𝑤(R𝑛) satisfy (aDec) and (A0). Then
every measurable set 𝐸 ⊂ R𝑛 satisfies |𝐸| ⩽ 𝑐𝐶𝜑(⋅)(𝐸), where𝑐 > 1 depends on (aDec) and (A0).
Proof. Let 𝑓 ∈ 𝑆1,𝜑(⋅)(𝐸). By (aDec), with exponent 𝑞 and
constant 𝐿, we conclude that 𝜑(𝑥, 𝑓(𝑥)/𝛽) ⩽ 𝐿𝛽−𝑞𝜑(𝑥, 𝑓(𝑥)).
We have 𝜑(𝑥, 𝑓(𝑥)/𝛽) ⩾ 1 for 𝑓(𝑥) ⩾ 1 by (A0). Therefore
|𝐸| ⩽ ∫
R𝑛
𝜑(𝑥, 𝑓𝛽)𝑑𝑥 ⩽ 𝐿𝛽−𝑞 ∫R𝑛 𝜑 (𝑥, 𝑓) 𝑑𝑥
⩽ 𝐿𝛽−𝑞󰜚1,𝜑(⋅) (𝑓) .
(40)
Taking infimum over 𝑓 ∈ 𝑆1,𝜑(⋅)(𝐸), we obtain the claim.
Proposition 12. Let 𝜑 ∈ Φ𝑤(R𝑛) satisfy (A0), (aInc)𝑝, 𝑝 > 1,
and (aDec). If 𝐶𝜑(⋅)(𝐸) = 0, then 𝐶𝑝(𝐸) = 0.
Proof. Let𝐵 = 𝐵(0, 𝑅+1) and 𝜂 ∈ 𝐶∞0 (𝐵) be a cut-off function
with 𝜂 = 1 in 𝐵(0, 𝑅), 0 ⩽ 𝜂 ⩽ 1 and |∇𝜂| ⩽ 2. Let 𝑓 ∈𝐿𝜑(⋅)(R𝑛). By Lemma 5, we obtain󵄩󵄩󵄩󵄩𝑓𝜂󵄩󵄩󵄩󵄩𝐿𝑝(R𝑛) = 󵄩󵄩󵄩󵄩𝑓𝜂󵄩󵄩󵄩󵄩𝐿𝑝(𝐵) ⩽ 𝑐 󵄩󵄩󵄩󵄩𝜂𝑓󵄩󵄩󵄩󵄩𝐿𝜑(⋅)(𝐵) ⩽ 𝑐 󵄩󵄩󵄩󵄩𝑓󵄩󵄩󵄩󵄩𝐿𝜑(⋅)(R𝑛) (41)
where c depends on (aInc) and |𝐵|.The same inequality holds
for |∇𝑓|. Now ∇(𝑓𝜂) = 𝜂∇𝑓+𝑓∇𝜂, and, thus |∇(𝑓𝜂)| ⩽ |∇𝑓| +2|𝑓| by the assumptions on 𝜂. Since ‖𝑓‖𝑊1,𝜑(⋅)(R𝑛) ≈ ‖𝑓‖𝐿𝑝(R𝑛)+‖∇𝑓‖𝐿𝑝(R𝑛) we obtain that󵄩󵄩󵄩󵄩𝑓𝜂󵄩󵄩󵄩󵄩𝑊1,𝑝(R𝑛) ≲ 󵄩󵄩󵄩󵄩𝑓𝜂󵄩󵄩󵄩󵄩𝐿𝑝(R𝑛) + 󵄩󵄩󵄩󵄩∇ (𝑓𝜂)󵄩󵄩󵄩󵄩𝐿𝑝(R𝑛)
≲ 󵄩󵄩󵄩󵄩𝑓󵄩󵄩󵄩󵄩𝑊1,𝜑(⋅)(R𝑛) . (42)
Since 𝐶𝜑(⋅)(𝐸) = 0, we can choose 𝑓𝑖 ∈ 𝑆1,𝜑(⋅)(𝐸 ∩𝐵(0, 𝑅)) with 󰜚1,𝜑(⋅)(𝑓𝑖) 󳨀→ 0. Since 𝜑 satisfies (aDec),
6 Journal of Function Spaces
‖𝑓𝑖‖𝑊1,𝜑(⋅)(R𝑛) 󳨀→ 0 [18, Lemma 2.1.11]. Thus the above
inequality implies that ‖𝑓𝑖𝜂‖𝑊1,𝑝(R𝑛) 󳨀→ 0. Since 𝑓𝑖𝜂 is a test-
function for the 𝑝-capacity of 𝐸 ∩ 𝐵(0, 𝑅), we get 𝐶𝑝(𝐸 ∩𝐵(0, 𝑅)) = 0, for every 𝑅 > 0. Since 𝐸 = ⋃∞𝑖=1(𝐸 ∩ 𝐵(0, 𝑖)),
the claim follows by the subadditivity of the 𝑝-capacity.
In the previous result the assumption (aInc)𝑝 is natural,
since 𝑝 gives the capacity to compare with. However, the
assumption (aDec) is surprising.Thenext example shows that
it is nevertheless needed.
Example 13. Let 𝜑(𝑡) = 0 when 𝑡 ∈ [0, 11/10] and 𝜑(𝑡) =𝑡 − 11/10 when 𝑡 > 11/10. Then 𝜑 satisfies (aInc) with 𝑝 = 1
and (A0) with 𝛽 = 10/21. Let 𝐵 be an open ball with a radius
one. Let𝑓 be a Lipschitz-continuous function that is one in 𝐵,
zero inR𝑛\2𝐵 and linear in 2𝐵\𝐵.Then |∇𝑓| = 1 in 2𝐵\𝐵 and
zero elsewhere. We obtain 󰜚1,𝜑(𝑓) = 0 and thus 𝐶𝜑(⋅)(𝐵) = 0.
On the other hand 𝐶𝑝(𝐵) > 0.
The 𝑠-dimensional Hausdorff measure of a set 𝐸 ⊂ R𝑛 is
denoted byH𝑠(𝐸).
Corollary 14. Let𝜑 ∈ Φ𝑤(R𝑛) satisfy (A0), (aInc)𝑝with𝑝 > 1
and (aDec).
(1) If𝑝 ⩽ 𝑛 and𝐸 ⊂ R𝑛 with𝐶𝜑(⋅)(𝐸) = 0, thenH𝑠(𝐸) = 0
for all 𝑠 > 𝑛 − 𝑝.
(2) If 𝑝 > 𝑛, then 𝐶𝜑(⋅)(𝐸) = 0 if and only if 𝐸 = 0, where𝐸 ⊂ R𝑛.
Proof. If 𝐶𝜑(⋅)(𝐸) = 0, we then obtain 𝐶𝑝(𝐸) = 0 by
Proposition 12. This leads to the first claim by [23, Theorem
4, p. 156].
If 𝑝 > 𝑛, then we may choose 𝑠 = 0 in (1) and soH0(𝐸) =0. SinceH0 is a counting measure, 𝐸 must be an empty set.
On the other hand, 𝐶𝜑(⋅)(0) = 0, by (S1).
Corollary 15. Let 𝜓 ∈ Φ𝑤(R𝑛) satisfy (A0), (aInc), and
(aDec)𝑞. Let 𝐸 ⊂ R𝑛 be bounded. If 𝐶𝑞(𝐸) = 0 orH𝑛−𝑞(𝐸) <∞, then 𝐶𝜓(⋅)(𝐸) = 0.
Proof. Let 𝐶𝑞(𝐸) = 0. By Lemma 5, 𝐿𝑞(𝐸) 󳨅→ 𝐿𝜓(⋅)(𝐸). The
remaining proof follows the same procedure as in the proof of
Proposition 12. IfH𝑛−𝑞(𝐸) < ∞, it follows from [23,Theorem
3, p. 154] that 𝐶𝑞(𝐸) = 0 and thus the claim follows from the
first part.
5. Quasicontinuity
In this section, we prove the existence of 𝜑(⋅)-quasicon-
tinuous representatives of generalized Orlicz functions. A
function 𝑓 : R𝑛 󳨀→ [−∞,∞] is 𝜑(⋅)-quasicontinuous if for
every 𝜀 > 0 there exists an open set 𝑈 with 𝐶𝜑(⋅)(𝑈) < 𝜀
such that 𝑓 restricted to R𝑛 \ 𝑈 is continuous. We say that
a claim holds 𝜑(⋅)-quasieverywhere if it holds except in a set
of Sobolev 𝜑(⋅)-capacity zero.
In this section we assume the density of continuous
functions. A sufficient condition for this can be found in
Theorem 6.5 of [16].
Lemma 16. Let 𝜑 ∈ Φ𝑤(R𝑛) satisfy (aInc) and (aDec). Then,
for each Cauchy sequence 𝐶(R𝑛) ∩ 𝑊1,𝜑(⋅)(R𝑛), there is a
subsequence which converges pointwise 𝜑(⋅)-quasieverywhere
in R𝑛. Moreover, the convergence is uniform outside a set of
arbitrarily small Sobolev 𝜑(⋅)-capacity.
Proof. Let (𝑓𝑖) be a Cauchy sequence in 𝐶(R𝑛) ∩𝑊1,𝜑(⋅)(R𝑛).
We assume without loss of generality, by considering a
subsequence if necessary, that ‖𝑓𝑖 − 𝑓𝑖+1‖1,𝜑(⋅) ⩽ 4−𝑖 for every𝑖 = 1, 2, . . . For 𝑖 ∈ N we denote 𝑔𝑖 fl 2𝑖|𝑓𝑖 − 𝑓𝑖+1| ∈𝑊1,𝜑(⋅)(R𝑛), 𝐸𝑖 fl {𝑔𝑖 > 1} and 𝐹𝑗 fl ⋃∞𝑖=𝑗 𝐸𝑖. Note that‖𝑔𝑖‖1,𝜑(⋅) ⩽ 2−𝑖. We obtain by Corollary 2.1.15(a) of [18] that
𝐶𝜑(⋅) (𝐸𝑖) ⩽ 󰜚1,𝜑(⋅) (𝑔𝑖) ⩽ 󵄩󵄩󵄩󵄩𝑔𝑖󵄩󵄩󵄩󵄩1,𝜑(⋅) ⩽ 2−𝑖. (43)
The subadditivity (S7) further implies
𝐶𝜑(⋅) (𝐹𝑗) ⩽ ∞∑
𝑖=𝑗
𝐶𝜑(⋅) (𝐸𝑖) ⩽ ∞∑
𝑖=𝑗
2−𝑖 = 21−𝑗. (44)
Since (𝐹𝑗)∞𝑗=1 is decreasing and 𝐶𝜑(⋅)(𝐹𝑗) ⩾ 0, the limit
lim𝑗󳨀→∞𝐶𝜑(⋅)(𝐹𝑗) exists. Since⋂∞𝑗=1 𝐹𝑗 ⊂ 𝐹𝑘 for all 𝑘 = 1, 2, . . .,
we obtain by (S2) that
𝐶𝜑(⋅)(∞⋂
𝑗=1
𝐹𝑗) ⩽ lim
𝑗󳨀→∞
𝐶𝜑(⋅) (𝐹𝑗) ⩽ lim
𝑗󳨀→∞
21−𝑗 = 0. (45)
Now (𝑓𝑗) converges pointwise in R𝑛 \ ⋂∞𝑗=1 𝐹𝑗 and𝐶𝜑(⋅)(⋂∞𝑗=1 𝐹𝑗) = 0, so the first claim of the lemma is
proved. Moreover, for 𝑥 ∈ R𝑛 \ 𝐹𝑗 and 𝑘 > 𝑙 > 𝑗,
󵄨󵄨󵄨󵄨𝑓𝑙 (𝑥) − 𝑓𝑘 (𝑥)󵄨󵄨󵄨󵄨 ⩽ 𝑘−1∑
𝑖=𝑙
󵄨󵄨󵄨󵄨𝑓𝑖 (𝑥) − 𝑓𝑖+1 (𝑥)󵄨󵄨󵄨󵄨 ⩽ 𝑘−1∑
𝑖=𝑙
2−𝑖
< 21−𝑙.
(46)
Therefore, the convergence is uniform in R𝑛 \ 𝐹𝑗, and the
second claim follows.
The existence of the so-called𝜑(⋅)-quasicontinuous repre-
sentative follows fromLemma 16by standard arguments (e.g.,
[18, Theorem 11.1.3]):
Theorem 17. Let 𝜑 ∈ Φ𝑤(R𝑛) satisfy (aInc) and (aDec).
Assume that 𝐶(R𝑛) ∩𝑊1,𝜑(⋅)(R𝑛) is dense in𝑊1,𝜑(⋅)(R𝑛). Then
for each 𝑓 ∈ 𝑊1,𝜑(⋅)(R𝑛), there exists a 𝜑(⋅)-quasicontinuous
function 𝑔 ∈ 𝑊1,𝜑(⋅)(R𝑛) such that 𝑓 = 𝑔 almost everywhere
in R𝑛.
Lemma 18. Let 𝜑 ∈ Φ𝑤(R𝑛) satisfy (aDec). Suppose that 𝑓 ∈𝑊1,𝜑(⋅)(R𝑛) is a nonnegative 𝜑(⋅)-quasicontinuous function
with 𝑓 ⩾ 1 in 𝐸 ⊂ R𝑛. Then for every 𝜀 > 0 there exists𝑔 ∈ 𝑆1,𝜑(⋅)(𝐸) such that 󰜚1,𝜑(⋅)(𝑓 − 𝑔) < 𝜀.
Proof. Let 𝜀 ∈ (0, 1) and 𝛿 fl 𝜀/2𝐿2𝑞(1 + 𝐿󰜚1,𝜑(⋅)(𝑓)), and
let 𝑈 ⊂ R𝑛 be an open set such that 𝑓 restricted to R𝑛 \ 𝑈
Journal of Function Spaces 7
is continuous and 𝐶𝜑(⋅)(𝑈) < 𝛿. Moreover, let us take ℎ ∈𝑆1,𝜑(⋅)(𝑈) such that 󰜚1,𝜑(⋅)(ℎ) < 𝛿, and denote 𝑔 fl (1+𝛿)𝑓+ℎ.
Then𝑔 ∈ 𝑊1,𝜑(⋅)(R𝑛).The set𝑉 fl {𝑥 ∈ R𝑛\𝑈 : (1+𝛿)𝑓(𝑥) >1}∪𝑈 is open and contains 𝐸. Since 𝑔 ⩾ 1 in𝑉, 𝑔 ∈ 𝑆1,𝜑(⋅)(𝐸).
It remains to estimate 󰜚1,𝜑(⋅)(𝑓 − 𝑔).
Now, 𝜑(𝑥, ℎ + 𝛿𝑓) ⩽ 𝜑(𝑥, 2ℎ) + 𝜑(𝑥, 2𝛿𝑓). By (aDec)𝑝 and(aInc)1 we have 𝜑(𝑥, 2𝛿𝑓) ⩽ 𝐿2𝑞𝜑(𝑥, 𝛿𝑓) ⩽ 𝐿22𝑞𝛿𝜑(𝑥, 𝑓), and
so 𝜑(𝑥, ℎ + 𝛿𝑓) ⩽ 𝐿2𝑞(𝜑(𝑥, ℎ) + 𝐿𝛿𝜑(𝑥, 𝑓)). Therefore󰜚𝜑(⋅) (𝑓 − 𝑔) = 󰜚𝜑(⋅) (ℎ + 𝛿𝑓)
⩽ 𝐿2𝑞 (󰜚𝜑(⋅) (ℎ) + 𝐿𝛿󰜚𝜑(⋅) (𝑓))
< 𝐿2𝑞 (1 + 𝐿󰜚𝜑(⋅) (𝑓)) 𝛿 = 12𝜀
(47)
by the assumption on 𝛿. An analogous inequality holds for
the gradient, and so the claim follows.
6. Relative Capacity
In this section, we introduce relative 𝜑(⋅)-capacity, analogous
to the relative 𝑝(⋅)-capacity of variable exponent spaces in
[18], taken with respect to an open set, Ω, in R𝑛.
Definition 19. Let Ω ⊂ R𝑛, 𝐾 ⊂ Ω be compact, and 𝜑 ∈Φ𝑤(Ω). Denote𝑅𝜑(⋅) (𝐾,Ω) fl {𝑓 ∈ 𝑊1,𝜑(⋅) (Ω) ∩ 𝐶0 (Ω) : 𝑓
> 1 in 𝐾 and 𝑓 ⩾ 0} . (48)
We define cap∗𝜑(⋅)(𝐾,Ω) fl inf𝑓∈𝑅𝜑(⋅)(𝐾,Ω)󰜚𝜑(⋅)(|∇𝑓|). Further,
for 𝑈 ⊂ Ω open, we set
cap𝜑(⋅) (𝑈,Ω) fl sup
𝐾⊂𝑈 compact
cap∗𝜑(⋅) (𝐾,Ω) , (49)
and, for an arbitrary set 𝐸 ⊂ Ω, we define
cap𝜑(⋅) (𝐸, Ω) fl inf
𝑈⊃𝐸 open
cap𝜑(⋅) (𝑈,Ω) . (50)
The number cap𝜑(⋅)(𝐸, Ω) is called the relative 𝜑(⋅)-capacity of𝐸 with respect toΩ.
In the next result we offer two alternate set of assump-
tions. It seems that (aDec) alone is not sufficient, although we
have not been able to prove this.
Proposition 20. If 𝜑 ∈ Φ𝑤(Ω) satisfies either (Dec) or (A0),
then
cap∗𝜑(⋅) (𝐾,Ω) = inf
𝑓∈?̃?𝜑(⋅)(𝐾,Ω)
∫
Ω
𝜑 (𝑥, 󵄨󵄨󵄨󵄨∇𝑓󵄨󵄨󵄨󵄨) 𝑑𝑥, (51)
where ?̃?𝜑(⋅)(𝐾,Ω) fl {𝑓 ∈ 𝑊1,𝜑(⋅)(Ω) ∩ 𝐶0(Ω) : 𝑓 ⩾ 1 𝑖𝑛𝐾 𝑎𝑛𝑑 𝑓 ⩾ 0}.
Proof. Since 𝑅𝜑(⋅)(𝐾,Ω) ⊂ ?̃?𝜑(⋅)(𝐾,Ω), the inequality “⩾” is
clear. Now, to prove the opposite inequality, we fix 𝜀 > 0 and
let 𝑓 ∈ ?̃?𝜑(⋅)(𝐾,Ω), be such that󰜚𝜑(⋅) (󵄨󵄨󵄨󵄨∇𝑓󵄨󵄨󵄨󵄨) ⩽ inf
𝑔∈?̃?𝜑(⋅)(𝐾,Ω)
󰜚𝜑(⋅) (󵄨󵄨󵄨󵄨∇𝑔󵄨󵄨󵄨󵄨) + 𝜀. (52)
If (Dec) holds, we set 𝑔 fl (1 + 𝜀)𝑓. Then 𝑔 > 1 in 𝐾, so
that
cap∗𝜑(⋅) (𝐾,Ω) ⩽ 󰜚𝜑(⋅) (∇𝑔) = ∫
Ω
𝜑 (𝑥, (1 + 𝜀) ∇𝑓) 𝑑𝑥
⩽ (1 + 𝜀)𝑞 ∫
Ω
𝜑 (𝑥, 󵄨󵄨󵄨󵄨∇𝑓󵄨󵄨󵄨󵄨) 𝑑𝑥,
(53)
where we used (Dec) for the last inequality. The claim follows
as 𝜀 󳨀→ 0+.
If, on the other hand, (A0) holds, then we fix 𝜂 ∈ 𝐶∞0 (Ω)
with 𝜂 = 1 in supp𝑓. Let 𝜆 = 𝛽‖∇𝜂‖−1∞ . We set 𝑔 fl 𝑓+ 𝜀󸀠𝜆𝜂 ∈𝑅𝜑(⋅)(𝐾,Ω). Since the supports of ∇𝑓 and ∇𝜂 are disjoint, we
observe that
󰜚𝜑(⋅) (∇𝑔) = 󰜚𝜑(⋅) (∇𝑓) + 󰜚𝜑(⋅) (𝜀󸀠𝜆∇𝜂)
⩽ 󰜚𝜑(⋅) (∇𝑓) + 𝐿𝜀󸀠󰜚𝜑(⋅) (𝜆∇𝜂) , (54)
where (aInc)1 has been used in the last inequality. The claim
follows as 𝜀, 𝜀󸀠 󳨀→ 0+.
Next, we show that cap∗𝜑(⋅)(𝐾,Ω) and cap𝜑(⋅)(𝐾,Ω) are the
same; that is, the relative capacity is well defined on compact
sets.
Proposition 21. Let 𝜑 ∈ Φ𝑤(Ω). Then cap∗𝜑(⋅)(𝐾,Ω) =
cap𝜑(⋅)(𝐾,Ω) for every compact 𝐾 ⊂ Ω.
Proof. The inequality cap∗𝜑(⋅)(𝐾,Ω) ⩽ cap𝜑(⋅)(𝐾,Ω) follows
directly from the definition.
Now, to prove the opposite inequality, fix 𝜀 > 0 and let𝑓 ∈ 𝑅𝜑(⋅)(𝐾,Ω) be such that 󰜚𝜑(⋅)(∇𝑓) ⩽ cap∗𝜑(⋅)(𝐾,Ω) + 𝜀.
Then, 𝑓 is greater than one in 𝑈 fl 𝑓−1(1,∞), which is open
since 𝑓 ∈ 𝐶0(Ω), and contains 𝐾. Thus, 𝑓 is also a valid test-
function for every compact set 𝐾󸀠 ⊂ 𝑈, so that
cap𝜑(⋅) (𝑈,Ω) = sup
𝐾󸀠⊂𝑈
cap∗𝜑(⋅) (𝐾󸀠, Ω) ⩽ 󰜚𝜑(⋅) (∇𝑓)
⩽ cap∗𝜑(⋅) (𝐾,Ω) + 𝜀. (55)
The result follows from this as 𝜀 󳨀→ 0+.
Next, we show that the relative capacity has the same basic
properties as the Sobolev capacity.
Proposition 22. Let Ω ⊂ R𝑛 be open and 𝜑 ∈ Φ𝑤(R𝑛).
(R1) cap𝜑(⋅)(0,Ω) = 0.
(R2) If 𝐸 ⊂ 𝐸󸀠 ⊂ Ω󸀠 ⊂ Ω, then cap𝜑(⋅)(𝐸,Ω) ⩽
cap𝜑(⋅)(𝐸󸀠, Ω󸀠).
(R3) If 𝐸 ⊂ Ω, then cap𝜑(⋅)(𝐸, Ω) = inf𝑈⊃𝐸 opencap𝜑(⋅)(𝑈,Ω).
(R4) If 𝐸, 𝐹 ⊂ Ω, then
cap𝜑(⋅) (𝐸 ∪ 𝐹,Ω) + cap𝜑(⋅) (𝐸 ∩ 𝐹,Ω)
⩽ cap𝜑(⋅) (𝐸, Ω) + cap𝜑(⋅) (𝐹, Ω) . (56)
8 Journal of Function Spaces
(R5) If𝐾1 ⊃ 𝐾2 ⊃ ⋅ ⋅ ⋅ are compact, then lim𝑖󳨀→∞cap𝜑(⋅)(𝐾𝑖,Ω) = cap𝜑(⋅)(⋂∞𝑖=1𝐾𝑖, Ω).
Proof. Properties (R1) and (R3) follow immediately from the
definition. For proving property (R2), we observe that if 𝐸󸀠 ⊂𝑈 ⊂ Ω󸀠 then also 𝐸 ⊂ 𝑈 ⊂ Ω. Thus
cap𝜑(⋅) (𝐸󸀠, Ω󸀠) = inf
𝐸󸀠⊂𝑈⊂Ω󸀠
cap𝜑(⋅) (𝑈,Ω󸀠)
⩾ cap𝜑(⋅) (𝐸, Ω) . (57)
Properties (R4) and (R5) are proved exactly like (S4) and (S5).
The proof of the following results follows from (R4) by
standard arguments, see, e.g., [18, Lemma 10.2.5].
Lemma 23. Suppose that 𝜑 ∈ Φ𝑤(Ω) and 𝐸1, 𝐸2, . . . , 𝐸𝑘 ⊂ Ω.
Then,
cap𝜑(⋅) ( 𝑘⋃
𝑖=1
𝐸𝑖, Ω) − cap𝜑(⋅)( 𝑘⋃
𝑖=1
𝐴 𝑖, Ω)
⩽ 𝑘∑
𝑖=1
(cap𝜑(⋅) (𝐸𝑖, Ω) − cap𝜑(⋅) (𝐴 𝑖, Ω)) ,
(58)
whenever 𝐴 𝑖 ⊂ 𝐸𝑖, 𝑖 = 1, 2, . . . , 𝑘 and cap𝜑(⋅)(⋃𝑘𝑖=1 𝐸𝑖, Ω) < ∞.
Furthermore, the previous lemma implies directly prop-
erties (R6) and (R7), see, e.g., [18, Theorem 10.2.6]. Note that
these properties hold for the relative capacity without any
extra assumptions on 𝜑.
Theorem 24. Let 𝜑 ∈ Φ𝑤(Ω).
(R6) If 𝐸1 ⊂ 𝐸2 ⊂ ⋅ ⋅ ⋅ ⊂ Ω, then lim𝑖󳨀→∞cap𝜑(⋅)(𝐸𝑖, Ω) =
cap𝜑(⋅)(⋃∞𝑖=1 𝐸𝑖, Ω).
(R7) If 𝐸𝑖 ⊂ Ω, 𝑖 = 1, 2, . . ., then cap𝜑(⋅)(⋃∞𝑖=1 𝐸𝑖, Ω) ⩽∑∞𝑖=1 cap𝜑(⋅)(𝐸𝑖, Ω).
Remark 25. A set function satisfying the properties (R1),
(R2), (R5), and (R6) is called a Choquet capacity [22]. They
imply for every Borel set 𝐸 ⊂ Ω that
cap𝜑(⋅) (𝐸, Ω)
= sup {cap𝜑(⋅) (𝐾,Ω) : 𝐾 is compact and 𝐾 ⊂ 𝐸} . (59)
7. Relationship between Capacities
Lemma 26. Assume that 𝜑 ∈ Φ𝑤(R𝑛) satisfies (A0) and
(aDec)𝑞. If Ω is bounded and 𝐾 ⊂ Ω is compact, then,
𝐶𝜑(⋅) (𝐾) ⩽ 𝐶max {cap𝜑(⋅) (𝐾,Ω)1/𝑞 , cap𝜑(⋅) (𝐾,Ω)} , (60)
where the constant 𝐶 depends on the dimension, |Ω| and the
constants in (A0) and (aDec).
Proof. We may assume that cap𝜑(⋅)(𝐾,Ω) < ∞. Let 𝑓 ∈𝑅𝜑(⋅)(𝐾,Ω)with 󰜚𝜑(⋅)(|∇𝑓|) < 2cap𝜑(⋅)(𝐾,Ω). Extend𝑓 by zero
outside of Ω and set 𝑔 fl min{1, 𝑓}. Since 𝑓 ∈ 𝐶0(Ω) and𝑓 > 1 in the compact set𝐾, 𝑈 = {𝑓 > 1} ⊃ 𝐾 is open. Hence𝑔 ∈ 𝑆1,𝜑(⋅)(𝐾) and so
𝐶𝜑(⋅) (𝐾) ⩽ ∫
R𝑛
𝜑 (𝑥, 𝑔) + 𝜑 (𝑥, 󵄨󵄨󵄨󵄨∇𝑔󵄨󵄨󵄨󵄨) 𝑑𝑥
⩽ ∫
R𝑛
𝜑 (𝑥, 𝑔) + 𝜑 (𝑥, 󵄨󵄨󵄨󵄨∇𝑓󵄨󵄨󵄨󵄨) 𝑑𝑥.
(61)
Using 0 ⩽ 𝑔 ⩽ 1, (aInc)1, (aDec)𝑞 and (A0), we obtain
𝜑 (𝑥, 𝑔 (𝑥)) ⩽ 𝐿𝑔 (𝑥) 𝜑 (𝑥, 1) ⩽ 𝐿2𝑔 (𝑥) 𝛽−𝑞𝜑 (𝑥, 𝛽)
⩽ 𝐿2𝑔 (𝑥) 𝛽−𝑞. (62)
Integrating over the bounded set Ω and using the classical
Poincare´ inequality, we find that
∫
Ω
𝜑 (𝑥, 𝑔) 𝑑𝑥 ⩽ 𝐶∫
Ω
𝑔𝑑𝑥 ⩽ 𝐶∫
Ω
𝑓𝑑𝑥
⩽ 𝐶∫
Ω
󵄨󵄨󵄨󵄨∇𝑓󵄨󵄨󵄨󵄨 𝑑𝑥.
(63)
Then the embedding 𝐿𝜑(⋅)(Ω) 󳨅→ 𝐿1(Ω) (Lemma 5) and
Lemma 6 for the function |∇𝑓| give that
∫
Ω
𝜑 (𝑥, 𝑔) 𝑑𝑥 ⩽ 𝐶max {󰜚𝜑(⋅) (󵄨󵄨󵄨󵄨∇𝑓󵄨󵄨󵄨󵄨)1/𝑞 , 󰜚𝜑(⋅) (󵄨󵄨󵄨󵄨∇𝑓󵄨󵄨󵄨󵄨)} . (64)
Combining this with (61) and taking the infimum over 𝑓, we
obtain the claim.
In the next proof the Choquet property for Sobolev
capacity is needed and hence we assume that 𝜑 is convex.
Theorem 27. Assume that 𝜑 ∈ Φ𝑐(R𝑛) satisfies (A0) and
(aDec)𝑞. IfΩ is bounded and 𝐸 ⊂ Ω, then
𝐶𝜑(⋅) (𝐸) ⩽ 𝐶max {cap𝜑(⋅) (𝐸, Ω)1/𝑞 , cap𝜑(⋅) (𝐸, Ω)} , (65)
where the constant 𝐶 depends on the dimension, |Ω|, and the
constants in (A0) and (aDec).
Proof. We may assume that cap𝜑(⋅)(𝐸,Ω) < ∞. By the
definition of cap𝜑(⋅)(𝐸, Ω), there exist open sets 𝑈𝑖 ⊃ 𝐸 with
cap𝜑(⋅)(𝑈𝑖, Ω) 󳨀→ cap𝜑(⋅)(𝐸, Ω), as 𝑖 󳨀→ ∞. Let 𝑈 fl ⋂∞𝑖=1𝑈𝑖.
Then 𝑈 is a Borel set, and hence, by the Choquet property
(Remark 10),𝐶𝜑(⋅) (𝐸) ⩽ 𝐶𝜑(⋅) (𝑈) = sup
𝐾⊂𝑈
𝐶𝜑(⋅) (𝐾) , (66)
where the supremum is taken over all compact sets 𝐾 ⊂ 𝑈.
By Lemma 26, we obtain
𝐶𝜑(⋅) (𝐸) ⩽ sup
𝐾
𝐶𝜑(⋅) (𝐾)
⩽ 𝐶 sup
𝐾⊂𝑈
max {cap𝜑(⋅) (𝐾,Ω)1/𝑞 , cap𝜑(⋅) (𝐾,Ω)}
⩽ 𝐶max {cap𝜑(⋅) (𝑈𝑖, Ω)1/𝑞 , cap𝜑(⋅) (𝑈𝑖, Ω)} .
(67)
The claim follows from this as 𝑖 󳨀→ ∞.
Journal of Function Spaces 9
From the above result, we can conclude that 𝐶𝜑(⋅)(𝐸) = 0
if cap𝜑(⋅)(𝐸, Ω) = 0. To get the converse implication, we first
prove the following results.
Lemma 28. Let 𝜑 ∈ Φ𝑤(Ω) satisfy (aDec). Assume that𝐶(R𝑛) ∩ 𝑊1,𝜑(⋅)(R𝑛) is dense in 𝑊1,𝜑(⋅)(R𝑛). If 𝐾 ⊂ R𝑛 is
compact, then
𝐶𝜑(⋅) (𝐾) = inf
𝑓∈𝑆𝑐
1,𝜑(⋅)
(𝐾)
∫
R𝑛
𝜑 (𝑥, 𝑓) + 𝜑 (𝑥, 󵄨󵄨󵄨󵄨∇𝑓󵄨󵄨󵄨󵄨) 𝑑𝑥, (68)
where 𝑆𝑐1,𝜑(⋅)(𝐾) = 𝑆1,𝜑(⋅)(𝐾) ∩ 𝐶(R𝑛).
Proof. Since𝐶𝜑(⋅)(𝐾) is defined as the infimumover the larger
set 𝑆1,𝜑(⋅)(𝐾), the inequality “⩽” is clear.
Suppose 𝑓 ∈ 𝑆1,𝜑(⋅)(𝐾), with 0 ⩽ 𝑓 ⩽ 1 and choose
functions 𝑓𝑗 ∈ 𝐶(R𝑛) ∩ 𝑊1,𝜑(⋅)(R𝑛) converging to 𝑓 in𝑊1,𝜑(⋅)(R𝑛), with 0 ⩽ 𝑓𝑗 ⩽ 1. Choose a bounded neighbor-
hood 𝑈 of 𝐾 such that 𝑓 = 1 in 𝑈. Let 𝜂 ∈ 𝐶1(R𝑛), 0 ⩽ 𝜂 ⩽ 1
with 𝜂 = 1 inR𝑛 \ 𝑈 and 𝜂 = 0 in a neighborhood of 𝐾.
Let 𝑔𝑗 fl 1 − (1 − 𝑓𝑗)𝜂 and note that 𝑔𝑗 ⩾ 0 since 0 ⩽𝑓𝑗, 𝜂 ⩽ 1. We find that 𝑓 − 𝑔𝑗 = 𝑓 − 1 + 𝜂 − 𝜂𝑓𝑗 = (𝑓 − 𝑓𝑗)𝜂 +(1 − 𝜂)(𝑓 − 1) = (𝑓 − 𝑓𝑗)𝜂, as 𝑓 = 1 in 𝑈 and 𝜂 = 1 inR𝑛 \ 𝑈.
Since 𝜂 ∈ 𝑊1,∞(R𝑛) and 𝑓𝑗 󳨀→ 𝑓 in𝑊1,𝜑(⋅)(R𝑛), we find that𝑔𝑗 󳨀→ 𝑓 in𝑊1,𝜑(⋅)(R𝑛). Further, 𝜂 = 0 in a neighborhood of𝐾, so 𝑔𝑗 = 1 in a neighborhood of 𝐾. Thus 𝑔𝑗 ∈ 𝑆𝑐1,𝜑(⋅)(𝐾).
This and 𝑔𝑗 󳨀→ 𝑓 imply the “⩾” inequality.
Proposition 29. Let Ω ⊂ R𝑛 be bounded and 𝜑 ∈ Φ𝑤(R𝑛)
satisfy (aDec). Assume that 𝐶(R𝑛) is dense in 𝑊1,𝜑(⋅)(R𝑛). If𝐸 ⊂ Ω with 𝐶𝜑(⋅)(𝐸) = 0, then cap𝜑(⋅)(𝐸, Ω) = 0.
Proof. Let 𝐾 ⊂ Ω be compact with 𝐶𝜑(⋅)(𝐾) = 0. By
Lemma 28 we may choose a sequence of functions (𝑓𝑖) from𝑆𝑐1,𝜑(⋅)(𝐾) such that ‖𝑓𝑖‖𝑊1,𝜑(⋅)(R𝑛) 󳨀→ 0. Let 𝜂 ∈ 𝐶∞0 (Ω) be a
cut-off function that equals two in 𝐾. Since 𝑓𝑖 ∈ 𝑆𝑐1,𝜑(⋅)(𝐾),
it is easy to conclude that 𝜂𝑓𝑖 > 1 in 𝐾 and 𝜂𝑓𝑖 ⩾ 0, hence𝜂𝑓𝑖 ∈ 𝑅𝜑(⋅)(𝐾,Ω). Since modular convergence and norm con-
vergence are equivalent [18, Theorem 2.1.11], we obtain
cap𝜑(⋅) (𝐾,Ω) ⩽ 󰜚𝜑(⋅) (∇ (𝜂𝑓𝑖))
⩽ 󰜚𝜑(⋅) (𝑐𝑓𝑖) + 󰜚𝜑(⋅) (𝑐 󵄨󵄨󵄨󵄨∇𝑓𝑖󵄨󵄨󵄨󵄨) 󳨀→ 0. (69)
Hence the claim holds for compact sets.
By (S3) there exists a sequence of open sets 𝑈𝑖 ⊃ 𝐸 with𝐶𝜑(⋅)(𝑈𝑖) 󳨀→ 0 as 𝑖 󳨀→ ∞. Let 𝑈 fl ⋂∞𝑖=1𝑈𝑖 ∩ Ω. Then, 𝑈 is
a Borel set containing 𝐸 which satisfies 𝐶𝜑(⋅)(𝑈) = 0. By the
Choquet property of the relative capacity, we obtain
cap𝜑(⋅) (𝐸, Ω) ⩽ cap𝜑(⋅) (𝑈,Ω) = sup
𝐾
cap𝜑(⋅) (𝐾,Ω) , (70)
where the supremum is taken over all compact sets 𝐾 ⊂ 𝑈.
By the first part of the proof, cap𝜑(⋅)(𝐾,Ω) = 0, and the claim
follows.
If 𝐸 ⊂ 𝐵 and Ω = 2𝐵 for a ball 𝐵, then the Lipschitz
constant of the cut-off function 𝜂 in the previous proof can
be chosen to be 𝑐(diam𝐵)−1. Then an similar proof gives the
following quantitative version of the previous result, cf. [18,
Theorem 10.3.5].
Theorem 30. Let 𝐵 ⊂ R𝑛 be a ball and 𝜑 ∈ Φ𝑤(R𝑛) satisfy
(aDec)𝑞. Assume that𝐶(R𝑛) is dense in𝑊1,𝜑(⋅)(R𝑛). For𝐸 ⊂ 𝐵,
cap𝜑(⋅) (𝐸, 2𝐵)
⩽ 𝐶 (1 +max {diam (𝐵) , diam (𝐵)−𝑞}) 𝐶𝜑(⋅) (𝐸) (71)
where the constant C depends on 𝑞 and 𝐿.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
Debangana Baruah was supported financially by the Univer-
sity of Turku Graduate School MATTI-program.
References
[1] P. Marcellini, “Regularity of minimizers of integrals of the
calculus of variations with nonstandard growth conditions,”
Archive for Rational Mechanics and Analysis, vol. 105, no. 3, pp.
267–284, 1989.
[2] V. V. Zhikov, “Averaging of functionals of the calculus of
variations and elasticity theory,” Izvestiya Rossiiskoi Akademii
Nauk, vol. 50, no. 4, pp. 675–710, 1986 (Russian).
[3] M. Ruzˇicˇka, Electrorheological Fluids: Modeling and Mathemat-
ical Theory, vol. 1748 of Lecture Notes in Mathematics, Springer,
Berlin, Germany, 2000.
[4] P. Baroni,M.Colombo, andG.Mingione, “Harnack inequalities
for double phase functionals,” Nonlinear Analysis, vol. 121, pp.
206–222, 2015.
[5] P. Baroni, M. Colombo, and G. Mingione, “Nonautonomous
functionals, borderline cases and related function classes,” St.
Petersburg Mathematical Journal, vol. 27, no. 3, pp. 347–379,
2015.
[6] P. Baroni, M. Colombo, and G. Mingione, “Regularity for gen-
eral functionals with double phase,” Calculus of Variations and
Partial Differential Equations, vol. 57, no. 62, 48 pages, 2018.
[7] M. Colombo and G. Mingione, “Regularity for double phase
variational problems,”Archive for RationalMechanics and Anal-
ysis, vol. 215, no. 2, pp. 443–496, 2015.
[8] S.-S. Byun and J. Oh, “Global gradient estimates for non-
uniformly elliptic equations,” Calculus of Variations and Partial
Differential Equations, vol. 56, no. 2, article no. 46, 36 pages,
2017.
[9] S.-S. Byun, S. Ryu, and P. Shin, “Calderon-Zygmund estimates
for omega-minimizers of double phase variational problems,”
Applied Mathematics Letters, vol. 86, pp. 256–263, 2018.
[10] A. Clop, R. Giova, and F. Hatami, “Passarelli di, Napoli: Con-
gested traffic dynamics and very degenerate elliptic equations
under supercritical Sobolev regularity,” Preprint, 2018.
10 Journal of Function Spaces
[11] G. Cupini, F. Giannetti, R. Giova, and A. Passarelli di Napoli,
“Regularity results for vectorial minimizers of a class of degen-
erate convex integrals,” Journal of Differential Equations, vol.
265, no. 9, pp. 4375–4416, 2018.
[12] J. Ok, “Regularity of𝜔-minimizers for a class of functionalswith
non-standard growth,” Calculus of Variations and Partial Dif-
ferential Equations, vol. 56, no. 2, article no. 48, 31 pages, 2017.
[13] P. Harjulehto, P. Ha¨sto¨, and O. Toivanen, “Ho¨lder regularity of
quasiminimizers under generalized growth conditions,” Calcu-
lus of Variations and Partial Differential Equations, vol. 56, no.
2, 26 pages, 2017.
[14] P. Gwiazda, I. Skrzypczak, and A. Zatorska-Goldstein, “Exis-
tence of renormalized solutions to elliptic equation inMusielak-
Orlicz space,” Journal of Differential Equations, vol. 264, no. 1,
pp. 341–377, 2018.
[15] P. Gwiazda, I. Skrzypczak, and A. Zatorska-Goldstein, “Well-
posedness of parabolic equations in the non-reflexive and
anisotropic Musielak-Orlicz spaces in the class of renormalized
solutions,” Journal ofDifferential Equations, vol. 265, no. 11, 2018.
[16] P. Harjulehto, P. Ha¨sto¨, and R. Kle´n, “Generalized Orlicz spaces
and related PDE,”Nonlinear Analysis, vol. 143, pp. 155–173, 2016.
[17] J. Heinonen, T. Kilpelainen, and O. Martio, Nonlinear Potential
Theory Of Degenerate Elliptic Equations, Oxford University
Press, New York, NY, USA, 1993.
[18] L. Diening, P. Harjulehto, P. Ha¨sto¨, and M. Ruzicka, Lebesgue
and Sobolev Spaces with Variable Exponents, Springer, Berlin,
Germany, 2011.
[19] T. Ohno and T. Shimomura, “Musielak-Orlicz-Sobolev spaces
with zero boundary values on metric measure spaces,” Czecho-
slovak Mathematical Journal, vol. 66(141), no. 2, pp. 371–394,
2016.
[20] P. Harjulehto and P. Ha¨sto¨, “Uniform convexity and associate
spaces,” Czechoslovak Mathematical Journal, pp. 1–10, 2018.
[21] P. A. Ha¨sto¨, “The maximal operator on generalized Orlicz
spaces,” Journal of Functional Analysis, vol. 269, no. 12, pp. 4038–
4048, 2015.
[22] G. Choquet, “Theory of capacities,”Annales de l’Institut Fourier,
vol. 5, pp. 131–295, 1954.
[23] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Pro-
perties of Functions, Studies in Advanced Mathematics, CRC
Press, Boca Raton, FL, USA, 1992.
Hindawi
www.hindawi.com Volume 2018
Mathematics
Journal of
Hindawi
www.hindawi.com Volume 2018
Mathematical Problems 
in Engineering
Applied Mathematics
Journal of
Hindawi
www.hindawi.com Volume 2018
Probability and Statistics
Hindawi
www.hindawi.com Volume 2018
Journal of
Hindawi
www.hindawi.com Volume 2018
Mathematical Physics
Advances in
Complex Analysis
Journal of
Hindawi
www.hindawi.com Volume 2018
Optimization
Journal of
Hindawi
www.hindawi.com Volume 2018
Hindawi
www.hindawi.com Volume 2018
Engineering  
 Mathematics
International Journal of
Hindawi
www.hindawi.com Volume 2018
Operations Research
Advances in
Journal of
Hindawi
www.hindawi.com Volume 2018
Function Spaces
Abstract and 
Applied Analysis
Hindawi
www.hindawi.com Volume 2018
International 
Journal of 
Mathematics and 
Mathematical 
Sciences
Hindawi
www.hindawi.com Volume 2018
Hindawi Publishing Corporation 
http://www.hindawi.com Volume 2013www.hindawi.com
The Scientific 
World Journal
8
Hindawi
www.hindawi.com Volume 2018
Numerical Analysis
Advances in Discrete Dynamics in 
Nature and Society
Hindawi
www.hindawi.com Volume 2018
Hindawi
www.hindawi.com
Dierential Equations
International Journal of
Volume 2018
Hindawi
www.hindawi.com Volume 2018
Decision Sciences
Advances in
Hindawi
www.hindawi.com Volume 2018
Analysis
International Journal of
Hindawi
www.hindawi.com Volume 2018
Stochastic Analysis
International Journal of
Submit your manuscripts at
www.hindawi.com